Properties

Label 15.8
Level 15
Weight 8
Dimension 36
Nonzero newspaces 3
Newform subspaces 5
Sturm bound 128
Trace bound 1

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Defining parameters

Level: \( N \) = \( 15 = 3 \cdot 5 \)
Weight: \( k \) = \( 8 \)
Nonzero newspaces: \( 3 \)
Newform subspaces: \( 5 \)
Sturm bound: \(128\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_1(15))\).

Total New Old
Modular forms 64 44 20
Cusp forms 48 36 12
Eisenstein series 16 8 8

Trace form

\( 36q - 28q^{2} + 78q^{3} - 200q^{4} - 444q^{5} + 1356q^{6} + 3608q^{7} - 2436q^{8} - 2916q^{9} + O(q^{10}) \) \( 36q - 28q^{2} + 78q^{3} - 200q^{4} - 444q^{5} + 1356q^{6} + 3608q^{7} - 2436q^{8} - 2916q^{9} - 4616q^{10} + 7952q^{11} + 8340q^{12} + 5360q^{13} - 34488q^{14} + 26814q^{15} + 38000q^{16} - 32032q^{17} - 126972q^{18} - 64176q^{19} + 46444q^{20} + 134904q^{21} + 146376q^{22} + 170088q^{23} - 237168q^{24} - 85644q^{25} - 445816q^{26} - 224802q^{27} + 436280q^{28} + 717688q^{29} + 774336q^{30} + 71568q^{31} - 1063916q^{32} - 774984q^{33} + 238760q^{34} + 636520q^{35} + 569016q^{36} - 789232q^{37} - 769432q^{38} - 1224396q^{39} - 2054392q^{40} + 883240q^{41} + 1622112q^{42} + 976328q^{43} + 537976q^{44} + 893916q^{45} + 4371848q^{46} + 1210520q^{47} + 1149684q^{48} - 2687956q^{49} - 5555236q^{50} - 4088820q^{51} - 8572528q^{52} + 1575488q^{53} - 393660q^{54} + 7062576q^{55} + 6479040q^{56} + 5320296q^{57} - 1200264q^{58} - 1197184q^{59} + 6515256q^{60} + 1179144q^{61} + 5573928q^{62} - 323496q^{63} + 9724488q^{64} - 3366592q^{65} - 30638208q^{66} - 21756808q^{67} - 9434504q^{68} - 5890968q^{69} - 1513560q^{70} - 4007216q^{71} + 34470036q^{72} + 6676736q^{73} + 20919712q^{74} + 19838094q^{75} + 37710128q^{76} + 6733248q^{77} - 10796952q^{78} - 10874880q^{79} - 39917876q^{80} - 24912684q^{81} - 40630392q^{82} - 17321256q^{83} - 43954272q^{84} - 33002768q^{85} - 6212848q^{86} + 50204484q^{87} + 97866504q^{88} + 58210824q^{89} + 80324424q^{90} + 88309648q^{91} + 15479424q^{92} - 54464928q^{93} - 78524776q^{94} - 33956656q^{95} - 104433168q^{96} - 63625168q^{97} - 21248492q^{98} - 9879408q^{99} + O(q^{100}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_1(15))\)

We only show spaces with even parity, since no modular forms exist when this condition is not satisfied. Within each space \( S_k^{\mathrm{new}}(N, \chi) \) we list the newforms together with their dimension.

Label \(\chi\) Newforms Dimension \(\chi\) degree
15.8.a \(\chi_{15}(1, \cdot)\) 15.8.a.a 1 1
15.8.a.b 1
15.8.a.c 2
15.8.b \(\chi_{15}(4, \cdot)\) 15.8.b.a 8 1
15.8.e \(\chi_{15}(2, \cdot)\) 15.8.e.a 24 2

