## 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

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

## 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

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