Properties

Label 48.24.a.a.1.1
Level $48$
Weight $24$
Character 48.1
Self dual yes
Analytic conductor $160.898$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 48.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(160.897937926\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 48.1

$q$-expansion

\(f(q)\) \(=\) \(q-177147. q^{3} -4.88637e7 q^{5} +1.72369e9 q^{7} +3.13811e10 q^{9} +O(q^{10})\) \(q-177147. q^{3} -4.88637e7 q^{5} +1.72369e9 q^{7} +3.13811e10 q^{9} +1.42826e12 q^{11} -8.22096e12 q^{13} +8.65606e12 q^{15} -5.98921e12 q^{17} -6.80005e14 q^{19} -3.05346e14 q^{21} -1.54406e13 q^{23} -9.53326e15 q^{25} -5.55906e15 q^{27} +1.15094e17 q^{29} +9.08297e16 q^{31} -2.53013e17 q^{33} -8.42259e16 q^{35} -1.29787e18 q^{37} +1.45632e18 q^{39} +5.21404e18 q^{41} +2.41043e18 q^{43} -1.53340e18 q^{45} +2.31327e19 q^{47} -2.43976e19 q^{49} +1.06097e18 q^{51} -4.45126e19 q^{53} -6.97903e19 q^{55} +1.20461e20 q^{57} +3.23974e20 q^{59} -1.99406e20 q^{61} +5.40912e19 q^{63} +4.01707e20 q^{65} +6.46393e20 q^{67} +2.73526e18 q^{69} -3.55146e21 q^{71} +3.35319e21 q^{73} +1.68879e21 q^{75} +2.46188e21 q^{77} +6.87213e21 q^{79} +9.84771e20 q^{81} +1.16977e21 q^{83} +2.92655e20 q^{85} -2.03886e22 q^{87} -2.34572e22 q^{89} -1.41704e22 q^{91} -1.60902e22 q^{93} +3.32276e22 q^{95} -3.06039e22 q^{97} +4.48204e22 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −177147. −0.577350
\(4\) 0 0
\(5\) −4.88637e7 −0.447540 −0.223770 0.974642i \(-0.571836\pi\)
−0.223770 + 0.974642i \(0.571836\pi\)
\(6\) 0 0
\(7\) 1.72369e9 0.329482 0.164741 0.986337i \(-0.447321\pi\)
0.164741 + 0.986337i \(0.447321\pi\)
\(8\) 0 0
\(9\) 3.13811e10 0.333333
\(10\) 0 0
\(11\) 1.42826e12 1.50936 0.754679 0.656094i \(-0.227793\pi\)
0.754679 + 0.656094i \(0.227793\pi\)
\(12\) 0 0
\(13\) −8.22096e12 −1.27226 −0.636128 0.771584i \(-0.719465\pi\)
−0.636128 + 0.771584i \(0.719465\pi\)
\(14\) 0 0
\(15\) 8.65606e12 0.258387
\(16\) 0 0
\(17\) −5.98921e12 −0.0423845 −0.0211922 0.999775i \(-0.506746\pi\)
−0.0211922 + 0.999775i \(0.506746\pi\)
\(18\) 0 0
\(19\) −6.80005e14 −1.33920 −0.669601 0.742721i \(-0.733535\pi\)
−0.669601 + 0.742721i \(0.733535\pi\)
\(20\) 0 0
\(21\) −3.05346e14 −0.190226
\(22\) 0 0
\(23\) −1.54406e13 −0.00337906 −0.00168953 0.999999i \(-0.500538\pi\)
−0.00168953 + 0.999999i \(0.500538\pi\)
\(24\) 0 0
\(25\) −9.53326e15 −0.799708
\(26\) 0 0
\(27\) −5.55906e15 −0.192450
\(28\) 0 0
\(29\) 1.15094e17 1.75177 0.875884 0.482522i \(-0.160279\pi\)
0.875884 + 0.482522i \(0.160279\pi\)
\(30\) 0 0
\(31\) 9.08297e16 0.642050 0.321025 0.947071i \(-0.395973\pi\)
0.321025 + 0.947071i \(0.395973\pi\)
\(32\) 0 0
\(33\) −2.53013e17 −0.871428
\(34\) 0 0
\(35\) −8.42259e16 −0.147456
\(36\) 0 0
\(37\) −1.29787e18 −1.19926 −0.599629 0.800278i \(-0.704685\pi\)
−0.599629 + 0.800278i \(0.704685\pi\)
\(38\) 0 0
\(39\) 1.45632e18 0.734537
\(40\) 0 0
\(41\) 5.21404e18 1.47965 0.739826 0.672798i \(-0.234908\pi\)
0.739826 + 0.672798i \(0.234908\pi\)
\(42\) 0 0
\(43\) 2.41043e18 0.395556 0.197778 0.980247i \(-0.436627\pi\)
0.197778 + 0.980247i \(0.436627\pi\)
\(44\) 0 0
\(45\) −1.53340e18 −0.149180
\(46\) 0 0
\(47\) 2.31327e19 1.36490 0.682449 0.730933i \(-0.260915\pi\)
0.682449 + 0.730933i \(0.260915\pi\)
\(48\) 0 0
\(49\) −2.43976e19 −0.891442
\(50\) 0 0
\(51\) 1.06097e18 0.0244707
\(52\) 0 0
\(53\) −4.45126e19 −0.659646 −0.329823 0.944043i \(-0.606989\pi\)
−0.329823 + 0.944043i \(0.606989\pi\)
\(54\) 0 0
\(55\) −6.97903e19 −0.675498
\(56\) 0 0
\(57\) 1.20461e20 0.773188
\(58\) 0 0
\(59\) 3.23974e20 1.39866 0.699331 0.714798i \(-0.253481\pi\)
0.699331 + 0.714798i \(0.253481\pi\)
\(60\) 0 0
\(61\) −1.99406e20 −0.586740 −0.293370 0.955999i \(-0.594777\pi\)
−0.293370 + 0.955999i \(0.594777\pi\)
\(62\) 0 0
\(63\) 5.40912e19 0.109827
\(64\) 0 0
\(65\) 4.01707e20 0.569385
\(66\) 0 0
\(67\) 6.46393e20 0.646601 0.323300 0.946296i \(-0.395208\pi\)
0.323300 + 0.946296i \(0.395208\pi\)
\(68\) 0 0
\(69\) 2.73526e18 0.00195090
\(70\) 0 0
\(71\) −3.55146e21 −1.82363 −0.911814 0.410604i \(-0.865318\pi\)
−0.911814 + 0.410604i \(0.865318\pi\)
\(72\) 0 0
\(73\) 3.35319e21 1.25096 0.625482 0.780238i \(-0.284902\pi\)
0.625482 + 0.780238i \(0.284902\pi\)
\(74\) 0 0
\(75\) 1.68879e21 0.461712
\(76\) 0 0
\(77\) 2.46188e21 0.497306
\(78\) 0 0
\(79\) 6.87213e21 1.03367 0.516833 0.856086i \(-0.327111\pi\)
