Properties

Label 828.2.q.c
Level $828$
Weight $2$
Character orbit 828.q
Analytic conductor $6.612$
Analytic rank $0$
Dimension $20$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 828 = 2^{2} \cdot 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 828.q (of order \(11\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(6.61161328736\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(2\) over \(\Q(\zeta_{11})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 8 x^{19} + 43 x^{18} - 165 x^{17} + 538 x^{16} - 1433 x^{15} + 3444 x^{14} - 7370 x^{13} + 15500 x^{12} - 28190 x^{11} + 41920 x^{10} - 33520 x^{9} - 13837 x^{8} + 78980 x^{7} - 92652 x^{6} - 52852 x^{5} + 177374 x^{4} + 151360 x^{3} + 115323 x^{2} + 12834 x + 529\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 276)
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( 1 - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} - \beta_{18} ) q^{5} + ( -1 + \beta_{7} + \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} + \beta_{18} ) q^{7} +O(q^{10})\) \( q + ( 1 - \beta_{8} - \beta_{9} - \beta_{10} + \beta_{11} + \beta_{12} + \beta_{13} - \beta_{14} + \beta_{15} - \beta_{16} - \beta_{18} ) q^{5} + ( -1 + \beta_{7} + \beta_{9} + \beta_{10} - 2 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} - \beta_{15} + \beta_{16} - \beta_{17} + \beta_{18} ) q^{7} + ( -1 + \beta_{1} - \beta_{8} - \beta_{11} - \beta_{12} + \beta_{13} + \beta_{14} - \beta_{15} - 2 \beta_{17} ) q^{11} + ( -2 + \beta_{1} + \beta_{4} + \beta_{7} + \beta_{10} - \beta_{11} - \beta_{12} - \beta_{13} - 2 \beta_{17} + \beta_{19} ) q^{13} + ( 2 + \beta_{2} - \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - 4 \beta_{10} + 4 \beta_{11} + 2 \beta_{12} - 2 \beta_{14} + 2 \beta_{15} - \beta_{16} + 3 \beta_{17} - 3 \beta_{18} - \beta_{19} ) q^{17} + ( 3 - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} + 3 \beta_{11} + 2 \beta_{12} - \beta_{13} - \beta_{14} - 2 \beta_{16} + \beta_{17} - \beta_{18} ) q^{19} + ( \beta_{1} - \beta_{2} - \beta_{6} - \beta_{7} + 2 \beta_{11} - \beta_{13} + \beta_{15} - \beta_{18} + \beta_{19} ) q^{23} + ( 2 + \beta_{1} + \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + 3 \beta_{11} - \beta_{12} + \beta_{14} + \beta_{15} - \beta_{16} - 3 \beta_{18} ) q^{25} + ( -2 + \beta_{4} - \beta_{5} + \beta_{7} + \beta_{10} - 2 \beta_{12} - \beta_{13} + \beta_{15} - \beta_{18} ) q^{29} + ( -1 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + 2 \beta_{5} - \beta_{6} - 2 \beta_{8} - \beta_{9} + 2 \beta_{13} - \beta_{14} - \beta_{15} + \beta_{16} - \beta_{19} ) q^{31} + ( -2 - \beta_{1} + 2 \beta_{6} + 2 \beta_{7} + \beta_{8} + 2 \beta_{9} + 3 \beta_{10} - 6 \beta_{11} - \beta_{12} - \beta_{13} + 2 \beta_{14} - 6 \beta_{15} + 5 \beta_{16} - \beta_{17} + 5 \beta_{18} ) q^{35} + ( -\beta_{1} - 3 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 2 \beta_{5} + \beta_{6} + 3 \beta_{7} - \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{12} - \beta_{13} + 2 \beta_{14} + 2 \beta_{15} - \beta_{16} + \beta_{17} - \beta_{18} + \beta_{19} ) q^{37} + ( 3 + 2 \beta_{1} + \beta_{2} - \beta_{3} + 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - \beta_{10} + 2 \beta_{11} + 3 \beta_{12} + \beta_{13} + 3 \beta_{15} - 2 \beta_{16} - 4 \beta_{18} - \beta_{19} ) q^{41} + ( -1 + \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} - \beta_{7} - \beta_{8} - \beta_{9} + 2 \beta_{10} - 2 \beta_{12} - \beta_{15} + 5 \beta_{16} + \beta_{17} - \beta_{19} ) q^{43} + ( 1 - 2 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{6} + \beta_{8} + \beta_{9} + \beta_{10} - \beta_{11} - \beta_{13} + 4 \beta_{14} - 3 \beta_{15} + \beta_{16} + \beta_{18} + \beta_{19} ) q^{47} + ( 2 + \beta_{1} - \beta_{3} - \beta_{6} - 3 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - 5 \beta_{10} + 6 \beta_{11} + 4 \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{15} - 5 \beta_{16} + \beta_{17} - 6 \beta_{18} - \beta_{19} ) q^{49} + ( 1 + 2 \beta_{2} + \beta_{5} - \beta_{6} + \beta_{7} - \beta_{9} + \beta_{10} - 2 \beta_{12} - \beta_{13} - \beta_{14} - \beta_{16} + 2 \beta_{18} ) q^{53} + ( -\beta_{2} - 2 \beta_{5} + \beta_{7} + \beta_{9} - 4 \beta_{10} + \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} + \beta_{16} - \beta_{17} - 4 \beta_{18} + \beta_{19} ) q^{55} + ( -2 + \beta_{1} - 2 \beta_{2} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{7} + 2 \beta_{8} + 5 \beta_{9} + 5 \beta_{10} - 2 \beta_{11} - 7 \beta_{12} - \beta_{13} + 