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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}
42.2.a.a 42.a 1.a 11 0.3350.335 Q\Q None 42.2.a.a 11 1-1 2-2 1-1 - SU(2)\mathrm{SU}(2) q+q2q3+q42q5q6q7+q+q^{2}-q^{3}+q^{4}-2q^{5}-q^{6}-q^{7}+\cdots
42.2.d.a 42.d 21.c 44 0.3350.335 Q(i,6)\Q(i, \sqrt{6}) None 42.2.d.a 00 00 00 4-4 SU(2)[C2]\mathrm{SU}(2)[C_{2}] qβ2q2+β1q3q4+(β1+β3)q5+q-\beta _{2}q^{2}+\beta _{1}q^{3}-q^{4}+(-\beta _{1}+\beta _{3})q^{5}+\cdots
42.2.e.a 42.e 7.c 22 0.3350.335 Q(3)\Q(\sqrt{-3}) None 42.2.e.a 1-1 11 1-1 11 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q+(1+ζ6)q2+ζ6q3ζ6q4+(1+)q5+q+(-1+\zeta_{6})q^{2}+\zeta_{6}q^{3}-\zeta_{6}q^{4}+(-1+\cdots)q^{5}+\cdots
42.2.e.b 42.e 7.c 22 0.3350.335 Q(3)\Q(\sqrt{-3}) None 42.2.e.b 11 1-1 3-3 55 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q+(1ζ6)q2ζ6q3ζ6q4+(3+)q5+q+(1-\zeta_{6})q^{2}-\zeta_{6}q^{3}-\zeta_{6}q^{4}+(-3+\cdots)q^{5}+\cdots
42.2.f.a 42.f 21.g 44 0.3350.335 Q(ζ12)\Q(\zeta_{12}) None 42.2.f.a 00 00 00 10-10 SU(2)[C6]\mathrm{SU}(2)[C_{6}] q+ζ12q2+(ζ12ζ123)q3+ζ122q4+q+\zeta_{12}q^{2}+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+\zeta_{12}^{2}q^{4}+\cdots
42.3.b.a 42.b 3.b 44 1.1441.144 Q(2,7)\Q(\sqrt{-2}, \sqrt{7}) None 42.3.b.a 00 00 00 00 SU(2)[C2]\mathrm{SU}(2)[C_{2}] qβ1q2+(β1+β3)q32q4+(2β1+)q5+q-\beta _{1}q^{2}+(-\beta _{1}+\beta _{3})q^{3}-2q^{4}+(-2\beta _{1}+\cdots)q^{5}+\cdots
42.3.c.a 42.c 7.b 44 1.1441.144 Q(2,3)\Q(\sqrt{2}, \sqrt{-3}) None 42.3.c.a 00 00 00 88 SU(2)[C2]\mathrm{SU}(2)[C_{2}] q+β1q2β2q3+2q4+(2β2β3)q5+q+\beta _{1}q^{2}-\beta _{2}q^{3}+2q^{4}+(-2\beta _{2}-\beta _{3})q^{5}+\cdots
42.3.g.a 42.g 7.d 44 1.1441.144 Q(2,3)\Q(\sqrt{2}, \sqrt{-3}) None 42.3.g.a 00 66 1212 10-10 SU(2)[C6]\mathrm{SU}(2)[C_{6}] q+β1q2+(2+β2)q3+2β2q4+(2+)q5+q+\beta _{1}q^{2}+(2+\beta _{2})q^{3}+2\beta _{2}q^{4}+(2+\cdots)q^{5}+\cdots
42.3.h.a 42.h 21.h 44 1.1441.144 Q(2,3)\Q(\sqrt{-2}, \sqrt{-3}) None 42.3.h.a 00 22 00 14-14 SU(2)[C6]\mathrm{SU}(2)[C_{6}] q+β1q2+(2β1+β22β3)q3+2β2q4+q+\beta _{1}q^{2}+(2\beta _{1}+\beta _{2}-2\beta _{3})q^{3}+2\beta _{2}q^{4}+\cdots
