## Defining parameters

 Level: $$N$$ = $$575 = 5^{2} \cdot 23$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$12$$ Newform subspaces: $$46$$ Sturm bound: $$52800$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(575))$$.

Total New Old
Modular forms 13816 12551 1265
Cusp forms 12585 11691 894
Eisenstein series 1231 860 371

## Trace form

 $$11691 q - 129 q^{2} - 131 q^{3} - 137 q^{4} - 166 q^{5} - 219 q^{6} - 139 q^{7} - 153 q^{8} - 149 q^{9} + O(q^{10})$$ $$11691 q - 129 q^{2} - 131 q^{3} - 137 q^{4} - 166 q^{5} - 219 q^{6} - 139 q^{7} - 153 q^{8} - 149 q^{9} - 186 q^{10} - 219 q^{11} - 179 q^{12} - 151 q^{13} - 171 q^{14} - 196 q^{15} - 247 q^{16} - 150 q^{17} - 195 q^{18} - 134 q^{19} - 156 q^{20} - 252 q^{21} - 148 q^{22} - 159 q^{23} - 312 q^{24} - 146 q^{25} - 461 q^{26} - 176 q^{27} - 159 q^{28} - 154 q^{29} - 196 q^{30} - 230 q^{31} - 201 q^{32} - 190 q^{33} - 203 q^{34} - 216 q^{35} - 320 q^{36} - 233 q^{37} - 238 q^{38} - 199 q^{39} - 206 q^{40} - 261 q^{41} - 225 q^{42} - 175 q^{43} - 250 q^{44} - 126 q^{45} - 313 q^{46} - 326 q^{47} - 181 q^{48} - 193 q^{49} - 126 q^{50} - 463 q^{51} - 209 q^{52} - 163 q^{53} - 213 q^{54} - 196 q^{55} - 372 q^{56} - 200 q^{57} - 262 q^{58} - 209 q^{59} - 156 q^{60} - 283 q^{61} - 161 q^{62} - 206 q^{63} - 245 q^{64} - 186 q^{65} - 388 q^{66} - 201 q^{67} - 278 q^{68} - 214 q^{69} - 412 q^{70} - 314 q^{71} - 338 q^{72} - 213 q^{73} - 230 q^{74} - 196 q^{75} - 594 q^{76} - 254 q^{77} - 375 q^{78} - 291 q^{79} - 186 q^{80} - 461 q^{81} - 245 q^{82} - 148 q^{83} - 391 q^{84} - 106 q^{85} - 351 q^{86} - 235 q^{87} + 137 q^{88} - 119 q^{89} - 126 q^{90} - 318 q^{91} - 260 q^{92} - 394 q^{93} - 212 q^{94} - 152 q^{95} - 204 q^{96} + 126 q^{97} + 75 q^{98} + 78 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(575))$$

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
575.2.a $$\chi_{575}(1, \cdot)$$ 575.2.a.a 1 1
575.2.a.b 1
575.2.a.c 1
575.2.a.d 1
575.2.a.e 1
575.2.a.f 2
575.2.a.g 2
575.2.a.h 4
575.2.a.i 4
575.2.a.j 4
575.2.a.k 7
575.2.a.l 7
575.2.b $$\chi_{575}(24, \cdot)$$ 575.2.b.a 2 1
575.2.b.b 2
575.2.b.c 4
575.2.b.d 4
575.2.b.e 8
575.2.b.f 14
575.2.e $$\chi_{575}(68, \cdot)$$ 575.2.e.a 4 2
575.2.e.b 8
575.2.e.c 12
575.2.e.d 20
575.2.e.e 24
575.2.g $$\chi_{575}(116, \cdot)$$ 575.2.g.a 4 4
575.2.g.b 108
575.2.g.c 112
575.2.i $$\chi_{575}(139, \cdot)$$ 575.2.i.a 216 4
575.2.k $$\chi_{575}(26, \cdot)$$ 575.2.k.a 10 10
575.2.k.b 10
575.2.k.c 20
575.2.k.d 50
575.2.k.e 80
575.2.k.f 80
575.2.k.g 100
575.2.m $$\chi_{575}(22, \cdot)$$ 575.2.m.a 464 8
575.2.p $$\chi_{575}(49, \cdot)$$ 575.2.p.a 20 10
575.2.p.b 20
575.2.p.c 40
575.2.p.d 100
575.2.p.e 160
575.2.r $$\chi_{575}(7, \cdot)$$ 575.2.r.a 160 20
575.2.r.b 200
575.2.r.c 320
575.2.s $$\chi_{575}(6, \cdot)$$ 575.2.s.a 2320 40
575.2.u $$\chi_{575}(4, \cdot)$$ 575.2.u.a 2320 40
575.2.w $$\chi_{575}(17, \cdot)$$ 575.2.w.a 4640 80

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(575))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(575)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(23))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(115))$$$$^{\oplus 2}$$