## 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

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

## 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

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