Properties

Label 1143.3.b.a.890.20
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 890.20
Character \(\chi\) \(=\) 1143.890
Dual form 1143.3.b.a.890.65

$q$-expansion

\(f(q)\) \(=\) \(q-2.40991i q^{2} -1.80767 q^{4} +4.01304i q^{5} -8.05053 q^{7} -5.28331i q^{8} +O(q^{10})\) \(q-2.40991i q^{2} -1.80767 q^{4} +4.01304i q^{5} -8.05053 q^{7} -5.28331i q^{8} +9.67106 q^{10} -7.16818i q^{11} +20.5325 q^{13} +19.4011i q^{14} -19.9630 q^{16} +13.1195i q^{17} -14.0782 q^{19} -7.25426i q^{20} -17.2747 q^{22} -26.8517i q^{23} +8.89554 q^{25} -49.4815i q^{26} +14.5527 q^{28} +1.48580i q^{29} -47.6593 q^{31} +26.9758i q^{32} +31.6169 q^{34} -32.3071i q^{35} -50.2868 q^{37} +33.9272i q^{38} +21.2021 q^{40} -54.4030i q^{41} +38.8264 q^{43} +12.9577i q^{44} -64.7103 q^{46} +59.0925i q^{47} +15.8110 q^{49} -21.4375i q^{50} -37.1160 q^{52} +50.8215i q^{53} +28.7662 q^{55} +42.5334i q^{56} +3.58064 q^{58} -36.7929i q^{59} -111.007 q^{61} +114.855i q^{62} -14.8426 q^{64} +82.3976i q^{65} -37.8794 q^{67} -23.7158i q^{68} -77.8571 q^{70} -60.7250i q^{71} -96.3723 q^{73} +121.187i q^{74} +25.4488 q^{76} +57.7076i q^{77} -148.728 q^{79} -80.1123i q^{80} -131.106 q^{82} -100.833i q^{83} -52.6491 q^{85} -93.5681i q^{86} -37.8717 q^{88} +123.265i q^{89} -165.297 q^{91} +48.5391i q^{92} +142.408 q^{94} -56.4963i q^{95} -69.7434 q^{97} -38.1031i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84q - 160q^{4} + O(q^{10}) \) \( 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.40991i − 1.20496i −0.798136 0.602478i \(-0.794180\pi\)
0.798136 0.602478i \(-0.205820\pi\)
\(3\) 0 0
\(4\) −1.80767 −0.451919
\(5\) 4.01304i 0.802607i 0.915945 + 0.401304i \(0.131443\pi\)
−0.915945 + 0.401304i \(0.868557\pi\)
\(6\) 0 0
\(7\) −8.05053 −1.15008 −0.575038 0.818127i \(-0.695013\pi\)
−0.575038 + 0.818127i \(0.695013\pi\)
\(8\) − 5.28331i − 0.660414i
\(9\) 0 0
\(10\) 9.67106 0.967106
\(11\) − 7.16818i − 0.651653i −0.945430 0.325826i \(-0.894358\pi\)
0.945430 0.325826i \(-0.105642\pi\)
\(12\) 0 0
\(13\) 20.5325 1.57942 0.789711 0.613480i \(-0.210231\pi\)
0.789711 + 0.613480i \(0.210231\pi\)
\(14\) 19.4011i 1.38579i
\(15\) 0 0
\(16\) −19.9630 −1.24769
\(17\) 13.1195i 0.771737i 0.922554 + 0.385869i \(0.126098\pi\)
−0.922554 + 0.385869i \(0.873902\pi\)
\(18\) 0 0
\(19\) −14.0782 −0.740957 −0.370478 0.928841i \(-0.620806\pi\)
−0.370478 + 0.928841i \(0.620806\pi\)
\(20\) − 7.25426i − 0.362713i
\(21\) 0 0
\(22\) −17.2747 −0.785213
\(23\) − 26.8517i − 1.16747i −0.811946 0.583733i \(-0.801592\pi\)
0.811946 0.583733i \(-0.198408\pi\)
\(24\) 0 0
\(25\) 8.89554 0.355822
\(26\) − 49.4815i − 1.90313i
\(27\) 0 0
\(28\) 14.5527 0.519740
\(29\) 1.48580i 0.0512344i 0.999672 + 0.0256172i \(0.00815510\pi\)
−0.999672 + 0.0256172i \(0.991845\pi\)
\(30\) 0 0
\(31\) −47.6593 −1.53740 −0.768698 0.639612i \(-0.779095\pi\)
−0.768698 + 0.639612i \(0.779095\pi\)
\(32\) 26.9758i 0.842995i
\(33\) 0 0
\(34\) 31.6169 0.929909
\(35\) − 32.3071i − 0.923059i
\(36\) 0 0
\(37\) −50.2868 −1.35910 −0.679551 0.733628i \(-0.737826\pi\)
−0.679551 + 0.733628i \(0.737826\pi\)
\(38\) 33.9272i 0.892820i
\(39\) 0 0
\(40\) 21.2021 0.530053
\(41\) − 54.4030i − 1.32690i −0.748220 0.663451i \(-0.769091\pi\)
0.748220 0.663451i \(-0.230909\pi\)
\(42\) 0 0
\(43\) 38.8264 0.902939 0.451469 0.892287i \(-0.350900\pi\)
0.451469 + 0.892287i \(0.350900\pi\)
\(44\) 12.9577i 0.294494i
\(45\) 0 0
\(46\) −64.7103 −1.40674
\(47\) 59.0925i 1.25729i 0.777694 + 0.628643i \(0.216389\pi\)
−0.777694 + 0.628643i \(0.783611\pi\)
\(48\) 0 0
\(49\) 15.8110 0.322673
\(50\) − 21.4375i − 0.428749i
\(51\) 0 0
\(52\) −37.1160 −0.713770
\(53\) 50.8215i 0.958896i 0.877570 + 0.479448i \(0.159163\pi\)
−0.877570 + 0.479448i \(0.840837\pi\)
\(54\) 0 0
\(55\) 28.7662 0.523021
\(56\) 42.5334i 0.759526i
\(57\) 0 0
\(58\) 3.58064 0.0617352
\(59\) − 36.7929i − 0.623609i −0.950146 0.311805i \(-0.899067\pi\)
0.950146 0.311805i \(-0.100933\pi\)
\(60\) 0 0
\(61\) −111.007 −1.81978 −0.909889 0.414851i \(-0.863834\pi\)
−0.909889 + 0.414851i \(0.863834\pi\)
\(62\) 114.855i 1.85249i
\(63\) 0 0
\(64\) −14.8426 −0.231916
\(65\) 82.3976i 1.26765i
\(66\) 0 0
\(67\) −37.8794 −0.565364 −0.282682 0.959214i \(-0.591224\pi\)
−0.282682 + 0.959214i \(0.591224\pi\)
\(68\) − 23.7158i − 0.348762i
\(69\) 0 0
\(70\) −77.8571 −1.11224
\(71\) − 60.7250i − 0.855282i −0.903949 0.427641i \(-0.859345\pi\)
0.903949 0.427641i \(-0.140655\pi\)
\(72\) 0 0
\(73\) −96.3723 −1.32017 −0.660085 0.751191i \(-0.729480\pi\)
−0.660085 + 0.751191i \(0.729480\pi\)
\(74\) 121.187i 1.63766i
