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Label Char Prim Dim $A$ Field CM RM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
312.1.b.a 312.b 312.b $4$ $0.156$ \(\Q(\zeta_{8})\) \(\Q(\sqrt{-39}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{8}q^{2}-\zeta_{8}^{2}q^{3}+\zeta_{8}^{2}q^{4}+(-\zeta_{8}+\cdots)q^{5}+\cdots\)
312.2.a.a 312.a 1.a $1$ $2.491$ \(\Q\) None None \(0\) \(-1\) \(-2\) \(4\) $-$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{5}+4q^{7}+q^{9}+q^{13}+2q^{15}+\cdots\)
312.2.a.b 312.a 1.a $1$ $2.491$ \(\Q\) None None \(0\) \(-1\) \(0\) \(-4\) $+$ $\mathrm{SU}(2)$ \(q-q^{3}-4q^{7}+q^{9}-2q^{11}-q^{13}+\cdots\)
312.2.a.c 312.a 1.a $1$ $2.491$ \(\Q\) None None \(0\) \(-1\) \(4\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-q^{3}+4q^{5}+q^{9}-2q^{11}-q^{13}+\cdots\)
312.2.a.d 312.a 1.a $1$ $2.491$ \(\Q\) None None \(0\) \(1\) \(-4\) \(-4\) $+$ $\mathrm{SU}(2)$ \(q+q^{3}-4q^{5}-4q^{7}+q^{9}-2q^{11}+\cdots\)
312.2.a.e 312.a 1.a $1$ $2.491$ \(\Q\) None None \(0\) \(1\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+q^{3}+q^{9}+6q^{11}-q^{13}+2q^{17}+\cdots\)
312.2.a.f 312.a 1.a $1$ $2.491$ \(\Q\) None None \(0\) \(1\) \(2\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{5}+q^{9}+q^{13}+2q^{15}+\cdots\)
312.2.c.a 312.c 13.b $2$ $2.491$ \(\Q(\sqrt{-1}) \) None None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-q^{3}+iq^{7}+q^{9}+iq^{11}+(3+i)q^{13}+\cdots\)
312.2.c.b 312.c 13.b $2$ $2.491$ \(\Q(\sqrt{-1}) \) None None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+q^{3}+iq^{5}+iq^{7}+q^{9}+2iq^{11}+\cdots\)
312.2.c.c 312.c 13.b $2$ $2.491$ \(\Q(\sqrt{-1}) \) None None \(0\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+q^{3}+2iq^{5}-iq^{7}+q^{9}+iq^{11}+\cdots\)
312.2.g.a 312.g 8.b $8$ $2.491$ \(\Q(\zeta_{20})\) None None \(2\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{2}]$ \(q+\zeta_{20}^{7}q^{2}-\zeta_{20}^{3}q^{3}+\zeta_{20}q^{4}+(-1+\cdots)q^{5}+\cdots\)
312.2.g.b 312.g 8.b $16$ $2.491$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(2\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{10}q^{2}+\beta _{9}q^{3}-\beta _{2}q^{4}+\beta _{14}q^{5}+\cdots\)
312.2.h.a 312.h 312.h $8$ $2.491$ 8.0.\(\cdots\).21 \(\Q(\sqrt{-39}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{1}q^{2}+\beta _{3}q^{3}+\beta _{2}q^{4}+(-\beta _{4}-\beta _{7})q^{5}+\cdots\)
312.2.h.b 312.h 312.h $12$ $2.491$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) \(\Q(\sqrt{-26}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{3}q^{2}+\beta _{7}q^{3}-2q^{4}-\beta _{9}q^{5}-\beta _{4}q^{6}+\cdots\)
312.2.h.c 312.h 312.h $32$ $2.491$ None None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
312.2.j.a 312.j 24.f $48$ $2.491$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
312.2.m.a 312.m 104.e $2$ $2.491$ \(\Q(\sqrt{-1}) \) None None \(-2\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1+i)q^{2}-iq^{3}-2iq^{4}-2q^{5}+\cdots\)
312.2.m.b 312.m 104.e $2$ $2.491$ \(\Q(\sqrt{-1}) \) None None \(2\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1+i)q^{2}+iq^{3}+2iq^{4}+2q^{5}+\cdots\)
312.2.m.c 312.m 104.e $24$ $2.491$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
312.2.q.a 312.q 13.c $2$ $2.491$ \(\Q(\sqrt{-3}) \) None None \(0\) \(-1\) \(6\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\zeta_{6})q^{3}+3q^{5}-\zeta_{6}q^{9}+(3+\cdots)q^{13}+\cdots\)
312.2.q.b 312.q 13.c $2$ $2.491$ \(\Q(\sqrt{-3}) \) None None \(0\) \(1\) \(-4\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}-2q^{5}-\zeta_{6}q^{7}-\zeta_{6}q^{9}+\cdots\)
312.2.q.c 312.q 13.c $2$ $2.491$ \(\Q(\sqrt{-3}) \) None None \(0\) \(1\) \(6\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{3}+3q^{5}+4\zeta_{6}q^{7}-\zeta_{6}q^{9}+\cdots\)
312.2.q.d 312.q 13.c $4$ $2.491$ \(\Q(\sqrt{-3}, \sqrt{13})\) None None \(0\) \(-2\) \(-4\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{1})q^{3}-q^{5}+(-\beta _{1}-\beta _{2}+\cdots)q^{7}+\cdots\)
312.2.q.e 312.q 13.c $6$ $2.491$ 6.0.2101707.2 None None \(0\) \(3\) \(0\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{3})q^{3}+\beta _{2}q^{5}+(-2-\beta _{4}+2\beta _{5})q^{7}+\cdots\)
