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Label Char Prim Dim $A$ Field CM RM Minimal twist Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
4000.1.b.a 4000.b 4.b $4$ $1.996$ \(\Q(i, \sqrt{5})\) None None 4000.1.b.a \(0\) \(0\) \(0\) \(0\) \(q+(-\beta _{1}-\beta _{3})q^{3}-\beta _{3}q^{7}+(-1-\beta _{2}+\cdots)q^{9}+\cdots\)
4000.1.b.b 4000.b 4.b $4$ $1.996$ \(\Q(i, \sqrt{5})\) None None 4000.1.b.a \(0\) \(0\) \(0\) \(0\) \(q-\beta _{1}q^{3}+\beta _{3}q^{7}+\beta _{2}q^{9}-\beta _{3}q^{11}+\cdots\)
4000.1.e.a 4000.e 40.e $2$ $1.996$ \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-10}) \) None 1000.1.g.a \(0\) \(0\) \(0\) \(-1\) \(q-\beta q^{7}+q^{9}+(1-\beta )q^{11}+(1-\beta )q^{13}+\cdots\)
4000.1.e.b 4000.e 40.e $2$ $1.996$ \(\Q(\sqrt{5}) \) \(\Q(\sqrt{-10}) \) None 1000.1.g.a \(0\) \(0\) \(0\) \(1\) \(q+\beta q^{7}+q^{9}+(1-\beta )q^{11}+(-1+\beta )q^{13}+\cdots\)
4000.1.g.a 4000.g 8.d $4$ $1.996$ \(\Q(i, \sqrt{5})\) \(\Q(\sqrt{-10}) \) None 1000.1.g.a \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{1}+\beta _{3})q^{7}-q^{9}-\beta _{2}q^{11}-\beta _{1}q^{13}+\cdots\)
4000.1.h.a 4000.h 20.d $4$ $1.996$ \(\Q(i, \sqrt{5})\) None None 4000.1.b.a \(0\) \(-2\) \(0\) \(4\) \(q+(-1-\beta _{2})q^{3}+q^{7}+(1+\beta _{2})q^{9}+\cdots\)
4000.1.h.b 4000.h 20.d $4$ $1.996$ \(\Q(i, \sqrt{5})\) None None 4000.1.b.a \(0\) \(2\) \(0\) \(-4\) \(q+(1+\beta _{2})q^{3}-q^{7}+(1+\beta _{2})q^{9}+\beta _{3}q^{11}+\cdots\)
4000.1.p.a 4000.p 5.c $8$ $1.996$ \(\Q(\zeta_{20})\) \(\Q(\sqrt{-5}) \) None 4000.1.p.a \(0\) \(-2\) \(0\) \(2\) \(q+(-\zeta_{20}+\zeta_{20}^{4})q^{3}+(\zeta_{20}^{7}-\zeta_{20}^{8}+\cdots)q^{7}+\cdots\)
4000.1.p.b 4000.p 5.c $8$ $1.996$ \(\Q(\zeta_{20})\) \(\Q(\sqrt{-5}) \) None 4000.1.p.a \(0\) \(2\) \(0\) \(-2\) \(q+(\zeta_{20}^{6}+\zeta_{20}^{9})q^{3}+(-\zeta_{20}^{2}-\zeta_{20}^{3}+\cdots)q^{7}+\cdots\)
4000.1.bf.a 4000.bf 100.h $8$ $1.996$ \(\Q(\zeta_{20})\) None None 800.1.bh.a \(0\) \(-2\) \(0\) \(4\) \(q-\zeta_{20}^{2}q^{3}+(-\zeta_{20}^{4}+\zeta_{20}^{6})q^{7}+\cdots\)
4000.1.bf.b 4000.bf 100.h $8$ $1.996$ \(\Q(\zeta_{20})\) None None 800.1.bh.a \(0\) \(2\) \(0\) \(-4\) \(q+\zeta_{20}^{2}q^{3}+(\zeta_{20}^{4}-\zeta_{20}^{6})q^{7}+(-\zeta_{20}^{3}+\cdots)q^{13}+\cdots\)
4000.1.bh.a 4000.bh 100.j $8$ $1.996$ \(\Q(\zeta_{20})\) None None 800.1.bh.a \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{20}q^{3}+(\zeta_{20}^{3}+\zeta_{20}^{7})q^{7}+(-1+\cdots)q^{13}+\cdots\)
4000.1.bo.a 4000.bo 25.f $8$ $1.996$ \(\Q(\zeta_{20})\) \(\Q(\sqrt{-1}) \) None 800.1.bo.a \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{20}^{9}q^{9}+(-\zeta_{20}^{6}-\zeta_{20}^{7})q^{13}+\cdots\)
