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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 188 matches)

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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
13.2.e.a	$13$	$2$	13.e	13.e	$6$	$2$	$1$	$0.104$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$-3$	$-2$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+(-1-\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots$
13.3.d.a	$13$	$3$	13.d	13.d	$4$	$4$	$2$	$0.354$	$\Q(i, \sqrt{10})$	None		$2$	$0$	$-4$	$-4$	$8$	$-12$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q+(-1+\beta _{1}-\beta _{2})q^{2}+(-1-\beta _{1}+\beta _{3})q^{3}+\cdots$
13.3.f.a	$13$	$3$	13.f	13.f	$12$	$4$	$1$	$0.354$	$\Q(\zeta_{12})$	None		$2$	$0$	$-2$	$-2$	$-14$	$16$		$1$	$\mathrm{SU}(2)[C_{12}]$	$q+(-1+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(\zeta_{12}-\zeta_{12}^{2}+\cdots)q^{3}+\cdots$
13.4.a.a	$13$	$4$	13.a	1.a	$1$	$1$	$1$	$0.767$	$\Q$	None	✓	$1$	$1$	$-5$	$-7$	$-7$	$-13$	$-$	$1$	$\mathrm{SU}(2)$	$q-5q^{2}-7q^{3}+17q^{4}-7q^{5}+35q^{6}+\cdots$
13.4.a.b	$13$	$4$	13.a	1.a	$1$	$2$	$2$	$0.767$	$\Q(\sqrt{17}) $	None	✓	$1$	$0$	$1$	$5$	$-3$	$-9$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}+(4-3\beta )q^{3}+(-4+\beta )q^{4}+\cdots$
13.4.b.a	$13$	$4$	13.b	13.b	$2$	$2$	$2$	$0.767$	$\Q(\sqrt{-1}) $	None		$2$	$0$	$0$	$-2$	$0$	$0$		$3$	$\mathrm{SU}(2)[C_{2}]$	$q+iq^{2}-q^{3}-q^{4}-3iq^{5}-iq^{6}+\cdots$
13.4.c.a	$13$	$4$	13.c	13.c	$3$	$2$	$1$	$0.767$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$-4$	$-2$	$34$	$-20$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-4+4\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}+\cdots$
13.4.c.b	$13$	$4$	13.c	13.c	$3$	$4$	$2$	$0.767$	$\Q(\sqrt{-3}, \sqrt{17})$	None		$2$	$0$	$5$	$-5$	$-30$	$-15$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(2-\beta _{1}-2\beta _{2}-\beta _{3})q^{2}+(-1+3\beta _{1}+\cdots)q^{3}+\cdots$
13.4.e.a	$13$	$4$	13.e	13.e	$6$	$2$	$1$	$0.767$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$-6$	$7$	$0$	$39$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+(-2-2\zeta_{6})q^{2}+(7-7\zeta_{6})q^{3}+4\zeta_{6}q^{4}+\cdots$
13.4.e.b	$13$	$4$	13.e	13.e	$6$	$2$	$1$	$0.767$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$3$	$-2$	$0$	$-24$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+(1+\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}-5\zeta_{6}q^{4}+\cdots$
13.5.d.a	$13$	$5$	13.d	13.d	$4$	$6$	$3$	$1.344$	6.0.$\cdots$.1	None		$2$	$0$	$-2$	$-4$	$-14$	$48$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q-\beta _{1}q^{2}+(-1+\beta _{4})q^{3}+(-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots$
