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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				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
414.2.a.a	$414$	$2$	414.a	1.a	$1$	$1$	$1$	$3.306$	$\Q$	None	✓	$1$	$1$	$-1$	$0$	$-2$	$0$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}-2q^{5}-q^{8}+2q^{10}-2q^{13}+\cdots$
414.2.a.b	$414$	$2$	414.a	1.a	$1$	$1$	$1$	$3.306$	$\Q$	None	✓	$1$	$1$	$1$	$0$	$-4$	$-4$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}-4q^{5}-4q^{7}+q^{8}-4q^{10}+\cdots$
414.2.a.c	$414$	$2$	414.a	1.a	$1$	$1$	$1$	$3.306$	$\Q$	None	✓	$1$	$0$	$1$	$0$	$0$	$2$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+2q^{7}+q^{8}+2q^{13}+2q^{14}+\cdots$
414.2.a.d	$414$	$2$	414.a	1.a	$1$	$1$	$1$	$3.306$	$\Q$	None	✓	$1$	$0$	$1$	$0$	$2$	$-2$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+2q^{5}-2q^{7}+q^{8}+2q^{10}+\cdots$
414.2.a.e	$414$	$2$	414.a	1.a	$1$	$2$	$2$	$3.306$	$\Q(\sqrt{7}) $	None	✓	$1$	$0$	$-2$	$0$	$-2$	$4$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(-1+\beta )q^{5}+2q^{7}-q^{8}+\cdots$
414.2.a.f	$414$	$2$	414.a	1.a	$1$	$2$	$2$	$3.306$	$\Q(\sqrt{5}) $	None	✓	$1$	$0$	$-2$	$0$	$2$	$0$	$-$	$2$	$\mathrm{SU}(2)$	$q-q^{2}+q^{4}+(1+\beta )q^{5}+2\beta q^{7}-q^{8}+\cdots$
414.2.a.g	$414$	$2$	414.a	1.a	$1$	$2$	$2$	$3.306$	$\Q(\sqrt{7}) $	None	✓	$1$	$0$	$2$	$0$	$2$	$4$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{2}+q^{4}+(1+\beta )q^{5}+2q^{7}+q^{8}+\cdots$
414.2.d.a	$414$	$2$	414.d	69.c	$2$	$8$	$8$	$3.306$	$\Q(\zeta_{16})$	None		$4$	$0$	$0$	$0$	$0$	$0$		$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	$q-\zeta_{16}q^{2}-q^{4}-\zeta_{16}^{5}q^{5}-2\zeta_{16}^{4}q^{7}+\cdots$
414.2.e.a	$414$	$2$	414.e	9.c	$3$	$2$	$1$	$3.306$	$\Q(\sqrt{-3}) $	None		$2$	$0$	$1$	$3$	$-4$	$0$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\zeta_{6})q^{2}+(1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots$
414.2.e.b	$414$	$2$	414.e	9.c	$3$	$10$	$5$	$3.306$	10.0.$\cdots$.1	None		$2$	$0$	$-5$	$1$	$-5$	$-3$		$3$	$\mathrm{SU}(2)[C_{3}]$	$q-\beta _{3}q^{2}-\beta _{7}q^{3}+(-1+\beta _{3})q^{4}+(-1+\cdots)q^{5}+\cdots$
414.2.e.c	$414$	$2$	414.e	9.c	$3$	$10$	$5$	$3.306$	10.0.$\cdots$.1	None		$2$	$0$	$-5$	$1$	$5$	$5$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1-\beta _{4})q^{2}+(1+\beta _{5}-\beta _{6}-\beta _{9})q^{3}+\cdots$
414.2.e.d	$414$	$2$	414.e	9.c	$3$	$10$	$5$	$3.306$	10.0.$\cdots$.1	None		$2$	$0$	$5$	$-3$	$-1$	$5$		$3^{2}$	$\mathrm{SU}(2)[C_{3}]$	$q+(1+\beta _{7})q^{2}+\beta _{1}q^{3}+\beta _{7}q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots$
414.2.e.e	$414$	$2$	414.e	9.c	$3$	$12$	$6$	$3.306$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	None		$2$	$0$	$6$	$0$	$5$	$-3$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\beta _{2}q^{2}-\beta _{1}q^{3}+(-1+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots$
414.2.f.a	$414$	$2$	414.f	207.g	$6$	$48$	$24$	$3.306$		None		$4$	$0$	$0$	$4$	$0$	$0$			$\mathrm{SU}(2)[C_{6}]$
414.2.i.a	$414$	$2$	414.i	23.c	$11$	$10$	$1$	$3.306$	$\Q(\zeta_{22})$	None		$2$	$0$	$-1$	$0$	$-2$	$2$		$1$	$\mathrm{SU}(2)[C_{11}]$	$q+\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(-1-\zeta_{22}^{2}+\cdots)q^{5}+\cdots$
