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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 (1-50 of 112 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
408.1.u.a	$408$	$1$	408.u	51.f	$4$	$4$	$2$	$0.204$	$\Q(\zeta_{8})$	$S_{4}$	None	None		✓	✓	✓	408.1.u.a	$4$	$0$	$0$	$0$	$0$	$-4$		$1$		$q-\zeta_{8}q^{3}-\zeta_{8}q^{5}+(-1-\zeta_{8}^{2})q^{7}+\cdots$
408.1.bg.a	$408$	$1$	408.bg	408.ag	$16$	$8$	$1$	$0.204$	$\Q(\zeta_{16})$	$D_{16}$	$\Q(\sqrt{-2}) $	None		✓			408.1.bg.a	$4$	$0$	$0$	$0$	$0$	$0$		$1$		$q-\zeta_{16}q^{2}+\zeta_{16}^{6}q^{3}+\zeta_{16}^{2}q^{4}-\zeta_{16}^{7}q^{6}+\cdots$
408.1.bg.b	$408$	$1$	408.bg	408.ag	$16$	$8$	$1$	$0.204$	$\Q(\zeta_{16})$	$D_{16}$	$\Q(\sqrt{-2}) $	None		✓			408.1.bg.a	$4$	$0$	$0$	$0$	$0$	$0$		$1$		$q+\zeta_{16}q^{2}+\zeta_{16}^{5}q^{3}+\zeta_{16}^{2}q^{4}+\zeta_{16}^{6}q^{6}+\cdots$
408.2.a.a	$408$	$2$	408.a	1.a	$1$	$1$	$1$	$3.258$	$\Q$	$_{}$	None	None	✓	✓	✓	✓	408.2.a.a	$1$	$0$	$0$	$-1$	$3$	$0$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{3}+3q^{5}+q^{9}-q^{11}+3q^{13}+\cdots$
408.2.a.b	$408$	$2$	408.a	1.a	$1$	$1$	$1$	$3.258$	$\Q$	$_{}$	None	None	✓	✓	✓	✓	408.2.a.b	$1$	$1$	$0$	$1$	$-3$	$-4$	$+$	$1$	$\mathrm{SU}(2)$	$q+q^{3}-3q^{5}-4q^{7}+q^{9}+q^{11}-5q^{13}+\cdots$
408.2.a.c	$408$	$2$	408.a	1.a	$1$	$1$	$1$	$3.258$	$\Q$	$_{}$	None	None	✓	✓	✓	✓	408.2.a.c	$1$	$0$	$0$	$1$	$0$	$2$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{3}+2q^{7}+q^{9}+2q^{13}-q^{17}+\cdots$
408.2.a.d	$408$	$2$	408.a	1.a	$1$	$1$	$1$	$3.258$	$\Q$	$_{}$	None	None	✓	✓			408.2.a.d	$1$	$0$	$0$	$1$	$2$	$-4$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{3}+2q^{5}-4q^{7}+q^{9}+4q^{11}+\cdots$
408.2.a.e	$408$	$2$	408.a	1.a	$1$	$2$	$2$	$3.258$	$\Q(\sqrt{17}) $	$_{}$	None	None	✓	✓	✓	✓	408.2.a.e	$1$	$1$	$0$	$-2$	$-1$	$-2$	$+$	$1$	$\mathrm{SU}(2)$	$q-q^{3}-\beta q^{5}+(-2+2\beta )q^{7}+q^{9}+\cdots$
408.2.a.f	$408$	$2$	408.a	1.a	$1$	$2$	$2$	$3.258$	$\Q(\sqrt{57}) $	$_{}$	None	None	✓	✓	✓		408.2.a.f	$1$	$0$	$0$	$2$	$-1$	$8$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{3}-\beta q^{5}+4q^{7}+q^{9}+(-2+\beta )q^{11}+\cdots$
408.2.c.a	$408$	$2$	408.c	17.b	$2$	$4$	$4$	$3.258$	$\Q(i, \sqrt{33})$	$_{}$	None	None		✓			408.2.c.a	$2$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{2}q^{3}+(\beta _{1}-\beta _{2})q^{5}+2\beta _{2}q^{7}-q^{9}+\cdots$
408.2.c.b	$408$	$2$	408.c	17.b	$2$	$6$	$6$	$3.258$	6.0.399424.1	$_{}$	None	None		✓	✓		408.2.c.b	$2$	$0$	$0$	$0$	$0$	$0$		$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{2}q^{3}+(\beta _{2}-\beta _{3})q^{5}+(-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots$
408.2.f.a	$408$	$2$	408.f	8.b	$2$	$2$	$2$	$3.258$	$\Q(\sqrt{-1}) $	$_{}$	None	None		✓			408.2.f.a	$2$	$1$	$-2$	$0$	$0$	$-8$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+(i-1)q^{2}-i q^{3}-2 i q^{4}+4 i q^{5}+\cdots$
