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Label Char Prim Dim $A$ Field CM RM Minimal twist Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
56.1.h.a 56.h 56.h $1$ $0.028$ \(\Q\) \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-14}) \) \(\Q(\sqrt{2}) \) 56.1.h.a \(-1\) \(0\) \(0\) \(-1\) \(q-q^{2}+q^{4}-q^{7}-q^{8}-q^{9}+q^{14}+\cdots\)
56.2.a.a 56.a 1.a $1$ $0.447$ \(\Q\) None None 56.2.a.a \(0\) \(0\) \(2\) \(-1\) $-$ $\mathrm{SU}(2)$ \(q+2q^{5}-q^{7}-3q^{9}-4q^{11}+2q^{13}+\cdots\)
56.2.a.b 56.a 1.a $1$ $0.447$ \(\Q\) None None 56.2.a.b \(0\) \(2\) \(-4\) \(1\) $-$ $\mathrm{SU}(2)$ \(q+2q^{3}-4q^{5}+q^{7}+q^{9}-8q^{15}+\cdots\)
56.2.b.a 56.b 8.b $2$ $0.447$ \(\Q(\sqrt{-2}) \) None None 56.2.b.a \(0\) \(0\) \(0\) \(2\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{2}+\beta q^{3}-2q^{4}-\beta q^{5}-2q^{6}+\cdots\)
56.2.b.b 56.b 8.b $4$ $0.447$ 4.0.2312.1 None None 56.2.b.b \(-1\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+(\beta _{1}-\beta _{2})q^{3}+\beta _{2}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
56.2.e.a 56.e 56.e $2$ $0.447$ \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{-7}) \) None 56.2.e.a \(1\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta q^{2}+(-2+\beta )q^{4}+(1-2\beta )q^{7}+\cdots\)
56.2.e.b 56.e 56.e $4$ $0.447$ \(\Q(i, \sqrt{6})\) None None 56.2.e.b \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1+\beta _{1})q^{2}+\beta _{2}q^{3}-2\beta _{1}q^{4}+\cdots\)
56.2.i.a 56.i 7.c $2$ $0.447$ \(\Q(\sqrt{-3}) \) None None 56.2.i.a \(0\) \(-3\) \(1\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q-3\zeta_{6}q^{3}+(1-\zeta_{6})q^{5}+(1+2\zeta_{6})q^{7}+\cdots\)
56.2.i.b 56.i 7.c $2$ $0.447$ \(\Q(\sqrt{-3}) \) None None 56.2.i.b \(0\) \(1\) \(1\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{3}+(1-\zeta_{6})q^{5}+(-3+2\zeta_{6})q^{7}+\cdots\)
56.2.m.a 56.m 56.m $12$ $0.447$ 12.0.\(\cdots\).2 None None 56.2.m.a \(0\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{1}-\beta _{8})q^{2}+(-1-\beta _{10})q^{3}+(\beta _{2}+\cdots)q^{4}+\cdots\)
56.2.p.a 56.p 56.p $12$ $0.447$ 12.0.\(\cdots\).1 None None 56.2.p.a \(-2\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{1}+\beta _{5})q^{2}+\beta _{6}q^{3}-\beta _{3}q^{4}+\cdots\)
56.3.c.a 56.c 7.b $4$ $1.526$ 4.0.2048.2 None None 56.3.c.a \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}+(\beta _{1}+\beta _{3})q^{5}+(1+\beta _{1}+\beta _{2}+\cdots)q^{7}+\cdots\)
56.3.g.a 56.g 8.d $4$ $1.526$ \(\Q(\sqrt{2}, \sqrt{-7})\) None None 56.3.g.a \(0\) \(8\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(2-\beta _{2})q^{3}+(-3+\beta _{3})q^{4}+\cdots\)
56.3.g.b 56.g 8.d $8$ $1.526$ 8.0.\(\cdots\).3 None None 56.3.g.b \(1\) \(-8\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{5}q^{2}+(-1+\beta _{3}-\beta _{4})q^{3}+(1+\beta _{6}+\cdots)q^{4}+\cdots\)
56.3.h.a 56.h 56.h $2$ $1.526$ \(\Q(\sqrt{7}) \) \(\Q(\sqrt{-14}) \) None 56.3.h.a \(-4\) \(0\) \(0\) \(14\) $\mathrm{U}(1)[D_{2}]$ \(q-2q^{2}+\beta q^{3}+4q^{4}-\beta q^{5}-2\beta q^{6}+\cdots\)
56.3.h.b 56.h 56.h $2$ $1.526$ \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{-7}) \) None 56.3.h.b \(3\) \(0\) \(0\) \(14\) $\mathrm{U}(1)[D_{2}]$ \(q+(1+\beta )q^{2}+(-1+3\beta )q^{4}+7q^{7}+\cdots\)