414.2.i.b	$414$	$2$	414.i	23.c	$11$	$10$	$1$	$3.306$	$\Q(\zeta_{22})$	None		$2$	$0$	$-1$	$0$	$0$	$-2$		$1$	$\mathrm{SU}(2)[C_{11}]$	$q+\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(1-2\zeta_{22}+\cdots)q^{5}+\cdots$
414.2.i.c	$414$	$2$	414.i	23.c	$11$	$10$	$1$	$3.306$	$\Q(\zeta_{22})$	None		$2$	$0$	$-1$	$0$	$4$	$-7$		$1$	$\mathrm{SU}(2)[C_{11}]$	$q+\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(\zeta_{22}+\zeta_{22}^{3}+\cdots)q^{5}+\cdots$
414.2.i.d	$414$	$2$	414.i	23.c	$11$	$10$	$1$	$3.306$	$\Q(\zeta_{22})$	None		$2$	$0$	$1$	$0$	$-8$	$8$		$1$	$\mathrm{SU}(2)[C_{11}]$	$q-\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(-1+\zeta_{22}^{2}+\cdots)q^{5}+\cdots$
414.2.i.e	$414$	$2$	414.i	23.c	$11$	$10$	$1$	$3.306$	$\Q(\zeta_{22})$	None		$2$	$0$	$1$	$0$	$2$	$0$		$1$	$\mathrm{SU}(2)[C_{11}]$	$q-\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(1-2\zeta_{22}+\cdots)q^{5}+\cdots$
414.2.i.f	$414$	$2$	414.i	23.c	$11$	$10$	$1$	$3.306$	$\Q(\zeta_{22})$	None		$2$	$0$	$1$	$0$	$6$	$3$		$1$	$\mathrm{SU}(2)[C_{11}]$	$q-\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(\zeta_{22}-2\zeta_{22}^{2}+\cdots)q^{5}+\cdots$
414.2.i.g	$414$	$2$	414.i	23.c	$11$	$20$	$2$	$3.306$	$\mathbb{Q}[x]/(x^{20} + \cdots)$	None		$2$	$0$	$-2$	$0$	$-2$	$-4$		$1$	$\mathrm{SU}(2)[C_{11}]$	$q+\beta _{3}q^{2}+\beta _{7}q^{4}+(\beta _{2}-\beta _{5}+\beta _{13}+\cdots)q^{5}+\cdots$
414.2.i.h	$414$	$2$	414.i	23.c	$11$	$20$	$2$	$3.306$	$\mathbb{Q}[x]/(x^{20} + \cdots)$	None		$2$	$0$	$2$	$0$	$2$	$-4$		$1$	$\mathrm{SU}(2)[C_{11}]$	$q-\beta _{3}q^{2}+\beta _{7}q^{4}+(-\beta _{2}+\beta _{5}-\beta _{13}+\cdots)q^{5}+\cdots$
414.2.j.a	$414$	$2$	414.j	69.g	$22$	$80$	$8$	$3.306$		None		$4$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{22}]$
414.2.m.a	$414$	$2$	414.m	207.m	$33$	$240$	$12$	$3.306$		None		$4$	$0$	$-12$	$0$	$0$	$-13$			$\mathrm{SU}(2)[C_{33}]$
414.2.m.b	$414$	$2$	414.m	207.m	$33$	$240$	$12$	$3.306$		None		$4$	$0$	$12$	$4$	$0$	$13$			$\mathrm{SU}(2)[C_{33}]$
414.2.p.a	$414$	$2$	414.p	207.o	$66$	$480$	$24$	$3.306$		None		$4$	$0$	$0$	$-4$	$0$	$0$			$\mathrm{SU}(2)[C_{66}]$
414.3.b.a	$414$	$3$	414.b	23.b	$2$	$4$	$4$	$11.281$	4.0.613376.1	None		$2$	$0$	$0$	$0$	$0$	$0$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{2}q^{2}+2q^{4}+\beta _{1}q^{5}+\beta _{3}q^{7}-2\beta _{2}q^{8}+\cdots$
414.3.b.b	$414$	$3$	414.b	23.b	$2$	$8$	$8$	$11.281$	8.0.$\cdots$.26	None		$4$	$0$	$0$	$0$	$0$	$0$		$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+2q^{4}+\beta _{6}q^{5}-\beta _{5}q^{7}+2\beta _{1}q^{8}+\cdots$
414.3.b.c	$414$	$3$	414.b	23.b	$2$	$8$	$8$	$11.281$	8.0.1358954496.3	None		$2$	$0$	$0$	$0$	$0$	$0$		$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{5}q^{2}+2q^{4}+(\beta _{2}-2\beta _{3}+\beta _{6})q^{5}+\cdots$
414.3.c.a	$414$	$3$	414.c	3.b	$2$	$4$	$4$	$11.281$	$\Q(\sqrt{-2}, \sqrt{-23})$	None		$2$	$0$	$0$	$0$	$0$	$0$		$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{1}q^{2}-2q^{4}-2\beta _{1}q^{5}-\beta _{2}q^{7}+\cdots$
414.3.c.b	$414$	$3$	414.c	3.b	$2$	$8$	$8$	$11.281$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	None		$2$	$0$	$0$	$0$	$0$	$16$		$2^{6}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}-2q^{4}-\beta _{7}q^{5}+(2-\beta _{2}-\beta _{6}+\cdots)q^{7}+\cdots$
