## Defining parameters

 Level: $$N$$ = $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newforms: $$34$$ Sturm bound: $$7200$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(225))$$.

Total New Old
Modular forms 2024 1406 618
Cusp forms 1577 1221 356
Eisenstein series 447 185 262

## Trace form

 $$1221q - 16q^{2} - 24q^{3} - 8q^{4} - 21q^{5} - 40q^{6} - 6q^{7} - 24q^{8} - 32q^{9} + O(q^{10})$$ $$1221q - 16q^{2} - 24q^{3} - 8q^{4} - 21q^{5} - 40q^{6} - 6q^{7} - 24q^{8} - 32q^{9} - 75q^{10} - 46q^{11} - 56q^{12} - 26q^{13} - 54q^{14} - 44q^{15} - 56q^{16} - 44q^{17} - 72q^{18} - 70q^{19} - 90q^{20} - 72q^{21} - 86q^{22} - 74q^{23} - 112q^{24} - 63q^{25} - 132q^{26} - 72q^{27} - 166q^{28} - 94q^{29} - 72q^{30} - 58q^{31} - 91q^{32} - 56q^{33} - 83q^{34} - 32q^{35} - 8q^{36} - 99q^{37} - 28q^{38} + 24q^{39} - 57q^{40} + 26q^{41} + 32q^{42} - 6q^{43} + 36q^{44} - 12q^{45} - 142q^{46} + 10q^{47} + 80q^{48} - 91q^{49} - 89q^{50} - 80q^{51} - 144q^{52} - 79q^{53} - 32q^{54} - 142q^{55} - 66q^{56} - 72q^{57} - 152q^{58} - 128q^{59} + 20q^{60} - 66q^{61} - 68q^{62} - 60q^{63} + 22q^{64} + 53q^{65} - 128q^{66} + 42q^{67} + 190q^{68} - 20q^{69} + 150q^{70} - 62q^{71} + 192q^{72} + 34q^{73} + 304q^{74} + 128q^{75} + 52q^{76} + 234q^{77} + 240q^{78} + 114q^{79} + 425q^{80} + 88q^{81} + 170q^{82} + 284q^{83} + 344q^{84} + 21q^{85} + 266q^{86} + 148q^{87} + 90q^{88} + 177q^{89} + 184q^{90} - 86q^{91} + 182q^{92} + 100q^{93} - 182q^{94} - 70q^{95} + 136q^{96} - 174q^{97} + 56q^{98} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(225))$$

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
225.2.a $$\chi_{225}(1, \cdot)$$ 225.2.a.a 1 1
225.2.a.b 1
225.2.a.c 1
225.2.a.d 1
225.2.a.e 1
225.2.a.f 2
225.2.b $$\chi_{225}(199, \cdot)$$ 225.2.b.a 2 1
225.2.b.b 2
225.2.b.c 2
225.2.e $$\chi_{225}(76, \cdot)$$ 225.2.e.a 2 2
225.2.e.b 6
225.2.e.c 8
225.2.e.d 8
225.2.e.e 8
225.2.f $$\chi_{225}(107, \cdot)$$ 225.2.f.a 4 2
225.2.f.b 8
225.2.h $$\chi_{225}(46, \cdot)$$ 225.2.h.a 4 4
225.2.h.b 4
225.2.h.c 8
225.2.h.d 12
225.2.h.e 16
225.2.k $$\chi_{225}(49, \cdot)$$ 225.2.k.a 4 2
225.2.k.b 12
225.2.k.c 16
225.2.m $$\chi_{225}(19, \cdot)$$ 225.2.m.a 8 4
225.2.m.b 16
225.2.m.c 24
225.2.p $$\chi_{225}(32, \cdot)$$ 225.2.p.a 16 4
225.2.p.b 16
225.2.p.c 32
225.2.q $$\chi_{225}(16, \cdot)$$ 225.2.q.a 224 8
225.2.s $$\chi_{225}(8, \cdot)$$ 225.2.s.a 80 8
225.2.u $$\chi_{225}(4, \cdot)$$ 225.2.u.a 224 8
225.2.w $$\chi_{225}(2, \cdot)$$ 225.2.w.a 448 16

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(225))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(225)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 2}$$