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Label Char Prim Dim $A$ Field CM RM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
585.1.o.a 585.o 65.h $8$ $0.292$ \(\Q(\zeta_{16})\) \(\Q(\sqrt{-39}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(\zeta_{16}^{5}-\zeta_{16}^{7})q^{2}+(-\zeta_{16}^{2}+\zeta_{16}^{4}+\cdots)q^{4}+\cdots\)
585.2.a.a 585.a 1.a $1$ $4.671$ \(\Q\) None None \(-2\) \(0\) \(-1\) \(-3\) $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}-q^{5}-3q^{7}+2q^{10}+\cdots\)
585.2.a.b 585.a 1.a $1$ $4.671$ \(\Q\) None None \(-2\) \(0\) \(-1\) \(3\) $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}-q^{5}+3q^{7}+2q^{10}+\cdots\)
585.2.a.c 585.a 1.a $1$ $4.671$ \(\Q\) None None \(-2\) \(0\) \(1\) \(-1\) $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+2q^{4}+q^{5}-q^{7}-2q^{10}+\cdots\)
585.2.a.d 585.a 1.a $1$ $4.671$ \(\Q\) None None \(-1\) \(0\) \(-1\) \(2\) $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-q^{5}+2q^{7}+3q^{8}+q^{10}+\cdots\)
585.2.a.e 585.a 1.a $1$ $4.671$ \(\Q\) None None \(0\) \(0\) \(-1\) \(-1\) $-$ $\mathrm{SU}(2)$ \(q-2q^{4}-q^{5}-q^{7}+3q^{11}+q^{13}+\cdots\)
585.2.a.f 585.a 1.a $1$ $4.671$ \(\Q\) None None \(0\) \(0\) \(1\) \(-1\) $+$ $\mathrm{SU}(2)$ \(q-2q^{4}+q^{5}-q^{7}-3q^{11}+q^{13}+\cdots\)
585.2.a.g 585.a 1.a $1$ $4.671$ \(\Q\) None None \(1\) \(0\) \(-1\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}-q^{5}-3q^{8}-q^{10}-4q^{11}+\cdots\)
585.2.a.h 585.a 1.a $1$ $4.671$ \(\Q\) None None \(1\) \(0\) \(1\) \(-4\) $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+q^{5}-4q^{7}-3q^{8}+q^{10}+\cdots\)
585.2.a.i 585.a 1.a $1$ $4.671$ \(\Q\) None None \(1\) \(0\) \(1\) \(2\) $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{4}+q^{5}+2q^{7}-3q^{8}+q^{10}+\cdots\)
585.2.a.j 585.a 1.a $2$ $4.671$ \(\Q(\sqrt{17}) \) None None \(-1\) \(0\) \(-2\) \(-5\) $+$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+(2+\beta )q^{4}-q^{5}+(-3+\beta )q^{7}+\cdots\)
585.2.a.k 585.a 1.a $2$ $4.671$ \(\Q(\sqrt{3}) \) None None \(0\) \(0\) \(2\) \(4\) $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{4}+q^{5}+2q^{7}-\beta q^{8}+\beta q^{10}+\cdots\)
585.2.a.l 585.a 1.a $2$ $4.671$ \(\Q(\sqrt{17}) \) None None \(1\) \(0\) \(2\) \(-5\) $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(2+\beta )q^{4}+q^{5}+(-3+\beta )q^{7}+\cdots\)
585.2.a.m 585.a 1.a $2$ $4.671$ \(\Q(\sqrt{2}) \) None None \(2\) \(0\) \(-2\) \(4\) $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}-q^{5}+(2+2\beta )q^{7}+\cdots\)
585.2.a.n 585.a 1.a $3$ $4.671$ 3.3.316.1 None None \(0\) \(0\) \(3\) \(1\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(3+\beta _{2})q^{4}+q^{5}-\beta _{2}q^{7}+\cdots\)
585.2.b.a 585.b 13.b $2$ $4.671$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+q^{4}+iq^{5}+2iq^{7}+3iq^{8}+\cdots\)
585.2.b.b 585.b 13.b $2$ $4.671$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+q^{4}+iq^{5}+4iq^{7}+3iq^{8}+\cdots\)
