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Label Char Prim Dim $A$ Field CM RM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
540.1.f.a 540.f 20.d $4$ $0.269$ \(\Q(\zeta_{12})\) \(\Q(\sqrt{-15}) \) None \(0\) \(0\) \(0\) \(0\) \(q-\zeta_{12}q^{2}+\zeta_{12}^{2}q^{4}+\zeta_{12}^{3}q^{5}-\zeta_{12}^{3}q^{8}+\cdots\)
540.1.p.a 540.p 180.p $2$ $0.269$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-5}) \) None \(-1\) \(0\) \(1\) \(-1\) \(q-\zeta_{6}q^{2}+\zeta_{6}^{2}q^{4}-\zeta_{6}^{2}q^{5}-\zeta_{6}q^{7}+\cdots\)
540.1.p.b 540.p 180.p $2$ $0.269$ \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-5}) \) None \(1\) \(0\) \(1\) \(1\) \(q+\zeta_{6}q^{2}+\zeta_{6}^{2}q^{4}-\zeta_{6}^{2}q^{5}+\zeta_{6}q^{7}+\cdots\)
540.1.bf.a 540.bf 540.af $6$ $0.269$ \(\Q(\zeta_{18})\) \(\Q(\sqrt{-5}) \) None \(0\) \(0\) \(0\) \(-3\) \(q-\zeta_{18}q^{2}-\zeta_{18}^{5}q^{3}+\zeta_{18}^{2}q^{4}-\zeta_{18}^{5}q^{5}+\cdots\)
540.1.bf.b 540.bf 540.af $6$ $0.269$ \(\Q(\zeta_{18})\) \(\Q(\sqrt{-5}) \) None \(0\) \(0\) \(0\) \(3\) \(q+\zeta_{18}q^{2}+\zeta_{18}^{5}q^{3}+\zeta_{18}^{2}q^{4}-\zeta_{18}^{5}q^{5}+\cdots\)
540.2.a.a 540.a 1.a $1$ $4.312$ \(\Q\) None None \(0\) \(0\) \(-1\) \(-4\) $+$ $\mathrm{SU}(2)$ \(q-q^{5}-4q^{7}+6q^{11}-4q^{13}-3q^{17}+\cdots\)
540.2.a.b 540.a 1.a $1$ $4.312$ \(\Q\) None None \(0\) \(0\) \(-1\) \(-1\) $+$ $\mathrm{SU}(2)$ \(q-q^{5}-q^{7}-6q^{11}-q^{13}-q^{19}+\cdots\)
540.2.a.c 540.a 1.a $1$ $4.312$ \(\Q\) None None \(0\) \(0\) \(-1\) \(2\) $-$ $\mathrm{SU}(2)$ \(q-q^{5}+2q^{7}+2q^{13}+3q^{17}+5q^{19}+\cdots\)
540.2.a.d 540.a 1.a $1$ $4.312$ \(\Q\) None None \(0\) \(0\) \(1\) \(-4\) $+$ $\mathrm{SU}(2)$ \(q+q^{5}-4q^{7}-6q^{11}-4q^{13}+3q^{17}+\cdots\)
540.2.a.e 540.a 1.a $1$ $4.312$ \(\Q\) None None \(0\) \(0\) \(1\) \(-1\) $-$ $\mathrm{SU}(2)$ \(q+q^{5}-q^{7}+6q^{11}-q^{13}-q^{19}+\cdots\)
540.2.a.f 540.a 1.a $1$ $4.312$ \(\Q\) None None \(0\) \(0\) \(1\) \(2\) $-$ $\mathrm{SU}(2)$ \(q+q^{5}+2q^{7}+2q^{13}-3q^{17}+5q^{19}+\cdots\)
540.2.d.a 540.d 5.b $2$ $4.312$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(-4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-2+i)q^{5}-2iq^{7}+2q^{11}-6iq^{13}+\cdots\)
540.2.d.b 540.d 5.b $2$ $4.312$ \(\Q(\sqrt{-1}) \) None None \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(2+i)q^{5}+2iq^{7}-2q^{11}+6iq^{13}+\cdots\)
540.2.d.c 540.d 5.b $4$ $4.312$ \(\Q(i, \sqrt{10})\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{5}+\beta _{2}q^{7}+(\beta _{1}-\beta _{3})q^{11}-3\beta _{2}q^{13}+\cdots\)
540.2.e.a 540.e 12.b $16$ $4.312$ 16.0.\(\cdots\).1 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{11}q^{2}+\beta _{2}q^{4}+\beta _{3}q^{5}+(\beta _{4}+\beta _{6}+\cdots)q^{7}+\cdots\)
