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Label Char Prim Dim $A$ Field CM RM Traces Fricke sign Sato-Tate $q$-expansion
$a_{2}$ $a_{3}$ $a_{5}$ $a_{7}$
68.1.d.a 68.d 68.d $1$ $0.034$ \(\Q\) \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-17}) \) \(\Q(\sqrt{17}) \) \(-1\) \(0\) \(0\) \(0\) \(q-q^{2}+q^{4}-q^{8}-q^{9}-2q^{13}+q^{16}+\cdots\)
68.1.f.a 68.f 68.f $2$ $0.034$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-2\) \(0\) \(q+iq^{2}-q^{4}+(-1-i)q^{5}-iq^{8}+\cdots\)
68.2.a.a 68.a 1.a $2$ $0.543$ \(\Q(\sqrt{3}) \) None None \(0\) \(2\) \(0\) \(-2\) $-$ $\mathrm{SU}(2)$ \(q+(1+\beta )q^{3}-2\beta q^{5}+(-1-\beta )q^{7}+\cdots\)
68.2.b.a 68.b 17.b $2$ $0.543$ \(\Q(\sqrt{-2}) \) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta q^{3}+2\beta q^{5}-3\beta q^{7}+q^{9}-\beta q^{11}+\cdots\)
68.2.e.a 68.e 17.c $4$ $0.543$ \(\Q(i, \sqrt{13})\) None None \(0\) \(-2\) \(-4\) \(2\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-1+\beta _{2})q^{3}+(-1-\beta _{1})q^{5}+(1+\cdots)q^{7}+\cdots\)
68.2.h.a 68.h 17.d $4$ $0.543$ \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(4\) \(0\) $\mathrm{SU}(2)[C_{8}]$ \(q+(\zeta_{8}^{2}-\zeta_{8}^{3})q^{3}+(1-\zeta_{8})q^{5}+(\zeta_{8}+\cdots)q^{7}+\cdots\)
68.2.i.a 68.i 68.i $8$ $0.543$ \(\Q(\zeta_{16})\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{16}]$ \(q+(-\zeta_{16}+\zeta_{16}^{5})q^{2}-2\zeta_{16}^{6}q^{4}+\cdots\)
68.2.i.b 68.i 68.i $48$ $0.543$ None None \(-8\) \(0\) \(-16\) \(0\) $\mathrm{SU}(2)[C_{16}]$
68.3.c.a 68.c 4.b $16$ $1.853$ \(\mathbb{Q}[x]/(x^{16} - \cdots)\) None None \(-2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{3}q^{2}-\beta _{12}q^{3}+\beta _{10}q^{4}-\beta _{2}q^{5}+\cdots\)
68.3.d.a 68.d 68.d $2$ $1.853$ \(\Q(\sqrt{34}) \) \(\Q(\sqrt{-17}) \) None \(-4\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-2q^{2}+\beta q^{3}+4q^{4}-2\beta q^{6}-\beta q^{7}+\cdots\)
68.3.d.b 68.d 68.d $2$ $1.853$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-17}) \) None \(4\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+2q^{2}+\beta q^{3}+4q^{4}+2\beta q^{6}-9\beta q^{7}+\cdots\)
68.3.d.c 68.d 68.d $12$ $1.853$ \(\mathbb{Q}[x]/(x^{12} + \cdots)\) None None \(-2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q-\beta _{1}q^{2}+\beta _{7}q^{3}+(-1-\beta _{3})q^{4}-\beta _{6}q^{5}+\cdots\)
68.3.f.a 68.f 68.f $32$ $1.853$ None None \(0\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{4}]$
68.3.g.a 68.g 68.g $4$ $1.853$ \(\Q(\zeta_{8})\) None None \(0\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{8}]$ \(q-2\zeta_{8}^{2}q^{2}+(-2\zeta_{8}^{2}-2\zeta_{8}^{3})q^{3}+\cdots\)
68.3.g.b 68.g 68.g $4$ $1.853$ \(\Q(\zeta_{8})\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-16\) \(0\) $\mathrm{U}(1)[D_{8}]$ \(q-2\zeta_{8}^{3}q^{2}-4\zeta_{8}^{2}q^{4}+(-4+4\zeta_{8}+\cdots)q^{5}+\cdots\)
