Properties

Label 6024.2.a.r
Level 6024
Weight 2
Character orbit 6024.a
Self dual Yes
Analytic conductor 48.102
Analytic rank 0
Dimension 20
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 6024 = 2^{3} \cdot 3 \cdot 251 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6024.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(48.1018821776\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{5} \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \(+ q^{3}\) \( + \beta_{1} q^{5} \) \( -\beta_{3} q^{7} \) \(+ q^{9}\) \(+O(q^{10})\) \( q\) \(+ q^{3}\) \( + \beta_{1} q^{5} \) \( -\beta_{3} q^{7} \) \(+ q^{9}\) \( -\beta_{13} q^{11} \) \( + ( 1 - \beta_{12} ) q^{13} \) \( + \beta_{1} q^{15} \) \( + ( 1 + \beta_{11} ) q^{17} \) \( -\beta_{17} q^{19} \) \( -\beta_{3} q^{21} \) \( + \beta_{4} q^{23} \) \( + ( 2 + \beta_{1} + \beta_{4} + \beta_{6} - \beta_{8} + \beta_{13} + \beta_{15} + \beta_{19} ) q^{25} \) \(+ q^{27}\) \( + ( 1 + \beta_{7} ) q^{29} \) \( + ( 1 + \beta_{9} ) q^{31} \) \( -\beta_{13} q^{33} \) \( + ( -2 + \beta_{1} - \beta_{3} + \beta_{8} + \beta_{10} - \beta_{11} + \beta_{12} + \beta_{16} - \beta_{19} ) q^{35} \) \( + ( 2 + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{13} + \beta_{14} ) q^{37} \) \( + ( 1 - \beta_{12} ) q^{39} \) \( + ( 1 - \beta_{4} - \beta_{6} + \beta_{8} + \beta_{16} ) q^{41} \) \( + ( -\beta_{3} + \beta_{6} - \beta_{8} - \beta_{9} - \beta_{10} - \beta_{15} - \beta_{16} ) q^{43} \) \( + \beta_{1} q^{45} \) \( + ( 2 - \beta_{2} - \beta_{8} - \beta_{10} - \beta_{13} + \beta_{17} - \beta_{18} ) q^{47} \) \( + ( 2 - \beta_{3} - \beta_{5} - \beta_{8} - \beta_{12} + \beta_{13} - \beta_{14} - \beta_{15} - \beta_{16} + \beta_{17} ) q^{49} \) \( + ( 1 + \beta_{11} ) q^{51} \) \( + ( 1 + \beta_{1} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{9} + \beta_{12} + \beta_{13} - \beta_{16} ) q^{53} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{10} - \beta_{14} + \beta_{18} ) q^{55} \) \( -\beta_{17} q^{57} \) \( + ( \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{10} - \beta_{16} + \beta_{18} ) q^{59} \) \( + ( 3 - \beta_{1} + \beta_{2} + \beta_{6} - \beta_{9} + \beta_{12} + \beta_{13} - \beta_{14} - \beta_{19} ) q^{61} \) \( -\beta_{3} q^{63} \) \( + ( 1 + \beta_{1} + \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} + \beta_{17} - \beta_{19} ) q^{65} \) \( + ( 1 - \beta_{2} - \beta_{4} + \beta_{5} - \beta_{6} + \beta_{8} - \beta_{10} - \beta_{11} - \beta_{13} + \beta_{14} + \beta_{15} ) q^{67} \) \( + \beta_{4} q^{69} \) \( + ( 3 - \beta_{1} - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{12} + \beta_{19} ) q^{71} \) \( + ( 1 + \beta_{1} - \beta_{2} - \beta_{6} + \beta_{9} - \beta_{11} - \beta_{12} + \beta_{14} + \beta_{17} ) q^{73} \) \( + ( 2 + \beta_{1} + \beta_{4} + \beta_{6} - \beta_{8} + \beta_{13} + \beta_{15} + \beta_{19} ) q^{75} \) \( + ( \beta_{1} - \beta_{2} + \beta_{6} - \beta_{7} + \beta_{12} - \beta_{13} + \beta_{14} - \beta_{16} + \beta_{18} ) q^{77} \) \( + ( 1 - \beta_{3} - \beta_{4} + \beta_{8} + \beta_{12} - \beta_{15} + \beta_{16} - \beta_{18} ) q^{79} \) \(+ q^{81}\) \( + ( -1 + \beta_{2} - \beta_{4} - \beta_{8} - \beta_{12} - \beta_{13} - 2 \beta_{15} - \beta_{16} + \beta_{17} ) q^{83} \) \( + ( 2 + \beta_{3} + \beta_{8} + \beta_{10} + 2 \beta_{16} - \beta_{17} ) q^{85} \) \( + ( 1 + \beta_{7} ) q^{87} \) \( + ( \beta_{1} + \beta_{2} + \beta_{4} - \beta_{5} - \beta_{8} + \beta_{10} + \beta_{12} + 2 \beta_{13} + \beta_{16} - \beta_{18} ) q^{89} \) \( + ( 1 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{10} + \beta_{13} + \beta_{16} - \beta_{18} - \beta_{19} ) q^{91} \) \( + ( 1 + \beta_{9} ) q^{93} \) \( + ( 2 - \beta_{2} + 2 \beta_{4} + 2 \beta_{6} + \beta_{7} - 2 \beta_{8} + 2 \beta_{12} + \beta_{13} + 2 \beta_{15} + \beta_{16} - \beta_{17} - \beta_{18} ) q^{95} \) \( + ( 2 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + 2 \beta_{6} + \beta_{7} - \beta_{8} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} + \beta_{17} - \beta_{18} ) q^{97} \) \( -\beta_{13} q^{99} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(20q \) \(\mathstrut +\mathstrut 20q^{3} \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut +\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut 20q^{9} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(20q \) \(\mathstrut +\mathstrut 20q^{3} \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut +\mathstrut 9q^{7} \) \(\mathstrut +\mathstrut 20q^{9} \) \(\mathstrut +\mathstrut 4q^{11} \) \(\mathstrut +\mathstrut 21q^{13} \) \(\mathstrut +\mathstrut 9q^{15} \) \(\mathstrut +\mathstrut 10q^{17} \) \(\mathstrut +\mathstrut 8q^{19} \) \(\mathstrut +\mathstrut 9q^{21} \) \(\mathstrut +\mathstrut 9q^{23} \) \(\mathstrut +\mathstrut 43q^{25} \) \(\mathstrut +\mathstrut 20q^{27} \) \(\mathstrut +\mathstrut 18q^{29} \) \(\mathstrut +\mathstrut 27q^{31} \) \(\mathstrut +\mathstrut 4q^{33} \) \(\mathstrut -\mathstrut 7q^{35} \) \(\mathstrut +\mathstrut 33q^{37} \) \(\mathstrut +\mathstrut 21q^{39} \) \(\mathstrut +\mathstrut 14q^{41} \) \(\mathstrut -\mathstrut 6q^{43} \) \(\mathstrut +\mathstrut 9q^{45} \) \(\mathstrut +\mathstrut 21q^{47} \) \(\mathstrut +\mathstrut 47q^{49} \) \(\mathstrut +\mathstrut 10q^{51} \) \(\mathstrut +\mathstrut 23q^{53} \) \(\mathstrut +\mathstrut 24q^{55} \) \(\mathstrut +\mathstrut 8q^{57} \) \(\mathstrut +\mathstrut 10q^{59} \) \(\mathstrut +\mathstrut 28q^{61} \) \(\mathstrut +\mathstrut 9q^{63} \) \(\mathstrut +\mathstrut 2q^{65} \) \(\mathstrut +\mathstrut 15q^{67} \) \(\mathstrut +\mathstrut 9q^{69} \) \(\mathstrut +\mathstrut 33q^{71} \) \(\mathstrut +\mathstrut 50q^{73} \) \(\mathstrut +\mathstrut 43q^{75} \) \(\mathstrut +\mathstrut 20q^{77} \) \(\mathstrut +\mathstrut 17q^{79} \) \(\mathstrut +\mathstrut 20q^{81} \) \(\mathstrut -\mathstrut 19q^{83} \) \(\mathstrut +\mathstrut 41q^{85} \) \(\mathstrut +\mathstrut 18q^{87} \) \(\mathstrut +\mathstrut 21q^{89} \) \(\mathstrut +\mathstrut 30q^{91} \) \(\mathstrut +\mathstrut 27q^{93} \) \(\mathstrut +\mathstrut 27q^{95} \) \(\mathstrut +\mathstrut 47q^{97} \) \(\mathstrut +\mathstrut 4q^{99} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20}\mathstrut -\mathstrut \) \(9\) \(x^{19}\mathstrut -\mathstrut \) \(31\) \(x^{18}\mathstrut +\mathstrut \) \(471\) \(x^{17}\mathstrut -\mathstrut \) \(82\) \(x^{16}\mathstrut -\mathstrut \) \(9476\) \(x^{15}\mathstrut +\mathstrut \) \(12881\) \(x^{14}\mathstrut +\mathstrut \) \(94079\) \(x^{13}\mathstrut -\mathstrut \) \(181272\) \(x^{12}\mathstrut -\mathstrut \) \(504914\) \(x^{11}\mathstrut +\mathstrut \) \(1108012\) \(x^{10}\mathstrut +\mathstrut \) \(1591114\) \(x^{9}\mathstrut -\mathstrut \) \(3399127\) \(x^{8}\mathstrut -\mathstrut \) \(3224089\) \(x^{7}\mathstrut +\mathstrut \) \(5089722\) \(x^{6}\mathstrut +\mathstrut \) \(4287384\) \(x^{5}\mathstrut -\mathstrut \) \(2840216\) \(x^{4}\mathstrut -\mathstrut \) \(2820864\) \(x^{3}\mathstrut -\mathstrut \) \(208576\) \(x^{2}\mathstrut +\mathstrut \) \(204992\) \(x\mathstrut +\mathstrut \) \(24832\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(11\!\cdots\!07\) \(\nu^{19}\mathstrut +\mathstrut \) \(10\!\cdots\!89\) \(\nu^{18}\mathstrut +\mathstrut \) \(38\!\cdots\!07\) \(\nu^{17}\mathstrut -\mathstrut \) \(52\!\cdots\!15\) \(\nu^{16}\mathstrut -\mathstrut \) \(34\!\cdots\!92\) \(\nu^{15}\mathstrut +\mathstrut \) \(10\!\cdots\!92\) \(\nu^{14}\mathstrut -\mathstrut \) \(12\!\cdots\!55\) \(\nu^{13}\mathstrut -\mathstrut \) \(98\!\cdots\!95\) \(\nu^{12}\mathstrut +\mathstrut \) \(17\!\cdots\!94\) \(\nu^{11}\mathstrut +\mathstrut \) \(48\!\cdots\!74\) \(\nu^{10}\mathstrut -\mathstrut \) \(10\!\cdots\!28\) \(\nu^{9}\mathstrut -\mathstrut \) \(13\!\cdots\!86\) \(\nu^{8}\mathstrut +\mathstrut \) \(30\!\cdots\!85\) \(\nu^{7}\mathstrut +\mathstrut \) \(19\!\cdots\!01\) \(\nu^{6}\mathstrut -\mathstrut \) \(42\!\cdots\!12\) \(\nu^{5}\mathstrut -\mathstrut \) \(18\!\cdots\!00\) \(\nu^{4}\mathstrut +\mathstrut \) \(24\!\cdots\!24\) \(\nu^{3}\mathstrut +\mathstrut \) \(89\!\cdots\!52\) \(\nu^{2}\mathstrut -\mathstrut \) \(19\!\cdots\!80\) \(\nu\mathstrut -\mathstrut \) \(40\!\cdots\!52\)\()/\)\(38\!\cdots\!84\)
\(\beta_{3}\)\(=\)\((\)\(-\)\(70\!\cdots\!27\) \(\nu^{19}\mathstrut +\mathstrut \) \(47\!\cdots\!41\) \(\nu^{18}\mathstrut +\mathstrut \) \(31\!\cdots\!95\) \(\nu^{17}\mathstrut -\mathstrut \) \(25\!\cdots\!39\) \(\nu^{16}\mathstrut -\mathstrut \) \(47\!\cdots\!04\) \(\nu^{15}\mathstrut +\mathstrut \) \(53\!\cdots\!56\) \(\nu^{14}\mathstrut +\mathstrut \) \(23\!\cdots\!53\) \(\nu^{13}\mathstrut -\mathstrut \) \(55\!\cdots\!43\) \(\nu^{12}\mathstrut +\mathstrut \) \(61\!\cdots\!26\) \(\nu^{11}\mathstrut +\mathstrut \) \(31\!\cdots\!86\) \(\nu^{10}\mathstrut -\mathstrut \) \(76\!\cdots\!08\) \(\nu^{9}\mathstrut -\mathstrut \) \(99\!\cdots\!14\) \(\nu^{8}\mathstrut +\mathstrut \) \(14\!\cdots\!33\) \(\nu^{7}\mathstrut +\mathstrut \) \(16\!\cdots\!81\) \(\nu^{6}\mathstrut +\mathstrut \) \(14\!\cdots\!56\) \(\nu^{5}\mathstrut -\mathstrut \) \(12\!\cdots\!96\) \(\nu^{4}\mathstrut -\mathstrut \) \(36\!\cdots\!20\) \(\nu^{3}\mathstrut +\mathstrut \) \(15\!\cdots\!28\) \(\nu^{2}\mathstrut +\mathstrut \) \(36\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(29\!\cdots\!24\)\()/\)\(19\!\cdots\!92\)
\(\beta_{4}\)\(=\)\((\)\(-\)\(18\!\cdots\!89\) \(\nu^{19}\mathstrut +\mathstrut \) \(16\!\cdots\!35\) \(\nu^{18}\mathstrut +\mathstrut \) \(50\!\cdots\!45\) \(\nu^{17}\mathstrut -\mathstrut \) \(85\!\cdots\!93\) \(\nu^{16}\mathstrut +\mathstrut \) \(44\!\cdots\!76\) \(\nu^{15}\mathstrut +\mathstrut \) \(16\!\cdots\!92\) \(\nu^{14}\mathstrut -\mathstrut \) \(28\!\cdots\!29\) \(\nu^{13}\mathstrut -\mathstrut \) \(15\!\cdots\!37\) \(\nu^{12}\mathstrut +\mathstrut \) \(37\!\cdots\!86\) \(\nu^{11}\mathstrut +\mathstrut \) \(68\!\cdots\!22\) \(\nu^{10}\mathstrut -\mathstrut \) \(21\!\cdots\!40\) \(\nu^{9}\mathstrut -\mathstrut \) \(16\!\cdots\!98\) \(\nu^{8}\mathstrut +\mathstrut \) \(63\!\cdots\!75\) \(\nu^{7}\mathstrut +\mathstrut \) \(22\!\cdots\!67\) \(\nu^{6}\mathstrut -\mathstrut \) \(92\!\cdots\!68\) \(\nu^{5}\mathstrut -\mathstrut \) \(24\!\cdots\!12\) \(\nu^{4}\mathstrut +\mathstrut \) \(55\!\cdots\!20\) \(\nu^{3}\mathstrut +\mathstrut \) \(18\!\cdots\!44\) \(\nu^{2}\mathstrut -\mathstrut \) \(45\!\cdots\!20\) \(\nu\mathstrut -\mathstrut \) \(99\!\cdots\!44\)\()/\)\(38\!\cdots\!84\)
\(\beta_{5}\)\(=\)\((\)\(10\!\cdots\!53\) \(\nu^{19}\mathstrut -\mathstrut \) \(60\!\cdots\!01\) \(\nu^{18}\mathstrut -\mathstrut \) \(51\!\cdots\!63\) \(\nu^{17}\mathstrut +\mathstrut \) \(33\!\cdots\!51\) \(\nu^{16}\mathstrut +\mathstrut \) \(10\!\cdots\!38\) \(\nu^{15}\mathstrut -\mathstrut \) \(71\!\cdots\!76\) \(\nu^{14}\mathstrut -\mathstrut \) \(98\!\cdots\!07\) \(\nu^{13}\mathstrut +\mathstrut \) \(77\!\cdots\!71\) \(\nu^{12}\mathstrut +\mathstrut \) \(58\!\cdots\!88\) \(\nu^{11}\mathstrut -\mathstrut \) \(46\!\cdots\!30\) \(\nu^{10}\mathstrut -\mathstrut \) \(26\!\cdots\!76\) \(\nu^{9}\mathstrut +\mathstrut \) \(15\!\cdots\!26\) \(\nu^{8}\mathstrut +\mathstrut \) \(93\!\cdots\!85\) \(\nu^{7}\mathstrut -\mathstrut \) \(25\!\cdots\!61\) \(\nu^{6}\mathstrut -\mathstrut \) \(20\!\cdots\!82\) \(\nu^{5}\mathstrut +\mathstrut \) \(15\!\cdots\!28\) \(\nu^{4}\mathstrut +\mathstrut \) \(17\!\cdots\!84\) \(\nu^{3}\mathstrut +\mathstrut \) \(22\!\cdots\!80\) \(\nu^{2}\mathstrut -\mathstrut \) \(11\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(20\!\cdots\!76\)\()/\)\(19\!\cdots\!92\)
