## Defining parameters

 Level: $$N$$ = $$70 = 2 \cdot 5 \cdot 7$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newform subspaces: $$10$$ Sturm bound: $$576$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(70))$$.

Total New Old
Modular forms 192 45 147
Cusp forms 97 45 52
Eisenstein series 95 0 95

## Trace form

 $$45 q + q^{2} - 3 q^{4} - 5 q^{5} - 8 q^{6} - 9 q^{7} + q^{8} - 15 q^{9} + O(q^{10})$$ $$45 q + q^{2} - 3 q^{4} - 5 q^{5} - 8 q^{6} - 9 q^{7} + q^{8} - 15 q^{9} - 5 q^{10} - 12 q^{11} - 6 q^{13} - 5 q^{14} - 20 q^{15} - 3 q^{16} - 30 q^{17} - 11 q^{18} - 24 q^{19} - 5 q^{20} - 24 q^{21} - 12 q^{22} - 24 q^{23} - 8 q^{24} + 9 q^{25} + 2 q^{26} + 24 q^{27} + 15 q^{28} + 30 q^{29} + 52 q^{30} + 24 q^{31} + q^{32} + 72 q^{33} + 42 q^{34} + 37 q^{35} + 33 q^{36} + 30 q^{37} + 32 q^{38} + 48 q^{39} - 5 q^{40} - 6 q^{41} + 40 q^{42} - 12 q^{43} + 12 q^{44} + 7 q^{45} - 51 q^{49} - 35 q^{50} - 48 q^{51} - 6 q^{52} - 66 q^{53} - 32 q^{54} - 36 q^{55} - 5 q^{56} - 24 q^{57} - 42 q^{58} - 20 q^{60} - 6 q^{61} - 40 q^{62} + 3 q^{63} - 3 q^{64} + 2 q^{65} - 48 q^{66} + 36 q^{67} - 30 q^{68} - 35 q^{70} - 24 q^{71} - 11 q^{72} + 18 q^{73} - 34 q^{74} - 24 q^{75} - 24 q^{76} - 12 q^{77} - 40 q^{78} - 5 q^{80} - 27 q^{81} - 30 q^{82} - 24 q^{83} - 24 q^{84} - 42 q^{85} - 4 q^{86} - 48 q^{87} + 12 q^{88} + 42 q^{89} + 31 q^{90} + 6 q^{91} + 24 q^{92} + 24 q^{93} + 24 q^{94} + 92 q^{95} + 16 q^{96} + 66 q^{97} + 97 q^{98} + 36 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(70))$$

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
70.2.a $$\chi_{70}(1, \cdot)$$ 70.2.a.a 1 1
70.2.c $$\chi_{70}(29, \cdot)$$ 70.2.c.a 4 1
70.2.e $$\chi_{70}(11, \cdot)$$ 70.2.e.a 2 2
70.2.e.b 2
70.2.e.c 2
70.2.e.d 2
70.2.g $$\chi_{70}(13, \cdot)$$ 70.2.g.a 8 2
70.2.i $$\chi_{70}(9, \cdot)$$ 70.2.i.a 4 2
70.2.i.b 4
70.2.k $$\chi_{70}(3, \cdot)$$ 70.2.k.a 16 4

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(70))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(70)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(14))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(35))$$$$^{\oplus 2}$$