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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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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
513.1.o.a	$513$	$1$	513.o	171.o	$6$	$4$	$2$	$0.256$	$\Q(\zeta_{12})$	$A_{4}$	None	None			✓	✓	171.1.o.a	$4$	$0$	$0$	$0$	$2$	$-2$		$1$		$q-\zeta_{12}q^{2}+\zeta_{12}^{2}q^{5}+\zeta_{12}^{4}q^{7}+\zeta_{12}^{3}q^{8}+\cdots$
513.1.br.a	$513$	$1$	513.br	19.f	$18$	$6$	$1$	$0.256$	$\Q(\zeta_{18})$	$D_{18}$	$\Q(\sqrt{-3}) $	None		✓	✓	✓	513.1.br.a	$4$	$0$	$0$	$0$	$0$	$0$		$1$		$q+\zeta_{18}^{4}q^{4}+(\zeta_{18}^{7}-\zeta_{18}^{8})q^{7}+(-1+\cdots)q^{13}+\cdots$
513.1.bs.a	$513$	$1$	513.bs	57.l	$18$	$6$	$1$	$0.256$	$\Q(\zeta_{18})$	$D_{9}$	$\Q(\sqrt{-3}) $	None		✓	✓	✓	513.1.bs.a	$4$	$0$	$0$	$0$	$0$	$0$		$1$		$q+\zeta_{18}^{4}q^{4}+(-\zeta_{18}^{7}+\zeta_{18}^{8})q^{7}+\cdots$
513.2.a.a	$513$	$2$	513.a	1.a	$1$	$1$	$1$	$4.096$	$\Q$	$_{}$	None	None	✓	✓			513.2.a.a	$1$	$1$	$-1$	$0$	$0$	$-2$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}-q^{4}-2q^{7}+3q^{8}+5q^{11}+\cdots$
513.2.a.b	$513$	$2$	513.a	1.a	$1$	$1$	$1$	$4.096$	$\Q$	$_{}$	None	None	✓	✓			513.2.a.a	$1$	$1$	$1$	$0$	$0$	$-2$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}-q^{4}-2q^{7}-3q^{8}-5q^{11}+\cdots$
513.2.a.c	$513$	$2$	513.a	1.a	$1$	$2$	$2$	$4.096$	$\Q(\sqrt{3}) $	$_{}$	None	None	✓	✓			513.2.a.c	$2$	$1$	$0$	$0$	$0$	$-8$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}+q^{4}-2\beta q^{5}-4q^{7}-\beta q^{8}+\cdots$
513.2.a.d	$513$	$2$	513.a	1.a	$1$	$3$	$3$	$4.096$	$\Q(\zeta_{18})^+$	$_{}$	None	None	✓	✓	✓		513.2.a.d	$1$	$1$	$-3$	$0$	$-3$	$-3$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots$
513.2.a.e	$513$	$2$	513.a	1.a	$1$	$3$	$3$	$4.096$	3.3.321.1	$_{}$	None	None	✓	✓	✓		513.2.a.e	$1$	$1$	$-1$	$0$	$-5$	$1$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(-2+\beta _{1}+\cdots)q^{5}+\cdots$
513.2.a.f	$513$	$2$	513.a	1.a	$1$	$3$	$3$	$4.096$	3.3.321.1	$_{}$	None	None	✓	✓			513.2.a.e	$1$	$0$	$1$	$0$	$5$	$1$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(2-\beta _{1}+\cdots)q^{5}+\cdots$
513.2.a.g	$513$	$2$	513.a	1.a	$1$	$3$	$3$	$4.096$	$\Q(\zeta_{18})^+$	$_{}$	None	None	✓	✓			513.2.a.d	$1$	$0$	$3$	$0$	$3$	$-3$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1-\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots$
513.2.a.h	$513$	$2$	513.a	1.a	$1$	$4$	$4$	$4.096$	4.4.27648.1	$_{}$	None	None	✓	✓	✓		513.2.a.h	$2$	$0$	$0$	$0$	$0$	$8$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+\beta _{3}q^{5}+(2-\beta _{2}+\cdots)q^{7}+\cdots$
513.2.a.i	$513$	$2$	513.a	1.a	$1$	$4$	$4$	$4.096$	4.4.29952.1	$_{}$	None	None	✓	✓	✓		513.2.a.i	$2$	$0$	$0$	$0$	$0$	$12$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{2}+(2+\beta _{2})q^{4}-\beta _{3}q^{5}+(3+\beta _{2}+\cdots)q^{7}+\cdots$
