Properties

Label 1143.3.b.a.890.1
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 890.1
Character \(\chi\) \(=\) 1143.890
Dual form 1143.3.b.a.890.84

$q$-expansion

\(f(q)\) \(=\) \(q-3.88526i q^{2} -11.0952 q^{4} +2.03195i q^{5} -5.71279 q^{7} +27.5669i q^{8} +O(q^{10})\) \(q-3.88526i q^{2} -11.0952 q^{4} +2.03195i q^{5} -5.71279 q^{7} +27.5669i q^{8} +7.89464 q^{10} -7.97333i q^{11} -22.6045 q^{13} +22.1957i q^{14} +62.7235 q^{16} +24.8287i q^{17} +15.9522 q^{19} -22.5449i q^{20} -30.9785 q^{22} -13.4635i q^{23} +20.8712 q^{25} +87.8245i q^{26} +63.3848 q^{28} +22.5281i q^{29} -1.55815 q^{31} -133.430i q^{32} +96.4661 q^{34} -11.6081i q^{35} +3.12185 q^{37} -61.9785i q^{38} -56.0144 q^{40} +4.15270i q^{41} +9.89437 q^{43} +88.4661i q^{44} -52.3091 q^{46} -35.6762i q^{47} -16.3640 q^{49} -81.0900i q^{50} +250.803 q^{52} -98.5702i q^{53} +16.2014 q^{55} -157.484i q^{56} +87.5273 q^{58} +70.4093i q^{59} +37.5028 q^{61} +6.05382i q^{62} -267.515 q^{64} -45.9312i q^{65} +102.001 q^{67} -275.481i q^{68} -45.1004 q^{70} -140.711i q^{71} -37.1066 q^{73} -12.1292i q^{74} -176.994 q^{76} +45.5500i q^{77} -69.2564 q^{79} +127.451i q^{80} +16.1343 q^{82} -82.8101i q^{83} -50.4507 q^{85} -38.4422i q^{86} +219.800 q^{88} +148.742i q^{89} +129.135 q^{91} +149.381i q^{92} -138.611 q^{94} +32.4140i q^{95} +25.7301 q^{97} +63.5784i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 84q - 160q^{4} + O(q^{10}) \) \( 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 3.88526i − 1.94263i −0.237796 0.971315i \(-0.576425\pi\)
0.237796 0.971315i \(-0.423575\pi\)
\(3\) 0 0
\(4\) −11.0952 −2.77381
\(5\) 2.03195i 0.406389i 0.979138 + 0.203195i \(0.0651324\pi\)
−0.979138 + 0.203195i \(0.934868\pi\)
\(6\) 0 0
\(7\) −5.71279 −0.816113 −0.408057 0.912957i \(-0.633793\pi\)
−0.408057 + 0.912957i \(0.633793\pi\)
\(8\) 27.5669i 3.44586i
\(9\) 0 0
\(10\) 7.89464 0.789464
\(11\) − 7.97333i − 0.724848i −0.932013 0.362424i \(-0.881949\pi\)
0.932013 0.362424i \(-0.118051\pi\)
\(12\) 0 0
\(13\) −22.6045 −1.73881 −0.869405 0.494100i \(-0.835498\pi\)
−0.869405 + 0.494100i \(0.835498\pi\)
\(14\) 22.1957i 1.58541i
\(15\) 0 0
\(16\) 62.7235 3.92022
\(17\) 24.8287i 1.46051i 0.683172 + 0.730257i \(0.260600\pi\)
−0.683172 + 0.730257i \(0.739400\pi\)
\(18\) 0 0
\(19\) 15.9522 0.839590 0.419795 0.907619i \(-0.362102\pi\)
0.419795 + 0.907619i \(0.362102\pi\)
\(20\) − 22.5449i − 1.12725i
\(21\) 0 0
\(22\) −30.9785 −1.40811
\(23\) − 13.4635i − 0.585369i −0.956209 0.292684i \(-0.905452\pi\)
0.956209 0.292684i \(-0.0945485\pi\)
\(24\) 0 0
\(25\) 20.8712 0.834848
\(26\) 87.8245i 3.37787i
\(27\) 0 0
\(28\) 63.3848 2.26374
\(29\) 22.5281i 0.776829i 0.921485 + 0.388415i \(0.126977\pi\)
−0.921485 + 0.388415i \(0.873023\pi\)
\(30\) 0 0
\(31\) −1.55815 −0.0502629 −0.0251315 0.999684i \(-0.508000\pi\)
−0.0251315 + 0.999684i \(0.508000\pi\)
\(32\) − 133.430i − 4.16968i
\(33\) 0 0
\(34\) 96.4661 2.83724
\(35\) − 11.6081i − 0.331660i
\(36\) 0 0
\(37\) 3.12185 0.0843744 0.0421872 0.999110i \(-0.486567\pi\)
0.0421872 + 0.999110i \(0.486567\pi\)
\(38\) − 61.9785i − 1.63101i
\(39\) 0 0
\(40\) −56.0144 −1.40036
\(41\) 4.15270i 0.101285i 0.998717 + 0.0506427i \(0.0161270\pi\)
−0.998717 + 0.0506427i \(0.983873\pi\)
\(42\) 0 0
\(43\) 9.89437 0.230102 0.115051 0.993360i \(-0.463297\pi\)
0.115051 + 0.993360i \(0.463297\pi\)
\(44\) 88.4661i 2.01059i
\(45\) 0 0
\(46\) −52.3091 −1.13715
\(47\) − 35.6762i − 0.759067i −0.925178 0.379534i \(-0.876084\pi\)
0.925178 0.379534i \(-0.123916\pi\)
\(48\) 0 0
\(49\) −16.3640 −0.333959
\(50\) − 81.0900i − 1.62180i
\(51\) 0 0
\(52\) 250.803 4.82313
\(53\) − 98.5702i − 1.85982i −0.367793 0.929908i \(-0.619886\pi\)
0.367793 0.929908i \(-0.380114\pi\)
\(54\) 0 0
\(55\) 16.2014 0.294571
\(56\) − 157.484i − 2.81221i
\(57\) 0 0
\(58\) 87.5273 1.50909
\(59\) 70.4093i 1.19338i 0.802472 + 0.596689i \(0.203517\pi\)
−0.802472 + 0.596689i \(0.796483\pi\)
\(60\) 0 0
\(61\) 37.5028 0.614801 0.307400 0.951580i \(-0.400541\pi\)
0.307400 + 0.951580i \(0.400541\pi\)
\(62\) 6.05382i 0.0976423i
\(63\) 0 0
\(64\) −267.515 −4.17992
\(65\) − 45.9312i − 0.706634i
\(66\) 0 0
\(67\) 102.001 1.52240 0.761198 0.648519i \(-0.224611\pi\)
0.761198 + 0.648519i \(0.224611\pi\)
\(68\) − 275.481i − 4.05119i
\(69\) 0 0
\(70\) −45.1004 −0.644292
\(71\) − 140.711i − 1.98185i −0.134410 0.990926i \(-0.542914\pi\)
0.134410 0.990926i \(-0.457086\pi\)
\(72\) 0 0
\(73\) −37.1066 −0.508310 −0.254155 0.967163i \(-0.581797\pi\)
−0.254155 + 0.967163i \(0.581797\pi\)
\(74\) − 12.1292i − 0.163908i
