# Properties

 Label 684.3.g Level $684$ Weight $3$ Character orbit 684.g Rep. character $\chi_{684}(343,\cdot)$ Character field $\Q$ Dimension $90$ Newform subspaces $4$ Sturm bound $360$ Trace bound $2$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 684.g (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$4$$ Character field: $$\Q$$ Newform subspaces: $$4$$ Sturm bound: $$360$$ Trace bound: $$2$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(684, [\chi])$$.

Total New Old
Modular forms 248 90 158
Cusp forms 232 90 142
Eisenstein series 16 0 16

## Trace form

 $$90 q - 2 q^{2} - 6 q^{4} - 4 q^{5} + 28 q^{8} + O(q^{10})$$ $$90 q - 2 q^{2} - 6 q^{4} - 4 q^{5} + 28 q^{8} + 8 q^{10} + 12 q^{13} - 24 q^{14} - 62 q^{16} + 20 q^{17} + 40 q^{20} - 72 q^{22} + 374 q^{25} + 86 q^{26} - 90 q^{28} - 100 q^{29} - 52 q^{32} - 76 q^{34} + 92 q^{37} + 180 q^{40} + 100 q^{41} - 120 q^{44} + 164 q^{46} - 710 q^{49} + 14 q^{50} + 132 q^{52} - 276 q^{53} + 180 q^{56} - 262 q^{58} - 20 q^{61} - 60 q^{62} - 174 q^{64} - 88 q^{65} + 46 q^{68} + 12 q^{70} - 44 q^{73} + 304 q^{74} + 104 q^{77} - 388 q^{80} + 564 q^{82} + 64 q^{85} - 400 q^{86} + 80 q^{88} + 500 q^{89} + 342 q^{92} - 224 q^{94} + 100 q^{97} + 578 q^{98} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(684, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
684.3.g.a $4$ $18.638$ $$\Q(\sqrt{-3}, \sqrt{-19})$$ None $$4$$ $$0$$ $$4$$ $$0$$ $$q+(1-\beta _{1})q^{2}+(-2-2\beta _{1})q^{4}+(2-2\beta _{2}+\cdots)q^{5}+\cdots$$
684.3.g.b $14$ $18.638$ $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ None $$-2$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{10}q^{5}+(\beta _{3}+\beta _{5}+\cdots)q^{7}+\cdots$$
684.3.g.c $36$ $18.638$ None $$-4$$ $$0$$ $$-8$$ $$0$$
684.3.g.d $36$ $18.638$ None $$0$$ $$0$$ $$0$$ $$0$$

## Decomposition of $$S_{3}^{\mathrm{old}}(684, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(684, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(12, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(76, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(228, [\chi])$$$$^{\oplus 2}$$