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Label Char Prim Dim $A$ Field CM RM Minimal twist Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
520.1.b.a 520.b 520.b $2$ $0.260$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{-26}) \) \(\Q(\sqrt{65}) \) 520.1.b.a \(0\) \(0\) \(0\) \(0\) \(q-iq^{2}-q^{4}+iq^{5}-iq^{7}+iq^{8}+\cdots\)
520.1.b.b 520.b 520.b $4$ $0.260$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-26}) \) None 520.1.b.b \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{12}^{3}q^{2}+(-\zeta_{12}^{2}-\zeta_{12}^{4})q^{3}+\cdots\)
520.1.bg.a 520.bg 520.ag $4$ $0.260$ \(\Q(\zeta_{8})\) None \(\Q(\sqrt{26}) \) 520.1.bg.a \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}-\zeta_{8}q^{5}-\zeta_{8}^{3}q^{8}+\cdots\)
520.1.bo.a 520.bo 520.ao $4$ $0.260$ \(\Q(\zeta_{8})\) None \(\Q(\sqrt{10}) \) 520.1.bo.a \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{8}^{3}q^{2}+(-\zeta_{8}+\zeta_{8}^{3})q^{3}-\zeta_{8}^{2}q^{4}+\cdots\)
520.1.bx.a 520.bx 520.ax $2$ $0.260$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-10}) \) None 520.1.bx.a \(-1\) \(0\) \(2\) \(1\) \(q-\zeta_{6}q^{2}+\zeta_{6}^{2}q^{4}+q^{5}-\zeta_{6}^{2}q^{7}+\cdots\)
520.1.bx.b 520.bx 520.ax $2$ $0.260$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-10}) \) None 520.1.bx.a \(1\) \(0\) \(-2\) \(-1\) \(q+\zeta_{6}q^{2}+\zeta_{6}^{2}q^{4}-q^{5}+\zeta_{6}^{2}q^{7}+\cdots\)
520.1.cd.a 520.cd 520.bd $4$ $0.260$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-10}) \) None 520.1.cd.a \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{12}^{5}q^{2}-\zeta_{12}^{4}q^{4}+\zeta_{12}^{3}q^{5}+\cdots\)
520.2.a.a 520.a 1.a $1$ $4.152$ \(\Q\) None None 520.2.a.a \(0\) \(0\) \(-1\) \(0\) $+$ $\mathrm{SU}(2)$ \(q-q^{5}-3q^{9}-4q^{11}-q^{13}-6q^{17}+\cdots\)
520.2.a.b 520.a 1.a $1$ $4.152$ \(\Q\) None None 520.2.a.b \(0\) \(2\) \(1\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{5}+q^{9}+2q^{11}+q^{13}+\cdots\)
520.2.a.c 520.a 1.a $2$ $4.152$ \(\Q(\sqrt{2}) \) None None 520.2.a.c \(0\) \(-4\) \(2\) \(-4\) $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta )q^{3}+q^{5}-2q^{7}+(3-4\beta )q^{9}+\cdots\)
520.2.a.d 520.a 1.a $2$ $4.152$ \(\Q(\sqrt{3}) \) None None 520.2.a.d \(0\) \(-2\) \(-2\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}-q^{5}-2\beta q^{7}+(1-2\beta )q^{9}+\cdots\)
520.2.a.e 520.a 1.a $2$ $4.152$ \(\Q(\sqrt{3}) \) None None 520.2.a.e \(0\) \(-2\) \(-2\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{3}-q^{5}+2\beta q^{7}+(1-2\beta )q^{9}+\cdots\)
520.2.a.f 520.a 1.a $2$ $4.152$ \(\Q(\sqrt{6}) \) None None 520.2.a.f \(0\) \(0\) \(2\) \(4\) $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+q^{5}+2q^{7}+3q^{9}+(-2+\beta )q^{11}+\cdots\)
520.2.a.g 520.a 1.a $2$ $4.152$ \(\Q(\sqrt{5}) \) None None 520.2.a.g \(0\) \(2\) \(-2\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}-q^{5}+(3+2\beta )q^{9}+(3-\beta )q^{11}+\cdots\)
520.2.d.a 520.d 5.b $2$ $4.152$ \(\Q(\sqrt{-1}) \) None None 520.2.d.a \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(2+i)q^{5}-4iq^{7}+3q^{9}-2q^{11}+\cdots\)
520.2.d.b 520.d 5.b $6$ $4.152$ 6.0.350464.1 None None 520.2.d.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{3}+\beta _{4})q^{3}+(\beta _{1}+\beta _{4})q^{5}+(-\beta _{3}+\cdots)q^{7}+\cdots\)