\(75\) 0 0
\(76\) −176.994 −2.32886
\(77\) 45.5500i 0.591558i
\(78\) 0 0
\(79\) −69.2564 −0.876663 −0.438331 0.898813i \(-0.644430\pi\)
−0.438331 + 0.898813i \(0.644430\pi\)
\(80\) 127.451i 1.59313i
\(81\) 0 0
\(82\) 16.1343 0.196760
\(83\) − 82.8101i − 0.997712i −0.866685 0.498856i \(-0.833754\pi\)
0.866685 0.498856i \(-0.166246\pi\)
\(84\) 0 0
\(85\) −50.4507 −0.593537
\(86\) − 38.4422i − 0.447002i
\(87\) 0 0
\(88\) 219.800 2.49773
\(89\) 148.742i 1.67125i 0.549298 + 0.835627i \(0.314895\pi\)
−0.549298 + 0.835627i \(0.685105\pi\)
\(90\) 0 0
\(91\) 129.135 1.41907
\(92\) 149.381i 1.62370i
\(93\) 0 0
\(94\) −138.611 −1.47459
\(95\) 32.4140i 0.341200i
\(96\) 0 0
\(97\) 25.7301 0.265258 0.132629 0.991166i \(-0.457658\pi\)
0.132629 + 0.991166i \(0.457658\pi\)
\(98\) 63.5784i 0.648760i
\(99\) 0 0
\(100\) −231.571 −2.31571
\(101\) 80.0970i 0.793040i 0.918026 + 0.396520i \(0.129782\pi\)
−0.918026 + 0.396520i \(0.870218\pi\)
\(102\) 0 0
\(103\) 89.2829 0.866824 0.433412 0.901196i \(-0.357309\pi\)
0.433412 + 0.901196i \(0.357309\pi\)
\(104\) − 623.137i − 5.99170i
\(105\) 0 0
\(106\) −382.971 −3.61293
\(107\) − 2.37443i − 0.0221910i −0.999938 0.0110955i \(-0.996468\pi\)
0.999938 0.0110955i \(-0.00353188\pi\)
\(108\) 0 0
\(109\) 2.17902 0.0199910 0.00999551 0.999950i \(-0.496818\pi\)
0.00999551 + 0.999950i \(0.496818\pi\)
\(110\) − 62.9466i − 0.572242i
\(111\) 0 0
\(112\) −358.326 −3.19934
\(113\) − 116.498i − 1.03096i −0.856902 0.515480i \(-0.827614\pi\)
0.856902 0.515480i \(-0.172386\pi\)
\(114\) 0 0
\(115\) 27.3571 0.237887
\(116\) − 249.954i − 2.15478i
\(117\) 0 0
\(118\) 273.559 2.31829
\(119\) − 141.841i − 1.19195i
\(120\) 0 0
\(121\) 57.4260 0.474595
\(122\) − 145.708i − 1.19433i
\(123\) 0 0
\(124\) 17.2881 0.139420
\(125\) 93.2078i 0.745662i
\(126\) 0 0
\(127\) −11.2694 −0.0887357
\(128\) 505.646i 3.95036i
\(129\) 0 0
\(130\) −178.455 −1.37273
\(131\) − 78.9935i − 0.603004i −0.953466 0.301502i \(-0.902512\pi\)
0.953466 0.301502i \(-0.0974879\pi\)
\(132\) 0 0
\(133\) −91.1316 −0.685200
\(134\) − 396.299i − 2.95745i
\(135\) 0 0
\(136\) −684.451 −5.03273
\(137\) 87.1459i 0.636102i 0.948074 + 0.318051i \(0.103028\pi\)
−0.948074 + 0.318051i \(0.896972\pi\)
\(138\) 0 0
\(139\) 232.830 1.67504 0.837519 0.546408i \(-0.184005\pi\)
0.837519 + 0.546408i \(0.184005\pi\)
\(140\) 128.795i 0.919961i
\(141\) 0 0
\(142\) −546.701 −3.85000
\(143\) 180.233i 1.26037i
\(144\) 0 0
\(145\) −45.7758 −0.315695
\(146\) 144.169i 0.987459i
\(147\) 0 0
\(148\) −34.6377 −0.234039
\(149\) 285.179i 1.91395i 0.290164 + 0.956977i \(0.406290\pi\)
−0.290164 + 0.956977i \(0.593710\pi\)
\(150\) 0 0
\(151\) 123.029 0.814759 0.407380 0.913259i \(-0.366443\pi\)
0.407380 + 0.913259i \(0.366443\pi\)
\(152\) 439.753i 2.89311i
\(153\) 0 0
\(154\) 176.974 1.14918
\(155\) − 3.16608i − 0.0204263i
\(156\) 0 0
\(157\) 209.967 1.33737 0.668685 0.743546i \(-0.266858\pi\)
0.668685 + 0.743546i \(0.266858\pi\)
\(158\) 269.079i 1.70303i
\(159\) 0 0
\(160\) 271.122 1.69451
\(161\) 76.9140i 0.477727i
\(162\) 0 0
\(163\) 222.316 1.36390 0.681952 0.731397i \(-0.261131\pi\)
0.681952 + 0.731397i \(0.261131\pi\)
\(164\) − 46.0753i − 0.280947i
\(165\) 0 0
\(166\) −321.739 −1.93819
\(167\) 172.780i 1.03461i 0.855801 + 0.517306i \(0.173065\pi\)
−0.855801 + 0.517306i \(0.826935\pi\)
\(168\) 0 0
\(169\) 341.965 2.02346
\(170\) 196.014i 1.15302i
\(171\) 0 0
\(172\) −109.780 −0.638259
\(173\) − 252.077i − 1.45709i −0.684996 0.728547i \(-0.740196\pi\)
0.684996 0.728547i \(-0.259804\pi\)
\(174\) 0 0
\(175\) −119.233 −0.681330
\(176\) − 500.115i − 2.84156i
\(177\) 0 0
\(178\) 577.899 3.24663
\(179\) − 26.4932i − 0.148007i −0.997258 0.0740034i \(-0.976422\pi\)
0.997258 0.0740034i \(-0.0235776\pi\)
\(180\) 0 0
\(181\) −63.2787 −0.349606 −0.174803 0.984603i \(-0.555929\pi\)
−0.174803 + 0.984603i \(0.555929\pi\)
\(182\) − 501.723i − 2.75672i
\(183\) 0 0
\(184\) 371.146 2.01710
\(185\) 6.34344i 0.0342888i
\(186\) 0 0
\(187\) 197.968 1.05865
\(188\) 395.836i 2.10551i
\(189\) 0 0
\(190\) 125.937 0.662826
\(191\) − 143.322i − 0.750379i −0.926948 0.375189i \(-0.877578\pi\)
0.926948 0.375189i \(-0.122422\pi\)
\(192\) 0 0
\(193\) 364.106 1.88656 0.943279 0.332002i \(-0.107724\pi\)
0.943279 + 0.332002i \(0.107724\pi\)
\(194\) − 99.9680i − 0.515299i
\(195\) 0 0
\(196\) 181.563 0.926340
\(197\) 236.066i 1.19830i 0.800635 + 0.599152i \(0.204495\pi\)
−0.800635 + 0.599152i \(0.795505\pi\)
\(198\) 0 0
\(199\) 278.694 1.40047 0.700236 0.713912i \(-0.253078\pi\)
0.700236 + 0.713912i \(0.253078\pi\)