513.2.d.a	$513$	$2$	513.d	57.d	$2$	$2$	$2$	$4.096$	$\Q(\sqrt{-3}) $	$_{}$	$\Q(\sqrt{-3}) $	None		✓			513.2.d.a	$4$	$0$	$0$	$0$	$0$	$2$		$3$	$\mathrm{U}(1)[D_{2}]$	$q-2 q^{4}+q^{7}+(-2\beta+1)q^{13}+4 q^{16}+\cdots$
513.2.d.b	$513$	$2$	513.d	57.d	$2$	$12$	$12$	$4.096$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	$_{}$	None	None		✓			513.2.d.b	$4$	$0$	$0$	$0$	$0$	$4$		$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+\beta _{5}q^{4}-\beta _{7}q^{5}-\beta _{4}q^{7}+\cdots$
513.2.d.c	$513$	$2$	513.d	57.d	$2$	$12$	$12$	$4.096$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	$_{}$	None	None		✓			513.2.d.c	$4$	$0$	$0$	$0$	$0$	$-8$		$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{7}q^{2}+(2+\beta _{1})q^{4}-\beta _{3}q^{5}+(-1+\cdots)q^{7}+\cdots$
513.2.e.a	$513$	$2$	513.e	9.c	$3$	$18$	$9$	$4.096$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	$_{}$	None	None					171.2.e.b	$2$	$0$	$-3$	$0$	$-7$	$0$		$3^{4}$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{3}q^{2}+(-\beta _{7}-\beta _{11}+\beta _{12})q^{4}+\cdots$
513.2.e.b	$513$	$2$	513.e	9.c	$3$	$18$	$9$	$4.096$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	$_{}$	None	None					171.2.e.a	$2$	$0$	$3$	$0$	$9$	$0$		$3^{6}$	$\mathrm{SU}(2)[C_{3}]$	$q+(-\beta _{1}+\beta _{4})q^{2}+(\beta _{3}+\beta _{7}-\beta _{16}+\cdots)q^{4}+\cdots$
513.2.f.a	$513$	$2$	513.f	19.c	$3$	$2$	$1$	$4.096$	$\Q(\sqrt{-3}) $	$_{}$	None	None		✓			513.2.f.a	$2$	$0$	$-2$	$0$	$4$	$0$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-2+2\zeta_{6})q^{2}-2\zeta_{6}q^{4}+(4-4\zeta_{6})q^{5}+\cdots$
513.2.f.b	$513$	$2$	513.f	19.c	$3$	$2$	$1$	$4.096$	$\Q(\sqrt{-3}) $	$_{}$	$\Q(\sqrt{-3}) $	None		✓			513.2.f.b	$4$	$0$	$0$	$0$	$0$	$-8$		$1$	$\mathrm{U}(1)[D_{3}]$	$q+2\zeta_{6}q^{4}-4q^{7}+7\zeta_{6}q^{13}+(-4+\cdots)q^{16}+\cdots$
513.2.f.c	$513$	$2$	513.f	19.c	$3$	$2$	$1$	$4.096$	$\Q(\sqrt{-3}) $	$_{}$	$\Q(\sqrt{-3}) $	None		✓			513.2.f.c	$4$	$0$	$0$	$0$	$0$	$10$		$1$	$\mathrm{U}(1)[D_{3}]$	$q+2\zeta_{6}q^{4}+5q^{7}-2\zeta_{6}q^{13}+(-4+\cdots)q^{16}+\cdots$
513.2.f.d	$513$	$2$	513.f	19.c	$3$	$2$	$1$	$4.096$	$\Q(\sqrt{-3}) $	$_{}$	None	None		✓			513.2.f.a	$2$	$0$	$2$	$0$	$-4$	$0$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(2-2\zeta_{6})q^{2}-2\zeta_{6}q^{4}+(-4+4\zeta_{6})q^{5}+\cdots$
513.2.f.e	$513$	$2$	513.f	19.c	$3$	$8$	$4$	$4.096$	8.0.$\cdots$.17	$_{}$	None	None		✓			513.2.f.e	$4$	$0$	$0$	$0$	$0$	$8$		$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{1}q^{2}+(-3+3\beta _{4}-\beta _{5})q^{4}+\beta _{1}q^{5}+\cdots$
513.2.f.f	$513$	$2$	513.f	19.c	$3$	$12$	$6$	$4.096$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	$_{}$	None	None		✓			513.2.f.f	$2$	$0$	$-1$	$0$	$-5$	$2$		$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q+(\beta _{1}+\beta _{6})q^{2}+(-\beta _{1}+\beta _{3}-\beta _{8})q^{4}+\cdots$
