Properties

Label 6045.2.a.bi
Level 6045
Weight 2
Character orbit 6045.a
Self dual Yes
Analytic conductor 48.270
Analytic rank 0
Dimension 18
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6045 = 3 \cdot 5 \cdot 13 \cdot 31 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6045.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.2695680219\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

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

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{18}\mathstrut -\mathstrut \) \(4\) \(x^{17}\mathstrut -\mathstrut \) \(21\) \(x^{16}\mathstrut +\mathstrut \) \(97\) \(x^{15}\mathstrut +\mathstrut \) \(156\) \(x^{14}\mathstrut -\mathstrut \) \(935\) \(x^{13}\mathstrut -\mathstrut \) \(411\) \(x^{12}\mathstrut +\mathstrut \) \(4582\) \(x^{11}\mathstrut -\mathstrut \) \(446\) \(x^{10}\mathstrut -\mathstrut \) \(12159\) \(x^{9}\mathstrut +\mathstrut \) \(4398\) \(x^{8}\mathstrut +\mathstrut \) \(17347\) \(x^{7}\mathstrut -\mathstrut \) \(7839\) \(x^{6}\mathstrut -\mathstrut \) \(12832\) \(x^{5}\mathstrut +\mathstrut \) \(4744\) \(x^{4}\mathstrut +\mathstrut \) \(5064\) \(x^{3}\mathstrut -\mathstrut \) \(746\) \(x^{2}\mathstrut -\mathstrut \) \(892\) \(x\mathstrut -\mathstrut \) \(112\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\( \nu^{3} - 5 \nu \)
\(\beta_{4}\)\(=\)\( \nu^{4} - 7 \nu^{2} + 6 \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(228355\) \(\nu^{17}\mathstrut -\mathstrut \) \(336512\) \(\nu^{16}\mathstrut +\mathstrut \) \(8567893\) \(\nu^{15}\mathstrut +\mathstrut \) \(6198769\) \(\nu^{14}\mathstrut -\mathstrut \) \(123630228\) \(\nu^{13}\mathstrut -\mathstrut \) \(31566349\) \(\nu^{12}\mathstrut +\mathstrut \) \(891219629\) \(\nu^{11}\mathstrut -\mathstrut \) \(59637650\) \(\nu^{10}\mathstrut -\mathstrut \) \(3413604116\) \(\nu^{9}\mathstrut +\mathstrut \) \(1071437101\) \(\nu^{8}\mathstrut +\mathstrut \) \(6645499448\) \(\nu^{7}\mathstrut -\mathstrut \) \(3608265451\) \(\nu^{6}\mathstrut -\mathstrut \) \(5229581577\) \(\nu^{5}\mathstrut +\mathstrut \) \(4661942660\) \(\nu^{4}\mathstrut -\mathstrut \) \(103549980\) \(\nu^{3}\mathstrut -\mathstrut \) \(1874991824\) \(\nu^{2}\mathstrut +\mathstrut \) \(727071450\) \(\nu\mathstrut +\mathstrut \) \(317132228\)\()/17310212\)
\(\beta_{6}\)\(=\)\((\)\(446299\) \(\nu^{17}\mathstrut +\mathstrut \) \(1332792\) \(\nu^{16}\mathstrut -\mathstrut \) \(19095959\) \(\nu^{15}\mathstrut -\mathstrut \) \(27974025\) \(\nu^{14}\mathstrut +\mathstrut \) \(299322856\) \(\nu^{13}\mathstrut +\mathstrut \) \(211957035\) \(\nu^{12}\mathstrut -\mathstrut \) \(2313633925\) \(\nu^{11}\mathstrut -\mathstrut \) \(639195438\) \(\nu^{10}\mathstrut +\mathstrut \) \(9631313854\) \(\nu^{9}\mathstrut +\mathstrut \) \(168689975\) \(\nu^{8}\mathstrut -\mathstrut \) \(21628657114\) \(\nu^{7}\mathstrut +\mathstrut \) \(2831804025\) \(\nu^{6}\mathstrut +\mathstrut \) \(24385040543\) \(\nu^{5}\mathstrut -\mathstrut \) \(4515463168\) \(\nu^{4}\mathstrut -\mathstrut \) \(11484187796\) \(\nu^{3}\mathstrut +\mathstrut \) \(1183377184\) \(\nu^{2}\mathstrut +\mathstrut \) \(1777245698\) \(\nu\mathstrut +\mathstrut \) \(152855836\)\()/17310212\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(740009\) \(\nu^{17}\mathstrut +\mathstrut \) \(2872262\) \(\nu^{16}\mathstrut +\mathstrut \) \(14791069\) \(\nu^{15}\mathstrut -\mathstrut \) \(67180803\) \(\nu^{14}\mathstrut -\mathstrut \) \(98215806\) \(\nu^{13}\mathstrut +\mathstrut \) \(613552943\) \(\nu^{12}\mathstrut +\mathstrut \) \(151793377\) \(\nu^{11}\mathstrut -\mathstrut \) \(2759586728\) \(\nu^{10}\mathstrut +\mathstrut \) \(976237358\) \(\nu^{9}\mathstrut +\mathstrut \) \(6315037203\) \(\nu^{8}\mathstrut -\mathstrut \) \(4581147060\) \(\nu^{7}\mathstrut -\mathstrut \) \(6769396543\) \(\nu^{6}\mathstrut +\mathstrut \) \(6955418053\) \(\nu^{5}\mathstrut +\mathstrut \) \(2731690702\) \(\nu^{4}\mathstrut -\mathstrut \) \(3873662520\) \(\nu^{3}\mathstrut -\mathstrut \) \(728115164\) \(\nu^{2}\mathstrut +\mathstrut \) \(862479718\) \(\nu\mathstrut +\mathstrut \) \(222862852\)\()/17310212\)
\(\beta_{8}\)\(=\)\((\)\(-\)\(610666\) \(\nu^{17}\mathstrut +\mathstrut \) \(354827\) \(\nu^{16}\mathstrut +\mathstrut \) \(18681397\) \(\nu^{15}\mathstrut -\mathstrut \) \(10014793\) \(\nu^{14}\mathstrut -\mathstrut \) \(234498267\) \(\nu^{13}\mathstrut +\mathstrut \) \(116139789\) \(\nu^{12}\mathstrut +\mathstrut \) \(1554418411\) \(\nu^{11}\mathstrut -\mathstrut \) \(711950221\) \(\nu^{10}\mathstrut -\mathstrut \) \(5830999753\) \(\nu^{9}\mathstrut +\mathstrut \) \(2463590886\) \(\nu^{8}\mathstrut +\mathstrut \) \(12312429494\) \(\nu^{7}\mathstrut -\mathstrut \) \(4696943709\) \(\nu^{6}\mathstrut -\mathstrut \) \(13698626968\) \(\nu^{5}\mathstrut +\mathstrut \) \(4235562551\) \(\nu^{4}\mathstrut +\mathstrut \) \(6955732814\) \(\nu^{3}\mathstrut -\mathstrut \) \(939434104\) \(\nu^{2}\mathstrut -\mathstrut \) \(1389743616\) \(\nu\mathstrut -\mathstrut \) \(175204944\)\()/8655106\)
\(\beta_{9}\)\(=\)\((\)\(658857\) \(\nu^{17}\mathstrut -\mathstrut \) \(2572882\) \(\nu^{16}\mathstrut -\mathstrut \) \(14528758\) \(\nu^{15}\mathstrut +\mathstrut \) \(64014013\) \(\nu^{14}\mathstrut +\mathstrut \) \(119878466\) \(\nu^{13}\mathstrut -\mathstrut \) \(639146514\) \(\nu^{12}\mathstrut -\mathstrut \) \(440926905\) \(\nu^{11}\mathstrut +\mathstrut \) \(3285967890\) \(\nu^{10}\mathstrut +\mathstrut \) \(584398593\) \(\nu^{9}\mathstrut -\mathstrut \) \(9283925003\) \(\nu^{8}\mathstrut +\mathstrut \) \(383710967\) \(\nu^{7}\mathstrut +\mathstrut \) \(14213130072\) \(\nu^{6}\mathstrut -\mathstrut \) \(1242628076\) \(\nu^{5}\mathstrut -\mathstrut \) \(10798785650\) \(\nu^{4}\mathstrut +\mathstrut \) \(121132724\) \(\nu^{3}\mathstrut +\mathstrut \) \(3275128816\) \(\nu^{2}\mathstrut +\mathstrut \) \(323408598\) \(\nu\mathstrut -\mathstrut \) \(101408846\)\()/8655106\)