4000.1.bo.b 4000.bo 25.f $8$ $1.996$ \(\Q(\zeta_{20})\) \(\Q(\sqrt{-1}) \) None 800.1.bo.a \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{20}^{9}q^{9}+(\zeta_{20}-\zeta_{20}^{2})q^{13}+(\zeta_{20}^{4}+\cdots)q^{17}+\cdots\)
4000.1.bo.c 4000.bo 25.f $8$ $1.996$ \(\Q(\zeta_{20})\) \(\Q(\sqrt{-1}) \) None 800.1.bo.a \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{20}^{9}q^{9}+(\zeta_{20}^{6}+\zeta_{20}^{7})q^{13}+(\zeta_{20}^{2}+\cdots)q^{17}+\cdots\)
4000.1.cq.a 4000.cq 125.i $40$ $1.996$ \(\Q(\zeta_{100})\) \(\Q(\sqrt{-1}) \) None 4000.1.cq.a \(0\) \(0\) \(0\) \(0\) \(q+\zeta_{100}^{13}q^{5}-\zeta_{100}^{49}q^{9}+(\zeta_{100}^{2}-\zeta_{100}^{21}+\cdots)q^{13}+\cdots\)
4000.2.a.a 4000.a 1.a $4$ $31.940$ \(\Q(\zeta_{20})^+\) \(\Q(\sqrt{-5}) \) None 4000.2.a.a \(0\) \(-6\) \(0\) \(-8\) $+$ $N(\mathrm{U}(1))$ \(q+(-2-\beta _{1}-\beta _{2}+\beta _{3})q^{3}+(-1+2\beta _{1}+\cdots)q^{7}+\cdots\)
4000.2.a.b 4000.a 1.a $4$ $31.940$ \(\Q(\zeta_{20})^+\) \(\Q(\sqrt{-5}) \) None 4000.2.a.b \(0\) \(-4\) \(0\) \(-2\) $+$ $N(\mathrm{U}(1))$ \(q+(-1+\beta _{1})q^{3}+(-1+\beta _{1}-\beta _{2}-\beta _{3})q^{7}+\cdots\)
4000.2.a.c 4000.a 1.a $4$ $31.940$ 4.4.16400.1 None None 4000.2.a.c \(0\) \(-2\) \(0\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{2}q^{3}+(-1-\beta _{2})q^{7}+(-2-\beta _{2}+\cdots)q^{9}+\cdots\)
4000.2.a.d 4000.a 1.a $4$ $31.940$ \(\Q(\zeta_{20})^+\) None None 4000.2.a.d \(0\) \(0\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{7}+\beta _{2}q^{9}+\cdots\)
4000.2.a.e 4000.a 1.a $4$ $31.940$ \(\Q(\zeta_{20})^+\) None None 4000.2.a.e \(0\) \(0\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+\beta _{3}q^{7}+\beta _{2}q^{9}-2\beta _{3}q^{11}+\cdots\)
4000.2.a.f 4000.a 1.a $4$ $31.940$ \(\Q(\zeta_{20})^+\) None None 4000.2.a.e \(0\) \(0\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+\beta _{3}q^{7}+\beta _{2}q^{9}+2\beta _{3}q^{11}+\cdots\)
4000.2.a.g 4000.a 1.a $4$ $31.940$ \(\Q(\zeta_{20})^+\) None None 4000.2.a.d \(0\) \(0\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-\beta _{1}-\beta _{3})q^{7}+\beta _{2}q^{9}+\cdots\)
4000.2.a.h 4000.a 1.a $4$ $31.940$ 4.4.16400.1 None None 4000.2.a.c \(0\) \(2\) \(0\) \(2\) $+$ $\mathrm{SU}(2)$ \(q-\beta _{2}q^{3}+(1+\beta _{2})q^{7}+(-2-\beta _{2})q^{9}+\cdots\)
4000.2.a.i 4000.a 1.a $4$ $31.940$ \(\Q(\zeta_{20})^+\) \(\Q(\sqrt{-5}) \) None 4000.2.a.b \(0\) \(4\) \(0\) \(2\) $-$ $N(\mathrm{U}(1))$ \(q+(1+\beta _{1})q^{3}+(1+\beta _{1}+\beta _{2}-\beta _{3})q^{7}+\cdots\)
