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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				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*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	Coefficient ring index	Sato-Tate	$q$-expansion
3.7.b.a	$3$	$7$	3.b	3.b	$2$	$1$	$1$	$0.690$	$\Q$	$\Q(\sqrt{-3}) $	✓	$2$	$0$	$0$	$-27$	$0$	$-286$	$1$	$\mathrm{U}(1)[D_{2}]$	$q-3^{3}q^{3}+2^{6}q^{4}-286q^{7}+3^{6}q^{9}+\cdots$
4.7.b.a	$4$	$7$	4.b	4.b	$2$	$2$	$2$	$0.920$	$\Q(\sqrt{-15}) $	None		$2$	$0$	$4$	$0$	$20$	$0$	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+(2+\beta )q^{2}-4\beta q^{3}+(-56+4\beta )q^{4}+\cdots$
5.7.c.a	$5$	$7$	5.c	5.c	$4$	$4$	$2$	$1.150$	$\Q(i, \sqrt{201})$	None		$2$	$0$	$-10$	$30$	$-70$	$550$	$2$	$\mathrm{SU}(2)[C_{4}]$	$q+(-2+2\beta _{1}+\beta _{3})q^{2}+(7+8\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots$
6.7.b.a	$6$	$7$	6.b	3.b	$2$	$2$	$2$	$1.380$	$\Q(\sqrt{-2}) $	None		$2$	$0$	$0$	$42$	$0$	$4$	$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta q^{2}+(21+3\beta )q^{3}-2^{5}q^{4}-30\beta q^{5}+\cdots$
7.7.b.a	$7$	$7$	7.b	7.b	$2$	$1$	$1$	$1.610$	$\Q$	$\Q(\sqrt{-7}) $	✓	$2$	$0$	$9$	$0$	$0$	$-343$	$1$	$\mathrm{U}(1)[D_{2}]$	$q+9q^{2}+17q^{4}-7^{3}q^{7}-423q^{8}+\cdots$
7.7.b.b	$7$	$7$	7.b	7.b	$2$	$2$	$2$	$1.610$	$\Q(\sqrt{-510}) $	None		$2$	$0$	$-16$	$0$	$0$	$266$	$2$	$\mathrm{SU}(2)[C_{2}]$	$q-8q^{2}+\beta q^{3}-\beta q^{5}-8\beta q^{6}+(133+\cdots)q^{7}+\cdots$
7.7.d.a	$7$	$7$	7.d	7.d	$6$	$2$	$1$	$1.610$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$12$	$-21$	$315$	$-686$	$1$	$\mathrm{SU}(2)[C_{6}]$	$q+12\zeta_{6}q^{2}+(-7-7\zeta_{6})q^{3}+(-80+\cdots)q^{4}+\cdots$
7.7.d.b	$7$	$7$	7.d	7.d	$6$	$4$	$2$	$1.610$	$\Q(\sqrt{2}, \sqrt{-3})$	None		$2$	$0$	$-8$	$18$	$-150$	$280$	$3$	$\mathrm{SU}(2)[C_{6}]$	$q+(-4-4\beta _{1}+\beta _{2}+\beta _{3})q^{2}+(6+3\beta _{1}+\cdots)q^{3}+\cdots$
8.7.d.a	$8$	$7$	8.d	8.d	$2$	$1$	$1$	$1.840$	$\Q$	$\Q(\sqrt{-2}) $	✓	$2$	$0$	$-8$	$46$	$0$	$0$	$1$	$\mathrm{U}(1)[D_{2}]$	$q-8q^{2}+46q^{3}+2^{6}q^{4}-368q^{6}+\cdots$
8.7.d.b	$8$	$7$	8.d	8.d	$2$	$4$	$4$	$1.840$	4.0.3803625.2	None		$2$	$0$	$2$	$-48$	$0$	$0$	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+(-13+2\beta _{1}+\beta _{2})q^{3}+(-12+\cdots)q^{4}+\cdots$
9.7.b.a	$9$	$7$	9.b	3.b	$2$	$2$	$2$	$2.070$	$\Q(\sqrt{-2}) $	None		$2$	$0$	$0$	$0$	$0$	$1048$	$3^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta q^{2}-98q^{4}+5\beta q^{5}+524q^{7}+\cdots$
9.7.d.a	$9$	$7$	9.d	9.d	$6$	$10$	$5$	$2.070$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	None		$2$	$0$	$-3$	$24$	$-219$	$-121$	$2^{2}\cdot 3^{10}$	$\mathrm{SU}(2)[C_{6}]$	$q-\beta _{1}q^{2}+(2+\beta _{1}-\beta _{2}+\beta _{3}+\beta _{6}+\cdots)q^{3}+\cdots$
