## Defining parameters

 Level: $$N$$ = $$310 = 2 \cdot 5 \cdot 31$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$31$$ Sturm bound: $$11520$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(310))$$.

Total New Old
Modular forms 3120 881 2239
Cusp forms 2641 881 1760
Eisenstein series 479 0 479

## Trace form

 $$881 q + q^{2} + 4 q^{3} + q^{4} + q^{5} + 4 q^{6} + 8 q^{7} + q^{8} + 13 q^{9} + O(q^{10})$$ $$881 q + q^{2} + 4 q^{3} + q^{4} + q^{5} + 4 q^{6} + 8 q^{7} + q^{8} + 13 q^{9} + q^{10} + 12 q^{11} + 4 q^{12} + 14 q^{13} + 8 q^{14} + 4 q^{15} + q^{16} + 18 q^{17} + 13 q^{18} + 20 q^{19} + q^{20} + 12 q^{21} - 48 q^{22} - 36 q^{23} + 4 q^{24} - 19 q^{25} - 46 q^{26} - 140 q^{27} - 72 q^{28} - 90 q^{29} - 56 q^{30} - 89 q^{31} + q^{32} - 72 q^{33} - 102 q^{34} - 52 q^{35} - 67 q^{36} - 142 q^{37} - 40 q^{38} - 84 q^{39} + q^{40} - 18 q^{41} - 28 q^{42} + 24 q^{43} + 12 q^{44} + 13 q^{45} + 24 q^{46} + 48 q^{47} + 4 q^{48} - 3 q^{49} + q^{50} - 108 q^{51} + 14 q^{52} - 6 q^{53} + 40 q^{54} - 48 q^{55} + 8 q^{56} - 100 q^{57} + 30 q^{58} + 4 q^{60} - 118 q^{61} + 31 q^{62} - 136 q^{63} + q^{64} - 76 q^{65} + 48 q^{66} + 8 q^{67} + 18 q^{68} - 84 q^{69} + 8 q^{70} - 48 q^{71} + 13 q^{72} + 14 q^{73} + 38 q^{74} - 116 q^{75} - 144 q^{77} - 124 q^{78} - 60 q^{79} - 29 q^{80} - 119 q^{81} - 78 q^{82} - 336 q^{83} - 88 q^{84} - 102 q^{85} - 196 q^{86} - 120 q^{87} - 108 q^{88} - 210 q^{89} - 167 q^{90} - 148 q^{91} + 24 q^{92} - 356 q^{93} - 72 q^{94} - 220 q^{95} + 4 q^{96} - 162 q^{97} - 303 q^{98} - 144 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(310))$$

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
310.2.a $$\chi_{310}(1, \cdot)$$ 310.2.a.a 1 1
310.2.a.b 1
310.2.a.c 2
310.2.a.d 2
310.2.a.e 3
310.2.b $$\chi_{310}(249, \cdot)$$ 310.2.b.a 8 1
310.2.b.b 8
310.2.e $$\chi_{310}(191, \cdot)$$ 310.2.e.a 6 2
310.2.e.b 6
310.2.e.c 6
310.2.e.d 6
310.2.f $$\chi_{310}(123, \cdot)$$ 310.2.f.a 32 2
310.2.h $$\chi_{310}(101, \cdot)$$ 310.2.h.a 4 4
310.2.h.b 4
310.2.h.c 8
310.2.h.d 8
310.2.h.e 8
310.2.k $$\chi_{310}(129, \cdot)$$ 310.2.k.a 32 2
310.2.n $$\chi_{310}(39, \cdot)$$ 310.2.n.a 64 4
310.2.p $$\chi_{310}(37, \cdot)$$ 310.2.p.a 64 4
310.2.q $$\chi_{310}(41, \cdot)$$ 310.2.q.a 8 8
310.2.q.b 8
310.2.q.c 8
310.2.q.d 8
310.2.q.e 8
310.2.q.f 8
310.2.q.g 24
310.2.q.h 24
310.2.s $$\chi_{310}(23, \cdot)$$ 310.2.s.a 128 8
310.2.t $$\chi_{310}(9, \cdot)$$ 310.2.t.a 128 8
310.2.w $$\chi_{310}(3, \cdot)$$ 310.2.w.a 256 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(310))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(310)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(31))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(62))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(155))$$$$^{\oplus 2}$$