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Label Char Prim Dim $A$ Field CM RM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
399.1.h.a 399.h 399.h $1$ $0.199$ \(\Q\) \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-399}) \) \(\Q(\sqrt{133}) \) \(0\) \(-1\) \(0\) \(1\) \(q-q^{3}-q^{4}+q^{7}+q^{9}+q^{12}+2q^{13}+\cdots\)
399.1.h.b 399.h 399.h $1$ $0.199$ \(\Q\) \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-399}) \) \(\Q(\sqrt{133}) \) \(0\) \(1\) \(0\) \(1\) \(q+q^{3}-q^{4}+q^{7}+q^{9}-q^{12}-2q^{13}+\cdots\)
399.1.h.c 399.h 399.h $2$ $0.199$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-399}) \) None \(0\) \(-2\) \(0\) \(-2\) \(q-\beta q^{2}-q^{3}+q^{4}-\beta q^{5}+\beta q^{6}-q^{7}+\cdots\)
399.1.h.d 399.h 399.h $2$ $0.199$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-399}) \) None \(0\) \(2\) \(0\) \(-2\) \(q-\beta q^{2}+q^{3}+q^{4}+\beta q^{5}-\beta q^{6}-q^{7}+\cdots\)
399.1.n.a 399.n 399.n $4$ $0.199$ \(\Q(\zeta_{12})\) None None \(0\) \(0\) \(0\) \(-4\) \(q-\zeta_{12}^{3}q^{2}+\zeta_{12}^{5}q^{3}-\zeta_{12}^{3}q^{5}+\cdots\)
399.1.q.a 399.q 399.q $2$ $0.199$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(-1\) \(0\) \(-1\) \(q-\zeta_{6}q^{3}-\zeta_{6}^{2}q^{4}-\zeta_{6}q^{7}+\zeta_{6}^{2}q^{9}+\cdots\)
399.1.q.b 399.q 399.q $2$ $0.199$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(1\) \(0\) \(-1\) \(q+\zeta_{6}q^{3}-\zeta_{6}^{2}q^{4}+\zeta_{6}^{2}q^{7}+\zeta_{6}^{2}q^{9}+\cdots\)
399.1.bi.a 399.bi 399.ai $4$ $0.199$ \(\Q(\zeta_{12})\) None None \(0\) \(0\) \(0\) \(-4\) \(q+\zeta_{12}^{5}q^{2}-\zeta_{12}^{3}q^{3}+\zeta_{12}^{5}q^{5}+\cdots\)
399.1.bs.a 399.bs 399.as $6$ $0.199$ \(\Q(\zeta_{18})\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{18}^{5}q^{3}-\zeta_{18}^{4}q^{4}+\zeta_{18}^{2}q^{7}+\cdots\)
399.1.by.a 399.by 399.ay $6$ $0.199$ \(\Q(\zeta_{18})\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{18}^{5}q^{3}+\zeta_{18}q^{4}-\zeta_{18}^{7}q^{7}-\zeta_{18}q^{9}+\cdots\)
399.1.ca.a 399.ca 399.ba $6$ $0.199$ \(\Q(\zeta_{18})\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{18}^{2}q^{3}-\zeta_{18}^{7}q^{4}+\zeta_{18}^{8}q^{7}+\cdots\)
399.1.ch.a 399.ch 399.bh $6$ $0.199$ \(\Q(\zeta_{18})\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{18}^{7}q^{3}-\zeta_{18}^{5}q^{4}+\zeta_{18}^{8}q^{7}+\cdots\)
399.2.a.a 399.a 1.a $1$ $3.186$ \(\Q\) None None \(-1\) \(-1\) \(0\) \(1\) $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{3}-q^{4}+q^{6}+q^{7}+3q^{8}+\cdots\)
399.2.a.b 399.a 1.a $1$ $3.186$ \(\Q\) None None \(-1\) \(1\) \(4\) \(-1\) $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}-q^{4}+4q^{5}-q^{6}-q^{7}+\cdots\)
399.2.a.c 399.a 1.a $1$ $3.186$ \(\Q\) None None \(1\) \(-1\) \(0\) \(-1\) $+$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}-q^{4}-q^{6}-q^{7}-3q^{8}+\cdots\)
399.2.a.d 399.a 1.a $3$ $3.186$ 3.3.148.1 None None \(1\) \(-3\) \(4\) \(-3\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-q^{3}+(\beta _{1}+\beta _{2})q^{4}+(1+\beta _{1}+\cdots)q^{5}+\cdots\)
