Properties

Label 9.6
Level 9
Weight 6
Dimension 9
Nonzero newspaces 2
Newform subspaces 2
Sturm bound 36
Trace bound 1

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Defining parameters

Level: \( N \) = \( 9\( 9 = 3^{2} \) \)
Weight: \( k \) = \( 6 \)
Nonzero newspaces: \( 2 \)
Newform subspaces: \( 2 \)
Sturm bound: \(36\)
Trace bound: \(1\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_1(9))\).

Total New Old
Modular forms 19 14 5
Cusp forms 11 9 2
Eisenstein series 8 5 3

Trace form

\( 9q + 9q^{2} - 12q^{3} - 45q^{4} + 72q^{5} + 171q^{6} - 12q^{7} - 918q^{8} - 414q^{9} + O(q^{10}) \) \( 9q + 9q^{2} - 12q^{3} - 45q^{4} + 72q^{5} + 171q^{6} - 12q^{7} - 918q^{8} - 414q^{9} + 24q^{10} + 1008q^{11} + 2724q^{12} + 456q^{13} + 1152q^{14} - 2052q^{15} - 1425q^{16} - 5238q^{17} - 8100q^{18} + 396q^{19} + 6660q^{20} + 8670q^{21} + 4395q^{22} + 9684q^{23} + 549q^{24} - 4743q^{25} - 21060q^{26} - 10152q^{27} - 1764q^{28} + 7380q^{29} + 22104q^{30} + 5532q^{31} + 7263q^{32} - 8820q^{33} + 4833q^{34} - 15984q^{35} - 14589q^{36} - 17586q^{37} - 14481q^{38} + 8220q^{39} - 7728q^{40} + 8118q^{41} + 10098q^{42} + 3552q^{43} + 51786q^{44} + 43038q^{45} + 51000q^{46} + 18612q^{47} - 57405q^{48} - 6117q^{49} - 75591q^{50} - 9396q^{51} - 29958q^{52} - 12798q^{53} + 8181q^{54} - 39504q^{55} - 54450q^{56} - 104298q^{57} - 19500q^{58} + 20160q^{59} + 88308q^{60} + 87900q^{61} + 268164q^{62} + 133524q^{63} + 6786q^{64} - 16974q^{65} - 100998q^{66} - 30216q^{67} - 126657q^{68} - 125982q^{69} + 786q^{70} - 67608q^{71} + 35451q^{72} + 81342q^{73} - 174780q^{74} - 18732q^{75} - 51795q^{76} + 16974q^{77} + 181422q^{78} - 37596q^{79} + 131328q^{80} + 194562q^{81} - 192378q^{82} + 35244q^{83} + 108354q^{84} + 34128q^{85} + 41949q^{86} - 142956q^{87} + 9771q^{88} - 357642q^{89} - 539892q^{90} + 85872q^{91} + 137394q^{92} + 113898q^{93} + 281496q^{94} + 298224q^{95} + 42768q^{96} + 83418q^{97} - 33588q^{98} + 33696q^{99} + O(q^{100}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_1(9))\)

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
9.6.a \(\chi_{9}(1, \cdot)\) 9.6.a.a 1 1
9.6.c \(\chi_{9}(4, \cdot)\) 9.6.c.a 8 2

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_1(9))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_1(9)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_1(3))\)\(^{\oplus 2}\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ (\( 1 - 6 T + 32 T^{2} \))(\( 1 - 3 T - 35 T^{2} + 300 T^{3} - 272 T^{4} - 7056 T^{5} + 47392 T^{6} + 37824 T^{7} - 1500608 T^{8} + 1210368 T^{9} + 48529408 T^{10} - 231211008 T^{11} - 285212672 T^{12} + 10066329600 T^{13} - 37580963840 T^{14} - 103079215104 T^{15} + 1099511627776 T^{16} \))
