Properties

Label 9.24.a.a.1.1
Level $9$
Weight $24$
Character 9.1
Self dual yes
Analytic conductor $30.168$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 9 = 3^{2} \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 9.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.1683633611\)
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 \(\chi\) \(=\) 9.1

$q$-expansion

\(f(q)\) \(=\) \(q-1128.00 q^{2} -7.11622e6 q^{4} +4.88637e7 q^{5} -1.72369e9 q^{7} +1.74895e10 q^{8} +O(q^{10})\) \(q-1128.00 q^{2} -7.11622e6 q^{4} +4.88637e7 q^{5} -1.72369e9 q^{7} +1.74895e10 q^{8} -5.51183e10 q^{10} +1.42826e12 q^{11} -8.22096e12 q^{13} +1.94432e12 q^{14} +3.99671e13 q^{16} +5.98921e12 q^{17} +6.80005e14 q^{19} -3.47725e14 q^{20} -1.61108e15 q^{22} -1.54406e13 q^{23} -9.53326e15 q^{25} +9.27325e15 q^{26} +1.22662e16 q^{28} -1.15094e17 q^{29} -9.08297e16 q^{31} -1.91795e17 q^{32} -6.75583e15 q^{34} -8.42259e16 q^{35} -1.29787e18 q^{37} -7.67046e17 q^{38} +8.54600e17 q^{40} -5.21404e18 q^{41} -2.41043e18 q^{43} -1.01638e19 q^{44} +1.74171e16 q^{46} +2.31327e19 q^{47} -2.43976e19 q^{49} +1.07535e19 q^{50} +5.85022e19 q^{52} +4.45126e19 q^{53} +6.97903e19 q^{55} -3.01464e19 q^{56} +1.29826e20 q^{58} +3.23974e20 q^{59} -1.99406e20 q^{61} +1.02456e20 q^{62} -1.18924e20 q^{64} -4.01707e20 q^{65} -6.46393e20 q^{67} -4.26206e19 q^{68} +9.50068e19 q^{70} -3.55146e21 q^{71} +3.35319e21 q^{73} +1.46400e21 q^{74} -4.83907e21 q^{76} -2.46188e21 q^{77} -6.87213e21 q^{79} +1.95294e21 q^{80} +5.88143e21 q^{82} +1.16977e21 q^{83} +2.92655e20 q^{85} +2.71897e21 q^{86} +2.49795e22 q^{88} +2.34572e22 q^{89} +1.41704e22 q^{91} +1.09879e20 q^{92} -2.60937e22 q^{94} +3.32276e22 q^{95} -3.06039e22 q^{97} +2.75205e22 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1128.00 −0.389461 −0.194731 0.980857i \(-0.562383\pi\)
−0.194731 + 0.980857i \(0.562383\pi\)
\(3\) 0 0
\(4\) −7.11622e6 −0.848320
\(5\) 4.88637e7 0.447540 0.223770 0.974642i \(-0.428164\pi\)
0.223770 + 0.974642i \(0.428164\pi\)
\(6\) 0 0
\(7\) −1.72369e9 −0.329482 −0.164741 0.986337i \(-0.552679\pi\)
−0.164741 + 0.986337i \(0.552679\pi\)
\(8\) 1.74895e10 0.719849
\(9\) 0 0
\(10\) −5.51183e10 −0.174299
\(11\) 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 \(0.719465\pi\)
\(14\) 1.94432e12 0.128320
\(15\) 0 0
\(16\) 3.99671e13 0.567967
\(17\) 5.98921e12 0.0423845 0.0211922 0.999775i \(-0.493254\pi\)
0.0211922 + 0.999775i \(0.493254\pi\)
\(18\) 0 0
\(19\) 6.80005e14 1.33920 0.669601 0.742721i \(-0.266465\pi\)
0.669601 + 0.742721i \(0.266465\pi\)
\(20\) −3.47725e14 −0.379657
\(21\) 0 0
\(22\) −1.61108e15 −0.587836
\(23\) −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
\(26\) 9.27325e15 0.495494
\(27\) 0 0
\(28\) 1.22662e16 0.279506
\(29\) −1.15094e17 −1.75177 −0.875884 0.482522i \(-0.839721\pi\)
−0.875884 + 0.482522i \(0.839721\pi\)
\(30\) 0 0
\(31\) −9.08297e16 −0.642050 −0.321025 0.947071i \(-0.604027\pi\)
−0.321025 + 0.947071i \(0.604027\pi\)
\(32\) −1.91795e17 −0.941050
\(33\) 0 0
\(34\) −6.75583e15 −0.0165071
\(35\) −8.42259e16 −0.147456
\(36\) 0 0
\(37\) −1.29787e18 −1.19926 −0.599629 0.800278i \(-0.704685\pi\)
−0.599629 + 0.800278i \(0.704685\pi\)
\(38\) −7.67046e17 −0.521567
\(39\) 0 0
\(40\) 8.54600e17 0.322161
\(41\) −5.21404e18 −1.47965 −0.739826 0.672798i \(-0.765092\pi\)
−0.739826 + 0.672798i \(0.765092\pi\)
\(42\) 0 0
\(43\) −2.41043e18 −0.395556 −0.197778 0.980247i \(-0.563373\pi\)
−0.197778 + 0.980247i \(0.563373\pi\)
\(44\) −1.01638e19 −1.28042
\(45\) 0 0
\(46\) 1.74171e16 0.00131601
\(47\) 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
\(50\) 1.07535e19 0.311455
\(51\) 0 0
\(52\) 5.85022e19 1.07928
\(53\) 4.45126e19 0.659646 0.329823 0.944043i \(-0.393011\pi\)
0.329823 + 0.944043i \(0.393011\pi\)
\(54\) 0 0
\(55\) 6.97903e19 0.675498
\(56\) −3.01464e19 −0.237177
\(57\) 0 0
\(58\) 1.29826e20 0.682245
\(59\) 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 \(0.594777\pi\)
\(62\) 1.02456e20 0.250053
\(63\) 0 0
\(64\) −1.18924e20 −0.201464
\(65\) −4.01707e20 −0.569385
\(66\) 0 0
\(67\) −6.46393e20 −0.646601 −0.323300 0.946296i \(-0.604792\pi\)
−0.323300 + 0.946296i \(0.604792\pi\)
\(68\) −4.26206e19 −0.0359556
\(69\) 0 0