399.2.a.e 399.a 1.a $3$ $3.186$ 3.3.404.1 None None \(1\) \(3\) \(0\) \(-3\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{2}+q^{3}+(3+\beta _{1}-\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
399.2.a.f 399.a 1.a $5$ $3.186$ 5.5.1240016.1 None None \(1\) \(5\) \(-2\) \(5\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{3}q^{2}+q^{3}+(1+\beta _{4})q^{4}+\beta _{2}q^{5}+\cdots\)
399.2.a.g 399.a 1.a $5$ $3.186$ 5.5.368464.1 None None \(3\) \(-5\) \(4\) \(5\) $-$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-q^{3}+(2-\beta _{1}+\beta _{2})q^{4}+\cdots\)
399.2.c.a 399.c 133.c $2$ $3.186$ \(\Q(\sqrt{-3}) \) None None \(0\) \(-2\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{6}q^{2}-q^{3}-q^{4}+2\zeta_{6}q^{5}+\zeta_{6}q^{6}+\cdots\)
399.2.c.b 399.c 133.c $2$ $3.186$ \(\Q(\sqrt{-3}) \) None None \(0\) \(2\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{6}q^{2}+q^{3}-q^{4}-2\zeta_{6}q^{5}-\zeta_{6}q^{6}+\cdots\)
399.2.c.c 399.c 133.c $12$ $3.186$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None None \(0\) \(-12\) \(0\) \(5\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}-q^{3}+(-1+\beta _{2})q^{4}-\beta _{6}q^{5}+\cdots\)
399.2.c.d 399.c 133.c $12$ $3.186$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None None \(0\) \(12\) \(0\) \(5\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+q^{3}+(-1+\beta _{2})q^{4}+\beta _{6}q^{5}+\cdots\)
399.2.d.a 399.d 21.c $48$ $3.186$ None None \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{2}]$
399.2.f.a 399.f 57.d $4$ $3.186$ \(\Q(\sqrt{-2}, \sqrt{3})\) None None \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}-\beta _{2}q^{3}+q^{4}+\beta _{1}q^{5}-3q^{6}+\cdots\)
399.2.f.b 399.f 57.d $16$ $3.186$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{14}q^{2}-\beta _{9}q^{3}+(1-\beta _{10})q^{4}-\beta _{4}q^{5}+\cdots\)
399.2.f.c 399.f 57.d $20$ $3.186$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None None \(0\) \(0\) \(0\) \(20\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{12}q^{2}+\beta _{1}q^{3}+(1-\beta _{10})q^{4}+\beta _{3}q^{5}+\cdots\)
399.2.i.a 399.i 133.g $2$ $3.186$ \(\Q(\sqrt{-3}) \) None None \(-2\) \(-2\) \(0\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}-q^{3}+(-2+2\zeta_{6})q^{4}+2\zeta_{6}q^{6}+\cdots\)
399.2.i.b 399.i 133.g $22$ $3.186$ None None \(2\) \(-22\) \(1\) \(-8\) $\mathrm{SU}(2)[C_{3}]$
399.2.i.c 399.i 133.g $28$ $3.186$ None None \(0\) \(28\) \(1\) \(2\) $\mathrm{SU}(2)[C_{3}]$
399.2.j.a 399.j 7.c $2$ $3.186$ \(\Q(\sqrt{-3}) \) None None \(-2\) \(-1\) \(-1\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(-1+\zeta_{6})q^{3}+(-2+2\zeta_{6})q^{4}+\cdots\)
399.2.j.b 399.j 7.c $2$ $3.186$ \(\Q(\sqrt{-3}) \) None None \(0\) \(-1\) \(-2\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+(2-2\zeta_{6})q^{4}-2\zeta_{6}q^{5}+\cdots\)
399.2.j.c 399.j 7.c $4$ $3.186$ \(\Q(\sqrt{2}, \sqrt{-3})\) None None \(2\) \(-2\) \(2\) \(-10\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1}+\beta _{2})q^{2}+\beta _{2}q^{3}+(2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
