Properties

Label 5.9.c
Level 5
Weight 9
Character orbit c
Rep. character \(\chi_{5}(2,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 6
Newform subspaces 1
Sturm bound 4
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 5 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 5.c (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(4\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{9}(5, [\chi])\).

Total New Old
Modular forms 10 10 0
Cusp forms 6 6 0
Eisenstein series 4 4 0

Trace form

\( 6q - 2q^{2} - 72q^{3} + 220q^{5} + 1752q^{6} - 2352q^{7} - 8220q^{8} + O(q^{10}) \) \( 6q - 2q^{2} - 72q^{3} + 220q^{5} + 1752q^{6} - 2352q^{7} - 8220q^{8} + 30870q^{10} + 23192q^{11} - 45912q^{12} - 119142q^{13} + 241440q^{15} + 218616q^{16} - 265502q^{17} - 454062q^{18} + 412260q^{20} + 231672q^{21} - 35664q^{22} + 28888q^{23} - 340350q^{25} - 801388q^{26} + 392040q^{27} + 1305192q^{28} - 2549760q^{30} - 747648q^{31} + 3033928q^{32} + 4269096q^{33} - 4971680q^{35} - 3972804q^{36} - 454002q^{37} + 1443720q^{38} + 2683500q^{40} + 2489432q^{41} + 4223856q^{42} + 792648q^{43} + 210690q^{45} - 3149928q^{46} - 15313352q^{47} - 21677712q^{48} + 29537650q^{50} + 35567712q^{51} - 735732q^{52} - 13509122q^{53} + 4448040q^{55} - 18454800q^{56} - 34625520q^{57} - 23903520q^{58} + 13688520q^{60} + 24111192q^{61} + 53913416q^{62} + 44837688q^{63} - 30943610q^{65} - 55047936q^{66} - 32827752q^{67} + 8118692q^{68} - 44156280q^{70} - 13992928q^{71} + 82596420q^{72} + 111859638q^{73} - 126793200q^{75} - 56470800q^{76} + 26260136q^{77} + 31125576q^{78} + 23045920q^{80} + 65834226q^{81} + 38023056q^{82} - 14768432q^{83} - 19713030q^{85} - 135560008q^{86} - 133207680q^{87} - 44555040q^{88} + 135147990q^{90} + 167542032q^{91} + 69931048q^{92} - 96798024q^{93} + 239661000q^{95} + 184867872q^{96} - 186656202q^{97} - 345959698q^{98} + O(q^{100}) \)

