## Defining parameters

 Level: $$N$$ = $$54 = 2 \cdot 3^{3}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$3$$ Newform subspaces: $$3$$ Sturm bound: $$486$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(54))$$.

Total New Old
Modular forms 192 42 150
Cusp forms 132 42 90
Eisenstein series 60 0 60

## Trace form

 $$42 q + 36 q^{5} + 12 q^{6} + 12 q^{7} - 12 q^{9} + O(q^{10})$$ $$42 q + 36 q^{5} + 12 q^{6} + 12 q^{7} - 12 q^{9} - 24 q^{10} - 36 q^{11} - 12 q^{12} - 12 q^{13} - 72 q^{14} - 18 q^{15} - 48 q^{18} + 18 q^{19} - 36 q^{20} - 228 q^{21} + 48 q^{22} - 198 q^{23} - 72 q^{25} + 54 q^{27} - 12 q^{28} + 126 q^{29} + 144 q^{30} - 72 q^{31} + 324 q^{33} + 24 q^{34} + 486 q^{35} + 168 q^{36} + 78 q^{37} + 252 q^{38} + 102 q^{39} + 48 q^{40} + 36 q^{41} + 48 q^{42} + 72 q^{43} - 378 q^{45} - 48 q^{46} - 432 q^{47} - 24 q^{48} + 12 q^{49} - 54 q^{51} + 24 q^{52} - 36 q^{54} + 36 q^{55} - 144 q^{56} + 72 q^{57} - 24 q^{58} + 126 q^{59} + 36 q^{60} - 36 q^{61} + 318 q^{63} + 96 q^{64} - 72 q^{65} - 432 q^{66} - 738 q^{67} - 252 q^{68} - 522 q^{69} - 588 q^{70} - 648 q^{71} - 192 q^{72} - 276 q^{73} - 576 q^{74} - 438 q^{75} - 228 q^{76} - 252 q^{77} - 288 q^{78} + 96 q^{79} + 324 q^{81} + 192 q^{82} + 972 q^{83} + 216 q^{84} + 900 q^{85} + 648 q^{86} + 1062 q^{87} + 120 q^{88} + 648 q^{89} + 720 q^{90} + 204 q^{91} + 360 q^{92} + 462 q^{93} + 612 q^{94} + 72 q^{95} + 96 q^{96} + 618 q^{97} + 648 q^{98} + 126 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(54))$$

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
54.3.b $$\chi_{54}(53, \cdot)$$ 54.3.b.a 2 1
54.3.d $$\chi_{54}(17, \cdot)$$ 54.3.d.a 4 2
54.3.f $$\chi_{54}(5, \cdot)$$ 54.3.f.a 36 6

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(54))$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(\Gamma_1(54)) \cong$$ $$S_{3}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(18))$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 2}$$