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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	Image	CM	RM	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	Image	CM	RM	Self-dual	Inner twists	Rank*	$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$	Fricke sign	Coefficient ring index	Sato-Tate	$q$-expansion
648.1.b.a	$648$	$1$	648.b	8.d	$2$	$1$	$1$	$0.323$	$\Q$	$D_{3}$	$\Q(\sqrt{-2}) $	None	✓	$2$	$0$	$-1$	$0$	$0$	$0$		$1$		$q-q^{2}+q^{4}-q^{8}+q^{11}+q^{16}+q^{17}+\cdots$
648.1.b.b	$648$	$1$	648.b	8.d	$2$	$1$	$1$	$0.323$	$\Q$	$D_{3}$	$\Q(\sqrt{-2}) $	None	✓	$2$	$0$	$1$	$0$	$0$	$0$		$1$		$q+q^{2}+q^{4}+q^{8}-q^{11}+q^{16}-q^{17}+\cdots$
648.1.j.a	$648$	$1$	648.j	72.j	$6$	$2$	$1$	$0.323$	$\Q(\sqrt{-3}) $	$D_{3}$	$\Q(\sqrt{-6}) $	None		$4$	$0$	$-1$	$0$	$1$	$1$		$1$		$q+\zeta_{6}^{2}q^{2}-\zeta_{6}q^{4}+\zeta_{6}q^{5}-\zeta_{6}^{2}q^{7}+\cdots$
648.1.j.b	$648$	$1$	648.j	72.j	$6$	$2$	$1$	$0.323$	$\Q(\sqrt{-3}) $	$D_{3}$	$\Q(\sqrt{-6}) $	None		$4$	$0$	$1$	$0$	$-1$	$1$		$1$		$q-\zeta_{6}^{2}q^{2}-\zeta_{6}q^{4}-\zeta_{6}q^{5}-\zeta_{6}^{2}q^{7}+\cdots$
648.1.r.a	$648$	$1$	648.r	216.r	$18$	$6$	$1$	$0.323$	$\Q(\zeta_{18})$	$D_{9}$	$\Q(\sqrt{-2}) $	None		$4$	$0$	$0$	$0$	$0$	$0$		$1$		$q-\zeta_{18}^{4}q^{2}+\zeta_{18}^{8}q^{4}+\zeta_{18}^{3}q^{8}+\cdots$
648.1.bf.a	$648$	$1$	648.bf	648.af	$54$	$18$	$1$	$0.323$	$\Q(\zeta_{54})$	$D_{27}$	$\Q(\sqrt{-2}) $	None		$4$	$0$	$0$	$0$	$0$	$0$		$1$		$q-\zeta_{54}^{7}q^{2}-\zeta_{54}^{19}q^{3}+\zeta_{54}^{14}q^{4}+\cdots$
648.2.a.a	$648$	$2$	648.a	1.a	$1$	$1$	$1$	$5.174$	$\Q$	$_{}$	None	None	✓	$1$	$1$	$0$	$0$	$-1$	$-3$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{5}-3q^{7}+5q^{11}-5q^{13}-2q^{17}+\cdots$
648.2.a.b	$648$	$2$	648.a	1.a	$1$	$1$	$1$	$5.174$	$\Q$	$_{}$	None	None	✓	$1$	$1$	$0$	$0$	$-1$	$0$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{5}-4q^{11}-5q^{13}-5q^{17}+8q^{19}+\cdots$
648.2.a.c	$648$	$2$	648.a	1.a	$1$	$1$	$1$	$5.174$	$\Q$	$_{}$	None	None	✓	$1$	$1$	$0$	$0$	$1$	$-3$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{5}-3q^{7}-5q^{11}-5q^{13}+2q^{17}+\cdots$
648.2.a.d	$648$	$2$	648.a	1.a	$1$	$1$	$1$	$5.174$	$\Q$	$_{}$	None	None	✓	$1$	$0$	$0$	$0$	$1$	$0$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{5}+4q^{11}-5q^{13}+5q^{17}+8q^{19}+\cdots$
648.2.a.e	$648$	$2$	648.a	1.a	$1$	$2$	$2$	$5.174$	$\Q(\sqrt{3}) $	$_{}$	None	None	✓	$1$	$1$	$0$	$0$	$-4$	$0$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-2+\beta )q^{5}-2\beta q^{7}-2q^{11}+(1+\cdots)q^{13}+\cdots$
648.2.a.f	$648$	$2$	648.a	1.a	$1$	$2$	$2$	$5.174$	$\Q(\sqrt{33}) $	$_{}$	None	None	✓	$1$	$0$	$0$	$0$	$-1$	$3$	$-$	$1$	$\mathrm{SU}(2)$	$q-\beta q^{5}+(2-\beta )q^{7}+q^{11}+(2+\beta )q^{13}+\cdots$
648.2.a.g	$648$	$2$	648.a	1.a	$1$	$2$	$2$	$5.174$	$\Q(\sqrt{33}) $	$_{}$	None	None	✓	$1$	$0$	$0$	$0$	$1$	$3$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{5}+(2-\beta )q^{7}-q^{11}+(2+\beta )q^{13}+\cdots$
