## Defining parameters

 Level: $$N$$ = $$33 = 3 \cdot 11$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newforms: $$5$$ 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

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

## 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

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