## Defining parameters

 Level: $$N$$ = $$675 = 3^{3} \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$18$$ Newform subspaces: $$76$$ Sturm bound: $$64800$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(675))$$.

Total New Old
Modular forms 17040 12125 4915
Cusp forms 15361 11469 3892
Eisenstein series 1679 656 1023

## Trace form

 $$11469q - 54q^{2} - 78q^{3} - 92q^{4} - 64q^{5} - 120q^{6} - 91q^{7} - 26q^{8} - 72q^{9} + O(q^{10})$$ $$11469q - 54q^{2} - 78q^{3} - 92q^{4} - 64q^{5} - 120q^{6} - 91q^{7} - 26q^{8} - 72q^{9} - 104q^{10} - 61q^{11} - 60q^{12} - 69q^{13} + 7q^{14} - 96q^{15} - 104q^{16} - 15q^{17} - 63q^{18} - 78q^{19} - 32q^{20} - 132q^{21} - 33q^{22} - 20q^{23} - 90q^{24} - 96q^{25} - 126q^{26} - 81q^{27} - 170q^{28} + 2q^{29} - 96q^{30} - 109q^{31} - 112q^{32} - 96q^{33} - 75q^{34} - 100q^{35} - 246q^{36} - 92q^{37} - 174q^{38} - 159q^{39} - 192q^{40} - 209q^{41} - 246q^{42} - 201q^{43} - 421q^{44} - 156q^{45} - 223q^{46} - 231q^{47} - 327q^{48} - 158q^{49} - 184q^{50} - 354q^{51} - 197q^{52} - 186q^{53} - 270q^{54} - 248q^{55} - 241q^{56} - 147q^{57} - 63q^{58} - 8q^{59} - 168q^{60} - 73q^{61} + 4q^{62} - 63q^{63} - 144q^{64} - 96q^{65} - 129q^{66} - 78q^{67} - 223q^{68} - 63q^{69} - 240q^{70} + 11q^{71} - 246q^{72} - 149q^{73} - 371q^{74} - 168q^{75} - 506q^{76} - 393q^{77} - 366q^{78} - 333q^{79} - 552q^{80} - 276q^{81} - 684q^{82} - 493q^{83} - 522q^{84} - 256q^{85} - 757q^{86} - 303q^{87} - 546q^{88} - 579q^{89} - 324q^{90} - 371q^{91} - 847q^{92} - 399q^{93} - 387q^{94} - 212q^{95} - 552q^{96} - 212q^{97} - 725q^{98} - 339q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(675))$$

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
675.2.a $$\chi_{675}(1, \cdot)$$ 675.2.a.a 1 1
675.2.a.b 1
675.2.a.c 1
675.2.a.d 1
675.2.a.e 1
675.2.a.f 1
675.2.a.g 1
675.2.a.h 1
675.2.a.i 1
675.2.a.j 2
675.2.a.k 2
675.2.a.l 2
675.2.a.m 2
675.2.a.n 2
675.2.a.o 2
675.2.a.p 2
675.2.a.q 2
675.2.b $$\chi_{675}(649, \cdot)$$ 675.2.b.a 2 1
675.2.b.b 2
675.2.b.c 2
675.2.b.d 2
675.2.b.e 2
675.2.b.f 2
675.2.b.g 4
675.2.b.h 4
675.2.b.i 4
675.2.e $$\chi_{675}(226, \cdot)$$ 675.2.e.a 2 2
675.2.e.b 6
675.2.e.c 8
675.2.e.d 8
675.2.e.e 8
675.2.f $$\chi_{675}(107, \cdot)$$ 675.2.f.a 4 2
675.2.f.b 4
675.2.f.c 4
675.2.f.d 4
675.2.f.e 4
675.2.f.f 4
675.2.f.g 8
675.2.f.h 8
675.2.f.i 8
675.2.h $$\chi_{675}(136, \cdot)$$ 675.2.h.a 40 4
675.2.h.b 40
675.2.h.c 40
675.2.h.d 40
675.2.k $$\chi_{675}(199, \cdot)$$ 675.2.k.a 4 2
675.2.k.b 12
675.2.k.c 16
675.2.l $$\chi_{675}(76, \cdot)$$ 675.2.l.a 6 6
675.2.l.b 6
675.2.l.c 12
675.2.l.d 30
675.2.l.e 42
675.2.l.f 66
675.2.l.g 66
675.2.l.h 96
675.2.n $$\chi_{675}(109, \cdot)$$ 675.2.n.a 80 4
675.2.n.b 80
675.2.q $$\chi_{675}(143, \cdot)$$ 675.2.q.a 16 4
675.2.q.b 16
675.2.q.c 32
675.2.r $$\chi_{675}(46, \cdot)$$ 675.2.r.a 224 8
675.2.u $$\chi_{675}(49, \cdot)$$ 675.2.u.a 12 6
675.2.u.b 24
675.2.u.c 60
675.2.u.d 84
675.2.u.e 132
675.2.w $$\chi_{675}(53, \cdot)$$ 675.2.w.a 160 8
675.2.w.b 160
675.2.y $$\chi_{675}(19, \cdot)$$ 675.2.y.a 224 8
675.2.ba $$\chi_{675}(32, \cdot)$$ 675.2.ba.a 144 12
675.2.ba.b 192
675.2.ba.c 288
675.2.bc $$\chi_{675}(16, \cdot)$$ 675.2.bc.a 2112 24
675.2.bd $$\chi_{675}(8, \cdot)$$ 675.2.bd.a 448 16
675.2.bg $$\chi_{675}(4, \cdot)$$ 675.2.bg.a 2112 24
675.2.bi $$\chi_{675}(2, \cdot)$$ 675.2.bi.a 4224 48

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(675))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(675)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(45))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(135))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(225))$$$$^{\oplus 2}$$