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Label Char Prim Dim $A$ Field CM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
20.2.a.a 20.a 1.a $1$ $0.160$ \(\Q\) None \(0\) \(-2\) \(-1\) \(2\) $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-q^{5}+2q^{7}+q^{9}+2q^{13}+\cdots\)
20.2.e.a 20.e 20.e $2$ $0.160$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(-2\) \(0\) \(-4\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(-1-i)q^{2}+2iq^{4}+(-2+i)q^{5}+\cdots\)
20.3.b.a 20.b 4.b $4$ $0.545$ \(\Q(\zeta_{10})\) None \(-2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\zeta_{10}^{2}q^{2}+\zeta_{10}^{3}q^{3}+(-2-\zeta_{10}+\cdots)q^{4}+\cdots\)
20.3.d.a 20.d 20.d $1$ $0.545$ \(\Q\) \(\Q(\sqrt{-5}) \) \(-2\) \(4\) \(-5\) \(-4\) $\mathrm{U}(1)[D_{2}]$ \(q-2q^{2}+4q^{3}+4q^{4}-5q^{5}-8q^{6}+\cdots\)
20.3.d.b 20.d 20.d $1$ $0.545$ \(\Q\) \(\Q(\sqrt{-5}) \) \(2\) \(-4\) \(-5\) \(4\) $\mathrm{U}(1)[D_{2}]$ \(q+2q^{2}-4q^{3}+4q^{4}-5q^{5}-8q^{6}+\cdots\)
20.3.d.c 20.d 20.d $2$ $0.545$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(6\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+iq^{2}-4q^{4}+(3-2i)q^{5}-4iq^{8}+\cdots\)
20.3.f.a 20.f 5.c $2$ $0.545$ \(\Q(\sqrt{-1}) \) None \(0\) \(2\) \(-6\) \(-14\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1-i)q^{3}+(-3+4i)q^{5}+(-7-7i)q^{7}+\cdots\)
20.4.a.a 20.a 1.a $1$ $1.180$ \(\Q\) None \(0\) \(4\) \(5\) \(-16\) $+$ $\mathrm{SU}(2)$ \(q+4q^{3}+5q^{5}-2^{4}q^{7}-11q^{9}-60q^{11}+\cdots\)
20.4.c.a 20.c 5.b $2$ $1.180$ \(\Q(\sqrt{-19}) \) None \(0\) \(0\) \(14\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta q^{3}+(7+\beta )q^{5}+\beta q^{7}-7^{2}q^{9}+\cdots\)
20.4.e.a 20.e 20.e $2$ $1.180$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(4\) \(0\) \(-4\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(2+2i)q^{2}+8iq^{4}+(-2-11i)q^{5}+\cdots\)
20.4.e.b 20.e 20.e $12$ $1.180$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-6\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{4}]$ \(q+(\beta _{1}-\beta _{5})q^{2}-\beta _{9}q^{3}+(2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
20.5.b.a 20.b 4.b $8$ $2.067$ 8.0.\(\cdots\).1 None \(6\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1+\beta _{1})q^{2}+\beta _{3}q^{3}+(-2+\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
20.5.d.a 20.d 20.d $1$ $2.067$ \(\Q\) \(\Q(\sqrt{-5}) \) \(-4\) \(2\) \(25\) \(82\) $\mathrm{U}(1)[D_{2}]$ \(q-4q^{2}+2q^{3}+2^{4}q^{4}+5^{2}q^{5}-8q^{6}+\cdots\)
20.5.d.b 20.d 20.d $1$ $2.067$ \(\Q\) \(\Q(\sqrt{-5}) \) \(4\) \(-2\) \(25\) \(-82\) $\mathrm{U}(1)[D_{2}]$ \(q+4q^{2}-2q^{3}+2^{4}q^{4}+5^{2}q^{5}-8q^{6}+\cdots\)
20.5.d.c 20.d 20.d $8$ $2.067$ 8.0.\(\cdots\).2 None \(0\) \(0\) \(-40\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}-\beta _{2}q^{3}+(-6+\beta _{3})q^{4}+(-5+\cdots)q^{5}+\cdots\)
20.5.f.a 20.f 5.c $4$ $2.067$ \(\Q(i, \sqrt{241})\) None \(0\) \(-10\) \(-6\) \(110\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2+2\beta _{1}+\beta _{3})q^{3}+(-6\beta _{1}-\beta _{2}+\cdots)q^{5}+\cdots\)
