# Properties

 Label 324.2.e Level $324$ Weight $2$ Character orbit 324.e Rep. character $\chi_{324}(109,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $8$ Newform subspaces $4$ Sturm bound $108$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 324.e (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$4$$ Sturm bound: $$108$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(324, [\chi])$$.

Total New Old
Modular forms 144 8 136
Cusp forms 72 8 64
Eisenstein series 72 0 72

## Trace form

 $$8 q - 5 q^{7} + O(q^{10})$$ $$8 q - 5 q^{7} - 5 q^{13} + 22 q^{19} + 2 q^{25} + 16 q^{31} - 2 q^{37} + 4 q^{43} - 21 q^{49} - 72 q^{55} - 11 q^{61} + q^{67} + 10 q^{73} + 37 q^{79} + 18 q^{85} - 46 q^{91} + q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(324, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
324.2.e.a $2$ $2.587$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-3$$ $$-2$$ $$q-3\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+(6-6\zeta_{6})q^{11}+\cdots$$
324.2.e.b $2$ $2.587$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-5$$ $$q+(-5+5\zeta_{6})q^{7}+7\zeta_{6}q^{13}-q^{19}+\cdots$$
324.2.e.c $2$ $2.587$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$4$$ $$q+(4-4\zeta_{6})q^{7}-2\zeta_{6}q^{13}+8q^{19}+\cdots$$
324.2.e.d $2$ $2.587$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$3$$ $$-2$$ $$q+3\zeta_{6}q^{5}+(-2+2\zeta_{6})q^{7}+(-6+6\zeta_{6})q^{11}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(324, [\chi])$$ into lower level spaces

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