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Label Char Prim Dim $A$ Field CM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
414.2.a.a 414.a 1.a $1$ $3.306$ \(\Q\) None \(-1\) \(0\) \(-2\) \(0\) $+$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}-2q^{5}-q^{8}+2q^{10}-2q^{13}+\cdots\)
414.2.a.b 414.a 1.a $1$ $3.306$ \(\Q\) None \(1\) \(0\) \(-4\) \(-4\) $+$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}-4q^{5}-4q^{7}+q^{8}-4q^{10}+\cdots\)
414.2.a.c 414.a 1.a $1$ $3.306$ \(\Q\) None \(1\) \(0\) \(0\) \(2\) $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+2q^{7}+q^{8}+2q^{13}+2q^{14}+\cdots\)
414.2.a.d 414.a 1.a $1$ $3.306$ \(\Q\) None \(1\) \(0\) \(2\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+2q^{5}-2q^{7}+q^{8}+2q^{10}+\cdots\)
414.2.a.e 414.a 1.a $2$ $3.306$ \(\Q(\sqrt{7}) \) None \(-2\) \(0\) \(-2\) \(4\) $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(-1+\beta )q^{5}+2q^{7}-q^{8}+\cdots\)
414.2.a.f 414.a 1.a $2$ $3.306$ \(\Q(\sqrt{5}) \) None \(-2\) \(0\) \(2\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{4}+(1+\beta )q^{5}+2\beta q^{7}-q^{8}+\cdots\)
414.2.a.g 414.a 1.a $2$ $3.306$ \(\Q(\sqrt{7}) \) None \(2\) \(0\) \(2\) \(4\) $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{4}+(1+\beta )q^{5}+2q^{7}+q^{8}+\cdots\)
414.2.d.a 414.d 69.c $8$ $3.306$ \(\Q(\zeta_{16})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{16}q^{2}-q^{4}-\zeta_{16}^{5}q^{5}-2\zeta_{16}^{4}q^{7}+\cdots\)
414.2.e.a 414.e 9.c $2$ $3.306$ \(\Q(\sqrt{-3}) \) None \(1\) \(3\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1-\zeta_{6})q^{2}+(1+\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots\)
414.2.e.b 414.e 9.c $10$ $3.306$ 10.0.\(\cdots\).1 None \(-5\) \(1\) \(-5\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{3}q^{2}-\beta _{7}q^{3}+(-1+\beta _{3})q^{4}+(-1+\cdots)q^{5}+\cdots\)
414.2.e.c 414.e 9.c $10$ $3.306$ 10.0.\(\cdots\).1 None \(-5\) \(1\) \(5\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{4})q^{2}+(1+\beta _{5}-\beta _{6}-\beta _{9})q^{3}+\cdots\)
414.2.e.d 414.e 9.c $10$ $3.306$ 10.0.\(\cdots\).1 None \(5\) \(-3\) \(-1\) \(5\) $\mathrm{SU}(2)[C_{3}]$ \(q+(1+\beta _{7})q^{2}+\beta _{1}q^{3}+\beta _{7}q^{4}+(\beta _{2}+\cdots)q^{5}+\cdots\)
414.2.e.e 414.e 9.c $12$ $3.306$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(6\) \(0\) \(5\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{2}q^{2}-\beta _{1}q^{3}+(-1+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots\)
414.2.f.a 414.f 207.g $48$ $3.306$ None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
414.2.i.a 414.i 23.c $10$ $3.306$ \(\Q(\zeta_{22})\) None \(-1\) \(0\) \(-2\) \(2\) $\mathrm{SU}(2)[C_{11}]$ \(q+\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(-1-\zeta_{22}^{2}+\cdots)q^{5}+\cdots\)
414.2.i.b 414.i 23.c $10$ $3.306$ \(\Q(\zeta_{22})\) None \(-1\) \(0\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{11}]$ \(q+\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(1-2\zeta_{22}+\cdots)q^{5}+\cdots\)
414.2.i.c 414.i 23.c $10$ $3.306$ \(\Q(\zeta_{22})\) None \(-1\) \(0\) \(4\) \(-7\) $\mathrm{SU}(2)[C_{11}]$ \(q+\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(\zeta_{22}+\zeta_{22}^{3}+\cdots)q^{5}+\cdots\)
414.2.i.d 414.i 23.c $10$ $3.306$ \(\Q(\zeta_{22})\) None \(1\) \(0\) \(-8\) \(8\) $\mathrm{SU}(2)[C_{11}]$ \(q-\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(-1+\zeta_{22}^{2}+\cdots)q^{5}+\cdots\)
414.2.i.e 414.i 23.c $10$ $3.306$ \(\Q(\zeta_{22})\) None \(1\) \(0\) \(2\) \(0\) $\mathrm{SU}(2)[C_{11}]$ \(q-\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(1-2\zeta_{22}+\cdots)q^{5}+\cdots\)
414.2.i.f 414.i 23.c $10$ $3.306$ \(\Q(\zeta_{22})\) None \(1\) \(0\) \(6\) \(3\) $\mathrm{SU}(2)[C_{11}]$ \(q-\zeta_{22}^{4}q^{2}+\zeta_{22}^{8}q^{4}+(\zeta_{22}-2\zeta_{22}^{2}+\cdots)q^{5}+\cdots\)
414.2.i.g 414.i 23.c $20$ $3.306$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(-2\) \(0\) \(-2\) \(-4\) $\mathrm{SU}(2)[C_{11}]$ \(q+\beta _{3}q^{2}+\beta _{7}q^{4}+(\beta _{2}-\beta _{5}+\beta _{13}+\cdots)q^{5}+\cdots\)
