## 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

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

## 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

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