540.2.e.b 540.e 12.b $16$ $4.312$ 16.0.\(\cdots\).1 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(\beta _{2}+\beta _{7})q^{4}+\beta _{3}q^{5}+(\beta _{6}+\cdots)q^{7}+\cdots\)
540.2.h.a 540.h 60.h $4$ $4.312$ \(\Q(\sqrt{-3}, \sqrt{5})\) \(\Q(\sqrt{-15}) \) None \(-3\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(-1+\beta _{1})q^{2}+(-\beta _{1}+\beta _{2})q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots\)
540.2.h.b 540.h 60.h $4$ $4.312$ \(\Q(\sqrt{-3}, \sqrt{5})\) \(\Q(\sqrt{-15}) \) None \(3\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(1-\beta _{1})q^{2}+(-\beta _{1}+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
540.2.h.c 540.h 60.h $16$ $4.312$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{7}q^{2}-\beta _{11}q^{4}+(-\beta _{6}-\beta _{10})q^{5}+\cdots\)
540.2.h.d 540.h 60.h $24$ $4.312$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
540.2.i.a 540.i 9.c $2$ $4.312$ \(\Q(\sqrt{-3}) \) None None \(0\) \(0\) \(1\) \(1\) $\mathrm{SU}(2)[C_{3}]$ \(q+\zeta_{6}q^{5}+(1-\zeta_{6})q^{7}+4\zeta_{6}q^{13}+6q^{17}+\cdots\)
540.2.i.b 540.i 9.c $6$ $4.312$ 6.0.954288.1 None None \(0\) \(0\) \(-3\) \(-3\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-1-\beta _{2})q^{5}+(\beta _{2}+\beta _{4})q^{7}+(\beta _{3}+\cdots)q^{11}+\cdots\)
540.2.j.a 540.j 15.e $8$ $4.312$ 8.0.\(\cdots\).8 None None \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+\beta _{2}q^{5}+(-1-\beta _{1}+\beta _{4})q^{7}+(-\beta _{5}+\cdots)q^{11}+\cdots\)
540.2.j.b 540.j 15.e $8$ $4.312$ 8.0.\(\cdots\).1 None None \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-\beta _{1}+\beta _{4}+\beta _{7})q^{5}+(1+\beta _{6})q^{7}+\cdots\)
540.2.k.a 540.k 20.e $4$ $4.312$ \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(0\) \(-8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\zeta_{8}+\zeta_{8}^{3})q^{2}-2q^{4}+(-2\zeta_{8}+\zeta_{8}^{3})q^{5}+\cdots\)
540.2.k.b 540.k 20.e $4$ $4.312$ \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(0\) \(4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\zeta_{8}+\zeta_{8}^{3})q^{2}-2q^{4}+(\zeta_{8}-2\zeta_{8}^{3})q^{5}+\cdots\)
540.2.k.c 540.k 20.e $4$ $4.312$ \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-\zeta_{8}+\zeta_{8}^{3})q^{2}+2q^{4}+(-\zeta_{8}+2\zeta_{8}^{3})q^{5}+\cdots\)
540.2.k.d 540.k 20.e $4$ $4.312$ \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(0\) \(8\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-\zeta_{8}+\zeta_{8}^{3})q^{2}+2q^{4}+(2\zeta_{8}-\zeta_{8}^{3})q^{5}+\cdots\)
540.2.k.e 540.k 20.e $32$ $4.312$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
540.2.k.f 540.k 20.e $48$ $4.312$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$
