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Label Char Prim Dim $A$ Field CM RM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
500.1.b.a 500.b 4.b $4$ $0.250$ \(\Q(i, \sqrt{5})\) \(\Q(\sqrt{-5}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{3}q^{2}-\beta _{1}q^{3}-q^{4}+\beta _{2}q^{6}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
500.1.d.a 500.d 20.d $2$ $0.250$ \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-5}) \) None \(-2\) \(1\) \(0\) \(1\) \(q-q^{2}+(1-\beta )q^{3}+q^{4}+(-1+\beta )q^{6}+\cdots\)
500.1.d.b 500.d 20.d $2$ $0.250$ \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-5}) \) None \(2\) \(-1\) \(0\) \(-1\) \(q+q^{2}+(-1+\beta )q^{3}+q^{4}+(-1+\beta )q^{6}+\cdots\)
500.1.h.a 500.h 100.h $8$ $0.250$ \(\Q(\zeta_{20})\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{20}q^{2}+\zeta_{20}^{2}q^{4}-\zeta_{20}^{3}q^{8}-\zeta_{20}^{4}q^{9}+\cdots\)
500.1.j.a 500.j 100.j $4$ $0.250$ \(\Q(\zeta_{10})\) \(\Q(\sqrt{-1}) \) None \(1\) \(0\) \(0\) \(0\) \(q-\zeta_{10}^{4}q^{2}-\zeta_{10}^{3}q^{4}-\zeta_{10}^{2}q^{8}+\cdots\)
500.1.p.a 500.p 500.p $20$ $0.250$ \(\Q(\zeta_{50})\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{50}^{13}q^{2}-\zeta_{50}q^{4}-\zeta_{50}^{9}q^{5}+\cdots\)
500.2.a.a 500.a 1.a $2$ $3.993$ \(\Q(\sqrt{5}) \) None None \(0\) \(-1\) \(0\) \(-4\) $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(-3+2\beta )q^{7}+(-2+\beta )q^{9}+\cdots\)
500.2.a.b 500.a 1.a $2$ $3.993$ \(\Q(\sqrt{5}) \) None None \(0\) \(1\) \(0\) \(4\) $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(3-2\beta )q^{7}+(-2+\beta )q^{9}+\cdots\)
500.2.a.c 500.a 1.a $4$ $3.993$ 4.4.12400.1 None None \(0\) \(0\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-\beta _{3}q^{3}+(-\beta _{1}-\beta _{3})q^{7}+(2-3\beta _{2}+\cdots)q^{9}+\cdots\)
500.2.c.a 500.c 5.b $4$ $3.993$ 4.0.12400.1 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{3}+(-\beta _{1}+\beta _{3})q^{7}+(-2-3\beta _{2}+\cdots)q^{9}+\cdots\)
500.2.c.b 500.c 5.b $4$ $3.993$ \(\Q(i, \sqrt{5})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}+(2\beta _{1}+3\beta _{3})q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)
500.2.e.a 500.e 20.e $8$ $3.993$ \(\Q(\zeta_{20})\) \(\Q(\sqrt{-5}) \) None \(-8\) \(2\) \(0\) \(-6\) $\mathrm{U}(1)[D_{4}]$ \(q+(-1+\zeta_{20}^{2})q^{2}-\zeta_{20}^{3}q^{3}-2\zeta_{20}^{2}q^{4}+\cdots\)
500.2.e.b 500.e 20.e $8$ $3.993$ \(\Q(\zeta_{20})\) \(\Q(\sqrt{-5}) \) None \(8\) \(-2\) \(0\) \(6\) $\mathrm{U}(1)[D_{4}]$ \(q+(1+\zeta_{20})q^{2}+\zeta_{20}^{4}q^{3}+2\zeta_{20}q^{4}+\cdots\)
500.2.e.c 500.e 20.e $32$ $3.993$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
500.2.e.d 500.e 20.e $48$ $3.993$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
500.2.g.a 500.g 25.d $12$ $3.993$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(0\) \(-2\) \(0\) \(2\) $\mathrm{SU}(2)[C_{5}]$ \(q+\beta _{4}q^{3}+(1+\beta _{3}+\beta _{6}+\beta _{7})q^{7}+(1+\cdots)q^{9}+\cdots\)
500.2.g.b 500.g 25.d $16$ $3.993$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-\beta _{1}-\beta _{11}-\beta _{13})q^{3}+(-\beta _{1}+\beta _{10}+\cdots)q^{7}+\cdots\)
500.2.i.a 500.i 25.e $8$ $3.993$ 8.0.58140625.2 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{10}]$ \(q+(\beta _{1}+\beta _{5}+\beta _{7})q^{3}+(-\beta _{2}-\beta _{3}+\beta _{4}+\cdots)q^{7}+\cdots\)
500.2.i.b 500.i 25.e $24$ $3.993$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{10}]$
500.2.l.a 500.l 100.l $8$ $3.993$ \(\Q(\zeta_{20})\) \(\Q(\sqrt{-1}) \) None \(-2\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{20}]$ \(q+(-1+\zeta_{20}^{2}-\zeta_{20}^{3}-\zeta_{20}^{4}+\zeta_{20}^{6}+\cdots)q^{2}+\cdots\)
500.2.l.b 500.l 100.l $8$ $3.993$ \(\Q(\zeta_{20})\) \(\Q(\sqrt{-1}) \) None \(-2\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{20}]$ \(q+(-1+\zeta_{20}^{2}-\zeta_{20}^{3}-\zeta_{20}^{4}+\zeta_{20}^{6}+\cdots)q^{2}+\cdots\)
500.2.l.c 500.l 100.l $8$ $3.993$ \(\Q(\zeta_{20})\) \(\Q(\sqrt{-1}) \) None \(2\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{20}]$ \(q+(1-\zeta_{20}^{2}+\zeta_{20}^{3}+\zeta_{20}^{4}-\zeta_{20}^{6}+\cdots)q^{2}+\cdots\)
