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Label Char Prim Dim $A$ Field CM Minimal twist Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
510.2.a.a 510.a 1.a $1$ $4.072$ \(\Q\) None 510.2.a.a \(-1\) \(-1\) \(-1\) \(2\) $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}-q^{5}+q^{6}+2q^{7}+\cdots\)
510.2.a.b 510.a 1.a $1$ $4.072$ \(\Q\) None 510.2.a.b \(-1\) \(1\) \(1\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}+q^{5}-q^{6}-2q^{7}+\cdots\)
510.2.a.c 510.a 1.a $1$ $4.072$ \(\Q\) None 510.2.a.c \(1\) \(-1\) \(-1\) \(-4\) $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}-4q^{7}+\cdots\)
510.2.a.d 510.a 1.a $1$ $4.072$ \(\Q\) None 510.2.a.d \(1\) \(-1\) \(-1\) \(2\) $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{5}-q^{6}+2q^{7}+\cdots\)
510.2.a.e 510.a 1.a $1$ $4.072$ \(\Q\) None 510.2.a.e \(1\) \(-1\) \(1\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}+q^{5}-q^{6}+q^{8}+\cdots\)
510.2.a.f 510.a 1.a $1$ $4.072$ \(\Q\) None 510.2.a.f \(1\) \(1\) \(-1\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}-q^{5}+q^{6}+q^{8}+\cdots\)
510.2.a.g 510.a 1.a $1$ $4.072$ \(\Q\) None 510.2.a.g \(1\) \(1\) \(1\) \(2\) $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}+q^{4}+q^{5}+q^{6}+2q^{7}+\cdots\)
510.2.a.h 510.a 1.a $2$ $4.072$ \(\Q(\sqrt{6}) \) None 510.2.a.h \(-2\) \(-2\) \(2\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}+q^{4}+q^{5}+q^{6}+\beta q^{7}+\cdots\)
510.2.c.a 510.c 17.b $2$ $4.072$ \(\Q(\sqrt{-1}) \) None 510.2.c.a \(-2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-q^{2}+i q^{3}+q^{4}-i q^{5}-i q^{6}+\cdots\)
510.2.c.b 510.c 17.b $2$ $4.072$ \(\Q(\sqrt{-1}) \) None 510.2.c.b \(2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+q^{2}+i q^{3}+q^{4}+i q^{5}+i q^{6}+\cdots\)
510.2.c.c 510.c 17.b $4$ $4.072$ \(\Q(i, \sqrt{13})\) None 510.2.c.c \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-q^{2}-\beta _{1}q^{3}+q^{4}-\beta _{1}q^{5}+\beta _{1}q^{6}+\cdots\)
510.2.c.d 510.c 17.b $4$ $4.072$ \(\Q(\zeta_{8})\) None 510.2.c.d \(4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+q^{2}+\beta_1 q^{3}+q^{4}-\beta_1 q^{5}+\beta_1 q^{6}+\cdots\)
510.2.d.a 510.d 5.b $2$ $4.072$ \(\Q(\sqrt{-1}) \) None 510.2.d.a \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+i q^{2}+i q^{3}-q^{4}+(i-2)q^{5}+\cdots\)
510.2.d.b 510.d 5.b $4$ $4.072$ \(\Q(i, \sqrt{5})\) None 510.2.d.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+\beta _{1}q^{3}-q^{4}-\beta _{2}q^{5}-q^{6}+\cdots\)
510.2.d.c 510.d 5.b $4$ $4.072$ \(\Q(i, \sqrt{6})\) None 510.2.d.c \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}+\beta _{2}q^{3}-q^{4}+(1+\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
510.2.d.d 510.d 5.b $6$ $4.072$ 6.0.5161984.1 None 510.2.d.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{2}+\beta _{3}q^{3}-q^{4}+\beta _{5}q^{5}+q^{6}+\cdots\)
510.2.f.a 510.f 85.c $4$ $4.072$ \(\Q(i, \sqrt{6})\) None 510.2.f.a \(0\) \(-4\) \(4\) \(-8\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}-q^{3}-q^{4}+(1+\beta _{2}-\beta _{3})q^{5}+\cdots\)
