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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 176 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
168.1.s.a	$168$	$1$	168.s	168.s	$6$	$2$	$1$	$0.084$	$\Q(\sqrt{-3}) $	$D_{3}$	$\Q(\sqrt{-6}) $	None		✓			168.1.s.a	$4$	$0$	$-1$	$-1$	$1$	$-1$		$1$		$q+\zeta_{6}^{2}q^{2}-\zeta_{6}q^{3}-\zeta_{6}q^{4}-\zeta_{6}^{2}q^{5}+\cdots$
168.1.s.b	$168$	$1$	168.s	168.s	$6$	$2$	$1$	$0.084$	$\Q(\sqrt{-3}) $	$D_{3}$	$\Q(\sqrt{-6}) $	None		✓			168.1.s.a	$4$	$0$	$1$	$1$	$-1$	$-1$		$1$		$q-\zeta_{6}^{2}q^{2}+\zeta_{6}q^{3}-\zeta_{6}q^{4}+\zeta_{6}^{2}q^{5}+\cdots$
168.2.a.a	$168$	$2$	168.a	1.a	$1$	$1$	$1$	$1.341$	$\Q$	$_{}$	None	None	✓	✓	✓	✓	168.2.a.a	$1$	$0$	$0$	$-1$	$2$	$1$	$-$	$1$	$\mathrm{SU}(2)$	$q-q^{3}+2q^{5}+q^{7}+q^{9}+6q^{13}-2q^{15}+\cdots$
168.2.a.b	$168$	$2$	168.a	1.a	$1$	$1$	$1$	$1.341$	$\Q$	$_{}$	None	None	✓	✓	✓	✓	168.2.a.b	$1$	$0$	$0$	$1$	$2$	$-1$	$-$	$1$	$\mathrm{SU}(2)$	$q+q^{3}+2q^{5}-q^{7}+q^{9}-2q^{13}+2q^{15}+\cdots$
168.2.c.a	$168$	$2$	168.c	8.b	$2$	$4$	$4$	$1.341$	$\Q(\zeta_{12})$	$_{}$	None	None		✓			168.2.c.a	$2$	$0$	$2$	$0$	$0$	$-4$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+(-\beta_{2}+\beta_1)q^{2}+\beta_{2} q^{3}+(\beta_{3}+\beta_1)q^{4}+\cdots$
168.2.c.b	$168$	$2$	168.c	8.b	$2$	$8$	$8$	$1.341$	8.0.386672896.3	$_{}$	None	None		✓	✓		168.2.c.b	$2$	$0$	$0$	$0$	$0$	$8$		$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{4}q^{2}-\beta _{2}q^{3}+\beta _{1}q^{4}+(-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots$
168.2.i.a	$168$	$2$	168.i	168.i	$2$	$4$	$4$	$1.341$	$\Q(\sqrt{-2}, \sqrt{-3})$	$_{}$	None	None		✓			168.2.i.a	$4$	$0$	$0$	$-4$	$0$	$-8$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+(-1+\beta _{2})q^{3}+(1+\beta _{3})q^{4}+\cdots$
168.2.i.b	$168$	$2$	168.i	168.i	$2$	$4$	$4$	$1.341$	$\Q(\sqrt{2}, \sqrt{-3})$	$_{}$	$\Q(\sqrt{-6}) $	None		✓			168.2.i.b	$8$	$0$	$0$	$0$	$0$	$4$		$2^{2}$	$\mathrm{U}(1)[D_{2}]$	$q+\beta _{1}q^{2}-\beta _{2}q^{3}+2q^{4}+2\beta _{2}q^{5}+\cdots$
168.2.i.c	$168$	$2$	168.i	168.i	$2$	$4$	$4$	$1.341$	$\Q(\sqrt{-2}, \sqrt{-3})$	$_{}$	None	None		✓			168.2.i.a	$4$	$0$	$0$	$4$	$0$	$-8$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{2}+(1-\beta _{2})q^{3}+(1+\beta _{3})q^{4}+\cdots$