4 \beta_{14} - 4 \beta_{15} + 5 \beta_{16} - 2 \beta_{17} + 6 \beta_{18} + 2 \beta_{19} ) q^{59} + ( 2 \beta_{1} + \beta_{2} - \beta_{6} - 2 \beta_{8} - 2 \beta_{9} + 2 \beta_{10} + \beta_{11} + \beta_{13} - \beta_{14} + 3 \beta_{16} - 6 \beta_{17} ) q^{61} + ( -3 - \beta_{1} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} + 2 \beta_{7} + 2 \beta_{8} - 3 \beta_{10} - 4 \beta_{11} - 4 \beta_{13} - 3 \beta_{14} + \beta_{16} - \beta_{17} + 2 \beta_{18} + \beta_{19} ) q^{65} + ( 2 - \beta_{1} - \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} - 2 \beta_{9} - \beta_{10} - \beta_{15} + 2 \beta_{16} + 4 \beta_{18} + \beta_{19} ) q^{67} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} - 3 \beta_{4} - 3 \beta_{6} - 2 \beta_{7} - 3 \beta_{8} - 2 \beta_{9} - 2 \beta_{10} + 2 \beta_{11} + 5 \beta_{12} - 5 \beta_{14} + 5 \beta_{15} - 5 \beta_{16} - 7 \beta_{18} ) q^{71} + ( -1 + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} + 4 \beta_{9} - 2 \beta_{10} - \beta_{11} + 3 \beta_{13} - 2 \beta_{14} - 2 \beta_{17} + 2 \beta_{18} ) q^{73} + ( -1 - \beta_{3} + 2 \beta_{6} + 6 \beta_{9} + 4 \beta_{10} - 5 \beta_{11} + \beta_{14} - 5 \beta_{15} + 3 \beta_{16} - 4 \beta_{17} + 4 \beta_{18} - \beta_{19} ) q^{77} + ( 2 + 2 \beta_{1} + 2 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{8} - 6 \beta_{9} - 2 \beta_{10} - \beta_{12} + 2 \beta_{13} - 4 \beta_{14} - 2 \beta_{16} + 2 \beta_{17} - 4 \beta_{18} ) q^{79} + ( -\beta_{2} - \beta_{5} + \beta_{7} - \beta_{9} + 2 \beta_{10} - 4 \beta_{12} + 2 \beta_{15} - \beta_{16} - 4 \beta_{17} + 2 \beta_{18} ) q^{83} + ( -3 \beta_{1} - 2 \beta_{2} + 3 \beta_{3} + 3 \beta_{4} + \beta_{7} + 3 \beta_{8} + \beta_{9} + \beta_{10} - 3 \beta_{11} - \beta_{12} - \beta_{13} + \beta_{14} + 3 \beta_{15} + 3 \beta_{17} + \beta_{18} ) q^{85} + ( 1 + 2 \beta_{2} - 2 \beta_{4} - 2 \beta_{6} - \beta_{7} - \beta_{8} - \beta_{9} - 2 \beta_{10} + 6 \beta_{11} + 2 \beta_{13} - 2 \beta_{14} + \beta_{15} - 3 \beta_{16} - \beta_{17} - 6 \beta_{18} - 2 \beta_{19} ) q^{89} + ( \beta_{1} + \beta_{2} - 3 \beta_{3} - 2 \beta_{4} + 4 \beta_{5} - \beta_{6} - 4 \beta_{7} - 2 \beta_{8} + \beta_{9} + \beta_{10} + 7 \beta_{11} + 6 \beta_{12} + 3 \beta_{13} - \beta_{14} - 2 \beta_{15} - 8 \beta_{16} + 7 \beta_{17} - \beta_{18} - \beta_{19} ) q^{91} + ( 8 - \beta_{1} - 3 \beta_{2} + \beta_{3} + 3 \beta_{4} - \beta_{7} - \beta_{8} - 3 \beta_{9} - 5 \beta_{10} + 3 \beta_{11} + 5 \beta_{12} + 8 \beta_{15} - 10 \beta_{16} + 3 \beta_{17} - 3 \beta_{18} + \beta_{19} ) q^{95} + ( -3 + \beta_{1} + 2 \beta_{2} - \beta_{3} + \beta_{5} - \beta_{6} - \beta_{7} + 4 \beta_{10} - 2 \beta_{11} + 6 \beta_{14} + 2 \beta_{16} - 4 \beta_{17} + \beta_{18} - 2 \beta_{19} ) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{5} + O(q^{10}) \) \( 20q + 4q^{5} - 22q^{13} - 7q^{17} + 19q^{19} - 20q^{23} + 20q^{25} - 32q^{29} - 3q^{31} + 26q^{35} - 10q^{37} + 40q^{41} + 8q^{43} + 18q^{47} - 34q^{49} + 34q^{53} - 17q^{55} + 32q^{59} + 32q^{61} - 49q^{65} + 35q^{67} - 33q^{71} - q^{73} + 50q^{77} + 22q^{79} + 14q^{83} - 9q^{85} - 10q^{89} - 72q^{91} + 51q^{95} - 4q^{97} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{20} - 8 x^{19} + 43 x^{18} - 165 x^{17} + 538 x^{16} - 1433 x^{15} + 3444 x^{14} - 7370 x^{13} + 15500 x^{12} - 28190 x^{11} + 41920 x^{10} - 33520 x^{9} - 13837 x^{8} + 78980 x^{7} - 92652 x^{6} - 52852 x^{5} + 177374 x^{4} + 151360 x^{3} + 115323 x^{2} + 12834 x + 529\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\(-\)\(17\!\cdots\!28\)\( \nu^{19} + \)\(15\!\cdots\!67\)\( \nu^{18} - \)\(86\!\cdots\!95\)\( \nu^{17} + \)\(35\!\cdots\!61\)\( \nu^{16} - \)\(11\!\cdots\!03\)\( \nu^{15} + \)\(33\!\cdots\!61\)\( \nu^{14} - \)\(85\!\cdots\!69\)\( \nu^{13} + \)\(19\!\cdots\!30\)\( \nu^{12} - \)\(40\!\cdots\!61\)\( \nu^{11} + \)\(78\!\cdots\!34\)\( \nu^{10} - \)\(13\!\cdots\!04\)\( \nu^{9} + \)\(15\!\cdots\!49\)\( \nu^{8} - \)\(82\!\cdots\!06\)\( \nu^{7} - \)\(93\!\cdots\!48\)\( \nu^{6} + \)\(24\!\cdots\!11\)\( \nu^{5} - \)\(97\!\cdots\!60\)\( \nu^{4} - \)\(23\!\cdots\!36\)\( \nu^{3} - \)\(42\!\cdots\!06\)\( \nu^{2} - \)\(19\!\cdots\!98\)\( \nu - \)\(23\!\cdots\!38\)\(\)\()/ \)\(18\!\cdots\!47\)\( \)