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_1(15))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_1(15)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_1(5))\)\(^{\oplus 2}\)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ (\( 1 + 22 T + 128 T^{2} \))(\( 1 + 13 T + 128 T^{2} \))(\( 1 - 7 T + 118 T^{2} - 896 T^{3} + 16384 T^{4} \))(\( 1 - 179 T^{2} + 15700 T^{4} - 2649728 T^{6} + 475871488 T^{8} - 43413143552 T^{10} + 4214436659200 T^{12} - 787250325487616 T^{14} + 72057594037927936 T^{16} \))
$3$ (\( 1 - 27 T \))(\( 1 + 27 T \))(\( ( 1 - 27 T )^{2} \))(\( ( 1 + 729 T^{2} )^{4} \))
$5$ (\( 1 + 125 T \))(\( 1 + 125 T \))(\( ( 1 - 125 T )^{2} \))(\( 1 + 444 T + 34760 T^{2} + 8158500 T^{3} + 7075218750 T^{4} + 637382812500 T^{5} + 212158203125000 T^{6} + 211715698242187500 T^{7} + 37252902984619140625 T^{8} \))
$7$ (\( 1 + 420 T + 823543 T^{2} \))(\( 1 - 1380 T + 823543 T^{2} \))(\( 1 - 1304 T + 1601006 T^{2} - 1073900072 T^{3} + 678223072849 T^{4} \))(\( 1 - 1660592 T^{2} + 2653149448732 T^{4} - 2679181372904542544 T^{6} + \)\(25\!\cdots\!34\)\( T^{8} - \)\(18\!\cdots\!56\)\( T^{10} + \)\(12\!\cdots\!32\)\( T^{12} - \)\(51\!\cdots\!08\)\( T^{14} + \)\(21\!\cdots\!01\)\( T^{16} \))
$11$ (\( 1 + 2944 T + 19487171 T^{2} \))(\( 1 + 3304 T + 19487171 T^{2} \))(\( 1 - 3448 T + 9598294 T^{2} - 67191765608 T^{3} + 379749833583241 T^{4} \))(\( ( 1 - 5376 T + 68059592 T^{2} - 265739821920 T^{3} + 1856153151437406 T^{4} - 5178517351264588320 T^{5} + \)\(25\!\cdots\!72\)\( T^{6} - \)\(39\!\cdots\!36\)\( T^{7} + \)\(14\!\cdots\!81\)\( T^{8} )^{2} \))
$13$ (\( 1 + 11006 T + 62748517 T^{2} \))(\( 1 - 8506 T + 62748517 T^{2} \))(\( 1 + 8988 T + 43676926 T^{2} + 563983670796 T^{3} + 3937376385699289 T^{4} \))(\( 1 - 184606016 T^{2} + 18820421105713180 T^{4} - \)\(14\!\cdots\!52\)\( T^{6} + \)\(10\!\cdots\!98\)\( T^{8} - \)\(58\!\cdots\!28\)\( T^{10} + \)\(29\!\cdots\!80\)\( T^{12} - \)\(11\!\cdots\!04\)\( T^{14} + \)\(24\!\cdots\!41\)\( T^{16} \))
$17$ (\( 1 + 16546 T + 410338673 T^{2} \))(\( 1 + 9994 T + 410338673 T^{2} \))(\( 1 + 5492 T + 533727862 T^{2} + 2253579992116 T^{3} + 168377826559400929 T^{4} \))(\( 1 - 1956231680 T^{2} + 1757320974428357116 T^{4} - \)\(10\!\cdots\!60\)\( T^{6} + \)\(44\!\cdots\!46\)\( T^{8} - \)\(16\!\cdots\!40\)\( T^{10} + \)\(49\!\cdots\!56\)\( T^{12} - \)\(93\!\cdots\!20\)\( T^{14} + \)\(80\!\cdots\!81\)\( T^{16} \))