510.2.f.b 510.f 85.c $4$ $4.072$ \(\Q(i, \sqrt{6})\) None 510.2.f.a \(0\) \(4\) \(-4\) \(8\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}+q^{3}-q^{4}+(-1-\beta _{2}-\beta _{3})q^{5}+\cdots\)
510.2.f.c 510.f 85.c $6$ $4.072$ 6.0.350464.1 None 510.2.f.c \(0\) \(-6\) \(-2\) \(4\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{4}q^{2}-q^{3}-q^{4}+(-\beta _{1}+\beta _{3})q^{5}+\cdots\)
510.2.f.d 510.f 85.c $6$ $4.072$ 6.0.350464.1 None 510.2.f.c \(0\) \(6\) \(2\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{4}q^{2}+q^{3}-q^{4}+(-\beta _{1}-\beta _{5})q^{5}+\cdots\)
510.2.i.a 510.i 255.k $72$ $4.072$ None 510.2.i.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
510.2.l.a 510.l 15.e $4$ $4.072$ \(\Q(\zeta_{8})\) None 510.2.l.a \(0\) \(4\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{2}+(1+\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+\zeta_{8}^{2}q^{4}+\cdots\)
510.2.l.b 510.l 15.e $4$ $4.072$ \(\Q(\zeta_{8})\) None 510.2.l.b \(0\) \(4\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{2}+(1+\zeta_{8}^{2}+\zeta_{8}^{3})q^{3}+\zeta_{8}^{2}q^{4}+\cdots\)
510.2.l.c 510.l 15.e $4$ $4.072$ \(\Q(\zeta_{8})\) None 510.2.l.c \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{2}+(1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+\zeta_{8}^{2}q^{4}+\cdots\)
510.2.l.d 510.l 15.e $8$ $4.072$ 8.0.4030726144.1 None 510.2.l.d \(0\) \(-8\) \(4\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{6}q^{2}+(-1-\beta _{4}+\beta _{6})q^{3}-\beta _{2}q^{4}+\cdots\)
510.2.l.e 510.l 15.e $8$ $4.072$ 8.0.4030726144.1 None 510.2.l.d \(0\) \(0\) \(-4\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{4}q^{2}+(-\beta _{2}+\beta _{4}+\beta _{6})q^{3}+\beta _{2}q^{4}+\cdots\)
510.2.l.f 510.l 15.e $8$ $4.072$ \(\Q(\zeta_{24})\) None 510.2.l.f \(0\) \(4\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\zeta_{24}-\zeta_{24}^{5})q^{2}+(1-\zeta_{24}^{2}-\zeta_{24}^{3}+\cdots)q^{3}+\cdots\)
510.2.l.g 510.l 15.e $28$ $4.072$ None 510.2.l.g \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{4}]$
510.2.m.a 510.m 85.j $8$ $4.072$ 8.0.110166016.2 None 510.2.m.a \(-8\) \(0\) \(-8\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q-q^{2}+\beta _{2}q^{3}+q^{4}+(-1+\beta _{3}-\beta _{7})q^{5}+\cdots\)
510.2.m.b 510.m 85.j $8$ $4.072$ 8.0.110166016.2 None 510.2.m.a \(8\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{4}]$ \(q+q^{2}-\beta _{3}q^{3}+q^{4}+(\beta _{1}-\beta _{3}-\beta _{5}+\cdots)q^{5}+\cdots\)
510.2.m.c 510.m 85.j $12$ $4.072$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 510.2.m.c \(-12\) \(0\) \(4\) \(4\) $\mathrm{SU}(2)[C_{4}]$ \(q-q^{2}-\beta _{7}q^{3}+q^{4}-\beta _{2}q^{5}+\beta _{7}q^{6}+\cdots\)
510.2.m.d 510.m 85.j $12$ $4.072$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None 510.2.m.c \(12\) \(0\) \(-4\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+q^{2}+\beta _{7}q^{3}+q^{4}+\beta _{5}q^{5}+\beta _{7}q^{6}+\cdots\)
