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Label Char Prim Dim $A$ Field CM Minimal twist Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
50.2.a.a 50.a 1.a $1$ $0.399$ \(\Q\) None 50.2.a.a \(-1\) \(1\) \(0\) \(2\) $-$ $\mathrm{SU}(2)$ \(q-q^{2}+q^{3}+q^{4}-q^{6}+2q^{7}-q^{8}+\cdots\)
50.2.a.b 50.a 1.a $1$ $0.399$ \(\Q\) None 50.2.a.a \(1\) \(-1\) \(0\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q+q^{2}-q^{3}+q^{4}-q^{6}-2q^{7}+q^{8}+\cdots\)
50.2.b.a 50.b 5.b $2$ $0.399$ \(\Q(\sqrt{-1}) \) None 50.2.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+iq^{3}-q^{4}-q^{6}-2iq^{7}+\cdots\)
50.2.d.a 50.d 25.d $4$ $0.399$ \(\Q(\zeta_{10})\) None 50.2.d.a \(1\) \(-1\) \(5\) \(-12\) $\mathrm{SU}(2)[C_{5}]$ \(q+(1-\zeta_{10}+\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+(-1+\cdots)q^{3}+\cdots\)
50.2.d.b 50.d 25.d $8$ $0.399$ 8.0.58140625.2 None 50.2.d.b \(-2\) \(-3\) \(0\) \(4\) $\mathrm{SU}(2)[C_{5}]$ \(q+(-1+\beta _{2}+\beta _{3}+\beta _{6})q^{2}+\beta _{4}q^{3}+\cdots\)
50.2.e.a 50.e 25.e $8$ $0.399$ \(\Q(\zeta_{20})\) None 50.2.e.a \(0\) \(0\) \(-10\) \(0\) $\mathrm{SU}(2)[C_{10}]$ \(q+\zeta_{20}q^{2}+(-1+\zeta_{20}^{2}-\zeta_{20}^{4}+2\zeta_{20}^{6}+\cdots)q^{3}+\cdots\)
50.3.c.a 50.c 5.c $2$ $1.362$ \(\Q(\sqrt{-1}) \) None 50.3.c.a \(-2\) \(6\) \(0\) \(-6\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1-i)q^{2}+(3-3i)q^{3}+2iq^{4}+\cdots\)
50.3.c.b 50.c 5.c $2$ $1.362$ \(\Q(\sqrt{-1}) \) None 50.3.c.a \(2\) \(-6\) \(0\) \(6\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{2}+(-3+3i)q^{3}+2iq^{4}+\cdots\)
50.3.c.c 50.c 5.c $2$ $1.362$ \(\Q(\sqrt{-1}) \) None 10.3.c.a \(2\) \(4\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+i)q^{2}+(2-2i)q^{3}+2iq^{4}+4q^{6}+\cdots\)
50.3.f.a 50.f 25.f $16$ $1.362$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None 50.3.f.a \(-4\) \(2\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{20}]$ \(q+(\beta _{1}+\beta _{9})q^{2}+(-1+\beta _{2}-\beta _{3}+\beta _{7}+\cdots)q^{3}+\cdots\)
50.3.f.b 50.f 25.f $24$ $1.362$ None 50.3.f.b \(6\) \(2\) \(0\) \(-2\) $\mathrm{SU}(2)[C_{20}]$
50.4.a.a 50.a 1.a $1$ $2.950$ \(\Q\) None 50.4.a.a \(-2\) \(-7\) \(0\) \(34\) $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-7q^{3}+4q^{4}+14q^{6}+34q^{7}+\cdots\)
50.4.a.b 50.a 1.a $1$ $2.950$ \(\Q\) None 10.4.b.a \(-2\) \(-2\) \(0\) \(-26\) $-$ $\mathrm{SU}(2)$ \(q-2q^{2}-2q^{3}+4q^{4}+4q^{6}-26q^{7}+\cdots\)
50.4.a.c 50.a 1.a $1$ $2.950$ \(\Q\) None 10.4.a.a \(-2\) \(8\) \(0\) \(4\) $+$ $\mathrm{SU}(2)$ \(q-2q^{2}+8q^{3}+4q^{4}-2^{4}q^{6}+4q^{7}+\cdots\)
50.4.a.d 50.a 1.a $1$ $2.950$ \(\Q\) None 10.4.b.a \(2\) \(2\) \(0\) \(26\) $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+2q^{3}+4q^{4}+4q^{6}+26q^{7}+\cdots\)
