## 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

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

## 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

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