## Defining parameters

 Level: $$N$$ = $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newform subspaces: $$7$$ Sturm bound: $$288$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(45))$$.

Total New Old
Modular forms 104 65 39
Cusp forms 41 39 2
Eisenstein series 63 26 37

## Trace form

 $$39q - 7q^{2} - 8q^{3} - 11q^{4} - 9q^{5} - 16q^{6} - 12q^{7} - 3q^{8} - 4q^{9} + O(q^{10})$$ $$39q - 7q^{2} - 8q^{3} - 11q^{4} - 9q^{5} - 16q^{6} - 12q^{7} - 3q^{8} - 4q^{9} - 15q^{10} - 4q^{11} + 8q^{12} - 2q^{13} + 12q^{14} + 4q^{15} - 7q^{16} + 2q^{17} + 16q^{18} - 20q^{19} + 25q^{20} + 8q^{22} + 12q^{23} + 36q^{24} + 3q^{25} + 14q^{26} + 16q^{27} + 8q^{28} + 22q^{29} + 32q^{30} - 8q^{31} + 13q^{32} + 8q^{33} + 14q^{34} - 8q^{35} - 32q^{36} + 2q^{37} - 16q^{38} - 32q^{39} - 23q^{40} - 50q^{41} - 36q^{42} - 32q^{43} - 68q^{44} - 28q^{45} - 40q^{46} - 40q^{47} - 40q^{48} + 3q^{49} - 31q^{50} - 16q^{51} + 2q^{52} + 2q^{53} - 4q^{54} + 12q^{55} + 24q^{56} + 16q^{57} + 26q^{58} + 44q^{59} + 20q^{60} + 26q^{61} + 84q^{62} + 60q^{63} + 49q^{64} + 62q^{65} + 88q^{66} + 24q^{67} + 70q^{68} + 60q^{69} + 60q^{70} + 64q^{71} + 24q^{72} - 2q^{73} - 2q^{74} - 8q^{75} + 36q^{76} - 12q^{77} - 40q^{78} - 12q^{79} - 35q^{80} - 40q^{81} - 30q^{82} - 72q^{83} - 72q^{84} - 6q^{85} - 100q^{86} - 44q^{87} - 66q^{89} - 104q^{90} - 48q^{91} - 96q^{92} - 60q^{93} - 4q^{94} - 40q^{95} - 56q^{96} + 2q^{97} - 83q^{98} - 20q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$

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
45.2.a $$\chi_{45}(1, \cdot)$$ 45.2.a.a 1 1
45.2.b $$\chi_{45}(19, \cdot)$$ 45.2.b.a 2 1
45.2.e $$\chi_{45}(16, \cdot)$$ 45.2.e.a 2 2
45.2.e.b 6
45.2.f $$\chi_{45}(8, \cdot)$$ 45.2.f.a 4 2
45.2.j $$\chi_{45}(4, \cdot)$$ 45.2.j.a 8 2
45.2.l $$\chi_{45}(2, \cdot)$$ 45.2.l.a 16 4

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(45))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(45)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 2}$$