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Label Char Prim Dim $A$ Field CM RM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
640.1.e.a 640.e 40.e $1$ $0.319$ \(\Q\) \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-10}) \) \(\Q(\sqrt{10}) \) \(0\) \(0\) \(-1\) \(0\) \(q-q^{5}+q^{9}+2q^{13}+q^{25}-2q^{37}+\cdots\)
640.1.e.b 640.e 40.e $1$ $0.319$ \(\Q\) \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-10}) \) \(\Q(\sqrt{10}) \) \(0\) \(0\) \(1\) \(0\) \(q+q^{5}+q^{9}-2q^{13}+q^{25}+2q^{37}+\cdots\)
640.1.e.c 640.e 40.e $4$ $0.319$ \(\Q(\zeta_{8})\) \(\Q(\sqrt{-5}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(-\zeta_{8}-\zeta_{8}^{3})q^{3}+\zeta_{8}^{2}q^{5}+(\zeta_{8}-\zeta_{8}^{3}+\cdots)q^{7}+\cdots\)
640.1.m.a 640.m 40.i $2$ $0.319$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-iq^{5}-iq^{9}+(-1-i)q^{13}+(1+i+\cdots)q^{17}+\cdots\)
640.1.m.b 640.m 40.i $2$ $0.319$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q+iq^{5}-iq^{9}+(1+i)q^{13}+(1+i+\cdots)q^{17}+\cdots\)
640.2.a.a 640.a 1.a $1$ $5.110$ \(\Q\) None None \(0\) \(-2\) \(-1\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-q^{5}+q^{9}-2q^{11}+2q^{13}+\cdots\)
640.2.a.b 640.a 1.a $1$ $5.110$ \(\Q\) None None \(0\) \(-2\) \(1\) \(0\) $+$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{5}+q^{9}-2q^{11}-2q^{13}+\cdots\)
640.2.a.c 640.a 1.a $1$ $5.110$ \(\Q\) None None \(0\) \(0\) \(-1\) \(-2\) $+$ $\mathrm{SU}(2)$ \(q-q^{5}-2q^{7}-3q^{9}+6q^{11}-2q^{13}+\cdots\)
640.2.a.d 640.a 1.a $1$ $5.110$ \(\Q\) None None \(0\) \(0\) \(-1\) \(2\) $+$ $\mathrm{SU}(2)$ \(q-q^{5}+2q^{7}-3q^{9}-6q^{11}-2q^{13}+\cdots\)
640.2.a.e 640.a 1.a $1$ $5.110$ \(\Q\) None None \(0\) \(0\) \(1\) \(-2\) $+$ $\mathrm{SU}(2)$ \(q+q^{5}-2q^{7}-3q^{9}-6q^{11}+2q^{13}+\cdots\)
640.2.a.f 640.a 1.a $1$ $5.110$ \(\Q\) None None \(0\) \(0\) \(1\) \(2\) $-$ $\mathrm{SU}(2)$ \(q+q^{5}+2q^{7}-3q^{9}+6q^{11}+2q^{13}+\cdots\)
640.2.a.g 640.a 1.a $1$ $5.110$ \(\Q\) None None \(0\) \(2\) \(-1\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+2q^{3}-q^{5}+q^{9}+2q^{11}+2q^{13}+\cdots\)
640.2.a.h 640.a 1.a $1$ $5.110$ \(\Q\) None None \(0\) \(2\) \(1\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+q^{5}+q^{9}+2q^{11}-2q^{13}+\cdots\)
640.2.a.i 640.a 1.a $2$ $5.110$ \(\Q(\sqrt{5}) \) None None \(0\) \(-2\) \(-2\) \(-2\) $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{3}-q^{5}+(-1+\beta )q^{7}+\cdots\)
640.2.a.j 640.a 1.a $2$ $5.110$ \(\Q(\sqrt{5}) \) None None \(0\) \(-2\) \(2\) \(2\) $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{3}+q^{5}+(1-\beta )q^{7}+(3+\cdots)q^{9}+\cdots\)
640.2.a.k 640.a 1.a $2$ $5.110$ \(\Q(\sqrt{5}) \) None None \(0\) \(2\) \(-2\) \(2\) $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}-q^{5}+(1-\beta )q^{7}+(3+2\beta )q^{9}+\cdots\)
640.2.a.l 640.a 1.a $2$ $5.110$ \(\Q(\sqrt{5}) \) None None \(0\) \(2\) \(2\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+q^{5}+(-1+\beta )q^{7}+(3+\cdots)q^{9}+\cdots\)
