# Properties

 Label 1680.2.f Level $1680$ Weight $2$ Character orbit 1680.f Rep. character $\chi_{1680}(881,\cdot)$ Character field $\Q$ Dimension $64$ Newform subspaces $12$ 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.f (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$21$$ Character field: $$\Q$$ Newform subspaces: $$12$$ Sturm bound: $$768$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$11$$, $$17$$, $$41$$

## Dimensions

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

Total New Old
Modular forms 408 64 344
Cusp forms 360 64 296
Eisenstein series 48 0 48

## Trace form

 $$64q - 4q^{7} + O(q^{10})$$ $$64q - 4q^{7} + 4q^{21} + 64q^{25} + 12q^{39} - 56q^{43} + 8q^{49} + 28q^{51} + 20q^{63} + 40q^{67} - 24q^{79} + 8q^{81} - 32q^{93} + 84q^{99} + O(q^{100})$$

## 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.f.a $$2$$ $$13.415$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$-3$$ $$2$$ $$-4$$ $$q+(-2+\zeta_{6})q^{3}+q^{5}+(-1-2\zeta_{6})q^{7}+\cdots$$
1680.2.f.b $$2$$ $$13.415$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$-2$$ $$-4$$ $$q+\zeta_{6}q^{3}-q^{5}+(-2-\zeta_{6})q^{7}-3q^{9}+\cdots$$
1680.2.f.c $$2$$ $$13.415$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$0$$ $$2$$ $$-4$$ $$q-\zeta_{6}q^{3}+q^{5}+(-2+\zeta_{6})q^{7}-3q^{9}+\cdots$$
1680.2.f.d $$2$$ $$13.415$$ $$\Q(\sqrt{-3})$$ None $$0$$ $$3$$ $$-2$$ $$-4$$ $$q+(2-\zeta_{6})q^{3}-q^{5}+(-3+2\zeta_{6})q^{7}+\cdots$$
1680.2.f.e $$4$$ $$13.415$$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$-2$$ $$-4$$ $$-6$$ $$q+(\beta _{1}+\beta _{2})q^{3}-q^{5}+(-1+\beta _{1}+2\beta _{2}+\cdots)q^{7}+\cdots$$
1680.2.f.f $$4$$ $$13.415$$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$-2$$ $$-4$$ $$6$$ $$q+(-1-\beta _{1})q^{3}-q^{5}+(1-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots$$
1680.2.f.g $$4$$ $$13.415$$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$0$$ $$-1$$ $$4$$ $$8$$ $$q-\beta _{1}q^{3}+q^{5}+(2+\beta _{2})q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots$$
1680.2.f.h $$4$$ $$13.415$$ $$\Q(\sqrt{-3}, \sqrt{-11})$$ None $$0$$ $$1$$ $$-4$$ $$8$$ $$q+\beta _{1}q^{3}-q^{5}+(2-\beta _{2})q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots$$
1680.2.f.i $$4$$ $$13.415$$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$2$$ $$4$$ $$-6$$ $$q+(-\beta _{1}-\beta _{2})q^{3}+q^{5}+(-1-2\beta _{2}+\cdots)q^{7}+\cdots$$
1680.2.f.j $$4$$ $$13.415$$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$2$$ $$4$$ $$6$$ $$q+(1+\beta _{1})q^{3}+q^{5}+(1+\beta _{2}+\beta _{3})q^{7}+\cdots$$
1680.2.f.k $$16$$ $$13.415$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$-16$$ $$-2$$ $$q+\beta _{3}q^{3}-q^{5}-\beta _{9}q^{7}+\beta _{7}q^{9}+\beta _{8}q^{11}+\cdots$$
1680.2.f.l $$16$$ $$13.415$$ $$\mathbb{Q}[x]/(x^{16} - \cdots)$$ None $$0$$ $$0$$ $$16$$ $$-2$$ $$q-\beta _{3}q^{3}+q^{5}-\beta _{4}q^{7}+\beta _{7}q^{9}+\beta _{8}q^{11}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(1680, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(42, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(84, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(105, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(168, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(210, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(336, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(420, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(840, [\chi])$$$$^{\oplus 2}$$