## Defining parameters

 Level: $$N$$ = $$81 = 3^{4}$$ Weight: $$k$$ = $$2$$ Nonzero newspaces: $$4$$ Newform subspaces: $$5$$ Sturm bound: $$972$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_1(81))$$.

Total New Old
Modular forms 297 220 77
Cusp forms 190 164 26
Eisenstein series 107 56 51

## Trace form

 $$164 q - 12 q^{2} - 18 q^{3} - 22 q^{4} - 15 q^{5} - 18 q^{6} - 23 q^{7} - 24 q^{8} - 18 q^{9} + O(q^{10})$$ $$164 q - 12 q^{2} - 18 q^{3} - 22 q^{4} - 15 q^{5} - 18 q^{6} - 23 q^{7} - 24 q^{8} - 18 q^{9} - 39 q^{10} - 21 q^{11} - 18 q^{12} - 29 q^{13} - 33 q^{14} - 18 q^{15} - 22 q^{16} - 27 q^{17} - 9 q^{18} - 23 q^{19} + 21 q^{20} + 9 q^{21} - 15 q^{22} + 21 q^{23} + 36 q^{24} - 10 q^{25} + 75 q^{26} + 9 q^{27} - 5 q^{28} + 15 q^{29} + 36 q^{30} - 11 q^{31} + 36 q^{32} + 9 q^{33} - 9 q^{34} - 3 q^{35} + 18 q^{36} - 41 q^{37} - 51 q^{38} - 18 q^{39} + 3 q^{40} - 15 q^{41} + 27 q^{42} - 23 q^{43} + 51 q^{44} + 36 q^{45} + 15 q^{46} + 51 q^{47} + 81 q^{48} + 132 q^{50} + 45 q^{51} + 19 q^{52} + 63 q^{53} + 108 q^{54} + 15 q^{55} + 159 q^{56} + 36 q^{57} + 3 q^{58} + 57 q^{59} + 99 q^{60} - 5 q^{61} + 93 q^{62} + 36 q^{63} - 16 q^{64} - 3 q^{65} - 18 q^{66} - 11 q^{67} - 108 q^{68} - 72 q^{69} + 21 q^{70} - 117 q^{71} - 234 q^{72} - 68 q^{73} - 195 q^{74} - 108 q^{75} + q^{76} - 177 q^{77} - 135 q^{78} - 23 q^{79} - 330 q^{80} - 90 q^{81} - 120 q^{82} - 129 q^{83} - 243 q^{84} + 9 q^{85} - 213 q^{86} - 162 q^{87} + 33 q^{88} - 90 q^{89} - 99 q^{90} - 34 q^{91} - 105 q^{92} + 21 q^{94} + 51 q^{95} - 27 q^{96} + 31 q^{97} + 126 q^{98} + 54 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_1(81))$$

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
81.2.a $$\chi_{81}(1, \cdot)$$ 81.2.a.a 2 1
81.2.c $$\chi_{81}(28, \cdot)$$ 81.2.c.a 2 2
81.2.c.b 4
81.2.e $$\chi_{81}(10, \cdot)$$ 81.2.e.a 12 6
81.2.g $$\chi_{81}(4, \cdot)$$ 81.2.g.a 144 18

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_1(81))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_1(81)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 2}$$