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Label Char Prim Dim $A$ Field CM RM Minimal twist Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
79.1.b.a 79.b 79.b $2$ $0.039$ \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-79}) \) None 79.1.b.a \(-1\) \(0\) \(-1\) \(0\) \(q+(-1+\beta )q^{2}+(1-\beta )q^{4}-\beta q^{5}-q^{8}+\cdots\)
79.2.a.a 79.a 1.a $1$ $0.631$ \(\Q\) None None 79.2.a.a \(-1\) \(-1\) \(-3\) \(-1\) $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}-3q^{5}+q^{6}-q^{7}+\cdots\)
79.2.a.b 79.a 1.a $5$ $0.631$ 5.5.81589.1 None None 79.2.a.b \(0\) \(1\) \(7\) \(-5\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1-\beta _{2}-\beta _{4})q^{3}+\beta _{2}q^{4}+\cdots\)
79.2.c.a 79.c 79.c $12$ $0.631$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None 79.2.c.a \(-2\) \(0\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{3}-\beta _{9})q^{3}+(-1+\cdots)q^{4}+\cdots\)
79.2.e.a 79.e 79.e $60$ $0.631$ None None 79.2.e.a \(-9\) \(-9\) \(-11\) \(1\) $\mathrm{SU}(2)[C_{13}]$
79.2.g.a 79.g 79.g $144$ $0.631$ None None 79.2.g.a \(-24\) \(-26\) \(-25\) \(-26\) $\mathrm{SU}(2)[C_{39}]$
79.3.b.a 79.b 79.b $5$ $2.153$ 5.5.19503125.1 \(\Q(\sqrt{-79}) \) None 79.3.b.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{2}q^{2}+(4+2\beta _{1}+\beta _{4})q^{4}+(-3\beta _{1}+\cdots)q^{5}+\cdots\)
79.3.b.b 79.b 79.b $8$ $2.153$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None None 79.3.b.b \(-4\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1-\beta _{4})q^{2}-\beta _{1}q^{3}+(1-\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
79.3.d.a 79.d 79.d $24$ $2.153$ None None 79.3.d.a \(-2\) \(0\) \(-1\) \(-6\) $\mathrm{SU}(2)[C_{6}]$
79.3.f.a 79.f 79.f $156$ $2.153$ None None 79.3.f.a \(-9\) \(-13\) \(-11\) \(-13\) $\mathrm{SU}(2)[C_{26}]$
79.3.h.a 79.h 79.h $288$ $2.153$ None None 79.3.h.a \(-24\) \(-26\) \(-25\) \(-20\) $\mathrm{SU}(2)[C_{78}]$
79.4.a.a 79.a 1.a $2$ $4.661$ \(\Q(\sqrt{17}) \) None None 79.4.a.a \(-1\) \(-2\) \(0\) \(8\) $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-q^{3}+(-4+\beta )q^{4}+(-3+6\beta )q^{5}+\cdots\)
79.4.a.b 79.a 1.a $5$ $4.661$ 5.5.4787257.1 None None 79.4.a.b \(-3\) \(-8\) \(-41\) \(-20\) $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{4})q^{2}+(-1-\beta _{2}+\beta _{4})q^{3}+\cdots\)
79.4.a.c 79.a 1.a $12$ $4.661$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None 79.4.a.c \(4\) \(8\) \(39\) \(16\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1-\beta _{9})q^{3}+(5+\beta _{2})q^{4}+\cdots\)
79.4.c.a 79.c 79.c $38$ $4.661$ None None 79.4.c.a \(0\) \(-7\) \(-1\) \(17\) $\mathrm{SU}(2)[C_{3}]$
79.4.e.a 79.e 79.e $228$ $4.661$ None None 79.4.e.a \(-13\) \(-11\) \(-11\) \(-17\) $\mathrm{SU}(2)[C_{13}]$
79.4.g.a 79.g 79.g $456$ $4.661$ None None 79.4.g.a \(-26\) \(-19\) \(-25\) \(-43\) $\mathrm{SU}(2)[C_{39}]$
79.5.b.a 79.b 79.b $5$ $8.166$ 5.5.19503125.1 \(\Q(\sqrt{-79}) \) None 79.5.b.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{2}q^{2}+(2^{4}+2\beta _{1}-5\beta _{3})q^{4}+(5\beta _{1}+\cdots)q^{5}+\cdots\)
79.5.b.b 79.b 79.b $20$ $8.166$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None None 79.5.b.b \(4\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}-\beta _{1}q^{3}+(6-\beta _{5})q^{4}-\beta _{4}q^{5}+\cdots\)
79.5.d.a 79.d 79.d $52$ $8.166$ None None 79.5.d.a \(2\) \(6\) \(-1\) \(120\) $\mathrm{SU}(2)[C_{6}]$
79.5.f.a 79.f 79.f $300$ $8.166$ None None 79.5.f.a \(-17\) \(-13\) \(-11\) \(-13\) $\mathrm{SU}(2)[C_{26}]$
79.5.h.a 79.h 79.h $624$ $8.166$ None None 79.5.h.a \(-28\) \(-32\) \(-25\) \(-146\) $\mathrm{SU}(2)[C_{78}]$
79.6.a.a 79.a 1.a $14$ $12.670$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None None 79.6.a.a \(-8\) \(-37\) \(-201\) \(-157\) $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(-3+\beta _{6})q^{3}+(14+\cdots)q^{4}+\cdots\)
