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

Downloads

Learn more

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 Char Prim Dim $A$ Field CM Traces A-L signs Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$ 2
256.4.a.a 256.a 1.a $1$ $15.104$ \(\Q\) \(\Q(\sqrt{-2}) \) \(0\) \(-10\) \(0\) \(0\) $-$ $N(\mathrm{U}(1))$ \(q-10q^{3}+73q^{9}+18q^{11}+90q^{17}+\cdots\)
256.4.a.b 256.a 1.a $1$ $15.104$ \(\Q\) None \(0\) \(-8\) \(-12\) \(32\) $-$ $\mathrm{SU}(2)$ \(q-8q^{3}-12q^{5}+2^{5}q^{7}+37q^{9}+8q^{11}+\cdots\)
256.4.a.c 256.a 1.a $1$ $15.104$ \(\Q\) None \(0\) \(-8\) \(12\) \(-32\) $+$ $\mathrm{SU}(2)$ \(q-8q^{3}+12q^{5}-2^{5}q^{7}+37q^{9}+8q^{11}+\cdots\)
256.4.a.d 256.a 1.a $1$ $15.104$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-4\) \(0\) $+$ $N(\mathrm{U}(1))$ \(q-4q^{5}-3^{3}q^{9}+92q^{13}+94q^{17}+\cdots\)
256.4.a.e 256.a 1.a $1$ $15.104$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(4\) \(0\) $-$ $N(\mathrm{U}(1))$ \(q+4q^{5}-3^{3}q^{9}-92q^{13}+94q^{17}+\cdots\)
256.4.a.f 256.a 1.a $1$ $15.104$ \(\Q\) None \(0\) \(8\) \(-12\) \(-32\) $-$ $\mathrm{SU}(2)$ \(q+8q^{3}-12q^{5}-2^{5}q^{7}+37q^{9}-8q^{11}+\cdots\)
256.4.a.g 256.a 1.a $1$ $15.104$ \(\Q\) None \(0\) \(8\) \(12\) \(32\) $+$ $\mathrm{SU}(2)$ \(q+8q^{3}+12q^{5}+2^{5}q^{7}+37q^{9}-8q^{11}+\cdots\)
256.4.a.h 256.a 1.a $1$ $15.104$ \(\Q\) \(\Q(\sqrt{-2}) \) \(0\) \(10\) \(0\) \(0\) $+$ $N(\mathrm{U}(1))$ \(q+10q^{3}+73q^{9}-18q^{11}+90q^{17}+\cdots\)
256.4.a.i 256.a 1.a $2$ $15.104$ \(\Q(\sqrt{3}) \) None \(0\) \(-4\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q-2q^{3}-\beta q^{5}-2\beta q^{7}-23q^{9}+42q^{11}+\cdots\)
256.4.a.j 256.a 1.a $2$ $15.104$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(0\) \(-16\) $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-2\beta q^{5}-8q^{7}+q^{9}+3\beta q^{11}+\cdots\)
256.4.a.k 256.a 1.a $2$ $15.104$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) $-$ $N(\mathrm{U}(1))$ \(q+\beta q^{3}-19q^{9}-5^{2}\beta q^{11}-90q^{17}+\cdots\)
256.4.a.l 256.a 1.a $2$ $15.104$ \(\Q(\sqrt{7}) \) None \(0\) \(0\) \(0\) \(16\) $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+2\beta q^{5}+8q^{7}+q^{9}+3\beta q^{11}+\cdots\)
256.4.a.m 256.a 1.a $2$ $15.104$ \(\Q(\sqrt{3}) \) None \(0\) \(4\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+2q^{3}-\beta q^{5}+2\beta q^{7}-23q^{9}-42q^{11}+\cdots\)
256.4.a.n 256.a 1.a $4$ $15.104$ \(\Q(\sqrt{2}, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}-\beta _{3}q^{5}-\beta _{2}q^{7}+13q^{9}+\cdots\)

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

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