## 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

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

## 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

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