Properties

Label 4010.2.a.m
Level 4010
Weight 2
Character orbit 4010.a
Self dual Yes
Analytic conductor 32.020
Analytic rank 0
Dimension 20
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4010 = 2 \cdot 5 \cdot 401 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4010.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(32.0200112105\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(- q^{2}\) \( + \beta_{1} q^{3} \) \(+ q^{4}\) \(+ q^{5}\) \( -\beta_{1} q^{6} \) \( + ( 1 + \beta_{16} ) q^{7} \) \(- q^{8}\) \( + ( 1 + \beta_{2} ) q^{9} \) \(+O(q^{10})\) \( q\) \(- q^{2}\) \( + \beta_{1} q^{3} \) \(+ q^{4}\) \(+ q^{5}\) \( -\beta_{1} q^{6} \) \( + ( 1 + \beta_{16} ) q^{7} \) \(- q^{8}\) \( + ( 1 + \beta_{2} ) q^{9} \) \(- q^{10}\) \( + ( 1 + \beta_{15} ) q^{11} \) \( + \beta_{1} q^{12} \) \( + ( -\beta_{4} + \beta_{16} ) q^{13} \) \( + ( -1 - \beta_{16} ) q^{14} \) \( + \beta_{1} q^{15} \) \(+ q^{16}\) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{7} - \beta_{8} + \beta_{15} - \beta_{18} + \beta_{19} ) q^{17} \) \( + ( -1 - \beta_{2} ) q^{18} \) \( + ( 1 - \beta_{3} + \beta_{6} ) q^{19} \) \(+ q^{20}\) \( + ( 1 - \beta_{3} + \beta_{6} + \beta_{8} - \beta_{9} - \beta_{12} ) q^{21} \) \( + ( -1 - \beta_{15} ) q^{22} \) \( + ( 1 - \beta_{2} - \beta_{4} - \beta_{7} + \beta_{15} - \beta_{17} + \beta_{19} ) q^{23} \) \( -\beta_{1} q^{24} \) \(+ q^{25}\) \( + ( \beta_{4} - \beta_{16} ) q^{26} \) \( + ( 2 \beta_{1} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{13} - \beta_{14} - 2 \beta_{18} + \beta_{19} ) q^{27} \) \( + ( 1 + \beta_{16} ) q^{28} \) \( + ( 2 - \beta_{2} - \beta_{5} - \beta_{9} + \beta_{15} ) q^{29} \) \( -\beta_{1} q^{30} \) \( + ( 1 - \beta_{2} - \beta_{5} - \beta_{6} - \beta_{8} + \beta_{10} + \beta_{11} ) q^{31} \) \(- q^{32}\) \( + ( -3 + \beta_{1} + 2 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{7} + \beta_{9} - \beta_{11} - \beta_{15} + \beta_{18} - \beta_{19} ) q^{33} \) \( + ( -1 - \beta_{1} + \beta_{2} + \beta_{7} + \beta_{8} - \beta_{15} + \beta_{18} - \beta_{19} ) q^{34} \) \( + ( 1 + \beta_{16} ) q^{35} \) \( + ( 1 + \beta_{2} ) q^{36} \) \( + ( 2 + \beta_{5} + \beta_{6} - \beta_{7} + \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} + \beta_{16} + \beta_{18} ) q^{37} \) \( + ( -1 + \beta_{3} - \beta_{6} ) q^{38} \) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{8} - \beta_{12} + \beta_{17} + \beta_{18} - \beta_{19} ) q^{39} \) \(- q^{40}\) \( + ( \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} - \beta_{11} + \beta_{12} - \beta_{15} - \beta_{19} ) q^{41} \) \( + ( -1 + \beta_{3} - \beta_{6} - \beta_{8} + \beta_{9} + \beta_{12} ) q^{42} \) \( + ( -1 + 3 \beta_{2} + \beta_{3} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} - 2 \beta_{15} + \beta_{16} + \beta_{17} + 2 \beta_{18} - 2 \beta_{19} ) q^{43} \) \( + ( 1 + \beta_{15} ) q^{44} \) \( + ( 1 + \beta_{2} ) q^{45} \) \( + ( -1 + \beta_{2} + \beta_{4} + \beta_{7} - \beta_{15} + \beta_{17} - \beta_{19} ) q^{46} \) \( + ( 1 + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{13} - \beta_{14} - \beta_{18} + \beta_{19} ) q^{47} \) \( + \beta_{1} q^{48} \) \( + ( 1 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{6} + \beta_{7} + \beta_{10} - \beta_{13} - \beta_{15} + \beta_{16} - \beta_{18} ) q^{49} \) \(- q^{50}\) \( + ( 2 \beta_{2} + \beta_{3} + \beta_{6} + \beta_{8} + 2 \beta_{14} - \beta_{15} - \beta_{16} + \beta_{18} - \beta_{19} ) q^{51} \) \( + ( -\beta_{4} + \beta_{16} ) q^{52} \) \( + ( 3 - \beta_{2} - 2 \beta_{3} - \beta_{5} + \beta_{6} - \beta_{10} + \beta_{11} - \beta_{12} - \beta_{14} + 2 \beta_{15} - \beta_{16} - \beta_{18} + 2 \beta_{19} ) q^{53} \) \( + ( -2 \beta_{1} + \beta_{6} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{13} + \beta_{14} + 2 \beta_{18} - \beta_{19} ) q^{54} \) \( + ( 1 + \beta_{15} ) q^{55} \) \( + ( -1 - \beta_{16} ) q^{56} \) \( + ( 2 \beta_{1} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} - \beta_{13} + \beta_{14} - \beta_{15} + \beta_{17} - \beta_{19} ) q^{57} \) \( + ( -2 + \beta_{2} + \beta_{5} + \beta_{9} - \beta_{15} ) q^{58} \) \( + ( 4 - \beta_{1} - 3 \beta_{2} + \beta_{3} + \beta_{4} - 2 \beta_{6} - 2 \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} + 2 \beta_{15} - \beta_{16} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{59} \) \( + \beta_{1} q^{60} \) \( + ( 1 - \beta_{3} - \beta_{6} - \beta_{8} + \beta_{9} - \beta_{10} + \beta_{12} - \beta_{14} + \beta_{16} + \beta_{18} ) q^{61} \) \( + ( -1 + \beta_{2} + \beta_{5} + \beta_{6} + \beta_{8} - \beta_{10} - \beta_{11} ) q^{62} \) \( + ( -2 + \beta_{3} - \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{10} - \beta_{11} - 2 \beta_{13} - \beta_{14} - 3 \beta_{15} + \beta_{16} + \beta_{18} - 2 \beta_{19} ) q^{63} \) \(+ q^{64}\) \( + ( -\beta_{4} + \beta_{16} ) q^{65} \) \( + ( 3 - \beta_{1} - 2 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{7} - \beta_{9} + \beta_{11} + \beta_{15} - \beta_{18} + \beta_{19} ) q^{66} \) \( + ( 3 - \beta_{1} + \beta_{2} - \beta_{5} + \beta_{7} - \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{14} - \beta_{15} ) q^{67} \) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{7} - \beta_{8} + \beta_{15} - \beta_{18} + \beta_{19} ) q^{68} \) \( + ( -2 + 2 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{7} + 2 \beta_{9} - \beta_{11} - \beta_{14} - 2 \beta_{15} - \beta_{17} + 2 \beta_{18} - \beta_{19} ) q^{69} \) \( + ( -1 - \beta_{16} ) q^{70} \) \( + ( 3 - \beta_{1} + \beta_{2} - 2 \beta_{3} + \beta_{5} + 2 \beta_{6} - \beta_{7} - \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} + \beta_{17} + \beta_{18} + \beta_{19} ) q^{71} \) \( + ( -1 - \beta_{2} ) q^{72} \) \( + ( 2 - 3 \beta_{2} - 2 \beta_{3} + \beta_{6} - \beta_{7} - \beta_{8} + 2 \beta_{11} + \beta_{13} + 2 \beta_{15} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{73} \) \( + ( -2 - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} - \beta_{18} ) q^{74} \) \( + \beta_{1} q^{75} \) \( + ( 1 - \beta_{3} + \beta_{6} ) q^{76} \) \( + ( -4 + \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{6} - \beta_{7} + 2 \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{13} - \beta_{15} + \beta_{16} - \beta_{17} - 2 \beta_{19} ) q^{77} \) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} - \beta_{8} + \beta_{12} - \beta_{17} - \beta_{18} + \beta_{19} ) q^{78} \) \( + ( 4 - \beta_{2} + \beta_{6} - \beta_{10} + \beta_{11} + \beta_{15} - 2 \beta_{18} + \beta_{19} ) q^{79} \) \(+ q^{80}\) \( + ( -1 + \beta_{1} + 4 \beta_{2} - \beta_{3} + 2 \beta_{6} + \beta_{7} + \beta_{8} + 2 \beta_{9} - \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{14} - 3 \beta_{15} - 2 \beta_{16} - \beta_{18} + \beta_{19} ) q^{81} \) \( + ( -\beta_{2} - \beta_{3} + \beta_{6} - \beta_{7} + \beta_{11} - \beta_{12} + \beta_{15} + \beta_{19} ) q^{82} \) \( + ( -1 + \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{9} + \beta_{10} + \beta_{12} + 2 \beta_{14} - \beta_{15} + \beta_{18} - \beta_{19} ) q^{83} \) \( + ( 1 - \beta_{3} + \beta_{6} + \beta_{8} - \beta_{9} - \beta_{12} ) q^{84} \) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{7} - \beta_{8} + \beta_{15} - \beta_{18} + \beta_{19} ) q^{85} \) \( + ( 1 - 3 \beta_{2} - \beta_{3} - \beta_{5} - \beta_{6} - \beta_{7} - 2 \beta_{8} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + 2 \beta_{15} - \beta_{16} - \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{86} \) \( + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 3 \beta_{9} + \beta_{10} - \beta_{12} + \beta_{13} + 2 \beta_{14} - \beta_{16} - \beta_{17} + 2 \beta_{18} - \beta_{19} ) q^{87} \) \( + ( -1 - \beta_{15} ) q^{88} \) \( + ( \beta_{1} + \beta_{2} + \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + 2 \beta_{8} - 2 \beta_{10} + \beta_{12} + \beta_{14} + \beta_{18} - \beta_{19} ) q^{89} \) \( + ( -1 - \beta_{2} ) q^{90} \) \( + ( 3 + \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{11} - \beta_{12} - 3 \beta_{13} - \beta_{14} - 2 \beta_{15} - \beta_{17} - 2 \beta_{18} + \beta_{19} ) q^{91} \) \( + ( 1 - \beta_{2} - \beta_{4} - \beta_{7} + \beta_{15} - \beta_{17} + \beta_{19} ) q^{92} \) \( + ( 5 + 2 \beta_{1} - 4 \beta_{2} - 3 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} - 2 \beta_{9} + 2 \beta_{11} - \beta_{12} + \beta_{13} + 3 \beta_{15} - 2 \beta_{16} + \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{93} \) \( + ( -1 - \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} - \beta_{13} + \beta_{14} + \beta_{18} - \beta_{19} ) q^{94} \) \( + ( 1 - \beta_{3} + \beta_{6} ) q^{95} \) \( -\beta_{1} q^{96} \) \( + ( 1 + \beta_{1} - 3 \beta_{2} - \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} + \beta_{15} + \beta_{18} + \beta_{19} ) q^{97} \) \( + ( -1 - \beta_{1} + \beta_{2} - 2 \beta_{3} - \beta_{4} + 2 \beta_{6} - \beta_{7} - \beta_{10} + \beta_{13} + \beta_{15} - \beta_{16} + \beta_{18} ) q^{98} \) \( + ( 5 - \beta_{1} + 2 \beta_{2} - \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + \beta_{8} + \beta_{9} - 2 \beta_{10} + \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{15} + \beta_{17} - 2 \beta_{18} + 2 \beta_{19} ) q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(20q \) \(\mathstrut -\mathstrut 20q^{2} \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 20q^{4} \) \(\mathstrut +\mathstrut 20q^{5} \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut 11q^{7} \) \(\mathstrut -\mathstrut 20q^{8} \) \(\mathstrut +\mathstrut 26q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(20q \) \(\mathstrut -\mathstrut 20q^{2} \) \(\mathstrut +\mathstrut 4q^{3} \) \(\mathstrut +\mathstrut 20q^{4} \) \(\mathstrut +\mathstrut 20q^{5} \) \(\mathstrut -\mathstrut 4q^{6} \) \(\mathstrut +\mathstrut 11q^{7} \) \(\mathstrut -\mathstrut 20q^{8} \) \(\mathstrut +\mathstrut 26q^{9} \) \(\mathstrut -\mathstrut 20q^{10} \) \(\mathstrut +\mathstrut 10q^{11} \) \(\mathstrut +\mathstrut 4q^{12} \) \(\mathstrut -\mathstrut 9q^{13} \) \(\mathstrut -\mathstrut 11q^{14} \) \(\mathstrut +\mathstrut 4q^{15} \) \(\mathstrut +\mathstrut 20q^{16} \) \(\mathstrut -\mathstrut 11q^{17} \) \(\mathstrut -\mathstrut 26q^{18} \) \(\mathstrut +\mathstrut 17q^{19} \) \(\mathstrut +\mathstrut 20q^{20} \) \(\mathstrut -\mathstrut 2q^{21} \) \(\mathstrut -\mathstrut 10q^{22} \) \(\mathstrut -\mathstrut 3q^{23} \) \(\mathstrut -\mathstrut 4q^{24} \) \(\mathstrut +\mathstrut 20q^{25} \) \(\mathstrut +\mathstrut 9q^{26} \) \(\mathstrut -\mathstrut 2q^{27} \) \(\mathstrut +\mathstrut 11q^{28} \) \(\mathstrut +\mathstrut 6q^{29} \) \(\mathstrut -\mathstrut 4q^{30} \) \(\mathstrut +\mathstrut 28q^{31} \) \(\mathstrut -\mathstrut 20q^{32} \) \(\mathstrut +\mathstrut 2q^{33} \) \(\mathstrut +\mathstrut 11q^{34} \) \(\mathstrut +\mathstrut 11q^{35} \) \(\mathstrut +\mathstrut 26q^{36} \) \(\mathstrut +\mathstrut 33q^{37} \) \(\mathstrut -\mathstrut 17q^{38} \) \(\mathstrut +\mathstrut 36q^{39} \) \(\mathstrut -\mathstrut 20q^{40} \) \(\mathstrut +\mathstrut 32q^{41} \) \(\mathstrut +\mathstrut 2q^{42} \) \(\mathstrut +\mathstrut 30q^{43} \) \(\mathstrut +\mathstrut 10q^{44} \) \(\mathstrut +\mathstrut 26q^{45} \) \(\mathstrut +\mathstrut 3q^{46} \) \(\mathstrut +\mathstrut 13q^{47} \) \(\mathstrut +\mathstrut 4q^{48} \) \(\mathstrut +\mathstrut 43q^{49} \) \(\mathstrut -\mathstrut 20q^{50} \) \(\mathstrut +\mathstrut 43q^{51} \) \(\mathstrut -\mathstrut 9q^{52} \) \(\mathstrut +\mathstrut 2q^{54} \) \(\mathstrut +\mathstrut 10q^{55} \) \(\mathstrut -\mathstrut 11q^{56} \) \(\mathstrut +\mathstrut 19q^{57} \) \(\mathstrut -\mathstrut 6q^{58} \) \(\mathstrut +\mathstrut 52q^{59} \) \(\mathstrut +\mathstrut 4q^{60} \) \(\mathstrut +\mathstrut 25q^{61} \) \(\mathstrut -\mathstrut 28q^{62} \) \(\mathstrut +\mathstrut 16q^{63} \) \(\mathstrut +\mathstrut 20q^{64} \) \(\mathstrut -\mathstrut 9q^{65} \) \(\mathstrut -\mathstrut 2q^{66} \) \(\mathstrut +\mathstrut 40q^{67} \) \(\mathstrut -\mathstrut 11q^{68} \) \(\mathstrut +\mathstrut 39q^{69} \) \(\mathstrut -\mathstrut 11q^{70} \) \(\mathstrut +\mathstrut 25q^{71} \) \(\mathstrut -\mathstrut 26q^{72} \) \(\mathstrut -\mathstrut 5q^{73} \) \(\mathstrut -\mathstrut 33q^{74} \) \(\mathstrut +\mathstrut 4q^{75} \) \(\mathstrut +\mathstrut 17q^{76} \) \(\mathstrut -\mathstrut 9q^{77} \) \(\mathstrut -\mathstrut 36q^{78} \) \(\mathstrut +\mathstrut 40q^{79} \) \(\mathstrut +\mathstrut 20q^{80} \) \(\mathstrut +\mathstrut 48q^{81} \) \(\mathstrut -\mathstrut 32q^{82} \) \(\mathstrut +\mathstrut 10q^{83} \) \(\mathstrut -\mathstrut 2q^{84} \) \(\mathstrut -\mathstrut 11q^{85} \) \(\mathstrut -\mathstrut 30q^{86} \) \(\mathstrut +\mathstrut 10q^{87} \) \(\mathstrut -\mathstrut 10q^{88} \) \(\mathstrut +\mathstrut 17q^{89} \) \(\mathstrut -\mathstrut 26q^{90} \) \(\mathstrut +\mathstrut 88q^{91} \) \(\mathstrut -\mathstrut 3q^{92} \) \(\mathstrut -\mathstrut 4q^{93} \) \(\mathstrut -\mathstrut 13q^{94} \) \(\mathstrut +\mathstrut 17q^{95} \) \(\mathstrut -\mathstrut 4q^{96} \) \(\mathstrut -\mathstrut 2q^{97} \) \(\mathstrut -\mathstrut 43q^{98} \) \(\mathstrut +\mathstrut 50q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20}\mathstrut -\mathstrut \) \(4\) \(x^{19}\mathstrut -\mathstrut \) \(35\) \(x^{18}\mathstrut +\mathstrut \) \(154\) \(x^{17}\mathstrut +\mathstrut \) \(460\) \(x^{16}\mathstrut -\mathstrut \) \(2392\) \(x^{15}\mathstrut -\mathstrut \) \(2591\) \(x^{14}\mathstrut +\mathstrut \) \(19157\) \(x^{13}\mathstrut +\mathstrut \) \(2776\) \(x^{12}\mathstrut -\mathstrut \) \(83577\) \(x^{11}\mathstrut +\mathstrut \) \(34362\) \(x^{10}\mathstrut +\mathstrut \) \(190617\) \(x^{9}\mathstrut -\mathstrut \) \(150697\) \(x^{8}\mathstrut -\mathstrut \) \(189098\) \(x^{7}\mathstrut +\mathstrut \) \(211360\) \(x^{6}\mathstrut +\mathstrut \) \(37365\) \(x^{5}\mathstrut -\mathstrut \) \(68876\) \(x^{4}\mathstrut -\mathstrut \) \(8984\) \(x^{3}\mathstrut +\mathstrut \) \(7920\) \(x^{2}\mathstrut +\mathstrut \) \(1972\) \(x\mathstrut +\mathstrut \) \(104\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 4 \)
