Space of modular forms of level 56, weight 2, and trivial character

• Label: 56.2.a
• Level: $56$
• Weight: $2$
• Character orbit: 56.a
• Rep. character: $\chi_{56}(1,\cdot)$
• Character field: $\Q$
• Dimension: $2$
• Newform subspaces: $2$
• Sturm bound: $16$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 56 = 2^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	56.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(16\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(56))\).

	Total	New	Old
Modular forms	12	2	10
Cusp forms	5	2	3
Eisenstein series	7	0	7

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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(1\)	\(0\)	\(1\)		\(0\)	\(0\)	\(0\)		\(1\)	\(0\)	\(1\)
\(+\)	\(-\)	\(-\)		\(5\)	\(1\)	\(4\)		\(3\)	\(1\)	\(2\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(3\)	\(1\)	\(2\)		\(1\)	\(1\)	\(0\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(3\)	\(0\)	\(3\)		\(1\)	\(0\)	\(1\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(4\)	\(0\)	\(4\)		\(1\)	\(0\)	\(1\)		\(3\)	\(0\)	\(3\)
Minus space		\(-\)		\(8\)	\(2\)	\(6\)		\(4\)	\(2\)	\(2\)		\(4\)	\(0\)	\(4\)

Trace form

2 * q + 2 * q^3 - 2 * q^5 - 2 * q^9 - 4 * q^11 + 2 * q^13 - 8 * q^15 - 8 * q^17 + 6 * q^19 + 2 * q^21 + 8 * q^23 + 10 * q^25 - 4 * q^27 + 8 * q^29 + 12 * q^31 - 6 * q^35 - 8 * q^37 + 4 * q^43 - 10 * q^45 - 12 * q^47 + 2 * q^49 - 4 * q^51 - 4 * q^53 - 8 * q^55 - 4 * q^57 + 6 * q^59 - 2 * q^61 + 4 * q^63 + 4 * q^65 - 16 * q^67 + 16 * q^69 - 8 * q^71 - 4 * q^73 + 22 * q^75 + 4 * q^77 + 8 * q^79 - 2 * q^81 + 14 * q^83 - 4 * q^85 + 4 * q^87 + 4 * q^89 - 2 * q^91 + 8 * q^93 + 24 * q^95 - 8 * q^97 + 12 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(56))\) 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
56.2.a.a	$56$	$2$	56.a	1.a	$1$	$1$	$1$	$0.447$	\(\Q\)	None	✓	✓	✓	✓	56.2.a.a	$1$	$0$	\(0\)	\(0\)	\(2\)	\(-1\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{5}-q^{7}-3q^{9}-4q^{11}+2q^{13}+\cdots\)
56.2.a.b	$56$	$2$	56.a	1.a	$1$	$1$	$1$	$0.447$	\(\Q\)	None	✓	✓	✓	✓	56.2.a.b	$1$	$0$	\(0\)	\(2\)	\(-4\)	\(1\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+2q^{3}-4q^{5}+q^{7}+q^{9}-8q^{15}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(56))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(56)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(14))\)\(^{\oplus 3}\)