\(\beta_{3}\)\(=\)\((\)\(\)\(28\!\cdots\!78\)\( \nu^{19} - \)\(23\!\cdots\!79\)\( \nu^{18} + \)\(13\!\cdots\!20\)\( \nu^{17} - \)\(51\!\cdots\!26\)\( \nu^{16} + \)\(17\!\cdots\!57\)\( \nu^{15} - \)\(46\!\cdots\!20\)\( \nu^{14} + \)\(11\!\cdots\!04\)\( \nu^{13} - \)\(24\!\cdots\!58\)\( \nu^{12} + \)\(51\!\cdots\!87\)\( \nu^{11} - \)\(96\!\cdots\!60\)\( \nu^{10} + \)\(14\!\cdots\!28\)\( \nu^{9} - \)\(14\!\cdots\!09\)\( \nu^{8} + \)\(26\!\cdots\!43\)\( \nu^{7} + \)\(22\!\cdots\!11\)\( \nu^{6} - \)\(30\!\cdots\!58\)\( \nu^{5} - \)\(12\!\cdots\!44\)\( \nu^{4} + \)\(64\!\cdots\!97\)\( \nu^{3} + \)\(24\!\cdots\!48\)\( \nu^{2} + \)\(12\!\cdots\!27\)\( \nu - \)\(62\!\cdots\!66\)\(\)\()/ \)\(18\!\cdots\!47\)\( \)
\(\beta_{4}\)\(=\)\((\)\(\)\(32\!\cdots\!78\)\( \nu^{19} - \)\(25\!\cdots\!20\)\( \nu^{18} + \)\(13\!\cdots\!66\)\( \nu^{17} - \)\(51\!\cdots\!92\)\( \nu^{16} + \)\(16\!\cdots\!38\)\( \nu^{15} - \)\(44\!\cdots\!86\)\( \nu^{14} + \)\(10\!\cdots\!88\)\( \nu^{13} - \)\(22\!\cdots\!04\)\( \nu^{12} + \)\(48\!\cdots\!38\)\( \nu^{11} - \)\(88\!\cdots\!42\)\( \nu^{10} + \)\(13\!\cdots\!14\)\( \nu^{9} - \)\(11\!\cdots\!75\)\( \nu^{8} - \)\(14\!\cdots\!94\)\( \nu^{7} + \)\(17\!\cdots\!78\)\( \nu^{6} - \)\(17\!\cdots\!39\)\( \nu^{5} - \)\(24\!\cdots\!28\)\( \nu^{4} + \)\(48\!\cdots\!43\)\( \nu^{3} + \)\(67\!\cdots\!90\)\( \nu^{2} + \)\(42\!\cdots\!45\)\( \nu + \)\(22\!\cdots\!33\)\(\)\()/ \)\(18\!\cdots\!47\)\( \)
\(\beta_{5}\)\(=\)\((\)\(-\)\(39\!\cdots\!69\)\( \nu^{19} + \)\(30\!\cdots\!46\)\( \nu^{18} - \)\(16\!\cdots\!59\)\( \nu^{17} + \)\(61\!\cdots\!59\)\( \nu^{16} - \)\(19\!\cdots\!92\)\( \nu^{15} + \)\(51\!\cdots\!83\)\( \nu^{14} - \)\(12\!\cdots\!75\)\( \nu^{13} + \)\(25\!\cdots\!26\)\( \nu^{12} - \)\(52\!\cdots\!23\)\( \nu^{11} + \)\(92\!\cdots\!49\)\( \nu^{10} - \)\(12\!\cdots\!95\)\( \nu^{9} + \)\(68\!\cdots\!61\)\( \nu^{8} + \)\(14\!\cdots\!67\)\( \nu^{7} - \)\(37\!\cdots\!12\)\( \nu^{6} + \)\(36\!\cdots\!58\)\( \nu^{5} + \)\(30\!\cdots\!36\)\( \nu^{4} - \)\(75\!\cdots\!88\)\( \nu^{3} - \)\(69\!\cdots\!01\)\( \nu^{2} - \)\(33\!\cdots\!76\)\( \nu - \)\(10\!\cdots\!84\)\(\)\()/ \)\(18\!\cdots\!47\)\( \)
\(\beta_{6}\)\(=\)\((\)\(-\)\(39\!\cdots\!27\)\( \nu^{19} + \)\(33\!\cdots\!00\)\( \nu^{18} - \)\(18\!\cdots\!73\)\( \nu^{17} + \)\(72\!\cdots\!73\)\( \nu^{16} - \)\(24\!\cdots\!57\)\( \nu^{15} + \)\(66\!\cdots\!76\)\( \nu^{14} - \)\(16\!\cdots\!69\)\( \nu^{13} + \)\(35\!\cdots\!21\)\( \nu^{12} - \)\(75\!\cdots\!05\)\( \nu^{11} + \)\(14\!\cdots\!71\)\( \nu^{10} - \)\(22\!\cdots\!60\)\( \nu^{9} + \)\(21\!\cdots\!33\)\( \nu^{8} - \)\(28\!\cdots\!24\)\( \nu^{7} - \)\(31\!\cdots\!92\)\( \nu^{6} + \)\(52\!\cdots\!07\)\( \nu^{5} - \)\(35\!\cdots\!45\)\( \nu^{4} - \)\(64\!\cdots\!38\)\( \nu^{3} - \)\(27\!\cdots\!02\)\( \nu^{2} - \)\(34\!\cdots\!72\)\( \nu - \)\(14\!\cdots\!85\)\(\)\()/ \)\(18\!\cdots\!47\)\( \)
\(\beta_{7}\)\(=\)\((\)\(-\)\(44\!\cdots\!45\)\( \nu^{19} + \)\(35\!\cdots\!52\)\( \nu^{18} - \)\(18\!\cdots\!95\)\( \nu^{17} + \)\(72\!\cdots\!61\)\( \nu^{16} - \)\(23\!\cdots\!41\)\( \nu^{15} + \)\(61\!\cdots\!97\)\( \nu^{14} - \)\(14\!\cdots\!04\)\( \nu^{13} + \)\(31\!\cdots\!12\)\( \nu^{12} - \)\(65\!\cdots\!64\)\( \nu^{11} + \)\(11\!\cdots\!52\)\( \nu^{10} - \)\(17\!\cdots\!95\)\( \nu^{9} + \)\(12\!\cdots\!38\)\( \nu^{8} + \)\(10\!\cdots\!13\)\( \nu^{7} - \)\(39\!\cdots\!56\)\( \nu^{6} + \)\(43\!\cdots\!91\)\( \nu^{5} + \)\(26\!\cdots\!06\)\( \nu^{4} - \)\(81\!\cdots\!56\)\( \nu^{3} - \)\(71\!\cdots\!18\)\( \nu^{2} - \)\(34\!\cdots\!58\)\( \nu - \)\(28\!\cdots\!56\)\(\)\()/ \)\(18\!\cdots\!47\)\( \)
\(\beta_{8}\)\(=\)\((\)\(-\)\(48\!\cdots\!89\)\( \nu^{19} + \)\(39\!\cdots\!13\)\( \nu^{18} - \)\(21\!\cdots\!50\)\( \nu^{17} + \)\(82\!\cdots\!99\)\( \nu^{16} - \)\(27\!\cdots\!01\)\( \nu^{15} + \)\(72\!\cdots\!06\)\( \nu^{14} - \)\(17\!\cdots\!74\)\( \nu^{13} + \)\(37\!\cdots\!61\)\( \nu^{12} - \)\(79\!\cdots\!48\)\( \nu^{11} + \)\(14\!\cdots\!05\)\( \nu^{10} - \)\(21\!\cdots\!17\)\( \nu^{9} + \)\(18\!\cdots\!34\)\( \nu^{8} + \)\(59\!\cdots\!66\)\( \nu^{7} - \)\(41\!\cdots\!73\)\( \nu^{6} + \)\(50\!\cdots\!13\)\( \nu^{5} + \)\(23\!\cdots\!87\)\( \nu^{4} - \)\(10\!\cdots\!40\)\( \nu^{3} - \)\(52\!\cdots\!60\)\( \nu^{2} - \)\(39\!\cdots\!32\)\( \nu - \)\(25\!\cdots\!91\)\(\)\()/ \)\(18\!\cdots\!47\)\( \)
\(\beta_{9}\)\(=\)\((\)\(\)\(11\!\cdots\!54\)\( \nu^{19} - \)\(91\!\cdots\!54\)\( \nu^{18} + \)\(48\!\cdots\!43\)\( \nu^{17} - \)\(18\!\cdots\!90\)\( \nu^{16} + \)\(58\!\cdots\!26\)\( \nu^{15} - \)\(15\!\cdots\!25\)\( \nu^{14} + \)\(36\!\cdots\!56\)\( \nu^{13} - \)\(75\!\cdots\!76\)\( \nu^{12} + \)\(15\!\cdots\!42\)\( \nu^{11} - \)\(28\!\cdots\!73\)\( \nu^{10} + \)\(39\!\cdots\!20\)\( \nu^{9} - \)\(24\!\cdots\!52\)\( \nu^{8} - \)\(30\!\cdots\!07\)\( \nu^{7} + \)\(93\!\cdots\!63\)\( \nu^{6} - \)\(87\!\cdots\!97\)\( \nu^{5} - \)\(92\!\cdots\!66\)\( \nu^{4} + \)\(19\!\cdots\!52\)\( \nu^{3} + \)\(24\!\cdots\!37\)\( \nu^{2} + \)\(16\!\cdots\!90\)\( \nu + \)\(27\!\cdots\!63\)\(\)\()/ \)\(18\!\cdots\!47\)\( \)
