Properties

Label 3.39.b
Level $3$
Weight $39$
Character orbit 3.b
Rep. character $\chi_{3}(2,\cdot)$
Character field $\Q$
Dimension $12$
Newform subspaces $1$
Sturm bound $13$
Trace bound $0$

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Defining parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 39 \)
Character orbit: \([\chi]\) \(=\) 3.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 3 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(13\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{39}(3, [\chi])\).

Total New Old
Modular forms 14 14 0
Cusp forms 12 12 0
Eisenstein series 2 2 0

Trace form

\( 12q - 114742404q^{3} - 1699274528448q^{4} - 483611204680128q^{6} + 8107872236538648q^{7} - 424319151461513940q^{9} + O(q^{10}) \) \( 12q - 114742404q^{3} - 1699274528448q^{4} - 483611204680128q^{6} + 8107872236538648q^{7} - 424319151461513940q^{9} + 8521437485093339520q^{10} - 2862564534392665536q^{12} + 1069098773333431501752q^{13} + 6325133233762551847680q^{15} + 67078537203201948051456q^{16} + 1420463738762719667349120q^{18} - 4607058619794992781108360q^{19} + 33320142533758881258920952q^{21} - 136880063896016789094648960q^{22} + 783927349783175843577206784q^{24} - 1296747019384761463507120020q^{25} + 3527882385497241000493515804q^{27} - 772965222736808266417757568q^{28} - 31212306699175351074822848640q^{30} + 62426311865853800230409578584q^{31} - 201583764927512776992433501440q^{33} + 279954989831379847783072264704q^{34} + 524806089486681368742761588544q^{36} - 1044821541252033349072004239752q^{37} + 3798009668755107990641466418968q^{39} - 8640808308189045028871484272640q^{40} + 3739179021877842175021096471680q^{42} + 10033244180304067819165288107192q^{43} - 61618154391458016749088317176320q^{45} + 133876649746070235210936111300864q^{46} - 169673882786170669254051959076864q^{48} - 74274509875804693019854807543452q^{49} + 717994133339248153525793387369472q^{51} - 993185387699150183190245387663232q^{52} + 1237237684228372752705008546718912q^{54} - 147621918295658178296158670231040q^{55} - 1942894316331924260346088046339112q^{57} + 5456952458981587224214732301397120q^{58} - 21422716753388621512702364409876480q^{60} + 19261150877798591818539348576756024q^{61} - 6850204156609697882698173452536488q^{63} - 3354811274780051137580237339295744q^{64} + 29491133770268183961996554653096320q^{66} - 12293160890248249992597347473356552q^{67} + 14368205812565646738112192879237632q^{69} + 131887910392782907972479210848551680q^{70} - 849782187269944662605305038851543040q^{72} + 904634266610182985560078196407011672q^{73} - 1950889856240531910659528579515703460q^{75} + 3737782451304773339414088480976030848q^{76} - 4783964522796872138966708226773738880q^{78} + 