\(\beta_{6}\)\(=\)\((\)\(30\!\cdots\!21\) \(\nu^{19}\mathstrut -\mathstrut \) \(21\!\cdots\!23\) \(\nu^{18}\mathstrut -\mathstrut \) \(13\!\cdots\!13\) \(\nu^{17}\mathstrut +\mathstrut \) \(11\!\cdots\!65\) \(\nu^{16}\mathstrut +\mathstrut \) \(17\!\cdots\!04\) \(\nu^{15}\mathstrut -\mathstrut \) \(23\!\cdots\!04\) \(\nu^{14}\mathstrut -\mathstrut \) \(25\!\cdots\!99\) \(\nu^{13}\mathstrut +\mathstrut \) \(23\!\cdots\!05\) \(\nu^{12}\mathstrut -\mathstrut \) \(10\!\cdots\!34\) \(\nu^{11}\mathstrut -\mathstrut \) \(12\!\cdots\!98\) \(\nu^{10}\mathstrut +\mathstrut \) \(70\!\cdots\!20\) \(\nu^{9}\mathstrut +\mathstrut \) \(38\!\cdots\!38\) \(\nu^{8}\mathstrut -\mathstrut \) \(16\!\cdots\!27\) \(\nu^{7}\mathstrut -\mathstrut \) \(61\!\cdots\!47\) \(\nu^{6}\mathstrut +\mathstrut \) \(89\!\cdots\!28\) \(\nu^{5}\mathstrut +\mathstrut \) \(42\!\cdots\!04\) \(\nu^{4}\mathstrut +\mathstrut \) \(59\!\cdots\!12\) \(\nu^{3}\mathstrut -\mathstrut \) \(50\!\cdots\!16\) \(\nu^{2}\mathstrut -\mathstrut \) \(42\!\cdots\!80\) \(\nu\mathstrut +\mathstrut \) \(81\!\cdots\!12\)\()/\)\(38\!\cdots\!84\)
\(\beta_{7}\)\(=\)\((\)\(17\!\cdots\!65\) \(\nu^{19}\mathstrut -\mathstrut \) \(11\!\cdots\!27\) \(\nu^{18}\mathstrut -\mathstrut \) \(84\!\cdots\!51\) \(\nu^{17}\mathstrut +\mathstrut \) \(60\!\cdots\!77\) \(\nu^{16}\mathstrut +\mathstrut \) \(14\!\cdots\!82\) \(\nu^{15}\mathstrut -\mathstrut \) \(12\!\cdots\!88\) \(\nu^{14}\mathstrut -\mathstrut \) \(11\!\cdots\!99\) \(\nu^{13}\mathstrut +\mathstrut \) \(13\!\cdots\!41\) \(\nu^{12}\mathstrut +\mathstrut \) \(49\!\cdots\!76\) \(\nu^{11}\mathstrut -\mathstrut \) \(79\!\cdots\!50\) \(\nu^{10}\mathstrut -\mathstrut \) \(18\!\cdots\!20\) \(\nu^{9}\mathstrut +\mathstrut \) \(25\!\cdots\!70\) \(\nu^{8}\mathstrut +\mathstrut \) \(84\!\cdots\!33\) \(\nu^{7}\mathstrut -\mathstrut \) \(42\!\cdots\!99\) \(\nu^{6}\mathstrut -\mathstrut \) \(23\!\cdots\!06\) \(\nu^{5}\mathstrut +\mathstrut \) \(26\!\cdots\!80\) \(\nu^{4}\mathstrut +\mathstrut \) \(23\!\cdots\!40\) \(\nu^{3}\mathstrut +\mathstrut \) \(17\!\cdots\!24\) \(\nu^{2}\mathstrut -\mathstrut \) \(20\!\cdots\!24\) \(\nu\mathstrut -\mathstrut \) \(14\!\cdots\!92\)\()/\)\(19\!\cdots\!92\)
\(\beta_{8}\)\(=\)\((\)\(93\!\cdots\!42\) \(\nu^{19}\mathstrut -\mathstrut \) \(57\!\cdots\!75\) \(\nu^{18}\mathstrut -\mathstrut \) \(46\!\cdots\!39\) \(\nu^{17}\mathstrut +\mathstrut \) \(31\!\cdots\!59\) \(\nu^{16}\mathstrut +\mathstrut \) \(86\!\cdots\!81\) \(\nu^{15}\mathstrut -\mathstrut \) \(67\!\cdots\!42\) \(\nu^{14}\mathstrut -\mathstrut \) \(78\!\cdots\!82\) \(\nu^{13}\mathstrut +\mathstrut \) \(72\!\cdots\!57\) \(\nu^{12}\mathstrut +\mathstrut \) \(40\!\cdots\!55\) \(\nu^{11}\mathstrut -\mathstrut \) \(42\!\cdots\!38\) \(\nu^{10}\mathstrut -\mathstrut \) \(17\!\cdots\!40\) \(\nu^{9}\mathstrut +\mathstrut \) \(13\!\cdots\!06\) \(\nu^{8}\mathstrut +\mathstrut \) \(68\!\cdots\!74\) \(\nu^{7}\mathstrut -\mathstrut \) \(22\!\cdots\!97\) \(\nu^{6}\mathstrut -\mathstrut \) \(16\!\cdots\!91\) \(\nu^{5}\mathstrut +\mathstrut \) \(13\!\cdots\!78\) \(\nu^{4}\mathstrut +\mathstrut \) \(15\!\cdots\!28\) \(\nu^{3}\mathstrut +\mathstrut \) \(17\!\cdots\!20\) \(\nu^{2}\mathstrut -\mathstrut \) \(13\!\cdots\!80\) \(\nu\mathstrut -\mathstrut \) \(19\!\cdots\!92\)\()/\)\(96\!\cdots\!96\)
\(\beta_{9}\)\(=\)\((\)\(32\!\cdots\!45\) \(\nu^{19}\mathstrut -\mathstrut \) \(21\!\cdots\!71\) \(\nu^{18}\mathstrut -\mathstrut \) \(14\!\cdots\!53\) \(\nu^{17}\mathstrut +\mathstrut \) \(11\!\cdots\!65\) \(\nu^{16}\mathstrut +\mathstrut \) \(23\!\cdots\!68\) \(\nu^{15}\mathstrut -\mathstrut \) \(24\!\cdots\!68\) \(\nu^{14}\mathstrut -\mathstrut \) \(12\!\cdots\!07\) \(\nu^{13}\mathstrut +\mathstrut \) \(25\!\cdots\!65\) \(\nu^{12}\mathstrut -\mathstrut \) \(70\!\cdots\!58\) \(\nu^{11}\mathstrut -\mathstrut \) \(14\!\cdots\!66\) \(\nu^{10}\mathstrut +\mathstrut \) \(21\!\cdots\!64\) \(\nu^{9}\mathstrut +\mathstrut \) \(44\!\cdots\!74\) \(\nu^{8}\mathstrut -\mathstrut \) \(15\!\cdots\!59\) \(\nu^{7}\mathstrut -\mathstrut \) \(71\!\cdots\!27\) \(\nu^{6}\mathstrut -\mathstrut \) \(15\!\cdots\!12\) \(\nu^{5}\mathstrut +\mathstrut \) \(47\!\cdots\!20\) \(\nu^{4}\mathstrut +\mathstrut \) \(23\!\cdots\!20\) \(\nu^{3}\mathstrut -\mathstrut \) \(13\!\cdots\!60\) \(\nu^{2}\mathstrut -\mathstrut \) \(17\!\cdots\!84\) \(\nu\mathstrut -\mathstrut \) \(40\!\cdots\!04\)\()/\)\(19\!\cdots\!92\)
