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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
																				$a_{2}$	$a_{3}$	$a_{5}$	$a_{7}$
1568.1.c.a	$1568$	$1$	1568.c	7.b	$2$	$4$	$4$	$0.783$	4.0.2048.2	$D_{8}$	$\Q(\sqrt{-1})$	None		✓	✓	✓	1568.1.c.a	$4$	$0$	$0$	$0$	$0$	$0$		$1$		$q-\beta _{1}q^{5}+q^{9}+\beta _{3}q^{13}-\beta _{3}q^{17}+\cdots$
1568.1.d.a	$1568$	$1$	1568.d	4.b	$2$	$2$	$2$	$0.783$	$\Q(\sqrt{-1})$	$A_{4}$	None	None					224.1.r.a	$2$	$0$	$0$	$0$	$-2$	$0$		$1$		$q-i q^{3}-q^{5}-i q^{11}+i q^{15}-q^{17}+\cdots$
1568.1.d.b	$1568$	$1$	1568.d	4.b	$2$	$2$	$2$	$0.783$	$\Q(\sqrt{-1})$	$A_{4}$	None	None					224.1.r.a	$2$	$0$	$0$	$0$	$2$	$0$		$1$		$q-i q^{3}+q^{5}+i q^{11}-i q^{15}+q^{17}+\cdots$
1568.1.g.a	$1568$	$1$	1568.g	8.d	$2$	$1$	$1$	$0.783$	$\Q$	$D_{2}$	$\Q(\sqrt{-7})$, $\Q(\sqrt{-2})$	$\Q(\sqrt{14})$	✓				392.1.g.a	$4$	$0$	$0$	$0$	$0$	$0$		$1$		$q-q^{9}+2q^{11}+q^{25}+2q^{43}-2q^{67}+\cdots$
1568.1.g.b	$1568$	$1$	1568.g	8.d	$2$	$2$	$2$	$0.783$	$\Q(\sqrt{2})$	$D_{4}$	$\Q(\sqrt{-2})$	None	✓		✓		392.1.g.b	$4$	$0$	$0$	$0$	$0$	$0$		$1$		$q-\beta q^{3}+q^{9}-\beta q^{17}+\beta q^{19}+q^{25}+\cdots$
1568.1.n.a	$1568$	$1$	1568.n	56.j	$6$	$2$	$1$	$0.783$	$\Q(\sqrt{-3})$	$D_{2}$	$\Q(\sqrt{-7})$, $\Q(\sqrt{-14})$	$\Q(\sqrt{2})$			✓	✓	56.1.h.a	$8$	$0$	$0$	$0$	$0$	$0$		$1$		$q-\zeta_{6}^{2}q^{9}-\zeta_{6}^{2}q^{23}+\zeta_{6}q^{25}+q^{71}+\cdots$
1568.1.o.a	$1568$	$1$	1568.o	56.k	$6$	$2$	$1$	$0.783$	$\Q(\sqrt{-3})$	$D_{2}$	$\Q(\sqrt{-7})$, $\Q(\sqrt{-2})$	$\Q(\sqrt{14})$					392.1.g.a	$8$	$0$	$0$	$0$	$0$	$0$		$1$		$q-\zeta_{6}^{2}q^{9}-\zeta_{6}q^{11}-\zeta_{6}q^{25}+q^{43}+\cdots$
1568.1.o.b	$1568$	$1$	1568.o	56.k	$6$	$4$	$2$	$0.783$	$\Q(\sqrt{2}, \sqrt{-3})$	$D_{4}$	$\Q(\sqrt{-2})$	None			✓		392.1.g.b	$8$	$0$	$0$	$0$	$0$	$0$		$1$		$q+(-\beta _{1}-\beta _{3})q^{3}+(-1-\beta _{2})q^{9}+(-\beta _{1}+\cdots)q^{17}+\cdots$
1568.1.r.a	$1568$	$1$	1568.r	28.g	$6$	$4$	$2$	$0.783$	$\Q(\zeta_{12})$	$A_{4}$	None	None			✓	✓	224.1.r.a	$4$	$0$	$0$	$0$	$2$	$0$		$1$		$q-\zeta_{12}q^{3}+\zeta_{12}^{2}q^{5}-\zeta_{12}q^{11}-\zeta_{12}^{3}q^{15}+\cdots$
1568.1.s.a	$1568$	$1$	1568.s	7.d	$6$	$8$	$4$	$0.783$	8.0.339738624.1	$D_{8}$	$\Q(\sqrt{-1})$	None		✓	✓	✓	1568.1.c.a	$8$	$0$	$0$	$0$	$0$	$0$		$1$		$q+(\beta _{3}+\beta _{7})q^{5}+\beta _{4}q^{9}+(\beta _{1}-\beta _{5})q^{13}+\cdots$