510.2.p.a 510.p 17.c $4$ $4.072$ \(\Q(\zeta_{8})\) None 510.2.p.a \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{8}^{2}q^{2}+\zeta_{8}q^{3}-q^{4}+\zeta_{8}q^{5}-\zeta_{8}^{3}q^{6}+\cdots\)
510.2.p.b 510.p 17.c $4$ $4.072$ \(\Q(\zeta_{8})\) None 510.2.p.b \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}^{2}q^{2}+\zeta_{8}q^{3}-q^{4}-\zeta_{8}q^{5}+\zeta_{8}^{3}q^{6}+\cdots\)
510.2.p.c 510.p 17.c $8$ $4.072$ 8.0.18939904.2 None 510.2.p.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{4}q^{2}+\beta _{2}q^{3}-q^{4}-\beta _{2}q^{5}-\beta _{3}q^{6}+\cdots\)
510.2.p.d 510.p 17.c $8$ $4.072$ \(\Q(\zeta_{16})\) None 510.2.p.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta_{2} q^{2}+\beta_{4} q^{3}-q^{4}+\beta_{4} q^{5}+\cdots\)
510.2.q.a 510.q 255.o $72$ $4.072$ None 510.2.q.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
510.2.t.a 510.t 255.r $72$ $4.072$ None 510.2.i.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
510.2.u.a 510.u 17.d $8$ $4.072$ \(\Q(\zeta_{16})\) None 510.2.u.a \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{8}]$ \(q-\zeta_{16}^{2}q^{2}+\zeta_{16}^{3}q^{3}+\zeta_{16}^{4}q^{4}+\cdots\)
510.2.u.b 510.u 17.d $8$ $4.072$ \(\Q(\zeta_{16})\) None 510.2.u.b \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{8}]$ \(q+\zeta_{16}^{6}q^{2}+\zeta_{16}q^{3}-\zeta_{16}^{4}q^{4}-\zeta_{16}^{5}q^{5}+\cdots\)
510.2.u.c 510.u 17.d $16$ $4.072$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None 510.2.u.c \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{8}]$ \(q-\beta _{5}q^{2}-\beta _{7}q^{3}+\beta _{1}q^{4}+\beta _{6}q^{5}+\cdots\)
510.2.u.d 510.u 17.d $16$ $4.072$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None 510.2.u.d \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{8}]$ \(q+\beta _{3}q^{2}-\beta _{9}q^{3}+\beta _{5}q^{4}-\beta _{1}q^{5}+\cdots\)
510.2.w.a 510.w 255.aa $4$ $4.072$ \(\Q(\zeta_{8})\) None 510.2.w.a \(-4\) \(0\) \(8\) \(-8\) $\mathrm{SU}(2)[C_{8}]$ \(q-q^{2}+(\zeta_{8}+\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+q^{4}+\cdots\)
510.2.w.b 510.w 255.aa $4$ $4.072$ \(\Q(\zeta_{8})\) None 510.2.w.a \(4\) \(-4\) \(-8\) \(-8\) $\mathrm{SU}(2)[C_{8}]$ \(q+q^{2}+(-1+\zeta_{8}+\zeta_{8}^{2})q^{3}+q^{4}+\cdots\)
510.2.w.c 510.w 255.aa $68$ $4.072$ None 510.2.w.c \(-68\) \(0\) \(-8\) \(8\) $\mathrm{SU}(2)[C_{8}]$
510.2.w.d 510.w 255.aa $68$ $4.072$ None 510.2.w.c \(68\) \(4\) \(8\) \(8\) $\mathrm{SU}(2)[C_{8}]$
510.2.z.a 510.z 255.v $4$ $4.072$ \(\Q(\zeta_{8})\) None 510.2.w.a \(0\) \(4\) \(0\) \(4\) $\mathrm{SU}(2)[C_{8}]$ \(q+\zeta_{8}^{2}q^{2}+(1+\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}-q^{4}+\cdots\)
510.2.z.b 510.z 255.v $4$ $4.072$ \(\Q(\zeta_{8})\) None 510.2.w.a \(0\) \(4\) \(0\) \(4\) $\mathrm{SU}(2)[C_{8}]$ \(q-\zeta_{8}^{2}q^{2}+(1-\zeta_{8}-\zeta_{8}^{3})q^{3}-q^{4}+\cdots\)
510.2.z.c 510.z 255.v $68$ $4.072$ None 510.2.w.c \(0\) \(-4\) \(-8\) \(-4\) $\mathrm{SU}(2)[C_{8}]$
510.2.z.d 510.z 255.v $68$ $4.072$ None 510.2.w.c \(0\) \(-4\) \(8\) \(-4\) $\mathrm{SU}(2)[C_{8}]$
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