\(70\) 9.50068e19 0.0574284
\(71\) −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 \(0.284902\pi\)
\(74\) 1.46400e21 0.467064
\(75\) 0 0
\(76\) −4.83907e21 −1.13607
\(77\) −2.46188e21 −0.497306
\(78\) 0 0
\(79\) −6.87213e21 −1.03367 −0.516833 0.856086i \(-0.672889\pi\)
−0.516833 + 0.856086i \(0.672889\pi\)
\(80\) 1.95294e21 0.254188
\(81\) 0 0
\(82\) 5.88143e21 0.576267
\(83\) 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
\(86\) 2.71897e21 0.154054
\(87\) 0 0
\(88\) 2.49795e22 1.08651
\(89\) 2.34572e22 0.895966 0.447983 0.894042i \(-0.352143\pi\)
0.447983 + 0.894042i \(0.352143\pi\)
\(90\) 0 0
\(91\) 1.41704e22 0.419185
\(92\) 1.09879e20 0.00286652
\(93\) 0 0
\(94\) −2.60937e22 −0.531575
\(95\) 3.32276e22 0.599346
\(96\) 0 0
\(97\) −3.06039e22 −0.434412 −0.217206 0.976126i \(-0.569694\pi\)
−0.217206 + 0.976126i \(0.569694\pi\)
\(98\) 2.75205e22 0.347182
\(99\) 0 0
\(100\) 6.78408e22 0.678408
\(101\) 2.39411e21 0.0213525 0.0106762 0.999943i \(-0.496602\pi\)
0.0106762 + 0.999943i \(0.496602\pi\)
\(102\) 0 0
\(103\) −2.98735e22 −0.212646 −0.106323 0.994332i \(-0.533908\pi\)
−0.106323 + 0.994332i \(0.533908\pi\)
\(104\) −1.43780e23 −0.915831
\(105\) 0 0
\(106\) −5.02102e22 −0.256906
\(107\) −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 \(0.591079\pi\)
\(110\) −7.87234e22 −0.263080
\(111\) 0 0
\(112\) −6.88909e22 −0.187135
\(113\) −5.18685e22 −0.127204 −0.0636021 0.997975i \(-0.520259\pi\)
−0.0636021 + 0.997975i \(0.520259\pi\)
\(114\) 0 0
\(115\) −7.54488e20 −0.00151226
\(116\) 8.19036e23 1.48606
\(117\) 0 0
\(118\) −3.65443e23 −0.544724
\(119\) −1.03235e22 −0.0139649
\(120\) 0 0
\(121\) 1.14451e24 1.27816
\(122\) 2.24930e23 0.228512
\(123\) 0 0
\(124\) 6.46365e23 0.544663
\(125\) −1.04833e24 −0.805441
\(126\) 0 0
\(127\) −3.34992e23 −0.214433 −0.107217 0.994236i \(-0.534194\pi\)
−0.107217 + 0.994236i \(0.534194\pi\)
\(128\) 1.74304e24 1.01951
\(129\) 0 0
\(130\) 4.53125e23 0.221753
\(131\) 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
\(134\) 7.29131e23 0.251826
\(135\) 0 0
\(136\) 1.04748e23 0.0305104
\(137\) 1.52550e24 0.408437 0.204219 0.978925i \(-0.434535\pi\)
0.204219 + 0.978925i \(0.434535\pi\)
\(138\) 0 0
\(139\) −2.87052e24 −0.650565 −0.325283 0.945617i \(-0.605459\pi\)
−0.325283 + 0.945617i \(0.605459\pi\)
\(140\) 5.99370e23 0.125090
\(141\) 0 0
\(142\) 4.00605e24 0.710232
\(143\) −1.17417e25 −1.92029
\(144\) 0 0
\(145\) −5.62393e24 −0.783985
\(146\) −3.78240e24 −0.487202
\(147\) 0 0
\(148\) 9.23596e24 1.01735
\(149\) −7.06708e24 −0.720440 −0.360220 0.932867i \(-0.617298\pi\)
−0.360220 + 0.932867i \(0.617298\pi\)
\(150\) 0 0
\(151\) −5.44882e24 −0.476505 −0.238253 0.971203i \(-0.576575\pi\)
−0.238253 + 0.971203i \(0.576575\pi\)
\(152\) 1.18929e25 0.964023
\(153\) 0 0
\(154\) 2.77700e24 0.193681
\(155\) −4.43828e24 −0.287343
\(156\) 0 0
\(157\) −2.79179e25 −1.55968 −0.779841 0.625977i \(-0.784700\pi\)
−0.779841 + 0.625977i \(0.784700\pi\)
\(158\) 7.75177e24 0.402573
\(159\) 0 0
\(160\) −9.37182e24 −0.421157
\(161\) 2.66149e22 0.00111334
\(162\) 0 0
\(163\) 4.83707e25 1.75560 0.877799 0.479029i \(-0.159011\pi\)
0.877799 + 0.479029i \(0.159011\pi\)
\(164\) 3.71042e25 1.25522
\(165\) 0 0
\(166\) −1.31950e24 −0.0388299
\(167\) −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
\(170\) −3.30115e23 −0.00738759
\(171\) 0 0
\(172\) 1.71532e25 0.335558
\(173\) 6.18040e25 1.13106 0.565531 0.824727i \(-0.308671\pi\)
0.565531 + 0.824727i \(0.308671\pi\)
\(174\) 0 0
\(175\) 1.64324e25 0.263489
\(176\) 5.70836e25 0.857265
\(177\) 0 0
\(178\) −2.64597e25 −0.348944
\(179\) −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 \(0.692822\pi\)
\(182\) −1.59842e25 −0.163256
\(183\) 0 0
\(184\) −2.70048e23 −0.00243241
\(185\) −6.34189e25 −0.536715
\(186\) 0 0
\(187\) 8.55417e24 0.0639734
\(188\) −1.64617e26 −1.15787
\(189\) 0 0
\(190\) −3.74807e25 −0.233422
\(191\) −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 \(0.571407\pi\)
\(194\) 3.45212e25 0.169187
\(195\) 0 0
\(196\) 1.73619e26 0.756228
\(197\) 9.41370e25 0.386722 0.193361 0.981128i \(-0.438061\pi\)
0.193361 + 0.981128i \(0.438061\pi\)
\(198\) 0 0
\(199\) 7.46484e25 0.273030 0.136515 0.990638i \(-0.456410\pi\)
