Properties

Label 4010.2.a.k
Level 4010
Weight 2
Character orbit 4010.a
Self dual Yes
Analytic conductor 32.020
Analytic rank 1
Dimension 15
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: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{15}\mathstrut -\mathstrut \) \(7\) \(x^{14}\mathstrut -\mathstrut \) \(7\) \(x^{13}\mathstrut +\mathstrut \) \(133\) \(x^{12}\mathstrut -\mathstrut \) \(99\) \(x^{11}\mathstrut -\mathstrut \) \(941\) \(x^{10}\mathstrut +\mathstrut \) \(1290\) \(x^{9}\mathstrut +\mathstrut \) \(3031\) \(x^{8}\mathstrut -\mathstrut \) \(5452\) \(x^{7}\mathstrut -\mathstrut \) \(4098\) \(x^{6}\mathstrut +\mathstrut \) \(9986\) \(x^{5}\mathstrut +\mathstrut \) \(850\) \(x^{4}\mathstrut -\mathstrut \) \(7216\) \(x^{3}\mathstrut +\mathstrut \) \(1688\) \(x^{2}\mathstrut +\mathstrut \) \(1441\) \(x\mathstrut -\mathstrut \) \(512\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(58437627\) \(\nu^{14}\mathstrut -\mathstrut \) \(534122576\) \(\nu^{13}\mathstrut +\mathstrut \) \(115180567\) \(\nu^{12}\mathstrut +\mathstrut \) \(10329491164\) \(\nu^{11}\mathstrut -\mathstrut \) \(16005728457\) \(\nu^{10}\mathstrut -\mathstrut \) \(76582202190\) \(\nu^{9}\mathstrut +\mathstrut \) \(150516117320\) \(\nu^{8}\mathstrut +\mathstrut \) \(276187879617\) \(\nu^{7}\mathstrut -\mathstrut \) \(577175461581\) \(\nu^{6}\mathstrut -\mathstrut \) \(500355197485\) \(\nu^{5}\mathstrut +\mathstrut \) \(991851100607\) \(\nu^{4}\mathstrut +\mathstrut \) \(406892526083\) \(\nu^{3}\mathstrut -\mathstrut \) \(654395507655\) \(\nu^{2}\mathstrut -\mathstrut \) \(84064848569\) \(\nu\mathstrut +\mathstrut \) \(117611134796\)\()/\)\(2559277300\)
\(\beta_{3}\)\(=\)\((\)\(20542792\) \(\nu^{14}\mathstrut -\mathstrut \) \(146543981\) \(\nu^{13}\mathstrut -\mathstrut \) \(164096893\) \(\nu^{12}\mathstrut +\mathstrut \) \(2932664074\) \(\nu^{11}\mathstrut -\mathstrut \) \(1662959242\) \(\nu^{10}\mathstrut -\mathstrut \) \(22602084955\) \(\nu^{9}\mathstrut +\mathstrut \) \(23806335705\) \(\nu^{8}\mathstrut +\mathstrut \) \(84798973897\) \(\nu^{7}\mathstrut -\mathstrut \) \(101567638106\) \(\nu^{6}\mathstrut -\mathstrut \) \(158908340525\) \(\nu^{5}\mathstrut +\mathstrut \) \(181723177022\) \(\nu^{4}\mathstrut +\mathstrut \) \(132349309753\) \(\nu^{3}\mathstrut -\mathstrut \) \(121796392900\) \(\nu^{2}\mathstrut -\mathstrut \) \(30937083869\) \(\nu\mathstrut +\mathstrut \) \(23702683016\)\()/\)\(255927730\)
