Space of modular forms of level 256, weight 5, and Character 256.c

• Label: 256.5.c
• Level: $256$
• Weight: $5$
• Character orbit: 256.c
• Rep. character: $\chi_{256}(255,\cdot)$
• Character field: $\Q$
• Dimension: $30$
• Newform subspaces: $11$
• Sturm bound: $160$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	256.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 11 \)
Sturm bound:			\(160\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(5\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{5}(256, [\chi])\).

	Total	New	Old
Modular forms	140	34	106
Cusp forms	116	30	86
Eisenstein series	24	4	20

Trace form

\( 30 q - 698 q^{9} + O(q^{10}) \)

Decomposition of \(S_{5}^{\mathrm{new}}(256, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
256.5.c.a	$256$	$5$	256.c	4.b	$2$	$1$	$1$	$26.463$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(-48\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-48q^{5}+3^{4}q^{9}-240q^{13}-322q^{17}+\cdots\)
256.5.c.b	$256$	$5$	256.c	4.b	$2$	$1$	$1$	$26.463$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(48\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+48q^{5}+3^{4}q^{9}+240q^{13}-322q^{17}+\cdots\)
256.5.c.c	$256$	$5$	256.c	4.b	$2$	$2$	$2$	$26.463$	\(\Q(\sqrt{-6}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-16\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}-8q^{5}+8\beta q^{7}-15q^{9}+11\beta q^{11}+\cdots\)
256.5.c.d	$256$	$5$	256.c	4.b	$2$	$2$	$2$	$26.463$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+7iq^{3}-115q^{9}-23iq^{11}-574q^{17}+\cdots\)
256.5.c.e	$256$	$5$	256.c	4.b	$2$	$2$	$2$	$26.463$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{3}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{3}-47q^{9}-21\beta q^{11}+574q^{17}+\cdots\)
256.5.c.f	$256$	$5$	256.c	4.b	$2$	$2$	$2$	$26.463$	\(\Q(\sqrt{-6}) \)	None		$2$	$0$	\(0\)	\(0\)	\(16\)	\(0\)	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta q^{3}+8q^{5}-8\beta q^{7}-15q^{9}+11\beta q^{11}+\cdots\)
256.5.c.g	$256$	$5$	256.c	4.b	$2$	$4$	$4$	$26.463$	\(\Q(\sqrt{-2}, \sqrt{13})\)	None		$2$	$0$	\(0\)	\(0\)	\(-48\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(-12-\beta _{3})q^{5}+(3\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
256.5.c.h	$256$	$5$	256.c	4.b	$2$	$4$	$4$	$26.463$	\(\Q(i, \sqrt{51})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+\beta _{1}q^{5}+7\beta _{2}q^{7}-123q^{9}+\cdots\)
256.5.c.i	$256$	$5$	256.c	4.b	$2$	$4$	$4$	$26.463$	\(\Q(i, \sqrt{15})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q+3\beta _{1}q^{3}-\beta _{2}q^{5}-\beta _{3}q^{7}+45q^{9}+\cdots\)
256.5.c.j	$256$	$5$	256.c	4.b	$2$	$4$	$4$	$26.463$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{12}^{2}q^{3}+3\zeta_{12}q^{5}-5\zeta_{12}^{3}q^{7}+\cdots\)
256.5.c.k	$256$	$5$	256.c	4.b	$2$	$4$	$4$	$26.463$	\(\Q(\sqrt{-2}, \sqrt{13})\)	None		$2$	$0$	\(0\)	\(0\)	\(48\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+(12+\beta _{3})q^{5}+(-3\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{5}^{\mathrm{old}}(256, [\chi])\) into lower level spaces

\( S_{5}^{\mathrm{old}}(256, [\chi]) \cong \) \(S_{5}^{\mathrm{new}}(4, [\chi])\)\(^{\oplus 7}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(8, [\chi])\)\(^{\oplus 6}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(16, [\chi])\)\(^{\oplus 5}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(32, [\chi])\)\(^{\oplus 4}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(64, [\chi])\)\(^{\oplus 3}\)\(\oplus\)\(S_{5}^{\mathrm{new}}(128, [\chi])\)\(^{\oplus 2}\)