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Label Char Prim Dim $A$ Field CM RM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
39.1.d.a 39.d 39.d $1$ $0.019$ \(\Q\) \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-39}) \) \(\Q(\sqrt{13}) \) \(0\) \(-1\) \(0\) \(0\) \(q-q^{3}-q^{4}+q^{9}+q^{12}-q^{13}+q^{16}+\cdots\)
39.2.a.a 39.a 1.a $1$ $0.311$ \(\Q\) None None \(1\) \(-1\) \(2\) \(-4\) $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}-q^{4}+2q^{5}-q^{6}-4q^{7}+\cdots\)
39.2.a.b 39.a 1.a $2$ $0.311$ \(\Q(\sqrt{2}) \) None None \(-2\) \(2\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}-2\beta q^{5}+\cdots\)
39.2.b.a 39.b 13.b $2$ $0.311$ \(\Q(\sqrt{-3}) \) None None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{6}q^{2}-q^{3}-q^{4}+\zeta_{6}q^{6}+2\zeta_{6}q^{7}+\cdots\)
39.2.e.a 39.e 13.c $2$ $0.311$ \(\Q(\sqrt{-3}) \) None None \(-1\) \(1\) \(-2\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
39.2.e.b 39.e 13.c $4$ $0.311$ \(\Q(\sqrt{-3}, \sqrt{17})\) None None \(-1\) \(-2\) \(-6\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}-\beta _{2}q^{3}+(-2+\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)
39.2.f.a 39.f 39.f $4$ $0.311$ \(\Q(\zeta_{8})\) None None \(0\) \(-4\) \(0\) \(4\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{8}q^{2}+(-1+\zeta_{8}+\zeta_{8}^{3})q^{3}-\zeta_{8}^{2}q^{4}+\cdots\)
39.2.j.a 39.j 13.e $2$ $0.311$ \(\Q(\sqrt{-3}) \) None None \(0\) \(1\) \(0\) \(-3\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1-\zeta_{6})q^{3}-2\zeta_{6}q^{4}+(-2+4\zeta_{6})q^{5}+\cdots\)
39.2.k.a 39.k 39.k $4$ $0.311$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-10\) $\mathrm{U}(1)[D_{12}]$ \(q+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+2\zeta_{12}q^{4}+(-2+\cdots)q^{7}+\cdots\)
39.2.k.b 39.k 39.k $8$ $0.311$ 8.0.56070144.2 None None \(0\) \(-2\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-\beta _{1}-\beta _{2}+\beta _{3}-\beta _{4}+2\beta _{5}+\beta _{7})q^{2}+\cdots\)
39.3.c.a 39.c 3.b $8$ $1.063$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None None \(0\) \(-2\) \(0\) \(8\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}+\beta _{3}q^{3}+(-2-\beta _{2}+\beta _{5}+\cdots)q^{4}+\cdots\)
39.3.d.a 39.d 39.d $2$ $1.063$ \(\Q(\sqrt{13}) \) \(\Q(\sqrt{-39}) \) None \(0\) \(-6\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\beta q^{2}-3q^{3}+9q^{4}+2\beta q^{5}+3\beta q^{6}+\cdots\)
39.3.d.b 39.d 39.d $2$ $1.063$ \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-39}) \) None \(0\) \(6\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta q^{2}+3q^{3}-q^{4}-4\beta q^{5}+3\beta q^{6}+\cdots\)
39.3.d.c 39.d 39.d $4$ $1.063$ \(\Q(\sqrt{3}, \sqrt{-35})\) None None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}+(-1+\beta _{1})q^{3}-q^{4}-3\beta _{2}q^{5}+\cdots\)
39.3.g.a 39.g 13.d $8$ $1.063$ 8.0.\(\cdots\).10 None None \(4\) \(0\) \(-20\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{4}q^{2}-\beta _{2}q^{3}+(-1-\beta _{1}-\beta _{4}+\cdots)q^{4}+\cdots\)
39.3.h.a 39.h 39.h $2$ $1.063$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(0\) \(9\) $\mathrm{U}(1)[D_{6}]$ \(q+(-3+3\zeta_{6})q^{3}+4\zeta_{6}q^{4}+(6-3\zeta_{6})q^{7}+\cdots\)
