Properties

Label 1143.3.b.a.890.10
Level $1143$
Weight $3$
Character 1143.890
Analytic conductor $31.144$
Analytic rank $0$
Dimension $84$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1143 = 3^{2} \cdot 127 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1143.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(31.1444942164\)
Analytic rank: \(0\)
Dimension: \(84\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 890.10
Character \(\chi\) \(=\) 1143.890
Dual form 1143.3.b.a.890.75

$q$-expansion

\(f(q)\) \(=\) \(q-3.21080i q^{2} -6.30922 q^{4} -0.600942i q^{5} +12.4847 q^{7} +7.41443i q^{8} +O(q^{10})\) \(q-3.21080i q^{2} -6.30922 q^{4} -0.600942i q^{5} +12.4847 q^{7} +7.41443i q^{8} -1.92950 q^{10} +13.0330i q^{11} -16.5720 q^{13} -40.0858i q^{14} -1.43064 q^{16} -8.91048i q^{17} +36.7009 q^{19} +3.79147i q^{20} +41.8463 q^{22} +24.0665i q^{23} +24.6389 q^{25} +53.2092i q^{26} -78.7685 q^{28} -31.9489i q^{29} -22.1728 q^{31} +34.2512i q^{32} -28.6097 q^{34} -7.50256i q^{35} +51.0182 q^{37} -117.839i q^{38} +4.45564 q^{40} -20.9151i q^{41} +23.4145 q^{43} -82.2281i q^{44} +77.2725 q^{46} -36.6261i q^{47} +106.867 q^{49} -79.1104i q^{50} +104.556 q^{52} +6.11869i q^{53} +7.83207 q^{55} +92.5668i q^{56} -102.582 q^{58} -48.2618i q^{59} +103.264 q^{61} +71.1923i q^{62} +104.251 q^{64} +9.95879i q^{65} +39.6988 q^{67} +56.2182i q^{68} -24.0892 q^{70} +19.8363i q^{71} +13.6967 q^{73} -163.809i q^{74} -231.554 q^{76} +162.713i q^{77} -81.4768 q^{79} +0.859729i q^{80} -67.1542 q^{82} -64.4091i q^{83} -5.35468 q^{85} -75.1792i q^{86} -96.6323 q^{88} -106.593i q^{89} -206.896 q^{91} -151.841i q^{92} -117.599 q^{94} -22.0551i q^{95} +97.3467 q^{97} -343.129i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84q - 160q^{4} + O(q^{10}) \) \( 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1143\mathbb{Z}\right)^\times\).

\(n\) \(128\) \(892\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.21080i − 1.60540i −0.596384 0.802699i \(-0.703396\pi\)
0.596384 0.802699i \(-0.296604\pi\)
\(3\) 0 0
\(4\) −6.30922 −1.57730
\(5\) − 0.600942i − 0.120188i −0.998193 0.0600942i \(-0.980860\pi\)
0.998193 0.0600942i \(-0.0191401\pi\)
\(6\) 0 0
\(7\) 12.4847 1.78352 0.891762 0.452504i \(-0.149469\pi\)
0.891762 + 0.452504i \(0.149469\pi\)
\(8\) 7.41443i 0.926804i
\(9\) 0 0
\(10\) −1.92950 −0.192950
\(11\) 13.0330i 1.18482i 0.805637 + 0.592409i \(0.201823\pi\)
−0.805637 + 0.592409i \(0.798177\pi\)
\(12\) 0 0
\(13\) −16.5720 −1.27477 −0.637384 0.770547i \(-0.719983\pi\)
−0.637384 + 0.770547i \(0.719983\pi\)
\(14\) − 40.0858i − 2.86327i
\(15\) 0 0
\(16\) −1.43064 −0.0894147
\(17\) − 8.91048i − 0.524146i −0.965048 0.262073i \(-0.915594\pi\)
0.965048 0.262073i \(-0.0844061\pi\)
\(18\) 0 0
\(19\) 36.7009 1.93163 0.965813 0.259240i \(-0.0834722\pi\)
0.965813 + 0.259240i \(0.0834722\pi\)
\(20\) 3.79147i 0.189574i
\(21\) 0 0
\(22\) 41.8463 1.90211
\(23\) 24.0665i 1.04637i 0.852220 + 0.523184i \(0.175256\pi\)
−0.852220 + 0.523184i \(0.824744\pi\)
\(24\) 0 0
\(25\) 24.6389 0.985555
\(26\) 53.2092i 2.04651i
\(27\) 0 0
\(28\) −78.7685 −2.81316
\(29\) − 31.9489i − 1.10169i −0.834608 0.550844i \(-0.814306\pi\)
0.834608 0.550844i \(-0.185694\pi\)
\(30\) 0 0
\(31\) −22.1728 −0.715251 −0.357625 0.933865i \(-0.616414\pi\)
−0.357625 + 0.933865i \(0.616414\pi\)
\(32\) 34.2512i 1.07035i
\(33\) 0 0
\(34\) −28.6097 −0.841463
\(35\) − 7.50256i − 0.214359i
\(36\) 0 0
\(37\) 51.0182 1.37887 0.689436 0.724347i \(-0.257859\pi\)
0.689436 + 0.724347i \(0.257859\pi\)
\(38\) − 117.839i − 3.10103i
\(39\) 0 0
\(40\) 4.45564 0.111391
\(41\) − 20.9151i − 0.510125i −0.966925 0.255062i \(-0.917904\pi\)
0.966925 0.255062i \(-0.0820960\pi\)
\(42\) 0 0
\(43\) 23.4145 0.544523 0.272262 0.962223i \(-0.412228\pi\)
0.272262 + 0.962223i \(0.412228\pi\)
\(44\) − 82.2281i − 1.86882i
\(45\) 0 0
\(46\) 77.2725 1.67984
\(47\) − 36.6261i − 0.779278i −0.920968 0.389639i \(-0.872600\pi\)
0.920968 0.389639i \(-0.127400\pi\)
\(48\) 0 0
\(49\) 106.867 2.18096
\(50\) − 79.1104i − 1.58221i
\(51\) 0 0
\(52\) 104.556 2.01070
\(53\) 6.11869i 0.115447i 0.998333 + 0.0577235i \(0.0183842\pi\)
−0.998333 + 0.0577235i \(0.981616\pi\)
\(54\) 0 0
\(55\) 7.83207 0.142401
\(56\) 92.5668i 1.65298i
\(57\) 0 0
\(58\) −102.582 −1.76865
\(59\) − 48.2618i − 0.817996i −0.912535 0.408998i \(-0.865878\pi\)
0.912535 0.408998i \(-0.134122\pi\)
\(60\) 0 0
\(61\) 103.264 1.69285 0.846424 0.532509i \(-0.178751\pi\)
0.846424 + 0.532509i \(0.178751\pi\)
\(62\) 71.1923i 1.14826i
\(63\) 0 0
\(64\) 104.251 1.62892
\(65\) 9.95879i 0.153212i
\(66\) 0 0
