# Properties

 Label 800.2.a Level $800$ Weight $2$ Character orbit 800.a Rep. character $\chi_{800}(1,\cdot)$ Character field $\Q$ Dimension $19$ 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.a (trivial) Character field: $$\Q$$ Newform subspaces: $$14$$ Sturm bound: $$240$$ Trace bound: $$13$$ Distinguishing $$T_p$$: $$3$$, $$11$$, $$13$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(800))$$.

Total New Old
Modular forms 144 19 125
Cusp forms 97 19 78
Eisenstein series 47 0 47

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$$2$$$$5$$FrickeDim.
$$+$$$$+$$$$+$$$$3$$
$$+$$$$-$$$$-$$$$6$$
$$-$$$$+$$$$-$$$$6$$
$$-$$$$-$$$$+$$$$4$$
Plus space$$+$$$$7$$
Minus space$$-$$$$12$$

## Trace form

 $$19 q + 15 q^{9} + O(q^{10})$$ $$19 q + 15 q^{9} + 10 q^{13} - 10 q^{17} + 16 q^{21} + 10 q^{29} + 16 q^{33} + 18 q^{37} + 14 q^{41} + 43 q^{49} - 30 q^{53} + 32 q^{57} + 10 q^{61} - 32 q^{69} - 2 q^{73} - 48 q^{77} + 11 q^{81} - 34 q^{89} - 16 q^{93} - 42 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(800))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 5
800.2.a.a $1$ $6.388$ $$\Q$$ None $$0$$ $$-2$$ $$0$$ $$-2$$ $+$ $+$ $$q-2q^{3}-2q^{7}+q^{9}+4q^{11}+6q^{13}+\cdots$$
800.2.a.b $1$ $6.388$ $$\Q$$ None $$0$$ $$-1$$ $$0$$ $$2$$ $-$ $-$ $$q-q^{3}+2q^{7}-2q^{9}-5q^{11}+5q^{17}+\cdots$$
800.2.a.c $1$ $6.388$ $$\Q$$ None $$0$$ $$-1$$ $$0$$ $$2$$ $-$ $+$ $$q-q^{3}+2q^{7}-2q^{9}+5q^{11}-5q^{17}+\cdots$$
800.2.a.d $1$ $6.388$ $$\Q$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $+$ $+$ $$q-3q^{9}-6q^{13}-2q^{17}-10q^{29}+\cdots$$
800.2.a.e $1$ $6.388$ $$\Q$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $-$ $-$ $$q-3q^{9}-4q^{13}-8q^{17}+10q^{29}+\cdots$$
800.2.a.f $1$ $6.388$ $$\Q$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$0$$ $$0$$ $+$ $-$ $$q-3q^{9}+4q^{13}+8q^{17}+10q^{29}+\cdots$$
800.2.a.g $1$ $6.388$ $$\Q$$ None $$0$$ $$1$$ $$0$$ $$-2$$ $+$ $+$ $$q+q^{3}-2q^{7}-2q^{9}-5q^{11}-5q^{17}+\cdots$$
800.2.a.h $1$ $6.388$ $$\Q$$ None $$0$$ $$1$$ $$0$$ $$-2$$ $+$ $-$ $$q+q^{3}-2q^{7}-2q^{9}+5q^{11}+5q^{17}+\cdots$$
800.2.a.i $1$ $6.388$ $$\Q$$ None $$0$$ $$2$$ $$0$$ $$2$$ $-$ $+$ $$q+2q^{3}+2q^{7}+q^{9}-4q^{11}+6q^{13}+\cdots$$
800.2.a.j $2$ $6.388$ $$\Q(\sqrt{5})$$ $$\Q(\sqrt{-5})$$ $$0$$ $$-2$$ $$0$$ $$-6$$ $-$ $-$ $$q+(-1-\beta )q^{3}+(-3+\beta )q^{7}+(3+2\beta )q^{9}+\cdots$$
800.2.a.k $2$ $6.388$ $$\Q(\sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $-$ $+$ $$q-\beta q^{3}-2\beta q^{7}+2q^{9}+\beta q^{11}-4q^{13}+\cdots$$
800.2.a.l $2$ $6.388$ $$\Q(\sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $+$ $-$ $$q-\beta q^{3}-2\beta q^{7}+2q^{9}-\beta q^{11}+4q^{13}+\cdots$$
800.2.a.m $2$ $6.388$ $$\Q(\sqrt{2})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $-$ $+$ $$q+\beta q^{3}-\beta q^{7}+5q^{9}+2\beta q^{11}+2q^{13}+\cdots$$
800.2.a.n $2$ $6.388$ $$\Q(\sqrt{5})$$ $$\Q(\sqrt{-5})$$ $$0$$ $$2$$ $$0$$ $$6$$ $+$ $-$ $$q+(1+\beta )q^{3}+(3-\beta )q^{7}+(3+2\beta )q^{9}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(800))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(800)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(20))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(32))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(40))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(50))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(80))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(100))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(160))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(200))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(400))$$$$^{\oplus 2}$$