Space of modular forms of level 256, weight 4, and trivial character

• 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: $7$

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.

\(2\)		Total				Cusp				Eisenstein
		All	New	Old		All	New	Old		All	New	Old
\(+\)		\(56\)	\(14\)	\(42\)		\(44\)	\(12\)	\(32\)		\(12\)	\(2\)	\(10\)
\(-\)		\(52\)	\(12\)	\(40\)		\(40\)	\(10\)	\(30\)		\(12\)	\(2\)	\(10\)

Trace form

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

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

Label	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
256.4.a.a	$256$	$4$	256.a	1.a	$1$	$1$	$1$	$15.104$	\(\Q\)	\(\Q(\sqrt{-2}) \)	✓				64.4.b.a	$2$	$1$	\(0\)	\(-10\)	\(0\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q-10q^{3}+73q^{9}+18q^{11}+90q^{17}+\cdots\)
256.4.a.b	$256$	$4$	256.a	1.a	$1$	$1$	$1$	$15.104$	\(\Q\)	None	✓				128.4.b.a	$1$	$1$	\(0\)	\(-8\)	\(-12\)	\(32\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}-12q^{5}+2^{5}q^{7}+37q^{9}+8q^{11}+\cdots\)
256.4.a.c	$256$	$4$	256.a	1.a	$1$	$1$	$1$	$15.104$	\(\Q\)	None	✓				128.4.b.a	$1$	$0$	\(0\)	\(-8\)	\(12\)	\(-32\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}+12q^{5}-2^{5}q^{7}+37q^{9}+8q^{11}+\cdots\)
256.4.a.d	$256$	$4$	256.a	1.a	$1$	$1$	$1$	$15.104$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				128.4.b.c	$2$	$0$	\(0\)	\(0\)	\(-4\)	\(0\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q-4q^{5}-3^{3}q^{9}+92q^{13}+94q^{17}+\cdots\)
256.4.a.e	$256$	$4$	256.a	1.a	$1$	$1$	$1$	$15.104$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓				128.4.b.c	$2$	$1$	\(0\)	\(0\)	\(4\)	\(0\)		$-$	$-$	$1$	$N(\mathrm{U}(1))$	\(q+4q^{5}-3^{3}q^{9}-92q^{13}+94q^{17}+\cdots\)
256.4.a.f	$256$	$4$	256.a	1.a	$1$	$1$	$1$	$15.104$	\(\Q\)	None	✓				128.4.b.a	$1$	$1$	\(0\)	\(8\)	\(-12\)	\(-32\)		$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}-12q^{5}-2^{5}q^{7}+37q^{9}-8q^{11}+\cdots\)
256.4.a.g	$256$	$4$	256.a	1.a	$1$	$1$	$1$	$15.104$	\(\Q\)	None	✓				128.4.b.a	$1$	$0$	\(0\)	\(8\)	\(12\)	\(32\)		$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+8q^{3}+12q^{5}+2^{5}q^{7}+37q^{9}-8q^{11}+\cdots\)
256.4.a.h	$256$	$4$	256.a	1.a	$1$	$1$	$1$	$15.104$	\(\Q\)	\(\Q(\sqrt{-2}) \)	✓				64.4.b.a	$2$	$0$	\(0\)	\(10\)	\(0\)	\(0\)		$+$	$+$	$1$	$N(\mathrm{U}(1))$	\(q+10q^{3}+73q^{9}-18q^{11}+90q^{17}+\cdots\)
256.4.a.i	$256$	$4$	256.a	1.a	$1$	$2$	$2$	$15.104$	\(\Q(\sqrt{3}) \)	None	✓				64.4.b.b	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)		$+$	$+$	$2^{3}$	$\mathrm{SU}(2)$	\(q-2q^{3}-\beta q^{5}-2\beta q^{7}-23q^{9}+42q^{11}+\cdots\)
256.4.a.j	$256$	$4$	256.a	1.a	$1$	$2$	$2$	$15.104$	\(\Q(\sqrt{7}) \)	None	✓				8.4.b.a	$2$	$1$	\(0\)	\(0\)	\(0\)	\(-16\)		$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}-2\beta q^{5}-8q^{7}+q^{9}+3\beta q^{11}+\cdots\)
256.4.a.k	$256$	$4$	256.a	1.a	$1$	$2$	$2$	$15.104$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-2}) \)	✓				128.4.b.b	$4$	$1$	\(0\)	\(0\)	\(0\)	\(0\)		$-$	$-$	$2$	$N(\mathrm{U}(1))$	\(q+\beta q^{3}-19q^{9}-5^{2}\beta q^{11}-90q^{17}+\cdots\)
256.4.a.l	$256$	$4$	256.a	1.a	$1$	$2$	$2$	$15.104$	\(\Q(\sqrt{7}) \)	None	✓				8.4.b.a	$2$	$0$	\(0\)	\(0\)	\(0\)	\(16\)		$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+\beta q^{3}+2\beta q^{5}+8q^{7}+q^{9}+3\beta q^{11}+\cdots\)
256.4.a.m	$256$	$4$	256.a	1.a	$1$	$2$	$2$	$15.104$	\(\Q(\sqrt{3}) \)	None	✓				64.4.b.b	$2$	$1$	\(0\)	\(4\)	\(0\)	\(0\)		$-$	$-$	$2^{3}$	$\mathrm{SU}(2)$	\(q+2q^{3}-\beta q^{5}+2\beta q^{7}-23q^{9}-42q^{11}+\cdots\)
256.4.a.n	$256$	$4$	256.a	1.a	$1$	$4$	$4$	$15.104$	\(\Q(\sqrt{2}, \sqrt{5})\)	None	✓		✓		128.4.b.e	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$+$	$+$	$2^{10}$	$\mathrm{SU}(2)$	\(q-\beta _{1}q^{3}-\beta _{3}q^{5}-\beta _{2}q^{7}+13q^{9}+\cdots\)

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

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