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Label Char Prim Dim $A$ Field CM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
19.2.a.a 19.a 1.a $1$ $0.152$ \(\Q\) None \(0\) \(-2\) \(3\) \(-1\) $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-2q^{4}+3q^{5}-q^{7}+q^{9}+3q^{11}+\cdots\)
19.2.e.a 19.e 19.e $6$ $0.152$ \(\Q(\zeta_{18})\) None \(-6\) \(-3\) \(-6\) \(0\) $\mathrm{SU}(2)[C_{9}]$ \(q+(-1+\zeta_{18}-\zeta_{18}^{2})q^{2}+(-1+\zeta_{18}^{2}+\cdots)q^{3}+\cdots\)
19.3.b.a 19.b 19.b $1$ $0.518$ \(\Q\) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(-9\) \(-5\) $\mathrm{U}(1)[D_{2}]$ \(q+4q^{4}-9q^{5}-5q^{7}+9q^{9}+3q^{11}+\cdots\)
19.3.b.b 19.b 19.b $2$ $0.518$ \(\Q(\sqrt{-13}) \) None \(0\) \(0\) \(8\) \(-10\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{2}-\beta q^{3}-9q^{4}+4q^{5}+13q^{6}+\cdots\)
19.3.d.a 19.d 19.d $6$ $0.518$ 6.0.6967728.1 None \(-3\) \(-9\) \(-2\) \(0\) $\mathrm{SU}(2)[C_{6}]$ \(q+(-1-\beta _{5})q^{2}+(\beta _{1}+2\beta _{2}+\beta _{3}+\beta _{5})q^{3}+\cdots\)
19.3.f.a 19.f 19.f $12$ $0.518$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(-6\) \(0\) \(-6\) \(6\) $\mathrm{SU}(2)[C_{18}]$ \(q+\beta _{10}q^{2}+(\beta _{1}-\beta _{4}-\beta _{5}+\beta _{6}-\beta _{7}+\cdots)q^{3}+\cdots\)
19.4.a.a 19.a 1.a $1$ $1.121$ \(\Q\) None \(-3\) \(-5\) \(-12\) \(11\) $-$ $\mathrm{SU}(2)$ \(q-3q^{2}-5q^{3}+q^{4}-12q^{5}+15q^{6}+\cdots\)
19.4.a.b 19.a 1.a $3$ $1.121$ 3.3.3144.1 None \(3\) \(1\) \(14\) \(-35\) $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1}-\beta _{2})q^{2}+(-\beta _{1}+2\beta _{2})q^{3}+\cdots\)
19.4.c.a 19.c 19.c $4$ $1.121$ \(\Q(\sqrt{-3}, \sqrt{55})\) None \(-2\) \(0\) \(14\) \(-28\) $\mathrm{SU}(2)[C_{3}]$ \(q+\beta _{2}q^{2}+(\beta _{1}+\beta _{3})q^{3}+(7+7\beta _{2})q^{4}+\cdots\)
19.4.c.b 19.c 19.c $4$ $1.121$ \(\Q(\sqrt{-3}, \sqrt{73})\) None \(-1\) \(-2\) \(-19\) \(40\) $\mathrm{SU}(2)[C_{3}]$ \(q-\beta _{1}q^{2}-\beta _{2}q^{3}+(-11+\beta _{1}+10\beta _{2}+\cdots)q^{4}+\cdots\)
19.4.e.a 19.e 19.e $24$ $1.121$ None \(-6\) \(-3\) \(-6\) \(3\) $\mathrm{SU}(2)[C_{9}]$
19.5.b.a 19.b 19.b $1$ $1.964$ \(\Q\) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(31\) \(-73\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{4}q^{4}+31q^{5}-73q^{7}+3^{4}q^{9}+\cdots\)
19.5.b.b 19.b 19.b $4$ $1.964$ 4.0.12107488.1 None \(0\) \(0\) \(-42\) \(136\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}-\beta _{3}q^{3}+(-3+3\beta _{2})q^{4}+\cdots\)
19.5.d.a 19.d 19.d $10$ $1.964$ \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(-3\) \(9\) \(8\) \(-24\) $\mathrm{SU}(2)[C_{6}]$ \(q-\beta _{1}q^{2}+(\beta _{3}+\beta _{5})q^{3}+(6\beta _{3}-\beta _{4}+\cdots)q^{4}+\cdots\)
19.5.f.a 19.f 19.f $36$ $1.964$ None \(-6\) \(-18\) \(-6\) \(-48\) $\mathrm{SU}(2)[C_{18}]$