414.3.g.a	$414$	$3$	414.g	9.d	$6$	$88$	$44$	$11.281$		None		$2$	$0$	$0$	$-4$	$0$	$-4$			$\mathrm{SU}(2)[C_{6}]$
414.3.h.a	$414$	$3$	414.h	207.f	$6$	$96$	$48$	$11.281$		None		$4$	$0$	$0$	$4$	$0$	$0$			$\mathrm{SU}(2)[C_{6}]$
414.3.k.a	$414$	$3$	414.k	69.h	$22$	$80$	$8$	$11.281$		None		$4$	$0$	$0$	$0$	$0$	$-16$			$\mathrm{SU}(2)[C_{22}]$
414.3.k.b	$414$	$3$	414.k	69.h	$22$	$80$	$8$	$11.281$		None		$4$	$0$	$0$	$0$	$0$	$16$			$\mathrm{SU}(2)[C_{22}]$
414.3.l.a	$414$	$3$	414.l	23.d	$22$	$40$	$4$	$11.281$		None		$2$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{22}]$
414.3.l.b	$414$	$3$	414.l	23.d	$22$	$80$	$8$	$11.281$		None		$2$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{22}]$
414.3.l.c	$414$	$3$	414.l	23.d	$22$	$80$	$8$	$11.281$		None		$4$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{22}]$
414.3.n.a	$414$	$3$	414.n	207.p	$66$	$960$	$48$	$11.281$		None		$4$	$0$	$0$	$-4$	$0$	$0$			$\mathrm{SU}(2)[C_{66}]$
414.3.o.a	$414$	$3$	414.o	207.n	$66$	$960$	$48$	$11.281$		None		$4$	$0$	$0$	$4$	$36$	$0$			$\mathrm{SU}(2)[C_{66}]$
414.4.a.a	$414$	$4$	414.a	1.a	$1$	$1$	$1$	$24.427$	$\Q$	None	✓	$1$	$0$	$-2$	$0$	$2$	$-34$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+4q^{4}+2q^{5}-34q^{7}-8q^{8}+\cdots$
414.4.a.b	$414$	$4$	414.a	1.a	$1$	$1$	$1$	$24.427$	$\Q$	None	✓	$1$	$0$	$-2$	$0$	$20$	$2$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+4q^{4}+20q^{5}+2q^{7}-8q^{8}+\cdots$
414.4.a.c	$414$	$4$	414.a	1.a	$1$	$1$	$1$	$24.427$	$\Q$	None	✓	$1$	$1$	$2$	$0$	$-2$	$-32$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+4q^{4}-2q^{5}-2^{5}q^{7}+8q^{8}+\cdots$
414.4.a.d	$414$	$4$	414.a	1.a	$1$	$1$	$1$	$24.427$	$\Q$	None	✓	$1$	$0$	$2$	$0$	$10$	$-12$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+4q^{4}+10q^{5}-12q^{7}+8q^{8}+\cdots$
414.4.a.e	$414$	$4$	414.a	1.a	$1$	$1$	$1$	$24.427$	$\Q$	None	✓	$1$	$0$	$2$	$0$	$10$	$32$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+4q^{4}+10q^{5}+2^{5}q^{7}+8q^{8}+\cdots$
414.4.a.f	$414$	$4$	414.a	1.a	$1$	$2$	$2$	$24.427$	$\Q(\sqrt{73}) $	None	✓	$1$	$1$	$-4$	$0$	$-10$	$12$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+4q^{4}+(-4-2\beta )q^{5}+(4+4\beta )q^{7}+\cdots$
414.4.a.g	$414$	$4$	414.a	1.a	$1$	$2$	$2$	$24.427$	$\Q(\sqrt{2}) $	None	✓	$1$	$1$	$-4$	$0$	$-8$	$12$	$-$	$2^{3}$	$\mathrm{SU}(2)$	$q-2q^{2}+4q^{4}+(-4+\beta )q^{5}+(6+\beta )q^{7}+\cdots$
414.4.a.h	$414$	$4$	414.a	1.a	$1$	$2$	$2$	$24.427$	$\Q(\sqrt{7}) $	None	✓	$1$	$1$	$-4$	$0$	$10$	$-10$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{2}+4q^{4}+(5+3\beta )q^{5}+(-5+5\beta )q^{7}+\cdots$
414.4.a.i	$414$	$4$	414.a	1.a	$1$	$2$	$2$	$24.427$	$\Q(\sqrt{7}) $	None	✓	$1$	$1$	$4$	$0$	$-10$	$-10$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+4q^{4}+(-5+3\beta )q^{5}+(-5+\cdots)q^{7}+\cdots$
414.4.a.j	$414$	$4$	414.a	1.a	$1$	$2$	$2$	$24.427$	$\Q(\sqrt{41}) $	None	✓	$1$	$1$	$4$	$0$	$-10$	$6$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{2}+4q^{4}+(-4-2\beta )q^{5}+(2+2\beta )q^{7}+\cdots$

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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.