\(\beta_{10}\)\(=\)\((\)\(-\)\(99\!\cdots\!39\) \(\nu^{19}\mathstrut +\mathstrut \) \(67\!\cdots\!61\) \(\nu^{18}\mathstrut +\mathstrut \) \(44\!\cdots\!58\) \(\nu^{17}\mathstrut -\mathstrut \) \(36\!\cdots\!72\) \(\nu^{16}\mathstrut -\mathstrut \) \(67\!\cdots\!76\) \(\nu^{15}\mathstrut +\mathstrut \) \(75\!\cdots\!32\) \(\nu^{14}\mathstrut +\mathstrut \) \(31\!\cdots\!17\) \(\nu^{13}\mathstrut -\mathstrut \) \(78\!\cdots\!21\) \(\nu^{12}\mathstrut +\mathstrut \) \(10\!\cdots\!49\) \(\nu^{11}\mathstrut +\mathstrut \) \(44\!\cdots\!91\) \(\nu^{10}\mathstrut -\mathstrut \) \(11\!\cdots\!31\) \(\nu^{9}\mathstrut -\mathstrut \) \(13\!\cdots\!73\) \(\nu^{8}\mathstrut +\mathstrut \) \(22\!\cdots\!30\) \(\nu^{7}\mathstrut +\mathstrut \) \(22\!\cdots\!92\) \(\nu^{6}\mathstrut +\mathstrut \) \(16\!\cdots\!60\) \(\nu^{5}\mathstrut -\mathstrut \) \(15\!\cdots\!66\) \(\nu^{4}\mathstrut -\mathstrut \) \(49\!\cdots\!72\) \(\nu^{3}\mathstrut +\mathstrut \) \(11\!\cdots\!60\) \(\nu^{2}\mathstrut +\mathstrut \) \(34\!\cdots\!00\) \(\nu\mathstrut -\mathstrut \) \(31\!\cdots\!48\)\()/\)\(48\!\cdots\!48\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(14\!\cdots\!70\) \(\nu^{19}\mathstrut +\mathstrut \) \(89\!\cdots\!74\) \(\nu^{18}\mathstrut +\mathstrut \) \(72\!\cdots\!51\) \(\nu^{17}\mathstrut -\mathstrut \) \(48\!\cdots\!01\) \(\nu^{16}\mathstrut -\mathstrut \) \(13\!\cdots\!04\) \(\nu^{15}\mathstrut +\mathstrut \) \(10\!\cdots\!48\) \(\nu^{14}\mathstrut +\mathstrut \) \(12\!\cdots\!70\) \(\nu^{13}\mathstrut -\mathstrut \) \(11\!\cdots\!16\) \(\nu^{12}\mathstrut -\mathstrut \) \(65\!\cdots\!75\) \(\nu^{11}\mathstrut +\mathstrut \) \(66\!\cdots\!27\) \(\nu^{10}\mathstrut +\mathstrut \) \(27\!\cdots\!67\) \(\nu^{9}\mathstrut -\mathstrut \) \(21\!\cdots\!49\) \(\nu^{8}\mathstrut -\mathstrut \) \(10\!\cdots\!23\) \(\nu^{7}\mathstrut +\mathstrut \) \(36\!\cdots\!95\) \(\nu^{6}\mathstrut +\mathstrut \) \(24\!\cdots\!04\) \(\nu^{5}\mathstrut -\mathstrut \) \(23\!\cdots\!14\) \(\nu^{4}\mathstrut -\mathstrut \) \(21\!\cdots\!88\) \(\nu^{3}\mathstrut -\mathstrut \) \(14\!\cdots\!96\) \(\nu^{2}\mathstrut +\mathstrut \) \(17\!\cdots\!92\) \(\nu\mathstrut +\mathstrut \) \(15\!\cdots\!52\)\()/\)\(68\!\cdots\!64\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(10\!\cdots\!05\) \(\nu^{19}\mathstrut +\mathstrut \) \(64\!\cdots\!99\) \(\nu^{18}\mathstrut +\mathstrut \) \(49\!\cdots\!25\) \(\nu^{17}\mathstrut -\mathstrut \) \(35\!\cdots\!57\) \(\nu^{16}\mathstrut -\mathstrut \) \(86\!\cdots\!32\) \(\nu^{15}\mathstrut +\mathstrut \) \(74\!\cdots\!44\) \(\nu^{14}\mathstrut +\mathstrut \) \(69\!\cdots\!35\) \(\nu^{13}\mathstrut -\mathstrut \) \(79\!\cdots\!77\) \(\nu^{12}\mathstrut -\mathstrut \) \(27\!\cdots\!94\) \(\nu^{11}\mathstrut +\mathstrut \) \(46\!\cdots\!66\) \(\nu^{10}\mathstrut +\mathstrut \) \(93\!\cdots\!76\) \(\nu^{9}\mathstrut -\mathstrut \) \(14\!\cdots\!26\) \(\nu^{8}\mathstrut -\mathstrut \) \(42\!\cdots\!81\) \(\nu^{7}\mathstrut +\mathstrut \) \(24\!\cdots\!27\) \(\nu^{6}\mathstrut +\mathstrut \) \(12\!\cdots\!12\) \(\nu^{5}\mathstrut -\mathstrut \) \(15\!\cdots\!64\) \(\nu^{4}\mathstrut -\mathstrut \) \(11\!\cdots\!40\) \(\nu^{3}\mathstrut -\mathstrut \) \(57\!\cdots\!04\) \(\nu^{2}\mathstrut +\mathstrut \) \(86\!\cdots\!40\) \(\nu\mathstrut +\mathstrut \) \(10\!\cdots\!56\)\()/\)\(38\!\cdots\!84\)
\(\beta_{13}\)\(=\)\((\)\(28\!\cdots\!13\) \(\nu^{19}\mathstrut -\mathstrut \) \(18\!\cdots\!77\) \(\nu^{18}\mathstrut -\mathstrut \) \(12\!\cdots\!33\) \(\nu^{17}\mathstrut +\mathstrut \) \(10\!\cdots\!05\) \(\nu^{16}\mathstrut +\mathstrut \) \(20\!\cdots\!06\) \(\nu^{15}\mathstrut -\mathstrut \) \(21\!\cdots\!32\) \(\nu^{14}\mathstrut -\mathstrut \) \(11\!\cdots\!63\) \(\nu^{13}\mathstrut +\mathstrut \) \(22\!\cdots\!15\) \(\nu^{12}\mathstrut -\mathstrut \) \(66\!\cdots\!70\) \(\nu^{11}\mathstrut -\mathstrut \) \(12\!\cdots\!96\) \(\nu^{10}\mathstrut +\mathstrut \) \(20\!\cdots\!18\) \(\nu^{9}\mathstrut +\mathstrut \) \(38\!\cdots\!80\) \(\nu^{8}\mathstrut -\mathstrut \) \(20\!\cdots\!65\) \(\nu^{7}\mathstrut -\mathstrut \) \(62\!\cdots\!15\) \(\nu^{6}\mathstrut -\mathstrut \) \(11\!\cdots\!98\) \(\nu^{5}\mathstrut +\mathstrut \) \(41\!\cdots\!60\) \(\nu^{4}\mathstrut +\mathstrut \) \(18\!\cdots\!84\) \(\nu^{3}\mathstrut -\mathstrut \) \(10\!\cdots\!60\) \(\nu^{2}\mathstrut -\mathstrut \) \(11\!\cdots\!12\) \(\nu\mathstrut -\mathstrut \) \(13\!\cdots\!16\)\()/\)\(96\!\cdots\!96\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(28\!\cdots\!89\) \(\nu^{19}\mathstrut +\mathstrut \) \(17\!\cdots\!54\) \(\nu^{18}\mathstrut +\mathstrut \) \(14\!\cdots\!05\) \(\nu^{17}\mathstrut -\mathstrut \) \(95\!\cdots\!28\) \(\nu^{16}\mathstrut -\mathstrut \) \(26\!\cdots\!26\) \(\nu^{15}\mathstrut +\mathstrut \) \(20\!\cdots\!54\) \(\nu^{14}\mathstrut +\mathstrut \) \(23\!\cdots\!01\) \(\nu^{13}\mathstrut -\mathstrut \) \(22\!\cdots\!48\) \(\nu^{12}\mathstrut -\mathstrut \) \(11\!\cdots\!08\) \(\nu^{11}\mathstrut +\mathstrut \) \(12\!\cdots\!38\) \(\nu^{10}\mathstrut +\mathstrut \) \(48\!\cdots\!58\) \(\nu^{9}\mathstrut -\mathstrut \) \(41\!\cdots\!80\) \(\nu^{8}\mathstrut -\mathstrut \) \(19\!\cdots\!09\) \(\nu^{7}\mathstrut +\mathstrut \) \(69\!\cdots\!78\) \(\nu^{6}\mathstrut +\mathstrut \) \(46\!\cdots\!24\) \(\nu^{5}\mathstrut -\mathstrut \) \(43\!\cdots\!76\) \(\nu^{4}\mathstrut -\mathstrut \) \(42\!\cdots\!64\) \(\nu^{3}\mathstrut -\mathstrut \) \(39\!\cdots\!32\) \(\nu^{2}\mathstrut +\mathstrut \) \(34\!\cdots\!16\) \(\nu\mathstrut +\mathstrut \) \(47\!\cdots\!28\)\()/\)\(96\!\cdots\!96\)
\(\beta_{15}\)\(=\)\((\)\(18\!\cdots\!69\) \(\nu^{19}\mathstrut -\mathstrut \) \(11\!\cdots\!11\) \(\nu^{18}\mathstrut -\mathstrut \) \(84\!\cdots\!21\) \(\nu^{17}\mathstrut +\mathstrut \) \(63\!\cdots\!73\) \(\nu^{16}\mathstrut +\mathstrut \) \(14\!\cdots\!88\) \(\nu^{15}\mathstrut -\mathstrut \) \(13\!\cdots\!08\) \(\nu^{14}\mathstrut -\mathstrut \) \(99\!\cdots\!87\) \(\nu^{13}\mathstrut +\mathstrut \) \(14\!\cdots\!01\) \(\nu^{12}\mathstrut +\mathstrut \) \(25\!\cdots\!50\) \(\nu^{11}\mathstrut -\mathstrut \) \(80\!\cdots\!02\) \(\nu^{10}\mathstrut -\mathstrut \) \(34\!\cdots\!68\) \(\nu^{9}\mathstrut +\mathstrut \) \(25\!\cdots\!46\) \(\nu^{8}\mathstrut +\mathstrut \) \(35\!\cdots\!85\) \(\nu^{7}\mathstrut -\mathstrut \) \(41\!\cdots\!15\) \(\nu^{6}\mathstrut -\mathstrut \) \(15\!\cdots\!12\) \(\nu^{5}\mathstrut +\mathstrut \) \(27\!\cdots\!80\) \(\nu^{4}\mathstrut +\mathstrut \) \(17\!\cdots\!84\) \(\nu^{3}\mathstrut +\mathstrut \) \(28\!\cdots\!60\) \(\nu^{2}\mathstrut -\mathstrut \) \(12\!\cdots\!12\) \(\nu\mathstrut -\mathstrut \) \(16\!\cdots\!04\)\()/\)\(38\!\cdots\!84\)