\(75\) 0 0
\(76\) 25.4488 0.334852
\(77\) 57.7076i 0.749450i
\(78\) 0 0
\(79\) −148.728 −1.88263 −0.941315 0.337530i \(-0.890409\pi\)
−0.941315 + 0.337530i \(0.890409\pi\)
\(80\) − 80.1123i − 1.00140i
\(81\) 0 0
\(82\) −131.106 −1.59886
\(83\) − 100.833i − 1.21486i −0.794374 0.607429i \(-0.792201\pi\)
0.794374 0.607429i \(-0.207799\pi\)
\(84\) 0 0
\(85\) −52.6491 −0.619402
\(86\) − 93.5681i − 1.08800i
\(87\) 0 0
\(88\) −37.8717 −0.430361
\(89\) 123.265i 1.38500i 0.721417 + 0.692501i \(0.243491\pi\)
−0.721417 + 0.692501i \(0.756509\pi\)
\(90\) 0 0
\(91\) −165.297 −1.81645
\(92\) 48.5391i 0.527599i
\(93\) 0 0
\(94\) 142.408 1.51498
\(95\) − 56.4963i − 0.594697i
\(96\) 0 0
\(97\) −69.7434 −0.719005 −0.359502 0.933144i \(-0.617054\pi\)
−0.359502 + 0.933144i \(0.617054\pi\)
\(98\) − 38.1031i − 0.388807i
\(99\) 0 0
\(100\) −16.0802 −0.160802
\(101\) 69.5740i 0.688851i 0.938814 + 0.344426i \(0.111926\pi\)
−0.938814 + 0.344426i \(0.888074\pi\)
\(102\) 0 0
\(103\) 47.2476 0.458714 0.229357 0.973342i \(-0.426338\pi\)
0.229357 + 0.973342i \(0.426338\pi\)
\(104\) − 108.479i − 1.04307i
\(105\) 0 0
\(106\) 122.475 1.15543
\(107\) 45.4403i 0.424676i 0.977196 + 0.212338i \(0.0681078\pi\)
−0.977196 + 0.212338i \(0.931892\pi\)
\(108\) 0 0
\(109\) −36.6880 −0.336587 −0.168294 0.985737i \(-0.553826\pi\)
−0.168294 + 0.985737i \(0.553826\pi\)
\(110\) − 69.3239i − 0.630217i
\(111\) 0 0
\(112\) 160.713 1.43494
\(113\) − 85.2429i − 0.754362i −0.926140 0.377181i \(-0.876893\pi\)
0.926140 0.377181i \(-0.123107\pi\)
\(114\) 0 0
\(115\) 107.757 0.937016
\(116\) − 2.68584i − 0.0231538i
\(117\) 0 0
\(118\) −88.6678 −0.751422
\(119\) − 105.619i − 0.887556i
\(120\) 0 0
\(121\) 69.6172 0.575349
\(122\) 267.516i 2.19275i
\(123\) 0 0
\(124\) 86.1524 0.694777
\(125\) 136.024i 1.08819i
\(126\) 0 0
\(127\) −11.2694 −0.0887357
\(128\) 143.673i 1.12244i
\(129\) 0 0
\(130\) 198.571 1.52747
\(131\) − 159.575i − 1.21813i −0.793120 0.609065i \(-0.791545\pi\)
0.793120 0.609065i \(-0.208455\pi\)
\(132\) 0 0
\(133\) 113.337 0.852156
\(134\) 91.2860i 0.681239i
\(135\) 0 0
\(136\) 69.3146 0.509666
\(137\) 134.857i 0.984358i 0.870494 + 0.492179i \(0.163799\pi\)
−0.870494 + 0.492179i \(0.836201\pi\)
\(138\) 0 0
\(139\) 160.927 1.15775 0.578874 0.815417i \(-0.303492\pi\)
0.578874 + 0.815417i \(0.303492\pi\)
\(140\) 58.4006i 0.417147i
\(141\) 0 0
\(142\) −146.342 −1.03058
\(143\) − 147.180i − 1.02923i
\(144\) 0 0
\(145\) −5.96256 −0.0411211
\(146\) 232.249i 1.59075i
\(147\) 0 0
\(148\) 90.9021 0.614204
\(149\) 33.7999i 0.226845i 0.993547 + 0.113423i \(0.0361814\pi\)
−0.993547 + 0.113423i \(0.963819\pi\)
\(150\) 0 0
\(151\) −227.193 −1.50459 −0.752294 0.658827i \(-0.771053\pi\)
−0.752294 + 0.658827i \(0.771053\pi\)
\(152\) 74.3794i 0.489338i
\(153\) 0 0
\(154\) 139.070 0.903054
\(155\) − 191.258i − 1.23392i
\(156\) 0 0
\(157\) −276.321 −1.76001 −0.880004 0.474966i \(-0.842460\pi\)
−0.880004 + 0.474966i \(0.842460\pi\)
\(158\) 358.421i 2.26849i
\(159\) 0 0
\(160\) −108.255 −0.676594
\(161\) 216.170i 1.34267i
\(162\) 0 0
\(163\) 96.7045 0.593279 0.296640 0.954990i \(-0.404134\pi\)
0.296640 + 0.954990i \(0.404134\pi\)
\(164\) 98.3429i 0.599652i
\(165\) 0 0
\(166\) −242.999 −1.46385
\(167\) − 285.957i − 1.71232i −0.516714 0.856158i \(-0.672845\pi\)
0.516714 0.856158i \(-0.327155\pi\)
\(168\) 0 0
\(169\) 252.583 1.49457
\(170\) 126.880i 0.746352i
\(171\) 0 0
\(172\) −70.1854 −0.408055
\(173\) − 51.8652i − 0.299799i −0.988701 0.149900i \(-0.952105\pi\)
0.988701 0.149900i \(-0.0478950\pi\)
\(174\) 0 0
\(175\) −71.6138 −0.409222
\(176\) 143.098i 0.813059i
\(177\) 0 0
\(178\) 297.058 1.66887
\(179\) 135.861i 0.758998i 0.925192 + 0.379499i \(0.123904\pi\)
−0.925192 + 0.379499i \(0.876096\pi\)
\(180\) 0 0
\(181\) 206.452 1.14062 0.570309 0.821430i \(-0.306824\pi\)
0.570309 + 0.821430i \(0.306824\pi\)
\(182\) 398.352i 2.18875i
\(183\) 0 0
\(184\) −141.866 −0.771011
\(185\) − 201.803i − 1.09083i
\(186\) 0 0
\(187\) 94.0431 0.502904
\(188\) − 106.820i − 0.568191i
\(189\) 0 0
\(190\) −136.151 −0.716584
\(191\) 1.82849i 0.00957327i 0.999989 + 0.00478663i \(0.00152364\pi\)
−0.999989 + 0.00478663i \(0.998476\pi\)
\(192\) 0 0
\(193\) −155.090 −0.803576 −0.401788 0.915733i \(-0.631611\pi\)
−0.401788 + 0.915733i \(0.631611\pi\)
\(194\) 168.076i 0.866369i
\(195\) 0 0
\(196\) −28.5811 −0.145822
\(197\) − 210.777i − 1.06993i −0.844874 0.534966i \(-0.820324\pi\)
0.844874 0.534966i \(-0.179676\pi\)
\(198\) 0 0
\(199\) 303.056 1.52290 0.761448 0.648226i \(-0.224489\pi\)
0.761448 + 0.648226i \(0.224489\pi\)