13.5.f.a	$13$	$5$	13.f	13.f	$12$	$16$	$4$	$1.344$	$\mathbb{Q}[x]/(x^{16} + \cdots)$	None		$2$	$0$	$-4$	$-2$	$8$	$56$		$2^{4}\cdot 3^{2}\cdot 13^{2}$	$\mathrm{SU}(2)[C_{12}]$	$q-\beta _{12}q^{2}+(-1-\beta _{8}+\beta _{9}+\beta _{12}+\cdots)q^{3}+\cdots$
13.6.a.a	$13$	$6$	13.a	1.a	$1$	$2$	$2$	$2.085$	$\Q(\sqrt{17}) $	None	✓	$1$	$1$	$-5$	$-28$	$-42$	$-36$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-2-\beta )q^{2}+(-17+6\beta )q^{3}+(-24+\cdots)q^{4}+\cdots$
13.6.a.b	$13$	$6$	13.a	1.a	$1$	$3$	$3$	$2.085$	3.3.168897.1	None	✓	$1$	$0$	$7$	$8$	$56$	$-60$	$-$	$1$	$\mathrm{SU}(2)$	$q+(2+\beta _{1})q^{2}+(3-\beta _{1}-\beta _{2})q^{3}+(40+\cdots)q^{4}+\cdots$
13.6.b.a	$13$	$6$	13.b	13.b	$2$	$6$	$6$	$2.085$	$\mathbb{Q}[x]/(x^{6} + \cdots)$	None		$2$	$0$	$0$	$16$	$0$	$0$		$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+(3+\beta _{2})q^{3}+(-22-\beta _{2}+\cdots)q^{4}+\cdots$
13.6.c.a	$13$	$6$	13.c	13.c	$3$	$8$	$4$	$2.085$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None		$2$	$0$	$-5$	$8$	$-20$	$68$		$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1-\beta _{1}+\beta _{3})q^{2}+(2-2\beta _{3}-\beta _{5}+\cdots)q^{3}+\cdots$
13.6.e.a	$13$	$6$	13.e	13.e	$6$	$10$	$5$	$2.085$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None		$2$	$0$	$-3$	$-10$	$0$	$-276$		$2^{6}\cdot 3\cdot 13^{2}$	$\mathrm{SU}(2)[C_{6}]$	$q-\beta _{3}q^{2}+(-2-2\beta _{2}-\beta _{5}-\beta _{6})q^{3}+\cdots$
13.7.d.a	$13$	$7$	13.d	13.d	$4$	$12$	$6$	$2.991$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None		$2$	$0$	$6$	$-4$	$108$	$398$		$2\cdot 5^{2}\cdot 13^{2}$	$\mathrm{SU}(2)[C_{4}]$	$q-\beta _{2}q^{2}+(\beta _{1}+\beta _{2}-\beta _{3}-\beta _{7})q^{3}+\cdots$
13.7.f.a	$13$	$7$	13.f	13.f	$12$	$24$	$6$	$2.991$		None		$2$	$0$	$-12$	$-2$	$-114$	$316$			$\mathrm{SU}(2)[C_{12}]$
13.8.a.a	$13$	$8$	13.a	1.a	$1$	$1$	$1$	$4.061$	$\Q$	None	✓	$1$	$1$	$10$	$-73$	$-295$	$1373$	$-$	$1$	$\mathrm{SU}(2)$	$q+10q^{2}-73q^{3}-28q^{4}-295q^{5}+\cdots$
13.8.a.b	$13$	$8$	13.a	1.a	$1$	$2$	$2$	$4.061$	$\Q(\sqrt{337}) $	None	✓	$1$	$1$	$-19$	$45$	$-353$	$-2009$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-9-\beta )q^{2}+(21+3\beta )q^{3}+(37+19\beta )q^{4}+\cdots$
13.8.a.c	$13$	$8$	13.a	1.a	$1$	$4$	$4$	$4.061$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	$1$	$0$	$15$	$80$	$258$	$1692$	$+$	$2$	$\mathrm{SU}(2)$	$q+(4-\beta _{1})q^{2}+(21-2\beta _{1}+\beta _{2})q^{3}+\cdots$
13.8.b.a	$13$	$8$	13.b	13.b	$2$	$6$	$6$	$4.061$	$\mathbb{Q}[x]/(x^{6} + \cdots)$	None		$2$	$0$	$0$	$-56$	$0$	$0$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+(-9-\beta _{2})q^{3}+(-22-\beta _{4}+\cdots)q^{4}+\cdots$
13.8.c.a	$13$	$8$	13.c	13.c	$3$	$16$	$8$	$4.061$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None		$2$	$0$	$-9$	$-28$	$384$	$196$		$2^{12}\cdot 3\cdot 13^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1-\beta _{1}-\beta _{2})q^{2}+(-4-4\beta _{2}+\cdots)q^{3}+\cdots$
