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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
1156.1.c.a	$1156$	$1$	1156.c	4.b	$2$	$1$	$1$	$0.577$	$\Q$	$D_{2}$	$\Q(\sqrt{-1}) $, $\Q(\sqrt{-17}) $	$\Q(\sqrt{17}) $	✓	$4$	$0$	$1$	$0$	$0$	$0$		$1$		$q+q^{2}+q^{4}+q^{8}+q^{9}-2q^{13}+q^{16}+\cdots$
1156.1.c.b	$1156$	$1$	1156.c	4.b	$2$	$2$	$2$	$0.577$	$\Q(\sqrt{2}) $	$D_{4}$	$\Q(\sqrt{-1}) $	None	✓	$4$	$0$	$-2$	$0$	$0$	$0$		$1$		$q-q^{2}+q^{4}-\beta q^{5}-q^{8}+q^{9}+\beta q^{10}+\cdots$
1156.1.d.a	$1156$	$1$	1156.d	68.d	$2$	$2$	$2$	$0.577$	$\Q(\sqrt{-2}) $	$D_{4}$	$\Q(\sqrt{-1}) $	None		$4$	$0$	$2$	$0$	$0$	$0$		$1$		$q+q^{2}+q^{4}-\beta q^{5}+q^{8}-q^{9}-\beta q^{10}+\cdots$
1156.1.f.a	$1156$	$1$	1156.f	68.f	$4$	$2$	$1$	$0.577$	$\Q(\sqrt{-1}) $	$D_{2}$	$\Q(\sqrt{-1}) $, $\Q(\sqrt{-17}) $	$\Q(\sqrt{17}) $		$8$	$0$	$0$	$0$	$0$	$0$		$1$		$q+iq^{2}-q^{4}-iq^{8}-iq^{9}+q^{13}+\cdots$
1156.1.f.b	$1156$	$1$	1156.f	68.f	$4$	$2$	$1$	$0.577$	$\Q(\sqrt{-1}) $	$D_{4}$	$\Q(\sqrt{-1}) $	None		$4$	$0$	$0$	$0$	$2$	$0$		$1$		$q-iq^{2}-q^{4}+(1-i)q^{5}+iq^{8}-iq^{9}+\cdots$
1156.1.g.a	$1156$	$1$	1156.g	68.g	$8$	$4$	$1$	$0.577$	$\Q(\zeta_{8})$	$D_{2}$	$\Q(\sqrt{-1}) $, $\Q(\sqrt{-17}) $	$\Q(\sqrt{17}) $		$16$	$0$	$0$	$0$	$0$	$0$		$1$		$q+\zeta_{8}^{3}q^{2}-\zeta_{8}^{2}q^{4}+\zeta_{8}q^{8}-\zeta_{8}q^{9}+\cdots$
1156.1.g.b	$1156$	$1$	1156.g	68.g	$8$	$8$	$2$	$0.577$	$\Q(\zeta_{16})$	$D_{4}$	$\Q(\sqrt{-1}) $	None		$16$	$0$	$0$	$0$	$0$	$0$		$2^{2}$		$q+\zeta_{16}^{6}q^{2}-\zeta_{16}^{4}q^{4}+(-\zeta_{16}^{3}-\zeta_{16}^{7}+\cdots)q^{5}+\cdots$
1156.1.l.a	$1156$	$1$	1156.l	1156.l	$34$	$16$	$1$	$0.577$	$\Q(\zeta_{34})$	$D_{34}$	$\Q(\sqrt{-1}) $	None		$4$	$0$	$1$	$0$	$-17$	$0$		$1$		$q-\zeta_{34}^{10}q^{2}-\zeta_{34}^{3}q^{4}+(-1+\zeta_{34}^{8}+\cdots)q^{5}+\cdots$
1156.1.m.a	$1156$	$1$	1156.m	1156.m	$34$	$16$	$1$	$0.577$	$\Q(\zeta_{34})$	$D_{17}$	$\Q(\sqrt{-1}) $	None		$4$	$0$	$-1$	$0$	$15$	$0$		$1$		$q+\zeta_{34}^{6}q^{2}+\zeta_{34}^{12}q^{4}+(1-\zeta_{34}^{15}+\cdots)q^{5}+\cdots$
1156.1.o.a	$1156$	$1$	1156.o	1156.o	$68$	$32$	$1$	$0.577$	$\Q(\zeta_{68})$	$D_{68}$	$\Q(\sqrt{-1}) $	None		$4$	$0$	$0$	$0$	$2$	$0$		$1$		$q+\zeta_{68}^{15}q^{2}+\zeta_{68}^{30}q^{4}+(-\zeta_{68}^{12}+\cdots)q^{5}+\cdots$