\(\beta_{10}\)\(=\)\((\)\(870813\) \(\nu^{17}\mathstrut -\mathstrut \) \(3033149\) \(\nu^{16}\mathstrut -\mathstrut \) \(19402563\) \(\nu^{15}\mathstrut +\mathstrut \) \(73271914\) \(\nu^{14}\mathstrut +\mathstrut \) \(163062311\) \(\nu^{13}\mathstrut -\mathstrut \) \(701176099\) \(\nu^{12}\mathstrut -\mathstrut \) \(623550578\) \(\nu^{11}\mathstrut +\mathstrut \) \(3385844207\) \(\nu^{10}\mathstrut +\mathstrut \) \(942614544\) \(\nu^{9}\mathstrut -\mathstrut \) \(8697601105\) \(\nu^{8}\mathstrut +\mathstrut \) \(175276449\) \(\nu^{7}\mathstrut +\mathstrut \) \(11465269759\) \(\nu^{6}\mathstrut -\mathstrut \) \(1412926912\) \(\nu^{5}\mathstrut -\mathstrut \) \(6833192315\) \(\nu^{4}\mathstrut +\mathstrut \) \(246169214\) \(\nu^{3}\mathstrut +\mathstrut \) \(1478840262\) \(\nu^{2}\mathstrut +\mathstrut \) \(262624594\) \(\nu\mathstrut +\mathstrut \) \(39621994\)\()/8655106\)
\(\beta_{11}\)\(=\)\((\)\(2172011\) \(\nu^{17}\mathstrut -\mathstrut \) \(4934604\) \(\nu^{16}\mathstrut -\mathstrut \) \(57254415\) \(\nu^{15}\mathstrut +\mathstrut \) \(122228351\) \(\nu^{14}\mathstrut +\mathstrut \) \(620355732\) \(\nu^{13}\mathstrut -\mathstrut \) \(1212202237\) \(\nu^{12}\mathstrut -\mathstrut \) \(3589511029\) \(\nu^{11}\mathstrut +\mathstrut \) \(6172190758\) \(\nu^{10}\mathstrut +\mathstrut \) \(12072754710\) \(\nu^{9}\mathstrut -\mathstrut \) \(17208963633\) \(\nu^{8}\mathstrut -\mathstrut \) \(24057580418\) \(\nu^{7}\mathstrut +\mathstrut \) \(25833441061\) \(\nu^{6}\mathstrut +\mathstrut \) \(27680230503\) \(\nu^{5}\mathstrut -\mathstrut \) \(18654446728\) \(\nu^{4}\mathstrut -\mathstrut \) \(16961256348\) \(\nu^{3}\mathstrut +\mathstrut \) \(4248411632\) \(\nu^{2}\mathstrut +\mathstrut \) \(4385523658\) \(\nu\mathstrut +\mathstrut \) \(547325784\)\()/17310212\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(1412139\) \(\nu^{17}\mathstrut +\mathstrut \) \(2759313\) \(\nu^{16}\mathstrut +\mathstrut \) \(37784959\) \(\nu^{15}\mathstrut -\mathstrut \) \(67272692\) \(\nu^{14}\mathstrut -\mathstrut \) \(415176825\) \(\nu^{13}\mathstrut +\mathstrut \) \(652347529\) \(\nu^{12}\mathstrut +\mathstrut \) \(2422989002\) \(\nu^{11}\mathstrut -\mathstrut \) \(3215790521\) \(\nu^{10}\mathstrut -\mathstrut \) \(8101395364\) \(\nu^{9}\mathstrut +\mathstrut \) \(8553876469\) \(\nu^{8}\mathstrut +\mathstrut \) \(15559974451\) \(\nu^{7}\mathstrut -\mathstrut \) \(11972992943\) \(\nu^{6}\mathstrut -\mathstrut \) \(16275952098\) \(\nu^{5}\mathstrut +\mathstrut \) \(7692601359\) \(\nu^{4}\mathstrut +\mathstrut \) \(8225920740\) \(\nu^{3}\mathstrut -\mathstrut \) \(1274068280\) \(\nu^{2}\mathstrut -\mathstrut \) \(1638615510\) \(\nu\mathstrut -\mathstrut \) \(238817406\)\()/8655106\)
\(\beta_{13}\)\(=\)\((\)\(-\)\(1412019\) \(\nu^{17}\mathstrut +\mathstrut \) \(5494307\) \(\nu^{16}\mathstrut +\mathstrut \) \(28955834\) \(\nu^{15}\mathstrut -\mathstrut \) \(129885278\) \(\nu^{14}\mathstrut -\mathstrut \) \(204157033\) \(\nu^{13}\mathstrut +\mathstrut \) \(1204272526\) \(\nu^{12}\mathstrut +\mathstrut \) \(436001482\) \(\nu^{11}\mathstrut -\mathstrut \) \(5541738709\) \(\nu^{10}\mathstrut +\mathstrut \) \(1252562487\) \(\nu^{9}\mathstrut +\mathstrut \) \(13173455103\) \(\nu^{8}\mathstrut -\mathstrut \) \(7494401204\) \(\nu^{7}\mathstrut -\mathstrut \) \(15183703738\) \(\nu^{6}\mathstrut +\mathstrut \) \(12043662899\) \(\nu^{5}\mathstrut +\mathstrut \) \(7115561525\) \(\nu^{4}\mathstrut -\mathstrut \) \(6641795912\) \(\nu^{3}\mathstrut -\mathstrut \) \(1548008000\) \(\nu^{2}\mathstrut +\mathstrut \) \(1216991632\) \(\nu\mathstrut +\mathstrut \) \(239590942\)\()/8655106\)
