## Defining parameters

 Level: $$N$$ = $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$6$$ Newform subspaces: $$7$$ Sturm bound: $$432$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(45))$$.

Total New Old
Modular forms 176 120 56
Cusp forms 112 92 20
Eisenstein series 64 28 36

## Trace form

 $$92q + 2q^{2} - 2q^{3} - 2q^{4} - 10q^{5} - 34q^{6} - 36q^{8} - 10q^{9} + O(q^{10})$$ $$92q + 2q^{2} - 2q^{3} - 2q^{4} - 10q^{5} - 34q^{6} - 36q^{8} - 10q^{9} - 68q^{10} - 50q^{11} - 52q^{12} - 28q^{13} - 24q^{14} + 4q^{15} + 34q^{16} + 68q^{17} + 40q^{18} + 20q^{19} + 66q^{20} - 12q^{21} - 106q^{22} - 28q^{23} + 6q^{24} - 94q^{25} - 200q^{26} - 164q^{27} - 120q^{28} - 168q^{29} + 14q^{30} + 120q^{31} + 278q^{32} + 158q^{33} + 290q^{34} + 392q^{35} + 610q^{36} + 208q^{37} + 678q^{38} + 400q^{39} + 222q^{40} + 274q^{41} + 228q^{42} - 142q^{43} - 94q^{45} - 336q^{46} - 280q^{47} - 682q^{48} - 494q^{49} - 1030q^{50} - 658q^{51} - 380q^{52} - 664q^{53} - 586q^{54} + 4q^{55} - 828q^{56} - 350q^{57} + 136q^{58} - 366q^{59} - 478q^{60} - 184q^{61} - 188q^{62} - 108q^{63} + 236q^{64} + 188q^{65} + 28q^{66} + 222q^{67} + 442q^{68} + 252q^{69} + 408q^{70} + 520q^{71} + 642q^{72} + 236q^{73} + 1044q^{74} + 1066q^{75} + 22q^{76} + 952q^{77} + 1352q^{78} - 168q^{79} + 1280q^{80} + 938q^{81} + 76q^{82} + 908q^{83} + 1164q^{84} + 178q^{85} + 1438q^{86} + 640q^{87} + 30q^{88} - 278q^{90} + 192q^{91} - 1048q^{92} - 972q^{93} - 412q^{94} - 1086q^{95} - 1184q^{96} - 122q^{97} - 1688q^{98} - 1292q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(45))$$

We only show spaces with odd 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
45.3.c $$\chi_{45}(26, \cdot)$$ 45.3.c.a 4 1
45.3.d $$\chi_{45}(44, \cdot)$$ 45.3.d.a 4 1
45.3.g $$\chi_{45}(28, \cdot)$$ 45.3.g.a 4 2
45.3.g.b 4
45.3.h $$\chi_{45}(14, \cdot)$$ 45.3.h.a 20 2
45.3.i $$\chi_{45}(11, \cdot)$$ 45.3.i.a 16 2
45.3.k $$\chi_{45}(7, \cdot)$$ 45.3.k.a 40 4

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(45))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(45)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 2}$$