1156.2.a.a	$1156$	$2$	1156.a	1.a	$1$	$2$	$2$	$9.231$	$\Q(\sqrt{3}) $	$_{}$	None	None	✓	$1$	$0$	$0$	$-2$	$0$	$2$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta )q^{3}-2\beta q^{5}+(1-\beta )q^{7}+\cdots$
1156.2.a.b	$1156$	$2$	1156.a	1.a	$1$	$2$	$2$	$9.231$	$\Q(\sqrt{21}) $	$_{}$	None	None	✓	$1$	$1$	$0$	$-1$	$-3$	$-5$	$+$	$1$	$\mathrm{SU}(2)$	$q-\beta q^{3}+(-1-\beta )q^{5}+(-3+\beta )q^{7}+\cdots$
1156.2.a.c	$1156$	$2$	1156.a	1.a	$1$	$2$	$2$	$9.231$	$\Q(\sqrt{2}) $	$_{}$	None	None	✓	$2$	$0$	$0$	$0$	$0$	$0$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}+2\beta q^{5}+3\beta q^{7}-q^{9}+\beta q^{11}+\cdots$
1156.2.a.d	$1156$	$2$	1156.a	1.a	$1$	$2$	$2$	$9.231$	$\Q(\sqrt{21}) $	$_{}$	None	None	✓	$1$	$0$	$0$	$1$	$3$	$5$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}+(1+\beta )q^{5}+(3-\beta )q^{7}+(2+\beta )q^{9}+\cdots$
1156.2.a.e	$1156$	$2$	1156.a	1.a	$1$	$3$	$3$	$9.231$	$\Q(\zeta_{18})^+$	$_{}$	None	None	✓	$1$	$1$	$0$	$-3$	$-6$	$6$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1-\beta _{1})q^{3}+(-2+\beta _{1}-\beta _{2})q^{5}+\cdots$
1156.2.a.f	$1156$	$2$	1156.a	1.a	$1$	$3$	$3$	$9.231$	$\Q(\zeta_{18})^+$	$_{}$	None	None	✓	$1$	$0$	$0$	$3$	$6$	$-6$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta _{1})q^{3}+(2-\beta _{1}+\beta _{2})q^{5}+(-2+\cdots)q^{7}+\cdots$
1156.2.a.g	$1156$	$2$	1156.a	1.a	$1$	$4$	$4$	$9.231$	$\Q(\zeta_{16})^+$	$_{}$	None	None	✓	$2$	$1$	$0$	$0$	$0$	$0$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}-\beta _{1}q^{5}+\beta _{3}q^{7}+(-1+\beta _{2}+\cdots)q^{9}+\cdots$
1156.2.a.h	$1156$	$2$	1156.a	1.a	$1$	$4$	$4$	$9.231$	$\Q(\sqrt{2}, \sqrt{13})$	$_{}$	None	None	✓	$2$	$0$	$0$	$0$	$0$	$0$	$-$	$2$	$\mathrm{SU}(2)$	$q+(\beta _{1}+\beta _{2})q^{3}-\beta _{2}q^{5}+(\beta _{1}+\beta _{2})q^{7}+\cdots$
1156.2.b.a	$1156$	$2$	1156.b	17.b	$2$	$4$	$4$	$9.231$	$\Q(\sqrt{-2}, \sqrt{13})$	$_{}$	None	None		$2$	$0$	$0$	$0$	$0$	$0$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{3}+\beta _{2}q^{5}-\beta _{1}q^{7}+(-4-\beta _{3})q^{9}+\cdots$
1156.2.b.b	$1156$	$2$	1156.b	17.b	$2$	$4$	$4$	$9.231$	$\Q(i, \sqrt{21})$	$_{}$	None	None		$2$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{3}+(\beta _{1}-\beta _{2})q^{5}+(\beta _{1}+3\beta _{2}+\cdots)q^{7}+\cdots$