19.6.a.a 19.a 1.a $1$ $3.047$ \(\Q\) None \(-6\) \(4\) \(54\) \(248\) $-$ $\mathrm{SU}(2)$ \(q-6q^{2}+4q^{3}+4q^{4}+54q^{5}-24q^{6}+\cdots\)
19.6.a.b 19.a 1.a $1$ $3.047$ \(\Q\) None \(-2\) \(-1\) \(-24\) \(-167\) $+$ $\mathrm{SU}(2)$ \(q-2q^{2}-q^{3}-28q^{4}-24q^{5}+2q^{6}+\cdots\)
19.6.a.c 19.a 1.a $2$ $3.047$ \(\Q(\sqrt{177}) \) None \(-7\) \(-7\) \(-133\) \(72\) $+$ $\mathrm{SU}(2)$ \(q+(-3-\beta )q^{2}+(-5+3\beta )q^{3}+(21+\cdots)q^{4}+\cdots\)
19.6.a.d 19.a 1.a $4$ $3.047$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(9\) \(6\) \(90\) \(-190\) $-$ $\mathrm{SU}(2)$ \(q+(3-\beta _{1}+\beta _{2})q^{2}+(3+3\beta _{2})q^{3}+(23+\cdots)q^{4}+\cdots\)
19.6.c.a 19.c 19.c $16$ $3.047$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(3\) \(28\) \(10\) \(208\) $\mathrm{SU}(2)[C_{3}]$ \(q+(-\beta _{1}-\beta _{3})q^{2}+(3+3\beta _{2}+\beta _{5})q^{3}+\cdots\)
19.6.e.a 19.e 19.e $42$ $3.047$ None \(-6\) \(-39\) \(-6\) \(-180\) $\mathrm{SU}(2)[C_{9}]$
19.7.b.a 19.b 19.b $1$ $4.371$ \(\Q\) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(-54\) \(610\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{6}q^{4}-54q^{5}+610q^{7}+3^{6}q^{9}+\cdots\)
19.7.b.b 19.b 19.b $8$ $4.371$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(108\) \(-140\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}-\beta _{2}q^{3}+(-57+\beta _{3}-\beta _{4}+\cdots)q^{4}+\cdots\)
19.7.d.a 19.d 19.d $18$ $4.371$ \(\mathbb{Q}[x]/(x^{18} + \cdots)\) None \(-3\) \(27\) \(-57\) \(-260\) $\mathrm{SU}(2)[C_{6}]$ \(q+(\beta _{1}-\beta _{3})q^{2}+(2+\beta _{2}-\beta _{7})q^{3}+(21+\cdots)q^{4}+\cdots\)
19.7.f.a 19.f 19.f $54$ $4.371$ None \(-6\) \(-36\) \(-6\) \(-219\) $\mathrm{SU}(2)[C_{18}]$
19.8.a.a 19.a 1.a $4$ $5.935$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-9\) \(-14\) \(-222\) \(-1246\) $-$ $\mathrm{SU}(2)$ \(q+(-2+\beta _{1})q^{2}+(-4-\beta _{1}-\beta _{2})q^{3}+\cdots\)
19.8.a.b 19.a 1.a $6$ $5.935$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(15\) \(40\) \(219\) \(2105\) $+$ $\mathrm{SU}(2)$ \(q+(2+\beta _{1})q^{2}+(6+\beta _{1}+\beta _{4})q^{3}+(57+\cdots)q^{4}+\cdots\)
19.8.c.a 19.c 19.c $20$ $5.935$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(-9\) \(-68\) \(0\) \(-1456\) $\mathrm{SU}(2)[C_{3}]$ \(q+(\beta _{1}+\beta _{2})q^{2}+(7\beta _{2}+\beta _{7})q^{3}+(-48+\cdots)q^{4}+\cdots\)
19.8.e.a 19.e 19.e $66$ $5.935$ None \(-6\) \(33\) \(-6\) \(588\) $\mathrm{SU}(2)[C_{9}]$
19.9.b.a 19.b 19.b $1$ $7.740$ \(\Q\) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(-289\) \(527\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{8}q^{4}-17^{2}q^{5}+527q^{7}+3^{8}q^{9}+\cdots\)
19.9.b.b 19.b 19.b $12$ $7.740$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None \(0\) \(0\) \(8\) \(3686\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}-\beta _{3}q^{3}+(-131+\beta _{2})q^{4}+\cdots\)
