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Label Dim. \(A\) Field CM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
51.2.a.a \(1\) \(0.407\) \(\Q\) None \(0\) \(1\) \(3\) \(-4\) \(-\) \(q+q^{3}-2q^{4}+3q^{5}-4q^{7}+q^{9}-3q^{11}+\cdots\)
51.2.a.b \(2\) \(0.407\) \(\Q(\sqrt{17}) \) None \(-1\) \(-2\) \(3\) \(0\) \(-\) \(q-\beta q^{2}-q^{3}+(2+\beta )q^{4}+(1+\beta )q^{5}+\cdots\)
51.2.d.a \(2\) \(0.407\) \(\Q(\sqrt{-1}) \) None \(-4\) \(0\) \(0\) \(0\) \(q-2q^{2}+iq^{3}+2q^{4}+3iq^{5}-2iq^{6}+\cdots\)
51.2.d.b \(2\) \(0.407\) \(\Q(\sqrt{-1}) \) None \(2\) \(0\) \(0\) \(0\) \(q+q^{2}+iq^{3}-q^{4}+iq^{6}-4iq^{7}+\cdots\)
51.2.e.a \(8\) \(0.407\) 8.0.836829184.2 None \(0\) \(0\) \(-4\) \(-4\) \(q+\beta _{1}q^{2}-\beta _{3}q^{3}+(-1-\beta _{4}-\beta _{5}+\cdots)q^{4}+\cdots\)
51.2.h.a \(8\) \(0.407\) \(\Q(\zeta_{16})\) None \(0\) \(0\) \(-8\) \(0\) \(q+(\zeta_{16}+\zeta_{16}^{3})q^{2}+\zeta_{16}^{7}q^{3}+(\zeta_{16}^{2}+\cdots)q^{4}+\cdots\)
51.2.i.a \(32\) \(0.407\) None \(0\) \(-8\) \(0\) \(-16\)
51.3.b.a \(10\) \(1.390\) \(\mathbb{Q}[x]/(x^{10} + \cdots)\) None \(0\) \(2\) \(0\) \(-4\) \(q+\beta _{1}q^{2}-\beta _{4}q^{3}+(-2+\beta _{2})q^{4}+\beta _{7}q^{5}+\cdots\)
51.3.c.a \(1\) \(1.390\) \(\Q\) \(\Q(\sqrt{-51}) \) \(0\) \(-3\) \(7\) \(0\) \(q-3q^{3}+4q^{4}+7q^{5}+9q^{9}-5q^{11}+\cdots\)
51.3.c.b \(1\) \(1.390\) \(\Q\) \(\Q(\sqrt{-51}) \) \(0\) \(3\) \(-7\) \(0\) \(q+3q^{3}+4q^{4}-7q^{5}+9q^{9}+5q^{11}+\cdots\)
51.3.c.c \(8\) \(1.390\) \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q-\beta _{3}q^{2}+(-\beta _{1}+\beta _{7})q^{3}+(-4-\beta _{2}+\cdots)q^{4}+\cdots\)
51.3.f.a \(20\) \(1.390\) \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(-6\) \(0\) \(-4\) \(q-\beta _{1}q^{2}-\beta _{11}q^{3}+(1-\beta _{2})q^{4}-\beta _{9}q^{5}+\cdots\)
51.3.g.a \(4\) \(1.390\) \(\Q(\zeta_{8})\) None \(-4\) \(12\) \(8\) \(-4\) \(q+(-1-\zeta_{8}-\zeta_{8}^{2})q^{2}+3q^{3}+(2\zeta_{8}+\cdots)q^{4}+\cdots\)
51.3.g.b \(4\) \(1.390\) \(\Q(\zeta_{8})\) None \(4\) \(0\) \(-8\) \(-4\) \(q+(1+\zeta_{8}+\zeta_{8}^{2})q^{2}+3\zeta_{8}q^{3}+(2\zeta_{8}+\cdots)q^{4}+\cdots\)
51.3.g.c \(32\) \(1.390\) None \(0\) \(-16\) \(0\) \(0\)
51.3.j.a \(48\) \(1.390\) None \(0\) \(0\) \(0\) \(0\)
51.4.a.a \(1\) \(3.009\) \(\Q\) None \(-1\) \(-3\) \(16\) \(34\) \(+\) \(q-q^{2}-3q^{3}-7q^{4}+2^{4}q^{5}+3q^{6}+\cdots\)
51.4.a.b \(1\) \(3.009\) \(\Q\) None \(-1\) \(3\) \(-20\) \(-2\) \(-\) \(q-q^{2}+3q^{3}-7q^{4}-20q^{5}-3q^{6}+\cdots\)
51.4.a.c \(1\) \(3.009\) \(\Q\) None \(1\) \(-3\) \(-10\) \(-8\) \(-\) \(q+q^{2}-3q^{3}-7q^{4}-10q^{5}-3q^{6}+\cdots\)
51.4.a.d \(2\) \(3.009\) \(\Q(\sqrt{2}) \) None \(0\) \(-6\) \(6\) \(-8\) \(+\) \(q+\beta q^{2}-3q^{3}+10q^{4}+(3+4\beta )q^{5}+\cdots\)
51.4.a.e \(3\) \(3.009\) 3.3.5912.1 None \(5\) \(9\) \(8\) \(-8\) \(+\) \(q+(2-\beta _{1})q^{2}+3q^{3}+(5-2\beta _{1}+\beta _{2})q^{4}+\cdots\)
51.4.d.a \(8\) \(3.009\) \(\mathbb{Q}[x]/(x^{8} + \cdots)\) None \(-4\) \(0\) \(0\) \(0\) \(q+(-1+\beta _{2})q^{2}-\beta _{4}q^{3}+(5-\beta _{2}+\beta _{3}+\cdots)q^{4}+\cdots\)
