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Label Char Prim Dim $A$ Field CM RM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
384.1.h.a 384.h 24.h $1$ $0.192$ \(\Q\) \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-6}) \) \(\Q(\sqrt{3}) \) \(0\) \(-1\) \(0\) \(0\) \(q-q^{3}+q^{9}+2q^{11}-q^{25}-q^{27}+\cdots\)
384.1.h.b 384.h 24.h $1$ $0.192$ \(\Q\) \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-6}) \) \(\Q(\sqrt{3}) \) \(0\) \(1\) \(0\) \(0\) \(q+q^{3}+q^{9}-2q^{11}-q^{25}+q^{27}+\cdots\)
384.2.a.a 384.a 1.a $1$ $3.066$ \(\Q\) None None \(0\) \(-1\) \(-4\) \(2\) $-$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{5}+2q^{7}+q^{9}+4q^{11}+\cdots\)
384.2.a.b 384.a 1.a $1$ $3.066$ \(\Q\) None None \(0\) \(-1\) \(0\) \(-2\) $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{7}+q^{9}-4q^{11}-6q^{13}+\cdots\)
384.2.a.c 384.a 1.a $1$ $3.066$ \(\Q\) None None \(0\) \(-1\) \(0\) \(2\) $-$ $\mathrm{SU}(2)$ \(q-q^{3}+2q^{7}+q^{9}-4q^{11}+6q^{13}+\cdots\)
384.2.a.d 384.a 1.a $1$ $3.066$ \(\Q\) None None \(0\) \(-1\) \(4\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q-q^{3}+4q^{5}-2q^{7}+q^{9}+4q^{11}+\cdots\)
384.2.a.e 384.a 1.a $1$ $3.066$ \(\Q\) None None \(0\) \(1\) \(-4\) \(-2\) $+$ $\mathrm{SU}(2)$ \(q+q^{3}-4q^{5}-2q^{7}+q^{9}-4q^{11}+\cdots\)
384.2.a.f 384.a 1.a $1$ $3.066$ \(\Q\) None None \(0\) \(1\) \(0\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q+q^{3}-2q^{7}+q^{9}+4q^{11}+6q^{13}+\cdots\)
384.2.a.g 384.a 1.a $1$ $3.066$ \(\Q\) None None \(0\) \(1\) \(0\) \(2\) $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{7}+q^{9}+4q^{11}-6q^{13}+\cdots\)
384.2.a.h 384.a 1.a $1$ $3.066$ \(\Q\) None None \(0\) \(1\) \(4\) \(2\) $-$ $\mathrm{SU}(2)$ \(q+q^{3}+4q^{5}+2q^{7}+q^{9}-4q^{11}+\cdots\)
384.2.c.a 384.c 12.b $4$ $3.066$ \(\Q(i, \sqrt{5})\) None None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1+\beta _{3})q^{3}+(\beta _{1}+\beta _{3})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
384.2.c.b 384.c 12.b $4$ $3.066$ \(\Q(i, \sqrt{5})\) None None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1+\beta _{3})q^{3}+(-\beta _{1}-\beta _{3})q^{5}+(\beta _{1}+\cdots)q^{7}+\cdots\)
384.2.c.c 384.c 12.b $4$ $3.066$ \(\Q(i, \sqrt{5})\) None None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1+\beta _{1})q^{3}+(\beta _{1}+\beta _{3})q^{5}+(\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
384.2.c.d 384.c 12.b $4$ $3.066$ \(\Q(i, \sqrt{5})\) None None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1+\beta _{1})q^{3}+(-\beta _{1}-\beta _{3})q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
384.2.d.a 384.d 8.b $2$ $3.066$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}-4q^{7}-q^{9}+4iq^{11}+4iq^{13}+\cdots\)
384.2.d.b 384.d 8.b $2$ $3.066$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+4q^{7}-q^{9}+4iq^{11}-4iq^{13}+\cdots\)
384.2.d.c 384.d 8.b $4$ $3.066$ \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{8}q^{3}+\zeta_{8}^{2}q^{5}+\zeta_{8}^{3}q^{7}-q^{9}+\cdots\)
