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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$		$8$	$1$	$7$		$6$	$1$	$5$		$2$	$0$	$2$
$+$	$-$	$-$		$9$	$2$	$7$		$7$	$2$	$5$		$2$	$0$	$2$
$-$	$+$	$-$		$9$	$1$	$8$		$7$	$1$	$6$		$2$	$0$	$2$
$-$	$-$	$+$		$8$	$1$	$7$		$6$	$1$	$5$		$2$	$0$	$2$
Plus space		$+$		$16$	$2$	$14$		$12$	$2$	$10$		$4$	$0$	$4$
Minus space		$-$		$18$	$3$	$15$		$14$	$3$	$11$		$4$	$0$	$4$

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}$