168.2.i.d	$168$	$2$	168.i	168.i	$2$	$8$	$8$	$1.341$	8.0.$\cdots$.11	$_{}$	$\Q(\sqrt{-14}) $	None		✓			168.2.i.d	$8$	$0$	$0$	$0$	$0$	$0$		$2^{3}\cdot 3$	$\mathrm{U}(1)[D_{2}]$	$q-\beta _{2}q^{2}-\beta _{1}q^{3}-2q^{4}+(\beta _{6}-\beta _{7})q^{5}+\cdots$
168.2.i.e	$168$	$2$	168.i	168.i	$2$	$8$	$8$	$1.341$	8.0.3317760000.1	$_{}$	None	None		✓			168.2.i.e	$8$	$0$	$0$	$0$	$0$	$8$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{5}q^{2}+(-\beta _{4}-\beta _{6})q^{3}+(-1-\beta _{2}+\cdots)q^{4}+\cdots$
168.2.j.a	$168$	$2$	168.j	24.f	$2$	$4$	$4$	$1.341$	$\Q(i, \sqrt{5})$	$_{}$	None	None		✓			168.2.j.a	$2$	$0$	$-4$	$-2$	$4$	$0$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+(-1-\beta _{2})q^{2}+(\beta _{1}+\beta _{2})q^{3}+2\beta _{2}q^{4}+\cdots$
168.2.j.b	$168$	$2$	168.j	24.f	$2$	$4$	$4$	$1.341$	$\Q(\zeta_{8})$	$_{}$	None	None		✓			168.2.j.b	$4$	$0$	$0$	$4$	$0$	$0$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta_{3} q^{2}+(-\beta_{2}+1)q^{3}+2 q^{4}+\cdots$
168.2.j.c	$168$	$2$	168.j	24.f	$2$	$4$	$4$	$1.341$	$\Q(i, \sqrt{5})$	$_{}$	None	None		✓			168.2.j.a	$2$	$0$	$4$	$-2$	$-4$	$0$		$2$	$\mathrm{SU}(2)[C_{2}]$	$q+(1+\beta _{2})q^{2}+(-\beta _{2}-\beta _{3})q^{3}+2\beta _{2}q^{4}+\cdots$
168.2.j.d	$168$	$2$	168.j	24.f	$2$	$12$	$12$	$1.341$	12.0.$\cdots$.2	$_{}$	None	None		✓	✓		168.2.j.d	$4$	$0$	$0$	$0$	$0$	$0$		$2^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{4}q^{2}-\beta _{6}q^{3}+(-\beta _{2}-\beta _{3})q^{4}+\cdots$
168.2.k.a	$168$	$2$	168.k	21.c	$2$	$8$	$8$	$1.341$	8.0.342102016.5	$_{}$	None	None		✓	✓	✓	168.2.k.a	$4$	$0$	$0$	$0$	$0$	$-4$		$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{3}-\beta _{2}q^{5}+(-1-\beta _{5}-\beta _{6}+\cdots)q^{7}+\cdots$
168.2.p.a	$168$	$2$	168.p	56.e	$2$	$16$	$16$	$1.341$	16.0.$\cdots$.1	$_{}$	None	None		✓	✓	✓	168.2.p.a	$4$	$0$	$2$	$0$	$0$	$0$		$2^{9}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{5}q^{2}-\beta _{3}q^{3}+\beta _{1}q^{4}+(\beta _{6}-\beta _{11}+\cdots)q^{5}+\cdots$
168.2.q.a	$168$	$2$	168.q	7.c	$3$	$2$	$1$	$1.341$	$\Q(\sqrt{-3}) $	$_{}$	None	None		✓			168.2.q.a	$2$	$0$	$0$	$-1$	$1$	$1$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q-\zeta_{6}q^{3}+(1-\zeta_{6})q^{5}+(2-3\zeta_{6})q^{7}+\cdots$