\(\beta_{3}\)\(=\)\((\)\(116014587172107068127\) \(\nu^{19}\mathstrut +\mathstrut \) \(640111674902866325002\) \(\nu^{18}\mathstrut -\mathstrut \) \(7785496031020130083777\) \(\nu^{17}\mathstrut -\mathstrut \) \(22774025091904881148580\) \(\nu^{16}\mathstrut +\mathstrut \) \(196621751166453406494308\) \(\nu^{15}\mathstrut +\mathstrut \) \(307904447861665559694004\) \(\nu^{14}\mathstrut -\mathstrut \) \(2523901525854569513964825\) \(\nu^{13}\mathstrut -\mathstrut \) \(1859977858221430827617547\) \(\nu^{12}\mathstrut +\mathstrut \) \(18123369656041119825105522\) \(\nu^{11}\mathstrut +\mathstrut \) \(3393190587424715294223925\) \(\nu^{10}\mathstrut -\mathstrut \) \(73746841119611087873283396\) \(\nu^{9}\mathstrut +\mathstrut \) \(14421889160306297367063167\) \(\nu^{8}\mathstrut +\mathstrut \) \(160504605873151474966194043\) \(\nu^{7}\mathstrut -\mathstrut \) \(75240595290174938068450304\) \(\nu^{6}\mathstrut -\mathstrut \) \(154547417955549848542242656\) \(\nu^{5}\mathstrut +\mathstrut \) \(100582635779098028690027499\) \(\nu^{4}\mathstrut +\mathstrut \) \(30317438886744979785281938\) \(\nu^{3}\mathstrut -\mathstrut \) \(15302329716260491274790272\) \(\nu^{2}\mathstrut -\mathstrut \) \(4243524215589906115557284\) \(\nu\mathstrut -\mathstrut \) \(192689250572281045604052\)\()/\)\(12\!\cdots\!76\)
\(\beta_{4}\)\(=\)\((\)\(381890322228470127811\) \(\nu^{19}\mathstrut -\mathstrut \) \(3713157145230569889126\) \(\nu^{18}\mathstrut -\mathstrut \) \(6105654787245623330365\) \(\nu^{17}\mathstrut +\mathstrut \) \(139473444254698661552340\) \(\nu^{16}\mathstrut -\mathstrut \) \(103781999398782250499212\) \(\nu^{15}\mathstrut -\mathstrut \) \(2080239065107749043928048\) \(\nu^{14}\mathstrut +\mathstrut \) \(3353365447882748541181147\) \(\nu^{13}\mathstrut +\mathstrut \) \(15512929663601058950255437\) \(\nu^{12}\mathstrut -\mathstrut \) \(33787734343004053799434818\) \(\nu^{11}\mathstrut -\mathstrut \) \(58676991135624483297555223\) \(\nu^{10}\mathstrut +\mathstrut \) \(165865851676723905907971088\) \(\nu^{9}\mathstrut +\mathstrut \) \(91242894083313732643391463\) \(\nu^{8}\mathstrut -\mathstrut \) \(409808669434183097900157517\) \(\nu^{7}\mathstrut +\mathstrut \) \(27830621819326022712984952\) \(\nu^{6}\mathstrut +\mathstrut \) \(441432831515493023022840972\) \(\nu^{5}\mathstrut -\mathstrut \) \(175380389696309148203994441\) \(\nu^{4}\mathstrut -\mathstrut \) \(111351583060334801827646554\) \(\nu^{3}\mathstrut +\mathstrut \) \(30692107501390467607144180\) \(\nu^{2}\mathstrut +\mathstrut \) \(13944746847373487468891484\) \(\nu\mathstrut +\mathstrut \) \(849481945617866039571448\)\()/\)\(12\!\cdots\!76\)
\(\beta_{5}\)\(=\)\((\)\(158358215677478871834\) \(\nu^{19}\mathstrut -\mathstrut \) \(165510479941526953732\) \(\nu^{18}\mathstrut -\mathstrut \) \(7052680498361920376066\) \(\nu^{17}\mathstrut +\mathstrut \) \(6947979307521970282602\) \(\nu^{16}\mathstrut +\mathstrut \) \(131042000829305332961906\) \(\nu^{15}\mathstrut -\mathstrut \) \(122430844130049582041055\) \(\nu^{14}\mathstrut -\mathstrut \) \(1316500543950236316611309\) \(\nu^{13}\mathstrut +\mathstrut \) \(1176238060510791095985110\) \(\nu^{12}\mathstrut +\mathstrut \) \(7734560140578400203662932\) \(\nu^{11}\mathstrut -\mathstrut \) \(6679433655930631630668038\) \(\nu^{10}\mathstrut -\mathstrut \) \(26717309844529810196229795\) \(\nu^{9}\mathstrut +\mathstrut \) \(22585670582814874853777716\) \(\nu^{8}\mathstrut +\mathstrut \) \(51173281785468624732206197\) \(\nu^{7}\mathstrut -\mathstrut \) \(42812984088725050790439808\) \(\nu^{6}\mathstrut -\mathstrut \) \(45795645395669983298002057\) \(\nu^{5}\mathstrut +\mathstrut \) \(37348188371556988720349338\) \(\nu^{4}\mathstrut +\mathstrut \) \(11232664459172077565024340\) \(\nu^{3}\mathstrut -\mathstrut \) \(5912694544531963846562872\) \(\nu^{2}\mathstrut -\mathstrut \) \(1845600069658615177331918\) \(\nu\mathstrut -\mathstrut \) \(97930463161670898984784\)\()/\)\(31\!\cdots\!19\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(999626326932643253811\) \(\nu^{19}\mathstrut +\mathstrut \) \(2885395428797447040394\) \(\nu^{18}\mathstrut +\mathstrut \) \(38333738780464460561397\) \(\nu^{17}\mathstrut -\mathstrut \) \(111572241785130709379400\) \(\nu^{16}\mathstrut -\mathstrut \) \(589292053025298126485872\) \(\nu^{15}\mathstrut +\mathstrut \) \(1747121076757043457794012\) \(\nu^{14}\mathstrut +\mathstrut \) \(4617869516112734577765049\) \(\nu^{13}\mathstrut -\mathstrut \) \(14199801235848615666407733\) \(\nu^{12}\mathstrut -\mathstrut \) \(19257689277831527744719262\) \(\nu^{11}\mathstrut +\mathstrut \) \(63680802429345329998519419\) \(\nu^{10}\mathstrut +\mathstrut \) \(39549692162843203014824044\) \(\nu^{9}\mathstrut -\mathstrut \) \(153750683910395021484611183\) \(\nu^{8}\mathstrut -\mathstrut \) \(27583030355109342007533899\) \(\nu^{7}\mathstrut +\mathstrut \) \(176608360040356555042609360\) \(\nu^{6}\mathstrut -\mathstrut \) \(7317961243969399947208616\) \(\nu^{5}\mathstrut -\mathstrut \) \(68619860824489316187794735\) \(\nu^{4}\mathstrut -\mathstrut \) \(9142653565077423376943002\) \(\nu^{3}\mathstrut +\mathstrut \) \(10241607847011599675503092\) \(\nu^{2}\mathstrut +\mathstrut \) \(3538350924370565587810236\) \(\nu\mathstrut +\mathstrut \) \(308889410401021481347968\)\()/\)\(12\!\cdots\!76\)
