Properties

Label 15.10.b
Level 15
Weight 10
Character orbit b
Rep. character \(\chi_{15}(4,\cdot)\)
Character field \(\Q\)
Dimension 8
Newform subspaces 1
Sturm bound 20
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 15.b (of order \(2\) and degree \(1\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q\)
Newform subspaces: \( 1 \)
Sturm bound: \(20\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(15, [\chi])\).

Total New Old
Modular forms 20 8 12
Cusp forms 16 8 8
Eisenstein series 4 0 4

Trace form

\( 8q - 1194q^{4} - 690q^{5} + 486q^{6} - 52488q^{9} + O(q^{10}) \) \( 8q - 1194q^{4} - 690q^{5} + 486q^{6} - 52488q^{9} + 67090q^{10} - 71988q^{11} + 416364q^{14} + 80190q^{15} - 1505630q^{16} + 851584q^{19} + 2078100q^{20} - 1593108q^{21} + 1242702q^{24} + 1695500q^{25} - 877524q^{26} - 73572q^{29} + 3086100q^{30} + 474088q^{31} - 8124388q^{34} - 36357180q^{35} + 7833834q^{36} + 12959676q^{39} - 15313390q^{40} + 93320088q^{41} - 74555892q^{44} + 4527090q^{45} - 9664072q^{46} + 51329600q^{49} + 67798200q^{50} - 108196236q^{51} - 3188646q^{54} + 64428480q^{55} - 67781220q^{56} + 236526036q^{59} + 63172710q^{60} - 357427760q^{61} - 12137026q^{64} + 19848300q^{65} + 23317308q^{66} + 167059584q^{69} + 200900520q^{70} - 156890664q^{71} - 1523381796q^{74} - 528573600q^{75} + 1098697344q^{76} + 863922280q^{79} + 630213180q^{80} + 344373768q^{81} + 529023636q^{84} - 2223350420q^{85} + 997642392q^{86} + 357382224q^{89} - 440177490q^{90} + 214754328q^{91} - 721679824q^{94} + 1698584640q^{95} - 475022718q^{96} + 472313268q^{99} + O(q^{100}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(15, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
15.10.b.a \(8\) \(7.726\) \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(-690\) \(0\) \(q-\beta _{3}q^{2}+\beta _{4}q^{3}+(-149-\beta _{1})q^{4}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(15, [\chi])\) into lower level spaces