\(200\) − 46.9979i − 0.234990i
\(201\) 0 0
\(202\) 167.667 0.830035
\(203\) − 11.9615i − 0.0589234i
\(204\) 0 0
\(205\) 218.321 1.06498
\(206\) − 113.862i − 0.552730i
\(207\) 0 0
\(208\) −409.890 −1.97063
\(209\) 100.915i 0.482847i
\(210\) 0 0
\(211\) 79.4793 0.376679 0.188340 0.982104i \(-0.439689\pi\)
0.188340 + 0.982104i \(0.439689\pi\)
\(212\) − 91.8687i − 0.433343i
\(213\) 0 0
\(214\) 109.507 0.511716
\(215\) 155.812i 0.724705i
\(216\) 0 0
\(217\) 383.682 1.76812
\(218\) 88.4148i 0.405573i
\(219\) 0 0
\(220\) −51.9998 −0.236363
\(221\) 269.376i 1.21890i
\(222\) 0 0
\(223\) 37.1959 0.166798 0.0833988 0.996516i \(-0.473422\pi\)
0.0833988 + 0.996516i \(0.473422\pi\)
\(224\) − 217.170i − 0.969508i
\(225\) 0 0
\(226\) −205.428 −0.908973
\(227\) − 67.1448i − 0.295792i −0.989003 0.147896i \(-0.952750\pi\)
0.989003 0.147896i \(-0.0472501\pi\)
\(228\) 0 0
\(229\) 20.3389 0.0888162 0.0444081 0.999013i \(-0.485860\pi\)
0.0444081 + 0.999013i \(0.485860\pi\)
\(230\) − 259.685i − 1.12906i
\(231\) 0 0
\(232\) 7.84993 0.0338359
\(233\) − 82.5152i − 0.354142i −0.984198 0.177071i \(-0.943338\pi\)
0.984198 0.177071i \(-0.0566623\pi\)
\(234\) 0 0
\(235\) −237.140 −1.00911
\(236\) 66.5097i 0.281821i
\(237\) 0 0
\(238\) −254.533 −1.06947
\(239\) − 142.959i − 0.598154i −0.954229 0.299077i \(-0.903321\pi\)
0.954229 0.299077i \(-0.0966788\pi\)
\(240\) 0 0
\(241\) 133.180 0.552616 0.276308 0.961069i \(-0.410889\pi\)
0.276308 + 0.961069i \(0.410889\pi\)
\(242\) − 167.771i − 0.693270i
\(243\) 0 0
\(244\) 200.664 0.822392
\(245\) 63.4501i 0.258980i
\(246\) 0 0
\(247\) −289.060 −1.17028
\(248\) 251.799i 1.01532i
\(249\) 0 0
\(250\) 327.806 1.31122
\(251\) − 112.331i − 0.447534i −0.974643 0.223767i \(-0.928165\pi\)
0.974643 0.223767i \(-0.0718354\pi\)
\(252\) 0 0
\(253\) −192.478 −0.760782
\(254\) 27.1583i 0.106923i
\(255\) 0 0
\(256\) 286.868 1.12058
\(257\) − 433.008i − 1.68485i −0.538810 0.842427i \(-0.681126\pi\)
0.538810 0.842427i \(-0.318874\pi\)
\(258\) 0 0
\(259\) 404.835 1.56307
\(260\) − 148.948i − 0.572877i
\(261\) 0 0
\(262\) −384.562 −1.46779
\(263\) 172.140i 0.654526i 0.944933 + 0.327263i \(0.106126\pi\)
−0.944933 + 0.327263i \(0.893874\pi\)
\(264\) 0 0
\(265\) −203.948 −0.769617
\(266\) − 273.132i − 1.02681i
\(267\) 0 0
\(268\) 68.4736 0.255498
\(269\) − 103.072i − 0.383167i −0.981476 0.191584i \(-0.938638\pi\)
0.981476 0.191584i \(-0.0613623\pi\)
\(270\) 0 0
\(271\) 197.651 0.729340 0.364670 0.931137i \(-0.381182\pi\)
0.364670 + 0.931137i \(0.381182\pi\)
\(272\) − 261.905i − 0.962887i
\(273\) 0 0
\(274\) 324.994 1.18611
\(275\) − 63.7648i − 0.231872i
\(276\) 0 0
\(277\) 101.105 0.365000 0.182500 0.983206i \(-0.441581\pi\)
0.182500 + 0.983206i \(0.441581\pi\)
\(278\) − 387.820i − 1.39503i
\(279\) 0 0
\(280\) −170.688 −0.609601
\(281\) 380.094i 1.35265i 0.736605 + 0.676323i \(0.236428\pi\)
−0.736605 + 0.676323i \(0.763572\pi\)
\(282\) 0 0
\(283\) 332.711 1.17566 0.587829 0.808986i \(-0.299983\pi\)
0.587829 + 0.808986i \(0.299983\pi\)
\(284\) 109.771i 0.386518i
\(285\) 0 0
\(286\) −354.692 −1.24018
\(287\) 437.973i 1.52604i
\(288\) 0 0
\(289\) 116.878 0.404422
\(290\) 14.3692i 0.0495491i
\(291\) 0 0
\(292\) 174.210 0.596609
\(293\) − 551.613i − 1.88264i −0.337517 0.941319i \(-0.609587\pi\)
0.337517 0.941319i \(-0.390413\pi\)
\(294\) 0 0
\(295\) 147.651 0.500513
\(296\) 265.681i 0.897570i
\(297\) 0 0
\(298\) 81.4548 0.273338
\(299\) − 551.332i − 1.84392i
\(300\) 0 0
\(301\) −312.573 −1.03845
\(302\) 547.515i 1.81296i
\(303\) 0 0
\(304\) 281.043 0.924483
\(305\) − 445.473i − 1.46057i
\(306\) 0 0
\(307\) −497.277 −1.61980 −0.809898 0.586571i \(-0.800478\pi\)
−0.809898 + 0.586571i \(0.800478\pi\)
\(308\) − 104.317i − 0.338690i
\(309\) 0 0
\(310\) −460.916 −1.48682
\(311\) 30.6422i 0.0985280i 0.998786 + 0.0492640i \(0.0156876\pi\)
−0.998786 + 0.0492640i \(0.984312\pi\)
\(312\) 0 0
\(313\) −327.233 −1.04547 −0.522737 0.852494i \(-0.675089\pi\)
−0.522737 + 0.852494i \(0.675089\pi\)
\(314\) 665.910i 2.12073i
\(315\) 0 0
\(316\) 268.851 0.850795
\(317\) − 457.919i − 1.44454i −0.691611 0.722270i \(-0.743099\pi\)
0.691611 0.722270i \(-0.256901\pi\)
\(318\) 0 0
\(319\) 10.6505 0.0333870
\(320\) − 59.5641i − 0.186138i
\(321\) 0 0
\(322\) 520.952 1.61786
\(323\) − 184.699i − 0.571824i
\(324\) 0 0
\(325\) 182.648 0.561992
\(326\) − 233.049i − 0.714875i
\(327\) 0 0
\(328\) −287.428 −0.876305
\(329\) − 475.726i − 1.44597i
\(330\) 0 0
\(331\) 391.290 1.18214 0.591072 0.806619i \(-0.298705\pi\)
0.591072 + 0.806619i \(0.298705\pi\)
