Properties

Label 4010.2.a.l
Level 4010
Weight 2
Character orbit 4010.a
Self dual Yes
Analytic conductor 32.020
Analytic rank 0
Dimension 17
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: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{17}\mathstrut -\mathstrut \) \(3\) \(x^{16}\mathstrut -\mathstrut \) \(24\) \(x^{15}\mathstrut +\mathstrut \) \(70\) \(x^{14}\mathstrut +\mathstrut \) \(228\) \(x^{13}\mathstrut -\mathstrut \) \(638\) \(x^{12}\mathstrut -\mathstrut \) \(1075\) \(x^{11}\mathstrut +\mathstrut \) \(2854\) \(x^{10}\mathstrut +\mathstrut \) \(2544\) \(x^{9}\mathstrut -\mathstrut \) \(6415\) \(x^{8}\mathstrut -\mathstrut \) \(2573\) \(x^{7}\mathstrut +\mathstrut \) \(6456\) \(x^{6}\mathstrut +\mathstrut \) \(485\) \(x^{5}\mathstrut -\mathstrut \) \(1839\) \(x^{4}\mathstrut +\mathstrut \) \(67\) \(x^{3}\mathstrut +\mathstrut \) \(136\) \(x^{2}\mathstrut -\mathstrut \) \(8\) \(x\mathstrut -\mathstrut \) \(2\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(517292\) \(\nu^{16}\mathstrut -\mathstrut \) \(36703983\) \(\nu^{15}\mathstrut +\mathstrut \) \(120153563\) \(\nu^{14}\mathstrut +\mathstrut \) \(741135014\) \(\nu^{13}\mathstrut -\mathstrut \) \(2742036456\) \(\nu^{12}\mathstrut -\mathstrut \) \(5467625787\) \(\nu^{11}\mathstrut +\mathstrut \) \(22855792597\) \(\nu^{10}\mathstrut +\mathstrut \) \(17099745144\) \(\nu^{9}\mathstrut -\mathstrut \) \(88325683716\) \(\nu^{8}\mathstrut -\mathstrut \) \(16507498270\) \(\nu^{7}\mathstrut +\mathstrut \) \(157200326808\) \(\nu^{6}\mathstrut -\mathstrut \) \(12946666199\) \(\nu^{5}\mathstrut -\mathstrut \) \(107935060143\) \(\nu^{4}\mathstrut +\mathstrut \) \(15957834852\) \(\nu^{3}\mathstrut +\mathstrut \) \(17753806708\) \(\nu^{2}\mathstrut -\mathstrut \) \(2951997384\) \(\nu\mathstrut -\mathstrut \) \(113062883\)\()/\)\(309515047\)
\(\beta_{4}\)\(=\)\((\)\(4837565\) \(\nu^{16}\mathstrut -\mathstrut \) \(18840353\) \(\nu^{15}\mathstrut -\mathstrut \) \(132711821\) \(\nu^{14}\mathstrut +\mathstrut \) \(583174545\) \(\nu^{13}\mathstrut +\mathstrut \) \(1249910253\) \(\nu^{12}\mathstrut -\mathstrut \) \(6920964417\) \(\nu^{11}\mathstrut -\mathstrut \) \(3832677742\) \(\nu^{10}\mathstrut +\mathstrut \) \(39522423700\) \(\nu^{9}\mathstrut -\mathstrut \) \(8919225821\) \(\nu^{8}\mathstrut -\mathstrut \) \(108891365197\) \(\nu^{7}\mathstrut +\mathstrut \) \(76557067027\) \(\nu^{6}\mathstrut +\mathstrut \) \(117093675538\) \(\nu^{5}\mathstrut -\mathstrut \) \(129782414532\) \(\nu^{4}\mathstrut -\mathstrut \) \(4830190853\) \(\nu^{3}\mathstrut +\mathstrut \) \(35278488729\) \(\nu^{2}\mathstrut -\mathstrut \) \(5708177003\) \(\nu\mathstrut -\mathstrut \) \(1191999696\)\()/\)\(309515047\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(13533000\) \(\nu^{16}\mathstrut +\mathstrut \) \(13049122\) \(\nu^{15}\mathstrut +\mathstrut \) \(426684545\) \(\nu^{14}\mathstrut -\mathstrut \) \(396189248\) \(\nu^{13}\mathstrut -\mathstrut \) \(5245102879\) \(\nu^{12}\mathstrut +\mathstrut \) \(4595130637\) \(\nu^{11}\mathstrut +\mathstrut \) \(31725390068\) \(\nu^{10}\mathstrut -\mathstrut \) \(25754222788\) \(\nu^{9}\mathstrut -\mathstrut \) \(96979101208\) \(\nu^{8}\mathstrut +\mathstrut \) \(70835731577\) \(\nu^{7}\mathstrut +\mathstrut \) \(135261243188\) \(\nu^{6}\mathstrut -\mathstrut \) \(80263510040\) \(\nu^{5}\mathstrut -\mathstrut \) \(62713131246\) \(\nu^{4}\mathstrut +\mathstrut \) \(13581871588\) \(\nu^{3}\mathstrut +\mathstrut \) \(9021163238\) \(\nu^{2}\mathstrut -\mathstrut \) \(3427129740\) \(\nu\mathstrut +\mathstrut \) \(30057660\)\()/\)\(309515047\)
\(\beta_{6}\)\(=\)\((\)\(29164450\) \(\nu^{16}\mathstrut -\mathstrut \) \(75243059\) \(\nu^{15}\mathstrut -\mathstrut \) \(712942292\) \(\nu^{14}\mathstrut +\mathstrut \) \(1640652448\) \(\nu^{13}\mathstrut +\mathstrut \) \(7106995572\) \(\nu^{12}\mathstrut -\mathstrut \) \(13595084080\) \(\nu^{11}\mathstrut -\mathstrut \) \(37253291795\) \(\nu^{10}\mathstrut +\mathstrut \) \(52998995529\) \(\nu^{9}\mathstrut +\mathstrut \) \(110121572033\) \(\nu^{8}\mathstrut -\mathstrut \) \(97602977359\) \(\nu^{7}\mathstrut -\mathstrut \) \(180587342672\) \(\nu^{6}\mathstrut +\mathstrut \) \(76514749485\) \(\nu^{5}\mathstrut +\mathstrut \) \(140382802694\) \(\nu^{4}\mathstrut -\mathstrut \) \(23538978861\) \(\nu^{3}\mathstrut -\mathstrut \) \(20618036438\) \(\nu^{2}\mathstrut +\mathstrut \) \(2530180189\) \(\nu\mathstrut -\mathstrut \) \(133252285\)\()/\)\(309515047\)
\(\beta_{7}\)\(=\)\((\)\(31959553\) \(\nu^{16}\mathstrut -\mathstrut \) \(24013167\) \(\nu^{15}\mathstrut -\mathstrut \) \(1004560849\) \(\nu^{14}\mathstrut +\mathstrut \) \(653567956\) \(\nu^{13}\mathstrut +\mathstrut \) \(12518001396\) \(\nu^{12}\mathstrut -\mathstrut \) \(6839928369\) \(\nu^{11}\mathstrut -\mathstrut \) \(78482109680\) \(\nu^{10}\mathstrut +\mathstrut \) \(34708011918\) \(\nu^{9}\mathstrut +\mathstrut \) \(258464542138\) \(\nu^{8}\mathstrut -\mathstrut \) \(86805681473\) \(\nu^{7}\mathstrut -\mathstrut \) \(422865996766\) \(\nu^{6}\mathstrut +\mathstrut \) \(88683444929\) \(\nu^{5}\mathstrut +\mathstrut \) \(289410490306\) \(\nu^{4}\mathstrut -\mathstrut \) \(5721251743\) \(\nu^{3}\mathstrut -\mathstrut \) \(57121635689\) \(\nu^{2}\mathstrut -\mathstrut \) \(2388994893\) \(\nu\mathstrut +\mathstrut \) \(2506119991\)\()/\)\(309515047\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(42578708\) \(\nu^{16}\mathstrut +\mathstrut \) \(160512035\) \(\nu^{15}\mathstrut +\mathstrut \) \(898763550\) \(\nu^{14}\mathstrut -\mathstrut \) \(3638281524\) \(\nu^{13}\mathstrut -\mathstrut \) \(7069307355\) \(\nu^{12}\mathstrut +\mathstrut \) \(32010911487\) \(\nu^{11}\mathstrut +\mathstrut \) \(24429820825\) \(\nu^{10}\mathstrut -\mathstrut \) \(137168820096\) \(\nu^{9}\mathstrut -\mathstrut \) \(28457584155\) \(\nu^{8}\mathstrut +\mathstrut \) \(293829879846\) \(\nu^{7}\mathstrut -\mathstrut \) \(25061014027\) \(\nu^{6}\mathstrut -\mathstrut \) \(288436995914\) \(\nu^{5}\mathstrut +\mathstrut \) \(61669212078\) \(\nu^{4}\mathstrut +\mathstrut \) \(99969508807\) \(\nu^{3}\mathstrut -\mathstrut \) \(15556414844\) \(\nu^{2}\mathstrut -\mathstrut \) \(9595715223\) \(\nu\mathstrut +\mathstrut \) \(734658957\)\()/\)\(309515047\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(42697450\) \(\nu^{16}\mathstrut +\mathstrut \) \(88292181\) \(\nu^{15}\mathstrut +\mathstrut \) \(1139626837\) \(\nu^{14}\mathstrut -\mathstrut \) \(2036841696\) \(\nu^{13}\mathstrut -\mathstrut \) \(12352098451\) \(\nu^{12}\mathstrut +\mathstrut \) \(18190214717\) \(\nu^{11}\mathstrut +\mathstrut \) \(68978681863\) \(\nu^{10}\mathstrut -\mathstrut \) \(78753218317\) \(\nu^{9}\mathstrut -\mathstrut \) \(207100673241\) \(\nu^{8}\mathstrut +\mathstrut \) \(168438708936\) \(\nu^{7}\mathstrut +\mathstrut \) \(315848585860\) \(\nu^{6}\mathstrut -\mathstrut \) \(156468744478\) \(\nu^{5}\mathstrut -\mathstrut \) \(203405448987\) \(\nu^{4}\mathstrut +\mathstrut \) \(34335215026\) \(\nu^{3}\mathstrut +\mathstrut \) \(31186774911\) \(\nu^{2}\mathstrut -\mathstrut \) \(1005069177\) \(\nu\mathstrut -\mathstrut \) \(455720149\)\()/\)\(309515047\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(80578876\) \(\nu^{16}\mathstrut +\mathstrut \) \(238642459\) \(\nu^{15}\mathstrut +\mathstrut \) \(1945071424\) \(\nu^{14}\mathstrut -\mathstrut \) \(5575538085\) \(\nu^{13}\mathstrut -\mathstrut \) \(18628834335\) \(\nu^{12}\mathstrut +\mathstrut \) \(50928586221\) \(\nu^{11}\mathstrut +\mathstrut \) \(88901872108\) \(\nu^{10}\mathstrut -\mathstrut \) \(228720423838\) \(\nu^{9}\mathstrut -\mathstrut \) \(214756568868\) \(\nu^{8}\mathstrut +\mathstrut \) \(518021909831\) \(\nu^{7}\mathstrut +\mathstrut \) \(227835002030\) \(\nu^{6}\mathstrut -\mathstrut \) \(530055382254\) \(\nu^{5}\mathstrut -\mathstrut \) \(57832557032\) \(\nu^{4}\mathstrut +\mathstrut \) \(158965030720\) \(\nu^{3}\mathstrut -\mathstrut \) \(276287293\) \(\nu^{2}\mathstrut -\mathstrut \) \(10017945931\) \(\nu\mathstrut -\mathstrut \) \(171215480\)\()/\)\(309515047\)
\(\beta_{11}\)\(=\)\((\)\(113055721\) \(\nu^{16}\mathstrut -\mathstrut \) \(299359609\) \(\nu^{15}\mathstrut -\mathstrut \) \(2829478710\) \(\nu^{14}\mathstrut +\mathstrut \) \(6970241055\) \(\nu^{13}\mathstrut +\mathstrut \) \(28404799275\) \(\nu^{12}\mathstrut -\mathstrut \) \(63236140377\) \(\nu^{11}\mathstrut -\mathstrut \) \(144528189191\) \(\nu^{10}\mathstrut +\mathstrut \) \(280528180900\) \(\nu^{9}\mathstrut +\mathstrut \) \(384895427290\) \(\nu^{8}\mathstrut -\mathstrut \) \(621335089574\) \(\nu^{7}\mathstrut -\mathstrut \) \(493500671988\) \(\nu^{6}\mathstrut +\mathstrut \) \(605792160984\) \(\nu^{5}\mathstrut +\mathstrut \) \(239307987195\) \(\nu^{4}\mathstrut -\mathstrut \) \(149037962658\) \(\nu^{3}\mathstrut -\mathstrut \) \(38782026641\) \(\nu^{2}\mathstrut +\mathstrut \) \(6224528889\) \(\nu\mathstrut +\mathstrut \) \(1635727447\)\()/\)\(309515047\)
