Space of modular forms of level 40, weight 6, and trivial character

• Label: 40.6.a
• Level: $40$
• Weight: $6$
• Character orbit: 40.a
• Rep. character: $\chi_{40}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $4$
• Sturm bound: $36$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 40 = 2^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	40.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(36\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{6}(\Gamma_0(40))\).

	Total	New	Old
Modular forms	34	5	29
Cusp forms	26	5	21
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
\(+\)	\(+\)	$+$	\(1\)
\(+\)	\(-\)	$-$	\(2\)
\(-\)	\(+\)	$-$	\(1\)
\(-\)	\(-\)	$+$	\(1\)
Plus space		\(+\)	\(2\)
Minus space		\(-\)	\(3\)

Trace form

\( 5 q - 40 q^{3} + 25 q^{5} + 124 q^{7} + 281 q^{9} + O(q^{10}) \)

Decomposition of \(S_{6}^{\mathrm{new}}(\Gamma_0(40))\) 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
40.6.a.a	$40$	$6$	40.a	1.a	$1$	$1$	$1$	$6.415$	\(\Q\)	None	✓	$1$	$0$	\(0\)	\(-18\)	\(-25\)	\(242\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q-18q^{3}-5^{2}q^{5}+242q^{7}+3^{4}q^{9}+\cdots\)
40.6.a.b	$40$	$6$	40.a	1.a	$1$	$1$	$1$	$6.415$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-8\)	\(25\)	\(-108\)		$-$	$-$	$+$	$1$	$\mathrm{SU}(2)$	\(q-8q^{3}+5^{2}q^{5}-108q^{7}-179q^{9}+\cdots\)
40.6.a.c	$40$	$6$	40.a	1.a	$1$	$1$	$1$	$6.415$	\(\Q\)	None	✓	$1$	$1$	\(0\)	\(-2\)	\(-25\)	\(-62\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q-2q^{3}-5^{2}q^{5}-62q^{7}-239q^{9}+\cdots\)
40.6.a.d	$40$	$6$	40.a	1.a	$1$	$2$	$2$	$6.415$	\(\Q(\sqrt{129}) \)	None	✓	$1$	$0$	\(0\)	\(-12\)	\(50\)	\(52\)		$+$	$-$	$-$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-6-\beta )q^{3}+5^{2}q^{5}+(26-3\beta )q^{7}+\cdots\)

Decomposition of \(S_{6}^{\mathrm{old}}(\Gamma_0(40))\) into lower level spaces

\( S_{6}^{\mathrm{old}}(\Gamma_0(40)) \cong \) \(S_{6}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(5))\)\(^{\oplus 4}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(8))\)\(^{\oplus 2}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(10))\)\(^{\oplus 3}\)\(\oplus\)\(S_{6}^{\mathrm{new}}(\Gamma_0(20))\)\(^{\oplus 2}\)