\(332\) 182.274i 0.549017i
\(333\) 0 0
\(334\) −689.130 −2.06326
\(335\) − 152.011i − 0.453765i
\(336\) 0 0
\(337\) −418.213 −1.24099 −0.620494 0.784212i \(-0.713068\pi\)
−0.620494 + 0.784212i \(0.713068\pi\)
\(338\) − 608.702i − 1.80089i
\(339\) 0 0
\(340\) 95.1725 0.279919
\(341\) 341.630i 1.00185i
\(342\) 0 0
\(343\) 267.189 0.778977
\(344\) − 205.132i − 0.596313i
\(345\) 0 0
\(346\) −124.991 −0.361245
\(347\) 436.901i 1.25908i 0.776967 + 0.629541i \(0.216757\pi\)
−0.776967 + 0.629541i \(0.783243\pi\)
\(348\) 0 0
\(349\) −297.279 −0.851802 −0.425901 0.904770i \(-0.640043\pi\)
−0.425901 + 0.904770i \(0.640043\pi\)
\(350\) 172.583i 0.493094i
\(351\) 0 0
\(352\) 193.368 0.549340
\(353\) 380.600i 1.07819i 0.842246 + 0.539094i \(0.181233\pi\)
−0.842246 + 0.539094i \(0.818767\pi\)
\(354\) 0 0
\(355\) 243.692 0.686455
\(356\) − 222.823i − 0.625908i
\(357\) 0 0
\(358\) 327.412 0.914559
\(359\) − 72.2876i − 0.201358i −0.994919 0.100679i \(-0.967898\pi\)
0.994919 0.100679i \(-0.0321015\pi\)
\(360\) 0 0
\(361\) −162.805 −0.450983
\(362\) − 497.531i − 1.37439i
\(363\) 0 0
\(364\) 298.804 0.820889
\(365\) − 386.746i − 1.05958i
\(366\) 0 0
\(367\) −567.460 −1.54621 −0.773106 0.634277i \(-0.781298\pi\)
−0.773106 + 0.634277i \(0.781298\pi\)
\(368\) 536.041i 1.45663i
\(369\) 0 0
\(370\) −486.327 −1.31440
\(371\) − 409.140i − 1.10280i
\(372\) 0 0
\(373\) 463.835 1.24353 0.621763 0.783206i \(-0.286417\pi\)
0.621763 + 0.783206i \(0.286417\pi\)
\(374\) − 226.636i − 0.605978i
\(375\) 0 0
\(376\) 312.204 0.830330
\(377\) 30.5071i 0.0809207i
\(378\) 0 0
\(379\) −427.077 −1.12685 −0.563427 0.826166i \(-0.690517\pi\)
−0.563427 + 0.826166i \(0.690517\pi\)
\(380\) 102.127i 0.268755i
\(381\) 0 0
\(382\) 4.40651 0.0115354
\(383\) − 410.113i − 1.07079i −0.844602 0.535395i \(-0.820163\pi\)
0.844602 0.535395i \(-0.179837\pi\)
\(384\) 0 0
\(385\) −231.583 −0.601514
\(386\) 373.753i 0.968273i
\(387\) 0 0
\(388\) 126.073 0.324931
\(389\) 176.669i 0.454162i 0.973876 + 0.227081i \(0.0729182\pi\)
−0.973876 + 0.227081i \(0.927082\pi\)
\(390\) 0 0
\(391\) 352.282 0.900977
\(392\) − 83.5344i − 0.213098i
\(393\) 0 0
\(394\) −507.953 −1.28922
\(395\) − 596.850i − 1.51101i
\(396\) 0 0
\(397\) 599.290 1.50955 0.754773 0.655986i \(-0.227747\pi\)
0.754773 + 0.655986i \(0.227747\pi\)
\(398\) − 730.339i − 1.83502i
\(399\) 0 0
\(400\) −177.582 −0.443955
\(401\) 677.386i 1.68924i 0.535366 + 0.844620i \(0.320174\pi\)
−0.535366 + 0.844620i \(0.679826\pi\)
\(402\) 0 0
\(403\) −978.563 −2.42820
\(404\) − 125.767i − 0.311305i
\(405\) 0 0
\(406\) −28.8261 −0.0710001
\(407\) 360.465i 0.885663i
\(408\) 0 0
\(409\) −187.238 −0.457796 −0.228898 0.973450i \(-0.573512\pi\)
−0.228898 + 0.973450i \(0.573512\pi\)
\(410\) − 526.135i − 1.28326i
\(411\) 0 0
\(412\) −85.4082 −0.207301
\(413\) 296.203i 0.717198i
\(414\) 0 0
\(415\) 404.647 0.975054
\(416\) 553.881i 1.33144i
\(417\) 0 0
\(418\) 243.196 0.581809
\(419\) − 111.510i − 0.266135i −0.991107 0.133067i \(-0.957517\pi\)
0.991107 0.133067i \(-0.0424827\pi\)
\(420\) 0 0
\(421\) 3.66404 0.00870319 0.00435159 0.999991i \(-0.498615\pi\)
0.00435159 + 0.999991i \(0.498615\pi\)
\(422\) − 191.538i − 0.453882i
\(423\) 0 0
\(424\) 268.506 0.633268
\(425\) 116.705i 0.274601i
\(426\) 0 0
\(427\) 893.661 2.09288
\(428\) − 82.1413i − 0.191919i
\(429\) 0 0
\(430\) 375.492 0.873237
\(431\) 256.037i 0.594054i 0.954869 + 0.297027i \(0.0959951\pi\)
−0.954869 + 0.297027i \(0.904005\pi\)
\(432\) 0 0
\(433\) 719.675 1.66207 0.831034 0.556222i \(-0.187750\pi\)
0.831034 + 0.556222i \(0.187750\pi\)
\(434\) − 924.640i − 2.13051i
\(435\) 0 0
\(436\) 66.3199 0.152110
\(437\) 378.023i 0.865042i
\(438\) 0 0
\(439\) −341.817 −0.778625 −0.389313 0.921106i \(-0.627287\pi\)
−0.389313 + 0.921106i \(0.627287\pi\)
\(440\) − 151.981i − 0.345410i
\(441\) 0 0
\(442\) 649.173 1.46872
\(443\) 534.762i 1.20714i 0.797311 + 0.603569i \(0.206255\pi\)
−0.797311 + 0.603569i \(0.793745\pi\)
\(444\) 0 0
\(445\) −494.668 −1.11161
\(446\) − 89.6388i − 0.200984i
\(447\) 0 0
\(448\) 119.491 0.266721
\(449\) 747.959i 1.66583i 0.553398 + 0.832917i \(0.313331\pi\)
−0.553398 + 0.832917i \(0.686669\pi\)
\(450\) 0 0
\(451\) −389.970 −0.864679
\(452\) 154.091i 0.340910i
\(453\) 0 0
\(454\) −161.813 −0.356416
\(455\) − 663.344i − 1.45790i
\(456\) 0 0
\(457\) 588.629 1.28803 0.644015 0.765013i \(-0.277268\pi\)
0.644015 + 0.765013i \(0.277268\pi\)
\(458\) − 49.0150i − 0.107020i
\(459\) 0 0
\(460\) −194.789 −0.423455
\(461\) − 521.154i − 1.13049i −0.824924 0.565243i \(-0.808782\pi\)
0.824924 0.565243i \(-0.191218\pi\)