\(\beta_{12}\)\(=\)\((\)\(127487096\) \(\nu^{16}\mathstrut -\mathstrut \) \(350251889\) \(\nu^{15}\mathstrut -\mathstrut \) \(3118269472\) \(\nu^{14}\mathstrut +\mathstrut \) \(7992434926\) \(\nu^{13}\mathstrut +\mathstrut \) \(30622076627\) \(\nu^{12}\mathstrut -\mathstrut \) \(70628441588\) \(\nu^{11}\mathstrut -\mathstrut \) \(153317794321\) \(\nu^{10}\mathstrut +\mathstrut \) \(302367912296\) \(\nu^{9}\mathstrut +\mathstrut \) \(408725147379\) \(\nu^{8}\mathstrut -\mathstrut \) \(637341894902\) \(\nu^{7}\mathstrut -\mathstrut \) \(549342878384\) \(\nu^{6}\mathstrut +\mathstrut \) \(579787781632\) \(\nu^{5}\mathstrut +\mathstrut \) \(316758674053\) \(\nu^{4}\mathstrut -\mathstrut \) \(127655594119\) \(\nu^{3}\mathstrut -\mathstrut \) \(61108503632\) \(\nu^{2}\mathstrut +\mathstrut \) \(3861941654\) \(\nu\mathstrut +\mathstrut \) \(2741181689\)\()/\)\(309515047\)
\(\beta_{13}\)\(=\)\((\)\(131491739\) \(\nu^{16}\mathstrut -\mathstrut \) \(366833857\) \(\nu^{15}\mathstrut -\mathstrut \) \(3252935367\) \(\nu^{14}\mathstrut +\mathstrut \) \(8598825523\) \(\nu^{13}\mathstrut +\mathstrut \) \(32151835641\) \(\nu^{12}\mathstrut -\mathstrut \) \(78734188184\) \(\nu^{11}\mathstrut -\mathstrut \) \(160079093236\) \(\nu^{10}\mathstrut +\mathstrut \) \(353717142183\) \(\nu^{9}\mathstrut +\mathstrut \) \(412232176906\) \(\nu^{8}\mathstrut -\mathstrut \) \(797062040583\) \(\nu^{7}\mathstrut -\mathstrut \) \(495886805261\) \(\nu^{6}\mathstrut +\mathstrut \) \(797125646421\) \(\nu^{5}\mathstrut +\mathstrut \) \(202932340288\) \(\nu^{4}\mathstrut -\mathstrut \) \(211409732500\) \(\nu^{3}\mathstrut -\mathstrut \) \(29094172573\) \(\nu^{2}\mathstrut +\mathstrut \) \(12492996538\) \(\nu\mathstrut +\mathstrut \) \(1300063361\)\()/\)\(309515047\)
\(\beta_{14}\)\(=\)\((\)\(143660013\) \(\nu^{16}\mathstrut -\mathstrut \) \(441098466\) \(\nu^{15}\mathstrut -\mathstrut \) \(3444899317\) \(\nu^{14}\mathstrut +\mathstrut \) \(10415391985\) \(\nu^{13}\mathstrut +\mathstrut \) \(32519190575\) \(\nu^{12}\mathstrut -\mathstrut \) \(96340887561\) \(\nu^{11}\mathstrut -\mathstrut \) \(150418300729\) \(\nu^{10}\mathstrut +\mathstrut \) \(438713481887\) \(\nu^{9}\mathstrut +\mathstrut \) \(337091453304\) \(\nu^{8}\mathstrut -\mathstrut \) \(1005248527892\) \(\nu^{7}\mathstrut -\mathstrut \) \(277759642068\) \(\nu^{6}\mathstrut +\mathstrut \) \(1022958505739\) \(\nu^{5}\mathstrut -\mathstrut \) \(49144481099\) \(\nu^{4}\mathstrut -\mathstrut \) \(272833266410\) \(\nu^{3}\mathstrut +\mathstrut \) \(34234490641\) \(\nu^{2}\mathstrut +\mathstrut \) \(14665985027\) \(\nu\mathstrut -\mathstrut \) \(820821813\)\()/\)\(309515047\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(171291908\) \(\nu^{16}\mathstrut +\mathstrut \) \(481721894\) \(\nu^{15}\mathstrut +\mathstrut \) \(4204915171\) \(\nu^{14}\mathstrut -\mathstrut \) \(11215905374\) \(\nu^{13}\mathstrut -\mathstrut \) \(41202594024\) \(\nu^{12}\mathstrut +\mathstrut \) \(101813111297\) \(\nu^{11}\mathstrut +\mathstrut \) \(203184397219\) \(\nu^{10}\mathstrut -\mathstrut \) \(452195502624\) \(\nu^{9}\mathstrut -\mathstrut \) \(517697609720\) \(\nu^{8}\mathstrut +\mathstrut \) \(1003050087593\) \(\nu^{7}\mathstrut +\mathstrut \) \(614763282936\) \(\nu^{6}\mathstrut -\mathstrut \) \(979513317111\) \(\nu^{5}\mathstrut -\mathstrut \) \(244022585342\) \(\nu^{4}\mathstrut +\mathstrut \) \(243290631232\) \(\nu^{3}\mathstrut +\mathstrut \) \(27086625562\) \(\nu^{2}\mathstrut -\mathstrut \) \(9885630770\) \(\nu\mathstrut -\mathstrut \) \(456913120\)\()/\)\(309515047\)