$3$ (\( 1 + 12 T + 279 T^{2} + 6156 T^{3} + 29160 T^{4} + 1495908 T^{5} + 16474671 T^{6} + 172186884 T^{7} + 3486784401 T^{8} \))
$5$ (\( 1 + 6 T + 3125 T^{2} \))(\( 1 - 78 T - 2381 T^{2} + 191094 T^{3} + 7253005 T^{4} - 66450636 T^{5} - 21330994286 T^{6} + 411629383632 T^{7} - 16622006693174 T^{8} + 1286341823850000 T^{9} - 208310491074218750 T^{10} - 2027912475585937500 T^{11} + \)\(69\!\cdots\!25\)\( T^{12} + \)\(56\!\cdots\!50\)\( T^{13} - \)\(22\!\cdots\!25\)\( T^{14} - \)\(22\!\cdots\!50\)\( T^{15} + \)\(90\!\cdots\!25\)\( T^{16} \))
$7$ (\( 1 + 40 T + 16807 T^{2} \))(\( 1 - 28 T - 37767 T^{2} - 688508 T^{3} + 752155661 T^{4} + 29978584680 T^{5} - 6360743348222 T^{6} - 326958516789136 T^{7} + 37987499418745374 T^{8} - 5495191791675008752 T^{9} - \)\(17\!\cdots\!78\)\( T^{10} + \)\(14\!\cdots\!40\)\( T^{11} + \)\(60\!\cdots\!61\)\( T^{12} - \)\(92\!\cdots\!56\)\( T^{13} - \)\(85\!\cdots\!83\)\( T^{14} - \)\(10\!\cdots\!04\)\( T^{15} + \)\(63\!\cdots\!01\)\( T^{16} \))
$11$ (\( 1 - 564 T + 161051 T^{2} \))(\( 1 - 444 T - 421874 T^{2} + 122372328 T^{3} + 147603288625 T^{4} - 24043789512072 T^{5} - 33881306185300610 T^{6} + 1343287267601416260 T^{7} + \)\(64\!\cdots\!56\)\( T^{8} + \)\(21\!\cdots\!60\)\( T^{9} - \)\(87\!\cdots\!10\)\( T^{10} - \)\(10\!\cdots\!72\)\( T^{11} + \)\(99\!\cdots\!25\)\( T^{12} + \)\(13\!\cdots\!28\)\( T^{13} - \)\(73\!\cdots\!74\)\( T^{14} - \)\(12\!\cdots\!44\)\( T^{15} + \)\(45\!\cdots\!01\)\( T^{16} \))
$13$ (\( 1 - 638 T + 371293 T^{2} \))(\( 1 + 182 T - 1009641 T^{2} - 162494414 T^{3} + 553084541681 T^{4} + 66263508549516 T^{5} - 217064082764017418 T^{6} - 12452829446314121656 T^{7} + \)\(75\!\cdots\!98\)\( T^{8} - \)\(46\!\cdots\!08\)\( T^{9} - \)\(29\!\cdots\!82\)\( T^{10} + \)\(33\!\cdots\!12\)\( T^{11} + \)\(10\!\cdots\!81\)\( T^{12} - \)\(11\!\cdots\!02\)\( T^{13} - \)\(26\!\cdots\!09\)\( T^{14} + \)\(17\!\cdots\!74\)\( T^{15} + \)\(36\!\cdots\!01\)\( T^{16} \))
$17$ (\( 1 + 882 T + 1419857 T^{2} \))(\( ( 1 + 2178 T + 4502417 T^{2} + 6473305458 T^{3} + 7835841277908 T^{4} + 9191168067679506 T^{5} + 9076845209277885233 T^{6} + \)\(62\!\cdots\!54\)\( T^{7} + \)\(40\!\cdots\!01\)\( T^{8} )^{2} \))
$19$ (\( 1 + 556 T + 2476099 T^{2} \))(\( ( 1 - 476 T + 5379895 T^{2} - 1785507140 T^{3} + 16120413258280 T^{4} - 4421092443846860 T^{5} + 32984492705012310895 T^{6} - \)\(72\!\cdots\!24\)\( T^{7} + \)\(37\!\cdots\!01\)\( T^{8} )^{2} \))