513.2.f.g	$513$	$2$	513.f	19.c	$3$	$12$	$6$	$4.096$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	$_{}$	None	None		✓			513.2.f.g	$4$	$0$	$0$	$0$	$0$	$-16$		$2^{4}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{7}q^{2}+(-1-\beta _{2}-\beta _{3})q^{4}+(\beta _{4}+\cdots)q^{5}+\cdots$
513.2.f.h	$513$	$2$	513.f	19.c	$3$	$12$	$6$	$4.096$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	$_{}$	None	None		✓			513.2.f.f	$2$	$0$	$1$	$0$	$5$	$2$		$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q+(-\beta _{1}-\beta _{6})q^{2}+(-\beta _{1}+\beta _{3}-\beta _{8}+\cdots)q^{4}+\cdots$
513.2.g.a	$513$	$2$	513.g	171.g	$3$	$2$	$1$	$4.096$	$\Q(\sqrt{-3}) $	$_{}$	None	None					171.2.g.a	$2$	$1$	$-1$	$0$	$-6$	$-1$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}-3q^{5}+(-1+\cdots)q^{7}+\cdots$
513.2.g.b	$513$	$2$	513.g	171.g	$3$	$2$	$1$	$4.096$	$\Q(\sqrt{-3}) $	$_{}$	None	None					171.2.g.b	$2$	$0$	$-1$	$0$	$2$	$-3$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}+q^{5}+(-3+3\zeta_{6})q^{7}+\cdots$
513.2.g.c	$513$	$2$	513.g	171.g	$3$	$32$	$16$	$4.096$		$_{}$	None	None			✓		171.2.g.c	$2$	$0$	$-1$	$0$	$6$	$1$			$\mathrm{SU}(2)[C_{3}]$
513.2.h.a	$513$	$2$	513.h	171.h	$3$	$2$	$1$	$4.096$	$\Q(\sqrt{-3}) $	$_{}$	None	None					171.2.g.b	$2$	$0$	$2$	$0$	$-1$	$-3$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+q^{2}-q^{4}-\zeta_{6}q^{5}-3\zeta_{6}q^{7}-3q^{8}+\cdots$
513.2.h.b	$513$	$2$	513.h	171.h	$3$	$2$	$1$	$4.096$	$\Q(\sqrt{-3}) $	$_{}$	None	None					171.2.g.a	$2$	$0$	$2$	$0$	$3$	$-1$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+q^{2}-q^{4}+3\zeta_{6}q^{5}-\zeta_{6}q^{7}-3q^{8}+\cdots$
513.2.h.c	$513$	$2$	513.h	171.h	$3$	$32$	$16$	$4.096$		$_{}$	None	None			✓		171.2.g.c	$2$	$0$	$2$	$0$	$-3$	$1$			$\mathrm{SU}(2)[C_{3}]$
513.2.k.a	$513$	$2$	513.k	171.k	$6$	$36$	$18$	$4.096$		$_{}$	None	None			✓	✓	171.2.k.a	$2$	$0$	$3$	$0$	$0$	$-1$			$\mathrm{SU}(2)[C_{6}]$
513.2.l.a	$513$	$2$	513.l	171.l	$6$	$36$	$18$	$4.096$		$_{}$	None	None			✓	✓	171.2.l.a	$4$	$0$	$0$	$0$	$6$	$-4$			$\mathrm{SU}(2)[C_{6}]$
513.2.m.a	$513$	$2$	513.m	57.f	$6$	$2$	$1$	$4.096$	$\Q(\sqrt{-3}) $	$_{}$	$\Q(\sqrt{-3}) $	None		✓			513.2.m.a	$4$	$0$	$0$	$0$	$0$	$-10$		$1$	$\mathrm{U}(1)[D_{6}]$	$q+2\zeta_{6}q^{4}-5q^{7}+(-8+4\zeta_{6})q^{13}+\cdots$
513.2.m.b	$513$	$2$	513.m	57.f	$6$	$2$	$1$	$4.096$	$\Q(\sqrt{-3}) $	$_{}$	$\Q(\sqrt{-3}) $	None		✓			513.2.m.b	$4$	$0$	$0$	$0$	$0$	$8$		$1$	$\mathrm{U}(1)[D_{6}]$	$q+2\zeta_{6}q^{4}+4q^{7}+(-2+\zeta_{6})q^{13}+\cdots$
513.2.m.c	$513$	$2$	513.m	57.f	$6$	$4$	$2$	$4.096$	$\Q(\sqrt{-2}, \sqrt{-3})$	$_{}$	None	None		✓			513.2.m.c	$4$	$0$	$0$	$0$	$0$	$8$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+(-\beta _{1}-\beta _{3})q^{2}+(-4+4\beta _{2})q^{4}+\cdots$
