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Label Char Prim Dim $A$ Field CM RM Minimal twist Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
684.1.h.a 684.h 19.b $1$ $0.341$ \(\Q\) \(\Q(\sqrt{-19}) \) None 76.1.c.a \(0\) \(0\) \(1\) \(-1\) \(q+q^{5}-q^{7}+q^{11}+q^{17}+q^{19}-2q^{23}+\cdots\)
684.1.h.b 684.h 19.b $2$ $0.341$ \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-19}) \) None 684.1.h.b \(0\) \(0\) \(0\) \(2\) \(q-\beta q^{5}+q^{7}+\beta q^{11}+\beta q^{17}-q^{19}+\cdots\)
684.1.q.a 684.q 76.g $8$ $0.341$ \(\Q(\zeta_{24})\) None None 684.1.q.a \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{24}^{5}q^{2}+\zeta_{24}^{10}q^{4}+(\zeta_{24}^{5}+\zeta_{24}^{11}+\cdots)q^{5}+\cdots\)
684.1.x.a 684.x 684.x $4$ $0.341$ \(\Q(\zeta_{12})\) None None 684.1.x.a \(-4\) \(0\) \(2\) \(0\) \(q-q^{2}-\zeta_{12}^{3}q^{3}+q^{4}+\zeta_{12}^{2}q^{5}+\cdots\)
684.1.bm.a 684.bm 684.am $4$ $0.341$ \(\Q(\zeta_{12})\) None None 684.1.x.a \(2\) \(0\) \(-4\) \(0\) \(q+\zeta_{12}^{2}q^{2}+\zeta_{12}^{5}q^{3}+\zeta_{12}^{4}q^{4}+\cdots\)
684.1.bx.a 684.bx 19.f $6$ $0.341$ \(\Q(\zeta_{18})\) \(\Q(\sqrt{-3}) \) None 684.1.bx.a \(0\) \(0\) \(0\) \(0\) \(q+(\zeta_{18}^{5}+\zeta_{18}^{7})q^{7}+(\zeta_{18}^{3}+\zeta_{18}^{4}+\cdots)q^{13}+\cdots\)
684.2.a.a 684.a 1.a $1$ $5.462$ \(\Q\) None None 228.2.a.b \(0\) \(0\) \(-2\) \(0\) $+$ $\mathrm{SU}(2)$ \(q-2q^{5}-2q^{11}+2q^{13}-6q^{17}-q^{19}+\cdots\)
684.2.a.b 684.a 1.a $1$ $5.462$ \(\Q\) None None 76.2.a.a \(0\) \(0\) \(1\) \(-3\) $+$ $\mathrm{SU}(2)$ \(q+q^{5}-3q^{7}-5q^{11}-4q^{13}+3q^{17}+\cdots\)
684.2.a.c 684.a 1.a $1$ $5.462$ \(\Q\) None None 228.2.a.a \(0\) \(0\) \(3\) \(1\) $-$ $\mathrm{SU}(2)$ \(q+3q^{5}+q^{7}+5q^{11}-6q^{13}+5q^{17}+\cdots\)
684.2.a.d 684.a 1.a $2$ $5.462$ \(\Q(\sqrt{33}) \) None None 228.2.a.c \(0\) \(0\) \(-3\) \(1\) $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{5}+(1-\beta )q^{7}+(1+\beta )q^{11}+\cdots\)
684.2.a.e 684.a 1.a $2$ $5.462$ \(\Q(\sqrt{7}) \) None None 684.2.a.e \(0\) \(0\) \(0\) \(6\) $-$ $\mathrm{SU}(2)$ \(q+\beta q^{5}+3q^{7}+\beta q^{11}+2q^{13}-3\beta q^{17}+\cdots\)
684.2.c.a 684.c 12.b $4$ $5.462$ \(\Q(\zeta_{8})\) None None 684.2.c.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta_{2} q^{2}-2 q^{4}-\beta_{2} q^{5}+4\beta_1 q^{7}+\cdots\)
684.2.c.b 684.c 12.b $32$ $5.462$ None None 684.2.c.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
684.2.d.a 684.d 57.d $8$ $5.462$ 8.0.\(\cdots\).4 \(\Q(\sqrt{-19}) \) None 684.2.d.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\beta _{3}q^{5}-\beta _{2}q^{7}-\beta _{6}q^{11}+(\beta _{3}-\beta _{5}+\cdots)q^{17}+\cdots\)
684.2.f.a 684.f 76.d $4$ $5.462$ \(\Q(\sqrt{-2}, \sqrt{19})\) \(\Q(\sqrt{-57}) \) None 684.2.f.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\beta _{1}q^{2}-2q^{4}+2\beta _{1}q^{8}-\beta _{2}q^{11}+\cdots\)
684.2.f.b 684.f 76.d $8$ $5.462$ 8.0.\(\cdots\).1 None None 76.2.d.a \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-1+\beta _{2})q^{4}+(1-\beta _{2}+\beta _{6}+\cdots)q^{5}+\cdots\)