0.516833 + 0.856086i \(0.327111\pi\)
\(80\) 0 0
\(81\) 9.84771e20 0.111111
\(82\) 0 0
\(83\) 1.16977e21 0.0997016 0.0498508 0.998757i \(-0.484125\pi\)
0.0498508 + 0.998757i \(0.484125\pi\)
\(84\) 0 0
\(85\) 2.92655e20 0.0189687
\(86\) 0 0
\(87\) −2.03886e22 −1.01138
\(88\) 0 0
\(89\) −2.34572e22 −0.895966 −0.447983 0.894042i \(-0.647857\pi\)
−0.447983 + 0.894042i \(0.647857\pi\)
\(90\) 0 0
\(91\) −1.41704e22 −0.419185
\(92\) 0 0
\(93\) −1.60902e22 −0.370687
\(94\) 0 0
\(95\) 3.32276e22 0.599346
\(96\) 0 0
\(97\) −3.06039e22 −0.434412 −0.217206 0.976126i \(-0.569694\pi\)
−0.217206 + 0.976126i \(0.569694\pi\)
\(98\) 0 0
\(99\) 4.48204e22 0.503119
\(100\) 0 0
\(101\) −2.39411e21 −0.0213525 −0.0106762 0.999943i \(-0.503398\pi\)
−0.0106762 + 0.999943i \(0.503398\pi\)
\(102\) 0 0
\(103\) 2.98735e22 0.212646 0.106323 0.994332i \(-0.466092\pi\)
0.106323 + 0.994332i \(0.466092\pi\)
\(104\) 0 0
\(105\) 1.49204e22 0.0851339
\(106\) 0 0
\(107\) −3.52639e23 −1.61963 −0.809817 0.586683i \(-0.800433\pi\)
−0.809817 + 0.586683i \(0.800433\pi\)
\(108\) 0 0
\(109\) −1.52076e23 −0.564489 −0.282245 0.959342i \(-0.591079\pi\)
−0.282245 + 0.959342i \(0.591079\pi\)
\(110\) 0 0
\(111\) 2.29914e23 0.692392
\(112\) 0 0
\(113\) 5.18685e22 0.127204 0.0636021 0.997975i \(-0.479741\pi\)
0.0636021 + 0.997975i \(0.479741\pi\)
\(114\) 0 0
\(115\) 7.54488e20 0.00151226
\(116\) 0 0
\(117\) −2.57983e23 −0.424085
\(118\) 0 0
\(119\) −1.03235e22 −0.0139649
\(120\) 0 0
\(121\) 1.14451e24 1.27816
\(122\) 0 0
\(123\) −9.23651e23 −0.854278
\(124\) 0 0
\(125\) 1.04833e24 0.805441
\(126\) 0 0
\(127\) 3.34992e23 0.214433 0.107217 0.994236i \(-0.465806\pi\)
0.107217 + 0.994236i \(0.465806\pi\)
\(128\) 0 0
\(129\) −4.27001e23 −0.228374
\(130\) 0 0
\(131\) 8.94767e23 0.400950 0.200475 0.979699i \(-0.435752\pi\)
0.200475 + 0.979699i \(0.435752\pi\)
\(132\) 0 0
\(133\) −1.17212e24 −0.441243
\(134\) 0 0
\(135\) 2.71636e23 0.0861291
\(136\) 0 0
\(137\) −1.52550e24 −0.408437 −0.204219 0.978925i \(-0.565465\pi\)
−0.204219 + 0.978925i \(0.565465\pi\)
\(138\) 0 0
\(139\) 2.87052e24 0.650565 0.325283 0.945617i \(-0.394541\pi\)
0.325283 + 0.945617i \(0.394541\pi\)
\(140\) 0 0
\(141\) −4.09788e24 −0.788025
\(142\) 0 0
\(143\) −1.17417e25 −1.92029
\(144\) 0 0
\(145\) −5.62393e24 −0.783985
\(146\) 0 0
\(147\) 4.32197e24 0.514674
\(148\) 0 0
\(149\) 7.06708e24 0.720440 0.360220 0.932867i \(-0.382702\pi\)
0.360220 + 0.932867i \(0.382702\pi\)
\(150\) 0 0
\(151\) 5.44882e24 0.476505 0.238253 0.971203i \(-0.423425\pi\)
0.238253 + 0.971203i \(0.423425\pi\)
\(152\) 0 0
\(153\) −1.87948e23 −0.0141282
\(154\) 0 0
\(155\) −4.43828e24 −0.287343
\(156\) 0 0
\(157\) −2.79179e25 −1.55968 −0.779841 0.625977i \(-0.784700\pi\)
−0.779841 + 0.625977i \(0.784700\pi\)
\(158\) 0 0
\(159\) 7.88528e24 0.380847
\(160\) 0 0
\(161\) −2.66149e22 −0.00111334
\(162\) 0 0
\(163\) −4.83707e25 −1.75560 −0.877799 0.479029i \(-0.840989\pi\)
−0.877799 + 0.479029i \(0.840989\pi\)
\(164\) 0 0
\(165\) 1.23631e25 0.389999
\(166\) 0 0
\(167\) −3.59666e25 −0.987779 −0.493890 0.869525i \(-0.664425\pi\)
−0.493890 + 0.869525i \(0.664425\pi\)
\(168\) 0 0
\(169\) 2.58303e25 0.618633
\(170\) 0 0
\(171\) −2.13393e25 −0.446401
\(172\) 0 0
\(173\) −6.18040e25 −1.13106 −0.565531 0.824727i \(-0.691329\pi\)
−0.565531 + 0.824727i \(0.691329\pi\)
\(174\) 0 0
\(175\) −1.64324e25 −0.263489
\(176\) 0 0
\(177\) −5.73911e25 −0.807518
\(178\) 0 0
\(179\) −4.18182e25 −0.517077 −0.258539 0.966001i \(-0.583241\pi\)
−0.258539 + 0.966001i \(0.583241\pi\)
\(180\) 0 0
\(181\) −1.04652e26 −1.13879 −0.569394 0.822065i \(-0.692822\pi\)
−0.569394 + 0.822065i \(0.692822\pi\)
\(182\) 0 0
\(183\) 3.53242e25 0.338754
\(184\) 0 0
\(185\) 6.34189e25 0.536715
\(186\) 0 0
\(187\) −8.55417e24 −0.0639734
\(188\) 0 0
\(189\) −9.58209e24 −0.0634088
\(190\) 0 0
\(191\) −1.25691e26 −0.736923 −0.368461 0.929643i \(-0.620115\pi\)
−0.368461 + 0.929643i \(0.620115\pi\)
\(192\) 0 0
\(193\) −8.55418e25 −0.444907 −0.222453 0.974943i \(-0.571407\pi\)
−0.222453 + 0.974943i \(0.571407\pi\)
\(194\) 0 0
\(195\) −7.11612e25 −0.328734
\(196\) 0 0
\(197\) −9.41370e25 −0.386722 −0.193361 0.981128i \(-0.561939\pi\)
−0.193361 + 0.981128i \(0.561939\pi\)
\(198\) 0 0
\(199\) −7.46484e25 −0.273030 −0.136515 0.990638i \(-0.543590\pi\)
−0.136515 + 0.990638i \(0.543590\pi\)