1568.1.x.a	$1568$	$1$	1568.x	32.h	$8$	$4$	$1$	$0.783$	$\Q(\zeta_{8})$	$D_{8}$	$\Q(\sqrt{-7})$	None		✓	✓	✓	1568.1.x.a	$4$	$0$	$0$	$0$	$0$	$0$		$1$		$q-\zeta_{8}^{3}q^{2}-\zeta_{8}^{2}q^{4}-\zeta_{8}q^{8}+\zeta_{8}q^{9}+\cdots$
1568.1.bl.a	$1568$	$1$	1568.bl	224.ac	$24$	$8$	$1$	$0.783$	$\Q(\zeta_{24})$	$D_{8}$	$\Q(\sqrt{-7})$	None			✓	✓	224.1.v.a	$8$	$0$	$0$	$0$	$0$	$0$		$1$		$q-\zeta_{24}^{5}q^{2}+\zeta_{24}^{10}q^{4}+\zeta_{24}^{3}q^{8}+\cdots$
1568.1.bo.a	$1568$	$1$	1568.bo	224.af	$24$	$8$	$1$	$0.783$	$\Q(\zeta_{24})$	$D_{8}$	$\Q(\sqrt{-7})$	None		✓	✓	✓	1568.1.x.a	$8$	$0$	$0$	$0$	$0$	$0$		$1$		$q+\zeta_{24}^{7}q^{2}-\zeta_{24}^{2}q^{4}-\zeta_{24}^{9}q^{8}+\cdots$
1568.2.a.a	$1568$	$2$	1568.a	1.a	$1$	$1$	$1$	$12.521$	$\Q$	$_{}$	None	None	✓	✓			1568.2.a.a	$1$	$0$	$0$	$-2$	$-2$	$0$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{3}-2q^{5}+q^{9}+4q^{11}-6q^{13}+\cdots$
1568.2.a.b	$1568$	$2$	1568.a	1.a	$1$	$1$	$1$	$12.521$	$\Q$	$_{}$	None	None	✓				224.2.a.a	$1$	$0$	$0$	$-2$	$0$	$0$	$-$	$1$	$\mathrm{SU}(2)$	$q-2q^{3}+q^{9}+4q^{11}+4q^{13}+2q^{17}+\cdots$
1568.2.a.c	$1568$	$2$	1568.a	1.a	$1$	$1$	$1$	$12.521$	$\Q$	$_{}$	None	None	✓	✓			1568.2.a.a	$1$	$1$	$0$	$-2$	$2$	$0$	$+$	$1$	$\mathrm{SU}(2)$	$q-2q^{3}+2q^{5}+q^{9}-4q^{11}+6q^{13}+\cdots$
1568.2.a.d	$1568$	$2$	1568.a	1.a	$1$	$1$	$1$	$12.521$	$\Q$	$_{}$	$\Q(\sqrt{-1})$	None	✓	✓			1568.2.a.d	$2$	$0$	$0$	$0$	$-4$	$0$	$-$	$1$	$N(\mathrm{U}(1))$	$q-4q^{5}-3q^{9}+4q^{13}-8q^{17}+11q^{25}+\cdots$
1568.2.a.e	$1568$	$2$	1568.a	1.a	$1$	$1$	$1$	$12.521$	$\Q$	$_{}$	$\Q(\sqrt{-1})$	None	✓				32.2.a.a	$2$	$1$	$0$	$0$	$2$	$0$	$+$	$1$	$N(\mathrm{U}(1))$	$q+2q^{5}-3q^{9}-6q^{13}-2q^{17}-q^{25}+\cdots$
1568.2.a.f	$1568$	$2$	1568.a	1.a	$1$	$1$	$1$	$12.521$	$\Q$	$_{}$	$\Q(\sqrt{-1})$	None	✓	✓			1568.2.a.d	$2$	$0$	$0$	$0$	$4$	$0$	$-$	$1$	$N(\mathrm{U}(1))$	$q+4q^{5}-3q^{9}-4q^{13}+8q^{17}+11q^{25}+\cdots$
1568.2.a.g	$1568$	$2$	1568.a	1.a	$1$	$1$	$1$	$12.521$	$\Q$	$_{}$	None	None	✓	✓			1568.2.a.a	$1$	$1$	$0$	$2$	$-2$	$0$	$+$	$1$	$\mathrm{SU}(2)$	$q+2q^{3}-2q^{5}+q^{9}-4q^{11}-6q^{13}+\cdots$
1568.2.a.h	$1568$	$2$	1568.a	1.a	$1$	$1$	$1$	$12.521$	$\Q$	$_{}$	None	None	✓				224.2.a.a	$1$	$0$	$0$	$2$	$0$	$0$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{3}+q^{9}-4q^{11}+4q^{13}+2q^{17}+\cdots$