\( S_{10}^{\mathrm{old}}(15, [\chi]) \cong \) \(S_{10}^{\mathrm{new}}(5, [\chi])\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 1451 T^{2} + 1581940 T^{4} - 1146579392 T^{6} + 685942325248 T^{8} - 300568908136448 T^{10} + 108710089027747840 T^{12} - 26138892237258358784 T^{14} + \)\(47\!\cdots\!96\)\( T^{16} \)
$3$ \( ( 1 + 6561 T^{2} )^{4} \)
$5$ \( 1 + 690 T - 609700 T^{2} - 1699931250 T^{3} - 2538785156250 T^{4} - 3320178222656250 T^{5} - 2325820922851562500 T^{6} + \)\(51\!\cdots\!50\)\( T^{7} + \)\(14\!\cdots\!25\)\( T^{8} \)
$7$ \( 1 - 187079228 T^{2} + 15253712953314292 T^{4} - \)\(75\!\cdots\!16\)\( T^{6} + \)\(30\!\cdots\!14\)\( T^{8} - \)\(12\!\cdots\!84\)\( T^{10} + \)\(40\!\cdots\!92\)\( T^{12} - \)\(80\!\cdots\!72\)\( T^{14} + \)\(70\!\cdots\!01\)\( T^{16} \)
$11$ \( ( 1 + 35994 T + 5523704672 T^{2} + 186668393765490 T^{3} + 15626722625814770286 T^{4} + \)\(44\!\cdots\!90\)\( T^{5} + \)\(30\!\cdots\!32\)\( T^{6} + \)\(47\!\cdots\!74\)\( T^{7} + \)\(30\!\cdots\!61\)\( T^{8} )^{2} \)
$13$ \( 1 - 57204710684 T^{2} + \)\(16\!\cdots\!00\)\( T^{4} - \)\(29\!\cdots\!48\)\( T^{6} + \)\(37\!\cdots\!18\)\( T^{8} - \)\(33\!\cdots\!92\)\( T^{10} + \)\(20\!\cdots\!00\)\( T^{12} - \)\(81\!\cdots\!76\)\( T^{14} + \)\(15\!\cdots\!81\)\( T^{16} \)
$17$ \( 1 - 118674378380 T^{2} + \)\(17\!\cdots\!36\)\( T^{4} - \)\(19\!\cdots\!60\)\( T^{6} + \)\(18\!\cdots\!86\)\( T^{8} - \)\(27\!\cdots\!40\)\( T^{10} + \)\(34\!\cdots\!16\)\( T^{12} - \)\(33\!\cdots\!20\)\( T^{14} + \)\(39\!\cdots\!61\)\( T^{16} \)
$19$ \( ( 1 - 425792 T + 640639588492 T^{2} - 117316391220409664 T^{3} + \)\(17\!\cdots\!54\)\( T^{4} - \)\(37\!\cdots\!56\)\( T^{5} + \)\(66\!\cdots\!72\)\( T^{6} - \)\(14\!\cdots\!88\)\( T^{7} + \)\(10\!\cdots\!81\)\( T^{8} )^{2} \)
$23$ \( 1 - 10272938895992 T^{2} + \)\(48\!\cdots\!32\)\( T^{4} - \)\(14\!\cdots\!84\)\( T^{6} + \)\(30\!\cdots\!94\)\( T^{8} - \)\(46\!\cdots\!96\)\( T^{10} + \)\(51\!\cdots\!52\)\( T^{12} - \)\(35\!\cdots\!28\)\( T^{14} + \)\(11\!\cdots\!21\)\( T^{16} \)
$29$ \( ( 1 + 36786 T + 11698955610980 T^{2} - 38232367405598309058 T^{3} + \)\(21\!\cdots\!18\)\( T^{4} - \)\(55\!\cdots\!02\)\( T^{5} + \)\(24\!\cdots\!80\)\( T^{6} + \)\(11\!\cdots\!74\)\( T^{7} + \)\(44\!\cdots\!21\)\( T^{8} )^{2} \)
$31$ \( ( 1 - 237044 T + 50199397014268 T^{2} - \)\(19\!\cdots\!72\)\( T^{3} + \)\(11\!\cdots\!74\)\( T^{4} - \)\(51\!\cdots\!12\)\( T^{5} + \)\(35\!\cdots\!88\)\( T^{6} - \)\(43\!\cdots\!84\)\( T^{7} + \)\(48\!\cdots\!81\)\( T^{8} )^{2} \)
$37$ \( 1 - 624655700715068 T^{2} + \)\(20\!\cdots\!12\)\( T^{4} - \)\(45\!\cdots\!76\)\( T^{6} + \)\(70\!\cdots\!14\)\( T^{8} - \)\(77\!\cdots\!04\)\( T^{10} + \)\(59\!\cdots\!92\)\( T^{12} - \)\(30\!\cdots\!52\)\( T^{14} + \)\(81\!\cdots\!81\)\( T^{16} \)
$41$ \( ( 1 - 46660044 T + 1629598509234068 T^{2} - \)\(39\!\cdots\!32\)\( T^{3} + \)\(78\!\cdots\!54\)\( T^{4} - \)\(12\!\cdots\!52\)\( T^{5} + \)\(17\!\cdots\!28\)\( T^{6} - \)\(16\!\cdots\!64\)\( T^{7} + \)\(11\!\cdots\!41\)\( T^{8} )^{2} \)
