Properties

Label 15.5.f
Level 15
Weight 5
Character orbit f
Rep. character \(\chi_{15}(7,\cdot)\)
Character field \(\Q(\zeta_{4})\)
Dimension 8
Newform subspaces 1
Sturm bound 10
Trace bound 0

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Defining parameters

Level: \( N \) \(=\) \( 15 = 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 15.f (of order \(4\) and degree \(2\))
Character conductor: \(\operatorname{cond}(\chi)\) \(=\) \( 5 \)
Character field: \(\Q(i)\)
Newform subspaces: \( 1 \)
Sturm bound: \(10\)
Trace bound: \(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(15, [\chi])\).

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

Trace form

\( 8q - 84q^{5} + 36q^{6} + 20q^{7} + 180q^{8} + O(q^{10}) \) \( 8q - 84q^{5} + 36q^{6} + 20q^{7} + 180q^{8} + 104q^{10} - 288q^{11} - 360q^{12} - 340q^{13} + 144q^{15} + 620q^{16} + 900q^{17} + 564q^{20} + 792q^{21} - 1100q^{22} - 1560q^{23} - 1204q^{25} - 3024q^{26} + 3580q^{28} - 2664q^{30} - 512q^{31} + 4980q^{32} + 2700q^{33} + 6600q^{35} + 2484q^{36} - 3820q^{37} - 7680q^{38} - 2952q^{40} - 2712q^{41} - 7380q^{42} - 1240q^{43} - 1944q^{45} + 13528q^{46} + 4800q^{47} + 3600q^{48} + 3744q^{50} + 6264q^{51} - 1240q^{52} + 1020q^{53} - 3644q^{55} - 30720q^{56} - 5400q^{57} + 2340q^{58} - 1044q^{60} - 4760q^{61} + 28680q^{62} + 540q^{63} - 1212q^{65} + 10008q^{66} - 8920q^{67} - 1920q^{68} + 7380q^{70} + 7536q^{71} - 4860q^{72} + 11600q^{73} - 5976q^{75} + 4344q^{76} - 360q^{77} - 4680q^{78} + 10644q^{80} - 5832q^{81} - 27200q^{82} - 32400q^{83} - 15628q^{85} + 14592q^{86} + 10620q^{87} - 14340q^{88} + 8964q^{90} + 16528q^{91} - 31800q^{92} + 14040q^{93} + 18864q^{95} - 4068q^{96} + 58640q^{97} + 46440q^{98} + O(q^{100}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(15, [\chi])\) into newform subspaces

Label Dim. \(A\) Field CM Traces $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
15.5.f.a \(8\) \(1.551\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(-84\) \(20\) \(q+\beta _{1}q^{2}+\beta _{5}q^{3}+(-\beta _{1}-12\beta _{2}-2\beta _{3}+\cdots)q^{4}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(15, [\chi])\) into lower level spaces

