# Properties

 Label 700.2.c Level $700$ Weight $2$ Character orbit 700.c Rep. character $\chi_{700}(699,\cdot)$ Character field $\Q$ Dimension $68$ Newform subspaces $11$ Sturm bound $240$ Trace bound $12$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$700 = 2^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 700.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$140$$ Character field: $$\Q$$ Newform subspaces: $$11$$ Sturm bound: $$240$$ Trace bound: $$12$$ Distinguishing $$T_p$$: $$3$$, $$11$$, $$13$$, $$19$$, $$23$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(700, [\chi])$$.

Total New Old
Modular forms 132 76 56
Cusp forms 108 68 40
Eisenstein series 24 8 16

## Trace form

 $$68 q + 4 q^{4} - 52 q^{9} + O(q^{10})$$ $$68 q + 4 q^{4} - 52 q^{9} + 12 q^{14} - 20 q^{16} - 24 q^{21} - 8 q^{29} - 12 q^{36} - 22 q^{44} + 50 q^{46} - 12 q^{49} + 50 q^{56} + 46 q^{64} - 82 q^{74} - 60 q^{81} - 56 q^{84} + 34 q^{86} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(700, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
700.2.c.a $4$ $5.590$ $$\Q(i, \sqrt{6})$$ None $$-4$$ $$0$$ $$0$$ $$-4$$ $$q+(-1+\beta _{1})q^{2}+\beta _{2}q^{3}-2\beta _{1}q^{4}+\cdots$$
700.2.c.b $4$ $5.590$ $$\Q(\zeta_{12})$$ None $$-2$$ $$0$$ $$0$$ $$-8$$ $$q+(-1-\zeta_{12}^{3})q^{2}+(\zeta_{12}^{2}-\zeta_{12}^{3})q^{3}+\cdots$$
700.2.c.c $4$ $5.590$ $$\Q(\zeta_{12})$$ None $$-2$$ $$0$$ $$0$$ $$-8$$ $$q+(-1-\zeta_{12}^{3})q^{2}+(-\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\cdots$$
700.2.c.d $4$ $5.590$ $$\Q(i, \sqrt{7})$$ $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-2\beta _{1}+\beta _{3})q^{7}+\cdots$$
700.2.c.e $4$ $5.590$ $$\Q(\zeta_{12})$$ None $$2$$ $$0$$ $$0$$ $$8$$ $$q+(1+\zeta_{12}^{3})q^{2}+(-\zeta_{12}^{2}+\zeta_{12}^{3})q^{3}+\cdots$$
700.2.c.f $4$ $5.590$ $$\Q(\zeta_{12})$$ None $$2$$ $$0$$ $$0$$ $$8$$ $$q+(1+\zeta_{12}^{3})q^{2}+(\zeta_{12}^{2}-\zeta_{12}^{3})q^{3}+\cdots$$
700.2.c.g $4$ $5.590$ $$\Q(i, \sqrt{6})$$ None $$4$$ $$0$$ $$0$$ $$4$$ $$q+(1-\beta _{1})q^{2}+\beta _{2}q^{3}-2\beta _{1}q^{4}+(\beta _{2}+\cdots)q^{6}+\cdots$$
700.2.c.h $8$ $5.590$ 8.0.49787136.1 $$\Q(\sqrt{-7})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{2}+(-1+\beta _{2})q^{4}+(\beta _{1}-\beta _{7})q^{7}+\cdots$$
700.2.c.i $8$ $5.590$ 8.0.342102016.5 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{5}q^{2}+(\beta _{3}+\beta _{4})q^{3}+(\beta _{3}+\beta _{4}+\beta _{7})q^{4}+\cdots$$
700.2.c.j $8$ $5.590$ 8.0.342102016.5 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{5}q^{2}+(-\beta _{3}-\beta _{4})q^{3}+(\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots$$
700.2.c.k $16$ $5.590$ 16.0.$$\cdots$$.7 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{7}q^{2}+(-\beta _{5}+\beta _{6}-\beta _{7}+\beta _{8}-\beta _{9}+\cdots)q^{3}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(700, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(700, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(140, [\chi])$$$$^{\oplus 2}$$