640.2.c.a 640.c 5.b $6$ $5.110$ 6.0.350464.1 None None \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{3}+(\beta _{2}+\beta _{5})q^{5}+(\beta _{2}-\beta _{3}-\beta _{4}+\cdots)q^{7}+\cdots\)
640.2.c.b 640.c 5.b $6$ $5.110$ 6.0.350464.1 None None \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{3}+(-\beta _{2}+\beta _{3})q^{5}+(\beta _{2}-\beta _{3}+\cdots)q^{7}+\cdots\)
640.2.c.c 640.c 5.b $6$ $5.110$ 6.0.350464.1 None None \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{3}+(-\beta _{2}-\beta _{5})q^{5}+(-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
640.2.c.d 640.c 5.b $6$ $5.110$ 6.0.350464.1 None None \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{3}+(\beta _{2}-\beta _{3})q^{5}+(-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
640.2.d.a 640.d 8.b $4$ $5.110$ \(\Q(i, \sqrt{10})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{3}+\beta _{1}q^{5}+\beta _{3}q^{7}-7q^{9}-6\beta _{1}q^{13}+\cdots\)
640.2.d.b 640.d 8.b $4$ $5.110$ \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{8}^{2}q^{3}+\zeta_{8}q^{5}+3\zeta_{8}^{3}q^{7}+q^{9}+\cdots\)
640.2.d.c 640.d 8.b $4$ $5.110$ \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{8}^{2}q^{3}+\zeta_{8}q^{5}-\zeta_{8}^{3}q^{7}+q^{9}+\cdots\)
640.2.d.d 640.d 8.b $4$ $5.110$ \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{8}^{2}q^{3}+\zeta_{8}q^{5}-3\zeta_{8}^{3}q^{7}+q^{9}+\cdots\)
640.2.f.a 640.f 40.f $2$ $5.110$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(-1+i)q^{5}-3q^{9}-6q^{13}-4iq^{17}+\cdots\)
640.2.f.b 640.f 40.f $2$ $5.110$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(2\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(1+i)q^{5}-3q^{9}+6q^{13}+4iq^{17}+\cdots\)
640.2.f.c 640.f 40.f $4$ $5.110$ \(\Q(i, \sqrt{6})\) None None \(0\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{3}+(-2-\beta _{1})q^{5}+\beta _{2}q^{7}+3q^{9}+\cdots\)
640.2.f.d 640.f 40.f $4$ $5.110$ \(\Q(\sqrt{-2}, \sqrt{5})\) \(\Q(\sqrt{-10}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{3}q^{5}-\beta _{1}q^{7}-3q^{9}-\beta _{2}q^{11}+\cdots\)
640.2.f.e 640.f 40.f $4$ $5.110$ \(\Q(\sqrt{2}, \sqrt{-5})\) \(\Q(\sqrt{-5}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{1}q^{3}-\beta _{3}q^{5}+\beta _{2}q^{7}-q^{9}+\beta _{2}q^{15}+\cdots\)
640.2.f.f 640.f 40.f $4$ $5.110$ \(\Q(\sqrt{-2}, \sqrt{-5})\) \(\Q(\sqrt{-5}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\beta _{3}q^{3}-\beta _{2}q^{5}-3\beta _{1}q^{7}+7q^{9}+\cdots\)
640.2.f.g 640.f 40.f $4$ $5.110$ \(\Q(i, \sqrt{6})\) None None \(0\) \(0\) \(8\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{3}+(2+\beta _{1})q^{5}+\beta _{2}q^{7}+3q^{9}+\cdots\)
640.2.j.a 640.j 80.j $2$ $5.110$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(-2\) \(-6\) $\mathrm{SU}(2)[C_{4}]$ \(q-2iq^{3}+(-1-2i)q^{5}+(-3-3i)q^{7}+\cdots\)
640.2.j.b 640.j 80.j $2$ $5.110$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(-2\) \(6\) $\mathrm{SU}(2)[C_{4}]$ \(q+2iq^{3}+(-1-2i)q^{5}+(3+3i)q^{7}+\cdots\)