\(200\) 0 0
\(201\) −1.14506e26 −0.373315
\(202\) 0 0
\(203\) 1.98387e26 0.577175
\(204\) 0 0
\(205\) −2.54777e26 −0.662203
\(206\) 0 0
\(207\) −4.84544e23 −0.00112635
\(208\) 0 0
\(209\) −9.71227e26 −2.02134
\(210\) 0 0
\(211\) −6.91338e26 −1.28956 −0.644781 0.764368i \(-0.723051\pi\)
−0.644781 + 0.764368i \(0.723051\pi\)
\(212\) 0 0
\(213\) 6.29131e26 1.05287
\(214\) 0 0
\(215\) −1.17783e26 −0.177027
\(216\) 0 0
\(217\) 1.56562e26 0.211544
\(218\) 0 0
\(219\) −5.94007e26 −0.722245
\(220\) 0 0
\(221\) 4.92371e25 0.0539239
\(222\) 0 0
\(223\) −7.98521e26 −0.788462 −0.394231 0.919011i \(-0.628989\pi\)
−0.394231 + 0.919011i \(0.628989\pi\)
\(224\) 0 0
\(225\) −2.99164e26 −0.266569
\(226\) 0 0
\(227\) −1.19285e27 −0.960042 −0.480021 0.877257i \(-0.659371\pi\)
−0.480021 + 0.877257i \(0.659371\pi\)
\(228\) 0 0
\(229\) −1.64063e27 −1.19372 −0.596860 0.802345i \(-0.703585\pi\)
−0.596860 + 0.802345i \(0.703585\pi\)
\(230\) 0 0
\(231\) −4.36115e26 −0.287120
\(232\) 0 0
\(233\) −1.47808e27 −0.881263 −0.440632 0.897688i \(-0.645246\pi\)
−0.440632 + 0.897688i \(0.645246\pi\)
\(234\) 0 0
\(235\) −1.13035e27 −0.610846
\(236\) 0 0
\(237\) −1.21738e27 −0.596787
\(238\) 0 0
\(239\) 1.00529e26 0.0447421 0.0223711 0.999750i \(-0.492878\pi\)
0.0223711 + 0.999750i \(0.492878\pi\)
\(240\) 0 0
\(241\) 2.89067e27 1.16897 0.584483 0.811406i \(-0.301297\pi\)
0.584483 + 0.811406i \(0.301297\pi\)
\(242\) 0 0
\(243\) −1.74449e26 −0.0641500
\(244\) 0 0
\(245\) 1.19216e27 0.398956
\(246\) 0 0
\(247\) 5.59030e27 1.70381
\(248\) 0 0
\(249\) −2.07221e26 −0.0575628
\(250\) 0 0
\(251\) 3.03848e26 0.0769855 0.0384928 0.999259i \(-0.487744\pi\)
0.0384928 + 0.999259i \(0.487744\pi\)
\(252\) 0 0
\(253\) −2.20533e25 −0.00510021
\(254\) 0 0
\(255\) −5.18430e25 −0.0109516
\(256\) 0 0
\(257\) −5.74761e27 −1.10983 −0.554915 0.831907i \(-0.687249\pi\)
−0.554915 + 0.831907i \(0.687249\pi\)
\(258\) 0 0
\(259\) −2.23713e27 −0.395134
\(260\) 0 0
\(261\) 3.61178e27 0.583922
\(262\) 0 0
\(263\) 9.08470e27 1.34530 0.672650 0.739961i \(-0.265156\pi\)
0.672650 + 0.739961i \(0.265156\pi\)
\(264\) 0 0
\(265\) 2.17505e27 0.295218
\(266\) 0 0
\(267\) 4.15537e27 0.517286
\(268\) 0 0
\(269\) 2.45445e27 0.280416 0.140208 0.990122i \(-0.455223\pi\)
0.140208 + 0.990122i \(0.455223\pi\)
\(270\) 0 0
\(271\) −5.65329e27 −0.593136 −0.296568 0.955012i \(-0.595842\pi\)
−0.296568 + 0.955012i \(0.595842\pi\)
\(272\) 0 0
\(273\) 2.51024e27 0.242016
\(274\) 0 0
\(275\) −1.36160e28 −1.20705
\(276\) 0 0
\(277\) 3.55531e27 0.289975 0.144987 0.989433i \(-0.453686\pi\)
0.144987 + 0.989433i \(0.453686\pi\)
\(278\) 0 0
\(279\) 2.85033e27 0.214017
\(280\) 0 0
\(281\) 2.83906e28 1.96359 0.981797 0.189931i \(-0.0608265\pi\)
0.981797 + 0.189931i \(0.0608265\pi\)
\(282\) 0 0
\(283\) 2.29202e28 1.46108 0.730542 0.682868i \(-0.239268\pi\)
0.730542 + 0.682868i \(0.239268\pi\)
\(284\) 0 0
\(285\) −5.88617e27 −0.346033
\(286\) 0 0
\(287\) 8.98738e27 0.487518
\(288\) 0 0
\(289\) −1.99317e28 −0.998204
\(290\) 0 0
\(291\) 5.42139e27 0.250808
\(292\) 0 0
\(293\) 3.43109e28 1.46708 0.733542 0.679644i \(-0.237866\pi\)
0.733542 + 0.679644i \(0.237866\pi\)
\(294\) 0 0
\(295\) −1.58306e28 −0.625957
\(296\) 0 0
\(297\) −7.93980e27 −0.290476
\(298\) 0 0
\(299\) 1.26937e26 0.00429902
\(300\) 0 0
\(301\) 4.15484e27 0.130329
\(302\) 0 0
\(303\) 4.24110e26 0.0123279
\(304\) 0 0
\(305\) 9.74373e27 0.262589
\(306\) 0 0
\(307\) 1.16886e28 0.292195 0.146097 0.989270i \(-0.453329\pi\)
0.146097 + 0.989270i \(0.453329\pi\)
\(308\) 0 0
\(309\) −5.29200e27 −0.122771
\(310\) 0 0
\(311\) 2.49825e28 0.538135 0.269067 0.963121i \(-0.413285\pi\)
0.269067 + 0.963121i \(0.413285\pi\)
\(312\) 0 0
\(313\) 6.03885e28 1.20835 0.604177 0.796850i \(-0.293502\pi\)
0.604177 + 0.796850i \(0.293502\pi\)
\(314\) 0 0
\(315\) −2.64310e27 −0.0491521
\(316\) 0 0
\(317\) −1.04255e29 −1.80266 −0.901330 0.433133i \(-0.857408\pi\)
−0.901330 + 0.433133i \(0.857408\pi\)
\(318\) 0 0
\(319\) 1.64385e29 2.64404
\(320\) 0 0
\(321\) 6.24689e28 0.935096
\(322\) 0 0
\(323\) 4.07270e27 0.0567614
\(324\) 0 0
\(325\) 7.83726e28 1.01743
\(326\) 0 0
\(327\) 2.69398e28 0.325908
\(328\) 0 0
\(329\) 3.98735e28 0.449709
\(330\) 0 0
\(331\) −1.38759e29 −1.45962 −0.729809 0.683651i \(-0.760391\pi\)
−0.729809 + 0.683651i \(0.760391\pi\)
\(332\) 0 0
\(333\) −4.07286e28 −0.399753
