Properties

Label 1344.4.p.d.223.13
Level $1344$
Weight $4$
Character 1344.223
Analytic conductor $79.299$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1344 = 2^{6} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1344.p (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(79.2985670477\)
Analytic rank: \(0\)
Dimension: \(32\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 223.13
Character \(\chi\) \(=\) 1344.223
Dual form 1344.4.p.d.223.14

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000i q^{3} -20.7003 q^{5} +(-11.8166 + 14.2607i) q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} -20.7003 q^{5} +(-11.8166 + 14.2607i) q^{7} -9.00000 q^{9} +39.2814 q^{11} +36.2449 q^{13} +62.1010i q^{15} +92.2048i q^{17} -87.5184i q^{19} +(42.7820 + 35.4499i) q^{21} -100.990i q^{23} +303.504 q^{25} +27.0000i q^{27} -36.2034i q^{29} -264.306 q^{31} -117.844i q^{33} +(244.609 - 295.201i) q^{35} +229.534i q^{37} -108.735i q^{39} -337.096i q^{41} +368.528 q^{43} +186.303 q^{45} +314.672 q^{47} +(-63.7338 - 337.027i) q^{49} +276.614 q^{51} +108.563i q^{53} -813.139 q^{55} -262.555 q^{57} +185.373i q^{59} +163.504 q^{61} +(106.350 - 128.346i) q^{63} -750.282 q^{65} +458.260 q^{67} -302.969 q^{69} -584.527i q^{71} +1099.24i q^{73} -910.512i q^{75} +(-464.175 + 560.180i) q^{77} +438.627i q^{79} +81.0000 q^{81} +722.114i q^{83} -1908.67i q^{85} -108.610 q^{87} -452.334i q^{89} +(-428.293 + 516.877i) q^{91} +792.917i q^{93} +1811.66i q^{95} -598.000i q^{97} -353.533 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q - 288q^{9} + O(q^{10}) \) \( 32q - 288q^{9} + 224q^{13} + 72q^{21} + 1120q^{25} - 752q^{49} - 672q^{57} + 544q^{61} + 1536q^{65} + 144q^{69} + 1632q^{77} + 2592q^{81} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1344\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(449\) \(577\) \(1093\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) −20.7003 −1.85149 −0.925747 0.378143i \(-0.876563\pi\)
−0.925747 + 0.378143i \(0.876563\pi\)
\(6\) 0 0
\(7\) −11.8166 + 14.2607i −0.638039 + 0.770004i
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 39.2814 1.07671 0.538354 0.842719i \(-0.319046\pi\)
0.538354 + 0.842719i \(0.319046\pi\)
\(12\) 0 0
\(13\) 36.2449 0.773271 0.386636 0.922233i \(-0.373637\pi\)
0.386636 + 0.922233i \(0.373637\pi\)
\(14\) 0 0
\(15\) 62.1010i 1.06896i
\(16\) 0 0
\(17\) 92.2048i 1.31547i 0.753251 + 0.657733i \(0.228485\pi\)
−0.753251 + 0.657733i \(0.771515\pi\)
\(18\) 0 0
\(19\) 87.5184i 1.05674i −0.849013 0.528371i \(-0.822803\pi\)
0.849013 0.528371i \(-0.177197\pi\)
\(20\) 0 0
\(21\) 42.7820 + 35.4499i 0.444562 + 0.368372i
\(22\) 0 0
\(23\) 100.990i 0.915558i −0.889066 0.457779i \(-0.848645\pi\)
0.889066 0.457779i \(-0.151355\pi\)
\(24\) 0 0
\(25\) 303.504 2.42803
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 36.2034i 0.231821i −0.993260 0.115911i \(-0.963021\pi\)
0.993260 0.115911i \(-0.0369786\pi\)
\(30\) 0 0
\(31\) −264.306 −1.53131 −0.765656 0.643250i \(-0.777586\pi\)
−0.765656 + 0.643250i \(0.777586\pi\)
\(32\) 0 0
\(33\) 117.844i 0.621638i
\(34\) 0 0
\(35\) 244.609 295.201i 1.18133 1.42566i
\(36\) 0 0
\(37\) 229.534i 1.01987i 0.860213 + 0.509936i \(0.170331\pi\)
−0.860213 + 0.509936i \(0.829669\pi\)
\(38\) 0 0
\(39\) 108.735i 0.446448i
\(40\) 0 0
\(41\) 337.096i 1.28404i −0.766688 0.642019i \(-0.778097\pi\)
0.766688 0.642019i \(-0.221903\pi\)
\(42\) 0 0
\(43\) 368.528 1.30697 0.653487 0.756937i \(-0.273305\pi\)
0.653487 + 0.756937i \(0.273305\pi\)
\(44\) 0 0
\(45\) 186.303 0.617165
\(46\) 0 0