0.136515 + 0.990638i \(0.456410\pi\)
\(200\) −1.66732e26 −0.575669
\(201\) 0 0
\(202\) −2.70056e24 −0.00831597
\(203\) 1.98387e26 0.577175
\(204\) 0 0
\(205\) −2.54777e26 −0.662203
\(206\) 3.36973e25 0.0828175
\(207\) 0 0
\(208\) −3.28568e26 −0.722599
\(209\) 9.71227e26 2.02134
\(210\) 0 0
\(211\) 6.91338e26 1.28956 0.644781 0.764368i \(-0.276949\pi\)
0.644781 + 0.764368i \(0.276949\pi\)
\(212\) −3.16762e26 −0.559591
\(213\) 0 0
\(214\) 3.97777e26 0.630784
\(215\) −1.17783e26 −0.177027
\(216\) 0 0
\(217\) 1.56562e26 0.211544
\(218\) 1.71542e26 0.219847
\(219\) 0 0
\(220\) −4.96643e26 −0.573038
\(221\) −4.92371e25 −0.0539239
\(222\) 0 0
\(223\) 7.98521e26 0.788462 0.394231 0.919011i \(-0.371011\pi\)
0.394231 + 0.919011i \(0.371011\pi\)
\(224\) 3.30595e26 0.310059
\(225\) 0 0
\(226\) 5.85077e25 0.0495411
\(227\) −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 \(0.703585\pi\)
\(230\) 8.51062e23 0.000588967 0
\(231\) 0 0
\(232\) −2.01293e27 −1.26101
\(233\) 1.47808e27 0.881263 0.440632 0.897688i \(-0.354754\pi\)
0.440632 + 0.897688i \(0.354754\pi\)
\(234\) 0 0
\(235\) 1.13035e27 0.610846
\(236\) −2.30547e27 −1.18651
\(237\) 0 0
\(238\) 1.16449e25 0.00543879
\(239\) 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 \(0.301297\pi\)
\(242\) −1.29100e27 −0.497795
\(243\) 0 0
\(244\) 1.41902e27 0.497743
\(245\) −1.19216e27 −0.398956
\(246\) 0 0
\(247\) −5.59030e27 −1.70381
\(248\) −1.58856e27 −0.462179
\(249\) 0 0
\(250\) 1.18252e27 0.313688
\(251\) 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
\(254\) 3.77872e26 0.0835134
\(255\) 0 0
\(256\) −9.68545e26 −0.195596
\(257\) 5.74761e27 1.10983 0.554915 0.831907i \(-0.312751\pi\)
0.554915 + 0.831907i \(0.312751\pi\)
\(258\) 0 0
\(259\) 2.23713e27 0.395134
\(260\) 2.85864e27 0.483020
\(261\) 0 0
\(262\) −1.00930e27 −0.156154
\(263\) 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
\(266\) 1.32215e27 0.171847
\(267\) 0 0
\(268\) 4.59987e27 0.548524
\(269\) −2.45445e27 −0.280416 −0.140208 0.990122i \(-0.544777\pi\)
−0.140208 + 0.990122i \(0.544777\pi\)
\(270\) 0 0
\(271\) 5.65329e27 0.593136 0.296568 0.955012i \(-0.404158\pi\)
0.296568 + 0.955012i \(0.404158\pi\)
\(272\) 2.39371e26 0.0240730
\(273\) 0 0
\(274\) −1.72076e27 −0.159070
\(275\) −1.36160e28 −1.20705
\(276\) 0 0
\(277\) 3.55531e27 0.289975 0.144987 0.989433i \(-0.453686\pi\)
0.144987 + 0.989433i \(0.453686\pi\)
\(278\) 3.23795e27 0.253370
\(279\) 0 0
\(280\) −1.47306e27 −0.106146
\(281\) −2.83906e28 −1.96359 −0.981797 0.189931i \(-0.939173\pi\)
−0.981797 + 0.189931i \(0.939173\pi\)
\(282\) 0 0
\(283\) −2.29202e28 −1.46108 −0.730542 0.682868i \(-0.760732\pi\)
−0.730542 + 0.682868i \(0.760732\pi\)
\(284\) 2.52730e28 1.54702
\(285\) 0 0
\(286\) 1.32446e28 0.747878
\(287\) 8.98738e27 0.487518
\(288\) 0 0
\(289\) −1.99317e28 −0.998204
\(290\) 6.34379e27 0.305332
\(291\) 0 0
\(292\) −2.38620e28 −1.06122
\(293\) −3.43109e28 −1.46708 −0.733542 0.679644i \(-0.762134\pi\)
−0.733542 + 0.679644i \(0.762134\pi\)
\(294\) 0 0
\(295\) 1.58306e28 0.625957
\(296\) −2.26991e28 −0.863284
\(297\) 0 0
\(298\) 7.97167e27 0.280584
\(299\) 1.26937e26 0.00429902
\(300\) 0 0
\(301\) 4.15484e27 0.130329
\(302\) 6.14627e27 0.185580
\(303\) 0 0
\(304\) 2.71779e28 0.760622
\(305\) −9.74373e27 −0.262589
\(306\) 0 0
\(307\) −1.16886e28 −0.292195 −0.146097 0.989270i \(-0.546671\pi\)
−0.146097 + 0.989270i \(0.546671\pi\)
\(308\) 1.75193e28 0.421875
\(309\) 0 0
\(310\) 5.00638e27 0.111909
\(311\) 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 \(0.293502\pi\)
\(314\) 3.14914e28 0.607436
\(315\) 0 0
\(316\) 4.89036e28 0.876879
\(317\) 1.04255e29 1.80266 0.901330 0.433133i \(-0.142592\pi\)
0.901330 + 0.433133i \(0.142592\pi\)
\(318\) 0 0
\(319\) −1.64385e29 −2.64404
\(320\) −5.81105e27 −0.0901633
\(321\) 0 0
\(322\) −3.00216e25 −0.000433602 0
\(323\) 4.07270e27 0.0567614
\(324\) 0 0
\(325\) 7.83726e28 1.01743
\(326\) −5.45622e28 −0.683737
\(327\) 0 0
\(328\) −9.11906e28 −1.06513
\(329\) −3.98735e28 −0.449709
\(330\) 0 0
\(331\) 1.38759e29 1.45962 0.729809 0.683651i \(-0.239609\pi\)
0.729809 + 0.683651i \(0.239609\pi\)
\(332\) −8.32434e27 −0.0845789
\(333\) 0 0
\(334\) 4.05703e28 0.384702