3380994931932561121624661644777634520q^{79} - 3842105375893366153709000791027089588q^{81} + 9707181808332667829526886407699144960q^{82} - 23112447300276128146951334880402529152q^{84} + 16191233322474057038450787601892136960q^{85} - 46435849480387750961684301913616613120q^{87} + 111257699492994117641050357545187921920q^{88} - 164148346324517317604081085509260974720q^{90} + 128973762284915213743998856562037216624q^{91} - 173596942201215499347713600620947459528q^{93} + 323629532832623192615016136649966897664q^{94} - 450074405332355244291177655815409434624q^{96} + 248679388944018942060778750016570824728q^{97} - 346605715634587898579110273074604669440q^{99} + O(q^{100}) \)

Decomposition of \(S_{39}^{\mathrm{new}}(3, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
3.39.b.a \(12\) \(27.439\) \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(-114742404\) \(0\) \(81\!\cdots\!48\) \(q+\beta _{1}q^{2}+(-9561867+97\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 799630177440 T^{2} + \)\(41\!\cdots\!64\)\( T^{4} - \)\(16\!\cdots\!80\)\( T^{6} + \)\(53\!\cdots\!20\)\( T^{8} - \)\(15\!\cdots\!80\)\( T^{10} + \)\(43\!\cdots\!60\)\( T^{12} - \)\(11\!\cdots\!80\)\( T^{14} + \)\(30\!\cdots\!20\)\( T^{16} - \)\(72\!\cdots\!80\)\( T^{18} + \)\(13\!\cdots\!24\)\( T^{20} - \)\(19\!\cdots\!40\)\( T^{22} + \)\(18\!\cdots\!36\)\( T^{24} \)
$3$ \( 1 + 114742404 T + 218742485368606578 T^{2} - \)\(11\!\cdots\!44\)\( T^{3} + \)\(84\!\cdots\!79\)\( T^{4} - \)\(20\!\cdots\!68\)\( T^{5} + \)\(15\!\cdots\!24\)\( T^{6} - \)\(27\!\cdots\!52\)\( T^{7} + \)\(15\!\cdots\!59\)\( T^{8} - \)\(28\!\cdots\!36\)\( T^{9} + \)\(72\!\cdots\!98\)\( T^{10} + \)\(51\!\cdots\!96\)\( T^{11} + \)\(60\!\cdots\!61\)\( T^{12} \)
$5$ \( 1 - \)\(15\!\cdots\!40\)\( T^{2} + \)\(13\!\cdots\!50\)\( T^{4} - \)\(88\!\cdots\!00\)\( T^{6} + \)\(47\!\cdots\!75\)\( T^{8} - \)\(21\!\cdots\!00\)\( T^{10} + \)\(85\!\cdots\!00\)\( T^{12} - \)\(28\!\cdots\!00\)\( T^{14} + \)\(83\!\cdots\!75\)\( T^{16} - \)\(20\!\cdots\!00\)\( T^{18} + \)\(41\!\cdots\!50\)\( T^{20} - \)\(62\!\cdots\!00\)\( T^{22} + \)\(53\!\cdots\!25\)\( T^{24} \)
$7$ \( ( 1 - 4053936118269324 T + \)\(41\!\cdots\!98\)\( T^{2} - \)\(33\!\cdots\!76\)\( T^{3} + \)\(90\!\cdots\!99\)\( T^{4} + \)\(61\!\cdots\!48\)\( T^{5} + \)\(13\!\cdots\!04\)\( T^{6} + \)\(79\!\cdots\!52\)\( T^{7} + \)\(15\!\cdots\!99\)\( T^{8} - \)\(73\!\cdots\!24\)\( T^{9} + \)\(11\!\cdots\!98\)\( T^{10} - \)\(15\!\cdots\!76\)\( T^{11} + \)\(48\!\cdots\!01\)\( T^{12} )^{2} \)
$11$ \( 1 - \)\(13\!\cdots\!32\)\( T^{2} + \)\(71\!\cdots\!26\)\( T^{4} - \)\(13\!\cdots\!20\)\( T^{6} - \)\(30\!\cdots\!05\)\( T^{8} + \)\(30\!\cdots\!08\)\( T^{10} - \)\(12\!\cdots\!36\)\( T^{12} + \)\(41\!\cdots\!88\)\( T^{14} - \)\(60\!\cdots\!05\)\( T^{16} - \)\(38\!\cdots\!20\)\( T^{18} + \)\(27\!\cdots\!66\)\( T^{20} - \)\(72\!\cdots\!32\)\( T^{22} + \)\(75\!\cdots\!61\)\( T^{24} \)