\(200\) 575.354i 2.87677i
\(201\) 0 0
\(202\) 311.198 1.54058
\(203\) − 128.698i − 0.633981i
\(204\) 0 0
\(205\) −8.43807 −0.0411613
\(206\) − 346.887i − 1.68392i
\(207\) 0 0
\(208\) −1417.84 −6.81652
\(209\) − 127.192i − 0.608575i
\(210\) 0 0
\(211\) −386.586 −1.83216 −0.916080 0.400996i \(-0.868664\pi\)
−0.916080 + 0.400996i \(0.868664\pi\)
\(212\) 1093.66i 5.15878i
\(213\) 0 0
\(214\) −9.22530 −0.0431089
\(215\) 20.1048i 0.0935108i
\(216\) 0 0
\(217\) 8.90139 0.0410202
\(218\) − 8.46606i − 0.0388352i
\(219\) 0 0
\(220\) −179.758 −0.817083
\(221\) − 561.242i − 2.53956i
\(222\) 0 0
\(223\) 7.23993 0.0324660 0.0162330 0.999868i \(-0.494833\pi\)
0.0162330 + 0.999868i \(0.494833\pi\)
\(224\) 762.256i 3.40293i
\(225\) 0 0
\(226\) −452.627 −2.00277
\(227\) 298.177i 1.31356i 0.754084 + 0.656778i \(0.228081\pi\)
−0.754084 + 0.656778i \(0.771919\pi\)
\(228\) 0 0
\(229\) −156.233 −0.682241 −0.341121 0.940019i \(-0.610806\pi\)
−0.341121 + 0.940019i \(0.610806\pi\)
\(230\) − 106.289i − 0.462127i
\(231\) 0 0
\(232\) −621.028 −2.67684
\(233\) 101.771i 0.436784i 0.975861 + 0.218392i \(0.0700811\pi\)
−0.975861 + 0.218392i \(0.929919\pi\)
\(234\) 0 0
\(235\) 72.4920 0.308477
\(236\) − 781.209i − 3.31021i
\(237\) 0 0
\(238\) −551.091 −2.31551
\(239\) 122.338i 0.511875i 0.966693 + 0.255937i \(0.0823841\pi\)
−0.966693 + 0.255937i \(0.917616\pi\)
\(240\) 0 0
\(241\) 0.637189 0.00264394 0.00132197 0.999999i \(-0.499579\pi\)
0.00132197 + 0.999999i \(0.499579\pi\)
\(242\) − 223.115i − 0.921962i
\(243\) 0 0
\(244\) −416.103 −1.70534
\(245\) − 33.2508i − 0.135717i
\(246\) 0 0
\(247\) −360.592 −1.45989
\(248\) − 42.9534i − 0.173199i
\(249\) 0 0
\(250\) 362.137 1.44855
\(251\) 135.183i 0.538577i 0.963060 + 0.269289i \(0.0867886\pi\)
−0.963060 + 0.269289i \(0.913211\pi\)
\(252\) 0 0
\(253\) −107.349 −0.424303
\(254\) 43.7847i 0.172381i
\(255\) 0 0
\(256\) 894.507 3.49417
\(257\) − 109.808i − 0.427267i −0.976914 0.213633i \(-0.931470\pi\)
0.976914 0.213633i \(-0.0685298\pi\)
\(258\) 0 0
\(259\) −17.8345 −0.0688591
\(260\) 509.618i 1.96007i
\(261\) 0 0
\(262\) −306.910 −1.17141
\(263\) − 266.392i − 1.01290i −0.862270 0.506448i \(-0.830958\pi\)
0.862270 0.506448i \(-0.169042\pi\)
\(264\) 0 0
\(265\) 200.289 0.755809
\(266\) 354.070i 1.33109i
\(267\) 0 0
\(268\) −1131.72 −4.22284
\(269\) 79.5758i 0.295821i 0.989001 + 0.147910i \(0.0472547\pi\)
−0.989001 + 0.147910i \(0.952745\pi\)
\(270\) 0 0
\(271\) 193.955 0.715700 0.357850 0.933779i \(-0.383510\pi\)
0.357850 + 0.933779i \(0.383510\pi\)
\(272\) 1557.35i 5.72554i
\(273\) 0 0
\(274\) 338.585 1.23571
\(275\) − 166.413i − 0.605138i
\(276\) 0 0
\(277\) 118.474 0.427702 0.213851 0.976866i \(-0.431399\pi\)
0.213851 + 0.976866i \(0.431399\pi\)
\(278\) − 904.607i − 3.25398i
\(279\) 0 0
\(280\) 319.999 1.14285
\(281\) − 57.3404i − 0.204058i −0.994781 0.102029i \(-0.967467\pi\)
0.994781 0.102029i \(-0.0325335\pi\)
\(282\) 0 0
\(283\) −179.085 −0.632810 −0.316405 0.948624i \(-0.602476\pi\)
−0.316405 + 0.948624i \(0.602476\pi\)
\(284\) 1561.23i 5.49728i
\(285\) 0 0
\(286\) 700.254 2.44844
\(287\) − 23.7235i − 0.0826604i
\(288\) 0 0
\(289\) −327.467 −1.13310
\(290\) 177.851i 0.613279i
\(291\) 0 0
\(292\) 411.707 1.40996
\(293\) − 423.884i − 1.44670i −0.690481 0.723351i \(-0.742601\pi\)
0.690481 0.723351i \(-0.257399\pi\)
\(294\) 0 0
\(295\) −143.068 −0.484976
\(296\) 86.0597i 0.290742i
\(297\) 0 0
\(298\) 1108.00 3.71810
\(299\) 304.336i 1.01784i
\(300\) 0 0
\(301\) −56.5245 −0.187789
\(302\) − 477.998i − 1.58278i
\(303\) 0 0
\(304\) 1000.58 3.29138
\(305\) 76.2037i 0.249848i
\(306\) 0 0
\(307\) −474.504 −1.54561 −0.772807 0.634641i \(-0.781148\pi\)
−0.772807 + 0.634641i \(0.781148\pi\)
\(308\) − 505.388i − 1.64087i
\(309\) 0 0
\(310\) −12.3010 −0.0396808
\(311\) 206.584i 0.664257i 0.943234 + 0.332128i \(0.107767\pi\)
−0.943234 + 0.332128i \(0.892233\pi\)
\(312\) 0 0
\(313\) 112.876 0.360626 0.180313 0.983609i \(-0.442289\pi\)
0.180313 + 0.983609i \(0.442289\pi\)
\(314\) − 815.776i − 2.59801i
\(315\) 0 0
\(316\) 768.416 2.43170
\(317\) − 434.770i − 1.37151i −0.727831 0.685756i \(-0.759472\pi\)
0.727831 0.685756i \(-0.240528\pi\)
\(318\) 0 0
\(319\) 179.624 0.563084
\(320\) − 543.576i − 1.69867i
\(321\) 0 0
\(322\) 298.831 0.928047
\(323\) 396.073i 1.22623i
\(324\) 0 0
\(325\) −471.784 −1.45164
\(326\) − 863.757i − 2.64956i
\(327\) 0 0
\(328\) −114.477 −0.349016
\(329\) 203.810i 0.619485i
\(330\) 0 0
\(331\) 261.919 0.791295 0.395647 0.918402i \(-0.370520\pi\)
0.395647 + 0.918402i \(0.370520\pi\)