13.8.e.a	$13$	$8$	13.e	13.e	$6$	$14$	$7$	$4.061$	$\mathbb{Q}[x]/(x^{14} + \cdots)$	None		$2$	$0$	$-3$	$26$	$0$	$-2772$		$2^{10}\cdot 3^{3}\cdot 13^{4}$	$\mathrm{SU}(2)[C_{6}]$	$q+\beta _{2}q^{2}+(4+4\beta _{3}-\beta _{4}-\beta _{6})q^{3}+\cdots$
13.9.d.a	$13$	$9$	13.d	13.d	$4$	$18$	$9$	$5.296$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	None		$2$	$0$	$-2$	$-4$	$166$	$5308$		$2^{16}\cdot 3^{2}\cdot 13^{4}$	$\mathrm{SU}(2)[C_{4}]$	$q-\beta _{3}q^{2}+\beta _{6}q^{3}+(-\beta _{1}-142\beta _{2}+\cdots)q^{4}+\cdots$
13.9.f.a	$13$	$9$	13.f	13.f	$12$	$32$	$8$	$5.296$		None		$2$	$0$	$-4$	$-2$	$-172$	$-2784$			$\mathrm{SU}(2)[C_{12}]$
13.10.a.a	$13$	$10$	13.a	1.a	$1$	$4$	$4$	$6.695$	$\mathbb{Q}[x]/(x^{4} - \cdots)$	None	✓	$1$	$1$	$-33$	$-163$	$471$	$-11241$	$+$	$2$	$\mathrm{SU}(2)$	$q+(-8-\beta _{1})q^{2}+(-41+2\beta _{1}-\beta _{3})q^{3}+\cdots$
13.10.a.b	$13$	$10$	13.a	1.a	$1$	$5$	$5$	$6.695$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓	$1$	$0$	$15$	$161$	$1803$	$10099$	$-$	$2^{2}\cdot 3$	$\mathrm{SU}(2)$	$q+(3+\beta _{1})q^{2}+(2^{5}+2\beta _{1}-\beta _{2})q^{3}+\cdots$
13.10.b.a	$13$	$10$	13.b	13.b	$2$	$10$	$10$	$6.695$	$\mathbb{Q}[x]/(x^{10} + \cdots)$	None		$2$	$0$	$0$	$-2$	$0$	$0$		$2^{7}\cdot 3^{4}\cdot 13^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+\beta _{3}q^{3}+(-2^{8}+\beta _{2})q^{4}+\cdots$
13.10.c.a	$13$	$10$	13.c	13.c	$3$	$18$	$9$	$6.695$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	None		$2$	$0$	$15$	$161$	$-2280$	$-1939$		$2^{16}\cdot 3^{3}\cdot 13^{4}$	$\mathrm{SU}(2)[C_{3}]$	$q+(-2\beta _{2}-\beta _{3})q^{2}+(-18\beta _{2}-\beta _{6}+\cdots)q^{3}+\cdots$
13.10.e.a	$13$	$10$	13.e	13.e	$6$	$20$	$10$	$6.695$	$\mathbb{Q}[x]/(x^{20} + \cdots)$	None		$2$	$0$	$-3$	$-163$	$0$	$1023$		$2^{22}\cdot 3^{10}\cdot 13^{6}$	$\mathrm{SU}(2)[C_{6}]$	$q+(\beta _{1}-\beta _{2})q^{2}+(2^{4}\beta _{5}-\beta _{6})q^{3}+(2^{8}+\cdots)q^{4}+\cdots$
13.11.d.a	$13$	$11$	13.d	13.d	$4$	$20$	$10$	$8.260$	$\mathbb{Q}[x]/(x^{20} - \cdots)$	None		$2$	$0$	$-34$	$-4$	$-7792$	$-38312$		$2^{20}\cdot 3^{6}\cdot 13^{6}$	$\mathrm{SU}(2)[C_{4}]$	$q+(-2-2\beta _{2}+\beta _{3})q^{2}+(-\beta _{1}-\beta _{3}+\cdots)q^{3}+\cdots$
13.11.f.a	$13$	$11$	13.f	13.f	$12$	$44$	$11$	$8.260$		None		$2$	$0$	$28$	$-2$	$7786$	$30856$			$\mathrm{SU}(2)[C_{12}]$
13.12.a.a	$13$	$12$	13.a	1.a	$1$	$5$	$5$	$9.988$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	None	✓	$1$	$1$	$-41$	$-496$	$-2542$	$-36296$	$-$	$2^{2}$	$\mathrm{SU}(2)$	$q+(-8-\beta _{1})q^{2}+(-99-2\beta _{1}-\beta _{4})q^{3}+\cdots$
13.12.a.b	$13$	$12$	13.a	1.a	$1$	$6$	$6$	$9.988$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓	$1$	$0$	$55$	$476$	$3312$	$-4176$	$+$	$2^{3}$	$\mathrm{SU}(2)$	$q+(9+\beta _{1})q^{2}+(80-2\beta _{1}+\beta _{4})q^{3}+\cdots$
