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Label Char Prim Dim $A$ Field CM RM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
539.1.c.a 539.c 11.b $1$ $0.269$ \(\Q\) \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-11}) \) \(\Q(\sqrt{77}) \) \(0\) \(0\) \(0\) \(0\) \(q+q^{4}-q^{9}+q^{11}+q^{16}-2q^{23}+\cdots\)
539.1.c.b 539.c 11.b $2$ $0.269$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-11}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\beta q^{3}+q^{4}+\beta q^{5}+q^{9}-q^{11}-\beta q^{12}+\cdots\)
539.1.h.a 539.h 77.h $2$ $0.269$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-11}) \) \(\Q(\sqrt{77}) \) \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{6}^{2}q^{4}+\zeta_{6}q^{9}+\zeta_{6}^{2}q^{11}-\zeta_{6}q^{16}+\cdots\)
539.1.h.b 539.h 77.h $4$ $0.269$ \(\Q(\sqrt{2}, \sqrt{-3})\) \(\Q(\sqrt{-11}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{3}+(-1-\beta _{2})q^{4}+(-\beta _{1}-\beta _{3})q^{5}+\cdots\)
539.1.l.a 539.l 11.d $4$ $0.269$ \(\Q(\zeta_{10})\) \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{10}+\zeta_{10}^{3})q^{2}+(-\zeta_{10}+\zeta_{10}^{2}+\cdots)q^{4}+\cdots\)
539.1.t.a 539.t 77.o $8$ $0.269$ \(\Q(\zeta_{15})\) \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{30}^{4}-\zeta_{30}^{13})q^{2}+(-\zeta_{30}^{2}+\cdots)q^{4}+\cdots\)
539.1.u.a 539.u 77.p $8$ $0.269$ \(\Q(\zeta_{15})\) \(\Q(\sqrt{-7}) \) None \(2\) \(0\) \(0\) \(0\) \(q+(\zeta_{30}^{8}-\zeta_{30}^{11})q^{2}+(-\zeta_{30}+\zeta_{30}^{4}+\cdots)q^{4}+\cdots\)
539.2.a.a 539.a 1.a $1$ $4.304$ \(\Q\) None None \(-2\) \(1\) \(-1\) \(0\) $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+q^{3}+2q^{4}-q^{5}-2q^{6}-2q^{9}+\cdots\)
539.2.a.b 539.a 1.a $1$ $4.304$ \(\Q\) None None \(0\) \(-1\) \(-3\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{4}-3q^{5}-2q^{9}-q^{11}+\cdots\)
539.2.a.c 539.a 1.a $1$ $4.304$ \(\Q\) None None \(0\) \(3\) \(1\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-2q^{4}+q^{5}+6q^{9}-q^{11}+\cdots\)
539.2.a.d 539.a 1.a $1$ $4.304$ \(\Q\) None None \(1\) \(-2\) \(2\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+q^{2}-2q^{3}-q^{4}+2q^{5}-2q^{6}-3q^{8}+\cdots\)
539.2.a.e 539.a 1.a $2$ $4.304$ \(\Q(\sqrt{2}) \) None None \(-2\) \(0\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q-q^{2}+\beta q^{3}-q^{4}-\beta q^{5}-\beta q^{6}+3q^{8}+\cdots\)
539.2.a.f 539.a 1.a $2$ $4.304$ \(\Q(\sqrt{5}) \) None None \(0\) \(-2\) \(4\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(-1-\beta )q^{3}+3q^{4}+2q^{5}+\cdots\)
539.2.a.g 539.a 1.a $3$ $4.304$ \(\Q(\zeta_{18})^+\) None None \(0\) \(-3\) \(-6\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(-1+\beta _{1}-\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots\)
539.2.a.h 539.a 1.a $3$ $4.304$ 3.3.257.1 None None \(0\) \(-1\) \(-2\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}-\beta _{1}q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+\cdots\)
539.2.a.i 539.a 1.a $3$ $4.304$ 3.3.257.1 None None \(0\) \(1\) \(2\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+\beta _{1}q^{3}+(1+\beta _{1}-\beta _{2})q^{4}+\cdots\)
539.2.a.j 539.a 1.a $3$ $4.304$ \(\Q(\zeta_{18})^+\) None None \(0\) \(3\) \(6\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1-\beta _{1}+\beta _{2})q^{3}+\beta _{2}q^{4}+\cdots\)
