Properties

Label 3.24.a.a.1.1
Level $3$
Weight $24$
Character 3.1
Self dual yes
Analytic conductor $10.056$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3 \)
Weight: \( k \) \(=\) \( 24 \)
Character orbit: \([\chi]\) \(=\) 3.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.0561211204\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 3.1

$q$-expansion

\(f(q)\) \(=\) \(q+1128.00 q^{2} +177147. q^{3} -7.11622e6 q^{4} -4.88637e7 q^{5} +1.99822e8 q^{6} -1.72369e9 q^{7} -1.74895e10 q^{8} +3.13811e10 q^{9} +O(q^{10})\) \(q+1128.00 q^{2} +177147. q^{3} -7.11622e6 q^{4} -4.88637e7 q^{5} +1.99822e8 q^{6} -1.72369e9 q^{7} -1.74895e10 q^{8} +3.13811e10 q^{9} -5.51183e10 q^{10} -1.42826e12 q^{11} -1.26062e12 q^{12} -8.22096e12 q^{13} -1.94432e12 q^{14} -8.65606e12 q^{15} +3.99671e13 q^{16} -5.98921e12 q^{17} +3.53978e13 q^{18} +6.80005e14 q^{19} +3.47725e14 q^{20} -3.05346e14 q^{21} -1.61108e15 q^{22} +1.54406e13 q^{23} -3.09820e15 q^{24} -9.53326e15 q^{25} -9.27325e15 q^{26} +5.55906e15 q^{27} +1.22662e16 q^{28} +1.15094e17 q^{29} -9.76404e15 q^{30} -9.08297e16 q^{31} +1.91795e17 q^{32} -2.53013e17 q^{33} -6.75583e15 q^{34} +8.42259e16 q^{35} -2.23315e17 q^{36} -1.29787e18 q^{37} +7.67046e17 q^{38} -1.45632e18 q^{39} +8.54600e17 q^{40} +5.21404e18 q^{41} -3.44431e17 q^{42} -2.41043e18 q^{43} +1.01638e19 q^{44} -1.53340e18 q^{45} +1.74171e16 q^{46} -2.31327e19 q^{47} +7.08005e18 q^{48} -2.43976e19 q^{49} -1.07535e19 q^{50} -1.06097e18 q^{51} +5.85022e19 q^{52} -4.45126e19 q^{53} +6.27062e18 q^{54} +6.97903e19 q^{55} +3.01464e19 q^{56} +1.20461e20 q^{57} +1.29826e20 q^{58} -3.23974e20 q^{59} +6.15985e19 q^{60} -1.99406e20 q^{61} -1.02456e20 q^{62} -5.40912e19 q^{63} -1.18924e20 q^{64} +4.01707e20 q^{65} -2.85398e20 q^{66} -6.46393e20 q^{67} +4.26206e19 q^{68} +2.73526e18 q^{69} +9.50068e19 q^{70} +3.55146e21 q^{71} -5.48837e20 q^{72} +3.35319e21 q^{73} -1.46400e21 q^{74} -1.68879e21 q^{75} -4.83907e21 q^{76} +2.46188e21 q^{77} -1.64273e21 q^{78} -6.87213e21 q^{79} -1.95294e21 q^{80} +9.84771e20 q^{81} +5.88143e21 q^{82} -1.16977e21 q^{83} +2.17291e21 q^{84} +2.92655e20 q^{85} -2.71897e21 q^{86} +2.03886e22 q^{87} +2.49795e22 q^{88} -2.34572e22 q^{89} -1.72967e21 q^{90} +1.41704e22 q^{91} -1.09879e20 q^{92} -1.60902e22 q^{93} -2.60937e22 q^{94} -3.32276e22 q^{95} +3.39759e22 q^{96} -3.06039e22 q^{97} -2.75205e22 q^{98} -4.48204e22 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1128.00 0.389461 0.194731 0.980857i \(-0.437617\pi\)
0.194731 + 0.980857i \(0.437617\pi\)
\(3\) 177147. 0.577350
\(4\) −7.11622e6 −0.848320
\(5\) −4.88637e7 −0.447540 −0.223770 0.974642i \(-0.571836\pi\)
−0.223770 + 0.974642i \(0.571836\pi\)
\(6\) 1.99822e8 0.224856
\(7\) −1.72369e9 −0.329482 −0.164741 0.986337i \(-0.552679\pi\)
−0.164741 + 0.986337i \(0.552679\pi\)
\(8\) −1.74895e10 −0.719849
\(9\) 3.13811e10 0.333333
\(10\) −5.51183e10 −0.174299
\(11\) −1.42826e12 −1.50936 −0.754679 0.656094i \(-0.772207\pi\)
−0.754679 + 0.656094i \(0.772207\pi\)
\(12\) −1.26062e12 −0.489778
\(13\) −8.22096e12 −1.27226 −0.636128 0.771584i \(-0.719465\pi\)
−0.636128 + 0.771584i \(0.719465\pi\)
\(14\) −1.94432e12 −0.128320
\(15\) −8.65606e12 −0.258387
\(16\) 3.99671e13 0.567967
\(17\) −5.98921e12 −0.0423845 −0.0211922 0.999775i \(-0.506746\pi\)
−0.0211922 + 0.999775i \(0.506746\pi\)
\(18\) 3.53978e13 0.129820
\(19\) 6.80005e14 1.33920 0.669601 0.742721i \(-0.266465\pi\)
0.669601 + 0.742721i \(0.266465\pi\)
\(20\) 3.47725e14 0.379657
\(21\) −3.05346e14 −0.190226
\(22\) −1.61108e15 −0.587836
\(23\) 1.54406e13 0.00337906 0.00168953 0.999999i \(-0.499462\pi\)
0.00168953 + 0.999999i \(0.499462\pi\)
\(24\) −3.09820e15 −0.415605
\(25\) −9.53326e15 −0.799708
\(26\) −9.27325e15 −0.495494
\(27\) 5.55906e15 0.192450
\(28\) 1.22662e16 0.279506
\(29\) 1.15094e17 1.75177 0.875884 0.482522i \(-0.160279\pi\)
0.875884 + 0.482522i \(0.160279\pi\)
\(30\) −9.76404e15 −0.100632
\(31\) −9.08297e16 −0.642050 −0.321025 0.947071i \(-0.604027\pi\)
−0.321025 + 0.947071i \(0.604027\pi\)
\(32\) 1.91795e17 0.941050
\(33\) −2.53013e17 −0.871428
\(34\) −6.75583e15 −0.0165071
\(35\) 8.42259e16 0.147456
\(36\) −2.23315e17 −0.282773
\(37\) −1.29787e18 −1.19926 −0.599629 0.800278i \(-0.704685\pi\)
−0.599629 + 0.800278i \(0.704685\pi\)
\(38\) 7.67046e17 0.521567
\(39\) −1.45632e18 −0.734537
\(40\) 8.54600e17 0.322161
\(41\) 5.21404e18 1.47965 0.739826 0.672798i \(-0.234908\pi\)
0.739826 + 0.672798i \(0.234908\pi\)
\(42\) −3.44431e17 −0.0740858
\(43\) −2.41043e18 −0.395556 −0.197778 0.980247i \(-0.563373\pi\)
−0.197778 + 0.980247i \(0.563373\pi\)
\(44\) 1.01638e19 1.28042
\(45\) −1.53340e18 −0.149180
\(46\) 1.74171e16 0.00131601
\(47\) −2.31327e19 −1.36490 −0.682449 0.730933i \(-0.739085\pi\)
−0.682449 + 0.730933i \(0.739085\pi\)
\(48\) 7.08005e18 0.327916
\(49\) −2.43976e19 −0.891442
\(50\) −1.07535e19 −0.311455
\(51\) −1.06097e18 −0.0244707
\(52\) 5.85022e19 1.07928
\(53\) −4.45126e19 −0.659646 −0.329823 0.944043i \(-0.606989\pi\)
−0.329823 + 0.944043i \(0.606989\pi\)
\(54\) 6.27062e18 0.0749518
\(55\) 6.97903e19 0.675498
\(56\) 3.01464e19 0.237177
\(57\) 1.20461e20 0.773188
\(58\) 1.29826e20 0.682245
\(59\) −3.23974e20 −1.39866 −0.699331 0.714798i \(-0.746519\pi\)
−0.699331 + 0.714798i \(0.746519\pi\)
\(60\) 6.15985e19 0.219195
\(61\) −1.99406e20 −0.586740 −0.293370 0.955999i \(-0.594777\pi\)
−0.293370 + 0.955999i \(0.594777\pi\)
\(62\) −1.02456e20 −0.250053
\(63\) −5.40912e19 −0.109827
\(64\) −1.18924e20 −0.201464
\(65\) 4.01707e20 0.569385
