Properties

Label 1344.4.p.c.223.17
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 223.17
Character \(\chi\) \(=\) 1344.223
Dual form 1344.4.p.c.223.18

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000i q^{3} +5.86484 q^{5} +(18.3297 - 2.64978i) q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} +5.86484 q^{5} +(18.3297 - 2.64978i) q^{7} -9.00000 q^{9} -34.5206 q^{11} +55.8600 q^{13} -17.5945i q^{15} -2.77104i q^{17} +67.8756i q^{19} +(-7.94934 - 54.9892i) q^{21} +176.293i q^{23} -90.6036 q^{25} +27.0000i q^{27} +116.050i q^{29} +312.323 q^{31} +103.562i q^{33} +(107.501 - 15.5406i) q^{35} -118.151i q^{37} -167.580i q^{39} +280.120i q^{41} -15.1530 q^{43} -52.7836 q^{45} +6.34315 q^{47} +(328.957 - 97.1395i) q^{49} -8.31311 q^{51} -23.2291i q^{53} -202.458 q^{55} +203.627 q^{57} +288.884i q^{59} -514.810 q^{61} +(-164.967 + 23.8480i) q^{63} +327.610 q^{65} +295.257 q^{67} +528.879 q^{69} +475.881i q^{71} -473.829i q^{73} +271.811i q^{75} +(-632.753 + 91.4720i) q^{77} +796.016i q^{79} +81.0000 q^{81} +877.767i q^{83} -16.2517i q^{85} +348.149 q^{87} -33.8577i q^{89} +(1023.90 - 148.017i) q^{91} -936.970i q^{93} +398.080i q^{95} +700.376i q^{97} +310.685 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 32q - 288q^{9} + O(q^{10}) \) \( 32q - 288q^{9} - 224q^{13} - 72q^{21} + 1120q^{25} - 752q^{49} - 672q^{57} - 544q^{61} + 1536q^{65} - 144q^{69} - 1632q^{77} + 2592q^{81} + O(q^{100}) \)