\(\beta_{4}\)\(=\)\((\)\(129009841\) \(\nu^{14}\mathstrut -\mathstrut \) \(883037733\) \(\nu^{13}\mathstrut -\mathstrut \) \(1197107414\) \(\nu^{12}\mathstrut +\mathstrut \) \(17651323112\) \(\nu^{11}\mathstrut -\mathstrut \) \(7047514031\) \(\nu^{10}\mathstrut -\mathstrut \) \(135332605845\) \(\nu^{9}\mathstrut +\mathstrut \) \(123099367535\) \(\nu^{8}\mathstrut +\mathstrut \) \(501129474486\) \(\nu^{7}\mathstrut -\mathstrut \) \(540954449473\) \(\nu^{6}\mathstrut -\mathstrut \) \(914114037680\) \(\nu^{5}\mathstrut +\mathstrut \) \(976805539231\) \(\nu^{4}\mathstrut +\mathstrut \) \(725876949064\) \(\nu^{3}\mathstrut -\mathstrut \) \(659616482965\) \(\nu^{2}\mathstrut -\mathstrut \) \(162102470402\) \(\nu\mathstrut +\mathstrut \) \(125216059318\)\()/\)\(1279638650\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(58921429\) \(\nu^{14}\mathstrut +\mathstrut \) \(416412812\) \(\nu^{13}\mathstrut +\mathstrut \) \(480260011\) \(\nu^{12}\mathstrut -\mathstrut \) \(8324831388\) \(\nu^{11}\mathstrut +\mathstrut \) \(4686009699\) \(\nu^{10}\mathstrut +\mathstrut \) \(63902469990\) \(\nu^{9}\mathstrut -\mathstrut \) \(68500177060\) \(\nu^{8}\mathstrut -\mathstrut \) \(237554782959\) \(\nu^{7}\mathstrut +\mathstrut \) \(294875044927\) \(\nu^{6}\mathstrut +\mathstrut \) \(437620917495\) \(\nu^{5}\mathstrut -\mathstrut \) \(530713167269\) \(\nu^{4}\mathstrut -\mathstrut \) \(354801041001\) \(\nu^{3}\mathstrut +\mathstrut \) \(355957229305\) \(\nu^{2}\mathstrut +\mathstrut \) \(80515602803\) \(\nu\mathstrut -\mathstrut \) \(67211679492\)\()/\)\(511855460\)
\(\beta_{6}\)\(=\)\((\)\(355634499\) \(\nu^{14}\mathstrut -\mathstrut \) \(2493292912\) \(\nu^{13}\mathstrut -\mathstrut \) \(2946774221\) \(\nu^{12}\mathstrut +\mathstrut \) \(49230821268\) \(\nu^{11}\mathstrut -\mathstrut \) \(25215809709\) \(\nu^{10}\mathstrut -\mathstrut \) \(372735449930\) \(\nu^{9}\mathstrut +\mathstrut \) \(372473016340\) \(\nu^{8}\mathstrut +\mathstrut \) \(1365149106929\) \(\nu^{7}\mathstrut -\mathstrut \) \(1568270885997\) \(\nu^{6}\mathstrut -\mathstrut \) \(2470692471645\) \(\nu^{5}\mathstrut +\mathstrut \) \(2745269900859\) \(\nu^{4}\mathstrut +\mathstrut \) \(1948867431071\) \(\nu^{3}\mathstrut -\mathstrut \) \(1792658570635\) \(\nu^{2}\mathstrut -\mathstrut \) \(414042980353\) \(\nu\mathstrut +\mathstrut \) \(330809703252\)\()/\)\(2559277300\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(357030859\) \(\nu^{14}\mathstrut +\mathstrut \) \(2078381242\) \(\nu^{13}\mathstrut +\mathstrut \) \(4813725111\) \(\nu^{12}\mathstrut -\mathstrut \) \(41262341138\) \(\nu^{11}\mathstrut -\mathstrut \) \(10365332281\) \(\nu^{10}\mathstrut +\mathstrut \) \(312320702630\) \(\nu^{9}\mathstrut -\mathstrut \) \(116638375190\) \(\nu^{8}\mathstrut -\mathstrut \) \(1130828000189\) \(\nu^{7}\mathstrut +\mathstrut \) \(709414616177\) \(\nu^{6}\mathstrut +\mathstrut \) \(1983178061645\) \(\nu^{5}\mathstrut -\mathstrut \) \(1413799065769\) \(\nu^{4}\mathstrut -\mathstrut \) \(1469719755011\) \(\nu^{3}\mathstrut +\mathstrut \) \(1004956026735\) \(\nu^{2}\mathstrut +\mathstrut \) \(305717451873\) \(\nu\mathstrut -\mathstrut \) \(192795588882\)\()/\)\(1279638650\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(91199013\) \(\nu^{14}\mathstrut +\mathstrut \) \(582879604\) \(\nu^{13}\mathstrut +\mathstrut \) \(1011640657\) \(\nu^{12}\mathstrut -\mathstrut \) \(11599830506\) \(\nu^{11}\mathstrut +\mathstrut \) \(1677789343\) \(\nu^{10}\mathstrut +\mathstrut \) \(88300145100\) \(\nu^{9}\mathstrut -\mathstrut \) \(62212304580\) \(\nu^{8}\mathstrut -\mathstrut \) \(323204937023\) \(\nu^{7}\mathstrut +\mathstrut \) \(295531472449\) \(\nu^{6}\mathstrut +\mathstrut \) \(577941929785\) \(\nu^{5}\mathstrut -\mathstrut \) \(550172826353\) \(\nu^{4}\mathstrut -\mathstrut \) \(442326826767\) \(\nu^{3}\mathstrut +\mathstrut \) \(380121257205\) \(\nu^{2}\mathstrut +\mathstrut \) \(92712343001\) \(\nu\mathstrut -\mathstrut \) \(72900927894\)\()/\)\(255927730\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(1161424079\) \(\nu^{14}\mathstrut +\mathstrut \) \(7214691402\) \(\nu^{13}\mathstrut +\mathstrut \) \(13595344791\) \(\nu^{12}\mathstrut -\mathstrut \) \(142730708728\) \(\nu^{11}\mathstrut +\mathstrut \) \(6317989589\) \(\nu^{10}\mathstrut +\mathstrut \) \(1078958237680\) \(\nu^{9}\mathstrut -\mathstrut \) \(673308909890\) \(\nu^{8}\mathstrut -\mathstrut \) \(3920595017759\) \(\nu^{7}\mathstrut +\mathstrut \) \(3327978734537\) \(\nu^{6}\mathstrut +\mathstrut \) \(6964442880895\) \(\nu^{5}\mathstrut -\mathstrut \) \(6271311559839\) \(\nu^{4}\mathstrut -\mathstrut \) \(5309427250941\) \(\nu^{3}\mathstrut +\mathstrut \) \(4356518712135\) \(\nu^{2}\mathstrut +\mathstrut \) \(1117219399263\) \(\nu\mathstrut -\mathstrut \) \(835598044392\)\()/\)\(2559277300\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(1173296833\) \(\nu^{14}\mathstrut +\mathstrut \) \(7231451704\) \(\nu^{13}\mathstrut +\mathstrut \) \(14132908607\) \(\nu^{12}\mathstrut -\mathstrut \) \(143965640456\) \(\nu^{11}\mathstrut +\mathstrut \) \(153207403\) \(\nu^{10}\mathstrut +\mathstrut \) \(1095593363710\) \(\nu^{9}\mathstrut -\mathstrut \) \(646517369380\) \(\nu^{8}\mathstrut -\mathstrut \) \(4006364293143\) \(\nu^{7}\mathstrut +\mathstrut \) \(3286835904099\) \(\nu^{6}\mathstrut +\mathstrut \) \(7154897368415\) \(\nu^{5}\mathstrut -\mathstrut \) \(6275827862253\) \(\nu^{4}\mathstrut -\mathstrut \) \(5476071340057\) \(\nu^{3}\mathstrut +\mathstrut \) \(4396936210545\) \(\nu^{2}\mathstrut +\mathstrut \) \(1159659716951\) \(\nu\mathstrut -\mathstrut \) \(845081247084\)\()/\)\(2559277300\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(371844021\) \(\nu^{14}\mathstrut +\mathstrut \) \(2450736548\) \(\nu^{13}\mathstrut +\mathstrut \) \(3766297784\) \(\nu^{12}\mathstrut -\mathstrut \) \(48621859147\) \(\nu^{11}\mathstrut +\mathstrut \) \(13776069811\) \(\nu^{10}\mathstrut +\mathstrut \) \(369500503845\) \(\nu^{9}\mathstrut -\mathstrut \) \(304198536685\) \(\nu^{8}\mathstrut -\mathstrut \) \(1355253643316\) \(\nu^{7}\mathstrut +\mathstrut \) \(1376687553763\) \(\nu^{6}\mathstrut +\mathstrut \) \(2447389181955\) \(\nu^{5}\mathstrut -\mathstrut \) \(2507543406986\) \(\nu^{4}\mathstrut -\mathstrut \) \(1918045318359\) \(\nu^{3}\mathstrut +\mathstrut \) \(1690697670940\) \(\nu^{2}\mathstrut +\mathstrut \) \(408737447487\) \(\nu\mathstrut -\mathstrut \) \(316360212508\)\()/\)\(639819325\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(2202353061\) \(\nu^{14}\mathstrut +\mathstrut \) \(13831059568\) \(\nu^{13}\mathstrut +\mathstrut \) \(25351582419\) \(\nu^{12}\mathstrut -\mathstrut \) \(274524650952\) \(\nu^{11}\mathstrut +\mathstrut \) \(20706458751\) \(\nu^{10}\mathstrut +\mathstrut \) \(2084529565470\) \(\nu^{9}\mathstrut -\mathstrut \) \(1341833245860\) \(\nu^{8}\mathstrut -\mathstrut \) \(7620298033831\) \(\nu^{7}\mathstrut +\mathstrut \) \(6526394884583\) \(\nu^{6}\mathstrut +\mathstrut \) \(13651845662955\) \(\nu^{5}\mathstrut -\mathstrut \) \(12187022546401\) \(\nu^{4}\mathstrut -\mathstrut \) \(10539653148669\) \(\nu^{3}\mathstrut +\mathstrut \) \(8354197042965\) \(\nu^{2}\mathstrut +\mathstrut \) \(2248242866867\) \(\nu\mathstrut -\mathstrut \) \(1574486438628\)\()/\)\(2559277300\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(1275103963\) \(\nu^{14}\mathstrut +\mathstrut \) \(8052863369\) \(\nu^{13}\mathstrut +\mathstrut \) \(14434590052\) \(\nu^{12}\mathstrut -\mathstrut \) \(159745442766\) \(\nu^{11}\mathstrut +\mathstrut \) \(17211325133\) \(\nu^{10}\mathstrut +\mathstrut \) \(1212073313185\) \(\nu^{9}\mathstrut -\mathstrut \) \(819621745905\) \(\nu^{8}\mathstrut -\mathstrut \) \(4427108700148\) \(\nu^{7}\mathstrut +\mathstrut \) \(3943631782139\) \(\nu^{6}\mathstrut +\mathstrut \) \(7925804013640\) \(\nu^{5}\mathstrut -\mathstrut \) \(7355352227083\) \(\nu^{4}\mathstrut -\mathstrut \) \(6119225069202\) \(\nu^{3}\mathstrut +\mathstrut \) \(5055329192595\) \(\nu^{2}\mathstrut +\mathstrut \) \(1302859800136\) \(\nu\mathstrut -\mathstrut \) \(960771925824\)\()/\)\(1279638650\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(1426055511\) \(\nu^{14}\mathstrut +\mathstrut \) \(8868477043\) \(\nu^{13}\mathstrut +\mathstrut \) \(16772024394\) \(\nu^{12}\mathstrut -\mathstrut \) \(176015990502\) \(\nu^{11}\mathstrut +\mathstrut \) \(6774287801\) \(\nu^{10}\mathstrut +\mathstrut \) \(1335565850995\) \(\nu^{9}\mathstrut -\mathstrut \) \(823368041735\) \(\nu^{8}\mathstrut -\mathstrut \) \(4872274029956\) \(\nu^{7}\mathstrut +\mathstrut \) \(4085111050933\) \(\nu^{6}\mathstrut +\mathstrut \) \(8687133324980\) \(\nu^{5}\mathstrut -\mathstrut \) \(7706263306751\) \(\nu^{4}\mathstrut -\mathstrut \) \(6637555814944\) \(\nu^{3}\mathstrut +\mathstrut \) \(5345561232915\) \(\nu^{2}\mathstrut +\mathstrut \) \(1390583026792\) \(\nu\mathstrut -\mathstrut \) \(1018945552978\)\()/\)\(1279638650\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\)\(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(3\) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(3\) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(2\) \(\beta_{2}\mathstrut +\mathstrut \) \(8\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(17\) \(\beta_{14}\mathstrut -\mathstrut \) \(3\) \(\beta_{13}\mathstrut +\mathstrut \) \(5\) \(\beta_{12}\mathstrut +\mathstrut \) \(13\) \(\beta_{11}\mathstrut +\mathstrut \) \(5\) \(\beta_{10}\mathstrut +\mathstrut \) \(15\) \(\beta_{9}\mathstrut +\mathstrut \) \(16\) \(\beta_{8}\mathstrut -\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(12\) \(\beta_{6}\mathstrut -\mathstrut \) \(12\) \(\beta_{5}\mathstrut +\mathstrut \) \(7\) \(\beta_{4}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut +\mathstrut \) \(11\) \(\beta_{2}\mathstrut +\mathstrut \) \(18\) \(\beta_{1}\mathstrut +\mathstrut \) \(17\)
\(\nu^{5}\)\(=\)\(-\)\(64\) \(\beta_{14}\mathstrut -\mathstrut \) \(8\) \(\beta_{13}\mathstrut +\mathstrut \) \(26\) \(\beta_{12}\mathstrut +\mathstrut \) \(38\) \(\beta_{11}\mathstrut +\mathstrut \) \(25\) \(\beta_{10}\mathstrut +\mathstrut \) \(47\) \(\beta_{9}\mathstrut +\mathstrut \) \(62\) \(\beta_{8}\mathstrut -\mathstrut \) \(16\) \(\beta_{7}\mathstrut +\mathstrut \) \(36\) \(\beta_{6}\mathstrut -\mathstrut \) \(36\) \(\beta_{5}\mathstrut +\mathstrut \) \(45\) \(\beta_{4}\mathstrut +\mathstrut \) \(14\) \(\beta_{3}\mathstrut +\mathstrut \) \(34\) \(\beta_{2}\mathstrut +\mathstrut \) \(92\) \(\beta_{1}\mathstrut +\mathstrut \) \(24\)
