## Defining parameters

 Level: $$N$$ = $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$12$$ Sturm bound: $$23040$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(304))$$.

Total New Old
Modular forms 8892 4909 3983
Cusp forms 8388 4757 3631
Eisenstein series 504 152 352

## Trace form

 $$4757q - 32q^{2} - 31q^{3} - 52q^{4} - 37q^{5} + 28q^{6} + 21q^{7} + 52q^{8} + 13q^{9} + O(q^{10})$$ $$4757q - 32q^{2} - 31q^{3} - 52q^{4} - 37q^{5} + 28q^{6} + 21q^{7} + 52q^{8} + 13q^{9} + 100q^{10} - 151q^{11} - 236q^{12} - 85q^{13} - 412q^{14} + 237q^{15} - 596q^{16} - 173q^{17} - 384q^{18} + 43q^{19} + 320q^{20} + 43q^{21} + 1140q^{22} - 139q^{23} + 1660q^{24} + 233q^{25} + 492q^{26} - 91q^{27} - 596q^{28} - 37q^{29} - 2508q^{30} - 1083q^{31} - 1972q^{32} - 425q^{33} - 908q^{34} - 1075q^{35} + 1156q^{36} + 290q^{37} + 1196q^{38} - 230q^{39} + 2636q^{40} + 387q^{41} + 1324q^{42} + 1753q^{43} - 1772q^{44} - 477q^{45} - 2300q^{46} + 2917q^{47} - 3572q^{48} - 735q^{49} - 1488q^{50} + 2573q^{51} + 436q^{52} + 1195q^{53} + 2716q^{54} + 149q^{55} + 940q^{56} + 167q^{57} - 88q^{58} - 4775q^{59} + 348q^{60} - 7825q^{61} + 124q^{62} - 11643q^{63} - 1060q^{64} - 1529q^{65} + 820q^{66} - 3211q^{67} + 1724q^{68} + 723q^{69} - 356q^{70} + 4399q^{71} - 2220q^{72} + 7135q^{73} + 868q^{74} + 16308q^{75} + 1192q^{76} + 10442q^{77} + 1508q^{78} + 15431q^{79} + 5260q^{80} + 4437q^{81} + 1372q^{82} + 6595q^{83} - 3956q^{84} - 901q^{85} - 7564q^{86} - 7767q^{87} - 3092q^{88} - 9141q^{89} - 3828q^{90} - 17515q^{91} - 1300q^{92} - 5257q^{93} + 6460q^{94} - 7015q^{95} + 8792q^{96} + 883q^{97} - 664q^{98} - 8975q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(304))$$

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
304.4.a $$\chi_{304}(1, \cdot)$$ 304.4.a.a 1 1
304.4.a.b 1
304.4.a.c 2
304.4.a.d 2
304.4.a.e 2
304.4.a.f 2
304.4.a.g 3
304.4.a.h 3
304.4.a.i 3
304.4.a.j 3
304.4.a.k 5
304.4.b $$\chi_{304}(151, \cdot)$$ None 0 1
304.4.c $$\chi_{304}(153, \cdot)$$ None 0 1
304.4.h $$\chi_{304}(303, \cdot)$$ 304.4.h.a 2 1
304.4.h.b 4
304.4.h.c 8
304.4.h.d 16
304.4.i $$\chi_{304}(49, \cdot)$$ 304.4.i.a 2 2
304.4.i.b 2
304.4.i.c 4
304.4.i.d 4
304.4.i.e 6
304.4.i.f 10
304.4.i.g 14
304.4.i.h 16
304.4.k $$\chi_{304}(77, \cdot)$$ n/a 216 2
304.4.m $$\chi_{304}(75, \cdot)$$ n/a 236 2
304.4.n $$\chi_{304}(31, \cdot)$$ 304.4.n.a 4 2
304.4.n.b 16
304.4.n.c 20
304.4.n.d 20
304.4.s $$\chi_{304}(103, \cdot)$$ None 0 2
304.4.t $$\chi_{304}(121, \cdot)$$ None 0 2
304.4.u $$\chi_{304}(17, \cdot)$$ n/a 174 6
304.4.v $$\chi_{304}(45, \cdot)$$ n/a 472 4
304.4.x $$\chi_{304}(27, \cdot)$$ n/a 472 4
304.4.bb $$\chi_{304}(9, \cdot)$$ None 0 6
304.4.bd $$\chi_{304}(71, \cdot)$$ None 0 6
304.4.be $$\chi_{304}(15, \cdot)$$ n/a 180 6
304.4.bg $$\chi_{304}(3, \cdot)$$ n/a 1416 12
304.4.bi $$\chi_{304}(5, \cdot)$$ n/a 1416 12

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

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(304))$$ into lower level spaces

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