\(47\) 314.672 0.976588 0.488294 0.872679i \(-0.337619\pi\)
0.488294 + 0.872679i \(0.337619\pi\)
\(48\) 0 0
\(49\) −63.7338 337.027i −0.185813 0.982585i
\(50\) 0 0
\(51\) 276.614 0.759485
\(52\) 0 0
\(53\) 108.563i 0.281362i 0.990055 + 0.140681i \(0.0449293\pi\)
−0.990055 + 0.140681i \(0.955071\pi\)
\(54\) 0 0
\(55\) −813.139 −1.99352
\(56\) 0 0
\(57\) −262.555 −0.610110
\(58\) 0 0
\(59\) 185.373i 0.409043i 0.978862 + 0.204521i \(0.0655638\pi\)
−0.978862 + 0.204521i \(0.934436\pi\)
\(60\) 0 0
\(61\) 163.504 0.343189 0.171594 0.985168i \(-0.445108\pi\)
0.171594 + 0.985168i \(0.445108\pi\)
\(62\) 0 0
\(63\) 106.350 128.346i 0.212680 0.256668i
\(64\) 0 0
\(65\) −750.282 −1.43171
\(66\) 0 0
\(67\) 458.260 0.835603 0.417802 0.908538i \(-0.362801\pi\)
0.417802 + 0.908538i \(0.362801\pi\)
\(68\) 0 0
\(69\) −302.969 −0.528597
\(70\) 0 0
\(71\) 584.527i 0.977051i −0.872550 0.488525i \(-0.837535\pi\)
0.872550 0.488525i \(-0.162465\pi\)
\(72\) 0 0
\(73\) 1099.24i 1.76241i 0.472736 + 0.881204i \(0.343266\pi\)
−0.472736 + 0.881204i \(0.656734\pi\)
\(74\) 0 0
\(75\) 910.512i 1.40183i
\(76\) 0 0
\(77\) −464.175 + 560.180i −0.686982 + 0.829070i
\(78\) 0 0
\(79\) 438.627i 0.624676i 0.949971 + 0.312338i \(0.101112\pi\)
−0.949971 + 0.312338i \(0.898888\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 722.114i 0.954967i 0.878641 + 0.477484i \(0.158451\pi\)
−0.878641 + 0.477484i \(0.841549\pi\)
\(84\) 0 0
\(85\) 1908.67i 2.43558i
\(86\) 0 0
\(87\) −108.610 −0.133842
\(88\) 0 0
\(89\) 452.334i 0.538734i −0.963038 0.269367i \(-0.913186\pi\)
0.963038 0.269367i \(-0.0868144\pi\)
\(90\) 0 0
\(91\) −428.293 + 516.877i −0.493377 + 0.595422i
\(92\) 0 0
\(93\) 792.917i 0.884103i
\(94\) 0 0
\(95\) 1811.66i 1.95655i
\(96\) 0 0
\(97\) 598.000i 0.625956i −0.949760 0.312978i \(-0.898673\pi\)
0.949760 0.312978i \(-0.101327\pi\)
\(98\) 0 0
\(99\) −353.533 −0.358903
\(100\) 0 0
\(101\) −1415.04 −1.39407 −0.697037 0.717035i \(-0.745499\pi\)
−0.697037 + 0.717035i \(0.745499\pi\)
\(102\) 0 0
\(103\) −1138.86 −1.08947 −0.544733 0.838609i \(-0.683369\pi\)
−0.544733 + 0.838609i \(0.683369\pi\)
\(104\) 0 0
\(105\) −885.602 733.826i −0.823104 0.682039i
\(106\) 0 0
\(107\) −1503.38 −1.35829 −0.679147 0.734002i \(-0.737650\pi\)
−0.679147 + 0.734002i \(0.737650\pi\)
\(108\) 0 0
\(109\) 359.283i 0.315716i 0.987462 + 0.157858i \(0.0504589\pi\)
−0.987462 + 0.157858i \(0.949541\pi\)
\(110\) 0 0
\(111\) 688.603 0.588823
\(112\) 0 0
\(113\) −2363.20 −1.96736 −0.983679 0.179932i \(-0.942412\pi\)
−0.983679 + 0.179932i \(0.942412\pi\)
\(114\) 0 0
\(115\) 2090.52i 1.69515i
\(116\) 0 0
\(117\) −326.204 −0.257757
\(118\) 0 0
\(119\) −1314.90 1089.55i −1.01291 0.839319i
\(120\) 0 0
\(121\) 212.030 0.159301
\(122\) 0 0
\(123\) −1011.29 −0.741340
\(124\) 0 0
\(125\) −3695.09 −2.64399
\(126\) 0 0
\(127\) 1620.69i 1.13238i 0.824274 + 0.566192i \(0.191584\pi\)
−0.824274 + 0.566192i \(0.808416\pi\)
\(128\) 0 0
\(129\) 1105.58i 0.754582i
\(130\) 0 0
\(131\) 503.229i 0.335629i 0.985819 + 0.167814i \(0.0536709\pi\)
−0.985819 + 0.167814i \(0.946329\pi\)
\(132\) 0 0
\(133\) 1248.07 + 1034.17i 0.813696 + 0.674243i
\(134\) 0 0
\(135\) 558.909i 0.356320i
\(136\) 0 0
\(137\) −577.769 −0.360308 −0.180154 0.983638i \(-0.557660\pi\)
−0.180154 + 0.983638i \(0.557660\pi\)
\(138\) 0 0
\(139\) 412.449i 0.251680i −0.992051 0.125840i \(-0.959837\pi\)
0.992051 0.125840i \(-0.0401625\pi\)
\(140\) 0 0
\(141\) 944.017i 0.563834i
\(142\) 0 0
\(143\) 1423.75 0.832588
\(144\) 0 0
\(145\) 749.423i 0.429215i
\(146\) 0 0
\(147\) −1011.08 + 191.201i −0.567296 + 0.107279i
\(148\) 0 0
