Properties

Label 18.16.a.b.1.1
Level $18$
Weight $16$
Character 18.1
Self dual yes
Analytic conductor $25.685$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 18.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(25.6848309180\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 6)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 18.1

$q$-expansion

\(f(q)\) \(=\) \(q-128.000 q^{2} +16384.0 q^{4} +114810. q^{5} -3.03453e6 q^{7} -2.09715e6 q^{8} +O(q^{10})\) \(q-128.000 q^{2} +16384.0 q^{4} +114810. q^{5} -3.03453e6 q^{7} -2.09715e6 q^{8} -1.46957e7 q^{10} +1.03452e8 q^{11} -1.04366e8 q^{13} +3.88420e8 q^{14} +2.68435e8 q^{16} -9.97690e8 q^{17} +4.93402e9 q^{19} +1.88105e9 q^{20} -1.32418e10 q^{22} -8.32492e9 q^{23} -1.73362e10 q^{25} +1.33588e10 q^{26} -4.97177e10 q^{28} -1.04128e11 q^{29} -2.96697e11 q^{31} -3.43597e10 q^{32} +1.27704e11 q^{34} -3.48394e11 q^{35} -1.78337e11 q^{37} -6.31554e11 q^{38} -2.40774e11 q^{40} +1.79088e12 q^{41} -2.86346e12 q^{43} +1.69495e12 q^{44} +1.06559e12 q^{46} -4.33291e12 q^{47} +4.46080e12 q^{49} +2.21904e12 q^{50} -1.70993e12 q^{52} -9.73232e12 q^{53} +1.18773e13 q^{55} +6.36387e12 q^{56} +1.33284e13 q^{58} +1.35148e13 q^{59} +5.35266e12 q^{61} +3.79772e13 q^{62} +4.39805e12 q^{64} -1.19822e13 q^{65} -5.32339e13 q^{67} -1.63461e13 q^{68} +4.45945e13 q^{70} +2.02297e13 q^{71} +2.62642e13 q^{73} +2.28272e13 q^{74} +8.08389e13 q^{76} -3.13927e14 q^{77} -3.39031e14 q^{79} +3.08191e13 q^{80} -2.29233e14 q^{82} -1.31685e14 q^{83} -1.14545e14 q^{85} +3.66523e14 q^{86} -2.16954e14 q^{88} +3.93521e13 q^{89} +3.16701e14 q^{91} -1.36395e14 q^{92} +5.54612e14 q^{94} +5.66474e14 q^{95} +1.12875e15 q^{97} -5.70982e14 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −128.000 −0.707107
\(3\) 0 0
\(4\) 16384.0 0.500000
\(5\) 114810. 0.657211 0.328605 0.944467i \(-0.393421\pi\)
0.328605 + 0.944467i \(0.393421\pi\)
\(6\) 0 0
\(7\) −3.03453e6 −1.39269 −0.696347 0.717705i \(-0.745193\pi\)
−0.696347 + 0.717705i \(0.745193\pi\)
\(8\) −2.09715e6 −0.353553
\(9\) 0 0
\(10\) −1.46957e7 −0.464718
\(11\) 1.03452e8 1.60064 0.800318 0.599576i \(-0.204664\pi\)
0.800318 + 0.599576i \(0.204664\pi\)
\(12\) 0 0
\(13\) −1.04366e8 −0.461300 −0.230650 0.973037i \(-0.574085\pi\)
−0.230650 + 0.973037i \(0.574085\pi\)
\(14\) 3.88420e8 0.984784
\(15\) 0 0
\(16\) 2.68435e8 0.250000
\(17\) −9.97690e8 −0.589697 −0.294848 0.955544i \(-0.595269\pi\)
−0.294848 + 0.955544i \(0.595269\pi\)
\(18\) 0 0
\(19\) 4.93402e9 1.26633 0.633167 0.774015i \(-0.281754\pi\)
0.633167 + 0.774015i \(0.281754\pi\)
\(20\) 1.88105e9 0.328605
\(21\) 0 0
\(22\) −1.32418e10 −1.13182
\(23\) −8.32492e9 −0.509825 −0.254913 0.966964i \(-0.582047\pi\)
−0.254913 + 0.966964i \(0.582047\pi\)
\(24\) 0 0
\(25\) −1.73362e10 −0.568074
\(26\) 1.33588e10 0.326188
\(27\) 0 0
\(28\) −4.97177e10 −0.696347
\(29\) −1.04128e11 −1.12094 −0.560472 0.828174i \(-0.689380\pi\)
−0.560472 + 0.828174i \(0.689380\pi\)
\(30\) 0 0
\(31\) −2.96697e11 −1.93687 −0.968434 0.249270i \(-0.919809\pi\)
−0.968434 + 0.249270i \(0.919809\pi\)
\(32\) −3.43597e10 −0.176777
\(33\) 0 0
\(34\) 1.27704e11 0.416978
\(35\) −3.48394e11 −0.915294
\(36\) 0 0
\(37\) −1.78337e11 −0.308837 −0.154419 0.988006i \(-0.549350\pi\)
−0.154419 + 0.988006i \(0.549350\pi\)
\(38\) −6.31554e11 −0.895434
\(39\) 0 0
\(40\) −2.40774e11 −0.232359
\(41\) 1.79088e12 1.43611 0.718056 0.695986i \(-0.245032\pi\)
0.718056 + 0.695986i \(0.245032\pi\)
\(42\) 0 0
\(43\) −2.86346e12 −1.60649 −0.803244 0.595650i \(-0.796895\pi\)
−0.803244 + 0.595650i \(0.796895\pi\)
\(44\) 1.69495e12 0.800318
\(45\) 0 0
\(46\) 1.06559e12 0.360501
\(47\) −4.33291e12 −1.24751 −0.623757 0.781618i \(-0.714395\pi\)
−0.623757 + 0.781618i \(0.714395\pi\)
\(48\) 0 0
\(49\) 4.46080e12 0.939598
\(50\) 2.21904e12 0.401689
\(51\) 0 0
\(52\) −1.70993e12 −0.230650
\(53\) −9.73232e12 −1.13801 −0.569007 0.822333i \(-0.692672\pi\)
−0.569007 + 0.822333i \(0.692672\pi\)
\(54\) 0 0
\(55\) 1.18773e13 1.05196
\(56\) 6.36387e12 0.492392
\(57\) 0 0
\(58\) 1.33284e13 0.792626
\(59\) 1.35148e13 0.707002 0.353501 0.935434i \(-0.384991\pi\)
0.353501 + 0.935434i \(0.384991\pi\)
\(60\) 0 0
\(61\) 5.35266e12 0.218070 0.109035 0.994038i \(-0.465224\pi\)
0.109035 + 0.994038i \(0.465224\pi\)
\(62\) 3.79772e13 1.36957
\(63\) 0 0
\(64\) 4.39805e12 0.125000
\(65\) −1.19822e13 −0.303171
\(66\) 0 0
\(67\) −5.32339e13 −1.07307 −0.536534 0.843879i \(-0.680267\pi\)
−0.536534 + 0.843879i \(0.680267\pi\)
\(68\) −1.63461e13 −0.294848
\(69\) 0 0
\(70\) 4.45945e13 0.647210
\(71\) 2.02297e13 0.263968 0.131984 0.991252i \(-0.457865\pi\)
0.131984 + 0.991252i \(0.457865\pi\)
\(72\) 0 0
\(73\) 2.62642e13 0.278254 0.139127 0.990275i \(-0.455570\pi\)
0.139127 + 0.990275i \(0.455570\pi\)