56.3.h.c 56.h 56.h $2$ $1.526$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-14}) \) None 56.3.h.c \(4\) \(0\) \(0\) \(-14\) $\mathrm{U}(1)[D_{2}]$ \(q+2q^{2}+\beta q^{3}+4q^{4}-3\beta q^{5}+2\beta q^{6}+\cdots\)
56.3.h.d 56.h 56.h $8$ $1.526$ 8.0.\(\cdots\).51 None None 56.3.h.d \(-4\) \(0\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1-\beta _{2})q^{2}+\beta _{6}q^{3}+(-2-\beta _{1}+\cdots)q^{4}+\cdots\)
56.3.j.a 56.j 56.j $28$ $1.526$ None None 56.3.j.a \(-2\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{6}]$
56.3.k.a 56.k 56.k $2$ $1.526$ \(\Q(\sqrt{-3}) \) None None 56.3.k.a \(-4\) \(-1\) \(-9\) \(-2\) $\mathrm{SU}(2)[C_{6}]$ \(q-2q^{2}-\zeta_{6}q^{3}+4q^{4}+(-3-3\zeta_{6})q^{5}+\cdots\)
56.3.k.b 56.k 56.k $2$ $1.526$ \(\Q(\sqrt{-3}) \) None None 56.3.k.a \(2\) \(-1\) \(9\) \(2\) $\mathrm{SU}(2)[C_{6}]$ \(q+2\zeta_{6}q^{2}-\zeta_{6}q^{3}+(-4+4\zeta_{6})q^{4}+\cdots\)
56.3.k.c 56.k 56.k $12$ $1.526$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None None 56.3.k.c \(0\) \(6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{7}q^{2}+(1+\beta _{1}+\beta _{4}+\beta _{8})q^{3}+(-1+\cdots)q^{4}+\cdots\)
56.3.k.d 56.k 56.k $12$ $1.526$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None 56.3.k.d \(2\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{3}q^{2}+(\beta _{8}-\beta _{9}-\beta _{10})q^{3}+(\beta _{2}+\cdots)q^{4}+\cdots\)
56.3.o.a 56.o 7.d $8$ $1.526$ 8.0.\(\cdots\).2 None None 56.3.o.a \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{5}q^{3}+\beta _{4}q^{5}+(1-3\beta _{1}-\beta _{2}+\beta _{3}+\cdots)q^{7}+\cdots\)
56.4.a.a 56.a 1.a $1$ $3.304$ \(\Q\) None None 56.4.a.a \(0\) \(-2\) \(-16\) \(-7\) $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-2^{4}q^{5}-7q^{7}-23q^{9}+24q^{11}+\cdots\)
56.4.a.b 56.a 1.a $1$ $3.304$ \(\Q\) None None 56.4.a.b \(0\) \(6\) \(8\) \(-7\) $+$ $\mathrm{SU}(2)$ \(q+6q^{3}+8q^{5}-7q^{7}+9q^{9}+56q^{11}+\cdots\)
56.4.a.c 56.a 1.a $2$ $3.304$ \(\Q(\sqrt{57}) \) None None 56.4.a.c \(0\) \(-2\) \(22\) \(14\) $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{3}+(11+\beta )q^{5}+7q^{7}+\cdots\)
56.4.b.a 56.b 8.b $8$ $3.304$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None None 56.4.b.a \(-2\) \(0\) \(0\) \(-56\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}-\beta _{4}q^{3}+(2+\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
56.4.b.b 56.b 8.b $10$ $3.304$ \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None None 56.4.b.b \(3\) \(0\) \(0\) \(70\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+\beta _{4}q^{3}+\beta _{2}q^{4}+(-\beta _{1}-\beta _{6}+\cdots)q^{5}+\cdots\)
56.4.e.a 56.e 56.e $2$ $3.304$ \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{-7}) \) None 56.4.e.a \(-5\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(-2-\beta )q^{2}+(2+5\beta )q^{4}+(-7+14\beta )q^{7}+\cdots\)
56.4.e.b 56.e 56.e $4$ $3.304$ \(\Q(\sqrt{-3}, \sqrt{-7})\) None None 56.4.e.b \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1+\beta _{2})q^{2}+\beta _{3}q^{3}+(-6-2\beta _{2}+\cdots)q^{4}+\cdots\)
56.4.e.c 56.e 56.e $16$ $3.304$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None 56.4.e.c \(8\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1+\beta _{2})q^{2}-\beta _{4}q^{3}-\beta _{8}q^{4}+\beta _{12}q^{5}+\cdots\)
