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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 222 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
36.2.a.a	$36$	$2$	36.a	1.a	$1$	$1$	$1$	$0.287$	$\Q$	$\Q(\sqrt{-3}) $	✓	$2$	$0$	$0$	$0$	$0$	$-4$	$-$	$1$	$N(\mathrm{U}(1))$	$q-4q^{7}+2q^{13}+8q^{19}-5q^{25}-4q^{31}+\cdots$
36.2.b.a	$36$	$2$	36.b	12.b	$2$	$2$	$2$	$0.287$	$\Q(\sqrt{-2}) $	$\Q(\sqrt{-1}) $		$4$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+\beta q^{2}-2q^{4}-\beta q^{5}-2\beta q^{8}+2q^{10}+\cdots$
36.2.e.a	$36$	$2$	36.e	9.c	$3$	$2$	$1$	$0.287$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$0$	$0$	$-3$	$1$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-2\zeta_{6})q^{3}+(-3+3\zeta_{6})q^{5}+\zeta_{6}q^{7}+\cdots$
36.2.h.a	$36$	$2$	36.h	36.h	$6$	$8$	$4$	$0.287$	8.0.170772624.1	None		$4$	$0$	$-3$	$0$	$-6$	$0$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+(1-\beta _{1}+\beta _{4}+\beta _{7})q^{2}+(-\beta _{2}-\beta _{3}+\cdots)q^{3}+\cdots$
36.3.d.a	$36$	$3$	36.d	4.b	$2$	$1$	$1$	$0.981$	$\Q$	$\Q(\sqrt{-1}) $	✓	$2$	$0$	$-2$	$0$	$8$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q-2q^{2}+4q^{4}+8q^{5}-8q^{8}-2^{4}q^{10}+\cdots$
36.3.d.b	$36$	$3$	36.d	4.b	$2$	$1$	$1$	$0.981$	$\Q$	$\Q(\sqrt{-1}) $	✓	$2$	$0$	$2$	$0$	$-8$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+2q^{2}+4q^{4}-8q^{5}+8q^{8}-2^{4}q^{10}+\cdots$
36.3.d.c	$36$	$3$	36.d	4.b	$2$	$2$	$2$	$0.981$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$2$	$0$	$4$	$0$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+(1+\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{4}+2q^{5}+\cdots$
36.3.f.a	$36$	$3$	36.f	36.f	$6$	$2$	$1$	$0.981$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$-2$	$3$	$-4$	$6$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+(-2+2\zeta_{6})q^{2}+3\zeta_{6}q^{3}-4\zeta_{6}q^{4}+\cdots$
36.3.f.b	$36$	$3$	36.f	36.f	$6$	$2$	$1$	$0.981$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$4$	$-3$	$-4$	$-6$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+2q^{2}-3\zeta_{6}q^{3}+4q^{4}+(-4+4\zeta_{6})q^{5}+\cdots$
36.3.f.c	$36$	$3$	36.f	36.f	$6$	$16$	$8$	$0.981$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	None		$4$	$0$	$-3$	$0$	$6$	$0$		$2^{8}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{6}]$	$q-\beta _{1}q^{2}+\beta _{14}q^{3}+(-\beta _{2}+\beta _{3})q^{4}+\cdots$
36.3.g.a	$36$	$3$	36.g	9.d	$6$	$4$	$2$	$0.981$	$\Q(\sqrt{-3}, \sqrt{-11})$	None		$2$	$0$	$0$	$3$	$9$	$-1$		$3$	$\mathrm{SU}(2)[C_{6}]$	$q+(1-\beta _{1}+\beta _{3})q^{3}+(2-2\beta _{1}-2\beta _{2}+\cdots)q^{5}+\cdots$