1156.2.b.c	$1156$	$2$	1156.b	17.b	$2$	$4$	$4$	$9.231$	$\Q(\zeta_{12})$	$_{}$	None	None		$2$	$0$	$0$	$0$	$0$	$0$		$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	$q-\zeta_{12}^{2}q^{3}+(\zeta_{12}+\zeta_{12}^{2})q^{5}-\zeta_{12}^{2}q^{7}+\cdots$
1156.2.b.d	$1156$	$2$	1156.b	17.b	$2$	$4$	$4$	$9.231$	4.0.2048.2	$_{}$	None	None		$2$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{3}-\beta _{1}q^{5}+\beta _{3}q^{7}+(1+\beta _{2}+\cdots)q^{9}+\cdots$
1156.2.b.e	$1156$	$2$	1156.b	17.b	$2$	$6$	$6$	$9.231$	6.0.419904.1	$_{}$	None	None		$2$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+(\beta _{1}+\beta _{3}+\beta _{5})q^{3}+(-\beta _{1}+2\beta _{3}+\cdots)q^{5}+\cdots$
1156.2.e.a	$1156$	$2$	1156.e	17.c	$4$	$2$	$1$	$9.231$	$\Q(\sqrt{-1}) $	$_{}$	None	None		$2$	$0$	$0$	$-2$	$4$	$6$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q+(-1-i)q^{3}+(2+2i)q^{5}+(3-3i)q^{7}+\cdots$
1156.2.e.b	$1156$	$2$	1156.e	17.c	$4$	$2$	$1$	$9.231$	$\Q(\sqrt{-1}) $	$_{}$	None	None		$2$	$0$	$0$	$2$	$-4$	$-6$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q+(1+i)q^{3}+(-2-2i)q^{5}+(-3+3i)q^{7}+\cdots$
1156.2.e.c	$1156$	$2$	1156.e	17.c	$4$	$4$	$2$	$9.231$	$\Q(i, \sqrt{13})$	$_{}$	None	None		$2$	$0$	$0$	$2$	$4$	$-2$		$2$	$\mathrm{SU}(2)[C_{4}]$	$q+(1-\beta _{1}+\beta _{3})q^{3}+(1-\beta _{1})q^{5}+(-1+\cdots)q^{7}+\cdots$
1156.2.e.d	$1156$	$2$	1156.e	17.c	$4$	$8$	$4$	$9.231$	$\Q(\zeta_{24})$	$_{}$	None	None		$4$	$0$	$0$	$0$	$0$	$0$		$2^{8}$	$\mathrm{SU}(2)[C_{4}]$	$q+(\zeta_{24}-\zeta_{24}^{2})q^{3}+(\zeta_{24}-2\zeta_{24}^{2}+\cdots)q^{5}+\cdots$
1156.2.e.e	$1156$	$2$	1156.e	17.c	$4$	$8$	$4$	$9.231$	8.0.$\cdots$.8	$_{}$	None	None		$4$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q+\beta _{1}q^{3}+(-\beta _{1}+\beta _{2})q^{5}+(-\beta _{6}-3\beta _{7})q^{7}+\cdots$
1156.2.e.f	$1156$	$2$	1156.e	17.c	$4$	$8$	$4$	$9.231$	$\Q(\zeta_{16})$	$_{}$	None	None		$4$	$0$	$0$	$0$	$0$	$0$		$2^{3}$	$\mathrm{SU}(2)[C_{4}]$	$q+\zeta_{16}q^{3}+\zeta_{16}q^{5}-\zeta_{16}^{5}q^{7}+(-\zeta_{16}^{2}+\cdots)q^{9}+\cdots$
1156.2.e.g	$1156$	$2$	1156.e	17.c	$4$	$12$	$6$	$9.231$	12.0.$\cdots$.1	$_{}$	None	None		$4$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{4}]$	$q+(-\beta _{1}-\beta _{3}+\beta _{5})q^{3}+(-\beta _{1}+2\beta _{3}+\cdots)q^{5}+\cdots$
