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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
322.2.a.a	$322$	$2$	322.a	1.a	$1$	$1$	$1$	$2.571$	$\Q$	None	✓	$1$	$1$	$-1$	$0$	$-2$	$1$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-2q^{5}+q^{7}-q^{8}-3q^{9}+\cdots$
322.2.a.b	$322$	$2$	322.a	1.a	$1$	$1$	$1$	$2.571$	$\Q$	None	✓	$1$	$0$	$-1$	$2$	$0$	$1$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+2q^{3}+q^{4}-2q^{6}+q^{7}-q^{8}+\cdots$
322.2.a.c	$322$	$2$	322.a	1.a	$1$	$1$	$1$	$2.571$	$\Q$	None	✓	$1$	$1$	$1$	$-2$	$-2$	$-1$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-2q^{3}+q^{4}-2q^{5}-2q^{6}-q^{7}+\cdots$
322.2.a.d	$322$	$2$	322.a	1.a	$1$	$1$	$1$	$2.571$	$\Q$	None	✓	$1$	$0$	$1$	$2$	$-2$	$1$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+2q^{3}+q^{4}-2q^{5}+2q^{6}+q^{7}+\cdots$
322.2.a.e	$322$	$2$	322.a	1.a	$1$	$2$	$2$	$2.571$	$\Q(\sqrt{5}) $	None	✓	$1$	$1$	$-2$	$-2$	$-2$	$-2$	$+$	$2$	$\mathrm{SU}(2)$	$q-q^{2}+(-1-\beta )q^{3}+q^{4}+(-1+\beta )q^{5}+\cdots$
322.2.a.f	$322$	$2$	322.a	1.a	$1$	$2$	$2$	$2.571$	$\Q(\sqrt{3}) $	None	✓	$1$	$0$	$2$	$-2$	$2$	$2$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+(-1+\beta )q^{3}+q^{4}+(1+\beta )q^{5}+\cdots$
322.2.a.g	$322$	$2$	322.a	1.a	$1$	$3$	$3$	$2.571$	3.3.316.1	None	✓	$1$	$0$	$3$	$2$	$4$	$-3$	$-$	$2$	$\mathrm{SU}(2)$	$q+q^{2}+(1+\beta _{2})q^{3}+q^{4}+(1-\beta _{2})q^{5}+\cdots$
322.2.c.a	$322$	$2$	322.c	161.c	$2$	$4$	$4$	$2.571$	$\Q(\sqrt{-2}, \sqrt{7})$	None		$4$	$0$	$-4$	$0$	$0$	$0$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q-q^{2}+\beta _{1}q^{3}+q^{4}-\beta _{1}q^{6}-\beta _{3}q^{7}+\cdots$
322.2.c.b	$322$	$2$	322.c	161.c	$2$	$4$	$4$	$2.571$	$\Q(\sqrt{-2}, \sqrt{-7})$	None		$4$	$0$	$-4$	$0$	$0$	$0$		$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q-q^{2}+\beta _{2}q^{3}+q^{4}+\beta _{1}q^{5}-\beta _{2}q^{6}+\cdots$
322.2.c.c	$322$	$2$	322.c	161.c	$2$	$4$	$4$	$2.571$	4.0.2312.1	None		$2$	$0$	$4$	$0$	$-8$	$-2$		$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+q^{2}-\beta _{2}q^{3}+q^{4}-2q^{5}-\beta _{2}q^{6}+\cdots$
322.2.c.d	$322$	$2$	322.c	161.c	$2$	$4$	$4$	$2.571$	4.0.2312.1	None		$2$	$0$	$4$	$0$	$8$	$2$		$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+q^{2}-\beta _{2}q^{3}+q^{4}+2q^{5}-\beta _{2}q^{6}+\cdots$
322.2.e.a	$322$	$2$	322.e	7.c	$3$	$8$	$4$	$2.571$	8.0.310217769.2	None		$2$	$0$	$-4$	$-3$	$-7$	$-1$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\beta _{5})q^{2}+(\beta _{1}+\beta _{2}-\beta _{5})q^{3}+\cdots$
322.2.e.b	$322$	$2$	322.e	7.c	$3$	$8$	$4$	$2.571$	8.0.1767277521.3	None		$2$	$0$	$-4$	$5$	$5$	$1$		$3$	$\mathrm{SU}(2)[C_{3}]$	$q-\beta _{5}q^{2}+(1+\beta _{2}-\beta _{5}+\beta _{6})q^{3}+(-1+\cdots)q^{4}+\cdots$
322.2.e.c	$322$	$2$	322.e	7.c	$3$	$8$	$4$	$2.571$	8.0.1767277521.3	None		$2$	$0$	$4$	$-1$	$-3$	$-1$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\beta _{6})q^{2}+(\beta _{4}+\beta _{5}-\beta _{6}-\beta _{7})q^{3}+\cdots$