384.2.f.a 384.f 24.f $4$ $3.066$ \(\Q(\sqrt{2}, \sqrt{-3})\) \(\Q(\sqrt{-6}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{2}q^{3}+\beta _{1}q^{5}-\beta _{3}q^{7}-3q^{9}-2\beta _{2}q^{11}+\cdots\)
384.2.f.b 384.f 24.f $4$ $3.066$ \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{8}q^{3}+(\zeta_{8}+\zeta_{8}^{2})q^{5}-\zeta_{8}^{3}q^{7}+\cdots\)
384.2.f.c 384.f 24.f $4$ $3.066$ \(\Q(\zeta_{8})\) \(\Q(\sqrt{-2}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\zeta_{8}q^{3}+(1-\zeta_{8}^{2})q^{9}+(-2\zeta_{8}+\zeta_{8}^{3})q^{11}+\cdots\)
384.2.f.d 384.f 24.f $4$ $3.066$ \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{8}q^{3}+(-\zeta_{8}-\zeta_{8}^{2})q^{5}+\zeta_{8}^{3}q^{7}+\cdots\)
384.2.j.a 384.j 16.e $8$ $3.066$ 8.0.18939904.2 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{2}q^{3}+(\beta _{3}-\beta _{5})q^{5}+(\beta _{4}+\beta _{5}+\beta _{6}+\cdots)q^{7}+\cdots\)
384.2.j.b 384.j 16.e $8$ $3.066$ 8.0.18939904.2 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{3}q^{3}+(-\beta _{2}+\beta _{6})q^{5}+(\beta _{4}+\beta _{5}+\cdots)q^{7}+\cdots\)
384.2.k.a 384.k 48.k $12$ $3.066$ 12.0.\(\cdots\).2 None None \(0\) \(-2\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{4}q^{3}+\beta _{6}q^{5}+(1-\beta _{11})q^{7}+(-\beta _{3}+\cdots)q^{9}+\cdots\)
384.2.k.b 384.k 48.k $12$ $3.066$ 12.0.\(\cdots\).2 None None \(0\) \(2\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{10}q^{3}-\beta _{7}q^{5}+(-1+\beta _{11})q^{7}+\cdots\)
384.2.n.a 384.n 32.g $32$ $3.066$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{8}]$
384.2.o.a 384.o 96.o $56$ $3.066$ None None \(0\) \(4\) \(0\) \(8\) $\mathrm{SU}(2)[C_{8}]$
384.2.v.a 384.v 128.k $512$ $3.066$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{32}]$
384.2.w.a 384.w 384.w $992$ $3.066$ None None \(0\) \(-16\) \(0\) \(-32\) $\mathrm{SU}(2)[C_{32}]$
384.3.b.a 384.b 8.d $4$ $10.463$ \(\Q(\zeta_{12})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{12}q^{3}+\zeta_{12}^{3}q^{5}+\zeta_{12}^{2}q^{7}+3q^{9}+\cdots\)
384.3.b.b 384.b 8.d $4$ $10.463$ \(\Q(\zeta_{12})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{12}q^{3}+\zeta_{12}^{2}q^{5}-\zeta_{12}^{3}q^{7}+3q^{9}+\cdots\)
384.3.b.c 384.b 8.d $8$ $10.463$ \(\Q(\zeta_{24})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{24}^{2}q^{3}+(-\zeta_{24}^{3}+\zeta_{24}^{5})q^{5}+\cdots\)
384.3.e.a 384.e 3.b $8$ $10.463$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None None \(0\) \(-4\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}+\beta _{3}q^{5}+(-1-\beta _{2})q^{7}+(\beta _{2}+\cdots)q^{9}+\cdots\)
384.3.e.b 384.e 3.b $8$ $10.463$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None None \(0\) \(-4\) \(0\) \(8\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}+\beta _{2}q^{5}+(1+\beta _{3})q^{7}+(\beta _{3}+\cdots)q^{9}+\cdots\)
384.3.e.c 384.e 3.b $8$ $10.463$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None None \(0\) \(4\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}+\beta _{2}q^{5}+(-1-\beta _{3})q^{7}+(\beta _{3}+\cdots)q^{9}+\cdots\)
