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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 115 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
31.1.b.a	$31$	$1$	31.b	31.b	$2$	$1$	$1$	$0.015$	$\Q$	$D_{3}$	$\Q(\sqrt{-31}) $	None	✓	✓	✓	✓	31.1.b.a	$2$	$0$	$-1$	$0$	$-1$	$-1$		$1$		$q-q^{2}-q^{5}-q^{7}+q^{8}+q^{9}+q^{10}+\cdots$
31.2.a.a	$31$	$2$	31.a	1.a	$1$	$2$	$2$	$0.248$	$\Q(\sqrt{5}) $	$_{}$	None	None	✓	✓	✓	✓	31.2.a.a	$1$	$0$	$1$	$-2$	$2$	$-4$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{2}-2\beta q^{3}+(-1+\beta )q^{4}+q^{5}+\cdots$
31.2.c.a	$31$	$2$	31.c	31.c	$3$	$4$	$2$	$0.248$	$\Q(\sqrt{2}, \sqrt{-3})$	$_{}$	None	None		✓	✓	✓	31.2.c.a	$2$	$0$	$-4$	$2$	$-2$	$-2$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(-1+\beta _{3})q^{2}+(1+\beta _{1}+\beta _{2})q^{3}+\cdots$
31.2.d.a	$31$	$2$	31.d	31.d	$5$	$4$	$1$	$0.248$	$\Q(\zeta_{10})$	$_{}$	None	None		✓	✓	✓	31.2.d.a	$2$	$0$	$-3$	$1$	$-6$	$-3$		$1$	$\mathrm{SU}(2)[C_{5}]$	$q+(-1-\zeta_{10}^{2})q^{2}+(1-\zeta_{10}+\zeta_{10}^{2}+\cdots)q^{3}+\cdots$
31.2.g.a	$31$	$2$	31.g	31.g	$15$	$16$	$2$	$0.248$	$\mathbb{Q}[x]/(x^{16} + \cdots)$	$_{}$	None	None		✓	✓	✓	31.2.g.a	$2$	$0$	$-6$	$-12$	$-3$	$2$		$1$	$\mathrm{SU}(2)[C_{15}]$	$q+(-1-\beta _{1}+\beta _{3}-\beta _{4}+\beta _{5}+\beta _{6}+\cdots)q^{2}+\cdots$
31.3.b.a	$31$	$3$	31.b	31.b	$2$	$2$	$2$	$0.845$	$\Q(\sqrt{-26}) $	$_{}$	None	None		✓			31.3.b.a	$2$	$0$	$-2$	$0$	$4$	$16$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q-q^{2}+\beta q^{3}-3q^{4}+2q^{5}-\beta q^{6}+\cdots$
31.3.b.b	$31$	$3$	31.b	31.b	$2$	$3$	$3$	$0.845$	3.3.837.1	$_{}$	$\Q(\sqrt{-31}) $	None	✓	✓	✓		31.3.b.b	$2$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+\beta _{2}q^{2}+(4+\beta _{1}-2\beta _{2})q^{4}+(-3\beta _{1}+\cdots)q^{5}+\cdots$
31.3.e.a	$31$	$3$	31.e	31.e	$6$	$2$	$1$	$0.845$	$\Q(\sqrt{-3}) $	$_{}$	None	None		✓			31.3.e.a	$2$	$0$	$-6$	$3$	$9$	$1$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q-3q^{2}+(1+\zeta_{6})q^{3}+5q^{4}+9\zeta_{6}q^{5}+\cdots$