322.2.e.d	$322$	$2$	322.e	7.c	$3$	$8$	$4$	$2.571$	8.0.6498455769.2	None		$2$	$0$	$4$	$-1$	$5$	$-3$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\beta _{5})q^{2}-\beta _{1}q^{3}-\beta _{5}q^{4}+(1+\beta _{2}+\cdots)q^{5}+\cdots$
322.2.g.a	$322$	$2$	322.g	161.g	$6$	$16$	$8$	$2.571$	$\mathbb{Q}[x]/(x^{16} + \cdots)$	None		$4$	$0$	$-8$	$6$	$0$	$0$		$7$	$\mathrm{SU}(2)[C_{6}]$	$q-\beta _{5}q^{2}-\beta _{1}q^{3}+(-1+\beta _{5})q^{4}+\beta _{6}q^{5}+\cdots$
322.2.g.b	$322$	$2$	322.g	161.g	$6$	$16$	$8$	$2.571$	$\mathbb{Q}[x]/(x^{16} + \cdots)$	None		$4$	$0$	$8$	$-6$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q-\beta _{6}q^{2}+\beta _{11}q^{3}+(-1-\beta _{6})q^{4}+\cdots$
322.2.i.a	$322$	$2$	322.i	23.c	$11$	$10$	$1$	$2.571$	$\Q(\zeta_{22})$	None		$2$	$0$	$1$	$-3$	$5$	$1$		$1$	$\mathrm{SU}(2)[C_{11}]$	$q-\zeta_{22}^{4}q^{2}+(-1-\zeta_{22}^{2}+\zeta_{22}^{3}+\cdots)q^{3}+\cdots$
322.2.i.b	$322$	$2$	322.i	23.c	$11$	$10$	$1$	$2.571$	$\Q(\zeta_{22})$	None		$2$	$0$	$1$	$5$	$8$	$1$		$1$	$\mathrm{SU}(2)[C_{11}]$	$q-\zeta_{22}^{4}q^{2}+(1+\zeta_{22}^{2}-\zeta_{22}^{3}+\zeta_{22}^{6}+\cdots)q^{3}+\cdots$
322.2.i.c	$322$	$2$	322.i	23.c	$11$	$20$	$2$	$2.571$	$\mathbb{Q}[x]/(x^{20} - \cdots)$	None		$2$	$0$	$-2$	$4$	$-5$	$-2$		$1$	$\mathrm{SU}(2)[C_{11}]$	$q+\beta _{19}q^{2}+(\beta _{1}+\beta _{2}+\beta _{4}+\beta _{5}-\beta _{6}+\cdots)q^{3}+\cdots$
322.2.i.d	$322$	$2$	322.i	23.c	$11$	$40$	$4$	$2.571$		None		$2$	$0$	$-4$	$0$	$9$	$4$			$\mathrm{SU}(2)[C_{11}]$
322.2.i.e	$322$	$2$	322.i	23.c	$11$	$40$	$4$	$2.571$		None		$2$	$0$	$4$	$2$	$-9$	$-4$			$\mathrm{SU}(2)[C_{11}]$
322.2.k.a	$322$	$2$	322.k	161.k	$22$	$80$	$8$	$2.571$		None		$4$	$0$	$-8$	$0$	$0$	$-11$			$\mathrm{SU}(2)[C_{22}]$
322.2.k.b	$322$	$2$	322.k	161.k	$22$	$80$	$8$	$2.571$		None		$4$	$0$	$8$	$0$	$0$	$11$			$\mathrm{SU}(2)[C_{22}]$
322.2.m.a	$322$	$2$	322.m	161.m	$33$	$160$	$8$	$2.571$		None		$4$	$0$	$-8$	$2$	$-2$	$15$			$\mathrm{SU}(2)[C_{33}]$
322.2.m.b	$322$	$2$	322.m	161.m	$33$	$160$	$8$	$2.571$		None		$4$	$0$	$8$	$-2$	$2$	$-11$			$\mathrm{SU}(2)[C_{33}]$
322.2.o.a	$322$	$2$	322.o	161.o	$66$	$160$	$8$	$2.571$		None		$4$	$0$	$-8$	$6$	$0$	$-11$			$\mathrm{SU}(2)[C_{66}]$
322.2.o.b	$322$	$2$	322.o	161.o	$66$	$160$	$8$	$2.571$		None		$4$	$0$	$8$	$-6$	$0$	$11$			$\mathrm{SU}(2)[C_{66}]$
322.3.b.a	$322$	$3$	322.b	7.b	$2$	$32$	$32$	$8.774$		None		$2$	$0$	$0$	$0$	$0$	$-12$			$\mathrm{SU}(2)[C_{2}]$
322.3.d.a	$322$	$3$	322.d	23.b	$2$	$24$	$24$	$8.774$		None		$2$	$0$	$0$	$8$	$0$	$0$			$\mathrm{SU}(2)[C_{2}]$
322.3.f.a	$322$	$3$	322.f	161.f	$6$	$64$	$32$	$8.774$		None		$4$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{6}]$
322.3.h.a	$322$	$3$	322.h	7.d	$6$	$56$	$28$	$8.774$		None		$2$	$0$	$0$	$12$	$0$	$-4$			$\mathrm{SU}(2)[C_{6}]$
322.3.j.a	$322$	$3$	322.j	23.d	$22$	$240$	$24$	$8.774$		None		$2$	$0$	$0$	$-8$	$0$	$0$			$\mathrm{SU}(2)[C_{22}]$
322.3.l.a	$322$	$3$	322.l	161.l	$22$	$320$	$32$	$8.774$		None		$4$	$0$	$0$	$0$	$0$	$12$			$\mathrm{SU}(2)[C_{22}]$