\(335\) −3.15851e28 −0.289379
\(336\) 0 0
\(337\) −1.50795e29 −1.29016 −0.645078 0.764117i \(-0.723175\pi\)
−0.645078 + 0.764117i \(0.723175\pi\)
\(338\) −2.91366e28 −0.240934
\(339\) 0 0
\(340\) −2.08260e27 −0.0160916
\(341\) −1.29729e29 −0.969083
\(342\) 0 0
\(343\) 8.92291e28 0.623196
\(344\) −4.21572e28 −0.284741
\(345\) 0 0
\(346\) −6.97149e28 −0.440504
\(347\) 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 \(0.404582\pi\)
\(350\) −1.85357e28 −0.102619
\(351\) 0 0
\(352\) −2.73934e29 −1.42038
\(353\) 9.96681e28 0.500204 0.250102 0.968220i \(-0.419536\pi\)
0.250102 + 0.968220i \(0.419536\pi\)
\(354\) 0 0
\(355\) −1.73538e29 −0.816146
\(356\) −1.66927e29 −0.760066
\(357\) 0 0
\(358\) 4.71709e28 0.201381
\(359\) −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
\(362\) 1.18047e29 0.443514
\(363\) 0 0
\(364\) −1.00840e29 −0.355603
\(365\) 1.63849e29 0.559856
\(366\) 0 0
\(367\) 5.60764e29 1.79937 0.899684 0.436541i \(-0.143797\pi\)
0.899684 + 0.436541i \(0.143797\pi\)
\(368\) −6.17118e26 −0.00191919
\(369\) 0 0
\(370\) 7.15366e28 0.209030
\(371\) −7.67259e28 −0.217341
\(372\) 0 0
\(373\) −4.46930e29 −1.19011 −0.595056 0.803684i \(-0.702870\pi\)
−0.595056 + 0.803684i \(0.702870\pi\)
\(374\) −9.64910e27 −0.0249151
\(375\) 0 0
\(376\) 4.04578e29 0.982521
\(377\) 9.46185e29 2.22869
\(378\) 0 0
\(379\) −3.66574e29 −0.812477 −0.406239 0.913767i \(-0.633160\pi\)
−0.406239 + 0.913767i \(0.633160\pi\)
\(380\) −2.36455e29 −0.508437
\(381\) 0 0
\(382\) 1.41780e29 0.287003
\(383\) 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
\(386\) 9.64911e28 0.173274
\(387\) 0 0
\(388\) 2.17784e29 0.368520
\(389\) 1.97534e29 0.324506 0.162253 0.986749i \(-0.448124\pi\)
0.162253 + 0.986749i \(0.448124\pi\)
\(390\) 0 0
\(391\) −9.24773e25 −0.000143220 0
\(392\) −4.26701e29 −0.641703
\(393\) 0 0
\(394\) −1.06187e29 −0.150613
\(395\) −3.35798e29 −0.462606
\(396\) 0 0
\(397\) −2.62393e28 −0.0341084 −0.0170542 0.999855i \(-0.505429\pi\)
−0.0170542 + 0.999855i \(0.505429\pi\)
\(398\) −8.42034e28 −0.106334
\(399\) 0 0
\(400\) −3.81017e29 −0.454208
\(401\) −4.34375e29 −0.503159 −0.251579 0.967837i \(-0.580950\pi\)
−0.251579 + 0.967837i \(0.580950\pi\)
\(402\) 0 0
\(403\) 7.46708e29 0.816851
\(404\) −1.70370e28 −0.0181137
\(405\) 0 0
\(406\) −2.23780e29 −0.224787
\(407\) −1.85370e30 −1.81011
\(408\) 0 0
\(409\) −8.14589e29 −0.751832 −0.375916 0.926654i \(-0.622672\pi\)
−0.375916 + 0.926654i \(0.622672\pi\)
\(410\) 2.87389e29 0.257902
\(411\) 0 0
\(412\) 2.12586e29 0.180392
\(413\) −5.58431e29 −0.460833
\(414\) 0 0
\(415\) 5.71593e28 0.0446204
\(416\) 1.57674e30 1.19726
\(417\) 0 0
\(418\) −1.09554e30 −0.787232
\(419\) −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 \(0.658974\pi\)
\(422\) −7.79829e29 −0.502234
\(423\) 0 0
\(424\) 7.78501e29 0.474845
\(425\) −5.70967e28 −0.0338952
\(426\) 0 0
\(427\) 3.43714e29 0.193320
\(428\) 2.50946e30 1.37397
\(429\) 0 0
\(430\) 1.32859e29 0.0689452
\(431\) 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 \(0.154843\pi\)
\(434\) −1.76602e29 −0.0823880
\(435\) 0 0
\(436\) 1.08221e30 0.478867
\(437\) −1.04997e28 −0.00452524
\(438\) 0 0
\(439\) 4.37828e29 0.179045 0.0895223 0.995985i \(-0.471466\pi\)
0.0895223 + 0.995985i \(0.471466\pi\)
\(440\) 1.22059e30 0.486256
\(441\) 0 0
\(442\) 5.55394e28 0.0210013
\(443\) −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
\(446\) −9.00732e29 −0.307075
\(447\) 0 0
\(448\) 2.04987e29 0.0663789
\(449\) 2.17842e29 0.0687558 0.0343779 0.999409i \(-0.489055\pi\)
0.0343779 + 0.999409i \(0.489055\pi\)
\(450\) 0 0
\(451\) −7.44702e30 −2.23333
\(452\) 3.69108e29 0.107910
\(453\) 0 0
\(454\) 1.34554e30 0.373899
\(455\) 6.92418e29 0.187602
\(456\) 0 0
\(457\) −1.90386e30 −0.490454 −0.245227 0.969466i \(-0.578862\pi\)
−0.245227 + 0.969466i \(0.578862\pi\)
\(458\) 1.85063e30 0.464908
\(459\) 0 0
\(460\) 5.36910e27 0.00128288
\(461\) 4.96919e30 1.15804 0.579022 0.815312i \(-0.303434\pi\)
0.579022 + 0.815312i \(0.303434\pi\)
\(462\) 0 0
\(463\) 1.11956e30 0.248237 0.124118 0.992267i \(-0.460390\pi\)
0.124118 + 0.992267i \(0.460390\pi\)
\(464\) −4.59998e30 −0.994946
\(465\) 0 0
\(466\) −1.66728e30 −0.343218
\(467\) −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