19.9.d.a 19.d 19.d $26$ $7.740$ None \(-3\) \(-171\) \(278\) \(-7504\) $\mathrm{SU}(2)[C_{6}]$
19.9.f.a 19.f 19.f $72$ $7.740$ None \(-6\) \(162\) \(-6\) \(3282\) $\mathrm{SU}(2)[C_{18}]$
19.10.a.a 19.a 1.a $6$ $9.786$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-33\) \(-155\) \(-3612\) \(4085\) $+$ $\mathrm{SU}(2)$ \(q+(-6+\beta _{1})q^{2}+(-26+\beta _{1}+\beta _{2}+\cdots)q^{3}+\cdots\)
19.10.a.b 19.a 1.a $8$ $9.786$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(15\) \(7\) \(3894\) \(-7133\) $-$ $\mathrm{SU}(2)$ \(q+(2-\beta _{1})q^{2}+(1+\beta _{1}-\beta _{3})q^{3}+(331+\cdots)q^{4}+\cdots\)
19.10.c.a 19.c 19.c $28$ $9.786$ None \(15\) \(-74\) \(-285\) \(-2676\) $\mathrm{SU}(2)[C_{3}]$
19.10.e.a 19.e 19.e $84$ $9.786$ None \(-6\) \(213\) \(-6\) \(5715\) $\mathrm{SU}(2)[C_{9}]$
19.11.b.a 19.b 19.b $1$ $12.072$ \(\Q\) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(3951\) \(-32525\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{10}q^{4}+3951q^{5}-32525q^{7}+\cdots\)
19.11.b.b 19.b 19.b $14$ $12.072$ \(\mathbb{Q}[x]/(x^{14} + \cdots)\) None \(0\) \(0\) \(-2842\) \(-3840\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}-\beta _{3}q^{3}+(-599+\beta _{2})q^{4}+\cdots\)
19.11.d.a 19.d 19.d $30$ $12.072$ None \(-3\) \(63\) \(-1112\) \(42200\) $\mathrm{SU}(2)[C_{6}]$
19.11.f.a 19.f 19.f $96$ $12.072$ None \(-6\) \(-72\) \(-6\) \(-5844\) $\mathrm{SU}(2)[C_{18}]$
19.12.a.a 19.a 1.a $7$ $14.599$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(-9\) \(10\) \(-14307\) \(-2209\) $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+(1+3\beta _{1}-\beta _{4})q^{3}+\cdots\)
19.12.a.b 19.a 1.a $9$ $14.599$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(87\) \(496\) \(2114\) \(-19080\) $+$ $\mathrm{SU}(2)$ \(q+(10-\beta _{1})q^{2}+(56-3\beta _{1}+\beta _{3})q^{3}+\cdots\)
19.12.c.a 19.c 19.c $36$ $14.599$ None \(-33\) \(-224\) \(2530\) \(112320\) $\mathrm{SU}(2)[C_{3}]$
19.12.e.a 19.e 19.e $102$ $14.599$ None \(-6\) \(-795\) \(-6\) \(-57552\) $\mathrm{SU}(2)[C_{9}]$
19.13.b.a 19.b 19.b $1$ $17.366$ \(\Q\) \(\Q(\sqrt{-19}) \) \(0\) \(0\) \(-28334\) \(136802\) $\mathrm{U}(1)[D_{2}]$ \(q+2^{12}q^{4}-28334q^{5}+136802q^{7}+\cdots\)
19.13.b.b 19.b 19.b $18$ $17.366$ \(\mathbb{Q}[x]/(x^{18} + \cdots)\) None \(0\) \(0\) \(25308\) \(-207284\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}-\beta _{3}q^{3}+(-2224+\beta _{2})q^{4}+\cdots\)
19.13.d.a 19.d 19.d $38$ $17.366$ None \(-3\) \(1377\) \(3023\) \(202236\) $\mathrm{SU}(2)[C_{6}]$
19.13.f.a 19.f 19.f $114$ $17.366$ None \(-6\) \(-1386\) \(-6\) \(-131763\) $\mathrm{SU}(2)[C_{18}]$
19.14.a.a 19.a 1.a $9$ $20.374$ \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None \(-129\) \(-1328\) \(-61942\) \(-435960\) $+$ $\mathrm{SU}(2)$ \(q+(-14-\beta _{1})q^{2}+(-148+\beta _{1}-\beta _{2}+\cdots)q^{3}+\cdots\)
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