# Properties

 Label 800.2.c Level $800$ Weight $2$ Character orbit 800.c Rep. character $\chi_{800}(449,\cdot)$ Character field $\Q$ Dimension $18$ Newform subspaces $7$ Sturm bound $240$ Trace bound $11$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 800.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$7$$ Sturm bound: $$240$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$3$$, $$11$$

## Dimensions

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

Total New Old
Modular forms 144 18 126
Cusp forms 96 18 78
Eisenstein series 48 0 48

## Trace form

 $$18 q - 18 q^{9} + O(q^{10})$$ $$18 q - 18 q^{9} + 16 q^{21} - 12 q^{29} + 28 q^{41} - 18 q^{49} - 20 q^{61} - 64 q^{69} - 62 q^{81} + 20 q^{89} + O(q^{100})$$

## 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.c.a $2$ $6.388$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-iq^{7}-q^{9}-4q^{11}+3iq^{13}+\cdots$$
800.2.c.b $2$ $6.388$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-iq^{7}-q^{9}+4q^{11}-3iq^{13}+\cdots$$
800.2.c.c $2$ $6.388$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+2iq^{7}+2q^{9}-5q^{11}+5iq^{17}+\cdots$$
800.2.c.d $2$ $6.388$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+2iq^{7}+2q^{9}+5q^{11}-5iq^{17}+\cdots$$
800.2.c.e $2$ $6.388$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+3q^{9}+3iq^{13}-iq^{17}+10q^{29}+\cdots$$
800.2.c.f $4$ $6.388$ $$\Q(\zeta_{8})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\zeta_{8}^{2}q^{3}+\zeta_{8}^{2}q^{7}-5q^{9}+\zeta_{8}^{3}q^{11}+\cdots$$
800.2.c.g $4$ $6.388$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}-2\beta _{2}q^{7}-2q^{9}+\beta _{3}q^{11}+\cdots$$

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

$$S_{2}^{\mathrm{old}}(800, [\chi]) \cong$$ $$S_{2}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(160, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(400, [\chi])$$$$^{\oplus 2}$$