$13$ \( ( 1 - \)\(53\!\cdots\!76\)\( T + \)\(59\!\cdots\!58\)\( T^{2} - \)\(45\!\cdots\!04\)\( T^{3} + \)\(21\!\cdots\!59\)\( T^{4} - \)\(16\!\cdots\!68\)\( T^{5} + \)\(55\!\cdots\!44\)\( T^{6} - \)\(35\!\cdots\!72\)\( T^{7} + \)\(99\!\cdots\!19\)\( T^{8} - \)\(44\!\cdots\!56\)\( T^{9} + \)\(12\!\cdots\!98\)\( T^{10} - \)\(23\!\cdots\!24\)\( T^{11} + \)\(95\!\cdots\!21\)\( T^{12} )^{2} \)
$17$ \( 1 - \)\(25\!\cdots\!60\)\( T^{2} + \)\(44\!\cdots\!34\)\( T^{4} - \)\(53\!\cdots\!20\)\( T^{6} + \)\(51\!\cdots\!35\)\( T^{8} - \)\(39\!\cdots\!20\)\( T^{10} + \)\(24\!\cdots\!00\)\( T^{12} - \)\(12\!\cdots\!20\)\( T^{14} + \)\(55\!\cdots\!35\)\( T^{16} - \)\(18\!\cdots\!20\)\( T^{18} + \)\(51\!\cdots\!14\)\( T^{20} - \)\(95\!\cdots\!60\)\( T^{22} + \)\(12\!\cdots\!81\)\( T^{24} \)
$19$ \( ( 1 + \)\(23\!\cdots\!80\)\( T + \)\(21\!\cdots\!94\)\( T^{2} + \)\(38\!\cdots\!60\)\( T^{3} + \)\(19\!\cdots\!55\)\( T^{4} + \)\(27\!\cdots\!60\)\( T^{5} + \)\(97\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!60\)\( T^{7} + \)\(29\!\cdots\!55\)\( T^{8} + \)\(22\!\cdots\!60\)\( T^{9} + \)\(49\!\cdots\!34\)\( T^{10} + \)\(21\!\cdots\!80\)\( T^{11} + \)\(35\!\cdots\!41\)\( T^{12} )^{2} \)
$23$ \( 1 - \)\(48\!\cdots\!40\)\( T^{2} + \)\(11\!\cdots\!14\)\( T^{4} - \)\(18\!\cdots\!80\)\( T^{6} + \)\(19\!\cdots\!95\)\( T^{8} - \)\(16\!\cdots\!80\)\( T^{10} + \)\(10\!\cdots\!60\)\( T^{12} - \)\(50\!\cdots\!80\)\( T^{14} + \)\(19\!\cdots\!95\)\( T^{16} - \)\(53\!\cdots\!80\)\( T^{18} + \)\(10\!\cdots\!74\)\( T^{20} - \)\(13\!\cdots\!40\)\( T^{22} + \)\(88\!\cdots\!61\)\( T^{24} \)
$29$ \( 1 - \)\(22\!\cdots\!72\)\( T^{2} + \)\(28\!\cdots\!86\)\( T^{4} - \)\(24\!\cdots\!20\)\( T^{6} + \)\(15\!\cdots\!95\)\( T^{8} - \)\(79\!\cdots\!92\)\( T^{10} + \)\(32\!\cdots\!44\)\( T^{12} - \)\(11\!\cdots\!32\)\( T^{14} + \)\(29\!\cdots\!95\)\( T^{16} - \)\(64\!\cdots\!20\)\( T^{18} + \)\(10\!\cdots\!66\)\( T^{20} - \)\(11\!\cdots\!72\)\( T^{22} + \)\(71\!\cdots\!21\)\( T^{24} \)
$31$ \( ( 1 - \)\(31\!\cdots\!92\)\( T + \)\(14\!\cdots\!06\)\( T^{2} - \)\(17\!\cdots\!20\)\( T^{3} + \)\(45\!\cdots\!95\)\( T^{4} + \)\(33\!\cdots\!08\)\( T^{5} + \)\(56\!\cdots\!84\)\( T^{6} + \)\(15\!\cdots\!28\)\( T^{7} + \)\(99\!\cdots\!95\)\( T^{8} - \)\(17\!\cdots\!20\)\( T^{9} + \)\(69\!\cdots\!66\)\( T^{10} - \)\(71\!\cdots\!92\)\( T^{11} + \)\(10\!\cdots\!41\)\( T^{12} )^{2} \)
