From 5eca4dcffba72ed6d88c003a8e184aca3aa78ddc Mon Sep 17 00:00:00 2001 From: renpengli01 Date: Tue, 26 Aug 2025 14:29:00 +0800 Subject: [PATCH 01/11] fix: update post main html README.md (#540) --- .../html/post_main_html_processer/README.md | 84 ++++++++++++++++-- .../post_main_html_processer/asserts/img.png | Bin 139133 -> 0 bytes .../post_main_html_processer/assets/img.png | Bin 0 -> 140328 bytes 3 files changed, 76 insertions(+), 8 deletions(-) delete mode 100644 llm_web_kit/extractor/html/post_main_html_processer/asserts/img.png create mode 100644 llm_web_kit/extractor/html/post_main_html_processer/assets/img.png diff --git a/llm_web_kit/extractor/html/post_main_html_processer/README.md b/llm_web_kit/extractor/html/post_main_html_processer/README.md index b6141c4e..f1973c42 100644 --- a/llm_web_kit/extractor/html/post_main_html_processer/README.md +++ b/llm_web_kit/extractor/html/post_main_html_processer/README.md @@ -2,14 +2,82 @@ ## 流程方案 -![img.png](asserts/img.png) +![img.png](assets/img.png) ## 执行步骤 -| filename | function | input & input_type | output_type | 实现功能 | -| :--------------- | :-------------------------- | :-------------------------------------------------------- | :------------------ | :------------- | -| choose_html.py | select_typical_html | html_strs: html迭代器 | str | 选出代表html | -| add_tags.py | process_html | input_html: str | str | 添加itemid | -| post_llm.py | get_llm_response | api_key: str, url: str, html_id_str: str, model_name: str | str | 模型打标 | -| generate_rule.py | restore_html_trim_ends_only | processed_html: str, llm_response: Dict\[str, int\] | Dict\[str, object\] | 生成删除规则 | -| post_mapping.py | mapping_html_by_rules | html_str: str, post_delete_node: List\[object\] | str | 推广到所有数据 | +| filename | function | input & input_type | output_type | 实现功能 | +| :--------------- | :-------------------------- | :------------------------------------------------------------------------------ | :------------------------- | :------------- | +| choose_html.py | select_typical_html | html_strs: html迭代器 | str | 选出代表html | +| add_tags.py | process_html | input_html: str | str | 添加itemid | +| post_llm.py | get_llm_response | api_key: str, url: str, html_id_str: str, model_name: str | Dict\[str, int\] \[^1\] | 模型打标 | +| generate_rule.py | restore_html_trim_ends_only | processed_html: str, llm_response: Dict\[str, int\](get_llm_response的输出结果) | Dict\[str, object\] \[^2\] | 生成删除规则 | +| post_mapping.py | mapping_html_by_rules | html_str: str, post_delete_node: List\[object\] \[^3\] | str | 推广到所有数据 | + +\[^1\] + +```jsonc + +{"item_id 1": 0, "item_id 2": 0, "item_id 3": 1, "item_id 4": 1, "item_id 5": 1, "item_id 6": 1, "item_id 7": 1} // 0：删除；1：保留 + +``` + +\[^2\] + +```jsonc + +{ + "html": "...", // 选出的代表main html经过处理之后的新html + "post_delete_node": [ // 删除规则 + { + "del_location": "start", // 删除的位置【start（头部） or end（尾部）】 + "xpath": "/div/div[1]/div/div/div[1]", // 删除节点的xpath + "tag": "div", // 删除节点的标签名称 + "attributes": {"class": "left-content"}, // 删除节点的属性 + "index_in_parent": 0, // 删除节点在父节点的索引 + "parent_xpath": "/div/div[1]/div/div", // 删除节点的父节点的xpath + "parent_tag": "div", // 删除节点的父节点的标签名称 + "parent_attributes": {"class": "main-content fl-clr"} // 删除节点的父节点的属性 + }, + { + "del_location": "end", + "xpath": "/div/div[3]", + "tag": "div", + "attributes": {"class": "footer"}, + "index_in_parent": 2, + "parent_xpath": "/div", + "parent_tag": "div", + "parent_attributes": {} + } + ] +} + +``` + +\[^3\] + +```jsonc +[ + { + "del_location": "start", // 删除的位置【start（头部） or end（尾部）】 + "xpath": "/div/div[1]/div/div/div[1]", // 删除节点的xpath + "tag": "div", // 删除节点的标签名称 + "attributes": {"class": "left-content"}, // 删除节点的属性 + "index_in_parent": 0, // 删除节点在父节点的索引 + "parent_xpath": "/div/div[1]/div/div", // 删除节点的父节点的xpath + "parent_tag": "div", // 删除节点的父节点的标签名称 + "parent_attributes": {"class": "main-content fl-clr"} // 删除节点的父节点的属性 + }, + { + "del_location": "end", + "xpath": "/div/div[3]", + "tag": "div", + "attributes": {"class": "footer"}, + "index_in_parent": 2, + "parent_xpath": "/div", + "parent_tag": "div", + "parent_attributes": {} + } +] + +``` diff --git a/llm_web_kit/extractor/html/post_main_html_processer/asserts/img.png b/llm_web_kit/extractor/html/post_main_html_processer/asserts/img.png deleted file mode 100644 index d516393fbeecee8eb96e6ac96d837693e8bb0aa8..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 139133 zcmeFZbzD^K);Eqyihu%wAR#RcGDtIYcZjsoq0-WWgmkA2-3?OGAX3r|(%sF_F!1it zyPo$s=XuV3-rrxp&-rWkFnji1*SfxIt?ydbT01~VK@#gO@m(Y&BrItuF%={v)O{qR z+Xm>jf&V0zJTw`rJuH-8B0m z!aQYbn7XfGa(RN?!Wht^dAS;VtFYl|dj}I6m{=6H z#8w|!@#mLhVgH|hxsj~s=Vi=onmx~@;!m@=b|`>i)|k1NrQea!a4ma z*kIlBk45au)UMTFLJ!z6S*VC9PSTN%I0DJ$Dlh_A2m0^3DkEJGiHBKD`m_?RoBB6A zMS=&XcW}KGS;RczXw}320FfA6N_b9jfO5g(+rVPB8`Ns$A9ZV>|EY`HFfDbX5Lk3u z{MNcDzCfcn9VIFu~YgFGX8^%{}maZt5QtMrJeQ*|BP&>LOsYk8LmL`gdb)zKGD4N7)=OFZoFsH 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+0800 Subject: [PATCH 02/11] =?UTF-8?q?fix:=20image=E6=8F=90=E5=8F=96caption=20(?= =?UTF-8?q?#542)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../extractor/html/recognizer/image.py | 3 +- llm_web_kit/input/datajson.py | 10 ++++-- .../extractor/html/recognizer/test_image.py | 33 +++++++++++++++++++ .../extractor/test_extractor_chain.py | 2 +- 4 files changed, 43 insertions(+), 5 deletions(-) diff --git a/llm_web_kit/extractor/html/recognizer/image.py b/llm_web_kit/extractor/html/recognizer/image.py index 30f8241d..53f612dc 100644 --- a/llm_web_kit/extractor/html/recognizer/image.py +++ b/llm_web_kit/extractor/html/recognizer/image.py @@ -157,8 +157,7 @@ def __parse_img_elements(self, base_url: str, img_elements: HtmlElement, html_ob 'html': raw_img_html, # 保留原始 标签作为属性值 'format': 'url', # 指定图片格式，url|base } - if elem.text and elem.text.strip(): - attributes['caption'] = elem.text.strip() + attributes['caption'] = elem.xpath('normalize-space()') if tag in ['embed', 'object', 'iframe', 'video', 'audio', 'canvas']: if not [img_elem for img_elem in self.IMG_LABEL if img_elem in raw_img_html.lower()]: diff --git a/llm_web_kit/input/datajson.py b/llm_web_kit/input/datajson.py index 3ded4c2b..bf970f93 100644 --- a/llm_web_kit/input/datajson.py +++ b/llm_web_kit/input/datajson.py @@ -298,7 +298,7 @@ def __content_lst_node_2_md(self, content_lst_node: dict, exclude_inline_types: else: image_caption = '' - image_des = image_title if image_title else image_caption if image_caption else '' + image_des = image_title if image_title else '' # 优先使用data, 其次path.其中data是base64编码的图片，path是图片的url if image_data: if image_des: @@ -310,7 +310,13 @@ def __content_lst_node_2_md(self, content_lst_node: dict, exclude_inline_types: image = f'![{image_alt}]({image_path} "{image_des}")' else: image = f'![{image_alt}]({image_path})' - return image + + if image_caption: + image_with_caption = f'{image}\n\n{image_caption}' + else: + image_with_caption = image + + return image_with_caption elif node_type == DocElementType.AUDIO: return '' # TODO: 音频格式 elif node_type == DocElementType.VIDEO: diff --git a/tests/llm_web_kit/extractor/html/recognizer/test_image.py b/tests/llm_web_kit/extractor/html/recognizer/test_image.py index 73a4c3a0..6396c6d1 100644 --- a/tests/llm_web_kit/extractor/html/recognizer/test_image.py +++ b/tests/llm_web_kit/extractor/html/recognizer/test_image.py @@ -350,3 +350,36 @@ def test_complex_heading_image_removal(self): img_in_p.extend(p.xpath('.//img')) self.assertEqual(len(img_in_p), 0, '段落中不应该有img标签') + + def test_image_caption(self): + complex_html = """ +
+
+ Roger Moore in + + For + Your Eyes Only + . Photo Courtesy: United + Artists/Everett Collection + +
+
+ """ + element = html_to_element(complex_html) + base_url = 'http://example.com' + parts = self.img_recognizer.recognize(base_url, [(element, element)], complex_html) + html = element_to_html(parts[0][0]) + self.assertIn('caption="Roger Moore in For Your Eyes Only . Photo Courtesy: United Artists/Everett Collection', html) diff --git a/tests/llm_web_kit/extractor/test_extractor_chain.py b/tests/llm_web_kit/extractor/test_extractor_chain.py index a5f68540..5255efe9 100644 --- a/tests/llm_web_kit/extractor/test_extractor_chain.py +++ b/tests/llm_web_kit/extractor/test_extractor_chain.py @@ -105,7 +105,7 @@ def test_html_pipeline(self): self.assertEqual(html_content['content']['title'], 'image-title') self.assertEqual(html_content['content']['alt'], 'image-alt') self.assertEqual(html_content['content']['url'], 'https://www.test.com/test.png') - self.assertEqual(html_content['content']['caption'], None) + self.assertEqual(html_content['content']['caption'], '') # 然后是simple table html_content = html_content_list[4] From b5e993672fcb73e159e0cf68ebec35adbc89d679 Mon Sep 17 00:00:00 2001 From: chupei Date: Wed, 3 Sep 2025 20:33:04 +0800 Subject: [PATCH 03/11] docs: update readme (#545) --- README.md | 10 ++++++++++ docs/images/extract_method.png | Bin 0 -> 259047 bytes 2 files changed, 10 insertions(+) create mode 100644 docs/images/extract_method.png diff --git a/README.md b/README.md index 4269b211..337a8b23 100644 --- a/README.md +++ b/README.md @@ -75,6 +75,16 @@ llm-web-kit is a python library that .. ## Quick Start +![extract_method picture](/docs/images/extract_method.png) + +This diagram shows three main HTML content extraction methods: + +1. **extract by magic_html+recognize**: Two-stage complete extraction that first uses magic-html to extract main content, then converts it to structured markdown. + +2. **only extract by recognize**: Direct content recognition that converts main_html to structured format without main content identification. + +3. **only extract main_html by magic-html**: First-stage only extraction that identifies and extracts main content area while preserving HTML structure. + ### extract by magic_html+recognize ```python diff --git a/docs/images/extract_method.png b/docs/images/extract_method.png new file 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z8Y}cM&g8WIBDH1d-o5HkrtQBS^n;E)nB{JLKD6D$IuHX}fvbsa5@O(A+_j}yY*g-T z3PETGmg@nZlHCuFOL^nWfJng)k|(EU2VBN%z;S_3fXBhY1l{uh!`dCow{IM774P4D zA+7AlY1+dXs2D4gb?qr!Ge&*XY-1SSOpxTnIDF~9ZzXKxTpQqX(UcIIqlS4 z6|Z>gX!d&^T|G{UVmsBIk-R2P8u$1 zoU3IdnQH?vH0QFq?f?J) literal 0 HcmV?d00001 From 3a5f143d54bfd023867179160046c8bdf6adbf42 Mon Sep 17 00:00:00 2001 From: Brown <128157150+1041206149@users.noreply.github.com> Date: Fri, 5 Sep 2025 17:07:13 +0800 Subject: [PATCH 04/11] fix: escape '%' in MathML formula v2 (#548) --- .../html/recognizer/cc_math/tag_math.py | 3 + .../assets/ccmath/math_percentage.html | 104 ++++++++++++++++++ .../assets/ccmath/math_percentage_1.html | 3 + .../ccmath/math_percentage_inline_1.html | 2 + .../extractor/html/recognizer/test_math.py | 8 ++ 5 files changed, 120 insertions(+) create mode 100644 tests/llm_web_kit/extractor/html/recognizer/assets/ccmath/math_percentage.html create mode 100644 tests/llm_web_kit/extractor/html/recognizer/assets/ccmath/math_percentage_1.html create mode 100644 tests/llm_web_kit/extractor/html/recognizer/assets/ccmath/math_percentage_inline_1.html diff --git a/llm_web_kit/extractor/html/recognizer/cc_math/tag_math.py b/llm_web_kit/extractor/html/recognizer/cc_math/tag_math.py index 66816c2d..aed792c9 100644 --- a/llm_web_kit/extractor/html/recognizer/cc_math/tag_math.py +++ b/llm_web_kit/extractor/html/recognizer/cc_math/tag_math.py @@ -52,6 +52,9 @@ def modify_tree(cm: CCMATH, math_render: str, o_html: str, node: HtmlElement, pa mathml = re.sub(r'([^\s])\s+([^\s])', r'\1 \2', mathml) # remove extra spaces latex = cm.mml_to_latex(mathml) + # 处理未转义的%为\% + if latex: + latex = re.sub(r'(? + + + + + Mathematical Formulas with Percent Symbol + + + + + +

Mathematical Formulas Containing % Symbol

+ +

1. LaTeX Format Examples

+ +

Inline LaTeX formulas:

+

The percentage increase is calculated as $\frac{new - old}{old} \times 100\%$.

+ +

Display LaTeX formulas:

+

Compound interest formula:

+$$A = P\left(1 + \frac{r\%}{n}\right)^{nt}$$ + +

2. MathML Format Examples

+ +

Percentage change in MathML:

+ + + Δ + % + = + + + new + - + old + + old + + × + 100 + % + + + +

Discount percentage:

+ + + Discount + % + = + + + Original + - + Sale + + Original + + × + 100 + % + + + +

Puyu badcase:

+ + + + + % + + Cell Death + = + + + ( + 1 + − + ( + Post treatment cell counts + + + + + + / + initial cell counts + ) + ) + * + 100 + + + + + + diff --git a/tests/llm_web_kit/extractor/html/recognizer/assets/ccmath/math_percentage_1.html b/tests/llm_web_kit/extractor/html/recognizer/assets/ccmath/math_percentage_1.html new file mode 100644 index 00000000..e29bbe09 --- /dev/null +++ b/tests/llm_web_kit/extractor/html/recognizer/assets/ccmath/math_percentage_1.html @@ -0,0 +1,3 @@ +A = P\left(1 + \frac{r\%}{n}\right)^{nt} +\Delta \%=\frac{\mathrm{new}-\mathrm{old}}{\mathrm{old}}×100\% +\begin{array}{ll}\%\text{Cell Death}=& \left(1-\left(\text{Post treatment cell counts}\\ /\text{initial cell counts}\right)\right)*100\end{array} \ No newline at end of file diff --git a/tests/llm_web_kit/extractor/html/recognizer/assets/ccmath/math_percentage_inline_1.html b/tests/llm_web_kit/extractor/html/recognizer/assets/ccmath/math_percentage_inline_1.html new file mode 100644 index 00000000..1fcfe99a --- /dev/null +++ b/tests/llm_web_kit/extractor/html/recognizer/assets/ccmath/math_percentage_inline_1.html @@ -0,0 +1,2 @@ +\frac{new - old}{old} \times 100\% +\mathrm{Discount}\%=\frac{\mathrm{Original}-\mathrm{Sale}}{\mathrm{Original}}×100\% \ No newline at end of file diff --git a/tests/llm_web_kit/extractor/html/recognizer/test_math.py b/tests/llm_web_kit/extractor/html/recognizer/test_math.py index f41af0c0..a6e1825e 100644 --- a/tests/llm_web_kit/extractor/html/recognizer/test_math.py +++ b/tests/llm_web_kit/extractor/html/recognizer/test_math.py @@ -216,6 +216,14 @@ 'base_url': 'https://convertoctopus.com/4-7-years-to-minutes', 'expected': 'assets/ccmath/math_class_math_1.html', 'expected_inline': 'assets/ccmath/math_class_math_inline_1.html' + }, + { + 'input': [ + 'assets/ccmath/math_percentage.html', + ], + 'base_url': 'https://test.com/', + 'expected': 'assets/ccmath/math_percentage_1.html', + 'expected_inline': 'assets/ccmath/math_percentage_inline_1.html' } ] From 980997a4a6ca293a69b4371c80387ed09de403df Mon Sep 17 00:00:00 2001 From: renpengli01 Date: Tue, 9 Sep 2025 11:05:31 +0800 Subject: [PATCH 05/11] fix: update post main html new plan & unit test (#546) --- .../html/post_main_html_processer/README.md | 136 +- .../html/post_main_html_processer/add_tags.py | 136 -- .../assets/html0.html | 1377 +++++++++++++++++ .../assets/html1.html | 201 +++ .../assets/html2.html | 132 ++ .../post_main_html_processer/assets/img.png | Bin 140328 -> 140676 bytes .../assets/llm_res.json | 26 + .../post_main_html_processer/assets/rule.json | 23 + .../post_main_html_processer/choose_html.py | 230 ++- .../post_main_html_processer/generate_rule.py | 340 ---- .../html/post_main_html_processer/post_llm.py | 178 +-- .../post_main_html_processer/post_mapping.py | 101 +- .../post_main_html_processer/assets/0.html | 392 +++++ .../post_main_html_processer/assets/1.html | 111 ++ .../post_main_html_processer/assets/2.html | 392 +++++ .../assets/html0.html | 392 +++++ .../assets/html1.html | 111 ++ .../assets/html2.html | 392 +++++ .../test_choose_html.py | 158 ++ .../post_main_html_processer/test_post_llm.py | 90 ++ .../test_post_mapping.py | 213 +++ 21 files changed, 4493 insertions(+), 638 deletions(-) delete mode 100644 llm_web_kit/extractor/html/post_main_html_processer/add_tags.py create mode 100644 llm_web_kit/extractor/html/post_main_html_processer/assets/html0.html create mode 100644 llm_web_kit/extractor/html/post_main_html_processer/assets/html1.html create mode 100644 llm_web_kit/extractor/html/post_main_html_processer/assets/html2.html create mode 100644 llm_web_kit/extractor/html/post_main_html_processer/assets/llm_res.json create mode 100644 llm_web_kit/extractor/html/post_main_html_processer/assets/rule.json delete mode 100644 llm_web_kit/extractor/html/post_main_html_processer/generate_rule.py create mode 100644 tests/llm_web_kit/extractor/html/post_main_html_processer/assets/0.html create mode 100644 tests/llm_web_kit/extractor/html/post_main_html_processer/assets/1.html create mode 100644 tests/llm_web_kit/extractor/html/post_main_html_processer/assets/2.html create mode 100644 tests/llm_web_kit/extractor/html/post_main_html_processer/assets/html0.html create mode 100644 tests/llm_web_kit/extractor/html/post_main_html_processer/assets/html1.html create mode 100644 tests/llm_web_kit/extractor/html/post_main_html_processer/assets/html2.html create mode 100644 tests/llm_web_kit/extractor/html/post_main_html_processer/test_choose_html.py create mode 100644 tests/llm_web_kit/extractor/html/post_main_html_processer/test_post_llm.py create mode 100644 tests/llm_web_kit/extractor/html/post_main_html_processer/test_post_mapping.py diff --git a/llm_web_kit/extractor/html/post_main_html_processer/README.md b/llm_web_kit/extractor/html/post_main_html_processer/README.md index f1973c42..ceb5a1c8 100644 --- a/llm_web_kit/extractor/html/post_main_html_processer/README.md +++ b/llm_web_kit/extractor/html/post_main_html_processer/README.md @@ -6,78 +6,90 @@ ## 执行步骤 -| filename | function | input & input_type | output_type | 实现功能 | -| :--------------- | :-------------------------- | :------------------------------------------------------------------------------ | :------------------------- | :------------- | -| choose_html.py | select_typical_html | html_strs: html迭代器 | str | 选出代表html | -| add_tags.py | process_html | input_html: str | str | 添加itemid | -| post_llm.py | get_llm_response | api_key: str, url: str, html_id_str: str, model_name: str | Dict\[str, int\] \[^1\] | 模型打标 | -| generate_rule.py | restore_html_trim_ends_only | processed_html: str, llm_response: Dict\[str, int\](get_llm_response的输出结果) | Dict\[str, object\] \[^2\] | 生成删除规则 | -| post_mapping.py | mapping_html_by_rules | html_str: str, post_delete_node: List\[object\] \[^3\] | str | 推广到所有数据 | +### choose_html.py 选出代表html -\[^1\] - -```jsonc +``` +func: select_typical_htmls -{"item_id 1": 0, "item_id 2": 0, "item_id 3": 1, "item_id 4": 1, "item_id 5": 1, "item_id 6": 1, "item_id 7": 1} // 0：删除；1：保留 +输入参数： + html_strings: List[dict] + [ + {"html": "html字符串","filename": "数据来源路径"} + ] + select_n: int (选出代表html的数量，default: 3) +输出参数： + List[dict] + [ + {"html": "html字符串","filename": "数据来源路径"} + ] ``` -\[^2\] +### post_llm.py 模型识别生成规则 -```jsonc +``` +func: get_llm_response -{ - "html": "...", // 选出的代表main html经过处理之后的新html - "post_delete_node": [ // 删除规则 - { - "del_location": "start", // 删除的位置【start（头部） or end（尾部）】 - "xpath": "/div/div[1]/div/div/div[1]", // 删除节点的xpath - "tag": "div", // 删除节点的标签名称 - "attributes": {"class": "left-content"}, // 删除节点的属性 - "index_in_parent": 0, // 删除节点在父节点的索引 - "parent_xpath": "/div/div[1]/div/div", // 删除节点的父节点的xpath - "parent_tag": "div", // 删除节点的父节点的标签名称 - "parent_attributes": {"class": "main-content fl-clr"} // 删除节点的父节点的属性 - }, - { - "del_location": "end", - "xpath": "/div/div[3]", - "tag": "div", - "attributes": {"class": "footer"}, - "index_in_parent": 2, - "parent_xpath": "/div", - "parent_tag": "div", - "parent_attributes": {} - } - ] -} +输入参数： + html_strings: List[dict] + ["html0", "html1", "html2"] + api_key: str (openai api key) + url: str (openai api url) + model_name: str (openai model name) +输出参数： + str + [ + { + "xpath": "//div[@class='et_pb_social_media_follow']", + "parent_tag": "div", + "parent_attributes": { + "class": "et_pb_column et_pb_column_2_3 et_pb_column_6 et_pb_css_mix_blend_mode_passthrough et-last-child" + }, + "reson": "Social media follow links are non-core content, typically used for sharing and external linking." + }, + { + "xpath": "//form[@class='et_pb_contact_form clearfix']", + "parent_tag": "div", + "parent_attributes": { + "class": "et_pb_column et_pb_column_2_3 et_pb_column_6 et_pb_css_mix_blend_mode_passthrough et-last-child" + }, + "reson": "Contact form is a footer widget, often considered as part of the contact section rather than main content." + } + ] ``` -\[^3\] +### post_mapping.py 推广到所有数据 + +``` +func: mapping_html_by_rules -```jsonc -[ - { - "del_location": "start", // 删除的位置【start（头部） or end（尾部）】 - "xpath": "/div/div[1]/div/div/div[1]", // 删除节点的xpath - "tag": "div", // 删除节点的标签名称 - "attributes": {"class": "left-content"}, // 删除节点的属性 - "index_in_parent": 0, // 删除节点在父节点的索引 - "parent_xpath": "/div/div[1]/div/div", // 删除节点的父节点的xpath - "parent_tag": "div", // 删除节点的父节点的标签名称 - "parent_attributes": {"class": "main-content fl-clr"} // 删除节点的父节点的属性 - }, - { - "del_location": "end", - "xpath": "/div/div[3]", - "tag": "div", - "attributes": {"class": "footer"}, - "index_in_parent": 2, - "parent_xpath": "/div", - "parent_tag": "div", - "parent_attributes": {} - } -] +输入参数： + html_content: str + xpaths_to_remove: List[dict] + [ + { + "xpath": "//div[@class='et_pb_social_media_follow']", + "parent_tag": "div", + "parent_attributes": { + "class": "et_pb_column et_pb_column_2_3 et_pb_column_6 et_pb_css_mix_blend_mode_passthrough et-last-child" + }, + "reson": "Social media follow links are non-core content, typically used for sharing and external linking." + }, + { + "xpath": "//form[@class='et_pb_contact_form clearfix']", + "parent_tag": "div", + "parent_attributes": { + "class": "et_pb_column et_pb_column_2_3 et_pb_column_6 et_pb_css_mix_blend_mode_passthrough et-last-child" + }, + "reson": "Contact form is a footer widget, often considered as part of the contact section rather than main content." + } + ] +输出参数： + tuple[str, bool] + ( + html_content, # html字符串 + is_success # 推广是否成功 + ) ``` diff --git a/llm_web_kit/extractor/html/post_main_html_processer/add_tags.py b/llm_web_kit/extractor/html/post_main_html_processer/add_tags.py deleted file mode 100644 index 63066cf8..00000000 --- a/llm_web_kit/extractor/html/post_main_html_processer/add_tags.py +++ /dev/null @@ -1,136 +0,0 @@ -from typing import Generator - -from lxml import etree - -from llm_web_kit.libs.html_utils import element_to_html, html_to_element - - -def process_html(input_html: str) -> str: - """处理HTML，为元素添加连续的_item_id属性. - - Args: - input_html: 输入的HTML字符串 - - Returns: - 处理后的HTML字符串，其中每个文本节点都有唯一的_item_id - """ - if not input_html: - return '' - - try: - tree = html_to_element(input_html) - root = tree.xpath('//body')[0] if tree.xpath('//body') else tree - except Exception as e: - raise ValueError(f'Invalid HTML input: {e}') - - # 使用ID生成器确保ID连续 - id_generator = __item_id_generator() - - # 遍历所有元素并处理 - __process_elements(root, id_generator) - - return element_to_html(root) - - -def __item_id_generator() -> Generator[int, None, None]: - """生成连续的ID序列. - - Yields: - 连续递增的整数ID - """ - counter = 1 - while True: - yield counter - counter += 1 - - -def __process_elements(tree: etree.Element, id_generator: Generator[int, None, None]) -> None: - """处理DOM树中的所有元素，为文本节点添加_item_id. - - Args: - tree: HTML DOM树根节点 - id_generator: ID生成器 - """ - # 创建静态列表避免在迭代时修改DOM结构 - elements_to_process = list(tree.iter()) - - for element in elements_to_process: - # 处理叶子节点（无子元素） - if len(element) == 0: - __process_leaf_element(element, id_generator) - else: - # 处理非叶子节点 - __process_non_leaf_element(element, id_generator) - - -def __process_leaf_element(element: etree.Element, - id_generator: Generator[int, None, None]) -> None: - """处理叶子节点元素. - - Args: - element: 叶子节点元素 - id_generator: ID生成器 - """ - # 为叶子节点分配_item_id - element.set('_item_id', str(next(id_generator))) - - # 处理tail文本 - if element.tail and element.tail.strip(): - parent = element.getparent() - if parent is not None: - # 创建custom_tail元素并插入到当前元素之后 - custom_tail = __create_custom_element( - 'custom_tail', element.tail, id_generator - ) - element.tail = None - - # 插入到正确位置 - parent_index = parent.index(element) - parent.insert(parent_index + 1, custom_tail) - - -def __process_non_leaf_element(element: etree.Element, - id_generator: Generator[int, None, None]) -> None: - """处理非叶子节点元素. - - Args: - element: 非叶子节点元素 - id_generator: ID生成器 - """ - parent = element.getparent() - parent_index = parent.index(element) if parent is not None else -1 - - # 处理元素的text内容 - if element.text and element.text.strip(): - custom_text = __create_custom_element( - 'custom_text', element.text, id_generator - ) - element.text = None - element.insert(0, custom_text) - - # 处理元素的tail内容 - if element.tail and element.tail.strip(): - if parent is not None: - custom_tail = __create_custom_element( - 'custom_tail', element.tail, id_generator - ) - element.tail = None - parent.insert(parent_index + 1, custom_tail) - - -def __create_custom_element(tag: str, text_content: str, - id_generator: Generator[int, None, None]) -> etree.Element: - """创建带_item_id的自定义元素. - - Args: - tag: 元素标签名 - text_content: 元素文本内容 - id_generator: ID生成器 - - Returns: - 带_item_id属性的自定义元素 - """ - custom_elem = etree.Element(tag) - custom_elem.text = text_content - custom_elem.set('_item_id', str(next(id_generator))) - return custom_elem diff --git a/llm_web_kit/extractor/html/post_main_html_processer/assets/html0.html b/llm_web_kit/extractor/html/post_main_html_processer/assets/html0.html new file mode 100644 index 00000000..5d353acb --- /dev/null +++ b/llm_web_kit/extractor/html/post_main_html_processer/assets/html0.html @@ -0,0 +1,1377 @@ + + + + +
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z({uR&Nj;`QkwfneU#7BiYYWMLY!Zyd>3Repk$y_<_;jQjv;KYUN-Aov)!7BQ|Jm6}qbjUhCH4PJ z759z~D3W<2aqYHFO%|y_i zHWnPguu|fCjNsw^4l8h+-F@)oL-Jg+wLfG}=h!XYJ1=9!OC`@|sXvV#@6+Lq`(ZIL z)c5RvN!=+vH$&$g&A0KZaynPC^b4qfb2!o!xBP;V-RWJOnUBJF&aVHN_Ei71#hLX+ zQx4lsKP*_K?R}7^S}gyPms>X5+U3VOwj16Rjhj{b@sG`8Kfkkc-+;qa+A!!u8pp9G zf8`}Etv2Lak`jOLWsmsla}~~PyU!k4KK(;--X90YWZR5Vr}7WsC#`RM>fpD^_-8J@ zu;zG`)ZLvawvbqBHkgWtwXTQmN4;0iPlwfvAWuI2&paoHfkUsv`2YhDc)I$ztaD0e F0s!tK8qNR! diff --git a/llm_web_kit/extractor/html/post_main_html_processer/assets/llm_res.json b/llm_web_kit/extractor/html/post_main_html_processer/assets/llm_res.json new file mode 100644 index 00000000..a4b6afac --- /dev/null +++ b/llm_web_kit/extractor/html/post_main_html_processer/assets/llm_res.json @@ -0,0 +1,26 @@ +[ + { + "xpath": "//div[@class='et_pb_section_1 et_pb_with_background et_section_regular']", + "parent_tag": "div", + "parent_attributes": { + "id": "main-content" + }, + "reson": "This section contains contact information and social media links, which are typically non-core content placed at the bottom of a webpage." + }, + { + "xpath": "//ul[@class='et_pb_module et_pb_social_media_follow et_pb_social_media_follow_0 clearfix et_pb_text_align_center et_pb_bg_layout_light']", + "parent_tag": "div", + "parent_attributes": { + "class": "et_pb_section_1 et_pb_with_background et_section_regular" + }, + "reson": "This is a social media follow block, commonly considered non-core content and usually found at the bottom of pages." + }, + { + "xpath": "//form[@class='et_pb_contact_form clearfix']", + "parent_tag": "div", + "parent_attributes": { + "class": "et_pb_section_1 et_pb_with_background et_section_regular" + }, + "reson": "This is a contact form, often used for user interaction but not central to the main page content." + } +] diff --git a/llm_web_kit/extractor/html/post_main_html_processer/assets/rule.json b/llm_web_kit/extractor/html/post_main_html_processer/assets/rule.json new file mode 100644 index 00000000..14f98c55 --- /dev/null +++ b/llm_web_kit/extractor/html/post_main_html_processer/assets/rule.json @@ -0,0 +1,23 @@ +[ + { + "xpath": "//div[@class='breadcrumb']", + "parent_tag": "div", + "parent_attributes": { + "class": "master-wrapper-content" + } + }, + { + "xpath": "//div[@class='product-selectors']", + "parent_tag": "div", + "parent_attributes": { + "class": "page-body" + } + }, + { + "xpath": "//div[@class='pager']", + "parent_tag": "div", + "parent_attributes": { + "class": "page-body" + } + } +] diff --git a/llm_web_kit/extractor/html/post_main_html_processer/choose_html.py b/llm_web_kit/extractor/html/post_main_html_processer/choose_html.py index 7e59d356..83beb5dd 100644 --- a/llm_web_kit/extractor/html/post_main_html_processer/choose_html.py +++ b/llm_web_kit/extractor/html/post_main_html_processer/choose_html.py @@ -1,6 +1,228 @@ -from typing import Generator +from collections import Counter +from typing import List +from lxml import etree -def select_typical_html(html_strs: Generator[str]) -> str: - """从多个HTML中选出头部和尾部最复杂的html.""" - pass +from llm_web_kit.libs.html_utils import html_to_element + +IGNORE_TAGS = {'script', 'style', 'meta', 'link', 'br', 'noscript'} +# 语义化标签 +SEMANTIC_TAGS = { + 'header', 'nav', 'main', 'article', 'section', 'aside', + 'footer', 'figure', 'figcaption', 'time', 'mark', 'summary', + 'details', 'h1', 'h2', 'h3', 'h4', 'h5', 'h6' +} +# 交互标签 +INTERACTIVE_TAGS = {'a', 'button', 'input', 'select', 'textarea', 'img', 'audio', 'video'} +# 各项指标的权重 +WEIGHTS = { + 'tag_diversity': 0.25, # 标签多样性权重 + 'total_elements': 0.2, # 元素总数权重 + 'max_depth': 0.15, # 嵌套深度权重 + 'semantic_tags': 0.25, # 语义标签权重 + 'styled_elements': 0.1, # 样式元素权重 + 'interactive_elements': 0.05 # 交互元素权重 +} + + +def select_typical_htmls(html_strings: List[dict], select_n: int = 3) -> List[dict]: + """从多个HTML中选择最具代表性的select_n个HTML. + + Args: + html_strings: + { + "html": "html字符串", + "filename": "html路径" + } + select_n: 需要选择的HTML数量，默认为3 + + Returns: + 选中的HTML字符串列表 + """ + if not html_strings: + return [] + + # 分析每个HTML + html_analysis = [] + for htmlstr_file in html_strings: + try: + analysis = __analyze_html_structure(htmlstr_file['html']) + if analysis: + analysis['html'] = htmlstr_file['html'] + analysis['filename'] = htmlstr_file['filename'] + html_analysis.append(analysis) + except Exception: + continue + + # 根据多个维度评分并排序 + scored_htmls = [] + for analysis in html_analysis: + score = __calculate_representativeness_score(analysis) + scored_htmls.append({ + 'html': analysis['html'], + 'filename': analysis['filename'], + 'score': score, + 'analysis': analysis + }) + + # 按分数排序并选择前select_n个 + scored_htmls.sort(key=lambda x: x['score'], reverse=True) + return scored_htmls[:select_n] if scored_htmls else [] + + +def __analyze_html_structure(html_str: str) -> dict: + """分析HTML结构特征. + + Args: + html_str: HTML字符串 + + Returns: + 包含分析结果的字典 + """ + try: + tree = html_to_element(html_str) + except Exception: + return None + + # 获取所有元素 + all_elements = list(tree.iter()) + + # 过滤有效标签 + valid_elements = [elem for elem in all_elements if __is_valid_tag(elem.tag)] + + if not valid_elements: + return None + + # 统计标签类型 + tag_counter = Counter(elem.tag for elem in valid_elements) + + # 计算结构复杂度指标 + metrics = { + # 标签多样性 + 'tag_diversity': len(tag_counter), + + # 总元素数 + 'total_elements': len(valid_elements), + + # 嵌套深度 + 'max_depth': __calculate_max_depth(tree), + + # 结构化语义标签使用情况 + 'semantic_tags': __count_semantic_tags(valid_elements), + + # CSS类和ID的使用 + 'styled_elements': __count_styled_elements(valid_elements), + + # 链接和媒体元素 + 'interactive_elements': __count_interactive_elements(valid_elements), + } + + return metrics + + +def __is_valid_tag(tag: str) -> bool: + """检查是否为有效的HTML标签.""" + return (tag and isinstance(tag, str) and + tag not in IGNORE_TAGS and + not tag.startswith(' int: + """计算DOM树的最大深度. + + Args: + element: 根元素 + + Returns: + 最大深度 + """ + if not element.getchildren(): + return 1 + + max_child_depth = 0 + for child in element.getchildren(): + if __is_valid_tag(child.tag): + child_depth = __calculate_max_depth(child) + max_child_depth = max(max_child_depth, child_depth) + + return max_child_depth + 1 + + +def __count_semantic_tags(elements: List[etree.Element]) -> int: + """计算语义化标签的数量. + + Args: + elements: 元素列表 + + Returns: + 语义化标签数量 + """ + return len([elem for elem in elements if elem.tag in SEMANTIC_TAGS]) + + +def __count_styled_elements(elements: List[etree.Element]) -> int: + """计算有样式属性的元素数量. + + Args: + elements: 元素列表 + + Returns: + 有样式属性的元素数量 + """ + count = 0 + for elem in elements: + if 'class' in elem.attrib or 'id' in elem.attrib: + count += 1 + return count + + +def __count_interactive_elements(elements: List[etree.Element]) -> int: + """计算交互元素数量. + + Args: + elements: 元素列表 + + Returns: + 交互元素数量 + """ + + return len([elem for elem in elements if elem.tag in INTERACTIVE_TAGS]) + + +def __calculate_representativeness_score(analysis: dict) -> float: + """计算HTML的代表性分数. + + Args: + analysis: HTML分析结果 + + Returns: + 代表性分数 + """ + if not analysis: + return 0.0 + + # 归一化各项指标（避免某些指标过大影响结果） + normalized_scores = {} + + # 标签多样性得分 (通常10-30种标签) + normalized_scores['tag_diversity'] = min(analysis.get('tag_diversity', 0) / 20.0, 1.0) + + # 元素总数得分 (通常几十到几百个元素) + normalized_scores['total_elements'] = min(analysis.get('total_elements', 0) / 100.0, 1.0) + + # 嵌套深度得分 (通常2-10层) + normalized_scores['max_depth'] = min(analysis.get('max_depth', 0) / 8.0, 1.0) + + # 语义标签得分 + normalized_scores['semantic_tags'] = min(analysis.get('semantic_tags', 0) / 10.0, 1.0) + + # 样式元素得分 + normalized_scores['styled_elements'] = min(analysis.get('styled_elements', 0) / 20.0, 1.0) + + # 交互元素得分 + normalized_scores['interactive_elements'] = min(analysis.get('interactive_elements', 0) / 10.0, 1.0) + + # 计算加权总分 + total_score = sum(normalized_scores[key] * WEIGHTS[key] for key in WEIGHTS) + + return total_score diff --git a/llm_web_kit/extractor/html/post_main_html_processer/generate_rule.py b/llm_web_kit/extractor/html/post_main_html_processer/generate_rule.py deleted file mode 100644 index 74d743a7..00000000 --- a/llm_web_kit/extractor/html/post_main_html_processer/generate_rule.py +++ /dev/null @@ -1,340 +0,0 @@ -import re -from typing import Dict, List, Optional - -from lxml import etree - -from llm_web_kit.html_layout.html_layout_cosin import (RE_MD5, RE_NUM, RE_SHA1, - RE_TIMESTAMP, RE_UUID) -from llm_web_kit.libs.html_utils import element_to_html, html_to_element - - -def restore_html_trim_ends_only(processed_html: str, llm_response: Dict[str, int]) -> Dict[str, object]: - """只删除HTML开头和结尾连续状态为0的元素，保留其他所有元素， 并删除所有的_item_id属性。同时将删除的节点信息记录在 - post_delete_node 字段中，删除规则。 - - Args: - processed_html: 带有_item_id属性的HTML字符串 - llm_response: LLM的响应，格式为 {'item_id 1': 0, 'item_id 2': 1, ...} - 0表示删除，1表示保留 - - Returns: - { 'html': 处理后的HTML字符串, 'post_delete_node': List[dict] } - """ - if not processed_html: - return {'html': '', 'post_delete_node': []} - if not llm_response: - # 如果没有LLM响应，则返回原始HTML - return {'html': processed_html, 'post_delete_node': []} - - try: - tree = html_to_element(processed_html) - except Exception as e: - raise ValueError(f'Invalid HTML input: {e}') - - # 预处理LLM响应：转换为{item_id: 状态}的字典 - item_status = {} - for key, status in llm_response.items(): - # 提取'item_id X'中的数字X作为item_id - item_id = int(key.split()[1]) - item_status[item_id] = status - - # 只处理开头和结尾的删除元素，并记录删除信息 - deletion_logger = _DeletionLogger() - __trim_ends_only(tree, item_status, deletion_logger) - - # 移除所有_item_id属性 - __remove_all_item_id_attributes(tree) - - return {'html': element_to_html(tree), 'post_delete_node': deletion_logger.records} - - -def __trim_ends_only( - tree: etree.Element, - item_status: Dict[int, int], - deletion_logger: '_DeletionLogger', -) -> None: - """只删除开头和结尾连续状态为0的元素. - - Args: - tree: HTML DOM树根节点 - item_status: 元素状态字典 - """ - # 获取所有带_item_id的元素 - elements = tree.xpath('//*[@_item_id]') - if not elements: - return - - # 从开头删除连续状态为0的元素 - start_index = 0 - while start_index < len(elements): - element = elements[start_index] - try: - item_id = int(element.get('_item_id', '')) - except ValueError: - start_index += 1 - continue - - status = item_status.get(item_id, 1) - if status == 0: - # 记录并删除元素及内容，并检查父节点是否需要删除 - __remove_element_and_check_parent( - root=tree, - element=element, - del_location='start', - deletion_logger=deletion_logger - ) - start_index += 1 - else: - break # 遇到保留状态的元素，停止删除 - - # 重新获取元素列表（因为可能有变化） - elements = tree.xpath('//*[@_item_id]') - if not elements: - return - - # 从结尾删除连续状态为0的元素 - end_index = len(elements) - 1 - while end_index >= 0: - element = elements[end_index] - try: - item_id = int(element.get('_item_id', '')) - except ValueError: - end_index -= 1 - continue - - status = item_status.get(item_id, 1) - if status == 0: - # 记录并删除元素及内容，并检查父节点是否需要删除 - __remove_element_and_check_parent( - root=tree, - element=element, - del_location='end', - deletion_logger=deletion_logger - ) - end_index -= 1 - else: - break # 遇到保留状态的元素，停止删除 - - -def __remove_element_and_check_parent( - *, - root: etree.Element, - element: etree.Element, - del_location: str, - deletion_logger: '_DeletionLogger', -) -> None: - """删除元素并检查其父节点是否需要删除. - - Args: - element: 要删除的元素 - """ - # 在移除前记录要删除的元素 - deletion_logger.record_element(root, element, del_location) - - parent = element.getparent() - if parent is None: - return - - # 记录父节点原始状态 - parent_has_item_id = '_item_id' in parent.attrib - - # 直接从父节点中移除元素 - parent.remove(element) - - # 如果父节点有_item_id，不需要进一步处理 - if parent_has_item_id: - return - - # 检查父节点是否还有子元素或者文本内容 - __check_and_remove_empty_parent(root, parent, del_location, deletion_logger) - - -def __check_and_remove_empty_parent( - root: etree.Element, parent: etree.Element, del_location: str, deletion_logger: '_DeletionLogger' -) -> None: - """检查父节点是否为空，如果为空则删除它（递归检查） - - Args: - parent: 要检查的父节点 - """ - # 检查父节点是否为空（没有子元素且没有文本内容） - last_snapshot: Optional[dict] = None - while parent is not None and __is_element_empty(parent): - grandparent = parent.getparent() - - # 如果父节点有_item_id，停止递归 - if '_item_id' in parent.attrib: - break - - # 如果没有父节点（已经是根节点），停止递归 - if grandparent is None: - break - - # 在移除前拍摄快照，用于最终仅记录最顶层被级联删除的父节点 - last_snapshot = deletion_logger.snapshot_element(root, parent, del_location) - # 移除空的父节点 - grandparent.remove(parent) - parent = grandparent - - # 仅在存在级联删除时，记录一次父节点删除，并清理其所有子节点的记录 - if last_snapshot is not None: - deletion_logger.prune_descendants_and_record_parent(last_snapshot) - - -class _DeletionLogger: - """记录被删除节点的信息，并在父节点被级联删除时，仅保留父节点记录。 - - 记录字段示例： - { - 'xpath': '/html/body/div[1]/p[2]', - 'tag': 'p', - 'attributes': {'class': 'note', 'id': 'note'}, - 'index_in_parent': 1, - 'parent_xpath': '/html/body/div[1]', - 'parent_tag': 'div', - 'parent_attributes': {'class': 'container', 'id': 'container'} - } - """ - - def __init__(self) -> None: - self.records: List[dict] = [] - self._xpaths: set[str] = set() - - def _compute_xpath(self, root: etree.Element, element: etree.Element) -> str: - # 使用 root 构建 ElementTree，并计算 element 的 XPath - try: - return etree.ElementTree(root).getpath(element) - except Exception: - # 尝试使用 element 自己的 roottree - try: - return element.getroottree().getpath(element) - except Exception: - return '' - - def snapshot_element(self, root: etree.Element, element: etree.Element, del_location: str) -> dict: - xpath = self._compute_xpath(root, element) - parent = element.getparent() - parent_xpath = self._compute_xpath(root, parent) if parent is not None else '' - parent_tag = parent.tag if parent is not None else '' - parent_attributes = self.parse_attrs(parent) if parent is not None else {} - - index_in_parent = -1 - if parent is not None: - try: - index_in_parent = list(parent).index(element) - except ValueError: - index_in_parent = -1 - - attrs = self.parse_attrs(element) - - snapshot = { - 'del_location': del_location, - 'xpath': xpath, - 'tag': element.tag, - 'attributes': attrs, - 'index_in_parent': index_in_parent, - 'parent_xpath': parent_xpath, - 'parent_tag': parent_tag, - 'parent_attributes': parent_attributes, - } - return snapshot - - def parse_attrs(self, element: etree.Element) -> Dict: - attrs = {k: self.dynamic_attributes_preprocess(v) for k, v in element.attrib.items() if - k in ['class', 'id']} if element.attrib else {} - return attrs - - def dynamic_attributes_preprocess(self, attr_str: str) -> str: - """动态属性值标准化处理.""" - res_attr_str = '' - if attr_str: - attr_lst = attr_str.split() - if len(attr_lst) > 1: - res_attr_str = ' '.join([i for i in attr_lst if not RE_NUM.search(i)]) - elif len(attr_lst) == 1: - res_attr_str = self.standardizing_dynamic_attributes(attr_lst[0]) - return res_attr_str - - def standardizing_dynamic_attributes(self, attr_value: str) -> str: - """将动态属性值标准化为统一表示.""" - if RE_MD5.fullmatch(attr_value): - return '[MD5]' - if RE_SHA1.fullmatch(attr_value): - return '[SHA1]' - if RE_UUID.fullmatch(attr_value): - return '[UUID]' - if RE_TIMESTAMP.fullmatch(attr_value): - return '[TIMESTAMP]' - if RE_NUM.search(attr_value): - return re.sub(r'\d+', '', attr_value) - - return attr_value - - def record_element(self, root: etree.Element, element: etree.Element, del_location: str) -> None: - snap = self.snapshot_element(root, element, del_location) - # 去重：相同 xpath 的记录只保留一次（优先保留先记录的） - if snap['xpath'] and snap['xpath'] not in self._xpaths: - self.records.append(snap) - self._xpaths.add(snap['xpath']) - - def prune_descendants_and_record_parent(self, parent_snapshot: dict) -> None: - """删除所有位于 parent_snapshot['xpath'] 之下的子节点记录，仅保留父节点记录。""" - parent_xpath = parent_snapshot.get('xpath', '') - if not parent_xpath: - return - - # 过滤掉所有子孙节点记录 - kept: List[dict] = [] - new_xpaths: set[str] = set() - prefix = parent_xpath + '/' - for rec in self.records: - xp = rec.get('xpath', '') - if xp == parent_xpath or xp.startswith(prefix): - # 丢弃，稍后添加父节点快照 - continue - kept.append(rec) - if xp: - new_xpaths.add(xp) - - # 添加父节点快照（若未存在） - if parent_xpath not in new_xpaths: - kept.append(parent_snapshot) - new_xpaths.add(parent_xpath) - - self.records = kept - self._xpaths = new_xpaths - - -def __is_element_empty(element: etree.Element) -> bool: - """检查元素是否为空（没有子元素且没有有意义的文本内容） - - Args: - element: 要检查的元素 - - Returns: - 如果元素为空返回True，否则返回False - """ - # 检查是否有子元素 - if len(element) > 0: - return False - - # 检查是否有文本内容 - if element.text and element.text.strip(): - return False - - # 检查是否有tail内容 - if element.tail and element.tail.strip(): - return False - - return True - - -def __remove_all_item_id_attributes(tree: etree.Element) -> None: - """移除DOM树中所有元素的_item_id属性. - - Args: - tree: HTML DOM树根节点 - """ - for element in tree.iter(): - if '_item_id' in element.attrib: - del element.attrib['_item_id'] diff --git a/llm_web_kit/extractor/html/post_main_html_processer/post_llm.py b/llm_web_kit/extractor/html/post_main_html_processer/post_llm.py index 579ce1cf..d30120cf 100644 --- a/llm_web_kit/extractor/html/post_main_html_processer/post_llm.py +++ b/llm_web_kit/extractor/html/post_main_html_processer/post_llm.py @@ -1,99 +1,100 @@ -from loguru import logger -from openai import BadRequestError, OpenAI +import re +from pathlib import Path +from typing import List from llm_web_kit.libs.standard_utils import json_loads -html_str = """ - -

 - -""" - -promtp = f"""你是文本识别专家，输入一个html字符串，且每个标签都有一个属性值不同的属性_item_id，你通过识别html能够解析出每个_item_id对应的内容是否是主体内容，主要在于去除以下两部分的内容： -1.去除头部导航栏、时间、作者、广告、推荐等非正文主体内容； -2.去除尾部链接、分享、翻页、广告、推荐等非正文主体内容。 -注意，主体内容链接保留 -识别出主体内容之后根据_item_id生成字典作为返回结果，无需解释生成依据，其中0代表非主体内容需要去除，1代表是主体内容要保留。示例如下： -输入: {html_str} -返回结果: {{'item_id 1': 1, 'item_id 2': 1, 'item_id 3': 1, 'item_id 4': 1, 'item_id 5': 1, 'item_id 6': 1, 'item_id 7': 1, 'item_id 8': 1, 'item_id 9': 1, 'item_id 10': 1, 'item_id 11': 1, 'item_id 12': 1, 'item_id 13': 1, 'item_id 14': 1, 'item_id 15': 1, 'item_id 16': 1, 'item_id 17': 1, 'item_id 18': 1, 'item_id 19': 0, 'item_id 20': 0, 'item_id 21': 0, 'item_id 22': 0, 'item_id 23': 0, 'item_id 24': 0, 'item_id 25': 0, 'item_id 26': 0, 'item_id 27': 0}} -""" - - -def get_llm_response(api_key: str, url: str, html_id_str: str, model_name: str) -> dict: - # Set OpenAI's API key and API base to use vLLM's API server. +base_dir = Path(__file__).parent + + +def __get_eg_data(): + eg_input_lst = [] + for i in range(3): + eg_input_lst.append(base_dir.joinpath(f'assets/html{i}.html').read_text(encoding='utf-8')) + + eg_output = json_loads(base_dir.joinpath('assets/rule.json').read_text(encoding='utf-8')) + return eg_input_lst, eg_output + + +output_format = '''[ + { + "xpath": "XPath of the node of the non-core content body", + "parent_tag": "The label name of the parent node of the node that is not the core content body", + "parent_attributes": "The class and id attributes of the parent node of the node that is not the core content body", + "reson": "Reasons for determining it as non-core content" + } +]''' + + +def clean_json_data(md_text: str) -> dict: + cleaned = re.sub(r'^```json|\```', '', md_text, flags=re.MULTILINE) + try: + json_data = json_loads(cleaned) + except Exception: + return None + return json_data + + +def get_llm_response(input_lst: List, api_key: str, url: str, model_name: str, is_llm: bool = True, + max_retry: int = 3) -> dict: + if not is_llm: + post_llm_response = base_dir.joinpath('assets/llm_res.json').read_text(encoding='utf-8') + return json_loads(post_llm_response) + + from openai import BadRequestError, OpenAI + client = OpenAI( # 若没有配置环境变量，请用百炼API Key将下行替换为：api_key='sk-xxx', api_key=api_key, base_url=url, ) - content = f"""{promtp}以下是需要判断的html代码： - ``` - {html_id_str} - ``` - 返回结果： - """ + html_count = len(input_lst) + eg_input_lst, eg_output = __get_eg_data() + + prompt = f""" +You are an expert in HTML semantics and will be assigned the following task: +Accept input:{input_lst}containing{html_count}HTML pages. +The input has the following characteristics: +1. These{html_count}HTML pages are from the same website. +2. These{html_count}pages use the same template, differing only in their main content. +################ +The tasks you need to complete are: +Deeply understand the{html_count}HTML input and find the node information and node paths for the non-core content at the page header (top section) and page footer (bottom section) of the {html_count} HTML. +################ +You need to follow the following rules when completing the task: +1. Identify and extract non-core content modules located in the page header (top section) and page footer (bottom section) of the HTML body. Do not identify the main title as non-core content. Non-core content includes breadcrumb navigation, related article links, advertisements, page turning, sharing, recommended content, etc. +2. If non-core content appears in the middle of the HTML, such as the time and author in a forum reply, it can be ignored. +3. If a node contains non-core content main nodes and core content main nodes, its internal elements need to be further analyzed; if a node is a wrapper for the entire page content or a container node containing multiple child elements, its internal elements need to be further analyzed. +4. Tables have semantic ambiguity. When analyzing table nodes, we need to consider the following: if they present structured data (such as product tables or data reports), they are classified as core content. If they are used for layout or display of simple lists (such as navigation menus or link lists), there are two cases: if it is a complete table structure, mark the entire table as non-core content. If it is an incomplete table structure or complex nesting, further analysis of its internal elements is required. +5. Non-core content should be carefully analyzed to prevent misjudgment. Uncertain elements should be excluded from non-core content. +6. It is necessary to consider the location of the HTML node of the non-core content body and the commonality of semantics in the web page. +7. When considering node paths, semantics should be prioritized. Avoid using indexes in node paths. Attribute values should be correct, especially those composed of multiple values. All attributes should be correctly matched. +8. Use '//' and '/' correctly when considering node paths. '//' is used for recursive searches, while '/' is used to locate direct children. +9. When considering node paths, always use the element's original tag name in the HTML source code. +10. Each node of the final non-core content body must have only one type of content, and the content of this node must be determined to be all non-core content bodies, without inclusion relationships or uncertain factors. +################ +The return data needs to follow the following rules: +1. Both node attributes and parent node attributes only consider the id attribute and class attribute. If both the id attribute and the class attribute are empty, they are ignored. +2. The returned node path must be unique and no duplicates are allowed. +3. The result is returned in JSON array format, requiring all strings to be enclosed in double quotes and not containing any additional information. The output format is as follows: +{output_format} +################ +Here are some examples for your reference: + +input:{eg_input_lst} +return:{eg_output} + +################ +Now return your result:""" try: completion = client.chat.completions.create( model=model_name, + # 此处以qwen-plus为例，可按需更换模型名称。模型列表：https://help.aliyun.com/zh/model-studio/getting-started/models extra_body={'enable_thinking': False}, messages=[ - {'role': 'system', 'content': 'You are a text recognition expert.'}, - {'role': 'user', 'content': content} + {'role': 'system', 'content': 'You are a HTML semantics expert.'}, + {'role': 'user', 'content': prompt} ], ) @@ -101,10 +102,11 @@ def get_llm_response(api_key: str, url: str, html_id_str: str, model_name: str) rtn = completion.model_dump_json() rtn_detail = json_loads(rtn) post_llm_response = rtn_detail.get('choices', [])[0].get('message', {}).get('content', '') - if '}' not in post_llm_response: - logger.exception(f'post_llm_response more than token limit, post_llm_response: {post_llm_response}') - return None - return post_llm_response - except BadRequestError as e: - logger.exception(e) + return clean_json_data(post_llm_response) + except BadRequestError: return None + except Exception: + if max_retry > 0: + return get_llm_response(input_lst, api_key, url, model_name, is_llm, max_retry - 1) + else: + return None diff --git a/llm_web_kit/extractor/html/post_main_html_processer/post_mapping.py b/llm_web_kit/extractor/html/post_main_html_processer/post_mapping.py index ba396d89..27dc876b 100644 --- a/llm_web_kit/extractor/html/post_main_html_processer/post_mapping.py +++ b/llm_web_kit/extractor/html/post_main_html_processer/post_mapping.py @@ -1,13 +1,98 @@ from typing import List +from lxml import html -def mapping_html_by_rules(html_str: str, post_delete_node: List[object]) -> str: +from llm_web_kit.libs.html_utils import element_to_html, html_to_element + + +def mapping_html_by_rules(html_content: str, xpaths_to_remove: List[dict]) -> tuple[str, bool]: + """从HTML中删除指定XPath匹配的所有节点. + + 参数: + html_content (str): 原始HTML内容 + xpaths_to_remove (list): 需要删除的元素列表 + + 返回: + str: 处理后的HTML + bool: 推广是否成功 """ - 根据删除规则推广到所有html - Args: - html_str: main html - post_delete_node: 删除规则 - Returns: - 处理之后的html + if not html_content: + return html_content, False + + is_success = False + tree = html_to_element(html_content) + + for remove_node in xpaths_to_remove: + xpath_content = remove_node.get('xpath') + # 获取所有元素节点 + all_elements = [element for element in tree.iter() if isinstance(element, html.HtmlElement)] + for node in tree.xpath(xpath_content): + # 获取节点内容占比 + content_rate = __calculate_node_content_ratio(tree, node) + if content_rate > 0.4: + continue + # 获取节点的位置 + node_position = __analyze_node_position(all_elements, node) + if node_position == 'middle': + continue + # 删除节点及其所有子节点 + node.getparent().remove(node) + is_success = True + + return element_to_html(tree), is_success + + +def __calculate_node_content_ratio(tree: html.HtmlElement, node: html.HtmlElement) -> float: + """计算节点内容占比. + + 参数: + tree(html.HtmlElement): 根节点对象 + node (html.HtmlElement): 节点对象 + + 返回: + float: 节点内容占比 """ - return html_str + # 获取节点的文本内容 + text_content = node.text_content() + + total_contents = tree.text_content() + content_rate = len(text_content) / len(total_contents) if total_contents else 0 + return content_rate + + +def __analyze_node_position(all_elements: List[html.HtmlElement], target_node: html.HtmlElement): + # 计算总节点数 + total_nodes = len(all_elements) + + # 新增逻辑：检查元素是否在
或
标签内 + parent = target_node.getparent() + while parent is not None: + if parent.tag == 'header': + return 'start' + elif parent.tag == 'footer': + return 'end' + parent = parent.getparent() + + # 查找当前节点在全部节点中的索引 + node_index = -1 + for idx, element in enumerate(all_elements): + if element == target_node: + node_index = idx + break + + if node_index == -1: + # 无法定位节点在文档中的位置 + return None + + # 计算位置比例 + position_ratio = (node_index + 1) / total_nodes + + # 判断位置 + if position_ratio < 0.4: + position = 'start' + elif position_ratio > 0.7: + position = 'end' + else: + position = 'middle' + + return position diff --git a/tests/llm_web_kit/extractor/html/post_main_html_processer/assets/0.html b/tests/llm_web_kit/extractor/html/post_main_html_processer/assets/0.html new file mode 100644 index 00000000..9099f6a7 --- /dev/null +++ b/tests/llm_web_kit/extractor/html/post_main_html_processer/assets/0.html @@ -0,0 +1,392 @@ + + + +
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Music, dance, sculpture, painting—all are said to have their own language, a living + structure that enables each form to express, inspire and engage. The language of + music, for example, in its rhythm, melody, composition not only helps us understand + the power of the art, but guides those who create it. Just as Christopher Alexander + describes good architecture, the best of artistic expressions become as rich and + complex as life itself—indeed they become somehow alive—”like ocean waves or blades + of grass [their] parts are governed by the endless play of repetition and variety + created in the presence of the fact that all things pass.”

+ +

Perhaps similar principles apply in the social creations of our life, the families, + organizations, communities and businesses in which we “live and breathe and have our + being.” There are organizations, gatherings, companies that are full of life and + those that seem lifeless. We are able to both experience this quality and + participate in releasing it.

+ +

The purpose of this blog is explore this language of participation as the integrated + application of content, tools and practices in specific contexts, that is, patterns + of engagement that consistently release insight, energy, and creativity. The intent + is to provide a useful guide for individuals, whether or not they show up in formal + leadership roles, to engage in the art of building organizations, communities and + business environments that are enlivening.

+ +

These reflections have been inspired by my involvement with Liberating Voices: A Pattern + Language for a Communications Revolution, and associated Web site involving an international collaboration + of academics and practitioners.

+ +

Entries are organized into categories with distinctive icons:

+ +

+ +

Markers and Milestones are stories of discovery that indicate the + presence of a pattern or patterns but not yet clearly defined.

+ +

+ +

Compass Bearings provide a point of reference for determining the + relative strengths or weaknesses in a given situation as well as guidance for which + kinds of patterns may be most effective.

+ +

+ +

Pattern Abstracts are preliminary discussions of patterns in + preparation for formally posting them to the Liberating Voices Web site.

+ +

+ +

Patterns in Action illustrate concrete ways in which typical sets of + patterns tend to arise in similar situations.

+ +

+ +

Inner Voices, in the sense of musical + harmony, highlight the role of “mid-range” or “mid-register” voices in adding depth, + richness, texture while reinforcing an overall “centeredness” to the presence and + interplay of patterns in a given situation.

+ +
+ +
+ +
+
+ + +
+ +
+ +
+ + +
+ + + + \ No newline at end of file diff --git a/tests/llm_web_kit/extractor/html/post_main_html_processer/assets/2.html b/tests/llm_web_kit/extractor/html/post_main_html_processer/assets/2.html new file mode 100644 index 00000000..05f2c498 --- /dev/null +++ b/tests/llm_web_kit/extractor/html/post_main_html_processer/assets/2.html @@ -0,0 +1,392 @@ + + + +
+ +
+
+
+
+
+
+
+
+
+
+ + +
+ + +
+
+
+ +

Gillgren Communication + Services Inc

+ + Cultivating Civic Imagination and Engagement + +
+ +
+ + +
+ +
+ + +
+
+ + +
+
+ + +
+ + +

Pioneering in Communication as + Engagement 

+
+ +
+ +
+
+ + +
+ + +

Discovering and mobilizing the + power of communication as engagement across sectors, + generations, and cultures

+
+ +
+ +
+ + +
+
+
+ + +
+ + +
+
+ +
+ +
+

Community + Development

+ + +
+ +
+ +
+ +
+
+ + +
+ + +
+
+ +
+ +
+

Corporate + Communications

+ + +
+ +
+ +
+ +
+
+ + +
+ + +
+
+ +
+ +
+

Citizen + Journalism

+ + +
+ +
+ +
+ +
+ + +
+ + +
+
+ + + + +
+
+ + +
+ + +

Connect with Us

+
+ +
+
+ + +

Ways for keeping in touch…

+
+ +
+
+ + +
+
+ +
+ +
+ +

1140 N  192nd + Street, Apt B226
+ Shoreline, WA 09133

+
+ +
+ +
+ +
+
+ + +
+
+ +
+ +
+ +

+1 (206) + 755-9578

+
+ +
+ +
+ +
+
+ + +
+
+ +
+ +
+ +

+ ken@gillgrencommunication.com

+
+ +
+ +
+ +
+ +
+
+ + +
+ + +

Leave A Message

+ +
+ + +
+
+

+ + + + +

+

+ + + + +

+

+ + + + +

+ + +
+ + +
+ + +
+
+ +
+ + +
+ + +
+ + +
+
+ +
+ +
+ +
+ +
+
+ + +
+ +
+ +
+ + +
+ + + + \ No newline at end of file diff --git a/tests/llm_web_kit/extractor/html/post_main_html_processer/assets/html0.html b/tests/llm_web_kit/extractor/html/post_main_html_processer/assets/html0.html new file mode 100644 index 00000000..05f2c498 --- /dev/null +++ b/tests/llm_web_kit/extractor/html/post_main_html_processer/assets/html0.html @@ -0,0 +1,392 @@ + + + +
+ +
+
+
+
+
+
+
+
+
+
+ + +
+ + +
+
+
+ +

Gillgren Communication + Services Inc

+ + Cultivating Civic Imagination and Engagement + +
+ +
+ + +
+ +
+ + +
+
+ + +
+
+ + +
+ + +

Pioneering in Communication as + Engagement 

+
+ +
+ +
+
+ + +
+ + +

Discovering and mobilizing the + power of communication as engagement across sectors, + generations, and cultures

+
+ +
+ +
+ + +
+
+
+ + +
+ + +
+
+ +
+ +
+

Community + Development

+ + +
+ +
+ +
+ +
+
+ + +
+ + +
+
+ +
+ +
+

Corporate + Communications

+ + +
+ +
+ +
+ +
+
+ + +
+ + +
+
+ +
+ +
+

Citizen + Journalism

+ + +
+ +
+ +
+ +
+ + +
+ + +
+
+ + + + +
+
+ + +
+ + +

Connect with Us

+
+ +
+
+ + +

Ways for keeping in touch…

+
+ +
+
+ + +
+
+ +
+ +
+ +

1140 N  192nd + Street, Apt B226
+ Shoreline, WA 09133

+
+ +
+ +
+ +
+
+ + +
+
+ +
+ +
+ +

+1 (206) + 755-9578

+
+ +
+ +
+ +
+
+ + +
+
+ +
+ +
+ +

+ ken@gillgrencommunication.com

+
+ +
+ +
+ +
+ +
+
+ + +
+ + +

Leave A Message

+ +
+ + +
+
+

+ + + + +

+

+ + + + +

+

+ + + + +

+ + +
+ + +
+ + +
+
+ +
+ + +
+ + +
+ + +
+
+ +
+ +
+ +
+ +
+
+ + +
+ +
+ +
+ + +
+ + + + \ No newline at end of file diff --git a/tests/llm_web_kit/extractor/html/post_main_html_processer/assets/html1.html b/tests/llm_web_kit/extractor/html/post_main_html_processer/assets/html1.html new file mode 100644 index 00000000..e98ec8ce --- /dev/null +++ b/tests/llm_web_kit/extractor/html/post_main_html_processer/assets/html1.html @@ -0,0 +1,111 @@ + + + +
+ +
+
+
+
+
+
+

Languages, Patterns, and Participation

+ +
+

Music, dance, sculpture, painting—all are said to have their own language, a living + structure that enables each form to express, inspire and engage. The language of + music, for example, in its rhythm, melody, composition not only helps us understand + the power of the art, but guides those who create it. Just as Christopher Alexander + describes good architecture, the best of artistic expressions become as rich and + complex as life itself—indeed they become somehow alive—”like ocean waves or blades + of grass [their] parts are governed by the endless play of repetition and variety + created in the presence of the fact that all things pass.”

+ +

Perhaps similar principles apply in the social creations of our life, the families, + organizations, communities and businesses in which we “live and breathe and have our + being.” There are organizations, gatherings, companies that are full of life and + those that seem lifeless. We are able to both experience this quality and + participate in releasing it.

+ +

The purpose of this blog is explore this language of participation as the integrated + application of content, tools and practices in specific contexts, that is, patterns + of engagement that consistently release insight, energy, and creativity. The intent + is to provide a useful guide for individuals, whether or not they show up in formal + leadership roles, to engage in the art of building organizations, communities and + business environments that are enlivening.

+ +

These reflections have been inspired by my involvement with Liberating Voices: A Pattern + Language for a Communications Revolution, and associated Web site involving an international collaboration + of academics and practitioners.

+ +

Entries are organized into categories with distinctive icons:

+ +

+ +

Markers and Milestones are stories of discovery that indicate the + presence of a pattern or patterns but not yet clearly defined.

+ +

+ +

Compass Bearings provide a point of reference for determining the + relative strengths or weaknesses in a given situation as well as guidance for which + kinds of patterns may be most effective.

+ +

+ +

Pattern Abstracts are preliminary discussions of patterns in + preparation for formally posting them to the Liberating Voices Web site.

+ +

+ +

Patterns in Action illustrate concrete ways in which typical sets of + patterns tend to arise in similar situations.

+ +

+ +

Inner Voices, in the sense of musical + harmony, highlight the role of “mid-range” or “mid-register” voices in adding depth, + richness, texture while reinforcing an overall “centeredness” to the presence and + interplay of patterns in a given situation.

+ +
+ +
+ +
+
+ + +
+ +
+ +
+ + +
+ + + + \ No newline at end of file diff --git a/tests/llm_web_kit/extractor/html/post_main_html_processer/assets/html2.html b/tests/llm_web_kit/extractor/html/post_main_html_processer/assets/html2.html new file mode 100644 index 00000000..9099f6a7 --- /dev/null +++ b/tests/llm_web_kit/extractor/html/post_main_html_processer/assets/html2.html @@ -0,0 +1,392 @@ + + + +
+ +
+
+
+
+
+
+
+
+
+
+ + +
+ + +
+
+
+ +

Gillgren Communication + Services Inc

+ + Cultivating Civic Imagination and Engagement + +
+ +
+ + +
+ +
+ + +
+
+ + +
+
+ + +
+ + +

Pioneering in Communication as + Engagement 

+
+ +
+ +
+
+ + +
+ + +

Discovering and mobilizing the + power of communication as engagement across sectors, + generations, and cultures

+
+ +
+ +
+ + +
+
+
+ + +
+ + +
+
+ +
+ +
+

Community + Development

+ + +
+ +
+ +
+ +
+
+ + +
+ + +
+
+ +
+ +
+

Corporate + Communications

+ + +
+ +
+ +
+ +
+
+ + +
+ + +
+
+ +
+ +
+

Citizen + Journalism

+ + +
+ +
+ +
+ +
+ + +
+ + +
+
+ + + + +
+
+ + +
+ + +

Connect with Us

+
+ +
+
+ + +

Ways for keeping in touch…

+
+ +
+
+ + +
+
+ +
+ +
+ +

1140 N  192nd + Street, Apt B226
+ Shoreline, WA 09133

+
+ +
+ +
+ +
+
+ + +
+
+ +
+ +
+ +

+1 (206) + 755-9578

+
+ +
+ +
+ +
+
+ + +
+
+ +
+ +
+ +

+ ken@gillgrencommunication.com

+
+ +
+ +
+ +
+ +
+
+ + +
+ + +

Leave A Message

+ +
+ + +
+
+

+ + + + +

+

+ + + + +

+

+ + + + +

+ + +
+ + +
+ + +
+
+ +
+ + +
+ + +
+ + +
+
+ +
+ +
+ +
+ +
+
+ + +
+ +
+ +
+ + +
+ + + + \ No newline at end of file diff --git a/tests/llm_web_kit/extractor/html/post_main_html_processer/test_choose_html.py b/tests/llm_web_kit/extractor/html/post_main_html_processer/test_choose_html.py new file mode 100644 index 00000000..56844695 --- /dev/null +++ b/tests/llm_web_kit/extractor/html/post_main_html_processer/test_choose_html.py @@ -0,0 +1,158 @@ +import unittest +from pathlib import Path + +from llm_web_kit.extractor.html.post_main_html_processer.choose_html import \ + select_typical_htmls + +base_dir = Path(__file__).parent + + +class TestChooseHtml(unittest.TestCase): + + def test_select_typical_html_with_complex_and_simple_html(self): + """测试select_typical_html函数能正确选择最复杂的HTML.""" + + # 简单HTML示例 + simple_html = """ + + + Simple + + +
+

Simple content1

+

Simple content2

+
+ + + """ + + # 复杂HTML示例 + complex_html = """ + + + Complex Test + + + + +
+ +
+
+
+

Title

+

Paragraph 1

+

Paragraph 2

+
+

Section Title

+

Section content

+
+ Text + More text +
+
+
+ +
+
+

Footer content

+
+ + + """ + + # 空HTML示例 + empty_html = """ + + + Empty + + + + + """ + + html_lst = list() + html_lst.append({'html': simple_html, 'filename': 'xxx0'}) + html_lst.append({'html': complex_html, 'filename': 'xxx1'}) + html_lst.append({'html': empty_html, 'filename': 'xxx2'}) + + result = select_typical_htmls(html_lst, 1) + + # 验证返回的不是None + self.assertIsNotNone(result) + # 应该返回最复杂的HTML + self.assertEqual(result[0]['html'], complex_html) + + def test_select_typical_html_with_similar_htmls(self): + """测试select_typical_html处理相似HTML的情况.""" + html2 = base_dir.joinpath('assets/0.html').read_text(encoding='utf-8') + + html_lst = list() + for i in range(3): + filename = f'assets/{i}.html' + html_lst.append({'html': base_dir.joinpath(filename).read_text(encoding='utf-8'), 'filename': filename}) + + result = select_typical_htmls(html_lst, 1) + + # 验证返回的不是None + self.assertIsNotNone(result) + # 由于HTML复杂度相似，应返回第一个HTML + self.assertEqual(result[0]['html'], html2) + + def test_select_typical_html_with_empty_input(self): + """测试select_typical_html处理空输入.""" + + html_lst = [{'html': '', 'filename': 'xxx'}] + + result = select_typical_htmls(html_lst, 1) + + self.assertEqual(result, []) + + def test_select_typical_html_with_single_html(self): + """测试select_typical_html处理只有一个HTML的情况.""" + + single_html = base_dir.joinpath('assets/1.html').read_text(encoding='utf-8') + + html_lst = [{'html': single_html, 'filename': 'xxx'}] + + result = select_typical_htmls(html_lst, 1) + + # 验证返回的就是这个HTML + self.assertIsNotNone(result) + self.assertEqual(result[0]['html'], single_html) + + def test_select_typical_html_with_invalid_html(self): + """测试select_typical_html处理无效HTML的情况.""" + + single_html = base_dir.joinpath('assets/1.html').read_text(encoding='utf-8') + invalid_html = '' # 无效的HTML + + html_lst = list() + html_lst.append({'html': invalid_html, 'filename': 'xxx0'}) + html_lst.append({'html': single_html, 'filename': 'xxx1'}) + + result = select_typical_htmls(html_lst, 1) + + # 应该跳过无效HTML，返回有效HTML + self.assertIsNotNone(result) + self.assertEqual(result[0]['html'], single_html) + + def test_select_typical_html_with_zero(self): + """测试输入为空的情况.""" + html_lst = list() + + result = select_typical_htmls(html_lst, 1) + + # 应该跳过无效HTML，返回有效HTML + self.assertIsNotNone(result) diff --git a/tests/llm_web_kit/extractor/html/post_main_html_processer/test_post_llm.py b/tests/llm_web_kit/extractor/html/post_main_html_processer/test_post_llm.py new file mode 100644 index 00000000..cbef314b --- /dev/null +++ b/tests/llm_web_kit/extractor/html/post_main_html_processer/test_post_llm.py @@ -0,0 +1,90 @@ +import unittest +from pathlib import Path + +from llm_web_kit.extractor.html.post_main_html_processer.post_llm import ( + clean_json_data, get_llm_response) + +base_dir = Path(__file__).parent + + +class TestGetLLMResponse(unittest.TestCase): + + def setUp(self): + """测试前的准备工作.""" + self.api_key = '' + self.url = '' + self.model_name = '' + + # 模拟的LLM输出 + self.mock_response_content = [ + { + 'xpath': "//div[@class='et_pb_section_1 et_pb_with_background et_section_regular']", + 'parent_tag': 'div', + 'parent_attributes': {'id': 'main-content'}, + 'reson': 'This section contains contact information and social media links, which are typically non-core content placed at the bottom of a webpage.' + }, + { + 'xpath': "//ul[@class='et_pb_module et_pb_social_media_follow et_pb_social_media_follow_0 clearfix et_pb_text_align_center et_pb_bg_layout_light']", + 'parent_tag': 'div', + 'parent_attributes': {'class': 'et_pb_section_1 et_pb_with_background et_section_regular'}, + 'reson': 'This is a social media follow block, commonly considered non-core content and usually found at the bottom of pages.' + }, + { + 'xpath': "//form[@class='et_pb_contact_form clearfix']", + 'parent_tag': 'div', + 'parent_attributes': {'class': 'et_pb_section_1 et_pb_with_background et_section_regular'}, + 'reson': 'This is a contact form, often used for user interaction but not central to the main page content.' + } + ] + + def test_get_llm_response_success(self): + """测试成功获取LLM响应的情况.""" + test_input = [] + for i in range(3): + filename = f'assets/html{i}.html' + test_input.append(base_dir.joinpath(filename).read_text(encoding='utf-8')) + + # 调用被测试的方法 + result = get_llm_response(test_input, self.api_key, self.url, self.model_name, is_llm=False) + + # 验证结果 + self.assertEqual(len(result), len(self.mock_response_content)) + self.assertEqual(True, bool("//form[@class='et_pb_contact_form clearfix']" in str(result))) + + def test_get_llm_response_fail(self): + """测试获取LLM响应失败的情况.""" + test_input = [] + for i in range(3): + filename = f'assets/html{i}.html' + test_input.append(base_dir.joinpath(filename).read_text(encoding='utf-8')) + + # 调用被测试的方法 + result = get_llm_response(test_input, self.api_key, self.url, self.model_name) + + # 验证结果 + self.assertIsNone(result) + + def test_valid_json_without_markdown_wrapping(self): + """测试不带 Markdown 包裹的合法 JSON.""" + input_text = '{"key": "value"}' + expected = {'key': 'value'} + result = clean_json_data(input_text) + self.assertEqual(result, expected) + + def test_invalid_json_syntax(self): + """测试语法错误的 JSON.""" + input_text = '{"key": value}' + result = clean_json_data(input_text) + self.assertIsNone(result) + + def test_empty_content(self): + """测试空内容.""" + input_text = '' + result = clean_json_data(input_text) + self.assertIsNone(result) + + def test_completely_invalid_string(self): + """测试完全不相关的字符串.""" + input_text = 'this is not json' + result = clean_json_data(input_text) + self.assertIsNone(result) diff --git a/tests/llm_web_kit/extractor/html/post_main_html_processer/test_post_mapping.py b/tests/llm_web_kit/extractor/html/post_main_html_processer/test_post_mapping.py new file mode 100644 index 00000000..137a69a4 --- /dev/null +++ b/tests/llm_web_kit/extractor/html/post_main_html_processer/test_post_mapping.py @@ -0,0 +1,213 @@ +import unittest + +from llm_web_kit.extractor.html.post_main_html_processer.post_mapping import \ + mapping_html_by_rules + + +class TestMappingHtmlByRules(unittest.TestCase): + + def setUp(self): + """测试前的准备工作.""" + self.html_content = """ + + + + Test Page + + +
+

Website Header

+ +
+
+

Main Content Title

+

This is the main content that should be preserved.

+ +

More main content here.

+
+
+

Footer content with copyright information.

+ +
+ + + """ + + # 用于测试的XPath规则 + self.xpaths_to_remove = [ + { + 'xpath': "//div[@class='header']", + 'parent_tag': 'body', + 'parent_attributes': {}, + 'reson': 'Header is non-core content at the beginning' + }, + { + 'xpath': "//div[@class='footer']", + 'parent_tag': 'body', + 'parent_attributes': {}, + 'reson': 'Footer is non-core content at the end' + } + ] + + def test_mapping_html_by_rules_with_valid_input(self): + """测试使用有效输入参数的情况.""" + result, is_success = mapping_html_by_rules(self.html_content, self.xpaths_to_remove) + + # 验证结果是字符串 + self.assertIsInstance(result, str) + self.assertEqual(is_success, True) + + # 验证header被删除（位于开始位置） + self.assertNotIn('Website Header', result) + self.assertNotIn('Navigation Menu', result) + self.assertNotIn('class="header"', result) + + # 验证footer被删除（位于结束位置） + self.assertNotIn('Footer content with copyright', result) + self.assertNotIn('Social media links', result) + self.assertNotIn('class="footer"', result) + + # 验证中间主要内容仍然存在 + self.assertIn('Main Content Title', result) + self.assertIn('main content that should be preserved', result) + self.assertIn('More main content here', result) + + # 验证中间元素未被删除（即使在规则中） + self.assertIn('advertisement', result) + self.assertIn('Ad content', result) + + def test_mapping_html_by_rules_with_empty_html_content(self): + """测试空HTML内容的情况.""" + empty_html = '' + result, is_success = mapping_html_by_rules(empty_html, self.xpaths_to_remove) + + # 空HTML应该返回空字符串或处理后的结果 + self.assertIsInstance(result, str) + self.assertEqual(is_success, False) + + def test_mapping_html_by_rules_with_none_html_content(self): + """测试None作为HTML内容的情况.""" + result, is_success = mapping_html_by_rules(None, self.xpaths_to_remove) + # 空HTML应该返回空字符串或处理后的结果 + self.assertIsNone(result) + self.assertEqual(is_success, False) + + def test_mapping_html_by_rules_with_empty_xpaths_list(self): + """测试空XPath规则列表.""" + empty_xpaths = [] + result, is_success = mapping_html_by_rules(self.html_content, empty_xpaths) + + # 结果应该与原始HTML基本一致（去除格式差异） + self.assertIsInstance(result, str) + self.assertEqual(is_success, False) + # 应该保留所有原始内容 + self.assertIn('Website Header', result) + self.assertIn('Main Content Title', result) + self.assertIn('Footer content', result) + + def test_mapping_html_by_rules_with_none_xpaths_list(self): + """测试None作为XPath规则列表的情况.""" + with self.assertRaises(TypeError): + mapping_html_by_rules(self.html_content, None) + + def test_mapping_html_by_rules_with_invalid_xpath(self): + """测试包含无效XPath的规则列表.""" + invalid_xpaths = [ + { + 'xpath': "//div[@class='nonexistent']", + 'parent_tag': 'body', + 'parent_attributes': {}, + 'reson': 'Non-existent element' + } + ] + + # 应该不会抛出异常 + try: + result, is_success = mapping_html_by_rules(self.html_content, invalid_xpaths) + self.assertIsInstance(result, str) + self.assertEqual(is_success, False) + # 内容应该基本保持不变 + self.assertIn('Main Content Title', result) + except Exception as e: + self.fail(f'mapping_html_by_rules raised Exception unexpectedly: {e}') + + def test_mapping_html_by_rules_with_malformed_html(self): + """测试格式不正确的HTML.""" + malformed_html = "
Unclosed div
Content
" + result, is_success = mapping_html_by_rules(malformed_html, self.xpaths_to_remove) + + # 应该返回处理后的字符串，不会抛出异常 + self.assertIsInstance(result, str) + self.assertEqual(is_success, False) + + def test_mapping_html_by_rules_partial_match_xpath(self): + """测试部分匹配的XPath.""" + partial_xpaths = [ + { + 'xpath': "//div[contains(@class, 'header')]", + 'parent_tag': 'body', + 'parent_attributes': {}, + 'reson': 'Partial match header' + } + ] + + result, is_success = mapping_html_by_rules(self.html_content, partial_xpaths) + self.assertEqual(is_success, True) + + # 验证匹配的header被删除 + self.assertNotIn('Website Header', result) + self.assertNotIn('Navigation Menu', result) + + def test_mapping_html_by_rules_multiple_matches_same_xpath(self): + """测试XPath匹配多个元素的情况.""" + html_with_duplicates = """ + + +
First div to remove
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Main content here

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Second div to remove
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More content

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+ + + """ + + remove_rules = [ + { + 'xpath': "//div[@class='remove-me']", + 'parent_tag': 'body', + 'parent_attributes': {}, + 'reson': 'Remove all elements with this class' + } + ] + + result, is_success = mapping_html_by_rules(html_with_duplicates, remove_rules) + self.assertEqual(is_success, True) + + # 验证所有匹配的元素都被删除 + self.assertNotIn('Second div to remove', result) + + # 验证未匹配的元素仍然存在 + self.assertIn('Main content here', result) + self.assertIn('More content', result) + self.assertIn('Div to keep', result) + + def test_mapping_html_by_rules_middle_position_elements_not_removed(self): + """测试中间位置的元素不会被删除.""" + middle_element_rules = [ + { + 'xpath': "//div[@class='advertisement']", + 'parent_tag': 'div', + 'parent_attributes': {'class': 'main-content'}, + 'reson': 'Advertisement is in middle position' + } + ] + + result, is_success = mapping_html_by_rules(self.html_content, middle_element_rules) + self.assertEqual(is_success, False) + + # 验证广告元素未被删除（因为它在中间位置） + self.assertIn('advertisement', result) + self.assertIn('Ad content', result) From 5b09976bfb1c519923dbcb08d5252d1c44e4f570 Mon Sep 17 00:00:00 2001 From: Kaiwen Liu Date: Wed, 10 Sep 2025 16:50:19 +0800 Subject: [PATCH 06/11] fix: fix tail content bug and improve multiple same first dynamic class id match (#553) --- .../parser/layout_batch_parser.py | 13 +- .../test_multi_same_first_class_id.html | 1598 +++++++++++++++++ .../test_multi_same_first_class_id.json | 31 + .../test_multi_same_first_class_id_tag.html | 1250 +++++++++++++ .../parser/test_layout_parser.py | 27 + 5 files changed, 2912 insertions(+), 7 deletions(-) create mode 100644 tests/llm_web_kit/main_html_parser/parser/assets/input_layout_batch_parser/test_multi_same_first_class_id.html create mode 100644 tests/llm_web_kit/main_html_parser/parser/assets/input_layout_batch_parser/test_multi_same_first_class_id.json create mode 100644 tests/llm_web_kit/main_html_parser/parser/assets/input_layout_batch_parser/test_multi_same_first_class_id_tag.html diff --git a/llm_web_kit/main_html_parser/parser/layout_batch_parser.py b/llm_web_kit/main_html_parser/parser/layout_batch_parser.py index 5259cea3..5d6ef2e4 100644 --- a/llm_web_kit/main_html_parser/parser/layout_batch_parser.py +++ b/llm_web_kit/main_html_parser/parser/layout_batch_parser.py @@ -115,8 +115,7 @@ def normalize_key(self, tup): return None tag, class_id, idd = tup if class_id: - class_id = re.sub(r' +', ' ', class_id) - + class_id = re.sub(r'[ \t\n]+', ' ', class_id) if idd: valid_id = self.ids.get(idd, True) idd = re.sub(r' +', ' ', idd) @@ -146,9 +145,9 @@ def find_blocks_drop(self, element, depth, element_dict, parent_keyy, parent_lab length = len(self.get_tokens(element.text_content().strip())) length_tail = 0 text = element.xpath('string()').strip() - is_natural_language = self.__is_natural_language(text) or length_tail >= 10 if element.tail: length_tail = len(element.tail.strip()) + is_natural_language = self.__is_natural_language(text) or length_tail >= 10 idd = element.get('id') tag = element.tag layer_nodes = element_dict.get(depth, {}) @@ -276,7 +275,7 @@ def find_blocks_drop(self, element, depth, element_dict, parent_keyy, parent_lab # 判断当前节点是否是红色节点 if keyy in layer_nodes_dict: if 'red' not in layer_nodes_dict[keyy]: - if self.more_noise_enable and tag in ['p', 'ul', 'br'] and not idd and is_natural_language: + if self.more_noise_enable and tag in ['p', 'ul', 'br', 'b'] and not idd and is_natural_language: label = 'red' else: parent = element.getparent() @@ -397,6 +396,7 @@ def __match_tag_class(self, layer_nodes, current_layer_key, parent_key, node_htm def __match_tag(self, layer_nodes, current_layer_key, parent_key, node_html, template_doc, class_must=False, id_exist=False): current_norm_key = (self.normalize_key((current_layer_key[0], None, None)), parent_key) + first_class_res = None, None, None for ele_keyy, ele_value in layer_nodes.items(): # class id要存在 if class_must and not ele_keyy[1]: @@ -432,9 +432,8 @@ def __match_tag(self, layer_nodes, current_layer_key, parent_key, node_html, tem norm_ele_keyy_with_first_class = self.normalize_key((ele_keyy[0], ele_keyy[1].strip().split(' ')[0], None)) norm_ele_keyy_parent_with_first_class = (norm_ele_keyy_with_first_class, ele_parent_keyy) if current_norm_key_with_first_class == norm_ele_keyy_parent_with_first_class: - return ele_label, self.normalize_key(ele_keyy[0:3]), is_drop_tail - - return None, None, None + first_class_res = ele_label, self.normalize_key(ele_keyy[0:3]), is_drop_tail + return first_class_res def __is_natural_language(self, text, min_words=10): """判断文本是否像自然语言. diff --git a/tests/llm_web_kit/main_html_parser/parser/assets/input_layout_batch_parser/test_multi_same_first_class_id.html b/tests/llm_web_kit/main_html_parser/parser/assets/input_layout_batch_parser/test_multi_same_first_class_id.html new file mode 100644 index 00000000..859f2c73 --- /dev/null +++ b/tests/llm_web_kit/main_html_parser/parser/assets/input_layout_batch_parser/test_multi_same_first_class_id.html @@ -0,0 +1,1598 @@ + + + + + + + + + + + + + + + + + Christopher Clements - Spredfast Social Suite and Saloon 2016 + + +
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Spredfast Social Suite and Saloon 2016

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Brief
Spredfast wanted to follow it's successful 2015 SXSW presence with a stand-out, expanded suite experience and equally enticing party at the Moody Theater.
Process
We leveraged a public focus on space exploration, along with the recently released NASA Apollo missions image archive to create a truly unique, on-trend event experience that earned Spredfast a seat on CNBC's best brand experiences at SXSW list. Also on the list were brands like McDonald's, Spotify, American Greetings, Gatorade, Visa and Deloitte Digital.
Competencies
Creative direction, Graphic design, Event design, UX design, Illustration
Key Metrics
Event attendance tied to pipeline influence, social engagement, hashtag usage, press coverage, number of sales meetings planned/completed.

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+ + + + + + diff --git a/tests/llm_web_kit/main_html_parser/parser/assets/input_layout_batch_parser/test_multi_same_first_class_id.json b/tests/llm_web_kit/main_html_parser/parser/assets/input_layout_batch_parser/test_multi_same_first_class_id.json new file mode 100644 index 00000000..35e642e1 --- /dev/null +++ b/tests/llm_web_kit/main_html_parser/parser/assets/input_layout_batch_parser/test_multi_same_first_class_id.json @@ -0,0 +1,31 @@ +{ + "item_id 1": 1, + "item_id 2": 1, + "item_id 3": 1, + "item_id 4": 1, + "item_id 5": 1, + "item_id 6": 1, + "item_id 7": 1, + "item_id 8": 1, + "item_id 9": 1, + "item_id 10": 1, + "item_id 11": 1, + "item_id 12": 1, + "item_id 13": 1, + "item_id 14": 1, + "item_id 15": 1, + "item_id 16": 1, + "item_id 17": 1, + "item_id 18": 1, + "item_id 19": 0, + "item_id 20": 0, + "item_id 21": 0, + "item_id 22": 0, + "item_id 23": 0, + "item_id 24": 0, + "item_id 25": 0, + "item_id 26": 1, + "item_id 27": 1, + "item_id 28": 1, + "item_id 29": 0 +} \ No newline at end of file diff --git a/tests/llm_web_kit/main_html_parser/parser/assets/input_layout_batch_parser/test_multi_same_first_class_id_tag.html b/tests/llm_web_kit/main_html_parser/parser/assets/input_layout_batch_parser/test_multi_same_first_class_id_tag.html new file mode 100644 index 00000000..504cec0e --- /dev/null +++ b/tests/llm_web_kit/main_html_parser/parser/assets/input_layout_batch_parser/test_multi_same_first_class_id_tag.html @@ -0,0 +1,1250 @@ + + + + + + + + + + + + + + + + +Christopher Clements - Upland Software brand refresh + + +
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Upland Software brand refresh

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When I arrived at Upland, the company was beginning a major transition in operating style and GTM strategy, and they needed a fresh brand to help re-introduce the company to the world. Additionally, the old brand system was virtually unusable, without many of the tools and trappings that a modern, extensible, and accessible has to have. Additional considerations were a wide variety of stakeholder groups (as Upland is a public company) as well as their decentralized, hub-and-spoke internal structure with certain functions sitting at a 'corporate' level, and individual business unit marketing teams.
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Solution
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Upon realizing the scope and scale of this effort, and considering the very limited internal resources, we knew we needed to bring our creative, design, and branding networks to bear on the project. Collaborating across key internal teams, we brought in critical partners Unfettered and The Graphic Standard to get things rolling. Together, we guided key stakeholders through a robust discovery process that helped us land at core company values, brand promise, and stack hands market positioning where we knew Upland could really be competitive. 

From this foundational information, we quickly explored visual expression, honing in on a new visual brand that fit Upland perfectly, and would continue to grow with the complexity and size of the company for years to come. 
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Creative direction, vendor management, project planning, outsourcing, graphic design, illustration, UX, education & training
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Completion of key brand assets (web experience, brand system and components, templates, guidance documents, etc), usability and access, buy-in from key stakeholders and executive leadership, competitive within peer set, clarify values and brand promise
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Here's a brief look at how Upland was showing up in the market pre-refresh. Drowning in a sea of same, the company's communications were uninspiring, bogged down with superfluous language, and unable to clearly communicate.
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Here's a look at some of that foundational work identified early on, focusing the key stakeholders in on values, market positioning, and value proposition / promise: 
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Moving into visual exploration, the design team was able to guide the stakeholders towards the most successful execution (also our favorite) and sold the new direction through with little meaningful friction. 
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Here's a closer look into a few of the refreshed brand system components, starting with the refreshed logo and hierarchy:
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Simple linework illustrations we call 'elements' act as grounding and foundational visuals that help reinforce feelings of order, intention, logic, and precision. These are used in a variety of ways across the brand system, further extending the number of visual options users have at their disposal. 
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Instead of leveraging traditional iconography sets, we opted to introduce 'geometries'...small, glyph-like visuals that unlock more conceptual storytelling capabilities, especially when used in the context of presentation. Rather than finding that 'perfect' icon from a set that aligns to your messaging or story, meaning is assigned to more abstract forms, again nodding to the flexibility of this system.
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Illustration plays a big part in the brand system. By leveraging the foundational components of line work, color, simple shape, and our unique slanted angle, we arrive at an illustration style that's own-able for the brand. Additionally, this approach allows us to quickly conceptualize and showcase product features for newly acquired tech, something Upland does 4-6 times per calendar year. 
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Turning up the fidelity on the illustrations one more notch, we can use the same direction to simplify and stylize screenshots. This is incredibly important to promote a sense of consistency across the products, as it takes time to transition a newly acquired product into the new design language system. In the mean time, we can leverage stylized screens to communicate functionality, while also reducing visual complexity and focusing the viewer on specific features:
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From the very start, we knew that this illustration direction opened up huge opportunities for motion. We've had a few opportunities to experiment with full videos as well as looping GIFs, with very promising results. 
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An important part of any brand system are it's templates, and there was certainly no shortage of template needs for Upland. Working with our newly defined brand system, we engaged internal and external designers to create a library of templates covering a wide variety of use cases.
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and of course all the obligatory office-type documents:
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One of the most signifiant changes was our web presence, as evidenced by the stark difference between old and new home pages:
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To get a taste for the various modules and layouts that we implemented for the new site, check out the Figma project and prototype below:
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With a project as massive as this, it 100% takes an army of talented folks to get it done. My sincerest thanks go out to key internal and external players who helped bring this new brand to life:

Always-resilient Uplanders:
Jim Rudden, Virginia Miracle, Meredith Begin, Rod Favaron, Kendell Kelton, Sara Whitwer, Justin Schiavoni, Rachel Quinn, Daiko Hachiya & the product design team.

Design and branding legends who lent their time and talents to build out system:
Brett Eaton and Sharon Arellano and the Unfettered team, Shane Bzdok, John Norton, Gray Luckett and The Graphic Standard team, Rex Burns, Tom Reardon, Courtney Boyle, Annalee Lanier, Paulo Selletti @ Hypnotic Design, Dustin Scott @ GreatJob.TV, Megan Willin, Maggie Moore, Mariella Krause, Scott McAfee, Zoe Randolph, Barry Epstein, Todd Kelgard and the Ovation Solutions team.

