# Properties

 Label 75.4 Level 75 Weight 4 Dimension 394 Nonzero newspaces 6 Newform subspaces 17 Sturm bound 1600 Trace bound 1

## 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

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

## 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

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