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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
670.2.a.a	$670$	$2$	670.a	1.a	$1$	$1$	$1$	$5.350$	$\Q$	None	✓	$1$	$1$	$-1$	$-2$	$1$	$-1$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-2q^{3}+q^{4}+q^{5}+2q^{6}-q^{7}+\cdots$
670.2.a.b	$670$	$2$	670.a	1.a	$1$	$1$	$1$	$5.350$	$\Q$	None	✓	$1$	$1$	$-1$	$0$	$1$	$1$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+q^{5}+q^{7}-q^{8}-3q^{9}+\cdots$
670.2.a.c	$670$	$2$	670.a	1.a	$1$	$1$	$1$	$5.350$	$\Q$	None	✓	$1$	$1$	$1$	$-2$	$-1$	$1$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-2q^{3}+q^{4}-q^{5}-2q^{6}+q^{7}+\cdots$
670.2.a.d	$670$	$2$	670.a	1.a	$1$	$1$	$1$	$5.350$	$\Q$	None	✓	$1$	$1$	$1$	$0$	$-1$	$-5$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-q^{5}-5q^{7}+q^{8}-3q^{9}+\cdots$
670.2.a.e	$670$	$2$	670.a	1.a	$1$	$2$	$2$	$5.350$	$\Q(\sqrt{2}) $	None	✓	$1$	$1$	$-2$	$0$	$-2$	$2$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+\beta q^{3}+q^{4}-q^{5}-\beta q^{6}+(1+\cdots)q^{7}+\cdots$
670.2.a.f	$670$	$2$	670.a	1.a	$1$	$2$	$2$	$5.350$	$\Q(\sqrt{2}) $	None	✓	$1$	$1$	$2$	$-4$	$2$	$-6$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(-2+\beta )q^{3}+q^{4}+q^{5}+(-2+\cdots)q^{6}+\cdots$
670.2.a.g	$670$	$2$	670.a	1.a	$1$	$3$	$3$	$5.350$	3.3.148.1	None	✓	$1$	$0$	$-3$	$4$	$3$	$1$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(1+\beta _{1})q^{3}+q^{4}+q^{5}+(-1+\cdots)q^{6}+\cdots$
670.2.a.h	$670$	$2$	670.a	1.a	$1$	$3$	$3$	$5.350$	3.3.756.1	None	✓	$1$	$0$	$3$	$0$	$-3$	$3$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-\beta _{1}q^{3}+q^{4}-q^{5}-\beta _{1}q^{6}+\cdots$
670.2.a.i	$670$	$2$	670.a	1.a	$1$	$3$	$3$	$5.350$	3.3.404.1	None	✓	$1$	$0$	$3$	$2$	$3$	$1$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(1-\beta _{1})q^{3}+q^{4}+q^{5}+(1-\beta _{1}+\cdots)q^{6}+\cdots$
670.2.a.j	$670$	$2$	670.a	1.a	$1$	$4$	$4$	$5.350$	4.4.15188.1	None	✓	$1$	$0$	$-4$	$-2$	$-4$	$-5$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+(-1-\beta _{2})q^{3}+q^{4}-q^{5}+(1+\cdots)q^{6}+\cdots$
670.2.c.a	$670$	$2$	670.c	5.b	$2$	$10$	$10$	$5.350$	10.0.$\cdots$.1	None		$2$	$0$	$0$	$0$	$-2$	$0$		$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{3}q^{2}-\beta _{8}q^{3}-q^{4}+(\beta _{1}-\beta _{7})q^{5}+\cdots$
670.2.c.b	$670$	$2$	670.c	5.b	$2$	$24$	$24$	$5.350$		None		$2$	$0$	$0$	$0$	$-2$	$0$			$\mathrm{SU}(2)[C_{2}]$
670.2.e.a	$670$	$2$	670.e	67.c	$3$	$2$	$1$	$5.350$	$\Q(\sqrt{-3}) $	None		$2$	$1$	$-1$	$-6$	$-2$	$2$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\zeta_{6})q^{2}-3q^{3}-\zeta_{6}q^{4}-q^{5}+\cdots$
670.2.e.b	$670$	$2$	670.e	67.c	$3$	$2$	$1$	$5.350$	$\Q(\sqrt{-3}) $	None		$2$	$1$	$-1$	$-6$	$2$	$-4$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\zeta_{6})q^{2}-3q^{3}-\zeta_{6}q^{4}+q^{5}+\cdots$
670.2.e.c	$670$	$2$	670.e	67.c	$3$	$2$	$1$	$5.350$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$-1$	$2$	$-2$	$5$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\zeta_{6})q^{2}+q^{3}-\zeta_{6}q^{4}-q^{5}+\cdots$
