## Defining parameters

 Level: $$N$$ = $$50 = 2 \cdot 5^{2}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$2$$ Newform subspaces: $$5$$ Sturm bound: $$450$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(50))$$.

Total New Old
Modular forms 178 46 132
Cusp forms 122 46 76
Eisenstein series 56 0 56

## Trace form

 $$46q + 4q^{2} + 8q^{3} - 16q^{6} - 8q^{7} - 8q^{8} + O(q^{10})$$ $$46q + 4q^{2} + 8q^{3} - 16q^{6} - 8q^{7} - 8q^{8} + 10q^{10} + 32q^{11} + 16q^{12} - 12q^{13} - 20q^{15} + 16q^{16} - 168q^{17} - 154q^{18} - 200q^{19} - 40q^{20} - 88q^{21} - 112q^{22} - 32q^{23} + 110q^{25} + 64q^{26} + 260q^{27} + 96q^{28} + 200q^{29} + 240q^{30} + 72q^{31} + 24q^{32} + 296q^{33} + 250q^{34} + 200q^{35} + 8q^{36} - 48q^{37} + 80q^{38} - 400q^{39} + 20q^{40} - 128q^{41} - 32q^{42} - 72q^{43} - 330q^{45} - 16q^{46} - 8q^{47} - 32q^{48} - 50q^{50} + 232q^{51} - 24q^{52} - 32q^{53} + 240q^{55} - 32q^{56} + 480q^{57} - 160q^{58} + 100q^{59} - 240q^{60} - 88q^{61} - 512q^{62} - 532q^{63} - 790q^{65} - 352q^{66} - 808q^{67} - 304q^{68} - 700q^{69} - 280q^{70} - 128q^{71} - 8q^{72} + 108q^{73} + 60q^{75} - 80q^{76} + 544q^{77} + 352q^{78} + 400q^{79} + 276q^{81} + 608q^{82} + 868q^{83} + 600q^{84} + 460q^{85} + 384q^{86} + 1140q^{87} + 64q^{88} + 1450q^{89} + 970q^{90} + 312q^{91} + 376q^{92} + 556q^{93} + 40q^{95} + 64q^{96} + 212q^{97} + 164q^{98} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(50))$$

We only show spaces with odd 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
50.3.c $$\chi_{50}(7, \cdot)$$ 50.3.c.a 2 2
50.3.c.b 2
50.3.c.c 2
50.3.f $$\chi_{50}(3, \cdot)$$ 50.3.f.a 16 8
50.3.f.b 24

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(50))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(50)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(10))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(25))$$$$^{\oplus 2}$$