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Label Char Prim Dim AA Field CM RM Minimal twist Traces Fricke sign Sato-Tate qq-expansion
a2a_{2} a3a_{3} a5a_{5} a7a_{7}
55.1.d.a 55.d 55.d 11 0.0270.027 Q\Q Q(11)\Q(\sqrt{-11}) , Q(55)\Q(\sqrt{-55}) Q(5)\Q(\sqrt{5}) 55.1.d.a 00 00 1-1 00 qq4q5+q9q11+q16+q20+q-q^{4}-q^{5}+q^{9}-q^{11}+q^{16}+q^{20}+\cdots
55.2.a.a 55.a 1.a 11 0.4390.439 Q\Q None None 55.2.a.a 11 00 11 00 - SU(2)\mathrm{SU}(2) q+q2q4+q53q83q9+q10+q+q^{2}-q^{4}+q^{5}-3q^{8}-3q^{9}+q^{10}+\cdots
55.2.a.b 55.a 1.a 22 0.4390.439 Q(2)\Q(\sqrt{2}) None None 55.2.a.b 22 00 2-2 4-4 - SU(2)\mathrm{SU}(2) q+(1+β)q22βq3+(1+2β)q4q5+q+(1+\beta )q^{2}-2\beta q^{3}+(1+2\beta )q^{4}-q^{5}+\cdots
55.2.b.a 55.b 5.b 44 0.4390.439 Q(3,11)\Q(\sqrt{-3}, \sqrt{-11}) None None 55.2.b.a 00 00 3-3 00 SU(2)[C2]\mathrm{SU}(2)[C_{2}] q+(β1β2+β3)q2+(β1β3)q3+q+(-\beta _{1}-\beta _{2}+\beta _{3})q^{2}+(\beta _{1}-\beta _{3})q^{3}+\cdots
55.2.e.a 55.e 55.e 44 0.4390.439 Q(i,10)\Q(i, \sqrt{10}) None None 55.2.e.a 00 4-4 8-8 00 SU(2)[C4]\mathrm{SU}(2)[C_{4}] q+β1q2+(1β2)q3+3β2q4+q+\beta _{1}q^{2}+(-1-\beta _{2})q^{3}+3\beta _{2}q^{4}+\cdots
55.2.e.b 55.e 55.e 44 0.4390.439 Q(i,11)\Q(i, \sqrt{11}) Q(11)\Q(\sqrt{-11}) None 55.2.e.b 00 2-2 00 00 U(1)[D4]\mathrm{U}(1)[D_{4}] q+(1+β1β3)q3+2β2q4+(β1+)q5+q+(-1+\beta _{1}-\beta _{3})q^{3}+2\beta _{2}q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots
55.2.g.a 55.g 11.c 88 0.4390.439 8.0.159390625.1 None None 55.2.g.a 4-4 11 2-2 3-3 SU(2)[C5]\mathrm{SU}(2)[C_{5}] q+(β1β2+β4)q2+β1q3+(2+)q4+q+(-\beta _{1}-\beta _{2}+\beta _{4})q^{2}+\beta _{1}q^{3}+(-2+\cdots)q^{4}+\cdots
55.2.g.b 55.g 11.c 88 0.4390.439 8.0.13140625.1 None None 55.2.g.b 2-2 5-5 22 1-1 SU(2)[C5]\mathrm{SU}(2)[C_{5}] qβ6q2+(1+β1+β3β4β5+)q3+q-\beta _{6}q^{2}+(-1+\beta _{1}+\beta _{3}-\beta _{4}-\beta _{5}+\cdots)q^{3}+\cdots
55.2.j.a 55.j 55.j 1616 0.4390.439 Q[x]/(x16)\mathbb{Q}[x]/(x^{16} - \cdots) None None 55.2.j.a 00 00 2-2 00 SU(2)[C10]\mathrm{SU}(2)[C_{10}] q+(β1β12β13)q2+(β5+β13+)q3+q+(\beta _{1}-\beta _{12}-\beta _{13})q^{2}+(\beta _{5}+\beta _{13}+\cdots)q^{3}+\cdots
55.2.l.a 55.l 55.l 3232 0.4390.439 None None 55.2.l.a 10-10 4-4 2-2 00 SU(2)[C20]\mathrm{SU}(2)[C_{20}]
55.3.c.a 55.c 11.b 88 1.4991.499 Q[x]/(x8+)\mathbb{Q}[x]/(x^{8} + \cdots) None None 55.3.c.a 00 88 00 00 SU(2)[C2]\mathrm{SU}(2)[C_{2}] q+β1q2+(1+β3)q3+(4+β2)q4+q+\beta _{1}q^{2}+(1+\beta _{3})q^{3}+(-4+\beta _{2})q^{4}+\cdots
