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Label Char Prim Dim $A$ Field CM RM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
768.1.e.a 768.e 3.b $1$ $0.383$ \(\Q\) \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \) \(\Q(\sqrt{6}) \) \(0\) \(-1\) \(0\) \(0\) \(q-q^{3}+q^{9}+2q^{19}+q^{25}-q^{27}+\cdots\)
768.1.e.b 768.e 3.b $1$ $0.383$ \(\Q\) \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-2}) \) \(\Q(\sqrt{6}) \) \(0\) \(1\) \(0\) \(0\) \(q+q^{3}+q^{9}-2q^{19}+q^{25}+q^{27}+\cdots\)
768.1.e.c 768.e 3.b $2$ $0.383$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-6}) \) \(\Q(\sqrt{3}) \) \(0\) \(0\) \(0\) \(0\) \(q-iq^{3}-q^{9}-iq^{11}+q^{25}+iq^{27}+\cdots\)
768.1.i.a 768.i 48.i $4$ $0.383$ \(\Q(\zeta_{8})\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{8}q^{3}+(\zeta_{8}+\zeta_{8}^{3})q^{7}+\zeta_{8}^{2}q^{9}+\cdots\)
768.1.i.b 768.i 48.i $4$ $0.383$ \(\Q(\zeta_{8})\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{8}q^{3}+(-\zeta_{8}-\zeta_{8}^{3})q^{7}+\zeta_{8}^{2}q^{9}+\cdots\)
768.2.a.a 768.a 1.a $1$ $6.133$ \(\Q\) None None \(0\) \(-1\) \(-2\) \(2\) $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{5}+2q^{7}+q^{9}-4q^{13}+\cdots\)
768.2.a.b 768.a 1.a $1$ $6.133$ \(\Q\) None None \(0\) \(-1\) \(0\) \(-4\) $+$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{7}+q^{9}+4q^{11}+4q^{13}+\cdots\)
768.2.a.c 768.a 1.a $1$ $6.133$ \(\Q\) None None \(0\) \(-1\) \(0\) \(4\) $-$ $\mathrm{SU}(2)$ \(q-q^{3}+4q^{7}+q^{9}+4q^{11}-4q^{13}+\cdots\)
768.2.a.d 768.a 1.a $1$ $6.133$ \(\Q\) None None \(0\) \(-1\) \(2\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{5}-2q^{7}+q^{9}+4q^{13}+\cdots\)
768.2.a.e 768.a 1.a $1$ $6.133$ \(\Q\) None None \(0\) \(1\) \(-2\) \(-2\) $+$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{5}-2q^{7}+q^{9}-4q^{13}+\cdots\)
768.2.a.f 768.a 1.a $1$ $6.133$ \(\Q\) None None \(0\) \(1\) \(0\) \(-4\) $+$ $\mathrm{SU}(2)$ \(q+q^{3}-4q^{7}+q^{9}-4q^{11}-4q^{13}+\cdots\)
768.2.a.g 768.a 1.a $1$ $6.133$ \(\Q\) None None \(0\) \(1\) \(0\) \(4\) $-$ $\mathrm{SU}(2)$ \(q+q^{3}+4q^{7}+q^{9}-4q^{11}+4q^{13}+\cdots\)
768.2.a.h 768.a 1.a $1$ $6.133$ \(\Q\) None None \(0\) \(1\) \(2\) \(2\) $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}+2q^{7}+q^{9}+4q^{13}+\cdots\)
768.2.a.i 768.a 1.a $2$ $6.133$ \(\Q(\sqrt{2}) \) None None \(0\) \(-2\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta q^{5}-\beta q^{7}+q^{9}-4q^{11}+\cdots\)
768.2.a.j 768.a 1.a $2$ $6.133$ \(\Q(\sqrt{3}) \) None None \(0\) \(-2\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-q^{3}+\beta q^{5}+\beta q^{7}+q^{9}-\beta q^{15}+\cdots\)
768.2.a.k 768.a 1.a $2$ $6.133$ \(\Q(\sqrt{3}) \) None None \(0\) \(2\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta q^{5}-\beta q^{7}+q^{9}+\beta q^{15}+\cdots\)