\(\beta_{7}\)\(=\)\((\)\(296221760166791911206\) \(\nu^{19}\mathstrut -\mathstrut \) \(1244183378690323911445\) \(\nu^{18}\mathstrut -\mathstrut \) \(10071449640775877619681\) \(\nu^{17}\mathstrut +\mathstrut \) \(47466929437924614832456\) \(\nu^{16}\mathstrut +\mathstrut \) \(125023760355703914577064\) \(\nu^{15}\mathstrut -\mathstrut \) \(727150517520315475409018\) \(\nu^{14}\mathstrut -\mathstrut \) \(596980868306248709499010\) \(\nu^{13}\mathstrut +\mathstrut \) \(5694219644280705889090321\) \(\nu^{12}\mathstrut -\mathstrut \) \(491328919934575895320239\) \(\nu^{11}\mathstrut -\mathstrut \) \(23857680923758730001730199\) \(\nu^{10}\mathstrut +\mathstrut \) \(15509572235364018390928239\) \(\nu^{9}\mathstrut +\mathstrut \) \(49857169321253674143592880\) \(\nu^{8}\mathstrut -\mathstrut \) \(54931703343995084463191829\) \(\nu^{7}\mathstrut -\mathstrut \) \(36975255093475330305977433\) \(\nu^{6}\mathstrut +\mathstrut \) \(67677154972691604550812769\) \(\nu^{5}\mathstrut -\mathstrut \) \(10652934185109545792409197\) \(\nu^{4}\mathstrut -\mathstrut \) \(14032845569575371093974027\) \(\nu^{3}\mathstrut +\mathstrut \) \(2035241542476548031276896\) \(\nu^{2}\mathstrut +\mathstrut \) \(1244849369564183836427184\) \(\nu\mathstrut +\mathstrut \) \(93583911115562606944987\)\()/\)\(31\!\cdots\!19\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(466793766632834090905\) \(\nu^{19}\mathstrut +\mathstrut \) \(2145576029452580751067\) \(\nu^{18}\mathstrut +\mathstrut \) \(15259569747340112833082\) \(\nu^{17}\mathstrut -\mathstrut \) \(81685678338935034438956\) \(\nu^{16}\mathstrut -\mathstrut \) \(173438948868376514272501\) \(\nu^{15}\mathstrut +\mathstrut \) \(1246937604319414623253837\) \(\nu^{14}\mathstrut +\mathstrut \) \(573279854859614309256400\) \(\nu^{13}\mathstrut -\mathstrut \) \(9703424217813463228470465\) \(\nu^{12}\mathstrut +\mathstrut \) \(3735812431564036078098954\) \(\nu^{11}\mathstrut +\mathstrut \) \(40154851697151314424725489\) \(\nu^{10}\mathstrut -\mathstrut \) \(37519361493098511832066873\) \(\nu^{9}\mathstrut -\mathstrut \) \(81409465321846185964993384\) \(\nu^{8}\mathstrut +\mathstrut \) \(117185728932173939708309293\) \(\nu^{7}\mathstrut +\mathstrut \) \(52681976363325771101082567\) \(\nu^{6}\mathstrut -\mathstrut \) \(139293281261527691706815030\) \(\nu^{5}\mathstrut +\mathstrut \) \(30085096393798773969719460\) \(\nu^{4}\mathstrut +\mathstrut \) \(31635282983204416494482189\) \(\nu^{3}\mathstrut -\mathstrut \) \(5656087873568691359199317\) \(\nu^{2}\mathstrut -\mathstrut \) \(3225994613560182407576658\) \(\nu\mathstrut -\mathstrut \) \(218947124132096447481041\)\()/\)\(31\!\cdots\!19\)
\(\beta_{9}\)\(=\)\((\)\(1884753628400079777355\) \(\nu^{19}\mathstrut -\mathstrut \) \(8888942055680230084478\) \(\nu^{18}\mathstrut -\mathstrut \) \(60790240391370671408381\) \(\nu^{17}\mathstrut +\mathstrut \) \(337841240953512241566252\) \(\nu^{16}\mathstrut +\mathstrut \) \(668813633126194984703628\) \(\nu^{15}\mathstrut -\mathstrut \) \(5143190390781541573654824\) \(\nu^{14}\mathstrut -\mathstrut \) \(1830244294205914165632057\) \(\nu^{13}\mathstrut +\mathstrut \) \(39837967846741924857698677\) \(\nu^{12}\mathstrut -\mathstrut \) \(18909251345769732840700930\) \(\nu^{11}\mathstrut -\mathstrut \) \(163388966210012147429801151\) \(\nu^{10}\mathstrut +\mathstrut \) \(167764158738860977503275800\) \(\nu^{9}\mathstrut +\mathstrut \) \(324059764027864699964818695\) \(\nu^{8}\mathstrut -\mathstrut \) \(508326460010982070729117693\) \(\nu^{7}\mathstrut -\mathstrut \) \(187674378859922962869132764\) \(\nu^{6}\mathstrut +\mathstrut \) \(591889518781073264019445620\) \(\nu^{5}\mathstrut -\mathstrut \) \(154815115746377582396262601\) \(\nu^{4}\mathstrut -\mathstrut \) \(125681247228007480688403482\) \(\nu^{3}\mathstrut +\mathstrut \) \(27708583065443687293291992\) \(\nu^{2}\mathstrut +\mathstrut \) \(12710119782824726532399896\) \(\nu\mathstrut +\mathstrut \) \(768531102416619265207312\)\()/\)\(12\!\cdots\!76\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(476462524614547076662\) \(\nu^{19}\mathstrut +\mathstrut \) \(2261150940496645137064\) \(\nu^{18}\mathstrut +\mathstrut \) \(15358368994970655094252\) \(\nu^{17}\mathstrut -\mathstrut \) \(86004232101683687400687\) \(\nu^{16}\mathstrut -\mathstrut \) \(168702378818310521319570\) \(\nu^{15}\mathstrut +\mathstrut \) \(1310730952554843700891871\) \(\nu^{14}\mathstrut +\mathstrut \) \(456367020084764491730201\) \(\nu^{13}\mathstrut -\mathstrut \) \(10170243128634954925798368\) \(\nu^{12}\mathstrut +\mathstrut \) \(4839230598661196559956655\) \(\nu^{11}\mathstrut +\mathstrut \) \(41844531593334313944906169\) \(\nu^{10}\mathstrut -\mathstrut \) \(42739873769995057284338591\) \(\nu^{9}\mathstrut -\mathstrut \) \(83625563617360858132525289\) \(\nu^{8}\mathstrut +\mathstrut \) \(129606374627203981236496536\) \(\nu^{7}\mathstrut +\mathstrut \) \(50366895624856063009545583\) \(\nu^{6}\mathstrut -\mathstrut \) \(151689817123212191455930115\) \(\nu^{5}\mathstrut +\mathstrut \) \(36907829246212195311101221\) \(\nu^{4}\mathstrut +\mathstrut \) \(33496443289456748286697156\) \(\nu^{3}\mathstrut -\mathstrut \) \(6701212743343895922814451\) \(\nu^{2}\mathstrut -\mathstrut \) \(3401240547623802135591091\) \(\nu\mathstrut -\mathstrut \) \(224710702256973187588725\)\()/\)\(31\!\cdots\!19\)
\(\beta_{11}\)\(=\)\((\)\(1207434850304812365853\) \(\nu^{19}\mathstrut -\mathstrut \) \(2247181342857484764768\) \(\nu^{18}\mathstrut -\mathstrut \) \(50414302737871564246701\) \(\nu^{17}\mathstrut +\mathstrut \) \(89232430953300415098274\) \(\nu^{16}\mathstrut +\mathstrut \) \(869790392702661565032060\) \(\nu^{15}\mathstrut -\mathstrut \) \(1455508542425966469741592\) \(\nu^{14}\mathstrut -\mathstrut \) \(8027508521515750685593553\) \(\nu^{13}\mathstrut +\mathstrut \) \(12603138058775366850027407\) \(\nu^{12}\mathstrut +\mathstrut \) \(42848505233762839661010308\) \(\nu^{11}\mathstrut -\mathstrut \) \(62509875650115713349464691\) \(\nu^{10}\mathstrut -\mathstrut \) \(133118622709114169808015760\) \(\nu^{9}\mathstrut +\mathstrut \) \(178427262859425888212445389\) \(\nu^{8}\mathstrut +\mathstrut \) \(227872220297451973399592059\) \(\nu^{7}\mathstrut -\mathstrut \) \(276670047108631459193912154\) \(\nu^{6}\mathstrut -\mathstrut \) \(184388049715590943196801386\) \(\nu^{5}\mathstrut +\mathstrut \) \(195466303198729993721916239\) \(\nu^{4}\mathstrut +\mathstrut \) \(48538361927323472489944564\) \(\nu^{3}\mathstrut -\mathstrut \) \(30774867882955314066751758\) \(\nu^{2}\mathstrut -\mathstrut \) \(8868469585737039803341956\) \(\nu\mathstrut -\mathstrut \) \(436036784036438682904452\)\()/\)\(63\!\cdots\!38\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(2640649899689473379501\) \(\nu^{19}\mathstrut +\mathstrut \) \(10649506491248448238214\) \(\nu^{18}\mathstrut +\mathstrut \) \(91288581180531005559555\) \(\nu^{17}\mathstrut -\mathstrut \) \(406992850767994113620008\) \(\nu^{16}\mathstrut -\mathstrut \) \(1172351233751431179109504\) \(\nu^{15}\mathstrut +\mathstrut \) \(6252075359393404533340596\) \(\nu^{14}\mathstrut +\mathstrut \) \(6216476889098677147808871\) \(\nu^{13}\mathstrut -\mathstrut \) \(49189944811880306907760159\) \(\nu^{12}\mathstrut -\mathstrut \) \(2749870479204769967432866\) \(\nu^{11}\mathstrut +\mathstrut \) \(207921164446610883661717853\) \(\nu^{10}\mathstrut -\mathstrut \) \(107367204487314058684431104\) \(\nu^{9}\mathstrut -\mathstrut \) \(443335161158479647005455289\) \(\nu^{8}\mathstrut +\mathstrut \) \(420010142839881237390997231\) \(\nu^{7}\mathstrut +\mathstrut \) \(354916157103821177416797104\) \(\nu^{6}\mathstrut -\mathstrut \) \(536002055774682948870807056\) \(\nu^{5}\mathstrut +\mathstrut \) \(53641870346786670161614791\) \(\nu^{4}\mathstrut +\mathstrut \) \(114546479174603106008396202\) \(\nu^{3}\mathstrut -\mathstrut \) \(13211705517797761836745184\) \(\nu^{2}\mathstrut -\mathstrut \) \(9609884254476182545264624\) \(\nu\mathstrut -\mathstrut \) \(604059387067058925497476\)\()/\)\(12\!\cdots\!76\)