414.2.i.h 414.i 23.c $20$ $3.306$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(2\) \(0\) \(2\) \(-4\) $\mathrm{SU}(2)[C_{11}]$ \(q-\beta _{3}q^{2}+\beta _{7}q^{4}+(-\beta _{2}+\beta _{5}-\beta _{13}+\cdots)q^{5}+\cdots\)
414.2.j.a 414.j 69.g $80$ $3.306$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{22}]$
414.2.m.a 414.m 207.m $240$ $3.306$ None \(-12\) \(0\) \(0\) \(-13\) $\mathrm{SU}(2)[C_{33}]$
414.2.m.b 414.m 207.m $240$ $3.306$ None \(12\) \(4\) \(0\) \(13\) $\mathrm{SU}(2)[C_{33}]$
414.2.p.a 414.p 207.o $480$ $3.306$ None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{66}]$
414.3.b.a 414.b 23.b $4$ $11.281$ 4.0.613376.1 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{2}q^{2}+2q^{4}+\beta _{1}q^{5}+\beta _{3}q^{7}-2\beta _{2}q^{8}+\cdots\)
414.3.b.b 414.b 23.b $8$ $11.281$ 8.0.\(\cdots\).26 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+2q^{4}+\beta _{6}q^{5}-\beta _{5}q^{7}+2\beta _{1}q^{8}+\cdots\)
414.3.b.c 414.b 23.b $8$ $11.281$ 8.0.1358954496.3 None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{5}q^{2}+2q^{4}+(\beta _{2}-2\beta _{3}+\beta _{6})q^{5}+\cdots\)
414.3.c.a 414.c 3.b $4$ $11.281$ \(\Q(\sqrt{-2}, \sqrt{-23})\) None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}-2q^{4}-2\beta _{1}q^{5}-\beta _{2}q^{7}+\cdots\)
414.3.c.b 414.c 3.b $8$ $11.281$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(0\) \(0\) \(16\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}-2q^{4}-\beta _{7}q^{5}+(2-\beta _{2}-\beta _{6}+\cdots)q^{7}+\cdots\)
414.3.g.a 414.g 9.d $88$ $11.281$ None \(0\) \(-4\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{6}]$
414.3.h.a 414.h 207.f $96$ $11.281$ None \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
414.3.k.a 414.k 69.h $80$ $11.281$ None \(0\) \(0\) \(0\) \(-16\) $\mathrm{SU}(2)[C_{22}]$
414.3.k.b 414.k 69.h $80$ $11.281$ None \(0\) \(0\) \(0\) \(16\) $\mathrm{SU}(2)[C_{22}]$
414.3.l.a 414.l 23.d $40$ $11.281$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{22}]$
414.3.l.b 414.l 23.d $80$ $11.281$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{22}]$
414.3.l.c 414.l 23.d $80$ $11.281$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{22}]$
414.3.n.a 414.n 207.p $960$ $11.281$ None \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{66}]$
414.3.o.a 414.o 207.n $960$ $11.281$ None \(0\) \(4\) \(36\) \(0\) $\mathrm{SU}(2)[C_{66}]$
414.4.a.a 414.a 1.a $1$ $24.427$ \(\Q\) None \(-2\) \(0\) \(2\) \(-34\) $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+2q^{5}-34q^{7}-8q^{8}+\cdots\)
414.4.a.b 414.a 1.a $1$ $24.427$ \(\Q\) None \(-2\) \(0\) \(20\) \(2\) $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+20q^{5}+2q^{7}-8q^{8}+\cdots\)
414.4.a.c 414.a 1.a $1$ $24.427$ \(\Q\) None \(2\) \(0\) \(-2\) \(-32\) $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}-2q^{5}-2^{5}q^{7}+8q^{8}+\cdots\)
414.4.a.d 414.a 1.a $1$ $24.427$ \(\Q\) None \(2\) \(0\) \(10\) \(-12\) $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+10q^{5}-12q^{7}+8q^{8}+\cdots\)
414.4.a.e 414.a 1.a $1$ $24.427$ \(\Q\) None \(2\) \(0\) \(10\) \(32\) $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+10q^{5}+2^{5}q^{7}+8q^{8}+\cdots\)
414.4.a.f 414.a 1.a $2$ $24.427$ \(\Q(\sqrt{73}) \) None \(-4\) \(0\) \(-10\) \(12\) $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-4-2\beta )q^{5}+(4+4\beta )q^{7}+\cdots\)
414.4.a.g 414.a 1.a $2$ $24.427$ \(\Q(\sqrt{2}) \) None \(-4\) \(0\) \(-8\) \(12\) $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(-4+\beta )q^{5}+(6+\beta )q^{7}+\cdots\)
414.4.a.h 414.a 1.a $2$ $24.427$ \(\Q(\sqrt{7}) \) None \(-4\) \(0\) \(10\) \(-10\) $-$ $\mathrm{SU}(2)$ \(q-2q^{2}+4q^{4}+(5+3\beta )q^{5}+(-5+5\beta )q^{7}+\cdots\)
414.4.a.i 414.a 1.a $2$ $24.427$ \(\Q(\sqrt{7}) \) None \(4\) \(0\) \(-10\) \(-10\) $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(-5+3\beta )q^{5}+(-5+\cdots)q^{7}+\cdots\)
414.4.a.j 414.a 1.a $2$ $24.427$ \(\Q(\sqrt{41}) \) None \(4\) \(0\) \(-10\) \(6\) $-$ $\mathrm{SU}(2)$ \(q+2q^{2}+4q^{4}+(-4-2\beta )q^{5}+(2+2\beta )q^{7}+\cdots\)
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