1156.2.h.a	$1156$	$2$	1156.h	17.d	$8$	$4$	$1$	$9.231$	$\Q(\zeta_{8})$	$_{}$	None	None		$2$	$0$	$0$	$-4$	$0$	$4$		$1$	$\mathrm{SU}(2)[C_{8}]$	$q+(-1-\zeta_{8})q^{3}+(\zeta_{8}^{2}+\zeta_{8}^{3})q^{5}+(1+\cdots)q^{7}+\cdots$
1156.2.h.b	$1156$	$2$	1156.h	17.d	$8$	$4$	$1$	$9.231$	$\Q(\zeta_{8})$	$_{}$	None	None		$2$	$0$	$0$	$0$	$-4$	$0$		$1$	$\mathrm{SU}(2)[C_{8}]$	$q+(-\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+(-1+\zeta_{8})q^{5}+\cdots$
1156.2.h.c	$1156$	$2$	1156.h	17.d	$8$	$4$	$1$	$9.231$	$\Q(\zeta_{8})$	$_{}$	None	None		$2$	$0$	$0$	$4$	$0$	$-4$		$1$	$\mathrm{SU}(2)[C_{8}]$	$q+(1+\zeta_{8})q^{3}+(-\zeta_{8}^{2}-\zeta_{8}^{3})q^{5}+(-1+\cdots)q^{7}+\cdots$
1156.2.h.d	$1156$	$2$	1156.h	17.d	$8$	$8$	$2$	$9.231$	$\Q(\zeta_{16})$	$_{}$	None	None		$8$	$0$	$0$	$0$	$0$	$0$		$2^{2}$	$\mathrm{SU}(2)[C_{8}]$	$q+\zeta_{16}^{3}q^{3}+2\zeta_{16}^{6}q^{5}+3\zeta_{16}^{7}q^{7}+\cdots$
1156.2.h.e	$1156$	$2$	1156.h	17.d	$8$	$16$	$4$	$9.231$	16.0.$\cdots$.1	$_{}$	None	None		$8$	$0$	$0$	$0$	$0$	$0$		$2^{8}$	$\mathrm{SU}(2)[C_{8}]$	$q-\beta _{10}q^{3}-\beta _{8}q^{5}-\beta _{15}q^{7}+(4\beta _{5}+\cdots)q^{9}+\cdots$
1156.2.h.f	$1156$	$2$	1156.h	17.d	$8$	$16$	$4$	$9.231$	$\Q(\zeta_{48})$	$_{}$	None	None		$8$	$0$	$0$	$0$	$0$	$0$		$2^{16}$	$\mathrm{SU}(2)[C_{8}]$	$q+\zeta_{48}^{15}q^{3}+(-\zeta_{48}^{3}+2\zeta_{48}^{5})q^{5}+\cdots$
1156.2.h.g	$1156$	$2$	1156.h	17.d	$8$	$16$	$4$	$9.231$	16.0.$\cdots$.4	$_{}$	None	None		$8$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{8}]$	$q+\beta _{6}q^{3}+(-\beta _{13}+\beta _{15})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots$
1156.2.h.h	$1156$	$2$	1156.h	17.d	$8$	$24$	$6$	$9.231$		$_{}$	None	None		$8$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{8}]$
1156.4.a.a	$1156$	$4$	1156.a	1.a	$1$	$1$	$1$	$68.206$	$\Q$	$_{}$	None	None	✓	$1$	$1$	$0$	$2$	$8$	$12$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{3}+8q^{5}+12q^{7}-23q^{9}+10q^{11}+\cdots$
1156.4.a.b	$1156$	$4$	1156.a	1.a	$1$	$2$	$2$	$68.206$	$\Q(\sqrt{21}) $	$_{}$	None	None	✓	$1$	$1$	$0$	$-5$	$15$	$5$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-2-\beta )q^{3}+(5+5\beta )q^{5}+(5-5\beta )q^{7}+\cdots$
1156.4.a.c	$1156$	$4$	1156.a	1.a	$1$	$2$	$2$	$68.206$	$\Q(\sqrt{229}) $	$_{}$	None	None	✓	$1$	$1$	$0$	$-3$	$-21$	$-25$	$-$	$1$	$\mathrm{SU}(2)$	$q+(-1-\beta )q^{3}+(-10-\beta )q^{5}+(-12+\cdots)q^{7}+\cdots$