168.2.q.b	$168$	$2$	168.q	7.c	$3$	$2$	$1$	$1.341$	$\Q(\sqrt{-3}) $	$_{}$	None	None		✓			168.2.q.b	$2$	$0$	$0$	$1$	$-2$	$5$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+\zeta_{6}q^{3}+(-2+2\zeta_{6})q^{5}+(3-\zeta_{6})q^{7}+\cdots$
168.2.q.c	$168$	$2$	168.q	7.c	$3$	$4$	$2$	$1.341$	$\Q(\sqrt{-3}, \sqrt{-19})$	$_{}$	None	None		✓	✓		168.2.q.c	$2$	$0$	$0$	$2$	$1$	$-6$		$1$	$\mathrm{SU}(2)[C_{3}]$	$q+(1-\beta _{2})q^{3}+(-1+2\beta _{1}+\beta _{2}-\beta _{3})q^{5}+\cdots$
168.2.t.a	$168$	$2$	168.t	56.m	$6$	$32$	$16$	$1.341$		$_{}$	None	None		✓	✓	✓	168.2.t.a	$4$	$0$	$-2$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{6}]$
168.2.u.a	$168$	$2$	168.u	21.g	$6$	$16$	$8$	$1.341$	$\mathbb{Q}[x]/(x^{16} - \cdots)$	$_{}$	None	None		✓	✓	✓	168.2.u.a	$4$	$0$	$0$	$0$	$0$	$4$		$2^{8}$	$\mathrm{SU}(2)[C_{6}]$	$q+(\beta _{3}+\beta _{9})q^{3}+\beta _{14}q^{5}-\beta _{10}q^{7}+\cdots$
168.2.v.a	$168$	$2$	168.v	168.v	$6$	$56$	$28$	$1.341$		$_{}$	None	None		✓	✓	✓	168.2.v.a	$8$	$0$	$0$	$-2$	$0$	$0$			$\mathrm{SU}(2)[C_{6}]$
168.2.ba.a	$168$	$2$	168.ba	168.aa	$6$	$4$	$2$	$1.341$	$\Q(\sqrt{2}, \sqrt{-3})$	$_{}$	$\Q(\sqrt{-6}) $	None		✓			168.2.ba.a	$4$	$0$	$0$	$-6$	$6$	$-2$		$1$	$\mathrm{U}(1)[D_{6}]$	$q+\beta _{1}q^{2}+(-2-\beta _{2})q^{3}+2\beta _{2}q^{4}+\cdots$
168.2.ba.b	$168$	$2$	168.ba	168.aa	$6$	$4$	$2$	$1.341$	$\Q(\sqrt{2}, \sqrt{-3})$	$_{}$	$\Q(\sqrt{-6}) $	None		✓			168.2.ba.a	$4$	$0$	$0$	$6$	$-6$	$-2$		$1$	$\mathrm{U}(1)[D_{6}]$	$q+\beta _{1}q^{2}+(2+\beta _{2})q^{3}+2\beta _{2}q^{4}+(-1+\cdots)q^{5}+\cdots$
168.2.ba.c	$168$	$2$	168.ba	168.aa	$6$	$48$	$24$	$1.341$		$_{}$	None	None		✓	✓		168.2.ba.c	$8$	$0$	$0$	$0$	$0$	$-4$			$\mathrm{SU}(2)[C_{6}]$
168.2.bc.a	$168$	$2$	168.bc	56.p	$6$	$32$	$16$	$1.341$		$_{}$	None	None		✓	✓	✓	168.2.bc.a	$4$	$0$	$2$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{6}]$
168.3.d.a	$168$	$3$	168.d	3.b	$2$	$12$	$12$	$4.578$	$\mathbb{Q}[x]/(x^{12} - \cdots)$	$_{}$	None	None		✓	✓	✓	168.3.d.a	$2$	$0$	$0$	$-4$	$0$	$0$		$2^{10}\cdot 3^{2}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{3}q^{3}+(-\beta _{3}-\beta _{11})q^{5}+\beta _{1}q^{7}+\cdots$
