## Defining parameters

 Level: $$N$$ = $$36 = 2^{2} \cdot 3^{2}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$3$$ Newform subspaces: $$7$$ Sturm bound: $$216$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(36))$$.

Total New Old
Modular forms 92 38 54
Cusp forms 52 28 24
Eisenstein series 40 10 30

## Trace form

 $$28q + q^{2} + 3q^{3} + 3q^{4} + 11q^{5} - 9q^{6} - q^{7} - 38q^{8} - 15q^{9} + O(q^{10})$$ $$28q + q^{2} + 3q^{3} + 3q^{4} + 11q^{5} - 9q^{6} - q^{7} - 38q^{8} - 15q^{9} - 24q^{10} - 36q^{11} - 36q^{12} - 13q^{13} - 45q^{15} + 15q^{16} - 52q^{17} + 12q^{18} + 2q^{19} + 44q^{20} + 69q^{21} + 15q^{22} + 99q^{23} + 129q^{24} + 37q^{25} + 164q^{26} + 12q^{28} - 37q^{29} + 156q^{30} - 7q^{31} + 151q^{32} - 78q^{33} + 33q^{34} - 45q^{36} - 160q^{37} - 225q^{38} + 57q^{39} - 156q^{40} - 76q^{41} - 354q^{42} - 46q^{43} - 342q^{44} + 3q^{45} - 192q^{46} - 81q^{47} - 333q^{48} + 33q^{49} - 279q^{50} - 27q^{51} - 66q^{52} + 284q^{53} + 57q^{54} + 90q^{55} + 270q^{56} + 255q^{57} + 264q^{58} + 126q^{59} + 576q^{60} + 47q^{61} + 540q^{62} + 141q^{63} + 258q^{64} + 325q^{65} + 510q^{66} + 116q^{67} + 371q^{68} + 45q^{69} + 150q^{70} - 189q^{72} + 230q^{73} - 352q^{74} - 297q^{75} - 51q^{76} - 621q^{77} - 690q^{78} + 83q^{79} - 880q^{80} - 327q^{81} - 234q^{82} - 81q^{83} - 642q^{84} - 330q^{85} - 279q^{86} - 63q^{87} + 75q^{88} - 556q^{89} + 180q^{90} - 302q^{91} + 234q^{92} - 645q^{93} + 288q^{94} - 144q^{95} + 900q^{96} - 514q^{97} + 1024q^{98} + 171q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(36))$$

We only show spaces with odd 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
36.3.c $$\chi_{36}(17, \cdot)$$ None 0 1
36.3.d $$\chi_{36}(19, \cdot)$$ 36.3.d.a 1 1
36.3.d.b 1
36.3.d.c 2
36.3.f $$\chi_{36}(7, \cdot)$$ 36.3.f.a 2 2
36.3.f.b 2
36.3.f.c 16
36.3.g $$\chi_{36}(5, \cdot)$$ 36.3.g.a 4 2

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(36))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(36)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(12))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 2}$$