## Defining parameters

 Level: $$N$$ = $$40 = 2^{3} \cdot 5$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$5$$ Newform subspaces: $$5$$ Sturm bound: $$192$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(40))$$.

Total New Old
Modular forms 72 31 41
Cusp forms 25 19 6
Eisenstein series 47 12 35

## Trace form

 $$19q - 4q^{2} - 4q^{3} - 4q^{4} - q^{5} - 8q^{6} - 8q^{7} - 4q^{8} - 13q^{9} + O(q^{10})$$ $$19q - 4q^{2} - 4q^{3} - 4q^{4} - q^{5} - 8q^{6} - 8q^{7} - 4q^{8} - 13q^{9} - 4q^{10} - 12q^{11} + 8q^{12} - 2q^{13} + 8q^{14} + 4q^{15} + 8q^{16} - 6q^{17} + 24q^{18} + 12q^{19} + 16q^{20} + 8q^{21} + 16q^{22} + 24q^{24} - 13q^{25} + 8q^{26} + 8q^{27} - 8q^{28} - 6q^{29} - 24q^{32} - 8q^{33} - 36q^{34} + 8q^{35} - 40q^{36} + 6q^{37} - 24q^{38} + 8q^{39} - 44q^{40} - 18q^{41} - 24q^{42} + 20q^{43} - 16q^{44} - q^{45} - 32q^{46} + 24q^{47} - 8q^{48} + 7q^{49} + 16q^{50} + 8q^{51} + 20q^{52} + 6q^{53} + 24q^{54} + 28q^{55} + 48q^{56} + 16q^{57} + 44q^{58} + 20q^{59} + 40q^{60} - 22q^{61} + 56q^{62} - 8q^{63} + 32q^{64} + 14q^{65} + 16q^{66} - 20q^{67} + 20q^{68} + 8q^{69} - 24q^{71} - 12q^{72} + 26q^{73} - 20q^{74} - 76q^{75} - 16q^{77} - 64q^{78} - 80q^{79} - 24q^{80} + 3q^{81} - 36q^{82} - 60q^{83} - 32q^{84} + 2q^{85} + 8q^{86} - 48q^{87} + 14q^{89} - 4q^{90} - 16q^{91} - 16q^{92} + 32q^{94} - 12q^{95} + 48q^{96} - 14q^{97} + 12q^{98} - 4q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(40))$$

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
40.2.a $$\chi_{40}(1, \cdot)$$ 40.2.a.a 1 1
40.2.c $$\chi_{40}(9, \cdot)$$ 40.2.c.a 2 1
40.2.d $$\chi_{40}(21, \cdot)$$ 40.2.d.a 4 1
40.2.f $$\chi_{40}(29, \cdot)$$ 40.2.f.a 4 1
40.2.j $$\chi_{40}(7, \cdot)$$ None 0 2
40.2.k $$\chi_{40}(3, \cdot)$$ 40.2.k.a 8 2

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(40))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(40)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(20))$$$$^{\oplus 2}$$