\(\beta_{13}\)\(=\)\((\)\(906457072838216964082\) \(\nu^{19}\mathstrut -\mathstrut \) \(3628977296486894225234\) \(\nu^{18}\mathstrut -\mathstrut \) \(31390865586963716408176\) \(\nu^{17}\mathstrut +\mathstrut \) \(138618825668018039168042\) \(\nu^{16}\mathstrut +\mathstrut \) \(404589169952395976706645\) \(\nu^{15}\mathstrut -\mathstrut \) \(2127899778284846061265956\) \(\nu^{14}\mathstrut -\mathstrut \) \(2169161541444692884268581\) \(\nu^{13}\mathstrut +\mathstrut \) \(16723705361063596849228437\) \(\nu^{12}\mathstrut +\mathstrut \) \(1249533456063875953741397\) \(\nu^{11}\mathstrut -\mathstrut \) \(70558132248270910576003880\) \(\nu^{10}\mathstrut +\mathstrut \) \(35339158994767751030520578\) \(\nu^{9}\mathstrut +\mathstrut \) \(149861361592849943117898630\) \(\nu^{8}\mathstrut -\mathstrut \) \(139857317203626593295263391\) \(\nu^{7}\mathstrut -\mathstrut \) \(118400990854284243160166845\) \(\nu^{6}\mathstrut +\mathstrut \) \(177362450703633048197133085\) \(\nu^{5}\mathstrut -\mathstrut \) \(20381918000793002939944745\) \(\nu^{4}\mathstrut -\mathstrut \) \(34766170116891192798108945\) \(\nu^{3}\mathstrut +\mathstrut \) \(4483077563939623557009137\) \(\nu^{2}\mathstrut +\mathstrut \) \(2631491299065404623481847\) \(\nu\mathstrut +\mathstrut \) \(140031249837851789305768\)\()/\)\(31\!\cdots\!19\)
\(\beta_{14}\)\(=\)\((\)\(3631258242295806765453\) \(\nu^{19}\mathstrut -\mathstrut \) \(17564545293762038295530\) \(\nu^{18}\mathstrut -\mathstrut \) \(115636723222934464488163\) \(\nu^{17}\mathstrut +\mathstrut \) \(667188840556892290729636\) \(\nu^{16}\mathstrut +\mathstrut \) \(1231316305717842238789600\) \(\nu^{15}\mathstrut -\mathstrut \) \(10147240693829979483090548\) \(\nu^{14}\mathstrut -\mathstrut \) \(2634367907510142021186183\) \(\nu^{13}\mathstrut +\mathstrut \) \(78462784018346905024421023\) \(\nu^{12}\mathstrut -\mathstrut \) \(43613024108359836061648198\) \(\nu^{11}\mathstrut -\mathstrut \) \(320693015892015725084839045\) \(\nu^{10}\mathstrut +\mathstrut \) \(354889209213996383882036604\) \(\nu^{9}\mathstrut +\mathstrut \) \(630419774302201966751461757\) \(\nu^{8}\mathstrut -\mathstrut \) \(1053311329079702402249544959\) \(\nu^{7}\mathstrut -\mathstrut \) \(346966847568919206376515488\) \(\nu^{6}\mathstrut +\mathstrut \) \(1218727562507744425200435480\) \(\nu^{5}\mathstrut -\mathstrut \) \(330977420585354991276338559\) \(\nu^{4}\mathstrut -\mathstrut \) \(264382287138187932221509550\) \(\nu^{3}\mathstrut +\mathstrut \) \(59277512190399136368455356\) \(\nu^{2}\mathstrut +\mathstrut \) \(27270917625149273833516268\) \(\nu\mathstrut +\mathstrut \) \(1592074908429308896508416\)\()/\)\(12\!\cdots\!76\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(3927785207346788767135\) \(\nu^{19}\mathstrut +\mathstrut \) \(14925956319362018390438\) \(\nu^{18}\mathstrut +\mathstrut \) \(138695996226101670224337\) \(\nu^{17}\mathstrut -\mathstrut \) \(571419830443223297042128\) \(\nu^{16}\mathstrut -\mathstrut \) \(1855859584849898459966444\) \(\nu^{15}\mathstrut +\mathstrut \) \(8803688782975893353907588\) \(\nu^{14}\mathstrut +\mathstrut \) \(10989958517098690230175977\) \(\nu^{13}\mathstrut -\mathstrut \) \(69620842627828531311759201\) \(\nu^{12}\mathstrut -\mathstrut \) \(18108209528281622156357818\) \(\nu^{11}\mathstrut +\mathstrut \) \(297161588029706448826147683\) \(\nu^{10}\mathstrut -\mathstrut \) \(98015177947681939839931512\) \(\nu^{9}\mathstrut -\mathstrut \) \(647802148768359369079590651\) \(\nu^{8}\mathstrut +\mathstrut \) \(481313367239573210807895753\) \(\nu^{7}\mathstrut +\mathstrut \) \(560935202377639861594338668\) \(\nu^{6}\mathstrut -\mathstrut \) \(647285058589129120834044824\) \(\nu^{5}\mathstrut +\mathstrut \) \(9139608244597743615157309\) \(\nu^{4}\mathstrut +\mathstrut \) \(130813845334220221247135166\) \(\nu^{3}\mathstrut -\mathstrut \) \(6832345975076365549363456\) \(\nu^{2}\mathstrut -\mathstrut \) \(8895736270030353418450328\) \(\nu\mathstrut -\mathstrut \) \(611989356765519066222632\)\()/\)\(12\!\cdots\!76\)
\(\beta_{16}\)\(=\)\((\)\(1264472658655511227494\) \(\nu^{19}\mathstrut -\mathstrut \) \(5614620561958718180829\) \(\nu^{18}\mathstrut -\mathstrut \) \(41989787193908721570809\) \(\nu^{17}\mathstrut +\mathstrut \) \(213893582685869905990658\) \(\nu^{16}\mathstrut +\mathstrut \) \(495060092922914141164018\) \(\nu^{15}\mathstrut -\mathstrut \) \(3268651599263988205081725\) \(\nu^{14}\mathstrut -\mathstrut \) \(1946728139186136232579558\) \(\nu^{13}\mathstrut +\mathstrut \) \(25485667188491878171747223\) \(\nu^{12}\mathstrut -\mathstrut \) \(6940209720507964758129297\) \(\nu^{11}\mathstrut -\mathstrut \) \(105875503238112161972589831\) \(\nu^{10}\mathstrut +\mathstrut \) \(87543514595202460841969015\) \(\nu^{9}\mathstrut +\mathstrut \) \(216735518039565914836472611\) \(\nu^{8}\mathstrut -\mathstrut \) \(284185995548277354367416141\) \(\nu^{7}\mathstrut -\mathstrut \) \(146866351638555918297252989\) \(\nu^{6}\mathstrut +\mathstrut \) \(341092711768412943785063238\) \(\nu^{5}\mathstrut -\mathstrut \) \(69210761944566981329902851\) \(\nu^{4}\mathstrut -\mathstrut \) \(74014035244757325495950890\) \(\nu^{3}\mathstrut +\mathstrut \) \(13193929518238796506121678\) \(\nu^{2}\mathstrut +\mathstrut \) \(7120300586889498935989802\) \(\nu\mathstrut +\mathstrut \) \(434790140985479929088453\)\()/\)\(31\!\cdots\!19\)