\(67\) 39.6988 0.592520 0.296260 0.955107i \(-0.404261\pi\)
0.296260 + 0.955107i \(0.404261\pi\)
\(68\) 56.2182i 0.826738i
\(69\) 0 0
\(70\) −24.0892 −0.344131
\(71\) 19.8363i 0.279384i 0.990195 + 0.139692i \(0.0446113\pi\)
−0.990195 + 0.139692i \(0.955389\pi\)
\(72\) 0 0
\(73\) 13.6967 0.187625 0.0938127 0.995590i \(-0.470095\pi\)
0.0938127 + 0.995590i \(0.470095\pi\)
\(74\) − 163.809i − 2.21364i
\(75\) 0 0
\(76\) −231.554 −3.04676
\(77\) 162.713i 2.11315i
\(78\) 0 0
\(79\) −81.4768 −1.03135 −0.515676 0.856784i \(-0.672459\pi\)
−0.515676 + 0.856784i \(0.672459\pi\)
\(80\) 0.859729i 0.0107466i
\(81\) 0 0
\(82\) −67.1542 −0.818954
\(83\) − 64.4091i − 0.776013i −0.921657 0.388007i \(-0.873164\pi\)
0.921657 0.388007i \(-0.126836\pi\)
\(84\) 0 0
\(85\) −5.35468 −0.0629962
\(86\) − 75.1792i − 0.874177i
\(87\) 0 0
\(88\) −96.6323 −1.09809
\(89\) − 106.593i − 1.19768i −0.800870 0.598838i \(-0.795629\pi\)
0.800870 0.598838i \(-0.204371\pi\)
\(90\) 0 0
\(91\) −206.896 −2.27358
\(92\) − 151.841i − 1.65044i
\(93\) 0 0
\(94\) −117.599 −1.25105
\(95\) − 22.0551i − 0.232159i
\(96\) 0 0
\(97\) 97.3467 1.00357 0.501787 0.864991i \(-0.332676\pi\)
0.501787 + 0.864991i \(0.332676\pi\)
\(98\) − 343.129i − 3.50131i
\(99\) 0 0
\(100\) −155.452 −1.55452
\(101\) − 149.590i − 1.48109i −0.672009 0.740543i \(-0.734569\pi\)
0.672009 0.740543i \(-0.265431\pi\)
\(102\) 0 0
\(103\) −162.609 −1.57873 −0.789364 0.613925i \(-0.789590\pi\)
−0.789364 + 0.613925i \(0.789590\pi\)
\(104\) − 122.872i − 1.18146i
\(105\) 0 0
\(106\) 19.6459 0.185338
\(107\) 87.0737i 0.813773i 0.913479 + 0.406886i \(0.133386\pi\)
−0.913479 + 0.406886i \(0.866614\pi\)
\(108\) 0 0
\(109\) −80.7161 −0.740515 −0.370257 0.928929i \(-0.620731\pi\)
−0.370257 + 0.928929i \(0.620731\pi\)
\(110\) − 25.1472i − 0.228611i
\(111\) 0 0
\(112\) −17.8610 −0.159473
\(113\) 184.031i 1.62859i 0.580450 + 0.814296i \(0.302876\pi\)
−0.580450 + 0.814296i \(0.697124\pi\)
\(114\) 0 0
\(115\) 14.4625 0.125761
\(116\) 201.573i 1.73770i
\(117\) 0 0
\(118\) −154.959 −1.31321
\(119\) − 111.244i − 0.934827i
\(120\) 0 0
\(121\) −48.8592 −0.403795
\(122\) − 331.559i − 2.71770i
\(123\) 0 0
\(124\) 139.893 1.12817
\(125\) − 29.8301i − 0.238640i
\(126\) 0 0
\(127\) −11.2694 −0.0887357
\(128\) − 197.724i − 1.54472i
\(129\) 0 0
\(130\) 31.9756 0.245967
\(131\) 220.741i 1.68505i 0.538659 + 0.842524i \(0.318931\pi\)
−0.538659 + 0.842524i \(0.681069\pi\)
\(132\) 0 0
\(133\) 458.199 3.44510
\(134\) − 127.465i − 0.951231i
\(135\) 0 0
\(136\) 66.0662 0.485781
\(137\) 200.417i 1.46290i 0.681896 + 0.731450i \(0.261156\pi\)
−0.681896 + 0.731450i \(0.738844\pi\)
\(138\) 0 0
\(139\) −196.038 −1.41035 −0.705173 0.709035i \(-0.749131\pi\)
−0.705173 + 0.709035i \(0.749131\pi\)
\(140\) 47.3353i 0.338109i
\(141\) 0 0
\(142\) 63.6903 0.448523
\(143\) − 215.983i − 1.51037i
\(144\) 0 0
\(145\) −19.1994 −0.132410
\(146\) − 43.9772i − 0.301214i
\(147\) 0 0
\(148\) −321.885 −2.17490
\(149\) 67.1955i 0.450976i 0.974246 + 0.225488i \(0.0723977\pi\)
−0.974246 + 0.225488i \(0.927602\pi\)
\(150\) 0 0
\(151\) −67.9887 −0.450257 −0.225128 0.974329i \(-0.572280\pi\)
−0.225128 + 0.974329i \(0.572280\pi\)
\(152\) 272.116i 1.79024i
\(153\) 0 0
\(154\) 522.438 3.39245
\(155\) 13.3245i 0.0859648i
\(156\) 0 0
\(157\) −32.1609 −0.204846 −0.102423 0.994741i \(-0.532660\pi\)
−0.102423 + 0.994741i \(0.532660\pi\)
\(158\) 261.606i 1.65573i
\(159\) 0 0
\(160\) 20.5830 0.128644
\(161\) 300.462i 1.86622i
\(162\) 0 0
\(163\) −125.911 −0.772461 −0.386231 0.922402i \(-0.626223\pi\)
−0.386231 + 0.922402i \(0.626223\pi\)
\(164\) 131.958i 0.804622i
\(165\) 0 0
\(166\) −206.805 −1.24581
\(167\) − 287.574i − 1.72200i −0.508603 0.861001i \(-0.669838\pi\)
0.508603 0.861001i \(-0.330162\pi\)
\(168\) 0 0
\(169\) 105.630 0.625031
\(170\) 17.1928i 0.101134i
\(171\) 0 0
\(172\) −147.727 −0.858879
\(173\) − 249.120i − 1.44000i −0.693973 0.720001i \(-0.744141\pi\)
0.693973 0.720001i \(-0.255859\pi\)
\(174\) 0 0
\(175\) 307.608 1.75776
\(176\) − 18.6455i − 0.105940i
\(177\) 0 0
\(178\) −342.249 −1.92275
\(179\) − 73.5806i − 0.411065i −0.978650 0.205532i \(-0.934107\pi\)
0.978650 0.205532i \(-0.0658926\pi\)
\(180\) 0 0
\(181\) −72.5518 −0.400839 −0.200419 0.979710i \(-0.564230\pi\)
−0.200419 + 0.979710i \(0.564230\pi\)
\(182\) 664.300i 3.65000i
\(183\) 0 0
\(184\) −178.439 −0.969778
\(185\) − 30.6590i − 0.165724i
\(186\) 0 0
\(187\) 116.130 0.621018
\(188\) 231.082i 1.22916i
\(189\) 0 0
\(190\) −70.8144 −0.372707
\(191\) 72.5095i 0.379631i 0.981820 + 0.189816i \(0.0607890\pi\)
−0.981820 + 0.189816i \(0.939211\pi\)
\(192\) 0 0
