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 + 468 * q^11 + 222 * q^13 + 1578 * q^17 - 3132 * q^19 - 584 * q^21 + 1076 * q^23 + 3125 * q^25 - 5920 * q^27 + 3446 * q^29 + 2392 * q^31 - 16240 * q^33 - 5900 * q^35 + 454 * q^37 - 13248 * q^39 + 10762 * q^41 + 30368 * q^43 + 14925 * q^45 - 28924 * q^47 + 677 * q^49 + 50320 * q^51 - 17306 * q^53 - 13900 * q^55 + 2880 * q^57 + 67388 * q^59 - 21202 * q^61 + 32668 * q^63 + 48550 * q^65 - 130848 * q^67 - 149336 * q^69 + 38432 * q^71 - 91454 * q^73 - 25000 * q^75 + 228896 * q^77 + 106304 * q^79 + 71765 * q^81 + 13016 * q^83 + 14450 * q^85 - 334160 * q^87 - 344862 * q^89 + 22680 * q^91 - 4400 * q^93 - 37900 * q^95 + 289834 * q^97 + 443012 * q^99

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

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

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