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Label Char Prim Dim $A$ Field CM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
81.2.a.a 81.a 1.a $2$ $0.647$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(4\) $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+q^{4}-\beta q^{5}+2q^{7}-\beta q^{8}+\cdots\)
81.2.c.a 81.c 9.c $2$ $0.647$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(1\) $\mathrm{U}(1)[D_{3}]$ \(q+2\zeta_{6}q^{4}+(1-\zeta_{6})q^{7}-5\zeta_{6}q^{13}+\cdots\)
81.2.c.b 81.c 9.c $4$ $0.647$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{12}^{2}q^{2}+(-1+\zeta_{12})q^{4}+(-\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)
81.2.e.a 81.e 27.e $12$ $0.647$ 12.0.\(\cdots\).1 None \(6\) \(0\) \(3\) \(-6\) $\mathrm{SU}(2)[C_{9}]$ \(q+(1+\beta _{3}-\beta _{8})q^{2}+(1+\beta _{1}+\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
81.2.g.a 81.g 81.g $144$ $0.647$ None \(-18\) \(-18\) \(-18\) \(-18\) $\mathrm{SU}(2)[C_{27}]$
81.3.b.a 81.b 3.b $2$ $2.207$ \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{6}q^{2}+q^{4}-2\zeta_{6}q^{5}+2q^{7}-5\zeta_{6}q^{8}+\cdots\)
81.3.b.b 81.b 3.b $4$ $2.207$ \(\Q(\sqrt{-2}, \sqrt{3})\) None \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-2+\beta _{3})q^{4}+(\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
81.3.d.a 81.d 9.d $2$ $2.207$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(13\) $\mathrm{U}(1)[D_{6}]$ \(q-4\zeta_{6}q^{4}+(13-13\zeta_{6})q^{7}+\zeta_{6}q^{13}+\cdots\)
81.3.d.b 81.d 9.d $4$ $2.207$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(-10\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}+5\zeta_{12}^{2}q^{4}+(\zeta_{12}-\zeta_{12}^{3})q^{5}+\cdots\)
81.3.d.c 81.d 9.d $8$ $2.207$ \(\Q(\zeta_{24})\) None \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{24}^{4}q^{2}+(2-2\zeta_{24}+\zeta_{24}^{2}-\zeta_{24}^{3}+\cdots)q^{4}+\cdots\)
81.3.f.a 81.f 27.f $30$ $2.207$ None \(6\) \(0\) \(15\) \(-6\) $\mathrm{SU}(2)[C_{18}]$
81.3.h.a 81.h 81.h $306$ $2.207$ None \(-18\) \(-18\) \(-18\) \(-18\) $\mathrm{SU}(2)[C_{54}]$
81.4.a.a 81.a 1.a $2$ $4.779$ \(\Q(\sqrt{33}) \) None \(-3\) \(0\) \(-15\) \(7\) $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+(1+3\beta )q^{4}+(-8+\beta )q^{5}+\cdots\)
81.4.a.b 81.a 1.a $2$ $4.779$ \(\Q(\sqrt{57}) \) None \(-3\) \(0\) \(12\) \(10\) $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+(7+3\beta )q^{4}+(7-2\beta )q^{5}+\cdots\)
81.4.a.c 81.a 1.a $2$ $4.779$ \(\Q(\sqrt{3}) \) None \(0\) \(0\) \(0\) \(-44\) $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}-5q^{4}-7\beta q^{5}-22q^{7}-13\beta q^{8}+\cdots\)
81.4.a.d 81.a 1.a $2$ $4.779$ \(\Q(\sqrt{33}) \) None \(3\) \(0\) \(15\) \(7\) $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(1+3\beta )q^{4}+(8-\beta )q^{5}+\cdots\)
81.4.a.e 81.a 1.a $2$ $4.779$ \(\Q(\sqrt{57}) \) None \(3\) \(0\) \(-12\) \(10\) $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(7+3\beta )q^{4}+(-7+2\beta )q^{5}+\cdots\)