Y'all moved mountains. Thank you.
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+
+
+ + + + + + + diff --git a/tests/llm_web_kit/main_html_parser/parser/test_layout_parser.py b/tests/llm_web_kit/main_html_parser/parser/test_layout_parser.py index d936fe89..c7a4df5f 100644 --- a/tests/llm_web_kit/main_html_parser/parser/test_layout_parser.py +++ b/tests/llm_web_kit/main_html_parser/parser/test_layout_parser.py @@ -420,3 +420,30 @@ def test_all_ids(self): parts = parser.parse(pre_data) main_html_body = parts[PreDataJsonKey.MAIN_HTML_BODY] assert '全部按定尺或倍尺供應,提高材料的利用率' in main_html_body and '在線留言' not in main_html_body and '批發兼零售' not in main_html_body + + def test_multi_same_first_class_id(self): + # 构造测试html + typical_raw_tag_html = base_dir.joinpath('assets/input_layout_batch_parser/test_multi_same_first_class_id_tag.html').read_text( + encoding='utf-8') + html_source = base_dir.joinpath('assets/input_layout_batch_parser/test_multi_same_first_class_id.html').read_text( + encoding='utf-8') + # 简化网页 + # 模型结果格式改写 + llm_path = 'assets/input_layout_batch_parser/test_multi_same_first_class_id.json' + llm_response = json.loads(base_dir.joinpath(llm_path).read_text(encoding='utf-8')) + pre_data = {'typical_raw_tag_html': typical_raw_tag_html, 'typical_raw_html': typical_raw_tag_html, + 'llm_response': llm_response, 'html_source': html_source} + pre_data = PreDataJson(pre_data) + # 映射 + parser = MapItemToHtmlTagsParser({}) + pre_data = parser.parse(pre_data) + + # 推广 + pre_data[PreDataJsonKey.DYNAMIC_ID_ENABLE] = True + pre_data[PreDataJsonKey.DYNAMIC_CLASSID_ENABLE] = True + pre_data[PreDataJsonKey.MORE_NOISE_ENABLE] = True + parser = LayoutBatchParser({}) + parts = parser.parse(pre_data) + main_html_body = parts[PreDataJsonKey.MAIN_HTML_BODY] + print(main_html_body) + assert 'Spredfast wanted to follow' in main_html_body and 'Photography' not in main_html_body From ada427b5ebb4fea108e62b67d6519f8efd6bc3e2 Mon Sep 17 00:00:00 2001 From: renpengli01 Date: Wed, 10 Sep 2025 16:52:16 +0800 Subject: [PATCH 07/11] fix: add to_plain_md by datajson.py & unit (#550) --- llm_web_kit/input/datajson.py | 32 +- llm_web_kit/libs/doc_element_type.py | 3 + llm_web_kit/simple.py | 28 +- .../input/assets/content_json.json | 297 ++++++++++++++++++ .../llm_web_kit/input/assets/to_plain_md.html | 119 +++++++ tests/llm_web_kit/input/test_datajson.py | 150 ++++++++- 6 files changed, 607 insertions(+), 22 deletions(-) create mode 100644 tests/llm_web_kit/input/assets/content_json.json create mode 100644 tests/llm_web_kit/input/assets/to_plain_md.html diff --git a/llm_web_kit/input/datajson.py b/llm_web_kit/input/datajson.py index bf970f93..e16d4ecf 100644 --- a/llm_web_kit/input/datajson.py +++ b/llm_web_kit/input/datajson.py @@ -48,6 +48,7 @@ class StructureMapper(ABC): Args: object (_type_): _description_ """ + def __init__(self): self.__txt_para_splitter = '\n' self.__md_para_splitter = '\n\n' @@ -55,12 +56,30 @@ def __init__(self): self.__list_item_start = '-' # md里的列表项前缀 self.__list_para_prefix = ' ' # 两个空格，md里的列表项非第一个段落的前缀：如果多个段落的情况，第二个以及之后的段落前缀 self.__md_special_chars = ['#', '`', '$'] # TODO 拼装table的时候还应该转义掉|符号 - self.__nodes_document_type = [DocElementType.MM_NODE_LIST, DocElementType.PARAGRAPH, DocElementType.LIST, DocElementType.SIMPLE_TABLE, DocElementType.COMPLEX_TABLE, DocElementType.TITLE, DocElementType.IMAGE, DocElementType.AUDIO, DocElementType.VIDEO, DocElementType.CODE, DocElementType.EQUATION_INTERLINE] + self.__nodes_document_type = [DocElementType.MM_NODE_LIST, DocElementType.PARAGRAPH, DocElementType.LIST, + DocElementType.SIMPLE_TABLE, DocElementType.COMPLEX_TABLE, DocElementType.TITLE, + DocElementType.IMAGE, DocElementType.AUDIO, DocElementType.VIDEO, + DocElementType.CODE, DocElementType.EQUATION_INTERLINE] self.__inline_types_document_type = [ParagraphTextType.EQUATION_INLINE, ParagraphTextType.CODE_INLINE] def to_html(self): raise NotImplementedError('This method must be implemented by the subclass.') + def to_plain_md(self, exclude_nodes=DocElementType.EXCLUDE_PLAIN_MD_LIST, + exclude_inline_types=DocElementType.EXCLUDE_PLAIN_MD_INLINE_LIST, use_raw_image_url=False): + """把content_list转化为md格式. + + Args: + exclude_nodes (list): 需要排除的节点类型 + exclude_inline_types: 需要排除的内联类型 + use_raw_image_url: 是否使用原始img url + Returns: + str: md格式的文本内容 + """ + self.__validate_exclude_nodes(exclude_nodes, exclude_inline_types) + md = self.__to_md(exclude_nodes, exclude_inline_types, use_raw_image_url) + return md + def to_txt(self, exclude_nodes=DocElementType.MM_NODE_LIST, exclude_inline_types=[]): """把content_list转化为txt格式. @@ -96,7 +115,8 @@ def __to_md(self, exclude_nodes=[], exclude_inline_types=[], use_raw_image_url=F for page in content_lst: for content_lst_node in page: if content_lst_node['type'] not in exclude_nodes: - txt_content = self.__content_lst_node_2_md(content_lst_node, exclude_inline_types, use_raw_image_url) + txt_content = self.__content_lst_node_2_md(content_lst_node, exclude_inline_types, + use_raw_image_url) if txt_content and len(txt_content) > 0: md_blocks.append(txt_content) @@ -243,7 +263,8 @@ def __process_nested_list(self, items, list_attribute, indent_level=0, exclude_i return result - def __content_lst_node_2_md(self, content_lst_node: dict, exclude_inline_types: list = [], use_raw_image_url=False) -> str: + def __content_lst_node_2_md(self, content_lst_node: dict, exclude_inline_types: list = [], + use_raw_image_url=False) -> str: """把content_list里定义的每种元素块转化为markdown格式. Args: @@ -253,7 +274,8 @@ def __content_lst_node_2_md(self, content_lst_node: dict, exclude_inline_types: """ node_type = content_lst_node['type'] if node_type == DocElementType.CODE: - code = content_lst_node['content']['code_content'] # 这里禁止有None的content, 如果有应该消灭在模块内部。模块应该处理更精细，防止因为拼装导致掩盖了错误。 + code = content_lst_node['content'][ + 'code_content'] # 这里禁止有None的content, 如果有应该消灭在模块内部。模块应该处理更精细，防止因为拼装导致掩盖了错误。 # 代码不可以 strip，因为首行可能有缩进，只能 rstrip code = code.rstrip() if not code: @@ -592,7 +614,7 @@ def get_content_list(self) -> ContentList: cl = self.__json_data[DataJsonKey.CONTENT_LIST] return cl - def get(self, key:str, default=None): + def get(self, key: str, default=None): return self.__json_data.get(key, default) def get_magic_html(self, page_layout_type=None): diff --git a/llm_web_kit/libs/doc_element_type.py b/llm_web_kit/libs/doc_element_type.py index dd962ed7..9d6e0953 100644 --- a/llm_web_kit/libs/doc_element_type.py +++ b/llm_web_kit/libs/doc_element_type.py @@ -21,3 +21,6 @@ class DocElementType(object): VIDEO = 'video' MM_NODE_LIST = [IMAGE, AUDIO, VIDEO] + + EXCLUDE_PLAIN_MD_LIST = [CODE, EQUATION_INTERLINE, IMAGE, COMPLEX_TABLE, AUDIO, VIDEO] + EXCLUDE_PLAIN_MD_INLINE_LIST = [ParagraphTextType.EQUATION_INLINE, ParagraphTextType.CODE_INLINE] diff --git a/llm_web_kit/simple.py b/llm_web_kit/simple.py index e8444636..fca6fbbb 100644 --- a/llm_web_kit/simple.py +++ b/llm_web_kit/simple.py @@ -96,7 +96,8 @@ def _extract_html(url: str, html_content: str, pipe_tpl: str, language: str = 'e # SDK方法（三种使用场景） # ======================================== -def extract_main_html_only(url: str, html_content: str, parser_type: str = PipeTpl.MAGIC_HTML, language: str = 'en') -> str: +def extract_main_html_only(url: str, html_content: str, parser_type: str = PipeTpl.MAGIC_HTML, + language: str = 'en') -> str: """场景1: 只执行第一阶段，抽取main_html. Args: @@ -118,7 +119,7 @@ def extract_content_from_main_html(url: str, main_html: str, output_format: str Args: url: 网页URL main_html: 已经抽取的主要HTML内容 - output_format: 输出格式，'md' 或 'mm_md' + output_format: 输出格式，'md' 或 'mm_md' 或 'plain_md' language: 语言，可选：'en' 或 'zh' Returns: @@ -131,19 +132,22 @@ def extract_content_from_main_html(url: str, main_html: str, output_format: str return content_list.to_nlp_md() elif output_format == 'mm_md': return content_list.to_mm_md() + elif output_format == 'plain_md': + return content_list.to_plain_md() elif output_format == 'json': return result.to_json() else: raise InvalidOutputFormatException(f'Invalid output format: {output_format}') -def extract_content_from_html_with_magic_html(url: str, html_content: str, output_format: str = 'md', language: str = 'en') -> str: +def extract_content_from_html_with_magic_html(url: str, html_content: str, output_format: str = 'md', + language: str = 'en') -> str: """场景3: 执行两个阶段，从magic_html抽取main_html，再从main_html抽取结构化内容. Args: url: 网页URL html_content: 原始HTML内容 - output_format: 输出格式，'md' 或 'mm_md' + output_format: 输出格式，'md' 或 'mm_md' 或 'plain_md' language: 语言，可选：'en' 或 'zh' Returns: @@ -156,19 +160,22 @@ def extract_content_from_html_with_magic_html(url: str, html_content: str, outpu return content_list.to_nlp_md() elif output_format == 'mm_md': return content_list.to_mm_md() + elif output_format == 'plain_md': + return content_list.to_plain_md() elif output_format == 'json': return result.to_json() else: raise InvalidOutputFormatException(f'Invalid output format: {output_format}') -def extract_content_from_html_with_llm(url: str, html_content: str, output_format: str = 'md', language: str = 'en') -> str: +def extract_content_from_html_with_llm(url: str, html_content: str, output_format: str = 'md', + language: str = 'en') -> str: """场景3: 执行两个阶段，从llm抽取main_html，再从main_html抽取结构化内容. Args: url: 网页URL html_content: 原始HTML内容 - output_format: 输出格式，'md' 或 'mm_md' + output_format: 输出格式，'md' 或 'mm_md' 或 'plain_md' language: 语言，可选：'en' 或 'zh' Returns: @@ -181,19 +188,22 @@ def extract_content_from_html_with_llm(url: str, html_content: str, output_forma return content_list.to_nlp_md() elif output_format == 'mm_md': return content_list.to_mm_md() + elif output_format == 'plain_md': + return content_list.to_plain_md() elif output_format == 'json': return result.to_json() else: raise InvalidOutputFormatException(f'Invalid output format: {output_format}') -def extract_content_from_html_with_layout_batch(url: str, html_content: str, output_format: str = 'md', language: str = 'en') -> str: +def extract_content_from_html_with_layout_batch(url: str, html_content: str, output_format: str = 'md', + language: str = 'en') -> str: """场景3: 执行两个阶段，从layout_batch抽取main_html，再从main_html抽取结构化内容. Args: url: 网页URL html_content: 原始HTML内容 - output_format: 输出格式，'md' 或 'mm_md' + output_format: 输出格式，'md' 或 'mm_md' 或 'plain_md' language: 语言，可选：'en' 或 'zh' Returns: @@ -206,6 +216,8 @@ def extract_content_from_html_with_layout_batch(url: str, html_content: str, out return content_list.to_nlp_md() elif output_format == 'mm_md': return content_list.to_mm_md() + elif output_format == 'plain_md': + return content_list.to_plain_md() elif output_format == 'json': return result.to_json() else: diff --git a/tests/llm_web_kit/input/assets/content_json.json b/tests/llm_web_kit/input/assets/content_json.json new file mode 100644 index 00000000..d8f038a2 --- /dev/null +++ b/tests/llm_web_kit/input/assets/content_json.json @@ -0,0 +1,297 @@ +{ + "track_id": "b68ba101-eaac-4eeb-83a2-761c43a91675", + "url": "http://example.com", + "html": "\n\n\n \n To Plain Markdown Test\n\n\n
==========================title====================================
\n

Title Test

\n
==========================code inline====================================
\n
\n
  • \n Dead simple\n Include prism.css and prism.js, use proper HTML5 code tags (code.language-xxxx), done!\n
    • \n
    \n
    ==========================code====================================
    \n
    \n
    Private Sub sitemenu_ItemCreated(ByVal sender As Object, ByVal e As Telerik.Web.UI.RadMenuEventArgs) Handles sitemenu.ItemCreated\n
    \n
            Dim item As RadMenuItem = sitemenu.FindItemByUrl(Request.Url.PathAndQuery)\n
    \n
            If item IsNot Nothing Then\n
    \n
                item.CssClass = \"focused\"\n
    \n
            End If\n
    \n
     
    \n
        End Sub\n
    \n
    \n\n
    ==========================simple table====================================
    \n
    \n \n \n \n \n \n \n \n \n \n
    1.12.1
    3.14.1
    \n
    \n
    ==========================complex table====================================
    \n
    \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    123
    4
    567
    \n     \n
    \n
    ==========================equation inline====================================
    \n

    测试行内公式$x=4$。

    \n
    ==========================equation interline====================================
    \n

    公式如下：

    $$a^2 + b^2 = c^2$$\n
    ==========================img====================================
    \n
    \n \"image\"\n
    \n
    ==========================list====================================
    \n
      \n
    • UL1 UL1.1
      • \n
    • UL2
      • \n
    \n
    \n
    HTML
    \n
    瓒呮枃鏈爣璁拌瑷€
    \n
    CSS
    \n
    灞傚彔鏍峰紡琛�
    \n
    \n
    ==========================paragraph====================================
    \n

    test paragraph

    \n
    ==========================audio====================================
    \n\n
    ==========================video====================================
    \n\n\n", + "dataset_name": "llm-web-kit-magic_html_noclip_html", + "data_source_category": "HTML", + "file_bytes": 4548, + "language": "en", + "meta_info": { + "input_datetime": "2025-09-10 15:59:01" + }, + "content_list": [ + [ + { + "type": "paragraph", + "raw_content": "
    ==========================title====================================
    ", + "content": [ + { + "c": "==========================title====================================", + "t": "text" + } + ] + }, + { + "type": "title", + "raw_content": "

    Title Test

    ", + "content": { + "title_content": "Title Test", + "level": "1" + } + }, + { + "type": "paragraph", + "raw_content": "
    ==========================code inline====================================
    ", + "content": [ + { + "c": "==========================code inline====================================", + "t": "text" + } + ] + }, + { + "type": "paragraph", + "raw_content": "
  • Dead simple\n Include prism.css and prism.js, use proper HTML5 code tags (code.language-xxxx), done!\n
    • ", + "content": [ + { + "c": "Dead simple Include prism.css and prism.js, use proper HTML5 code tags (", + "t": "text" + }, + { + "c": "code.language-xxxx", + "t": "code-inline" + }, + { + "c": "), done!", + "t": "text" + } + ] + }, + { + "type": "paragraph", + "raw_content": "
    ==========================code====================================
    ", + "content": [ + { + "c": "==========================code====================================", + "t": "text" + } + ] + }, + { + "type": "code", + "raw_content": "
    \n
    Private Sub sitemenu_ItemCreated(ByVal sender As Object, ByVal e As Telerik.Web.UI.RadMenuEventArgs) Handles sitemenu.ItemCreated\n
    \n
            Dim item As RadMenuItem = sitemenu.FindItemByUrl(Request.Url.PathAndQuery)\n
    \n
            If item IsNot Nothing Then\n
    \n
                item.CssClass = \"focused\"\n
    \n
            End If\n
    \n
     
    \n
        End Sub\n
    \n
    \n\n", + "inline": false, + "content": { + "code_content": "Private Sub sitemenu_ItemCreated(ByVal sender As Object, ByVal e As Telerik.Web.UI.RadMenuEventArgs) Handles sitemenu.ItemCreated\n Dim item As RadMenuItem = sitemenu.FindItemByUrl(Request.Url.PathAndQuery)\n If item IsNot Nothing Then\n item.CssClass = \"focused\"\n End If\n\n End Sub", + "by": "tag_code" + } + }, + { + "type": "paragraph", + "raw_content": "
    ==========================simple table====================================
    ", + "content": [ + { + "c": "==========================simple table====================================", + "t": "text" + } + ] + }, + { + "type": "simple_table", + "raw_content": "
    1.12.1
    3.14.1
    ", + "content": { + "html": "
    1.12.1
    3.14.1
    ", + "is_complex": false, + "table_nest_level": "1" + } + }, + { + "type": "paragraph", + "raw_content": "
    ==========================complex table====================================
    ", + "content": [ + { + "c": "==========================complex table====================================", + "t": "text" + } + ] + }, + { + "type": "complex_table", + "raw_content": "
    123
    4
    567
    ", + "content": { + "html": "
    123
    4
    567
    ", + "is_complex": true, + "table_nest_level": "1" + } + }, + { + "type": "paragraph", + "raw_content": "
    ==========================equation inline====================================
    ", + "content": [ + { + "c": "==========================equation inline====================================", + "t": "text" + } + ] + }, + { + "type": "paragraph", + "raw_content": "

    测试行内公式x=4。

    ", + "content": [ + { + "c": "测试行内公式", + "t": "text" + }, + { + "c": "x=4", + "t": "equation-inline" + }, + { + "c": "。", + "t": "text" + } + ] + }, + { + "type": "paragraph", + "raw_content": "
    ==========================equation interline====================================
    ", + "content": [ + { + "c": "==========================equation interline====================================", + "t": "text" + } + ] + }, + { + "type": "paragraph", + "raw_content": "

    公式如下：

    ", + "content": [ + { + "c": "公式如下：", + "t": "text" + } + ] + }, + { + "type": "equation-interline", + "raw_content": "

    $$a^2 + b^2 = c^2$$

    ", + "content": { + "math_content": "a^2 + b^2 = c^2", + "math_type": "latex", + "by": "None" + } + }, + { + "type": "paragraph", + "raw_content": "
    ==========================img====================================
    ", + "content": [ + { + "c": "==========================img====================================", + "t": "text" + } + ] + }, + { + "type": "image", + "raw_content": "\"image\"", + "content": { + "url": "http://example.com/image.png", + "data": null, + "alt": "image", + "title": null, + "caption": "" + } + }, + { + "type": "paragraph", + "raw_content": "
    ==========================list====================================
    ", + "content": [ + { + "c": "==========================list====================================", + "t": "text" + } + ] + }, + { + "type": "list", + "raw_content": "
    • UL1 UL1.1
    • UL2
    ", + "content": { + "items": [ + { + "c": "UL1 UL1.1" + }, + { + "c": "UL2" + } + ], + "list_attribute": "unordered", + "list_nest_level": "1" + } + }, + { + "type": "list", + "raw_content": "
    HTML
    瓒呮枃鏈爣璁拌瑷€
    CSS
    灞傚彔鏍峰紡琛�
    ", + "content": { + "items": [ + { + "c": "HTML" + }, + { + "c": "瓒呮枃鏈爣璁拌瑷€" + }, + { + "c": "CSS" + }, + { + "c": "灞傚彔鏍峰紡琛" + } + ], + "list_attribute": "definition", + "list_nest_level": "1" + } + }, + { + "type": "paragraph", + "raw_content": "
    ==========================paragraph====================================
    ", + "content": [ + { + "c": "==========================paragraph====================================", + "t": "text" + } + ] + }, + { + "type": "paragraph", + "raw_content": "

    test paragraph

    ", + "content": [ + { + "c": "test paragraph", + "t": "text" + } + ] + }, + { + "type": "paragraph", + "raw_content": "
    ==========================audio====================================
    ", + "content": [ + { + "c": "==========================audio====================================", + "t": "text" + } + ] + }, + { + "type": "paragraph", + "raw_content": "
    ==========================video====================================
    ", + "content": [ + { + "c": "==========================video====================================", + "t": "text" + } + ] + }, + { + "type": "paragraph", + "raw_content": "
    ", + "content": [ + { + "c": "Download the WEBM or MP4 video.", + "t": "text" + } + ] + } + ] + ], + "main_html": "
    \n
    ==========================title====================================
    \n

    Title Test

    \n
    ==========================code inline====================================
    \n
    \n
  • \n Dead simple\n Include prism.css and prism.js, use proper HTML5 code tags (code.language-xxxx), done!\n
    • \n
    \n
    ==========================code====================================
    \n
    \n
    Private Sub sitemenu_ItemCreated(ByVal sender As Object, ByVal e As Telerik.Web.UI.RadMenuEventArgs) Handles sitemenu.ItemCreated\n
    \n
            Dim item As RadMenuItem = sitemenu.FindItemByUrl(Request.Url.PathAndQuery)\n
    \n
            If item IsNot Nothing Then\n
    \n
                item.CssClass = \"focused\"\n
    \n
            End If\n
    \n
     
    \n
        End Sub\n
    \n
    \n\n
    ==========================simple table====================================
    \n
    \n \n \n \n \n \n \n \n \n \n
    1.12.1
    3.14.1
    \n
    \n
    ==========================complex table====================================
    \n
    \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
    123
    4
    567
    \n     \n
    \n
    ==========================equation inline====================================
    \n

    测试行内公式$x=4$。

    \n
    ==========================equation interline====================================
    \n

    公式如下：

    $$a^2 + b^2 = c^2$$\n
    ==========================img====================================
    \n
    \n \"image\"\n
    \n
    ==========================list====================================
    \n
      \n
    • UL1 UL1.1
      • \n
    • UL2
      • \n
    \n
    \n
    HTML
    \n
    瓒呮枃鏈爣璁拌瑷€
    \n
    CSS
    \n
    灞傚彔鏍峰紡琛�
    \n
    \n
    ==========================paragraph====================================
    \n

    test paragraph

    \n
    ==========================audio====================================
    \n\n
    ==========================video====================================
    \n\n\n
    ", + "title": " Test" +} \ No newline at end of file diff --git a/tests/llm_web_kit/input/assets/to_plain_md.html b/tests/llm_web_kit/input/assets/to_plain_md.html new file mode 100644 index 00000000..4881e86c --- /dev/null +++ b/tests/llm_web_kit/input/assets/to_plain_md.html @@ -0,0 +1,119 @@ + + + + + To Plain Markdown Test + + +
    ==========================title====================================
    +

    Title Test

    +
    ==========================code inline====================================
    +
    +
  • + Dead simple + Include prism.css and prism.js, use proper HTML5 code tags (code.language-xxxx), done! +
    • +
    +
    ==========================code====================================
    +
    +
    Private Sub sitemenu_ItemCreated(ByVal sender As Object, ByVal e As Telerik.Web.UI.RadMenuEventArgs) Handles sitemenu.ItemCreated +
    +
            Dim item As RadMenuItem = sitemenu.FindItemByUrl(Request.Url.PathAndQuery) +
    +
            If item IsNot Nothing Then +
    +
                item.CssClass = "focused" +
    +
            End If +
    +
     
    +
        End Sub +
    +
    + +
    ==========================simple table====================================
    +
    + + + + + + + + + +
    1.12.1
    3.14.1
    +
    +
    ==========================complex table====================================
    +
    + + + + + + + + + + + + + + + + +
    123
    4
    567
    +     +
    +
    ==========================equation inline====================================
    +

    测试行内公式$x=4$。

    +
    ==========================equation interline====================================
    +

    公式如下：

    $$a^2 + b^2 = c^2$$ +
    ==========================img====================================
    +
    + image +
    +
    ==========================list====================================
    +
      +
    • UL1 UL1.1
      • +
    • UL2
      • +
    +
    +
    HTML
    +
    瓒呮枃鏈爣璁拌瑷€
    +
    CSS
    +
    灞傚彔鏍峰紡琛�
    +
    +
    ==========================paragraph====================================
    +

    test paragraph

    +
    ==========================audio====================================
    + +
    ==========================video====================================
    + + + \ No newline at end of file diff --git a/tests/llm_web_kit/input/test_datajson.py b/tests/llm_web_kit/input/test_datajson.py index a9f1c7e8..6996fc38 100644 --- a/tests/llm_web_kit/input/test_datajson.py +++ b/tests/llm_web_kit/input/test_datajson.py @@ -168,13 +168,13 @@ def test_datajson_exclude_nodes_to_mmd(self): def test_data_json_deepcopy(self): """从一个外部dict构建datajson, 改变datajson，不改变外部dict.""" d = {'track_id': '32266dfa-c335-45c5-896e-56f057889d28', - 'url': 'http://mathematica.stackexchange.com/users/1931/ywdr1987?tab=activity&sort=all', - 'html': '', - 'page_layout_type': 'forum', - 'domain': 'mathematica.stackexchange.com', - 'dataset_name': 'math', - 'data_source_category': 'HTML', - 'meta_info': {'warc_headers': {'WARC-IP-Address': '104.16.12.13'}}} + 'url': 'http://mathematica.stackexchange.com/users/1931/ywdr1987?tab=activity&sort=all', + 'html': '', + 'page_layout_type': 'forum', + 'domain': 'mathematica.stackexchange.com', + 'dataset_name': 'math', + 'data_source_category': 'HTML', + 'meta_info': {'warc_headers': {'WARC-IP-Address': '104.16.12.13'}}} copied = copy.deepcopy(d) _ = DataJson(copied) cl = copied.get('content_list') # 不该变外部变量d @@ -263,7 +263,8 @@ def test_default_exclude(): def test_custom_exclude(): datajson = DataJson(d) - md = datajson.get_content_list().to_nlp_md(exclude_nodes=[DocElementType.COMPLEX_TABLE, DocElementType.SIMPLE_TABLE]) + md = datajson.get_content_list().to_nlp_md( + exclude_nodes=[DocElementType.COMPLEX_TABLE, DocElementType.SIMPLE_TABLE]) assert 'Ziet u iets wat niet hoort of niet klopt?' in md assert 'Openingstijden' in md assert 'Maandag' not in md @@ -390,7 +391,8 @@ def test_to_txt_with_math_delimiters(self): for case in test_cases: doc = DataJson(case['data']) result = doc.get_content_list().to_txt() - assert result.strip() == case['expected'].strip(), f"测试失败: 期望得到 '{case['expected']}' 但得到 '{result.strip()}'" + assert result.strip() == case[ + 'expected'].strip(), f"测试失败: 期望得到 '{case['expected']}' 但得到 '{result.strip()}'" def test_to_nlp_md_with_math_delimiters(self): """测试 to_nlp_md 方法对数学特殊公式分隔符的处理.""" @@ -521,6 +523,136 @@ def test_to_nlp_md_with_math_delimiters(self): result = doc.get_content_list().to_nlp_md() assert result == case['expected'], f"测试失败: 期望得到 '{case['expected']}' 但得到 '{result}'" + def test_to_plain_md(self): + """测试to_plain_md方法调用.""" + from llm_web_kit.libs.standard_utils import json_loads + from llm_web_kit.simple import ( + extract_content_from_html_with_layout_batch, + extract_content_from_html_with_llm, + extract_content_from_html_with_magic_html, + extract_content_from_main_html) + + base_dir = Path(__file__).parent + raw_html = base_dir.joinpath('assets/to_plain_md.html').read_text(encoding='utf-8') + url = 'http://example.com' + + plain_md = extract_content_from_html_with_magic_html(url, raw_html, 'plain_md') + mm_md = extract_content_from_html_with_magic_html(url, raw_html, 'mm_md') + json_json = json_loads(extract_content_from_html_with_magic_html(url, raw_html, 'json')) + + self.assertNotIn('code.language', plain_md) + self.assertNotIn('Private Sub sitemenu_ItemCreated', plain_md) + self.assertNotIn('', plain_md) + self.assertNotIn('a^2', plain_md) + self.assertNotIn('x=4', plain_md) + self.assertNotIn('image', plain_md) + self.assertNotIn('test.mp3', plain_md) + self.assertNotIn('flower.mp4', plain_md) + + self.assertIn('Title Test', plain_md) + self.assertIn('1.1', plain_md) + self.assertIn('UL1.1', plain_md) + self.assertIn('test paragraph', plain_md) + + self.assertIn('code.language', mm_md) + self.assertIn('Private Sub sitemenu_ItemCreated', mm_md) + self.assertIn('
    ', mm_md) + self.assertIn('a^2', mm_md) + self.assertIn('x=4', mm_md) + self.assertIn('image', mm_md) + self.assertNotIn('test.mp3', mm_md) + self.assertNotIn('flower.mp4', mm_md) + + content_json = json_loads(base_dir.joinpath('assets/content_json.json').read_text(encoding='utf-8')) + self.assertEqual(json_json['content_list'], content_json['content_list']) + + plain_md_main = extract_content_from_main_html(url, raw_html, 'plain_md') + mm_md_main = extract_content_from_html_with_magic_html(url, raw_html, 'mm_md') + json_main = json_loads(extract_content_from_html_with_magic_html(url, raw_html, 'json')) + + self.assertNotIn('code.language', plain_md_main) + self.assertNotIn('Private Sub sitemenu_ItemCreated', plain_md_main) + self.assertNotIn('
    ', plain_md_main) + self.assertNotIn('a^2', plain_md_main) + self.assertNotIn('x=4', plain_md_main) + self.assertNotIn('image', plain_md_main) + self.assertNotIn('test.mp3', plain_md_main) + self.assertNotIn('flower.mp4', plain_md_main) + + self.assertIn('Title Test', plain_md_main) + self.assertIn('1.1', plain_md_main) + self.assertIn('UL1.1', plain_md_main) + self.assertIn('test paragraph', plain_md_main) + + self.assertIn('code.language', mm_md_main) + self.assertIn('Private Sub sitemenu_ItemCreated', mm_md_main) + self.assertIn('
    ', mm_md_main) + self.assertIn('a^2', mm_md_main) + self.assertIn('x=4', mm_md_main) + self.assertIn('image', mm_md_main) + self.assertNotIn('test.mp3', mm_md_main) + self.assertNotIn('flower.mp4', mm_md_main) + + self.assertEqual(json_main['content_list'], content_json['content_list']) + + plain_md_llm = extract_content_from_html_with_llm(url, raw_html, 'plain_md') + mm_md_llm = extract_content_from_html_with_magic_html(url, raw_html, 'mm_md') + json_llm = json_loads(extract_content_from_html_with_magic_html(url, raw_html, 'json')) + + self.assertNotIn('code.language', plain_md_llm) + self.assertNotIn('Private Sub sitemenu_ItemCreated', plain_md_llm) + self.assertNotIn('
    ', plain_md_llm) + self.assertNotIn('a^2', plain_md_llm) + self.assertNotIn('x=4', plain_md_llm) + self.assertNotIn('image', plain_md_llm) + self.assertNotIn('test.mp3', plain_md_llm) + self.assertNotIn('flower.mp4', plain_md_llm) + + self.assertIn('Title Test', plain_md_llm) + self.assertIn('1.1', plain_md_llm) + self.assertIn('UL1.1', plain_md_llm) + self.assertIn('test paragraph', plain_md_llm) + + self.assertIn('code.language', mm_md_llm) + self.assertIn('Private Sub sitemenu_ItemCreated', mm_md_llm) + self.assertIn('
    ', mm_md_llm) + self.assertIn('a^2', mm_md_llm) + self.assertIn('x=4', mm_md_llm) + self.assertIn('image', mm_md_llm) + self.assertNotIn('test.mp3', mm_md_llm) + self.assertNotIn('flower.mp4', mm_md_llm) + + self.assertEqual(json_llm['content_list'], content_json['content_list']) + + plain_md_layout = extract_content_from_html_with_layout_batch(url, raw_html, 'plain_md') + mm_md_layout = extract_content_from_html_with_magic_html(url, raw_html, 'mm_md') + json_layout = json_loads(extract_content_from_html_with_magic_html(url, raw_html, 'json')) + + self.assertNotIn('code.language', plain_md_layout) + self.assertNotIn('Private Sub sitemenu_ItemCreated', plain_md_layout) + self.assertNotIn('
    ', plain_md_layout) + self.assertNotIn('a^2', plain_md_layout) + self.assertNotIn('x=4', plain_md_layout) + self.assertNotIn('image', plain_md_layout) + self.assertNotIn('test.mp3', plain_md_layout) + self.assertNotIn('flower.mp4', plain_md_layout) + + self.assertIn('Title Test', plain_md_layout) + self.assertIn('1.1', plain_md_layout) + self.assertIn('UL1.1', plain_md_layout) + self.assertIn('test paragraph', plain_md_layout) + + self.assertIn('code.language', mm_md_layout) + self.assertIn('Private Sub sitemenu_ItemCreated', mm_md_layout) + self.assertIn('
    ', mm_md_layout) + self.assertIn('a^2', mm_md_layout) + self.assertIn('x=4', mm_md_layout) + self.assertIn('image', mm_md_layout) + self.assertNotIn('test.mp3', mm_md_layout) + self.assertNotIn('flower.mp4', mm_md_layout) + + self.assertEqual(json_layout['content_list'], content_json['content_list']) + class TestDataJsonGetMagicHtml: base_dir = Path(__file__).parent From 803441fc14aaf46d42a25bf4b8719c69b55abe11 Mon Sep 17 00:00:00 2001 From: idea_overflow <793884420@qq.com> Date: Wed, 10 Sep 2025 18:13:32 +0800 Subject: [PATCH 08/11] new simplify for dripper v1.5 (#554) Co-authored-by: ningwenchang --- .../main_html_parser/parser/tag_simplifier.py | 4 +- .../simplify_html/simplify_html.py | 329 +- .../template_www.wdi.it_llm.json | 11 +- .../assets/test_html_data/normal_dl.html | 2 +- .../simplify_cases/abnormal_comment.html | 1 + .../simplify_cases/block_select.html | 70 + .../simplify_cases/data_table.html | 1700 ++++++ .../simplify_cases/header_tag.html | 842 +++ .../simplify_cases/inline_block.html | 422 ++ .../test_html_data/simplify_cases/list.html | 208 + .../simplify_cases/nav_class.html | 442 ++ .../simplify_cases/nested_table_caption.html | 217 + .../simplify_cases/nested_table_colgroup.html | 389 ++ .../simplify_cases/nested_table_headers.html | 388 ++ .../simplify_cases/non_list_child.html | 5343 +++++++++++++++++ .../test_html_data/simplify_cases/table.html | 677 +++ .../test_html_data/special_table_1.html | 11 +- .../parser/test_layout_parser.py | 12 +- .../parser/test_tag_simplifier.py | 428 +- 19 files changed, 11291 insertions(+), 205 deletions(-) create mode 100644 tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/abnormal_comment.html create mode 100644 tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/block_select.html create mode 100644 tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/data_table.html create mode 100644 tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/header_tag.html create mode 100644 tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/inline_block.html create mode 100644 tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/list.html create mode 100644 tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/nav_class.html create mode 100644 tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/nested_table_caption.html create mode 100644 tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/nested_table_colgroup.html create mode 100644 tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/nested_table_headers.html create mode 100644 tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/non_list_child.html create mode 100644 tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/table.html diff --git a/llm_web_kit/main_html_parser/parser/tag_simplifier.py b/llm_web_kit/main_html_parser/parser/tag_simplifier.py index eede6cfb..493d46cc 100644 --- a/llm_web_kit/main_html_parser/parser/tag_simplifier.py +++ b/llm_web_kit/main_html_parser/parser/tag_simplifier.py @@ -19,12 +19,11 @@ def parse(self, pre_data: PreDataJson) -> PreDataJson: """ # 获取输入数据 typical_raw_html = pre_data.get(PreDataJsonKey.TYPICAL_RAW_HTML, '') - is_xpath = pre_data.get(PreDataJsonKey.IS_XPATH, True) # layout_file_list = pre_data.get(PreDataJsonKey.LAYOUT_FILE_LIST, []) # 执行HTML标签简化逻辑 try: - simplified_html, original_html, _ = simplify_html(typical_raw_html, is_xpath=is_xpath) + simplified_html, original_html = simplify_html(typical_raw_html) except TagSimplifiedParserException as e1: raise e1 except Exception as e2: @@ -33,6 +32,5 @@ def parse(self, pre_data: PreDataJson) -> PreDataJson: # 设置输出数据 pre_data[PreDataJsonKey.TYPICAL_RAW_TAG_HTML] = original_html # 保存原始标签HTML pre_data[PreDataJsonKey.TYPICAL_SIMPLIFIED_HTML] = simplified_html # 保存简化后的HTML - pre_data[PreDataJsonKey.XPATH_MAPPING] = _ # 保存xpath return pre_data diff --git a/llm_web_kit/main_html_parser/simplify_html/simplify_html.py b/llm_web_kit/main_html_parser/simplify_html/simplify_html.py index bfff8f29..3c37da8e 100644 --- a/llm_web_kit/main_html_parser/simplify_html/simplify_html.py +++ b/llm_web_kit/main_html_parser/simplify_html/simplify_html.py @@ -3,33 +3,34 @@ import uuid from typing import Dict, List, Tuple -from bs4 import BeautifulSoup from lxml import etree, html +from selectolax.parser import HTMLParser # 行内标签 inline_tags = { 'map', 'optgroup', 'span', 'br', 'input', 'time', 'u', 'strong', 'textarea', 'small', 'sub', 'samp', 'blink', 'b', 'code', 'nobr', 'strike', 'bdo', 'basefont', 'abbr', 'var', 'i', 'cccode-inline', - 'select', 's', 'pic', 'label', 'mark', 'object', 'dd', 'dt', 'ccmath-inline', 'svg', 'li', + 'select', 's', 'pic', 'label', 'mark', 'object', 'ccmath-inline', 'svg', 'button', 'a', 'font', 'dfn', 'sup', 'kbd', 'q', 'script', 'acronym', 'option', 'img', 'big', 'cite', 'em', 'marked-tail', 'marked-text' - # 'td', 'th' + # 'td', 'th', 'dd', 'dt', 'li' } +# 表格内部可能包含的跟表格相关的标签 +table_tags_set = {"caption", "colgroup", "col", "thead", "tbody", "tfoot", "tr", "td", "th"} + # 需要删除的标签 tags_to_remove = { + 'title', 'head', - 'header', - 'footer', 'nav', - 'aside', 'style', 'script', 'noscript', 'link', 'meta', 'iframe', - 'frame' + 'frame', } # 需要保留的特殊标签（即使它们是行内标签） @@ -37,7 +38,7 @@ # 需要删除的属性名模式（独立单词） ATTR_PATTERNS_TO_REMOVE = { - 'nav', 'footer', 'header', # 独立单词 + 'nav', # 'footer', 'header', # 独立单词 } # 需要删除的属性名模式（特定前缀/后缀） @@ -72,90 +73,76 @@ def build_uid_map(dom: html.HtmlElement) -> Dict[str, html.HtmlElement]: return {node.get('data-uid'): node for node in dom.iter() if node.get('data-uid')} -def is_unique_attribute(tree, attr_name, attr_value): - """检查给定的属性名和值组合是否在文档中唯一。""" - elements = tree.xpath(f"//*[@{attr_name}='{attr_value}']") - return len(elements) == 1 - - -def get_relative_xpath(element): - root_tree = element.getroottree() - current_element = element - path_from_element = [] - found_unique_ancestor = False - - # 从当前元素开始向上查找 - while current_element is not None and current_element.getparent() is not None: - siblings = [sib for sib in current_element.getparent() if sib.tag == current_element.tag] - - # 检查当前元素是否有唯一属性 - unique_attr = None - candidate_attrs = [ - attr for attr in current_element.attrib - if not (attr.startswith('data-') or attr == 'style' or - attr == '_item_id' or - (current_element.attrib[attr].startswith('{') and current_element.attrib[attr].endswith('}'))) - ] - - for attr in candidate_attrs: - if is_unique_attribute(root_tree, attr, current_element.attrib[attr]): - unique_attr = attr +def judge_table_parent(table_element, node_list): + for node in node_list: + ancestor = node.getparent() + while ancestor is not None: + if ancestor is table_element: + return True + elif ancestor.tag == 'table': break - - # 如果有唯一属性，构建相对路径并停止向上查找 - if unique_attr is not None: - path_from_element.insert(0, f'*[@{unique_attr}="{current_element.attrib[unique_attr]}"]') - found_unique_ancestor = True - break - else: - # 没有唯一属性，使用常规方式 - if len(siblings) > 1: - index = siblings.index(current_element) + 1 - path_from_element.insert(0, f'{current_element.tag}[{index}]') - else: - path_from_element.insert(0, current_element.tag) - - current_element = current_element.getparent() - - # 构建最终的XPath - if found_unique_ancestor: - return f'//{"/".join(path_from_element)}' - else: - # 如果没有找到唯一属性祖先，返回完整路径 - return f'//{"/".join(path_from_element)}' + ancestor = ancestor.getparent() + return False def is_data_table(table_element: html.HtmlElement) -> bool: """判断表格是否是数据表格而非布局表格.""" - # 检查表格是否有 caption 标签 - if table_element.xpath('.//caption'): - return True - - # 检查是否有 th 标签 - if table_element.xpath('.//th'): + # 检查当前表格（不包括内部嵌套表格）是否有 caption 标签 + caption_nodes = table_element.xpath('.//caption') + if judge_table_parent(table_element, caption_nodes): return True - # 检查是否有 thead 或 tfoot 标签 - if table_element.xpath('.//thead') or table_element.xpath('.//tfoot'): + # 检查当前表格（不包括内部嵌套表格）是否有 colgroup 或 col 标签 + col_nodes = table_element.xpath('.//col') + colgroup_nodes = table_element.xpath('.//colgroup') + if judge_table_parent(table_element, col_nodes) or judge_table_parent(table_element, colgroup_nodes): return True - # 检查是否有 colgroup 或 col 标签 - if table_element.xpath('.//colgroup') or table_element.xpath('.//col'): - return True - - # 检查是否有 summary 属性 - if table_element.get('summary'): + # 检查当前表格（不包括内部嵌套表格）单元格是否有 headers 属性 + cell_nodes = table_element.xpath(".//*[self::td or self::th][@headers]") + if judge_table_parent(table_element, cell_nodes): return True # 检查是否有 role="table" 或 data-table 属性 if table_element.get('role') == 'table' or table_element.get('data-table'): return True - # 检查单元格是否有 headers 属性 - if table_element.xpath('.//*[@headers]'): + for node in table_element.iterdescendants(): + if node.tag in table_tags_set: + continue + if node.tag not in inline_tags: + return False + + return True + + +def has_non_listitem_children(list_element): + """检查列表元素是否包含非列表项的直接子节点. + + :param list_element: lxml元素对象 (ul, ol, dl) + :return: True 如果存在非列表项的直接子节点，否则 False + """ + + # 根据列表类型确定允许的子元素标签 + if list_element.tag in ['ul', 'ol']: + allowed_tags = {'li'} + elif list_element.tag == 'dl': + allowed_tags = {'dt', 'dd'} + + # 使用XPath直接查找是否存在不允许的直接子元素 + # 例如，对于
      ，查找所有不是
    • 的直接子元素 + # 对于
      ，查找所有不是
      或
      的直接子元素 + exclude_conditions = " and ".join([f"name()!='{tag}'" for tag in allowed_tags]) + disallowed_children_xpath = f"./*[{exclude_conditions}]" + + if list_element.xpath(disallowed_children_xpath): return True - return False + # 检查是否存在非空白文本节点 + text_children = list_element.xpath("./text()") + non_whitespace_text = any(text.strip() for text in text_children) + + return non_whitespace_text def extract_paragraphs(processing_dom: html.HtmlElement, uid_map: Dict[str, html.HtmlElement], @@ -185,33 +172,71 @@ def extract_paragraphs(processing_dom: html.HtmlElement, uid_map: Dict[str, html for table in processing_dom.xpath('.//table'): table_types[table.get('data-uid')] = is_data_table(table) + # 创建列表类型映射，记录每个列表是内容列表还是布局列表 + list_types = {} + def is_block_element(node) -> bool: """判断是否为块级元素.""" - # 处理表格单元格特殊情况 - if node.tag in ('td', 'th'): - # 找到最近的祖先table元素 - table_ancestor = node - while table_ancestor is not None and table_ancestor.tag != 'table': - table_ancestor = table_ancestor.getparent() - - # 如果是表格单元格，根据表格类型决定是否为块级元素 - if table_ancestor is not None: - table_uid = table_ancestor.get('data-uid') - if table_types.get(table_uid, False): - # 数据表格的td/th不作为块级元素 + def judge_special_case(node, expected_tags, types_map): + ancestor = node + while ancestor is not None and ancestor.tag not in expected_tags: + ancestor = ancestor.getparent() + + if ancestor is not None: + ancestor_uid = ancestor.get('data-uid') + if types_map.get(ancestor_uid, False): + # 数据表格/内容列表的子元素不作为块级元素 return False else: - # 布局表格的td/th作为块级元素 + # 布局表格/列表的子元素作为块级元素 return True + # 处理表格和列表的特殊情况 + if node.tag in ('td', 'th'): + return judge_special_case(node, ['table'], table_types) + + if node.tag == "li": + return judge_special_case(node, ['ul', 'ol'], list_types) + + if node.tag == "dt" or node.tag == "dd": + return judge_special_case(node, ['dl'], list_types) + # 默认处理其他元素 if node.tag in inline_tags: return False return isinstance(node, html.HtmlElement) - def has_block_children(node) -> bool: - """判断是否有块级子元素.""" - return any(is_block_element(child) for child in node.iterchildren()) + def has_block_descendants(node): + for child in node.iterdescendants(): + if is_block_element(child): + if node.tag in inline_tags: + original_element = uid_map.get(node.get('data-uid')) + original_element.set('cc-block-type', "true") + return True + return False + + def is_content_list(list_element): + # 获取列表项（支持多种列表类型） + items = list_element.xpath("li | dt | dd") + + # 不包含列表项，则不是内容列表 + if len(items) == 0: + return False + # 列表包含非列表项子元素视为布局列表 + if has_non_listitem_children(list_element): + return False + + # 列表内任意子项存在块级元素，则视为布局列表 + for item in items: + if has_block_descendants(item): + return False + + # 默认视为内容列表 + return True + + # 先分析所有列表的类型 + for list_element in processing_dom.xpath('.//ul | .//ol | .//dl'): + list_types[list_element.get('data-uid')] = is_content_list(list_element) def clone_structure(path: List[html.HtmlElement]) -> Tuple[html.HtmlElement, html.HtmlElement]: """克隆节点结构.""" @@ -245,7 +270,7 @@ def process_node(node: html.HtmlElement, path: List[html.HtmlElement]): # 处理子节点 for child in node: - if is_block_element(child): + if is_block_element(child) or has_block_descendants(child): # 处理累积的内联内容 if inline_content: try: @@ -271,7 +296,7 @@ def process_node(node: html.HtmlElement, path: List[html.HtmlElement]): content_sources = [] # 处理块级元素 - if not has_block_children(child): + if table_types.get(child.get('data-uid')) or (not has_block_descendants(child)): try: root, last_node = clone_structure(current_path + [child]) last_node.text = child.text if child.text else None @@ -362,8 +387,7 @@ def remove_xml_declaration(html_string): # 正则表达式匹配 或 （没有问号结尾的情况） pattern = r'<\?xml\s+.*?\??>' html_content = re.sub(pattern, '', html_string, flags=re.DOTALL) - # 1. 删除HTML注释 - html_content = re.sub(r'', '', html_content, flags=re.DOTALL) + return html_content @@ -372,10 +396,7 @@ def post_process_html(html_content: str) -> str: if not html_content: return html_content - # 1. 删除HTML注释 - html_content = re.sub(r'', '', html_content, flags=re.DOTALL) - - # 2. 处理标签外的空白（保留标签内文本的换行） + # 处理标签外的空白（保留标签内文本的换行） def replace_outside_tag_space(match): """只替换标签外的连续空白.""" if match.group(1): # 如果是标签内容 @@ -574,31 +595,14 @@ def simplify_list(element): def should_remove_element(element) -> bool: """判断元素的class或id属性是否匹配需要删除的模式.""" - # 检查class属性 - class_name = element.get('class', '') - if class_name: - class_parts = class_name.strip().split() - for part in class_parts: - # 检查是否完全匹配独立单词 - if part in ATTR_PATTERNS_TO_REMOVE: - return True - # 检查是否包含特定前缀/后缀 - # for pattern in ATTR_SUFFIX_TO_REMOVE: - # if part.endswith(pattern): - # return True - # 检查id属性 + class_name = element.get('class', '') id_name = element.get('id', '') - if id_name: - id_parts = id_name.strip().split('-') # id通常用连字符分隔 - for part in id_parts: - # 检查是否完全匹配独立单词 - if part in ATTR_PATTERNS_TO_REMOVE: - return True - # 检查是否包含特定前缀/后缀 - # for pattern in ATTR_SUFFIX_TO_REMOVE: - # if part.endswith(pattern): - # return True + + if class_name in ATTR_PATTERNS_TO_REMOVE or id_name in ATTR_PATTERNS_TO_REMOVE: + parent = element.getparent() + if parent is not None and parent.tag == 'body': + return True # 检查style属性 style_attr = element.get('style', '') @@ -665,7 +669,7 @@ def truncate_text_content(element, max_length=500): remaining -= len(text) -def process_paragraphs(paragraphs: List[Dict[str, str]], uid_map: Dict[str, html.HtmlElement], is_xpath: bool = True) -> Tuple[str, html.HtmlElement]: +def process_paragraphs(paragraphs: List[Dict[str, str]], uid_map: Dict[str, html.HtmlElement]) -> Tuple[str, html.HtmlElement]: """处理段落并添加 _item_id，同时在原始DOM的对应元素上添加相同ID. Args: @@ -680,9 +684,7 @@ def process_paragraphs(paragraphs: List[Dict[str, str]], uid_map: Dict[str, html for para in paragraphs: try: - html_content = re.sub(r'', '', para['html'], flags=re.DOTALL) - # 解析段落HTML - root = html.fromstring(html_content) + root = html.fragment_fromstring(para['html'], create_parent=False) root_for_xpath = copy.deepcopy(root) content_type = para.get('content_type', 'block_element') @@ -698,29 +700,6 @@ def process_paragraphs(paragraphs: List[Dict[str, str]], uid_map: Dict[str, html # 截断过长的文本内容 truncate_text_content(root, max_length=1000) - para_xpath = [] - if is_xpath: - if content_type in ('inline_elements', 'mixed'): - for child in root_for_xpath.iterchildren(): - original_element = uid_map.get(child.get('data-uid')) - try: - _xpath = get_relative_xpath(original_element) - except Exception: - _xpath = None - para_xpath.append(_xpath) - elif content_type == 'block_element': - try: - _xpath = get_relative_xpath(para['_original_element']) - except Exception: - _xpath = None - para_xpath.append(_xpath) - else: - try: - _xpath = get_relative_xpath(para['_original_element']) - except Exception: - _xpath = None - para_xpath.append(_xpath) - # 为当前段落和原始元素添加相同的 _item_id current_id = str(item_id) root.set('_item_id', current_id) @@ -730,9 +709,9 @@ def process_paragraphs(paragraphs: List[Dict[str, str]], uid_map: Dict[str, html if content_type != 'block_element': if original_parent is not None: # root_for_xpath有子元素 + original_element = uid_map.get(root_for_xpath.get('data-uid')) if len(root_for_xpath) > 0: - if root_for_xpath.tag in inline_tags and uid_map.get(root_for_xpath.get('data-uid')).tag != 'body': - original_element = uid_map.get(root_for_xpath.get('data-uid')) + if root_for_xpath.tag in inline_tags and original_element.tag != 'body' and original_element.get('cc-block-type') != "true": original_element.set('_item_id', current_id) else: # 收集需要包裹的子元素 @@ -767,6 +746,9 @@ def process_paragraphs(paragraphs: List[Dict[str, str]], uid_map: Dict[str, html # 创建wrapper元素 wrapper = etree.Element(tail_block_tag) wrapper.set('_item_id', current_id) + # 如果父元素包含cc-select，那么包裹的wrapper元素也应该包含cc-select，避免_item_id和cc-select不在同一层级中 + if original_parent.get("cc-select") is not None: + wrapper.set("cc-select", original_parent.get("cc-select")) # 设置前面的文本 if leading_text: @@ -790,7 +772,6 @@ def process_paragraphs(paragraphs: List[Dict[str, str]], uid_map: Dict[str, html # last_child.tail = None else: if content_type == 'inline_elements': - original_element = uid_map.get(root_for_xpath.get('data-uid')) original_element.set('_item_id', current_id) else: # root_for_xpath只有文本内容 @@ -804,7 +785,9 @@ def process_paragraphs(paragraphs: List[Dict[str, str]], uid_map: Dict[str, html wrapper = etree.Element(tail_block_tag) wrapper.set('_item_id', current_id) wrapper.text = original_parent.text - + # 如果父元素包含cc-select，那么包裹的wrapper元素也应该包含cc-select + if original_parent.get("cc-select") is not None: + wrapper.set("cc-select", original_parent.get("cc-select")) # 替换父节点的text original_parent.text = None @@ -824,7 +807,9 @@ def process_paragraphs(paragraphs: List[Dict[str, str]], uid_map: Dict[str, html wrapper = etree.Element(tail_block_tag) wrapper.set('_item_id', current_id) wrapper.text = child.tail - + # 如果父元素包含cc-select，那么包裹的wrapper元素也应该包含cc-select + if original_parent.get("cc-select") is not None: + wrapper.set("cc-select", original_parent.get("cc-select")) # 替换tail child.tail = None @@ -838,6 +823,10 @@ def process_paragraphs(paragraphs: List[Dict[str, str]], uid_map: Dict[str, html else: # 块级元素直接设置属性 original_parent.set('_item_id', current_id) + for child in original_parent.iterdescendants(): + if child.get("cc-select") is not None: + original_parent.set("cc-select", child.get("cc-select")) + break item_id += 1 @@ -846,7 +835,6 @@ def process_paragraphs(paragraphs: List[Dict[str, str]], uid_map: Dict[str, html result.append({ 'html': cleaned_html, '_item_id': current_id, - '_xpath': para_xpath, 'content_type': content_type }) @@ -859,25 +847,25 @@ def process_paragraphs(paragraphs: List[Dict[str, str]], uid_map: Dict[str, html simplified_html = '' + ''.join( p['html'] for p in result) + '' - return post_process_html(simplified_html), result + return post_process_html(simplified_html) -def simplify_html(html_str, is_xpath: bool = True) -> etree.Element: +def simplify_html(html_str) -> etree.Element: """ :return: simplified_html: 精简HTML original_html: 添加_item_id的原始HTML _xpath_mapping: xpath映射 """ - # 预处理 - preprocessed_html = remove_xml_declaration(html_str) - - # 用 BeautifulSoup 修复未闭合标签，lxml 无法完全修复 - soup = BeautifulSoup(preprocessed_html, 'html.parser') - fixed_html = str(soup) - - # 解析原始DOM - original_dom = html.fromstring(fixed_html) + # 使用selectolax的HTMLParser来修复html + soup = HTMLParser(html_str) + fixed_html = soup.html + + preprocessed_html = remove_xml_declaration(fixed_html) + # 注释通过lxml的HTMLParser的remove_comments参数处理 + parser = html.HTMLParser(remove_comments=True) + original_dom = html.fromstring(preprocessed_html, parser=parser) + # 添加data_uid add_data_uids(original_dom) original_uid_map = build_uid_map(original_dom) @@ -891,14 +879,9 @@ def simplify_html(html_str, is_xpath: bool = True) -> etree.Element: paragraphs = extract_paragraphs(processing_dom, original_uid_map, include_parents=False) # 处理段落（同步添加ID） - simplified_html, result = process_paragraphs(paragraphs, original_uid_map, is_xpath) + simplified_html = process_paragraphs(paragraphs, original_uid_map) remove_all_uids(original_dom) original_html = etree.tostring(original_dom, pretty_print=True, method='html', encoding='unicode') - _xpath_mapping = {item['_item_id']: { - '_xpath': item['_xpath'], - 'content_type': item['content_type'] - } for item in result} - - return simplified_html, original_html, _xpath_mapping + return simplified_html, original_html diff --git a/tests/llm_web_kit/main_html_parser/parser/assets/input_layout_batch_parser/template_www.wdi.it_llm.json b/tests/llm_web_kit/main_html_parser/parser/assets/input_layout_batch_parser/template_www.wdi.it_llm.json index abb40deb..917c500b 100644 --- a/tests/llm_web_kit/main_html_parser/parser/assets/input_layout_batch_parser/template_www.wdi.it_llm.json +++ b/tests/llm_web_kit/main_html_parser/parser/assets/input_layout_batch_parser/template_www.wdi.it_llm.json @@ -7,8 +7,8 @@ "item_id 6": "No", "item_id 7": "No", "item_id 8": "No", - "item_id 9": "Yes", - "item_id 10": "No", + "item_id 9": "No", + "item_id 10": "Yes", "item_id 11": "No", "item_id 12": "No", "item_id 13": "No", @@ -30,10 +30,5 @@ "item_id 29": "No", "item_id 30": "No", "item_id 31": "No", - "item_id 32": "No", - "item_id 33": "No", - "item_id 34": "No", - "item_id 35": "No", - "item_id 36": "No", - "item_id 37": "No" + "item_id 32": "No" } \ No newline at end of file diff --git a/tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/normal_dl.html b/tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/normal_dl.html index cbd81a5b..f465ccff 100644 --- a/tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/normal_dl.html +++ b/tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/normal_dl.html @@ -647,7 +647,7 @@

      -
      +

      Wow!  Number 400.  What a milestone for CAS(E) This Sketch.  Congratulations!  I used this week's sketch for another Christmas card.

      The images are from Altenew's Holiday Bow and the gray glitter strips are from a paper pack by DCWV.  I used a darker gray from the same pack for the branches.  The sentiment is from HLS.

      Thanks for dropping by.  I love it when you do.




      diff --git a/tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/abnormal_comment.html b/tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/abnormal_comment.html new file mode 100644 index 00000000..840fc308 --- /dev/null +++ b/tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/abnormal_comment.html @@ -0,0 +1 @@ + Appell sequence

       

      .

      In mathematics, an Appell sequence, named after Paul Émile Appell, is any polynomial sequence {pn(x)}n = 0, 1, 2, ... satisfying the identity

      \( {d \over dx} p_n(x) = np_{n-1}(x), \)

      and in which p0(x) is a non-zero constant.

      Among the most notable Appell sequences besides the trivial example { xn } are the Hermite polynomials, the Bernoulli polynomials, and the Euler polynomials. Every Appell sequence is a Sheffer sequence, but most Sheffer sequences are not Appell sequences.

      Equivalent characterizations of Appell sequences

      The following conditions on polynomial sequences can easily be seen to be equivalent:

      For n = 1, 2, 3, ...,

      \( {d \over dx} p_n(x) = np_{n-1}(x) \)

      and p0(x) is a non-zero constant;

      For some sequence {cn}n = 0, 1, 2, ... of scalars with c0 ≠ 0,

      \( p_n(x) = \sum_{k=0}^n {n \choose k} c_k x^{n-k}; \)

      For the same sequence of scalars,

      \( p_n(x) = \left(\sum_{k=0}^\infty {c_k \over k!} D^k\right) x^n, \)

      where

      D = {d \over dx};

      For n = 0, 1, 2, ...,

      p_n(x+y) = \sum_{k=0}^n {n \choose k} p_k(x) y^{n-k}.

      Recursion formula

      Suppose

      \( p_n(x) = \left(\sum_{k=0}^\infty {c_k \over k!} D^k\right) x^n = Sx^n, \)

      where the last equality is taken to define the linear operator S on the space of polynomials in x. Let

      \( T = S^{-1} = \left(\sum_{k=0}^\infty {c_k \over k!} D^k\right)^{-1} = \sum_{k=1}^\infty {a_k \over k!} D^k \)

      be the inverse operator, the coefficients ak being those of the usual reciprocal of a formal power series, so that

      \( Tp_n(x) = x^n.\, \)

      In the conventions of the umbral calculus, one often treats this formal power series T as representing the Appell sequence {pn}. One can define

      \( \log T = \log\left(\sum_{k=0}^\infty {a_k \over k!} D^k \right) \)

      by using the usual power series expansion of the log(1 + x) and the usual definition of composition of formal power series. Then we have

      \( p_{n+1}(x) = (x - (\log T)')p_n(x).\, \)

      (This formal differentiation of a power series in the differential operator D is an instance of Pincherle differentiation.)

      In the case of Hermite polynomials, this reduces to the conventional recursion formula for that sequence.
      Subgroup of the Sheffer polynomials

      The set of all Appell sequences is closed under the operation of umbral composition of polynomial sequences, defined as follows. Suppose { pn(x) : n = 0, 1, 2, 3, ... } and { qn(x) : n = 0, 1, 2, 3, ... } are polynomial sequences, given by

      \( p_n(x)=\sum_{k=0}^n a_{n,k}x^k\ \mbox{and}\ q_n(x)=\sum_{k=0}^n b_{n,k}x^k. \)

      Then the umbral composition p o q is the polynomial sequence whose nth term is

      \( (p_n\circ q)(x)=\sum_{k=0}^n a_{n,k}q_k(x)=\sum_{0\le k \le \ell \le n} a_{n,k}b_{k,\ell}x^\ell \)

      (the subscript n appears in pn, since this is the n term of that sequence, but not in q, since this refers to the sequence as a whole rather than one of its terms).

      Under this operation, the set of all Sheffer sequences is a non-abelian group, but the set of all Appell sequences is an abelian subgroup. That it is abelian can be seen by considering the fact that every Appell sequence is of the form

      \( p_n(x) = \left(\sum_{k=0}^\infty {c_k \over k!} D^k\right) x^n, \)

      and that umbral composition of Appell sequences corresponds to multiplication of these formal power series in the operator D.
      Different convention

      Another convention followed by some authors (see Chihara) defines this concept in a different way, conflicting with Appell's original definition, by using the identity

      \( {d \over dx} p_n(x) = p_{n-1}(x) \)

      instead.
      See also

      Sheffer sequence
      Umbral calculus
      Generalized Appell polynomials
      Wick product

      References

      Paul Appell, "Sur une classe de polynômes", Annales scientifiques de l'École Normale Supérieure 2e série, tome 9, 1880.
      Steven Roman and Gian-Carlo Rota, "The Umbral Calculus", Advances in Mathematics, volume 27, pages 95 – 188, (1978).
      G.-C. Rota, D. Kahaner, and A. Odlyzko, "Finite Operator Calculus", Journal of Mathematical Analysis and its Applications, vol. 42, no. 3, June 1973. Reprinted in the book with the same title, Academic Press, New York, 1975.
      Steven Roman. The Umbral Calculus. Dover Publications.
      Theodore Seio Chihara (1978). An Introduction to Orthogonal Polynomials. Gordon and Breach, New York. ISBN 0-677-04150-0.

      External links

      Appell Sequence at MathWorld

      Retrieved from "http://en.wikipedia.org/"
      All text is available under the terms of the GNU Free Documentation License

      Home - Hellenica World


      \ No newline at end of file diff --git a/tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/block_select.html b/tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/block_select.html new file mode 100644 index 00000000..ac222e70 --- /dev/null +++ b/tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/block_select.html @@ -0,0 +1,70 @@ + + + +Ikea - Term Papers - Business + + + + + + + + + + + + + + + + +
      +
      + +
      + + +
      +
      + + + +
      +
      +full version Ikea Essay +

      Ikea

      +

      Category: Business

      +

      Autor: jessica85 01 June 2010

      +

      Words: 1191 | Pages: 5

      +

      A. What are the cultural factors which make expansion abroad in retailing difficult? What has made it possible in IKEA's case?

      Retailing expansions can be difficult, because of differences in culture in the global market. When entering a new market, corporations tend to do considerable studies catered towards local tastes. There are many factors to consider when expanding into a new area or culture, because culture can have a great impact on merchandising, and promotion of products. (Hibbert, Edgar 2000;)
      The retailing difficulties are not only limited to merchandising and promotion but the cross-over of store brands and brand images. The social systems and social behavior also affects the corporation as different management styles and company cultures may be difficult for employees to adjust to and their maybe clashes which can make the whole process less effective and also less efficient. If there are major differences in the existing culture and language difficulties it may establish greater cultural barriers.
      Culture also affects the four P’s in marketing the new product abroad, there can be difficulties and adjustments if the new market is price sensitive, has a high context culture and the corporation came from a country with a low context culture or vise-versa which can affect the promotion of the product. In another case study presented it was highlighted, that some cultures associate the price of a product with the quality of the product. (Hibbert, Edgar 2000;)The culture of the country also dictates if the target markets will be living in urban, rural or suburban areas and research must be done to show if they would be willing to travel out of their area to come to a different place to obtain the product.
      In expanding to global markets retaining corporations will have to take into consideration the import, duties and taxes, and also government rules and regulations. The cost and availability of land, for example in when Toys �R’ Us moved to Japan, land in city areas were scare, limited and very expensive. (Hibbert, Edgar 2000;)
      IKEA is an European store, more specifically, a Swedish furniture store. Expanding throughout Europe brought about less challenging difficulties because being an European store, there were many cultural similarities and cost advantages due to economies of scale. However, in there early expansion to the United States they faced many hurdles as they failed to adapt there strategies to the American culture and instead of imposed their own.
      IKEA ultimately recognized their mistakes, for example Ikea tried to impose their European standard bed, which were longer and thinner, while selling American standardize bed sheets to the American customers. IKEA soon redesigned its American product range which immediately increased there sales. They also reduced their dependence on outside suppliers and recruited American suppliers. They also had their own people working alongside the manufactures to give technical tips and to find the better quality or lower cost materials. IKEA also had to change the way they did promotions, because the United States did not have a homogenous culture so the traditional forms of promotion would not have been as effective as elsewhere.


      B. How does the TV advertising campaign initiated by IKEA overcome the entry barrier of high advertising expenditures?

      IKEA could no longer use their strategy since America has a very diverse population with a variety of sub cultures and the “word of mouth” strategy would have been less effective than it was Europe and other countries. Because of the culturally diversity that exist in the United States, social norms and interpersonal communication are less reliable, foreign companies coming to the United States often find that corporate advertising done here far outstrips what they have used elsewhere. ” ( Johansson, Johny K. 2006; )
      Therefore, IKEA came up with a new slogan and advertising message that would have the same effect and be consistent with previous marketing strategies used in countries with a more homogenous population. To implement this strategy IKEA’s advertising company created eight, thirty second ads that showed people in the different stages in their life. This focus allowed them to capitalize on reaching diverse markets at a fairly low cost.
      IKEA’s TV advertising campaign overcame the entry barrier of high advertising by studying the American advertising, where they realized “in Europe you advertise to fain business; in the United States you advertise to stay in business The role of advertising in the United States is much greater than in other countries, according to statistics in 2004, the United States spent 242.5 billion in advertising, Canada 5.2 billion and Sweden 2.7 billion. In order to be on an even playing field IKEA found a strategy that was at the same time low cost and very effective. (Johansson, Johny K. 2006; )



      C. Should IKEA expand further in the United States or focus on other countries? Be specific. What should they do and why?

      As a marketing manager for IKEA, the suggestion of expanding further in the United States would be the best recommendation. As stated in the opening of the case the United States is potentially a very large market for IKEA which has not been taken advantage of, seeing that IKEA has been in the United States for twenty years and only has twenty stores. IKEA has not recognized its full potential in the United States; they need to build many smaller stores shifting away from its current strategy of a few large stores. They need to target communities to who they can cater to specifically.
      The United States has many urban areas where IKEA products would be in high demand. IKEA products are user friendly in design, which includes furniture -in -a -box model and other low cost varieties of furniture. However, people in urban areas would not necessarily want to travel to suburban areas many miles from home to choose their furniture. Even with delivery, the locations of most of IKEA stores are inconvenient to people who do not have cars to get to the stores.
      IKEA needs to be better promoted in the United States; commercials that may have worked with their original entrance cannot compete with new advertising aimed at several of their target markets. Retailers often redo their advertising every year, or have commercials made for specific targets. With low cost, packaged kits, IKEA should for example, have advertising to target college kids during the months of August and September, and advertising targeting new graduates during the months of May and June, young adults who are now starting off with their first homes.
      In conclusion, IKEA products which are well designed and low-cost, suit very many consumers, however the locations maybe too out of the way to attract costumers who value convenience and ease over price. In addition to their huge warehouse stores IKEA needs to open smaller retail outlets in key urban areas where their product are needed, only then will they have taken advantage of the markets in the United States and can focus on expanding into other countries.


      References
      Hibbert, E. (2000). Globalisation in retailing -The impact of culture on market entry. Retrieved 12/8, 2007, from http://www.mubs.mdx.ac.uk/Research/Discussion_Papers/Marketing/dpap_mkt_no14.pdf
      Johansson, J. K. (2006). Global marketing (4th edition ed.). New York: McGraw Hill Irwin.

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        + It is always best to you hire an asbestos removal professional when you need to remove asbestos from your abode. But if you only need to remove a small amount of firmly-bound asbestos that does not require you to obtain an asbestos removal permit, then you can take on the project yourself provided you take the necessary safety measures. Here are a number of effective safety practices to follow when removing non-friable asbestos from your house.
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      + + + + + + + + \ No newline at end of file diff --git a/tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/inline_block.html b/tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/inline_block.html new file mode 100644 index 00000000..40b59109 --- /dev/null +++ b/tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/simplify_cases/inline_block.html @@ -0,0 +1,422 @@ + + + + + + + +WEBINAR Replay: Wellbeing in Schools: Discussion and Solutions + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
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       In this webinar, recorded on Wednesday 28 July 2021, CEO and Founder Nikki Bonus invites Cheryl Edward, Senior Psychologist/Pastoral Care and Wellbeing Officer: Catholic Education (NT) and Lisa Franks, Assistant Principal at Kurrajong Public School (NSW), to discuss their whole-school approaches to wellbeing and its benefits so every child can be supported and connect, thrive and learn.

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      In this webinar, our experts will discuss:

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      • How Life Skills GO supports the teaching and measurement of wellbeing
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      About our experts:

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      Cheryl Edward is a psychologist currently working in the Northern Territory, supporting Inclusion Support Services and Pastoral Care and Wellbeing in Catholic Education, NT.  She started as a School Counsellor with Katherine Group School, in the Department of Education, in 2010, working in 8 schools east of Katherine.  By the time Cheryl moved to Catholic Education, NT in Term 2, 2017, she was working across what is known as the Big Rivers region, covering 28 schools and 4,000 students across 400,000 square km’s. Throughout her work over the past 12 years, she has seen the most difference when schools work with a trauma-informed lens, incorporating neuroscience with social emotional learning. 

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      Who is in your Top 3 Mentalists of all time?

      • This topic has 4 replies, 2 voices, and was last updated 1 year ago by Kenny.
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        • 2017-03-07 at 12:05 pm #1515
          Kenny
          Keymaster

          If you want to – tell us who the top mentalist in your country is.

        • 2021-05-21 at 3:49 pm #7011
          Bernie Amler
          Participant

          Obviously top billing goes to Banachek (Steve Shaw)
          and in no particular order…
          Max Maven (Phil Goldstein)
          The Amazing Kreskin (yes, he’s still around)
          Richard Osterlind
          Marc Salem
          and many who has passed on.

        • 2021-05-24 at 10:55 am #7018
          Kenny
          Keymaster

          Great answer Bernie.

        • 2023-05-17 at 5:20 pm #8617
          Bernie Amler
          Participant

          I just recently found out that Banacheck originally lived in Soth Africa. He is from Port Elizabeth.

        • 2023-05-21 at 7:03 pm #8624
          Kenny
          Keymaster

          Hi Bernie, yes he is from SA. Good observation.

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      + + +【澎湃新闻】谢贵安：张居正对万历皇帝的儒学教育-国学院 + + + + + + + + + + + +
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      +【澎湃新闻】谢贵安：张居正对万历皇帝的儒学教育 +
      + + 2017-04-06 17:46 +国学院宣传部  + + + +
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      武汉大学历史学院的谢贵安教授做客湖北国学大讲堂，做了题为“帝师张居正对万历皇帝的儒学教育”的讲座，本次活动为湖北国学大讲堂2017年度讲座计划系列之一。讲座中，谢贵安教授主要谈及了张居正对万历皇帝的教育形式、内容、技巧以及效果。现将本次讲座的内容整理成文，以飨读者。

      +

      +

      讲座中的谢贵安教授

      +

      有关张居正

      +

      +

      张居正

      +

      谢贵安首先介绍了张居正的生平事迹。张居正（1525－1582年），字叔大，号太岳，幼名张白圭。湖北江陵人（今荆州），所以人称“张江陵”。13岁考举人时受乡试主考官湖广巡抚顾璘赏识，但因年纪过小没能中举；三年之后他以16岁中举；22岁（嘉靖二十六年）中进士；由庶吉士至翰林院编修，可谓少年得志。

      +

      世宗后期，升任右中允，与国子监祭酒高拱、重臣严嵩、徐阶都保持良好关系。穆宗隆庆元年任吏部左侍郎兼东阁大学士，入阁后与高拱并为宰辅，为吏部尚书、建极殿大学士。

      +

      神宗万历初年，与宦官冯保合谋驱逐高拱，成为首辅，与李太后、冯保构成“铁三角”，建立了稳定的权力核心，继而推动了长达十年的“张居正改革”。张居正的作品汇编有《张太岳集》，后张舜徽主编《张居正集》时曾将其收录在内。

      +

      教育形式

      +

      下面就进入了本次讲座的正题——张居正对万历皇帝的教育。万历皇帝（神宗）朱翊钧，继位时年方10岁。张居正教育的对象就是这样一个年少的小皇帝。

      +

      +

      万历皇帝的课表

      +

      谢贵安首先展示了一张万历皇帝的课表，继而详细介绍这份课表中各项教育形式的具体情况。万历皇帝除上朝外，其他时间分为经筵和日讲。经筵每月三次，逢二举行（二、十二、二十二），因为参与人数众多，形式严肃，所以教育效果并不理想，可见经筵的仪式性色彩更为浓厚。相比之下，日讲更为实用，但是非常频繁，除去逢三、六、九上朝之日（初三、六、九、十三、十六、十九、二十三、二十六、二十九）外，其余每天都要举行为皇帝讲授经史的教学活动。张居正担任“知经筵”（主持经筵的官员）后，为神宗制定了明代皇帝中最为严苛的经筵日讲标准。他还亲自制定了神宗参加日讲和经筵的仪式，并规定了春季经筵（春讲）和秋季经筵（秋讲）的期限，即“一学年，两学期”的学制。

      +

      张居正制定的学习计划数年如一日地执行着，根据谢贵安的不完全统计，从隆庆六年八月十八日第一次日讲开始，到万历十七年二月二日最后一次日讲截止，神宗参加的经筵是99次，日讲是601次，共计700次。其中张居正在世时（万历十年六月二十日前），举行的经筵是72次，日讲是551次，共623次；分别占到全部经筵的73%，全部日讲的92%，所有经筵日讲的89%。可见其教学计划执行的有效性。除了经筵日讲之外，张居正平日对万历皇帝的耳提面命也是不可忽略的重要的教育形式。

      +

      从这里可以看出，万历皇帝的学习任务非常繁重，这对一个还只有十几岁的孩子而言，亦可说是某种程度的摧残，不可避免会产生逆反心理。讲到这里，谢贵安教授也提醒大家注意孩子教育过程中给予适当玩耍娱乐的空间，帮助其更好的成长。

      +

      教育内容

      +

      谢贵安认为，张居正教育万历皇帝的目的是希望通过提高神宗“圣学”，培养其“圣德”，这也是经筵日讲教育的最终目的，因此他特别重视用儒学思想教育神宗，竭力使之成为儒家期待的“尧舜之君”，使其具有“大德”，心怀天下。他以儒家思想为中心，儒家经典为教材，给神宗制定了三门功课，并亲自编写教材，可谓用心良苦。第一门课为《四书》课，教材即《大学》、《中庸》、《论语》、《孟子》，以及其他相关书籍，比如真德秀《大学衍义》。如此选择体现了当时的时代特色，明代非常重视《四书》，甚至科举考试的标准参考答案亦常出于其中。当然除此之外也有入门较易这层考虑，《四书》比较通俗易懂，相对《五经》而言更为浅显。

      +

      第二门课为《五经》课，教材为《尚书》、《诗经》、《周易》、《礼记》、《春秋》，缘何顺序如此今已不可考，因为如果从由易入难来考虑，显然《诗经》朗朗上口，更容易学习才对，而张居正选择把《尚书》放在第一位，或许有其他方面的考虑。

      +

      第三门课为《史鉴》课，张居正为皇帝编写了《通鉴》、《训录类编》、《贞观政要》等教材。《通鉴》不是司马光《资治通鉴》的原版，而是其节要精缩版，张居正删去众多繁琐且无关紧要内容，精选其中有价值的部分浓缩而成，同时内容相对容易，符合皇帝的接受程度。《训录类编》中的“训”是指《大明宝训》等皇帝宝训，录自各位皇帝的实录，一般皇帝都是先修实录，继而从中摘录出一部宝训，即皇帝的精华语录。其中实录收藏起来秘不示人，宝训则面向公众发行。张居正从这两类书中择其精华加以编订，按类编写，共40个类目，以供皇帝学习之用。谢贵安认为，这一教材的编写可以看出张居正希望皇帝在《通鉴》之外，还必须对当代的历史经验教训加以总结，他不但崇尚先贤，同时亦注重现实问题的解决，如此可以培养皇帝的现代意识，注重解决现代问题。如此思想也是他能主持长达十年改革的原因之一。《贞观政要》主要记载唐太宗李世民与大臣讨论如何治理国家、弘扬道德，以达成盛世。张居正把它作为学习教材，是希望皇帝能接受劝诫，培养起纳谏的宏大胸怀。

      +

      明神宗所上三门课程，《四书》、《五经》和史鉴课，都得以完成，所讲的经书如《尚书》甚至讲读过两次以上。对于日理万机的皇帝来说，忠实而完整地执行儒生们设计的经筵、日讲制度，并非易事。神宗之所以执行得相对较好，与其冲龄继位，以及张居正强力执行有关。

      +

      谢贵安接下来详细解读了张居正安排这三门功课的用意。第一，提倡重德爱民的观念。儒家提倡仁者爱人，重视民本思想，希望皇帝能成为圣人，把百姓当作自己的小孩那般爱护。张居正不断向小皇帝朱翊钧灌输这种思想，告诫其曰：“为治之道，唯在布德修政以固结民心，乃安宁长久之道，不在事事防范。”并用太祖朱元璋“及将仙逝之年，犹下令劝课农桑，各处里老粮长，有以事至京者，皆亲召见赐食，问以民间疾苦。事事念念，惟在于安民，未尝有一毫受享逸豫之意。深仁厚泽，结于民心”的例子，教育神宗要爱民如子。

      +

      第二，强化重俭轻奢的思想。谢贵安举皇帝想办烟火会被张居正拒之一例加以说明。万历二年闰十二月，神宗想办“鳌山烟火”，张居正却回答说以前是事母和奉神，“非为游观。隆庆以来，乃岁供元夕之娱，糜费无益，是在新政所当节省。”并教育其道：“今天下民力殚诎，有司计无所出，仅得数十铢，亦输京师，其窘可知。及今无事，时加意撙节，稍稍蓄以待用，不然，臣恐浚民脂膏，不给也。”神宗也只得附和：“朕极知民穷如先生言。”于是“明年元夕，罢烟火鳌山”。

      +

      第三，树立重勤轻玩的观念。儒家有“业精于勤，荒于嬉”的观念，张居正常以此教育神宗。如万历四年三月，神宗出御文华殿讲读，辅臣张居正进讲《帝鉴图说》，“至唐玄宗于勤政楼设宴宠幸安禄山事”时，神宗“览其图”，见有“勤政楼”三字，便指出：“楼名甚佳，乃不于此勤理政事，而佚乐宴饮，何也？”张居正便指出：“人情靡不有初，鲜克有终。故有始治而终乱，由圣而入狂者。古圣帝明王，兢兢业业，日慎一日，盖虑克终之难也。玄宗不能常持此心，故及于乱。”是以提醒神宗要有始有终，兢业勤恳。

      +

      第四，强调重道轻艺的观念。张居正要求神宗不要总是练习书法，并以此沾沾自喜，指出：“帝王之学，当务其大。自尧舜以来，至于唐宋所称英贤之主，皆以其修德行政、治世安民，不闻有技艺之巧也。”他还列举“汉成帝知音律，能吹萧度曲，六朝梁元帝、陈后主、隋炀帝、宋徽宗、宁宗皆能文章，善画，然皆无救于乱亡”。进一步提醒神宗皇帝，“君德之大，不在技艺间也。今皇上圣聪日开，宜及时讲求治理，留心政务，以自古圣帝明王为法，若写字一事，不过假此以收放心，虽殚精费神，直逼钟、王，亦有何益？”

      +

      +

      讲座现场

      +

      教育的经验与技巧

      +

      张居正教育万历皇帝的经验和技巧也是本次讲座的核心内容。在谢贵安看来，第一，张居正重视对皇帝朗读和背诵等基本功的训练。他规定神宗要“先读《大学》十遍，次读《尚书》十遍”。后来万历皇帝甚至能背诵唐代张蕴古所撰的《大宝箴》，“居正受册，北面立。上覆诵终篇，不失一字”，可见其基本功之扎实。

      +

      第二，重视循序渐进、逐步提高的方法。虽然张居正雷厉风行，做事讲究执行效率，可是在对皇帝的教育上却有着足够的耐心。他指导神宗学习从低级到高级、从小学（蒙学）到大学（治国大道）逐步提升。“皇上每日日讲经书，以前起止不过四五句，盖以为学工夫当以渐而进，故不敢骤加。今圣学日进，睿质日开，前项经书，似宜稍加增益。”直至最后领悟治国大义。

      +

      第三，照顾蒙学特点，注重图文并茂。一味的经典阅读会令人产生抵触心理，张居正考虑到小孩子对图画的喜以及对此接受能力更高，因此主编了《帝鉴图说》，配合图片讲解历史上发生的故事和道理。这本书是前面所讲《史鉴》课里的一本书，放在此处是强调其教育特点。

      +

      第四，注重灌输与启发相贯穿，学习与提问相结合。他亲自给神宗讲解《帝鉴图说》、《训录类编》，又重视启发，培养其问题意识。“每日各官讲读毕，或圣心于书义有疑，乞即下问，臣等再用俗说讲解，务求明白。”万历六年十二月，神宗在文华殿讲读时，张居正建议：“今所讲经书……其中或有疑义，皇上不妨面问，令讲官通俗讲解，务期明白，或讲官一时仓卒，辞理未畅，臣等当从旁代对，罄竭其愚。”鼓励神宗对不懂的地方加以提问。

      +

      第五，注重温习的方法。学习需要“学而时习之”，因此张居正特别重视“温习”和“温讲”。他要求“将讲读过经书从容温习”；而对于像《大学》及《尚书·舜典》这样的重要内容，还要“温讲”。

      +

      第六，打通课程，一箭双雕。张居正还注重将诵读与练字相结合，把书法课与经史课相贯通。如他一方面要求皇帝诵读《大宝箴》，一方面“以《大宝箴》为习字影格”。这样皇帝在练字的过程中即可学习经典，可谓一举两得。

      +

      第七，严厉与鼓励并用的方法。小皇帝读《论语》“色勃如也”一句时，将“勃”念成“背”。张居正厉声纠正，使神宗“悚然而惊”，“同列皆失色”。但他同时对皇帝亦多有鼓励。如宦官曾拿着神宗读过的《尚书》告诉张居正等人“上于宫中读书，日夕有程，常二四遍覆背，须精熟乃已”。张居正便与讲官“相顾嗟异，以为上好学如此，儒生家所不及！”

      +

      第八，教师与家长相配合的方法。张居正与慈圣李太后经常配合。张居正建议神宗“停罢写字”，改为每日早讲之后学习处理各衙门紧要章奏。神宗说：“先生每说的是。朕回宫奏知圣母，待来年行。”可见张居正在对皇帝的教育上与李太后取得了一致的意见，两者共同配合以达成既定计划。

      +

      教育效果

      +

      在讲座的最后，谢贵安从正反两方面总结了张居正对万历皇帝儒学教育的效果。

      +

      正面而言：神宗坚持经筵和日讲的学习，直到万历十七年二月才中辍面对面的讲读，但以进呈日讲讲章的“函授”形式仍在继续，最终到万历四十三年停止。这些教育导致神宗儒家观念和中庸人格的形成：他虽对大臣“冷战”，但对不满的官员处置较轻，不残忍，可以说有一些仁爱之心存在。同时，神宗不喜游乐，极少出宫，一生主要都待在皇宫之内。张居正的教育也影响了他的学习习惯：他阅读过《击壤集》、《类证本草》、《古文真宝》、《皇明典礼》、《明实录》等大量典籍，成为皇帝接受国学教育的典范之一。

      +

      而负面影响也很明显：万历皇帝是出了名的懒政，甚至长达20余年不上朝（也有一说30年）。万历皇帝极为重利，非常贪婪，如设置矿监税使，任命太监大肆搜刮，横征暴敛，给当时较为活跃的商业经济带来了毁灭性的恶果。还有就是其重男轻女观念的形成，如万历元年十月八日，宦官对张居正等人说：“上……每见史书上王莽及吕太后、武则天、萧太后，即以手指而骂之。”体现了万历皇帝对女性当政者的偏见。

      +

      来源：澎湃新闻 2017-02-26 15:18 作者：武汉大学历史学院郭伦

      +

      （本文经澎湃新闻授权转载）

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      Heating methodAerodynamic
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      31-Mar-2012DCTおよびVQを用いた画像電子透かし柴田, 且崇; 沢田, 克敏; 中村, 栄治; SHIBATA, Katsutaka; SAWADA, Katsutoshi; NAKAMURA, Eiji
      30-Sep-2015ESLによるハンズフリーセキュリティシステム中村, 栄治; 森, 雅斗; 伊藤, 朔太; NAKAMURA, Eiji; MORI, Masato; ITO, Shota
      31-Aug-2021FDSシミュレーション結果を取り入れた避難シミュレーションの試み中村, 栄治; NAKAMURA, Eiji
      31-Mar-2003サブバンドおよびベクトル量子化と組み合わせたフラクタル画像符号化石川, 敬介; 中村, 栄治; 沢田, 克敏; ISHIKAWA, Keisuke; NAKAMURA, Eiji; SAWADA, Katsutoshi
      31-Mar-2013サブバンド処理を用いた画像電子透かし栗本, 裕巳; 沢田, 克敏; 中村, 栄治; KURIMOTO, Hiromi; SAWADA, Katsutoshi; NAKAMURA, Eiji
      31-Mar-2001サブバンド分割と組み合わせたフラクタル画像符号化永井, 進也; 中村, 栄治; 沢田, 克敏; NAGAI, Shinya; NAKAMURA, Eiji; SAWADA, Katsutoshi
      31-Mar-2000サブブロック輝度シフトを用いたフラクタル画像符号化平岩, 裕樹; 中村, 栄治; 沢田, 克敏; HIRAIWA, Yuuki; NAKAMURA, Eiji; SAWADA, Katsutoshi
      30-Sep-2019サリエンシーと人工知能による景観に配慮した防災サインの設置検討山本, 義幸; 中村, 栄治; 倉橋, 奨; YAMAMOTO, Yoshiyuki; NAKAMURA, Eiji; KURAHASHI, Susumu
      31-Oct-2016トンネル災害調査を想定した調査ロボットシステム奥川, 雅之; 中村, 栄治; 山本, 義幸; 倉橋, 奨; 落合, 鋭充; OKUGAWA, Masayuki; NAKAMURA, Eiji; YAMAMOTO, Yoshiyuki; KURAHASHI, Susumu; OCHIAI, Toshimichi
      31-Mar-2005ブロックの分散・平均値を用いた画像電子透かし桃井, 秀人; 中村, 栄治; 沢田, 克敏; MOMOI, Hideto; NAKAMURA, Eiji; SAWADA, Katsutoshi
      31-Jul-2015衛星動画による波の挙動解析ー津波モニタリングに向けてー山本, 義幸; 田中, 純; 中村, 栄治; YAMAMOTO, Yoshiyuki; TANAKA, Jyun; NAKAMURA, Eiji
      23-Sep-2016下水マンホールデータビューワの開発中村, 栄治; 蟹江, 秀俊; NAKAMURA, Eiji; KANIE, Hidetoshi
      31-Aug-2022火災時における階段施設が主避難路となる場合の避難シミュレーション中村, 栄治; NAKAMURA, Eiji
      31-Jul-2014海水浴場における津波避難行動に関する研究森田, 匡俊; 小池, 則満; 小林, 哲郎; 山本, 義幸; 中村, 栄治; 正木, 和明; MORITA, Masatoshi; KOIKE, Norimitsu; KOBAYASHI, Tetsurou; YAMAMOTO, Yoshiyuki; NAKAMURA, Eiji; MASAKI, Kazuaki
      31-Jul-2015環境情報取得におけるUAV活用の検討中村, 栄治; 山本, 義幸; NAKAMURA, Eiji; YAMAMOTO, Yoshiyuki
      31-Mar-2003輝度・色差成分間の相関を利用したカラー画像のサブバンド・フラクタル符号化中根, 勇樹; 中村, 栄治; 沢田, 克敏; NAKANE, Yuki; NAKAMURA, Eiji; SAWADA, Katsutoshi
      31-Oct-2016巨大津波を想定した海上ハザードマップの作成に関する研究 : 三重県南伊勢町を事例として小池, 則満; 服部, 亜由未; 森田, 匡俊; 岩見, 麻子; 江見, 友作; 中村, 栄治; KOIKE, Norimitsu; HATTORI, Ayumi; MORITA, Masatoshi; IWAMI, Asako; EMI, Yusaku; NAKAMURA, Eiji
      31-Aug-2023交通信号機の機能停止時における車両渋滞の回避対策中村, 栄治; 中井, 俊; NAKAMURA, Eiji; NAKAI, Shun
      31-Jul-2013自然災害に対する意思決定支援システムの構築正木, 和明; 小池, 則満; 森田, 匡俊; 中村, 栄治; 奥川, 雅之; 山本, 義幸; 倉橋, 奨; 落合, 鋭充; MASAKI, Kazuaki; KOIKE, Norimitsu; MORITA, Masatoshi; NAKAMURA, Eiji; OKUGAWA, Masayuki; YAMAMOTO, Yoshiyuki; KURAHASHI, Susumu; OCHIAI, Toshimichi
      30-Sep-2020大規模屋内施設からの避難シミュレーション中村, 栄治; 小池, 則満; NAKAMURA, Eiji; KOIKE, Norimitsu
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      Issue DateTitleAuthor(s)
      31-Mar-2012DCTおよびVQを用いた画像電子透かし柴田, 且崇; 沢田, 克敏; 中村, 栄治; SHIBATA, Katsutaka; SAWADA, Katsutoshi; NAKAMURA, Eiji
      30-Sep-2015ESLによるハンズフリーセキュリティシステム中村, 栄治; 森, 雅斗; 伊藤, 朔太; NAKAMURA, Eiji; MORI, Masato; ITO, Shota
      31-Aug-2021FDSシミュレーション結果を取り入れた避難シミュレーションの試み中村, 栄治; NAKAMURA, Eiji
      31-Mar-2003サブバンドおよびベクトル量子化と組み合わせたフラクタル画像符号化石川, 敬介; 中村, 栄治; 沢田, 克敏; ISHIKAWA, Keisuke; NAKAMURA, Eiji; SAWADA, Katsutoshi
      31-Mar-2013サブバンド処理を用いた画像電子透かし栗本, 裕巳; 沢田, 克敏; 中村, 栄治; KURIMOTO, Hiromi; SAWADA, Katsutoshi; NAKAMURA, Eiji
      31-Mar-2001サブバンド分割と組み合わせたフラクタル画像符号化永井, 進也; 中村, 栄治; 沢田, 克敏; NAGAI, Shinya; NAKAMURA, Eiji; SAWADA, Katsutoshi
      31-Mar-2000サブブロック輝度シフトを用いたフラクタル画像符号化平岩, 裕樹; 中村, 栄治; 沢田, 克敏; HIRAIWA, Yuuki; NAKAMURA, Eiji; SAWADA, Katsutoshi
      30-Sep-2019サリエンシーと人工知能による景観に配慮した防災サインの設置検討山本, 義幸; 中村, 栄治; 倉橋, 奨; YAMAMOTO, Yoshiyuki; NAKAMURA, Eiji; KURAHASHI, Susumu
      31-Oct-2016トンネル災害調査を想定した調査ロボットシステム奥川, 雅之; 中村, 栄治; 山本, 義幸; 倉橋, 奨; 落合, 鋭充; OKUGAWA, Masayuki; NAKAMURA, Eiji; YAMAMOTO, Yoshiyuki; KURAHASHI, Susumu; OCHIAI, Toshimichi
      31-Mar-2005ブロックの分散・平均値を用いた画像電子透かし桃井, 秀人; 中村, 栄治; 沢田, 克敏; MOMOI, Hideto; NAKAMURA, Eiji; SAWADA, Katsutoshi
      31-Jul-2015衛星動画による波の挙動解析ー津波モニタリングに向けてー山本, 義幸; 田中, 純; 中村, 栄治; YAMAMOTO, Yoshiyuki; TANAKA, Jyun; NAKAMURA, Eiji
      23-Sep-2016下水マンホールデータビューワの開発中村, 栄治; 蟹江, 秀俊; NAKAMURA, Eiji; KANIE, Hidetoshi
      31-Aug-2022火災時における階段施設が主避難路となる場合の避難シミュレーション中村, 栄治; NAKAMURA, Eiji
      31-Jul-2014海水浴場における津波避難行動に関する研究森田, 匡俊; 小池, 則満; 小林, 哲郎; 山本, 義幸; 中村, 栄治; 正木, 和明; MORITA, Masatoshi; KOIKE, Norimitsu; KOBAYASHI, Tetsurou; YAMAMOTO, Yoshiyuki; NAKAMURA, Eiji; MASAKI, Kazuaki
      31-Jul-2015環境情報取得におけるUAV活用の検討中村, 栄治; 山本, 義幸; NAKAMURA, Eiji; YAMAMOTO, Yoshiyuki
      31-Mar-2003輝度・色差成分間の相関を利用したカラー画像のサブバンド・フラクタル符号化中根, 勇樹; 中村, 栄治; 沢田, 克敏; NAKANE, Yuki; NAKAMURA, Eiji; SAWADA, Katsutoshi
      31-Oct-2016巨大津波を想定した海上ハザードマップの作成に関する研究 : 三重県南伊勢町を事例として小池, 則満; 服部, 亜由未; 森田, 匡俊; 岩見, 麻子; 江見, 友作; 中村, 栄治; KOIKE, Norimitsu; HATTORI, Ayumi; MORITA, Masatoshi; IWAMI, Asako; EMI, Yusaku; NAKAMURA, Eiji
      31-Aug-2023交通信号機の機能停止時における車両渋滞の回避対策中村, 栄治; 中井, 俊; NAKAMURA, Eiji; NAKAI, Shun
      31-Jul-2013自然災害に対する意思決定支援システムの構築正木, 和明; 小池, 則満; 森田, 匡俊; 中村, 栄治; 奥川, 雅之; 山本, 義幸; 倉橋, 奨; 落合, 鋭充; MASAKI, Kazuaki; KOIKE, Norimitsu; MORITA, Masatoshi; NAKAMURA, Eiji; OKUGAWA, Masayuki; YAMAMOTO, Yoshiyuki; KURAHASHI, Susumu; OCHIAI, Toshimichi
      30-Sep-2020大規模屋内施設からの避難シミュレーション中村, 栄治; 小池, 則満; NAKAMURA, Eiji; KOIKE, Norimitsu
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      泡泡详情
      • henryspace @henryspace
        #Crypto + +发现个端到端加密的聊天和文件共享工具，挺有意思～ +
        https://keybase.io/
        发布于 2023-07-06 23:12 浙江省杭州市 最后回复 2023-12-13 01:04
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      • henryspace @henryspace
        浙江省杭州市
        发展大了的电报毕竟也要受监管，凡是中心化的东西都逃不脱要被控制的命运；但是太自由了也很容易玩脱了，成为作恶和犯罪的温床，所以区块链这种东西注定也只是小众的，只要有人群在的地方，人们不允许有不能被控制的东西任意发展。
        2023-07-06 23:19
        0
        alimy:
        上海市
        个人观点，端密聊天 本身就是小众的所需，也是小众受用，应该是，有人群的地方，小众虽小，却有很强的关注度与话语权，容易被这部分小众的视角带偏思维，以偏概全。大众某种程度上来说，是有限空间内无限自由的！
        2023-07-07 03:56
        0
      • 北野 @alimy
        上海市
        好熟悉的app啊，看👀其logo，就好像哪见过，之前好像有开源版本来着？后来被zoom给收购了？ 忘了，可能搞混了吧～
        2023-07-07 02:41
        0
        henryspace:
        浙江省杭州市
        这个是开源的：https://github.com/keybase/book-content
        2023-07-07 09:03
        0
      • 北野 @alimy
        上海市
        @henryspace 端到端加密 说实话也并不是大众群体的刚性需求，这玩意说以开发商的角度思考，就TM的一个好噱头，方便讲故事拉投资顺便拉点用户。 +站在普通大众群体，端密聊天，可有可无，老子又不是名人，也没人说我是帅哥美女，除了对面的聊天对象，谁tm在乎老子啊，都不在乎了，还怕你吖的看老子聊啥啊，不存在的～ 所以端密聊天，大众群体是概念模糊的，有选择当然更好，没得选也不太在乎～ +
        2023-07-07 02:46
        0
        alimy:
        上海市
        但是那一小撮小众群体，确实非常需要 端密聊天，一是上层精英认识，有时这个确实是刚需，还有一个群体，就是需要逃避某种潜在审核的聊天，对这个也是刚需～
        2023-07-07 02:48
        0
        alimy:
        上海市
        要我说，端密聊天，作为运营者角度来看，就是一个需要多投入精力去开发，还需要更多算力与带宽来支撑的鸡肋产品特性，大环境下其本质是满足那一小撮小众群体的隐形刚需而已
        2023-07-07 02:50
        0
        alimy:
        上海市
        但是作为老板，为了在资本市场有素材讲好故事拉来投资亦或给陌生用户留个好印象端密这玩意确实好使，就像一个摄影白痴买手机却去在意拍照功能要多么牛逼，其实一年到头也拍不了几张照片，够用就OK了
        2023-07-07 02:54
        0
        alimy:
        上海市
        - 但用户还是会为潜在需求买单。端密聊天这个特性和上面👆说的买相机送手机非常类似，你其实要的是手机，却盯着相机特性去买；这里的话，作为普通用户，他喵的老子就想愉快的聊个天，其他的还真不太在乎～
        2023-07-07 02:57
        0
        alimy:
        上海市
        - 关键是他喵的普通大众的老子，也没啥太值得被在乎啊，想到这，就莫名的有种安全感，是不是呢～
        2023-07-07 02:58
        0
        alimy:
        上海市
        - 这让我想到一个段子，长相普通甚至有点丑的女生，走在昏暗的大街上，也是非常安全的，有时候丑也是一种安全保障，很奇特是吧，但从某种角度来说，是说的通的～
        2023-07-07 03:01
        0
        alimy:
        上海市
        - 接上面👆 反过来说，漂亮美丽，有的时候在某些场景下是优势，在另外一些场景，就可能是危机本身的来源(因为漂亮所以容易被惦记啥的)，所以深入想想看，漂亮也是一种负担哦，至少有些情况是这样的～
        2023-07-07 03:04
        0
        alimy:
        上海市
        - 回到我们的端密聊天，端密聊天其实本身就是那一小撮小众群体的刚需，大众群体对这玩意，真不是特别在意，除非～你让这些人有感觉自己非常非常重要～
        2023-07-07 03:07
        0
        alimy:
        上海市
        - 闲的没事瞎琢磨哈，以上纯属个人观点，老铁们当笑话看就OK了，不要太在意哈🤣～
        2023-07-07 03:09
        0
        henryspace:
        浙江省杭州市
        单从技术角度来讲，聊天软件本来就应该不受监管，谁也不希望自己信息能被第三方摊开来看
        2023-07-07 09:06
        1
        henryspace:
        浙江省杭州市
        而社会意识又要求为了群体利益而放弃掉一部分个体利益，所以适合群体的总能大行其道
        2023-07-07 09:08
        0
        henryspace:
        浙江省杭州市
        不过我发现这个软件在00后里倒是很受欢迎╮(╯▽╰)╭
        2023-07-07 09:09
        0
        yanjun回复alimy
        广东省深圳市
        主要是不赚钱
        2023-12-13 01:04
        1
      • yanjun @yanjun
        广东省深圳市
        新的练手项目get
        2023-12-13 01:02
        0
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      +K O'Connor +Offline +
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      +#1 +
      + +04-13-2023, 01:42 PM +
      +
      + Hi Guys
      +
      +The idle condition of a power amplifier output stage is referred to as its "operating class". It is important to note that the bias condition is signal-dependent and has nothing to do with how the output stage devices are controlled. Again, bias condition is a universally-applicable concept.
      +
      +Class-A: All of the output devices contribute to the signal over the full audio cycle (360-degrees)
      +
      +Class-A2: Tubes only, the "2" indicates that grid conduction occurs in the output devices. This is simply class-AB below with a low-impedance drive  circuit and very high idle current.
      +
      +Limiting Class-A: The peak signal current never exceeds the total idle current. This term was common in tube days, but still applies universally although it may be considered redundant to the class-A definition above.
      +
      +Sliding Class-A: Solid-state, a method of varying the idle condition so that neither half of the circuit ever turns off even though transference of signal control may occur.
      +
      +Class-B: In push-pull, each half of the output stage contributes exactly half the output signal (180-degrees)
      +
      +Class-B2: Tubes only, the "2" indicates grid conduction made possible by a low-impedance drive circuit.
      +
      +Class-AB: In push-pull, each half of the output stage contributes to slightly more than half of the signal output. Most "class-B" output stages are actually biased this way, with a slight overlap of conduction between circuit halves.
      +
      +Class-C: The output device conducts for only half the signal cycle (180-degrees) with a tuned load providing the remainder. Used in RF.
      +
      +Class-D: Solid-state only, a method of using a nonlinear output stage where the devices switch 'on' and 'off' in a pulse-width-modulated (PWM) format, and the output signal is integrated using LC filters. This approach is highly load noncompliant inasmuch as the load should be of fixed value versus frequency (resistive rather than inductive or capacitive). Class-D allows cold operation of the output devices but is only suitable for driving subwoofers in audio.
      +
      +Class-E: Solid-state, where parallel-driven output stages supported by different supply values contribute to the signal. The low-voltage stage amplifies the signal up to its limits with the high-voltage stage contributing higher amplitude signals as required. The low-voltage output stage can be biased class-A,-B or -AB.
      +
      +Class-G: Solid-state, a multi-tier output stage uses multiple supply voltages,switching between them as the signal requires. The transition shifts the burden of output heat from the low-tier device to the next higher-tier device. Overall dissipation is generally reduced by the number of tiers.
      +
      +Class-H: Solid-state, a multi-tier output stage supported by multiple supply voltages, switching between them as the signal requires. The supply switches turn 'on' hard and the burden of heat dissipation remains with the lowest-tier devices. Overall dissipation is reduced by the number of tiers.
      +
      +Class-I: Similar to sliding class-A.
      +
      +Class-T: A variation of class-D, with all of the same inherent issues.
      +
      +Class-Z: A method of power transfer using saturable coils "steered" by tubes with output stage power provided by a switching supply. designed by Lundahl (SE) in the 1960s, then revised and patented by Berning in the 1990s. +
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      A warm welcome to tube amp modding fans and those interested in hi-fi audio! Readers of Kevin O'Connor's The Ultimate Tone (TUT) book series form a part of our population. Kevin O'Connor is the creator of the popular Power Scaling methodology for amplifiers.
      Please remember these three principles: respect, sharing, community.
      Not familiar with The Ultimate Tone book series? See discussion topics, or click here to visit London Power/Power Press Publishing.

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      + + + \ No newline at end of file diff --git a/tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/special_table_1.html b/tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/special_table_1.html index 9a84ea19..f201aceb 100644 --- a/tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/special_table_1.html +++ b/tests/llm_web_kit/main_html_parser/parser/assets/test_html_data/special_table_1.html @@ -156,7 +156,7 @@

      Вас могут заинтересовать:
      Подснежник, стихотворения для детей и юношества (1897 г.)

