Properties

Label 256.2.a
Level $256$
Weight $2$
Character orbit 256.a
Rep. character $\chi_{256}(1,\cdot)$
Character field $\Q$
Dimension $6$
Newform subspaces $5$
Sturm bound $64$
Trace bound $5$

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Defining parameters

Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 256.a (trivial)
Character field: \(\Q\)
Newform subspaces: \( 5 \)
Sturm bound: \(64\)
Trace bound: \(5\)
Distinguishing \(T_p\): \(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(256))\).

Total New Old
Modular forms 44 10 34
Cusp forms 21 6 15
Eisenstein series 23 4 19

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.
\(+\)\(2\)
\(-\)\(4\)

Trace form

\( 6 q + 6 q^{9} + O(q^{10}) \) \( 6 q + 6 q^{9} - 4 q^{17} + 2 q^{25} + 8 q^{33} + 4 q^{41} - 42 q^{49} - 40 q^{57} + 32 q^{65} - 12 q^{73} - 2 q^{81} + 20 q^{89} - 36 q^{97} + O(q^{100}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(256))\) 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
256.2.a.a 256.a 1.a $1$ $2.044$ \(\Q\) \(\Q(\sqrt{-2}) \) \(0\) \(-2\) \(0\) \(0\) $+$ $N(\mathrm{U}(1))$ \(q-2q^{3}+q^{9}-6q^{11}-6q^{17}-2q^{19}+\cdots\)
256.2.a.b 256.a 1.a $1$ $2.044$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-4\) \(0\) $+$ $N(\mathrm{U}(1))$ \(q-4q^{5}-3q^{9}-4q^{13}-2q^{17}+11q^{25}+\cdots\)
256.2.a.c 256.a 1.a $1$ $2.044$ \(\Q\) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(4\) \(0\) $-$ $N(\mathrm{U}(1))$ \(q+4q^{5}-3q^{9}+4q^{13}-2q^{17}+11q^{25}+\cdots\)
256.2.a.d 256.a 1.a $1$ $2.044$ \(\Q\) \(\Q(\sqrt{-2}) \) \(0\) \(2\) \(0\) \(0\) $-$ $N(\mathrm{U}(1))$ \(q+2q^{3}+q^{9}+6q^{11}-6q^{17}+2q^{19}+\cdots\)
256.2.a.e 256.a 1.a $2$ $2.044$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-2}) \) \(0\) \(0\) \(0\) \(0\) $-$ $N(\mathrm{U}(1))$ \(q+\beta q^{3}+5q^{9}-\beta q^{11}+6q^{17}-3\beta q^{19}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(256))\) into lower level spaces

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