768.2.a.l 768.a 1.a $2$ $6.133$ \(\Q(\sqrt{2}) \) None None \(0\) \(2\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+q^{3}+\beta q^{5}+\beta q^{7}+q^{9}+4q^{11}+\cdots\)
768.2.c.a 768.c 12.b $2$ $6.133$ \(\Q(\sqrt{-2}) \) None None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1+\beta )q^{3}+2\beta q^{5}-2\beta q^{7}+(-1+\cdots)q^{9}+\cdots\)
768.2.c.b 768.c 12.b $2$ $6.133$ \(\Q(\sqrt{-2}) \) None None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1-\beta )q^{3}+2\beta q^{5}-2\beta q^{7}+(-1+\cdots)q^{9}+\cdots\)
768.2.c.c 768.c 12.b $2$ $6.133$ \(\Q(\sqrt{-2}) \) \(\Q(\sqrt{-2}) \) None \(0\) \(-2\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(-1-\beta )q^{3}+(-1+2\beta )q^{9}+6q^{11}+\cdots\)
768.2.c.d 768.c 12.b $2$ $6.133$ \(\Q(\sqrt{-2}) \) \(\Q(\sqrt{-2}) \) None \(0\) \(2\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(1+\beta )q^{3}+(-1+2\beta )q^{9}-6q^{11}+\cdots\)
768.2.c.e 768.c 12.b $2$ $6.133$ \(\Q(\sqrt{-2}) \) None None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1-\beta )q^{3}+2\beta q^{5}+2\beta q^{7}+(-1+\cdots)q^{9}+\cdots\)
768.2.c.f 768.c 12.b $2$ $6.133$ \(\Q(\sqrt{-2}) \) None None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1+\beta )q^{3}+2\beta q^{5}+2\beta q^{7}+(-1+\cdots)q^{9}+\cdots\)
768.2.c.g 768.c 12.b $4$ $6.133$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\zeta_{12}^{2}q^{3}-\zeta_{12}q^{7}-3q^{9}+\zeta_{12}^{3}q^{13}+\cdots\)
768.2.c.h 768.c 12.b $4$ $6.133$ \(\Q(\zeta_{8})\) \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\zeta_{8}^{2}q^{3}+(1+\zeta_{8}^{3})q^{9}+(\zeta_{8}+\zeta_{8}^{2}+\cdots)q^{11}+\cdots\)
768.2.c.i 768.c 12.b $4$ $6.133$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-6}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\zeta_{12}q^{3}-\zeta_{12}^{2}q^{5}+\zeta_{12}^{3}q^{7}+3q^{9}+\cdots\)
768.2.c.j 768.c 12.b $4$ $6.133$ \(\Q(\sqrt{-2}, \sqrt{3})\) \(\Q(\sqrt{-6}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\beta _{2}q^{3}-\beta _{1}q^{5}-\beta _{3}q^{7}+3q^{9}-2\beta _{2}q^{11}+\cdots\)
768.2.d.a 768.d 8.b $2$ $6.133$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+2iq^{5}-4q^{7}-q^{9}-4iq^{11}+\cdots\)
768.2.d.b 768.d 8.b $2$ $6.133$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}-2q^{7}-q^{9}-4iq^{11}-6iq^{13}+\cdots\)
768.2.d.c 768.d 8.b $2$ $6.133$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{3}+4iq^{5}-2q^{7}-q^{9}-4iq^{11}+\cdots\)
768.2.d.d 768.d 8.b $2$ $6.133$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+2iq^{5}-q^{9}+4iq^{11}-2iq^{13}+\cdots\)
768.2.d.e 768.d 8.b $2$ $6.133$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{3}+2iq^{5}-q^{9}-4iq^{11}-2iq^{13}+\cdots\)
768.2.d.f 768.d 8.b $2$ $6.133$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+4iq^{5}+2q^{7}-q^{9}+4iq^{11}+\cdots\)
768.2.d.g 768.d 8.b $2$ $6.133$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{3}+2q^{7}-q^{9}+4iq^{11}-6iq^{13}+\cdots\)