20.6.a.a 20.a 1.a $1$ $3.208$ \(\Q\) None \(0\) \(22\) \(-25\) \(218\) $-$ $\mathrm{SU}(2)$ \(q+22q^{3}-5^{2}q^{5}+218q^{7}+241q^{9}+\cdots\)
20.6.c.a 20.c 5.b $2$ $3.208$ \(\Q(\sqrt{-31}) \) None \(0\) \(0\) \(-10\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta q^{3}+(-5-5\beta )q^{5}-11\beta q^{7}+119q^{9}+\cdots\)
20.6.e.a 20.e 20.e $2$ $3.208$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(8\) \(0\) \(76\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(4+4i)q^{2}+2^{5}iq^{4}+(38+41i)q^{5}+\cdots\)
20.6.e.b 20.e 20.e $24$ $3.208$ None \(-10\) \(0\) \(-80\) \(0\) $\mathrm{SU}(2)[C_{4}]$
20.7.b.a 20.b 4.b $12$ $4.601$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None \(-10\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-1-\beta _{2})q^{2}+(-\beta _{2}-\beta _{6})q^{3}+(13+\cdots)q^{4}+\cdots\)
20.7.d.a 20.d 20.d $1$ $4.601$ \(\Q\) \(\Q(\sqrt{-5}) \) \(-8\) \(-44\) \(-125\) \(524\) $\mathrm{U}(1)[D_{2}]$ \(q-8q^{2}-44q^{3}+2^{6}q^{4}-5^{3}q^{5}+352q^{6}+\cdots\)
20.7.d.b 20.d 20.d $1$ $4.601$ \(\Q\) \(\Q(\sqrt{-5}) \) \(8\) \(44\) \(-125\) \(-524\) $\mathrm{U}(1)[D_{2}]$ \(q+8q^{2}+44q^{3}+2^{6}q^{4}-5^{3}q^{5}+352q^{6}+\cdots\)
20.7.d.c 20.d 20.d $2$ $4.601$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-234\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+2iq^{2}-2^{6}q^{4}+(-117+11i)q^{5}+\cdots\)
20.7.d.d 20.d 20.d $12$ $4.601$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(460\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{2})q^{3}+(6+\beta _{4}-\beta _{8}+\cdots)q^{4}+\cdots\)
20.7.f.a 20.f 5.c $6$ $4.601$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(0\) \(32\) \(-156\) \(-264\) $\mathrm{SU}(2)[C_{4}]$ \(q+(5+5\beta _{1}+\beta _{4})q^{3}+(-5^{2}+20\beta _{1}+\cdots)q^{5}+\cdots\)
20.8.a.a 20.a 1.a $1$ $6.248$ \(\Q\) None \(0\) \(-6\) \(-125\) \(-706\) $-$ $\mathrm{SU}(2)$ \(q-6q^{3}-5^{3}q^{5}-706q^{7}-2151q^{9}+\cdots\)
20.8.a.b 20.a 1.a $2$ $6.248$ \(\Q(\sqrt{1129}) \) None \(0\) \(-20\) \(250\) \(1660\) $+$ $\mathrm{SU}(2)$ \(q+(-10-\beta )q^{3}+5^{3}q^{5}+(830+9\beta )q^{7}+\cdots\)
20.8.c.a 20.c 5.b $4$ $6.248$ \(\mathbb{Q}[x]/(x^{4} + \cdots)\) None \(0\) \(0\) \(-156\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}+(-39-\beta _{1}+\beta _{2})q^{5}+(-5\beta _{1}+\cdots)q^{7}+\cdots\)
20.8.e.a 20.e 20.e $2$ $6.248$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(-16\) \(0\) \(556\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(-8-8i)q^{2}+2^{7}iq^{4}+(278+29i)q^{5}+\cdots\)
20.8.e.b 20.e 20.e $36$ $6.248$ None \(14\) \(0\) \(-560\) \(0\) $\mathrm{SU}(2)[C_{4}]$
20.9.b.a 20.b 4.b $16$ $8.148$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(6\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+(\beta _{1}+\beta _{3})q^{3}+(-3+\beta _{2}+\cdots)q^{4}+\cdots\)
20.9.d.a 20.d 20.d $1$ $8.148$ \(\Q\) \(\Q(\sqrt{-5}) \) \(-16\) \(158\) \(625\) \(-1922\) $\mathrm{U}(1)[D_{2}]$ \(q-2^{4}q^{2}+158q^{3}+2^{8}q^{4}+5^{4}q^{5}+\cdots\)
20.9.d.b 20.d 20.d $1$ $8.148$ \(\Q\) \(\Q(\sqrt{-5}) \) \(16\) \(-158\) \(625\) \(1922\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{4}q^{2}-158q^{3}+2^{8}q^{4}+5^{4}q^{5}+\cdots\)