\(\beta_{10}\)\(=\)\((\)\(-\)\(19\!\cdots\!96\)\( \nu^{19} + \)\(16\!\cdots\!37\)\( \nu^{18} - \)\(87\!\cdots\!74\)\( \nu^{17} + \)\(33\!\cdots\!99\)\( \nu^{16} - \)\(11\!\cdots\!07\)\( \nu^{15} + \)\(30\!\cdots\!60\)\( \nu^{14} - \)\(72\!\cdots\!07\)\( \nu^{13} + \)\(15\!\cdots\!95\)\( \nu^{12} - \)\(32\!\cdots\!26\)\( \nu^{11} + \)\(60\!\cdots\!63\)\( \nu^{10} - \)\(91\!\cdots\!69\)\( \nu^{9} + \)\(78\!\cdots\!15\)\( \nu^{8} + \)\(20\!\cdots\!91\)\( \nu^{7} - \)\(16\!\cdots\!47\)\( \nu^{6} + \)\(21\!\cdots\!04\)\( \nu^{5} + \)\(67\!\cdots\!34\)\( \nu^{4} - \)\(37\!\cdots\!40\)\( \nu^{3} - \)\(22\!\cdots\!72\)\( \nu^{2} - \)\(15\!\cdots\!07\)\( \nu + \)\(88\!\cdots\!12\)\(\)\()/ \)\(18\!\cdots\!47\)\( \)
\(\beta_{11}\)\(=\)\((\)\(\)\(24\!\cdots\!93\)\( \nu^{19} - \)\(19\!\cdots\!65\)\( \nu^{18} + \)\(10\!\cdots\!97\)\( \nu^{17} - \)\(40\!\cdots\!91\)\( \nu^{16} + \)\(13\!\cdots\!17\)\( \nu^{15} - \)\(35\!\cdots\!08\)\( \nu^{14} + \)\(85\!\cdots\!30\)\( \nu^{13} - \)\(18\!\cdots\!44\)\( \nu^{12} + \)\(38\!\cdots\!11\)\( \nu^{11} - \)\(70\!\cdots\!17\)\( \nu^{10} + \)\(10\!\cdots\!09\)\( \nu^{9} - \)\(84\!\cdots\!68\)\( \nu^{8} - \)\(33\!\cdots\!86\)\( \nu^{7} + \)\(19\!\cdots\!44\)\( \nu^{6} - \)\(23\!\cdots\!26\)\( \nu^{5} - \)\(11\!\cdots\!44\)\( \nu^{4} + \)\(43\!\cdots\!82\)\( \nu^{3} + \)\(34\!\cdots\!24\)\( \nu^{2} + \)\(29\!\cdots\!43\)\( \nu + \)\(33\!\cdots\!91\)\(\)\()/ \)\(18\!\cdots\!47\)\( \)
\(\beta_{12}\)\(=\)\((\)\(-\)\(27\!\cdots\!65\)\( \nu^{19} + \)\(22\!\cdots\!47\)\( \nu^{18} - \)\(12\!\cdots\!95\)\( \nu^{17} + \)\(47\!\cdots\!98\)\( \nu^{16} - \)\(15\!\cdots\!43\)\( \nu^{15} + \)\(42\!\cdots\!02\)\( \nu^{14} - \)\(10\!\cdots\!36\)\( \nu^{13} + \)\(22\!\cdots\!19\)\( \nu^{12} - \)\(46\!\cdots\!21\)\( \nu^{11} + \)\(86\!\cdots\!55\)\( \nu^{10} - \)\(13\!\cdots\!71\)\( \nu^{9} + \)\(11\!\cdots\!60\)\( \nu^{8} + \)\(16\!\cdots\!72\)\( \nu^{7} - \)\(21\!\cdots\!76\)\( \nu^{6} + \)\(29\!\cdots\!72\)\( \nu^{5} + \)\(95\!\cdots\!73\)\( \nu^{4} - \)\(49\!\cdots\!65\)\( \nu^{3} - \)\(35\!\cdots\!62\)\( \nu^{2} - \)\(29\!\cdots\!93\)\( \nu - \)\(15\!\cdots\!38\)\(\)\()/ \)\(18\!\cdots\!47\)\( \)
\(\beta_{13}\)\(=\)\((\)\(\)\(32\!\cdots\!22\)\( \nu^{19} - \)\(24\!\cdots\!27\)\( \nu^{18} + \)\(13\!\cdots\!07\)\( \nu^{17} - \)\(49\!\cdots\!24\)\( \nu^{16} + \)\(15\!\cdots\!92\)\( \nu^{15} - \)\(41\!\cdots\!17\)\( \nu^{14} + \)\(98\!\cdots\!78\)\( \nu^{13} - \)\(20\!\cdots\!68\)\( \nu^{12} + \)\(43\!\cdots\!49\)\( \nu^{11} - \)\(78\!\cdots\!11\)\( \nu^{10} + \)\(11\!\cdots\!50\)\( \nu^{9} - \)\(73\!\cdots\!27\)\( \nu^{8} - \)\(70\!\cdots\!23\)\( \nu^{7} + \)\(23\!\cdots\!01\)\( \nu^{6} - \)\(21\!\cdots\!32\)\( \nu^{5} - \)\(26\!\cdots\!49\)\( \nu^{4} + \)\(54\!\cdots\!24\)\( \nu^{3} + \)\(64\!\cdots\!16\)\( \nu^{2} + \)\(44\!\cdots\!56\)\( \nu + \)\(75\!\cdots\!61\)\(\)\()/ \)\(18\!\cdots\!47\)\( \)
\(\beta_{14}\)\(=\)\((\)\(\)\(39\!\cdots\!53\)\( \nu^{19} - \)\(31\!\cdots\!35\)\( \nu^{18} + \)\(17\!\cdots\!54\)\( \nu^{17} - \)\(65\!\cdots\!29\)\( \nu^{16} + \)\(21\!\cdots\!18\)\( \nu^{15} - \)\(56\!\cdots\!04\)\( \nu^{14} + \)\(13\!\cdots\!80\)\( \nu^{13} - \)\(29\!\cdots\!59\)\( \nu^{12} + \)\(61\!\cdots\!28\)\( \nu^{11} - \)\(11\!\cdots\!75\)\( \nu^{10} + \)\(16\!\cdots\!86\)\( \nu^{9} - \)\(13\!\cdots\!73\)\( \nu^{8} - \)\(57\!\cdots\!44\)\( \nu^{7} + \)\(31\!\cdots\!03\)\( \nu^{6} - \)\(36\!\cdots\!38\)\( \nu^{5} - \)\(22\!\cdots\!55\)\( \nu^{4} + \)\(72\!\cdots\!84\)\( \nu^{3} + \)\(59\!\cdots\!52\)\( \nu^{2} + \)\(42\!\cdots\!20\)\( \nu + \)\(36\!\cdots\!49\)\(\)\()/ \)\(18\!\cdots\!47\)\( \)
\(\beta_{15}\)\(=\)\((\)\(\)\(42\!\cdots\!77\)\( \nu^{19} - \)\(34\!\cdots\!94\)\( \nu^{18} + \)\(18\!\cdots\!31\)\( \nu^{17} - \)\(71\!\cdots\!71\)\( \nu^{16} + \)\(23\!\cdots\!18\)\( \nu^{15} - \)\(62\!\cdots\!79\)\( \nu^{14} + \)\(15\!\cdots\!74\)\( \nu^{13} - \)\(32\!\cdots\!78\)\( \nu^{12} + \)\(68\!\cdots\!04\)\( \nu^{11} - \)\(12\!\cdots\!68\)\( \nu^{10} + \)\(18\!\cdots\!82\)\( \nu^{9} - \)\(15\!\cdots\!54\)\( \nu^{8} - \)\(47\!\cdots\!74\)\( \nu^{7} + \)\(33\!\cdots\!54\)\( \nu^{6} - \)\(41\!\cdots\!82\)\( \nu^{5} - \)\(20\!\cdots\!65\)\( \nu^{4} + \)\(77\!\cdots\!26\)\( \nu^{3} + \)\(59\!\cdots\!77\)\( \nu^{2} + \)\(42\!\cdots\!81\)\( \nu + \)\(12\!\cdots\!73\)\(\)\()/ \)\(18\!\cdots\!47\)\( \)