\(\beta_{16}\)\(=\)\((\)\(377743308\) \(\nu^{16}\mathstrut -\mathstrut \) \(1097265728\) \(\nu^{15}\mathstrut -\mathstrut \) \(9106434527\) \(\nu^{14}\mathstrut +\mathstrut \) \(25302804816\) \(\nu^{13}\mathstrut +\mathstrut \) \(87385432777\) \(\nu^{12}\mathstrut -\mathstrut \) \(226952470094\) \(\nu^{11}\mathstrut -\mathstrut \) \(421019731065\) \(\nu^{10}\mathstrut +\mathstrut \) \(992835992437\) \(\nu^{9}\mathstrut +\mathstrut \) \(1047036217357\) \(\nu^{8}\mathstrut -\mathstrut \) \(2161251749155\) \(\nu^{7}\mathstrut -\mathstrut \) \(1218459358638\) \(\nu^{6}\mathstrut +\mathstrut \) \(2069300872805\) \(\nu^{5}\mathstrut +\mathstrut \) \(489226814098\) \(\nu^{4}\mathstrut -\mathstrut \) \(519085786871\) \(\nu^{3}\mathstrut -\mathstrut \) \(64166588488\) \(\nu^{2}\mathstrut +\mathstrut \) \(26265785258\) \(\nu\mathstrut +\mathstrut \) \(2910598480\)\()/\)\(309515047\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{16}\mathstrut +\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(2\) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(16\)
\(\nu^{5}\)\(=\)\(\beta_{16}\mathstrut +\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(11\) \(\beta_{11}\mathstrut -\mathstrut \) \(9\) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(11\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(10\) \(\beta_{3}\mathstrut +\mathstrut \) \(12\) \(\beta_{2}\mathstrut +\mathstrut \) \(30\) \(\beta_{1}\mathstrut +\mathstrut \) \(3\)
\(\nu^{6}\)\(=\)\(11\) \(\beta_{16}\mathstrut +\mathstrut \) \(9\) \(\beta_{15}\mathstrut +\mathstrut \) \(12\) \(\beta_{14}\mathstrut -\mathstrut \) \(2\) \(\beta_{13}\mathstrut -\mathstrut \) \(11\) \(\beta_{12}\mathstrut -\mathstrut \) \(24\) \(\beta_{11}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut +\mathstrut \) \(11\) \(\beta_{8}\mathstrut +\mathstrut \) \(24\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(11\) \(\beta_{5}\mathstrut -\mathstrut \) \(12\) \(\beta_{4}\mathstrut +\mathstrut \) \(11\) \(\beta_{3}\mathstrut +\mathstrut \) \(62\) \(\beta_{2}\mathstrut +\mathstrut \) \(16\) \(\beta_{1}\mathstrut +\mathstrut \) \(100\)
\(\nu^{7}\)\(=\)\(16\) \(\beta_{16}\mathstrut +\mathstrut \) \(13\) \(\beta_{15}\mathstrut +\mathstrut \) \(16\) \(\beta_{14}\mathstrut -\mathstrut \) \(2\) \(\beta_{13}\mathstrut -\mathstrut \) \(18\) \(\beta_{12}\mathstrut -\mathstrut \) \(99\) \(\beta_{11}\mathstrut -\mathstrut \) \(68\) \(\beta_{10}\mathstrut +\mathstrut \) \(16\) \(\beta_{9}\mathstrut +\mathstrut \) \(17\) \(\beta_{8}\mathstrut +\mathstrut \) \(102\) \(\beta_{7}\mathstrut +\mathstrut \) \(14\) \(\beta_{6}\mathstrut +\mathstrut \) \(5\) \(\beta_{5}\mathstrut -\mathstrut \) \(17\) \(\beta_{4}\mathstrut +\mathstrut \) \(85\) \(\beta_{3}\mathstrut +\mathstrut \) \(118\) \(\beta_{2}\mathstrut +\mathstrut \) \(199\) \(\beta_{1}\mathstrut +\mathstrut \) \(47\)
