Properties

Label 8037.2.a.o
Level 8037
Weight 2
Character orbit 8037.a
Self dual Yes
Analytic conductor 64.176
Analytic rank 0
Dimension 18
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8037 = 3^{2} \cdot 19 \cdot 47 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8037.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(64.1757681045\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18}\mathstrut -\mathstrut \) \(x^{17}\mathstrut -\mathstrut \) \(28\) \(x^{16}\mathstrut +\mathstrut \) \(25\) \(x^{15}\mathstrut +\mathstrut \) \(322\) \(x^{14}\mathstrut -\mathstrut \) \(247\) \(x^{13}\mathstrut -\mathstrut \) \(1971\) \(x^{12}\mathstrut +\mathstrut \) \(1231\) \(x^{11}\mathstrut +\mathstrut \) \(6953\) \(x^{10}\mathstrut -\mathstrut \) \(3283\) \(x^{9}\mathstrut -\mathstrut \) \(14235\) \(x^{8}\mathstrut +\mathstrut \) \(4562\) \(x^{7}\mathstrut +\mathstrut \) \(15962\) \(x^{6}\mathstrut -\mathstrut \) \(2882\) \(x^{5}\mathstrut -\mathstrut \) \(8159\) \(x^{4}\mathstrut +\mathstrut \) \(606\) \(x^{3}\mathstrut +\mathstrut \) \(890\) \(x^{2}\mathstrut -\mathstrut \) \(179\) \(x\mathstrut +\mathstrut \) \(9\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(1375\) \(\nu^{17}\mathstrut +\mathstrut \) \(5084\) \(\nu^{16}\mathstrut +\mathstrut \) \(36270\) \(\nu^{15}\mathstrut -\mathstrut \) \(122133\) \(\nu^{14}\mathstrut -\mathstrut \) \(385293\) \(\nu^{13}\mathstrut +\mathstrut \) \(1142502\) \(\nu^{12}\mathstrut +\mathstrut \) \(2121571\) \(\nu^{11}\mathstrut -\mathstrut \) \(5271936\) \(\nu^{10}\mathstrut -\mathstrut \) \(6496241\) \(\nu^{9}\mathstrut +\mathstrut \) \(12574330\) \(\nu^{8}\mathstrut +\mathstrut \) \(10966627\) \(\nu^{7}\mathstrut -\mathstrut \) \(14789013\) \(\nu^{6}\mathstrut -\mathstrut \) \(9398813\) \(\nu^{5}\mathstrut +\mathstrut \) \(6930877\) \(\nu^{4}\mathstrut +\mathstrut \) \(3224026\) \(\nu^{3}\mathstrut -\mathstrut \) \(357518\) \(\nu^{2}\mathstrut -\mathstrut \) \(175528\) \(\nu\mathstrut +\mathstrut \) \(15159\)\()/25060\)
\(\beta_{4}\)\(=\)\((\)\(1245\) \(\nu^{17}\mathstrut -\mathstrut \) \(14641\) \(\nu^{16}\mathstrut -\mathstrut \) \(40655\) \(\nu^{15}\mathstrut +\mathstrut \) \(363742\) \(\nu^{14}\mathstrut +\mathstrut \) \(548917\) \(\nu^{13}\mathstrut -\mathstrut \) \(3562623\) \(\nu^{12}\mathstrut -\mathstrut \) \(3965914\) \(\nu^{11}\mathstrut +\mathstrut \) \(17502894\) \(\nu^{10}\mathstrut +\mathstrut \) \(16545409\) \(\nu^{9}\mathstrut -\mathstrut \) \(45330325\) \(\nu^{8}\mathstrut -\mathstrut \) \(39853058\) \(\nu^{7}\mathstrut +\mathstrut \) \(58721567\) \(\nu^{6}\mathstrut +\mathstrut \) \(51873662\) \(\nu^{5}\mathstrut -\mathstrut \) \(29528923\) \(\nu^{4}\mathstrut -\mathstrut \) \(29497239\) \(\nu^{3}\mathstrut -\mathstrut \) \(499933\) \(\nu^{2}\mathstrut +\mathstrut \) \(2194037\) \(\nu\mathstrut -\mathstrut \) \(125826\)\()/25060\)