81.4.c.a 81.c 9.c $2$ $4.779$ \(\Q(\sqrt{-3}) \) None \(-3\) \(0\) \(-15\) \(25\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-3+3\zeta_{6})q^{2}-\zeta_{6}q^{4}-15\zeta_{6}q^{5}+\cdots\)
81.4.c.b 81.c 9.c $2$ $4.779$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-20\) $\mathrm{U}(1)[D_{3}]$ \(q+8\zeta_{6}q^{4}+(-20+20\zeta_{6})q^{7}+70\zeta_{6}q^{13}+\cdots\)
81.4.c.c 81.c 9.c $2$ $4.779$ \(\Q(\sqrt{-3}) \) None \(3\) \(0\) \(15\) \(25\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\zeta_{6})q^{2}-\zeta_{6}q^{4}+15\zeta_{6}q^{5}+\cdots\)
81.4.c.d 81.c 9.c $4$ $4.779$ \(\Q(\sqrt{-3}, \sqrt{-19})\) None \(-3\) \(0\) \(12\) \(-10\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{1}-\beta _{3})q^{2}+(7\beta _{1}-3\beta _{2}+\cdots)q^{4}+\cdots\)
81.4.c.e 81.c 9.c $4$ $4.779$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-22\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+10\beta _{2}q^{4}+(4\beta _{1}+4\beta _{3})q^{5}+\cdots\)
81.4.c.f 81.c 9.c $4$ $4.779$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(44\) $\mathrm{SU}(2)[C_{3}]$ \(q-\zeta_{12}^{2}q^{2}+(5-5\zeta_{12})q^{4}+(-7\zeta_{12}^{2}+\cdots)q^{5}+\cdots\)
81.4.c.g 81.c 9.c $4$ $4.779$ \(\Q(\sqrt{-3}, \sqrt{-19})\) None \(3\) \(0\) \(-12\) \(-10\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1}+\beta _{3})q^{2}+(7\beta _{1}-3\beta _{2}+3\beta _{3})q^{4}+\cdots\)
81.4.e.a 81.e 27.e $48$ $4.779$ None \(6\) \(0\) \(-6\) \(-6\) $\mathrm{SU}(2)[C_{9}]$
81.4.g.a 81.g 81.g $468$ $4.779$ None \(-18\) \(-18\) \(-18\) \(-18\) $\mathrm{SU}(2)[C_{27}]$
81.5.b.a 81.b 3.b $6$ $8.373$ 6.0.39400128.1 None \(0\) \(0\) \(0\) \(-24\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(-5+\beta _{2})q^{4}+\beta _{4}q^{5}+(-4+\cdots)q^{7}+\cdots\)
81.5.b.b 81.b 3.b $8$ $8.373$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(52\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+(-8+\beta _{5})q^{4}+(\beta _{1}-\beta _{4}+\cdots)q^{5}+\cdots\)
81.5.d.a 81.d 9.d $2$ $8.373$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(-71\) $\mathrm{U}(1)[D_{6}]$ \(q-2^{4}\zeta_{6}q^{4}+(-71+71\zeta_{6})q^{7}+337\zeta_{6}q^{13}+\cdots\)
81.5.d.b 81.d 9.d $4$ $8.373$ \(\Q(\zeta_{12})\) None \(0\) \(0\) \(0\) \(38\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}q^{2}-7\zeta_{12}^{2}q^{4}+(-11\zeta_{12}+\cdots)q^{5}+\cdots\)
81.5.d.c 81.d 9.d $4$ $8.373$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(56\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}+2\beta _{2}q^{4}+(7\beta _{1}-7\beta _{3})q^{5}+\cdots\)
81.5.d.d 81.d 9.d $4$ $8.373$ \(\Q(\sqrt{2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-34\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{3}q^{2}+(38+38\beta _{1})q^{4}+(2\beta _{2}-2\beta _{3})q^{5}+\cdots\)
81.5.d.e 81.d 9.d $16$ $8.373$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(0\) \(0\) \(-52\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{7}q^{2}+(8-8\beta _{1}-\beta _{4}-\beta _{9})q^{4}+\cdots\)
81.5.f.a 81.f 27.f $66$ $8.373$ None \(6\) \(0\) \(-3\) \(-6\) $\mathrm{SU}(2)[C_{18}]$
