# Properties

 Label 648.2.i Level $648$ Weight $2$ Character orbit 648.i Rep. character $\chi_{648}(217,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $24$ Newform subspaces $10$ Sturm bound $216$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$648 = 2^{3} \cdot 3^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 648.i (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$10$$ Sturm bound: $$216$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$5$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(648, [\chi])$$.

Total New Old
Modular forms 264 24 240
Cusp forms 168 24 144
Eisenstein series 96 0 96

## Trace form

 $$24 q + O(q^{10})$$ $$24 q - 12 q^{19} - 12 q^{25} + 30 q^{31} + 36 q^{43} + 60 q^{55} + 12 q^{61} + 6 q^{67} + 36 q^{73} - 30 q^{79} - 6 q^{85} - 60 q^{91} - 24 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(648, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$$a_2$$ $$a_3$$ $$a_5$$ $$a_7$$
648.2.i.a $$2$$ $$5.174$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-4$$ $$3$$ $$q-4\zeta_{6}q^{5}+(3-3\zeta_{6})q^{7}+(-4+4\zeta_{6})q^{11}+\cdots$$
648.2.i.b $$2$$ $$5.174$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-2$$ $$0$$ $$q-2\zeta_{6}q^{5}+(4-4\zeta_{6})q^{11}+2\zeta_{6}q^{13}+\cdots$$
648.2.i.c $$2$$ $$5.174$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$-3$$ $$q-\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+(5-5\zeta_{6})q^{11}+\cdots$$
648.2.i.d $$2$$ $$5.174$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-1$$ $$0$$ $$q-\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{11}+5\zeta_{6}q^{13}+\cdots$$
648.2.i.e $$2$$ $$5.174$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$-3$$ $$q+\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+(-5+5\zeta_{6})q^{11}+\cdots$$
648.2.i.f $$2$$ $$5.174$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$1$$ $$0$$ $$q+\zeta_{6}q^{5}+(4-4\zeta_{6})q^{11}+5\zeta_{6}q^{13}+\cdots$$
648.2.i.g $$2$$ $$5.174$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$2$$ $$0$$ $$q+2\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{11}+2\zeta_{6}q^{13}+\cdots$$
648.2.i.h $$2$$ $$5.174$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$4$$ $$3$$ $$q+4\zeta_{6}q^{5}+(3-3\zeta_{6})q^{7}+(4-4\zeta_{6})q^{11}+\cdots$$
648.2.i.i $$4$$ $$5.174$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$-4$$ $$0$$ $$q+(-2\zeta_{12}+\zeta_{12}^{2})q^{5}+(-2\zeta_{12}^{2}+\cdots)q^{7}+\cdots$$
648.2.i.j $$4$$ $$5.174$$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+(2-2\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{5}-2\zeta_{12}^{2}q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(648, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(648, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(54, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(72, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(81, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(108, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(162, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(216, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(324, [\chi])$$$$^{\oplus 2}$$