\(\beta_{5}\)\(=\)\((\)\(3027\) \(\nu^{17}\mathstrut +\mathstrut \) \(6697\) \(\nu^{16}\mathstrut -\mathstrut \) \(74329\) \(\nu^{15}\mathstrut -\mathstrut \) \(168948\) \(\nu^{14}\mathstrut +\mathstrut \) \(714237\) \(\nu^{13}\mathstrut +\mathstrut \) \(1691399\) \(\nu^{12}\mathstrut -\mathstrut \) \(3386660\) \(\nu^{11}\mathstrut -\mathstrut \) \(8576416\) \(\nu^{10}\mathstrut +\mathstrut \) \(8111547\) \(\nu^{9}\mathstrut +\mathstrut \) \(23254771\) \(\nu^{8}\mathstrut -\mathstrut \) \(8377838\) \(\nu^{7}\mathstrut -\mathstrut \) \(32405463\) \(\nu^{6}\mathstrut +\mathstrut \) \(180072\) \(\nu^{5}\mathstrut +\mathstrut \) \(19092659\) \(\nu^{4}\mathstrut +\mathstrut \) \(4021219\) \(\nu^{3}\mathstrut -\mathstrut \) \(1587523\) \(\nu^{2}\mathstrut -\mathstrut \) \(104647\) \(\nu\mathstrut +\mathstrut \) \(23292\)\()/25060\)
\(\beta_{6}\)\(=\)\((\)\(2588\) \(\nu^{17}\mathstrut -\mathstrut \) \(5615\) \(\nu^{16}\mathstrut -\mathstrut \) \(79161\) \(\nu^{15}\mathstrut +\mathstrut \) \(139029\) \(\nu^{14}\mathstrut +\mathstrut \) \(1002284\) \(\nu^{13}\mathstrut -\mathstrut \) \(1353473\) \(\nu^{12}\mathstrut -\mathstrut \) \(6792347\) \(\nu^{11}\mathstrut +\mathstrut \) \(6572488\) \(\nu^{10}\mathstrut +\mathstrut \) \(26570780\) \(\nu^{9}\mathstrut -\mathstrut \) \(16607951\) \(\nu^{8}\mathstrut -\mathstrut \) \(60094951\) \(\nu^{7}\mathstrut +\mathstrut \) \(20184294\) \(\nu^{6}\mathstrut +\mathstrut \) \(73715119\) \(\nu^{5}\mathstrut -\mathstrut \) \(7638688\) \(\nu^{4}\mathstrut -\mathstrut \) \(40208151\) \(\nu^{3}\mathstrut -\mathstrut \) \(2452891\) \(\nu^{2}\mathstrut +\mathstrut \) \(3890843\) \(\nu\mathstrut -\mathstrut \) \(358605\)\()/25060\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(1240\) \(\nu^{17}\mathstrut -\mathstrut \) \(3931\) \(\nu^{16}\mathstrut +\mathstrut \) \(38655\) \(\nu^{15}\mathstrut +\mathstrut \) \(100262\) \(\nu^{14}\mathstrut -\mathstrut \) \(492753\) \(\nu^{13}\mathstrut -\mathstrut \) \(1019383\) \(\nu^{12}\mathstrut +\mathstrut \) \(3303026\) \(\nu^{11}\mathstrut +\mathstrut \) \(5294509\) \(\nu^{10}\mathstrut -\mathstrut \) \(12477886\) \(\nu^{9}\mathstrut -\mathstrut \) \(14969960\) \(\nu^{8}\mathstrut +\mathstrut \) \(26490527\) \(\nu^{7}\mathstrut +\mathstrut \) \(22628692\) \(\nu^{6}\mathstrut -\mathstrut \) \(29599058\) \(\nu^{5}\mathstrut -\mathstrut \) \(16352508\) \(\nu^{4}\mathstrut +\mathstrut \) \(14323951\) \(\nu^{3}\mathstrut +\mathstrut \) \(3935337\) \(\nu^{2}\mathstrut -\mathstrut \) \(1330428\) \(\nu\mathstrut +\mathstrut \) \(66324\)\()/12530\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(1023\) \(\nu^{17}\mathstrut +\mathstrut \) \(7752\) \(\nu^{16}\mathstrut +\mathstrut \) \(35806\) \(\nu^{15}\mathstrut -\mathstrut \) \(193633\) \(\nu^{14}\mathstrut -\mathstrut \) \(510113\) \(\nu^{13}\mathstrut +\mathstrut \) \(1911944\) \(\nu^{12}\mathstrut +\mathstrut \) \(3828075\) \(\nu^{11}\mathstrut -\mathstrut \) \(9513046\) \(\nu^{10}\mathstrut -\mathstrut \) \(16327263\) \(\nu^{9}\mathstrut +\mathstrut \) \(25165856\) \(\nu^{8}\mathstrut +\mathstrut \) \(39653957\) \(\nu^{7}\mathstrut -\mathstrut \) \(33965223\) \(\nu^{6}\mathstrut -\mathstrut \) \(51556133\) \(\nu^{5}\mathstrut +\mathstrut \) \(19134419\) \(\nu^{4}\mathstrut +\mathstrut \) \(29575684\) \(\nu^{3}\mathstrut -\mathstrut \) \(1483328\) \(\nu^{2}\mathstrut -\mathstrut \) \(3129092\) \(\nu\mathstrut +\mathstrut \) \(361577\)\()/12530\)
