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Label Char Prim Dim $A$ Field CM RM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
252.1.h.a 252.h 84.h $4$ $0.126$ \(\Q(\zeta_{8})\) \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{8}q^{2}+\zeta_{8}^{2}q^{4}-\zeta_{8}^{2}q^{7}-\zeta_{8}^{3}q^{8}+\cdots\)
252.1.z.a 252.z 7.d $2$ $0.126$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(1\) \(q-\zeta_{6}^{2}q^{7}+(\zeta_{6}+\zeta_{6}^{2})q^{13}+(-1-\zeta_{6}+\cdots)q^{19}+\cdots\)
252.2.a.a 252.a 1.a $1$ $2.012$ \(\Q\) None None \(0\) \(0\) \(-4\) \(-1\) $+$ $\mathrm{SU}(2)$ \(q-4q^{5}-q^{7}-2q^{11}-6q^{13}+4q^{17}+\cdots\)
252.2.a.b 252.a 1.a $1$ $2.012$ \(\Q\) None None \(0\) \(0\) \(0\) \(1\) $-$ $\mathrm{SU}(2)$ \(q+q^{7}+6q^{11}+2q^{13}-4q^{19}+6q^{23}+\cdots\)
252.2.b.a 252.b 28.d $2$ $2.012$ \(\Q(\sqrt{-7}) \) \(\Q(\sqrt{-7}) \) None \(1\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta q^{2}+(-2+\beta )q^{4}+(-1+2\beta )q^{7}+\cdots\)
252.2.b.b 252.b 28.d $4$ $2.012$ \(\Q(\sqrt{-2}, \sqrt{7})\) \(\Q(\sqrt{-21}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{1}q^{2}-2q^{4}-\beta _{2}q^{5}+\beta _{3}q^{7}-2\beta _{1}q^{8}+\cdots\)
252.2.b.c 252.b 28.d $4$ $2.012$ \(\Q(i, \sqrt{7})\) \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(-1+2\beta _{2}+\cdots)q^{7}+\cdots\)
252.2.b.d 252.b 28.d $4$ $2.012$ 4.0.2312.1 None None \(1\) \(0\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(-\beta _{1}-\beta _{3})q^{5}+\cdots\)
252.2.b.e 252.b 28.d $4$ $2.012$ 4.0.2312.1 None None \(1\) \(0\) \(0\) \(2\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+(\beta _{1}+\beta _{3})q^{5}+(1+\cdots)q^{7}+\cdots\)
252.2.e.a 252.e 12.b $12$ $2.012$ 12.0.\(\cdots\).1 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(1+\beta _{2})q^{4}+(\beta _{7}-\beta _{9}-\beta _{10}+\cdots)q^{5}+\cdots\)
252.2.f.a 252.f 21.c $4$ $2.012$ \(\Q(\sqrt{-2}, \sqrt{3})\) None None \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{5}+(-1-\beta _{1})q^{7}-\beta _{2}q^{11}+\cdots\)
252.2.i.a 252.i 63.h $2$ $2.012$ \(\Q(\sqrt{-3}) \) None None \(0\) \(-3\) \(-2\) \(-5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\zeta_{6})q^{3}-2\zeta_{6}q^{5}+(-3+\zeta_{6})q^{7}+\cdots\)
252.2.i.b 252.i 63.h $14$ $2.012$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None None \(0\) \(3\) \(-2\) \(6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}+\beta _{3})q^{3}-\beta _{9}q^{5}+\beta _{10}q^{7}+\cdots\)
252.2.j.a 252.j 9.c $6$ $2.012$ 6.0.309123.1 None None \(0\) \(-2\) \(-1\) \(3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{2}+\beta _{4})q^{3}+(-\beta _{1}+\beta _{5})q^{5}+(1+\cdots)q^{7}+\cdots\)
252.2.j.b 252.j 9.c $6$ $2.012$ 6.0.309123.1 None None \(0\) \(2\) \(3\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{3}+\beta _{4})q^{3}+(-\beta _{1}-\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots\)
252.2.k.a 252.k 7.c $2$ $2.012$ \(\Q(\sqrt{-3}) \) None None \(0\) \(0\) \(-2\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{5}+(-1+3\zeta_{6})q^{7}+\cdots\)
