## Defining parameters

 Level: $$N$$ = $$36 = 2^{2} \cdot 3^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newforms: $$4$$ Sturm bound: $$144$$ Trace bound: $$4$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(36))$$.

Total New Old
Modular forms 56 21 35
Cusp forms 17 13 4
Eisenstein series 39 8 31

## Trace form

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

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(36))$$

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
36.2.a $$\chi_{36}(1, \cdot)$$ 36.2.a.a 1 1
36.2.b $$\chi_{36}(35, \cdot)$$ 36.2.b.a 2 1
36.2.e $$\chi_{36}(13, \cdot)$$ 36.2.e.a 2 2
36.2.h $$\chi_{36}(11, \cdot)$$ 36.2.h.a 8 2

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(36))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(36)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 2}$$