\(66\) −2.85398e20 −0.339388
\(67\) −6.46393e20 −0.646601 −0.323300 0.946296i \(-0.604792\pi\)
−0.323300 + 0.946296i \(0.604792\pi\)
\(68\) 4.26206e19 0.0359556
\(69\) 2.73526e18 0.00195090
\(70\) 9.50068e19 0.0574284
\(71\) 3.55146e21 1.82363 0.911814 0.410604i \(-0.134682\pi\)
0.911814 + 0.410604i \(0.134682\pi\)
\(72\) −5.48837e20 −0.239950
\(73\) 3.35319e21 1.25096 0.625482 0.780238i \(-0.284902\pi\)
0.625482 + 0.780238i \(0.284902\pi\)
\(74\) −1.46400e21 −0.467064
\(75\) −1.68879e21 −0.461712
\(76\) −4.83907e21 −1.13607
\(77\) 2.46188e21 0.497306
\(78\) −1.64273e21 −0.286074
\(79\) −6.87213e21 −1.03367 −0.516833 0.856086i \(-0.672889\pi\)
−0.516833 + 0.856086i \(0.672889\pi\)
\(80\) −1.95294e21 −0.254188
\(81\) 9.84771e20 0.111111
\(82\) 5.88143e21 0.576267
\(83\) −1.16977e21 −0.0997016 −0.0498508 0.998757i \(-0.515875\pi\)
−0.0498508 + 0.998757i \(0.515875\pi\)
\(84\) 2.17291e21 0.161373
\(85\) 2.92655e20 0.0189687
\(86\) −2.71897e21 −0.154054
\(87\) 2.03886e22 1.01138
\(88\) 2.49795e22 1.08651
\(89\) −2.34572e22 −0.895966 −0.447983 0.894042i \(-0.647857\pi\)
−0.447983 + 0.894042i \(0.647857\pi\)
\(90\) −1.72967e21 −0.0580998
\(91\) 1.41704e22 0.419185
\(92\) −1.09879e20 −0.00286652
\(93\) −1.60902e22 −0.370687
\(94\) −2.60937e22 −0.531575
\(95\) −3.32276e22 −0.599346
\(96\) 3.39759e22 0.543315
\(97\) −3.06039e22 −0.434412 −0.217206 0.976126i \(-0.569694\pi\)
−0.217206 + 0.976126i \(0.569694\pi\)
\(98\) −2.75205e22 −0.347182
\(99\) −4.48204e22 −0.503119
\(100\) 6.78408e22 0.678408
\(101\) −2.39411e21 −0.0213525 −0.0106762 0.999943i \(-0.503398\pi\)
−0.0106762 + 0.999943i \(0.503398\pi\)
\(102\) −1.19677e21 −0.00953039
\(103\) −2.98735e22 −0.212646 −0.106323 0.994332i \(-0.533908\pi\)
−0.106323 + 0.994332i \(0.533908\pi\)
\(104\) 1.43780e23 0.915831
\(105\) 1.49204e22 0.0851339
\(106\) −5.02102e22 −0.256906
\(107\) 3.52639e23 1.61963 0.809817 0.586683i \(-0.199567\pi\)
0.809817 + 0.586683i \(0.199567\pi\)
\(108\) −3.95595e22 −0.163259
\(109\) −1.52076e23 −0.564489 −0.282245 0.959342i \(-0.591079\pi\)
−0.282245 + 0.959342i \(0.591079\pi\)
\(110\) 7.87234e22 0.263080
\(111\) −2.29914e23 −0.692392
\(112\) −6.88909e22 −0.187135
\(113\) 5.18685e22 0.127204 0.0636021 0.997975i \(-0.479741\pi\)
0.0636021 + 0.997975i \(0.479741\pi\)
\(114\) 1.35880e23 0.301127
\(115\) −7.54488e20 −0.00151226
\(116\) −8.19036e23 −1.48606
\(117\) −2.57983e23 −0.424085
\(118\) −3.65443e23 −0.544724
\(119\) 1.03235e22 0.0139649
\(120\) 1.51390e23 0.186000
\(121\) 1.14451e24 1.27816
\(122\) −2.24930e23 −0.228512
\(123\) 9.23651e23 0.854278
\(124\) 6.46365e23 0.544663
\(125\) 1.04833e24 0.805441
\(126\) −6.10148e22 −0.0427734
\(127\) −3.34992e23 −0.214433 −0.107217 0.994236i \(-0.534194\pi\)
−0.107217 + 0.994236i \(0.534194\pi\)
\(128\) −1.74304e24 −1.01951
\(129\) −4.27001e23 −0.228374
\(130\) 4.53125e23 0.221753
\(131\) −8.94767e23 −0.400950 −0.200475 0.979699i \(-0.564248\pi\)
−0.200475 + 0.979699i \(0.564248\pi\)
\(132\) 1.80049e24 0.739250
\(133\) −1.17212e24 −0.441243
\(134\) −7.29131e23 −0.251826
\(135\) −2.71636e23 −0.0861291
\(136\) 1.04748e23 0.0305104
\(137\) −1.52550e24 −0.408437 −0.204219 0.978925i \(-0.565465\pi\)
−0.204219 + 0.978925i \(0.565465\pi\)
\(138\) 3.08538e21 0.000759799 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
\(141\) −4.09788e24 −0.788025
\(142\) 4.00605e24 0.710232
\(143\) 1.17417e25 1.92029
\(144\) 1.25421e24 0.189322
\(145\) −5.62393e24 −0.783985
\(146\) 3.78240e24 0.487202
\(147\) −4.32197e24 −0.514674
\(148\) 9.23596e24 1.01735
\(149\) 7.06708e24 0.720440 0.360220 0.932867i \(-0.382702\pi\)
0.360220 + 0.932867i \(0.382702\pi\)
\(150\) −1.90495e24 −0.179819
\(151\) −5.44882e24 −0.476505 −0.238253 0.971203i \(-0.576575\pi\)
−0.238253 + 0.971203i \(0.576575\pi\)
\(152\) −1.18929e25 −0.964023
\(153\) −1.87948e23 −0.0141282
\(154\) 2.77700e24 0.193681
\(155\) 4.43828e24 0.287343
\(156\) 1.03635e25 0.623122
\(157\) −2.79179e25 −1.55968 −0.779841 0.625977i \(-0.784700\pi\)
−0.779841 + 0.625977i \(0.784700\pi\)
\(158\) −7.75177e24 −0.402573
\(159\) −7.88528e24 −0.380847
\(160\) −9.37182e24 −0.421157
\(161\) −2.66149e22 −0.00111334
\(162\) 1.11082e24 0.0432735
\(163\) 4.83707e25 1.75560 0.877799 0.479029i \(-0.159011\pi\)
0.877799 + 0.479029i \(0.159011\pi\)
\(164\) −3.71042e25 −1.25522
\(165\) 1.23631e25 0.389999
\(166\) −1.31950e24 −0.0388299
\(167\) 3.59666e25 0.987779 0.493890 0.869525i \(-0.335575\pi\)
0.493890 + 0.869525i \(0.335575\pi\)
\(168\) 5.34034e24 0.136934
\(169\) 2.58303e25 0.618633
\(170\) 3.30115e23 0.00738759
\(171\) 2.13393e25 0.446401
\(172\) 1.71532e25 0.335558
\(173\) −6.18040e25 −1.13106 −0.565531 0.824727i \(-0.691329\pi\)
−0.565531 + 0.824727i \(0.691329\pi\)
\(174\) 2.29983e25 0.393895
\(175\) 1.64324e25 0.263489
\(176\) −5.70836e25 −0.857265
\(177\) −5.73911e25 −0.807518
\(178\) −2.64597e25 −0.348944
\(179\) 4.18182e25 0.517077 0.258539 0.966001i \(-0.416759\pi\)
0.258539 + 0.966001i \(0.416759\pi\)
\(180\) 1.09120e25 0.126552
\(181\) −1.04652e26 −1.13879 −0.569394 0.822065i \(-0.692822\pi\)
−0.569394 + 0.822065i \(0.692822\pi\)
\(182\) 1.59842e25 0.163256
\(183\) −3.53242e25 −0.338754
\(184\) −2.70048e23 −0.00243241
\(185\) 6.34189e25 0.536715
\(186\) −1.81498e25 −0.144368
\(187\) 8.55417e24 0.0639734
\(188\) 1.64617e26 1.15787
\(189\) −9.58209e24 −0.0634088
\(190\) −3.74807e25 −0.233422
\(191\) 1.25691e26 0.736923 0.368461 0.929643i \(-0.379885\pi\)
0.368461 + 0.929643i \(0.379885\pi\)
\(192\) −2.10670e25 −0.116316
