# Properties

 Label 1216.4 Level 1216 Weight 4 Dimension 76954 Nonzero newspaces 24 Sturm bound 368640 Trace bound 49

## Defining parameters

 Level: $$N$$ = $$1216 = 2^{6} \cdot 19$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$24$$ Sturm bound: $$368640$$ Trace bound: $$49$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1216))$$.

Total New Old
Modular forms 139536 77702 61834
Cusp forms 136944 76954 59990
Eisenstein series 2592 748 1844

## Trace form

 $$76954 q - 128 q^{2} - 96 q^{3} - 128 q^{4} - 128 q^{5} - 128 q^{6} - 92 q^{7} - 128 q^{8} - 106 q^{9} + O(q^{10})$$ $$76954 q - 128 q^{2} - 96 q^{3} - 128 q^{4} - 128 q^{5} - 128 q^{6} - 92 q^{7} - 128 q^{8} - 106 q^{9} - 128 q^{10} - 56 q^{11} - 128 q^{12} - 272 q^{13} - 128 q^{14} - 340 q^{15} - 128 q^{16} - 432 q^{17} - 128 q^{18} - 126 q^{19} - 272 q^{20} - 104 q^{21} - 1072 q^{22} - 92 q^{23} - 2128 q^{24} - 262 q^{25} - 48 q^{26} + 276 q^{27} + 1392 q^{28} + 672 q^{29} + 4512 q^{30} + 628 q^{31} + 2352 q^{32} + 1868 q^{33} + 1872 q^{34} + 860 q^{35} + 1632 q^{36} + 912 q^{37} - 576 q^{38} - 200 q^{39} - 3408 q^{40} - 2208 q^{41} - 6448 q^{42} - 1768 q^{43} - 2128 q^{44} - 3096 q^{45} - 128 q^{46} - 1980 q^{47} - 128 q^{48} - 2350 q^{49} + 5584 q^{50} - 9044 q^{51} + 6496 q^{52} - 944 q^{53} + 3328 q^{54} - 1244 q^{55} - 912 q^{56} + 908 q^{57} - 5024 q^{58} + 8824 q^{59} - 9920 q^{60} + 2032 q^{61} - 6112 q^{62} + 15260 q^{63} - 12224 q^{64} + 4196 q^{65} - 11200 q^{66} + 11952 q^{67} - 4256 q^{68} + 1144 q^{69} - 4160 q^{70} + 804 q^{71} + 1168 q^{72} - 2208 q^{73} + 5136 q^{74} - 14832 q^{75} + 5816 q^{76} - 6520 q^{77} + 3856 q^{78} - 20252 q^{79} - 8656 q^{80} - 13862 q^{81} - 14048 q^{82} - 5216 q^{83} - 8416 q^{84} - 3440 q^{85} + 912 q^{86} - 92 q^{87} + 6112 q^{88} + 6976 q^{89} + 18592 q^{90} + 6572 q^{91} + 25104 q^{92} + 16624 q^{93} + 17728 q^{94} + 6792 q^{95} + 25568 q^{96} + 18608 q^{97} + 24080 q^{98} + 9392 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1216))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
1216.4.a $$\chi_{1216}(1, \cdot)$$ 1216.4.a.a 1 1
1216.4.a.b 1
1216.4.a.c 1
1216.4.a.d 1
1216.4.a.e 1
1216.4.a.f 1
1216.4.a.g 2
1216.4.a.h 2
1216.4.a.i 2
1216.4.a.j 2
1216.4.a.k 2
1216.4.a.l 2
1216.4.a.m 2
1216.4.a.n 2
1216.4.a.o 2
1216.4.a.p 2
1216.4.a.q 3
1216.4.a.r 3
1216.4.a.s 3
1216.4.a.t 3
1216.4.a.u 3
1216.4.a.v 3
1216.4.a.w 3
1216.4.a.x 3
1216.4.a.y 5
1216.4.a.z 5
1216.4.a.ba 5
1216.4.a.bb 5
1216.4.a.bc 5
1216.4.a.bd 5
1216.4.a.be 7
1216.4.a.bf 7
1216.4.a.bg 7
1216.4.a.bh 7
1216.4.b $$\chi_{1216}(607, \cdot)$$ n/a 120 1
1216.4.c $$\chi_{1216}(609, \cdot)$$ n/a 108 1
1216.4.h $$\chi_{1216}(1215, \cdot)$$ n/a 118 1
1216.4.i $$\chi_{1216}(577, \cdot)$$ n/a 236 2
1216.4.k $$\chi_{1216}(305, \cdot)$$ n/a 216 2
1216.4.m $$\chi_{1216}(303, \cdot)$$ n/a 236 2
1216.4.n $$\chi_{1216}(255, \cdot)$$ n/a 236 2
1216.4.s $$\chi_{1216}(31, \cdot)$$ n/a 240 2
1216.4.t $$\chi_{1216}(353, \cdot)$$ n/a 240 2
1216.4.u $$\chi_{1216}(151, \cdot)$$ None 0 4
1216.4.v $$\chi_{1216}(153, \cdot)$$ None 0 4
1216.4.y $$\chi_{1216}(321, \cdot)$$ n/a 708 6
1216.4.z $$\chi_{1216}(49, \cdot)$$ n/a 472 4
1216.4.bb $$\chi_{1216}(335, \cdot)$$ n/a 472 4
1216.4.bd $$\chi_{1216}(77, \cdot)$$ n/a 3456 8
1216.4.be $$\chi_{1216}(75, \cdot)$$ n/a 3824 8
1216.4.bj $$\chi_{1216}(161, \cdot)$$ n/a 720 6
1216.4.bl $$\chi_{1216}(223, \cdot)$$ n/a 720 6
1216.4.bm $$\chi_{1216}(127, \cdot)$$ n/a 708 6
1216.4.bq $$\chi_{1216}(121, \cdot)$$ None 0 8
1216.4.br $$\chi_{1216}(103, \cdot)$$ None 0 8
1216.4.bs $$\chi_{1216}(15, \cdot)$$ n/a 1416 12
1216.4.bu $$\chi_{1216}(17, \cdot)$$ n/a 1416 12
1216.4.bw $$\chi_{1216}(27, \cdot)$$ n/a 7648 16
1216.4.bx $$\chi_{1216}(45, \cdot)$$ n/a 7648 16
1216.4.ca $$\chi_{1216}(9, \cdot)$$ None 0 24
1216.4.cb $$\chi_{1216}(71, \cdot)$$ None 0 24
1216.4.cg $$\chi_{1216}(5, \cdot)$$ n/a 22944 48
1216.4.ch $$\chi_{1216}(3, \cdot)$$ n/a 22944 48

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(1216))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1216)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 14}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(2))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(4))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(19))$$$$^{\oplus 7}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(32))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(38))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(64))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(76))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(152))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(304))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(608))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(1216))$$$$^{\oplus 1}$$