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Label Char Prim Dim $A$ Field CM RM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
400.1.b.a 400.b 4.b $1$ $0.200$ \(\Q\) \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-5}) \) \(\Q(\sqrt{5}) \) \(0\) \(0\) \(0\) \(0\) \(q+q^{9}-2q^{29}-2q^{41}+q^{49}-2q^{61}+\cdots\)
400.1.x.a 400.x 100.h $4$ $0.200$ \(\Q(\zeta_{10})\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(1\) \(0\) \(q+\zeta_{10}^{3}q^{5}-\zeta_{10}^{4}q^{9}+(\zeta_{10}+\zeta_{10}^{2}+\cdots)q^{13}+\cdots\)
400.2.a.a 400.a 1.a $1$ $3.194$ \(\Q\) None None \(0\) \(-3\) \(0\) \(2\) $+$ $\mathrm{SU}(2)$ \(q-3q^{3}+2q^{7}+6q^{9}-q^{11}-4q^{13}+\cdots\)
400.2.a.b 400.a 1.a $1$ $3.194$ \(\Q\) None None \(0\) \(-2\) \(0\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-2q^{7}+q^{9}+4q^{11}+4q^{13}+\cdots\)
400.2.a.c 400.a 1.a $1$ $3.194$ \(\Q\) None None \(0\) \(-2\) \(0\) \(2\) $-$ $\mathrm{SU}(2)$ \(q-2q^{3}+2q^{7}+q^{9}-2q^{13}+6q^{17}+\cdots\)
400.2.a.d 400.a 1.a $1$ $3.194$ \(\Q\) None None \(0\) \(-1\) \(0\) \(-2\) $+$ $\mathrm{SU}(2)$ \(q-q^{3}-2q^{7}-2q^{9}+3q^{11}-4q^{13}+\cdots\)
400.2.a.e 400.a 1.a $1$ $3.194$ \(\Q\) None None \(0\) \(0\) \(0\) \(-4\) $+$ $\mathrm{SU}(2)$ \(q-4q^{7}-3q^{9}-4q^{11}+2q^{13}-2q^{17}+\cdots\)
400.2.a.f 400.a 1.a $1$ $3.194$ \(\Q\) None None \(0\) \(1\) \(0\) \(2\) $-$ $\mathrm{SU}(2)$ \(q+q^{3}+2q^{7}-2q^{9}+3q^{11}+4q^{13}+\cdots\)
400.2.a.g 400.a 1.a $1$ $3.194$ \(\Q\) None None \(0\) \(2\) \(0\) \(2\) $-$ $\mathrm{SU}(2)$ \(q+2q^{3}+2q^{7}+q^{9}+4q^{11}-4q^{13}+\cdots\)
400.2.a.h 400.a 1.a $1$ $3.194$ \(\Q\) None None \(0\) \(3\) \(0\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q+3q^{3}-2q^{7}+6q^{9}-q^{11}+4q^{13}+\cdots\)
400.2.c.a 400.c 5.b $2$ $3.194$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+3iq^{3}+2iq^{7}-6q^{9}-q^{11}+4iq^{13}+\cdots\)
400.2.c.b 400.c 5.b $2$ $3.194$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}+iq^{7}-q^{9}+iq^{13}+3iq^{17}+\cdots\)
400.2.c.c 400.c 5.b $2$ $3.194$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{3}-2iq^{7}+2q^{9}+3q^{11}+4iq^{13}+\cdots\)
400.2.c.d 400.c 5.b $2$ $3.194$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{7}+3q^{9}-4q^{11}+iq^{13}+iq^{17}+\cdots\)
400.2.j.a 400.j 80.j $2$ $3.194$ \(\Q(\sqrt{-1}) \) None None \(-2\) \(0\) \(0\) \(6\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+i)q^{2}-2iq^{3}-2iq^{4}+(2+\cdots)q^{6}+\cdots\)
400.2.j.b 400.j 80.j $8$ $3.194$ \(\Q(\zeta_{24})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-\zeta_{24}-\zeta_{24}^{5})q^{2}+(\zeta_{24}^{4}-\zeta_{24}^{5}+\cdots)q^{3}+\cdots\)
400.2.j.c 400.j 80.j $16$ $3.194$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{12}q^{2}-\beta _{8}q^{3}+\beta _{15}q^{4}+(-1+\cdots)q^{6}+\cdots\)
400.2.j.d 400.j 80.j $18$ $3.194$ \(\mathbb{Q}[x]/(x^{18} + \cdots)\) None None \(4\) \(0\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{7}q^{2}+\beta _{16}q^{3}+\beta _{14}q^{4}+(-1+\cdots)q^{6}+\cdots\)