\(\beta_{14}\)\(=\)\((\)\(1613979\) \(\nu^{17}\mathstrut -\mathstrut \) \(7015797\) \(\nu^{16}\mathstrut -\mathstrut \) \(31883760\) \(\nu^{15}\mathstrut +\mathstrut \) \(167766567\) \(\nu^{14}\mathstrut +\mathstrut \) \(205502471\) \(\nu^{13}\mathstrut -\mathstrut \) \(1582940924\) \(\nu^{12}\mathstrut -\mathstrut \) \(250642325\) \(\nu^{11}\mathstrut +\mathstrut \) \(7492954365\) \(\nu^{10}\mathstrut -\mathstrut \) \(2503879729\) \(\nu^{9}\mathstrut -\mathstrut \) \(18709447578\) \(\nu^{8}\mathstrut +\mathstrut \) \(10879951810\) \(\nu^{7}\mathstrut +\mathstrut \) \(23716389037\) \(\nu^{6}\mathstrut -\mathstrut \) \(15972796554\) \(\nu^{5}\mathstrut -\mathstrut \) \(13649055788\) \(\nu^{4}\mathstrut +\mathstrut \) \(8029654174\) \(\nu^{3}\mathstrut +\mathstrut \) \(3521009816\) \(\nu^{2}\mathstrut -\mathstrut \) \(1072984312\) \(\nu\mathstrut -\mathstrut \) \(297472862\)\()/8655106\)
\(\beta_{15}\)\(=\)\((\)\(-\)\(1740463\) \(\nu^{17}\mathstrut +\mathstrut \) \(6102749\) \(\nu^{16}\mathstrut +\mathstrut \) \(38885676\) \(\nu^{15}\mathstrut -\mathstrut \) \(147290343\) \(\nu^{14}\mathstrut -\mathstrut \) \(329077721\) \(\nu^{13}\mathstrut +\mathstrut \) \(1408087416\) \(\nu^{12}\mathstrut +\mathstrut \) \(1283554233\) \(\nu^{11}\mathstrut -\mathstrut \) \(6795292397\) \(\nu^{10}\mathstrut -\mathstrut \) \(2108436645\) \(\nu^{9}\mathstrut +\mathstrut \) \(17479801312\) \(\nu^{8}\mathstrut +\mathstrut \) \(400301490\) \(\nu^{7}\mathstrut -\mathstrut \) \(23233156855\) \(\nu^{6}\mathstrut +\mathstrut \) \(1419906454\) \(\nu^{5}\mathstrut +\mathstrut \) \(14283007824\) \(\nu^{4}\mathstrut +\mathstrut \) \(836788936\) \(\nu^{3}\mathstrut -\mathstrut \) \(3418543606\) \(\nu^{2}\mathstrut -\mathstrut \) \(973053192\) \(\nu\mathstrut -\mathstrut \) \(43868350\)\()/8655106\)
\(\beta_{16}\)\(=\)\((\)\(-\)\(1814482\) \(\nu^{17}\mathstrut +\mathstrut \) \(6186612\) \(\nu^{16}\mathstrut +\mathstrut \) \(40308086\) \(\nu^{15}\mathstrut -\mathstrut \) \(148177865\) \(\nu^{14}\mathstrut -\mathstrut \) \(336742404\) \(\nu^{13}\mathstrut +\mathstrut \) \(1402238426\) \(\nu^{12}\mathstrut +\mathstrut \) \(1266225703\) \(\nu^{11}\mathstrut -\mathstrut \) \(6674822568\) \(\nu^{10}\mathstrut -\mathstrut \) \(1758453438\) \(\nu^{9}\mathstrut +\mathstrut \) \(16857338927\) \(\nu^{8}\mathstrut -\mathstrut \) \(1134599382\) \(\nu^{7}\mathstrut -\mathstrut \) \(21905743275\) \(\nu^{6}\mathstrut +\mathstrut \) \(4577755507\) \(\nu^{5}\mathstrut +\mathstrut \) \(13292663783\) \(\nu^{4}\mathstrut -\mathstrut \) \(2217653358\) \(\nu^{3}\mathstrut -\mathstrut \) \(3544809292\) \(\nu^{2}\mathstrut +\mathstrut \) \(50031272\) \(\nu\mathstrut +\mathstrut \) \(194458614\)\()/8655106\)