10.7.c.a	$10$	$7$	10.c	5.c	$4$	$2$	$1$	$2.301$	$\Q(\sqrt{-1}) $	None		$2$	$0$	$-8$	$-46$	$-150$	$-494$	$1$	$\mathrm{SU}(2)[C_{4}]$	$q+(-4-4i)q^{2}+(-23+23i)q^{3}+\cdots$
10.7.c.b	$10$	$7$	10.c	5.c	$4$	$4$	$2$	$2.301$	$\Q(i, \sqrt{129})$	None		$2$	$0$	$16$	$-18$	$330$	$-202$	$2\cdot 5^{2}$	$\mathrm{SU}(2)[C_{4}]$	$q+(4+4\beta _{1})q^{2}+(-5+4\beta _{1}-\beta _{3})q^{3}+\cdots$
11.7.b.a	$11$	$7$	11.b	11.b	$2$	$1$	$1$	$2.531$	$\Q$	$\Q(\sqrt{-11}) $	✓	$2$	$0$	$0$	$10$	$74$	$0$	$1$	$\mathrm{U}(1)[D_{2}]$	$q+10q^{3}+2^{6}q^{4}+74q^{5}-629q^{9}+\cdots$
11.7.b.b	$11$	$7$	11.b	11.b	$2$	$4$	$4$	$2.531$	$\mathbb{Q}[x]/(x^{4} + \cdots)$	None		$2$	$0$	$0$	$24$	$-260$	$0$	$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+(6-\beta _{2})q^{3}+(-71+\beta _{2}+\cdots)q^{4}+\cdots$
11.7.d.a	$11$	$7$	11.d	11.d	$10$	$20$	$5$	$2.531$	$\mathbb{Q}[x]/(x^{20} + \cdots)$	None		$2$	$0$	$-5$	$-39$	$181$	$-365$	$2^{4}\cdot 5\cdot 11^{2}$	$\mathrm{SU}(2)[C_{10}]$	$q+(-\beta _{4}+\beta _{6}-\beta _{9})q^{2}+(3\beta _{4}+2\beta _{5}+\cdots)q^{3}+\cdots$
12.7.c.a	$12$	$7$	12.c	3.b	$2$	$2$	$2$	$2.761$	$\Q(\sqrt{-5}) $	None		$2$	$0$	$0$	$-6$	$0$	$484$	$2^{2}\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	$q+(-3+\beta )q^{3}+6\beta q^{5}+242q^{7}+(-711+\cdots)q^{9}+\cdots$
12.7.d.a	$12$	$7$	12.d	4.b	$2$	$6$	$6$	$2.761$	6.0.50898483.1	None		$2$	$0$	$-10$	$0$	$-44$	$0$	$2^{12}\cdot 3^{5}$	$\mathrm{SU}(2)[C_{2}]$	$q+(-2-\beta _{1})q^{2}-\beta _{2}q^{3}+(3^{3}+2\beta _{1}+\cdots)q^{4}+\cdots$
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		$2$	$0$	$-12$	$-2$	$-114$	$316$		$\mathrm{SU}(2)[C_{12}]$
14.7.b.a	$14$	$7$	14.b	7.b	$2$	$4$	$4$	$3.221$	4.0.211968.1	None		$2$	$0$	$0$	$0$	$0$	$308$	$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{3}q^{2}-\beta _{1}q^{3}+2^{5}q^{4}-5\beta _{2}q^{5}+\cdots$
14.7.d.a	$14$	$7$	14.d	7.d	$6$	$8$	$4$	$3.221$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None		$2$	$0$	$0$	$0$	$-336$	$652$	$2^{6}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{6}]$	$q+(\beta _{2}-\beta _{3})q^{2}+(-\beta _{2}+2\beta _{3}-\beta _{6}+\cdots)q^{3}+\cdots$
15.7.c.a	$15$	$7$	15.c	3.b	$2$	$8$	$8$	$3.451$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None		$2$	$0$	$0$	$20$	$0$	$160$	$2\cdot 3^{4}\cdot 5^{4}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{2}q^{2}+(2-\beta _{5})q^{3}+(-41-\beta _{3}+\cdots)q^{4}+\cdots$
15.7.d.a	$15$	$7$	15.d	15.d	$2$	$1$	$1$	$3.451$	$\Q$	$\Q(\sqrt{-15}) $	✓	$2$	$0$	$-11$	$-27$	$125$	$0$	$1$	$\mathrm{U}(1)[D_{2}]$	$q-11q^{2}-3^{3}q^{3}+57q^{4}+5^{3}q^{5}+\cdots$
15.7.d.b	$15$	$7$	15.d	15.d	$2$	$1$	$1$	$3.451$	$\Q$	$\Q(\sqrt{-15}) $	✓	$2$	$0$	$11$	$27$	$-125$	$0$	$1$	$\mathrm{U}(1)[D_{2}]$	$q+11q^{2}+3^{3}q^{3}+57q^{4}-5^{3}q^{5}+\cdots$