55.3.d.a 55.d 55.d 22 1.4991.499 Q(11)\Q(\sqrt{-11}) Q(11)\Q(\sqrt{-11}) None 55.3.d.a 00 00 11 00 U(1)[D2]\mathrm{U}(1)[D_{2}] q+(12β)q34q4+(23β)q52q9+q+(1-2\beta )q^{3}-4q^{4}+(2-3\beta )q^{5}-2q^{9}+\cdots
55.3.d.b 55.d 55.d 22 1.4991.499 Q(5)\Q(\sqrt{5}) Q(55)\Q(\sqrt{-55}) None 55.3.d.b 00 00 1010 00 U(1)[D2]\mathrm{U}(1)[D_{2}] qβq2+q4+5q5+2βq7+3βq8+q-\beta q^{2}+q^{4}+5q^{5}+2\beta q^{7}+3\beta q^{8}+\cdots
55.3.d.c 55.d 55.d 22 1.4991.499 Q(11)\Q(\sqrt{11}) Q(55)\Q(\sqrt{-55}) None 55.3.d.c 00 00 10-10 00 U(1)[D2]\mathrm{U}(1)[D_{2}] q+βq2+7q45q54βq7+3βq8+q+\beta q^{2}+7q^{4}-5q^{5}-4\beta q^{7}+3\beta q^{8}+\cdots
55.3.d.d 55.d 55.d 44 1.4991.499 Q(5,21)\Q(\sqrt{5}, \sqrt{-21}) None None 55.3.d.d 00 00 8-8 00 SU(2)[C2]\mathrm{SU}(2)[C_{2}] qβ2q2β1q3+q4+(2+β1+)q5+q-\beta _{2}q^{2}-\beta _{1}q^{3}+q^{4}+(-2+\beta _{1}+\cdots)q^{5}+\cdots
55.3.f.a 55.f 5.c 2020 1.4991.499 Q[x]/(x20)\mathbb{Q}[x]/(x^{20} - \cdots) None None 55.3.f.a 4-4 2-2 8-8 00 SU(2)[C4]\mathrm{SU}(2)[C_{4}] qβ5q2β4q3+(2β8β16)q4+q-\beta _{5}q^{2}-\beta _{4}q^{3}+(-2\beta _{8}-\beta _{16})q^{4}+\cdots
55.3.h.a 55.h 55.h 4040 1.4991.499 None None 55.3.h.a 00 00 22 00 SU(2)[C10]\mathrm{SU}(2)[C_{10}]
55.3.i.a 55.i 11.d 44 1.4991.499 Q(ζ10)\Q(\zeta_{10}) None None 55.3.i.a 5-5 44 5-5 1515 SU(2)[C10]\mathrm{SU}(2)[C_{10}] q+(2ζ10+2ζ102ζ103)q2+(4+)q3+q+(-2\zeta_{10}+2\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+(4+\cdots)q^{3}+\cdots
55.3.i.b 55.i 11.d 44 1.4991.499 Q(ζ10)\Q(\zeta_{10}) None None 55.3.i.b 55 6-6 55 3030 SU(2)[C10]\mathrm{SU}(2)[C_{10}] q+(2+2ζ102ζ103)q2+(4+3ζ10+)q3+q+(2+2\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+(-4+3\zeta_{10}+\cdots)q^{3}+\cdots
55.3.i.c 55.i 11.d 1212 1.4991.499 Q[x]/(x12)\mathbb{Q}[x]/(x^{12} - \cdots) None None 55.3.i.c 5-5 77 1515 40-40 SU(2)[C10]\mathrm{SU}(2)[C_{10}] q+(β1+β2β3+β4+β7β11)q2+q+(\beta _{1}+\beta _{2}-\beta _{3}+\beta _{4}+\beta _{7}-\beta _{11})q^{2}+\cdots
55.3.i.d 55.i 11.d 1212 1.4991.499 Q[x]/(x12+)\mathbb{Q}[x]/(x^{12} + \cdots) None None 55.3.i.d 55 13-13 15-15 5-5 SU(2)[C10]\mathrm{SU}(2)[C_{10}] q+(1+β1β2β3β4β10)q2+q+(1+\beta _{1}-\beta _{2}-\beta _{3}-\beta _{4}-\beta _{10})q^{2}+\cdots
55.3.k.a 55.k 55.k 8080 1.4991.499 None None 55.3.k.a 6-6 8-8 2-2 40-40 SU(2)[C20]\mathrm{SU}(2)[C_{20}]
55.4.a.a 55.a 1.a 11 3.2453.245 Q\Q None None 55.4.a.a 11 3-3 5-5 9-9 - SU(2)\mathrm{SU}(2) q+q23q37q45q53q6+q+q^{2}-3q^{3}-7q^{4}-5q^{5}-3q^{6}+\cdots