\(149\) 2985.68i 1.64158i −0.571227 0.820792i \(-0.693532\pi\)
0.571227 0.820792i \(-0.306468\pi\)
\(150\) 0 0
\(151\) 2788.11i 1.50261i 0.659957 + 0.751303i \(0.270574\pi\)
−0.659957 + 0.751303i \(0.729426\pi\)
\(152\) 0 0
\(153\) 829.843i 0.438489i
\(154\) 0 0
\(155\) 5471.21 2.83522
\(156\) 0 0
\(157\) −1616.65 −0.821801 −0.410900 0.911680i \(-0.634786\pi\)
−0.410900 + 0.911680i \(0.634786\pi\)
\(158\) 0 0
\(159\) 325.688 0.162445
\(160\) 0 0
\(161\) 1440.18 + 1193.36i 0.704983 + 0.584161i
\(162\) 0 0
\(163\) 1934.76 0.929708 0.464854 0.885387i \(-0.346107\pi\)
0.464854 + 0.885387i \(0.346107\pi\)
\(164\) 0 0
\(165\) 2439.42i 1.15096i
\(166\) 0 0
\(167\) −816.453 −0.378318 −0.189159 0.981947i \(-0.560576\pi\)
−0.189159 + 0.981947i \(0.560576\pi\)
\(168\) 0 0
\(169\) −883.307 −0.402051
\(170\) 0 0
\(171\) 787.666i 0.352247i
\(172\) 0 0
\(173\) 3644.18 1.60151 0.800757 0.598989i \(-0.204431\pi\)
0.800757 + 0.598989i \(0.204431\pi\)
\(174\) 0 0
\(175\) −3586.40 + 4328.17i −1.54918 + 1.86959i
\(176\) 0 0
\(177\) 556.119 0.236161
\(178\) 0 0
\(179\) −3415.80 −1.42631 −0.713154 0.701008i \(-0.752734\pi\)
−0.713154 + 0.701008i \(0.752734\pi\)
\(180\) 0 0
\(181\) 3513.95 1.44304 0.721519 0.692395i \(-0.243444\pi\)
0.721519 + 0.692395i \(0.243444\pi\)
\(182\) 0 0
\(183\) 490.512i 0.198140i
\(184\) 0 0
\(185\) 4751.44i 1.88829i
\(186\) 0 0
\(187\) 3621.93i 1.41637i
\(188\) 0 0
\(189\) −385.038 319.049i −0.148187 0.122791i
\(190\) 0 0
\(191\) 3058.15i 1.15854i 0.815137 + 0.579268i \(0.196661\pi\)
−0.815137 + 0.579268i \(0.803339\pi\)
\(192\) 0 0
\(193\) −1894.89 −0.706721 −0.353361 0.935487i \(-0.614961\pi\)
−0.353361 + 0.935487i \(0.614961\pi\)
\(194\) 0 0
\(195\) 2250.85i 0.826597i
\(196\) 0 0
\(197\) 109.557i 0.0396224i 0.999804 + 0.0198112i \(0.00630651\pi\)
−0.999804 + 0.0198112i \(0.993693\pi\)
\(198\) 0 0
\(199\) −1525.71 −0.543492 −0.271746 0.962369i \(-0.587601\pi\)
−0.271746 + 0.962369i \(0.587601\pi\)
\(200\) 0 0
\(201\) 1374.78i 0.482436i
\(202\) 0 0
\(203\) 516.286 + 427.803i 0.178503 + 0.147911i
\(204\) 0 0
\(205\) 6978.01i 2.37739i
\(206\) 0 0
\(207\) 908.908i 0.305186i
\(208\) 0 0
\(209\) 3437.85i 1.13780i
\(210\) 0 0
\(211\) 5626.27 1.83568 0.917841 0.396949i \(-0.129931\pi\)
0.917841 + 0.396949i \(0.129931\pi\)
\(212\) 0 0
\(213\) −1753.58 −0.564101
\(214\) 0 0
\(215\) −7628.64 −2.41986
\(216\) 0 0
\(217\) 3123.21 3769.18i 0.977037 1.17912i
\(218\) 0 0
\(219\) 3297.71 1.01753
\(220\) 0 0
\(221\) 3341.95i 1.01721i
\(222\) 0 0
\(223\) 1409.96 0.423398 0.211699 0.977335i \(-0.432100\pi\)
0.211699 + 0.977335i \(0.432100\pi\)
\(224\) 0 0
\(225\) −2731.54 −0.809344
\(226\) 0 0
\(227\) 848.622i 0.248128i −0.992274 0.124064i \(-0.960407\pi\)
0.992274 0.124064i \(-0.0395928\pi\)
\(228\) 0 0
\(229\) −5880.63 −1.69696 −0.848479 0.529230i \(-0.822481\pi\)
−0.848479 + 0.529230i \(0.822481\pi\)
\(230\) 0 0
\(231\) 1680.54 + 1392.52i 0.478664 + 0.396629i
\(232\) 0 0
\(233\) −3101.62 −0.872077 −0.436038 0.899928i \(-0.643619\pi\)
−0.436038 + 0.899928i \(0.643619\pi\)
\(234\) 0 0
\(235\) −6513.82 −1.80815
\(236\) 0 0
\(237\) 1315.88 0.360657
\(238\) 0 0
\(239\) 6225.18i 1.68483i −0.538831 0.842414i \(-0.681134\pi\)
0.538831 0.842414i \(-0.318866\pi\)
\(240\) 0 0
\(241\) 201.092i 0.0537489i −0.999639 0.0268744i \(-0.991445\pi\)
0.999639 0.0268744i \(-0.00855543\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) 1319.31 + 6976.57i 0.344031 + 1.81925i
\(246\) 0 0
\(247\) 3172.10i 0.817149i
\(248\) 0 0
\(249\) 2166.34 0.551351
\(250\) 0 0
\(251\) 4561.95i 1.14720i −0.819135 0.573601i \(-0.805546\pi\)