42.3.h.b 42.h 21.h 88 1.1441.144 8.0.4857532416.2 None 42.3.h.b 00 2-2 00 00 SU(2)[C6]\mathrm{SU}(2)[C_{6}] q+(β2β6)q2+(β1β2β3+β4+)q3+q+(\beta _{2}-\beta _{6})q^{2}+(\beta _{1}-\beta _{2}-\beta _{3}+\beta _{4}+\cdots)q^{3}+\cdots
42.4.a.a 42.a 1.a 11 2.4782.478 Q\Q None 42.4.a.a 22 3-3 1818 77 ++ SU(2)\mathrm{SU}(2) q+2q23q3+4q4+18q56q6+q+2q^{2}-3q^{3}+4q^{4}+18q^{5}-6q^{6}+\cdots
42.4.a.b 42.a 1.a 11 2.4782.478 Q\Q None 42.4.a.b 22 33 22 7-7 ++ SU(2)\mathrm{SU}(2) q+2q2+3q3+4q4+2q5+6q6+q+2q^{2}+3q^{3}+4q^{4}+2q^{5}+6q^{6}+\cdots
42.4.d.a 42.d 21.c 88 2.4782.478 8.0.\cdots.13 None 42.4.d.a 00 00 00 44 SU(2)[C2]\mathrm{SU}(2)[C_{2}] q+β3q2+β5q34q4+(2β52β6+)q5+q+\beta _{3}q^{2}+\beta _{5}q^{3}-4q^{4}+(2\beta _{5}-2\beta _{6}+\cdots)q^{5}+\cdots
42.4.e.a 42.e 7.c 22 2.4782.478 Q(3)\Q(\sqrt{-3}) None 42.4.e.a 22 3-3 66 7-7 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q+(22ζ6)q23ζ6q34ζ6q4+q+(2-2\zeta_{6})q^{2}-3\zeta_{6}q^{3}-4\zeta_{6}q^{4}+\cdots
42.4.e.b 42.e 7.c 22 2.4782.478 Q(3)\Q(\sqrt{-3}) None 42.4.e.b 22 33 1515 3535 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q+(22ζ6)q2+3ζ6q34ζ6q4+q+(2-2\zeta_{6})q^{2}+3\zeta_{6}q^{3}-4\zeta_{6}q^{4}+\cdots
42.4.e.c 42.e 7.c 44 2.4782.478 Q(3,1345)\Q(\sqrt{-3}, \sqrt{1345}) None 42.4.e.c 4-4 6-6 5-5 00 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q2β2q2+(3+3β2)q3+(4+4β2+)q4+q-2\beta _{2}q^{2}+(-3+3\beta _{2})q^{3}+(-4+4\beta _{2}+\cdots)q^{4}+\cdots
42.4.f.a 42.f 21.g 1616 2.4782.478 Q[x]/(x16)\mathbb{Q}[x]/(x^{16} - \cdots) None 42.4.f.a 00 00 00 8080 SU(2)[C6]\mathrm{SU}(2)[C_{6}] qβ2q2+β1q3+(4+4β5)q4+(β2+)q5+q-\beta _{2}q^{2}+\beta _{1}q^{3}+(4+4\beta _{5})q^{4}+(-\beta _{2}+\cdots)q^{5}+\cdots
42.5.b.a 42.b 3.b 88 4.3424.342 Q[x]/(x8)\mathbb{Q}[x]/(x^{8} - \cdots) None 42.5.b.a 00 1212 00 00 SU(2)[C2]\mathrm{SU}(2)[C_{2}] q+β1q2+(2β3)q38q4+(4β1+)q5+q+\beta _{1}q^{2}+(2-\beta _{3})q^{3}-8q^{4}+(-4\beta _{1}+\cdots)q^{5}+\cdots
42.5.c.a 42.c 7.b 44 4.3424.342 Q(2,3)\Q(\sqrt{2}, \sqrt{-3}) None 42.5.c.a 00 00 00 2020 SU(2)[C2]\mathrm{SU}(2)[C_{2}] q+β1q23β2q3+8q4+(4β2+)q5+q+\beta _{1}q^{2}-3\beta _{2}q^{3}+8q^{4}+(-4\beta _{2}+\cdots)q^{5}+\cdots
