## Defining parameters

 Level: $$N$$ = $$54 = 2 \cdot 3^{3}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$3$$ Newform subspaces: $$5$$ Sturm bound: $$324$$ Trace bound: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(54))$$.

Total New Old
Modular forms 111 22 89
Cusp forms 52 22 30
Eisenstein series 59 0 59

## Trace form

 $$22q + q^{2} + q^{4} - 6q^{5} - 6q^{6} - 4q^{7} - 5q^{8} - 12q^{9} + O(q^{10})$$ $$22q + q^{2} + q^{4} - 6q^{5} - 6q^{6} - 4q^{7} - 5q^{8} - 12q^{9} - 6q^{10} - 18q^{11} - 3q^{12} - 10q^{13} - 4q^{14} - 18q^{15} + q^{16} - 6q^{17} + 6q^{18} + 2q^{19} + 12q^{20} + 24q^{21} + 18q^{23} - 5q^{25} + 32q^{26} + 27q^{27} + 2q^{28} + 36q^{29} + 36q^{30} - 4q^{31} + q^{32} + 27q^{33} - 6q^{34} + 6q^{35} + 6q^{36} - 4q^{37} - 13q^{38} - 42q^{39} - 6q^{40} - 6q^{41} - 24q^{42} - 28q^{43} - 6q^{47} - 6q^{48} - 27q^{49} - 41q^{50} - 10q^{52} - 48q^{53} - 36q^{54} - 18q^{55} - 4q^{56} - 9q^{57} - 18q^{58} + 9q^{59} - 18q^{60} - 10q^{61} - 16q^{62} - 6q^{63} - 5q^{64} + 6q^{65} + 50q^{67} - 15q^{68} + 18q^{69} + 42q^{70} + 48q^{71} + 24q^{72} - 10q^{73} + 26q^{74} + 12q^{75} + 23q^{76} + 36q^{77} + 36q^{78} + 92q^{79} + 12q^{80} + 30q^{82} + 12q^{83} + 72q^{85} + 26q^{86} + 36q^{87} + 27q^{88} - 15q^{89} + 18q^{90} - 2q^{91} - 6q^{93} + 30q^{94} + 6q^{95} + 6q^{96} + 20q^{97} - 9q^{98} - 36q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(54))$$

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
54.2.a $$\chi_{54}(1, \cdot)$$ 54.2.a.a 1 1
54.2.a.b 1
54.2.c $$\chi_{54}(19, \cdot)$$ 54.2.c.a 2 2
54.2.e $$\chi_{54}(7, \cdot)$$ 54.2.e.a 6 6
54.2.e.b 12

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(54))$$ into lower level spaces

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