36.4.a.a	$36$	$4$	36.a	1.a	$1$	$1$	$1$	$2.124$	$\Q$	None	✓	$1$	$0$	$0$	$0$	$18$	$8$	$+$	$1$	$\mathrm{SU}(2)$	$q+18q^{5}+8q^{7}-6^{2}q^{11}-10q^{13}+\cdots$
36.4.b.a	$36$	$4$	36.b	12.b	$2$	$2$	$2$	$2.124$	$\Q(\sqrt{-2}) $	$\Q(\sqrt{-1}) $		$4$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+2\beta q^{2}-8q^{4}+13\beta q^{5}-2^{4}\beta q^{8}+\cdots$
36.4.b.b	$36$	$4$	36.b	12.b	$2$	$4$	$4$	$2.124$	$\Q(\sqrt{-2}, \sqrt{-15})$	None		$4$	$0$	$0$	$0$	$0$	$0$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+(-\beta _{1}-\beta _{2})q^{2}+(7-\beta _{3})q^{4}-7\beta _{2}q^{5}+\cdots$
36.4.e.a	$36$	$4$	36.e	9.c	$3$	$6$	$3$	$2.124$	6.0.6831243.2	None		$2$	$0$	$0$	$-3$	$6$	$-6$		$2^{2}\cdot 3^{3}$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1-\beta _{2})q^{3}+(1-3\beta _{1}-2\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots$
36.4.h.a	$36$	$4$	36.h	36.h	$6$	$8$	$4$	$2.124$	8.0.$\cdots$.1	None		$4$	$0$	$-3$	$0$	$66$	$0$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q-\beta _{7}q^{2}+(-\beta _{2}+\beta _{5})q^{3}+(2\beta _{2}-2\beta _{3}+\cdots)q^{4}+\cdots$
36.4.h.b	$36$	$4$	36.h	36.h	$6$	$24$	$12$	$2.124$		None		$4$	$0$	$0$	$0$	$-72$	$0$			$\mathrm{SU}(2)[C_{6}]$
36.5.c.a	$36$	$5$	36.c	3.b	$2$	$2$	$2$	$3.721$	$\Q(\sqrt{-2}) $	None		$2$	$0$	$0$	$0$	$0$	$136$		$3^{3}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta q^{5}+68q^{7}+4\beta q^{11}-2^{4}q^{13}+\cdots$
36.5.d.a	$36$	$5$	36.d	4.b	$2$	$1$	$1$	$3.721$	$\Q$	$\Q(\sqrt{-1}) $	✓	$2$	$0$	$4$	$0$	$14$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+4q^{2}+2^{4}q^{4}+14q^{5}+2^{6}q^{8}+56q^{10}+\cdots$
36.5.d.b	$36$	$5$	36.d	4.b	$2$	$4$	$4$	$3.721$	$\Q(\sqrt{-3}, \sqrt{13})$	None		$2$	$0$	$-6$	$0$	$-24$	$0$		$2^{6}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	$q+(-2+\beta _{1})q^{2}+(-4-2\beta _{1}+\beta _{3})q^{4}+\cdots$
36.5.d.c	$36$	$5$	36.d	4.b	$2$	$4$	$4$	$3.721$	$\Q(i, \sqrt{7})$	None		$4$	$0$	$0$	$0$	$0$	$0$		$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+(-2+\beta _{3})q^{4}+(4\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots$
36.5.f.a	$36$	$5$	36.f	36.f	$6$	$44$	$22$	$3.721$		None		$4$	$0$	$-1$	$0$	$-2$	$0$			$\mathrm{SU}(2)[C_{6}]$
36.5.g.a	$36$	$5$	36.g	9.d	$6$	$8$	$4$	$3.721$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None		$2$	$0$	$0$	$-9$	$-9$	$13$		$2^{2}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{6}]$	$q+(-1+\beta _{2})q^{3}+(-1+\beta _{1}-\beta _{2}+\beta _{4}+\cdots)q^{5}+\cdots$
36.6.a.a	$36$	$6$	36.a	1.a	$1$	$1$	$1$	$5.774$	$\Q$	None	✓	$1$	$1$	$0$	$0$	$-54$	$-88$	$+$	$1$	$\mathrm{SU}(2)$	$q-54q^{5}-88q^{7}-540q^{11}-418q^{13}+\cdots$