\(332\) 918.799i 2.76747i
\(333\) 0 0
\(334\) 671.296 2.00987
\(335\) 207.260i 0.618685i
\(336\) 0 0
\(337\) 364.685 1.08215 0.541076 0.840974i \(-0.318017\pi\)
0.541076 + 0.840974i \(0.318017\pi\)
\(338\) − 1328.62i − 3.93084i
\(339\) 0 0
\(340\) 559.763 1.64636
\(341\) 12.4237i 0.0364330i
\(342\) 0 0
\(343\) 373.411 1.08866
\(344\) 272.757i 0.792898i
\(345\) 0 0
\(346\) −979.385 −2.83059
\(347\) 149.497i 0.430828i 0.976523 + 0.215414i \(0.0691101\pi\)
−0.976523 + 0.215414i \(0.930890\pi\)
\(348\) 0 0
\(349\) −463.218 −1.32727 −0.663636 0.748056i \(-0.730988\pi\)
−0.663636 + 0.748056i \(0.730988\pi\)
\(350\) 463.250i 1.32357i
\(351\) 0 0
\(352\) −1063.88 −3.02238
\(353\) 419.892i 1.18950i 0.803912 + 0.594748i \(0.202748\pi\)
−0.803912 + 0.594748i \(0.797252\pi\)
\(354\) 0 0
\(355\) 285.918 0.805403
\(356\) − 1650.32i − 4.63574i
\(357\) 0 0
\(358\) −102.933 −0.287523
\(359\) 78.8625i 0.219673i 0.993950 + 0.109836i \(0.0350327\pi\)
−0.993950 + 0.109836i \(0.964967\pi\)
\(360\) 0 0
\(361\) −106.527 −0.295089
\(362\) 245.854i 0.679155i
\(363\) 0 0
\(364\) −1432.78 −3.93622
\(365\) − 75.3987i − 0.206572i
\(366\) 0 0
\(367\) 584.688 1.59316 0.796578 0.604536i \(-0.206642\pi\)
0.796578 + 0.604536i \(0.206642\pi\)
\(368\) − 844.476i − 2.29477i
\(369\) 0 0
\(370\) 24.6459 0.0666105
\(371\) 563.111i 1.51782i
\(372\) 0 0
\(373\) 24.0162 0.0643866 0.0321933 0.999482i \(-0.489751\pi\)
0.0321933 + 0.999482i \(0.489751\pi\)
\(374\) − 769.157i − 2.05657i
\(375\) 0 0
\(376\) 983.480 2.61564
\(377\) − 509.236i − 1.35076i
\(378\) 0 0
\(379\) 609.893 1.60922 0.804609 0.593806i \(-0.202375\pi\)
0.804609 + 0.593806i \(0.202375\pi\)
\(380\) − 359.642i − 0.946425i
\(381\) 0 0
\(382\) −556.845 −1.45771
\(383\) − 545.175i − 1.42343i −0.702467 0.711716i \(-0.747918\pi\)
0.702467 0.711716i \(-0.252082\pi\)
\(384\) 0 0
\(385\) −92.5551 −0.240403
\(386\) − 1414.64i − 3.66488i
\(387\) 0 0
\(388\) −285.481 −0.735777
\(389\) 106.310i 0.273291i 0.990620 + 0.136646i \(0.0436322\pi\)
−0.990620 + 0.136646i \(0.956368\pi\)
\(390\) 0 0
\(391\) 334.281 0.854939
\(392\) − 451.105i − 1.15078i
\(393\) 0 0
\(394\) 917.177 2.32786
\(395\) − 140.725i − 0.356266i
\(396\) 0 0
\(397\) 710.850 1.79056 0.895278 0.445509i \(-0.146977\pi\)
0.895278 + 0.445509i \(0.146977\pi\)
\(398\) − 1082.80i − 2.72060i
\(399\) 0 0
\(400\) 1309.11 3.27279
\(401\) 208.003i 0.518711i 0.965782 + 0.259356i \(0.0835102\pi\)
−0.965782 + 0.259356i \(0.916490\pi\)
\(402\) 0 0
\(403\) 35.2213 0.0873977
\(404\) − 888.696i − 2.19974i
\(405\) 0 0
\(406\) −500.025 −1.23159
\(407\) − 24.8916i − 0.0611587i
\(408\) 0 0
\(409\) −655.639 −1.60303 −0.801515 0.597974i \(-0.795972\pi\)
−0.801515 + 0.597974i \(0.795972\pi\)
\(410\) 32.7841i 0.0799612i
\(411\) 0 0
\(412\) −990.616 −2.40441
\(413\) − 402.234i − 0.973932i
\(414\) 0 0
\(415\) 168.266 0.405460
\(416\) 3016.12i 7.25028i
\(417\) 0 0
\(418\) −494.175 −1.18224
\(419\) 69.6645i 0.166264i 0.996539 + 0.0831318i \(0.0264923\pi\)
−0.996539 + 0.0831318i \(0.973508\pi\)
\(420\) 0 0
\(421\) 280.499 0.666268 0.333134 0.942880i \(-0.391894\pi\)
0.333134 + 0.942880i \(0.391894\pi\)
\(422\) 1501.99i 3.55921i
\(423\) 0 0
\(424\) 2717.27 6.40866
\(425\) 518.206i 1.21931i
\(426\) 0 0
\(427\) −214.246 −0.501747
\(428\) 26.3449i 0.0615536i
\(429\) 0 0
\(430\) 78.1125 0.181657
\(431\) − 507.000i − 1.17633i −0.808740 0.588167i \(-0.799850\pi\)
0.808740 0.588167i \(-0.200150\pi\)
\(432\) 0 0
\(433\) 221.867 0.512396 0.256198 0.966624i \(-0.417530\pi\)
0.256198 + 0.966624i \(0.417530\pi\)
\(434\) − 34.5842i − 0.0796872i
\(435\) 0 0
\(436\) −24.1768 −0.0554513
\(437\) − 214.772i − 0.491470i
\(438\) 0 0
\(439\) −243.190 −0.553964 −0.276982 0.960875i \(-0.589334\pi\)
−0.276982 + 0.960875i \(0.589334\pi\)
\(440\) 446.621i 1.01505i
\(441\) 0 0
\(442\) −2180.57 −4.93342
\(443\) − 317.504i − 0.716714i −0.933585 0.358357i \(-0.883337\pi\)
0.933585 0.358357i \(-0.116663\pi\)
\(444\) 0 0
\(445\) −302.235 −0.679179
\(446\) − 28.1290i − 0.0630695i
\(447\) 0 0
\(448\) 1528.26 3.41129
\(449\) − 706.223i − 1.57288i −0.617666 0.786440i \(-0.711922\pi\)
0.617666 0.786440i \(-0.288078\pi\)
\(450\) 0 0
\(451\) 33.1109 0.0734166
\(452\) 1292.58i 2.85969i
\(453\) 0 0
\(454\) 1158.50 2.55175
\(455\) 262.395i 0.576693i
\(456\) 0 0
\(457\) 657.374 1.43846 0.719228 0.694774i \(-0.244496\pi\)
0.719228 + 0.694774i \(0.244496\pi\)
\(458\) 607.007i 1.32534i
\(459\) 0 0
\(460\) −303.533 −0.659855
\(461\) − 558.636i − 1.21179i −0.795544 0.605896i \(-0.792815\pi\)
0.795544 0.605896i \(-0.207185\pi\)
