Space of modular forms of level 85, weight 2, and trivial character

• Label: 85.2.a
• Level: $85$
• Weight: $2$
• Character orbit: 85.a
• Rep. character: $\chi_{85}(1,\cdot)$
• Character field: $\Q$
• Dimension: $5$
• Newform subspaces: $3$
• Sturm bound: $18$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	85.a (trivial)
Character field:			\(\Q\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(18\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(\Gamma_0(85))\).

	Total	New	Old
Modular forms	10	5	5
Cusp forms	7	5	2
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.

\(5\)	\(17\)	Fricke	Dim
\(+\)	\(+\)	$+$	\(2\)
\(+\)	\(-\)	$-$	\(1\)
\(-\)	\(+\)	$-$	\(2\)
Plus space		\(+\)	\(2\)
Minus space		\(-\)	\(3\)

Trace form

\( 5 q - q^{2} + 3 q^{4} - q^{5} - 4 q^{6} - 8 q^{7} - 9 q^{8} + 9 q^{9} + O(q^{10}) \)

Decomposition of \(S_{2}^{\mathrm{new}}(\Gamma_0(85))\) 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
														$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	17
85.2.a.a	$85$	$2$	85.a	1.a	$1$	$1$	$1$	$0.679$	\(\Q\)	None	✓	$1$	$0$	\(1\)	\(2\)	\(-1\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}-q^{4}-q^{5}+2q^{6}-2q^{7}+\cdots\)
85.2.a.b	$85$	$2$	85.a	1.a	$1$	$2$	$2$	$0.679$	\(\Q(\sqrt{2}) \)	None	✓	$1$	$1$	\(-2\)	\(-4\)	\(-2\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(-2-\beta )q^{3}+(1-2\beta )q^{4}+\cdots\)
85.2.a.c	$85$	$2$	85.a	1.a	$1$	$2$	$2$	$0.679$	\(\Q(\sqrt{3}) \)	None	✓	$1$	$0$	\(0\)	\(2\)	\(2\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1-\beta )q^{3}+q^{4}+q^{5}+(-3+\cdots)q^{6}+\cdots\)

Decomposition of \(S_{2}^{\mathrm{old}}(\Gamma_0(85))\) into lower level spaces

\( S_{2}^{\mathrm{old}}(\Gamma_0(85)) \cong \) \(S_{2}^{\mathrm{new}}(\Gamma_0(17))\)\(^{\oplus 2}\)