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Level	Weight	Analytic conductor	Analytic rank	Dim.

Bad $p$	Char.	Primitive character	Character order	Is maximal/largest

Coefficient field	Self-twists	CM/RM discriminant	Inner twist count	Is self-dual

Coefficient ring index	Coefficient ring gens.	Is twist minimal	Projective image

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Results (46 matches)

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Label	Level	Weight	Char	Prim	Char order	Dim	Rel. Dim	$A$	Field	Image	CM	RM	Self-dual	Twist minimal	Largest	Maximal	Minimal twist	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	Twist minimal	Largest	Maximal	Minimal twist	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
3971.1.c.a	$3971$	$1$	3971.c	11.b	$2$	$1$	$1$	$1.982$	$\Q$	$D_{2}$	$\Q(\sqrt{-11}) $, $\Q(\sqrt{-19}) $	$\Q(\sqrt{209}) $	✓	✓			3971.1.c.a	$4$	$0$	$0$	$0$	$-2$	$0$		$1$		$q+q^{4}-2q^{5}-q^{9}-q^{11}+q^{16}-2q^{20}+\cdots$
3971.1.c.b	$3971$	$1$	3971.c	11.b	$2$	$2$	$2$	$1.982$	$\Q(\sqrt{-1}) $	$A_{4}$	None	None					209.1.h.a	$2$	$0$	$0$	$-2$	$2$	$0$		$1$		$q-i q^{2}-q^{3}+q^{5}+i q^{6}-i q^{8}+\cdots$
3971.1.c.c	$3971$	$1$	3971.c	11.b	$2$	$2$	$2$	$1.982$	$\Q(\sqrt{5}) $	$D_{5}$	$\Q(\sqrt{-11}) $	None	✓	✓			3971.1.c.c	$2$	$0$	$0$	$-1$	$-1$	$0$		$1$		$q+(-1+\beta )q^{3}+q^{4}+(-1+\beta )q^{5}+\cdots$
3971.1.c.d	$3971$	$1$	3971.c	11.b	$2$	$2$	$2$	$1.982$	$\Q(\sqrt{5}) $	$D_{5}$	$\Q(\sqrt{-11}) $	None	✓	✓			3971.1.c.c	$2$	$0$	$0$	$1$	$-1$	$0$		$1$		$q+(1-\beta )q^{3}+q^{4}+(-1+\beta )q^{5}+(1+\cdots)q^{9}+\cdots$
3971.1.c.e	$3971$	$1$	3971.c	11.b	$2$	$2$	$2$	$1.982$	$\Q(\sqrt{-1}) $	$A_{4}$	None	None					209.1.h.a	$2$	$0$	$0$	$2$	$2$	$0$		$1$		$q-i q^{2}+q^{3}+q^{5}-i q^{6}-i q^{8}+\cdots$
3971.1.c.f	$3971$	$1$	3971.c	11.b	$2$	$4$	$4$	$1.982$	$\Q(\zeta_{20})^+$	$D_{10}$	$\Q(\sqrt{-11}) $	None	✓	✓	✓		3971.1.c.f	$4$	$0$	$0$	$0$	$2$	$0$		$1$		$q-\beta _{1}q^{3}+q^{4}-\beta _{2}q^{5}+(2+\beta _{2})q^{9}+\cdots$
3971.1.h.a	$3971$	$1$	3971.h	209.h	$6$	$2$	$1$	$1.982$	$\Q(\sqrt{-3}) $	$D_{2}$	$\Q(\sqrt{-11}) $, $\Q(\sqrt{-19}) $	$\Q(\sqrt{209}) $		✓			3971.1.c.a	$8$	$0$	$0$	$0$	$2$	$0$		$1$		$q+\zeta_{6}^{2}q^{4}+\zeta_{6}q^{5}-\zeta_{6}^{2}q^{9}-q^{11}+\cdots$