\(193\) −8.55418e25 −0.444907 −0.222453 0.974943i \(-0.571407\pi\)
−0.222453 + 0.974943i \(0.571407\pi\)
\(194\) −3.45212e25 −0.169187
\(195\) 7.11612e25 0.328734
\(196\) 1.73619e26 0.756228
\(197\) −9.41370e25 −0.386722 −0.193361 0.981128i \(-0.561939\pi\)
−0.193361 + 0.981128i \(0.561939\pi\)
\(198\) −5.05574e25 −0.195945
\(199\) 7.46484e25 0.273030 0.136515 0.990638i \(-0.456410\pi\)
0.136515 + 0.990638i \(0.456410\pi\)
\(200\) 1.66732e26 0.575669
\(201\) −1.14506e26 −0.373315
\(202\) −2.70056e24 −0.00831597
\(203\) −1.98387e26 −0.577175
\(204\) 7.55010e24 0.0207590
\(205\) −2.54777e26 −0.662203
\(206\) −3.36973e25 −0.0828175
\(207\) 4.84544e23 0.00112635
\(208\) −3.28568e26 −0.722599
\(209\) −9.71227e26 −2.02134
\(210\) 1.68302e25 0.0331563
\(211\) 6.91338e26 1.28956 0.644781 0.764368i \(-0.276949\pi\)
0.644781 + 0.764368i \(0.276949\pi\)
\(212\) 3.16762e26 0.559591
\(213\) 6.29131e26 1.05287
\(214\) 3.97777e26 0.630784
\(215\) 1.17783e26 0.177027
\(216\) −9.72249e25 −0.138535
\(217\) 1.56562e26 0.211544
\(218\) −1.71542e26 −0.219847
\(219\) 5.94007e26 0.722245
\(220\) −4.96643e26 −0.573038
\(221\) 4.92371e25 0.0539239
\(222\) −2.59343e26 −0.269660
\(223\) 7.98521e26 0.788462 0.394231 0.919011i \(-0.371011\pi\)
0.394231 + 0.919011i \(0.371011\pi\)
\(224\) −3.30595e26 −0.310059
\(225\) −2.99164e26 −0.266569
\(226\) 5.85077e25 0.0495411
\(227\) 1.19285e27 0.960042 0.480021 0.877257i \(-0.340629\pi\)
0.480021 + 0.877257i \(0.340629\pi\)
\(228\) −8.57227e26 −0.655911
\(229\) −1.64063e27 −1.19372 −0.596860 0.802345i \(-0.703585\pi\)
−0.596860 + 0.802345i \(0.703585\pi\)
\(230\) −8.51062e23 −0.000588967 0
\(231\) 4.36115e26 0.287120
\(232\) −2.01293e27 −1.26101
\(233\) −1.47808e27 −0.881263 −0.440632 0.897688i \(-0.645246\pi\)
−0.440632 + 0.897688i \(0.645246\pi\)
\(234\) −2.91004e26 −0.165165
\(235\) 1.13035e27 0.610846
\(236\) 2.30547e27 1.18651
\(237\) −1.21738e27 −0.596787
\(238\) 1.16449e25 0.00543879
\(239\) −1.00529e26 −0.0447421 −0.0223711 0.999750i \(-0.507122\pi\)
−0.0223711 + 0.999750i \(0.507122\pi\)
\(240\) −3.45958e26 −0.146755
\(241\) 2.89067e27 1.16897 0.584483 0.811406i \(-0.301297\pi\)
0.584483 + 0.811406i \(0.301297\pi\)
\(242\) 1.29100e27 0.497795
\(243\) 1.74449e26 0.0641500
\(244\) 1.41902e27 0.497743
\(245\) 1.19216e27 0.398956
\(246\) 1.04188e27 0.332708
\(247\) −5.59030e27 −1.70381
\(248\) 1.58856e27 0.462179
\(249\) −2.07221e26 −0.0575628
\(250\) 1.18252e27 0.313688
\(251\) −3.03848e26 −0.0769855 −0.0384928 0.999259i \(-0.512256\pi\)
−0.0384928 + 0.999259i \(0.512256\pi\)
\(252\) 3.84925e26 0.0931687
\(253\) −2.20533e25 −0.00510021
\(254\) −3.77872e26 −0.0835134
\(255\) 5.18430e25 0.0109516
\(256\) −9.68545e26 −0.195596
\(257\) −5.74761e27 −1.10983 −0.554915 0.831907i \(-0.687249\pi\)
−0.554915 + 0.831907i \(0.687249\pi\)
\(258\) −4.81657e26 −0.0889430
\(259\) 2.23713e27 0.395134
\(260\) −2.85864e27 −0.483020
\(261\) 3.61178e27 0.583922
\(262\) −1.00930e27 −0.156154
\(263\) −9.08470e27 −1.34530 −0.672650 0.739961i \(-0.734844\pi\)
−0.672650 + 0.739961i \(0.734844\pi\)
\(264\) 4.42505e27 0.627297
\(265\) 2.17505e27 0.295218
\(266\) −1.32215e27 −0.171847
\(267\) −4.15537e27 −0.517286
\(268\) 4.59987e27 0.548524
\(269\) 2.45445e27 0.280416 0.140208 0.990122i \(-0.455223\pi\)
0.140208 + 0.990122i \(0.455223\pi\)
\(270\) −3.06406e26 −0.0335439
\(271\) 5.65329e27 0.593136 0.296568 0.955012i \(-0.404158\pi\)
0.296568 + 0.955012i \(0.404158\pi\)
\(272\) −2.39371e26 −0.0240730
\(273\) 2.51024e27 0.242016
\(274\) −1.72076e27 −0.159070
\(275\) 1.36160e28 1.20705
\(276\) −1.94648e25 −0.00165499
\(277\) 3.55531e27 0.289975 0.144987 0.989433i \(-0.453686\pi\)
0.144987 + 0.989433i \(0.453686\pi\)
\(278\) −3.23795e27 −0.253370
\(279\) −2.85033e27 −0.214017
\(280\) −1.47306e27 −0.106146
\(281\) 2.83906e28 1.96359 0.981797 0.189931i \(-0.0608265\pi\)
0.981797 + 0.189931i \(0.0608265\pi\)
\(282\) −4.62241e27 −0.306905
\(283\) −2.29202e28 −1.46108 −0.730542 0.682868i \(-0.760732\pi\)
−0.730542 + 0.682868i \(0.760732\pi\)
\(284\) −2.52730e28 −1.54702
\(285\) −5.88617e27 −0.346033
\(286\) 1.32446e28 0.747878
\(287\) −8.98738e27 −0.487518
\(288\) 6.01873e27 0.313683
\(289\) −1.99317e28 −0.998204
\(290\) −6.34379e27 −0.305332
\(291\) −5.42139e27 −0.250808
\(292\) −2.38620e28 −1.06122
\(293\) 3.43109e28 1.46708 0.733542 0.679644i \(-0.237866\pi\)
0.733542 + 0.679644i \(0.237866\pi\)
\(294\) −4.87518e27 −0.200446
\(295\) 1.58306e28 0.625957
\(296\) 2.26991e28 0.863284
\(297\) −7.93980e27 −0.290476
\(298\) 7.97167e27 0.280584
\(299\) −1.26937e26 −0.00429902
\(300\) 1.20178e28 0.391679
\(301\) 4.15484e27 0.130329
\(302\) −6.14627e27 −0.185580
\(303\) −4.24110e26 −0.0123279
\(304\) 2.71779e28 0.760622
\(305\) 9.74373e27 0.262589
\(306\) −2.12005e26 −0.00550237
\(307\) −1.16886e28 −0.292195 −0.146097 0.989270i \(-0.546671\pi\)
−0.146097 + 0.989270i \(0.546671\pi\)
\(308\) −1.75193e28 −0.421875
\(309\) −5.29200e27 −0.122771
\(310\) 5.00638e27 0.111909
\(311\) −2.49825e28 −0.538135 −0.269067 0.963121i \(-0.586715\pi\)
−0.269067 + 0.963121i \(0.586715\pi\)
\(312\) 2.54702e28 0.528755
\(313\) 6.03885e28 1.20835 0.604177 0.796850i \(-0.293502\pi\)
0.604177 + 0.796850i \(0.293502\pi\)
\(314\) −3.14914e28 −0.607436
\(315\) 2.64310e27 0.0491521
\(316\) 4.89036e28 0.876879
\(317\) −1.04255e29 −1.80266 −0.901330 0.433133i \(-0.857408\pi\)
−0.901330 + 0.433133i \(0.857408\pi\)
\(318\) −8.89459e27 −0.148325
\(319\) −1.64385e29 −2.64404
\(320\) 5.81105e27 0.0901633
\(321\) 6.24689e28 0.935096