\(\beta_{17}\)\(=\)\((\)\(5418404036695537037889\) \(\nu^{19}\mathstrut -\mathstrut \) \(21028882424189033473846\) \(\nu^{18}\mathstrut -\mathstrut \) \(189937094677161838202411\) \(\nu^{17}\mathstrut +\mathstrut \) \(804798273735627353154448\) \(\nu^{16}\mathstrut +\mathstrut \) \(2506202389660641475350108\) \(\nu^{15}\mathstrut -\mathstrut \) \(12391881937746586803605320\) \(\nu^{14}\mathstrut -\mathstrut \) \(14315694482819121711656027\) \(\nu^{13}\mathstrut +\mathstrut \) \(97889157905570962471010043\) \(\nu^{12}\mathstrut +\mathstrut \) \(18120464240992583538283722\) \(\nu^{11}\mathstrut -\mathstrut \) \(416911015802940054516558769\) \(\nu^{10}\mathstrut +\mathstrut \) \(165875733340926319740473736\) \(\nu^{9}\mathstrut +\mathstrut \) \(904241211612978215609104833\) \(\nu^{8}\mathstrut -\mathstrut \) \(737414111761402929820796123\) \(\nu^{7}\mathstrut -\mathstrut \) \(768988415983968580414341056\) \(\nu^{6}\mathstrut +\mathstrut \) \(975470236198143672379935044\) \(\nu^{5}\mathstrut -\mathstrut \) \(36563629108135351662405247\) \(\nu^{4}\mathstrut -\mathstrut \) \(209919950187223911336578014\) \(\nu^{3}\mathstrut +\mathstrut \) \(14691710720940172080593128\) \(\nu^{2}\mathstrut +\mathstrut \) \(16294502047601695803672512\) \(\nu\mathstrut +\mathstrut \) \(1112816409198791104770760\)\()/\)\(12\!\cdots\!76\)
\(\beta_{18}\)\(=\)\((\)\(2758753361192571885413\) \(\nu^{19}\mathstrut -\mathstrut \) \(11431752151240422468314\) \(\nu^{18}\mathstrut -\mathstrut \) \(94397143893885790930495\) \(\nu^{17}\mathstrut +\mathstrut \) \(436634595061272592855124\) \(\nu^{16}\mathstrut +\mathstrut \) \(1187274336422850598138538\) \(\nu^{15}\mathstrut -\mathstrut \) \(6700748010287355896296002\) \(\nu^{14}\mathstrut -\mathstrut \) \(5911219384637382066178013\) \(\nu^{13}\mathstrut +\mathstrut \) \(52626949058217068420362635\) \(\nu^{12}\mathstrut -\mathstrut \) \(1813842703208030192357358\) \(\nu^{11}\mathstrut -\mathstrut \) \(221689602467094470043371235\) \(\nu^{10}\mathstrut +\mathstrut \) \(132771934441679099734661132\) \(\nu^{9}\mathstrut +\mathstrut \) \(468939029513702380312779877\) \(\nu^{8}\mathstrut -\mathstrut \) \(486641734722183014335399537\) \(\nu^{7}\mathstrut -\mathstrut \) \(364231038870957598427223512\) \(\nu^{6}\mathstrut +\mathstrut \) \(609995251439267990222801910\) \(\nu^{5}\mathstrut -\mathstrut \) \(74734627074977725735521141\) \(\nu^{4}\mathstrut -\mathstrut \) \(132983362883294444683458648\) \(\nu^{3}\mathstrut +\mathstrut \) \(16388938799762510418203468\) \(\nu^{2}\mathstrut +\mathstrut \) \(11773889608624996970981568\) \(\nu\mathstrut +\mathstrut \) \(788415911390967734805626\)\()/\)\(63\!\cdots\!38\)
\(\beta_{19}\)\(=\)\((\)\(11634694858096714461999\) \(\nu^{19}\mathstrut -\mathstrut \) \(48406217720396724392326\) \(\nu^{18}\mathstrut -\mathstrut \) \(397193027005254290163733\) \(\nu^{17}\mathstrut +\mathstrut \) \(1848029398972936980518076\) \(\nu^{16}\mathstrut +\mathstrut \) \(4972099334320587284612548\) \(\nu^{15}\mathstrut -\mathstrut \) \(28340846153083168985306712\) \(\nu^{14}\mathstrut -\mathstrut \) \(24389190301742277875629049\) \(\nu^{13}\mathstrut +\mathstrut \) \(222332963227085352961603569\) \(\nu^{12}\mathstrut -\mathstrut \) \(11918526908097317367907938\) \(\nu^{11}\mathstrut -\mathstrut \) \(934616652952177256655492303\) \(\nu^{10}\mathstrut +\mathstrut \) \(578082175331391735988689832\) \(\nu^{9}\mathstrut +\mathstrut \) \(1967670775033323362634671215\) \(\nu^{8}\mathstrut -\mathstrut \) \(2091395692889556420271389561\) \(\nu^{7}\mathstrut -\mathstrut \) \(1501016739616879718371822192\) \(\nu^{6}\mathstrut +\mathstrut \) \(2605018497515346382262540644\) \(\nu^{5}\mathstrut -\mathstrut \) \(357276643406505762646451513\) \(\nu^{4}\mathstrut -\mathstrut \) \(558302259460451814779557586\) \(\nu^{3}\mathstrut +\mathstrut \) \(75869399959125235523461804\) \(\nu^{2}\mathstrut +\mathstrut \) \(49110444353489173637753464\) \(\nu\mathstrut +\mathstrut \) \(3071590555772710668815584\)\()/\)\(12\!\cdots\!76\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(4\)
\(\nu^{3}\)\(=\)\(\beta_{19}\mathstrut -\mathstrut \) \(2\) \(\beta_{18}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{19}\mathstrut -\mathstrut \) \(\beta_{18}\mathstrut -\mathstrut \) \(2\) \(\beta_{16}\mathstrut -\mathstrut \) \(3\) \(\beta_{15}\mathstrut -\mathstrut \) \(2\) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(13\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(26\)
\(\nu^{5}\)\(=\)\(16\) \(\beta_{19}\mathstrut -\mathstrut \) \(28\) \(\beta_{18}\mathstrut -\mathstrut \) \(\beta_{17}\mathstrut -\mathstrut \) \(2\) \(\beta_{16}\mathstrut +\mathstrut \) \(2\) \(\beta_{15}\mathstrut -\mathstrut \) \(14\) \(\beta_{14}\mathstrut -\mathstrut \) \(14\) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(12\) \(\beta_{10}\mathstrut +\mathstrut \) \(15\) \(\beta_{9}\mathstrut -\mathstrut \) \(14\) \(\beta_{8}\mathstrut -\mathstrut \) \(14\) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(72\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{6}\)\(=\)\(15\) \(\beta_{19}\mathstrut -\mathstrut \) \(16\) \(\beta_{18}\mathstrut -\mathstrut \) \(32\) \(\beta_{16}\mathstrut -\mathstrut \) \(45\) \(\beta_{15}\mathstrut -\mathstrut \) \(32\) \(\beta_{14}\mathstrut -\mathstrut \) \(15\) \(\beta_{13}\mathstrut -\mathstrut \) \(13\) \(\beta_{12}\mathstrut -\mathstrut \) \(15\) \(\beta_{11}\mathstrut -\mathstrut \) \(13\) \(\beta_{10}\mathstrut +\mathstrut \) \(37\) \(\beta_{9}\mathstrut +\mathstrut \) \(10\) \(\beta_{8}\mathstrut +\mathstrut \) \(16\) \(\beta_{7}\mathstrut +\mathstrut \) \(24\) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(4\) \(\beta_{4}\mathstrut -\mathstrut \) \(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(142\) \(\beta_{2}\mathstrut +\mathstrut \) \(22\) \(\beta_{1}\mathstrut +\mathstrut \) \(190\)
\(\nu^{7}\)\(=\)\(197\) \(\beta_{19}\mathstrut -\mathstrut \) \(324\) \(\beta_{18}\mathstrut -\mathstrut \) \(13\) \(\beta_{17}\mathstrut -\mathstrut \) \(40\) \(\beta_{16}\mathstrut +\mathstrut \) \(34\) \(\beta_{15}\mathstrut -\mathstrut \) \(162\) \(\beta_{14}\mathstrut -\mathstrut \) \(158\) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(14\) \(\beta_{11}\mathstrut -\mathstrut \) \(132\) \(\beta_{10}\mathstrut +\mathstrut \) \(180\) \(\beta_{9}\mathstrut -\mathstrut \) \(153\) \(\beta_{8}\mathstrut +\mathstrut \) \(7\) \(\beta_{7}\mathstrut -\mathstrut \) \(150\) \(\beta_{6}\mathstrut -\mathstrut \) \(11\) \(\beta_{5}\mathstrut +\mathstrut \) \(16\) \(\beta_{4}\mathstrut -\mathstrut \) \(17\) \(\beta_{3}\mathstrut -\mathstrut \) \(14\) \(\beta_{2}\mathstrut +\mathstrut \) \(681\) \(\beta_{1}\mathstrut +\mathstrut \) \(56\)