400.2.j.e 400.j 80.j $24$ $3.194$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
400.2.l.a 400.l 16.e $2$ $3.194$ \(\Q(\sqrt{-1}) \) None None \(-2\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+i)q^{2}+(-1+i)q^{3}-2iq^{4}+\cdots\)
400.2.l.b 400.l 16.e $2$ $3.194$ \(\Q(\sqrt{-1}) \) None None \(2\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-i)q^{2}+(1-i)q^{3}-2iq^{4}-2iq^{6}+\cdots\)
400.2.l.c 400.l 16.e $2$ $3.194$ \(\Q(\sqrt{-1}) \) None None \(2\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{2}+(1-i)q^{3}+2iq^{4}+2q^{6}+\cdots\)
400.2.l.d 400.l 16.e $4$ $3.194$ \(\Q(i, \sqrt{11})\) None None \(-4\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-\beta _{1})q^{2}+(-\beta _{1}+\beta _{2})q^{3}+2\beta _{1}q^{4}+\cdots\)
400.2.l.e 400.l 16.e $4$ $3.194$ \(\Q(i, \sqrt{11})\) None None \(4\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+\beta _{1})q^{2}+(\beta _{1}-\beta _{2})q^{3}+2\beta _{1}q^{4}+\cdots\)
400.2.l.f 400.l 16.e $12$ $3.194$ 12.0.\(\cdots\).1 None None \(-4\) \(-2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{1}q^{2}-\beta _{6}q^{3}+\beta _{2}q^{4}+(\beta _{1}-\beta _{2}+\cdots)q^{6}+\cdots\)
400.2.l.g 400.l 16.e $12$ $3.194$ 12.0.\(\cdots\).1 None None \(4\) \(2\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{1}q^{2}+\beta _{6}q^{3}+\beta _{2}q^{4}+(\beta _{1}-\beta _{2}+\cdots)q^{6}+\cdots\)
400.2.l.h 400.l 16.e $16$ $3.194$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{8}q^{2}+(\beta _{4}+\beta _{11})q^{3}+(\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
400.2.l.i 400.l 16.e $16$ $3.194$ 16.0.\(\cdots\).1 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{10}q^{2}+(-\beta _{2}+\beta _{9}+\beta _{12}-\beta _{14}+\cdots)q^{3}+\cdots\)
400.2.n.a 400.n 20.e $2$ $3.194$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q-3iq^{9}+(5-5i)q^{13}+(5+5i)q^{17}+\cdots\)
400.2.n.b 400.n 20.e $4$ $3.194$ \(\Q(\zeta_{12})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\zeta_{12}^{2}q^{3}+\zeta_{12}^{3}q^{7}+3\zeta_{12}q^{9}+\cdots\)
400.2.n.c 400.n 20.e $4$ $3.194$ \(\Q(i, \sqrt{5})\) \(\Q(\sqrt{-5}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q-\beta _{3}q^{3}+\beta _{2}q^{7}-7\beta _{1}q^{9}+10q^{21}+\cdots\)
400.2.n.d 400.n 20.e $8$ $3.194$ \(\Q(\zeta_{24})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+\zeta_{24}q^{3}+4\zeta_{24}^{5}q^{7}-2\zeta_{24}^{3}q^{9}+\cdots\)
400.2.q.a 400.q 80.q $2$ $3.194$ \(\Q(\sqrt{-1}) \) None None \(-2\) \(-2\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+i)q^{2}+(-1-i)q^{3}-2iq^{4}+\cdots\)
400.2.q.b 400.q 80.q $2$ $3.194$ \(\Q(\sqrt{-1}) \) None None \(2\) \(2\) \(0\) \(4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-i)q^{2}+(1+i)q^{3}-2iq^{4}+2q^{6}+\cdots\)
400.2.q.c 400.q 80.q $4$ $3.194$ \(\Q(i, \sqrt{11})\) None None \(-4\) \(2\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-\beta _{1})q^{2}+(\beta _{1}-\beta _{2})q^{3}+2\beta _{1}q^{4}+\cdots\)
