# Properties

 Label 1216.3 Level 1216 Weight 3 Dimension 51178 Nonzero newspaces 24 Sturm bound 276480 Trace bound 49

## Defining parameters

 Level: $$N$$ = $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$24$$ Sturm bound: $$276480$$ Trace bound: $$49$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(1216))$$.

Total New Old
Modular forms 93456 51926 41530
Cusp forms 90864 51178 39686
Eisenstein series 2592 748 1844

## Trace form

 $$51178q - 128q^{2} - 96q^{3} - 128q^{4} - 128q^{5} - 128q^{6} - 100q^{7} - 128q^{8} - 178q^{9} + O(q^{10})$$ $$51178q - 128q^{2} - 96q^{3} - 128q^{4} - 128q^{5} - 128q^{6} - 100q^{7} - 128q^{8} - 178q^{9} - 128q^{10} - 128q^{11} - 128q^{12} - 160q^{13} - 128q^{14} - 92q^{15} - 128q^{16} - 192q^{17} - 128q^{18} - 70q^{19} - 272q^{20} + 40q^{21} + 16q^{22} + 28q^{23} + 432q^{24} + 82q^{25} + 272q^{26} + 36q^{27} + 112q^{28} - 64q^{29} + 32q^{30} - 76q^{31} - 208q^{32} - 260q^{33} - 368q^{34} - 292q^{35} - 928q^{36} - 480q^{37} - 416q^{38} - 592q^{39} - 848q^{40} - 672q^{41} - 1008q^{42} - 320q^{43} - 336q^{44} - 424q^{45} - 128q^{46} - 92q^{47} - 128q^{48} - 62q^{49} - 752q^{50} + 988q^{51} - 1184q^{52} + 160q^{53} - 1280q^{54} + 1436q^{55} - 912q^{56} + 68q^{57} - 992q^{58} + 832q^{59} - 704q^{60} + 96q^{61} - 160q^{62} - 108q^{63} + 64q^{64} - 540q^{65} + 384q^{66} - 1184q^{67} + 352q^{68} - 536q^{69} + 1216q^{70} - 1636q^{71} + 1168q^{72} - 672q^{73} + 1104q^{74} - 2104q^{75} + 696q^{76} + 24q^{77} + 2128q^{78} - 1116q^{79} + 2608q^{80} + 602q^{81} + 1952q^{82} + 544q^{83} + 2336q^{84} + 800q^{85} + 1744q^{86} + 796q^{87} + 992q^{88} + 928q^{89} + 1312q^{90} + 284q^{91} + 784q^{92} + 592q^{93} + 64q^{94} - 92q^{95} - 544q^{96} - 512q^{97} - 944q^{98} - 688q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(1216))$$

We only show spaces with odd 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
1216.3.d $$\chi_{1216}(191, \cdot)$$ 1216.3.d.a 4 1
1216.3.d.b 6
1216.3.d.c 12
1216.3.d.d 14
1216.3.d.e 16
1216.3.d.f 20
1216.3.e $$\chi_{1216}(1025, \cdot)$$ 1216.3.e.a 1 1
1216.3.e.b 1
1216.3.e.c 2
1216.3.e.d 2
1216.3.e.e 2
1216.3.e.f 2
1216.3.e.g 2
1216.3.e.h 2
1216.3.e.i 2
1216.3.e.j 2
1216.3.e.k 2
1216.3.e.l 2
1216.3.e.m 8
1216.3.e.n 8
1216.3.e.o 20
1216.3.e.p 20
1216.3.f $$\chi_{1216}(799, \cdot)$$ 1216.3.f.a 24 1
1216.3.f.b 48
1216.3.g $$\chi_{1216}(417, \cdot)$$ 1216.3.g.a 4 1
1216.3.g.b 4
1216.3.g.c 8
1216.3.g.d 16
1216.3.g.e 48
1216.3.j $$\chi_{1216}(113, \cdot)$$ n/a 156 2
1216.3.l $$\chi_{1216}(495, \cdot)$$ n/a 144 2
1216.3.o $$\chi_{1216}(159, \cdot)$$ n/a 160 2
1216.3.p $$\chi_{1216}(673, \cdot)$$ n/a 160 2
1216.3.q $$\chi_{1216}(767, \cdot)$$ n/a 156 2
1216.3.r $$\chi_{1216}(65, \cdot)$$ n/a 156 2
1216.3.w $$\chi_{1216}(265, \cdot)$$ None 0 4
1216.3.x $$\chi_{1216}(39, \cdot)$$ None 0 4
1216.3.ba $$\chi_{1216}(145, \cdot)$$ n/a 312 4
1216.3.bc $$\chi_{1216}(239, \cdot)$$ n/a 312 4
1216.3.bf $$\chi_{1216}(115, \cdot)$$ n/a 2304 8
1216.3.bg $$\chi_{1216}(37, \cdot)$$ n/a 2544 8
1216.3.bh $$\chi_{1216}(129, \cdot)$$ n/a 468 6
1216.3.bi $$\chi_{1216}(33, \cdot)$$ n/a 480 6
1216.3.bk $$\chi_{1216}(351, \cdot)$$ n/a 480 6
1216.3.bn $$\chi_{1216}(63, \cdot)$$ n/a 468 6
1216.3.bo $$\chi_{1216}(7, \cdot)$$ None 0 8
1216.3.bp $$\chi_{1216}(217, \cdot)$$ None 0 8
1216.3.bt $$\chi_{1216}(47, \cdot)$$ n/a 936 12
1216.3.bv $$\chi_{1216}(241, \cdot)$$ n/a 936 12
1216.3.by $$\chi_{1216}(69, \cdot)$$ n/a 5088 16
1216.3.bz $$\chi_{1216}(11, \cdot)$$ n/a 5088 16
1216.3.cc $$\chi_{1216}(23, \cdot)$$ None 0 24
1216.3.cd $$\chi_{1216}(41, \cdot)$$ None 0 24
1216.3.ce $$\chi_{1216}(35, \cdot)$$ n/a 15264 48
1216.3.cf $$\chi_{1216}(13, \cdot)$$ n/a 15264 48

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(1216))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(1216)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 7}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(76))$$$$^{\oplus 5}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(152))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(304))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(608))$$$$^{\oplus 2}$$