## 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

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

## 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

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