Decomposition of \(S_{9}^{\mathrm{new}}(5, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
5.9.c.a \(6\) \(2.037\) \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-2\) \(-72\) \(220\) \(-2352\) \(q-\beta _{3}q^{2}+(-12-12\beta _{1}-\beta _{2}-\beta _{5})q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + 2 T + 2 T^{2} + 2912 T^{3} - 51136 T^{4} - 1135744 T^{5} + 2070656 T^{6} - 290750464 T^{7} - 3351248896 T^{8} + 48855252992 T^{9} + 8589934592 T^{10} + 2199023255552 T^{11} + 281474976710656 T^{12} \)
$3$ \( 1 + 72 T + 2592 T^{2} + 88992 T^{3} - 43043121 T^{4} - 3659977224 T^{5} - 147990802464 T^{6} - 24013110566664 T^{7} - 1852865220656241 T^{8} + 25133969310517152 T^{9} + 4803028329503971872 T^{10} + \)\(87\!\cdots\!72\)\( T^{11} + \)\(79\!\cdots\!61\)\( T^{12} \)
$5$ \( 1 - 220 T + 194375 T^{2} - 199375000 T^{3} + 75927734375 T^{4} - 33569335937500 T^{5} + 59604644775390625 T^{6} \)
$7$ \( 1 + 2352 T + 2765952 T^{2} - 2361533048 T^{3} + 6289666320399 T^{4} + 128493496173721896 T^{5} + \)\(28\!\cdots\!96\)\( T^{6} + \)\(74\!\cdots\!96\)\( T^{7} + \)\(20\!\cdots\!99\)\( T^{8} - \)\(45\!\cdots\!48\)\( T^{9} + \)\(30\!\cdots\!52\)\( T^{10} + \)\(14\!\cdots\!52\)\( T^{11} + \)\(36\!\cdots\!01\)\( T^{12} \)
$11$ \( ( 1 - 11596 T + 493098215 T^{2} - 5106545516920 T^{3} + 105699981590497415 T^{4} - \)\(53\!\cdots\!56\)\( T^{5} + \)\(98\!\cdots\!41\)\( T^{6} )^{2} \)
$13$ \( 1 + 119142 T + 7097408082 T^{2} + 332686420223782 T^{3} + 13680551086291514559 T^{4} + \)\(46\!\cdots\!56\)\( T^{5} + \)\(13\!\cdots\!76\)\( T^{6} + \)\(38\!\cdots\!76\)\( T^{7} + \)\(91\!\cdots\!19\)\( T^{8} + \)\(18\!\cdots\!02\)\( T^{9} + \)\(31\!\cdots\!42\)\( T^{10} + \)\(43\!\cdots\!42\)\( T^{11} + \)\(29\!\cdots\!21\)\( T^{12} \)
$17$ \( 1 + 265502 T + 35245656002 T^{2} + 3505767378301982 T^{3} + \)\(37\!\cdots\!19\)\( T^{4} + \)\(40\!\cdots\!76\)\( T^{5} + \)\(37\!\cdots\!76\)\( T^{6} + \)\(28\!\cdots\!16\)\( T^{7} + \)\(18\!\cdots\!39\)\( T^{8} + \)\(11\!\cdots\!22\)\( T^{9} + \)\(83\!\cdots\!22\)\( T^{10} + \)\(43\!\cdots\!02\)\( T^{11} + \)\(11\!\cdots\!41\)\( T^{12} \)
$19$ \( 1 - 66391003446 T^{2} + \)\(21\!\cdots\!15\)\( T^{4} - \)\(42\!\cdots\!20\)\( T^{6} + \)\(60\!\cdots\!15\)\( T^{8} - \)\(55\!\cdots\!06\)\( T^{10} + \)\(23\!\cdots\!41\)\( T^{12} \)
$23$ \( 1 - 28888 T + 417258272 T^{2} - 2207314762520128 T^{3} + \)\(86\!\cdots\!39\)\( T^{4} - \)\(83\!\cdots\!64\)\( T^{5} + \)\(12\!\cdots\!16\)\( T^{6} - \)\(65\!\cdots\!84\)\( T^{7} + \)\(52\!\cdots\!79\)\( T^{8} - \)\(10\!\cdots\!48\)\( T^{9} + \)\(15\!\cdots\!12\)\( T^{10} - \)\(85\!\cdots\!88\)\( T^{11} + \)\(23\!\cdots\!81\)\( T^{12} \)
$29$ \( 1 - 1726260912966 T^{2} + \)\(13\!\cdots\!15\)\( T^{4} - \)\(77\!\cdots\!20\)\( T^{6} + \)\(34\!\cdots\!15\)\( T^{8} - \)\(10\!\cdots\!06\)\( T^{10} + \)\(15\!\cdots\!61\)\( T^{12} \)
$31$ \( ( 1 + 373824 T + 1438821265815 T^{2} + 103488660558742480 T^{3} + \)\(12\!\cdots\!15\)\( T^{4} + \)\(27\!\cdots\!44\)\( T^{5} + \)\(62\!\cdots\!21\)\( T^{6} )^{2} \)
$37$ \( 1 + 454002 T + 103058908002 T^{2} + 1591257258997412242 T^{3} + \)\(36\!\cdots\!59\)\( T^{4} + \)\(11\!\cdots\!36\)\( T^{5} + \)\(25\!\cdots\!36\)\( T^{6} + \)\(39\!\cdots\!56\)\( T^{7} + \)\(45\!\cdots\!19\)\( T^{8} + \)\(68\!\cdots\!62\)\( T^{9} + \)\(15\!\cdots\!62\)\( T^{10} + \)\(24\!\cdots\!02\)\( T^{11} + \)\(18\!\cdots\!21\)\( T^{12} \)
$41$ \( ( 1 - 1244716 T + 19779856453415 T^{2} - 17348876621060070520 T^{3} + \)\(15\!\cdots\!15\)\( T^{4} - \)\(79\!\cdots\!56\)\( T^{5} + \)\(50\!\cdots\!61\)\( T^{6} )^{2} \)
