# Properties

 Label 1680.2.bg Level $1680$ Weight $2$ Character orbit 1680.bg Rep. character $\chi_{1680}(961,\cdot)$ Character field $\Q(\zeta_{3})$ Dimension $64$ Newform subspaces $22$ Sturm bound $768$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1680.bg (of order $$3$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$7$$ Character field: $$\Q(\zeta_{3})$$ Newform subspaces: $$22$$ Sturm bound: $$768$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$11$$, $$13$$, $$17$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(1680, [\chi])$$.

Total New Old
Modular forms 816 64 752
Cusp forms 720 64 656
Eisenstein series 96 0 96

## Trace form

 $$64 q + 4 q^{3} + 4 q^{7} - 32 q^{9} + O(q^{10})$$ $$64 q + 4 q^{3} + 4 q^{7} - 32 q^{9} - 8 q^{11} - 4 q^{19} - 8 q^{23} - 32 q^{25} - 8 q^{27} - 32 q^{29} - 4 q^{31} + 16 q^{37} - 4 q^{39} - 16 q^{41} + 24 q^{43} + 24 q^{49} + 32 q^{53} + 32 q^{55} + 16 q^{57} + 16 q^{59} + 16 q^{61} + 4 q^{63} + 8 q^{65} - 12 q^{67} - 16 q^{71} - 8 q^{73} + 4 q^{75} - 16 q^{77} - 12 q^{79} - 32 q^{81} - 24 q^{87} - 16 q^{89} - 44 q^{91} - 32 q^{97} + 16 q^{99} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(1680, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
1680.2.bg.a $2$ $13.415$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$-1$$ $$-5$$ $$q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots$$
1680.2.bg.b $2$ $13.415$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$-1$$ $$-4$$ $$q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-3+2\zeta_{6})q^{7}+\cdots$$
1680.2.bg.c $2$ $13.415$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$-1$$ $$1$$ $$q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(2-3\zeta_{6})q^{7}+\cdots$$
1680.2.bg.d $2$ $13.415$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$-1$$ $$4$$ $$q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(1+2\zeta_{6})q^{7}+\cdots$$
1680.2.bg.e $2$ $13.415$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$-1$$ $$5$$ $$q+(-1+\zeta_{6})q^{3}-\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+\cdots$$
1680.2.bg.f $2$ $13.415$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$1$$ $$-5$$ $$q+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots$$
1680.2.bg.g $2$ $13.415$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$1$$ $$-4$$ $$q+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-1-2\zeta_{6})q^{7}+\cdots$$
1680.2.bg.h $2$ $13.415$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$1$$ $$-1$$ $$q+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-2+3\zeta_{6})q^{7}+\cdots$$
1680.2.bg.i $2$ $13.415$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$1$$ $$4$$ $$q+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(3-2\zeta_{6})q^{7}+\cdots$$
1680.2.bg.j $2$ $13.415$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-1$$ $$1$$ $$5$$ $$q+(-1+\zeta_{6})q^{3}+\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+\cdots$$
1680.2.bg.k $2$ $13.415$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$-1$$ $$-4$$ $$q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+(-3+2\zeta_{6})q^{7}+\cdots$$
1680.2.bg.l $2$ $13.415$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$-1$$ $$1$$ $$q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+(2-3\zeta_{6})q^{7}+\cdots$$
1680.2.bg.m $2$ $13.415$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$1$$ $$-5$$ $$q+(1-\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-2-\zeta_{6})q^{7}+\cdots$$
1680.2.bg.n $2$ $13.415$ $$\Q(\sqrt{-3})$$ None $$0$$ $$1$$ $$1$$ $$4$$ $$q+(1-\zeta_{6})q^{3}+\zeta_{6}q^{5}+(3-2\zeta_{6})q^{7}+\cdots$$
1680.2.bg.o $4$ $13.415$ $$\Q(\zeta_{12})$$ None $$0$$ $$-2$$ $$-2$$ $$0$$ $$q-\zeta_{12}^{2}q^{3}+(-1+\zeta_{12}^{2})q^{5}+(-\zeta_{12}+\cdots)q^{7}+\cdots$$
1680.2.bg.p $4$ $13.415$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$-2$$ $$2$$ $$2$$ $$q+\beta _{2}q^{3}+(1+\beta _{2})q^{5}+(1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots$$
1680.2.bg.q $4$ $13.415$ $$\Q(\sqrt{-3}, \sqrt{7})$$ None $$0$$ $$2$$ $$-2$$ $$0$$ $$q+(1+\beta _{2})q^{3}+\beta _{2}q^{5}+(\beta _{1}+\beta _{3})q^{7}+\cdots$$
1680.2.bg.r $4$ $13.415$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$2$$ $$-2$$ $$2$$ $$q+(1+\beta _{2})q^{3}+\beta _{2}q^{5}+(-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots$$
1680.2.bg.s $4$ $13.415$ $$\Q(\sqrt{-3}, \sqrt{7})$$ None $$0$$ $$2$$ $$2$$ $$0$$ $$q+(1+\beta _{2})q^{3}-\beta _{2}q^{5}+(\beta _{1}+\beta _{3})q^{7}+\cdots$$
1680.2.bg.t $4$ $13.415$ $$\Q(\sqrt{2}, \sqrt{-3})$$ None $$0$$ $$2$$ $$2$$ $$2$$ $$q+(1+\beta _{1})q^{3}-\beta _{1}q^{5}+(-\beta _{1}+\beta _{3})q^{7}+\cdots$$
1680.2.bg.u $6$ $13.415$ 6.0.38363328.2 None $$0$$ $$3$$ $$-3$$ $$2$$ $$q+(1+\beta _{3})q^{3}+\beta _{3}q^{5}+(\beta _{1}-\beta _{4})q^{7}+\cdots$$
1680.2.bg.v $6$ $13.415$ 6.0.29428272.1 None $$0$$ $$3$$ $$3$$ $$0$$ $$q-\beta _{3}q^{3}+(1+\beta _{3})q^{5}+(-\beta _{1}-\beta _{5})q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(1680, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(1680, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(21, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(28, [\chi])$$$$^{\oplus 12}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(35, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(56, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(70, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(112, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(140, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(210, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(280, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(336, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(420, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(560, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(840, [\chi])$$$$^{\oplus 2}$$