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Label Char Prim Dim $A$ Field CM RM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
567.1.d.a 567.d 7.b $1$ $0.283$ \(\Q\) \(\Q(\sqrt{-7}) \) None \(-1\) \(0\) \(0\) \(1\) \(q-q^{2}+q^{7}+q^{8}-q^{11}-q^{14}-q^{16}+\cdots\)
567.1.d.b 567.d 7.b $1$ $0.283$ \(\Q\) \(\Q(\sqrt{-7}) \) None \(1\) \(0\) \(0\) \(1\) \(q+q^{2}+q^{7}-q^{8}+q^{11}+q^{14}-q^{16}+\cdots\)
567.1.d.c 567.d 7.b $2$ $0.283$ \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(-2\) \(q-\beta q^{2}+2q^{4}-q^{7}-\beta q^{8}+\beta q^{11}+\cdots\)
567.1.j.a 567.j 63.j $2$ $0.283$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-1\) \(q+q^{4}+\zeta_{6}^{2}q^{7}+\zeta_{6}q^{13}+q^{16}-\zeta_{6}q^{19}+\cdots\)
567.1.k.a 567.k 63.k $2$ $0.283$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(1\) \(q-\zeta_{6}^{2}q^{4}-\zeta_{6}^{2}q^{7}+(-1+\zeta_{6}^{2})q^{13}+\cdots\)
567.1.l.a 567.l 63.l $2$ $0.283$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-7}) \) None \(-1\) \(0\) \(0\) \(-1\) \(q+\zeta_{6}^{2}q^{2}+\zeta_{6}^{2}q^{7}-q^{8}+\zeta_{6}^{2}q^{11}+\cdots\)
567.1.l.b 567.l 63.l $2$ $0.283$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{21}) \) \(0\) \(0\) \(0\) \(1\) \(q+\zeta_{6}q^{4}-\zeta_{6}^{2}q^{7}+\zeta_{6}^{2}q^{16}+\zeta_{6}^{2}q^{25}+\cdots\)
567.1.l.c 567.l 63.l $2$ $0.283$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-7}) \) None \(1\) \(0\) \(0\) \(-1\) \(q-\zeta_{6}^{2}q^{2}+\zeta_{6}^{2}q^{7}+q^{8}-\zeta_{6}^{2}q^{11}+\cdots\)
567.1.l.d 567.l 63.l $4$ $0.283$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(2\) \(q+(\zeta_{12}+\zeta_{12}^{3})q^{2}+(-1+\zeta_{12}^{2}+\zeta_{12}^{4}+\cdots)q^{4}+\cdots\)
567.1.n.a 567.n 63.n $2$ $0.283$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-1\) \(q-\zeta_{6}q^{4}-\zeta_{6}q^{7}-\zeta_{6}^{2}q^{13}+\zeta_{6}^{2}q^{16}+\cdots\)
567.1.t.a 567.t 63.t $2$ $0.283$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(1\) \(q-q^{4}+\zeta_{6}q^{7}+(1+\zeta_{6})q^{13}+q^{16}+\cdots\)
567.2.a.a 567.a 1.a $1$ $4.528$ \(\Q\) None None \(-1\) \(0\) \(1\) \(-1\) $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}+q^{5}-q^{7}+3q^{8}-q^{10}+\cdots\)
567.2.a.b 567.a 1.a $1$ $4.528$ \(\Q\) None None \(1\) \(0\) \(-1\) \(-1\) $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-q^{5}-q^{7}-3q^{8}-q^{10}+\cdots\)
567.2.a.c 567.a 1.a $3$ $4.528$ \(\Q(\zeta_{18})^+\) None None \(-3\) \(0\) \(-3\) \(3\) $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
567.2.a.d 567.a 1.a $3$ $4.528$ 3.3.321.1 None None \(-1\) \(0\) \(-5\) \(-3\) $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(-2-\beta _{2})q^{5}+\cdots\)
567.2.a.e 567.a 1.a $3$ $4.528$ 3.3.621.1 None None \(0\) \(0\) \(-3\) \(3\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}+(-1-\beta _{2})q^{5}+\cdots\)
567.2.a.f 567.a 1.a $3$ $4.528$ 3.3.621.1 None None \(0\) \(0\) \(3\) \(3\) $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(2+\beta _{1}+\beta _{2})q^{4}+(1+\beta _{2})q^{5}+\cdots\)
567.2.a.g 567.a 1.a $3$ $4.528$ 3.3.321.1 None None \(1\) \(0\) \(5\) \(-3\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{1}+\beta _{2})q^{4}+(2+\beta _{2})q^{5}+\cdots\)
