# Properties

 Label 624.4 Level 624 Weight 4 Dimension 13346 Nonzero newspaces 28 Sturm bound 86016 Trace bound 13

## Defining parameters

 Level: $$N$$ = $$624 = 2^{4} \cdot 3 \cdot 13$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$28$$ Sturm bound: $$86016$$ Trace bound: $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(624))$$.

Total New Old
Modular forms 32928 13546 19382
Cusp forms 31584 13346 18238
Eisenstein series 1344 200 1144

## Trace form

 $$13346 q - 8 q^{3} - 4 q^{5} - 80 q^{6} - 84 q^{7} - 168 q^{8} - 120 q^{9} + O(q^{10})$$ $$13346 q - 8 q^{3} - 4 q^{5} - 80 q^{6} - 84 q^{7} - 168 q^{8} - 120 q^{9} - 304 q^{10} + 120 q^{11} + 184 q^{12} - 46 q^{13} + 696 q^{14} - 318 q^{15} + 560 q^{16} + 52 q^{17} - 56 q^{18} - 100 q^{19} - 160 q^{20} + 490 q^{21} - 1376 q^{22} + 496 q^{23} - 240 q^{24} + 442 q^{25} + 20 q^{26} + 286 q^{27} + 1232 q^{28} - 116 q^{29} + 1456 q^{30} + 1404 q^{31} + 1920 q^{32} - 1174 q^{33} + 320 q^{34} - 240 q^{35} + 968 q^{36} - 2288 q^{37} - 2512 q^{38} + 280 q^{39} - 4848 q^{40} + 2004 q^{41} - 2736 q^{42} + 1180 q^{43} + 400 q^{44} + 2526 q^{45} + 1488 q^{46} - 1080 q^{47} - 1368 q^{48} - 1582 q^{49} - 1416 q^{50} - 2532 q^{51} - 1376 q^{52} - 7172 q^{53} - 3424 q^{54} - 8148 q^{55} - 2688 q^{56} - 5574 q^{57} - 3072 q^{58} + 1616 q^{59} + 824 q^{60} + 5992 q^{61} + 1992 q^{62} + 1386 q^{63} + 4176 q^{64} + 5940 q^{65} + 4024 q^{66} + 13844 q^{67} + 3136 q^{68} + 8362 q^{69} + 13856 q^{70} + 2576 q^{71} + 8608 q^{72} + 9632 q^{73} + 5480 q^{74} - 3142 q^{75} + 4848 q^{76} - 5344 q^{77} + 3216 q^{78} - 12136 q^{79} - 1424 q^{80} - 784 q^{81} - 1504 q^{82} - 11736 q^{83} - 7664 q^{84} - 21876 q^{85} - 35552 q^{86} - 11082 q^{87} - 42784 q^{88} - 20412 q^{89} - 31312 q^{90} - 964 q^{91} - 18944 q^{92} + 4690 q^{93} - 16832 q^{94} + 25592 q^{95} - 1496 q^{96} + 12928 q^{97} + 18016 q^{98} + 6638 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(624))$$

