Space of modular forms of level 56, weight 4, and trivial character

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

Defining parameters

Level:	\( N \)	\(=\)	\( 56 = 2^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	56.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(32\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(56))\).

	Total	New	Old
Modular forms	28	4	24
Cusp forms	20	4	16
Eisenstein series	8	0	8

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

\(2\)	\(7\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(1\)
\(-\)	\(+\)	$-$	\(1\)
\(-\)	\(-\)	$+$	\(2\)
Plus space		\(+\)	\(3\)
Minus space		\(-\)	\(1\)

Trace form

\( 4 q + 2 q^{3} + 14 q^{5} + 48 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(56))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		2	7
56.4.a.a	$56$	$4$	56.a	1.a	$1$	$1$	$1$	$3.304$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-16\)	\(-7\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-2^{4}q^{5}-7q^{7}-23q^{9}+24q^{11}+\cdots\)
56.4.a.b	$56$	$4$	56.a	1.a	$1$	$1$	$1$	$3.304$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(6\)	\(8\)	\(-7\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+6q^{3}+8q^{5}-7q^{7}+9q^{9}+56q^{11}+\cdots\)
56.4.a.c	$56$	$4$	56.a	1.a	$1$	$2$	$2$	$3.304$	\(\Q(\sqrt{57}) \)	None	✓	$1$	$0$	\(0\)	\(-2\)	\(22\)	\(14\)		$-$	$-$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-1-\beta )q^{3}+(11+\beta )q^{5}+7q^{7}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(56))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(56)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 2}\)