539.2.a.k 539.a 1.a $4$ $4.304$ 4.4.9248.1 None None \(2\) \(0\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-\beta _{3}q^{2}+\beta _{2}q^{3}+(2-\beta _{3})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
539.2.a.l 539.a 1.a $10$ $4.304$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None None \(2\) \(0\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{5}q^{2}+\beta _{1}q^{3}+(2+\beta _{9})q^{4}+\beta _{7}q^{5}+\cdots\)
539.2.b.a 539.b 77.b $8$ $4.304$ 8.0.\(\cdots\).1 \(\Q(\sqrt{-11}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{1}q^{3}+2q^{4}+\beta _{5}q^{5}+(-3+\beta _{3}+\cdots)q^{9}+\cdots\)
539.2.b.b 539.b 77.b $12$ $4.304$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{6}q^{2}-\beta _{7}q^{3}+(-1+\beta _{2})q^{4}+(\beta _{5}+\cdots)q^{5}+\cdots\)
539.2.b.c 539.b 77.b $16$ $4.304$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{8}q^{2}-\beta _{7}q^{3}+(-2+\beta _{6})q^{4}-\beta _{4}q^{5}+\cdots\)
539.2.e.a 539.e 7.c $2$ $4.304$ \(\Q(\sqrt{-3}) \) None None \(-1\) \(-2\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
539.2.e.b 539.e 7.c $2$ $4.304$ \(\Q(\sqrt{-3}) \) None None \(-1\) \(2\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{6}q^{2}+(2-2\zeta_{6})q^{3}+(1-\zeta_{6})q^{4}+\cdots\)
539.2.e.c 539.e 7.c $2$ $4.304$ \(\Q(\sqrt{-3}) \) None None \(0\) \(-3\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}-\zeta_{6}q^{5}+\cdots\)
539.2.e.d 539.e 7.c $2$ $4.304$ \(\Q(\sqrt{-3}) \) None None \(0\) \(-1\) \(-3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}-3\zeta_{6}q^{5}+\cdots\)
539.2.e.e 539.e 7.c $2$ $4.304$ \(\Q(\sqrt{-3}) \) None None \(0\) \(1\) \(3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}+3\zeta_{6}q^{5}+\cdots\)
539.2.e.f 539.e 7.c $2$ $4.304$ \(\Q(\sqrt{-3}) \) None None \(0\) \(3\) \(1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}+\zeta_{6}q^{5}+\cdots\)
539.2.e.g 539.e 7.c $2$ $4.304$ \(\Q(\sqrt{-3}) \) None None \(2\) \(-1\) \(1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+\cdots\)
539.2.e.h 539.e 7.c $2$ $4.304$ \(\Q(\sqrt{-3}) \) None None \(2\) \(1\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{2}+(1-\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+\cdots\)
539.2.e.i 539.e 7.c $4$ $4.304$ \(\Q(\sqrt{-3}, \sqrt{5})\) None None \(0\) \(-2\) \(4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{2}q^{2}+(\beta _{1}-\beta _{2}-\beta _{3})q^{3}+3\beta _{1}q^{4}+\cdots\)
539.2.e.j 539.e 7.c $4$ $4.304$ \(\Q(\sqrt{-3}, \sqrt{5})\) None None \(0\) \(2\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{2}q^{2}+(-\beta _{1}+\beta _{2}+\beta _{3})q^{3}+3\beta _{1}q^{4}+\cdots\)
539.2.e.k 539.e 7.c $4$ $4.304$ \(\Q(\sqrt{2}, \sqrt{-3})\) None None \(2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{2}q^{2}-\beta _{1}q^{3}+(1+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
539.2.e.l 539.e 7.c $6$ $4.304$ 6.0.1783323.2 None None \(0\) \(-1\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{3}+\beta _{5})q^{2}-\beta _{1}q^{3}+(-1-\beta _{1}+\cdots)q^{4}+\cdots\)
539.2.e.m 539.e 7.c $6$ $4.304$ \(\Q(\zeta_{18})\) None None \(0\) \(3\) \(6\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\zeta_{18}^{2}+\zeta_{18}^{3}+\zeta_{18}^{4}-\zeta_{18}^{5})q^{2}+\cdots\)
