## Defining parameters

 Level: $$N$$ = $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$6$$ Newform subspaces: $$17$$ Sturm bound: $$1600$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(75))$$.

Total New Old
Modular forms 656 434 222
Cusp forms 544 394 150
Eisenstein series 112 40 72

## Trace form

 $$394q - 8q^{2} + 2q^{3} + 28q^{4} - 6q^{5} - 18q^{6} + 20q^{7} + 72q^{8} + 26q^{9} + O(q^{10})$$ $$394q - 8q^{2} + 2q^{3} + 28q^{4} - 6q^{5} - 18q^{6} + 20q^{7} + 72q^{8} + 26q^{9} + 76q^{10} + 112q^{11} - 226q^{12} - 348q^{13} - 528q^{14} - 184q^{15} - 132q^{16} + 560q^{17} + 578q^{18} + 732q^{19} + 156q^{20} + 918q^{21} - 420q^{22} - 304q^{23} - 864q^{24} - 1546q^{25} - 1344q^{26} - 1414q^{27} - 2852q^{28} - 1736q^{29} - 886q^{30} - 580q^{31} + 2008q^{32} + 974q^{33} + 4416q^{34} + 2240q^{35} + 1886q^{36} + 3570q^{37} + 5764q^{38} + 3678q^{39} + 4232q^{40} + 1648q^{41} + 114q^{42} - 1484q^{43} - 4116q^{44} - 2456q^{45} - 5260q^{46} - 3904q^{47} - 4640q^{48} - 4710q^{49} - 8224q^{50} - 1960q^{51} - 7888q^{52} - 3554q^{53} - 3348q^{54} - 1516q^{55} - 840q^{56} - 1442q^{57} + 3088q^{58} + 2288q^{59} + 74q^{60} + 1052q^{61} + 10076q^{62} - 374q^{63} + 7192q^{64} + 1222q^{65} - 762q^{66} - 268q^{67} - 304q^{68} + 2782q^{69} + 1540q^{70} + 1376q^{71} + 13758q^{72} + 5940q^{73} + 6752q^{74} + 9396q^{75} + 8808q^{76} + 3456q^{77} + 8380q^{78} + 3020q^{79} - 1004q^{80} - 286q^{81} + 8716q^{82} + 11104q^{83} + 4358q^{84} + 12538q^{85} + 1192q^{86} + 1962q^{87} + 3532q^{88} + 3402q^{89} - 10894q^{90} - 7540q^{91} - 10712q^{92} - 13418q^{93} - 28932q^{94} - 15944q^{95} - 15150q^{96} - 31028q^{97} - 27968q^{98} - 2016q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(75))$$

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
75.4.a $$\chi_{75}(1, \cdot)$$ 75.4.a.a 1 1
75.4.a.b 1
75.4.a.c 2
75.4.a.d 2
75.4.a.e 2
75.4.a.f 2
75.4.b $$\chi_{75}(49, \cdot)$$ 75.4.b.a 2 1
75.4.b.b 2
75.4.b.c 4
75.4.e $$\chi_{75}(32, \cdot)$$ 75.4.e.a 4 2
75.4.e.b 4
75.4.e.c 8
75.4.e.d 16
75.4.g $$\chi_{75}(16, \cdot)$$ 75.4.g.a 28 4
75.4.g.b 28
75.4.i $$\chi_{75}(4, \cdot)$$ 75.4.i.a 64 4
75.4.l $$\chi_{75}(2, \cdot)$$ 75.4.l.a 224 8

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(75))$$ into lower level spaces

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