0.819135 0.573601i \(-0.194454\pi\)
\(252\) 0 0
\(253\) 3967.02i 0.985789i
\(254\) 0 0
\(255\) −5726.01 −1.40618
\(256\) 0 0
\(257\) 1939.24i 0.470687i −0.971912 0.235343i \(-0.924379\pi\)
0.971912 0.235343i \(-0.0756215\pi\)
\(258\) 0 0
\(259\) −3273.32 2712.33i −0.785305 0.650718i
\(260\) 0 0
\(261\) 325.831i 0.0772737i
\(262\) 0 0
\(263\) 1244.55i 0.291796i 0.989300 + 0.145898i \(0.0466072\pi\)
−0.989300 + 0.145898i \(0.953393\pi\)
\(264\) 0 0
\(265\) 2247.28i 0.520941i
\(266\) 0 0
\(267\) −1357.00 −0.311038
\(268\) 0 0
\(269\) −188.083 −0.0426305 −0.0213153 0.999773i \(-0.506785\pi\)
−0.0213153 + 0.999773i \(0.506785\pi\)
\(270\) 0 0
\(271\) −504.590 −0.113106 −0.0565529 0.998400i \(-0.518011\pi\)
−0.0565529 + 0.998400i \(0.518011\pi\)
\(272\) 0 0
\(273\) 1550.63 + 1284.88i 0.343767 + 0.284851i
\(274\) 0 0
\(275\) 11922.1 2.61428
\(276\) 0 0
\(277\) 4147.02i 0.899532i 0.893146 + 0.449766i \(0.148493\pi\)
−0.893146 + 0.449766i \(0.851507\pi\)
\(278\) 0 0
\(279\) 2378.75 0.510437
\(280\) 0 0
\(281\) −2018.75 −0.428572 −0.214286 0.976771i \(-0.568742\pi\)
−0.214286 + 0.976771i \(0.568742\pi\)
\(282\) 0 0
\(283\) 50.2649i 0.0105581i −0.999986 0.00527905i \(-0.998320\pi\)
0.999986 0.00527905i \(-0.00168038\pi\)
\(284\) 0 0
\(285\) 5434.98 1.12962
\(286\) 0 0
\(287\) 4807.22 + 3983.35i 0.988715 + 0.819267i
\(288\) 0 0
\(289\) −3588.72 −0.730453
\(290\) 0 0
\(291\) −1794.00 −0.361396
\(292\) 0 0
\(293\) −2247.98 −0.448221 −0.224110 0.974564i \(-0.571948\pi\)
−0.224110 + 0.974564i \(0.571948\pi\)
\(294\) 0 0
\(295\) 3837.29i 0.757340i
\(296\) 0 0
\(297\) 1060.60i 0.207213i
\(298\) 0 0
\(299\) 3660.36i 0.707974i
\(300\) 0 0
\(301\) −4354.76 + 5255.45i −0.833901 + 1.00638i
\(302\) 0 0
\(303\) 4245.11i 0.804869i
\(304\) 0 0
\(305\) −3384.59 −0.635413
\(306\) 0 0
\(307\) 1759.91i 0.327177i 0.986529 + 0.163588i \(0.0523069\pi\)
−0.986529 + 0.163588i \(0.947693\pi\)
\(308\) 0 0
\(309\) 3416.57i 0.629004i
\(310\) 0 0
\(311\) −8517.10 −1.55293 −0.776464 0.630162i \(-0.782988\pi\)
−0.776464 + 0.630162i \(0.782988\pi\)
\(312\) 0 0
\(313\) 10113.5i 1.82636i −0.407556 0.913180i \(-0.633619\pi\)
0.407556 0.913180i \(-0.366381\pi\)
\(314\) 0 0
\(315\) −2201.48 + 2656.81i −0.393775 + 0.475220i
\(316\) 0 0
\(317\) 6511.01i 1.15361i −0.816882 0.576805i \(-0.804299\pi\)
0.816882 0.576805i \(-0.195701\pi\)
\(318\) 0 0
\(319\) 1422.12i 0.249604i
\(320\) 0 0
\(321\) 4510.15i 0.784212i
\(322\) 0 0
\(323\) 8069.62 1.39011
\(324\) 0 0
\(325\) 11000.5 1.87753
\(326\) 0 0
\(327\) 1077.85 0.182279
\(328\) 0 0
\(329\) −3718.37 + 4487.44i −0.623101 + 0.751977i
\(330\) 0 0
\(331\) −2031.11 −0.337280 −0.168640 0.985678i \(-0.553938\pi\)
−0.168640 + 0.985678i \(0.553938\pi\)
\(332\) 0 0
\(333\) 2065.81i 0.339957i
\(334\) 0 0
\(335\) −9486.15 −1.54712
\(336\) 0 0
\(337\) −9702.01 −1.56826 −0.784128 0.620599i \(-0.786889\pi\)
−0.784128 + 0.620599i \(0.786889\pi\)
\(338\) 0 0
\(339\) 7089.61i 1.13585i
\(340\) 0 0
\(341\) −10382.3 −1.64878
\(342\) 0 0
\(343\) 5559.35 + 3073.64i 0.875150 + 0.483851i
\(344\) 0 0
\(345\) 6271.57 0.978695
\(346\) 0 0
\(347\) −4652.17 −0.719716 −0.359858 0.933007i \(-0.617175\pi\)
−0.359858 + 0.933007i \(0.617175\pi\)
\(348\) 0 0
\(349\) −5507.67 −0.844753 −0.422376 0.906421i \(-0.638804\pi\)
−0.422376 + 0.906421i \(0.638804\pi\)
\(350\) 0 0
\(351\) 978.612i 0.148816i
\(352\) 0 0
\(353\) 8066.25i 1.21621i 0.793856 + 0.608106i \(0.208071\pi\)
−0.793856 + 0.608106i \(0.791929\pi\)
\(354\) 0 0
\(355\) 12099.9i 1.80900i
\(356\) 0 0
\(357\) −3268.65 + 3944.71i −0.484581 + 0.584807i
