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Label Char Prim Dim $A$ Field CM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
690.2.a.a 690.a 1.a $1$ $5.510$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(-2\) $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}-2q^{7}+\cdots\)
690.2.a.b 690.a 1.a $1$ $5.510$ \(\Q\) None \(-1\) \(-1\) \(-1\) \(4\) $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+4q^{7}+\cdots\)
690.2.a.c 690.a 1.a $1$ $5.510$ \(\Q\) None \(-1\) \(-1\) \(1\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}-2q^{7}+\cdots\)
690.2.a.d 690.a 1.a $1$ $5.510$ \(\Q\) None \(-1\) \(-1\) \(1\) \(4\) $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+4q^{7}+\cdots\)
690.2.a.e 690.a 1.a $1$ $5.510$ \(\Q\) None \(-1\) \(1\) \(-1\) \(0\) $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{5}-q^{6}-q^{8}+\cdots\)
690.2.a.f 690.a 1.a $1$ $5.510$ \(\Q\) None \(-1\) \(1\) \(1\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-q^{8}+\cdots\)
690.2.a.g 690.a 1.a $1$ $5.510$ \(\Q\) None \(1\) \(-1\) \(-1\) \(-2\) $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-2q^{7}+\cdots\)
690.2.a.h 690.a 1.a $1$ $5.510$ \(\Q\) None \(1\) \(-1\) \(-1\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+q^{8}+\cdots\)
690.2.a.i 690.a 1.a $1$ $5.510$ \(\Q\) None \(1\) \(1\) \(-1\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{8}+\cdots\)
690.2.a.j 690.a 1.a $1$ $5.510$ \(\Q\) None \(1\) \(1\) \(1\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+q^{8}+\cdots\)
690.2.a.k 690.a 1.a $1$ $5.510$ \(\Q\) None \(1\) \(1\) \(1\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+q^{8}+\cdots\)
690.2.a.l 690.a 1.a $2$ $5.510$ \(\Q(\sqrt{17}) \) None \(2\) \(-2\) \(2\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+(-1+\cdots)q^{7}+\cdots\)
690.2.d.a 690.d 5.b $4$ $5.510$ \(\Q(i, \sqrt{5})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+\beta _{1}q^{3}-q^{4}+\beta _{3}q^{5}+q^{6}+\cdots\)
690.2.d.b 690.d 5.b $4$ $5.510$ \(\Q(\zeta_{8})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}^{2}q^{2}-\zeta_{8}^{2}q^{3}-q^{4}+(2\zeta_{8}+\zeta_{8}^{3})q^{5}+\cdots\)
690.2.d.c 690.d 5.b $6$ $5.510$ 6.0.350464.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+\beta _{1}q^{3}-q^{4}+\beta _{3}q^{5}-q^{6}+\cdots\)
690.2.d.d 690.d 5.b $6$ $5.510$ 6.0.5161984.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{2}-\beta _{3}q^{3}-q^{4}-\beta _{4}q^{5}-q^{6}+\cdots\)
690.2.e.a 690.e 69.c $16$ $5.510$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(-16\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{9}q^{2}+\beta _{2}q^{3}-q^{4}-q^{5}+\beta _{1}q^{6}+\cdots\)
690.2.e.b 690.e 69.c $16$ $5.510$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(16\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{9}q^{2}+\beta _{2}q^{3}-q^{4}+q^{5}+\beta _{1}q^{6}+\cdots\)
690.2.h.a 690.h 345.h $24$ $5.510$ None \(-24\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
690.2.h.b 690.h 345.h $24$ $5.510$ None \(24\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
690.2.i.a 690.i 15.e $4$ $5.510$ \(\Q(\zeta_{8})\) None \(0\) \(4\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{2}+(1+\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+\zeta_{8}^{2}q^{4}+\cdots\)
690.2.i.b 690.i 15.e $4$ $5.510$ \(\Q(\zeta_{8})\) None \(0\) \(4\) \(0\) \(12\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{2}+(1+\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+\zeta_{8}^{2}q^{4}+\cdots\)