50.4.a.e 50.a 1.a $1$ $2.950$ \(\Q\) None 50.4.a.a \(2\) \(7\) \(0\) \(-34\) $+$ $\mathrm{SU}(2)$ \(q+2q^{2}+7q^{3}+4q^{4}+14q^{6}-34q^{7}+\cdots\)
50.4.b.a 50.b 5.b $2$ $2.950$ \(\Q(\sqrt{-1}) \) None 10.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+iq^{2}+4iq^{3}-4q^{4}-2^{4}q^{6}-2iq^{7}+\cdots\)
50.4.b.b 50.b 5.b $2$ $2.950$ \(\Q(\sqrt{-1}) \) None 50.4.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}-7iq^{3}-4q^{4}+14q^{6}-34iq^{7}+\cdots\)
50.4.d.a 50.d 25.d $12$ $2.950$ \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None 50.4.d.a \(6\) \(1\) \(20\) \(58\) $\mathrm{SU}(2)[C_{5}]$ \(q+2\beta _{5}q^{2}+(-\beta _{2}+\beta _{3}-\beta _{4}+\beta _{5}+\cdots)q^{3}+\cdots\)
50.4.d.b 50.d 25.d $16$ $2.950$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None 50.4.d.b \(-8\) \(7\) \(15\) \(-54\) $\mathrm{SU}(2)[C_{5}]$ \(q+2\beta _{6}q^{2}+(1-\beta _{1}+\beta _{2}-\beta _{3}-\beta _{4}+\cdots)q^{3}+\cdots\)
50.4.e.a 50.e 25.e $32$ $2.950$ None 50.4.e.a \(0\) \(0\) \(-30\) \(0\) $\mathrm{SU}(2)[C_{10}]$
50.5.c.a 50.c 5.c $2$ $5.168$ \(\Q(\sqrt{-1}) \) None 10.5.c.b \(-4\) \(-2\) \(0\) \(38\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2-2i)q^{2}+(-1+i)q^{3}+8iq^{4}+\cdots\)
50.5.c.b 50.c 5.c $2$ $5.168$ \(\Q(\sqrt{-1}) \) None 10.5.c.a \(4\) \(-18\) \(0\) \(-58\) $\mathrm{SU}(2)[C_{4}]$ \(q+(2+2i)q^{2}+(-9+9i)q^{3}+8iq^{4}+\cdots\)
50.5.c.c 50.c 5.c $4$ $5.168$ \(\Q(i, \sqrt{6})\) None 50.5.c.c \(-8\) \(-24\) \(0\) \(-144\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-2+2\beta _{2})q^{2}+(-6+\beta _{1}-6\beta _{2}+\cdots)q^{3}+\cdots\)
50.5.c.d 50.c 5.c $4$ $5.168$ \(\Q(i, \sqrt{6})\) None 50.5.c.c \(8\) \(24\) \(0\) \(144\) $\mathrm{SU}(2)[C_{4}]$ \(q+(2-2\beta _{2})q^{2}+(6+\beta _{1}+6\beta _{2})q^{3}+\cdots\)
50.5.f.a 50.f 25.f $40$ $5.168$ None 50.5.f.a \(-20\) \(-10\) \(30\) \(-10\) $\mathrm{SU}(2)[C_{20}]$
50.5.f.b 50.f 25.f $40$ $5.168$ None 50.5.f.b \(20\) \(-10\) \(30\) \(-10\) $\mathrm{SU}(2)[C_{20}]$
50.6.a.a 50.a 1.a $1$ $8.019$ \(\Q\) None 50.6.a.a \(-4\) \(-11\) \(0\) \(-142\) $-$ $\mathrm{SU}(2)$ \(q-4q^{2}-11q^{3}+2^{4}q^{4}+44q^{6}-142q^{7}+\cdots\)
50.6.a.b 50.a 1.a $1$ $8.019$ \(\Q\) None 10.6.a.c \(-4\) \(-6\) \(0\) \(118\) $+$ $\mathrm{SU}(2)$ \(q-4q^{2}-6q^{3}+2^{4}q^{4}+24q^{6}+118q^{7}+\cdots\)
50.6.a.c 50.a 1.a $1$ $8.019$ \(\Q\) None 10.6.b.a \(-4\) \(14\) \(0\) \(158\) $-$ $\mathrm{SU}(2)$ \(q-4q^{2}+14q^{3}+2^{4}q^{4}-56q^{6}+158q^{7}+\cdots\)
50.6.a.d 50.a 1.a $1$ $8.019$ \(\Q\) None 10.6.a.b \(4\) \(-24\) \(0\) \(172\) $-$ $\mathrm{SU}(2)$ \(q+4q^{2}-24q^{3}+2^{4}q^{4}-96q^{6}+172q^{7}+\cdots\)
50.6.a.e 50.a 1.a $1$ $8.019$ \(\Q\) None 10.6.b.a \(4\) \(-14\) \(0\) \(-158\) $+$ $\mathrm{SU}(2)$ \(q+4q^{2}-14q^{3}+2^{4}q^{4}-56q^{6}-158q^{7}+\cdots\)
