# Properties

 Label 324.2.h Level $324$ Weight $2$ Character orbit 324.h Rep. character $\chi_{324}(107,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $44$ Newform subspaces $6$ 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.h (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$36$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$6$$ 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 132 52 80
Cusp forms 84 44 40
Eisenstein series 48 8 40

## Trace form

 $$44 q + 2 q^{4} + O(q^{10})$$ $$44 q + 2 q^{4} + 4 q^{10} + 4 q^{13} + 2 q^{16} + 6 q^{22} + 18 q^{25} + 12 q^{28} - 32 q^{34} - 8 q^{37} - 38 q^{40} - 72 q^{46} + 14 q^{49} - 8 q^{52} - 20 q^{58} + 4 q^{61} - 100 q^{64} - 18 q^{70} - 32 q^{73} + 24 q^{76} - 32 q^{82} - 16 q^{85} + 78 q^{88} + 84 q^{94} - 8 q^{97} + O(q^{100})$$

## 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.h.a $$4$$ $$2.587$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$6$$ $$q-\beta _{3}q^{2}-2q^{4}+2\beta _{1}q^{5}+(2-\beta _{2}+\cdots)q^{7}+\cdots$$
324.2.h.b $$4$$ $$2.587$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-6$$ $$q+\beta _{1}q^{2}+2\beta _{2}q^{4}+2\beta _{1}q^{5}+(-2+\cdots)q^{7}+\cdots$$
324.2.h.c $$4$$ $$2.587$$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+(\beta _{1}-\beta _{3})q^{2}+(2-2\beta _{2})q^{4}+\beta _{1}q^{5}+\cdots$$
324.2.h.d $$8$$ $$2.587$$ 8.0.12960000.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{3}+\beta _{7})q^{2}+(-\beta _{1}-\beta _{2})q^{4}+\cdots$$
324.2.h.e $$8$$ $$2.587$$ $$\Q(\zeta_{24})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\zeta_{24}^{3}q^{2}+(2-2\zeta_{24})q^{4}+(\zeta_{24}^{5}+\cdots)q^{5}+\cdots$$
324.2.h.f $$16$$ $$2.587$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{15}q^{2}+(-1+\beta _{8}-\beta _{9}-\beta _{11}+\cdots)q^{4}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(324, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(108, [\chi])$$$$^{\oplus 2}$$