\(\beta_{16}\)\(=\)\((\)\(-\)\(43\!\cdots\!22\)\( \nu^{19} + \)\(35\!\cdots\!04\)\( \nu^{18} - \)\(18\!\cdots\!13\)\( \nu^{17} + \)\(73\!\cdots\!25\)\( \nu^{16} - \)\(23\!\cdots\!97\)\( \nu^{15} + \)\(63\!\cdots\!29\)\( \nu^{14} - \)\(15\!\cdots\!29\)\( \nu^{13} + \)\(33\!\cdots\!09\)\( \nu^{12} - \)\(69\!\cdots\!30\)\( \nu^{11} + \)\(12\!\cdots\!41\)\( \nu^{10} - \)\(19\!\cdots\!74\)\( \nu^{9} + \)\(15\!\cdots\!44\)\( \nu^{8} + \)\(45\!\cdots\!65\)\( \nu^{7} - \)\(33\!\cdots\!54\)\( \nu^{6} + \)\(41\!\cdots\!92\)\( \nu^{5} + \)\(20\!\cdots\!33\)\( \nu^{4} - \)\(76\!\cdots\!68\)\( \nu^{3} - \)\(63\!\cdots\!84\)\( \nu^{2} - \)\(50\!\cdots\!00\)\( \nu - \)\(37\!\cdots\!50\)\(\)\()/ \)\(18\!\cdots\!47\)\( \)
\(\beta_{17}\)\(=\)\((\)\(-\)\(47\!\cdots\!79\)\( \nu^{19} + \)\(38\!\cdots\!21\)\( \nu^{18} - \)\(20\!\cdots\!10\)\( \nu^{17} + \)\(80\!\cdots\!85\)\( \nu^{16} - \)\(26\!\cdots\!01\)\( \nu^{15} + \)\(70\!\cdots\!08\)\( \nu^{14} - \)\(17\!\cdots\!82\)\( \nu^{13} + \)\(36\!\cdots\!04\)\( \nu^{12} - \)\(77\!\cdots\!61\)\( \nu^{11} + \)\(14\!\cdots\!58\)\( \nu^{10} - \)\(21\!\cdots\!85\)\( \nu^{9} + \)\(18\!\cdots\!97\)\( \nu^{8} + \)\(47\!\cdots\!89\)\( \nu^{7} - \)\(38\!\cdots\!86\)\( \nu^{6} + \)\(48\!\cdots\!81\)\( \nu^{5} + \)\(20\!\cdots\!95\)\( \nu^{4} - \)\(86\!\cdots\!33\)\( \nu^{3} - \)\(61\!\cdots\!00\)\( \nu^{2} - \)\(49\!\cdots\!57\)\( \nu - \)\(21\!\cdots\!54\)\(\)\()/ \)\(18\!\cdots\!47\)\( \)
\(\beta_{18}\)\(=\)\((\)\(-\)\(55\!\cdots\!03\)\( \nu^{19} + \)\(44\!\cdots\!46\)\( \nu^{18} - \)\(24\!\cdots\!99\)\( \nu^{17} + \)\(92\!\cdots\!19\)\( \nu^{16} - \)\(30\!\cdots\!38\)\( \nu^{15} + \)\(80\!\cdots\!24\)\( \nu^{14} - \)\(19\!\cdots\!03\)\( \nu^{13} + \)\(41\!\cdots\!24\)\( \nu^{12} - \)\(87\!\cdots\!77\)\( \nu^{11} + \)\(15\!\cdots\!44\)\( \nu^{10} - \)\(23\!\cdots\!40\)\( \nu^{9} + \)\(19\!\cdots\!64\)\( \nu^{8} + \)\(75\!\cdots\!14\)\( \nu^{7} - \)\(44\!\cdots\!18\)\( \nu^{6} + \)\(52\!\cdots\!52\)\( \nu^{5} + \)\(28\!\cdots\!50\)\( \nu^{4} - \)\(99\!\cdots\!40\)\( \nu^{3} - \)\(82\!\cdots\!42\)\( \nu^{2} - \)\(61\!\cdots\!87\)\( \nu - \)\(32\!\cdots\!97\)\(\)\()/ \)\(20\!\cdots\!83\)\( \)
\(\beta_{19}\)\(=\)\((\)\(\)\(39\!\cdots\!80\)\( \nu^{19} - \)\(32\!\cdots\!71\)\( \nu^{18} + \)\(17\!\cdots\!39\)\( \nu^{17} - \)\(66\!\cdots\!01\)\( \nu^{16} + \)\(21\!\cdots\!27\)\( \nu^{15} - \)\(57\!\cdots\!64\)\( \nu^{14} + \)\(13\!\cdots\!04\)\( \nu^{13} - \)\(29\!\cdots\!54\)\( \nu^{12} + \)\(62\!\cdots\!00\)\( \nu^{11} - \)\(11\!\cdots\!58\)\( \nu^{10} + \)\(17\!\cdots\!93\)\( \nu^{9} - \)\(13\!\cdots\!87\)\( \nu^{8} - \)\(50\!\cdots\!93\)\( \nu^{7} + \)\(31\!\cdots\!08\)\( \nu^{6} - \)\(38\!\cdots\!80\)\( \nu^{5} - \)\(19\!\cdots\!63\)\( \nu^{4} + \)\(70\!\cdots\!84\)\( \nu^{3} + \)\(58\!\cdots\!08\)\( \nu^{2} + \)\(45\!\cdots\!57\)\( \nu + \)\(34\!\cdots\!99\)\(\)\()/ \)\(62\!\cdots\!49\)\( \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(-\beta_{18} + 4 \beta_{17} - \beta_{16} + \beta_{15} + 2 \beta_{11} - \beta_{10} - 2 \beta_{9} - \beta_{8} - \beta_{6} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(\beta_{19} + 2 \beta_{18} + 2 \beta_{16} + \beta_{14} - \beta_{13} - 2 \beta_{12} - \beta_{11} + 2 \beta_{10} + 3 \beta_{9} - 5 \beta_{8} - \beta_{5} + \beta_{4} - \beta_{2} - \beta_{1} - 1\)
\(\nu^{4}\)\(=\)\(2 \beta_{19} - 3 \beta_{18} - 2 \beta_{17} - 3 \beta_{16} + 13 \beta_{15} - 29 \beta_{14} + 6 \beta_{13} + 13 \beta_{12} + 3 \beta_{11} - 4 \beta_{10} - 6 \beta_{9} - 6 \beta_{8} - 7 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{3} - 2 \beta_{2} - \beta_{1} + 5\)
\(\nu^{5}\)\(=\)\(-\beta_{19} - 11 \beta_{18} + 10 \beta_{17} - 24 \beta_{16} - 58 \beta_{14} + 41 \beta_{13} + 23 \beta_{12} + 7 \beta_{11} - 58 \beta_{10} - 61 \beta_{9} - \beta_{8} - 11 \beta_{7} - 10 \beta_{6} + \beta_{5} - 10 \beta_{4} - 11 \beta_{3} + \beta_{1} + 6\)
\(\nu^{6}\)\(=\)\(-15 \beta_{19} + 92 \beta_{18} - 54 \beta_{17} - 15 \beta_{16} - 86 \beta_{15} + 44 \beta_{14} + 63 \beta_{13} - 59 \beta_{12} - 39 \beta_{11} - 124 \beta_{10} + 23 \beta_{9} - 15 \beta_{8} - 15 \beta_{7} - 27 \beta_{6} + 63 \beta_{5} - 15 \beta_{4} - 76 \beta_{3} + 27 \beta_{2} - 59\)