540.2.n.a 540.n 180.n $4$ $4.312$ \(\Q(\sqrt{2}, \sqrt{-3})\) None None \(0\) \(0\) \(-6\) \(-6\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}+2\beta _{2}q^{4}+(-2-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
540.2.n.b 540.n 180.n $4$ $4.312$ \(\Q(\sqrt{2}, \sqrt{-3})\) None None \(0\) \(0\) \(-6\) \(6\) $\mathrm{SU}(2)[C_{6}]$ \(q+\beta _{1}q^{2}+2\beta _{2}q^{4}+(-2-\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
540.2.n.c 540.n 180.n $8$ $4.312$ 8.0.3317760000.8 \(\Q(\sqrt{-5}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{6}]$ \(q+(\beta _{4}+\beta _{5})q^{2}+(2-2\beta _{3})q^{4}+\beta _{6}q^{5}+\cdots\)
540.2.n.d 540.n 180.n $48$ $4.312$ None None \(0\) \(0\) \(18\) \(0\) $\mathrm{SU}(2)[C_{6}]$
540.2.q.a 540.q 36.h $48$ $4.312$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{6}]$
540.2.r.a 540.r 45.j $12$ $4.312$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(0\) \(0\) \(-1\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-\beta _{5}+\beta _{10})q^{5}+(\beta _{7}-\beta _{9}+\beta _{10}+\cdots)q^{7}+\cdots\)
540.2.u.a 540.u 27.e $30$ $4.312$ None None \(0\) \(-3\) \(0\) \(0\) $\mathrm{SU}(2)[C_{9}]$
540.2.u.b 540.u 27.e $42$ $4.312$ None None \(0\) \(3\) \(0\) \(0\) $\mathrm{SU}(2)[C_{9}]$
540.2.x.a 540.x 45.l $24$ $4.312$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{12}]$
540.2.y.a 540.y 180.x $128$ $4.312$ None None \(2\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{12}]$
540.2.bb.a 540.bb 540.ab $24$ $4.312$ \(\Q(\sqrt{-5}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{18}]$
540.2.bb.b 540.bb 540.ab $600$ $4.312$ None None \(0\) \(0\) \(-12\) \(0\) $\mathrm{SU}(2)[C_{18}]$
540.2.bd.a 540.bd 135.p $108$ $4.312$ None None \(0\) \(0\) \(-3\) \(0\) $\mathrm{SU}(2)[C_{18}]$
540.2.be.a 540.be 108.l $432$ $4.312$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{18}]$
540.2.bh.a 540.bh 540.ah $1248$ $4.312$ None None \(-12\) \(0\) \(-24\) \(0\) $\mathrm{SU}(2)[C_{36}]$
540.2.bj.a 540.bj 135.q $216$ $4.312$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{36}]$
540.3.b.a 540.b 15.d $4$ $14.714$ \(\Q(\sqrt{-11}, \sqrt{-19})\) None None \(0\) \(0\) \(-3\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1-\beta _{3})q^{5}-\beta _{2}q^{7}+(-2\beta _{1}-3\beta _{2}+\cdots)q^{11}+\cdots\)
540.3.b.b 540.b 15.d $4$ $14.714$ \(\Q(\sqrt{-11}, \sqrt{-19})\) None None \(0\) \(0\) \(3\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1+\beta _{3})q^{5}-\beta _{2}q^{7}+(2\beta _{1}+3\beta _{2}+\cdots)q^{11}+\cdots\)
540.3.b.c 540.b 15.d $8$ $14.714$ 8.0.\(\cdots\).13 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{2}q^{5}+(-\beta _{1}-\beta _{3})q^{7}-\beta _{6}q^{11}+\cdots\)
540.3.c.a 540.c 4.b $32$ $14.714$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
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