\(\beta_{16}\)\(=\)\((\)\(-\)\(94\!\cdots\!53\) \(\nu^{19}\mathstrut +\mathstrut \) \(59\!\cdots\!15\) \(\nu^{18}\mathstrut +\mathstrut \) \(45\!\cdots\!61\) \(\nu^{17}\mathstrut -\mathstrut \) \(32\!\cdots\!65\) \(\nu^{16}\mathstrut -\mathstrut \) \(81\!\cdots\!96\) \(\nu^{15}\mathstrut +\mathstrut \) \(68\!\cdots\!84\) \(\nu^{14}\mathstrut +\mathstrut \) \(67\!\cdots\!71\) \(\nu^{13}\mathstrut -\mathstrut \) \(73\!\cdots\!65\) \(\nu^{12}\mathstrut -\mathstrut \) \(29\!\cdots\!42\) \(\nu^{11}\mathstrut +\mathstrut \) \(42\!\cdots\!10\) \(\nu^{10}\mathstrut +\mathstrut \) \(10\!\cdots\!48\) \(\nu^{9}\mathstrut -\mathstrut \) \(13\!\cdots\!38\) \(\nu^{8}\mathstrut -\mathstrut \) \(46\!\cdots\!05\) \(\nu^{7}\mathstrut +\mathstrut \) \(22\!\cdots\!87\) \(\nu^{6}\mathstrut +\mathstrut \) \(12\!\cdots\!72\) \(\nu^{5}\mathstrut -\mathstrut \) \(14\!\cdots\!96\) \(\nu^{4}\mathstrut -\mathstrut \) \(12\!\cdots\!52\) \(\nu^{3}\mathstrut -\mathstrut \) \(92\!\cdots\!28\) \(\nu^{2}\mathstrut +\mathstrut \) \(91\!\cdots\!92\) \(\nu\mathstrut +\mathstrut \) \(11\!\cdots\!72\)\()/\)\(19\!\cdots\!92\)
\(\beta_{17}\)\(=\)\((\)\(-\)\(98\!\cdots\!37\) \(\nu^{19}\mathstrut +\mathstrut \) \(61\!\cdots\!97\) \(\nu^{18}\mathstrut +\mathstrut \) \(47\!\cdots\!39\) \(\nu^{17}\mathstrut -\mathstrut \) \(33\!\cdots\!07\) \(\nu^{16}\mathstrut -\mathstrut \) \(82\!\cdots\!22\) \(\nu^{15}\mathstrut +\mathstrut \) \(71\!\cdots\!48\) \(\nu^{14}\mathstrut +\mathstrut \) \(66\!\cdots\!07\) \(\nu^{13}\mathstrut -\mathstrut \) \(76\!\cdots\!23\) \(\nu^{12}\mathstrut -\mathstrut \) \(26\!\cdots\!68\) \(\nu^{11}\mathstrut +\mathstrut \) \(44\!\cdots\!98\) \(\nu^{10}\mathstrut +\mathstrut \) \(86\!\cdots\!16\) \(\nu^{9}\mathstrut -\mathstrut \) \(14\!\cdots\!06\) \(\nu^{8}\mathstrut -\mathstrut \) \(39\!\cdots\!29\) \(\nu^{7}\mathstrut +\mathstrut \) \(23\!\cdots\!97\) \(\nu^{6}\mathstrut +\mathstrut \) \(11\!\cdots\!10\) \(\nu^{5}\mathstrut -\mathstrut \) \(15\!\cdots\!72\) \(\nu^{4}\mathstrut -\mathstrut \) \(11\!\cdots\!28\) \(\nu^{3}\mathstrut -\mathstrut \) \(49\!\cdots\!64\) \(\nu^{2}\mathstrut +\mathstrut \) \(85\!\cdots\!56\) \(\nu\mathstrut +\mathstrut \) \(86\!\cdots\!08\)\()/\)\(19\!\cdots\!92\)
\(\beta_{18}\)\(=\)\((\)\(-\)\(25\!\cdots\!59\) \(\nu^{19}\mathstrut +\mathstrut \) \(15\!\cdots\!69\) \(\nu^{18}\mathstrut +\mathstrut \) \(12\!\cdots\!27\) \(\nu^{17}\mathstrut -\mathstrut \) \(83\!\cdots\!35\) \(\nu^{16}\mathstrut -\mathstrut \) \(22\!\cdots\!72\) \(\nu^{15}\mathstrut +\mathstrut \) \(17\!\cdots\!68\) \(\nu^{14}\mathstrut +\mathstrut \) \(20\!\cdots\!29\) \(\nu^{13}\mathstrut -\mathstrut \) \(19\!\cdots\!83\) \(\nu^{12}\mathstrut -\mathstrut \) \(10\!\cdots\!34\) \(\nu^{11}\mathstrut +\mathstrut \) \(11\!\cdots\!66\) \(\nu^{10}\mathstrut +\mathstrut \) \(44\!\cdots\!44\) \(\nu^{9}\mathstrut -\mathstrut \) \(36\!\cdots\!02\) \(\nu^{8}\mathstrut -\mathstrut \) \(17\!\cdots\!11\) \(\nu^{7}\mathstrut +\mathstrut \) \(60\!\cdots\!09\) \(\nu^{6}\mathstrut +\mathstrut \) \(40\!\cdots\!52\) \(\nu^{5}\mathstrut -\mathstrut \) \(37\!\cdots\!96\) \(\nu^{4}\mathstrut -\mathstrut \) \(37\!\cdots\!92\) \(\nu^{3}\mathstrut -\mathstrut \) \(35\!\cdots\!76\) \(\nu^{2}\mathstrut +\mathstrut \) \(29\!\cdots\!76\) \(\nu\mathstrut +\mathstrut \) \(37\!\cdots\!44\)\()/\)\(38\!\cdots\!84\)
\(\beta_{19}\)\(=\)\((\)\(-\)\(26\!\cdots\!85\) \(\nu^{19}\mathstrut +\mathstrut \) \(17\!\cdots\!07\) \(\nu^{18}\mathstrut +\mathstrut \) \(12\!\cdots\!65\) \(\nu^{17}\mathstrut -\mathstrut \) \(94\!\cdots\!29\) \(\nu^{16}\mathstrut -\mathstrut \) \(20\!\cdots\!68\) \(\nu^{15}\mathstrut +\mathstrut \) \(19\!\cdots\!80\) \(\nu^{14}\mathstrut +\mathstrut \) \(14\!\cdots\!39\) \(\nu^{13}\mathstrut -\mathstrut \) \(20\!\cdots\!01\) \(\nu^{12}\mathstrut -\mathstrut \) \(33\!\cdots\!02\) \(\nu^{11}\mathstrut +\mathstrut \) \(11\!\cdots\!10\) \(\nu^{10}\mathstrut +\mathstrut \) \(29\!\cdots\!56\) \(\nu^{9}\mathstrut -\mathstrut \) \(37\!\cdots\!82\) \(\nu^{8}\mathstrut -\mathstrut \) \(46\!\cdots\!77\) \(\nu^{7}\mathstrut +\mathstrut \) \(61\!\cdots\!67\) \(\nu^{6}\mathstrut +\mathstrut \) \(21\!\cdots\!80\) \(\nu^{5}\mathstrut -\mathstrut \) \(39\!\cdots\!00\) \(\nu^{4}\mathstrut -\mathstrut \) \(24\!\cdots\!40\) \(\nu^{3}\mathstrut -\mathstrut \) \(48\!\cdots\!84\) \(\nu^{2}\mathstrut +\mathstrut \) \(16\!\cdots\!56\) \(\nu\mathstrut +\mathstrut \) \(19\!\cdots\!44\)\()/\)\(38\!\cdots\!84\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{19}\mathstrut +\mathstrut \) \(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut -\mathstrut \) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(7\)
\(\nu^{3}\)\(=\)\(2\) \(\beta_{19}\mathstrut +\mathstrut \) \(\beta_{18}\mathstrut -\mathstrut \) \(2\) \(\beta_{17}\mathstrut -\mathstrut \) \(2\) \(\beta_{16}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(2\) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(13\) \(\beta_{1}\mathstrut +\mathstrut \) \(2\)
