Space of modular forms of level 85, weight 2, and Character 85.j

• Label: 85.2.j
• Level: $85$
• Weight: $2$
• Character orbit: 85.j
• Rep. character: $\chi_{85}(4,\cdot)$
• Character field: $\Q(\zeta_{4})$
• Dimension: $16$
• Newform subspaces: $3$
• Sturm bound: $18$
• Trace bound: $2$

Defining parameters

Level:	\( N \)	\(=\)	\( 85 = 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	85.j (of order \(4\) and degree \(2\))
Character conductor:	\(\operatorname{cond}(\chi)\)	\(=\)	\( 85 \)
Character field:			\(\Q(i)\)
Newform subspaces:			\( 3 \)
Sturm bound:			\(18\)
Trace bound:			\(2\)
Distinguishing \(T_p\):			\(2\)

Dimensions

The following table gives the dimensions of various subspaces of \(M_{2}(85, [\chi])\).

	Total	New	Old
Modular forms	24	24	0
Cusp forms	16	16	0
Eisenstein series	8	8	0

Trace form

16 * q + 8 * q^4 - 6 * q^5 - 16 * q^6 - 2 * q^10 + 4 * q^11 - 4 * q^14 - 26 * q^20 - 44 * q^24 - 8 * q^29 + 40 * q^30 + 4 * q^34 + 32 * q^35 + 12 * q^39 + 18 * q^40 + 4 * q^41 + 40 * q^44 - 14 * q^45 - 8 * q^46 - 52 * q^50 + 32 * q^51 + 116 * q^54 + 24 * q^55 - 32 * q^56 - 44 * q^61 - 48 * q^64 + 8 * q^65 - 96 * q^69 - 28 * q^71 + 16 * q^74 + 52 * q^75 - 32 * q^79 + 18 * q^80 - 40 * q^81 + 16 * q^84 - 58 * q^85 + 32 * q^86 + 8 * q^89 - 82 * q^90 + 36 * q^91 - 36 * q^95 + 16 * q^96 + 24 * q^99

Decomposition of \(S_{2}^{\mathrm{new}}(85, [\chi])\) 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				Coefficient ring index	Sato-Tate	$q$-expansion
																		$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
85.2.j.a	$85$	$2$	85.j	85.j	$4$	$2$	$1$	$0.679$	\(\Q(\sqrt{-1}) \)	None		✓			85.2.j.a	$2$	$1$	\(-2\)	\(-2\)	\(-4\)	\(-6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q-q^{2}+(i-1)q^{3}-q^{4}+(-i-2)q^{5}+\cdots\)
85.2.j.b	$85$	$2$	85.j	85.j	$4$	$2$	$1$	$0.679$	\(\Q(\sqrt{-1}) \)	None		✓			85.2.j.a	$2$	$0$	\(2\)	\(2\)	\(-2\)	\(6\)	$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+q^{2}+(-i+1)q^{3}-q^{4}+(-2 i-1)q^{5}+\cdots\)
85.2.j.c	$85$	$2$	85.j	85.j	$4$	$12$	$6$	$0.679$	\(\mathbb{Q}[x]/(x^{12} + \cdots)\)	None		✓	✓		85.2.j.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)	$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+\beta _{2}q^{2}+(-\beta _{2}-\beta _{5}+\beta _{7})q^{3}+(1+\cdots)q^{4}+\cdots\)