# Properties

 Label 312.2 Level 312 Weight 2 Dimension 1076 Nonzero newspaces 18 Newform subspaces 46 Sturm bound 10752 Trace bound 10

## Defining parameters

 Level: $$N$$ = $$312 = 2^{3} \cdot 3 \cdot 13$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$18$$ Newform subspaces: $$46$$ Sturm bound: $$10752$$ Trace bound: $$10$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(312))$$.

Total New Old
Modular forms 2976 1164 1812
Cusp forms 2401 1076 1325
Eisenstein series 575 88 487

## Trace form

 $$1076 q + 4 q^{2} - 6 q^{3} - 16 q^{4} + 4 q^{5} - 16 q^{6} - 16 q^{7} - 8 q^{8} - 18 q^{9} + O(q^{10})$$ $$1076 q + 4 q^{2} - 6 q^{3} - 16 q^{4} + 4 q^{5} - 16 q^{6} - 16 q^{7} - 8 q^{8} - 18 q^{9} - 32 q^{10} - 8 q^{11} - 28 q^{12} + 2 q^{13} - 8 q^{14} - 24 q^{15} - 24 q^{16} + 10 q^{17} + 16 q^{20} + 24 q^{21} - 8 q^{22} + 24 q^{23} + 12 q^{24} + 8 q^{26} - 6 q^{27} - 24 q^{28} + 18 q^{29} - 4 q^{30} - 24 q^{31} - 16 q^{32} - 8 q^{33} - 64 q^{34} + 24 q^{35} - 20 q^{36} - 6 q^{37} - 16 q^{38} - 18 q^{39} - 160 q^{40} - 14 q^{41} - 92 q^{42} - 88 q^{43} - 120 q^{44} - 14 q^{45} - 248 q^{46} - 24 q^{47} - 92 q^{48} - 74 q^{49} - 212 q^{50} - 60 q^{51} - 256 q^{52} - 68 q^{53} - 24 q^{54} - 152 q^{55} - 200 q^{56} - 52 q^{57} - 192 q^{58} - 80 q^{59} - 132 q^{60} - 86 q^{61} - 112 q^{62} - 56 q^{63} - 136 q^{64} - 38 q^{65} - 88 q^{66} - 96 q^{67} - 52 q^{69} - 128 q^{70} - 64 q^{71} - 36 q^{72} - 52 q^{73} - 32 q^{74} - 130 q^{75} + 24 q^{76} - 20 q^{78} - 72 q^{79} - 50 q^{81} + 48 q^{82} + 8 q^{83} + 88 q^{84} - 10 q^{85} + 112 q^{86} - 96 q^{87} + 64 q^{88} - 20 q^{89} + 152 q^{90} - 216 q^{91} + 144 q^{92} - 92 q^{93} + 120 q^{94} - 224 q^{95} + 156 q^{96} - 76 q^{97} + 204 q^{98} - 180 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(312))$$

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
312.2.a $$\chi_{312}(1, \cdot)$$ 312.2.a.a 1 1
312.2.a.b 1
312.2.a.c 1
312.2.a.d 1
312.2.a.e 1
312.2.a.f 1
312.2.c $$\chi_{312}(25, \cdot)$$ 312.2.c.a 2 1
312.2.c.b 2
312.2.c.c 2
312.2.d $$\chi_{312}(287, \cdot)$$ None 0 1
312.2.g $$\chi_{312}(157, \cdot)$$ 312.2.g.a 8 1
312.2.g.b 16
312.2.h $$\chi_{312}(155, \cdot)$$ 312.2.h.a 8 1
312.2.h.b 12
312.2.h.c 32
312.2.j $$\chi_{312}(131, \cdot)$$ 312.2.j.a 48 1
312.2.m $$\chi_{312}(181, \cdot)$$ 312.2.m.a 2 1
312.2.m.b 2
312.2.m.c 24
312.2.n $$\chi_{312}(311, \cdot)$$ None 0 1
312.2.q $$\chi_{312}(217, \cdot)$$ 312.2.q.a 2 2
312.2.q.b 2
312.2.q.c 2
312.2.q.d 4
312.2.q.e 6
312.2.t $$\chi_{312}(187, \cdot)$$ 312.2.t.a 2 2
312.2.t.b 2
312.2.t.c 2
312.2.t.d 2
312.2.t.e 24
312.2.t.f 24
312.2.u $$\chi_{312}(31, \cdot)$$ None 0 2
312.2.x $$\chi_{312}(161, \cdot)$$ 312.2.x.a 4 2
312.2.x.b 8
312.2.x.c 16
312.2.y $$\chi_{312}(5, \cdot)$$ 312.2.y.a 104 2
312.2.ba $$\chi_{312}(179, \cdot)$$ 312.2.ba.a 104 2
312.2.bb $$\chi_{312}(61, \cdot)$$ 312.2.bb.a 56 2
312.2.be $$\chi_{312}(191, \cdot)$$ None 0 2
312.2.bf $$\chi_{312}(49, \cdot)$$ 312.2.bf.a 4 2
312.2.bf.b 8
312.2.bj $$\chi_{312}(23, \cdot)$$ None 0 2
312.2.bk $$\chi_{312}(205, \cdot)$$ 312.2.bk.a 8 2
312.2.bk.b 48
312.2.bn $$\chi_{312}(35, \cdot)$$ 312.2.bn.a 104 2
312.2.bo $$\chi_{312}(149, \cdot)$$ 312.2.bo.a 208 4
312.2.bp $$\chi_{312}(41, \cdot)$$ 312.2.bp.a 56 4
312.2.bs $$\chi_{312}(7, \cdot)$$ None 0 4
312.2.bt $$\chi_{312}(19, \cdot)$$ 312.2.bt.a 4 4
312.2.bt.b 4
312.2.bt.c 48
312.2.bt.d 56

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(312))$$ into lower level spaces

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