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Label Char Prim Dim $A$ Field CM RM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
847.1.d.a 847.d 7.b $2$ $0.423$ \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-7}) \) None \(-1\) \(0\) \(0\) \(2\) \(q+(-1+\beta )q^{2}+(1-\beta )q^{4}+q^{7}-q^{8}+\cdots\)
847.1.d.b 847.d 7.b $2$ $0.423$ \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-7}) \) None \(1\) \(0\) \(0\) \(-2\) \(q+(1-\beta )q^{2}+(1-\beta )q^{4}-q^{7}+q^{8}+\cdots\)
847.1.j.a 847.j 77.j $4$ $0.423$ \(\Q(\zeta_{10})\) \(\Q(\sqrt{-7}) \) None \(-3\) \(0\) \(0\) \(1\) \(q+(-1-\zeta_{10}^{4})q^{2}+(1-\zeta_{10}^{3}+\zeta_{10}^{4}+\cdots)q^{4}+\cdots\)
847.1.j.b 847.j 77.j $4$ $0.423$ \(\Q(\zeta_{10})\) \(\Q(\sqrt{-7}) \) None \(2\) \(0\) \(0\) \(1\) \(q+(\zeta_{10}+\zeta_{10}^{3})q^{2}+(-\zeta_{10}+\zeta_{10}^{2}+\cdots)q^{4}+\cdots\)
847.1.j.c 847.j 77.j $4$ $0.423$ \(\Q(\zeta_{10})\) \(\Q(\sqrt{-7}) \) None \(3\) \(0\) \(0\) \(-1\) \(q+(1+\zeta_{10}^{4})q^{2}+(1-\zeta_{10}^{3}+\zeta_{10}^{4}+\cdots)q^{4}+\cdots\)
847.1.o.a 847.o 847.o $10$ $0.423$ \(\Q(\zeta_{22})\) \(\Q(\sqrt{-7}) \) None \(-2\) \(0\) \(0\) \(-1\) \(q+(-\zeta_{22}^{9}+\zeta_{22}^{10})q^{2}+(-\zeta_{22}^{7}+\cdots)q^{4}+\cdots\)
847.1.bb.a 847.bb 847.ab $40$ $0.423$ \(\Q(\zeta_{55})\) \(\Q(\sqrt{-7}) \) None \(2\) \(0\) \(0\) \(1\) \(q+(\zeta_{110}^{8}-\zeta_{110}^{49})q^{2}+(\zeta_{110}^{2}+\zeta_{110}^{16}+\cdots)q^{4}+\cdots\)
847.2.a.a 847.a 1.a $1$ $6.763$ \(\Q\) None None \(-1\) \(2\) \(-2\) \(1\) $+$ $\mathrm{SU}(2)$ \(q-q^{2}+2q^{3}-q^{4}-2q^{5}-2q^{6}+q^{7}+\cdots\)
847.2.a.b 847.a 1.a $1$ $6.763$ \(\Q\) None None \(0\) \(-3\) \(-1\) \(1\) $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-2q^{4}-q^{5}+q^{7}+6q^{9}+6q^{12}+\cdots\)
847.2.a.c 847.a 1.a $1$ $6.763$ \(\Q\) None None \(0\) \(1\) \(3\) \(-1\) $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{4}+3q^{5}-q^{7}-2q^{9}-2q^{12}+\cdots\)
847.2.a.d 847.a 1.a $2$ $6.763$ \(\Q(\sqrt{5}) \) None None \(-3\) \(-1\) \(2\) \(2\) $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+(-1+\beta )q^{3}+3\beta q^{4}+\cdots\)
847.2.a.e 847.a 1.a $2$ $6.763$ \(\Q(\sqrt{13}) \) None None \(-1\) \(-1\) \(0\) \(2\) $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-\beta q^{3}+(1+\beta )q^{4}+(-1+2\beta )q^{5}+\cdots\)
847.2.a.f 847.a 1.a $2$ $6.763$ \(\Q(\sqrt{5}) \) None None \(0\) \(2\) \(-4\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(1-\beta )q^{3}+3q^{4}-2q^{5}+(5+\cdots)q^{6}+\cdots\)
847.2.a.g 847.a 1.a $2$ $6.763$ \(\Q(\sqrt{13}) \) None None \(1\) \(-1\) \(0\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-\beta q^{3}+(1+\beta )q^{4}+(-1+2\beta )q^{5}+\cdots\)
847.2.a.h 847.a 1.a $2$ $6.763$ \(\Q(\sqrt{5}) \) None None \(3\) \(-1\) \(2\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(-1+\beta )q^{3}+3\beta q^{4}+\cdots\)
847.2.a.i 847.a 1.a $3$ $6.763$ 3.3.568.1 None None \(-2\) \(-1\) \(1\) \(-3\) $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}-\beta _{1}q^{3}+(3+\beta _{2})q^{4}+\cdots\)
847.2.a.j 847.a 1.a $3$ $6.763$ 3.3.568.1 None None \(2\) \(-1\) \(1\) \(3\) $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-\beta _{1}q^{3}+(3+\beta _{2})q^{4}+\cdots\)