\(322\) −3.00216e25 −0.000433602 0
\(323\) −4.07270e27 −0.0567614
\(324\) −7.00785e27 −0.0942578
\(325\) 7.83726e28 1.01743
\(326\) 5.45622e28 0.683737
\(327\) −2.69398e28 −0.325908
\(328\) −9.11906e28 −1.06513
\(329\) 3.98735e28 0.449709
\(330\) 1.39456e28 0.151889
\(331\) 1.38759e29 1.45962 0.729809 0.683651i \(-0.239609\pi\)
0.729809 + 0.683651i \(0.239609\pi\)
\(332\) 8.32434e27 0.0845789
\(333\) −4.07286e28 −0.399753
\(334\) 4.05703e28 0.384702
\(335\) 3.15851e28 0.289379
\(336\) −1.22038e28 −0.108042
\(337\) −1.50795e29 −1.29016 −0.645078 0.764117i \(-0.723175\pi\)
−0.645078 + 0.764117i \(0.723175\pi\)
\(338\) 2.91366e28 0.240934
\(339\) 9.18835e27 0.0734414
\(340\) −2.08260e27 −0.0160916
\(341\) 1.29729e29 0.969083
\(342\) 2.40707e28 0.173856
\(343\) 8.92291e28 0.623196
\(344\) 4.21572e28 0.284741
\(345\) −1.33655e26 −0.000873105 0
\(346\) −6.97149e28 −0.440504
\(347\) −1.45030e29 −0.886482 −0.443241 0.896403i \(-0.646171\pi\)
−0.443241 + 0.896403i \(0.646171\pi\)
\(348\) −1.45090e29 −0.857977
\(349\) 1.03224e29 0.590592 0.295296 0.955406i \(-0.404582\pi\)
0.295296 + 0.955406i \(0.404582\pi\)
\(350\) 1.85357e28 0.102619
\(351\) −4.57008e28 −0.244846
\(352\) −2.73934e29 −1.42038
\(353\) −9.96681e28 −0.500204 −0.250102 0.968220i \(-0.580464\pi\)
−0.250102 + 0.968220i \(0.580464\pi\)
\(354\) −6.47372e28 −0.314497
\(355\) −1.73538e29 −0.816146
\(356\) 1.66927e29 0.760066
\(357\) 1.82878e27 0.00806265
\(358\) 4.71709e28 0.201381
\(359\) 1.99929e29 0.826589 0.413294 0.910597i \(-0.364378\pi\)
0.413294 + 0.910597i \(0.364378\pi\)
\(360\) 2.68182e28 0.107387
\(361\) 2.04578e29 0.793461
\(362\) −1.18047e29 −0.443514
\(363\) 2.02746e29 0.737948
\(364\) −1.00840e29 −0.355603
\(365\) −1.63849e29 −0.559856
\(366\) −3.98457e28 −0.131932
\(367\) 5.60764e29 1.79937 0.899684 0.436541i \(-0.143797\pi\)
0.899684 + 0.436541i \(0.143797\pi\)
\(368\) 6.17118e26 0.00191919
\(369\) 1.63622e29 0.493217
\(370\) 7.15366e28 0.209030
\(371\) 7.67259e28 0.217341
\(372\) 1.14502e29 0.314462
\(373\) −4.46930e29 −1.19011 −0.595056 0.803684i \(-0.702870\pi\)
−0.595056 + 0.803684i \(0.702870\pi\)
\(374\) 9.64910e27 0.0249151
\(375\) 1.85709e29 0.465022
\(376\) 4.04578e29 0.982521
\(377\) −9.46185e29 −2.22869
\(378\) −1.08086e28 −0.0246953
\(379\) −3.66574e29 −0.812477 −0.406239 0.913767i \(-0.633160\pi\)
−0.406239 + 0.913767i \(0.633160\pi\)
\(380\) 2.36455e29 0.508437
\(381\) −5.93429e28 −0.123803
\(382\) 1.41780e29 0.287003
\(383\) −1.49291e29 −0.293257 −0.146628 0.989192i \(-0.546842\pi\)
−0.146628 + 0.989192i \(0.546842\pi\)
\(384\) −3.08774e29 −0.588616
\(385\) −1.20297e29 −0.222564
\(386\) −9.64911e28 −0.173274
\(387\) −7.56420e28 −0.131852
\(388\) 2.17784e29 0.368520
\(389\) −1.97534e29 −0.324506 −0.162253 0.986749i \(-0.551876\pi\)
−0.162253 + 0.986749i \(0.551876\pi\)
\(390\) 8.02698e28 0.128029
\(391\) −9.24773e25 −0.000143220 0
\(392\) 4.26701e29 0.641703
\(393\) −1.58505e29 −0.231488
\(394\) −1.06187e29 −0.150613
\(395\) 3.35798e29 0.462606
\(396\) 3.18952e29 0.426806
\(397\) −2.62393e28 −0.0341084 −0.0170542 0.999855i \(-0.505429\pi\)
−0.0170542 + 0.999855i \(0.505429\pi\)
\(398\) 8.42034e28 0.106334
\(399\) −2.07637e29 −0.254751
\(400\) −3.81017e29 −0.454208
\(401\) 4.34375e29 0.503159 0.251579 0.967837i \(-0.419050\pi\)
0.251579 + 0.967837i \(0.419050\pi\)
\(402\) −1.29163e29 −0.145392
\(403\) 7.46708e29 0.816851
\(404\) 1.70370e28 0.0181137
\(405\) −4.81196e28 −0.0497266
\(406\) −2.23780e29 −0.224787
\(407\) 1.85370e30 1.81011
\(408\) 1.85558e28 0.0176152
\(409\) −8.14589e29 −0.751832 −0.375916 0.926654i \(-0.622672\pi\)
−0.375916 + 0.926654i \(0.622672\pi\)
\(410\) −2.87389e29 −0.257902
\(411\) −2.70238e29 −0.235811
\(412\) 2.12586e29 0.180392
\(413\) 5.58431e29 0.460833
\(414\) 5.46566e26 0.000438670 0
\(415\) 5.71593e28 0.0446204
\(416\) −1.57674e30 −1.19726
\(417\) −5.08505e29 −0.375604
\(418\) −1.09554e30 −0.787232
\(419\) 1.28769e29 0.0900221 0.0450111 0.998986i \(-0.485668\pi\)
0.0450111 + 0.998986i \(0.485668\pi\)
\(420\) −1.06177e29 −0.0722208
\(421\) −1.44725e30 −0.957852 −0.478926 0.877855i \(-0.658974\pi\)
−0.478926 + 0.877855i \(0.658974\pi\)
\(422\) 7.79829e29 0.502234
\(423\) −7.25928e29 −0.454966
\(424\) 7.78501e29 0.474845
\(425\) 5.70967e28 0.0338952
\(426\) 7.09659e29 0.410053
\(427\) 3.43714e29 0.193320
\(428\) −2.50946e30 −1.37397
\(429\) 2.08001e30 1.10868
\(430\) 1.32859e29 0.0689452
\(431\) −1.55229e30 −0.784303 −0.392152 0.919901i \(-0.628269\pi\)
−0.392152 + 0.919901i \(0.628269\pi\)
\(432\) 2.22180e29 0.109305
\(433\) 3.69055e30 1.76799 0.883997 0.467493i \(-0.154843\pi\)
0.883997 + 0.467493i \(0.154843\pi\)
\(434\) 1.76602e29 0.0823880
\(435\) −9.96263e29 −0.452634
\(436\) 1.08221e30 0.478867
\(437\) 1.04997e28 0.00452524
\(438\) 6.70040e29 0.281286
\(439\) 4.37828e29 0.179045 0.0895223 0.995985i \(-0.471466\pi\)
0.0895223 + 0.995985i \(0.471466\pi\)
\(440\) −1.22059e30 −0.486256
\(441\) −7.65624e29 −0.297147
\(442\) 5.55394e28 0.0210013
\(443\) 1.47008e30 0.541624 0.270812 0.962632i \(-0.412708\pi\)
0.270812 + 0.962632i \(0.412708\pi\)
\(444\) 1.63612e30 0.587370
\(445\) 1.14621e30 0.400980
\(446\) 9.00732e29 0.307075
\(447\) 1.25191e30 0.415946
\(448\) 2.04987e29 0.0663789
\(449\) −2.17842e29 −0.0687558 −0.0343779 0.999409i \(-0.510945\pi\)
−0.0343779 + 0.999409i \(0.510945\pi\)
\(450\) −3.37457e29 −0.103818
\(451\) −7.44702e30 −2.23333
\(452\) −3.69108e29 −0.107910
\(453\) −9.65243e29 −0.275111
\(454\) 1.34554e30 0.373899
\(455\) −6.92418e29 −0.187602