36.6.a.b	$36$	$6$	36.a	1.a	$1$	$1$	$1$	$5.774$	$\Q$	$\Q(\sqrt{-3}) $	✓	$2$	$0$	$0$	$0$	$0$	$236$	$-$	$1$	$N(\mathrm{U}(1))$	$q+236q^{7}+1202q^{13}-1432q^{19}+\cdots$
36.6.b.a	$36$	$6$	36.b	12.b	$2$	$2$	$2$	$5.774$	$\Q(\sqrt{-2}) $	$\Q(\sqrt{-1}) $		$4$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+4\beta q^{2}-2^{5}q^{4}-79\beta q^{5}-2^{7}\beta q^{8}+\cdots$
36.6.b.b	$36$	$6$	36.b	12.b	$2$	$8$	$8$	$5.774$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None		$4$	$0$	$0$	$0$	$0$	$0$		$2^{10}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{2}q^{2}+(6+\beta _{7})q^{4}+(6\beta _{1}+3\beta _{2}+\cdots)q^{5}+\cdots$
36.6.e.a	$36$	$6$	36.e	9.c	$3$	$10$	$5$	$5.774$	$\mathbb{Q}[x]/(x^{10} + \cdots)$	None		$2$	$0$	$0$	$12$	$-21$	$29$		$2^{8}\cdot 3^{10}$	$\mathrm{SU}(2)[C_{3}]$	$q+(2+2\beta _{1}-\beta _{3}-\beta _{4})q^{3}+(4\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots$
36.6.h.a	$36$	$6$	36.h	36.h	$6$	$56$	$28$	$5.774$		None		$4$	$0$	$-3$	$0$	$-6$	$0$			$\mathrm{SU}(2)[C_{6}]$
36.7.c.a	$36$	$7$	36.c	3.b	$2$	$2$	$2$	$8.282$	$\Q(\sqrt{-2}) $	None		$2$	$0$	$0$	$0$	$0$	$-488$		$3^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+5\beta q^{5}-244q^{7}+188\beta q^{11}-2728q^{13}+\cdots$
36.7.d.a	$36$	$7$	36.d	4.b	$2$	$1$	$1$	$8.282$	$\Q$	$\Q(\sqrt{-1}) $	✓	$2$	$0$	$-8$	$0$	$-88$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q-8q^{2}+2^{6}q^{4}-88q^{5}-2^{9}q^{8}+704q^{10}+\cdots$
36.7.d.b	$36$	$7$	36.d	4.b	$2$	$1$	$1$	$8.282$	$\Q$	$\Q(\sqrt{-1}) $	✓	$2$	$0$	$8$	$0$	$88$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+8q^{2}+2^{6}q^{4}+88q^{5}+2^{9}q^{8}+704q^{10}+\cdots$
36.7.d.c	$36$	$7$	36.d	4.b	$2$	$2$	$2$	$8.282$	$\Q(\sqrt{-15}) $	None		$2$	$0$	$-4$	$0$	$-20$	$0$		$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+(-2-\beta )q^{2}+(-56+4\beta )q^{4}-10q^{5}+\cdots$
36.7.d.d	$36$	$7$	36.d	4.b	$2$	$4$	$4$	$8.282$	$\Q(\sqrt{13}, \sqrt{-51})$	None		$4$	$0$	$0$	$0$	$0$	$0$		$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+(-38+\beta _{3})q^{4}+(-10\beta _{1}+\cdots)q^{5}+\cdots$
36.7.d.e	$36$	$7$	36.d	4.b	$2$	$6$	$6$	$8.282$	6.0.50898483.1	None		$2$	$0$	$10$	$0$	$44$	$0$		$2^{15}\cdot 3^{5}$	$\mathrm{SU}(2)[C_{2}]$	$q+(2-\beta _{1})q^{2}+(26-\beta _{1}-\beta _{2})q^{4}+(7+\cdots)q^{5}+\cdots$
36.7.f.a	$36$	$7$	36.f	36.f	$6$	$68$	$34$	$8.282$		None		$4$	$0$	$-1$	$0$	$-2$	$0$			$\mathrm{SU}(2)[C_{6}]$
36.7.g.a	$36$	$7$	36.g	9.d	$6$	$12$	$6$	$8.282$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None		$2$	$0$	$0$	$6$	$-216$	$-120$		$2^{12}\cdot 3^{15}$	$\mathrm{SU}(2)[C_{6}]$	$q+(-2+4\beta _{1}+\beta _{3})q^{3}+(-24+12\beta _{1}+\cdots)q^{5}+\cdots$
