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Label Char Prim Dim $A$ Field CM RM Minimal twist Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
315.1.bg.a 315.bg 315.ag $2$ $0.157$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-35}) \) None 315.1.bg.a \(0\) \(-2\) \(1\) \(1\) \(q-q^{3}+\zeta_{6}^{2}q^{4}-\zeta_{6}^{2}q^{5}+\zeta_{6}q^{7}+\cdots\)
315.1.bg.b 315.bg 315.ag $2$ $0.157$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-35}) \) None 315.1.bg.a \(0\) \(2\) \(-1\) \(-1\) \(q+q^{3}+\zeta_{6}^{2}q^{4}+\zeta_{6}^{2}q^{5}-\zeta_{6}q^{7}+\cdots\)
315.2.a.a 315.a 1.a $1$ $2.515$ \(\Q\) None None 105.2.a.a \(-1\) \(0\) \(-1\) \(1\) $+$ $\mathrm{SU}(2)$ \(q-q^{2}-q^{4}-q^{5}+q^{7}+3q^{8}+q^{10}+\cdots\)
315.2.a.b 315.a 1.a $1$ $2.515$ \(\Q\) None None 35.2.a.a \(0\) \(0\) \(1\) \(1\) $-$ $\mathrm{SU}(2)$ \(q-2q^{4}+q^{5}+q^{7}+3q^{11}+5q^{13}+\cdots\)
315.2.a.c 315.a 1.a $2$ $2.515$ \(\Q(\sqrt{2}) \) None None 315.2.a.c \(-2\) \(0\) \(-2\) \(-2\) $+$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}+(1-2\beta )q^{4}-q^{5}-q^{7}+\cdots\)
315.2.a.d 315.a 1.a $2$ $2.515$ \(\Q(\sqrt{5}) \) None None 105.2.a.b \(0\) \(0\) \(2\) \(2\) $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}+3q^{4}+q^{5}+q^{7}-\beta q^{8}-\beta q^{10}+\cdots\)
315.2.a.e 315.a 1.a $2$ $2.515$ \(\Q(\sqrt{17}) \) None None 35.2.a.b \(1\) \(0\) \(-2\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q+\beta q^{2}+(2+\beta )q^{4}-q^{5}-q^{7}+(4+\beta )q^{8}+\cdots\)
315.2.a.f 315.a 1.a $2$ $2.515$ \(\Q(\sqrt{2}) \) None None 315.2.a.c \(2\) \(0\) \(2\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{2}+(1+2\beta )q^{4}+q^{5}-q^{7}+\cdots\)
315.2.b.a 315.b 21.c $4$ $2.515$ \(\Q(\sqrt{-2}, \sqrt{3})\) None None 315.2.b.a \(0\) \(0\) \(-4\) \(4\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+\beta _{2}q^{4}-q^{5}+(1+\beta _{1}+\beta _{3})q^{7}+\cdots\)
315.2.b.b 315.b 21.c $4$ $2.515$ \(\Q(\sqrt{-2}, \sqrt{3})\) None None 315.2.b.a \(0\) \(0\) \(4\) \(4\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+\beta _{2}q^{4}+q^{5}+(1-\beta _{1}-\beta _{3})q^{7}+\cdots\)
315.2.d.a 315.d 5.b $2$ $2.515$ \(\Q(\sqrt{-1}) \) None None 35.2.b.a \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}-2q^{4}+(2-i)q^{5}+iq^{7}+\cdots\)
315.2.d.b 315.d 5.b $2$ $2.515$ \(\Q(\sqrt{-1}) \) None None 315.2.d.b \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+q^{4}+(-2+i)q^{5}+iq^{7}+\cdots\)
315.2.d.c 315.d 5.b $2$ $2.515$ \(\Q(\sqrt{-1}) \) None None 105.2.d.a \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+q^{4}+(-1-2i)q^{5}-iq^{7}+\cdots\)
315.2.d.d 315.d 5.b $2$ $2.515$ \(\Q(\sqrt{-1}) \) None None 315.2.d.b \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+q^{4}+(2+i)q^{5}-iq^{7}+3iq^{8}+\cdots\)
315.2.d.e 315.d 5.b $6$ $2.515$ 6.0.350464.1 None None 105.2.d.b \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+(-1+\beta _{3}+\beta _{5})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
315.2.g.a 315.g 105.g $16$ $2.515$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None 315.2.g.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{3}q^{2}+(1+\beta _{14})q^{4}-\beta _{4}q^{5}-\beta _{12}q^{7}+\cdots\)