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - 60 T^{3} - 523 T^{4} - 420 T^{5} + 1800 T^{6} + 9480 T^{7} + 180804 T^{8} + 151680 T^{9} + 460800 T^{10} - 1720320 T^{11} - 34275328 T^{12} - 62914560 T^{13} + 4294967296 T^{16} \)
$3$ \( ( 1 + 729 T^{4} )^{2} \)
$5$ \( 1 + 84 T + 4130 T^{2} + 145500 T^{3} + 4037250 T^{4} + 90937500 T^{5} + 1613281250 T^{6} + 20507812500 T^{7} + 152587890625 T^{8} \)
$7$ \( 1 - 20 T + 200 T^{2} + 199700 T^{3} - 6227200 T^{4} + 79376300 T^{5} + 19597959000 T^{6} - 40112935980 T^{7} + 30640225075198 T^{8} - 96311159287980 T^{9} + 112978333641159000 T^{10} + 1098670165252736300 T^{11} - \)\(20\!\cdots\!00\)\( T^{12} + \)\(15\!\cdots\!00\)\( T^{13} + \)\(38\!\cdots\!00\)\( T^{14} - \)\(91\!\cdots\!20\)\( T^{15} + \)\(11\!\cdots\!01\)\( T^{16} \)
$11$ \( ( 1 + 144 T + 41822 T^{2} + 6150240 T^{3} + 823068186 T^{4} + 90045663840 T^{5} + 8964917121182 T^{6} + 451933686247824 T^{7} + 45949729863572161 T^{8} )^{2} \)
$13$ \( 1 + 340 T + 57800 T^{2} + 10577660 T^{3} + 3418178672 T^{4} + 809516994020 T^{5} + 133608496263000 T^{6} + 23843087441136780 T^{7} + 4251050341381084894 T^{8} + \)\(68\!\cdots\!80\)\( T^{9} + \)\(10\!\cdots\!00\)\( T^{10} + \)\(18\!\cdots\!20\)\( T^{11} + \)\(22\!\cdots\!52\)\( T^{12} + \)\(20\!\cdots\!60\)\( T^{13} + \)\(31\!\cdots\!00\)\( T^{14} + \)\(52\!\cdots\!40\)\( T^{15} + \)\(44\!\cdots\!81\)\( T^{16} \)
$17$ \( 1 - 900 T + 405000 T^{2} - 162344700 T^{3} + 70846529936 T^{4} - 26992676052900 T^{5} + 8778464632575000 T^{6} - 2859179891204281500 T^{7} + \)\(88\!\cdots\!86\)\( T^{8} - \)\(23\!\cdots\!00\)\( T^{9} + \)\(61\!\cdots\!00\)\( T^{10} - \)\(15\!\cdots\!00\)\( T^{11} + \)\(34\!\cdots\!16\)\( T^{12} - \)\(65\!\cdots\!00\)\( T^{13} + \)\(13\!\cdots\!00\)\( T^{14} - \)\(25\!\cdots\!00\)\( T^{15} + \)\(23\!\cdots\!61\)\( T^{16} \)
$19$ \( 1 - 475280 T^{2} + 118243003996 T^{4} - 19617731389431920 T^{6} + \)\(27\!\cdots\!66\)\( T^{8} - \)\(33\!\cdots\!20\)\( T^{10} + \)\(34\!\cdots\!76\)\( T^{12} - \)\(23\!\cdots\!80\)\( T^{14} + \)\(83\!\cdots\!61\)\( T^{16} \)
$23$ \( 1 + 1560 T + 1216800 T^{2} + 797095800 T^{3} + 382745999300 T^{4} + 92493403026600 T^{5} - 3754766037924000 T^{6} - 28706182737150138360 T^{7} - \)\(24\!\cdots\!22\)\( T^{8} - \)\(80\!\cdots\!60\)\( T^{9} - \)\(29\!\cdots\!00\)\( T^{10} + \)\(20\!\cdots\!00\)\( T^{11} + \)\(23\!\cdots\!00\)\( T^{12} + \)\(13\!\cdots\!00\)\( T^{13} + \)\(58\!\cdots\!00\)\( T^{14} + \)\(20\!\cdots\!60\)\( T^{15} + \)\(37\!\cdots\!21\)\( T^{16} \)
$29$ \( 1 - 3993236 T^{2} + 7931637401512 T^{4} - 9869586507702691580 T^{6} + \)\(83\!\cdots\!06\)\( T^{8} - \)\(49\!\cdots\!80\)\( T^{10} + \)\(19\!\cdots\!52\)\( T^{12} - \)\(49\!\cdots\!16\)\( T^{14} + \)\(62\!\cdots\!41\)\( T^{16} \)