567.2.a.h 567.a 1.a $3$ $4.528$ \(\Q(\zeta_{18})^+\) None None \(3\) \(0\) \(3\) \(3\) $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(1-2\beta _{1}+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)
567.2.a.i 567.a 1.a $4$ $4.528$ \(\Q(\sqrt{3}, \sqrt{7})\) None None \(0\) \(0\) \(0\) \(-4\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{2}-\beta _{2}q^{4}-2\beta _{1}q^{5}-q^{7}+(\beta _{1}+\cdots)q^{8}+\cdots\)
567.2.c.a 567.c 21.c $8$ $4.528$ 8.0.\(\cdots\).3 \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{1}q^{2}+(-2+\beta _{2})q^{4}+\beta _{4}q^{7}+(-2\beta _{1}+\cdots)q^{8}+\cdots\)
567.2.c.b 567.c 21.c $8$ $4.528$ 8.0.\(\cdots\).6 None None \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{3}-\beta _{5})q^{2}-\beta _{1}q^{4}+\beta _{2}q^{5}+\cdots\)
567.2.c.c 567.c 21.c $12$ $4.528$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}-\beta _{4}q^{4}-\beta _{3}q^{5}+(\beta _{1}-\beta _{6}+\cdots)q^{7}+\cdots\)
567.2.e.a 567.e 7.c $2$ $4.528$ \(\Q(\sqrt{-3}) \) None None \(-1\) \(0\) \(1\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}+\zeta_{6}q^{5}+(-2+\cdots)q^{7}+\cdots\)
567.2.e.b 567.e 7.c $2$ $4.528$ \(\Q(\sqrt{-3}) \) None None \(1\) \(0\) \(-1\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{2}+(1-\zeta_{6})q^{4}-\zeta_{6}q^{5}+(-2+\cdots)q^{7}+\cdots\)
567.2.e.c 567.e 7.c $8$ $4.528$ 8.0.1767277521.3 None None \(-1\) \(0\) \(-2\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{2}-\beta _{6})q^{2}+(-\beta _{5}+\beta _{7})q^{4}+\cdots\)
567.2.e.d 567.e 7.c $8$ $4.528$ 8.0.1767277521.3 None None \(1\) \(0\) \(2\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{2}+\beta _{6})q^{2}+(-\beta _{5}+\beta _{7})q^{4}+(1+\cdots)q^{5}+\cdots\)
567.2.e.e 567.e 7.c $10$ $4.528$ 10.0.\(\cdots\).1 None None \(-2\) \(0\) \(-4\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(-\beta _{3}+\beta _{6}+\beta _{7})q^{4}+(-1+\cdots)q^{5}+\cdots\)
567.2.e.f 567.e 7.c $10$ $4.528$ 10.0.\(\cdots\).1 None None \(2\) \(0\) \(4\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+(-\beta _{3}+\beta _{6}+\beta _{7})q^{4}+(1+\cdots)q^{5}+\cdots\)
567.2.e.g 567.e 7.c $16$ $4.528$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(6\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(-\beta _{5}-\beta _{7}-\beta _{8}-\beta _{12}+\cdots)q^{4}+\cdots\)
567.2.f.a 567.f 9.c $2$ $4.528$ \(\Q(\sqrt{-3}) \) None None \(-2\) \(0\) \(-1\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}-2\zeta_{6}q^{4}-\zeta_{6}q^{5}+\cdots\)
567.2.f.b 567.f 9.c $2$ $4.528$ \(\Q(\sqrt{-3}) \) None None \(-1\) \(0\) \(-2\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}+\zeta_{6}q^{4}-2\zeta_{6}q^{5}+\cdots\)
567.2.f.c 567.f 9.c $2$ $4.528$ \(\Q(\sqrt{-3}) \) None None \(-1\) \(0\) \(1\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}+\zeta_{6}q^{4}+\zeta_{6}q^{5}+(1+\cdots)q^{7}+\cdots\)
567.2.f.d 567.f 9.c $2$ $4.528$ \(\Q(\sqrt{-3}) \) None None \(0\) \(0\) \(-3\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{4}-3\zeta_{6}q^{5}+(-1+\zeta_{6})q^{7}+\cdots\)