      -
      +
      Автор:

      @@ -180,8 +180,10 @@

      П Р О Д А Н О

      Формат книги: 150 х 220 мм
      Твердый переплет

      -

      Плещеев Алексей Николаевич (1825-1893 гг.) - русский писатель, поэт, переводчик; литературный и театральный критик. В 1846 году первый же сборник стихов сделал Плещеева знаменитым в революционной молодежной среде; как участник кружка Петрашевского он был в 1849 году арестован и некоторое время спустя отправлен в ссылку, где провел на военной службе почти десять лет. По возвращении из ссылки Плещеев продолжил литературную деятельность; пройдя через годы бедности и лишений, он стал авторитетным литератором, критиком, издателем, а в конце жизни и меценатом. Многие произведения поэта (особенно - стихи для детей) стали хрестоматийными, считаются классикой. На стихи Плещеева известнейшими русскими композиторами написаны более ста романсов.
      -
      +

      Плещеев Алексей Николаевич (1825-1893 гг.) - русский писатель, поэт, переводчик; литературный и театральный критик. В 1846 году первый же сборник стихов сделал Плещеева знаменитым в революционной молодежной среде; как участник кружка Петрашевского он был в 1849 году арестован и некоторое время спустя отправлен в ссылку, где провел на военной службе почти десять лет. По возвращении из ссылки Плещеев продолжил литературную деятельность; пройдя через годы бедности и лишений, он стал авторитетным литератором, критиком, издателем, а в конце жизни и меценатом. Многие произведения поэта (особенно - стихи для детей) стали хрестоматийными, считаются классикой. На стихи Плещеева известнейшими русскими композиторами написаны более ста романсов. +
      + +
      Купив старинную книгу, изданную более 100 лет назад, вы станете обладателем антикварного издания, которое не подлежит вывозу за пределы Российской Федерации

      @@ -243,7 +245,8 @@

      Му - Антикварная книга во владельческом составном переплете. Золотое тиснением по корешку. Потертости переплета. Следы реставрации двух страниц (изображения Колизея). На титульном листе старой книги библиотечный штамп. Следы загрязнения страниц. Лисьи пятна. Следы загрязнения страниц от перелистывания. Редкие следы залития. Мелкие надрывы и фрагментарные утраты по краям некоторых страниц (текст не задет). Книготорговые штампы и пометы на нахзаце. Реставрация бумажной полосой стр. 53 (текст незначительно задет). Издание шестое. Исторический рассказ для детей. Орфография и пунктуация приближены к современным нормам. Первая публикация произведения состоялась в 1890 году, "Мученики Колизея" тогда были напечатаны в Университетской типографии.
      + Антикварная книга во владельческом составном переплете. Золотое тиснением по корешку. Потертости переплета. Следы реставрации двух страниц (изображения Колизея). На титульном листе старой книги библиотечный штамп. Следы загрязнения страниц. Лисьи пятна. Следы загрязнения страниц от перелистывания. Редкие следы залития. Мелкие надрывы и фрагментарные утраты по краям некоторых страниц (текст не задет). Книготорговые штампы и пометы на нахзаце. Реставрация бумажной полосой стр. 53 (текст незначительно задет). Издание шестое. Исторический рассказ для детей. Орфография и пунктуация приближены к современным нормам. Первая публикация произведения состоялась в 1890 году, "Мученики Колизея" тогда были напечатаны в Университетской типографии. +


      Издательство: Типография Г. Лисснера и Д. Собко (Москва)

      Язык: русский

      diff --git a/tests/llm_web_kit/main_html_parser/parser/test_layout_parser.py b/tests/llm_web_kit/main_html_parser/parser/test_layout_parser.py index c7a4df5f..3eaf8471 100644 --- a/tests/llm_web_kit/main_html_parser/parser/test_layout_parser.py +++ b/tests/llm_web_kit/main_html_parser/parser/test_layout_parser.py @@ -163,7 +163,7 @@ def test_dynamic_id(self): template_source = re.sub('post-37041', 'test-37041', template_source) expand_source = re.sub('test-37041', 'test-25031', template_source) # 简化网页 - simplified_html, typical_raw_tag_html, _ = simplify_html(template_source) + simplified_html, typical_raw_tag_html = simplify_html(template_source) # 模型结果格式改写 llm_path = base_dir.joinpath(TEST_CASES[0]['input'][2][0]) llm_response = json.loads(llm_path.read_text(encoding='utf-8')) @@ -203,7 +203,7 @@ def test_dynamic_classid(self): expand_source2 = re.sub('testid-25031', '', expand_source) template_source2 = re.sub('testid-37041', '', template_source) # 简化网页 - simplified_html, typical_raw_tag_html, _ = simplify_html(template_source) + simplified_html, typical_raw_tag_html = simplify_html(template_source) # 模型结果格式改写 llm_path = base_dir.joinpath(TEST_CASES[0]['input'][2][0]) llm_response = json.loads(llm_path.read_text(encoding='utf-8')) @@ -226,7 +226,7 @@ def test_dynamic_classid(self): assert 'Permalink link a questo articolo' not in main_html_body and 'Con la stesura di un' in main_html_body # 简化网页 - simplified_html, typical_raw_tag_html, _ = simplify_html(template_source2) + simplified_html, typical_raw_tag_html = simplify_html(template_source2) # 模型结果格式改写 llm_path = base_dir.joinpath(TEST_CASES[0]['input'][2][0]) llm_response = json.loads(llm_path.read_text(encoding='utf-8')) @@ -268,7 +268,7 @@ def test_more_noise_enable(self): new_p.text = 'test more noise' expand_source = html.tostring(tree, encoding='utf-8').decode() # 简化网页 - simplified_html, typical_raw_tag_html, _ = simplify_html(template_source) + simplified_html, typical_raw_tag_html = simplify_html(template_source) # 模型结果格式改写 llm_path = base_dir.joinpath(TEST_CASES[0]['input'][2][0]) llm_response = json.loads(llm_path.read_text(encoding='utf-8')) @@ -305,7 +305,7 @@ def test_classid_with_first_class(self): template_source = re.sub('post-37041', '', template_source) expand_source = re.sub('post-classid', 'post-classid template-classid', template_source) # 简化网页 - simplified_html, typical_raw_tag_html, _ = simplify_html(template_source) + simplified_html, typical_raw_tag_html = simplify_html(template_source) # 模型结果格式改写 llm_path = base_dir.joinpath(TEST_CASES[0]['input'][2][0]) llm_response = json.loads(llm_path.read_text(encoding='utf-8')) @@ -385,7 +385,7 @@ def test_incomplete_tag(self): # 模型结果格式改写 llm_path = 'assets/input_layout_batch_parser/test_incomplete_tag.json' llm_response = json.loads(base_dir.joinpath(llm_path).read_text(encoding='utf-8')) - simplified_html, typical_raw_tag_html, _ = simplify_html(html_source) + simplified_html, typical_raw_tag_html = simplify_html(html_source) pre_data = {'typical_raw_tag_html': typical_raw_tag_html, 'typical_raw_html': html_source, 'llm_response': llm_response} pre_data = PreDataJson(pre_data) diff --git a/tests/llm_web_kit/main_html_parser/parser/test_tag_simplifier.py b/tests/llm_web_kit/main_html_parser/parser/test_tag_simplifier.py index ea5d57e9..45962b1e 100644 --- a/tests/llm_web_kit/main_html_parser/parser/test_tag_simplifier.py +++ b/tests/llm_web_kit/main_html_parser/parser/test_tag_simplifier.py @@ -1,6 +1,9 @@ +import re import unittest from pathlib import Path +from lxml import html + from llm_web_kit.input.pre_data_json import PreDataJson, PreDataJsonKey from llm_web_kit.main_html_parser.parser.tag_simplifier import \ HtmlTagSimplifierParser @@ -9,60 +12,465 @@ class MyTestCase(unittest.TestCase): + def check_and_find_max_item_id(self, input_str: str) -> int: + # 匹配所有 _item_id="XXX" 的模式，提取XXX部分 + pattern = "_item_id" + r'="(\d+)"' + matches = re.findall(pattern, input_str) + + # 至少匹配一个 + if len(matches) == 0: + return 0 + + # 匹配到的对象全部转化为int + int_list = [] + for match in matches: + try: + int_list.append(int(match)) + except Exception: + raise ValueError(f'error while convert match {match} to int') + + # 检查是否为从1开始的连续整数 + target_value = 1 + for int_id in int_list: + if int_id == target_value: + target_value += 1 + else: + raise ValueError( + f'mistake find in int list, current target value is {target_value}, but find {int_id}' + '\n' + input_str + ) + + # 都没有问题的情况下，返回最大的数 + return int_list[-1] + def test_tag_simplifier(self): file_path = base_dir / 'assets/test_html_data/test_tah_simplifier.html' with open(file_path, 'r', encoding='utf-8') as file: raw_html = file.read() + data_dict = {PreDataJsonKey.TYPICAL_RAW_HTML: raw_html} pre_data = PreDataJson(data_dict) + pre_data_result = HtmlTagSimplifierParser({}).parse(pre_data) + simplifier_raw_html = pre_data_result.get(PreDataJsonKey.TYPICAL_SIMPLIFIED_HTML, '') - _item_id_count = simplifier_raw_html.count('_item_id') - self.assertEqual(_item_id_count, 32) + simple_id_count = self.check_and_find_max_item_id(simplifier_raw_html) + self.assertEqual(simple_id_count, 34) + + raw_tag_html = pre_data_result.get(PreDataJsonKey.TYPICAL_RAW_TAG_HTML, '') + tag_id_count = self.check_and_find_max_item_id(raw_tag_html) + self.assertEqual(tag_id_count, simple_id_count) def test_tag_simplifier1(self): file_path = base_dir / 'assets/test_html_data/normal_dl.html' with open(file_path, 'r', encoding='utf-8') as file: raw_html = file.read() + data_dict = {PreDataJsonKey.TYPICAL_RAW_HTML: raw_html} pre_data = PreDataJson(data_dict) + pre_data_result = HtmlTagSimplifierParser({}).parse(pre_data) + simplifier_raw_html = pre_data_result.get(PreDataJsonKey.TYPICAL_SIMPLIFIED_HTML, '') - _item_id_count = simplifier_raw_html.count('_item_id') - self.assertEqual(_item_id_count, 18) + simple_id_count = self.check_and_find_max_item_id(simplifier_raw_html) + self.assertEqual(simple_id_count, 48) + + raw_tag_html = pre_data_result.get(PreDataJsonKey.TYPICAL_RAW_TAG_HTML, '') + tag_id_count = self.check_and_find_max_item_id(raw_tag_html) + self.assertEqual(tag_id_count, simple_id_count) def test_tag_simplifier2(self): file_path = base_dir / 'assets/test_html_data/normal_table.html' with open(file_path, 'r', encoding='utf-8') as file: raw_html = file.read() + data_dict = {PreDataJsonKey.TYPICAL_RAW_HTML: raw_html} pre_data = PreDataJson(data_dict) + pre_data_result = HtmlTagSimplifierParser({}).parse(pre_data) + simplifier_raw_html = pre_data_result.get(PreDataJsonKey.TYPICAL_SIMPLIFIED_HTML, '') - _item_id_count = simplifier_raw_html.count('_item_id') - self.assertEqual(_item_id_count, 30) + simple_id_count = self.check_and_find_max_item_id(simplifier_raw_html) + self.assertEqual(simple_id_count, 11) + + raw_tag_html = pre_data_result.get(PreDataJsonKey.TYPICAL_RAW_TAG_HTML, '') + tag_id_count = self.check_and_find_max_item_id(raw_tag_html) + self.assertEqual(tag_id_count, simple_id_count) def test_tag_simplifier3(self): file_path = base_dir / 'assets/test_html_data/special_table_1.html' with open(file_path, 'r', encoding='utf-8') as file: raw_html = file.read() + data_dict = {PreDataJsonKey.TYPICAL_RAW_HTML: raw_html} pre_data = PreDataJson(data_dict) + pre_data_result = HtmlTagSimplifierParser({}).parse(pre_data) + simplifier_raw_html = pre_data_result.get(PreDataJsonKey.TYPICAL_SIMPLIFIED_HTML, '') - _item_id_count = simplifier_raw_html.count('_item_id') - self.assertEqual(_item_id_count, 66) + simple_id_count = self.check_and_find_max_item_id(simplifier_raw_html) + self.assertEqual(simple_id_count, 41) + + raw_tag_html = pre_data_result.get(PreDataJsonKey.TYPICAL_RAW_TAG_HTML, '') + tag_id_count = self.check_and_find_max_item_id(raw_tag_html) + self.assertEqual(tag_id_count, simple_id_count) def test_tag_simplifier4(self): file_path = base_dir / 'assets/test_html_data/1.html' with open(file_path, 'r', encoding='utf-8') as file: raw_html = file.read() + data_dict = {PreDataJsonKey.TYPICAL_RAW_HTML: raw_html, PreDataJsonKey.IS_XPATH: False} pre_data = PreDataJson(data_dict) + + pre_data_result = HtmlTagSimplifierParser({}).parse(pre_data) + + simplifier_raw_html = pre_data_result.get(PreDataJsonKey.TYPICAL_SIMPLIFIED_HTML, '') + simple_id_count = self.check_and_find_max_item_id(simplifier_raw_html) + self.assertEqual(simple_id_count, 113) + + raw_tag_html = pre_data_result.get(PreDataJsonKey.TYPICAL_RAW_TAG_HTML, '') + tag_id_count = self.check_and_find_max_item_id(raw_tag_html) + self.assertEqual(tag_id_count, simple_id_count) + + def test_tag_simplifier_table(self): + file_path = base_dir / 'assets/test_html_data/simplify_cases/table.html' + with open(file_path, 'r', encoding='utf-8') as file: + raw_html = file.read() + + data_dict = {PreDataJsonKey.TYPICAL_RAW_HTML: raw_html} + pre_data = PreDataJson(data_dict) + + pre_data_result = HtmlTagSimplifierParser({}).parse(pre_data) + + simplifier_raw_html = pre_data_result.get(PreDataJsonKey.TYPICAL_SIMPLIFIED_HTML, '') + simple_id_count = self.check_and_find_max_item_id(simplifier_raw_html) + self.assertEqual(simple_id_count, 35) + + raw_tag_html = pre_data_result.get(PreDataJsonKey.TYPICAL_RAW_TAG_HTML, '') + tag_id_count = self.check_and_find_max_item_id(raw_tag_html) + self.assertEqual(tag_id_count, simple_id_count) + + id_dom = html.fromstring(raw_tag_html) + # 用xpath定位data-anno-uid="anno-uid-3vtzg9uxee4"的table元素，该table用于布局 + table_element = id_dom.xpath('//table[@data-anno-uid="anno-uid-3vtzg9uxee4"]')[0] + # 确认该table元素没有_item_id属性 + self.assertEqual(table_element.get('_item_id'), None) + # 确认该table的3个td元素的内部都包含若干个存在_item_id属性的元素 + for td_element in table_element.xpath('./tbody/tr/td'): + td_item_count = 0 + for child in td_element.iter(): + if child.get('_item_id') is not None: + td_item_count += 1 + self.assertNotEqual(td_item_count, 0) + + def test_tag_simplifier_data_table(self): + file_path = base_dir / 'assets/test_html_data/simplify_cases/data_table.html' + with open(file_path, 'r', encoding='utf-8') as file: + raw_html = file.read() + + data_dict = {PreDataJsonKey.TYPICAL_RAW_HTML: raw_html} + pre_data = PreDataJson(data_dict) + + pre_data_result = HtmlTagSimplifierParser({}).parse(pre_data) + + simplifier_raw_html = pre_data_result.get(PreDataJsonKey.TYPICAL_SIMPLIFIED_HTML, '') + simple_id_count = self.check_and_find_max_item_id(simplifier_raw_html) + self.assertEqual(simple_id_count, 105) + + raw_tag_html = pre_data_result.get(PreDataJsonKey.TYPICAL_RAW_TAG_HTML, '') + tag_id_count = self.check_and_find_max_item_id(raw_tag_html) + self.assertEqual(tag_id_count, simple_id_count) + + id_dom = html.fromstring(raw_tag_html) + # 用xpath定位带有data-table属性的table元素 + table_element = id_dom.xpath('//table[@data-table]')[0] + # 确认该table元素有_item_id属性 + self.assertIsNotNone(table_element.get('_item_id')) + + def test_tag_simplifier_nested_table_colgroup(self): + file_path = base_dir / 'assets/test_html_data/simplify_cases/nested_table_colgroup.html' + with open(file_path, 'r', encoding='utf-8') as file: + raw_html = file.read() + + data_dict = {PreDataJsonKey.TYPICAL_RAW_HTML: raw_html} + pre_data = PreDataJson(data_dict) + + pre_data_result = HtmlTagSimplifierParser({}).parse(pre_data) + + simplifier_raw_html = pre_data_result.get(PreDataJsonKey.TYPICAL_SIMPLIFIED_HTML, '') + simple_id_count = self.check_and_find_max_item_id(simplifier_raw_html) + self.assertEqual(simple_id_count, 13) + + raw_tag_html = pre_data_result.get(PreDataJsonKey.TYPICAL_RAW_TAG_HTML, '') + tag_id_count = self.check_and_find_max_item_id(raw_tag_html) + self.assertEqual(tag_id_count, simple_id_count) + + id_dom = html.fromstring(raw_tag_html) + # 用xpath定位外层table元素，该table用于布局 + table_element = id_dom.xpath('//table[@class="centralPane"]')[0] + # 确认该table元素没有_item_id属性 + self.assertIsNone(table_element.get('_item_id')) + + # 用xpath定位内层table元素，该table是数据表格，

      标签内存在colgroup元素 + table_element = id_dom.xpath('//table[@class="miscTable"]')[0] + # 确认该table元素有_item_id属性 + self.assertIsNotNone(table_element.get('_item_id')) + + def test_tag_simplifier_nested_table_headers(self): + # 测试表格的单元格中包含headers属性的情况，这个测试用例中的表格单元格存在headers属性，但是
      标签内不包含colgroup元素 + file_path = base_dir / 'assets/test_html_data/simplify_cases/nested_table_headers.html' + with open(file_path, 'r', encoding='utf-8') as file: + raw_html = file.read() + + data_dict = {PreDataJsonKey.TYPICAL_RAW_HTML: raw_html} + pre_data = PreDataJson(data_dict) + + pre_data_result = HtmlTagSimplifierParser({}).parse(pre_data) + + simplifier_raw_html = pre_data_result.get(PreDataJsonKey.TYPICAL_SIMPLIFIED_HTML, '') + simple_id_count = self.check_and_find_max_item_id(simplifier_raw_html) + self.assertEqual(simple_id_count, 13) + + raw_tag_html = pre_data_result.get(PreDataJsonKey.TYPICAL_RAW_TAG_HTML, '') + tag_id_count = self.check_and_find_max_item_id(raw_tag_html) + self.assertEqual(tag_id_count, simple_id_count) + + id_dom = html.fromstring(raw_tag_html) + # 用xpath定位外层table元素，该table用于布局 + table_element = id_dom.xpath('//table[@class="centralPane"]')[0] + # 确认该table元素没有_item_id属性 + self.assertIsNone(table_element.get('_item_id')) + + # 用xpath定位内层table元素，该table是数据表格，其单元格包含headers属性 + table_element = id_dom.xpath('//table[@class="miscTable"]')[0] + # 确认该table元素有_item_id属性 + self.assertIsNotNone(table_element.get('_item_id')) + + def test_tag_simplifier_nested_table_caption(self): + file_path = base_dir / 'assets/test_html_data/simplify_cases/nested_table_caption.html' + with open(file_path, 'r', encoding='utf-8') as file: + raw_html = file.read() + + data_dict = {PreDataJsonKey.TYPICAL_RAW_HTML: raw_html} + pre_data = PreDataJson(data_dict) + + pre_data_result = HtmlTagSimplifierParser({}).parse(pre_data) + + simplifier_raw_html = pre_data_result.get(PreDataJsonKey.TYPICAL_SIMPLIFIED_HTML, '') + simple_id_count = self.check_and_find_max_item_id(simplifier_raw_html) + self.assertEqual(simple_id_count, 14) + + raw_tag_html = pre_data_result.get(PreDataJsonKey.TYPICAL_RAW_TAG_HTML, '') + tag_id_count = self.check_and_find_max_item_id(raw_tag_html) + self.assertEqual(tag_id_count, simple_id_count) + + id_dom = html.fromstring(raw_tag_html) + # 用xpath定位外层table元素，该table用于布局 + table_element = id_dom.xpath('//table[@data-anno-uid="anno-uid-xgzpvn8fnqk"]')[0] + # 确认该table元素没有_item_id属性 + self.assertIsNone(table_element.get('_item_id')) + + # 用xpath定位内层table元素，该table是数据表格，其包含caption元素 + table_element = id_dom.xpath('//table[@data-anno-uid="anno-uid-olo3onur84"]')[0] + # 确认该table元素有_item_id属性 + self.assertIsNotNone(table_element.get('_item_id')) + + def test_tag_simplifier_list(self): + file_path = base_dir / 'assets/test_html_data/simplify_cases/list.html' + with open(file_path, 'r', encoding='utf-8') as file: + raw_html = file.read() + + data_dict = {PreDataJsonKey.TYPICAL_RAW_HTML: raw_html} + pre_data = PreDataJson(data_dict) + + pre_data_result = HtmlTagSimplifierParser({}).parse(pre_data) + + simplifier_raw_html = pre_data_result.get(PreDataJsonKey.TYPICAL_SIMPLIFIED_HTML, '') + simple_id_count = self.check_and_find_max_item_id(simplifier_raw_html) + self.assertEqual(simple_id_count, 118) + + raw_tag_html = pre_data_result.get(PreDataJsonKey.TYPICAL_RAW_TAG_HTML, '') + tag_id_count = self.check_and_find_max_item_id(raw_tag_html) + self.assertEqual(tag_id_count, simple_id_count) + + id_dom = html.fromstring(raw_tag_html) + # 用xpath定位ul元素，该ul用于布局 + list_element = id_dom.xpath('//ul[@data-anno-uid="anno-uid-7s58m3hrcz5"]')[0] + # 确认该ul元素没有_item_id属性 + self.assertIsNone(list_element.get('_item_id')) + # 确认该ul元素下的li元素内均包含有_item_id属性的元素 + for li_element in list_element.xpath('./li'): + li_item_count = 0 + for child in li_element.iter(): + if child.get('_item_id') is not None: + li_item_count += 1 + self.assertNotEqual(li_item_count, 0) + + def test_tag_simplifier_non_list_child(self): + file_path = base_dir / 'assets/test_html_data/simplify_cases/non_list_child.html' + with open(file_path, 'r', encoding='utf-8') as file: + raw_html = file.read() + + data_dict = {PreDataJsonKey.TYPICAL_RAW_HTML: raw_html} + pre_data = PreDataJson(data_dict) + + pre_data_result = HtmlTagSimplifierParser({}).parse(pre_data) + + simplifier_raw_html = pre_data_result.get(PreDataJsonKey.TYPICAL_SIMPLIFIED_HTML, '') + simple_id_count = self.check_and_find_max_item_id(simplifier_raw_html) + self.assertEqual(simple_id_count, 151) + + raw_tag_html = pre_data_result.get(PreDataJsonKey.TYPICAL_RAW_TAG_HTML, '') + tag_id_count = self.check_and_find_max_item_id(raw_tag_html) + self.assertEqual(tag_id_count, simple_id_count) + + id_dom = html.fromstring(raw_tag_html) + # 用xpath定位ul元素，该ul用于布局 + list_element = id_dom.xpath('//ul[@data-anno-uid="anno-uid-myobddy8ord"]')[0] + # 确认该ul元素没有_item_id属性 + self.assertIsNone(list_element.get('_item_id')) + # 用xpath定位上述ul内部的一个li，该li内部结构复杂，应该包含多个_item_id + li_element = id_dom.xpath('//li[@data-anno-uid="anno-uid-7wux77fqc7t"]')[0] + li_item_count = 0 + for child in li_element.iter(): + if child.get('_item_id') is not None: + li_item_count += 1 + self.assertNotEqual(li_item_count, 0) + + def test_tag_simplifier_inline_block(self): + file_path = base_dir / 'assets/test_html_data/simplify_cases/inline_block.html' + with open(file_path, 'r', encoding='utf-8') as file: + raw_html = file.read() + + data_dict = {PreDataJsonKey.TYPICAL_RAW_HTML: raw_html} + pre_data = PreDataJson(data_dict) + pre_data_result = HtmlTagSimplifierParser({}).parse(pre_data) + simplifier_raw_html = pre_data_result.get(PreDataJsonKey.TYPICAL_SIMPLIFIED_HTML, '') - _item_id_count = simplifier_raw_html.count('_item_id') - self.assertEqual(_item_id_count, 51) + simple_id_count = self.check_and_find_max_item_id(simplifier_raw_html) + self.assertEqual(simple_id_count, 12) + + raw_tag_html = pre_data_result.get(PreDataJsonKey.TYPICAL_RAW_TAG_HTML, '') + tag_id_count = self.check_and_find_max_item_id(raw_tag_html) + self.assertEqual(tag_id_count, simple_id_count) + + id_dom = html.fromstring(raw_tag_html) + # 用xpath定位span元素，该span内部包含了块级元素 + span_element = id_dom.xpath('//span[@data-anno-uid="anno-uid-yrlyp4ay0l"]')[0] + # 确认该span元素没有_item_id属性 + self.assertIsNone(span_element.get('_item_id')) + # 该span元素内部包含多个块级元素，每个块级元素都包含_item_id属性 + for child in span_element.iterchildren(): + self.assertIsNotNone(child.get("_item_id")) + + def test_tag_simplifier_abnormal_comment(self): + file_path = base_dir / 'assets/test_html_data/simplify_cases/abnormal_comment.html' + with open(file_path, 'r', encoding='utf-8') as file: + raw_html = file.read() + + data_dict = {PreDataJsonKey.TYPICAL_RAW_HTML: raw_html} + pre_data = PreDataJson(data_dict) + + pre_data_result = HtmlTagSimplifierParser({}).parse(pre_data) + + simplifier_raw_html = pre_data_result.get(PreDataJsonKey.TYPICAL_SIMPLIFIED_HTML, '') + simple_id_count = self.check_and_find_max_item_id(simplifier_raw_html) + self.assertEqual(simple_id_count, 53) + + raw_tag_html = pre_data_result.get(PreDataJsonKey.TYPICAL_RAW_TAG_HTML, '') + tag_id_count = self.check_and_find_max_item_id(raw_tag_html) + self.assertEqual(tag_id_count, simple_id_count) + + # 验证不规范的注释内包含的有效内容没有被删除 + self.assertIn('', raw_tag_html) + # 验证规范的注释都已被删除 + comment_res = re.search(r'', raw_tag_html, flags=re.DOTALL) + self.assertIsNone(comment_res) + + def test_tag_simplifier_header_tag(self): + file_path = base_dir / 'assets/test_html_data/simplify_cases/header_tag.html' + with open(file_path, 'r', encoding='utf-8') as file: + raw_html = file.read() + + data_dict = {PreDataJsonKey.TYPICAL_RAW_HTML: raw_html} + pre_data = PreDataJson(data_dict) + + pre_data_result = HtmlTagSimplifierParser({}).parse(pre_data) + + simplifier_raw_html = pre_data_result.get(PreDataJsonKey.TYPICAL_SIMPLIFIED_HTML, '') + simple_id_count = self.check_and_find_max_item_id(simplifier_raw_html) + self.assertEqual(simple_id_count, 35) + + raw_tag_html = pre_data_result.get(PreDataJsonKey.TYPICAL_RAW_TAG_HTML, '') + tag_id_count = self.check_and_find_max_item_id(raw_tag_html) + self.assertEqual(tag_id_count, simple_id_count) + + id_dom = html.fromstring(raw_tag_html) + # 用xpath定位元素，该元素位于id名为header的元素内部，且这个'header'是body的直接子元素 + header_element = id_dom.xpath('//h1[@data-anno-uid="anno-uid-g513k4pfha8"]')[0] + # 确认该元素有_item_id属性，也就是被保留了 + self.assertIsNotNone(header_element.get('_item_id')) + # 用xpath定位元素，该元素位于header标签内部，但这个header不是body的直接子元素 + header_element = id_dom.xpath('//h2[@data-anno-uid="anno-uid-g8cyd0j0kn6"]')[0] + # 确认该元素有_item_id属性，也被保留了（目前的simplify是所有的header都保留） + self.assertIsNotNone(header_element.get('_item_id')) + + def test_tag_simplifier_nav_class(self): + file_path = base_dir / 'assets/test_html_data/simplify_cases/nav_class.html' + with open(file_path, 'r', encoding='utf-8') as file: + raw_html = file.read() + + data_dict = {PreDataJsonKey.TYPICAL_RAW_HTML: raw_html} + pre_data = PreDataJson(data_dict) + + pre_data_result = HtmlTagSimplifierParser({}).parse(pre_data) + + simplifier_raw_html = pre_data_result.get(PreDataJsonKey.TYPICAL_SIMPLIFIED_HTML, '') + simple_id_count = self.check_and_find_max_item_id(simplifier_raw_html) + self.assertEqual(simple_id_count, 58) + + raw_tag_html = pre_data_result.get(PreDataJsonKey.TYPICAL_RAW_TAG_HTML, '') + tag_id_count = self.check_and_find_max_item_id(raw_tag_html) + self.assertEqual(tag_id_count, simple_id_count) + + id_dom = html.fromstring(raw_tag_html) + # 用xpath定位元素，该元素的class是nav，但不是body的直接子元素，应该保留 + nav_element = id_dom.xpath('//div[@data-anno-uid="anno-uid-v6mugwj7iv"]')[0] + # 验证nav内部有_item_id属性的元素，证明nav没有被删除 + nav_item_count = 0 + for child in nav_element.iter(): + if child.get('_item_id') is not None: + nav_item_count += 1 + self.assertNotEqual(nav_item_count, 0) + + nav_element = id_dom.xpath('//div[@data-anno-uid="anno-uid-189mlskr0fc"]')[0] + self.assertIsNone(nav_element.get("_item_id")) + + def test_tag_simplifier_block_select(self): + file_path = base_dir / 'assets/test_html_data/simplify_cases/block_select.html' + with open(file_path, 'r', encoding='utf-8') as file: + raw_html = file.read() + + data_dict = {PreDataJsonKey.TYPICAL_RAW_HTML: raw_html} + pre_data = PreDataJson(data_dict) + + pre_data_result = HtmlTagSimplifierParser({}).parse(pre_data) + + simplifier_raw_html = pre_data_result.get(PreDataJsonKey.TYPICAL_SIMPLIFIED_HTML, '') + simple_id_count = self.check_and_find_max_item_id(simplifier_raw_html) + self.assertEqual(simple_id_count, 7) + + raw_tag_html = pre_data_result.get(PreDataJsonKey.TYPICAL_RAW_TAG_HTML, '') + tag_id_count = self.check_and_find_max_item_id(raw_tag_html) + self.assertEqual(tag_id_count, simple_id_count) + + id_dom = html.fromstring(raw_tag_html) + # 用xpath定位元素，该元素是块级元素且内部不包含块级元素，但该元素本身没有cc-select，只是其内部元素有cc-select + p_element = id_dom.xpath('//p[@data-anno-uid="anno-uid-tnbktgx26s"]')[0] + # 验证该元素被加上了_item_id和cc-select + self.assertIsNotNone(p_element.get("_item_id")) + self.assertIsNotNone(p_element.get("cc-select")) if __name__ == '__main__': From cac1743cd6eeae72d7b37be54e1cb210ce4cbcca Mon Sep 17 00:00:00 2001 From: chupei Date: Thu, 11 Sep 2025 11:18:00 +0800 Subject: [PATCH 09/11] =?UTF-8?q?=E9=BB=98=E8=AE=A4math=E5=85=A8=E9=87=8F?= =?UTF-8?q?=E6=89=AB=E6=8F=8F=E6=96=87=E6=9C=AC=E6=AD=A3=E5=88=99=E5=8C=B9?= =?UTF-8?q?=E9=85=8D=20(#556)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-authored-by: Yanggq <1041206149@qq.com> --- .../data/groundtruth/math_katex_latex_1.jsonl | 2 +- .../data/groundtruth/math_katex_latex_3.jsonl | 2 +- .../extractor/html/recognizer/ccmath.py | 19 +--- .../extractor/html/recognizer/table.py | 3 + .../good_data/html/math_dollar.html | 2 +- .../assets/ccmath/katex_mathjax_1.html | 1 + .../extractor/html/recognizer/test_math.py | 5 + tests/llm_web_kit/simple/test_simple.py | 95 +++++++++++++++++++ 8 files changed, 109 insertions(+), 20 deletions(-) diff --git a/bench/data/groundtruth/math_katex_latex_1.jsonl b/bench/data/groundtruth/math_katex_latex_1.jsonl index 7de24ba4..9cde9573 100644 --- a/bench/data/groundtruth/math_katex_latex_1.jsonl +++ b/bench/data/groundtruth/math_katex_latex_1.jsonl @@ -1 +1 @@ -{"content_list": [[{"type": "title", "raw_content": "

      Solve the cubic equation:

      ", "content": {"title_content": "Solve the cubic equation:", "level": "1"}}, {"type": "title", "raw_content": "

      $$x^3+2x^2+8x+1=0 $$

      ", "content": {"title_content": "$$x^3+2x^2+8x+1=0 $$", "level": "2"}}, {"type": "paragraph", "raw_content": "

      Quick Answer

      ", "content": [{"c": "Quick Answer", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      Since the discriminant

      ", "content": [{"c": "Since the discriminant", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\Delta >0", "content": {"math_content": "\\Delta >0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      , the cubic equation has one real root and two conjugate complex roots.

      ", "content": [{"c": ", the cubic equation has one real root and two conjugate complex roots.", "t": "text"}]}, {"type": "equation-interline", "raw_content": " \\Delta=14.472222222222", "content": {"math_content": "\\Delta=14.472222222222", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}", "content": {"math_content": "\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      In decimals,

      ", "content": [{"c": "In decimals,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}", "content": {"math_content": "\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Detailed Steps on Solution

      ", "content": [{"c": "Detailed Steps on Solution", "t": "text"}]}, {"type": "title", "raw_content": "

      1. Convert to depressed cubic equation

      ", "content": {"title_content": "1. Convert to depressed cubic equation", "level": "2"}}, {"type": "paragraph", "raw_content": "

      The idea is to convert general form of cubic equation

      ", "content": [{"c": "The idea is to convert general form of cubic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "ax^3+bx^2+cx+d = 0", "content": {"math_content": "ax^3+bx^2+cx+d = 0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      to the form without quadratic term.

      ", "content": [{"c": "to the form without quadratic term.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t^3+pt+q = 0", "content": {"math_content": "t^3+pt+q = 0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      By substituting

      ", "content": [{"c": "By substituting", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x", "content": {"math_content": "x", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      with

      ", "content": [{"c": "with", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t - \\dfrac{b}{3a}", "content": {"math_content": "t - \\dfrac{b}{3a}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      , the general cubic equation could be transformed to

      ", "content": [{"c": ", the general cubic equation could be transformed to", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t^3+\\dfrac{3ac-b^2}{3a^2}t+\\dfrac{2b^3-9abc+27a^2d}{27a^3} = 0 ", "content": {"math_content": "t^3+\\dfrac{3ac-b^2}{3a^2}t+\\dfrac{2b^3-9abc+27a^2d}{27a^3} = 0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Compare with the depressed cubic equation. Then,

      ", "content": [{"c": "Compare with the depressed cubic equation. Then,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "p = \\dfrac{3ac-b^2}{3a^2}", "content": {"math_content": "p = \\dfrac{3ac-b^2}{3a^2}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "q = \\dfrac{2b^3-9abc+27a^2d}{27a^3} ", "content": {"math_content": "q = \\dfrac{2b^3-9abc+27a^2d}{27a^3}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Substitute the values of coefficients,

      ", "content": [{"c": "Substitute the values of coefficients,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "p, q", "content": {"math_content": "p, q", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      is obtained as

      ", "content": [{"c": "is obtained as", "t": "text"}]}, {"type": "equation-interline", "raw_content": "p = \\dfrac{3\\cdot 1\\cdot 8-2^2}{3\\cdot 1^2}=\\dfrac{20}{3}", "content": {"math_content": "p = \\dfrac{3\\cdot 1\\cdot 8-2^2}{3\\cdot 1^2}=\\dfrac{20}{3}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "q = \\dfrac{2\\cdot 2^3-9\\cdot1\\cdot 2\\cdot 8+27\\cdot 1^2\\cdot1}{27\\cdot 1^3}=-\\dfrac{101}{27}", "content": {"math_content": "q = \\dfrac{2\\cdot 2^3-9\\cdot1\\cdot 2\\cdot 8+27\\cdot 1^2\\cdot1}{27\\cdot 1^3}=-\\dfrac{101}{27}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

      Use the substitution to transform

      ", "content": {"title_content": "Use the substitution to transform", "level": "3"}}, {"type": "paragraph", "raw_content": "

      Let

      ", "content": [{"c": "Let", "t": "text"}]}, {"type": "equation-interline", "raw_content": "p", "content": {"math_content": "p", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      and

      ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "q", "content": {"math_content": "q", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      being the coefficient of the linean and constant terms, the depressed cubic equation is expressed as.

      ", "content": [{"c": "being the coefficient of the linean and constant terms, the depressed cubic equation is expressed as.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t^3 +pt+q=0", "content": {"math_content": "t^3 +pt+q=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Let

      ", "content": [{"c": "Let", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x=t-\\dfrac{2}{3}", "content": {"math_content": "x=t-\\dfrac{2}{3}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      The cubic equation

      ", "content": [{"c": "The cubic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x\u00b3 + 2x\u00b2 + 8x + 1=0", "content": {"math_content": "x\u00b3 + 2x\u00b2 + 8x + 1=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      is transformed to

      ", "content": [{"c": "is transformed to", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}=0", "content": {"math_content": "t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

      2. Cardano's solution

      ", "content": {"title_content": "2. Cardano's solution", "level": "2"}}, {"type": "paragraph", "raw_content": "

      Let

      ", "content": [{"c": "Let", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t=u-v", "content": {"math_content": "t=u-v", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Cube both sides and extract common factor from two middle terms after expanding the bracket.

      ", "content": [{"c": "Cube both sides and extract common factor from two middle terms after expanding the bracket.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\t^3&=(u-v)^3\\\\ & =u^3-3u^2v+3uv^2-v^3\\\\ & =-3uv(u-v)+u^3-v^3\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\t^3&=(u-v)^3\\\\ & =u^3-3u^2v+3uv^2-v^3\\\\ & =-3uv(u-v)+u^3-v^3\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Since

      ", "content": [{"c": "Since", "t": "text"}]}, {"type": "equation-interline", "raw_content": "u-v=t", "content": {"math_content": "u-v=t", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      , substitution gives a linear term for the equation.\n Rearrange terms.

      ", "content": [{"c": ", substitution gives a linear term for the equation.\n Rearrange terms.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x^3+3uvx-u^3+v^3=0", "content": {"math_content": "x^3+3uvx-u^3+v^3=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Compare the cubic equation with the original one (1)

      ", "content": [{"c": "Compare the cubic equation with the original one (1)", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{cases} 3uv=\\dfrac{20}{3}\\quad\\text{or}\\quad v=\\dfrac{20}{9u}\\\\ v^3-u^3=-\\dfrac{101}{27}\\\\ \\end{cases}", "content": {"math_content": "\\begin{cases} 3uv=\\dfrac{20}{3}\\quad\\text{or}\\quad v=\\dfrac{20}{9u}\\\\ v^3-u^3=-\\dfrac{101}{27}\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "v=\\dfrac{20}{9u}", "content": {"math_content": "v=\\dfrac{20}{9u}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      gives relationship between the two variables. Substitute the value of

      ", "content": [{"c": "gives relationship between the two variables. Substitute the value of", "t": "text"}]}, {"type": "equation-interline", "raw_content": "v", "content": {"math_content": "v", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      to the second equation

      ", "content": [{"c": "to the second equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\Big(\\dfrac{20}{9u}\\Big)^3-u^3=-\\dfrac{101}{27}", "content": {"math_content": "\\Big(\\dfrac{20}{9u}\\Big)^3-u^3=-\\dfrac{101}{27}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Simplifying gives,

      ", "content": [{"c": "Simplifying gives,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0", "content": {"math_content": "u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      2

      ", "content": [{"c": "2", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      Let

      ", "content": [{"c": "Let", "t": "text"}]}, {"type": "equation-interline", "raw_content": "m=u^3", "content": {"math_content": "m=u^3", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      , then the equation is transformed to a quadratic equation in terms of $$m$$.\n Once the value of $$m$$ is determined, $$v^3$$ could be determined by

      ", "content": [{"c": ", then the equation is transformed to a quadratic equation in terms of $$m$$.\n Once the value of $$m$$ is determined, $$v^3$$ could be determined by", "t": "text"}]}, {"type": "equation-interline", "raw_content": "v^3=-\\dfrac{101}{27}+u^3", "content": {"math_content": "v^3=-\\dfrac{101}{27}+u^3", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      .

      ", "content": [{"c": ".", "t": "text"}]}, {"type": "equation-interline", "raw_content": "m^2-\\dfrac{101}{27}m-\\dfrac{8000}{729}=0", "content": {"math_content": "m^2-\\dfrac{101}{27}m-\\dfrac{8000}{729}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Sovling the quadratic euqation will give two roots (some may be equal). Here we only cosider\n one case with positive sign before the square root radical since the negative case will produce the same result.

      ", "content": [{"c": "Sovling the quadratic euqation will give two roots (some may be equal). Here we only cosider\n one case with positive sign before the square root radical since the negative case will produce the same result.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\u^3=m&=\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\Big(-\\dfrac{101}{27}^2\\Big)-4\\cdot \\Big(-\\dfrac{8000}{729}\\Big)}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{10201}{729}+\\dfrac{32000}{729}}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{521}{9}}\\\\ & =\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\u^3=m&=\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\Big(-\\dfrac{101}{27}^2\\Big)-4\\cdot \\Big(-\\dfrac{8000}{729}\\Big)}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{10201}{729}+\\dfrac{32000}{729}}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{521}{9}}\\\\ & =\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "v^3", "content": {"math_content": "v^3", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      can be determined by the equation we deduced

      ", "content": [{"c": "can be determined by the equation we deduced", "t": "text"}]}, {"type": "equation-interline", "raw_content": "v^3-u^3=-\\dfrac{101}{27}", "content": {"math_content": "v^3-u^3=-\\dfrac{101}{27}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      . Then,

      ", "content": [{"c": ". Then,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\v^3&=-\\dfrac{101}{27}+u^3\\\\ & =-\\dfrac{101}{27}+\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ & =-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\v^3&=-\\dfrac{101}{27}+u^3\\\\ & =-\\dfrac{101}{27}+\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ & =-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Now we have,

      ", "content": [{"c": "Now we have,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "u^3=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}", "content": {"math_content": "u^3=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      and

      ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "v^3=-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}", "content": {"math_content": "v^3=-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Evaluating the simplest cubic equation

      ", "content": [{"c": "Evaluating the simplest cubic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x^3-A=0", "content": {"math_content": "x^3-A=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      ,\n it has 3 roots, in which the first root is a real number . The second and third are\n expressed in the product of cubic root of unity and the first one.

      ", "content": [{"c": ",\n it has 3 roots, in which the first root is a real number . The second and third are\n expressed in the product of cubic root of unity and the first one.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      If

      ", "content": [{"c": "If", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}", "content": {"math_content": "\u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      , then its reciprocal is equal to its conjugate,

      ", "content": [{"c": ", then its reciprocal is equal to its conjugate,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\dfrac{1}{\u03c9}=\\overline{\u03c9}", "content": {"math_content": "\\dfrac{1}{\u03c9}=\\overline{\u03c9}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      .

      ", "content": [{"c": ".", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{cases} r_1=\\sqrt[3]{A}\\\\ r_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ r_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ \\end{cases}", "content": {"math_content": "\\begin{cases} r_1=\\sqrt[3]{A}\\\\ r_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ r_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Similary, taking cubic root for

      ", "content": [{"c": "Similary, taking cubic root for", "t": "text"}]}, {"type": "equation-interline", "raw_content": "u^3", "content": {"math_content": "u^3", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      and

      ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "v^3", "content": {"math_content": "v^3", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      also gives 3 roots.

      ", "content": [{"c": "also gives 3 roots.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{cases} u_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}", "content": {"math_content": "\\begin{cases} u_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      For

      ", "content": [{"c": "For", "t": "text"}]}, {"type": "equation-interline", "raw_content": "v_2", "content": {"math_content": "v_2", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      and $$v_3$$, the complex numbers before radicals are the conjugates of\n those for $$u_2$$ and $$u_3$$, which can be verified by the reciprocal property\n of the cubic root of unity from the equation $$v=\\dfrac{20}{9u}$$. The radicand can be taken as the\n negative conjugate of that in $$u_1$$, $$u_2$$ and

      ", "content": [{"c": "and $$v_3$$, the complex numbers before radicals are the conjugates of\n those for $$u_2$$ and $$u_3$$, which can be verified by the reciprocal property\n of the cubic root of unity from the equation $$v=\\dfrac{20}{9u}$$. The radicand can be taken as the\n negative conjugate of that in $$u_1$$, $$u_2$$ and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "u_3", "content": {"math_content": "u_3", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      , which is the same in value.

      ", "content": [{"c": ", which is the same in value.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{cases} v_1=\\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_2=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_3=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}", "content": {"math_content": "\\begin{cases} v_1=\\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_2=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_3=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Verification for the redicand in

      ", "content": [{"c": "Verification for the redicand in", "t": "text"}]}, {"type": "equation-interline", "raw_content": "v", "content": {"math_content": "v", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      .

      ", "content": [{"c": ".", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\v_1&=\\dfrac{20}{9u_1}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\Big(\\dfrac{101}{54}\\Big)^2-\\Big(\\dfrac{\\sqrt{521}}{6}\\Big)^2}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{10201}{2916}-\\dfrac{521}{36}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{-1\\cdot 20^3}{9^3}}}\\\\ & =-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\v_1&=\\dfrac{20}{9u_1}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\Big(\\dfrac{101}{54}\\Big)^2-\\Big(\\dfrac{\\sqrt{521}}{6}\\Big)^2}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{10201}{2916}-\\dfrac{521}{36}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{-1\\cdot 20^3}{9^3}}}\\\\ & =-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Since

      ", "content": [{"c": "Since", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x=u-v", "content": {"math_content": "x=u-v", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      , combining the real and imaginary parts gives\n 3 results for

      ", "content": [{"c": ", combining the real and imaginary parts gives\n 3 results for", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t", "content": {"math_content": "t", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\t_1&=u_1-v_1\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\t_1&=u_1-v_1\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\t_2&=u_2-v_2\\\\ & =\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\t_2&=u_2-v_2\\\\ & =\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\t_3&=u_3-v_3\\\\ & =\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\t_3&=u_3-v_3\\\\ & =\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

      3. Vieta's Substitution

      ", "content": {"title_content": "3. Vieta's Substitution", "level": "2"}}, {"type": "paragraph", "raw_content": "

      In Cardano' solution,

      ", "content": [{"c": "In Cardano' solution,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t", "content": {"math_content": "t", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      is defined as the difference of $$u$$ and\n $$v$$. If we substitute the value of $$v$$ (4) into (2), we get the\n equation. $$t=u-\\dfrac{20}{9u}$$. And then substitute the equation to the cubic equation\n

      ", "content": [{"c": "is defined as the difference of $$u$$ and\n $$v$$. If we substitute the value of $$v$$ (4) into (2), we get the\n equation. $$t=u-\\dfrac{20}{9u}$$. And then substitute the equation to the cubic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0", "content": {"math_content": "t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      . This method is called Vieta's Substitution\n for solving a cubic equation, which simplied the Cardano' solution. The substitution\n expression can be obtained by the following formula directly.

      ", "content": [{"c": ". This method is called Vieta's Substitution\n for solving a cubic equation, which simplied the Cardano' solution. The substitution\n expression can be obtained by the following formula directly.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t=u-\\dfrac{p}{3u}", "content": {"math_content": "t=u-\\dfrac{p}{3u}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Substitute the expression

      ", "content": [{"c": "Substitute the expression", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t=u-\\dfrac{20}{9u}", "content": {"math_content": "t=u-\\dfrac{20}{9u}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      to the cubic equation

      ", "content": [{"c": "to the cubic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\Big(u-\\dfrac{20}{9u}\\Big)^3+\\dfrac{20}{3}\\Big(u-\\dfrac{20}{9u}\\Big)-\\dfrac{101}{27}=0", "content": {"math_content": "\\Big(u-\\dfrac{20}{9u}\\Big)^3+\\dfrac{20}{3}\\Big(u-\\dfrac{20}{9u}\\Big)-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Expand brackets and cancel the like terms

      ", "content": [{"c": "Expand brackets and cancel the like terms", "t": "text"}]}, {"type": "equation-interline", "raw_content": "u^3-\\cancel{\\dfrac{20}{3}u^2\\dfrac{1}{u}}+\\cancel{\\dfrac{400}{27}u\\dfrac{1}{u^2}}-\\dfrac{8000}{729}\\dfrac{1}{u^3}+\\cancel{\\dfrac{20}{3}u}-\\cancel{\\dfrac{400}{27}\\dfrac{1}{u}}-\\dfrac{101}{27}=0", "content": {"math_content": "u^3-\\cancel{\\dfrac{20}{3}u^2\\dfrac{1}{u}}+\\cancel{\\dfrac{400}{27}u\\dfrac{1}{u^2}}-\\dfrac{8000}{729}\\dfrac{1}{u^3}+\\cancel{\\dfrac{20}{3}u}-\\cancel{\\dfrac{400}{27}\\dfrac{1}{u}}-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Then we get the same equation as (2)

      ", "content": [{"c": "Then we get the same equation as (2)", "t": "text"}]}, {"type": "equation-interline", "raw_content": "u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0", "content": {"math_content": "u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      The rest of the steps will be the same as those of Cardano's solution

      ", "content": [{"c": "The rest of the steps will be the same as those of Cardano's solution", "t": "text"}]}, {"type": "title", "raw_content": "

      4. Euler's Solution

      ", "content": {"title_content": "4. Euler's Solution", "level": "2"}}, {"type": "title", "raw_content": "

      $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

      ", "content": {"title_content": "$$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$", "level": "2"}}, {"type": "paragraph", "raw_content": "

      Move the linear term and constant of (1) to its right hand side.\n We get the following form of the equation.

      ", "content": [{"c": "Move the linear term and constant of (1) to its right hand side.\n We get the following form of the equation.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t^3=-\\dfrac{20}{3}t+\\dfrac{101}{27} ", "content": {"math_content": "t^3=-\\dfrac{20}{3}t+\\dfrac{101}{27}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      3

      ", "content": [{"c": "3", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      Let the root of the cubic equation be the sum of two cubic roots

      ", "content": [{"c": "Let the root of the cubic equation be the sum of two cubic roots", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t=\\sqrt[3]{r_1}+\\sqrt[3]{r_2} ", "content": {"math_content": "t=\\sqrt[3]{r_1}+\\sqrt[3]{r_2}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      4

      ", "content": [{"c": "4", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      in which

      ", "content": [{"c": "in which", "t": "text"}]}, {"type": "equation-interline", "raw_content": "r_1", "content": {"math_content": "r_1", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      and

      ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "r_2", "content": {"math_content": "r_2", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      are two roots of a quadratic equation

      ", "content": [{"c": "are two roots of a quadratic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "z^2-\\alpha z+ \u03b2=0 ", "content": {"math_content": "z^2-\\alpha z+ \u03b2=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      5

      ", "content": [{"c": "5", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      Using Vieta's Formula, the following equations are established.

      ", "content": [{"c": "Using Vieta's Formula, the following equations are established.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "r_1+r_2 = \\alpha \\quad \\text{and} \\quad r_1r_2 = \u03b2 ", "content": {"math_content": "r_1+r_2 = \\alpha \\quad \\text{and} \\quad r_1r_2 = \u03b2", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      To determine

      ", "content": [{"c": "To determine", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\alpha", "content": {"math_content": "\\alpha", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      ,

      ", "content": [{"c": ",", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\u03b2", "content": {"math_content": "\u03b2", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      , cube both sides of the equation (4)

      ", "content": [{"c": ", cube both sides of the equation (4)", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t^3=3\\sqrt[3]{r_1r_2}(\\sqrt[3]{r_1}+\\sqrt[3]{r_2})+r_1+r_2 ", "content": {"math_content": "t^3=3\\sqrt[3]{r_1r_2}(\\sqrt[3]{r_1}+\\sqrt[3]{r_2})+r_1+r_2", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Substituting, the equation is simplified to

      ", "content": [{"c": "Substituting, the equation is simplified to", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t^3=3\\sqrt[3]{\u03b2}t+\\alpha ", "content": {"math_content": "t^3=3\\sqrt[3]{\u03b2}t+\\alpha", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Compare the cubic equation with (3), the following equations are established

      ", "content": [{"c": "Compare the cubic equation with (3), the following equations are established", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{cases} 3\\sqrt[3]{\u03b2}=-\\dfrac{20}{3}\\\\ \\alpha=\\dfrac{101}{27}\\\\ \\end{cases}", "content": {"math_content": "\\begin{cases} 3\\sqrt[3]{\u03b2}=-\\dfrac{20}{3}\\\\ \\alpha=\\dfrac{101}{27}\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Solving for

      ", "content": [{"c": "Solving for", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\u03b2", "content": {"math_content": "\u03b2", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      gives

      ", "content": [{"c": "gives", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\u03b2=-\\dfrac{8000}{729} ", "content": {"math_content": "\u03b2=-\\dfrac{8000}{729}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      So the quadratic equation (5) is determined as

      ", "content": [{"c": "So the quadratic equation (5) is determined as", "t": "text"}]}, {"type": "equation-interline", "raw_content": "z^2-\\dfrac{101}{27}z-\\dfrac{8000}{729}=0", "content": {"math_content": "z^2-\\dfrac{101}{27}z-\\dfrac{8000}{729}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      6

      ", "content": [{"c": "6", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      Solving the quadratic equation yields

      ", "content": [{"c": "Solving the quadratic equation yields", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{cases} r_1=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\approx5.6746077738748\\\\ r_2=\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}\\approx-1.9338670331341\\\\ \\end{cases}", "content": {"math_content": "\\begin{cases} r_1=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\approx5.6746077738748\\\\ r_2=\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}\\approx-1.9338670331341\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Therefore, one of the roots of the cubic equation could be obtained from (4).

      ", "content": [{"c": "Therefore, one of the roots of the cubic equation could be obtained from (4).", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} ", "content": {"math_content": "t_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      in decimals,

      ", "content": [{"c": "in decimals,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t_1=0.53778143658824 ", "content": {"math_content": "t_1=0.53778143658824", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      However, since the cube root of a quantity has triple values,

      ", "content": [{"c": "However, since the cube root of a quantity has triple values,", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      The other two roots could be determined as,

      ", "content": [{"c": "The other two roots could be determined as,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t_2=\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} ", "content": {"math_content": "t_2=\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "t_3=\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} ", "content": {"math_content": "t_3=\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Combining the real and imaginary parts\n results in the same result as that obtained by Cardano's solution.

      ", "content": [{"c": "Combining the real and imaginary parts\n results in the same result as that obtained by Cardano's solution.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      For the equation

      ", "content": [{"c": "For the equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}", "content": {"math_content": "t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      , we have $$p=\\dfrac{20}{3}$$ and

      ", "content": [{"c": ", we have $$p=\\dfrac{20}{3}$$ and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "q = -\\dfrac{101}{27}", "content": {"math_content": "q = -\\dfrac{101}{27}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

      Calculate the discriminant

      ", "content": {"title_content": "Calculate the discriminant", "level": "3"}}, {"type": "paragraph", "raw_content": "

      The nature of the roots are determined by the sign of the discriminant.

      ", "content": [{"c": "The nature of the roots are determined by the sign of the discriminant.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\\\Delta&=\\dfrac{q^2}{4}+\\dfrac{p^3}{27}\\\\ & =\\dfrac{\\Big(-\\dfrac{101}{27}\\Big)^2}{4}+\\dfrac{\\Big(\\dfrac{20}{3}\\Big)^3}{27}\\\\ & =\\dfrac{10201}{2916}+\\dfrac{8000}{729}\\\\ & =\\dfrac{10201\\cdot 1+8000\\cdot 4}{2916}\\\\ & =14.472222222222\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\\\Delta&=\\dfrac{q^2}{4}+\\dfrac{p^3}{27}\\\\ & =\\dfrac{\\Big(-\\dfrac{101}{27}\\Big)^2}{4}+\\dfrac{\\Big(\\dfrac{20}{3}\\Big)^3}{27}\\\\ & =\\dfrac{10201}{2916}+\\dfrac{8000}{729}\\\\ & =\\dfrac{10201\\cdot 1+8000\\cdot 4}{2916}\\\\ & =14.472222222222\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

      4.1 Use the root formula directly

      ", "content": {"title_content": "4.1 Use the root formula directly", "level": "3"}}, {"type": "paragraph", "raw_content": "

      If the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.

      ", "content": [{"c": "If the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "t_{1,2,3} =\\begin{cases} \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } +\\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}& \\\\ \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \\overline{\u03c9} \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}&\\ \\\\ \\overline{\u03c9}\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }} \\end{cases}", "content": {"math_content": "t_{1,2,3} =\\begin{cases} \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } +\\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}& \\\\ \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \\overline{\u03c9} \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}&\\ \\\\ \\overline{\u03c9}\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }} \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      in which,

      ", "content": [{"c": "in which,", "t": "text"}]}, {"type": "equation-interline", "raw_content": " \u03c9 = \\dfrac{-1+i\\sqrt{3}}{2} ", "content": {"math_content": "\u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      and

      ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": " \\overline{\u03c9} =\\dfrac{-1-i\\sqrt{3}}{2}", "content": {"math_content": "\\overline{\u03c9} =\\dfrac{-1-i\\sqrt{3}}{2}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Substitute the values of

      ", "content": [{"c": "Substitute the values of", "t": "text"}]}, {"type": "equation-interline", "raw_content": "p, q", "content": {"math_content": "p, q", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      and

      ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\Delta", "content": {"math_content": "\\Delta", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      which we have calculated. Then,

      ", "content": [{"c": "which we have calculated. Then,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\t_1&=\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{42201}{2916}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{42201}{2916}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\t_1&=\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{42201}{2916}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{42201}{2916}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      If we denote

      ", "content": [{"c": "If we denote", "t": "text"}]}, {"type": "equation-interline", "raw_content": "R = -\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }", "content": {"math_content": "R = -\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "\\overline{R} = -\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }", "content": {"math_content": "\\overline{R} = -\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      then,

      ", "content": [{"c": "then,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\sqrt[3]{R} = \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}", "content": {"math_content": "\\sqrt[3]{R} = \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      ,

      ", "content": [{"c": ",", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\sqrt[3]{\\overline{R}} =\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}", "content": {"math_content": "\\sqrt[3]{\\overline{R}} =\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\t_2&= \u03c9\\cdotp \\sqrt[3]{R}+ \\overline{\u03c9} \\sqrt[3]{\\overline{R} }\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}( \\sqrt[3]{R} - \\sqrt[3]{\\overline{R} }) }{2} i\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\t_2&= \u03c9\\cdotp \\sqrt[3]{R}+ \\overline{\u03c9} \\sqrt[3]{\\overline{R} }\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}( \\sqrt[3]{R} - \\sqrt[3]{\\overline{R} }) }{2} i\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "\\begin{aligned} \\\\t_3&= \\overline{\u03c9}\\cdotp \\sqrt[3]{R}+ \u03c9\\cdotp \\sqrt[3]{\\overline{R}}\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}(- \\sqrt[3]{R} + \\sqrt[3]{\\overline{R} }) }{2}i \\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "content": {"math_content": "\\begin{aligned} \\\\t_3&= \\overline{\u03c9}\\cdotp \\sqrt[3]{R}+ \u03c9\\cdotp \\sqrt[3]{\\overline{R}}\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}(- \\sqrt[3]{R} + \\sqrt[3]{\\overline{R} }) }{2}i \\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

      Roots of the general cubic equation

      ", "content": {"title_content": "Roots of the general cubic equation", "level": "2"}}, {"type": "paragraph", "raw_content": "

      Since

      ", "content": [{"c": "Since", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x = t - \\dfrac{b}{3a}", "content": {"math_content": "x = t - \\dfrac{b}{3a}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      , substituting the values of $$t$$, $$a$$ and

      ", "content": [{"c": ", substituting the values of $$t$$, $$a$$ and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "b", "content": {"math_content": "b", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      gives

      ", "content": [{"c": "gives", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x_1 = t_1-\\dfrac{2}{3}", "content": {"math_content": "x_1 = t_1-\\dfrac{2}{3}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "x_2 = t_2-\\dfrac{2}{3}", "content": {"math_content": "x_2 = t_2-\\dfrac{2}{3}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "x_3 = t_3-\\dfrac{2}{3}", "content": {"math_content": "x_3 = t_3-\\dfrac{2}{3}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

      5. Summary

      ", "content": {"title_content": "5. Summary", "level": "2"}}, {"type": "paragraph", "raw_content": "

      In summary, we have tried the method of cubic root formula to explore the solutions of the equation. The cubic equation

      ", "content": [{"c": "In summary, we have tried the method of cubic root formula to explore the solutions of the equation. The cubic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "x\u00b3 + 2x\u00b2 + 8x + 1=0", "content": {"math_content": "x\u00b3 + 2x\u00b2 + 8x + 1=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      is found to have one real root and two complex roots. Exact values and approximations are given below.

      ", "content": [{"c": "is found to have one real root and two complex roots. Exact values and approximations are given below.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}", "content": {"math_content": "\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      in decimal notation,

      ", "content": [{"c": "in decimal notation,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}", "content": {"math_content": "\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

      6. Graph for the function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$

      ", "content": {"title_content": "6. Graph for the function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$", "level": "2"}}, {"type": "paragraph", "raw_content": "

      Since the discriminat is greater than zero, the curve of the cubic function

      ", "content": [{"c": "Since the discriminat is greater than zero, the curve of the cubic function", "t": "text"}]}, {"type": "equation-interline", "raw_content": "f(x) = x\u00b3 + 2x\u00b2 + 8x + 1", "content": {"math_content": "f(x) = x\u00b3 + 2x\u00b2 + 8x + 1", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      has one intersection point with the x-axis.

      ", "content": [{"c": "has one intersection point with the x-axis.", "t": "text"}]}, {"type": "title", "raw_content": "

      More cubic equations

      ", "content": {"title_content": "More cubic equations", "level": "2"}}]], "main_html": "

      Solve the cubic equation:

      $$x^3+2x^2+8x+1=0 $$

      Quick Answer

      Since the discriminant

      \\Delta >0

      , the cubic equation has one real root and two conjugate complex roots.

      \\Delta=14.472222222222\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}

      In decimals,

      \\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}

      Detailed Steps on Solution

      1. Convert to depressed cubic equation

      The idea is to convert general form of cubic equation

      ax^3+bx^2+cx+d = 0

      to the form without quadratic term.

      t^3+pt+q = 0

      By substituting

      x

      with

      t - \\dfrac{b}{3a}

      , the general cubic equation could be transformed to

      t^3+\\dfrac{3ac-b^2}{3a^2}t+\\dfrac{2b^3-9abc+27a^2d}{27a^3} = 0

      Compare with the depressed cubic equation. Then,

      p = \\dfrac{3ac-b^2}{3a^2}q = \\dfrac{2b^3-9abc+27a^2d}{27a^3}

      Substitute the values of coefficients,

      p, q

      is obtained as

      p = \\dfrac{3\\cdot 1\\cdot 8-2^2}{3\\cdot 1^2}=\\dfrac{20}{3}q = \\dfrac{2\\cdot 2^3-9\\cdot1\\cdot 2\\cdot 8+27\\cdot 1^2\\cdot1}{27\\cdot 1^3}=-\\dfrac{101}{27}

      Use the substitution to transform

      Let

      p

      and

      q

      being the coefficient of the linean and constant terms, the depressed cubic equation is expressed as.

      t^3 +pt+q=0

      Let

      x=t-\\dfrac{2}{3}

      The cubic equation

      x\u00b3 + 2x\u00b2 + 8x + 1=0

      is transformed to

      t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}=0

      2. Cardano's solution

      Let

      t=u-v

      Cube both sides and extract common factor from two middle terms after expanding the bracket.

      \\begin{aligned} \\\\t^3&=(u-v)^3\\\\ & =u^3-3u^2v+3uv^2-v^3\\\\ & =-3uv(u-v)+u^3-v^3\\\\ \\end{aligned}

      Since

      u-v=t

      , substitution gives a linear term for the equation.\n Rearrange terms.

      x^3+3uvx-u^3+v^3=0

      Compare the cubic equation with the original one (1)

      \\begin{cases} 3uv=\\dfrac{20}{3}\\quad\\text{or}\\quad v=\\dfrac{20}{9u}\\\\ v^3-u^3=-\\dfrac{101}{27}\\\\ \\end{cases}v=\\dfrac{20}{9u}

      gives relationship between the two variables. Substitute the value of

      v

      to the second equation

      \\Big(\\dfrac{20}{9u}\\Big)^3-u^3=-\\dfrac{101}{27}

      Simplifying gives,

      u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0

      2

      Let

      m=u^3

      , then the equation is transformed to a quadratic equation in terms of $$m$$.\n Once the value of $$m$$ is determined, $$v^3$$ could be determined by

      v^3=-\\dfrac{101}{27}+u^3

      .

      m^2-\\dfrac{101}{27}m-\\dfrac{8000}{729}=0

      Sovling the quadratic euqation will give two roots (some may be equal). Here we only cosider\n one case with positive sign before the square root radical since the negative case will produce the same result.

      \\begin{aligned} \\\\u^3=m&=\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\Big(-\\dfrac{101}{27}^2\\Big)-4\\cdot \\Big(-\\dfrac{8000}{729}\\Big)}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{10201}{729}+\\dfrac{32000}{729}}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{521}{9}}\\\\ & =\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}v^3

      can be determined by the equation we deduced

      v^3-u^3=-\\dfrac{101}{27}

      . Then,

      \\begin{aligned} \\\\v^3&=-\\dfrac{101}{27}+u^3\\\\ & =-\\dfrac{101}{27}+\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ & =-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}

      Now we have,

      u^3=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}

      and

      v^3=-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}

      Evaluating the simplest cubic equation

      x^3-A=0

      ,\n it has 3 roots, in which the first root is a real number . The second and third are\n expressed in the product of cubic root of unity and the first one.

      If

      \u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}

      , then its reciprocal is equal to its conjugate,

      \\dfrac{1}{\u03c9}=\\overline{\u03c9}

      .

      \\begin{cases} r_1=\\sqrt[3]{A}\\\\ r_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ r_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ \\end{cases}

      Similary, taking cubic root for

      u^3

      and

      v^3

      also gives 3 roots.

      \\begin{cases} u_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}

      For

      v_2

      and $$v_3$$, the complex numbers before radicals are the conjugates of\n those for $$u_2$$ and $$u_3$$, which can be verified by the reciprocal property\n of the cubic root of unity from the equation $$v=\\dfrac{20}{9u}$$. The radicand can be taken as the\n negative conjugate of that in $$u_1$$, $$u_2$$ and

      u_3

      , which is the same in value.

      \\begin{cases} v_1=\\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_2=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_3=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}

      Verification for the redicand in

      v

      .

      \\begin{aligned} \\\\v_1&=\\dfrac{20}{9u_1}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\Big(\\dfrac{101}{54}\\Big)^2-\\Big(\\dfrac{\\sqrt{521}}{6}\\Big)^2}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{10201}{2916}-\\dfrac{521}{36}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{-1\\cdot 20^3}{9^3}}}\\\\ & =-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}

      Since

      x=u-v

      , combining the real and imaginary parts gives\n 3 results for

      t\\begin{aligned} \\\\t_1&=u_1-v_1\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}\\begin{aligned} \\\\t_2&=u_2-v_2\\\\ & =\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}\\begin{aligned} \\\\t_3&=u_3-v_3\\\\ & =\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}

      3. Vieta's Substitution

      In Cardano' solution,

      t

      is defined as the difference of $$u$$ and\n $$v$$. If we substitute the value of $$v$$ (4) into (2), we get the\n equation. $$t=u-\\dfrac{20}{9u}$$. And then substitute the equation to the cubic equation\n

      t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0

      . This method is called Vieta's Substitution\n for solving a cubic equation, which simplied the Cardano' solution. The substitution\n expression can be obtained by the following formula directly.

      t=u-\\dfrac{p}{3u}

      Substitute the expression

      t=u-\\dfrac{20}{9u}

      to the cubic equation

      \\Big(u-\\dfrac{20}{9u}\\Big)^3+\\dfrac{20}{3}\\Big(u-\\dfrac{20}{9u}\\Big)-\\dfrac{101}{27}=0

      Expand brackets and cancel the like terms

      u^3-\\cancel{\\dfrac{20}{3}u^2\\dfrac{1}{u}}+\\cancel{\\dfrac{400}{27}u\\dfrac{1}{u^2}}-\\dfrac{8000}{729}\\dfrac{1}{u^3}+\\cancel{\\dfrac{20}{3}u}-\\cancel{\\dfrac{400}{27}\\dfrac{1}{u}}-\\dfrac{101}{27}=0

      Then we get the same equation as (2)

      u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0

      The rest of the steps will be the same as those of Cardano's solution

      4. Euler's Solution

      $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

      Move the linear term and constant of (1) to its right hand side.\n We get the following form of the equation.

      t^3=-\\dfrac{20}{3}t+\\dfrac{101}{27}

      3

      Let the root of the cubic equation be the sum of two cubic roots

      t=\\sqrt[3]{r_1}+\\sqrt[3]{r_2}

      4

      in which

      r_1

      and

      r_2

      are two roots of a quadratic equation

      z^2-\\alpha z+ \u03b2=0

      5

      Using Vieta's Formula, the following equations are established.

      r_1+r_2 = \\alpha \\quad \\text{and} \\quad r_1r_2 = \u03b2

      To determine

      \\alpha

      ,

      \u03b2

      , cube both sides of the equation (4)

      t^3=3\\sqrt[3]{r_1r_2}(\\sqrt[3]{r_1}+\\sqrt[3]{r_2})+r_1+r_2

      Substituting, the equation is simplified to

      t^3=3\\sqrt[3]{\u03b2}t+\\alpha

      Compare the cubic equation with (3), the following equations are established

      \\begin{cases} 3\\sqrt[3]{\u03b2}=-\\dfrac{20}{3}\\\\ \\alpha=\\dfrac{101}{27}\\\\ \\end{cases}

      Solving for

      \u03b2

      gives

      \u03b2=-\\dfrac{8000}{729}

      So the quadratic equation (5) is determined as

      z^2-\\dfrac{101}{27}z-\\dfrac{8000}{729}=0

      6

      Solving the quadratic equation yields

      \\begin{cases} r_1=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\approx5.6746077738748\\\\ r_2=\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}\\approx-1.9338670331341\\\\ \\end{cases}

      Therefore, one of the roots of the cubic equation could be obtained from (4).

      t_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}

      in decimals,

      t_1=0.53778143658824

      However, since the cube root of a quantity has triple values,

      The other two roots could be determined as,

      t_2=\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} t_3=\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}

      Combining the real and imaginary parts\n results in the same result as that obtained by Cardano's solution.

      For the equation

      t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}

      , we have $$p=\\dfrac{20}{3}$$ and

      q = -\\dfrac{101}{27}

      Calculate the discriminant

      The nature of the roots are determined by the sign of the discriminant.

      \\begin{aligned} \\\\\\Delta&=\\dfrac{q^2}{4}+\\dfrac{p^3}{27}\\\\ & =\\dfrac{\\Big(-\\dfrac{101}{27}\\Big)^2}{4}+\\dfrac{\\Big(\\dfrac{20}{3}\\Big)^3}{27}\\\\ & =\\dfrac{10201}{2916}+\\dfrac{8000}{729}\\\\ & =\\dfrac{10201\\cdot 1+8000\\cdot 4}{2916}\\\\ & =14.472222222222\\\\ \\end{aligned}

      4.1 Use the root formula directly

      If the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.

      t_{1,2,3} =\\begin{cases} \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } +\\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}& \\\\ \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \\overline{\u03c9} \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}&\\ \\\\ \\overline{\u03c9}\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }} \\end{cases}

      in which,

      \u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}

      and

      \\overline{\u03c9} =\\dfrac{-1-i\\sqrt{3}}{2}

      Substitute the values of

      p, q

      and

      \\Delta

      which we have calculated. Then,

      \\begin{aligned} \\\\t_1&=\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{42201}{2916}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{42201}{2916}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}

      If we denote

      R = -\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }\\overline{R} = -\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }

      then,

      \\sqrt[3]{R} = \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}

      ,

      \\sqrt[3]{\\overline{R}} =\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\begin{aligned} \\\\t_2&= \u03c9\\cdotp \\sqrt[3]{R}+ \\overline{\u03c9} \\sqrt[3]{\\overline{R} }\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}( \\sqrt[3]{R} - \\sqrt[3]{\\overline{R} }) }{2} i\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}\\begin{aligned} \\\\t_3&= \\overline{\u03c9}\\cdotp \\sqrt[3]{R}+ \u03c9\\cdotp \\sqrt[3]{\\overline{R}}\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}(- \\sqrt[3]{R} + \\sqrt[3]{\\overline{R} }) }{2}i \\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}

      Roots of the general cubic equation

      Since

      x = t - \\dfrac{b}{3a}

      , substituting the values of $$t$$, $$a$$ and

      b

      gives

      x_1 = t_1-\\dfrac{2}{3}x_2 = t_2-\\dfrac{2}{3}x_3 = t_3-\\dfrac{2}{3}

      5. Summary

      In summary, we have tried the method of cubic root formula to explore the solutions of the equation. The cubic equation

      x\u00b3 + 2x\u00b2 + 8x + 1=0

      is found to have one real root and two complex roots. Exact values and approximations are given below.

      \\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}

      in decimal notation,

      \\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}

      6. Graph for the function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$

      Since the discriminat is greater than zero, the curve of the cubic function

      f(x) = x\u00b3 + 2x\u00b2 + 8x + 1

      has one intersection point with the x-axis.

      More cubic equations

      ", "statics": {"title": 14, "paragraph": 103, "paragraph.text": 103, "equation-interline": 105}, "url": "https://uniteasy.com/solver/cubicequation/x%5E3%2B2x%5E2%2B8x%2B1%3D0/", "content": "# Solve the cubic equation:\n\n## $$x^3+2x^2+8x+1=0 $$\n\nQuick Answer\n\nSince the discriminant\n\n$$\n\\Delta >0\n$$\n\n, the cubic equation has one real root and two conjugate complex roots.\n\n$$\n\\Delta=14.472222222222\n$$\n\n$$\n\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}\n$$\n\nIn decimals,\n\n$$\n\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}\n$$\n\nDetailed Steps on Solution\n\n## 1. Convert to depressed cubic equation\n\nThe idea is to convert general form of cubic equation\n\n$$\nax^3+bx^2+cx+d = 0\n$$\n\nto the form without quadratic term.\n\n$$\nt^3+pt+q = 0\n$$\n\nBy substituting\n\n$$\nx\n$$\n\nwith\n\n$$\nt - \\dfrac{b}{3a}\n$$\n\n, the general cubic equation could be transformed to\n\n$$\nt^3+\\dfrac{3ac-b^2}{3a^2}t+\\dfrac{2b^3-9abc+27a^2d}{27a^3} = 0\n$$\n\nCompare with the depressed cubic equation. Then,\n\n$$\np = \\dfrac{3ac-b^2}{3a^2}\n$$\n\n$$\nq = \\dfrac{2b^3-9abc+27a^2d}{27a^3}\n$$\n\nSubstitute the values of coefficients,\n\n$$\np, q\n$$\n\nis obtained as\n\n$$\np = \\dfrac{3\\cdot 1\\cdot 8-2^2}{3\\cdot 1^2}=\\dfrac{20}{3}\n$$\n\n$$\nq = \\dfrac{2\\cdot 2^3-9\\cdot1\\cdot 2\\cdot 8+27\\cdot 1^2\\cdot1}{27\\cdot 1^3}=-\\dfrac{101}{27}\n$$\n\n### Use the substitution to transform\n\nLet\n\n$$\np\n$$\n\nand\n\n$$\nq\n$$\n\nbeing the coefficient of the linean and constant terms, the depressed cubic equation is expressed as.\n\n$$\nt^3 +pt+q=0\n$$\n\nLet\n\n$$\nx=t-\\dfrac{2}{3}\n$$\n\nThe cubic equation\n\n$$\nx\u00b3 + 2x\u00b2 + 8x + 1=0\n$$\n\nis transformed to\n\n$$\nt^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}=0\n$$\n\n## 2. Cardano's solution\n\nLet\n\n$$\nt=u-v\n$$\n\nCube both sides and extract common factor from two middle terms after expanding the bracket.\n\n$$\n\\begin{aligned} \\\\t^3&=(u-v)^3\\\\ & =u^3-3u^2v+3uv^2-v^3\\\\ & =-3uv(u-v)+u^3-v^3\\\\ \\end{aligned}\n$$\n\nSince\n\n$$\nu-v=t\n$$\n\n, substitution gives a linear term for the equation.\n Rearrange terms.\n\n$$\nx^3+3uvx-u^3+v^3=0\n$$\n\nCompare the cubic equation with the original one (1)\n\n$$\n\\begin{cases} 3uv=\\dfrac{20}{3}\\quad\\text{or}\\quad v=\\dfrac{20}{9u}\\\\ v^3-u^3=-\\dfrac{101}{27}\\\\ \\end{cases}\n$$\n\n$$\nv=\\dfrac{20}{9u}\n$$\n\ngives relationship between the two variables. Substitute the value of\n\n$$\nv\n$$\n\nto the second equation\n\n$$\n\\Big(\\dfrac{20}{9u}\\Big)^3-u^3=-\\dfrac{101}{27}\n$$\n\nSimplifying gives,\n\n$$\nu^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0\n$$\n\n2\n\nLet\n\n$$\nm=u^3\n$$\n\n, then the equation is transformed to a quadratic equation in terms of $$m$$.\n Once the value of $$m$$ is determined, $$v^3$$ could be determined by\n\n$$\nv^3=-\\dfrac{101}{27}+u^3\n$$\n\n.\n\n$$\nm^2-\\dfrac{101}{27}m-\\dfrac{8000}{729}=0\n$$\n\nSovling the quadratic euqation will give two roots (some may be equal). Here we only cosider\n one case with positive sign before the square root radical since the negative case will produce the same result.\n\n$$\n\\begin{aligned} \\\\u^3=m&=\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\Big(-\\dfrac{101}{27}^2\\Big)-4\\cdot \\Big(-\\dfrac{8000}{729}\\Big)}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{10201}{729}+\\dfrac{32000}{729}}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{521}{9}}\\\\ & =\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}\n$$\n\n$$\nv^3\n$$\n\ncan be determined by the equation we deduced\n\n$$\nv^3-u^3=-\\dfrac{101}{27}\n$$\n\n. Then,\n\n$$\n\\begin{aligned} \\\\v^3&=-\\dfrac{101}{27}+u^3\\\\ & =-\\dfrac{101}{27}+\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ & =-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}\n$$\n\nNow we have,\n\n$$\nu^3=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\n$$\n\nand\n\n$$\nv^3=-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\n$$\n\nEvaluating the simplest cubic equation\n\n$$\nx^3-A=0\n$$\n\n,\n it has 3 roots, in which the first root is a real number . The second and third are\n expressed in the product of cubic root of unity and the first one.\n\nIf\n\n$$\n\u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}\n$$\n\n, then its reciprocal is equal to its conjugate,\n\n$$\n\\dfrac{1}{\u03c9}=\\overline{\u03c9}\n$$\n\n.\n\n$$\n\\begin{cases} r_1=\\sqrt[3]{A}\\\\ r_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ r_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ \\end{cases}\n$$\n\nSimilary, taking cubic root for\n\n$$\nu^3\n$$\n\nand\n\n$$\nv^3\n$$\n\nalso gives 3 roots.\n\n$$\n\\begin{cases} u_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}\n$$\n\nFor\n\n$$\nv_2\n$$\n\nand $$v_3$$, the complex numbers before radicals are the conjugates of\n those for $$u_2$$ and $$u_3$$, which can be verified by the reciprocal property\n of the cubic root of unity from the equation $$v=\\dfrac{20}{9u}$$. The radicand can be taken as the\n negative conjugate of that in $$u_1$$, $$u_2$$ and\n\n$$\nu_3\n$$\n\n, which is the same in value.\n\n$$\n\\begin{cases} v_1=\\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_2=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_3=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}\n$$\n\nVerification for the redicand in\n\n$$\nv\n$$\n\n.\n\n$$\n\\begin{aligned} \\\\v_1&=\\dfrac{20}{9u_1}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\Big(\\dfrac{101}{54}\\Big)^2-\\Big(\\dfrac{\\sqrt{521}}{6}\\Big)^2}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{10201}{2916}-\\dfrac{521}{36}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{-1\\cdot 20^3}{9^3}}}\\\\ & =-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}\n$$\n\nSince\n\n$$\nx=u-v\n$$\n\n, combining the real and imaginary parts gives\n 3 results for\n\n$$\nt\n$$\n\n$$\n\\begin{aligned} \\\\t_1&=u_1-v_1\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}\n$$\n\n$$\n\\begin{aligned} \\\\t_2&=u_2-v_2\\\\ & =\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}\n$$\n\n$$\n\\begin{aligned} \\\\t_3&=u_3-v_3\\\\ & =\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}\n$$\n\n## 3. Vieta's Substitution\n\nIn Cardano' solution,\n\n$$\nt\n$$\n\nis defined as the difference of $$u$$ and\n $$v$$. If we substitute the value of $$v$$ (4) into (2), we get the\n equation. $$t=u-\\dfrac{20}{9u}$$. And then substitute the equation to the cubic equation\n\n$$\nt^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0\n$$\n\n. This method is called Vieta's Substitution\n for solving a cubic equation, which simplied the Cardano' solution. The substitution\n expression can be obtained by the following formula directly.\n\n$$\nt=u-\\dfrac{p}{3u}\n$$\n\nSubstitute the expression\n\n$$\nt=u-\\dfrac{20}{9u}\n$$\n\nto the cubic equation\n\n$$\n\\Big(u-\\dfrac{20}{9u}\\Big)^3+\\dfrac{20}{3}\\Big(u-\\dfrac{20}{9u}\\Big)-\\dfrac{101}{27}=0\n$$\n\nExpand brackets and cancel the like terms\n\n$$\nu^3-\\cancel{\\dfrac{20}{3}u^2\\dfrac{1}{u}}+\\cancel{\\dfrac{400}{27}u\\dfrac{1}{u^2}}-\\dfrac{8000}{729}\\dfrac{1}{u^3}+\\cancel{\\dfrac{20}{3}u}-\\cancel{\\dfrac{400}{27}\\dfrac{1}{u}}-\\dfrac{101}{27}=0\n$$\n\nThen we get the same equation as (2)\n\n$$\nu^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0\n$$\n\nThe rest of the steps will be the same as those of Cardano's solution\n\n## 4. Euler's Solution\n\n## $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$\n\nMove the linear term and constant of (1) to its right hand side.\n We get the following form of the equation.\n\n$$\nt^3=-\\dfrac{20}{3}t+\\dfrac{101}{27}\n$$\n\n3\n\nLet the root of the cubic equation be the sum of two cubic roots\n\n$$\nt=\\sqrt[3]{r_1}+\\sqrt[3]{r_2}\n$$\n\n4\n\nin which\n\n$$\nr_1\n$$\n\nand\n\n$$\nr_2\n$$\n\nare two roots of a quadratic equation\n\n$$\nz^2-\\alpha z+ \u03b2=0\n$$\n\n5\n\nUsing Vieta's Formula, the following equations are established.\n\n$$\nr_1+r_2 = \\alpha \\quad \\text{and} \\quad r_1r_2 = \u03b2\n$$\n\nTo determine\n\n$$\n\\alpha\n$$\n\n,\n\n$$\n\u03b2\n$$\n\n, cube both sides of the equation (4)\n\n$$\nt^3=3\\sqrt[3]{r_1r_2}(\\sqrt[3]{r_1}+\\sqrt[3]{r_2})+r_1+r_2\n$$\n\nSubstituting, the equation is simplified to\n\n$$\nt^3=3\\sqrt[3]{\u03b2}t+\\alpha\n$$\n\nCompare the cubic equation with (3), the following equations are established\n\n$$\n\\begin{cases} 3\\sqrt[3]{\u03b2}=-\\dfrac{20}{3}\\\\ \\alpha=\\dfrac{101}{27}\\\\ \\end{cases}\n$$\n\nSolving for\n\n$$\n\u03b2\n$$\n\ngives\n\n$$\n\u03b2=-\\dfrac{8000}{729}\n$$\n\nSo the quadratic equation (5) is determined as\n\n$$\nz^2-\\dfrac{101}{27}z-\\dfrac{8000}{729}=0\n$$\n\n6\n\nSolving the quadratic equation yields\n\n$$\n\\begin{cases} r_1=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\approx5.6746077738748\\\\ r_2=\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}\\approx-1.9338670331341\\\\ \\end{cases}\n$$\n\nTherefore, one of the roots of the cubic equation could be obtained from (4).\n\n$$\nt_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\n$$\n\nin decimals,\n\n$$\nt_1=0.53778143658824\n$$\n\nHowever, since the cube root of a quantity has triple values,\n\nThe other two roots could be determined as,\n\n$$\nt_2=\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\n$$\n\n$$\nt_3=\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\n$$\n\nCombining the real and imaginary parts\n results in the same result as that obtained by Cardano's solution.\n\nFor the equation\n\n$$\nt^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}\n$$\n\n, we have $$p=\\dfrac{20}{3}$$ and\n\n$$\nq = -\\dfrac{101}{27}\n$$\n\n### Calculate the discriminant\n\nThe nature of the roots are determined by the sign of the discriminant.\n\n$$\n\\begin{aligned} \\\\\\Delta&=\\dfrac{q^2}{4}+\\dfrac{p^3}{27}\\\\ & =\\dfrac{\\Big(-\\dfrac{101}{27}\\Big)^2}{4}+\\dfrac{\\Big(\\dfrac{20}{3}\\Big)^3}{27}\\\\ & =\\dfrac{10201}{2916}+\\dfrac{8000}{729}\\\\ & =\\dfrac{10201\\cdot 1+8000\\cdot 4}{2916}\\\\ & =14.472222222222\\\\ \\end{aligned}\n$$\n\n### 4.1 Use the root formula directly\n\nIf the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.\n\n$$\nt_{1,2,3} =\\begin{cases} \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } +\\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}& \\\\ \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \\overline{\u03c9} \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}&\\ \\\\ \\overline{\u03c9}\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }} \\end{cases}\n$$\n\nin which,\n\n$$\n\u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}\n$$\n\nand\n\n$$\n\\overline{\u03c9} =\\dfrac{-1-i\\sqrt{3}}{2}\n$$\n\nSubstitute the values of\n\n$$\np, q\n$$\n\nand\n\n$$\n\\Delta\n$$\n\nwhich we have calculated. Then,\n\n$$\n\\begin{aligned} \\\\t_1&=\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{42201}{2916}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{42201}{2916}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}\n$$\n\nIf we denote\n\n$$\nR = -\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }\n$$\n\n$$\n\\overline{R} = -\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }\n$$\n\nthen,\n\n$$\n\\sqrt[3]{R} = \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\n$$\n\n,\n\n$$\n\\sqrt[3]{\\overline{R}} =\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\n$$\n\n$$\n\\begin{aligned} \\\\t_2&= \u03c9\\cdotp \\sqrt[3]{R}+ \\overline{\u03c9} \\sqrt[3]{\\overline{R} }\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}( \\sqrt[3]{R} - \\sqrt[3]{\\overline{R} }) }{2} i\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}\n$$\n\n$$\n\\begin{aligned} \\\\t_3&= \\overline{\u03c9}\\cdotp \\sqrt[3]{R}+ \u03c9\\cdotp \\sqrt[3]{\\overline{R}}\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}(- \\sqrt[3]{R} + \\sqrt[3]{\\overline{R} }) }{2}i \\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}\n$$\n\n## Roots of the general cubic equation\n\nSince\n\n$$\nx = t - \\dfrac{b}{3a}\n$$\n\n, substituting the values of $$t$$, $$a$$ and\n\n$$\nb\n$$\n\ngives\n\n$$\nx_1 = t_1-\\dfrac{2}{3}\n$$\n\n$$\nx_2 = t_2-\\dfrac{2}{3}\n$$\n\n$$\nx_3 = t_3-\\dfrac{2}{3}\n$$\n\n## 5. Summary\n\nIn summary, we have tried the method of cubic root formula to explore the solutions of the equation. The cubic equation\n\n$$\nx\u00b3 + 2x\u00b2 + 8x + 1=0\n$$\n\nis found to have one real root and two complex roots. Exact values and approximations are given below.\n\n$$\n\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}\n$$\n\nin decimal notation,\n\n$$\n\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}\n$$\n\n## 6. Graph for the function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$\n\nSince the discriminat is greater than zero, the curve of the cubic function\n\n$$\nf(x) = x\u00b3 + 2x\u00b2 + 8x + 1\n$$\n\nhas one intersection point with the x-axis.\n\n## More cubic equations\n", "html": "\n\n\n\n\n\nSolve x^3+2x^2+8x+1=0 | Uniteasy.com\n\n\n\n\n\n\n\n\n\n\n\n \n
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      \n

      Solve the cubic equation:

      \n

      $$x^3+2x^2+8x+1=0 $$

      \n

      Quick Answer

      \n

      Since the discriminant $$\\Delta >0$$, the cubic equation has one real root and two conjugate complex roots.

      $$ \\Delta=14.472222222222$$

      $$\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}$$

      In decimals,

      $$\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}$$

      Detailed Steps on Solution

      1. Convert to depressed cubic equation

      The idea is to convert general form of cubic equation

      $$ax^3+bx^2+cx+d = 0$$

      to the form without quadratic term.

      $$t^3+pt+q = 0$$

      By substituting $$x$$ with $$t - \\dfrac{b}{3a}$$, the general cubic equation could be transformed to

      $$t^3+\\dfrac{3ac-b^2}{3a^2}t+\\dfrac{2b^3-9abc+27a^2d}{27a^3} = 0 $$

      Compare with the depressed cubic equation. Then,

      $$p = \\dfrac{3ac-b^2}{3a^2}$$

      $$q = \\dfrac{2b^3-9abc+27a^2d}{27a^3} $$

      Substitute the values of coefficients, $$p, q$$ is obtained as

      $$p = \\dfrac{3\\cdot 1\\cdot 8-2^2}{3\\cdot 1^2}=\\dfrac{20}{3}$$

      $$q = \\dfrac{2\\cdot 2^3-9\\cdot1\\cdot 2\\cdot 8+27\\cdot 1^2\\cdot1}{27\\cdot 1^3}=-\\dfrac{101}{27}$$

      Use the substitution to transform

      Let $$p$$ and $$q$$ being the coefficient of the linean and constant terms, the depressed cubic equation is expressed as.

      $$t^3 +pt+q=0$$

      Let $$x=t-\\dfrac{2}{3}$$

      The cubic equation $$x\u00b3 + 2x\u00b2 + 8x + 1=0$$ is transformed to

      $$t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

      2. Cardano's solution

      Let $$t=u-v$$

      Cube both sides and extract common factor from two middle terms after expanding the bracket.

      $$\\begin{aligned} \\\\t^3&=(u-v)^3\\\\ & =u^3-3u^2v+3uv^2-v^3\\\\ & =-3uv(u-v)+u^3-v^3\\\\ \\end{aligned}$$

      Since $$u-v=t$$, substitution gives a linear term for the equation.\n Rearrange terms.

      $$x^3+3uvx-u^3+v^3=0$$

      Compare the cubic equation with the original one (1)

      $$\\begin{cases} 3uv=\\dfrac{20}{3}\\quad\\text{or}\\quad v=\\dfrac{20}{9u}\\\\ v^3-u^3=-\\dfrac{101}{27}\\\\ \\end{cases}$$

      $$v=\\dfrac{20}{9u}$$ gives relationship between the two variables. Substitute the value of $$v$$ to the second equation

      $$\\Big(\\dfrac{20}{9u}\\Big)^3-u^3=-\\dfrac{101}{27}$$

      Simplifying gives,

      $$u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0$$2

      Let $$m=u^3$$, then the equation is transformed to a quadratic equation in terms of $$m$$.\n Once the value of $$m$$ is determined, $$v^3$$ could be determined by $$v^3=-\\dfrac{101}{27}+u^3$$.

      $$m^2-\\dfrac{101}{27}m-\\dfrac{8000}{729}=0$$

      Sovling the quadratic euqation will give two roots (some may be equal). Here we only cosider\n one case with positive sign before the square root radical since the negative case will produce the same result.

      $$\\begin{aligned} \\\\u^3=m&=\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\Big(-\\dfrac{101}{27}^2\\Big)-4\\cdot \\Big(-\\dfrac{8000}{729}\\Big)}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{10201}{729}+\\dfrac{32000}{729}}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{521}{9}}\\\\ & =\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}$$

      $$v^3$$ can be determined by the equation we deduced $$v^3-u^3=-\\dfrac{101}{27}$$. Then,

      $$\\begin{aligned} \\\\v^3&=-\\dfrac{101}{27}+u^3\\\\ & =-\\dfrac{101}{27}+\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ & =-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}$$

      Now we have,

      $$u^3=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}$$ and $$v^3=-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}$$

      Evaluating the simplest cubic equation $$x^3-A=0$$,\n it has 3 roots, in which the first root is a real number . The second and third are\n expressed in the product of cubic root of unity and the first one.

      If $$\u03c9 = \\dfrac{-1+i\\sqrt{3}}{2}$$, then its reciprocal is equal to its conjugate, $$\\dfrac{1}{\u03c9}=\\overline{\u03c9}$$.

      $$\\begin{cases} r_1=\\sqrt[3]{A}\\\\ r_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ r_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ \\end{cases}$$

      Similary, taking cubic root for $$u^3$$ and $$v^3$$ also gives 3 roots.

      $$\\begin{cases} u_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}$$

      For $$v_2$$ and $$v_3$$, the complex numbers before radicals are the conjugates of\n those for $$u_2$$ and $$u_3$$, which can be verified by the reciprocal property\n of the cubic root of unity from the equation $$v=\\dfrac{20}{9u}$$. The radicand can be taken as the\n negative conjugate of that in $$u_1$$, $$u_2$$ and $$u_3$$, which is the same in value.

      $$\\begin{cases} v_1=\\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_2=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_3=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}$$

      Verification for the redicand in $$v$$.

      $$\\begin{aligned} \\\\v_1&=\\dfrac{20}{9u_1}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\Big(\\dfrac{101}{54}\\Big)^2-\\Big(\\dfrac{\\sqrt{521}}{6}\\Big)^2}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{10201}{2916}-\\dfrac{521}{36}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{-1\\cdot 20^3}{9^3}}}\\\\ & =-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

      Since $$x=u-v$$, combining the real and imaginary parts gives\n 3 results for $$t$$

      $$\\begin{aligned} \\\\t_1&=u_1-v_1\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

      $$\\begin{aligned} \\\\t_2&=u_2-v_2\\\\ & =\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

      $$\\begin{aligned} \\\\t_3&=u_3-v_3\\\\ & =\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

      3. Vieta's Substitution

      In Cardano' solution, $$t$$ is defined as the difference of $$u$$ and\n $$v$$. If we substitute the value of $$v$$ (4) into (2), we get the\n equation. $$t=u-\\dfrac{20}{9u}$$. And then substitute the equation to the cubic equation\n $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$. This method is called Vieta's Substitution\n for solving a cubic equation, which simplied the Cardano' solution. The substitution\n expression can be obtained by the following formula directly.

      $$t=u-\\dfrac{p}{3u}$$

      Substitute the expression $$t=u-\\dfrac{20}{9u}$$ to the cubic equation

      $$\\Big(u-\\dfrac{20}{9u}\\Big)^3+\\dfrac{20}{3}\\Big(u-\\dfrac{20}{9u}\\Big)-\\dfrac{101}{27}=0$$

      Expand brackets and cancel the like terms

      $$u^3-\\cancel{\\dfrac{20}{3}u^2\\dfrac{1}{u}}+\\cancel{\\dfrac{400}{27}u\\dfrac{1}{u^2}}-\\dfrac{8000}{729}\\dfrac{1}{u^3}+\\cancel{\\dfrac{20}{3}u}-\\cancel{\\dfrac{400}{27}\\dfrac{1}{u}}-\\dfrac{101}{27}=0$$

      Then we get the same equation as (2)

      $$u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0$$

      The rest of the steps will be the same as those of Cardano's solution

      4. Euler's Solution

      $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

      Move the linear term and constant of (1) to its right hand side.\n We get the following form of the equation.

      $$t^3=-\\dfrac{20}{3}t+\\dfrac{101}{27} $$3

      Let the root of the cubic equation be the sum of two cubic roots

      $$t=\\sqrt[3]{r_1}+\\sqrt[3]{r_2} $$4

      in which $$r_1$$ and $$r_2$$ are two roots of a quadratic equation

      $$z^2-\\alpha z+ \u03b2=0 $$5

      Using Vieta's Formula, the following equations are established.

      $$r_1+r_2 = \\alpha \\quad \\text{and} \\quad r_1r_2 = \u03b2 $$

      To determine $$\\alpha$$, $$\u03b2$$, cube both sides of the equation (4)

      $$t^3=3\\sqrt[3]{r_1r_2}(\\sqrt[3]{r_1}+\\sqrt[3]{r_2})+r_1+r_2 $$

      Substituting, the equation is simplified to

      $$t^3=3\\sqrt[3]{\u03b2}t+\\alpha $$

      Compare the cubic equation with (3), the following equations are established

      $$\\begin{cases} 3\\sqrt[3]{\u03b2}=-\\dfrac{20}{3}\\\\ \\alpha=\\dfrac{101}{27}\\\\ \\end{cases}$$

      Solving for $$\u03b2$$ gives

      $$\u03b2=-\\dfrac{8000}{729} $$

      So the quadratic equation (5) is determined as

      $$z^2-\\dfrac{101}{27}z-\\dfrac{8000}{729}=0$$6

      Solving the quadratic equation yields

      $$\\begin{cases} r_1=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\approx5.6746077738748\\\\ r_2=\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}\\approx-1.9338670331341\\\\ \\end{cases}$$

      Therefore, one of the roots of the cubic equation could be obtained from (4).

      $$t_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

      in decimals,

      $$t_1=0.53778143658824 $$

      However, since the cube root of a quantity has triple values,

      The other two roots could be determined as,

      $$t_2=\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

      $$t_3=\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

      Combining the real and imaginary parts\n results in the same result as that obtained by Cardano's solution.

      For the equation $$t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}$$, we have $$p=\\dfrac{20}{3}$$ and $$q = -\\dfrac{101}{27}$$

      Calculate the discriminant

      The nature of the roots are determined by the sign of the discriminant.

      $$\\begin{aligned} \\\\\\Delta&=\\dfrac{q^2}{4}+\\dfrac{p^3}{27}\\\\ & =\\dfrac{\\Big(-\\dfrac{101}{27}\\Big)^2}{4}+\\dfrac{\\Big(\\dfrac{20}{3}\\Big)^3}{27}\\\\ & =\\dfrac{10201}{2916}+\\dfrac{8000}{729}\\\\ & =\\dfrac{10201\\cdot 1+8000\\cdot 4}{2916}\\\\ & =14.472222222222\\\\ \\end{aligned}$$

      4.1 Use the root formula directly

      If the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.

      $$t_{1,2,3} =\\begin{cases} \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } +\\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}& \\\\ \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \\overline{\u03c9} \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}&\\ \\\\ \\overline{\u03c9}\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \u03c9\\cdotp \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }} \\end{cases}$$

      in which, $$ \u03c9 = \\dfrac{-1+i\\sqrt{3}}{2} $$ and $$ \\overline{\u03c9} =\\dfrac{-1-i\\sqrt{3}}{2}$$

      Substitute the values of $$p, q$$ and $$\\Delta$$ which we have calculated. Then,

      $$\\begin{aligned} \\\\t_1&=\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{42201}{2916}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{42201}{2916}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

      If we denote

      $$R = -\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }$$

      $$\\overline{R} = -\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }$$

      then,

      $$\\sqrt[3]{R} = \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}$$, $$\\sqrt[3]{\\overline{R}} =\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}$$

      $$\\begin{aligned} \\\\t_2&= \u03c9\\cdotp \\sqrt[3]{R}+ \\overline{\u03c9} \\sqrt[3]{\\overline{R} }\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}( \\sqrt[3]{R} - \\sqrt[3]{\\overline{R} }) }{2} i\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

      $$\\begin{aligned} \\\\t_3&= \\overline{\u03c9}\\cdotp \\sqrt[3]{R}+ \u03c9\\cdotp \\sqrt[3]{\\overline{R}}\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}(- \\sqrt[3]{R} + \\sqrt[3]{\\overline{R} }) }{2}i \\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

      Roots of the general cubic equation

      Since $$x = t - \\dfrac{b}{3a}$$, substituting the values of $$t$$, $$a$$ and $$b$$ gives

      $$x_1 = t_1-\\dfrac{2}{3}$$

      $$x_2 = t_2-\\dfrac{2}{3}$$

      $$x_3 = t_3-\\dfrac{2}{3}$$

      5. Summary

      In summary, we have tried the method of cubic root formula to explore the solutions of the equation. The cubic equation $$x\u00b3 + 2x\u00b2 + 8x + 1=0$$ is found to have one real root and two complex roots. Exact values and approximations are given below.

      $$\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}$$

      in decimal notation,

      $$\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}$$

      6. Graph for the function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$

      Since the discriminat is greater than zero, the curve of the cubic function $$f(x) = x\u00b3 + 2x\u00b2 + 8x + 1$$ has one intersection point with the x-axis.

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      \n\n\n"} +{"url": "https://uniteasy.com/solver/cubicequation/x%5E3%2B2x%5E2%2B8x%2B1%3D0/", "content": "# Solve the cubic equation:\n\n$$\nx^3+2x^2+8x+1=0\n$$\n\nQuick Answer\n\nSince the discriminant\n\n$$\n\\Delta >0\n$$\n\n, the cubic equation has one real root and two conjugate complex roots.\n\n$$\n\\Delta=14.472222222222\n$$\n\n$$\n\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}\n$$\n\nIn decimals,\n\n$$\n\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}\n$$\n\nDetailed Steps on Solution\n\n## 1. Convert to depressed cubic equation\n\nThe idea is to convert general form of cubic equation\n\n$$\nax^3+bx^2+cx+d = 0\n$$\n\nto the form without quadratic term.\n\n$$\nt^3+pt+q = 0\n$$\n\nBy substituting\n\n$$\nx\n$$\n\nwith\n\n$$\nt - \\dfrac{b}{3a}\n$$\n\n, the general cubic equation could be transformed to\n\n$$\nt^3+\\dfrac{3ac-b^2}{3a^2}t+\\dfrac{2b^3-9abc+27a^2d}{27a^3} = 0\n$$\n\nCompare with the depressed cubic equation. Then,\n\n$$\np = \\dfrac{3ac-b^2}{3a^2}\n$$\n\n$$\nq = \\dfrac{2b^3-9abc+27a^2d}{27a^3}\n$$\n\nSubstitute the values of coefficients,\n\n$$\np, q\n$$\n\nis obtained as\n\n$$\np = \\dfrac{3\\cdot 1\\cdot 8-2^2}{3\\cdot 1^2}=\\dfrac{20}{3}\n$$\n\n$$\nq = \\dfrac{2\\cdot 2^3-9\\cdot1\\cdot 2\\cdot 8+27\\cdot 1^2\\cdot1}{27\\cdot 1^3}=-\\dfrac{101}{27}\n$$\n\n### Use the substitution to transform\n\nLet\n\n$$\np\n$$\n\nand\n\n$$\nq\n$$\n\nbeing the coefficient of the linean and constant terms, the depressed cubic equation is expressed as.\n\n$$\nt^3 +pt+q=0\n$$\n\nLet\n\n$$\nx=t-\\dfrac{2}{3}\n$$\n\nThe cubic equation\n\n$$\nx³ + 2x² + 8x + 1=0\n$$\n\nis transformed to\n\n$$\nt^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}=0\n$$\n\n## 2. Cardano's solution\n\nLet\n\n$$\nt=u-v\n$$\n\nCube both sides and extract common factor from two middle terms after expanding the bracket.\n\n$$\n\\begin{aligned} \\\\t^3&=(u-v)^3\\\\ & =u^3-3u^2v+3uv^2-v^3\\\\ & =-3uv(u-v)+u^3-v^3\\\\ \\end{aligned}\n$$\n\nSince\n\n$$\nu-v=t\n$$\n\n, substitution gives a linear term for the equation. Rearrange terms.\n\n$$\nx^3+3uvx-u^3+v^3=0\n$$\n\nCompare the cubic equation with the original one (1)\n\n$$\n\\begin{cases} 3uv=\\dfrac{20}{3}\\quad\\text{or}\\quad v=\\dfrac{20}{9u}\\\\ v^3-u^3=-\\dfrac{101}{27}\\\\ \\end{cases}\n$$\n\n$$\nv=\\dfrac{20}{9u}\n$$\n\ngives relationship between the two variables. Substitute the value of\n\n$$\nv\n$$\n\nto the second equation\n\n$$\n\\Big(\\dfrac{20}{9u}\\Big)^3-u^3=-\\dfrac{101}{27}\n$$\n\nSimplifying gives,\n\n$$\nu^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0\n$$\n\n2\n\nLet\n\n$$\nm=u^3\n$$\n\n, then the equation is transformed to a quadratic equation in terms of\n\n$$\nm\n$$\n\n. Once the value of\n\n$$\nm\n$$\n\nis determined,\n\n$$\nv^3\n$$\n\ncould be determined by\n\n$$\nv^3=-\\dfrac{101}{27}+u^3\n$$\n\n.\n\n$$\nm^2-\\dfrac{101}{27}m-\\dfrac{8000}{729}=0\n$$\n\nSovling the quadratic euqation will give two roots (some may be equal). Here we only cosider one case with positive sign before the square root radical since the negative case will produce the same result.\n\n$$\n\\begin{aligned} \\\\u^3=m&=\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\Big(-\\dfrac{101}{27}^2\\Big)-4\\cdot \\Big(-\\dfrac{8000}{729}\\Big)}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{10201}{729}+\\dfrac{32000}{729}}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{521}{9}}\\\\ & =\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}\n$$\n\n$$\nv^3\n$$\n\ncan be determined by the equation we deduced\n\n$$\nv^3-u^3=-\\dfrac{101}{27}\n$$\n\n. Then,\n\n$$\n\\begin{aligned} \\\\v^3&=-\\dfrac{101}{27}+u^3\\\\ & =-\\dfrac{101}{27}+\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ & =-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}\n$$\n\nNow we have,\n\n$$\nu^3=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\n$$\n\nand\n\n$$\nv^3=-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\n$$\n\nEvaluating the simplest cubic equation\n\n$$\nx^3-A=0\n$$\n\n, it has 3 roots, in which the first root is a real number . The second and third are expressed in the product of cubic root of unity and the first one.\n\nIf\n\n$$\nω = \\dfrac{-1+i\\sqrt{3}}{2}\n$$\n\n, then its reciprocal is equal to its conjugate,\n\n$$\n\\dfrac{1}{ω}=\\overline{ω}\n$$\n\n.\n\n$$\n\\begin{cases} r_1=\\sqrt[3]{A}\\\\ r_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ r_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ \\end{cases}\n$$\n\nSimilary, taking cubic root for\n\n$$\nu^3\n$$\n\nand\n\n$$\nv^3\n$$\n\nalso gives 3 roots.\n\n$$\n\\begin{cases} u_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}\n$$\n\nFor\n\n$$\nv_2\n$$\n\nand\n\n$$\nv_3\n$$\n\n, the complex numbers before radicals are the conjugates of those for\n\n$$\nu_2\n$$\n\nand\n\n$$\nu_3\n$$\n\n, which can be verified by the reciprocal property of the cubic root of unity from the equation\n\n$$\nv=\\dfrac{20}{9u}\n$$\n\n. The radicand can be taken as the negative conjugate of that in\n\n$$\nu_1\n$$\n\n,\n\n$$\nu_2\n$$\n\nand\n\n$$\nu_3\n$$\n\n, which is the same in value.\n\n$$\n\\begin{cases} v_1=\\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_2=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_3=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}\n$$\n\nVerification for the redicand in\n\n$$\nv\n$$\n\n.\n\n$$\n\\begin{aligned} \\\\v_1&=\\dfrac{20}{9u_1}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\Big(\\dfrac{101}{54}\\Big)^2-\\Big(\\dfrac{\\sqrt{521}}{6}\\Big)^2}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{10201}{2916}-\\dfrac{521}{36}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{-1\\cdot 20^3}{9^3}}}\\\\ & =-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}\n$$\n\nSince\n\n$$\nx=u-v\n$$\n\n, combining the real and imaginary parts gives 3 results for\n\n$$\nt\n$$\n\n$$\n\\begin{aligned} \\\\t_1&=u_1-v_1\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}\n$$\n\n$$\n\\begin{aligned} \\\\t_2&=u_2-v_2\\\\ & =\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}\n$$\n\n$$\n\\begin{aligned} \\\\t_3&=u_3-v_3\\\\ & =\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}\n$$\n\n## 3. Vieta's Substitution\n\nIn Cardano' solution,\n\n$$\nt\n$$\n\nis defined as the difference of\n\n$$\nu\n$$\n\nand\n\n$$\nv\n$$\n\n. If we substitute the value of\n\n$$\nv\n$$\n\n(4) into (2), we get the equation.\n\n$$\nt=u-\\dfrac{20}{9u}\n$$\n\n. And then substitute the equation to the cubic equation\n\n$$\nt^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0\n$$\n\n. This method is called Vieta's Substitution for solving a cubic equation, which simplied the Cardano' solution. The substitution expression can be obtained by the following formula directly.\n\n$$\nt=u-\\dfrac{p}{3u}\n$$\n\nSubstitute the expression\n\n$$\nt=u-\\dfrac{20}{9u}\n$$\n\nto the cubic equation\n\n$$\n\\Big(u-\\dfrac{20}{9u}\\Big)^3+\\dfrac{20}{3}\\Big(u-\\dfrac{20}{9u}\\Big)-\\dfrac{101}{27}=0\n$$\n\nExpand brackets and cancel the like terms\n\n$$\nu^3-\\cancel{\\dfrac{20}{3}u^2\\dfrac{1}{u}}+\\cancel{\\dfrac{400}{27}u\\dfrac{1}{u^2}}-\\dfrac{8000}{729}\\dfrac{1}{u^3}+\\cancel{\\dfrac{20}{3}u}-\\cancel{\\dfrac{400}{27}\\dfrac{1}{u}}-\\dfrac{101}{27}=0\n$$\n\nThen we get the same equation as (2)\n\n$$\nu^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0\n$$\n\nThe rest of the steps will be the same as those of Cardano's solution\n\n## 4. Euler's Solution\n\n$$\nt^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0\n$$\n\nMove the linear term and constant of (1) to its right hand side. We get the following form of the equation.\n\n$$\nt^3=-\\dfrac{20}{3}t+\\dfrac{101}{27}\n$$\n\n3\n\nLet the root of the cubic equation be the sum of two cubic roots\n\n$$\nt=\\sqrt[3]{r_1}+\\sqrt[3]{r_2}\n$$\n\n4\n\nin which\n\n$$\nr_1\n$$\n\nand\n\n$$\nr_2\n$$\n\nare two roots of a quadratic equation\n\n$$\nz^2-\\alpha z+ β=0\n$$\n\n5\n\nUsing Vieta's Formula, the following equations are established.\n\n$$\nr_1+r_2 = \\alpha \\quad \\text{and} \\quad r_1r_2 = β\n$$\n\nTo determine\n\n$$\n\\alpha\n$$\n\n,\n\n$$\nβ\n$$\n\n, cube both sides of the equation (4)\n\n$$\nt^3=3\\sqrt[3]{r_1r_2}(\\sqrt[3]{r_1}+\\sqrt[3]{r_2})+r_1+r_2\n$$\n\nSubstituting, the equation is simplified to\n\n$$\nt^3=3\\sqrt[3]{β}t+\\alpha\n$$\n\nCompare the cubic equation with (3), the following equations are established\n\n$$\n\\begin{cases} 3\\sqrt[3]{β}=-\\dfrac{20}{3}\\\\ \\alpha=\\dfrac{101}{27}\\\\ \\end{cases}\n$$\n\nSolving for\n\n$$\nβ\n$$\n\ngives\n\n$$\nβ=-\\dfrac{8000}{729}\n$$\n\nSo the quadratic equation (5) is determined as\n\n$$\nz^2-\\dfrac{101}{27}z-\\dfrac{8000}{729}=0\n$$\n\n6\n\nSolving the quadratic equation yields\n\n$$\n\\begin{cases} r_1=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\approx5.6746077738748\\\\ r_2=\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}\\approx-1.9338670331341\\\\ \\end{cases}\n$$\n\nTherefore, one of the roots of the cubic equation could be obtained from (4).\n\n$$\nt_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\n$$\n\nin decimals,\n\n$$\nt_1=0.53778143658824\n$$\n\nHowever, since the cube root of a quantity has triple values,\n\nThe other two roots could be determined as,\n\n$$\nt_2=\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\n$$\n\n$$\nt_3=\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\n$$\n\nCombining the real and imaginary parts results in the same result as that obtained by Cardano's solution.\n\nFor the equation\n\n$$\nt^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}\n$$\n\n, we have\n\n$$\np=\\dfrac{20}{3}\n$$\n\nand\n\n$$\nq = -\\dfrac{101}{27}\n$$\n\n### Calculate the discriminant\n\nThe nature of the roots are determined by the sign of the discriminant.\n\n$$\n\\begin{aligned} \\\\\\Delta&=\\dfrac{q^2}{4}+\\dfrac{p^3}{27}\\\\ & =\\dfrac{\\Big(-\\dfrac{101}{27}\\Big)^2}{4}+\\dfrac{\\Big(\\dfrac{20}{3}\\Big)^3}{27}\\\\ & =\\dfrac{10201}{2916}+\\dfrac{8000}{729}\\\\ & =\\dfrac{10201\\cdot 1+8000\\cdot 4}{2916}\\\\ & =14.472222222222\\\\ \\end{aligned}\n$$\n\n### 4.1 Use the root formula directly\n\nIf the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.\n\n$$\nt_{1,2,3} =\\begin{cases} \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } +\\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}& \\\\ ω\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \\overline{ω} \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}&\\ \\\\ \\overline{ω}\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + ω\\cdotp \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }} \\end{cases}\n$$\n\nin which,\n\n$$\nω = \\dfrac{-1+i\\sqrt{3}}{2}\n$$\n\nand\n\n$$\n\\overline{ω} =\\dfrac{-1-i\\sqrt{3}}{2}\n$$\n\nSubstitute the values of\n\n$$\np, q\n$$\n\nand\n\n$$\n\\Delta\n$$\n\nwhich we have calculated. Then,\n\n$$\n\\begin{aligned} \\\\t_1&=\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{42201}{2916}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{42201}{2916}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}\n$$\n\nIf we denote\n\n$$\nR = -\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }\n$$\n\n$$\n\\overline{R} = -\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }\n$$\n\nthen,\n\n$$\n\\sqrt[3]{R} = \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\n$$\n\n,\n\n$$\n\\sqrt[3]{\\overline{R}} =\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\n$$\n\n$$\n\\begin{aligned} \\\\t_2&= ω\\cdotp \\sqrt[3]{R}+ \\overline{ω} \\sqrt[3]{\\overline{R} }\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}( \\sqrt[3]{R} - \\sqrt[3]{\\overline{R} }) }{2} i\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}\n$$\n\n$$\n\\begin{aligned} \\\\t_3&= \\overline{ω}\\cdotp \\sqrt[3]{R}+ ω\\cdotp \\sqrt[3]{\\overline{R}}\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}(- \\sqrt[3]{R} + \\sqrt[3]{\\overline{R} }) }{2}i \\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}\n$$\n\n## Roots of the general cubic equation\n\nSince\n\n$$\nx = t - \\dfrac{b}{3a}\n$$\n\n, substituting the values of\n\n$$\nt\n$$\n\n,\n\n$$\na\n$$\n\nand\n\n$$\nb\n$$\n\ngives\n\n$$\nx_1 = t_1-\\dfrac{2}{3}\n$$\n\n$$\nx_2 = t_2-\\dfrac{2}{3}\n$$\n\n$$\nx_3 = t_3-\\dfrac{2}{3}\n$$\n\n## 5. Summary\n\nIn summary, we have tried the method of cubic root formula to explore the solutions of the equation. The cubic equation\n\n$$\nx³ + 2x² + 8x + 1=0\n$$\n\nis found to have one real root and two complex roots. Exact values and approximations are given below.\n\n$$\n\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}\n$$\n\nin decimal notation,\n\n$$\n\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}\n$$\n\n## 6. Graph for the function\n\n$$\nf(x) = x³ + 2x² + 8x + 1\n$$\n\nSince the discriminat is greater than zero, the curve of the cubic function\n\n$$\nf(x) = x³ + 2x² + 8x + 1\n$$\n\nhas one intersection point with the x-axis.\n\n## More cubic equations\n", "main_html": "
      \n
      \n

      Solve the cubic equation:

      \n

      $$x^3+2x^2+8x+1=0 $$

      \n

      Quick Answer

      \n

      Since the discriminant $$\\Delta >0$$, the cubic equation has one real root and two conjugate complex roots.

      $$ \\Delta=14.472222222222$$

      $$\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}$$

      In decimals,

      $$\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}$$

      Detailed Steps on Solution

      1. Convert to depressed cubic equation

      The idea is to convert general form of cubic equation

      $$ax^3+bx^2+cx+d = 0$$

      to the form without quadratic term.

      $$t^3+pt+q = 0$$

      By substituting $$x$$ with $$t - \\dfrac{b}{3a}$$, the general cubic equation could be transformed to

      $$t^3+\\dfrac{3ac-b^2}{3a^2}t+\\dfrac{2b^3-9abc+27a^2d}{27a^3} = 0 $$

      Compare with the depressed cubic equation. Then,

      $$p = \\dfrac{3ac-b^2}{3a^2}$$

      $$q = \\dfrac{2b^3-9abc+27a^2d}{27a^3} $$

      Substitute the values of coefficients, $$p, q$$ is obtained as

      $$p = \\dfrac{3\\cdot 1\\cdot 8-2^2}{3\\cdot 1^2}=\\dfrac{20}{3}$$

      $$q = \\dfrac{2\\cdot 2^3-9\\cdot1\\cdot 2\\cdot 8+27\\cdot 1^2\\cdot1}{27\\cdot 1^3}=-\\dfrac{101}{27}$$

      Use the substitution to transform

      Let $$p$$ and $$q$$ being the coefficient of the linean and constant terms, the depressed cubic equation is expressed as.

      $$t^3 +pt+q=0$$

      Let $$x=t-\\dfrac{2}{3}$$

      The cubic equation $$x³ + 2x² + 8x + 1=0$$ is transformed to

      $$t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

      2. Cardano's solution

      Let $$t=u-v$$

      Cube both sides and extract common factor from two middle terms after expanding the bracket.

      $$\\begin{aligned} \\\\t^3&=(u-v)^3\\\\ & =u^3-3u^2v+3uv^2-v^3\\\\ & =-3uv(u-v)+u^3-v^3\\\\ \\end{aligned}$$

      Since $$u-v=t$$, substitution gives a linear term for the equation.\n Rearrange terms.

      $$x^3+3uvx-u^3+v^3=0$$

      Compare the cubic equation with the original one (1)

      $$\\begin{cases} 3uv=\\dfrac{20}{3}\\quad\\text{or}\\quad v=\\dfrac{20}{9u}\\\\ v^3-u^3=-\\dfrac{101}{27}\\\\ \\end{cases}$$

      $$v=\\dfrac{20}{9u}$$ gives relationship between the two variables. Substitute the value of $$v$$ to the second equation

      $$\\Big(\\dfrac{20}{9u}\\Big)^3-u^3=-\\dfrac{101}{27}$$

      Simplifying gives,

      $$u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0$$2

      Let $$m=u^3$$, then the equation is transformed to a quadratic equation in terms of $$m$$.\n Once the value of $$m$$ is determined, $$v^3$$ could be determined by $$v^3=-\\dfrac{101}{27}+u^3$$.

      $$m^2-\\dfrac{101}{27}m-\\dfrac{8000}{729}=0$$

      Sovling the quadratic euqation will give two roots (some may be equal). Here we only cosider\n one case with positive sign before the square root radical since the negative case will produce the same result.

      $$\\begin{aligned} \\\\u^3=m&=\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\Big(-\\dfrac{101}{27}^2\\Big)-4\\cdot \\Big(-\\dfrac{8000}{729}\\Big)}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{10201}{729}+\\dfrac{32000}{729}}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{521}{9}}\\\\ & =\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}$$

      $$v^3$$ can be determined by the equation we deduced $$v^3-u^3=-\\dfrac{101}{27}$$. Then,

      $$\\begin{aligned} \\\\v^3&=-\\dfrac{101}{27}+u^3\\\\ & =-\\dfrac{101}{27}+\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ & =-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}$$

      Now we have,

      $$u^3=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}$$ and $$v^3=-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}$$

      Evaluating the simplest cubic equation $$x^3-A=0$$,\n it has 3 roots, in which the first root is a real number . The second and third are\n expressed in the product of cubic root of unity and the first one.

