# Properties

 Label 324.2.b Level $324$ Weight $2$ Character orbit 324.b Rep. character $\chi_{324}(323,\cdot)$ Character field $\Q$ Dimension $20$ Newform subspaces $3$ Sturm bound $108$ Trace bound $4$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$324 = 2^{2} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 324.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$12$$ Character field: $$\Q$$ Newform subspaces: $$3$$ Sturm bound: $$108$$ Trace bound: $$4$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

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

Total New Old
Modular forms 66 28 38
Cusp forms 42 20 22
Eisenstein series 24 8 16

## Trace form

 $$20q + 2q^{4} + O(q^{10})$$ $$20q + 2q^{4} - 8q^{10} + 4q^{13} + 14q^{16} + 6q^{22} - 24q^{28} + 10q^{34} + 4q^{37} + 4q^{40} - 24q^{46} + 8q^{49} - 32q^{52} - 20q^{58} - 8q^{61} + 2q^{64} - 24q^{70} + 4q^{73} - 54q^{76} + 46q^{82} - 28q^{85} - 18q^{88} - 12q^{94} - 8q^{97} + O(q^{100})$$

## 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.b.a $$4$$ $$2.587$$ $$\Q(\sqrt{-2}, \sqrt{3})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}-2q^{4}-\beta _{2}q^{5}-2\beta _{1}q^{8}+\cdots$$
324.2.b.b $$8$$ $$2.587$$ 8.0.170772624.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{2}-\beta _{2}q^{4}+(\beta _{3}-\beta _{4}+\beta _{7})q^{5}+\cdots$$
324.2.b.c $$8$$ $$2.587$$ 8.0.5780865024.3 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(\beta _{3}+\beta _{5})q^{5}+\cdots$$

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

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