# Properties

 Label 800.2.o Level $800$ Weight $2$ Character orbit 800.o Rep. character $\chi_{800}(143,\cdot)$ Character field $\Q(\zeta_{4})$ Dimension $32$ Newform subspaces $8$ Sturm bound $240$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 800.o (of order $$4$$ and degree $$2$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$40$$ Character field: $$\Q(i)$$ Newform subspaces: $$8$$ Sturm bound: $$240$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$3$$, $$7$$

## Dimensions

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

Total New Old
Modular forms 288 40 248
Cusp forms 192 32 160
Eisenstein series 96 8 88

## Trace form

 $$32q - 4q^{3} + O(q^{10})$$ $$32q - 4q^{3} + 8q^{11} + 8q^{17} + 8q^{27} + 16q^{33} - 8q^{41} + 28q^{43} + 40q^{51} - 8q^{57} - 28q^{67} - 16q^{73} - 40q^{81} - 44q^{83} - 56q^{91} - 16q^{97} + O(q^{100})$$

## 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.o.a $$2$$ $$6.388$$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$-4$$ $$0$$ $$0$$ $$q+(-2+2i)q^{3}-5iq^{9}+6q^{11}+(4+\cdots)q^{17}+\cdots$$
800.2.o.b $$2$$ $$6.388$$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-10})$$ $$0$$ $$0$$ $$0$$ $$-4$$ $$q+(-2+2i)q^{7}+3iq^{9}-2q^{11}+(-4+\cdots)q^{13}+\cdots$$
800.2.o.c $$2$$ $$6.388$$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-10})$$ $$0$$ $$0$$ $$0$$ $$4$$ $$q+(2-2i)q^{7}+3iq^{9}-2q^{11}+(4+4i)q^{13}+\cdots$$
800.2.o.d $$2$$ $$6.388$$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$4$$ $$0$$ $$0$$ $$q+(2-2i)q^{3}-5iq^{9}+6q^{11}+(-4+\cdots)q^{17}+\cdots$$
800.2.o.e $$4$$ $$6.388$$ $$\Q(i, \sqrt{6})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$-4$$ $$0$$ $$0$$ $$q+(-1+\beta _{1}-\beta _{2})q^{3}+(-2\beta _{1}+2\beta _{2}+\cdots)q^{9}+\cdots$$
800.2.o.f $$4$$ $$6.388$$ $$\Q(i, \sqrt{6})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$4$$ $$0$$ $$0$$ $$q+(1+\beta _{1}+\beta _{2})q^{3}+(2\beta _{1}+2\beta _{2}+2\beta _{3})q^{9}+\cdots$$
800.2.o.g $$8$$ $$6.388$$ $$\Q(\zeta_{20})$$ None $$0$$ $$-4$$ $$0$$ $$0$$ $$q+(-1+\zeta_{20}^{2})q^{3}-\zeta_{20}^{7}q^{7}+(1+\zeta_{20}+\cdots)q^{9}+\cdots$$
800.2.o.h $$8$$ $$6.388$$ 8.0.3317760000.5 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{3}+\beta _{6}q^{7}+q^{11}-\beta _{4}q^{13}+\cdots$$

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

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