\(358\) 0 0
\(359\) 2334.87i 0.343258i −0.985162 0.171629i \(-0.945097\pi\)
0.985162 0.171629i \(-0.0549030\pi\)
\(360\) 0 0
\(361\) −800.476 −0.116704
\(362\) 0 0
\(363\) 636.090i 0.0919726i
\(364\) 0 0
\(365\) 22754.6i 3.26309i
\(366\) 0 0
\(367\) 8870.84 1.26173 0.630864 0.775894i \(-0.282701\pi\)
0.630864 + 0.775894i \(0.282701\pi\)
\(368\) 0 0
\(369\) 3033.87i 0.428013i
\(370\) 0 0
\(371\) −1548.17 1282.84i −0.216650 0.179520i
\(372\) 0 0
\(373\) 8302.86i 1.15256i −0.817251 0.576281i \(-0.804503\pi\)
0.817251 0.576281i \(-0.195497\pi\)
\(374\) 0 0
\(375\) 11085.3i 1.52651i
\(376\) 0 0
\(377\) 1312.19i 0.179261i
\(378\) 0 0
\(379\) −530.766 −0.0719357 −0.0359679 0.999353i \(-0.511451\pi\)
−0.0359679 + 0.999353i \(0.511451\pi\)
\(380\) 0 0
\(381\) 4862.06 0.653782
\(382\) 0 0
\(383\) −534.478 −0.0713069 −0.0356535 0.999364i \(-0.511351\pi\)
−0.0356535 + 0.999364i \(0.511351\pi\)
\(384\) 0 0
\(385\) 9608.57 11595.9i 1.27194 1.53502i
\(386\) 0 0
\(387\) −3316.75 −0.435658
\(388\) 0 0
\(389\) 11253.1i 1.46672i −0.679842 0.733358i \(-0.737952\pi\)
0.679842 0.733358i \(-0.262048\pi\)
\(390\) 0 0
\(391\) 9311.74 1.20439
\(392\) 0 0
\(393\) 1509.69 0.193775
\(394\) 0 0
\(395\) 9079.73i 1.15658i
\(396\) 0 0
\(397\) −8896.66 −1.12471 −0.562356 0.826895i \(-0.690105\pi\)
−0.562356 + 0.826895i \(0.690105\pi\)
\(398\) 0 0
\(399\) 3102.52 3744.22i 0.389274 0.469788i
\(400\) 0 0
\(401\) 6210.10 0.773360 0.386680 0.922214i \(-0.373622\pi\)
0.386680 + 0.922214i \(0.373622\pi\)
\(402\) 0 0
\(403\) −9579.73 −1.18412
\(404\) 0 0
\(405\) −1676.73 −0.205722
\(406\) 0 0
\(407\) 9016.44i 1.09810i
\(408\) 0 0
\(409\) 3899.47i 0.471434i 0.971822 + 0.235717i \(0.0757439\pi\)
−0.971822 + 0.235717i \(0.924256\pi\)
\(410\) 0 0
\(411\) 1733.31i 0.208024i
\(412\) 0 0
\(413\) −2643.54 2190.49i −0.314965 0.260985i
\(414\) 0 0
\(415\) 14948.0i 1.76812i
\(416\) 0 0
\(417\) −1237.35 −0.145307
\(418\) 0 0
\(419\) 12136.9i 1.41510i 0.706663 + 0.707550i \(0.250200\pi\)
−0.706663 + 0.707550i \(0.749800\pi\)
\(420\) 0 0
\(421\) 5388.65i 0.623817i −0.950112 0.311908i \(-0.899032\pi\)
0.950112 0.311908i \(-0.100968\pi\)
\(422\) 0 0
\(423\) −2832.05 −0.325529
\(424\) 0 0
\(425\) 27984.5i 3.19400i
\(426\) 0 0
\(427\) −1932.07 + 2331.68i −0.218968 + 0.264257i
\(428\) 0 0
\(429\) 4271.25i 0.480695i
\(430\) 0 0
\(431\) 2284.44i 0.255308i 0.991819 + 0.127654i \(0.0407447\pi\)
−0.991819 + 0.127654i \(0.959255\pi\)
\(432\) 0 0
\(433\) 1711.79i 0.189985i 0.995478 + 0.0949925i \(0.0302827\pi\)
−0.995478 + 0.0949925i \(0.969717\pi\)
\(434\) 0 0
\(435\) 2248.27 0.247808
\(436\) 0 0
\(437\) −8838.47 −0.967509
\(438\) 0 0
\(439\) −2820.52 −0.306643 −0.153322 0.988176i \(-0.548997\pi\)
−0.153322 + 0.988176i \(0.548997\pi\)
\(440\) 0 0
\(441\) 573.604 + 3033.24i 0.0619376 + 0.327528i
\(442\) 0 0
\(443\) 13392.2 1.43630 0.718150 0.695888i \(-0.244989\pi\)
0.718150 + 0.695888i \(0.244989\pi\)
\(444\) 0 0
\(445\) 9363.47i 0.997463i
\(446\) 0 0
\(447\) −8957.03 −0.947769
\(448\) 0 0
\(449\) −15186.9 −1.59624 −0.798120 0.602498i \(-0.794172\pi\)
−0.798120 + 0.602498i \(0.794172\pi\)
\(450\) 0 0
\(451\) 13241.6i 1.38254i
\(452\) 0 0
\(453\) 8364.34 0.867530
\(454\) 0 0
\(455\) 8865.81 10699.5i 0.913485 1.10242i
\(456\) 0 0
\(457\) −19132.7 −1.95840 −0.979202 0.202888i \(-0.934967\pi\)
−0.979202 + 0.202888i \(0.934967\pi\)
\(458\) 0 0
\(459\) −2489.53 −0.253162
\(460\) 0 0
\(461\) −237.892 −0.0240341 −0.0120170 0.999928i \(-0.503825\pi\)
−0.0120170 + 0.999928i \(0.503825\pi\)
\(462\) 0 0
\(463\) 6527.48i 0.655200i 0.944816 + 0.327600i \(0.106240\pi\)