\(74\) 2.28272e13 0.218381
\(75\) 0 0
\(76\) 8.08389e13 0.633167
\(77\) −3.13927e14 −2.22920
\(78\) 0 0
\(79\) −3.39031e14 −1.98626 −0.993131 0.117005i \(-0.962671\pi\)
−0.993131 + 0.117005i \(0.962671\pi\)
\(80\) 3.08191e13 0.164303
\(81\) 0 0
\(82\) −2.29233e14 −1.01548
\(83\) −1.31685e14 −0.532660 −0.266330 0.963882i \(-0.585811\pi\)
−0.266330 + 0.963882i \(0.585811\pi\)
\(84\) 0 0
\(85\) −1.14545e14 −0.387555
\(86\) 3.66523e14 1.13596
\(87\) 0 0
\(88\) −2.16954e14 −0.565910
\(89\) 3.93521e13 0.0943069 0.0471534 0.998888i \(-0.484985\pi\)
0.0471534 + 0.998888i \(0.484985\pi\)
\(90\) 0 0
\(91\) 3.16701e14 0.642450
\(92\) −1.36395e14 −0.254913
\(93\) 0 0
\(94\) 5.54612e14 0.882126
\(95\) 5.66474e14 0.832249
\(96\) 0 0
\(97\) 1.12875e15 1.41844 0.709219 0.704989i \(-0.249048\pi\)
0.709219 + 0.704989i \(0.249048\pi\)
\(98\) −5.70982e14 −0.664396
\(99\) 0 0
\(100\) −2.84037e14 −0.284037
\(101\) −3.79528e13 −0.0352236 −0.0176118 0.999845i \(-0.505606\pi\)
−0.0176118 + 0.999845i \(0.505606\pi\)
\(102\) 0 0
\(103\) −2.07297e14 −0.166079 −0.0830393 0.996546i \(-0.526463\pi\)
−0.0830393 + 0.996546i \(0.526463\pi\)
\(104\) 2.18871e14 0.163094
\(105\) 0 0
\(106\) 1.24574e15 0.804697
\(107\) 1.99692e15 1.20221 0.601107 0.799169i \(-0.294727\pi\)
0.601107 + 0.799169i \(0.294727\pi\)
\(108\) 0 0
\(109\) 1.35603e13 0.00710510 0.00355255 0.999994i \(-0.498869\pi\)
0.00355255 + 0.999994i \(0.498869\pi\)
\(110\) −1.52029e15 −0.743845
\(111\) 0 0
\(112\) −8.14575e14 −0.348174
\(113\) 7.08794e14 0.283421 0.141710 0.989908i \(-0.454740\pi\)
0.141710 + 0.989908i \(0.454740\pi\)
\(114\) 0 0
\(115\) −9.55784e14 −0.335063
\(116\) −1.70604e15 −0.560472
\(117\) 0 0
\(118\) −1.72990e15 −0.499926
\(119\) 3.02752e15 0.821267
\(120\) 0 0
\(121\) 6.52501e15 1.56203
\(122\) −6.85141e14 −0.154199
\(123\) 0 0
\(124\) −4.86108e15 −0.968434
\(125\) −5.49410e15 −1.03056
\(126\) 0 0
\(127\) −1.23021e15 −0.204858 −0.102429 0.994740i \(-0.532661\pi\)
−0.102429 + 0.994740i \(0.532661\pi\)
\(128\) −5.62950e14 −0.0883883
\(129\) 0 0
\(130\) 1.53373e15 0.214374
\(131\) −7.94836e15 −1.04892 −0.524460 0.851435i \(-0.675733\pi\)
−0.524460 + 0.851435i \(0.675733\pi\)
\(132\) 0 0
\(133\) −1.49724e16 −1.76362
\(134\) 6.81394e15 0.758774
\(135\) 0 0
\(136\) 2.09231e15 0.208489
\(137\) 1.78131e16 1.68010 0.840050 0.542508i \(-0.182525\pi\)
0.840050 + 0.542508i \(0.182525\pi\)
\(138\) 0 0
\(139\) 3.89941e15 0.329904 0.164952 0.986302i \(-0.447253\pi\)
0.164952 + 0.986302i \(0.447253\pi\)
\(140\) −5.70809e15 −0.457647
\(141\) 0 0
\(142\) −2.58940e15 −0.186653
\(143\) −1.07968e16 −0.738373
\(144\) 0 0
\(145\) −1.19550e16 −0.736696
\(146\) −3.36181e15 −0.196756
\(147\) 0 0
\(148\) −2.92188e15 −0.154419
\(149\) 3.48726e15 0.175222 0.0876108 0.996155i \(-0.472077\pi\)
0.0876108 + 0.996155i \(0.472077\pi\)
\(150\) 0 0
\(151\) 6.85712e15 0.311756 0.155878 0.987776i \(-0.450179\pi\)
0.155878 + 0.987776i \(0.450179\pi\)
\(152\) −1.03474e16 −0.447717
\(153\) 0 0
\(154\) 4.01827e16 1.57628
\(155\) −3.40637e16 −1.27293
\(156\) 0 0
\(157\) 3.69836e16 1.25534 0.627670 0.778480i \(-0.284009\pi\)
0.627670 + 0.778480i \(0.284009\pi\)
\(158\) 4.33960e16 1.40450
\(159\) 0 0
\(160\) −3.94484e15 −0.116180
\(161\) 2.52622e16 0.710031
\(162\) 0 0
\(163\) 7.42535e15 0.190244 0.0951218 0.995466i \(-0.469676\pi\)
0.0951218 + 0.995466i \(0.469676\pi\)
\(164\) 2.93418e16 0.718056
\(165\) 0 0
\(166\) 1.68557e16 0.376647
\(167\) 1.47365e16 0.314789 0.157395 0.987536i \(-0.449691\pi\)
0.157395 + 0.987536i \(0.449691\pi\)
\(168\) 0 0
\(169\) −4.02937e16 −0.787203
\(170\) 1.46617e16 0.274043
\(171\) 0 0
\(172\) −4.69149e16 −0.803244
\(173\) −3.40039e16 −0.557421 −0.278710 0.960375i \(-0.589907\pi\)
−0.278710 + 0.960375i \(0.589907\pi\)
\(174\) 0 0
\(175\) 5.26073e16 0.791153
\(176\) 2.77701e16 0.400159
\(177\) 0 0
\(178\) −5.03707e15 −0.0666850
\(179\) −3.81276e16 −0.483996 −0.241998 0.970277i \(-0.577803\pi\)
−0.241998 + 0.970277i \(0.577803\pi\)
\(180\) 0 0
\(181\) −5.14124e16 −0.600452 −0.300226 0.953868i \(-0.597062\pi\)
−0.300226 + 0.953868i \(0.597062\pi\)
\(182\) −4.05377e16 −0.454280
\(183\) 0 0
\(184\) 1.74586e16 0.180250
\(185\) −2.04749e16 −0.202971
\(186\) 0 0
\(187\) −1.03213e17 −0.943889
\(188\) −7.09904e16 −0.623757
\(189\) 0 0
\(190\) −7.25087e16 −0.588489
\(191\) −6.75568e16 −0.527131 −0.263566 0.964641i \(-0.584899\pi\)
−0.263566 + 0.964641i \(0.584899\pi\)
\(192\) 0 0
\(193\) −2.07235e16 −0.149549 −0.0747746 0.997200i \(-0.523824\pi\)
−0.0747746 + 0.997200i \(0.523824\pi\)
\(194\) −1.44480e17 −1.00299
\(195\) 0 0
\(196\) 7.30857e16 0.469799
\(197\) −1.71244e16 −0.105954 −0.0529772 0.998596i \(-0.516871\pi\)
−0.0529772 + 0.998596i \(0.516871\pi\)
\(198\) 0 0
