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Label Dim. \(A\) Field CM RM Traces Fricke sign $q$-expansion
\(a_2\) \(a_3\) \(a_5\) \(a_7\)
59.1.b.a \(1\) \(0.029\) \(\Q\) \(\Q(\sqrt{-59}) \) None \(0\) \(-1\) \(-1\) \(-1\) \(q-q^{3}+q^{4}-q^{5}-q^{7}-q^{12}+q^{15}+\cdots\)
59.2.a.a \(5\) \(0.471\) 5.5.138136.1 None None \(0\) \(-2\) \(2\) \(2\) \(-\) \(q+(\beta _{1}-\beta _{3})q^{2}+(-1-\beta _{2}+\beta _{4})q^{3}+\cdots\)
59.2.c.a \(112\) \(0.471\) None None \(-26\) \(-23\) \(-25\) \(-23\)
59.3.b.a \(3\) \(1.608\) 3.3.1593.1 \(\Q(\sqrt{-59}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(\beta _{1}+\beta _{2})q^{3}+4q^{4}+(-\beta _{1}-2\beta _{2})q^{5}+\cdots\)
59.3.b.b \(6\) \(1.608\) 6.0.7196038400.1 None None \(0\) \(4\) \(-8\) \(6\) \(q+\beta _{1}q^{2}+(1+\beta _{3})q^{3}+(-5-\beta _{3}+\beta _{4}+\cdots)q^{4}+\cdots\)
59.3.d.a \(252\) \(1.608\) None None \(-29\) \(-33\) \(-21\) \(-35\)
59.4.a.a \(2\) \(3.481\) \(\Q(\sqrt{5}) \) None None \(-4\) \(-2\) \(6\) \(-22\) \(-\) \(q+(-2-\beta )q^{2}-q^{3}+(1+4\beta )q^{4}+(3+\cdots)q^{5}+\cdots\)
59.4.a.b \(2\) \(3.481\) \(\Q(\sqrt{17}) \) None None \(1\) \(-1\) \(-31\) \(-8\) \(-\) \(q+\beta q^{2}+(1-3\beta )q^{3}+(-4+\beta )q^{4}+\cdots\)
59.4.a.c \(10\) \(3.481\) \(\mathbb{Q}[x]/(x^{10} - \cdots)\) None None \(3\) \(3\) \(15\) \(40\) \(+\) \(q+\beta _{1}q^{2}-\beta _{6}q^{3}+(6+\beta _{2})q^{4}+(2+\beta _{1}+\cdots)q^{5}+\cdots\)
59.4.c.a \(392\) \(3.481\) None None \(-29\) \(-29\) \(-19\) \(-39\)
59.5.b.a \(3\) \(6.099\) 3.3.1593.1 \(\Q(\sqrt{-59}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(4\beta _{1}-3\beta _{2})q^{3}+2^{4}q^{4}+(5\beta _{1}-11\beta _{2})q^{5}+\cdots\)
59.5.b.b \(16\) \(6.099\) \(\mathbb{Q}[x]/(x^{16} + \cdots)\) None None \(0\) \(-20\) \(4\) \(-82\) \(q+\beta _{1}q^{2}+(-1-\beta _{4})q^{3}+(-12+\beta _{2}+\cdots)q^{4}+\cdots\)
59.5.d.a \(532\) \(6.099\) None None \(-29\) \(-9\) \(-33\) \(53\)
59.6.a.a \(9\) \(9.463\) \(\mathbb{Q}[x]/(x^{9} - \cdots)\) None None \(-9\) \(0\) \(-72\) \(-208\) \(+\) \(q+(-1+\beta _{1})q^{2}+(-\beta _{1}+\beta _{5})q^{3}+(7+\cdots)q^{4}+\cdots\)
59.6.a.b \(15\) \(9.463\) \(\mathbb{Q}[x]/(x^{15} - \cdots)\) None None \(3\) \(18\) \(128\) \(282\) \(-\) \(q+\beta _{1}q^{2}+(1-\beta _{7})q^{3}+(20+\beta _{2})q^{4}+\cdots\)
59.6.c.a \(672\) \(9.463\) None None \(-23\) \(-47\) \(-85\) \(-103\)
59.7.b.a \(1\) \(13.573\) \(\Q\) \(\Q(\sqrt{-59}) \) None \(0\) \(-5\) \(191\) \(155\) \(q-5q^{3}+2^{6}q^{4}+191q^{5}+155q^{7}+\cdots\)
59.7.b.b \(2\) \(13.573\) \(\Q(\sqrt{177}) \) \(\Q(\sqrt{-59}) \) None \(0\) \(5\) \(-191\) \(-155\) \(q+(-1+7\beta )q^{3}+2^{6}q^{4}+(-85-21\beta )q^{5}+\cdots\)
59.7.b.c \(26\) \(13.573\) None None \(0\) \(10\) \(142\) \(406\)
59.7.d.a \(812\) \(13.573\) None None \(-29\) \(-39\) \(-171\) \(-435\)
59.8.a.a \(14\) \(18.431\) \(\mathbb{Q}[x]/(x^{14} - \cdots)\) None None \(-9\) \(-54\) \(-430\) \(-2390\) \(-\) \(q+(-1+\beta _{1})q^{2}+(-4+\beta _{3})q^{3}+(41+\cdots)q^{4}+\cdots\)
59.8.a.b \(20\) \(18.431\) \(\mathbb{Q}[x]/(x^{20} - \cdots)\) None None \(15\) \(0\) \(570\) \(1040\) \(+\) \(q+(1-\beta _{1})q^{2}+\beta _{4}q^{3}+(77-\beta _{1}+\beta _{2}+\cdots)q^{4}+\cdots\)