\(462\) 0 0
\(463\) 248.378 0.536454 0.268227 0.963356i \(-0.413562\pi\)
0.268227 + 0.963356i \(0.413562\pi\)
\(464\) 1413.04i 3.04534i
\(465\) 0 0
\(466\) 395.405 0.848510
\(467\) 449.563i 0.962662i 0.876539 + 0.481331i \(0.159847\pi\)
−0.876539 + 0.481331i \(0.840153\pi\)
\(468\) 0 0
\(469\) −582.708 −1.24245
\(470\) − 281.650i − 0.599256i
\(471\) 0 0
\(472\) −1940.97 −4.11221
\(473\) − 78.8911i − 0.166789i
\(474\) 0 0
\(475\) 332.942 0.700930
\(476\) 1573.77i 3.30623i
\(477\) 0 0
\(478\) 475.315 0.994383
\(479\) 581.495i 1.21398i 0.794711 + 0.606988i \(0.207622\pi\)
−0.794711 + 0.606988i \(0.792378\pi\)
\(480\) 0 0
\(481\) −70.5680 −0.146711
\(482\) − 2.47564i − 0.00513619i
\(483\) 0 0
\(484\) −637.155 −1.31644
\(485\) 52.2821i 0.107798i
\(486\) 0 0
\(487\) 117.988 0.242276 0.121138 0.992636i \(-0.461346\pi\)
0.121138 + 0.992636i \(0.461346\pi\)
\(488\) 1033.84i 2.11852i
\(489\) 0 0
\(490\) −129.188 −0.263649
\(491\) 536.142i 1.09194i 0.837805 + 0.545969i \(0.183838\pi\)
−0.837805 + 0.545969i \(0.816162\pi\)
\(492\) 0 0
\(493\) −559.343 −1.13457
\(494\) 1400.99i 2.83602i
\(495\) 0 0
\(496\) −97.7327 −0.197042
\(497\) 803.855i 1.61741i
\(498\) 0 0
\(499\) 162.152 0.324954 0.162477 0.986712i \(-0.448052\pi\)
0.162477 + 0.986712i \(0.448052\pi\)
\(500\) − 1034.16i − 2.06833i
\(501\) 0 0
\(502\) 525.221 1.04626
\(503\) 80.4958i 0.160031i 0.996794 + 0.0800157i \(0.0254970\pi\)
−0.996794 + 0.0800157i \(0.974503\pi\)
\(504\) 0 0
\(505\) −162.753 −0.322283
\(506\) 417.078i 0.824265i
\(507\) 0 0
\(508\) 125.037 0.246136
\(509\) − 662.979i − 1.30251i −0.758858 0.651256i \(-0.774242\pi\)
0.758858 0.651256i \(-0.225758\pi\)
\(510\) 0 0
\(511\) 211.983 0.414839
\(512\) − 1452.81i − 2.83752i
\(513\) 0 0
\(514\) −426.631 −0.830021
\(515\) 181.418i 0.352268i
\(516\) 0 0
\(517\) −284.458 −0.550209
\(518\) 69.2917i 0.133768i
\(519\) 0 0
\(520\) 1266.18 2.43496
\(521\) 619.817i 1.18967i 0.803848 + 0.594834i \(0.202782\pi\)
−0.803848 + 0.594834i \(0.797218\pi\)
\(522\) 0 0
\(523\) 114.109 0.218182 0.109091 0.994032i \(-0.465206\pi\)
0.109091 + 0.994032i \(0.465206\pi\)
\(524\) 876.452i 1.67262i
\(525\) 0 0
\(526\) −1035.00 −1.96768
\(527\) − 38.6869i − 0.0734098i
\(528\) 0 0
\(529\) 347.735 0.657344
\(530\) − 778.176i − 1.46826i
\(531\) 0 0
\(532\) 1011.13 1.90062
\(533\) − 93.8700i − 0.176116i
\(534\) 0 0
\(535\) 4.82472 0.00901817
\(536\) 2811.84i 5.24596i
\(537\) 0 0
\(538\) 309.173 0.574670
\(539\) 130.476i 0.242070i
\(540\) 0 0
\(541\) 18.1841 0.0336120 0.0168060 0.999859i \(-0.494650\pi\)
0.0168060 + 0.999859i \(0.494650\pi\)
\(542\) − 753.565i − 1.39034i
\(543\) 0 0
\(544\) 3312.89 6.08987
\(545\) 4.42765i 0.00812413i
\(546\) 0 0
\(547\) −946.157 −1.72972 −0.864860 0.502013i \(-0.832593\pi\)
−0.864860 + 0.502013i \(0.832593\pi\)
\(548\) − 966.906i − 1.76443i
\(549\) 0 0
\(550\) −646.558 −1.17556
\(551\) 359.372i 0.652218i
\(552\) 0 0
\(553\) 395.647 0.715456
\(554\) − 460.301i − 0.830868i
\(555\) 0 0
\(556\) −2583.31 −4.64624
\(557\) 226.354i 0.406380i 0.979139 + 0.203190i \(0.0651309\pi\)
−0.979139 + 0.203190i \(0.934869\pi\)
\(558\) 0 0
\(559\) −223.658 −0.400103
\(560\) − 728.100i − 1.30018i
\(561\) 0 0
\(562\) −222.782 −0.396410
\(563\) 813.544i 1.44502i 0.691363 + 0.722508i \(0.257011\pi\)
−0.691363 + 0.722508i \(0.742989\pi\)
\(564\) 0 0
\(565\) 236.719 0.418971
\(566\) 695.793i 1.22932i
\(567\) 0 0
\(568\) 3878.98 6.82918
\(569\) − 672.821i − 1.18246i −0.806502 0.591231i \(-0.798642\pi\)
0.806502 0.591231i \(-0.201358\pi\)
\(570\) 0 0
\(571\) 335.830 0.588144 0.294072 0.955783i \(-0.404989\pi\)
0.294072 + 0.955783i \(0.404989\pi\)
\(572\) − 1999.73i − 3.49604i
\(573\) 0 0
\(574\) −92.1721 −0.160579
\(575\) − 280.999i − 0.488694i
\(576\) 0 0
\(577\) −124.392 −0.215584 −0.107792 0.994173i \(-0.534378\pi\)
−0.107792 + 0.994173i \(0.534378\pi\)
\(578\) 1272.29i 2.20120i
\(579\) 0 0
\(580\) 507.894 0.875679
\(581\) 473.077i 0.814246i
\(582\) 0 0
\(583\) −785.933 −1.34808
\(584\) − 1022.91i − 1.75157i
\(585\) 0 0
\(586\) −1646.90 −2.81041
\(587\) 170.171i 0.289899i 0.989439 + 0.144950i \(0.0463020\pi\)
−0.989439 + 0.144950i \(0.953698\pi\)
\(588\) 0 0
\(589\) −24.8560 −0.0422003
\(590\) 555.856i 0.942129i
\(591\) 0 0
\(592\) 195.814 0.330766
\(593\) − 233.420i − 0.393625i −0.980441 0.196812i \(-0.936941\pi\)
0.980441 0.196812i \(-0.0630590\pi\)
\(594\) 0 0
\(595\) 288.214 0.484394
\(596\) − 3164.13i − 5.30895i
\(597\) 0 0
\(598\) 1182.42 1.97730
\(599\) 876.407i 1.46312i 0.681779 + 0.731559i \(0.261207\pi\)
