## Defining parameters

 Level: $$N$$ = $$308 = 2^{2} \cdot 7 \cdot 11$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$16$$ Newforms: $$29$$ Sturm bound: $$11520$$ Trace bound: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(308))$$.

Total New Old
Modular forms 3180 1670 1510
Cusp forms 2581 1494 1087
Eisenstein series 599 176 423

## Trace form

 $$1494q - 14q^{2} + 2q^{3} - 14q^{4} - 22q^{5} - 20q^{6} + 13q^{7} - 44q^{8} - 12q^{9} + O(q^{10})$$ $$1494q - 14q^{2} + 2q^{3} - 14q^{4} - 22q^{5} - 20q^{6} + 13q^{7} - 44q^{8} - 12q^{9} - 42q^{10} + 7q^{11} - 64q^{12} - 38q^{13} - 48q^{14} - 12q^{15} - 78q^{16} - 32q^{17} - 76q^{18} - 32q^{19} - 70q^{20} - 90q^{21} - 92q^{22} - 4q^{23} - 56q^{24} - 64q^{25} - 46q^{26} + 20q^{27} + 4q^{28} - 90q^{29} - 18q^{30} - 14q^{31} - 24q^{32} - 29q^{33} - 15q^{35} - 18q^{36} - 114q^{37} - 6q^{38} - 6q^{39} + 26q^{40} - 94q^{41} - 4q^{42} + 36q^{43} + 32q^{44} - 192q^{45} + 6q^{46} - 8q^{47} + 90q^{48} - 23q^{49} - 36q^{50} - 4q^{51} - 10q^{52} - 88q^{53} + 6q^{54} - 2q^{55} - 98q^{56} - 94q^{57} - 78q^{58} - 22q^{59} - 86q^{60} - 26q^{61} - 170q^{62} - 70q^{63} - 188q^{64} - 182q^{65} - 144q^{66} - 104q^{67} - 150q^{68} - 272q^{69} - 62q^{70} - 160q^{71} - 90q^{72} - 182q^{73} - 86q^{74} - 222q^{75} - 176q^{77} - 128q^{78} - 126q^{79} + 58q^{80} - 260q^{81} + 16q^{82} - 88q^{83} - 20q^{84} - 150q^{85} + 116q^{86} - 62q^{87} + 122q^{88} - 128q^{89} + 70q^{90} - 39q^{91} - 48q^{92} - 96q^{93} + 90q^{94} + 14q^{95} + 18q^{96} + 20q^{97} - 32q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(308))$$

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
308.2.a $$\chi_{308}(1, \cdot)$$ 308.2.a.a 1 1
308.2.a.b 2
308.2.a.c 3
308.2.c $$\chi_{308}(153, \cdot)$$ 308.2.c.a 4 1
308.2.c.b 4
308.2.d $$\chi_{308}(43, \cdot)$$ 308.2.d.a 18 1
308.2.d.b 18
308.2.f $$\chi_{308}(111, \cdot)$$ 308.2.f.a 40 1
308.2.i $$\chi_{308}(177, \cdot)$$ 308.2.i.a 6 2
308.2.i.b 6
308.2.j $$\chi_{308}(113, \cdot)$$ 308.2.j.a 4 4
308.2.j.b 8
308.2.j.c 12
308.2.l $$\chi_{308}(199, \cdot)$$ 308.2.l.a 80 2
308.2.n $$\chi_{308}(219, \cdot)$$ 308.2.n.a 4 2
308.2.n.b 4
308.2.n.c 80
308.2.q $$\chi_{308}(241, \cdot)$$ 308.2.q.a 16 2
308.2.t $$\chi_{308}(27, \cdot)$$ 308.2.t.a 8 4
308.2.t.b 8
308.2.t.c 160
308.2.v $$\chi_{308}(127, \cdot)$$ 308.2.v.a 72 4
308.2.v.b 72
308.2.w $$\chi_{308}(13, \cdot)$$ 308.2.w.a 32 4
308.2.y $$\chi_{308}(9, \cdot)$$ 308.2.y.a 16 8
308.2.y.b 48
308.2.z $$\chi_{308}(17, \cdot)$$ 308.2.z.a 64 8
308.2.bc $$\chi_{308}(39, \cdot)$$ 308.2.bc.a 352 8
308.2.be $$\chi_{308}(3, \cdot)$$ 308.2.be.a 352 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(308))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(308)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(22))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(28))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(44))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(77))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(154))$$$$^{\oplus 2}$$