## Defining parameters

 Level: $$N$$ = $$39 = 3 \cdot 13$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newform subspaces: $$9$$ Sturm bound: $$224$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(39))$$.

Total New Old
Modular forms 80 53 27
Cusp forms 33 29 4
Eisenstein series 47 24 23

## Trace form

 $$29 q - 3 q^{2} - 7 q^{3} - 19 q^{4} - 6 q^{5} - 9 q^{6} - 16 q^{7} + 3 q^{8} - 5 q^{9} + O(q^{10})$$ $$29 q - 3 q^{2} - 7 q^{3} - 19 q^{4} - 6 q^{5} - 9 q^{6} - 16 q^{7} + 3 q^{8} - 5 q^{9} + 13 q^{12} - q^{13} - 3 q^{16} + 15 q^{18} + 8 q^{19} + 24 q^{20} + 12 q^{21} + 12 q^{22} + 15 q^{24} - q^{25} + 21 q^{26} + 23 q^{27} + 16 q^{28} - 20 q^{31} - 3 q^{32} - 12 q^{33} - 18 q^{34} - 24 q^{35} - 43 q^{36} - 44 q^{37} - 60 q^{38} - 35 q^{39} - 66 q^{40} - 12 q^{42} + 8 q^{43} - 24 q^{44} - 24 q^{45} + 36 q^{46} - q^{48} + 19 q^{49} + 21 q^{50} + 30 q^{51} + 47 q^{52} + 42 q^{53} + 39 q^{54} + 48 q^{55} + 60 q^{56} + 48 q^{57} + 48 q^{58} + 24 q^{59} + 54 q^{60} + 40 q^{61} + 36 q^{62} + 12 q^{63} + 35 q^{64} + 12 q^{65} - 36 q^{66} - 16 q^{67} - 36 q^{68} - 24 q^{69} - 24 q^{70} - 48 q^{71} - 27 q^{72} - 46 q^{73} - 36 q^{74} - 35 q^{75} - 40 q^{76} - 24 q^{77} - 99 q^{78} - 32 q^{79} - 48 q^{80} - 41 q^{81} - 36 q^{82} - 48 q^{83} - 12 q^{84} - 30 q^{85} - 36 q^{86} - 12 q^{87} + 12 q^{88} + 30 q^{89} + 6 q^{90} + 32 q^{91} + 24 q^{93} + 60 q^{94} + 48 q^{95} + 45 q^{96} + 22 q^{97} - 15 q^{98} + 60 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(39))$$

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
39.2.a $$\chi_{39}(1, \cdot)$$ 39.2.a.a 1 1
39.2.a.b 2
39.2.b $$\chi_{39}(25, \cdot)$$ 39.2.b.a 2 1
39.2.e $$\chi_{39}(16, \cdot)$$ 39.2.e.a 2 2
39.2.e.b 4
39.2.f $$\chi_{39}(5, \cdot)$$ 39.2.f.a 4 2
39.2.j $$\chi_{39}(4, \cdot)$$ 39.2.j.a 2 2
39.2.k $$\chi_{39}(2, \cdot)$$ 39.2.k.a 4 4
39.2.k.b 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(39))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(39)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(13))$$$$^{\oplus 2}$$