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
624.4.a $$\chi_{624}(1, \cdot)$$ 624.4.a.a 1 1
624.4.a.b 1
624.4.a.c 1
624.4.a.d 1
624.4.a.e 1
624.4.a.f 1
624.4.a.g 1
624.4.a.h 1
624.4.a.i 1
624.4.a.j 2
624.4.a.k 2
624.4.a.l 2
624.4.a.m 2
624.4.a.n 2
624.4.a.o 2
624.4.a.p 2
624.4.a.q 2
624.4.a.r 2
624.4.a.s 3
624.4.a.t 3
624.4.a.u 3
624.4.c $$\chi_{624}(337, \cdot)$$ 624.4.c.a 2 1
624.4.c.b 2
624.4.c.c 4
624.4.c.d 4
624.4.c.e 4
624.4.c.f 4
624.4.c.g 10
624.4.c.h 12
624.4.d $$\chi_{624}(287, \cdot)$$ 624.4.d.a 12 1
624.4.d.b 12
624.4.d.c 24
624.4.d.d 24
624.4.g $$\chi_{624}(313, \cdot)$$ None 0 1
624.4.h $$\chi_{624}(311, \cdot)$$ None 0 1
624.4.j $$\chi_{624}(599, \cdot)$$ None 0 1
624.4.m $$\chi_{624}(25, \cdot)$$ None 0 1
624.4.n $$\chi_{624}(623, \cdot)$$ 624.4.n.a 2 1
624.4.n.b 2
624.4.n.c 24
624.4.n.d 56
624.4.q $$\chi_{624}(289, \cdot)$$ 624.4.q.a 2 2
624.4.q.b 2
624.4.q.c 2
624.4.q.d 4
624.4.q.e 4
624.4.q.f 4
624.4.q.g 6
624.4.q.h 6
624.4.q.i 8
624.4.q.j 8
624.4.q.k 8
624.4.q.l 8
624.4.q.m 10
624.4.q.n 12
624.4.r $$\chi_{624}(499, \cdot)$$ n/a 336 2
624.4.u $$\chi_{624}(5, \cdot)$$ n/a 664 2
624.4.v $$\chi_{624}(155, \cdot)$$ n/a 664 2
624.4.x $$\chi_{624}(157, \cdot)$$ n/a 288 2
624.4.bb $$\chi_{624}(151, \cdot)$$ None 0 2
624.4.bc $$\chi_{624}(31, \cdot)$$ 624.4.bc.a 14 2
624.4.bc.b 14
624.4.bc.c 28
624.4.bc.d 28
624.4.bf $$\chi_{624}(161, \cdot)$$ n/a 164 2
624.4.bg $$\chi_{624}(281, \cdot)$$ None 0 2
624.4.bh $$\chi_{624}(131, \cdot)$$ n/a 576 2
624.4.bj $$\chi_{624}(181, \cdot)$$ n/a 336 2
624.4.bm $$\chi_{624}(317, \cdot)$$ n/a 664 2
624.4.bn $$\chi_{624}(187, \cdot)$$ n/a 336 2
624.4.bq $$\chi_{624}(23, \cdot)$$ None 0 2
624.4.br $$\chi_{624}(217, \cdot)$$ None 0 2
624.4.bu $$\chi_{624}(191, \cdot)$$ n/a 168 2
624.4.bv $$\chi_{624}(49, \cdot)$$ 624.4.bv.a 2 2
624.4.bv.b 2
624.4.bv.c 4
624.4.bv.d 4
624.4.bv.e 4
624.4.bv.f 6
624.4.bv.g 8
624.4.bv.h 10
624.4.bv.i 20
624.4.bv.j 24
624.4.bz $$\chi_{624}(95, \cdot)$$ n/a 168 2
624.4.ca $$\chi_{624}(121, \cdot)$$ None 0 2
624.4.cd $$\chi_{624}(263, \cdot)$$ None 0 2
624.4.ce $$\chi_{624}(149, \cdot)$$ n/a 1328 4
624.4.ch $$\chi_{624}(19, \cdot)$$ n/a 672 4
624.4.cj $$\chi_{624}(205, \cdot)$$ n/a 672 4
624.4.cl $$\chi_{624}(35, \cdot)$$ n/a 1328 4
624.4.cm $$\chi_{624}(41, \cdot)$$ None 0 4
624.4.cn $$\chi_{624}(305, \cdot)$$ n/a 328 4
624.4.cq $$\chi_{624}(175, \cdot)$$ n/a 168 4
624.4.cr $$\chi_{624}(7, \cdot)$$ None 0 4
624.4.cv $$\chi_{624}(61, \cdot)$$ n/a 672 4
624.4.cx $$\chi_{624}(179, \cdot)$$ n/a 1328 4
624.4.cz $$\chi_{624}(115, \cdot)$$ n/a 672 4
624.4.da $$\chi_{624}(245, \cdot)$$ n/a 1328 4

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(624))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(624)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(6))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(8))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(24))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(26))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(48))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(52))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(78))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(104))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(156))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(208))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(312))$$$$^{\oplus 2}$$