400.2.q.d 400.q 80.q $4$ $3.194$ \(\Q(i, \sqrt{11})\) None None \(4\) \(-2\) \(0\) \(4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+\beta _{1})q^{2}+(-1-\beta _{2})q^{3}+2\beta _{1}q^{4}+\cdots\)
400.2.q.e 400.q 80.q $12$ $3.194$ 12.0.\(\cdots\).1 None None \(-2\) \(2\) \(0\) \(-12\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{6}q^{2}-\beta _{7}q^{3}+\beta _{10}q^{4}+(-\beta _{3}+\cdots)q^{6}+\cdots\)
400.2.q.f 400.q 80.q $12$ $3.194$ 12.0.\(\cdots\).1 None None \(2\) \(-2\) \(0\) \(12\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{6}q^{2}+\beta _{7}q^{3}+\beta _{10}q^{4}+(-\beta _{3}+\cdots)q^{6}+\cdots\)
400.2.q.g 400.q 80.q $16$ $3.194$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(-4\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{2}q^{2}+(-\beta _{6}-\beta _{11})q^{3}+(-\beta _{2}+\cdots)q^{4}+\cdots\)
400.2.q.h 400.q 80.q $16$ $3.194$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(4\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{2}q^{2}+(\beta _{6}+\beta _{11})q^{3}+(-\beta _{2}-\beta _{12}+\cdots)q^{4}+\cdots\)
400.2.s.a 400.s 80.s $2$ $3.194$ \(\Q(\sqrt{-1}) \) None None \(2\) \(4\) \(0\) \(6\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-i)q^{2}+2q^{3}-2iq^{4}+(2-2i)q^{6}+\cdots\)
400.2.s.b 400.s 80.s $8$ $3.194$ \(\Q(\zeta_{24})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\zeta_{24}-\zeta_{24}^{5})q^{2}+\zeta_{24}^{7}q^{3}-2q^{4}+\cdots\)
400.2.s.c 400.s 80.s $16$ $3.194$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q-\beta _{6}q^{2}+\beta _{5}q^{3}-\beta _{15}q^{4}+(-1-\beta _{11}+\cdots)q^{6}+\cdots\)
400.2.s.d 400.s 80.s $18$ $3.194$ \(\mathbb{Q}[x]/(x^{18} + \cdots)\) None None \(0\) \(0\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{2}q^{2}+\beta _{4}q^{3}-\beta _{14}q^{4}+(-1-\beta _{3}+\cdots)q^{6}+\cdots\)
400.2.s.e 400.s 80.s $24$ $3.194$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
400.2.u.a 400.u 25.d $4$ $3.194$ \(\Q(\zeta_{10})\) None None \(0\) \(-5\) \(-5\) \(-6\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-2+2\zeta_{10}+\zeta_{10}^{3})q^{3}+(-2+\zeta_{10}+\cdots)q^{5}+\cdots\)
400.2.u.b 400.u 25.d $4$ $3.194$ \(\Q(\zeta_{10})\) None None \(0\) \(1\) \(-5\) \(2\) $\mathrm{SU}(2)[C_{5}]$ \(q+\zeta_{10}^{3}q^{3}+(-2+\zeta_{10}-2\zeta_{10}^{2}+\cdots)q^{5}+\cdots\)
400.2.u.c 400.u 25.d $4$ $3.194$ \(\Q(\zeta_{10})\) None None \(0\) \(1\) \(5\) \(12\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1-\zeta_{10}-2\zeta_{10}^{3})q^{3}+(2-2\zeta_{10}+\cdots)q^{5}+\cdots\)
400.2.u.d 400.u 25.d $8$ $3.194$ 8.0.58140625.2 None None \(0\) \(3\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{5}]$ \(q+(\beta _{2}+\beta _{3}+\beta _{5}+\beta _{6})q^{3}+(\beta _{1}+\beta _{3}+\cdots)q^{5}+\cdots\)
400.2.u.e 400.u 25.d $8$ $3.194$ 8.0.58140625.2 None None \(0\) \(6\) \(5\) \(-4\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1-\beta _{3})q^{3}+(-\beta _{4}+\beta _{6})q^{5}+(-1+\cdots)q^{7}+\cdots\)
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