408.2.f.b	$408$	$2$	408.f	8.b	$2$	$4$	$4$	$3.258$	$\Q(i, \sqrt{5})$	$_{}$	None	None		✓			408.2.f.b	$2$	$0$	$4$	$0$	$0$	$4$		$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+(1+\beta _{1})q^{2}+\beta _{1}q^{3}+2\beta _{1}q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots$
408.2.f.c	$408$	$2$	408.f	8.b	$2$	$12$	$12$	$3.258$	12.0.$\cdots$.1	$_{}$	None	None		✓			408.2.f.c	$2$	$0$	$-2$	$0$	$0$	$0$		$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{7}q^{2}+\beta _{5}q^{3}+\beta _{10}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$
408.2.f.d	$408$	$2$	408.f	8.b	$2$	$14$	$14$	$3.258$	$\mathbb{Q}[x]/(x^{14} - \cdots)$	$_{}$	None	None		✓	✓		408.2.f.d	$2$	$0$	$4$	$0$	$0$	$12$		$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{8}q^{2}+\beta _{6}q^{3}+\beta _{2}q^{4}+\beta _{3}q^{5}+\cdots$
408.2.h.a	$408$	$2$	408.h	408.h	$2$	$2$	$2$	$3.258$	$\Q(\sqrt{-2}) $	$_{}$	$\Q(\sqrt{-2}) $	None		✓			408.2.h.a	$4$	$1$	$0$	$-2$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+\beta q^{2}+(-1+\beta )q^{3}-2q^{4}+(-2+\cdots)q^{6}+\cdots$
408.2.h.b	$408$	$2$	408.h	408.h	$2$	$2$	$2$	$3.258$	$\Q(\sqrt{-2}) $	$_{}$	$\Q(\sqrt{-2}) $	None		✓			408.2.h.a	$4$	$0$	$0$	$2$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+\beta q^{2}+(1-\beta )q^{3}-2q^{4}+(2+\beta )q^{6}+\cdots$
408.2.h.c	$408$	$2$	408.h	408.h	$2$	$64$	$64$	$3.258$		$_{}$	None	None		✓	✓		408.2.h.c	$8$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{2}]$
408.2.j.a	$408$	$2$	408.j	24.f	$2$	$4$	$4$	$3.258$	$\Q(\zeta_{8})$	$_{}$	None	None		✓			408.2.j.a	$2$	$1$	$-4$	$-4$	$-8$	$0$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+(-1-\zeta_{8}^{2})q^{2}+(-1-\zeta_{8}+\zeta_{8}^{2}+\cdots)q^{3}+\cdots$
408.2.j.b	$408$	$2$	408.j	24.f	$2$	$4$	$4$	$3.258$	$\Q(i, \sqrt{6})$	$_{}$	None	None		✓			408.2.j.b	$2$	$0$	$-4$	$0$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+(-1+\beta _{2})q^{2}+\beta _{1}q^{3}-2\beta _{2}q^{4}+\cdots$
408.2.j.c	$408$	$2$	408.j	24.f	$2$	$4$	$4$	$3.258$	$\Q(\zeta_{8})$	$_{}$	None	None		✓			408.2.j.a	$2$	$0$	$4$	$-4$	$8$	$0$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+(1+\zeta_{8}^{2})q^{2}+(-1-\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+\cdots$
408.2.j.d	$408$	$2$	408.j	24.f	$2$	$4$	$4$	$3.258$	$\Q(i, \sqrt{6})$	$_{}$	None	None		✓			408.2.j.b	$2$	$0$	$4$	$0$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+(1-\beta _{2})q^{2}+\beta _{3}q^{3}-2\beta _{2}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots$
408.2.j.e	$408$	$2$	408.j	24.f	$2$	$48$	$48$	$3.258$		$_{}$	None	None		✓	✓		408.2.j.e	$4$	$0$	$0$	$8$	$0$	$0$			$\mathrm{SU}(2)[C_{2}]$
408.2.l.a	$408$	$2$	408.l	136.h	$2$	$18$	$18$	$3.258$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	$_{}$	None	None		✓			408.2.l.a	$2$	$0$	$0$	$-18$	$4$	$0$		$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{5}q^{2}-q^{3}-\beta _{2}q^{4}-\beta _{9}q^{5}+\beta _{5}q^{6}+\cdots$