\(\nu^{6}\)\(=\)\(-\)\(291\) \(\beta_{14}\mathstrut -\mathstrut \) \(63\) \(\beta_{13}\mathstrut +\mathstrut \) \(113\) \(\beta_{12}\mathstrut +\mathstrut \) \(190\) \(\beta_{11}\mathstrut +\mathstrut \) \(118\) \(\beta_{10}\mathstrut +\mathstrut \) \(239\) \(\beta_{9}\mathstrut +\mathstrut \) \(272\) \(\beta_{8}\mathstrut -\mathstrut \) \(46\) \(\beta_{7}\mathstrut +\mathstrut \) \(166\) \(\beta_{6}\mathstrut -\mathstrut \) \(176\) \(\beta_{5}\mathstrut +\mathstrut \) \(175\) \(\beta_{4}\mathstrut +\mathstrut \) \(36\) \(\beta_{3}\mathstrut +\mathstrut \) \(148\) \(\beta_{2}\mathstrut +\mathstrut \) \(305\) \(\beta_{1}\mathstrut +\mathstrut \) \(165\)
\(\nu^{7}\)\(=\)\(-\)\(1180\) \(\beta_{14}\mathstrut -\mathstrut \) \(218\) \(\beta_{13}\mathstrut +\mathstrut \) \(503\) \(\beta_{12}\mathstrut +\mathstrut \) \(688\) \(\beta_{11}\mathstrut +\mathstrut \) \(503\) \(\beta_{10}\mathstrut +\mathstrut \) \(896\) \(\beta_{9}\mathstrut +\mathstrut \) \(1128\) \(\beta_{8}\mathstrut -\mathstrut \) \(240\) \(\beta_{7}\mathstrut +\mathstrut \) \(614\) \(\beta_{6}\mathstrut -\mathstrut \) \(648\) \(\beta_{5}\mathstrut +\mathstrut \) \(850\) \(\beta_{4}\mathstrut +\mathstrut \) \(180\) \(\beta_{3}\mathstrut +\mathstrut \) \(552\) \(\beta_{2}\mathstrut +\mathstrut \) \(1360\) \(\beta_{1}\mathstrut +\mathstrut \) \(468\)
\(\nu^{8}\)\(=\)\(-\)\(5098\) \(\beta_{14}\mathstrut -\mathstrut \) \(1146\) \(\beta_{13}\mathstrut +\mathstrut \) \(2134\) \(\beta_{12}\mathstrut +\mathstrut \) \(3105\) \(\beta_{11}\mathstrut +\mathstrut \) \(2248\) \(\beta_{10}\mathstrut +\mathstrut \) \(4040\) \(\beta_{9}\mathstrut +\mathstrut \) \(4785\) \(\beta_{8}\mathstrut -\mathstrut \) \(843\) \(\beta_{7}\mathstrut +\mathstrut \) \(2647\) \(\beta_{6}\mathstrut -\mathstrut \) \(2917\) \(\beta_{5}\mathstrut +\mathstrut \) \(3467\) \(\beta_{4}\mathstrut +\mathstrut \) \(569\) \(\beta_{3}\mathstrut +\mathstrut \) \(2310\) \(\beta_{2}\mathstrut +\mathstrut \) \(5265\) \(\beta_{1}\mathstrut +\mathstrut \) \(2333\)
\(\nu^{9}\)\(=\)\(-\)\(21215\) \(\beta_{14}\mathstrut -\mathstrut \) \(4459\) \(\beta_{13}\mathstrut +\mathstrut \) \(9167\) \(\beta_{12}\mathstrut +\mathstrut \) \(12348\) \(\beta_{11}\mathstrut +\mathstrut \) \(9418\) \(\beta_{10}\mathstrut +\mathstrut \) \(16284\) \(\beta_{9}\mathstrut +\mathstrut \) \(20152\) \(\beta_{8}\mathstrut -\mathstrut \) \(3831\) \(\beta_{7}\mathstrut +\mathstrut \) \(10663\) \(\beta_{6}\mathstrut -\mathstrut \) \(11692\) \(\beta_{5}\mathstrut +\mathstrut \) \(15430\) \(\beta_{4}\mathstrut +\mathstrut \) \(2536\) \(\beta_{3}\mathstrut +\mathstrut \) \(9290\) \(\beta_{2}\mathstrut +\mathstrut \) \(22573\) \(\beta_{1}\mathstrut +\mathstrut \) \(8640\)