1568.2.a.i	$1568$	$2$	1568.a	1.a	$1$	$1$	$1$	$12.521$	$\Q$	$_{}$	None	None	✓	✓			1568.2.a.a	$1$	$0$	$0$	$2$	$2$	$0$	$-$	$1$	$\mathrm{SU}(2)$	$q+2q^{3}+2q^{5}+q^{9}+4q^{11}+6q^{13}+\cdots$
1568.2.a.j	$1568$	$2$	1568.a	1.a	$1$	$2$	$2$	$12.521$	$\Q(\sqrt{2})$	$_{}$	None	None	✓				224.2.i.a	$1$	$1$	$0$	$-2$	$-2$	$0$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta )q^{3}+(-1+2\beta )q^{5}-2\beta q^{9}+\cdots$
1568.2.a.k	$1568$	$2$	1568.a	1.a	$1$	$2$	$2$	$12.521$	$\Q(\sqrt{5})$	$_{}$	None	None	✓				224.2.a.c	$1$	$1$	$0$	$-2$	$-2$	$0$	$+$	$2$	$\mathrm{SU}(2)$	$q+(-1-\beta )q^{3}+(-1+\beta )q^{5}+(3+2\beta )q^{9}+\cdots$
1568.2.a.l	$1568$	$2$	1568.a	1.a	$1$	$2$	$2$	$12.521$	$\Q(\sqrt{2})$	$_{}$	None	None	✓				224.2.i.a	$1$	$1$	$0$	$-2$	$2$	$0$	$+$	$1$	$\mathrm{SU}(2)$	$q+(-1+\beta )q^{3}+(1-2\beta )q^{5}-2\beta q^{9}+\cdots$
1568.2.a.m	$1568$	$2$	1568.a	1.a	$1$	$2$	$2$	$12.521$	$\Q(\sqrt{7})$	$_{}$	None	None	✓				224.2.i.c	$2$	$1$	$0$	$0$	$-6$	$0$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}-3q^{5}+4q^{9}-\beta q^{11}-4q^{13}+\cdots$
1568.2.a.n	$1568$	$2$	1568.a	1.a	$1$	$2$	$2$	$12.521$	$\Q(\sqrt{3})$	$_{}$	None	None	✓				224.2.i.b	$2$	$1$	$0$	$0$	$-2$	$0$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}-q^{5}-3\beta q^{11}-\beta q^{15}-5q^{17}+\cdots$
1568.2.a.o	$1568$	$2$	1568.a	1.a	$1$	$2$	$2$	$12.521$	$\Q(\sqrt{2})$	$_{}$	$\Q(\sqrt{-1})$	None	✓	✓			1568.2.a.o	$4$	$1$	$0$	$0$	$0$	$0$	$+$	$1$	$N(\mathrm{U}(1))$	$q+\beta q^{5}-3q^{9}-\beta q^{13}-5\beta q^{17}-3q^{25}+\cdots$
1568.2.a.p	$1568$	$2$	1568.a	1.a	$1$	$2$	$2$	$12.521$	$\Q(\sqrt{2})$	$_{}$	$\Q(\sqrt{-1})$	None	✓	✓			1568.2.a.p	$4$	$0$	$0$	$0$	$0$	$0$	$-$	$1$	$N(\mathrm{U}(1))$	$q+3\beta q^{5}-3q^{9}+5\beta q^{13}-3\beta q^{17}+\cdots$
1568.2.a.q	$1568$	$2$	1568.a	1.a	$1$	$2$	$2$	$12.521$	$\Q(\sqrt{2})$	$_{}$	None	None	✓	✓			1568.2.a.q	$2$	$1$	$0$	$0$	$0$	$0$	$+$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}-q^{9}-2q^{11}-2\beta q^{13}-3\beta q^{17}+\cdots$
1568.2.a.r	$1568$	$2$	1568.a	1.a	$1$	$2$	$2$	$12.521$	$\Q(\sqrt{2})$	$_{}$	None	None	✓	✓			1568.2.a.q	$2$	$0$	$0$	$0$	$0$	$0$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}-q^{9}+2q^{11}+2\beta q^{13}+3\beta q^{17}+\cdots$
