Properties

Label 1155.2.c.e
Level 1155
Weight 2
Character orbit 1155.c
Analytic conductor 9.223
Analytic rank 0
Dimension 20
CM No
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1155.c (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20}\mathstrut +\mathstrut \) \(33\) \(x^{18}\mathstrut +\mathstrut \) \(456\) \(x^{16}\mathstrut +\mathstrut \) \(3426\) \(x^{14}\mathstrut +\mathstrut \) \(15194\) \(x^{12}\mathstrut +\mathstrut \) \(40320\) \(x^{10}\mathstrut +\mathstrut \) \(61593\) \(x^{8}\mathstrut +\mathstrut \) \(48545\) \(x^{6}\mathstrut +\mathstrut \) \(15624\) \(x^{4}\mathstrut +\mathstrut \) \(2116\) \(x^{2}\mathstrut +\mathstrut \) \(100\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} + 3 \)
\(\beta_{3}\)\(=\)\((\)\( -2 \nu^{18} - 65 \nu^{16} - 883 \nu^{14} - 6504 \nu^{12} - 28141 \nu^{10} - 72149 \nu^{8} - 104134 \nu^{6} - 72770 \nu^{4} - 15852 \nu^{2} - 1000 \)\()/20\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(5\) \(\nu^{19}\mathstrut -\mathstrut \) \(2\) \(\nu^{18}\mathstrut -\mathstrut \) \(167\) \(\nu^{17}\mathstrut -\mathstrut \) \(72\) \(\nu^{16}\mathstrut -\mathstrut \) \(2337\) \(\nu^{15}\mathstrut -\mathstrut \) \(1090\) \(\nu^{14}\mathstrut -\mathstrut \) \(17770\) \(\nu^{13}\mathstrut -\mathstrut \) \(8984\) \(\nu^{12}\mathstrut -\mathstrut \) \(79489\) \(\nu^{11}\mathstrut -\mathstrut \) \(43540\) \(\nu^{10}\mathstrut -\mathstrut \) \(210785\) \(\nu^{9}\mathstrut -\mathstrut \) \(124604\) \(\nu^{8}\mathstrut -\mathstrut \) \(314269\) \(\nu^{7}\mathstrut -\mathstrut \) \(198808\) \(\nu^{6}\mathstrut -\mathstrut \) \(225998\) \(\nu^{5}\mathstrut -\mathstrut \) \(150428\) \(\nu^{4}\mathstrut -\mathstrut \) \(50148\) \(\nu^{3}\mathstrut -\mathstrut \) \(33220\) \(\nu^{2}\mathstrut -\mathstrut \) \(3220\) \(\nu\mathstrut -\mathstrut \) \(2040\)\()/20\)
\(\beta_{5}\)\(=\)\((\)\(-\)\(10\) \(\nu^{19}\mathstrut -\mathstrut \) \(\nu^{18}\mathstrut -\mathstrut \) \(337\) \(\nu^{17}\mathstrut -\mathstrut \) \(23\) \(\nu^{16}\mathstrut -\mathstrut \) \(4762\) \(\nu^{15}\mathstrut -\mathstrut \) \(157\) \(\nu^{14}\mathstrut -\mathstrut \) \(36585\) \(\nu^{13}\mathstrut +\mathstrut \) \(198\) \(\nu^{12}\mathstrut -\mathstrut \) \(165399\) \(\nu^{11}\mathstrut +\mathstrut \) \(7591\) \(\nu^{10}\mathstrut -\mathstrut \) \(443130\) \(\nu^{9}\mathstrut +\mathstrut \) \(38433\) \(\nu^{8}\mathstrut -\mathstrut \) \(666509\) \(\nu^{7}\mathstrut +\mathstrut \) \(83507\) \(\nu^{6}\mathstrut -\mathstrut \) \(481448\) \(\nu^{5}\mathstrut +\mathstrut \) \(75668\) \(\nu^{4}\mathstrut -\mathstrut \) \(105408\) \(\nu^{3}\mathstrut +\mathstrut \) \(17612\) \(\nu^{2}\mathstrut -\mathstrut \) \(6620\) \(\nu\mathstrut +\mathstrut \) \(1100\)\()/40\)