$19$ (\( 1 + 25364 T + 893871739 T^{2} \))(\( 1 - 41236 T + 893871739 T^{2} \))(\( 1 + 49584 T + 1767259558 T^{2} + 44321736306576 T^{3} + 799006685782884121 T^{4} \))(\( ( 1 + 15232 T + 392774572 T^{2} + 22311843627904 T^{3} + 1443987068893239574 T^{4} + \)\(19\!\cdots\!56\)\( T^{5} + \)\(31\!\cdots\!12\)\( T^{6} + \)\(10\!\cdots\!08\)\( T^{7} + \)\(63\!\cdots\!41\)\( T^{8} )^{2} \))
$23$ (\( 1 + 5880 T + 3404825447 T^{2} \))(\( 1 - 84120 T + 3404825447 T^{2} \))(\( 1 - 91848 T + 4843394254 T^{2} - 312726407656056 T^{3} + 11592836324538749809 T^{4} \))(\( 1 - 17784361208 T^{2} + \)\(15\!\cdots\!32\)\( T^{4} - \)\(86\!\cdots\!36\)\( T^{6} + \)\(34\!\cdots\!14\)\( T^{8} - \)\(10\!\cdots\!24\)\( T^{10} + \)\(20\!\cdots\!92\)\( T^{12} - \)\(27\!\cdots\!32\)\( T^{14} + \)\(18\!\cdots\!61\)\( T^{16} \))
$29$ (\( 1 - 163042 T + 17249876309 T^{2} \))(\( 1 - 132802 T + 17249876309 T^{2} \))(\( 1 - 181772 T + 32439203278 T^{2} - 3135544516439548 T^{3} + \)\(29\!\cdots\!81\)\( T^{4} \))(\( ( 1 - 120036 T + 67036689080 T^{2} - 5675627737126332 T^{3} + \)\(17\!\cdots\!78\)\( T^{4} - \)\(97\!\cdots\!88\)\( T^{5} + \)\(19\!\cdots\!80\)\( T^{6} - \)\(61\!\cdots\!44\)\( T^{7} + \)\(88\!\cdots\!61\)\( T^{8} )^{2} \))
$31$ (\( 1 + 201600 T + 27512614111 T^{2} \))(\( 1 + 55800 T + 27512614111 T^{2} \))(\( 1 - 304232 T + 77458297022 T^{2} - 8370217616217752 T^{3} + \)\(75\!\cdots\!21\)\( T^{4} \))(\( ( 1 - 116864 T + 81093883468 T^{2} - 6882918957523712 T^{3} + \)\(30\!\cdots\!54\)\( T^{4} - \)\(18\!\cdots\!32\)\( T^{5} + \)\(61\!\cdots\!28\)\( T^{6} - \)\(24\!\cdots\!84\)\( T^{7} + \)\(57\!\cdots\!41\)\( T^{8} )^{2} \))
$37$ (\( 1 - 120530 T + 94931877133 T^{2} \))(\( 1 - 228170 T + 94931877133 T^{2} \))(\( 1 + 502316 T + 221684315886 T^{2} + 47685800793940028 T^{3} + \)\(90\!\cdots\!89\)\( T^{4} \))(\( 1 - 277591519232 T^{2} + \)\(45\!\cdots\!12\)\( T^{4} - \)\(56\!\cdots\!44\)\( T^{6} + \)\(56\!\cdots\!94\)\( T^{8} - \)\(50\!\cdots\!16\)\( T^{10} + \)\(36\!\cdots\!52\)\( T^{12} - \)\(20\!\cdots\!08\)\( T^{14} + \)\(65\!\cdots\!41\)\( T^{16} \))
$41$ (\( 1 + 115910 T + 194754273881 T^{2} \))(\( 1 + 139670 T + 194754273881 T^{2} \))(\( 1 - 631172 T + 420346017142 T^{2} - 122923444554018532 T^{3} + \)\(37\!\cdots\!61\)\( T^{4} \))(\( ( 1 - 253824 T + 334342452188 T^{2} - 27270984718092672 T^{3} + \)\(66\!\cdots\!34\)\( T^{4} - \)\(53\!\cdots\!32\)\( T^{5} + \)\(12\!\cdots\!68\)\( T^{6} - \)\(18\!\cdots\!84\)\( T^{7} + \)\(14\!\cdots\!21\)\( T^{8} )^{2} \))
$43$ (\( 1 + 19148 T + 271818611107 T^{2} \))(\( 1 + 755492 T + 271818611107 T^{2} \))(\( 1 - 353640 T + 567251429590 T^{2} - 96125933631879480 T^{3} + \)\(73\!\cdots\!49\)\( T^{4} \))(\( 1 - 898135198040 T^{2} + \)\(48\!\cdots\!96\)\( T^{4} - \)\(17\!\cdots\!80\)\( T^{6} + \)\(53\!\cdots\!06\)\( T^{8} - \)\(13\!\cdots\!20\)\( T^{10} + \)\(26\!\cdots\!96\)\( T^{12} - \)\(36\!\cdots\!60\)\( T^{14} + \)\(29\!\cdots\!01\)\( T^{16} \))