847.2.a.k 847.a 1.a $4$ $6.763$ 4.4.2525.1 None None \(-2\) \(-2\) \(-6\) \(4\) $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(-1-\beta _{2})q^{3}+(1+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.2.a.l 847.a 1.a $4$ $6.763$ 4.4.2525.1 None None \(2\) \(-2\) \(-6\) \(-4\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(-1-\beta _{2})q^{3}+(1+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.2.a.m 847.a 1.a $6$ $6.763$ 6.6.7674048.1 None None \(-4\) \(-2\) \(-4\) \(-6\) $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+\beta _{5}q^{3}+(1-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.2.a.n 847.a 1.a $6$ $6.763$ 6.6.7674048.1 None None \(4\) \(-2\) \(-4\) \(6\) $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+\beta _{5}q^{3}+(1-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
847.2.a.o 847.a 1.a $8$ $6.763$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None None \(-1\) \(4\) \(10\) \(-8\) $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{2}+(1-\beta _{4}-\beta _{7})q^{3}+(1+\beta _{5}+\cdots)q^{4}+\cdots\)
847.2.a.p 847.a 1.a $8$ $6.763$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None None \(1\) \(4\) \(10\) \(8\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1-\beta _{4}-\beta _{7})q^{3}+(1+\beta _{5}+\cdots)q^{4}+\cdots\)
847.2.b.a 847.b 77.b $8$ $6.763$ 8.0.\(\cdots\).11 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{2}+\beta _{4})q^{2}-\beta _{3}q^{3}+(-2+\beta _{6}+\cdots)q^{4}+\cdots\)
847.2.b.b 847.b 77.b $8$ $6.763$ 8.0.37515625.1 \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{1}q^{2}+(-1+\beta _{2})q^{4}+(-\beta _{1}+\beta _{4}+\cdots)q^{7}+\cdots\)
847.2.b.c 847.b 77.b $8$ $6.763$ 8.0.4956160000.2 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{3}+\beta _{5})q^{2}+\beta _{6}q^{3}+(-1-\beta _{4}+\cdots)q^{4}+\cdots\)
847.2.b.d 847.b 77.b $8$ $6.763$ 8.0.\(\cdots\).2 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{6}q^{2}-\beta _{1}q^{4}+\beta _{7}q^{5}+(-\beta _{3}-\beta _{5}+\cdots)q^{7}+\cdots\)
847.2.b.e 847.b 77.b $16$ $6.763$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{3}+\beta _{10})q^{2}+(-\beta _{5}+\beta _{14}+\beta _{15})q^{3}+\cdots\)
847.2.b.f 847.b 77.b $16$ $6.763$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{1}+\beta _{5})q^{2}-\beta _{14}q^{3}+(1-\beta _{6}-\beta _{8}+\cdots)q^{4}+\cdots\)
847.2.e.a 847.e 7.c $4$ $6.763$ \(\Q(\sqrt{-3}, \sqrt{5})\) None None \(-1\) \(-2\) \(0\) \(10\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+\beta _{3}q^{3}+(\beta _{1}+\beta _{2}-\beta _{3})q^{4}+\cdots\)
847.2.e.b 847.e 7.c $4$ $6.763$ \(\Q(\sqrt{-3}, \sqrt{5})\) None None \(1\) \(-2\) \(0\) \(-10\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+\beta _{3}q^{3}+(\beta _{1}+\beta _{2}-\beta _{3})q^{4}+\cdots\)
847.2.e.c 847.e 7.c $6$ $6.763$ \(\Q(\zeta_{18})\) None None \(0\) \(-3\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\zeta_{18}-\zeta_{18}^{2}-\zeta_{18}^{4}+\zeta_{18}^{5})q^{2}+\cdots\)
847.2.e.d 847.e 7.c $6$ $6.763$ 6.0.1783323.2 None None \(0\) \(1\) \(2\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{3}-\beta _{5})q^{2}+\beta _{1}q^{3}+(-1-\beta _{1}+\cdots)q^{4}+\cdots\)
847.2.e.e 847.e 7.c $12$ $6.763$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None None \(0\) \(2\) \(4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{4}q^{2}-\beta _{6}q^{3}+(-1-\beta _{5}-\beta _{7}+\cdots)q^{4}+\cdots\)