\(\nu^{8}\)\(=\)\(99\) \(\beta_{16}\mathstrut +\mathstrut \) \(64\) \(\beta_{15}\mathstrut +\mathstrut \) \(117\) \(\beta_{14}\mathstrut -\mathstrut \) \(34\) \(\beta_{13}\mathstrut -\mathstrut \) \(102\) \(\beta_{12}\mathstrut -\mathstrut \) \(233\) \(\beta_{11}\mathstrut -\mathstrut \) \(2\) \(\beta_{10}\mathstrut +\mathstrut \) \(37\) \(\beta_{9}\mathstrut +\mathstrut \) \(100\) \(\beta_{8}\mathstrut +\mathstrut \) \(238\) \(\beta_{7}\mathstrut +\mathstrut \) \(17\) \(\beta_{6}\mathstrut +\mathstrut \) \(98\) \(\beta_{5}\mathstrut -\mathstrut \) \(120\) \(\beta_{4}\mathstrut +\mathstrut \) \(107\) \(\beta_{3}\mathstrut +\mathstrut \) \(489\) \(\beta_{2}\mathstrut +\mathstrut \) \(176\) \(\beta_{1}\mathstrut +\mathstrut \) \(682\)
\(\nu^{9}\)\(=\)\(184\) \(\beta_{16}\mathstrut +\mathstrut \) \(125\) \(\beta_{15}\mathstrut +\mathstrut \) \(195\) \(\beta_{14}\mathstrut -\mathstrut \) \(43\) \(\beta_{13}\mathstrut -\mathstrut \) \(224\) \(\beta_{12}\mathstrut -\mathstrut \) \(849\) \(\beta_{11}\mathstrut -\mathstrut \) \(486\) \(\beta_{10}\mathstrut +\mathstrut \) \(186\) \(\beta_{9}\mathstrut +\mathstrut \) \(202\) \(\beta_{8}\mathstrut +\mathstrut \) \(908\) \(\beta_{7}\mathstrut +\mathstrut \) \(149\) \(\beta_{6}\mathstrut +\mathstrut \) \(90\) \(\beta_{5}\mathstrut -\mathstrut \) \(215\) \(\beta_{4}\mathstrut +\mathstrut \) \(702\) \(\beta_{3}\mathstrut +\mathstrut \) \(1084\) \(\beta_{2}\mathstrut +\mathstrut \) \(1396\) \(\beta_{1}\mathstrut +\mathstrut \) \(532\)
\(\nu^{10}\)\(=\)\(849\) \(\beta_{16}\mathstrut +\mathstrut \) \(424\) \(\beta_{15}\mathstrut +\mathstrut \) \(1080\) \(\beta_{14}\mathstrut -\mathstrut \) \(407\) \(\beta_{13}\mathstrut -\mathstrut \) \(922\) \(\beta_{12}\mathstrut -\mathstrut \) \(2116\) \(\beta_{11}\mathstrut -\mathstrut \) \(38\) \(\beta_{10}\mathstrut +\mathstrut \) \(468\) \(\beta_{9}\mathstrut +\mathstrut \) \(878\) \(\beta_{8}\mathstrut +\mathstrut \) \(2230\) \(\beta_{7}\mathstrut +\mathstrut \) \(216\) \(\beta_{6}\mathstrut +\mathstrut \) \(829\) \(\beta_{5}\mathstrut -\mathstrut \) \(1141\) \(\beta_{4}\mathstrut +\mathstrut \) \(1026\) \(\beta_{3}\mathstrut +\mathstrut \) \(3940\) \(\beta_{2}\mathstrut +\mathstrut \) \(1685\) \(\beta_{1}\mathstrut +\mathstrut \) \(4905\)
\(\nu^{11}\)\(=\)\(1848\) \(\beta_{16}\mathstrut +\mathstrut \) \(1058\) \(\beta_{15}\mathstrut +\mathstrut \) \(2117\) \(\beta_{14}\mathstrut -\mathstrut \) \(622\) \(\beta_{13}\mathstrut -\mathstrut \) \(2383\) \(\beta_{12}\mathstrut -\mathstrut \) \(7176\) \(\beta_{11}\mathstrut -\mathstrut \) \(3392\) \(\beta_{10}\mathstrut +\mathstrut \) \(1920\) \(\beta_{9}\mathstrut +\mathstrut \) \(2081\) \(\beta_{8}\mathstrut +\mathstrut \) \(7968\) \(\beta_{7}\mathstrut +\mathstrut \) \(1443\) \(\beta_{6}\mathstrut +\mathstrut \) \(1112\) \(\beta_{5}\mathstrut -\mathstrut \) \(2381\) \(\beta_{4}\mathstrut +\mathstrut \) \(5794\) \(\beta_{3}\mathstrut +\mathstrut \) \(9644\) \(\beta_{2}\mathstrut +\mathstrut \) \(10140\) \(\beta_{1}\mathstrut +\mathstrut \) \(5326\)