$47$ (\( 1 - 841016 T + 506623120463 T^{2} \))(\( 1 - 836984 T + 506623120463 T^{2} \))(\( 1 + 467480 T + 1062629128990 T^{2} + 236836176354043240 T^{3} + \)\(25\!\cdots\!69\)\( T^{4} \))(\( 1 - 1825527695960 T^{2} + \)\(18\!\cdots\!76\)\( T^{4} - \)\(13\!\cdots\!20\)\( T^{6} + \)\(78\!\cdots\!66\)\( T^{8} - \)\(35\!\cdots\!80\)\( T^{10} + \)\(12\!\cdots\!36\)\( T^{12} - \)\(30\!\cdots\!40\)\( T^{14} + \)\(43\!\cdots\!21\)\( T^{16} \))
$53$ (\( 1 - 501890 T + 1174711139837 T^{2} \))(\( 1 - 1641650 T + 1174711139837 T^{2} \))(\( 1 + 568052 T + 2403786403294 T^{2} + 667297012406687524 T^{3} + \)\(13\!\cdots\!69\)\( T^{4} \))(\( 1 - 5984391843968 T^{2} + \)\(16\!\cdots\!12\)\( T^{4} - \)\(30\!\cdots\!96\)\( T^{6} + \)\(40\!\cdots\!34\)\( T^{8} - \)\(42\!\cdots\!24\)\( T^{10} + \)\(32\!\cdots\!32\)\( T^{12} - \)\(15\!\cdots\!12\)\( T^{14} + \)\(36\!\cdots\!21\)\( T^{16} \))
$59$ (\( 1 + 1586176 T + 2488651484819 T^{2} \))(\( 1 + 989656 T + 2488651484819 T^{2} \))(\( 1 - 287224 T + 1627391637238 T^{2} - 714800434075652456 T^{3} + \)\(61\!\cdots\!61\)\( T^{4} \))(\( ( 1 - 545712 T + 6864766660712 T^{2} - 4026585148405309584 T^{3} + \)\(21\!\cdots\!34\)\( T^{4} - \)\(10\!\cdots\!96\)\( T^{5} + \)\(42\!\cdots\!32\)\( T^{6} - \)\(84\!\cdots\!08\)\( T^{7} + \)\(38\!\cdots\!21\)\( T^{8} )^{2} \))
$61$ (\( 1 + 372962 T + 3142742836021 T^{2} \))(\( 1 + 1658162 T + 3142742836021 T^{2} \))(\( 1 + 2514180 T + 7865442419758 T^{2} + 7901421183467277780 T^{3} + \)\(98\!\cdots\!41\)\( T^{4} \))(\( ( 1 + 3216760 T + 11354186987116 T^{2} + 29259400663039022440 T^{3} + \)\(51\!\cdots\!46\)\( T^{4} + \)\(91\!\cdots\!40\)\( T^{5} + \)\(11\!\cdots\!56\)\( T^{6} + \)\(99\!\cdots\!60\)\( T^{7} + \)\(97\!\cdots\!81\)\( T^{8} )^{2} \))
$67$ (\( 1 - 4561044 T + 6060711605323 T^{2} \))(\( 1 + 4523844 T + 6060711605323 T^{2} \))(\( 1 + 5073832 T + 16021265240102 T^{2} + 30751032485859207736 T^{3} + \)\(36\!\cdots\!29\)\( T^{4} \))(\( 1 - 24610217983160 T^{2} + \)\(30\!\cdots\!16\)\( T^{4} - \)\(27\!\cdots\!20\)\( T^{6} + \)\(18\!\cdots\!46\)\( T^{8} - \)\(99\!\cdots\!80\)\( T^{10} + \)\(41\!\cdots\!56\)\( T^{12} - \)\(12\!\cdots\!40\)\( T^{14} + \)\(18\!\cdots\!81\)\( T^{16} \))
