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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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Results (1-50 of 117 matches)

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Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	Image	CM	RM	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	Image	CM	RM	Self-dual	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
688.1.b.a	$688$	$1$	688.b	43.b	$2$	$1$	$1$	$0.343$	$\Q$	$D_{3}$	$\Q(\sqrt{-43}) $	None	✓	$2$	$0$	$0$	$0$	$0$	$0$		$1$		$q+q^{9}+q^{11}-q^{13}-q^{17}+q^{23}+\cdots$
688.1.b.b	$688$	$1$	688.b	43.b	$2$	$2$	$2$	$0.343$	$\Q(\sqrt{-2}) $	$S_{4}$	None	None		$2$	$0$	$0$	$0$	$0$	$0$		$1$		$q+\beta q^{3}-\beta q^{5}-\beta q^{7}-q^{9}+q^{11}+\cdots$
688.1.r.a	$688$	$1$	688.r	172.g	$6$	$4$	$2$	$0.343$	$\Q(\zeta_{12})$	$A_{4}$	None	None		$4$	$0$	$0$	$0$	$-2$	$0$		$1$		$q-\zeta_{12}q^{3}+\zeta_{12}^{4}q^{5}+\zeta_{12}^{5}q^{7}-\zeta_{12}^{3}q^{11}+\cdots$
688.2.a.a	$688$	$2$	688.a	1.a	$1$	$1$	$1$	$5.494$	$\Q$	$_{}$	None	None	✓	$1$	$1$	$0$	$0$	$-2$	$2$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{5}+2q^{7}-3q^{9}-q^{11}-q^{13}+\cdots$
688.2.a.b	$688$	$2$	688.a	1.a	$1$	$1$	$1$	$5.494$	$\Q$	$_{}$	None	None	✓	$1$	$1$	$0$	$2$	$-4$	$0$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{3}-4q^{5}+q^{9}-3q^{11}-5q^{13}+\cdots$
688.2.a.c	$688$	$2$	688.a	1.a	$1$	$1$	$1$	$5.494$	$\Q$	$_{}$	None	None	✓	$1$	$0$	$0$	$2$	$0$	$4$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{3}+4q^{7}+q^{9}+3q^{11}-q^{13}+\cdots$
688.2.a.d	$688$	$2$	688.a	1.a	$1$	$2$	$2$	$5.494$	$\Q(\sqrt{2}) $	$_{}$	None	None	✓	$1$	$1$	$0$	$-4$	$0$	$0$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-2+\beta )q^{3}+\beta q^{5}-\beta q^{7}+(3-4\beta )q^{9}+\cdots$
688.2.a.e	$688$	$2$	688.a	1.a	$1$	$2$	$2$	$5.494$	$\Q(\sqrt{5}) $	$_{}$	None	None	✓	$1$	$1$	$0$	$-1$	$-3$	$0$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta q^{3}+(-1-\beta )q^{5}+(-2+4\beta )q^{7}+\cdots$
688.2.a.f	$688$	$2$	688.a	1.a	$1$	$2$	$2$	$5.494$	$\Q(\sqrt{2}) $	$_{}$	None	None	✓	$1$	$0$	$0$	$0$	$4$	$4$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}+(2-\beta )q^{5}+(2-\beta )q^{7}-q^{9}+\cdots$
688.2.a.g	$688$	$2$	688.a	1.a	$1$	$2$	$2$	$5.494$	$\Q(\sqrt{21}) $	$_{}$	None	None	✓	$1$	$0$	$0$	$1$	$3$	$-4$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}+(1+\beta )q^{5}-2q^{7}+(2+\beta )q^{9}+\cdots$
688.2.a.h	$688$	$2$	688.a	1.a	$1$	$2$	$2$	$5.494$	$\Q(\sqrt{3}) $	$_{}$	None	None	✓	$1$	$0$	$0$	$2$	$-2$	$2$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{3}+(-1+\beta )q^{5}+(1-\beta )q^{7}+\cdots$
688.2.a.i	$688$	$2$	688.a	1.a	$1$	$3$	$3$	$5.494$	3.3.229.1	$_{}$	None	None	✓	$1$	$1$	$0$	$-3$	$1$	$-6$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta _{1})q^{3}-\beta _{2}q^{5}-2q^{7}+(1+\cdots)q^{9}+\cdots$
