Space of modular forms of level 256, weight 3, and Character 256.d

• Label: 256.3.d
• Level: $256$
• Weight: $3$
• Character orbit: 256.d
• Rep. character: $\chi_{256}(127,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $5$
• Sturm bound: $96$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	256.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(96\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(256, [\chi])\).

	Total	New	Old
Modular forms	76	18	58
Cusp forms	52	14	38
Eisenstein series	24	4	20

Trace form

\( 14 q + 34 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(256, [\chi])\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
256.3.d.a	$256$	$3$	256.d	8.d	$2$	$2$	$2$	$6.975$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-8\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4q^{3}+iq^{5}-4iq^{7}+7q^{9}-4q^{11}+\cdots\)
256.3.d.b	$256$	$3$	256.d	8.d	$2$	$2$	$2$	$6.975$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+3iq^{5}-9q^{9}+5iq^{13}-30q^{17}+\cdots\)
256.3.d.c	$256$	$3$	256.d	8.d	$2$	$2$	$2$	$6.975$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(8\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4q^{3}+iq^{5}+4iq^{7}+7q^{9}+4q^{11}+\cdots\)
256.3.d.d	$256$	$3$	256.d	8.d	$2$	$4$	$4$	$6.975$	\(\Q(\zeta_{8})\)	None		$2$	$0$	\(0\)	\(-8\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-2-\zeta_{8}^{2})q^{3}+(-\zeta_{8}-\zeta_{8}^{3})q^{5}+\cdots\)
256.3.d.e	$256$	$3$	256.d	8.d	$2$	$4$	$4$	$6.975$	\(\Q(\zeta_{8})\)	None		$2$	$0$	\(0\)	\(8\)	\(0\)	\(0\)	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(2-\zeta_{8}^{2})q^{3}+(\zeta_{8}-\zeta_{8}^{3})q^{5}+(2\zeta_{8}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{3}^{\mathrm{old}}(256, [\chi])\) into lower level spaces

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