42.5.g.a 42.g 7.d 44 4.3424.342 Q(2,3)\Q(\sqrt{2}, \sqrt{-3}) None 42.5.g.a 00 18-18 66-66 70-70 SU(2)[C6]\mathrm{SU}(2)[C_{6}] q+β1q2+(63β2)q3+8β2q4+q+\beta _{1}q^{2}+(-6-3\beta _{2})q^{3}+8\beta _{2}q^{4}+\cdots
42.5.g.b 42.g 7.d 88 4.3424.342 Q[x]/(x8)\mathbb{Q}[x]/(x^{8} - \cdots) None 42.5.g.b 00 3636 42-42 7676 SU(2)[C6]\mathrm{SU}(2)[C_{6}] q+β3q2+(6+3β1)q3+8β1q4+(3+)q5+q+\beta _{3}q^{2}+(6+3\beta _{1})q^{3}+8\beta _{1}q^{4}+(-3+\cdots)q^{5}+\cdots
42.5.h.a 42.h 21.h 2020 4.3424.342 Q[x]/(x20)\mathbb{Q}[x]/(x^{20} - \cdots) None 42.5.h.a 00 00 00 130130 SU(2)[C6]\mathrm{SU}(2)[C_{6}] q+(β1+β9)q2β10q3+(8+8β2+)q4+q+(\beta _{1}+\beta _{9})q^{2}-\beta _{10}q^{3}+(8+8\beta _{2}+\cdots)q^{4}+\cdots
42.6.a.a 42.a 1.a 11 6.7366.736 Q\Q None 42.6.a.a 4-4 9-9 54-54 4949 - SU(2)\mathrm{SU}(2) q4q29q3+24q454q5+62q6+q-4q^{2}-9q^{3}+2^{4}q^{4}-54q^{5}+6^{2}q^{6}+\cdots
42.6.a.b 42.a 1.a 11 6.7366.736 Q\Q None 42.6.a.b 4-4 9-9 4444 49-49 ++ SU(2)\mathrm{SU}(2) q4q29q3+24q4+44q5+62q6+q-4q^{2}-9q^{3}+2^{4}q^{4}+44q^{5}+6^{2}q^{6}+\cdots
42.6.a.c 42.a 1.a 11 6.7366.736 Q\Q None 42.6.a.c 4-4 99 72-72 4949 ++ SU(2)\mathrm{SU}(2) q4q2+9q3+24q472q562q6+q-4q^{2}+9q^{3}+2^{4}q^{4}-72q^{5}-6^{2}q^{6}+\cdots
42.6.a.d 42.a 1.a 11 6.7366.736 Q\Q None 42.6.a.d 4-4 99 2626 49-49 - SU(2)\mathrm{SU}(2) q4q2+9q3+24q4+26q562q6+q-4q^{2}+9q^{3}+2^{4}q^{4}+26q^{5}-6^{2}q^{6}+\cdots
42.6.a.e 42.a 1.a 11 6.7366.736 Q\Q None 42.6.a.e 44 9-9 7676 49-49 - SU(2)\mathrm{SU}(2) q+4q29q3+24q4+76q562q6+q+4q^{2}-9q^{3}+2^{4}q^{4}+76q^{5}-6^{2}q^{6}+\cdots
42.6.a.f 42.a 1.a 11 6.7366.736 Q\Q None 42.6.a.f 44 99 2424 4949 - SU(2)\mathrm{SU}(2) q+4q2+9q3+24q4+24q5+62q6+q+4q^{2}+9q^{3}+2^{4}q^{4}+24q^{5}+6^{2}q^{6}+\cdots
42.6.d.a 42.d 21.c 1212 6.7366.736 Q[x]/(x12+)\mathbb{Q}[x]/(x^{12} + \cdots) None 42.6.d.a 00 00 00 252-252 SU(2)[C2]\mathrm{SU}(2)[C_{2}] qβ1q2+β3q324q4+(β2β3+)q5+q-\beta _{1}q^{2}+\beta _{3}q^{3}-2^{4}q^{4}+(\beta _{2}-\beta _{3}+\cdots)q^{5}+\cdots
42.6.e.a 42.e 7.c 22 6.7366.736 Q(3)\Q(\sqrt{-3}) None 42.6.e.a 4-4 9-9 66 119119 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q+(4+4ζ6)q29ζ6q324ζ6q4+q+(-4+4\zeta_{6})q^{2}-9\zeta_{6}q^{3}-2^{4}\zeta_{6}q^{4}+\cdots