\(456\) −2.10680e30 −0.556579
\(457\) −1.90386e30 −0.490454 −0.245227 0.969466i \(-0.578862\pi\)
−0.245227 + 0.969466i \(0.578862\pi\)
\(458\) −1.85063e30 −0.464908
\(459\) −3.32944e28 −0.00815690
\(460\) 5.36910e27 0.00128288
\(461\) −4.96919e30 −1.15804 −0.579022 0.815312i \(-0.696566\pi\)
−0.579022 + 0.815312i \(0.696566\pi\)
\(462\) 4.91938e29 0.111822
\(463\) 1.11956e30 0.248237 0.124118 0.992267i \(-0.460390\pi\)
0.124118 + 0.992267i \(0.460390\pi\)
\(464\) 4.59998e30 0.994946
\(465\) 7.86228e29 0.165897
\(466\) −1.66728e30 −0.343218
\(467\) 8.69961e30 1.74725 0.873626 0.486598i \(-0.161762\pi\)
0.873626 + 0.486598i \(0.161762\pi\)
\(468\) 1.83586e30 0.359760
\(469\) 1.11418e30 0.213043
\(470\) 1.27503e30 0.237901
\(471\) −4.94557e30 −0.900483
\(472\) 5.66614e30 1.00682
\(473\) 3.44273e30 0.597036
\(474\) −1.37320e30 −0.232425
\(475\) −6.48267e30 −1.07097
\(476\) −7.34646e28 −0.0118467
\(477\) −1.39685e30 −0.219882
\(478\) −1.13397e29 −0.0174253
\(479\) −6.06140e29 −0.0909315 −0.0454658 0.998966i \(-0.514477\pi\)
−0.0454658 + 0.998966i \(0.514477\pi\)
\(480\) −1.66019e30 −0.243155
\(481\) 1.06698e31 1.52576
\(482\) 3.26067e30 0.455267
\(483\) −4.71474e27 −0.000642786 0
\(484\) −8.14456e30 −1.08429
\(485\) 1.49542e30 0.194417
\(486\) 1.96779e29 0.0249839
\(487\) −4.30348e30 −0.533626 −0.266813 0.963748i \(-0.585971\pi\)
−0.266813 + 0.963748i \(0.585971\pi\)
\(488\) 3.48750e30 0.422364
\(489\) 8.56873e30 1.01360
\(490\) 1.34476e30 0.155378
\(491\) −1.13566e31 −1.28177 −0.640885 0.767637i \(-0.721432\pi\)
−0.640885 + 0.767637i \(0.721432\pi\)
\(492\) −6.57291e30 −0.724701
\(493\) −6.89323e29 −0.0742478
\(494\) −6.30586e30 −0.663566
\(495\) 2.19009e30 0.225166
\(496\) −3.63020e30 −0.364663
\(497\) −6.12161e30 −0.600852
\(498\) −2.33746e29 −0.0224185
\(499\) 9.99815e30 0.937052 0.468526 0.883450i \(-0.344785\pi\)
0.468526 + 0.883450i \(0.344785\pi\)
\(500\) −7.46016e30 −0.683272
\(501\) 6.37137e30 0.570295
\(502\) −3.42741e29 −0.0299829
\(503\) −6.16806e30 −0.527372 −0.263686 0.964609i \(-0.584938\pi\)
−0.263686 + 0.964609i \(0.584938\pi\)
\(504\) 9.46025e29 0.0790590
\(505\) 1.16985e29 0.00955609
\(506\) −2.48761e28 −0.00198633
\(507\) 4.57577e30 0.357168
\(508\) 2.38388e30 0.181908
\(509\) 2.98637e30 0.222786 0.111393 0.993776i \(-0.464469\pi\)
0.111393 + 0.993776i \(0.464469\pi\)
\(510\) 5.84789e28 0.00426523
\(511\) −5.77985e30 −0.412170
\(512\) 1.35292e31 0.943335
\(513\) 3.78019e30 0.257729
\(514\) −6.48331e30 −0.432236
\(515\) 1.45973e30 0.0951677
\(516\) 3.03864e30 0.193735
\(517\) 3.30395e31 2.06012
\(518\) 2.52348e30 0.153889
\(519\) −1.09484e31 −0.653019
\(520\) −7.02563e30 −0.409871
\(521\) −8.51705e30 −0.486022 −0.243011 0.970024i \(-0.578135\pi\)
−0.243011 + 0.970024i \(0.578135\pi\)
\(522\) 4.07409e30 0.227415
\(523\) 1.27472e31 0.696059 0.348029 0.937484i \(-0.386851\pi\)
0.348029 + 0.937484i \(0.386851\pi\)
\(524\) 6.36736e30 0.340134
\(525\) 2.91095e30 0.152126
\(526\) −1.02475e31 −0.523942
\(527\) 5.43998e29 0.0272129
\(528\) −1.01122e31 −0.494942
\(529\) −2.08802e31 −0.999989
\(530\) 2.45346e30 0.114976
\(531\) −1.01667e31 −0.466221
\(532\) 8.34105e30 0.374315
\(533\) −4.28644e31 −1.88250
\(534\) −4.68726e30 −0.201463
\(535\) −1.72312e31 −0.724850
\(536\) 1.13050e31 0.465455
\(537\) 7.40796e30 0.298535
\(538\) 2.76862e30 0.109211
\(539\) 3.48463e31 1.34551
\(540\) 1.93303e30 0.0730650
\(541\) −1.00626e31 −0.372342 −0.186171 0.982517i \(-0.559608\pi\)
−0.186171 + 0.982517i \(0.559608\pi\)
\(542\) 6.37691e30 0.231004
\(543\) −1.85387e31 −0.657479
\(544\) −1.14870e30 −0.0398859
\(545\) 7.43099e30 0.252631
\(546\) 2.83155e30 0.0942560
\(547\) −4.27216e31 −1.39249 −0.696247 0.717802i \(-0.745148\pi\)
−0.696247 + 0.717802i \(0.745148\pi\)
\(548\) 1.08558e31 0.346485
\(549\) −6.25758e30 −0.195580
\(550\) 1.53589e31 0.470098
\(551\) 7.82647e31 2.34597
\(552\) −4.78383e28 −0.00140435
\(553\) 1.18454e31 0.340574
\(554\) 4.01039e30 0.112934
\(555\) 1.12345e31 0.309873
\(556\) 2.04273e31 0.551887
\(557\) −6.81696e31 −1.80408 −0.902040 0.431652i \(-0.857931\pi\)
−0.902040 + 0.431652i \(0.857931\pi\)
\(558\) −3.21518e30 −0.0833511
\(559\) 1.98161e31 0.503248
\(560\) 3.36626e30 0.0837502
\(561\) 1.51535e30 0.0369351
\(562\) 3.20246e31 0.764744
\(563\) 6.17906e30 0.144569 0.0722846 0.997384i \(-0.476971\pi\)
0.0722846 + 0.997384i \(0.476971\pi\)
\(564\) 2.91615e31 0.668497
\(565\) −2.53449e30 −0.0569289
\(566\) −2.58540e31 −0.569035
\(567\) −1.69744e30 −0.0366091
\(568\) −6.21131e31 −1.31274
\(569\) −1.87031e31 −0.387366 −0.193683 0.981064i \(-0.562043\pi\)
−0.193683 + 0.981064i \(0.562043\pi\)
\(570\) −6.63960e30 −0.134766
\(571\) 2.56599e31 0.510433 0.255217 0.966884i \(-0.417853\pi\)
0.255217 + 0.966884i \(0.417853\pi\)
\(572\) −8.35566e31 −1.62902
\(573\) 2.22659e31 0.425463
\(574\) −1.01378e31 −0.189870
\(575\) −1.47200e29 −0.00270226
\(576\) −3.73195e30 −0.0671548
\(577\) −2.35477e31 −0.415362 −0.207681 0.978197i \(-0.566592\pi\)
−0.207681 + 0.978197i \(0.566592\pi\)
\(578\) −2.24830e31 −0.388762
\(579\) −1.51535e31 −0.256867
\(580\) 4.00212e31 0.665071
\(581\) 2.01632e30 0.0328499
\(582\) −6.11532e30 −0.0976799
\(583\) 6.35758e31 0.995642
\(584\) −5.86454e31 −0.900505
\(585\) 1.26060e31 0.189795
\(586\) 3.87027e31 0.571372
\(587\) 7.54955e31 1.09291 0.546453 0.837490i \(-0.315978\pi\)
0.546453 + 0.837490i \(0.315978\pi\)
\(588\) 3.07561e31 0.436608
\(589\) −6.17647e31 −0.859834
\(590\) 1.78569e31 0.243786
\(591\) −1.66761e31 −0.223274
