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Label Char Prim Dim $A$ Field CM RM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
784.1.d.a 784.d 4.b $2$ $0.391$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\beta q^{5}+q^{9}+\beta q^{13}+\beta q^{17}+q^{25}+\cdots\)
784.1.k.a 784.k 16.f $2$ $0.391$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) \(q-iq^{2}-q^{4}+iq^{8}-iq^{9}+(1-i+\cdots)q^{11}+\cdots\)
784.1.r.a 784.r 28.g $4$ $0.391$ \(\Q(\sqrt{2}, \sqrt{-3})\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{5}+(-1-\beta _{2})q^{9}+\beta _{3}q^{13}+\cdots\)
784.1.v.a 784.v 112.u $4$ $0.391$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+\zeta_{12}^{3}q^{8}+\zeta_{12}q^{9}+\cdots\)
784.1.y.a 784.y 112.x $4$ $0.391$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{12}^{5}q^{2}-\zeta_{12}^{4}q^{4}-\zeta_{12}^{3}q^{8}+\cdots\)
784.2.a.a 784.a 1.a $1$ $6.260$ \(\Q\) None None \(0\) \(-3\) \(-1\) \(0\) $+$ $\mathrm{SU}(2)$ \(q-3q^{3}-q^{5}+6q^{9}+q^{11}+2q^{13}+\cdots\)
784.2.a.b 784.a 1.a $1$ $6.260$ \(\Q\) None None \(0\) \(-2\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q-2q^{3}+q^{9}+4q^{13}-6q^{17}+2q^{19}+\cdots\)
784.2.a.c 784.a 1.a $1$ $6.260$ \(\Q\) None None \(0\) \(-1\) \(1\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-q^{3}+q^{5}-2q^{9}-3q^{11}+6q^{13}+\cdots\)
784.2.a.d 784.a 1.a $1$ $6.260$ \(\Q\) None None \(0\) \(-1\) \(3\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-q^{3}+3q^{5}-2q^{9}+3q^{11}+2q^{13}+\cdots\)
784.2.a.e 784.a 1.a $1$ $6.260$ \(\Q\) None None \(0\) \(0\) \(-2\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-2q^{5}-3q^{9}+4q^{11}-2q^{13}+6q^{17}+\cdots\)
784.2.a.f 784.a 1.a $1$ $6.260$ \(\Q\) \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) $+$ $N(\mathrm{U}(1))$ \(q-3q^{9}-4q^{11}-8q^{23}-5q^{25}+2q^{29}+\cdots\)
784.2.a.g 784.a 1.a $1$ $6.260$ \(\Q\) None None \(0\) \(1\) \(-3\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+q^{3}-3q^{5}-2q^{9}+3q^{11}-2q^{13}+\cdots\)
784.2.a.h 784.a 1.a $1$ $6.260$ \(\Q\) None None \(0\) \(1\) \(-1\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+q^{3}-q^{5}-2q^{9}-3q^{11}-6q^{13}+\cdots\)
784.2.a.i 784.a 1.a $1$ $6.260$ \(\Q\) None None \(0\) \(2\) \(4\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+4q^{5}+q^{9}+8q^{15}+2q^{17}+\cdots\)
784.2.a.j 784.a 1.a $1$ $6.260$ \(\Q\) None None \(0\) \(3\) \(1\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+3q^{3}+q^{5}+6q^{9}+q^{11}-2q^{13}+\cdots\)
784.2.a.k 784.a 1.a $2$ $6.260$ \(\Q(\sqrt{2}) \) None None \(0\) \(0\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-2\beta q^{5}-q^{9}-6q^{11}+4\beta q^{13}+\cdots\)
784.2.a.l 784.a 1.a $2$ $6.260$ \(\Q(\sqrt{2}) \) None None \(0\) \(0\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}+2\beta q^{5}-q^{9}+2q^{11}+4q^{15}+\cdots\)
784.2.a.m 784.a 1.a $2$ $6.260$ \(\Q(\sqrt{2}) \) None None \(0\) \(0\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+2\beta q^{3}+\beta q^{5}+5q^{9}-4q^{11}+3\beta q^{13}+\cdots\)
784.2.a.n 784.a 1.a $2$ $6.260$ \(\Q(\sqrt{2}) \) None None \(0\) \(0\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+\beta q^{3}-\beta q^{5}+5q^{9}+4q^{11}+\beta q^{13}+\cdots\)
784.2.f.a 784.f 28.d $2$ $6.260$ \(\Q(\sqrt{-3}) \) None None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-q^{3}+\zeta_{6}q^{5}-2q^{9}-\zeta_{6}q^{11}-\zeta_{6}q^{15}+\cdots\)
784.2.f.b 784.f 28.d $2$ $6.260$ \(\Q(\sqrt{-3}) \) None None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+q^{3}+\zeta_{6}q^{5}-2q^{9}+\zeta_{6}q^{11}+\zeta_{6}q^{15}+\cdots\)
784.2.f.c 784.f 28.d $4$ $6.260$ 4.0.2048.2 \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{2}q^{5}-3q^{9}+(\beta _{1}+\beta _{2})q^{13}+(\beta _{1}+\cdots)q^{17}+\cdots\)
784.2.f.d 784.f 28.d $4$ $6.260$ \(\Q(\sqrt{-3}, \sqrt{7})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}-\beta _{2}q^{5}+4q^{9}+\beta _{3}q^{11}+\cdots\)
784.2.f.e 784.f 28.d $8$ $6.260$ 8.0.339738624.1 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{3}+(\beta _{2}-\beta _{4})q^{5}+(3-3\beta _{1})q^{9}+\cdots\)
784.2.i.a 784.i 7.c $2$ $6.260$ \(\Q(\sqrt{-3}) \) None None \(0\) \(-3\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{3}-\zeta_{6}q^{5}-6\zeta_{6}q^{9}+\cdots\)