252.2.k.b 252.k 7.c $2$ $2.012$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(5\) $\mathrm{U}(1)[D_{3}]$ \(q+(3-\zeta_{6})q^{7}+5q^{13}+(1-\zeta_{6})q^{19}+\cdots\)
252.2.k.c 252.k 7.c $2$ $2.012$ \(\Q(\sqrt{-3}) \) None None \(0\) \(0\) \(3\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(3-3\zeta_{6})q^{5}+(-1-2\zeta_{6})q^{7}-3\zeta_{6}q^{11}+\cdots\)
252.2.l.a 252.l 63.g $2$ $2.012$ \(\Q(\sqrt{-3}) \) None None \(0\) \(0\) \(4\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-2\zeta_{6})q^{3}+2q^{5}+(1+2\zeta_{6})q^{7}+\cdots\)
252.2.l.b 252.l 63.g $14$ $2.012$ \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None None \(0\) \(0\) \(4\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{3}+(-\beta _{5}+\beta _{9})q^{5}+\beta _{6}q^{7}+\cdots\)
252.2.n.a 252.n 252.n $4$ $2.012$ \(\Q(\zeta_{12})\) None None \(2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(-2\zeta_{12}+\cdots)q^{3}+\cdots\)
252.2.n.b 252.n 252.n $84$ $2.012$ None None \(-1\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
252.2.o.a 252.o 252.o $88$ $2.012$ None None \(-3\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
252.2.t.a 252.t 21.g $4$ $2.012$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None None \(0\) \(0\) \(0\) \(2\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{2}q^{5}+(2-3\beta _{1})q^{7}+(-\beta _{2}-\beta _{3})q^{11}+\cdots\)
252.2.w.a 252.w 63.i $16$ $2.012$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{10}q^{3}+\beta _{7}q^{5}+(\beta _{3}-\beta _{8})q^{7}-\beta _{4}q^{9}+\cdots\)
252.2.x.a 252.x 63.o $16$ $2.012$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{1}-\beta _{8})q^{3}+\beta _{15}q^{5}+(-\beta _{1}+\cdots)q^{7}+\cdots\)
252.2.ba.a 252.ba 36.h $72$ $2.012$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
252.2.bb.a 252.bb 252.ab $88$ $2.012$ None None \(0\) \(0\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{6}]$
252.2.be.a 252.be 84.n $32$ $2.012$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
252.2.bf.a 252.bf 28.f $4$ $2.012$ \(\Q(\sqrt{-3}, \sqrt{-7})\) None None \(-1\) \(0\) \(6\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{3}q^{2}+(1+\beta _{1}-\beta _{2})q^{4}+(2-\beta _{2}+\cdots)q^{5}+\cdots\)
252.2.bf.b 252.bf 28.f $4$ $2.012$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None None \(0\) \(0\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{3}q^{2}-2q^{4}-2\beta _{1}q^{5}+(1-3\beta _{2}+\cdots)q^{7}+\cdots\)
252.2.bf.c 252.bf 28.f $4$ $2.012$ \(\Q(\sqrt{-2}, \sqrt{-3})\) None None \(0\) \(0\) \(0\) \(2\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}+2\beta _{2}q^{4}+(2\beta _{1}-2\beta _{3})q^{5}+\cdots\)
252.2.bf.d 252.bf 28.f $4$ $2.012$ \(\Q(\sqrt{-3}, \sqrt{-7})\) None None \(1\) \(0\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{3}q^{2}+(1+\beta _{1}-\beta _{2})q^{4}+(-2+\beta _{2}+\cdots)q^{5}+\cdots\)
252.2.bf.e 252.bf 28.f $4$ $2.012$ \(\Q(\zeta_{12})\) None None \(2\) \(0\) \(6\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+2\zeta_{12}q^{4}+\cdots\)