\(\nu^{7}\)\(=\)\(2 \beta_{19} + 81 \beta_{18} - 180 \beta_{17} + 216 \beta_{16} - 39 \beta_{15} + 127 \beta_{14} - 2 \beta_{13} - 180 \beta_{12} - 127 \beta_{11} + 81 \beta_{10} + 216 \beta_{9} - 69 \beta_{7} + 20 \beta_{6} + 275 \beta_{5} - 80 \beta_{4} - 80 \beta_{3} + 69 \beta_{2} - 20 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-332 \beta_{19} - 1181 \beta_{18} - 190 \beta_{16} - 142 \beta_{15} - 95 \beta_{14} + 95 \beta_{12} + 4 \beta_{11} + 328 \beta_{10} - 446 \beta_{9} + 162 \beta_{8} - 107 \beta_{7} + 162 \beta_{6} + 332 \beta_{5} - 454 \beta_{4} + 9 \beta_{3} - 9 \beta_{2} + 107 \beta_{1} + 446\)
\(\nu^{9}\)\(=\)\(-1972 \beta_{19} - 2836 \beta_{18} + 1640 \beta_{17} - 3235 \beta_{16} - 203 \beta_{15} - 45 \beta_{14} + 45 \beta_{13} + 571 \beta_{12} + 2836 \beta_{11} - 616 \beta_{10} - 1685 \beta_{9} + 212 \beta_{8} + 212 \beta_{7} - 45 \beta_{6} - 640 \beta_{4} - 549 \beta_{3} + 640 \beta_{2} + 549 \beta_{1} - 203\)
\(\nu^{10}\)\(=\)\(3473 \beta_{18} + 2239 \beta_{17} + 1891 \beta_{16} + 2239 \beta_{15} + 5064 \beta_{14} - 3855 \beta_{13} - 3473 \beta_{12} + 5039 \beta_{11} + 3855 \beta_{10} + 5064 \beta_{9} + 1454 \beta_{8} + 3855 \beta_{7} + 2311 \beta_{6} - 2311 \beta_{5} + 2490 \beta_{4} + 1454 \beta_{3} + 1039 \beta_{2} - 2490 \beta_{1} - 1891\)
\(\nu^{11}\)\(=\)\(14352 \beta_{19} + 14352 \beta_{18} - 6649 \beta_{17} + 21001 \beta_{16} + 8645 \beta_{15} + 7806 \beta_{14} - 16451 \beta_{13} + 1768 \beta_{12} - 12584 \beta_{11} + 21793 \beta_{10} + 21793 \beta_{9} + 10094 \beta_{8} + 13698 \beta_{7} + 14352 \beta_{6} - 13698 \beta_{5} + 10094 \beta_{4} + 16451 \beta_{3} - 9420 \beta_{2} - 9420 \beta_{1} + 5425\)
\(\nu^{12}\)\(=\)\(16140 \beta_{19} - 12711 \beta_{18} + 13271 \beta_{17} - 26763 \beta_{16} + 15857 \beta_{15} - 33309 \beta_{14} - 9594 \beta_{13} + 63071 \beta_{12} - 16140 \beta_{11} - 6263 \beta_{10} - 19257 \beta_{9} + 18511 \beta_{8} + 16140 \beta_{7} - 29938 \beta_{5} + 9594 \beta_{4} + 29938 \beta_{3} - 18511 \beta_{2} + 8589 \beta_{1} + 13271\)
\(\nu^{13}\)\(=\)\(-7830 \beta_{19} + 49326 \beta_{18} + 32552 \beta_{17} - 66024 \beta_{16} - 57156 \beta_{15} + 32974 \beta_{14} + 33309 \beta_{13} - 1853 \beta_{11} - 91503 \beta_{10} - 23626 \beta_{9} - 38000 \beta_{8} - 106311 \beta_{6} - 18962 \beta_{5} + 18962 \beta_{4} - 7830 \beta_{3} + 33309 \beta_{2} + 38000 \beta_{1} - 66283\)
\(\nu^{14}\)\(=\)\(118706 \beta_{19} + 481841 \beta_{18} - 365341 \beta_{17} + 481841 \beta_{16} - 235616 \beta_{15} + 391771 \beta_{14} - 26430 \beta_{13} - 475963 \beta_{12} - 354322 \beta_{11} + 175573 \beta_{10} + 502393 \beta_{9} - 183407 \beta_{8} - 112855 \beta_{7} - 112855 \beta_{6} + 48915 \beta_{5} + 118706 \beta_{4} + 48915 \beta_{2} - 26430 \beta_{1} - 149143\)
\(\nu^{15}\)\(=\)\(246225 \beta_{19} - 234914 \beta_{18} - 521536 \beta_{17} + 699115 \beta_{16} - 43151 \beta_{15} - 496051 \beta_{14} + 209898 \beta_{13} - 43151 \beta_{12} - 699115 \beta_{11} + 311638 \beta_{10} - 209898 \beta_{9} - 209898 \beta_{8} - 714524 \beta_{7} + 84131 \beta_{6} + 178749 \beta_{5} + 84131 \beta_{3} - 246225 \beta_{2} + 178749 \beta_{1} + 481139\)
\(\nu^{16}\)\(=\)\(-1359003 \beta_{19} - 3875898 \beta_{18} + 2516895 \beta_{17} - 3358624 \beta_{16} - 3284227 \beta_{14} + 2365813 \beta_{13} + 1999621 \beta_{12} + 2174313 \beta_{11} - 3284227 \beta_{10} - 6007240 \beta_{9} - 728320 \beta_{8} - 1785631 \beta_{7} - 996807 \beta_{6} + 728320 \beta_{5} - 996807 \beta_{4} - 1785631 \beta_{3} + 1359003 \beta_{1} + 815310\)
\(\nu^{17}\)\(=\)\(-3426712 \beta_{19} + 593231 \beta_{18} + 4013236 \beta_{17} - 3426712 \beta_{16} + 84365 \beta_{15} + 396078 \beta_{14} + 3623865 \beta_{13} - 3472966 \beta_{12} + 7439948 \beta_{11} - 6203736 \beta_{10} - 3708230 \beta_{9} - 3426712 \beta_{8} - 1336445 \beta_{7} - 2181118 \beta_{6} + 3623865 \beta_{5} - 1336445 \beta_{4} - 7404189 \beta_{3} + 2181118 \beta_{2} - 3472966\)
\(\nu^{18}\)\(=\)\(5253987 \beta_{19} + 14241220 \beta_{18} - 10610971 \beta_{17} + 21841851 \beta_{16} + 11560191 \beta_{15} + 2296270 \beta_{14} - 5253987 \beta_{13} - 10610971 \beta_{12} - 2296270 \beta_{11} + 14241220 \beta_{10} + 21841851 \beta_{9} + 2676791 \beta_{7} + 10912914 \beta_{6} + 7610724 \beta_{5} - 2827798 \beta_{4} - 2827798 \beta_{3} - 2676791 \beta_{2} - 10912914 \beta_{1}\)
\(\nu^{19}\)\(=\)\(-1170363 \beta_{19} - 50581146 \beta_{18} - 28971841 \beta_{16} + 27801478 \beta_{15} - 59664219 \beta_{14} + 59664219 \beta_{12} - 3358630 \beta_{11} + 4528993 \beta_{10} - 29086355 \beta_{9} + 27463738 \beta_{8} + 8616644 \beta_{7} + 27463738 \beta_{6} + 1170363 \beta_{5} - 29799572 \beta_{4} + 16717783 \beta_{3} - 16717783 \beta_{2} - 8616644 \beta_{1} + 29086355\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/828\mathbb{Z}\right)^\times\).

