## Defining parameters

 Level: $$N$$ = $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$12$$ Newform subspaces: $$32$$ Sturm bound: $$10800$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(225))$$.

Total New Old
Modular forms 3824 2718 1106
Cusp forms 3376 2534 842
Eisenstein series 448 184 264

## Trace form

 $$2534q - 25q^{2} - 27q^{3} - 23q^{4} - 26q^{5} - 31q^{6} - 24q^{7} + 42q^{8} + 7q^{9} + O(q^{10})$$ $$2534q - 25q^{2} - 27q^{3} - 23q^{4} - 26q^{5} - 31q^{6} - 24q^{7} + 42q^{8} + 7q^{9} - 10q^{10} + 55q^{11} + 46q^{12} + 14q^{13} - 44q^{15} - 91q^{16} - 96q^{17} - 120q^{18} - 96q^{19} + 50q^{20} - 30q^{21} + 253q^{22} + 190q^{23} + 83q^{24} + 72q^{25} + 276q^{26} + 126q^{27} - 102q^{28} - 28q^{29} - 72q^{30} - 338q^{31} - 947q^{32} - 383q^{33} - 779q^{34} - 532q^{35} - 1265q^{36} - 242q^{37} - 957q^{38} - 768q^{39} + 168q^{40} - 379q^{41} - 496q^{42} + 311q^{43} + 320q^{44} + 108q^{45} + 402q^{46} + 862q^{47} + 1145q^{48} + 833q^{49} + 796q^{50} + 961q^{51} + 408q^{52} + 1118q^{53} + 1375q^{54} - 322q^{55} + 888q^{56} + 759q^{57} - 698q^{58} - 409q^{59} - 700q^{60} - 538q^{61} - 2464q^{62} - 1032q^{63} - 2986q^{64} - 1672q^{65} - 1034q^{66} - 1407q^{67} - 2993q^{68} - 1244q^{69} - 1410q^{70} - 1454q^{71} - 2289q^{72} - 1072q^{73} - 2454q^{74} - 832q^{75} + 513q^{76} - 1232q^{77} - 2052q^{78} + 1020q^{79} + 1210q^{80} - 1145q^{81} + 2232q^{82} + 936q^{83} + 50q^{84} + 2216q^{85} + 559q^{86} + 682q^{87} + 4545q^{88} + 2970q^{89} + 2104q^{90} + 1162q^{91} + 4782q^{92} + 1708q^{93} + 1342q^{94} + 1310q^{95} + 2872q^{96} + 149q^{97} + 3366q^{98} + 2544q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(225))$$

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
225.3.c $$\chi_{225}(26, \cdot)$$ 225.3.c.a 2 1
225.3.c.b 2
225.3.c.c 4
225.3.c.d 4
225.3.d $$\chi_{225}(224, \cdot)$$ 225.3.d.a 4 1
225.3.d.b 8
225.3.g $$\chi_{225}(82, \cdot)$$ 225.3.g.a 4 2
225.3.g.b 4
225.3.g.c 4
225.3.g.d 4
225.3.g.e 4
225.3.g.f 4
225.3.g.g 4
225.3.i $$\chi_{225}(74, \cdot)$$ 225.3.i.a 4 2
225.3.i.b 32
225.3.i.c 32
225.3.j $$\chi_{225}(101, \cdot)$$ 225.3.j.a 2 2
225.3.j.b 16
225.3.j.c 16
225.3.j.d 16
225.3.j.e 20
225.3.l $$\chi_{225}(44, \cdot)$$ 225.3.l.a 80 4
225.3.n $$\chi_{225}(71, \cdot)$$ 225.3.n.a 80 4
225.3.o $$\chi_{225}(7, \cdot)$$ 225.3.o.a 32 4
225.3.o.b 40
225.3.o.c 64
225.3.r $$\chi_{225}(28, \cdot)$$ 225.3.r.a 32 8
225.3.r.b 80
225.3.r.c 80
225.3.t $$\chi_{225}(11, \cdot)$$ 225.3.t.a 464 8
225.3.v $$\chi_{225}(14, \cdot)$$ 225.3.v.a 464 8
225.3.x $$\chi_{225}(13, \cdot)$$ 225.3.x.a 928 16

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(225))$$ into lower level spaces

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