−0.681779 + 0.731559i \(0.738793\pi\)
\(600\) 0 0
\(601\) −821.070 −1.36617 −0.683087 0.730337i \(-0.739363\pi\)
−0.683087 + 0.730337i \(0.739363\pi\)
\(602\) 219.612i 0.364805i
\(603\) 0 0
\(604\) −1365.03 −2.25999
\(605\) 116.686i 0.192870i
\(606\) 0 0
\(607\) −243.151 −0.400579 −0.200289 0.979737i \(-0.564188\pi\)
−0.200289 + 0.979737i \(0.564188\pi\)
\(608\) − 2128.50i − 3.50082i
\(609\) 0 0
\(610\) 296.071 0.485363
\(611\) 806.443i 1.31987i
\(612\) 0 0
\(613\) −496.958 −0.810698 −0.405349 0.914162i \(-0.632850\pi\)
−0.405349 + 0.914162i \(0.632850\pi\)
\(614\) 1843.57i 3.00256i
\(615\) 0 0
\(616\) −1255.67 −2.03843
\(617\) − 866.834i − 1.40492i −0.711724 0.702459i \(-0.752086\pi\)
0.711724 0.702459i \(-0.247914\pi\)
\(618\) 0 0
\(619\) −766.155 −1.23773 −0.618866 0.785497i \(-0.712407\pi\)
−0.618866 + 0.785497i \(0.712407\pi\)
\(620\) 35.1284i 0.0566588i
\(621\) 0 0
\(622\) 802.632 1.29041
\(623\) − 849.729i − 1.36393i
\(624\) 0 0
\(625\) 332.387 0.531819
\(626\) − 438.553i − 0.700564i
\(627\) 0 0
\(628\) −2329.64 −3.70961
\(629\) 77.5117i 0.123230i
\(630\) 0 0
\(631\) −878.466 −1.39218 −0.696090 0.717954i \(-0.745079\pi\)
−0.696090 + 0.717954i \(0.745079\pi\)
\(632\) − 1909.18i − 3.02086i
\(633\) 0 0
\(634\) −1689.19 −2.66434
\(635\) − 22.8989i − 0.0360612i
\(636\) 0 0
\(637\) 369.901 0.580692
\(638\) − 697.885i − 1.09386i
\(639\) 0 0
\(640\) −1027.45 −1.60538
\(641\) 901.514i 1.40642i 0.710983 + 0.703209i \(0.248250\pi\)
−0.710983 + 0.703209i \(0.751750\pi\)
\(642\) 0 0
\(643\) −96.0218 −0.149334 −0.0746671 0.997209i \(-0.523789\pi\)
−0.0746671 + 0.997209i \(0.523789\pi\)
\(644\) − 853.380i − 1.32512i
\(645\) 0 0
\(646\) 1538.85 2.38212
\(647\) 932.456i 1.44120i 0.693351 + 0.720600i \(0.256134\pi\)
−0.693351 + 0.720600i \(0.743866\pi\)
\(648\) 0 0
\(649\) 561.397 0.865018
\(650\) 1833.00i 2.82000i
\(651\) 0 0
\(652\) −2466.66 −3.78321
\(653\) 75.4491i 0.115542i 0.998330 + 0.0577711i \(0.0183994\pi\)
−0.998330 + 0.0577711i \(0.981601\pi\)
\(654\) 0 0
\(655\) 160.510 0.245054
\(656\) 260.472i 0.397061i
\(657\) 0 0
\(658\) 791.857 1.20343
\(659\) 383.689i 0.582229i 0.956688 + 0.291114i \(0.0940260\pi\)
−0.956688 + 0.291114i \(0.905974\pi\)
\(660\) 0 0
\(661\) 134.541 0.203542 0.101771 0.994808i \(-0.467549\pi\)
0.101771 + 0.994808i \(0.467549\pi\)
\(662\) − 1017.62i − 1.53719i
\(663\) 0 0
\(664\) 2282.82 3.43798
\(665\) − 185.175i − 0.278458i
\(666\) 0 0
\(667\) 303.306 0.454731
\(668\) − 1917.04i − 2.86982i
\(669\) 0 0
\(670\) 805.257 1.20188
\(671\) − 299.023i − 0.445637i
\(672\) 0 0
\(673\) 8.81088 0.0130920 0.00654598 0.999979i \(-0.497916\pi\)
0.00654598 + 0.999979i \(0.497916\pi\)
\(674\) − 1416.90i − 2.10222i
\(675\) 0 0
\(676\) −3794.19 −5.61270
\(677\) − 1222.18i − 1.80529i −0.430388 0.902644i \(-0.641623\pi\)
0.430388 0.902644i \(-0.358377\pi\)
\(678\) 0 0
\(679\) −146.990 −0.216481
\(680\) − 1390.77i − 2.04525i
\(681\) 0 0
\(682\) 48.2691 0.0707759
\(683\) − 650.368i − 0.952222i −0.879385 0.476111i \(-0.842046\pi\)
0.879385 0.476111i \(-0.157954\pi\)
\(684\) 0 0
\(685\) −177.076 −0.258505
\(686\) − 1450.80i − 2.11487i
\(687\) 0 0
\(688\) 620.610 0.902049
\(689\) 2228.13i 3.23387i
\(690\) 0 0
\(691\) 709.417 1.02665 0.513326 0.858194i \(-0.328413\pi\)
0.513326 + 0.858194i \(0.328413\pi\)
\(692\) 2796.86i 4.04170i
\(693\) 0 0
\(694\) 580.836 0.836939
\(695\) 473.099i 0.680718i
\(696\) 0 0
\(697\) −103.106 −0.147929
\(698\) 1799.72i 2.57840i
\(699\) 0 0
\(700\) 1322.92 1.88988
\(701\) 850.556i 1.21335i 0.794951 + 0.606673i \(0.207496\pi\)
−0.794951 + 0.606673i \(0.792504\pi\)
\(702\) 0 0
\(703\) 49.8005 0.0708399
\(704\) 2132.98i 3.02981i
\(705\) 0 0
\(706\) 1631.39 2.31075
\(707\) − 457.578i − 0.647210i
\(708\) 0 0
\(709\) −1199.05 −1.69118 −0.845591 0.533831i \(-0.820752\pi\)
−0.845591 + 0.533831i \(0.820752\pi\)
\(710\) − 1110.87i − 1.56460i
\(711\) 0 0
\(712\) −4100.34 −5.75890
\(713\) 20.9781i 0.0294223i
\(714\) 0 0
\(715\) −366.225 −0.512202
\(716\) 293.949i 0.410543i
\(717\) 0 0
\(718\) 306.401 0.426743
\(719\) − 255.580i − 0.355466i −0.984079 0.177733i \(-0.943124\pi\)
0.984079 0.177733i \(-0.0568763\pi\)
\(720\) 0 0
\(721\) −510.055 −0.707427
\(722\) 413.885i 0.573248i
\(723\) 0 0
\(724\) 702.093 0.969741
\(725\) 470.187i 0.648534i
\(726\) 0 0
\(727\) −481.234 −0.661945 −0.330972 0.943640i \(-0.607377\pi\)
−0.330972 + 0.943640i \(0.607377\pi\)
\(728\) 3559.85i 4.88990i
\(729\) 0 0
\(730\) −292.944 −0.401293
\(731\) 245.665i 0.336067i
\(732\) 0 0
\(733\) −245.297 −0.334648 −0.167324 0.985902i \(-0.553513\pi\)