\(\beta_{17}\)\(=\)\((\)\(3825863\) \(\nu^{17}\mathstrut -\mathstrut \) \(11750022\) \(\nu^{16}\mathstrut -\mathstrut \) \(91459471\) \(\nu^{15}\mathstrut +\mathstrut \) \(285997777\) \(\nu^{14}\mathstrut +\mathstrut \) \(868124882\) \(\nu^{13}\mathstrut -\mathstrut \) \(2766904757\) \(\nu^{12}\mathstrut -\mathstrut \) \(4205477675\) \(\nu^{11}\mathstrut +\mathstrut \) \(13590072516\) \(\nu^{10}\mathstrut +\mathstrut \) \(11275251570\) \(\nu^{9}\mathstrut -\mathstrut \) \(35930753321\) \(\nu^{8}\mathstrut -\mathstrut \) \(17639487940\) \(\nu^{7}\mathstrut +\mathstrut \) \(49918468081\) \(\nu^{6}\mathstrut +\mathstrut \) \(18090271893\) \(\nu^{5}\mathstrut -\mathstrut \) \(32683891182\) \(\nu^{4}\mathstrut -\mathstrut \) \(13449627508\) \(\nu^{3}\mathstrut +\mathstrut \) \(7516525160\) \(\nu^{2}\mathstrut +\mathstrut \) \(4490857882\) \(\nu\mathstrut +\mathstrut \) \(543688764\)\()/17310212\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{3}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\)
\(\nu^{4}\)\(=\)\(\beta_{4}\mathstrut +\mathstrut \) \(7\) \(\beta_{2}\mathstrut +\mathstrut \) \(15\)
\(\nu^{5}\)\(=\)\(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(9\) \(\beta_{3}\mathstrut +\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(30\) \(\beta_{1}\mathstrut +\mathstrut \) \(1\)
\(\nu^{6}\)\(=\)\(-\)\(\beta_{15}\mathstrut -\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(11\) \(\beta_{4}\mathstrut +\mathstrut \) \(48\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(88\)
\(\nu^{7}\)\(=\)\(\beta_{17}\mathstrut +\mathstrut \) \(2\) \(\beta_{16}\mathstrut -\mathstrut \) \(2\) \(\beta_{14}\mathstrut -\mathstrut \) \(3\) \(\beta_{13}\mathstrut +\mathstrut \) \(13\) \(\beta_{12}\mathstrut +\mathstrut \) \(13\) \(\beta_{11}\mathstrut +\mathstrut \) \(12\) \(\beta_{10}\mathstrut -\mathstrut \) \(11\) \(\beta_{9}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(14\) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(67\) \(\beta_{3}\mathstrut +\mathstrut \) \(14\) \(\beta_{2}\mathstrut +\mathstrut \) \(193\) \(\beta_{1}\mathstrut +\mathstrut \) \(16\)
\(\nu^{8}\)\(=\)\(\beta_{17}\mathstrut +\mathstrut \) \(2\) \(\beta_{16}\mathstrut -\mathstrut \) \(16\) \(\beta_{15}\mathstrut -\mathstrut \) \(16\) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(15\) \(\beta_{12}\mathstrut +\mathstrut \) \(14\) \(\beta_{11}\mathstrut +\mathstrut \) \(16\) \(\beta_{10}\mathstrut -\mathstrut \) \(15\) \(\beta_{9}\mathstrut +\mathstrut \) \(14\) \(\beta_{8}\mathstrut -\mathstrut \) \(15\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(32\) \(\beta_{5}\mathstrut +\mathstrut \) \(95\) \(\beta_{4}\mathstrut +\mathstrut \) \(3\) \(\beta_{3}\mathstrut +\mathstrut \) \(334\) \(\beta_{2}\mathstrut +\mathstrut \) \(15\) \(\beta_{1}\mathstrut +\mathstrut \) \(555\)