252.2.bf.f 252.bf 28.f $8$ $2.012$ 8.0.562828176.1 None None \(-1\) \(0\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{5}q^{2}+(-\beta _{4}+\beta _{6}-\beta _{7})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
252.2.bf.g 252.bf 28.f $8$ $2.012$ 8.0.562828176.1 None None \(-1\) \(0\) \(0\) \(2\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{1}-\beta _{5})q^{2}+(-\beta _{2}-\beta _{6}+\beta _{7})q^{4}+\cdots\)
252.2.bi.a 252.bi 252.ai $4$ $2.012$ \(\Q(\zeta_{12})\) None None \(2\) \(0\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(2\zeta_{12}+\cdots)q^{3}+\cdots\)
252.2.bi.b 252.bi 252.ai $4$ $2.012$ \(\Q(\zeta_{12})\) None None \(2\) \(0\) \(6\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\zeta_{12}+\zeta_{12}^{2}-\zeta_{12}^{3})q^{2}+(-2\zeta_{12}+\cdots)q^{3}+\cdots\)
252.2.bi.c 252.bi 252.ai $80$ $2.012$ None None \(-6\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
252.2.bj.a 252.bj 252.aj $4$ $2.012$ \(\Q(\zeta_{12})\) None None \(-4\) \(0\) \(-12\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1+\zeta_{12}^{3})q^{2}+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+\cdots\)
252.2.bj.b 252.bj 252.aj $84$ $2.012$ None None \(2\) \(0\) \(6\) \(0\) $\mathrm{SU}(2)[C_{6}]$
252.2.bm.a 252.bm 63.s $16$ $2.012$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(-1\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{5}-\beta _{10})q^{3}+(\beta _{7}-\beta _{11})q^{5}+\cdots\)
252.3.c.a 252.c 3.b $4$ $6.867$ \(\Q(\sqrt{-2}, \sqrt{7})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{5}+\beta _{3}q^{7}+(2\beta _{1}+\beta _{2})q^{11}+\cdots\)
252.3.d.a 252.d 7.b $2$ $6.867$ \(\Q(\sqrt{-3}) \) None None \(0\) \(0\) \(0\) \(-14\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{6}q^{5}-7q^{7}-18q^{11}+3\zeta_{6}q^{13}+\cdots\)
252.3.d.b 252.d 7.b $2$ $6.867$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-2\) $\mathrm{U}(1)[D_{2}]$ \(q+(-1-\zeta_{6})q^{7}-2\zeta_{6}q^{13}-4\zeta_{6}q^{19}+\cdots\)
252.3.d.c 252.d 7.b $2$ $6.867$ \(\Q(\sqrt{-6}) \) None None \(0\) \(0\) \(0\) \(10\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{5}+(5-\beta )q^{7}+6q^{11}+\beta q^{13}+\cdots\)
252.3.g.a 252.g 4.b $6$ $6.867$ 6.0.1539727.2 None None \(1\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{2}+(-\beta _{1}-\beta _{4})q^{4}+(1-\beta _{2}+\cdots)q^{5}+\cdots\)
252.3.g.b 252.g 4.b $12$ $6.867$ 12.0.\(\cdots\).1 None None \(-2\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{2}-\beta _{5}q^{4}+(-\beta _{3}+\beta _{8})q^{5}+\cdots\)
252.3.g.c 252.g 4.b $12$ $6.867$ 12.0.\(\cdots\).1 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}+(-1-\beta _{6})q^{4}+(-\beta _{2}+\beta _{7}+\cdots)q^{5}+\cdots\)
252.3.h.a 252.h 84.h $8$ $6.867$ 8.0.157351936.1 \(\Q(\sqrt{-7}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{7}q^{2}+(\beta _{3}+\beta _{6})q^{4}+7\beta _{3}q^{7}+(\beta _{2}+\cdots)q^{8}+\cdots\)
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