585.2.b.c 585.b 13.b $2$ $4.671$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+q^{4}+iq^{5}-4iq^{7}+3iq^{8}+\cdots\)
585.2.b.d 585.b 13.b $2$ $4.671$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2q^{4}+iq^{5}-3iq^{7}-3iq^{11}+(2+\cdots)q^{13}+\cdots\)
585.2.b.e 585.b 13.b $4$ $4.671$ \(\Q(i, \sqrt{17})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-3+\beta _{3})q^{4}-\beta _{2}q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
585.2.b.f 585.b 13.b $4$ $4.671$ \(\Q(i, \sqrt{13})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-2\beta _{1}q^{2}-2q^{4}+\beta _{1}q^{5}-\beta _{2}q^{7}+\cdots\)
585.2.b.g 585.b 13.b $6$ $4.671$ 6.0.5089536.1 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{5}q^{2}+(-2+\beta _{3})q^{4}+\beta _{4}q^{5}+(-\beta _{2}+\cdots)q^{7}+\cdots\)
585.2.c.a 585.c 5.b $2$ $4.671$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2q^{4}+(1+2i)q^{5}-iq^{7}+q^{11}+\cdots\)
585.2.c.b 585.c 5.b $6$ $4.671$ 6.0.350464.1 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(\beta _{3}-\beta _{5})q^{2}+(-2-\beta _{1}-\beta _{2})q^{4}+\cdots\)
585.2.c.c 585.c 5.b $10$ $4.671$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None None \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-1-\beta _{4}+\beta _{6}+\beta _{7})q^{4}+\cdots\)
585.2.c.d 585.c 5.b $12$ $4.671$ 12.0.\(\cdots\).1 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{8}q^{2}+(-1-\beta _{9})q^{4}+(-\beta _{5}+\beta _{8}+\cdots)q^{5}+\cdots\)
585.2.h.a 585.h 65.d $2$ $4.671$ \(\Q(\sqrt{-1}) \) None None \(-4\) \(0\) \(-4\) \(-6\) $\mathrm{SU}(2)[C_{2}]$ \(q-2q^{2}+2q^{4}+(-2+i)q^{5}-3q^{7}+\cdots\)
585.2.h.b 585.h 65.d $2$ $4.671$ \(\Q(\sqrt{-1}) \) None None \(-2\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-q^{2}-q^{4}+(-1-i)q^{5}+3q^{8}+(1+\cdots)q^{10}+\cdots\)
585.2.h.c 585.h 65.d $2$ $4.671$ \(\Q(\sqrt{-1}) \) None None \(2\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+q^{2}-q^{4}+(1+i)q^{5}-3q^{8}+(1+i)q^{10}+\cdots\)
585.2.h.d 585.h 65.d $2$ $4.671$ \(\Q(\sqrt{-1}) \) None None \(4\) \(0\) \(4\) \(6\) $\mathrm{SU}(2)[C_{2}]$ \(q+2q^{2}+2q^{4}+(2-i)q^{5}+3q^{7}+(4+\cdots)q^{10}+\cdots\)
585.2.h.e 585.h 65.d $4$ $4.671$ \(\Q(\sqrt{-5}, \sqrt{13})\) \(\Q(\sqrt{-195}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-2q^{4}-\beta _{1}q^{5}+\beta _{3}q^{7}+\beta _{1}q^{11}+\cdots\)
585.2.h.f 585.h 65.d $8$ $4.671$ 8.0.\(\cdots\).21 \(\Q(\sqrt{-39}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\beta _{4}q^{2}+(2-\beta _{2})q^{4}+\beta _{1}q^{5}+(\beta _{1}+\cdots)q^{8}+\cdots\)
585.2.h.g 585.h 65.d $12$ $4.671$ 12.0.\(\cdots\).1 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{10}q^{2}+(1+\beta _{5}-\beta _{11})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
585.2.i.a 585.i 9.c $2$ $4.671$ \(\Q(\sqrt{-3}) \) None None \(0\) \(3\) \(-1\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-\zeta_{6})q^{3}+2\zeta_{6}q^{4}-\zeta_{6}q^{5}+(4+\cdots)q^{7}+\cdots\)