$23$ (\( 1 - 840 T + 6436343 T^{2} \))(\( 1 - 8844 T + 29643313 T^{2} - 58247545548 T^{3} + 172348926961837 T^{4} - 574153269020383320 T^{5} + \)\(14\!\cdots\!02\)\( T^{6} - \)\(52\!\cdots\!88\)\( T^{7} + \)\(17\!\cdots\!06\)\( T^{8} - \)\(33\!\cdots\!84\)\( T^{9} + \)\(60\!\cdots\!98\)\( T^{10} - \)\(15\!\cdots\!40\)\( T^{11} + \)\(29\!\cdots\!37\)\( T^{12} - \)\(64\!\cdots\!64\)\( T^{13} + \)\(21\!\cdots\!37\)\( T^{14} - \)\(40\!\cdots\!08\)\( T^{15} + \)\(29\!\cdots\!01\)\( T^{16} \))
$29$ (\( 1 + 4638 T + 20511149 T^{2} \))(\( 1 - 12018 T + 15291415 T^{2} + 58796364810 T^{3} + 2512265879646529 T^{4} - 12711396421291774644 T^{5} - \)\(20\!\cdots\!74\)\( T^{6} - \)\(10\!\cdots\!36\)\( T^{7} + \)\(21\!\cdots\!82\)\( T^{8} - \)\(21\!\cdots\!64\)\( T^{9} - \)\(86\!\cdots\!74\)\( T^{10} - \)\(10\!\cdots\!56\)\( T^{11} + \)\(44\!\cdots\!29\)\( T^{12} + \)\(21\!\cdots\!90\)\( T^{13} + \)\(11\!\cdots\!15\)\( T^{14} - \)\(18\!\cdots\!82\)\( T^{15} + \)\(31\!\cdots\!01\)\( T^{16} \))
$31$ (\( 1 - 4400 T + 28629151 T^{2} \))(\( 1 - 1132 T - 60658635 T^{2} + 62431167700 T^{3} + 1166392900733297 T^{4} - 126919079777751576 T^{5} - \)\(56\!\cdots\!38\)\( T^{6} - \)\(20\!\cdots\!72\)\( T^{7} + \)\(28\!\cdots\!06\)\( T^{8} - \)\(60\!\cdots\!72\)\( T^{9} - \)\(46\!\cdots\!38\)\( T^{10} - \)\(29\!\cdots\!76\)\( T^{11} + \)\(78\!\cdots\!97\)\( T^{12} + \)\(12\!\cdots\!00\)\( T^{13} - \)\(33\!\cdots\!35\)\( T^{14} - \)\(17\!\cdots\!32\)\( T^{15} + \)\(45\!\cdots\!01\)\( T^{16} \))
$37$ (\( 1 + 2410 T + 69343957 T^{2} \))(\( ( 1 + 7588 T + 181007992 T^{2} + 1702118220604 T^{3} + 15462039197985406 T^{4} + \)\(11\!\cdots\!28\)\( T^{5} + \)\(87\!\cdots\!08\)\( T^{6} + \)\(25\!\cdots\!84\)\( T^{7} + \)\(23\!\cdots\!01\)\( T^{8} )^{2} \))
$41$ (\( 1 - 6870 T + 115856201 T^{2} \))(\( 1 - 1248 T - 412882754 T^{2} + 256860139200 T^{3} + 101750473497083809 T^{4} - 31706402029277721984 T^{5} - \)\(17\!\cdots\!98\)\( T^{6} + \)\(13\!\cdots\!52\)\( T^{7} + \)\(23\!\cdots\!00\)\( T^{8} + \)\(15\!\cdots\!52\)\( T^{9} - \)\(23\!\cdots\!98\)\( T^{10} - \)\(49\!\cdots\!84\)\( T^{11} + \)\(18\!\cdots\!09\)\( T^{12} + \)\(53\!\cdots\!00\)\( T^{13} - \)\(99\!\cdots\!54\)\( T^{14} - \)\(34\!\cdots\!48\)\( T^{15} + \)\(32\!\cdots\!01\)\( T^{16} \))
