## Defining parameters

 Level: $$N$$ = $$130 = 2 \cdot 5 \cdot 13$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$30$$ Sturm bound: $$2016$$ Trace bound: $$9$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(130))$$.

Total New Old
Modular forms 600 155 445
Cusp forms 409 155 254
Eisenstein series 191 0 191

## Trace form

 $$155 q + q^{2} + 4 q^{3} + q^{4} + q^{5} + 4 q^{6} - 5 q^{8} - 19 q^{9} + O(q^{10})$$ $$155 q + q^{2} + 4 q^{3} + q^{4} + q^{5} + 4 q^{6} - 5 q^{8} - 19 q^{9} - 14 q^{10} - 12 q^{11} - 4 q^{12} - 35 q^{13} - 16 q^{14} - 20 q^{15} - 7 q^{16} - 12 q^{17} - 17 q^{18} - 36 q^{19} - 2 q^{20} - 24 q^{21} + 12 q^{22} + 4 q^{24} + q^{25} + 13 q^{26} - 32 q^{27} - 48 q^{29} - 20 q^{30} - 48 q^{31} + q^{32} - 72 q^{33} - 30 q^{34} - 64 q^{35} - 35 q^{36} - 24 q^{37} - 52 q^{38} - 52 q^{39} + q^{40} - 60 q^{41} - 40 q^{42} - 44 q^{43} - 36 q^{44} - 62 q^{45} - 24 q^{46} - 48 q^{47} + 4 q^{48} - 7 q^{49} - 14 q^{50} + 24 q^{51} + 11 q^{52} + 78 q^{53} + 112 q^{54} + 96 q^{55} + 32 q^{56} + 192 q^{57} + 84 q^{58} + 108 q^{59} + 76 q^{60} + 80 q^{61} + 104 q^{62} + 192 q^{63} + 7 q^{64} + 118 q^{65} + 144 q^{66} + 108 q^{67} + 24 q^{68} + 192 q^{69} + 116 q^{70} + 24 q^{71} + 61 q^{72} + 42 q^{73} + 68 q^{74} + 116 q^{75} + 12 q^{76} + 76 q^{78} - 16 q^{79} + 10 q^{80} - 31 q^{81} - 48 q^{82} - 108 q^{83} - 24 q^{84} - 105 q^{85} - 4 q^{86} - 120 q^{87} + 12 q^{88} - 102 q^{89} - 47 q^{90} - 48 q^{91} - 24 q^{92} - 96 q^{93} - 72 q^{94} - 52 q^{95} + 4 q^{96} - 102 q^{97} - 39 q^{98} - 60 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(130))$$

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
130.2.a $$\chi_{130}(1, \cdot)$$ 130.2.a.a 1 1
130.2.a.b 1
130.2.a.c 1
130.2.b $$\chi_{130}(79, \cdot)$$ 130.2.b.a 6 1
130.2.c $$\chi_{130}(129, \cdot)$$ 130.2.c.a 4 1
130.2.c.b 4
130.2.d $$\chi_{130}(51, \cdot)$$ 130.2.d.a 2 1
130.2.e $$\chi_{130}(61, \cdot)$$ 130.2.e.a 2 2
130.2.e.b 2
130.2.e.c 4
130.2.e.d 4
130.2.g $$\chi_{130}(57, \cdot)$$ 130.2.g.a 2 2
130.2.g.b 2
130.2.g.c 2
130.2.g.d 4
130.2.g.e 4
130.2.j $$\chi_{130}(47, \cdot)$$ 130.2.j.a 2 2
130.2.j.b 2
130.2.j.c 2
130.2.j.d 4
130.2.j.e 4
130.2.l $$\chi_{130}(101, \cdot)$$ 130.2.l.a 4 2
130.2.l.b 8
130.2.m $$\chi_{130}(49, \cdot)$$ 130.2.m.a 8 2
130.2.m.b 8
130.2.n $$\chi_{130}(9, \cdot)$$ 130.2.n.a 12 2
130.2.p $$\chi_{130}(7, \cdot)$$ 130.2.p.a 12 4
130.2.p.b 16
130.2.s $$\chi_{130}(33, \cdot)$$ 130.2.s.a 12 4
130.2.s.b 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(130))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(130)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(26))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(65))$$$$^{\oplus 2}$$