      If $$ω = \\dfrac{-1+i\\sqrt{3}}{2}$$, then its reciprocal is equal to its conjugate, $$\\dfrac{1}{ω}=\\overline{ω}$$.

      $$\\begin{cases} r_1=\\sqrt[3]{A}\\\\ r_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ r_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ \\end{cases}$$

      Similary, taking cubic root for $$u^3$$ and $$v^3$$ also gives 3 roots.

      $$\\begin{cases} u_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}$$

      For $$v_2$$ and $$v_3$$, the complex numbers before radicals are the conjugates of\n those for $$u_2$$ and $$u_3$$, which can be verified by the reciprocal property\n of the cubic root of unity from the equation $$v=\\dfrac{20}{9u}$$. The radicand can be taken as the\n negative conjugate of that in $$u_1$$, $$u_2$$ and $$u_3$$, which is the same in value.

      $$\\begin{cases} v_1=\\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_2=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_3=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}$$

      Verification for the redicand in $$v$$.

      $$\\begin{aligned} \\\\v_1&=\\dfrac{20}{9u_1}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\Big(\\dfrac{101}{54}\\Big)^2-\\Big(\\dfrac{\\sqrt{521}}{6}\\Big)^2}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{10201}{2916}-\\dfrac{521}{36}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{-1\\cdot 20^3}{9^3}}}\\\\ & =-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

      Since $$x=u-v$$, combining the real and imaginary parts gives\n 3 results for $$t$$

      $$\\begin{aligned} \\\\t_1&=u_1-v_1\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

      $$\\begin{aligned} \\\\t_2&=u_2-v_2\\\\ & =\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

      $$\\begin{aligned} \\\\t_3&=u_3-v_3\\\\ & =\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

      3. Vieta's Substitution

      In Cardano' solution, $$t$$ is defined as the difference of $$u$$ and\n $$v$$. If we substitute the value of $$v$$ (4) into (2), we get the\n equation. $$t=u-\\dfrac{20}{9u}$$. And then substitute the equation to the cubic equation\n $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$. This method is called Vieta's Substitution\n for solving a cubic equation, which simplied the Cardano' solution. The substitution\n expression can be obtained by the following formula directly.

      $$t=u-\\dfrac{p}{3u}$$

      Substitute the expression $$t=u-\\dfrac{20}{9u}$$ to the cubic equation

      $$\\Big(u-\\dfrac{20}{9u}\\Big)^3+\\dfrac{20}{3}\\Big(u-\\dfrac{20}{9u}\\Big)-\\dfrac{101}{27}=0$$

      Expand brackets and cancel the like terms

      $$u^3-\\cancel{\\dfrac{20}{3}u^2\\dfrac{1}{u}}+\\cancel{\\dfrac{400}{27}u\\dfrac{1}{u^2}}-\\dfrac{8000}{729}\\dfrac{1}{u^3}+\\cancel{\\dfrac{20}{3}u}-\\cancel{\\dfrac{400}{27}\\dfrac{1}{u}}-\\dfrac{101}{27}=0$$

      Then we get the same equation as (2)

      $$u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0$$

      The rest of the steps will be the same as those of Cardano's solution

      4. Euler's Solution

      $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

      Move the linear term and constant of (1) to its right hand side.\n We get the following form of the equation.

      $$t^3=-\\dfrac{20}{3}t+\\dfrac{101}{27} $$3

      Let the root of the cubic equation be the sum of two cubic roots

      $$t=\\sqrt[3]{r_1}+\\sqrt[3]{r_2} $$4

      in which $$r_1$$ and $$r_2$$ are two roots of a quadratic equation

      $$z^2-\\alpha z+ β=0 $$5

      Using Vieta's Formula, the following equations are established.

      $$r_1+r_2 = \\alpha \\quad \\text{and} \\quad r_1r_2 = β $$

      To determine $$\\alpha$$, $$β$$, cube both sides of the equation (4)

      $$t^3=3\\sqrt[3]{r_1r_2}(\\sqrt[3]{r_1}+\\sqrt[3]{r_2})+r_1+r_2 $$

      Substituting, the equation is simplified to

      $$t^3=3\\sqrt[3]{β}t+\\alpha $$

      Compare the cubic equation with (3), the following equations are established

      $$\\begin{cases} 3\\sqrt[3]{β}=-\\dfrac{20}{3}\\\\ \\alpha=\\dfrac{101}{27}\\\\ \\end{cases}$$

      Solving for $$β$$ gives

      $$β=-\\dfrac{8000}{729} $$

      So the quadratic equation (5) is determined as

      $$z^2-\\dfrac{101}{27}z-\\dfrac{8000}{729}=0$$6

      Solving the quadratic equation yields

      $$\\begin{cases} r_1=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\approx5.6746077738748\\\\ r_2=\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}\\approx-1.9338670331341\\\\ \\end{cases}$$

      Therefore, one of the roots of the cubic equation could be obtained from (4).

      $$t_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

      in decimals,

      $$t_1=0.53778143658824 $$

      However, since the cube root of a quantity has triple values,

      The other two roots could be determined as,

      $$t_2=\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

      $$t_3=\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

      Combining the real and imaginary parts\n results in the same result as that obtained by Cardano's solution.

      For the equation $$t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}$$, we have $$p=\\dfrac{20}{3}$$ and $$q = -\\dfrac{101}{27}$$

      Calculate the discriminant

      The nature of the roots are determined by the sign of the discriminant.

      $$\\begin{aligned} \\\\\\Delta&=\\dfrac{q^2}{4}+\\dfrac{p^3}{27}\\\\ & =\\dfrac{\\Big(-\\dfrac{101}{27}\\Big)^2}{4}+\\dfrac{\\Big(\\dfrac{20}{3}\\Big)^3}{27}\\\\ & =\\dfrac{10201}{2916}+\\dfrac{8000}{729}\\\\ & =\\dfrac{10201\\cdot 1+8000\\cdot 4}{2916}\\\\ & =14.472222222222\\\\ \\end{aligned}$$

      4.1 Use the root formula directly

      If the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.

      $$t_{1,2,3} =\\begin{cases} \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } +\\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}& \\\\ ω\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \\overline{ω} \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}&\\ \\\\ \\overline{ω}\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + ω\\cdotp \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }} \\end{cases}$$

      in which, $$ ω = \\dfrac{-1+i\\sqrt{3}}{2} $$ and $$ \\overline{ω} =\\dfrac{-1-i\\sqrt{3}}{2}$$

      Substitute the values of $$p, q$$ and $$\\Delta$$ which we have calculated. Then,

      $$\\begin{aligned} \\\\t_1&=\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{42201}{2916}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{42201}{2916}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

      If we denote

      $$R = -\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }$$

      $$\\overline{R} = -\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }$$

      then,

      $$\\sqrt[3]{R} = \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}$$, $$\\sqrt[3]{\\overline{R}} =\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}$$

      $$\\begin{aligned} \\\\t_2&= ω\\cdotp \\sqrt[3]{R}+ \\overline{ω} \\sqrt[3]{\\overline{R} }\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}( \\sqrt[3]{R} - \\sqrt[3]{\\overline{R} }) }{2} i\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

      $$\\begin{aligned} \\\\t_3&= \\overline{ω}\\cdotp \\sqrt[3]{R}+ ω\\cdotp \\sqrt[3]{\\overline{R}}\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}(- \\sqrt[3]{R} + \\sqrt[3]{\\overline{R} }) }{2}i \\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

      Roots of the general cubic equation

      Since $$x = t - \\dfrac{b}{3a}$$, substituting the values of $$t$$, $$a$$ and $$b$$ gives

      $$x_1 = t_1-\\dfrac{2}{3}$$

      $$x_2 = t_2-\\dfrac{2}{3}$$

      $$x_3 = t_3-\\dfrac{2}{3}$$

      5. Summary

      In summary, we have tried the method of cubic root formula to explore the solutions of the equation. The cubic equation $$x³ + 2x² + 8x + 1=0$$ is found to have one real root and two complex roots. Exact values and approximations are given below.

      $$\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}$$

      in decimal notation,

      $$\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}$$

      6. Graph for the function $$f(x) = x³ + 2x² + 8x + 1$$

      Since the discriminat is greater than zero, the curve of the cubic function $$f(x) = x³ + 2x² + 8x + 1$$ has one intersection point with the x-axis.

      \n\n\n\n\n\n\n
      \n
      \n\n
      \n

      More cubic equations

      \n\n
      \n
      \n
      \n", "content_list": [[{"type": "title", "raw_content": "

      Solve the cubic equation:

      ", "content": {"title_content": "Solve the cubic equation:", "level": "1"}}, {"type": "equation-interline", "raw_content": "

      $$x^3+2x^2+8x+1=0 $$

      ", "content": {"math_content": "x^3+2x^2+8x+1=0", "math_type": "latex", "by": "mathjax_mock"}}, {"type": "paragraph", "raw_content": "

      Quick Answer

      ", "content": [{"c": "Quick Answer", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      Since the discriminant

      ", "content": [{"c": "Since the discriminant", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\Delta >0$$

      ", "content": {"math_content": "\\Delta >0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      , the cubic equation has one real root and two conjugate complex roots.

      ", "content": [{"c": ", the cubic equation has one real root and two conjugate complex roots.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$ \\Delta=14.472222222222$$

      ", "content": {"math_content": "\\Delta=14.472222222222", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

      $$\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}$$

      ", "content": {"math_content": "\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      In decimals,

      ", "content": [{"c": "In decimals,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}$$

      ", "content": {"math_content": "\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Detailed Steps on Solution

      ", "content": [{"c": "Detailed Steps on Solution", "t": "text"}]}, {"type": "title", "raw_content": "

      1. Convert to depressed cubic equation

      ", "content": {"title_content": "1. Convert to depressed cubic equation", "level": "2"}}, {"type": "paragraph", "raw_content": "

      The idea is to convert general form of cubic equation

      ", "content": [{"c": "The idea is to convert general form of cubic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$ax^3+bx^2+cx+d = 0$$

      ", "content": {"math_content": "ax^3+bx^2+cx+d = 0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      to the form without quadratic term.

      ", "content": [{"c": "to the form without quadratic term.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t^3+pt+q = 0$$

      ", "content": {"math_content": "t^3+pt+q = 0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      By substituting

      ", "content": [{"c": "By substituting", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$x$$

      ", "content": {"math_content": "x", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      with

      ", "content": [{"c": "with", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t - \\dfrac{b}{3a}$$

      ", "content": {"math_content": "t - \\dfrac{b}{3a}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      , the general cubic equation could be transformed to

      ", "content": [{"c": ", the general cubic equation could be transformed to", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t^3+\\dfrac{3ac-b^2}{3a^2}t+\\dfrac{2b^3-9abc+27a^2d}{27a^3} = 0 $$

      ", "content": {"math_content": "t^3+\\dfrac{3ac-b^2}{3a^2}t+\\dfrac{2b^3-9abc+27a^2d}{27a^3} = 0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Compare with the depressed cubic equation. Then,

      ", "content": [{"c": "Compare with the depressed cubic equation. Then,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$p = \\dfrac{3ac-b^2}{3a^2}$$

      ", "content": {"math_content": "p = \\dfrac{3ac-b^2}{3a^2}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

      $$q = \\dfrac{2b^3-9abc+27a^2d}{27a^3} $$

      ", "content": {"math_content": "q = \\dfrac{2b^3-9abc+27a^2d}{27a^3}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Substitute the values of coefficients,

      ", "content": [{"c": "Substitute the values of coefficients,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$p, q$$

      ", "content": {"math_content": "p, q", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      is obtained as

      ", "content": [{"c": "is obtained as", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$p = \\dfrac{3\\cdot 1\\cdot 8-2^2}{3\\cdot 1^2}=\\dfrac{20}{3}$$

      ", "content": {"math_content": "p = \\dfrac{3\\cdot 1\\cdot 8-2^2}{3\\cdot 1^2}=\\dfrac{20}{3}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

      $$q = \\dfrac{2\\cdot 2^3-9\\cdot1\\cdot 2\\cdot 8+27\\cdot 1^2\\cdot1}{27\\cdot 1^3}=-\\dfrac{101}{27}$$

      ", "content": {"math_content": "q = \\dfrac{2\\cdot 2^3-9\\cdot1\\cdot 2\\cdot 8+27\\cdot 1^2\\cdot1}{27\\cdot 1^3}=-\\dfrac{101}{27}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

      Use the substitution to transform

      ", "content": {"title_content": "Use the substitution to transform", "level": "3"}}, {"type": "paragraph", "raw_content": "

      Let

      ", "content": [{"c": "Let", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$p$$

      ", "content": {"math_content": "p", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      and

      ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$q$$

      ", "content": {"math_content": "q", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      being the coefficient of the linean and constant terms, the depressed cubic equation is expressed as.

      ", "content": [{"c": "being the coefficient of the linean and constant terms, the depressed cubic equation is expressed as.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t^3 +pt+q=0$$

      ", "content": {"math_content": "t^3 +pt+q=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Let

      ", "content": [{"c": "Let", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$x=t-\\dfrac{2}{3}$$

      ", "content": {"math_content": "x=t-\\dfrac{2}{3}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      The cubic equation

      ", "content": [{"c": "The cubic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$x³ + 2x² + 8x + 1=0$$

      ", "content": {"math_content": "x³ + 2x² + 8x + 1=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      is transformed to

      ", "content": [{"c": "is transformed to", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

      ", "content": {"math_content": "t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

      2. Cardano's solution

      ", "content": {"title_content": "2. Cardano's solution", "level": "2"}}, {"type": "paragraph", "raw_content": "

      Let

      ", "content": [{"c": "Let", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t=u-v$$

      ", "content": {"math_content": "t=u-v", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Cube both sides and extract common factor from two middle terms after expanding the bracket.

      ", "content": [{"c": "Cube both sides and extract common factor from two middle terms after expanding the bracket.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\begin{aligned} \\\\t^3&=(u-v)^3\\\\ & =u^3-3u^2v+3uv^2-v^3\\\\ & =-3uv(u-v)+u^3-v^3\\\\ \\end{aligned}$$

      ", "content": {"math_content": "\\begin{aligned} \\\\t^3&=(u-v)^3\\\\ & =u^3-3u^2v+3uv^2-v^3\\\\ & =-3uv(u-v)+u^3-v^3\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Since

      ", "content": [{"c": "Since", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$u-v=t$$

      ", "content": {"math_content": "u-v=t", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      , substitution gives a linear term for the equation.\n Rearrange terms.

      ", "content": [{"c": ", substitution gives a linear term for the equation. Rearrange terms.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$x^3+3uvx-u^3+v^3=0$$

      ", "content": {"math_content": "x^3+3uvx-u^3+v^3=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Compare the cubic equation with the original one (1)

      ", "content": [{"c": "Compare the cubic equation with the original one (1)", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\begin{cases} 3uv=\\dfrac{20}{3}\\quad\\text{or}\\quad v=\\dfrac{20}{9u}\\\\ v^3-u^3=-\\dfrac{101}{27}\\\\ \\end{cases}$$

      ", "content": {"math_content": "\\begin{cases} 3uv=\\dfrac{20}{3}\\quad\\text{or}\\quad v=\\dfrac{20}{9u}\\\\ v^3-u^3=-\\dfrac{101}{27}\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

      $$v=\\dfrac{20}{9u}$$

      ", "content": {"math_content": "v=\\dfrac{20}{9u}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      gives relationship between the two variables. Substitute the value of

      ", "content": [{"c": "gives relationship between the two variables. Substitute the value of", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$v$$

      ", "content": {"math_content": "v", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      to the second equation

      ", "content": [{"c": "to the second equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\Big(\\dfrac{20}{9u}\\Big)^3-u^3=-\\dfrac{101}{27}$$

      ", "content": {"math_content": "\\Big(\\dfrac{20}{9u}\\Big)^3-u^3=-\\dfrac{101}{27}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Simplifying gives,

      ", "content": [{"c": "Simplifying gives,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0$$

      ", "content": {"math_content": "u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      2

      ", "content": [{"c": "2", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      Let

      ", "content": [{"c": "Let", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$m=u^3$$

      ", "content": {"math_content": "m=u^3", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      , then the equation is transformed to a quadratic equation in terms of

      ", "content": [{"c": ", then the equation is transformed to a quadratic equation in terms of", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$m$$

      ", "content": {"math_content": "m", "math_type": "latex", "by": "mathjax_mock"}}, {"type": "paragraph", "raw_content": "

      .\n Once the value of

      ", "content": [{"c": ". Once the value of", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$m$$

      ", "content": {"math_content": "m", "math_type": "latex", "by": "mathjax_mock"}}, {"type": "paragraph", "raw_content": "

      is determined,

      ", "content": [{"c": "is determined,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$v^3$$

      ", "content": {"math_content": "v^3", "math_type": "latex", "by": "mathjax_mock"}}, {"type": "paragraph", "raw_content": "

      could be determined by

      ", "content": [{"c": "could be determined by", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$v^3=-\\dfrac{101}{27}+u^3$$

      ", "content": {"math_content": "v^3=-\\dfrac{101}{27}+u^3", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      .

      ", "content": [{"c": ".", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$m^2-\\dfrac{101}{27}m-\\dfrac{8000}{729}=0$$

      ", "content": {"math_content": "m^2-\\dfrac{101}{27}m-\\dfrac{8000}{729}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Sovling the quadratic euqation will give two roots (some may be equal). Here we only cosider\n one case with positive sign before the square root radical since the negative case will produce the same result.

      ", "content": [{"c": "Sovling the quadratic euqation will give two roots (some may be equal). Here we only cosider one case with positive sign before the square root radical since the negative case will produce the same result.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\begin{aligned} \\\\u^3=m&=\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\Big(-\\dfrac{101}{27}^2\\Big)-4\\cdot \\Big(-\\dfrac{8000}{729}\\Big)}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{10201}{729}+\\dfrac{32000}{729}}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{521}{9}}\\\\ & =\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}$$

      ", "content": {"math_content": "\\begin{aligned} \\\\u^3=m&=\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\Big(-\\dfrac{101}{27}^2\\Big)-4\\cdot \\Big(-\\dfrac{8000}{729}\\Big)}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{10201}{729}+\\dfrac{32000}{729}}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{521}{9}}\\\\ & =\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

      $$v^3$$

      ", "content": {"math_content": "v^3", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      can be determined by the equation we deduced

      ", "content": [{"c": "can be determined by the equation we deduced", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$v^3-u^3=-\\dfrac{101}{27}$$

      ", "content": {"math_content": "v^3-u^3=-\\dfrac{101}{27}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      . Then,

      ", "content": [{"c": ". Then,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\begin{aligned} \\\\v^3&=-\\dfrac{101}{27}+u^3\\\\ & =-\\dfrac{101}{27}+\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ & =-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}$$

      ", "content": {"math_content": "\\begin{aligned} \\\\v^3&=-\\dfrac{101}{27}+u^3\\\\ & =-\\dfrac{101}{27}+\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ & =-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Now we have,

      ", "content": [{"c": "Now we have,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$u^3=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}$$

      ", "content": {"math_content": "u^3=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      and

      ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$v^3=-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}$$

      ", "content": {"math_content": "v^3=-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Evaluating the simplest cubic equation

      ", "content": [{"c": "Evaluating the simplest cubic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$x^3-A=0$$

      ", "content": {"math_content": "x^3-A=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      ,\n it has 3 roots, in which the first root is a real number . The second and third are\n expressed in the product of cubic root of unity and the first one.

      ", "content": [{"c": ", it has 3 roots, in which the first root is a real number . The second and third are expressed in the product of cubic root of unity and the first one.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      If

      ", "content": [{"c": "If", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$ω = \\dfrac{-1+i\\sqrt{3}}{2}$$

      ", "content": {"math_content": "ω = \\dfrac{-1+i\\sqrt{3}}{2}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      , then its reciprocal is equal to its conjugate,

      ", "content": [{"c": ", then its reciprocal is equal to its conjugate,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\dfrac{1}{ω}=\\overline{ω}$$

      ", "content": {"math_content": "\\dfrac{1}{ω}=\\overline{ω}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      .

      ", "content": [{"c": ".", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\begin{cases} r_1=\\sqrt[3]{A}\\\\ r_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ r_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ \\end{cases}$$

      ", "content": {"math_content": "\\begin{cases} r_1=\\sqrt[3]{A}\\\\ r_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ r_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Similary, taking cubic root for

      ", "content": [{"c": "Similary, taking cubic root for", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$u^3$$

      ", "content": {"math_content": "u^3", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      and

      ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$v^3$$

      ", "content": {"math_content": "v^3", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      also gives 3 roots.

      ", "content": [{"c": "also gives 3 roots.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\begin{cases} u_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}$$

      ", "content": {"math_content": "\\begin{cases} u_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      For

      ", "content": [{"c": "For", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$v_2$$

      ", "content": {"math_content": "v_2", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      and

      ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$v_3$$

      ", "content": {"math_content": "v_3", "math_type": "latex", "by": "mathjax_mock"}}, {"type": "paragraph", "raw_content": "

      , the complex numbers before radicals are the conjugates of\n those for

      ", "content": [{"c": ", the complex numbers before radicals are the conjugates of those for", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$u_2$$

      ", "content": {"math_content": "u_2", "math_type": "latex", "by": "mathjax_mock"}}, {"type": "paragraph", "raw_content": "

      and

      ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$u_3$$

      ", "content": {"math_content": "u_3", "math_type": "latex", "by": "mathjax_mock"}}, {"type": "paragraph", "raw_content": "

      , which can be verified by the reciprocal property\n of the cubic root of unity from the equation

      ", "content": [{"c": ", which can be verified by the reciprocal property of the cubic root of unity from the equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$v=\\dfrac{20}{9u}$$

      ", "content": {"math_content": "v=\\dfrac{20}{9u}", "math_type": "latex", "by": "mathjax_mock"}}, {"type": "paragraph", "raw_content": "

      . The radicand can be taken as the\n negative conjugate of that in

      ", "content": [{"c": ". The radicand can be taken as the negative conjugate of that in", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$u_1$$

      ", "content": {"math_content": "u_1", "math_type": "latex", "by": "mathjax_mock"}}, {"type": "paragraph", "raw_content": "

      ,

      ", "content": [{"c": ",", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$u_2$$

      ", "content": {"math_content": "u_2", "math_type": "latex", "by": "mathjax_mock"}}, {"type": "paragraph", "raw_content": "

      and

      ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$u_3$$

      ", "content": {"math_content": "u_3", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      , which is the same in value.

      ", "content": [{"c": ", which is the same in value.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\begin{cases} v_1=\\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_2=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_3=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}$$

      ", "content": {"math_content": "\\begin{cases} v_1=\\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_2=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_3=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Verification for the redicand in

      ", "content": [{"c": "Verification for the redicand in", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$v$$

      ", "content": {"math_content": "v", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      .

      ", "content": [{"c": ".", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\begin{aligned} \\\\v_1&=\\dfrac{20}{9u_1}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\Big(\\dfrac{101}{54}\\Big)^2-\\Big(\\dfrac{\\sqrt{521}}{6}\\Big)^2}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{10201}{2916}-\\dfrac{521}{36}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{-1\\cdot 20^3}{9^3}}}\\\\ & =-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

      ", "content": {"math_content": "\\begin{aligned} \\\\v_1&=\\dfrac{20}{9u_1}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\Big(\\dfrac{101}{54}\\Big)^2-\\Big(\\dfrac{\\sqrt{521}}{6}\\Big)^2}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{10201}{2916}-\\dfrac{521}{36}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{-1\\cdot 20^3}{9^3}}}\\\\ & =-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Since

      ", "content": [{"c": "Since", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$x=u-v$$

      ", "content": {"math_content": "x=u-v", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      , combining the real and imaginary parts gives\n 3 results for

      ", "content": [{"c": ", combining the real and imaginary parts gives 3 results for", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t$$

      ", "content": {"math_content": "t", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

      $$\\begin{aligned} \\\\t_1&=u_1-v_1\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

      ", "content": {"math_content": "\\begin{aligned} \\\\t_1&=u_1-v_1\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

      $$\\begin{aligned} \\\\t_2&=u_2-v_2\\\\ & =\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

      ", "content": {"math_content": "\\begin{aligned} \\\\t_2&=u_2-v_2\\\\ & =\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

      $$\\begin{aligned} \\\\t_3&=u_3-v_3\\\\ & =\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

      ", "content": {"math_content": "\\begin{aligned} \\\\t_3&=u_3-v_3\\\\ & =\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

      3. Vieta's Substitution

      ", "content": {"title_content": "3. Vieta's Substitution", "level": "2"}}, {"type": "paragraph", "raw_content": "

      In Cardano' solution,

      ", "content": [{"c": "In Cardano' solution,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t$$

      ", "content": {"math_content": "t", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      is defined as the difference of

      ", "content": [{"c": "is defined as the difference of", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$u$$

      ", "content": {"math_content": "u", "math_type": "latex", "by": "mathjax_mock"}}, {"type": "paragraph", "raw_content": "

      and\n

      ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$v$$

      ", "content": {"math_content": "v", "math_type": "latex", "by": "mathjax_mock"}}, {"type": "paragraph", "raw_content": "

      . If we substitute the value of

      ", "content": [{"c": ". If we substitute the value of", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$v$$

      ", "content": {"math_content": "v", "math_type": "latex", "by": "mathjax_mock"}}, {"type": "paragraph", "raw_content": "

      (4) into (2), we get the\n equation.

      ", "content": [{"c": "(4) into (2), we get the equation.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t=u-\\dfrac{20}{9u}$$

      ", "content": {"math_content": "t=u-\\dfrac{20}{9u}", "math_type": "latex", "by": "mathjax_mock"}}, {"type": "paragraph", "raw_content": "

      . And then substitute the equation to the cubic equation\n

      ", "content": [{"c": ". And then substitute the equation to the cubic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

      ", "content": {"math_content": "t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      . This method is called Vieta's Substitution\n for solving a cubic equation, which simplied the Cardano' solution. The substitution\n expression can be obtained by the following formula directly.

      ", "content": [{"c": ". This method is called Vieta's Substitution for solving a cubic equation, which simplied the Cardano' solution. The substitution expression can be obtained by the following formula directly.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t=u-\\dfrac{p}{3u}$$

      ", "content": {"math_content": "t=u-\\dfrac{p}{3u}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Substitute the expression

      ", "content": [{"c": "Substitute the expression", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t=u-\\dfrac{20}{9u}$$

      ", "content": {"math_content": "t=u-\\dfrac{20}{9u}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      to the cubic equation

      ", "content": [{"c": "to the cubic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\Big(u-\\dfrac{20}{9u}\\Big)^3+\\dfrac{20}{3}\\Big(u-\\dfrac{20}{9u}\\Big)-\\dfrac{101}{27}=0$$

      ", "content": {"math_content": "\\Big(u-\\dfrac{20}{9u}\\Big)^3+\\dfrac{20}{3}\\Big(u-\\dfrac{20}{9u}\\Big)-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Expand brackets and cancel the like terms

      ", "content": [{"c": "Expand brackets and cancel the like terms", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$u^3-\\cancel{\\dfrac{20}{3}u^2\\dfrac{1}{u}}+\\cancel{\\dfrac{400}{27}u\\dfrac{1}{u^2}}-\\dfrac{8000}{729}\\dfrac{1}{u^3}+\\cancel{\\dfrac{20}{3}u}-\\cancel{\\dfrac{400}{27}\\dfrac{1}{u}}-\\dfrac{101}{27}=0$$

      ", "content": {"math_content": "u^3-\\cancel{\\dfrac{20}{3}u^2\\dfrac{1}{u}}+\\cancel{\\dfrac{400}{27}u\\dfrac{1}{u^2}}-\\dfrac{8000}{729}\\dfrac{1}{u^3}+\\cancel{\\dfrac{20}{3}u}-\\cancel{\\dfrac{400}{27}\\dfrac{1}{u}}-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Then we get the same equation as (2)

      ", "content": [{"c": "Then we get the same equation as (2)", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0$$

      ", "content": {"math_content": "u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      The rest of the steps will be the same as those of Cardano's solution

      ", "content": [{"c": "The rest of the steps will be the same as those of Cardano's solution", "t": "text"}]}, {"type": "title", "raw_content": "

      4. Euler's Solution

      ", "content": {"title_content": "4. Euler's Solution", "level": "2"}}, {"type": "equation-interline", "raw_content": "

      $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

      ", "content": {"math_content": "t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0", "math_type": "latex", "by": "mathjax_mock"}}, {"type": "paragraph", "raw_content": "

      Move the linear term and constant of (1) to its right hand side.\n We get the following form of the equation.

      ", "content": [{"c": "Move the linear term and constant of (1) to its right hand side. We get the following form of the equation.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t^3=-\\dfrac{20}{3}t+\\dfrac{101}{27} $$

      ", "content": {"math_content": "t^3=-\\dfrac{20}{3}t+\\dfrac{101}{27}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      3

      ", "content": [{"c": "3", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      Let the root of the cubic equation be the sum of two cubic roots

      ", "content": [{"c": "Let the root of the cubic equation be the sum of two cubic roots", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t=\\sqrt[3]{r_1}+\\sqrt[3]{r_2} $$

      ", "content": {"math_content": "t=\\sqrt[3]{r_1}+\\sqrt[3]{r_2}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      4

      ", "content": [{"c": "4", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      in which

      ", "content": [{"c": "in which", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$r_1$$

      ", "content": {"math_content": "r_1", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      and

      ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$r_2$$

      ", "content": {"math_content": "r_2", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      are two roots of a quadratic equation

      ", "content": [{"c": "are two roots of a quadratic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$z^2-\\alpha z+ β=0 $$

      ", "content": {"math_content": "z^2-\\alpha z+ β=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      5

      ", "content": [{"c": "5", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      Using Vieta's Formula, the following equations are established.

      ", "content": [{"c": "Using Vieta's Formula, the following equations are established.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$r_1+r_2 = \\alpha \\quad \\text{and} \\quad r_1r_2 = β $$

      ", "content": {"math_content": "r_1+r_2 = \\alpha \\quad \\text{and} \\quad r_1r_2 = β", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      To determine

      ", "content": [{"c": "To determine", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\alpha$$

      ", "content": {"math_content": "\\alpha", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      ,

      ", "content": [{"c": ",", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$β$$

      ", "content": {"math_content": "β", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      , cube both sides of the equation (4)

      ", "content": [{"c": ", cube both sides of the equation (4)", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t^3=3\\sqrt[3]{r_1r_2}(\\sqrt[3]{r_1}+\\sqrt[3]{r_2})+r_1+r_2 $$

      ", "content": {"math_content": "t^3=3\\sqrt[3]{r_1r_2}(\\sqrt[3]{r_1}+\\sqrt[3]{r_2})+r_1+r_2", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Substituting, the equation is simplified to

      ", "content": [{"c": "Substituting, the equation is simplified to", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t^3=3\\sqrt[3]{β}t+\\alpha $$

      ", "content": {"math_content": "t^3=3\\sqrt[3]{β}t+\\alpha", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Compare the cubic equation with (3), the following equations are established

      ", "content": [{"c": "Compare the cubic equation with (3), the following equations are established", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\begin{cases} 3\\sqrt[3]{β}=-\\dfrac{20}{3}\\\\ \\alpha=\\dfrac{101}{27}\\\\ \\end{cases}$$

      ", "content": {"math_content": "\\begin{cases} 3\\sqrt[3]{β}=-\\dfrac{20}{3}\\\\ \\alpha=\\dfrac{101}{27}\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Solving for

      ", "content": [{"c": "Solving for", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$β$$

      ", "content": {"math_content": "β", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      gives

      ", "content": [{"c": "gives", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$β=-\\dfrac{8000}{729} $$

      ", "content": {"math_content": "β=-\\dfrac{8000}{729}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      So the quadratic equation (5) is determined as

      ", "content": [{"c": "So the quadratic equation (5) is determined as", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$z^2-\\dfrac{101}{27}z-\\dfrac{8000}{729}=0$$

      ", "content": {"math_content": "z^2-\\dfrac{101}{27}z-\\dfrac{8000}{729}=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      6

      ", "content": [{"c": "6", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      Solving the quadratic equation yields

      ", "content": [{"c": "Solving the quadratic equation yields", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\begin{cases} r_1=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\approx5.6746077738748\\\\ r_2=\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}\\approx-1.9338670331341\\\\ \\end{cases}$$

      ", "content": {"math_content": "\\begin{cases} r_1=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\approx5.6746077738748\\\\ r_2=\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}\\approx-1.9338670331341\\\\ \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Therefore, one of the roots of the cubic equation could be obtained from (4).

      ", "content": [{"c": "Therefore, one of the roots of the cubic equation could be obtained from (4).", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

      ", "content": {"math_content": "t_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      in decimals,

      ", "content": [{"c": "in decimals,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t_1=0.53778143658824 $$

      ", "content": {"math_content": "t_1=0.53778143658824", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      However, since the cube root of a quantity has triple values,

      ", "content": [{"c": "However, since the cube root of a quantity has triple values,", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      The other two roots could be determined as,

      ", "content": [{"c": "The other two roots could be determined as,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t_2=\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

      ", "content": {"math_content": "t_2=\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

      $$t_3=\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

      ", "content": {"math_content": "t_3=\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Combining the real and imaginary parts\n results in the same result as that obtained by Cardano's solution.

      ", "content": [{"c": "Combining the real and imaginary parts results in the same result as that obtained by Cardano's solution.", "t": "text"}]}, {"type": "paragraph", "raw_content": "

      For the equation

      ", "content": [{"c": "For the equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}$$

      ", "content": {"math_content": "t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      , we have

      ", "content": [{"c": ", we have", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$p=\\dfrac{20}{3}$$

      ", "content": {"math_content": "p=\\dfrac{20}{3}", "math_type": "latex", "by": "mathjax_mock"}}, {"type": "paragraph", "raw_content": "

      and

      ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$q = -\\dfrac{101}{27}$$

      ", "content": {"math_content": "q = -\\dfrac{101}{27}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

      Calculate the discriminant

      ", "content": {"title_content": "Calculate the discriminant", "level": "3"}}, {"type": "paragraph", "raw_content": "

      The nature of the roots are determined by the sign of the discriminant.

      ", "content": [{"c": "The nature of the roots are determined by the sign of the discriminant.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\begin{aligned} \\\\\\Delta&=\\dfrac{q^2}{4}+\\dfrac{p^3}{27}\\\\ & =\\dfrac{\\Big(-\\dfrac{101}{27}\\Big)^2}{4}+\\dfrac{\\Big(\\dfrac{20}{3}\\Big)^3}{27}\\\\ & =\\dfrac{10201}{2916}+\\dfrac{8000}{729}\\\\ & =\\dfrac{10201\\cdot 1+8000\\cdot 4}{2916}\\\\ & =14.472222222222\\\\ \\end{aligned}$$

      ", "content": {"math_content": "\\begin{aligned} \\\\\\Delta&=\\dfrac{q^2}{4}+\\dfrac{p^3}{27}\\\\ & =\\dfrac{\\Big(-\\dfrac{101}{27}\\Big)^2}{4}+\\dfrac{\\Big(\\dfrac{20}{3}\\Big)^3}{27}\\\\ & =\\dfrac{10201}{2916}+\\dfrac{8000}{729}\\\\ & =\\dfrac{10201\\cdot 1+8000\\cdot 4}{2916}\\\\ & =14.472222222222\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

      4.1 Use the root formula directly

      ", "content": {"title_content": "4.1 Use the root formula directly", "level": "3"}}, {"type": "paragraph", "raw_content": "

      If the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.

      ", "content": [{"c": "If the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t_{1,2,3} =\\begin{cases} \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } +\\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}& \\\\ ω\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \\overline{ω} \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}&\\ \\\\ \\overline{ω}\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + ω\\cdotp \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }} \\end{cases}$$

      ", "content": {"math_content": "t_{1,2,3} =\\begin{cases} \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } +\\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}& \\\\ ω\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \\overline{ω} \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}&\\ \\\\ \\overline{ω}\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + ω\\cdotp \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }} \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      in which,

      ", "content": [{"c": "in which,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$ ω = \\dfrac{-1+i\\sqrt{3}}{2} $$

      ", "content": {"math_content": "ω = \\dfrac{-1+i\\sqrt{3}}{2}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      and

      ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$ \\overline{ω} =\\dfrac{-1-i\\sqrt{3}}{2}$$

      ", "content": {"math_content": "\\overline{ω} =\\dfrac{-1-i\\sqrt{3}}{2}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      Substitute the values of

      ", "content": [{"c": "Substitute the values of", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$p, q$$

      ", "content": {"math_content": "p, q", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      and

      ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\Delta$$

      ", "content": {"math_content": "\\Delta", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      which we have calculated. Then,

      ", "content": [{"c": "which we have calculated. Then,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\begin{aligned} \\\\t_1&=\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{42201}{2916}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{42201}{2916}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

      ", "content": {"math_content": "\\begin{aligned} \\\\t_1&=\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{42201}{2916}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{42201}{2916}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      If we denote

      ", "content": [{"c": "If we denote", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$R = -\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }$$

      ", "content": {"math_content": "R = -\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

      $$\\overline{R} = -\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }$$

      ", "content": {"math_content": "\\overline{R} = -\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      then,

      ", "content": [{"c": "then,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\sqrt[3]{R} = \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}$$

      ", "content": {"math_content": "\\sqrt[3]{R} = \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      ,

      ", "content": [{"c": ",", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\sqrt[3]{\\overline{R}} =\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}$$

      ", "content": {"math_content": "\\sqrt[3]{\\overline{R}} =\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

      $$\\begin{aligned} \\\\t_2&= ω\\cdotp \\sqrt[3]{R}+ \\overline{ω} \\sqrt[3]{\\overline{R} }\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}( \\sqrt[3]{R} - \\sqrt[3]{\\overline{R} }) }{2} i\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

      ", "content": {"math_content": "\\begin{aligned} \\\\t_2&= ω\\cdotp \\sqrt[3]{R}+ \\overline{ω} \\sqrt[3]{\\overline{R} }\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}( \\sqrt[3]{R} - \\sqrt[3]{\\overline{R} }) }{2} i\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

      $$\\begin{aligned} \\\\t_3&= \\overline{ω}\\cdotp \\sqrt[3]{R}+ ω\\cdotp \\sqrt[3]{\\overline{R}}\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}(- \\sqrt[3]{R} + \\sqrt[3]{\\overline{R} }) }{2}i \\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

      ", "content": {"math_content": "\\begin{aligned} \\\\t_3&= \\overline{ω}\\cdotp \\sqrt[3]{R}+ ω\\cdotp \\sqrt[3]{\\overline{R}}\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}(- \\sqrt[3]{R} + \\sqrt[3]{\\overline{R} }) }{2}i \\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

      Roots of the general cubic equation

      ", "content": {"title_content": "Roots of the general cubic equation", "level": "2"}}, {"type": "paragraph", "raw_content": "

      Since

      ", "content": [{"c": "Since", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$x = t - \\dfrac{b}{3a}$$

      ", "content": {"math_content": "x = t - \\dfrac{b}{3a}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      , substituting the values of

      ", "content": [{"c": ", substituting the values of", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$t$$

      ", "content": {"math_content": "t", "math_type": "latex", "by": "mathjax_mock"}}, {"type": "paragraph", "raw_content": "

      ,

      ", "content": [{"c": ",", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$a$$

      ", "content": {"math_content": "a", "math_type": "latex", "by": "mathjax_mock"}}, {"type": "paragraph", "raw_content": "

      and

      ", "content": [{"c": "and", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$b$$

      ", "content": {"math_content": "b", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      gives

      ", "content": [{"c": "gives", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$x_1 = t_1-\\dfrac{2}{3}$$

      ", "content": {"math_content": "x_1 = t_1-\\dfrac{2}{3}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

      $$x_2 = t_2-\\dfrac{2}{3}$$

      ", "content": {"math_content": "x_2 = t_2-\\dfrac{2}{3}", "math_type": "latex", "by": "katex"}}, {"type": "equation-interline", "raw_content": "

      $$x_3 = t_3-\\dfrac{2}{3}$$

      ", "content": {"math_content": "x_3 = t_3-\\dfrac{2}{3}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

      5. Summary

      ", "content": {"title_content": "5. Summary", "level": "2"}}, {"type": "paragraph", "raw_content": "

      In summary, we have tried the method of cubic root formula to explore the solutions of the equation. The cubic equation

      ", "content": [{"c": "In summary, we have tried the method of cubic root formula to explore the solutions of the equation. The cubic equation", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$x³ + 2x² + 8x + 1=0$$

      ", "content": {"math_content": "x³ + 2x² + 8x + 1=0", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      is found to have one real root and two complex roots. Exact values and approximations are given below.

      ", "content": [{"c": "is found to have one real root and two complex roots. Exact values and approximations are given below.", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}$$

      ", "content": {"math_content": "\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      in decimal notation,

      ", "content": [{"c": "in decimal notation,", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}$$

      ", "content": {"math_content": "\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}", "math_type": "latex", "by": "katex"}}, {"type": "title", "raw_content": "

      6. Graph for the function

      ", "content": {"title_content": "6. Graph for the function", "level": "2"}}, {"type": "equation-interline", "raw_content": "

      $$f(x) = x³ + 2x² + 8x + 1$$

      ", "content": {"math_content": "f(x) = x³ + 2x² + 8x + 1", "math_type": "latex", "by": "mathjax_mock"}}, {"type": "paragraph", "raw_content": "

      Since the discriminat is greater than zero, the curve of the cubic function

      ", "content": [{"c": "Since the discriminat is greater than zero, the curve of the cubic function", "t": "text"}]}, {"type": "equation-interline", "raw_content": "

      $$f(x) = x³ + 2x² + 8x + 1$$

      ", "content": {"math_content": "f(x) = x³ + 2x² + 8x + 1", "math_type": "latex", "by": "katex"}}, {"type": "paragraph", "raw_content": "

      has one intersection point with the x-axis.

      ", "content": [{"c": "has one intersection point with the x-axis.", "t": "text"}]}, {"type": "title", "raw_content": "

      More cubic equations

      ", "content": {"title_content": "More cubic equations", "level": "2"}}]], "html": "\n\n\n\n\n\nSolve x^3+2x^2+8x+1=0 | Uniteasy.com\n\n\n\n\n\n\n\n\n\n\n\n \n
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      \n

      Solve the cubic equation:

      \n

      $$x^3+2x^2+8x+1=0 $$

      \n

      Quick Answer

      \n

      Since the discriminant $$\\Delta >0$$, the cubic equation has one real root and two conjugate complex roots.

      $$ \\Delta=14.472222222222$$

      $$\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}$$

      In decimals,

      $$\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}$$

      Detailed Steps on Solution

      1. Convert to depressed cubic equation

      The idea is to convert general form of cubic equation

      $$ax^3+bx^2+cx+d = 0$$

      to the form without quadratic term.

      $$t^3+pt+q = 0$$

      By substituting $$x$$ with $$t - \\dfrac{b}{3a}$$, the general cubic equation could be transformed to

      $$t^3+\\dfrac{3ac-b^2}{3a^2}t+\\dfrac{2b^3-9abc+27a^2d}{27a^3} = 0 $$

      Compare with the depressed cubic equation. Then,

      $$p = \\dfrac{3ac-b^2}{3a^2}$$

      $$q = \\dfrac{2b^3-9abc+27a^2d}{27a^3} $$

      Substitute the values of coefficients, $$p, q$$ is obtained as

      $$p = \\dfrac{3\\cdot 1\\cdot 8-2^2}{3\\cdot 1^2}=\\dfrac{20}{3}$$

      $$q = \\dfrac{2\\cdot 2^3-9\\cdot1\\cdot 2\\cdot 8+27\\cdot 1^2\\cdot1}{27\\cdot 1^3}=-\\dfrac{101}{27}$$

      Use the substitution to transform

      Let $$p$$ and $$q$$ being the coefficient of the linean and constant terms, the depressed cubic equation is expressed as.

      $$t^3 +pt+q=0$$

      Let $$x=t-\\dfrac{2}{3}$$

      The cubic equation $$x³ + 2x² + 8x + 1=0$$ is transformed to

      $$t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

      2. Cardano's solution

      Let $$t=u-v$$

      Cube both sides and extract common factor from two middle terms after expanding the bracket.

      $$\\begin{aligned} \\\\t^3&=(u-v)^3\\\\ & =u^3-3u^2v+3uv^2-v^3\\\\ & =-3uv(u-v)+u^3-v^3\\\\ \\end{aligned}$$

      Since $$u-v=t$$, substitution gives a linear term for the equation.\n Rearrange terms.

      $$x^3+3uvx-u^3+v^3=0$$

      Compare the cubic equation with the original one (1)

      $$\\begin{cases} 3uv=\\dfrac{20}{3}\\quad\\text{or}\\quad v=\\dfrac{20}{9u}\\\\ v^3-u^3=-\\dfrac{101}{27}\\\\ \\end{cases}$$

      $$v=\\dfrac{20}{9u}$$ gives relationship between the two variables. Substitute the value of $$v$$ to the second equation

      $$\\Big(\\dfrac{20}{9u}\\Big)^3-u^3=-\\dfrac{101}{27}$$

      Simplifying gives,

      $$u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0$$2

      Let $$m=u^3$$, then the equation is transformed to a quadratic equation in terms of $$m$$.\n Once the value of $$m$$ is determined, $$v^3$$ could be determined by $$v^3=-\\dfrac{101}{27}+u^3$$.

      $$m^2-\\dfrac{101}{27}m-\\dfrac{8000}{729}=0$$

      Sovling the quadratic euqation will give two roots (some may be equal). Here we only cosider\n one case with positive sign before the square root radical since the negative case will produce the same result.

      $$\\begin{aligned} \\\\u^3=m&=\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\Big(-\\dfrac{101}{27}^2\\Big)-4\\cdot \\Big(-\\dfrac{8000}{729}\\Big)}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{10201}{729}+\\dfrac{32000}{729}}\\\\ & =\\dfrac{101}{54}+\\dfrac{1}{2}\\sqrt{\\dfrac{521}{9}}\\\\ & =\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}$$

      $$v^3$$ can be determined by the equation we deduced $$v^3-u^3=-\\dfrac{101}{27}$$. Then,

      $$\\begin{aligned} \\\\v^3&=-\\dfrac{101}{27}+u^3\\\\ & =-\\dfrac{101}{27}+\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ & =-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\\\ \\end{aligned}$$

      Now we have,

      $$u^3=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}$$ and $$v^3=-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}$$

      Evaluating the simplest cubic equation $$x^3-A=0$$,\n it has 3 roots, in which the first root is a real number . The second and third are\n expressed in the product of cubic root of unity and the first one.

      If $$ω = \\dfrac{-1+i\\sqrt{3}}{2}$$, then its reciprocal is equal to its conjugate, $$\\dfrac{1}{ω}=\\overline{ω}$$.

      $$\\begin{cases} r_1=\\sqrt[3]{A}\\\\ r_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ r_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{A}\\\\ \\end{cases}$$

      Similary, taking cubic root for $$u^3$$ and $$v^3$$ also gives 3 roots.

      $$\\begin{cases} u_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_2=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ u_3=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}$$

      For $$v_2$$ and $$v_3$$, the complex numbers before radicals are the conjugates of\n those for $$u_2$$ and $$u_3$$, which can be verified by the reciprocal property\n of the cubic root of unity from the equation $$v=\\dfrac{20}{9u}$$. The radicand can be taken as the\n negative conjugate of that in $$u_1$$, $$u_2$$ and $$u_3$$, which is the same in value.

      $$\\begin{cases} v_1=\\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_2=\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ v_3=\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{cases}$$

      Verification for the redicand in $$v$$.

      $$\\begin{aligned} \\\\v_1&=\\dfrac{20}{9u_1}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{1}{\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\Big(\\dfrac{101}{54}\\Big)^2-\\Big(\\dfrac{\\sqrt{521}}{6}\\Big)^2}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{10201}{2916}-\\dfrac{521}{36}}}\\\\ & =\\dfrac{20}{9}\\cdot \\dfrac{\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}}{\\sqrt[3]{\\dfrac{-1\\cdot 20^3}{9^3}}}\\\\ & =-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

      Since $$x=u-v$$, combining the real and imaginary parts gives\n 3 results for $$t$$

      $$\\begin{aligned} \\\\t_1&=u_1-v_1\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

      $$\\begin{aligned} \\\\t_2&=u_2-v_2\\\\ & =\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

      $$\\begin{aligned} \\\\t_3&=u_3-v_3\\\\ & =\\dfrac{-1-i\\sqrt{3}}{2}\\cdot \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\Big(\\dfrac{-1+i\\sqrt{3}}{2}\\cdot \\sqrt[3]{-\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

      3. Vieta's Substitution

      In Cardano' solution, $$t$$ is defined as the difference of $$u$$ and\n $$v$$. If we substitute the value of $$v$$ (4) into (2), we get the\n equation. $$t=u-\\dfrac{20}{9u}$$. And then substitute the equation to the cubic equation\n $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$. This method is called Vieta's Substitution\n for solving a cubic equation, which simplied the Cardano' solution. The substitution\n expression can be obtained by the following formula directly.

      $$t=u-\\dfrac{p}{3u}$$

      Substitute the expression $$t=u-\\dfrac{20}{9u}$$ to the cubic equation

      $$\\Big(u-\\dfrac{20}{9u}\\Big)^3+\\dfrac{20}{3}\\Big(u-\\dfrac{20}{9u}\\Big)-\\dfrac{101}{27}=0$$

      Expand brackets and cancel the like terms

      $$u^3-\\cancel{\\dfrac{20}{3}u^2\\dfrac{1}{u}}+\\cancel{\\dfrac{400}{27}u\\dfrac{1}{u^2}}-\\dfrac{8000}{729}\\dfrac{1}{u^3}+\\cancel{\\dfrac{20}{3}u}-\\cancel{\\dfrac{400}{27}\\dfrac{1}{u}}-\\dfrac{101}{27}=0$$

      Then we get the same equation as (2)

      $$u^3-\\dfrac{8000}{729}\\dfrac{1}{u^3}-\\dfrac{101}{27}=0$$

      The rest of the steps will be the same as those of Cardano's solution

      4. Euler's Solution

      $$t^3+\\dfrac{20}{3}t-\\dfrac{101}{27}=0$$

      Move the linear term and constant of (1) to its right hand side.\n We get the following form of the equation.

      $$t^3=-\\dfrac{20}{3}t+\\dfrac{101}{27} $$3

      Let the root of the cubic equation be the sum of two cubic roots

      $$t=\\sqrt[3]{r_1}+\\sqrt[3]{r_2} $$4

      in which $$r_1$$ and $$r_2$$ are two roots of a quadratic equation

      $$z^2-\\alpha z+ β=0 $$5

      Using Vieta's Formula, the following equations are established.

      $$r_1+r_2 = \\alpha \\quad \\text{and} \\quad r_1r_2 = β $$

      To determine $$\\alpha$$, $$β$$, cube both sides of the equation (4)

      $$t^3=3\\sqrt[3]{r_1r_2}(\\sqrt[3]{r_1}+\\sqrt[3]{r_2})+r_1+r_2 $$

      Substituting, the equation is simplified to

      $$t^3=3\\sqrt[3]{β}t+\\alpha $$

      Compare the cubic equation with (3), the following equations are established

      $$\\begin{cases} 3\\sqrt[3]{β}=-\\dfrac{20}{3}\\\\ \\alpha=\\dfrac{101}{27}\\\\ \\end{cases}$$

      Solving for $$β$$ gives

      $$β=-\\dfrac{8000}{729} $$

      So the quadratic equation (5) is determined as

      $$z^2-\\dfrac{101}{27}z-\\dfrac{8000}{729}=0$$6

      Solving the quadratic equation yields

      $$\\begin{cases} r_1=\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}\\approx5.6746077738748\\\\ r_2=\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}\\approx-1.9338670331341\\\\ \\end{cases}$$

      Therefore, one of the roots of the cubic equation could be obtained from (4).

      $$t_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

      in decimals,

      $$t_1=0.53778143658824 $$

      However, since the cube root of a quantity has triple values,

      The other two roots could be determined as,

      $$t_2=\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

      $$t_3=\\dfrac{-1-i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\dfrac{-1+i\\sqrt{3}}{2}\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}} $$

      Combining the real and imaginary parts\n results in the same result as that obtained by Cardano's solution.

      For the equation $$t^3 +\\dfrac{20}{3}t-\\dfrac{101}{27}$$, we have $$p=\\dfrac{20}{3}$$ and $$q = -\\dfrac{101}{27}$$

      Calculate the discriminant

      The nature of the roots are determined by the sign of the discriminant.

      $$\\begin{aligned} \\\\\\Delta&=\\dfrac{q^2}{4}+\\dfrac{p^3}{27}\\\\ & =\\dfrac{\\Big(-\\dfrac{101}{27}\\Big)^2}{4}+\\dfrac{\\Big(\\dfrac{20}{3}\\Big)^3}{27}\\\\ & =\\dfrac{10201}{2916}+\\dfrac{8000}{729}\\\\ & =\\dfrac{10201\\cdot 1+8000\\cdot 4}{2916}\\\\ & =14.472222222222\\\\ \\end{aligned}$$

      4.1 Use the root formula directly

      If the discriminant is greater than zero, we can use the root formula to determine the roots of the cubic equation.

      $$t_{1,2,3} =\\begin{cases} \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } +\\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}& \\\\ ω\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + \\overline{ω} \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }}&\\ \\\\ \\overline{ω}\\cdotp \\sqrt[3]{-\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} } } + ω\\cdotp \\sqrt[3]{-\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }} \\end{cases}$$

      in which, $$ ω = \\dfrac{-1+i\\sqrt{3}}{2} $$ and $$ \\overline{ω} =\\dfrac{-1-i\\sqrt{3}}{2}$$

      Substitute the values of $$p, q$$ and $$\\Delta$$ which we have calculated. Then,

      $$\\begin{aligned} \\\\t_1&=\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{42201}{2916}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{42201}{2916}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}+\\sqrt[3]{\\dfrac{101}{54}-\\sqrt{\\dfrac{521\\cdot\\cancel{81}}{36\\cdot\\cancel{81}}}}\\\\ & =\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\\\ \\end{aligned}$$

      If we denote

      $$R = -\\dfrac{q}{2}+\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }$$

      $$\\overline{R} = -\\dfrac{q}{2} -\\sqrt{\\dfrac{q^2}{4}+\\dfrac{p^3}{27} }$$

      then,

      $$\\sqrt[3]{R} = \\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}$$, $$\\sqrt[3]{\\overline{R}} =\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}$$

      $$\\begin{aligned} \\\\t_2&= ω\\cdotp \\sqrt[3]{R}+ \\overline{ω} \\sqrt[3]{\\overline{R} }\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}( \\sqrt[3]{R} - \\sqrt[3]{\\overline{R} }) }{2} i\\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

      $$\\begin{aligned} \\\\t_3&= \\overline{ω}\\cdotp \\sqrt[3]{R}+ ω\\cdotp \\sqrt[3]{\\overline{R}}\\\\ & =\\dfrac{-\\sqrt[3]{R}-\\sqrt[3]{\\overline{R} }}{2} +\\dfrac{\\sqrt{3}(- \\sqrt[3]{R} + \\sqrt[3]{\\overline{R} }) }{2}i \\\\ & =\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)\\\\&-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i\\\\ \\end{aligned}$$

      Roots of the general cubic equation

      Since $$x = t - \\dfrac{b}{3a}$$, substituting the values of $$t$$, $$a$$ and $$b$$ gives

      $$x_1 = t_1-\\dfrac{2}{3}$$

      $$x_2 = t_2-\\dfrac{2}{3}$$

      $$x_3 = t_3-\\dfrac{2}{3}$$

      5. Summary

      In summary, we have tried the method of cubic root formula to explore the solutions of the equation. The cubic equation $$x³ + 2x² + 8x + 1=0$$ is found to have one real root and two complex roots. Exact values and approximations are given below.

      $$\\begin{cases} x_1=\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}+\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}-\\dfrac{2}{3} \\\\ x_2=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)+\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\\\ x_3=\\dfrac{1}{2}\\Big(-\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)-\\dfrac{\\sqrt{3}}{2}\\Big(\\sqrt[3]{\\dfrac{101}{54}+\\dfrac{\\sqrt{521}}{6}}-\\sqrt[3]{\\dfrac{101}{54}-\\dfrac{\\sqrt{521}}{6}}\\Big)i-\\dfrac{2}{3} \\end{cases}$$

      in decimal notation,

      $$\\begin{cases} x_1=-0.12888523007843 \\\\ x_2=-0.93555738496079+2.6236564793854i \\\\ x_3=-0.93555738496079-2.6236564793854i \\end{cases}$$

      6. Graph for the function $$f(x) = x³ + 2x² + 8x + 1$$

      Since the discriminat is greater than zero, the curve of the cubic function $$f(x) = x³ + 2x² + 8x + 1$$ has one intersection point with the x-axis.

      \n\n\n\n\n\n\n
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      \n\n\n", "statics": {"title": 12, "equation-interline": 124, "paragraph": 119, "paragraph.text": 119}} diff --git a/bench/data/groundtruth/math_katex_latex_3.jsonl b/bench/data/groundtruth/math_katex_latex_3.jsonl index ac465409..132af8f9 100644 --- a/bench/data/groundtruth/math_katex_latex_3.jsonl +++ b/bench/data/groundtruth/math_katex_latex_3.jsonl @@ -1 +1 @@ -{"content_list": [[{"type": "paragraph", "raw_content": "
      \n Show commands:\n Magma\n / PariGP\n / SageMath
      ", "content": [{"c": "Show commands: Magma/ PariGP/ SageMath", "t": "text"}]}, {"type": "paragraph", "raw_content": "
      [N,k,chi] = [3332,1,Mod(667,3332)]
      mf = mfinit([N,k,chi],0)
      lf = mfeigenbasis(mf)
      ", "content": [{"c": "[N,k,chi] = [3332,1,Mod(667,3332)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)", "t": "text"}]}, {"type": "paragraph", "raw_content": "
      from sage.modular.dirichlet import DirichletCharacter
      H = DirichletGroup(3332, base_ring=CyclotomicField(12))
      chi = DirichletCharacter(H, H._module([6, 4, 9]))
      N = Newforms(chi, 1, names=\"a\")
      ", "content": [{"c": "from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3332, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 4, 9])) N = Newforms(chi, 1, names=\"a\")", "t": "text"}]}, {"type": "paragraph", "raw_content": "
      //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
      chi := DirichletCharacter(\"3332.667\");
      S:= CuspForms(chi, 1);
      N := Newforms(S);
      ", "content": [{"c": "//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter(\"3332.667\"); S:= CuspForms(chi, 1); N := Newforms(S);", "t": "text"}]}, {"type": "table", "raw_content": "
      Level: \\( N \\) \\(=\\)\\( 3332 = 2^{2} \\cdot 7^{2} \\cdot 17 \\)
      Weight: \\( k \\) \\(=\\)\\( 1 \\)
      Character orbit: \\([\\chi]\\) \\(=\\)3332.bc (of order \\(12\\), degree \\(4\\), not minimal)
      ", "content": {"html": "
      Level\\( N \\)\\(=\\)\\( 3332 = 2^{2} \\cdot 7^{2} \\cdot 17 \\)
      Weight\\( k \\)\\(=\\)\\( 1 \\)
      Character orbit\\([\\chi]\\)\\(=\\)3332.bc<br>order<br>degree<br>minimal
      ", "is_complex": false, "table_nest_level": "1"}}, {"type": "title", "raw_content": "

      Newform invariants

      ", "content": {"title_content": "Newform invariants", "level": "2"}}, {"type": "paragraph", "raw_content": "
      sage:\u00a0f = N[0] # Warning: the index may be different
      ", "content": [{"c": "sage:\u00a0f = N[0] # Warning: the index may be different", "t": "text"}]}, {"type": "paragraph", "raw_content": "
      gp:\u00a0f = lf[1] \\\\ Warning: the index may be different
      ", "content": [{"c": "gp:\u00a0f = lf[1] \\\\ Warning: the index may be different", "t": "text"}]}, {"type": "table", "raw_content": "
      Self dual: no
      Analytic conductor: \\(1.66288462209\\)
      Analytic rank: \\(0\\)
      Dimension: \\(4\\)
      Coefficient field: \\(\\Q(\\zeta_{12})\\)
      gp:\u00a0f.mod \\\\ as an extension of the character field
      Defining polynomial: \\( x^{4} - x^{2} + 1 \\)\"Copy\"Toggle
      Coefficient ring: \\(\\Z[a_1, a_2]\\)
      Coefficient ring index: \\( 1 \\)
      Twist minimal: no (minimal twist has level 68)
      Projective image:\\(D_{4}\\)
      Projective field:Galois closure of 4.2.19652.1
      Artin image:$C_4\\wr C_2\\times C_6$
      Artin field:Galois closure of \\(\\mathbb{Q}[x]/(x^{48} - \\cdots)\\)
      ", "content": {"html": "
      Self dualno
      Analytic conductor\\(1.66288462209\\)
      Analytic rank\\(0\\)
      Dimension\\(4\\)
      Coefficient field\\(\\Q(\\zeta_{12})\\)
      gp:\u00a0f.mod \\\\ as an extension of the character field
      Defining polynomial\\( x^{4} - x^{2} + 1 \\)
      Coefficient ring\\(\\Z[a_1, a_2]\\)
      Coefficient ring index\\( 1 \\)
      Twist minimalno (minimal twist has level 68)
      Projective image\\(D_{4}\\)
      Projective fieldGalois closure of<br>4.2.19652.1
      Artin image$C_4\\wr C_2\\times C_6$
      Artin fieldGalois closure of<br>\\(\\mathbb{Q}[x]/(x^{48} - \\cdots)\\)
      ", "is_complex": true, "table_nest_level": "1"}}, {"type": "title", "raw_content": "

      Embedding invariants

      ", "content": {"title_content": "Embedding invariants", "level": "2"}}, {"type": "table", "raw_content": "
      Embedding label 2027.1
      Root\\(0.866025 - 0.500000i\\) of defining polynomial
      Character\\(\\chi\\)\\(=\\)3332.2027
      Dual form3332.1.bc.b.863.1
      ", "content": {"html": "
      Embedding label2027.1
      Root\\(0.866025 - 0.500000i\\) of defining polynomial
      Character\\(\\chi\\)\\(=\\)3332.2027
      Dual form3332.1.bc.b.863.1
      ", "is_complex": false, "table_nest_level": "1"}}, {"type": "paragraph", "raw_content": "
      sage:\u00a0f.q_expansion() # note that sage often uses an isomorphic number field
      ", "content": [{"c": "sage:\u00a0f.q_expansion() # note that sage often uses an isomorphic number field", "t": "text"}]}, {"type": "paragraph", "raw_content": "
      gp:\u00a0mfcoefs(f, 20)
      ", "content": [{"c": "gp:\u00a0mfcoefs(f, 20)", "t": "text"}]}, {"type": "table", "raw_content": "
      \\(f(q)\\)\\(=\\)\\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.36603 - 0.366025i) q^{5} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.00000i) q^{20} +(0.866025 + 0.500000i) q^{25} +(1.00000 - 1.00000i) q^{29} +(0.866025 + 0.500000i) q^{32} -1.00000i q^{34} +1.00000i q^{36} +(1.36603 + 0.366025i) q^{37} +(0.366025 - 1.36603i) q^{40} +(-1.00000 - 1.00000i) q^{41} +(1.36603 - 0.366025i) q^{45} -1.00000 q^{50} +(-0.366025 + 1.36603i) q^{58} +(0.366025 - 1.36603i) q^{61} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(-0.366025 - 1.36603i) q^{73} +(-1.36603 + 0.366025i) q^{74} +(0.366025 + 1.36603i) q^{80} +(0.500000 - 0.866025i) q^{81} +(1.36603 + 0.366025i) q^{82} +(1.00000 - 1.00000i) q^{85} +(-1.00000 + 1.00000i) q^{90} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\\)
      \\(\\operatorname{Tr}(f)(q)\\)\\(=\\)\\( 4 q + 2 q^{4} - 2 q^{5} + 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 4 q^{20} + 4 q^{29} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 2 q^{45} - 4 q^{50} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{68} - 2 q^{72} + 2 q^{73}+ \\cdots + 4 q^{97}+O(q^{100}) \\)\"Copy\"Toggle
      ", "content": {"html": "
      \\(f(q)\\)\\(=\\)\\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.36603 - 0.366025i) q^{5} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.00000i) q^{20} +(0.866025 + 0.500000i) q^{25} +(1.00000 - 1.00000i) q^{29} +(0.866025 + 0.500000i) q^{32} -1.00000i q^{34} +1.00000i q^{36} +(1.36603 + 0.366025i) q^{37} +(0.366025 - 1.36603i) q^{40} +(-1.00000 - 1.00000i) q^{41} +(1.36603 - 0.366025i) q^{45} -1.00000 q^{50} +(-0.366025 + 1.36603i) q^{58} +(0.366025 - 1.36603i) q^{61} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(-0.366025 - 1.36603i) q^{73} +(-1.36603 + 0.366025i) q^{74} +(0.366025 + 1.36603i) q^{80} +(0.500000 - 0.866025i) q^{81} +(1.36603 + 0.366025i) q^{82} +(1.00000 - 1.00000i) q^{85} +(-1.00000 + 1.00000i) q^{90} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\\)
      \\(\\operatorname{Tr}(f)(q)\\)\\(=\\)\\( 4 q + 2 q^{4} - 2 q^{5} + 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 4 q^{20} + 4 q^{29} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 2 q^{45} - 4 q^{50} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{68} - 2 q^{72} + 2 q^{73}+ \\cdots + 4 q^{97}+O(q^{100}) \\)
      ", "is_complex": false, "table_nest_level": "1"}}, {"type": "title", "raw_content": "

      Character values

      ", "content": {"title_content": "Character values", "level": "2"}}, {"type": "paragraph", "raw_content": "

      We give the values of \\chi on generators for \\left(\\mathbb{Z}/3332\\mathbb{Z}\\right)^\\times.

      ", "content": [{"c": "We give the values of", "t": "text"}, {"c": "\\chi", "t": "equation-inline"}, {"c": "on generators for", "t": "text"}, {"c": "\\left(\\mathbb{Z}/3332\\mathbb{Z}\\right)^\\times", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "table", "raw_content": "
      \\(n\\)\\(785\\)\\(885\\)\\(1667\\)
      \\(\\chi(n)\\)\\(e\\left(\\frac{3}{4}\\right)\\)\\(e\\left(\\frac{2}{3}\\right)\\)\\(-1\\)
      ", "content": {"html": "
      \\(n\\)\\(785\\)\\(885\\)\\(1667\\)
      \\(\\chi(n)\\)\\(e\\left(\\frac{3}{4}\\right)\\)\\(e\\left(\\frac{2}{3}\\right)\\)\\(-1\\)
      ", "is_complex": false, "table_nest_level": "1"}}, {"type": "title", "raw_content": "

      Coefficient data

      ", "content": {"title_content": "Coefficient data", "level": "2"}}, {"type": "paragraph", "raw_content": "

      For each \\(n\\) we display the coefficients of the \\(q\\)-expansion \\(a_n\\), the\nSatake parameters \\(\\alpha_p\\),\nand the Satake angles \\(\\theta_p = \\textrm{Arg}(\\alpha_p)\\).