−0.944816 + 0.327600i \(0.893760\pi\)
\(464\) 0 0
\(465\) 16413.6i 1.63691i
\(466\) 0 0
\(467\) 488.599i 0.0484147i −0.999707 0.0242073i \(-0.992294\pi\)
0.999707 0.0242073i \(-0.00770619\pi\)
\(468\) 0 0
\(469\) −5415.10 + 6535.10i −0.533147 + 0.643418i
\(470\) 0 0
\(471\) 4849.95i 0.474467i
\(472\) 0 0
\(473\) 14476.3 1.40723
\(474\) 0 0
\(475\) 26562.2i 2.56580i
\(476\) 0 0
\(477\) 977.063i 0.0937875i
\(478\) 0 0
\(479\) −5872.67 −0.560186 −0.280093 0.959973i \(-0.590365\pi\)
−0.280093 + 0.959973i \(0.590365\pi\)
\(480\) 0 0
\(481\) 8319.45i 0.788637i
\(482\) 0 0
\(483\) 3580.08 4320.55i 0.337266 0.407022i
\(484\) 0 0
\(485\) 12378.8i 1.15895i
\(486\) 0 0
\(487\) 14307.7i 1.33130i 0.746263 + 0.665652i \(0.231846\pi\)
−0.746263 + 0.665652i \(0.768154\pi\)
\(488\) 0 0
\(489\) 5804.29i 0.536767i
\(490\) 0 0
\(491\) −10665.1 −0.980264 −0.490132 0.871648i \(-0.663052\pi\)
−0.490132 + 0.871648i \(0.663052\pi\)
\(492\) 0 0
\(493\) 3338.13 0.304953
\(494\) 0 0
\(495\) 7318.25 0.664507
\(496\) 0 0
\(497\) 8335.75 + 6907.15i 0.752333 + 0.623397i
\(498\) 0 0
\(499\) −11905.8 −1.06809 −0.534046 0.845455i \(-0.679329\pi\)
−0.534046 + 0.845455i \(0.679329\pi\)
\(500\) 0 0
\(501\) 2449.36i 0.218422i
\(502\) 0 0
\(503\) 11100.0 0.983941 0.491971 0.870612i \(-0.336277\pi\)
0.491971 + 0.870612i \(0.336277\pi\)
\(504\) 0 0
\(505\) 29291.8 2.58112
\(506\) 0 0
\(507\) 2649.92i 0.232125i
\(508\) 0 0
\(509\) 8580.71 0.747216 0.373608 0.927587i \(-0.378120\pi\)
0.373608 + 0.927587i \(0.378120\pi\)
\(510\) 0 0
\(511\) −15675.8 12989.3i −1.35706 1.12448i
\(512\) 0 0
\(513\) 2363.00 0.203370
\(514\) 0 0
\(515\) 23574.7 2.01714
\(516\) 0 0
\(517\) 12360.8 1.05150
\(518\) 0 0
\(519\) 10932.5i 0.924635i
\(520\) 0 0
\(521\) 4351.09i 0.365882i −0.983124 0.182941i \(-0.941438\pi\)
0.983124 0.182941i \(-0.0585618\pi\)
\(522\) 0 0
\(523\) 3685.85i 0.308166i 0.988058 + 0.154083i \(0.0492424\pi\)
−0.988058 + 0.154083i \(0.950758\pi\)
\(524\) 0 0
\(525\) 12984.5 + 10759.2i 1.07941 + 0.894419i
\(526\) 0 0
\(527\) 24370.2i 2.01439i
\(528\) 0 0
\(529\) 1968.06 0.161754
\(530\) 0 0
\(531\) 1668.36i 0.136348i
\(532\) 0 0
\(533\) 12218.0i 0.992910i
\(534\) 0 0
\(535\) 31120.5 2.51487
\(536\) 0 0
\(537\) 10247.4i 0.823479i
\(538\) 0 0
\(539\) −2503.55 13238.9i −0.200066 1.05796i
\(540\) 0 0
\(541\) 21536.0i 1.71147i 0.517413 + 0.855736i \(0.326895\pi\)
−0.517413 + 0.855736i \(0.673105\pi\)
\(542\) 0 0
\(543\) 10541.8i 0.833138i
\(544\) 0 0
\(545\) 7437.28i 0.584547i
\(546\) 0 0
\(547\) 5550.29 0.433845 0.216923 0.976189i \(-0.430398\pi\)
0.216923 + 0.976189i \(0.430398\pi\)
\(548\) 0 0
\(549\) −1471.54 −0.114396
\(550\) 0 0
\(551\) −3168.47 −0.244975
\(552\) 0 0
\(553\) −6255.12 5183.10i −0.481003 0.398568i
\(554\) 0 0
\(555\) −14254.3 −1.09020
\(556\) 0 0
\(557\) 14848.3i 1.12952i −0.825256 0.564758i \(-0.808969\pi\)
0.825256 0.564758i \(-0.191031\pi\)
\(558\) 0 0
\(559\) 13357.2 1.01065
\(560\) 0 0
\(561\) 10865.8 0.817744
\(562\) 0 0
\(563\) 1189.55i 0.0890469i −0.999008 0.0445235i \(-0.985823\pi\)
0.999008 0.0445235i \(-0.0141769\pi\)
\(564\) 0 0
\(565\) 48919.1 3.64255
\(566\) 0 0
\(567\) −957.148 + 1155.11i −0.0708932 + 0.0855560i
\(568\) 0 0
\(569\) 9572.43 0.705267 0.352634 0.935761i \(-0.385286\pi\)
0.352634 + 0.935761i \(0.385286\pi\)
\(570\) 0 0
\(571\) −19487.5 −1.42825 −0.714123 0.700021i \(-0.753174\pi\)
−0.714123 + 0.700021i \(0.753174\pi\)
\(572\) 0 0
\(573\) 9174.46 0.668881
\(574\) 0 0
\(575\) 30650.8i 2.22300i
\(576\) 0 0
\(577\) 24751.1i 1.78579i 0.450263 + 0.892896i \(0.351330\pi\)
