Space of modular forms of level 76, weight 2, and trivial character

• Label: 76.2.a
• Level: $76$
• Weight: $2$
• Character orbit: 76.a
• Rep. character: $\chi_{76}(1,\cdot)$
• Character field: $\Q$
• Dimension: $1$
• Newform subspaces: $1$
• Sturm bound: $20$
• Trace bound: $0$

Defining parameters

Level:	$N$	$=$	$76 = 2^{2} \cdot 19$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	76.a (trivial)
Character field:			$\Q$
Newform subspaces:			$1$
Sturm bound:			$20$
Trace bound:			$0$

Dimensions

The following table gives the dimensions of various subspaces of $M_{2}(\Gamma_0(76))$.

	Total	New	Old
Modular forms	13	1	12
Cusp forms	8	1	7
Eisenstein series	5	0	5

The following table gives the dimensions of the cuspidal new subspaces with specified eigenvalues for the Atkin-Lehner operators and the Fricke involution.

$2$	$19$	Fricke		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
$+$	$+$	$+$		$2$	$0$	$2$		$1$	$0$	$1$		$1$	$0$	$1$
$+$	$-$	$-$		$5$	$0$	$5$		$3$	$0$	$3$		$2$	$0$	$2$
$-$	$+$	$-$		$3$	$1$	$2$		$2$	$1$	$1$		$1$	$0$	$1$
$-$	$-$	$+$		$3$	$0$	$3$		$2$	$0$	$2$		$1$	$0$	$1$
Plus space		$+$		$5$	$0$	$5$		$3$	$0$	$3$		$2$	$0$	$2$
Minus space		$-$		$8$	$1$	$7$		$5$	$1$	$4$		$3$	$0$	$3$

Trace form

q + 2 * q^3 - q^5 - 3 * q^7 + q^9 + 5 * q^11 - 4 * q^13 - 2 * q^15 - 3 * q^17 - q^19 - 6 * q^21 + 8 * q^23 - 4 * q^25 - 4 * q^27 - 2 * q^29 + 4 * q^31 + 10 * q^33 + 3 * q^35 + 10 * q^37 - 8 * q^39 + 10 * q^41 + q^43 - q^45 - q^47 + 2 * q^49 - 6 * q^51 - 4 * q^53 - 5 * q^55 - 2 * q^57 + 6 * q^59 - 13 * q^61 - 3 * q^63 + 4 * q^65 - 12 * q^67 + 16 * q^69 + 2 * q^71 + 9 * q^73 - 8 * q^75 - 15 * q^77 + 8 * q^79 - 11 * q^81 - 12 * q^83 + 3 * q^85 - 4 * q^87 + 12 * q^89 + 12 * q^91 + 8 * q^93 + q^95 - 8 * q^97 + 5 * q^99

Decomposition of $S_{2}^{\mathrm{new}}(\Gamma_0(76))$ 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	19
76.2.a.a	$76$	$2$	76.a	1.a	$1$	$1$	$1$	$0.607$	$\Q$	None	✓	✓	✓	✓	76.2.a.a	$1$	$0$	$0$	$2$	$-1$	$-3$		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{3}-q^{5}-3q^{7}+q^{9}+5q^{11}+\cdots$

Decomposition of $S_{2}^{\mathrm{old}}(\Gamma_0(76))$ into lower level spaces

$S_{2}^{\mathrm{old}}(\Gamma_0(76)) \simeq$ $S_{2}^{\mathrm{new}}(\Gamma_0(19))$$^{\oplus 3}$$\oplus$$S_{2}^{\mathrm{new}}(\Gamma_0(38))$$^{\oplus 2}$