−0.167324 + 0.985902i \(0.553513\pi\)
\(734\) − 2271.67i − 3.09491i
\(735\) 0 0
\(736\) −1796.43 −2.44080
\(737\) − 813.284i − 1.10351i
\(738\) 0 0
\(739\) −761.537 −1.03050 −0.515248 0.857041i \(-0.672300\pi\)
−0.515248 + 0.857041i \(0.672300\pi\)
\(740\) − 70.3820i − 0.0951108i
\(741\) 0 0
\(742\) 2187.83 2.94856
\(743\) − 1378.21i − 1.85493i −0.373912 0.927464i \(-0.621984\pi\)
0.373912 0.927464i \(-0.378016\pi\)
\(744\) 0 0
\(745\) −579.469 −0.777810
\(746\) − 93.3092i − 0.125079i
\(747\) 0 0
\(748\) −2196.50 −2.93650
\(749\) 13.5647i 0.0181103i
\(750\) 0 0
\(751\) −85.4470 −0.113778 −0.0568888 0.998381i \(-0.518118\pi\)
−0.0568888 + 0.998381i \(0.518118\pi\)
\(752\) − 2237.73i − 2.97571i
\(753\) 0 0
\(754\) −1978.52 −2.62403
\(755\) 249.988i 0.331109i
\(756\) 0 0
\(757\) 89.9427 0.118815 0.0594074 0.998234i \(-0.481079\pi\)
0.0594074 + 0.998234i \(0.481079\pi\)
\(758\) − 2369.59i − 3.12611i
\(759\) 0 0
\(760\) −893.554 −1.17573
\(761\) 1194.58i 1.56975i 0.619653 + 0.784876i \(0.287273\pi\)
−0.619653 + 0.784876i \(0.712727\pi\)
\(762\) 0 0
\(763\) −12.4483 −0.0163149
\(764\) 1590.20i 2.08141i
\(765\) 0 0
\(766\) −2118.15 −2.76520
\(767\) − 1591.57i − 2.07506i
\(768\) 0 0
\(769\) 1240.46 1.61308 0.806542 0.591177i \(-0.201337\pi\)
0.806542 + 0.591177i \(0.201337\pi\)
\(770\) 359.601i 0.467014i
\(771\) 0 0
\(772\) −4039.84 −5.23295
\(773\) 1146.78i 1.48354i 0.670655 + 0.741770i \(0.266013\pi\)
−0.670655 + 0.741770i \(0.733987\pi\)
\(774\) 0 0
\(775\) −32.5205 −0.0419619
\(776\) 709.297i 0.914043i
\(777\) 0 0
\(778\) 413.044 0.530904
\(779\) 66.2448i 0.0850383i
\(780\) 0 0
\(781\) −1121.94 −1.43654
\(782\) − 1298.77i − 1.66083i
\(783\) 0 0
\(784\) −1026.41 −1.30919
\(785\) 426.642i 0.543492i
\(786\) 0 0
\(787\) 1153.33 1.46548 0.732740 0.680509i \(-0.238241\pi\)
0.732740 + 0.680509i \(0.238241\pi\)
\(788\) − 2619.21i − 3.32387i
\(789\) 0 0
\(790\) −546.754 −0.692094
\(791\) 665.532i 0.841380i
\(792\) 0 0
\(793\) −847.734 −1.06902
\(794\) − 2761.84i − 3.47839i
\(795\) 0 0
\(796\) −3092.18 −3.88464
\(797\) 934.090i 1.17201i 0.810308 + 0.586004i \(0.199300\pi\)
−0.810308 + 0.586004i \(0.800700\pi\)
\(798\) 0 0
\(799\) 885.794 1.10863
\(800\) − 2784.84i − 3.48104i
\(801\) 0 0
\(802\) 808.147 1.00766
\(803\) 295.864i 0.368448i
\(804\) 0 0
\(805\) −156.285 −0.194143
\(806\) − 136.844i − 0.169781i
\(807\) 0 0
\(808\) −2208.03 −2.73270
\(809\) − 10.3335i − 0.0127732i −0.999980 0.00638658i \(-0.997967\pi\)
0.999980 0.00638658i \(-0.00203293\pi\)
\(810\) 0 0
\(811\) 1442.35 1.77848 0.889240 0.457440i \(-0.151234\pi\)
0.889240 + 0.457440i \(0.151234\pi\)
\(812\) 1427.94i 1.75854i
\(813\) 0 0
\(814\) −96.7102 −0.118809
\(815\) 451.735i 0.554276i
\(816\) 0 0
\(817\) 157.837 0.193191
\(818\) 2547.33i 3.11410i
\(819\) 0 0
\(820\) 93.6225 0.114174
\(821\) − 1546.21i − 1.88333i −0.336552 0.941665i \(-0.609261\pi\)
0.336552 0.941665i \(-0.390739\pi\)
\(822\) 0 0
\(823\) 1582.02 1.92226 0.961129 0.276100i \(-0.0890421\pi\)
0.961129 + 0.276100i \(0.0890421\pi\)
\(824\) 2461.25i 2.98695i
\(825\) 0 0
\(826\) −1562.78 −1.89199
\(827\) − 58.0629i − 0.0702091i −0.999384 0.0351045i \(-0.988824\pi\)
0.999384 0.0351045i \(-0.0111764\pi\)
\(828\) 0 0
\(829\) −279.841 −0.337565 −0.168783 0.985653i \(-0.553984\pi\)
−0.168783 + 0.985653i \(0.553984\pi\)
\(830\) − 653.756i − 0.787658i
\(831\) 0 0
\(832\) 6047.05 7.26809
\(833\) − 406.298i − 0.487753i
\(834\) 0 0
\(835\) −351.080 −0.420455
\(836\) 1411.23i 1.68807i
\(837\) 0 0
\(838\) 270.665 0.322989
\(839\) 611.537i 0.728888i 0.931225 + 0.364444i \(0.118741\pi\)
−0.931225 + 0.364444i \(0.881259\pi\)
\(840\) 0 0
\(841\) 333.487 0.396536
\(842\) − 1089.81i − 1.29431i
\(843\) 0 0
\(844\) 4289.26 5.08207
\(845\) 694.855i 0.822313i
\(846\) 0 0
\(847\) −328.063 −0.387323
\(848\) − 6182.67i − 7.29088i
\(849\) 0 0
\(850\) 2013.36 2.36866
\(851\) − 42.0310i − 0.0493901i
\(852\) 0 0
\(853\) −780.661 −0.915195 −0.457597 0.889159i \(-0.651290\pi\)
−0.457597 + 0.889159i \(0.651290\pi\)
\(854\) 832.401i 0.974708i
\(855\) 0 0
\(856\) 65.4558 0.0764670
\(857\) − 91.5738i − 0.106854i −0.998572 0.0534269i \(-0.982986\pi\)
0.998572 0.0534269i \(-0.0170144\pi\)
\(858\) 0 0
\(859\) −976.219 −1.13646 −0.568230 0.822870i \(-0.692372\pi\)
−0.568230 + 0.822870i \(0.692372\pi\)
\(860\) − 223.068i − 0.259381i
\(861\) 0 0
\(862\) −1969.83 −2.28518
\(863\) − 514.017i − 0.595616i −0.954626 0.297808i \(-0.903744\pi\)
0.954626 0.297808i \(-0.0962555\pi\)
\(864\) 0 0
\(865\) 512.207 0.592147
\(866\) − 862.012i − 0.995395i
