## Defining parameters

 Level: $$N$$ = $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$12$$ Newform subspaces: $$54$$ Sturm bound: $$14400$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(225))$$.

Total New Old
Modular forms 5624 4033 1591
Cusp forms 5176 3848 1328
Eisenstein series 448 185 263

## Trace form

 $$3848q - 17q^{2} - 27q^{3} - 51q^{4} - 21q^{5} - 31q^{6} - 21q^{7} - 84q^{8} - 107q^{9} + O(q^{10})$$ $$3848q - 17q^{2} - 27q^{3} - 51q^{4} - 21q^{5} - 31q^{6} - 21q^{7} - 84q^{8} - 107q^{9} - 190q^{10} - 104q^{11} + 76q^{12} + 379q^{13} + 846q^{14} + 136q^{15} + 673q^{16} + 188q^{17} + 192q^{18} - 668q^{19} - 870q^{20} - 1155q^{21} - 2085q^{22} - 1219q^{23} - 691q^{24} + 957q^{25} + 1716q^{26} + 648q^{27} + 4066q^{28} + 2977q^{29} + 1128q^{30} + 1811q^{31} + 2833q^{32} + 2398q^{33} - 1665q^{34} - 352q^{35} + 1591q^{36} - 3419q^{37} - 6191q^{38} - 2337q^{39} - 4092q^{40} - 4994q^{41} - 7318q^{42} + 110q^{43} - 5226q^{44} - 2292q^{45} + 1066q^{46} - 5401q^{47} - 9373q^{48} - 771q^{49} + 1556q^{50} - 743q^{51} + 5610q^{52} + 5587q^{53} + 8623q^{54} + 1758q^{55} + 14556q^{56} + 9219q^{57} + 6636q^{58} + 11504q^{59} + 13940q^{60} + 5945q^{61} + 13472q^{62} + 8571q^{63} + 4666q^{64} + 3373q^{65} + 826q^{66} - 2766q^{67} - 8827q^{68} - 5309q^{69} - 14730q^{70} - 4990q^{71} - 10125q^{72} - 16790q^{73} - 17512q^{74} - 6592q^{75} - 14203q^{76} - 9057q^{77} - 1302q^{78} - 4239q^{79} - 15350q^{80} - 2783q^{81} - 3380q^{82} - 12275q^{83} - 17254q^{84} - 1189q^{85} - 20915q^{86} - 13601q^{87} + 4749q^{88} - 11235q^{89} - 1016q^{90} - 660q^{91} - 4184q^{92} - 13589q^{93} + 15078q^{94} + 4330q^{95} - 8912q^{96} + 18464q^{97} + 14200q^{98} - 663q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(225))$$

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
225.4.a $$\chi_{225}(1, \cdot)$$ 225.4.a.a 1 1
225.4.a.b 1
225.4.a.c 1
225.4.a.d 1
225.4.a.e 1
225.4.a.f 1
225.4.a.g 1
225.4.a.h 1
225.4.a.i 2
225.4.a.j 2
225.4.a.k 2
225.4.a.l 2
225.4.a.m 2
225.4.a.n 2
225.4.a.o 2
225.4.b $$\chi_{225}(199, \cdot)$$ 225.4.b.a 2 1
225.4.b.b 2
225.4.b.c 2
225.4.b.d 2
225.4.b.e 2
225.4.b.f 2
225.4.b.g 2
225.4.b.h 4
225.4.b.i 4
225.4.e $$\chi_{225}(76, \cdot)$$ 225.4.e.a 4 2
225.4.e.b 4
225.4.e.c 6
225.4.e.d 14
225.4.e.e 24
225.4.e.f 24
225.4.e.g 32
225.4.f $$\chi_{225}(107, \cdot)$$ 225.4.f.a 4 2
225.4.f.b 4
225.4.f.c 12
225.4.f.d 16
225.4.h $$\chi_{225}(46, \cdot)$$ 225.4.h.a 28 4
225.4.h.b 28
225.4.h.c 28
225.4.h.d 64
225.4.k $$\chi_{225}(49, \cdot)$$ 225.4.k.a 8 2
225.4.k.b 8
225.4.k.c 12
225.4.k.d 28
225.4.k.e 48
225.4.m $$\chi_{225}(19, \cdot)$$ 225.4.m.a 24 4
225.4.m.b 56
225.4.m.c 64
225.4.p $$\chi_{225}(32, \cdot)$$ 225.4.p.a 48 4
225.4.p.b 64
225.4.p.c 96
225.4.q $$\chi_{225}(16, \cdot)$$ 225.4.q.a 704 8
225.4.s $$\chi_{225}(8, \cdot)$$ 225.4.s.a 240 8
225.4.u $$\chi_{225}(4, \cdot)$$ 225.4.u.a 704 8
225.4.w $$\chi_{225}(2, \cdot)$$ 225.4.w.a 1408 16

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(225))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(225)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 2}$$