Space of modular forms of level 56, weight 3, and Character 56.h

• Label: 56.3.h
• Level: $56$
• Weight: $3$
• Character orbit: 56.h
• Rep. character: $\chi_{56}(13,\cdot)$
• Character field: $\Q$
• Dimension: $14$
• Newform subspaces: $4$
• Sturm bound: $24$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 56 = 2^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	56.h (of order \(2\) and degree \(1\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 56 \)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(24\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{3}(56, [\chi])\).

	Total	New	Old
Modular forms	18	18	0
Cusp forms	14	14	0
Eisenstein series	4	4	0

Trace form

\( 14 q - q^{2} + q^{4} - 2 q^{7} - q^{8} + 26 q^{9} + O(q^{10}) \)

Decomposition of \(S_{3}^{\mathrm{new}}(56, [\chi])\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
56.3.h.a	$56$	$3$	56.h	56.h	$2$	$2$	$2$	$1.526$	\(\Q(\sqrt{7}) \)	\(\Q(\sqrt{-14}) \)	✓	$4$	$0$	\(-4\)	\(0\)	\(0\)	\(14\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q-2q^{2}+\beta q^{3}+4q^{4}-\beta q^{5}-2\beta q^{6}+\cdots\)
56.3.h.b	$56$	$3$	56.h	56.h	$2$	$2$	$2$	$1.526$	\(\Q(\sqrt{-7}) \)	\(\Q(\sqrt{-7}) \)		$4$	$0$	\(3\)	\(0\)	\(0\)	\(14\)	$1$	$\mathrm{U}(1)[D_{2}]$	\(q+(1+\beta )q^{2}+(-1+3\beta )q^{4}+7q^{7}+\cdots\)
56.3.h.c	$56$	$3$	56.h	56.h	$2$	$2$	$2$	$1.526$	\(\Q(\sqrt{2}) \)	\(\Q(\sqrt{-14}) \)	✓	$4$	$0$	\(4\)	\(0\)	\(0\)	\(-14\)	$2$	$\mathrm{U}(1)[D_{2}]$	\(q+2q^{2}+\beta q^{3}+4q^{4}-3\beta q^{5}+2\beta q^{6}+\cdots\)
56.3.h.d	$56$	$3$	56.h	56.h	$2$	$8$	$8$	$1.526$	8.0.\(\cdots\).51	None		$4$	$0$	\(-4\)	\(0\)	\(0\)	\(-16\)	$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+(-1-\beta _{2})q^{2}+\beta _{6}q^{3}+(-2-\beta _{1}+\cdots)q^{4}+\cdots\)