688.2.a.j	$688$	$2$	688.a	1.a	$1$	$5$	$5$	$5.494$	5.5.7998268.1	$_{}$	None	None	✓	$1$	$0$	$0$	$1$	$1$	$-2$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}-\beta _{4}q^{5}+(-1+\beta _{2}-\beta _{4})q^{7}+\cdots$
688.2.g.a	$688$	$2$	688.g	172.d	$2$	$2$	$2$	$5.494$	$\Q(\sqrt{-3}) $	$_{}$	None	None		$2$	$1$	$0$	$-4$	$0$	$-4$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q-2q^{3}-2\zeta_{6}q^{5}-2q^{7}+q^{9}-\zeta_{6}q^{11}+\cdots$
688.2.g.b	$688$	$2$	688.g	172.d	$2$	$2$	$2$	$5.494$	$\Q(\sqrt{-43}) $	$_{}$	$\Q(\sqrt{-43}) $	None		$4$	$0$	$0$	$0$	$0$	$0$		$2$	$\mathrm{U}(1)[D_{2}]$	$q-3q^{9}-\beta q^{11}+3q^{13}-5q^{17}-\beta q^{23}+\cdots$
688.2.g.c	$688$	$2$	688.g	172.d	$2$	$2$	$2$	$5.494$	$\Q(\sqrt{-3}) $	$_{}$	None	None		$2$	$0$	$0$	$4$	$0$	$4$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+2q^{3}+2\zeta_{6}q^{5}+2q^{7}+q^{9}-\zeta_{6}q^{11}+\cdots$
688.2.g.d	$688$	$2$	688.g	172.d	$2$	$4$	$4$	$5.494$	$\Q(\sqrt{-3}, \sqrt{-43})$	$_{}$	$\Q(\sqrt{-43}) $	None		$4$	$0$	$0$	$0$	$0$	$0$		$2^{2}$	$\mathrm{U}(1)[D_{2}]$	$q-3q^{9}+(-\beta _{1}-\beta _{3})q^{11}+(-2+\beta _{2}+\cdots)q^{13}+\cdots$
688.2.g.e	$688$	$2$	688.g	172.d	$2$	$12$	$12$	$5.494$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	$_{}$	None	None		$4$	$0$	$0$	$0$	$0$	$0$		$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{3}q^{3}+\beta _{8}q^{5}+\beta _{10}q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots$
688.2.i.a	$688$	$2$	688.i	43.c	$3$	$2$	$1$	$5.494$	$\Q(\sqrt{-3}) $	$_{}$	None	None		$2$	$1$	$0$	$-3$	$-1$	$1$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-3+3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{5}+\zeta_{6}q^{7}+\cdots$
688.2.i.b	$688$	$2$	688.i	43.c	$3$	$2$	$1$	$5.494$	$\Q(\sqrt{-3}) $	$_{}$	None	None		$2$	$0$	$0$	$-1$	$1$	$5$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\zeta_{6})q^{3}+(1-\zeta_{6})q^{5}+5\zeta_{6}q^{7}+\cdots$
688.2.i.c	$688$	$2$	688.i	43.c	$3$	$2$	$1$	$5.494$	$\Q(\sqrt{-3}) $	$_{}$	None	None		$2$	$0$	$0$	$1$	$-3$	$-1$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\zeta_{6})q^{3}+(-3+3\zeta_{6})q^{5}-\zeta_{6}q^{7}+\cdots$
688.2.i.d	$688$	$2$	688.i	43.c	$3$	$2$	$1$	$5.494$	$\Q(\sqrt{-3}) $	$_{}$	None	None		$2$	$0$	$0$	$1$	$1$	$3$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\zeta_{6})q^{3}+(1-\zeta_{6})q^{5}+3\zeta_{6}q^{7}+\cdots$
688.2.i.e	$688$	$2$	688.i	43.c	$3$	$4$	$2$	$5.494$	$\Q(\sqrt{-3}, \sqrt{5})$	$_{}$	None	None		$2$	$0$	$0$	$-3$	$2$	$-4$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1-\beta _{1}-\beta _{3})q^{3}+(2-2\beta _{1}+2\beta _{3})q^{5}+\cdots$
688.2.i.f	$688$	$2$	688.i	43.c	$3$	$4$	$2$	$5.494$	$\Q(\sqrt{-3}, \sqrt{17})$	$_{}$	None	None		$2$	$0$	$0$	$1$	$-3$	$3$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{1}q^{3}+(\beta _{1}-2\beta _{2})q^{5}+(2+\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots$