768.2.d.h 768.d 8.b $2$ $6.133$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{2}]$ \(q-iq^{3}+2iq^{5}+4q^{7}-q^{9}+4iq^{11}+\cdots\)
768.2.f.a 768.f 24.f $4$ $6.133$ \(\Q(\zeta_{8})\) None None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1-\zeta_{8}^{2})q^{3}-\zeta_{8}^{3}q^{5}+\zeta_{8}q^{7}+\cdots\)
768.2.f.b 768.f 24.f $4$ $6.133$ \(\Q(i, \sqrt{5})\) None None \(0\) \(-2\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1+\beta _{3})q^{3}+(-2-\beta _{1}+\beta _{3})q^{5}+\cdots\)
768.2.f.c 768.f 24.f $4$ $6.133$ \(\Q(i, \sqrt{5})\) None None \(0\) \(-2\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1-\beta _{1})q^{3}+(2+\beta _{1}-\beta _{3})q^{5}+\cdots\)
768.2.f.d 768.f 24.f $4$ $6.133$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\zeta_{12}q^{3}+\zeta_{12}^{2}q^{7}+3q^{9}+\zeta_{12}^{3}q^{13}+\cdots\)
768.2.f.e 768.f 24.f $4$ $6.133$ \(\Q(i, \sqrt{5})\) None None \(0\) \(2\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1-\beta _{3})q^{3}+(-2-\beta _{1}+\beta _{3})q^{5}+\cdots\)
768.2.f.f 768.f 24.f $4$ $6.133$ \(\Q(i, \sqrt{5})\) None None \(0\) \(2\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1+\beta _{1})q^{3}+(2+\beta _{1}-\beta _{3})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)
768.2.f.g 768.f 24.f $4$ $6.133$ \(\Q(\zeta_{8})\) None None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1+\zeta_{8}^{2})q^{3}-\zeta_{8}^{3}q^{5}-\zeta_{8}q^{7}+(-1+\cdots)q^{9}+\cdots\)
768.2.j.a 768.j 16.e $4$ $6.133$ \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{8}q^{3}+(-2+2\zeta_{8}^{2})q^{5}+(3\zeta_{8}+3\zeta_{8}^{3})q^{7}+\cdots\)
768.2.j.b 768.j 16.e $4$ $6.133$ \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}^{3}q^{3}+(\zeta_{8}+\zeta_{8}^{3})q^{7}-\zeta_{8}^{2}q^{9}+\cdots\)
768.2.j.c 768.j 16.e $4$ $6.133$ \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}^{3}q^{3}+(-\zeta_{8}-\zeta_{8}^{3})q^{7}-\zeta_{8}^{2}q^{9}+\cdots\)
768.2.j.d 768.j 16.e $4$ $6.133$ \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(8\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{3}+(2-2\zeta_{8}^{2})q^{5}+(3\zeta_{8}+3\zeta_{8}^{3})q^{7}+\cdots\)
768.2.j.e 768.j 16.e $8$ $6.133$ \(\Q(\zeta_{24})\) None None \(0\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{24}q^{3}+(-1+\zeta_{24}^{2}-\zeta_{24}^{6})q^{5}+\cdots\)
768.2.j.f 768.j 16.e $8$ $6.133$ \(\Q(\zeta_{24})\) None None \(0\) \(0\) \(8\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{24}q^{3}+(1-\zeta_{24}^{2}+\zeta_{24}^{6})q^{5}+\cdots\)
768.2.k.a 768.k 48.k $4$ $6.133$ \(\Q(\zeta_{8})\) None None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+2\zeta_{8}q^{5}+\cdots\)
768.2.k.b 768.k 48.k $4$ $6.133$ \(\Q(\zeta_{8})\) None None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+2\zeta_{8}q^{5}+\cdots\)
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