# Properties

 Label 81.9.d Level $81$ Weight $9$ Character orbit 81.d Rep. character $\chi_{81}(26,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $62$ Newform subspaces $7$ Sturm bound $81$ Trace bound $4$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 81.d (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$7$$ Sturm bound: $$81$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{9}(81, [\chi])$$.

Total New Old
Modular forms 156 66 90
Cusp forms 132 62 70
Eisenstein series 24 4 20

## Trace form

 $$62 q + 3842 q^{4} - 4613 q^{7} + O(q^{10})$$ $$62 q + 3842 q^{4} - 4613 q^{7} + 1020 q^{10} - 8423 q^{13} - 459262 q^{16} + 816910 q^{19} - 124158 q^{22} + 2447771 q^{25} - 4986884 q^{28} - 100088 q^{31} - 3935448 q^{34} + 1702366 q^{37} - 2094438 q^{40} - 8371598 q^{43} - 28201272 q^{46} - 11920596 q^{49} + 14086816 q^{52} - 55794516 q^{55} - 51983580 q^{58} + 23577385 q^{61} - 24463252 q^{64} - 7154315 q^{67} - 102009918 q^{70} + 98885482 q^{73} + 111921640 q^{76} + 129475165 q^{79} - 332207808 q^{82} - 92230884 q^{85} - 144881538 q^{88} + 459559442 q^{91} + 334432932 q^{94} - 131846513 q^{97} + O(q^{100})$$

## Decomposition of $$S_{9}^{\mathrm{new}}(81, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
81.9.d.a $2$ $32.998$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-239$$ $$q-2^{8}\zeta_{6}q^{4}+(-239+239\zeta_{6})q^{7}+\cdots$$
81.9.d.b $4$ $32.998$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-3304$$ $$q+\beta _{1}q^{2}-238\beta _{2}q^{4}+(-233\beta _{1}+233\beta _{3})q^{5}+\cdots$$
81.9.d.c $4$ $32.998$ $$\Q(\sqrt{-3}, \sqrt{10})$$ None $$0$$ $$0$$ $$0$$ $$1358$$ $$q+\beta _{3}q^{2}+(14+14\beta _{1})q^{4}+(26\beta _{2}-26\beta _{3})q^{5}+\cdots$$
81.9.d.d $4$ $32.998$ $$\Q(\sqrt{-3}, \sqrt{-14})$$ None $$0$$ $$0$$ $$0$$ $$3500$$ $$q+\beta _{1}q^{2}+248\beta _{2}q^{4}+(10\beta _{1}-10\beta _{3})q^{5}+\cdots$$
81.9.d.e $4$ $32.998$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-3934$$ $$q+\beta _{3}q^{2}+(608+608\beta _{1})q^{4}+(-28\beta _{2}+\cdots)q^{5}+\cdots$$
81.9.d.f $12$ $32.998$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$1698$$ $$q-\beta _{4}q^{2}+(-131\beta _{1}+\beta _{7})q^{4}+(20\beta _{2}+\cdots)q^{5}+\cdots$$
81.9.d.g $32$ $32.998$ None $$0$$ $$0$$ $$0$$ $$-3692$$

## Decomposition of $$S_{9}^{\mathrm{old}}(81, [\chi])$$ into lower level spaces

$$S_{9}^{\mathrm{old}}(81, [\chi]) \simeq$$ $$S_{9}^{\mathrm{new}}(9, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(27, [\chi])$$$$^{\oplus 2}$$