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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(6\)	\(1\)	\(5\)		\(4\)	\(1\)	\(3\)		\(2\)	\(0\)	\(2\)
\(+\)	\(-\)	\(-\)		\(5\)	\(0\)	\(5\)		\(3\)	\(0\)	\(3\)		\(2\)	\(0\)	\(2\)
\(-\)	\(+\)	\(-\)		\(5\)	\(1\)	\(4\)		\(3\)	\(1\)	\(2\)		\(2\)	\(0\)	\(2\)
\(-\)	\(-\)	\(+\)		\(6\)	\(1\)	\(5\)		\(4\)	\(1\)	\(3\)		\(2\)	\(0\)	\(2\)
Plus space		\(+\)		\(12\)	\(2\)	\(10\)		\(8\)	\(2\)	\(6\)		\(4\)	\(0\)	\(4\)
Minus space		\(-\)		\(10\)	\(1\)	\(9\)		\(6\)	\(1\)	\(5\)		\(4\)	\(0\)	\(4\)

Trace form

3 * q + 8 * q^3 - 5 * q^5 - 36 * q^7 + 71 * q^9 + 36 * q^11 + 10 * q^13 - 186 * q^17 - 236 * q^19 + 88 * q^21 - 76 * q^23 + 75 * q^25 + 416 * q^27 + 98 * q^29 + 152 * q^31 - 112 * q^33 + 340 * q^35 - 494 * q^37 - 576 * q^39 + 358 * q^41 - 48 * q^43 - 465 * q^45 + 452 * q^47 + 707 * q^49 - 80 * q^51 + 1042 * q^53 + 180 * q^55 - 1728 * q^57 - 52 * q^59 - 582 * q^61 - 1796 * q^63 - 470 * q^65 + 560 * q^67 + 1288 * q^69 - 544 * q^71 + 1054 * q^73 + 200 * q^75 + 320 * q^77 - 832 * q^79 + 1427 * q^81 + 1640 * q^83 - 170 * q^85 + 3664 * q^87 - 546 * q^89 - 2536 * q^91 - 1520 * q^93 + 20 * q^95 + 518 * q^97 - 1420 * q^99

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

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

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