\(\beta_{6}\)\(=\)\((\)\(6\) \(\nu^{19}\mathstrut +\mathstrut \) \(5\) \(\nu^{18}\mathstrut +\mathstrut \) \(194\) \(\nu^{17}\mathstrut +\mathstrut \) \(167\) \(\nu^{16}\mathstrut +\mathstrut \) \(2614\) \(\nu^{15}\mathstrut +\mathstrut \) \(2337\) \(\nu^{14}\mathstrut +\mathstrut \) \(19028\) \(\nu^{13}\mathstrut +\mathstrut \) \(17770\) \(\nu^{12}\mathstrut +\mathstrut \) \(81026\) \(\nu^{11}\mathstrut +\mathstrut \) \(79489\) \(\nu^{10}\mathstrut +\mathstrut \) \(203584\) \(\nu^{9}\mathstrut +\mathstrut \) \(210785\) \(\nu^{8}\mathstrut +\mathstrut \) \(286860\) \(\nu^{7}\mathstrut +\mathstrut \) \(314269\) \(\nu^{6}\mathstrut +\mathstrut \) \(195158\) \(\nu^{5}\mathstrut +\mathstrut \) \(225998\) \(\nu^{4}\mathstrut +\mathstrut \) \(41326\) \(\nu^{3}\mathstrut +\mathstrut \) \(50148\) \(\nu^{2}\mathstrut +\mathstrut \) \(2564\) \(\nu\mathstrut +\mathstrut \) \(3220\)\()/20\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(11\) \(\nu^{19}\mathstrut -\mathstrut \) \(10\) \(\nu^{18}\mathstrut -\mathstrut \) \(364\) \(\nu^{17}\mathstrut -\mathstrut \) \(337\) \(\nu^{16}\mathstrut -\mathstrut \) \(5039\) \(\nu^{15}\mathstrut -\mathstrut \) \(4762\) \(\nu^{14}\mathstrut -\mathstrut \) \(37843\) \(\nu^{13}\mathstrut -\mathstrut \) \(36585\) \(\nu^{12}\mathstrut -\mathstrut \) \(166936\) \(\nu^{11}\mathstrut -\mathstrut \) \(165399\) \(\nu^{10}\mathstrut -\mathstrut \) \(435929\) \(\nu^{9}\mathstrut -\mathstrut \) \(443130\) \(\nu^{8}\mathstrut -\mathstrut \) \(639090\) \(\nu^{7}\mathstrut -\mathstrut \) \(666509\) \(\nu^{6}\mathstrut -\mathstrut \) \(450488\) \(\nu^{5}\mathstrut -\mathstrut \) \(481448\) \(\nu^{4}\mathstrut -\mathstrut \) \(96196\) \(\nu^{3}\mathstrut -\mathstrut \) \(105408\) \(\nu^{2}\mathstrut -\mathstrut \) \(5664\) \(\nu\mathstrut -\mathstrut \) \(6620\)\()/40\)
\(\beta_{8}\)\(=\)\((\)\( -9 \nu^{18} - 296 \nu^{16} - 4067 \nu^{14} - 30273 \nu^{12} - 132214 \nu^{10} - 341693 \nu^{8} - 496350 \nu^{6} - 348454 \nu^{4} - 76008 \nu^{2} - 4740 \)\()/20\)
\(\beta_{9}\)\(=\)\((\)\( 13 \nu^{18} + 413 \nu^{16} + 5445 \nu^{14} + 38586 \nu^{12} + 159025 \nu^{10} + 384351 \nu^{8} + 518257 \nu^{6} + 337002 \nu^{4} + 69240 \nu^{2} + 4100 \)\()/20\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(5\) \(\nu^{19}\mathstrut +\mathstrut \) \(2\) \(\nu^{18}\mathstrut -\mathstrut \) \(167\) \(\nu^{17}\mathstrut +\mathstrut \) \(72\) \(\nu^{16}\mathstrut -\mathstrut \) \(2337\) \(\nu^{15}\mathstrut +\mathstrut \) \(1090\) \(\nu^{14}\mathstrut -\mathstrut \) \(17770\) \(\nu^{13}\mathstrut +\mathstrut \) \(8984\) \(\nu^{12}\mathstrut -\mathstrut \) \(79489\) \(\nu^{11}\mathstrut +\mathstrut \) \(43540\) \(\nu^{10}\mathstrut -\mathstrut \) \(210785\) \(\nu^{9}\mathstrut +\mathstrut \) \(124604\) \(\nu^{8}\mathstrut -\mathstrut \) \(314269\) \(\nu^{7}\mathstrut +\mathstrut \) \(198808\) \(\nu^{6}\mathstrut -\mathstrut \) \(225998\) \(\nu^{5}\mathstrut +\mathstrut \) \(150428\) \(\nu^{4}\mathstrut -\mathstrut \) \(50148\) \(\nu^{3}\mathstrut +\mathstrut \) \(33220\) \(\nu^{2}\mathstrut -\mathstrut \) \(3220\) \(\nu\mathstrut +\mathstrut \) \(2040\)\()/20\)
