# Properties

 Label 81.3 Level 81 Weight 3 Dimension 356 Nonzero newspaces 4 Newform subspaces 7 Sturm bound 1458 Trace bound 1

## Defining parameters

 Level: $$N$$ = $$81 = 3^{4}$$ Weight: $$k$$ = $$3$$ Nonzero newspaces: $$4$$ Newform subspaces: $$7$$ Sturm bound: $$1458$$ Trace bound: $$1$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(\Gamma_1(81))$$.

Total New Old
Modular forms 540 412 128
Cusp forms 432 356 76
Eisenstein series 108 56 52

## Trace form

 $$356 q - 12 q^{2} - 18 q^{3} - 16 q^{4} - 3 q^{5} - 18 q^{6} - 17 q^{7} - 9 q^{8} - 18 q^{9} + O(q^{10})$$ $$356 q - 12 q^{2} - 18 q^{3} - 16 q^{4} - 3 q^{5} - 18 q^{6} - 17 q^{7} - 9 q^{8} - 18 q^{9} - 33 q^{10} - 12 q^{11} - 18 q^{12} - 11 q^{13} - 3 q^{14} - 18 q^{15} - 52 q^{16} - 9 q^{17} - 90 q^{18} - 125 q^{19} - 447 q^{20} - 153 q^{21} - 156 q^{22} - 219 q^{23} - 126 q^{24} - 64 q^{25} - 27 q^{26} + 9 q^{27} + 127 q^{28} + 231 q^{29} + 198 q^{30} + 133 q^{31} + 666 q^{32} + 171 q^{33} + 198 q^{34} + 477 q^{35} + 342 q^{36} + 109 q^{37} + 174 q^{38} - 18 q^{39} - 93 q^{40} - 462 q^{41} - 513 q^{42} - 314 q^{43} - 1305 q^{44} - 450 q^{45} - 681 q^{46} - 741 q^{47} - 513 q^{48} - 258 q^{49} - 672 q^{50} - 144 q^{51} - 167 q^{52} - 27 q^{53} + 108 q^{54} + 183 q^{55} + 789 q^{56} + 198 q^{57} + 447 q^{58} + 834 q^{59} + 801 q^{60} + 505 q^{61} + 1773 q^{62} + 522 q^{63} + 833 q^{64} + 1581 q^{65} + 1926 q^{66} + 52 q^{67} + 2070 q^{68} + 1062 q^{69} - 183 q^{70} + 315 q^{71} + 1710 q^{72} - 80 q^{73} + 525 q^{74} + 432 q^{75} - 266 q^{76} + 213 q^{77} + 189 q^{78} - 125 q^{79} - 90 q^{81} + 102 q^{82} - 201 q^{83} - 945 q^{84} - 9 q^{85} - 624 q^{86} - 1026 q^{87} + 48 q^{88} - 1386 q^{89} - 2529 q^{90} - 892 q^{91} - 3957 q^{92} - 2214 q^{93} - 327 q^{94} - 3075 q^{95} - 3321 q^{96} - 368 q^{97} - 3735 q^{98} - 1566 q^{99} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(\Gamma_1(81))$$

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
81.3.b $$\chi_{81}(80, \cdot)$$ 81.3.b.a 2 1
81.3.b.b 4
81.3.d $$\chi_{81}(26, \cdot)$$ 81.3.d.a 2 2
81.3.d.b 4
81.3.d.c 8
81.3.f $$\chi_{81}(8, \cdot)$$ 81.3.f.a 30 6
81.3.h $$\chi_{81}(2, \cdot)$$ 81.3.h.a 306 18

## Decomposition of $$S_{3}^{\mathrm{old}}(\Gamma_1(81))$$ into lower level spaces

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