\(\beta_{9}\)\(=\)\((\)\(1455\) \(\nu^{17}\mathstrut -\mathstrut \) \(6528\) \(\nu^{16}\mathstrut -\mathstrut \) \(49475\) \(\nu^{15}\mathstrut +\mathstrut \) \(157256\) \(\nu^{14}\mathstrut +\mathstrut \) \(679971\) \(\nu^{13}\mathstrut -\mathstrut \) \(1476119\) \(\nu^{12}\mathstrut -\mathstrut \) \(4872722\) \(\nu^{11}\mathstrut +\mathstrut \) \(6830792\) \(\nu^{10}\mathstrut +\mathstrut \) \(19578362\) \(\nu^{9}\mathstrut -\mathstrut \) \(16240180\) \(\nu^{8}\mathstrut -\mathstrut \) \(44099559\) \(\nu^{7}\mathstrut +\mathstrut \) \(18552046\) \(\nu^{6}\mathstrut +\mathstrut \) \(52477321\) \(\nu^{5}\mathstrut -\mathstrut \) \(7241889\) \(\nu^{4}\mathstrut -\mathstrut \) \(27430832\) \(\nu^{3}\mathstrut -\mathstrut \) \(969549\) \(\nu^{2}\mathstrut +\mathstrut \) \(2832801\) \(\nu\mathstrut -\mathstrut \) \(270523\)\()/12530\)
\(\beta_{10}\)\(=\)\((\)\(2073\) \(\nu^{17}\mathstrut +\mathstrut \) \(1488\) \(\nu^{16}\mathstrut -\mathstrut \) \(54846\) \(\nu^{15}\mathstrut -\mathstrut \) \(36877\) \(\nu^{14}\mathstrut +\mathstrut \) \(589003\) \(\nu^{13}\mathstrut +\mathstrut \) \(363546\) \(\nu^{12}\mathstrut -\mathstrut \) \(3325055\) \(\nu^{11}\mathstrut -\mathstrut \) \(1834374\) \(\nu^{10}\mathstrut +\mathstrut \) \(10698493\) \(\nu^{9}\mathstrut +\mathstrut \) \(5102844\) \(\nu^{8}\mathstrut -\mathstrut \) \(19844447\) \(\nu^{7}\mathstrut -\mathstrut \) \(7933067\) \(\nu^{6}\mathstrut +\mathstrut \) \(20060543\) \(\nu^{5}\mathstrut +\mathstrut \) \(6557961\) \(\nu^{4}\mathstrut -\mathstrut \) \(9138204\) \(\nu^{3}\mathstrut -\mathstrut \) \(2305702\) \(\nu^{2}\mathstrut +\mathstrut \) \(809712\) \(\nu\mathstrut -\mathstrut \) \(13747\)\()/12530\)
\(\beta_{11}\)\(=\)\((\)\(4595\) \(\nu^{17}\mathstrut +\mathstrut \) \(9469\) \(\nu^{16}\mathstrut -\mathstrut \) \(115125\) \(\nu^{15}\mathstrut -\mathstrut \) \(236848\) \(\nu^{14}\mathstrut +\mathstrut \) \(1140117\) \(\nu^{13}\mathstrut +\mathstrut \) \(2345227\) \(\nu^{12}\mathstrut -\mathstrut \) \(5679104\) \(\nu^{11}\mathstrut -\mathstrut \) \(11727536\) \(\nu^{10}\mathstrut +\mathstrut \) \(14932039\) \(\nu^{9}\mathstrut +\mathstrut \) \(31280355\) \(\nu^{8}\mathstrut -\mathstrut \) \(19577278\) \(\nu^{7}\mathstrut -\mathstrut \) \(42921283\) \(\nu^{6}\mathstrut +\mathstrut \) \(9713092\) \(\nu^{5}\mathstrut +\mathstrut \) \(25344387\) \(\nu^{4}\mathstrut +\mathstrut \) \(374051\) \(\nu^{3}\mathstrut -\mathstrut \) \(2682883\) \(\nu^{2}\mathstrut +\mathstrut \) \(434997\) \(\nu\mathstrut -\mathstrut \) \(33996\)\()/25060\)
\(\beta_{12}\)\(=\)\((\)\(6239\) \(\nu^{17}\mathstrut +\mathstrut \) \(4103\) \(\nu^{16}\mathstrut -\mathstrut \) \(153743\) \(\nu^{15}\mathstrut -\mathstrut \) \(100284\) \(\nu^{14}\mathstrut +\mathstrut \) \(1492971\) \(\nu^{13}\mathstrut +\mathstrut \) \(964945\) \(\nu^{12}\mathstrut -\mathstrut \) \(7287684\) \(\nu^{11}\mathstrut -\mathstrut \) \(4649798\) \(\nu^{10}\mathstrut +\mathstrut \) \(18950143\) \(\nu^{9}\mathstrut +\mathstrut \) \(11770167\) \(\nu^{8}\mathstrut -\mathstrut \) \(25628064\) \(\nu^{7}\mathstrut -\mathstrut \) \(14928519\) \(\nu^{6}\mathstrut +\mathstrut \) \(15819256\) \(\nu^{5}\mathstrut +\mathstrut \) \(7398175\) \(\nu^{4}\mathstrut -\mathstrut \) \(3077931\) \(\nu^{3}\mathstrut +\mathstrut \) \(255031\) \(\nu^{2}\mathstrut +\mathstrut \) \(441793\) \(\nu\mathstrut -\mathstrut \) \(142672\)\()/25060\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(3067\) \(\nu^{17}\mathstrut +\mathstrut \) \(290\) \(\nu^{16}\mathstrut +\mathstrut \) \(84064\) \(\nu^{15}\mathstrut -\mathstrut \) \(8371\) \(\nu^{14}\mathstrut -\mathstrut \) \(943021\) \(\nu^{13}\mathstrut +\mathstrut \) \(94912\) \(\nu^{12}\mathstrut +\mathstrut \) \(5611143\) \(\nu^{11}\mathstrut -\mathstrut \) \(535462\) \(\nu^{10}\mathstrut -\mathstrut \) \(19185335\) \(\nu^{9}\mathstrut +\mathstrut \) \(1570704\) \(\nu^{8}\mathstrut +\mathstrut \) \(38006829\) \(\nu^{7}\mathstrut -\mathstrut \) \(2201281\) \(\nu^{6}\mathstrut -\mathstrut \) \(41184681\) \(\nu^{5}\mathstrut +\mathstrut \) \(1060727\) \(\nu^{4}\mathstrut +\mathstrut \) \(20386644\) \(\nu^{3}\mathstrut -\mathstrut \) \(7476\) \(\nu^{2}\mathstrut -\mathstrut \) \(2317232\) \(\nu\mathstrut +\mathstrut \) \(261015\)\()/12530\)
\(\beta_{14}\)\(=\)\((\)\(3252\) \(\nu^{17}\mathstrut +\mathstrut \) \(2949\) \(\nu^{16}\mathstrut -\mathstrut \) \(82884\) \(\nu^{15}\mathstrut -\mathstrut \) \(73532\) \(\nu^{14}\mathstrut +\mathstrut \) \(844628\) \(\nu^{13}\mathstrut +\mathstrut \) \(726240\) \(\nu^{12}\mathstrut -\mathstrut \) \(4427692\) \(\nu^{11}\mathstrut -\mathstrut \) \(3626404\) \(\nu^{10}\mathstrut +\mathstrut \) \(12846929\) \(\nu^{9}\mathstrut +\mathstrut \) \(9675021\) \(\nu^{8}\mathstrut -\mathstrut \) \(20754167\) \(\nu^{7}\mathstrut -\mathstrut \) \(13366767\) \(\nu^{6}\mathstrut +\mathstrut \) \(17725613\) \(\nu^{5}\mathstrut +\mathstrut \) \(8147890\) \(\nu^{4}\mathstrut -\mathstrut \) \(7023393\) \(\nu^{3}\mathstrut -\mathstrut \) \(1092042\) \(\nu^{2}\mathstrut +\mathstrut \) \(1032434\) \(\nu\mathstrut -\mathstrut \) \(74371\)\()/12530\)
\(\beta_{15}\)\(=\)\((\)\(7089\) \(\nu^{17}\mathstrut +\mathstrut \) \(664\) \(\nu^{16}\mathstrut -\mathstrut \) \(186758\) \(\nu^{15}\mathstrut -\mathstrut \) \(19521\) \(\nu^{14}\mathstrut +\mathstrut \) \(1989179\) \(\nu^{13}\mathstrut +\mathstrut \) \(238418\) \(\nu^{12}\mathstrut -\mathstrut \) \(11064125\) \(\nu^{11}\mathstrut -\mathstrut \) \(1547932\) \(\nu^{10}\mathstrut +\mathstrut \) \(34713719\) \(\nu^{9}\mathstrut +\mathstrut \) \(5656942\) \(\nu^{8}\mathstrut -\mathstrut \) \(61854961\) \(\nu^{7}\mathstrut -\mathstrut \) \(11592941\) \(\nu^{6}\mathstrut +\mathstrut \) \(59027859\) \(\nu^{5}\mathstrut +\mathstrut \) \(12461833\) \(\nu^{4}\mathstrut -\mathstrut \) \(24922062\) \(\nu^{3}\mathstrut -\mathstrut \) \(5524666\) \(\nu^{2}\mathstrut +\mathstrut \) \(1933636\) \(\nu\mathstrut +\mathstrut \) \(299\)\()/25060\)
\(\beta_{16}\)\(=\)\((\)\(-\)\(1383\) \(\nu^{17}\mathstrut +\mathstrut \) \(1899\) \(\nu^{16}\mathstrut +\mathstrut \) \(41081\) \(\nu^{15}\mathstrut -\mathstrut \) \(46402\) \(\nu^{14}\mathstrut -\mathstrut \) \(503867\) \(\nu^{13}\mathstrut +\mathstrut \) \(442805\) \(\nu^{12}\mathstrut +\mathstrut \) \(3299938\) \(\nu^{11}\mathstrut -\mathstrut \) \(2083314\) \(\nu^{10}\mathstrut -\mathstrut \) \(12447731\) \(\nu^{9}\mathstrut +\mathstrut \) \(4986871\) \(\nu^{8}\mathstrut +\mathstrut \) \(27095998\) \(\nu^{7}\mathstrut -\mathstrut \) \(5408617\) \(\nu^{6}\mathstrut -\mathstrut \) \(31944222\) \(\nu^{5}\mathstrut +\mathstrut \) \(1121065\) \(\nu^{4}\mathstrut +\mathstrut \) \(16760177\) \(\nu^{3}\mathstrut +\mathstrut \) \(1331003\) \(\nu^{2}\mathstrut -\mathstrut \) \(1611611\) \(\nu\mathstrut +\mathstrut \) \(121854\)\()/3580\)