81.5.h.a 81.h 81.h $630$ $8.373$ None \(-18\) \(-18\) \(-18\) \(-18\) $\mathrm{SU}(2)[C_{54}]$
81.6.a.a 81.a 1.a $2$ $12.991$ \(\Q(\sqrt{129}) \) None \(-3\) \(0\) \(30\) \(-128\) $+$ $\mathrm{SU}(2)$ \(q+(-1-\beta )q^{2}+(1+3\beta )q^{4}+(13+4\beta )q^{5}+\cdots\)
81.6.a.b 81.a 1.a $2$ $12.991$ \(\Q(\sqrt{129}) \) None \(3\) \(0\) \(-30\) \(-128\) $+$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(1+3\beta )q^{4}+(-13-4\beta )q^{5}+\cdots\)
81.6.a.c 81.a 1.a $4$ $12.991$ 4.4.4875021.1 None \(-3\) \(0\) \(-78\) \(-28\) $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta _{1})q^{2}+(13-2\beta _{1}+\beta _{3})q^{4}+\cdots\)
81.6.a.d 81.a 1.a $4$ $12.991$ 4.4.4875021.1 None \(3\) \(0\) \(78\) \(-28\) $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+(13-2\beta _{1}+\beta _{3})q^{4}+\cdots\)
81.6.a.e 81.a 1.a $6$ $12.991$ 6.6.6100947648.1 None \(0\) \(0\) \(0\) \(372\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}+(5^{2}-\beta _{3})q^{4}+(2\beta _{1}+\beta _{2}+\cdots)q^{5}+\cdots\)
81.6.c.a 81.c 9.c $2$ $12.991$ \(\Q(\sqrt{-3}) \) None \(-6\) \(0\) \(6\) \(40\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-6+6\zeta_{6})q^{2}-4\zeta_{6}q^{4}+6\zeta_{6}q^{5}+\cdots\)
81.6.c.b 81.c 9.c $2$ $12.991$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) \(0\) \(0\) \(0\) \(211\) $\mathrm{U}(1)[D_{3}]$ \(q+2^{5}\zeta_{6}q^{4}+(211-211\zeta_{6})q^{7}+775\zeta_{6}q^{13}+\cdots\)
81.6.c.c 81.c 9.c $2$ $12.991$ \(\Q(\sqrt{-3}) \) None \(6\) \(0\) \(-6\) \(40\) $\mathrm{SU}(2)[C_{3}]$ \(q+(6-6\zeta_{6})q^{2}-4\zeta_{6}q^{4}-6\zeta_{6}q^{5}+\cdots\)
81.6.c.d 81.c 9.c $4$ $12.991$ \(\Q(\sqrt{-3}, \sqrt{17})\) None \(-9\) \(0\) \(-72\) \(8\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-4\beta _{1}-\beta _{2})q^{2}+(-31+22\beta _{1}+\cdots)q^{4}+\cdots\)
81.6.c.e 81.c 9.c $4$ $12.991$ \(\Q(\sqrt{-3}, \sqrt{-43})\) None \(-3\) \(0\) \(30\) \(128\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2-2\beta _{1}-\beta _{3})q^{2}+(4\beta _{1}-3\beta _{2}+\cdots)q^{4}+\cdots\)
81.6.c.f 81.c 9.c $4$ $12.991$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None \(0\) \(0\) \(0\) \(-334\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{2}q^{2}+(-22+22\beta _{1})q^{4}+(8\beta _{2}+\cdots)q^{5}+\cdots\)
81.6.c.g 81.c 9.c $4$ $12.991$ \(\Q(\sqrt{-3}, \sqrt{-43})\) None \(3\) \(0\) \(-30\) \(128\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1}-\beta _{3})q^{2}+(\beta _{1}+3\beta _{2}-3\beta _{3})q^{4}+\cdots\)
81.6.c.h 81.c 9.c $4$ $12.991$ \(\Q(\sqrt{-3}, \sqrt{17})\) None \(9\) \(0\) \(72\) \(8\) $\mathrm{SU}(2)[C_{3}]$ \(q+(4\beta _{1}+\beta _{2})q^{2}+(-31+22\beta _{1}+9\beta _{2}+\cdots)q^{4}+\cdots\)
81.6.c.i 81.c 9.c $12$ $12.991$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(0\) \(0\) \(0\) \(-372\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{5}q^{2}+(\beta _{1}-5^{2}\beta _{2}-\beta _{8})q^{4}+(2\beta _{4}+\cdots)q^{5}+\cdots\)
81.6.e.a 81.e 27.e $84$ $12.991$ None \(6\) \(0\) \(93\) \(-6\) $\mathrm{SU}(2)[C_{9}]$
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