Space of modular forms of level 40, weight 2, and trivial character

• Label: 40.2.a
• Level: $40$
• Weight: $2$
• Character orbit: 40.a
• Rep. character: $\chi_{40}(1,\cdot)$
• Character field: $\Q$
• Dimension: $1$
• Newform subspaces: $1$
• Sturm bound: $12$
• Trace bound: $0$

Defining parameters

Level:	\( N \)	\(=\)	\( 40 = 2^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	40.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 1 \)
Sturm bound:			\(12\)
Trace bound:			\(0\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(40))\).

	Total	New	Old
Modular forms	10	1	9
Cusp forms	3	1	2
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.

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

Trace form

q + q^5 - 4 * q^7 - 3 * q^9 + 4 * q^11 - 2 * q^13 + 2 * q^17 + 4 * q^19 + 4 * q^23 + q^25 - 2 * q^29 - 8 * q^31 - 4 * q^35 + 6 * q^37 - 6 * q^41 - 8 * q^43 - 3 * q^45 + 4 * q^47 + 9 * q^49 + 6 * q^53 + 4 * q^55 - 4 * q^59 - 2 * q^61 + 12 * q^63 - 2 * q^65 + 8 * q^67 - 6 * q^73 - 16 * q^77 + 9 * q^81 - 16 * q^83 + 2 * q^85 - 6 * q^89 + 8 * q^91 + 4 * q^95 - 14 * q^97 - 12 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(40))\) 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	5
40.2.a.a	$40$	$2$	40.a	1.a	$1$	$1$	$1$	$0.319$	\(\Q\)	None	✓	✓	✓	✓	40.2.a.a	$1$	$0$	\(0\)	\(0\)	\(1\)	\(-4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{5}-4q^{7}-3q^{9}+4q^{11}-2q^{13}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(40))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(40)) \simeq \) \(S_{2}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)