\(193\) 326.580 1.69212 0.846062 0.533084i \(-0.178967\pi\)
0.846062 + 0.533084i \(0.178967\pi\)
\(194\) − 312.560i − 1.61114i
\(195\) 0 0
\(196\) −674.248 −3.44004
\(197\) 71.2897i 0.361877i 0.983494 + 0.180938i \(0.0579134\pi\)
−0.983494 + 0.180938i \(0.942087\pi\)
\(198\) 0 0
\(199\) −29.7643 −0.149569 −0.0747847 0.997200i \(-0.523827\pi\)
−0.0747847 + 0.997200i \(0.523827\pi\)
\(200\) 182.683i 0.913416i
\(201\) 0 0
\(202\) −480.302 −2.37773
\(203\) − 398.872i − 1.96489i
\(204\) 0 0
\(205\) −12.5688 −0.0613110
\(206\) 522.105i 2.53449i
\(207\) 0 0
\(208\) 23.7085 0.113983
\(209\) 478.323i 2.28863i
\(210\) 0 0
\(211\) 338.681 1.60513 0.802563 0.596568i \(-0.203469\pi\)
0.802563 + 0.596568i \(0.203469\pi\)
\(212\) − 38.6042i − 0.182095i
\(213\) 0 0
\(214\) 279.576 1.30643
\(215\) − 14.0707i − 0.0654453i
\(216\) 0 0
\(217\) −276.820 −1.27567
\(218\) 259.163i 1.18882i
\(219\) 0 0
\(220\) −49.4143 −0.224610
\(221\) 147.664i 0.668164i
\(222\) 0 0
\(223\) 264.383 1.18557 0.592786 0.805360i \(-0.298028\pi\)
0.592786 + 0.805360i \(0.298028\pi\)
\(224\) 427.615i 1.90900i
\(225\) 0 0
\(226\) 590.886 2.61454
\(227\) 378.068i 1.66550i 0.553649 + 0.832750i \(0.313235\pi\)
−0.553649 + 0.832750i \(0.686765\pi\)
\(228\) 0 0
\(229\) 207.425 0.905785 0.452893 0.891565i \(-0.350392\pi\)
0.452893 + 0.891565i \(0.350392\pi\)
\(230\) − 46.4363i − 0.201897i
\(231\) 0 0
\(232\) 236.883 1.02105
\(233\) − 285.957i − 1.22728i −0.789584 0.613642i \(-0.789704\pi\)
0.789584 0.613642i \(-0.210296\pi\)
\(234\) 0 0
\(235\) −22.0101 −0.0936601
\(236\) 304.494i 1.29023i
\(237\) 0 0
\(238\) −357.183 −1.50077
\(239\) − 337.604i − 1.41257i −0.707927 0.706285i \(-0.750370\pi\)
0.707927 0.706285i \(-0.249630\pi\)
\(240\) 0 0
\(241\) 298.039 1.23667 0.618337 0.785913i \(-0.287807\pi\)
0.618337 + 0.785913i \(0.287807\pi\)
\(242\) 156.877i 0.648252i
\(243\) 0 0
\(244\) −651.514 −2.67014
\(245\) − 64.2209i − 0.262126i
\(246\) 0 0
\(247\) −608.206 −2.46237
\(248\) − 164.399i − 0.662897i
\(249\) 0 0
\(250\) −95.7783 −0.383113
\(251\) 30.2125i 0.120368i 0.998187 + 0.0601842i \(0.0191688\pi\)
−0.998187 + 0.0601842i \(0.980831\pi\)
\(252\) 0 0
\(253\) −313.658 −1.23976
\(254\) 36.1838i 0.142456i
\(255\) 0 0
\(256\) −217.849 −0.850971
\(257\) − 428.708i − 1.66813i −0.551670 0.834063i \(-0.686009\pi\)
0.551670 0.834063i \(-0.313991\pi\)
\(258\) 0 0
\(259\) 636.946 2.45925
\(260\) − 62.8322i − 0.241662i
\(261\) 0 0
\(262\) 708.756 2.70517
\(263\) 129.896i 0.493902i 0.969028 + 0.246951i \(0.0794287\pi\)
−0.969028 + 0.246951i \(0.920571\pi\)
\(264\) 0 0
\(265\) 3.67698 0.0138754
\(266\) − 1471.18i − 5.53076i
\(267\) 0 0
\(268\) −250.469 −0.934584
\(269\) − 105.796i − 0.393293i −0.980474 0.196647i \(-0.936995\pi\)
0.980474 0.196647i \(-0.0630052\pi\)
\(270\) 0 0
\(271\) −154.578 −0.570399 −0.285200 0.958468i \(-0.592060\pi\)
−0.285200 + 0.958468i \(0.592060\pi\)
\(272\) 12.7477i 0.0468664i
\(273\) 0 0
\(274\) 643.499 2.34854
\(275\) 321.118i 1.16770i
\(276\) 0 0
\(277\) −286.356 −1.03377 −0.516887 0.856054i \(-0.672909\pi\)
−0.516887 + 0.856054i \(0.672909\pi\)
\(278\) 629.439i 2.26417i
\(279\) 0 0
\(280\) 55.6272 0.198669
\(281\) 185.828i 0.661311i 0.943751 + 0.330656i \(0.107270\pi\)
−0.943751 + 0.330656i \(0.892730\pi\)
\(282\) 0 0
\(283\) 72.1935 0.255101 0.127550 0.991832i \(-0.459289\pi\)
0.127550 + 0.991832i \(0.459289\pi\)
\(284\) − 125.151i − 0.440674i
\(285\) 0 0
\(286\) −693.476 −2.42474
\(287\) − 261.118i − 0.909820i
\(288\) 0 0
\(289\) 209.603 0.725271
\(290\) 61.6455i 0.212571i
\(291\) 0 0
\(292\) −86.4152 −0.295942
\(293\) 345.539i 1.17931i 0.807654 + 0.589657i \(0.200737\pi\)
−0.807654 + 0.589657i \(0.799263\pi\)
\(294\) 0 0
\(295\) −29.0025 −0.0983136
\(296\) 378.271i 1.27794i
\(297\) 0 0
\(298\) 215.751 0.723997
\(299\) − 398.829i − 1.33388i
\(300\) 0 0
\(301\) 292.322 0.971171
\(302\) 218.298i 0.722841i
\(303\) 0 0
\(304\) −52.5056 −0.172716
\(305\) − 62.0555i − 0.203461i
\(306\) 0 0
\(307\) −252.762 −0.823328 −0.411664 0.911336i \(-0.635052\pi\)
−0.411664 + 0.911336i \(0.635052\pi\)
\(308\) − 1026.59i − 3.33309i
\(309\) 0 0
\(310\) 42.7824 0.138008
\(311\) 410.767i 1.32079i 0.750917 + 0.660397i \(0.229612\pi\)
−0.750917 + 0.660397i \(0.770388\pi\)
\(312\) 0 0
\(313\) 309.912 0.990134 0.495067 0.868855i \(-0.335144\pi\)
0.495067 + 0.868855i \(0.335144\pi\)
\(314\) 103.262i 0.328860i
\(315\) 0 0
\(316\) 514.055 1.62676
\(317\) − 354.995i − 1.11986i −0.828541 0.559928i \(-0.810829\pi\)
0.828541 0.559928i \(-0.189171\pi\)
\(318\) 0 0
\(319\) 416.391 1.30530
\(320\) − 62.6488i − 0.195778i
\(321\) 0 0
\(322\) 964.722 2.99603
