# Properties

 Label 81.4 Level 81 Weight 4 Dimension 548 Nonzero newspaces 4 Newform subspaces 14 Sturm bound 1944 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$81 = 3^{4}$$ Weight: $$k$$ = $$4$$ Nonzero newspaces: $$4$$ Newform subspaces: $$14$$ Sturm bound: $$1944$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_1(81))$$.

Total New Old
Modular forms 783 604 179
Cusp forms 675 548 127
Eisenstein series 108 56 52

## Trace form

 $$548 q - 12 q^{2} - 18 q^{3} - 28 q^{4} - 24 q^{5} - 18 q^{6} - 2 q^{7} + 57 q^{8} - 18 q^{9} + O(q^{10})$$ $$548 q - 12 q^{2} - 18 q^{3} - 28 q^{4} - 24 q^{5} - 18 q^{6} - 2 q^{7} + 57 q^{8} - 18 q^{9} - 9 q^{10} - 75 q^{11} - 18 q^{12} - 74 q^{13} - 69 q^{14} - 18 q^{15} + 152 q^{16} + 189 q^{17} + 396 q^{18} + 421 q^{19} + 1155 q^{20} + 90 q^{21} - 198 q^{22} - 798 q^{23} - 1098 q^{24} - 883 q^{25} - 3327 q^{26} - 720 q^{27} - 1577 q^{28} - 1038 q^{29} - 774 q^{30} + 34 q^{31} + 846 q^{32} + 414 q^{33} + 864 q^{34} + 2481 q^{35} + 2286 q^{36} + 1501 q^{37} + 2460 q^{38} - 18 q^{39} + 1989 q^{40} + 3567 q^{41} + 3537 q^{42} + 1429 q^{43} + 4857 q^{44} + 1008 q^{45} - 225 q^{46} - 1596 q^{47} - 2295 q^{48} - 1821 q^{49} - 9174 q^{50} - 2979 q^{51} - 4097 q^{52} - 5607 q^{53} - 6696 q^{54} - 4275 q^{55} - 11775 q^{56} - 2232 q^{57} - 3519 q^{58} - 3705 q^{59} - 1305 q^{60} - 56 q^{61} + 3981 q^{62} + 1980 q^{63} + 5705 q^{64} + 5478 q^{65} - 1962 q^{66} + 6433 q^{67} + 8640 q^{68} - 396 q^{69} + 6039 q^{70} + 2367 q^{71} + 1710 q^{72} - 389 q^{73} + 1911 q^{74} + 4482 q^{75} - 4646 q^{76} + 5052 q^{77} + 10881 q^{78} - 3944 q^{79} + 10254 q^{80} + 5742 q^{81} - 630 q^{82} + 1842 q^{83} + 17361 q^{84} - 3429 q^{85} + 3558 q^{86} + 5454 q^{87} + 2412 q^{88} + 12249 q^{89} + 11079 q^{90} + 9707 q^{91} + 19173 q^{92} + 3780 q^{93} + 13329 q^{94} + 6027 q^{95} - 17253 q^{96} + 5155 q^{97} - 14805 q^{98} - 13230 q^{99} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_1(81))$$

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 available newforms together with their dimension.

Label $$\chi$$ Newforms Dimension $$\chi$$ degree
81.4.a $$\chi_{81}(1, \cdot)$$ 81.4.a.a 2 1
81.4.a.b 2
81.4.a.c 2
81.4.a.d 2
81.4.a.e 2
81.4.c $$\chi_{81}(28, \cdot)$$ 81.4.c.a 2 2
81.4.c.b 2
81.4.c.c 2
81.4.c.d 4
81.4.c.e 4
81.4.c.f 4
81.4.c.g 4
81.4.e $$\chi_{81}(10, \cdot)$$ 81.4.e.a 48 6
81.4.g $$\chi_{81}(4, \cdot)$$ 81.4.g.a 468 18

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_1(81))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_1(81)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_1(1))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(3))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(9))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(27))$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_1(81))$$$$^{\oplus 1}$$