## Defining parameters

 Level: $$N$$ = $$1232 = 2^{4} \cdot 7 \cdot 11$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$32$$ Sturm bound: $$368640$$ Trace bound: $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1232))$$.

Total New Old
Modular forms 139920 73684 66236
Cusp forms 136560 72872 63688
Eisenstein series 3360 812 2548

## Trace form

 $$72872 q - 64 q^{2} - 34 q^{3} - 24 q^{4} - 86 q^{5} - 184 q^{6} - 105 q^{7} - 328 q^{8} - 58 q^{9} + O(q^{10})$$ $$72872 q - 64 q^{2} - 34 q^{3} - 24 q^{4} - 86 q^{5} - 184 q^{6} - 105 q^{7} - 328 q^{8} - 58 q^{9} - 328 q^{10} + 73 q^{11} + 264 q^{12} + 16 q^{13} + 300 q^{14} - 642 q^{15} + 1064 q^{16} + 58 q^{17} + 640 q^{18} + 218 q^{19} - 840 q^{20} + 182 q^{21} - 1352 q^{22} - 130 q^{23} - 3448 q^{24} - 1506 q^{25} - 1112 q^{26} - 1216 q^{27} + 484 q^{28} - 806 q^{29} + 4888 q^{30} + 1350 q^{31} + 3816 q^{32} + 2943 q^{33} + 1608 q^{34} + 2905 q^{35} - 2536 q^{36} + 714 q^{37} - 4984 q^{38} - 1352 q^{39} - 5400 q^{40} - 3760 q^{41} - 6620 q^{42} - 7476 q^{43} - 2028 q^{44} - 9344 q^{45} + 272 q^{46} - 8402 q^{47} + 6776 q^{48} + 7385 q^{49} + 6952 q^{50} - 1582 q^{51} + 8288 q^{52} + 4034 q^{53} + 12248 q^{54} + 6418 q^{55} + 8088 q^{56} + 14994 q^{57} + 8208 q^{58} + 11518 q^{59} + 10936 q^{60} + 8170 q^{61} + 2432 q^{62} + 5532 q^{63} - 960 q^{64} - 4870 q^{65} - 7524 q^{66} + 6410 q^{67} - 15696 q^{68} - 2674 q^{69} - 21496 q^{70} - 3690 q^{71} - 38576 q^{72} - 17782 q^{73} - 20864 q^{74} - 21252 q^{75} - 19368 q^{76} - 5242 q^{77} - 4880 q^{78} - 18874 q^{79} + 2992 q^{80} + 4128 q^{81} + 19928 q^{82} - 13504 q^{83} + 21844 q^{84} + 17506 q^{85} + 42200 q^{86} + 5686 q^{87} + 58668 q^{88} + 11334 q^{89} + 64552 q^{90} - 2507 q^{91} + 30272 q^{92} + 7090 q^{93} + 36632 q^{94} + 7322 q^{95} + 19176 q^{96} + 1984 q^{97} + 6048 q^{98} + 3218 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1232))$$