3971.1.h.b	$3971$	$1$	3971.h	209.h	$6$	$4$	$2$	$1.982$	$\Q(\sqrt{-3}, \sqrt{5})$	$D_{5}$	$\Q(\sqrt{-11}) $	None		✓			3971.1.c.c	$4$	$0$	$0$	$-1$	$1$	$0$		$1$		$q+(-\beta _{1}-\beta _{2}+\beta _{3})q^{3}+(-1-\beta _{3})q^{4}+\cdots$
3971.1.h.c	$3971$	$1$	3971.h	209.h	$6$	$4$	$2$	$1.982$	$\Q(\sqrt{-3}, \sqrt{5})$	$D_{5}$	$\Q(\sqrt{-11}) $	None		✓			3971.1.c.c	$4$	$0$	$0$	$1$	$1$	$0$		$1$		$q+(\beta _{1}+\beta _{2}-\beta _{3})q^{3}+(-1-\beta _{3})q^{4}+\cdots$
3971.1.h.d	$3971$	$1$	3971.h	209.h	$6$	$4$	$2$	$1.982$	$\Q(\zeta_{12})$	$A_{4}$	None	None					209.1.h.a	$4$	$0$	$0$	$2$	$-2$	$0$		$1$		$q-\zeta_{12}q^{2}-\zeta_{12}^{4}q^{3}+\zeta_{12}^{4}q^{5}+\zeta_{12}^{5}q^{6}+\cdots$
3971.1.h.e	$3971$	$1$	3971.h	209.h	$6$	$8$	$4$	$1.982$	8.0.324000000.2	$D_{10}$	$\Q(\sqrt{-11}) $	None		✓	✓		3971.1.c.f	$8$	$0$	$0$	$0$	$-2$	$0$		$1$		$q+(-\beta _{1}-\beta _{5})q^{3}+(-1-\beta _{4})q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots$
3971.1.l.a	$3971$	$1$	3971.l	11.d	$10$	$4$	$1$	$1.982$	$\Q(\zeta_{10})$	$D_{10}$	$\Q(\sqrt{-19}) $	None		✓	✓	✓	3971.1.l.a	$4$	$0$	$0$	$0$	$2$	$-5$		$1$		$q+\zeta_{10}^{4}q^{4}+(-\zeta_{10}^{2}-\zeta_{10}^{4})q^{5}+\cdots$
3971.1.q.a	$3971$	$1$	3971.q	209.q	$18$	$6$	$1$	$1.982$	$\Q(\zeta_{18})$	$D_{2}$	$\Q(\sqrt{-11}) $, $\Q(\sqrt{-19}) $	$\Q(\sqrt{209}) $		✓			3971.1.c.a	$24$	$0$	$0$	$0$	$0$	$0$		$1$		$q+\zeta_{18}^{2}q^{4}+\zeta_{18}^{7}q^{5}-\zeta_{18}^{8}q^{9}+\cdots$
3971.1.q.b	$3971$	$1$	3971.q	209.q	$18$	$12$	$2$	$1.982$	12.0.$\cdots$.1	$D_{5}$	$\Q(\sqrt{-11}) $	None		✓			3971.1.c.c	$12$	$0$	$0$	$0$	$0$	$0$		$1$		$q+\beta _{4}q^{3}+\beta _{3}q^{4}+(\beta _{4}+\beta _{10})q^{5}+(\beta _{1}+\cdots)q^{9}+\cdots$
3971.1.q.c	$3971$	$1$	3971.q	209.q	$18$	$12$	$2$	$1.982$	12.0.$\cdots$.1	$D_{5}$	$\Q(\sqrt{-11}) $	None		✓			3971.1.c.c	$12$	$0$	$0$	$0$	$0$	$0$		$1$		$q+(\beta _{4}-\beta _{5})q^{3}+\beta _{3}q^{4}+(-\beta _{4}-\beta _{10}+\cdots)q^{5}+\cdots$
3971.1.q.d	$3971$	$1$	3971.q	209.q	$18$	$12$	$2$	$1.982$	$\Q(\zeta_{36})$	$A_{4}$	None	None					209.1.h.a	$12$	$0$	$0$	$0$	$0$	$0$		$1$		$q+\zeta_{36}^{11}q^{2}-\zeta_{36}^{8}q^{3}-\zeta_{36}^{14}q^{5}+\cdots$
3971.1.q.e	$3971$	$1$	3971.q	209.q	$18$	$12$	$2$	$1.982$	$\Q(\zeta_{36})$	$A_{4}$	None	None					209.1.h.a	$12$	$0$	$0$	$0$	$0$	$0$		$1$		$q+\zeta_{36}^{11}q^{2}+\zeta_{36}^{8}q^{3}-\zeta_{36}^{14}q^{5}+\cdots$