322.3.n.a	$322$	$3$	322.n	161.n	$66$	$640$	$32$	$8.774$		None		$4$	$0$	$0$	$0$	$12$	$-12$			$\mathrm{SU}(2)[C_{66}]$
322.3.p.a	$322$	$3$	322.p	161.p	$66$	$640$	$32$	$8.774$		None		$4$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{66}]$
322.4.a.a	$322$	$4$	322.a	1.a	$1$	$2$	$2$	$18.999$	$\Q(\sqrt{2}) $	None	✓	$1$	$0$	$-4$	$-4$	$16$	$14$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+(-2+3\beta )q^{3}+4q^{4}+(8-3\beta )q^{5}+\cdots$
322.4.a.b	$322$	$4$	322.a	1.a	$1$	$2$	$2$	$18.999$	$\Q(\sqrt{73}) $	None	✓	$1$	$0$	$-4$	$9$	$-8$	$14$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+(5-\beta )q^{3}+4q^{4}+(-2-4\beta )q^{5}+\cdots$
322.4.a.c	$322$	$4$	322.a	1.a	$1$	$2$	$2$	$18.999$	$\Q(\sqrt{57}) $	None	✓	$1$	$1$	$4$	$1$	$-18$	$14$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+\beta q^{3}+4q^{4}+(-8-2\beta )q^{5}+\cdots$
322.4.a.d	$322$	$4$	322.a	1.a	$1$	$3$	$3$	$18.999$	3.3.15384.1	None	✓	$1$	$0$	$-6$	$-3$	$8$	$-21$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+(-1+\beta _{1})q^{3}+4q^{4}+(3+2\beta _{1}+\cdots)q^{5}+\cdots$
322.4.a.e	$322$	$4$	322.a	1.a	$1$	$3$	$3$	$18.999$	3.3.3368.1	None	✓	$1$	$1$	$6$	$-7$	$-8$	$-21$	$-$	$2$	$\mathrm{SU}(2)$	$q+2q^{2}+(-2-\beta _{2})q^{3}+4q^{4}+(-3+\cdots)q^{5}+\cdots$
322.4.a.f	$322$	$4$	322.a	1.a	$1$	$4$	$4$	$18.999$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	$1$	$1$	$-8$	$-1$	$-2$	$28$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}-\beta _{1}q^{3}+4q^{4}-\beta _{2}q^{5}+2\beta _{1}q^{6}+\cdots$
322.4.a.g	$322$	$4$	322.a	1.a	$1$	$5$	$5$	$18.999$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓	$1$	$1$	$-10$	$-9$	$-2$	$-35$	$-$	$2\cdot 3$	$\mathrm{SU}(2)$	$q-2q^{2}+(-2+\beta _{4})q^{3}+4q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots$
322.4.a.h	$322$	$4$	322.a	1.a	$1$	$5$	$5$	$18.999$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓	$1$	$0$	$10$	$5$	$22$	$-35$	$+$	$2$	$\mathrm{SU}(2)$	$q+2q^{2}+(1-\beta _{1})q^{3}+4q^{4}+(4-\beta _{1}+\cdots)q^{5}+\cdots$
322.4.a.i	$322$	$4$	322.a	1.a	$1$	$6$	$6$	$18.999$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓	$1$	$0$	$12$	$13$	$12$	$42$	$+$	$2^{2}$	$\mathrm{SU}(2)$	$q+2q^{2}+(2+\beta _{1})q^{3}+4q^{4}+(2-\beta _{1}+\cdots)q^{5}+\cdots$
322.4.c.a	$322$	$4$	322.c	161.c	$2$	$24$	$24$	$18.999$		None		$4$	$0$	$-48$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{2}]$
322.4.c.b	$322$	$4$	322.c	161.c	$2$	$24$	$24$	$18.999$		None		$4$	$0$	$48$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{2}]$
322.4.e.a	$322$	$4$	322.e	7.c	$3$	$22$	$11$	$18.999$		None		$2$	$0$	$-22$	$-18$	$-33$	$33$			$\mathrm{SU}(2)[C_{3}]$
322.4.e.b	$322$	$4$	322.e	7.c	$3$	$22$	$11$	$18.999$		None		$2$	$0$	$-22$	$6$	$27$	$1$			$\mathrm{SU}(2)[C_{3}]$
322.4.e.c	$322$	$4$	322.e	7.c	$3$	$22$	$11$	$18.999$		None		$2$	$0$	$22$	$0$	$-17$	$11$			$\mathrm{SU}(2)[C_{3}]$

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Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 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.