## Defining parameters

 Level: $$N$$ = $$115 = 5 \cdot 23$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newform subspaces: $$11$$ Sturm bound: $$2112$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(115))$$.

Total New Old
Modular forms 616 545 71
Cusp forms 441 417 24
Eisenstein series 175 128 47

## Trace form

 $$417q - 25q^{2} - 26q^{3} - 29q^{4} - 34q^{5} - 78q^{6} - 30q^{7} - 37q^{8} - 35q^{9} + O(q^{10})$$ $$417q - 25q^{2} - 26q^{3} - 29q^{4} - 34q^{5} - 78q^{6} - 30q^{7} - 37q^{8} - 35q^{9} - 36q^{10} - 78q^{11} - 50q^{12} - 36q^{13} - 46q^{14} - 26q^{15} - 53q^{16} - 18q^{17} + 27q^{18} - 20q^{19} + 4q^{20} - 32q^{21} + 8q^{22} - q^{23} + 28q^{24} - 12q^{25} - 64q^{26} + 4q^{27} + 10q^{28} - 30q^{29} - q^{30} - 76q^{31} - 41q^{32} - 48q^{33} - 32q^{34} - 19q^{35} - 47q^{36} + 28q^{37} + 28q^{38} + 10q^{39} + 40q^{40} - 64q^{41} + 102q^{42} + 22q^{43} + 92q^{44} + 31q^{45} + 41q^{46} - 4q^{47} + 74q^{48} + 53q^{49} - 14q^{50} - 50q^{51} + 100q^{52} - 32q^{53} + 78q^{54} + 21q^{55} + 56q^{56} + 52q^{57} + 86q^{58} + 50q^{59} + 71q^{60} - 40q^{61} + 14q^{62} - 16q^{63} + 27q^{64} + 30q^{65} + 32q^{66} - 46q^{67} + 138q^{68} - 4q^{69} - 2q^{70} - 28q^{71} + 113q^{72} - 52q^{73} + 62q^{74} + 84q^{75} + 80q^{76} + 80q^{77} + 162q^{78} + 74q^{79} + 145q^{80} + 165q^{81} + 72q^{82} + 48q^{83} + 194q^{84} + 81q^{85} + 66q^{86} + 122q^{87} - 114q^{88} + 20q^{89} + 159q^{90} + 20q^{91} + 15q^{92} + 92q^{93} + 76q^{94} + 2q^{95} + 100q^{96} - 54q^{97} - 105q^{98} - 24q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(115))$$

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
115.2.a $$\chi_{115}(1, \cdot)$$ 115.2.a.a 1 1
115.2.a.b 2
115.2.a.c 4
115.2.b $$\chi_{115}(24, \cdot)$$ 115.2.b.a 2 1
115.2.b.b 8
115.2.e $$\chi_{115}(22, \cdot)$$ 115.2.e.a 20 2
115.2.g $$\chi_{115}(6, \cdot)$$ 115.2.g.a 10 10
115.2.g.b 20
115.2.g.c 50
115.2.j $$\chi_{115}(4, \cdot)$$ 115.2.j.a 100 10
115.2.l $$\chi_{115}(7, \cdot)$$ 115.2.l.a 200 20

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(115))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(115)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(23))$$$$^{\oplus 2}$$