\(\nu^{9}\)\(=\)\(17\) \(\beta_{17}\mathstrut +\mathstrut \) \(32\) \(\beta_{16}\mathstrut -\mathstrut \) \(\beta_{15}\mathstrut -\mathstrut \) \(37\) \(\beta_{14}\mathstrut -\mathstrut \) \(52\) \(\beta_{13}\mathstrut +\mathstrut \) \(126\) \(\beta_{12}\mathstrut +\mathstrut \) \(126\) \(\beta_{11}\mathstrut +\mathstrut \) \(110\) \(\beta_{10}\mathstrut -\mathstrut \) \(92\) \(\beta_{9}\mathstrut +\mathstrut \) \(2\) \(\beta_{8}\mathstrut -\mathstrut \) \(13\) \(\beta_{7}\mathstrut -\mathstrut \) \(18\) \(\beta_{6}\mathstrut -\mathstrut \) \(144\) \(\beta_{5}\mathstrut +\mathstrut \) \(18\) \(\beta_{4}\mathstrut +\mathstrut \) \(474\) \(\beta_{3}\mathstrut +\mathstrut \) \(152\) \(\beta_{2}\mathstrut +\mathstrut \) \(1285\) \(\beta_{1}\mathstrut +\mathstrut \) \(174\)
\(\nu^{10}\)\(=\)\(24\) \(\beta_{17}\mathstrut +\mathstrut \) \(35\) \(\beta_{16}\mathstrut -\mathstrut \) \(173\) \(\beta_{15}\mathstrut -\mathstrut \) \(180\) \(\beta_{14}\mathstrut -\mathstrut \) \(22\) \(\beta_{13}\mathstrut +\mathstrut \) \(164\) \(\beta_{12}\mathstrut +\mathstrut \) \(141\) \(\beta_{11}\mathstrut +\mathstrut \) \(177\) \(\beta_{10}\mathstrut -\mathstrut \) \(160\) \(\beta_{9}\mathstrut +\mathstrut \) \(140\) \(\beta_{8}\mathstrut -\mathstrut \) \(158\) \(\beta_{7}\mathstrut -\mathstrut \) \(36\) \(\beta_{6}\mathstrut -\mathstrut \) \(357\) \(\beta_{5}\mathstrut +\mathstrut \) \(758\) \(\beta_{4}\mathstrut +\mathstrut \) \(58\) \(\beta_{3}\mathstrut +\mathstrut \) \(2360\) \(\beta_{2}\mathstrut +\mathstrut \) \(168\) \(\beta_{1}\mathstrut +\mathstrut \) \(3646\)
\(\nu^{11}\)\(=\)\(199\) \(\beta_{17}\mathstrut +\mathstrut \) \(348\) \(\beta_{16}\mathstrut -\mathstrut \) \(17\) \(\beta_{15}\mathstrut -\mathstrut \) \(457\) \(\beta_{14}\mathstrut -\mathstrut \) \(608\) \(\beta_{13}\mathstrut +\mathstrut \) \(1098\) \(\beta_{12}\mathstrut +\mathstrut \) \(1097\) \(\beta_{11}\mathstrut +\mathstrut \) \(929\) \(\beta_{10}\mathstrut -\mathstrut \) \(707\) \(\beta_{9}\mathstrut +\mathstrut \) \(46\) \(\beta_{8}\mathstrut -\mathstrut \) \(114\) \(\beta_{7}\mathstrut -\mathstrut \) \(217\) \(\beta_{6}\mathstrut -\mathstrut \) \(1319\) \(\beta_{5}\mathstrut +\mathstrut \) \(223\) \(\beta_{4}\mathstrut +\mathstrut \) \(3304\) \(\beta_{3}\mathstrut +\mathstrut \) \(1491\) \(\beta_{2}\mathstrut +\mathstrut \) \(8735\) \(\beta_{1}\mathstrut +\mathstrut \) \(1658\)
\(\nu^{12}\)\(=\)\(352\) \(\beta_{17}\mathstrut +\mathstrut \) \(411\) \(\beta_{16}\mathstrut -\mathstrut \) \(1586\) \(\beta_{15}\mathstrut -\mathstrut \) \(1761\) \(\beta_{14}\mathstrut -\mathstrut \) \(317\) \(\beta_{13}\mathstrut +\mathstrut \) \(1584\) \(\beta_{12}\mathstrut +\mathstrut \) \(1261\) \(\beta_{11}\mathstrut +\mathstrut \) \(1703\) \(\beta_{10}\mathstrut -\mathstrut \) \(1490\) \(\beta_{9}\mathstrut +\mathstrut \) \(1237\) \(\beta_{8}\mathstrut -\mathstrut \) \(1438\) \(\beta_{7}\mathstrut -\mathstrut \) \(434\) \(\beta_{6}\mathstrut -\mathstrut \) \(3441\) \(\beta_{5}\mathstrut +\mathstrut \) \(5843\) \(\beta_{4}\mathstrut +\mathstrut \) \(734\) \(\beta_{3}\mathstrut +\mathstrut \) \(16914\) \(\beta_{2}\mathstrut +\mathstrut \) \(1705\) \(\beta_{1}\mathstrut +\mathstrut \) \(24615\)