\(\nu^{4}\)\(=\)\(21\) \(\beta_{19}\mathstrut -\mathstrut \) \(\beta_{18}\mathstrut -\mathstrut \) \(\beta_{17}\mathstrut -\mathstrut \) \(4\) \(\beta_{16}\mathstrut +\mathstrut \) \(20\) \(\beta_{15}\mathstrut +\mathstrut \) \(4\) \(\beta_{14}\mathstrut +\mathstrut \) \(16\) \(\beta_{13}\mathstrut +\mathstrut \) \(3\) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut -\mathstrut \) \(5\) \(\beta_{10}\mathstrut -\mathstrut \) \(22\) \(\beta_{8}\mathstrut +\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(21\) \(\beta_{6}\mathstrut +\mathstrut \) \(2\) \(\beta_{5}\mathstrut +\mathstrut \) \(21\) \(\beta_{4}\mathstrut +\mathstrut \) \(6\) \(\beta_{3}\mathstrut -\mathstrut \) \(3\) \(\beta_{2}\mathstrut +\mathstrut \) \(19\) \(\beta_{1}\mathstrut +\mathstrut \) \(96\)
\(\nu^{5}\)\(=\)\(53\) \(\beta_{19}\mathstrut +\mathstrut \) \(18\) \(\beta_{18}\mathstrut -\mathstrut \) \(44\) \(\beta_{17}\mathstrut -\mathstrut \) \(45\) \(\beta_{16}\mathstrut +\mathstrut \) \(3\) \(\beta_{15}\mathstrut +\mathstrut \) \(2\) \(\beta_{14}\mathstrut +\mathstrut \) \(35\) \(\beta_{13}\mathstrut +\mathstrut \) \(23\) \(\beta_{11}\mathstrut -\mathstrut \) \(5\) \(\beta_{10}\mathstrut -\mathstrut \) \(5\) \(\beta_{9}\mathstrut +\mathstrut \) \(6\) \(\beta_{8}\mathstrut -\mathstrut \) \(18\) \(\beta_{7}\mathstrut +\mathstrut \) \(9\) \(\beta_{6}\mathstrut -\mathstrut \) \(7\) \(\beta_{5}\mathstrut +\mathstrut \) \(9\) \(\beta_{4}\mathstrut +\mathstrut \) \(48\) \(\beta_{3}\mathstrut -\mathstrut \) \(22\) \(\beta_{2}\mathstrut +\mathstrut \) \(205\) \(\beta_{1}\mathstrut +\mathstrut \) \(48\)
\(\nu^{6}\)\(=\)\(399\) \(\beta_{19}\mathstrut -\mathstrut \) \(30\) \(\beta_{18}\mathstrut -\mathstrut \) \(20\) \(\beta_{17}\mathstrut -\mathstrut \) \(107\) \(\beta_{16}\mathstrut +\mathstrut \) \(366\) \(\beta_{15}\mathstrut +\mathstrut \) \(100\) \(\beta_{14}\mathstrut +\mathstrut \) \(261\) \(\beta_{13}\mathstrut +\mathstrut \) \(70\) \(\beta_{12}\mathstrut +\mathstrut \) \(37\) \(\beta_{11}\mathstrut -\mathstrut \) \(151\) \(\beta_{10}\mathstrut +\mathstrut \) \(2\) \(\beta_{9}\mathstrut -\mathstrut \) \(427\) \(\beta_{8}\mathstrut +\mathstrut \) \(18\) \(\beta_{7}\mathstrut +\mathstrut \) \(397\) \(\beta_{6}\mathstrut +\mathstrut \) \(57\) \(\beta_{5}\mathstrut +\mathstrut \) \(407\) \(\beta_{4}\mathstrut +\mathstrut \) \(163\) \(\beta_{3}\mathstrut -\mathstrut \) \(91\) \(\beta_{2}\mathstrut +\mathstrut \) \(360\) \(\beta_{1}\mathstrut +\mathstrut \) \(1556\)
\(\nu^{7}\)\(=\)\(1147\) \(\beta_{19}\mathstrut +\mathstrut \) \(272\) \(\beta_{18}\mathstrut -\mathstrut \) \(818\) \(\beta_{17}\mathstrut -\mathstrut \) \(874\) \(\beta_{16}\mathstrut +\mathstrut \) \(118\) \(\beta_{15}\mathstrut +\mathstrut \) \(107\) \(\beta_{14}\mathstrut +\mathstrut \) \(831\) \(\beta_{13}\mathstrut -\mathstrut \) \(25\) \(\beta_{12}\mathstrut +\mathstrut \) \(449\) \(\beta_{11}\mathstrut -\mathstrut \) \(162\) \(\beta_{10}\mathstrut -\mathstrut \) \(125\) \(\beta_{9}\mathstrut -\mathstrut \) \(74\) \(\beta_{8}\mathstrut -\mathstrut \) \(317\) \(\beta_{7}\mathstrut +\mathstrut \) \(298\) \(\beta_{6}\mathstrut -\mathstrut \) \(218\) \(\beta_{5}\mathstrut +\mathstrut \) \(311\) \(\beta_{4}\mathstrut +\mathstrut \) \(962\) \(\beta_{3}\mathstrut -\mathstrut \) \(455\) \(\beta_{2}\mathstrut +\mathstrut \) \(3525\) \(\beta_{1}\mathstrut +\mathstrut \) \(1093\)
\(\nu^{8}\)\(=\)\(7433\) \(\beta_{19}\mathstrut -\mathstrut \) \(758\) \(\beta_{18}\mathstrut -\mathstrut \) \(250\) \(\beta_{17}\mathstrut -\mathstrut \) \(2307\) \(\beta_{16}\mathstrut +\mathstrut \) \(6564\) \(\beta_{15}\mathstrut +\mathstrut \) \(2107\) \(\beta_{14}\mathstrut +\mathstrut \) \(4443\) \(\beta_{13}\mathstrut +\mathstrut \) \(1273\) \(\beta_{12}\mathstrut +\mathstrut \) \(920\) \(\beta_{11}\mathstrut -\mathstrut \) \(3499\) \(\beta_{10}\mathstrut +\mathstrut \) \(118\) \(\beta_{9}\mathstrut -\mathstrut \) \(8047\) \(\beta_{8}\mathstrut +\mathstrut \) \(243\) \(\beta_{7}\mathstrut +\mathstrut \) \(7286\) \(\beta_{6}\mathstrut +\mathstrut \) \(1279\) \(\beta_{5}\mathstrut +\mathstrut \) \(7681\) \(\beta_{4}\mathstrut +\mathstrut \) \(3478\) \(\beta_{3}\mathstrut -\mathstrut \) \(2155\) \(\beta_{2}\mathstrut +\mathstrut \) \(7005\) \(\beta_{1}\mathstrut +\mathstrut \) \(26808\)
\(\nu^{9}\)\(=\)\(23242\) \(\beta_{19}\mathstrut +\mathstrut \) \(3879\) \(\beta_{18}\mathstrut -\mathstrut \) \(14550\) \(\beta_{17}\mathstrut -\mathstrut \) \(16454\) \(\beta_{16}\mathstrut +\mathstrut \) \(3076\) \(\beta_{15}\mathstrut +\mathstrut \) \(3277\) \(\beta_{14}\mathstrut +\mathstrut \) \(17583\) \(\beta_{13}\mathstrut -\mathstrut \) \(1099\) \(\beta_{12}\mathstrut +\mathstrut \) \(8368\) \(\beta_{11}\mathstrut -\mathstrut \) \(4092\) \(\beta_{10}\mathstrut -\mathstrut \) \(2434\) \(\beta_{9}\mathstrut -\mathstrut \) \(3749\) \(\beta_{8}\mathstrut -\mathstrut \) \(5844\) \(\beta_{7}\mathstrut +\mathstrut \) \(7395\) \(\beta_{6}\mathstrut -\mathstrut \) \(4912\) \(\beta_{5}\mathstrut +\mathstrut \) \(7852\) \(\beta_{4}\mathstrut +\mathstrut \) \(18441\) \(\beta_{3}\mathstrut -\mathstrut \) \(9193\) \(\beta_{2}\mathstrut +\mathstrut \) \(63102\) \(\beta_{1}\mathstrut +\mathstrut \) \(24000\)