15.7.d.c	$15$	$7$	15.d	15.d	$2$	$8$	$8$	$3.451$	$\mathbb{Q}[x]/(x^{8} + \cdots)$	None		$4$	$0$	$0$	$0$	$0$	$0$	$2^{7}\cdot 3^{6}\cdot 5^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{5}q^{2}+(\beta _{1}+\beta _{3}+\beta _{5})q^{3}+(9+\beta _{2}+\cdots)q^{4}+\cdots$
15.7.f.a	$15$	$7$	15.f	5.c	$4$	$12$	$6$	$3.451$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None		$2$	$0$	$16$	$0$	$136$	$-696$	$2\cdot 3^{10}\cdot 5^{3}$	$\mathrm{SU}(2)[C_{4}]$	$q+(1+\beta _{1}-\beta _{4})q^{2}-\beta _{6}q^{3}+(18\beta _{1}+\cdots)q^{4}+\cdots$
16.7.c.a	$16$	$7$	16.c	4.b	$2$	$1$	$1$	$3.681$	$\Q$	$\Q(\sqrt{-1}) $	✓	$2$	$0$	$0$	$0$	$234$	$0$	$1$	$\mathrm{U}(1)[D_{2}]$	$q+234q^{5}+3^{6}q^{9}-4070q^{13}-990q^{17}+\cdots$
16.7.c.b	$16$	$7$	16.c	4.b	$2$	$2$	$2$	$3.681$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$0$	$0$	$-300$	$0$	$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	$q-\zeta_{6}q^{3}-150q^{5}-22\zeta_{6}q^{7}-39q^{9}+\cdots$
16.7.f.a	$16$	$7$	16.f	16.f	$4$	$22$	$11$	$3.681$		None		$2$	$0$	$-2$	$-2$	$-2$	$-4$		$\mathrm{SU}(2)[C_{4}]$
17.7.e.a	$17$	$7$	17.e	17.e	$16$	$64$	$8$	$3.911$		None		$2$	$0$	$-8$	$-8$	$-8$	$-8$		$\mathrm{SU}(2)[C_{16}]$
18.7.b.a	$18$	$7$	18.b	3.b	$2$	$2$	$2$	$4.141$	$\Q(\sqrt{-2}) $	None		$2$	$0$	$0$	$0$	$0$	$-968$	$1$	$\mathrm{SU}(2)[C_{2}]$	$q+4\beta q^{2}-2^{5}q^{4}+123\beta q^{5}-22^{2}q^{7}+\cdots$
18.7.d.a	$18$	$7$	18.d	9.d	$6$	$12$	$6$	$4.141$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	None		$2$	$0$	$0$	$-42$	$432$	$240$	$2^{10}\cdot 3^{10}$	$\mathrm{SU}(2)[C_{6}]$	$q-\beta _{3}q^{2}+(-6+4\beta _{1}+\beta _{2}-\beta _{3}+\beta _{7}+\cdots)q^{3}+\cdots$
19.7.b.a	$19$	$7$	19.b	19.b	$2$	$1$	$1$	$4.371$	$\Q$	$\Q(\sqrt{-19}) $	✓	$2$	$0$	$0$	$0$	$-54$	$610$	$1$	$\mathrm{U}(1)[D_{2}]$	$q+2^{6}q^{4}-54q^{5}+610q^{7}+3^{6}q^{9}+\cdots$
19.7.b.b	$19$	$7$	19.b	19.b	$2$	$8$	$8$	$4.371$	$\mathbb{Q}[x]/(x^{8} + \cdots)$	None		$2$	$0$	$0$	$0$	$108$	$-140$	$2^{3}\cdot 29$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}-\beta _{2}q^{3}+(-57+\beta _{3}-\beta _{4}+\cdots)q^{4}+\cdots$
19.7.d.a	$19$	$7$	19.d	19.d	$6$	$18$	$9$	$4.371$	$\mathbb{Q}[x]/(x^{18} + \cdots)$	None		$2$	$0$	$-3$	$27$	$-57$	$-260$	$2^{2}\cdot 3^{3}\cdot 5^{2}\cdot 7^{2}\cdot 19^{2}$	$\mathrm{SU}(2)[C_{6}]$	$q+(\beta _{1}-\beta _{3})q^{2}+(2+\beta _{2}-\beta _{7})q^{3}+(21+\cdots)q^{4}+\cdots$
19.7.f.a	$19$	$7$	19.f	19.f	$18$	$54$	$9$	$4.371$		None		$2$	$0$	$-6$	$-36$	$-6$	$-219$		$\mathrm{SU}(2)[C_{18}]$
20.7.b.a	$20$	$7$	20.b	4.b	$2$	$12$	$12$	$4.601$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	None		$2$	$0$	$-10$	$0$	$0$	$0$	$2^{33}\cdot 5^{7}$	$\mathrm{SU}(2)[C_{2}]$	$q+(-1-\beta _{2})q^{2}+(-\beta _{2}-\beta _{6})q^{3}+(13+\cdots)q^{4}+\cdots$
