## Defining parameters

 Level: $$N$$ = $$50 = 2 \cdot 5^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newform subspaces: $$6$$ Sturm bound: $$300$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(50))$$.

Total New Old
Modular forms 103 24 79
Cusp forms 48 24 24
Eisenstein series 55 0 55

## Trace form

 $$24 q - q^{2} - 4 q^{3} - q^{4} - 5 q^{5} - 4 q^{6} - 8 q^{7} - q^{8} - 13 q^{9} + O(q^{10})$$ $$24 q - q^{2} - 4 q^{3} - q^{4} - 5 q^{5} - 4 q^{6} - 8 q^{7} - q^{8} - 13 q^{9} - 5 q^{10} - 12 q^{11} - 4 q^{12} - 14 q^{13} - 8 q^{14} - 20 q^{15} - q^{16} + 2 q^{17} + 12 q^{18} + 20 q^{19} + 8 q^{21} + 28 q^{22} + 16 q^{23} + 16 q^{24} + 35 q^{25} + 6 q^{26} + 20 q^{27} + 12 q^{28} + 10 q^{29} + 20 q^{30} + 8 q^{31} + 4 q^{32} - 8 q^{33} + 7 q^{34} - 20 q^{35} - 13 q^{36} - 33 q^{37} - 20 q^{38} - 16 q^{39} - 5 q^{40} - 22 q^{41} - 32 q^{42} - 4 q^{43} - 12 q^{44} + 15 q^{45} - 24 q^{46} - 8 q^{47} - 4 q^{48} - 12 q^{49} - 25 q^{50} - 12 q^{51} - 14 q^{52} - 9 q^{53} - 40 q^{54} - 20 q^{55} - 8 q^{56} - 30 q^{58} - 2 q^{61} + 28 q^{62} + 36 q^{63} - q^{64} + 35 q^{65} + 12 q^{66} + 12 q^{67} + 42 q^{68} + 44 q^{69} + 40 q^{70} + 8 q^{71} - 13 q^{72} + 6 q^{73} + 62 q^{74} + 60 q^{75} + 20 q^{76} + 64 q^{77} + 44 q^{78} - 5 q^{80} - 41 q^{81} + 38 q^{82} + 76 q^{83} + 28 q^{84} + 15 q^{85} + 16 q^{86} + 20 q^{87} - 12 q^{88} + 15 q^{89} - 5 q^{90} - 32 q^{91} - 4 q^{92} - 68 q^{93} - 48 q^{94} - 60 q^{95} - 4 q^{96} - 78 q^{97} - 57 q^{98} - 96 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(50))$$

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
50.2.a $$\chi_{50}(1, \cdot)$$ 50.2.a.a 1 1
50.2.a.b 1
50.2.b $$\chi_{50}(49, \cdot)$$ 50.2.b.a 2 1
50.2.d $$\chi_{50}(11, \cdot)$$ 50.2.d.a 4 4
50.2.d.b 8
50.2.e $$\chi_{50}(9, \cdot)$$ 50.2.e.a 8 4

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(50))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(50)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 2}$$