312.2.t.a 312.t 104.m $2$ $2.491$ \(\Q(\sqrt{-1}) \) None None \(-2\) \(-2\) \(-4\) \(2\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+i)q^{2}-q^{3}-2iq^{4}+(-2+\cdots)q^{5}+\cdots\)
312.2.t.b 312.t 104.m $2$ $2.491$ \(\Q(\sqrt{-1}) \) None None \(-2\) \(2\) \(-4\) \(-6\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{2}+q^{3}+2iq^{4}+(-2+\cdots)q^{5}+\cdots\)
312.2.t.c 312.t 104.m $2$ $2.491$ \(\Q(\sqrt{-1}) \) None None \(-2\) \(2\) \(4\) \(6\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{2}+q^{3}+2iq^{4}+(2-2i)q^{5}+\cdots\)
312.2.t.d 312.t 104.m $2$ $2.491$ \(\Q(\sqrt{-1}) \) None None \(2\) \(-2\) \(4\) \(-2\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-i)q^{2}-q^{3}-2iq^{4}+(2-2i)q^{5}+\cdots\)
312.2.t.e 312.t 104.m $24$ $2.491$ None None \(0\) \(-24\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
312.2.t.f 312.t 104.m $24$ $2.491$ None None \(4\) \(24\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
312.2.x.a 312.x 39.f $4$ $2.491$ \(\Q(\zeta_{8})\) None None \(0\) \(4\) \(0\) \(12\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+\zeta_{8}^{2})q^{3}+(-\zeta_{8}^{2}-\zeta_{8}^{3})q^{5}+\cdots\)
312.2.x.b 312.x 39.f $8$ $2.491$ \(\Q(\zeta_{24})\) None None \(0\) \(-8\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-\zeta_{24}^{2})q^{3}-\zeta_{24}^{7}q^{5}+(1+\zeta_{24}+\cdots)q^{7}+\cdots\)
312.2.x.c 312.x 39.f $16$ $2.491$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(4\) \(0\) \(-24\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{12}q^{3}+(\beta _{6}-\beta _{7}-\beta _{13}+\beta _{14}+\cdots)q^{5}+\cdots\)
312.2.y.a 312.y 312.y $104$ $2.491$ None None \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$
312.2.ba.a 312.ba 312.aa $104$ $2.491$ None None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
312.2.bb.a 312.bb 104.r $56$ $2.491$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
312.2.bf.a 312.bf 13.e $4$ $2.491$ \(\Q(\zeta_{12})\) None None \(0\) \(2\) \(0\) \(6\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1-\zeta_{12}^{2})q^{3}+(1-2\zeta_{12}^{2})q^{5}+(2+\cdots)q^{7}+\cdots\)
312.2.bf.b 312.bf 13.e $8$ $2.491$ 8.0.649638144.4 None None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{2}q^{3}+(-\beta _{2}-\beta _{6})q^{5}+(1+\beta _{1}+\cdots)q^{7}+\cdots\)
312.2.bk.a 312.bk 104.s $8$ $2.491$ 8.0.12960000.1 None None \(0\) \(0\) \(0\) \(12\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{6}+\beta _{7})q^{2}+(\beta _{3}+\beta _{7})q^{3}+(-1+\cdots)q^{4}+\cdots\)
312.2.bk.b 312.bk 104.s $48$ $2.491$ None None \(0\) \(0\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{6}]$
312.2.bn.a 312.bn 312.an $104$ $2.491$ None None \(0\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
312.2.bo.a 312.bo 312.ao $208$ $2.491$ None None \(0\) \(0\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{12}]$
312.2.bp.a 312.bp 39.k $56$ $2.491$ None None \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{12}]$
312.2.bt.a 312.bt 104.u $4$ $2.491$ \(\Q(\zeta_{12})\) None None \(-4\) \(-2\) \(-2\) \(-6\) $\mathrm{SU}(2)[C_{12}]$ \(q+(-1+\zeta_{12}^{3})q^{2}-\zeta_{12}^{2}q^{3}-2\zeta_{12}^{3}q^{4}+\cdots\)
312.2.bt.b 312.bt 104.u $4$ $2.491$ \(\Q(\zeta_{12})\) None None \(2\) \(-2\) \(2\) \(6\) $\mathrm{SU}(2)[C_{12}]$ \(q+(1-\zeta_{12}-\zeta_{12}^{2})q^{2}-\zeta_{12}^{2}q^{3}+\cdots\)
312.2.bt.c 312.bt 104.u $48$ $2.491$ None None \(2\) \(-24\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$
312.2.bt.d 312.bt 104.u $56$ $2.491$ None None \(0\) \(28\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$
312.3.b.a 312.b 312.b $1$ $8.501$ \(\Q\) \(\Q(\sqrt{-78}) \) None \(-2\) \(-3\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-2q^{2}-3q^{3}+4q^{4}+6q^{6}-8q^{8}+\cdots\)
312.3.b.b 312.b 312.b $1$ $8.501$ \(\Q\) \(\Q(\sqrt{-78}) \) None \(-2\) \(3\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-2q^{2}+3q^{3}+4q^{4}-6q^{6}-8q^{8}+\cdots\)
312.3.b.c 312.b 312.b $1$ $8.501$ \(\Q\) \(\Q(\sqrt{-78}) \) None \(2\) \(-3\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+2q^{2}-3q^{3}+4q^{4}-6q^{6}+8q^{8}+\cdots\)
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