\(199\) −1.05743e17 −0.606533 −0.303267 0.952906i \(-0.598077\pi\)
−0.303267 + 0.952906i \(0.598077\pi\)
\(200\) 3.63567e16 0.200844
\(201\) 0 0
\(202\) 4.85796e15 0.0249069
\(203\) 3.15980e17 1.56113
\(204\) 0 0
\(205\) 2.05611e17 0.943828
\(206\) 2.65340e16 0.117435
\(207\) 0 0
\(208\) −2.80155e16 −0.115325
\(209\) 5.10432e17 2.02694
\(210\) 0 0
\(211\) −3.13093e17 −1.15759 −0.578795 0.815473i \(-0.696477\pi\)
−0.578795 + 0.815473i \(0.696477\pi\)
\(212\) −1.59454e17 −0.569007
\(213\) 0 0
\(214\) −2.55605e17 −0.850093
\(215\) −3.28754e17 −1.05580
\(216\) 0 0
\(217\) 9.00334e17 2.69747
\(218\) −1.73572e15 −0.00502406
\(219\) 0 0
\(220\) 1.94598e17 0.525978
\(221\) 1.04125e17 0.272027
\(222\) 0 0
\(223\) 3.35017e17 0.818052 0.409026 0.912523i \(-0.365869\pi\)
0.409026 + 0.912523i \(0.365869\pi\)
\(224\) 1.04266e17 0.246196
\(225\) 0 0
\(226\) −9.07256e16 −0.200409
\(227\) −4.67560e17 −0.999180 −0.499590 0.866262i \(-0.666516\pi\)
−0.499590 + 0.866262i \(0.666516\pi\)
\(228\) 0 0
\(229\) 3.61186e17 0.722711 0.361355 0.932428i \(-0.382314\pi\)
0.361355 + 0.932428i \(0.382314\pi\)
\(230\) 1.22340e17 0.236925
\(231\) 0 0
\(232\) 2.18373e17 0.396313
\(233\) −3.61787e17 −0.635747 −0.317873 0.948133i \(-0.602969\pi\)
−0.317873 + 0.948133i \(0.602969\pi\)
\(234\) 0 0
\(235\) −4.97461e17 −0.819880
\(236\) 2.21427e17 0.353501
\(237\) 0 0
\(238\) −3.87522e17 −0.580724
\(239\) 1.06842e18 1.55152 0.775762 0.631026i \(-0.217366\pi\)
0.775762 + 0.631026i \(0.217366\pi\)
\(240\) 0 0
\(241\) −1.00588e18 −1.37220 −0.686102 0.727505i \(-0.740680\pi\)
−0.686102 + 0.727505i \(0.740680\pi\)
\(242\) −8.35201e17 −1.10453
\(243\) 0 0
\(244\) 8.76980e16 0.109035
\(245\) 5.12144e17 0.617514
\(246\) 0 0
\(247\) −5.14943e17 −0.584160
\(248\) 6.22218e17 0.684786
\(249\) 0 0
\(250\) 7.03244e17 0.728713
\(251\) −1.74766e18 −1.75754 −0.878768 0.477249i \(-0.841634\pi\)
−0.878768 + 0.477249i \(0.841634\pi\)
\(252\) 0 0
\(253\) −8.61227e17 −0.816044
\(254\) 1.57467e17 0.144856
\(255\) 0 0
\(256\) 7.20576e16 0.0625000
\(257\) 2.77264e17 0.233558 0.116779 0.993158i \(-0.462743\pi\)
0.116779 + 0.993158i \(0.462743\pi\)
\(258\) 0 0
\(259\) 5.41170e17 0.430116
\(260\) −1.96317e17 −0.151586
\(261\) 0 0
\(262\) 1.01739e18 0.741699
\(263\) −1.73303e18 −1.22783 −0.613914 0.789373i \(-0.710406\pi\)
−0.613914 + 0.789373i \(0.710406\pi\)
\(264\) 0 0
\(265\) −1.11737e18 −0.747914
\(266\) 1.91647e18 1.24707
\(267\) 0 0
\(268\) −8.72184e17 −0.536534
\(269\) 5.11271e16 0.0305850 0.0152925 0.999883i \(-0.495132\pi\)
0.0152925 + 0.999883i \(0.495132\pi\)
\(270\) 0 0
\(271\) 2.08455e17 0.117962 0.0589812 0.998259i \(-0.481215\pi\)
0.0589812 + 0.998259i \(0.481215\pi\)
\(272\) −2.67815e17 −0.147424
\(273\) 0 0
\(274\) −2.28008e18 −1.18801
\(275\) −1.79346e18 −0.909279
\(276\) 0 0
\(277\) 3.29723e18 1.58326 0.791628 0.611003i \(-0.209234\pi\)
0.791628 + 0.611003i \(0.209234\pi\)
\(278\) −4.99124e17 −0.233277
\(279\) 0 0
\(280\) 7.30636e17 0.323605
\(281\) 3.98328e18 1.71768 0.858841 0.512242i \(-0.171185\pi\)
0.858841 + 0.512242i \(0.171185\pi\)
\(282\) 0 0
\(283\) −2.19051e18 −0.895668 −0.447834 0.894117i \(-0.647804\pi\)
−0.447834 + 0.894117i \(0.647804\pi\)
\(284\) 3.31443e17 0.131984
\(285\) 0 0
\(286\) 1.38199e18 0.522108
\(287\) −5.43448e18 −2.00006
\(288\) 0 0
\(289\) −1.86704e18 −0.652258
\(290\) 1.53024e18 0.520923
\(291\) 0 0
\(292\) 4.30312e17 0.139127
\(293\) 5.27183e18 1.66132 0.830662 0.556778i \(-0.187962\pi\)
0.830662 + 0.556778i \(0.187962\pi\)
\(294\) 0 0
\(295\) 1.55164e18 0.464649
\(296\) 3.74001e17 0.109190
\(297\) 0 0
\(298\) −4.46369e17 −0.123900
\(299\) 8.68837e17 0.235182
\(300\) 0 0
\(301\) 8.68925e18 2.23735
\(302\) −8.77712e17 −0.220445
\(303\) 0 0
\(304\) 1.32446e18 0.316584
\(305\) 6.14539e17 0.143318
\(306\) 0 0
\(307\) 4.77067e18 1.05935 0.529677 0.848199i \(-0.322313\pi\)
0.529677 + 0.848199i \(0.322313\pi\)
\(308\) −5.14338e18 −1.11460
\(309\) 0 0
\(310\) 4.36016e18 0.900098
\(311\) 6.46666e18 1.30310 0.651549 0.758607i \(-0.274120\pi\)
0.651549 + 0.758607i \(0.274120\pi\)
\(312\) 0 0
\(313\) −6.00129e18 −1.15256 −0.576278 0.817254i \(-0.695496\pi\)
−0.576278 + 0.817254i \(0.695496\pi\)
\(314\) −4.73389e18 −0.887659
\(315\) 0 0
\(316\) −5.55469e18 −0.993131
\(317\) 9.53943e17 0.166563 0.0832814 0.996526i \(-0.473460\pi\)
0.0832814 + 0.996526i \(0.473460\pi\)
\(318\) 0 0
\(319\) −1.07722e19 −1.79422
\(320\) 5.04940e17 0.0821513
\(321\) 0 0
\(322\) −3.23356e18 −0.502068
\(323\) −4.92262e18 −0.746753
\(324\) 0 0
\(325\) 1.80931e18 0.262052
\(326\) −9.50445e17 −0.134523
\(327\) 0 0
\(328\) −3.75575e18 −0.507742
\(329\) 1.31483e19 1.73741
\(330\) 0 0
\(331\) −5.29071e18 −0.668042 −0.334021 0.942566i \(-0.608406\pi\)
−0.334021 + 0.942566i \(0.608406\pi\)
