## Defining parameters

 Level: $$N$$ = $$1875 = 3 \cdot 5^{4}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$12$$ Sturm bound: $$1000000$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(1875))$$.

Total New Old
Modular forms 377200 259968 117232
Cusp forms 372800 258432 114368
Eisenstein series 4400 1536 2864

## Trace form

 $$258432 q - 160 q^{3} - 320 q^{4} - 288 q^{6} - 320 q^{7} - 160 q^{9} + O(q^{10})$$ $$258432 q - 160 q^{3} - 320 q^{4} - 288 q^{6} - 320 q^{7} - 160 q^{9} - 400 q^{10} - 160 q^{12} - 320 q^{13} - 200 q^{15} - 64 q^{16} + 640 q^{17} + 20 q^{18} - 80 q^{19} - 528 q^{21} - 1760 q^{22} - 880 q^{23} - 1590 q^{24} - 400 q^{25} - 880 q^{26} - 160 q^{27} - 1760 q^{28} - 160 q^{29} - 200 q^{30} + 144 q^{31} + 2240 q^{32} + 800 q^{33} + 2980 q^{34} + 224 q^{36} + 750 q^{37} + 4980 q^{38} + 2280 q^{39} - 400 q^{40} + 920 q^{41} + 540 q^{42} - 880 q^{43} - 2660 q^{44} - 200 q^{45} - 5016 q^{46} - 3440 q^{47} - 6350 q^{48} - 4710 q^{49} - 3238 q^{51} - 7900 q^{52} - 3330 q^{53} - 3930 q^{54} - 400 q^{55} - 840 q^{56} + 400 q^{57} + 2900 q^{58} + 3280 q^{59} - 200 q^{60} + 2744 q^{61} + 10700 q^{62} + 2260 q^{63} + 9340 q^{64} + 1560 q^{66} - 320 q^{67} + 2920 q^{69} - 400 q^{70} + 9360 q^{72} - 320 q^{73} - 200 q^{75} + 188 q^{76} + 6310 q^{78} - 320 q^{79} + 992 q^{81} + 16240 q^{82} + 13840 q^{83} + 9200 q^{84} - 400 q^{85} + 9960 q^{86} + 300 q^{87} + 7600 q^{88} + 1650 q^{89} - 200 q^{90} - 4896 q^{91} - 8600 q^{92} - 9680 q^{93} - 22400 q^{94} - 14684 q^{96} - 26360 q^{97} - 27160 q^{98} - 150 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(1875))$$

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
1875.4.a $$\chi_{1875}(1, \cdot)$$ 1875.4.a.a 10 1
1875.4.a.b 10
1875.4.a.c 10
1875.4.a.d 10
1875.4.a.e 14
1875.4.a.f 14
1875.4.a.g 14
1875.4.a.h 14
1875.4.a.i 16
1875.4.a.j 16
1875.4.a.k 24
1875.4.a.l 24
1875.4.a.m 32
1875.4.a.n 32
1875.4.b $$\chi_{1875}(1249, \cdot)$$ n/a 240 1
1875.4.e $$\chi_{1875}(182, \cdot)$$ n/a 928 2
1875.4.g $$\chi_{1875}(376, \cdot)$$ n/a 960 4
1875.4.i $$\chi_{1875}(124, \cdot)$$ n/a 960 4
1875.4.l $$\chi_{1875}(68, \cdot)$$ n/a 3744 8
1875.4.m $$\chi_{1875}(76, \cdot)$$ n/a 4520 20
1875.4.o $$\chi_{1875}(49, \cdot)$$ n/a 4480 20
1875.4.r $$\chi_{1875}(32, \cdot)$$ n/a 17760 40
1875.4.s $$\chi_{1875}(16, \cdot)$$ n/a 37400 100
1875.4.v $$\chi_{1875}(4, \cdot)$$ n/a 37600 100
1875.4.x $$\chi_{1875}(2, \cdot)$$ n/a 149600 200

"n/a" means that newforms for that character have not been added to the database yet

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(1875))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(1875)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(5))$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(15))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(75))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(125))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(375))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(625))$$$$^{\oplus 2}$$