36.8.a.a	$36$	$8$	36.a	1.a	$1$	$1$	$1$	$11.246$	$\Q$	None	✓	$1$	$0$	$0$	$0$	$-270$	$1112$	$+$	$1$	$\mathrm{SU}(2)$	$q-270q^{5}+1112q^{7}+5724q^{11}+\cdots$
36.8.a.b	$36$	$8$	36.a	1.a	$1$	$1$	$1$	$11.246$	$\Q$	$\Q(\sqrt{-3}) $	✓	$2$	$1$	$0$	$0$	$0$	$-508$	$-$	$1$	$N(\mathrm{U}(1))$	$q-508q^{7}-14614q^{13}-57448q^{19}+\cdots$
36.8.a.c	$36$	$8$	36.a	1.a	$1$	$1$	$1$	$11.246$	$\Q$	None	✓	$1$	$0$	$0$	$0$	$378$	$-832$	$+$	$1$	$\mathrm{SU}(2)$	$q+378q^{5}-832q^{7}+2484q^{11}+14870q^{13}+\cdots$
36.8.b.a	$36$	$8$	36.b	12.b	$2$	$2$	$2$	$11.246$	$\Q(\sqrt{-2}) $	$\Q(\sqrt{-1}) $		$4$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+8\beta q^{2}-2^{7}q^{4}+307\beta q^{5}-2^{10}\beta q^{8}+\cdots$
36.8.b.b	$36$	$8$	36.b	12.b	$2$	$12$	$12$	$11.246$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	None		$4$	$0$	$0$	$0$	$0$	$0$		$2^{27}\cdot 3^{20}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+(17+\beta _{3})q^{4}+(-9\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$
36.8.e.a	$36$	$8$	36.e	9.c	$3$	$14$	$7$	$11.246$	$\mathbb{Q}[x]/(x^{14} - \cdots)$	None		$2$	$0$	$0$	$0$	$321$	$-83$		$2^{20}\cdot 3^{21}$	$\mathrm{SU}(2)[C_{3}]$	$q+(-2-4\beta _{2}-\beta _{4})q^{3}+(\beta _{1}-46\beta _{2}+\cdots)q^{5}+\cdots$
36.8.h.a	$36$	$8$	36.h	36.h	$6$	$80$	$40$	$11.246$		None		$4$	$0$	$-3$	$0$	$-6$	$0$			$\mathrm{SU}(2)[C_{6}]$
36.9.c.a	$36$	$9$	36.c	3.b	$2$	$2$	$2$	$14.666$	$\Q(\sqrt{-2}) $	None		$2$	$0$	$0$	$0$	$0$	$616$		$3^{3}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta q^{5}+308q^{7}+244\beta q^{11}+18464q^{13}+\cdots$
36.9.d.a	$36$	$9$	36.d	4.b	$2$	$1$	$1$	$14.666$	$\Q$	$\Q(\sqrt{-1}) $	✓	$2$	$0$	$-16$	$0$	$1054$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q-2^{4}q^{2}+2^{8}q^{4}+1054q^{5}-2^{12}q^{8}+\cdots$
36.9.d.b	$36$	$9$	36.d	4.b	$2$	$2$	$2$	$14.666$	$\Q(\sqrt{-39}) $	None		$2$	$0$	$20$	$0$	$-1220$	$0$		$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+(10-\beta )q^{2}+(-56-20\beta )q^{4}-610q^{5}+\cdots$
36.9.d.c	$36$	$9$	36.d	4.b	$2$	$8$	$8$	$14.666$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None		$2$	$0$	$-6$	$0$	$336$	$0$		$2^{26}\cdot 3^{10}$	$\mathrm{SU}(2)[C_{2}]$	$q+(-1-\beta _{1})q^{2}+(-6+\beta _{1}+\beta _{3})q^{4}+\cdots$
36.9.d.d	$36$	$9$	36.d	4.b	$2$	$8$	$8$	$14.666$	$\mathbb{Q}[x]/(x^{8} + \cdots)$	None		$4$	$0$	$0$	$0$	$0$	$0$		$2^{28}\cdot 3^{6}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+(-53-\beta _{5})q^{4}+(3\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots$
36.9.f.a	$36$	$9$	36.f	36.f	$6$	$92$	$46$	$14.666$		None		$4$	$0$	$-1$	$0$	$-2$	$0$			$\mathrm{SU}(2)[C_{6}]$

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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.