\(470\) −1.27503e30 −0.237901
\(471\) 0 0
\(472\) 5.66614e30 1.00682
\(473\) −3.44273e30 −0.597036
\(474\) 0 0
\(475\) −6.48267e30 −1.07097
\(476\) 7.34646e28 0.0118467
\(477\) 0 0
\(478\) −1.13397e29 −0.0174253
\(479\) 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
\(482\) −3.26067e30 −0.455267
\(483\) 0 0
\(484\) −8.14456e30 −1.08429
\(485\) −1.49542e30 −0.194417
\(486\) 0 0
\(487\) −4.30348e30 −0.533626 −0.266813 0.963748i \(-0.585971\pi\)
−0.266813 + 0.963748i \(0.585971\pi\)
\(488\) −3.48750e30 −0.422364
\(489\) 0 0
\(490\) 1.34476e30 0.155378
\(491\) 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
\(494\) 6.30586e30 0.663566
\(495\) 0 0
\(496\) −3.63020e30 −0.364663
\(497\) 6.12161e30 0.600852
\(498\) 0 0
\(499\) 9.99815e30 0.937052 0.468526 0.883450i \(-0.344785\pi\)
0.468526 + 0.883450i \(0.344785\pi\)
\(500\) 7.46016e30 0.683272
\(501\) 0 0
\(502\) −3.42741e29 −0.0299829
\(503\) 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
\(506\) 2.48761e28 0.00198633
\(507\) 0 0
\(508\) 2.38388e30 0.181908
\(509\) −2.98637e30 −0.222786 −0.111393 0.993776i \(-0.535531\pi\)
−0.111393 + 0.993776i \(0.535531\pi\)
\(510\) 0 0
\(511\) −5.77985e30 −0.412170
\(512\) −1.35292e31 −0.943335
\(513\) 0 0
\(514\) −6.48331e30 −0.432236
\(515\) −1.45973e30 −0.0951677
\(516\) 0 0
\(517\) 3.30395e31 2.06012
\(518\) −2.52348e30 −0.153889
\(519\) 0 0
\(520\) −7.02563e30 −0.409871
\(521\) 8.51705e30 0.486022 0.243011 0.970024i \(-0.421865\pi\)
0.243011 + 0.970024i \(0.421865\pi\)
\(522\) 0 0
\(523\) 1.27472e31 0.696059 0.348029 0.937484i \(-0.386851\pi\)
0.348029 + 0.937484i \(0.386851\pi\)
\(524\) −6.36736e30 −0.340134
\(525\) 0 0
\(526\) −1.02475e31 −0.523942
\(527\) −5.43998e29 −0.0272129
\(528\) 0 0
\(529\) −2.08802e31 −0.999989
\(530\) −2.45346e30 −0.114976
\(531\) 0 0
\(532\) 8.34105e30 0.374315
\(533\) 4.28644e31 1.88250
\(534\) 0 0
\(535\) −1.72312e31 −0.724850
\(536\) −1.13050e31 −0.465455
\(537\) 0 0
\(538\) 2.76862e30 0.109211
\(539\) −3.48463e31 −1.34551
\(540\) 0 0
\(541\) −1.00626e31 −0.372342 −0.186171 0.982517i \(-0.559608\pi\)
−0.186171 + 0.982517i \(0.559608\pi\)
\(542\) −6.37691e30 −0.231004
\(543\) 0 0
\(544\) −1.14870e30 −0.0398859
\(545\) −7.43099e30 −0.252631
\(546\) 0 0
\(547\) −4.27216e31 −1.39249 −0.696247 0.717802i \(-0.745148\pi\)
−0.696247 + 0.717802i \(0.745148\pi\)
\(548\) −1.08558e31 −0.346485
\(549\) 0 0
\(550\) 1.53589e31 0.470098
\(551\) −7.82647e31 −2.34597
\(552\) 0 0
\(553\) 1.18454e31 0.340574
\(554\) −4.01039e30 −0.112934
\(555\) 0 0
\(556\) 2.04273e31 0.551887
\(557\) 6.81696e31 1.80408 0.902040 0.431652i \(-0.142069\pi\)
0.902040 + 0.431652i \(0.142069\pi\)
\(558\) 0 0
\(559\) 1.98161e31 0.503248
\(560\) −3.36626e30 −0.0837502
\(561\) 0 0
\(562\) 3.20246e31 0.764744
\(563\) −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
\(566\) 2.58540e31 0.569035
\(567\) 0 0
\(568\) −6.21131e31 −1.31274
\(569\) 1.87031e31 0.387366 0.193683 0.981064i \(-0.437957\pi\)
0.193683 + 0.981064i \(0.437957\pi\)
\(570\) 0 0
\(571\) 2.56599e31 0.510433 0.255217 0.966884i \(-0.417853\pi\)
0.255217 + 0.966884i \(0.417853\pi\)
\(572\) 8.35566e31 1.62902
\(573\) 0 0
\(574\) −1.01378e31 −0.189870
\(575\) 1.47200e29 0.00270226
\(576\) 0 0
\(577\) −2.35477e31 −0.415362 −0.207681 0.978197i \(-0.566592\pi\)
−0.207681 + 0.978197i \(0.566592\pi\)
\(578\) 2.24830e31 0.388762
\(579\) 0 0
\(580\) 4.00212e31 0.665071
\(581\) −2.01632e30 −0.0328499
\(582\) 0 0
\(583\) 6.35758e31 0.995642
\(584\) 5.86454e31 0.900505
\(585\) 0 0
\(586\) 3.87027e31 0.571372
\(587\) −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
\(590\) −1.78569e31 −0.243786
\(591\) 0 0
\(592\) −5.18723e31 −0.681139
\(593\) −3.26412e31 −0.420376 −0.210188 0.977661i \(-0.567408\pi\)
−0.210188 + 0.977661i \(0.567408\pi\)
\(594\) 0 0
\(595\) −5.04446e29 −0.00624985
\(596\) 5.02909e31 0.611164
\(597\) 0 0
\(598\) −1.43185e29 −0.00167430
\(599\) −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 \(0.211515\pi\)
\(602\) −4.68666e30 −0.0507579
\(603\) 0 0
\(604\) 3.87750e31 0.404229
\(605\) 5.59248e31 0.572029
\(606\) 0 0
\(607\) −1.51222e32 −1.48917 −0.744586 0.667526i \(-0.767353\pi\)