168.3.e.a	$168$	$3$	168.e	168.e	$2$	$1$	$1$	$4.578$	$\Q$	$_{}$	$\Q(\sqrt{-42}) $	None	✓	✓			168.3.e.a	$2$	$0$	$-2$	$-3$	$0$	$-7$		$1$	$\mathrm{U}(1)[D_{2}]$	$q-2q^{2}-3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots$
168.3.e.b	$168$	$3$	168.e	168.e	$2$	$1$	$1$	$4.578$	$\Q$	$_{}$	$\Q(\sqrt{-42}) $	None	✓	✓			168.3.e.a	$2$	$0$	$-2$	$3$	$0$	$7$		$1$	$\mathrm{U}(1)[D_{2}]$	$q-2q^{2}+3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots$
168.3.e.c	$168$	$3$	168.e	168.e	$2$	$1$	$1$	$4.578$	$\Q$	$_{}$	$\Q(\sqrt{-42}) $	None	✓	✓			168.3.e.a	$2$	$0$	$2$	$-3$	$0$	$7$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+7q^{7}+\cdots$
168.3.e.d	$168$	$3$	168.e	168.e	$2$	$1$	$1$	$4.578$	$\Q$	$_{}$	$\Q(\sqrt{-42}) $	None	✓	✓			168.3.e.a	$2$	$0$	$2$	$3$	$0$	$-7$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+2q^{2}+3q^{3}+4q^{4}+6q^{6}-7q^{7}+\cdots$
168.3.e.e	$168$	$3$	168.e	168.e	$2$	$8$	$8$	$4.578$	8.0.$\cdots$.2	$_{}$	None	None		✓			168.3.e.e	$8$	$0$	$0$	$0$	$0$	$0$		$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	$q+(\beta _{1}+\beta _{3})q^{2}+(\beta _{2}+\beta _{6})q^{3}+(3-\beta _{7})q^{4}+\cdots$
168.3.e.f	$168$	$3$	168.e	168.e	$2$	$48$	$48$	$4.578$		$_{}$	None	None		✓	✓		168.3.e.f	$8$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{2}]$
168.3.f.a	$168$	$3$	168.f	7.b	$2$	$8$	$8$	$4.578$	8.0.$\cdots$.1	$_{}$	None	None		✓	✓	✓	168.3.f.a	$2$	$0$	$0$	$0$	$0$	$-4$		$2^{8}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{3}+(-\beta _{1}+\beta _{3})q^{5}+(-\beta _{1}+\beta _{3}+\cdots)q^{7}+\cdots$
168.3.g.a	$168$	$3$	168.g	8.d	$2$	$24$	$24$	$4.578$		$_{}$	None	None		✓	✓	✓	168.3.g.a	$2$	$0$	$-2$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{2}]$
168.3.l.a	$168$	$3$	168.l	56.h	$2$	$32$	$32$	$4.578$		$_{}$	None	None		✓	✓	✓	168.3.l.a	$4$	$0$	$-2$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{2}]$
168.3.n.a	$168$	$3$	168.n	24.h	$2$	$48$	$48$	$4.578$		$_{}$	None	None		✓	✓	✓	168.3.n.a	$4$	$0$	$0$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{2}]$
168.3.s.a	$168$	$3$	168.s	168.s	$6$	$4$	$2$	$4.578$	$\Q(\sqrt{2}, \sqrt{-3})$	$_{}$	$\Q(\sqrt{-6}) $	None		✓			168.3.s.a	$4$	$0$	$-4$	$6$	$-2$	$10$		$2^{2}\cdot 3$	$\mathrm{U}(1)[D_{6}]$	$q+(-2-2\beta _{1})q^{2}-3\beta _{1}q^{3}+4\beta _{1}q^{4}+\cdots$