3971.1.q.f	$3971$	$1$	3971.q	209.q	$18$	$24$	$4$	$1.982$	24.0.$\cdots$.3	$D_{10}$	$\Q(\sqrt{-11}) $	None		✓	✓		3971.1.c.f	$24$	$0$	$0$	$0$	$0$	$0$		$1$		$q+(-\beta _{11}+\beta _{23})q^{3}-\beta _{8}q^{4}-\beta _{16}q^{5}+\cdots$
3971.1.s.a	$3971$	$1$	3971.s	209.r	$30$	$8$	$1$	$1.982$	$\Q(\zeta_{15})$	$D_{5}$	$\Q(\sqrt{-19}) $	None			✓	✓	209.1.m.a	$8$	$0$	$0$	$0$	$2$	$6$		$1$		$q-\zeta_{30}^{13}q^{4}+(\zeta_{30}^{8}+\zeta_{30}^{14})q^{5}+\cdots$
3971.1.t.a	$3971$	$1$	3971.t	209.s	$30$	$8$	$1$	$1.982$	$\Q(\zeta_{15})$	$D_{10}$	$\Q(\sqrt{-19}) $	None		✓	✓	✓	3971.1.l.a	$8$	$0$	$0$	$0$	$-2$	$-10$		$1$		$q-\zeta_{30}^{11}q^{4}+(\zeta_{30}+\zeta_{30}^{13})q^{5}+(-1+\cdots)q^{7}+\cdots$
3971.1.ba.a	$3971$	$1$	3971.ba	209.v	$90$	$24$	$1$	$1.982$	$\Q(\zeta_{45})$	$D_{10}$	$\Q(\sqrt{-19}) $	None		✓	✓	✓	3971.1.l.a	$24$	$0$	$0$	$0$	$0$	$15$		$1$		$q+\zeta_{90}^{22}q^{4}+(-\zeta_{90}^{32}+\zeta_{90}^{41}+\cdots)q^{5}+\cdots$
3971.1.bc.a	$3971$	$1$	3971.bc	209.x	$90$	$24$	$1$	$1.982$	$\Q(\zeta_{45})$	$D_{5}$	$\Q(\sqrt{-19}) $	None			✓	✓	209.1.m.a	$24$	$0$	$0$	$0$	$0$	$-9$		$1$		$q+\zeta_{90}^{28}q^{4}+(\zeta_{90}^{8}+\zeta_{90}^{44})q^{5}+\cdots$
3971.2.a.a	$3971$	$2$	3971.a	1.a	$1$	$1$	$1$	$31.709$	$\Q$	$_{}$	None	None	✓				209.2.a.a	$1$	$1$	$0$	$-1$	$-3$	$-4$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{3}-2q^{4}-3q^{5}-4q^{7}-2q^{9}+\cdots$
3971.2.a.b	$3971$	$2$	3971.a	1.a	$1$	$1$	$1$	$31.709$	$\Q$	$_{}$	None	None	✓				11.2.a.a	$1$	$1$	$2$	$1$	$1$	$-2$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+q^{3}+2q^{4}+q^{5}+2q^{6}-2q^{7}+\cdots$
3971.2.a.c	$3971$	$2$	3971.a	1.a	$1$	$2$	$2$	$31.709$	$\Q(\sqrt{7}) $	$_{}$	None	None	✓	✓			3971.2.a.c	$2$	$1$	$0$	$0$	$-2$	$-8$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}-2q^{4}-q^{5}-4q^{7}+4q^{9}+\cdots$
3971.2.a.d	$3971$	$2$	3971.a	1.a	$1$	$2$	$2$	$31.709$	$\Q(\sqrt{2}) $	$_{}$	None	None	✓				209.2.a.b	$1$	$0$	$0$	$2$	$-2$	$-4$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}+(1-\beta )q^{3}-q^{5}+(-2+\beta )q^{6}+\cdots$
3971.2.a.e	$3971$	$2$	3971.a	1.a	$1$	$3$	$3$	$31.709$	$\Q(\zeta_{18})^+$	$_{}$	None	None	✓				209.2.j.a	$1$	$2$	$-3$	$-6$	$-6$	$-6$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta _{1})q^{2}-2q^{3}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots$