−0.744586 + 0.667526i \(0.767353\pi\)
\(608\) −1.30422e32 −1.26026
\(609\) 0 0
\(610\) 1.09909e31 0.102268
\(611\) −1.90173e32 −1.73650
\(612\) 0 0
\(613\) −1.24132e32 −1.09166 −0.545831 0.837895i \(-0.683786\pi\)
−0.545831 + 0.837895i \(0.683786\pi\)
\(614\) 1.31848e31 0.113798
\(615\) 0 0
\(616\) −4.30569e31 −0.357985
\(617\) 1.89291e32 1.54472 0.772361 0.635184i \(-0.219076\pi\)
0.772361 + 0.635184i \(0.219076\pi\)
\(618\) 0 0
\(619\) −1.33375e32 −1.04865 −0.524327 0.851517i \(-0.675683\pi\)
−0.524327 + 0.851517i \(0.675683\pi\)
\(620\) 3.15838e31 0.243759
\(621\) 0 0
\(622\) −2.81802e31 −0.209583
\(623\) −4.04329e31 −0.295204
\(624\) 0 0
\(625\) 6.24200e31 0.439241
\(626\) −6.81182e31 −0.470607
\(627\) 0 0
\(628\) 1.98670e32 1.32311
\(629\) −7.77324e30 −0.0508299
\(630\) 0 0
\(631\) 2.80913e31 0.177107 0.0885533 0.996071i \(-0.471776\pi\)
0.0885533 + 0.996071i \(0.471776\pi\)
\(632\) −1.20190e32 −0.744083
\(633\) 0 0
\(634\) −1.17599e32 −0.702066
\(635\) −1.63690e31 −0.0959674
\(636\) 0 0
\(637\) 2.00572e32 1.13414
\(638\) 1.85426e32 1.02975
\(639\) 0 0
\(640\) 8.51714e31 0.456272
\(641\) 3.78628e31 0.199225 0.0996127 0.995026i \(-0.468240\pi\)
0.0996127 + 0.995026i \(0.468240\pi\)
\(642\) 0 0
\(643\) −1.43262e32 −0.727283 −0.363641 0.931539i \(-0.618467\pi\)
−0.363641 + 0.931539i \(0.618467\pi\)
\(644\) −1.89397e29 −0.000944466 0
\(645\) 0 0
\(646\) −4.59400e30 −0.0221064
\(647\) 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
\(650\) −8.84043e31 −0.396251
\(651\) 0 0
\(652\) −3.44217e32 −1.48931
\(653\) 4.07291e32 1.73142 0.865712 0.500543i \(-0.166866\pi\)
0.865712 + 0.500543i \(0.166866\pi\)
\(654\) 0 0
\(655\) 4.37216e31 0.179441
\(656\) −2.08390e32 −0.840393
\(657\) 0 0
\(658\) 4.49773e31 0.175144
\(659\) 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 \(0.784990\pi\)
\(662\) −1.56520e32 −0.568465
\(663\) 0 0
\(664\) 2.04586e31 0.0717701
\(665\) −5.72740e31 −0.197474
\(666\) 0 0
\(667\) 1.77713e30 0.00591932
\(668\) 2.55946e32 0.837953
\(669\) 0 0
\(670\) 3.56280e31 0.112702
\(671\) −2.84805e32 −0.885601
\(672\) 0 0
\(673\) 3.27631e32 0.984491 0.492245 0.870456i \(-0.336176\pi\)
0.492245 + 0.870456i \(0.336176\pi\)
\(674\) 1.70096e32 0.502466
\(675\) 0 0
\(676\) −1.83815e32 −0.524799
\(677\) −1.65241e32 −0.463820 −0.231910 0.972737i \(-0.574497\pi\)
−0.231910 + 0.972737i \(0.574497\pi\)
\(678\) 0 0
\(679\) 5.27516e31 0.143131
\(680\) 5.11838e30 0.0136546
\(681\) 0 0
\(682\) 1.46334e32 0.377420
\(683\) −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
\(686\) −1.00650e32 −0.242710
\(687\) 0 0
\(688\) −9.63381e31 −0.224663
\(689\) −3.65937e32 −0.839238
\(690\) 0 0
\(691\) 2.43790e32 0.540778 0.270389 0.962751i \(-0.412848\pi\)
0.270389 + 0.962751i \(0.412848\pi\)
\(692\) −4.39811e32 −0.959502
\(693\) 0 0
\(694\) −1.63594e32 −0.345250
\(695\) −1.40264e32 −0.291154
\(696\) 0 0
\(697\) −3.12280e31 −0.0627143
\(698\) −1.16437e32 −0.230013
\(699\) 0 0
\(700\) −1.16937e32 −0.223523
\(701\) 3.79931e32 0.714409 0.357205 0.934026i \(-0.383730\pi\)
0.357205 + 0.934026i \(0.383730\pi\)
\(702\) 0 0
\(703\) −8.82561e32 −1.60605
\(704\) −1.69854e32 −0.304082
\(705\) 0 0
\(706\) −1.12426e32 −0.194810
\(707\) −4.12670e30 −0.00703526
\(708\) 0 0
\(709\) −1.44569e31 −0.0238586 −0.0119293 0.999929i \(-0.503797\pi\)
−0.0119293 + 0.999929i \(0.503797\pi\)
\(710\) 1.95750e32 0.317857
\(711\) 0 0
\(712\) 4.10254e32 0.644960
\(713\) 1.40247e30 0.00216952
\(714\) 0 0
\(715\) −5.73743e32 −0.859405
\(716\) 2.97588e32 0.438647
\(717\) 0 0
\(718\) 2.25520e32 0.321924
\(719\) 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
\(722\) −2.30764e32 −0.309022
\(723\) 0 0
\(724\) 7.44725e32 0.966057
\(725\) 1.09722e33 1.40090
\(726\) 0 0
\(727\) 4.93859e31 0.0610882 0.0305441 0.999533i \(-0.490276\pi\)
0.0305441 + 0.999533i \(0.490276\pi\)
\(728\) 2.47832e32 0.301750
\(729\) 0 0
\(730\) −1.84822e32 −0.218042
\(731\) −1.44366e31 −0.0167654
\(732\) 0 0
\(733\) 1.05389e33 1.18604 0.593019 0.805189i \(-0.297936\pi\)
0.593019 + 0.805189i \(0.297936\pi\)
\(734\) −6.32541e32 −0.700784
\(735\) 0 0
\(736\) 2.96144e30 0.00317986
\(737\) −9.23219e32 −0.975952
\(738\) 0 0
\(739\) 1.09776e33 1.12486 0.562428 0.826846i \(-0.309867\pi\)