$31$ \( ( 1 + 256 T + 1349068 T^{2} - 180990272 T^{3} + 1174676381974 T^{4} - 167148316987712 T^{5} + 1150608006098454988 T^{6} + \)\(20\!\cdots\!16\)\( T^{7} + \)\(72\!\cdots\!81\)\( T^{8} )^{2} \)
$37$ \( 1 + 3820 T + 7296200 T^{2} + 10974405380 T^{3} + 13446367756400 T^{4} + 10277980840235420 T^{5} + 1373285107740096600 T^{6} - \)\(20\!\cdots\!20\)\( T^{7} - \)\(46\!\cdots\!82\)\( T^{8} - \)\(38\!\cdots\!20\)\( T^{9} + \)\(48\!\cdots\!00\)\( T^{10} + \)\(67\!\cdots\!20\)\( T^{11} + \)\(16\!\cdots\!00\)\( T^{12} + \)\(25\!\cdots\!80\)\( T^{13} + \)\(31\!\cdots\!00\)\( T^{14} + \)\(31\!\cdots\!20\)\( T^{15} + \)\(15\!\cdots\!81\)\( T^{16} \)
$41$ \( ( 1 + 1356 T + 8325968 T^{2} + 6593365668 T^{3} + 29888933841054 T^{4} + 18631275563373348 T^{5} + 66482231940054114128 T^{6} + \)\(30\!\cdots\!36\)\( T^{7} + \)\(63\!\cdots\!41\)\( T^{8} )^{2} \)
$43$ \( 1 + 1240 T + 768800 T^{2} + 4155585560 T^{3} + 45689035751396 T^{4} + 43012988909739080 T^{5} + 26844821235643471200 T^{6} + \)\(14\!\cdots\!20\)\( T^{7} + \)\(79\!\cdots\!06\)\( T^{8} + \)\(49\!\cdots\!20\)\( T^{9} + \)\(31\!\cdots\!00\)\( T^{10} + \)\(17\!\cdots\!80\)\( T^{11} + \)\(62\!\cdots\!96\)\( T^{12} + \)\(19\!\cdots\!60\)\( T^{13} + \)\(12\!\cdots\!00\)\( T^{14} + \)\(67\!\cdots\!40\)\( T^{15} + \)\(18\!\cdots\!01\)\( T^{16} \)
$47$ \( 1 - 4800 T + 11520000 T^{2} - 34026427200 T^{3} + 139196198193956 T^{4} - 374777003457297600 T^{5} + \)\(77\!\cdots\!00\)\( T^{6} - \)\(20\!\cdots\!00\)\( T^{7} + \)\(52\!\cdots\!26\)\( T^{8} - \)\(98\!\cdots\!00\)\( T^{9} + \)\(18\!\cdots\!00\)\( T^{10} - \)\(43\!\cdots\!00\)\( T^{11} + \)\(78\!\cdots\!76\)\( T^{12} - \)\(94\!\cdots\!00\)\( T^{13} + \)\(15\!\cdots\!00\)\( T^{14} - \)\(31\!\cdots\!00\)\( T^{15} + \)\(32\!\cdots\!41\)\( T^{16} \)
$53$ \( 1 - 1020 T + 520200 T^{2} - 3711679620 T^{3} + 27373065594800 T^{4} - 31066466654514780 T^{5} + 24336610065951787800 T^{6} - \)\(12\!\cdots\!80\)\( T^{7} - \)\(51\!\cdots\!42\)\( T^{8} - \)\(95\!\cdots\!80\)\( T^{9} + \)\(15\!\cdots\!00\)\( T^{10} - \)\(15\!\cdots\!80\)\( T^{11} + \)\(10\!\cdots\!00\)\( T^{12} - \)\(11\!\cdots\!20\)\( T^{13} + \)\(12\!\cdots\!00\)\( T^{14} - \)\(19\!\cdots\!20\)\( T^{15} + \)\(15\!\cdots\!41\)\( T^{16} \)
$59$ \( 1 - 68266460 T^{2} + 2308748573454136 T^{4} - \)\(49\!\cdots\!40\)\( T^{6} + \)\(71\!\cdots\!06\)\( T^{8} - \)\(72\!\cdots\!40\)\( T^{10} + \)\(49\!\cdots\!76\)\( T^{12} - \)\(21\!\cdots\!60\)\( T^{14} + \)\(46\!\cdots\!81\)\( T^{16} \)
$61$ \( ( 1 + 2380 T + 35645776 T^{2} + 28234432420 T^{3} + 552631006355806 T^{4} + 390929462012565220 T^{5} + \)\(68\!\cdots\!56\)\( T^{6} + \)\(63\!\cdots\!80\)\( T^{7} + \)\(36\!\cdots\!61\)\( T^{8} )^{2} \)
