Space of modular forms of level 56, weight 4, and Character 56.e

• Label: 56.4.e
• Level: $56$
• Weight: $4$
• Character orbit: 56.e
• Rep. character: $\chi_{56}(27,\cdot)$
• Character field: $\Q$
• Dimension: $22$
• Newform subspaces: $3$
• Sturm bound: $32$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 56 = 2^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	56.e (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 56 \)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(32\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(56, [\chi])\).

	Total	New	Old
Modular forms	26	26	0
Cusp forms	22	22	0
Eisenstein series	4	4	0

Trace form

\( 22 q - q^{2} - 7 q^{4} - 61 q^{8} - 166 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(56, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
56.4.e.a	$56$	$4$	56.e	56.e	$2$	$2$	$2$	$3.304$	\(\Q(\sqrt{-7}) \)	\(\Q(\sqrt{-7}) \)		$4$	$0$	\(-5\)	\(0\)	\(0\)	\(0\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(-2-\beta )q^{2}+(2+5\beta )q^{4}+(-7+14\beta )q^{7}+\cdots\)
56.4.e.b	$56$	$4$	56.e	56.e	$2$	$4$	$4$	$3.304$	\(\Q(\sqrt{-3}, \sqrt{-7})\)	None		$4$	$0$	\(-4\)	\(0\)	\(0\)	\(0\)	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1+\beta _{2})q^{2}+\beta _{3}q^{3}+(-6-2\beta _{2}+\cdots)q^{4}+\cdots\)
56.4.e.c	$56$	$4$	56.e	56.e	$2$	$16$	$16$	$3.304$	\(\mathbb{Q}[x]/(x^{16} + \cdots)\)	None		$4$	$0$	\(8\)	\(0\)	\(0\)	\(0\)	$2^{22}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(1+\beta _{2})q^{2}-\beta _{4}q^{3}-\beta _{8}q^{4}+\beta _{12}q^{5}+\cdots\)