0.562428 + 0.826846i \(0.309867\pi\)
\(740\) 4.51303e32 0.455306
\(741\) 0 0
\(742\) 8.65468e31 0.0846460
\(743\) 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
\(746\) 5.04137e32 0.463502
\(747\) 0 0
\(748\) −6.08734e31 −0.0542699
\(749\) 6.07840e32 0.533640
\(750\) 0 0
\(751\) −1.01215e33 −0.861759 −0.430880 0.902409i \(-0.641797\pi\)
−0.430880 + 0.902409i \(0.641797\pi\)
\(752\) 9.24546e32 0.775217
\(753\) 0 0
\(754\) −1.06730e33 −0.867990
\(755\) −2.66250e32 −0.213255
\(756\) 0 0
\(757\) −9.48447e32 −0.736903 −0.368452 0.929647i \(-0.620112\pi\)
−0.368452 + 0.929647i \(0.620112\pi\)
\(758\) 4.13496e32 0.316428
\(759\) 0 0
\(760\) 5.81133e32 0.431438
\(761\) −9.81408e32 −0.717672 −0.358836 0.933401i \(-0.616826\pi\)
−0.358836 + 0.933401i \(0.616826\pi\)
\(762\) 0 0
\(763\) 2.62131e32 0.185989
\(764\) 8.94448e32 0.625146
\(765\) 0 0
\(766\) −1.68400e32 −0.114212
\(767\) −2.66338e33 −1.77945
\(768\) 0 0
\(769\) 8.57155e32 0.555785 0.277892 0.960612i \(-0.410364\pi\)
0.277892 + 0.960612i \(0.410364\pi\)
\(770\) 1.35695e32 0.0866801
\(771\) 0 0
\(772\) 6.08734e32 0.377423
\(773\) 9.41593e31 0.0575174 0.0287587 0.999586i \(-0.490845\pi\)
0.0287587 + 0.999586i \(0.490845\pi\)
\(774\) 0 0
\(775\) 8.65904e32 0.513452
\(776\) −5.35245e32 −0.312711
\(777\) 0 0
\(778\) −2.22819e32 −0.126382
\(779\) −3.54557e33 −1.98155
\(780\) 0 0
\(781\) −5.07242e33 −2.75251
\(782\) 1.04314e29 5.57785e−5 0
\(783\) 0 0
\(784\) −9.75103e32 −0.506309
\(785\) −1.36417e33 −0.698020
\(786\) 0 0
\(787\) 2.72161e33 1.35243 0.676216 0.736703i \(-0.263618\pi\)
0.676216 + 0.736703i \(0.263618\pi\)
\(788\) −6.69900e32 −0.328064
\(789\) 0 0
\(790\) 3.78780e32 0.180167
\(791\) 8.94051e31 0.0419115
\(792\) 0 0
\(793\) 1.63931e33 0.746483
\(794\) 2.95979e31 0.0132839
\(795\) 0 0
\(796\) −5.31215e32 −0.231617
\(797\) 3.49544e33 1.50221 0.751106 0.660182i \(-0.229521\pi\)
0.751106 + 0.660182i \(0.229521\pi\)
\(798\) 0 0
\(799\) 1.38546e32 0.0578505
\(800\) 1.82843e33 0.752565
\(801\) 0 0
\(802\) 4.89975e32 0.195961
\(803\) 4.78923e33 1.88815
\(804\) 0 0
\(805\) 1.30050e30 0.000498263 0
\(806\) −8.42287e32 −0.318132
\(807\) 0 0
\(808\) 4.18717e31 0.0153706
\(809\) −1.04226e33 −0.377196 −0.188598 0.982054i \(-0.560394\pi\)
−0.188598 + 0.982054i \(0.560394\pi\)
\(810\) 0 0
\(811\) −2.22771e33 −0.783643 −0.391821 0.920041i \(-0.628155\pi\)
−0.391821 + 0.920041i \(0.628155\pi\)
\(812\) −1.41176e33 −0.489629
\(813\) 0 0
\(814\) 2.09098e33 0.704967
\(815\) 2.36357e33 0.785700
\(816\) 0 0
\(817\) −1.63911e33 −0.529729
\(818\) 9.18857e32 0.292809
\(819\) 0 0
\(820\) 1.81305e33 0.561760
\(821\) 7.36864e31 0.0225134 0.0112567 0.999937i \(-0.496417\pi\)
0.0112567 + 0.999937i \(0.496417\pi\)
\(822\) 0 0
\(823\) 4.08369e33 1.21326 0.606631 0.794984i \(-0.292521\pi\)
0.606631 + 0.794984i \(0.292521\pi\)
\(824\) −5.22471e32 −0.153073
\(825\) 0 0
\(826\) 6.29910e32 0.179477
\(827\) −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 \(0.672308\pi\)
\(830\) −6.44757e31 −0.0173779
\(831\) 0 0
\(832\) 9.77667e32 0.256314
\(833\) −1.46123e32 −0.0377833
\(834\) 0 0
\(835\) −1.75746e33 −0.442070
\(836\) −6.91147e33 −1.71474
\(837\) 0 0
\(838\) 1.45251e32 0.0350601
\(839\) 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
\(842\) 1.63249e33 0.373046
\(843\) 0 0
\(844\) −4.91971e33 −1.09396
\(845\) 1.26217e33 0.276863
\(846\) 0 0
\(847\) −1.97277e33 −0.421131
\(848\) 1.77904e33 0.374657
\(849\) 0 0
\(850\) 6.44051e31 0.0132009
\(851\) 2.00400e31 0.00405236
\(852\) 0 0
\(853\) −2.52826e33 −0.497631 −0.248815 0.968551i \(-0.580041\pi\)
−0.248815 + 0.968551i \(0.580041\pi\)
\(854\) −3.87710e32 −0.0752907
\(855\) 0 0
\(856\) −6.16746e33 −1.16589
\(857\) −7.45015e33 −1.38959 −0.694793 0.719209i \(-0.744504\pi\)
−0.694793 + 0.719209i \(0.744504\pi\)
\(858\) 0 0
\(859\) −5.27466e32 −0.0957797 −0.0478898 0.998853i \(-0.515250\pi\)
−0.0478898 + 0.998853i \(0.515250\pi\)
\(860\) 8.38169e32 0.150176
\(861\) 0 0
\(862\) −1.75098e33 −0.305456
\(863\) 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
\(866\) −4.16294e33 −0.688565
\(867\) 0 0
\(868\) −1.11413e33 −0.179457
\(869\) −9.81522e33 −1.56017
\(870\) 0 0
\(871\) 5.31397e33 0.822641
