# Properties

 Label 800.2.n Level $800$ Weight $2$ Character orbit 800.n Rep. character $\chi_{800}(543,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $36$ Newform subspaces $14$ Sturm bound $240$ Trace bound $13$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 800.n (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$20$$ Character field: $$\Q(i)$$ Newform subspaces: $$14$$ Sturm bound: $$240$$ Trace bound: $$13$$ Distinguishing $$T_p$$: $$3$$, $$7$$, $$11$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(800, [\chi])$$.

Total New Old
Modular forms 288 36 252
Cusp forms 192 36 156
Eisenstein series 96 0 96

## Trace form

 $$36 q + O(q^{10})$$ $$36 q - 4 q^{13} + 12 q^{17} - 32 q^{21} + 16 q^{33} + 20 q^{37} + 32 q^{41} + 52 q^{53} - 32 q^{57} - 28 q^{73} - 48 q^{77} - 132 q^{81} - 80 q^{93} + 36 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(800, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
800.2.n.a $2$ $6.388$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-4$$ $$0$$ $$-4$$ $$q+(-2-2i)q^{3}+(-2+2i)q^{7}+5iq^{9}+\cdots$$
800.2.n.b $2$ $6.388$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$-2$$ $$q+(-1-i)q^{3}+(-1+i)q^{7}-iq^{9}+\cdots$$
800.2.n.c $2$ $6.388$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$-2$$ $$q+(-1-i)q^{3}+(-1+i)q^{7}-iq^{9}+\cdots$$
800.2.n.d $2$ $6.388$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$-2$$ $$q+(-1-i)q^{3}+(-1+i)q^{7}-iq^{9}+\cdots$$
800.2.n.e $2$ $6.388$ $$\Q(\sqrt{-1})$$ None $$0$$ $$-2$$ $$0$$ $$6$$ $$q+(-1-i)q^{3}+(3-3i)q^{7}-iq^{9}+\cdots$$
800.2.n.f $2$ $6.388$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$0$$ $$-6$$ $$q+(1+i)q^{3}+(-3+3i)q^{7}-iq^{9}+\cdots$$
800.2.n.g $2$ $6.388$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$0$$ $$2$$ $$q+(1+i)q^{3}+(1-i)q^{7}-iq^{9}+4iq^{11}+\cdots$$
800.2.n.h $2$ $6.388$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$0$$ $$2$$ $$q+(1+i)q^{3}+(1-i)q^{7}-iq^{9}-6iq^{11}+\cdots$$
800.2.n.i $2$ $6.388$ $$\Q(\sqrt{-1})$$ None $$0$$ $$2$$ $$0$$ $$2$$ $$q+(1+i)q^{3}+(1-i)q^{7}-iq^{9}-4iq^{11}+\cdots$$
800.2.n.j $2$ $6.388$ $$\Q(\sqrt{-1})$$ None $$0$$ $$4$$ $$0$$ $$4$$ $$q+(2+2i)q^{3}+(2-2i)q^{7}+5iq^{9}+\cdots$$
800.2.n.k $4$ $6.388$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$-4$$ $$0$$ $$8$$ $$q+(-1+\beta _{1}-\beta _{2})q^{3}+(2-2\beta _{2})q^{7}+\cdots$$
800.2.n.l $4$ $6.388$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$-4$$ $$0$$ $$8$$ $$q+(-1+\beta _{1}-\beta _{2})q^{3}+(2-2\beta _{2})q^{7}+\cdots$$
800.2.n.m $4$ $6.388$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$4$$ $$0$$ $$-8$$ $$q+(1+\beta _{1}+\beta _{2})q^{3}+(-2+2\beta _{2})q^{7}+\cdots$$
800.2.n.n $4$ $6.388$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$4$$ $$0$$ $$-8$$ $$q+(1+\beta _{1}+\beta _{2})q^{3}+(-2+2\beta _{2})q^{7}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(800, [\chi])$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(800, [\chi]) \simeq$$ $$S_{2}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(160, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(400, [\chi])$$$$^{\oplus 2}$$