68.3.g.c 68.g 68.g $4$ $1.853$ \(\Q(\zeta_{8})\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(16\) \(0\) $\mathrm{U}(1)[D_{8}]$ \(q+2\zeta_{8}^{3}q^{2}-4\zeta_{8}^{2}q^{4}+(4-4\zeta_{8}-3\zeta_{8}^{2}+\cdots)q^{5}+\cdots\)
68.3.g.d 68.g 68.g $4$ $1.853$ \(\Q(\zeta_{8})\) None None \(8\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{8}]$ \(q+2q^{2}+(2\zeta_{8}^{2}+2\zeta_{8}^{3})q^{3}+4q^{4}+\cdots\)
68.3.g.e 68.g 68.g $48$ $1.853$ None None \(-12\) \(0\) \(8\) \(0\) $\mathrm{SU}(2)[C_{8}]$
68.3.j.a 68.j 17.e $24$ $1.853$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{16}]$
68.4.a.a 68.a 1.a $1$ $4.012$ \(\Q\) None None \(0\) \(-2\) \(-8\) \(-12\) $-$ $\mathrm{SU}(2)$ \(q-2q^{3}-8q^{5}-12q^{7}-23q^{9}-10q^{11}+\cdots\)
68.4.a.b 68.a 1.a $3$ $4.012$ 3.3.1524.1 None None \(0\) \(4\) \(26\) \(8\) $+$ $\mathrm{SU}(2)$ \(q+(1+\beta _{2})q^{3}+(9-\beta _{1})q^{5}+(2+3\beta _{1}+\cdots)q^{7}+\cdots\)
68.4.b.a 68.b 17.b $4$ $4.012$ 4.0.1499912.1 None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}-\beta _{2}q^{5}+(\beta _{1}-\beta _{2})q^{7}+(-15+\cdots)q^{9}+\cdots\)
68.4.e.a 68.e 17.c $2$ $4.012$ \(\Q(\sqrt{-1}) \) None None \(0\) \(-8\) \(-2\) \(-40\) $\mathrm{SU}(2)[C_{4}]$ \(q+(-4-4i)q^{3}+(-1-i)q^{5}+(-20+\cdots)q^{7}+\cdots\)
68.4.e.b 68.e 17.c $6$ $4.012$ 6.0.27793984.1 None None \(0\) \(6\) \(0\) \(44\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+\beta _{2}+\beta _{3})q^{3}+(1-\beta _{2}+\beta _{3}-3\beta _{5})q^{5}+\cdots\)
68.4.h.a 68.h 17.d $20$ $4.012$ \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None None \(0\) \(0\) \(-16\) \(0\) $\mathrm{SU}(2)[C_{8}]$ \(q+\beta _{6}q^{3}+(-1-\beta _{8}-\beta _{9}+\beta _{10}+\beta _{14}+\cdots)q^{5}+\cdots\)
68.4.i.a 68.i 68.i $8$ $4.012$ \(\Q(\zeta_{16})\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{16}]$ \(q+(2\zeta_{16}-2\zeta_{16}^{5})q^{2}-8\zeta_{16}^{6}q^{4}+\cdots\)
68.4.i.b 68.i 68.i $192$ $4.012$ None None \(-8\) \(0\) \(-16\) \(0\) $\mathrm{SU}(2)[C_{16}]$
68.5.c.a 68.c 4.b $32$ $7.029$ None None \(6\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
68.5.d.a 68.d 68.d $1$ $7.029$ \(\Q\) \(\Q(\sqrt{-17}) \) None \(4\) \(-16\) \(0\) \(64\) $\mathrm{U}(1)[D_{2}]$ \(q+4q^{2}-2^{4}q^{3}+2^{4}q^{4}-2^{6}q^{6}+2^{6}q^{7}+\cdots\)
68.5.d.b 68.d 68.d $1$ $7.029$ \(\Q\) \(\Q(\sqrt{-17}) \) None \(4\) \(16\) \(0\) \(-64\) $\mathrm{U}(1)[D_{2}]$ \(q+4q^{2}+2^{4}q^{3}+2^{4}q^{4}+2^{6}q^{6}-2^{6}q^{7}+\cdots\)
68.5.d.c 68.d 68.d $2$ $7.029$ \(\Q(\sqrt{17}) \) \(\Q(\sqrt{-17}) \) None \(-8\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-4q^{2}-\beta q^{3}+2^{4}q^{4}+4\beta q^{6}-9\beta q^{7}+\cdots\)
68.5.d.d 68.d 68.d $2$ $7.029$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) None \(8\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+4q^{2}+2^{4}q^{4}+iq^{5}+2^{6}q^{8}-3^{4}q^{9}+\cdots\)