\(\beta_{11}\)\(=\)\((\)\(6\) \(\nu^{19}\mathstrut -\mathstrut \) \(5\) \(\nu^{18}\mathstrut +\mathstrut \) \(194\) \(\nu^{17}\mathstrut -\mathstrut \) \(167\) \(\nu^{16}\mathstrut +\mathstrut \) \(2614\) \(\nu^{15}\mathstrut -\mathstrut \) \(2337\) \(\nu^{14}\mathstrut +\mathstrut \) \(19028\) \(\nu^{13}\mathstrut -\mathstrut \) \(17770\) \(\nu^{12}\mathstrut +\mathstrut \) \(81026\) \(\nu^{11}\mathstrut -\mathstrut \) \(79489\) \(\nu^{10}\mathstrut +\mathstrut \) \(203584\) \(\nu^{9}\mathstrut -\mathstrut \) \(210785\) \(\nu^{8}\mathstrut +\mathstrut \) \(286860\) \(\nu^{7}\mathstrut -\mathstrut \) \(314269\) \(\nu^{6}\mathstrut +\mathstrut \) \(195158\) \(\nu^{5}\mathstrut -\mathstrut \) \(225998\) \(\nu^{4}\mathstrut +\mathstrut \) \(41326\) \(\nu^{3}\mathstrut -\mathstrut \) \(50148\) \(\nu^{2}\mathstrut +\mathstrut \) \(2564\) \(\nu\mathstrut -\mathstrut \) \(3220\)\()/20\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(11\) \(\nu^{19}\mathstrut +\mathstrut \) \(10\) \(\nu^{18}\mathstrut -\mathstrut \) \(364\) \(\nu^{17}\mathstrut +\mathstrut \) \(337\) \(\nu^{16}\mathstrut -\mathstrut \) \(5039\) \(\nu^{15}\mathstrut +\mathstrut \) \(4762\) \(\nu^{14}\mathstrut -\mathstrut \) \(37843\) \(\nu^{13}\mathstrut +\mathstrut \) \(36585\) \(\nu^{12}\mathstrut -\mathstrut \) \(166936\) \(\nu^{11}\mathstrut +\mathstrut \) \(165399\) \(\nu^{10}\mathstrut -\mathstrut \) \(435929\) \(\nu^{9}\mathstrut +\mathstrut \) \(443130\) \(\nu^{8}\mathstrut -\mathstrut \) \(639090\) \(\nu^{7}\mathstrut +\mathstrut \) \(666509\) \(\nu^{6}\mathstrut -\mathstrut \) \(450488\) \(\nu^{5}\mathstrut +\mathstrut \) \(481448\) \(\nu^{4}\mathstrut -\mathstrut \) \(96196\) \(\nu^{3}\mathstrut +\mathstrut \) \(105408\) \(\nu^{2}\mathstrut -\mathstrut \) \(5664\) \(\nu\mathstrut +\mathstrut \) \(6620\)\()/40\)
\(\beta_{13}\)\(=\)\((\)\( -10 \nu^{19} - 328 \nu^{17} - 4495 \nu^{15} - 33377 \nu^{13} - 145436 \nu^{11} - 375059 \nu^{9} - 543781 \nu^{7} - 381316 \nu^{5} - 83470 \nu^{3} - 5308 \nu \)\()/20\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(11\) \(\nu^{19}\mathstrut -\mathstrut \) \(32\) \(\nu^{18}\mathstrut -\mathstrut \) \(364\) \(\nu^{17}\mathstrut -\mathstrut \) \(1023\) \(\nu^{16}\mathstrut -\mathstrut \) \(5039\) \(\nu^{15}\mathstrut -\mathstrut \) \(13596\) \(\nu^{14}\mathstrut -\mathstrut \) \(37843\) \(\nu^{13}\mathstrut -\mathstrut \) \(97349\) \(\nu^{12}\mathstrut -\mathstrut \) \(166936\) \(\nu^{11}\mathstrut -\mathstrut \) \(406577\) \(\nu^{10}\mathstrut -\mathstrut \) \(435929\) \(\nu^{9}\mathstrut -\mathstrut \) \(999584\) \(\nu^{8}\mathstrut -\mathstrut \) \(639090\) \(\nu^{7}\mathstrut -\mathstrut \) \(1377875\) \(\nu^{6}\mathstrut -\mathstrut \) \(450488\) \(\nu^{5}\mathstrut -\mathstrut \) \(923432\) \(\nu^{4}\mathstrut -\mathstrut \) \(96196\) \(\nu^{3}\mathstrut -\mathstrut \) \(201024\) \(\nu^{2}\mathstrut -\mathstrut \) \(5664\) \(\nu\mathstrut -\mathstrut \) \(12860\)\()/40\)