      ", "content": [{"c": "For each \\(n\\) we display the coefficients of the \\(q\\)-expansion \\(a_n\\), the Satake parameters\\(\\alpha_p\\),\nand the Satake angles \\(\\theta_p = \\textrm{Arg}(\\alpha_p)\\).", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See \\(a_n\\) instead)", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See \\(a_n\\) instead)", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See \\(a_n\\) instead)", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See only \\(a_p\\))", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See only \\(a_p\\))", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See only \\(a_p\\))", "t": "text"}]}, {"type": "table", "raw_content": "
      \n \\(n\\)\n \n \\(a_n\\)\n \n \\(a_n / n^{(k-1)/2}\\)\n \n \\( \\alpha_n \\)\n \n \\( \\theta_n \\)\n
      \n \\(p\\)\n \n \\(a_p\\)\n \n \\(a_p / p^{(k-1)/2}\\)\n \n \\( \\alpha_p\\)\n \n \\( \\theta_p \\)\n
      \n \\(2\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
      \n \\(3\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(4\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
      \n \\(5\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(6\\)\n \n 0\n \n 0\n
      \n \\(7\\)\n \n 0\n \n 0\n
      \n \\(8\\)\n \n 1.00000i\n 1.00000i
      \n \\(9\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
      \n \\(10\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
      \n \\(11\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(12\\)\n \n 0\n \n 0\n
      \n \\(13\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(14\\)\n \n 0\n \n 0\n
      \n \\(15\\)\n \n 0\n \n 0\n
      \n \\(16\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
      \n \\(17\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
      \n \\(18\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
      \n \\(19\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(20\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
      \n \\(21\\)\n \n 0\n \n 0\n
      \n \\(22\\)\n \n 0\n \n 0\n
      \n \\(23\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
      \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
      \n \\(24\\)\n \n 0\n \n 0\n
      \n \\(25\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
      \n \\(26\\)\n \n 0\n \n 0\n
      \n \\(27\\)\n \n 0\n \n 0\n
      \n \\(28\\)\n \n 0\n \n 0\n
      \n \\(29\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n \\(0\\)\n
      \n \\(30\\)\n \n 0\n \n 0\n
      \n \\(31\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(32\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
      \n \\(33\\)\n \n 0\n \n 0\n
      \n \\(34\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
      \n \\(35\\)\n \n 0\n \n 0\n
      \n \\(36\\)\n \n 1.00000i\n 1.00000i
      \n \\(37\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(38\\)\n \n 0\n \n 0\n
      \n \\(39\\)\n \n 0\n \n 0\n
      \n \\(40\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
      \n \\(41\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(42\\)\n \n 0\n \n 0\n
      \n \\(43\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(44\\)\n \n 0\n \n 0\n
      \n \\(45\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
      \n \\(46\\)\n \n 0\n \n 0\n
      \n \\(47\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(48\\)\n \n 0\n \n 0\n
      \n \\(49\\)\n \n 0\n \n 0\n
      \n \\(50\\)\n \n \u22121.00000\n \n \u22121.00000\n
      \n \\(51\\)\n \n 0\n \n 0\n
      \n \\(52\\)\n \n 0\n \n 0\n
      \n \\(53\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(54\\)\n \n 0\n \n 0\n
      \n \\(55\\)\n \n 0\n \n 0\n
      \n \\(56\\)\n \n 0\n \n 0\n
      \n \\(57\\)\n \n 0\n \n 0\n
      \n \\(58\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
      \n \\(59\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(60\\)\n \n 0\n \n 0\n
      \n \\(61\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \\(62\\)\n \n 0\n \n 0\n
      \n \\(63\\)\n \n 0\n \n 0\n
      \n \\(64\\)\n \n \u22121.00000\n \n \u22121.00000\n
      \n \\(65\\)\n \n 0\n \n 0\n
      \n \\(66\\)\n \n 0\n \n 0\n
      \n \\(67\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(68\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
      \n \\(69\\)\n \n 0\n \n 0\n
      \n \\(70\\)\n \n 0\n \n 0\n
      \n \\(71\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(72\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
      \n \\(73\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \\(74\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
      \n \\(75\\)\n \n 0\n \n 0\n
      \n \\(76\\)\n \n 0\n \n 0\n
      \n \\(77\\)\n \n 0\n \n 0\n
      \n \\(78\\)\n \n 0\n \n 0\n
      \n \\(79\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
      \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
      \n \\(80\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(81\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
      \n \\(82\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(83\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(84\\)\n \n 0\n \n 0\n
      \n \\(85\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
      \n \\(86\\)\n \n 0\n \n 0\n
      \n \\(87\\)\n \n 0\n \n 0\n
      \n \\(88\\)\n \n 0\n \n 0\n
      \n \\(89\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(90\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
      \n \\(91\\)\n \n 0\n \n 0\n
      \n \\(92\\)\n \n 0\n \n 0\n
      \n \\(93\\)\n \n 0\n \n 0\n
      \n \\(94\\)\n \n 0\n \n 0\n
      \n \\(95\\)\n \n 0\n \n 0\n
      \n \\(96\\)\n \n 0\n \n 0\n
      \n \\(97\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n \\(0\\)\n
      \n \\(98\\)\n \n 0\n \n 0\n
      \n \\(99\\)\n \n 0\n \n 0\n
      \n \\(100\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
      \n \\(101\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(102\\)\n \n 0\n \n 0\n
      \n \\(103\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(104\\)\n \n 0\n \n 0\n
      \n \\(105\\)\n \n 0\n \n 0\n
      \n \\(106\\)\n \n 0\n \n 0\n
      \n \\(107\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(108\\)\n \n 0\n \n 0\n
      \n \\(109\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(110\\)\n \n 0\n \n 0\n
      \n \\(111\\)\n \n 0\n \n 0\n
      \n \\(112\\)\n \n 0\n \n 0\n
      \n \\(113\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(114\\)\n \n 0\n \n 0\n
      \n \\(115\\)\n \n 0\n \n 0\n
      \n \\(116\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
      \n \\(117\\)\n \n 0\n \n 0\n
      \n \\(118\\)\n \n 0\n \n 0\n
      \n \\(119\\)\n \n 0\n \n 0\n
      \n \\(120\\)\n \n 0\n \n 0\n
      \n \\(121\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
      \n \\(122\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(123\\)\n \n 0\n \n 0\n
      \n \\(124\\)\n \n 0\n \n 0\n
      \n \\(125\\)\n \n 0\n \n 0\n
      \n \\(126\\)\n \n 0\n \n 0\n
      \n \\(127\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(128\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
      \n \\(129\\)\n \n 0\n \n 0\n
      \n \\(130\\)\n \n 0\n \n 0\n
      \n \\(131\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(132\\)\n \n 0\n \n 0\n
      \n \\(133\\)\n \n 0\n \n 0\n
      \n \\(134\\)\n \n 0\n \n 0\n
      \n \\(135\\)\n \n 0\n \n 0\n
      \n \\(136\\)\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \u22120.866025\n \n \u2212\n \n 0.500000i
      \n \\(137\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(138\\)\n \n 0\n \n 0\n
      \n \\(139\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(140\\)\n \n 0\n \n 0\n
      \n \\(141\\)\n \n 0\n \n 0\n
      \n \\(142\\)\n \n 0\n \n 0\n
      \n \\(143\\)\n \n 0\n \n 0\n
      \n \\(144\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
      \n \\(145\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
      \n \\(146\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(147\\)\n \n 0\n \n 0\n
      \n \\(148\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
      \n \\(149\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n \\(150\\)\n \n 0\n \n 0\n
      \n \\(151\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(152\\)\n \n 0\n \n 0\n
      \n \\(153\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
      \n \\(154\\)\n \n 0\n \n 0\n
      \n \\(155\\)\n \n 0\n \n 0\n
      \n \\(156\\)\n \n 0\n \n 0\n
      \n \\(157\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \\(158\\)\n \n 0\n \n 0\n
      \n \\(159\\)\n \n 0\n \n 0\n
      \n \\(160\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
      \n \\(161\\)\n \n 0\n \n 0\n
      \n \\(162\\)\n \n 1.00000i\n 1.00000i
      \n \\(163\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
      \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
      \n \\(164\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
      \n \\(165\\)\n \n 0\n \n 0\n
      \n \\(166\\)\n \n 0\n \n 0\n
      \n \\(167\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(168\\)\n \n 0\n \n 0\n
      \n \\(169\\)\n \n \u22121.00000\n \n \u22121.00000\n
      \n \\(170\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
      \n \\(171\\)\n \n 0\n \n 0\n
      \n \\(172\\)\n \n 0\n \n 0\n
      \n \\(173\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(174\\)\n \n 0\n \n 0\n
      \n \\(175\\)\n \n 0\n \n 0\n
      \n \\(176\\)\n \n 0\n \n 0\n
      \n \\(177\\)\n \n 0\n \n 0\n
      \n \\(178\\)\n \n 0\n \n 0\n
      \n \\(179\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(180\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
      \n \\(181\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n \\(0\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(182\\)\n \n 0\n \n 0\n
      \n \\(183\\)\n \n 0\n \n 0\n
      \n \\(184\\)\n \n 0\n \n 0\n
      \n \\(185\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i
      \n \\(186\\)\n \n 0\n \n 0\n
      \n \\(187\\)\n \n 0\n \n 0\n
      \n \\(188\\)\n \n 0\n \n 0\n
      \n \\(189\\)\n \n 0\n \n 0\n
      \n \\(190\\)\n \n 0\n \n 0\n
      \n \\(191\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(192\\)\n \n 0\n \n 0\n
      \n \\(193\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(194\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
      \n \\(195\\)\n \n 0\n \n 0\n
      \n \\(196\\)\n \n 0\n \n 0\n
      \n \\(197\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n \\(0\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(198\\)\n \n 0\n \n 0\n
      \n \\(199\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(200\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
      \n \\(201\\)\n \n 0\n \n 0\n
      \n \\(202\\)\n \n 0\n \n 0\n
      \n \\(203\\)\n \n 0\n \n 0\n
      \n \\(204\\)\n \n 0\n \n 0\n
      \n \\(205\\)\n \n 1.00000\n \n +\n \n 1.73205i\n 1.00000\n \n +\n \n 1.73205i
      \n \\(206\\)\n \n 0\n \n 0\n
      \n \\(207\\)\n \n 0\n \n 0\n
      \n \\(208\\)\n \n 0\n \n 0\n
      \n \\(209\\)\n \n 0\n \n 0\n
      \n \\(210\\)\n \n 0\n \n 0\n
      \n \\(211\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
      \n \\(212\\)\n \n 0\n \n 0\n
      \n \\(213\\)\n \n 0\n \n 0\n
      \n \\(214\\)\n \n 0\n \n 0\n
      \n \\(215\\)\n \n 0\n \n 0\n
      \n \\(216\\)\n \n 0\n \n 0\n
      \n \\(217\\)\n \n 0\n \n 0\n
      \n \\(218\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
      \n \\(219\\)\n \n 0\n \n 0\n
      \n \\(220\\)\n \n 0\n \n 0\n
      \n \\(221\\)\n \n 0\n \n 0\n
      \n \\(222\\)\n \n 0\n \n 0\n
      \n \\(223\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(224\\)\n \n 0\n \n 0\n
      \n \\(225\\)\n \n \u22121.00000\n \n \u22121.00000\n
      \n \\(226\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(227\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(228\\)\n \n 0\n \n 0\n
      \n \\(229\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i\n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(230\\)\n \n 0\n \n 0\n
      \n \\(231\\)\n \n 0\n \n 0\n
      \n \\(232\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(233\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(234\\)\n \n 0\n \n 0\n
      \n \\(235\\)\n \n 0\n \n 0\n
      \n \\(236\\)\n \n 0\n \n 0\n
      \n \\(237\\)\n \n 0\n \n 0\n
      \n \\(238\\)\n \n 0\n \n 0\n
      \n \\(239\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(240\\)\n \n 0\n \n 0\n
      \n \\(241\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \\(242\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
      \n \\(243\\)\n \n 0\n \n 0\n
      \n \\(244\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
      \n \\(245\\)\n \n 0\n \n 0\n
      \n \\(246\\)\n \n 0\n \n 0\n
      \n \\(247\\)\n \n 0\n \n 0\n
      \n \\(248\\)\n \n 0\n \n 0\n
      \n \\(249\\)\n \n 0\n \n 0\n
      \n \\(250\\)\n \n 0\n \n 0\n
      \n \\(251\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(252\\)\n \n 0\n \n 0\n
      \n \\(253\\)\n \n 0\n \n 0\n
      \n \\(254\\)\n \n 0\n \n 0\n
      \n \\(255\\)\n \n 0\n \n 0\n
      \n \\(256\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
      \n \\(257\\)\n \n 1.73205\n \n \u2212\n \n 1.00000i\n 1.73205\n \n \u2212\n \n 1.00000i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \\(258\\)\n \n 0\n \n 0\n
      \n \\(259\\)\n \n 0\n \n 0\n
      \n \\(260\\)\n \n 0\n \n 0\n
      \n \\(261\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
      \n \\(262\\)\n \n 0\n \n 0\n
      \n \\(263\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(264\\)\n \n 0\n \n 0\n
      \n \\(265\\)\n \n 0\n \n 0\n
      \n \\(266\\)\n \n 0\n \n 0\n
      \n \\(267\\)\n \n 0\n \n 0\n
      \n \\(268\\)\n \n 0\n \n 0\n
      \n \\(269\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i\n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(270\\)\n \n 0\n \n 0\n
      \n \\(271\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(272\\)\n \n 1.00000\n \n 1.00000\n
      \n \\(273\\)\n \n 0\n \n 0\n
      \n \\(274\\)\n \n 0\n \n 0\n
      \n \\(275\\)\n \n 0\n \n 0\n
      \n \\(276\\)\n \n 0\n \n 0\n
      \n \\(277\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i\n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(278\\)\n \n 0\n \n 0\n
      \n \\(279\\)\n \n 0\n \n 0\n
      \n \\(280\\)\n \n 0\n \n 0\n
      \n \\(281\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(282\\)\n \n 0\n \n 0\n
      \n \\(283\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(284\\)\n \n 0\n \n 0\n
      \n \\(285\\)\n \n 0\n \n 0\n
      \n \\(286\\)\n \n 0\n \n 0\n
      \n \\(287\\)\n \n 0\n \n 0\n
      \n \\(288\\)\n \n \u22121.00000\n \n \u22121.00000\n
      \n \\(289\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
      \n \\(290\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
      \n \\(291\\)\n \n 0\n \n 0\n
      \n \\(292\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
      \n \\(293\\)\n \n 2.00000\n \n 2.00000\n \n 1.00000\n \n \\(0\\)\n
      \n 1.00000\n \n \\(0\\)\n
      \n \\(294\\)\n \n 0\n \n 0\n
      \n \\(295\\)\n \n 0\n \n 0\n
      \n \\(296\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
      \n \\(297\\)\n \n 0\n \n 0\n
      \n \\(298\\)\n \n 1.73205\n \n +\n \n 1.00000i\n 1.73205\n \n +\n \n 1.00000i
      \n \\(299\\)\n \n 0\n \n 0\n
      \n \\(300\\)\n \n 0\n \n 0\n
      \n \\(301\\)\n \n 0\n \n 0\n
      \n \\(302\\)\n \n 0\n \n 0\n
      \n \\(303\\)\n \n 0\n \n 0\n
      \n \\(304\\)\n \n 0\n \n 0\n
      \n \\(305\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i
      \n \\(306\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
      \n \\(307\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(308\\)\n \n 0\n \n 0\n
      \n \\(309\\)\n \n 0\n \n 0\n
      \n \\(310\\)\n \n 0\n \n 0\n
      \n \\(311\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(312\\)\n \n 0\n \n 0\n
      \n \\(313\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(314\\)\n \n 2.00000i\n 2.00000i
      \n \\(315\\)\n \n 0\n \n 0\n
      \n \\(316\\)\n \n 0\n \n 0\n
      \n \\(317\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \\(318\\)\n \n 0\n \n 0\n
      \n \\(319\\)\n \n 0\n \n 0\n
      \n \\(320\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(321\\)\n \n 0\n \n 0\n
      \n \\(322\\)\n \n 0\n \n 0\n
      \n \\(323\\)\n \n 0\n \n 0\n
      \n \\(324\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
      \n \\(325\\)\n \n 0\n \n 0\n
      \n \\(326\\)\n \n 0\n \n 0\n
      \n \\(327\\)\n \n 0\n \n 0\n
      \n \\(328\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
      \n \\(329\\)\n \n 0\n \n 0\n
      \n \\(330\\)\n \n 0\n \n 0\n
      \n \\(331\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(332\\)\n \n 0\n \n 0\n
      \n \\(333\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
      \n \\(334\\)\n \n 0\n \n 0\n
      \n \\(335\\)\n \n 0\n \n 0\n
      \n \\(336\\)\n \n 0\n \n 0\n
      \n \\(337\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n \\(0\\)\n
      \n \\(338\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
      \n \\(339\\)\n \n 0\n \n 0\n
      \n \\(340\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
      \n \\(341\\)\n \n 0\n \n 0\n
      \n \\(342\\)\n \n 0\n \n 0\n
      \n \\(343\\)\n \n 0\n \n 0\n
      \n \\(344\\)\n \n 0\n \n 0\n
      \n \\(345\\)\n \n 0\n \n 0\n
      \n \\(346\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
      \n \\(347\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(348\\)\n \n 0\n \n 0\n
      \n \\(349\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(350\\)\n \n 0\n \n 0\n
      \n \\(351\\)\n \n 0\n \n 0\n
      \n \\(352\\)\n \n 0\n \n 0\n
      \n \\(353\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i\n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(354\\)\n \n 0\n \n 0\n
      \n \\(355\\)\n \n 0\n \n 0\n
      \n \\(356\\)\n \n 0\n \n 0\n
      \n \\(357\\)\n \n 0\n \n 0\n
      \n \\(358\\)\n \n 0\n \n 0\n
      \n \\(359\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(360\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(361\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
      \n \\(362\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
      \n \\(363\\)\n \n 0\n \n 0\n
      \n \\(364\\)\n \n 0\n \n 0\n
      \n \\(365\\)\n \n 2.00000i\n 2.00000i
      \n \\(366\\)\n \n 0\n \n 0\n
      \n \\(367\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(368\\)\n \n 0\n \n 0\n
      \n \\(369\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(370\\)\n \n 2.00000\n \n 2.00000\n
      \n \\(371\\)\n \n 0\n \n 0\n
      \n \\(372\\)\n \n 0\n \n 0\n
      \n \\(373\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(374\\)\n \n 0\n \n 0\n
      \n \\(375\\)\n \n 0\n \n 0\n
      \n \\(376\\)\n \n 0\n \n 0\n
      \n \\(377\\)\n \n 0\n \n 0\n
      \n \\(378\\)\n \n 0\n \n 0\n
      \n \\(379\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(380\\)\n \n 0\n \n 0\n
      \n \\(381\\)\n \n 0\n \n 0\n
      \n \\(382\\)\n \n 0\n \n 0\n
      \n \\(383\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(384\\)\n \n 0\n \n 0\n
      \n \\(385\\)\n \n 0\n \n 0\n
      \n \\(386\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
      \n \\(387\\)\n \n 0\n \n 0\n
      \n \\(388\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
      \n \\(389\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n \\(390\\)\n \n 0\n \n 0\n
      \n \\(391\\)\n \n 0\n \n 0\n
      \n \\(392\\)\n \n 0\n \n 0\n
      \n \\(393\\)\n \n 0\n \n 0\n
      \n \\(394\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
      \n \\(395\\)\n \n 0\n \n 0\n
      \n \\(396\\)\n \n 0\n \n 0\n
      \n \\(397\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \\(398\\)\n \n 0\n \n 0\n
      \n \\(399\\)\n \n 0\n \n 0\n
      \n \\(400\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
      \n \\(401\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \\(402\\)\n \n 0\n \n 0\n
      \n \\(403\\)\n \n 0\n \n 0\n
      \n \\(404\\)\n \n 0\n \n 0\n
      \n \\(405\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
      \n \\(406\\)\n \n 0\n \n 0\n
      \n \\(407\\)\n \n 0\n \n 0\n
      \n \\(408\\)\n \n 0\n \n 0\n
      \n \\(409\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i\n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(410\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i
      \n \\(411\\)\n \n 0\n \n 0\n
      \n \\(412\\)\n \n 0\n \n 0\n
      \n \\(413\\)\n \n 0\n \n 0\n
      \n \\(414\\)\n \n 0\n \n 0\n
      \n \\(415\\)\n \n 0\n \n 0\n
      \n \\(416\\)\n \n 0\n \n 0\n
      \n \\(417\\)\n \n 0\n \n 0\n
      \n \\(418\\)\n \n 0\n \n 0\n
      \n \\(419\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
      \n \\(420\\)\n \n 0\n \n 0\n
      \n \\(421\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(422\\)\n \n 0\n \n 0\n
      \n \\(423\\)\n \n 0\n \n 0\n
      \n \\(424\\)\n \n 0\n \n 0\n
      \n \\(425\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
      \n \\(426\\)\n \n 0\n \n 0\n
      \n \\(427\\)\n \n 0\n \n 0\n
      \n \\(428\\)\n \n 0\n \n 0\n
      \n \\(429\\)\n \n 0\n \n 0\n
      \n \\(430\\)\n \n 0\n \n 0\n
      \n \\(431\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(432\\)\n \n 0\n \n 0\n
      \n \\(433\\)\n \n 2.00000i\n 2.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(434\\)\n \n 0\n \n 0\n
      \n \\(435\\)\n \n 0\n \n 0\n
      \n \\(436\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
      \n \\(437\\)\n \n 0\n \n 0\n
      \n \\(438\\)\n \n 0\n \n 0\n
      \n \\(439\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(440\\)\n \n 0\n \n 0\n
      \n \\(441\\)\n \n 0\n \n 0\n
      \n \\(442\\)\n \n 0\n \n 0\n
      \n \\(443\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(444\\)\n \n 0\n \n 0\n
      \n \\(445\\)\n \n 0\n \n 0\n
      \n \\(446\\)\n \n 0\n \n 0\n
      \n \\(447\\)\n \n 0\n \n 0\n
      \n \\(448\\)\n \n 0\n \n 0\n
      \n \\(449\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(450\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
      \n \\(451\\)\n \n 0\n \n 0\n
      \n \\(452\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
      \n \\(453\\)\n \n 0\n \n 0\n
      \n \\(454\\)\n \n 0\n \n 0\n
      \n \\(455\\)\n \n 0\n \n 0\n
      \n \\(456\\)\n \n 0\n \n 0\n
      \n \\(457\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i\n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(458\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
      \n \\(459\\)\n \n 0\n \n 0\n
      \n \\(460\\)\n \n 0\n \n 0\n
      \n \\(461\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(462\\)\n \n 0\n \n 0\n
      \n \\(463\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(464\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
      \n \\(465\\)\n \n 0\n \n 0\n
      \n \\(466\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
      \n \\(467\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(468\\)\n \n 0\n \n 0\n
      \n \\(469\\)\n \n 0\n \n 0\n
      \n \\(470\\)\n \n 0\n \n 0\n
      \n \\(471\\)\n \n 0\n \n 0\n
      \n \\(472\\)\n \n 0\n \n 0\n
      \n \\(473\\)\n \n 0\n \n 0\n
      \n \\(474\\)\n \n 0\n \n 0\n
      \n \\(475\\)\n \n 0\n \n 0\n
      \n \\(476\\)\n \n 0\n \n 0\n
      \n \\(477\\)\n \n 0\n \n 0\n
      \n \\(478\\)\n \n 0\n \n 0\n
      \n \\(479\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(480\\)\n \n 0\n \n 0\n
      \n \\(481\\)\n \n 0\n \n 0\n
      \n \\(482\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(483\\)\n \n 0\n \n 0\n
      \n \\(484\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
      \n \\(485\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
      \n \\(486\\)\n \n 0\n \n 0\n
      \n \\(487\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(488\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(489\\)\n \n 0\n \n 0\n
      \n \\(490\\)\n \n 0\n \n 0\n
      \n \\(491\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(492\\)\n \n 0\n \n 0\n
      \n \\(493\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(494\\)\n \n 0\n \n 0\n
      \n \\(495\\)\n \n 0\n \n 0\n
      \n \\(496\\)\n \n 0\n \n 0\n
      \n \\(497\\)\n \n 0\n \n 0\n
      \n \\(498\\)\n \n 0\n \n 0\n
      \n \\(499\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
      \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
      \n \\(500\\)\n \n 0\n \n 0\n
      \n \\(501\\)\n \n 0\n \n 0\n
      \n \\(502\\)\n \n 0\n \n 0\n
      \n \\(503\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
      \n \\(504\\)\n \n 0\n \n 0\n
      \n \\(505\\)\n \n 0\n \n 0\n
      \n \\(506\\)\n \n 0\n \n 0\n
      \n \\(507\\)\n \n 0\n \n 0\n
      \n \\(508\\)\n \n 0\n \n 0\n
      \n \\(509\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n \\(510\\)\n \n 0\n \n 0\n
      \n \\(511\\)\n \n 0\n \n 0\n
      \n \\(512\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
      \n \\(513\\)\n \n 0\n \n 0\n
      \n \\(514\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i
      \n \\(515\\)\n \n 0\n \n 0\n
      \n \\(516\\)\n \n 0\n \n 0\n
      \n \\(517\\)\n \n 0\n \n 0\n
      \n \\(518\\)\n \n 0\n \n 0\n
      \n \\(519\\)\n \n 0\n \n 0\n
      \n \\(520\\)\n \n 0\n \n 0\n
      \n \\(521\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(522\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
      \n \\(523\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(524\\)\n \n 0\n \n 0\n
      \n \\(525\\)\n \n 0\n \n 0\n
      \n \\(526\\)\n \n 0\n \n 0\n
      \n \\(527\\)\n \n 0\n \n 0\n
      \n \\(528\\)\n \n 0\n \n 0\n
      \n \\(529\\)\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \u22120.866025\n \n \u2212\n \n 0.500000i
      \n \\(530\\)\n \n 0\n \n 0\n
      \n \\(531\\)\n \n 0\n \n 0\n
      \n \\(532\\)\n \n 0\n \n 0\n
      \n \\(533\\)\n \n 0\n \n 0\n
      \n \\(534\\)\n \n 0\n \n 0\n
      \n \\(535\\)\n \n 0\n \n 0\n
      \n \\(536\\)\n \n 0\n \n 0\n
      \n \\(537\\)\n \n 0\n \n 0\n
      \n \\(538\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
      \n \\(539\\)\n \n 0\n \n 0\n
      \n \\(540\\)\n \n 0\n \n 0\n
      \n \\(541\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(542\\)\n \n 0\n \n 0\n
      \n \\(543\\)\n \n 0\n \n 0\n
      \n \\(544\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
      \n \\(545\\)\n \n \u22122.00000\n \n \u22122.00000\n
      \n \\(546\\)\n \n 0\n \n 0\n
      \n \\(547\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(548\\)\n \n 0\n \n 0\n
      \n \\(549\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(550\\)\n \n 0\n \n 0\n
      \n \\(551\\)\n \n 0\n \n 0\n
      \n \\(552\\)\n \n 0\n \n 0\n
      \n \\(553\\)\n \n 0\n \n 0\n
      \n \\(554\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
      \n \\(555\\)\n \n 0\n \n 0\n
      \n \\(556\\)\n \n 0\n \n 0\n
      \n \\(557\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(558\\)\n \n 0\n \n 0\n
      \n \\(559\\)\n \n 0\n \n 0\n
      \n \\(560\\)\n \n 0\n \n 0\n
      \n \\(561\\)\n \n 0\n \n 0\n
      \n \\(562\\)\n \n 0\n \n 0\n
      \n \\(563\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(564\\)\n \n 0\n \n 0\n
      \n \\(565\\)\n \n 1.00000\n \n +\n \n 1.73205i\n 1.00000\n \n +\n \n 1.73205i
      \n \\(566\\)\n \n 0\n \n 0\n
      \n \\(567\\)\n \n 0\n \n 0\n
      \n \\(568\\)\n \n 0\n \n 0\n
      \n \\(569\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(570\\)\n \n 0\n \n 0\n
      \n \\(571\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(572\\)\n \n 0\n \n 0\n
      \n \\(573\\)\n \n 0\n \n 0\n
      \n \\(574\\)\n \n 0\n \n 0\n
      \n \\(575\\)\n \n 0\n \n 0\n
      \n \\(576\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
      \n \\(577\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(578\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
      \n \\(579\\)\n \n 0\n \n 0\n
      \n \\(580\\)\n \n 2.00000i\n 2.00000i
      \n \\(581\\)\n \n 0\n \n 0\n
      \n \\(582\\)\n \n 0\n \n 0\n
      \n \\(583\\)\n \n 0\n \n 0\n
      \n \\(584\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
      \n \\(585\\)\n \n 0\n \n 0\n
      \n \\(586\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
      \n \\(587\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(588\\)\n \n 0\n \n 0\n
      \n \\(589\\)\n \n 0\n \n 0\n
      \n \\(590\\)\n \n 0\n \n 0\n
      \n \\(591\\)\n \n 0\n \n 0\n
      \n \\(592\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
      \n \\(593\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(594\\)\n \n 0\n \n 0\n
      \n \\(595\\)\n \n 0\n \n 0\n
      \n \\(596\\)\n \n \u22122.00000\n \n \u22122.00000\n
      \n \\(597\\)\n \n 0\n \n 0\n
      \n \\(598\\)\n \n 0\n \n 0\n
      \n \\(599\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(600\\)\n \n 0\n \n 0\n
      \n \\(601\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(602\\)\n \n 0\n \n 0\n
      \n \\(603\\)\n \n 0\n \n 0\n
      \n \\(604\\)\n \n 0\n \n 0\n
      \n \\(605\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
      \n \\(606\\)\n \n 0\n \n 0\n
      \n \\(607\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(608\\)\n \n 0\n \n 0\n
      \n \\(609\\)\n \n 0\n \n 0\n
      \n \\(610\\)\n \n \u2212\n \n 2.00000i\n \u2212\n \n 2.00000i
      \n \\(611\\)\n \n 0\n \n 0\n
      \n \\(612\\)\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \u22120.866025\n \n \u2212\n \n 0.500000i
      \n \\(613\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i\n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(614\\)\n \n 0\n \n 0\n
      \n \\(615\\)\n \n 0\n \n 0\n
      \n \\(616\\)\n \n 0\n \n 0\n
      \n \\(617\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(618\\)\n \n 0\n \n 0\n
      \n \\(619\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(620\\)\n \n 0\n \n 0\n
      \n \\(621\\)\n \n 0\n \n 0\n
      \n \\(622\\)\n \n 0\n \n 0\n
      \n \\(623\\)\n \n 0\n \n 0\n
      \n \\(624\\)\n \n 0\n \n 0\n
      \n \\(625\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
      \n \\(626\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
      \n \\(627\\)\n \n 0\n \n 0\n
      \n \\(628\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
      \n \\(629\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
      \n \\(630\\)\n \n 0\n \n 0\n
      \n \\(631\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(632\\)\n \n 0\n \n 0\n
      \n \\(633\\)\n \n 0\n \n 0\n
      \n \\(634\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(635\\)\n \n 0\n \n 0\n
      \n \\(636\\)\n \n 0\n \n 0\n
      \n \\(637\\)\n \n 0\n \n 0\n
      \n \\(638\\)\n \n 0\n \n 0\n
      \n \\(639\\)\n \n 0\n \n 0\n
      \n \\(640\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
      \n \\(641\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \\(642\\)\n \n 0\n \n 0\n
      \n \\(643\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(644\\)\n \n 0\n \n 0\n
      \n \\(645\\)\n \n 0\n \n 0\n
      \n \\(646\\)\n \n 0\n \n 0\n
      \n \\(647\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(648\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
      \n \\(649\\)\n \n 0\n \n 0\n
      \n \\(650\\)\n \n 0\n \n 0\n
      \n \\(651\\)\n \n 0\n \n 0\n
      \n \\(652\\)\n \n 0\n \n 0\n
      \n \\(653\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \\(654\\)\n \n 0\n \n 0\n
      \n \\(655\\)\n \n 0\n \n 0\n
      \n \\(656\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
      \n \\(657\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(658\\)\n \n 0\n \n 0\n
      \n \\(659\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(660\\)\n \n 0\n \n 0\n
      \n \\(661\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(662\\)\n \n 0\n \n 0\n
      \n \\(663\\)\n \n 0\n \n 0\n
      \n \\(664\\)\n \n 0\n \n 0\n
      \n \\(665\\)\n \n 0\n \n 0\n
      \n \\(666\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
      \n \\(667\\)\n \n 0\n \n 0\n
      \n \\(668\\)\n \n 0\n \n 0\n
      \n \\(669\\)\n \n 0\n \n 0\n
      \n \\(670\\)\n \n 0\n \n 0\n
      \n \\(671\\)\n \n 0\n \n 0\n
      \n \\(672\\)\n \n 0\n \n 0\n
      \n \\(673\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(674\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
      \n \\(675\\)\n \n 0\n \n 0\n
      \n \\(676\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
      \n \\(677\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(678\\)\n \n 0\n \n 0\n
      \n \\(679\\)\n \n 0\n \n 0\n
      \n \\(680\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(681\\)\n \n 0\n \n 0\n
      \n \\(682\\)\n \n 0\n \n 0\n
      \n \\(683\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(684\\)\n \n 0\n \n 0\n
      \n \\(685\\)\n \n 0\n \n 0\n
      \n \\(686\\)\n \n 0\n \n 0\n
      \n \\(687\\)\n \n 0\n \n 0\n
      \n \\(688\\)\n \n 0\n \n 0\n
      \n \\(689\\)\n \n 0\n \n 0\n
      \n \\(690\\)\n \n 0\n \n 0\n
      \n \\(691\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
      \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
      \n \\(692\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
      \n \\(693\\)\n \n 0\n \n 0\n
      \n \\(694\\)\n \n 0\n \n 0\n
      \n \\(695\\)\n \n 0\n \n 0\n
      \n \\(696\\)\n \n 0\n \n 0\n
      \n \\(697\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
      \n \\(698\\)\n \n 0\n \n 0\n
      \n \\(699\\)\n \n 0\n \n 0\n
      \n \\(700\\)\n \n 0\n \n 0\n
      \n \\(701\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(702\\)\n \n 0\n \n 0\n
      \n \\(703\\)\n \n 0\n \n 0\n
      \n \\(704\\)\n \n 0\n \n 0\n
      \n \\(705\\)\n \n 0\n \n 0\n
      \n \\(706\\)\n \n \u2212\n \n 2.00000i\n \u2212\n \n 2.00000i
      \n \\(707\\)\n \n 0\n \n 0\n
      \n \\(708\\)\n \n 0\n \n 0\n
      \n \\(709\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(710\\)\n \n 0\n \n 0\n
      \n \\(711\\)\n \n 0\n \n 0\n
      \n \\(712\\)\n \n 0\n \n 0\n
      \n \\(713\\)\n \n 0\n \n 0\n
      \n \\(714\\)\n \n 0\n \n 0\n
      \n \\(715\\)\n \n 0\n \n 0\n
      \n \\(716\\)\n \n 0\n \n 0\n
      \n \\(717\\)\n \n 0\n \n 0\n
      \n \\(718\\)\n \n 0\n \n 0\n
      \n \\(719\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(720\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
      \n \\(721\\)\n \n 0\n \n 0\n
      \n \\(722\\)\n \n 1.00000i\n 1.00000i
      \n \\(723\\)\n \n 0\n \n 0\n
      \n \\(724\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
      \n \\(725\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
      \n \\(726\\)\n \n 0\n \n 0\n
      \n \\(727\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(728\\)\n \n 0\n \n 0\n
      \n \\(729\\)\n \n 1.00000i\n 1.00000i
      \n \\(730\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
      \n \\(731\\)\n \n 0\n \n 0\n
      \n \\(732\\)\n \n 0\n \n 0\n
      \n \\(733\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(734\\)\n \n 0\n \n 0\n
      \n \\(735\\)\n \n 0\n \n 0\n
      \n \\(736\\)\n \n 0\n \n 0\n
      \n \\(737\\)\n \n 0\n \n 0\n
      \n \\(738\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
      \n \\(739\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(740\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
      \n \\(741\\)\n \n 0\n \n 0\n
      \n \\(742\\)\n \n 0\n \n 0\n
      \n \\(743\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(744\\)\n \n 0\n \n 0\n
      \n \\(745\\)\n \n 0.732051\n \n +\n \n 2.73205i\n 0.732051\n \n +\n \n 2.73205i
      \n \\(746\\)\n \n 0\n \n 0\n
      \n \\(747\\)\n \n 0\n \n 0\n
      \n \\(748\\)\n \n 0\n \n 0\n
      \n \\(749\\)\n \n 0\n \n 0\n
      \n \\(750\\)\n \n 0\n \n 0\n
      \n \\(751\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(752\\)\n \n 0\n \n 0\n
      \n \\(753\\)\n \n 0\n \n 0\n
      \n \\(754\\)\n \n 0\n \n 0\n
      \n \\(755\\)\n \n 0\n \n 0\n
      \n \\(756\\)\n \n 0\n \n 0\n
      \n \\(757\\)\n \n 2.00000i\n 2.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(758\\)\n \n 0\n \n 0\n
      \n \\(759\\)\n \n 0\n \n 0\n
      \n \\(760\\)\n \n 0\n \n 0\n
      \n \\(761\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(762\\)\n \n 0\n \n 0\n
      \n \\(763\\)\n \n 0\n \n 0\n
      \n \\(764\\)\n \n 0\n \n 0\n
      \n \\(765\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
      \n \\(766\\)\n \n 0\n \n 0\n
      \n \\(767\\)\n \n 0\n \n 0\n
      \n \\(768\\)\n \n 0\n \n 0\n
      \n \\(769\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(770\\)\n \n 0\n \n 0\n
      \n \\(771\\)\n \n 0\n \n 0\n
      \n \\(772\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
      \n \\(773\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n \\(774\\)\n \n 0\n \n 0\n
      \n \\(775\\)\n \n 0\n \n 0\n
      \n \\(776\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(777\\)\n \n 0\n \n 0\n
      \n \\(778\\)\n \n 2.00000\n \n 2.00000\n
      \n \\(779\\)\n \n 0\n \n 0\n
      \n \\(780\\)\n \n 0\n \n 0\n
      \n \\(781\\)\n \n 0\n \n 0\n
      \n \\(782\\)\n \n 0\n \n 0\n
      \n \\(783\\)\n \n 0\n \n 0\n
      \n \\(784\\)\n \n 0\n \n 0\n
      \n \\(785\\)\n \n \u22122.00000\n \n +\n \n 2.00000i\n \u22122.00000\n \n +\n \n 2.00000i
      \n \\(786\\)\n \n 0\n \n 0\n
      \n \\(787\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(788\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
      \n \\(789\\)\n \n 0\n \n 0\n
      \n \\(790\\)\n \n 0\n \n 0\n
      \n \\(791\\)\n \n 0\n \n 0\n
      \n \\(792\\)\n \n 0\n \n 0\n
      \n \\(793\\)\n \n 0\n \n 0\n
      \n \\(794\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(795\\)\n \n 0\n \n 0\n
      \n \\(796\\)\n \n 0\n \n 0\n
      \n \\(797\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(798\\)\n \n 0\n \n 0\n
      \n \\(799\\)\n \n 0\n \n 0\n
      \n \\(800\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
      \n \\(801\\)\n \n 0\n \n 0\n
      \n \\(802\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(803\\)\n \n 0\n \n 0\n
      \n \\(804\\)\n \n 0\n \n 0\n
      \n \\(805\\)\n \n 0\n \n 0\n
      \n \\(806\\)\n \n 0\n \n 0\n
      \n \\(807\\)\n \n 0\n \n 0\n
      \n \\(808\\)\n \n 0\n \n 0\n
      \n \\(809\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(810\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
      \n \\(811\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(812\\)\n \n 0\n \n 0\n
      \n \\(813\\)\n \n 0\n \n 0\n
      \n \\(814\\)\n \n 0\n \n 0\n
      \n \\(815\\)\n \n 0\n \n 0\n
      \n \\(816\\)\n \n 0\n \n 0\n
      \n \\(817\\)\n \n 0\n \n 0\n
      \n \\(818\\)\n \n \u2212\n \n 2.00000i\n \u2212\n \n 2.00000i
      \n \\(819\\)\n \n 0\n \n 0\n
      \n \\(820\\)\n \n 2.00000\n \n 2.00000\n
      \n \\(821\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(822\\)\n \n 0\n \n 0\n
      \n \\(823\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(824\\)\n \n 0\n \n 0\n
      \n \\(825\\)\n \n 0\n \n 0\n
      \n \\(826\\)\n \n 0\n \n 0\n
      \n \\(827\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
      \n \\(828\\)\n \n 0\n \n 0\n
      \n \\(829\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \\(830\\)\n \n 0\n \n 0\n
      \n \\(831\\)\n \n 0\n \n 0\n
      \n \\(832\\)\n \n 0\n \n 0\n
      \n \\(833\\)\n \n 0\n \n 0\n
      \n \\(834\\)\n \n 0\n \n 0\n
      \n \\(835\\)\n \n 0\n \n 0\n
      \n \\(836\\)\n \n 0\n \n 0\n
      \n \\(837\\)\n \n 0\n \n 0\n
      \n \\(838\\)\n \n 0\n \n 0\n
      \n \\(839\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
      \n \\(840\\)\n \n 0\n \n 0\n
      \n \\(841\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
      \n \\(842\\)\n \n 0\n \n 0\n
      \n \\(843\\)\n \n 0\n \n 0\n
      \n \\(844\\)\n \n 0\n \n 0\n
      \n \\(845\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(846\\)\n \n 0\n \n 0\n
      \n \\(847\\)\n \n 0\n \n 0\n
      \n \\(848\\)\n \n 0\n \n 0\n
      \n \\(849\\)\n \n 0\n \n 0\n
      \n \\(850\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
      \n \\(851\\)\n \n 0\n \n 0\n
      \n \\(852\\)\n \n 0\n \n 0\n
      \n \\(853\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n \\(0\\)\n
      \n \\(854\\)\n \n 0\n \n 0\n
      \n \\(855\\)\n \n 0\n \n 0\n
      \n \\(856\\)\n \n 0\n \n 0\n
      \n \\(857\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(858\\)\n \n 0\n \n 0\n
      \n \\(859\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(860\\)\n \n 0\n \n 0\n
      \n \\(861\\)\n \n 0\n \n 0\n
      \n \\(862\\)\n \n 0\n \n 0\n
      \n \\(863\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(864\\)\n \n 0\n \n 0\n
      \n \\(865\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i
      \n \\(866\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
      \n \\(867\\)\n \n 0\n \n 0\n
      \n \\(868\\)\n \n 0\n \n 0\n
      \n \\(869\\)\n \n 0\n \n 0\n
      \n \\(870\\)\n \n 0\n \n 0\n
      \n \\(871\\)\n \n 0\n \n 0\n
      \n \\(872\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(873\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
      \n \\(874\\)\n \n 0\n \n 0\n
      \n \\(875\\)\n \n 0\n \n 0\n
      \n \\(876\\)\n \n 0\n \n 0\n
      \n \\(877\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(878\\)\n \n 0\n \n 0\n
      \n \\(879\\)\n \n 0\n \n 0\n
      \n \\(880\\)\n \n 0\n \n 0\n
      \n \\(881\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(882\\)\n \n 0\n \n 0\n
      \n \\(883\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(884\\)\n \n 0\n \n 0\n
      \n \\(885\\)\n \n 0\n \n 0\n
      \n \\(886\\)\n \n 0\n \n 0\n
      \n \\(887\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(888\\)\n \n 0\n \n 0\n
      \n \\(889\\)\n \n 0\n \n 0\n
      \n \\(890\\)\n \n 0\n \n 0\n
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      \n \\(894\\)\n \n 0\n \n 0\n
      \n \\(895\\)\n \n 0\n \n 0\n
      \n \\(896\\)\n \n 0\n \n 0\n
      \n \\(897\\)\n \n 0\n \n 0\n
      \n \\(898\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(899\\)\n \n 0\n \n 0\n
      \n \\(900\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
      \n \\(901\\)\n \n 0\n \n 0\n
      \n \\(902\\)\n \n 0\n \n 0\n
      \n \\(903\\)\n \n 0\n \n 0\n
      \n \\(904\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
      \n \\(905\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
      \n \\(906\\)\n \n 0\n \n 0\n
      \n \\(907\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(908\\)\n \n 0\n \n 0\n
      \n \\(909\\)\n \n 0\n \n 0\n
      \n \\(910\\)\n \n 0\n \n 0\n
      \n \\(911\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
      \n \\(912\\)\n \n 0\n \n 0\n
      \n \\(913\\)\n \n 0\n \n 0\n
      \n \\(914\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
      \n \\(915\\)\n \n 0\n \n 0\n
      \n \\(916\\)\n \n 2.00000i\n 2.00000i
      \n \\(917\\)\n \n 0\n \n 0\n
      \n \\(918\\)\n \n 0\n \n 0\n
      \n \\(919\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(920\\)\n \n 0\n \n 0\n
      \n \\(921\\)\n \n 0\n \n 0\n
      \n \\(922\\)\n \n 0\n \n 0\n
      \n \\(923\\)\n \n 0\n \n 0\n
      \n \\(924\\)\n \n 0\n \n 0\n
      \n \\(925\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(926\\)\n \n 0\n \n 0\n
      \n \\(927\\)\n \n 0\n \n 0\n
      \n \\(928\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
      \n \\(929\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(930\\)\n \n 0\n \n 0\n
      \n \\(931\\)\n \n 0\n \n 0\n
      \n \\(932\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
      \n \\(933\\)\n \n 0\n \n 0\n
      \n \\(934\\)\n \n 0\n \n 0\n
      \n \\(935\\)\n \n 0\n \n 0\n
      \n \\(936\\)\n \n 0\n \n 0\n
      \n \\(937\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(938\\)\n \n 0\n \n 0\n
      \n \\(939\\)\n \n 0\n \n 0\n
      \n \\(940\\)\n \n 0\n \n 0\n
      \n \\(941\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(942\\)\n \n 0\n \n 0\n
      \n \\(943\\)\n \n 0\n \n 0\n
      \n \\(944\\)\n \n 0\n \n 0\n
      \n \\(945\\)\n \n 0\n \n 0\n
      \n \\(946\\)\n \n 0\n \n 0\n
      \n \\(947\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(948\\)\n \n 0\n \n 0\n
      \n \\(949\\)\n \n 0\n \n 0\n
      \n \\(950\\)\n \n 0\n \n 0\n
      \n \\(951\\)\n \n 0\n \n 0\n
      \n \\(952\\)\n \n 0\n \n 0\n
      \n \\(953\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(954\\)\n \n 0\n \n 0\n
      \n \\(955\\)\n \n 0\n \n 0\n
      \n \\(956\\)\n \n 0\n \n 0\n
      \n \\(957\\)\n \n 0\n \n 0\n
      \n \\(958\\)\n \n 0\n \n 0\n
      \n \\(959\\)\n \n 0\n \n 0\n
      \n \\(960\\)\n \n 0\n \n 0\n
      \n \\(961\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
      \n \\(962\\)\n \n 0\n \n 0\n
      \n \\(963\\)\n \n 0\n \n 0\n
      \n \\(964\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
      \n \\(965\\)\n \n 2.00000\n \n 2.00000\n
      \n \\(966\\)\n \n 0\n \n 0\n
      \n \\(967\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(968\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
      \n \\(969\\)\n \n 0\n \n 0\n
      \n \\(970\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
      \n \\(971\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(972\\)\n \n 0\n \n 0\n
      \n \\(973\\)\n \n 0\n \n 0\n
      \n \\(974\\)\n \n 0\n \n 0\n
      \n \\(975\\)\n \n 0\n \n 0\n
      \n \\(976\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
      \n \\(977\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(978\\)\n \n 0\n \n 0\n
      \n \\(979\\)\n \n 0\n \n 0\n
      \n \\(980\\)\n \n 0\n \n 0\n
      \n \\(981\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
      \n \\(982\\)\n \n 0\n \n 0\n
      \n \\(983\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(984\\)\n \n 0\n \n 0\n
      \n \\(985\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
      \n \\(986\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
      \n \\(987\\)\n \n 0\n \n 0\n
      \n \\(988\\)\n \n 0\n \n 0\n
      \n \\(989\\)\n \n 0\n \n 0\n
      \n \\(990\\)\n \n 0\n \n 0\n
      \n \\(991\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(992\\)\n \n 0\n \n 0\n
      \n \\(993\\)\n \n 0\n \n 0\n
      \n \\(994\\)\n \n 0\n \n 0\n
      \n \\(995\\)\n \n 0\n \n 0\n
      \n \\(996\\)\n \n 0\n \n 0\n
      \n \\(997\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(998\\)\n \n 0\n \n 0\n
      \n \\(999\\)\n \n 0\n \n 0\n
      ", "content": {"html": "
      \\(n\\)\\(a_n\\)\\(a_n / n^{(k-1)/2}\\)\\( \\alpha_n \\)\\( \\theta_n \\)
      \\(p\\)\\(a_p\\)\\(a_p / p^{(k-1)/2}\\)\\( \\alpha_p\\)\\( \\theta_p \\)
      \\(2\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
      \\(3\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
      0.258819+0.965926<br>i\\(0.416667\\pi\\)
      \\(4\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
      \\(5\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(6\\)00
      \\(7\\)00
      \\(8\\)1.00000<br>i1.00000<br>i
      \\(9\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
      \\(10\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
      \\(11\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(12\\)00
      \\(13\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(14\\)00
      \\(15\\)00
      \\(16\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
      \\(17\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
      \\(18\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
      \\(19\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(20\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
      \\(21\\)00
      \\(22\\)00
      \\(23\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
      \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
      \\(24\\)00
      \\(25\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
      \\(26\\)00
      \\(27\\)00
      \\(28\\)00
      \\(29\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000\\(0\\)
      \\(30\\)00
      \\(31\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
      0.258819+0.965926<br>i\\(0.416667\\pi\\)
      \\(32\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
      \\(33\\)00
      \\(34\\)\u22121.00000<br>i\u22121.00000<br>i
      \\(35\\)00
      \\(36\\)1.00000<br>i1.00000<br>i
      \\(37\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(38\\)00
      \\(39\\)00
      \\(40\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
      \\(41\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
      \u22121.00000\\(\\pi\\)
      \\(42\\)00
      \\(43\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(44\\)00
      \\(45\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
      \\(46\\)00
      \\(47\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(48\\)00
      \\(49\\)00
      \\(50\\)\u22121.00000\u22121.00000
      \\(51\\)00
      \\(52\\)00
      \\(53\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(54\\)00
      \\(55\\)00
      \\(56\\)00
      \\(57\\)00
      \\(58\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
      \\(59\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(60\\)00
      \\(61\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \\(62\\)00
      \\(63\\)00
      \\(64\\)\u22121.00000\u22121.00000
      \\(65\\)00
      \\(66\\)00
      \\(67\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(68\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
      \\(69\\)00
      \\(70\\)00
      \\(71\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
      \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
      \\(72\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
      \\(73\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \\(74\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
      \\(75\\)00
      \\(76\\)00
      \\(77\\)00
      \\(78\\)00
      \\(79\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
      \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
      \\(80\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
      \\(81\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
      \\(82\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
      \\(83\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(84\\)00
      \\(85\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
      \\(86\\)00
      \\(87\\)00
      \\(88\\)00
      \\(89\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(90\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
      \\(91\\)00
      \\(92\\)00
      \\(93\\)00
      \\(94\\)00
      \\(95\\)00
      \\(96\\)00
      \\(97\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000\\(0\\)
      \\(98\\)00
      \\(99\\)00
      \\(100\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
      \\(101\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(102\\)00
      \\(103\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(104\\)00
      \\(105\\)00
      \\(106\\)00
      \\(107\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
      0.965926+0.258819<br>i\\(0.0833333\\pi\\)
      \\(108\\)00
      \\(109\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(110\\)00
      \\(111\\)00
      \\(112\\)00
      \\(113\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
      \u22121.00000\\(\\pi\\)
      \\(114\\)00
      \\(115\\)00
      \\(116\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
      \\(117\\)00
      \\(118\\)00
      \\(119\\)00
      \\(120\\)00
      \\(121\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
      \\(122\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
      \\(123\\)00
      \\(124\\)00
      \\(125\\)00
      \\(126\\)00
      \\(127\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(128\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
      \\(129\\)00
      \\(130\\)00
      \\(131\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
      0.965926+0.258819<br>i\\(0.0833333\\pi\\)
      \\(132\\)00
      \\(133\\)00
      \\(134\\)00
      \\(135\\)00
      \\(136\\)\u22120.866025\u22120.500000<br>i\u22120.866025\u22120.500000<br>i
      \\(137\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(138\\)00
      \\(139\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
      \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
      \\(140\\)00
      \\(141\\)00
      \\(142\\)00
      \\(143\\)00
      \\(144\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
      \\(145\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
      \\(146\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
      \\(147\\)00
      \\(148\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
      \\(149\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      \u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      \\(150\\)00
      \\(151\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(152\\)00
      \\(153\\)\u22121.00000<br>i\u22121.00000<br>i
      \\(154\\)00
      \\(155\\)00
      \\(156\\)00
      \\(157\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \\(158\\)00
      \\(159\\)00
      \\(160\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
      \\(161\\)00
      \\(162\\)1.00000<br>i1.00000<br>i
      \\(163\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
      \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
      \\(164\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
      \\(165\\)00
      \\(166\\)00
      \\(167\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
      \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
      \\(168\\)00
      \\(169\\)\u22121.00000\u22121.00000
      \\(170\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
      \\(171\\)00
      \\(172\\)00
      \\(173\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(174\\)00
      \\(175\\)00
      \\(176\\)00
      \\(177\\)00
      \\(178\\)00
      \\(179\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(180\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
      \\(181\\)1.00000+1.00000<br>i1.00000+1.00000<br>i1.00000\\(0\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(182\\)00
      \\(183\\)00
      \\(184\\)00
      \\(185\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i
      \\(186\\)00
      \\(187\\)00
      \\(188\\)00
      \\(189\\)00
      \\(190\\)00
      \\(191\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(192\\)00
      \\(193\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(194\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
      \\(195\\)00
      \\(196\\)00
      \\(197\\)1.00000+1.00000<br>i1.00000+1.00000<br>i1.00000\\(0\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(198\\)00
      \\(199\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
      0.258819+0.965926<br>i\\(0.416667\\pi\\)
      \\(200\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
      \\(201\\)00
      \\(202\\)00
      \\(203\\)00
      \\(204\\)00
      \\(205\\)1.00000+1.73205<br>i1.00000+1.73205<br>i
      \\(206\\)00
      \\(207\\)00
      \\(208\\)00
      \\(209\\)00
      \\(210\\)00
      \\(211\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
      0.707107+0.707107<br>i\\(0.250000\\pi\\)
      \\(212\\)00
      \\(213\\)00
      \\(214\\)00
      \\(215\\)00
      \\(216\\)00
      \\(217\\)00
      \\(218\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
      \\(219\\)00
      \\(220\\)00
      \\(221\\)00
      \\(222\\)00
      \\(223\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(224\\)00
      \\(225\\)\u22121.00000\u22121.00000
      \\(226\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
      \\(227\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(228\\)00
      \\(229\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i\u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(230\\)00
      \\(231\\)00
      \\(232\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
      \\(233\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(234\\)00
      \\(235\\)00
      \\(236\\)00
      \\(237\\)00
      \\(238\\)00
      \\(239\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(240\\)00
      \\(241\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \\(242\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
      \\(243\\)00
      \\(244\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
      \\(245\\)00
      \\(246\\)00
      \\(247\\)00
      \\(248\\)00
      \\(249\\)00
      \\(250\\)00
      \\(251\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(252\\)00
      \\(253\\)00
      \\(254\\)00
      \\(255\\)00
      \\(256\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
      \\(257\\)1.73205\u22121.00000<br>i1.73205\u22121.00000<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \\(258\\)00
      \\(259\\)00
      \\(260\\)00
      \\(261\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
      \\(262\\)00
      \\(263\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(264\\)00
      \\(265\\)00
      \\(266\\)00
      \\(267\\)00
      \\(268\\)00
      \\(269\\)0.366025+1.36603<br>i0.366025+1.36603<br>i0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(270\\)00
      \\(271\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(272\\)1.000001.00000
      \\(273\\)00
      \\(274\\)00
      \\(275\\)00
      \\(276\\)00
      \\(277\\)0.366025+1.36603<br>i0.366025+1.36603<br>i0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(278\\)00
      \\(279\\)00
      \\(280\\)00
      \\(281\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(282\\)00
      \\(283\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(284\\)00
      \\(285\\)00
      \\(286\\)00
      \\(287\\)00
      \\(288\\)\u22121.00000\u22121.00000
      \\(289\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
      \\(290\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
      \\(291\\)00
      \\(292\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
      \\(293\\)2.000002.000001.00000\\(0\\)
      1.00000\\(0\\)
      \\(294\\)00
      \\(295\\)00
      \\(296\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
      \\(297\\)00
      \\(298\\)1.73205+1.00000<br>i1.73205+1.00000<br>i
      \\(299\\)00
      \\(300\\)00
      \\(301\\)00
      \\(302\\)00
      \\(303\\)00
      \\(304\\)00
      \\(305\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i
      \\(306\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
      \\(307\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(308\\)00
      \\(309\\)00
      \\(310\\)00
      \\(311\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
      0.258819+0.965926<br>i\\(0.416667\\pi\\)
      \\(312\\)00
      \\(313\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(314\\)2.00000<br>i2.00000<br>i
      \\(315\\)00
      \\(316\\)00
      \\(317\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \\(318\\)00
      \\(319\\)00
      \\(320\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
      \\(321\\)00
      \\(322\\)00
      \\(323\\)00
      \\(324\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
      \\(325\\)00
      \\(326\\)00
      \\(327\\)00
      \\(328\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
      \\(329\\)00
      \\(330\\)00
      \\(331\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(332\\)00
      \\(333\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
      \\(334\\)00
      \\(335\\)00
      \\(336\\)00
      \\(337\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000\\(0\\)
      \\(338\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
      \\(339\\)00
      \\(340\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
      \\(341\\)00
      \\(342\\)00
      \\(343\\)00
      \\(344\\)00
      \\(345\\)00
      \\(346\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
      \\(347\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(348\\)00
      \\(349\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(350\\)00
      \\(351\\)00
      \\(352\\)00
      \\(353\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i\u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(354\\)00
      \\(355\\)00
      \\(356\\)00
      \\(357\\)00
      \\(358\\)00
      \\(359\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(360\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
      \\(361\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
      \\(362\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
      \\(363\\)00
      \\(364\\)00
      \\(365\\)2.00000<br>i2.00000<br>i
      \\(366\\)00
      \\(367\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(368\\)00
      \\(369\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
      \\(370\\)2.000002.00000
      \\(371\\)00
      \\(372\\)00
      \\(373\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(374\\)00
      \\(375\\)00
      \\(376\\)00
      \\(377\\)00
      \\(378\\)00
      \\(379\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
      \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
      \\(380\\)00
      \\(381\\)00
      \\(382\\)00
      \\(383\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(384\\)00
      \\(385\\)00
      \\(386\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
      \\(387\\)00
      \\(388\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
      \\(389\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      \u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      \\(390\\)00
      \\(391\\)00
      \\(392\\)00
      \\(393\\)00
      \\(394\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
      \\(395\\)00
      \\(396\\)00
      \\(397\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \\(398\\)00
      \\(399\\)00
      \\(400\\)\u22121.00000<br>i\u22121.00000<br>i
      \\(401\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \\(402\\)00
      \\(403\\)00
      \\(404\\)00
      \\(405\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
      \\(406\\)00
      \\(407\\)00
      \\(408\\)00
      \\(409\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i\u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(410\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i
      \\(411\\)00
      \\(412\\)00
      \\(413\\)00
      \\(414\\)00
      \\(415\\)00
      \\(416\\)00
      \\(417\\)00
      \\(418\\)00
      \\(419\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
      0.707107+0.707107<br>i\\(0.250000\\pi\\)
      \\(420\\)00
      \\(421\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(422\\)00
      \\(423\\)00
      \\(424\\)00
      \\(425\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
      \\(426\\)00
      \\(427\\)00
      \\(428\\)00
      \\(429\\)00
      \\(430\\)00
      \\(431\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(432\\)00
      \\(433\\)2.00000<br>i2.00000<br>i1.00000<br>i\\(0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(434\\)00
      \\(435\\)00
      \\(436\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
      \\(437\\)00
      \\(438\\)00
      \\(439\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
      0.965926+0.258819<br>i\\(0.0833333\\pi\\)
      \\(440\\)00
      \\(441\\)00
      \\(442\\)00
      \\(443\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(444\\)00
      \\(445\\)00
      \\(446\\)00
      \\(447\\)00
      \\(448\\)00
      \\(449\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
      \u22121.00000\\(\\pi\\)
      \\(450\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
      \\(451\\)00
      \\(452\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
      \\(453\\)00
      \\(454\\)00
      \\(455\\)00
      \\(456\\)00
      \\(457\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i\u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(458\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
      \\(459\\)00
      \\(460\\)00
      \\(461\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(462\\)00
      \\(463\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(464\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
      \\(465\\)00
      \\(466\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
      \\(467\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(468\\)00
      \\(469\\)00
      \\(470\\)00
      \\(471\\)00
      \\(472\\)00
      \\(473\\)00
      \\(474\\)00
      \\(475\\)00
      \\(476\\)00
      \\(477\\)00
      \\(478\\)00
      \\(479\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
      0.258819+0.965926<br>i\\(0.416667\\pi\\)
      \\(480\\)00
      \\(481\\)00
      \\(482\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
      \\(483\\)00
      \\(484\\)\u22121.00000<br>i\u22121.00000<br>i
      \\(485\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
      \\(486\\)00
      \\(487\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(488\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
      \\(489\\)00
      \\(490\\)00
      \\(491\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(492\\)00
      \\(493\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
      \\(494\\)00
      \\(495\\)00
      \\(496\\)00
      \\(497\\)00
      \\(498\\)00
      \\(499\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
      \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
      \\(500\\)00
      \\(501\\)00
      \\(502\\)00
      \\(503\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
      0.707107+0.707107<br>i\\(0.250000\\pi\\)
      \\(504\\)00
      \\(505\\)00
      \\(506\\)00
      \\(507\\)00
      \\(508\\)00
      \\(509\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      \u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      \\(510\\)00
      \\(511\\)00
      \\(512\\)\u22121.00000<br>i\u22121.00000<br>i
      \\(513\\)00
      \\(514\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i
      \\(515\\)00
      \\(516\\)00
      \\(517\\)00
      \\(518\\)00
      \\(519\\)00
      \\(520\\)00
      \\(521\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(522\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
      \\(523\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(524\\)00
      \\(525\\)00
      \\(526\\)00
      \\(527\\)00
      \\(528\\)00
      \\(529\\)\u22120.866025\u22120.500000<br>i\u22120.866025\u22120.500000<br>i
      \\(530\\)00
      \\(531\\)00
      \\(532\\)00
      \\(533\\)00
      \\(534\\)00
      \\(535\\)00
      \\(536\\)00
      \\(537\\)00
      \\(538\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
      \\(539\\)00
      \\(540\\)00
      \\(541\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(542\\)00
      \\(543\\)00
      \\(544\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
      \\(545\\)\u22122.00000\u22122.00000
      \\(546\\)00
      \\(547\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
      \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
      \\(548\\)00
      \\(549\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
      \\(550\\)00
      \\(551\\)00
      \\(552\\)00
      \\(553\\)00
      \\(554\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
      \\(555\\)00
      \\(556\\)00
      \\(557\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(558\\)00
      \\(559\\)00
      \\(560\\)00
      \\(561\\)00
      \\(562\\)00
      \\(563\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(564\\)00
      \\(565\\)1.00000+1.73205<br>i1.00000+1.73205<br>i
      \\(566\\)00
      \\(567\\)00
      \\(568\\)00
      \\(569\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(570\\)00
      \\(571\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(572\\)00
      \\(573\\)00
      \\(574\\)00
      \\(575\\)00
      \\(576\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
      \\(577\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(578\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
      \\(579\\)00
      \\(580\\)2.00000<br>i2.00000<br>i
      \\(581\\)00
      \\(582\\)00
      \\(583\\)00
      \\(584\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
      \\(585\\)00
      \\(586\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
      \\(587\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(588\\)00
      \\(589\\)00
      \\(590\\)00
      \\(591\\)00
      \\(592\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
      \\(593\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(594\\)00
      \\(595\\)00
      \\(596\\)\u22122.00000\u22122.00000
      \\(597\\)00
      \\(598\\)00
      \\(599\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(600\\)00
      \\(601\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
      \u22121.00000\\(\\pi\\)
      \\(602\\)00
      \\(603\\)00
      \\(604\\)00
      \\(605\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
      \\(606\\)00
      \\(607\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
      0.965926+0.258819<br>i\\(0.0833333\\pi\\)
      \\(608\\)00
      \\(609\\)00
      \\(610\\)\u22122.00000<br>i\u22122.00000<br>i
      \\(611\\)00
      \\(612\\)\u22120.866025\u22120.500000<br>i\u22120.866025\u22120.500000<br>i
      \\(613\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i\u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(614\\)00
      \\(615\\)00
      \\(616\\)00
      \\(617\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i1.00000<br>i\\(0.5\\pi\\)
      \u22121.00000\\(\\pi\\)
      \\(618\\)00
      \\(619\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(620\\)00
      \\(621\\)00
      \\(622\\)00
      \\(623\\)00
      \\(624\\)00
      \\(625\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
      \\(626\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
      \\(627\\)00
      \\(628\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
      \\(629\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
      \\(630\\)00
      \\(631\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(632\\)00
      \\(633\\)00
      \\(634\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
      \\(635\\)00
      \\(636\\)00
      \\(637\\)00
      \\(638\\)00
      \\(639\\)00
      \\(640\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
      \\(641\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \\(642\\)00
      \\(643\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
      \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
      \\(644\\)00
      \\(645\\)00
      \\(646\\)00
      \\(647\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(648\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
      \\(649\\)00
      \\(650\\)00
      \\(651\\)00
      \\(652\\)00
      \\(653\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \\(654\\)00
      \\(655\\)00
      \\(656\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
      \\(657\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
      \\(658\\)00
      \\(659\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(660\\)00
      \\(661\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(662\\)00
      \\(663\\)00
      \\(664\\)00
      \\(665\\)00
      \\(666\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
      \\(667\\)00
      \\(668\\)00
      \\(669\\)00
      \\(670\\)00
      \\(671\\)00
      \\(672\\)00
      \\(673\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
      \u22121.00000\\(\\pi\\)
      \\(674\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
      \\(675\\)00
      \\(676\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
      \\(677\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(678\\)00
      \\(679\\)00
      \\(680\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
      \\(681\\)00
      \\(682\\)00
      \\(683\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
      0.258819+0.965926<br>i\\(0.416667\\pi\\)
      \\(684\\)00
      \\(685\\)00
      \\(686\\)00
      \\(687\\)00
      \\(688\\)00
      \\(689\\)00
      \\(690\\)00
      \\(691\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
      \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
      \\(692\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
      \\(693\\)00
      \\(694\\)00
      \\(695\\)00
      \\(696\\)00
      \\(697\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
      \\(698\\)00
      \\(699\\)00
      \\(700\\)00
      \\(701\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(702\\)00
      \\(703\\)00
      \\(704\\)00
      \\(705\\)00
      \\(706\\)\u22122.00000<br>i\u22122.00000<br>i
      \\(707\\)00
      \\(708\\)00
      \\(709\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(710\\)00
      \\(711\\)00
      \\(712\\)00
      \\(713\\)00
      \\(714\\)00
      \\(715\\)00
      \\(716\\)00
      \\(717\\)00
      \\(718\\)00
      \\(719\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
      0.965926+0.258819<br>i\\(0.0833333\\pi\\)
      \\(720\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
      \\(721\\)00
      \\(722\\)1.00000<br>i1.00000<br>i
      \\(723\\)00
      \\(724\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
      \\(725\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
      \\(726\\)00
      \\(727\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(728\\)00
      \\(729\\)1.00000<br>i1.00000<br>i
      \\(730\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
      \\(731\\)00
      \\(732\\)00
      \\(733\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(734\\)00
      \\(735\\)00
      \\(736\\)00
      \\(737\\)00
      \\(738\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
      \\(739\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(740\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
      \\(741\\)00
      \\(742\\)00
      \\(743\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
      \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
      \\(744\\)00
      \\(745\\)0.732051+2.73205<br>i0.732051+2.73205<br>i
      \\(746\\)00
      \\(747\\)00
      \\(748\\)00
      \\(749\\)00
      \\(750\\)00
      \\(751\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
      0.965926+0.258819<br>i\\(0.0833333\\pi\\)
      \\(752\\)00
      \\(753\\)00
      \\(754\\)00
      \\(755\\)00
      \\(756\\)00
      \\(757\\)2.00000<br>i2.00000<br>i1.00000<br>i\\(0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(758\\)00
      \\(759\\)00
      \\(760\\)00
      \\(761\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(762\\)00
      \\(763\\)00
      \\(764\\)00
      \\(765\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
      \\(766\\)00
      \\(767\\)00
      \\(768\\)00
      \\(769\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(770\\)00
      \\(771\\)00
      \\(772\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
      \\(773\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      \u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      \\(774\\)00
      \\(775\\)00
      \\(776\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
      \\(777\\)00
      \\(778\\)2.000002.00000
      \\(779\\)00
      \\(780\\)00
      \\(781\\)00
      \\(782\\)00
      \\(783\\)00
      \\(784\\)00
      \\(785\\)\u22122.00000+2.00000<br>i\u22122.00000+2.00000<br>i
      \\(786\\)00
      \\(787\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
      0.258819+0.965926<br>i\\(0.416667\\pi\\)
      \\(788\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
      \\(789\\)00
      \\(790\\)00
      \\(791\\)00
      \\(792\\)00
      \\(793\\)00
      \\(794\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
      \\(795\\)00
      \\(796\\)00
      \\(797\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(798\\)00
      \\(799\\)00
      \\(800\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
      \\(801\\)00
      \\(802\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
      \\(803\\)00
      \\(804\\)00
      \\(805\\)00
      \\(806\\)00
      \\(807\\)00
      \\(808\\)00
      \\(809\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(810\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
      \\(811\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
      \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
      \\(812\\)00
      \\(813\\)00
      \\(814\\)00
      \\(815\\)00
      \\(816\\)00
      \\(817\\)00
      \\(818\\)\u22122.00000<br>i\u22122.00000<br>i
      \\(819\\)00
      \\(820\\)2.000002.00000
      \\(821\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(822\\)00
      \\(823\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(824\\)00
      \\(825\\)00
      \\(826\\)00
      \\(827\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
      0.707107+0.707107<br>i\\(0.250000\\pi\\)
      \\(828\\)00
      \\(829\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \\(830\\)00
      \\(831\\)00
      \\(832\\)00
      \\(833\\)00
      \\(834\\)00
      \\(835\\)00
      \\(836\\)00
      \\(837\\)00
      \\(838\\)00
      \\(839\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
      0.707107+0.707107<br>i\\(0.250000\\pi\\)
      \\(840\\)00
      \\(841\\)\u22121.00000<br>i\u22121.00000<br>i
      \\(842\\)00
      \\(843\\)00
      \\(844\\)00
      \\(845\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
      \\(846\\)00
      \\(847\\)00
      \\(848\\)00
      \\(849\\)00
      \\(850\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
      \\(851\\)00
      \\(852\\)00
      \\(853\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000\\(0\\)
      \\(854\\)00
      \\(855\\)00
      \\(856\\)00
      \\(857\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(858\\)00
      \\(859\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(860\\)00
      \\(861\\)00
      \\(862\\)00
      \\(863\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(864\\)00
      \\(865\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i
      \\(866\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
      \\(867\\)00
      \\(868\\)00
      \\(869\\)00
      \\(870\\)00
      \\(871\\)00
      \\(872\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
      \\(873\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
      \\(874\\)00
      \\(875\\)00
      \\(876\\)00
      \\(877\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(878\\)00
      \\(879\\)00
      \\(880\\)00
      \\(881\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i1.00000<br>i\\(0.5\\pi\\)
      \u22121.00000\\(\\pi\\)
      \\(882\\)00
      \\(883\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(884\\)00
      \\(885\\)00
      \\(886\\)00
      \\(887\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
      0.965926+0.258819<br>i\\(0.0833333\\pi\\)
      \\(888\\)00
      \\(889\\)00
      \\(890\\)00
      \\(891\\)00
      \\(892\\)00
      \\(893\\)00
      \\(894\\)00
      \\(895\\)00
      \\(896\\)00
      \\(897\\)00
      \\(898\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
      \\(899\\)00
      \\(900\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
      \\(901\\)00
      \\(902\\)00
      \\(903\\)00
      \\(904\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
      \\(905\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
      \\(906\\)00
      \\(907\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(908\\)00
      \\(909\\)00
      \\(910\\)00
      \\(911\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
      0.707107+0.707107<br>i\\(0.250000\\pi\\)
      \\(912\\)00
      \\(913\\)00
      \\(914\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
      \\(915\\)00
      \\(916\\)2.00000<br>i2.00000<br>i
      \\(917\\)00
      \\(918\\)00
      \\(919\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(920\\)00
      \\(921\\)00
      \\(922\\)00
      \\(923\\)00
      \\(924\\)00
      \\(925\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
      \\(926\\)00
      \\(927\\)00
      \\(928\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
      \\(929\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(930\\)00
      \\(931\\)00
      \\(932\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
      \\(933\\)00
      \\(934\\)00
      \\(935\\)00
      \\(936\\)00
      \\(937\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(938\\)00
      \\(939\\)00
      \\(940\\)00
      \\(941\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(942\\)00
      \\(943\\)00
      \\(944\\)00
      \\(945\\)00
      \\(946\\)00
      \\(947\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
      0.965926+0.258819<br>i\\(0.0833333\\pi\\)
      \\(948\\)00
      \\(949\\)00
      \\(950\\)00
      \\(951\\)00
      \\(952\\)00
      \\(953\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(954\\)00
      \\(955\\)00
      \\(956\\)00
      \\(957\\)00
      \\(958\\)00
      \\(959\\)00
      \\(960\\)00
      \\(961\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
      \\(962\\)00
      \\(963\\)00
      \\(964\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
      \\(965\\)2.000002.00000
      \\(966\\)00
      \\(967\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(968\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
      \\(969\\)00
      \\(970\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
      \\(971\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(972\\)00
      \\(973\\)00
      \\(974\\)00
      \\(975\\)00
      \\(976\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
      \\(977\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(978\\)00
      \\(979\\)00
      \\(980\\)00
      \\(981\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
      \\(982\\)00
      \\(983\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
      0.258819+0.965926<br>i\\(0.416667\\pi\\)
      \\(984\\)00
      \\(985\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
      \\(986\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
      \\(987\\)00
      \\(988\\)00
      \\(989\\)00
      \\(990\\)00
      \\(991\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
      0.258819+0.965926<br>i\\(0.416667\\pi\\)
      \\(992\\)00
      \\(993\\)00
      \\(994\\)00
      \\(995\\)00
      \\(996\\)00
      \\(997\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(998\\)00
      \\(999\\)00
      ", "is_complex": true, "table_nest_level": "1"}}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See \\(a_n\\) instead)", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See \\(a_n\\) instead)", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See \\(a_n\\) instead)", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See only \\(a_p\\))", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See only \\(a_p\\))", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "( See only \\(a_p\\))", "t": "text"}]}, {"type": "table", "raw_content": "
      \u2003\u2003\u2003\u2003\u2003\u2003\u2003By twisting character
      CharParityOrdTypeTwistMinDim
      1.1even1trivial3332.1.bc.b.2027.14
      4.3odd2CM3332.1.bc.b.2027.14
      7.2even3inner3332.1.bc.b.667.14
      7.3odd668.1.f.a.55.1yes2
      7.4even33332.1.m.b.2843.12
      7.5odd63332.1.bc.c.667.14
      7.6odd23332.1.bc.c.2027.14
      17.13even4inner3332.1.bc.b.2223.14
      21.17even6612.1.l.a.55.12
      28.3even668.1.f.a.55.1yes2
      28.11odd63332.1.m.b.2843.12
      28.19even63332.1.bc.c.667.14
      28.23odd6inner3332.1.bc.b.667.14
      28.27even23332.1.bc.c.2027.14
      35.3even121700.1.n.a.599.12
      35.17even121700.1.n.b.599.12
      35.24odd61700.1.p.a.1551.12
      56.3even61088.1.p.a.191.12
      56.45odd61088.1.p.a.191.12
      68.47odd4inner3332.1.bc.b.2223.14
      84.59odd6612.1.l.a.55.12
      119.3even481156.1.g.b.155.18
      119.10even481156.1.g.b.179.18
      119.13odd43332.1.bc.c.2223.14
      119.24even481156.1.g.b.179.28
      119.30even12inner3332.1.bc.b.863.14
      119.31even481156.1.g.b.155.28
      119.38odd121156.1.f.b.251.12
      119.45even481156.1.g.b.399.18
      119.47odd123332.1.bc.c.863.14
      119.59odd241156.1.c.b.579.12
      119.66odd241156.1.d.a.1155.22
      119.73even481156.1.g.b.423.28
      119.80even481156.1.g.b.423.18
      119.81even123332.1.m.b.3039.12
      119.87odd241156.1.d.a.1155.12
      119.94odd241156.1.c.b.579.22
      119.101odd61156.1.f.b.327.12
      119.108even481156.1.g.b.399.28
      119.115odd1268.1.f.a.47.1&check;2
      140.3odd121700.1.n.a.599.12
      140.59even61700.1.p.a.1551.12
      140.87odd121700.1.n.b.599.12
      357.353even12612.1.l.a.523.12
      476.3odd481156.1.g.b.155.18
      476.31odd481156.1.g.b.155.28
      476.47even123332.1.bc.c.863.14
      476.59even241156.1.c.b.579.12
      476.87even241156.1.d.a.1155.12
      476.115even1268.1.f.a.47.1&check;2
      476.143odd481156.1.g.b.179.28
      476.199odd481156.1.g.b.423.18
      476.227odd481156.1.g.b.399.28
      476.251even43332.1.bc.c.2223.14
      476.283odd481156.1.g.b.399.18
      476.311odd481156.1.g.b.423.28
      476.319odd123332.1.m.b.3039.12
      476.339even61156.1.f.b.327.12
      476.367odd481156.1.g.b.179.18
      476.387odd12inner3332.1.bc.b.863.14
      476.395even121156.1.f.b.251.12
      476.423even241156.1.d.a.1155.22
      476.451even241156.1.c.b.579.22
      595.234odd121700.1.p.a.251.12
      595.353even121700.1.n.b.999.12
      595.472even121700.1.n.a.999.12
      952.115even121088.1.p.a.319.12
      952.829odd121088.1.p.a.319.12
      1428.1067odd12612.1.l.a.523.12
      2380.1067odd121700.1.n.a.999.12
      2380.1543odd121700.1.n.b.999.12
      2380.2019even121700.1.p.a.251.12
      ", "content": {"html": "
      By<br>twisting character
      CharParityOrdTypeTwistMinDim
      1.1even1trivial3332.1.bc.b.2027.14
      4.3odd2CM3332.1.bc.b.2027.14
      7.2even3inner3332.1.bc.b.667.14
      7.3odd668.1.f.a.55.1yes2
      7.4even33332.1.m.b.2843.12
      7.5odd63332.1.bc.c.667.14
      7.6odd23332.1.bc.c.2027.14
      17.13even4inner3332.1.bc.b.2223.14
      21.17even6612.1.l.a.55.12
      28.3even668.1.f.a.55.1yes2
      28.11odd63332.1.m.b.2843.12
      28.19even63332.1.bc.c.667.14
      28.23odd6inner3332.1.bc.b.667.14
      28.27even23332.1.bc.c.2027.14
      35.3even121700.1.n.a.599.12
      35.17even121700.1.n.b.599.12
      35.24odd61700.1.p.a.1551.12
      56.3even61088.1.p.a.191.12
      56.45odd61088.1.p.a.191.12
      68.47odd4inner3332.1.bc.b.2223.14
      84.59odd6612.1.l.a.55.12
      119.3even481156.1.g.b.155.18
      119.10even481156.1.g.b.179.18
      119.13odd43332.1.bc.c.2223.14
      119.24even481156.1.g.b.179.28
      119.30even12inner3332.1.bc.b.863.14
      119.31even481156.1.g.b.155.28
      119.38odd121156.1.f.b.251.12
      119.45even481156.1.g.b.399.18
      119.47odd123332.1.bc.c.863.14
      119.59odd241156.1.c.b.579.12
      119.66odd241156.1.d.a.1155.22
      119.73even481156.1.g.b.423.28
      119.80even481156.1.g.b.423.18
      119.81even123332.1.m.b.3039.12
      119.87odd241156.1.d.a.1155.12
      119.94odd241156.1.c.b.579.22
      119.101odd61156.1.f.b.327.12
      119.108even481156.1.g.b.399.28
      119.115odd1268.1.f.a.47.1&check;2
      140.3odd121700.1.n.a.599.12
      140.59even61700.1.p.a.1551.12
      140.87odd121700.1.n.b.599.12
      357.353even12612.1.l.a.523.12
      476.3odd481156.1.g.b.155.18
      476.31odd481156.1.g.b.155.28
      476.47even123332.1.bc.c.863.14
      476.59even241156.1.c.b.579.12
      476.87even241156.1.d.a.1155.12
      476.115even1268.1.f.a.47.1&check;2
      476.143odd481156.1.g.b.179.28
      476.199odd481156.1.g.b.423.18
      476.227odd481156.1.g.b.399.28
      476.251even43332.1.bc.c.2223.14
      476.283odd481156.1.g.b.399.18
      476.311odd481156.1.g.b.423.28
      476.319odd123332.1.m.b.3039.12
      476.339even61156.1.f.b.327.12
      476.367odd481156.1.g.b.179.18
      476.387odd12inner3332.1.bc.b.863.14
      476.395even121156.1.f.b.251.12
      476.423even241156.1.d.a.1155.22
      476.451even241156.1.c.b.579.22
      595.234odd121700.1.p.a.251.12
      595.353even121700.1.n.b.999.12
      595.472even121700.1.n.a.999.12
      952.115even121088.1.p.a.319.12
      952.829odd121088.1.p.a.319.12
      1428.1067odd12612.1.l.a.523.12
      2380.1067odd121700.1.n.a.999.12
      2380.1543odd121700.1.n.b.999.12
      2380.2019even121700.1.p.a.251.12
      ", "is_complex": true, "table_nest_level": "1"}}, {"type": "table", "raw_content": "
      \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003By twisted newform
      TwistMinDimCharParityOrdType
      68.1.f.a.47.1&check;2119.115odd12
      68.1.f.a.47.1&check;2476.115even12
      68.1.f.a.55.1yes27.3odd6
      68.1.f.a.55.1yes228.3even6
      612.1.l.a.55.1221.17even6
      612.1.l.a.55.1284.59odd6
      612.1.l.a.523.12357.353even12
      612.1.l.a.523.121428.1067odd12
      1088.1.p.a.191.1256.3even6
      1088.1.p.a.191.1256.45odd6
      1088.1.p.a.319.12952.115even12
      1088.1.p.a.319.12952.829odd12
      1156.1.c.b.579.12119.59odd24
      1156.1.c.b.579.12476.59even24
      1156.1.c.b.579.22119.94odd24
      1156.1.c.b.579.22476.451even24
      1156.1.d.a.1155.12119.87odd24
      1156.1.d.a.1155.12476.87even24
      1156.1.d.a.1155.22119.66odd24
      1156.1.d.a.1155.22476.423even24
      1156.1.f.b.251.12119.38odd12
      1156.1.f.b.251.12476.395even12
      1156.1.f.b.327.12119.101odd6
      1156.1.f.b.327.12476.339even6
      1156.1.g.b.155.18119.3even48
      1156.1.g.b.155.18476.3odd48
      1156.1.g.b.155.28119.31even48
      1156.1.g.b.155.28476.31odd48
      1156.1.g.b.179.18119.10even48
      1156.1.g.b.179.18476.367odd48
      1156.1.g.b.179.28119.24even48
      1156.1.g.b.179.28476.143odd48
      1156.1.g.b.399.18119.45even48
      1156.1.g.b.399.18476.283odd48
      1156.1.g.b.399.28119.108even48
      1156.1.g.b.399.28476.227odd48
      1156.1.g.b.423.18119.80even48
      1156.1.g.b.423.18476.199odd48
      1156.1.g.b.423.28119.73even48
      1156.1.g.b.423.28476.311odd48
      1700.1.n.a.599.1235.3even12
      1700.1.n.a.599.12140.3odd12
      1700.1.n.a.999.12595.472even12
      1700.1.n.a.999.122380.1067odd12
      1700.1.n.b.599.1235.17even12
      1700.1.n.b.599.12140.87odd12
      1700.1.n.b.999.12595.353even12
      1700.1.n.b.999.122380.1543odd12
      1700.1.p.a.251.12595.234odd12
      1700.1.p.a.251.122380.2019even12
      1700.1.p.a.1551.1235.24odd6
      1700.1.p.a.1551.12140.59even6
      3332.1.m.b.2843.127.4even3
      3332.1.m.b.2843.1228.11odd6
      3332.1.m.b.3039.12119.81even12
      3332.1.m.b.3039.12476.319odd12
      3332.1.bc.b.667.147.2even3inner
      3332.1.bc.b.667.1428.23odd6inner
      3332.1.bc.b.863.14119.30even12inner
      3332.1.bc.b.863.14476.387odd12inner
      3332.1.bc.b.2027.141.1even1trivial
      3332.1.bc.b.2027.144.3odd2CM
      3332.1.bc.b.2223.1417.13even4inner
      3332.1.bc.b.2223.1468.47odd4inner
      3332.1.bc.c.667.147.5odd6
      3332.1.bc.c.667.1428.19even6
      3332.1.bc.c.863.14119.47odd12
      3332.1.bc.c.863.14476.47even12
      3332.1.bc.c.2027.147.6odd2
      3332.1.bc.c.2027.1428.27even2
      3332.1.bc.c.2223.14119.13odd4
      3332.1.bc.c.2223.14476.251even4
      ", "content": {"html": "
      By<br>twisted newform
      TwistMinDimCharParityOrdType
      68.1.f.a.47.1&check;2119.115odd12
      68.1.f.a.47.1&check;2476.115even12
      68.1.f.a.55.1yes27.3odd6
      68.1.f.a.55.1yes228.3even6
      612.1.l.a.55.1221.17even6
      612.1.l.a.55.1284.59odd6
      612.1.l.a.523.12357.353even12
      612.1.l.a.523.121428.1067odd12
      1088.1.p.a.191.1256.3even6
      1088.1.p.a.191.1256.45odd6
      1088.1.p.a.319.12952.115even12
      1088.1.p.a.319.12952.829odd12
      1156.1.c.b.579.12119.59odd24
      1156.1.c.b.579.12476.59even24
      1156.1.c.b.579.22119.94odd24
      1156.1.c.b.579.22476.451even24
      1156.1.d.a.1155.12119.87odd24
      1156.1.d.a.1155.12476.87even24
      1156.1.d.a.1155.22119.66odd24
      1156.1.d.a.1155.22476.423even24
      1156.1.f.b.251.12119.38odd12
      1156.1.f.b.251.12476.395even12
      1156.1.f.b.327.12119.101odd6
      1156.1.f.b.327.12476.339even6
      1156.1.g.b.155.18119.3even48
      1156.1.g.b.155.18476.3odd48
      1156.1.g.b.155.28119.31even48
      1156.1.g.b.155.28476.31odd48
      1156.1.g.b.179.18119.10even48
      1156.1.g.b.179.18476.367odd48
      1156.1.g.b.179.28119.24even48
      1156.1.g.b.179.28476.143odd48
      1156.1.g.b.399.18119.45even48
      1156.1.g.b.399.18476.283odd48
      1156.1.g.b.399.28119.108even48
      1156.1.g.b.399.28476.227odd48
      1156.1.g.b.423.18119.80even48
      1156.1.g.b.423.18476.199odd48
      1156.1.g.b.423.28119.73even48
      1156.1.g.b.423.28476.311odd48
      1700.1.n.a.599.1235.3even12
      1700.1.n.a.599.12140.3odd12
      1700.1.n.a.999.12595.472even12
      1700.1.n.a.999.122380.1067odd12
      1700.1.n.b.599.1235.17even12
      1700.1.n.b.599.12140.87odd12
      1700.1.n.b.999.12595.353even12
      1700.1.n.b.999.122380.1543odd12
      1700.1.p.a.251.12595.234odd12
      1700.1.p.a.251.122380.2019even12
      1700.1.p.a.1551.1235.24odd6
      1700.1.p.a.1551.12140.59even6
      3332.1.m.b.2843.127.4even3
      3332.1.m.b.2843.1228.11odd6
      3332.1.m.b.3039.12119.81even12
      3332.1.m.b.3039.12476.319odd12
      3332.1.bc.b.667.147.2even3inner
      3332.1.bc.b.667.1428.23odd6inner
      3332.1.bc.b.863.14119.30even12inner
      3332.1.bc.b.863.14476.387odd12inner
      3332.1.bc.b.2027.141.1even1trivial
      3332.1.bc.b.2027.144.3odd2CM
      3332.1.bc.b.2223.1417.13even4inner
      3332.1.bc.b.2223.1468.47odd4inner
      3332.1.bc.c.667.147.5odd6
      3332.1.bc.c.667.1428.19even6
      3332.1.bc.c.863.14119.47odd12
      3332.1.bc.c.863.14476.47even12
      3332.1.bc.c.2027.147.6odd2
      3332.1.bc.c.2027.1428.27even2
      3332.1.bc.c.2223.14119.13odd4
      3332.1.bc.c.2223.14476.251even4
      ", "is_complex": true, "table_nest_level": "1"}}]], "main_html": "
      \n Show commands:\n Magma\n / PariGP\n / SageMath
      [N,k,chi] = [3332,1,Mod(667,3332)]
      mf = mfinit([N,k,chi],0)
      lf = mfeigenbasis(mf)
      from sage.modular.dirichlet import DirichletCharacter
      H = DirichletGroup(3332, base_ring=CyclotomicField(12))
      chi = DirichletCharacter(H, H._module([6, 4, 9]))
      N = Newforms(chi, 1, names=\"a\")
      //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
      chi := DirichletCharacter(\"3332.667\");
      S:= CuspForms(chi, 1);
      N := Newforms(S);
      Level: \\( N \\) \\(=\\)\\( 3332 = 2^{2} \\cdot 7^{2} \\cdot 17 \\)
      Weight: \\( k \\) \\(=\\)\\( 1 \\)
      Character orbit: \\([\\chi]\\) \\(=\\)3332.bc (of order \\(12\\), degree \\(4\\), not minimal)

      Newform invariants

      sage:\u00a0f = N[0] # Warning: the index may be different
      gp:\u00a0f = lf[1] \\\\ Warning: the index may be different
      Self dual: no
      Analytic conductor: \\(1.66288462209\\)
      Analytic rank: \\(0\\)
      Dimension: \\(4\\)
      Coefficient field: \\(\\Q(\\zeta_{12})\\)
      gp:\u00a0f.mod \\\\ as an extension of the character field
      Defining polynomial: \\( x^{4} - x^{2} + 1 \\)\"Copy\"Toggle
      Coefficient ring: \\(\\Z[a_1, a_2]\\)
      Coefficient ring index: \\( 1 \\)
      Twist minimal: no (minimal twist has level 68)
      Projective image:\\(D_{4}\\)
      Projective field:Galois closure of 4.2.19652.1
      Artin image:$C_4\\wr C_2\\times C_6$
      Artin field:Galois closure of \\(\\mathbb{Q}[x]/(x^{48} - \\cdots)\\)

      Embedding invariants

      Embedding label 2027.1
      Root\\(0.866025 - 0.500000i\\) of defining polynomial
      Character\\(\\chi\\)\\(=\\)3332.2027
      Dual form3332.1.bc.b.863.1
      sage:\u00a0f.q_expansion() # note that sage often uses an isomorphic number field
      gp:\u00a0mfcoefs(f, 20)
      \\(f(q)\\)\\(=\\)\\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.36603 - 0.366025i) q^{5} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.00000i) q^{20} +(0.866025 + 0.500000i) q^{25} +(1.00000 - 1.00000i) q^{29} +(0.866025 + 0.500000i) q^{32} -1.00000i q^{34} +1.00000i q^{36} +(1.36603 + 0.366025i) q^{37} +(0.366025 - 1.36603i) q^{40} +(-1.00000 - 1.00000i) q^{41} +(1.36603 - 0.366025i) q^{45} -1.00000 q^{50} +(-0.366025 + 1.36603i) q^{58} +(0.366025 - 1.36603i) q^{61} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(-0.366025 - 1.36603i) q^{73} +(-1.36603 + 0.366025i) q^{74} +(0.366025 + 1.36603i) q^{80} +(0.500000 - 0.866025i) q^{81} +(1.36603 + 0.366025i) q^{82} +(1.00000 - 1.00000i) q^{85} +(-1.00000 + 1.00000i) q^{90} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\\)
      \\(\\operatorname{Tr}(f)(q)\\)\\(=\\)\\( 4 q + 2 q^{4} - 2 q^{5} + 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 4 q^{20} + 4 q^{29} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 2 q^{45} - 4 q^{50} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{68} - 2 q^{72} + 2 q^{73}+ \\cdots + 4 q^{97}+O(q^{100}) \\)\"Copy\"Toggle

      Character values

      We give the values of \\chi on generators for \\left(\\mathbb{Z}/3332\\mathbb{Z}\\right)^\\times.

      \\(n\\)\\(785\\)\\(885\\)\\(1667\\)
      \\(\\chi(n)\\)\\(e\\left(\\frac{3}{4}\\right)\\)\\(e\\left(\\frac{2}{3}\\right)\\)\\(-1\\)

      Coefficient data

      For each \\(n\\) we display the coefficients of the \\(q\\)-expansion \\(a_n\\), the\nSatake parameters \\(\\alpha_p\\),\nand the Satake angles \\(\\theta_p = \\textrm{Arg}(\\alpha_p)\\).

      \n \\(n\\)\n \n \\(a_n\\)\n \n \\(a_n / n^{(k-1)/2}\\)\n \n \\( \\alpha_n \\)\n \n \\( \\theta_n \\)\n
      \n \\(p\\)\n \n \\(a_p\\)\n \n \\(a_p / p^{(k-1)/2}\\)\n \n \\( \\alpha_p\\)\n \n \\( \\theta_p \\)\n
      \n \\(2\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
      \n \\(3\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(4\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
      \n \\(5\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(6\\)\n \n 0\n \n 0\n
      \n \\(7\\)\n \n 0\n \n 0\n
      \n \\(8\\)\n \n 1.00000i\n 1.00000i
      \n \\(9\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
      \n \\(10\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
      \n \\(11\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(12\\)\n \n 0\n \n 0\n
      \n \\(13\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(14\\)\n \n 0\n \n 0\n
      \n \\(15\\)\n \n 0\n \n 0\n
      \n \\(16\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
      \n \\(17\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
      \n \\(18\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
      \n \\(19\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(20\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
      \n \\(21\\)\n \n 0\n \n 0\n
      \n \\(22\\)\n \n 0\n \n 0\n
      \n \\(23\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
      \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
      \n \\(24\\)\n \n 0\n \n 0\n
      \n \\(25\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
      \n \\(26\\)\n \n 0\n \n 0\n
      \n \\(27\\)\n \n 0\n \n 0\n
      \n \\(28\\)\n \n 0\n \n 0\n
      \n \\(29\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n \\(0\\)\n
      \n \\(30\\)\n \n 0\n \n 0\n
      \n \\(31\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(32\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
      \n \\(33\\)\n \n 0\n \n 0\n
      \n \\(34\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
      \n \\(35\\)\n \n 0\n \n 0\n
      \n \\(36\\)\n \n 1.00000i\n 1.00000i
      \n \\(37\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(38\\)\n \n 0\n \n 0\n
      \n \\(39\\)\n \n 0\n \n 0\n
      \n \\(40\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
      \n \\(41\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(42\\)\n \n 0\n \n 0\n
      \n \\(43\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(44\\)\n \n 0\n \n 0\n
      \n \\(45\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
      \n \\(46\\)\n \n 0\n \n 0\n
      \n \\(47\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(48\\)\n \n 0\n \n 0\n
      \n \\(49\\)\n \n 0\n \n 0\n
      \n \\(50\\)\n \n \u22121.00000\n \n \u22121.00000\n
      \n \\(51\\)\n \n 0\n \n 0\n
      \n \\(52\\)\n \n 0\n \n 0\n
      \n \\(53\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(54\\)\n \n 0\n \n 0\n
      \n \\(55\\)\n \n 0\n \n 0\n
      \n \\(56\\)\n \n 0\n \n 0\n
      \n \\(57\\)\n \n 0\n \n 0\n
      \n \\(58\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
      \n \\(59\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(60\\)\n \n 0\n \n 0\n
      \n \\(61\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \\(62\\)\n \n 0\n \n 0\n
      \n \\(63\\)\n \n 0\n \n 0\n
      \n \\(64\\)\n \n \u22121.00000\n \n \u22121.00000\n
      \n \\(65\\)\n \n 0\n \n 0\n
      \n \\(66\\)\n \n 0\n \n 0\n
      \n \\(67\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(68\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
      \n \\(69\\)\n \n 0\n \n 0\n
      \n \\(70\\)\n \n 0\n \n 0\n
      \n \\(71\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(72\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
      \n \\(73\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \\(74\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
      \n \\(75\\)\n \n 0\n \n 0\n
      \n \\(76\\)\n \n 0\n \n 0\n
      \n \\(77\\)\n \n 0\n \n 0\n
      \n \\(78\\)\n \n 0\n \n 0\n
      \n \\(79\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
      \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
      \n \\(80\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(81\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
      \n \\(82\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(83\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(84\\)\n \n 0\n \n 0\n
      \n \\(85\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
      \n \\(86\\)\n \n 0\n \n 0\n
      \n \\(87\\)\n \n 0\n \n 0\n
      \n \\(88\\)\n \n 0\n \n 0\n
      \n \\(89\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(90\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
      \n \\(91\\)\n \n 0\n \n 0\n
      \n \\(92\\)\n \n 0\n \n 0\n
      \n \\(93\\)\n \n 0\n \n 0\n
      \n \\(94\\)\n \n 0\n \n 0\n
      \n \\(95\\)\n \n 0\n \n 0\n
      \n \\(96\\)\n \n 0\n \n 0\n
      \n \\(97\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n \\(0\\)\n
      \n \\(98\\)\n \n 0\n \n 0\n
      \n \\(99\\)\n \n 0\n \n 0\n
      \n \\(100\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
      \n \\(101\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(102\\)\n \n 0\n \n 0\n
      \n \\(103\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(104\\)\n \n 0\n \n 0\n
      \n \\(105\\)\n \n 0\n \n 0\n
      \n \\(106\\)\n \n 0\n \n 0\n
      \n \\(107\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(108\\)\n \n 0\n \n 0\n
      \n \\(109\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(110\\)\n \n 0\n \n 0\n
      \n \\(111\\)\n \n 0\n \n 0\n
      \n \\(112\\)\n \n 0\n \n 0\n
      \n \\(113\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(114\\)\n \n 0\n \n 0\n
      \n \\(115\\)\n \n 0\n \n 0\n
      \n \\(116\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
      \n \\(117\\)\n \n 0\n \n 0\n
      \n \\(118\\)\n \n 0\n \n 0\n
      \n \\(119\\)\n \n 0\n \n 0\n
      \n \\(120\\)\n \n 0\n \n 0\n
      \n \\(121\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
      \n \\(122\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(123\\)\n \n 0\n \n 0\n
      \n \\(124\\)\n \n 0\n \n 0\n
      \n \\(125\\)\n \n 0\n \n 0\n
      \n \\(126\\)\n \n 0\n \n 0\n
      \n \\(127\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(128\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
      \n \\(129\\)\n \n 0\n \n 0\n
      \n \\(130\\)\n \n 0\n \n 0\n
      \n \\(131\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(132\\)\n \n 0\n \n 0\n
      \n \\(133\\)\n \n 0\n \n 0\n
      \n \\(134\\)\n \n 0\n \n 0\n
      \n \\(135\\)\n \n 0\n \n 0\n
      \n \\(136\\)\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \u22120.866025\n \n \u2212\n \n 0.500000i
      \n \\(137\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(138\\)\n \n 0\n \n 0\n
      \n \\(139\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(140\\)\n \n 0\n \n 0\n
      \n \\(141\\)\n \n 0\n \n 0\n
      \n \\(142\\)\n \n 0\n \n 0\n
      \n \\(143\\)\n \n 0\n \n 0\n
      \n \\(144\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
      \n \\(145\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
      \n \\(146\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(147\\)\n \n 0\n \n 0\n
      \n \\(148\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
      \n \\(149\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n \\(150\\)\n \n 0\n \n 0\n
      \n \\(151\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(152\\)\n \n 0\n \n 0\n
      \n \\(153\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
      \n \\(154\\)\n \n 0\n \n 0\n
      \n \\(155\\)\n \n 0\n \n 0\n
      \n \\(156\\)\n \n 0\n \n 0\n
      \n \\(157\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \\(158\\)\n \n 0\n \n 0\n
      \n \\(159\\)\n \n 0\n \n 0\n
      \n \\(160\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
      \n \\(161\\)\n \n 0\n \n 0\n
      \n \\(162\\)\n \n 1.00000i\n 1.00000i
      \n \\(163\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
      \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
      \n \\(164\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
      \n \\(165\\)\n \n 0\n \n 0\n
      \n \\(166\\)\n \n 0\n \n 0\n
      \n \\(167\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(168\\)\n \n 0\n \n 0\n
      \n \\(169\\)\n \n \u22121.00000\n \n \u22121.00000\n
      \n \\(170\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
      \n \\(171\\)\n \n 0\n \n 0\n
      \n \\(172\\)\n \n 0\n \n 0\n
      \n \\(173\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(174\\)\n \n 0\n \n 0\n
      \n \\(175\\)\n \n 0\n \n 0\n
      \n \\(176\\)\n \n 0\n \n 0\n
      \n \\(177\\)\n \n 0\n \n 0\n
      \n \\(178\\)\n \n 0\n \n 0\n
      \n \\(179\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(180\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
      \n \\(181\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n \\(0\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(182\\)\n \n 0\n \n 0\n
      \n \\(183\\)\n \n 0\n \n 0\n
      \n \\(184\\)\n \n 0\n \n 0\n
      \n \\(185\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i
      \n \\(186\\)\n \n 0\n \n 0\n
      \n \\(187\\)\n \n 0\n \n 0\n
      \n \\(188\\)\n \n 0\n \n 0\n
      \n \\(189\\)\n \n 0\n \n 0\n
      \n \\(190\\)\n \n 0\n \n 0\n
      \n \\(191\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(192\\)\n \n 0\n \n 0\n
      \n \\(193\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(194\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
      \n \\(195\\)\n \n 0\n \n 0\n
      \n \\(196\\)\n \n 0\n \n 0\n
      \n \\(197\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n \\(0\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(198\\)\n \n 0\n \n 0\n
      \n \\(199\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(200\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
      \n \\(201\\)\n \n 0\n \n 0\n
      \n \\(202\\)\n \n 0\n \n 0\n
      \n \\(203\\)\n \n 0\n \n 0\n
      \n \\(204\\)\n \n 0\n \n 0\n
      \n \\(205\\)\n \n 1.00000\n \n +\n \n 1.73205i\n 1.00000\n \n +\n \n 1.73205i
      \n \\(206\\)\n \n 0\n \n 0\n
      \n \\(207\\)\n \n 0\n \n 0\n
      \n \\(208\\)\n \n 0\n \n 0\n
      \n \\(209\\)\n \n 0\n \n 0\n
      \n \\(210\\)\n \n 0\n \n 0\n
      \n \\(211\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
      \n \\(212\\)\n \n 0\n \n 0\n
      \n \\(213\\)\n \n 0\n \n 0\n
      \n \\(214\\)\n \n 0\n \n 0\n
      \n \\(215\\)\n \n 0\n \n 0\n
      \n \\(216\\)\n \n 0\n \n 0\n
      \n \\(217\\)\n \n 0\n \n 0\n
      \n \\(218\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
      \n \\(219\\)\n \n 0\n \n 0\n
      \n \\(220\\)\n \n 0\n \n 0\n
      \n \\(221\\)\n \n 0\n \n 0\n
      \n \\(222\\)\n \n 0\n \n 0\n
      \n \\(223\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(224\\)\n \n 0\n \n 0\n
      \n \\(225\\)\n \n \u22121.00000\n \n \u22121.00000\n
      \n \\(226\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(227\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(228\\)\n \n 0\n \n 0\n
      \n \\(229\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i\n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(230\\)\n \n 0\n \n 0\n
      \n \\(231\\)\n \n 0\n \n 0\n
      \n \\(232\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(233\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(234\\)\n \n 0\n \n 0\n
      \n \\(235\\)\n \n 0\n \n 0\n
      \n \\(236\\)\n \n 0\n \n 0\n
      \n \\(237\\)\n \n 0\n \n 0\n
      \n \\(238\\)\n \n 0\n \n 0\n
      \n \\(239\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(240\\)\n \n 0\n \n 0\n
      \n \\(241\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \\(242\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
      \n \\(243\\)\n \n 0\n \n 0\n
      \n \\(244\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
      \n \\(245\\)\n \n 0\n \n 0\n
      \n \\(246\\)\n \n 0\n \n 0\n
      \n \\(247\\)\n \n 0\n \n 0\n
      \n \\(248\\)\n \n 0\n \n 0\n
      \n \\(249\\)\n \n 0\n \n 0\n
      \n \\(250\\)\n \n 0\n \n 0\n
      \n \\(251\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(252\\)\n \n 0\n \n 0\n
      \n \\(253\\)\n \n 0\n \n 0\n
      \n \\(254\\)\n \n 0\n \n 0\n
      \n \\(255\\)\n \n 0\n \n 0\n
      \n \\(256\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
      \n \\(257\\)\n \n 1.73205\n \n \u2212\n \n 1.00000i\n 1.73205\n \n \u2212\n \n 1.00000i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \\(258\\)\n \n 0\n \n 0\n
      \n \\(259\\)\n \n 0\n \n 0\n
      \n \\(260\\)\n \n 0\n \n 0\n
      \n \\(261\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
      \n \\(262\\)\n \n 0\n \n 0\n
      \n \\(263\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(264\\)\n \n 0\n \n 0\n
      \n \\(265\\)\n \n 0\n \n 0\n
      \n \\(266\\)\n \n 0\n \n 0\n
      \n \\(267\\)\n \n 0\n \n 0\n
      \n \\(268\\)\n \n 0\n \n 0\n
      \n \\(269\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i\n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(270\\)\n \n 0\n \n 0\n
      \n \\(271\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(272\\)\n \n 1.00000\n \n 1.00000\n
      \n \\(273\\)\n \n 0\n \n 0\n
      \n \\(274\\)\n \n 0\n \n 0\n
      \n \\(275\\)\n \n 0\n \n 0\n
      \n \\(276\\)\n \n 0\n \n 0\n
      \n \\(277\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i\n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(278\\)\n \n 0\n \n 0\n
      \n \\(279\\)\n \n 0\n \n 0\n
      \n \\(280\\)\n \n 0\n \n 0\n
      \n \\(281\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(282\\)\n \n 0\n \n 0\n
      \n \\(283\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(284\\)\n \n 0\n \n 0\n
      \n \\(285\\)\n \n 0\n \n 0\n
      \n \\(286\\)\n \n 0\n \n 0\n
      \n \\(287\\)\n \n 0\n \n 0\n
      \n \\(288\\)\n \n \u22121.00000\n \n \u22121.00000\n
      \n \\(289\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
      \n \\(290\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
      \n \\(291\\)\n \n 0\n \n 0\n
      \n \\(292\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
      \n \\(293\\)\n \n 2.00000\n \n 2.00000\n \n 1.00000\n \n \\(0\\)\n
      \n 1.00000\n \n \\(0\\)\n
      \n \\(294\\)\n \n 0\n \n 0\n
      \n \\(295\\)\n \n 0\n \n 0\n
      \n \\(296\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
      \n \\(297\\)\n \n 0\n \n 0\n
      \n \\(298\\)\n \n 1.73205\n \n +\n \n 1.00000i\n 1.73205\n \n +\n \n 1.00000i
      \n \\(299\\)\n \n 0\n \n 0\n
      \n \\(300\\)\n \n 0\n \n 0\n
      \n \\(301\\)\n \n 0\n \n 0\n
      \n \\(302\\)\n \n 0\n \n 0\n
      \n \\(303\\)\n \n 0\n \n 0\n
      \n \\(304\\)\n \n 0\n \n 0\n
      \n \\(305\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i
      \n \\(306\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
      \n \\(307\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(308\\)\n \n 0\n \n 0\n
      \n \\(309\\)\n \n 0\n \n 0\n
      \n \\(310\\)\n \n 0\n \n 0\n
      \n \\(311\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(312\\)\n \n 0\n \n 0\n
      \n \\(313\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(314\\)\n \n 2.00000i\n 2.00000i
      \n \\(315\\)\n \n 0\n \n 0\n
      \n \\(316\\)\n \n 0\n \n 0\n
      \n \\(317\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \\(318\\)\n \n 0\n \n 0\n
      \n \\(319\\)\n \n 0\n \n 0\n
      \n \\(320\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(321\\)\n \n 0\n \n 0\n
      \n \\(322\\)\n \n 0\n \n 0\n
      \n \\(323\\)\n \n 0\n \n 0\n
      \n \\(324\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
      \n \\(325\\)\n \n 0\n \n 0\n
      \n \\(326\\)\n \n 0\n \n 0\n
      \n \\(327\\)\n \n 0\n \n 0\n
      \n \\(328\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
      \n \\(329\\)\n \n 0\n \n 0\n
      \n \\(330\\)\n \n 0\n \n 0\n
      \n \\(331\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(332\\)\n \n 0\n \n 0\n
      \n \\(333\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
      \n \\(334\\)\n \n 0\n \n 0\n
      \n \\(335\\)\n \n 0\n \n 0\n
      \n \\(336\\)\n \n 0\n \n 0\n
      \n \\(337\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n \\(0\\)\n
      \n \\(338\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
      \n \\(339\\)\n \n 0\n \n 0\n
      \n \\(340\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
      \n \\(341\\)\n \n 0\n \n 0\n
      \n \\(342\\)\n \n 0\n \n 0\n
      \n \\(343\\)\n \n 0\n \n 0\n
      \n \\(344\\)\n \n 0\n \n 0\n
      \n \\(345\\)\n \n 0\n \n 0\n
      \n \\(346\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
      \n \\(347\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(348\\)\n \n 0\n \n 0\n
      \n \\(349\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(350\\)\n \n 0\n \n 0\n
      \n \\(351\\)\n \n 0\n \n 0\n
      \n \\(352\\)\n \n 0\n \n 0\n
      \n \\(353\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i\n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(354\\)\n \n 0\n \n 0\n
      \n \\(355\\)\n \n 0\n \n 0\n
      \n \\(356\\)\n \n 0\n \n 0\n
      \n \\(357\\)\n \n 0\n \n 0\n
      \n \\(358\\)\n \n 0\n \n 0\n
      \n \\(359\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(360\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(361\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
      \n \\(362\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
      \n \\(363\\)\n \n 0\n \n 0\n
      \n \\(364\\)\n \n 0\n \n 0\n
      \n \\(365\\)\n \n 2.00000i\n 2.00000i
      \n \\(366\\)\n \n 0\n \n 0\n
      \n \\(367\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(368\\)\n \n 0\n \n 0\n
      \n \\(369\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(370\\)\n \n 2.00000\n \n 2.00000\n
      \n \\(371\\)\n \n 0\n \n 0\n
      \n \\(372\\)\n \n 0\n \n 0\n
      \n \\(373\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(374\\)\n \n 0\n \n 0\n
      \n \\(375\\)\n \n 0\n \n 0\n
      \n \\(376\\)\n \n 0\n \n 0\n
      \n \\(377\\)\n \n 0\n \n 0\n
      \n \\(378\\)\n \n 0\n \n 0\n
      \n \\(379\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(380\\)\n \n 0\n \n 0\n
      \n \\(381\\)\n \n 0\n \n 0\n
      \n \\(382\\)\n \n 0\n \n 0\n
      \n \\(383\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(384\\)\n \n 0\n \n 0\n
      \n \\(385\\)\n \n 0\n \n 0\n
      \n \\(386\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
      \n \\(387\\)\n \n 0\n \n 0\n
      \n \\(388\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
      \n \\(389\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n \\(390\\)\n \n 0\n \n 0\n
      \n \\(391\\)\n \n 0\n \n 0\n
      \n \\(392\\)\n \n 0\n \n 0\n
      \n \\(393\\)\n \n 0\n \n 0\n
      \n \\(394\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
      \n \\(395\\)\n \n 0\n \n 0\n
      \n \\(396\\)\n \n 0\n \n 0\n
      \n \\(397\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \\(398\\)\n \n 0\n \n 0\n
      \n \\(399\\)\n \n 0\n \n 0\n
      \n \\(400\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
      \n \\(401\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \\(402\\)\n \n 0\n \n 0\n
      \n \\(403\\)\n \n 0\n \n 0\n
      \n \\(404\\)\n \n 0\n \n 0\n
      \n \\(405\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
      \n \\(406\\)\n \n 0\n \n 0\n
      \n \\(407\\)\n \n 0\n \n 0\n
      \n \\(408\\)\n \n 0\n \n 0\n
      \n \\(409\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i\n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(410\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i
      \n \\(411\\)\n \n 0\n \n 0\n
      \n \\(412\\)\n \n 0\n \n 0\n
      \n \\(413\\)\n \n 0\n \n 0\n
      \n \\(414\\)\n \n 0\n \n 0\n
      \n \\(415\\)\n \n 0\n \n 0\n
      \n \\(416\\)\n \n 0\n \n 0\n
      \n \\(417\\)\n \n 0\n \n 0\n
      \n \\(418\\)\n \n 0\n \n 0\n
      \n \\(419\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
      \n \\(420\\)\n \n 0\n \n 0\n
      \n \\(421\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(422\\)\n \n 0\n \n 0\n
      \n \\(423\\)\n \n 0\n \n 0\n
      \n \\(424\\)\n \n 0\n \n 0\n
      \n \\(425\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
      \n \\(426\\)\n \n 0\n \n 0\n
      \n \\(427\\)\n \n 0\n \n 0\n
      \n \\(428\\)\n \n 0\n \n 0\n
      \n \\(429\\)\n \n 0\n \n 0\n
      \n \\(430\\)\n \n 0\n \n 0\n
      \n \\(431\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(432\\)\n \n 0\n \n 0\n
      \n \\(433\\)\n \n 2.00000i\n 2.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(434\\)\n \n 0\n \n 0\n
      \n \\(435\\)\n \n 0\n \n 0\n
      \n \\(436\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
      \n \\(437\\)\n \n 0\n \n 0\n
      \n \\(438\\)\n \n 0\n \n 0\n
      \n \\(439\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(440\\)\n \n 0\n \n 0\n
      \n \\(441\\)\n \n 0\n \n 0\n
      \n \\(442\\)\n \n 0\n \n 0\n
      \n \\(443\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(444\\)\n \n 0\n \n 0\n
      \n \\(445\\)\n \n 0\n \n 0\n
      \n \\(446\\)\n \n 0\n \n 0\n
      \n \\(447\\)\n \n 0\n \n 0\n
      \n \\(448\\)\n \n 0\n \n 0\n
      \n \\(449\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(450\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
      \n \\(451\\)\n \n 0\n \n 0\n
      \n \\(452\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
      \n \\(453\\)\n \n 0\n \n 0\n
      \n \\(454\\)\n \n 0\n \n 0\n
      \n \\(455\\)\n \n 0\n \n 0\n
      \n \\(456\\)\n \n 0\n \n 0\n
      \n \\(457\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i\n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(458\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
      \n \\(459\\)\n \n 0\n \n 0\n
      \n \\(460\\)\n \n 0\n \n 0\n
      \n \\(461\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(462\\)\n \n 0\n \n 0\n
      \n \\(463\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(464\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
      \n \\(465\\)\n \n 0\n \n 0\n
      \n \\(466\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
      \n \\(467\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(468\\)\n \n 0\n \n 0\n
      \n \\(469\\)\n \n 0\n \n 0\n
      \n \\(470\\)\n \n 0\n \n 0\n
      \n \\(471\\)\n \n 0\n \n 0\n
      \n \\(472\\)\n \n 0\n \n 0\n
      \n \\(473\\)\n \n 0\n \n 0\n
      \n \\(474\\)\n \n 0\n \n 0\n
      \n \\(475\\)\n \n 0\n \n 0\n
      \n \\(476\\)\n \n 0\n \n 0\n
      \n \\(477\\)\n \n 0\n \n 0\n
      \n \\(478\\)\n \n 0\n \n 0\n
      \n \\(479\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(480\\)\n \n 0\n \n 0\n
      \n \\(481\\)\n \n 0\n \n 0\n
      \n \\(482\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(483\\)\n \n 0\n \n 0\n
      \n \\(484\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
      \n \\(485\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
      \n \\(486\\)\n \n 0\n \n 0\n
      \n \\(487\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(488\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(489\\)\n \n 0\n \n 0\n
      \n \\(490\\)\n \n 0\n \n 0\n
      \n \\(491\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(492\\)\n \n 0\n \n 0\n
      \n \\(493\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(494\\)\n \n 0\n \n 0\n
      \n \\(495\\)\n \n 0\n \n 0\n
      \n \\(496\\)\n \n 0\n \n 0\n
      \n \\(497\\)\n \n 0\n \n 0\n
      \n \\(498\\)\n \n 0\n \n 0\n
      \n \\(499\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
      \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
      \n \\(500\\)\n \n 0\n \n 0\n
      \n \\(501\\)\n \n 0\n \n 0\n
      \n \\(502\\)\n \n 0\n \n 0\n
      \n \\(503\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
      \n \\(504\\)\n \n 0\n \n 0\n
      \n \\(505\\)\n \n 0\n \n 0\n
      \n \\(506\\)\n \n 0\n \n 0\n
      \n \\(507\\)\n \n 0\n \n 0\n
      \n \\(508\\)\n \n 0\n \n 0\n
      \n \\(509\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n \\(510\\)\n \n 0\n \n 0\n
      \n \\(511\\)\n \n 0\n \n 0\n
      \n \\(512\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
      \n \\(513\\)\n \n 0\n \n 0\n
      \n \\(514\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i
      \n \\(515\\)\n \n 0\n \n 0\n
      \n \\(516\\)\n \n 0\n \n 0\n
      \n \\(517\\)\n \n 0\n \n 0\n
      \n \\(518\\)\n \n 0\n \n 0\n
      \n \\(519\\)\n \n 0\n \n 0\n
      \n \\(520\\)\n \n 0\n \n 0\n
      \n \\(521\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(522\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
      \n \\(523\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(524\\)\n \n 0\n \n 0\n
      \n \\(525\\)\n \n 0\n \n 0\n
      \n \\(526\\)\n \n 0\n \n 0\n
      \n \\(527\\)\n \n 0\n \n 0\n
      \n \\(528\\)\n \n 0\n \n 0\n
      \n \\(529\\)\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \u22120.866025\n \n \u2212\n \n 0.500000i
      \n \\(530\\)\n \n 0\n \n 0\n
      \n \\(531\\)\n \n 0\n \n 0\n
      \n \\(532\\)\n \n 0\n \n 0\n
      \n \\(533\\)\n \n 0\n \n 0\n
      \n \\(534\\)\n \n 0\n \n 0\n
      \n \\(535\\)\n \n 0\n \n 0\n
      \n \\(536\\)\n \n 0\n \n 0\n
      \n \\(537\\)\n \n 0\n \n 0\n
      \n \\(538\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
      \n \\(539\\)\n \n 0\n \n 0\n
      \n \\(540\\)\n \n 0\n \n 0\n
      \n \\(541\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(542\\)\n \n 0\n \n 0\n
      \n \\(543\\)\n \n 0\n \n 0\n
      \n \\(544\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
      \n \\(545\\)\n \n \u22122.00000\n \n \u22122.00000\n
      \n \\(546\\)\n \n 0\n \n 0\n
      \n \\(547\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(548\\)\n \n 0\n \n 0\n
      \n \\(549\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(550\\)\n \n 0\n \n 0\n
      \n \\(551\\)\n \n 0\n \n 0\n
      \n \\(552\\)\n \n 0\n \n 0\n
      \n \\(553\\)\n \n 0\n \n 0\n
      \n \\(554\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
      \n \\(555\\)\n \n 0\n \n 0\n
      \n \\(556\\)\n \n 0\n \n 0\n
      \n \\(557\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(558\\)\n \n 0\n \n 0\n
      \n \\(559\\)\n \n 0\n \n 0\n
      \n \\(560\\)\n \n 0\n \n 0\n
      \n \\(561\\)\n \n 0\n \n 0\n
      \n \\(562\\)\n \n 0\n \n 0\n
      \n \\(563\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(564\\)\n \n 0\n \n 0\n
      \n \\(565\\)\n \n 1.00000\n \n +\n \n 1.73205i\n 1.00000\n \n +\n \n 1.73205i
      \n \\(566\\)\n \n 0\n \n 0\n
      \n \\(567\\)\n \n 0\n \n 0\n
      \n \\(568\\)\n \n 0\n \n 0\n
      \n \\(569\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(570\\)\n \n 0\n \n 0\n
      \n \\(571\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(572\\)\n \n 0\n \n 0\n
      \n \\(573\\)\n \n 0\n \n 0\n
      \n \\(574\\)\n \n 0\n \n 0\n
      \n \\(575\\)\n \n 0\n \n 0\n
      \n \\(576\\)\n \n 0.866025\n \n \u2212\n \n 0.500000i\n 0.866025\n \n \u2212\n \n 0.500000i
      \n \\(577\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(578\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
      \n \\(579\\)\n \n 0\n \n 0\n
      \n \\(580\\)\n \n 2.00000i\n 2.00000i
      \n \\(581\\)\n \n 0\n \n 0\n
      \n \\(582\\)\n \n 0\n \n 0\n
      \n \\(583\\)\n \n 0\n \n 0\n
      \n \\(584\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
      \n \\(585\\)\n \n 0\n \n 0\n
      \n \\(586\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
      \n \\(587\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(588\\)\n \n 0\n \n 0\n
      \n \\(589\\)\n \n 0\n \n 0\n
      \n \\(590\\)\n \n 0\n \n 0\n
      \n \\(591\\)\n \n 0\n \n 0\n
      \n \\(592\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
      \n \\(593\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(594\\)\n \n 0\n \n 0\n
      \n \\(595\\)\n \n 0\n \n 0\n
      \n \\(596\\)\n \n \u22122.00000\n \n \u22122.00000\n
      \n \\(597\\)\n \n 0\n \n 0\n
      \n \\(598\\)\n \n 0\n \n 0\n
      \n \\(599\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(600\\)\n \n 0\n \n 0\n
      \n \\(601\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(602\\)\n \n 0\n \n 0\n
      \n \\(603\\)\n \n 0\n \n 0\n
      \n \\(604\\)\n \n 0\n \n 0\n
      \n \\(605\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
      \n \\(606\\)\n \n 0\n \n 0\n
      \n \\(607\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(608\\)\n \n 0\n \n 0\n
      \n \\(609\\)\n \n 0\n \n 0\n
      \n \\(610\\)\n \n \u2212\n \n 2.00000i\n \u2212\n \n 2.00000i
      \n \\(611\\)\n \n 0\n \n 0\n
      \n \\(612\\)\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \u22120.866025\n \n \u2212\n \n 0.500000i
      \n \\(613\\)\n \n \u22121.00000\n \n +\n \n 1.73205i\n \u22121.00000\n \n +\n \n 1.73205i\n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(614\\)\n \n 0\n \n 0\n
      \n \\(615\\)\n \n 0\n \n 0\n
      \n \\(616\\)\n \n 0\n \n 0\n
      \n \\(617\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(618\\)\n \n 0\n \n 0\n
      \n \\(619\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(620\\)\n \n 0\n \n 0\n
      \n \\(621\\)\n \n 0\n \n 0\n
      \n \\(622\\)\n \n 0\n \n 0\n
      \n \\(623\\)\n \n 0\n \n 0\n
      \n \\(624\\)\n \n 0\n \n 0\n
      \n \\(625\\)\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \u22120.500000\n \n \u2212\n \n 0.866025i
      \n \\(626\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i
      \n \\(627\\)\n \n 0\n \n 0\n
      \n \\(628\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
      \n \\(629\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
      \n \\(630\\)\n \n 0\n \n 0\n
      \n \\(631\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(632\\)\n \n 0\n \n 0\n
      \n \\(633\\)\n \n 0\n \n 0\n
      \n \\(634\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(635\\)\n \n 0\n \n 0\n
      \n \\(636\\)\n \n 0\n \n 0\n
      \n \\(637\\)\n \n 0\n \n 0\n
      \n \\(638\\)\n \n 0\n \n 0\n
      \n \\(639\\)\n \n 0\n \n 0\n
      \n \\(640\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
      \n \\(641\\)\n \n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.366025\n \n \u2212\n \n 1.36603i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \\(642\\)\n \n 0\n \n 0\n
      \n \\(643\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(644\\)\n \n 0\n \n 0\n
      \n \\(645\\)\n \n 0\n \n 0\n
      \n \\(646\\)\n \n 0\n \n 0\n
      \n \\(647\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(648\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
      \n \\(649\\)\n \n 0\n \n 0\n
      \n \\(650\\)\n \n 0\n \n 0\n
      \n \\(651\\)\n \n 0\n \n 0\n
      \n \\(652\\)\n \n 0\n \n 0\n
      \n \\(653\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \\(654\\)\n \n 0\n \n 0\n
      \n \\(655\\)\n \n 0\n \n 0\n
      \n \\(656\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
      \n \\(657\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(658\\)\n \n 0\n \n 0\n
      \n \\(659\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(660\\)\n \n 0\n \n 0\n
      \n \\(661\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(662\\)\n \n 0\n \n 0\n
      \n \\(663\\)\n \n 0\n \n 0\n
      \n \\(664\\)\n \n 0\n \n 0\n
      \n \\(665\\)\n \n 0\n \n 0\n
      \n \\(666\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
      \n \\(667\\)\n \n 0\n \n 0\n
      \n \\(668\\)\n \n 0\n \n 0\n
      \n \\(669\\)\n \n 0\n \n 0\n
      \n \\(670\\)\n \n 0\n \n 0\n
      \n \\(671\\)\n \n 0\n \n 0\n
      \n \\(672\\)\n \n 0\n \n 0\n
      \n \\(673\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(674\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
      \n \\(675\\)\n \n 0\n \n 0\n
      \n \\(676\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
      \n \\(677\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(678\\)\n \n 0\n \n 0\n
      \n \\(679\\)\n \n 0\n \n 0\n
      \n \\(680\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(681\\)\n \n 0\n \n 0\n
      \n \\(682\\)\n \n 0\n \n 0\n
      \n \\(683\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(684\\)\n \n 0\n \n 0\n
      \n \\(685\\)\n \n 0\n \n 0\n
      \n \\(686\\)\n \n 0\n \n 0\n
      \n \\(687\\)\n \n 0\n \n 0\n
      \n \\(688\\)\n \n 0\n \n 0\n
      \n \\(689\\)\n \n 0\n \n 0\n
      \n \\(690\\)\n \n 0\n \n 0\n
      \n \\(691\\)\n \n 0\n \n 0\n \n 0.258819\n \n \u2212\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
      \n \u22120.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
      \n \\(692\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
      \n \\(693\\)\n \n 0\n \n 0\n
      \n \\(694\\)\n \n 0\n \n 0\n
      \n \\(695\\)\n \n 0\n \n 0\n
      \n \\(696\\)\n \n 0\n \n 0\n
      \n \\(697\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
      \n \\(698\\)\n \n 0\n \n 0\n
      \n \\(699\\)\n \n 0\n \n 0\n
      \n \\(700\\)\n \n 0\n \n 0\n
      \n \\(701\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(702\\)\n \n 0\n \n 0\n
      \n \\(703\\)\n \n 0\n \n 0\n
      \n \\(704\\)\n \n 0\n \n 0\n
      \n \\(705\\)\n \n 0\n \n 0\n
      \n \\(706\\)\n \n \u2212\n \n 2.00000i\n \u2212\n \n 2.00000i
      \n \\(707\\)\n \n 0\n \n 0\n
      \n \\(708\\)\n \n 0\n \n 0\n
      \n \\(709\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(710\\)\n \n 0\n \n 0\n
      \n \\(711\\)\n \n 0\n \n 0\n
      \n \\(712\\)\n \n 0\n \n 0\n
      \n \\(713\\)\n \n 0\n \n 0\n
      \n \\(714\\)\n \n 0\n \n 0\n
      \n \\(715\\)\n \n 0\n \n 0\n
      \n \\(716\\)\n \n 0\n \n 0\n
      \n \\(717\\)\n \n 0\n \n 0\n
      \n \\(718\\)\n \n 0\n \n 0\n
      \n \\(719\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(720\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
      \n \\(721\\)\n \n 0\n \n 0\n
      \n \\(722\\)\n \n 1.00000i\n 1.00000i
      \n \\(723\\)\n \n 0\n \n 0\n
      \n \\(724\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
      \n \\(725\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
      \n \\(726\\)\n \n 0\n \n 0\n
      \n \\(727\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(728\\)\n \n 0\n \n 0\n
      \n \\(729\\)\n \n 1.00000i\n 1.00000i
      \n \\(730\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
      \n \\(731\\)\n \n 0\n \n 0\n
      \n \\(732\\)\n \n 0\n \n 0\n
      \n \\(733\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(734\\)\n \n 0\n \n 0\n
      \n \\(735\\)\n \n 0\n \n 0\n
      \n \\(736\\)\n \n 0\n \n 0\n
      \n \\(737\\)\n \n 0\n \n 0\n
      \n \\(738\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
      \n \\(739\\)\n \n 0\n \n 0\n \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(740\\)\n \n \u22121.73205\n \n +\n \n 1.00000i\n \u22121.73205\n \n +\n \n 1.00000i
      \n \\(741\\)\n \n 0\n \n 0\n
      \n \\(742\\)\n \n 0\n \n 0\n
      \n \\(743\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(744\\)\n \n 0\n \n 0\n
      \n \\(745\\)\n \n 0.732051\n \n +\n \n 2.73205i\n 0.732051\n \n +\n \n 2.73205i
      \n \\(746\\)\n \n 0\n \n 0\n
      \n \\(747\\)\n \n 0\n \n 0\n
      \n \\(748\\)\n \n 0\n \n 0\n
      \n \\(749\\)\n \n 0\n \n 0\n
      \n \\(750\\)\n \n 0\n \n 0\n
      \n \\(751\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(752\\)\n \n 0\n \n 0\n
      \n \\(753\\)\n \n 0\n \n 0\n
      \n \\(754\\)\n \n 0\n \n 0\n
      \n \\(755\\)\n \n 0\n \n 0\n
      \n \\(756\\)\n \n 0\n \n 0\n
      \n \\(757\\)\n \n 2.00000i\n 2.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(758\\)\n \n 0\n \n 0\n
      \n \\(759\\)\n \n 0\n \n 0\n
      \n \\(760\\)\n \n 0\n \n 0\n
      \n \\(761\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(762\\)\n \n 0\n \n 0\n
      \n \\(763\\)\n \n 0\n \n 0\n
      \n \\(764\\)\n \n 0\n \n 0\n
      \n \\(765\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
      \n \\(766\\)\n \n 0\n \n 0\n
      \n \\(767\\)\n \n 0\n \n 0\n
      \n \\(768\\)\n \n 0\n \n 0\n
      \n \\(769\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(770\\)\n \n 0\n \n 0\n
      \n \\(771\\)\n \n 0\n \n 0\n
      \n \\(772\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
      \n \\(773\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n \u22120.866025\n \n \u2212\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n \\(774\\)\n \n 0\n \n 0\n
      \n \\(775\\)\n \n 0\n \n 0\n
      \n \\(776\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(777\\)\n \n 0\n \n 0\n
      \n \\(778\\)\n \n 2.00000\n \n 2.00000\n
      \n \\(779\\)\n \n 0\n \n 0\n
      \n \\(780\\)\n \n 0\n \n 0\n
      \n \\(781\\)\n \n 0\n \n 0\n
      \n \\(782\\)\n \n 0\n \n 0\n
      \n \\(783\\)\n \n 0\n \n 0\n
      \n \\(784\\)\n \n 0\n \n 0\n
      \n \\(785\\)\n \n \u22122.00000\n \n +\n \n 2.00000i\n \u22122.00000\n \n +\n \n 2.00000i
      \n \\(786\\)\n \n 0\n \n 0\n
      \n \\(787\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(788\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
      \n \\(789\\)\n \n 0\n \n 0\n
      \n \\(790\\)\n \n 0\n \n 0\n
      \n \\(791\\)\n \n 0\n \n 0\n
      \n \\(792\\)\n \n 0\n \n 0\n
      \n \\(793\\)\n \n 0\n \n 0\n
      \n \\(794\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(795\\)\n \n 0\n \n 0\n
      \n \\(796\\)\n \n 0\n \n 0\n
      \n \\(797\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(798\\)\n \n 0\n \n 0\n
      \n \\(799\\)\n \n 0\n \n 0\n
      \n \\(800\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
      \n \\(801\\)\n \n 0\n \n 0\n
      \n \\(802\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(803\\)\n \n 0\n \n 0\n
      \n \\(804\\)\n \n 0\n \n 0\n
      \n \\(805\\)\n \n 0\n \n 0\n
      \n \\(806\\)\n \n 0\n \n 0\n
      \n \\(807\\)\n \n 0\n \n 0\n
      \n \\(808\\)\n \n 0\n \n 0\n
      \n \\(809\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(810\\)\n \n 0.366025\n \n \u2212\n \n 1.36603i\n 0.366025\n \n \u2212\n \n 1.36603i
      \n \\(811\\)\n \n 0\n \n 0\n \n 0.707107\n \n \u2212\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n \u22120.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(812\\)\n \n 0\n \n 0\n
      \n \\(813\\)\n \n 0\n \n 0\n
      \n \\(814\\)\n \n 0\n \n 0\n
      \n \\(815\\)\n \n 0\n \n 0\n
      \n \\(816\\)\n \n 0\n \n 0\n
      \n \\(817\\)\n \n 0\n \n 0\n
      \n \\(818\\)\n \n \u2212\n \n 2.00000i\n \u2212\n \n 2.00000i
      \n \\(819\\)\n \n 0\n \n 0\n
      \n \\(820\\)\n \n 2.00000\n \n 2.00000\n
      \n \\(821\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(822\\)\n \n 0\n \n 0\n
      \n \\(823\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(824\\)\n \n 0\n \n 0\n
      \n \\(825\\)\n \n 0\n \n 0\n
      \n \\(826\\)\n \n 0\n \n 0\n
      \n \\(827\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
      \n \\(828\\)\n \n 0\n \n 0\n
      \n \\(829\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \\(830\\)\n \n 0\n \n 0\n
      \n \\(831\\)\n \n 0\n \n 0\n
      \n \\(832\\)\n \n 0\n \n 0\n
      \n \\(833\\)\n \n 0\n \n 0\n
      \n \\(834\\)\n \n 0\n \n 0\n
      \n \\(835\\)\n \n 0\n \n 0\n
      \n \\(836\\)\n \n 0\n \n 0\n
      \n \\(837\\)\n \n 0\n \n 0\n
      \n \\(838\\)\n \n 0\n \n 0\n
      \n \\(839\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
      \n \\(840\\)\n \n 0\n \n 0\n
      \n \\(841\\)\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i
      \n \\(842\\)\n \n 0\n \n 0\n
      \n \\(843\\)\n \n 0\n \n 0\n
      \n \\(844\\)\n \n 0\n \n 0\n
      \n \\(845\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(846\\)\n \n 0\n \n 0\n
      \n \\(847\\)\n \n 0\n \n 0\n
      \n \\(848\\)\n \n 0\n \n 0\n
      \n \\(849\\)\n \n 0\n \n 0\n
      \n \\(850\\)\n \n 0.500000\n \n \u2212\n \n 0.866025i\n 0.500000\n \n \u2212\n \n 0.866025i
      \n \\(851\\)\n \n 0\n \n 0\n
      \n \\(852\\)\n \n 0\n \n 0\n
      \n \\(853\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i\n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n \\(0\\)\n
      \n \\(854\\)\n \n 0\n \n 0\n
      \n \\(855\\)\n \n 0\n \n 0\n
      \n \\(856\\)\n \n 0\n \n 0\n
      \n \\(857\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(858\\)\n \n 0\n \n 0\n
      \n \\(859\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(860\\)\n \n 0\n \n 0\n
      \n \\(861\\)\n \n 0\n \n 0\n
      \n \\(862\\)\n \n 0\n \n 0\n
      \n \\(863\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(864\\)\n \n 0\n \n 0\n
      \n \\(865\\)\n \n \u22121.73205\n \n \u2212\n \n 1.00000i\n \u22121.73205\n \n \u2212\n \n 1.00000i
      \n \\(866\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
      \n \\(867\\)\n \n 0\n \n 0\n
      \n \\(868\\)\n \n 0\n \n 0\n
      \n \\(869\\)\n \n 0\n \n 0\n
      \n \\(870\\)\n \n 0\n \n 0\n
      \n \\(871\\)\n \n 0\n \n 0\n
      \n \\(872\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(873\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i
      \n \\(874\\)\n \n 0\n \n 0\n
      \n \\(875\\)\n \n 0\n \n 0\n
      \n \\(876\\)\n \n 0\n \n 0\n
      \n \\(877\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(878\\)\n \n 0\n \n 0\n
      \n \\(879\\)\n \n 0\n \n 0\n
      \n \\(880\\)\n \n 0\n \n 0\n
      \n \\(881\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(882\\)\n \n 0\n \n 0\n
      \n \\(883\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(884\\)\n \n 0\n \n 0\n
      \n \\(885\\)\n \n 0\n \n 0\n
      \n \\(886\\)\n \n 0\n \n 0\n
      \n \\(887\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(888\\)\n \n 0\n \n 0\n
      \n \\(889\\)\n \n 0\n \n 0\n
      \n \\(890\\)\n \n 0\n \n 0\n
      \n \\(891\\)\n \n 0\n \n 0\n
      \n \\(892\\)\n \n 0\n \n 0\n
      \n \\(893\\)\n \n 0\n \n 0\n
      \n \\(894\\)\n \n 0\n \n 0\n
      \n \\(895\\)\n \n 0\n \n 0\n
      \n \\(896\\)\n \n 0\n \n 0\n
      \n \\(897\\)\n \n 0\n \n 0\n
      \n \\(898\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(899\\)\n \n 0\n \n 0\n
      \n \\(900\\)\n \n \u22120.500000\n \n +\n \n 0.866025i\n \u22120.500000\n \n +\n \n 0.866025i
      \n \\(901\\)\n \n 0\n \n 0\n
      \n \\(902\\)\n \n 0\n \n 0\n
      \n \\(903\\)\n \n 0\n \n 0\n
      \n \\(904\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
      \n \\(905\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
      \n \\(906\\)\n \n 0\n \n 0\n
      \n \\(907\\)\n \n 0\n \n 0\n \n 0.965926\n \n \u2212\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n \u22120.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(908\\)\n \n 0\n \n 0\n
      \n \\(909\\)\n \n 0\n \n 0\n
      \n \\(910\\)\n \n 0\n \n 0\n
      \n \\(911\\)\n \n 0\n \n 0\n \n \u22120.707107\n \n \u2212\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
      \n \\(912\\)\n \n 0\n \n 0\n
      \n \\(913\\)\n \n 0\n \n 0\n
      \n \\(914\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
      \n \\(915\\)\n \n 0\n \n 0\n
      \n \\(916\\)\n \n 2.00000i\n 2.00000i
      \n \\(917\\)\n \n 0\n \n 0\n
      \n \\(918\\)\n \n 0\n \n 0\n
      \n \\(919\\)\n \n 0\n \n 0\n \n \u22120.500000\n \n \u2212\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(920\\)\n \n 0\n \n 0\n
      \n \\(921\\)\n \n 0\n \n 0\n
      \n \\(922\\)\n \n 0\n \n 0\n
      \n \\(923\\)\n \n 0\n \n 0\n
      \n \\(924\\)\n \n 0\n \n 0\n
      \n \\(925\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(926\\)\n \n 0\n \n 0\n
      \n \\(927\\)\n \n 0\n \n 0\n
      \n \\(928\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i
      \n \\(929\\)\n \n \u22120.366025\n \n +\n \n 1.36603i\n \u22120.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(930\\)\n \n 0\n \n 0\n
      \n \\(931\\)\n \n 0\n \n 0\n
      \n \\(932\\)\n \n 1.00000\n \n \u2212\n \n 1.00000i\n 1.00000\n \n \u2212\n \n 1.00000i
      \n \\(933\\)\n \n 0\n \n 0\n
      \n \\(934\\)\n \n 0\n \n 0\n
      \n \\(935\\)\n \n 0\n \n 0\n
      \n \\(936\\)\n \n 0\n \n 0\n
      \n \\(937\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n \u22121.00000\n \n \\(\\pi\\)\n
      \n \\(938\\)\n \n 0\n \n 0\n
      \n \\(939\\)\n \n 0\n \n 0\n
      \n \\(940\\)\n \n 0\n \n 0\n
      \n \\(941\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(942\\)\n \n 0\n \n 0\n
      \n \\(943\\)\n \n 0\n \n 0\n
      \n \\(944\\)\n \n 0\n \n 0\n
      \n \\(945\\)\n \n 0\n \n 0\n
      \n \\(946\\)\n \n 0\n \n 0\n
      \n \\(947\\)\n \n 0\n \n 0\n \n \u22120.965926\n \n \u2212\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(948\\)\n \n 0\n \n 0\n
      \n \\(949\\)\n \n 0\n \n 0\n
      \n \\(950\\)\n \n 0\n \n 0\n
      \n \\(951\\)\n \n 0\n \n 0\n
      \n \\(952\\)\n \n 0\n \n 0\n
      \n \\(953\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(954\\)\n \n 0\n \n 0\n
      \n \\(955\\)\n \n 0\n \n 0\n
      \n \\(956\\)\n \n 0\n \n 0\n
      \n \\(957\\)\n \n 0\n \n 0\n
      \n \\(958\\)\n \n 0\n \n 0\n
      \n \\(959\\)\n \n 0\n \n 0\n
      \n \\(960\\)\n \n 0\n \n 0\n
      \n \\(961\\)\n \n \u22120.866025\n \n +\n \n 0.500000i\n \u22120.866025\n \n +\n \n 0.500000i
      \n \\(962\\)\n \n 0\n \n 0\n
      \n \\(963\\)\n \n 0\n \n 0\n
      \n \\(964\\)\n \n \u22121.36603\n \n \u2212\n \n 0.366025i\n \u22121.36603\n \n \u2212\n \n 0.366025i
      \n \\(965\\)\n \n 2.00000\n \n 2.00000\n
      \n \\(966\\)\n \n 0\n \n 0\n
      \n \\(967\\)\n \n 0\n \n 0\n \n \u2212\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(968\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
      \n \\(969\\)\n \n 0\n \n 0\n
      \n \\(970\\)\n \n 1.00000\n \n \u2212\n \n 1.73205i\n 1.00000\n \n \u2212\n \n 1.73205i
      \n \\(971\\)\n \n 0\n \n 0\n \n 0.866025\n \n \u2212\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \u22120.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(972\\)\n \n 0\n \n 0\n
      \n \\(973\\)\n \n 0\n \n 0\n
      \n \\(974\\)\n \n 0\n \n 0\n
      \n \\(975\\)\n \n 0\n \n 0\n
      \n \\(976\\)\n \n \u22121.36603\n \n +\n \n 0.366025i\n \u22121.36603\n \n +\n \n 0.366025i
      \n \\(977\\)\n \n 0\n \n 0\n \n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \u22120.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(978\\)\n \n 0\n \n 0\n
      \n \\(979\\)\n \n 0\n \n 0\n
      \n \\(980\\)\n \n 0\n \n 0\n
      \n \\(981\\)\n \n \u22121.00000\n \n +\n \n 1.00000i\n \u22121.00000\n \n +\n \n 1.00000i
      \n \\(982\\)\n \n 0\n \n 0\n
      \n \\(983\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(984\\)\n \n 0\n \n 0\n
      \n \\(985\\)\n \n \u22121.00000\n \n \u2212\n \n 1.73205i\n \u22121.00000\n \n \u2212\n \n 1.73205i
      \n \\(986\\)\n \n \u22121.00000\n \n \u2212\n \n 1.00000i\n \u22121.00000\n \n \u2212\n \n 1.00000i
      \n \\(987\\)\n \n 0\n \n 0\n
      \n \\(988\\)\n \n 0\n \n 0\n
      \n \\(989\\)\n \n 0\n \n 0\n
      \n \\(990\\)\n \n 0\n \n 0\n
      \n \\(991\\)\n \n 0\n \n 0\n \n \u22120.258819\n \n \u2212\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(992\\)\n \n 0\n \n 0\n
      \n \\(993\\)\n \n 0\n \n 0\n
      \n \\(994\\)\n \n 0\n \n 0\n
      \n \\(995\\)\n \n 0\n \n 0\n
      \n \\(996\\)\n \n 0\n \n 0\n
      \n \\(997\\)\n \n 1.36603\n \n \u2212\n \n 0.366025i\n 1.36603\n \n \u2212\n \n 0.366025i\n 0.500000\n \n \u2212\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(998\\)\n \n 0\n \n 0\n
      \n \\(999\\)\n \n 0\n \n 0\n
      \u2003\u2003\u2003\u2003\u2003\u2003\u2003By twisting character
      CharParityOrdTypeTwistMinDim
      1.1even1trivial3332.1.bc.b.2027.14
      4.3odd2CM3332.1.bc.b.2027.14
      7.2even3inner3332.1.bc.b.667.14
      7.3odd668.1.f.a.55.1yes2
      7.4even33332.1.m.b.2843.12
      7.5odd63332.1.bc.c.667.14
      7.6odd23332.1.bc.c.2027.14
      17.13even4inner3332.1.bc.b.2223.14
      21.17even6612.1.l.a.55.12
      28.3even668.1.f.a.55.1yes2
      28.11odd63332.1.m.b.2843.12
      28.19even63332.1.bc.c.667.14
      28.23odd6inner3332.1.bc.b.667.14
      28.27even23332.1.bc.c.2027.14
      35.3even121700.1.n.a.599.12
      35.17even121700.1.n.b.599.12
      35.24odd61700.1.p.a.1551.12
      56.3even61088.1.p.a.191.12
      56.45odd61088.1.p.a.191.12
      68.47odd4inner3332.1.bc.b.2223.14
      84.59odd6612.1.l.a.55.12
      119.3even481156.1.g.b.155.18
      119.10even481156.1.g.b.179.18
      119.13odd43332.1.bc.c.2223.14
      119.24even481156.1.g.b.179.28
      119.30even12inner3332.1.bc.b.863.14
      119.31even481156.1.g.b.155.28
      119.38odd121156.1.f.b.251.12
      119.45even481156.1.g.b.399.18
      119.47odd123332.1.bc.c.863.14
      119.59odd241156.1.c.b.579.12
      119.66odd241156.1.d.a.1155.22
      119.73even481156.1.g.b.423.28
      119.80even481156.1.g.b.423.18
      119.81even123332.1.m.b.3039.12
      119.87odd241156.1.d.a.1155.12
      119.94odd241156.1.c.b.579.22
      119.101odd61156.1.f.b.327.12
      119.108even481156.1.g.b.399.28
      119.115odd1268.1.f.a.47.1&check;2
      140.3odd121700.1.n.a.599.12
      140.59even61700.1.p.a.1551.12
      140.87odd121700.1.n.b.599.12
      357.353even12612.1.l.a.523.12
      476.3odd481156.1.g.b.155.18
      476.31odd481156.1.g.b.155.28
      476.47even123332.1.bc.c.863.14
      476.59even241156.1.c.b.579.12
      476.87even241156.1.d.a.1155.12
      476.115even1268.1.f.a.47.1&check;2
      476.143odd481156.1.g.b.179.28
      476.199odd481156.1.g.b.423.18
      476.227odd481156.1.g.b.399.28
      476.251even43332.1.bc.c.2223.14
      476.283odd481156.1.g.b.399.18
      476.311odd481156.1.g.b.423.28
      476.319odd123332.1.m.b.3039.12
      476.339even61156.1.f.b.327.12
      476.367odd481156.1.g.b.179.18
      476.387odd12inner3332.1.bc.b.863.14
      476.395even121156.1.f.b.251.12
      476.423even241156.1.d.a.1155.22
      476.451even241156.1.c.b.579.22
      595.234odd121700.1.p.a.251.12
      595.353even121700.1.n.b.999.12
      595.472even121700.1.n.a.999.12
      952.115even121088.1.p.a.319.12
      952.829odd121088.1.p.a.319.12
      1428.1067odd12612.1.l.a.523.12
      2380.1067odd121700.1.n.a.999.12
      2380.1543odd121700.1.n.b.999.12
      2380.2019even121700.1.p.a.251.12
      \u2003\u2003\u2003\u2003\u2003\u2003\u2003\u2003By twisted newform
      TwistMinDimCharParityOrdType
      68.1.f.a.47.1&check;2119.115odd12
      68.1.f.a.47.1&check;2476.115even12
      68.1.f.a.55.1yes27.3odd6
      68.1.f.a.55.1yes228.3even6
      612.1.l.a.55.1221.17even6
      612.1.l.a.55.1284.59odd6
      612.1.l.a.523.12357.353even12
      612.1.l.a.523.121428.1067odd12
      1088.1.p.a.191.1256.3even6
      1088.1.p.a.191.1256.45odd6
      1088.1.p.a.319.12952.115even12
      1088.1.p.a.319.12952.829odd12
      1156.1.c.b.579.12119.59odd24
      1156.1.c.b.579.12476.59even24
      1156.1.c.b.579.22119.94odd24
      1156.1.c.b.579.22476.451even24
      1156.1.d.a.1155.12119.87odd24
      1156.1.d.a.1155.12476.87even24
      1156.1.d.a.1155.22119.66odd24
      1156.1.d.a.1155.22476.423even24
      1156.1.f.b.251.12119.38odd12
      1156.1.f.b.251.12476.395even12
      1156.1.f.b.327.12119.101odd6
      1156.1.f.b.327.12476.339even6
      1156.1.g.b.155.18119.3even48
      1156.1.g.b.155.18476.3odd48
      1156.1.g.b.155.28119.31even48
      1156.1.g.b.155.28476.31odd48
      1156.1.g.b.179.18119.10even48
      1156.1.g.b.179.18476.367odd48
      1156.1.g.b.179.28119.24even48
      1156.1.g.b.179.28476.143odd48
      1156.1.g.b.399.18119.45even48
      1156.1.g.b.399.18476.283odd48
      1156.1.g.b.399.28119.108even48
      1156.1.g.b.399.28476.227odd48
      1156.1.g.b.423.18119.80even48
      1156.1.g.b.423.18476.199odd48
      1156.1.g.b.423.28119.73even48
      1156.1.g.b.423.28476.311odd48
      1700.1.n.a.599.1235.3even12
      1700.1.n.a.599.12140.3odd12
      1700.1.n.a.999.12595.472even12
      1700.1.n.a.999.122380.1067odd12
      1700.1.n.b.599.1235.17even12
      1700.1.n.b.599.12140.87odd12
      1700.1.n.b.999.12595.353even12
      1700.1.n.b.999.122380.1543odd12
      1700.1.p.a.251.12595.234odd12
      1700.1.p.a.251.122380.2019even12
      1700.1.p.a.1551.1235.24odd6
      1700.1.p.a.1551.12140.59even6
      3332.1.m.b.2843.127.4even3
      3332.1.m.b.2843.1228.11odd6
      3332.1.m.b.3039.12119.81even12
      3332.1.m.b.3039.12476.319odd12
      3332.1.bc.b.667.147.2even3inner
      3332.1.bc.b.667.1428.23odd6inner
      3332.1.bc.b.863.14119.30even12inner
      3332.1.bc.b.863.14476.387odd12inner
      3332.1.bc.b.2027.141.1even1trivial
      3332.1.bc.b.2027.144.3odd2CM
      3332.1.bc.b.2223.1417.13even4inner
      3332.1.bc.b.2223.1468.47odd4inner
      3332.1.bc.c.667.147.5odd6
      3332.1.bc.c.667.1428.19even6
      3332.1.bc.c.863.14119.47odd12
      3332.1.bc.c.863.14476.47even12
      3332.1.bc.c.2027.147.6odd2
      3332.1.bc.c.2027.1428.27even2
      3332.1.bc.c.2223.14119.13odd4
      3332.1.bc.c.2223.14476.251even4
      ", "statics": {"paragraph": 22, "paragraph.text": 24, "table": 8, "title": 4, "table.complex": 4, "paragraph.equation-inline": 2}, "url": "https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/3332/1/bc/b/2027/1/", "content": "Show commands: Magma/ PariGP/ SageMath\n\n[N,k,chi] = [3332,1,Mod(667,3332)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)\n\nfrom sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3332, base_ring=CyclotomicField(12)) chi = DirichletCharacter(H, H._module([6, 4, 9])) N = Newforms(chi, 1, names=\"a\")\n\n//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter(\"3332.667\"); S:= CuspForms(chi, 1); N := Newforms(S);\n\n| Level | \\( N \\) | \\(=\\) | \\( 3332 = 2^{2} \\cdot 7^{2} \\cdot 17 \\) |\n|---|---|---|---|\n| Weight | \\( k \\) | \\(=\\) | \\( 1 \\) |\n| Character orbit | \\([\\chi]\\) | \\(=\\) | 3332.bc
      order
      degree
      minimal |\n\n## Newform invariants\n\nsage:\u00a0f = N[0] \\# Warning: the index may be different\n\ngp:\u00a0f = lf[1] \\\\ Warning: the index may be different\n\n
      Self dualno
      Analytic conductor\\(1.66288462209\\)
      Analytic rank\\(0\\)
      Dimension\\(4\\)
      Coefficient field\\(\\Q(\\zeta_{12})\\)
      gp:\u00a0f.mod \\\\ as an extension of the character field
      Defining polynomial\\( x^{4} - x^{2} + 1 \\)
      Coefficient ring\\(\\Z[a_1, a_2]\\)
      Coefficient ring index\\( 1 \\)
      Twist minimalno (minimal twist has level 68)
      Projective image\\(D_{4}\\)
      Projective fieldGalois closure of<br>4.2.19652.1
      Artin image$C_4\\wr C_2\\times C_6$
      Artin fieldGalois closure of<br>\\(\\mathbb{Q}[x]/(x^{48} - \\cdots)\\)
      \n\n## Embedding invariants\n\n| Embedding label | | | 2027.1 |\n|---|---|---|---|\n| Root | | | \\(0.866025 - 0.500000i\\) of defining polynomial |\n| Character | \\(\\chi\\) | \\(=\\) | 3332.2027 |\n| Dual form | | | 3332.1.bc.b.863.1 |\n\nsage:\u00a0f.q_expansion() \\# note that sage often uses an isomorphic number field\n\ngp:\u00a0mfcoefs(f, 20)\n\n| \\(f(q)\\) | \\(=\\) | \\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.36603 - 0.366025i) q^{5} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.00000i) q^{20} +(0.866025 + 0.500000i) q^{25} +(1.00000 - 1.00000i) q^{29} +(0.866025 + 0.500000i) q^{32} -1.00000i q^{34} +1.00000i q^{36} +(1.36603 + 0.366025i) q^{37} +(0.366025 - 1.36603i) q^{40} +(-1.00000 - 1.00000i) q^{41} +(1.36603 - 0.366025i) q^{45} -1.00000 q^{50} +(-0.366025 + 1.36603i) q^{58} +(0.366025 - 1.36603i) q^{61} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(-0.366025 - 1.36603i) q^{73} +(-1.36603 + 0.366025i) q^{74} +(0.366025 + 1.36603i) q^{80} +(0.500000 - 0.866025i) q^{81} +(1.36603 + 0.366025i) q^{82} +(1.00000 - 1.00000i) q^{85} +(-1.00000 + 1.00000i) q^{90} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\\) |\n|---|---|---|\n| \\(\\operatorname{Tr}(f)(q)\\) | \\(=\\) | \\( 4 q + 2 q^{4} - 2 q^{5} + 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 4 q^{20} + 4 q^{29} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 2 q^{45} - 4 q^{50} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{68} - 2 q^{72} + 2 q^{73}+ \\cdots + 4 q^{97}+O(q^{100}) \\) |\n\n## Character values\n\nWe give the values of $\\chi$ on generators for $\\left(\\mathbb{Z}/3332\\mathbb{Z}\\right)^\\times$ .\n\n| \\(n\\) | \\(785\\) | \\(885\\) | \\(1667\\) |\n|---|---|---|---|\n| \\(\\chi(n)\\) | \\(e\\left(\\frac{3}{4}\\right)\\) | \\(e\\left(\\frac{2}{3}\\right)\\) | \\(-1\\) |\n\n## Coefficient data\n\nFor each \\(n\\) we display the coefficients of the \\(q\\)-expansion \\(a_n\\), the Satake parameters\\(\\alpha_p\\),\nand the Satake angles \\(\\theta_p = \\textrm{Arg}(\\alpha_p)\\).\n\n( See \\(a_n\\) instead)\n\n( See \\(a_n\\) instead)\n\n( See \\(a_n\\) instead)\n\n( See only \\(a_p\\))\n\n( See only \\(a_p\\))\n\n( See only \\(a_p\\))\n\n
      \\(n\\)\\(a_n\\)\\(a_n / n^{(k-1)/2}\\)\\( \\alpha_n \\)\\( \\theta_n \\)
      \\(p\\)\\(a_p\\)\\(a_p / p^{(k-1)/2}\\)\\( \\alpha_p\\)\\( \\theta_p \\)
      \\(2\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
      \\(3\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
      0.258819+0.965926<br>i\\(0.416667\\pi\\)
      \\(4\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
      \\(5\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(6\\)00
      \\(7\\)00
      \\(8\\)1.00000<br>i1.00000<br>i
      \\(9\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
      \\(10\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
      \\(11\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(12\\)00
      \\(13\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(14\\)00
      \\(15\\)00
      \\(16\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
      \\(17\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
      \\(18\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
      \\(19\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(20\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
      \\(21\\)00
      \\(22\\)00
      \\(23\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
      \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
      \\(24\\)00
      \\(25\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
      \\(26\\)00
      \\(27\\)00
      \\(28\\)00
      \\(29\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000\\(0\\)
      \\(30\\)00
      \\(31\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
      0.258819+0.965926<br>i\\(0.416667\\pi\\)
      \\(32\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
      \\(33\\)00
      \\(34\\)\u22121.00000<br>i\u22121.00000<br>i
      \\(35\\)00
      \\(36\\)1.00000<br>i1.00000<br>i
      \\(37\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(38\\)00
      \\(39\\)00
      \\(40\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
      \\(41\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
      \u22121.00000\\(\\pi\\)
      \\(42\\)00
      \\(43\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(44\\)00
      \\(45\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
      \\(46\\)00
      \\(47\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(48\\)00
      \\(49\\)00
      \\(50\\)\u22121.00000\u22121.00000
      \\(51\\)00
      \\(52\\)00
      \\(53\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(54\\)00
      \\(55\\)00
      \\(56\\)00
      \\(57\\)00
      \\(58\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
      \\(59\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(60\\)00
      \\(61\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \\(62\\)00
      \\(63\\)00
      \\(64\\)\u22121.00000\u22121.00000
      \\(65\\)00
      \\(66\\)00
      \\(67\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(68\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
      \\(69\\)00
      \\(70\\)00
      \\(71\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
      \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
      \\(72\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
      \\(73\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \\(74\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
      \\(75\\)00
      \\(76\\)00
      \\(77\\)00
      \\(78\\)00
      \\(79\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
      \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
      \\(80\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
      \\(81\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
      \\(82\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
      \\(83\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(84\\)00
      \\(85\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
      \\(86\\)00
      \\(87\\)00
      \\(88\\)00
      \\(89\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(90\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
      \\(91\\)00
      \\(92\\)00
      \\(93\\)00
      \\(94\\)00
      \\(95\\)00
      \\(96\\)00
      \\(97\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000\\(0\\)
      \\(98\\)00
      \\(99\\)00
      \\(100\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
      \\(101\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(102\\)00
      \\(103\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(104\\)00
      \\(105\\)00
      \\(106\\)00
      \\(107\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
      0.965926+0.258819<br>i\\(0.0833333\\pi\\)
      \\(108\\)00
      \\(109\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(110\\)00
      \\(111\\)00
      \\(112\\)00
      \\(113\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
      \u22121.00000\\(\\pi\\)
      \\(114\\)00
      \\(115\\)00
      \\(116\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
      \\(117\\)00
      \\(118\\)00
      \\(119\\)00
      \\(120\\)00
      \\(121\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
      \\(122\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
      \\(123\\)00
      \\(124\\)00
      \\(125\\)00
      \\(126\\)00
      \\(127\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(128\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
      \\(129\\)00
      \\(130\\)00
      \\(131\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
      0.965926+0.258819<br>i\\(0.0833333\\pi\\)
      \\(132\\)00
      \\(133\\)00
      \\(134\\)00
      \\(135\\)00
      \\(136\\)\u22120.866025\u22120.500000<br>i\u22120.866025\u22120.500000<br>i
      \\(137\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(138\\)00
      \\(139\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
      \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
      \\(140\\)00
      \\(141\\)00
      \\(142\\)00
      \\(143\\)00
      \\(144\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
      \\(145\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
      \\(146\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
      \\(147\\)00
      \\(148\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
      \\(149\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      \u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      \\(150\\)00
      \\(151\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(152\\)00
      \\(153\\)\u22121.00000<br>i\u22121.00000<br>i
      \\(154\\)00
      \\(155\\)00
      \\(156\\)00
      \\(157\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \\(158\\)00
      \\(159\\)00
      \\(160\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
      \\(161\\)00
      \\(162\\)1.00000<br>i1.00000<br>i
      \\(163\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
      \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
      \\(164\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
      \\(165\\)00
      \\(166\\)00
      \\(167\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
      \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
      \\(168\\)00
      \\(169\\)\u22121.00000\u22121.00000
      \\(170\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
      \\(171\\)00
      \\(172\\)00
      \\(173\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(174\\)00
      \\(175\\)00
      \\(176\\)00
      \\(177\\)00
      \\(178\\)00
      \\(179\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(180\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
      \\(181\\)1.00000+1.00000<br>i1.00000+1.00000<br>i1.00000\\(0\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(182\\)00
      \\(183\\)00
      \\(184\\)00
      \\(185\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i
      \\(186\\)00
      \\(187\\)00
      \\(188\\)00
      \\(189\\)00
      \\(190\\)00
      \\(191\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(192\\)00
      \\(193\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(194\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
      \\(195\\)00
      \\(196\\)00
      \\(197\\)1.00000+1.00000<br>i1.00000+1.00000<br>i1.00000\\(0\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(198\\)00
      \\(199\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
      0.258819+0.965926<br>i\\(0.416667\\pi\\)
      \\(200\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
      \\(201\\)00
      \\(202\\)00
      \\(203\\)00
      \\(204\\)00
      \\(205\\)1.00000+1.73205<br>i1.00000+1.73205<br>i
      \\(206\\)00
      \\(207\\)00
      \\(208\\)00
      \\(209\\)00
      \\(210\\)00
      \\(211\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
      0.707107+0.707107<br>i\\(0.250000\\pi\\)
      \\(212\\)00
      \\(213\\)00
      \\(214\\)00
      \\(215\\)00
      \\(216\\)00
      \\(217\\)00
      \\(218\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
      \\(219\\)00
      \\(220\\)00
      \\(221\\)00
      \\(222\\)00
      \\(223\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(224\\)00
      \\(225\\)\u22121.00000\u22121.00000
      \\(226\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
      \\(227\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(228\\)00
      \\(229\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i\u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(230\\)00
      \\(231\\)00
      \\(232\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
      \\(233\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(234\\)00
      \\(235\\)00
      \\(236\\)00
      \\(237\\)00
      \\(238\\)00
      \\(239\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(240\\)00
      \\(241\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \\(242\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
      \\(243\\)00
      \\(244\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
      \\(245\\)00
      \\(246\\)00
      \\(247\\)00
      \\(248\\)00
      \\(249\\)00
      \\(250\\)00
      \\(251\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(252\\)00
      \\(253\\)00
      \\(254\\)00
      \\(255\\)00
      \\(256\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
      \\(257\\)1.73205\u22121.00000<br>i1.73205\u22121.00000<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \\(258\\)00
      \\(259\\)00
      \\(260\\)00
      \\(261\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
      \\(262\\)00
      \\(263\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(264\\)00
      \\(265\\)00
      \\(266\\)00
      \\(267\\)00
      \\(268\\)00
      \\(269\\)0.366025+1.36603<br>i0.366025+1.36603<br>i0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(270\\)00
      \\(271\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(272\\)1.000001.00000
      \\(273\\)00
      \\(274\\)00
      \\(275\\)00
      \\(276\\)00
      \\(277\\)0.366025+1.36603<br>i0.366025+1.36603<br>i0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(278\\)00
      \\(279\\)00
      \\(280\\)00
      \\(281\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(282\\)00
      \\(283\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(284\\)00
      \\(285\\)00
      \\(286\\)00
      \\(287\\)00
      \\(288\\)\u22121.00000\u22121.00000
      \\(289\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
      \\(290\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
      \\(291\\)00
      \\(292\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
      \\(293\\)2.000002.000001.00000\\(0\\)
      1.00000\\(0\\)
      \\(294\\)00
      \\(295\\)00
      \\(296\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
      \\(297\\)00
      \\(298\\)1.73205+1.00000<br>i1.73205+1.00000<br>i
      \\(299\\)00
      \\(300\\)00
      \\(301\\)00
      \\(302\\)00
      \\(303\\)00
      \\(304\\)00
      \\(305\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i
      \\(306\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
      \\(307\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(308\\)00
      \\(309\\)00
      \\(310\\)00
      \\(311\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
      0.258819+0.965926<br>i\\(0.416667\\pi\\)
      \\(312\\)00
      \\(313\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(314\\)2.00000<br>i2.00000<br>i
      \\(315\\)00
      \\(316\\)00
      \\(317\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \\(318\\)00
      \\(319\\)00
      \\(320\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
      \\(321\\)00
      \\(322\\)00
      \\(323\\)00
      \\(324\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
      \\(325\\)00
      \\(326\\)00
      \\(327\\)00
      \\(328\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
      \\(329\\)00
      \\(330\\)00
      \\(331\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(332\\)00
      \\(333\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
      \\(334\\)00
      \\(335\\)00
      \\(336\\)00
      \\(337\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000\\(0\\)
      \\(338\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
      \\(339\\)00
      \\(340\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
      \\(341\\)00
      \\(342\\)00
      \\(343\\)00
      \\(344\\)00
      \\(345\\)00
      \\(346\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
      \\(347\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(348\\)00
      \\(349\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(350\\)00
      \\(351\\)00
      \\(352\\)00
      \\(353\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i\u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(354\\)00
      \\(355\\)00
      \\(356\\)00
      \\(357\\)00
      \\(358\\)00
      \\(359\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(360\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
      \\(361\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
      \\(362\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
      \\(363\\)00
      \\(364\\)00
      \\(365\\)2.00000<br>i2.00000<br>i
      \\(366\\)00
      \\(367\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(368\\)00
      \\(369\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
      \\(370\\)2.000002.00000
      \\(371\\)00
      \\(372\\)00
      \\(373\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(374\\)00
      \\(375\\)00
      \\(376\\)00
      \\(377\\)00
      \\(378\\)00
      \\(379\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
      \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
      \\(380\\)00
      \\(381\\)00
      \\(382\\)00
      \\(383\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(384\\)00
      \\(385\\)00
      \\(386\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
      \\(387\\)00
      \\(388\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
      \\(389\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      \u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      \\(390\\)00
      \\(391\\)00
      \\(392\\)00
      \\(393\\)00
      \\(394\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
      \\(395\\)00
      \\(396\\)00
      \\(397\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \\(398\\)00
      \\(399\\)00
      \\(400\\)\u22121.00000<br>i\u22121.00000<br>i
      \\(401\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \\(402\\)00
      \\(403\\)00
      \\(404\\)00
      \\(405\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
      \\(406\\)00
      \\(407\\)00
      \\(408\\)00
      \\(409\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i\u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(410\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i
      \\(411\\)00
      \\(412\\)00
      \\(413\\)00
      \\(414\\)00
      \\(415\\)00
      \\(416\\)00
      \\(417\\)00
      \\(418\\)00
      \\(419\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
      0.707107+0.707107<br>i\\(0.250000\\pi\\)
      \\(420\\)00
      \\(421\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(422\\)00
      \\(423\\)00
      \\(424\\)00
      \\(425\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
      \\(426\\)00
      \\(427\\)00
      \\(428\\)00
      \\(429\\)00
      \\(430\\)00
      \\(431\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(432\\)00
      \\(433\\)2.00000<br>i2.00000<br>i1.00000<br>i\\(0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(434\\)00
      \\(435\\)00
      \\(436\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
      \\(437\\)00
      \\(438\\)00
      \\(439\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
      0.965926+0.258819<br>i\\(0.0833333\\pi\\)
      \\(440\\)00
      \\(441\\)00
      \\(442\\)00
      \\(443\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(444\\)00
      \\(445\\)00
      \\(446\\)00
      \\(447\\)00
      \\(448\\)00
      \\(449\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
      \u22121.00000\\(\\pi\\)
      \\(450\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
      \\(451\\)00
      \\(452\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
      \\(453\\)00
      \\(454\\)00
      \\(455\\)00
      \\(456\\)00
      \\(457\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i\u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(458\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
      \\(459\\)00
      \\(460\\)00
      \\(461\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(462\\)00
      \\(463\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(464\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
      \\(465\\)00
      \\(466\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
      \\(467\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(468\\)00
      \\(469\\)00
      \\(470\\)00
      \\(471\\)00
      \\(472\\)00
      \\(473\\)00
      \\(474\\)00
      \\(475\\)00
      \\(476\\)00
      \\(477\\)00
      \\(478\\)00
      \\(479\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
      0.258819+0.965926<br>i\\(0.416667\\pi\\)
      \\(480\\)00
      \\(481\\)00
      \\(482\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
      \\(483\\)00
      \\(484\\)\u22121.00000<br>i\u22121.00000<br>i
      \\(485\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
      \\(486\\)00
      \\(487\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(488\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
      \\(489\\)00
      \\(490\\)00
      \\(491\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(492\\)00
      \\(493\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
      \\(494\\)00
      \\(495\\)00
      \\(496\\)00
      \\(497\\)00
      \\(498\\)00
      \\(499\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
      \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
      \\(500\\)00
      \\(501\\)00
      \\(502\\)00
      \\(503\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
      0.707107+0.707107<br>i\\(0.250000\\pi\\)
      \\(504\\)00
      \\(505\\)00
      \\(506\\)00
      \\(507\\)00
      \\(508\\)00
      \\(509\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      \u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      \\(510\\)00
      \\(511\\)00
      \\(512\\)\u22121.00000<br>i\u22121.00000<br>i
      \\(513\\)00
      \\(514\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i
      \\(515\\)00
      \\(516\\)00
      \\(517\\)00
      \\(518\\)00
      \\(519\\)00
      \\(520\\)00
      \\(521\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(522\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
      \\(523\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(524\\)00
      \\(525\\)00
      \\(526\\)00
      \\(527\\)00
      \\(528\\)00
      \\(529\\)\u22120.866025\u22120.500000<br>i\u22120.866025\u22120.500000<br>i
      \\(530\\)00
      \\(531\\)00
      \\(532\\)00
      \\(533\\)00
      \\(534\\)00
      \\(535\\)00
      \\(536\\)00
      \\(537\\)00
      \\(538\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
      \\(539\\)00
      \\(540\\)00
      \\(541\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(542\\)00
      \\(543\\)00
      \\(544\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
      \\(545\\)\u22122.00000\u22122.00000
      \\(546\\)00
      \\(547\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
      \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
      \\(548\\)00
      \\(549\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
      \\(550\\)00
      \\(551\\)00
      \\(552\\)00
      \\(553\\)00
      \\(554\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
      \\(555\\)00
      \\(556\\)00
      \\(557\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(558\\)00
      \\(559\\)00
      \\(560\\)00
      \\(561\\)00
      \\(562\\)00
      \\(563\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(564\\)00
      \\(565\\)1.00000+1.73205<br>i1.00000+1.73205<br>i
      \\(566\\)00
      \\(567\\)00
      \\(568\\)00
      \\(569\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(570\\)00
      \\(571\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(572\\)00
      \\(573\\)00
      \\(574\\)00
      \\(575\\)00
      \\(576\\)0.866025\u22120.500000<br>i0.866025\u22120.500000<br>i
      \\(577\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(578\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
      \\(579\\)00
      \\(580\\)2.00000<br>i2.00000<br>i
      \\(581\\)00
      \\(582\\)00
      \\(583\\)00
      \\(584\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
      \\(585\\)00
      \\(586\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
      \\(587\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(588\\)00
      \\(589\\)00
      \\(590\\)00
      \\(591\\)00
      \\(592\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
      \\(593\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(594\\)00
      \\(595\\)00
      \\(596\\)\u22122.00000\u22122.00000
      \\(597\\)00
      \\(598\\)00
      \\(599\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(600\\)00
      \\(601\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
      \u22121.00000\\(\\pi\\)
      \\(602\\)00
      \\(603\\)00
      \\(604\\)00
      \\(605\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
      \\(606\\)00
      \\(607\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
      0.965926+0.258819<br>i\\(0.0833333\\pi\\)
      \\(608\\)00
      \\(609\\)00
      \\(610\\)\u22122.00000<br>i\u22122.00000<br>i
      \\(611\\)00
      \\(612\\)\u22120.866025\u22120.500000<br>i\u22120.866025\u22120.500000<br>i
      \\(613\\)\u22121.00000+1.73205<br>i\u22121.00000+1.73205<br>i\u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(614\\)00
      \\(615\\)00
      \\(616\\)00
      \\(617\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i1.00000<br>i\\(0.5\\pi\\)
      \u22121.00000\\(\\pi\\)
      \\(618\\)00
      \\(619\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(620\\)00
      \\(621\\)00
      \\(622\\)00
      \\(623\\)00
      \\(624\\)00
      \\(625\\)\u22120.500000\u22120.866025<br>i\u22120.500000\u22120.866025<br>i
      \\(626\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i
      \\(627\\)00
      \\(628\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
      \\(629\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
      \\(630\\)00
      \\(631\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(632\\)00
      \\(633\\)00
      \\(634\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
      \\(635\\)00
      \\(636\\)00
      \\(637\\)00
      \\(638\\)00
      \\(639\\)00
      \\(640\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
      \\(641\\)\u22120.366025\u22121.36603<br>i\u22120.366025\u22121.36603<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \\(642\\)00
      \\(643\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
      \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
      \\(644\\)00
      \\(645\\)00
      \\(646\\)00
      \\(647\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(648\\)0.866025+0.500000<br>i0.866025+0.500000<br>i
      \\(649\\)00
      \\(650\\)00
      \\(651\\)00
      \\(652\\)00
      \\(653\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \\(654\\)00
      \\(655\\)00
      \\(656\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
      \\(657\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
      \\(658\\)00
      \\(659\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(660\\)00
      \\(661\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(662\\)00
      \\(663\\)00
      \\(664\\)00
      \\(665\\)00
      \\(666\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
      \\(667\\)00
      \\(668\\)00
      \\(669\\)00
      \\(670\\)00
      \\(671\\)00
      \\(672\\)00
      \\(673\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
      \u22121.00000\\(\\pi\\)
      \\(674\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
      \\(675\\)00
      \\(676\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
      \\(677\\)1.36603+0.366025<br>i1.36603+0.366025<br>i0.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(678\\)00
      \\(679\\)00
      \\(680\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
      \\(681\\)00
      \\(682\\)00
      \\(683\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
      0.258819+0.965926<br>i\\(0.416667\\pi\\)
      \\(684\\)00
      \\(685\\)00
      \\(686\\)00
      \\(687\\)00
      \\(688\\)00
      \\(689\\)00
      \\(690\\)00
      \\(691\\)000.258819\u22120.965926<br>i\\(-0.416667\\pi\\)
      \u22120.258819+0.965926<br>i\\(0.583333\\pi\\)
      \\(692\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
      \\(693\\)00
      \\(694\\)00
      \\(695\\)00
      \\(696\\)00
      \\(697\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
      \\(698\\)00
      \\(699\\)00
      \\(700\\)00
      \\(701\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(702\\)00
      \\(703\\)00
      \\(704\\)00
      \\(705\\)00
      \\(706\\)\u22122.00000<br>i\u22122.00000<br>i
      \\(707\\)00
      \\(708\\)00
      \\(709\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(710\\)00
      \\(711\\)00
      \\(712\\)00
      \\(713\\)00
      \\(714\\)00
      \\(715\\)00
      \\(716\\)00
      \\(717\\)00
      \\(718\\)00
      \\(719\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
      0.965926+0.258819<br>i\\(0.0833333\\pi\\)
      \\(720\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
      \\(721\\)00
      \\(722\\)1.00000<br>i1.00000<br>i
      \\(723\\)00
      \\(724\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
      \\(725\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
      \\(726\\)00
      \\(727\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(728\\)00
      \\(729\\)1.00000<br>i1.00000<br>i
      \\(730\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
      \\(731\\)00
      \\(732\\)00
      \\(733\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(734\\)00
      \\(735\\)00
      \\(736\\)00
      \\(737\\)00
      \\(738\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
      \\(739\\)00\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(740\\)\u22121.73205+1.00000<br>i\u22121.73205+1.00000<br>i
      \\(741\\)00
      \\(742\\)00
      \\(743\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
      \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
      \\(744\\)00
      \\(745\\)0.732051+2.73205<br>i0.732051+2.73205<br>i
      \\(746\\)00
      \\(747\\)00
      \\(748\\)00
      \\(749\\)00
      \\(750\\)00
      \\(751\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
      0.965926+0.258819<br>i\\(0.0833333\\pi\\)
      \\(752\\)00
      \\(753\\)00
      \\(754\\)00
      \\(755\\)00
      \\(756\\)00
      \\(757\\)2.00000<br>i2.00000<br>i1.00000<br>i\\(0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(758\\)00
      \\(759\\)00
      \\(760\\)00
      \\(761\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(762\\)00
      \\(763\\)00
      \\(764\\)00
      \\(765\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
      \\(766\\)00
      \\(767\\)00
      \\(768\\)00
      \\(769\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(770\\)00
      \\(771\\)00
      \\(772\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
      \\(773\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i\u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      \u22120.866025\u22120.500000<br>i\\(-0.833333\\pi\\)
      \\(774\\)00
      \\(775\\)00
      \\(776\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
      \\(777\\)00
      \\(778\\)2.000002.00000
      \\(779\\)00
      \\(780\\)00
      \\(781\\)00
      \\(782\\)00
      \\(783\\)00
      \\(784\\)00
      \\(785\\)\u22122.00000+2.00000<br>i\u22122.00000+2.00000<br>i
      \\(786\\)00
      \\(787\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
      0.258819+0.965926<br>i\\(0.416667\\pi\\)
      \\(788\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
      \\(789\\)00
      \\(790\\)00
      \\(791\\)00
      \\(792\\)00
      \\(793\\)00
      \\(794\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
      \\(795\\)00
      \\(796\\)00
      \\(797\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(798\\)00
      \\(799\\)00
      \\(800\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
      \\(801\\)00
      \\(802\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
      \\(803\\)00
      \\(804\\)00
      \\(805\\)00
      \\(806\\)00
      \\(807\\)00
      \\(808\\)00
      \\(809\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(810\\)0.366025\u22121.36603<br>i0.366025\u22121.36603<br>i
      \\(811\\)000.707107\u22120.707107<br>i\\(-0.250000\\pi\\)
      \u22120.707107+0.707107<br>i\\(0.750000\\pi\\)
      \\(812\\)00
      \\(813\\)00
      \\(814\\)00
      \\(815\\)00
      \\(816\\)00
      \\(817\\)00
      \\(818\\)\u22122.00000<br>i\u22122.00000<br>i
      \\(819\\)00
      \\(820\\)2.000002.00000
      \\(821\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(822\\)00
      \\(823\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(824\\)00
      \\(825\\)00
      \\(826\\)00
      \\(827\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
      0.707107+0.707107<br>i\\(0.250000\\pi\\)
      \\(828\\)00
      \\(829\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \\(830\\)00
      \\(831\\)00
      \\(832\\)00
      \\(833\\)00
      \\(834\\)00
      \\(835\\)00
      \\(836\\)00
      \\(837\\)00
      \\(838\\)00
      \\(839\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
      0.707107+0.707107<br>i\\(0.250000\\pi\\)
      \\(840\\)00
      \\(841\\)\u22121.00000<br>i\u22121.00000<br>i
      \\(842\\)00
      \\(843\\)00
      \\(844\\)00
      \\(845\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
      \\(846\\)00
      \\(847\\)00
      \\(848\\)00
      \\(849\\)00
      \\(850\\)0.500000\u22120.866025<br>i0.500000\u22120.866025<br>i
      \\(851\\)00
      \\(852\\)00
      \\(853\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000\\(0\\)
      \\(854\\)00
      \\(855\\)00
      \\(856\\)00
      \\(857\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(858\\)00
      \\(859\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(860\\)00
      \\(861\\)00
      \\(862\\)00
      \\(863\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(864\\)00
      \\(865\\)\u22121.73205\u22121.00000<br>i\u22121.73205\u22121.00000<br>i
      \\(866\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
      \\(867\\)00
      \\(868\\)00
      \\(869\\)00
      \\(870\\)00
      \\(871\\)00
      \\(872\\)0.366025+1.36603<br>i0.366025+1.36603<br>i
      \\(873\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i
      \\(874\\)00
      \\(875\\)00
      \\(876\\)00
      \\(877\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(878\\)00
      \\(879\\)00
      \\(880\\)00
      \\(881\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i1.00000<br>i\\(0.5\\pi\\)
      \u22121.00000\\(\\pi\\)
      \\(882\\)00
      \\(883\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(884\\)00
      \\(885\\)00
      \\(886\\)00
      \\(887\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
      0.965926+0.258819<br>i\\(0.0833333\\pi\\)
      \\(888\\)00
      \\(889\\)00
      \\(890\\)00
      \\(891\\)00
      \\(892\\)00
      \\(893\\)00
      \\(894\\)00
      \\(895\\)00
      \\(896\\)00
      \\(897\\)00
      \\(898\\)1.36603+0.366025<br>i1.36603+0.366025<br>i
      \\(899\\)00
      \\(900\\)\u22120.500000+0.866025<br>i\u22120.500000+0.866025<br>i
      \\(901\\)00
      \\(902\\)00
      \\(903\\)00
      \\(904\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
      \\(905\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
      \\(906\\)00
      \\(907\\)000.965926\u22120.258819<br>i\\(-0.0833333\\pi\\)
      \u22120.965926+0.258819<br>i\\(0.916667\\pi\\)
      \\(908\\)00
      \\(909\\)00
      \\(910\\)00
      \\(911\\)00\u22120.707107\u22120.707107<br>i\\(-0.750000\\pi\\)
      0.707107+0.707107<br>i\\(0.250000\\pi\\)
      \\(912\\)00
      \\(913\\)00
      \\(914\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
      \\(915\\)00
      \\(916\\)2.00000<br>i2.00000<br>i
      \\(917\\)00
      \\(918\\)00
      \\(919\\)00\u22120.500000\u22120.866025<br>i\\(-0.666667\\pi\\)
      0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \\(920\\)00
      \\(921\\)00
      \\(922\\)00
      \\(923\\)00
      \\(924\\)00
      \\(925\\)1.00000+1.00000<br>i1.00000+1.00000<br>i
      \\(926\\)00
      \\(927\\)00
      \\(928\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i
      \\(929\\)\u22120.366025+1.36603<br>i\u22120.366025+1.36603<br>i0.500000+0.866025<br>i\\(0.333333\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(930\\)00
      \\(931\\)00
      \\(932\\)1.00000\u22121.00000<br>i1.00000\u22121.00000<br>i
      \\(933\\)00
      \\(934\\)00
      \\(935\\)00
      \\(936\\)00
      \\(937\\)001.00000\\(0\\)
      \u22121.00000\\(\\pi\\)
      \\(938\\)00
      \\(939\\)00
      \\(940\\)00
      \\(941\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(942\\)00
      \\(943\\)00
      \\(944\\)00
      \\(945\\)00
      \\(946\\)00
      \\(947\\)00\u22120.965926\u22120.258819<br>i\\(-0.916667\\pi\\)
      0.965926+0.258819<br>i\\(0.0833333\\pi\\)
      \\(948\\)00
      \\(949\\)00
      \\(950\\)00
      \\(951\\)00
      \\(952\\)00
      \\(953\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(954\\)00
      \\(955\\)00
      \\(956\\)00
      \\(957\\)00
      \\(958\\)00
      \\(959\\)00
      \\(960\\)00
      \\(961\\)\u22120.866025+0.500000<br>i\u22120.866025+0.500000<br>i
      \\(962\\)00
      \\(963\\)00
      \\(964\\)\u22121.36603\u22120.366025<br>i\u22121.36603\u22120.366025<br>i
      \\(965\\)2.000002.00000
      \\(966\\)00
      \\(967\\)00\u22121.00000<br>i\\(-0.5\\pi\\)
      1.00000<br>i\\(0.5\\pi\\)
      \\(968\\)0.500000+0.866025<br>i0.500000+0.866025<br>i
      \\(969\\)00
      \\(970\\)1.00000\u22121.73205<br>i1.00000\u22121.73205<br>i
      \\(971\\)000.866025\u22120.500000<br>i\\(-0.166667\\pi\\)
      \u22120.866025+0.500000<br>i\\(0.833333\\pi\\)
      \\(972\\)00
      \\(973\\)00
      \\(974\\)00
      \\(975\\)00
      \\(976\\)\u22121.36603+0.366025<br>i\u22121.36603+0.366025<br>i
      \\(977\\)000.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      \u22120.500000+0.866025<br>i\\(0.666667\\pi\\)
      \\(978\\)00
      \\(979\\)00
      \\(980\\)00
      \\(981\\)\u22121.00000+1.00000<br>i\u22121.00000+1.00000<br>i
      \\(982\\)00
      \\(983\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
      0.258819+0.965926<br>i\\(0.416667\\pi\\)
      \\(984\\)00
      \\(985\\)\u22121.00000\u22121.73205<br>i\u22121.00000\u22121.73205<br>i
      \\(986\\)\u22121.00000\u22121.00000<br>i\u22121.00000\u22121.00000<br>i
      \\(987\\)00
      \\(988\\)00
      \\(989\\)00
      \\(990\\)00
      \\(991\\)00\u22120.258819\u22120.965926<br>i\\(-0.583333\\pi\\)
      0.258819+0.965926<br>i\\(0.416667\\pi\\)
      \\(992\\)00
      \\(993\\)00
      \\(994\\)00
      \\(995\\)00
      \\(996\\)00
      \\(997\\)1.36603\u22120.366025<br>i1.36603\u22120.366025<br>i0.500000\u22120.866025<br>i\\(-0.333333\\pi\\)
      0.866025+0.500000<br>i\\(0.166667\\pi\\)
      \\(998\\)00
      \\(999\\)00
      \n\n( See \\(a_n\\) instead)\n\n( See \\(a_n\\) instead)\n\n( See \\(a_n\\) instead)\n\n( See only \\(a_p\\))\n\n( See only \\(a_p\\))\n\n( See only \\(a_p\\))\n\n
      By<br>twisting character
      CharParityOrdTypeTwistMinDim
      1.1even1trivial3332.1.bc.b.2027.14
      4.3odd2CM3332.1.bc.b.2027.14
      7.2even3inner3332.1.bc.b.667.14
      7.3odd668.1.f.a.55.1yes2
      7.4even33332.1.m.b.2843.12
      7.5odd63332.1.bc.c.667.14
      7.6odd23332.1.bc.c.2027.14
      17.13even4inner3332.1.bc.b.2223.14
      21.17even6612.1.l.a.55.12
      28.3even668.1.f.a.55.1yes2
      28.11odd63332.1.m.b.2843.12
      28.19even63332.1.bc.c.667.14
      28.23odd6inner3332.1.bc.b.667.14
      28.27even23332.1.bc.c.2027.14
      35.3even121700.1.n.a.599.12
      35.17even121700.1.n.b.599.12
      35.24odd61700.1.p.a.1551.12
      56.3even61088.1.p.a.191.12
      56.45odd61088.1.p.a.191.12
      68.47odd4inner3332.1.bc.b.2223.14
      84.59odd6612.1.l.a.55.12
      119.3even481156.1.g.b.155.18
      119.10even481156.1.g.b.179.18
      119.13odd43332.1.bc.c.2223.14
      119.24even481156.1.g.b.179.28
      119.30even12inner3332.1.bc.b.863.14
      119.31even481156.1.g.b.155.28
      119.38odd121156.1.f.b.251.12
      119.45even481156.1.g.b.399.18
      119.47odd123332.1.bc.c.863.14
      119.59odd241156.1.c.b.579.12
      119.66odd241156.1.d.a.1155.22
      119.73even481156.1.g.b.423.28
      119.80even481156.1.g.b.423.18
      119.81even123332.1.m.b.3039.12
      119.87odd241156.1.d.a.1155.12
      119.94odd241156.1.c.b.579.22
      119.101odd61156.1.f.b.327.12
      119.108even481156.1.g.b.399.28
      119.115odd1268.1.f.a.47.1&check;2
      140.3odd121700.1.n.a.599.12
      140.59even61700.1.p.a.1551.12
      140.87odd121700.1.n.b.599.12
      357.353even12612.1.l.a.523.12
      476.3odd481156.1.g.b.155.18
      476.31odd481156.1.g.b.155.28
      476.47even123332.1.bc.c.863.14
      476.59even241156.1.c.b.579.12
      476.87even241156.1.d.a.1155.12
      476.115even1268.1.f.a.47.1&check;2
      476.143odd481156.1.g.b.179.28
      476.199odd481156.1.g.b.423.18
      476.227odd481156.1.g.b.399.28
      476.251even43332.1.bc.c.2223.14
      476.283odd481156.1.g.b.399.18
      476.311odd481156.1.g.b.423.28
      476.319odd123332.1.m.b.3039.12
      476.339even61156.1.f.b.327.12
      476.367odd481156.1.g.b.179.18
      476.387odd12inner3332.1.bc.b.863.14
      476.395even121156.1.f.b.251.12
      476.423even241156.1.d.a.1155.22
      476.451even241156.1.c.b.579.22
      595.234odd121700.1.p.a.251.12
      595.353even121700.1.n.b.999.12
      595.472even121700.1.n.a.999.12
      952.115even121088.1.p.a.319.12
      952.829odd121088.1.p.a.319.12
      1428.1067odd12612.1.l.a.523.12
      2380.1067odd121700.1.n.a.999.12
      2380.1543odd121700.1.n.b.999.12
      2380.2019even121700.1.p.a.251.12
      \n\n
      By<br>twisted newform
      TwistMinDimCharParityOrdType
      68.1.f.a.47.1&check;2119.115odd12
      68.1.f.a.47.1&check;2476.115even12
      68.1.f.a.55.1yes27.3odd6
      68.1.f.a.55.1yes228.3even6
      612.1.l.a.55.1221.17even6
      612.1.l.a.55.1284.59odd6
      612.1.l.a.523.12357.353even12
      612.1.l.a.523.121428.1067odd12
      1088.1.p.a.191.1256.3even6
      1088.1.p.a.191.1256.45odd6
      1088.1.p.a.319.12952.115even12
      1088.1.p.a.319.12952.829odd12
      1156.1.c.b.579.12119.59odd24
      1156.1.c.b.579.12476.59even24
      1156.1.c.b.579.22119.94odd24
      1156.1.c.b.579.22476.451even24
      1156.1.d.a.1155.12119.87odd24
      1156.1.d.a.1155.12476.87even24
      1156.1.d.a.1155.22119.66odd24
      1156.1.d.a.1155.22476.423even24
      1156.1.f.b.251.12119.38odd12
      1156.1.f.b.251.12476.395even12
      1156.1.f.b.327.12119.101odd6
      1156.1.f.b.327.12476.339even6
      1156.1.g.b.155.18119.3even48
      1156.1.g.b.155.18476.3odd48
      1156.1.g.b.155.28119.31even48
      1156.1.g.b.155.28476.31odd48
      1156.1.g.b.179.18119.10even48
      1156.1.g.b.179.18476.367odd48
      1156.1.g.b.179.28119.24even48
      1156.1.g.b.179.28476.143odd48
      1156.1.g.b.399.18119.45even48
      1156.1.g.b.399.18476.283odd48
      1156.1.g.b.399.28119.108even48
      1156.1.g.b.399.28476.227odd48
      1156.1.g.b.423.18119.80even48
      1156.1.g.b.423.18476.199odd48
      1156.1.g.b.423.28119.73even48
      1156.1.g.b.423.28476.311odd48
      1700.1.n.a.599.1235.3even12
      1700.1.n.a.599.12140.3odd12
      1700.1.n.a.999.12595.472even12
      1700.1.n.a.999.122380.1067odd12
      1700.1.n.b.599.1235.17even12
      1700.1.n.b.599.12140.87odd12
      1700.1.n.b.999.12595.353even12
      1700.1.n.b.999.122380.1543odd12
      1700.1.p.a.251.12595.234odd12
      1700.1.p.a.251.122380.2019even12
      1700.1.p.a.1551.1235.24odd6
      1700.1.p.a.1551.12140.59even6
      3332.1.m.b.2843.127.4even3
      3332.1.m.b.2843.1228.11odd6
      3332.1.m.b.3039.12119.81even12
      3332.1.m.b.3039.12476.319odd12
      3332.1.bc.b.667.147.2even3inner
      3332.1.bc.b.667.1428.23odd6inner
      3332.1.bc.b.863.14119.30even12inner
      3332.1.bc.b.863.14476.387odd12inner
      3332.1.bc.b.2027.141.1even1trivial
      3332.1.bc.b.2027.144.3odd2CM
      3332.1.bc.b.2223.1417.13even4inner
      3332.1.bc.b.2223.1468.47odd4inner
      3332.1.bc.c.667.147.5odd6
      3332.1.bc.c.667.1428.19even6
      3332.1.bc.c.863.14119.47odd12
      3332.1.bc.c.863.14476.47even12
      3332.1.bc.c.2027.147.6odd2
      3332.1.bc.c.2027.1428.27even2
      3332.1.bc.c.2223.14119.13odd4
      3332.1.bc.c.2223.14476.251even4
      \n", "html": "\n\n\n \n \n \n \n LMFDB - Embedded newform 3332.1.bc.b.2027.1 \n \n\n \n \n \n \n \n\n \n \n\n\n\n\n \n\n \n \n\n \n \n\n\n \n \n \n \n \n \n\n\n\n\n\n \n \n\n
      \n \n
      \n
      \n
      \n ⌂\n → Modular forms\n → Classical\n → Level 3332\n → Weight 1\n → Character orbit bc\n → Newform orbit b\n → Embedding 2027.1\n
      \n
      \n \n
      \n Citation\n ·\n Feedback\n ·\n Hide Menu\n \n
      \n
      \n\n
      \n
      Embedded newform 3332.1.bc.b.2027.1
      \n\n
      \n
      \n\n\n\n\n\n
      \n
      ↑
      \n