585.2.i.b 585.i 9.c $2$ $4.671$ \(\Q(\sqrt{-3}) \) None None \(1\) \(0\) \(1\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(-1+2\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
585.2.i.c 585.i 9.c $2$ $4.671$ \(\Q(\sqrt{-3}) \) None None \(1\) \(3\) \(1\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(2-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
585.2.i.d 585.i 9.c $2$ $4.671$ \(\Q(\sqrt{-3}) \) None None \(2\) \(-3\) \(1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}-2\zeta_{6}q^{4}+\cdots\)
585.2.i.e 585.i 9.c $16$ $4.671$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(-3\) \(1\) \(8\) \(11\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{6}q^{2}+(-\beta _{7}+\beta _{14})q^{3}+(-\beta _{1}+\cdots)q^{4}+\cdots\)
585.2.i.f 585.i 9.c $16$ $4.671$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(1\) \(-2\) \(-8\) \(6\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+(-\beta _{1}-\beta _{3}-\beta _{4}-\beta _{5}-\beta _{6}+\cdots)q^{3}+\cdots\)
585.2.i.g 585.i 9.c $26$ $4.671$ None None \(1\) \(1\) \(-13\) \(-10\) $\mathrm{SU}(2)[C_{3}]$
585.2.i.h 585.i 9.c $30$ $4.671$ None None \(1\) \(1\) \(15\) \(-10\) $\mathrm{SU}(2)[C_{3}]$
585.2.j.a 585.j 13.c $2$ $4.671$ \(\Q(\sqrt{-3}) \) None None \(0\) \(0\) \(2\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{4}+q^{5}+\zeta_{6}q^{7}+(6-6\zeta_{6})q^{11}+\cdots\)
585.2.j.b 585.j 13.c $2$ $4.671$ \(\Q(\sqrt{-3}) \) None None \(2\) \(0\) \(2\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{2}-2\zeta_{6}q^{4}+q^{5}-5\zeta_{6}q^{7}+\cdots\)
585.2.j.c 585.j 13.c $4$ $4.671$ \(\Q(\zeta_{12})\) None None \(-2\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\zeta_{12}+\zeta_{12}^{2})q^{2}+(-2+2\zeta_{12}+\cdots)q^{4}+\cdots\)
585.2.j.d 585.j 13.c $4$ $4.671$ \(\Q(\sqrt{-3}, \sqrt{13})\) None None \(-1\) \(0\) \(4\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(-1+\beta _{1}+\beta _{2}+\beta _{3})q^{4}+\cdots\)
585.2.j.e 585.j 13.c $4$ $4.671$ \(\Q(\sqrt{-3}, \sqrt{5})\) None None \(1\) \(0\) \(-4\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{2}-\beta _{3})q^{4}-q^{5}+\cdots\)
585.2.j.f 585.j 13.c $6$ $4.671$ 6.0.1714608.1 None None \(0\) \(0\) \(-6\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{3}-\beta _{5})q^{2}+(-2\beta _{1}-\beta _{2}+\beta _{4}+\cdots)q^{4}+\cdots\)
585.2.j.g 585.j 13.c $6$ $4.671$ 6.0.591408.1 None None \(0\) \(0\) \(6\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{5}q^{2}+(\beta _{1}-\beta _{3}-\beta _{4}+\beta _{5})q^{4}+\cdots\)
585.2.j.h 585.j 13.c $10$ $4.671$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None None \(-2\) \(0\) \(10\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}+\beta _{3})q^{2}+(-1+\beta _{4}-\beta _{8})q^{4}+\cdots\)
585.2.j.i 585.j 13.c $10$ $4.671$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None None \(2\) \(0\) \(-10\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}-\beta _{3})q^{2}+(-1+\beta _{4}-\beta _{8}+\cdots)q^{4}+\cdots\)
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