$43$ \( 1 - 1288760154444440 T^{2} + \)\(10\!\cdots\!96\)\( T^{4} - \)\(57\!\cdots\!80\)\( T^{6} + \)\(31\!\cdots\!06\)\( T^{8} - \)\(14\!\cdots\!20\)\( T^{10} + \)\(65\!\cdots\!96\)\( T^{12} - \)\(20\!\cdots\!60\)\( T^{14} + \)\(40\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 - 4836473602067240 T^{2} + \)\(12\!\cdots\!56\)\( T^{4} - \)\(21\!\cdots\!80\)\( T^{6} + \)\(28\!\cdots\!26\)\( T^{8} - \)\(27\!\cdots\!20\)\( T^{10} + \)\(19\!\cdots\!76\)\( T^{12} - \)\(95\!\cdots\!60\)\( T^{14} + \)\(24\!\cdots\!41\)\( T^{16} \)
$53$ \( 1 - 4341506689340012 T^{2} + \)\(22\!\cdots\!72\)\( T^{4} - \)\(59\!\cdots\!84\)\( T^{6} + \)\(28\!\cdots\!74\)\( T^{8} - \)\(64\!\cdots\!76\)\( T^{10} + \)\(26\!\cdots\!12\)\( T^{12} - \)\(56\!\cdots\!28\)\( T^{14} + \)\(14\!\cdots\!41\)\( T^{16} \)
$59$ \( ( 1 - 118263018 T + 24386862278408432 T^{2} - \)\(17\!\cdots\!86\)\( T^{3} + \)\(26\!\cdots\!54\)\( T^{4} - \)\(15\!\cdots\!54\)\( T^{5} + \)\(18\!\cdots\!72\)\( T^{6} - \)\(76\!\cdots\!42\)\( T^{7} + \)\(56\!\cdots\!41\)\( T^{8} )^{2} \)
$61$ \( ( 1 + 178713880 T + 47272829372304076 T^{2} + \)\(59\!\cdots\!20\)\( T^{3} + \)\(82\!\cdots\!06\)\( T^{4} + \)\(69\!\cdots\!20\)\( T^{5} + \)\(64\!\cdots\!56\)\( T^{6} + \)\(28\!\cdots\!80\)\( T^{7} + \)\(18\!\cdots\!61\)\( T^{8} )^{2} \)
$67$ \( 1 - 128915336297443400 T^{2} + \)\(88\!\cdots\!36\)\( T^{4} - \)\(39\!\cdots\!00\)\( T^{6} + \)\(12\!\cdots\!86\)\( T^{8} - \)\(29\!\cdots\!00\)\( T^{10} + \)\(48\!\cdots\!16\)\( T^{12} - \)\(52\!\cdots\!00\)\( T^{14} + \)\(30\!\cdots\!61\)\( T^{16} \)
$71$ \( ( 1 + 78445332 T + 132742682733898508 T^{2} + \)\(14\!\cdots\!24\)\( T^{3} + \)\(78\!\cdots\!70\)\( T^{4} + \)\(65\!\cdots\!44\)\( T^{5} + \)\(27\!\cdots\!88\)\( T^{6} + \)\(75\!\cdots\!12\)\( T^{7} + \)\(44\!\cdots\!21\)\( T^{8} )^{2} \)
$73$ \( 1 - 112790784848235992 T^{2} + \)\(16\!\cdots\!32\)\( T^{4} - \)\(11\!\cdots\!84\)\( T^{6} + \)\(89\!\cdots\!94\)\( T^{8} - \)\(39\!\cdots\!96\)\( T^{10} + \)\(19\!\cdots\!52\)\( T^{12} - \)\(46\!\cdots\!28\)\( T^{14} + \)\(14\!\cdots\!21\)\( T^{16} \)
$79$ \( ( 1 - 431961140 T + 452072920533003676 T^{2} - \)\(14\!\cdots\!80\)\( T^{3} + \)\(79\!\cdots\!66\)\( T^{4} - \)\(17\!\cdots\!20\)\( T^{5} + \)\(64\!\cdots\!36\)\( T^{6} - \)\(74\!\cdots\!60\)\( T^{7} + \)\(20\!\cdots\!21\)\( T^{8} )^{2} \)
$83$ \( 1 - 981210392397464024 T^{2} + \)\(42\!\cdots\!20\)\( T^{4} - \)\(11\!\cdots\!28\)\( T^{6} + \)\(22\!\cdots\!98\)\( T^{8} - \)\(38\!\cdots\!52\)\( T^{10} + \)\(51\!\cdots\!20\)\( T^{12} - \)\(41\!\cdots\!96\)\( T^{14} + \)\(14\!\cdots\!61\)\( T^{16} \)
$89$ \( ( 1 - 178691112 T + 708605882924008892 T^{2} - \)\(39\!\cdots\!44\)\( T^{3} + \)\(23\!\cdots\!94\)\( T^{4} - \)\(13\!\cdots\!96\)\( T^{5} + \)\(86\!\cdots\!52\)\( T^{6} - \)\(76\!\cdots\!48\)\( T^{7} + \)\(15\!\cdots\!61\)\( T^{8} )^{2} \)
$97$ \( 1 - 1313769052010852360 T^{2} + \)\(18\!\cdots\!56\)\( T^{4} - \)\(20\!\cdots\!20\)\( T^{6} + \)\(15\!\cdots\!26\)\( T^{8} - \)\(11\!\cdots\!80\)\( T^{10} + \)\(62\!\cdots\!76\)\( T^{12} - \)\(25\!\cdots\!40\)\( T^{14} + \)\(11\!\cdots\!41\)\( T^{16} \)
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