## Defining parameters

 Level: $$N$$ = $$68 = 2^{2} \cdot 17$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$5$$ Newform subspaces: $$6$$ Sturm bound: $$576$$ Trace bound: $$3$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(68))$$.

Total New Old
Modular forms 184 100 84
Cusp forms 105 68 37
Eisenstein series 79 32 47

## Trace form

 $$68 q - 8 q^{2} - 8 q^{4} - 16 q^{5} - 8 q^{6} - 8 q^{8} - 16 q^{9} + O(q^{10})$$ $$68 q - 8 q^{2} - 8 q^{4} - 16 q^{5} - 8 q^{6} - 8 q^{8} - 16 q^{9} - 8 q^{10} - 8 q^{11} - 8 q^{12} - 24 q^{13} - 8 q^{14} - 24 q^{15} - 24 q^{17} - 16 q^{18} - 8 q^{19} - 8 q^{20} - 40 q^{21} - 8 q^{22} - 8 q^{23} + 8 q^{24} - 4 q^{25} + 24 q^{26} + 24 q^{27} + 40 q^{28} + 4 q^{29} + 56 q^{30} + 32 q^{31} + 32 q^{32} + 32 q^{33} + 56 q^{34} + 32 q^{35} + 56 q^{36} + 16 q^{37} + 32 q^{38} + 16 q^{39} + 56 q^{40} - 20 q^{41} + 40 q^{42} + 8 q^{43} + 24 q^{44} - 28 q^{45} + 8 q^{46} - 40 q^{47} - 32 q^{48} - 48 q^{49} - 32 q^{51} - 16 q^{52} - 36 q^{53} - 24 q^{54} - 16 q^{55} - 48 q^{56} + 32 q^{57} - 64 q^{58} - 16 q^{59} - 112 q^{60} + 16 q^{61} - 64 q^{62} + 24 q^{63} - 56 q^{64} + 76 q^{65} - 96 q^{66} + 16 q^{67} - 96 q^{68} + 48 q^{69} - 80 q^{70} - 112 q^{72} + 52 q^{73} - 56 q^{74} + 16 q^{75} - 64 q^{76} - 112 q^{78} - 16 q^{79} - 24 q^{80} + 24 q^{81} - 48 q^{82} - 20 q^{85} + 64 q^{86} + 56 q^{88} + 8 q^{89} + 72 q^{90} + 16 q^{91} + 104 q^{92} + 16 q^{93} + 88 q^{94} + 48 q^{95} + 144 q^{96} - 16 q^{97} + 128 q^{98} + 16 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(68))$$

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
68.2.a $$\chi_{68}(1, \cdot)$$ 68.2.a.a 2 1
68.2.b $$\chi_{68}(33, \cdot)$$ 68.2.b.a 2 1
68.2.e $$\chi_{68}(13, \cdot)$$ 68.2.e.a 4 2
68.2.h $$\chi_{68}(9, \cdot)$$ 68.2.h.a 4 4
68.2.i $$\chi_{68}(3, \cdot)$$ 68.2.i.a 8 8
68.2.i.b 48

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(68))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(68)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(17))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(34))$$$$^{\oplus 2}$$