\(592\) −5.18723e31 −0.681139
\(593\) 3.26412e31 0.420376 0.210188 0.977661i \(-0.432592\pi\)
0.210188 + 0.977661i \(0.432592\pi\)
\(594\) −8.95610e30 −0.113129
\(595\) −5.04446e29 −0.00624985
\(596\) −5.02909e31 −0.611164
\(597\) 1.32237e31 0.157634
\(598\) −1.43185e29 −0.00167430
\(599\) 5.33038e31 0.611434 0.305717 0.952122i \(-0.401104\pi\)
0.305717 + 0.952122i \(0.401104\pi\)
\(600\) 2.95360e31 0.332363
\(601\) 1.42622e32 1.57446 0.787229 0.616661i \(-0.211515\pi\)
0.787229 + 0.616661i \(0.211515\pi\)
\(602\) 4.68666e30 0.0507579
\(603\) −2.02845e31 −0.215534
\(604\) 3.87750e31 0.404229
\(605\) −5.59248e31 −0.572029
\(606\) −4.78396e29 −0.00480123
\(607\) −1.51222e32 −1.48917 −0.744586 0.667526i \(-0.767353\pi\)
−0.744586 + 0.667526i \(0.767353\pi\)
\(608\) 1.30422e32 1.26026
\(609\) −3.51436e31 −0.333232
\(610\) 1.09909e31 0.102268
\(611\) 1.90173e32 1.73650
\(612\) 1.33748e30 0.0119852
\(613\) −1.24132e32 −1.09166 −0.545831 0.837895i \(-0.683786\pi\)
−0.545831 + 0.837895i \(0.683786\pi\)
\(614\) −1.31848e31 −0.113798
\(615\) −4.51330e31 −0.382323
\(616\) −4.30569e31 −0.357985
\(617\) −1.89291e32 −1.54472 −0.772361 0.635184i \(-0.780924\pi\)
−0.772361 + 0.635184i \(0.780924\pi\)
\(618\) −5.96937e30 −0.0478147
\(619\) −1.33375e32 −1.04865 −0.524327 0.851517i \(-0.675683\pi\)
−0.524327 + 0.851517i \(0.675683\pi\)
\(620\) −3.15838e31 −0.243759
\(621\) 8.58355e28 0.000650300 0
\(622\) −2.81802e31 −0.209583
\(623\) 4.04329e31 0.295204
\(624\) −5.82049e31 −0.417193
\(625\) 6.24200e31 0.439241
\(626\) 6.81182e31 0.470607
\(627\) −1.72050e32 −1.16702
\(628\) 1.98670e32 1.32311
\(629\) 7.77324e30 0.0508299
\(630\) 2.98141e30 0.0191428
\(631\) 2.80913e31 0.177107 0.0885533 0.996071i \(-0.471776\pi\)
0.0885533 + 0.996071i \(0.471776\pi\)
\(632\) 1.20190e32 0.744083
\(633\) 1.22468e32 0.744529
\(634\) −1.17599e32 −0.702066
\(635\) 1.63690e31 0.0959674
\(636\) 5.61134e31 0.323080
\(637\) 2.00572e32 1.13414
\(638\) −1.85426e32 −1.02975
\(639\) 1.11449e32 0.607876
\(640\) 8.51714e31 0.456272
\(641\) −3.78628e31 −0.199225 −0.0996127 0.995026i \(-0.531760\pi\)
−0.0996127 + 0.995026i \(0.531760\pi\)
\(642\) 7.04649e31 0.364183
\(643\) −1.43262e32 −0.727283 −0.363641 0.931539i \(-0.618467\pi\)
−0.363641 + 0.931539i \(0.618467\pi\)
\(644\) 1.89397e29 0.000944466 0
\(645\) 2.08649e31 0.102207
\(646\) −4.59400e30 −0.0221064
\(647\) −2.32752e31 −0.110026 −0.0550129 0.998486i \(-0.517520\pi\)
−0.0550129 + 0.998486i \(0.517520\pi\)
\(648\) −1.72231e31 −0.0799832
\(649\) 4.62721e32 2.11108
\(650\) 8.84043e31 0.396251
\(651\) 2.77345e31 0.122135
\(652\) −3.44217e32 −1.48931
\(653\) −4.07291e32 −1.73142 −0.865712 0.500543i \(-0.833134\pi\)
−0.865712 + 0.500543i \(0.833134\pi\)
\(654\) −3.03881e31 −0.126928
\(655\) 4.37216e31 0.179441
\(656\) 2.08390e32 0.840393
\(657\) 1.05227e32 0.416988
\(658\) 4.49773e31 0.175144
\(659\) −4.35631e32 −1.66700 −0.833501 0.552518i \(-0.813667\pi\)
−0.833501 + 0.552518i \(0.813667\pi\)
\(660\) −8.79788e31 −0.330844
\(661\) −4.22347e32 −1.56082 −0.780410 0.625268i \(-0.784990\pi\)
−0.780410 + 0.625268i \(0.784990\pi\)
\(662\) 1.56520e32 0.568465
\(663\) 8.72220e30 0.0311330
\(664\) 2.04586e31 0.0717701
\(665\) 5.72740e31 0.197474
\(666\) −4.59419e31 −0.155688
\(667\) 1.77713e30 0.00591932
\(668\) −2.55946e32 −0.837953
\(669\) 1.41456e32 0.455218
\(670\) 3.56280e31 0.112702
\(671\) 2.84805e32 0.885601
\(672\) −5.85639e31 −0.179013
\(673\) 3.27631e32 0.984491 0.492245 0.870456i \(-0.336176\pi\)
0.492245 + 0.870456i \(0.336176\pi\)
\(674\) −1.70096e32 −0.502466
\(675\) −5.29960e31 −0.153904
\(676\) −1.83815e32 −0.524799
\(677\) 1.65241e32 0.463820 0.231910 0.972737i \(-0.425503\pi\)
0.231910 + 0.972737i \(0.425503\pi\)
\(678\) 1.03645e31 0.0286026
\(679\) 5.27516e31 0.143131
\(680\) −5.11838e30 −0.0136546
\(681\) 2.11311e32 0.554281
\(682\) 1.46334e32 0.377420
\(683\) 1.17826e32 0.298816 0.149408 0.988776i \(-0.452263\pi\)
0.149408 + 0.988776i \(0.452263\pi\)
\(684\) −1.51855e32 −0.378691
\(685\) 7.45416e31 0.182792
\(686\) 1.00650e32 0.242710
\(687\) −2.90633e32 −0.689194
\(688\) −9.63381e31 −0.224663
\(689\) 3.65937e32 0.839238
\(690\) −1.50763e29 −0.000340040 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
\(693\) 7.72564e31 0.165769
\(694\) −1.63594e32 −0.345250
\(695\) 1.40264e32 0.291154
\(696\) −3.56585e32 −0.728043
\(697\) −3.12280e31 −0.0627143
\(698\) 1.16437e32 0.230013
\(699\) −2.61838e32 −0.508798
\(700\) −1.16937e32 −0.223523
\(701\) −3.79931e32 −0.714409 −0.357205 0.934026i \(-0.616270\pi\)
−0.357205 + 0.934026i \(0.616270\pi\)
\(702\) −5.15505e31 −0.0953579
\(703\) −8.82561e32 −1.60605
\(704\) 1.69854e32 0.304082
\(705\) 2.00238e32 0.352672
\(706\) −1.12426e32 −0.194810
\(707\) 4.12670e30 0.00703526
\(708\) 4.08408e32 0.685033
\(709\) −1.44569e31 −0.0238586 −0.0119293 0.999929i \(-0.503797\pi\)
−0.0119293 + 0.999929i \(0.503797\pi\)
\(710\) −1.95750e32 −0.317857
\(711\) −2.15655e32 −0.344555
\(712\) 4.10254e32 0.644960
\(713\) −1.40247e30 −0.00216952
\(714\) 2.06287e30 0.00314009
\(715\) −5.73743e32 −0.859405
\(716\) −2.97588e32 −0.438647
\(717\) −1.78085e31 −0.0258319
\(718\) 2.25520e32 0.321924
\(719\) −1.33387e32 −0.187383 −0.0936915 0.995601i \(-0.529867\pi\)
−0.0936915 + 0.995601i \(0.529867\pi\)
\(720\) −6.12854e31 −0.0847292
\(721\) 5.14926e31 0.0700631
\(722\) 2.30764e32 0.309022
\(723\) 5.12073e32 0.674903
\(724\) 7.44725e32 0.966057
\(725\) −1.09722e33 −1.40090
\(726\) 2.28697e32 0.287402
\(727\) 4.93859e31 0.0610882 0.0305441 0.999533i \(-0.490276\pi\)