315.2.i.a 315.i 9.c $2$ $2.515$ \(\Q(\sqrt{-3}) \) None None 315.2.i.a \(-2\) \(-3\) \(1\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-2+2\zeta_{6})q^{2}+(-1-\zeta_{6})q^{3}-2\zeta_{6}q^{4}+\cdots\)
315.2.i.b 315.i 9.c $2$ $2.515$ \(\Q(\sqrt{-3}) \) None None 315.2.i.b \(0\) \(3\) \(1\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-\zeta_{6})q^{3}+2\zeta_{6}q^{4}+\zeta_{6}q^{5}+(-1+\cdots)q^{7}+\cdots\)
315.2.i.c 315.i 9.c $8$ $2.515$ 8.0.142635249.1 None None 315.2.i.c \(1\) \(-1\) \(-4\) \(-4\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{7}q^{2}+(-\beta _{1}+\beta _{2}+\beta _{4}+\beta _{6})q^{3}+\cdots\)
315.2.i.d 315.i 9.c $8$ $2.515$ \(\Q(\zeta_{15})\) None None 315.2.i.d \(1\) \(3\) \(4\) \(4\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\zeta_{15}-\zeta_{15}^{2}+\zeta_{15}^{3})q^{2}+(1+\cdots)q^{3}+\cdots\)
315.2.i.e 315.i 9.c $12$ $2.515$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None 315.2.i.e \(3\) \(1\) \(6\) \(-6\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\beta _{2}-\beta _{4})q^{2}+\beta _{9}q^{3}+(-\beta _{1}+\cdots)q^{4}+\cdots\)
315.2.i.f 315.i 9.c $16$ $2.515$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None 315.2.i.f \(1\) \(1\) \(-8\) \(8\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{2}+\beta _{13})q^{2}+\beta _{5}q^{3}+(-2-2\beta _{3}+\cdots)q^{4}+\cdots\)
315.2.j.a 315.j 7.c $2$ $2.515$ \(\Q(\sqrt{-3}) \) None None 105.2.i.b \(-2\) \(0\) \(1\) \(-1\) $\mathrm{SU}(2)[C_{3}]$ \(q-2\zeta_{6}q^{2}+(-2+2\zeta_{6})q^{4}+\zeta_{6}q^{5}+\cdots\)
315.2.j.b 315.j 7.c $2$ $2.515$ \(\Q(\sqrt{-3}) \) None None 105.2.i.a \(0\) \(0\) \(-1\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(2-2\zeta_{6})q^{4}-\zeta_{6}q^{5}+(2+\zeta_{6})q^{7}+\cdots\)
315.2.j.c 315.j 7.c $4$ $2.515$ \(\Q(\zeta_{12})\) None None 105.2.i.d \(-2\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\zeta_{12}-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(-2+2\zeta_{12}+\cdots)q^{4}+\cdots\)
315.2.j.d 315.j 7.c $4$ $2.515$ \(\Q(\sqrt{2}, \sqrt{-3})\) None None 105.2.i.c \(0\) \(0\) \(-2\) \(-2\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+(-1-\beta _{2})q^{5}+(-1-\beta _{1}+\cdots)q^{7}+\cdots\)
315.2.j.e 315.j 7.c $4$ $2.515$ \(\Q(\sqrt{2}, \sqrt{-3})\) None None 35.2.e.a \(2\) \(0\) \(2\) \(2\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{1}+\beta _{2})q^{2}+(2\beta _{1}+\beta _{2}+2\beta _{3})q^{4}+\cdots\)
315.2.j.f 315.j 7.c $6$ $2.515$ 6.0.4406832.1 None None 315.2.j.f \(-2\) \(0\) \(-3\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}+\beta _{2}-\beta _{4})q^{2}+(-2+\beta _{1}+\cdots)q^{4}+\cdots\)
315.2.j.g 315.j 7.c $6$ $2.515$ 6.0.4406832.1 None None 315.2.j.f \(2\) \(0\) \(3\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}-\beta _{2}+\beta _{4})q^{2}+(-2+\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
315.2.k.a 315.k 63.g $4$ $2.515$ \(\Q(\sqrt{-3}, \sqrt{13})\) None None 315.2.k.a \(1\) \(0\) \(-4\) \(-8\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{1}q^{2}+(1+2\beta _{2})q^{3}+(-1+\beta _{1}+\cdots)q^{4}+\cdots\)
315.2.k.b 315.k 63.g $24$ $2.515$ None None 315.2.k.b \(-1\) \(1\) \(-24\) \(7\) $\mathrm{SU}(2)[C_{3}]$
315.2.k.c 315.k 63.g $36$ $2.515$ None None 315.2.k.c \(0\) \(-1\) \(36\) \(-1\) $\mathrm{SU}(2)[C_{3}]$