\(\nu^{10}\)\(=\)\(-\)\(90328\) \(\beta_{14}\mathstrut -\mathstrut \) \(20460\) \(\beta_{13}\mathstrut +\mathstrut \) \(38737\) \(\beta_{12}\mathstrut +\mathstrut \) \(53525\) \(\beta_{11}\mathstrut +\mathstrut \) \(40822\) \(\beta_{10}\mathstrut +\mathstrut \) \(70440\) \(\beta_{9}\mathstrut +\mathstrut \) \(85094\) \(\beta_{8}\mathstrut -\mathstrut \) \(14902\) \(\beta_{7}\mathstrut +\mathstrut \) \(45240\) \(\beta_{6}\mathstrut -\mathstrut \) \(50749\) \(\beta_{5}\mathstrut +\mathstrut \) \(64270\) \(\beta_{4}\mathstrut +\mathstrut \) \(9244\) \(\beta_{3}\mathstrut +\mathstrut \) \(38858\) \(\beta_{2}\mathstrut +\mathstrut \) \(92417\) \(\beta_{1}\mathstrut +\mathstrut \) \(38650\)
\(\nu^{11}\)\(=\)\(-\)\(379511\) \(\beta_{14}\mathstrut -\mathstrut \) \(83664\) \(\beta_{13}\mathstrut +\mathstrut \) \(164653\) \(\beta_{12}\mathstrut +\mathstrut \) \(220960\) \(\beta_{11}\mathstrut +\mathstrut \) \(171450\) \(\beta_{10}\mathstrut +\mathstrut \) \(292222\) \(\beta_{9}\mathstrut +\mathstrut \) \(359487\) \(\beta_{8}\mathstrut -\mathstrut \) \(64557\) \(\beta_{7}\mathstrut +\mathstrut \) \(187961\) \(\beta_{6}\mathstrut -\mathstrut \) \(210223\) \(\beta_{5}\mathstrut +\mathstrut \) \(277076\) \(\beta_{4}\mathstrut +\mathstrut \) \(39605\) \(\beta_{3}\mathstrut +\mathstrut \) \(160979\) \(\beta_{2}\mathstrut +\mathstrut \) \(392001\) \(\beta_{1}\mathstrut +\mathstrut \) \(156303\)
\(\nu^{12}\)\(=\)\(-\)\(1608230\) \(\beta_{14}\mathstrut -\mathstrut \) \(364938\) \(\beta_{13}\mathstrut +\mathstrut \) \(695590\) \(\beta_{12}\mathstrut +\mathstrut \) \(942964\) \(\beta_{11}\mathstrut +\mathstrut \) \(732462\) \(\beta_{10}\mathstrut +\mathstrut \) \(1245514\) \(\beta_{9}\mathstrut +\mathstrut \) \(1518069\) \(\beta_{8}\mathstrut -\mathstrut \) \(263329\) \(\beta_{7}\mathstrut +\mathstrut \) \(794776\) \(\beta_{6}\mathstrut -\mathstrut \) \(898311\) \(\beta_{5}\mathstrut +\mathstrut \) \(1164318\) \(\beta_{4}\mathstrut +\mathstrut \) \(156360\) \(\beta_{3}\mathstrut +\mathstrut \) \(676249\) \(\beta_{2}\mathstrut +\mathstrut \) \(1637252\) \(\beta_{1}\mathstrut +\mathstrut \) \(674056\)
\(\nu^{13}\)\(=\)\(-\)\(6781632\) \(\beta_{14}\mathstrut -\mathstrut \) \(1522673\) \(\beta_{13}\mathstrut +\mathstrut \) \(2946121\) \(\beta_{12}\mathstrut +\mathstrut \) \(3949413\) \(\beta_{11}\mathstrut +\mathstrut \) \(3086141\) \(\beta_{10}\mathstrut +\mathstrut \) \(5225929\) \(\beta_{9}\mathstrut +\mathstrut \) \(6416439\) \(\beta_{8}\mathstrut -\mathstrut \) \(1122120\) \(\beta_{7}\mathstrut +\mathstrut \) \(3338232\) \(\beta_{6}\mathstrut -\mathstrut \) \(3768020\) \(\beta_{5}\mathstrut +\mathstrut \) \(4959102\) \(\beta_{4}\mathstrut +\mathstrut \) \(662156\) \(\beta_{3}\mathstrut +\mathstrut \) \(2834799\) \(\beta_{2}\mathstrut +\mathstrut \) \(6925183\) \(\beta_{1}\mathstrut +\mathstrut \) \(2805548\)
