# Properties

 Label 684.2.k Level $684$ Weight $2$ Character orbit 684.k Rep. character $\chi_{684}(505,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $18$ Newform subspaces $7$ Sturm bound $240$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 684.k (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$19$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$7$$ Sturm bound: $$240$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$, $$7$$

## Dimensions

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

Total New Old
Modular forms 264 18 246
Cusp forms 216 18 198
Eisenstein series 48 0 48

## Trace form

 $$18 q + q^{5} + O(q^{10})$$ $$18 q + q^{5} - 4 q^{11} - 7 q^{13} + 7 q^{17} - 4 q^{19} - 13 q^{23} - 16 q^{25} + 11 q^{29} + 6 q^{35} - 4 q^{37} + 3 q^{41} + 9 q^{43} + 5 q^{47} + 34 q^{49} + 9 q^{53} - 16 q^{55} + 13 q^{59} + 9 q^{61} + 50 q^{65} - 17 q^{67} + 3 q^{71} + 21 q^{73} - 20 q^{77} + q^{79} + 8 q^{83} + 3 q^{85} + 13 q^{89} + 20 q^{91} - 61 q^{95} - 11 q^{97} + O(q^{100})$$

## 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.k.a $2$ $5.462$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-4$$ $$-6$$ $$q+(-4+4\zeta_{6})q^{5}-3q^{7}+4q^{11}-5\zeta_{6}q^{13}+\cdots$$
684.2.k.b $2$ $5.462$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$0$$ $$q+(-1+\zeta_{6})q^{5}+4q^{11}+\zeta_{6}q^{13}+\cdots$$
684.2.k.c $2$ $5.462$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-2$$ $$q-q^{7}+7\zeta_{6}q^{13}+(5-2\zeta_{6})q^{19}+5\zeta_{6}q^{25}+\cdots$$
684.2.k.d $2$ $5.462$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$10$$ $$q+5q^{7}-5\zeta_{6}q^{13}+(3+2\zeta_{6})q^{19}+\cdots$$
684.2.k.e $2$ $5.462$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$4$$ $$-6$$ $$q+(4-4\zeta_{6})q^{5}-3q^{7}-4q^{11}-5\zeta_{6}q^{13}+\cdots$$
684.2.k.f $4$ $5.462$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$0$$ $$0$$ $$-4$$ $$q-\beta _{2}q^{5}+(-1+\beta _{3})q^{7}-\beta _{3}q^{11}+\cdots$$
684.2.k.g $4$ $5.462$ $$\Q(\sqrt{-3}, \sqrt{7})$$ None $$0$$ $$0$$ $$2$$ $$8$$ $$q+(1+\beta _{1}+\beta _{2})q^{5}+(2-\beta _{3})q^{7}+(-3+\cdots)q^{11}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(684, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(38, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(57, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(76, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(114, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(171, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(228, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(342, [\chi])$$$$^{\oplus 2}$$