\(334\) 0 0
\(335\) −3.15851e28 −0.289379
\(336\) 0 0
\(337\) −1.50795e29 −1.29016 −0.645078 0.764117i \(-0.723175\pi\)
−0.645078 + 0.764117i \(0.723175\pi\)
\(338\) 0 0
\(339\) −9.18835e27 −0.0734414
\(340\) 0 0
\(341\) 1.29729e29 0.969083
\(342\) 0 0
\(343\) −8.92291e28 −0.623196
\(344\) 0 0
\(345\) −1.33655e26 −0.000873105 0
\(346\) 0 0
\(347\) 1.45030e29 0.886482 0.443241 0.896403i \(-0.353829\pi\)
0.443241 + 0.896403i \(0.353829\pi\)
\(348\) 0 0
\(349\) 1.03224e29 0.590592 0.295296 0.955406i \(-0.404582\pi\)
0.295296 + 0.955406i \(0.404582\pi\)
\(350\) 0 0
\(351\) 4.57008e28 0.244846
\(352\) 0 0
\(353\) −9.96681e28 −0.500204 −0.250102 0.968220i \(-0.580464\pi\)
−0.250102 + 0.968220i \(0.580464\pi\)
\(354\) 0 0
\(355\) 1.73538e29 0.816146
\(356\) 0 0
\(357\) 1.82878e27 0.00806265
\(358\) 0 0
\(359\) −1.99929e29 −0.826589 −0.413294 0.910597i \(-0.635622\pi\)
−0.413294 + 0.910597i \(0.635622\pi\)
\(360\) 0 0
\(361\) 2.04578e29 0.793461
\(362\) 0 0
\(363\) −2.02746e29 −0.737948
\(364\) 0 0
\(365\) −1.63849e29 −0.559856
\(366\) 0 0
\(367\) −5.60764e29 −1.79937 −0.899684 0.436541i \(-0.856203\pi\)
−0.899684 + 0.436541i \(0.856203\pi\)
\(368\) 0 0
\(369\) 1.63622e29 0.493217
\(370\) 0 0
\(371\) −7.67259e28 −0.217341
\(372\) 0 0
\(373\) −4.46930e29 −1.19011 −0.595056 0.803684i \(-0.702870\pi\)
−0.595056 + 0.803684i \(0.702870\pi\)
\(374\) 0 0
\(375\) −1.85709e29 −0.465022
\(376\) 0 0
\(377\) −9.46185e29 −2.22869
\(378\) 0 0
\(379\) 3.66574e29 0.812477 0.406239 0.913767i \(-0.366840\pi\)
0.406239 + 0.913767i \(0.366840\pi\)
\(380\) 0 0
\(381\) −5.93429e28 −0.123803
\(382\) 0 0
\(383\) 1.49291e29 0.293257 0.146628 0.989192i \(-0.453158\pi\)
0.146628 + 0.989192i \(0.453158\pi\)
\(384\) 0 0
\(385\) −1.20297e29 −0.222564
\(386\) 0 0
\(387\) 7.56420e28 0.131852
\(388\) 0 0
\(389\) −1.97534e29 −0.324506 −0.162253 0.986749i \(-0.551876\pi\)
−0.162253 + 0.986749i \(0.551876\pi\)
\(390\) 0 0
\(391\) 9.24773e25 0.000143220 0
\(392\) 0 0
\(393\) −1.58505e29 −0.231488
\(394\) 0 0
\(395\) −3.35798e29 −0.462606
\(396\) 0 0
\(397\) −2.62393e28 −0.0341084 −0.0170542 0.999855i \(-0.505429\pi\)
−0.0170542 + 0.999855i \(0.505429\pi\)
\(398\) 0 0
\(399\) 2.07637e29 0.254751
\(400\) 0 0
\(401\) 4.34375e29 0.503159 0.251579 0.967837i \(-0.419050\pi\)
0.251579 + 0.967837i \(0.419050\pi\)
\(402\) 0 0
\(403\) −7.46708e29 −0.816851
\(404\) 0 0
\(405\) −4.81196e28 −0.0497266
\(406\) 0 0
\(407\) −1.85370e30 −1.81011
\(408\) 0 0
\(409\) −8.14589e29 −0.751832 −0.375916 0.926654i \(-0.622672\pi\)
−0.375916 + 0.926654i \(0.622672\pi\)
\(410\) 0 0
\(411\) 2.70238e29 0.235811
\(412\) 0 0
\(413\) 5.58431e29 0.460833
\(414\) 0 0
\(415\) −5.71593e28 −0.0446204
\(416\) 0 0
\(417\) −5.08505e29 −0.375604
\(418\) 0 0
\(419\) −1.28769e29 −0.0900221 −0.0450111 0.998986i \(-0.514332\pi\)
−0.0450111 + 0.998986i \(0.514332\pi\)
\(420\) 0 0
\(421\) −1.44725e30 −0.957852 −0.478926 0.877855i \(-0.658974\pi\)
−0.478926 + 0.877855i \(0.658974\pi\)
\(422\) 0 0
\(423\) 7.25928e29 0.454966
\(424\) 0 0
\(425\) 5.70967e28 0.0338952
\(426\) 0 0
\(427\) −3.43714e29 −0.193320
\(428\) 0 0
\(429\) 2.08001e30 1.10868
\(430\) 0 0
\(431\) 1.55229e30 0.784303 0.392152 0.919901i \(-0.371731\pi\)
0.392152 + 0.919901i \(0.371731\pi\)
\(432\) 0 0
\(433\) 3.69055e30 1.76799 0.883997 0.467493i \(-0.154843\pi\)
0.883997 + 0.467493i \(0.154843\pi\)
\(434\) 0 0
\(435\) 9.96263e29 0.452634
\(436\) 0 0
\(437\) 1.04997e28 0.00452524
\(438\) 0 0
\(439\) −4.37828e29 −0.179045 −0.0895223 0.995985i \(-0.528534\pi\)
−0.0895223 + 0.995985i \(0.528534\pi\)
\(440\) 0 0
\(441\) −7.65624e29 −0.297147
\(442\) 0 0
\(443\) −1.47008e30 −0.541624 −0.270812 0.962632i \(-0.587292\pi\)
−0.270812 + 0.962632i \(0.587292\pi\)
\(444\) 0 0
\(445\) 1.14621e30 0.400980
\(446\) 0 0
\(447\) −1.25191e30 −0.415946
\(448\) 0 0
\(449\) −2.17842e29 −0.0687558 −0.0343779 0.999409i \(-0.510945\pi\)
−0.0343779 + 0.999409i \(0.510945\pi\)
\(450\) 0 0
\(451\) 7.44702e30 2.23333
\(452\) 0 0
\(453\) −9.65243e29 −0.275111
\(454\) 0 0
\(455\) 6.92418e29 0.187602
\(456\) 0 0
\(457\) −1.90386e30 −0.490454 −0.245227 0.969466i \(-0.578862\pi\)
−0.245227 + 0.969466i \(0.578862\pi\)
\(458\) 0 0
\(459\) 3.32944e28 0.00815690
\(460\) 0 0
\(461\) −4.96919e30 −1.15804 −0.579022 0.815312i \(-0.696566\pi\)
−0.579022 + 0.815312i \(0.696566\pi\)