\(332\) −2.15752e18 −0.266330
\(333\) 0 0
\(334\) −1.88627e18 −0.222590
\(335\) −6.11179e18 −0.705232
\(336\) 0 0
\(337\) 6.22409e18 0.686834 0.343417 0.939183i \(-0.388416\pi\)
0.343417 + 0.939183i \(0.388416\pi\)
\(338\) 5.15759e18 0.556636
\(339\) 0 0
\(340\) −1.87670e18 −0.193777
\(341\) −3.06938e19 −3.10022
\(342\) 0 0
\(343\) 8.70190e17 0.0841217
\(344\) 6.00511e18 0.567979
\(345\) 0 0
\(346\) 4.35250e18 0.394156
\(347\) 3.22035e18 0.285386 0.142693 0.989767i \(-0.454424\pi\)
0.142693 + 0.989767i \(0.454424\pi\)
\(348\) 0 0
\(349\) −6.17407e18 −0.524060 −0.262030 0.965060i \(-0.584392\pi\)
−0.262030 + 0.965060i \(0.584392\pi\)
\(350\) −6.73374e18 −0.559430
\(351\) 0 0
\(352\) −3.55457e18 −0.282955
\(353\) 6.14267e18 0.478681 0.239341 0.970936i \(-0.423069\pi\)
0.239341 + 0.970936i \(0.423069\pi\)
\(354\) 0 0
\(355\) 2.32257e18 0.173483
\(356\) 6.44746e17 0.0471534
\(357\) 0 0
\(358\) 4.88033e18 0.342237
\(359\) 5.81103e18 0.399066 0.199533 0.979891i \(-0.436057\pi\)
0.199533 + 0.979891i \(0.436057\pi\)
\(360\) 0 0
\(361\) 9.16338e18 0.603603
\(362\) 6.58079e18 0.424584
\(363\) 0 0
\(364\) 5.18883e18 0.321225
\(365\) 3.01539e18 0.182872
\(366\) 0 0
\(367\) −2.77147e17 −0.0161330 −0.00806650 0.999967i \(-0.502568\pi\)
−0.00806650 + 0.999967i \(0.502568\pi\)
\(368\) −2.23470e18 −0.127456
\(369\) 0 0
\(370\) 2.62079e18 0.143522
\(371\) 2.95330e19 1.58490
\(372\) 0 0
\(373\) 2.30498e19 1.18809 0.594047 0.804430i \(-0.297529\pi\)
0.594047 + 0.804430i \(0.297529\pi\)
\(374\) 1.32112e19 0.667431
\(375\) 0 0
\(376\) 9.08677e18 0.441063
\(377\) 1.08674e19 0.517091
\(378\) 0 0
\(379\) −1.34397e19 −0.614604 −0.307302 0.951612i \(-0.599426\pi\)
−0.307302 + 0.951612i \(0.599426\pi\)
\(380\) 9.28112e18 0.416124
\(381\) 0 0
\(382\) 8.64727e18 0.372738
\(383\) −2.64151e19 −1.11651 −0.558254 0.829670i \(-0.688529\pi\)
−0.558254 + 0.829670i \(0.688529\pi\)
\(384\) 0 0
\(385\) −3.60420e19 −1.46505
\(386\) 2.65261e18 0.105747
\(387\) 0 0
\(388\) 1.84935e19 0.709219
\(389\) −3.95275e19 −1.48688 −0.743442 0.668800i \(-0.766808\pi\)
−0.743442 + 0.668800i \(0.766808\pi\)
\(390\) 0 0
\(391\) 8.30569e18 0.300642
\(392\) −9.35497e18 −0.332198
\(393\) 0 0
\(394\) 2.19192e18 0.0749210
\(395\) −3.89242e19 −1.30539
\(396\) 0 0
\(397\) 2.82890e18 0.0913457 0.0456729 0.998956i \(-0.485457\pi\)
0.0456729 + 0.998956i \(0.485457\pi\)
\(398\) 1.35351e19 0.428884
\(399\) 0 0
\(400\) −4.65366e18 −0.142018
\(401\) 4.63681e19 1.38879 0.694395 0.719594i \(-0.255672\pi\)
0.694395 + 0.719594i \(0.255672\pi\)
\(402\) 0 0
\(403\) 3.09650e19 0.893477
\(404\) −6.21819e17 −0.0176118
\(405\) 0 0
\(406\) −4.04454e19 −1.10389
\(407\) −1.84493e19 −0.494336
\(408\) 0 0
\(409\) −1.17092e19 −0.302415 −0.151207 0.988502i \(-0.548316\pi\)
−0.151207 + 0.988502i \(0.548316\pi\)
\(410\) −2.63182e19 −0.667387
\(411\) 0 0
\(412\) −3.39635e18 −0.0830393
\(413\) −4.10112e19 −0.984638
\(414\) 0 0
\(415\) −1.51187e19 −0.350070
\(416\) 3.58598e18 0.0815470
\(417\) 0 0
\(418\) −6.53353e19 −1.43326
\(419\) 4.20102e19 0.905210 0.452605 0.891711i \(-0.350495\pi\)
0.452605 + 0.891711i \(0.350495\pi\)
\(420\) 0 0
\(421\) −1.59718e19 −0.332077 −0.166039 0.986119i \(-0.553098\pi\)
−0.166039 + 0.986119i \(0.553098\pi\)
\(422\) 4.00759e19 0.818540
\(423\) 0 0
\(424\) 2.04101e19 0.402348
\(425\) 1.72962e19 0.334991
\(426\) 0 0
\(427\) −1.62428e19 −0.303705
\(428\) 3.27175e19 0.601107
\(429\) 0 0
\(430\) 4.20805e19 0.746564
\(431\) 9.76365e19 1.70229 0.851143 0.524934i \(-0.175910\pi\)
0.851143 + 0.524934i \(0.175910\pi\)
\(432\) 0 0
\(433\) −6.91455e19 −1.16441 −0.582204 0.813043i \(-0.697809\pi\)
−0.582204 + 0.813043i \(0.697809\pi\)
\(434\) −1.15243e20 −1.90740
\(435\) 0 0
\(436\) 2.22172e17 0.00355255
\(437\) −4.10753e19 −0.645609
\(438\) 0 0
\(439\) 5.48990e19 0.833836 0.416918 0.908944i \(-0.363110\pi\)
0.416918 + 0.908944i \(0.363110\pi\)
\(440\) −2.49085e19 −0.371922
\(441\) 0 0
\(442\) −1.33280e19 −0.192352
\(443\) −3.80504e19 −0.539923 −0.269961 0.962871i \(-0.587011\pi\)
−0.269961 + 0.962871i \(0.587011\pi\)
\(444\) 0 0
\(445\) 4.51802e18 0.0619795
\(446\) −4.28822e19 −0.578450
\(447\) 0 0
\(448\) −1.33460e19 −0.174087
\(449\) −1.11994e18 −0.0143663 −0.00718317 0.999974i \(-0.502286\pi\)
−0.00718317 + 0.999974i \(0.502286\pi\)
\(450\) 0 0
\(451\) 1.85270e20 2.29869
\(452\) 1.16129e19 0.141710
\(453\) 0 0
\(454\) 5.98477e19 0.706527
\(455\) 3.63604e19 0.422225
\(456\) 0 0
\(457\) −9.70734e19 −1.09076 −0.545379 0.838189i \(-0.683614\pi\)
−0.545379 + 0.838189i \(0.683614\pi\)
\(458\) −4.62318e19 −0.511034
\(459\) 0 0
\(460\) −1.56596e19 −0.167531
\(461\) 9.78165e19 1.02957 0.514784 0.857320i \(-0.327872\pi\)
0.514784 + 0.857320i \(0.327872\pi\)
\(462\) 0 0
