## Defining parameters

 Level: $$N$$ = $$32 = 2^{5}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$2$$ Newform subspaces: $$3$$ Sturm bound: $$128$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(32))$$.

Total New Old
Modular forms 48 23 25
Cusp forms 17 13 4
Eisenstein series 31 10 21

## Trace form

 $$13q - 4q^{2} - 4q^{3} - 4q^{4} - 6q^{5} - 4q^{6} - 4q^{7} - 4q^{8} - 7q^{9} + O(q^{10})$$ $$13q - 4q^{2} - 4q^{3} - 4q^{4} - 6q^{5} - 4q^{6} - 4q^{7} - 4q^{8} - 7q^{9} + 4q^{10} - 4q^{11} + 12q^{12} + 2q^{13} + 12q^{14} + 16q^{16} + 2q^{17} + 16q^{18} - 4q^{19} + 12q^{20} - 4q^{21} + 8q^{22} + 4q^{23} - 16q^{24} - 5q^{25} - 24q^{26} + 20q^{27} - 24q^{28} - 14q^{29} - 36q^{30} + 16q^{31} - 24q^{32} - 8q^{33} - 16q^{34} + 20q^{35} - 32q^{36} - 6q^{37} + 4q^{38} + 20q^{39} + 8q^{40} + 6q^{41} + 16q^{42} + 4q^{43} + 36q^{44} + 14q^{45} + 28q^{46} + 48q^{48} - 7q^{49} + 36q^{50} - 8q^{51} + 4q^{52} + 26q^{53} + 8q^{54} - 36q^{55} + 8q^{56} - 4q^{57} - 8q^{58} - 36q^{59} - 8q^{60} + 18q^{61} - 24q^{62} - 48q^{63} - 40q^{64} - 20q^{65} - 28q^{66} - 44q^{67} + 16q^{68} + 28q^{69} - 16q^{70} - 36q^{71} + 20q^{72} - 10q^{73} + 12q^{74} - 16q^{75} - 4q^{76} + 12q^{77} + 36q^{78} - 8q^{80} + 9q^{81} - 4q^{82} + 36q^{83} + 16q^{84} + 12q^{85} - 24q^{86} + 52q^{87} + 6q^{89} + 8q^{90} + 44q^{91} - 40q^{92} - 16q^{93} + 8q^{94} + 56q^{95} + 10q^{97} - 24q^{98} + 48q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(32))$$

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
32.2.a $$\chi_{32}(1, \cdot)$$ 32.2.a.a 1 1
32.2.b $$\chi_{32}(17, \cdot)$$ None 0 1
32.2.e $$\chi_{32}(9, \cdot)$$ None 0 2
32.2.g $$\chi_{32}(5, \cdot)$$ 32.2.g.a 4 4
32.2.g.b 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(32))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(32)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(16))$$$$^{\oplus 2}$$