20.7.d.a	$20$	$7$	20.d	20.d	$2$	$1$	$1$	$4.601$	$\Q$	$\Q(\sqrt{-5}) $	✓	$2$	$0$	$-8$	$-44$	$-125$	$524$	$1$	$\mathrm{U}(1)[D_{2}]$	$q-8q^{2}-44q^{3}+2^{6}q^{4}-5^{3}q^{5}+352q^{6}+\cdots$
20.7.d.b	$20$	$7$	20.d	20.d	$2$	$1$	$1$	$4.601$	$\Q$	$\Q(\sqrt{-5}) $	✓	$2$	$0$	$8$	$44$	$-125$	$-524$	$1$	$\mathrm{U}(1)[D_{2}]$	$q+8q^{2}+44q^{3}+2^{6}q^{4}-5^{3}q^{5}+352q^{6}+\cdots$
20.7.d.c	$20$	$7$	20.d	20.d	$2$	$2$	$2$	$4.601$	$\Q(\sqrt{-1}) $	$\Q(\sqrt{-1}) $		$4$	$0$	$0$	$0$	$-234$	$0$	$2^{2}$	$\mathrm{U}(1)[D_{2}]$	$q+2iq^{2}-2^{6}q^{4}+(-117+11i)q^{5}+\cdots$
20.7.d.d	$20$	$7$	20.d	20.d	$2$	$12$	$12$	$4.601$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	None		$4$	$0$	$0$	$0$	$460$	$0$	$2^{32}\cdot 3^{2}\cdot 5^{2}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{3}+(6+\beta _{4}-\beta _{8}+\cdots)q^{4}+\cdots$
20.7.f.a	$20$	$7$	20.f	5.c	$4$	$6$	$3$	$4.601$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None		$2$	$0$	$0$	$32$	$-156$	$-264$	$2^{7}\cdot 5$	$\mathrm{SU}(2)[C_{4}]$	$q+(5+5\beta _{1}+\beta _{4})q^{3}+(-5^{2}+20\beta _{1}+\cdots)q^{5}+\cdots$
21.7.b.a	$21$	$7$	21.b	3.b	$2$	$12$	$12$	$4.831$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	None		$2$	$0$	$0$	$52$	$0$	$0$	$2^{2}\cdot 3^{8}\cdot 7^{6}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+(4-\beta _{3})q^{3}+(-43+\beta _{2}+\cdots)q^{4}+\cdots$
21.7.d.a	$21$	$7$	21.d	7.b	$2$	$8$	$8$	$4.831$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None		$2$	$0$	$10$	$0$	$0$	$28$	$2^{5}\cdot 3^{7}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+(1-\beta _{1})q^{2}-\beta _{4}q^{3}+(44+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots$
21.7.f.a	$21$	$7$	21.f	7.d	$6$	$8$	$4$	$4.831$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None		$2$	$0$	$-5$	$-108$	$-294$	$-656$	$2^{2}\cdot 3\cdot 7$	$\mathrm{SU}(2)[C_{6}]$	$q+(-1-\beta _{1}+\beta _{2})q^{2}+(-18+9\beta _{2}+\cdots)q^{3}+\cdots$
21.7.f.b	$21$	$7$	21.f	7.d	$6$	$8$	$4$	$4.831$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None		$2$	$0$	$-5$	$108$	$-42$	$748$	$3\cdot 7^{2}$	$\mathrm{SU}(2)[C_{6}]$	$q+(-\beta _{1}-\beta _{2})q^{2}+(9+9\beta _{2})q^{3}+(-43+\cdots)q^{4}+\cdots$
21.7.h.a	$21$	$7$	21.h	21.h	$6$	$28$	$14$	$4.831$		None		$4$	$0$	$0$	$-1$	$0$	$1120$		$\mathrm{SU}(2)[C_{6}]$
22.7.b.a	$22$	$7$	22.b	11.b	$2$	$6$	$6$	$5.061$	$\mathbb{Q}[x]/(x^{6} - \cdots)$	None		$2$	$0$	$0$	$-52$	$368$	$0$	$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{2}q^{2}+(-9+\beta _{1})q^{3}-2^{5}q^{4}+(61+\cdots)q^{5}+\cdots$

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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.