# Properties

 Label 800.4.c Level $800$ Weight $4$ Character orbit 800.c Rep. character $\chi_{800}(449,\cdot)$ Character field $\Q$ Dimension $54$ Newform subspaces $15$ Sturm bound $480$ Trace bound $11$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 800.c (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$5$$ Character field: $$\Q$$ Newform subspaces: $$15$$ Sturm bound: $$480$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$3$$, $$11$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(800, [\chi])$$.

Total New Old
Modular forms 384 54 330
Cusp forms 336 54 282
Eisenstein series 48 0 48

## Trace form

 $$54 q - 486 q^{9} + O(q^{10})$$ $$54 q - 486 q^{9} - 272 q^{21} + 172 q^{29} + 116 q^{41} - 1622 q^{49} - 780 q^{61} - 3520 q^{69} + 7462 q^{81} - 4388 q^{89} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(800, [\chi])$$ into newform subspaces

Label Dim. $$A$$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
800.4.c.a $2$ $47.202$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4iq^{3}-8iq^{7}-37q^{9}-40q^{11}+\cdots$$
800.4.c.b $2$ $47.202$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+4iq^{3}-8iq^{7}-37q^{9}+40q^{11}+\cdots$$
800.4.c.c $2$ $47.202$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+5iq^{3}+10iq^{7}+2q^{9}-15q^{11}+\cdots$$
800.4.c.d $2$ $47.202$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+5iq^{3}+10iq^{7}+2q^{9}+15q^{11}+\cdots$$
800.4.c.e $2$ $47.202$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+3iq^{7}+23q^{9}-60q^{11}+\cdots$$
800.4.c.f $2$ $47.202$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}+3iq^{7}+23q^{9}+60q^{11}+\cdots$$
800.4.c.g $2$ $47.202$ $$\Q(\sqrt{-1})$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $$q+3^{3}q^{9}-9iq^{13}+47iq^{17}+130q^{29}+\cdots$$
800.4.c.h $4$ $47.202$ $$\Q(i, \sqrt{13})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{3}+\beta _{2}q^{7}-5^{2}q^{9}+3\beta _{3}q^{11}+\cdots$$
800.4.c.i $4$ $47.202$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(2\beta _{1}-\beta _{2})q^{3}+(-2\beta _{1}-5\beta _{2})q^{7}+\cdots$$
800.4.c.j $4$ $47.202$ $$\Q(i, \sqrt{10})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{3}+3\beta _{2}q^{7}-13q^{9}+\beta _{3}q^{11}+\cdots$$
800.4.c.k $4$ $47.202$ $$\Q(i, \sqrt{6})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(2\beta _{1}-\beta _{2})q^{3}+(-2\beta _{1}-5\beta _{2})q^{7}+\cdots$$
800.4.c.l $4$ $47.202$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{2}q^{3}+7\beta _{2}q^{7}+7q^{9}-\beta _{3}q^{11}+\cdots$$
800.4.c.m $6$ $47.202$ 6.0.2068430400.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{1}+2\beta _{2})q^{3}+(-\beta _{1}+\beta _{2}-\beta _{5})q^{7}+\cdots$$
800.4.c.n $6$ $47.202$ 6.0.2068430400.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+(-\beta _{1}+2\beta _{2})q^{3}+(-\beta _{1}+\beta _{2}-\beta _{5})q^{7}+\cdots$$
800.4.c.o $8$ $47.202$ 8.0.$$\cdots$$.12 None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{3}+(\beta _{3}+\beta _{7})q^{7}+(-24+\beta _{4}+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(800, [\chi])$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(800, [\chi]) \cong$$ $$S_{4}^{\mathrm{new}}(10, [\chi])$$$$^{\oplus 10}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(25, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(40, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(50, [\chi])$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(160, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(200, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(400, [\chi])$$$$^{\oplus 2}$$