4000.2.a.j 4000.a 1.a $4$ $31.940$ \(\Q(\zeta_{20})^+\) \(\Q(\sqrt{-5}) \) None 4000.2.a.a \(0\) \(6\) \(0\) \(8\) $-$ $N(\mathrm{U}(1))$ \(q+(2-\beta _{1}+\beta _{2}+\beta _{3})q^{3}+(1+2\beta _{1}+\cdots)q^{7}+\cdots\)
4000.2.a.k 4000.a 1.a $6$ $31.940$ 6.6.30040000.1 None None 4000.2.a.k \(0\) \(0\) \(0\) \(-3\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-1+\beta _{5})q^{7}+(1+\beta _{1}+\beta _{3}+\cdots)q^{9}+\cdots\)
4000.2.a.l 4000.a 1.a $6$ $31.940$ 6.6.30040000.1 None None 4000.2.a.k \(0\) \(0\) \(0\) \(-3\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{3}+(-1+\beta _{5})q^{7}+(1+\beta _{1}+\beta _{3}+\cdots)q^{9}+\cdots\)
4000.2.a.m 4000.a 1.a $6$ $31.940$ 6.6.30040000.1 None None 4000.2.a.k \(0\) \(0\) \(0\) \(3\) $+$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(1-\beta _{5})q^{7}+(1+\beta _{1}+\beta _{3}+\cdots)q^{9}+\cdots\)
4000.2.a.n 4000.a 1.a $6$ $31.940$ 6.6.30040000.1 None None 4000.2.a.k \(0\) \(0\) \(0\) \(3\) $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+(1-\beta _{5})q^{7}+(1+\beta _{1}+\beta _{3}+\cdots)q^{9}+\cdots\)
4000.2.a.o 4000.a 1.a $8$ $31.940$ 8.8.\(\cdots\).1 None None 4000.2.a.o \(0\) \(0\) \(0\) \(0\) $+$ $\mathrm{SU}(2)$ \(q-\beta _{4}q^{3}-\beta _{6}q^{7}+(1-\beta _{5})q^{9}+(\beta _{4}+\cdots)q^{11}+\cdots\)
4000.2.a.p 4000.a 1.a $8$ $31.940$ 8.8.\(\cdots\).1 None None 4000.2.a.o \(0\) \(0\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-\beta _{4}q^{3}-\beta _{6}q^{7}+(1-\beta _{5})q^{9}+(-\beta _{4}+\cdots)q^{11}+\cdots\)
4000.2.a.q 4000.a 1.a $8$ $31.940$ 8.8.\(\cdots\).1 None None 4000.2.a.q \(0\) \(0\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+\beta _{7}q^{7}+(2+\beta _{2})q^{9}+(-\beta _{1}+\cdots)q^{11}+\cdots\)
4000.2.a.r 4000.a 1.a $8$ $31.940$ 8.8.\(\cdots\).1 None None 4000.2.a.q \(0\) \(0\) \(0\) \(0\) $-$ $\mathrm{SU}(2)$ \(q-\beta _{1}q^{3}+\beta _{7}q^{7}+(2+\beta _{2})q^{9}+(\beta _{1}+\cdots)q^{11}+\cdots\)
4000.2.c.a 4000.c 5.b $8$ $31.940$ \(\Q(\zeta_{20})\) \(\Q(\sqrt{-5}) \) None 4000.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(\beta_{7}+\beta_{3})q^{3}+(2\beta_{7}-2\beta_{3}+\beta_{2})q^{7}+\cdots\)
4000.2.c.b 4000.c 5.b $8$ $31.940$ \(\Q(\zeta_{20})\) \(\Q(\sqrt{-5}) \) None 4000.2.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\beta_{3} q^{3}+(-\beta_{7}+\beta_{3})q^{7}+(-\beta_{6}-\beta_1-1)q^{9}+\cdots\)
4000.2.c.c 4000.c 5.b $8$ $31.940$ \(\Q(\zeta_{20})\) None None 4000.2.a.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta_{7} q^{3}+(-\beta_{7}+\beta_{2})q^{7}+(\beta_{5}+1)q^{9}+\cdots\)