539.2.e.n 539.e 7.c $8$ $4.304$ 8.0.6927565824.3 None None \(-2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{3}-\beta _{4}-\beta _{5})q^{2}+\beta _{2}q^{3}+\cdots\)
539.2.e.o 539.e 7.c $20$ $4.304$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None None \(-2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{8}q^{2}+(-\beta _{1}+\beta _{10})q^{3}+(-2\beta _{2}+\cdots)q^{4}+\cdots\)
539.2.f.a 539.f 11.c $4$ $4.304$ \(\Q(\zeta_{10})\) None None \(2\) \(-1\) \(5\) \(0\) $\mathrm{SU}(2)[C_{5}]$ \(q+(\zeta_{10}-\zeta_{10}^{2})q^{2}+\zeta_{10}^{2}q^{3}+(-1+\cdots)q^{4}+\cdots\)
539.2.f.b 539.f 11.c $4$ $4.304$ \(\Q(\zeta_{10})\) None None \(2\) \(1\) \(-5\) \(0\) $\mathrm{SU}(2)[C_{5}]$ \(q+(\zeta_{10}-\zeta_{10}^{2})q^{2}-\zeta_{10}^{2}q^{3}+(-1+\cdots)q^{4}+\cdots\)
539.2.f.c 539.f 11.c $8$ $4.304$ 8.0.37515625.1 \(\Q(\sqrt{-7}) \) None \(-2\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{5}]$ \(q+(-1+\beta _{1}+2\beta _{2}-\beta _{4}-\beta _{5}+\beta _{6}+\cdots)q^{2}+\cdots\)
539.2.f.d 539.f 11.c $8$ $4.304$ 8.0.159390625.1 None None \(-1\) \(4\) \(-3\) \(0\) $\mathrm{SU}(2)[C_{5}]$ \(q+\beta _{4}q^{2}+(\beta _{2}+\beta _{3})q^{3}+(1-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
539.2.f.e 539.f 11.c $16$ $4.304$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(-3\) \(2\) \(5\) \(0\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-\beta _{5}-\beta _{6})q^{2}+(-\beta _{9}+\beta _{11}-\beta _{13}+\cdots)q^{3}+\cdots\)
539.2.f.f 539.f 11.c $16$ $4.304$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(-2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1-\beta _{2}+\beta _{5}-\beta _{7})q^{2}+(\beta _{11}-\beta _{12}+\cdots)q^{3}+\cdots\)
539.2.f.g 539.f 11.c $20$ $4.304$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None None \(3\) \(-4\) \(4\) \(0\) $\mathrm{SU}(2)[C_{5}]$ \(q+\beta _{6}q^{2}+(-\beta _{1}-\beta _{7}-\beta _{18}+\beta _{19})q^{3}+\cdots\)
539.2.f.h 539.f 11.c $20$ $4.304$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None None \(3\) \(4\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{5}]$ \(q+\beta _{6}q^{2}+(\beta _{1}+\beta _{7}+\beta _{18}-\beta _{19})q^{3}+\cdots\)
539.2.f.i 539.f 11.c $48$ $4.304$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{5}]$
539.2.i.a 539.i 77.i $4$ $4.304$ \(\Q(\sqrt{-3}, \sqrt{-7})\) \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+\beta _{1}q^{2}+5\beta _{2}q^{4}+(3\beta _{1}-3\beta _{3})q^{8}+\cdots\)
539.2.i.b 539.i 77.i $8$ $4.304$ 8.0.3317760000.8 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{3}-\beta _{4})q^{2}+\beta _{5}q^{3}-\beta _{1}q^{5}+(\beta _{6}+\cdots)q^{6}+\cdots\)
539.2.i.c 539.i 77.i $12$ $4.304$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(0\) \(6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{1}+\beta _{7})q^{2}+(1-\beta _{2}+\beta _{8}+\beta _{9}+\cdots)q^{3}+\cdots\)
539.2.i.d 539.i 77.i $16$ $4.304$ 16.0.\(\cdots\).7 \(\Q(\sqrt{-11}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+(\beta _{1}+\beta _{8})q^{3}-2\beta _{7}q^{4}+\beta _{5}q^{5}+(3+\cdots)q^{9}+\cdots\)
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