\(462\) 0 0
\(463\) −752.738 −1.62578 −0.812892 0.582414i \(-0.802108\pi\)
−0.812892 + 0.582414i \(0.802108\pi\)
\(464\) − 29.6610i − 0.0639246i
\(465\) 0 0
\(466\) −198.854 −0.426726
\(467\) − 867.632i − 1.85789i −0.370224 0.928943i \(-0.620719\pi\)
0.370224 0.928943i \(-0.379281\pi\)
\(468\) 0 0
\(469\) 304.949 0.650211
\(470\) 571.487i 1.21593i
\(471\) 0 0
\(472\) −194.389 −0.411840
\(473\) − 278.314i − 0.588402i
\(474\) 0 0
\(475\) −125.233 −0.263649
\(476\) 190.925i 0.401103i
\(477\) 0 0
\(478\) −344.518 −0.720750
\(479\) − 672.423i − 1.40381i −0.712272 0.701903i \(-0.752334\pi\)
0.712272 0.701903i \(-0.247666\pi\)
\(480\) 0 0
\(481\) −1032.51 −2.14660
\(482\) − 320.953i − 0.665877i
\(483\) 0 0
\(484\) −125.845 −0.260011
\(485\) − 279.883i − 0.577078i
\(486\) 0 0
\(487\) −570.328 −1.17111 −0.585553 0.810634i \(-0.699123\pi\)
−0.585553 + 0.810634i \(0.699123\pi\)
\(488\) 586.482i 1.20181i
\(489\) 0 0
\(490\) 152.909 0.312059
\(491\) 84.6117i 0.172325i 0.996281 + 0.0861626i \(0.0274605\pi\)
−0.996281 + 0.0861626i \(0.972540\pi\)
\(492\) 0 0
\(493\) −19.4930 −0.0395395
\(494\) 696.609i 1.41014i
\(495\) 0 0
\(496\) 951.422 1.91819
\(497\) 488.868i 0.983639i
\(498\) 0 0
\(499\) 515.637 1.03334 0.516670 0.856185i \(-0.327171\pi\)
0.516670 + 0.856185i \(0.327171\pi\)
\(500\) − 245.887i − 0.491774i
\(501\) 0 0
\(502\) −270.708 −0.539258
\(503\) 804.967i 1.60033i 0.599779 + 0.800166i \(0.295255\pi\)
−0.599779 + 0.800166i \(0.704745\pi\)
\(504\) 0 0
\(505\) −279.203 −0.552877
\(506\) 463.855i 0.916709i
\(507\) 0 0
\(508\) 20.3715 0.0401013
\(509\) − 213.386i − 0.419225i −0.977785 0.209612i \(-0.932780\pi\)
0.977785 0.209612i \(-0.0672202\pi\)
\(510\) 0 0
\(511\) 775.848 1.51829
\(512\) − 116.636i − 0.227804i
\(513\) 0 0
\(514\) −1043.51 −2.03018
\(515\) 189.606i 0.368167i
\(516\) 0 0
\(517\) 423.585 0.819314
\(518\) − 975.617i − 1.88343i
\(519\) 0 0
\(520\) 435.332 0.837177
\(521\) − 643.878i − 1.23585i −0.786237 0.617925i \(-0.787973\pi\)
0.786237 0.617925i \(-0.212027\pi\)
\(522\) 0 0
\(523\) −94.4710 −0.180633 −0.0903164 0.995913i \(-0.528788\pi\)
−0.0903164 + 0.995913i \(0.528788\pi\)
\(524\) 288.460i 0.550496i
\(525\) 0 0
\(526\) 414.843 0.788675
\(527\) − 625.267i − 1.18647i
\(528\) 0 0
\(529\) −192.015 −0.362977
\(530\) 491.498i 0.927354i
\(531\) 0 0
\(532\) −204.876 −0.385105
\(533\) − 1117.03i − 2.09574i
\(534\) 0 0
\(535\) −182.354 −0.340848
\(536\) 200.129i 0.373374i
\(537\) 0 0
\(538\) −248.394 −0.461699
\(539\) − 113.336i − 0.210271i
\(540\) 0 0
\(541\) 52.3339 0.0967355 0.0483678 0.998830i \(-0.484598\pi\)
0.0483678 + 0.998830i \(0.484598\pi\)
\(542\) − 476.322i − 0.878823i
\(543\) 0 0
\(544\) −353.910 −0.650571
\(545\) − 147.230i − 0.270147i
\(546\) 0 0
\(547\) −726.667 −1.32846 −0.664230 0.747528i \(-0.731240\pi\)
−0.664230 + 0.747528i \(0.731240\pi\)
\(548\) − 243.778i − 0.444850i
\(549\) 0 0
\(550\) −153.668 −0.279396
\(551\) − 20.9173i − 0.0379625i
\(552\) 0 0
\(553\) 1197.34 2.16517
\(554\) − 243.654i − 0.439808i
\(555\) 0 0
\(556\) −290.903 −0.523207
\(557\) − 363.285i − 0.652217i −0.945332 0.326108i \(-0.894263\pi\)
0.945332 0.326108i \(-0.105737\pi\)
\(558\) 0 0
\(559\) 797.201 1.42612
\(560\) 644.946i 1.15169i
\(561\) 0 0
\(562\) 915.992 1.62988
\(563\) − 719.685i − 1.27830i −0.769080 0.639152i \(-0.779285\pi\)
0.769080 0.639152i \(-0.220715\pi\)
\(564\) 0 0
\(565\) 342.083 0.605456
\(566\) − 801.804i − 1.41661i
\(567\) 0 0
\(568\) −320.829 −0.564840
\(569\) − 432.080i − 0.759368i −0.925116 0.379684i \(-0.876033\pi\)
0.925116 0.379684i \(-0.123967\pi\)
\(570\) 0 0
\(571\) 521.210 0.912802 0.456401 0.889774i \(-0.349138\pi\)
0.456401 + 0.889774i \(0.349138\pi\)
\(572\) 266.054i 0.465130i
\(573\) 0 0
\(574\) 1055.48 1.83881
\(575\) − 238.861i − 0.415410i
\(576\) 0 0
\(577\) −1027.18 −1.78021 −0.890103 0.455760i \(-0.849368\pi\)
−0.890103 + 0.455760i \(0.849368\pi\)
\(578\) − 281.666i − 0.487311i
\(579\) 0 0
\(580\) 10.7784 0.0185834
\(581\) 811.761i 1.39718i
\(582\) 0 0
\(583\) 364.297 0.624867
\(584\) 509.165i 0.871858i
\(585\) 0 0
\(586\) −1329.34 −2.26850
\(587\) 106.115i 0.180775i 0.995907 + 0.0903876i \(0.0288106\pi\)
−0.995907 + 0.0903876i \(0.971189\pi\)
\(588\) 0 0
\(589\) 670.956 1.13914
\(590\) − 355.827i − 0.603096i
\(591\) 0 0
\(592\) 1003.88 1.69574
\(593\) − 705.404i − 1.18955i −0.803892 0.594776i \(-0.797241\pi\)
0.803892 0.594776i \(-0.202759\pi\)
\(594\) 0 0
\(595\) 423.853 0.712359
\(596\) − 61.0992i − 0.102516i
\(597\) 0 0
\(598\) −1328.66 −2.22184
