# Properties

 Label 720.2.q Level $720$ Weight $2$ Character orbit 720.q Rep. character $\chi_{720}(241,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $48$ Newform subspaces $12$ Sturm bound $288$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$720 = 2^{4} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 720.q (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$9$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$12$$ Sturm bound: $$288$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$7$$, $$11$$

## Dimensions

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

Total New Old
Modular forms 312 48 264
Cusp forms 264 48 216
Eisenstein series 48 0 48

## Trace form

 $$48 q + 4 q^{9} + O(q^{10})$$ $$48 q + 4 q^{9} + 8 q^{17} + 4 q^{21} + 12 q^{23} - 24 q^{25} + 12 q^{27} - 4 q^{29} - 12 q^{31} + 12 q^{33} + 24 q^{35} + 36 q^{39} + 4 q^{41} - 12 q^{43} - 8 q^{45} + 20 q^{47} - 24 q^{49} + 8 q^{51} - 12 q^{57} - 12 q^{59} - 40 q^{63} - 28 q^{69} - 24 q^{71} + 24 q^{73} - 16 q^{77} - 8 q^{81} - 56 q^{83} - 60 q^{87} - 24 q^{89} + 24 q^{91} - 8 q^{93} + 16 q^{95} - 12 q^{97} + 76 q^{99} + O(q^{100})$$

## 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.q.a $2$ $5.749$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$-1$$ $$-1$$ $$q+(-2+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-1+\zeta_{6})q^{7}+\cdots$$
720.2.q.b $2$ $5.749$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$1$$ $$-1$$ $$q+(-2+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-1+\zeta_{6})q^{7}+\cdots$$
720.2.q.c $2$ $5.749$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$-1$$ $$-4$$ $$q+(1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{7}+\cdots$$
720.2.q.d $2$ $5.749$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$1$$ $$-3$$ $$q+(2-\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+\cdots$$
720.2.q.e $2$ $5.749$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$1$$ $$0$$ $$q+(1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+3\zeta_{6}q^{9}+(-5+\cdots)q^{11}+\cdots$$
720.2.q.f $4$ $5.749$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$0$$ $$-2$$ $$2$$ $$1$$ $$q+(-\beta _{1}+\beta _{3})q^{3}+(1-\beta _{2})q^{5}+(-1+\cdots)q^{7}+\cdots$$
720.2.q.g $4$ $5.749$ $$\Q(\sqrt{-2}, \sqrt{-3})$$ None $$0$$ $$-2$$ $$2$$ $$2$$ $$q+(\beta _{1}-\beta _{2}-\beta _{3})q^{3}+\beta _{2}q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots$$
720.2.q.h $4$ $5.749$ $$\Q(\zeta_{12})$$ None $$0$$ $$0$$ $$2$$ $$4$$ $$q-\zeta_{12}^{2}q^{3}+\zeta_{12}q^{5}+(2-2\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{7}+\cdots$$
720.2.q.i $6$ $5.749$ 6.0.954288.1 None $$0$$ $$-1$$ $$-3$$ $$5$$ $$q-\beta _{4}q^{3}+\beta _{3}q^{5}+(2-\beta _{1}+\beta _{2}+2\beta _{3}+\cdots)q^{7}+\cdots$$
720.2.q.j $6$ $5.749$ 6.0.954288.1 None $$0$$ $$1$$ $$-3$$ $$-5$$ $$q+(\beta _{2}-\beta _{5})q^{3}+\beta _{3}q^{5}+(-2-2\beta _{3}+\cdots)q^{7}+\cdots$$
720.2.q.k $6$ $5.749$ 6.0.954288.1 None $$0$$ $$1$$ $$3$$ $$3$$ $$q-\beta _{4}q^{3}+(1+\beta _{2})q^{5}+(-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots$$
720.2.q.l $8$ $5.749$ 8.0.856615824.2 None $$0$$ $$0$$ $$-4$$ $$-1$$ $$q+\beta _{4}q^{3}-\beta _{1}q^{5}+(\beta _{2}-\beta _{4})q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(720, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(18, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(36, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(45, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(72, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(90, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(144, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(180, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(360, [\chi])$$$$^{\oplus 2}$$