\(462\) 0 0
\(463\) −1.11956e30 −0.248237 −0.124118 0.992267i \(-0.539610\pi\)
−0.124118 + 0.992267i \(0.539610\pi\)
\(464\) 0 0
\(465\) 7.86228e29 0.165897
\(466\) 0 0
\(467\) −8.69961e30 −1.74725 −0.873626 0.486598i \(-0.838238\pi\)
−0.873626 + 0.486598i \(0.838238\pi\)
\(468\) 0 0
\(469\) 1.11418e30 0.213043
\(470\) 0 0
\(471\) 4.94557e30 0.900483
\(472\) 0 0
\(473\) 3.44273e30 0.597036
\(474\) 0 0
\(475\) 6.48267e30 1.07097
\(476\) 0 0
\(477\) −1.39685e30 −0.219882
\(478\) 0 0
\(479\) 6.06140e29 0.0909315 0.0454658 0.998966i \(-0.485523\pi\)
0.0454658 + 0.998966i \(0.485523\pi\)
\(480\) 0 0
\(481\) 1.06698e31 1.52576
\(482\) 0 0
\(483\) 4.71474e27 0.000642786 0
\(484\) 0 0
\(485\) 1.49542e30 0.194417
\(486\) 0 0
\(487\) 4.30348e30 0.533626 0.266813 0.963748i \(-0.414029\pi\)
0.266813 + 0.963748i \(0.414029\pi\)
\(488\) 0 0
\(489\) 8.56873e30 1.01360
\(490\) 0 0
\(491\) 1.13566e31 1.28177 0.640885 0.767637i \(-0.278568\pi\)
0.640885 + 0.767637i \(0.278568\pi\)
\(492\) 0 0
\(493\) −6.89323e29 −0.0742478
\(494\) 0 0
\(495\) −2.19009e30 −0.225166
\(496\) 0 0
\(497\) −6.12161e30 −0.600852
\(498\) 0 0
\(499\) −9.99815e30 −0.937052 −0.468526 0.883450i \(-0.655215\pi\)
−0.468526 + 0.883450i \(0.655215\pi\)
\(500\) 0 0
\(501\) 6.37137e30 0.570295
\(502\) 0 0
\(503\) 6.16806e30 0.527372 0.263686 0.964609i \(-0.415062\pi\)
0.263686 + 0.964609i \(0.415062\pi\)
\(504\) 0 0
\(505\) 1.16985e29 0.00955609
\(506\) 0 0
\(507\) −4.57577e30 −0.357168
\(508\) 0 0
\(509\) 2.98637e30 0.222786 0.111393 0.993776i \(-0.464469\pi\)
0.111393 + 0.993776i \(0.464469\pi\)
\(510\) 0 0
\(511\) 5.77985e30 0.412170
\(512\) 0 0
\(513\) 3.78019e30 0.257729
\(514\) 0 0
\(515\) −1.45973e30 −0.0951677
\(516\) 0 0
\(517\) 3.30395e31 2.06012
\(518\) 0 0
\(519\) 1.09484e31 0.653019
\(520\) 0 0
\(521\) −8.51705e30 −0.486022 −0.243011 0.970024i \(-0.578135\pi\)
−0.243011 + 0.970024i \(0.578135\pi\)
\(522\) 0 0
\(523\) −1.27472e31 −0.696059 −0.348029 0.937484i \(-0.613149\pi\)
−0.348029 + 0.937484i \(0.613149\pi\)
\(524\) 0 0
\(525\) 2.91095e30 0.152126
\(526\) 0 0
\(527\) −5.43998e29 −0.0272129
\(528\) 0 0
\(529\) −2.08802e31 −0.999989
\(530\) 0 0
\(531\) 1.01667e31 0.466221
\(532\) 0 0
\(533\) −4.28644e31 −1.88250
\(534\) 0 0
\(535\) 1.72312e31 0.724850
\(536\) 0 0
\(537\) 7.40796e30 0.298535
\(538\) 0 0
\(539\) −3.48463e31 −1.34551
\(540\) 0 0
\(541\) −1.00626e31 −0.372342 −0.186171 0.982517i \(-0.559608\pi\)
−0.186171 + 0.982517i \(0.559608\pi\)
\(542\) 0 0
\(543\) 1.85387e31 0.657479
\(544\) 0 0
\(545\) 7.43099e30 0.252631
\(546\) 0 0
\(547\) 4.27216e31 1.39249 0.696247 0.717802i \(-0.254852\pi\)
0.696247 + 0.717802i \(0.254852\pi\)
\(548\) 0 0
\(549\) −6.25758e30 −0.195580
\(550\) 0 0
\(551\) −7.82647e31 −2.34597
\(552\) 0 0
\(553\) 1.18454e31 0.340574
\(554\) 0 0
\(555\) −1.12345e31 −0.309873
\(556\) 0 0
\(557\) −6.81696e31 −1.80408 −0.902040 0.431652i \(-0.857931\pi\)
−0.902040 + 0.431652i \(0.857931\pi\)
\(558\) 0 0
\(559\) −1.98161e31 −0.503248
\(560\) 0 0
\(561\) 1.51535e30 0.0369351
\(562\) 0 0
\(563\) −6.17906e30 −0.144569 −0.0722846 0.997384i \(-0.523029\pi\)
−0.0722846 + 0.997384i \(0.523029\pi\)
\(564\) 0 0
\(565\) −2.53449e30 −0.0569289
\(566\) 0 0
\(567\) 1.69744e30 0.0366091
\(568\) 0 0
\(569\) −1.87031e31 −0.387366 −0.193683 0.981064i \(-0.562043\pi\)
−0.193683 + 0.981064i \(0.562043\pi\)
\(570\) 0 0
\(571\) −2.56599e31 −0.510433 −0.255217 0.966884i \(-0.582147\pi\)
−0.255217 + 0.966884i \(0.582147\pi\)
\(572\) 0 0
\(573\) 2.22659e31 0.425463
\(574\) 0 0
\(575\) 1.47200e29 0.00270226
\(576\) 0 0
\(577\) −2.35477e31 −0.415362 −0.207681 0.978197i \(-0.566592\pi\)
−0.207681 + 0.978197i \(0.566592\pi\)
\(578\) 0 0
\(579\) 1.51535e31 0.256867
\(580\) 0 0
\(581\) 2.01632e30 0.0328499
\(582\) 0 0
\(583\) −6.35758e31 −0.995642
\(584\) 0 0
\(585\) 1.26060e31 0.189795
\(586\) 0 0
\(587\) −7.54955e31 −1.09291 −0.546453 0.837490i \(-0.684022\pi\)
−0.546453 + 0.837490i \(0.684022\pi\)
\(588\) 0 0
\(589\) −6.17647e31 −0.859834
\(590\) 0 0
\(591\) 1.66761e31 0.223274
\(592\) 0 0
\(593\) 3.26412e31 0.420376 0.210188 0.977661i \(-0.432592\pi\)
0.210188 + 0.977661i \(0.432592\pi\)
\(594\) 0 0
\(595\) 5.04446e29 0.00624985
\(596\) 0 0
\(597\) 1.32237e31 0.157634
\(598\) 0 0
\(599\) −5.33038e31 −0.611434 −0.305717 0.952122i \(-0.598896\pi\)