513.2.m.d	$513$	$2$	513.m	57.f	$6$	$4$	$2$	$4.096$	$\Q(\sqrt{-3}, \sqrt{-14})$	$_{}$	None	None		✓			513.2.m.d	$4$	$0$	$0$	$0$	$0$	$8$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+(2-2\beta _{2})q^{4}+(\beta _{1}-\beta _{3})q^{5}+2q^{7}+\cdots$
513.2.m.e	$513$	$2$	513.m	57.f	$6$	$8$	$4$	$4.096$	8.0.2569273344.7	$_{}$	None	None		✓			513.2.m.e	$4$	$0$	$0$	$0$	$0$	$8$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q-\beta _{7}q^{2}+(2\beta _{2}+3\beta _{5}+\beta _{6})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots$
513.2.m.f	$513$	$2$	513.m	57.f	$6$	$8$	$4$	$4.096$	8.0.$\cdots$.5	$_{}$	None	None		✓			513.2.m.f	$4$	$0$	$0$	$0$	$0$	$-16$		$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	$q+(-\beta _{2}+\beta _{5})q^{2}+(-\beta _{3}+\beta _{4}-\beta _{5}+\cdots)q^{5}+\cdots$
513.2.m.g	$513$	$2$	513.m	57.f	$6$	$24$	$12$	$4.096$		$_{}$	None	None		✓	✓		513.2.m.g	$4$	$0$	$0$	$0$	$0$	$-4$			$\mathrm{SU}(2)[C_{6}]$
513.2.t.a	$513$	$2$	513.t	171.t	$6$	$36$	$18$	$4.096$		$_{}$	None	None			✓	✓	171.2.k.a	$2$	$0$	$6$	$0$	$3$	$-1$			$\mathrm{SU}(2)[C_{6}]$
513.2.u.a	$513$	$2$	513.u	513.u	$9$	$348$	$58$	$4.096$		$_{}$	None	None		✓	✓	✓	513.2.u.a	$2$	$0$	$3$	$-6$	$-6$	$-3$			$\mathrm{SU}(2)[C_{9}]$
513.2.v.a	$513$	$2$	513.v	513.v	$9$	$348$	$58$	$4.096$		$_{}$	None	None		✓	✓	✓	513.2.v.a	$2$	$0$	$3$	$-6$	$-6$	$-3$			$\mathrm{SU}(2)[C_{9}]$
513.2.w.a	$513$	$2$	513.w	513.w	$9$	$348$	$58$	$4.096$		$_{}$	None	None		✓	✓	✓	513.2.w.a	$2$	$0$	$-6$	$-12$	$-3$	$-3$			$\mathrm{SU}(2)[C_{9}]$
513.2.x.a	$513$	$2$	513.x	513.x	$9$	$348$	$58$	$4.096$		$_{}$	None	None		✓	✓	✓	513.2.u.a	$2$	$0$	$-6$	$-3$	$-3$	$-3$			$\mathrm{SU}(2)[C_{9}]$
513.2.y.a	$513$	$2$	513.y	19.e	$9$	$6$	$1$	$4.096$	$\Q(\zeta_{18})$	$_{}$	None	None		✓			513.2.y.a	$2$	$0$	$-3$	$0$	$-3$	$9$		$1$	$\mathrm{SU}(2)[C_{9}]$	$q+(-\zeta_{18}-\zeta_{18}^{2}-\zeta_{18}^{3}+\zeta_{18}^{5})q^{2}+\cdots$
513.2.y.b	$513$	$2$	513.y	19.e	$9$	$6$	$1$	$4.096$	$\Q(\zeta_{18})$	$_{}$	$\Q(\sqrt{-3}) $	None		✓			513.2.y.b	$4$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{9}]$	$q-2\zeta_{18}^{2}q^{4}+(-3\zeta_{18}+2\zeta_{18}^{2}+\zeta_{18}^{4}+\cdots)q^{7}+\cdots$
513.2.y.c	$513$	$2$	513.y	19.e	$9$	$6$	$1$	$4.096$	$\Q(\zeta_{18})$	$_{}$	None	None		✓			513.2.y.a	$2$	$0$	$3$	$0$	$3$	$9$		$1$	$\mathrm{SU}(2)[C_{9}]$	$q+(\zeta_{18}+\zeta_{18}^{2}+\zeta_{18}^{3}-\zeta_{18}^{5})q^{2}+\cdots$
513.2.y.d	$513$	$2$	513.y	19.e	$9$	$24$	$4$	$4.096$		$_{}$	None	None		✓			513.2.y.d	$4$	$0$	$0$	$0$	$0$	$12$			$\mathrm{SU}(2)[C_{9}]$
513.2.y.e	$513$	$2$	513.y	19.e	$9$	$36$	$6$	$4.096$		$_{}$	None	None		✓			513.2.y.e	$2$	$0$	$0$	$0$	$-9$	$-9$			$\mathrm{SU}(2)[C_{9}]$

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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.