# Properties

 Label 684.5.h Level $684$ Weight $5$ Character orbit 684.h Rep. character $\chi_{684}(37,\cdot)$ Character field $\Q$ Dimension $34$ Newform subspaces $6$ Sturm bound $600$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$684 = 2^{2} \cdot 3^{2} \cdot 19$$ Weight: $$k$$ $$=$$ $$5$$ Character orbit: $$[\chi]$$ $$=$$ 684.h (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$19$$ Character field: $$\Q$$ Newform subspaces: $$6$$ Sturm bound: $$600$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$5$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{5}(684, [\chi])$$.

Total New Old
Modular forms 492 34 458
Cusp forms 468 34 434
Eisenstein series 24 0 24

## Trace form

 $$34q - 9q^{5} - 15q^{7} + O(q^{10})$$ $$34q - 9q^{5} - 15q^{7} - 297q^{11} + 225q^{17} - 36q^{19} - 186q^{23} + 3421q^{25} + 327q^{35} - 5977q^{43} - 3825q^{47} + 13875q^{49} + 9071q^{55} + 63q^{61} - 7195q^{73} - 13161q^{77} - 10500q^{83} - 3325q^{85} + 19107q^{95} + O(q^{100})$$

## Decomposition of $$S_{5}^{\mathrm{new}}(684, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
684.5.h.a $$2$$ $$70.705$$ $$\Q(\sqrt{-3})$$ $$\Q(\sqrt{-3})$$ $$0$$ $$0$$ $$0$$ $$188$$ $$q+94q^{7}+11\zeta_{6}q^{13}+(23+13\zeta_{6})q^{19}+\cdots$$
684.5.h.b $$2$$ $$70.705$$ $$\Q(\sqrt{57})$$ $$\Q(\sqrt{-19})$$ $$0$$ $$0$$ $$31$$ $$73$$ $$q+(17+3\beta )q^{5}+(39+5\beta )q^{7}+(-119+\cdots)q^{11}+\cdots$$
684.5.h.c $$4$$ $$70.705$$ $$\mathbb{Q}[x]/(x^{4} + \cdots)$$ None $$0$$ $$0$$ $$-22$$ $$24$$ $$q+(-6+\beta _{3})q^{5}+(7-2\beta _{3})q^{7}+(48+\cdots)q^{11}+\cdots$$
684.5.h.d $$4$$ $$70.705$$ $$\Q(\sqrt{3}, \sqrt{19})$$ $$\Q(\sqrt{-19})$$ $$0$$ $$0$$ $$0$$ $$-146$$ $$q+(-2\beta _{1}-\beta _{2})q^{5}+(-34+5\beta _{3})q^{7}+\cdots$$
684.5.h.e $$8$$ $$70.705$$ $$\mathbb{Q}[x]/(x^{8} + \cdots)$$ None $$0$$ $$0$$ $$0$$ $$-68$$ $$q+\beta _{4}q^{5}+(-9+\beta _{1})q^{7}+(-2\beta _{4}+\beta _{5}+\cdots)q^{11}+\cdots$$
684.5.h.f $$14$$ $$70.705$$ $$\mathbb{Q}[x]/(x^{14} - \cdots)$$ None $$0$$ $$0$$ $$-18$$ $$-86$$ $$q+(-1-\beta _{1})q^{5}+(-6+\beta _{2})q^{7}+(-18+\cdots)q^{11}+\cdots$$

## Decomposition of $$S_{5}^{\mathrm{old}}(684, [\chi])$$ into lower level spaces

$$S_{5}^{\mathrm{old}}(684, [\chi]) \cong$$ $$S_{5}^{\mathrm{new}}(19, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(38, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(57, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(76, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(114, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(171, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(228, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{5}^{\mathrm{new}}(342, [\chi])$$$$^{\oplus 2}$$