\(323\) − 327.023i − 1.01245i
\(324\) 0 0
\(325\) −408.315 −1.25635
\(326\) 404.275i 1.24011i
\(327\) 0 0
\(328\) 155.074 0.472786
\(329\) − 457.265i − 1.38986i
\(330\) 0 0
\(331\) −234.603 −0.708772 −0.354386 0.935099i \(-0.615310\pi\)
−0.354386 + 0.935099i \(0.615310\pi\)
\(332\) 406.371i 1.22401i
\(333\) 0 0
\(334\) −923.343 −2.76450
\(335\) − 23.8567i − 0.0712140i
\(336\) 0 0
\(337\) −613.452 −1.82033 −0.910167 0.414242i \(-0.864047\pi\)
−0.910167 + 0.414242i \(0.864047\pi\)
\(338\) − 339.157i − 1.00342i
\(339\) 0 0
\(340\) 33.7838 0.0993642
\(341\) − 288.978i − 0.847442i
\(342\) 0 0
\(343\) 722.452 2.10627
\(344\) 173.605i 0.504666i
\(345\) 0 0
\(346\) −799.875 −2.31178
\(347\) 19.2791i 0.0555592i 0.999614 + 0.0277796i \(0.00884366\pi\)
−0.999614 + 0.0277796i \(0.991156\pi\)
\(348\) 0 0
\(349\) 85.9984 0.246414 0.123207 0.992381i \(-0.460682\pi\)
0.123207 + 0.992381i \(0.460682\pi\)
\(350\) − 987.668i − 2.82191i
\(351\) 0 0
\(352\) −446.396 −1.26817
\(353\) 16.0427i 0.0454467i 0.999742 + 0.0227233i \(0.00723369\pi\)
−0.999742 + 0.0227233i \(0.992766\pi\)
\(354\) 0 0
\(355\) 11.9204 0.0335787
\(356\) 672.520i 1.88910i
\(357\) 0 0
\(358\) −236.252 −0.659923
\(359\) 570.789i 1.58994i 0.606648 + 0.794971i \(0.292514\pi\)
−0.606648 + 0.794971i \(0.707486\pi\)
\(360\) 0 0
\(361\) 985.955 2.73118
\(362\) 232.949i 0.643506i
\(363\) 0 0
\(364\) 1305.35 3.58613
\(365\) − 8.23089i − 0.0225504i
\(366\) 0 0
\(367\) −89.6682 −0.244328 −0.122164 0.992510i \(-0.538983\pi\)
−0.122164 + 0.992510i \(0.538983\pi\)
\(368\) − 34.4304i − 0.0935607i
\(369\) 0 0
\(370\) −98.4398 −0.266053
\(371\) 76.3899i 0.205903i
\(372\) 0 0
\(373\) −198.369 −0.531820 −0.265910 0.963998i \(-0.585672\pi\)
−0.265910 + 0.963998i \(0.585672\pi\)
\(374\) − 372.871i − 0.996981i
\(375\) 0 0
\(376\) 271.562 0.722238
\(377\) 529.457i 1.40439i
\(378\) 0 0
\(379\) −186.729 −0.492689 −0.246345 0.969182i \(-0.579230\pi\)
−0.246345 + 0.969182i \(0.579230\pi\)
\(380\) 139.150i 0.366185i
\(381\) 0 0
\(382\) 232.813 0.609459
\(383\) 210.931i 0.550734i 0.961339 + 0.275367i \(0.0887994\pi\)
−0.961339 + 0.275367i \(0.911201\pi\)
\(384\) 0 0
\(385\) 97.7809 0.253976
\(386\) − 1048.58i − 2.71653i
\(387\) 0 0
\(388\) −614.181 −1.58294
\(389\) 396.261i 1.01867i 0.860570 + 0.509333i \(0.170108\pi\)
−0.860570 + 0.509333i \(0.829892\pi\)
\(390\) 0 0
\(391\) 214.444 0.548450
\(392\) 792.359i 2.02132i
\(393\) 0 0
\(394\) 228.897 0.580956
\(395\) 48.9628i 0.123956i
\(396\) 0 0
\(397\) −326.905 −0.823437 −0.411719 0.911311i \(-0.635071\pi\)
−0.411719 + 0.911311i \(0.635071\pi\)
\(398\) 95.5672i 0.240119i
\(399\) 0 0
\(400\) −35.2493 −0.0881231
\(401\) − 721.209i − 1.79853i −0.437407 0.899264i \(-0.644103\pi\)
0.437407 0.899264i \(-0.355897\pi\)
\(402\) 0 0
\(403\) 367.447 0.911778
\(404\) 943.793i 2.33612i
\(405\) 0 0
\(406\) −1280.70 −3.15443
\(407\) 664.921i 1.63371i
\(408\) 0 0
\(409\) 152.630 0.373178 0.186589 0.982438i \(-0.440257\pi\)
0.186589 + 0.982438i \(0.440257\pi\)
\(410\) 40.3557i 0.0984287i
\(411\) 0 0
\(412\) 1025.94 2.49014
\(413\) − 602.533i − 1.45892i
\(414\) 0 0
\(415\) −38.7061 −0.0932678
\(416\) − 567.610i − 1.36445i
\(417\) 0 0
\(418\) 1535.80 3.67416
\(419\) 145.228i 0.346605i 0.984869 + 0.173303i \(0.0554439\pi\)
−0.984869 + 0.173303i \(0.944556\pi\)
\(420\) 0 0
\(421\) −146.196 −0.347259 −0.173630 0.984811i \(-0.555550\pi\)
−0.173630 + 0.984811i \(0.555550\pi\)
\(422\) − 1087.44i − 2.57687i
\(423\) 0 0
\(424\) −45.3666 −0.106997
\(425\) − 219.544i − 0.516575i
\(426\) 0 0
\(427\) 1289.21 3.01924
\(428\) − 549.367i − 1.28357i
\(429\) 0 0
\(430\) −45.1783 −0.105066
\(431\) 616.911i 1.43135i 0.698435 + 0.715674i \(0.253880\pi\)
−0.698435 + 0.715674i \(0.746120\pi\)
\(432\) 0 0
\(433\) 81.3788 0.187942 0.0939709 0.995575i \(-0.470044\pi\)
0.0939709 + 0.995575i \(0.470044\pi\)
\(434\) 888.813i 2.04796i
\(435\) 0 0
\(436\) 509.256 1.16802
\(437\) 883.261i 2.02119i
\(438\) 0 0
\(439\) 105.870 0.241162 0.120581 0.992703i \(-0.461524\pi\)
0.120581 + 0.992703i \(0.461524\pi\)
\(440\) 58.0704i 0.131978i
\(441\) 0 0
\(442\) 474.120 1.07267
\(443\) − 143.172i − 0.323188i −0.986857 0.161594i \(-0.948336\pi\)
0.986857 0.161594i \(-0.0516635\pi\)
\(444\) 0 0
\(445\) −64.0563 −0.143947
\(446\) − 848.879i − 1.90332i
\(447\) 0 0
\(448\) 1301.54 2.90523
\(449\) 868.777i 1.93492i 0.253032 + 0.967458i \(0.418572\pi\)
−0.253032 + 0.967458i \(0.581428\pi\)
\(450\) 0 0
\(451\) 272.587 0.604405
\(452\) − 1161.09i − 2.56878i
\(453\) 0 0
\(454\) 1213.90 2.67379
\(455\) 124.332i 0.273258i
\(456\) 0 0
\(457\) 427.913 0.936353 0.468176 0.883635i \(-0.344911\pi\)