56.4.i.a 56.i 7.c $6$ $3.304$ 6.0.11163123.4 None None 56.4.i.a \(0\) \(-1\) \(-13\) \(-20\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{3}q^{3}+(-5+5\beta _{1}-\beta _{3}-\beta _{4}-2\beta _{5})q^{5}+\cdots\)
56.4.i.b 56.i 7.c $6$ $3.304$ 6.0.11163123.4 None None 56.4.i.b \(0\) \(7\) \(3\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\beta _{1}-\beta _{3}+\beta _{5})q^{3}+(\beta _{1}+\beta _{4}+\cdots)q^{5}+\cdots\)
56.4.m.a 56.m 56.m $44$ $3.304$ None None 56.4.m.a \(-2\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
56.4.p.a 56.p 56.p $44$ $3.304$ None None 56.4.p.a \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{6}]$
56.5.c.a 56.c 7.b $8$ $5.789$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None None 56.5.c.a \(0\) \(0\) \(0\) \(56\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}-\beta _{2}q^{5}+(7-\beta _{5})q^{7}+(-31+\cdots)q^{9}+\cdots\)
56.5.g.a 56.g 8.d $2$ $5.789$ \(\Q(\sqrt{-7}) \) None None 56.5.g.a \(6\) \(-12\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(3+\beta )q^{2}-6q^{3}+(2+6\beta )q^{4}+4\beta q^{5}+\cdots\)
56.5.g.b 56.g 8.d $22$ $5.789$ None None 56.5.g.b \(-9\) \(12\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
56.5.h.a 56.h 56.h $1$ $5.789$ \(\Q\) \(\Q(\sqrt{-14}) \) None 56.5.h.a \(4\) \(-10\) \(22\) \(49\) $\mathrm{U}(1)[D_{2}]$ \(q+4q^{2}-10q^{3}+2^{4}q^{4}+22q^{5}-40q^{6}+\cdots\)
56.5.h.b 56.h 56.h $1$ $5.789$ \(\Q\) \(\Q(\sqrt{-14}) \) None 56.5.h.a \(4\) \(10\) \(-22\) \(49\) $\mathrm{U}(1)[D_{2}]$ \(q+4q^{2}+10q^{3}+2^{4}q^{4}-22q^{5}+40q^{6}+\cdots\)
56.5.h.c 56.h 56.h $2$ $5.789$ \(\Q(\sqrt{14}) \) \(\Q(\sqrt{-14}) \) None 56.5.h.c \(-8\) \(0\) \(0\) \(-98\) $\mathrm{U}(1)[D_{2}]$ \(q-4q^{2}+\beta q^{3}+2^{4}q^{4}+3\beta q^{5}-4\beta q^{6}+\cdots\)
56.5.h.d 56.h 56.h $2$ $5.789$ \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{-7}) \) None 56.5.h.d \(-1\) \(0\) \(0\) \(-98\) $\mathrm{U}(1)[D_{2}]$ \(q-\beta q^{2}+(-2^{4}+\beta )q^{4}-7^{2}q^{7}+(2^{4}+\cdots)q^{8}+\cdots\)
56.5.h.e 56.h 56.h $24$ $5.789$ None None 56.5.h.e \(-4\) \(0\) \(0\) \(96\) $\mathrm{SU}(2)[C_{2}]$
56.5.j.a 56.j 56.j $60$ $5.789$ None None 56.5.j.a \(2\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{6}]$
56.5.k.a 56.k 56.k $60$ $5.789$ None None 56.5.k.a \(-4\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
56.5.o.a 56.o 7.d $16$ $5.789$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None 56.5.o.a \(0\) \(0\) \(0\) \(-56\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{2}-\beta _{6})q^{3}+\beta _{4}q^{5}+(-5-3\beta _{1}+\cdots)q^{7}+\cdots\)
56.6.a.a 56.a 1.a $1$ $8.981$ \(\Q\) None None 56.6.a.a \(0\) \(-6\) \(4\) \(49\) $+$ $\mathrm{SU}(2)$ \(q-6q^{3}+4q^{5}+7^{2}q^{7}-207q^{9}-240q^{11}+\cdots\)
56.6.a.b 56.a 1.a $1$ $8.981$ \(\Q\) None None 56.6.a.b \(0\) \(30\) \(32\) \(49\) $-$ $\mathrm{SU}(2)$ \(q+30q^{3}+2^{5}q^{5}+7^{2}q^{7}+657q^{9}+\cdots\)
56.6.a.c 56.a 1.a $2$ $8.981$ \(\Q(\sqrt{177}) \) None None 56.6.a.c \(0\) \(-26\) \(-62\) \(98\) $-$ $\mathrm{SU}(2)$ \(q+(-13-\beta )q^{3}+(-31-5\beta )q^{5}+7^{2}q^{7}+\cdots\)
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