# Properties

 Label 684.3.h Level $684$ Weight $3$ Character orbit 684.h Rep. character $\chi_{684}(37,\cdot)$ Character field $\Q$ Dimension $16$ Newform subspaces $5$ Sturm bound $360$ Trace bound $5$

# Related objects

## Defining parameters

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

## Dimensions

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

Total New Old
Modular forms 252 16 236
Cusp forms 228 16 212
Eisenstein series 24 0 24

## Trace form

 $$16 q + q^{5} - 13 q^{7} + O(q^{10})$$ $$16 q + q^{5} - 13 q^{7} + q^{11} + 19 q^{17} + 22 q^{19} + 34 q^{23} + 141 q^{25} + 53 q^{35} + 137 q^{43} + q^{47} + 291 q^{49} + 13 q^{55} + 89 q^{61} + 71 q^{73} + 197 q^{77} + 160 q^{83} + 457 q^{85} - 107 q^{95} + O(q^{100})$$

## 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.h.a $2$ $18.638$ $$\Q(\sqrt{57})$$ $$\Q(\sqrt{-19})$$ $$0$$ $$0$$ $$-9$$ $$5$$ $$q+(-4-\beta )q^{5}+(4-3\beta )q^{7}+(4-5\beta )q^{11}+\cdots$$
684.3.h.b $2$ $18.638$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$-4$$ $$q-2q^{7}+\zeta_{6}q^{13}+(-13-\zeta_{6})q^{19}+\cdots$$
684.3.h.c $2$ $18.638$ $$\Q(\sqrt{-29})$$ None $$0$$ $$0$$ $$8$$ $$-2$$ $$q+4q^{5}-q^{7}-14q^{11}-\beta q^{13}-23q^{17}+\cdots$$
684.3.h.d $4$ $18.638$ $$\Q(\sqrt{3}, \sqrt{19})$$ $$\Q(\sqrt{-19})$$ $$0$$ $$0$$ $$0$$ $$-10$$ $$q+\beta _{1}q^{5}+(-3-\beta _{3})q^{7}-\beta _{2}q^{11}+\cdots$$
684.3.h.e $6$ $18.638$ 6.0.219615408.1 None $$0$$ $$0$$ $$2$$ $$-2$$ $$q+\beta _{3}q^{5}-\beta _{2}q^{7}+(4+\beta _{2})q^{11}+(-\beta _{1}+\cdots)q^{13}+\cdots$$

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

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