\(463\) −2.17159e19 −0.221269 −0.110634 0.993861i \(-0.535288\pi\)
−0.110634 + 0.993861i \(0.535288\pi\)
\(464\) −2.79517e19 −0.280236
\(465\) 0 0
\(466\) 4.63088e19 0.449541
\(467\) 1.98443e20 1.89565 0.947826 0.318788i \(-0.103276\pi\)
0.947826 + 0.318788i \(0.103276\pi\)
\(468\) 0 0
\(469\) 1.61540e20 1.49446
\(470\) 6.36750e19 0.579743
\(471\) 0 0
\(472\) −2.83427e19 −0.249963
\(473\) −2.96230e20 −2.57140
\(474\) 0 0
\(475\) −8.55373e19 −0.719372
\(476\) 4.96028e19 0.410634
\(477\) 0 0
\(478\) −1.36758e20 −1.09709
\(479\) 3.78661e19 0.299043 0.149522 0.988758i \(-0.452227\pi\)
0.149522 + 0.988758i \(0.452227\pi\)
\(480\) 0 0
\(481\) 1.86123e19 0.142467
\(482\) 1.28753e20 0.970295
\(483\) 0 0
\(484\) 1.06906e20 0.781017
\(485\) 1.29592e20 0.932212
\(486\) 0 0
\(487\) 3.89265e19 0.271505 0.135752 0.990743i \(-0.456655\pi\)
0.135752 + 0.990743i \(0.456655\pi\)
\(488\) −1.12253e19 −0.0770994
\(489\) 0 0
\(490\) −6.55545e19 −0.436648
\(491\) −1.40516e20 −0.921753 −0.460876 0.887464i \(-0.652465\pi\)
−0.460876 + 0.887464i \(0.652465\pi\)
\(492\) 0 0
\(493\) 1.03888e20 0.661016
\(494\) 6.59127e19 0.413063
\(495\) 0 0
\(496\) −7.96439e19 −0.484217
\(497\) −6.13875e19 −0.367627
\(498\) 0 0
\(499\) −1.53703e20 −0.893157 −0.446578 0.894745i \(-0.647358\pi\)
−0.446578 + 0.894745i \(0.647358\pi\)
\(500\) −9.00153e19 −0.515278
\(501\) 0 0
\(502\) 2.23701e20 1.24277
\(503\) −5.62265e18 −0.0307738 −0.0153869 0.999882i \(-0.504898\pi\)
−0.0153869 + 0.999882i \(0.504898\pi\)
\(504\) 0 0
\(505\) −4.35737e18 −0.0231493
\(506\) 1.10237e20 0.577031
\(507\) 0 0
\(508\) −2.01558e19 −0.102429
\(509\) −3.29307e20 −1.64899 −0.824495 0.565869i \(-0.808541\pi\)
−0.824495 + 0.565869i \(0.808541\pi\)
\(510\) 0 0
\(511\) −7.96993e19 −0.387523
\(512\) −9.22337e18 −0.0441942
\(513\) 0 0
\(514\) −3.54898e19 −0.165151
\(515\) −2.37998e19 −0.109149
\(516\) 0 0
\(517\) −4.48247e20 −1.99682
\(518\) −6.92698e19 −0.304138
\(519\) 0 0
\(520\) 2.51286e19 0.107187
\(521\) −1.71775e20 −0.722231 −0.361116 0.932521i \(-0.617604\pi\)
−0.361116 + 0.932521i \(0.617604\pi\)
\(522\) 0 0
\(523\) −1.59414e20 −0.651275 −0.325637 0.945495i \(-0.605579\pi\)
−0.325637 + 0.945495i \(0.605579\pi\)
\(524\) −1.30226e20 −0.524460
\(525\) 0 0
\(526\) 2.21828e20 0.868205
\(527\) 2.96011e20 1.14216
\(528\) 0 0
\(529\) −1.97331e20 −0.740078
\(530\) 1.43023e20 0.528855
\(531\) 0 0
\(532\) −2.45308e20 −0.881809
\(533\) −1.86907e20 −0.662478
\(534\) 0 0
\(535\) 2.29266e20 0.790108
\(536\) 1.11640e20 0.379387
\(537\) 0 0
\(538\) −6.54427e18 −0.0216269
\(539\) 4.61477e20 1.50395
\(540\) 0 0
\(541\) 4.74121e19 0.150283 0.0751415 0.997173i \(-0.476059\pi\)
0.0751415 + 0.997173i \(0.476059\pi\)
\(542\) −2.66823e19 −0.0834120
\(543\) 0 0
\(544\) 3.42804e19 0.104245
\(545\) 1.55686e18 0.00466955
\(546\) 0 0
\(547\) −2.80986e20 −0.819934 −0.409967 0.912100i \(-0.634460\pi\)
−0.409967 + 0.912100i \(0.634460\pi\)
\(548\) 2.91850e20 0.840050
\(549\) 0 0
\(550\) 2.29563e20 0.642958
\(551\) −5.13770e20 −1.41949
\(552\) 0 0
\(553\) 1.02880e21 2.76626
\(554\) −4.22046e20 −1.11953
\(555\) 0 0
\(556\) 6.38879e19 0.164952
\(557\) 3.49642e20 0.890654 0.445327 0.895368i \(-0.353087\pi\)
0.445327 + 0.895368i \(0.353087\pi\)
\(558\) 0 0
\(559\) 2.98847e20 0.741073
\(560\) −9.35213e19 −0.228823
\(561\) 0 0
\(562\) −5.09859e20 −1.21459
\(563\) −2.68215e20 −0.630479 −0.315240 0.949012i \(-0.602085\pi\)
−0.315240 + 0.949012i \(0.602085\pi\)
\(564\) 0 0
\(565\) 8.13766e19 0.186267
\(566\) 2.80385e20 0.633333
\(567\) 0 0
\(568\) −4.24247e19 −0.0933267
\(569\) 7.95691e20 1.72744 0.863719 0.503974i \(-0.168129\pi\)
0.863719 + 0.503974i \(0.168129\pi\)
\(570\) 0 0
\(571\) 8.84470e20 1.87030 0.935152 0.354246i \(-0.115262\pi\)
0.935152 + 0.354246i \(0.115262\pi\)
\(572\) −1.76895e20 −0.369186
\(573\) 0 0
\(574\) 6.95614e20 1.41426
\(575\) 1.44323e20 0.289618
\(576\) 0 0
\(577\) 3.15722e20 0.617286 0.308643 0.951178i \(-0.400125\pi\)
0.308643 + 0.951178i \(0.400125\pi\)
\(578\) 2.38981e20 0.461216
\(579\) 0 0
\(580\) −1.95870e20 −0.368348
\(581\) 3.99601e20 0.741832
\(582\) 0 0
\(583\) −1.00682e21 −1.82154
\(584\) −5.50799e19 −0.0983778
\(585\) 0 0
\(586\) −6.74794e20 −1.17473
\(587\) −7.33397e20 −1.26053 −0.630265 0.776380i \(-0.717054\pi\)
−0.630265 + 0.776380i \(0.717054\pi\)
\(588\) 0 0
\(589\) −1.46391e21 −2.45272
\(590\) −1.98610e20 −0.328557
\(591\) 0 0
\(592\) −4.78721e19 −0.0772093
\(593\) 3.25571e20 0.518485 0.259242 0.965812i \(-0.416527\pi\)
0.259242 + 0.965812i \(0.416527\pi\)
\(594\) 0 0
\(595\) 3.47589e20 0.539746
\(596\) 5.71353e19 0.0876108
\(597\) 0 0
\(598\) −1.11211e20 −0.166299
\(599\) −6.61173e20 −0.976368 −0.488184 0.872741i \(-0.662341\pi\)
−0.488184 + 0.872741i \(0.662341\pi\)
\(600\) 0 0