\(\nu^{10}\)\(=\)\(138196\) \(\beta_{19}\mathstrut -\mathstrut \) \(17646\) \(\beta_{18}\mathstrut -\mathstrut \) \(1693\) \(\beta_{17}\mathstrut -\mathstrut \) \(46816\) \(\beta_{16}\mathstrut +\mathstrut \) \(117174\) \(\beta_{15}\mathstrut +\mathstrut \) \(42550\) \(\beta_{14}\mathstrut +\mathstrut \) \(78258\) \(\beta_{13}\mathstrut +\mathstrut \) \(21085\) \(\beta_{12}\mathstrut +\mathstrut \) \(20250\) \(\beta_{11}\mathstrut -\mathstrut \) \(74027\) \(\beta_{10}\mathstrut +\mathstrut \) \(3577\) \(\beta_{9}\mathstrut -\mathstrut \) \(150019\) \(\beta_{8}\mathstrut +\mathstrut \) \(2450\) \(\beta_{7}\mathstrut +\mathstrut \) \(132710\) \(\beta_{6}\mathstrut +\mathstrut \) \(26538\) \(\beta_{5}\mathstrut +\mathstrut \) \(143316\) \(\beta_{4}\mathstrut +\mathstrut \) \(68967\) \(\beta_{3}\mathstrut -\mathstrut \) \(46659\) \(\beta_{2}\mathstrut +\mathstrut \) \(137512\) \(\beta_{1}\mathstrut +\mathstrut \) \(475381\)
\(\nu^{11}\)\(=\)\(457878\) \(\beta_{19}\mathstrut +\mathstrut \) \(52126\) \(\beta_{18}\mathstrut -\mathstrut \) \(255043\) \(\beta_{17}\mathstrut -\mathstrut \) \(307432\) \(\beta_{16}\mathstrut +\mathstrut \) \(69205\) \(\beta_{15}\mathstrut +\mathstrut \) \(82767\) \(\beta_{14}\mathstrut +\mathstrut \) \(354044\) \(\beta_{13}\mathstrut -\mathstrut \) \(32468\) \(\beta_{12}\mathstrut +\mathstrut \) \(153971\) \(\beta_{11}\mathstrut -\mathstrut \) \(95249\) \(\beta_{10}\mathstrut -\mathstrut \) \(44046\) \(\beta_{9}\mathstrut -\mathstrut \) \(101338\) \(\beta_{8}\mathstrut -\mathstrut \) \(110913\) \(\beta_{7}\mathstrut +\mathstrut \) \(164897\) \(\beta_{6}\mathstrut -\mathstrut \) \(97388\) \(\beta_{5}\mathstrut +\mathstrut \) \(176346\) \(\beta_{4}\mathstrut +\mathstrut \) \(349447\) \(\beta_{3}\mathstrut -\mathstrut \) \(183634\) \(\beta_{2}\mathstrut +\mathstrut \) \(1152930\) \(\beta_{1}\mathstrut +\mathstrut \) \(510755\)
\(\nu^{12}\)\(=\)\(2576151\) \(\beta_{19}\mathstrut -\mathstrut \) \(389933\) \(\beta_{18}\mathstrut +\mathstrut \) \(20627\) \(\beta_{17}\mathstrut -\mathstrut \) \(926883\) \(\beta_{16}\mathstrut +\mathstrut \) \(2092657\) \(\beta_{15}\mathstrut +\mathstrut \) \(845770\) \(\beta_{14}\mathstrut +\mathstrut \) \(1414194\) \(\beta_{13}\mathstrut +\mathstrut \) \(330983\) \(\beta_{12}\mathstrut +\mathstrut \) \(423046\) \(\beta_{11}\mathstrut -\mathstrut \) \(1501680\) \(\beta_{10}\mathstrut +\mathstrut \) \(85810\) \(\beta_{9}\mathstrut -\mathstrut \) \(2784362\) \(\beta_{8}\mathstrut +\mathstrut \) \(6023\) \(\beta_{7}\mathstrut +\mathstrut \) \(2417927\) \(\beta_{6}\mathstrut +\mathstrut \) \(530923\) \(\beta_{5}\mathstrut +\mathstrut \) \(2662067\) \(\beta_{4}\mathstrut +\mathstrut \) \(1331119\) \(\beta_{3}\mathstrut -\mathstrut \) \(967096\) \(\beta_{2}\mathstrut +\mathstrut \) \(2700252\) \(\beta_{1}\mathstrut +\mathstrut \) \(8568397\)
\(\nu^{13}\)\(=\)\(8899212\) \(\beta_{19}\mathstrut +\mathstrut \) \(628325\) \(\beta_{18}\mathstrut -\mathstrut \) \(4449474\) \(\beta_{17}\mathstrut -\mathstrut \) \(5739069\) \(\beta_{16}\mathstrut +\mathstrut \) \(1458467\) \(\beta_{15}\mathstrut +\mathstrut \) \(1910001\) \(\beta_{14}\mathstrut +\mathstrut \) \(6952399\) \(\beta_{13}\mathstrut -\mathstrut \) \(812059\) \(\beta_{12}\mathstrut +\mathstrut \) \(2834611\) \(\beta_{11}\mathstrut -\mathstrut \) \(2128605\) \(\beta_{10}\mathstrut -\mathstrut \) \(776096\) \(\beta_{9}\mathstrut -\mathstrut \) \(2343169\) \(\beta_{8}\mathstrut -\mathstrut \) \(2132226\) \(\beta_{7}\mathstrut +\mathstrut \) \(3491437\) \(\beta_{6}\mathstrut -\mathstrut \) \(1807561\) \(\beta_{5}\mathstrut +\mathstrut \) \(3749010\) \(\beta_{4}\mathstrut +\mathstrut \) \(6605438\) \(\beta_{3}\mathstrut -\mathstrut \) \(3648714\) \(\beta_{2}\mathstrut +\mathstrut \) \(21308839\) \(\beta_{1}\mathstrut +\mathstrut \) \(10620443\)
\(\nu^{14}\)\(=\)\(48195391\) \(\beta_{19}\mathstrut -\mathstrut \) \(8321865\) \(\beta_{18}\mathstrut +\mathstrut \) \(1205585\) \(\beta_{17}\mathstrut -\mathstrut \) \(18114234\) \(\beta_{16}\mathstrut +\mathstrut \) \(37473372\) \(\beta_{15}\mathstrut +\mathstrut \) \(16682893\) \(\beta_{14}\mathstrut +\mathstrut \) \(26044890\) \(\beta_{13}\mathstrut +\mathstrut \) \(4983224\) \(\beta_{12}\mathstrut +\mathstrut \) \(8597808\) \(\beta_{11}\mathstrut -\mathstrut \) \(29776367\) \(\beta_{10}\mathstrut +\mathstrut \) \(1843524\) \(\beta_{9}\mathstrut -\mathstrut \) \(51598682\) \(\beta_{8}\mathstrut -\mathstrut \) \(611632\) \(\beta_{7}\mathstrut +\mathstrut \) \(44199048\) \(\beta_{6}\mathstrut +\mathstrut \) \(10401402\) \(\beta_{5}\mathstrut +\mathstrut \) \(49404747\) \(\beta_{4}\mathstrut +\mathstrut \) \(25422665\) \(\beta_{3}\mathstrut -\mathstrut \) \(19563253\) \(\beta_{2}\mathstrut +\mathstrut \) \(52914848\) \(\beta_{1}\mathstrut +\mathstrut \) \(156044666\)
\(\nu^{15}\)\(=\)\(171778708\) \(\beta_{19}\mathstrut +\mathstrut \) \(5758273\) \(\beta_{18}\mathstrut -\mathstrut \) \(77561636\) \(\beta_{17}\mathstrut -\mathstrut \) \(107254757\) \(\beta_{16}\mathstrut +\mathstrut \) \(29778817\) \(\beta_{15}\mathstrut +\mathstrut \) \(41931110\) \(\beta_{14}\mathstrut +\mathstrut \) \(134681307\) \(\beta_{13}\mathstrut -\mathstrut \) \(18550584\) \(\beta_{12}\mathstrut +\mathstrut \) \(52476292\) \(\beta_{11}\mathstrut -\mathstrut \) \(46325687\) \(\beta_{10}\mathstrut -\mathstrut \) \(13541102\) \(\beta_{9}\mathstrut -\mathstrut \) \(50845393\) \(\beta_{8}\mathstrut -\mathstrut \) \(41153443\) \(\beta_{7}\mathstrut +\mathstrut \) \(71847496\) \(\beta_{6}\mathstrut -\mathstrut \) \(32247461\) \(\beta_{5}\mathstrut +\mathstrut \) \(77443118\) \(\beta_{4}\mathstrut +\mathstrut \) \(124832165\) \(\beta_{3}\mathstrut -\mathstrut \) \(72301148\) \(\beta_{2}\mathstrut +\mathstrut \) \(396664560\) \(\beta_{1}\mathstrut +\mathstrut \) \(217260447\)