684.2.f.c 684.f 76.d $10$ $5.462$ 10.0.\(\cdots\).1 None None 228.2.f.a \(-2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{4}q^{5}-\beta _{5}q^{7}+\cdots\)
684.2.f.d 684.f 76.d $10$ $5.462$ 10.0.\(\cdots\).1 None None 228.2.f.a \(2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+\beta _{2}q^{4}-\beta _{4}q^{5}-\beta _{5}q^{7}+\cdots\)
684.2.f.e 684.f 76.d $16$ $5.462$ 16.0.\(\cdots\).4 None None 684.2.f.e \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{2}+(1+\beta _{6})q^{4}-\beta _{11}q^{5}+(\beta _{7}+\cdots)q^{7}+\cdots\)
684.2.i.a 684.i 9.c $2$ $5.462$ \(\Q(\sqrt{-3}) \) None None 684.2.i.a \(0\) \(0\) \(-3\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-2\zeta_{6})q^{3}-3\zeta_{6}q^{5}+(5-5\zeta_{6})q^{7}+\cdots\)
684.2.i.b 684.i 9.c $2$ $5.462$ \(\Q(\sqrt{-3}) \) None None 684.2.i.b \(0\) \(0\) \(1\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-2\zeta_{6})q^{3}+\zeta_{6}q^{5}+(-3+3\zeta_{6})q^{7}+\cdots\)
684.2.i.c 684.i 9.c $16$ $5.462$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None 684.2.i.c \(0\) \(-1\) \(2\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{3}-\beta _{13}q^{5}+(-\beta _{3}+\beta _{8}+\beta _{11}+\cdots)q^{7}+\cdots\)
684.2.i.d 684.i 9.c $16$ $5.462$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None 684.2.i.d \(0\) \(-1\) \(6\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{2}+\beta _{13})q^{3}+(\beta _{7}+\beta _{8})q^{5}+(-1+\cdots)q^{7}+\cdots\)
684.2.j.a 684.j 171.h $40$ $5.462$ None None 684.2.j.a \(0\) \(-2\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{3}]$
684.2.k.a 684.k 19.c $2$ $5.462$ \(\Q(\sqrt{-3}) \) None None 684.2.k.a \(0\) \(0\) \(-4\) \(-6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-4+4\zeta_{6})q^{5}-3q^{7}+4q^{11}-5\zeta_{6}q^{13}+\cdots\)
684.2.k.b 684.k 19.c $2$ $5.462$ \(\Q(\sqrt{-3}) \) None None 76.2.e.a \(0\) \(0\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{5}+4q^{11}+\zeta_{6}q^{13}+\cdots\)
684.2.k.c 684.k 19.c $2$ $5.462$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None 684.2.k.c \(0\) \(0\) \(0\) \(-2\) $\mathrm{U}(1)[D_{3}]$ \(q-q^{7}+7\zeta_{6}q^{13}+(5-2\zeta_{6})q^{19}+5\zeta_{6}q^{25}+\cdots\)
684.2.k.d 684.k 19.c $2$ $5.462$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None 684.2.k.d \(0\) \(0\) \(0\) \(10\) $\mathrm{U}(1)[D_{3}]$ \(q+5q^{7}-5\zeta_{6}q^{13}+(3+2\zeta_{6})q^{19}+\cdots\)
684.2.k.e 684.k 19.c $2$ $5.462$ \(\Q(\sqrt{-3}) \) None None 684.2.k.a \(0\) \(0\) \(4\) \(-6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(4-4\zeta_{6})q^{5}-3q^{7}-4q^{11}-5\zeta_{6}q^{13}+\cdots\)
684.2.k.f 684.k 19.c $4$ $5.462$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None None 228.2.i.a \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{2}q^{5}+(-1+\beta _{3})q^{7}-\beta _{3}q^{11}+\cdots\)
684.2.k.g 684.k 19.c $4$ $5.462$ \(\Q(\sqrt{-3}, \sqrt{7})\) None None 228.2.i.b \(0\) \(0\) \(2\) \(8\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1}+\beta _{2})q^{5}+(2-\beta _{3})q^{7}+(-3+\cdots)q^{11}+\cdots\)
