## Defining parameters

 Level: $$N$$ = $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$34$$ 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

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

## 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

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