520.2.d.c 520.d 5.b $10$ $4.152$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None None 520.2.d.c \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{9}q^{3}-\beta _{1}q^{5}-\beta _{4}q^{7}+(-4-\beta _{5}+\cdots)q^{9}+\cdots\)
520.2.e.a 520.e 104.e $28$ $4.152$ None None 520.2.e.a \(0\) \(0\) \(-28\) \(0\) $\mathrm{SU}(2)[C_{2}]$
520.2.e.b 520.e 104.e $28$ $4.152$ None None 520.2.e.a \(0\) \(0\) \(28\) \(0\) $\mathrm{SU}(2)[C_{2}]$
520.2.f.a 520.f 65.d $10$ $4.152$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None None 520.2.f.a \(0\) \(0\) \(-3\) \(2\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{3}+\beta _{6}q^{5}-\beta _{2}q^{7}+(-1-\beta _{1}+\cdots)q^{9}+\cdots\)
520.2.f.b 520.f 65.d $10$ $4.152$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None None 520.2.f.a \(0\) \(0\) \(3\) \(-2\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{3}-\beta _{6}q^{5}+\beta _{2}q^{7}+(-1-\beta _{1}+\cdots)q^{9}+\cdots\)
520.2.g.a 520.g 8.b $24$ $4.152$ None None 520.2.g.a \(2\) \(0\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{2}]$
520.2.g.b 520.g 8.b $24$ $4.152$ None None 520.2.g.b \(2\) \(0\) \(0\) \(24\) $\mathrm{SU}(2)[C_{2}]$
520.2.j.a 520.j 40.f $36$ $4.152$ None None 520.2.j.a \(0\) \(-8\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
520.2.j.b 520.j 40.f $36$ $4.152$ None None 520.2.j.a \(0\) \(8\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
520.2.k.a 520.k 13.b $6$ $4.152$ 6.0.350464.1 None None 520.2.k.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}-\beta _{3}q^{5}+(\beta _{3}-\beta _{4}+\beta _{5})q^{7}+\cdots\)
520.2.k.b 520.k 13.b $8$ $4.152$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None None 520.2.k.b \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{3}-\beta _{2}q^{5}+(\beta _{2}+\beta _{4}-\beta _{5})q^{7}+\cdots\)
520.2.p.a 520.p 520.p $8$ $4.152$ 8.0.207360000.1 None None 520.2.p.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-\beta _{4}+\beta _{7})q^{3}-\beta _{5}q^{4}+\cdots\)
520.2.p.b 520.p 520.p $72$ $4.152$ None None 520.2.p.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
520.2.q.a 520.q 13.c $2$ $4.152$ \(\Q(\sqrt{-3}) \) None None 520.2.q.a \(0\) \(-2\) \(-2\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{3}-q^{5}-3\zeta_{6}q^{7}-\zeta_{6}q^{9}+\cdots\)
520.2.q.b 520.q 13.c $2$ $4.152$ \(\Q(\sqrt{-3}) \) None None 520.2.q.b \(0\) \(1\) \(-2\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-q^{5}+3\zeta_{6}q^{7}+2\zeta_{6}q^{9}+\cdots\)
520.2.q.c 520.q 13.c $2$ $4.152$ \(\Q(\sqrt{-3}) \) None None 520.2.q.c \(0\) \(1\) \(-2\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-q^{5}+3\zeta_{6}q^{7}+2\zeta_{6}q^{9}+\cdots\)
520.2.q.d 520.q 13.c $2$ $4.152$ \(\Q(\sqrt{-3}) \) None None 520.2.q.d \(0\) \(1\) \(2\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+q^{5}+3\zeta_{6}q^{7}+2\zeta_{6}q^{9}+\cdots\)
520.2.q.e 520.q 13.c $2$ $4.152$ \(\Q(\sqrt{-3}) \) None None 520.2.q.e \(0\) \(3\) \(-2\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\zeta_{6})q^{3}-q^{5}-3\zeta_{6}q^{7}-6\zeta_{6}q^{9}+\cdots\)