688.2.i.g	$688$	$2$	688.i	43.c	$3$	$8$	$4$	$5.494$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	$_{}$	None	None		$2$	$0$	$0$	$-1$	$0$	$-4$		$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q+(\beta _{4}-\beta _{6})q^{3}+\beta _{5}q^{5}+(-1+\beta _{2}+\cdots)q^{7}+\cdots$
688.2.i.h	$688$	$2$	688.i	43.c	$3$	$8$	$4$	$5.494$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	$_{}$	None	None		$2$	$0$	$0$	$3$	$0$	$-2$		$3$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\beta _{1}+\beta _{2})q^{3}-\beta _{5}q^{5}+(\beta _{4}+\beta _{7})q^{7}+\cdots$
688.2.i.i	$688$	$2$	688.i	43.c	$3$	$10$	$5$	$5.494$	$\mathbb{Q}[x]/(x^{10} + \cdots)$	$_{}$	None	None		$2$	$0$	$0$	$5$	$2$	$-4$		$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\beta _{1}+\beta _{7})q^{3}+(\beta _{1}-\beta _{6})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots$
688.2.j.a	$688$	$2$	688.j	688.j	$4$	$172$	$86$	$5.494$		$_{}$	None	None		$4$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{4}]$
688.2.k.a	$688$	$2$	688.k	16.e	$4$	$168$	$84$	$5.494$		$_{}$	None	None		$2$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{4}]$
688.2.o.a	$688$	$2$	688.o	172.f	$6$	$2$	$1$	$5.494$	$\Q(\sqrt{-3}) $	$_{}$	None	None		$2$	$1$	$0$	$-1$	$-3$	$-1$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+(-1+\zeta_{6})q^{3}+(-1-\zeta_{6})q^{5}-\zeta_{6}q^{7}+\cdots$
688.2.o.b	$688$	$2$	688.o	172.f	$6$	$2$	$1$	$5.494$	$\Q(\sqrt{-3}) $	$_{}$	None	None		$2$	$0$	$0$	$1$	$-3$	$1$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+(1-\zeta_{6})q^{3}+(-1-\zeta_{6})q^{5}+\zeta_{6}q^{7}+\cdots$
688.2.o.c	$688$	$2$	688.o	172.f	$6$	$6$	$3$	$5.494$	6.0.1714608.1	$_{}$	None	None		$2$	$0$	$0$	$-3$	$3$	$-3$		$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	$q+\beta _{4}q^{3}+(1-\beta _{1}+\beta _{2}+\beta _{5})q^{5}+(-1+\cdots)q^{7}+\cdots$
688.2.o.d	$688$	$2$	688.o	172.f	$6$	$6$	$3$	$5.494$	6.0.3636603.4	$_{}$	None	None		$2$	$0$	$0$	$-1$	$0$	$2$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+(-\beta _{1}-\beta _{2}+\beta _{4})q^{3}+(\beta _{1}+\beta _{4})q^{7}+\cdots$
688.2.o.e	$688$	$2$	688.o	172.f	$6$	$6$	$3$	$5.494$	6.0.3636603.4	$_{}$	None	None		$2$	$0$	$0$	$1$	$0$	$-2$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+(\beta _{1}+\beta _{2}-\beta _{4})q^{3}+(-\beta _{1}-\beta _{4})q^{7}+\cdots$
688.2.o.f	$688$	$2$	688.o	172.f	$6$	$6$	$3$	$5.494$	6.0.1714608.1	$_{}$	None	None		$2$	$0$	$0$	$3$	$3$	$3$		$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	$q-\beta _{4}q^{3}+(1-\beta _{1}+\beta _{2}+\beta _{5})q^{5}+(1+\cdots)q^{7}+\cdots$
688.2.o.g	$688$	$2$	688.o	172.f	$6$	$16$	$8$	$5.494$	$\mathbb{Q}[x]/(x^{16} + \cdots)$	$_{}$	None	None		$4$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+\beta _{7}q^{3}+\beta _{4}q^{5}+(-\beta _{9}-\beta _{10})q^{7}+\cdots$
