Space of modular forms of level 256, weight 7, and Character 256.d

• Label: 256.7.d
• Level: $256$
• Weight: $7$
• Character orbit: 256.d
• Rep. character: $\chi_{256}(127,\cdot)$
• Character field: $\Q$
• Dimension: $46$
• Newform subspaces: $9$
• Sturm bound: $224$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	256.d (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 8 \)
Character field:			\(\Q\)
Newform subspaces:			\( 9 \)
Sturm bound:			\(224\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{7}(256, [\chi])\).

	Total	New	Old
Modular forms	204	50	154
Cusp forms	180	46	134
Eisenstein series	24	4	20

Trace form

\( 46 q + 10210 q^{9} + O(q^{10}) \)

Decomposition of \(S_{7}^{\mathrm{new}}(256, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
256.7.d.a	$256$	$7$	256.d	8.d	$2$	$2$	$2$	$58.894$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(-8\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-4q^{3}-5^{2}iq^{5}-92iq^{7}-713q^{9}+\cdots\)
256.7.d.b	$256$	$7$	256.d	8.d	$2$	$2$	$2$	$58.894$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-1}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-117iq^{5}-3^{6}q^{9}-2035iq^{13}+\cdots\)
256.7.d.c	$256$	$7$	256.d	8.d	$2$	$2$	$2$	$58.894$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(8\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+4q^{3}+5^{2}iq^{5}-92iq^{7}-713q^{9}+\cdots\)
256.7.d.d	$256$	$7$	256.d	8.d	$2$	$4$	$4$	$58.894$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(-48\)	\(0\)	\(0\)	$2^{11}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-12+\beta _{2})q^{3}+(-7\beta _{1}-2\beta _{3})q^{5}+\cdots\)
256.7.d.e	$256$	$7$	256.d	8.d	$2$	$4$	$4$	$58.894$	\(\Q(\zeta_{12})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\zeta_{12}^{2}q^{3}-75\zeta_{12}q^{5}-11\zeta_{12}^{3}q^{7}+\cdots\)
256.7.d.f	$256$	$7$	256.d	8.d	$2$	$4$	$4$	$58.894$	\(\Q(i, \sqrt{15})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{2}q^{3}-5\beta _{1}q^{5}+5\beta _{3}q^{7}+231q^{9}+\cdots\)
256.7.d.g	$256$	$7$	256.d	8.d	$2$	$4$	$4$	$58.894$	\(\Q(i, \sqrt{6})\)	None		$2$	$0$	\(0\)	\(48\)	\(0\)	\(0\)	$2^{11}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(12+\beta _{2})q^{3}+(7\beta _{1}-2\beta _{3})q^{5}+(148\beta _{1}+\cdots)q^{7}+\cdots\)
256.7.d.h	$256$	$7$	256.d	8.d	$2$	$12$	$12$	$58.894$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(0\)	\(-40\)	\(0\)	\(0\)	$2^{83}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-3+\beta _{2})q^{3}+\beta _{5}q^{5}+(2\beta _{4}-\beta _{5}+\cdots)q^{7}+\cdots\)
256.7.d.i	$256$	$7$	256.d	8.d	$2$	$12$	$12$	$58.894$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		$2$	$0$	\(0\)	\(40\)	\(0\)	\(0\)	$2^{83}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(3-\beta _{2})q^{3}-\beta _{5}q^{5}+(2\beta _{4}-\beta _{5}-\beta _{10}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{7}^{\mathrm{old}}(256, [\chi])\) into lower level spaces

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