39.3.h.b 39.h 39.h $12$ $1.063$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(0\) \(2\) \(0\) \(-36\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{5}q^{2}+\beta _{11}q^{3}+(-3+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
39.3.i.a 39.i 39.i $2$ $1.063$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(0\) \(13\) $\mathrm{U}(1)[D_{6}]$ \(q+3\zeta_{6}q^{3}+(-4+4\zeta_{6})q^{4}+(13-13\zeta_{6})q^{7}+\cdots\)
39.3.i.b 39.i 39.i $12$ $1.063$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(0\) \(-4\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{4}q^{2}+(-1+\beta _{1}+\beta _{3}-\beta _{5}-\beta _{7}+\cdots)q^{3}+\cdots\)
39.3.l.a 39.l 13.f $8$ $1.063$ 8.0.\(\cdots\).10 None None \(-2\) \(0\) \(16\) \(14\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-\beta _{1}-\beta _{5})q^{2}+(2\beta _{4}-\beta _{5})q^{3}+(1+\cdots)q^{4}+\cdots\)
39.3.l.b 39.l 13.f $12$ $1.063$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(-2\) \(0\) \(4\) \(-32\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-\beta _{1}-\beta _{2})q^{2}+(\beta _{4}+\beta _{5})q^{3}+(-2+\cdots)q^{4}+\cdots\)
39.4.a.a 39.a 1.a $1$ $2.301$ \(\Q\) None None \(0\) \(-3\) \(-12\) \(2\) $-$ $\mathrm{SU}(2)$ \(q-3q^{3}-8q^{4}-12q^{5}+2q^{7}+9q^{9}+\cdots\)
39.4.a.b 39.a 1.a $2$ $2.301$ \(\Q(\sqrt{14}) \) None None \(2\) \(-6\) \(24\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}-3q^{3}+(7+2\beta )q^{4}+(12+\cdots)q^{5}+\cdots\)
39.4.a.c 39.a 1.a $3$ $2.301$ 3.3.3144.1 None None \(2\) \(9\) \(4\) \(30\) $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+3q^{3}+(3+\beta _{2})q^{4}+(2+\cdots)q^{5}+\cdots\)
39.4.b.a 39.b 13.b $4$ $2.301$ 4.0.5054412.1 None None \(0\) \(-12\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}-3q^{3}+(-7+\beta _{3})q^{4}+\beta _{2}q^{5}+\cdots\)
39.4.b.b 39.b 13.b $4$ $2.301$ 4.0.1362828.1 None None \(0\) \(12\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+3q^{3}+(-4+\beta _{3})q^{4}+(2\beta _{1}+\cdots)q^{5}+\cdots\)
39.4.e.a 39.e 13.c $2$ $2.301$ \(\Q(\sqrt{-3}) \) None None \(-3\) \(-3\) \(-18\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{2}+(-3+3\zeta_{6})q^{3}+\cdots\)
39.4.e.b 39.e 13.c $2$ $2.301$ \(\Q(\sqrt{-3}) \) None None \(1\) \(-3\) \(14\) \(10\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(-3+3\zeta_{6})q^{3}+7\zeta_{6}q^{4}+\cdots\)
39.4.e.c 39.e 13.c $8$ $2.301$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None None \(-2\) \(12\) \(-12\) \(14\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(3+3\beta _{2})q^{3}+(-1+\beta _{1}+\cdots)q^{4}+\cdots\)
39.4.f.a 39.f 39.f $4$ $2.301$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-40\) $\mathrm{U}(1)[D_{4}]$ \(q-\zeta_{12}^{3}q^{3}+8\zeta_{12}q^{4}+(-10+10\zeta_{12}+\cdots)q^{7}+\cdots\)
39.4.f.b 39.f 39.f $20$ $2.301$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None None \(0\) \(-4\) \(0\) \(44\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{8}q^{2}-\beta _{5}q^{3}+(-6\beta _{4}-\beta _{14})q^{4}+\cdots\)
39.4.j.a 39.j 13.e $2$ $2.301$ \(\Q(\sqrt{-3}) \) None None \(0\) \(3\) \(0\) \(18\) $\mathrm{SU}(2)[C_{6}]$ \(q+(3-3\zeta_{6})q^{3}-8\zeta_{6}q^{4}+(3-6\zeta_{6})q^{5}+\cdots\)
39.4.j.b 39.j 13.e $4$ $2.301$ \(\Q(\sqrt{-3}, \sqrt{-17})\) None None \(0\) \(6\) \(0\) \(-66\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}+(3-3\beta _{2})q^{3}+9\beta _{2}q^{4}+(3+\cdots)q^{5}+\cdots\)