−0.450263 + 0.892896i \(0.648670\pi\)
\(578\) 0 0
\(579\) 5684.67i 0.408026i
\(580\) 0 0
\(581\) −10297.8 8532.96i −0.735329 0.609306i
\(582\) 0 0
\(583\) 4264.49i 0.302945i
\(584\) 0 0
\(585\) 6752.54 0.477236
\(586\) 0 0
\(587\) 15015.1i 1.05577i −0.849315 0.527886i \(-0.822985\pi\)
0.849315 0.527886i \(-0.177015\pi\)
\(588\) 0 0
\(589\) 23131.6i 1.61820i
\(590\) 0 0
\(591\) 328.671 0.0228760
\(592\) 0 0
\(593\) 18080.4i 1.25206i 0.779799 + 0.626030i \(0.215321\pi\)
−0.779799 + 0.626030i \(0.784679\pi\)
\(594\) 0 0
\(595\) 27218.9 + 22554.1i 1.87541 + 1.55399i
\(596\) 0 0
\(597\) 4577.13i 0.313785i
\(598\) 0 0
\(599\) 18896.2i 1.28894i −0.764628 0.644472i \(-0.777077\pi\)
0.764628 0.644472i \(-0.222923\pi\)
\(600\) 0 0
\(601\) 316.737i 0.0214974i −0.999942 0.0107487i \(-0.996579\pi\)
0.999942 0.0107487i \(-0.00342149\pi\)
\(602\) 0 0
\(603\) −4124.34 −0.278534
\(604\) 0 0
\(605\) −4389.09 −0.294945
\(606\) 0 0
\(607\) −24428.2 −1.63346 −0.816730 0.577020i \(-0.804216\pi\)
−0.816730 + 0.577020i \(0.804216\pi\)
\(608\) 0 0
\(609\) 1283.41 1548.86i 0.0853964 0.103059i
\(610\) 0 0
\(611\) 11405.3 0.755168
\(612\) 0 0
\(613\) 24938.5i 1.64316i −0.570092 0.821581i \(-0.693092\pi\)
0.570092 0.821581i \(-0.306908\pi\)
\(614\) 0 0
\(615\) 20934.0 1.37259
\(616\) 0 0
\(617\) −10603.6 −0.691873 −0.345936 0.938258i \(-0.612439\pi\)
−0.345936 + 0.938258i \(0.612439\pi\)
\(618\) 0 0
\(619\) 17013.3i 1.10472i −0.833604 0.552362i \(-0.813727\pi\)
0.833604 0.552362i \(-0.186273\pi\)
\(620\) 0 0
\(621\) 2726.72 0.176199
\(622\) 0 0
\(623\) 6450.59 + 5345.07i 0.414827 + 0.343733i
\(624\) 0 0
\(625\) 38551.7 2.46731
\(626\) 0 0
\(627\) −10313.5 −0.656911
\(628\) 0 0
\(629\) −21164.2 −1.34161
\(630\) 0 0
\(631\) 2157.13i 0.136092i 0.997682 + 0.0680460i \(0.0216765\pi\)
−0.997682 + 0.0680460i \(0.978324\pi\)
\(632\) 0 0
\(633\) 16878.8i 1.05983i
\(634\) 0 0
\(635\) 33548.8i 2.09660i
\(636\) 0 0
\(637\) −2310.02 12215.5i −0.143684 0.759805i
\(638\) 0 0
\(639\) 5260.75i 0.325684i
\(640\) 0 0
\(641\) −22030.9 −1.35752 −0.678758 0.734362i \(-0.737481\pi\)
−0.678758 + 0.734362i \(0.737481\pi\)
\(642\) 0 0
\(643\) 30067.6i 1.84409i 0.387081 + 0.922046i \(0.373483\pi\)
−0.387081 + 0.922046i \(0.626517\pi\)
\(644\) 0 0
\(645\) 22885.9i 1.39711i
\(646\) 0 0
\(647\) −6954.93 −0.422607 −0.211303 0.977421i \(-0.567771\pi\)
−0.211303 + 0.977421i \(0.567771\pi\)
\(648\) 0 0
\(649\) 7281.72i 0.440420i
\(650\) 0 0
\(651\) −11307.5 9369.62i −0.680763 0.564092i
\(652\) 0 0
\(653\) 1199.25i 0.0718686i −0.999354 0.0359343i \(-0.988559\pi\)
0.999354 0.0359343i \(-0.0114407\pi\)
\(654\) 0 0
\(655\) 10417.0i 0.621415i
\(656\) 0 0
\(657\) 9893.12i 0.587469i
\(658\) 0 0
\(659\) −13611.9 −0.804619 −0.402310 0.915504i \(-0.631792\pi\)
−0.402310 + 0.915504i \(0.631792\pi\)
\(660\) 0 0
\(661\) 30065.6 1.76916 0.884582 0.466385i \(-0.154444\pi\)
0.884582 + 0.466385i \(0.154444\pi\)
\(662\) 0 0
\(663\) 10025.9 0.587288
\(664\) 0 0
\(665\) −25835.5 21407.8i −1.50655 1.24836i
\(666\) 0 0
\(667\) −3656.18 −0.212246
\(668\) 0 0
\(669\) 4229.88i 0.244449i
\(670\) 0 0
\(671\) 6422.67 0.369515
\(672\) 0 0
\(673\) 10262.1 0.587777 0.293888 0.955840i \(-0.405051\pi\)
0.293888 + 0.955840i \(0.405051\pi\)
\(674\) 0 0
\(675\) 8194.61i 0.467275i
\(676\) 0 0
\(677\) 426.182 0.0241943 0.0120971 0.999927i \(-0.496149\pi\)
0.0120971 + 0.999927i \(0.496149\pi\)
\(678\) 0 0
\(679\) 8527.88 + 7066.35i 0.481988 + 0.399384i
\(680\) 0 0
\(681\) −2545.87 −0.143257
\(682\) 0 0
\(683\) −2385.48 −0.133643 −0.0668213 0.997765i \(-0.521286\pi\)
−0.0668213 + 0.997765i \(0.521286\pi\)