\(n\) \(415\) \(461\) \(649\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{18}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
73.1
−0.404188 + 2.81119i
0.302381 2.10310i
−0.962045 + 0.282482i
1.84381 0.541390i
2.31834 + 1.48991i
−0.0616736 0.0396352i
−0.962045 0.282482i
1.84381 + 0.541390i
−0.404188 2.81119i
0.302381 + 2.10310i
1.02355 + 2.24127i
−0.262998 0.575885i
1.74521 2.01408i
−1.54238 + 1.78001i
1.74521 + 2.01408i
−1.54238 1.78001i
1.02355 2.24127i
−0.262998 + 0.575885i
2.31834 1.48991i
−0.0616736 + 0.0396352i
0 0 0 0.332496 + 0.213682i 0 −0.440112 3.06105i 0 0 0
73.2 0 0 0 1.52130 + 0.977682i 0 0.485296 + 3.37531i 0 0 0
289.1 0 0 0 −1.11647 + 2.44474i 0 0.161591 + 0.0474474i 0 0 0
289.2 0 0 0 1.21471 2.65985i 0 0.960219 + 0.281946i 0 0 0
325.1 0 0 0 −0.735636 + 0.848969i 0 0.891451 0.572901i 0 0 0
325.2 0 0 0 2.38152 2.74842i 0 −3.67577 + 2.36227i 0 0 0
361.1 0 0 0 −1.11647 2.44474i 0 0.161591 0.0474474i 0 0 0
361.2 0 0 0 1.21471 + 2.65985i 0 0.960219 0.281946i 0 0 0
397.1 0 0 0 0.332496 0.213682i 0 −0.440112 + 3.06105i 0 0 0
397.2 0 0 0 1.52130 0.977682i 0 0.485296 3.37531i 0 0 0
469.1 0 0 0 −0.217172 1.51046i 0 −1.55685 + 3.40903i 0 0 0
469.2 0 0 0 0.149019 + 1.03645i 0 0.607780 1.33085i 0 0 0
541.1 0 0 0 −3.91931 + 1.15081i 0 −0.0825209 0.0952342i 0 0 0
541.2 0 0 0 2.38954 0.701632i 0 2.64891 + 3.05701i 0 0 0
577.1 0 0 0 −3.91931 1.15081i 0 −0.0825209 + 0.0952342i 0 0 0
577.2 0 0 0 2.38954 + 0.701632i 0 2.64891 3.05701i 0 0 0
685.1 0 0 0 −0.217172 + 1.51046i 0 −1.55685 3.40903i 0 0 0
685.2 0 0 0 0.149019 1.03645i 0 0.607780 + 1.33085i 0 0 0
721.1 0 0 0 −0.735636 0.848969i 0 0.891451 + 0.572901i 0 0 0
721.2 0 0 0 2.38152 + 2.74842i 0 −3.67577 2.36227i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 721.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 828.2.q.c 20
3.b odd 2 1 276.2.i.a 20
23.c even 11 1 inner 828.2.q.c 20
69.g even 22 1 6348.2.a.t 10
69.h odd 22 1 276.2.i.a 20
69.h odd 22 1 6348.2.a.s 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
276.2.i.a 20 3.b odd 2 1
276.2.i.a 20 69.h odd 22 1
828.2.q.c 20 1.a even 1 1 trivial
828.2.q.c 20 23.c even 11 1 inner
6348.2.a.s 10 69.h odd 22 1
6348.2.a.t 10 69.g even 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \(T_{5}^{20} - \cdots\) acting on \(S_{2}^{\mathrm{new}}(828, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} \)
$3$ \( T^{20} \)
$5$ \( 139129 - 659837 T + 1370675 T^{2} - 1372212 T^{3} + 1879946 T^{4} - 1476498 T^{5} + 1518254 T^{6} - 1324154 T^{7} + 944325 T^{8} - 576264 T^{9} + 312388 T^{10} - 117918 T^{11} + 32177 T^{12} - 7254 T^{13} + 1043 T^{14} + 188 T^{15} - 199 T^{16} + 62 T^{17} + 3 T^{18} - 4 T^{19} + T^{20} \)
$7$ \( 529 - 2530 T - 5263 T^{2} - 152240 T^{3} + 1813793 T^{4} - 5575526 T^{5} + 8765093 T^{6} - 8532689 T^{7} + 5696271 T^{8} - 2768359 T^{9} + 1036815 T^{10} - 337579 T^{11} + 105066 T^{12} - 15873 T^{13} + 7686 T^{14} - 231 T^{15} + 455 T^{16} + 24 T^{18} + T^{20} \)
$11$ \( 36590401 + 22357104 T - 15648931 T^{2} + 6714697 T^{3} + 13901977 T^{4} + 905597 T^{5} + 6461861 T^{6} + 2908906 T^{7} + 1511093 T^{8} + 1024793 T^{9} + 285229 T^{10} + 140525 T^{11} + 59566 T^{12} + 8591 T^{13} + 8678 T^{14} - 330 T^{15} + 595 T^{16} - 66 T^{17} + 10 T^{18} + T^{20} \)
$13$ \( 746546329 - 3899675175 T + 7491372877 T^{2} - 6423196593 T^{3} + 2968964780 T^{4} - 595120405 T^{5} + 50964803 T^{6} + 119149558 T^{7} + 22526164 T^{8} + 16471851 T^{9} + 13758942 T^{10} + 6486040 T^{11} + 2508837 T^{12} + 893288 T^{13} + 290806 T^{14} + 75185 T^{15} + 15621 T^{16} + 2453 T^{17} + 285 T^{18} + 22 T^{19} + T^{20} \)
$17$ \( 181252369 + 425848153 T + 296568609 T^{2} - 439386189 T^{3} - 867895682 T^{4} - 360956995 T^{5} + 424708084 T^{6} + 767459429 T^{7} + 637891631 T^{8} + 349625082 T^{9} + 141436283 T^{10} + 43334088 T^{11} + 10151453 T^{12} + 1754049 T^{13} + 217607 T^{14} + 16943 T^{15} + 1289 T^{16} + 187 T^{17} + 63 T^{18} + 7 T^{19} + T^{20} \)
$19$ \( 512524321 - 3123978249 T + 26405217311 T^{2} - 21295938332 T^{3} + 80259141223 T^{4} - 75137752954 T^{5} + 19751255693 T^{6} - 688924957 T^{7} + 1218325768 T^{8} - 451626915 T^{9} + 58792986 T^{10} - 13803937 T^{11} + 5269935 T^{12} - 1151262 T^{13} + 195803 T^{14} - 36190 T^{15} + 7465 T^{16} - 1358 T^{17} + 198 T^{18} - 19 T^{19} + T^{20} \)
$23$ \( 41426511213649 + 36023053229260 T + 20909033070027 T^{2} + 8900213718458 T^{3} + 3127258155125 T^{4} + 934659985088 T^{5} + 247062384147 T^{6} + 59304415734 T^{7} + 13279102417 T^{8} + 2853515424 T^{9} + 598445177 T^{10} + 124065888 T^{11} + 25102273 T^{12} + 4874202 T^{13} + 882867 T^{14} + 145216 T^{15} + 21125 T^{16} + 2614 T^{17} + 267 T^{18} + 20 T^{19} + T^{20} \)
$29$ \( 94249 - 519444 T + 7821200 T^{2} - 23219658 T^{3} + 55182717 T^{4} - 89529410 T^{5} + 94828913 T^{6} - 86003885 T^{7} - 24528037 T^{8} + 118089797 T^{9} + 114501529 T^{10} + 48304863 T^{11} + 12578894 T^{12} + 2739088 T^{13} + 688692 T^{14} + 176564 T^{15} + 35791 T^{16} + 5181 T^{17} + 512 T^{18} + 32 T^{19} + T^{20} \)
$31$ \( 1366115521 - 1978004876 T + 14271050678 T^{2} + 25924290097 T^{3} + 28775697562 T^{4} + 6940917303 T^{5} + 1899672010 T^{6} + 940463831 T^{7} - 359440379 T^{8} - 120133918 T^{9} + 29345997 T^{10} - 6790926 T^{11} + 1962384 T^{12} + 250094 T^{13} + 132664 T^{14} + 17126 T^{15} + 2558 T^{16} - 197 T^{17} - 10 T^{18} + 3 T^{19} + T^{20} \)