0.468176 + 0.883635i \(0.344911\pi\)
\(458\) − 665.999i − 1.45415i
\(459\) 0 0
\(460\) −91.2473 −0.198364
\(461\) − 53.8113i − 0.116727i −0.998295 0.0583637i \(-0.981412\pi\)
0.998295 0.0583637i \(-0.0185883\pi\)
\(462\) 0 0
\(463\) −222.272 −0.480068 −0.240034 0.970764i \(-0.577159\pi\)
−0.240034 + 0.970764i \(0.577159\pi\)
\(464\) 45.7073i 0.0985071i
\(465\) 0 0
\(466\) −918.151 −1.97028
\(467\) − 59.2741i − 0.126925i −0.997984 0.0634626i \(-0.979786\pi\)
0.997984 0.0634626i \(-0.0202144\pi\)
\(468\) 0 0
\(469\) 495.627 1.05677
\(470\) 70.6701i 0.150362i
\(471\) 0 0
\(472\) 357.834 0.758122
\(473\) 305.161i 0.645161i
\(474\) 0 0
\(475\) 904.268 1.90372
\(476\) 701.866i 1.47451i
\(477\) 0 0
\(478\) −1083.98 −2.26774
\(479\) 726.560i 1.51683i 0.651774 + 0.758413i \(0.274025\pi\)
−0.651774 + 0.758413i \(0.725975\pi\)
\(480\) 0 0
\(481\) −845.473 −1.75774
\(482\) − 956.942i − 1.98536i
\(483\) 0 0
\(484\) 308.263 0.636907
\(485\) − 58.4997i − 0.120618i
\(486\) 0 0
\(487\) −384.609 −0.789752 −0.394876 0.918734i \(-0.629213\pi\)
−0.394876 + 0.918734i \(0.629213\pi\)
\(488\) 765.642i 1.56894i
\(489\) 0 0
\(490\) −206.200 −0.420817
\(491\) 266.424i 0.542616i 0.962493 + 0.271308i \(0.0874562\pi\)
−0.962493 + 0.271308i \(0.912544\pi\)
\(492\) 0 0
\(493\) −284.680 −0.577445
\(494\) 1952.83i 3.95309i
\(495\) 0 0
\(496\) 31.7212 0.0639540
\(497\) 247.650i 0.498289i
\(498\) 0 0
\(499\) −331.733 −0.664795 −0.332398 0.943139i \(-0.607858\pi\)
−0.332398 + 0.943139i \(0.607858\pi\)
\(500\) 188.204i 0.376409i
\(501\) 0 0
\(502\) 97.0061 0.193239
\(503\) − 356.706i − 0.709157i −0.935026 0.354579i \(-0.884624\pi\)
0.935026 0.354579i \(-0.115376\pi\)
\(504\) 0 0
\(505\) −89.8946 −0.178009
\(506\) 1007.09i 1.99030i
\(507\) 0 0
\(508\) 71.1013 0.139963
\(509\) 0.634209i 0.00124599i 1.00000 0.000622995i \(0.000198306\pi\)
−1.00000 0.000622995i \(0.999802\pi\)
\(510\) 0 0
\(511\) 170.998 0.334635
\(512\) − 91.4304i − 0.178575i
\(513\) 0 0
\(514\) −1376.49 −2.67801
\(515\) 97.7186i 0.189745i
\(516\) 0 0
\(517\) 477.348 0.923303
\(518\) − 2045.10i − 3.94808i
\(519\) 0 0
\(520\) −73.8388 −0.141998
\(521\) 674.406i 1.29445i 0.762301 + 0.647223i \(0.224070\pi\)
−0.762301 + 0.647223i \(0.775930\pi\)
\(522\) 0 0
\(523\) −177.818 −0.339995 −0.169998 0.985444i \(-0.554376\pi\)
−0.169998 + 0.985444i \(0.554376\pi\)
\(524\) − 1392.71i − 2.65783i
\(525\) 0 0
\(526\) 417.071 0.792910
\(527\) 197.570i 0.374896i
\(528\) 0 0
\(529\) −50.1948 −0.0948861
\(530\) − 11.8060i − 0.0222755i
\(531\) 0 0
\(532\) −2890.88 −5.43398
\(533\) 346.605i 0.650290i
\(534\) 0 0
\(535\) 52.3262 0.0978060
\(536\) 294.344i 0.549150i
\(537\) 0 0
\(538\) −339.689 −0.631393
\(539\) 1392.80i 2.58404i
\(540\) 0 0
\(541\) 82.7287 0.152918 0.0764590 0.997073i \(-0.475639\pi\)
0.0764590 + 0.997073i \(0.475639\pi\)
\(542\) 496.319i 0.915718i
\(543\) 0 0
\(544\) 305.195 0.561020
\(545\) 48.5057i 0.0890012i
\(546\) 0 0
\(547\) −352.774 −0.644925 −0.322462 0.946582i \(-0.604511\pi\)
−0.322462 + 0.946582i \(0.604511\pi\)
\(548\) − 1264.48i − 2.30744i
\(549\) 0 0
\(550\) 1031.05 1.87463
\(551\) − 1172.55i − 2.12805i
\(552\) 0 0
\(553\) −1017.21 −1.83944
\(554\) 919.429i 1.65962i
\(555\) 0 0
\(556\) 1236.85 2.22455
\(557\) − 758.420i − 1.36162i −0.732462 0.680808i \(-0.761629\pi\)
0.732462 0.680808i \(-0.238371\pi\)
\(558\) 0 0
\(559\) −388.024 −0.694140
\(560\) 10.7334i 0.0191668i
\(561\) 0 0
\(562\) 596.657 1.06167
\(563\) − 32.9364i − 0.0585017i −0.999572 0.0292508i \(-0.990688\pi\)
0.999572 0.0292508i \(-0.00931216\pi\)
\(564\) 0 0
\(565\) 110.592 0.195738
\(566\) − 231.799i − 0.409538i
\(567\) 0 0
\(568\) −147.075 −0.258934
\(569\) − 182.967i − 0.321559i −0.986990 0.160780i \(-0.948599\pi\)
0.986990 0.160780i \(-0.0514008\pi\)
\(570\) 0 0
\(571\) −474.017 −0.830152 −0.415076 0.909787i \(-0.636245\pi\)
−0.415076 + 0.909787i \(0.636245\pi\)
\(572\) 1362.68i 2.38231i
\(573\) 0 0
\(574\) −838.398 −1.46062
\(575\) 592.970i 1.03125i
\(576\) 0 0
\(577\) −874.351 −1.51534 −0.757670 0.652638i \(-0.773662\pi\)
−0.757670 + 0.652638i \(0.773662\pi\)
\(578\) − 672.994i − 1.16435i
\(579\) 0 0
\(580\) 121.133 0.208851
\(581\) − 804.127i − 1.38404i
\(582\) 0 0
\(583\) −79.7449 −0.136784
\(584\) 101.553i 0.173892i
\(585\) 0 0
\(586\) 1109.46 1.89327
\(587\) 4.68898i 0.00798804i 0.999992 + 0.00399402i \(0.00127134\pi\)
−0.999992 + 0.00399402i \(0.998729\pi\)
\(588\) 0 0
\(589\) −813.761 −1.38160
\(590\) 93.1212i 0.157833i
\(591\) 0 0
\(592\) −72.9885 −0.123291
\(593\) 582.642i 0.982533i 0.871009 + 0.491267i \(0.163466\pi\)
−0.871009 + 0.491267i \(0.836534\pi\)
