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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
																				$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
312.1.b.a	$312$	$1$	312.b	312.b	$2$	$4$	$4$	$0.156$	\(\Q(\zeta_{8})\)	$D_{4}$	\(\Q(\sqrt{-39}) \)	None		✓	✓	✓	312.1.b.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$1$		\(q-\zeta_{8}q^{2}-\zeta_{8}^{2}q^{3}+\zeta_{8}^{2}q^{4}+(-\zeta_{8}+\cdots)q^{5}+\cdots\)
312.2.a.a	$312$	$2$	312.a	1.a	$1$	$1$	$1$	$2.491$	\(\Q\)	$_{}$	None	None	✓	✓	✓	✓	312.2.a.a	$1$	$0$	\(0\)	\(-1\)	\(-2\)	\(4\)	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-2q^{5}+4q^{7}+q^{9}+q^{13}+2q^{15}+\cdots\)
312.2.a.b	$312$	$2$	312.a	1.a	$1$	$1$	$1$	$2.491$	\(\Q\)	$_{}$	None	None	✓	✓	✓	✓	312.2.a.b	$1$	$1$	\(0\)	\(-1\)	\(0\)	\(-4\)	$+$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}-4q^{7}+q^{9}-2q^{11}-q^{13}+\cdots\)
312.2.a.c	$312$	$2$	312.a	1.a	$1$	$1$	$1$	$2.491$	\(\Q\)	$_{}$	None	None	✓	✓	✓	✓	312.2.a.c	$1$	$0$	\(0\)	\(-1\)	\(4\)	\(0\)	$-$	$1$	$\mathrm{SU}(2)$	\(q-q^{3}+4q^{5}+q^{9}-2q^{11}-q^{13}+\cdots\)
312.2.a.d	$312$	$2$	312.a	1.a	$1$	$1$	$1$	$2.491$	\(\Q\)	$_{}$	None	None	✓	✓	✓	✓	312.2.a.d	$1$	$1$	\(0\)	\(1\)	\(-4\)	\(-4\)	$+$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}-4q^{5}-4q^{7}+q^{9}-2q^{11}+\cdots\)
312.2.a.e	$312$	$2$	312.a	1.a	$1$	$1$	$1$	$2.491$	\(\Q\)	$_{}$	None	None	✓	✓	✓	✓	312.2.a.e	$1$	$0$	\(0\)	\(1\)	\(0\)	\(0\)	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+q^{9}+6q^{11}-q^{13}+2q^{17}+\cdots\)
312.2.a.f	$312$	$2$	312.a	1.a	$1$	$1$	$1$	$2.491$	\(\Q\)	$_{}$	None	None	✓	✓	✓	✓	312.2.a.f	$1$	$0$	\(0\)	\(1\)	\(2\)	\(0\)	$-$	$1$	$\mathrm{SU}(2)$	\(q+q^{3}+2q^{5}+q^{9}+q^{13}+2q^{15}+\cdots\)
312.2.c.a	$312$	$2$	312.c	13.b	$2$	$2$	$2$	$2.491$	\(\Q(\sqrt{-1}) \)	$_{}$	None	None		✓			312.2.c.a	$2$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)		$2$	$\mathrm{SU}(2)[C_{2}]$	\(q-q^{3}+\beta q^{7}+q^{9}+\beta q^{11}+(\beta+3)q^{13}+\cdots\)
312.2.c.b	$312$	$2$	312.c	13.b	$2$	$2$	$2$	$2.491$	\(\Q(\sqrt{-1}) \)	$_{}$	None	None		✓			312.2.c.b	$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}+\beta q^{5}+\beta q^{7}+q^{9}+2\beta q^{11}+\cdots\)
