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Label Char Prim Dim $A$ Field CM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
69.2.a.a 69.a 1.a $1$ $0.551$ \(\Q\) None \(1\) \(1\) \(0\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q+q^{2}+q^{3}-q^{4}+q^{6}-2q^{7}-3q^{8}+\cdots\)
69.2.a.b 69.a 1.a $2$ $0.551$ \(\Q(\sqrt{5}) \) None \(0\) \(-2\) \(-2\) \(2\) $-$ $\mathrm{SU}(2)$ \(q-\beta q^{2}-q^{3}+3q^{4}+(-1+\beta )q^{5}+\cdots\)
69.2.c.a 69.c 69.c $6$ $0.551$ 6.0.8869743.1 \(\Q(\sqrt{-23}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+\beta _{2}q^{2}-\beta _{1}q^{3}+(-2+\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
69.2.e.a 69.e 23.c $10$ $0.551$ \(\Q(\zeta_{22})\) None \(-4\) \(1\) \(5\) \(-8\) $\mathrm{SU}(2)[C_{11}]$ \(q+(-1+\zeta_{22}-\zeta_{22}^{2}-\zeta_{22}^{4}-\zeta_{22}^{6}+\cdots)q^{2}+\cdots\)
69.2.e.b 69.e 23.c $10$ $0.551$ \(\Q(\zeta_{22})\) None \(4\) \(1\) \(-3\) \(6\) $\mathrm{SU}(2)[C_{11}]$ \(q+(-\zeta_{22}^{4}+\zeta_{22}^{5}-\zeta_{22}^{8}+\zeta_{22}^{9})q^{2}+\cdots\)
69.2.e.c 69.e 23.c $20$ $0.551$ \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None \(-4\) \(-2\) \(-6\) \(-6\) $\mathrm{SU}(2)[C_{11}]$ \(q+(\beta _{1}+\beta _{3}-\beta _{4}+\beta _{5}+2\beta _{6}+2\beta _{7}+\cdots)q^{2}+\cdots\)
69.2.g.a 69.g 69.g $60$ $0.551$ None \(0\) \(-11\) \(0\) \(-22\) $\mathrm{SU}(2)[C_{22}]$
69.3.b.a 69.b 3.b $14$ $1.880$ \(\mathbb{Q}[x]/(x^{14} + \cdots)\) None \(0\) \(-4\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{2}+\beta _{8}q^{3}+(-2+\beta _{2})q^{4}+\beta _{3}q^{5}+\cdots\)
69.3.d.a 69.d 23.b $8$ $1.880$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(-4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(-\beta _{2}+\beta _{3})q^{2}+\beta _{3}q^{3}+(-1+\beta _{2}+\cdots)q^{4}+\cdots\)
69.3.f.a 69.f 23.d $80$ $1.880$ None \(4\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{22}]$
69.3.h.a 69.h 69.h $140$ $1.880$ None \(0\) \(-7\) \(0\) \(-18\) $\mathrm{SU}(2)[C_{22}]$
69.4.a.a 69.a 1.a $2$ $4.071$ \(\Q(\sqrt{5}) \) None \(-4\) \(6\) \(-26\) \(-10\) $-$ $\mathrm{SU}(2)$ \(q+(-2-\beta )q^{2}+3q^{3}+(1+4\beta )q^{4}+\cdots\)
69.4.a.b 69.a 1.a $2$ $4.071$ \(\Q(\sqrt{2}) \) None \(-2\) \(-6\) \(8\) \(-28\) $-$ $\mathrm{SU}(2)$ \(q+(-1+\beta )q^{2}-3q^{3}+(1-2\beta )q^{4}+\cdots\)
69.4.a.c 69.a 1.a $4$ $4.071$ 4.4.1140200.1 None \(-2\) \(-12\) \(-2\) \(32\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3q^{3}+(6-\beta _{1}-2\beta _{2})q^{4}+\cdots\)
69.4.a.d 69.a 1.a $4$ $4.071$ 4.4.2009704.1 None \(4\) \(12\) \(4\) \(-14\) $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}+3q^{3}+(7-2\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
69.4.c.a 69.c 69.c $2$ $4.071$ \(\Q(\sqrt{-23}) \) \(\Q(\sqrt{-23}) \) \(0\) \(4\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-\beta q^{2}+(2-\beta )q^{3}-15q^{4}+(-23+\cdots)q^{6}+\cdots\)
69.4.c.b 69.c 69.c $4$ $4.071$ \(\Q(\sqrt{-3}, \sqrt{-23})\) \(\Q(\sqrt{-23}) \) \(0\) \(-4\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(-1-\beta _{2}-\beta _{3})q^{2}+(-1+2\beta _{1}+\cdots)q^{3}+\cdots\)
69.4.c.c 69.c 69.c $16$ $4.071$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None \(0\) \(-6\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{11}q^{2}-\beta _{10}q^{3}+(-2+\beta _{1})q^{4}+\cdots\)
69.4.e.a 69.e 23.c $60$ $4.071$ None \(0\) \(-18\) \(22\) \(24\) $\mathrm{SU}(2)[C_{11}]$
69.4.e.b 69.e 23.c $60$ $4.071$ None \(4\) \(18\) \(-6\) \(-4\) $\mathrm{SU}(2)[C_{11}]$
69.4.g.a 69.g 69.g $220$ $4.071$ None \(0\) \(-5\) \(0\) \(-22\) $\mathrm{SU}(2)[C_{22}]$
69.5.b.a 69.b 3.b $30$ $7.133$ None \(0\) \(8\) \(0\) \(-4\) $\mathrm{SU}(2)[C_{2}]$
69.5.d.a 69.d 23.b $16$ $7.133$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(12\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+(1+\beta _{2})q^{2}+\beta _{3}q^{3}+(9+\beta _{2}+\beta _{5}+\cdots)q^{4}+\cdots\)