784.2.i.b 784.i 7.c $2$ $6.260$ \(\Q(\sqrt{-3}) \) None None \(0\) \(-2\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{3}-4\zeta_{6}q^{5}-\zeta_{6}q^{9}+\cdots\)
784.2.i.c 784.i 7.c $2$ $6.260$ \(\Q(\sqrt{-3}) \) None None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{3}-\zeta_{6}q^{9}-4q^{13}+\cdots\)
784.2.i.d 784.i 7.c $2$ $6.260$ \(\Q(\sqrt{-3}) \) None None \(0\) \(-1\) \(3\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+3\zeta_{6}q^{5}+2\zeta_{6}q^{9}+\cdots\)
784.2.i.e 784.i 7.c $2$ $6.260$ \(\Q(\sqrt{-3}) \) None None \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+(-4+4\zeta_{6})q^{11}+\cdots\)
784.2.i.f 784.i 7.c $2$ $6.260$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{3}]$ \(q+3\zeta_{6}q^{9}+(4-4\zeta_{6})q^{11}+8\zeta_{6}q^{23}+\cdots\)
784.2.i.g 784.i 7.c $2$ $6.260$ \(\Q(\sqrt{-3}) \) None None \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\zeta_{6}q^{5}+3\zeta_{6}q^{9}+(-4+4\zeta_{6})q^{11}+\cdots\)
784.2.i.h 784.i 7.c $2$ $6.260$ \(\Q(\sqrt{-3}) \) None None \(0\) \(1\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-\zeta_{6}q^{5}+2\zeta_{6}q^{9}+(3+\cdots)q^{11}+\cdots\)
784.2.i.i 784.i 7.c $2$ $6.260$ \(\Q(\sqrt{-3}) \) None None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{3}-\zeta_{6}q^{9}+4q^{13}+(6+\cdots)q^{17}+\cdots\)
784.2.i.j 784.i 7.c $2$ $6.260$ \(\Q(\sqrt{-3}) \) None None \(0\) \(2\) \(4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{3}+4\zeta_{6}q^{5}-\zeta_{6}q^{9}+8q^{15}+\cdots\)
784.2.i.k 784.i 7.c $4$ $6.260$ \(\Q(\sqrt{2}, \sqrt{-3})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{3})q^{5}+5\beta _{2}q^{9}+(-4+\cdots)q^{11}+\cdots\)
784.2.i.l 784.i 7.c $4$ $6.260$ \(\Q(\sqrt{2}, \sqrt{-3})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+2\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{5}+5\beta _{2}q^{9}+\cdots\)
784.2.i.m 784.i 7.c $4$ $6.260$ \(\Q(\sqrt{2}, \sqrt{-3})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{3}+(-2\beta _{1}-2\beta _{3})q^{5}-\beta _{2}q^{9}+\cdots\)
784.2.i.n 784.i 7.c $4$ $6.260$ \(\Q(\sqrt{2}, \sqrt{-3})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{3}+(2\beta _{1}+2\beta _{3})q^{5}-\beta _{2}q^{9}+\cdots\)
784.2.j.a 784.j 112.j $56$ $6.260$ None None \(4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
784.2.j.b 784.j 112.j $96$ $6.260$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
784.2.m.a 784.m 16.e $2$ $6.260$ \(\Q(\sqrt{-1}) \) None None \(-2\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+i)q^{2}-2iq^{4}+(-2-2i)q^{5}+\cdots\)
784.2.m.b 784.m 16.e $2$ $6.260$ \(\Q(\sqrt{-1}) \) None None \(-2\) \(2\) \(2\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{2}+(1-i)q^{3}+2iq^{4}+\cdots\)
784.2.m.c 784.m 16.e $2$ $6.260$ \(\Q(\sqrt{-1}) \) None None \(-2\) \(4\) \(4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+i)q^{2}+(2-2i)q^{3}-2iq^{4}+\cdots\)
784.2.m.d 784.m 16.e $4$ $6.260$ \(\Q(\zeta_{12})\) None None \(-2\) \(-2\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+\zeta_{12}^{3}q^{3}+\cdots\)
784.2.m.e 784.m 16.e $4$ $6.260$ \(\Q(\zeta_{12})\) None None \(-2\) \(2\) \(6\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\zeta_{12})q^{2}+(1-\zeta_{12}+\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
784.2.m.f 784.m 16.e $4$ $6.260$ \(\Q(i, \sqrt{7})\) \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q-\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-\beta _{1}-2\beta _{3})q^{8}+\cdots\)
784.2.m.g 784.m 16.e $8$ $6.260$ 8.0.214798336.3 None None \(2\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{3}q^{2}+(\beta _{1}+\beta _{4})q^{3}+(1+\beta _{4}-\beta _{6}+\cdots)q^{4}+\cdots\)
784.2.m.h 784.m 16.e $12$ $6.260$ 12.0.\(\cdots\).1 None None \(2\) \(-4\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{2}q^{2}+(\beta _{2}-\beta _{10})q^{3}+\beta _{1}q^{4}+(-1+\cdots)q^{5}+\cdots\)
784.2.m.i 784.m 16.e $20$ $6.260$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{4}q^{2}+\beta _{9}q^{3}-\beta _{17}q^{4}+\beta _{2}q^{5}+\cdots\)
784.2.m.j 784.m 16.e $24$ $6.260$ None None \(4\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{4}]$
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