$37$ \( 41662309043667841 + 41731143250060526 T + 22356141446478578 T^{2} + 8075906329777803 T^{3} + 2177449951797787 T^{4} + 444412175882650 T^{5} + 70750852182783 T^{6} + 9064644220885 T^{7} + 1012395922334 T^{8} + 103024379921 T^{9} + 9793034624 T^{10} + 743266086 T^{11} + 32345102 T^{12} - 1764653 T^{13} - 220832 T^{14} - 2681 T^{15} - 1125 T^{16} + 77 T^{17} + 72 T^{18} + 10 T^{19} + T^{20} \)
$41$ \( 389752131263881 - 669630738999991 T + 707953600583513 T^{2} - 461805388747059 T^{3} + 205218730340075 T^{4} - 66531457171570 T^{5} + 17142140167902 T^{6} - 3598689054226 T^{7} + 625389775992 T^{8} - 84863330246 T^{9} + 7549111603 T^{10} + 70954108 T^{11} - 163923425 T^{12} + 28078887 T^{13} - 1519224 T^{14} - 275017 T^{15} + 76244 T^{16} - 9659 T^{17} + 782 T^{18} - 40 T^{19} + T^{20} \)
$43$ \( 577670697717961 - 463909200184418 T + 171338146440788 T^{2} - 54905861248707 T^{3} + 21299481357280 T^{4} - 6810443215059 T^{5} + 1474974796583 T^{6} - 296529410945 T^{7} + 82779882905 T^{8} - 18687977058 T^{9} + 2443057990 T^{10} - 268560425 T^{11} + 44026667 T^{12} - 6573659 T^{13} + 1135437 T^{14} - 172293 T^{15} + 19380 T^{16} - 1740 T^{17} + 237 T^{18} - 8 T^{19} + T^{20} \)
$47$ \( ( -337853 + 112793 T + 691965 T^{2} - 815546 T^{3} + 361609 T^{4} - 63239 T^{5} - 1114 T^{6} + 1665 T^{7} - 125 T^{8} - 9 T^{9} + T^{10} )^{2} \)
$53$ \( 11896440939631249 - 11128281313853122 T + 5587027563331885 T^{2} - 1697746128917211 T^{3} + 315796719707061 T^{4} - 46237293611795 T^{5} + 13028620028722 T^{6} - 5127709123725 T^{7} + 1514610526385 T^{8} - 329258348550 T^{9} + 57187682551 T^{10} - 8609367447 T^{11} + 1201045131 T^{12} - 158550866 T^{13} + 19247658 T^{14} - 2040236 T^{15} + 181290 T^{16} - 13133 T^{17} + 759 T^{18} - 34 T^{19} + T^{20} \)
$59$ \( 1812553308721 + 8949419274204 T - 5693007031864 T^{2} - 41298924756925 T^{3} + 93785300758035 T^{4} - 99417124595049 T^{5} + 65713808784726 T^{6} - 29908664797847 T^{7} + 9846622188804 T^{8} - 2420978984634 T^{9} + 458543484068 T^{10} - 69203826067 T^{11} + 8577394846 T^{12} - 882744313 T^{13} + 77386621 T^{14} - 5717924 T^{15} + 375515 T^{16} - 19965 T^{17} + 996 T^{18} - 32 T^{19} + T^{20} \)
$61$ \( 16910706981169 + 13521351030728 T + 21181259906611 T^{2} + 9841256141673 T^{3} + 6007237816983 T^{4} + 848503312286 T^{5} + 222488054748 T^{6} - 266115937989 T^{7} + 94408392138 T^{8} - 33711904722 T^{9} + 16246854502 T^{10} - 5657273338 T^{11} + 1230757372 T^{12} - 168619605 T^{13} + 13914476 T^{14} - 517190 T^{15} - 7791 T^{16} - 44 T^{17} + 325 T^{18} - 32 T^{19} + T^{20} \)
$67$ \( 24475292457001 - 60492442340986 T + 86621718869308 T^{2} - 80656859182347 T^{3} + 54227478592951 T^{4} - 26887554432959 T^{5} + 10283893312668 T^{6} - 3112931992534 T^{7} + 770962969712 T^{8} - 158363476676 T^{9} + 27292918179 T^{10} - 4043670382 T^{11} + 512642541 T^{12} - 54006967 T^{13} + 4731822 T^{14} - 282469 T^{15} + 13060 T^{16} - 2749 T^{17} + 472 T^{18} - 35 T^{19} + T^{20} \)
$71$ \( 14637426640321 + 136783569942789 T + 501278829617156 T^{2} + 1715569725290344 T^{3} + 5756998707742049 T^{4} + 3342206430928013 T^{5} + 974602274541561 T^{6} + 177912397479575 T^{7} + 29255301652229 T^{8} + 4256202216101 T^{9} + 542325677266 T^{10} + 57807076855 T^{11} + 5637296701 T^{12} + 498105432 T^{13} + 42031926 T^{14} + 3243427 T^{15} + 227001 T^{16} + 12496 T^{17} + 623 T^{18} + 33 T^{19} + T^{20} \)
$73$ \( 41788181960853241 - 15955582587444860 T + 2794954307710516 T^{2} - 247841669832781 T^{3} + 53783136580094 T^{4} + 43932000343253 T^{5} - 5330147265514 T^{6} - 611684060090 T^{7} + 230744116727 T^{8} - 9377005991 T^{9} + 1316336031 T^{10} - 714559816 T^{11} + 71257576 T^{12} + 1289334 T^{13} + 749861 T^{14} - 142120 T^{15} + 23864 T^{16} - 315 T^{17} + 261 T^{18} + T^{19} + T^{20} \)
$79$ \( 684944324016481 + 1007307698589533 T + 725468196081510 T^{2} + 342451363704934 T^{3} + 120765990289091 T^{4} + 35536820378609 T^{5} + 9892003594800 T^{6} + 2685392761623 T^{7} + 663475474095 T^{8} + 136476335369 T^{9} + 22517102486 T^{10} + 2784334663 T^{11} + 263299925 T^{12} + 14781987 T^{13} + 649788 T^{14} + 17842 T^{15} + 9705 T^{16} - 308 T^{17} + 105 T^{18} - 22 T^{19} + T^{20} \)
$83$ \( 1251350934564721 - 3235983602956341 T + 3122798983535958 T^{2} - 503910130702060 T^{3} + 120454263688919 T^{4} - 23737763861453 T^{5} + 4049123011748 T^{6} - 1012207001133 T^{7} + 141016540959 T^{8} - 9760198929 T^{9} + 1124967876 T^{10} + 68876847 T^{11} + 25858491 T^{12} + 523297 T^{13} + 92020 T^{14} + 49328 T^{15} + 1143 T^{16} - 618 T^{17} + 179 T^{18} - 14 T^{19} + T^{20} \)
$89$ \( 6872782557649009 + 37488648447839868 T + 115032853652783955 T^{2} + 70824988505802990 T^{3} + 21897041167232970 T^{4} + 4270498507709518 T^{5} + 621137417849182 T^{6} + 72213650609129 T^{7} + 6413842447184 T^{8} + 437657925376 T^{9} + 24791006899 T^{10} - 760176289 T^{11} - 278929550 T^{12} - 6284913 T^{13} + 337071 T^{14} - 14593 T^{15} + 41819 T^{16} + 1709 T^{17} + 246 T^{18} + 10 T^{19} + T^{20} \)
$97$ \( 58917139761417961 + 23869085839650988 T + 9607634304263708 T^{2} + 1190266860993040 T^{3} + 871800404775500 T^{4} + 104153120045272 T^{5} + 25398571393977 T^{6} + 8302568888676 T^{7} + 1021825258220 T^{8} - 17799470814 T^{9} + 7724302783 T^{10} + 1242332430 T^{11} - 9391303 T^{12} + 38333828 T^{13} + 5241069 T^{14} - 113103 T^{15} - 10979 T^{16} - 4143 T^{17} + 124 T^{18} + 4 T^{19} + T^{20} \)
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