$71$ (\( 1 - 1512832 T + 9095120158391 T^{2} \))(\( 1 + 389408 T + 9095120158391 T^{2} \))(\( 1 + 3748816 T + 20824804809646 T^{2} + 34095931971698715056 T^{3} + \)\(82\!\cdots\!81\)\( T^{4} \))(\( ( 1 + 690912 T + 24146587182668 T^{2} + 3374261752345932384 T^{3} + \)\(28\!\cdots\!70\)\( T^{4} + \)\(30\!\cdots\!44\)\( T^{5} + \)\(19\!\cdots\!08\)\( T^{6} + \)\(51\!\cdots\!52\)\( T^{7} + \)\(68\!\cdots\!61\)\( T^{8} )^{2} \))
$73$ (\( 1 + 1522910 T + 11047398519097 T^{2} \))(\( 1 - 5617330 T + 11047398519097 T^{2} \))(\( 1 + 1477212 T - 3158190986 T^{2} + 16319349661192317564 T^{3} + \)\(12\!\cdots\!09\)\( T^{4} \))(\( 1 - 51485713155368 T^{2} + \)\(13\!\cdots\!12\)\( T^{4} - \)\(23\!\cdots\!16\)\( T^{6} + \)\(30\!\cdots\!54\)\( T^{8} - \)\(29\!\cdots\!44\)\( T^{10} + \)\(20\!\cdots\!72\)\( T^{12} - \)\(93\!\cdots\!72\)\( T^{14} + \)\(22\!\cdots\!61\)\( T^{16} \))
$79$ (\( 1 - 4231920 T + 19203908986159 T^{2} \))(\( 1 - 3901080 T + 19203908986159 T^{2} \))(\( 1 + 4627720 T + 42789559383518 T^{2} + 88870313693427727480 T^{3} + \)\(36\!\cdots\!81\)\( T^{4} \))(\( ( 1 + 7190080 T + 89811290861836 T^{2} + \)\(40\!\cdots\!60\)\( T^{3} + \)\(26\!\cdots\!86\)\( T^{4} + \)\(78\!\cdots\!40\)\( T^{5} + \)\(33\!\cdots\!16\)\( T^{6} + \)\(50\!\cdots\!20\)\( T^{7} + \)\(13\!\cdots\!61\)\( T^{8} )^{2} \))
$83$ (\( 1 + 1854204 T + 27136050989627 T^{2} \))(\( 1 + 9394116 T + 27136050989627 T^{2} \))(\( 1 + 6072936 T + 62224060120582 T^{2} + \)\(16\!\cdots\!72\)\( T^{3} + \)\(73\!\cdots\!29\)\( T^{4} \))(\( 1 - 169657932472856 T^{2} + \)\(13\!\cdots\!80\)\( T^{4} - \)\(67\!\cdots\!12\)\( T^{6} + \)\(22\!\cdots\!58\)\( T^{8} - \)\(49\!\cdots\!48\)\( T^{10} + \)\(74\!\cdots\!80\)\( T^{12} - \)\(67\!\cdots\!84\)\( T^{14} + \)\(29\!\cdots\!81\)\( T^{16} \))
$89$ (\( 1 + 6888174 T + 44231334895529 T^{2} \))(\( 1 - 2803746 T + 44231334895529 T^{2} \))(\( 1 - 16516356 T + 156597055746838 T^{2} - \)\(73\!\cdots\!24\)\( T^{3} + \)\(19\!\cdots\!41\)\( T^{4} \))(\( ( 1 - 22889448 T + 297109023037052 T^{2} - \)\(26\!\cdots\!16\)\( T^{3} + \)\(19\!\cdots\!34\)\( T^{4} - \)\(11\!\cdots\!64\)\( T^{5} + \)\(58\!\cdots\!32\)\( T^{6} - \)\(19\!\cdots\!72\)\( T^{7} + \)\(38\!\cdots\!81\)\( T^{8} )^{2} \))
$97$ (\( 1 - 3700034 T + 80798284478113 T^{2} \))(\( 1 - 5099426 T + 80798284478113 T^{2} \))(\( 1 - 2723428 T + 135845063471622 T^{2} - \)\(22\!\cdots\!64\)\( T^{3} + \)\(65\!\cdots\!69\)\( T^{4} \))(\( 1 - 563431252495880 T^{2} + \)\(14\!\cdots\!76\)\( T^{4} - \)\(22\!\cdots\!60\)\( T^{6} + \)\(22\!\cdots\!66\)\( T^{8} - \)\(14\!\cdots\!40\)\( T^{10} + \)\(61\!\cdots\!36\)\( T^{12} - \)\(15\!\cdots\!20\)\( T^{14} + \)\(18\!\cdots\!21\)\( T^{16} \))
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