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		Total				Cusp				Eisenstein
				All	New	Old		All	New	Old		All	New	Old
\(+\)	\(+\)	\(+\)		\(2\)	\(2\)	\(0\)		\(2\)	\(2\)	\(0\)		\(0\)	\(0\)	\(0\)
\(+\)	\(-\)	\(-\)		\(3\)	\(1\)	\(2\)		\(2\)	\(1\)	\(1\)		\(1\)	\(0\)	\(1\)
\(-\)	\(+\)	\(-\)		\(3\)	\(2\)	\(1\)		\(2\)	\(2\)	\(0\)		\(1\)	\(0\)	\(1\)
\(-\)	\(-\)	\(+\)		\(2\)	\(0\)	\(2\)		\(1\)	\(0\)	\(1\)		\(1\)	\(0\)	\(1\)
Plus space		\(+\)		\(4\)	\(2\)	\(2\)		\(3\)	\(2\)	\(1\)		\(1\)	\(0\)	\(1\)
Minus space		\(-\)		\(6\)	\(3\)	\(3\)		\(4\)	\(3\)	\(1\)		\(2\)	\(0\)	\(2\)

Trace form

5 * q - q^2 + 3 * q^4 - q^5 - 4 * q^6 - 8 * q^7 - 9 * q^8 + 9 * q^9 + q^10 + 4 * q^12 - 6 * q^13 + 12 * q^14 + 4 * q^15 - 5 * q^16 - 3 * q^17 - q^18 + 4 * q^19 + q^20 - 8 * q^21 + 8 * q^22 - 4 * q^23 + 8 * q^24 + 5 * q^25 - 6 * q^26 - 12 * q^27 - 12 * q^28 - 10 * q^29 - 8 * q^30 + 11 * q^32 + 28 * q^33 + 3 * q^34 + 4 * q^35 - 25 * q^36 - 10 * q^37 + 4 * q^38 + 4 * q^39 + 9 * q^40 + 14 * q^41 + 4 * q^42 - 12 * q^44 - 5 * q^45 + 24 * q^46 + 20 * q^47 - 24 * q^48 - 11 * q^49 - q^50 + 4 * q^51 + 6 * q^52 + 14 * q^53 - 20 * q^54 + 12 * q^55 + 16 * q^56 + 2 * q^58 - 4 * q^59 - 6 * q^61 + 8 * q^62 - 12 * q^63 - 5 * q^64 - 10 * q^65 - 24 * q^66 - 24 * q^67 - 5 * q^68 + 4 * q^71 + 7 * q^72 - 26 * q^73 + 18 * q^74 + 20 * q^76 + 4 * q^77 + 36 * q^78 - 8 * q^79 - 15 * q^80 + 37 * q^81 - 6 * q^82 + 24 * q^83 - q^85 + 4 * q^86 - 8 * q^87 + 28 * q^88 - 22 * q^89 - 23 * q^90 - 4 * q^91 - 8 * q^92 - 28 * q^93 - 16 * q^94 + 4 * q^95 + 12 * q^96 + 2 * q^97 - 29 * q^98 + 12 * q^99

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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	Traces					A-L signs		Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$		5	17
85.2.a.a	$85$	$2$	85.a	1.a	$1$	$1$	$1$	$0.679$	\(\Q\)	None	✓	✓	✓	✓	85.2.a.a	$1$	$0$	\(1\)	\(2\)	\(-1\)	\(-2\)		$+$	$-$	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{2}+2q^{3}-q^{4}-q^{5}+2q^{6}-2q^{7}+\cdots\)
85.2.a.b	$85$	$2$	85.a	1.a	$1$	$2$	$2$	$0.679$	\(\Q(\sqrt{2}) \)	None	✓	✓	✓	✓	85.2.a.b	$1$	$1$	\(-2\)	\(-4\)	\(-2\)	\(-4\)		$+$	$+$	$+$	$1$	$\mathrm{SU}(2)$	\(q+(-1+\beta )q^{2}+(-2-\beta )q^{3}+(1-2\beta )q^{4}+\cdots\)
85.2.a.c	$85$	$2$	85.a	1.a	$1$	$2$	$2$	$0.679$	\(\Q(\sqrt{3}) \)	None	✓	✓	✓	✓	85.2.a.c	$1$	$0$	\(0\)	\(2\)	\(2\)	\(-2\)		$-$	$+$	$-$	$1$	$\mathrm{SU}(2)$	\(q+\beta q^{2}+(1-\beta )q^{3}+q^{4}+q^{5}+(-3+\cdots)q^{6}+\cdots\)

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

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