\(601\) −2.49467e20 −0.359297 −0.179649 0.983731i \(-0.557496\pi\)
−0.179649 + 0.983731i \(0.557496\pi\)
\(602\) −1.11222e21 −1.58204
\(603\) 0 0
\(604\) 1.12347e20 0.155878
\(605\) 7.49136e20 1.02659
\(606\) 0 0
\(607\) 1.08917e21 1.45606 0.728031 0.685544i \(-0.240436\pi\)
0.728031 + 0.685544i \(0.240436\pi\)
\(608\) −1.69531e20 −0.223858
\(609\) 0 0
\(610\) −7.86610e19 −0.101341
\(611\) 4.52208e20 0.575478
\(612\) 0 0
\(613\) 1.99450e20 0.247674 0.123837 0.992303i \(-0.460480\pi\)
0.123837 + 0.992303i \(0.460480\pi\)
\(614\) −6.10645e20 −0.749076
\(615\) 0 0
\(616\) 6.58353e20 0.788140
\(617\) −4.21127e20 −0.498052 −0.249026 0.968497i \(-0.580110\pi\)
−0.249026 + 0.968497i \(0.580110\pi\)
\(618\) 0 0
\(619\) −1.27272e21 −1.46910 −0.734552 0.678552i \(-0.762608\pi\)
−0.734552 + 0.678552i \(0.762608\pi\)
\(620\) −5.58100e20 −0.636465
\(621\) 0 0
\(622\) −8.27732e20 −0.921429
\(623\) −1.19415e20 −0.131341
\(624\) 0 0
\(625\) −1.01717e20 −0.109218
\(626\) 7.68165e20 0.814980
\(627\) 0 0
\(628\) 6.05938e20 0.627670
\(629\) 1.77925e20 0.182120
\(630\) 0 0
\(631\) −3.42884e20 −0.342711 −0.171355 0.985209i \(-0.554815\pi\)
−0.171355 + 0.985209i \(0.554815\pi\)
\(632\) 7.11000e20 0.702250
\(633\) 0 0
\(634\) −1.22105e20 −0.117778
\(635\) −1.41241e20 −0.134635
\(636\) 0 0
\(637\) −4.65555e20 −0.433436
\(638\) 1.37885e21 1.26871
\(639\) 0 0
\(640\) −6.46323e19 −0.0580898
\(641\) −1.00075e21 −0.888980 −0.444490 0.895784i \(-0.646615\pi\)
−0.444490 + 0.895784i \(0.646615\pi\)
\(642\) 0 0
\(643\) −1.35729e20 −0.117785 −0.0588927 0.998264i \(-0.518757\pi\)
−0.0588927 + 0.998264i \(0.518757\pi\)
\(644\) 4.13896e20 0.355015
\(645\) 0 0
\(646\) 6.30095e20 0.528034
\(647\) −2.15462e21 −1.78479 −0.892397 0.451251i \(-0.850978\pi\)
−0.892397 + 0.451251i \(0.850978\pi\)
\(648\) 0 0
\(649\) 1.39813e21 1.13165
\(650\) −2.31592e20 −0.185299
\(651\) 0 0
\(652\) 1.21657e20 0.0951218
\(653\) 2.06101e21 1.59306 0.796529 0.604601i \(-0.206667\pi\)
0.796529 + 0.604601i \(0.206667\pi\)
\(654\) 0 0
\(655\) −9.12551e20 −0.689362
\(656\) 4.80736e20 0.359028
\(657\) 0 0
\(658\) −1.68299e21 −1.22853
\(659\) 1.61309e21 1.16418 0.582089 0.813125i \(-0.302236\pi\)
0.582089 + 0.813125i \(0.302236\pi\)
\(660\) 0 0
\(661\) 1.60888e21 1.13504 0.567521 0.823359i \(-0.307903\pi\)
0.567521 + 0.823359i \(0.307903\pi\)
\(662\) 6.77211e20 0.472377
\(663\) 0 0
\(664\) 2.76163e20 0.188324
\(665\) −1.71898e21 −1.15907
\(666\) 0 0
\(667\) 8.66859e20 0.571485
\(668\) 2.41442e20 0.157395
\(669\) 0 0
\(670\) 7.82309e20 0.498674
\(671\) 5.53742e20 0.349051
\(672\) 0 0
\(673\) 1.67294e21 1.03126 0.515628 0.856812i \(-0.327559\pi\)
0.515628 + 0.856812i \(0.327559\pi\)
\(674\) −7.96684e20 −0.485665
\(675\) 0 0
\(676\) −6.60171e20 −0.393601
\(677\) −7.27000e20 −0.428666 −0.214333 0.976761i \(-0.568758\pi\)
−0.214333 + 0.976761i \(0.568758\pi\)
\(678\) 0 0
\(679\) −3.42523e21 −1.97545
\(680\) 2.40218e20 0.137021
\(681\) 0 0
\(682\) 3.92880e21 2.19219
\(683\) −2.85909e21 −1.57787 −0.788936 0.614475i \(-0.789368\pi\)
−0.788936 + 0.614475i \(0.789368\pi\)
\(684\) 0 0
\(685\) 2.04512e21 1.10418
\(686\) −1.11384e20 −0.0594830
\(687\) 0 0
\(688\) −7.68654e20 −0.401622
\(689\) 1.01572e21 0.524965
\(690\) 0 0
\(691\) 1.71274e21 0.866178 0.433089 0.901351i \(-0.357424\pi\)
0.433089 + 0.901351i \(0.357424\pi\)
\(692\) −5.57120e20 −0.278710
\(693\) 0 0
\(694\) −4.12205e20 −0.201798
\(695\) 4.47691e20 0.216817
\(696\) 0 0
\(697\) −1.78675e21 −0.846870
\(698\) 7.90281e20 0.370566
\(699\) 0 0
\(700\) 8.61918e20 0.395577
\(701\) 8.95519e20 0.406621 0.203311 0.979114i \(-0.434830\pi\)
0.203311 + 0.979114i \(0.434830\pi\)
\(702\) 0 0
\(703\) −8.79920e20 −0.391091
\(704\) 4.54985e20 0.200079
\(705\) 0 0
\(706\) −7.86261e20 −0.338479
\(707\) 1.15169e20 0.0490557
\(708\) 0 0
\(709\) −3.99642e21 −1.66657 −0.833286 0.552842i \(-0.813543\pi\)
−0.833286 + 0.552842i \(0.813543\pi\)
\(710\) −2.97289e20 −0.122671
\(711\) 0 0
\(712\) −8.25274e19 −0.0333425
\(713\) 2.46998e21 0.987464
\(714\) 0 0
\(715\) −1.23958e21 −0.485267
\(716\) −6.24683e20 −0.241998
\(717\) 0 0
\(718\) −7.43812e20 −0.282183
\(719\) 2.66669e20 0.100117 0.0500583 0.998746i \(-0.484059\pi\)
0.0500583 + 0.998746i \(0.484059\pi\)
\(720\) 0 0
\(721\) 6.29049e20 0.231297
\(722\) −1.17291e21 −0.426812
\(723\) 0 0
\(724\) −8.42341e20 −0.300226
\(725\) 1.80519e21 0.636779
\(726\) 0 0
\(727\) −1.69447e21 −0.585499 −0.292750 0.956189i \(-0.594570\pi\)
−0.292750 + 0.956189i \(0.594570\pi\)
\(728\) −6.64170e20 −0.227140
\(729\) 0 0
\(730\) −3.85970e20 −0.129310
\(731\) 2.85684e21 0.947341
\(732\) 0 0
\(733\) 3.16239e21 1.02739 0.513695 0.857973i \(-0.328276\pi\)
0.513695 + 0.857973i \(0.328276\pi\)
\(734\) 3.54749e19 0.0114078
\(735\) 0 0
\(736\) 2.86042e20 0.0901252
