## Defining parameters

 Level: $$N$$ = $$36 = 2^{2} \cdot 3^{2}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newform subspaces: $$4$$ 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

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

## 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

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