384.3.e.d 384.e 3.b $8$ $10.463$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None None \(0\) \(4\) \(0\) \(8\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}+\beta _{3}q^{5}+(1+\beta _{2})q^{7}+(\beta _{2}+\cdots)q^{9}+\cdots\)
384.3.g.a 384.g 4.b $8$ $10.463$ \(\Q(\zeta_{24})\) None None \(0\) \(0\) \(-16\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{24}q^{3}+(-2+\zeta_{24}^{3})q^{5}+(-\zeta_{24}^{2}+\cdots)q^{7}+\cdots\)
384.3.g.b 384.g 4.b $8$ $10.463$ \(\Q(\zeta_{24})\) None None \(0\) \(0\) \(16\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{24}q^{3}+(2-\zeta_{24}^{3})q^{5}+(-\zeta_{24}^{2}+\cdots)q^{7}+\cdots\)
384.3.h.a 384.h 24.h $2$ $10.463$ \(\Q(\sqrt{6}) \) \(\Q(\sqrt{-6}) \) None \(0\) \(-6\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-3q^{3}+\beta q^{5}+\beta q^{7}+9q^{9}-10q^{11}+\cdots\)
384.3.h.b 384.h 24.h $2$ $10.463$ \(\Q(\sqrt{-2}) \) \(\Q(\sqrt{-2}) \) None \(0\) \(-2\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(-1+\beta )q^{3}+(-7-2\beta )q^{9}-14q^{11}+\cdots\)
384.3.h.c 384.h 24.h $2$ $10.463$ \(\Q(\sqrt{-2}) \) \(\Q(\sqrt{-2}) \) None \(0\) \(2\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(1+\beta )q^{3}+(-7+2\beta )q^{9}+14q^{11}+\cdots\)
384.3.h.d 384.h 24.h $2$ $10.463$ \(\Q(\sqrt{6}) \) \(\Q(\sqrt{-6}) \) None \(0\) \(6\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+3q^{3}+\beta q^{5}-\beta q^{7}+9q^{9}+10q^{11}+\cdots\)
384.3.h.e 384.h 24.h $4$ $10.463$ \(\Q(i, \sqrt{5})\) None None \(0\) \(0\) \(-16\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{3}-4q^{5}+(-2\beta _{1}-2\beta _{3})q^{7}+\cdots\)
384.3.h.f 384.h 24.h $4$ $10.463$ \(\Q(i, \sqrt{5})\) None None \(0\) \(0\) \(16\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}+4q^{5}+(-2\beta _{1}-2\beta _{3})q^{7}+\cdots\)
384.3.h.g 384.h 24.h $16$ $10.463$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{8}q^{3}+\beta _{2}q^{5}+\beta _{1}q^{7}+(-1-\beta _{5}+\cdots)q^{9}+\cdots\)
384.3.i.a 384.i 48.i $8$ $10.463$ 8.0.629407744.1 None None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\beta _{1}-\beta _{2}-\beta _{3})q^{3}+(3\beta _{4}-\beta _{6}-\beta _{7})q^{5}+\cdots\)
384.3.i.b 384.i 48.i $8$ $10.463$ 8.0.629407744.1 None None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-\beta _{1}+\beta _{2}+\beta _{3})q^{3}+(3\beta _{4}-\beta _{6}+\cdots)q^{5}+\cdots\)
384.3.i.c 384.i 48.i $20$ $10.463$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None None \(0\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{5}q^{3}+(\beta _{8}-\beta _{9}-\beta _{10})q^{5}-\beta _{6}q^{7}+\cdots\)
384.3.i.d 384.i 48.i $20$ $10.463$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None None \(0\) \(6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{5}q^{3}+(\beta _{8}-\beta _{9}-\beta _{10})q^{5}+\beta _{6}q^{7}+\cdots\)
384.3.l.a 384.l 16.f $16$ $10.463$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{2}q^{3}+\beta _{9}q^{5}-\beta _{5}q^{7}-3\beta _{4}q^{9}+\cdots\)
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