315.2.l.a 315.l 63.h $4$ $2.515$ \(\Q(\sqrt{-3}, \sqrt{13})\) None None 315.2.k.a \(-2\) \(6\) \(2\) \(10\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1+\beta _{3})q^{2}+(1-\beta _{2})q^{3}+(2-\beta _{3})q^{4}+\cdots\)
315.2.l.b 315.l 63.h $24$ $2.515$ None None 315.2.k.b \(2\) \(-5\) \(12\) \(-11\) $\mathrm{SU}(2)[C_{3}]$
315.2.l.c 315.l 63.h $36$ $2.515$ None None 315.2.k.c \(0\) \(-1\) \(-18\) \(-1\) $\mathrm{SU}(2)[C_{3}]$
315.2.m.a 315.m 15.e $12$ $2.515$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None None 315.2.m.a \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{2}q^{2}+(\beta _{5}+\beta _{6}-\beta _{8})q^{4}+(-\beta _{3}+\cdots)q^{5}+\cdots\)
315.2.m.b 315.m 15.e $12$ $2.515$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None None 315.2.m.a \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\beta _{1}+\beta _{8}+\beta _{9})q^{2}+(-\beta _{5}-\beta _{6}+\beta _{8}+\cdots)q^{4}+\cdots\)
315.2.p.a 315.p 35.f $4$ $2.515$ \(\Q(i, \sqrt{10})\) None None 315.2.p.a \(0\) \(0\) \(-8\) \(4\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}+3\beta _{2}q^{4}+(-2+\beta _{2})q^{5}+\cdots\)
315.2.p.b 315.p 35.f $4$ $2.515$ \(\Q(i, \sqrt{10})\) None None 315.2.p.a \(0\) \(0\) \(8\) \(4\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}+3\beta _{2}q^{4}+(2-\beta _{2})q^{5}+(1+\cdots)q^{7}+\cdots\)
315.2.p.c 315.p 35.f $4$ $2.515$ \(\Q(i, \sqrt{10})\) None None 35.2.f.a \(4\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+\beta _{2})q^{2}+\beta _{1}q^{5}+(1+\beta _{2}+\beta _{3})q^{7}+\cdots\)
315.2.p.d 315.p 35.f $8$ $2.515$ 8.0.40960000.1 None None 315.2.p.d \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}-\beta _{3}q^{4}+\beta _{5}q^{5}+(-1-\beta _{3}+\cdots)q^{7}+\cdots\)
315.2.p.e 315.p 35.f $16$ $2.515$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None 105.2.m.a \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{2}q^{2}+(-\beta _{6}-\beta _{7}+\beta _{13})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
315.2.r.a 315.r 315.r $4$ $2.515$ \(\Q(\zeta_{12})\) None None 315.2.r.a \(0\) \(0\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+\zeta_{12}^{3}q^{2}+(\zeta_{12}+\zeta_{12}^{3})q^{3}+q^{4}+\cdots\)
315.2.r.b 315.r 315.r $84$ $2.515$ None None 315.2.r.b \(0\) \(0\) \(3\) \(0\) $\mathrm{SU}(2)[C_{6}]$
315.2.t.a 315.t 63.i $2$ $2.515$ \(\Q(\sqrt{-3}) \) None None 315.2.t.a \(0\) \(-3\) \(1\) \(4\) $\mathrm{SU}(2)[C_{6}]$ \(q+(1-2\zeta_{6})q^{2}+(-2+\zeta_{6})q^{3}-q^{4}+\cdots\)
315.2.t.b 315.t 63.i $30$ $2.515$ None None 315.2.t.b \(0\) \(4\) \(15\) \(-3\) $\mathrm{SU}(2)[C_{6}]$
315.2.t.c 315.t 63.i $32$ $2.515$ None None 315.2.t.c \(0\) \(-1\) \(-16\) \(1\) $\mathrm{SU}(2)[C_{6}]$
315.2.u.a 315.u 315.u $88$ $2.515$ None None 315.2.u.a \(0\) \(0\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{6}]$
315.2.z.a 315.z 315.z $8$ $2.515$ 8.0.\(\cdots\).2 \(\Q(\sqrt{-35}) \) None 315.2.z.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+\beta _{5}q^{3}+(2+2\beta _{4})q^{4}+(\beta _{1}-\beta _{6})q^{5}+\cdots\)
315.2.z.b 315.z 315.z $80$ $2.515$ None None 315.2.z.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
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