\(\nu^{16}\)\(=\)\(904774360\) \(\beta_{19}\mathstrut -\mathstrut \) \(173406944\) \(\beta_{18}\mathstrut +\mathstrut \) \(34663333\) \(\beta_{17}\mathstrut -\mathstrut \) \(351199008\) \(\beta_{16}\mathstrut +\mathstrut \) \(673499668\) \(\beta_{15}\mathstrut +\mathstrut \) \(327626383\) \(\beta_{14}\mathstrut +\mathstrut \) \(486391204\) \(\beta_{13}\mathstrut +\mathstrut \) \(71994566\) \(\beta_{12}\mathstrut +\mathstrut \) \(171859421\) \(\beta_{11}\mathstrut -\mathstrut \) \(582447683\) \(\beta_{10}\mathstrut +\mathstrut \) \(37256943\) \(\beta_{9}\mathstrut -\mathstrut \) \(956209649\) \(\beta_{8}\mathstrut -\mathstrut \) \(24441369\) \(\beta_{7}\mathstrut +\mathstrut \) \(811479539\) \(\beta_{6}\mathstrut +\mathstrut \) \(201025576\) \(\beta_{5}\mathstrut +\mathstrut \) \(917860076\) \(\beta_{4}\mathstrut +\mathstrut \) \(483566362\) \(\beta_{3}\mathstrut -\mathstrut \) \(389968667\) \(\beta_{2}\mathstrut +\mathstrut \) \(1034624429\) \(\beta_{1}\mathstrut +\mathstrut \) \(2862083727\)
\(\nu^{17}\)\(=\)\(3303997969\) \(\beta_{19}\mathstrut +\mathstrut \) \(2308883\) \(\beta_{18}\mathstrut -\mathstrut \) \(1353331821\) \(\beta_{17}\mathstrut -\mathstrut \) \(2007632198\) \(\beta_{16}\mathstrut +\mathstrut \) \(598504767\) \(\beta_{15}\mathstrut +\mathstrut \) \(892641411\) \(\beta_{14}\mathstrut +\mathstrut \) \(2589462042\) \(\beta_{13}\mathstrut -\mathstrut \) \(400344782\) \(\beta_{12}\mathstrut +\mathstrut \) \(978142016\) \(\beta_{11}\mathstrut -\mathstrut \) \(988414071\) \(\beta_{10}\mathstrut -\mathstrut \) \(235763199\) \(\beta_{9}\mathstrut -\mathstrut \) \(1069398731\) \(\beta_{8}\mathstrut -\mathstrut \) \(794213472\) \(\beta_{7}\mathstrut +\mathstrut \) \(1453639668\) \(\beta_{6}\mathstrut -\mathstrut \) \(560031524\) \(\beta_{5}\mathstrut +\mathstrut \) \(1573943871\) \(\beta_{4}\mathstrut +\mathstrut \) \(2359507302\) \(\beta_{3}\mathstrut -\mathstrut \) \(1430086744\) \(\beta_{2}\mathstrut +\mathstrut \) \(7419879187\) \(\beta_{1}\mathstrut +\mathstrut \) \(4393716710\)
\(\nu^{18}\)\(=\)\(17037005368\) \(\beta_{19}\mathstrut -\mathstrut \) \(3552823207\) \(\beta_{18}\mathstrut +\mathstrut \) \(819203440\) \(\beta_{17}\mathstrut -\mathstrut \) \(6772878174\) \(\beta_{16}\mathstrut +\mathstrut \) \(12153775821\) \(\beta_{15}\mathstrut +\mathstrut \) \(6415705459\) \(\beta_{14}\mathstrut +\mathstrut \) \(9176544885\) \(\beta_{13}\mathstrut +\mathstrut \) \(989942585\) \(\beta_{12}\mathstrut +\mathstrut \) \(3398326070\) \(\beta_{11}\mathstrut -\mathstrut \) \(11294830162\) \(\beta_{10}\mathstrut +\mathstrut \) \(724613107\) \(\beta_{9}\mathstrut -\mathstrut \) \(17735149640\) \(\beta_{8}\mathstrut -\mathstrut \) \(689580002\) \(\beta_{7}\mathstrut +\mathstrut \) \(14966689555\) \(\beta_{6}\mathstrut +\mathstrut \) \(3849065276\) \(\beta_{5}\mathstrut +\mathstrut \) \(17086401612\) \(\beta_{4}\mathstrut +\mathstrut \) \(9184474226\) \(\beta_{3}\mathstrut -\mathstrut \) \(7700755730\) \(\beta_{2}\mathstrut +\mathstrut \) \(20192073469\) \(\beta_{1}\mathstrut +\mathstrut \) \(52767726574\)
\(\nu^{19}\)\(=\)\(63433494478\) \(\beta_{19}\mathstrut -\mathstrut \) \(1833217772\) \(\beta_{18}\mathstrut -\mathstrut \) \(23659744477\) \(\beta_{17}\mathstrut -\mathstrut \) \(37638208977\) \(\beta_{16}\mathstrut +\mathstrut \) \(11935554272\) \(\beta_{15}\mathstrut +\mathstrut \) \(18617520975\) \(\beta_{14}\mathstrut +\mathstrut \) \(49582810074\) \(\beta_{13}\mathstrut -\mathstrut \) \(8308544438\) \(\beta_{12}\mathstrut +\mathstrut \) \(18351519232\) \(\beta_{11}\mathstrut -\mathstrut \) \(20758462390\) \(\beta_{10}\mathstrut -\mathstrut \) \(4114394754\) \(\beta_{9}\mathstrut -\mathstrut \) \(22102259018\) \(\beta_{8}\mathstrut -\mathstrut \) \(15301595159\) \(\beta_{7}\mathstrut +\mathstrut \) \(29100109449\) \(\beta_{6}\mathstrut -\mathstrut \) \(9525524455\) \(\beta_{5}\mathstrut +\mathstrut \) \(31675591950\) \(\beta_{4}\mathstrut +\mathstrut \) \(44603406604\) \(\beta_{3}\mathstrut -\mathstrut \) \(28241505883\) \(\beta_{2}\mathstrut +\mathstrut \) \(139288065851\) \(\beta_{1}\mathstrut +\mathstrut \) \(88126279520\)

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
−4.22324
−3.69479
−3.19440
−1.98459
−1.83693
−1.23330
−1.06426
−0.553985
−0.458931
−0.129395
0.273378
1.32210
2.20464
2.35321
2.60908
2.61062
3.18186
4.06601
4.35600
4.39693
0 1.00000 0 −4.22324 0 1.38831 0 1.00000 0
1.2 0 1.00000 0 −3.69479 0 −1.23254 0 1.00000 0
1.3 0 1.00000 0 −3.19440 0 5.21224 0 1.00000 0
1.4 0 1.00000 0 −1.98459 0 −4.60442 0 1.00000 0
1.5 0 1.00000 0 −1.83693 0 2.82191 0 1.00000 0
1.6 0 1.00000 0 −1.23330 0 3.99613 0 1.00000 0
1.7 0 1.00000 0 −1.06426 0 1.22980 0 1.00000 0
1.8 0 1.00000 0 −0.553985 0 −2.25153 0 1.00000 0
1.9 0 1.00000 0 −0.458931 0 −2.87812 0 1.00000 0
1.10 0 1.00000 0 −0.129395 0 2.39454 0 1.00000 0
1.11 0 1.00000 0 0.273378 0 −2.92800 0 1.00000 0
1.12 0 1.00000 0 1.32210 0 −1.93279 0 1.00000 0
1.13 0 1.00000 0 2.20464 0 −1.71268 0 1.00000 0
1.14 0 1.00000 0 2.35321 0 2.49191 0 1.00000 0
1.15 0 1.00000 0 2.60908 0 3.89020 0 1.00000 0
1.16 0 1.00000 0 2.61062 0 4.62670 0 1.00000 0
1.17 0 1.00000 0 3.18186 0 2.54652 0 1.00000 0
1.18 0 1.00000 0 4.06601 0 1.32234 0 1.00000 0
1.19 0 1.00000 0 4.35600 0 −4.92117 0 1.00000 0
1.20 0 1.00000 0 4.39693 0 −0.459336 0 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.20
Significant digits:
Format:

Inner twists

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

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(251\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(6024))\):

\(T_{5}^{20} - \cdots\)
\(T_{7}^{20} - \cdots\)