\(\nu^{12}\)\(=\)\(7182\) \(\beta_{16}\mathstrut +\mathstrut \) \(2736\) \(\beta_{15}\mathstrut +\mathstrut \) \(9779\) \(\beta_{14}\mathstrut -\mathstrut \) \(4253\) \(\beta_{13}\mathstrut -\mathstrut \) \(8293\) \(\beta_{12}\mathstrut -\mathstrut \) \(18641\) \(\beta_{11}\mathstrut -\mathstrut \) \(469\) \(\beta_{10}\mathstrut +\mathstrut \) \(5097\) \(\beta_{9}\mathstrut +\mathstrut \) \(7675\) \(\beta_{8}\mathstrut +\mathstrut \) \(20349\) \(\beta_{7}\mathstrut +\mathstrut \) \(2437\) \(\beta_{6}\mathstrut +\mathstrut \) \(6928\) \(\beta_{5}\mathstrut -\mathstrut \) \(10596\) \(\beta_{4}\mathstrut +\mathstrut \) \(9730\) \(\beta_{3}\mathstrut +\mathstrut \) \(32306\) \(\beta_{2}\mathstrut +\mathstrut \) \(15121\) \(\beta_{1}\mathstrut +\mathstrut \) \(36564\)
\(\nu^{13}\)\(=\)\(17293\) \(\beta_{16}\mathstrut +\mathstrut \) \(8332\) \(\beta_{15}\mathstrut +\mathstrut \) \(21517\) \(\beta_{14}\mathstrut -\mathstrut \) \(7563\) \(\beta_{13}\mathstrut -\mathstrut \) \(23325\) \(\beta_{12}\mathstrut -\mathstrut \) \(60345\) \(\beta_{11}\mathstrut -\mathstrut \) \(23398\) \(\beta_{10}\mathstrut +\mathstrut \) \(18715\) \(\beta_{9}\mathstrut +\mathstrut \) \(19951\) \(\beta_{8}\mathstrut +\mathstrut \) \(69425\) \(\beta_{7}\mathstrut +\mathstrut \) \(13414\) \(\beta_{6}\mathstrut +\mathstrut \) \(11771\) \(\beta_{5}\mathstrut -\mathstrut \) \(24430\) \(\beta_{4}\mathstrut +\mathstrut \) \(48084\) \(\beta_{3}\mathstrut +\mathstrut \) \(84323\) \(\beta_{2}\mathstrut +\mathstrut \) \(75513\) \(\beta_{1}\mathstrut +\mathstrut \) \(50206\)
\(\nu^{14}\)\(=\)\(60478\) \(\beta_{16}\mathstrut +\mathstrut \) \(17465\) \(\beta_{15}\mathstrut +\mathstrut \) \(87776\) \(\beta_{14}\mathstrut -\mathstrut \) \(41619\) \(\beta_{13}\mathstrut -\mathstrut \) \(74174\) \(\beta_{12}\mathstrut -\mathstrut \) \(161446\) \(\beta_{11}\mathstrut -\mathstrut \) \(4795\) \(\beta_{10}\mathstrut +\mathstrut \) \(51463\) \(\beta_{9}\mathstrut +\mathstrut \) \(67073\) \(\beta_{8}\mathstrut +\mathstrut \) \(182709\) \(\beta_{7}\mathstrut +\mathstrut \) \(25706\) \(\beta_{6}\mathstrut +\mathstrut \) \(57852\) \(\beta_{5}\mathstrut -\mathstrut \) \(96852\) \(\beta_{4}\mathstrut +\mathstrut \) \(90883\) \(\beta_{3}\mathstrut +\mathstrut \) \(268314\) \(\beta_{2}\mathstrut +\mathstrut \) \(131210\) \(\beta_{1}\mathstrut +\mathstrut \) \(279909\)
\(\nu^{15}\)\(=\)\(155277\) \(\beta_{16}\mathstrut +\mathstrut \) \(62631\) \(\beta_{15}\mathstrut +\mathstrut \) \(209671\) \(\beta_{14}\mathstrut -\mathstrut \) \(83436\) \(\beta_{13}\mathstrut -\mathstrut \) \(217305\) \(\beta_{12}\mathstrut -\mathstrut \) \(506337\) \(\beta_{11}\mathstrut -\mathstrut \) \(160396\) \(\beta_{10}\mathstrut +\mathstrut \) \(176640\) \(\beta_{9}\mathstrut +\mathstrut \) \(183676\) \(\beta_{8}\mathstrut +\mathstrut \) \(602168\) \(\beta_{7}\mathstrut +\mathstrut \) \(122294\) \(\beta_{6}\mathstrut +\mathstrut \) \(114914\) \(\beta_{5}\mathstrut -\mathstrut \) \(238750\) \(\beta_{4}\mathstrut +\mathstrut \) \(401509\) \(\beta_{3}\mathstrut +\mathstrut \) \(729972\) \(\beta_{2}\mathstrut +\mathstrut \) \(573602\) \(\beta_{1}\mathstrut +\mathstrut \) \(457476\)