\(\nu^{13}\)\(=\)\(2008\) \(\beta_{17}\mathstrut +\mathstrut \) \(3234\) \(\beta_{16}\mathstrut -\mathstrut \) \(187\) \(\beta_{15}\mathstrut -\mathstrut \) \(4759\) \(\beta_{14}\mathstrut -\mathstrut \) \(6033\) \(\beta_{13}\mathstrut +\mathstrut \) \(9098\) \(\beta_{12}\mathstrut +\mathstrut \) \(9065\) \(\beta_{11}\mathstrut +\mathstrut \) \(7605\) \(\beta_{10}\mathstrut -\mathstrut \) \(5270\) \(\beta_{9}\mathstrut +\mathstrut \) \(667\) \(\beta_{8}\mathstrut -\mathstrut \) \(851\) \(\beta_{7}\mathstrut -\mathstrut \) \(2212\) \(\beta_{6}\mathstrut -\mathstrut \) \(11406\) \(\beta_{5}\mathstrut +\mathstrut \) \(2356\) \(\beta_{4}\mathstrut +\mathstrut \) \(22982\) \(\beta_{3}\mathstrut +\mathstrut \) \(13815\) \(\beta_{2}\mathstrut +\mathstrut \) \(60272\) \(\beta_{1}\mathstrut +\mathstrut \) \(14882\)
\(\nu^{14}\)\(=\)\(4132\) \(\beta_{17}\mathstrut +\mathstrut \) \(4088\) \(\beta_{16}\mathstrut -\mathstrut \) \(13322\) \(\beta_{15}\mathstrut -\mathstrut \) \(16046\) \(\beta_{14}\mathstrut -\mathstrut \) \(3777\) \(\beta_{13}\mathstrut +\mathstrut \) \(14333\) \(\beta_{12}\mathstrut +\mathstrut \) \(10699\) \(\beta_{11}\mathstrut +\mathstrut \) \(15311\) \(\beta_{10}\mathstrut -\mathstrut \) \(12924\) \(\beta_{9}\mathstrut +\mathstrut \) \(10325\) \(\beta_{8}\mathstrut -\mathstrut \) \(12104\) \(\beta_{7}\mathstrut -\mathstrut \) \(4435\) \(\beta_{6}\mathstrut -\mathstrut \) \(30750\) \(\beta_{5}\mathstrut +\mathstrut \) \(44276\) \(\beta_{4}\mathstrut +\mathstrut \) \(7752\) \(\beta_{3}\mathstrut +\mathstrut \) \(122797\) \(\beta_{2}\mathstrut +\mathstrut \) \(16463\) \(\beta_{1}\mathstrut +\mathstrut \) \(169687\)
\(\nu^{15}\)\(=\)\(18770\) \(\beta_{17}\mathstrut +\mathstrut \) \(27735\) \(\beta_{16}\mathstrut -\mathstrut \) \(1720\) \(\beta_{15}\mathstrut -\mathstrut \) \(45215\) \(\beta_{14}\mathstrut -\mathstrut \) \(54884\) \(\beta_{13}\mathstrut +\mathstrut \) \(73369\) \(\beta_{12}\mathstrut +\mathstrut \) \(72728\) \(\beta_{11}\mathstrut +\mathstrut \) \(61411\) \(\beta_{10}\mathstrut -\mathstrut \) \(38911\) \(\beta_{9}\mathstrut +\mathstrut \) \(7822\) \(\beta_{8}\mathstrut -\mathstrut \) \(5865\) \(\beta_{7}\mathstrut -\mathstrut \) \(20629\) \(\beta_{6}\mathstrut -\mathstrut \) \(95444\) \(\beta_{5}\mathstrut +\mathstrut \) \(22823\) \(\beta_{4}\mathstrut +\mathstrut \) \(160374\) \(\beta_{3}\mathstrut +\mathstrut \) \(123295\) \(\beta_{2}\mathstrut +\mathstrut \) \(420988\) \(\beta_{1}\mathstrut +\mathstrut \) \(129274\)
\(\nu^{16}\)\(=\)\(42802\) \(\beta_{17}\mathstrut +\mathstrut \) \(37277\) \(\beta_{16}\mathstrut -\mathstrut \) \(106255\) \(\beta_{15}\mathstrut -\mathstrut \) \(140290\) \(\beta_{14}\mathstrut -\mathstrut \) \(40349\) \(\beta_{13}\mathstrut +\mathstrut \) \(124643\) \(\beta_{12}\mathstrut +\mathstrut \) \(88514\) \(\beta_{11}\mathstrut +\mathstrut \) \(132495\) \(\beta_{10}\mathstrut -\mathstrut \) \(107445\) \(\beta_{9}\mathstrut +\mathstrut \) \(83696\) \(\beta_{8}\mathstrut -\mathstrut \) \(97178\) \(\beta_{7}\mathstrut -\mathstrut \) \(41625\) \(\beta_{6}\mathstrut -\mathstrut \) \(262937\) \(\beta_{5}\mathstrut +\mathstrut \) \(332557\) \(\beta_{4}\mathstrut +\mathstrut \) \(74348\) \(\beta_{3}\mathstrut +\mathstrut \) \(901853\) \(\beta_{2}\mathstrut +\mathstrut \) \(153689\) \(\beta_{1}\mathstrut +\mathstrut \) \(1190213\)
