# Properties

 Label 324.9.g Level $324$ Weight $9$ Character orbit 324.g Rep. character $\chi_{324}(53,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $64$ Newform subspaces $8$ Sturm bound $486$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 324.g (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$8$$ Sturm bound: $$486$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{9}(324, [\chi])$$.

Total New Old
Modular forms 900 64 836
Cusp forms 828 64 764
Eisenstein series 72 0 72

## Trace form

 $$64 q - 4615 q^{7} + O(q^{10})$$ $$64 q - 4615 q^{7} - 8425 q^{13} + 108626 q^{19} + 2838394 q^{25} + 970790 q^{31} + 1702370 q^{37} + 2714960 q^{43} - 32815383 q^{49} + 74574828 q^{55} - 21835225 q^{61} + 46321427 q^{67} - 30700306 q^{73} - 187831237 q^{79} - 22955364 q^{85} - 304930790 q^{91} + 117284045 q^{97} + O(q^{100})$$

## Decomposition of $$S_{9}^{\mathrm{new}}(324, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
324.9.g.a $2$ $131.991$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-4034$$ $$q+(-4034+4034\zeta_{6})q^{7}+35806\zeta_{6}q^{13}+\cdots$$
324.9.g.b $2$ $131.991$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$4273$$ $$q+(4273-4273\zeta_{6})q^{7}+20641\zeta_{6}q^{13}+\cdots$$
324.9.g.c $4$ $131.991$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-5278$$ $$q+\beta _{3}q^{5}+(-2639-2639\beta _{1})q^{7}+\cdots$$
324.9.g.d $4$ $131.991$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-616$$ $$q+\beta _{1}q^{5}-308\beta _{2}q^{7}+(-244\beta _{1}+244\beta _{3})q^{11}+\cdots$$
324.9.g.e $4$ $131.991$ $$\Q(\sqrt{-3}, \sqrt{10})$$ None $$0$$ $$0$$ $$0$$ $$14$$ $$q+\beta _{3}q^{5}+(7+7\beta _{1})q^{7}+(19\beta _{2}-19\beta _{3})q^{11}+\cdots$$
324.9.g.f $4$ $131.991$ $$\Q(\sqrt{-3}, \sqrt{-110})$$ None $$0$$ $$0$$ $$0$$ $$6188$$ $$q+\beta _{1}q^{5}+3094\beta _{2}q^{7}+(-\beta _{1}+\beta _{3})q^{11}+\cdots$$
324.9.g.g $12$ $131.991$ $$\mathbb{Q}[x]/(x^{12} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-1470$$ $$q+\beta _{5}q^{5}+(-245+245\beta _{1}-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots$$
324.9.g.h $32$ $131.991$ None $$0$$ $$0$$ $$0$$ $$-3692$$

## Decomposition of $$S_{9}^{\mathrm{old}}(324, [\chi])$$ into lower level spaces

$$S_{9}^{\mathrm{old}}(324, [\chi]) \simeq$$ $$S_{9}^{\mathrm{new}}(9, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(27, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(54, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(81, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(108, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{9}^{\mathrm{new}}(162, [\chi])$$$$^{\oplus 2}$$