1568.2.a.s	$1568$	$2$	1568.a	1.a	$1$	$2$	$2$	$12.521$	$\Q(\sqrt{3})$	$_{}$	None	None	✓				224.2.i.b	$2$	$0$	$0$	$0$	$2$	$0$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}+q^{5}+3\beta q^{11}+\beta q^{15}+5q^{17}+\cdots$
1568.2.a.t	$1568$	$2$	1568.a	1.a	$1$	$2$	$2$	$12.521$	$\Q(\sqrt{7})$	$_{}$	None	None	✓				224.2.i.c	$2$	$0$	$0$	$0$	$6$	$0$	$-$	$1$	$\mathrm{SU}(2)$	$q+\beta q^{3}+3q^{5}+4q^{9}+\beta q^{11}+4q^{13}+\cdots$
1568.2.a.u	$1568$	$2$	1568.a	1.a	$1$	$2$	$2$	$12.521$	$\Q(\sqrt{2})$	$_{}$	None	None	✓				224.2.i.a	$1$	$0$	$0$	$2$	$-2$	$0$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{3}+(-1-2\beta )q^{5}+2\beta q^{9}+\cdots$
1568.2.a.v	$1568$	$2$	1568.a	1.a	$1$	$2$	$2$	$12.521$	$\Q(\sqrt{5})$	$_{}$	None	None	✓				224.2.a.c	$1$	$0$	$0$	$2$	$-2$	$0$	$-$	$2$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{3}+(-1+\beta )q^{5}+(3+2\beta )q^{9}+\cdots$
1568.2.a.w	$1568$	$2$	1568.a	1.a	$1$	$2$	$2$	$12.521$	$\Q(\sqrt{2})$	$_{}$	None	None	✓				224.2.i.a	$1$	$0$	$0$	$2$	$2$	$0$	$-$	$1$	$\mathrm{SU}(2)$	$q+(1+\beta )q^{3}+(1+2\beta )q^{5}+2\beta q^{9}+(-1+\cdots)q^{11}+\cdots$
1568.2.a.x	$1568$	$2$	1568.a	1.a	$1$	$4$	$4$	$12.521$	$\Q(\sqrt{2}, \sqrt{5})$	$_{}$	None	None	✓	✓	✓		1568.2.a.x	$4$	$0$	$0$	$0$	$0$	$0$	$-$	$2^{3}$	$\mathrm{SU}(2)$	$q-\beta _{2}q^{3}-2\beta _{1}q^{5}+7q^{9}-\beta _{3}q^{11}+\cdots$
1568.2.b.a	$1568$	$2$	1568.b	8.b	$2$	$2$	$2$	$12.521$	$\Q(\sqrt{-2})$	$_{}$	None	None					56.2.b.a	$2$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta q^{3}+\beta q^{5}+q^{9}-2\beta q^{11}+3\beta q^{13}+\cdots$
1568.2.b.b	$1568$	$2$	1568.b	8.b	$2$	$2$	$2$	$12.521$	$\Q(\sqrt{-7})$	$_{}$	$\Q(\sqrt{-7})$	None					392.2.b.a	$4$	$0$	$0$	$0$	$0$	$0$		$2^{2}$	$\mathrm{U}(1)[D_{2}]$	$q+3q^{9}-\beta q^{11}-8q^{23}+5q^{25}-2\beta q^{29}+\cdots$
1568.2.b.c	$1568$	$2$	1568.b	8.b	$2$	$4$	$4$	$12.521$	4.0.7168.1	$_{}$	$\Q(\sqrt{-14})$	None					392.2.b.d	$4$	$0$	$0$	$0$	$0$	$0$		$2^{2}$	$\mathrm{U}(1)[D_{2}]$	$q+\beta _{1}q^{3}+\beta _{2}q^{5}+(-3+\beta _{3})q^{9}+(-2\beta _{1}+\cdots)q^{13}+\cdots$
1568.2.b.d	$1568$	$2$	1568.b	8.b	$2$	$4$	$4$	$12.521$	4.0.2312.1	$_{}$	None	None					56.2.b.b	$2$	$0$	$0$	$0$	$0$	$0$		$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{3}+\beta _{2}q^{5}+(-2+\beta _{3})q^{9}+(\beta _{1}+\cdots)q^{11}+\cdots$
