## 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

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

## 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

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