79.6.a.b 79.a 1.a $19$ $12.670$ \(\mathbb{Q}[x]/(x^{19} - \cdots)\) None None 79.6.a.b \(8\) \(17\) \(199\) \(39\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(1+\beta _{5})q^{3}+(19+\beta _{2})q^{4}+\cdots\)
79.6.c.a 79.c 79.c $64$ $12.670$ None None 79.6.c.a \(0\) \(8\) \(-1\) \(-36\) $\mathrm{SU}(2)[C_{3}]$
79.6.e.a 79.e 79.e $396$ $12.670$ None None 79.6.e.a \(-13\) \(7\) \(-11\) \(105\) $\mathrm{SU}(2)[C_{13}]$
79.6.g.a 79.g 79.g $768$ $12.670$ None None 79.6.g.a \(-26\) \(-34\) \(-25\) \(10\) $\mathrm{SU}(2)[C_{39}]$
79.7.b.a 79.b 79.b $5$ $18.174$ 5.5.19503125.1 \(\Q(\sqrt{-79}) \) None 79.7.b.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(3\beta _{1}-2\beta _{4})q^{2}+(2^{6}-11\beta _{2}-28\beta _{3}+\cdots)q^{4}+\cdots\)
79.7.b.b 79.b 79.b $34$ $18.174$ None None 79.7.b.b \(-12\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$
79.7.d.a 79.d 79.d $78$ $18.174$ None None 79.7.d.a \(-6\) \(-3\) \(-1\) \(225\) $\mathrm{SU}(2)[C_{6}]$
79.7.f.a 79.f 79.f $468$ $18.174$ None None 79.7.f.a \(-1\) \(-13\) \(-11\) \(-13\) $\mathrm{SU}(2)[C_{26}]$
79.7.h.a 79.h 79.h $936$ $18.174$ None None 79.7.h.a \(-20\) \(-23\) \(-25\) \(-251\) $\mathrm{SU}(2)[C_{78}]$
79.8.a.a 79.a 1.a $20$ $24.678$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None None 79.8.a.a \(-24\) \(-55\) \(-1001\) \(-489\) $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+(-3-\beta _{4})q^{3}+(54+\cdots)q^{4}+\cdots\)
79.8.a.b 79.a 1.a $25$ $24.678$ None None 79.8.a.b \(8\) \(107\) \(999\) \(883\) $+$ $\mathrm{SU}(2)$
79.8.c.a 79.c 79.c $92$ $24.678$ None None 79.8.c.a \(-8\) \(-28\) \(-1\) \(-2844\) $\mathrm{SU}(2)[C_{3}]$
79.8.e.a 79.e 79.e $540$ $24.678$ None None 79.8.e.a \(3\) \(-65\) \(-11\) \(-407\) $\mathrm{SU}(2)[C_{13}]$
79.8.g.a 79.g 79.g $1104$ $24.678$ None None 79.8.g.a \(-18\) \(2\) \(-25\) \(2818\) $\mathrm{SU}(2)[C_{39}]$
79.9.b.a 79.b 79.b $5$ $32.183$ 5.5.19503125.1 \(\Q(\sqrt{-79}) \) None 79.9.b.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(-\beta _{2}+\beta _{3})q^{2}+(2^{8}-59\beta _{1}-5\beta _{4})q^{4}+\cdots\)
79.9.b.b 79.b 79.b $48$ $32.183$ None None 79.9.b.b \(4\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$
79.9.d.a 79.d 79.d $104$ $32.183$ None None 79.9.d.a \(2\) \(-84\) \(-1\) \(-2190\) $\mathrm{SU}(2)[C_{6}]$
79.9.f.a 79.f 79.f $636$ $32.183$ None None 79.9.f.a \(-17\) \(-13\) \(-11\) \(-13\) $\mathrm{SU}(2)[C_{26}]$
79.9.h.a 79.h 79.h $1248$ $32.183$ None None 79.9.h.a \(-28\) \(58\) \(-25\) \(2164\) $\mathrm{SU}(2)[C_{78}]$
79.10.a.a 79.a 1.a $27$ $40.688$ None None 79.10.a.a \(-16\) \(-244\) \(-5001\) \(-2896\) $+$ $\mathrm{SU}(2)$
79.10.a.b 79.a 1.a $32$ $40.688$ None None 79.10.a.b \(48\) \(242\) \(4999\) \(6708\) $-$ $\mathrm{SU}(2)$
79.10.c.a 79.c 79.c $118$ $40.688$ None None 79.10.c.a \(16\) \(161\) \(-1\) \(11161\) $\mathrm{SU}(2)[C_{3}]$
79.10.e.a 79.e 79.e $708$ $40.688$ None None 79.10.e.a \(-45\) \(-11\) \(-11\) \(-3825\) $\mathrm{SU}(2)[C_{13}]$
79.10.g.a 79.g 79.g $1416$ $40.688$ None None 79.10.g.a \(-42\) \(-187\) \(-25\) \(-11187\) $\mathrm{SU}(2)[C_{39}]$
79.11.b.a 79.b 79.b $1$ $50.193$ \(\Q\) \(\Q(\sqrt{-79}) \) None 79.11.b.a \(-15\) \(0\) \(4986\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-15q^{2}-799q^{4}+4986q^{5}+27345q^{8}+\cdots\)
79.11.b.b 79.b 79.b $4$ $50.193$ 4.4.780125.1 \(\Q(\sqrt{-79}) \) None 79.11.b.b \(15\) \(0\) \(-4986\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(4+4\beta _{2}-7\beta _{3})q^{2}+(2^{10}+105\beta _{1}+\cdots)q^{4}+\cdots\)
79.11.b.c 79.b 79.b $60$ $50.193$ None None 79.11.b.c \(20\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$
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