\(872\) −2.65972e33 −0.406347
\(873\) 0 0
\(874\) 1.18437e31 0.00176240
\(875\) 1.80700e33 0.265378
\(876\) 0 0
\(877\) −7.45124e33 −1.06594 −0.532970 0.846134i \(-0.678924\pi\)
−0.532970 + 0.846134i \(0.678924\pi\)
\(878\) −4.93870e32 −0.0697310
\(879\) 0 0
\(880\) 2.78932e33 0.383660
\(881\) −1.17245e34 −1.59174 −0.795870 0.605468i \(-0.792986\pi\)
−0.795870 + 0.605468i \(0.792986\pi\)
\(882\) 0 0
\(883\) 1.05295e34 1.39271 0.696353 0.717699i \(-0.254805\pi\)
0.696353 + 0.717699i \(0.254805\pi\)
\(884\) 3.50382e32 0.0457447
\(885\) 0 0
\(886\) 1.65825e33 0.210941
\(887\) 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
\(890\) −1.29292e33 −0.156166
\(891\) 0 0
\(892\) −5.68246e33 −0.668868
\(893\) 1.57303e34 1.82787
\(894\) 0 0
\(895\) −2.04339e33 −0.231412
\(896\) −3.00446e33 −0.335911
\(897\) 0 0
\(898\) −2.45726e32 −0.0267777
\(899\) 1.04540e34 1.12472
\(900\) 0 0
\(901\) 2.66596e32 0.0279587
\(902\) 8.40023e33 0.869794
\(903\) 0 0
\(904\) −9.07152e32 −0.0915678
\(905\) −5.11367e33 −0.509653
\(906\) 0 0
\(907\) −9.01545e33 −0.875999 −0.438000 0.898975i \(-0.644313\pi\)
−0.438000 + 0.898975i \(0.644313\pi\)
\(908\) 8.48862e33 0.814423
\(909\) 0 0
\(910\) −7.81047e32 −0.0730636
\(911\) −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
\(914\) 2.14755e33 0.191013
\(915\) 0 0
\(916\) 1.16751e34 1.01266
\(917\) −1.54230e33 −0.132106
\(918\) 0 0
\(919\) 3.21779e32 0.0268800 0.0134400 0.999910i \(-0.495722\pi\)
0.0134400 + 0.999910i \(0.495722\pi\)
\(920\) −1.31956e31 −0.00108860
\(921\) 0 0
\(922\) −5.60525e33 −0.451013
\(923\) 2.91964e34 2.32012
\(924\) 0 0
\(925\) 1.23730e34 0.959056
\(926\) −1.26286e33 −0.0966786
\(927\) 0 0
\(928\) 2.20745e34 1.64850
\(929\) −6.89145e33 −0.508311 −0.254156 0.967163i \(-0.581798\pi\)
−0.254156 + 0.967163i \(0.581798\pi\)
\(930\) 0 0
\(931\) −1.65905e34 −1.19382
\(932\) −1.05184e34 −0.747593
\(933\) 0 0
\(934\) 9.81316e33 0.680487
\(935\) 4.17989e32 0.0286306
\(936\) 0 0
\(937\) 6.25703e32 0.0418180 0.0209090 0.999781i \(-0.493344\pi\)
0.0209090 + 0.999781i \(0.493344\pi\)
\(938\) −1.25679e33 −0.0829720
\(939\) 0 0
\(940\) −8.04381e33 −0.518193
\(941\) −2.43281e34 −1.54820 −0.774101 0.633062i \(-0.781798\pi\)
−0.774101 + 0.633062i \(0.781798\pi\)
\(942\) 0 0
\(943\) 8.05081e31 0.00499983
\(944\) 1.29483e34 0.794393
\(945\) 0 0
\(946\) 3.88340e33 0.232522
\(947\) −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
\(950\) 7.31245e33 0.417101
\(951\) 0 0
\(952\) −1.80553e32 −0.0100526
\(953\) −2.99818e33 −0.164926 −0.0824632 0.996594i \(-0.526279\pi\)
−0.0824632 + 0.996594i \(0.526279\pi\)
\(954\) 0 0
\(955\) −6.14175e33 −0.329802
\(956\) −7.15389e32 −0.0379556
\(957\) 0 0
\(958\) −6.83726e32 −0.0354143
\(959\) −2.62948e33 −0.134573
\(960\) 0 0
\(961\) −1.17633e34 −0.587772
\(962\) −1.20355e34 −0.594225
\(963\) 0 0
\(964\) −2.05706e34 −0.991658
\(965\) −4.17989e33 −0.199114
\(966\) 0 0
\(967\) 5.37477e33 0.250009 0.125004 0.992156i \(-0.460106\pi\)
0.125004 + 0.992156i \(0.460106\pi\)
\(968\) 2.00168e34 0.920084
\(969\) 0 0
\(970\) 1.68683e33 0.0757177
\(971\) −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
\(974\) 4.85433e33 0.207827
\(975\) 0 0
\(976\) −7.96969e33 −0.333249
\(977\) −2.41460e34 −0.997831 −0.498915 0.866651i \(-0.666268\pi\)
−0.498915 + 0.866651i \(0.666268\pi\)
\(978\) 0 0
\(979\) 3.35031e34 1.35233
\(980\) 8.48368e33 0.338442
\(981\) 0 0
\(982\) −1.28102e34 −0.499199
\(983\) 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
\(986\) 7.77557e32 0.0289166
\(987\) 0 0
\(988\) 3.97818e34 1.44537
\(989\) 3.72187e31 0.00133661
\(990\) 0 0
\(991\) 1.83350e34 0.643329 0.321664 0.946854i \(-0.395758\pi\)
0.321664 + 0.946854i \(0.395758\pi\)
\(992\) 1.74207e34 0.604201
\(993\) 0 0
\(994\) −6.90518e33 −0.234008
\(995\) 3.64760e33 0.122192
\(996\) 0 0
\(997\) 1.59227e34 0.521222 0.260611 0.965444i \(-0.416076\pi\)
0.260611 + 0.965444i \(0.416076\pi\)
\(998\) −1.12779e34 −0.364945
\(999\) 0 0
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 9.24.a.a.1.1 1
3.2 odd 2 3.24.a.a.1.1 1
12.11 even 2 48.24.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.24.a.a.1.1 1 3.2 odd 2
9.24.a.a.1.1 1 1.1 even 1 trivial
48.24.a.a.1.1 1 12.11 even 2