# Properties

 Label 720.2.x Level $720$ Weight $2$ Character orbit 720.x Rep. character $\chi_{720}(127,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $30$ Newform subspaces $7$ Sturm bound $288$ Trace bound $5$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 720.x (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$20$$ Character field: $$\Q(i)$$ Newform subspaces: $$7$$ Sturm bound: $$288$$ Trace bound: $$5$$ Distinguishing $$T_p$$: $$7$$, $$11$$, $$13$$, $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(720, [\chi])$$.

Total New Old
Modular forms 336 30 306
Cusp forms 240 30 210
Eisenstein series 96 0 96

## Trace form

 $$30 q + O(q^{10})$$ $$30 q - 6 q^{13} - 6 q^{17} - 6 q^{25} + 6 q^{37} + 24 q^{41} + 30 q^{53} + 54 q^{65} + 30 q^{73} + 24 q^{77} + 18 q^{85} + 30 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(720, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
720.2.x.a $2$ $5.749$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$-4$$ $$0$$ $$q+(-2-i)q^{5}+(-5+5i)q^{13}+(5+\cdots)q^{17}+\cdots$$
720.2.x.b $2$ $5.749$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$-2$$ $$0$$ $$q+(-1-2i)q^{5}+(1-i)q^{13}+(-5+\cdots)q^{17}+\cdots$$
720.2.x.c $2$ $5.749$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$2$$ $$0$$ $$q+(1+2i)q^{5}+(1-i)q^{13}+(5+5i)q^{17}+\cdots$$
720.2.x.d $4$ $5.749$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$4$$ $$0$$ $$q+(1+2\zeta_{12})q^{5}-\zeta_{12}^{3}q^{7}+(\zeta_{12}^{2}+\cdots)q^{11}+\cdots$$
720.2.x.e $4$ $5.749$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$8$$ $$0$$ $$q+(2+\zeta_{8}^{2})q^{5}+\zeta_{8}q^{7}+(-\zeta_{8}-\zeta_{8}^{3})q^{11}+\cdots$$
720.2.x.f $8$ $5.749$ $$\Q(\zeta_{24})$$ None $$0$$ $$0$$ $$-8$$ $$0$$ $$q+(-1-\zeta_{24}^{2}-\zeta_{24}^{3})q^{5}+(\zeta_{24}^{4}+\cdots)q^{7}+\cdots$$
720.2.x.g $8$ $5.749$ 8.0.3317760000.2 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{5}+\beta _{6}q^{7}-\beta _{7}q^{11}+(-2+2\beta _{2}+\cdots)q^{13}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(720, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(720, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 9}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(60, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(120, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(180, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(240, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(360, [\chi])$$$$^{\oplus 2}$$