567.2.f.e 567.f 9.c $2$ $4.528$ \(\Q(\sqrt{-3}) \) None None \(0\) \(0\) \(3\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{4}+3\zeta_{6}q^{5}+(-1+\zeta_{6})q^{7}+\cdots\)
567.2.f.f 567.f 9.c $2$ $4.528$ \(\Q(\sqrt{-3}) \) None None \(1\) \(0\) \(-1\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+\zeta_{6}q^{4}-\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+\cdots\)
567.2.f.g 567.f 9.c $2$ $4.528$ \(\Q(\sqrt{-3}) \) None None \(1\) \(0\) \(2\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+\zeta_{6}q^{4}+2\zeta_{6}q^{5}+(1+\cdots)q^{7}+\cdots\)
567.2.f.h 567.f 9.c $2$ $4.528$ \(\Q(\sqrt{-3}) \) None None \(2\) \(0\) \(1\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}-2\zeta_{6}q^{4}+\zeta_{6}q^{5}+(1+\cdots)q^{7}+\cdots\)
567.2.f.i 567.f 9.c $4$ $4.528$ \(\Q(\sqrt{-3}, \sqrt{7})\) None None \(0\) \(0\) \(0\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+5\beta _{2}q^{4}+(\beta _{1}+\beta _{3})q^{5}+(1+\cdots)q^{7}+\cdots\)
567.2.f.j 567.f 9.c $4$ $4.528$ \(\Q(\zeta_{12})\) None None \(0\) \(0\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{12}^{2}q^{2}+(-1+\zeta_{12})q^{4}+(-2\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)
567.2.f.k 567.f 9.c $4$ $4.528$ \(\Q(\zeta_{12})\) None None \(0\) \(0\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{12}^{2}q^{2}+(-1+\zeta_{12})q^{4}+(\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)
567.2.f.l 567.f 9.c $6$ $4.528$ 6.0.1156923.1 None None \(0\) \(0\) \(-3\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{4}q^{2}+(-\beta _{1}+\beta _{2}-2\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
567.2.f.m 567.f 9.c $6$ $4.528$ 6.0.1156923.1 None None \(0\) \(0\) \(3\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{4}q^{2}+(-\beta _{1}+\beta _{2}-2\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
567.2.f.n 567.f 9.c $8$ $4.528$ 8.0.49787136.1 None None \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{3}q^{2}+\beta _{4}q^{4}-2\beta _{7}q^{5}+(1-\beta _{5}+\cdots)q^{7}+\cdots\)
567.2.g.a 567.g 63.g $2$ $4.528$ \(\Q(\sqrt{-3}) \) None None \(-2\) \(0\) \(-4\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}-2\zeta_{6}q^{4}-2q^{5}+\cdots\)
567.2.g.b 567.g 63.g $2$ $4.528$ \(\Q(\sqrt{-3}) \) None None \(-1\) \(0\) \(-8\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}+\zeta_{6}q^{4}-4q^{5}+(2+\cdots)q^{7}+\cdots\)
567.2.g.c 567.g 63.g $2$ $4.528$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-4\) $\mathrm{U}(1)[D_{3}]$ \(q+2\zeta_{6}q^{4}+(-3+2\zeta_{6})q^{7}+(-2+2\zeta_{6})q^{13}+\cdots\)
567.2.g.d 567.g 63.g $2$ $4.528$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(5\) $\mathrm{U}(1)[D_{3}]$ \(q+2\zeta_{6}q^{4}+(3-\zeta_{6})q^{7}+(7-7\zeta_{6})q^{13}+\cdots\)
567.2.g.e 567.g 63.g $2$ $4.528$ \(\Q(\sqrt{-3}) \) None None \(1\) \(0\) \(8\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+\zeta_{6}q^{4}+4q^{5}+(2-3\zeta_{6})q^{7}+\cdots\)
567.2.g.f 567.g 63.g $2$ $4.528$ \(\Q(\sqrt{-3}) \) None None \(2\) \(0\) \(4\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}-2\zeta_{6}q^{4}+2q^{5}+(-1+\cdots)q^{7}+\cdots\)
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