## Defining parameters

 Level: $$N$$ = $$125 = 5^{3}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newform subspaces: $$11$$ Sturm bound: $$2500$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(125))$$.

Total New Old
Modular forms 715 640 75
Cusp forms 536 512 24
Eisenstein series 179 128 51

## Trace form

 $$512 q - 33 q^{2} - 34 q^{3} - 37 q^{4} - 40 q^{5} - 66 q^{6} - 38 q^{7} - 45 q^{8} - 43 q^{9} + O(q^{10})$$ $$512 q - 33 q^{2} - 34 q^{3} - 37 q^{4} - 40 q^{5} - 66 q^{6} - 38 q^{7} - 45 q^{8} - 43 q^{9} - 40 q^{10} - 66 q^{11} - 58 q^{12} - 44 q^{13} - 54 q^{14} - 40 q^{15} - 53 q^{16} - 28 q^{17} - 19 q^{18} - 10 q^{19} - 5 q^{20} - 46 q^{21} + 14 q^{22} - 14 q^{23} + 50 q^{24} - 10 q^{25} - 76 q^{26} - 10 q^{27} + 34 q^{28} - 20 q^{29} - 46 q^{31} - 23 q^{32} - 38 q^{33} - 34 q^{34} - 30 q^{35} - 33 q^{36} - 58 q^{37} - 30 q^{38} - 6 q^{39} + 10 q^{40} - 56 q^{41} + 74 q^{42} + 6 q^{43} + 26 q^{44} + 40 q^{45} - 6 q^{46} + 2 q^{47} + 146 q^{48} + 23 q^{49} + 50 q^{50} - 26 q^{51} + 92 q^{52} + 6 q^{53} + 110 q^{54} + 10 q^{56} + 50 q^{57} + 20 q^{58} + 10 q^{59} + 100 q^{60} - 36 q^{61} + 94 q^{62} + 46 q^{63} + 63 q^{64} + 15 q^{65} + 18 q^{66} - 18 q^{67} + 84 q^{68} + 14 q^{69} + 40 q^{70} - 46 q^{71} + 115 q^{72} - 24 q^{73} + 76 q^{74} + 40 q^{75} + 10 q^{76} + 34 q^{77} + 62 q^{78} - 30 q^{79} + 100 q^{80} - 23 q^{81} + 84 q^{82} + 106 q^{83} + 146 q^{84} + 75 q^{85} + 54 q^{86} + 110 q^{87} + 190 q^{88} + 130 q^{89} + 170 q^{90} + 34 q^{91} + 82 q^{92} + 102 q^{93} + 146 q^{94} + 40 q^{95} - 26 q^{96} + 152 q^{97} + 119 q^{98} + 74 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(125))$$

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
125.2.a $$\chi_{125}(1, \cdot)$$ 125.2.a.a 2 1
125.2.a.b 2
125.2.a.c 4
125.2.b $$\chi_{125}(124, \cdot)$$ 125.2.b.a 4 1
125.2.b.b 4
125.2.d $$\chi_{125}(26, \cdot)$$ 125.2.d.a 4 4
125.2.d.b 16
125.2.e $$\chi_{125}(24, \cdot)$$ 125.2.e.a 8 4
125.2.e.b 8
125.2.g $$\chi_{125}(6, \cdot)$$ 125.2.g.a 220 20
125.2.h $$\chi_{125}(4, \cdot)$$ 125.2.h.a 240 20

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(125))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(125)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 2}$$