−0.305717 + 0.952122i \(0.598896\pi\)
\(600\) 0 0
\(601\) 1.42622e32 1.57446 0.787229 0.616661i \(-0.211515\pi\)
0.787229 + 0.616661i \(0.211515\pi\)
\(602\) 0 0
\(603\) 2.02845e31 0.215534
\(604\) 0 0
\(605\) −5.59248e31 −0.572029
\(606\) 0 0
\(607\) 1.51222e32 1.48917 0.744586 0.667526i \(-0.232647\pi\)
0.744586 + 0.667526i \(0.232647\pi\)
\(608\) 0 0
\(609\) −3.51436e31 −0.333232
\(610\) 0 0
\(611\) −1.90173e32 −1.73650
\(612\) 0 0
\(613\) −1.24132e32 −1.09166 −0.545831 0.837895i \(-0.683786\pi\)
−0.545831 + 0.837895i \(0.683786\pi\)
\(614\) 0 0
\(615\) 4.51330e31 0.382323
\(616\) 0 0
\(617\) −1.89291e32 −1.54472 −0.772361 0.635184i \(-0.780924\pi\)
−0.772361 + 0.635184i \(0.780924\pi\)
\(618\) 0 0
\(619\) 1.33375e32 1.04865 0.524327 0.851517i \(-0.324317\pi\)
0.524327 + 0.851517i \(0.324317\pi\)
\(620\) 0 0
\(621\) 8.58355e28 0.000650300 0
\(622\) 0 0
\(623\) −4.04329e31 −0.295204
\(624\) 0 0
\(625\) 6.24200e31 0.439241
\(626\) 0 0
\(627\) 1.72050e32 1.16702
\(628\) 0 0
\(629\) 7.77324e30 0.0508299
\(630\) 0 0
\(631\) −2.80913e31 −0.177107 −0.0885533 0.996071i \(-0.528224\pi\)
−0.0885533 + 0.996071i \(0.528224\pi\)
\(632\) 0 0
\(633\) 1.22468e32 0.744529
\(634\) 0 0
\(635\) −1.63690e31 −0.0959674
\(636\) 0 0
\(637\) 2.00572e32 1.13414
\(638\) 0 0
\(639\) −1.11449e32 −0.607876
\(640\) 0 0
\(641\) −3.78628e31 −0.199225 −0.0996127 0.995026i \(-0.531760\pi\)
−0.0996127 + 0.995026i \(0.531760\pi\)
\(642\) 0 0
\(643\) 1.43262e32 0.727283 0.363641 0.931539i \(-0.381533\pi\)
0.363641 + 0.931539i \(0.381533\pi\)
\(644\) 0 0
\(645\) 2.08649e31 0.102207
\(646\) 0 0
\(647\) 2.32752e31 0.110026 0.0550129 0.998486i \(-0.482480\pi\)
0.0550129 + 0.998486i \(0.482480\pi\)
\(648\) 0 0
\(649\) 4.62721e32 2.11108
\(650\) 0 0
\(651\) −2.77345e31 −0.122135
\(652\) 0 0
\(653\) −4.07291e32 −1.73142 −0.865712 0.500543i \(-0.833134\pi\)
−0.865712 + 0.500543i \(0.833134\pi\)
\(654\) 0 0
\(655\) −4.37216e31 −0.179441
\(656\) 0 0
\(657\) 1.05227e32 0.416988
\(658\) 0 0
\(659\) 4.35631e32 1.66700 0.833501 0.552518i \(-0.186333\pi\)
0.833501 + 0.552518i \(0.186333\pi\)
\(660\) 0 0
\(661\) −4.22347e32 −1.56082 −0.780410 0.625268i \(-0.784990\pi\)
−0.780410 + 0.625268i \(0.784990\pi\)
\(662\) 0 0
\(663\) −8.72220e30 −0.0311330
\(664\) 0 0
\(665\) 5.72740e31 0.197474
\(666\) 0 0
\(667\) −1.77713e30 −0.00591932
\(668\) 0 0
\(669\) 1.41456e32 0.455218
\(670\) 0 0
\(671\) −2.84805e32 −0.885601
\(672\) 0 0
\(673\) 3.27631e32 0.984491 0.492245 0.870456i \(-0.336176\pi\)
0.492245 + 0.870456i \(0.336176\pi\)
\(674\) 0 0
\(675\) 5.29960e31 0.153904
\(676\) 0 0
\(677\) 1.65241e32 0.463820 0.231910 0.972737i \(-0.425503\pi\)
0.231910 + 0.972737i \(0.425503\pi\)
\(678\) 0 0
\(679\) −5.27516e31 −0.143131
\(680\) 0 0
\(681\) 2.11311e32 0.554281
\(682\) 0 0
\(683\) −1.17826e32 −0.298816 −0.149408 0.988776i \(-0.547737\pi\)
−0.149408 + 0.988776i \(0.547737\pi\)
\(684\) 0 0
\(685\) 7.45416e31 0.182792
\(686\) 0 0
\(687\) 2.90633e32 0.689194
\(688\) 0 0
\(689\) 3.65937e32 0.839238
\(690\) 0 0
\(691\) −2.43790e32 −0.540778 −0.270389 0.962751i \(-0.587152\pi\)
−0.270389 + 0.962751i \(0.587152\pi\)
\(692\) 0 0
\(693\) 7.72564e31 0.165769
\(694\) 0 0
\(695\) −1.40264e32 −0.291154
\(696\) 0 0
\(697\) −3.12280e31 −0.0627143
\(698\) 0 0
\(699\) 2.61838e32 0.508798
\(700\) 0 0
\(701\) −3.79931e32 −0.714409 −0.357205 0.934026i \(-0.616270\pi\)
−0.357205 + 0.934026i \(0.616270\pi\)
\(702\) 0 0
\(703\) 8.82561e32 1.60605
\(704\) 0 0
\(705\) 2.00238e32 0.352672
\(706\) 0 0
\(707\) −4.12670e30 −0.00703526
\(708\) 0 0
\(709\) −1.44569e31 −0.0238586 −0.0119293 0.999929i \(-0.503797\pi\)
−0.0119293 + 0.999929i \(0.503797\pi\)
\(710\) 0 0
\(711\) 2.15655e32 0.344555
\(712\) 0 0
\(713\) −1.40247e30 −0.00216952
\(714\) 0 0
\(715\) 5.73743e32 0.859405
\(716\) 0 0
\(717\) −1.78085e31 −0.0258319
\(718\) 0 0
\(719\) 1.33387e32 0.187383 0.0936915 0.995601i \(-0.470133\pi\)
0.0936915 + 0.995601i \(0.470133\pi\)
\(720\) 0 0
\(721\) 5.14926e31 0.0700631
\(722\) 0 0
\(723\) −5.12073e32 −0.674903
\(724\) 0 0
\(725\) −1.09722e33 −1.40090
\(726\) 0 0
\(727\) −4.93859e31 −0.0610882 −0.0305441 0.999533i \(-0.509724\pi\)
−0.0305441 + 0.999533i \(0.509724\pi\)
\(728\) 0 0
\(729\) 3.09032e31 0.0370370
\(730\) 0 0
\(731\) −1.44366e31 −0.0167654
\(732\) 0 0
\(733\) 1.05389e33 1.18604 0.593019 0.805189i \(-0.297936\pi\)