684.2.l.a 684.l 171.g $40$ $5.462$ None None 684.2.j.a \(0\) \(1\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{3}]$
684.2.n.a 684.n 171.k $2$ $5.462$ \(\Q(\sqrt{-3}) \) None None 684.2.n.a \(0\) \(0\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1-2\zeta_{6})q^{3}+(-1+\zeta_{6})q^{7}-3q^{9}+\cdots\)
684.2.n.b 684.n 171.k $38$ $5.462$ None None 684.2.n.b \(0\) \(3\) \(0\) \(2\) $\mathrm{SU}(2)[C_{6}]$
684.2.o.a 684.o 684.o $232$ $5.462$ None None 684.2.o.a \(-3\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
684.2.r.a 684.r 76.f $16$ $5.462$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None 76.2.f.a \(3\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{12}q^{2}+(-1-\beta _{1}+\beta _{2}-2\beta _{3}+\cdots)q^{4}+\cdots\)
684.2.r.b 684.r 76.f $20$ $5.462$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None None 228.2.k.a \(-1\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{1}+\beta _{4})q^{2}+(-\beta _{2}-\beta _{9})q^{4}+\cdots\)
684.2.r.c 684.r 76.f $20$ $5.462$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None None 228.2.k.a \(1\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{16}q^{2}+(\beta _{14}-\beta _{17})q^{4}-\beta _{6}q^{5}+\cdots\)
684.2.r.d 684.r 76.f $40$ $5.462$ None None 684.2.r.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
684.2.u.a 684.u 684.u $232$ $5.462$ None None 684.2.u.a \(0\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{6}]$
684.2.w.a 684.w 684.w $232$ $5.462$ None None 684.2.w.a \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{6}]$
684.2.z.a 684.z 228.m $8$ $5.462$ 8.0.\(\cdots\).3 None None 684.2.z.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{3}q^{2}-2\beta _{4}q^{4}-2\beta _{3}q^{5}+(-\beta _{5}+\cdots)q^{7}+\cdots\)
684.2.z.b 684.z 228.m $72$ $5.462$ None None 684.2.z.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
684.2.bb.a 684.bb 171.t $2$ $5.462$ \(\Q(\sqrt{-3}) \) None None 684.2.n.a \(0\) \(-3\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-2+\zeta_{6})q^{3}-\zeta_{6}q^{7}+(3-3\zeta_{6})q^{9}+\cdots\)
684.2.bb.b 684.bb 171.t $38$ $5.462$ None None 684.2.n.b \(0\) \(3\) \(0\) \(2\) $\mathrm{SU}(2)[C_{6}]$
684.2.bd.a 684.bd 171.l $40$ $5.462$ None None 684.2.bd.a \(0\) \(0\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{6}]$
684.2.bg.a 684.bg 36.h $4$ $5.462$ \(\Q(\zeta_{12})\) None None 684.2.bg.a \(-2\) \(0\) \(-6\) \(-12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(1-2\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
684.2.bg.b 684.bg 36.h $4$ $5.462$ \(\Q(\zeta_{12})\) None None 684.2.bg.a \(2\) \(0\) \(-6\) \(12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(-1+2\zeta_{12}^{2}+\cdots)q^{3}+\cdots\)
684.2.bg.c 684.bg 36.h $208$ $5.462$ None None 684.2.bg.c \(0\) \(0\) \(12\) \(0\) $\mathrm{SU}(2)[C_{6}]$
684.2.bi.a 684.bi 684.ai $232$ $5.462$ None None 684.2.o.a \(0\) \(0\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{6}]$
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