## Defining parameters

 Level: $$N$$ = $$304 = 2^{4} \cdot 19$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$40$$ Sturm bound: $$11520$$ Trace bound: $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(304))$$.

Total New Old
Modular forms 3132 1687 1445
Cusp forms 2629 1535 1094
Eisenstein series 503 152 351

## Trace form

 $$1535q - 32q^{2} - 23q^{3} - 36q^{4} - 41q^{5} - 44q^{6} - 27q^{7} - 44q^{8} - 9q^{9} + O(q^{10})$$ $$1535q - 32q^{2} - 23q^{3} - 36q^{4} - 41q^{5} - 44q^{6} - 27q^{7} - 44q^{8} - 9q^{9} - 36q^{10} - 31q^{11} - 28q^{12} - 41q^{13} - 28q^{14} - 35q^{15} - 20q^{16} - 73q^{17} - 40q^{18} - 33q^{19} - 80q^{20} - 53q^{21} - 36q^{22} - 27q^{23} - 36q^{24} - 9q^{25} - 44q^{26} - 11q^{27} - 52q^{28} - 57q^{29} - 28q^{30} + 5q^{31} - 52q^{32} - 73q^{33} - 44q^{34} - 19q^{35} - 28q^{36} - 66q^{37} - 24q^{38} - 54q^{39} - 20q^{40} - 9q^{41} - 36q^{42} - 47q^{43} - 28q^{44} - 49q^{45} - 60q^{46} - 59q^{47} - 52q^{48} - 93q^{49} - 24q^{50} - 35q^{51} - 28q^{52} - 25q^{53} - 36q^{54} - 27q^{55} - 20q^{56} - 9q^{57} - 48q^{58} - 15q^{59} - 36q^{60} - 45q^{61} - 68q^{62} - 107q^{63} - 36q^{64} - 161q^{65} - 44q^{66} - 115q^{67} - 36q^{68} - 93q^{69} - 52q^{70} - 81q^{71} - 44q^{72} - 117q^{73} - 36q^{74} - 156q^{75} - 48q^{76} - 206q^{77} - 28q^{78} - 153q^{79} - 52q^{80} - 209q^{81} - 36q^{82} - 77q^{83} - 20q^{84} - 125q^{85} - 36q^{86} - 135q^{87} - 52q^{88} - 81q^{89} - 28q^{90} - 107q^{91} + 12q^{92} - 113q^{93} - 4q^{94} - 15q^{95} - 40q^{96} - 73q^{97} - 24q^{98} - 23q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(304))$$

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
304.2.a $$\chi_{304}(1, \cdot)$$ 304.2.a.a 1 1
304.2.a.b 1
304.2.a.c 1
304.2.a.d 1
304.2.a.e 1
304.2.a.f 1
304.2.a.g 3
304.2.b $$\chi_{304}(151, \cdot)$$ None 0 1
304.2.c $$\chi_{304}(153, \cdot)$$ None 0 1
304.2.h $$\chi_{304}(303, \cdot)$$ 304.2.h.a 2 1
304.2.h.b 4
304.2.h.c 4
304.2.i $$\chi_{304}(49, \cdot)$$ 304.2.i.a 2 2
304.2.i.b 2
304.2.i.c 2
304.2.i.d 2
304.2.i.e 4
304.2.i.f 6
304.2.k $$\chi_{304}(77, \cdot)$$ 304.2.k.a 4 2
304.2.k.b 68
304.2.m $$\chi_{304}(75, \cdot)$$ 304.2.m.a 76 2
304.2.n $$\chi_{304}(31, \cdot)$$ 304.2.n.a 2 2
304.2.n.b 2
304.2.n.c 4
304.2.n.d 6
304.2.n.e 6
304.2.s $$\chi_{304}(103, \cdot)$$ None 0 2
304.2.t $$\chi_{304}(121, \cdot)$$ None 0 2
304.2.u $$\chi_{304}(17, \cdot)$$ 304.2.u.a 6 6
304.2.u.b 6
304.2.u.c 6
304.2.u.d 6
304.2.u.e 12
304.2.u.f 18
304.2.v $$\chi_{304}(45, \cdot)$$ 304.2.v.a 152 4
304.2.x $$\chi_{304}(27, \cdot)$$ 304.2.x.a 4 4
304.2.x.b 4
304.2.x.c 144
304.2.bb $$\chi_{304}(9, \cdot)$$ None 0 6
304.2.bd $$\chi_{304}(71, \cdot)$$ None 0 6
304.2.be $$\chi_{304}(15, \cdot)$$ 304.2.be.a 12 6
304.2.be.b 12
304.2.be.c 18
304.2.be.d 18
304.2.bg $$\chi_{304}(3, \cdot)$$ 304.2.bg.a 456 12
304.2.bi $$\chi_{304}(5, \cdot)$$ 304.2.bi.a 456 12

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(304))$$ into lower level spaces

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