312.2.c.c	$312$	$2$	312.c	13.b	$2$	$2$	$2$	$2.491$	\(\Q(\sqrt{-1}) \)	$_{}$	None	None		✓			312.2.c.c	$2$	$0$	\(0\)	\(2\)	\(0\)	\(0\)		$2$	$\mathrm{SU}(2)[C_{2}]$	\(q+q^{3}+2\beta q^{5}-\beta q^{7}+q^{9}+\beta q^{11}+\cdots\)
312.2.g.a	$312$	$2$	312.g	8.b	$2$	$8$	$8$	$2.491$	\(\Q(\zeta_{20})\)	$_{}$	None	None		✓			312.2.g.a	$2$	$0$	\(2\)	\(0\)	\(0\)	\(4\)		$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta_{7} q^{2}-\beta_{3} q^{3}+\beta_1 q^{4}+(\beta_{7}+\beta_{4}-\beta_{3}+\cdots-1)q^{5}+\cdots\)
312.2.g.b	$312$	$2$	312.g	8.b	$2$	$16$	$16$	$2.491$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	$_{}$	None	None		✓	✓		312.2.g.b	$2$	$0$	\(2\)	\(0\)	\(0\)	\(4\)		$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	\(q+\beta _{10}q^{2}+\beta _{9}q^{3}-\beta _{2}q^{4}+\beta _{14}q^{5}+\cdots\)
312.2.h.a	$312$	$2$	312.h	312.h	$2$	$8$	$8$	$2.491$	8.0.\(\cdots\).21	$_{}$	\(\Q(\sqrt{-39}) \)	None		✓			312.2.h.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$2$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{1}q^{2}+\beta _{3}q^{3}+\beta _{2}q^{4}+(-\beta _{4}-\beta _{7})q^{5}+\cdots\)
312.2.h.b	$312$	$2$	312.h	312.h	$2$	$12$	$12$	$2.491$	\(\mathbb{Q}[x]/(x^{12} - \cdots)\)	$_{}$	\(\Q(\sqrt{-26}) \)	None		✓			312.2.h.b	$8$	$0$	\(0\)	\(0\)	\(0\)	\(0\)		$2^{6}\cdot 3^{2}$	$\mathrm{U}(1)[D_{2}]$	\(q+\beta _{3}q^{2}+\beta _{7}q^{3}-2q^{4}-\beta _{9}q^{5}-\beta _{4}q^{6}+\cdots\)
312.2.h.c	$312$	$2$	312.h	312.h	$2$	$32$	$32$	$2.491$		$_{}$	None	None		✓	✓		312.2.h.c	$8$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)			$\mathrm{SU}(2)[C_{2}]$
312.2.j.a	$312$	$2$	312.j	24.f	$2$	$48$	$48$	$2.491$		$_{}$	None	None		✓	✓	✓	312.2.j.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)			$\mathrm{SU}(2)[C_{2}]$
312.2.m.a	$312$	$2$	312.m	104.e	$2$	$2$	$2$	$2.491$	\(\Q(\sqrt{-1}) \)	$_{}$	None	None		✓			312.2.m.a	$2$	$1$	\(-2\)	\(0\)	\(-4\)	\(0\)		$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(i-1)q^{2}-i q^{3}-2 i q^{4}-2 q^{5}+\cdots\)
312.2.m.b	$312$	$2$	312.m	104.e	$2$	$2$	$2$	$2.491$	\(\Q(\sqrt{-1}) \)	$_{}$	None	None		✓			312.2.m.a	$2$	$0$	\(2\)	\(0\)	\(4\)	\(0\)		$1$	$\mathrm{SU}(2)[C_{2}]$	\(q+(i+1)q^{2}+i q^{3}+2 i q^{4}+2 q^{5}+\cdots\)
312.2.m.c	$312$	$2$	312.m	104.e	$2$	$24$	$24$	$2.491$		$_{}$	None	None		✓	✓		312.2.m.c	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)			$\mathrm{SU}(2)[C_{2}]$