\(\beta_{15}\)\(=\)\((\)\(24\) \(\nu^{19}\mathstrut -\mathstrut \) \(\nu^{18}\mathstrut +\mathstrut \) \(795\) \(\nu^{17}\mathstrut -\mathstrut \) \(23\) \(\nu^{16}\mathstrut +\mathstrut \) \(11018\) \(\nu^{15}\mathstrut -\mathstrut \) \(157\) \(\nu^{14}\mathstrut +\mathstrut \) \(82855\) \(\nu^{13}\mathstrut +\mathstrut \) \(198\) \(\nu^{12}\mathstrut +\mathstrut \) \(366117\) \(\nu^{11}\mathstrut +\mathstrut \) \(7591\) \(\nu^{10}\mathstrut +\mathstrut \) \(958442\) \(\nu^{9}\mathstrut +\mathstrut \) \(38433\) \(\nu^{8}\mathstrut +\mathstrut \) \(1411273\) \(\nu^{7}\mathstrut +\mathstrut \) \(83507\) \(\nu^{6}\mathstrut +\mathstrut \) \(1004904\) \(\nu^{5}\mathstrut +\mathstrut \) \(75668\) \(\nu^{4}\mathstrut +\mathstrut \) \(223552\) \(\nu^{3}\mathstrut +\mathstrut \) \(17612\) \(\nu^{2}\mathstrut +\mathstrut \) \(14468\) \(\nu\mathstrut +\mathstrut \) \(1100\)\()/40\)
\(\beta_{16}\)\(=\)\((\)\(-\)\(24\) \(\nu^{19}\mathstrut -\mathstrut \) \(\nu^{18}\mathstrut -\mathstrut \) \(795\) \(\nu^{17}\mathstrut -\mathstrut \) \(23\) \(\nu^{16}\mathstrut -\mathstrut \) \(11018\) \(\nu^{15}\mathstrut -\mathstrut \) \(157\) \(\nu^{14}\mathstrut -\mathstrut \) \(82855\) \(\nu^{13}\mathstrut +\mathstrut \) \(198\) \(\nu^{12}\mathstrut -\mathstrut \) \(366117\) \(\nu^{11}\mathstrut +\mathstrut \) \(7591\) \(\nu^{10}\mathstrut -\mathstrut \) \(958442\) \(\nu^{9}\mathstrut +\mathstrut \) \(38433\) \(\nu^{8}\mathstrut -\mathstrut \) \(1411273\) \(\nu^{7}\mathstrut +\mathstrut \) \(83507\) \(\nu^{6}\mathstrut -\mathstrut \) \(1004904\) \(\nu^{5}\mathstrut +\mathstrut \) \(75668\) \(\nu^{4}\mathstrut -\mathstrut \) \(223552\) \(\nu^{3}\mathstrut +\mathstrut \) \(17612\) \(\nu^{2}\mathstrut -\mathstrut \) \(14468\) \(\nu\mathstrut +\mathstrut \) \(1100\)\()/40\)
\(\beta_{17}\)\(=\)\((\)\( 28 \nu^{19} + 919 \nu^{17} + 12602 \nu^{15} + 93627 \nu^{13} + 408167 \nu^{11} + 1053028 \nu^{9} + 1527209 \nu^{7} + 1071178 \nu^{5} + 234558 \nu^{3} + 14924 \nu \)\()/20\)
\(\beta_{18}\)\(=\)\((\)\( -28 \nu^{19} - 919 \nu^{17} - 12602 \nu^{15} - 93627 \nu^{13} - 408167 \nu^{11} - 1053028 \nu^{9} - 1527209 \nu^{7} - 1071178 \nu^{5} - 234538 \nu^{3} - 14824 \nu \)\()/20\)
\(\beta_{19}\)\(=\)\((\)\( 33 \nu^{19} + 1090 \nu^{17} + 15057 \nu^{15} + 112801 \nu^{13} + 496244 \nu^{11} + 1292171 \nu^{9} + 1889226 \nu^{7} + 1329130 \nu^{5} + 284588 \nu^{3} + 17340 \nu \)\()/20\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut -\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{18}\mathstrut +\mathstrut \) \(\beta_{17}\mathstrut -\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(14\)
\(\nu^{5}\)\(=\)\(\beta_{19}\mathstrut -\mathstrut \) \(9\) \(\beta_{18}\mathstrut -\mathstrut \) \(10\) \(\beta_{17}\mathstrut -\mathstrut \) \(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(2\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(29\) \(\beta_{1}\)
