Space of modular forms of level 56, weight 6, and trivial character

• Label: 56.6.a
• Level: $56$
• Weight: $6$
• Character orbit: 56.a
• Rep. character: $\chi_{56}(1,\cdot)$
• Character field: $\Q$
• Dimension: $8$
• Newform subspaces: $5$
• Sturm bound: $48$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 56 = 2^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	56.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 5 \)
Sturm bound:			\(48\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(56))\).

	Total	New	Old
Modular forms	44	8	36
Cusp forms	36	8	28
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
\(+\)	\(+\)	\(+\)	\(2\)
\(+\)	\(-\)	\(-\)	\(3\)
\(-\)	\(+\)	\(-\)	\(2\)
\(-\)	\(-\)	\(+\)	\(1\)
Plus space		\(+\)	\(3\)
Minus space		\(-\)	\(5\)

Trace form

\( 8 q - 22 q^{3} + 98 q^{5} + 876 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(56))\) into newform subspaces

Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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
56.6.a.a	$56$	$6$	56.a	1.a	$1$	$1$	$1$	$8.981$	\(\Q\)	None	✓	✓	✓	✓	56.6.a.a	$1$	$1$	\(0\)	\(-6\)	\(4\)	\(49\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-6q^{3}+4q^{5}+7^{2}q^{7}-207q^{9}-240q^{11}+\cdots\)
56.6.a.b	$56$	$6$	56.a	1.a	$1$	$1$	$1$	$8.981$	\(\Q\)	None	✓	✓			56.6.a.b	$1$	$0$	\(0\)	\(30\)	\(32\)	\(49\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+30q^{3}+2^{5}q^{5}+7^{2}q^{7}+657q^{9}+\cdots\)
56.6.a.c	$56$	$6$	56.a	1.a	$1$	$2$	$2$	$8.981$	\(\Q(\sqrt{177}) \)	None	✓	✓	✓		56.6.a.c	$1$	$0$	\(0\)	\(-26\)	\(-62\)	\(98\)		$+$	$-$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-13-\beta )q^{3}+(-31-5\beta )q^{5}+7^{2}q^{7}+\cdots\)
56.6.a.d	$56$	$6$	56.a	1.a	$1$	$2$	$2$	$8.981$	\(\Q(\sqrt{193}) \)	None	✓	✓	✓	✓	56.6.a.d	$1$	$1$	\(0\)	\(-14\)	\(42\)	\(-98\)		$+$	$+$	$+$	$2$	$\mathrm{SU}(2)$	\(q+(-7-\beta )q^{3}+(21+5\beta )q^{5}-7^{2}q^{7}+\cdots\)
56.6.a.e	$56$	$6$	56.a	1.a	$1$	$2$	$2$	$8.981$	\(\Q(\sqrt{345}) \)	None	✓	✓	✓	✓	56.6.a.e	$1$	$0$	\(0\)	\(-6\)	\(82\)	\(-98\)		$-$	$+$	$-$	$2$	$\mathrm{SU}(2)$	\(q+(-3-\beta )q^{3}+(41-3\beta )q^{5}-7^{2}q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(56))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(56)) \simeq \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(7))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(28))\)\(^{\oplus 2}\)