      Properties

      \n
      \n \n
      Label\n 3332.1.bc.b.2027.1
      \n
      Level\n $3332$
      Weight\n $1$
      Character\n 3332.2027
      Analytic conductor\n $1.663$
      Analytic rank\n $0$
      Dimension\n $4$
      Projective image\n $D_{4}$
      CM discriminant\n -4
      Inner twists\n $8$
      \n
      \n\n\n\n

      Related objects

      \n
      \n \n
      \n\n\n\n

      Downloads

      \n
      \n \n
      \n\n

      Learn more

      \n
      \n \n
      \n\n
      \n
      \n
      \n
      \n
      \n \n
      \n Show commands:\n Magma\n / PariGP\n / SageMath\n
      \n\n\n\n\n

      Newspace parameters

      \n\n
      comment: Compute space of new eigenforms
       
      \n
      [N,k,chi] = [3332,1,Mod(667,3332)]
       
      mf = mfinit([N,k,chi],0)
       
      lf = mfeigenbasis(mf)
       
      \n
      from sage.modular.dirichlet import DirichletCharacter
       
      H = DirichletGroup(3332, base_ring=CyclotomicField(12))
       
      chi = DirichletCharacter(H, H._module([6, 4, 9]))
       
      N = Newforms(chi, 1, names="a")
       
      \n
      //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
       
      chi := DirichletCharacter("3332.667");
       
      S:= CuspForms(chi, 1);
       
      N := Newforms(S);
       
      \n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
      Level: \\( N \\) \\(=\\)\\( 3332 = 2^{2} \\cdot 7^{2} \\cdot 17 \\)
      Weight: \\( k \\) \\(=\\)\\( 1 \\)
      Character orbit: \\([\\chi]\\) \\(=\\) 3332.bc (of order \\(12\\), degree \\(4\\), not minimal)
      \n\n

      Newform invariants

      \n\n
      comment: select newform
       
      \n
      sage: f = N[0] # Warning: the index may be different
       
      \n
      gp: f = lf[1] \\\\ Warning: the index may be different
       
      \n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\t\n \n \t\n \t\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
      Self dual: no
      Analytic conductor: \\(1.66288462209\\)
      Analytic rank: \\(0\\)
      Dimension: \\(4\\)
      Coefficient field: \\(\\Q(\\zeta_{12})\\)
      \n
      comment: defining polynomial
       
      \n
      gp: f.mod \\\\ as an extension of the character field
       
      \n\n
      Defining polynomial: \n\n \\( x^{4} - x^{2} + 1 \\)\n \n\n \n \"Copy\n \n \n \"Toggle\n \n
      Coefficient ring: \\(\\Z[a_1, a_2]\\)
      Coefficient ring index: \\( 1 \\)
      Twist minimal: no (minimal twist has level 68)
      Projective image:\\(D_{4}\\)
      Projective field:Galois closure of 4.2.19652.1
      Artin image:$C_4\\wr C_2\\times C_6$
      Artin field:Galois closure of \\(\\mathbb{Q}[x]/(x^{48} - \\cdots)\\)
      \n\n\n

      Embedding invariants

      \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
      Embedding label 2027.1
      Root\\(0.866025 - 0.500000i\\) of defining polynomial
      Character\\(\\chi\\)\\(=\\)3332.2027
      Dual form 3332.1.bc.b.863.1
      \n\n\n

      $q$-expansion

      \n
      \n
      comment: q-expansion
       
      \n
      sage: f.q_expansion() # note that sage often uses an isomorphic number field
       
      \n
      gp: mfcoefs(f, 20)
       
      \n\n
      \n \n \n \n \n \n \n \n \n \n \n \n
      \\(f(q)\\)\\(=\\)\\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.36603 - 0.366025i) q^{5} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.00000i) q^{20} +(0.866025 + 0.500000i) q^{25} +(1.00000 - 1.00000i) q^{29} +(0.866025 + 0.500000i) q^{32} -1.00000i q^{34} +1.00000i q^{36} +(1.36603 + 0.366025i) q^{37} +(0.366025 - 1.36603i) q^{40} +(-1.00000 - 1.00000i) q^{41} +(1.36603 - 0.366025i) q^{45} -1.00000 q^{50} +(-0.366025 + 1.36603i) q^{58} +(0.366025 - 1.36603i) q^{61} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(-0.366025 - 1.36603i) q^{73} +(-1.36603 + 0.366025i) q^{74} +(0.366025 + 1.36603i) q^{80} +(0.500000 - 0.866025i) q^{81} +(1.36603 + 0.366025i) q^{82} +(1.00000 - 1.00000i) q^{85} +(-1.00000 + 1.00000i) q^{90} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\\)
      \\(\\operatorname{Tr}(f)(q)\\)\\(=\\)\n\n \\( 4 q + 2 q^{4} - 2 q^{5} + 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 4 q^{20} + 4 q^{29} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 2 q^{45} - 4 q^{50} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{68} - 2 q^{72} + 2 q^{73}+ \\cdots + 4 q^{97}+O(q^{100}) \\)\n \n\n \n \"Copy\n \n \n \"Toggle\n \n
      \n
      \n\n
      \n\n\n

      Character values

      \n

      We give the values of \\(\\chi\\) on generators for \\(\\left(\\mathbb{Z}/3332\\mathbb{Z}\\right)^\\times\\).

      \n\n \n \n \n \n \n \n \n \n \n \n \n \n
      \\(n\\)\\(785\\)\\(885\\)\\(1667\\)
      \\(\\chi(n)\\)\\(e\\left(\\frac{3}{4}\\right)\\)\\(e\\left(\\frac{2}{3}\\right)\\)\\(-1\\)
      \n\n\n

      Coefficient data

      \n\n

      For each \\(n\\) we display the coefficients of the \\(q\\)-expansion \\(a_n\\), the\nSatake parameters \\(\\alpha_p\\),\nand the Satake angles \\(\\theta_p = \\textrm{Arg}(\\alpha_p)\\).

      \n\n\n\n\n

      \n
      \n \n Display \\(a_p\\) with \\(p\\) up to:\n 50\n 250\n 1000\n \n\n \n (See \\(a_n\\) instead)\n \n\n \n (See \\(a_n\\) instead)\n \n\n \n (See \\(a_n\\) instead)\n \n \n Display \\(a_n\\) with \\(n\\) up to:\n 50\n 250\n 1000\n \n\n \n (See only \\(a_p\\))\n \n\n \n (See only \\(a_p\\))\n \n\n \n (See only \\(a_p\\))\n \n
      \n\n\n
      \n \n \n \n \n \n \n
      Significant digits:
      \n
      \n\n
      \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n 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\n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n
      \n \\(n\\)\n \n \\(a_n\\)\n \n \\(a_n / n^{(k-1)/2}\\)\n \n \\( \\alpha_n \\)\n \n \\( \\theta_n \\)\n
      \n \\(p\\)\n \n \\(a_p\\)\n \n \\(a_p / p^{(k-1)/2}\\)\n \n \\( \\alpha_p\\)\n \n \\( \\theta_p \\)\n
      \n \\(2\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n \n\n \n\n \n\n \n\n
      \n\n \n\n \n\n \n\n
      \n \\(3\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(4\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
      \n \\(5\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(6\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(7\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n\n \n\n \n\n
      \n\n \n\n \n\n \n\n
      \n \\(8\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
      \n \\(9\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n
      \n \\(10\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(11\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(12\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(13\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(14\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(15\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(16\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
      \n \\(17\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n \n\n \n\n \n\n \n\n
      \n\n \n\n \n\n \n\n
      \n \\(18\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
      \n \\(19\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(20\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
      \n \\(21\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(22\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(23\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
      \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
      \n \\(24\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(25\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
      \n \\(26\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(27\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(28\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(29\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n \\(30\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(31\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(32\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
      \n \\(33\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(34\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
      \n \\(35\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(36\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
      \n \\(37\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(38\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(39\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(40\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n
      \n \\(41\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(42\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(43\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(44\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(45\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(46\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(47\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(48\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(49\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(50\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
      \n \\(51\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(52\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(53\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(54\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(55\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(56\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(57\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(58\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(59\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(60\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(61\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n \\(62\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(63\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(64\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
      \n \\(65\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(66\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(67\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(68\\)\n \n 0.500000\n \n +\n \n 0.866025i\n \n 0.500000\n \n +\n \n 0.866025i\n
      \n \\(69\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(70\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(71\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(72\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
      \n \\(73\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n \\(74\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(75\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(76\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(77\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(78\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(79\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
      \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
      \n \\(80\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(81\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
      \n \\(82\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(83\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(84\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(85\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(86\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(87\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(88\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(89\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(90\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
      \n \\(91\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(92\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(93\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(94\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(95\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(96\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(97\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n \\(98\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(99\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(100\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
      \n \\(101\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(102\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(103\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(104\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(105\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(106\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(107\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(108\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(109\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(110\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(111\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(112\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(113\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(114\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(115\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(116\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
      \n \\(117\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(118\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(119\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(120\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(121\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
      \n \\(122\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(123\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(124\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(125\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(126\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(127\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(128\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
      \n \\(129\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(130\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(131\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(132\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(133\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(134\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(135\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(136\\)\n \n −0.866025\n \n −\n \n 0.500000i\n \n −0.866025\n \n −\n \n 0.500000i\n
      \n \\(137\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(138\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(139\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(140\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(141\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(142\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(143\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(144\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
      \n \\(145\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n
      \n \\(146\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(147\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(148\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(149\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n \\(150\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(151\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(152\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(153\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
      \n \\(154\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(155\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(156\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(157\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n \\(158\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(159\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(160\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
      \n \\(161\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(162\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
      \n \\(163\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
      \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
      \n \\(164\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(165\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(166\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(167\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(168\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(169\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
      \n \\(170\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(171\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(172\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(173\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(174\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(175\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(176\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(177\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(178\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(179\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(180\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n
      \n \\(181\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(182\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(183\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(184\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(185\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n
      \n \\(186\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(187\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(188\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(189\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(190\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(191\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(192\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(193\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(194\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(195\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(196\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(197\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(198\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(199\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(200\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
      \n \\(201\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(202\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(203\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(204\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(205\\)\n \n 1.00000\n \n +\n \n 1.73205i\n \n 1.00000\n \n +\n \n 1.73205i\n
      \n \\(206\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(207\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(208\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(209\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(210\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(211\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
      \n \\(212\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(213\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(214\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(215\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(216\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(217\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(218\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
      \n \\(219\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(220\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(221\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(222\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(223\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(224\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(225\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
      \n \\(226\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(227\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(228\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(229\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(230\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(231\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(232\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(233\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(234\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(235\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(236\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(237\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(238\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(239\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(240\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(241\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n \\(242\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
      \n \\(243\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(244\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
      \n \\(245\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(246\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(247\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(248\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(249\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(250\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(251\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(252\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(253\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(254\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(255\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(256\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
      \n \\(257\\)\n \n 1.73205\n \n −\n \n 1.00000i\n \n 1.73205\n \n −\n \n 1.00000i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n \\(258\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(259\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(260\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(261\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(262\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(263\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(264\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(265\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(266\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(267\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(268\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(269\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(270\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(271\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(272\\)\n \n 1.00000\n \n\n \n\n \n 1.00000\n \n\n \n\n
      \n \\(273\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(274\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(275\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(276\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(277\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(278\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(279\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(280\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(281\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(282\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(283\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(284\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(285\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(286\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(287\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(288\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
      \n \\(289\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
      \n \\(290\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n
      \n \\(291\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(292\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
      \n \\(293\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n \\(294\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(295\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(296\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(297\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(298\\)\n \n 1.73205\n \n +\n \n 1.00000i\n \n 1.73205\n \n +\n \n 1.00000i\n
      \n \\(299\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(300\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(301\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(302\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(303\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(304\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(305\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n
      \n \\(306\\)\n \n 0.500000\n \n +\n \n 0.866025i\n \n 0.500000\n \n +\n \n 0.866025i\n
      \n \\(307\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(308\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(309\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(310\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(311\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(312\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(313\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(314\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n
      \n \\(315\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(316\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(317\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n \\(318\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(319\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(320\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(321\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(322\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(323\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(324\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
      \n \\(325\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(326\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(327\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(328\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(329\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(330\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(331\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(332\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(333\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(334\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(335\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(336\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(337\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n \\(338\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
      \n \\(339\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(340\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
      \n \\(341\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(342\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(343\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(344\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(345\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(346\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(347\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(348\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(349\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(350\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(351\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(352\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(353\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(354\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(355\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(356\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(357\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(358\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(359\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(360\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(361\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
      \n \\(362\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
      \n \\(363\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(364\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(365\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n
      \n \\(366\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(367\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(368\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(369\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(370\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n
      \n \\(371\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(372\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(373\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(374\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(375\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(376\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(377\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(378\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(379\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(380\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(381\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(382\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(383\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(384\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(385\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(386\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(387\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(388\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
      \n \\(389\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n \\(390\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(391\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(392\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(393\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(394\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
      \n \\(395\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(396\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(397\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n \\(398\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(399\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(400\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
      \n \\(401\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n \\(402\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(403\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(404\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(405\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
      \n \\(406\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(407\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(408\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(409\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(410\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n
      \n \\(411\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(412\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(413\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(414\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(415\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(416\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(417\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(418\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(419\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
      \n \\(420\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(421\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(422\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(423\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(424\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(425\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n
      \n \\(426\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(427\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(428\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(429\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(430\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(431\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(432\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(433\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(434\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(435\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(436\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n
      \n \\(437\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(438\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(439\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(440\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(441\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(442\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(443\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(444\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(445\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(446\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(447\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(448\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(449\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(450\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
      \n \\(451\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(452\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(453\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(454\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(455\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(456\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(457\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(458\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n
      \n \\(459\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(460\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(461\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(462\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(463\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(464\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
      \n \\(465\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(466\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(467\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(468\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(469\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(470\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(471\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(472\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(473\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(474\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(475\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(476\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(477\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(478\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(479\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(480\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(481\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(482\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(483\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(484\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
      \n \\(485\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n
      \n \\(486\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(487\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(488\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(489\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(490\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(491\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(492\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(493\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(494\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(495\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(496\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(497\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(498\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(499\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
      \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
      \n \\(500\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(501\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(502\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(503\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
      \n \\(504\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(505\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(506\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(507\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(508\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(509\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n \\(510\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(511\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(512\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
      \n \\(513\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(514\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n
      \n \\(515\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(516\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(517\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(518\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(519\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(520\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(521\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(522\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
      \n \\(523\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(524\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(525\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(526\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(527\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(528\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(529\\)\n \n −0.866025\n \n −\n \n 0.500000i\n \n −0.866025\n \n −\n \n 0.500000i\n
      \n \\(530\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(531\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(532\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(533\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(534\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(535\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(536\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(537\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(538\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
      \n \\(539\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(540\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(541\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(542\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(543\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(544\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n
      \n \\(545\\)\n \n −2.00000\n \n\n \n\n \n −2.00000\n \n\n \n\n
      \n \\(546\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(547\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(548\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(549\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(550\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(551\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(552\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(553\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(554\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
      \n \\(555\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(556\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(557\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(558\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(559\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(560\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(561\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(562\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(563\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(564\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(565\\)\n \n 1.00000\n \n +\n \n 1.73205i\n \n 1.00000\n \n +\n \n 1.73205i\n
      \n \\(566\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(567\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(568\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(569\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(570\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(571\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(572\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(573\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(574\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(575\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(576\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
      \n \\(577\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(578\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
      \n \\(579\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(580\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n
      \n \\(581\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(582\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(583\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(584\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(585\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(586\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n
      \n \\(587\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(588\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(589\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(590\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(591\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(592\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
      \n \\(593\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(594\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(595\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(596\\)\n \n −2.00000\n \n\n \n\n \n −2.00000\n \n\n \n\n
      \n \\(597\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(598\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(599\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(600\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(601\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(602\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(603\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(604\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(605\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(606\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(607\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(608\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(609\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(610\\)\n \n\n \n −\n \n 2.00000i\n \n\n \n −\n \n 2.00000i\n
      \n \\(611\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(612\\)\n \n −0.866025\n \n −\n \n 0.500000i\n \n −0.866025\n \n −\n \n 0.500000i\n
      \n \\(613\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(614\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(615\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(616\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(617\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(618\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(619\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(620\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(621\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(622\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(623\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(624\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(625\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
      \n \\(626\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
      \n \\(627\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(628\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
      \n \\(629\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
      \n \\(630\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(631\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(632\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(633\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(634\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(635\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(636\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(637\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(638\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(639\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(640\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(641\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n \\(642\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(643\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(644\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(645\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(646\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(647\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(648\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
      \n \\(649\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(650\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(651\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(652\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(653\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n \\(654\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(655\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(656\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(657\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(658\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(659\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(660\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(661\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(662\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(663\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(664\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(665\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(666\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(667\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(668\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(669\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(670\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(671\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(672\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(673\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(674\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(675\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(676\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
      \n \\(677\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(678\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(679\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(680\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(681\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(682\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(683\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(684\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(685\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(686\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(687\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(688\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(689\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(690\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(691\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
      \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
      \n \\(692\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(693\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(694\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(695\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(696\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(697\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(698\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(699\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(700\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(701\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(702\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(703\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(704\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(705\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(706\\)\n \n\n \n −\n \n 2.00000i\n \n\n \n −\n \n 2.00000i\n
      \n \\(707\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(708\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(709\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(710\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(711\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(712\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(713\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(714\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(715\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(716\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(717\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(718\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(719\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(720\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
      \n \\(721\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(722\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
      \n \\(723\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(724\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(725\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(726\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(727\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(728\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(729\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
      \n \\(730\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
      \n \\(731\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(732\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(733\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(734\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(735\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(736\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(737\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(738\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(739\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(740\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n
      \n \\(741\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(742\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(743\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(744\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(745\\)\n \n 0.732051\n \n +\n \n 2.73205i\n \n 0.732051\n \n +\n \n 2.73205i\n
      \n \\(746\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(747\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(748\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(749\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(750\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(751\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(752\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(753\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(754\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(755\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(756\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(757\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(758\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(759\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(760\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(761\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(762\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(763\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(764\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(765\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(766\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(767\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(768\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(769\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(770\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(771\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(772\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(773\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n \\(774\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(775\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(776\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(777\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(778\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n
      \n \\(779\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(780\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(781\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(782\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(783\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(784\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(785\\)\n \n −2.00000\n \n +\n \n 2.00000i\n \n −2.00000\n \n +\n \n 2.00000i\n
      \n \\(786\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(787\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(788\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(789\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(790\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(791\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(792\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(793\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(794\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(795\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(796\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(797\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(798\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(799\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(800\\)\n \n 0.500000\n \n +\n \n 0.866025i\n \n 0.500000\n \n +\n \n 0.866025i\n
      \n \\(801\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(802\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(803\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(804\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(805\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(806\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(807\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(808\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(809\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(810\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n
      \n \\(811\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(812\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(813\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(814\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(815\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(816\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(817\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(818\\)\n \n\n \n −\n \n 2.00000i\n \n\n \n −\n \n 2.00000i\n
      \n \\(819\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(820\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n
      \n \\(821\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(822\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(823\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(824\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(825\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(826\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(827\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
      \n \\(828\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(829\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n \\(830\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(831\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(832\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(833\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(834\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(835\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(836\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(837\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(838\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(839\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
      \n \\(840\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(841\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
      \n \\(842\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(843\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(844\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(845\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(846\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(847\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(848\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(849\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(850\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
      \n \\(851\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(852\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(853\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n \\(854\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(855\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(856\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(857\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(858\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(859\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(860\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(861\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(862\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(863\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(864\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(865\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n
      \n \\(866\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
      \n \\(867\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(868\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(869\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(870\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(871\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(872\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(873\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(874\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(875\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(876\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(877\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(878\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(879\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(880\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(881\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(882\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(883\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(884\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(885\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(886\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(887\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(888\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(889\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(890\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(891\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(892\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(893\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(894\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(895\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(896\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(897\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(898\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(899\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(900\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
      \n \\(901\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(902\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(903\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(904\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(905\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
      \n \\(906\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(907\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(908\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(909\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(910\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(911\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
      \n \\(912\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(913\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(914\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n
      \n \\(915\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(916\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n
      \n \\(917\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(918\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(919\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(920\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(921\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(922\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(923\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(924\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(925\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(926\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(927\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(928\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(929\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(930\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(931\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(932\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(933\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(934\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(935\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(936\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(937\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(938\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(939\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(940\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(941\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(942\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(943\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(944\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(945\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(946\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(947\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(948\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(949\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(950\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(951\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(952\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(953\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(954\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(955\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(956\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(957\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(958\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(959\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(960\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(961\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n
      \n \\(962\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(963\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(964\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
      \n \\(965\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n
      \n \\(966\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(967\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(968\\)\n \n 0.500000\n \n +\n \n 0.866025i\n \n 0.500000\n \n +\n \n 0.866025i\n
      \n \\(969\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(970\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n
      \n \\(971\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(972\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(973\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(974\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(975\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(976\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(977\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(978\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(979\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(980\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(981\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
      \n \\(982\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(983\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(984\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(985\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
      \n \\(986\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
      \n \\(987\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(988\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(989\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(990\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(991\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(992\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(993\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(994\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(995\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(996\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(997\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(998\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(999\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n
      \n
      \n \n Display \\(a_p\\) with \\(p\\) up to:\n 50\n 250\n 1000\n \n\n \n (See \\(a_n\\) instead)\n \n\n \n (See \\(a_n\\) instead)\n \n\n \n (See \\(a_n\\) instead)\n \n \n Display \\(a_n\\) with \\(n\\) up to:\n 50\n 250\n 1000\n \n\n \n (See only \\(a_p\\))\n \n\n \n (See only \\(a_p\\))\n \n\n \n (See only \\(a_p\\))\n \n
      \n\n\n

      Twists

      \n\n\n\n\n \n \n \n \n \n \n \n\n\n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n
             By twisting character
      CharParityOrdTypeTwistMinDim
      1.1even1trivial3332.1.bc.b.2027.14
      4.3odd2CM3332.1.bc.b.2027.14
      7.2even3inner3332.1.bc.b.667.14
      7.3odd668.1.f.a.55.1yes2
      7.4even33332.1.m.b.2843.12
      7.5odd63332.1.bc.c.667.14
      7.6odd23332.1.bc.c.2027.14
      17.13even4inner3332.1.bc.b.2223.14
      21.17even6612.1.l.a.55.12
      28.3even668.1.f.a.55.1yes2
      28.11odd63332.1.m.b.2843.12
      28.19even63332.1.bc.c.667.14
      28.23odd6inner3332.1.bc.b.667.14
      28.27even23332.1.bc.c.2027.14
      35.3even121700.1.n.a.599.12
      35.17even121700.1.n.b.599.12
      35.24odd61700.1.p.a.1551.12
      56.3even61088.1.p.a.191.12
      56.45odd61088.1.p.a.191.12
      68.47odd4inner3332.1.bc.b.2223.14
      84.59odd6612.1.l.a.55.12
      119.3even481156.1.g.b.155.18
      119.10even481156.1.g.b.179.18
      119.13odd43332.1.bc.c.2223.14
      119.24even481156.1.g.b.179.28
      119.30even12inner3332.1.bc.b.863.14
      119.31even481156.1.g.b.155.28
      119.38odd121156.1.f.b.251.12
      119.45even481156.1.g.b.399.18
      119.47odd123332.1.bc.c.863.14
      119.59odd241156.1.c.b.579.12
      119.66odd241156.1.d.a.1155.22
      119.73even481156.1.g.b.423.28
      119.80even481156.1.g.b.423.18
      119.81even123332.1.m.b.3039.12
      119.87odd241156.1.d.a.1155.12
      119.94odd241156.1.c.b.579.22
      119.101odd61156.1.f.b.327.12
      119.108even481156.1.g.b.399.28
      119.115odd1268.1.f.a.47.1✓2
      140.3odd121700.1.n.a.599.12
      140.59even61700.1.p.a.1551.12
      140.87odd121700.1.n.b.599.12
      357.353even12612.1.l.a.523.12
      476.3odd481156.1.g.b.155.18
      476.31odd481156.1.g.b.155.28
      476.47even123332.1.bc.c.863.14
      476.59even241156.1.c.b.579.12
      476.87even241156.1.d.a.1155.12
      476.115even1268.1.f.a.47.1✓2
      476.143odd481156.1.g.b.179.28
      476.199odd481156.1.g.b.423.18
      476.227odd481156.1.g.b.399.28
      476.251even43332.1.bc.c.2223.14
      476.283odd481156.1.g.b.399.18
      476.311odd481156.1.g.b.423.28
      476.319odd123332.1.m.b.3039.12
      476.339even61156.1.f.b.327.12
      476.367odd481156.1.g.b.179.18
      476.387odd12inner3332.1.bc.b.863.14
      476.395even121156.1.f.b.251.12
      476.423even241156.1.d.a.1155.22
      476.451even241156.1.c.b.579.22
      595.234odd121700.1.p.a.251.12
      595.353even121700.1.n.b.999.12
      595.472even121700.1.n.a.999.12
      952.115even121088.1.p.a.319.12
      952.829odd121088.1.p.a.319.12
      1428.1067odd12612.1.l.a.523.12
      2380.1067odd121700.1.n.a.999.12
      2380.1543odd121700.1.n.b.999.12
      2380.2019even121700.1.p.a.251.12
      \n
          
      \n\n\n\n\n \n \n \n \n \n \n \n\n\n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n
              By twisted newform
      TwistMinDimCharParityOrdType
      68.1.f.a.47.1✓2119.115odd12
      68.1.f.a.47.1✓2476.115even12
      68.1.f.a.55.1yes27.3odd6
      68.1.f.a.55.1yes228.3even6
      612.1.l.a.55.1221.17even6
      612.1.l.a.55.1284.59odd6
      612.1.l.a.523.12357.353even12
      612.1.l.a.523.121428.1067odd12
      1088.1.p.a.191.1256.3even6
      1088.1.p.a.191.1256.45odd6
      1088.1.p.a.319.12952.115even12
      1088.1.p.a.319.12952.829odd12
      1156.1.c.b.579.12119.59odd24
      1156.1.c.b.579.12476.59even24
      1156.1.c.b.579.22119.94odd24
      1156.1.c.b.579.22476.451even24
      1156.1.d.a.1155.12119.87odd24
      1156.1.d.a.1155.12476.87even24
      1156.1.d.a.1155.22119.66odd24
      1156.1.d.a.1155.22476.423even24
      1156.1.f.b.251.12119.38odd12
      1156.1.f.b.251.12476.395even12
      1156.1.f.b.327.12119.101odd6
      1156.1.f.b.327.12476.339even6
      1156.1.g.b.155.18119.3even48
      1156.1.g.b.155.18476.3odd48
      1156.1.g.b.155.28119.31even48
      1156.1.g.b.155.28476.31odd48
      1156.1.g.b.179.18119.10even48
      1156.1.g.b.179.18476.367odd48
      1156.1.g.b.179.28119.24even48
      1156.1.g.b.179.28476.143odd48
      1156.1.g.b.399.18119.45even48
      1156.1.g.b.399.18476.283odd48
      1156.1.g.b.399.28119.108even48
      1156.1.g.b.399.28476.227odd48
      1156.1.g.b.423.18119.80even48
      1156.1.g.b.423.18476.199odd48
      1156.1.g.b.423.28119.73even48
      1156.1.g.b.423.28476.311odd48
      1700.1.n.a.599.1235.3even12
      1700.1.n.a.599.12140.3odd12
      1700.1.n.a.999.12595.472even12
      1700.1.n.a.999.122380.1067odd12
      1700.1.n.b.599.1235.17even12
      1700.1.n.b.599.12140.87odd12
      1700.1.n.b.999.12595.353even12
      1700.1.n.b.999.122380.1543odd12
      1700.1.p.a.251.12595.234odd12
      1700.1.p.a.251.122380.2019even12
      1700.1.p.a.1551.1235.24odd6
      1700.1.p.a.1551.12140.59even6
      3332.1.m.b.2843.127.4even3
      3332.1.m.b.2843.1228.11odd6
      3332.1.m.b.3039.12119.81even12
      3332.1.m.b.3039.12476.319odd12
      3332.1.bc.b.667.147.2even3inner
      3332.1.bc.b.667.1428.23odd6inner
      3332.1.bc.b.863.14119.30even12inner
      3332.1.bc.b.863.14476.387odd12inner
      3332.1.bc.b.2027.141.1even1trivial
      3332.1.bc.b.2027.144.3odd2CM
      3332.1.bc.b.2223.1417.13even4inner
      3332.1.bc.b.2223.1468.47odd4inner
      3332.1.bc.c.667.147.5odd6
      3332.1.bc.c.667.1428.19even6
      3332.1.bc.c.863.14119.47odd12
      3332.1.bc.c.863.14476.47even12
      3332.1.bc.c.2027.147.6odd2
      3332.1.bc.c.2027.1428.27even2
      3332.1.bc.c.2223.14119.13odd4
      3332.1.bc.c.2223.14476.251even4
      \n
      \n\n\n\n

      \n
      \n
      \n\n\n
      \n\n

      This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

      \n
      \n Contact\n ·\n Citation\n ·\n Acknowledgments\n ·\n Editorial Board\n ·\n Source\n ·\n SageMath version 10.1\n ·\n LMFDB Release 1.2.1\n
      \n
      \n\n"} +{"url": "https://www.lmfdb.org/ModularForm/GL2/Q/holomorphic/3332/1/bc/b/2027/1/", "content": "Show commands: Magma/ PariGP/ SageMath\n\n```\n[N,k,chi] = [3332,1,Mod(667,3332)]\n\nmf = mfinit([N,k,chi],0)\n\nlf = mfeigenbasis(mf)\n```\n\n```\nfrom sage.modular.dirichlet import DirichletCharacter\n\nH = DirichletGroup(3332, base_ring=CyclotomicField(12))\n\nchi = DirichletCharacter(H, H._module([6, 4, 9]))\n\nN = Newforms(chi, 1, names=\"a\")\n```\n\n```\n//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code\n\nchi := DirichletCharacter(\"3332.667\");\n\nS:= CuspForms(chi, 1);\n\nN := Newforms(S);\n```\n\n| Level : | $N$ | $=$ | $3332 = 2^{2} \\cdot 7^{2} \\cdot 17$ |\n|---|---|---|---|\n| Weight : | $k$ | $=$ | $1$ |\n| Character orbit : | $[\\chi]$ | $=$ | 3332.bc (oforder12, degree4, not minimal ) |\n\n## Newform invariants\n\n```\nsage: f = N[0] # Warning: the index may be different\n```\n\n```\ngp: f = lf[1] \\\\ Warning: the index may be different\n```\n\n
      Self dual :no
      Analytic conductor :$1.66288462209$
      Analytic rank :$0$
      Dimension :$4$
      Coefficient field :$\\Q(\\zeta_{12})$
      \n\n```\ngp: f.mod \\\\ as an extension of the character field\n```\n\n
      Defining polynomial :$x^{4} - x^{2} + 1$
      Coefficient ring :$\\Z[a_1, a_2]$
      Coefficient ring index :$1$
      Twist minimal :no (minimal twist has level 68)
      Projective image :$D_{4}$
      Projective field :Galois closure of 4.2.19652.1
      Artin image :$C_4\\wr C_2\\times C_6$
      Artin field :Galois closure of $\\mathbb{Q}[x]/(x^{48} - \\cdots)$
      \n\n## Embedding invariants\n\n| Embedding label | | | 2027.1 |\n|---|---|---|---|\n| Root | | | $0.866025 - 0.500000i$ of defining polynomial |\n| Character | $\\chi$ | $=$ | 3332.2027 |\n| Dual form | | | 3332.1.bc.b.863.1 |\n\n```\nsage: f.q_expansion() # note that sage often uses an isomorphic number field\n```\n\n```\ngp: mfcoefs(f, 20)\n```\n\n| $f(q)$ | $=$ | $q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.36603 - 0.366025i) q^{5} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.00000i) q^{20} +(0.866025 + 0.500000i) q^{25} +(1.00000 - 1.00000i) q^{29} +(0.866025 + 0.500000i) q^{32} -1.00000i q^{34} +1.00000i q^{36} +(1.36603 + 0.366025i) q^{37} +(0.366025 - 1.36603i) q^{40} +(-1.00000 - 1.00000i) q^{41} +(1.36603 - 0.366025i) q^{45} -1.00000 q^{50} +(-0.366025 + 1.36603i) q^{58} +(0.366025 - 1.36603i) q^{61} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(-0.366025 - 1.36603i) q^{73} +(-1.36603 + 0.366025i) q^{74} +(0.366025 + 1.36603i) q^{80} +(0.500000 - 0.866025i) q^{81} +(1.36603 + 0.366025i) q^{82} +(1.00000 - 1.00000i) q^{85} +(-1.00000 + 1.00000i) q^{90} +(1.00000 - 1.00000i) q^{97} +O(q^{100})$ |\n|---|---|---|\n| $\\operatorname{Tr}(f)(q)$ | $=$ | $4 q + 2 q^{4} - 2 q^{5} + 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 4 q^{20} + 4 q^{29} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 2 q^{45} - 4 q^{50} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{68} - 2 q^{72} + 2 q^{73}+ \\cdots + 4 q^{97}+O(q^{100})$ |\n\n## Character values\n\nWe give the values of $\\chi$ on generators for $\\left(\\mathbb{Z}/3332\\mathbb{Z}\\right)^\\times$ .\n\n| $n$ | $785$ | $885$ | $1667$ |\n|---|---|---|---|\n| $\\chi(n)$ | $e\\left(\\frac{3}{4}\\right)$ | $e\\left(\\frac{2}{3}\\right)$ | $-1$ |\n\n## Coefficient data\n\nFor each $n$ we display the coefficients of the $q$ -expansion $a_n$ , the Satake parameters $\\alpha_p$ , and the Satake angles $\\theta_p = \\textrm{Arg}(\\alpha_p)$ .\n\n(See $a_n$ instead)(See $a_n$ instead)(See $a_n$ instead)(See only $a_p$ )(See only $a_p$ )(See only $a_p$ )\n\n
      $n$$a_n$$a_n / n^{(k-1)/2}$$\\alpha_n$$\\theta_n$
      $p$$a_p$$a_p / p^{(k-1)/2}$$\\alpha_p$$\\theta_p$
      $2$−0.866025+0.500000i−0.866025+0.500000i
      $3$00−0.258819−0.965926i$-0.583333\\pi$
      0.258819+0.965926i$0.416667\\pi$
      $4$0.500000−0.866025i0.500000−0.866025i
      $5$−1.36603−0.366025i−1.36603−0.366025i−0.500000−0.866025i$-0.666667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $6$00
      $7$00
      $8$1.00000i1.00000i
      $9$−0.866025+0.500000i−0.866025+0.500000i
      $10$1.36603−0.366025i1.36603−0.366025i
      $11$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $12$00
      $13$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $14$00
      $15$00
      $16$−0.500000−0.866025i−0.500000−0.866025i
      $17$−0.500000+0.866025i−0.500000+0.866025i
      $18$0.500000−0.866025i0.500000−0.866025i
      $19$000.866025−0.500000i$-0.166667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $20$−1.00000+1.00000i−1.00000+1.00000i
      $21$00
      $22$00
      $23$000.258819−0.965926i$-0.416667\\pi$
      −0.258819+0.965926i$0.583333\\pi$
      $24$00
      $25$0.866025+0.500000i0.866025+0.500000i
      $26$00
      $27$00
      $28$00
      $29$1.00000−1.00000i1.00000−1.00000i−1.00000i$-0.5\\pi$
      1.00000$0$
      $30$00
      $31$00−0.258819−0.965926i$-0.583333\\pi$
      0.258819+0.965926i$0.416667\\pi$
      $32$0.866025+0.500000i0.866025+0.500000i
      $33$00
      $34$−1.00000i−1.00000i
      $35$00
      $36$1.00000i1.00000i
      $37$1.36603+0.366025i1.36603+0.366025i0.866025−0.500000i$-0.166667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $38$00
      $39$00
      $40$0.366025−1.36603i0.366025−1.36603i
      $41$−1.00000−1.00000i−1.00000−1.00000i−1.00000i$-0.5\\pi$
      −1.00000$\\pi$
      $42$00
      $43$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $44$00
      $45$1.36603−0.366025i1.36603−0.366025i
      $46$00
      $47$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $48$00
      $49$00
      $50$−1.00000−1.00000
      $51$00
      $52$00
      $53$000.500000−0.866025i$-0.333333\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $54$00
      $55$00
      $56$00
      $57$00
      $58$−0.366025+1.36603i−0.366025+1.36603i
      $59$00−0.866025−0.500000i$-0.833333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $60$00
      $61$0.366025−1.36603i0.366025−1.36603i−0.500000−0.866025i$-0.666667\\pi$
      0.866025−0.500000i$-0.166667\\pi$
      $62$00
      $63$00
      $64$−1.00000−1.00000
      $65$00
      $66$00
      $67$000.500000−0.866025i$-0.333333\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $68$0.500000+0.866025i0.500000+0.866025i
      $69$00
      $70$00
      $71$000.707107−0.707107i$-0.250000\\pi$
      −0.707107+0.707107i$0.750000\\pi$
      $72$−0.500000−0.866025i−0.500000−0.866025i
      $73$−0.366025−1.36603i−0.366025−1.36603i−0.866025−0.500000i$-0.833333\\pi$
      0.500000−0.866025i$-0.333333\\pi$
      $74$−1.36603+0.366025i−1.36603+0.366025i
      $75$00
      $76$00
      $77$00
      $78$00
      $79$000.258819−0.965926i$-0.416667\\pi$
      −0.258819+0.965926i$0.583333\\pi$
      $80$0.366025+1.36603i0.366025+1.36603i
      $81$0.500000−0.866025i0.500000−0.866025i
      $82$1.36603+0.366025i1.36603+0.366025i
      $83$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $84$00
      $85$1.00000−1.00000i1.00000−1.00000i
      $86$00
      $87$00
      $88$00
      $89$000.866025−0.500000i$-0.166667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $90$−1.00000+1.00000i−1.00000+1.00000i
      $91$00
      $92$00
      $93$00
      $94$00
      $95$00
      $96$00
      $97$1.00000−1.00000i1.00000−1.00000i−1.00000i$-0.5\\pi$
      1.00000$0$
      $98$00
      $99$00
      $100$0.866025−0.500000i0.866025−0.500000i
      $101$00−0.866025−0.500000i$-0.833333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $102$00
      $103$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $104$00
      $105$00
      $106$00
      $107$00−0.965926−0.258819i$-0.916667\\pi$
      0.965926+0.258819i$0.0833333\\pi$
      $108$00
      $109$1.36603−0.366025i1.36603−0.366025i0.500000−0.866025i$-0.333333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $110$00
      $111$00
      $112$00
      $113$−1.00000−1.00000i−1.00000−1.00000i−1.00000i$-0.5\\pi$
      −1.00000$\\pi$
      $114$00
      $115$00
      $116$−0.366025−1.36603i−0.366025−1.36603i
      $117$00
      $118$00
      $119$00
      $120$00
      $121$0.866025−0.500000i0.866025−0.500000i
      $122$0.366025+1.36603i0.366025+1.36603i
      $123$00
      $124$00
      $125$00
      $126$00
      $127$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $128$0.866025−0.500000i0.866025−0.500000i
      $129$00
      $130$00
      $131$00−0.965926−0.258819i$-0.916667\\pi$
      0.965926+0.258819i$0.0833333\\pi$
      $132$00
      $133$00
      $134$00
      $135$00
      $136$−0.866025−0.500000i−0.866025−0.500000i
      $137$00−0.866025−0.500000i$-0.833333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $138$00
      $139$000.707107−0.707107i$-0.250000\\pi$
      −0.707107+0.707107i$0.750000\\pi$
      $140$00
      $141$00
      $142$00
      $143$00
      $144$0.866025+0.500000i0.866025+0.500000i
      $145$−1.73205+1.00000i−1.73205+1.00000i
      $146$1.00000+1.00000i1.00000+1.00000i
      $147$00
      $148$1.00000−1.00000i1.00000−1.00000i
      $149$−1.00000−1.73205i−1.00000−1.73205i−0.500000−0.866025i$-0.666667\\pi$
      −0.500000−0.866025i$-0.666667\\pi$
      $150$00
      $151$00−0.866025−0.500000i$-0.833333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $152$00
      $153$−1.00000i−1.00000i
      $154$00
      $155$00
      $156$00
      $157$1.00000−1.73205i1.00000−1.73205i0.500000−0.866025i$-0.333333\\pi$
      0.500000−0.866025i$-0.333333\\pi$
      $158$00
      $159$00
      $160$−1.00000−1.00000i−1.00000−1.00000i
      $161$00
      $162$1.00000i1.00000i
      $163$000.258819−0.965926i$-0.416667\\pi$
      −0.258819+0.965926i$0.583333\\pi$
      $164$−1.36603+0.366025i−1.36603+0.366025i
      $165$00
      $166$00
      $167$000.707107−0.707107i$-0.250000\\pi$
      −0.707107+0.707107i$0.750000\\pi$
      $168$00
      $169$−1.00000−1.00000
      $170$−0.366025+1.36603i−0.366025+1.36603i
      $171$00
      $172$00
      $173$1.36603+0.366025i1.36603+0.366025i0.866025−0.500000i$-0.166667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $174$00
      $175$00
      $176$00
      $177$00
      $178$00
      $179$00−0.866025−0.500000i$-0.833333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $180$0.366025−1.36603i0.366025−1.36603i
      $181$1.00000+1.00000i1.00000+1.00000i1.00000$0$
      1.00000i$0.5\\pi$
      $182$00
      $183$00
      $184$00
      $185$−1.73205−1.00000i−1.73205−1.00000i
      $186$00
      $187$00
      $188$00
      $189$00
      $190$00
      $191$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $192$00
      $193$−1.36603+0.366025i−1.36603+0.366025i−0.866025−0.500000i$-0.833333\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $194$−0.366025+1.36603i−0.366025+1.36603i
      $195$00
      $196$00
      $197$1.00000+1.00000i1.00000+1.00000i1.00000$0$
      1.00000i$0.5\\pi$
      $198$00
      $199$00−0.258819−0.965926i$-0.583333\\pi$
      0.258819+0.965926i$0.416667\\pi$
      $200$−0.500000+0.866025i−0.500000+0.866025i
      $201$00
      $202$00
      $203$00
      $204$00
      $205$1.00000+1.73205i1.00000+1.73205i
      $206$00
      $207$00
      $208$00
      $209$00
      $210$00
      $211$00−0.707107−0.707107i$-0.750000\\pi$
      0.707107+0.707107i$0.250000\\pi$
      $212$00
      $213$00
      $214$00
      $215$00
      $216$00
      $217$00
      $218$−1.00000+1.00000i−1.00000+1.00000i
      $219$00
      $220$00
      $221$00
      $222$00
      $223$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $224$00
      $225$−1.00000−1.00000
      $226$1.36603+0.366025i1.36603+0.366025i
      $227$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $228$00
      $229$−1.73205+1.00000i−1.73205+1.00000i−0.866025+0.500000i$0.833333\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $230$00
      $231$00
      $232$1.00000+1.00000i1.00000+1.00000i
      $233$1.36603+0.366025i1.36603+0.366025i0.866025−0.500000i$-0.166667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $234$00
      $235$00
      $236$00
      $237$00
      $238$00
      $239$001.00000$0$
      −1.00000$\\pi$
      $240$00
      $241$−0.366025−1.36603i−0.366025−1.36603i−0.866025−0.500000i$-0.833333\\pi$
      0.500000−0.866025i$-0.333333\\pi$
      $242$−0.500000+0.866025i−0.500000+0.866025i
      $243$00
      $244$−1.00000−1.00000i−1.00000−1.00000i
      $245$00
      $246$00
      $247$00
      $248$00
      $249$00
      $250$00
      $251$001.00000$0$
      −1.00000$\\pi$
      $252$00
      $253$00
      $254$00
      $255$00
      $256$−0.500000+0.866025i−0.500000+0.866025i
      $257$1.73205−1.00000i1.73205−1.00000i0.866025−0.500000i$-0.166667\\pi$
      0.866025−0.500000i$-0.166667\\pi$
      $258$00
      $259$00
      $260$00
      $261$−0.366025+1.36603i−0.366025+1.36603i
      $262$00
      $263$00−0.866025−0.500000i$-0.833333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $264$00
      $265$00
      $266$00
      $267$00
      $268$00
      $269$0.366025+1.36603i0.366025+1.36603i0.866025+0.500000i$0.166667\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $270$00
      $271$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $272$1.000001.00000
      $273$00
      $274$00
      $275$00
      $276$00
      $277$0.366025+1.36603i0.366025+1.36603i0.866025+0.500000i$0.166667\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $278$00
      $279$00
      $280$00
      $281$001.00000$0$
      −1.00000$\\pi$
      $282$00
      $283$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $284$00
      $285$00
      $286$00
      $287$00
      $288$−1.00000−1.00000
      $289$−0.500000−0.866025i−0.500000−0.866025i
      $290$1.00000−1.73205i1.00000−1.73205i
      $291$00
      $292$−1.36603−0.366025i−1.36603−0.366025i
      $293$2.000002.000001.00000$0$
      1.00000$0$
      $294$00
      $295$00
      $296$−0.366025+1.36603i−0.366025+1.36603i
      $297$00
      $298$1.73205+1.00000i1.73205+1.00000i
      $299$00
      $300$00
      $301$00
      $302$00
      $303$00
      $304$00
      $305$−1.00000+1.73205i−1.00000+1.73205i
      $306$0.500000+0.866025i0.500000+0.866025i
      $307$001.00000$0$
      −1.00000$\\pi$
      $308$00
      $309$00
      $310$00
      $311$00−0.258819−0.965926i$-0.583333\\pi$
      0.258819+0.965926i$0.416667\\pi$
      $312$00
      $313$−0.366025+1.36603i−0.366025+1.36603i0.500000+0.866025i$0.333333\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $314$2.00000i2.00000i
      $315$00
      $316$00
      $317$0.366025−1.36603i0.366025−1.36603i−0.500000−0.866025i$-0.666667\\pi$
      0.866025−0.500000i$-0.166667\\pi$
      $318$00
      $319$00
      $320$1.36603+0.366025i1.36603+0.366025i
      $321$00
      $322$00
      $323$00
      $324$−0.500000−0.866025i−0.500000−0.866025i
      $325$00
      $326$00
      $327$00
      $328$1.00000−1.00000i1.00000−1.00000i
      $329$00
      $330$00
      $331$000.866025−0.500000i$-0.166667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $332$00
      $333$−1.36603+0.366025i−1.36603+0.366025i
      $334$00
      $335$00
      $336$00
      $337$1.00000−1.00000i1.00000−1.00000i−1.00000i$-0.5\\pi$
      1.00000$0$
      $338$0.866025−0.500000i0.866025−0.500000i
      $339$00
      $340$−0.366025−1.36603i−0.366025−1.36603i
      $341$00
      $342$00
      $343$00
      $344$00
      $345$00
      $346$−1.36603+0.366025i−1.36603+0.366025i
      $347$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $348$00
      $349$001.00000$0$
      −1.00000$\\pi$
      $350$00
      $351$00
      $352$00
      $353$−1.00000+1.73205i−1.00000+1.73205i−0.500000+0.866025i$0.666667\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $354$00
      $355$00
      $356$00
      $357$00
      $358$00
      $359$000.866025−0.500000i$-0.166667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $360$0.366025+1.36603i0.366025+1.36603i
      $361$0.500000−0.866025i0.500000−0.866025i
      $362$−1.36603−0.366025i−1.36603−0.366025i
      $363$00
      $364$00
      $365$2.00000i2.00000i
      $366$00
      $367$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $368$00
      $369$1.36603+0.366025i1.36603+0.366025i
      $370$2.000002.00000
      $371$00
      $372$00
      $373$000.866025−0.500000i$-0.166667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $374$00
      $375$00
      $376$00
      $377$00
      $378$00
      $379$000.707107−0.707107i$-0.250000\\pi$
      −0.707107+0.707107i$0.750000\\pi$
      $380$00
      $381$00
      $382$00
      $383$000.866025−0.500000i$-0.166667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $384$00
      $385$00
      $386$1.00000−1.00000i1.00000−1.00000i
      $387$00
      $388$−0.366025−1.36603i−0.366025−1.36603i
      $389$−1.73205−1.00000i−1.73205−1.00000i−0.866025−0.500000i$-0.833333\\pi$
      −0.866025−0.500000i$-0.833333\\pi$
      $390$00
      $391$00
      $392$00
      $393$00
      $394$−1.36603−0.366025i−1.36603−0.366025i
      $395$00
      $396$00
      $397$0.366025−1.36603i0.366025−1.36603i−0.500000−0.866025i$-0.666667\\pi$
      0.866025−0.500000i$-0.166667\\pi$
      $398$00
      $399$00
      $400$−1.00000i−1.00000i
      $401$0.366025−1.36603i0.366025−1.36603i−0.500000−0.866025i$-0.666667\\pi$
      0.866025−0.500000i$-0.166667\\pi$
      $402$00
      $403$00
      $404$00
      $405$−1.00000+1.00000i−1.00000+1.00000i
      $406$00
      $407$00
      $408$00
      $409$−1.00000+1.73205i−1.00000+1.73205i−0.500000+0.866025i$0.666667\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $410$−1.73205−1.00000i−1.73205−1.00000i
      $411$00
      $412$00
      $413$00
      $414$00
      $415$00
      $416$00
      $417$00
      $418$00
      $419$00−0.707107−0.707107i$-0.750000\\pi$
      0.707107+0.707107i$0.250000\\pi$
      $420$00
      $421$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $422$00
      $423$00
      $424$00
      $425$−0.866025+0.500000i−0.866025+0.500000i
      $426$00
      $427$00
      $428$00
      $429$00
      $430$00
      $431$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $432$00
      $433$2.00000i2.00000i1.00000i$0.5\\pi$
      1.00000i$0.5\\pi$
      $434$00
      $435$00
      $436$0.366025−1.36603i0.366025−1.36603i
      $437$00
      $438$00
      $439$00−0.965926−0.258819i$-0.916667\\pi$
      0.965926+0.258819i$0.0833333\\pi$
      $440$00
      $441$00
      $442$00
      $443$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $444$00
      $445$00
      $446$00
      $447$00
      $448$00
      $449$−1.00000−1.00000i−1.00000−1.00000i−1.00000i$-0.5\\pi$
      −1.00000$\\pi$
      $450$0.866025−0.500000i0.866025−0.500000i
      $451$00
      $452$−1.36603+0.366025i−1.36603+0.366025i
      $453$00
      $454$00
      $455$00
      $456$00
      $457$−1.73205+1.00000i−1.73205+1.00000i−0.866025+0.500000i$0.833333\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $458$1.00000−1.73205i1.00000−1.73205i
      $459$00
      $460$00
      $461$001.00000$0$
      −1.00000$\\pi$
      $462$00
      $463$001.00000$0$
      −1.00000$\\pi$
      $464$−1.36603−0.366025i−1.36603−0.366025i
      $465$00
      $466$−1.36603+0.366025i−1.36603+0.366025i
      $467$000.866025−0.500000i$-0.166667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $468$00
      $469$00
      $470$00
      $471$00
      $472$00
      $473$00
      $474$00
      $475$00
      $476$00
      $477$00
      $478$00
      $479$00−0.258819−0.965926i$-0.583333\\pi$
      0.258819+0.965926i$0.416667\\pi$
      $480$00
      $481$00
      $482$1.00000+1.00000i1.00000+1.00000i
      $483$00
      $484$−1.00000i−1.00000i
      $485$−1.73205+1.00000i−1.73205+1.00000i
      $486$00
      $487$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $488$1.36603+0.366025i1.36603+0.366025i
      $489$00
      $490$00
      $491$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $492$00
      $493$0.366025+1.36603i0.366025+1.36603i
      $494$00
      $495$00
      $496$00
      $497$00
      $498$00
      $499$000.258819−0.965926i$-0.416667\\pi$
      −0.258819+0.965926i$0.583333\\pi$
      $500$00
      $501$00
      $502$00
      $503$00−0.707107−0.707107i$-0.750000\\pi$
      0.707107+0.707107i$0.250000\\pi$
      $504$00
      $505$00
      $506$00
      $507$00
      $508$00
      $509$−1.00000−1.73205i−1.00000−1.73205i−0.500000−0.866025i$-0.666667\\pi$
      −0.500000−0.866025i$-0.666667\\pi$
      $510$00
      $511$00
      $512$−1.00000i−1.00000i
      $513$00
      $514$−1.00000+1.73205i−1.00000+1.73205i
      $515$00
      $516$00
      $517$00
      $518$00
      $519$00
      $520$00
      $521$1.36603−0.366025i1.36603−0.366025i0.500000−0.866025i$-0.333333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $522$−0.366025−1.36603i−0.366025−1.36603i
      $523$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $524$00
      $525$00
      $526$00
      $527$00
      $528$00
      $529$−0.866025−0.500000i−0.866025−0.500000i
      $530$00
      $531$00
      $532$00
      $533$00
      $534$00
      $535$00
      $536$00
      $537$00
      $538$−1.00000−1.00000i−1.00000−1.00000i
      $539$00
      $540$00
      $541$1.36603+0.366025i1.36603+0.366025i0.866025−0.500000i$-0.166667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $542$00
      $543$00
      $544$−0.866025+0.500000i−0.866025+0.500000i
      $545$−2.00000−2.00000
      $546$00
      $547$000.707107−0.707107i$-0.250000\\pi$
      −0.707107+0.707107i$0.750000\\pi$
      $548$00
      $549$0.366025+1.36603i0.366025+1.36603i
      $550$00
      $551$00
      $552$00
      $553$00
      $554$−1.00000−1.00000i−1.00000−1.00000i
      $555$00
      $556$00
      $557$00−0.866025−0.500000i$-0.833333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $558$00
      $559$00
      $560$00
      $561$00
      $562$00
      $563$00−0.866025−0.500000i$-0.833333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $564$00
      $565$1.00000+1.73205i1.00000+1.73205i
      $566$00
      $567$00
      $568$00
      $569$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $570$00
      $571$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $572$00
      $573$00
      $574$00
      $575$00
      $576$0.866025−0.500000i0.866025−0.500000i
      $577$00−0.866025−0.500000i$-0.833333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $578$0.866025+0.500000i0.866025+0.500000i
      $579$00
      $580$2.00000i2.00000i
      $581$00
      $582$00
      $583$00
      $584$1.36603−0.366025i1.36603−0.366025i
      $585$00
      $586$−1.73205+1.00000i−1.73205+1.00000i
      $587$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $588$00
      $589$00
      $590$00
      $591$00
      $592$−0.366025−1.36603i−0.366025−1.36603i
      $593$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $594$00
      $595$00
      $596$−2.00000−2.00000
      $597$00
      $598$00
      $599$000.500000−0.866025i$-0.333333\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $600$00
      $601$−1.00000−1.00000i−1.00000−1.00000i−1.00000i$-0.5\\pi$
      −1.00000$\\pi$
      $602$00
      $603$00
      $604$00
      $605$−1.36603+0.366025i−1.36603+0.366025i
      $606$00
      $607$00−0.965926−0.258819i$-0.916667\\pi$
      0.965926+0.258819i$0.0833333\\pi$
      $608$00
      $609$00
      $610$−2.00000i−2.00000i
      $611$00
      $612$−0.866025−0.500000i−0.866025−0.500000i
      $613$−1.00000+1.73205i−1.00000+1.73205i−0.500000+0.866025i$0.666667\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $614$00
      $615$00
      $616$00
      $617$−1.00000+1.00000i−1.00000+1.00000i1.00000i$0.5\\pi$
      −1.00000$\\pi$
      $618$00
      $619$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $620$00
      $621$00
      $622$00
      $623$00
      $624$00
      $625$−0.500000−0.866025i−0.500000−0.866025i
      $626$−0.366025−1.36603i−0.366025−1.36603i
      $627$00
      $628$−1.00000−1.73205i−1.00000−1.73205i
      $629$−1.00000+1.00000i−1.00000+1.00000i
      $630$00
      $631$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $632$00
      $633$00
      $634$0.366025+1.36603i0.366025+1.36603i
      $635$00
      $636$00
      $637$00
      $638$00
      $639$00
      $640$−1.36603+0.366025i−1.36603+0.366025i
      $641$−0.366025−1.36603i−0.366025−1.36603i−0.866025−0.500000i$-0.833333\\pi$
      0.500000−0.866025i$-0.333333\\pi$
      $642$00
      $643$000.707107−0.707107i$-0.250000\\pi$
      −0.707107+0.707107i$0.750000\\pi$
      $644$00
      $645$00
      $646$00
      $647$000.500000−0.866025i$-0.333333\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $648$0.866025+0.500000i0.866025+0.500000i
      $649$00
      $650$00
      $651$00
      $652$00
      $653$0.366025−1.36603i0.366025−1.36603i−0.500000−0.866025i$-0.666667\\pi$
      0.866025−0.500000i$-0.166667\\pi$
      $654$00
      $655$00
      $656$−0.366025+1.36603i−0.366025+1.36603i
      $657$1.00000+1.00000i1.00000+1.00000i
      $658$00
      $659$001.00000$0$
      −1.00000$\\pi$
      $660$00
      $661$000.500000−0.866025i$-0.333333\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $662$00
      $663$00
      $664$00
      $665$00
      $666$1.00000−1.00000i1.00000−1.00000i
      $667$00
      $668$00
      $669$00
      $670$00
      $671$00
      $672$00
      $673$−1.00000−1.00000i−1.00000−1.00000i−1.00000i$-0.5\\pi$
      −1.00000$\\pi$
      $674$−0.366025+1.36603i−0.366025+1.36603i
      $675$00
      $676$−0.500000+0.866025i−0.500000+0.866025i
      $677$1.36603+0.366025i1.36603+0.366025i0.866025−0.500000i$-0.166667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $678$00
      $679$00
      $680$1.00000+1.00000i1.00000+1.00000i
      $681$00
      $682$00
      $683$00−0.258819−0.965926i$-0.583333\\pi$
      0.258819+0.965926i$0.416667\\pi$
      $684$00
      $685$00
      $686$00
      $687$00
      $688$00
      $689$00
      $690$00
      $691$000.258819−0.965926i$-0.416667\\pi$
      −0.258819+0.965926i$0.583333\\pi$
      $692$1.00000−1.00000i1.00000−1.00000i
      $693$00
      $694$00
      $695$00
      $696$00
      $697$1.36603−0.366025i1.36603−0.366025i
      $698$00
      $699$00
      $700$00
      $701$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $702$00
      $703$00
      $704$00
      $705$00
      $706$−2.00000i−2.00000i
      $707$00
      $708$00
      $709$−1.36603−0.366025i−1.36603−0.366025i−0.500000−0.866025i$-0.666667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $710$00
      $711$00
      $712$00
      $713$00
      $714$00
      $715$00
      $716$00
      $717$00
      $718$00
      $719$00−0.965926−0.258819i$-0.916667\\pi$
      0.965926+0.258819i$0.0833333\\pi$
      $720$−1.00000−1.00000i−1.00000−1.00000i
      $721$00
      $722$1.00000i1.00000i
      $723$00
      $724$1.36603−0.366025i1.36603−0.366025i
      $725$1.36603−0.366025i1.36603−0.366025i
      $726$00
      $727$001.00000$0$
      −1.00000$\\pi$
      $728$00
      $729$1.00000i1.00000i
      $730$−1.00000−1.73205i−1.00000−1.73205i
      $731$00
      $732$00
      $733$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $734$00
      $735$00
      $736$00
      $737$00
      $738$−1.36603+0.366025i−1.36603+0.366025i
      $739$00−0.866025−0.500000i$-0.833333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $740$−1.73205+1.00000i−1.73205+1.00000i
      $741$00
      $742$00
      $743$000.707107−0.707107i$-0.250000\\pi$
      −0.707107+0.707107i$0.750000\\pi$
      $744$00
      $745$0.732051+2.73205i0.732051+2.73205i
      $746$00
      $747$00
      $748$00
      $749$00
      $750$00
      $751$00−0.965926−0.258819i$-0.916667\\pi$
      0.965926+0.258819i$0.0833333\\pi$
      $752$00
      $753$00
      $754$00
      $755$00
      $756$00
      $757$2.00000i2.00000i1.00000i$0.5\\pi$
      1.00000i$0.5\\pi$
      $758$00
      $759$00
      $760$00
      $761$000.866025−0.500000i$-0.166667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $762$00
      $763$00
      $764$00
      $765$−0.366025+1.36603i−0.366025+1.36603i
      $766$00
      $767$00
      $768$00
      $769$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $770$00
      $771$00
      $772$−0.366025+1.36603i−0.366025+1.36603i
      $773$−1.73205−1.00000i−1.73205−1.00000i−0.866025−0.500000i$-0.833333\\pi$
      −0.866025−0.500000i$-0.833333\\pi$
      $774$00
      $775$00
      $776$1.00000+1.00000i1.00000+1.00000i
      $777$00
      $778$2.000002.00000
      $779$00
      $780$00
      $781$00
      $782$00
      $783$00
      $784$00
      $785$−2.00000+2.00000i−2.00000+2.00000i
      $786$00
      $787$00−0.258819−0.965926i$-0.583333\\pi$
      0.258819+0.965926i$0.416667\\pi$
      $788$1.36603−0.366025i1.36603−0.366025i
      $789$00
      $790$00
      $791$00
      $792$00
      $793$00
      $794$0.366025+1.36603i0.366025+1.36603i
      $795$00
      $796$00
      $797$001.00000$0$
      −1.00000$\\pi$
      $798$00
      $799$00
      $800$0.500000+0.866025i0.500000+0.866025i
      $801$00
      $802$0.366025+1.36603i0.366025+1.36603i
      $803$00
      $804$00
      $805$00
      $806$00
      $807$00
      $808$00
      $809$1.36603−0.366025i1.36603−0.366025i0.500000−0.866025i$-0.333333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $810$0.366025−1.36603i0.366025−1.36603i
      $811$000.707107−0.707107i$-0.250000\\pi$
      −0.707107+0.707107i$0.750000\\pi$
      $812$00
      $813$00
      $814$00
      $815$00
      $816$00
      $817$00
      $818$−2.00000i−2.00000i
      $819$00
      $820$2.000002.00000
      $821$−1.36603−0.366025i−1.36603−0.366025i−0.500000−0.866025i$-0.666667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $822$00
      $823$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $824$00
      $825$00
      $826$00
      $827$00−0.707107−0.707107i$-0.750000\\pi$
      0.707107+0.707107i$0.250000\\pi$
      $828$00
      $829$1.00000−1.73205i1.00000−1.73205i0.500000−0.866025i$-0.333333\\pi$
      0.500000−0.866025i$-0.333333\\pi$
      $830$00
      $831$00
      $832$00
      $833$00
      $834$00
      $835$00
      $836$00
      $837$00
      $838$00
      $839$00−0.707107−0.707107i$-0.750000\\pi$
      0.707107+0.707107i$0.250000\\pi$
      $840$00
      $841$−1.00000i−1.00000i
      $842$00
      $843$00
      $844$00
      $845$1.36603+0.366025i1.36603+0.366025i
      $846$00
      $847$00
      $848$00
      $849$00
      $850$0.500000−0.866025i0.500000−0.866025i
      $851$00
      $852$00
      $853$1.00000−1.00000i1.00000−1.00000i−1.00000i$-0.5\\pi$
      1.00000$0$
      $854$00
      $855$00
      $856$00
      $857$1.36603−0.366025i1.36603−0.366025i0.500000−0.866025i$-0.333333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $858$00
      $859$000.866025−0.500000i$-0.166667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $860$00
      $861$00
      $862$00
      $863$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $864$00
      $865$−1.73205−1.00000i−1.73205−1.00000i
      $866$−1.00000−1.73205i−1.00000−1.73205i
      $867$00
      $868$00
      $869$00
      $870$00
      $871$00
      $872$0.366025+1.36603i0.366025+1.36603i
      $873$−0.366025+1.36603i−0.366025+1.36603i
      $874$00
      $875$00
      $876$00
      $877$−0.366025+1.36603i−0.366025+1.36603i0.500000+0.866025i$0.333333\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $878$00
      $879$00
      $880$00
      $881$−1.00000+1.00000i−1.00000+1.00000i1.00000i$0.5\\pi$
      −1.00000$\\pi$
      $882$00
      $883$001.00000$0$
      −1.00000$\\pi$
      $884$00
      $885$00
      $886$00
      $887$00−0.965926−0.258819i$-0.916667\\pi$
      0.965926+0.258819i$0.0833333\\pi$
      $888$00
      $889$00
      $890$00
      $891$00
      $892$00
      $893$00
      $894$00
      $895$00
      $896$00
      $897$00
      $898$1.36603+0.366025i1.36603+0.366025i
      $899$00
      $900$−0.500000+0.866025i−0.500000+0.866025i
      $901$00
      $902$00
      $903$00
      $904$1.00000−1.00000i1.00000−1.00000i
      $905$−1.00000−1.73205i−1.00000−1.73205i
      $906$00
      $907$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $908$00
      $909$00
      $910$00
      $911$00−0.707107−0.707107i$-0.750000\\pi$
      0.707107+0.707107i$0.250000\\pi$
      $912$00
      $913$00
      $914$1.00000−1.73205i1.00000−1.73205i
      $915$00
      $916$2.00000i2.00000i
      $917$00
      $918$00
      $919$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $920$00
      $921$00
      $922$00
      $923$00
      $924$00
      $925$1.00000+1.00000i1.00000+1.00000i
      $926$00
      $927$00
      $928$1.36603−0.366025i1.36603−0.366025i
      $929$−0.366025+1.36603i−0.366025+1.36603i0.500000+0.866025i$0.333333\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $930$00
      $931$00
      $932$1.00000−1.00000i1.00000−1.00000i
      $933$00
      $934$00
      $935$00
      $936$00
      $937$001.00000$0$
      −1.00000$\\pi$
      $938$00
      $939$00
      $940$00
      $941$1.36603−0.366025i1.36603−0.366025i0.500000−0.866025i$-0.333333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $942$00
      $943$00
      $944$00
      $945$00
      $946$00
      $947$00−0.965926−0.258819i$-0.916667\\pi$
      0.965926+0.258819i$0.0833333\\pi$
      $948$00
      $949$00
      $950$00
      $951$00
      $952$00
      $953$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $954$00
      $955$00
      $956$00
      $957$00
      $958$00
      $959$00
      $960$00
      $961$−0.866025+0.500000i−0.866025+0.500000i
      $962$00
      $963$00
      $964$−1.36603−0.366025i−1.36603−0.366025i
      $965$2.000002.00000
      $966$00
      $967$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $968$0.500000+0.866025i0.500000+0.866025i
      $969$00
      $970$1.00000−1.73205i1.00000−1.73205i
      $971$000.866025−0.500000i$-0.166667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $972$00
      $973$00
      $974$00
      $975$00
      $976$−1.36603+0.366025i−1.36603+0.366025i
      $977$000.500000−0.866025i$-0.333333\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $978$00
      $979$00
      $980$00
      $981$−1.00000+1.00000i−1.00000+1.00000i
      $982$00
      $983$00−0.258819−0.965926i$-0.583333\\pi$
      0.258819+0.965926i$0.416667\\pi$
      $984$00
      $985$−1.00000−1.73205i−1.00000−1.73205i
      $986$−1.00000−1.00000i−1.00000−1.00000i
      $987$00
      $988$00
      $989$00
      $990$00
      $991$00−0.258819−0.965926i$-0.583333\\pi$
      0.258819+0.965926i$0.416667\\pi$
      $992$00
      $993$00
      $994$00
      $995$00
      $996$00
      $997$1.36603−0.366025i1.36603−0.366025i0.500000−0.866025i$-0.333333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $998$00
      $999$00
      \n\n(See $a_n$ instead)(See $a_n$ instead)(See $a_n$ instead)(See only $a_p$ )(See only $a_p$ )(See only $a_p$ )\n\n
             By twisting character
      CharParityOrdTypeTwistMinDim
      1.1even1trivial3332.1.bc.b.2027.14
      4.3odd2CM3332.1.bc.b.2027.14
      7.2even3inner3332.1.bc.b.667.14
      7.3odd668.1.f.a.55.1yes2
      7.4even33332.1.m.b.2843.12
      7.5odd63332.1.bc.c.667.14
      7.6odd23332.1.bc.c.2027.14
      17.13even4inner3332.1.bc.b.2223.14
      21.17even6612.1.l.a.55.12
      28.3even668.1.f.a.55.1yes2
      28.11odd63332.1.m.b.2843.12
      28.19even63332.1.bc.c.667.14
      28.23odd6inner3332.1.bc.b.667.14
      28.27even23332.1.bc.c.2027.14
      35.3even121700.1.n.a.599.12
      35.17even121700.1.n.b.599.12
      35.24odd61700.1.p.a.1551.12
      56.3even61088.1.p.a.191.12
      56.45odd61088.1.p.a.191.12
      68.47odd4inner3332.1.bc.b.2223.14
      84.59odd6612.1.l.a.55.12
      119.3even481156.1.g.b.155.18
      119.10even481156.1.g.b.179.18
      119.13odd43332.1.bc.c.2223.14
      119.24even481156.1.g.b.179.28
      119.30even12inner3332.1.bc.b.863.14
      119.31even481156.1.g.b.155.28
      119.38odd121156.1.f.b.251.12
      119.45even481156.1.g.b.399.18
      119.47odd123332.1.bc.c.863.14
      119.59odd241156.1.c.b.579.12
      119.66odd241156.1.d.a.1155.22
      119.73even481156.1.g.b.423.28
      119.80even481156.1.g.b.423.18
      119.81even123332.1.m.b.3039.12
      119.87odd241156.1.d.a.1155.12
      119.94odd241156.1.c.b.579.22
      119.101odd61156.1.f.b.327.12
      119.108even481156.1.g.b.399.28
      119.115odd1268.1.f.a.47.1✓2
      140.3odd121700.1.n.a.599.12
      140.59even61700.1.p.a.1551.12
      140.87odd121700.1.n.b.599.12
      357.353even12612.1.l.a.523.12
      476.3odd481156.1.g.b.155.18
      476.31odd481156.1.g.b.155.28
      476.47even123332.1.bc.c.863.14
      476.59even241156.1.c.b.579.12
      476.87even241156.1.d.a.1155.12
      476.115even1268.1.f.a.47.1✓2
      476.143odd481156.1.g.b.179.28
      476.199odd481156.1.g.b.423.18
      476.227odd481156.1.g.b.399.28
      476.251even43332.1.bc.c.2223.14
      476.283odd481156.1.g.b.399.18
      476.311odd481156.1.g.b.423.28
      476.319odd123332.1.m.b.3039.12
      476.339even61156.1.f.b.327.12
      476.367odd481156.1.g.b.179.18
      476.387odd12inner3332.1.bc.b.863.14
      476.395even121156.1.f.b.251.12
      476.423even241156.1.d.a.1155.22
      476.451even241156.1.c.b.579.22
      595.234odd121700.1.p.a.251.12
      595.353even121700.1.n.b.999.12
      595.472even121700.1.n.a.999.12
      952.115even121088.1.p.a.319.12
      952.829odd121088.1.p.a.319.12
      1428.1067odd12612.1.l.a.523.12
      2380.1067odd121700.1.n.a.999.12
      2380.1543odd121700.1.n.b.999.12
      2380.2019even121700.1.p.a.251.12
      \n\n
              By twisted newform
      TwistMinDimCharParityOrdType
      68.1.f.a.47.1✓2119.115odd12
      68.1.f.a.47.1✓2476.115even12
      68.1.f.a.55.1yes27.3odd6
      68.1.f.a.55.1yes228.3even6
      612.1.l.a.55.1221.17even6
      612.1.l.a.55.1284.59odd6
      612.1.l.a.523.12357.353even12
      612.1.l.a.523.121428.1067odd12
      1088.1.p.a.191.1256.3even6
      1088.1.p.a.191.1256.45odd6
      1088.1.p.a.319.12952.115even12
      1088.1.p.a.319.12952.829odd12
      1156.1.c.b.579.12119.59odd24
      1156.1.c.b.579.12476.59even24
      1156.1.c.b.579.22119.94odd24
      1156.1.c.b.579.22476.451even24
      1156.1.d.a.1155.12119.87odd24
      1156.1.d.a.1155.12476.87even24
      1156.1.d.a.1155.22119.66odd24
      1156.1.d.a.1155.22476.423even24
      1156.1.f.b.251.12119.38odd12
      1156.1.f.b.251.12476.395even12
      1156.1.f.b.327.12119.101odd6
      1156.1.f.b.327.12476.339even6
      1156.1.g.b.155.18119.3even48
      1156.1.g.b.155.18476.3odd48
      1156.1.g.b.155.28119.31even48
      1156.1.g.b.155.28476.31odd48
      1156.1.g.b.179.18119.10even48
      1156.1.g.b.179.18476.367odd48
      1156.1.g.b.179.28119.24even48
      1156.1.g.b.179.28476.143odd48
      1156.1.g.b.399.18119.45even48
      1156.1.g.b.399.18476.283odd48
      1156.1.g.b.399.28119.108even48
      1156.1.g.b.399.28476.227odd48
      1156.1.g.b.423.18119.80even48
      1156.1.g.b.423.18476.199odd48
      1156.1.g.b.423.28119.73even48
      1156.1.g.b.423.28476.311odd48
      1700.1.n.a.599.1235.3even12
      1700.1.n.a.599.12140.3odd12
      1700.1.n.a.999.12595.472even12
      1700.1.n.a.999.122380.1067odd12
      1700.1.n.b.599.1235.17even12
      1700.1.n.b.599.12140.87odd12
      1700.1.n.b.999.12595.353even12
      1700.1.n.b.999.122380.1543odd12
      1700.1.p.a.251.12595.234odd12
      1700.1.p.a.251.122380.2019even12
      1700.1.p.a.1551.1235.24odd6
      1700.1.p.a.1551.12140.59even6
      3332.1.m.b.2843.127.4even3
      3332.1.m.b.2843.1228.11odd6
      3332.1.m.b.3039.12119.81even12
      3332.1.m.b.3039.12476.319odd12
      3332.1.bc.b.667.147.2even3inner
      3332.1.bc.b.667.1428.23odd6inner
      3332.1.bc.b.863.14119.30even12inner
      3332.1.bc.b.863.14476.387odd12inner
      3332.1.bc.b.2027.141.1even1trivial
      3332.1.bc.b.2027.144.3odd2CM
      3332.1.bc.b.2223.1417.13even4inner
      3332.1.bc.b.2223.1468.47odd4inner
      3332.1.bc.c.667.147.5odd6
      3332.1.bc.c.667.1428.19even6
      3332.1.bc.c.863.14119.47odd12
      3332.1.bc.c.863.14476.47even12
      3332.1.bc.c.2027.147.6odd2
      3332.1.bc.c.2027.1428.27even2
      3332.1.bc.c.2223.14119.13odd4
      3332.1.bc.c.2223.14476.251even4
      \n", "main_html": "
      \n
      \n
      \n \n
      \n Show commands:\n Magma\n / PariGP\n / SageMath\n
      \n\n\n\n\n\n\n\n
      [N,k,chi] = [3332,1,Mod(667,3332)]
       
      mf = mfinit([N,k,chi],0)
       
      lf = mfeigenbasis(mf)
       
      \n
      from sage.modular.dirichlet import DirichletCharacter
       
      H = DirichletGroup(3332, base_ring=CyclotomicField(12))
       
      chi = DirichletCharacter(H, H._module([6, 4, 9]))
       
      N = Newforms(chi, 1, names=\"a\")
       
      \n
      //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
       
      chi := DirichletCharacter(\"3332.667\");
       
      S:= CuspForms(chi, 1);
       
      N := Newforms(S);
       
      \n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
      Level: \\( N \\) \\(=\\)\\( 3332 = 2^{2} \\cdot 7^{2} \\cdot 17 \\)
      Weight: \\( k \\) \\(=\\)\\( 1 \\)
      Character orbit: \\([\\chi]\\) \\(=\\) 3332.bc (of order \\(12\\), degree \\(4\\), not minimal)
      \n\n

      Newform invariants

      \n\n\n
      sage: f = N[0] # Warning: the index may be different
       
      \n
      gp: f = lf[1] \\\\ Warning: the index may be different
       
      \n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\t\n \n \t\n \t\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
      Self dual: no
      Analytic conductor: \\(1.66288462209\\)
      Analytic rank: \\(0\\)
      Dimension: \\(4\\)
      Coefficient field: \\(\\Q(\\zeta_{12})\\)
      \n \n
      gp: f.mod \\\\ as an extension of the character field
       
      \n\n
      Defining polynomial: \n\n \\( x^{4} - x^{2} + 1 \\)\n \n\n \n \"Copy\n \n \n \"Toggle\n \n
      Coefficient ring: \\(\\Z[a_1, a_2]\\)
      Coefficient ring index: \\( 1 \\)
      Twist minimal: no (minimal twist has level 68)
      Projective image:\\(D_{4}\\)
      Projective field:Galois closure of 4.2.19652.1
      Artin image:$C_4\\wr C_2\\times C_6$
      Artin field:Galois closure of \\(\\mathbb{Q}[x]/(x^{48} - \\cdots)\\)
      \n\n\n

      Embedding invariants

      \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
      Embedding label 2027.1
      Root\\(0.866025 - 0.500000i\\) of defining polynomial
      Character\\(\\chi\\)\\(=\\)3332.2027
      Dual form 3332.1.bc.b.863.1
      \n\n\n\n
      \n \n
      sage: f.q_expansion() # note that sage often uses an isomorphic number field
       
      \n
      gp: mfcoefs(f, 20)
       
      \n\n
      \n \n \n \n \n \n \n \n \n \n \n \n
      \\(f(q)\\)\\(=\\)\\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.36603 - 0.366025i) q^{5} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.00000i) q^{20} +(0.866025 + 0.500000i) q^{25} +(1.00000 - 1.00000i) q^{29} +(0.866025 + 0.500000i) q^{32} -1.00000i q^{34} +1.00000i q^{36} +(1.36603 + 0.366025i) q^{37} +(0.366025 - 1.36603i) q^{40} +(-1.00000 - 1.00000i) q^{41} +(1.36603 - 0.366025i) q^{45} -1.00000 q^{50} +(-0.366025 + 1.36603i) q^{58} +(0.366025 - 1.36603i) q^{61} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(-0.366025 - 1.36603i) q^{73} +(-1.36603 + 0.366025i) q^{74} +(0.366025 + 1.36603i) q^{80} +(0.500000 - 0.866025i) q^{81} +(1.36603 + 0.366025i) q^{82} +(1.00000 - 1.00000i) q^{85} +(-1.00000 + 1.00000i) q^{90} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\\)
      \\(\\operatorname{Tr}(f)(q)\\)\\(=\\)\n\n \\( 4 q + 2 q^{4} - 2 q^{5} + 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 4 q^{20} + 4 q^{29} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 2 q^{45} - 4 q^{50} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{68} - 2 q^{72} + 2 q^{73}+ \\cdots + 4 q^{97}+O(q^{100}) \\)\n \n\n \n \"Copy\n \n \n \"Toggle\n \n
      \n
      \n\n
      \n\n\n

      Character values

      \n

      We give the values of \\(\\chi\\) on generators for \\(\\left(\\mathbb{Z}/3332\\mathbb{Z}\\right)^\\times\\).