\(599\) 129.503i 0.216199i 0.994140 + 0.108100i \(0.0344766\pi\)
−0.994140 + 0.108100i \(0.965523\pi\)
\(600\) 0 0
\(601\) 759.823 1.26427 0.632133 0.774860i \(-0.282180\pi\)
0.632133 + 0.774860i \(0.282180\pi\)
\(602\) 753.272i 1.25128i
\(603\) 0 0
\(604\) 410.691 0.679951
\(605\) 279.376i 0.461779i
\(606\) 0 0
\(607\) 862.669 1.42120 0.710601 0.703595i \(-0.248423\pi\)
0.710601 + 0.703595i \(0.248423\pi\)
\(608\) − 379.771i − 0.624623i
\(609\) 0 0
\(610\) −1073.55 −1.75992
\(611\) 1213.31i 1.98579i
\(612\) 0 0
\(613\) 443.227 0.723046 0.361523 0.932363i \(-0.382257\pi\)
0.361523 + 0.932363i \(0.382257\pi\)
\(614\) 1198.39i 1.95178i
\(615\) 0 0
\(616\) 304.887 0.494947
\(617\) 681.056i 1.10382i 0.833904 + 0.551909i \(0.186101\pi\)
−0.833904 + 0.551909i \(0.813899\pi\)
\(618\) 0 0
\(619\) −401.105 −0.647988 −0.323994 0.946059i \(-0.605026\pi\)
−0.323994 + 0.946059i \(0.605026\pi\)
\(620\) 345.733i 0.557633i
\(621\) 0 0
\(622\) 73.8450 0.118722
\(623\) − 992.350i − 1.59286i
\(624\) 0 0
\(625\) −323.481 −0.517569
\(626\) 788.603i 1.25975i
\(627\) 0 0
\(628\) 499.499 0.795380
\(629\) − 659.739i − 1.04887i
\(630\) 0 0
\(631\) −697.284 −1.10505 −0.552523 0.833498i \(-0.686335\pi\)
−0.552523 + 0.833498i \(0.686335\pi\)
\(632\) 785.775i 1.24331i
\(633\) 0 0
\(634\) −1103.55 −1.74061
\(635\) − 45.2246i − 0.0712199i
\(636\) 0 0
\(637\) 324.639 0.509637
\(638\) − 25.6667i − 0.0402299i
\(639\) 0 0
\(640\) −576.564 −0.900882
\(641\) 1045.27i 1.63068i 0.578981 + 0.815341i \(0.303451\pi\)
−0.578981 + 0.815341i \(0.696549\pi\)
\(642\) 0 0
\(643\) 109.777 0.170726 0.0853630 0.996350i \(-0.472795\pi\)
0.0853630 + 0.996350i \(0.472795\pi\)
\(644\) − 390.766i − 0.606779i
\(645\) 0 0
\(646\) −445.109 −0.689023
\(647\) 8.76887i 0.0135531i 0.999977 + 0.00677656i \(0.00215706\pi\)
−0.999977 + 0.00677656i \(0.997843\pi\)
\(648\) 0 0
\(649\) −263.738 −0.406377
\(650\) − 440.164i − 0.677176i
\(651\) 0 0
\(652\) −174.810 −0.268114
\(653\) − 239.411i − 0.366632i −0.983054 0.183316i \(-0.941317\pi\)
0.983054 0.183316i \(-0.0586832\pi\)
\(654\) 0 0
\(655\) 640.381 0.977680
\(656\) 1086.05i 1.65556i
\(657\) 0 0
\(658\) −1146.46 −1.74234
\(659\) 32.6693i 0.0495741i 0.999693 + 0.0247870i \(0.00789077\pi\)
−0.999693 + 0.0247870i \(0.992109\pi\)
\(660\) 0 0
\(661\) 248.759 0.376337 0.188169 0.982137i \(-0.439745\pi\)
0.188169 + 0.982137i \(0.439745\pi\)
\(662\) − 942.973i − 1.42443i
\(663\) 0 0
\(664\) −532.733 −0.802309
\(665\) 454.825i 0.683947i
\(666\) 0 0
\(667\) 39.8962 0.0598144
\(668\) 516.917i 0.773827i
\(669\) 0 0
\(670\) −366.334 −0.546767
\(671\) 795.715i 1.18586i
\(672\) 0 0
\(673\) 38.2884 0.0568921 0.0284461 0.999595i \(-0.490944\pi\)
0.0284461 + 0.999595i \(0.490944\pi\)
\(674\) 1007.86i 1.49533i
\(675\) 0 0
\(676\) −456.587 −0.675424
\(677\) 824.364i 1.21767i 0.793296 + 0.608836i \(0.208363\pi\)
−0.793296 + 0.608836i \(0.791637\pi\)
\(678\) 0 0
\(679\) 561.472 0.826909
\(680\) 278.162i 0.409062i
\(681\) 0 0
\(682\) 823.298 1.20718
\(683\) 1040.88i 1.52398i 0.647590 + 0.761989i \(0.275777\pi\)
−0.647590 + 0.761989i \(0.724223\pi\)
\(684\) 0 0
\(685\) −541.186 −0.790053
\(686\) − 643.902i − 0.938633i
\(687\) 0 0
\(688\) −775.091 −1.12659
\(689\) 1043.49i 1.51450i
\(690\) 0 0
\(691\) 133.361 0.192998 0.0964988 0.995333i \(-0.469236\pi\)
0.0964988 + 0.995333i \(0.469236\pi\)
\(692\) 93.7555i 0.135485i
\(693\) 0 0
\(694\) 1052.89 1.51714
\(695\) 645.805i 0.929216i
\(696\) 0 0
\(697\) 713.742 1.02402
\(698\) 716.416i 1.02638i
\(699\) 0 0
\(700\) 129.454 0.184935
\(701\) 617.809i 0.881325i 0.897673 + 0.440663i \(0.145256\pi\)
−0.897673 + 0.440663i \(0.854744\pi\)
\(702\) 0 0
\(703\) 707.947 1.00704
\(704\) 106.395i 0.151129i
\(705\) 0 0
\(706\) 917.213 1.29917
\(707\) − 560.107i − 0.792231i
\(708\) 0 0
\(709\) 582.110 0.821030 0.410515 0.911854i \(-0.365349\pi\)
0.410515 + 0.911854i \(0.365349\pi\)
\(710\) − 587.275i − 0.827148i
\(711\) 0 0
\(712\) 651.249 0.914675
\(713\) 1279.73i 1.79486i
\(714\) 0 0
\(715\) 590.640 0.826071
\(716\) − 245.592i − 0.343005i
\(717\) 0 0
\(718\) −174.207 −0.242628
\(719\) − 76.0348i − 0.105751i −0.998601 0.0528754i \(-0.983161\pi\)
0.998601 0.0528754i \(-0.0168386\pi\)
\(720\) 0 0
\(721\) −380.368 −0.527556
\(722\) 392.345i 0.543414i
\(723\) 0 0
\(724\) −373.197 −0.515466
\(725\) 13.2170i 0.0182303i
\(726\) 0 0
\(727\) 335.337 0.461261 0.230631 0.973041i \(-0.425921\pi\)
0.230631 + 0.973041i \(0.425921\pi\)
\(728\) 873.317i 1.19961i
\(729\) 0 0
\(730\) −932.023 −1.27674
\(731\) 509.384i 0.696831i
\(732\) 0 0
\(733\) 474.692 0.647602 0.323801 0.946125i \(-0.395039\pi\)
