## Defining parameters

 Level: $$N$$ = $$108 = 2^{2} \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$6$$ Newform subspaces: $$7$$ Sturm bound: $$1296$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(108))$$.

Total New Old
Modular forms 399 165 234
Cusp forms 250 133 117
Eisenstein series 149 32 117

## Trace form

 $$133q - 3q^{2} - 5q^{4} - 6q^{6} + 6q^{7} - 9q^{8} - 6q^{9} + O(q^{10})$$ $$133q - 3q^{2} - 5q^{4} - 6q^{6} + 6q^{7} - 9q^{8} - 6q^{9} - 13q^{10} + 6q^{11} - 15q^{12} - 16q^{13} - 33q^{14} - 9q^{15} - 29q^{16} - 42q^{17} - 27q^{18} - 9q^{19} - 45q^{20} - 42q^{21} - 21q^{22} - 33q^{23} - 12q^{24} - 30q^{25} - 27q^{27} - 6q^{28} - 51q^{29} + 9q^{30} + 57q^{32} - 60q^{33} + 23q^{34} - 15q^{35} + 24q^{36} - 31q^{37} + 45q^{38} + 3q^{39} + 11q^{40} - 42q^{41} + 54q^{42} + 63q^{44} + 21q^{45} + 9q^{46} + 36q^{47} + 69q^{48} - 32q^{49} + 66q^{50} + 63q^{51} - 25q^{52} + 78q^{53} + 78q^{54} + 18q^{55} + 81q^{56} + 30q^{57} - 25q^{58} + 57q^{59} + 102q^{60} - 4q^{61} + 90q^{62} + 57q^{63} + 13q^{64} + 72q^{65} + 87q^{66} - 9q^{67} + 66q^{68} + 3q^{69} + 27q^{70} + 12q^{71} + 12q^{72} + 20q^{73} + 51q^{74} - 33q^{75} + 15q^{76} - 36q^{77} - 24q^{78} - 60q^{79} - 42q^{81} - 58q^{82} - 54q^{83} - 12q^{84} - 80q^{85} - 51q^{86} - 63q^{87} - 21q^{88} - 60q^{89} - 78q^{90} - 24q^{91} - 147q^{92} + 27q^{93} - 33q^{94} - 6q^{95} - 138q^{96} - 67q^{97} - 180q^{98} + 27q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(108))$$

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
108.2.a $$\chi_{108}(1, \cdot)$$ 108.2.a.a 1 1
108.2.b $$\chi_{108}(107, \cdot)$$ 108.2.b.a 4 1
108.2.b.b 4
108.2.e $$\chi_{108}(37, \cdot)$$ 108.2.e.a 2 2
108.2.h $$\chi_{108}(35, \cdot)$$ 108.2.h.a 8 2
108.2.i $$\chi_{108}(13, \cdot)$$ 108.2.i.a 18 6
108.2.l $$\chi_{108}(11, \cdot)$$ 108.2.l.a 96 6

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(108))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(108)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(36))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_1(54))$$$$^{\oplus 2}$$