\(867\) 0 0
\(868\) −98.7631 −0.113782
\(869\) 552.204i 0.635448i
\(870\) 0 0
\(871\) −2305.68 −2.64716
\(872\) 60.0688i 0.0688862i
\(873\) 0 0
\(874\) −834.446 −0.954743
\(875\) − 532.477i − 0.608545i
\(876\) 0 0
\(877\) 609.520 0.695006 0.347503 0.937679i \(-0.387030\pi\)
0.347503 + 0.937679i \(0.387030\pi\)
\(878\) 944.858i 1.07615i
\(879\) 0 0
\(880\) 1016.21 1.15478
\(881\) − 981.132i − 1.11366i −0.830628 0.556828i \(-0.812018\pi\)
0.830628 0.556828i \(-0.187982\pi\)
\(882\) 0 0
\(883\) 549.991 0.622867 0.311433 0.950268i \(-0.399191\pi\)
0.311433 + 0.950268i \(0.399191\pi\)
\(884\) 6227.12i 7.04426i
\(885\) 0 0
\(886\) −1233.59 −1.39231
\(887\) − 381.280i − 0.429853i −0.976630 0.214927i \(-0.931049\pi\)
0.976630 0.214927i \(-0.0689512\pi\)
\(888\) 0 0
\(889\) 64.3799 0.0724183
\(890\) 1174.26i 1.31939i
\(891\) 0 0
\(892\) −80.3288 −0.0900547
\(893\) − 569.114i − 0.637305i
\(894\) 0 0
\(895\) 53.8328 0.0601484
\(896\) − 2888.65i − 3.22394i
\(897\) 0 0
\(898\) −2743.86 −3.05553
\(899\) − 35.1021i − 0.0390457i
\(900\) 0 0
\(901\) 2447.38 2.71629
\(902\) − 128.644i − 0.142621i
\(903\) 0 0
\(904\) 3211.50 3.55254
\(905\) − 128.579i − 0.142076i
\(906\) 0 0
\(907\) −701.779 −0.773737 −0.386869 0.922135i \(-0.626443\pi\)
−0.386869 + 0.922135i \(0.626443\pi\)
\(908\) − 3308.35i − 3.64356i
\(909\) 0 0
\(910\) 1019.47 1.12030
\(911\) 514.496i 0.564760i 0.959303 + 0.282380i \(0.0911239\pi\)
−0.959303 + 0.282380i \(0.908876\pi\)
\(912\) 0 0
\(913\) −660.273 −0.723190
\(914\) − 2554.07i − 2.79439i
\(915\) 0 0
\(916\) 1733.45 1.89241
\(917\) 451.273i 0.492119i
\(918\) 0 0
\(919\) 753.168 0.819551 0.409776 0.912186i \(-0.365607\pi\)
0.409776 + 0.912186i \(0.365607\pi\)
\(920\) 754.149i 0.819727i
\(921\) 0 0
\(922\) −2170.45 −2.35406
\(923\) 3180.72i 3.44606i
\(924\) 0 0
\(925\) 65.1568 0.0704398
\(926\) − 965.015i − 1.04213i
\(927\) 0 0
\(928\) 3005.91 3.23913
\(929\) 646.826i 0.696261i 0.937446 + 0.348130i \(0.113183\pi\)
−0.937446 + 0.348130i \(0.886817\pi\)
\(930\) 0 0
\(931\) −261.042 −0.280389
\(932\) − 1129.17i − 1.21156i
\(933\) 0 0
\(934\) 1746.67 1.87010
\(935\) 402.260i 0.430225i
\(936\) 0 0
\(937\) 1059.11 1.13032 0.565160 0.824981i \(-0.308814\pi\)
0.565160 + 0.824981i \(0.308814\pi\)
\(938\) 2263.97i 2.41362i
\(939\) 0 0
\(940\) −804.317 −0.855656
\(941\) 66.0163i 0.0701555i 0.999385 + 0.0350777i \(0.0111679\pi\)
−0.999385 + 0.0350777i \(0.988832\pi\)
\(942\) 0 0
\(943\) 55.9098 0.0592893
\(944\) 4416.32i 4.67830i
\(945\) 0 0
\(946\) −306.512 −0.324009
\(947\) − 1136.85i − 1.20048i −0.799821 0.600238i \(-0.795073\pi\)
0.799821 0.600238i \(-0.204927\pi\)
\(948\) 0 0
\(949\) 838.778 0.883855
\(950\) − 1293.57i − 1.36165i
\(951\) 0 0
\(952\) 3910.13 4.10728
\(953\) 35.9360i 0.0377083i 0.999822 + 0.0188542i \(0.00600182\pi\)
−0.999822 + 0.0188542i \(0.993998\pi\)
\(954\) 0 0
\(955\) 291.223 0.304946
\(956\) − 1357.37i − 1.41984i
\(957\) 0 0
\(958\) 2259.26 2.35831
\(959\) − 497.847i − 0.519131i
\(960\) 0 0
\(961\) −958.572 −0.997474
\(962\) 274.175i 0.285005i
\(963\) 0 0
\(964\) −7.06977 −0.00733378
\(965\) 739.843i 0.766676i
\(966\) 0 0
\(967\) −773.061 −0.799443 −0.399721 0.916637i \(-0.630893\pi\)
−0.399721 + 0.916637i \(0.630893\pi\)
\(968\) 1583.05i 1.63539i
\(969\) 0 0
\(970\) 203.129 0.209412
\(971\) 745.333i 0.767593i 0.923418 + 0.383796i \(0.125384\pi\)
−0.923418 + 0.383796i \(0.874616\pi\)
\(972\) 0 0
\(973\) −1330.11 −1.36702
\(974\) − 458.415i − 0.470652i
\(975\) 0 0
\(976\) 2352.31 2.41015
\(977\) 187.891i 0.192314i 0.995366 + 0.0961572i \(0.0306551\pi\)
−0.995366 + 0.0961572i \(0.969345\pi\)
\(978\) 0 0
\(979\) 1185.97 1.21141
\(980\) 368.926i 0.376455i
\(981\) 0 0
\(982\) 2083.05 2.12123
\(983\) 970.266i 0.987046i 0.869733 + 0.493523i \(0.164291\pi\)
−0.869733 + 0.493523i \(0.835709\pi\)
\(984\) 0 0
\(985\) −479.673 −0.486978
\(986\) 2173.19i 2.20405i
\(987\) 0 0
\(988\) 4000.86 4.04945
\(989\) − 133.213i − 0.134694i
\(990\) 0 0
\(991\) 654.491 0.660435 0.330218 0.943905i \(-0.392878\pi\)
0.330218 + 0.943905i \(0.392878\pi\)
\(992\) 207.904i 0.209580i
\(993\) 0 0
\(994\) 3123.19 3.14204
\(995\) 566.291i 0.569137i
\(996\) 0 0
\(997\) 1089.23 1.09251 0.546253 0.837620i \(-0.316054\pi\)
0.546253 + 0.837620i \(0.316054\pi\)
\(998\) − 630.003i − 0.631266i
\(999\) 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.890.1 84
3.2 odd 2 inner 1143.3.b.a.890.84 yes 84
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1143.3.b.a.890.1 84 1.1 even 1 trivial
1143.3.b.a.890.84 yes 84 3.2 odd 2 inner