\(\nu^{14}\)\(=\)\(-\)\(28690546\) \(\beta_{14}\mathstrut -\mathstrut \) \(6514636\) \(\beta_{13}\mathstrut +\mathstrut \) \(12448701\) \(\beta_{12}\mathstrut +\mathstrut \) \(16753801\) \(\beta_{11}\mathstrut +\mathstrut \) \(13101570\) \(\beta_{10}\mathstrut +\mathstrut \) \(22155352\) \(\beta_{9}\mathstrut +\mathstrut \) \(27107353\) \(\beta_{8}\mathstrut -\mathstrut \) \(4672966\) \(\beta_{7}\mathstrut +\mathstrut \) \(14107397\) \(\beta_{6}\mathstrut -\mathstrut \) \(15996587\) \(\beta_{5}\mathstrut +\mathstrut \) \(20913321\) \(\beta_{4}\mathstrut +\mathstrut \) \(2716879\) \(\beta_{3}\mathstrut +\mathstrut \) \(11944811\) \(\beta_{2}\mathstrut +\mathstrut \) \(29136294\) \(\beta_{1}\mathstrut +\mathstrut \) \(11945030\)

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
−2.07374
0.679191
2.09866
4.22659
−2.62108
−1.71302
1.95650
1.81559
2.79590
−1.99921
0.698269
−0.560392
−1.05167
2.25248
0.495927
−1.00000 −3.34293 1.00000 −1.00000 3.34293 3.45446 −1.00000 8.17517 1.00000
1.2 −1.00000 −3.00420 1.00000 −1.00000 3.00420 −4.38135 −1.00000 6.02524 1.00000
1.3 −1.00000 −2.86623 1.00000 −1.00000 2.86623 0.981408 −1.00000 5.21525 1.00000
1.4 −1.00000 −2.24378 1.00000 −1.00000 2.24378 −2.96737 −1.00000 2.03457 1.00000
1.5 −1.00000 −1.88769 1.00000 −1.00000 1.88769 1.11871 −1.00000 0.563364 1.00000
1.6 −1.00000 −1.21181 1.00000 −1.00000 1.21181 0.441519 −1.00000 −1.53151 1.00000
1.7 −1.00000 −0.918908 1.00000 −1.00000 0.918908 −3.92602 −1.00000 −2.15561 1.00000
1.8 −1.00000 −0.876107 1.00000 −1.00000 0.876107 4.75824 −1.00000 −2.23244 1.00000
1.9 −1.00000 −0.154802 1.00000 −1.00000 0.154802 −4.41535 −1.00000 −2.97604 1.00000
1.10 −1.00000 0.271531 1.00000 −1.00000 −0.271531 1.75980 −1.00000 −2.92627 1.00000
1.11 −1.00000 1.23027 1.00000 −1.00000 −1.23027 3.50539 −1.00000 −1.48643 1.00000
1.12 −1.00000 1.40727 1.00000 −1.00000 −1.40727 −2.31473 −1.00000 −1.01959 1.00000
1.13 −1.00000 1.72412 1.00000 −1.00000 −1.72412 −1.94532 −1.00000 −0.0274146 1.00000
1.14 −1.00000 2.71977 1.00000 −1.00000 −2.71977 −0.594829 −1.00000 4.39715 1.00000
1.15 −1.00000 3.15350 1.00000 −1.00000 −3.15350 −0.474552 −1.00000 6.94456 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\(T_{3}^{15} + \cdots\)
\(T_{7}^{15} + \cdots\)
\(T_{11}^{15} + \cdots\)