312.2.q.a	$312$	$2$	312.q	13.c	$3$	$2$	$1$	$2.491$	\(\Q(\sqrt{-3}) \)	$_{}$	None	None		✓			312.2.q.a	$2$	$0$	\(0\)	\(-1\)	\(6\)	\(0\)		$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\zeta_{6})q^{3}+3q^{5}-\zeta_{6}q^{9}+(3+\cdots)q^{13}+\cdots\)
312.2.q.b	$312$	$2$	312.q	13.c	$3$	$2$	$1$	$2.491$	\(\Q(\sqrt{-3}) \)	$_{}$	None	None		✓			312.2.q.b	$2$	$0$	\(0\)	\(1\)	\(-4\)	\(-1\)		$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}-2q^{5}-\zeta_{6}q^{7}-\zeta_{6}q^{9}+\cdots\)
312.2.q.c	$312$	$2$	312.q	13.c	$3$	$2$	$1$	$2.491$	\(\Q(\sqrt{-3}) \)	$_{}$	None	None		✓			312.2.q.c	$2$	$0$	\(0\)	\(1\)	\(6\)	\(4\)		$1$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1-\zeta_{6})q^{3}+3q^{5}+4\zeta_{6}q^{7}-\zeta_{6}q^{9}+\cdots\)
312.2.q.d	$312$	$2$	312.q	13.c	$3$	$4$	$2$	$2.491$	\(\Q(\sqrt{-3}, \sqrt{13})\)	$_{}$	None	None		✓			312.2.q.d	$2$	$0$	\(0\)	\(-2\)	\(-4\)	\(-2\)		$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(-1+\beta _{1})q^{3}-q^{5}+(-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
312.2.q.e	$312$	$2$	312.q	13.c	$3$	$6$	$3$	$2.491$	6.0.2101707.2	$_{}$	None	None		✓	✓		312.2.q.e	$2$	$0$	\(0\)	\(3\)	\(0\)	\(-3\)		$2^{2}$	$\mathrm{SU}(2)[C_{3}]$	\(q+(1+\beta _{3})q^{3}+\beta _{2}q^{5}+(-2-\beta _{4}+2\beta _{5})q^{7}+\cdots\)
312.2.t.a	$312$	$2$	312.t	104.m	$4$	$2$	$1$	$2.491$	\(\Q(\sqrt{-1}) \)	$_{}$	None	None		✓			312.2.t.a	$2$	$1$	\(-2\)	\(-2\)	\(-4\)	\(2\)		$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(i-1)q^{2}-q^{3}-2 i q^{4}+(2 i-2)q^{5}+\cdots\)
312.2.t.b	$312$	$2$	312.t	104.m	$4$	$2$	$1$	$2.491$	\(\Q(\sqrt{-1}) \)	$_{}$	None	None		✓			312.2.t.b	$2$	$1$	\(-2\)	\(2\)	\(-4\)	\(-6\)		$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i-1)q^{2}+q^{3}+2 i q^{4}+(2 i-2)q^{5}+\cdots\)
312.2.t.c	$312$	$2$	312.t	104.m	$4$	$2$	$1$	$2.491$	\(\Q(\sqrt{-1}) \)	$_{}$	None	None		✓			312.2.t.b	$2$	$0$	\(-2\)	\(2\)	\(4\)	\(6\)		$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i-1)q^{2}+q^{3}+2 i q^{4}+(-2 i+2)q^{5}+\cdots\)
312.2.t.d	$312$	$2$	312.t	104.m	$4$	$2$	$1$	$2.491$	\(\Q(\sqrt{-1}) \)	$_{}$	None	None		✓			312.2.t.a	$2$	$0$	\(2\)	\(-2\)	\(4\)	\(-2\)		$1$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-i+1)q^{2}-q^{3}-2 i q^{4}+(-2 i+2)q^{5}+\cdots\)
