## Defining parameters

 Level: $$N$$ = $$45 = 3^{2} \cdot 5$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$6$$ Newform subspaces: $$13$$ Sturm bound: $$576$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(45))$$.

Total New Old
Modular forms 248 169 79
Cusp forms 184 143 41
Eisenstein series 64 26 38

## Trace form

 $$143 q - 2 q^{2} - 2 q^{3} + 30 q^{4} + 6 q^{5} - 34 q^{6} - 24 q^{7} - 72 q^{8} - 34 q^{9} + O(q^{10})$$ $$143 q - 2 q^{2} - 2 q^{3} + 30 q^{4} + 6 q^{5} - 34 q^{6} - 24 q^{7} - 72 q^{8} - 34 q^{9} + 88 q^{10} + 124 q^{11} + 176 q^{12} - 116 q^{13} - 348 q^{14} - 203 q^{15} - 598 q^{16} - 566 q^{17} - 440 q^{18} + 376 q^{19} + 968 q^{20} + 510 q^{21} + 1338 q^{22} + 864 q^{23} + 474 q^{24} - 388 q^{25} + 112 q^{26} + 304 q^{27} - 1976 q^{28} - 1540 q^{29} - 1456 q^{30} - 1510 q^{31} - 2626 q^{32} - 1516 q^{33} + 438 q^{34} - 544 q^{35} - 494 q^{36} + 1382 q^{37} + 994 q^{38} + 10 q^{39} + 2276 q^{40} + 2438 q^{41} + 4368 q^{42} + 982 q^{43} + 3956 q^{44} + 2927 q^{45} + 1288 q^{46} + 4144 q^{47} + 4778 q^{48} + 613 q^{49} + 2020 q^{50} - 130 q^{51} - 2920 q^{52} - 2138 q^{53} - 6154 q^{54} - 2342 q^{55} - 6228 q^{56} - 2798 q^{57} - 4228 q^{58} - 2900 q^{59} - 3064 q^{60} - 204 q^{61} - 2388 q^{62} - 2310 q^{63} + 2476 q^{64} - 1793 q^{65} + 916 q^{66} + 3090 q^{67} - 322 q^{68} - 1782 q^{69} + 2058 q^{70} - 5356 q^{71} - 5814 q^{72} + 3610 q^{73} - 1120 q^{74} - 4175 q^{75} - 3458 q^{76} - 3894 q^{77} - 4084 q^{78} - 3666 q^{79} - 3352 q^{80} + 926 q^{81} - 10100 q^{82} + 3672 q^{83} + 9564 q^{84} - 2252 q^{85} + 13210 q^{86} + 11110 q^{87} + 10422 q^{88} + 16578 q^{89} + 21652 q^{90} + 11712 q^{91} + 16860 q^{92} + 9774 q^{93} + 1116 q^{94} + 1828 q^{95} + 4528 q^{96} + 362 q^{97} + 1064 q^{98} - 134 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(45))$$

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
45.4.a $$\chi_{45}(1, \cdot)$$ 45.4.a.a 1 1
45.4.a.b 1
45.4.a.c 1
45.4.a.d 1
45.4.a.e 1
45.4.b $$\chi_{45}(19, \cdot)$$ 45.4.b.a 2 1
45.4.b.b 4
45.4.e $$\chi_{45}(16, \cdot)$$ 45.4.e.a 4 2
45.4.e.b 6
45.4.e.c 14
45.4.f $$\chi_{45}(8, \cdot)$$ 45.4.f.a 12 2
45.4.j $$\chi_{45}(4, \cdot)$$ 45.4.j.a 32 2
45.4.l $$\chi_{45}(2, \cdot)$$ 45.4.l.a 64 4

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(45))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(45)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 2}$$