      \n\n \n \n \n \n \n \n \n \n \n \n \n \n
      \\(n\\)\\(785\\)\\(885\\)\\(1667\\)
      \\(\\chi(n)\\)\\(e\\left(\\frac{3}{4}\\right)\\)\\(e\\left(\\frac{2}{3}\\right)\\)\\(-1\\)
      \n\n\n

      Coefficient data

      \n\n

      For each \\(n\\) we display the coefficients of the \\(q\\)-expansion \\(a_n\\), the\nSatake parameters \\(\\alpha_p\\),\nand the Satake angles \\(\\theta_p = \\textrm{Arg}(\\alpha_p)\\).

      \n\n\n\n\n

      \n
      \n \n\n \n (See \\(a_n\\) instead)\n \n\n \n (See \\(a_n\\) instead)\n \n\n \n (See \\(a_n\\) instead)\n \n \n\n \n (See only \\(a_p\\))\n \n\n \n (See only \\(a_p\\))\n \n\n \n (See only \\(a_p\\))\n \n
      \n\n\n\n\n
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\n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n
      \n \\(n\\)\n \n \\(a_n\\)\n \n \\(a_n / n^{(k-1)/2}\\)\n \n \\( \\alpha_n \\)\n \n \\( \\theta_n \\)\n
      \n \\(p\\)\n \n \\(a_p\\)\n \n \\(a_p / p^{(k-1)/2}\\)\n \n \\( \\alpha_p\\)\n \n \\( \\theta_p \\)\n
      \n \\(2\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n \n\n \n\n \n\n \n\n
      \n\n \n\n \n\n \n\n
      \n \\(3\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(4\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
      \n \\(5\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(6\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(7\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n\n \n\n \n\n
      \n\n \n\n \n\n \n\n
      \n \\(8\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
      \n \\(9\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n
      \n \\(10\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(11\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(12\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(13\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(14\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(15\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(16\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
      \n \\(17\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n \n\n \n\n \n\n \n\n
      \n\n \n\n \n\n \n\n
      \n \\(18\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
      \n \\(19\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(20\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
      \n \\(21\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(22\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(23\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
      \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
      \n \\(24\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(25\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
      \n \\(26\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(27\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(28\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(29\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n \\(30\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(31\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(32\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
      \n \\(33\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(34\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
      \n \\(35\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(36\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
      \n \\(37\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(38\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(39\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(40\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n
      \n \\(41\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(42\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(43\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(44\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(45\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(46\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(47\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(48\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(49\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(50\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
      \n \\(51\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(52\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(53\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(54\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(55\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(56\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(57\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(58\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(59\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(60\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(61\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n \\(62\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(63\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(64\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
      \n \\(65\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(66\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(67\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(68\\)\n \n 0.500000\n \n +\n \n 0.866025i\n \n 0.500000\n \n +\n \n 0.866025i\n
      \n \\(69\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(70\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(71\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(72\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
      \n \\(73\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n \\(74\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(75\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(76\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(77\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(78\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(79\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
      \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
      \n \\(80\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(81\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
      \n \\(82\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(83\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(84\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(85\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(86\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(87\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(88\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(89\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(90\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
      \n \\(91\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(92\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(93\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(94\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(95\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(96\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(97\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n \\(98\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(99\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(100\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
      \n \\(101\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(102\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(103\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(104\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(105\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(106\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(107\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(108\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(109\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(110\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(111\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(112\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(113\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(114\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(115\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(116\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
      \n \\(117\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(118\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(119\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(120\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(121\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
      \n \\(122\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(123\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(124\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(125\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(126\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(127\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(128\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
      \n \\(129\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(130\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(131\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(132\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(133\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(134\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(135\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(136\\)\n \n −0.866025\n \n −\n \n 0.500000i\n \n −0.866025\n \n −\n \n 0.500000i\n
      \n \\(137\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(138\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(139\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(140\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(141\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(142\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(143\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(144\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
      \n \\(145\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n
      \n \\(146\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(147\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(148\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(149\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n \\(150\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(151\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(152\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(153\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
      \n \\(154\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(155\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(156\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(157\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n \\(158\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(159\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(160\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
      \n \\(161\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(162\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
      \n \\(163\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
      \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
      \n \\(164\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(165\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(166\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(167\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(168\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(169\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
      \n \\(170\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(171\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(172\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(173\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(174\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(175\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(176\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(177\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(178\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(179\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(180\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n
      \n \\(181\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(182\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(183\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(184\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(185\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n
      \n \\(186\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(187\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(188\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(189\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(190\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(191\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(192\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(193\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(194\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(195\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(196\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(197\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(198\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(199\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(200\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
      \n \\(201\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(202\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(203\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(204\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(205\\)\n \n 1.00000\n \n +\n \n 1.73205i\n \n 1.00000\n \n +\n \n 1.73205i\n
      \n \\(206\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(207\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(208\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(209\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(210\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(211\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
      \n \\(212\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(213\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(214\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(215\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(216\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(217\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(218\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
      \n \\(219\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(220\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(221\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(222\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(223\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(224\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(225\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
      \n \\(226\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(227\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(228\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(229\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(230\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(231\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(232\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(233\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(234\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(235\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(236\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(237\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(238\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(239\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(240\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(241\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n \\(242\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
      \n \\(243\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(244\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
      \n \\(245\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(246\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(247\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(248\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(249\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(250\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(251\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(252\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(253\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(254\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(255\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(256\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
      \n \\(257\\)\n \n 1.73205\n \n −\n \n 1.00000i\n \n 1.73205\n \n −\n \n 1.00000i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n \\(258\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(259\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(260\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(261\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(262\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(263\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(264\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(265\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(266\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(267\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(268\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(269\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(270\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(271\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(272\\)\n \n 1.00000\n \n\n \n\n \n 1.00000\n \n\n \n\n
      \n \\(273\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(274\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(275\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(276\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(277\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(278\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(279\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(280\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(281\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(282\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(283\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(284\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(285\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(286\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(287\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(288\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
      \n \\(289\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
      \n \\(290\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n
      \n \\(291\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(292\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
      \n \\(293\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n 1.00000\n \n\n \n\n \n \\(0\\)\n
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      \n \\(295\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(296\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(297\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(298\\)\n \n 1.73205\n \n +\n \n 1.00000i\n \n 1.73205\n \n +\n \n 1.00000i\n
      \n \\(299\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(300\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(301\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(302\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(303\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(304\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(305\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n
      \n \\(306\\)\n \n 0.500000\n \n +\n \n 0.866025i\n \n 0.500000\n \n +\n \n 0.866025i\n
      \n \\(307\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(308\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(309\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(310\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(311\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(312\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(313\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(314\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n
      \n \\(315\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(316\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(317\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n \\(318\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(319\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(320\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(321\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(322\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(323\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(324\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
      \n \\(325\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(326\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(327\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(328\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(329\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(330\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(331\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(332\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(333\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(334\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(335\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(336\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(337\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n \\(338\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
      \n \\(339\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(340\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
      \n \\(341\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(342\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(343\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(344\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(345\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(346\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(347\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(348\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(349\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(350\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(351\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(352\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(353\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(354\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(355\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(356\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(357\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(358\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
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      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(360\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(361\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
      \n \\(362\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
      \n \\(363\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(364\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(365\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n
      \n \\(366\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(367\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(368\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(369\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(370\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n
      \n \\(371\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(372\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(373\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(374\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(375\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(376\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(377\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(378\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(379\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(380\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(381\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(382\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(383\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(384\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(385\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(386\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(387\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(388\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
      \n \\(389\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n \\(390\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(391\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(392\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(393\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(394\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
      \n \\(395\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(396\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(397\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n \\(398\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(399\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(400\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
      \n \\(401\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n \\(402\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(403\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(404\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(405\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
      \n \\(406\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(407\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(408\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(409\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(410\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n
      \n \\(411\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(412\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(413\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(414\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(415\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(416\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(417\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(418\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(419\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
      \n \\(420\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(421\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(422\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(423\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(424\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(425\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n
      \n \\(426\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(427\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(428\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(429\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(430\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(431\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(432\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(433\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(434\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(435\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(436\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n
      \n \\(437\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(438\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(439\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(440\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(441\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(442\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(443\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(444\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(445\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(446\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(447\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(448\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(449\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(450\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
      \n \\(451\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(452\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(453\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(454\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(455\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(456\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(457\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(458\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n
      \n \\(459\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(460\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(461\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(462\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(463\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(464\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
      \n \\(465\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(466\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(467\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(468\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(469\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(470\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(471\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(472\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(473\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(474\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(475\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(476\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(477\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(478\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(479\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(480\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(481\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(482\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(483\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(484\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
      \n \\(485\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n
      \n \\(486\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(487\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(488\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(489\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(490\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(491\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(492\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(493\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(494\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(495\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(496\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(497\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(498\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(499\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
      \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
      \n \\(500\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(501\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(502\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(503\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
      \n \\(504\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(505\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(506\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(507\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(508\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(509\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n \\(510\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(511\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(512\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
      \n \\(513\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(514\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n
      \n \\(515\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(516\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(517\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(518\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(519\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(520\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(521\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(522\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
      \n \\(523\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(524\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(525\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(526\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(527\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(528\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(529\\)\n \n −0.866025\n \n −\n \n 0.500000i\n \n −0.866025\n \n −\n \n 0.500000i\n
      \n \\(530\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(531\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(532\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(533\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(534\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(535\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(536\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(537\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(538\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
      \n \\(539\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(540\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(541\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(542\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(543\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(544\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n
      \n \\(545\\)\n \n −2.00000\n \n\n \n\n \n −2.00000\n \n\n \n\n
      \n \\(546\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(547\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(548\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(549\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(550\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(551\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(552\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(553\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(554\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
      \n \\(555\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(556\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(557\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(558\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(559\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(560\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(561\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(562\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(563\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(564\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(565\\)\n \n 1.00000\n \n +\n \n 1.73205i\n \n 1.00000\n \n +\n \n 1.73205i\n
      \n \\(566\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(567\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(568\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(569\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(570\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(571\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(572\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(573\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(574\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(575\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(576\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
      \n \\(577\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(578\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
      \n \\(579\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(580\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n
      \n \\(581\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(582\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(583\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(584\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(585\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(586\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n
      \n \\(587\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(588\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(589\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(590\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(591\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(592\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
      \n \\(593\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(594\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(595\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(596\\)\n \n −2.00000\n \n\n \n\n \n −2.00000\n \n\n \n\n
      \n \\(597\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(598\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(599\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(600\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(601\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(602\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(603\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(604\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(605\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(606\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(607\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(608\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(609\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(610\\)\n \n\n \n −\n \n 2.00000i\n \n\n \n −\n \n 2.00000i\n
      \n \\(611\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(612\\)\n \n −0.866025\n \n −\n \n 0.500000i\n \n −0.866025\n \n −\n \n 0.500000i\n
      \n \\(613\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(614\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(615\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(616\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(617\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(618\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(619\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(620\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(621\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(622\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(623\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(624\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(625\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
      \n \\(626\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
      \n \\(627\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(628\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
      \n \\(629\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
      \n \\(630\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(631\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(632\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(633\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(634\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(635\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(636\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(637\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(638\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(639\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(640\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(641\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n \\(642\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(643\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(644\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(645\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(646\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(647\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(648\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
      \n \\(649\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(650\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(651\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(652\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(653\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n \\(654\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(655\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(656\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(657\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(658\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(659\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(660\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(661\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(662\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(663\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(664\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(665\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(666\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(667\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(668\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(669\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(670\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(671\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(672\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(673\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(674\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(675\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(676\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
      \n \\(677\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(678\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(679\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(680\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(681\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(682\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(683\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(684\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(685\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(686\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(687\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(688\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(689\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(690\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(691\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
      \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
      \n \\(692\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(693\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(694\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(695\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(696\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(697\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(698\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(699\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(700\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(701\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(702\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(703\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(704\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(705\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(706\\)\n \n\n \n −\n \n 2.00000i\n \n\n \n −\n \n 2.00000i\n
      \n \\(707\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(708\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(709\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(710\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(711\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(712\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(713\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(714\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(715\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(716\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(717\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(718\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(719\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(720\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
      \n \\(721\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(722\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
      \n \\(723\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(724\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(725\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(726\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(727\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(728\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(729\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
      \n \\(730\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
      \n \\(731\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(732\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(733\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(734\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(735\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(736\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(737\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(738\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(739\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(740\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n
      \n \\(741\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(742\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(743\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(744\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(745\\)\n \n 0.732051\n \n +\n \n 2.73205i\n \n 0.732051\n \n +\n \n 2.73205i\n
      \n \\(746\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(747\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(748\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(749\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(750\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(751\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(752\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(753\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(754\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(755\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(756\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(757\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(758\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(759\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(760\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(761\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(762\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(763\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(764\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(765\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(766\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(767\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(768\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(769\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(770\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(771\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(772\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(773\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n \\(774\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(775\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(776\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(777\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(778\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n
      \n \\(779\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(780\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(781\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(782\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(783\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(784\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(785\\)\n \n −2.00000\n \n +\n \n 2.00000i\n \n −2.00000\n \n +\n \n 2.00000i\n
      \n \\(786\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(787\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(788\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(789\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(790\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(791\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(792\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(793\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(794\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(795\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(796\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(797\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(798\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(799\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(800\\)\n \n 0.500000\n \n +\n \n 0.866025i\n \n 0.500000\n \n +\n \n 0.866025i\n
      \n \\(801\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(802\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(803\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(804\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(805\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(806\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(807\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(808\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(809\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(810\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n
      \n \\(811\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(812\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(813\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(814\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(815\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(816\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(817\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(818\\)\n \n\n \n −\n \n 2.00000i\n \n\n \n −\n \n 2.00000i\n
      \n \\(819\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(820\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n
      \n \\(821\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(822\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(823\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(824\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(825\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(826\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(827\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
      \n \\(828\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(829\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n \\(830\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(831\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(832\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(833\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(834\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(835\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(836\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(837\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(838\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(839\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
      \n \\(840\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(841\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
      \n \\(842\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(843\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(844\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(845\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(846\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(847\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(848\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(849\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(850\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
      \n \\(851\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(852\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(853\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n \\(854\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(855\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(856\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(857\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(858\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(859\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(860\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(861\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(862\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(863\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(864\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(865\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n
      \n \\(866\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
      \n \\(867\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(868\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(869\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(870\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(871\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(872\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(873\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(874\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(875\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(876\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(877\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(878\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(879\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(880\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(881\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(882\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(883\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(884\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(885\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(886\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(887\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(888\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(889\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(890\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(891\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(892\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(893\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(894\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(895\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(896\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(897\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(898\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(899\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(900\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
      \n \\(901\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(902\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(903\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(904\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(905\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
      \n \\(906\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(907\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(908\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(909\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(910\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(911\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
      \n \\(912\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(913\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(914\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n
      \n \\(915\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(916\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n
      \n \\(917\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(918\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(919\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(920\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(921\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(922\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(923\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(924\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(925\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(926\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(927\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(928\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(929\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(930\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(931\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(932\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(933\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(934\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(935\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(936\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(937\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(938\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(939\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(940\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(941\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(942\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(943\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(944\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(945\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(946\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(947\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(948\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(949\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(950\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(951\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(952\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(953\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(954\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(955\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(956\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(957\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(958\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(959\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(960\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(961\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n
      \n \\(962\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(963\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(964\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
      \n \\(965\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n
      \n \\(966\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(967\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(968\\)\n \n 0.500000\n \n +\n \n 0.866025i\n \n 0.500000\n \n +\n \n 0.866025i\n
      \n \\(969\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(970\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n
      \n \\(971\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(972\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(973\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(974\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(975\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(976\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(977\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(978\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(979\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(980\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(981\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
      \n \\(982\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(983\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(984\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(985\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
      \n \\(986\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
      \n \\(987\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(988\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(989\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(990\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(991\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(992\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(993\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(994\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(995\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(996\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(997\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(998\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(999\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n
      \n
      \n \n\n \n (See \\(a_n\\) instead)\n \n\n \n (See \\(a_n\\) instead)\n \n\n \n (See \\(a_n\\) instead)\n \n \n\n \n (See only \\(a_p\\))\n \n\n \n (See only \\(a_p\\))\n \n\n \n (See only \\(a_p\\))\n \n
      \n\n\n\n\n\n\n\n \n \n \n \n \n \n \n\n\n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n
             By twisting character
      CharParityOrdTypeTwistMinDim
      1.1even1trivial3332.1.bc.b.2027.14
      4.3odd2CM3332.1.bc.b.2027.14
      7.2even3inner3332.1.bc.b.667.14
      7.3odd668.1.f.a.55.1yes2
      7.4even33332.1.m.b.2843.12
      7.5odd63332.1.bc.c.667.14
      7.6odd23332.1.bc.c.2027.14
      17.13even4inner3332.1.bc.b.2223.14
      21.17even6612.1.l.a.55.12
      28.3even668.1.f.a.55.1yes2
      28.11odd63332.1.m.b.2843.12
      28.19even63332.1.bc.c.667.14
      28.23odd6inner3332.1.bc.b.667.14
      28.27even23332.1.bc.c.2027.14
      35.3even121700.1.n.a.599.12
      35.17even121700.1.n.b.599.12
      35.24odd61700.1.p.a.1551.12
      56.3even61088.1.p.a.191.12
      56.45odd61088.1.p.a.191.12
      68.47odd4inner3332.1.bc.b.2223.14
      84.59odd6612.1.l.a.55.12
      119.3even481156.1.g.b.155.18
      119.10even481156.1.g.b.179.18
      119.13odd43332.1.bc.c.2223.14
      119.24even481156.1.g.b.179.28
      119.30even12inner3332.1.bc.b.863.14
      119.31even481156.1.g.b.155.28
      119.38odd121156.1.f.b.251.12
      119.45even481156.1.g.b.399.18
      119.47odd123332.1.bc.c.863.14
      119.59odd241156.1.c.b.579.12
      119.66odd241156.1.d.a.1155.22
      119.73even481156.1.g.b.423.28
      119.80even481156.1.g.b.423.18
      119.81even123332.1.m.b.3039.12
      119.87odd241156.1.d.a.1155.12
      119.94odd241156.1.c.b.579.22
      119.101odd61156.1.f.b.327.12
      119.108even481156.1.g.b.399.28
      119.115odd1268.1.f.a.47.1&check;2
      140.3odd121700.1.n.a.599.12
      140.59even61700.1.p.a.1551.12
      140.87odd121700.1.n.b.599.12
      357.353even12612.1.l.a.523.12
      476.3odd481156.1.g.b.155.18
      476.31odd481156.1.g.b.155.28
      476.47even123332.1.bc.c.863.14
      476.59even241156.1.c.b.579.12
      476.87even241156.1.d.a.1155.12
      476.115even1268.1.f.a.47.1&check;2
      476.143odd481156.1.g.b.179.28
      476.199odd481156.1.g.b.423.18
      476.227odd481156.1.g.b.399.28
      476.251even43332.1.bc.c.2223.14
      476.283odd481156.1.g.b.399.18
      476.311odd481156.1.g.b.423.28
      476.319odd123332.1.m.b.3039.12
      476.339even61156.1.f.b.327.12
      476.367odd481156.1.g.b.179.18
      476.387odd12inner3332.1.bc.b.863.14
      476.395even121156.1.f.b.251.12
      476.423even241156.1.d.a.1155.22
      476.451even241156.1.c.b.579.22
      595.234odd121700.1.p.a.251.12
      595.353even121700.1.n.b.999.12
      595.472even121700.1.n.a.999.12
      952.115even121088.1.p.a.319.12
      952.829odd121088.1.p.a.319.12
      1428.1067odd12612.1.l.a.523.12
      2380.1067odd121700.1.n.a.999.12
      2380.1543odd121700.1.n.b.999.12
      2380.2019even121700.1.p.a.251.12
      \n
          
      \n\n\n\n\n \n \n \n \n \n \n \n\n\n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n
              By twisted newform
      TwistMinDimCharParityOrdType
      68.1.f.a.47.1&check;2119.115odd12
      68.1.f.a.47.1&check;2476.115even12
      68.1.f.a.55.1yes27.3odd6
      68.1.f.a.55.1yes228.3even6
      612.1.l.a.55.1221.17even6
      612.1.l.a.55.1284.59odd6
      612.1.l.a.523.12357.353even12
      612.1.l.a.523.121428.1067odd12
      1088.1.p.a.191.1256.3even6
      1088.1.p.a.191.1256.45odd6
      1088.1.p.a.319.12952.115even12
      1088.1.p.a.319.12952.829odd12
      1156.1.c.b.579.12119.59odd24
      1156.1.c.b.579.12476.59even24
      1156.1.c.b.579.22119.94odd24
      1156.1.c.b.579.22476.451even24
      1156.1.d.a.1155.12119.87odd24
      1156.1.d.a.1155.12476.87even24
      1156.1.d.a.1155.22119.66odd24
      1156.1.d.a.1155.22476.423even24
      1156.1.f.b.251.12119.38odd12
      1156.1.f.b.251.12476.395even12
      1156.1.f.b.327.12119.101odd6
      1156.1.f.b.327.12476.339even6
      1156.1.g.b.155.18119.3even48
      1156.1.g.b.155.18476.3odd48
      1156.1.g.b.155.28119.31even48
      1156.1.g.b.155.28476.31odd48
      1156.1.g.b.179.18119.10even48
      1156.1.g.b.179.18476.367odd48
      1156.1.g.b.179.28119.24even48
      1156.1.g.b.179.28476.143odd48
      1156.1.g.b.399.18119.45even48
      1156.1.g.b.399.18476.283odd48
      1156.1.g.b.399.28119.108even48
      1156.1.g.b.399.28476.227odd48
      1156.1.g.b.423.18119.80even48
      1156.1.g.b.423.18476.199odd48
      1156.1.g.b.423.28119.73even48
      1156.1.g.b.423.28476.311odd48
      1700.1.n.a.599.1235.3even12
      1700.1.n.a.599.12140.3odd12
      1700.1.n.a.999.12595.472even12
      1700.1.n.a.999.122380.1067odd12
      1700.1.n.b.599.1235.17even12
      1700.1.n.b.599.12140.87odd12
      1700.1.n.b.999.12595.353even12
      1700.1.n.b.999.122380.1543odd12
      1700.1.p.a.251.12595.234odd12
      1700.1.p.a.251.122380.2019even12
      1700.1.p.a.1551.1235.24odd6
      1700.1.p.a.1551.12140.59even6
      3332.1.m.b.2843.127.4even3
      3332.1.m.b.2843.1228.11odd6
      3332.1.m.b.3039.12119.81even12
      3332.1.m.b.3039.12476.319odd12
      3332.1.bc.b.667.147.2even3inner
      3332.1.bc.b.667.1428.23odd6inner
      3332.1.bc.b.863.14119.30even12inner
      3332.1.bc.b.863.14476.387odd12inner
      3332.1.bc.b.2027.141.1even1trivial
      3332.1.bc.b.2027.144.3odd2CM
      3332.1.bc.b.2223.1417.13even4inner
      3332.1.bc.b.2223.1468.47odd4inner
      3332.1.bc.c.667.147.5odd6
      3332.1.bc.c.667.1428.19even6
      3332.1.bc.c.863.14119.47odd12
      3332.1.bc.c.863.14476.47even12
      3332.1.bc.c.2027.147.6odd2
      3332.1.bc.c.2027.1428.27even2
      3332.1.bc.c.2223.14119.13odd4
      3332.1.bc.c.2223.14476.251even4
      \n
      \n\n\n\n

      \n
      \n", "content_list": [[{"type": "paragraph", "raw_content": "
      \n Show commands:\n Magma\n / PariGP\n / SageMath
      ", "content": [{"c": "Show commands: Magma/ PariGP/ SageMath", "t": "text"}]}, {"type": "code", "raw_content": "
      [N,k,chi] = [3332,1,Mod(667,3332)]
       
      mf = mfinit([N,k,chi],0)
       
      lf = mfeigenbasis(mf)
       
      \n", "inline": false, "content": {"code_content": "[N,k,chi] = [3332,1,Mod(667,3332)]\n\nmf = mfinit([N,k,chi],0)\n\nlf = mfeigenbasis(mf)", "by": "classname"}}, {"type": "code", "raw_content": "
      from sage.modular.dirichlet import DirichletCharacter
       
      H = DirichletGroup(3332, base_ring=CyclotomicField(12))
       
      chi = DirichletCharacter(H, H._module([6, 4, 9]))
       
      N = Newforms(chi, 1, names=\"a\")
       
      \n", "inline": false, "content": {"code_content": "from sage.modular.dirichlet import DirichletCharacter\n\nH = DirichletGroup(3332, base_ring=CyclotomicField(12))\n\nchi = DirichletCharacter(H, H._module([6, 4, 9]))\n\nN = Newforms(chi, 1, names=\"a\")", "by": "classname"}}, {"type": "code", "raw_content": "
      //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
       
      chi := DirichletCharacter(\"3332.667\");
       
      S:= CuspForms(chi, 1);
       
      N := Newforms(S);
       
      \n\n\n", "inline": false, "content": {"code_content": "//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code\n\nchi := DirichletCharacter(\"3332.667\");\n\nS:= CuspForms(chi, 1);\n\nN := Newforms(S);", "by": "classname"}}, {"type": "simple_table", "raw_content": "
      Level: \\( N \\) \\(=\\)\\( 3332 = 2^{2} \\cdot 7^{2} \\cdot 17 \\)
      Weight: \\( k \\) \\(=\\)\\( 1 \\)
      Character orbit: \\([\\chi]\\) \\(=\\)3332.bc (of order \\(12\\), degree \\(4\\), not minimal)
      ", "content": {"html": "
      Level :$N$$=$$3332 = 2^{2} \\cdot 7^{2} \\cdot 17$
      Weight :$k$$=$$1$
      Character orbit :$[\\chi]$$=$3332.bc (oforder12, degree4, not minimal )
      ", "is_complex": false, "table_nest_level": "1"}}, {"type": "title", "raw_content": "

      Newform invariants

      ", "content": {"title_content": "Newform invariants", "level": "2"}}, {"type": "code", "raw_content": "
      sage: f = N[0] # Warning: the index may be different
       
      \n", "inline": false, "content": {"code_content": "sage: f = N[0] # Warning: the index may be different", "by": "classname"}}, {"type": "code", "raw_content": "
      gp: f = lf[1] \\\\ Warning: the index may be different
       
      \n\n\n", "inline": false, "content": {"code_content": "gp: f = lf[1] \\\\ Warning: the index may be different", "by": "classname"}}, {"type": "complex_table", "raw_content": "
      Self dual: no
      Analytic conductor: \\(1.66288462209\\)
      Analytic rank: \\(0\\)
      Dimension: \\(4\\)
      Coefficient field: \\(\\Q(\\zeta_{12})\\)
      ", "content": {"html": "
      Self dual :no
      Analytic conductor :$1.66288462209$
      Analytic rank :$0$
      Dimension :$4$
      Coefficient field :$\\Q(\\zeta_{12})$
      ", "is_complex": true, "table_nest_level": "1"}}, {"type": "code", "raw_content": "
      gp: f.mod \\\\ as an extension of the character field
       
      \n\n ", "inline": false, "content": {"code_content": "gp: f.mod \\\\ as an extension of the character field", "by": "classname"}}, {"type": "complex_table", "raw_content": "
      Defining polynomial: \\( x^{4} - x^{2} + 1 \\)\"Copy\"Toggle
      Coefficient ring: \\(\\Z[a_1, a_2]\\)
      Coefficient ring index: \\( 1 \\)
      Twist minimal: no (minimal twist has level 68)
      Projective image:\\(D_{4}\\)
      Projective field:Galois closure of 4.2.19652.1
      Artin image:$C_4\\wr C_2\\times C_6$
      Artin field:Galois closure of \\(\\mathbb{Q}[x]/(x^{48} - \\cdots)\\)
      ", "content": {"html": "
      Defining polynomial :$x^{4} - x^{2} + 1$
      Coefficient ring :$\\Z[a_1, a_2]$
      Coefficient ring index :$1$
      Twist minimal :no (minimal twist has level 68)
      Projective image :$D_{4}$
      Projective field :Galois closure of 4.2.19652.1
      Artin image :$C_4\\wr C_2\\times C_6$
      Artin field :Galois closure of $\\mathbb{Q}[x]/(x^{48} - \\cdots)$
      ", "is_complex": true, "table_nest_level": "1"}}, {"type": "title", "raw_content": "

      Embedding invariants

      ", "content": {"title_content": "Embedding invariants", "level": "2"}}, {"type": "simple_table", "raw_content": "
      Embedding label 2027.1
      Root\\(0.866025 - 0.500000i\\) of defining polynomial
      Character\\(\\chi\\)\\(=\\)3332.2027
      Dual form3332.1.bc.b.863.1
      ", "content": {"html": "
      Embedding label2027.1
      Root$0.866025 - 0.500000i$ of defining polynomial
      Character$\\chi$$=$3332.2027
      Dual form3332.1.bc.b.863.1
      ", "is_complex": false, "table_nest_level": "1"}}, {"type": "code", "raw_content": "
      sage: f.q_expansion() # note that sage often uses an isomorphic number field
       
      \n", "inline": false, "content": {"code_content": "sage: f.q_expansion() # note that sage often uses an isomorphic number field", "by": "classname"}}, {"type": "code", "raw_content": "
      gp: mfcoefs(f, 20)
       
      \n\n ", "inline": false, "content": {"code_content": "gp: mfcoefs(f, 20)", "by": "classname"}}, {"type": "simple_table", "raw_content": "
      \\(f(q)\\)\\(=\\)\\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.36603 - 0.366025i) q^{5} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.00000i) q^{20} +(0.866025 + 0.500000i) q^{25} +(1.00000 - 1.00000i) q^{29} +(0.866025 + 0.500000i) q^{32} -1.00000i q^{34} +1.00000i q^{36} +(1.36603 + 0.366025i) q^{37} +(0.366025 - 1.36603i) q^{40} +(-1.00000 - 1.00000i) q^{41} +(1.36603 - 0.366025i) q^{45} -1.00000 q^{50} +(-0.366025 + 1.36603i) q^{58} +(0.366025 - 1.36603i) q^{61} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(-0.366025 - 1.36603i) q^{73} +(-1.36603 + 0.366025i) q^{74} +(0.366025 + 1.36603i) q^{80} +(0.500000 - 0.866025i) q^{81} +(1.36603 + 0.366025i) q^{82} +(1.00000 - 1.00000i) q^{85} +(-1.00000 + 1.00000i) q^{90} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\\)
      \\(\\operatorname{Tr}(f)(q)\\)\\(=\\)\\( 4 q + 2 q^{4} - 2 q^{5} + 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 4 q^{20} + 4 q^{29} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 2 q^{45} - 4 q^{50} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{68} - 2 q^{72} + 2 q^{73}+ \\cdots + 4 q^{97}+O(q^{100}) \\)\"Copy\"Toggle
      ", "content": {"html": "
      $f(q)$$=$$q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.36603 - 0.366025i) q^{5} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.00000i) q^{20} +(0.866025 + 0.500000i) q^{25} +(1.00000 - 1.00000i) q^{29} +(0.866025 + 0.500000i) q^{32} -1.00000i q^{34} +1.00000i q^{36} +(1.36603 + 0.366025i) q^{37} +(0.366025 - 1.36603i) q^{40} +(-1.00000 - 1.00000i) q^{41} +(1.36603 - 0.366025i) q^{45} -1.00000 q^{50} +(-0.366025 + 1.36603i) q^{58} +(0.366025 - 1.36603i) q^{61} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(-0.366025 - 1.36603i) q^{73} +(-1.36603 + 0.366025i) q^{74} +(0.366025 + 1.36603i) q^{80} +(0.500000 - 0.866025i) q^{81} +(1.36603 + 0.366025i) q^{82} +(1.00000 - 1.00000i) q^{85} +(-1.00000 + 1.00000i) q^{90} +(1.00000 - 1.00000i) q^{97} +O(q^{100})$
      $\\operatorname{Tr}(f)(q)$$=$$4 q + 2 q^{4} - 2 q^{5} + 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 4 q^{20} + 4 q^{29} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 2 q^{45} - 4 q^{50} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{68} - 2 q^{72} + 2 q^{73}+ \\cdots + 4 q^{97}+O(q^{100})$
      ", "is_complex": false, "table_nest_level": "1"}}, {"type": "title", "raw_content": "

      Character values

      ", "content": {"title_content": "Character values", "level": "2"}}, {"type": "paragraph", "raw_content": "

      We give the values of \\chi on generators for \\left(\\mathbb{Z}/3332\\mathbb{Z}\\right)^\\times.

      ", "content": [{"c": "We give the values of", "t": "text"}, {"c": "\\chi", "t": "equation-inline"}, {"c": "on generators for", "t": "text"}, {"c": "\\left(\\mathbb{Z}/3332\\mathbb{Z}\\right)^\\times", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "simple_table", "raw_content": "
      \\(n\\)\\(785\\)\\(885\\)\\(1667\\)
      \\(\\chi(n)\\)\\(e\\left(\\frac{3}{4}\\right)\\)\\(e\\left(\\frac{2}{3}\\right)\\)\\(-1\\)
      ", "content": {"html": "
      $n$$785$$885$$1667$
      $\\chi(n)$$e\\left(\\frac{3}{4}\\right)$$e\\left(\\frac{2}{3}\\right)$$-1$
      ", "is_complex": false, "table_nest_level": "1"}}, {"type": "title", "raw_content": "

      Coefficient data

      ", "content": {"title_content": "Coefficient data", "level": "2"}}, {"type": "paragraph", "raw_content": "

      For each n we display the coefficients of the q-expansion a_n, the\nSatake parameters\\alpha_p,\nand the Satake angles \\theta_p = \\textrm{Arg}(\\alpha_p).