31.3.e.b	$31$	$3$	31.e	31.e	$6$	$6$	$3$	$0.845$	6.0.954288.1	$_{}$	None	None		✓	✓		31.3.e.b	$2$	$0$	$2$	$-3$	$-7$	$-9$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+(-\beta _{2}+\beta _{4})q^{2}+(-\beta _{1}-\beta _{2}-\beta _{4}+\cdots)q^{3}+\cdots$
31.3.f.a	$31$	$3$	31.f	31.f	$10$	$20$	$5$	$0.845$	$\mathbb{Q}[x]/(x^{20} - \cdots)$	$_{}$	None	None		✓	✓	✓	31.3.f.a	$2$	$0$	$-3$	$-5$	$-14$	$-1$		$1$	$\mathrm{SU}(2)[C_{10}]$	$q-\beta _{10}q^{2}-\beta _{13}q^{3}+(3\beta _{11}+\beta _{19})q^{4}+\cdots$
31.3.h.a	$31$	$3$	31.h	31.h	$30$	$32$	$4$	$0.845$		$_{}$	None	None		✓	✓	✓	31.3.h.a	$2$	$0$	$-6$	$-10$	$-7$	$-22$			$\mathrm{SU}(2)[C_{30}]$
31.4.a.a	$31$	$4$	31.a	1.a	$1$	$2$	$2$	$1.829$	$\Q(\sqrt{17}) $	$_{}$	None	None	✓	✓	✓	✓	31.4.a.a	$1$	$1$	$-5$	$-2$	$-25$	$-19$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-2-\beta )q^{2}+(-2+2\beta )q^{3}+5\beta q^{4}+\cdots$
31.4.a.b	$31$	$4$	31.a	1.a	$1$	$5$	$5$	$1.829$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	$_{}$	None	None	✓	✓	✓	✓	31.4.a.b	$1$	$0$	$3$	$4$	$15$	$9$	$+$	$1$	$\mathrm{SU}(2)$	$q+(1-\beta _{1})q^{2}+(-\beta _{2}+\beta _{3}-\beta _{4})q^{3}+\cdots$
31.4.c.a	$31$	$4$	31.c	31.c	$3$	$14$	$7$	$1.829$	$\mathbb{Q}[x]/(x^{14} - \cdots)$	$_{}$	None	None		✓	✓	✓	31.4.c.a	$2$	$0$	$-4$	$-11$	$-5$	$31$		$2^{6}$	$\mathrm{SU}(2)[C_{3}]$	$q+(-\beta _{1}+\beta _{4})q^{2}+(-2+2\beta _{3}-\beta _{8}+\cdots)q^{3}+\cdots$
31.4.d.a	$31$	$4$	31.d	31.d	$5$	$28$	$7$	$1.829$		$_{}$	None	None		✓	✓	✓	31.4.d.a	$2$	$0$	$-3$	$-7$	$0$	$35$			$\mathrm{SU}(2)[C_{5}]$
31.4.g.a	$31$	$4$	31.g	31.g	$15$	$56$	$7$	$1.829$		$_{}$	None	None		✓	✓	✓	31.4.g.a	$2$	$0$	$-6$	$1$	$0$	$-71$			$\mathrm{SU}(2)[C_{15}]$
31.5.b.a	$31$	$5$	31.b	31.b	$2$	$3$	$3$	$3.204$	3.3.837.1	$_{}$	$\Q(\sqrt{-31}) $	None	✓	✓			31.5.b.a	$2$	$0$	$0$	$0$	$0$	$0$		$3$	$\mathrm{U}(1)[D_{2}]$	$q+(-\beta _{1}+\beta _{2})q^{2}+(2^{4}-6\beta _{1}+\beta _{2})q^{4}+\cdots$
31.5.b.b	$31$	$5$	31.b	31.b	$2$	$6$	$6$	$3.204$	$\mathbb{Q}[x]/(x^{6} + \cdots)$	$_{}$	None	None		✓	✓		31.5.b.b	$2$	$0$	$-2$	$0$	$-8$	$-132$		$2\cdot 3$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{3}q^{2}+\beta _{1}q^{3}+(4+\beta _{3}+\beta _{4})q^{4}+\cdots$