68.5.d.e 68.d 68.d $28$ $7.029$ None None \(-10\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
68.5.f.a 68.f 68.f $2$ $7.029$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(-34\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q-4iq^{2}-2^{4}q^{4}+(-17-17i)q^{5}+\cdots\)
68.5.f.b 68.f 68.f $2$ $7.029$ \(\Q(\sqrt{-1}) \) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(62\) \(0\) $\mathrm{U}(1)[D_{4}]$ \(q-4iq^{2}-2^{4}q^{4}+(31+31i)q^{5}+2^{6}iq^{8}+\cdots\)
68.5.f.c 68.f 68.f $64$ $7.029$ None None \(0\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{4}]$
68.5.g.a 68.g 68.g $136$ $7.029$ None None \(-4\) \(0\) \(-8\) \(0\) $\mathrm{SU}(2)[C_{8}]$
68.5.j.a 68.j 17.e $48$ $7.029$ None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{16}]$
68.6.a.a 68.a 1.a $2$ $10.906$ \(\Q(\sqrt{13}) \) None None \(0\) \(-8\) \(-20\) \(128\) $+$ $\mathrm{SU}(2)$ \(q+(-4-\beta )q^{3}+(-10+2\beta )q^{5}+(2^{6}+\cdots)q^{7}+\cdots\)
68.6.a.b 68.a 1.a $4$ $10.906$ \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None None \(0\) \(10\) \(-88\) \(166\) $-$ $\mathrm{SU}(2)$ \(q+(2-\beta _{3})q^{3}+(-22+\beta _{1}-\beta _{2}-\beta _{3})q^{5}+\cdots\)
68.6.b.a 68.b 17.b $8$ $10.906$ \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None None \(0\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$ \(q+\beta _{1}q^{3}+\beta _{4}q^{5}+(-2\beta _{1}-\beta _{6})q^{7}+\cdots\)
68.6.e.a 68.e 17.c $16$ $10.906$ \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(0\) \(22\) \(-44\) \(-118\) $\mathrm{SU}(2)[C_{4}]$ \(q+(1+\beta _{1}+\beta _{5})q^{3}+(-3-3\beta _{5}+\beta _{6}+\cdots)q^{5}+\cdots\)
68.6.h.a 68.h 17.d $28$ $10.906$ None None \(0\) \(0\) \(44\) \(0\) $\mathrm{SU}(2)[C_{8}]$
68.6.i.a 68.i 68.i $8$ $10.906$ \(\Q(\zeta_{16})\) \(\Q(\sqrt{-1}) \) None \(0\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{16}]$ \(q+(4\zeta_{16}-4\zeta_{16}^{5})q^{2}-2^{5}\zeta_{16}^{6}q^{4}+\cdots\)
68.6.i.b 68.i 68.i $336$ $10.906$ None None \(-8\) \(0\) \(-16\) \(0\) $\mathrm{SU}(2)[C_{16}]$
68.7.c.a 68.c 4.b $48$ $15.644$ None None \(-10\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
68.7.d.a 68.d 68.d $2$ $15.644$ \(\Q(\sqrt{34}) \) \(\Q(\sqrt{-17}) \) None \(-16\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q-8q^{2}+7\beta q^{3}+2^{6}q^{4}-56\beta q^{6}+\cdots\)
68.7.d.b 68.d 68.d $2$ $15.644$ \(\Q(\sqrt{2}) \) \(\Q(\sqrt{-17}) \) None \(16\) \(0\) \(0\) \(0\) $\mathrm{U}(1)[D_{2}]$ \(q+8q^{2}+5\beta q^{3}+2^{6}q^{4}+40\beta q^{6}+\cdots\)
68.7.d.c 68.d 68.d $48$ $15.644$ None None \(-2\) \(0\) \(0\) \(0\) $\mathrm{SU}(2)[C_{2}]$
68.7.f.a 68.f 68.f $104$ $15.644$ None None \(0\) \(0\) \(-48\) \(0\) $\mathrm{SU}(2)[C_{4}]$
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