\(\nu^{8}\)\(=\)\(179\) \(\beta_{19}\mathstrut -\mathstrut \) \(212\) \(\beta_{18}\mathstrut +\mathstrut \) \(\beta_{17}\mathstrut -\mathstrut \) \(394\) \(\beta_{16}\mathstrut -\mathstrut \) \(537\) \(\beta_{15}\mathstrut -\mathstrut \) \(400\) \(\beta_{14}\mathstrut -\mathstrut \) \(188\) \(\beta_{13}\mathstrut -\mathstrut \) \(133\) \(\beta_{12}\mathstrut -\mathstrut \) \(181\) \(\beta_{11}\mathstrut -\mathstrut \) \(152\) \(\beta_{10}\mathstrut +\mathstrut \) \(521\) \(\beta_{9}\mathstrut +\mathstrut \) \(69\) \(\beta_{8}\mathstrut +\mathstrut \) \(202\) \(\beta_{7}\mathstrut +\mathstrut \) \(230\) \(\beta_{6}\mathstrut +\mathstrut \) \(33\) \(\beta_{5}\mathstrut +\mathstrut \) \(75\) \(\beta_{4}\mathstrut -\mathstrut \) \(142\) \(\beta_{3}\mathstrut +\mathstrut \) \(1504\) \(\beta_{2}\mathstrut +\mathstrut \) \(345\) \(\beta_{1}\mathstrut +\mathstrut \) \(1474\)
\(\nu^{9}\)\(=\)\(2242\) \(\beta_{19}\mathstrut -\mathstrut \) \(3548\) \(\beta_{18}\mathstrut -\mathstrut \) \(130\) \(\beta_{17}\mathstrut -\mathstrut \) \(589\) \(\beta_{16}\mathstrut +\mathstrut \) \(458\) \(\beta_{15}\mathstrut -\mathstrut \) \(1775\) \(\beta_{14}\mathstrut -\mathstrut \) \(1666\) \(\beta_{13}\mathstrut +\mathstrut \) \(27\) \(\beta_{12}\mathstrut +\mathstrut \) \(154\) \(\beta_{11}\mathstrut -\mathstrut \) \(1434\) \(\beta_{10}\mathstrut +\mathstrut \) \(2046\) \(\beta_{9}\mathstrut -\mathstrut \) \(1569\) \(\beta_{8}\mathstrut +\mathstrut \) \(136\) \(\beta_{7}\mathstrut -\mathstrut \) \(1467\) \(\beta_{6}\mathstrut -\mathstrut \) \(56\) \(\beta_{5}\mathstrut +\mathstrut \) \(192\) \(\beta_{4}\mathstrut -\mathstrut \) \(217\) \(\beta_{3}\mathstrut +\mathstrut \) \(59\) \(\beta_{2}\mathstrut +\mathstrut \) \(6643\) \(\beta_{1}\mathstrut +\mathstrut \) \(792\)
\(\nu^{10}\)\(=\)\(2022\) \(\beta_{19}\mathstrut -\mathstrut \) \(2637\) \(\beta_{18}\mathstrut +\mathstrut \) \(\beta_{17}\mathstrut -\mathstrut \) \(4475\) \(\beta_{16}\mathstrut -\mathstrut \) \(5936\) \(\beta_{15}\mathstrut -\mathstrut \) \(4580\) \(\beta_{14}\mathstrut -\mathstrut \) \(2215\) \(\beta_{13}\mathstrut -\mathstrut \) \(1242\) \(\beta_{12}\mathstrut -\mathstrut \) \(2029\) \(\beta_{11}\mathstrut -\mathstrut \) \(1758\) \(\beta_{10}\mathstrut +\mathstrut \) \(6612\) \(\beta_{9}\mathstrut +\mathstrut \) \(296\) \(\beta_{8}\mathstrut +\mathstrut \) \(2331\) \(\beta_{7}\mathstrut +\mathstrut \) \(2070\) \(\beta_{6}\mathstrut +\mathstrut \) \(662\) \(\beta_{5}\mathstrut +\mathstrut \) \(1023\) \(\beta_{4}\mathstrut -\mathstrut \) \(1228\) \(\beta_{3}\mathstrut +\mathstrut \) \(15831\) \(\beta_{2}\mathstrut +\mathstrut \) \(4760\) \(\beta_{1}\mathstrut +\mathstrut \) \(11839\)
\(\nu^{11}\)\(=\)\(24694\) \(\beta_{19}\mathstrut -\mathstrut \) \(38052\) \(\beta_{18}\mathstrut -\mathstrut \) \(1220\) \(\beta_{17}\mathstrut -\mathstrut \) \(7703\) \(\beta_{16}\mathstrut +\mathstrut \) \(5632\) \(\beta_{15}\mathstrut -\mathstrut \) \(19042\) \(\beta_{14}\mathstrut -\mathstrut \) \(17090\) \(\beta_{13}\mathstrut +\mathstrut \) \(557\) \(\beta_{12}\mathstrut +\mathstrut \) \(1576\) \(\beta_{11}\mathstrut -\mathstrut \) \(15534\) \(\beta_{10}\mathstrut +\mathstrut \) \(22922\) \(\beta_{9}\mathstrut -\mathstrut \) \(15828\) \(\beta_{8}\mathstrut +\mathstrut \) \(1847\) \(\beta_{7}\mathstrut -\mathstrut \) \(13777\) \(\beta_{6}\mathstrut +\mathstrut \) \(406\) \(\beta_{5}\mathstrut +\mathstrut \) \(2133\) \(\beta_{4}\mathstrut -\mathstrut \) \(2405\) \(\beta_{3}\mathstrut +\mathstrut \) \(3310\) \(\beta_{2}\mathstrut +\mathstrut \) \(66233\) \(\beta_{1}\mathstrut +\mathstrut \) \(10042\)
\(\nu^{12}\)\(=\)\(22669\) \(\beta_{19}\mathstrut -\mathstrut \) \(31860\) \(\beta_{18}\mathstrut -\mathstrut \) \(353\) \(\beta_{17}\mathstrut -\mathstrut \) \(49411\) \(\beta_{16}\mathstrut -\mathstrut \) \(63248\) \(\beta_{15}\mathstrut -\mathstrut \) \(50359\) \(\beta_{14}\mathstrut -\mathstrut \) \(25164\) \(\beta_{13}\mathstrut -\mathstrut \) \(10933\) \(\beta_{12}\mathstrut -\mathstrut \) \(21906\) \(\beta_{11}\mathstrut -\mathstrut \) \(20358\) \(\beta_{10}\mathstrut +\mathstrut \) \(79596\) \(\beta_{9}\mathstrut -\mathstrut \) \(1148\) \(\beta_{8}\mathstrut +\mathstrut \) \(25731\) \(\beta_{7}\mathstrut +\mathstrut \) \(18378\) \(\beta_{6}\mathstrut +\mathstrut \) \(10727\) \(\beta_{5}\mathstrut +\mathstrut \) \(12516\) \(\beta_{4}\mathstrut -\mathstrut \) \(9242\) \(\beta_{3}\mathstrut +\mathstrut \) \(166584\) \(\beta_{2}\mathstrut +\mathstrut \) \(61363\) \(\beta_{1}\mathstrut +\mathstrut \) \(97185\)
\(\nu^{13}\)\(=\)\(267779\) \(\beta_{19}\mathstrut -\mathstrut \) \(404748\) \(\beta_{18}\mathstrut -\mathstrut \) \(11574\) \(\beta_{17}\mathstrut -\mathstrut \) \(94960\) \(\beta_{16}\mathstrut +\mathstrut \) \(65865\) \(\beta_{15}\mathstrut -\mathstrut \) \(202402\) \(\beta_{14}\mathstrut -\mathstrut \) \(173219\) \(\beta_{13}\mathstrut +\mathstrut \) \(9779\) \(\beta_{12}\mathstrut +\mathstrut \) \(15735\) \(\beta_{11}\mathstrut -\mathstrut \) \(168085\) \(\beta_{10}\mathstrut +\mathstrut \) \(255934\) \(\beta_{9}\mathstrut -\mathstrut \) \(159750\) \(\beta_{8}\mathstrut +\mathstrut \) \(21890\) \(\beta_{7}\mathstrut -\mathstrut \) \(126656\) \(\beta_{6}\mathstrut +\mathstrut \) \(16685\) \(\beta_{5}\mathstrut +\mathstrut \) \(23371\) \(\beta_{4}\mathstrut -\mathstrut \) \(24056\) \(\beta_{3}\mathstrut +\mathstrut \) \(64731\) \(\beta_{2}\mathstrut +\mathstrut \) \(671333\) \(\beta_{1}\mathstrut +\mathstrut \) \(119688\)
\(\nu^{14}\)\(=\)\(255922\) \(\beta_{19}\mathstrut -\mathstrut \) \(378970\) \(\beta_{18}\mathstrut -\mathstrut \) \(9832\) \(\beta_{17}\mathstrut -\mathstrut \) \(540829\) \(\beta_{16}\mathstrut -\mathstrut \) \(659313\) \(\beta_{15}\mathstrut -\mathstrut \) \(542483\) \(\beta_{14}\mathstrut -\mathstrut \) \(278787\) \(\beta_{13}\mathstrut -\mathstrut \) \(90757\) \(\beta_{12}\mathstrut -\mathstrut \) \(230996\) \(\beta_{11}\mathstrut -\mathstrut \) \(236055\) \(\beta_{10}\mathstrut +\mathstrut \) \(929323\) \(\beta_{9}\mathstrut -\mathstrut \) \(52810\) \(\beta_{8}\mathstrut +\mathstrut \) \(277141\) \(\beta_{7}\mathstrut +\mathstrut \) \(164453\) \(\beta_{6}\mathstrut +\mathstrut \) \(154678\) \(\beta_{5}\mathstrut +\mathstrut \) \(146344\) \(\beta_{4}\mathstrut -\mathstrut \) \(56115\) \(\beta_{3}\mathstrut +\mathstrut \) \(1755226\) \(\beta_{2}\mathstrut +\mathstrut \) \(758547\) \(\beta_{1}\mathstrut +\mathstrut \) \(809827\)