31.5.e.a	$31$	$5$	31.e	31.e	$6$	$20$	$10$	$3.204$	$\mathbb{Q}[x]/(x^{20} - \cdots)$	$_{}$	None	None		✓	✓	✓	31.5.e.a	$2$	$0$	$-4$	$6$	$-4$	$-98$		$2^{2}\cdot 3^{11}$	$\mathrm{SU}(2)[C_{6}]$	$q-\beta _{3}q^{2}+\beta _{8}q^{3}+(6+\beta _{2})q^{4}+(-\beta _{11}+\cdots)q^{5}+\cdots$
31.5.f.a	$31$	$5$	31.f	31.f	$10$	$36$	$9$	$3.204$		$_{}$	None	None		✓	✓	✓	31.5.f.a	$2$	$0$	$-3$	$-5$	$-2$	$-73$			$\mathrm{SU}(2)[C_{10}]$
31.5.h.a	$31$	$5$	31.h	31.h	$30$	$80$	$10$	$3.204$		$_{}$	None	None		✓	✓	✓	31.5.h.a	$2$	$0$	$-6$	$-16$	$-1$	$288$			$\mathrm{SU}(2)[C_{30}]$
31.6.a.a	$31$	$6$	31.a	1.a	$1$	$5$	$5$	$4.972$	$\mathbb{Q}[x]/(x^{5} - \cdots)$	$_{}$	None	None	✓	✓	✓	✓	31.6.a.a	$1$	$1$	$-9$	$-20$	$-72$	$-108$	$+$	$2$	$\mathrm{SU}(2)$	$q+(-2+\beta _{1})q^{2}+(-4-\beta _{1}-\beta _{3})q^{3}+\cdots$
31.6.a.b	$31$	$6$	31.a	1.a	$1$	$8$	$8$	$4.972$	$\mathbb{Q}[x]/(x^{8} - \cdots)$	$_{}$	None	None	✓	✓	✓	✓	31.6.a.b	$1$	$0$	$7$	$-2$	$128$	$88$	$-$	$2^{2}\cdot 5\cdot 13$	$\mathrm{SU}(2)$	$q+(1-\beta _{1})q^{2}+\beta _{6}q^{3}+(19-3\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots$
31.6.c.a	$31$	$6$	31.c	31.c	$3$	$24$	$12$	$4.972$		$_{}$	None	None		✓	✓	✓	31.6.c.a	$2$	$0$	$-4$	$10$	$28$	$-134$			$\mathrm{SU}(2)[C_{3}]$
31.6.d.a	$31$	$6$	31.d	31.d	$5$	$52$	$13$	$4.972$		$_{}$	None	None		✓	✓	✓	31.6.d.a	$2$	$0$	$-3$	$17$	$-66$	$205$			$\mathrm{SU}(2)[C_{5}]$
31.6.g.a	$31$	$6$	31.g	31.g	$15$	$96$	$12$	$4.972$		$_{}$	None	None		✓	✓	✓	31.6.g.a	$2$	$0$	$-6$	$-20$	$-33$	$-66$			$\mathrm{SU}(2)[C_{15}]$
31.7.b.a	$31$	$7$	31.b	31.b	$2$	$1$	$1$	$7.132$	$\Q$	$_{}$	$\Q(\sqrt{-31}) $	None	✓	✓			31.7.b.a	$2$	$0$	$-15$	$0$	$-246$	$-430$		$1$	$\mathrm{U}(1)[D_{2}]$	$q-15q^{2}+161q^{4}-246q^{5}-430q^{7}+\cdots$
31.7.b.b	$31$	$7$	31.b	31.b	$2$	$2$	$2$	$7.132$	$\Q(\sqrt{93}) $	$_{}$	$\Q(\sqrt{-31}) $	None	✓	✓			31.7.b.b	$2$	$0$	$15$	$0$	$246$	$430$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+(8-\beta )q^{2}+(23-15\beta )q^{4}+(127-8\beta )q^{5}+\cdots$