20.9.d.c 20.d 20.d $20$ $8.148$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(0\) \(-1420\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+(\beta _{1}+\beta _{6})q^{3}+(-38+\beta _{2}+\cdots)q^{4}+\cdots\)
20.9.f.a 20.f 5.c $8$ $8.148$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(0\) \(-70\) \(894\) \(-2030\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-9-9\beta _{1}+\beta _{2})q^{3}+(112-58\beta _{1}+\cdots)q^{5}+\cdots\)
20.10.a.a 20.a 1.a $1$ $10.301$ \(\Q\) None \(0\) \(-48\) \(625\) \(-532\) $+$ $\mathrm{SU}(2)$ \(q-48q^{3}+5^{4}q^{5}-532q^{7}-17379q^{9}+\cdots\)
20.10.a.b 20.a 1.a $2$ $10.301$ \(\Q(\sqrt{79}) \) None \(0\) \(-260\) \(-1250\) \(-380\) $-$ $\mathrm{SU}(2)$ \(q+(-130+\beta )q^{3}-5^{4}q^{5}+(-190+\cdots)q^{7}+\cdots\)
20.10.c.a 20.c 5.b $4$ $10.301$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(0\) \(0\) \(660\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}+(165+\beta _{1}-\beta _{3})q^{5}+(3\beta _{1}+\cdots)q^{7}+\cdots\)
20.10.e.a 20.e 20.e $2$ $10.301$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(-32\) \(0\) \(1436\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q+(-2^{4}-2^{4}i)q^{2}+2^{9}iq^{4}+(718+\cdots)q^{5}+\cdots\)
20.10.e.b 20.e 20.e $48$ $10.301$ None \(30\) \(0\) \(-1440\) \(0\) $\mathrm{SU}(2)[C_{4}]$
20.11.b.a 20.b 4.b $20$ $12.707$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(22\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1-\beta _{1})q^{2}+(-\beta _{1}+\beta _{2})q^{3}+(-2^{5}+\cdots)q^{4}+\cdots\)
20.11.d.a 20.d 20.d $1$ $12.707$ \(\Q\) \(\Q(\sqrt{-5}) \) \(-32\) \(-236\) \(-3125\) \(-33364\) $\mathrm{U}(1)[D_{2}]$ \(q-2^{5}q^{2}-236q^{3}+2^{10}q^{4}-5^{5}q^{5}+\cdots\)
20.11.d.b 20.d 20.d $1$ $12.707$ \(\Q\) \(\Q(\sqrt{-5}) \) \(32\) \(236\) \(-3125\) \(33364\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{5}q^{2}+236q^{3}+2^{10}q^{4}-5^{5}q^{5}+\cdots\)
20.11.d.c 20.d 20.d $2$ $12.707$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) \(0\) \(0\) \(-474\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+8iq^{2}-2^{10}q^{4}+(-237+779i)q^{5}+\cdots\)
20.11.d.d 20.d 20.d $24$ $12.707$ None \(0\) \(0\) \(8280\) \(0\) $\mathrm{SU}(2)[C_{2}]$
20.11.f.a 20.f 5.c $10$ $12.707$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(62\) \(894\) \(22286\) $\mathrm{SU}(2)[C_{4}]$ \(q+(6+6\beta _{1}-\beta _{2})q^{3}+(90-174\beta _{1}+\cdots)q^{5}+\cdots\)
20.12.a.a 20.a 1.a $1$ $15.367$ \(\Q\) None \(0\) \(306\) \(-3125\) \(-32074\) $-$ $\mathrm{SU}(2)$ \(q+306q^{3}-5^{5}q^{5}-32074q^{7}-83511q^{9}+\cdots\)
20.12.a.b 20.a 1.a $2$ $15.367$ \(\Q(\sqrt{46729}) \) None \(0\) \(220\) \(6250\) \(3340\) $+$ $\mathrm{SU}(2)$ \(q+(110-\beta )q^{3}+5^{5}q^{5}+(1670-111\beta )q^{7}+\cdots\)
20.12.c.a 20.c 5.b $6$ $15.367$ \(\mathbb{Q}[x]/(x^{6} + \cdots)\) None \(0\) \(0\) \(-4126\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}+(-688+2\beta _{1}-\beta _{2})q^{5}+\cdots\)
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