## Defining parameters

 Level: $$N$$ = $$1224 = 2^{3} \cdot 3^{2} \cdot 17$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$30$$ Sturm bound: $$331776$$ Trace bound: $$10$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1224))$$.

Total New Old
Modular forms 125952 55288 70664
Cusp forms 122880 54748 68132
Eisenstein series 3072 540 2532

## Trace form

 $$54748 q - 40 q^{2} - 58 q^{3} - 20 q^{4} + 44 q^{5} - 24 q^{6} + 36 q^{7} - 100 q^{8} - 170 q^{9} + O(q^{10})$$ $$54748 q - 40 q^{2} - 58 q^{3} - 20 q^{4} + 44 q^{5} - 24 q^{6} + 36 q^{7} - 100 q^{8} - 170 q^{9} - 304 q^{10} - 262 q^{11} - 276 q^{12} - 128 q^{13} - 340 q^{14} - 112 q^{15} - 316 q^{16} + 46 q^{17} + 112 q^{18} + 228 q^{19} + 860 q^{20} - 456 q^{21} + 1196 q^{22} - 540 q^{23} + 464 q^{24} - 894 q^{25} - 112 q^{26} - 640 q^{27} + 160 q^{28} + 644 q^{29} - 300 q^{30} + 880 q^{31} - 380 q^{32} + 810 q^{33} - 1430 q^{34} + 1784 q^{35} + 1468 q^{36} + 1008 q^{37} - 836 q^{38} + 488 q^{39} - 1668 q^{40} + 838 q^{41} + 172 q^{42} + 1598 q^{43} + 2028 q^{44} - 892 q^{45} + 5248 q^{46} + 2604 q^{47} + 1528 q^{48} - 1502 q^{49} + 6188 q^{50} - 409 q^{51} + 3028 q^{52} - 2164 q^{53} - 1312 q^{54} + 528 q^{55} - 96 q^{56} - 3202 q^{57} - 7068 q^{58} - 8506 q^{59} - 1660 q^{60} + 4 q^{61} - 12792 q^{62} - 2432 q^{63} - 9224 q^{64} - 536 q^{65} - 5064 q^{66} - 1622 q^{67} - 9980 q^{68} + 7420 q^{69} + 2020 q^{70} + 9376 q^{71} - 1972 q^{72} + 2088 q^{73} + 548 q^{74} + 11694 q^{75} + 2996 q^{76} + 3624 q^{77} - 7956 q^{78} - 4520 q^{79} - 6400 q^{80} - 4002 q^{81} - 6512 q^{82} - 704 q^{83} - 368 q^{84} + 18040 q^{85} - 388 q^{86} - 3700 q^{87} - 8908 q^{88} - 1408 q^{89} - 3804 q^{90} + 5760 q^{91} + 1860 q^{92} - 11204 q^{93} + 10056 q^{94} - 4192 q^{95} + 8000 q^{96} - 6750 q^{97} + 20604 q^{98} + 1172 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1224))$$

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
1224.4.a $$\chi_{1224}(1, \cdot)$$ 1224.4.a.a 1 1
1224.4.a.b 1
1224.4.a.c 2
1224.4.a.d 2
1224.4.a.e 3
1224.4.a.f 3
1224.4.a.g 3
1224.4.a.h 3
1224.4.a.i 3
1224.4.a.j 3
1224.4.a.k 4
1224.4.a.l 4
1224.4.a.m 4
1224.4.a.n 5
1224.4.a.o 5
1224.4.a.p 7
1224.4.a.q 7
1224.4.c $$\chi_{1224}(577, \cdot)$$ 1224.4.c.a 2 1
1224.4.c.b 6
1224.4.c.c 6
1224.4.c.d 6
1224.4.c.e 8
1224.4.c.f 12
1224.4.c.g 14
1224.4.c.h 14
1224.4.e $$\chi_{1224}(647, \cdot)$$ None 0 1
1224.4.f $$\chi_{1224}(613, \cdot)$$ n/a 240 1
1224.4.h $$\chi_{1224}(611, \cdot)$$ n/a 216 1
1224.4.j $$\chi_{1224}(35, \cdot)$$ n/a 192 1
1224.4.l $$\chi_{1224}(1189, \cdot)$$ n/a 268 1
1224.4.o $$\chi_{1224}(1223, \cdot)$$ None 0 1
1224.4.q $$\chi_{1224}(409, \cdot)$$ n/a 288 2
1224.4.r $$\chi_{1224}(251, \cdot)$$ n/a 432 2
1224.4.t $$\chi_{1224}(829, \cdot)$$ n/a 536 2
1224.4.w $$\chi_{1224}(217, \cdot)$$ n/a 136 2
1224.4.y $$\chi_{1224}(863, \cdot)$$ None 0 2
1224.4.ba $$\chi_{1224}(407, \cdot)$$ None 0 2
1224.4.bd $$\chi_{1224}(373, \cdot)$$ n/a 1288 2
1224.4.bf $$\chi_{1224}(443, \cdot)$$ n/a 1152 2
1224.4.bh $$\chi_{1224}(203, \cdot)$$ n/a 1288 2
1224.4.bj $$\chi_{1224}(205, \cdot)$$ n/a 1152 2
1224.4.bk $$\chi_{1224}(239, \cdot)$$ None 0 2
1224.4.bm $$\chi_{1224}(169, \cdot)$$ n/a 324 2
1224.4.bq $$\chi_{1224}(145, \cdot)$$ n/a 268 4
1224.4.br $$\chi_{1224}(287, \cdot)$$ None 0 4
1224.4.bs $$\chi_{1224}(253, \cdot)$$ n/a 1072 4
1224.4.bt $$\chi_{1224}(179, \cdot)$$ n/a 864 4
1224.4.bx $$\chi_{1224}(625, \cdot)$$ n/a 648 4
1224.4.bz $$\chi_{1224}(47, \cdot)$$ None 0 4
1224.4.ca $$\chi_{1224}(659, \cdot)$$ n/a 2576 4
1224.4.cc $$\chi_{1224}(13, \cdot)$$ n/a 2576 4
1224.4.cf $$\chi_{1224}(233, \cdot)$$ n/a 432 8
1224.4.cg $$\chi_{1224}(199, \cdot)$$ None 0 8
1224.4.cj $$\chi_{1224}(91, \cdot)$$ n/a 2144 8
1224.4.ck $$\chi_{1224}(125, \cdot)$$ n/a 1728 8
1224.4.cm $$\chi_{1224}(229, \cdot)$$ n/a 5152 8
1224.4.cn $$\chi_{1224}(59, \cdot)$$ n/a 5152 8
1224.4.cs $$\chi_{1224}(25, \cdot)$$ n/a 1296 8
1224.4.ct $$\chi_{1224}(263, \cdot)$$ None 0 8
1224.4.cv $$\chi_{1224}(7, \cdot)$$ None 0 16
1224.4.cw $$\chi_{1224}(41, \cdot)$$ n/a 2592 16
1224.4.cz $$\chi_{1224}(5, \cdot)$$ n/a 10304 16
1224.4.da $$\chi_{1224}(139, \cdot)$$ n/a 10304 16

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(1224))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1224)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(17))$$$$^{\oplus 12}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(34))$$$$^{\oplus 9}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(51))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(68))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(72))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(102))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(136))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(153))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(204))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(306))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(408))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(612))$$$$^{\oplus 2}$$