31.7.b.c	$31$	$7$	31.b	31.b	$2$	$12$	$12$	$7.132$	$\mathbb{Q}[x]/(x^{12} + \cdots)$	$_{}$	None	None		✓	✓		31.7.b.c	$2$	$0$	$-2$	$0$	$-146$	$6$		$2^{10}\cdot 3^{2}\cdot 5$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{8}q^{2}-\beta _{1}q^{3}+(10+\beta _{6}-\beta _{8})q^{4}+\cdots$
31.7.e.a	$31$	$7$	31.e	31.e	$6$	$30$	$15$	$7.132$		$_{}$	None	None		✓	✓	✓	31.7.e.a	$2$	$0$	$-4$	$-3$	$-73$	$723$			$\mathrm{SU}(2)[C_{6}]$
31.7.f.a	$31$	$7$	31.f	31.f	$10$	$60$	$15$	$7.132$		$_{}$	None	None		✓	✓	✓	31.7.f.a	$2$	$0$	$-3$	$-5$	$136$	$1009$			$\mathrm{SU}(2)[C_{10}]$
31.7.h.a	$31$	$7$	31.h	31.h	$30$	$120$	$15$	$7.132$		$_{}$	None	None		✓	✓	✓	31.7.h.a	$2$	$0$	$-6$	$-7$	$68$	$-1753$			$\mathrm{SU}(2)[C_{30}]$
31.8.a.a	$31$	$8$	31.a	1.a	$1$	$7$	$7$	$9.684$	$\mathbb{Q}[x]/(x^{7} - \cdots)$	$_{}$	None	None	✓	✓	✓	✓	31.8.a.a	$1$	$1$	$-17$	$-14$	$-430$	$-832$	$-$	$2^{4}$	$\mathrm{SU}(2)$	$q+(-2-\beta _{1})q^{2}+(-2+\beta _{1}+\beta _{3})q^{3}+\cdots$
31.8.a.b	$31$	$8$	31.a	1.a	$1$	$10$	$10$	$9.684$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	$_{}$	None	None	✓	✓	✓	✓	31.8.a.b	$1$	$0$	$15$	$40$	$570$	$540$	$+$	$2^{6}$	$\mathrm{SU}(2)$	$q+(2-\beta _{1})q^{2}+(4+\beta _{2})q^{3}+(61+\beta _{1}+\cdots)q^{4}+\cdots$
31.8.c.a	$31$	$8$	31.c	31.c	$3$	$36$	$18$	$9.684$		$_{}$	None	None		✓	✓	✓	31.8.c.a	$2$	$0$	$-4$	$-2$	$70$	$-2158$			$\mathrm{SU}(2)[C_{3}]$
31.8.d.a	$31$	$8$	31.d	31.d	$5$	$68$	$17$	$9.684$		$_{}$	None	None		✓	✓	✓	31.8.d.a	$2$	$0$	$-3$	$-31$	$-150$	$-3083$			$\mathrm{SU}(2)[C_{5}]$
31.8.g.a	$31$	$8$	31.g	31.g	$15$	$144$	$18$	$9.684$		$_{}$	None	None		✓	✓	✓	31.8.g.a	$2$	$0$	$-6$	$-8$	$-75$	$5518$			$\mathrm{SU}(2)[C_{15}]$
31.9.b.a	$31$	$9$	31.b	31.b	$2$	$3$	$3$	$12.629$	3.3.837.1	$_{}$	$\Q(\sqrt{-31}) $	None	✓	✓			31.9.b.a	$2$	$0$	$0$	$0$	$0$	$0$		$3$	$\mathrm{U}(1)[D_{2}]$	$q+(-6\beta _{1}+\beta _{2})q^{2}+(2^{8}-\beta _{1}+3^{4}\beta _{2})q^{4}+\cdots$
31.9.b.b	$31$	$9$	31.b	31.b	$2$	$18$	$18$	$12.629$	$\mathbb{Q}[x]/(x^{18} + \cdots)$	$_{}$	None	None		✓	✓		31.9.b.b	$2$	$0$	$-2$	$0$	$292$	$-5472$		$2^{20}\cdot 3^{8}\cdot 7^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{2}q^{2}+\beta _{1}q^{3}+(140+\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots$