1568.2.b.e	$1568$	$2$	1568.b	8.b	$2$	$6$	$6$	$12.521$	6.0.1142512.1	$_{}$	None	None					56.2.p.a	$2$	$0$	$0$	$0$	$0$	$0$		$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{3}q^{3}+(-\beta _{1}-\beta _{3})q^{5}+(-\beta _{2}+\beta _{5})q^{9}+\cdots$
1568.2.b.f	$1568$	$2$	1568.b	8.b	$2$	$6$	$6$	$12.521$	6.0.1142512.1	$_{}$	None	None					56.2.p.a	$2$	$0$	$0$	$0$	$0$	$0$		$2^{3}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{3}q^{3}+(-\beta _{1}-\beta _{3})q^{5}+(-\beta _{2}+\beta _{5})q^{9}+\cdots$
1568.2.b.g	$1568$	$2$	1568.b	8.b	$2$	$12$	$12$	$12.521$	12.0.$\cdots$.1	$_{}$	None	None			✓		392.2.b.g	$4$	$0$	$0$	$0$	$0$	$0$		$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{9}q^{3}+\beta _{11}q^{5}+(-1-\beta _{4})q^{9}+\cdots$
1568.2.e.a	$1568$	$2$	1568.e	56.e	$2$	$4$	$4$	$12.521$	4.0.2048.2	$_{}$	$\Q(\sqrt{-2})$	None					392.2.e.a	$4$	$0$	$0$	$0$	$0$	$0$		$7$	$\mathrm{U}(1)[D_{2}]$	$q+\beta _{1}q^{3}+(-7-\beta _{2})q^{9}+(2-3\beta _{2}+\cdots)q^{11}+\cdots$
1568.2.e.b	$1568$	$2$	1568.e	56.e	$2$	$4$	$4$	$12.521$	4.0.2048.2	$_{}$	$\Q(\sqrt{-2})$	None					392.2.e.b	$4$	$0$	$0$	$0$	$0$	$0$		$1$	$\mathrm{U}(1)[D_{2}]$	$q+\beta _{1}q^{3}+(1+\beta _{2})q^{9}+(-2+3\beta _{2}+\cdots)q^{11}+\cdots$
1568.2.e.c	$1568$	$2$	1568.e	56.e	$2$	$8$	$8$	$12.521$	8.0.339738624.1	$_{}$	None	None					392.2.e.c	$4$	$0$	$0$	$0$	$0$	$0$		$2^{5}$	$\mathrm{SU}(2)[C_{2}]$	$q+(\beta _{2}-\beta _{4})q^{3}+\beta _{3}q^{5}+(-1+2\beta _{1}+\cdots)q^{9}+\cdots$
1568.2.e.d	$1568$	$2$	1568.e	56.e	$2$	$8$	$8$	$12.521$	8.0.$\cdots$.10	$_{}$	None	None					392.2.e.d	$4$	$0$	$0$	$0$	$0$	$0$		$2^{7}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta _{1}q^{3}+\beta _{6}q^{5}+(1-\beta _{3})q^{9}+3\beta _{3}q^{11}+\cdots$
1568.2.e.e	$1568$	$2$	1568.e	56.e	$2$	$12$	$12$	$12.521$	12.0.$\cdots$.2	$_{}$	None	None			✓		56.2.m.a	$4$	$0$	$0$	$0$	$0$	$0$		$2^{12}$	$\mathrm{SU}(2)[C_{2}]$	$q-\beta _{9}q^{3}+\beta _{3}q^{5}-\beta _{2}q^{9}+(-1+\beta _{1}+\cdots)q^{11}+\cdots$
1568.2.f.a	$1568$	$2$	1568.f	28.d	$2$	$8$	$8$	$12.521$	$\Q(\zeta_{16})$	$_{}$	None	None		✓			1568.2.f.a	$4$	$0$	$0$	$0$	$0$	$0$		$2^{4}$	$\mathrm{SU}(2)[C_{2}]$	$q+\beta_{7} q^{3}+(\beta_{4}-\beta_{2})q^{5}+(-\beta_{6}-1)q^{9}+\cdots$

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