# Properties

 Label 375.2 Level 375 Weight 2 Dimension 3328 Nonzero newspaces 9 Newform subspaces 28 Sturm bound 20000 Trace bound 4

## Defining parameters

 Level: $$N$$ = $$375 = 3 \cdot 5^{3}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$9$$ Newform subspaces: $$28$$ Sturm bound: $$20000$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(375))$$.

Total New Old
Modular forms 5360 3584 1776
Cusp forms 4641 3328 1313
Eisenstein series 719 256 463

## Trace form

 $$3328 q + 2 q^{2} - 30 q^{3} - 54 q^{4} - 50 q^{6} - 52 q^{7} + 18 q^{8} - 28 q^{9} + O(q^{10})$$ $$3328 q + 2 q^{2} - 30 q^{3} - 54 q^{4} - 50 q^{6} - 52 q^{7} + 18 q^{8} - 28 q^{9} - 80 q^{10} + 8 q^{11} - 22 q^{12} - 48 q^{13} + 24 q^{14} - 40 q^{15} - 142 q^{16} - 20 q^{17} - 48 q^{18} - 116 q^{19} - 70 q^{20} - 86 q^{21} - 180 q^{22} - 56 q^{23} - 158 q^{24} - 140 q^{25} - 36 q^{26} - 30 q^{27} - 244 q^{28} - 52 q^{29} - 80 q^{30} - 156 q^{31} - 82 q^{32} - 54 q^{33} - 108 q^{34} - 20 q^{35} - 112 q^{36} - 52 q^{37} - 64 q^{38} - 94 q^{39} - 180 q^{40} - 28 q^{41} - 206 q^{42} - 172 q^{43} - 192 q^{44} - 120 q^{45} - 276 q^{46} - 104 q^{47} - 218 q^{48} - 230 q^{49} - 180 q^{50} - 178 q^{51} - 400 q^{52} - 136 q^{53} - 286 q^{54} - 160 q^{55} - 120 q^{56} - 174 q^{57} - 248 q^{58} - 104 q^{59} - 100 q^{60} - 128 q^{61} - 104 q^{62} - 2 q^{63} - 46 q^{64} - 10 q^{65} + 2 q^{66} + 20 q^{67} + 124 q^{68} + 134 q^{69} - 80 q^{70} + 64 q^{71} + 268 q^{72} + 24 q^{73} + 124 q^{74} + 40 q^{75} - 140 q^{76} + 96 q^{77} + 150 q^{78} + 20 q^{79} + 28 q^{81} - 104 q^{82} - 64 q^{83} + 106 q^{84} - 230 q^{85} - 112 q^{86} - 58 q^{87} - 372 q^{88} - 216 q^{89} - 40 q^{90} - 316 q^{91} - 232 q^{92} - 218 q^{93} - 564 q^{94} - 160 q^{95} - 74 q^{96} - 520 q^{97} - 382 q^{98} - 42 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(375))$$

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
375.2.a $$\chi_{375}(1, \cdot)$$ 375.2.a.a 2 1
375.2.a.b 2
375.2.a.c 2
375.2.a.d 2
375.2.a.e 4
375.2.a.f 4
375.2.b $$\chi_{375}(124, \cdot)$$ 375.2.b.a 4 1
375.2.b.b 4
375.2.b.c 8
375.2.e $$\chi_{375}(68, \cdot)$$ 375.2.e.a 16 2
375.2.e.b 16
375.2.e.c 32
375.2.g $$\chi_{375}(76, \cdot)$$ 375.2.g.a 4 4
375.2.g.b 8
375.2.g.c 12
375.2.g.d 16
375.2.g.e 16
375.2.i $$\chi_{375}(49, \cdot)$$ 375.2.i.a 8 4
375.2.i.b 16
375.2.i.c 16
375.2.i.d 24
375.2.l $$\chi_{375}(32, \cdot)$$ 375.2.l.a 64 8
375.2.l.b 64
375.2.l.c 64
375.2.m $$\chi_{375}(16, \cdot)$$ 375.2.m.a 260 20
375.2.m.b 260
375.2.o $$\chi_{375}(4, \cdot)$$ 375.2.o.a 480 20
375.2.r $$\chi_{375}(2, \cdot)$$ 375.2.r.a 1920 40

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(375))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(375)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(125))$$$$^{\oplus 2}$$