$37$ \( ( 1 + \)\(52\!\cdots\!76\)\( T + \)\(22\!\cdots\!58\)\( T^{2} + \)\(91\!\cdots\!04\)\( T^{3} + \)\(20\!\cdots\!59\)\( T^{4} + \)\(67\!\cdots\!68\)\( T^{5} + \)\(10\!\cdots\!44\)\( T^{6} + \)\(26\!\cdots\!72\)\( T^{7} + \)\(31\!\cdots\!19\)\( T^{8} + \)\(54\!\cdots\!56\)\( T^{9} + \)\(51\!\cdots\!98\)\( T^{10} + \)\(47\!\cdots\!24\)\( T^{11} + \)\(35\!\cdots\!21\)\( T^{12} )^{2} \)
$41$ \( 1 - \)\(69\!\cdots\!12\)\( T^{2} + \)\(25\!\cdots\!06\)\( T^{4} - \)\(62\!\cdots\!20\)\( T^{6} + \)\(13\!\cdots\!95\)\( T^{8} - \)\(30\!\cdots\!92\)\( T^{10} + \)\(66\!\cdots\!24\)\( T^{12} - \)\(11\!\cdots\!72\)\( T^{14} + \)\(19\!\cdots\!95\)\( T^{16} - \)\(32\!\cdots\!20\)\( T^{18} + \)\(49\!\cdots\!66\)\( T^{20} - \)\(50\!\cdots\!12\)\( T^{22} + \)\(26\!\cdots\!41\)\( T^{24} \)
$43$ \( ( 1 - \)\(50\!\cdots\!96\)\( T + \)\(42\!\cdots\!98\)\( T^{2} - \)\(15\!\cdots\!04\)\( T^{3} + \)\(97\!\cdots\!99\)\( T^{4} - \)\(27\!\cdots\!08\)\( T^{5} + \)\(13\!\cdots\!04\)\( T^{6} - \)\(32\!\cdots\!92\)\( T^{7} + \)\(13\!\cdots\!99\)\( T^{8} - \)\(25\!\cdots\!96\)\( T^{9} + \)\(83\!\cdots\!98\)\( T^{10} - \)\(11\!\cdots\!04\)\( T^{11} + \)\(26\!\cdots\!01\)\( T^{12} )^{2} \)
$47$ \( 1 - \)\(20\!\cdots\!60\)\( T^{2} + \)\(22\!\cdots\!94\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!35\)\( T^{8} - \)\(46\!\cdots\!20\)\( T^{10} + \)\(17\!\cdots\!00\)\( T^{12} - \)\(56\!\cdots\!20\)\( T^{14} + \)\(14\!\cdots\!35\)\( T^{16} - \)\(29\!\cdots\!20\)\( T^{18} + \)\(46\!\cdots\!14\)\( T^{20} - \)\(51\!\cdots\!60\)\( T^{22} + \)\(29\!\cdots\!21\)\( T^{24} \)
$53$ \( 1 - \)\(21\!\cdots\!60\)\( T^{2} + \)\(21\!\cdots\!94\)\( T^{4} - \)\(14\!\cdots\!20\)\( T^{6} + \)\(66\!\cdots\!35\)\( T^{8} - \)\(26\!\cdots\!20\)\( T^{10} + \)\(90\!\cdots\!00\)\( T^{12} - \)\(28\!\cdots\!20\)\( T^{14} + \)\(82\!\cdots\!35\)\( T^{16} - \)\(19\!\cdots\!20\)\( T^{18} + \)\(32\!\cdots\!14\)\( T^{20} - \)\(35\!\cdots\!60\)\( T^{22} + \)\(18\!\cdots\!21\)\( T^{24} \)
$59$ \( 1 - \)\(11\!\cdots\!12\)\( T^{2} + \)\(61\!\cdots\!06\)\( T^{4} - \)\(24\!\cdots\!20\)\( T^{6} + \)\(72\!\cdots\!95\)\( T^{8} - \)\(18\!\cdots\!92\)\( T^{10} + \)\(38\!\cdots\!24\)\( T^{12} - \)\(69\!\cdots\!72\)\( T^{14} + \)\(10\!\cdots\!95\)\( T^{16} - \)\(13\!\cdots\!20\)\( T^{18} + \)\(13\!\cdots\!66\)\( T^{20} - \)\(92\!\cdots\!12\)\( T^{22} + \)\(32\!\cdots\!41\)\( T^{24} \)
$61$ \( ( 1 - \)\(96\!\cdots\!12\)\( T + \)\(25\!\cdots\!46\)\( T^{2} - \)\(16\!\cdots\!20\)\( T^{3} + \)\(28\!\cdots\!95\)\( T^{4} - \)\(14\!\cdots\!92\)\( T^{5} + \)\(21\!\cdots\!24\)\( T^{6} - \)\(98\!\cdots\!52\)\( T^{7} + \)\(13\!\cdots\!95\)\( T^{8} - \)\(57\!\cdots\!20\)\( T^{9} + \)\(59\!\cdots\!66\)\( T^{10} - \)\(15\!\cdots\!12\)\( T^{11} + \)\(11\!\cdots\!81\)\( T^{12} )^{2} \)