\(594\) 0 0
\(595\) −66.8514 −0.112355
\(596\) − 423.951i − 0.711327i
\(597\) 0 0
\(598\) −1280.56 −2.14140
\(599\) 725.223i 1.21072i 0.795951 + 0.605361i \(0.206971\pi\)
−0.795951 + 0.605361i \(0.793029\pi\)
\(600\) 0 0
\(601\) 1108.14 1.84382 0.921910 0.387403i \(-0.126628\pi\)
0.921910 + 0.387403i \(0.126628\pi\)
\(602\) − 938.588i − 1.55912i
\(603\) 0 0
\(604\) 428.956 0.710192
\(605\) 29.3615i 0.0485314i
\(606\) 0 0
\(607\) −280.661 −0.462375 −0.231187 0.972909i \(-0.574261\pi\)
−0.231187 + 0.972909i \(0.574261\pi\)
\(608\) 1257.05i 2.06752i
\(609\) 0 0
\(610\) −199.248 −0.326635
\(611\) 606.966i 0.993398i
\(612\) 0 0
\(613\) 933.143 1.52226 0.761128 0.648602i \(-0.224646\pi\)
0.761128 + 0.648602i \(0.224646\pi\)
\(614\) 811.567i 1.32177i
\(615\) 0 0
\(616\) −1206.42 −1.95848
\(617\) 456.102i 0.739225i 0.929186 + 0.369613i \(0.120510\pi\)
−0.929186 + 0.369613i \(0.879490\pi\)
\(618\) 0 0
\(619\) −817.095 −1.32002 −0.660012 0.751255i \(-0.729449\pi\)
−0.660012 + 0.751255i \(0.729449\pi\)
\(620\) − 84.0675i − 0.135593i
\(621\) 0 0
\(622\) 1318.89 2.12040
\(623\) − 1330.78i − 2.13609i
\(624\) 0 0
\(625\) 598.046 0.956873
\(626\) − 995.064i − 1.58956i
\(627\) 0 0
\(628\) 202.910 0.323105
\(629\) − 454.597i − 0.722730i
\(630\) 0 0
\(631\) 350.841 0.556007 0.278004 0.960580i \(-0.410327\pi\)
0.278004 + 0.960580i \(0.410327\pi\)
\(632\) − 604.104i − 0.955861i
\(633\) 0 0
\(634\) −1139.82 −1.79782
\(635\) 6.77227i 0.0106650i
\(636\) 0 0
\(637\) −1771.00 −2.78022
\(638\) − 1336.95i − 2.09553i
\(639\) 0 0
\(640\) −118.821 −0.185658
\(641\) 379.458i 0.591979i 0.955191 + 0.295989i \(0.0956493\pi\)
−0.955191 + 0.295989i \(0.904351\pi\)
\(642\) 0 0
\(643\) −118.904 −0.184920 −0.0924601 0.995716i \(-0.529473\pi\)
−0.0924601 + 0.995716i \(0.529473\pi\)
\(644\) − 1895.68i − 2.94360i
\(645\) 0 0
\(646\) −1050.00 −1.62539
\(647\) 230.598i 0.356411i 0.983993 + 0.178205i \(0.0570291\pi\)
−0.983993 + 0.178205i \(0.942971\pi\)
\(648\) 0 0
\(649\) 628.996 0.969177
\(650\) 1311.02i 2.01695i
\(651\) 0 0
\(652\) 794.401 1.21841
\(653\) − 1163.78i − 1.78221i −0.453801 0.891103i \(-0.649932\pi\)
0.453801 0.891103i \(-0.350068\pi\)
\(654\) 0 0
\(655\) 132.653 0.202523
\(656\) 29.9219i 0.0456127i
\(657\) 0 0
\(658\) −1468.18 −2.23128
\(659\) − 578.497i − 0.877841i −0.898526 0.438920i \(-0.855361\pi\)
0.898526 0.438920i \(-0.144639\pi\)
\(660\) 0 0
\(661\) 91.1558 0.137906 0.0689530 0.997620i \(-0.478034\pi\)
0.0689530 + 0.997620i \(0.478034\pi\)
\(662\) 753.264i 1.13786i
\(663\) 0 0
\(664\) 477.557 0.719212
\(665\) − 275.351i − 0.414061i
\(666\) 0 0
\(667\) 768.898 1.15277
\(668\) 1814.37i 2.71612i
\(669\) 0 0
\(670\) −76.5990 −0.114327
\(671\) 1345.84i 2.00572i
\(672\) 0 0
\(673\) −517.141 −0.768412 −0.384206 0.923247i \(-0.625525\pi\)
−0.384206 + 0.923247i \(0.625525\pi\)
\(674\) 1969.67i 2.92236i
\(675\) 0 0
\(676\) −666.444 −0.985864
\(677\) − 934.397i − 1.38020i −0.723713 0.690101i \(-0.757566\pi\)
0.723713 0.690101i \(-0.242434\pi\)
\(678\) 0 0
\(679\) 1215.34 1.78990
\(680\) − 39.7019i − 0.0583851i
\(681\) 0 0
\(682\) −927.849 −1.36048
\(683\) − 637.476i − 0.933346i −0.884430 0.466673i \(-0.845452\pi\)
0.884430 0.466673i \(-0.154548\pi\)
\(684\) 0 0
\(685\) 120.439 0.175823
\(686\) − 2319.65i − 3.38141i
\(687\) 0 0
\(688\) −33.4976 −0.0486884
\(689\) − 101.399i − 0.147168i
\(690\) 0 0
\(691\) −259.071 −0.374922 −0.187461 0.982272i \(-0.560026\pi\)
−0.187461 + 0.982272i \(0.560026\pi\)
\(692\) 1571.75i 2.27132i
\(693\) 0 0
\(694\) 61.9011 0.0891947
\(695\) 117.807i 0.169507i
\(696\) 0 0
\(697\) −186.364 −0.267380
\(698\) − 276.123i − 0.395592i
\(699\) 0 0
\(700\) −1940.77 −2.77253
\(701\) − 472.819i − 0.674492i −0.941417 0.337246i \(-0.890505\pi\)
0.941417 0.337246i \(-0.109495\pi\)
\(702\) 0 0
\(703\) 1872.41 2.66346
\(704\) 1358.71i 1.92998i
\(705\) 0 0
\(706\) 51.5098 0.0729600
\(707\) − 1867.58i − 2.64155i
\(708\) 0 0
\(709\) −1096.90 −1.54710 −0.773552 0.633732i \(-0.781522\pi\)
−0.773552 + 0.633732i \(0.781522\pi\)
\(710\) − 38.2741i − 0.0539072i
\(711\) 0 0
\(712\) 790.328 1.11001
\(713\) − 533.620i − 0.748416i
\(714\) 0 0
\(715\) −129.793 −0.181529
\(716\) 464.236i 0.648375i
\(717\) 0 0
\(718\) 1832.69 2.55249
\(719\) 1144.75i 1.59214i 0.605202 + 0.796072i \(0.293092\pi\)
−0.605202 + 0.796072i \(0.706908\pi\)
\(720\) 0 0
\(721\) −2030.12 −2.81570
\(722\) − 3165.70i − 4.38463i
\(723\) 0 0
\(724\) 457.745 0.632245
\(725\) − 787.186i − 1.08577i
\(726\) 0 0
\(727\) 58.4006 0.0803309 0.0401654 0.999193i \(-0.487212\pi\)
0.0401654 + 0.999193i \(0.487212\pi\)
\(728\) − 1534.01i − 2.10716i