0.323801 + 0.946125i \(0.395039\pi\)
\(734\) 1367.53i 1.86312i
\(735\) 0 0
\(736\) 724.348 0.984168
\(737\) 271.526i 0.368421i
\(738\) 0 0
\(739\) −181.712 −0.245889 −0.122944 0.992414i \(-0.539234\pi\)
−0.122944 + 0.992414i \(0.539234\pi\)
\(740\) 364.794i 0.492964i
\(741\) 0 0
\(742\) −985.991 −1.32883
\(743\) 917.281i 1.23456i 0.786742 + 0.617282i \(0.211766\pi\)
−0.786742 + 0.617282i \(0.788234\pi\)
\(744\) 0 0
\(745\) −135.640 −0.182068
\(746\) − 1117.80i − 1.49839i
\(747\) 0 0
\(748\) −169.999 −0.227272
\(749\) − 365.819i − 0.488409i
\(750\) 0 0
\(751\) −551.998 −0.735017 −0.367508 0.930020i \(-0.619789\pi\)
−0.367508 + 0.930020i \(0.619789\pi\)
\(752\) − 1179.66i − 1.56870i
\(753\) 0 0
\(754\) 73.5194 0.0975059
\(755\) − 911.733i − 1.20759i
\(756\) 0 0
\(757\) 137.805 0.182041 0.0910203 0.995849i \(-0.470987\pi\)
0.0910203 + 0.995849i \(0.470987\pi\)
\(758\) 1029.22i 1.35781i
\(759\) 0 0
\(760\) −298.487 −0.392746
\(761\) − 762.095i − 1.00144i −0.865610 0.500719i \(-0.833069\pi\)
0.865610 0.500719i \(-0.166931\pi\)
\(762\) 0 0
\(763\) 295.358 0.387100
\(764\) − 3.30532i − 0.00432634i
\(765\) 0 0
\(766\) −988.336 −1.29026
\(767\) − 755.450i − 0.984942i
\(768\) 0 0
\(769\) 245.712 0.319522 0.159761 0.987156i \(-0.448928\pi\)
0.159761 + 0.987156i \(0.448928\pi\)
\(770\) 558.094i 0.724797i
\(771\) 0 0
\(772\) 280.352 0.363151
\(773\) − 293.377i − 0.379531i −0.981829 0.189765i \(-0.939227\pi\)
0.981829 0.189765i \(-0.0607727\pi\)
\(774\) 0 0
\(775\) −423.955 −0.547039
\(776\) 368.476i 0.474841i
\(777\) 0 0
\(778\) 425.757 0.547245
\(779\) 765.895i 0.983177i
\(780\) 0 0
\(781\) −435.288 −0.557347
\(782\) − 848.968i − 1.08564i
\(783\) 0 0
\(784\) −315.635 −0.402596
\(785\) − 1108.89i − 1.41260i
\(786\) 0 0
\(787\) −488.046 −0.620135 −0.310068 0.950715i \(-0.600352\pi\)
−0.310068 + 0.950715i \(0.600352\pi\)
\(788\) 381.015i 0.483522i
\(789\) 0 0
\(790\) −1438.36 −1.82070
\(791\) 686.250i 0.867573i
\(792\) 0 0
\(793\) −2279.24 −2.87420
\(794\) − 1444.24i − 1.81894i
\(795\) 0 0
\(796\) −547.827 −0.688225
\(797\) − 430.259i − 0.539849i −0.962882 0.269924i \(-0.913001\pi\)
0.962882 0.269924i \(-0.0869987\pi\)
\(798\) 0 0
\(799\) −775.266 −0.970295
\(800\) 239.965i 0.299956i
\(801\) 0 0
\(802\) 1632.44 2.03546
\(803\) 690.814i 0.860292i
\(804\) 0 0
\(805\) −867.500 −1.07764
\(806\) 2358.25i 2.92587i
\(807\) 0 0
\(808\) 367.581 0.454927
\(809\) 144.409i 0.178503i 0.996009 + 0.0892516i \(0.0284475\pi\)
−0.996009 + 0.0892516i \(0.971552\pi\)
\(810\) 0 0
\(811\) 1008.89 1.24401 0.622005 0.783014i \(-0.286319\pi\)
0.622005 + 0.783014i \(0.286319\pi\)
\(812\) 21.6224i 0.0266286i
\(813\) 0 0
\(814\) 868.688 1.06718
\(815\) 388.079i 0.476170i
\(816\) 0 0
\(817\) −546.605 −0.669039
\(818\) 451.228i 0.551624i
\(819\) 0 0
\(820\) −394.653 −0.481285
\(821\) 719.054i 0.875827i 0.899017 + 0.437914i \(0.144282\pi\)
−0.899017 + 0.437914i \(0.855718\pi\)
\(822\) 0 0
\(823\) 695.297 0.844832 0.422416 0.906402i \(-0.361182\pi\)
0.422416 + 0.906402i \(0.361182\pi\)
\(824\) − 249.624i − 0.302941i
\(825\) 0 0
\(826\) 713.822 0.864191
\(827\) − 1201.83i − 1.45324i −0.687041 0.726618i \(-0.741091\pi\)
0.687041 0.726618i \(-0.258909\pi\)
\(828\) 0 0
\(829\) −129.209 −0.155861 −0.0779307 0.996959i \(-0.524831\pi\)
−0.0779307 + 0.996959i \(0.524831\pi\)
\(830\) − 975.164i − 1.17490i
\(831\) 0 0
\(832\) −304.756 −0.366294
\(833\) 207.433i 0.249019i
\(834\) 0 0
\(835\) 1147.55 1.37432
\(836\) − 182.421i − 0.218207i
\(837\) 0 0
\(838\) −268.730 −0.320681
\(839\) − 598.182i − 0.712970i −0.934301 0.356485i \(-0.883975\pi\)
0.934301 0.356485i \(-0.116025\pi\)
\(840\) 0 0
\(841\) 838.792 0.997375
\(842\) − 8.83002i − 0.0104870i
\(843\) 0 0
\(844\) −143.673 −0.170228
\(845\) 1013.62i 1.19955i
\(846\) 0 0
\(847\) −560.455 −0.661695
\(848\) − 1014.55i − 1.19640i
\(849\) 0 0
\(850\) 281.250 0.330882
\(851\) 1350.29i 1.58671i
\(852\) 0 0
\(853\) −50.4253 −0.0591153 −0.0295576 0.999563i \(-0.509410\pi\)
−0.0295576 + 0.999563i \(0.509410\pi\)
\(854\) − 2153.64i − 2.52183i
\(855\) 0 0
\(856\) 240.075 0.280462
\(857\) − 1015.39i − 1.18482i −0.805637 0.592409i \(-0.798177\pi\)
0.805637 0.592409i \(-0.201823\pi\)
\(858\) 0 0
\(859\) 554.440 0.645448 0.322724 0.946493i \(-0.395401\pi\)
0.322724 + 0.946493i \(0.395401\pi\)
\(860\) − 281.657i − 0.327508i
\(861\) 0 0
\(862\) 617.027 0.715808
\(863\) 249.538i 0.289151i 0.989494 + 0.144576i \(0.0461817\pi\)
−0.989494 + 0.144576i \(0.953818\pi\)
\(864\) 0 0
\(865\) 208.137 0.240621
\(866\) − 1734.35i − 2.00272i