42.6.e.b 42.e 7.c 22 6.7366.736 Q(3)\Q(\sqrt{-3}) None 42.6.e.b 44 99 86-86 4949 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q+(44ζ6)q2+9ζ6q324ζ6q4+q+(4-4\zeta_{6})q^{2}+9\zeta_{6}q^{3}-2^{4}\zeta_{6}q^{4}+\cdots
42.6.e.c 42.e 7.c 44 6.7366.736 Q(3,9601)\Q(\sqrt{-3}, \sqrt{9601}) None 42.6.e.c 8-8 1818 5353 66 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q4β2q2+(99β2)q3+(24+24β2+)q4+q-4\beta _{2}q^{2}+(9-9\beta _{2})q^{3}+(-2^{4}+2^{4}\beta _{2}+\cdots)q^{4}+\cdots
42.6.e.d 42.e 7.c 44 6.7366.736 Q(3,505)\Q(\sqrt{-3}, \sqrt{505}) None 42.6.e.d 88 18-18 17-17 408-408 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q+(44β1)q29β1q324β1q4+q+(4-4\beta _{1})q^{2}-9\beta _{1}q^{3}-2^{4}\beta _{1}q^{4}+\cdots
42.6.f.a 42.f 21.g 2828 6.7366.736 None 42.6.f.a 00 00 00 154-154 SU(2)[C6]\mathrm{SU}(2)[C_{6}]
42.7.b.a 42.b 3.b 1212 9.6629.662 Q[x]/(x12+)\mathbb{Q}[x]/(x^{12} + \cdots) None 42.7.b.a 00 84-84 00 00 SU(2)[C2]\mathrm{SU}(2)[C_{2}] qβ3q2+(7+β1)q325q4+(β1+)q5+q-\beta _{3}q^{2}+(-7+\beta _{1})q^{3}-2^{5}q^{4}+(\beta _{1}+\cdots)q^{5}+\cdots
42.7.c.a 42.c 7.b 88 9.6629.662 Q[x]/(x8)\mathbb{Q}[x]/(x^{8} - \cdots) None 42.7.c.a 00 00 00 212-212 SU(2)[C2]\mathrm{SU}(2)[C_{2}] qβ2q2+β1q3+25q4+(2β1+)q5+q-\beta _{2}q^{2}+\beta _{1}q^{3}+2^{5}q^{4}+(-2\beta _{1}+\cdots)q^{5}+\cdots
42.7.g.a 42.g 7.d 88 9.6629.662 Q[x]/(x8)\mathbb{Q}[x]/(x^{8} - \cdots) None 42.7.g.a 00 108-108 462462 580580 SU(2)[C6]\mathrm{SU}(2)[C_{6}] q+β3q2+(18+9β1)q325β1q4+q+\beta _{3}q^{2}+(-18+9\beta _{1})q^{3}-2^{5}\beta _{1}q^{4}+\cdots
42.7.g.b 42.g 7.d 88 9.6629.662 Q[x]/(x8)\mathbb{Q}[x]/(x^{8} - \cdots) None 42.7.g.b 00 108108 210210 608-608 SU(2)[C6]\mathrm{SU}(2)[C_{6}] qβ5q2+(9+9β1)q3+(25+25β1+)q4+q-\beta _{5}q^{2}+(9+9\beta _{1})q^{3}+(-2^{5}+2^{5}\beta _{1}+\cdots)q^{4}+\cdots
42.7.h.a 42.h 21.h 3232 9.6629.662 None 42.7.h.a 00 00 00 1684-1684 SU(2)[C6]\mathrm{SU}(2)[C_{6}]
42.8.a.a 42.a 1.a 11 13.12013.120 Q\Q None 42.8.a.a 8-8 27-27 410-410 343-343 ++ SU(2)\mathrm{SU}(2) q8q233q3+26q4410q5+q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}-410q^{5}+\cdots