\(\beta_{17}\)\(=\)\((\)\(-\)\(8478\) \(\nu^{17}\mathstrut +\mathstrut \) \(1039\) \(\nu^{16}\mathstrut +\mathstrut \) \(223616\) \(\nu^{15}\mathstrut -\mathstrut \) \(23022\) \(\nu^{14}\mathstrut -\mathstrut \) \(2385882\) \(\nu^{13}\mathstrut +\mathstrut \) \(184260\) \(\nu^{12}\mathstrut +\mathstrut \) \(13306628\) \(\nu^{11}\mathstrut -\mathstrut \) \(605434\) \(\nu^{10}\mathstrut -\mathstrut \) \(41938751\) \(\nu^{9}\mathstrut +\mathstrut \) \(457221\) \(\nu^{8}\mathstrut +\mathstrut \) \(75327343\) \(\nu^{7}\mathstrut +\mathstrut \) \(1634383\) \(\nu^{6}\mathstrut -\mathstrut \) \(73063087\) \(\nu^{5}\mathstrut -\mathstrut \) \(3294680\) \(\nu^{4}\mathstrut +\mathstrut \) \(32332347\) \(\nu^{3}\mathstrut +\mathstrut \) \(1362438\) \(\nu^{2}\mathstrut -\mathstrut \) \(3544416\) \(\nu\mathstrut +\mathstrut \) \(341599\)\()/12530\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(-\)\(\beta_{17}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(4\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(-\)\(\beta_{17}\mathstrut -\mathstrut \) \(2\) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(8\) \(\beta_{2}\mathstrut +\mathstrut \) \(16\)
\(\nu^{5}\)\(=\)\(-\)\(10\) \(\beta_{17}\mathstrut -\mathstrut \) \(10\) \(\beta_{15}\mathstrut -\mathstrut \) \(9\) \(\beta_{14}\mathstrut +\mathstrut \) \(9\) \(\beta_{13}\mathstrut -\mathstrut \) \(2\) \(\beta_{12}\mathstrut +\mathstrut \) \(10\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(8\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(20\) \(\beta_{1}\mathstrut +\mathstrut \) \(8\)
\(\nu^{6}\)\(=\)\(-\)\(12\) \(\beta_{17}\mathstrut -\mathstrut \) \(2\) \(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(23\) \(\beta_{12}\mathstrut -\mathstrut \) \(10\) \(\beta_{11}\mathstrut -\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(12\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut +\mathstrut \) \(3\) \(\beta_{6}\mathstrut +\mathstrut \) \(11\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut -\mathstrut \) \(\beta_{3}\mathstrut +\mathstrut \) \(60\) \(\beta_{2}\mathstrut -\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(98\)
\(\nu^{7}\)\(=\)\(-\)\(82\) \(\beta_{17}\mathstrut -\mathstrut \) \(81\) \(\beta_{15}\mathstrut -\mathstrut \) \(69\) \(\beta_{14}\mathstrut +\mathstrut \) \(67\) \(\beta_{13}\mathstrut -\mathstrut \) \(27\) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(79\) \(\beta_{10}\mathstrut -\mathstrut \) \(15\) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{6}\mathstrut +\mathstrut \) \(53\) \(\beta_{3}\mathstrut +\mathstrut \) \(14\) \(\beta_{2}\mathstrut +\mathstrut \) \(114\) \(\beta_{1}\mathstrut +\mathstrut \) \(54\)
