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