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Label Char Prim Dim $A$ Field CM RM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
688.1.b.a 688.b 43.b $1$ $0.343$ \(\Q\) \(\Q(\sqrt{-43}) \) None \(0\) \(0\) \(0\) \(0\) \(q+q^{9}+q^{11}-q^{13}-q^{17}+q^{23}+\cdots\)
688.1.b.b 688.b 43.b $2$ $0.343$ \(\Q(\sqrt{-2}) \) None None \(0\) \(0\) \(0\) \(0\) \(q+\beta q^{3}-\beta q^{5}-\beta q^{7}-q^{9}+q^{11}+\cdots\)
688.1.r.a 688.r 172.g $4$ $0.343$ \(\Q(\zeta_{12})\) None None \(0\) \(0\) \(-2\) \(0\) \(q-\zeta_{12}q^{3}+\zeta_{12}^{4}q^{5}+\zeta_{12}^{5}q^{7}-\zeta_{12}^{3}q^{11}+\cdots\)
688.2.a.a 688.a 1.a $1$ $5.494$ \(\Q\) None None \(0\) \(0\) \(-2\) \(2\) $+$ $\mathrm{SU}(2)$ \(q-2q^{5}+2q^{7}-3q^{9}-q^{11}-q^{13}+\cdots\)
688.2.a.b 688.a 1.a $1$ $5.494$ \(\Q\) None None \(0\) \(2\) \(-4\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+2q^{3}-4q^{5}+q^{9}-3q^{11}-5q^{13}+\cdots\)
688.2.a.c 688.a 1.a $1$ $5.494$ \(\Q\) None None \(0\) \(2\) \(0\) \(4\) $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+4q^{7}+q^{9}+3q^{11}-q^{13}+\cdots\)
688.2.a.d 688.a 1.a $2$ $5.494$ \(\Q(\sqrt{2}) \) None None \(0\) \(-4\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta )q^{3}+\beta q^{5}-\beta q^{7}+(3-4\beta )q^{9}+\cdots\)
688.2.a.e 688.a 1.a $2$ $5.494$ \(\Q(\sqrt{5}) \) None None \(0\) \(-1\) \(-3\) \(0\) $+$ $\mathrm{SU}(2)$ \(q-\beta q^{3}+(-1-\beta )q^{5}+(-2+4\beta )q^{7}+\cdots\)
688.2.a.f 688.a 1.a $2$ $5.494$ \(\Q(\sqrt{2}) \) None None \(0\) \(0\) \(4\) \(4\) $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(2-\beta )q^{5}+(2-\beta )q^{7}-q^{9}+\cdots\)
688.2.a.g 688.a 1.a $2$ $5.494$ \(\Q(\sqrt{21}) \) None None \(0\) \(1\) \(3\) \(-4\) $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+(1+\beta )q^{5}-2q^{7}+(2+\beta )q^{9}+\cdots\)
688.2.a.h 688.a 1.a $2$ $5.494$ \(\Q(\sqrt{3}) \) None None \(0\) \(2\) \(-2\) \(2\) $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}+(-1+\beta )q^{5}+(1-\beta )q^{7}+\cdots\)
688.2.a.i 688.a 1.a $3$ $5.494$ 3.3.229.1 None None \(0\) \(-3\) \(1\) \(-6\) $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{3}-\beta _{2}q^{5}-2q^{7}+(1+\cdots)q^{9}+\cdots\)
688.2.a.j 688.a 1.a $5$ $5.494$ 5.5.7998268.1 None None \(0\) \(1\) \(1\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}-\beta _{4}q^{5}+(-1+\beta _{2}-\beta _{4})q^{7}+\cdots\)
688.2.g.a 688.g 172.d $2$ $5.494$ \(\Q(\sqrt{-3}) \) None None \(0\) \(-4\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q-2q^{3}-2\zeta_{6}q^{5}-2q^{7}+q^{9}-\zeta_{6}q^{11}+\cdots\)
688.2.g.b 688.g 172.d $2$ $5.494$ \(\Q(\sqrt{-43}) \) \(\Q(\sqrt{-43}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-3q^{9}-\beta q^{11}+3q^{13}-5q^{17}-\beta q^{23}+\cdots\)
688.2.g.c 688.g 172.d $2$ $5.494$ \(\Q(\sqrt{-3}) \) None None \(0\) \(4\) \(0\) \(4\) $\mathrm{SU}(2)[C_{2}]$ \(q+2q^{3}+2\zeta_{6}q^{5}+2q^{7}+q^{9}-\zeta_{6}q^{11}+\cdots\)
