# Properties

 Label 768.2.a Level $768$ Weight $2$ Character orbit 768.a Rep. character $\chi_{768}(1,\cdot)$ Character field $\Q$ Dimension $16$ Newform subspaces $12$ Sturm bound $256$ Trace bound $11$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$768 = 2^{8} \cdot 3$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 768.a (trivial) Character field: $$\Q$$ Newform subspaces: $$12$$ Sturm bound: $$256$$ Trace bound: $$11$$ Distinguishing $$T_p$$: $$5$$, $$7$$, $$11$$, $$19$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{2}(\Gamma_0(768))$$.

Total New Old
Modular forms 152 16 136
Cusp forms 105 16 89
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.

$$2$$$$3$$FrickeDim
$$+$$$$+$$$+$$$4$$
$$+$$$$-$$$-$$$6$$
$$-$$$$+$$$-$$$4$$
$$-$$$$-$$$+$$$2$$
Plus space$$+$$$$6$$
Minus space$$-$$$$10$$

## Trace form

 $$16 q + 16 q^{9} + O(q^{10})$$ $$16 q + 16 q^{9} + 16 q^{25} + 48 q^{49} + 32 q^{57} - 32 q^{65} + 16 q^{81} - 32 q^{89} + 32 q^{97} + O(q^{100})$$

## Decomposition of $$S_{2}^{\mathrm{new}}(\Gamma_0(768))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2 3
768.2.a.a $1$ $6.133$ $$\Q$$ None $$0$$ $$-1$$ $$-2$$ $$2$$ $+$ $+$ $$q-q^{3}-2q^{5}+2q^{7}+q^{9}-4q^{13}+\cdots$$
768.2.a.b $1$ $6.133$ $$\Q$$ None $$0$$ $$-1$$ $$0$$ $$-4$$ $+$ $+$ $$q-q^{3}-4q^{7}+q^{9}+4q^{11}+4q^{13}+\cdots$$
768.2.a.c $1$ $6.133$ $$\Q$$ None $$0$$ $$-1$$ $$0$$ $$4$$ $-$ $+$ $$q-q^{3}+4q^{7}+q^{9}+4q^{11}-4q^{13}+\cdots$$
768.2.a.d $1$ $6.133$ $$\Q$$ None $$0$$ $$-1$$ $$2$$ $$-2$$ $-$ $+$ $$q-q^{3}+2q^{5}-2q^{7}+q^{9}+4q^{13}+\cdots$$
768.2.a.e $1$ $6.133$ $$\Q$$ None $$0$$ $$1$$ $$-2$$ $$-2$$ $-$ $-$ $$q+q^{3}-2q^{5}-2q^{7}+q^{9}-4q^{13}+\cdots$$
768.2.a.f $1$ $6.133$ $$\Q$$ None $$0$$ $$1$$ $$0$$ $$-4$$ $-$ $-$ $$q+q^{3}-4q^{7}+q^{9}-4q^{11}-4q^{13}+\cdots$$
768.2.a.g $1$ $6.133$ $$\Q$$ None $$0$$ $$1$$ $$0$$ $$4$$ $+$ $-$ $$q+q^{3}+4q^{7}+q^{9}-4q^{11}+4q^{13}+\cdots$$
768.2.a.h $1$ $6.133$ $$\Q$$ None $$0$$ $$1$$ $$2$$ $$2$$ $+$ $-$ $$q+q^{3}+2q^{5}+2q^{7}+q^{9}+4q^{13}+\cdots$$
768.2.a.i $2$ $6.133$ $$\Q(\sqrt{2})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $+$ $+$ $$q-q^{3}+\beta q^{5}-\beta q^{7}+q^{9}-4q^{11}+\cdots$$
768.2.a.j $2$ $6.133$ $$\Q(\sqrt{3})$$ None $$0$$ $$-2$$ $$0$$ $$0$$ $-$ $+$ $$q-q^{3}+\beta q^{5}+\beta q^{7}+q^{9}-\beta q^{15}+\cdots$$
768.2.a.k $2$ $6.133$ $$\Q(\sqrt{3})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $+$ $-$ $$q+q^{3}+\beta q^{5}-\beta q^{7}+q^{9}+\beta q^{15}+\cdots$$
768.2.a.l $2$ $6.133$ $$\Q(\sqrt{2})$$ None $$0$$ $$2$$ $$0$$ $$0$$ $+$ $-$ $$q+q^{3}+\beta q^{5}+\beta q^{7}+q^{9}+4q^{11}+\cdots$$

## Decomposition of $$S_{2}^{\mathrm{old}}(\Gamma_0(768))$$ into lower level spaces

$$S_{2}^{\mathrm{old}}(\Gamma_0(768)) \cong$$ $$S_{2}^{\mathrm{new}}(\Gamma_0(24))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(32))$$$$^{\oplus 8}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(48))$$$$^{\oplus 5}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(64))$$$$^{\oplus 6}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(96))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(128))$$$$^{\oplus 4}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(192))$$$$^{\oplus 3}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(256))$$$$^{\oplus 2}$$$$\oplus$$$$S_{2}^{\mathrm{new}}(\Gamma_0(384))$$$$^{\oplus 2}$$