Space of modular forms of level 256, weight 2, and trivial character

• 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: $5$

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.

\(2\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(20\)	\(4\)	\(16\)		\(9\)	\(2\)	\(7\)		\(11\)	\(2\)	\(9\)
\(-\)		\(24\)	\(6\)	\(18\)		\(12\)	\(4\)	\(8\)		\(12\)	\(2\)	\(10\)

Trace form

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

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(256))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces					A-L signs	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2
256.2.a.a	$256$	$2$	256.a	1.a	$1$	$1$	$1$	$2.044$	\(\Q\)	\(\Q(\sqrt{-2}) \)	✓				64.2.b.a	$2$	$1$	\(0\)	\(-2\)	\(0\)	\(0\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-2q^{3}+q^{9}-6q^{11}-6q^{17}-2q^{19}+\cdots\)
256.2.a.b	$256$	$2$	256.a	1.a	$1$	$1$	$1$	$2.044$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				128.2.b.b	$2$	$1$	\(0\)	\(0\)	\(-4\)	\(0\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-4q^{5}-3q^{9}-4q^{13}-2q^{17}+11q^{25}+\cdots\)
256.2.a.c	$256$	$2$	256.a	1.a	$1$	$1$	$1$	$2.044$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				128.2.b.b	$2$	$0$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+4q^{5}-3q^{9}+4q^{13}-2q^{17}+11q^{25}+\cdots\)
256.2.a.d	$256$	$2$	256.a	1.a	$1$	$1$	$1$	$2.044$	\(\Q\)	\(\Q(\sqrt{-2}) \)	✓				64.2.b.a	$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+2q^{3}+q^{9}+6q^{11}-6q^{17}+2q^{19}+\cdots\)
256.2.a.e	$256$	$2$	256.a	1.a	$1$	$2$	$2$	$2.044$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-2}) \)	✓		✓		128.2.b.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$2$	$N(\mathrm{U}(1))$	\(q+\beta q^{3}+5q^{9}-\beta q^{11}+6q^{17}-3\beta q^{19}+\cdots\)

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

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