670.2.e.d	$670$	$2$	670.e	67.c	$3$	$2$	$1$	$5.350$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$1$	$2$	$-2$	$-2$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\zeta_{6})q^{2}+q^{3}-\zeta_{6}q^{4}-q^{5}+(1+\cdots)q^{6}+\cdots$
670.2.e.e	$670$	$2$	670.e	67.c	$3$	$4$	$2$	$5.350$	$\Q(\sqrt{-3}, \sqrt{193})$	None		$2$	$0$	$-2$	$4$	$-4$	$-4$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q-\beta _{2}q^{2}+q^{3}+(-1+\beta _{2})q^{4}-q^{5}+\cdots$
670.2.e.f	$670$	$2$	670.e	67.c	$3$	$4$	$2$	$5.350$	$\Q(\sqrt{-2}, \sqrt{-3})$	None		$2$	$0$	$-2$	$4$	$-4$	$-4$		$3$	$\mathrm{SU}(2)[C_{3}]$	$q-\beta _{1}q^{2}+(1-\beta _{3})q^{3}+(-1+\beta _{1})q^{4}+\cdots$
670.2.e.g	$670$	$2$	670.e	67.c	$3$	$6$	$3$	$5.350$	6.0.954288.1	None		$2$	$0$	$3$	$-2$	$6$	$-2$		$3$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\beta _{3})q^{2}+\beta _{4}q^{3}-\beta _{3}q^{4}+q^{5}+\cdots$
670.2.e.h	$670$	$2$	670.e	67.c	$3$	$6$	$3$	$5.350$	6.0.2696112.1	None		$2$	$0$	$3$	$2$	$6$	$2$		$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{3}q^{2}-\beta _{2}q^{3}+(-1+\beta _{3})q^{4}+q^{5}+\cdots$
670.2.e.i	$670$	$2$	670.e	67.c	$3$	$8$	$4$	$5.350$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None		$2$	$0$	$-4$	$8$	$8$	$0$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\beta _{2})q^{2}+(1+\beta _{3})q^{3}-\beta _{2}q^{4}+\cdots$
670.2.e.j	$670$	$2$	670.e	67.c	$3$	$12$	$6$	$5.350$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None		$2$	$0$	$6$	$-4$	$-12$	$3$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q-\beta _{5}q^{2}-\beta _{2}q^{3}+(-1-\beta _{5})q^{4}-q^{5}+\cdots$
670.2.f.a	$670$	$2$	670.f	335.f	$4$	$68$	$34$	$5.350$		None		$4$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{4}]$
670.2.h.a	$670$	$2$	670.h	335.i	$6$	$4$	$2$	$5.350$	$\Q(\zeta_{12})$	None		$4$	$0$	$0$	$0$	$-4$	$0$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+\zeta_{12}q^{2}-\zeta_{12}^{3}q^{3}+\zeta_{12}^{2}q^{4}+(-1+\cdots)q^{5}+\cdots$
670.2.h.b	$670$	$2$	670.h	335.i	$6$	$12$	$6$	$5.350$	12.0.$\cdots$.1	None		$4$	$0$	$0$	$0$	$24$	$0$		$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	$q+(\beta _{3}+\beta _{4})q^{2}+(-\beta _{4}-\beta _{9})q^{3}+(1+\cdots)q^{4}+\cdots$
670.2.h.c	$670$	$2$	670.h	335.i	$6$	$52$	$26$	$5.350$		None		$4$	$0$	$0$	$0$	$-16$	$0$			$\mathrm{SU}(2)[C_{6}]$
670.2.k.a	$670$	$2$	670.k	67.e	$11$	$40$	$4$	$5.350$		None		$2$	$0$	$-4$	$6$	$-4$	$13$			$\mathrm{SU}(2)[C_{11}]$
670.2.k.b	$670$	$2$	670.k	67.e	$11$	$50$	$5$	$5.350$		None		$2$	$0$	$-5$	$2$	$5$	$1$			$\mathrm{SU}(2)[C_{11}]$
670.2.k.c	$670$	$2$	670.k	67.e	$11$	$50$	$5$	$5.350$		None		$2$	$0$	$5$	$-2$	$-5$	$-1$			$\mathrm{SU}(2)[C_{11}]$
670.2.k.d	$670$	$2$	670.k	67.e	$11$	$60$	$6$	$5.350$		None		$2$	$0$	$6$	$2$	$6$	$3$			$\mathrm{SU}(2)[C_{11}]$
670.2.m.a	$670$	$2$	670.m	335.m	$12$	$136$	$34$	$5.350$		None		$4$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{12}]$
670.2.o.a	$670$	$2$	670.o	335.o	$22$	$340$	$34$	$5.350$		None		$4$	$0$	$0$	$0$	$4$	$0$			$\mathrm{SU}(2)[C_{22}]$
670.2.q.a	$670$	$2$	670.q	67.g	$33$	$100$	$5$	$5.350$		None		$2$	$0$	$5$	$-2$	$-10$	$4$			$\mathrm{SU}(2)[C_{33}]$
