## 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

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

## 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

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