168.3.s.b	$168$	$3$	168.s	168.s	$6$	$4$	$2$	$4.578$	$\Q(\sqrt{2}, \sqrt{-3})$	$_{}$	$\Q(\sqrt{-6}) $	None		✓			168.3.s.a	$4$	$0$	$4$	$-6$	$2$	$10$		$2^{2}\cdot 3$	$\mathrm{U}(1)[D_{6}]$	$q+(2+2\beta _{1})q^{2}+3\beta _{1}q^{3}+4\beta _{1}q^{4}+\cdots$
168.3.s.c	$168$	$3$	168.s	168.s	$6$	$112$	$56$	$4.578$		$_{}$	None	None		✓	✓		168.3.s.c	$8$	$0$	$0$	$0$	$0$	$-28$			$\mathrm{SU}(2)[C_{6}]$
168.3.x.a	$168$	$3$	168.x	56.j	$6$	$4$	$2$	$4.578$	$\Q(\zeta_{12})$	$_{}$	None	None		✓			168.3.x.a	$4$	$0$	$0$	$0$	$0$	$-26$		$1$	$\mathrm{SU}(2)[C_{6}]$	$q+2\zeta_{12}q^{2}+(\zeta_{12}+\zeta_{12}^{3})q^{3}+4\zeta_{12}^{2}q^{4}+\cdots$
168.3.x.b	$168$	$3$	168.x	56.j	$6$	$60$	$30$	$4.578$		$_{}$	None	None		✓	✓		168.3.x.b	$4$	$0$	$2$	$0$	$0$	$26$			$\mathrm{SU}(2)[C_{6}]$
168.3.y.a	$168$	$3$	168.y	56.k	$6$	$64$	$32$	$4.578$		$_{}$	None	None		✓	✓	✓	168.3.y.a	$4$	$0$	$-2$	$0$	$0$	$0$			$\mathrm{SU}(2)[C_{6}]$
168.3.z.a	$168$	$3$	168.z	7.d	$6$	$8$	$4$	$4.578$	8.0.$\cdots$.2	$_{}$	None	None		✓			168.3.z.a	$2$	$0$	$0$	$-12$	$6$	$-4$		$2^{6}$	$\mathrm{SU}(2)[C_{6}]$	$q+(-2+\beta _{4})q^{3}+(1+\beta _{6})q^{5}+(1-3\beta _{4}+\cdots)q^{7}+\cdots$
168.3.z.b	$168$	$3$	168.z	7.d	$6$	$8$	$4$	$4.578$	8.0.$\cdots$.9	$_{}$	None	None		✓			168.3.z.b	$2$	$0$	$0$	$12$	$-6$	$8$		$2^{4}\cdot 7$	$\mathrm{SU}(2)[C_{6}]$	$q+(1-\beta _{2})q^{3}+(-1+\beta _{7})q^{5}+(1-\beta _{2}+\cdots)q^{7}+\cdots$
168.3.be.a	$168$	$3$	168.be	168.ae	$6$	$120$	$60$	$4.578$		$_{}$	None	None		✓	✓	✓	168.3.be.a	$8$	$0$	$0$	$-6$	$0$	$0$			$\mathrm{SU}(2)[C_{6}]$
168.3.bf.a	$168$	$3$	168.bf	21.h	$6$	$32$	$16$	$4.578$		$_{}$	None	None		✓	✓	✓	168.3.bf.a	$4$	$0$	$0$	$0$	$0$	$12$			$\mathrm{SU}(2)[C_{6}]$
168.4.a.a	$168$	$4$	168.a	1.a	$1$	$1$	$1$	$9.912$	$\Q$	$_{}$	None	None	✓	✓	✓	✓	168.4.a.a	$1$	$0$	$0$	$-3$	$-10$	$7$	$+$	$1$	$\mathrm{SU}(2)$	$q-3q^{3}-10q^{5}+7q^{7}+9q^{9}-12q^{11}+\cdots$
168.4.a.b	$168$	$4$	168.a	1.a	$1$	$1$	$1$	$9.912$	$\Q$	$_{}$	None	None	✓	✓	✓	✓	168.4.a.b	$1$	$1$	$0$	$-3$	$-2$	$7$	$-$	$1$	$\mathrm{SU}(2)$	$q-3q^{3}-2q^{5}+7q^{7}+9q^{9}+12q^{11}+\cdots$

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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.