312.2.t.e	$312$	$2$	312.t	104.m	$4$	$24$	$12$	$2.491$		$_{}$	None	None		✓			312.2.t.e	$4$	$0$	\(0\)	\(-24\)	\(0\)	\(0\)			$\mathrm{SU}(2)[C_{4}]$
312.2.t.f	$312$	$2$	312.t	104.m	$4$	$24$	$12$	$2.491$		$_{}$	None	None		✓			312.2.t.f	$4$	$0$	\(4\)	\(24\)	\(0\)	\(0\)			$\mathrm{SU}(2)[C_{4}]$
312.2.x.a	$312$	$2$	312.x	39.f	$4$	$4$	$2$	$2.491$	\(\Q(\zeta_{8})\)	$_{}$	None	None		✓			312.2.x.a	$4$	$0$	\(0\)	\(4\)	\(0\)	\(12\)		$2$	$\mathrm{SU}(2)[C_{4}]$	\(q+(\beta_{2}+1)q^{3}+(-\beta_{3}-\beta_{2})q^{5}+(3\beta_1+3)q^{7}+\cdots\)
312.2.x.b	$312$	$2$	312.x	39.f	$4$	$8$	$4$	$2.491$	\(\Q(\zeta_{24})\)	$_{}$	None	None		✓			312.2.x.b	$4$	$0$	\(0\)	\(-8\)	\(0\)	\(8\)		$2^{9}$	$\mathrm{SU}(2)[C_{4}]$	\(q+(-\beta_{2}-1)q^{3}-\beta_{7} q^{5}+(\beta_1+1)q^{7}+\cdots\)
312.2.x.c	$312$	$2$	312.x	39.f	$4$	$16$	$8$	$2.491$	\(\mathbb{Q}[x]/(x^{16} - \cdots)\)	$_{}$	None	None		✓	✓		312.2.x.c	$4$	$0$	\(0\)	\(4\)	\(0\)	\(-24\)		$2^{8}$	$\mathrm{SU}(2)[C_{4}]$	\(q-\beta _{12}q^{3}+(\beta _{6}-\beta _{7}-\beta _{13}+\beta _{14}+\cdots)q^{5}+\cdots\)
312.2.y.a	$312$	$2$	312.y	312.y	$4$	$104$	$52$	$2.491$		$_{}$	None	None		✓	✓	✓	312.2.y.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(-8\)			$\mathrm{SU}(2)[C_{4}]$
312.2.ba.a	$312$	$2$	312.ba	312.aa	$6$	$104$	$52$	$2.491$		$_{}$	None	None		✓	✓	✓	312.2.ba.a	$8$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)			$\mathrm{SU}(2)[C_{6}]$
312.2.bb.a	$312$	$2$	312.bb	104.r	$6$	$56$	$28$	$2.491$		$_{}$	None	None		✓	✓	✓	312.2.bb.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(0\)			$\mathrm{SU}(2)[C_{6}]$
312.2.bf.a	$312$	$2$	312.bf	13.e	$6$	$4$	$2$	$2.491$	\(\Q(\zeta_{12})\)	$_{}$	None	None		✓			312.2.bf.a	$2$	$0$	\(0\)	\(2\)	\(0\)	\(6\)		$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(1-\zeta_{12}^{2})q^{3}+(1-2\zeta_{12}^{2})q^{5}+(2+\cdots)q^{7}+\cdots\)
312.2.bf.b	$312$	$2$	312.bf	13.e	$6$	$8$	$4$	$2.491$	8.0.649638144.4	$_{}$	None	None		✓	✓		312.2.bf.b	$2$	$0$	\(0\)	\(-4\)	\(0\)	\(0\)		$2^{4}$	$\mathrm{SU}(2)[C_{6}]$	\(q+\beta _{2}q^{3}+(-\beta _{2}-\beta _{6})q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
312.2.bk.a	$312$	$2$	312.bk	104.s	$6$	$8$	$4$	$2.491$	8.0.12960000.1	$_{}$	None	None		✓			312.2.bk.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(12\)		$1$	$\mathrm{SU}(2)[C_{6}]$	\(q+(\beta _{6}+\beta _{7})q^{2}+(\beta _{3}+\beta _{7})q^{3}+(-1+\cdots)q^{4}+\cdots\)