\(\nu^{8}\)\(=\)\(-\)\(112\) \(\beta_{17}\mathstrut -\mathstrut \) \(30\) \(\beta_{15}\mathstrut -\mathstrut \) \(16\) \(\beta_{14}\mathstrut -\mathstrut \) \(13\) \(\beta_{13}\mathstrut -\mathstrut \) \(208\) \(\beta_{12}\mathstrut -\mathstrut \) \(80\) \(\beta_{11}\mathstrut -\mathstrut \) \(12\) \(\beta_{10}\mathstrut -\mathstrut \) \(111\) \(\beta_{9}\mathstrut +\mathstrut \) \(29\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(45\) \(\beta_{6}\mathstrut +\mathstrut \) \(94\) \(\beta_{5}\mathstrut +\mathstrut \) \(13\) \(\beta_{4}\mathstrut -\mathstrut \) \(15\) \(\beta_{3}\mathstrut +\mathstrut \) \(441\) \(\beta_{2}\mathstrut -\mathstrut \) \(16\) \(\beta_{1}\mathstrut +\mathstrut \) \(639\)
\(\nu^{9}\)\(=\)\(-\)\(637\) \(\beta_{17}\mathstrut +\mathstrut \) \(2\) \(\beta_{16}\mathstrut -\mathstrut \) \(620\) \(\beta_{15}\mathstrut -\mathstrut \) \(507\) \(\beta_{14}\mathstrut +\mathstrut \) \(476\) \(\beta_{13}\mathstrut -\mathstrut \) \(272\) \(\beta_{12}\mathstrut -\mathstrut \) \(20\) \(\beta_{11}\mathstrut +\mathstrut \) \(587\) \(\beta_{10}\mathstrut -\mathstrut \) \(158\) \(\beta_{9}\mathstrut +\mathstrut \) \(18\) \(\beta_{8}\mathstrut -\mathstrut \) \(16\) \(\beta_{7}\mathstrut +\mathstrut \) \(36\) \(\beta_{6}\mathstrut +\mathstrut \) \(4\) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(338\) \(\beta_{3}\mathstrut +\mathstrut \) \(146\) \(\beta_{2}\mathstrut +\mathstrut \) \(699\) \(\beta_{1}\mathstrut +\mathstrut \) \(358\)
\(\nu^{10}\)\(=\)\(-\)\(955\) \(\beta_{17}\mathstrut -\mathstrut \) \(3\) \(\beta_{16}\mathstrut -\mathstrut \) \(321\) \(\beta_{15}\mathstrut -\mathstrut \) \(180\) \(\beta_{14}\mathstrut -\mathstrut \) \(116\) \(\beta_{13}\mathstrut -\mathstrut \) \(1722\) \(\beta_{12}\mathstrut -\mathstrut \) \(601\) \(\beta_{11}\mathstrut -\mathstrut \) \(99\) \(\beta_{10}\mathstrut -\mathstrut \) \(932\) \(\beta_{9}\mathstrut +\mathstrut \) \(298\) \(\beta_{8}\mathstrut +\mathstrut \) \(19\) \(\beta_{7}\mathstrut +\mathstrut \) \(469\) \(\beta_{6}\mathstrut +\mathstrut \) \(735\) \(\beta_{5}\mathstrut +\mathstrut \) \(118\) \(\beta_{4}\mathstrut -\mathstrut \) \(154\) \(\beta_{3}\mathstrut +\mathstrut \) \(3216\) \(\beta_{2}\mathstrut -\mathstrut \) \(174\) \(\beta_{1}\mathstrut +\mathstrut \) \(4302\)
\(\nu^{11}\)\(=\)\(-\)\(4845\) \(\beta_{17}\mathstrut +\mathstrut \) \(41\) \(\beta_{16}\mathstrut -\mathstrut \) \(4645\) \(\beta_{15}\mathstrut -\mathstrut \) \(3680\) \(\beta_{14}\mathstrut +\mathstrut \) \(3342\) \(\beta_{13}\mathstrut -\mathstrut \) \(2440\) \(\beta_{12}\mathstrut -\mathstrut \) \(254\) \(\beta_{11}\mathstrut +\mathstrut \) \(4280\) \(\beta_{10}\mathstrut -\mathstrut \) \(1444\) \(\beta_{9}\mathstrut +\mathstrut \) \(212\) \(\beta_{8}\mathstrut -\mathstrut \) \(175\) \(\beta_{7}\mathstrut +\mathstrut \) \(431\) \(\beta_{6}\mathstrut +\mathstrut \) \(84\) \(\beta_{5}\mathstrut +\mathstrut \) \(38\) \(\beta_{4}\mathstrut +\mathstrut \) \(2154\) \(\beta_{3}\mathstrut +\mathstrut \) \(1357\) \(\beta_{2}\mathstrut +\mathstrut \) \(4472\) \(\beta_{1}\mathstrut +\mathstrut \) \(2427\)