\(\nu^{6}\)\(=\)\(10\) \(\beta_{16}\mathstrut +\mathstrut \) \(10\) \(\beta_{15}\mathstrut -\mathstrut \) \(12\) \(\beta_{14}\mathstrut +\mathstrut \) \(3\) \(\beta_{12}\mathstrut +\mathstrut \) \(13\) \(\beta_{11}\mathstrut +\mathstrut \) \(11\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(9\) \(\beta_{7}\mathstrut -\mathstrut \) \(13\) \(\beta_{6}\mathstrut -\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(22\) \(\beta_{3}\mathstrut +\mathstrut \) \(48\) \(\beta_{2}\mathstrut -\mathstrut \) \(75\)
\(\nu^{7}\)\(=\)\(-\)\(12\) \(\beta_{19}\mathstrut +\mathstrut \) \(69\) \(\beta_{18}\mathstrut +\mathstrut \) \(81\) \(\beta_{17}\mathstrut +\mathstrut \) \(14\) \(\beta_{16}\mathstrut +\mathstrut \) \(2\) \(\beta_{15}\mathstrut +\mathstrut \) \(12\) \(\beta_{13}\mathstrut -\mathstrut \) \(22\) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(22\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(16\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(183\) \(\beta_{1}\)
\(\nu^{8}\)\(=\)\(-\)\(77\) \(\beta_{16}\mathstrut -\mathstrut \) \(77\) \(\beta_{15}\mathstrut +\mathstrut \) \(111\) \(\beta_{14}\mathstrut -\mathstrut \) \(48\) \(\beta_{12}\mathstrut -\mathstrut \) \(128\) \(\beta_{11}\mathstrut -\mathstrut \) \(97\) \(\beta_{10}\mathstrut +\mathstrut \) \(16\) \(\beta_{9}\mathstrut -\mathstrut \) \(16\) \(\beta_{8}\mathstrut -\mathstrut \) \(63\) \(\beta_{7}\mathstrut +\mathstrut \) \(128\) \(\beta_{6}\mathstrut +\mathstrut \) \(97\) \(\beta_{4}\mathstrut -\mathstrut \) \(190\) \(\beta_{3}\mathstrut -\mathstrut \) \(333\) \(\beta_{2}\mathstrut +\mathstrut \) \(438\)
\(\nu^{9}\)\(=\)\(111\) \(\beta_{19}\mathstrut -\mathstrut \) \(503\) \(\beta_{18}\mathstrut -\mathstrut \) \(618\) \(\beta_{17}\mathstrut -\mathstrut \) \(149\) \(\beta_{16}\mathstrut -\mathstrut \) \(30\) \(\beta_{15}\mathstrut -\mathstrut \) \(111\) \(\beta_{13}\mathstrut +\mathstrut \) \(188\) \(\beta_{12}\mathstrut -\mathstrut \) \(14\) \(\beta_{11}\mathstrut -\mathstrut \) \(15\) \(\beta_{10}\mathstrut +\mathstrut \) \(188\) \(\beta_{7}\mathstrut -\mathstrut \) \(14\) \(\beta_{6}\mathstrut +\mathstrut \) \(179\) \(\beta_{5}\mathstrut -\mathstrut \) \(15\) \(\beta_{4}\mathstrut +\mathstrut \) \(1215\) \(\beta_{1}\)
\(\nu^{10}\)\(=\)\(542\) \(\beta_{16}\mathstrut +\mathstrut \) \(542\) \(\beta_{15}\mathstrut -\mathstrut \) \(940\) \(\beta_{14}\mathstrut +\mathstrut \) \(535\) \(\beta_{12}\mathstrut +\mathstrut \) \(1134\) \(\beta_{11}\mathstrut +\mathstrut \) \(800\) \(\beta_{10}\mathstrut -\mathstrut \) \(179\) \(\beta_{9}\mathstrut +\mathstrut \) \(181\) \(\beta_{8}\mathstrut +\mathstrut \) \(405\) \(\beta_{7}\mathstrut -\mathstrut \) \(1134\) \(\beta_{6}\mathstrut -\mathstrut \) \(800\) \(\beta_{4}\mathstrut +\mathstrut \) \(1526\) \(\beta_{3}\mathstrut +\mathstrut \) \(2336\) \(\beta_{2}\mathstrut -\mathstrut \) \(2703\)