\(729\) 0 0
\(730\) −26.4277 −0.0362023
\(731\) − 208.634i − 0.285410i
\(732\) 0 0
\(733\) −470.767 −0.642247 −0.321123 0.947037i \(-0.604060\pi\)
−0.321123 + 0.947037i \(0.604060\pi\)
\(734\) 287.907i 0.392243i
\(735\) 0 0
\(736\) −824.306 −1.11998
\(737\) 517.395i 0.702029i
\(738\) 0 0
\(739\) −666.625 −0.902063 −0.451031 0.892508i \(-0.648944\pi\)
−0.451031 + 0.892508i \(0.648944\pi\)
\(740\) 193.434i 0.261398i
\(741\) 0 0
\(742\) 245.272 0.330556
\(743\) − 441.655i − 0.594421i −0.954812 0.297210i \(-0.903944\pi\)
0.954812 0.297210i \(-0.0960563\pi\)
\(744\) 0 0
\(745\) 40.3806 0.0542021
\(746\) 636.923i 0.853784i
\(747\) 0 0
\(748\) −732.692 −0.979534
\(749\) 1087.09i 1.45138i
\(750\) 0 0
\(751\) −521.623 −0.694572 −0.347286 0.937759i \(-0.612897\pi\)
−0.347286 + 0.937759i \(0.612897\pi\)
\(752\) 52.3986i 0.0696790i
\(753\) 0 0
\(754\) 1699.98 2.25461
\(755\) 40.8573i 0.0541156i
\(756\) 0 0
\(757\) 1155.22 1.52605 0.763023 0.646371i \(-0.223714\pi\)
0.763023 + 0.646371i \(0.223714\pi\)
\(758\) 599.550i 0.790963i
\(759\) 0 0
\(760\) 163.526 0.215166
\(761\) − 355.823i − 0.467573i −0.972288 0.233787i \(-0.924888\pi\)
0.972288 0.233787i \(-0.0751117\pi\)
\(762\) 0 0
\(763\) −1007.71 −1.32073
\(764\) − 457.479i − 0.598794i
\(765\) 0 0
\(766\) 677.257 0.884148
\(767\) 799.793i 1.04275i
\(768\) 0 0
\(769\) −572.981 −0.745099 −0.372549 0.928012i \(-0.621516\pi\)
−0.372549 + 0.928012i \(0.621516\pi\)
\(770\) − 313.955i − 0.407733i
\(771\) 0 0
\(772\) −2060.46 −2.66900
\(773\) 1135.65i 1.46915i 0.678530 + 0.734573i \(0.262617\pi\)
−0.678530 + 0.734573i \(0.737383\pi\)
\(774\) 0 0
\(775\) −546.312 −0.704919
\(776\) 721.770i 0.930116i
\(777\) 0 0
\(778\) 1272.31 1.63536
\(779\) − 767.603i − 0.985370i
\(780\) 0 0
\(781\) −258.526 −0.331020
\(782\) − 688.535i − 0.880480i
\(783\) 0 0
\(784\) −152.888 −0.195010
\(785\) 19.3268i 0.0246201i
\(786\) 0 0
\(787\) −1434.07 −1.82220 −0.911101 0.412184i \(-0.864766\pi\)
−0.911101 + 0.412184i \(0.864766\pi\)
\(788\) − 449.782i − 0.570790i
\(789\) 0 0
\(790\) 157.210 0.199000
\(791\) 2297.56i 2.90463i
\(792\) 0 0
\(793\) −1711.28 −2.15799
\(794\) 1049.62i 1.32195i
\(795\) 0 0
\(796\) 187.790 0.235917
\(797\) 882.755i 1.10760i 0.832651 + 0.553799i \(0.186822\pi\)
−0.832651 + 0.553799i \(0.813178\pi\)
\(798\) 0 0
\(799\) −326.356 −0.408456
\(800\) 843.911i 1.05489i
\(801\) 0 0
\(802\) −2315.66 −2.88735
\(803\) 178.509i 0.222302i
\(804\) 0 0
\(805\) 180.560 0.224298
\(806\) − 1179.80i − 1.46377i
\(807\) 0 0
\(808\) 1109.12 1.37268
\(809\) − 346.433i − 0.428223i −0.976809 0.214112i \(-0.931314\pi\)
0.976809 0.214112i \(-0.0686856\pi\)
\(810\) 0 0
\(811\) −32.2606 −0.0397788 −0.0198894 0.999802i \(-0.506331\pi\)
−0.0198894 + 0.999802i \(0.506331\pi\)
\(812\) 2516.57i 3.09923i
\(813\) 0 0
\(814\) 2134.93 2.62276
\(815\) 75.6653i 0.0928408i
\(816\) 0 0
\(817\) 859.333 1.05182
\(818\) − 490.064i − 0.599100i
\(819\) 0 0
\(820\) 79.2991 0.0967062
\(821\) 198.978i 0.242360i 0.992631 + 0.121180i \(0.0386679\pi\)
−0.992631 + 0.121180i \(0.961332\pi\)
\(822\) 0 0
\(823\) 27.0628 0.0328831 0.0164415 0.999865i \(-0.494766\pi\)
0.0164415 + 0.999865i \(0.494766\pi\)
\(824\) − 1205.65i − 1.46317i
\(825\) 0 0
\(826\) −1934.61 −2.34214
\(827\) − 614.445i − 0.742980i −0.928437 0.371490i \(-0.878847\pi\)
0.928437 0.371490i \(-0.121153\pi\)
\(828\) 0 0
\(829\) −1310.05 −1.58028 −0.790138 0.612928i \(-0.789991\pi\)
−0.790138 + 0.612928i \(0.789991\pi\)
\(830\) 124.277i 0.149732i
\(831\) 0 0
\(832\) −1727.65 −2.07650
\(833\) − 952.237i − 1.14314i
\(834\) 0 0
\(835\) −172.815 −0.206964
\(836\) − 3017.84i − 3.60986i
\(837\) 0 0
\(838\) 466.297 0.556440
\(839\) − 783.420i − 0.933755i −0.884322 0.466877i \(-0.845379\pi\)
0.884322 0.466877i \(-0.154621\pi\)
\(840\) 0 0
\(841\) −179.735 −0.213715
\(842\) 469.406i 0.557490i
\(843\) 0 0
\(844\) −2136.82 −2.53177
\(845\) − 63.4776i − 0.0751214i
\(846\) 0 0
\(847\) −609.991 −0.720178
\(848\) − 8.75362i − 0.0103227i
\(849\) 0 0
\(850\) −704.912 −0.829308
\(851\) 1227.83i 1.44281i
\(852\) 0 0
\(853\) −1370.99 −1.60726 −0.803629 0.595131i \(-0.797100\pi\)
−0.803629 + 0.595131i \(0.797100\pi\)
\(854\) − 4139.41i − 4.84708i
\(855\) 0 0
\(856\) −645.602 −0.754208
\(857\) 1542.77i 1.80020i 0.435685 + 0.900099i \(0.356506\pi\)
−0.435685 + 0.900099i \(0.643494\pi\)
\(858\) 0 0
\(859\) −1041.40 −1.21234 −0.606169 0.795336i \(-0.707294\pi\)
−0.606169 + 0.795336i \(0.707294\pi\)
\(860\) 88.7754i 0.103227i
\(861\) 0 0
\(862\) 1980.78 2.29788
\(863\) − 341.420i − 0.395620i −0.980240 0.197810i \(-0.936617\pi\)
0.980240 0.197810i \(-0.0633829\pi\)
