# Properties

 Label 256.4.a Level $256$ Weight $4$ Character orbit 256.a Rep. character $\chi_{256}(1,\cdot)$ Character field $\Q$ Dimension $22$ Newform subspaces $14$ Sturm bound $128$ Trace bound $7$

# Related objects

## Defining parameters

 Level: $$N$$ $$=$$ $$256 = 2^{8}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 256.a (trivial) Character field: $$\Q$$ Newform subspaces: $$14$$ Sturm bound: $$128$$ Trace bound: $$7$$ Distinguishing $$T_p$$: $$3$$, $$5$$, $$7$$

## Dimensions

The following table gives the dimensions of various subspaces of $$M_{4}(\Gamma_0(256))$$.

Total New Old
Modular forms 108 26 82
Cusp forms 84 22 62
Eisenstein series 24 4 20

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

$$2$$Dim
$$+$$$$12$$
$$-$$$$10$$

## Trace form

 $$22 q + 166 q^{9} + O(q^{10})$$ $$22 q + 166 q^{9} - 4 q^{17} + 354 q^{25} + 104 q^{33} + 4 q^{41} + 1926 q^{49} + 568 q^{57} - 480 q^{65} - 1452 q^{73} + 590 q^{81} + 180 q^{89} - 3172 q^{97} + O(q^{100})$$

## Decomposition of $$S_{4}^{\mathrm{new}}(\Gamma_0(256))$$ into newform subspaces

Label Dim $A$ Field CM Traces A-L signs $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2
256.4.a.a $1$ $15.104$ $$\Q$$ $$\Q(\sqrt{-2})$$ $$0$$ $$-10$$ $$0$$ $$0$$ $-$ $$q-10q^{3}+73q^{9}+18q^{11}+90q^{17}+\cdots$$
256.4.a.b $1$ $15.104$ $$\Q$$ None $$0$$ $$-8$$ $$-12$$ $$32$$ $-$ $$q-8q^{3}-12q^{5}+2^{5}q^{7}+37q^{9}+8q^{11}+\cdots$$
256.4.a.c $1$ $15.104$ $$\Q$$ None $$0$$ $$-8$$ $$12$$ $$-32$$ $+$ $$q-8q^{3}+12q^{5}-2^{5}q^{7}+37q^{9}+8q^{11}+\cdots$$
256.4.a.d $1$ $15.104$ $$\Q$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$-4$$ $$0$$ $+$ $$q-4q^{5}-3^{3}q^{9}+92q^{13}+94q^{17}+\cdots$$
256.4.a.e $1$ $15.104$ $$\Q$$ $$\Q(\sqrt{-1})$$ $$0$$ $$0$$ $$4$$ $$0$$ $-$ $$q+4q^{5}-3^{3}q^{9}-92q^{13}+94q^{17}+\cdots$$
256.4.a.f $1$ $15.104$ $$\Q$$ None $$0$$ $$8$$ $$-12$$ $$-32$$ $-$ $$q+8q^{3}-12q^{5}-2^{5}q^{7}+37q^{9}-8q^{11}+\cdots$$
256.4.a.g $1$ $15.104$ $$\Q$$ None $$0$$ $$8$$ $$12$$ $$32$$ $+$ $$q+8q^{3}+12q^{5}+2^{5}q^{7}+37q^{9}-8q^{11}+\cdots$$
256.4.a.h $1$ $15.104$ $$\Q$$ $$\Q(\sqrt{-2})$$ $$0$$ $$10$$ $$0$$ $$0$$ $+$ $$q+10q^{3}+73q^{9}-18q^{11}+90q^{17}+\cdots$$
256.4.a.i $2$ $15.104$ $$\Q(\sqrt{3})$$ None $$0$$ $$-4$$ $$0$$ $$0$$ $+$ $$q-2q^{3}-\beta q^{5}-2\beta q^{7}-23q^{9}+42q^{11}+\cdots$$
256.4.a.j $2$ $15.104$ $$\Q(\sqrt{7})$$ None $$0$$ $$0$$ $$0$$ $$-16$$ $-$ $$q+\beta q^{3}-2\beta q^{5}-8q^{7}+q^{9}+3\beta q^{11}+\cdots$$
256.4.a.k $2$ $15.104$ $$\Q(\sqrt{2})$$ $$\Q(\sqrt{-2})$$ $$0$$ $$0$$ $$0$$ $$0$$ $-$ $$q+\beta q^{3}-19q^{9}-5^{2}\beta q^{11}-90q^{17}+\cdots$$
256.4.a.l $2$ $15.104$ $$\Q(\sqrt{7})$$ None $$0$$ $$0$$ $$0$$ $$16$$ $+$ $$q+\beta q^{3}+2\beta q^{5}+8q^{7}+q^{9}+3\beta q^{11}+\cdots$$
256.4.a.m $2$ $15.104$ $$\Q(\sqrt{3})$$ None $$0$$ $$4$$ $$0$$ $$0$$ $-$ $$q+2q^{3}-\beta q^{5}+2\beta q^{7}-23q^{9}-42q^{11}+\cdots$$
256.4.a.n $4$ $15.104$ $$\Q(\sqrt{2}, \sqrt{5})$$ None $$0$$ $$0$$ $$0$$ $$0$$ $+$ $$q-\beta _{1}q^{3}-\beta _{3}q^{5}-\beta _{2}q^{7}+13q^{9}+\cdots$$

## Decomposition of $$S_{4}^{\mathrm{old}}(\Gamma_0(256))$$ into lower level spaces

$$S_{4}^{\mathrm{old}}(\Gamma_0(256)) \cong$$ $$S_{4}^{\mathrm{new}}(\Gamma_0(8))$$$$^{\oplus 6}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(16))$$$$^{\oplus 5}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(32))$$$$^{\oplus 4}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(64))$$$$^{\oplus 3}$$$$\oplus$$$$S_{4}^{\mathrm{new}}(\Gamma_0(128))$$$$^{\oplus 2}$$