31.9.e.a	$31$	$9$	31.e	31.e	$6$	$40$	$20$	$12.629$		$_{}$	None	None		✓	✓	✓	31.9.e.a	$2$	$0$	$-4$	$-84$	$146$	$272$			$\mathrm{SU}(2)[C_{6}]$
31.9.f.a	$31$	$9$	31.f	31.f	$10$	$84$	$21$	$12.629$		$_{}$	None	None		✓	✓	✓	31.9.f.a	$2$	$0$	$-3$	$-5$	$-302$	$5067$			$\mathrm{SU}(2)[C_{10}]$
31.9.h.a	$31$	$9$	31.h	31.h	$30$	$160$	$20$	$12.629$		$_{}$	None	None		✓	✓	✓	31.9.h.a	$2$	$0$	$-6$	$74$	$-151$	$118$			$\mathrm{SU}(2)[C_{30}]$
31.10.a.a	$31$	$10$	31.a	1.a	$1$	$10$	$10$	$15.966$	$\mathbb{Q}[x]/(x^{10} - \cdots)$	$_{}$	None	None	✓	✓	✓	✓	31.10.a.a	$1$	$1$	$-33$	$-8$	$-3297$	$-495$	$+$	$2^{10}\cdot 23$	$\mathrm{SU}(2)$	$q+(-3-\beta _{1})q^{2}+(-1+\beta _{1}-\beta _{3})q^{3}+\cdots$
31.10.a.b	$31$	$10$	31.a	1.a	$1$	$13$	$13$	$15.966$	$\mathbb{Q}[x]/(x^{13} - \cdots)$	$_{}$	None	None	✓	✓	✓	✓	31.10.a.b	$1$	$0$	$31$	$154$	$1703$	$9109$	$-$	$2^{10}\cdot 3^{5}\cdot 5$	$\mathrm{SU}(2)$	$q+(2+\beta _{1})q^{2}+(12+\beta _{3})q^{3}+(300+6\beta _{1}+\cdots)q^{4}+\cdots$
31.10.c.a	$31$	$10$	31.c	31.c	$3$	$46$	$23$	$15.966$		$_{}$	None	None		✓	✓	✓	31.10.c.a	$2$	$0$	$-4$	$13$	$-797$	$6359$			$\mathrm{SU}(2)[C_{3}]$
31.10.d.a	$31$	$10$	31.d	31.d	$5$	$92$	$23$	$15.966$		$_{}$	None	None		✓	✓	✓	31.10.d.a	$2$	$0$	$-3$	$-151$	$1584$	$15771$			$\mathrm{SU}(2)[C_{5}]$
31.10.g.a	$31$	$10$	31.g	31.g	$15$	$184$	$23$	$15.966$		$_{}$	None	None		✓	✓	✓	31.10.g.a	$2$	$0$	$-6$	$-23$	$792$	$-30759$			$\mathrm{SU}(2)[C_{15}]$
31.11.b.a	$31$	$11$	31.b	31.b	$2$	$3$	$3$	$19.696$	3.3.837.1	$_{}$	$\Q(\sqrt{-31}) $	None	✓	✓			31.11.b.a	$2$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+(-15\beta _{1}+14\beta _{2})q^{2}+(2^{10}-644\beta _{1}+\cdots)q^{4}+\cdots$
31.11.b.b	$31$	$11$	31.b	31.b	$2$	$22$	$22$	$19.696$		$_{}$	None	None		✓	✓		31.11.b.b	$2$	$0$	$-2$	$0$	$3304$	$19916$			$\mathrm{SU}(2)[C_{2}]$
31.11.e.a	$31$	$11$	31.e	31.e	$6$	$52$	$26$	$19.696$		$_{}$	None	None		✓	✓	✓	31.11.e.a	$2$	$0$	$-4$	$-246$	$1652$	$-19306$			$\mathrm{SU}(2)[C_{6}]$

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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.