688.2.g.d 688.g 172.d $4$ $5.494$ \(\Q(\sqrt{-3}, \sqrt{-43})\) \(\Q(\sqrt{-43}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-3q^{9}+(-\beta _{1}-\beta _{3})q^{11}+(-2+\beta _{2}+\cdots)q^{13}+\cdots\)
688.2.g.e 688.g 172.d $12$ $5.494$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{3}+\beta _{8}q^{5}+\beta _{10}q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)
688.2.i.a 688.i 43.c $2$ $5.494$ \(\Q(\sqrt{-3}) \) None None \(0\) \(-3\) \(-1\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{3}+(-1+\zeta_{6})q^{5}+\zeta_{6}q^{7}+\cdots\)
688.2.i.b 688.i 43.c $2$ $5.494$ \(\Q(\sqrt{-3}) \) None None \(0\) \(-1\) \(1\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+(1-\zeta_{6})q^{5}+5\zeta_{6}q^{7}+\cdots\)
688.2.i.c 688.i 43.c $2$ $5.494$ \(\Q(\sqrt{-3}) \) None None \(0\) \(1\) \(-3\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+(-3+3\zeta_{6})q^{5}-\zeta_{6}q^{7}+\cdots\)
688.2.i.d 688.i 43.c $2$ $5.494$ \(\Q(\sqrt{-3}) \) None None \(0\) \(1\) \(1\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+(1-\zeta_{6})q^{5}+3\zeta_{6}q^{7}+\cdots\)
688.2.i.e 688.i 43.c $4$ $5.494$ \(\Q(\sqrt{-3}, \sqrt{5})\) None None \(0\) \(-3\) \(2\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{1}-\beta _{3})q^{3}+(2-2\beta _{1}+2\beta _{3})q^{5}+\cdots\)
688.2.i.f 688.i 43.c $4$ $5.494$ \(\Q(\sqrt{-3}, \sqrt{17})\) None None \(0\) \(1\) \(-3\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{3}+(\beta _{1}-2\beta _{2})q^{5}+(2+\beta _{1}-2\beta _{2}+\cdots)q^{7}+\cdots\)
688.2.i.g 688.i 43.c $8$ $5.494$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None None \(0\) \(-1\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{4}-\beta _{6})q^{3}+\beta _{5}q^{5}+(-1+\beta _{2}+\cdots)q^{7}+\cdots\)
688.2.i.h 688.i 43.c $8$ $5.494$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None None \(0\) \(3\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\beta _{1}+\beta _{2})q^{3}-\beta _{5}q^{5}+(\beta _{4}+\beta _{7})q^{7}+\cdots\)
688.2.i.i 688.i 43.c $10$ $5.494$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None None \(0\) \(5\) \(2\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\beta _{1}+\beta _{7})q^{3}+(\beta _{1}-\beta _{6})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
688.2.j.a 688.j 688.j $172$ $5.494$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
688.2.k.a 688.k 16.e $168$ $5.494$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
688.2.o.a 688.o 172.f $2$ $5.494$ \(\Q(\sqrt{-3}) \) None None \(0\) \(-1\) \(-3\) \(-1\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1+\zeta_{6})q^{3}+(-1-\zeta_{6})q^{5}-\zeta_{6}q^{7}+\cdots\)
688.2.o.b 688.o 172.f $2$ $5.494$ \(\Q(\sqrt{-3}) \) None None \(0\) \(1\) \(-3\) \(1\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1-\zeta_{6})q^{3}+(-1-\zeta_{6})q^{5}+\zeta_{6}q^{7}+\cdots\)
688.2.o.c 688.o 172.f $6$ $5.494$ 6.0.1714608.1 None None \(0\) \(-3\) \(3\) \(-3\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{4}q^{3}+(1-\beta _{1}+\beta _{2}+\beta _{5})q^{5}+(-1+\cdots)q^{7}+\cdots\)
