## Defining parameters

 Level: $$N$$ = $$117 = 3^{2} \cdot 13$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$15$$ Newform subspaces: $$31$$ Sturm bound: $$2016$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(117))$$.

Total New Old
Modular forms 600 443 157
Cusp forms 409 345 64
Eisenstein series 191 98 93

## Trace form

 $$345 q - 18 q^{2} - 24 q^{3} - 18 q^{4} - 18 q^{5} - 24 q^{6} - 20 q^{7} - 27 q^{8} - 24 q^{9} + O(q^{10})$$ $$345 q - 18 q^{2} - 24 q^{3} - 18 q^{4} - 18 q^{5} - 24 q^{6} - 20 q^{7} - 27 q^{8} - 24 q^{9} - 69 q^{10} - 24 q^{11} - 24 q^{12} - 30 q^{13} - 48 q^{14} - 24 q^{15} - 38 q^{16} - 27 q^{17} - 24 q^{18} - 74 q^{19} - 51 q^{20} - 24 q^{21} - 48 q^{22} - 30 q^{23} - 39 q^{25} - 48 q^{26} - 48 q^{27} - 72 q^{28} + 3 q^{29} + 24 q^{30} + 12 q^{31} + 60 q^{32} + 12 q^{33} + 30 q^{34} + 60 q^{35} + 72 q^{36} - 15 q^{37} + 108 q^{38} + 24 q^{39} + 84 q^{40} + 33 q^{41} + 48 q^{42} - 8 q^{43} + 96 q^{44} + 36 q^{45} - 42 q^{46} + 12 q^{47} + 48 q^{48} - 20 q^{49} + 33 q^{50} - 69 q^{52} - 84 q^{53} - 24 q^{54} - 120 q^{55} - 36 q^{56} - 24 q^{57} - 93 q^{58} - 60 q^{59} + 36 q^{60} - 75 q^{61} - 30 q^{62} + 24 q^{63} - 33 q^{64} + 45 q^{65} + 72 q^{66} + 22 q^{67} + 171 q^{68} + 48 q^{69} + 132 q^{70} + 114 q^{71} + 132 q^{72} + 70 q^{73} + 195 q^{74} + 96 q^{75} + 130 q^{76} + 108 q^{77} + 204 q^{78} + 291 q^{80} + 72 q^{81} + 75 q^{82} + 144 q^{83} + 144 q^{84} + 81 q^{85} + 168 q^{86} + 72 q^{87} + 96 q^{88} + 30 q^{89} + 132 q^{90} - 56 q^{91} + 60 q^{92} + 48 q^{93} - 18 q^{94} - 30 q^{95} - 180 q^{96} - 78 q^{97} - 42 q^{98} - 96 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(117))$$

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
117.2.a $$\chi_{117}(1, \cdot)$$ 117.2.a.a 1 1
117.2.a.b 2
117.2.a.c 2
117.2.b $$\chi_{117}(64, \cdot)$$ 117.2.b.a 2 1
117.2.b.b 4
117.2.e $$\chi_{117}(40, \cdot)$$ 117.2.e.a 2 2
117.2.e.b 10
117.2.e.c 12
117.2.f $$\chi_{117}(61, \cdot)$$ 117.2.f.a 24 2
117.2.g $$\chi_{117}(55, \cdot)$$ 117.2.g.a 2 2
117.2.g.b 2
117.2.g.c 4
117.2.h $$\chi_{117}(16, \cdot)$$ 117.2.h.a 24 2
117.2.i $$\chi_{117}(8, \cdot)$$ 117.2.i.a 12 2
117.2.l $$\chi_{117}(4, \cdot)$$ 117.2.l.a 2 2
117.2.l.b 22
117.2.q $$\chi_{117}(10, \cdot)$$ 117.2.q.a 2 2
117.2.q.b 2
117.2.q.c 2
117.2.q.d 4
117.2.r $$\chi_{117}(43, \cdot)$$ 117.2.r.a 2 2
117.2.r.b 22
117.2.t $$\chi_{117}(25, \cdot)$$ 117.2.t.a 2 2
117.2.t.b 2
117.2.t.c 20
117.2.x $$\chi_{117}(2, \cdot)$$ 117.2.x.a 48 4
117.2.z $$\chi_{117}(5, \cdot)$$ 117.2.z.a 4 4
117.2.z.b 44
117.2.ba $$\chi_{117}(71, \cdot)$$ 117.2.ba.a 8 4
117.2.ba.b 8
117.2.bc $$\chi_{117}(20, \cdot)$$ 117.2.bc.a 48 4

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(117))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(117)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(39))$$$$^{\oplus 2}$$