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Level	Weight	Analytic conductor	$Nk^2$	Dim.

Bad $p$	Char.	Primitive character	Character order	Coefficient field

Self-twists	CM/RM discriminant	Inner twist count	Is self-dual	Analytic rank

Coefficient ring index	Coefficient ring gens.		Projective image	Projective image type
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Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	Traces				Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	CM	Self-dual	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
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	✓	$1$	$0$	$2$	$1$	$-3$	$0$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+\beta q^{3}+q^{4}+(-1-\beta )q^{5}+\beta q^{6}+\cdots$
86.2.c.a	$86$	$2$	86.c	43.c	$3$	$2$	$1$	$0.687$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$-2$	$-1$	$-3$	$1$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q-q^{2}+(-1+\zeta_{6})q^{3}+q^{4}+(-3+3\zeta_{6})q^{5}+\cdots$
86.2.c.b	$86$	$2$	86.c	43.c	$3$	$4$	$2$	$0.687$	$\Q(\sqrt{-3}, \sqrt{17})$	None		$2$	$0$	$4$	$-1$	$-3$	$-3$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+q^{2}-\beta _{1}q^{3}+q^{4}+(\beta _{1}-2\beta _{2})q^{5}+\cdots$
86.2.e.a	$86$	$2$	86.e	43.e	$7$	$12$	$2$	$0.687$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None		$2$	$0$	$2$	$-6$	$-3$	$-4$		$7$	$\mathrm{SU}(2)[C_{7}]$	$q+\beta _{6}q^{2}+(-\beta _{7}-\beta _{11})q^{3}-\beta _{3}q^{4}+\cdots$
86.2.e.b	$86$	$2$	86.e	43.e	$7$	$18$	$3$	$0.687$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	None		$2$	$0$	$-3$	$2$	$-3$	$-8$		$1$	$\mathrm{SU}(2)[C_{7}]$	$q+\beta _{8}q^{2}+(\beta _{1}+\beta _{2}+\beta _{3}-\beta _{5}+\beta _{6}+\cdots)q^{3}+\cdots$
86.2.g.a	$86$	$2$	86.g	43.g	$21$	$12$	$1$	$0.687$	$\Q(\zeta_{21})$	None		$2$	$0$	$2$	$8$	$3$	$-1$		$1$	$\mathrm{SU}(2)[C_{21}]$	$q+(1-\zeta_{21}+\zeta_{21}^{3}-\zeta_{21}^{4}+\zeta_{21}^{6}+\cdots)q^{2}+\cdots$
86.2.g.b	$86$	$2$	86.g	43.g	$21$	$24$	$2$	$0.687$		None		$2$	$0$	$-4$	$-6$	$3$	$3$			$\mathrm{SU}(2)[C_{21}]$
86.3.b.a	$86$	$3$	86.b	43.b	$2$	$2$	$2$	$2.343$	$\Q(\sqrt{-2}) $	None		$2$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta q^{2}-\beta q^{3}-2q^{4}+3\beta q^{5}+2q^{6}+\cdots$
86.3.b.b	$86$	$3$	86.b	43.b	$2$	$4$	$4$	$2.343$	$\Q(\sqrt{-2}, \sqrt{21})$	None		$2$	$0$	$0$	$0$	$0$	$0$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{2}q^{2}+\beta _{1}q^{3}-2q^{4}+(\beta _{1}+2\beta _{2}+\cdots)q^{5}+\cdots$
86.3.d.a	$86$	$3$	86.d	43.d	$6$	$16$	$8$	$2.343$	$\mathbb{Q}[x]/(x^{16} + \cdots)$	None		$2$	$0$	$0$	$-6$	$0$	$0$		$3^{2}$	$\mathrm{SU}(2)[C_{6}]$	$q+(-\beta _{5}+\beta _{6})q^{2}+(\beta _{3}-\beta _{6})q^{3}-2q^{4}+\cdots$
86.3.f.a	$86$	$3$	86.f	43.f	$14$	$36$	$6$	$2.343$		None		$2$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{14}]$
86.3.h.a	$86$	$3$	86.h	43.h	$42$	$96$	$8$	$2.343$		None		$2$	$0$	$0$	$6$	$0$	$0$			$\mathrm{SU}(2)[C_{42}]$
86.4.a.a	$86$	$4$	86.a	1.a	$1$	$1$	$1$	$5.074$	$\Q$	None	✓	$1$	$0$	$-2$	$8$	$6$	$14$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+8q^{3}+4q^{4}+6q^{5}-2^{4}q^{6}+\cdots$
