Space of modular forms of level 76, weight 8, and trivial character

• Label: 76.8.a
• Level: $76$
• Weight: $8$
• Character orbit: 76.a
• Rep. character: $\chi_{76}(1,\cdot)$
• Character field: $\Q$
• Dimension: $11$
• Newform subspaces: $2$
• Sturm bound: $80$
• Trace bound: $1$

Defining parameters

Level:	\( N \)	\(=\)	\( 76 = 2^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	76.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(80\)
Trace bound:			\(1\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{8}(\Gamma_0(76))\).

	Total	New	Old
Modular forms	73	11	62
Cusp forms	67	11	56
Eisenstein series	6	0	6

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	Dim
\(-\)	\(+\)	$-$	\(5\)
\(-\)	\(-\)	$+$	\(6\)
Plus space		\(+\)	\(6\)
Minus space		\(-\)	\(5\)

Trace form

\( 11 q + 26 q^{3} - q^{5} - 1151 q^{7} + 7679 q^{9} + O(q^{10}) \)

Decomposition of \(S_{8}^{\mathrm{new}}(\Gamma_0(76))\) 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	19
76.8.a.a	$76$	$8$	76.a	1.a	$1$	$5$	$5$	$23.741$	\(\mathbb{Q}[x]/(x^{5} - \cdots)\)	None	✓	$1$	$1$	\(0\)	\(-14\)	\(-280\)	\(414\)		$-$	$+$	$-$	$2^{3}\cdot 3^{2}\cdot 7$	$\mathrm{SU}(2)$	\(q+(-3-\beta _{1})q^{3}+(-56+\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
76.8.a.b	$76$	$8$	76.a	1.a	$1$	$6$	$6$	$23.741$	\(\mathbb{Q}[x]/(x^{6} - \cdots)\)	None	✓	$1$	$0$	\(0\)	\(40\)	\(279\)	\(-1565\)		$-$	$-$	$+$	$2^{6}$	$\mathrm{SU}(2)$	\(q+(7-\beta _{1})q^{3}+(47-\beta _{1}-\beta _{2})q^{5}+(-262+\cdots)q^{7}+\cdots\)

Decomposition of \(S_{8}^{\mathrm{old}}(\Gamma_0(76))\) into lower level spaces

\( S_{8}^{\mathrm{old}}(\Gamma_0(76)) \cong \) \(S_{8}^{\mathrm{new}}(\Gamma_0(2))\)\(^{\oplus 4}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(4))\)\(^{\oplus 2}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(19))\)\(^{\oplus 3}\)\(\oplus\)\(S_{8}^{\mathrm{new}}(\Gamma_0(38))\)\(^{\oplus 2}\)