\(684\) 0 0
\(685\) 11960.0 0.667108
\(686\) 0 0
\(687\) 17641.9i 0.979739i
\(688\) 0 0
\(689\) 3934.84i 0.217570i
\(690\) 0 0
\(691\) 9485.21i 0.522192i 0.965313 + 0.261096i \(0.0840839\pi\)
−0.965313 + 0.261096i \(0.915916\pi\)
\(692\) 0 0
\(693\) 4177.57 5041.62i 0.228994 0.276357i
\(694\) 0 0
\(695\) 8537.84i 0.465984i
\(696\) 0 0
\(697\) 31081.9 1.68911
\(698\) 0 0
\(699\) 9304.86i 0.503494i
\(700\) 0 0
\(701\) 13463.5i 0.725404i 0.931905 + 0.362702i \(0.118146\pi\)
−0.931905 + 0.362702i \(0.881854\pi\)
\(702\) 0 0
\(703\) 20088.5 1.07774
\(704\) 0 0
\(705\) 19541.5i 1.04393i
\(706\) 0 0
\(707\) 16721.0 20179.4i 0.889473 1.07344i
\(708\) 0 0
\(709\) 1482.11i 0.0785075i −0.999229 0.0392538i \(-0.987502\pi\)
0.999229 0.0392538i \(-0.0124981\pi\)
\(710\) 0 0
\(711\) 3947.65i 0.208225i
\(712\) 0 0
\(713\) 26692.2i 1.40200i
\(714\) 0 0
\(715\) −29472.1 −1.54153
\(716\) 0 0
\(717\) −18675.6 −0.972735
\(718\) 0 0
\(719\) 33887.8 1.75772 0.878862 0.477077i \(-0.158304\pi\)
0.878862 + 0.477077i \(0.158304\pi\)
\(720\) 0 0
\(721\) 13457.5 16240.9i 0.695122 0.838893i
\(722\) 0 0
\(723\) −603.276 −0.0310319
\(724\) 0 0
\(725\) 10987.9i 0.562869i
\(726\) 0 0
\(727\) 9840.41 0.502009 0.251004 0.967986i \(-0.419239\pi\)
0.251004 + 0.967986i \(0.419239\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 33980.0i 1.71928i
\(732\) 0 0
\(733\) −13241.0 −0.667212 −0.333606 0.942713i \(-0.608265\pi\)
−0.333606 + 0.942713i \(0.608265\pi\)
\(734\) 0 0
\(735\) 20929.7 3957.93i 1.05035 0.198627i
\(736\) 0 0
\(737\) 18001.1 0.899701
\(738\) 0 0
\(739\) −6185.80 −0.307914 −0.153957 0.988078i \(-0.549202\pi\)
−0.153957 + 0.988078i \(0.549202\pi\)
\(740\) 0 0
\(741\) −9516.29 −0.471781
\(742\) 0 0
\(743\) 11911.6i 0.588148i 0.955783 + 0.294074i \(0.0950112\pi\)
−0.955783 + 0.294074i \(0.904989\pi\)
\(744\) 0 0
\(745\) 61804.5i 3.03939i
\(746\) 0 0
\(747\) 6499.02i 0.318322i
\(748\) 0 0
\(749\) 17764.9 21439.3i 0.866645 1.04589i
\(750\) 0 0
\(751\) 28902.1i 1.40433i 0.712015 + 0.702165i \(0.247783\pi\)
−0.712015 + 0.702165i \(0.752217\pi\)
\(752\) 0 0
\(753\) −13685.8 −0.662337
\(754\) 0 0
\(755\) 57714.9i 2.78207i
\(756\) 0 0
\(757\) 7827.05i 0.375798i −0.982188 0.187899i \(-0.939832\pi\)
0.982188 0.187899i \(-0.0601678\pi\)
\(758\) 0 0
\(759\) −11901.1 −0.569145
\(760\) 0 0
\(761\) 4882.67i 0.232584i 0.993215 + 0.116292i \(0.0371008\pi\)
−0.993215 + 0.116292i \(0.962899\pi\)
\(762\) 0 0
\(763\) −5123.62 4245.52i −0.243103 0.201439i
\(764\) 0 0
\(765\) 17178.0i 0.811860i
\(766\) 0 0
\(767\) 6718.83i 0.316301i
\(768\) 0 0
\(769\) 20071.4i 0.941215i 0.882343 + 0.470607i \(0.155965\pi\)
−0.882343 + 0.470607i \(0.844035\pi\)
\(770\) 0 0
\(771\) −5817.72 −0.271751
\(772\) 0 0
\(773\) −4965.70 −0.231053 −0.115526 0.993304i \(-0.536855\pi\)
−0.115526 + 0.993304i \(0.536855\pi\)
\(774\) 0 0
\(775\) −80217.8 −3.71808
\(776\) 0 0
\(777\) −8136.98 + 9819.95i −0.375692 + 0.453396i
\(778\) 0 0
\(779\) −29502.1 −1.35690
\(780\) 0 0
\(781\) 22961.1i 1.05200i
\(782\) 0 0
\(783\) 977.493 0.0446140
\(784\) 0 0
\(785\) 33465.2 1.52156
\(786\) 0 0
\(787\) 40092.6i 1.81594i 0.419035 + 0.907970i \(0.362369\pi\)
−0.419035 + 0.907970i \(0.637631\pi\)
\(788\) 0 0
\(789\) 3733.66 0.168469
\(790\) 0 0
\(791\) 27925.1 33700.9i 1.25525 1.51487i
\(792\) 0 0
\(793\) 5926.18 0.265378
\(794\) 0 0
\(795\) −6741.84 −0.300765
\(796\) 0 0
\(797\) −26919.4 −1.19640 −0.598202 0.801345i \(-0.704118\pi\)
−0.598202 + 0.801345i \(0.704118\pi\)
\(798\) 0 0
\(799\) 29014.3i 1.28467i
\(800\) 0 0
\(801\) 4071.01i 0.179578i
\(802\)