690.2.i.c 690.i 15.e $8$ $5.510$ \(\Q(\zeta_{24})\) None \(0\) \(-8\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{24}^{5}q^{2}+(-1+\zeta_{24}+\zeta_{24}^{3})q^{3}+\cdots\)
690.2.i.d 690.i 15.e $8$ $5.510$ 8.0.40960000.1 None \(0\) \(-8\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{6}q^{2}+(-1+\beta _{1}+\beta _{3})q^{3}-\beta _{3}q^{4}+\cdots\)
690.2.i.e 690.i 15.e $32$ $5.510$ None \(0\) \(4\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$
690.2.i.f 690.i 15.e $32$ $5.510$ None \(0\) \(4\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$
690.2.j.a 690.j 115.e $24$ $5.510$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
690.2.j.b 690.j 115.e $24$ $5.510$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
690.2.m.a 690.m 23.c $10$ $5.510$ \(\Q(\zeta_{22})\) None \(-1\) \(-1\) \(1\) \(0\) $\mathrm{SU}(2)[C_{11}]$ \(q+\zeta_{22}^{4}q^{2}+(-1+\zeta_{22}-\zeta_{22}^{2}+\zeta_{22}^{3}+\cdots)q^{3}+\cdots\)
690.2.m.b 690.m 23.c $10$ $5.510$ \(\Q(\zeta_{22})\) None \(1\) \(-1\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{11}]$ \(q-\zeta_{22}^{4}q^{2}+(-1+\zeta_{22}-\zeta_{22}^{2}+\zeta_{22}^{3}+\cdots)q^{3}+\cdots\)
690.2.m.c 690.m 23.c $20$ $5.510$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(-2\) \(2\) \(-2\) \(2\) $\mathrm{SU}(2)[C_{11}]$ \(q+\beta _{16}q^{2}-\beta _{14}q^{3}-\beta _{12}q^{4}+\beta _{11}q^{5}+\cdots\)
690.2.m.d 690.m 23.c $20$ $5.510$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(-2\) \(2\) \(2\) \(2\) $\mathrm{SU}(2)[C_{11}]$ \(q+\beta _{7}q^{2}-\beta _{12}q^{3}+\beta _{10}q^{4}-\beta _{6}q^{5}+\cdots\)
690.2.m.e 690.m 23.c $20$ $5.510$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(2\) \(2\) \(-2\) \(-2\) $\mathrm{SU}(2)[C_{11}]$ \(q-\beta _{13}q^{2}-\beta _{16}q^{3}-\beta _{2}q^{4}+(-1+\cdots)q^{5}+\cdots\)
690.2.m.f 690.m 23.c $20$ $5.510$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(2\) \(2\) \(2\) \(-2\) $\mathrm{SU}(2)[C_{11}]$ \(q+\beta _{8}q^{2}+(1+\beta _{5}+\beta _{6}-\beta _{7}-\beta _{8}+\cdots)q^{3}+\cdots\)
690.2.m.g 690.m 23.c $30$ $5.510$ None \(-3\) \(-3\) \(-3\) \(-8\) $\mathrm{SU}(2)[C_{11}]$
690.2.m.h 690.m 23.c $30$ $5.510$ None \(3\) \(-3\) \(3\) \(8\) $\mathrm{SU}(2)[C_{11}]$
690.2.n.a 690.n 345.n $240$ $5.510$ None \(-24\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{22}]$
690.2.n.b 690.n 345.n $240$ $5.510$ None \(24\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{22}]$
690.2.q.a 690.q 69.g $160$ $5.510$ None \(0\) \(0\) \(-16\) \(0\) $\mathrm{SU}(2)[C_{22}]$
690.2.q.b 690.q 69.g $160$ $5.510$ None \(0\) \(0\) \(16\) \(0\) $\mathrm{SU}(2)[C_{22}]$
690.2.r.a 690.r 115.j $120$ $5.510$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{22}]$
690.2.r.b 690.r 115.j $120$ $5.510$ None \(0\) \(0\) \(8\) \(0\) $\mathrm{SU}(2)[C_{22}]$
690.2.w.a 690.w 115.l $240$ $5.510$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{44}]$
690.2.w.b 690.w 115.l $240$ $5.510$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{44}]$
690.2.x.a 690.x 345.x $960$ $5.510$ None \(0\) \(8\) \(0\) \(0\) $\mathrm{SU}(2)[C_{44}]$
690.3.b.a 690.b 15.d $88$ $18.801$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
690.3.c.a 690.c 23.b $32$ $18.801$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
690.3.f.a 690.f 115.c $48$ $18.801$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
690.3.g.a 690.g 3.b $56$ $18.801$ None \(0\) \(-8\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{2}]$
690.3.k.a 690.k 5.c $40$ $18.801$ None \(40\) \(0\) \(-8\) \(-8\) $\mathrm{SU}(2)[C_{4}]$
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