Space of modular forms of level 40, weight 10, and trivial character

• Label: 40.10.a
• Level: $40$
• Weight: $10$
• Character orbit: 40.a
• Rep. character: $\chi_{40}(1,\cdot)$
• Character field: $\Q$
• Dimension: $9$
• Newform subspaces: $4$
• Sturm bound: $60$
• Trace bound: $3$

Defining parameters

Level:	\( N \)	\(=\)	\( 40 = 2^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	40.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 4 \)
Sturm bound:			\(60\)
Trace bound:			\(3\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{10}(\Gamma_0(40))\).

	Total	New	Old
Modular forms	58	9	49
Cusp forms	50	9	41
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
\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(-\)	$-$	\(3\)
\(-\)	\(+\)	$-$	\(2\)
\(-\)	\(-\)	$+$	\(2\)
Plus space		\(+\)	\(4\)
Minus space		\(-\)	\(5\)

Trace form

\( 9 q - 16 q^{3} + 625 q^{5} - 2052 q^{7} + 117557 q^{9} + O(q^{10}) \)

Decomposition of \(S_{10}^{\mathrm{new}}(\Gamma_0(40))\) 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
40.10.a.a	$40$	$10$	40.a	1.a	$1$	$2$	$2$	$20.601$	\(\Q(\sqrt{22}) \)	None	✓	$1$	$1$	\(0\)	\(-116\)	\(-1250\)	\(11284\)		$+$	$+$	$+$	$2^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q+(-58+\beta )q^{3}-5^{4}q^{5}+(5642-35\beta )q^{7}+\cdots\)
40.10.a.b	$40$	$10$	40.a	1.a	$1$	$2$	$2$	$20.601$	\(\Q(\sqrt{6049}) \)	None	✓	$1$	$1$	\(0\)	\(-92\)	\(1250\)	\(-6908\)		$-$	$-$	$+$	$2^{2}$	$\mathrm{SU}(2)$	\(q+(-46-\beta )q^{3}+5^{4}q^{5}+(-3454+\cdots)q^{7}+\cdots\)
40.10.a.c	$40$	$10$	40.a	1.a	$1$	$2$	$2$	$20.601$	\(\Q(\sqrt{46}) \)	None	✓	$1$	$0$	\(0\)	\(108\)	\(-1250\)	\(-908\)		$-$	$+$	$-$	$2^{3}\cdot 3$	$\mathrm{SU}(2)$	\(q+(54+\beta )q^{3}-5^{4}q^{5}+(-454+13\beta )q^{7}+\cdots\)
40.10.a.d	$40$	$10$	40.a	1.a	$1$	$3$	$3$	$20.601$	3.3.7117.1	None	✓	$1$	$0$	\(0\)	\(84\)	\(1875\)	\(-5520\)		$+$	$-$	$-$	$2^{9}\cdot 3^{3}\cdot 5$	$\mathrm{SU}(2)$	\(q+(28+\beta _{1})q^{3}+5^{4}q^{5}+(-1840+17\beta _{1}+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{10}^{\mathrm{old}}(\Gamma_0(40))\) into lower level spaces

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