688.2.u.a	$688$	$2$	688.u	43.e	$7$	$6$	$1$	$5.494$	$\Q(\zeta_{14})$	$_{}$	None	None		$2$	$1$	$0$	$-8$	$-3$	$-4$		$1$	$\mathrm{SU}(2)[C_{7}]$	$q+(-2+2\zeta_{14}+\zeta_{14}^{3}+\zeta_{14}^{5})q^{3}+\cdots$
688.2.u.b	$688$	$2$	688.u	43.e	$7$	$6$	$1$	$5.494$	$\Q(\zeta_{14})$	$_{}$	None	None		$2$	$0$	$0$	$-2$	$-3$	$0$		$1$	$\mathrm{SU}(2)[C_{7}]$	$q+(-\zeta_{14}^{3}-\zeta_{14}^{5})q^{3}+(-\zeta_{14}+\zeta_{14}^{2}+\cdots)q^{5}+\cdots$
688.2.u.c	$688$	$2$	688.u	43.e	$7$	$6$	$1$	$5.494$	$\Q(\zeta_{14})$	$_{}$	None	None		$2$	$0$	$0$	$3$	$2$	$16$		$1$	$\mathrm{SU}(2)[C_{7}]$	$q+(1-\zeta_{14}+2\zeta_{14}^{4})q^{3}-2\zeta_{14}^{2}q^{5}+\cdots$
688.2.u.d	$688$	$2$	688.u	43.e	$7$	$12$	$2$	$5.494$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	$_{}$	None	None		$2$	$0$	$0$	$6$	$-3$	$4$		$7$	$\mathrm{SU}(2)[C_{7}]$	$q+(\beta _{6}+\beta _{9})q^{3}+(-1+\beta _{3}+\beta _{6}+\beta _{7}+\cdots)q^{5}+\cdots$
688.2.u.e	$688$	$2$	688.u	43.e	$7$	$12$	$2$	$5.494$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	$_{}$	None	None		$2$	$0$	$0$	$10$	$3$	$0$		$1$	$\mathrm{SU}(2)[C_{7}]$	$q+(1-\beta _{10})q^{3}+(-1+\beta _{1}-\beta _{2}-2\beta _{3}+\cdots)q^{5}+\cdots$
688.2.u.f	$688$	$2$	688.u	43.e	$7$	$18$	$3$	$5.494$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	$_{}$	None	None		$2$	$0$	$0$	$-2$	$-3$	$8$		$1$	$\mathrm{SU}(2)[C_{7}]$	$q+(-\beta _{1}-\beta _{2}-\beta _{3}+\beta _{5}-\beta _{6}+\beta _{7}+\cdots)q^{3}+\cdots$
688.2.u.g	$688$	$2$	688.u	43.e	$7$	$30$	$5$	$5.494$		$_{}$	None	None		$2$	$0$	$0$	$-6$	$1$	$4$			$\mathrm{SU}(2)[C_{7}]$
688.2.u.h	$688$	$2$	688.u	43.e	$7$	$36$	$6$	$5.494$		$_{}$	None	None		$2$	$0$	$0$	$6$	$1$	$0$			$\mathrm{SU}(2)[C_{7}]$
688.2.x.a	$688$	$2$	688.x	688.x	$12$	$344$	$86$	$5.494$		$_{}$	None	None		$4$	$0$	$-8$	$-2$	$-2$	$0$			$\mathrm{SU}(2)[C_{12}]$
688.2.y.a	$688$	$2$	688.y	688.y	$12$	$344$	$86$	$5.494$		$_{}$	None	None		$4$	$0$	$0$	$-6$	$-6$	$-12$			$\mathrm{SU}(2)[C_{12}]$
688.2.bb.a	$688$	$2$	688.bb	172.j	$14$	$36$	$6$	$5.494$		$_{}$	None	None		$4$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{14}]$
688.2.bb.b	$688$	$2$	688.bb	172.j	$14$	$96$	$16$	$5.494$		$_{}$	None	None		$4$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{14}]$
688.2.bg.a	$688$	$2$	688.bg	43.g	$21$	$12$	$1$	$5.494$	$\Q(\zeta_{21})$	$_{}$	None	None		$2$	$0$	$0$	$-8$	$3$	$1$		$1$	$\mathrm{SU}(2)[C_{21}]$	$q+(-1-\zeta_{21}^{7}+\zeta_{21}^{9})q^{3}+(1-\zeta_{21}+\cdots)q^{5}+\cdots$
688.2.bg.b	$688$	$2$	688.bg	43.g	$21$	$24$	$2$	$5.494$		$_{}$	None	None		$2$	$0$	$0$	$6$	$3$	$-3$			$\mathrm{SU}(2)[C_{21}]$

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Elliptic curves over $\Q$
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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.