42.8.a.b 42.a 1.a 11 13.12013.120 Q\Q None 42.8.a.b 8-8 27-27 18-18 343343 - SU(2)\mathrm{SU}(2) q8q233q3+26q418q5+63q6+q-8q^{2}-3^{3}q^{3}+2^{6}q^{4}-18q^{5}+6^{3}q^{6}+\cdots
42.8.a.c 42.a 1.a 11 13.12013.120 Q\Q None 42.8.a.c 8-8 2727 122-122 343-343 - SU(2)\mathrm{SU}(2) q8q2+33q3+26q4122q5+q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}-122q^{5}+\cdots
42.8.a.d 42.a 1.a 11 13.12013.120 Q\Q None 42.8.a.d 8-8 2727 270270 343343 ++ SU(2)\mathrm{SU}(2) q8q2+33q3+26q4+270q5+q-8q^{2}+3^{3}q^{3}+2^{6}q^{4}+270q^{5}+\cdots
42.8.a.e 42.a 1.a 11 13.12013.120 Q\Q None 42.8.a.e 88 27-27 3030 343343 ++ SU(2)\mathrm{SU}(2) q+8q233q3+26q4+30q563q6+q+8q^{2}-3^{3}q^{3}+2^{6}q^{4}+30q^{5}-6^{3}q^{6}+\cdots
42.8.a.f 42.a 1.a 11 13.12013.120 Q\Q None 42.8.a.f 88 2727 470470 343-343 ++ SU(2)\mathrm{SU}(2) q+8q2+33q3+26q4+470q5+q+8q^{2}+3^{3}q^{3}+2^{6}q^{4}+470q^{5}+\cdots
42.8.d.a 42.d 21.c 2020 13.12013.120 Q[x]/(x20+)\mathbb{Q}[x]/(x^{20} + \cdots) None 42.8.d.a 00 00 00 892892 SU(2)[C2]\mathrm{SU}(2)[C_{2}] q+β4q2+β3q326q4+β1q5+q+\beta _{4}q^{2}+\beta _{3}q^{3}-2^{6}q^{4}+\beta _{1}q^{5}+\cdots
42.8.e.a 42.e 7.c 22 13.12013.120 Q(3)\Q(\sqrt{-3}) None 42.8.e.a 88 27-27 165-165 343-343 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q+(88ζ6)q233ζ6q326ζ6q4+q+(8-8\zeta_{6})q^{2}-3^{3}\zeta_{6}q^{3}-2^{6}\zeta_{6}q^{4}+\cdots
42.8.e.b 42.e 7.c 22 13.12013.120 Q(3)\Q(\sqrt{-3}) None 42.8.e.b 88 27-27 290290 14771477 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q+(88ζ6)q233ζ6q326ζ6q4+q+(8-8\zeta_{6})q^{2}-3^{3}\zeta_{6}q^{3}-2^{6}\zeta_{6}q^{4}+\cdots
42.8.e.c 42.e 7.c 44 13.12013.120 Q(3,2881)\Q(\sqrt{-3}, \sqrt{2881}) None 42.8.e.c 16-16 5454 309-309 868868 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q8β1q2+(3333β1)q3+(26+)q4+q-8\beta _{1}q^{2}+(3^{3}-3^{3}\beta _{1})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots
42.8.e.d 42.e 7.c 66 13.12013.120 Q[x]/(x6+)\mathbb{Q}[x]/(x^{6} + \cdots) None 42.8.e.d 24-24 81-81 7070 895-895 SU(2)[C3]\mathrm{SU}(2)[C_{3}] q8β1q2+(33+33β1)q3+(26+)q4+q-8\beta _{1}q^{2}+(-3^{3}+3^{3}\beta _{1})q^{3}+(-2^{6}+\cdots)q^{4}+\cdots
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