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http://functions.wolfram.com/01.09.03.0041.01
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Cot[(4 Pi)/7] == (14^(1/3) (4 I (1 - 3 I Sqrt[3])^(1/3) -
2 2^(1/3) 7^(2/3) (-I + Sqrt[3]) + 7^(1/3) (2 - 6 I Sqrt[3])^(2/3)
(I + Sqrt[3])))/(-4 I 7^(5/6) (2 - 6 I Sqrt[3])^(1/3) +
2^(1/3) (2 Sqrt[7] (I + Sqrt[3]) (14 - I Sqrt[7] - 3 Sqrt[21])^(1/3) -
2 (14 - I Sqrt[7] - 3 Sqrt[21])^(2/3) (28 + 2 I Sqrt[7] + 6 Sqrt[21])^
(1/3) - I (98 + 7 I Sqrt[7] + 21 Sqrt[21])^(1/3)
(4 7^(1/6) + 2^(2/3) (1 - 3 I Sqrt[3])^(1/3) (-I + Sqrt[3]))))
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Cell[BoxData[RowBox[List[RowBox[List["Cot", "[", FractionBox[RowBox[List["4", " ", "\[Pi]"]], "7"], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["14", RowBox[List["1", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "-", RowBox[List["2", " ", SuperscriptBox["2", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["7", RowBox[List["2", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", SqrtBox["3"]]], ")"]]]], "+", RowBox[List[SuperscriptBox["7", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", "-", RowBox[List["6", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]]]], ")"]], RowBox[List["2", "/", "3"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SqrtBox["3"]]], ")"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["7", RowBox[List["5", "/", "6"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", "-", RowBox[List["6", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox["7"], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["14", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "-", RowBox[List["3", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "-", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["14", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "-", RowBox[List["3", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["28", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox["7"]]], "+", RowBox[List["6", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["98", "+", RowBox[List["7", " ", "\[ImaginaryI]", " ", SqrtBox["7"]]], "+", RowBox[List["21", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["1", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["7", RowBox[List["1", "/", "6"]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]]]], ")"]], RowBox[List["1", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", SqrtBox["3"]]], ")"]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> cot </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mn> 7 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mroot> <mn> 14 </mn> <mn> 3 </mn> </mroot> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> </mrow> <mn> 3 </mn> </mroot> </mrow> <mo> + </mo> <mrow> <mroot> <mn> 7 </mn> <mn> 3 </mn> </mroot> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> + </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> - </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mroot> <mn> 2 </mn> <mn> 3 </mn> </mroot> <mo> ⁢ </mo> <msup> <mn> 7 </mn> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> + </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mroot> <mn> 2 </mn> <mn> 3 </mn> </mroot> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 14 </mn> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 7 </mn> </msqrt> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msqrt> <mn> 21 </mn> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> ⁢ </mo> <mroot> <mrow> <mn> 28 </mn> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 7 </mn> </msqrt> </mrow> <mo> + </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <msqrt> <mn> 21 </mn> </msqrt> </mrow> </mrow> <mn> 3 </mn> </mroot> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mn> 7 </mn> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> + </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mroot> <mrow> <mn> 14 </mn> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 7 </mn> </msqrt> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msqrt> <mn> 21 </mn> </msqrt> </mrow> </mrow> <mn> 3 </mn> </mroot> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mroot> <mrow> <mn> 98 </mn> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 7 </mn> </msqrt> </mrow> <mo> + </mo> <mrow> <mn> 21 </mn> <mo> ⁢ </mo> <msqrt> <mn> 21 </mn> </msqrt> </mrow> </mrow> <mn> 3 </mn> </mroot> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mn> 2 </mn> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> + </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> </mrow> <mn> 3 </mn> </mroot> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mroot> <mn> 7 </mn> <mn> 6 </mn> </mroot> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mn> 7 </mn> <mrow> <mn> 5 </mn> <mo> / </mo> <mn> 6 </mn> </mrow> </msup> <mo> ⁢ </mo> <mroot> <mrow> <mn> 2 </mn> <mo> - </mo> <mrow> <mn> 6 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> </mrow> <mn> 3 </mn> </mroot> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <cot /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 14 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <imaginaryi /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <plus /> <imaginaryi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <imaginaryi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 14 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <cn type='integer'> 21 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 28 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <cn type='integer'> 21 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <imaginaryi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 14 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <cn type='integer'> 21 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <cn type='integer'> 98 </cn> <apply> <times /> <cn type='integer'> 7 </cn> <imaginaryi /> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 21 </cn> <apply> <power /> <cn type='integer'> 21 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='rational'> 1 <sep /> 6 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 4 </cn> <imaginaryi /> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='rational'> 5 <sep /> 6 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <imaginaryi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Cot", "[", FractionBox[RowBox[List["4", " ", "\[Pi]"]], "7"], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["14", RowBox[List["1", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "-", RowBox[List["2", " ", SuperscriptBox["2", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox["7", RowBox[List["2", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", SqrtBox["3"]]], ")"]]]], "+", RowBox[List[SuperscriptBox["7", RowBox[List["1", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", "-", RowBox[List["6", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]]]], ")"]], RowBox[List["2", "/", "3"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SqrtBox["3"]]], ")"]]]]]], ")"]]]], RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", "\[ImaginaryI]", " ", SuperscriptBox["7", RowBox[List["5", "/", "6"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", "-", RowBox[List["6", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["1", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox["7"], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["14", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "-", RowBox[List["3", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "-", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["14", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "-", RowBox[List["3", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["28", "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", SqrtBox["7"]]], "+", RowBox[List["6", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "-", RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["98", "+", RowBox[List["7", " ", "\[ImaginaryI]", " ", SqrtBox["7"]]], "+", RowBox[List["21", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["1", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SuperscriptBox["7", RowBox[List["1", "/", "6"]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]]]], ")"]], RowBox[List["1", "/", "3"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], "+", SqrtBox["3"]]], ")"]]]]]], ")"]]]]]], ")"]]]]]]]]]]] |
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){kind=small}