86.4.a.b	$86$	$4$	86.a	1.a	$1$	$1$	$1$	$5.074$	$\Q$	None	✓	$1$	$1$	$2$	$-4$	$-14$	$-14$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}-4q^{3}+4q^{4}-14q^{5}-8q^{6}+\cdots$
86.4.a.c	$86$	$4$	86.a	1.a	$1$	$2$	$2$	$5.074$	$\Q(\sqrt{3}) $	None	✓	$1$	$1$	$-4$	$2$	$-14$	$-22$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+(1+3\beta )q^{3}+4q^{4}+(-7-7\beta )q^{5}+\cdots$
86.4.a.d	$86$	$4$	86.a	1.a	$1$	$3$	$3$	$5.074$	3.3.85685.1	None	✓	$1$	$0$	$-6$	$-9$	$-5$	$-8$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+(-3-\beta _{1})q^{3}+4q^{4}+(-2+\cdots)q^{5}+\cdots$
86.4.a.e	$86$	$4$	86.a	1.a	$1$	$4$	$4$	$5.074$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	$1$	$0$	$8$	$11$	$1$	$42$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+(3-\beta _{1})q^{3}+4q^{4}+(\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$
86.4.c.a	$86$	$4$	86.c	43.c	$3$	$4$	$2$	$5.074$	$\Q(\sqrt{-3}, \sqrt{10})$	None		$2$	$0$	$8$	$-6$	$2$	$-34$		$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q+2q^{2}+(-3-\beta _{1}-3\beta _{2})q^{3}+4q^{4}+\cdots$
86.4.c.b	$86$	$4$	86.c	43.c	$3$	$6$	$3$	$5.074$	6.0.$\cdots$.1	None		$2$	$0$	$12$	$2$	$11$	$39$		$3$	$\mathrm{SU}(2)[C_{3}]$	$q+2q^{2}+(1-\beta _{1}-\beta _{3})q^{3}+4q^{4}+(3+\cdots)q^{5}+\cdots$
86.4.c.c	$86$	$4$	86.c	43.c	$3$	$12$	$6$	$5.074$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None		$2$	$0$	$-24$	$8$	$13$	$49$		$2^{4}\cdot 3$	$\mathrm{SU}(2)[C_{3}]$	$q-2q^{2}+(1+\beta _{1}-\beta _{3})q^{3}+4q^{4}+(2+\cdots)q^{5}+\cdots$
86.4.e.a	$86$	$4$	86.e	43.e	$7$	$30$	$5$	$5.074$		None		$2$	$0$	$-10$	$-14$	$13$	$-28$			$\mathrm{SU}(2)[C_{7}]$
86.4.e.b	$86$	$4$	86.e	43.e	$7$	$36$	$6$	$5.074$		None		$2$	$0$	$12$	$6$	$13$	$16$			$\mathrm{SU}(2)[C_{7}]$
86.4.g.a	$86$	$4$	86.g	43.g	$21$	$60$	$5$	$5.074$		None		$2$	$0$	$-20$	$11$	$-13$	$-5$			$\mathrm{SU}(2)[C_{21}]$
86.4.g.b	$86$	$4$	86.g	43.g	$21$	$72$	$6$	$5.074$		None		$2$	$0$	$24$	$-15$	$-13$	$-49$			$\mathrm{SU}(2)[C_{21}]$
86.5.b.a	$86$	$5$	86.b	43.b	$2$	$16$	$16$	$8.890$	$\mathbb{Q}[x]/(x^{16} + \cdots)$	None		$2$	$0$	$0$	$0$	$0$	$0$		$2^{14}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{9}q^{2}+(\beta _{8}+\beta _{9})q^{3}-8q^{4}+\beta _{10}q^{5}+\cdots$
86.5.d.a	$86$	$5$	86.d	43.d	$6$	$28$	$14$	$8.890$		None		$2$	$0$	$0$	$-18$	$0$	$-216$			$\mathrm{SU}(2)[C_{6}]$
86.5.f.a	$86$	$5$	86.f	43.f	$14$	$96$	$16$	$8.890$		None		$2$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{14}]$
86.5.h.a	$86$	$5$	86.h	43.h	$42$	$168$	$14$	$8.890$		None		$2$	$0$	$0$	$18$	$0$	$216$			$\mathrm{SU}(2)[C_{42}]$
86.6.a.a	$86$	$6$	86.a	1.a	$1$	$3$	$3$	$13.793$	3.3.159992.1	None	✓	$1$	$1$	$-12$	$8$	$-14$	$-74$	$+$	$2$	$\mathrm{SU}(2)$	$q-4q^{2}+(3+\beta _{2})q^{3}+2^{4}q^{4}+(-5+\cdots)q^{5}+\cdots$
86.6.a.b	$86$	$6$	86.a	1.a	$1$	$3$	$3$	$13.793$	3.3.146508.1	None	✓	$1$	$1$	$12$	$-28$	$-14$	$-182$	$+$	$2$	$\mathrm{SU}(2)$	$q+4q^{2}+(-9+\beta _{2})q^{3}+2^{4}q^{4}+(-5+\cdots)q^{5}+\cdots$
86.6.a.c	$86$	$6$	86.a	1.a	$1$	$5$	$5$	$13.793$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓	$1$	$0$	$-20$	$-1$	$61$	$122$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	$q-4q^{2}+\beta _{1}q^{3}+2^{4}q^{4}+(12-2\beta _{1}+\cdots)q^{5}+\cdots$