648.2.a.h	$648$	$2$	648.a	1.a	$1$	$2$	$2$	$5.174$	$\Q(\sqrt{3}) $	$_{}$	None	None	✓	$1$	$0$	$0$	$0$	$4$	$0$	$-$	$1$	$\mathrm{SU}(2)$	$q+(2+\beta )q^{5}+2\beta q^{7}+2q^{11}+(1-2\beta )q^{13}+\cdots$
648.2.d.a	$648$	$2$	648.d	8.b	$2$	$2$	$2$	$5.174$	$\Q(\sqrt{-1}) $	$_{}$	None	None		$2$	$0$	$-2$	$0$	$0$	$8$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+(-1+i)q^{2}-2iq^{4}-2iq^{5}+4q^{7}+\cdots$
648.2.d.b	$648$	$2$	648.d	8.b	$2$	$2$	$2$	$5.174$	$\Q(\sqrt{-7}) $	$_{}$	None	None		$2$	$1$	$-1$	$0$	$0$	$-4$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta q^{2}+(-2+\beta )q^{4}+(-1+2\beta )q^{5}+\cdots$
648.2.d.c	$648$	$2$	648.d	8.b	$2$	$2$	$2$	$5.174$	$\Q(\sqrt{-7}) $	$_{}$	None	None		$2$	$0$	$1$	$0$	$0$	$-4$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta q^{2}+(-2+\beta )q^{4}+(1-2\beta )q^{5}+\cdots$
648.2.d.d	$648$	$2$	648.d	8.b	$2$	$2$	$2$	$5.174$	$\Q(\sqrt{-1}) $	$_{}$	None	None		$2$	$0$	$2$	$0$	$0$	$8$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+(1+i)q^{2}+2iq^{4}-2iq^{5}+4q^{7}+\cdots$
648.2.d.e	$648$	$2$	648.d	8.b	$2$	$4$	$4$	$5.174$	$\Q(\zeta_{12})$	$_{}$	None	None		$2$	$0$	$-2$	$0$	$0$	$4$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+(-\zeta_{12}+\zeta_{12}^{2})q^{2}+(\zeta_{12}+\zeta_{12}^{3})q^{4}+\cdots$
648.2.d.f	$648$	$2$	648.d	8.b	$2$	$4$	$4$	$5.174$	$\Q(\sqrt{3}, \sqrt{-5})$	$_{}$	None	None		$4$	$0$	$0$	$0$	$0$	$-16$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{1}+\beta _{3})q^{5}-4q^{7}+\cdots$
648.2.d.g	$648$	$2$	648.d	8.b	$2$	$4$	$4$	$5.174$	$\Q(\sqrt{-2}, \sqrt{-3})$	$_{}$	None	None		$2$	$0$	$0$	$0$	$0$	$8$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+(1+\beta _{3})q^{4}+(\beta _{2}-\beta _{3})q^{5}+\cdots$
648.2.d.h	$648$	$2$	648.d	8.b	$2$	$4$	$4$	$5.174$	$\Q(\sqrt{-2}, \sqrt{-3})$	$_{}$	None	None		$2$	$0$	$0$	$0$	$0$	$8$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+(1+\beta _{3})q^{4}+(\beta _{2}+\beta _{3})q^{5}+\cdots$
648.2.d.i	$648$	$2$	648.d	8.b	$2$	$4$	$4$	$5.174$	$\Q(\zeta_{12})$	$_{}$	None	None		$2$	$0$	$2$	$0$	$0$	$4$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+(\zeta_{12}-\zeta_{12}^{2})q^{2}+(\zeta_{12}+\zeta_{12}^{3})q^{4}+\cdots$
648.2.d.j	$648$	$2$	648.d	8.b	$2$	$8$	$8$	$5.174$	8.0.$\cdots$.1	$_{}$	None	None		$2$	$0$	$-1$	$0$	$0$	$-6$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{4}q^{2}-\beta _{5}q^{4}+(\beta _{1}+\beta _{4}-\beta _{7})q^{5}+\cdots$
648.2.d.k	$648$	$2$	648.d	8.b	$2$	$8$	$8$	$5.174$	8.0.$\cdots$.1	$_{}$	None	None		$2$	$0$	$1$	$0$	$0$	$-6$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{4}q^{2}-\beta _{5}q^{4}+(-\beta _{1}-\beta _{4}+\beta _{7})q^{5}+\cdots$
