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

• Label: 256.7.c
• Level: $256$
• Weight: $7$
• Character orbit: 256.c
• Rep. character: $\chi_{256}(255,\cdot)$
• Character field: $\Q$
• Dimension: $46$
• Newform subspaces: $12$
• Sturm bound: $224$
• Trace bound: $9$

Defining parameters

Level:	\( N \)	\(=\)	\( 256 = 2^{8} \)
Weight:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	256.c (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 4 \)
Character field:			\(\Q\)
Newform subspaces:			\( 12 \)
Sturm bound:			\(224\)
Trace bound:			\(9\)
Distinguishing \(T_p\):			\(3\), \(5\)

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

\( 46 q - 10202 q^{9} + O(q^{10}) \)

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.c.a	$256$	$7$	256.c	4.b	$2$	$1$	$1$	$58.894$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(-88\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q-88q^{5}+3^{6}q^{9}-1656q^{13}+990q^{17}+\cdots\)
256.7.c.b	$256$	$7$	256.c	4.b	$2$	$1$	$1$	$58.894$	\(\Q\)	\(\Q(\sqrt{-1}) \)	✓	$2$	$0$	\(0\)	\(0\)	\(88\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+88q^{5}+3^{6}q^{9}+1656q^{13}+990q^{17}+\cdots\)
256.7.c.c	$256$	$7$	256.c	4.b	$2$	$2$	$2$	$58.894$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(-400\)	\(0\)	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}-200q^{5}+8iq^{7}-1575q^{9}+\cdots\)
256.7.c.d	$256$	$7$	256.c	4.b	$2$	$2$	$2$	$58.894$	\(\Q(\sqrt{-1}) \)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+23iq^{3}-1387q^{9}+1169iq^{11}+\cdots\)
256.7.c.e	$256$	$7$	256.c	4.b	$2$	$2$	$2$	$58.894$	\(\Q(\sqrt{-2}) \)	\(\Q(\sqrt{-2}) \)		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{2}\cdot 5$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta q^{3}-71q^{9}-45\beta q^{11}+1726q^{17}+\cdots\)
256.7.c.f	$256$	$7$	256.c	4.b	$2$	$2$	$2$	$58.894$	\(\Q(\sqrt{-1}) \)	None		$2$	$0$	\(0\)	\(0\)	\(400\)	\(0\)	$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q+iq^{3}+200q^{5}-8iq^{7}-1575q^{9}+\cdots\)
256.7.c.g	$256$	$7$	256.c	4.b	$2$	$4$	$4$	$58.894$	\(\Q(\sqrt{-10}, \sqrt{-42})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{14}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	\(q-\beta _{1}q^{3}+\beta _{3}q^{5}+\beta _{2}q^{7}+57q^{9}+\cdots\)
256.7.c.h	$256$	$7$	256.c	4.b	$2$	$4$	$4$	$58.894$	\(\Q(i, \sqrt{11})\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{8}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{3}q^{3}+5\beta _{2}q^{5}+11\beta _{1}q^{7}+333q^{9}+\cdots\)
256.7.c.i	$256$	$7$	256.c	4.b	$2$	$6$	$6$	$58.894$	6.0.\(\cdots\).1	None		$2$	$0$	\(0\)	\(0\)	\(-88\)	\(0\)	$2^{27}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(-15-\beta _{1})q^{5}+(3\beta _{2}-\beta _{3}+\cdots)q^{7}+\cdots\)
256.7.c.j	$256$	$7$	256.c	4.b	$2$	$6$	$6$	$58.894$	6.0.\(\cdots\).1	None		$2$	$0$	\(0\)	\(0\)	\(88\)	\(0\)	$2^{27}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{2}q^{3}+(15+\beta _{1})q^{5}+(-3\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
256.7.c.k	$256$	$7$	256.c	4.b	$2$	$8$	$8$	$58.894$	8.0.\(\cdots\).9	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{40}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{4}q^{3}-\beta _{3}q^{5}+(-\beta _{5}+\beta _{7})q^{7}+\cdots\)
256.7.c.l	$256$	$7$	256.c	4.b	$2$	$8$	$8$	$58.894$	\(\mathbb{Q}[x]/(x^{8} - \cdots)\)	None		$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2^{42}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{1}q^{3}-\beta _{5}q^{5}+\beta _{4}q^{7}+(165+3\beta _{3}+\cdots)q^{9}+\cdots\)

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

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