688.2.o.d 688.o 172.f $6$ $5.494$ 6.0.3636603.4 None None \(0\) \(-1\) \(0\) \(2\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{1}-\beta _{2}+\beta _{4})q^{3}+(\beta _{1}+\beta _{4})q^{7}+\cdots\)
688.2.o.e 688.o 172.f $6$ $5.494$ 6.0.3636603.4 None None \(0\) \(1\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{1}+\beta _{2}-\beta _{4})q^{3}+(-\beta _{1}-\beta _{4})q^{7}+\cdots\)
688.2.o.f 688.o 172.f $6$ $5.494$ 6.0.1714608.1 None None \(0\) \(3\) \(3\) \(3\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{4}q^{3}+(1-\beta _{1}+\beta _{2}+\beta _{5})q^{5}+(1+\cdots)q^{7}+\cdots\)
688.2.o.g 688.o 172.f $16$ $5.494$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{7}q^{3}+\beta _{4}q^{5}+(-\beta _{9}-\beta _{10})q^{7}+\cdots\)
688.2.u.a 688.u 43.e $6$ $5.494$ \(\Q(\zeta_{14})\) None None \(0\) \(-8\) \(-3\) \(-4\) $\mathrm{SU}(2)[C_{7}]$ \(q+(-2+2\zeta_{14}+\zeta_{14}^{3}+\zeta_{14}^{5})q^{3}+\cdots\)
688.2.u.b 688.u 43.e $6$ $5.494$ \(\Q(\zeta_{14})\) None None \(0\) \(-2\) \(-3\) \(0\) $\mathrm{SU}(2)[C_{7}]$ \(q+(-\zeta_{14}^{3}-\zeta_{14}^{5})q^{3}+(-\zeta_{14}+\zeta_{14}^{2}+\cdots)q^{5}+\cdots\)
688.2.u.c 688.u 43.e $6$ $5.494$ \(\Q(\zeta_{14})\) None None \(0\) \(3\) \(2\) \(16\) $\mathrm{SU}(2)[C_{7}]$ \(q+(1-\zeta_{14}+2\zeta_{14}^{4})q^{3}-2\zeta_{14}^{2}q^{5}+\cdots\)
688.2.u.d 688.u 43.e $12$ $5.494$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(0\) \(6\) \(-3\) \(4\) $\mathrm{SU}(2)[C_{7}]$ \(q+(\beta _{6}+\beta _{9})q^{3}+(-1+\beta _{3}+\beta _{6}+\beta _{7}+\cdots)q^{5}+\cdots\)
688.2.u.e 688.u 43.e $12$ $5.494$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(0\) \(10\) \(3\) \(0\) $\mathrm{SU}(2)[C_{7}]$ \(q+(1-\beta _{10})q^{3}+(-1+\beta _{1}-\beta _{2}-2\beta _{3}+\cdots)q^{5}+\cdots\)
688.2.u.f 688.u 43.e $18$ $5.494$ \(\mathbb{Q}[x]/(x^{18} - \cdots)\) None None \(0\) \(-2\) \(-3\) \(8\) $\mathrm{SU}(2)[C_{7}]$ \(q+(-\beta _{1}-\beta _{2}-\beta _{3}+\beta _{5}-\beta _{6}+\beta _{7}+\cdots)q^{3}+\cdots\)
688.2.u.g 688.u 43.e $30$ $5.494$ None None \(0\) \(-6\) \(1\) \(4\) $\mathrm{SU}(2)[C_{7}]$
688.2.u.h 688.u 43.e $36$ $5.494$ None None \(0\) \(6\) \(1\) \(0\) $\mathrm{SU}(2)[C_{7}]$
688.2.x.a 688.x 688.x $344$ $5.494$ None None \(-8\) \(-2\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{12}]$
688.2.y.a 688.y 688.y $344$ $5.494$ None None \(0\) \(-6\) \(-6\) \(-12\) $\mathrm{SU}(2)[C_{12}]$
688.2.bb.a 688.bb 172.j $36$ $5.494$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{14}]$
688.2.bb.b 688.bb 172.j $96$ $5.494$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{14}]$
688.2.bg.a 688.bg 43.g $12$ $5.494$ \(\Q(\zeta_{21})\) None None \(0\) \(-8\) \(3\) \(1\) $\mathrm{SU}(2)[C_{21}]$ \(q+(-1-\zeta_{21}^{7}+\zeta_{21}^{9})q^{3}+(1-\zeta_{21}+\cdots)q^{5}+\cdots\)
688.2.bg.b 688.bg 43.g $24$ $5.494$ None None \(0\) \(6\) \(3\) \(-3\) $\mathrm{SU}(2)[C_{21}]$
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