# Properties

 Label 81.9.b Level $81$ Weight $9$ Character orbit 81.b Rep. character $\chi_{81}(80,\cdot)$ Character field $\Q$ Dimension $30$ Newform subspaces $2$ Sturm bound $81$ Trace bound $1$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$81 = 3^{4}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 81.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$3$$ Character field: $$\Q$$ Newform subspaces: $$2$$ Sturm bound: $$81$$ Trace bound: $$1$$ Distinguishing $$T_p$$: $$2$$

## Dimensions

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

Total New Old
Modular forms 78 34 44
Cusp forms 66 30 36
Eisenstein series 12 4 8

## Trace form

 $$30 q - 3582 q^{4} + 1848 q^{7} + O(q^{10})$$ $$30 q - 3582 q^{4} + 1848 q^{7} + 10236 q^{10} + 60492 q^{13} + 225678 q^{16} - 84522 q^{19} - 814602 q^{22} - 672066 q^{25} + 78576 q^{28} + 899400 q^{31} - 282330 q^{34} + 2682600 q^{37} - 4200420 q^{40} + 3869454 q^{43} - 3044508 q^{46} + 24255834 q^{49} - 14279520 q^{52} - 6345096 q^{55} - 27210576 q^{58} + 41549196 q^{61} - 2560830 q^{64} - 45779118 q^{67} - 44552388 q^{70} - 57995346 q^{73} + 155239794 q^{76} - 85501392 q^{79} + 195898278 q^{82} - 33976728 q^{85} + 317064414 q^{88} + 172166292 q^{91} - 471345096 q^{94} - 387977010 q^{97} + O(q^{100})$$

## 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.b.a $14$ $32.998$ $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-1844$$ $$q-\beta _{1}q^{2}+(-110+\beta _{2})q^{4}+\beta _{6}q^{5}+\cdots$$
81.9.b.b $16$ $32.998$ $$\mathbb{Q}[x]/(x^{16} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$3692$$ $$q+\beta _{8}q^{2}+(-2^{7}+\beta _{1})q^{4}+(-2\beta _{8}+\cdots)q^{5}+\cdots$$

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

$$S_{9}^{\mathrm{old}}(81, [\chi]) \cong$$ $$S_{9}^{\mathrm{new}}(3, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(9, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(27, [\chi])$$$$^{\oplus 2}$$