500.2.l.d 500.l 100.l $96$ $3.993$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{20}]$
500.2.l.e 500.l 100.l $96$ $3.993$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{20}]$
500.2.l.f 500.l 100.l $96$ $3.993$ None None \(10\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{20}]$
500.2.m.a 500.m 125.g $260$ $3.993$ None None \(0\) \(0\) \(5\) \(0\) $\mathrm{SU}(2)[C_{25}]$
500.2.o.a 500.o 125.h $240$ $3.993$ None None \(0\) \(0\) \(-5\) \(0\) $\mathrm{SU}(2)[C_{50}]$
500.2.r.a 500.r 500.r $40$ $3.993$ \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{100}]$
500.2.r.b 500.r 500.r $2880$ $3.993$ None None \(-40\) \(0\) \(-80\) \(0\) $\mathrm{SU}(2)[C_{100}]$
500.3.b.a 500.b 4.b $8$ $13.624$ \(\Q(\zeta_{20})\) \(\Q(\sqrt{-5}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\zeta_{20}q^{2}+(-\zeta_{20}-\zeta_{20}^{3})q^{3}-4q^{4}+\cdots\)
500.3.b.b 500.b 4.b $16$ $13.624$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}+\beta _{5}q^{3}+(1-\beta _{3})q^{4}+(-2+\cdots)q^{6}+\cdots\)
500.3.b.c 500.b 4.b $24$ $13.624$ None None \(-5\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
500.3.b.d 500.b 4.b $24$ $13.624$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
500.3.b.e 500.b 4.b $24$ $13.624$ None None \(5\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
500.3.d.a 500.d 20.d $4$ $13.624$ \(\Q(\zeta_{20})^+\) \(\Q(\sqrt{-5}) \) None \(-8\) \(-4\) \(0\) \(4\) $\mathrm{U}(1)[D_{2}]$ \(q-2q^{2}+(-1-\beta _{2})q^{3}+4q^{4}+(2+2\beta _{2}+\cdots)q^{6}+\cdots\)
500.3.d.b 500.d 20.d $4$ $13.624$ \(\Q(\zeta_{20})^+\) \(\Q(\sqrt{-5}) \) None \(8\) \(4\) \(0\) \(-4\) $\mathrm{U}(1)[D_{2}]$ \(q+2q^{2}+(1-\beta _{2})q^{3}+4q^{4}+(2-2\beta _{2}+\cdots)q^{6}+\cdots\)
500.3.d.c 500.d 20.d $16$ $13.624$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+\beta _{4}q^{3}+(-2+\beta _{3})q^{4}+(-1+\cdots)q^{6}+\cdots\)
500.3.d.d 500.d 20.d $24$ $13.624$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
500.3.d.e 500.d 20.d $48$ $13.624$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
500.3.f.a 500.f 5.c $16$ $13.624$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{7}q^{3}+\beta _{12}q^{7}+(\beta _{1}+\beta _{11})q^{9}+\cdots\)
500.3.f.b 500.f 5.c $16$ $13.624$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{5}q^{3}-\beta _{12}q^{7}+(-5\beta _{7}-\beta _{10}+\cdots)q^{9}+\cdots\)
500.4.a.a 500.a 1.a $4$ $29.501$ 4.4.13525.1 None None \(0\) \(0\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+(2\beta _{2}-\beta _{3})q^{7}+(-2-3\beta _{1}+\cdots)q^{9}+\cdots\)
500.4.a.b 500.a 1.a $6$ $29.501$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None None \(0\) \(-2\) \(0\) \(43\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+(7+\beta _{1}-\beta _{2}+\beta _{3}+\beta _{4}+\cdots)q^{7}+\cdots\)
500.4.a.c 500.a 1.a $6$ $29.501$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None None \(0\) \(2\) \(0\) \(-43\) $-$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+(-7-\beta _{1}+\beta _{2}-\beta _{3}-\beta _{4}+\cdots)q^{7}+\cdots\)
500.4.a.d 500.a 1.a $8$ $29.501$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+(-\beta _{2}-\beta _{5})q^{7}+(2^{4}+\beta _{4}+\cdots)q^{9}+\cdots\)
500.4.c.a 500.c 5.b $4$ $29.501$ 4.0.216400.1 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}+(2\beta _{1}-\beta _{3})q^{7}+(2+3\beta _{2}+\cdots)q^{9}+\cdots\)
500.4.c.b 500.c 5.b $8$ $29.501$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}+(\beta _{1}+\beta _{5})q^{7}+(-17-\beta _{4}+\cdots)q^{9}+\cdots\)
500.4.c.c 500.c 5.b $12$ $29.501$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}+(-\beta _{1}+\beta _{3})q^{7}+(-10-\beta _{4}+\cdots)q^{9}+\cdots\)
500.4.g.a 500.g 25.d $28$ $29.501$ None None \(0\) \(4\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{5}]$
500.4.g.b 500.g 25.d $64$ $29.501$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{5}]$
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