670.2.q.b	$670$	$2$	670.q	67.g	$33$	$120$	$6$	$5.350$		None		$2$	$0$	$-6$	$0$	$-12$	$0$			$\mathrm{SU}(2)[C_{33}]$
670.2.q.c	$670$	$2$	670.q	67.g	$33$	$120$	$6$	$5.350$		None		$2$	$0$	$6$	$-4$	$12$	$1$			$\mathrm{SU}(2)[C_{33}]$
670.2.q.d	$670$	$2$	670.q	67.g	$33$	$140$	$7$	$5.350$		None		$2$	$0$	$-7$	$2$	$14$	$-1$			$\mathrm{SU}(2)[C_{33}]$
670.2.s.a	$670$	$2$	670.s	335.s	$44$	$680$	$34$	$5.350$		None		$4$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{44}]$
670.2.v.a	$670$	$2$	670.v	335.u	$66$	$680$	$34$	$5.350$		None		$4$	$0$	$0$	$0$	$-4$	$0$			$\mathrm{SU}(2)[C_{66}]$
670.2.w.a	$670$	$2$	670.w	335.w	$132$	$1360$	$34$	$5.350$		None		$4$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{132}]$
670.3.b.a	$670$	$3$	670.b	67.b	$2$	$48$	$48$	$18.256$		None		$2$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{2}]$
670.3.d.a	$670$	$3$	670.d	335.d	$2$	$68$	$68$	$18.256$		None		$4$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{2}]$
670.3.i.a	$670$	$3$	670.i	67.d	$6$	$88$	$44$	$18.256$		None		$2$	$0$	$0$	$0$	$0$	$-36$			$\mathrm{SU}(2)[C_{6}]$
670.4.a.a	$670$	$4$	670.a	1.a	$1$	$1$	$1$	$39.531$	$\Q$	None	✓	$1$	$1$	$-2$	$-6$	$-5$	$31$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-6q^{3}+4q^{4}-5q^{5}+12q^{6}+\cdots$
670.4.a.b	$670$	$4$	670.a	1.a	$1$	$1$	$1$	$39.531$	$\Q$	None	✓	$1$	$0$	$-2$	$7$	$-5$	$31$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+7q^{3}+4q^{4}-5q^{5}-14q^{6}+\cdots$
670.4.a.c	$670$	$4$	670.a	1.a	$1$	$6$	$6$	$39.531$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓	$1$	$1$	$-12$	$7$	$-30$	$-10$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+(1+\beta _{1})q^{3}+4q^{4}-5q^{5}+\cdots$
670.4.a.d	$670$	$4$	670.a	1.a	$1$	$7$	$7$	$39.531$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓	$1$	$1$	$14$	$-13$	$35$	$-49$	$-$	$2$	$\mathrm{SU}(2)$	$q+2q^{2}+(-2+\beta _{5})q^{3}+4q^{4}+5q^{5}+\cdots$
670.4.a.e	$670$	$4$	670.a	1.a	$1$	$8$	$8$	$39.531$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓	$1$	$0$	$-16$	$-12$	$-40$	$-59$	$+$	$3$	$\mathrm{SU}(2)$	$q-2q^{2}+(-2+\beta _{1})q^{3}+4q^{4}-5q^{5}+\cdots$
670.4.a.f	$670$	$4$	670.a	1.a	$1$	$8$	$8$	$39.531$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓	$1$	$1$	$-16$	$-7$	$40$	$-1$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+(-1+\beta _{1})q^{3}+4q^{4}+5q^{5}+\cdots$
670.4.a.g	$670$	$4$	670.a	1.a	$1$	$8$	$8$	$39.531$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓	$1$	$1$	$16$	$-5$	$-40$	$-27$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+(-1+\beta _{1})q^{3}+4q^{4}-5q^{5}+\cdots$
670.4.a.h	$670$	$4$	670.a	1.a	$1$	$9$	$9$	$39.531$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	None	✓	$1$	$0$	$-18$	$11$	$45$	$6$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+(1+\beta _{1})q^{3}+4q^{4}+5q^{5}+\cdots$

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Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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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.