\(737\) −5.50714e21 −1.71759
\(738\) 0 0
\(739\) 2.01381e21 0.615440 0.307720 0.951477i \(-0.400434\pi\)
0.307720 + 0.951477i \(0.400434\pi\)
\(740\) −3.35461e20 −0.101486
\(741\) 0 0
\(742\) −3.78022e21 −1.12070
\(743\) −2.11411e21 −0.620458 −0.310229 0.950662i \(-0.600406\pi\)
−0.310229 + 0.950662i \(0.600406\pi\)
\(744\) 0 0
\(745\) 4.00372e20 0.115157
\(746\) −2.95038e21 −0.840110
\(747\) 0 0
\(748\) −1.69104e21 −0.471945
\(749\) −6.05970e21 −1.67432
\(750\) 0 0
\(751\) 4.95087e21 1.34086 0.670428 0.741974i \(-0.266110\pi\)
0.670428 + 0.741974i \(0.266110\pi\)
\(752\) −1.16311e21 −0.311879
\(753\) 0 0
\(754\) −1.39103e21 −0.365638
\(755\) 7.87266e20 0.204889
\(756\) 0 0
\(757\) −4.85879e21 −1.23968 −0.619839 0.784729i \(-0.712802\pi\)
−0.619839 + 0.784729i \(0.712802\pi\)
\(758\) 1.72028e21 0.434590
\(759\) 0 0
\(760\) −1.18798e21 −0.294244
\(761\) 3.69092e21 0.905211 0.452605 0.891711i \(-0.350495\pi\)
0.452605 + 0.891711i \(0.350495\pi\)
\(762\) 0 0
\(763\) −4.11491e19 −0.00989523
\(764\) −1.10685e21 −0.263566
\(765\) 0 0
\(766\) 3.38114e21 0.789491
\(767\) −1.41049e21 −0.326140
\(768\) 0 0
\(769\) −2.33815e21 −0.530181 −0.265091 0.964224i \(-0.585402\pi\)
−0.265091 + 0.964224i \(0.585402\pi\)
\(770\) 4.61337e21 1.03595
\(771\) 0 0
\(772\) −3.39534e20 −0.0747746
\(773\) 5.20682e21 1.13560 0.567801 0.823166i \(-0.307794\pi\)
0.567801 + 0.823166i \(0.307794\pi\)
\(774\) 0 0
\(775\) 5.14361e21 1.10028
\(776\) −2.36716e21 −0.501493
\(777\) 0 0
\(778\) 5.05952e21 1.05139
\(779\) 8.83624e21 1.81860
\(780\) 0 0
\(781\) 2.09279e21 0.422516
\(782\) −1.06313e21 −0.212586
\(783\) 0 0
\(784\) 1.19744e21 0.234899
\(785\) 4.24608e21 0.825023
\(786\) 0 0
\(787\) −1.92196e21 −0.366381 −0.183191 0.983077i \(-0.558643\pi\)
−0.183191 + 0.983077i \(0.558643\pi\)
\(788\) −2.80566e20 −0.0529772
\(789\) 0 0
\(790\) 4.98230e21 0.923052
\(791\) −2.15085e21 −0.394718
\(792\) 0 0
\(793\) −5.58635e20 −0.100596
\(794\) −3.62099e20 −0.0645912
\(795\) 0 0
\(796\) −1.73250e21 −0.303267
\(797\) 6.23880e20 0.108184 0.0540921 0.998536i \(-0.482774\pi\)
0.0540921 + 0.998536i \(0.482774\pi\)
\(798\) 0 0
\(799\) 4.32290e21 0.735655
\(800\) 5.95669e20 0.100422
\(801\) 0 0
\(802\) −5.93511e21 −0.982022
\(803\) 2.71707e21 0.445384
\(804\) 0 0
\(805\) 2.90035e21 0.466640
\(806\) −3.96352e21 −0.631783
\(807\) 0 0
\(808\) 7.95929e19 0.0124534
\(809\) −5.72858e21 −0.888042 −0.444021 0.896016i \(-0.646448\pi\)
−0.444021 + 0.896016i \(0.646448\pi\)
\(810\) 0 0
\(811\) 4.80014e21 0.730462 0.365231 0.930917i \(-0.380990\pi\)
0.365231 + 0.930917i \(0.380990\pi\)
\(812\) 5.17702e21 0.780566
\(813\) 0 0
\(814\) 2.36151e21 0.349548
\(815\) 8.52505e20 0.125030
\(816\) 0 0
\(817\) −1.41284e22 −2.03435
\(818\) 1.49878e21 0.213840
\(819\) 0 0
\(820\) 3.36873e21 0.471914
\(821\) −8.89775e21 −1.23511 −0.617556 0.786527i \(-0.711877\pi\)
−0.617556 + 0.786527i \(0.711877\pi\)
\(822\) 0 0
\(823\) −1.39946e21 −0.190749 −0.0953744 0.995441i \(-0.530405\pi\)
−0.0953744 + 0.995441i \(0.530405\pi\)
\(824\) 4.34733e20 0.0587177
\(825\) 0 0
\(826\) 5.24943e21 0.696244
\(827\) −6.61343e21 −0.869232 −0.434616 0.900616i \(-0.643116\pi\)
−0.434616 + 0.900616i \(0.643116\pi\)
\(828\) 0 0
\(829\) 4.14735e21 0.535318 0.267659 0.963514i \(-0.413750\pi\)
0.267659 + 0.963514i \(0.413750\pi\)
\(830\) 1.93520e21 0.247537
\(831\) 0 0
\(832\) −4.59006e20 −0.0576625
\(833\) −4.45049e21 −0.554078
\(834\) 0 0
\(835\) 1.69190e21 0.206883
\(836\) 8.36292e21 1.01347
\(837\) 0 0
\(838\) −5.37730e21 −0.640080
\(839\) −9.57811e21 −1.12997 −0.564983 0.825103i \(-0.691117\pi\)
−0.564983 + 0.825103i \(0.691117\pi\)
\(840\) 0 0
\(841\) 2.21350e21 0.256513
\(842\) 2.04440e21 0.234814
\(843\) 0 0
\(844\) −5.12972e21 −0.578795
\(845\) −4.62612e21 −0.517358
\(846\) 0 0
\(847\) −1.98003e22 −2.17544
\(848\) −2.61250e21 −0.284503
\(849\) 0 0
\(850\) −2.21391e21 −0.236875
\(851\) 1.48465e21 0.157453
\(852\) 0 0
\(853\) −8.15361e21 −0.849634 −0.424817 0.905279i \(-0.639662\pi\)
−0.424817 + 0.905279i \(0.639662\pi\)
\(854\) 2.07908e21 0.214752
\(855\) 0 0
\(856\) −4.18784e21 −0.425047
\(857\) 1.53552e22 1.54490 0.772449 0.635076i \(-0.219031\pi\)
0.772449 + 0.635076i \(0.219031\pi\)
\(858\) 0 0
\(859\) −4.87617e21 −0.482092 −0.241046 0.970514i \(-0.577490\pi\)
−0.241046 + 0.970514i \(0.577490\pi\)
\(860\) −5.38630e21 −0.527901
\(861\) 0 0
\(862\) −1.24975e22 −1.20370
\(863\) 4.76667e21 0.455128 0.227564 0.973763i \(-0.426924\pi\)
0.227564 + 0.973763i \(0.426924\pi\)
\(864\) 0 0
\(865\) −3.90399e21 −0.366343
\(866\) 8.85063e21 0.823360
\(867\) 0 0
\(868\) 1.47511e22 1.34873
\(869\) −3.50734e22 −3.17928
\(870\) 0 0
\(871\) 5.55580e21 0.495006
\(872\) −2.84380e19 −0.00251203
\(873\) 0 0
\(874\) 5.25764e21 0.456515
