# Properties

 Label 684.3.y Level $684$ Weight $3$ Character orbit 684.y Rep. character $\chi_{684}(145,\cdot)$ Character field $\Q(\zeta_{6})$ Dimension $32$ Newform subspaces $8$ Sturm bound $360$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 684.y (of order $$6$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$19$$ Character field: $$\Q(\zeta_{6})$$ Newform subspaces: $$8$$ Sturm bound: $$360$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$, $$7$$

## Dimensions

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

Total New Old
Modular forms 504 32 472
Cusp forms 456 32 424
Eisenstein series 48 0 48

## Trace form

 $$32 q - q^{5} + 4 q^{7} + O(q^{10})$$ $$32 q - q^{5} + 4 q^{7} + 14 q^{11} + 21 q^{13} - 19 q^{17} - 13 q^{19} + 17 q^{23} - 33 q^{25} - 51 q^{29} - 14 q^{35} + 54 q^{41} - 77 q^{43} - 79 q^{47} + 120 q^{49} - 33 q^{53} - 46 q^{55} - 30 q^{59} + 43 q^{61} - 12 q^{67} + 243 q^{71} + 16 q^{73} - 116 q^{77} + 63 q^{79} + 74 q^{83} + 101 q^{85} + 219 q^{89} - 102 q^{91} + 335 q^{95} + 216 q^{97} + O(q^{100})$$

## 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.y.a $2$ $18.638$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-2$$ $$-2$$ $$q+(-2+2\zeta_{6})q^{5}-q^{7}-2^{4}q^{11}+(18+\cdots)q^{13}+\cdots$$
684.3.y.b $2$ $18.638$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-2$$ $$22$$ $$q+(-2+2\zeta_{6})q^{5}+11q^{7}+8q^{11}+\cdots$$
684.3.y.c $2$ $18.638$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-22$$ $$q-11q^{7}+(30-15\zeta_{6})q^{13}+(-21+\cdots)q^{19}+\cdots$$
684.3.y.d $2$ $18.638$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$26$$ $$q+13q^{7}+(14-7\zeta_{6})q^{13}+(-5-2^{4}\zeta_{6})q^{19}+\cdots$$
684.3.y.e $2$ $18.638$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$6$$ $$-10$$ $$q+(6-6\zeta_{6})q^{5}-5q^{7}+(-22+11\zeta_{6})q^{13}+\cdots$$
684.3.y.f $6$ $18.638$ 6.0.954288.1 None $$0$$ $$0$$ $$-4$$ $$10$$ $$q+(-\beta _{1}-2\beta _{2}+\beta _{4}-\beta _{5})q^{5}+(2+\beta _{1}+\cdots)q^{7}+\cdots$$
684.3.y.g $8$ $18.638$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-8$$ $$q+(\beta _{1}-\beta _{5})q^{5}+(-1+\beta _{3})q^{7}-\beta _{4}q^{11}+\cdots$$
684.3.y.h $8$ $18.638$ $$\mathbb{Q}[x]/(x^{8} - \cdots)$$ None $$0$$ $$0$$ $$1$$ $$-12$$ $$q+\beta _{5}q^{5}+(-2+\beta _{1})q^{7}+(1-\beta _{1}-\beta _{4}+\cdots)q^{11}+\cdots$$

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

$$S_{3}^{\mathrm{old}}(684, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(19, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(38, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(57, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(76, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(114, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(171, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(228, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(342, [\chi])$$$$^{\oplus 2}$$