520.2.q.f 520.q 13.c $4$ $4.152$ \(\Q(\sqrt{-3}, \sqrt{17})\) None None 520.2.q.f \(0\) \(1\) \(4\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{3}+q^{5}+(-1+\beta _{2})q^{7}+(-1+\cdots)q^{9}+\cdots\)
520.2.q.g 520.q 13.c $6$ $4.152$ 6.0.27870912.1 None None 520.2.q.g \(0\) \(-1\) \(-6\) \(-9\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{3}-q^{5}+(-3+3\beta _{4})q^{7}+(-1+\cdots)q^{9}+\cdots\)
520.2.q.h 520.q 13.c $8$ $4.152$ 8.0.649638144.4 None None 520.2.q.h \(0\) \(0\) \(8\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{4}q^{3}+q^{5}+(1+\beta _{3}+\beta _{4}-\beta _{5}+\cdots)q^{7}+\cdots\)
520.2.t.a 520.t 520.t $4$ $4.152$ \(\Q(i, \sqrt{10})\) \(\Q(\sqrt{-10}) \) None 520.2.t.a \(-4\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(-1-\beta _{2})q^{2}+2\beta _{2}q^{4}+\beta _{1}q^{5}+\cdots\)
520.2.t.b 520.t 520.t $4$ $4.152$ \(\Q(i, \sqrt{10})\) \(\Q(\sqrt{-10}) \) None 520.2.t.a \(4\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(1-\beta _{2})q^{2}-2\beta _{2}q^{4}-\beta _{3}q^{5}+2\beta _{1}q^{7}+\cdots\)
520.2.t.c 520.t 520.t $152$ $4.152$ None None 520.2.t.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
520.2.w.a 520.w 65.k $2$ $4.152$ \(\Q(\sqrt{-1}) \) None None 520.2.w.a \(0\) \(-2\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{3}+(-2-i)q^{5}-iq^{9}+\cdots\)
520.2.w.b 520.w 65.k $2$ $4.152$ \(\Q(\sqrt{-1}) \) None None 520.2.w.b \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2-i)q^{5}+4iq^{7}-3iq^{9}+(4+\cdots)q^{11}+\cdots\)
520.2.w.c 520.w 65.k $2$ $4.152$ \(\Q(\sqrt{-1}) \) None None 520.2.w.c \(0\) \(2\) \(2\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{3}+(1-2i)q^{5}-2iq^{7}-iq^{9}+\cdots\)
520.2.w.d 520.w 65.k $4$ $4.152$ \(\Q(\zeta_{8})\) None None 520.2.w.d \(0\) \(-4\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\zeta_{8}-\zeta_{8}^{2})q^{3}+(-1+2\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
520.2.w.e 520.w 65.k $12$ $4.152$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None 520.2.w.e \(0\) \(4\) \(8\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{3}+(1-\beta _{7})q^{5}-\beta _{10}q^{7}+(\beta _{1}+\cdots)q^{9}+\cdots\)
520.2.w.f 520.w 65.k $20$ $4.152$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None None 520.2.w.f \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{3}q^{3}+\beta _{7}q^{5}-\beta _{11}q^{7}+(-\beta _{1}+\cdots)q^{9}+\cdots\)
520.2.y.a 520.y 520.y $160$ $4.152$ None None 520.2.y.a \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
520.2.bc.a 520.bc 520.ac $12$ $4.152$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) \(\Q(\sqrt{-26}) \) None 520.2.bc.a \(-12\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(-1-\beta _{4})q^{2}+\beta _{3}q^{3}+2\beta _{4}q^{4}+\cdots\)
520.2.bc.b 520.bc 520.ac $12$ $4.152$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) \(\Q(\sqrt{-26}) \) None 520.2.bc.a \(12\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(1-\beta _{3})q^{2}-\beta _{10}q^{3}-2\beta _{3}q^{4}-\beta _{9}q^{5}+\cdots\)
520.2.bc.c 520.bc 520.ac $136$ $4.152$ None None 520.2.bc.c \(0\) \(-8\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
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