640.2.j.c 640.j 80.j $18$ $5.110$ \(\mathbb{Q}[x]/(x^{18} + \cdots)\) None None \(0\) \(0\) \(4\) \(-2\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{12}q^{3}-\beta _{6}q^{5}-\beta _{11}q^{7}+(-1+\cdots)q^{9}+\cdots\)
640.2.j.d 640.j 80.j $18$ $5.110$ \(\mathbb{Q}[x]/(x^{18} + \cdots)\) None None \(0\) \(0\) \(4\) \(2\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{12}q^{3}+\beta _{4}q^{5}-\beta _{10}q^{7}+(-1+\cdots)q^{9}+\cdots\)
640.2.l.a 640.l 16.e $16$ $5.110$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\beta _{5}-\beta _{7})q^{3}-\beta _{1}q^{5}+(\beta _{1}-\beta _{4}-\beta _{7}+\cdots)q^{7}+\cdots\)
640.2.l.b 640.l 16.e $16$ $5.110$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-\beta _{5}+\beta _{7})q^{3}-\beta _{1}q^{5}+(-\beta _{1}+\beta _{4}+\cdots)q^{7}+\cdots\)
640.2.n.a 640.n 20.e $12$ $5.110$ 12.0.\(\cdots\).1 None None \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{2}q^{3}+\beta _{9}q^{5}+(-1-\beta _{3}-\beta _{5}+\cdots)q^{7}+\cdots\)
640.2.n.b 640.n 20.e $12$ $5.110$ 12.0.\(\cdots\).1 None None \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{3}q^{3}-\beta _{8}q^{5}+(-1-\beta _{2}+\beta _{4}+\cdots)q^{7}+\cdots\)
640.2.n.c 640.n 20.e $12$ $5.110$ 12.0.\(\cdots\).1 None None \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{3}q^{3}+\beta _{8}q^{5}+(1+\beta _{2}-\beta _{4}-\beta _{6}+\cdots)q^{7}+\cdots\)
640.2.n.d 640.n 20.e $12$ $5.110$ 12.0.\(\cdots\).1 None None \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{2}q^{3}-\beta _{9}q^{5}+(1+\beta _{3}+\beta _{5}-\beta _{7}+\cdots)q^{7}+\cdots\)
640.2.o.a 640.o 40.k $2$ $5.110$ \(\Q(\sqrt{-1}) \) None None \(0\) \(-4\) \(-4\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2+2i)q^{3}+(-2+i)q^{5}+(-2+\cdots)q^{7}+\cdots\)
640.2.o.b 640.o 40.k $2$ $5.110$ \(\Q(\sqrt{-1}) \) None None \(0\) \(-4\) \(4\) \(4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2+2i)q^{3}+(2-i)q^{5}+(2-2i)q^{7}+\cdots\)
640.2.o.c 640.o 40.k $2$ $5.110$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-4\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(-2+i)q^{5}+3iq^{9}+(-5-5i)q^{13}+\cdots\)
640.2.o.d 640.o 40.k $2$ $5.110$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-4\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(-2-i)q^{5}+3iq^{9}+(1+i)q^{13}+\cdots\)
640.2.o.e 640.o 40.k $2$ $5.110$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(2+i)q^{5}+3iq^{9}+(-1-i)q^{13}+\cdots\)
640.2.o.f 640.o 40.k $2$ $5.110$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(4\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(2-i)q^{5}+3iq^{9}+(5+5i)q^{13}+\cdots\)
640.2.o.g 640.o 40.k $2$ $5.110$ \(\Q(\sqrt{-1}) \) None None \(0\) \(4\) \(-4\) \(4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(2-2i)q^{3}+(-2+i)q^{5}+(2-2i)q^{7}+\cdots\)
640.2.o.h 640.o 40.k $2$ $5.110$ \(\Q(\sqrt{-1}) \) None None \(0\) \(4\) \(4\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(2-2i)q^{3}+(2-i)q^{5}+(-2+2i)q^{7}+\cdots\)
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