\(\nu^{11}\)\(=\)\(-\)\(940\) \(\beta_{19}\mathstrut +\mathstrut \) \(3599\) \(\beta_{18}\mathstrut +\mathstrut \) \(4621\) \(\beta_{17}\mathstrut +\mathstrut \) \(1412\) \(\beta_{16}\mathstrut +\mathstrut \) \(316\) \(\beta_{15}\mathstrut +\mathstrut \) \(956\) \(\beta_{13}\mathstrut -\mathstrut \) \(1482\) \(\beta_{12}\mathstrut +\mathstrut \) \(142\) \(\beta_{11}\mathstrut +\mathstrut \) \(153\) \(\beta_{10}\mathstrut -\mathstrut \) \(1482\) \(\beta_{7}\mathstrut +\mathstrut \) \(142\) \(\beta_{6}\mathstrut -\mathstrut \) \(1728\) \(\beta_{5}\mathstrut +\mathstrut \) \(153\) \(\beta_{4}\mathstrut -\mathstrut \) \(8315\) \(\beta_{1}\)
\(\nu^{12}\)\(=\)\(-\)\(3669\) \(\beta_{16}\mathstrut -\mathstrut \) \(3669\) \(\beta_{15}\mathstrut +\mathstrut \) \(7647\) \(\beta_{14}\mathstrut -\mathstrut \) \(5152\) \(\beta_{12}\mathstrut -\mathstrut \) \(9528\) \(\beta_{11}\mathstrut -\mathstrut \) \(6412\) \(\beta_{10}\mathstrut +\mathstrut \) \(1728\) \(\beta_{9}\mathstrut -\mathstrut \) \(1780\) \(\beta_{8}\mathstrut -\mathstrut \) \(2495\) \(\beta_{7}\mathstrut +\mathstrut \) \(9528\) \(\beta_{6}\mathstrut +\mathstrut \) \(6412\) \(\beta_{4}\mathstrut -\mathstrut \) \(11926\) \(\beta_{3}\mathstrut -\mathstrut \) \(16535\) \(\beta_{2}\mathstrut +\mathstrut \) \(17276\)
\(\nu^{13}\)\(=\)\(7647\) \(\beta_{19}\mathstrut -\mathstrut \) \(25601\) \(\beta_{18}\mathstrut -\mathstrut \) \(34328\) \(\beta_{17}\mathstrut -\mathstrut \) \(12527\) \(\beta_{16}\mathstrut -\mathstrut \) \(2902\) \(\beta_{15}\mathstrut -\mathstrut \) \(8021\) \(\beta_{13}\mathstrut +\mathstrut \) \(11316\) \(\beta_{12}\mathstrut -\mathstrut \) \(1287\) \(\beta_{11}\mathstrut -\mathstrut \) \(1336\) \(\beta_{10}\mathstrut +\mathstrut \) \(11316\) \(\beta_{7}\mathstrut -\mathstrut \) \(1287\) \(\beta_{6}\mathstrut +\mathstrut \) \(15429\) \(\beta_{5}\mathstrut -\mathstrut \) \(1336\) \(\beta_{4}\mathstrut +\mathstrut \) \(57993\) \(\beta_{1}\)
\(\nu^{14}\)\(=\)\(24390\) \(\beta_{16}\mathstrut +\mathstrut \) \(24390\) \(\beta_{15}\mathstrut -\mathstrut \) \(60876\) \(\beta_{14}\mathstrut +\mathstrut \) \(45959\) \(\beta_{12}\mathstrut +\mathstrut \) \(77641\) \(\beta_{11}\mathstrut +\mathstrut \) \(50606\) \(\beta_{10}\mathstrut -\mathstrut \) \(15429\) \(\beta_{9}\mathstrut +\mathstrut \) \(16229\) \(\beta_{8}\mathstrut +\mathstrut \) \(14917\) \(\beta_{7}\mathstrut -\mathstrut \) \(77641\) \(\beta_{6}\mathstrut -\mathstrut \) \(50606\) \(\beta_{4}\mathstrut +\mathstrut \) \(92106\) \(\beta_{3}\mathstrut +\mathstrut \) \(117922\) \(\beta_{2}\mathstrut -\mathstrut \) \(113005\)
\(\nu^{15}\)\(=\)\(-\)\(60876\) \(\beta_{19}\mathstrut +\mathstrut \) \(182131\) \(\beta_{18}\mathstrut +\mathstrut \) \(254675\) \(\beta_{17}\mathstrut +\mathstrut \) \(106552\) \(\beta_{16}\mathstrut +\mathstrut \) \(24902\) \(\beta_{15}\mathstrut +\mathstrut \) \(66250\) \(\beta_{13}\mathstrut -\mathstrut \) \(85266\) \(\beta_{12}\mathstrut +\mathstrut \) \(11070\) \(\beta_{11}\mathstrut +\mathstrut \) \(10806\) \(\beta_{10}\mathstrut -\mathstrut \) \(85266\) \(\beta_{7}\mathstrut +\mathstrut \) \(11070\) \(\beta_{6}\mathstrut -\mathstrut \) \(131454\) \(\beta_{5}\mathstrut +\mathstrut \) \(10806\) \(\beta_{4}\mathstrut -\mathstrut \) \(409725\) \(\beta_{1}\)