0.593019 + 0.805189i \(0.297936\pi\)
\(734\) 0 0
\(735\) −2.11188e32 −0.230337
\(736\) 0 0
\(737\) 9.23219e32 0.975952
\(738\) 0 0
\(739\) −1.09776e33 −1.12486 −0.562428 0.826846i \(-0.690133\pi\)
−0.562428 + 0.826846i \(0.690133\pi\)
\(740\) 0 0
\(741\) −9.90305e32 −0.983693
\(742\) 0 0
\(743\) 5.60678e32 0.539936 0.269968 0.962869i \(-0.412987\pi\)
0.269968 + 0.962869i \(0.412987\pi\)
\(744\) 0 0
\(745\) −3.45324e32 −0.322426
\(746\) 0 0
\(747\) 3.67086e31 0.0332339
\(748\) 0 0
\(749\) −6.07840e32 −0.533640
\(750\) 0 0
\(751\) 1.01215e33 0.861759 0.430880 0.902409i \(-0.358203\pi\)
0.430880 + 0.902409i \(0.358203\pi\)
\(752\) 0 0
\(753\) −5.38258e31 −0.0444476
\(754\) 0 0
\(755\) −2.66250e32 −0.213255
\(756\) 0 0
\(757\) −9.48447e32 −0.736903 −0.368452 0.929647i \(-0.620112\pi\)
−0.368452 + 0.929647i \(0.620112\pi\)
\(758\) 0 0
\(759\) 3.90668e30 0.00294461
\(760\) 0 0
\(761\) 9.81408e32 0.717672 0.358836 0.933401i \(-0.383174\pi\)
0.358836 + 0.933401i \(0.383174\pi\)
\(762\) 0 0
\(763\) −2.62131e32 −0.185989
\(764\) 0 0
\(765\) 9.18383e30 0.00632291
\(766\) 0 0
\(767\) −2.66338e33 −1.77945
\(768\) 0 0
\(769\) 8.57155e32 0.555785 0.277892 0.960612i \(-0.410364\pi\)
0.277892 + 0.960612i \(0.410364\pi\)
\(770\) 0 0
\(771\) 1.01817e33 0.640761
\(772\) 0 0
\(773\) −9.41593e31 −0.0575174 −0.0287587 0.999586i \(-0.509155\pi\)
−0.0287587 + 0.999586i \(0.509155\pi\)
\(774\) 0 0
\(775\) −8.65904e32 −0.513452
\(776\) 0 0
\(777\) 3.96301e32 0.228130
\(778\) 0 0
\(779\) −3.54557e33 −1.98155
\(780\) 0 0
\(781\) −5.07242e33 −2.75251
\(782\) 0 0
\(783\) −6.39816e32 −0.337128
\(784\) 0 0
\(785\) 1.36417e33 0.698020
\(786\) 0 0
\(787\) −2.72161e33 −1.35243 −0.676216 0.736703i \(-0.736382\pi\)
−0.676216 + 0.736703i \(0.736382\pi\)
\(788\) 0 0
\(789\) −1.60933e33 −0.776710
\(790\) 0 0
\(791\) 8.94051e31 0.0419115
\(792\) 0 0
\(793\) 1.63931e33 0.746483
\(794\) 0 0
\(795\) −3.85304e32 −0.170444
\(796\) 0 0
\(797\) −3.49544e33 −1.50221 −0.751106 0.660182i \(-0.770479\pi\)
−0.751106 + 0.660182i \(0.770479\pi\)
\(798\) 0 0
\(799\) −1.38546e32 −0.0578505
\(800\) 0 0
\(801\) −7.36112e32 −0.298655
\(802\) 0 0
\(803\) 4.78923e33 1.88815
\(804\) 0 0
\(805\) 1.30050e30 0.000498263 0
\(806\) 0 0
\(807\) −4.34799e32 −0.161898
\(808\) 0 0
\(809\) 1.04226e33 0.377196 0.188598 0.982054i \(-0.439606\pi\)
0.188598 + 0.982054i \(0.439606\pi\)
\(810\) 0 0
\(811\) 2.22771e33 0.783643 0.391821 0.920041i \(-0.371845\pi\)
0.391821 + 0.920041i \(0.371845\pi\)
\(812\) 0 0
\(813\) 1.00146e33 0.342447
\(814\) 0 0
\(815\) 2.36357e33 0.785700
\(816\) 0 0
\(817\) −1.63911e33 −0.529729
\(818\) 0 0
\(819\) −4.44682e32 −0.139728
\(820\) 0 0
\(821\) −7.36864e31 −0.0225134 −0.0112567 0.999937i \(-0.503583\pi\)
−0.0112567 + 0.999937i \(0.503583\pi\)
\(822\) 0 0
\(823\) −4.08369e33 −1.21326 −0.606631 0.794984i \(-0.707479\pi\)
−0.606631 + 0.794984i \(0.707479\pi\)
\(824\) 0 0
\(825\) 2.41204e33 0.696888
\(826\) 0 0
\(827\) −6.73455e32 −0.189232 −0.0946161 0.995514i \(-0.530162\pi\)
−0.0946161 + 0.995514i \(0.530162\pi\)
\(828\) 0 0
\(829\) −3.77086e33 −1.03054 −0.515268 0.857029i \(-0.672308\pi\)
−0.515268 + 0.857029i \(0.672308\pi\)
\(830\) 0 0
\(831\) −6.29813e32 −0.167417
\(832\) 0 0
\(833\) 1.46123e32 0.0377833
\(834\) 0 0
\(835\) 1.75746e33 0.442070
\(836\) 0 0
\(837\) −5.04928e32 −0.123562
\(838\) 0 0
\(839\) 4.99200e33 1.18853 0.594267 0.804268i \(-0.297442\pi\)
0.594267 + 0.804268i \(0.297442\pi\)
\(840\) 0 0
\(841\) 8.92995e33 2.06869
\(842\) 0 0
\(843\) −5.02931e33 −1.13368
\(844\) 0 0
\(845\) −1.26217e33 −0.276863
\(846\) 0 0
\(847\) 1.97277e33 0.421131
\(848\) 0 0
\(849\) −4.06025e33 −0.843557
\(850\) 0 0
\(851\) 2.00400e31 0.00405236
\(852\) 0 0
\(853\) −2.52826e33 −0.497631 −0.248815 0.968551i \(-0.580041\pi\)
−0.248815 + 0.968551i \(0.580041\pi\)
\(854\) 0 0
\(855\) 1.04272e33 0.199782
\(856\) 0 0
\(857\) 7.45015e33 1.38959 0.694793 0.719209i \(-0.255496\pi\)
0.694793 + 0.719209i \(0.255496\pi\)
\(858\) 0 0
\(859\) 5.27466e32 0.0957797 0.0478898 0.998853i \(-0.484750\pi\)
0.0478898 + 0.998853i \(0.484750\pi\)
\(860\) 0 0
\(861\) −1.59209e33 −0.281469
\(862\) 0 0
\(863\) 6.72847e33 1.15822 0.579112 0.815248i \(-0.303400\pi\)
0.579112 + 0.815248i \(0.303400\pi\)
\(864\) 0 0
\(865\) 3.01997e33 0.506195
\(866\) 0 0
\(867\) 3.53084e33 0.576313