0.0305441 + 0.999533i \(0.490276\pi\)
\(728\) −2.47832e32 −0.301750
\(729\) 3.09032e31 0.0370370
\(730\) −1.84822e32 −0.218042
\(731\) 1.44366e31 0.0167654
\(732\) 2.51375e32 0.287372
\(733\) 1.05389e33 1.18604 0.593019 0.805189i \(-0.297936\pi\)
0.593019 + 0.805189i \(0.297936\pi\)
\(734\) 6.32541e32 0.700784
\(735\) 2.11188e32 0.230337
\(736\) 2.96144e30 0.00317986
\(737\) 9.23219e32 0.975952
\(738\) 1.84566e32 0.192089
\(739\) 1.09776e33 1.12486 0.562428 0.826846i \(-0.309867\pi\)
0.562428 + 0.826846i \(0.309867\pi\)
\(740\) −4.51303e32 −0.455306
\(741\) −9.90305e32 −0.983693
\(742\) 8.65468e31 0.0846460
\(743\) −5.60678e32 −0.539936 −0.269968 0.962869i \(-0.587013\pi\)
−0.269968 + 0.962869i \(0.587013\pi\)
\(744\) 2.81409e32 0.266839
\(745\) −3.45324e32 −0.322426
\(746\) −5.04137e32 −0.463502
\(747\) −3.67086e31 −0.0332339
\(748\) −6.08734e31 −0.0542699
\(749\) −6.07840e32 −0.533640
\(750\) 2.09480e32 0.181108
\(751\) −1.01215e33 −0.861759 −0.430880 0.902409i \(-0.641797\pi\)
−0.430880 + 0.902409i \(0.641797\pi\)
\(752\) −9.24546e32 −0.775217
\(753\) −5.38258e31 −0.0444476
\(754\) −1.06730e33 −0.867990
\(755\) 2.66250e32 0.213255
\(756\) 6.81883e31 0.0537909
\(757\) −9.48447e32 −0.736903 −0.368452 0.929647i \(-0.620112\pi\)
−0.368452 + 0.929647i \(0.620112\pi\)
\(758\) −4.13496e32 −0.316428
\(759\) −3.90668e30 −0.00294461
\(760\) 5.81133e32 0.431438
\(761\) 9.81408e32 0.717672 0.358836 0.933401i \(-0.383174\pi\)
0.358836 + 0.933401i \(0.383174\pi\)
\(762\) −6.69388e31 −0.0482165
\(763\) 2.62131e32 0.185989
\(764\) −8.94448e32 −0.625146
\(765\) 9.18383e30 0.00632291
\(766\) −1.68400e32 −0.114212
\(767\) 2.66338e33 1.77945
\(768\) −1.71575e32 −0.112927
\(769\) 8.57155e32 0.555785 0.277892 0.960612i \(-0.410364\pi\)
0.277892 + 0.960612i \(0.410364\pi\)
\(770\) −1.35695e32 −0.0866801
\(771\) −1.01817e33 −0.640761
\(772\) 6.08734e32 0.377423
\(773\) −9.41593e31 −0.0575174 −0.0287587 0.999586i \(-0.509155\pi\)
−0.0287587 + 0.999586i \(0.509155\pi\)
\(774\) −8.53242e31 −0.0513512
\(775\) 8.65904e32 0.513452
\(776\) 5.35245e32 0.312711
\(777\) 3.96301e32 0.228130
\(778\) −2.22819e32 −0.126382
\(779\) 3.54557e33 1.98155
\(780\) −5.06399e32 −0.278872
\(781\) −5.07242e33 −2.75251
\(782\) −1.04314e29 −5.57785e−5 0
\(783\) 6.39816e32 0.337128
\(784\) −9.75103e32 −0.506309
\(785\) 1.36417e33 0.698020
\(786\) −1.78794e32 −0.0901557
\(787\) 2.72161e33 1.35243 0.676216 0.736703i \(-0.263618\pi\)
0.676216 + 0.736703i \(0.263618\pi\)
\(788\) 6.69900e32 0.328064
\(789\) −1.60933e33 −0.776710
\(790\) 3.78780e32 0.180167
\(791\) −8.94051e31 −0.0419115
\(792\) 7.83884e32 0.362170
\(793\) 1.63931e33 0.746483
\(794\) −2.95979e31 −0.0132839
\(795\) 3.85304e32 0.170444
\(796\) −5.31215e32 −0.231617
\(797\) −3.49544e33 −1.50221 −0.751106 0.660182i \(-0.770479\pi\)
−0.751106 + 0.660182i \(0.770479\pi\)
\(798\) −2.34215e32 −0.0992158
\(799\) 1.38546e32 0.0578505
\(800\) −1.82843e33 −0.752565
\(801\) −7.36112e32 −0.298655
\(802\) 4.89975e32 0.195961
\(803\) −4.78923e33 −1.88815
\(804\) 8.14854e32 0.316691
\(805\) 1.30050e30 0.000498263 0
\(806\) 8.42287e32 0.318132
\(807\) 4.34799e32 0.161898
\(808\) 4.18717e31 0.0153706
\(809\) 1.04226e33 0.377196 0.188598 0.982054i \(-0.439606\pi\)
0.188598 + 0.982054i \(0.439606\pi\)
\(810\) −5.42789e31 −0.0193666
\(811\) −2.22771e33 −0.783643 −0.391821 0.920041i \(-0.628155\pi\)
−0.391821 + 0.920041i \(0.628155\pi\)
\(812\) 1.41176e33 0.489629
\(813\) 1.00146e33 0.342447
\(814\) 2.09098e33 0.704967
\(815\) −2.36357e33 −0.785700
\(816\) −4.24039e31 −0.0138985
\(817\) −1.63911e33 −0.529729
\(818\) −9.18857e32 −0.292809
\(819\) 4.44682e32 0.139728
\(820\) 1.81305e33 0.561760
\(821\) −7.36864e31 −0.0225134 −0.0112567 0.999937i \(-0.503583\pi\)
−0.0112567 + 0.999937i \(0.503583\pi\)
\(822\) −3.04828e32 −0.0918394
\(823\) 4.08369e33 1.21326 0.606631 0.794984i \(-0.292521\pi\)
0.606631 + 0.794984i \(0.292521\pi\)
\(824\) 5.22471e32 0.153073
\(825\) 2.41204e33 0.696888
\(826\) 6.29910e32 0.179477
\(827\) 6.73455e32 0.189232 0.0946161 0.995514i \(-0.469838\pi\)
0.0946161 + 0.995514i \(0.469838\pi\)
\(828\) −3.44812e30 −0.000955507 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
\(831\) 6.29813e32 0.167417
\(832\) 9.77667e32 0.256314
\(833\) 1.46123e32 0.0377833
\(834\) −5.73593e32 −0.146283
\(835\) −1.75746e33 −0.442070
\(836\) 6.91147e33 1.71474
\(837\) −5.04928e32 −0.123562
\(838\) 1.45251e32 0.0350601
\(839\) −4.99200e33 −1.18853 −0.594267 0.804268i \(-0.702558\pi\)
−0.594267 + 0.804268i \(0.702558\pi\)
\(840\) −2.60949e32 −0.0612835
\(841\) 8.92995e33 2.06869
\(842\) −1.63249e33 −0.373046
\(843\) 5.02931e33 1.13368
\(844\) −4.91971e33 −1.09396
\(845\) −1.26217e33 −0.276863
\(846\) −8.18846e32 −0.177192
\(847\) −1.97277e33 −0.421131
\(848\) −1.77904e33 −0.374657
\(849\) −4.06025e33 −0.843557
\(850\) 6.44051e31 0.0132009
\(851\) −2.00400e31 −0.00405236
\(852\) −4.47704e33 −0.893172
\(853\) −2.52826e33 −0.497631 −0.248815 0.968551i \(-0.580041\pi\)
−0.248815 + 0.968551i \(0.580041\pi\)
\(854\) 3.87710e32 0.0752907
\(855\) −1.04272e33 −0.199782
\(856\) −6.16746e33 −1.16589
\(857\) 7.45015e33 1.38959 0.694793 0.719209i \(-0.255496\pi\)
0.694793 + 0.719209i \(0.255496\pi\)
\(858\) 2.34625e33 0.431788
\(859\) −5.27466e32 −0.0957797 −0.0478898 0.998853i \(-0.515250\pi\)
−0.0478898 + 0.998853i \(0.515250\pi\)
\(860\) −8.38169e32 −0.150176
\(861\) −1.59209e33 −0.281469
\(862\) −1.75098e33 −0.305456
\(863\) −6.72847e33 −1.15822 −0.579112 0.815248i \(-0.696600\pi\)