408.2.l.b	$408$	$2$	408.l	136.h	$2$	$18$	$18$	$3.258$	$\mathbb{Q}[x]/(x^{18} - \cdots)$	$_{}$	None	None		✓			408.2.l.a	$2$	$0$	$0$	$18$	$-4$	$0$		$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{5}q^{2}+q^{3}-\beta _{2}q^{4}+\beta _{9}q^{5}-\beta _{5}q^{6}+\cdots$
408.2.q.a	$408$	$2$	408.q	408.q	$4$	$4$	$2$	$3.258$	$\Q(\zeta_{8})$	$_{}$	$\Q(\sqrt{-2}) $	None		✓			408.2.q.a	$4$	$0$	$0$	$0$	$0$	$0$		$2$	$\mathrm{U}(1)[D_{4}]$	$q-\beta_{3} q^{2}+(\beta_{3}+\beta_1)q^{3}+2 q^{4}+\cdots$
408.2.q.b	$408$	$2$	408.q	408.q	$4$	$4$	$2$	$3.258$	$\Q(\zeta_{8})$	$_{}$	$\Q(\sqrt{-2}) $	None		✓			408.2.q.a	$4$	$0$	$0$	$4$	$0$	$0$		$2$	$\mathrm{U}(1)[D_{4}]$	$q-\beta_{3} q^{2}+(-\beta_{2}+1)q^{3}+2 q^{4}+\cdots$
408.2.q.c	$408$	$2$	408.q	408.q	$4$	$128$	$64$	$3.258$		$_{}$	None	None		✓	✓		408.2.q.c	$8$	$0$	$0$	$-8$	$0$	$0$			$\mathrm{SU}(2)[C_{4}]$
408.2.s.a	$408$	$2$	408.s	136.i	$4$	$4$	$2$	$3.258$	$\Q(\zeta_{8})$	$_{}$	None	None		✓			408.2.s.a	$4$	$0$	$0$	$0$	$0$	$4$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q+(-\zeta_{8}+\zeta_{8}^{3})q^{2}-\zeta_{8}^{3}q^{3}+2q^{4}+\cdots$
408.2.s.b	$408$	$2$	408.s	136.i	$4$	$68$	$34$	$3.258$		$_{}$	None	None		✓	✓		408.2.s.b	$4$	$0$	$0$	$0$	$0$	$-4$			$\mathrm{SU}(2)[C_{4}]$
408.2.v.a	$408$	$2$	408.v	17.c	$4$	$4$	$2$	$3.258$	$\Q(\zeta_{8})$	$_{}$	None	None		✓			408.2.v.a	$2$	$0$	$0$	$0$	$0$	$-4$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q+\zeta_{8}q^{3}-\zeta_{8}q^{5}+(-1+\zeta_{8}^{2}-2\zeta_{8}^{3})q^{7}+\cdots$
408.2.v.b	$408$	$2$	408.v	17.c	$4$	$4$	$2$	$3.258$	$\Q(\zeta_{8})$	$_{}$	None	None		✓			408.2.v.b	$2$	$0$	$0$	$0$	$4$	$0$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q+\zeta_{8}q^{3}+(1+2\zeta_{8}+\zeta_{8}^{2})q^{5}+2\zeta_{8}^{3}q^{7}+\cdots$
408.2.v.c	$408$	$2$	408.v	17.c	$4$	$12$	$6$	$3.258$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	$_{}$	None	None		✓	✓		408.2.v.c	$2$	$0$	$0$	$0$	$0$	$4$		$2^{6}$	$\mathrm{SU}(2)[C_{4}]$	$q+\beta _{4}q^{3}+\beta _{7}q^{5}+(-\beta _{3}+\beta _{9})q^{7}+\cdots$
408.2.ba.a	$408$	$2$	408.ba	17.d	$8$	$16$	$4$	$3.258$	$\mathbb{Q}[x]/(x^{16} + \cdots)$	$_{}$	None	None		✓			408.2.ba.a	$2$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{8}]$	$q+\beta _{12}q^{3}+(\beta _{6}+\beta _{11}+\beta _{14})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots$
408.2.ba.b	$408$	$2$	408.ba	17.d	$8$	$16$	$4$	$3.258$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	$_{}$	None	None		✓			408.2.ba.b	$2$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{8}]$	$q+\beta _{3}q^{3}+(\beta _{1}-\beta _{8}-\beta _{12}-\beta _{15})q^{5}+\cdots$
408.2.bc.a	$408$	$2$	408.bc	136.o	$8$	$144$	$36$	$3.258$		$_{}$	None	None		✓	✓	✓	408.2.bc.a	$4$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{8}]$