1156.4.a.d	$1156$	$4$	1156.a	1.a	$1$	$2$	$2$	$68.206$	$\Q(\sqrt{2}) $	$_{}$	None	None	✓	$2$	$1$	$0$	$0$	$0$	$0$	$-$	$1$	$\mathrm{SU}(2)$	$q+4\beta q^{3}-\beta q^{5}-20\beta q^{7}+5q^{9}+2^{5}\beta q^{11}+\cdots$
1156.4.a.e	$1156$	$4$	1156.a	1.a	$1$	$2$	$2$	$68.206$	$\Q(\sqrt{229}) $	$_{}$	None	None	✓	$1$	$0$	$0$	$3$	$21$	$25$	$+$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{3}+(10+\beta )q^{5}+(12+\beta )q^{7}+\cdots$
1156.4.a.f	$1156$	$4$	1156.a	1.a	$1$	$2$	$2$	$68.206$	$\Q(\sqrt{21}) $	$_{}$	None	None	✓	$1$	$0$	$0$	$5$	$-15$	$-5$	$+$	$1$	$\mathrm{SU}(2)$	$q+(3-\beta )q^{3}+(-10+5\beta )q^{5}-5\beta q^{7}+\cdots$
1156.4.a.g	$1156$	$4$	1156.a	1.a	$1$	$3$	$3$	$68.206$	3.3.1524.1	$_{}$	None	None	✓	$1$	$1$	$0$	$-4$	$-26$	$-8$	$-$	$2^{3}$	$\mathrm{SU}(2)$	$q+(-1-\beta _{2})q^{3}+(-9+\beta _{1})q^{5}+(-2+\cdots)q^{7}+\cdots$
1156.4.a.h	$1156$	$4$	1156.a	1.a	$1$	$4$	$4$	$68.206$	4.4.5999648.1	$_{}$	None	None	✓	$2$	$1$	$0$	$0$	$0$	$0$	$-$	$2^{4}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}+\beta _{2}q^{5}+(-\beta _{1}-\beta _{2})q^{7}+\cdots$
1156.4.a.i	$1156$	$4$	1156.a	1.a	$1$	$6$	$6$	$68.206$	6.6.889407488.2	$_{}$	None	None	✓	$2$	$1$	$0$	$0$	$0$	$0$	$-$	$2^{3}$	$\mathrm{SU}(2)$	$q+(-\beta _{1}-\beta _{2})q^{3}+(-\beta _{1}+\beta _{2}+3\beta _{5})q^{5}+\cdots$
1156.4.a.j	$1156$	$4$	1156.a	1.a	$1$	$12$	$12$	$68.206$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	$_{}$	None	None	✓	$1$	$1$	$0$	$0$	$-30$	$18$	$-$	$2^{6}\cdot 3\cdot 17^{2}$	$\mathrm{SU}(2)$	$q-\beta _{1}q^{3}+(-2-\beta _{6})q^{5}+(2+\beta _{3}+\beta _{5}+\cdots)q^{7}+\cdots$
1156.4.a.k	$1156$	$4$	1156.a	1.a	$1$	$12$	$12$	$68.206$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	$_{}$	None	None	✓	$1$	$0$	$0$	$0$	$30$	$-18$	$+$	$2^{6}\cdot 3\cdot 17^{2}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}+(2+\beta _{6})q^{5}+(-2-\beta _{3}-\beta _{5}+\cdots)q^{7}+\cdots$
1156.4.a.l	$1156$	$4$	1156.a	1.a	$1$	$20$	$20$	$68.206$	$\mathbb{Q}[x]/(x^{20} - \cdots)$	$_{}$	None	None	✓	$2$	$0$	$0$	$0$	$0$	$0$	$+$	$2^{16}\cdot 17^{6}$	$\mathrm{SU}(2)$	$q+\beta _{1}q^{3}-\beta _{13}q^{5}-\beta _{15}q^{7}+(13+\beta _{2}+\cdots)q^{9}+\cdots$

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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.