\(864\) 0 0
\(865\) −149.707 −0.173071
\(866\) − 261.291i − 0.301722i
\(867\) 0 0
\(868\) 1746.52 2.01212
\(869\) − 1061.89i − 1.22196i
\(870\) 0 0
\(871\) −657.888 −0.755325
\(872\) − 598.464i − 0.686312i
\(873\) 0 0
\(874\) 2835.97 3.24482
\(875\) − 372.419i − 0.425621i
\(876\) 0 0
\(877\) 1453.08 1.65687 0.828436 0.560084i \(-0.189231\pi\)
0.828436 + 0.560084i \(0.189231\pi\)
\(878\) − 339.927i − 0.387161i
\(879\) 0 0
\(880\) −11.2048 −0.0127328
\(881\) 799.175i 0.907123i 0.891225 + 0.453561i \(0.149847\pi\)
−0.891225 + 0.453561i \(0.850153\pi\)
\(882\) 0 0
\(883\) 103.793 0.117546 0.0587729 0.998271i \(-0.481281\pi\)
0.0587729 + 0.998271i \(0.481281\pi\)
\(884\) − 931.646i − 1.05390i
\(885\) 0 0
\(886\) −459.697 −0.518846
\(887\) 792.458i 0.893414i 0.894680 + 0.446707i \(0.147403\pi\)
−0.894680 + 0.446707i \(0.852597\pi\)
\(888\) 0 0
\(889\) −140.695 −0.158262
\(890\) 205.672i 0.231092i
\(891\) 0 0
\(892\) −1668.05 −1.87001
\(893\) − 1344.21i − 1.50527i
\(894\) 0 0
\(895\) −44.2177 −0.0494052
\(896\) − 2468.53i − 2.75505i
\(897\) 0 0
\(898\) 2789.47 3.10631
\(899\) 708.397i 0.787983i
\(900\) 0 0
\(901\) 54.5205 0.0605111
\(902\) − 875.221i − 0.970311i
\(903\) 0 0
\(904\) −1364.48 −1.50938
\(905\) 43.5994i 0.0481761i
\(906\) 0 0
\(907\) 710.256 0.783083 0.391542 0.920160i \(-0.371942\pi\)
0.391542 + 0.920160i \(0.371942\pi\)
\(908\) − 2385.32i − 2.62700i
\(909\) 0 0
\(910\) 399.206 0.438687
\(911\) − 215.745i − 0.236822i −0.992965 0.118411i \(-0.962220\pi\)
0.992965 0.118411i \(-0.0377800\pi\)
\(912\) 0 0
\(913\) 839.444 0.919435
\(914\) − 1373.94i − 1.50322i
\(915\) 0 0
\(916\) −1308.69 −1.42870
\(917\) 2755.88i 3.00533i
\(918\) 0 0
\(919\) 1340.47 1.45862 0.729311 0.684182i \(-0.239841\pi\)
0.729311 + 0.684182i \(0.239841\pi\)
\(920\) 107.232i 0.116556i
\(921\) 0 0
\(922\) −172.777 −0.187394
\(923\) − 328.726i − 0.356150i
\(924\) 0 0
\(925\) 1257.03 1.35895
\(926\) 713.669i 0.770701i
\(927\) 0 0
\(928\) 1094.29 1.17919
\(929\) 415.150i 0.446878i 0.974718 + 0.223439i \(0.0717284\pi\)
−0.974718 + 0.223439i \(0.928272\pi\)
\(930\) 0 0
\(931\) 3922.12 4.21280
\(932\) 1804.17i 1.93580i
\(933\) 0 0
\(934\) −190.317 −0.203766
\(935\) − 69.7875i − 0.0746391i
\(936\) 0 0
\(937\) 527.696 0.563176 0.281588 0.959535i \(-0.409139\pi\)
0.281588 + 0.959535i \(0.409139\pi\)
\(938\) − 1591.36i − 1.69654i
\(939\) 0 0
\(940\) 138.867 0.147731
\(941\) − 599.517i − 0.637107i −0.947905 0.318553i \(-0.896803\pi\)
0.947905 0.318553i \(-0.103197\pi\)
\(942\) 0 0
\(943\) 503.353 0.533778
\(944\) 69.0450i 0.0731409i
\(945\) 0 0
\(946\) 979.811 1.03574
\(947\) − 1193.07i − 1.25985i −0.776658 0.629923i \(-0.783087\pi\)
0.776658 0.629923i \(-0.216913\pi\)
\(948\) 0 0
\(949\) −226.981 −0.239179
\(950\) − 2903.42i − 3.05623i
\(951\) 0 0
\(952\) 824.814 0.866402
\(953\) − 774.833i − 0.813046i −0.913640 0.406523i \(-0.866741\pi\)
0.913640 0.406523i \(-0.133259\pi\)
\(954\) 0 0
\(955\) 43.5740 0.0456272
\(956\) 2130.02i 2.22805i
\(957\) 0 0
\(958\) 2332.84 2.43511
\(959\) 2502.14i 2.60912i
\(960\) 0 0
\(961\) −469.368 −0.488416
\(962\) 2714.64i 2.82187i
\(963\) 0 0
\(964\) −1880.39 −1.95061
\(965\) − 196.255i − 0.203374i
\(966\) 0 0
\(967\) 41.7164 0.0431400 0.0215700 0.999767i \(-0.493134\pi\)
0.0215700 + 0.999767i \(0.493134\pi\)
\(968\) − 362.263i − 0.374239i
\(969\) 0 0
\(970\) −187.831 −0.193640
\(971\) − 1043.46i − 1.07463i −0.843382 0.537315i \(-0.819439\pi\)
0.843382 0.537315i \(-0.180561\pi\)
\(972\) 0 0
\(973\) −2447.47 −2.51539
\(974\) 1234.90i 1.26787i
\(975\) 0 0
\(976\) −147.733 −0.151366
\(977\) 255.000i 0.261003i 0.991448 + 0.130501i \(0.0416587\pi\)
−0.991448 + 0.130501i \(0.958341\pi\)
\(978\) 0 0
\(979\) 1389.23 1.41903
\(980\) 405.184i 0.413453i
\(981\) 0 0
\(982\) 855.435 0.871115
\(983\) 1287.67i 1.30994i 0.755654 + 0.654971i \(0.227319\pi\)
−0.755654 + 0.654971i \(0.772681\pi\)
\(984\) 0 0
\(985\) 42.8409 0.0434933
\(986\) 914.051i 0.927029i
\(987\) 0 0
\(988\) 3837.30 3.88391
\(989\) 563.504i 0.569772i
\(990\) 0 0
\(991\) −1586.51 −1.60092 −0.800461 0.599384i \(-0.795412\pi\)
−0.800461 + 0.599384i \(0.795412\pi\)
\(992\) − 759.444i − 0.765569i
\(993\) 0 0
\(994\) 795.152 0.799952
\(995\) 17.8866i 0.0179765i
\(996\) 0 0
\(997\) 1221.04 1.22471 0.612356 0.790582i \(-0.290222\pi\)
0.612356 + 0.790582i \(0.290222\pi\)
\(998\) 1065.13i 1.06726i
\(999\) 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 1143.3.b.a.890.10 84
3.2 odd 2 inner 1143.3.b.a.890.75 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.10 84 1.1 even 1 trivial
1143.3.b.a.890.75 yes 84 3.2 odd 2 inner