# Properties

 Label 800.3.b Level $800$ Weight $3$ Character orbit 800.b Rep. character $\chi_{800}(351,\cdot)$ Character field $\Q$ Dimension $38$ Newform subspaces $9$ Sturm bound $360$ Trace bound $17$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$800 = 2^{5} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 800.b (of order $$2$$ and degree $$1$$) Character conductor: $$\operatorname{cond}(\chi)$$ $$=$$ $$4$$ Character field: $$\Q$$ Newform subspaces: $$9$$ Sturm bound: $$360$$ Trace bound: $$17$$ Distinguishing $$T_p$$: $$3$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{3}(800, [\chi])$$.

Total New Old
Modular forms 264 38 226
Cusp forms 216 38 178
Eisenstein series 48 0 48

## Trace form

 $$38 q - 122 q^{9} + O(q^{10})$$ $$38 q - 122 q^{9} - 4 q^{13} + 12 q^{17} + 32 q^{21} + 28 q^{29} - 64 q^{33} - 100 q^{37} - 116 q^{41} - 218 q^{49} - 100 q^{53} + 160 q^{57} - 36 q^{61} + 384 q^{69} - 20 q^{73} + 32 q^{77} + 358 q^{81} - 116 q^{89} - 224 q^{93} - 340 q^{97} + O(q^{100})$$

## Decomposition of $$S_{3}^{\mathrm{new}}(800, [\chi])$$ into newform subspaces

Label Dim $A$ Field CM Traces $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
800.3.b.a $2$ $21.798$ $$\Q(\sqrt{-1})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+iq^{3}-2iq^{7}-7q^{9}+iq^{11}+14q^{13}+\cdots$$
800.3.b.b $4$ $21.798$ $$\Q(i, \sqrt{11})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{3}+2\beta _{2}q^{7}+(-6-\beta _{1})q^{9}+\cdots$$
800.3.b.c $4$ $21.798$ $$\Q(i, \sqrt{11})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{2}q^{3}+2\beta _{2}q^{7}+(-6-\beta _{1})q^{9}+\cdots$$
800.3.b.d $4$ $21.798$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{1}q^{3}+(-\beta _{1}+4\beta _{2})q^{7}+(-5+3\beta _{3})q^{9}+\cdots$$
800.3.b.e $4$ $21.798$ $$\Q(i, \sqrt{11})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q-\beta _{3}q^{3}+(-\beta _{1}-2\beta _{3})q^{7}-2q^{9}+\cdots$$
800.3.b.f $4$ $21.798$ $$\Q(i, \sqrt{11})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{3}q^{3}+(\beta _{1}+2\beta _{3})q^{7}-2q^{9}+(2\beta _{1}+\cdots)q^{11}+\cdots$$
800.3.b.g $4$ $21.798$ $$\Q(i, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}+\beta _{1}q^{7}+(3+\beta _{3})q^{9}+(-5\beta _{1}+\cdots)q^{11}+\cdots$$
800.3.b.h $6$ $21.798$ 6.0.1827904.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}-\beta _{5}q^{7}+(-3-\beta _{2}+\beta _{4}+\cdots)q^{9}+\cdots$$
800.3.b.i $6$ $21.798$ 6.0.1827904.1 None $$0$$ $$0$$ $$0$$ $$0$$ $$q+\beta _{1}q^{3}-\beta _{5}q^{7}+(-3-\beta _{2}+\beta _{4}+\cdots)q^{9}+\cdots$$

## Decomposition of $$S_{3}^{\mathrm{old}}(800, [\chi])$$ into lower level spaces

$$S_{3}^{\mathrm{old}}(800, [\chi]) \cong$$ $$S_{3}^{\mathrm{new}}(16, [\chi])$$$$^{\oplus 6}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(20, [\chi])$$$$^{\oplus 8}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(32, [\chi])$$$$^{\oplus 3}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(80, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(100, [\chi])$$$$^{\oplus 4}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(160, [\chi])$$$$^{\oplus 2}$$$$\oplus$$$$S_{3}^{\mathrm{new}}(400, [\chi])$$$$^{\oplus 2}$$