39.4.j.c 39.j 13.e $10$ $2.301$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None None \(0\) \(-15\) \(0\) \(30\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{3}q^{2}-3\beta _{2}q^{3}+(6-6\beta _{2}-\beta _{4}+\cdots)q^{4}+\cdots\)
39.4.k.a 39.k 39.k $48$ $2.301$ None None \(0\) \(-2\) \(0\) \(20\) $\mathrm{SU}(2)[C_{12}]$
39.5.c.a 39.c 3.b $16$ $4.031$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(0\) \(-2\) \(0\) \(-80\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+\beta _{6}q^{3}+(-6+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
39.5.d.a 39.d 39.d $1$ $4.031$ \(\Q\) \(\Q(\sqrt{-39}) \) None \(-5\) \(9\) \(-2\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-5q^{2}+9q^{3}+9q^{4}-2q^{5}-45q^{6}+\cdots\)
39.5.d.b 39.d 39.d $1$ $4.031$ \(\Q\) \(\Q(\sqrt{-39}) \) None \(5\) \(9\) \(2\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+5q^{2}+9q^{3}+9q^{4}+2q^{5}+45q^{6}+\cdots\)
39.5.d.c 39.d 39.d $2$ $4.031$ \(\Q(\sqrt{39}) \) \(\Q(\sqrt{-39}) \) None \(0\) \(-18\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta q^{2}-9q^{3}+23q^{4}+8\beta q^{5}-9\beta q^{6}+\cdots\)
39.5.d.d 39.d 39.d $12$ $4.031$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{2}+\beta _{4}q^{3}+(2+\beta _{1})q^{4}+(2\beta _{3}+\cdots)q^{5}+\cdots\)
39.5.g.a 39.g 13.d $20$ $4.031$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None None \(0\) \(0\) \(24\) \(-24\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{1}q^{2}+\beta _{6}q^{3}+(10\beta _{4}+\beta _{9}+\beta _{15}+\cdots)q^{4}+\cdots\)
39.5.h.a 39.h 39.h $2$ $4.031$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(9\) \(0\) \(117\) $\mathrm{U}(1)[D_{6}]$ \(q+(9-9\zeta_{6})q^{3}+2^{4}\zeta_{6}q^{4}+(78-39\zeta_{6})q^{7}+\cdots\)
39.5.h.b 39.h 39.h $32$ $4.031$ None None \(0\) \(-10\) \(0\) \(-210\) $\mathrm{SU}(2)[C_{6}]$
39.5.i.a 39.i 39.i $2$ $4.031$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(-9\) \(0\) \(-71\) $\mathrm{U}(1)[D_{6}]$ \(q-9\zeta_{6}q^{3}+(-2^{4}+2^{4}\zeta_{6})q^{4}+(-71+\cdots)q^{7}+\cdots\)
39.5.i.b 39.i 39.i $32$ $4.031$ None None \(0\) \(8\) \(0\) \(122\) $\mathrm{SU}(2)[C_{6}]$
39.5.l.a 39.l 13.f $16$ $4.031$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(0\) \(0\) \(-54\) \(-44\) $\mathrm{SU}(2)[C_{12}]$ \(q+\beta _{6}q^{2}+(-6\beta _{4}+3\beta _{5})q^{3}+(-5+\cdots)q^{4}+\cdots\)
39.5.l.b 39.l 13.f $20$ $4.031$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None None \(0\) \(0\) \(30\) \(-42\) $\mathrm{SU}(2)[C_{12}]$ \(q+\beta _{5}q^{2}+(-3\beta _{4}-3\beta _{7})q^{3}+(4-\beta _{1}+\cdots)q^{4}+\cdots\)
39.6.a.a 39.a 1.a $1$ $6.255$ \(\Q\) None None \(2\) \(9\) \(-74\) \(-112\) $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+9q^{3}-28q^{4}-74q^{5}+18q^{6}+\cdots\)
39.6.a.b 39.a 1.a $2$ $6.255$ \(\Q(\sqrt{14}) \) None None \(-4\) \(-18\) \(-48\) \(72\) $+$ $\mathrm{SU}(2)$ \(q+(-2+\beta )q^{2}-9q^{3}+(28-4\beta )q^{4}+\cdots\)
39.6.a.c 39.a 1.a $3$ $6.255$ 3.3.125308.1 None None \(0\) \(-27\) \(54\) \(84\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-9q^{3}+(5+2\beta _{1}+\beta _{2})q^{4}+\cdots\)
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