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Label Char Prim Dim AA Field CM Minimal twist Traces Fricke sign Sato-Tate qq-expansion
a2a_{2} a3a_{3} a5a_{5} a7a_{7}
169.2.a.a 169.a 1.a 22 1.3491.349 Q(3)\Q(\sqrt{3}) None 13.2.e.a 00 44 00 00 - SU(2)\mathrm{SU}(2) q+βq2+2q3+q4βq5+2βq6+q+\beta q^{2}+2q^{3}+q^{4}-\beta q^{5}+2\beta q^{6}+\cdots
169.2.a.b 169.a 1.a 33 1.3491.349 Q(ζ14)+\Q(\zeta_{14})^+ None 169.2.a.b 2-2 2-2 4-4 3-3 ++ SU(2)\mathrm{SU}(2) q+(1β2)q2+(β1+β2)q3+(β1+)q4+q+(-1-\beta _{2})q^{2}+(-\beta _{1}+\beta _{2})q^{3}+(\beta _{1}+\cdots)q^{4}+\cdots
169.2.a.c 169.a 1.a 33 1.3491.349 Q(ζ14)+\Q(\zeta_{14})^+ None 169.2.a.b 22 2-2 44 33 - SU(2)\mathrm{SU}(2) q+(1β1)q2+(1β2)q3+(12β1+)q4+q+(1-\beta _{1})q^{2}+(-1-\beta _{2})q^{3}+(1-2\beta _{1}+\cdots)q^{4}+\cdots
169.2.b.a 169.b 13.b 22 1.3491.349 Q(3)\Q(\sqrt{-3}) None 13.2.e.a 00 44 00 00 SU(2)[C2]\mathrm{SU}(2)[C_{2}] qβq2+2q3q4+βq52βq6+q-\beta q^{2}+2 q^{3}-q^{4}+\beta q^{5}-2\beta q^{6}+\cdots
169.2.b.b 169.b 13.b 66 1.3491.349 6.0.153664.1 None 169.2.a.b 00 4-4 00 00 SU(2)[C2]\mathrm{SU}(2)[C_{2}] q+(β3+β5)q2+(1+β4)q3+(12β2+)q4+q+(\beta _{3}+\beta _{5})q^{2}+(-1+\beta _{4})q^{3}+(1-2\beta _{2}+\cdots)q^{4}+\cdots
169.2.c.a 169.c 13.c 44 1.3491.349 Q(ζ12)\Q(\zeta_{12}) None 13.2.e.a 00 4-4 00 00 SU(2)[C3]\mathrm{SU}(2)[C_{3}] qβ2q22β1q3+(β11)q4+q-\beta_{2} q^{2}-2\beta_1 q^{3}+(\beta_1-1)q^{4}+\cdots
169.2.c.b 169.c 13.c 66 1.3491.349 6.0.64827.1 None 169.2.a.b 2-2 22 88 3-3 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q+(1+β4+β5)q2+(1β1β5)q3+q+(-1+\beta _{4}+\beta _{5})q^{2}+(1-\beta _{1}-\beta _{5})q^{3}+\cdots
169.2.c.c 169.c 13.c 66 1.3491.349 6.0.64827.1 None 169.2.a.b 22 22 8-8 33 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q+(β1+β4)q2+(1β4β5)q3+(2β1+)q4+q+(\beta _{1}+\beta _{4})q^{2}+(1-\beta _{4}-\beta _{5})q^{3}+(2\beta _{1}+\cdots)q^{4}+\cdots
169.2.e.a 169.e 13.e 22 1.3491.349 Q(3)\Q(\sqrt{-3}) None 13.2.e.a 33 2-2 00 00 SU(2)[C6]\mathrm{SU}(2)[C_{6}] q+(1+ζ6)q2+(2+2ζ6)q3+ζ6q4+q+(1+\zeta_{6})q^{2}+(-2+2\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots
169.2.e.b 169.e 13.e 1212 1.3491.349 12.0.\cdots.1 None 169.2.a.b 00 44 00 00 SU(2)[C6]\mathrm{SU}(2)[C_{6}] q+(β1+β8+β10)q2+(1β3+β4+)q3+q+(-\beta _{1}+\beta _{8}+\beta _{10})q^{2}+(1-\beta _{3}+\beta _{4}+\cdots)q^{3}+\cdots
169.2.g.a 169.g 169.g 156156 1.3491.349 None 169.2.g.a 10-10 9-9 7-7 5-5 SU(2)[C13]\mathrm{SU}(2)[C_{13}]