\(867\) 0 0
\(868\) −693.572 −0.799046
\(869\) 1066.11i 1.22682i
\(870\) 0 0
\(871\) −777.758 −0.892948
\(872\) 193.834i 0.222287i
\(873\) 0 0
\(874\) 911.003 1.04234
\(875\) − 1095.07i − 1.25150i
\(876\) 0 0
\(877\) 462.780 0.527685 0.263843 0.964566i \(-0.415010\pi\)
0.263843 + 0.964566i \(0.415010\pi\)
\(878\) 823.748i 0.938209i
\(879\) 0 0
\(880\) −574.259 −0.652567
\(881\) − 650.354i − 0.738200i −0.929390 0.369100i \(-0.879666\pi\)
0.929390 0.369100i \(-0.120334\pi\)
\(882\) 0 0
\(883\) 94.7771 0.107335 0.0536677 0.998559i \(-0.482909\pi\)
0.0536677 + 0.998559i \(0.482909\pi\)
\(884\) − 486.945i − 0.550842i
\(885\) 0 0
\(886\) 1288.73 1.45455
\(887\) − 330.723i − 0.372855i −0.982469 0.186428i \(-0.940309\pi\)
0.982469 0.186428i \(-0.0596910\pi\)
\(888\) 0 0
\(889\) 90.7248 0.102053
\(890\) 1192.11i 1.33944i
\(891\) 0 0
\(892\) −67.2380 −0.0753790
\(893\) − 831.915i − 0.931595i
\(894\) 0 0
\(895\) −545.214 −0.609177
\(896\) − 1156.64i − 1.29090i
\(897\) 0 0
\(898\) 1802.52 2.00726
\(899\) − 70.8120i − 0.0787676i
\(900\) 0 0
\(901\) −666.754 −0.740015
\(902\) 939.794i 1.04190i
\(903\) 0 0
\(904\) −450.365 −0.498191
\(905\) 828.498i 0.915468i
\(906\) 0 0
\(907\) 975.660 1.07570 0.537850 0.843040i \(-0.319237\pi\)
0.537850 + 0.843040i \(0.319237\pi\)
\(908\) 121.376i 0.133674i
\(909\) 0 0
\(910\) −1598.60 −1.75670
\(911\) 861.914i 0.946119i 0.881030 + 0.473059i \(0.156850\pi\)
−0.881030 + 0.473059i \(0.843150\pi\)
\(912\) 0 0
\(913\) −722.791 −0.791666
\(914\) − 1418.54i − 1.55202i
\(915\) 0 0
\(916\) −36.7661 −0.0401377
\(917\) 1284.66i 1.40094i
\(918\) 0 0
\(919\) −788.050 −0.857508 −0.428754 0.903421i \(-0.641047\pi\)
−0.428754 + 0.903421i \(0.641047\pi\)
\(920\) − 569.313i − 0.618819i
\(921\) 0 0
\(922\) −1255.94 −1.36219
\(923\) − 1246.83i − 1.35085i
\(924\) 0 0
\(925\) −447.328 −0.483598
\(926\) 1814.03i 1.95900i
\(927\) 0 0
\(928\) −40.0807 −0.0431904
\(929\) 301.848i 0.324917i 0.986715 + 0.162459i \(0.0519424\pi\)
−0.986715 + 0.162459i \(0.948058\pi\)
\(930\) 0 0
\(931\) −222.590 −0.239087
\(932\) 149.161i 0.160044i
\(933\) 0 0
\(934\) −2090.92 −2.23867
\(935\) 377.398i 0.403635i
\(936\) 0 0
\(937\) 1469.40 1.56819 0.784096 0.620640i \(-0.213127\pi\)
0.784096 + 0.620640i \(0.213127\pi\)
\(938\) − 734.900i − 0.783476i
\(939\) 0 0
\(940\) 428.672 0.456034
\(941\) 887.430i 0.943071i 0.881847 + 0.471536i \(0.156300\pi\)
−0.881847 + 0.471536i \(0.843700\pi\)
\(942\) 0 0
\(943\) −1460.81 −1.54911
\(944\) 734.498i 0.778070i
\(945\) 0 0
\(946\) −670.713 −0.708999
\(947\) 707.982i 0.747605i 0.927508 + 0.373802i \(0.121946\pi\)
−0.927508 + 0.373802i \(0.878054\pi\)
\(948\) 0 0
\(949\) −1978.76 −2.08510
\(950\) 301.801i 0.317685i
\(951\) 0 0
\(952\) −558.019 −0.586154
\(953\) 512.218i 0.537480i 0.963213 + 0.268740i \(0.0866072\pi\)
−0.963213 + 0.268740i \(0.913393\pi\)
\(954\) 0 0
\(955\) −7.33781 −0.00768357
\(956\) 258.423i 0.270317i
\(957\) 0 0
\(958\) −1620.48 −1.69152
\(959\) − 1085.67i − 1.13209i
\(960\) 0 0
\(961\) 1310.41 1.36358
\(962\) 2488.26i 2.58655i
\(963\) 0 0
\(964\) −240.747 −0.249737
\(965\) − 622.382i − 0.644956i
\(966\) 0 0
\(967\) 0.643507 0.000665467 0 0.000332734 1.00000i \(-0.499894\pi\)
0.000332734 1.00000i \(0.499894\pi\)
\(968\) − 367.809i − 0.379968i
\(969\) 0 0
\(970\) −674.493 −0.695354
\(971\) − 706.216i − 0.727308i −0.931534 0.363654i \(-0.881529\pi\)
0.931534 0.363654i \(-0.118471\pi\)
\(972\) 0 0
\(973\) −1295.55 −1.33150
\(974\) 1374.44i 1.41113i
\(975\) 0 0
\(976\) 2216.02 2.27052
\(977\) − 860.514i − 0.880772i −0.897808 0.440386i \(-0.854842\pi\)
0.897808 0.440386i \(-0.145158\pi\)
\(978\) 0 0
\(979\) 883.587 0.902541
\(980\) − 114.697i − 0.117038i
\(981\) 0 0
\(982\) 203.907 0.207644
\(983\) 1605.30i 1.63306i 0.577304 + 0.816529i \(0.304105\pi\)
−0.577304 + 0.816529i \(0.695895\pi\)
\(984\) 0 0
\(985\) 845.854 0.858735
\(986\) 46.9763i 0.0476433i
\(987\) 0 0
\(988\) 522.526 0.528873
\(989\) − 1042.55i − 1.05415i
\(990\) 0 0
\(991\) −542.923 −0.547854 −0.273927 0.961750i \(-0.588323\pi\)
−0.273927 + 0.961750i \(0.588323\pi\)
\(992\) − 1285.65i − 1.29602i
\(993\) 0 0
\(994\) 1178.13 1.18524
\(995\) 1216.18i 1.22229i
\(996\) 0 0
\(997\) −953.401 −0.956269 −0.478135 0.878287i \(-0.658687\pi\)
−0.478135 + 0.878287i \(0.658687\pi\)
\(998\) − 1242.64i − 1.24513i
\(999\) 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.890.20 84
3.2 odd 2 inner 1143.3.b.a.890.65 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.20 84 1.1 even 1 trivial
1143.3.b.a.890.65 yes 84 3.2 odd 2 inner