# Properties

 Label 684.2.f Level $684$ Weight $2$ Character orbit 684.f Rep. character $\chi_{684}(379,\cdot)$ Character field $\Q$ Dimension $48$ Newform subspaces $5$ Sturm bound $240$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 684.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$76$$ Character field: $$\Q$$ Newform subspaces: $$5$$ Sturm bound: $$240$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$5$$, $$31$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(684, [\chi])$$.

Total New Old
Modular forms 128 52 76
Cusp forms 112 48 64
Eisenstein series 16 4 12

## Trace form

 $$48q + 2q^{4} + 4q^{5} + O(q^{10})$$ $$48q + 2q^{4} + 4q^{5} + 2q^{16} + 8q^{17} + 12q^{20} + 36q^{25} + 6q^{26} + 22q^{28} - 10q^{38} + 32q^{44} - 8q^{49} + 42q^{58} - 36q^{61} + 36q^{62} - 10q^{64} + 26q^{68} - 16q^{73} - 44q^{74} - 24q^{76} - 20q^{77} - 40q^{80} + 20q^{82} - 28q^{85} - 34q^{92} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(684, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
684.2.f.a $$4$$ $$5.462$$ $$\Q(\sqrt{-2}, \sqrt{19})$$ $$\Q(\sqrt{-57})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}-2q^{4}+2\beta _{1}q^{8}-\beta _{2}q^{11}+\cdots$$
684.2.f.b $$8$$ $$5.462$$ 8.0.$$\cdots$$.1 None $$0$$ $$0$$ $$4$$ $$0$$ $$q+\beta _{1}q^{2}+(-1+\beta _{2})q^{4}+(1-\beta _{2}+\beta _{6}+\cdots)q^{5}+\cdots$$
684.2.f.c $$10$$ $$5.462$$ 10.0.$$\cdots$$.1 None $$-2$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{4}q^{5}-\beta _{5}q^{7}+\cdots$$
684.2.f.d $$10$$ $$5.462$$ 10.0.$$\cdots$$.1 None $$2$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{4}q^{5}-\beta _{5}q^{7}+\cdots$$
684.2.f.e $$16$$ $$5.462$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{2}+(1+\beta _{6})q^{4}-\beta _{11}q^{5}+(\beta _{7}+\cdots)q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(684, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(684, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(76, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(228, [\chi])$$$$^{\oplus 2}$$