\(\nu^{15}\)\(=\)\(2881358\) \(\beta_{19}\mathstrut -\mathstrut \) \(4293131\) \(\beta_{18}\mathstrut -\mathstrut \) \(115183\) \(\beta_{17}\mathstrut -\mathstrut \) \(1132664\) \(\beta_{16}\mathstrut +\mathstrut \) \(746471\) \(\beta_{15}\mathstrut -\mathstrut \) \(2142698\) \(\beta_{14}\mathstrut -\mathstrut \) \(1746963\) \(\beta_{13}\mathstrut +\mathstrut \) \(153024\) \(\beta_{12}\mathstrut +\mathstrut \) \(156000\) \(\beta_{11}\mathstrut -\mathstrut \) \(1817868\) \(\beta_{10}\mathstrut +\mathstrut \) \(2856511\) \(\beta_{9}\mathstrut -\mathstrut \) \(1623913\) \(\beta_{8}\mathstrut +\mathstrut \) \(243658\) \(\beta_{7}\mathstrut -\mathstrut \) \(1148814\) \(\beta_{6}\mathstrut +\mathstrut \) \(315763\) \(\beta_{5}\mathstrut +\mathstrut \) \(258149\) \(\beta_{4}\mathstrut -\mathstrut \) \(218500\) \(\beta_{3}\mathstrut +\mathstrut \) \(996727\) \(\beta_{2}\mathstrut +\mathstrut \) \(6892946\) \(\beta_{1}\mathstrut +\mathstrut \) \(1369666\)
\(\nu^{16}\)\(=\)\(2916094\) \(\beta_{19}\mathstrut -\mathstrut \) \(4463886\) \(\beta_{18}\mathstrut -\mathstrut \) \(185101\) \(\beta_{17}\mathstrut -\mathstrut \) \(5912648\) \(\beta_{16}\mathstrut -\mathstrut \) \(6771585\) \(\beta_{15}\mathstrut -\mathstrut \) \(5783070\) \(\beta_{14}\mathstrut -\mathstrut \) \(3034233\) \(\beta_{13}\mathstrut -\mathstrut \) \(694271\) \(\beta_{12}\mathstrut -\mathstrut \) \(2396598\) \(\beta_{11}\mathstrut -\mathstrut \) \(2735823\) \(\beta_{10}\mathstrut +\mathstrut \) \(10646296\) \(\beta_{9}\mathstrut -\mathstrut \) \(945847\) \(\beta_{8}\mathstrut +\mathstrut \) \(2942256\) \(\beta_{7}\mathstrut +\mathstrut \) \(1500291\) \(\beta_{6}\mathstrut +\mathstrut \) \(2076332\) \(\beta_{5}\mathstrut +\mathstrut \) \(1675575\) \(\beta_{4}\mathstrut -\mathstrut \) \(163863\) \(\beta_{3}\mathstrut +\mathstrut \) \(18529245\) \(\beta_{2}\mathstrut +\mathstrut \) \(9118033\) \(\beta_{1}\mathstrut +\mathstrut \) \(6824720\)
\(\nu^{17}\)\(=\)\(30888593\) \(\beta_{19}\mathstrut -\mathstrut \) \(45526694\) \(\beta_{18}\mathstrut -\mathstrut \) \(1217038\) \(\beta_{17}\mathstrut -\mathstrut \) \(13245715\) \(\beta_{16}\mathstrut +\mathstrut \) \(8285536\) \(\beta_{15}\mathstrut -\mathstrut \) \(22650576\) \(\beta_{14}\mathstrut -\mathstrut \) \(17593238\) \(\beta_{13}\mathstrut +\mathstrut \) \(2205910\) \(\beta_{12}\mathstrut +\mathstrut \) \(1546697\) \(\beta_{11}\mathstrut -\mathstrut \) \(19660193\) \(\beta_{10}\mathstrut +\mathstrut \) \(31889147\) \(\beta_{9}\mathstrut -\mathstrut \) \(16664782\) \(\beta_{8}\mathstrut +\mathstrut \) \(2631900\) \(\beta_{7}\mathstrut -\mathstrut \) \(10308512\) \(\beta_{6}\mathstrut +\mathstrut \) \(4841333\) \(\beta_{5}\mathstrut +\mathstrut \) \(2890837\) \(\beta_{4}\mathstrut -\mathstrut \) \(1760551\) \(\beta_{3}\mathstrut +\mathstrut \) \(13762848\) \(\beta_{2}\mathstrut +\mathstrut \) \(71512985\) \(\beta_{1}\mathstrut +\mathstrut \) \(15231619\)
\(\nu^{18}\)\(=\)\(33465540\) \(\beta_{19}\mathstrut -\mathstrut \) \(52204864\) \(\beta_{18}\mathstrut -\mathstrut \) \(2926840\) \(\beta_{17}\mathstrut -\mathstrut \) \(64734513\) \(\beta_{16}\mathstrut -\mathstrut \) \(68796280\) \(\beta_{15}\mathstrut -\mathstrut \) \(61350429\) \(\beta_{14}\mathstrut -\mathstrut \) \(32615584\) \(\beta_{13}\mathstrut -\mathstrut \) \(4563528\) \(\beta_{12}\mathstrut -\mathstrut \) \(24577691\) \(\beta_{11}\mathstrut -\mathstrut \) \(31644668\) \(\beta_{10}\mathstrut +\mathstrut \) \(120478540\) \(\beta_{9}\mathstrut -\mathstrut \) \(13742616\) \(\beta_{8}\mathstrut +\mathstrut \) \(30965684\) \(\beta_{7}\mathstrut +\mathstrut \) \(14047429\) \(\beta_{6}\mathstrut +\mathstrut \) \(26590683\) \(\beta_{5}\mathstrut +\mathstrut \) \(18977414\) \(\beta_{4}\mathstrut +\mathstrut \) \(2574068\) \(\beta_{3}\mathstrut +\mathstrut \) \(196036179\) \(\beta_{2}\mathstrut +\mathstrut \) \(107483928\) \(\beta_{1}\mathstrut +\mathstrut \) \(58044469\)
\(\nu^{19}\)\(=\)\(330626168\) \(\beta_{19}\mathstrut -\mathstrut \) \(483298046\) \(\beta_{18}\mathstrut -\mathstrut \) \(13558852\) \(\beta_{17}\mathstrut -\mathstrut \) \(152965455\) \(\beta_{16}\mathstrut +\mathstrut \) \(90657185\) \(\beta_{15}\mathstrut -\mathstrut \) \(239435481\) \(\beta_{14}\mathstrut -\mathstrut \) \(177270201\) \(\beta_{13}\mathstrut +\mathstrut \) \(29974767\) \(\beta_{12}\mathstrut +\mathstrut \) \(15376435\) \(\beta_{11}\mathstrut -\mathstrut \) \(212705635\) \(\beta_{10}\mathstrut +\mathstrut \) \(356064094\) \(\beta_{9}\mathstrut -\mathstrut \) \(172693043\) \(\beta_{8}\mathstrut +\mathstrut \) \(28054584\) \(\beta_{7}\mathstrut -\mathstrut \) \(91494216\) \(\beta_{6}\mathstrut +\mathstrut \) \(67104395\) \(\beta_{5}\mathstrut +\mathstrut \) \(32782410\) \(\beta_{4}\mathstrut -\mathstrut \) \(11477953\) \(\beta_{3}\mathstrut +\mathstrut \) \(178706955\) \(\beta_{2}\mathstrut +\mathstrut \) \(748301832\) \(\beta_{1}\mathstrut +\mathstrut \) \(165879535\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−3.16561
−2.99276
−2.78165
−2.20278
−1.97232
−1.72088
−0.330073
−0.323343
−0.241588
−0.0780369
0.622776
1.02602
1.03584
1.30183
1.96074
2.32194
2.35112
2.72853
3.13315
3.32710
−1.00000 −3.16561 1.00000 1.00000 3.16561 0.575814 −1.00000 7.02106 −1.00000
1.2 −1.00000 −2.99276 1.00000 1.00000 2.99276 0.839081 −1.00000 5.95662 −1.00000
1.3 −1.00000 −2.78165 1.00000 1.00000 2.78165 −3.35211 −1.00000 4.73759 −1.00000
1.4 −1.00000 −2.20278 1.00000 1.00000 2.20278 4.44205 −1.00000 1.85224 −1.00000
1.5 −1.00000 −1.97232 1.00000 1.00000 1.97232 4.46416 −1.00000 0.890065 −1.00000
1.6 −1.00000 −1.72088 1.00000 1.00000 1.72088 −0.251451 −1.00000 −0.0385865 −1.00000
1.7 −1.00000 −0.330073 1.00000 1.00000 0.330073 −4.66703 −1.00000 −2.89105 −1.00000
1.8 −1.00000 −0.323343 1.00000 1.00000 0.323343 4.27073 −1.00000 −2.89545 −1.00000
1.9 −1.00000 −0.241588 1.00000 1.00000 0.241588 0.516163 −1.00000 −2.94164 −1.00000
1.10 −1.00000 −0.0780369 1.00000 1.00000 0.0780369 −1.81697 −1.00000 −2.99391 −1.00000
1.11 −1.00000 0.622776 1.00000 1.00000 −0.622776 3.30687 −1.00000 −2.61215 −1.00000
1.12 −1.00000 1.02602 1.00000 1.00000 −1.02602 −3.81201 −1.00000 −1.94728 −1.00000
1.13 −1.00000 1.03584 1.00000 1.00000 −1.03584 −1.66553 −1.00000 −1.92705 −1.00000
1.14 −1.00000 1.30183 1.00000 1.00000 −1.30183 5.09963 −1.00000 −1.30525 −1.00000
1.15 −1.00000 1.96074 1.00000 1.00000 −1.96074 −2.08760 −1.00000 0.844520 −1.00000
1.16 −1.00000 2.32194 1.00000 1.00000 −2.32194 1.00226 −1.00000 2.39142 −1.00000
1.17 −1.00000 2.35112 1.00000 1.00000 −2.35112 2.99774 −1.00000 2.52776 −1.00000
1.18 −1.00000 2.72853 1.00000 1.00000 −2.72853 0.894798 −1.00000 4.44487 −1.00000
1.19 −1.00000 3.13315 1.00000 1.00000 −3.13315 3.12083 −1.00000 6.81662 −1.00000
1.20 −1.00000 3.32710 1.00000 1.00000 −3.32710 −2.87743 −1.00000 8.06959 −1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(5\) \(-1\)
\(401\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\(T_{3}^{20} - \cdots\)
\(T_{7}^{20} - \cdots\)
\(T_{11}^{20} - \cdots\)