\(\nu^{12}\)\(=\)\(-\)\(7786\) \(\beta_{17}\mathstrut -\mathstrut \) \(67\) \(\beta_{16}\mathstrut -\mathstrut \) \(3009\) \(\beta_{15}\mathstrut -\mathstrut \) \(1748\) \(\beta_{14}\mathstrut -\mathstrut \) \(880\) \(\beta_{13}\mathstrut -\mathstrut \) \(13644\) \(\beta_{12}\mathstrut -\mathstrut \) \(4417\) \(\beta_{11}\mathstrut -\mathstrut \) \(679\) \(\beta_{10}\mathstrut -\mathstrut \) \(7457\) \(\beta_{9}\mathstrut +\mathstrut \) \(2677\) \(\beta_{8}\mathstrut +\mathstrut \) \(237\) \(\beta_{7}\mathstrut +\mathstrut \) \(4219\) \(\beta_{6}\mathstrut +\mathstrut \) \(5515\) \(\beta_{5}\mathstrut +\mathstrut \) \(932\) \(\beta_{4}\mathstrut -\mathstrut \) \(1354\) \(\beta_{3}\mathstrut +\mathstrut \) \(23384\) \(\beta_{2}\mathstrut -\mathstrut \) \(1596\) \(\beta_{1}\mathstrut +\mathstrut \) \(29520\)
\(\nu^{13}\)\(=\)\(-\)\(36459\) \(\beta_{17}\mathstrut +\mathstrut \) \(538\) \(\beta_{16}\mathstrut -\mathstrut \) \(34447\) \(\beta_{15}\mathstrut -\mathstrut \) \(26632\) \(\beta_{14}\mathstrut +\mathstrut \) \(23438\) \(\beta_{13}\mathstrut -\mathstrut \) \(20592\) \(\beta_{12}\mathstrut -\mathstrut \) \(2657\) \(\beta_{11}\mathstrut +\mathstrut \) \(31024\) \(\beta_{10}\mathstrut -\mathstrut \) \(12270\) \(\beta_{9}\mathstrut +\mathstrut \) \(2088\) \(\beta_{8}\mathstrut -\mathstrut \) \(1636\) \(\beta_{7}\mathstrut +\mathstrut \) \(4335\) \(\beta_{6}\mathstrut +\mathstrut \) \(1132\) \(\beta_{5}\mathstrut +\mathstrut \) \(470\) \(\beta_{4}\mathstrut +\mathstrut \) \(13866\) \(\beta_{3}\mathstrut +\mathstrut \) \(11874\) \(\beta_{2}\mathstrut +\mathstrut \) \(29412\) \(\beta_{1}\mathstrut +\mathstrut \) \(17018\)
\(\nu^{14}\)\(=\)\(-\)\(61837\) \(\beta_{17}\mathstrut -\mathstrut \) \(934\) \(\beta_{16}\mathstrut -\mathstrut \) \(26339\) \(\beta_{15}\mathstrut -\mathstrut \) \(15697\) \(\beta_{14}\mathstrut -\mathstrut \) \(6050\) \(\beta_{13}\mathstrut -\mathstrut \) \(105420\) \(\beta_{12}\mathstrut -\mathstrut \) \(32186\) \(\beta_{11}\mathstrut -\mathstrut \) \(4009\) \(\beta_{10}\mathstrut -\mathstrut \) \(58034\) \(\beta_{9}\mathstrut +\mathstrut \) \(22463\) \(\beta_{8}\mathstrut +\mathstrut \) \(2459\) \(\beta_{7}\mathstrut +\mathstrut \) \(35200\) \(\beta_{6}\mathstrut +\mathstrut \) \(40495\) \(\beta_{5}\mathstrut +\mathstrut \) \(6877\) \(\beta_{4}\mathstrut -\mathstrut \) \(10967\) \(\beta_{3}\mathstrut +\mathstrut \) \(169875\) \(\beta_{2}\mathstrut -\mathstrut \) \(13303\) \(\beta_{1}\mathstrut +\mathstrut \) \(205229\)
\(\nu^{15}\)\(=\)\(-\)\(272599\) \(\beta_{17}\mathstrut +\mathstrut \) \(5784\) \(\beta_{16}\mathstrut -\mathstrut \) \(254027\) \(\beta_{15}\mathstrut -\mathstrut \) \(192725\) \(\beta_{14}\mathstrut +\mathstrut \) \(164755\) \(\beta_{13}\mathstrut -\mathstrut \) \(167527\) \(\beta_{12}\mathstrut -\mathstrut \) \(25047\) \(\beta_{11}\mathstrut +\mathstrut \) \(224540\) \(\beta_{10}\mathstrut -\mathstrut \) \(99951\) \(\beta_{9}\mathstrut +\mathstrut \) \(18714\) \(\beta_{8}\mathstrut -\mathstrut \) \(14093\) \(\beta_{7}\mathstrut +\mathstrut \) \(39671\) \(\beta_{6}\mathstrut +\mathstrut \) \(12501\) \(\beta_{5}\mathstrut +\mathstrut \) \(4830\) \(\beta_{4}\mathstrut +\mathstrut \) \(90388\) \(\beta_{3}\mathstrut +\mathstrut \) \(100171\) \(\beta_{2}\mathstrut +\mathstrut \) \(197322\) \(\beta_{1}\mathstrut +\mathstrut \) \(123317\)