\(875\) 1.66720e22 1.43525
\(876\) 0 0
\(877\) 1.57944e22 1.33661 0.668306 0.743887i \(-0.267020\pi\)
0.668306 + 0.743887i \(0.267020\pi\)
\(878\) −7.02707e21 −0.589611
\(879\) 0 0
\(880\) 3.18829e21 0.262989
\(881\) 1.12789e22 0.922461 0.461230 0.887280i \(-0.347408\pi\)
0.461230 + 0.887280i \(0.347408\pi\)
\(882\) 0 0
\(883\) 4.48307e21 0.360471 0.180236 0.983623i \(-0.442314\pi\)
0.180236 + 0.983623i \(0.442314\pi\)
\(884\) 1.70598e21 0.136013
\(885\) 0 0
\(886\) 4.87045e21 0.381783
\(887\) 8.56246e21 0.665536 0.332768 0.943009i \(-0.392017\pi\)
0.332768 + 0.943009i \(0.392017\pi\)
\(888\) 0 0
\(889\) 3.73312e21 0.285304
\(890\) −5.78307e20 −0.0438261
\(891\) 0 0
\(892\) 5.48893e21 0.409026
\(893\) −2.13786e22 −1.57977
\(894\) 0 0
\(895\) −4.37743e21 −0.318087
\(896\) 1.70829e21 0.123098
\(897\) 0 0
\(898\) 1.43352e20 0.0101585
\(899\) 3.08945e22 2.17112
\(900\) 0 0
\(901\) 9.70983e21 0.671082
\(902\) −2.37145e22 −1.62542
\(903\) 0 0
\(904\) −1.48645e21 −0.100204
\(905\) −5.90266e21 −0.394624
\(906\) 0 0
\(907\) 7.34638e21 0.483079 0.241540 0.970391i \(-0.422348\pi\)
0.241540 + 0.970391i \(0.422348\pi\)
\(908\) −7.66051e21 −0.499590
\(909\) 0 0
\(910\) −4.65414e21 −0.298558
\(911\) 1.18999e22 0.757106 0.378553 0.925580i \(-0.376422\pi\)
0.378553 + 0.925580i \(0.376422\pi\)
\(912\) 0 0
\(913\) −1.36230e22 −0.852594
\(914\) 1.24254e22 0.771283
\(915\) 0 0
\(916\) 5.91767e21 0.361355
\(917\) 2.41195e22 1.46083
\(918\) 0 0
\(919\) 1.92629e22 1.14777 0.573886 0.818935i \(-0.305435\pi\)
0.573886 + 0.818935i \(0.305435\pi\)
\(920\) 2.00442e21 0.118463
\(921\) 0 0
\(922\) −1.25205e22 −0.728015
\(923\) −2.11129e21 −0.121768
\(924\) 0 0
\(925\) 3.09170e21 0.175442
\(926\) 2.77964e21 0.156461
\(927\) 0 0
\(928\) 3.57782e21 0.198157
\(929\) −8.78109e21 −0.482426 −0.241213 0.970472i \(-0.577545\pi\)
−0.241213 + 0.970472i \(0.577545\pi\)
\(930\) 0 0
\(931\) 2.20096e22 1.18985
\(932\) −5.92753e21 −0.317873
\(933\) 0 0
\(934\) −2.54007e22 −1.34043
\(935\) −1.18499e22 −0.620334
\(936\) 0 0
\(937\) −2.35424e22 −1.21284 −0.606421 0.795144i \(-0.707395\pi\)
−0.606421 + 0.795144i \(0.707395\pi\)
\(938\) −2.06771e22 −1.05674
\(939\) 0 0
\(940\) −8.15040e21 −0.409940
\(941\) 2.31660e22 1.15592 0.577961 0.816064i \(-0.303848\pi\)
0.577961 + 0.816064i \(0.303848\pi\)
\(942\) 0 0
\(943\) −1.49090e22 −0.732166
\(944\) 3.62786e21 0.176751
\(945\) 0 0
\(946\) 3.79174e22 1.81826
\(947\) −2.02578e22 −0.963755 −0.481877 0.876239i \(-0.660045\pi\)
−0.481877 + 0.876239i \(0.660045\pi\)
\(948\) 0 0
\(949\) −2.74108e21 −0.128359
\(950\) 1.09488e22 0.508673
\(951\) 0 0
\(952\) −6.34916e21 −0.290362
\(953\) 1.66351e22 0.754794 0.377397 0.926052i \(-0.376819\pi\)
0.377397 + 0.926052i \(0.376819\pi\)
\(954\) 0 0
\(955\) −7.75620e21 −0.346436
\(956\) 1.75050e22 0.775762
\(957\) 0 0
\(958\) −4.84686e21 −0.211456
\(959\) −5.40544e22 −2.33987
\(960\) 0 0
\(961\) 6.45637e22 2.75146
\(962\) −2.38238e21 −0.100739
\(963\) 0 0
\(964\) −1.64804e22 −0.686102
\(965\) −2.37927e21 −0.0982853
\(966\) 0 0
\(967\) −2.10794e22 −0.857355 −0.428677 0.903458i \(-0.641020\pi\)
−0.428677 + 0.903458i \(0.641020\pi\)
\(968\) −1.36839e22 −0.552263
\(969\) 0 0
\(970\) −1.65878e22 −0.659174
\(971\) 3.09261e22 1.21950 0.609749 0.792594i \(-0.291270\pi\)
0.609749 + 0.792594i \(0.291270\pi\)
\(972\) 0 0
\(973\) −1.18329e22 −0.459456
\(974\) −4.98259e21 −0.191983
\(975\) 0 0
\(976\) 1.43684e21 0.0545175
\(977\) −7.85725e21 −0.295843 −0.147922 0.988999i \(-0.547258\pi\)
−0.147922 + 0.988999i \(0.547258\pi\)
\(978\) 0 0
\(979\) 4.07105e21 0.150951
\(980\) 8.39097e21 0.308757
\(981\) 0 0
\(982\) 1.79860e22 0.651778
\(983\) −1.27981e22 −0.460251 −0.230125 0.973161i \(-0.573914\pi\)
−0.230125 + 0.973161i \(0.573914\pi\)
\(984\) 0 0
\(985\) −1.96605e21 −0.0696343
\(986\) −1.32976e22 −0.467409
\(987\) 0 0
\(988\) −8.43682e21 −0.292080
\(989\) 2.38381e22 0.819028
\(990\) 0 0
\(991\) −5.39385e22 −1.82535 −0.912675 0.408686i \(-0.865987\pi\)
−0.912675 + 0.408686i \(0.865987\pi\)
\(992\) 1.01944e22 0.342393
\(993\) 0 0
\(994\) 7.85760e21 0.259951
\(995\) −1.21404e22 −0.398620
\(996\) 0 0
\(997\) −1.99678e22 −0.645828 −0.322914 0.946428i \(-0.604663\pi\)
−0.322914 + 0.946428i \(0.604663\pi\)
\(998\) 1.96740e22 0.631557
\(999\) 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 18.16.a.b.1.1 1
3.2 odd 2 6.16.a.b.1.1 1
4.3 odd 2 144.16.a.j.1.1 1
12.11 even 2 48.16.a.d.1.1 1
15.2 even 4 150.16.c.a.49.2 2
15.8 even 4 150.16.c.a.49.1 2
15.14 odd 2 150.16.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.16.a.b.1.1 1 3.2 odd 2
18.16.a.b.1.1 1 1.1 even 1 trivial
48.16.a.d.1.1 1 12.11 even 2
144.16.a.j.1.1 1 4.3 odd 2
150.16.a.f.1.1 1 15.14 odd 2
150.16.c.a.49.1 2 15.8 even 4
150.16.c.a.49.2 2 15.2 even 4