86.6.a.d	$86$	$6$	86.a	1.a	$1$	$6$	$6$	$13.793$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓	$1$	$0$	$24$	$17$	$61$	$210$	$-$	$2$	$\mathrm{SU}(2)$	$q+4q^{2}+(3-\beta _{1})q^{3}+2^{4}q^{4}+(10-\beta _{1}+\cdots)q^{5}+\cdots$
86.6.c.a	$86$	$6$	86.c	43.c	$3$	$18$	$9$	$13.793$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	None		$2$	$0$	$-72$	$17$	$-47$	$-107$		$2^{12}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q-4q^{2}+(-\beta _{1}-2\beta _{2})q^{3}+2^{4}q^{4}+\cdots$
86.6.c.b	$86$	$6$	86.c	43.c	$3$	$20$	$10$	$13.793$	$\mathbb{Q}[x]/(x^{20} - \cdots)$	None		$2$	$0$	$80$	$5$	$-47$	$-87$		$2^{12}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q+4q^{2}+(1+\beta _{1}-\beta _{2}-\beta _{3})q^{3}+2^{4}q^{4}+\cdots$
86.6.e.a	$86$	$6$	86.e	43.e	$7$	$48$	$8$	$13.793$		None		$2$	$0$	$32$	$28$	$-47$	$-48$			$\mathrm{SU}(2)[C_{7}]$
86.6.e.b	$86$	$6$	86.e	43.e	$7$	$54$	$9$	$13.793$		None		$2$	$0$	$-36$	$-24$	$-47$	$-28$			$\mathrm{SU}(2)[C_{7}]$
86.6.g.a	$86$	$6$	86.g	43.g	$21$	$108$	$9$	$13.793$		None		$2$	$0$	$72$	$-52$	$47$	$107$			$\mathrm{SU}(2)[C_{21}]$
86.6.g.b	$86$	$6$	86.g	43.g	$21$	$120$	$10$	$13.793$		None		$2$	$0$	$-80$	$30$	$47$	$87$			$\mathrm{SU}(2)[C_{21}]$
86.7.b.a	$86$	$7$	86.b	43.b	$2$	$22$	$22$	$19.785$		None		$2$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{2}]$
86.7.d.a	$86$	$7$	86.d	43.d	$6$	$44$	$22$	$19.785$		None		$2$	$0$	$0$	$0$	$0$	$-792$			$\mathrm{SU}(2)[C_{6}]$
86.7.f.a	$86$	$7$	86.f	43.f	$14$	$132$	$22$	$19.785$		None		$2$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{14}]$
86.7.h.a	$86$	$7$	86.h	43.h	$42$	$264$	$22$	$19.785$		None		$2$	$0$	$0$	$0$	$0$	$792$			$\mathrm{SU}(2)[C_{42}]$
86.8.a.a	$86$	$8$	86.a	1.a	$1$	$5$	$5$	$26.865$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓	$1$	$1$	$-40$	$-13$	$-39$	$-972$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	$q-8q^{2}+(-3+\beta _{4})q^{3}+2^{6}q^{4}+(-9+\cdots)q^{5}+\cdots$
86.8.a.b	$86$	$8$	86.a	1.a	$1$	$5$	$5$	$26.865$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓	$1$	$1$	$40$	$-67$	$-249$	$-1760$	$-$	$2$	$\mathrm{SU}(2)$	$q+8q^{2}+(-13-\beta _{2})q^{3}+2^{6}q^{4}+(-51+\cdots)q^{5}+\cdots$
86.8.a.c	$86$	$8$	86.a	1.a	$1$	$7$	$7$	$26.865$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓	$1$	$0$	$-56$	$-40$	$336$	$400$	$+$	$2^{3}$	$\mathrm{SU}(2)$	$q-8q^{2}+(-6+\beta _{1})q^{3}+2^{6}q^{4}+(48+\cdots)q^{5}+\cdots$
86.8.a.d	$86$	$8$	86.a	1.a	$1$	$8$	$8$	$26.865$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓	$1$	$0$	$64$	$68$	$126$	$984$	$+$	$2^{3}$	$\mathrm{SU}(2)$	$q+8q^{2}+(9-\beta _{1})q^{3}+2^{6}q^{4}+(17-2\beta _{1}+\cdots)q^{5}+\cdots$
86.8.c.a	$86$	$8$	86.c	43.c	$3$	$24$	$12$	$26.865$		None		$2$	$0$	$192$	$59$	$123$	$1621$			$\mathrm{SU}(2)[C_{3}]$
86.8.c.b	$86$	$8$	86.c	43.c	$3$	$26$	$13$	$26.865$		None		$2$	$0$	$-208$	$-85$	$123$	$-615$			$\mathrm{SU}(2)[C_{3}]$
86.8.e.a	$86$	$8$	86.e	43.e	$7$	$78$	$13$	$26.865$		None		$2$	$0$	$-104$	$202$	$123$	$776$			$\mathrm{SU}(2)[C_{7}]$

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Elliptic curves over $\Q$
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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.