51.4.e.a \(16\) \(3.009\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(0\) \(0\) \(-32\) \(-8\) \(q+(\beta _{1}-\beta _{2})q^{2}+\beta _{4}q^{3}+(-4-\beta _{7}+\cdots)q^{4}+\cdots\)
51.4.h.a \(40\) \(3.009\) None \(0\) \(0\) \(32\) \(0\)
51.4.i.a \(128\) \(3.009\) None \(0\) \(-8\) \(0\) \(-16\)
51.5.b.a \(22\) \(5.272\) None \(0\) \(-10\) \(0\) \(-4\)
51.5.c.a \(1\) \(5.272\) \(\Q\) \(\Q(\sqrt{-51}) \) \(0\) \(-9\) \(1\) \(0\) \(q-9q^{3}+2^{4}q^{4}+q^{5}+3^{4}q^{9}+217q^{11}+\cdots\)
51.5.c.b \(1\) \(5.272\) \(\Q\) \(\Q(\sqrt{-51}) \) \(0\) \(9\) \(-1\) \(0\) \(q+9q^{3}+2^{4}q^{4}-q^{5}+3^{4}q^{9}-217q^{11}+\cdots\)
51.5.c.c \(20\) \(5.272\) \(\mathbb{Q}[x]/(x^{20} + \cdots)\) None \(0\) \(0\) \(0\) \(0\) \(q+\beta _{1}q^{2}+\beta _{6}q^{3}+(-8-\beta _{4})q^{4}-\beta _{14}q^{5}+\cdots\)
51.5.f.a \(44\) \(5.272\) None \(0\) \(6\) \(0\) \(-4\)
51.5.g.a \(88\) \(5.272\) None \(0\) \(-4\) \(0\) \(-8\)
51.5.j.a \(96\) \(5.272\) None \(0\) \(0\) \(0\) \(0\)
51.6.a.a \(2\) \(8.180\) \(\Q(\sqrt{145}) \) None \(-7\) \(18\) \(-113\) \(0\) \(+\) \(q+(-3-\beta )q^{2}+9q^{3}+(13+7\beta )q^{4}+\cdots\)
51.6.a.b \(3\) \(8.180\) 3.3.76361.1 None \(5\) \(-27\) \(-37\) \(-176\) \(+\) \(q+(2-\beta _{1})q^{2}-9q^{3}+(12+\beta _{1}+2\beta _{2})q^{4}+\cdots\)
51.6.a.c \(4\) \(8.180\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(5\) \(36\) \(146\) \(60\) \(-\) \(q+(1+\beta _{1})q^{2}+9q^{3}+(3^{3}+4\beta _{2}+\beta _{3})q^{4}+\cdots\)
51.6.a.d \(5\) \(8.180\) \(\mathbb{Q}[x]/(x^{5} - \cdots)\) None \(1\) \(-45\) \(4\) \(-40\) \(-\) \(q+\beta _{1}q^{2}-9q^{3}+(24+\beta _{1}+2\beta _{2}+\beta _{4})q^{4}+\cdots\)
51.6.d.a \(16\) \(8.180\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None \(8\) \(0\) \(0\) \(0\) \(q-\beta _{3}q^{2}+\beta _{9}q^{3}+(2^{4}-\beta _{2}-\beta _{3})q^{4}+\cdots\)
51.6.e.a \(32\) \(8.180\) None \(0\) \(0\) \(76\) \(236\)
51.6.h.a \(56\) \(8.180\) None \(0\) \(0\) \(-88\) \(0\)
51.6.i.a \(224\) \(8.180\) None \(0\) \(-8\) \(0\) \(-16\)
51.7.b.a \(32\) \(11.733\) None \(0\) \(32\) \(0\) \(568\)
51.7.c.a \(1\) \(11.733\) \(\Q\) \(\Q(\sqrt{-51}) \) \(0\) \(-27\) \(-182\) \(0\) \(q-3^{3}q^{3}+2^{6}q^{4}-182q^{5}+3^{6}q^{9}+\cdots\)
51.7.c.b \(1\) \(11.733\) \(\Q\) \(\Q(\sqrt{-51}) \) \(0\) \(27\) \(182\) \(0\) \(q+3^{3}q^{3}+2^{6}q^{4}+182q^{5}+3^{6}q^{9}+\cdots\)
51.7.c.c \(32\) \(11.733\) None \(0\) \(0\) \(0\) \(0\)
51.7.f.a \(68\) \(11.733\) None \(0\) \(18\) \(0\) \(-4\)
51.7.g.a \(136\) \(11.733\) None \(0\) \(-4\) \(0\) \(-8\)
51.7.j.a \(144\) \(11.733\) None \(0\) \(0\) \(0\) \(0\)
51.8.a.a \(2\) \(15.932\) \(\Q(\sqrt{1177}) \) None \(7\) \(54\) \(320\) \(-1500\) \(+\) \(q+(4-\beta )q^{2}+3^{3}q^{3}+(182-7\beta )q^{4}+\cdots\)
51.8.a.b \(3\) \(15.932\) 3.3.1514860.1 None \(-7\) \(-81\) \(-440\) \(808\) \(-\) \(q+(-2-\beta _{1})q^{2}-3^{3}q^{3}+(78-2\beta _{1}+\cdots)q^{4}+\cdots\)
51.8.a.c \(4\) \(15.932\) \(\mathbb{Q}[x]/(x^{4} - \cdots)\) None \(-15\) \(108\) \(-310\) \(-1036\) \(-\) \(q+(-4+\beta _{1})q^{2}+3^{3}q^{3}+(39-7\beta _{1}+\cdots)q^{4}+\cdots\)
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