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Label Dim. \(A\) Field CM RM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
39.1.d.a \(1\) \(0.019\) \(\Q\) \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-39}) \) \(\Q(\sqrt{13}) \) \(0\) \(-1\) \(0\) \(0\) \(q-q^{3}-q^{4}+q^{9}+q^{12}-q^{13}+q^{16}+\cdots\)
39.2.a.a \(1\) \(0.311\) \(\Q\) None None \(1\) \(-1\) \(2\) \(-4\) \(-\) \(q+q^{2}-q^{3}-q^{4}+2q^{5}-q^{6}-4q^{7}+\cdots\)
39.2.a.b \(2\) \(0.311\) \(\Q(\sqrt{2}) \) None None \(-2\) \(2\) \(0\) \(0\) \(-\) \(q+(-1+\beta )q^{2}+q^{3}+(1-2\beta )q^{4}-2\beta q^{5}+\cdots\)
39.2.b.a \(2\) \(0.311\) \(\Q(\sqrt{-3}) \) None None \(0\) \(-2\) \(0\) \(0\) \(q-\zeta_{6}q^{2}-q^{3}-q^{4}+\zeta_{6}q^{6}+2\zeta_{6}q^{7}+\cdots\)
39.2.e.a \(2\) \(0.311\) \(\Q(\sqrt{-3}) \) None None \(-1\) \(1\) \(-2\) \(-2\) \(q+(-1+\zeta_{6})q^{2}+(1-\zeta_{6})q^{3}+\zeta_{6}q^{4}+\cdots\)
39.2.e.b \(4\) \(0.311\) \(\Q(\sqrt{-3}, \sqrt{17})\) None None \(-1\) \(-2\) \(-6\) \(3\) \(q-\beta _{1}q^{2}-\beta _{2}q^{3}+(-2+\beta _{1}+2\beta _{2}+\cdots)q^{4}+\cdots\)
39.2.f.a \(4\) \(0.311\) \(\Q(\zeta_{8})\) None None \(0\) \(-4\) \(0\) \(4\) \(q+\zeta_{8}q^{2}+(-1+\zeta_{8}+\zeta_{8}^{3})q^{3}-\zeta_{8}^{2}q^{4}+\cdots\)
39.2.j.a \(2\) \(0.311\) \(\Q(\sqrt{-3}) \) None None \(0\) \(1\) \(0\) \(-3\) \(q+(1-\zeta_{6})q^{3}-2\zeta_{6}q^{4}+(-2+4\zeta_{6})q^{5}+\cdots\)
39.2.k.a \(4\) \(0.311\) \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-10\) \(q+(-\zeta_{12}-\zeta_{12}^{3})q^{3}+2\zeta_{12}q^{4}+(-2+\cdots)q^{7}+\cdots\)
39.2.k.b \(8\) \(0.311\) 8.0.56070144.2 None None \(0\) \(-2\) \(0\) \(-4\) \(q+(-\beta _{1}-\beta _{2}+\beta _{3}-\beta _{4}+2\beta _{5}+\beta _{7})q^{2}+\cdots\)
39.3.c.a \(8\) \(1.063\) \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None None \(0\) \(-2\) \(0\) \(8\) \(q+\beta _{2}q^{2}+\beta _{3}q^{3}+(-2-\beta _{2}+\beta _{5}+\cdots)q^{4}+\cdots\)
39.3.d.a \(2\) \(1.063\) \(\Q(\sqrt{13}) \) \(\Q(\sqrt{-39}) \) None \(0\) \(-6\) \(0\) \(0\) \(q-\beta q^{2}-3q^{3}+9q^{4}+2\beta q^{5}+3\beta q^{6}+\cdots\)
39.3.d.b \(2\) \(1.063\) \(\Q(\sqrt{3}) \) \(\Q(\sqrt{-39}) \) None \(0\) \(6\) \(0\) \(0\) \(q+\beta q^{2}+3q^{3}-q^{4}-4\beta q^{5}+3\beta q^{6}+\cdots\)
39.3.d.c \(4\) \(1.063\) \(\Q(\sqrt{3}, \sqrt{-35})\) None None \(0\) \(-2\) \(0\) \(0\) \(q-\beta _{2}q^{2}+(-1+\beta _{1})q^{3}-q^{4}-3\beta _{2}q^{5}+\cdots\)
39.3.g.a \(8\) \(1.063\) 8.0.\(\cdots\).10 None None \(4\) \(0\) \(-20\) \(8\) \(q-\beta _{4}q^{2}-\beta _{2}q^{3}+(-1-\beta _{1}-\beta _{4}+\cdots)q^{4}+\cdots\)
39.3.h.a \(2\) \(1.063\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(-3\) \(0\) \(9\) \(q+(-3+3\zeta_{6})q^{3}+4\zeta_{6}q^{4}+(6-3\zeta_{6})q^{7}+\cdots\)