55.4.a.b 55.a 1.a 22 3.2453.245 Q(17)\Q(\sqrt{17}) None None 55.4.a.b 7-7 3-3 1010 25-25 - SU(2)\mathrm{SU}(2) q+(3β)q2+(1β)q3+(5+7β)q4+q+(-3-\beta )q^{2}+(-1-\beta )q^{3}+(5+7\beta )q^{4}+\cdots
55.4.a.c 55.a 1.a 33 3.2453.245 3.3.568.1 None None 55.4.a.c 55 3-3 15-15 15-15 ++ SU(2)\mathrm{SU}(2) q+(1+β1β2)q2+(23β2)q3+q+(1+\beta _{1}-\beta _{2})q^{2}+(-2-3\beta _{2})q^{3}+\cdots
55.4.a.d 55.a 1.a 44 3.2453.245 4.4.1539480.1 None None 55.4.a.d 11 99 2020 99 ++ SU(2)\mathrm{SU}(2) q+β1q2+(2β2)q3+(5+β1+β3)q4+q+\beta _{1}q^{2}+(2-\beta _{2})q^{3}+(5+\beta _{1}+\beta _{3})q^{4}+\cdots
55.4.b.a 55.b 5.b 66 3.2453.245 Q[x]/(x6+)\mathbb{Q}[x]/(x^{6} + \cdots) None None 55.4.b.a 00 00 4-4 00 SU(2)[C2]\mathrm{SU}(2)[C_{2}] q+β1q2+(β1+β5)q3+(1+β3)q4+q+\beta _{1}q^{2}+(\beta _{1}+\beta _{5})q^{3}+(1+\beta _{3})q^{4}+\cdots
55.4.b.b 55.b 5.b 1010 3.2453.245 Q[x]/(x10+)\mathbb{Q}[x]/(x^{10} + \cdots) None None 55.4.b.b 00 00 1414 00 SU(2)[C2]\mathrm{SU}(2)[C_{2}] q+β1q2β3q3+(6+β2)q4+(1+)q5+q+\beta _{1}q^{2}-\beta _{3}q^{3}+(-6+\beta _{2})q^{4}+(1+\cdots)q^{5}+\cdots
55.4.e.a 55.e 55.e 44 3.2453.245 Q(i,11)\Q(i, \sqrt{11}) Q(11)\Q(\sqrt{-11}) None 55.4.e.a 00 1616 00 00 U(1)[D4]\mathrm{U}(1)[D_{4}] q+(4+4β1+β2+β3)q38β1q4+q+(4+4\beta _{1}+\beta _{2}+\beta _{3})q^{3}-8\beta _{1}q^{4}+\cdots
55.4.e.b 55.e 55.e 2828 3.2453.245 None None 55.4.e.b 00 16-16 1212 00 SU(2)[C4]\mathrm{SU}(2)[C_{4}]
55.4.g.a 55.g 11.c 2424 3.2453.245 None None 55.4.g.a 22 8-8 3030 53-53 SU(2)[C5]\mathrm{SU}(2)[C_{5}]
55.4.g.b 55.g 11.c 2424 3.2453.245 None None 55.4.g.b 66 44 30-30 8181 SU(2)[C5]\mathrm{SU}(2)[C_{5}]
55.4.j.a 55.j 55.j 6464 3.2453.245 None None 55.4.j.a 00 00 15-15 00 SU(2)[C10]\mathrm{SU}(2)[C_{10}]
55.4.l.a 55.l 55.l 128128 3.2453.245 None None 55.4.l.a 10-10 10-10 22-22 1010 SU(2)[C20]\mathrm{SU}(2)[C_{20}]
55.5.c.a 55.c 11.b 1616 5.6855.685 Q[x]/(x16+)\mathbb{Q}[x]/(x^{16} + \cdots) None None 55.5.c.a 00 16-16 00 00 SU(2)[C2]\mathrm{SU}(2)[C_{2}] q+β1q2+(1+β5)q3+(9+β2+)q4+q+\beta _{1}q^{2}+(-1+\beta _{5})q^{3}+(-9+\beta _{2}+\cdots)q^{4}+\cdots
55.5.d.a 55.d 55.d 11 5.6855.685 Q\Q Q(55)\Q(\sqrt{-55}) None 55.5.d.a 3-3 00 2525 78-78 U(1)[D2]\mathrm{U}(1)[D_{2}] q3q27q4+52q578q7+69q8+q-3q^{2}-7q^{4}+5^{2}q^{5}-78q^{7}+69q^{8}+\cdots
55.5.d.b 55.d 55.d 11 5.6855.685 Q\Q Q(55)\Q(\sqrt{-55}) None 55.5.d.a 33 00 2525 7878 U(1)[D2]\mathrm{U}(1)[D_{2}] q+3q27q4+52q5+78q769q8+q+3q^{2}-7q^{4}+5^{2}q^{5}+78q^{7}-69q^{8}+\cdots