50.6.a.f 50.a 1.a $1$ $8.019$ \(\Q\) None 50.6.a.a \(4\) \(11\) \(0\) \(142\) $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+11q^{3}+2^{4}q^{4}+44q^{6}+142q^{7}+\cdots\)
50.6.a.g 50.a 1.a $1$ $8.019$ \(\Q\) None 10.6.a.a \(4\) \(26\) \(0\) \(22\) $-$ $\mathrm{SU}(2)$ \(q+4q^{2}+26q^{3}+2^{4}q^{4}+104q^{6}+\cdots\)
50.6.b.a 50.b 5.b $2$ $8.019$ \(\Q(\sqrt{-1}) \) None 10.6.a.b \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}+12iq^{3}-2^{4}q^{4}-96q^{6}+\cdots\)
50.6.b.b 50.b 5.b $2$ $8.019$ \(\Q(\sqrt{-1}) \) None 10.6.a.c \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}-3iq^{3}-2^{4}q^{4}+24q^{6}+\cdots\)
50.6.b.c 50.b 5.b $2$ $8.019$ \(\Q(\sqrt{-1}) \) None 50.6.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+4iq^{2}-11iq^{3}-2^{4}q^{4}+44q^{6}+\cdots\)
50.6.b.d 50.b 5.b $2$ $8.019$ \(\Q(\sqrt{-1}) \) None 10.6.a.a \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+2iq^{2}-13iq^{3}-2^{4}q^{4}+104q^{6}+\cdots\)
50.6.d.a 50.d 25.d $24$ $8.019$ None 50.6.d.a \(-24\) \(-11\) \(-120\) \(548\) $\mathrm{SU}(2)[C_{5}]$
50.6.d.b 50.d 25.d $28$ $8.019$ None 50.6.d.b \(28\) \(7\) \(-145\) \(-236\) $\mathrm{SU}(2)[C_{5}]$
50.6.e.a 50.e 25.e $48$ $8.019$ None 50.6.e.a \(0\) \(0\) \(180\) \(0\) $\mathrm{SU}(2)[C_{10}]$
50.7.c.a 50.c 5.c $2$ $11.503$ \(\Q(\sqrt{-1}) \) None 50.7.c.a \(-8\) \(-6\) \(0\) \(-234\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-4-4i)q^{2}+(-3+3i)q^{3}+2^{5}iq^{4}+\cdots\)
50.7.c.b 50.c 5.c $2$ $11.503$ \(\Q(\sqrt{-1}) \) None 50.7.c.a \(8\) \(6\) \(0\) \(234\) $\mathrm{SU}(2)[C_{4}]$ \(q+(4+4i)q^{2}+(3-3i)q^{3}+2^{5}iq^{4}+\cdots\)
50.7.c.c 50.c 5.c $2$ $11.503$ \(\Q(\sqrt{-1}) \) None 10.7.c.a \(8\) \(46\) \(0\) \(494\) $\mathrm{SU}(2)[C_{4}]$ \(q+(4+4i)q^{2}+(23-23i)q^{3}+2^{5}iq^{4}+\cdots\)
50.7.c.d 50.c 5.c $4$ $11.503$ \(\Q(i, \sqrt{129})\) None 10.7.c.b \(-16\) \(18\) \(0\) \(202\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-4+4\beta _{1})q^{2}+(5+5\beta _{1}-\beta _{2})q^{3}+\cdots\)
50.7.c.e 50.c 5.c $4$ $11.503$ \(\Q(i, \sqrt{6})\) None 50.7.c.e \(-16\) \(48\) \(0\) \(672\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-4-4\beta _{2})q^{2}+(12-12\beta _{2}-\beta _{3})q^{3}+\cdots\)
50.7.c.f 50.c 5.c $4$ $11.503$ \(\Q(i, \sqrt{6})\) None 50.7.c.e \(16\) \(-48\) \(0\) \(-672\) $\mathrm{SU}(2)[C_{4}]$ \(q+(4+4\beta _{2})q^{2}+(-12+12\beta _{2}-\beta _{3})q^{3}+\cdots\)
50.7.f.a 50.f 25.f $56$ $11.503$ None 50.7.f.a \(56\) \(32\) \(150\) \(348\) $\mathrm{SU}(2)[C_{20}]$
50.7.f.b 50.f 25.f $64$ $11.503$ None 50.7.f.b \(-64\) \(32\) \(-330\) \(348\) $\mathrm{SU}(2)[C_{20}]$
50.8.a.a 50.a 1.a $1$ $15.619$ \(\Q\) None 50.8.a.a \(-8\) \(-43\) \(0\) \(-974\) $+$ $\mathrm{SU}(2)$ \(q-8q^{2}-43q^{3}+2^{6}q^{4}+344q^{6}+\cdots\)
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