169.2.h.a 169.h 169.h 168168 1.3491.349 None 169.2.h.a 13-13 13-13 13-13 13-13 SU(2)[C26]\mathrm{SU}(2)[C_{26}]
169.2.i.a 169.i 169.i 336336 1.3491.349 None 169.2.i.a 26-26 26-26 26-26 26-26 SU(2)[C39]\mathrm{SU}(2)[C_{39}]
169.2.k.a 169.k 169.k 360360 1.3491.349 None 169.2.k.a 23-23 24-24 26-26 26-26 SU(2)[C78]\mathrm{SU}(2)[C_{78}]
169.3.d.a 169.d 13.d 44 4.6054.605 Q(ζ12)\Q(\zeta_{12}) None 13.3.f.a 2-2 44 14-14 44 SU(2)[C4]\mathrm{SU}(2)[C_{4}] q+β3q2+(β3+β2β1+1)q3+q+\beta_{3} q^{2}+(-\beta_{3}+\beta_{2}-\beta_1+1)q^{3}+\cdots
169.3.d.b 169.d 13.d 44 4.6054.605 Q(i,22)\Q(i, \sqrt{22}) None 169.3.d.b 00 12-12 00 00 SU(2)[C4]\mathrm{SU}(2)[C_{4}] q+β1q23q3+7β2q4β1q5+q+\beta _{1}q^{2}-3q^{3}+7\beta _{2}q^{4}-\beta _{1}q^{5}+\cdots
169.3.d.c 169.d 13.d 44 4.6054.605 Q(ζ12)\Q(\zeta_{12}) None 13.3.f.a 22 44 1414 4-4 SU(2)[C4]\mathrm{SU}(2)[C_{4}] q+(β2β1+1)q2+(β3+β2β1+1)q3+q+(\beta_{2}-\beta_1+1)q^{2}+(-\beta_{3}+\beta_{2}-\beta_1+1)q^{3}+\cdots
169.3.d.d 169.d 13.d 44 4.6054.605 Q(i,10)\Q(i, \sqrt{10}) None 13.3.d.a 44 4-4 8-8 1212 SU(2)[C4]\mathrm{SU}(2)[C_{4}] q+(1+β1+β2)q2+(1+β1β3)q3+q+(1+\beta _{1}+\beta _{2})q^{2}+(-1+\beta _{1}-\beta _{3})q^{3}+\cdots
169.3.d.e 169.d 13.d 2424 4.6054.605 None 169.3.d.e 00 1212 00 00 SU(2)[C4]\mathrm{SU}(2)[C_{4}]
169.3.f.a 169.f 13.f 44 4.6054.605 Q(ζ12)\Q(\zeta_{12}) None 13.3.f.a 4-4 2-2 1414 2020 SU(2)[C12]\mathrm{SU}(2)[C_{12}] q+(1ζ12)q2+(ζ12ζ122+)q3+q+(-1-\zeta_{12})q^{2}+(-\zeta_{12}-\zeta_{12}^{2}+\cdots)q^{3}+\cdots
169.3.f.b 169.f 13.f 44 4.6054.605 Q(ζ12)\Q(\zeta_{12}) None 13.3.f.a 22 2-2 1414 16-16 SU(2)[C12]\mathrm{SU}(2)[C_{12}] q+(1ζ122+ζ123)q2+(ζ12ζ122+)q3+q+(1-\zeta_{12}^{2}+\zeta_{12}^{3})q^{2}+(\zeta_{12}-\zeta_{12}^{2}+\cdots)q^{3}+\cdots
169.3.f.c 169.f 13.f 44 4.6054.605 Q(ζ12)\Q(\zeta_{12}) None 13.3.f.a 44 2-2 14-14 20-20 SU(2)[C12]\mathrm{SU}(2)[C_{12}] q+(1+ζ12)q2+(ζ12ζ122ζ123)q3+q+(1+\zeta_{12})q^{2}+(-\zeta_{12}-\zeta_{12}^{2}-\zeta_{12}^{3})q^{3}+\cdots
169.3.f.d 169.f 13.f 88 4.6054.605 8.0.3317760000.2 None 13.3.d.a 4-4 44 16-16 12-12 SU(2)[C12]\mathrm{SU}(2)[C_{12}] q+(1β2β3+β4+β7)q2+(1+)q3+q+(-1-\beta _{2}-\beta _{3}+\beta _{4}+\beta _{7})q^{2}+(1+\cdots)q^{3}+\cdots
169.3.f.e 169.f 13.f 88 4.6054.605 8.0.\cdots.9 None 169.3.d.b 00 1212 00 00 SU(2)[C12]\mathrm{SU}(2)[C_{12}] q+β1q2+3β4q3+7β2q4+(β1β5+)q5+q+\beta _{1}q^{2}+3\beta _{4}q^{3}+7\beta _{2}q^{4}+(\beta _{1}-\beta _{5}+\cdots)q^{5}+\cdots
