## Defining parameters

 Level: $$N$$ = $$33 = 3 \cdot 11$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newform subspaces: $$5$$ Sturm bound: $$160$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(33))$$.

Total New Old
Modular forms 60 39 21
Cusp forms 21 19 2
Eisenstein series 39 20 19

## Trace form

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

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(33))$$

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
33.2.a $$\chi_{33}(1, \cdot)$$ 33.2.a.a 1 1
33.2.d $$\chi_{33}(32, \cdot)$$ 33.2.d.a 2 1
33.2.e $$\chi_{33}(4, \cdot)$$ 33.2.e.a 4 4
33.2.e.b 4
33.2.f $$\chi_{33}(2, \cdot)$$ 33.2.f.a 8 4

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(33))$$ into lower level spaces

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