\(\nu^{16}\)\(=\)\(-\)\(482938\) \(\beta_{17}\mathstrut -\mathstrut \) \(10494\) \(\beta_{16}\mathstrut -\mathstrut \) \(221286\) \(\beta_{15}\mathstrut -\mathstrut \) \(134481\) \(\beta_{14}\mathstrut -\mathstrut \) \(38421\) \(\beta_{13}\mathstrut -\mathstrut \) \(801916\) \(\beta_{12}\mathstrut -\mathstrut \) \(233707\) \(\beta_{11}\mathstrut -\mathstrut \) \(19374\) \(\beta_{10}\mathstrut -\mathstrut \) \(443929\) \(\beta_{9}\mathstrut +\mathstrut \) \(181158\) \(\beta_{8}\mathstrut +\mathstrut \) \(23040\) \(\beta_{7}\mathstrut +\mathstrut \) \(281274\) \(\beta_{6}\mathstrut +\mathstrut \) \(293813\) \(\beta_{5}\mathstrut +\mathstrut \) \(48888\) \(\beta_{4}\mathstrut -\mathstrut \) \(84514\) \(\beta_{3}\mathstrut +\mathstrut \) \(1234044\) \(\beta_{2}\mathstrut -\mathstrut \) \(104350\) \(\beta_{1}\mathstrut +\mathstrut \) \(1440869\)
\(\nu^{17}\)\(=\)\(-\)\(2029601\) \(\beta_{17}\mathstrut +\mathstrut \) \(55621\) \(\beta_{16}\mathstrut -\mathstrut \) \(1867082\) \(\beta_{15}\mathstrut -\mathstrut \) \(1395823\) \(\beta_{14}\mathstrut +\mathstrut \) \(1161951\) \(\beta_{13}\mathstrut -\mathstrut \) \(1330972\) \(\beta_{12}\mathstrut -\mathstrut \) \(221695\) \(\beta_{11}\mathstrut +\mathstrut \) \(1625134\) \(\beta_{10}\mathstrut -\mathstrut \) \(793015\) \(\beta_{9}\mathstrut +\mathstrut \) \(158636\) \(\beta_{8}\mathstrut -\mathstrut \) \(115727\) \(\beta_{7}\mathstrut +\mathstrut \) \(342661\) \(\beta_{6}\mathstrut +\mathstrut \) \(123433\) \(\beta_{5}\mathstrut +\mathstrut \) \(44918\) \(\beta_{4}\mathstrut +\mathstrut \) \(596467\) \(\beta_{3}\mathstrut +\mathstrut \) \(824853\) \(\beta_{2}\mathstrut +\mathstrut \) \(1344049\) \(\beta_{1}\mathstrut +\mathstrut \) \(917721\)

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
2.72555
2.37327
2.33471
2.33085
1.52472
1.43818
1.35283
0.230018
0.146947
0.0898969
−0.494084
−1.15272
−1.40846
−1.48566
−1.61470
−2.13101
−2.60234
−2.65800
−2.72555 0 5.42860 0.150400 0 −4.90968 −9.34479 0 −0.409922
1.2 −2.37327 0 3.63241 2.08614 0 4.17378 −3.87415 0 −4.95098
1.3 −2.33471 0 3.45089 −4.12904 0 −0.743390 −3.38741 0 9.64013
1.4 −2.33085 0 3.43286 −1.49796 0 0.334615 −3.33977 0 3.49151
1.5 −1.52472 0 0.324761 1.59729 0 −1.34206 2.55426 0 −2.43541
1.6 −1.43818 0 0.0683575 −1.27662 0 4.10362 2.77805 0 1.83600
1.7 −1.35283 0 −0.169850 −0.632563 0 −0.940374 2.93544 0 0.855750
1.8 −0.230018 0 −1.94709 2.49148 0 −0.810750 0.907901 0 −0.573085
1.9 −0.146947 0 −1.97841 −2.45635 0 −1.38215 0.584613 0 0.360953
1.10 −0.0898969 0 −1.99192 −2.73841 0 4.94566 0.358861 0 0.246175
1.11 0.494084 0 −1.75588 3.82270 0 −3.40967 −1.85572 0 1.88874
1.12 1.15272 0 −0.671235 1.54521 0 2.91562 −3.07919 0 1.78119
1.13 1.40846 0 −0.0162457 −0.976908 0 −4.66797 −2.83980 0 −1.37593
1.14 1.48566 0 0.207175 −2.74559 0 0.513695 −2.66352 0 −4.07900
1.15 1.61470 0 0.607250 −2.02448 0 −1.29797 −2.24887 0 −3.26892
1.16 2.13101 0 2.54118 1.76024 0 2.12155 1.15327 0 3.75109
1.17 2.60234 0 4.77220 −3.17530 0 1.94962 7.21421 0 −8.26324
1.18 2.65800 0 5.06495 3.19976 0 1.44585 8.14662 0 8.50495
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.18
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(19\) \(1\)
\(47\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8037))\):

\(T_{2}^{18} + \cdots\)
\(T_{5}^{18} + \cdots\)