−0.579112 + 0.815248i \(0.696600\pi\)
\(864\) 1.06620e33 0.181105
\(865\) 3.01997e33 0.506195
\(866\) 4.16294e33 0.688565
\(867\) −3.53084e33 −0.576313
\(868\) −1.11413e33 −0.179457
\(869\) 9.81522e33 1.56017
\(870\) −1.12378e33 −0.176283
\(871\) 5.31397e33 0.822641
\(872\) 2.65972e33 0.406347
\(873\) −9.60382e32 −0.144804
\(874\) 1.18437e31 0.00176240
\(875\) −1.80700e33 −0.265378
\(876\) −4.22709e33 −0.612695
\(877\) −7.45124e33 −1.06594 −0.532970 0.846134i \(-0.678924\pi\)
−0.532970 + 0.846134i \(0.678924\pi\)
\(878\) 4.93870e32 0.0697310
\(879\) 6.07808e33 0.847021
\(880\) 2.78932e33 0.383660
\(881\) 1.17245e34 1.59174 0.795870 0.605468i \(-0.207014\pi\)
0.795870 + 0.605468i \(0.207014\pi\)
\(882\) −8.63624e32 −0.115727
\(883\) 1.05295e34 1.39271 0.696353 0.717699i \(-0.254805\pi\)
0.696353 + 0.717699i \(0.254805\pi\)
\(884\) −3.50382e32 −0.0457447
\(885\) 2.80434e33 0.361396
\(886\) 1.65825e33 0.210941
\(887\) −6.86475e33 −0.861992 −0.430996 0.902354i \(-0.641838\pi\)
−0.430996 + 0.902354i \(0.641838\pi\)
\(888\) 4.02108e33 0.498417
\(889\) 5.77423e32 0.0706519
\(890\) 1.29292e33 0.156166
\(891\) −1.40651e33 −0.167706
\(892\) −5.68246e33 −0.668868
\(893\) −1.57303e34 −1.82787
\(894\) 1.41216e33 0.161995
\(895\) −2.04339e33 −0.231412
\(896\) 3.00446e33 0.335911
\(897\) −2.24865e31 −0.00248204
\(898\) −2.45726e32 −0.0267777
\(899\) −1.04540e34 −1.12472
\(900\) 2.12892e33 0.226136
\(901\) 2.66596e32 0.0279587
\(902\) −8.40023e33 −0.869794
\(903\) 7.36017e32 0.0752452
\(904\) −9.07152e32 −0.0915678
\(905\) 5.11367e33 0.509653
\(906\) −1.08879e33 −0.107145
\(907\) −9.01545e33 −0.875999 −0.438000 0.898975i \(-0.644313\pi\)
−0.438000 + 0.898975i \(0.644313\pi\)
\(908\) −8.48862e33 −0.814423
\(909\) −7.51298e31 −0.00711750
\(910\) −7.81047e32 −0.0730636
\(911\) 1.54259e34 1.42492 0.712459 0.701714i \(-0.247582\pi\)
0.712459 + 0.701714i \(0.247582\pi\)
\(912\) 4.81448e33 0.439145
\(913\) 1.67074e33 0.150486
\(914\) −2.14755e33 −0.191013
\(915\) 1.72607e33 0.151606
\(916\) 1.16751e34 1.01266
\(917\) 1.54230e33 0.132106
\(918\) −3.75561e31 −0.00317680
\(919\) 3.21779e32 0.0268800 0.0134400 0.999910i \(-0.495722\pi\)
0.0134400 + 0.999910i \(0.495722\pi\)
\(920\) 1.31956e31 0.00108860
\(921\) −2.07060e33 −0.168699
\(922\) −5.60525e33 −0.451013
\(923\) −2.91964e34 −2.32012
\(924\) −3.10349e33 −0.243569
\(925\) 1.23730e34 0.959056
\(926\) 1.26286e33 0.0966786
\(927\) −9.37462e32 −0.0708821
\(928\) 2.20745e34 1.64850
\(929\) 6.89145e33 0.508311 0.254156 0.967163i \(-0.418202\pi\)
0.254156 + 0.967163i \(0.418202\pi\)
\(930\) 8.86865e32 0.0646106
\(931\) −1.65905e34 −1.19382
\(932\) 1.05184e34 0.747593
\(933\) −4.42557e33 −0.310692
\(934\) 9.81316e33 0.680487
\(935\) −4.17989e32 −0.0286306
\(936\) 4.51197e33 0.305277
\(937\) 6.25703e32 0.0418180 0.0209090 0.999781i \(-0.493344\pi\)
0.0209090 + 0.999781i \(0.493344\pi\)
\(938\) 1.25679e33 0.0829720
\(939\) 1.06976e34 0.697644
\(940\) −8.04381e33 −0.518193
\(941\) 2.43281e34 1.54820 0.774101 0.633062i \(-0.218202\pi\)
0.774101 + 0.633062i \(0.218202\pi\)
\(942\) −5.57860e33 −0.350703
\(943\) 8.05081e31 0.00499983
\(944\) −1.29483e34 −0.794393
\(945\) 4.68217e32 0.0283780
\(946\) 3.88340e33 0.232522
\(947\) 1.95908e34 1.15885 0.579424 0.815026i \(-0.303277\pi\)
0.579424 + 0.815026i \(0.303277\pi\)
\(948\) 8.66313e33 0.506266
\(949\) −2.75664e34 −1.59155
\(950\) −7.31245e33 −0.417101
\(951\) −1.84684e34 −1.04077
\(952\) −1.80553e32 −0.0100526
\(953\) 2.99818e33 0.164926 0.0824632 0.996594i \(-0.473721\pi\)
0.0824632 + 0.996594i \(0.473721\pi\)
\(954\) −1.57565e33 −0.0856355
\(955\) −6.14175e33 −0.329802
\(956\) 7.15389e32 0.0379556
\(957\) −2.91203e34 −1.52654
\(958\) −6.83726e32 −0.0354143
\(959\) 2.62948e33 0.134573
\(960\) 1.02941e33 0.0520558
\(961\) −1.17633e34 −0.587772
\(962\) 1.20355e34 0.594225
\(963\) 1.10662e34 0.539878
\(964\) −2.05706e34 −0.991658
\(965\) 4.17989e33 0.199114
\(966\) −5.31823e30 −0.000250340 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
\(969\) −7.21466e32 −0.0327712
\(970\) 1.68683e33 0.0757177
\(971\) 1.12238e34 0.497872 0.248936 0.968520i \(-0.419919\pi\)
0.248936 + 0.968520i \(0.419919\pi\)
\(972\) −1.24142e33 −0.0544198
\(973\) 4.94789e33 0.214349
\(974\) −4.85433e33 −0.207827
\(975\) 1.38835e34 0.587415
\(976\) −7.96969e33 −0.333249
\(977\) 2.41460e34 0.997831 0.498915 0.866651i \(-0.333732\pi\)
0.498915 + 0.866651i \(0.333732\pi\)
\(978\) 9.66553e33 0.394756
\(979\) 3.35031e34 1.35233
\(980\) −8.48368e33 −0.338442
\(981\) −4.77230e33 −0.188163
\(982\) −1.28102e34 −0.499199
\(983\) −1.21263e34 −0.467048 −0.233524 0.972351i \(-0.575026\pi\)
−0.233524 + 0.972351i \(0.575026\pi\)
\(984\) −1.61541e34 −0.614951
\(985\) 4.59989e33 0.173073
\(986\) −7.77557e32 −0.0289166
\(987\) 7.06347e33 0.259640
\(988\) 3.97818e34 1.44537
\(989\) −3.72187e31 −0.00133661
\(990\) 2.47042e33 0.0876934
\(991\) 1.83350e34 0.643329 0.321664 0.946854i \(-0.395758\pi\)
0.321664 + 0.946854i \(0.395758\pi\)
\(992\) −1.74207e34 −0.604201
\(993\) 2.45807e34 0.842711
\(994\) −6.90518e33 −0.234008
\(995\) −3.64760e33 −0.122192
\(996\) 1.47463e33 0.0488316
\(997\) 1.59227e34 0.521222 0.260611 0.965444i \(-0.416076\pi\)
0.260611 + 0.965444i \(0.416076\pi\)
\(998\) 1.12779e34 0.364945
\(999\) −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 3.24.a.a.1.1 1
3.2 odd 2 9.24.a.a.1.1 1
4.3 odd 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 1.1 even 1 trivial
9.24.a.a.1.1 1 3.2 odd 2
48.24.a.a.1.1 1 4.3 odd 2