$43$ \( 1 - 792648 T + 314145425952 T^{2} - 6701894073526462448 T^{3} + \)\(27\!\cdots\!99\)\( T^{4} - \)\(18\!\cdots\!04\)\( T^{5} + \)\(86\!\cdots\!96\)\( T^{6} - \)\(22\!\cdots\!04\)\( T^{7} + \)\(37\!\cdots\!99\)\( T^{8} - \)\(10\!\cdots\!48\)\( T^{9} + \)\(58\!\cdots\!52\)\( T^{10} - \)\(17\!\cdots\!48\)\( T^{11} + \)\(25\!\cdots\!01\)\( T^{12} \)
$47$ \( 1 + 15313352 T + 117249374737952 T^{2} + \)\(85\!\cdots\!72\)\( T^{3} + \)\(63\!\cdots\!79\)\( T^{4} + \)\(35\!\cdots\!16\)\( T^{5} + \)\(17\!\cdots\!16\)\( T^{6} + \)\(85\!\cdots\!76\)\( T^{7} + \)\(36\!\cdots\!59\)\( T^{8} + \)\(11\!\cdots\!32\)\( T^{9} + \)\(37\!\cdots\!32\)\( T^{10} + \)\(11\!\cdots\!52\)\( T^{11} + \)\(18\!\cdots\!61\)\( T^{12} \)
$53$ \( 1 + 13509122 T + 91248188605442 T^{2} + \)\(98\!\cdots\!42\)\( T^{3} + \)\(11\!\cdots\!79\)\( T^{4} + \)\(81\!\cdots\!76\)\( T^{5} + \)\(50\!\cdots\!36\)\( T^{6} + \)\(51\!\cdots\!36\)\( T^{7} + \)\(46\!\cdots\!59\)\( T^{8} + \)\(23\!\cdots\!02\)\( T^{9} + \)\(13\!\cdots\!22\)\( T^{10} + \)\(12\!\cdots\!22\)\( T^{11} + \)\(58\!\cdots\!61\)\( T^{12} \)
$59$ \( 1 - 413223229068726 T^{2} + \)\(98\!\cdots\!15\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(21\!\cdots\!15\)\( T^{8} - \)\(19\!\cdots\!06\)\( T^{10} + \)\(10\!\cdots\!21\)\( T^{12} \)
$61$ \( ( 1 - 12055596 T + 200152007609415 T^{2} + \)\(24\!\cdots\!80\)\( T^{3} + \)\(38\!\cdots\!15\)\( T^{4} - \)\(44\!\cdots\!56\)\( T^{5} + \)\(70\!\cdots\!41\)\( T^{6} )^{2} \)
$67$ \( 1 + 32827752 T + 538830650686752 T^{2} + \)\(16\!\cdots\!32\)\( T^{3} + \)\(54\!\cdots\!19\)\( T^{4} + \)\(88\!\cdots\!76\)\( T^{5} + \)\(12\!\cdots\!76\)\( T^{6} + \)\(36\!\cdots\!16\)\( T^{7} + \)\(90\!\cdots\!39\)\( T^{8} + \)\(10\!\cdots\!72\)\( T^{9} + \)\(14\!\cdots\!72\)\( T^{10} + \)\(36\!\cdots\!52\)\( T^{11} + \)\(44\!\cdots\!41\)\( T^{12} \)
$71$ \( ( 1 + 6996464 T + 1071469029384215 T^{2} + \)\(14\!\cdots\!80\)\( T^{3} + \)\(69\!\cdots\!15\)\( T^{4} + \)\(29\!\cdots\!44\)\( T^{5} + \)\(26\!\cdots\!81\)\( T^{6} )^{2} \)
$73$ \( 1 - 111859638 T + 6256289306745522 T^{2} - \)\(29\!\cdots\!78\)\( T^{3} + \)\(12\!\cdots\!39\)\( T^{4} - \)\(42\!\cdots\!64\)\( T^{5} + \)\(12\!\cdots\!16\)\( T^{6} - \)\(33\!\cdots\!84\)\( T^{7} + \)\(79\!\cdots\!79\)\( T^{8} - \)\(15\!\cdots\!98\)\( T^{9} + \)\(26\!\cdots\!62\)\( T^{10} - \)\(38\!\cdots\!38\)\( T^{11} + \)\(27\!\cdots\!81\)\( T^{12} \)
$79$ \( 1 - 4185433961698566 T^{2} + \)\(55\!\cdots\!15\)\( T^{4} - \)\(46\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!15\)\( T^{8} - \)\(22\!\cdots\!06\)\( T^{10} + \)\(12\!\cdots\!61\)\( T^{12} \)
$83$ \( 1 + 14768432 T + 109053291869312 T^{2} + \)\(28\!\cdots\!12\)\( T^{3} + \)\(10\!\cdots\!19\)\( T^{4} + \)\(10\!\cdots\!16\)\( T^{5} + \)\(83\!\cdots\!56\)\( T^{6} + \)\(23\!\cdots\!56\)\( T^{7} + \)\(51\!\cdots\!39\)\( T^{8} + \)\(32\!\cdots\!52\)\( T^{9} + \)\(28\!\cdots\!32\)\( T^{10} + \)\(85\!\cdots\!32\)\( T^{11} + \)\(13\!\cdots\!41\)\( T^{12} \)
$89$ \( 1 - 8165894455313286 T^{2} + \)\(62\!\cdots\!15\)\( T^{4} - \)\(26\!\cdots\!20\)\( T^{6} + \)\(97\!\cdots\!15\)\( T^{8} - \)\(19\!\cdots\!06\)\( T^{10} + \)\(37\!\cdots\!81\)\( T^{12} \)
$97$ \( 1 + 186656202 T + 17420268872532402 T^{2} + \)\(93\!\cdots\!22\)\( T^{3} + \)\(66\!\cdots\!79\)\( T^{4} + \)\(11\!\cdots\!16\)\( T^{5} + \)\(13\!\cdots\!16\)\( T^{6} + \)\(87\!\cdots\!76\)\( T^{7} + \)\(41\!\cdots\!59\)\( T^{8} + \)\(45\!\cdots\!82\)\( T^{9} + \)\(65\!\cdots\!82\)\( T^{10} + \)\(55\!\cdots\!02\)\( T^{11} + \)\(23\!\cdots\!61\)\( T^{12} \)
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