Space of modular forms of level 40, weight 4, and trivial character

• Label: 40.4.a
• Level: $40$
• Weight: $4$
• Character orbit: 40.a
• Rep. character: $\chi_{40}(1,\cdot)$
• Character field: $\Q$
• Dimension: $3$
• Newform subspaces: $3$
• Sturm bound: $24$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 40 = 2^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	40.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(24\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{4}(\Gamma_0(40))\).

	Total	New	Old
Modular forms	22	3	19
Cusp forms	14	3	11
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.

\(2\)	\(5\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(1\)
\(-\)	\(+\)	$-$	\(1\)
\(-\)	\(-\)	$+$	\(1\)
Plus space		\(+\)	\(2\)
Minus space		\(-\)	\(1\)

Trace form

\( 3 q + 8 q^{3} - 5 q^{5} - 36 q^{7} + 71 q^{9} + O(q^{10}) \)

Decomposition of \(S_{4}^{\mathrm{new}}(\Gamma_0(40))\) 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	5
40.4.a.a	$40$	$4$	40.a	1.a	$1$	$1$	$1$	$2.360$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-6\)	\(-5\)	\(-34\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-6q^{3}-5q^{5}-34q^{7}+9q^{9}+2^{4}q^{11}+\cdots\)
40.4.a.b	$40$	$4$	40.a	1.a	$1$	$1$	$1$	$2.360$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(4\)	\(5\)	\(16\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q+4q^{3}+5q^{5}+2^{4}q^{7}-11q^{9}+6^{2}q^{11}+\cdots\)
40.4.a.c	$40$	$4$	40.a	1.a	$1$	$1$	$1$	$2.360$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(10\)	\(-5\)	\(-18\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+10q^{3}-5q^{5}-18q^{7}+73q^{9}-2^{4}q^{11}+\cdots\)

Decomposition of \(S_{4}^{\mathrm{old}}(\Gamma_0(40))\) into lower level spaces

\( S_{4}^{\mathrm{old}}(\Gamma_0(40)) \cong \) \(S_{4}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{4}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)