## Defining parameters

 Level: $$N$$ = $$69 = 3 \cdot 23$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newform subspaces: $$7$$ Sturm bound: $$704$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(69))$$.

Total New Old
Modular forms 220 153 67
Cusp forms 133 109 24
Eisenstein series 87 44 43

## Trace form

 $$109 q - 3 q^{2} - 12 q^{3} - 29 q^{4} - 6 q^{5} - 14 q^{6} - 30 q^{7} - 15 q^{8} - 12 q^{9} + O(q^{10})$$ $$109 q - 3 q^{2} - 12 q^{3} - 29 q^{4} - 6 q^{5} - 14 q^{6} - 30 q^{7} - 15 q^{8} - 12 q^{9} - 40 q^{10} - 12 q^{11} - 18 q^{12} - 36 q^{13} - 24 q^{14} - 6 q^{15} - 9 q^{16} + 4 q^{17} + 30 q^{18} - 20 q^{19} + 46 q^{20} + 14 q^{21} + 8 q^{22} + 21 q^{23} + 29 q^{24} - 9 q^{25} + 2 q^{26} + 21 q^{27} + 10 q^{28} - 8 q^{29} + 15 q^{30} - 32 q^{31} - 19 q^{32} - 12 q^{33} - 32 q^{34} - 4 q^{35} + 4 q^{36} + 28 q^{37} + 50 q^{38} + 19 q^{39} + 64 q^{40} + 2 q^{41} + 75 q^{42} + 22 q^{43} + 70 q^{44} - 6 q^{45} + 85 q^{46} + 40 q^{47} + 68 q^{48} + 53 q^{49} + 61 q^{50} + 15 q^{51} + 100 q^{52} - 10 q^{53} - 14 q^{54} - 6 q^{55} - 10 q^{56} - 20 q^{57} - 2 q^{58} - 16 q^{59} - 141 q^{60} - 84 q^{61} - 96 q^{62} - 74 q^{63} - 149 q^{64} - 84 q^{65} - 102 q^{66} - 90 q^{67} - 126 q^{68} - 89 q^{69} - 188 q^{70} - 72 q^{71} - 158 q^{72} - 96 q^{73} - 92 q^{74} - 75 q^{75} - 52 q^{76} - 8 q^{77} - 9 q^{78} - 14 q^{79} + 12 q^{80} + 76 q^{81} - 16 q^{82} + 4 q^{83} + 21 q^{84} + 90 q^{85} + 88 q^{86} + 91 q^{87} + 62 q^{88} + 42 q^{89} + 158 q^{90} + 64 q^{91} + 37 q^{92} + 78 q^{93} + 76 q^{94} + 78 q^{95} + 135 q^{96} + 78 q^{97} + 93 q^{98} + 131 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(69))$$

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
69.2.a $$\chi_{69}(1, \cdot)$$ 69.2.a.a 1 1
69.2.a.b 2
69.2.c $$\chi_{69}(68, \cdot)$$ 69.2.c.a 6 1
69.2.e $$\chi_{69}(4, \cdot)$$ 69.2.e.a 10 10
69.2.e.b 10
69.2.e.c 20
69.2.g $$\chi_{69}(5, \cdot)$$ 69.2.g.a 60 10

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(69))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(69)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(23))$$$$^{\oplus 2}$$