\(868\) 0 0
\(869\) 9.81522e33 1.56017
\(870\) 0 0
\(871\) −5.31397e33 −0.822641
\(872\) 0 0
\(873\) −9.60382e32 −0.144804
\(874\) 0 0
\(875\) 1.80700e33 0.265378
\(876\) 0 0
\(877\) −7.45124e33 −1.06594 −0.532970 0.846134i \(-0.678924\pi\)
−0.532970 + 0.846134i \(0.678924\pi\)
\(878\) 0 0
\(879\) −6.07808e33 −0.847021
\(880\) 0 0
\(881\) 1.17245e34 1.59174 0.795870 0.605468i \(-0.207014\pi\)
0.795870 + 0.605468i \(0.207014\pi\)
\(882\) 0 0
\(883\) −1.05295e34 −1.39271 −0.696353 0.717699i \(-0.745195\pi\)
−0.696353 + 0.717699i \(0.745195\pi\)
\(884\) 0 0
\(885\) 2.80434e33 0.361396
\(886\) 0 0
\(887\) 6.86475e33 0.861992 0.430996 0.902354i \(-0.358162\pi\)
0.430996 + 0.902354i \(0.358162\pi\)
\(888\) 0 0
\(889\) 5.77423e32 0.0706519
\(890\) 0 0
\(891\) 1.40651e33 0.167706
\(892\) 0 0
\(893\) −1.57303e34 −1.82787
\(894\) 0 0
\(895\) 2.04339e33 0.231412
\(896\) 0 0
\(897\) −2.24865e31 −0.00248204
\(898\) 0 0
\(899\) 1.04540e34 1.12472
\(900\) 0 0
\(901\) 2.66596e32 0.0279587
\(902\) 0 0
\(903\) −7.36017e32 −0.0752452
\(904\) 0 0
\(905\) 5.11367e33 0.509653
\(906\) 0 0
\(907\) 9.01545e33 0.875999 0.438000 0.898975i \(-0.355687\pi\)
0.438000 + 0.898975i \(0.355687\pi\)
\(908\) 0 0
\(909\) −7.51298e31 −0.00711750
\(910\) 0 0
\(911\) −1.54259e34 −1.42492 −0.712459 0.701714i \(-0.752418\pi\)
−0.712459 + 0.701714i \(0.752418\pi\)
\(912\) 0 0
\(913\) 1.67074e33 0.150486
\(914\) 0 0
\(915\) −1.72607e33 −0.151606
\(916\) 0 0
\(917\) 1.54230e33 0.132106
\(918\) 0 0
\(919\) −3.21779e32 −0.0268800 −0.0134400 0.999910i \(-0.504278\pi\)
−0.0134400 + 0.999910i \(0.504278\pi\)
\(920\) 0 0
\(921\) −2.07060e33 −0.168699
\(922\) 0 0
\(923\) 2.91964e34 2.32012
\(924\) 0 0
\(925\) 1.23730e34 0.959056
\(926\) 0 0
\(927\) 9.37462e32 0.0708821
\(928\) 0 0
\(929\) 6.89145e33 0.508311 0.254156 0.967163i \(-0.418202\pi\)
0.254156 + 0.967163i \(0.418202\pi\)
\(930\) 0 0
\(931\) 1.65905e34 1.19382
\(932\) 0 0
\(933\) −4.42557e33 −0.310692
\(934\) 0 0
\(935\) 4.17989e32 0.0286306
\(936\) 0 0
\(937\) 6.25703e32 0.0418180 0.0209090 0.999781i \(-0.493344\pi\)
0.0209090 + 0.999781i \(0.493344\pi\)
\(938\) 0 0
\(939\) −1.06976e34 −0.697644
\(940\) 0 0
\(941\) 2.43281e34 1.54820 0.774101 0.633062i \(-0.218202\pi\)
0.774101 + 0.633062i \(0.218202\pi\)
\(942\) 0 0
\(943\) −8.05081e31 −0.00499983
\(944\) 0 0
\(945\) 4.68217e32 0.0283780
\(946\) 0 0
\(947\) −1.95908e34 −1.15885 −0.579424 0.815026i \(-0.696723\pi\)
−0.579424 + 0.815026i \(0.696723\pi\)
\(948\) 0 0
\(949\) −2.75664e34 −1.59155
\(950\) 0 0
\(951\) 1.84684e34 1.04077
\(952\) 0 0
\(953\) 2.99818e33 0.164926 0.0824632 0.996594i \(-0.473721\pi\)
0.0824632 + 0.996594i \(0.473721\pi\)
\(954\) 0 0
\(955\) 6.14175e33 0.329802
\(956\) 0 0
\(957\) −2.91203e34 −1.52654
\(958\) 0 0
\(959\) −2.62948e33 −0.134573
\(960\) 0 0
\(961\) −1.17633e34 −0.587772
\(962\) 0 0
\(963\) −1.10662e34 −0.539878
\(964\) 0 0
\(965\) 4.17989e33 0.199114
\(966\) 0 0
\(967\) −5.37477e33 −0.250009 −0.125004 0.992156i \(-0.539894\pi\)
−0.125004 + 0.992156i \(0.539894\pi\)
\(968\) 0 0
\(969\) −7.21466e32 −0.0327712
\(970\) 0 0
\(971\) −1.12238e34 −0.497872 −0.248936 0.968520i \(-0.580081\pi\)
−0.248936 + 0.968520i \(0.580081\pi\)
\(972\) 0 0
\(973\) 4.94789e33 0.214349
\(974\) 0 0
\(975\) −1.38835e34 −0.587415
\(976\) 0 0
\(977\) 2.41460e34 0.997831 0.498915 0.866651i \(-0.333732\pi\)
0.498915 + 0.866651i \(0.333732\pi\)
\(978\) 0 0
\(979\) −3.35031e34 −1.35233
\(980\) 0 0
\(981\) −4.77230e33 −0.188163
\(982\) 0 0
\(983\) 1.21263e34 0.467048 0.233524 0.972351i \(-0.424974\pi\)
0.233524 + 0.972351i \(0.424974\pi\)
\(984\) 0 0
\(985\) 4.59989e33 0.173073
\(986\) 0 0
\(987\) −7.06347e33 −0.259640
\(988\) 0 0
\(989\) −3.72187e31 −0.00133661
\(990\) 0 0
\(991\) −1.83350e34 −0.643329 −0.321664 0.946854i \(-0.604242\pi\)
−0.321664 + 0.946854i \(0.604242\pi\)
\(992\) 0 0
\(993\) 2.45807e34 0.842711
\(994\) 0 0
\(995\) 3.64760e33 0.122192
\(996\) 0 0
\(997\) 1.59227e34 0.521222 0.260611 0.965444i \(-0.416076\pi\)
0.260611 + 0.965444i \(0.416076\pi\)
\(998\) 0 0
\(999\) 7.21496e33 0.230797
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 48.24.a.a.1.1 1
4.3 odd 2 3.24.a.a.1.1 1
12.11 even 2 9.24.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.24.a.a.1.1 1 4.3 odd 2
9.24.a.a.1.1 1 12.11 even 2
48.24.a.a.1.1 1 1.1 even 1 trivial