55.5.d.c 55.d 55.d 22 5.6855.685 Q(11)\Q(\sqrt{-11}) Q(11)\Q(\sqrt{-11}) None 55.5.d.c 00 00 4949 00 U(1)[D2]\mathrm{U}(1)[D_{2}] q+(510β)q324q4+(263β)q5+q+(5-10\beta )q^{3}-2^{4}q^{4}+(26-3\beta )q^{5}+\cdots
55.5.d.d 55.d 55.d 22 5.6855.685 Q(55)\Q(\sqrt{55}) Q(55)\Q(\sqrt{-55}) None 55.5.d.d 00 00 50-50 00 U(1)[D2]\mathrm{U}(1)[D_{2}] q+βq2+39q452q5+8βq7+23βq8+q+\beta q^{2}+39q^{4}-5^{2}q^{5}+8\beta q^{7}+23\beta q^{8}+\cdots
55.5.d.e 55.d 55.d 1616 5.6855.685 Q[x]/(x16+)\mathbb{Q}[x]/(x^{16} + \cdots) None None 55.5.d.e 00 00 30-30 00 SU(2)[C2]\mathrm{SU}(2)[C_{2}] qβ8q2β7q3+(8β3)q4+(1+)q5+q-\beta _{8}q^{2}-\beta _{7}q^{3}+(8-\beta _{3})q^{4}+(-1+\cdots)q^{5}+\cdots
55.5.f.a 55.f 5.c 4040 5.6855.685 None None 55.5.f.a 00 1010 48-48 00 SU(2)[C4]\mathrm{SU}(2)[C_{4}]
55.5.h.a 55.h 55.h 8888 5.6855.685 None None 55.5.h.a 00 00 24-24 00 SU(2)[C10]\mathrm{SU}(2)[C_{10}]
55.5.i.a 55.i 11.d 44 5.6855.685 Q(ζ10)\Q(\zeta_{10}) None None 55.5.i.a 55 30-30 25-25 150150 SU(2)[C10]\mathrm{SU}(2)[C_{10}] q+(2+2ζ102ζ103)q2+(49ζ10+)q3+q+(2+2\zeta_{10}^{2}-\zeta_{10}^{3})q^{2}+(-4-9\zeta_{10}+\cdots)q^{3}+\cdots
55.5.i.b 55.i 11.d 2828 5.6855.685 None None 55.5.i.b 5-5 4343 175-175 90-90 SU(2)[C10]\mathrm{SU}(2)[C_{10}]
55.5.i.c 55.i 11.d 3232 5.6855.685 None None 55.5.i.c 00 33 200200 60-60 SU(2)[C10]\mathrm{SU}(2)[C_{10}]
55.5.k.a 55.k 55.k 176176 5.6855.685 None None 55.5.k.a 6-6 44 42-42 244244 SU(2)[C20]\mathrm{SU}(2)[C_{20}]
55.6.a.a 55.a 1.a 33 8.8218.821 3.3.21865.1 None None 55.6.a.a 7-7 36-36 7575 102-102 ++ SU(2)\mathrm{SU}(2) q+(2β2)q2+(113β1)q3+q+(-2-\beta _{2})q^{2}+(-11-3\beta _{1})q^{3}+\cdots
55.6.a.b 55.a 1.a 44 8.8218.821 Q[x]/(x4)\mathbb{Q}[x]/(x^{4} - \cdots) None None 55.6.a.b 5-5 00 100-100 90-90 ++ SU(2)\mathrm{SU}(2) q+(1β1)q2+(β1β3)q3+(15+)q4+q+(-1-\beta _{1})q^{2}+(\beta _{1}-\beta _{3})q^{3}+(15+\cdots)q^{4}+\cdots
55.6.a.c 55.a 1.a 55 8.8218.821 Q[x]/(x5)\mathbb{Q}[x]/(x^{5} - \cdots) None None 55.6.a.c 99 00 125125 7070 - SU(2)\mathrm{SU}(2) q+(2β1)q2+(β1β3)q3+(243β1+)q4+q+(2-\beta _{1})q^{2}+(\beta _{1}-\beta _{3})q^{3}+(24-3\beta _{1}+\cdots)q^{4}+\cdots
55.6.a.d 55.a 1.a 66 8.8218.821 Q[x]/(x6)\mathbb{Q}[x]/(x^{6} - \cdots) None None 55.6.a.d 33 00 150-150 66-66 - SU(2)\mathrm{SU}(2) q+(1β1)q2+(β1β3)q3+(20+)q4+q+(1-\beta _{1})q^{2}+(-\beta _{1}-\beta _{3})q^{3}+(20+\cdots)q^{4}+\cdots
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