3971.2.a.f	$3971$	$2$	3971.a	1.a	$1$	$3$	$3$	$31.709$	$\Q(\zeta_{18})^+$	$_{}$	None	None	✓				209.2.j.a	$1$	$1$	$3$	$6$	$-6$	$-6$	$+$	$1$	$\mathrm{SU}(2)$	$q+(1-\beta _{1})q^{2}+2q^{3}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots$
3971.2.a.g	$3971$	$2$	3971.a	1.a	$1$	$4$	$4$	$31.709$	$\Q(\sqrt{3}, \sqrt{7})$	$_{}$	None	None	✓	✓			3971.2.a.g	$2$	$1$	$0$	$0$	$2$	$2$	$+$	$1$	$\mathrm{SU}(2)$	$q+(\beta _{1}+\beta _{3})q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{2}q^{5}+\cdots$
3971.2.a.h	$3971$	$2$	3971.a	1.a	$1$	$5$	$5$	$31.709$	5.5.246832.1	$_{}$	None	None	✓				209.2.a.c	$1$	$1$	$-2$	$-1$	$-5$	$6$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{2}q^{2}-\beta _{3}q^{3}+(\beta _{1}+\beta _{2}+\beta _{3}+\beta _{4})q^{4}+\cdots$
3971.2.a.i	$3971$	$2$	3971.a	1.a	$1$	$7$	$7$	$31.709$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	$_{}$	None	None	✓				209.2.a.d	$1$	$0$	$1$	$-2$	$2$	$10$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+\beta _{2}q^{3}+(2+\beta _{2}+\beta _{3})q^{4}+\cdots$
3971.2.a.j	$3971$	$2$	3971.a	1.a	$1$	$9$	$9$	$31.709$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	$_{}$	None	None	✓				209.2.e.a	$1$	$1$	$-1$	$-4$	$0$	$-3$	$+$	$3$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}-\beta _{3}q^{3}+(1+\beta _{2})q^{4}+\beta _{8}q^{5}+\cdots$
3971.2.a.k	$3971$	$2$	3971.a	1.a	$1$	$9$	$9$	$31.709$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	$_{}$	None	None	✓				209.2.e.b	$1$	$1$	$-1$	$0$	$0$	$-3$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}-\beta _{3}q^{3}+(1+\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots$
3971.2.a.l	$3971$	$2$	3971.a	1.a	$1$	$9$	$9$	$31.709$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	$_{}$	None	None	✓				209.2.e.b	$1$	$0$	$1$	$0$	$0$	$-3$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+\beta _{3}q^{3}+(1+\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots$
3971.2.a.m	$3971$	$2$	3971.a	1.a	$1$	$9$	$9$	$31.709$	$\mathbb{Q}[x]/(x^{9} - \cdots)$	$_{}$	None	None	✓				209.2.e.a	$1$	$0$	$1$	$4$	$0$	$-3$	$-$	$3$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+\beta _{3}q^{3}+(1+\beta _{2})q^{4}+\beta _{8}q^{5}+\cdots$