408.2.bd.a	$408$	$2$	408.bd	408.ad	$8$	$4$	$1$	$3.258$	$\Q(\zeta_{8})$	$_{}$	$\Q(\sqrt{-2}) $	None		✓			408.2.bd.a	$4$	$1$	$-4$	$-4$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{8}]$	$q+(-1-\zeta_{8}^{2})q^{2}+(-1-\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+\cdots$
408.2.bd.b	$408$	$2$	408.bd	408.ad	$8$	$4$	$1$	$3.258$	$\Q(\zeta_{8})$	$_{}$	$\Q(\sqrt{-2}) $	None		✓			408.2.bd.b	$4$	$0$	$-4$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{8}]$	$q+(-1-\zeta_{8}^{2})q^{2}+(\zeta_{8}-\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+\cdots$
408.2.bd.c	$408$	$2$	408.bd	408.ad	$8$	$4$	$1$	$3.258$	$\Q(\zeta_{8})$	$_{}$	$\Q(\sqrt{-2}) $	None		✓			408.2.bd.a	$4$	$0$	$4$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{8}]$	$q+(1+\zeta_{8}^{2})q^{2}+(-\zeta_{8}-\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+\cdots$
408.2.bd.d	$408$	$2$	408.bd	408.ad	$8$	$4$	$1$	$3.258$	$\Q(\zeta_{8})$	$_{}$	$\Q(\sqrt{-2}) $	None		✓			408.2.bd.b	$4$	$0$	$4$	$4$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{8}]$	$q+(1+\zeta_{8}^{2})q^{2}+(1+\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+\cdots$
408.2.bd.e	$408$	$2$	408.bd	408.ad	$8$	$256$	$64$	$3.258$		$_{}$	None	None		✓	✓		408.2.bd.e	$8$	$0$	$0$	$-8$	$0$	$0$			$\mathrm{SU}(2)[C_{8}]$
408.2.bh.a	$408$	$2$	408.bh	51.i	$16$	$72$	$9$	$3.258$		$_{}$	None	None		✓			408.2.bh.a	$2$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{16}]$
408.2.bh.b	$408$	$2$	408.bh	51.i	$16$	$72$	$9$	$3.258$		$_{}$	None	None		✓			408.2.bh.a	$2$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{16}]$
408.2.bl.a	$408$	$2$	408.bl	136.s	$16$	$288$	$36$	$3.258$		$_{}$	None	None		✓	✓	✓	408.2.bl.a	$4$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{16}]$
408.2.bm.a	$408$	$2$	408.bm	408.am	$16$	$544$	$68$	$3.258$		$_{}$	None	None		✓	✓	✓	408.2.bm.a	$8$	$0$	$0$	$0$	$0$	$-32$			$\mathrm{SU}(2)[C_{16}]$
408.3.b.a	$408$	$3$	408.b	408.b	$2$	$1$	$1$	$11.117$	$\Q$	$_{}$	$\Q(\sqrt{-102}) $	None	✓	✓			408.3.b.a	$2$	$0$	$-2$	$-3$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q-2q^{2}-3q^{3}+4q^{4}+6q^{6}-8q^{8}+\cdots$
408.3.b.b	$408$	$3$	408.b	408.b	$2$	$1$	$1$	$11.117$	$\Q$	$_{}$	$\Q(\sqrt{-102}) $	None	✓	✓			408.3.b.a	$2$	$0$	$-2$	$3$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q-2q^{2}+3q^{3}+4q^{4}-6q^{6}-8q^{8}+\cdots$
408.3.b.c	$408$	$3$	408.b	408.b	$2$	$1$	$1$	$11.117$	$\Q$	$_{}$	$\Q(\sqrt{-102}) $	None	✓	✓			408.3.b.a	$2$	$0$	$2$	$-3$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+8q^{8}+\cdots$
408.3.b.d	$408$	$3$	408.b	408.b	$2$	$1$	$1$	$11.117$	$\Q$	$_{}$	$\Q(\sqrt{-102}) $	None	✓	✓			408.3.b.a	$2$	$0$	$2$	$3$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+2q^{2}+3q^{3}+4q^{4}+6q^{6}+8q^{8}+\cdots$
408.3.b.e	$408$	$3$	408.b	408.b	$2$	$136$	$136$	$11.117$		$_{}$	None	None		✓	✓		408.3.b.e	$8$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{2}]$

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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.