$67$ \( 1 + 8920 T + 39783200 T^{2} + 240429760280 T^{3} + 743487179691236 T^{4} - 1082655630131333560 T^{5} - \)\(10\!\cdots\!00\)\( T^{6} - \)\(87\!\cdots\!40\)\( T^{7} - \)\(61\!\cdots\!14\)\( T^{8} - \)\(17\!\cdots\!40\)\( T^{9} - \)\(41\!\cdots\!00\)\( T^{10} - \)\(88\!\cdots\!60\)\( T^{11} + \)\(12\!\cdots\!16\)\( T^{12} + \)\(79\!\cdots\!80\)\( T^{13} + \)\(26\!\cdots\!00\)\( T^{14} + \)\(12\!\cdots\!20\)\( T^{15} + \)\(27\!\cdots\!61\)\( T^{16} \)
$71$ \( ( 1 - 3768 T + 59990708 T^{2} - 219866356776 T^{3} + 1753029620362470 T^{4} - 5587173721023900456 T^{5} + \)\(38\!\cdots\!88\)\( T^{6} - \)\(61\!\cdots\!88\)\( T^{7} + \)\(41\!\cdots\!21\)\( T^{8} )^{2} \)
$73$ \( 1 - 11600 T + 67280000 T^{2} - 460834857520 T^{3} + 3747670062415100 T^{4} - 21684328079009172880 T^{5} + \)\(10\!\cdots\!00\)\( T^{6} - \)\(63\!\cdots\!00\)\( T^{7} + \)\(38\!\cdots\!78\)\( T^{8} - \)\(18\!\cdots\!00\)\( T^{9} + \)\(85\!\cdots\!00\)\( T^{10} - \)\(49\!\cdots\!80\)\( T^{11} + \)\(24\!\cdots\!00\)\( T^{12} - \)\(85\!\cdots\!20\)\( T^{13} + \)\(35\!\cdots\!00\)\( T^{14} - \)\(17\!\cdots\!00\)\( T^{15} + \)\(42\!\cdots\!21\)\( T^{16} \)
$79$ \( 1 - 192979448 T^{2} + 19833375036960508 T^{4} - \)\(13\!\cdots\!96\)\( T^{6} + \)\(60\!\cdots\!70\)\( T^{8} - \)\(19\!\cdots\!56\)\( T^{10} + \)\(45\!\cdots\!68\)\( T^{12} - \)\(67\!\cdots\!88\)\( T^{14} + \)\(52\!\cdots\!41\)\( T^{16} \)
$83$ \( 1 + 32400 T + 524880000 T^{2} + 6014129561040 T^{3} + 60088627758116132 T^{4} + \)\(56\!\cdots\!80\)\( T^{5} + \)\(48\!\cdots\!00\)\( T^{6} + \)\(37\!\cdots\!80\)\( T^{7} + \)\(27\!\cdots\!14\)\( T^{8} + \)\(18\!\cdots\!80\)\( T^{9} + \)\(11\!\cdots\!00\)\( T^{10} + \)\(60\!\cdots\!80\)\( T^{11} + \)\(30\!\cdots\!92\)\( T^{12} + \)\(14\!\cdots\!40\)\( T^{13} + \)\(59\!\cdots\!00\)\( T^{14} + \)\(17\!\cdots\!00\)\( T^{15} + \)\(25\!\cdots\!61\)\( T^{16} \)
$89$ \( 1 - 372973760 T^{2} + 67192056713379196 T^{4} - \)\(75\!\cdots\!40\)\( T^{6} + \)\(56\!\cdots\!26\)\( T^{8} - \)\(29\!\cdots\!40\)\( T^{10} + \)\(10\!\cdots\!56\)\( T^{12} - \)\(22\!\cdots\!60\)\( T^{14} + \)\(24\!\cdots\!21\)\( T^{16} \)
$97$ \( 1 - 58640 T + 1719324800 T^{2} - 34474607566960 T^{3} + 547427197512783356 T^{4} - \)\(74\!\cdots\!80\)\( T^{5} + \)\(91\!\cdots\!00\)\( T^{6} - \)\(10\!\cdots\!20\)\( T^{7} + \)\(10\!\cdots\!26\)\( T^{8} - \)\(89\!\cdots\!20\)\( T^{9} + \)\(71\!\cdots\!00\)\( T^{10} - \)\(51\!\cdots\!80\)\( T^{11} + \)\(33\!\cdots\!76\)\( T^{12} - \)\(18\!\cdots\!60\)\( T^{13} + \)\(82\!\cdots\!00\)\( T^{14} - \)\(24\!\cdots\!40\)\( T^{15} + \)\(37\!\cdots\!41\)\( T^{16} \)
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