Space of modular forms of level 86, weight 2, and trivial character

• Label: 86.2.a
• Level: $86$
• Weight: $2$
• Character orbit: 86.a
• Rep. character: $\chi_{86}(1,\cdot)$
• Character field: $\Q$
• Dimension: $4$
• Newform subspaces: $2$
• Sturm bound: $22$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 86 = 2 \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	86.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 2 \)
Sturm bound:			\(22\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(3\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(86))\).

	Total	New	Old
Modular forms	13	4	9
Cusp forms	10	4	6
Eisenstein series	3	0	3

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

\(2\)	\(43\)	Fricke	Dim
\(+\)	\(-\)	$-$	\(2\)
\(-\)	\(+\)	$-$	\(2\)
Plus space		\(+\)	\(0\)
Minus space		\(-\)	\(4\)

Trace form

\( 4 q + 4 q^{4} + 2 q^{6} + 4 q^{7} + 2 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(86))\) 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	43
86.2.a.a	$86$	$2$	86.a	1.a	$1$	$2$	$2$	$0.687$	\(\Q(\sqrt{21}) \)	None	✓	$1$	$0$	\(-2\)	\(-1\)	\(3\)	\(4\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{2}-\beta q^{3}+q^{4}+(1+\beta )q^{5}+\beta q^{6}+\cdots\)
86.2.a.b	$86$	$2$	86.a	1.a	$1$	$2$	$2$	$0.687$	\(\Q(\sqrt{5}) \)	None	✓	$1$	$0$	\(2\)	\(1\)	\(-3\)	\(0\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+\beta q^{3}+q^{4}+(-1-\beta )q^{5}+\beta q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(86))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(86)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(43))\)\(^{\oplus 2}\)