39.3.h.b \(12\) \(1.063\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(0\) \(2\) \(0\) \(-36\) \(q+\beta _{5}q^{2}+\beta _{11}q^{3}+(-3+\beta _{1}-\beta _{2}+\cdots)q^{4}+\cdots\)
39.3.i.a \(2\) \(1.063\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(3\) \(0\) \(13\) \(q+3\zeta_{6}q^{3}+(-4+4\zeta_{6})q^{4}+(13-13\zeta_{6})q^{7}+\cdots\)
39.3.i.b \(12\) \(1.063\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(0\) \(-4\) \(0\) \(-16\) \(q+\beta _{4}q^{2}+(-1+\beta _{1}+\beta _{3}-\beta _{5}-\beta _{7}+\cdots)q^{3}+\cdots\)
39.3.l.a \(8\) \(1.063\) 8.0.\(\cdots\).10 None None \(-2\) \(0\) \(16\) \(14\) \(q+(-\beta _{1}-\beta _{5})q^{2}+(2\beta _{4}-\beta _{5})q^{3}+(1+\cdots)q^{4}+\cdots\)
39.3.l.b \(12\) \(1.063\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(-2\) \(0\) \(4\) \(-32\) \(q+(-\beta _{1}-\beta _{2})q^{2}+(\beta _{4}+\beta _{5})q^{3}+(-2+\cdots)q^{4}+\cdots\)
39.4.a.a \(1\) \(2.301\) \(\Q\) None None \(0\) \(-3\) \(-12\) \(2\) \(-\) \(q-3q^{3}-8q^{4}-12q^{5}+2q^{7}+9q^{9}+\cdots\)
39.4.a.b \(2\) \(2.301\) \(\Q(\sqrt{14}) \) None None \(2\) \(-6\) \(24\) \(0\) \(+\) \(q+(1+\beta )q^{2}-3q^{3}+(7+2\beta )q^{4}+(12+\cdots)q^{5}+\cdots\)
39.4.a.c \(3\) \(2.301\) 3.3.3144.1 None None \(2\) \(9\) \(4\) \(30\) \(+\) \(q+(1-\beta _{1})q^{2}+3q^{3}+(3+\beta _{2})q^{4}+(2+\cdots)q^{5}+\cdots\)
39.4.b.a \(4\) \(2.301\) 4.0.5054412.1 None None \(0\) \(-12\) \(0\) \(0\) \(q+\beta _{1}q^{2}-3q^{3}+(-7+\beta _{3})q^{4}+\beta _{2}q^{5}+\cdots\)
39.4.b.b \(4\) \(2.301\) 4.0.1362828.1 None None \(0\) \(12\) \(0\) \(0\) \(q+\beta _{1}q^{2}+3q^{3}+(-4+\beta _{3})q^{4}+(2\beta _{1}+\cdots)q^{5}+\cdots\)
39.4.e.a \(2\) \(2.301\) \(\Q(\sqrt{-3}) \) None None \(-3\) \(-3\) \(-18\) \(-2\) \(q+(-3+3\zeta_{6})q^{2}+(-3+3\zeta_{6})q^{3}+\cdots\)
39.4.e.b \(2\) \(2.301\) \(\Q(\sqrt{-3}) \) None None \(1\) \(-3\) \(14\) \(10\) \(q+(1-\zeta_{6})q^{2}+(-3+3\zeta_{6})q^{3}+7\zeta_{6}q^{4}+\cdots\)
39.4.e.c \(8\) \(2.301\) \(\mathbb{Q}[x]/(x^{8} - \cdots)\) None None \(-2\) \(12\) \(-12\) \(14\) \(q-\beta _{1}q^{2}+(3+3\beta _{2})q^{3}+(-1+\beta _{1}+\cdots)q^{4}+\cdots\)
39.4.f.a \(4\) \(2.301\) \(\Q(\zeta_{12})\) \(\Q(\sqrt{-3}) \) None \(0\) \(0\) \(0\) \(-40\) \(q-\zeta_{12}^{3}q^{3}+8\zeta_{12}q^{4}+(-10+10\zeta_{12}+\cdots)q^{7}+\cdots\)
39.4.f.b \(20\) \(2.301\) \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None None \(0\) \(-4\) \(0\) \(44\) \(q+\beta _{8}q^{2}-\beta _{5}q^{3}+(-6\beta _{4}-\beta _{14})q^{4}+\cdots\)
39.4.j.a \(2\) \(2.301\) \(\Q(\sqrt{-3}) \) None None \(0\) \(3\) \(0\) \(18\) \(q+(3-3\zeta_{6})q^{3}-8\zeta_{6}q^{4}+(3-6\zeta_{6})q^{5}+\cdots\)