13.12.b.a	$13$	$12$	13.b	13.b	$2$	$12$	$12$	$9.988$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	None		$2$	$0$	$0$	$-488$	$0$	$0$		$2^{9}\cdot 3^{4}\cdot 13^{4}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+(-41-\beta _{2})q^{3}+(-2^{10}+\cdots)q^{4}+\cdots$
13.12.c.a	$13$	$12$	13.c	13.c	$3$	$24$	$12$	$9.988$		None		$2$	$0$	$31$	$-244$	$-10436$	$-2928$			$\mathrm{SU}(2)[C_{3}]$
13.12.e.a	$13$	$12$	13.e	13.e	$6$	$22$	$11$	$9.988$		None		$2$	$0$	$-3$	$242$	$0$	$128496$			$\mathrm{SU}(2)[C_{6}]$
13.13.d.a	$13$	$13$	13.d	13.d	$4$	$26$	$13$	$11.882$		None		$2$	$0$	$-2$	$-4$	$5146$	$-114402$			$\mathrm{SU}(2)[C_{4}]$
13.13.f.a	$13$	$13$	13.f	13.f	$12$	$52$	$13$	$11.882$		None		$2$	$0$	$-4$	$-2$	$-5152$	$-91524$			$\mathrm{SU}(2)[C_{12}]$
13.14.a.a	$13$	$14$	13.a	1.a	$1$	$6$	$6$	$13.940$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None	✓	$1$	$1$	$-65$	$-2188$	$-76752$	$271440$	$+$	$2^{4}$	$\mathrm{SU}(2)$	$q+(-11+\beta _{1})q^{2}+(-365+2\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots$
13.14.a.b	$13$	$14$	13.a	1.a	$1$	$7$	$7$	$13.940$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓	$1$	$0$	$127$	$728$	$34886$	$131864$	$-$	$2^{5}\cdot 3^{2}$	$\mathrm{SU}(2)$	$q+(18+\beta _{1})q^{2}+(104+\beta _{1}+\beta _{2})q^{3}+\cdots$
13.14.b.a	$13$	$14$	13.b	13.b	$2$	$14$	$14$	$13.940$	$\mathbb{Q}[x]/(x^{14} + \cdots)$	None		$2$	$0$	$0$	$1456$	$0$	$0$		$2^{15}\cdot 3^{6}\cdot 13^{6}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+(104+\beta _{3})q^{3}+(-3511+\cdots)q^{4}+\cdots$
13.14.c.a	$13$	$14$	13.c	13.c	$3$	$28$	$14$	$13.940$		None		$2$	$0$	$-65$	$728$	$41860$	$-173992$			$\mathrm{SU}(2)[C_{3}]$
13.14.e.a	$13$	$14$	13.e	13.e	$6$	$30$	$15$	$13.940$		None		$2$	$0$	$-3$	$-730$	$0$	$438984$			$\mathrm{SU}(2)[C_{6}]$
13.15.d.a	$13$	$15$	13.d	13.d	$4$	$32$	$16$	$16.163$		None		$2$	$0$	$126$	$-4$	$40308$	$-342172$			$\mathrm{SU}(2)[C_{4}]$
13.15.f.a	$13$	$15$	13.f	13.f	$12$	$60$	$15$	$16.163$		None		$2$	$0$	$-132$	$-2$	$-40314$	$-1557904$			$\mathrm{SU}(2)[C_{12}]$
13.16.a.a	$13$	$16$	13.a	1.a	$1$	$7$	$7$	$18.550$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	None	✓	$1$	$1$	$-345$	$-1027$	$-169503$	$-3685093$	$-$	$2^{8}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)$	$q+(-7^{2}-\beta _{1})q^{2}+(-149+7\beta _{1}+\beta _{4}+\cdots)q^{3}+\cdots$
13.16.a.b	$13$	$16$	13.a	1.a	$1$	$8$	$8$	$18.550$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None	✓	$1$	$0$	$39$	$7721$	$161733$	$719199$	$+$	$2^{11}\cdot 3^{2}\cdot 5^{2}$	$\mathrm{SU}(2)$	$q+(5-\beta _{1})q^{2}+(966-6\beta _{1}-\beta _{4})q^{3}+\cdots$

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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.