\(\nu^{16}\)\(=\)\(508006\) \(\beta_{16}\mathstrut +\mathstrut \) \(110854\) \(\beta_{15}\mathstrut +\mathstrut \) \(783879\) \(\beta_{14}\mathstrut -\mathstrut \) \(393189\) \(\beta_{13}\mathstrut -\mathstrut \) \(658089\) \(\beta_{12}\mathstrut -\mathstrut \) \(1383296\) \(\beta_{11}\mathstrut -\mathstrut \) \(44245\) \(\beta_{10}\mathstrut +\mathstrut \) \(497284\) \(\beta_{9}\mathstrut +\mathstrut \) \(585346\) \(\beta_{8}\mathstrut +\mathstrut \) \(1622062\) \(\beta_{7}\mathstrut +\mathstrut \) \(259478\) \(\beta_{6}\mathstrut +\mathstrut \) \(483987\) \(\beta_{5}\mathstrut -\mathstrut \) \(874463\) \(\beta_{4}\mathstrut +\mathstrut \) \(835576\) \(\beta_{3}\mathstrut +\mathstrut \) \(2248610\) \(\beta_{2}\mathstrut +\mathstrut \) \(1117763\) \(\beta_{1}\mathstrut +\mathstrut \) \(2188243\)

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
−2.56168
−2.39414
−2.24738
−1.71266
−1.70497
−0.548857
−0.291319
−0.103424
0.204745
0.278065
0.507814
1.40160
1.51896
2.15213
2.65670
2.89857
2.94584
−1.00000 −2.56168 1.00000 −1.00000 2.56168 0.101571 −1.00000 3.56218 1.00000
1.2 −1.00000 −2.39414 1.00000 −1.00000 2.39414 4.25395 −1.00000 2.73189 1.00000
1.3 −1.00000 −2.24738 1.00000 −1.00000 2.24738 2.64455 −1.00000 2.05073 1.00000
1.4 −1.00000 −1.71266 1.00000 −1.00000 1.71266 −2.30195 −1.00000 −0.0668018 1.00000
1.5 −1.00000 −1.70497 1.00000 −1.00000 1.70497 −1.19610 −1.00000 −0.0930693 1.00000
1.6 −1.00000 −0.548857 1.00000 −1.00000 0.548857 −1.86676 −1.00000 −2.69876 1.00000
1.7 −1.00000 −0.291319 1.00000 −1.00000 0.291319 −2.64183 −1.00000 −2.91513 1.00000
1.8 −1.00000 −0.103424 1.00000 −1.00000 0.103424 1.84281 −1.00000 −2.98930 1.00000
1.9 −1.00000 0.204745 1.00000 −1.00000 −0.204745 0.0646599 −1.00000 −2.95808 1.00000
1.10 −1.00000 0.278065 1.00000 −1.00000 −0.278065 4.59715 −1.00000 −2.92268 1.00000
1.11 −1.00000 0.507814 1.00000 −1.00000 −0.507814 1.00840 −1.00000 −2.74213 1.00000
1.12 −1.00000 1.40160 1.00000 −1.00000 −1.40160 −1.83860 −1.00000 −1.03552 1.00000
1.13 −1.00000 1.51896 1.00000 −1.00000 −1.51896 1.25913 −1.00000 −0.692760 1.00000
1.14 −1.00000 2.15213 1.00000 −1.00000 −2.15213 −4.40972 −1.00000 1.63168 1.00000
1.15 −1.00000 2.65670 1.00000 −1.00000 −2.65670 4.81257 −1.00000 4.05806 1.00000
1.16 −1.00000 2.89857 1.00000 −1.00000 −2.89857 −4.89781 −1.00000 5.40172 1.00000
1.17 −1.00000 2.94584 1.00000 −1.00000 −2.94584 2.56799 −1.00000 5.67796 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
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 \(S_{2}^{\mathrm{new}}(\Gamma_0(4010))\):

\(T_{3}^{17} - \cdots\)
\(T_{7}^{17} - \cdots\)
\(T_{11}^{17} + \cdots\)