      ", "content": [{"c": "For each", "t": "text"}, {"c": "n", "t": "equation-inline"}, {"c": "we display the coefficients of the", "t": "text"}, {"c": "q", "t": "equation-inline"}, {"c": "-expansion", "t": "text"}, {"c": "a_n", "t": "equation-inline"}, {"c": ", the Satake parameters", "t": "text"}, {"c": "\\alpha_p", "t": "equation-inline"}, {"c": ", and the Satake angles", "t": "text"}, {"c": "\\theta_p = \\textrm{Arg}(\\alpha_p)", "t": "equation-inline"}, {"c": ".", "t": "text"}]}, {"type": "paragraph", "raw_content": "", "content": [{"c": "(See", "t": "text"}, {"c": "a_n", "t": "equation-inline"}, {"c": "instead)(See", "t": "text"}, {"c": "a_n", "t": "equation-inline"}, {"c": "instead)(See", "t": "text"}, {"c": "a_n", "t": "equation-inline"}, {"c": "instead)(See only", "t": "text"}, {"c": "a_p", "t": "equation-inline"}, {"c": ")(See only", "t": "text"}, {"c": "a_p", "t": "equation-inline"}, {"c": ")(See only", "t": "text"}, {"c": "a_p", "t": "equation-inline"}, {"c": ")", "t": "text"}]}, {"type": "complex_table", "raw_content": "
      \n \\(n\\)\n \n \\(a_n\\)\n \n \\(a_n / n^{(k-1)/2}\\)\n \n \\( \\alpha_n \\)\n \n \\( \\theta_n \\)\n
      \n \\(p\\)\n \n \\(a_p\\)\n \n \\(a_p / p^{(k-1)/2}\\)\n \n \\( \\alpha_p\\)\n \n \\( \\theta_p \\)\n
      \n \\(2\\)\n \n −0.866025\n \n +\n \n 0.500000i\n −0.866025\n \n +\n \n 0.500000i
      \n \\(3\\)\n \n 0\n \n 0\n \n −0.258819\n \n −\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(4\\)\n \n 0.500000\n \n −\n \n 0.866025i\n 0.500000\n \n −\n \n 0.866025i
      \n \\(5\\)\n \n −1.36603\n \n −\n \n 0.366025i\n −1.36603\n \n −\n \n 0.366025i\n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(6\\)\n \n 0\n \n 0\n
      \n \\(7\\)\n \n 0\n \n 0\n
      \n \\(8\\)\n \n 1.00000i\n 1.00000i
      \n \\(9\\)\n \n −0.866025\n \n +\n \n 0.500000i\n −0.866025\n \n +\n \n 0.500000i
      \n \\(10\\)\n \n 1.36603\n \n −\n \n 0.366025i\n 1.36603\n \n −\n \n 0.366025i
      \n \\(11\\)\n \n 0\n \n 0\n \n 0.965926\n \n −\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(12\\)\n \n 0\n \n 0\n
      \n \\(13\\)\n \n 0\n \n 0\n \n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(14\\)\n \n 0\n \n 0\n
      \n \\(15\\)\n \n 0\n \n 0\n
      \n \\(16\\)\n \n −0.500000\n \n −\n \n 0.866025i\n −0.500000\n \n −\n \n 0.866025i
      \n \\(17\\)\n \n −0.500000\n \n +\n \n 0.866025i\n −0.500000\n \n +\n \n 0.866025i
      \n \\(18\\)\n \n 0.500000\n \n −\n \n 0.866025i\n 0.500000\n \n −\n \n 0.866025i
      \n \\(19\\)\n \n 0\n \n 0\n \n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(20\\)\n \n −1.00000\n \n +\n \n 1.00000i\n −1.00000\n \n +\n \n 1.00000i
      \n \\(21\\)\n \n 0\n \n 0\n
      \n \\(22\\)\n \n 0\n \n 0\n
      \n \\(23\\)\n \n 0\n \n 0\n \n 0.258819\n \n −\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
      \n −0.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
      \n \\(24\\)\n \n 0\n \n 0\n
      \n \\(25\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
      \n \\(26\\)\n \n 0\n \n 0\n
      \n \\(27\\)\n \n 0\n \n 0\n
      \n \\(28\\)\n \n 0\n \n 0\n
      \n \\(29\\)\n \n 1.00000\n \n −\n \n 1.00000i\n 1.00000\n \n −\n \n 1.00000i\n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n \\(0\\)\n
      \n \\(30\\)\n \n 0\n \n 0\n
      \n \\(31\\)\n \n 0\n \n 0\n \n −0.258819\n \n −\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(32\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
      \n \\(33\\)\n \n 0\n \n 0\n
      \n \\(34\\)\n \n −\n \n 1.00000i\n −\n \n 1.00000i
      \n \\(35\\)\n \n 0\n \n 0\n
      \n \\(36\\)\n \n 1.00000i\n 1.00000i
      \n \\(37\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(38\\)\n \n 0\n \n 0\n
      \n \\(39\\)\n \n 0\n \n 0\n
      \n \\(40\\)\n \n 0.366025\n \n −\n \n 1.36603i\n 0.366025\n \n −\n \n 1.36603i
      \n \\(41\\)\n \n −1.00000\n \n −\n \n 1.00000i\n −1.00000\n \n −\n \n 1.00000i\n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n −1.00000\n \n \\(\\pi\\)\n
      \n \\(42\\)\n \n 0\n \n 0\n
      \n \\(43\\)\n \n 0\n \n 0\n \n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(44\\)\n \n 0\n \n 0\n
      \n \\(45\\)\n \n 1.36603\n \n −\n \n 0.366025i\n 1.36603\n \n −\n \n 0.366025i
      \n \\(46\\)\n \n 0\n \n 0\n
      \n \\(47\\)\n \n 0\n \n 0\n \n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(48\\)\n \n 0\n \n 0\n
      \n \\(49\\)\n \n 0\n \n 0\n
      \n \\(50\\)\n \n −1.00000\n \n −1.00000\n
      \n \\(51\\)\n \n 0\n \n 0\n
      \n \\(52\\)\n \n 0\n \n 0\n
      \n \\(53\\)\n \n 0\n \n 0\n \n 0.500000\n \n −\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(54\\)\n \n 0\n \n 0\n
      \n \\(55\\)\n \n 0\n \n 0\n
      \n \\(56\\)\n \n 0\n \n 0\n
      \n \\(57\\)\n \n 0\n \n 0\n
      \n \\(58\\)\n \n −0.366025\n \n +\n \n 1.36603i\n −0.366025\n \n +\n \n 1.36603i
      \n \\(59\\)\n \n 0\n \n 0\n \n −0.866025\n \n −\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(60\\)\n \n 0\n \n 0\n
      \n \\(61\\)\n \n 0.366025\n \n −\n \n 1.36603i\n 0.366025\n \n −\n \n 1.36603i\n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \\(62\\)\n \n 0\n \n 0\n
      \n \\(63\\)\n \n 0\n \n 0\n
      \n \\(64\\)\n \n −1.00000\n \n −1.00000\n
      \n \\(65\\)\n \n 0\n \n 0\n
      \n \\(66\\)\n \n 0\n \n 0\n
      \n \\(67\\)\n \n 0\n \n 0\n \n 0.500000\n \n −\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(68\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
      \n \\(69\\)\n \n 0\n \n 0\n
      \n \\(70\\)\n \n 0\n \n 0\n
      \n \\(71\\)\n \n 0\n \n 0\n \n 0.707107\n \n −\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(72\\)\n \n −0.500000\n \n −\n \n 0.866025i\n −0.500000\n \n −\n \n 0.866025i
      \n \\(73\\)\n \n −0.366025\n \n −\n \n 1.36603i\n −0.366025\n \n −\n \n 1.36603i\n −0.866025\n \n −\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.500000\n \n −\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \\(74\\)\n \n −1.36603\n \n +\n \n 0.366025i\n −1.36603\n \n +\n \n 0.366025i
      \n \\(75\\)\n \n 0\n \n 0\n
      \n \\(76\\)\n \n 0\n \n 0\n
      \n \\(77\\)\n \n 0\n \n 0\n
      \n \\(78\\)\n \n 0\n \n 0\n
      \n \\(79\\)\n \n 0\n \n 0\n \n 0.258819\n \n −\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
      \n −0.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
      \n \\(80\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(81\\)\n \n 0.500000\n \n −\n \n 0.866025i\n 0.500000\n \n −\n \n 0.866025i
      \n \\(82\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(83\\)\n \n 0\n \n 0\n \n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(84\\)\n \n 0\n \n 0\n
      \n \\(85\\)\n \n 1.00000\n \n −\n \n 1.00000i\n 1.00000\n \n −\n \n 1.00000i
      \n \\(86\\)\n \n 0\n \n 0\n
      \n \\(87\\)\n \n 0\n \n 0\n
      \n \\(88\\)\n \n 0\n \n 0\n
      \n \\(89\\)\n \n 0\n \n 0\n \n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(90\\)\n \n −1.00000\n \n +\n \n 1.00000i\n −1.00000\n \n +\n \n 1.00000i
      \n \\(91\\)\n \n 0\n \n 0\n
      \n \\(92\\)\n \n 0\n \n 0\n
      \n \\(93\\)\n \n 0\n \n 0\n
      \n \\(94\\)\n \n 0\n \n 0\n
      \n \\(95\\)\n \n 0\n \n 0\n
      \n \\(96\\)\n \n 0\n \n 0\n
      \n \\(97\\)\n \n 1.00000\n \n −\n \n 1.00000i\n 1.00000\n \n −\n \n 1.00000i\n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n \\(0\\)\n
      \n \\(98\\)\n \n 0\n \n 0\n
      \n \\(99\\)\n \n 0\n \n 0\n
      \n \\(100\\)\n \n 0.866025\n \n −\n \n 0.500000i\n 0.866025\n \n −\n \n 0.500000i
      \n \\(101\\)\n \n 0\n \n 0\n \n −0.866025\n \n −\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(102\\)\n \n 0\n \n 0\n
      \n \\(103\\)\n \n 0\n \n 0\n \n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(104\\)\n \n 0\n \n 0\n
      \n \\(105\\)\n \n 0\n \n 0\n
      \n \\(106\\)\n \n 0\n \n 0\n
      \n \\(107\\)\n \n 0\n \n 0\n \n −0.965926\n \n −\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(108\\)\n \n 0\n \n 0\n
      \n \\(109\\)\n \n 1.36603\n \n −\n \n 0.366025i\n 1.36603\n \n −\n \n 0.366025i\n 0.500000\n \n −\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(110\\)\n \n 0\n \n 0\n
      \n \\(111\\)\n \n 0\n \n 0\n
      \n \\(112\\)\n \n 0\n \n 0\n
      \n \\(113\\)\n \n −1.00000\n \n −\n \n 1.00000i\n −1.00000\n \n −\n \n 1.00000i\n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n −1.00000\n \n \\(\\pi\\)\n
      \n \\(114\\)\n \n 0\n \n 0\n
      \n \\(115\\)\n \n 0\n \n 0\n
      \n \\(116\\)\n \n −0.366025\n \n −\n \n 1.36603i\n −0.366025\n \n −\n \n 1.36603i
      \n \\(117\\)\n \n 0\n \n 0\n
      \n \\(118\\)\n \n 0\n \n 0\n
      \n \\(119\\)\n \n 0\n \n 0\n
      \n \\(120\\)\n \n 0\n \n 0\n
      \n \\(121\\)\n \n 0.866025\n \n −\n \n 0.500000i\n 0.866025\n \n −\n \n 0.500000i
      \n \\(122\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(123\\)\n \n 0\n \n 0\n
      \n \\(124\\)\n \n 0\n \n 0\n
      \n \\(125\\)\n \n 0\n \n 0\n
      \n \\(126\\)\n \n 0\n \n 0\n
      \n \\(127\\)\n \n 0\n \n 0\n \n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(128\\)\n \n 0.866025\n \n −\n \n 0.500000i\n 0.866025\n \n −\n \n 0.500000i
      \n \\(129\\)\n \n 0\n \n 0\n
      \n \\(130\\)\n \n 0\n \n 0\n
      \n \\(131\\)\n \n 0\n \n 0\n \n −0.965926\n \n −\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(132\\)\n \n 0\n \n 0\n
      \n \\(133\\)\n \n 0\n \n 0\n
      \n \\(134\\)\n \n 0\n \n 0\n
      \n \\(135\\)\n \n 0\n \n 0\n
      \n \\(136\\)\n \n −0.866025\n \n −\n \n 0.500000i\n −0.866025\n \n −\n \n 0.500000i
      \n \\(137\\)\n \n 0\n \n 0\n \n −0.866025\n \n −\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(138\\)\n \n 0\n \n 0\n
      \n \\(139\\)\n \n 0\n \n 0\n \n 0.707107\n \n −\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(140\\)\n \n 0\n \n 0\n
      \n \\(141\\)\n \n 0\n \n 0\n
      \n \\(142\\)\n \n 0\n \n 0\n
      \n \\(143\\)\n \n 0\n \n 0\n
      \n \\(144\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
      \n \\(145\\)\n \n −1.73205\n \n +\n \n 1.00000i\n −1.73205\n \n +\n \n 1.00000i
      \n \\(146\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(147\\)\n \n 0\n \n 0\n
      \n \\(148\\)\n \n 1.00000\n \n −\n \n 1.00000i\n 1.00000\n \n −\n \n 1.00000i
      \n \\(149\\)\n \n −1.00000\n \n −\n \n 1.73205i\n −1.00000\n \n −\n \n 1.73205i\n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n \\(150\\)\n \n 0\n \n 0\n
      \n \\(151\\)\n \n 0\n \n 0\n \n −0.866025\n \n −\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(152\\)\n \n 0\n \n 0\n
      \n \\(153\\)\n \n −\n \n 1.00000i\n −\n \n 1.00000i
      \n \\(154\\)\n \n 0\n \n 0\n
      \n \\(155\\)\n \n 0\n \n 0\n
      \n \\(156\\)\n \n 0\n \n 0\n
      \n \\(157\\)\n \n 1.00000\n \n −\n \n 1.73205i\n 1.00000\n \n −\n \n 1.73205i\n 0.500000\n \n −\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.500000\n \n −\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \\(158\\)\n \n 0\n \n 0\n
      \n \\(159\\)\n \n 0\n \n 0\n
      \n \\(160\\)\n \n −1.00000\n \n −\n \n 1.00000i\n −1.00000\n \n −\n \n 1.00000i
      \n \\(161\\)\n \n 0\n \n 0\n
      \n \\(162\\)\n \n 1.00000i\n 1.00000i
      \n \\(163\\)\n \n 0\n \n 0\n \n 0.258819\n \n −\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
      \n −0.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
      \n \\(164\\)\n \n −1.36603\n \n +\n \n 0.366025i\n −1.36603\n \n +\n \n 0.366025i
      \n \\(165\\)\n \n 0\n \n 0\n
      \n \\(166\\)\n \n 0\n \n 0\n
      \n \\(167\\)\n \n 0\n \n 0\n \n 0.707107\n \n −\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(168\\)\n \n 0\n \n 0\n
      \n \\(169\\)\n \n −1.00000\n \n −1.00000\n
      \n \\(170\\)\n \n −0.366025\n \n +\n \n 1.36603i\n −0.366025\n \n +\n \n 1.36603i
      \n \\(171\\)\n \n 0\n \n 0\n
      \n \\(172\\)\n \n 0\n \n 0\n
      \n \\(173\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(174\\)\n \n 0\n \n 0\n
      \n \\(175\\)\n \n 0\n \n 0\n
      \n \\(176\\)\n \n 0\n \n 0\n
      \n \\(177\\)\n \n 0\n \n 0\n
      \n \\(178\\)\n \n 0\n \n 0\n
      \n \\(179\\)\n \n 0\n \n 0\n \n −0.866025\n \n −\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(180\\)\n \n 0.366025\n \n −\n \n 1.36603i\n 0.366025\n \n −\n \n 1.36603i
      \n \\(181\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n \\(0\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(182\\)\n \n 0\n \n 0\n
      \n \\(183\\)\n \n 0\n \n 0\n
      \n \\(184\\)\n \n 0\n \n 0\n
      \n \\(185\\)\n \n −1.73205\n \n −\n \n 1.00000i\n −1.73205\n \n −\n \n 1.00000i
      \n \\(186\\)\n \n 0\n \n 0\n
      \n \\(187\\)\n \n 0\n \n 0\n
      \n \\(188\\)\n \n 0\n \n 0\n
      \n \\(189\\)\n \n 0\n \n 0\n
      \n \\(190\\)\n \n 0\n \n 0\n
      \n \\(191\\)\n \n 0\n \n 0\n \n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(192\\)\n \n 0\n \n 0\n
      \n \\(193\\)\n \n −1.36603\n \n +\n \n 0.366025i\n −1.36603\n \n +\n \n 0.366025i\n −0.866025\n \n −\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(194\\)\n \n −0.366025\n \n +\n \n 1.36603i\n −0.366025\n \n +\n \n 1.36603i
      \n \\(195\\)\n \n 0\n \n 0\n
      \n \\(196\\)\n \n 0\n \n 0\n
      \n \\(197\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n \\(0\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(198\\)\n \n 0\n \n 0\n
      \n \\(199\\)\n \n 0\n \n 0\n \n −0.258819\n \n −\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(200\\)\n \n −0.500000\n \n +\n \n 0.866025i\n −0.500000\n \n +\n \n 0.866025i
      \n \\(201\\)\n \n 0\n \n 0\n
      \n \\(202\\)\n \n 0\n \n 0\n
      \n \\(203\\)\n \n 0\n \n 0\n
      \n \\(204\\)\n \n 0\n \n 0\n
      \n \\(205\\)\n \n 1.00000\n \n +\n \n 1.73205i\n 1.00000\n \n +\n \n 1.73205i
      \n \\(206\\)\n \n 0\n \n 0\n
      \n \\(207\\)\n \n 0\n \n 0\n
      \n \\(208\\)\n \n 0\n \n 0\n
      \n \\(209\\)\n \n 0\n \n 0\n
      \n \\(210\\)\n \n 0\n \n 0\n
      \n \\(211\\)\n \n 0\n \n 0\n \n −0.707107\n \n −\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
      \n \\(212\\)\n \n 0\n \n 0\n
      \n \\(213\\)\n \n 0\n \n 0\n
      \n \\(214\\)\n \n 0\n \n 0\n
      \n \\(215\\)\n \n 0\n \n 0\n
      \n \\(216\\)\n \n 0\n \n 0\n
      \n \\(217\\)\n \n 0\n \n 0\n
      \n \\(218\\)\n \n −1.00000\n \n +\n \n 1.00000i\n −1.00000\n \n +\n \n 1.00000i
      \n \\(219\\)\n \n 0\n \n 0\n
      \n \\(220\\)\n \n 0\n \n 0\n
      \n \\(221\\)\n \n 0\n \n 0\n
      \n \\(222\\)\n \n 0\n \n 0\n
      \n \\(223\\)\n \n 0\n \n 0\n \n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(224\\)\n \n 0\n \n 0\n
      \n \\(225\\)\n \n −1.00000\n \n −1.00000\n
      \n \\(226\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(227\\)\n \n 0\n \n 0\n \n 0.965926\n \n −\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(228\\)\n \n 0\n \n 0\n
      \n \\(229\\)\n \n −1.73205\n \n +\n \n 1.00000i\n −1.73205\n \n +\n \n 1.00000i\n −0.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(230\\)\n \n 0\n \n 0\n
      \n \\(231\\)\n \n 0\n \n 0\n
      \n \\(232\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(233\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(234\\)\n \n 0\n \n 0\n
      \n \\(235\\)\n \n 0\n \n 0\n
      \n \\(236\\)\n \n 0\n \n 0\n
      \n \\(237\\)\n \n 0\n \n 0\n
      \n \\(238\\)\n \n 0\n \n 0\n
      \n \\(239\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n −1.00000\n \n \\(\\pi\\)\n
      \n \\(240\\)\n \n 0\n \n 0\n
      \n \\(241\\)\n \n −0.366025\n \n −\n \n 1.36603i\n −0.366025\n \n −\n \n 1.36603i\n −0.866025\n \n −\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.500000\n \n −\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \\(242\\)\n \n −0.500000\n \n +\n \n 0.866025i\n −0.500000\n \n +\n \n 0.866025i
      \n \\(243\\)\n \n 0\n \n 0\n
      \n \\(244\\)\n \n −1.00000\n \n −\n \n 1.00000i\n −1.00000\n \n −\n \n 1.00000i
      \n \\(245\\)\n \n 0\n \n 0\n
      \n \\(246\\)\n \n 0\n \n 0\n
      \n \\(247\\)\n \n 0\n \n 0\n
      \n \\(248\\)\n \n 0\n \n 0\n
      \n \\(249\\)\n \n 0\n \n 0\n
      \n \\(250\\)\n \n 0\n \n 0\n
      \n \\(251\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n −1.00000\n \n \\(\\pi\\)\n
      \n \\(252\\)\n \n 0\n \n 0\n
      \n \\(253\\)\n \n 0\n \n 0\n
      \n \\(254\\)\n \n 0\n \n 0\n
      \n \\(255\\)\n \n 0\n \n 0\n
      \n \\(256\\)\n \n −0.500000\n \n +\n \n 0.866025i\n −0.500000\n \n +\n \n 0.866025i
      \n \\(257\\)\n \n 1.73205\n \n −\n \n 1.00000i\n 1.73205\n \n −\n \n 1.00000i\n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \\(258\\)\n \n 0\n \n 0\n
      \n \\(259\\)\n \n 0\n \n 0\n
      \n \\(260\\)\n \n 0\n \n 0\n
      \n \\(261\\)\n \n −0.366025\n \n +\n \n 1.36603i\n −0.366025\n \n +\n \n 1.36603i
      \n \\(262\\)\n \n 0\n \n 0\n
      \n \\(263\\)\n \n 0\n \n 0\n \n −0.866025\n \n −\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(264\\)\n \n 0\n \n 0\n
      \n \\(265\\)\n \n 0\n \n 0\n
      \n \\(266\\)\n \n 0\n \n 0\n
      \n \\(267\\)\n \n 0\n \n 0\n
      \n \\(268\\)\n \n 0\n \n 0\n
      \n \\(269\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i\n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(270\\)\n \n 0\n \n 0\n
      \n \\(271\\)\n \n 0\n \n 0\n \n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(272\\)\n \n 1.00000\n \n 1.00000\n
      \n \\(273\\)\n \n 0\n \n 0\n
      \n \\(274\\)\n \n 0\n \n 0\n
      \n \\(275\\)\n \n 0\n \n 0\n
      \n \\(276\\)\n \n 0\n \n 0\n
      \n \\(277\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i\n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(278\\)\n \n 0\n \n 0\n
      \n \\(279\\)\n \n 0\n \n 0\n
      \n \\(280\\)\n \n 0\n \n 0\n
      \n \\(281\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n −1.00000\n \n \\(\\pi\\)\n
      \n \\(282\\)\n \n 0\n \n 0\n
      \n \\(283\\)\n \n 0\n \n 0\n \n 0.965926\n \n −\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(284\\)\n \n 0\n \n 0\n
      \n \\(285\\)\n \n 0\n \n 0\n
      \n \\(286\\)\n \n 0\n \n 0\n
      \n \\(287\\)\n \n 0\n \n 0\n
      \n \\(288\\)\n \n −1.00000\n \n −1.00000\n
      \n \\(289\\)\n \n −0.500000\n \n −\n \n 0.866025i\n −0.500000\n \n −\n \n 0.866025i
      \n \\(290\\)\n \n 1.00000\n \n −\n \n 1.73205i\n 1.00000\n \n −\n \n 1.73205i
      \n \\(291\\)\n \n 0\n \n 0\n
      \n \\(292\\)\n \n −1.36603\n \n −\n \n 0.366025i\n −1.36603\n \n −\n \n 0.366025i
      \n \\(293\\)\n \n 2.00000\n \n 2.00000\n \n 1.00000\n \n \\(0\\)\n
      \n 1.00000\n \n \\(0\\)\n
      \n \\(294\\)\n \n 0\n \n 0\n
      \n \\(295\\)\n \n 0\n \n 0\n
      \n \\(296\\)\n \n −0.366025\n \n +\n \n 1.36603i\n −0.366025\n \n +\n \n 1.36603i
      \n \\(297\\)\n \n 0\n \n 0\n
      \n \\(298\\)\n \n 1.73205\n \n +\n \n 1.00000i\n 1.73205\n \n +\n \n 1.00000i
      \n \\(299\\)\n \n 0\n \n 0\n
      \n \\(300\\)\n \n 0\n \n 0\n
      \n \\(301\\)\n \n 0\n \n 0\n
      \n \\(302\\)\n \n 0\n \n 0\n
      \n \\(303\\)\n \n 0\n \n 0\n
      \n \\(304\\)\n \n 0\n \n 0\n
      \n \\(305\\)\n \n −1.00000\n \n +\n \n 1.73205i\n −1.00000\n \n +\n \n 1.73205i
      \n \\(306\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
      \n \\(307\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n −1.00000\n \n \\(\\pi\\)\n
      \n \\(308\\)\n \n 0\n \n 0\n
      \n \\(309\\)\n \n 0\n \n 0\n
      \n \\(310\\)\n \n 0\n \n 0\n
      \n \\(311\\)\n \n 0\n \n 0\n \n −0.258819\n \n −\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(312\\)\n \n 0\n \n 0\n
      \n \\(313\\)\n \n −0.366025\n \n +\n \n 1.36603i\n −0.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(314\\)\n \n 2.00000i\n 2.00000i
      \n \\(315\\)\n \n 0\n \n 0\n
      \n \\(316\\)\n \n 0\n \n 0\n
      \n \\(317\\)\n \n 0.366025\n \n −\n \n 1.36603i\n 0.366025\n \n −\n \n 1.36603i\n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \\(318\\)\n \n 0\n \n 0\n
      \n \\(319\\)\n \n 0\n \n 0\n
      \n \\(320\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(321\\)\n \n 0\n \n 0\n
      \n \\(322\\)\n \n 0\n \n 0\n
      \n \\(323\\)\n \n 0\n \n 0\n
      \n \\(324\\)\n \n −0.500000\n \n −\n \n 0.866025i\n −0.500000\n \n −\n \n 0.866025i
      \n \\(325\\)\n \n 0\n \n 0\n
      \n \\(326\\)\n \n 0\n \n 0\n
      \n \\(327\\)\n \n 0\n \n 0\n
      \n \\(328\\)\n \n 1.00000\n \n −\n \n 1.00000i\n 1.00000\n \n −\n \n 1.00000i
      \n \\(329\\)\n \n 0\n \n 0\n
      \n \\(330\\)\n \n 0\n \n 0\n
      \n \\(331\\)\n \n 0\n \n 0\n \n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(332\\)\n \n 0\n \n 0\n
      \n \\(333\\)\n \n −1.36603\n \n +\n \n 0.366025i\n −1.36603\n \n +\n \n 0.366025i
      \n \\(334\\)\n \n 0\n \n 0\n
      \n \\(335\\)\n \n 0\n \n 0\n
      \n \\(336\\)\n \n 0\n \n 0\n
      \n \\(337\\)\n \n 1.00000\n \n −\n \n 1.00000i\n 1.00000\n \n −\n \n 1.00000i\n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n \\(0\\)\n
      \n \\(338\\)\n \n 0.866025\n \n −\n \n 0.500000i\n 0.866025\n \n −\n \n 0.500000i
      \n \\(339\\)\n \n 0\n \n 0\n
      \n \\(340\\)\n \n −0.366025\n \n −\n \n 1.36603i\n −0.366025\n \n −\n \n 1.36603i
      \n \\(341\\)\n \n 0\n \n 0\n
      \n \\(342\\)\n \n 0\n \n 0\n
      \n \\(343\\)\n \n 0\n \n 0\n
      \n \\(344\\)\n \n 0\n \n 0\n
      \n \\(345\\)\n \n 0\n \n 0\n
      \n \\(346\\)\n \n −1.36603\n \n +\n \n 0.366025i\n −1.36603\n \n +\n \n 0.366025i
      \n \\(347\\)\n \n 0\n \n 0\n \n 0.965926\n \n −\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(348\\)\n \n 0\n \n 0\n
      \n \\(349\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n −1.00000\n \n \\(\\pi\\)\n
      \n \\(350\\)\n \n 0\n \n 0\n
      \n \\(351\\)\n \n 0\n \n 0\n
      \n \\(352\\)\n \n 0\n \n 0\n
      \n \\(353\\)\n \n −1.00000\n \n +\n \n 1.73205i\n −1.00000\n \n +\n \n 1.73205i\n −0.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(354\\)\n \n 0\n \n 0\n
      \n \\(355\\)\n \n 0\n \n 0\n
      \n \\(356\\)\n \n 0\n \n 0\n
      \n \\(357\\)\n \n 0\n \n 0\n
      \n \\(358\\)\n \n 0\n \n 0\n
      \n \\(359\\)\n \n 0\n \n 0\n \n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(360\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(361\\)\n \n 0.500000\n \n −\n \n 0.866025i\n 0.500000\n \n −\n \n 0.866025i
      \n \\(362\\)\n \n −1.36603\n \n −\n \n 0.366025i\n −1.36603\n \n −\n \n 0.366025i
      \n \\(363\\)\n \n 0\n \n 0\n
      \n \\(364\\)\n \n 0\n \n 0\n
      \n \\(365\\)\n \n 2.00000i\n 2.00000i
      \n \\(366\\)\n \n 0\n \n 0\n
      \n \\(367\\)\n \n 0\n \n 0\n \n 0.965926\n \n −\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(368\\)\n \n 0\n \n 0\n
      \n \\(369\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(370\\)\n \n 2.00000\n \n 2.00000\n
      \n \\(371\\)\n \n 0\n \n 0\n
      \n \\(372\\)\n \n 0\n \n 0\n
      \n \\(373\\)\n \n 0\n \n 0\n \n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(374\\)\n \n 0\n \n 0\n
      \n \\(375\\)\n \n 0\n \n 0\n
      \n \\(376\\)\n \n 0\n \n 0\n
      \n \\(377\\)\n \n 0\n \n 0\n
      \n \\(378\\)\n \n 0\n \n 0\n
      \n \\(379\\)\n \n 0\n \n 0\n \n 0.707107\n \n −\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(380\\)\n \n 0\n \n 0\n
      \n \\(381\\)\n \n 0\n \n 0\n
      \n \\(382\\)\n \n 0\n \n 0\n
      \n \\(383\\)\n \n 0\n \n 0\n \n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(384\\)\n \n 0\n \n 0\n
      \n \\(385\\)\n \n 0\n \n 0\n
      \n \\(386\\)\n \n 1.00000\n \n −\n \n 1.00000i\n 1.00000\n \n −\n \n 1.00000i
      \n \\(387\\)\n \n 0\n \n 0\n
      \n \\(388\\)\n \n −0.366025\n \n −\n \n 1.36603i\n −0.366025\n \n −\n \n 1.36603i
      \n \\(389\\)\n \n −1.73205\n \n −\n \n 1.00000i\n −1.73205\n \n −\n \n 1.00000i\n −0.866025\n \n −\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n −0.866025\n \n −\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n \\(390\\)\n \n 0\n \n 0\n
      \n \\(391\\)\n \n 0\n \n 0\n
      \n \\(392\\)\n \n 0\n \n 0\n
      \n \\(393\\)\n \n 0\n \n 0\n
      \n \\(394\\)\n \n −1.36603\n \n −\n \n 0.366025i\n −1.36603\n \n −\n \n 0.366025i
      \n \\(395\\)\n \n 0\n \n 0\n
      \n \\(396\\)\n \n 0\n \n 0\n
      \n \\(397\\)\n \n 0.366025\n \n −\n \n 1.36603i\n 0.366025\n \n −\n \n 1.36603i\n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \\(398\\)\n \n 0\n \n 0\n
      \n \\(399\\)\n \n 0\n \n 0\n
      \n \\(400\\)\n \n −\n \n 1.00000i\n −\n \n 1.00000i
      \n \\(401\\)\n \n 0.366025\n \n −\n \n 1.36603i\n 0.366025\n \n −\n \n 1.36603i\n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \\(402\\)\n \n 0\n \n 0\n
      \n \\(403\\)\n \n 0\n \n 0\n
      \n \\(404\\)\n \n 0\n \n 0\n
      \n \\(405\\)\n \n −1.00000\n \n +\n \n 1.00000i\n −1.00000\n \n +\n \n 1.00000i
      \n \\(406\\)\n \n 0\n \n 0\n
      \n \\(407\\)\n \n 0\n \n 0\n
      \n \\(408\\)\n \n 0\n \n 0\n
      \n \\(409\\)\n \n −1.00000\n \n +\n \n 1.73205i\n −1.00000\n \n +\n \n 1.73205i\n −0.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(410\\)\n \n −1.73205\n \n −\n \n 1.00000i\n −1.73205\n \n −\n \n 1.00000i
      \n \\(411\\)\n \n 0\n \n 0\n
      \n \\(412\\)\n \n 0\n \n 0\n
      \n \\(413\\)\n \n 0\n \n 0\n
      \n \\(414\\)\n \n 0\n \n 0\n
      \n \\(415\\)\n \n 0\n \n 0\n
      \n \\(416\\)\n \n 0\n \n 0\n
      \n \\(417\\)\n \n 0\n \n 0\n
      \n \\(418\\)\n \n 0\n \n 0\n
      \n \\(419\\)\n \n 0\n \n 0\n \n −0.707107\n \n −\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
      \n \\(420\\)\n \n 0\n \n 0\n
      \n \\(421\\)\n \n 0\n \n 0\n \n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(422\\)\n \n 0\n \n 0\n
      \n \\(423\\)\n \n 0\n \n 0\n
      \n \\(424\\)\n \n 0\n \n 0\n
      \n \\(425\\)\n \n −0.866025\n \n +\n \n 0.500000i\n −0.866025\n \n +\n \n 0.500000i
      \n \\(426\\)\n \n 0\n \n 0\n
      \n \\(427\\)\n \n 0\n \n 0\n
      \n \\(428\\)\n \n 0\n \n 0\n
      \n \\(429\\)\n \n 0\n \n 0\n
      \n \\(430\\)\n \n 0\n \n 0\n
      \n \\(431\\)\n \n 0\n \n 0\n \n 0.965926\n \n −\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(432\\)\n \n 0\n \n 0\n
      \n \\(433\\)\n \n 2.00000i\n 2.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(434\\)\n \n 0\n \n 0\n
      \n \\(435\\)\n \n 0\n \n 0\n
      \n \\(436\\)\n \n 0.366025\n \n −\n \n 1.36603i\n 0.366025\n \n −\n \n 1.36603i
      \n \\(437\\)\n \n 0\n \n 0\n
      \n \\(438\\)\n \n 0\n \n 0\n
      \n \\(439\\)\n \n 0\n \n 0\n \n −0.965926\n \n −\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(440\\)\n \n 0\n \n 0\n
      \n \\(441\\)\n \n 0\n \n 0\n
      \n \\(442\\)\n \n 0\n \n 0\n
      \n \\(443\\)\n \n 0\n \n 0\n \n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(444\\)\n \n 0\n \n 0\n
      \n \\(445\\)\n \n 0\n \n 0\n
      \n \\(446\\)\n \n 0\n \n 0\n
      \n \\(447\\)\n \n 0\n \n 0\n
      \n \\(448\\)\n \n 0\n \n 0\n
      \n \\(449\\)\n \n −1.00000\n \n −\n \n 1.00000i\n −1.00000\n \n −\n \n 1.00000i\n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n −1.00000\n \n \\(\\pi\\)\n
      \n \\(450\\)\n \n 0.866025\n \n −\n \n 0.500000i\n 0.866025\n \n −\n \n 0.500000i
      \n \\(451\\)\n \n 0\n \n 0\n
      \n \\(452\\)\n \n −1.36603\n \n +\n \n 0.366025i\n −1.36603\n \n +\n \n 0.366025i
      \n \\(453\\)\n \n 0\n \n 0\n
      \n \\(454\\)\n \n 0\n \n 0\n
      \n \\(455\\)\n \n 0\n \n 0\n
      \n \\(456\\)\n \n 0\n \n 0\n
      \n \\(457\\)\n \n −1.73205\n \n +\n \n 1.00000i\n −1.73205\n \n +\n \n 1.00000i\n −0.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(458\\)\n \n 1.00000\n \n −\n \n 1.73205i\n 1.00000\n \n −\n \n 1.73205i
      \n \\(459\\)\n \n 0\n \n 0\n
      \n \\(460\\)\n \n 0\n \n 0\n
      \n \\(461\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n −1.00000\n \n \\(\\pi\\)\n
      \n \\(462\\)\n \n 0\n \n 0\n
      \n \\(463\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n −1.00000\n \n \\(\\pi\\)\n
      \n \\(464\\)\n \n −1.36603\n \n −\n \n 0.366025i\n −1.36603\n \n −\n \n 0.366025i
      \n \\(465\\)\n \n 0\n \n 0\n
      \n \\(466\\)\n \n −1.36603\n \n +\n \n 0.366025i\n −1.36603\n \n +\n \n 0.366025i
      \n \\(467\\)\n \n 0\n \n 0\n \n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(468\\)\n \n 0\n \n 0\n
      \n \\(469\\)\n \n 0\n \n 0\n
      \n \\(470\\)\n \n 0\n \n 0\n
      \n \\(471\\)\n \n 0\n \n 0\n
      \n \\(472\\)\n \n 0\n \n 0\n
      \n \\(473\\)\n \n 0\n \n 0\n
      \n \\(474\\)\n \n 0\n \n 0\n
      \n \\(475\\)\n \n 0\n \n 0\n
      \n \\(476\\)\n \n 0\n \n 0\n
      \n \\(477\\)\n \n 0\n \n 0\n
      \n \\(478\\)\n \n 0\n \n 0\n
      \n \\(479\\)\n \n 0\n \n 0\n \n −0.258819\n \n −\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(480\\)\n \n 0\n \n 0\n
      \n \\(481\\)\n \n 0\n \n 0\n
      \n \\(482\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(483\\)\n \n 0\n \n 0\n
      \n \\(484\\)\n \n −\n \n 1.00000i\n −\n \n 1.00000i
      \n \\(485\\)\n \n −1.73205\n \n +\n \n 1.00000i\n −1.73205\n \n +\n \n 1.00000i
      \n \\(486\\)\n \n 0\n \n 0\n
      \n \\(487\\)\n \n 0\n \n 0\n \n 0.965926\n \n −\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(488\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(489\\)\n \n 0\n \n 0\n
      \n \\(490\\)\n \n 0\n \n 0\n
      \n \\(491\\)\n \n 0\n \n 0\n \n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(492\\)\n \n 0\n \n 0\n
      \n \\(493\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(494\\)\n \n 0\n \n 0\n
      \n \\(495\\)\n \n 0\n \n 0\n
      \n \\(496\\)\n \n 0\n \n 0\n
      \n \\(497\\)\n \n 0\n \n 0\n
      \n \\(498\\)\n \n 0\n \n 0\n
      \n \\(499\\)\n \n 0\n \n 0\n \n 0.258819\n \n −\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
      \n −0.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
      \n \\(500\\)\n \n 0\n \n 0\n
      \n \\(501\\)\n \n 0\n \n 0\n
      \n \\(502\\)\n \n 0\n \n 0\n
      \n \\(503\\)\n \n 0\n \n 0\n \n −0.707107\n \n −\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
      \n \\(504\\)\n \n 0\n \n 0\n
      \n \\(505\\)\n \n 0\n \n 0\n
      \n \\(506\\)\n \n 0\n \n 0\n
      \n \\(507\\)\n \n 0\n \n 0\n
      \n \\(508\\)\n \n 0\n \n 0\n
      \n \\(509\\)\n \n −1.00000\n \n −\n \n 1.73205i\n −1.00000\n \n −\n \n 1.73205i\n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n \\(510\\)\n \n 0\n \n 0\n
      \n \\(511\\)\n \n 0\n \n 0\n
      \n \\(512\\)\n \n −\n \n 1.00000i\n −\n \n 1.00000i
      \n \\(513\\)\n \n 0\n \n 0\n
      \n \\(514\\)\n \n −1.00000\n \n +\n \n 1.73205i\n −1.00000\n \n +\n \n 1.73205i
      \n \\(515\\)\n \n 0\n \n 0\n
      \n \\(516\\)\n \n 0\n \n 0\n
      \n \\(517\\)\n \n 0\n \n 0\n
      \n \\(518\\)\n \n 0\n \n 0\n
      \n \\(519\\)\n \n 0\n \n 0\n
      \n \\(520\\)\n \n 0\n \n 0\n
      \n \\(521\\)\n \n 1.36603\n \n −\n \n 0.366025i\n 1.36603\n \n −\n \n 0.366025i\n 0.500000\n \n −\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(522\\)\n \n −0.366025\n \n −\n \n 1.36603i\n −0.366025\n \n −\n \n 1.36603i
      \n \\(523\\)\n \n 0\n \n 0\n \n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(524\\)\n \n 0\n \n 0\n
      \n \\(525\\)\n \n 0\n \n 0\n
      \n \\(526\\)\n \n 0\n \n 0\n
      \n \\(527\\)\n \n 0\n \n 0\n
      \n \\(528\\)\n \n 0\n \n 0\n
      \n \\(529\\)\n \n −0.866025\n \n −\n \n 0.500000i\n −0.866025\n \n −\n \n 0.500000i
      \n \\(530\\)\n \n 0\n \n 0\n
      \n \\(531\\)\n \n 0\n \n 0\n
      \n \\(532\\)\n \n 0\n \n 0\n
      \n \\(533\\)\n \n 0\n \n 0\n
      \n \\(534\\)\n \n 0\n \n 0\n
      \n \\(535\\)\n \n 0\n \n 0\n
      \n \\(536\\)\n \n 0\n \n 0\n
      \n \\(537\\)\n \n 0\n \n 0\n
      \n \\(538\\)\n \n −1.00000\n \n −\n \n 1.00000i\n −1.00000\n \n −\n \n 1.00000i
      \n \\(539\\)\n \n 0\n \n 0\n
      \n \\(540\\)\n \n 0\n \n 0\n
      \n \\(541\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(542\\)\n \n 0\n \n 0\n
      \n \\(543\\)\n \n 0\n \n 0\n
      \n \\(544\\)\n \n −0.866025\n \n +\n \n 0.500000i\n −0.866025\n \n +\n \n 0.500000i
      \n \\(545\\)\n \n −2.00000\n \n −2.00000\n
      \n \\(546\\)\n \n 0\n \n 0\n
      \n \\(547\\)\n \n 0\n \n 0\n \n 0.707107\n \n −\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(548\\)\n \n 0\n \n 0\n
      \n \\(549\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(550\\)\n \n 0\n \n 0\n
      \n \\(551\\)\n \n 0\n \n 0\n
      \n \\(552\\)\n \n 0\n \n 0\n
      \n \\(553\\)\n \n 0\n \n 0\n
      \n \\(554\\)\n \n −1.00000\n \n −\n \n 1.00000i\n −1.00000\n \n −\n \n 1.00000i
      \n \\(555\\)\n \n 0\n \n 0\n
      \n \\(556\\)\n \n 0\n \n 0\n
      \n \\(557\\)\n \n 0\n \n 0\n \n −0.866025\n \n −\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(558\\)\n \n 0\n \n 0\n
      \n \\(559\\)\n \n 0\n \n 0\n
      \n \\(560\\)\n \n 0\n \n 0\n
      \n \\(561\\)\n \n 0\n \n 0\n
      \n \\(562\\)\n \n 0\n \n 0\n
      \n \\(563\\)\n \n 0\n \n 0\n \n −0.866025\n \n −\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(564\\)\n \n 0\n \n 0\n
      \n \\(565\\)\n \n 1.00000\n \n +\n \n 1.73205i\n 1.00000\n \n +\n \n 1.73205i
      \n \\(566\\)\n \n 0\n \n 0\n
      \n \\(567\\)\n \n 0\n \n 0\n
      \n \\(568\\)\n \n 0\n \n 0\n
      \n \\(569\\)\n \n 0\n \n 0\n \n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(570\\)\n \n 0\n \n 0\n
      \n \\(571\\)\n \n 0\n \n 0\n \n 0.965926\n \n −\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(572\\)\n \n 0\n \n 0\n
      \n \\(573\\)\n \n 0\n \n 0\n
      \n \\(574\\)\n \n 0\n \n 0\n
      \n \\(575\\)\n \n 0\n \n 0\n
      \n \\(576\\)\n \n 0.866025\n \n −\n \n 0.500000i\n 0.866025\n \n −\n \n 0.500000i
      \n \\(577\\)\n \n 0\n \n 0\n \n −0.866025\n \n −\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(578\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
      \n \\(579\\)\n \n 0\n \n 0\n
      \n \\(580\\)\n \n 2.00000i\n 2.00000i
      \n \\(581\\)\n \n 0\n \n 0\n
      \n \\(582\\)\n \n 0\n \n 0\n
      \n \\(583\\)\n \n 0\n \n 0\n
      \n \\(584\\)\n \n 1.36603\n \n −\n \n 0.366025i\n 1.36603\n \n −\n \n 0.366025i
      \n \\(585\\)\n \n 0\n \n 0\n
      \n \\(586\\)\n \n −1.73205\n \n +\n \n 1.00000i\n −1.73205\n \n +\n \n 1.00000i
      \n \\(587\\)\n \n 0\n \n 0\n \n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(588\\)\n \n 0\n \n 0\n
      \n \\(589\\)\n \n 0\n \n 0\n
      \n \\(590\\)\n \n 0\n \n 0\n
      \n \\(591\\)\n \n 0\n \n 0\n
      \n \\(592\\)\n \n −0.366025\n \n −\n \n 1.36603i\n −0.366025\n \n −\n \n 1.36603i
      \n \\(593\\)\n \n 0\n \n 0\n \n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(594\\)\n \n 0\n \n 0\n
      \n \\(595\\)\n \n 0\n \n 0\n
      \n \\(596\\)\n \n −2.00000\n \n −2.00000\n
      \n \\(597\\)\n \n 0\n \n 0\n
      \n \\(598\\)\n \n 0\n \n 0\n
      \n \\(599\\)\n \n 0\n \n 0\n \n 0.500000\n \n −\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(600\\)\n \n 0\n \n 0\n
      \n \\(601\\)\n \n −1.00000\n \n −\n \n 1.00000i\n −1.00000\n \n −\n \n 1.00000i\n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n −1.00000\n \n \\(\\pi\\)\n
      \n \\(602\\)\n \n 0\n \n 0\n
      \n \\(603\\)\n \n 0\n \n 0\n
      \n \\(604\\)\n \n 0\n \n 0\n
      \n \\(605\\)\n \n −1.36603\n \n +\n \n 0.366025i\n −1.36603\n \n +\n \n 0.366025i
      \n \\(606\\)\n \n 0\n \n 0\n
      \n \\(607\\)\n \n 0\n \n 0\n \n −0.965926\n \n −\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(608\\)\n \n 0\n \n 0\n
      \n \\(609\\)\n \n 0\n \n 0\n
      \n \\(610\\)\n \n −\n \n 2.00000i\n −\n \n 2.00000i
      \n \\(611\\)\n \n 0\n \n 0\n
      \n \\(612\\)\n \n −0.866025\n \n −\n \n 0.500000i\n −0.866025\n \n −\n \n 0.500000i
      \n \\(613\\)\n \n −1.00000\n \n +\n \n 1.73205i\n −1.00000\n \n +\n \n 1.73205i\n −0.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(614\\)\n \n 0\n \n 0\n
      \n \\(615\\)\n \n 0\n \n 0\n
      \n \\(616\\)\n \n 0\n \n 0\n
      \n \\(617\\)\n \n −1.00000\n \n +\n \n 1.00000i\n −1.00000\n \n +\n \n 1.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
      \n −1.00000\n \n \\(\\pi\\)\n
      \n \\(618\\)\n \n 0\n \n 0\n
      \n \\(619\\)\n \n 0\n \n 0\n \n 0.965926\n \n −\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(620\\)\n \n 0\n \n 0\n
      \n \\(621\\)\n \n 0\n \n 0\n
      \n \\(622\\)\n \n 0\n \n 0\n
      \n \\(623\\)\n \n 0\n \n 0\n
      \n \\(624\\)\n \n 0\n \n 0\n
      \n \\(625\\)\n \n −0.500000\n \n −\n \n 0.866025i\n −0.500000\n \n −\n \n 0.866025i
      \n \\(626\\)\n \n −0.366025\n \n −\n \n 1.36603i\n −0.366025\n \n −\n \n 1.36603i
      \n \\(627\\)\n \n 0\n \n 0\n
      \n \\(628\\)\n \n −1.00000\n \n −\n \n 1.73205i\n −1.00000\n \n −\n \n 1.73205i
      \n \\(629\\)\n \n −1.00000\n \n +\n \n 1.00000i\n −1.00000\n \n +\n \n 1.00000i
      \n \\(630\\)\n \n 0\n \n 0\n
      \n \\(631\\)\n \n 0\n \n 0\n \n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(632\\)\n \n 0\n \n 0\n
      \n \\(633\\)\n \n 0\n \n 0\n
      \n \\(634\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(635\\)\n \n 0\n \n 0\n
      \n \\(636\\)\n \n 0\n \n 0\n
      \n \\(637\\)\n \n 0\n \n 0\n
      \n \\(638\\)\n \n 0\n \n 0\n
      \n \\(639\\)\n \n 0\n \n 0\n
      \n \\(640\\)\n \n −1.36603\n \n +\n \n 0.366025i\n −1.36603\n \n +\n \n 0.366025i
      \n \\(641\\)\n \n −0.366025\n \n −\n \n 1.36603i\n −0.366025\n \n −\n \n 1.36603i\n −0.866025\n \n −\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.500000\n \n −\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \\(642\\)\n \n 0\n \n 0\n
      \n \\(643\\)\n \n 0\n \n 0\n \n 0.707107\n \n −\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(644\\)\n \n 0\n \n 0\n
      \n \\(645\\)\n \n 0\n \n 0\n
      \n \\(646\\)\n \n 0\n \n 0\n
      \n \\(647\\)\n \n 0\n \n 0\n \n 0.500000\n \n −\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(648\\)\n \n 0.866025\n \n +\n \n 0.500000i\n 0.866025\n \n +\n \n 0.500000i
      \n \\(649\\)\n \n 0\n \n 0\n
      \n \\(650\\)\n \n 0\n \n 0\n
      \n \\(651\\)\n \n 0\n \n 0\n
      \n \\(652\\)\n \n 0\n \n 0\n
      \n \\(653\\)\n \n 0.366025\n \n −\n \n 1.36603i\n 0.366025\n \n −\n \n 1.36603i\n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n \\(654\\)\n \n 0\n \n 0\n
      \n \\(655\\)\n \n 0\n \n 0\n
      \n \\(656\\)\n \n −0.366025\n \n +\n \n 1.36603i\n −0.366025\n \n +\n \n 1.36603i
      \n \\(657\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(658\\)\n \n 0\n \n 0\n
      \n \\(659\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n −1.00000\n \n \\(\\pi\\)\n
      \n \\(660\\)\n \n 0\n \n 0\n
      \n \\(661\\)\n \n 0\n \n 0\n \n 0.500000\n \n −\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(662\\)\n \n 0\n \n 0\n
      \n \\(663\\)\n \n 0\n \n 0\n
      \n \\(664\\)\n \n 0\n \n 0\n
      \n \\(665\\)\n \n 0\n \n 0\n
      \n \\(666\\)\n \n 1.00000\n \n −\n \n 1.00000i\n 1.00000\n \n −\n \n 1.00000i
      \n \\(667\\)\n \n 0\n \n 0\n
      \n \\(668\\)\n \n 0\n \n 0\n
      \n \\(669\\)\n \n 0\n \n 0\n
      \n \\(670\\)\n \n 0\n \n 0\n
      \n \\(671\\)\n \n 0\n \n 0\n
      \n \\(672\\)\n \n 0\n \n 0\n
      \n \\(673\\)\n \n −1.00000\n \n −\n \n 1.00000i\n −1.00000\n \n −\n \n 1.00000i\n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n −1.00000\n \n \\(\\pi\\)\n
      \n \\(674\\)\n \n −0.366025\n \n +\n \n 1.36603i\n −0.366025\n \n +\n \n 1.36603i
      \n \\(675\\)\n \n 0\n \n 0\n
      \n \\(676\\)\n \n −0.500000\n \n +\n \n 0.866025i\n −0.500000\n \n +\n \n 0.866025i
      \n \\(677\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i\n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(678\\)\n \n 0\n \n 0\n
      \n \\(679\\)\n \n 0\n \n 0\n
      \n \\(680\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(681\\)\n \n 0\n \n 0\n
      \n \\(682\\)\n \n 0\n \n 0\n
      \n \\(683\\)\n \n 0\n \n 0\n \n −0.258819\n \n −\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(684\\)\n \n 0\n \n 0\n
      \n \\(685\\)\n \n 0\n \n 0\n
      \n \\(686\\)\n \n 0\n \n 0\n
      \n \\(687\\)\n \n 0\n \n 0\n
      \n \\(688\\)\n \n 0\n \n 0\n
      \n \\(689\\)\n \n 0\n \n 0\n
      \n \\(690\\)\n \n 0\n \n 0\n
      \n \\(691\\)\n \n 0\n \n 0\n \n 0.258819\n \n −\n \n 0.965926i\n \\(-0.416667\\pi\\)\n
      \n −0.258819\n \n +\n \n 0.965926i\n \\(0.583333\\pi\\)\n
      \n \\(692\\)\n \n 1.00000\n \n −\n \n 1.00000i\n 1.00000\n \n −\n \n 1.00000i
      \n \\(693\\)\n \n 0\n \n 0\n
      \n \\(694\\)\n \n 0\n \n 0\n
      \n \\(695\\)\n \n 0\n \n 0\n
      \n \\(696\\)\n \n 0\n \n 0\n
      \n \\(697\\)\n \n 1.36603\n \n −\n \n 0.366025i\n 1.36603\n \n −\n \n 0.366025i
      \n \\(698\\)\n \n 0\n \n 0\n
      \n \\(699\\)\n \n 0\n \n 0\n
      \n \\(700\\)\n \n 0\n \n 0\n
      \n \\(701\\)\n \n 0\n \n 0\n \n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(702\\)\n \n 0\n \n 0\n
      \n \\(703\\)\n \n 0\n \n 0\n
      \n \\(704\\)\n \n 0\n \n 0\n
      \n \\(705\\)\n \n 0\n \n 0\n
      \n \\(706\\)\n \n −\n \n 2.00000i\n −\n \n 2.00000i
      \n \\(707\\)\n \n 0\n \n 0\n
      \n \\(708\\)\n \n 0\n \n 0\n
      \n \\(709\\)\n \n −1.36603\n \n −\n \n 0.366025i\n −1.36603\n \n −\n \n 0.366025i\n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(710\\)\n \n 0\n \n 0\n
      \n \\(711\\)\n \n 0\n \n 0\n
      \n \\(712\\)\n \n 0\n \n 0\n
      \n \\(713\\)\n \n 0\n \n 0\n
      \n \\(714\\)\n \n 0\n \n 0\n
      \n \\(715\\)\n \n 0\n \n 0\n
      \n \\(716\\)\n \n 0\n \n 0\n
      \n \\(717\\)\n \n 0\n \n 0\n
      \n \\(718\\)\n \n 0\n \n 0\n
      \n \\(719\\)\n \n 0\n \n 0\n \n −0.965926\n \n −\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(720\\)\n \n −1.00000\n \n −\n \n 1.00000i\n −1.00000\n \n −\n \n 1.00000i
      \n \\(721\\)\n \n 0\n \n 0\n
      \n \\(722\\)\n \n 1.00000i\n 1.00000i
      \n \\(723\\)\n \n 0\n \n 0\n
      \n \\(724\\)\n \n 1.36603\n \n −\n \n 0.366025i\n 1.36603\n \n −\n \n 0.366025i
      \n \\(725\\)\n \n 1.36603\n \n −\n \n 0.366025i\n 1.36603\n \n −\n \n 0.366025i
      \n \\(726\\)\n \n 0\n \n 0\n
      \n \\(727\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n −1.00000\n \n \\(\\pi\\)\n
      \n \\(728\\)\n \n 0\n \n 0\n
      \n \\(729\\)\n \n 1.00000i\n 1.00000i
      \n \\(730\\)\n \n −1.00000\n \n −\n \n 1.73205i\n −1.00000\n \n −\n \n 1.73205i
      \n \\(731\\)\n \n 0\n \n 0\n
      \n \\(732\\)\n \n 0\n \n 0\n
      \n \\(733\\)\n \n 0\n \n 0\n \n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(734\\)\n \n 0\n \n 0\n
      \n \\(735\\)\n \n 0\n \n 0\n
      \n \\(736\\)\n \n 0\n \n 0\n
      \n \\(737\\)\n \n 0\n \n 0\n
      \n \\(738\\)\n \n −1.36603\n \n +\n \n 0.366025i\n −1.36603\n \n +\n \n 0.366025i
      \n \\(739\\)\n \n 0\n \n 0\n \n −0.866025\n \n −\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(740\\)\n \n −1.73205\n \n +\n \n 1.00000i\n −1.73205\n \n +\n \n 1.00000i
      \n \\(741\\)\n \n 0\n \n 0\n
      \n \\(742\\)\n \n 0\n \n 0\n
      \n \\(743\\)\n \n 0\n \n 0\n \n 0.707107\n \n −\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(744\\)\n \n 0\n \n 0\n
      \n \\(745\\)\n \n 0.732051\n \n +\n \n 2.73205i\n 0.732051\n \n +\n \n 2.73205i
      \n \\(746\\)\n \n 0\n \n 0\n
      \n \\(747\\)\n \n 0\n \n 0\n
      \n \\(748\\)\n \n 0\n \n 0\n
      \n \\(749\\)\n \n 0\n \n 0\n
      \n \\(750\\)\n \n 0\n \n 0\n
      \n \\(751\\)\n \n 0\n \n 0\n \n −0.965926\n \n −\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(752\\)\n \n 0\n \n 0\n
      \n \\(753\\)\n \n 0\n \n 0\n
      \n \\(754\\)\n \n 0\n \n 0\n
      \n \\(755\\)\n \n 0\n \n 0\n
      \n \\(756\\)\n \n 0\n \n 0\n
      \n \\(757\\)\n \n 2.00000i\n 2.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(758\\)\n \n 0\n \n 0\n
      \n \\(759\\)\n \n 0\n \n 0\n
      \n \\(760\\)\n \n 0\n \n 0\n
      \n \\(761\\)\n \n 0\n \n 0\n \n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(762\\)\n \n 0\n \n 0\n
      \n \\(763\\)\n \n 0\n \n 0\n
      \n \\(764\\)\n \n 0\n \n 0\n
      \n \\(765\\)\n \n −0.366025\n \n +\n \n 1.36603i\n −0.366025\n \n +\n \n 1.36603i
      \n \\(766\\)\n \n 0\n \n 0\n
      \n \\(767\\)\n \n 0\n \n 0\n
      \n \\(768\\)\n \n 0\n \n 0\n
      \n \\(769\\)\n \n 0\n \n 0\n \n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(770\\)\n \n 0\n \n 0\n
      \n \\(771\\)\n \n 0\n \n 0\n
      \n \\(772\\)\n \n −0.366025\n \n +\n \n 1.36603i\n −0.366025\n \n +\n \n 1.36603i
      \n \\(773\\)\n \n −1.73205\n \n −\n \n 1.00000i\n −1.73205\n \n −\n \n 1.00000i\n −0.866025\n \n −\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n −0.866025\n \n −\n \n 0.500000i\n \\(-0.833333\\pi\\)\n
      \n \\(774\\)\n \n 0\n \n 0\n
      \n \\(775\\)\n \n 0\n \n 0\n
      \n \\(776\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(777\\)\n \n 0\n \n 0\n
      \n \\(778\\)\n \n 2.00000\n \n 2.00000\n
      \n \\(779\\)\n \n 0\n \n 0\n
      \n \\(780\\)\n \n 0\n \n 0\n
      \n \\(781\\)\n \n 0\n \n 0\n
      \n \\(782\\)\n \n 0\n \n 0\n
      \n \\(783\\)\n \n 0\n \n 0\n
      \n \\(784\\)\n \n 0\n \n 0\n
      \n \\(785\\)\n \n −2.00000\n \n +\n \n 2.00000i\n −2.00000\n \n +\n \n 2.00000i
      \n \\(786\\)\n \n 0\n \n 0\n
      \n \\(787\\)\n \n 0\n \n 0\n \n −0.258819\n \n −\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(788\\)\n \n 1.36603\n \n −\n \n 0.366025i\n 1.36603\n \n −\n \n 0.366025i
      \n \\(789\\)\n \n 0\n \n 0\n
      \n \\(790\\)\n \n 0\n \n 0\n
      \n \\(791\\)\n \n 0\n \n 0\n
      \n \\(792\\)\n \n 0\n \n 0\n
      \n \\(793\\)\n \n 0\n \n 0\n
      \n \\(794\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(795\\)\n \n 0\n \n 0\n
      \n \\(796\\)\n \n 0\n \n 0\n
      \n \\(797\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n −1.00000\n \n \\(\\pi\\)\n
      \n \\(798\\)\n \n 0\n \n 0\n
      \n \\(799\\)\n \n 0\n \n 0\n
      \n \\(800\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
      \n \\(801\\)\n \n 0\n \n 0\n
      \n \\(802\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(803\\)\n \n 0\n \n 0\n
      \n \\(804\\)\n \n 0\n \n 0\n
      \n \\(805\\)\n \n 0\n \n 0\n
      \n \\(806\\)\n \n 0\n \n 0\n
      \n \\(807\\)\n \n 0\n \n 0\n
      \n \\(808\\)\n \n 0\n \n 0\n
      \n \\(809\\)\n \n 1.36603\n \n −\n \n 0.366025i\n 1.36603\n \n −\n \n 0.366025i\n 0.500000\n \n −\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(810\\)\n \n 0.366025\n \n −\n \n 1.36603i\n 0.366025\n \n −\n \n 1.36603i
      \n \\(811\\)\n \n 0\n \n 0\n \n 0.707107\n \n −\n \n 0.707107i\n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \\(0.750000\\pi\\)\n
      \n \\(812\\)\n \n 0\n \n 0\n
      \n \\(813\\)\n \n 0\n \n 0\n
      \n \\(814\\)\n \n 0\n \n 0\n
      \n \\(815\\)\n \n 0\n \n 0\n
      \n \\(816\\)\n \n 0\n \n 0\n
      \n \\(817\\)\n \n 0\n \n 0\n
      \n \\(818\\)\n \n −\n \n 2.00000i\n −\n \n 2.00000i
      \n \\(819\\)\n \n 0\n \n 0\n
      \n \\(820\\)\n \n 2.00000\n \n 2.00000\n
      \n \\(821\\)\n \n −1.36603\n \n −\n \n 0.366025i\n −1.36603\n \n −\n \n 0.366025i\n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(822\\)\n \n 0\n \n 0\n
      \n \\(823\\)\n \n 0\n \n 0\n \n 0.965926\n \n −\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(824\\)\n \n 0\n \n 0\n
      \n \\(825\\)\n \n 0\n \n 0\n
      \n \\(826\\)\n \n 0\n \n 0\n
      \n \\(827\\)\n \n 0\n \n 0\n \n −0.707107\n \n −\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
      \n \\(828\\)\n \n 0\n \n 0\n
      \n \\(829\\)\n \n 1.00000\n \n −\n \n 1.73205i\n 1.00000\n \n −\n \n 1.73205i\n 0.500000\n \n −\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.500000\n \n −\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n \\(830\\)\n \n 0\n \n 0\n
      \n \\(831\\)\n \n 0\n \n 0\n
      \n \\(832\\)\n \n 0\n \n 0\n
      \n \\(833\\)\n \n 0\n \n 0\n
      \n \\(834\\)\n \n 0\n \n 0\n
      \n \\(835\\)\n \n 0\n \n 0\n
      \n \\(836\\)\n \n 0\n \n 0\n
      \n \\(837\\)\n \n 0\n \n 0\n
      \n \\(838\\)\n \n 0\n \n 0\n
      \n \\(839\\)\n \n 0\n \n 0\n \n −0.707107\n \n −\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
      \n \\(840\\)\n \n 0\n \n 0\n
      \n \\(841\\)\n \n −\n \n 1.00000i\n −\n \n 1.00000i
      \n \\(842\\)\n \n 0\n \n 0\n
      \n \\(843\\)\n \n 0\n \n 0\n
      \n \\(844\\)\n \n 0\n \n 0\n
      \n \\(845\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(846\\)\n \n 0\n \n 0\n
      \n \\(847\\)\n \n 0\n \n 0\n
      \n \\(848\\)\n \n 0\n \n 0\n
      \n \\(849\\)\n \n 0\n \n 0\n
      \n \\(850\\)\n \n 0.500000\n \n −\n \n 0.866025i\n 0.500000\n \n −\n \n 0.866025i
      \n \\(851\\)\n \n 0\n \n 0\n
      \n \\(852\\)\n \n 0\n \n 0\n
      \n \\(853\\)\n \n 1.00000\n \n −\n \n 1.00000i\n 1.00000\n \n −\n \n 1.00000i\n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n \\(0\\)\n
      \n \\(854\\)\n \n 0\n \n 0\n
      \n \\(855\\)\n \n 0\n \n 0\n
      \n \\(856\\)\n \n 0\n \n 0\n
      \n \\(857\\)\n \n 1.36603\n \n −\n \n 0.366025i\n 1.36603\n \n −\n \n 0.366025i\n 0.500000\n \n −\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(858\\)\n \n 0\n \n 0\n
      \n \\(859\\)\n \n 0\n \n 0\n \n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(860\\)\n \n 0\n \n 0\n
      \n \\(861\\)\n \n 0\n \n 0\n
      \n \\(862\\)\n \n 0\n \n 0\n
      \n \\(863\\)\n \n 0\n \n 0\n \n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(864\\)\n \n 0\n \n 0\n
      \n \\(865\\)\n \n −1.73205\n \n −\n \n 1.00000i\n −1.73205\n \n −\n \n 1.00000i
      \n \\(866\\)\n \n −1.00000\n \n −\n \n 1.73205i\n −1.00000\n \n −\n \n 1.73205i
      \n \\(867\\)\n \n 0\n \n 0\n
      \n \\(868\\)\n \n 0\n \n 0\n
      \n \\(869\\)\n \n 0\n \n 0\n
      \n \\(870\\)\n \n 0\n \n 0\n
      \n \\(871\\)\n \n 0\n \n 0\n
      \n \\(872\\)\n \n 0.366025\n \n +\n \n 1.36603i\n 0.366025\n \n +\n \n 1.36603i
      \n \\(873\\)\n \n −0.366025\n \n +\n \n 1.36603i\n −0.366025\n \n +\n \n 1.36603i
      \n \\(874\\)\n \n 0\n \n 0\n
      \n \\(875\\)\n \n 0\n \n 0\n
      \n \\(876\\)\n \n 0\n \n 0\n
      \n \\(877\\)\n \n −0.366025\n \n +\n \n 1.36603i\n −0.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(878\\)\n \n 0\n \n 0\n
      \n \\(879\\)\n \n 0\n \n 0\n
      \n \\(880\\)\n \n 0\n \n 0\n
      \n \\(881\\)\n \n −1.00000\n \n +\n \n 1.00000i\n −1.00000\n \n +\n \n 1.00000i\n 1.00000i\n \\(0.5\\pi\\)\n
      \n −1.00000\n \n \\(\\pi\\)\n
      \n \\(882\\)\n \n 0\n \n 0\n
      \n \\(883\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n −1.00000\n \n \\(\\pi\\)\n
      \n \\(884\\)\n \n 0\n \n 0\n
      \n \\(885\\)\n \n 0\n \n 0\n
      \n \\(886\\)\n \n 0\n \n 0\n
      \n \\(887\\)\n \n 0\n \n 0\n \n −0.965926\n \n −\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(888\\)\n \n 0\n \n 0\n
      \n \\(889\\)\n \n 0\n \n 0\n
      \n \\(890\\)\n \n 0\n \n 0\n
      \n \\(891\\)\n \n 0\n \n 0\n
      \n \\(892\\)\n \n 0\n \n 0\n
      \n \\(893\\)\n \n 0\n \n 0\n
      \n \\(894\\)\n \n 0\n \n 0\n
      \n \\(895\\)\n \n 0\n \n 0\n
      \n \\(896\\)\n \n 0\n \n 0\n
      \n \\(897\\)\n \n 0\n \n 0\n
      \n \\(898\\)\n \n 1.36603\n \n +\n \n 0.366025i\n 1.36603\n \n +\n \n 0.366025i
      \n \\(899\\)\n \n 0\n \n 0\n
      \n \\(900\\)\n \n −0.500000\n \n +\n \n 0.866025i\n −0.500000\n \n +\n \n 0.866025i
      \n \\(901\\)\n \n 0\n \n 0\n
      \n \\(902\\)\n \n 0\n \n 0\n
      \n \\(903\\)\n \n 0\n \n 0\n
      \n \\(904\\)\n \n 1.00000\n \n −\n \n 1.00000i\n 1.00000\n \n −\n \n 1.00000i
      \n \\(905\\)\n \n −1.00000\n \n −\n \n 1.73205i\n −1.00000\n \n −\n \n 1.73205i
      \n \\(906\\)\n \n 0\n \n 0\n
      \n \\(907\\)\n \n 0\n \n 0\n \n 0.965926\n \n −\n \n 0.258819i\n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \\(0.916667\\pi\\)\n
      \n \\(908\\)\n \n 0\n \n 0\n
      \n \\(909\\)\n \n 0\n \n 0\n
      \n \\(910\\)\n \n 0\n \n 0\n
      \n \\(911\\)\n \n 0\n \n 0\n \n −0.707107\n \n −\n \n 0.707107i\n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \\(0.250000\\pi\\)\n
      \n \\(912\\)\n \n 0\n \n 0\n
      \n \\(913\\)\n \n 0\n \n 0\n
      \n \\(914\\)\n \n 1.00000\n \n −\n \n 1.73205i\n 1.00000\n \n −\n \n 1.73205i
      \n \\(915\\)\n \n 0\n \n 0\n
      \n \\(916\\)\n \n 2.00000i\n 2.00000i
      \n \\(917\\)\n \n 0\n \n 0\n
      \n \\(918\\)\n \n 0\n \n 0\n
      \n \\(919\\)\n \n 0\n \n 0\n \n −0.500000\n \n −\n \n 0.866025i\n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n \\(920\\)\n \n 0\n \n 0\n
      \n \\(921\\)\n \n 0\n \n 0\n
      \n \\(922\\)\n \n 0\n \n 0\n
      \n \\(923\\)\n \n 0\n \n 0\n
      \n \\(924\\)\n \n 0\n \n 0\n
      \n \\(925\\)\n \n 1.00000\n \n +\n \n 1.00000i\n 1.00000\n \n +\n \n 1.00000i
      \n \\(926\\)\n \n 0\n \n 0\n
      \n \\(927\\)\n \n 0\n \n 0\n
      \n \\(928\\)\n \n 1.36603\n \n −\n \n 0.366025i\n 1.36603\n \n −\n \n 0.366025i
      \n \\(929\\)\n \n −0.366025\n \n +\n \n 1.36603i\n −0.366025\n \n +\n \n 1.36603i\n 0.500000\n \n +\n \n 0.866025i\n \\(0.333333\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(930\\)\n \n 0\n \n 0\n
      \n \\(931\\)\n \n 0\n \n 0\n
      \n \\(932\\)\n \n 1.00000\n \n −\n \n 1.00000i\n 1.00000\n \n −\n \n 1.00000i
      \n \\(933\\)\n \n 0\n \n 0\n
      \n \\(934\\)\n \n 0\n \n 0\n
      \n \\(935\\)\n \n 0\n \n 0\n
      \n \\(936\\)\n \n 0\n \n 0\n
      \n \\(937\\)\n \n 0\n \n 0\n \n 1.00000\n \n \\(0\\)\n
      \n −1.00000\n \n \\(\\pi\\)\n
      \n \\(938\\)\n \n 0\n \n 0\n
      \n \\(939\\)\n \n 0\n \n 0\n
      \n \\(940\\)\n \n 0\n \n 0\n
      \n \\(941\\)\n \n 1.36603\n \n −\n \n 0.366025i\n 1.36603\n \n −\n \n 0.366025i\n 0.500000\n \n −\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(942\\)\n \n 0\n \n 0\n
      \n \\(943\\)\n \n 0\n \n 0\n
      \n \\(944\\)\n \n 0\n \n 0\n
      \n \\(945\\)\n \n 0\n \n 0\n
      \n \\(946\\)\n \n 0\n \n 0\n
      \n \\(947\\)\n \n 0\n \n 0\n \n −0.965926\n \n −\n \n 0.258819i\n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \\(0.0833333\\pi\\)\n
      \n \\(948\\)\n \n 0\n \n 0\n
      \n \\(949\\)\n \n 0\n \n 0\n
      \n \\(950\\)\n \n 0\n \n 0\n
      \n \\(951\\)\n \n 0\n \n 0\n
      \n \\(952\\)\n \n 0\n \n 0\n
      \n \\(953\\)\n \n 0\n \n 0\n \n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(954\\)\n \n 0\n \n 0\n
      \n \\(955\\)\n \n 0\n \n 0\n
      \n \\(956\\)\n \n 0\n \n 0\n
      \n \\(957\\)\n \n 0\n \n 0\n
      \n \\(958\\)\n \n 0\n \n 0\n
      \n \\(959\\)\n \n 0\n \n 0\n
      \n \\(960\\)\n \n 0\n \n 0\n
      \n \\(961\\)\n \n −0.866025\n \n +\n \n 0.500000i\n −0.866025\n \n +\n \n 0.500000i
      \n \\(962\\)\n \n 0\n \n 0\n
      \n \\(963\\)\n \n 0\n \n 0\n
      \n \\(964\\)\n \n −1.36603\n \n −\n \n 0.366025i\n −1.36603\n \n −\n \n 0.366025i
      \n \\(965\\)\n \n 2.00000\n \n 2.00000\n
      \n \\(966\\)\n \n 0\n \n 0\n
      \n \\(967\\)\n \n 0\n \n 0\n \n −\n \n 1.00000i\n \\(-0.5\\pi\\)\n
      \n 1.00000i\n \\(0.5\\pi\\)\n
      \n \\(968\\)\n \n 0.500000\n \n +\n \n 0.866025i\n 0.500000\n \n +\n \n 0.866025i
      \n \\(969\\)\n \n 0\n \n 0\n
      \n \\(970\\)\n \n 1.00000\n \n −\n \n 1.73205i\n 1.00000\n \n −\n \n 1.73205i
      \n \\(971\\)\n \n 0\n \n 0\n \n 0.866025\n \n −\n \n 0.500000i\n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \\(0.833333\\pi\\)\n
      \n \\(972\\)\n \n 0\n \n 0\n
      \n \\(973\\)\n \n 0\n \n 0\n
      \n \\(974\\)\n \n 0\n \n 0\n
      \n \\(975\\)\n \n 0\n \n 0\n
      \n \\(976\\)\n \n −1.36603\n \n +\n \n 0.366025i\n −1.36603\n \n +\n \n 0.366025i
      \n \\(977\\)\n \n 0\n \n 0\n \n 0.500000\n \n −\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \\(0.666667\\pi\\)\n
      \n \\(978\\)\n \n 0\n \n 0\n
      \n \\(979\\)\n \n 0\n \n 0\n
      \n \\(980\\)\n \n 0\n \n 0\n
      \n \\(981\\)\n \n −1.00000\n \n +\n \n 1.00000i\n −1.00000\n \n +\n \n 1.00000i
      \n \\(982\\)\n \n 0\n \n 0\n
      \n \\(983\\)\n \n 0\n \n 0\n \n −0.258819\n \n −\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(984\\)\n \n 0\n \n 0\n
      \n \\(985\\)\n \n −1.00000\n \n −\n \n 1.73205i\n −1.00000\n \n −\n \n 1.73205i
      \n \\(986\\)\n \n −1.00000\n \n −\n \n 1.00000i\n −1.00000\n \n −\n \n 1.00000i
      \n \\(987\\)\n \n 0\n \n 0\n
      \n \\(988\\)\n \n 0\n \n 0\n
      \n \\(989\\)\n \n 0\n \n 0\n
      \n \\(990\\)\n \n 0\n \n 0\n
      \n \\(991\\)\n \n 0\n \n 0\n \n −0.258819\n \n −\n \n 0.965926i\n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \\(0.416667\\pi\\)\n
      \n \\(992\\)\n \n 0\n \n 0\n
      \n \\(993\\)\n \n 0\n \n 0\n
      \n \\(994\\)\n \n 0\n \n 0\n
      \n \\(995\\)\n \n 0\n \n 0\n
      \n \\(996\\)\n \n 0\n \n 0\n
      \n \\(997\\)\n \n 1.36603\n \n −\n \n 0.366025i\n 1.36603\n \n −\n \n 0.366025i\n 0.500000\n \n −\n \n 0.866025i\n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \\(0.166667\\pi\\)\n
      \n \\(998\\)\n \n 0\n \n 0\n
      \n \\(999\\)\n \n 0\n \n 0\n
      ", "content": {"html": "
      $n$$a_n$$a_n / n^{(k-1)/2}$$\\alpha_n$$\\theta_n$
      $p$$a_p$$a_p / p^{(k-1)/2}$$\\alpha_p$$\\theta_p$
      $2$−0.866025+0.500000i−0.866025+0.500000i
      $3$00−0.258819−0.965926i$-0.583333\\pi$
      0.258819+0.965926i$0.416667\\pi$
      $4$0.500000−0.866025i0.500000−0.866025i
      $5$−1.36603−0.366025i−1.36603−0.366025i−0.500000−0.866025i$-0.666667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $6$00
      $7$00
      $8$1.00000i1.00000i
      $9$−0.866025+0.500000i−0.866025+0.500000i
      $10$1.36603−0.366025i1.36603−0.366025i
      $11$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $12$00
      $13$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $14$00
      $15$00
      $16$−0.500000−0.866025i−0.500000−0.866025i
      $17$−0.500000+0.866025i−0.500000+0.866025i
      $18$0.500000−0.866025i0.500000−0.866025i
      $19$000.866025−0.500000i$-0.166667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $20$−1.00000+1.00000i−1.00000+1.00000i
      $21$00
      $22$00
      $23$000.258819−0.965926i$-0.416667\\pi$
      −0.258819+0.965926i$0.583333\\pi$
      $24$00
      $25$0.866025+0.500000i0.866025+0.500000i
      $26$00
      $27$00
      $28$00
      $29$1.00000−1.00000i1.00000−1.00000i−1.00000i$-0.5\\pi$
      1.00000$0$
      $30$00
      $31$00−0.258819−0.965926i$-0.583333\\pi$
      0.258819+0.965926i$0.416667\\pi$
      $32$0.866025+0.500000i0.866025+0.500000i
      $33$00
      $34$−1.00000i−1.00000i
      $35$00
      $36$1.00000i1.00000i
      $37$1.36603+0.366025i1.36603+0.366025i0.866025−0.500000i$-0.166667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $38$00
      $39$00
      $40$0.366025−1.36603i0.366025−1.36603i
      $41$−1.00000−1.00000i−1.00000−1.00000i−1.00000i$-0.5\\pi$
      −1.00000$\\pi$
      $42$00
      $43$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $44$00
      $45$1.36603−0.366025i1.36603−0.366025i
      $46$00
      $47$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $48$00
      $49$00
      $50$−1.00000−1.00000
      $51$00
      $52$00
      $53$000.500000−0.866025i$-0.333333\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $54$00
      $55$00
      $56$00
      $57$00
      $58$−0.366025+1.36603i−0.366025+1.36603i
      $59$00−0.866025−0.500000i$-0.833333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $60$00
      $61$0.366025−1.36603i0.366025−1.36603i−0.500000−0.866025i$-0.666667\\pi$
      0.866025−0.500000i$-0.166667\\pi$
      $62$00
      $63$00
      $64$−1.00000−1.00000
      $65$00
      $66$00
      $67$000.500000−0.866025i$-0.333333\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $68$0.500000+0.866025i0.500000+0.866025i
      $69$00
      $70$00
      $71$000.707107−0.707107i$-0.250000\\pi$
      −0.707107+0.707107i$0.750000\\pi$
      $72$−0.500000−0.866025i−0.500000−0.866025i
      $73$−0.366025−1.36603i−0.366025−1.36603i−0.866025−0.500000i$-0.833333\\pi$
      0.500000−0.866025i$-0.333333\\pi$
      $74$−1.36603+0.366025i−1.36603+0.366025i
      $75$00
      $76$00
      $77$00
      $78$00
      $79$000.258819−0.965926i$-0.416667\\pi$
      −0.258819+0.965926i$0.583333\\pi$
      $80$0.366025+1.36603i0.366025+1.36603i
      $81$0.500000−0.866025i0.500000−0.866025i
      $82$1.36603+0.366025i1.36603+0.366025i
      $83$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $84$00
      $85$1.00000−1.00000i1.00000−1.00000i
      $86$00
      $87$00
      $88$00
      $89$000.866025−0.500000i$-0.166667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $90$−1.00000+1.00000i−1.00000+1.00000i
      $91$00
      $92$00
      $93$00
      $94$00
      $95$00
      $96$00
      $97$1.00000−1.00000i1.00000−1.00000i−1.00000i$-0.5\\pi$
      1.00000$0$
      $98$00
      $99$00
      $100$0.866025−0.500000i0.866025−0.500000i
      $101$00−0.866025−0.500000i$-0.833333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $102$00
      $103$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $104$00
      $105$00
      $106$00
      $107$00−0.965926−0.258819i$-0.916667\\pi$
      0.965926+0.258819i$0.0833333\\pi$
      $108$00
      $109$1.36603−0.366025i1.36603−0.366025i0.500000−0.866025i$-0.333333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $110$00
      $111$00
      $112$00
      $113$−1.00000−1.00000i−1.00000−1.00000i−1.00000i$-0.5\\pi$
      −1.00000$\\pi$
      $114$00
      $115$00
      $116$−0.366025−1.36603i−0.366025−1.36603i
      $117$00
      $118$00
      $119$00
      $120$00
      $121$0.866025−0.500000i0.866025−0.500000i
      $122$0.366025+1.36603i0.366025+1.36603i
      $123$00
      $124$00
      $125$00
      $126$00
      $127$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $128$0.866025−0.500000i0.866025−0.500000i
      $129$00
      $130$00
      $131$00−0.965926−0.258819i$-0.916667\\pi$
      0.965926+0.258819i$0.0833333\\pi$
      $132$00
      $133$00
      $134$00
      $135$00
      $136$−0.866025−0.500000i−0.866025−0.500000i
      $137$00−0.866025−0.500000i$-0.833333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $138$00
      $139$000.707107−0.707107i$-0.250000\\pi$
      −0.707107+0.707107i$0.750000\\pi$
      $140$00
      $141$00
      $142$00
      $143$00
      $144$0.866025+0.500000i0.866025+0.500000i
      $145$−1.73205+1.00000i−1.73205+1.00000i
      $146$1.00000+1.00000i1.00000+1.00000i
      $147$00
      $148$1.00000−1.00000i1.00000−1.00000i
      $149$−1.00000−1.73205i−1.00000−1.73205i−0.500000−0.866025i$-0.666667\\pi$
      −0.500000−0.866025i$-0.666667\\pi$
      $150$00
      $151$00−0.866025−0.500000i$-0.833333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $152$00
      $153$−1.00000i−1.00000i
      $154$00
      $155$00
      $156$00
      $157$1.00000−1.73205i1.00000−1.73205i0.500000−0.866025i$-0.333333\\pi$
      0.500000−0.866025i$-0.333333\\pi$
      $158$00
      $159$00
      $160$−1.00000−1.00000i−1.00000−1.00000i
      $161$00
      $162$1.00000i1.00000i
      $163$000.258819−0.965926i$-0.416667\\pi$
      −0.258819+0.965926i$0.583333\\pi$
      $164$−1.36603+0.366025i−1.36603+0.366025i
      $165$00
      $166$00
      $167$000.707107−0.707107i$-0.250000\\pi$
      −0.707107+0.707107i$0.750000\\pi$
      $168$00
      $169$−1.00000−1.00000
      $170$−0.366025+1.36603i−0.366025+1.36603i
      $171$00
      $172$00
      $173$1.36603+0.366025i1.36603+0.366025i0.866025−0.500000i$-0.166667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $174$00
      $175$00
      $176$00
      $177$00
      $178$00
      $179$00−0.866025−0.500000i$-0.833333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $180$0.366025−1.36603i0.366025−1.36603i
      $181$1.00000+1.00000i1.00000+1.00000i1.00000$0$
      1.00000i$0.5\\pi$
      $182$00
      $183$00
      $184$00
      $185$−1.73205−1.00000i−1.73205−1.00000i
      $186$00
      $187$00
      $188$00
      $189$00
      $190$00
      $191$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $192$00
      $193$−1.36603+0.366025i−1.36603+0.366025i−0.866025−0.500000i$-0.833333\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $194$−0.366025+1.36603i−0.366025+1.36603i
      $195$00
      $196$00
      $197$1.00000+1.00000i1.00000+1.00000i1.00000$0$
      1.00000i$0.5\\pi$
      $198$00
      $199$00−0.258819−0.965926i$-0.583333\\pi$
      0.258819+0.965926i$0.416667\\pi$
      $200$−0.500000+0.866025i−0.500000+0.866025i
      $201$00
      $202$00
      $203$00
      $204$00
      $205$1.00000+1.73205i1.00000+1.73205i
      $206$00
      $207$00
      $208$00
      $209$00
      $210$00
      $211$00−0.707107−0.707107i$-0.750000\\pi$
      0.707107+0.707107i$0.250000\\pi$
      $212$00
      $213$00
      $214$00
      $215$00
      $216$00
      $217$00
      $218$−1.00000+1.00000i−1.00000+1.00000i
      $219$00
      $220$00
      $221$00
      $222$00
      $223$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $224$00
      $225$−1.00000−1.00000
      $226$1.36603+0.366025i1.36603+0.366025i
      $227$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $228$00
      $229$−1.73205+1.00000i−1.73205+1.00000i−0.866025+0.500000i$0.833333\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $230$00
      $231$00
      $232$1.00000+1.00000i1.00000+1.00000i
      $233$1.36603+0.366025i1.36603+0.366025i0.866025−0.500000i$-0.166667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $234$00
      $235$00
      $236$00
      $237$00
      $238$00
      $239$001.00000$0$
      −1.00000$\\pi$
      $240$00
      $241$−0.366025−1.36603i−0.366025−1.36603i−0.866025−0.500000i$-0.833333\\pi$
      0.500000−0.866025i$-0.333333\\pi$
      $242$−0.500000+0.866025i−0.500000+0.866025i
      $243$00
      $244$−1.00000−1.00000i−1.00000−1.00000i
      $245$00
      $246$00
      $247$00
      $248$00
      $249$00
      $250$00
      $251$001.00000$0$
      −1.00000$\\pi$
      $252$00
      $253$00
      $254$00
      $255$00
      $256$−0.500000+0.866025i−0.500000+0.866025i
      $257$1.73205−1.00000i1.73205−1.00000i0.866025−0.500000i$-0.166667\\pi$
      0.866025−0.500000i$-0.166667\\pi$
      $258$00
      $259$00
      $260$00
      $261$−0.366025+1.36603i−0.366025+1.36603i
      $262$00
      $263$00−0.866025−0.500000i$-0.833333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $264$00
      $265$00
      $266$00
      $267$00
      $268$00
      $269$0.366025+1.36603i0.366025+1.36603i0.866025+0.500000i$0.166667\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $270$00
      $271$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $272$1.000001.00000
      $273$00
      $274$00
      $275$00
      $276$00
      $277$0.366025+1.36603i0.366025+1.36603i0.866025+0.500000i$0.166667\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $278$00
      $279$00
      $280$00
      $281$001.00000$0$
      −1.00000$\\pi$
      $282$00
      $283$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $284$00
      $285$00
      $286$00
      $287$00
      $288$−1.00000−1.00000
      $289$−0.500000−0.866025i−0.500000−0.866025i
      $290$1.00000−1.73205i1.00000−1.73205i
      $291$00
      $292$−1.36603−0.366025i−1.36603−0.366025i
      $293$2.000002.000001.00000$0$
      1.00000$0$
      $294$00
      $295$00
      $296$−0.366025+1.36603i−0.366025+1.36603i
      $297$00
      $298$1.73205+1.00000i1.73205+1.00000i
      $299$00
      $300$00
      $301$00
      $302$00
      $303$00
      $304$00
      $305$−1.00000+1.73205i−1.00000+1.73205i
      $306$0.500000+0.866025i0.500000+0.866025i
      $307$001.00000$0$
      −1.00000$\\pi$
      $308$00
      $309$00
      $310$00
      $311$00−0.258819−0.965926i$-0.583333\\pi$
      0.258819+0.965926i$0.416667\\pi$
      $312$00
      $313$−0.366025+1.36603i−0.366025+1.36603i0.500000+0.866025i$0.333333\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $314$2.00000i2.00000i
      $315$00
      $316$00
      $317$0.366025−1.36603i0.366025−1.36603i−0.500000−0.866025i$-0.666667\\pi$
      0.866025−0.500000i$-0.166667\\pi$
      $318$00
      $319$00
      $320$1.36603+0.366025i1.36603+0.366025i
      $321$00
      $322$00
      $323$00
      $324$−0.500000−0.866025i−0.500000−0.866025i
      $325$00
      $326$00
      $327$00
      $328$1.00000−1.00000i1.00000−1.00000i
      $329$00
      $330$00
      $331$000.866025−0.500000i$-0.166667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $332$00
      $333$−1.36603+0.366025i−1.36603+0.366025i
      $334$00
      $335$00
      $336$00
      $337$1.00000−1.00000i1.00000−1.00000i−1.00000i$-0.5\\pi$
      1.00000$0$
      $338$0.866025−0.500000i0.866025−0.500000i
      $339$00
      $340$−0.366025−1.36603i−0.366025−1.36603i
      $341$00
      $342$00
      $343$00
      $344$00
      $345$00
      $346$−1.36603+0.366025i−1.36603+0.366025i
      $347$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $348$00
      $349$001.00000$0$
      −1.00000$\\pi$
      $350$00
      $351$00
      $352$00
      $353$−1.00000+1.73205i−1.00000+1.73205i−0.500000+0.866025i$0.666667\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $354$00
      $355$00
      $356$00
      $357$00
      $358$00
      $359$000.866025−0.500000i$-0.166667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $360$0.366025+1.36603i0.366025+1.36603i
      $361$0.500000−0.866025i0.500000−0.866025i
      $362$−1.36603−0.366025i−1.36603−0.366025i
      $363$00
      $364$00
      $365$2.00000i2.00000i
      $366$00
      $367$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $368$00
      $369$1.36603+0.366025i1.36603+0.366025i
      $370$2.000002.00000
      $371$00
      $372$00
      $373$000.866025−0.500000i$-0.166667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $374$00
      $375$00
      $376$00
      $377$00
      $378$00
      $379$000.707107−0.707107i$-0.250000\\pi$
      −0.707107+0.707107i$0.750000\\pi$
      $380$00
      $381$00
      $382$00
      $383$000.866025−0.500000i$-0.166667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $384$00
      $385$00
      $386$1.00000−1.00000i1.00000−1.00000i
      $387$00
      $388$−0.366025−1.36603i−0.366025−1.36603i
      $389$−1.73205−1.00000i−1.73205−1.00000i−0.866025−0.500000i$-0.833333\\pi$
      −0.866025−0.500000i$-0.833333\\pi$
      $390$00
      $391$00
      $392$00
      $393$00
      $394$−1.36603−0.366025i−1.36603−0.366025i
      $395$00
      $396$00
      $397$0.366025−1.36603i0.366025−1.36603i−0.500000−0.866025i$-0.666667\\pi$
      0.866025−0.500000i$-0.166667\\pi$
      $398$00
      $399$00
      $400$−1.00000i−1.00000i
      $401$0.366025−1.36603i0.366025−1.36603i−0.500000−0.866025i$-0.666667\\pi$
      0.866025−0.500000i$-0.166667\\pi$
      $402$00
      $403$00
      $404$00
      $405$−1.00000+1.00000i−1.00000+1.00000i
      $406$00
      $407$00
      $408$00
      $409$−1.00000+1.73205i−1.00000+1.73205i−0.500000+0.866025i$0.666667\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $410$−1.73205−1.00000i−1.73205−1.00000i
      $411$00
      $412$00
      $413$00
      $414$00
      $415$00
      $416$00
      $417$00
      $418$00
      $419$00−0.707107−0.707107i$-0.750000\\pi$
      0.707107+0.707107i$0.250000\\pi$
      $420$00
      $421$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $422$00
      $423$00
      $424$00
      $425$−0.866025+0.500000i−0.866025+0.500000i
      $426$00
      $427$00
      $428$00
      $429$00
      $430$00
      $431$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $432$00
      $433$2.00000i2.00000i1.00000i$0.5\\pi$
      1.00000i$0.5\\pi$
      $434$00
      $435$00
      $436$0.366025−1.36603i0.366025−1.36603i
      $437$00
      $438$00
      $439$00−0.965926−0.258819i$-0.916667\\pi$
      0.965926+0.258819i$0.0833333\\pi$
      $440$00
      $441$00
      $442$00
      $443$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $444$00
      $445$00
      $446$00
      $447$00
      $448$00
      $449$−1.00000−1.00000i−1.00000−1.00000i−1.00000i$-0.5\\pi$
      −1.00000$\\pi$
      $450$0.866025−0.500000i0.866025−0.500000i
      $451$00
      $452$−1.36603+0.366025i−1.36603+0.366025i
      $453$00
      $454$00
      $455$00
      $456$00
      $457$−1.73205+1.00000i−1.73205+1.00000i−0.866025+0.500000i$0.833333\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $458$1.00000−1.73205i1.00000−1.73205i
      $459$00
      $460$00
      $461$001.00000$0$
      −1.00000$\\pi$
      $462$00
      $463$001.00000$0$
      −1.00000$\\pi$
      $464$−1.36603−0.366025i−1.36603−0.366025i
      $465$00
      $466$−1.36603+0.366025i−1.36603+0.366025i
      $467$000.866025−0.500000i$-0.166667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $468$00
      $469$00
      $470$00
      $471$00
      $472$00
      $473$00
      $474$00
      $475$00
      $476$00
      $477$00
      $478$00
      $479$00−0.258819−0.965926i$-0.583333\\pi$
      0.258819+0.965926i$0.416667\\pi$
      $480$00
      $481$00
      $482$1.00000+1.00000i1.00000+1.00000i
      $483$00
      $484$−1.00000i−1.00000i
      $485$−1.73205+1.00000i−1.73205+1.00000i
      $486$00
      $487$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $488$1.36603+0.366025i1.36603+0.366025i
      $489$00
      $490$00
      $491$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $492$00
      $493$0.366025+1.36603i0.366025+1.36603i
      $494$00
      $495$00
      $496$00
      $497$00
      $498$00
      $499$000.258819−0.965926i$-0.416667\\pi$
      −0.258819+0.965926i$0.583333\\pi$
      $500$00
      $501$00
      $502$00
      $503$00−0.707107−0.707107i$-0.750000\\pi$
      0.707107+0.707107i$0.250000\\pi$
      $504$00
      $505$00
      $506$00
      $507$00
      $508$00
      $509$−1.00000−1.73205i−1.00000−1.73205i−0.500000−0.866025i$-0.666667\\pi$
      −0.500000−0.866025i$-0.666667\\pi$
      $510$00
      $511$00
      $512$−1.00000i−1.00000i
      $513$00
      $514$−1.00000+1.73205i−1.00000+1.73205i
      $515$00
      $516$00
      $517$00
      $518$00
      $519$00
      $520$00
      $521$1.36603−0.366025i1.36603−0.366025i0.500000−0.866025i$-0.333333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $522$−0.366025−1.36603i−0.366025−1.36603i
      $523$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $524$00
      $525$00
      $526$00
      $527$00
      $528$00
      $529$−0.866025−0.500000i−0.866025−0.500000i
      $530$00
      $531$00
      $532$00
      $533$00
      $534$00
      $535$00
      $536$00
      $537$00
      $538$−1.00000−1.00000i−1.00000−1.00000i
      $539$00
      $540$00
      $541$1.36603+0.366025i1.36603+0.366025i0.866025−0.500000i$-0.166667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $542$00
      $543$00
      $544$−0.866025+0.500000i−0.866025+0.500000i
      $545$−2.00000−2.00000
      $546$00
      $547$000.707107−0.707107i$-0.250000\\pi$
      −0.707107+0.707107i$0.750000\\pi$
      $548$00
      $549$0.366025+1.36603i0.366025+1.36603i
      $550$00
      $551$00
      $552$00
      $553$00
      $554$−1.00000−1.00000i−1.00000−1.00000i
      $555$00
      $556$00
      $557$00−0.866025−0.500000i$-0.833333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $558$00
      $559$00
      $560$00
      $561$00
      $562$00
      $563$00−0.866025−0.500000i$-0.833333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $564$00
      $565$1.00000+1.73205i1.00000+1.73205i
      $566$00
      $567$00
      $568$00
      $569$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $570$00
      $571$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $572$00
      $573$00
      $574$00
      $575$00
      $576$0.866025−0.500000i0.866025−0.500000i
      $577$00−0.866025−0.500000i$-0.833333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $578$0.866025+0.500000i0.866025+0.500000i
      $579$00
      $580$2.00000i2.00000i
      $581$00
      $582$00
      $583$00
      $584$1.36603−0.366025i1.36603−0.366025i
      $585$00
      $586$−1.73205+1.00000i−1.73205+1.00000i
      $587$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $588$00
      $589$00
      $590$00
      $591$00
      $592$−0.366025−1.36603i−0.366025−1.36603i
      $593$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $594$00
      $595$00
      $596$−2.00000−2.00000
      $597$00
      $598$00
      $599$000.500000−0.866025i$-0.333333\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $600$00
      $601$−1.00000−1.00000i−1.00000−1.00000i−1.00000i$-0.5\\pi$
      −1.00000$\\pi$
      $602$00
      $603$00
      $604$00
      $605$−1.36603+0.366025i−1.36603+0.366025i
      $606$00
      $607$00−0.965926−0.258819i$-0.916667\\pi$
      0.965926+0.258819i$0.0833333\\pi$
      $608$00
      $609$00
      $610$−2.00000i−2.00000i
      $611$00
      $612$−0.866025−0.500000i−0.866025−0.500000i
      $613$−1.00000+1.73205i−1.00000+1.73205i−0.500000+0.866025i$0.666667\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $614$00
      $615$00
      $616$00
      $617$−1.00000+1.00000i−1.00000+1.00000i1.00000i$0.5\\pi$
      −1.00000$\\pi$
      $618$00
      $619$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $620$00
      $621$00
      $622$00
      $623$00
      $624$00
      $625$−0.500000−0.866025i−0.500000−0.866025i
      $626$−0.366025−1.36603i−0.366025−1.36603i
      $627$00
      $628$−1.00000−1.73205i−1.00000−1.73205i
      $629$−1.00000+1.00000i−1.00000+1.00000i
      $630$00
      $631$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $632$00
      $633$00
      $634$0.366025+1.36603i0.366025+1.36603i
      $635$00
      $636$00
      $637$00
      $638$00
      $639$00
      $640$−1.36603+0.366025i−1.36603+0.366025i
      $641$−0.366025−1.36603i−0.366025−1.36603i−0.866025−0.500000i$-0.833333\\pi$
      0.500000−0.866025i$-0.333333\\pi$
      $642$00
      $643$000.707107−0.707107i$-0.250000\\pi$
      −0.707107+0.707107i$0.750000\\pi$
      $644$00
      $645$00
      $646$00
      $647$000.500000−0.866025i$-0.333333\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $648$0.866025+0.500000i0.866025+0.500000i
      $649$00
      $650$00
      $651$00
      $652$00
      $653$0.366025−1.36603i0.366025−1.36603i−0.500000−0.866025i$-0.666667\\pi$
      0.866025−0.500000i$-0.166667\\pi$
      $654$00
      $655$00
      $656$−0.366025+1.36603i−0.366025+1.36603i
      $657$1.00000+1.00000i1.00000+1.00000i
      $658$00
      $659$001.00000$0$
      −1.00000$\\pi$
      $660$00
      $661$000.500000−0.866025i$-0.333333\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $662$00
      $663$00
      $664$00
      $665$00
      $666$1.00000−1.00000i1.00000−1.00000i
      $667$00
      $668$00
      $669$00
      $670$00
      $671$00
      $672$00
      $673$−1.00000−1.00000i−1.00000−1.00000i−1.00000i$-0.5\\pi$
      −1.00000$\\pi$
      $674$−0.366025+1.36603i−0.366025+1.36603i
      $675$00
      $676$−0.500000+0.866025i−0.500000+0.866025i
      $677$1.36603+0.366025i1.36603+0.366025i0.866025−0.500000i$-0.166667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $678$00
      $679$00
      $680$1.00000+1.00000i1.00000+1.00000i
      $681$00
      $682$00
      $683$00−0.258819−0.965926i$-0.583333\\pi$
      0.258819+0.965926i$0.416667\\pi$
      $684$00
      $685$00
      $686$00
      $687$00
      $688$00
      $689$00
      $690$00
      $691$000.258819−0.965926i$-0.416667\\pi$
      −0.258819+0.965926i$0.583333\\pi$
      $692$1.00000−1.00000i1.00000−1.00000i
      $693$00
      $694$00
      $695$00
      $696$00
      $697$1.36603−0.366025i1.36603−0.366025i
      $698$00
      $699$00
      $700$00
      $701$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $702$00
      $703$00
      $704$00
      $705$00
      $706$−2.00000i−2.00000i
      $707$00
      $708$00
      $709$−1.36603−0.366025i−1.36603−0.366025i−0.500000−0.866025i$-0.666667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $710$00
      $711$00
      $712$00
      $713$00
      $714$00
      $715$00
      $716$00
      $717$00
      $718$00
      $719$00−0.965926−0.258819i$-0.916667\\pi$
      0.965926+0.258819i$0.0833333\\pi$
      $720$−1.00000−1.00000i−1.00000−1.00000i
      $721$00
      $722$1.00000i1.00000i
      $723$00
      $724$1.36603−0.366025i1.36603−0.366025i
      $725$1.36603−0.366025i1.36603−0.366025i
      $726$00
      $727$001.00000$0$
      −1.00000$\\pi$
      $728$00
      $729$1.00000i1.00000i
      $730$−1.00000−1.73205i−1.00000−1.73205i
      $731$00
      $732$00
      $733$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $734$00
      $735$00
      $736$00
      $737$00
      $738$−1.36603+0.366025i−1.36603+0.366025i
      $739$00−0.866025−0.500000i$-0.833333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $740$−1.73205+1.00000i−1.73205+1.00000i
      $741$00
      $742$00
      $743$000.707107−0.707107i$-0.250000\\pi$
      −0.707107+0.707107i$0.750000\\pi$
      $744$00
      $745$0.732051+2.73205i0.732051+2.73205i
      $746$00
      $747$00
      $748$00
      $749$00
      $750$00
      $751$00−0.965926−0.258819i$-0.916667\\pi$
      0.965926+0.258819i$0.0833333\\pi$
      $752$00
      $753$00
      $754$00
      $755$00
      $756$00
      $757$2.00000i2.00000i1.00000i$0.5\\pi$
      1.00000i$0.5\\pi$
      $758$00
      $759$00
      $760$00
      $761$000.866025−0.500000i$-0.166667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $762$00
      $763$00
      $764$00
      $765$−0.366025+1.36603i−0.366025+1.36603i
      $766$00
      $767$00
      $768$00
      $769$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $770$00
      $771$00
      $772$−0.366025+1.36603i−0.366025+1.36603i
      $773$−1.73205−1.00000i−1.73205−1.00000i−0.866025−0.500000i$-0.833333\\pi$
      −0.866025−0.500000i$-0.833333\\pi$
      $774$00
      $775$00
      $776$1.00000+1.00000i1.00000+1.00000i
      $777$00
      $778$2.000002.00000
      $779$00
      $780$00
      $781$00
      $782$00
      $783$00
      $784$00
      $785$−2.00000+2.00000i−2.00000+2.00000i
      $786$00
      $787$00−0.258819−0.965926i$-0.583333\\pi$
      0.258819+0.965926i$0.416667\\pi$
      $788$1.36603−0.366025i1.36603−0.366025i
      $789$00
      $790$00
      $791$00
      $792$00
      $793$00
      $794$0.366025+1.36603i0.366025+1.36603i
      $795$00
      $796$00
      $797$001.00000$0$
      −1.00000$\\pi$
      $798$00
      $799$00
      $800$0.500000+0.866025i0.500000+0.866025i
      $801$00
      $802$0.366025+1.36603i0.366025+1.36603i
      $803$00
      $804$00
      $805$00
      $806$00
      $807$00
      $808$00
      $809$1.36603−0.366025i1.36603−0.366025i0.500000−0.866025i$-0.333333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $810$0.366025−1.36603i0.366025−1.36603i
      $811$000.707107−0.707107i$-0.250000\\pi$
      −0.707107+0.707107i$0.750000\\pi$
      $812$00
      $813$00
      $814$00
      $815$00
      $816$00
      $817$00
      $818$−2.00000i−2.00000i
      $819$00
      $820$2.000002.00000
      $821$−1.36603−0.366025i−1.36603−0.366025i−0.500000−0.866025i$-0.666667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $822$00
      $823$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $824$00
      $825$00
      $826$00
      $827$00−0.707107−0.707107i$-0.750000\\pi$
      0.707107+0.707107i$0.250000\\pi$
      $828$00
      $829$1.00000−1.73205i1.00000−1.73205i0.500000−0.866025i$-0.333333\\pi$
      0.500000−0.866025i$-0.333333\\pi$
      $830$00
      $831$00
      $832$00
      $833$00
      $834$00
      $835$00
      $836$00
      $837$00
      $838$00
      $839$00−0.707107−0.707107i$-0.750000\\pi$
      0.707107+0.707107i$0.250000\\pi$
      $840$00
      $841$−1.00000i−1.00000i
      $842$00
      $843$00
      $844$00
      $845$1.36603+0.366025i1.36603+0.366025i
      $846$00
      $847$00
      $848$00
      $849$00
      $850$0.500000−0.866025i0.500000−0.866025i
      $851$00
      $852$00
      $853$1.00000−1.00000i1.00000−1.00000i−1.00000i$-0.5\\pi$
      1.00000$0$
      $854$00
      $855$00
      $856$00
      $857$1.36603−0.366025i1.36603−0.366025i0.500000−0.866025i$-0.333333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $858$00
      $859$000.866025−0.500000i$-0.166667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $860$00
      $861$00
      $862$00
      $863$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $864$00
      $865$−1.73205−1.00000i−1.73205−1.00000i
      $866$−1.00000−1.73205i−1.00000−1.73205i
      $867$00
      $868$00
      $869$00
      $870$00
      $871$00
      $872$0.366025+1.36603i0.366025+1.36603i
      $873$−0.366025+1.36603i−0.366025+1.36603i
      $874$00
      $875$00
      $876$00
      $877$−0.366025+1.36603i−0.366025+1.36603i0.500000+0.866025i$0.333333\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $878$00
      $879$00
      $880$00
      $881$−1.00000+1.00000i−1.00000+1.00000i1.00000i$0.5\\pi$
      −1.00000$\\pi$
      $882$00
      $883$001.00000$0$
      −1.00000$\\pi$
      $884$00
      $885$00
      $886$00
      $887$00−0.965926−0.258819i$-0.916667\\pi$
      0.965926+0.258819i$0.0833333\\pi$
      $888$00
      $889$00
      $890$00
      $891$00
      $892$00
      $893$00
      $894$00
      $895$00
      $896$00
      $897$00
      $898$1.36603+0.366025i1.36603+0.366025i
      $899$00
      $900$−0.500000+0.866025i−0.500000+0.866025i
      $901$00
      $902$00
      $903$00
      $904$1.00000−1.00000i1.00000−1.00000i
      $905$−1.00000−1.73205i−1.00000−1.73205i
      $906$00
      $907$000.965926−0.258819i$-0.0833333\\pi$
      −0.965926+0.258819i$0.916667\\pi$
      $908$00
      $909$00
      $910$00
      $911$00−0.707107−0.707107i$-0.750000\\pi$
      0.707107+0.707107i$0.250000\\pi$
      $912$00
      $913$00
      $914$1.00000−1.73205i1.00000−1.73205i
      $915$00
      $916$2.00000i2.00000i
      $917$00
      $918$00
      $919$00−0.500000−0.866025i$-0.666667\\pi$
      0.500000+0.866025i$0.333333\\pi$
      $920$00
      $921$00
      $922$00
      $923$00
      $924$00
      $925$1.00000+1.00000i1.00000+1.00000i
      $926$00
      $927$00
      $928$1.36603−0.366025i1.36603−0.366025i
      $929$−0.366025+1.36603i−0.366025+1.36603i0.500000+0.866025i$0.333333\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $930$00
      $931$00
      $932$1.00000−1.00000i1.00000−1.00000i
      $933$00
      $934$00
      $935$00
      $936$00
      $937$001.00000$0$
      −1.00000$\\pi$
      $938$00
      $939$00
      $940$00
      $941$1.36603−0.366025i1.36603−0.366025i0.500000−0.866025i$-0.333333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $942$00
      $943$00
      $944$00
      $945$00
      $946$00
      $947$00−0.965926−0.258819i$-0.916667\\pi$
      0.965926+0.258819i$0.0833333\\pi$
      $948$00
      $949$00
      $950$00
      $951$00
      $952$00
      $953$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $954$00
      $955$00
      $956$00
      $957$00
      $958$00
      $959$00
      $960$00
      $961$−0.866025+0.500000i−0.866025+0.500000i
      $962$00
      $963$00
      $964$−1.36603−0.366025i−1.36603−0.366025i
      $965$2.000002.00000
      $966$00
      $967$00−1.00000i$-0.5\\pi$
      1.00000i$0.5\\pi$
      $968$0.500000+0.866025i0.500000+0.866025i
      $969$00
      $970$1.00000−1.73205i1.00000−1.73205i
      $971$000.866025−0.500000i$-0.166667\\pi$
      −0.866025+0.500000i$0.833333\\pi$
      $972$00
      $973$00
      $974$00
      $975$00
      $976$−1.36603+0.366025i−1.36603+0.366025i
      $977$000.500000−0.866025i$-0.333333\\pi$
      −0.500000+0.866025i$0.666667\\pi$
      $978$00
      $979$00
      $980$00
      $981$−1.00000+1.00000i−1.00000+1.00000i
      $982$00
      $983$00−0.258819−0.965926i$-0.583333\\pi$
      0.258819+0.965926i$0.416667\\pi$
      $984$00
      $985$−1.00000−1.73205i−1.00000−1.73205i
      $986$−1.00000−1.00000i−1.00000−1.00000i
      $987$00
      $988$00
      $989$00
      $990$00
      $991$00−0.258819−0.965926i$-0.583333\\pi$
      0.258819+0.965926i$0.416667\\pi$
      $992$00
      $993$00
      $994$00
      $995$00
      $996$00
      $997$1.36603−0.366025i1.36603−0.366025i0.500000−0.866025i$-0.333333\\pi$
      0.866025+0.500000i$0.166667\\pi$
      $998$00
      $999$00
      ", "is_complex": true, "table_nest_level": "1"}}, {"type": "paragraph", "raw_content": "", "content": [{"c": "(See", "t": "text"}, {"c": "a_n", "t": "equation-inline"}, {"c": "instead)(See", "t": "text"}, {"c": "a_n", "t": "equation-inline"}, {"c": "instead)(See", "t": "text"}, {"c": "a_n", "t": "equation-inline"}, {"c": "instead)(See only", "t": "text"}, {"c": "a_p", "t": "equation-inline"}, {"c": ")(See only", "t": "text"}, {"c": "a_p", "t": "equation-inline"}, {"c": ")(See only", "t": "text"}, {"c": "a_p", "t": "equation-inline"}, {"c": ")", "t": "text"}]}, {"type": "complex_table", "raw_content": "
             By twisting character
      CharParityOrdTypeTwistMinDim
      1.1even1trivial3332.1.bc.b.2027.14
      4.3odd2CM3332.1.bc.b.2027.14
      7.2even3inner3332.1.bc.b.667.14
      7.3odd668.1.f.a.55.1yes2
      7.4even33332.1.m.b.2843.12
      7.5odd63332.1.bc.c.667.14
      7.6odd23332.1.bc.c.2027.14
      17.13even4inner3332.1.bc.b.2223.14
      21.17even6612.1.l.a.55.12
      28.3even668.1.f.a.55.1yes2
      28.11odd63332.1.m.b.2843.12
      28.19even63332.1.bc.c.667.14
      28.23odd6inner3332.1.bc.b.667.14
      28.27even23332.1.bc.c.2027.14
      35.3even121700.1.n.a.599.12
      35.17even121700.1.n.b.599.12
      35.24odd61700.1.p.a.1551.12
      56.3even61088.1.p.a.191.12
      56.45odd61088.1.p.a.191.12
      68.47odd4inner3332.1.bc.b.2223.14
      84.59odd6612.1.l.a.55.12
      119.3even481156.1.g.b.155.18
      119.10even481156.1.g.b.179.18
      119.13odd43332.1.bc.c.2223.14
      119.24even481156.1.g.b.179.28
      119.30even12inner3332.1.bc.b.863.14
      119.31even481156.1.g.b.155.28
      119.38odd121156.1.f.b.251.12
      119.45even481156.1.g.b.399.18
      119.47odd123332.1.bc.c.863.14
      119.59odd241156.1.c.b.579.12
      119.66odd241156.1.d.a.1155.22
      119.73even481156.1.g.b.423.28
      119.80even481156.1.g.b.423.18
      119.81even123332.1.m.b.3039.12
      119.87odd241156.1.d.a.1155.12
      119.94odd241156.1.c.b.579.22
      119.101odd61156.1.f.b.327.12
      119.108even481156.1.g.b.399.28
      119.115odd1268.1.f.a.47.1&check;2
      140.3odd121700.1.n.a.599.12
      140.59even61700.1.p.a.1551.12
      140.87odd121700.1.n.b.599.12
      357.353even12612.1.l.a.523.12
      476.3odd481156.1.g.b.155.18
      476.31odd481156.1.g.b.155.28
      476.47even123332.1.bc.c.863.14
      476.59even241156.1.c.b.579.12
      476.87even241156.1.d.a.1155.12
      476.115even1268.1.f.a.47.1&check;2
      476.143odd481156.1.g.b.179.28
      476.199odd481156.1.g.b.423.18
      476.227odd481156.1.g.b.399.28
      476.251even43332.1.bc.c.2223.14
      476.283odd481156.1.g.b.399.18
      476.311odd481156.1.g.b.423.28
      476.319odd123332.1.m.b.3039.12
      476.339even61156.1.f.b.327.12
      476.367odd481156.1.g.b.179.18
      476.387odd12inner3332.1.bc.b.863.14
      476.395even121156.1.f.b.251.12
      476.423even241156.1.d.a.1155.22
      476.451even241156.1.c.b.579.22
      595.234odd121700.1.p.a.251.12
      595.353even121700.1.n.b.999.12
      595.472even121700.1.n.a.999.12
      952.115even121088.1.p.a.319.12
      952.829odd121088.1.p.a.319.12
      1428.1067odd12612.1.l.a.523.12
      2380.1067odd121700.1.n.a.999.12
      2380.1543odd121700.1.n.b.999.12
      2380.2019even121700.1.p.a.251.12
      ", "content": {"html": "
             By twisting character
      CharParityOrdTypeTwistMinDim
      1.1even1trivial3332.1.bc.b.2027.14
      4.3odd2CM3332.1.bc.b.2027.14
      7.2even3inner3332.1.bc.b.667.14
      7.3odd668.1.f.a.55.1yes2
      7.4even33332.1.m.b.2843.12
      7.5odd63332.1.bc.c.667.14
      7.6odd23332.1.bc.c.2027.14
      17.13even4inner3332.1.bc.b.2223.14
      21.17even6612.1.l.a.55.12
      28.3even668.1.f.a.55.1yes2
      28.11odd63332.1.m.b.2843.12
      28.19even63332.1.bc.c.667.14
      28.23odd6inner3332.1.bc.b.667.14
      28.27even23332.1.bc.c.2027.14
      35.3even121700.1.n.a.599.12
      35.17even121700.1.n.b.599.12
      35.24odd61700.1.p.a.1551.12
      56.3even61088.1.p.a.191.12
      56.45odd61088.1.p.a.191.12
      68.47odd4inner3332.1.bc.b.2223.14
      84.59odd6612.1.l.a.55.12
      119.3even481156.1.g.b.155.18
      119.10even481156.1.g.b.179.18
      119.13odd43332.1.bc.c.2223.14
      119.24even481156.1.g.b.179.28
      119.30even12inner3332.1.bc.b.863.14
      119.31even481156.1.g.b.155.28
      119.38odd121156.1.f.b.251.12
      119.45even481156.1.g.b.399.18
      119.47odd123332.1.bc.c.863.14
      119.59odd241156.1.c.b.579.12
      119.66odd241156.1.d.a.1155.22
      119.73even481156.1.g.b.423.28
      119.80even481156.1.g.b.423.18
      119.81even123332.1.m.b.3039.12
      119.87odd241156.1.d.a.1155.12
      119.94odd241156.1.c.b.579.22
      119.101odd61156.1.f.b.327.12
      119.108even481156.1.g.b.399.28
      119.115odd1268.1.f.a.47.1✓2
      140.3odd121700.1.n.a.599.12
      140.59even61700.1.p.a.1551.12
      140.87odd121700.1.n.b.599.12
      357.353even12612.1.l.a.523.12
      476.3odd481156.1.g.b.155.18
      476.31odd481156.1.g.b.155.28
      476.47even123332.1.bc.c.863.14
      476.59even241156.1.c.b.579.12
      476.87even241156.1.d.a.1155.12
      476.115even1268.1.f.a.47.1✓2
      476.143odd481156.1.g.b.179.28
      476.199odd481156.1.g.b.423.18
      476.227odd481156.1.g.b.399.28
      476.251even43332.1.bc.c.2223.14
      476.283odd481156.1.g.b.399.18
      476.311odd481156.1.g.b.423.28
      476.319odd123332.1.m.b.3039.12
      476.339even61156.1.f.b.327.12
      476.367odd481156.1.g.b.179.18
      476.387odd12inner3332.1.bc.b.863.14
      476.395even121156.1.f.b.251.12
      476.423even241156.1.d.a.1155.22
      476.451even241156.1.c.b.579.22
      595.234odd121700.1.p.a.251.12
      595.353even121700.1.n.b.999.12
      595.472even121700.1.n.a.999.12
      952.115even121088.1.p.a.319.12
      952.829odd121088.1.p.a.319.12
      1428.1067odd12612.1.l.a.523.12
      2380.1067odd121700.1.n.a.999.12
      2380.1543odd121700.1.n.b.999.12
      2380.2019even121700.1.p.a.251.12
      ", "is_complex": true, "table_nest_level": "1"}}, {"type": "complex_table", "raw_content": "
              By twisted newform
      TwistMinDimCharParityOrdType
      68.1.f.a.47.1&check;2119.115odd12
      68.1.f.a.47.1&check;2476.115even12
      68.1.f.a.55.1yes27.3odd6
      68.1.f.a.55.1yes228.3even6
      612.1.l.a.55.1221.17even6
      612.1.l.a.55.1284.59odd6
      612.1.l.a.523.12357.353even12
      612.1.l.a.523.121428.1067odd12
      1088.1.p.a.191.1256.3even6
      1088.1.p.a.191.1256.45odd6
      1088.1.p.a.319.12952.115even12
      1088.1.p.a.319.12952.829odd12
      1156.1.c.b.579.12119.59odd24
      1156.1.c.b.579.12476.59even24
      1156.1.c.b.579.22119.94odd24
      1156.1.c.b.579.22476.451even24
      1156.1.d.a.1155.12119.87odd24
      1156.1.d.a.1155.12476.87even24
      1156.1.d.a.1155.22119.66odd24
      1156.1.d.a.1155.22476.423even24
      1156.1.f.b.251.12119.38odd12
      1156.1.f.b.251.12476.395even12
      1156.1.f.b.327.12119.101odd6
      1156.1.f.b.327.12476.339even6
      1156.1.g.b.155.18119.3even48
      1156.1.g.b.155.18476.3odd48
      1156.1.g.b.155.28119.31even48
      1156.1.g.b.155.28476.31odd48
      1156.1.g.b.179.18119.10even48
      1156.1.g.b.179.18476.367odd48
      1156.1.g.b.179.28119.24even48
      1156.1.g.b.179.28476.143odd48
      1156.1.g.b.399.18119.45even48
      1156.1.g.b.399.18476.283odd48
      1156.1.g.b.399.28119.108even48
      1156.1.g.b.399.28476.227odd48
      1156.1.g.b.423.18119.80even48
      1156.1.g.b.423.18476.199odd48
      1156.1.g.b.423.28119.73even48
      1156.1.g.b.423.28476.311odd48
      1700.1.n.a.599.1235.3even12
      1700.1.n.a.599.12140.3odd12
      1700.1.n.a.999.12595.472even12
      1700.1.n.a.999.122380.1067odd12
      1700.1.n.b.599.1235.17even12
      1700.1.n.b.599.12140.87odd12
      1700.1.n.b.999.12595.353even12
      1700.1.n.b.999.122380.1543odd12
      1700.1.p.a.251.12595.234odd12
      1700.1.p.a.251.122380.2019even12
      1700.1.p.a.1551.1235.24odd6
      1700.1.p.a.1551.12140.59even6
      3332.1.m.b.2843.127.4even3
      3332.1.m.b.2843.1228.11odd6
      3332.1.m.b.3039.12119.81even12
      3332.1.m.b.3039.12476.319odd12
      3332.1.bc.b.667.147.2even3inner
      3332.1.bc.b.667.1428.23odd6inner
      3332.1.bc.b.863.14119.30even12inner
      3332.1.bc.b.863.14476.387odd12inner
      3332.1.bc.b.2027.141.1even1trivial
      3332.1.bc.b.2027.144.3odd2CM
      3332.1.bc.b.2223.1417.13even4inner
      3332.1.bc.b.2223.1468.47odd4inner
      3332.1.bc.c.667.147.5odd6
      3332.1.bc.c.667.1428.19even6
      3332.1.bc.c.863.14119.47odd12
      3332.1.bc.c.863.14476.47even12
      3332.1.bc.c.2027.147.6odd2
      3332.1.bc.c.2027.1428.27even2
      3332.1.bc.c.2223.14119.13odd4
      3332.1.bc.c.2223.14476.251even4
      ", "content": {"html": "
              By twisted newform
      TwistMinDimCharParityOrdType
      68.1.f.a.47.1✓2119.115odd12
      68.1.f.a.47.1✓2476.115even12
      68.1.f.a.55.1yes27.3odd6
      68.1.f.a.55.1yes228.3even6
      612.1.l.a.55.1221.17even6
      612.1.l.a.55.1284.59odd6
      612.1.l.a.523.12357.353even12
      612.1.l.a.523.121428.1067odd12
      1088.1.p.a.191.1256.3even6
      1088.1.p.a.191.1256.45odd6
      1088.1.p.a.319.12952.115even12
      1088.1.p.a.319.12952.829odd12
      1156.1.c.b.579.12119.59odd24
      1156.1.c.b.579.12476.59even24
      1156.1.c.b.579.22119.94odd24
      1156.1.c.b.579.22476.451even24
      1156.1.d.a.1155.12119.87odd24
      1156.1.d.a.1155.12476.87even24
      1156.1.d.a.1155.22119.66odd24
      1156.1.d.a.1155.22476.423even24
      1156.1.f.b.251.12119.38odd12
      1156.1.f.b.251.12476.395even12
      1156.1.f.b.327.12119.101odd6
      1156.1.f.b.327.12476.339even6
      1156.1.g.b.155.18119.3even48
      1156.1.g.b.155.18476.3odd48
      1156.1.g.b.155.28119.31even48
      1156.1.g.b.155.28476.31odd48
      1156.1.g.b.179.18119.10even48
      1156.1.g.b.179.18476.367odd48
      1156.1.g.b.179.28119.24even48
      1156.1.g.b.179.28476.143odd48
      1156.1.g.b.399.18119.45even48
      1156.1.g.b.399.18476.283odd48
      1156.1.g.b.399.28119.108even48
      1156.1.g.b.399.28476.227odd48
      1156.1.g.b.423.18119.80even48
      1156.1.g.b.423.18476.199odd48
      1156.1.g.b.423.28119.73even48
      1156.1.g.b.423.28476.311odd48
      1700.1.n.a.599.1235.3even12
      1700.1.n.a.599.12140.3odd12
      1700.1.n.a.999.12595.472even12
      1700.1.n.a.999.122380.1067odd12
      1700.1.n.b.599.1235.17even12
      1700.1.n.b.599.12140.87odd12
      1700.1.n.b.999.12595.353even12
      1700.1.n.b.999.122380.1543odd12
      1700.1.p.a.251.12595.234odd12
      1700.1.p.a.251.122380.2019even12
      1700.1.p.a.1551.1235.24odd6
      1700.1.p.a.1551.12140.59even6
      3332.1.m.b.2843.127.4even3
      3332.1.m.b.2843.1228.11odd6
      3332.1.m.b.3039.12119.81even12
      3332.1.m.b.3039.12476.319odd12
      3332.1.bc.b.667.147.2even3inner
      3332.1.bc.b.667.1428.23odd6inner
      3332.1.bc.b.863.14119.30even12inner
      3332.1.bc.b.863.14476.387odd12inner
      3332.1.bc.b.2027.141.1even1trivial
      3332.1.bc.b.2027.144.3odd2CM
      3332.1.bc.b.2223.1417.13even4inner
      3332.1.bc.b.2223.1468.47odd4inner
      3332.1.bc.c.667.147.5odd6
      3332.1.bc.c.667.1428.19even6
      3332.1.bc.c.863.14119.47odd12
      3332.1.bc.c.863.14476.47even12
      3332.1.bc.c.2027.147.6odd2
      3332.1.bc.c.2027.1428.27even2
      3332.1.bc.c.2223.14119.13odd4
      3332.1.bc.c.2223.14476.251even4
      ", "is_complex": true, "table_nest_level": "1"}}]], "html": "\n\n\n \n \n \n \n LMFDB - Embedded newform 3332.1.bc.b.2027.1 \n \n\n \n \n \n \n \n\n \n \n\n\n\n\n \n\n \n \n\n \n \n\n\n \n \n \n \n \n \n\n\n\n\n\n \n \n\n
      \n \n
      \n
      \n
      \n ⌂\n → Modular forms\n → Classical\n → Level 3332\n → Weight 1\n → Character orbit bc\n → Newform orbit b\n → Embedding 2027.1\n
      \n
      \n \n
      \n Citation\n ·\n Feedback\n ·\n Hide Menu\n \n
      \n
      \n\n
      \n
      Embedded newform 3332.1.bc.b.2027.1
      \n\n
      \n
      \n\n\n\n\n\n
      \n
      ↑
      \n

      Properties

      \n
      \n \n
      Label\n 3332.1.bc.b.2027.1
      \n
      Level\n $3332$
      Weight\n $1$
      Character\n 3332.2027
      Analytic conductor\n $1.663$
      Analytic rank\n $0$
      Dimension\n $4$
      Projective image\n $D_{4}$
      CM discriminant\n -4
      Inner twists\n $8$
      \n
      \n\n\n\n

      Related objects

      \n
      \n \n
      \n\n\n\n

      Downloads

      \n
      \n \n
      \n\n

      Learn more

      \n
      \n \n
      \n\n
      \n
      \n
      \n
      \n
      \n \n
      \n Show commands:\n Magma\n / PariGP\n / SageMath\n
      \n\n\n\n\n

      Newspace parameters

      \n\n
      comment: Compute space of new eigenforms
       
      \n
      [N,k,chi] = [3332,1,Mod(667,3332)]
       
      mf = mfinit([N,k,chi],0)
       
      lf = mfeigenbasis(mf)
       
      \n
      from sage.modular.dirichlet import DirichletCharacter
       
      H = DirichletGroup(3332, base_ring=CyclotomicField(12))
       
      chi = DirichletCharacter(H, H._module([6, 4, 9]))
       
      N = Newforms(chi, 1, names="a")
       
      \n
      //Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
       
      chi := DirichletCharacter("3332.667");
       
      S:= CuspForms(chi, 1);
       
      N := Newforms(S);
       
      \n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
      Level: \\( N \\) \\(=\\)\\( 3332 = 2^{2} \\cdot 7^{2} \\cdot 17 \\)
      Weight: \\( k \\) \\(=\\)\\( 1 \\)
      Character orbit: \\([\\chi]\\) \\(=\\) 3332.bc (of order \\(12\\), degree \\(4\\), not minimal)
      \n\n

      Newform invariants

      \n\n
      comment: select newform
       
      \n
      sage: f = N[0] # Warning: the index may be different
       
      \n
      gp: f = lf[1] \\\\ Warning: the index may be different
       
      \n\n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\t\n \n \t\n \t\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
      Self dual: no
      Analytic conductor: \\(1.66288462209\\)
      Analytic rank: \\(0\\)
      Dimension: \\(4\\)
      Coefficient field: \\(\\Q(\\zeta_{12})\\)
      \n
      comment: defining polynomial
       
      \n
      gp: f.mod \\\\ as an extension of the character field
       
      \n\n
      Defining polynomial: \n\n \\( x^{4} - x^{2} + 1 \\)\n \n\n \n \"Copy\n \n \n \"Toggle\n \n
      Coefficient ring: \\(\\Z[a_1, a_2]\\)
      Coefficient ring index: \\( 1 \\)
      Twist minimal: no (minimal twist has level 68)
      Projective image:\\(D_{4}\\)
      Projective field:Galois closure of 4.2.19652.1
      Artin image:$C_4\\wr C_2\\times C_6$
      Artin field:Galois closure of \\(\\mathbb{Q}[x]/(x^{48} - \\cdots)\\)
      \n\n\n

      Embedding invariants

      \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n
      Embedding label 2027.1
      Root\\(0.866025 - 0.500000i\\) of defining polynomial
      Character\\(\\chi\\)\\(=\\)3332.2027
      Dual form 3332.1.bc.b.863.1
      \n\n\n

      $q$-expansion

      \n
      \n
      comment: q-expansion
       
      \n
      sage: f.q_expansion() # note that sage often uses an isomorphic number field
       
      \n
      gp: mfcoefs(f, 20)
       
      \n\n
      \n \n \n \n \n \n \n \n \n \n \n \n
      \\(f(q)\\)\\(=\\)\\(q+(-0.866025 + 0.500000i) q^{2} +(0.500000 - 0.866025i) q^{4} +(-1.36603 - 0.366025i) q^{5} +1.00000i q^{8} +(-0.866025 + 0.500000i) q^{9} +(1.36603 - 0.366025i) q^{10} +(-0.500000 - 0.866025i) q^{16} +(-0.500000 + 0.866025i) q^{17} +(0.500000 - 0.866025i) q^{18} +(-1.00000 + 1.00000i) q^{20} +(0.866025 + 0.500000i) q^{25} +(1.00000 - 1.00000i) q^{29} +(0.866025 + 0.500000i) q^{32} -1.00000i q^{34} +1.00000i q^{36} +(1.36603 + 0.366025i) q^{37} +(0.366025 - 1.36603i) q^{40} +(-1.00000 - 1.00000i) q^{41} +(1.36603 - 0.366025i) q^{45} -1.00000 q^{50} +(-0.366025 + 1.36603i) q^{58} +(0.366025 - 1.36603i) q^{61} -1.00000 q^{64} +(0.500000 + 0.866025i) q^{68} +(-0.500000 - 0.866025i) q^{72} +(-0.366025 - 1.36603i) q^{73} +(-1.36603 + 0.366025i) q^{74} +(0.366025 + 1.36603i) q^{80} +(0.500000 - 0.866025i) q^{81} +(1.36603 + 0.366025i) q^{82} +(1.00000 - 1.00000i) q^{85} +(-1.00000 + 1.00000i) q^{90} +(1.00000 - 1.00000i) q^{97} +O(q^{100})\\)
      \\(\\operatorname{Tr}(f)(q)\\)\\(=\\)\n\n \\( 4 q + 2 q^{4} - 2 q^{5} + 2 q^{10} - 2 q^{16} - 2 q^{17} + 2 q^{18} - 4 q^{20} + 4 q^{29} + 2 q^{37} - 2 q^{40} - 4 q^{41} + 2 q^{45} - 4 q^{50} + 2 q^{58} - 2 q^{61} - 4 q^{64} + 2 q^{68} - 2 q^{72} + 2 q^{73}+ \\cdots + 4 q^{97}+O(q^{100}) \\)\n \n\n \n \"Copy\n \n \n \"Toggle\n \n
      \n
      \n\n
      \n\n\n

      Character values

      \n

      We give the values of \\(\\chi\\) on generators for \\(\\left(\\mathbb{Z}/3332\\mathbb{Z}\\right)^\\times\\).

      \n\n \n \n \n \n \n \n \n \n \n \n \n \n
      \\(n\\)\\(785\\)\\(885\\)\\(1667\\)
      \\(\\chi(n)\\)\\(e\\left(\\frac{3}{4}\\right)\\)\\(e\\left(\\frac{2}{3}\\right)\\)\\(-1\\)
      \n\n\n

      Coefficient data

      \n\n

      For each \\(n\\) we display the coefficients of the \\(q\\)-expansion \\(a_n\\), the\nSatake parameters \\(\\alpha_p\\),\nand the Satake angles \\(\\theta_p = \\textrm{Arg}(\\alpha_p)\\).

      \n\n\n\n\n

      \n
      \n \n Display \\(a_p\\) with \\(p\\) up to:\n 50\n 250\n 1000\n \n\n \n (See \\(a_n\\) instead)\n \n\n \n (See \\(a_n\\) instead)\n \n\n \n (See \\(a_n\\) instead)\n \n \n Display \\(a_n\\) with \\(n\\) up to:\n 50\n 250\n 1000\n \n\n \n (See only \\(a_p\\))\n \n\n \n (See only \\(a_p\\))\n \n\n \n (See only \\(a_p\\))\n \n
      \n\n\n
      \n \n \n \n \n \n \n
      Significant digits:
      \n
      \n\n
      \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n 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\n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n 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\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n 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\n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n \n \n \n \n
      \n \\(n\\)\n \n \\(a_n\\)\n \n \\(a_n / n^{(k-1)/2}\\)\n \n \\( \\alpha_n \\)\n \n \\( \\theta_n \\)\n
      \n \\(p\\)\n \n \\(a_p\\)\n \n \\(a_p / p^{(k-1)/2}\\)\n \n \\( \\alpha_p\\)\n \n \\( \\theta_p \\)\n
      \n \\(2\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n \n\n \n\n \n\n \n\n
      \n\n \n\n \n\n \n\n
      \n \\(3\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(4\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
      \n \\(5\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(6\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(7\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n\n \n\n \n\n
      \n\n \n\n \n\n \n\n
      \n \\(8\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
      \n \\(9\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n
      \n \\(10\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(11\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(12\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(13\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(14\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(15\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(16\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
      \n \\(17\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n \n\n \n\n \n\n \n\n
      \n\n \n\n \n\n \n\n
      \n \\(18\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
      \n \\(19\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(20\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
      \n \\(21\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(22\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(23\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
      \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
      \n \\(24\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(25\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
      \n \\(26\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(27\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(28\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(29\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n \\(30\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(31\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(32\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
      \n \\(33\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(34\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
      \n \\(35\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(36\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
      \n \\(37\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(38\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(39\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(40\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n
      \n \\(41\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(42\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(43\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(44\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(45\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(46\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(47\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(48\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(49\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(50\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
      \n \\(51\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(52\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(53\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(54\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(55\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(56\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(57\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(58\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(59\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(60\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(61\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n \\(62\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(63\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(64\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
      \n \\(65\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(66\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(67\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(68\\)\n \n 0.500000\n \n +\n \n 0.866025i\n \n 0.500000\n \n +\n \n 0.866025i\n
      \n \\(69\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(70\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(71\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(72\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
      \n \\(73\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n \\(74\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(75\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(76\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(77\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(78\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(79\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
      \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
      \n \\(80\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(81\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
      \n \\(82\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(83\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(84\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(85\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(86\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(87\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(88\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(89\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(90\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
      \n \\(91\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(92\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(93\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(94\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(95\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(96\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(97\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n \\(98\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(99\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(100\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
      \n \\(101\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(102\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(103\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(104\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(105\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(106\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(107\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(108\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(109\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(110\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(111\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(112\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(113\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(114\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(115\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(116\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
      \n \\(117\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(118\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(119\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(120\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(121\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
      \n \\(122\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(123\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(124\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(125\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(126\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(127\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(128\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
      \n \\(129\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(130\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(131\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(132\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(133\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(134\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(135\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(136\\)\n \n −0.866025\n \n −\n \n 0.500000i\n \n −0.866025\n \n −\n \n 0.500000i\n
      \n \\(137\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(138\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(139\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(140\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(141\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(142\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(143\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(144\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
      \n \\(145\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n
      \n \\(146\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(147\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(148\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(149\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n \\(150\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(151\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(152\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(153\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
      \n \\(154\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(155\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(156\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(157\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n \\(158\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(159\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(160\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
      \n \\(161\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(162\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
      \n \\(163\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
      \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
      \n \\(164\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(165\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(166\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(167\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(168\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(169\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
      \n \\(170\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(171\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(172\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(173\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(174\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(175\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(176\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(177\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(178\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(179\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(180\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n
      \n \\(181\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(182\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(183\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(184\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(185\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n
      \n \\(186\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(187\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(188\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(189\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(190\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(191\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(192\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(193\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(194\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(195\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(196\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(197\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(198\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(199\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(200\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
      \n \\(201\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(202\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(203\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(204\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(205\\)\n \n 1.00000\n \n +\n \n 1.73205i\n \n 1.00000\n \n +\n \n 1.73205i\n
      \n \\(206\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(207\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(208\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(209\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(210\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(211\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
      \n \\(212\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(213\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(214\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(215\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(216\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(217\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(218\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
      \n \\(219\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(220\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(221\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(222\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(223\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(224\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(225\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
      \n \\(226\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(227\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(228\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(229\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(230\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(231\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(232\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(233\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(234\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(235\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(236\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(237\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(238\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(239\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(240\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(241\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n \\(242\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
      \n \\(243\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(244\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
      \n \\(245\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(246\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(247\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(248\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(249\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(250\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(251\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(252\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(253\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(254\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(255\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(256\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
      \n \\(257\\)\n \n 1.73205\n \n −\n \n 1.00000i\n \n 1.73205\n \n −\n \n 1.00000i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n \\(258\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(259\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(260\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(261\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(262\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(263\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(264\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(265\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(266\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(267\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(268\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(269\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(270\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(271\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(272\\)\n \n 1.00000\n \n\n \n\n \n 1.00000\n \n\n \n\n
      \n \\(273\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(274\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(275\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(276\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(277\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(278\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(279\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(280\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(281\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(282\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(283\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(284\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(285\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(286\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(287\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(288\\)\n \n −1.00000\n \n\n \n\n \n −1.00000\n \n\n \n\n
      \n \\(289\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
      \n \\(290\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n
      \n \\(291\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(292\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
      \n \\(293\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n 1.00000\n \n\n \n\n \n \\(0\\)\n
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      \n \\(295\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(296\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(297\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(298\\)\n \n 1.73205\n \n +\n \n 1.00000i\n \n 1.73205\n \n +\n \n 1.00000i\n
      \n \\(299\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(300\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(301\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(302\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(303\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(304\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(305\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n
      \n \\(306\\)\n \n 0.500000\n \n +\n \n 0.866025i\n \n 0.500000\n \n +\n \n 0.866025i\n
      \n \\(307\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(308\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(309\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(310\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(311\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(312\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(313\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(314\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n
      \n \\(315\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(316\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(317\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n \\(318\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(319\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(320\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(321\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(322\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(323\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(324\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
      \n \\(325\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(326\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(327\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(328\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(329\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(330\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(331\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(332\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(333\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(334\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(335\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(336\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(337\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n \\(338\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
      \n \\(339\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(340\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
      \n \\(341\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(342\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(343\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(344\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(345\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(346\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(347\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(348\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(349\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(350\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(351\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(352\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(353\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(354\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(355\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(356\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(357\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(358\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
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      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(360\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(361\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
      \n \\(362\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
      \n \\(363\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(364\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(365\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n
      \n \\(366\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(367\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(368\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(369\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(370\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n
      \n \\(371\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(372\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(373\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(374\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(375\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(376\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(377\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(378\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(379\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(380\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(381\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(382\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(383\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(384\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(385\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(386\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(387\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(388\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
      \n \\(389\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n \\(390\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(391\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(392\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(393\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(394\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
      \n \\(395\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(396\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(397\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n \\(398\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(399\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(400\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
      \n \\(401\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n \\(402\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(403\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(404\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(405\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
      \n \\(406\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(407\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(408\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(409\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(410\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n
      \n \\(411\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(412\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(413\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(414\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(415\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(416\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(417\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(418\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(419\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
      \n \\(420\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(421\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(422\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(423\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(424\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(425\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n
      \n \\(426\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(427\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(428\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(429\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(430\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(431\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(432\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(433\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(434\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(435\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(436\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n
      \n \\(437\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(438\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(439\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(440\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(441\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(442\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(443\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(444\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(445\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(446\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(447\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(448\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(449\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(450\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
      \n \\(451\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(452\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(453\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(454\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(455\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(456\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(457\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(458\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n
      \n \\(459\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(460\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(461\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(462\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(463\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(464\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
      \n \\(465\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(466\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(467\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(468\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(469\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(470\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(471\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(472\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(473\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(474\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(475\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(476\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(477\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(478\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(479\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(480\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(481\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(482\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(483\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(484\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
      \n \\(485\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n
      \n \\(486\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(487\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(488\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(489\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(490\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(491\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(492\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(493\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(494\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(495\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(496\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(497\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(498\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(499\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
      \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
      \n \\(500\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(501\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(502\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(503\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
      \n \\(504\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(505\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(506\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(507\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(508\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(509\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n \\(510\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(511\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(512\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
      \n \\(513\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(514\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n
      \n \\(515\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(516\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(517\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(518\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(519\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(520\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(521\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(522\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
      \n \\(523\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(524\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(525\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(526\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(527\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(528\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(529\\)\n \n −0.866025\n \n −\n \n 0.500000i\n \n −0.866025\n \n −\n \n 0.500000i\n
      \n \\(530\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(531\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(532\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(533\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(534\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(535\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(536\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(537\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(538\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
      \n \\(539\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(540\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(541\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(542\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(543\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(544\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n
      \n \\(545\\)\n \n −2.00000\n \n\n \n\n \n −2.00000\n \n\n \n\n
      \n \\(546\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(547\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(548\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(549\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(550\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(551\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(552\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(553\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(554\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
      \n \\(555\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(556\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(557\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(558\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(559\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(560\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(561\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(562\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(563\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(564\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(565\\)\n \n 1.00000\n \n +\n \n 1.73205i\n \n 1.00000\n \n +\n \n 1.73205i\n
      \n \\(566\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(567\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(568\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(569\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(570\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(571\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(572\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(573\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(574\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(575\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(576\\)\n \n 0.866025\n \n −\n \n 0.500000i\n \n 0.866025\n \n −\n \n 0.500000i\n
      \n \\(577\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(578\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
      \n \\(579\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(580\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n
      \n \\(581\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(582\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(583\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(584\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(585\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(586\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n
      \n \\(587\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(588\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(589\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(590\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(591\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(592\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
      \n \\(593\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(594\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(595\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(596\\)\n \n −2.00000\n \n\n \n\n \n −2.00000\n \n\n \n\n
      \n \\(597\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(598\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(599\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(600\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(601\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(602\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(603\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(604\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(605\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(606\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(607\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(608\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(609\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(610\\)\n \n\n \n −\n \n 2.00000i\n \n\n \n −\n \n 2.00000i\n
      \n \\(611\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(612\\)\n \n −0.866025\n \n −\n \n 0.500000i\n \n −0.866025\n \n −\n \n 0.500000i\n
      \n \\(613\\)\n \n −1.00000\n \n +\n \n 1.73205i\n \n −1.00000\n \n +\n \n 1.73205i\n \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(614\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(615\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(616\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(617\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(618\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(619\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(620\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(621\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(622\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(623\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(624\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(625\\)\n \n −0.500000\n \n −\n \n 0.866025i\n \n −0.500000\n \n −\n \n 0.866025i\n
      \n \\(626\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n
      \n \\(627\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(628\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
      \n \\(629\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
      \n \\(630\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(631\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(632\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(633\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(634\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(635\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(636\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(637\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(638\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(639\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(640\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(641\\)\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.366025\n \n −\n \n 1.36603i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n \\(642\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(643\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(644\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(645\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(646\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(647\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(648\\)\n \n 0.866025\n \n +\n \n 0.500000i\n \n 0.866025\n \n +\n \n 0.500000i\n
      \n \\(649\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(650\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(651\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(652\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(653\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n \\(654\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(655\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(656\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(657\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(658\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(659\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(660\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(661\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(662\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(663\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(664\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(665\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(666\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(667\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(668\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(669\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(670\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(671\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(672\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(673\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(674\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(675\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(676\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
      \n \\(677\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(678\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(679\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(680\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(681\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(682\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(683\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(684\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(685\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(686\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(687\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(688\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(689\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(690\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(691\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.258819\n \n −\n \n 0.965926i\n \n \\(-0.416667\\pi\\)\n
      \n −0.258819\n \n +\n \n 0.965926i\n \n \\(0.583333\\pi\\)\n
      \n \\(692\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(693\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(694\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(695\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(696\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(697\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(698\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(699\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(700\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(701\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(702\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(703\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(704\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(705\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(706\\)\n \n\n \n −\n \n 2.00000i\n \n\n \n −\n \n 2.00000i\n
      \n \\(707\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(708\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(709\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(710\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(711\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(712\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(713\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(714\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(715\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(716\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(717\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(718\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(719\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(720\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
      \n \\(721\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(722\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
      \n \\(723\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(724\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(725\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(726\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(727\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(728\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(729\\)\n \n\n \n\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n
      \n \\(730\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
      \n \\(731\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(732\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(733\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(734\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(735\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(736\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(737\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(738\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(739\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(740\\)\n \n −1.73205\n \n +\n \n 1.00000i\n \n −1.73205\n \n +\n \n 1.00000i\n
      \n \\(741\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(742\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(743\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(744\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(745\\)\n \n 0.732051\n \n +\n \n 2.73205i\n \n 0.732051\n \n +\n \n 2.73205i\n
      \n \\(746\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(747\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(748\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(749\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(750\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(751\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(752\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(753\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(754\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(755\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(756\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(757\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(758\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(759\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(760\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(761\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(762\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(763\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(764\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(765\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(766\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(767\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(768\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(769\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(770\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(771\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(772\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(773\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n −0.866025\n \n −\n \n 0.500000i\n \n \\(-0.833333\\pi\\)\n
      \n \\(774\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(775\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(776\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(777\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(778\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n
      \n \\(779\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(780\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(781\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(782\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(783\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(784\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(785\\)\n \n −2.00000\n \n +\n \n 2.00000i\n \n −2.00000\n \n +\n \n 2.00000i\n
      \n \\(786\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(787\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(788\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(789\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(790\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(791\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(792\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(793\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(794\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(795\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(796\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(797\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(798\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(799\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(800\\)\n \n 0.500000\n \n +\n \n 0.866025i\n \n 0.500000\n \n +\n \n 0.866025i\n
      \n \\(801\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(802\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(803\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(804\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(805\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(806\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(807\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(808\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(809\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(810\\)\n \n 0.366025\n \n −\n \n 1.36603i\n \n 0.366025\n \n −\n \n 1.36603i\n
      \n \\(811\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.707107\n \n −\n \n 0.707107i\n \n \\(-0.250000\\pi\\)\n
      \n −0.707107\n \n +\n \n 0.707107i\n \n \\(0.750000\\pi\\)\n
      \n \\(812\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(813\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(814\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(815\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(816\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(817\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(818\\)\n \n\n \n −\n \n 2.00000i\n \n\n \n −\n \n 2.00000i\n
      \n \\(819\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(820\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n
      \n \\(821\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(822\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(823\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(824\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(825\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(826\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(827\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
      \n \\(828\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(829\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n \\(830\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(831\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(832\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(833\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(834\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(835\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(836\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(837\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(838\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(839\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
      \n \\(840\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(841\\)\n \n\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n
      \n \\(842\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(843\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(844\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(845\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(846\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(847\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(848\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(849\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(850\\)\n \n 0.500000\n \n −\n \n 0.866025i\n \n 0.500000\n \n −\n \n 0.866025i\n
      \n \\(851\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(852\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(853\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n \\(854\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(855\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(856\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(857\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(858\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(859\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(860\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(861\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(862\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(863\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(864\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(865\\)\n \n −1.73205\n \n −\n \n 1.00000i\n \n −1.73205\n \n −\n \n 1.00000i\n
      \n \\(866\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
      \n \\(867\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(868\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(869\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(870\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(871\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(872\\)\n \n 0.366025\n \n +\n \n 1.36603i\n \n 0.366025\n \n +\n \n 1.36603i\n
      \n \\(873\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n
      \n \\(874\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(875\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(876\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(877\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(878\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(879\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(880\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(881\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(882\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(883\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(884\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(885\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(886\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(887\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(888\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(889\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(890\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(891\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(892\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(893\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(894\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(895\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(896\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(897\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(898\\)\n \n 1.36603\n \n +\n \n 0.366025i\n \n 1.36603\n \n +\n \n 0.366025i\n
      \n \\(899\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(900\\)\n \n −0.500000\n \n +\n \n 0.866025i\n \n −0.500000\n \n +\n \n 0.866025i\n
      \n \\(901\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(902\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(903\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(904\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(905\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
      \n \\(906\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(907\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.965926\n \n −\n \n 0.258819i\n \n \\(-0.0833333\\pi\\)\n
      \n −0.965926\n \n +\n \n 0.258819i\n \n \\(0.916667\\pi\\)\n
      \n \\(908\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(909\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(910\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(911\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.707107\n \n −\n \n 0.707107i\n \n \\(-0.750000\\pi\\)\n
      \n 0.707107\n \n +\n \n 0.707107i\n \n \\(0.250000\\pi\\)\n
      \n \\(912\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(913\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(914\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n
      \n \\(915\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(916\\)\n \n\n \n\n \n 2.00000i\n \n\n \n\n \n 2.00000i\n
      \n \\(917\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(918\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(919\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.500000\n \n −\n \n 0.866025i\n \n \\(-0.666667\\pi\\)\n
      \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n \\(920\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(921\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(922\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(923\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(924\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(925\\)\n \n 1.00000\n \n +\n \n 1.00000i\n \n 1.00000\n \n +\n \n 1.00000i\n
      \n \\(926\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(927\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(928\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n
      \n \\(929\\)\n \n −0.366025\n \n +\n \n 1.36603i\n \n −0.366025\n \n +\n \n 1.36603i\n \n 0.500000\n \n +\n \n 0.866025i\n \n \\(0.333333\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(930\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(931\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(932\\)\n \n 1.00000\n \n −\n \n 1.00000i\n \n 1.00000\n \n −\n \n 1.00000i\n
      \n \\(933\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(934\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(935\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(936\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(937\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 1.00000\n \n\n \n\n \n \\(0\\)\n
      \n −1.00000\n \n\n \n\n \n \\(\\pi\\)\n
      \n \\(938\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(939\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(940\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(941\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(942\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(943\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(944\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(945\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(946\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(947\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.965926\n \n −\n \n 0.258819i\n \n \\(-0.916667\\pi\\)\n
      \n 0.965926\n \n +\n \n 0.258819i\n \n \\(0.0833333\\pi\\)\n
      \n \\(948\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(949\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(950\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(951\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(952\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(953\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(954\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(955\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(956\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(957\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(958\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(959\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(960\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(961\\)\n \n −0.866025\n \n +\n \n 0.500000i\n \n −0.866025\n \n +\n \n 0.500000i\n
      \n \\(962\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(963\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(964\\)\n \n −1.36603\n \n −\n \n 0.366025i\n \n −1.36603\n \n −\n \n 0.366025i\n
      \n \\(965\\)\n \n 2.00000\n \n\n \n\n \n 2.00000\n \n\n \n\n
      \n \\(966\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(967\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n\n \n −\n \n 1.00000i\n \n \\(-0.5\\pi\\)\n
      \n\n \n\n \n 1.00000i\n \n \\(0.5\\pi\\)\n
      \n \\(968\\)\n \n 0.500000\n \n +\n \n 0.866025i\n \n 0.500000\n \n +\n \n 0.866025i\n
      \n \\(969\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(970\\)\n \n 1.00000\n \n −\n \n 1.73205i\n \n 1.00000\n \n −\n \n 1.73205i\n
      \n \\(971\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.866025\n \n −\n \n 0.500000i\n \n \\(-0.166667\\pi\\)\n
      \n −0.866025\n \n +\n \n 0.500000i\n \n \\(0.833333\\pi\\)\n
      \n \\(972\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(973\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(974\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(975\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(976\\)\n \n −1.36603\n \n +\n \n 0.366025i\n \n −1.36603\n \n +\n \n 0.366025i\n
      \n \\(977\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n −0.500000\n \n +\n \n 0.866025i\n \n \\(0.666667\\pi\\)\n
      \n \\(978\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(979\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(980\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(981\\)\n \n −1.00000\n \n +\n \n 1.00000i\n \n −1.00000\n \n +\n \n 1.00000i\n
      \n \\(982\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(983\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(984\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(985\\)\n \n −1.00000\n \n −\n \n 1.73205i\n \n −1.00000\n \n −\n \n 1.73205i\n
      \n \\(986\\)\n \n −1.00000\n \n −\n \n 1.00000i\n \n −1.00000\n \n −\n \n 1.00000i\n
      \n \\(987\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(988\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(989\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(990\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(991\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n \n −0.258819\n \n −\n \n 0.965926i\n \n \\(-0.583333\\pi\\)\n
      \n 0.258819\n \n +\n \n 0.965926i\n \n \\(0.416667\\pi\\)\n
      \n \\(992\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(993\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(994\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(995\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(996\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(997\\)\n \n 1.36603\n \n −\n \n 0.366025i\n \n 1.36603\n \n −\n \n 0.366025i\n \n 0.500000\n \n −\n \n 0.866025i\n \n \\(-0.333333\\pi\\)\n
      \n 0.866025\n \n +\n \n 0.500000i\n \n \\(0.166667\\pi\\)\n
      \n \\(998\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n \\(999\\)\n \n\n \n 0\n \n\n \n\n \n 0\n \n\n
      \n
      \n
      \n \n Display \\(a_p\\) with \\(p\\) up to:\n 50\n 250\n 1000\n \n\n \n (See \\(a_n\\) instead)\n \n\n \n (See \\(a_n\\) instead)\n \n\n \n (See \\(a_n\\) instead)\n \n \n Display \\(a_n\\) with \\(n\\) up to:\n 50\n 250\n 1000\n \n\n \n (See only \\(a_p\\))\n \n\n \n (See only \\(a_p\\))\n \n\n \n (See only \\(a_p\\))\n \n
      \n\n\n

      Twists

      \n\n\n\n\n \n \n \n \n \n \n \n\n\n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n
             By twisting character
      CharParityOrdTypeTwistMinDim
      1.1even1trivial3332.1.bc.b.2027.14
      4.3odd2CM3332.1.bc.b.2027.14
      7.2even3inner3332.1.bc.b.667.14
      7.3odd668.1.f.a.55.1yes2
      7.4even33332.1.m.b.2843.12
      7.5odd63332.1.bc.c.667.14
      7.6odd23332.1.bc.c.2027.14
      17.13even4inner3332.1.bc.b.2223.14
      21.17even6612.1.l.a.55.12
      28.3even668.1.f.a.55.1yes2
      28.11odd63332.1.m.b.2843.12
      28.19even63332.1.bc.c.667.14
      28.23odd6inner3332.1.bc.b.667.14
      28.27even23332.1.bc.c.2027.14
      35.3even121700.1.n.a.599.12
      35.17even121700.1.n.b.599.12
      35.24odd61700.1.p.a.1551.12
      56.3even61088.1.p.a.191.12
      56.45odd61088.1.p.a.191.12
      68.47odd4inner3332.1.bc.b.2223.14
      84.59odd6612.1.l.a.55.12
      119.3even481156.1.g.b.155.18
      119.10even481156.1.g.b.179.18
      119.13odd43332.1.bc.c.2223.14
      119.24even481156.1.g.b.179.28
      119.30even12inner3332.1.bc.b.863.14
      119.31even481156.1.g.b.155.28
      119.38odd121156.1.f.b.251.12
      119.45even481156.1.g.b.399.18
      119.47odd123332.1.bc.c.863.14
      119.59odd241156.1.c.b.579.12
      119.66odd241156.1.d.a.1155.22
      119.73even481156.1.g.b.423.28
      119.80even481156.1.g.b.423.18
      119.81even123332.1.m.b.3039.12
      119.87odd241156.1.d.a.1155.12
      119.94odd241156.1.c.b.579.22
      119.101odd61156.1.f.b.327.12
      119.108even481156.1.g.b.399.28
      119.115odd1268.1.f.a.47.1✓2
      140.3odd121700.1.n.a.599.12
      140.59even61700.1.p.a.1551.12
      140.87odd121700.1.n.b.599.12
      357.353even12612.1.l.a.523.12
      476.3odd481156.1.g.b.155.18
      476.31odd481156.1.g.b.155.28
      476.47even123332.1.bc.c.863.14
      476.59even241156.1.c.b.579.12
      476.87even241156.1.d.a.1155.12
      476.115even1268.1.f.a.47.1✓2
      476.143odd481156.1.g.b.179.28
      476.199odd481156.1.g.b.423.18
      476.227odd481156.1.g.b.399.28
      476.251even43332.1.bc.c.2223.14
      476.283odd481156.1.g.b.399.18
      476.311odd481156.1.g.b.423.28
      476.319odd123332.1.m.b.3039.12
      476.339even61156.1.f.b.327.12
      476.367odd481156.1.g.b.179.18
      476.387odd12inner3332.1.bc.b.863.14
      476.395even121156.1.f.b.251.12
      476.423even241156.1.d.a.1155.22
      476.451even241156.1.c.b.579.22
      595.234odd121700.1.p.a.251.12
      595.353even121700.1.n.b.999.12
      595.472even121700.1.n.a.999.12
      952.115even121088.1.p.a.319.12
      952.829odd121088.1.p.a.319.12
      1428.1067odd12612.1.l.a.523.12
      2380.1067odd121700.1.n.a.999.12
      2380.1543odd121700.1.n.b.999.12
      2380.2019even121700.1.p.a.251.12
      \n
          
      \n\n\n\n\n \n \n \n \n \n \n \n\n\n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n \n \n \n \n \n \n \n\n\n
              By twisted newform
      TwistMinDimCharParityOrdType
      68.1.f.a.47.1✓2119.115odd12
      68.1.f.a.47.1✓2476.115even12
      68.1.f.a.55.1yes27.3odd6
      68.1.f.a.55.1yes228.3even6
      612.1.l.a.55.1221.17even6
      612.1.l.a.55.1284.59odd6
      612.1.l.a.523.12357.353even12
      612.1.l.a.523.121428.1067odd12
      1088.1.p.a.191.1256.3even6
      1088.1.p.a.191.1256.45odd6
      1088.1.p.a.319.12952.115even12
      1088.1.p.a.319.12952.829odd12
      1156.1.c.b.579.12119.59odd24
      1156.1.c.b.579.12476.59even24
      1156.1.c.b.579.22119.94odd24
      1156.1.c.b.579.22476.451even24
      1156.1.d.a.1155.12119.87odd24
      1156.1.d.a.1155.12476.87even24
      1156.1.d.a.1155.22119.66odd24
      1156.1.d.a.1155.22476.423even24
      1156.1.f.b.251.12119.38odd12
      1156.1.f.b.251.12476.395even12
      1156.1.f.b.327.12119.101odd6
      1156.1.f.b.327.12476.339even6
      1156.1.g.b.155.18119.3even48
      1156.1.g.b.155.18476.3odd48
      1156.1.g.b.155.28119.31even48
      1156.1.g.b.155.28476.31odd48
      1156.1.g.b.179.18119.10even48
      1156.1.g.b.179.18476.367odd48
      1156.1.g.b.179.28119.24even48
      1156.1.g.b.179.28476.143odd48
      1156.1.g.b.399.18119.45even48
      1156.1.g.b.399.18476.283odd48
      1156.1.g.b.399.28119.108even48
      1156.1.g.b.399.28476.227odd48
      1156.1.g.b.423.18119.80even48
      1156.1.g.b.423.18476.199odd48
      1156.1.g.b.423.28119.73even48
      1156.1.g.b.423.28476.311odd48
      1700.1.n.a.599.1235.3even12
      1700.1.n.a.599.12140.3odd12
      1700.1.n.a.999.12595.472even12
      1700.1.n.a.999.122380.1067odd12
      1700.1.n.b.599.1235.17even12
      1700.1.n.b.599.12140.87odd12
      1700.1.n.b.999.12595.353even12
      1700.1.n.b.999.122380.1543odd12
      1700.1.p.a.251.12595.234odd12
      1700.1.p.a.251.122380.2019even12
      1700.1.p.a.1551.1235.24odd6
      1700.1.p.a.1551.12140.59even6
      3332.1.m.b.2843.127.4even3
      3332.1.m.b.2843.1228.11odd6
      3332.1.m.b.3039.12119.81even12
      3332.1.m.b.3039.12476.319odd12
      3332.1.bc.b.667.147.2even3inner
      3332.1.bc.b.667.1428.23odd6inner
      3332.1.bc.b.863.14119.30even12inner
      3332.1.bc.b.863.14476.387odd12inner
      3332.1.bc.b.2027.141.1even1trivial
      3332.1.bc.b.2027.144.3odd2CM
      3332.1.bc.b.2223.1417.13even4inner
      3332.1.bc.b.2223.1468.47odd4inner
      3332.1.bc.c.667.147.5odd6
      3332.1.bc.c.667.1428.19even6
      3332.1.bc.c.863.14119.47odd12
      3332.1.bc.c.863.14476.47even12
      3332.1.bc.c.2027.147.6odd2
      3332.1.bc.c.2027.1428.27even2
      3332.1.bc.c.2223.14119.13odd4
      3332.1.bc.c.2223.14476.251even4
      \n
      \n\n\n\n

      \n
      \n
      \n\n\n
      \n\n

      This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

      \n
      \n Contact\n ·\n Citation\n ·\n Acknowledgments\n ·\n Editorial Board\n ·\n Source\n ·\n SageMath version 10.1\n ·\n LMFDB Release 1.2.1\n
      \n
      \n\n", "statics": {"paragraph": 5, "paragraph.text": 24, "code": 8, "simple_table": 4, "title": 4, "complex_table": 5, "complex_table.complex": 5, "paragraph.equation-inline": 19}} diff --git a/llm_web_kit/extractor/html/recognizer/ccmath.py b/llm_web_kit/extractor/html/recognizer/ccmath.py index 28078250..780ba583 100644 --- a/llm_web_kit/extractor/html/recognizer/ccmath.py +++ b/llm_web_kit/extractor/html/recognizer/ccmath.py @@ -123,7 +123,6 @@ def process_ccmath_html(self, cc_html: str, o_html: str, math_render: BaseMathRe self.cm.url = base_url tree = cc_html math_render_type = math_render.get_render_type() - self.mathjax_detected = False # 重置标记 # process1: node循环逻辑 for node in iter_node(tree): @@ -136,11 +135,9 @@ def process_ccmath_html(self, cc_html: str, o_html: str, math_render: BaseMathRe node.tag == 'span' and node.get('class') in [CSDN.INLINE, CSDN.DISPLAY]): tag_script.process_katex_mathml(self.cm, math_render_type, node) - self.mathjax_detected = True if ZHIHU.DOMAIN in self.cm.url and node.tag == 'span' and node.get('class') == ZHIHU.MATH: tag_script.process_zhihu_custom_tag(self.cm, math_render_type, node) - self.mathjax_detected = True # tag = span， class 为 math-containerm， 或者 mathjax 或者 wp-katex-eq if node.tag == 'span' and node.get('class') and ( @@ -151,44 +148,32 @@ def process_ccmath_html(self, cc_html: str, o_html: str, math_render: BaseMathRe 'tex' in node.get('class') ): tag_common_modify.modify_tree(self.cm, math_render_type, original_html, node, parent) - self.mathjax_detected = True # math tags if node.tag == 'math' or node.tag.endswith(':math'): # print(f"匹配到数学标签: {node.tag}") # print(f"标签内容: {original_html}") tag_math.modify_tree(self.cm, math_render_type, original_html, node, parent) - self.mathjax_detected = True if node.tag == 'mjx-container': tag_mjx.modify_tree(self.cm, math_render, original_html, node) - self.mathjax_detected = True # img中的latex if node.tag == 'img': tag_img.modify_tree(self.cm, math_render_type, original_html, node, parent) - self.mathjax_detected = True # span.katex if node.tag == 'script' or 'math' == node.get('class') or 'katex' == node.get('class'): # print('匹配到script/math/katex标签: ', original_html) tag_script.modify_tree(self.cm, math_render_type, original_html, node, parent) - self.mathjax_detected = True - # 只有有渲染器的网站才会走下面文本匹配逻辑 - if math_render_type: - # 14. 只处理只有一层的p标签 - if node.tag == 'p' and len(node.getchildren()) == 0: - # print('匹配到p标签: ', original_html) - tag_common_modify.modify_tree(self.cm, math_render_type, original_html, node, parent) - self.mathjax_detected = True # procsee2: mathjax渲染器逻辑 try: # case1：有mathjax配置 if math_render_type == MathRenderType.MATHJAX: math_render.find_math(tree) - # case2：无Mathjax配置但是开启Mathjax逻辑开关（node循环抽到公式的情况） - elif math_render_type is None and self.mathjax_detected: + # case2：其他情况默认开启 Mathjax配置 + else: from llm_web_kit.extractor.html.recognizer.cc_math.render.mathjax import \ MathJaxRenderMock math_render = MathJaxRenderMock() diff --git a/llm_web_kit/extractor/html/recognizer/table.py b/llm_web_kit/extractor/html/recognizer/table.py index aa96060c..e4cc78ce 100644 --- a/llm_web_kit/extractor/html/recognizer/table.py +++ b/llm_web_kit/extractor/html/recognizer/table.py @@ -267,6 +267,9 @@ def __simplify_td_th_content(self, table_nest_level, elem: HtmlElement) -> None: else: math_res = self.__check_table_include_math_code(elem) math_res_text = ' '.join(normalize_text_segment(item) for item in math_res) + # 清除math和code元素 + if any(child.tag in [CCTag.CC_MATH_INLINE, CCTag.CC_MATH_INTERLINE, CCTag.CC_CODE, CCTag.CC_CODE_INLINE] for child in elem.iterchildren()): + elem.clear() elem.text = math_res_text else: math_res = self.__check_table_include_math_code(elem) diff --git a/tests/llm_web_kit/extractor/assets/extractor_chain_input/good_data/html/math_dollar.html b/tests/llm_web_kit/extractor/assets/extractor_chain_input/good_data/html/math_dollar.html index af86ebbb..aaf64c70 100644 --- a/tests/llm_web_kit/extractor/assets/extractor_chain_input/good_data/html/math_dollar.html +++ b/tests/llm_web_kit/extractor/assets/extractor_chain_input/good_data/html/math_dollar.html @@ -673,7 +673,7 @@

      If Cowboys want to lock up Tony Romo, they will ha

      -

      On why the Cowboys haven’t extended Tony #Romo yet#: “The problem with Tony Romo is this. His cap number is $16.8 million. The only way you can reduce that is by extending the contract, but here is the problem. Next year, if he becomes a franchise quarterback if they use the franchise tag, they will have to tender him a 1-year deal in the amount of $21.6 million. If you’re the agent for Tony Romo, you say ok we’ll do a long-term deal, but we have to use these numbers. $16.8 million this year, $20.16 million next year and then the following year it goes up another 20 percent to $24 or 25 million. You’re looking at at least $60 million over three years if they want to lock Tony Romo up. That’s the dance that’s going on between the Cowboys and Tony Romo. If Romo is willing to keep the risk of injury, if he’s willing to roll the dice and go through this year and see what happens, the Cowboys could be backed into a corner next year.” On why the Cowboys haven’t made moves in free agency: “The problem is you can’t do anything without creating cap space, and you’re not going to create cap space without cutting guys unless you extend Anthony Spencer’s deal. Now the problem is moving to the 4-3. His agent Jordan Woy is going to be saying you need to be paying him more like a 4-3 defensive not a 3-4 outside linebacker because a 4-3 defensive end makes more money.”

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      All right... We have returned from Romania again! It was a pleasure - to slash you - with quality metal once again. Thanks to Manu, the - beginning of the - performance at TATTOO & MUSIC FEST II in Iasi could be - seen below... -
      - Thanks to all, who managed to attend the events, meet us and help us on the way! - You know who you are!
      - Stay tuned! News coming soon...

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