3971.2.a.n	$3971$	$2$	3971.a	1.a	$1$	$10$	$10$	$31.709$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	$_{}$	None	None	✓	✓			3971.2.a.n	$1$	$1$	$-2$	$1$	$-8$	$-7$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+\beta _{7}q^{3}+(-\beta _{5}+\beta _{6}+\beta _{7}+\cdots)q^{4}+\cdots$
3971.2.a.o	$3971$	$2$	3971.a	1.a	$1$	$10$	$10$	$31.709$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	$_{}$	None	None	✓	✓			3971.2.a.o	$2$	$0$	$0$	$0$	$2$	$6$	$-$	$2^{3}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+\beta _{4}q^{3}+(2+\beta _{2})q^{4}+\beta _{8}q^{5}+\cdots$
3971.2.a.p	$3971$	$2$	3971.a	1.a	$1$	$10$	$10$	$31.709$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	$_{}$	None	None	✓	✓			3971.2.a.n	$1$	$1$	$2$	$-1$	$-8$	$-7$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}-\beta _{7}q^{3}+(-\beta _{5}+\beta _{6}+\beta _{7}+\cdots)q^{4}+\cdots$
3971.2.a.q	$3971$	$2$	3971.a	1.a	$1$	$18$	$18$	$31.709$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	$_{}$	None	None	✓	✓			3971.2.a.q	$1$	$0$	$-2$	$0$	$5$	$7$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}-\beta _{6}q^{3}+(1+\beta _{2})q^{4}-\beta _{4}q^{5}+\cdots$
3971.2.a.r	$3971$	$2$	3971.a	1.a	$1$	$18$	$18$	$31.709$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	$_{}$	None	None	✓	✓			3971.2.a.q	$1$	$0$	$2$	$0$	$5$	$7$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+\beta _{6}q^{3}+(1+\beta _{2})q^{4}-\beta _{4}q^{5}+\cdots$
3971.2.a.s	$3971$	$2$	3971.a	1.a	$1$	$21$	$21$	$31.709$		$_{}$	None	None	✓		✓		209.2.j.b	$1$	$1$	$-6$	$-15$	$6$	$6$	$+$		$\mathrm{SU}(2)$
3971.2.a.t	$3971$	$2$	3971.a	1.a	$1$	$21$	$21$	$31.709$		$_{}$	None	None	✓				209.2.j.b	$1$	$0$	$6$	$15$	$6$	$6$	$-$		$\mathrm{SU}(2)$
3971.2.a.u	$3971$	$2$	3971.a	1.a	$1$	$24$	$24$	$31.709$		$_{}$	None	None	✓				209.2.j.c	$1$	$1$	$-3$	$-3$	$0$	$0$	$+$		$\mathrm{SU}(2)$
3971.2.a.v	$3971$	$2$	3971.a	1.a	$1$	$24$	$24$	$31.709$		$_{}$	None	None	✓	✓			3971.2.a.v	$2$	$1$	$0$	$0$	$-10$	$-14$	$+$		$\mathrm{SU}(2)$
3971.2.a.w	$3971$	$2$	3971.a	1.a	$1$	$24$	$24$	$31.709$		$_{}$	None	None	✓		✓		209.2.j.c	$1$	$0$	$3$	$3$	$0$	$0$	$-$		$\mathrm{SU}(2)$
3971.2.a.x	$3971$	$2$	3971.a	1.a	$1$	$40$	$40$	$31.709$		$_{}$	None	None	✓	✓	✓		3971.2.a.x	$2$	$0$	$0$	$0$	$28$	$14$	$-$		$\mathrm{SU}(2)$

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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.