312.2.bk.b	$312$	$2$	312.bk	104.s	$6$	$48$	$24$	$2.491$		$_{}$	None	None		✓	✓		312.2.bk.b	$4$	$0$	\(0\)	\(0\)	\(0\)	\(-12\)			$\mathrm{SU}(2)[C_{6}]$
312.2.bn.a	$312$	$2$	312.bn	312.an	$6$	$104$	$52$	$2.491$		$_{}$	None	None		✓	✓	✓	312.2.bn.a	$8$	$0$	\(0\)	\(-2\)	\(0\)	\(0\)			$\mathrm{SU}(2)[C_{6}]$
312.2.bo.a	$312$	$2$	312.bo	312.ao	$12$	$208$	$52$	$2.491$		$_{}$	None	None		✓	✓	✓	312.2.bo.a	$8$	$0$	\(0\)	\(0\)	\(0\)	\(-16\)			$\mathrm{SU}(2)[C_{12}]$
312.2.bp.a	$312$	$2$	312.bp	39.k	$12$	$56$	$14$	$2.491$		$_{}$	None	None		✓	✓	✓	312.2.bp.a	$4$	$0$	\(0\)	\(0\)	\(0\)	\(4\)			$\mathrm{SU}(2)[C_{12}]$
312.2.bt.a	$312$	$2$	312.bt	104.u	$12$	$4$	$1$	$2.491$	\(\Q(\zeta_{12})\)	$_{}$	None	None		✓			312.2.bt.a	$2$	$1$	\(-4\)	\(-2\)	\(-2\)	\(-6\)		$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(-1+\zeta_{12}^{3})q^{2}-\zeta_{12}^{2}q^{3}-2\zeta_{12}^{3}q^{4}+\cdots\)
312.2.bt.b	$312$	$2$	312.bt	104.u	$12$	$4$	$1$	$2.491$	\(\Q(\zeta_{12})\)	$_{}$	None	None		✓			312.2.bt.a	$2$	$0$	\(2\)	\(-2\)	\(2\)	\(6\)		$1$	$\mathrm{SU}(2)[C_{12}]$	\(q+(1-\zeta_{12}-\zeta_{12}^{2})q^{2}-\zeta_{12}^{2}q^{3}+\cdots\)
312.2.bt.c	$312$	$2$	312.bt	104.u	$12$	$48$	$12$	$2.491$		$_{}$	None	None		✓			312.2.bt.c	$4$	$0$	\(2\)	\(-24\)	\(0\)	\(0\)			$\mathrm{SU}(2)[C_{12}]$
312.2.bt.d	$312$	$2$	312.bt	104.u	$12$	$56$	$14$	$2.491$		$_{}$	None	None		✓	✓		312.2.bt.d	$4$	$0$	\(0\)	\(28\)	\(0\)	\(0\)			$\mathrm{SU}(2)[C_{12}]$
312.3.b.a	$312$	$3$	312.b	312.b	$2$	$1$	$1$	$8.501$	\(\Q\)	$_{}$	\(\Q(\sqrt{-78}) \)	None	✓	✓			312.3.b.a	$2$	$0$	\(-2\)	\(-3\)	\(0\)	\(0\)		$1$	$\mathrm{U}(1)[D_{2}]$	\(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}-8q^{8}+\cdots\)
312.3.b.b	$312$	$3$	312.b	312.b	$2$	$1$	$1$	$8.501$	\(\Q\)	$_{}$	\(\Q(\sqrt{-78}) \)	None	✓	✓			312.3.b.a	$2$	$0$	\(-2\)	\(3\)	\(0\)	\(0\)		$1$	$\mathrm{U}(1)[D_{2}]$	\(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}-8q^{8}+\cdots\)
312.3.b.c	$312$	$3$	312.b	312.b	$2$	$1$	$1$	$8.501$	\(\Q\)	$_{}$	\(\Q(\sqrt{-78}) \)	None	✓	✓			312.3.b.a	$2$	$0$	\(2\)	\(-3\)	\(0\)	\(0\)		$1$	$\mathrm{U}(1)[D_{2}]$	\(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+8q^{8}+\cdots\)

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