39.4.j.b \(4\) \(2.301\) \(\Q(\sqrt{-3}, \sqrt{-17})\) None None \(0\) \(6\) \(0\) \(-66\) \(q+\beta _{1}q^{2}+(3-3\beta _{2})q^{3}+9\beta _{2}q^{4}+(3+\cdots)q^{5}+\cdots\)
39.4.j.c \(10\) \(2.301\) \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None None \(0\) \(-15\) \(0\) \(30\) \(q+\beta _{3}q^{2}-3\beta _{2}q^{3}+(6-6\beta _{2}-\beta _{4}+\cdots)q^{4}+\cdots\)
39.4.k.a \(48\) \(2.301\) None None \(0\) \(-2\) \(0\) \(20\)
39.5.c.a \(16\) \(4.031\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(0\) \(-2\) \(0\) \(-80\) \(q+\beta _{1}q^{2}+\beta _{6}q^{3}+(-6+\beta _{2})q^{4}+(-\beta _{1}+\cdots)q^{5}+\cdots\)
39.5.d.a \(1\) \(4.031\) \(\Q\) \(\Q(\sqrt{-39}) \) None \(-5\) \(9\) \(-2\) \(0\) \(q-5q^{2}+9q^{3}+9q^{4}-2q^{5}-45q^{6}+\cdots\)
39.5.d.b \(1\) \(4.031\) \(\Q\) \(\Q(\sqrt{-39}) \) None \(5\) \(9\) \(2\) \(0\) \(q+5q^{2}+9q^{3}+9q^{4}+2q^{5}+45q^{6}+\cdots\)
39.5.d.c \(2\) \(4.031\) \(\Q(\sqrt{39}) \) \(\Q(\sqrt{-39}) \) None \(0\) \(-18\) \(0\) \(0\) \(q+\beta q^{2}-9q^{3}+23q^{4}+8\beta q^{5}-9\beta q^{6}+\cdots\)
39.5.d.d \(12\) \(4.031\) \(\mathbb{Q}[x]/(x^{12} - \cdots)\) None None \(0\) \(-2\) \(0\) \(0\) \(q-\beta _{3}q^{2}+\beta _{4}q^{3}+(2+\beta _{1})q^{4}+(2\beta _{3}+\cdots)q^{5}+\cdots\)
39.5.g.a \(20\) \(4.031\) \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None None \(0\) \(0\) \(24\) \(-24\) \(q-\beta _{1}q^{2}+\beta _{6}q^{3}+(10\beta _{4}+\beta _{9}+\beta _{15}+\cdots)q^{4}+\cdots\)
39.5.h.a \(2\) \(4.031\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(9\) \(0\) \(117\) \(q+(9-9\zeta_{6})q^{3}+2^{4}\zeta_{6}q^{4}+(78-39\zeta_{6})q^{7}+\cdots\)
39.5.h.b \(32\) \(4.031\) None None \(0\) \(-10\) \(0\) \(-210\)
39.5.i.a \(2\) \(4.031\) \(\Q(\sqrt{-3}) \) \(\Q(\sqrt{-3}) \) None \(0\) \(-9\) \(0\) \(-71\) \(q-9\zeta_{6}q^{3}+(-2^{4}+2^{4}\zeta_{6})q^{4}+(-71+\cdots)q^{7}+\cdots\)
39.5.i.b \(32\) \(4.031\) None None \(0\) \(8\) \(0\) \(122\)
39.5.l.a \(16\) \(4.031\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(0\) \(0\) \(-54\) \(-44\) \(q+\beta _{6}q^{2}+(-6\beta _{4}+3\beta _{5})q^{3}+(-5+\cdots)q^{4}+\cdots\)
39.5.l.b \(20\) \(4.031\) \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None None \(0\) \(0\) \(30\) \(-42\) \(q+\beta _{5}q^{2}+(-3\beta _{4}-3\beta _{7})q^{3}+(4-\beta _{1}+\cdots)q^{4}+\cdots\)
39.6.a.a \(1\) \(6.255\) \(\Q\) None None \(2\) \(9\) \(-74\) \(-112\) \(+\) \(q+2q^{2}+9q^{3}-28q^{4}-74q^{5}+18q^{6}+\cdots\)
39.6.a.b \(2\) \(6.255\) \(\Q(\sqrt{14}) \) None None \(-4\) \(-18\) \(-48\) \(72\) \(+\) \(q+(-2+\beta )q^{2}-9q^{3}+(28-4\beta )q^{4}+\cdots\)
39.6.a.c \(3\) \(6.255\) 3.3.125308.1 None None \(0\) \(-27\) \(54\) \(84\) \(-\) \(q+\beta _{1}q^{2}-9q^{3}+(5+2\beta _{1}+\beta _{2})q^{4}+\cdots\)
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