## Defining parameters

 Level: $$N$$ = $$66 = 2 \cdot 3 \cdot 11$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newforms: $$9$$ Sturm bound: $$480$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(66))$$.

Total New Old
Modular forms 160 31 129
Cusp forms 81 31 50
Eisenstein series 79 0 79

## Trace form

 $$31q + q^{2} + q^{3} + q^{4} + 6q^{5} - 4q^{6} - 12q^{7} + q^{8} - 19q^{9} + O(q^{10})$$ $$31q + q^{2} + q^{3} + q^{4} + 6q^{5} - 4q^{6} - 12q^{7} + q^{8} - 19q^{9} - 14q^{10} - 9q^{11} - 9q^{12} - 6q^{13} - 12q^{14} - 24q^{15} + q^{16} - 22q^{17} - 4q^{18} - 10q^{19} + 6q^{20} - 12q^{21} + 11q^{22} + 4q^{23} + 6q^{24} - 9q^{25} + 14q^{26} + 31q^{27} + 8q^{28} + 10q^{29} + 36q^{30} + 12q^{31} + q^{32} + 56q^{33} + 18q^{34} + 28q^{35} + 6q^{36} + 38q^{37} + 20q^{38} + 24q^{39} + 6q^{40} + 22q^{41} + 18q^{42} - 16q^{43} - 9q^{44} + 6q^{45} - 16q^{46} - 32q^{47} + q^{48} - 63q^{49} - 9q^{50} - 37q^{51} - 26q^{52} - 6q^{53} - 9q^{54} - 34q^{55} + 8q^{56} + 15q^{57} - 50q^{58} - 14q^{60} - 38q^{61} - 28q^{62} - 2q^{63} + q^{64} - 36q^{65} - 49q^{66} - 32q^{67} - 22q^{68} - 6q^{69} - 32q^{70} - 8q^{71} - 19q^{72} - 6q^{73} - 2q^{74} - 14q^{75} + 8q^{77} - 26q^{78} + 20q^{79} - 14q^{80} - 39q^{81} + 32q^{82} + 24q^{83} + 8q^{84} + 48q^{85} - 16q^{86} + 30q^{87} - 9q^{88} + 30q^{89} + 16q^{90} + 152q^{91} + 24q^{92} + 2q^{93} + 88q^{94} + 60q^{95} + q^{96} + 68q^{97} + 57q^{98} + 41q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(66))$$

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
66.2.a $$\chi_{66}(1, \cdot)$$ 66.2.a.a 1 1
66.2.a.b 1
66.2.a.c 1
66.2.b $$\chi_{66}(65, \cdot)$$ 66.2.b.a 2 1
66.2.b.b 2
66.2.e $$\chi_{66}(25, \cdot)$$ 66.2.e.a 4 4
66.2.e.b 4
66.2.h $$\chi_{66}(17, \cdot)$$ 66.2.h.a 8 4
66.2.h.b 8

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(66))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(66)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(11))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(22))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(33))$$$$^{\oplus 2}$$