59.8.c.a \(952\) \(18.431\) None None \(-35\) \(25\) \(-169\) \(1321\)
59.9.b.a \(3\) \(24.035\) 3.3.1593.1 \(\Q(\sqrt{-59}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(-41\beta _{1}+22\beta _{2})q^{3}+2^{8}q^{4}+(-316\beta _{1}+\cdots)q^{5}+\cdots\)
59.9.b.b \(36\) \(24.035\) None None \(0\) \(40\) \(-296\) \(-162\)
59.9.d.a \(1092\) \(24.035\) None None \(-29\) \(-69\) \(267\) \(133\)
59.10.a.a \(19\) \(30.387\) \(\mathbb{Q}[x]/(x^{19} - \cdots)\) None None \(-33\) \(-81\) \(-3297\) \(-7128\) \(+\) \(q+(-2+\beta _{1})q^{2}+(-4-\beta _{4})q^{3}+(189+\cdots)q^{4}+\cdots\)
59.10.a.b \(25\) \(30.387\) None None \(15\) \(81\) \(1703\) \(16882\) \(-\)
59.10.c.a \(1232\) \(30.387\) None None \(-11\) \(-29\) \(1565\) \(-9783\)
59.11.b.a \(3\) \(37.486\) 3.3.1593.1 \(\Q(\sqrt{-59}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(-101\beta _{1}-66\beta _{2})q^{3}+2^{10}q^{4}+\cdots\)
59.11.b.b \(46\) \(37.486\) None None \(0\) \(16\) \(-3308\) \(-18394\)
59.11.d.a \(1372\) \(37.486\) None None \(-29\) \(-45\) \(3279\) \(18365\)
59.12.a.a \(23\) \(45.332\) None None \(-9\) \(-486\) \(-18310\) \(-83046\) \(-\)
59.12.a.b \(29\) \(45.332\) None None \(87\) \(0\) \(6690\) \(85024\) \(+\)
59.12.c.a \(1512\) \(45.332\) None None \(-59\) \(-47\) \(1931\) \(31481\)
59.13.b.a \(1\) \(53.926\) \(\Q\) \(\Q(\sqrt{-59}) \) None \(0\) \(-1433\) \(5231\) \(-211273\) \(q-1433q^{3}+2^{12}q^{4}+5231q^{5}-211273q^{7}+\cdots\)
59.13.b.b \(2\) \(53.926\) \(\Q(\sqrt{177}) \) \(\Q(\sqrt{-59}) \) None \(0\) \(1433\) \(-5231\) \(211273\) \(q+(734-35\beta )q^{3}+2^{12}q^{4}+(-610+\cdots)q^{5}+\cdots\)
59.13.b.c \(56\) \(53.926\) None None \(0\) \(-602\) \(10654\) \(191758\)
59.14.a.a \(29\) \(63.266\) None None \(-129\) \(0\) \(-41622\) \(-669728\) \(+\)
59.14.a.b \(35\) \(63.266\) None None \(63\) \(1458\) \(83378\) \(506762\) \(-\)
59.15.b.a \(3\) \(73.354\) 3.3.1593.1 \(\Q(\sqrt{-59}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(250\beta _{1}+113\beta _{2})q^{3}+2^{14}q^{4}+(8659\beta _{1}+\cdots)q^{5}+\cdots\)
59.15.b.b \(66\) \(73.354\) None None \(0\) \(76\) \(71992\) \(-1016394\)
59.16.a.a \(33\) \(84.189\) None None \(-345\) \(-2187\) \(-361015\) \(-5353406\) \(-\)
59.16.a.b \(39\) \(84.189\) None None \(39\) \(2187\) \(263985\) \(2882024\) \(+\)
59.17.b.a \(3\) \(95.771\) 3.3.1593.1 \(\Q(\sqrt{-59}) \) None \(0\) \(0\) \(0\) \(0\) \(q+(-413\beta _{1}-586\beta _{2})q^{3}+2^{16}q^{4}+\cdots\)
59.17.b.b \(76\) \(95.771\) None None \(0\) \(11716\) \(-338396\) \(767678\)
59.18.a.a \(38\) \(108.101\) None None \(15\) \(0\) \(-1062222\) \(-43925040\) \(+\)
59.18.a.b \(44\) \(108.101\) None None \(783\) \(13122\) \(2062778\) \(13722970\) \(-\)
59.19.b.a \(1\) \(121.178\) \(\Q\) \(\Q(\sqrt{-59}) \) None \(0\) \(10810\) \(-1985254\) \(-50982910\) \(q+10810q^{3}+2^{18}q^{4}-1985254q^{5}+\cdots\)
59.19.b.b \(2\) \(121.178\) \(\Q(\sqrt{177}) \) \(\Q(\sqrt{-59}) \) None \(0\) \(-10810\) \(1985254\) \(50982910\) \(q+(-5405-616\beta )q^{3}+2^{18}q^{4}+(992627+\cdots)q^{5}+\cdots\)
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