\(\nu^{16}\)\(=\)\(-\)\(160845\) \(\beta_{16}\mathstrut -\mathstrut \) \(160845\) \(\beta_{15}\mathstrut +\mathstrut \) \(478251\) \(\beta_{14}\mathstrut -\mathstrut \) \(391600\) \(\beta_{12}\mathstrut -\mathstrut \) \(620511\) \(\beta_{11}\mathstrut -\mathstrut \) \(395499\) \(\beta_{10}\mathstrut +\mathstrut \) \(131454\) \(\beta_{9}\mathstrut -\mathstrut \) \(141088\) \(\beta_{8}\mathstrut -\mathstrut \) \(86651\) \(\beta_{7}\mathstrut +\mathstrut \) \(620511\) \(\beta_{6}\mathstrut +\mathstrut \) \(395499\) \(\beta_{4}\mathstrut -\mathstrut \) \(707054\) \(\beta_{3}\mathstrut -\mathstrut \) \(846531\) \(\beta_{2}\mathstrut +\mathstrut \) \(751452\)
\(\nu^{17}\)\(=\)\(478251\) \(\beta_{19}\mathstrut -\mathstrut \) \(1299675\) \(\beta_{18}\mathstrut -\mathstrut \) \(1890748\) \(\beta_{17}\mathstrut -\mathstrut \) \(880665\) \(\beta_{16}\mathstrut -\mathstrut \) \(205648\) \(\beta_{15}\mathstrut -\mathstrut \) \(539857\) \(\beta_{13}\mathstrut +\mathstrut \) \(639096\) \(\beta_{12}\mathstrut -\mathstrut \) \(92386\) \(\beta_{11}\mathstrut -\mathstrut \) \(83924\) \(\beta_{10}\mathstrut +\mathstrut \) \(639096\) \(\beta_{7}\mathstrut -\mathstrut \) \(92386\) \(\beta_{6}\mathstrut +\mathstrut \) \(1086313\) \(\beta_{5}\mathstrut -\mathstrut \) \(83924\) \(\beta_{4}\mathstrut +\mathstrut \) \(2922765\) \(\beta_{1}\)
\(\nu^{18}\)\(=\)\(1058106\) \(\beta_{16}\mathstrut +\mathstrut \) \(1058106\) \(\beta_{15}\mathstrut -\mathstrut \) \(3724028\) \(\beta_{14}\mathstrut +\mathstrut \) \(3238063\) \(\beta_{12}\mathstrut +\mathstrut \) \(4894265\) \(\beta_{11}\mathstrut +\mathstrut \) \(3069467\) \(\beta_{10}\mathstrut -\mathstrut \) \(1086313\) \(\beta_{9}\mathstrut +\mathstrut \) \(1187181\) \(\beta_{8}\mathstrut +\mathstrut \) \(485965\) \(\beta_{7}\mathstrut -\mathstrut \) \(4894265\) \(\beta_{6}\mathstrut -\mathstrut \) \(3069467\) \(\beta_{4}\mathstrut +\mathstrut \) \(5407666\) \(\beta_{3}\mathstrut +\mathstrut \) \(6113188\) \(\beta_{2}\mathstrut -\mathstrut \) \(5061191\)
\(\nu^{19}\)\(=\)\(-\)\(3724028\) \(\beta_{19}\mathstrut +\mathstrut \) \(9315713\) \(\beta_{18}\mathstrut +\mathstrut \) \(14057701\) \(\beta_{17}\mathstrut +\mathstrut \) \(7131710\) \(\beta_{16}\mathstrut +\mathstrut \) \(1658454\) \(\beta_{15}\mathstrut +\mathstrut \) \(4344692\) \(\beta_{13}\mathstrut -\mathstrut \) \(4782134\) \(\beta_{12}\mathstrut +\mathstrut \) \(755429\) \(\beta_{11}\mathstrut +\mathstrut \) \(637617\) \(\beta_{10}\mathstrut -\mathstrut \) \(4782134\) \(\beta_{7}\mathstrut +\mathstrut \) \(755429\) \(\beta_{6}\mathstrut -\mathstrut \) \(8790164\) \(\beta_{5}\mathstrut +\mathstrut \) \(637617\) \(\beta_{4}\mathstrut -\mathstrut \) \(21011595\) \(\beta_{1}\)

Character Values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1155\mathbb{Z}\right)^\times\).