648.2.f.a	$648$	$2$	648.f	24.f	$2$	$4$	$4$	$5.174$	$\Q(\sqrt{-2}, \sqrt{-3})$	$_{}$	$\Q(\sqrt{-2}) $	None		$4$	$0$	$0$	$0$	$0$	$0$		$2^{2}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	$q+\beta _{1}q^{2}-2q^{4}-2\beta _{1}q^{8}+(-\beta _{1}-\beta _{2}+\cdots)q^{11}+\cdots$
648.2.f.b	$648$	$2$	648.f	24.f	$2$	$16$	$16$	$5.174$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	$_{}$	None	None		$4$	$0$	$0$	$0$	$0$	$0$		$2^{10}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{3}q^{2}+(1-\beta _{9})q^{4}-\beta _{4}q^{5}+(-\beta _{7}+\cdots)q^{7}+\cdots$
648.2.f.c	$648$	$2$	648.f	24.f	$2$	$24$	$24$	$5.174$		$_{}$	None	None		$4$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{2}]$
648.2.i.a	$648$	$2$	648.i	9.c	$3$	$2$	$1$	$5.174$	$\Q(\sqrt{-3}) $	$_{}$	None	None		$2$	$0$	$0$	$0$	$-4$	$3$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q-4\zeta_{6}q^{5}+(3-3\zeta_{6})q^{7}+(-4+4\zeta_{6})q^{11}+\cdots$
648.2.i.b	$648$	$2$	648.i	9.c	$3$	$2$	$1$	$5.174$	$\Q(\sqrt{-3}) $	$_{}$	None	None		$2$	$0$	$0$	$0$	$-2$	$0$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q-2\zeta_{6}q^{5}+(4-4\zeta_{6})q^{11}+2\zeta_{6}q^{13}+\cdots$
648.2.i.c	$648$	$2$	648.i	9.c	$3$	$2$	$1$	$5.174$	$\Q(\sqrt{-3}) $	$_{}$	None	None		$2$	$0$	$0$	$0$	$-1$	$-3$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q-\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+(5-5\zeta_{6})q^{11}+\cdots$
648.2.i.d	$648$	$2$	648.i	9.c	$3$	$2$	$1$	$5.174$	$\Q(\sqrt{-3}) $	$_{}$	None	None		$2$	$0$	$0$	$0$	$-1$	$0$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q-\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{11}+5\zeta_{6}q^{13}+\cdots$
648.2.i.e	$648$	$2$	648.i	9.c	$3$	$2$	$1$	$5.174$	$\Q(\sqrt{-3}) $	$_{}$	None	None		$2$	$0$	$0$	$0$	$1$	$-3$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+(-5+5\zeta_{6})q^{11}+\cdots$
648.2.i.f	$648$	$2$	648.i	9.c	$3$	$2$	$1$	$5.174$	$\Q(\sqrt{-3}) $	$_{}$	None	None		$2$	$0$	$0$	$0$	$1$	$0$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\zeta_{6}q^{5}+(4-4\zeta_{6})q^{11}+5\zeta_{6}q^{13}+\cdots$
648.2.i.g	$648$	$2$	648.i	9.c	$3$	$2$	$1$	$5.174$	$\Q(\sqrt{-3}) $	$_{}$	None	None		$2$	$0$	$0$	$0$	$2$	$0$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+2\zeta_{6}q^{5}+(-4+4\zeta_{6})q^{11}+2\zeta_{6}q^{13}+\cdots$
648.2.i.h	$648$	$2$	648.i	9.c	$3$	$2$	$1$	$5.174$	$\Q(\sqrt{-3}) $	$_{}$	None	None		$2$	$0$	$0$	$0$	$4$	$3$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+4\zeta_{6}q^{5}+(3-3\zeta_{6})q^{7}+(4-4\zeta_{6})q^{11}+\cdots$
648.2.i.i	$648$	$2$	648.i	9.c	$3$	$4$	$2$	$5.174$	$\Q(\zeta_{12})$	$_{}$	None	None		$2$	$0$	$0$	$0$	$-4$	$0$		$3$	$\mathrm{SU}(2)[C_{3}]$	$q+(-2\zeta_{12}+\zeta_{12}^{2})q^{5}+(-2\zeta_{12}^{2}+\cdots)q^{7}+\cdots$
648.2.i.j	$648$	$2$	648.i	9.c	$3$	$4$	$2$	$5.174$	$\Q(\zeta_{12})$	$_{}$	None	None		$2$	$0$	$0$	$0$	$4$	$0$		$3$	$\mathrm{SU}(2)[C_{3}]$	$q+(2-2\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{5}-2\zeta_{12}^{2}q^{7}+\cdots$