169.3.f.f 169.f 13.f 88 4.6054.605 8.0.3317760000.2 None 13.3.d.a 44 44 1616 1212 SU(2)[C12]\mathrm{SU}(2)[C_{12}] q+(1+β2β3β4+β7)q2+(1+β3+)q3+q+(1+\beta _{2}-\beta _{3}-\beta _{4}+\beta _{7})q^{2}+(1+\beta _{3}+\cdots)q^{3}+\cdots
169.3.f.g 169.f 13.f 4848 4.6054.605 None 169.3.d.e 00 12-12 00 00 SU(2)[C12]\mathrm{SU}(2)[C_{12}]
169.3.j.a 169.j 169.j 720720 4.6054.605 None 169.3.j.a 22-22 22-22 34-34 14-14 SU(2)[C52]\mathrm{SU}(2)[C_{52}]
169.3.l.a 169.l 169.l 13921392 4.6054.605 None 169.3.l.a 50-50 50-50 38-38 68-68 SU(2)[C156]\mathrm{SU}(2)[C_{156}]
169.4.a.a 169.a 1.a 11 9.9719.971 Q\Q None 13.4.c.a 4-4 22 17-17 20-20 ++ SU(2)\mathrm{SU}(2) q4q2+2q3+8q417q58q6+q-4q^{2}+2q^{3}+8q^{4}-17q^{5}-8q^{6}+\cdots
169.4.a.b 169.a 1.a 11 9.9719.971 Q\Q None 13.4.b.a 3-3 1-1 99 15-15 - SU(2)\mathrm{SU}(2) q3q2q3+q4+9q5+3q615q7+q-3q^{2}-q^{3}+q^{4}+9q^{5}+3q^{6}-15q^{7}+\cdots
169.4.a.c 169.a 1.a 11 9.9719.971 Q\Q None 13.4.b.a 33 1-1 9-9 1515 - SU(2)\mathrm{SU}(2) q+3q2q3+q49q53q6+15q7+q+3q^{2}-q^{3}+q^{4}-9q^{5}-3q^{6}+15q^{7}+\cdots
169.4.a.d 169.a 1.a 11 9.9719.971 Q\Q None 13.4.c.a 44 22 1717 2020 ++ SU(2)\mathrm{SU}(2) q+4q2+2q3+8q4+17q5+8q6+q+4q^{2}+2q^{3}+8q^{4}+17q^{5}+8q^{6}+\cdots
169.4.a.e 169.a 1.a 11 9.9719.971 Q\Q None 13.4.a.a 55 7-7 77 1313 ++ SU(2)\mathrm{SU}(2) q+5q27q3+17q4+7q535q6+q+5q^{2}-7q^{3}+17q^{4}+7q^{5}-35q^{6}+\cdots
169.4.a.f 169.a 1.a 22 9.9719.971 Q(17)\Q(\sqrt{17}) None 13.4.c.b 5-5 55 15-15 1515 ++ SU(2)\mathrm{SU}(2) q+(2β)q2+(1+3β)q3+5βq4+q+(-2-\beta )q^{2}+(1+3\beta )q^{3}+5\beta q^{4}+\cdots
169.4.a.g 169.a 1.a 22 9.9719.971 Q(17)\Q(\sqrt{17}) None 13.4.a.b 1-1 55 33 99 ++ SU(2)\mathrm{SU}(2) qβq2+(43β)q3+(4+β)q4+q-\beta q^{2}+(4-3\beta )q^{3}+(-4+\beta )q^{4}+\cdots
169.4.a.h 169.a 1.a 22 9.9719.971 Q(3)\Q(\sqrt{3}) None 13.4.e.a 00 14-14 00 00 - SU(2)\mathrm{SU}(2) q+2βq27q3+4q4+8βq514βq6+q+2\beta q^{2}-7q^{3}+4q^{4}+8\beta q^{5}-14\beta q^{6}+\cdots
169.4.a.i 169.a 1.a 22 9.9719.971 Q(3)\Q(\sqrt{3}) None 13.4.e.b 00 44 00 00 - SU(2)\mathrm{SU}(2) q+βq2+2q35q4+βq5+2βq6+q+\beta q^{2}+2q^{3}-5q^{4}+\beta q^{5}+2\beta q^{6}+\cdots
169.4.a.j 169.a 1.a 22 9.9719.971 Q(17)\Q(\sqrt{17}) None 13.4.c.b 55 55 1515 15-15 ++ SU(2)\mathrm{SU}(2) q+(3β)q2+(43β)q3+(55β)q4+q+(3-\beta )q^{2}+(4-3\beta )q^{3}+(5-5\beta )q^{4}+\cdots
169.4.a.k 169.a 1.a 99 9.9719.971 Q[x]/(x9)\mathbb{Q}[x]/(x^{9} - \cdots) None 169.4.a.k 5-5 11 30-30 38-38 - SU(2)\mathrm{SU}(2) q+(1+β1)q2+β2q3+(5+β5+β6+)q4+q+(-1+\beta _{1})q^{2}+\beta _{2}q^{3}+(5+\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots
169.4.a.l 169.a 1.a 99 9.9719.971 Q[x]/(x9)\mathbb{Q}[x]/(x^{9} - \cdots) None 169.4.a.k 55 11 3030 3838 ++ SU(2)\mathrm{SU}(2) q+(1β1)q2+β2q3+(5+β5+β6+)q4+q+(1-\beta _{1})q^{2}+\beta _{2}q^{3}+(5+\beta _{5}+\beta _{6}+\cdots)q^{4}+\cdots
169.4.b.a 169.b 13.b 22 9.9719.971 Q(1)\Q(\sqrt{-1}) None 13.4.a.a 00 14-14 00 00 SU(2)[C2]\mathrm{SU}(2)[C_{2}] q+5iq27q317q4+7iq5+q+5 i q^{2}-7 q^{3}-17 q^{4}+7 i q^{5}+\cdots
169.4.b.b 169.b 13.b 22 9.9719.971 Q(3)\Q(\sqrt{-3}) None 13.4.e.a 00 14-14 00 00 SU(2)[C2]\mathrm{SU}(2)[C_{2}] q2βq27q34q48βq5+q-2\beta q^{2}-7 q^{3}-4 q^{4}-8\beta q^{5}+\cdots
169.4.b.c 169.b 13.b 22 9.9719.971 Q(1)\Q(\sqrt{-1}) None 13.4.c.a 00 44 00 00 SU(2)[C2]\mathrm{SU}(2)[C_{2}] q+4iq2+2q38q4+17iq5+q+4 i q^{2}+2 q^{3}-8 q^{4}+17 i q^{5}+\cdots
169.4.b.d 169.b 13.b 22 9.9719.971 Q(3)\Q(\sqrt{-3}) None 13.4.e.b 00 44 00 00 SU(2)[C2]\mathrm{SU}(2)[C_{2}] qβq2+2q3+5q4βq52βq6+q-\beta q^{2}+2 q^{3}+5 q^{4}-\beta q^{5}-2\beta q^{6}+\cdots
169.4.b.e 169.b 13.b 44 9.9719.971 Q(i,17)\Q(i, \sqrt{17}) None 13.4.c.b 00 1010 00 00 SU(2)[C2]\mathrm{SU}(2)[C_{2}] q+(β1+3β2)q2+(1+3β3)q35β3q4+q+(\beta _{1}+3\beta _{2})q^{2}+(1+3\beta _{3})q^{3}-5\beta _{3}q^{4}+\cdots
169.4.b.f 169.b 13.b 44 9.9719.971 Q(i,17)\Q(i, \sqrt{17}) None 13.4.a.b 00 1010 00 00 SU(2)[C2]\mathrm{SU}(2)[C_{2}] q+β1q2+(1+3β3)q3+(3+β3)q4+q+\beta _{1}q^{2}+(1+3\beta _{3})q^{3}+(3+\beta _{3})q^{4}+\cdots
169.4.b.g 169.b 13.b 1818 9.9719.971 Q[x]/(x18+)\mathbb{Q}[x]/(x^{18} + \cdots) None 169.4.a.k 00 22 00 00 SU(2)[C2]\mathrm{SU}(2)[C_{2}] q+β9q2β2q3+(4+β1)q4+(β9+)q5+q+\beta _{9}q^{2}-\beta _{2}q^{3}+(-4+\beta _{1})q^{4}+(-\beta _{9}+\cdots)q^{5}+\cdots
169.4.c.a 169.c 13.c 22 9.9719.971 Q(3)\Q(\sqrt{-3}) None 13.4.a.a 5-5 77 1414 13-13 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q+(5+5ζ6)q2+(77ζ6)q317ζ6q4+q+(-5+5\zeta_{6})q^{2}+(7-7\zeta_{6})q^{3}-17\zeta_{6}q^{4}+\cdots
169.4.c.b 169.c 13.c 22 9.9719.971 Q(3)\Q(\sqrt{-3}) None 13.4.b.a 3-3 11 18-18 15-15 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q+(3+3ζ6)q2+(1ζ6)q3ζ6q4+q+(-3+3\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots
169.4.c.c 169.c 13.c 22 9.9719.971 Q(3)\Q(\sqrt{-3}) None 13.4.b.a 33 11 1818 1515 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q+(33ζ6)q2+(1ζ6)q3ζ6q4+q+(3-3\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}-\zeta_{6}q^{4}+\cdots
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