4000.2.c.d 4000.c 5.b $8$ $31.940$ \(\Q(\zeta_{20})\) None None 4000.2.a.e \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta_{2} q^{3}-\beta_{7} q^{7}-\beta_{5} q^{9}+\beta_{4} q^{11}+\cdots\)
4000.2.c.e 4000.c 5.b $8$ $31.940$ 8.0.268960000.3 None None 4000.2.a.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{1}+\beta _{3})q^{3}+\beta _{3}q^{7}+(2+\beta _{2}+\cdots)q^{9}+\cdots\)
4000.2.c.f 4000.c 5.b $12$ $31.940$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None None 4000.2.a.k \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}-\beta _{11}q^{7}+(\beta _{3}+\beta _{4}+\beta _{6}+\cdots)q^{9}+\cdots\)
4000.2.c.g 4000.c 5.b $12$ $31.940$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None None 4000.2.a.k \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}-\beta _{11}q^{7}+(\beta _{3}+\beta _{4}+\beta _{6}+\cdots)q^{9}+\cdots\)
4000.2.c.h 4000.c 5.b $16$ $31.940$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None 4000.2.a.q \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{3}-\beta _{15}q^{7}+(-2+\beta _{4})q^{9}+\cdots\)
4000.2.c.i 4000.c 5.b $16$ $31.940$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None 4000.2.a.o \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{3}-\beta _{12}q^{7}+(-1+\beta _{6})q^{9}+\cdots\)
4000.2.d.a 4000.d 8.b $4$ $31.940$ \(\Q(i, \sqrt{5})\) None None 1000.2.d.a \(0\) \(0\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{2}]$ \(q+2\beta _{1}q^{3}+(1+3\beta _{2})q^{7}+(-1+4\beta _{2}+\cdots)q^{9}+\cdots\)
4000.2.d.b 4000.d 8.b $4$ $31.940$ \(\Q(i, \sqrt{5})\) None None 1000.2.d.a \(0\) \(0\) \(0\) \(2\) $\mathrm{SU}(2)[C_{2}]$ \(q+2\beta _{1}q^{3}+(-1-3\beta _{2})q^{7}+(-1+4\beta _{2}+\cdots)q^{9}+\cdots\)
4000.2.d.c 4000.d 8.b $40$ $31.940$ None None 1000.2.d.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
4000.2.d.d 4000.d 8.b $48$ $31.940$ None None 1000.2.d.d \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
4000.2.f.a 4000.f 40.f $4$ $31.940$ \(\Q(i, \sqrt{5})\) None None 1000.2.d.a \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-2-2\beta _{2})q^{3}+(3\beta _{1}+2\beta _{3})q^{7}+\cdots\)
4000.2.f.b 4000.f 40.f $4$ $31.940$ \(\Q(i, \sqrt{5})\) None None 1000.2.d.a \(0\) \(4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(2+2\beta _{2})q^{3}+(3\beta _{1}+2\beta _{3})q^{7}+(5+\cdots)q^{9}+\cdots\)
4000.2.f.c 4000.f 40.f $20$ $31.940$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None None 1000.2.d.c \(0\) \(-4\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{9}q^{3}+\beta _{5}q^{7}+(\beta _{1}-\beta _{7}+\beta _{10}+\cdots)q^{9}+\cdots\)
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