\(n\) \(211\) \(232\) \(386\) \(661\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
694.1
2.74619i
2.51841i
2.51027i
2.02181i
1.76958i
1.76011i
1.41336i
0.445112i
0.437361i
0.332438i
0.332438i
0.437361i
0.445112i
1.41336i
1.76011i
1.76958i
2.02181i
2.51027i
2.51841i
2.74619i
2.74619i 1.00000i −5.54154 2.23561 + 0.0452669i 2.74619 1.00000i 9.72574i −1.00000 0.124311 6.13940i
694.2 2.51841i 1.00000i −4.34238 −2.17741 0.508790i 2.51841 1.00000i 5.89907i −1.00000 −1.28134 + 5.48362i
694.3 2.51027i 1.00000i −4.30144 −1.80593 1.31856i −2.51027 1.00000i 5.77723i −1.00000 −3.30994 + 4.53337i
694.4 2.02181i 1.00000i −2.08773 −0.549616 + 2.16747i 2.02181 1.00000i 0.177380i −1.00000 4.38222 + 1.11122i
694.5 1.76958i 1.00000i −1.13141 −1.09533 + 1.94942i −1.76958 1.00000i 1.53703i −1.00000 3.44966 + 1.93828i
694.6 1.76011i 1.00000i −1.09799 2.22415 0.230563i −1.76011 1.00000i 1.58764i −1.00000 −0.405817 3.91475i
694.7 1.41336i 1.00000i 0.00241967 1.69889 1.45388i 1.41336 1.00000i 2.83014i −1.00000 −2.05485 2.40114i
694.8 0.445112i 1.00000i 1.80187 −2.13786 0.655413i 0.445112 1.00000i 1.69226i −1.00000 −0.291732 + 0.951587i
694.9 0.437361i 1.00000i 1.80872 0.978157 2.01077i −0.437361 1.00000i 1.66578i −1.00000 −0.879433 0.427807i
694.10 0.332438i 1.00000i 1.88948 −0.370655 2.20513i 0.332438 1.00000i 1.29301i −1.00000 −0.733071 + 0.123220i
694.11 0.332438i 1.00000i 1.88948 −0.370655 + 2.20513i 0.332438 1.00000i 1.29301i −1.00000 −0.733071 0.123220i
694.12 0.437361i 1.00000i 1.80872 0.978157 + 2.01077i −0.437361 1.00000i 1.66578i −1.00000 −0.879433 + 0.427807i
694.13 0.445112i 1.00000i 1.80187 −2.13786 + 0.655413i 0.445112 1.00000i 1.69226i −1.00000 −0.291732 0.951587i
694.14 1.41336i 1.00000i 0.00241967 1.69889 + 1.45388i 1.41336 1.00000i 2.83014i −1.00000 −2.05485 + 2.40114i
694.15 1.76011i 1.00000i −1.09799 2.22415 + 0.230563i −1.76011 1.00000i 1.58764i −1.00000 −0.405817 + 3.91475i
694.16 1.76958i 1.00000i −1.13141 −1.09533 1.94942i −1.76958 1.00000i 1.53703i −1.00000 3.44966 1.93828i
694.17 2.02181i 1.00000i −2.08773 −0.549616 2.16747i 2.02181 1.00000i 0.177380i −1.00000 4.38222 1.11122i
694.18 2.51027i 1.00000i −4.30144 −1.80593 + 1.31856i −2.51027 1.00000i 5.77723i −1.00000 −3.30994 4.53337i
694.19 2.51841i 1.00000i −4.34238 −2.17741 + 0.508790i 2.51841 1.00000i 5.89907i −1.00000 −1.28134 5.48362i
694.20 2.74619i 1.00000i −5.54154 2.23561 0.0452669i 2.74619 1.00000i 9.72574i −1.00000 0.124311 + 6.13940i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 694.20
Significant digits:
Format:

Inner twists

Char. orbit Parity Mult. Self Twist Proved
1.a Even 1 trivial yes
5.b Even 1 yes

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1155, [\chi])\):

\(T_{2}^{20} + \cdots\)
\(T_{13}^{20} + \cdots\)