847.2.e.f 847.e 7.c $14$ $6.763$ \(\mathbb{Q}[x]/(x^{14} + \cdots)\) None None \(0\) \(-3\) \(-4\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(\beta _{4}+\beta _{10})q^{3}+(-1-\beta _{2}+\cdots)q^{4}+\cdots\)
847.2.e.g 847.e 7.c $14$ $6.763$ \(\mathbb{Q}[x]/(x^{14} + \cdots)\) None None \(0\) \(-3\) \(-4\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+(\beta _{4}+\beta _{10})q^{3}+(-1-\beta _{2}+\cdots)q^{4}+\cdots\)
847.2.e.h 847.e 7.c $20$ $6.763$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None None \(-1\) \(3\) \(2\) \(11\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+\beta _{17}q^{3}+(-1-\beta _{4}-\beta _{6}+\cdots)q^{4}+\cdots\)
847.2.e.i 847.e 7.c $20$ $6.763$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None None \(1\) \(3\) \(2\) \(-11\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+\beta _{17}q^{3}+(-1-\beta _{4}-\beta _{6}+\cdots)q^{4}+\cdots\)
847.2.e.j 847.e 7.c $28$ $6.763$ None None \(0\) \(6\) \(8\) \(0\) $\mathrm{SU}(2)[C_{3}]$
847.2.f.a 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None None \(-5\) \(-6\) \(2\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1-\zeta_{10}+\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
847.2.f.b 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None None \(-5\) \(4\) \(2\) \(1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1-\zeta_{10}+\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
847.2.f.c 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None None \(-4\) \(3\) \(-1\) \(1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-2+\zeta_{10}-\zeta_{10}^{2}+2\zeta_{10}^{3})q^{2}+\cdots\)
847.2.f.d 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None None \(-1\) \(-2\) \(-1\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1-2\zeta_{10}+2\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+\cdots\)
847.2.f.e 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None None \(-1\) \(-2\) \(2\) \(1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+\zeta_{10}-\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
847.2.f.f 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None None \(0\) \(-1\) \(-3\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q+\zeta_{10}^{2}q^{3}+2\zeta_{10}^{3}q^{4}-3\zeta_{10}q^{5}+\cdots\)
847.2.f.g 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None None \(0\) \(-1\) \(-3\) \(1\) $\mathrm{SU}(2)[C_{5}]$ \(q+\zeta_{10}^{2}q^{3}+2\zeta_{10}^{3}q^{4}-3\zeta_{10}q^{5}+\cdots\)
847.2.f.h 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None None \(0\) \(3\) \(1\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q-3\zeta_{10}^{2}q^{3}+2\zeta_{10}^{3}q^{4}+\zeta_{10}q^{5}+\cdots\)
847.2.f.i 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None None \(0\) \(3\) \(1\) \(1\) $\mathrm{SU}(2)[C_{5}]$ \(q-3\zeta_{10}^{2}q^{3}+2\zeta_{10}^{3}q^{4}+\zeta_{10}q^{5}+\cdots\)
847.2.f.j 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None None \(1\) \(-2\) \(-1\) \(1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+2\zeta_{10}-2\zeta_{10}^{2}+\zeta_{10}^{3})q^{2}+\cdots\)
847.2.f.k 847.f 11.c $4$ $6.763$ \(\Q(\zeta_{10})\) None None \(1\) \(-2\) \(2\) \(-1\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+2\zeta_{10}^{2}q^{3}+\cdots\)
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