399.2.j.d 399.j 7.c $8$ $3.186$ 8.0.310217769.2 None None \(0\) \(4\) \(-4\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(1-\beta _{5})q^{3}+(-\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
399.2.j.e 399.j 7.c $8$ $3.186$ 8.0.542936601.2 None None \(2\) \(-4\) \(-4\) \(6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{3}+\beta _{5})q^{2}-\beta _{4}q^{3}+(2-\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)
399.2.j.f 399.j 7.c $8$ $3.186$ 8.0.42575625.1 None None \(2\) \(-4\) \(2\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{5}q^{2}+(-1-\beta _{6})q^{3}+(-\beta _{2}-\beta _{3}+\cdots)q^{4}+\cdots\)
399.2.j.g 399.j 7.c $16$ $3.186$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(0\) \(8\) \(5\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+(1-\beta _{5})q^{3}+(-1+\beta _{5}+\beta _{9}+\cdots)q^{4}+\cdots\)
399.2.k.a 399.k 19.c $8$ $3.186$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None None \(-1\) \(-4\) \(2\) \(8\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}+\beta _{4}q^{3}+(-1-\beta _{3}-\beta _{4}+\cdots)q^{4}+\cdots\)
399.2.k.b 399.k 19.c $8$ $3.186$ 8.0.310217769.2 None None \(1\) \(4\) \(2\) \(-8\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{4}-\beta _{6})q^{2}+(1-\beta _{5})q^{3}+(-\beta _{1}+\cdots)q^{4}+\cdots\)
399.2.k.c 399.k 19.c $12$ $3.186$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(-1\) \(-6\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}-\beta _{7}q^{3}+(-1+\beta _{7}-\beta _{10}+\cdots)q^{4}+\cdots\)
399.2.k.d 399.k 19.c $12$ $3.186$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(1\) \(6\) \(-4\) \(12\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+\beta _{4}q^{3}+(-1+\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
399.2.l.a 399.l 133.h $2$ $3.186$ \(\Q(\sqrt{-3}) \) None None \(4\) \(1\) \(0\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+2q^{2}+\zeta_{6}q^{3}+2q^{4}+2\zeta_{6}q^{6}+(3+\cdots)q^{7}+\cdots\)
399.2.l.b 399.l 133.h $22$ $3.186$ None None \(-4\) \(11\) \(-2\) \(-8\) $\mathrm{SU}(2)[C_{3}]$
399.2.l.c 399.l 133.h $28$ $3.186$ None None \(0\) \(-14\) \(-2\) \(2\) $\mathrm{SU}(2)[C_{3}]$
399.2.m.a 399.m 133.i $2$ $3.186$ \(\Q(\sqrt{-3}) \) None None \(0\) \(-1\) \(0\) \(5\) $\mathrm{SU}(2)[C_{6}]$ \(q-\zeta_{6}q^{3}+2q^{4}+(-2+4\zeta_{6})q^{5}+(3+\cdots)q^{7}+\cdots\)
399.2.m.b 399.m 133.i $4$ $3.186$ \(\Q(\zeta_{12})\) None None \(0\) \(2\) \(0\) \(-10\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1-2\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+\zeta_{12}^{2}q^{3}+\cdots\)
399.2.m.c 399.m 133.i $22$ $3.186$ None None \(0\) \(11\) \(0\) \(10\) $\mathrm{SU}(2)[C_{6}]$
399.2.m.d 399.m 133.i $24$ $3.186$ None None \(0\) \(-12\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{6}]$
399.2.p.a 399.p 399.p $2$ $3.186$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(0\) \(-5\) $\mathrm{U}(1)[D_{6}]$ \(q+(1+\zeta_{6})q^{3}+2q^{4}+(-2-\zeta_{6})q^{7}+\cdots\)
399.2.p.b 399.p 399.p $2$ $3.186$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(0\) \(4\) $\mathrm{U}(1)[D_{6}]$ \(q+(1+\zeta_{6})q^{3}+2q^{4}+(3-2\zeta_{6})q^{7}+\cdots\)
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