# Properties

 Label 684.2.r Level $684$ Weight $2$ Character orbit 684.r Rep. character $\chi_{684}(487,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $96$ Newform subspaces $4$ 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.r (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$76$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$4$$ Sturm bound: $$240$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$5$$, $$23$$

## Dimensions

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

Total New Old
Modular forms 256 104 152
Cusp forms 224 96 128
Eisenstein series 32 8 24

## Trace form

 $$96 q + 3 q^{2} + q^{4} + 2 q^{5} + O(q^{10})$$ $$96 q + 3 q^{2} + q^{4} + 2 q^{5} - 18 q^{10} + 6 q^{13} - 12 q^{14} + q^{16} - 2 q^{17} + 24 q^{20} + 3 q^{22} - 42 q^{25} + 24 q^{26} - 4 q^{28} + 6 q^{29} + 33 q^{32} + 24 q^{34} + 10 q^{38} + 48 q^{40} + 24 q^{41} + 7 q^{44} - 112 q^{49} - 42 q^{52} + 18 q^{53} + 24 q^{58} + 30 q^{61} - 6 q^{62} - 2 q^{64} - 44 q^{68} + 90 q^{70} - 8 q^{73} - 46 q^{74} - 9 q^{76} - 16 q^{77} - 32 q^{80} - 17 q^{82} - 2 q^{85} + 66 q^{86} + 30 q^{89} + 34 q^{92} - 12 q^{97} - 39 q^{98} + O(q^{100})$$

## 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.r.a $16$ $5.462$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$3$$ $$0$$ $$2$$ $$0$$ $$q-\beta _{12}q^{2}+(-1-\beta _{1}+\beta _{2}-2\beta _{3}+\cdots)q^{4}+\cdots$$
684.2.r.b $20$ $5.462$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$-1$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{1}+\beta _{4})q^{2}+(-\beta _{2}-\beta _{9})q^{4}+\cdots$$
684.2.r.c $20$ $5.462$ $$\mathbb{Q}[x]/(x^{20} - \cdots)$$ None $$1$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{16}q^{2}+(\beta _{14}-\beta _{17})q^{4}-\beta _{6}q^{5}+\cdots$$
684.2.r.d $40$ $5.462$ None $$0$$ $$0$$ $$0$$ $$0$$

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

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