$43$ (\( 1 - 9644 T + 147008443 T^{2} \))(\( 1 + 6092 T - 493633434 T^{2} - 1717030281896 T^{3} + 152938627936979465 T^{4} + \)\(29\!\cdots\!88\)\( T^{5} - \)\(33\!\cdots\!22\)\( T^{6} - \)\(15\!\cdots\!68\)\( T^{7} + \)\(57\!\cdots\!20\)\( T^{8} - \)\(23\!\cdots\!24\)\( T^{9} - \)\(72\!\cdots\!78\)\( T^{10} + \)\(92\!\cdots\!16\)\( T^{11} + \)\(71\!\cdots\!65\)\( T^{12} - \)\(11\!\cdots\!28\)\( T^{13} - \)\(49\!\cdots\!66\)\( T^{14} + \)\(90\!\cdots\!44\)\( T^{15} + \)\(21\!\cdots\!01\)\( T^{16} \))
$47$ (\( 1 - 18672 T + 229345007 T^{2} \))(\( 1 + 60 T - 695011823 T^{2} + 3293074754652 T^{3} + 272693815319492413 T^{4} - \)\(15\!\cdots\!00\)\( T^{5} - \)\(70\!\cdots\!78\)\( T^{6} + \)\(17\!\cdots\!56\)\( T^{7} + \)\(15\!\cdots\!82\)\( T^{8} + \)\(41\!\cdots\!92\)\( T^{9} - \)\(37\!\cdots\!22\)\( T^{10} - \)\(18\!\cdots\!00\)\( T^{11} + \)\(75\!\cdots\!13\)\( T^{12} + \)\(20\!\cdots\!64\)\( T^{13} - \)\(10\!\cdots\!27\)\( T^{14} + \)\(20\!\cdots\!80\)\( T^{15} + \)\(76\!\cdots\!01\)\( T^{16} \))
$53$ (\( 1 + 33750 T + 418195493 T^{2} \))(\( ( 1 - 10476 T + 1064413976 T^{2} - 16431260960628 T^{3} + 549720542476264830 T^{4} - \)\(68\!\cdots\!04\)\( T^{5} + \)\(18\!\cdots\!24\)\( T^{6} - \)\(76\!\cdots\!32\)\( T^{7} + \)\(30\!\cdots\!01\)\( T^{8} )^{2} \))
$59$ (\( 1 - 18084 T + 714924299 T^{2} \))(\( 1 - 2076 T - 2248193858 T^{2} + 15695201332392 T^{3} + 2863253985413117089 T^{4} - \)\(21\!\cdots\!96\)\( T^{5} - \)\(25\!\cdots\!86\)\( T^{6} + \)\(73\!\cdots\!56\)\( T^{7} + \)\(18\!\cdots\!76\)\( T^{8} + \)\(52\!\cdots\!44\)\( T^{9} - \)\(12\!\cdots\!86\)\( T^{10} - \)\(77\!\cdots\!04\)\( T^{11} + \)\(74\!\cdots\!89\)\( T^{12} + \)\(29\!\cdots\!08\)\( T^{13} - \)\(30\!\cdots\!58\)\( T^{14} - \)\(19\!\cdots\!24\)\( T^{15} + \)\(68\!\cdots\!01\)\( T^{16} \))
$61$ (\( 1 - 39758 T + 844596301 T^{2} \))(\( 1 - 48142 T - 755431725 T^{2} + 43590251951830 T^{3} + 1116615498545065565 T^{4} - \)\(17\!\cdots\!08\)\( T^{5} - \)\(17\!\cdots\!86\)\( T^{6} + \)\(66\!\cdots\!20\)\( T^{7} + \)\(16\!\cdots\!34\)\( T^{8} + \)\(56\!\cdots\!20\)\( T^{9} - \)\(12\!\cdots\!86\)\( T^{10} - \)\(10\!\cdots\!08\)\( T^{11} + \)\(56\!\cdots\!65\)\( T^{12} + \)\(18\!\cdots\!30\)\( T^{13} - \)\(27\!\cdots\!25\)\( T^{14} - \)\(14\!\cdots\!42\)\( T^{15} + \)\(25\!\cdots\!01\)\( T^{16} \))
$67$ (\( 1 + 23068 T + 1350125107 T^{2} \))(\( 1 + 7148 T - 3268771122 T^{2} - 16753922588168 T^{3} + 4825996342110423185 T^{4} + \)\(80\!\cdots\!92\)\( T^{5} - \)\(77\!\cdots\!58\)\( T^{6} + \)\(18\!\cdots\!60\)\( T^{7} + \)\(12\!\cdots\!20\)\( T^{8} + \)\(25\!\cdots\!20\)\( T^{9} - \)\(14\!\cdots\!42\)\( T^{10} + \)\(19\!\cdots\!56\)\( T^{11} + \)\(16\!\cdots\!85\)\( T^{12} - \)\(75\!\cdots\!76\)\( T^{13} - \)\(19\!\cdots\!78\)\( T^{14} + \)\(58\!\cdots\!64\)\( T^{15} + \)\(11\!\cdots\!01\)\( T^{16} \))