69.5.f.a 69.f 23.d $160$ $7.133$ None \(-12\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{22}]$
69.5.h.a 69.h 69.h $300$ $7.133$ None \(0\) \(-19\) \(0\) \(-18\) $\mathrm{SU}(2)[C_{22}]$
69.6.a.a 69.a 1.a $2$ $11.066$ \(\Q(\sqrt{29}) \) None \(0\) \(-18\) \(94\) \(-118\) $-$ $\mathrm{SU}(2)$ \(q-2\beta q^{2}-9q^{3}+84q^{4}+(47-\beta )q^{5}+\cdots\)
69.6.a.b 69.a 1.a $3$ $11.066$ 3.3.5333.1 None \(-8\) \(27\) \(-56\) \(-114\) $+$ $\mathrm{SU}(2)$ \(q+(-3+\beta _{2})q^{2}+9q^{3}+(9-4\beta _{1}-9\beta _{2})q^{4}+\cdots\)
69.6.a.c 69.a 1.a $4$ $11.066$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(4\) \(-36\) \(-122\) \(62\) $-$ $\mathrm{SU}(2)$ \(q+(1+\beta _{1})q^{2}-9q^{3}+(-12+4\beta _{1}+\cdots)q^{4}+\cdots\)
69.6.a.d 69.a 1.a $4$ $11.066$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(4\) \(-36\) \(22\) \(-62\) $+$ $\mathrm{SU}(2)$ \(q+(1-\beta _{1})q^{2}-9q^{3}+(7-\beta _{1}+\beta _{3})q^{4}+\cdots\)
69.6.a.e 69.a 1.a $5$ $11.066$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(8\) \(45\) \(94\) \(272\) $-$ $\mathrm{SU}(2)$ \(q+(2-\beta _{2})q^{2}+9q^{3}+(24-\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
69.6.c.a 69.c 69.c $6$ $11.066$ 6.0.8869743.1 \(\Q(\sqrt{-23}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(-\beta _{1}-3\beta _{2}-2\beta _{3}+\beta _{4}+\beta _{5})q^{2}+\cdots\)
69.6.c.b 69.c 69.c $32$ $11.066$ None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
69.6.e.a 69.e 23.c $100$ $11.066$ None \(-8\) \(90\) \(6\) \(118\) $\mathrm{SU}(2)[C_{11}]$
69.6.e.b 69.e 23.c $100$ $11.066$ None \(12\) \(-90\) \(-50\) \(-78\) $\mathrm{SU}(2)[C_{11}]$
69.6.g.a 69.g 69.g $380$ $11.066$ None \(0\) \(-11\) \(0\) \(-22\) $\mathrm{SU}(2)[C_{22}]$
69.7.b.a 69.b 3.b $44$ $15.874$ None \(0\) \(20\) \(0\) \(568\) $\mathrm{SU}(2)[C_{2}]$
69.7.d.a 69.d 23.b $24$ $15.874$ None \(-20\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
69.7.f.a 69.f 23.d $240$ $15.874$ None \(20\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{22}]$
69.7.h.a 69.h 69.h $460$ $15.874$ None \(0\) \(23\) \(0\) \(-18\) $\mathrm{SU}(2)[C_{22}]$
69.8.a.a 69.a 1.a $5$ $21.555$ \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(0\) \(-135\) \(-266\) \(-496\) $-$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3^{3}q^{3}+(54+2\beta _{1}+\beta _{3}+\cdots)q^{4}+\cdots\)
69.8.a.b 69.a 1.a $6$ $21.555$ \(\mathbb{Q}[x]/(x^{6} - \cdots)\) None \(-8\) \(162\) \(-372\) \(-1104\) $-$ $\mathrm{SU}(2)$ \(q+(-1-\beta _{1})q^{2}+3^{3}q^{3}+(29+3\beta _{1}+\cdots)q^{4}+\cdots\)
69.8.a.c 69.a 1.a $7$ $21.555$ \(\mathbb{Q}[x]/(x^{7} - \cdots)\) None \(0\) \(-189\) \(-516\) \(1018\) $+$ $\mathrm{SU}(2)$ \(q+\beta _{1}q^{2}-3^{3}q^{3}+(93+\beta _{1}+\beta _{2})q^{4}+\cdots\)
69.8.a.d 69.a 1.a $8$ $21.555$ \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None \(24\) \(216\) \(378\) \(126\) $+$ $\mathrm{SU}(2)$ \(q+(3-\beta _{1})q^{2}+3^{3}q^{3}+(70-4\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
69.8.c.a 69.c 69.c $6$ $21.555$ 6.0.8869743.1 \(\Q(\sqrt{-23}) \) \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+(-2\beta _{1}-6\beta _{2}+3\beta _{3}+2\beta _{4}+2\beta _{5})q^{2}+\cdots\)
69.8.c.b 69.c 69.c $48$ $21.555$ None \(0\) \(24\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
69.8.e.a 69.e 23.c $140$ $21.555$ None \(-16\) \(-378\) \(-6\) \(978\) $\mathrm{SU}(2)[C_{11}]$
69.8.e.b 69.e 23.c $140$ $21.555$ None \(-12\) \(378\) \(2\) \(-394\) $\mathrm{SU}(2)[C_{11}]$
69.8.g.a 69.g 69.g $540$ $21.555$ None \(0\) \(-35\) \(0\) \(-22\) $\mathrm{SU}(2)[C_{22}]$
69.9.b.a 69.b 3.b $58$ $28.109$ None \(0\) \(-112\) \(0\) \(6996\) $\mathrm{SU}(2)[C_{2}]$
69.9.d.a 69.d 23.b $32$ $28.109$ None \(12\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
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