$67$ \( ( 1 + \)\(61\!\cdots\!76\)\( T + \)\(73\!\cdots\!18\)\( T^{2} - \)\(18\!\cdots\!16\)\( T^{3} + \)\(30\!\cdots\!19\)\( T^{4} - \)\(14\!\cdots\!12\)\( T^{5} + \)\(89\!\cdots\!84\)\( T^{6} - \)\(36\!\cdots\!08\)\( T^{7} + \)\(18\!\cdots\!39\)\( T^{8} - \)\(26\!\cdots\!64\)\( T^{9} + \)\(26\!\cdots\!98\)\( T^{10} + \)\(55\!\cdots\!24\)\( T^{11} + \)\(22\!\cdots\!41\)\( T^{12} )^{2} \)
$71$ \( 1 - \)\(13\!\cdots\!72\)\( T^{2} + \)\(88\!\cdots\!86\)\( T^{4} - \)\(35\!\cdots\!20\)\( T^{6} + \)\(10\!\cdots\!95\)\( T^{8} - \)\(25\!\cdots\!92\)\( T^{10} + \)\(58\!\cdots\!44\)\( T^{12} - \)\(12\!\cdots\!32\)\( T^{14} + \)\(25\!\cdots\!95\)\( T^{16} - \)\(43\!\cdots\!20\)\( T^{18} + \)\(53\!\cdots\!66\)\( T^{20} - \)\(41\!\cdots\!72\)\( T^{22} + \)\(14\!\cdots\!21\)\( T^{24} \)
$73$ \( ( 1 - \)\(45\!\cdots\!36\)\( T + \)\(29\!\cdots\!38\)\( T^{2} - \)\(11\!\cdots\!84\)\( T^{3} + \)\(43\!\cdots\!39\)\( T^{4} - \)\(12\!\cdots\!08\)\( T^{5} + \)\(35\!\cdots\!64\)\( T^{6} - \)\(80\!\cdots\!52\)\( T^{7} + \)\(17\!\cdots\!79\)\( T^{8} - \)\(28\!\cdots\!56\)\( T^{9} + \)\(50\!\cdots\!98\)\( T^{10} - \)\(48\!\cdots\!64\)\( T^{11} + \)\(68\!\cdots\!81\)\( T^{12} )^{2} \)
$79$ \( ( 1 - \)\(16\!\cdots\!60\)\( T + \)\(60\!\cdots\!54\)\( T^{2} - \)\(81\!\cdots\!20\)\( T^{3} + \)\(16\!\cdots\!35\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{5} + \)\(27\!\cdots\!00\)\( T^{6} - \)\(22\!\cdots\!20\)\( T^{7} + \)\(27\!\cdots\!35\)\( T^{8} - \)\(17\!\cdots\!20\)\( T^{9} + \)\(16\!\cdots\!14\)\( T^{10} - \)\(59\!\cdots\!60\)\( T^{11} + \)\(45\!\cdots\!61\)\( T^{12} )^{2} \)
$83$ \( 1 - \)\(52\!\cdots\!60\)\( T^{2} + \)\(14\!\cdots\!34\)\( T^{4} - \)\(26\!\cdots\!20\)\( T^{6} + \)\(37\!\cdots\!35\)\( T^{8} - \)\(42\!\cdots\!20\)\( T^{10} + \)\(38\!\cdots\!00\)\( T^{12} - \)\(29\!\cdots\!20\)\( T^{14} + \)\(18\!\cdots\!35\)\( T^{16} - \)\(94\!\cdots\!20\)\( T^{18} + \)\(36\!\cdots\!14\)\( T^{20} - \)\(94\!\cdots\!60\)\( T^{22} + \)\(12\!\cdots\!81\)\( T^{24} \)
$89$ \( 1 - \)\(10\!\cdots\!32\)\( T^{2} + \)\(50\!\cdots\!26\)\( T^{4} - \)\(16\!\cdots\!20\)\( T^{6} + \)\(36\!\cdots\!95\)\( T^{8} - \)\(63\!\cdots\!92\)\( T^{10} + \)\(85\!\cdots\!64\)\( T^{12} - \)\(90\!\cdots\!12\)\( T^{14} + \)\(74\!\cdots\!95\)\( T^{16} - \)\(46\!\cdots\!20\)\( T^{18} + \)\(20\!\cdots\!66\)\( T^{20} - \)\(60\!\cdots\!32\)\( T^{22} + \)\(83\!\cdots\!61\)\( T^{24} \)
$97$ \( ( 1 - \)\(12\!\cdots\!64\)\( T + \)\(14\!\cdots\!78\)\( T^{2} - \)\(10\!\cdots\!96\)\( T^{3} + \)\(79\!\cdots\!79\)\( T^{4} - \)\(46\!\cdots\!12\)\( T^{5} + \)\(30\!\cdots\!24\)\( T^{6} - \)\(14\!\cdots\!68\)\( T^{7} + \)\(78\!\cdots\!59\)\( T^{8} - \)\(31\!\cdots\!24\)\( T^{9} + \)\(14\!\cdots\!98\)\( T^{10} - \)\(38\!\cdots\!36\)\( T^{11} + \)\(96\!\cdots\!61\)\( T^{12} )^{2} \)
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