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
1232.4.a $$\chi_{1232}(1, \cdot)$$ 1232.4.a.a 1 1
1232.4.a.b 1
1232.4.a.c 1
1232.4.a.d 1
1232.4.a.e 1
1232.4.a.f 1
1232.4.a.g 1
1232.4.a.h 1
1232.4.a.i 1
1232.4.a.j 2
1232.4.a.k 2
1232.4.a.l 2
1232.4.a.m 2
1232.4.a.n 2
1232.4.a.o 2
1232.4.a.p 2
1232.4.a.q 3
1232.4.a.r 3
1232.4.a.s 4
1232.4.a.t 4
1232.4.a.u 4
1232.4.a.v 4
1232.4.a.w 4
1232.4.a.x 5
1232.4.a.y 5
1232.4.a.z 5
1232.4.a.ba 6
1232.4.a.bb 6
1232.4.a.bc 7
1232.4.a.bd 7
1232.4.c $$\chi_{1232}(617, \cdot)$$ None 0 1
1232.4.e $$\chi_{1232}(769, \cdot)$$ n/a 142 1
1232.4.f $$\chi_{1232}(351, \cdot)$$ n/a 108 1
1232.4.h $$\chi_{1232}(727, \cdot)$$ None 0 1
1232.4.j $$\chi_{1232}(111, \cdot)$$ n/a 120 1
1232.4.l $$\chi_{1232}(967, \cdot)$$ None 0 1
1232.4.o $$\chi_{1232}(153, \cdot)$$ None 0 1
1232.4.q $$\chi_{1232}(177, \cdot)$$ n/a 240 2
1232.4.r $$\chi_{1232}(43, \cdot)$$ n/a 864 2
1232.4.s $$\chi_{1232}(419, \cdot)$$ n/a 960 2
1232.4.x $$\chi_{1232}(461, \cdot)$$ n/a 1144 2
1232.4.y $$\chi_{1232}(309, \cdot)$$ n/a 720 2
1232.4.z $$\chi_{1232}(113, \cdot)$$ n/a 432 4
1232.4.ba $$\chi_{1232}(857, \cdot)$$ None 0 2
1232.4.be $$\chi_{1232}(815, \cdot)$$ n/a 240 2
1232.4.bg $$\chi_{1232}(263, \cdot)$$ None 0 2
1232.4.bi $$\chi_{1232}(527, \cdot)$$ n/a 288 2
1232.4.bk $$\chi_{1232}(199, \cdot)$$ None 0 2
1232.4.bl $$\chi_{1232}(793, \cdot)$$ None 0 2
1232.4.bn $$\chi_{1232}(241, \cdot)$$ n/a 284 2
1232.4.bq $$\chi_{1232}(41, \cdot)$$ None 0 4
1232.4.bt $$\chi_{1232}(183, \cdot)$$ None 0 4
1232.4.bv $$\chi_{1232}(223, \cdot)$$ n/a 576 4
1232.4.bx $$\chi_{1232}(279, \cdot)$$ None 0 4
1232.4.bz $$\chi_{1232}(127, \cdot)$$ n/a 432 4
1232.4.ca $$\chi_{1232}(321, \cdot)$$ n/a 568 4
1232.4.cc $$\chi_{1232}(169, \cdot)$$ None 0 4
1232.4.cg $$\chi_{1232}(243, \cdot)$$ n/a 1920 4
1232.4.ch $$\chi_{1232}(219, \cdot)$$ n/a 2288 4
1232.4.ci $$\chi_{1232}(221, \cdot)$$ n/a 1920 4
1232.4.cj $$\chi_{1232}(285, \cdot)$$ n/a 2288 4
1232.4.cm $$\chi_{1232}(81, \cdot)$$ n/a 1136 8
1232.4.cn $$\chi_{1232}(141, \cdot)$$ n/a 3456 8
1232.4.co $$\chi_{1232}(13, \cdot)$$ n/a 4576 8
1232.4.ct $$\chi_{1232}(27, \cdot)$$ n/a 4576 8
1232.4.cu $$\chi_{1232}(211, \cdot)$$ n/a 3456 8
1232.4.cw $$\chi_{1232}(17, \cdot)$$ n/a 1136 8
1232.4.cy $$\chi_{1232}(9, \cdot)$$ None 0 8
1232.4.cz $$\chi_{1232}(103, \cdot)$$ None 0 8
1232.4.db $$\chi_{1232}(79, \cdot)$$ n/a 1152 8
1232.4.dd $$\chi_{1232}(39, \cdot)$$ None 0 8
1232.4.df $$\chi_{1232}(31, \cdot)$$ n/a 1152 8
1232.4.dj $$\chi_{1232}(73, \cdot)$$ None 0 8
1232.4.dm $$\chi_{1232}(61, \cdot)$$ n/a 9152 16
1232.4.dn $$\chi_{1232}(37, \cdot)$$ n/a 9152 16
1232.4.do $$\chi_{1232}(51, \cdot)$$ n/a 9152 16
1232.4.dp $$\chi_{1232}(3, \cdot)$$ n/a 9152 16

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(1232))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1232)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(7))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(22))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(44))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(56))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(77))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(88))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(112))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(154))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(176))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(308))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(616))$$$$^{\oplus 2}$$