648.2.l.a	$648$	$2$	648.l	72.l	$6$	$4$	$2$	$5.174$	$\Q(\sqrt{-2}, \sqrt{-3})$	$_{}$	None	None		$4$	$1$	$0$	$0$	$0$	$-12$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+\beta _{3}q^{2}-2q^{4}+(-\beta _{1}+2\beta _{3})q^{5}+\cdots$
648.2.l.b	$648$	$2$	648.l	72.l	$6$	$4$	$2$	$5.174$	$\Q(\sqrt{-2}, \sqrt{-3})$	$_{}$	$\Q(\sqrt{-2}) $	None		$8$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{6}]$	$q+\beta _{1}q^{2}+2\beta _{2}q^{4}+2\beta _{3}q^{8}+2\beta _{1}q^{11}+\cdots$
648.2.l.c	$648$	$2$	648.l	72.l	$6$	$4$	$2$	$5.174$	$\Q(\sqrt{-2}, \sqrt{-3})$	$_{}$	None	None		$4$	$0$	$0$	$0$	$0$	$12$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+\beta _{1}q^{2}+2\beta _{2}q^{4}+(-\beta _{1}+2\beta _{3})q^{5}+\cdots$
648.2.l.d	$648$	$2$	648.l	72.l	$6$	$8$	$4$	$5.174$	8.0.170772624.1	$_{}$	None	None		$4$	$0$	$-3$	$0$	$0$	$0$		$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	$q+(1-\beta _{1}-\beta _{5}+\beta _{6})q^{2}+(-\beta _{2}-\beta _{7})q^{4}+\cdots$
648.2.l.e	$648$	$2$	648.l	72.l	$6$	$8$	$4$	$5.174$	8.0.170772624.1	$_{}$	None	None		$4$	$0$	$3$	$0$	$0$	$0$		$2^{2}$	$\mathrm{SU}(2)[C_{6}]$	$q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{3}+\beta _{7})q^{5}+(2+\cdots)q^{7}+\cdots$
648.2.l.f	$648$	$2$	648.l	72.l	$6$	$16$	$8$	$5.174$	16.0.$\cdots$.1	$_{}$	None	None		$8$	$0$	$0$	$0$	$0$	$0$		$2^{4}\cdot 3^{4}$	$\mathrm{SU}(2)[C_{6}]$	$q+(\beta _{6}+\beta _{12})q^{2}+(1-\beta _{3}-\beta _{8})q^{4}+\cdots$
648.2.l.g	$648$	$2$	648.l	72.l	$6$	$48$	$24$	$5.174$		$_{}$	None	None		$8$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{6}]$
648.2.n.a	$648$	$2$	648.n	72.n	$6$	$4$	$2$	$5.174$	$\Q(\zeta_{12})$	$_{}$	None	None		$2$	$1$	$-4$	$0$	$-6$	$-2$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+(-1+\zeta_{12}^{3})q^{2}-2\zeta_{12}^{3}q^{4}+(-1+\cdots)q^{5}+\cdots$
648.2.n.b	$648$	$2$	648.n	72.n	$6$	$4$	$2$	$5.174$	$\Q(\zeta_{12})$	$_{}$	None	None		$2$	$0$	$-2$	$0$	$-6$	$-2$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots$
648.2.n.c	$648$	$2$	648.n	72.n	$6$	$4$	$2$	$5.174$	$\Q(\zeta_{12})$	$_{}$	None	None		$4$	$0$	$-2$	$0$	$0$	$4$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots$
648.2.n.d	$648$	$2$	648.n	72.n	$6$	$4$	$2$	$5.174$	$\Q(\sqrt{-3}, \sqrt{-7})$	$_{}$	None	None		$4$	$0$	$-1$	$0$	$0$	$4$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q-\beta _{3}q^{2}+(1+\beta _{1}-\beta _{2})q^{4}+(1-2\beta _{1}+\cdots)q^{5}+\cdots$
648.2.n.e	$648$	$2$	648.n	72.n	$6$	$4$	$2$	$5.174$	$\Q(\sqrt{-2}, \sqrt{-3})$	$_{}$	None	None		$2$	$1$	$0$	$0$	$-6$	$-4$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+\beta _{3}q^{2}-2q^{4}+(-2-\beta _{1}+\beta _{2}+\beta _{3})q^{5}+\cdots$

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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.