\(\nu^{17}\)\(=\)\(167614\) \(\beta_{17}\mathstrut +\mathstrut \) \(227228\) \(\beta_{16}\mathstrut -\mathstrut \) \(14592\) \(\beta_{15}\mathstrut -\mathstrut \) \(406314\) \(\beta_{14}\mathstrut -\mathstrut \) \(474075\) \(\beta_{13}\mathstrut +\mathstrut \) \(582521\) \(\beta_{12}\mathstrut +\mathstrut \) \(572993\) \(\beta_{11}\mathstrut +\mathstrut \) \(492490\) \(\beta_{10}\mathstrut -\mathstrut \) \(287313\) \(\beta_{9}\mathstrut +\mathstrut \) \(81269\) \(\beta_{8}\mathstrut -\mathstrut \) \(38911\) \(\beta_{7}\mathstrut -\mathstrut \) \(182403\) \(\beta_{6}\mathstrut -\mathstrut \) \(782561\) \(\beta_{5}\mathstrut +\mathstrut \) \(209639\) \(\beta_{4}\mathstrut +\mathstrut \) \(1125602\) \(\beta_{3}\mathstrut +\mathstrut \) \(1071334\) \(\beta_{2}\mathstrut +\mathstrut \) \(2972215\) \(\beta_{1}\mathstrut +\mathstrut \) \(1099782\)

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
−2.62138
−2.48986
−2.10676
−1.73135
−1.55076
−0.698229
−0.603996
−0.396252
−0.173863
0.844677
0.925440
1.33141
1.46257
1.69075
2.34662
2.43391
2.53544
2.80164
−2.62138 1.00000 4.87165 1.00000 −2.62138 2.00076 −7.52769 1.00000 −2.62138
1.2 −2.48986 1.00000 4.19938 1.00000 −2.48986 −2.76131 −5.47615 1.00000 −2.48986
1.3 −2.10676 1.00000 2.43846 1.00000 −2.10676 3.31927 −0.923726 1.00000 −2.10676
1.4 −1.73135 1.00000 0.997580 1.00000 −1.73135 0.816002 1.73554 1.00000 −1.73135
1.5 −1.55076 1.00000 0.404854 1.00000 −1.55076 −3.31793 2.47369 1.00000 −1.55076
1.6 −0.698229 1.00000 −1.51248 1.00000 −0.698229 4.99716 2.45251 1.00000 −0.698229
1.7 −0.603996 1.00000 −1.63519 1.00000 −0.603996 0.896809 2.19564 1.00000 −0.603996
1.8 −0.396252 1.00000 −1.84298 1.00000 −0.396252 0.499526 1.52279 1.00000 −0.396252
1.9 −0.173863 1.00000 −1.96977 1.00000 −0.173863 −4.18769 0.690195 1.00000 −0.173863
1.10 0.844677 1.00000 −1.28652 1.00000 0.844677 −2.49791 −2.77605 1.00000 0.844677
1.11 0.925440 1.00000 −1.14356 1.00000 0.925440 −0.122708 −2.90918 1.00000 0.925440
1.12 1.33141 1.00000 −0.227354 1.00000 1.33141 4.86785 −2.96552 1.00000 1.33141
1.13 1.46257 1.00000 0.139121 1.00000 1.46257 2.97636 −2.72167 1.00000 1.46257
1.14 1.69075 1.00000 0.858621 1.00000 1.69075 0.189464 −1.92978 1.00000 1.69075
1.15 2.34662 1.00000 3.50662 1.00000 2.34662 −4.52573 3.53546 1.00000 2.34662
1.16 2.43391 1.00000 3.92391 1.00000 2.43391 2.58128 4.68261 1.00000 2.43391
1.17 2.53544 1.00000 4.42846 1.00000 2.53544 3.07784 6.15722 1.00000 2.53544
1.18 2.80164 1.00000 5.84921 1.00000 2.80164 −0.809034 10.7841 1.00000 2.80164
\(n\): 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
\(3\) \(-1\)
\(5\) \(-1\)
\(13\) \(1\)
\(31\) \(-1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6045))\):

\(T_{2}^{18} - \cdots\)
\(T_{7}^{18} - \cdots\)
\(T_{11}^{18} - \cdots\)