$71$ (\( 1 - 4248 T + 1804229351 T^{2} \))(\( ( 1 + 35928 T + 5285725772 T^{2} + 127659705990840 T^{3} + 12554595869440882758 T^{4} + \)\(23\!\cdots\!40\)\( T^{5} + \)\(17\!\cdots\!72\)\( T^{6} + \)\(21\!\cdots\!28\)\( T^{7} + \)\(10\!\cdots\!01\)\( T^{8} )^{2} \))
$73$ (\( 1 + 41110 T + 2073071593 T^{2} \))(\( ( 1 - 61226 T + 6621012721 T^{2} - 212085627051026 T^{3} + 16124371439127547300 T^{4} - \)\(43\!\cdots\!18\)\( T^{5} + \)\(28\!\cdots\!29\)\( T^{6} - \)\(54\!\cdots\!82\)\( T^{7} + \)\(18\!\cdots\!01\)\( T^{8} )^{2} \))
$79$ (\( 1 - 21920 T + 3077056399 T^{2} \))(\( 1 + 59516 T - 3717013179 T^{2} - 374196053042180 T^{3} - 3765094998545504239 T^{4} + \)\(30\!\cdots\!72\)\( T^{5} - \)\(95\!\cdots\!50\)\( T^{6} + \)\(90\!\cdots\!24\)\( T^{7} + \)\(17\!\cdots\!98\)\( T^{8} + \)\(27\!\cdots\!76\)\( T^{9} - \)\(90\!\cdots\!50\)\( T^{10} + \)\(87\!\cdots\!28\)\( T^{11} - \)\(33\!\cdots\!39\)\( T^{12} - \)\(10\!\cdots\!20\)\( T^{13} - \)\(31\!\cdots\!79\)\( T^{14} + \)\(15\!\cdots\!84\)\( T^{15} + \)\(80\!\cdots\!01\)\( T^{16} \))
$83$ (\( 1 + 82452 T + 3939040643 T^{2} \))(\( 1 - 117696 T - 5262215903 T^{2} + 507938387214576 T^{3} + 83906673955765181161 T^{4} - \)\(37\!\cdots\!00\)\( T^{5} - \)\(42\!\cdots\!38\)\( T^{6} + \)\(12\!\cdots\!52\)\( T^{7} + \)\(24\!\cdots\!34\)\( T^{8} + \)\(47\!\cdots\!36\)\( T^{9} - \)\(65\!\cdots\!62\)\( T^{10} - \)\(22\!\cdots\!00\)\( T^{11} + \)\(20\!\cdots\!61\)\( T^{12} + \)\(48\!\cdots\!68\)\( T^{13} - \)\(19\!\cdots\!47\)\( T^{14} - \)\(17\!\cdots\!72\)\( T^{15} + \)\(57\!\cdots\!01\)\( T^{16} \))
$89$ (\( 1 - 94086 T + 5584059449 T^{2} \))(\( ( 1 + 225864 T + 38022042668 T^{2} + 4189883418729720 T^{3} + \)\(36\!\cdots\!22\)\( T^{4} + \)\(23\!\cdots\!80\)\( T^{5} + \)\(11\!\cdots\!68\)\( T^{6} + \)\(39\!\cdots\!36\)\( T^{7} + \)\(97\!\cdots\!01\)\( T^{8} )^{2} \))
$97$ (\( 1 - 49442 T + 8587340257 T^{2} \))(\( 1 - 33976 T - 11979920730 T^{2} + 1870850391619312 T^{3} + 3904959771682438313 T^{4} - \)\(16\!\cdots\!64\)\( T^{5} + \)\(12\!\cdots\!22\)\( T^{6} + \)\(57\!\cdots\!68\)\( T^{7} - \)\(11\!\cdots\!00\)\( T^{8} + \)\(49\!\cdots\!76\)\( T^{9} + \)\(91\!\cdots\!78\)\( T^{10} - \)\(10\!\cdots\!52\)\( T^{11} + \)\(21\!\cdots\!13\)\( T^{12} + \)\(87\!\cdots\!84\)\( T^{13} - \)\(48\!\cdots\!70\)\( T^{14} - \)\(11\!\cdots\!68\)\( T^{15} + \)\(29\!\cdots\!01\)\( T^{16} \))
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