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
\(5\) 5.86484 0.524568 0.262284 0.964991i \(-0.415524\pi\)
0.262284 + 0.964991i \(0.415524\pi\)
\(6\) 0 0
\(7\) 18.3297 2.64978i 0.989712 0.143075i
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −34.5206 −0.946214 −0.473107 0.881005i \(-0.656868\pi\)
−0.473107 + 0.881005i \(0.656868\pi\)
\(12\) 0 0
\(13\) 55.8600 1.19175 0.595876 0.803076i \(-0.296805\pi\)
0.595876 + 0.803076i \(0.296805\pi\)
\(14\) 0 0
\(15\) 17.5945i 0.302859i
\(16\) 0 0
\(17\) 2.77104i 0.0395338i −0.999805 0.0197669i \(-0.993708\pi\)
0.999805 0.0197669i \(-0.00629241\pi\)
\(18\) 0 0
\(19\) 67.8756i 0.819565i 0.912183 + 0.409782i \(0.134395\pi\)
−0.912183 + 0.409782i \(0.865605\pi\)
\(20\) 0 0
\(21\) −7.94934 54.9892i −0.0826042 0.571410i
\(22\) 0 0
\(23\) 176.293i 1.59825i 0.601167 + 0.799123i \(0.294703\pi\)
−0.601167 + 0.799123i \(0.705297\pi\)
\(24\) 0 0
\(25\) −90.6036 −0.724829
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 116.050i 0.743099i 0.928413 + 0.371550i \(0.121173\pi\)
−0.928413 + 0.371550i \(0.878827\pi\)
\(30\) 0 0
\(31\) 312.323 1.80951 0.904757 0.425927i \(-0.140052\pi\)
0.904757 + 0.425927i \(0.140052\pi\)
\(32\) 0 0
\(33\) 103.562i 0.546297i
\(34\) 0 0
\(35\) 107.501 15.5406i 0.519171 0.0750524i
\(36\) 0 0
\(37\) 118.151i 0.524973i −0.964936 0.262486i \(-0.915458\pi\)
0.964936 0.262486i \(-0.0845424\pi\)
\(38\) 0 0
\(39\) 167.580i 0.688059i
\(40\) 0 0
\(41\) 280.120i 1.06701i 0.845797 + 0.533505i \(0.179125\pi\)
−0.845797 + 0.533505i \(0.820875\pi\)
\(42\) 0 0
\(43\) −15.1530 −0.0537396 −0.0268698 0.999639i \(-0.508554\pi\)
−0.0268698 + 0.999639i \(0.508554\pi\)
\(44\) 0 0
\(45\) −52.7836 −0.174856
\(46\) 0 0
\(47\) 6.34315 0.0196860 0.00984302 0.999952i \(-0.496867\pi\)
0.00984302 + 0.999952i \(0.496867\pi\)
\(48\) 0 0
\(49\) 328.957 97.1395i 0.959059 0.283206i
\(50\) 0 0
\(51\) −8.31311 −0.0228249
\(52\) 0 0
\(53\) 23.2291i 0.0602029i −0.999547 0.0301015i \(-0.990417\pi\)
0.999547 0.0301015i \(-0.00958304\pi\)
\(54\) 0 0
\(55\) −202.458 −0.496353
\(56\) 0 0
\(57\) 203.627 0.473176
\(58\) 0 0
\(59\) 288.884i 0.637450i 0.947847 + 0.318725i \(0.103255\pi\)
−0.947847 + 0.318725i \(0.896745\pi\)
\(60\) 0 0
\(61\) −514.810 −1.08057 −0.540285 0.841482i \(-0.681683\pi\)
−0.540285 + 0.841482i \(0.681683\pi\)
\(62\) 0 0
\(63\) −164.967 + 23.8480i −0.329904 + 0.0476916i
\(64\) 0 0
\(65\) 327.610 0.625155
\(66\) 0 0
\(67\) 295.257 0.538379 0.269190 0.963087i \(-0.413244\pi\)
0.269190 + 0.963087i \(0.413244\pi\)
\(68\) 0 0
\(69\) 528.879 0.922748
\(70\) 0 0
\(71\) 475.881i 0.795447i 0.917505 + 0.397724i \(0.130200\pi\)
−0.917505 + 0.397724i \(0.869800\pi\)
\(72\) 0 0
\(73\) 473.829i 0.759691i −0.925050 0.379845i \(-0.875977\pi\)
0.925050 0.379845i \(-0.124023\pi\)
\(74\) 0 0
\(75\) 271.811i 0.418480i
\(76\) 0 0
\(77\) −632.753 + 91.4720i −0.936479 + 0.135379i
\(78\) 0 0
\(79\) 796.016i 1.13366i 0.823836 + 0.566828i \(0.191829\pi\)
−0.823836 + 0.566828i \(0.808171\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 877.767i 1.16081i 0.814327 + 0.580406i \(0.197106\pi\)
−0.814327 + 0.580406i \(0.802894\pi\)
\(84\) 0 0
\(85\) 16.2517i 0.0207382i
\(86\) 0 0
\(87\) 348.149 0.429029
\(88\) 0 0
\(89\) 33.8577i 0.0403249i −0.999797 0.0201624i \(-0.993582\pi\)
0.999797 0.0201624i \(-0.00641833\pi\)
\(90\) 0 0
\(91\) 1023.90 148.017i 1.17949 0.170510i
\(92\) 0 0
\(93\) 936.970i 1.04472i
\(94\) 0 0
\(95\) 398.080i 0.429917i
\(96\) 0 0
\(97\) 700.376i 0.733118i 0.930395 + 0.366559i \(0.119464\pi\)
−0.930395 + 0.366559i \(0.880536\pi\)
\(98\) 0 0
\(99\) 310.685 0.315405
\(100\) 0 0
\(101\) 292.370 0.288038 0.144019 0.989575i \(-0.453997\pi\)
0.144019 + 0.989575i \(0.453997\pi\)
\(102\) 0 0
\(103\) −491.442 −0.470128 −0.235064 0.971980i \(-0.575530\pi\)
−0.235064 + 0.971980i \(0.575530\pi\)
\(104\) 0 0
\(105\) −46.6217 322.503i −0.0433315 0.299743i
\(106\) 0 0
\(107\) −1953.70 −1.76516 −0.882578 0.470166i \(-0.844194\pi\)
−0.882578 + 0.470166i \(0.844194\pi\)
\(108\) 0 0
\(109\) 1887.60i 1.65871i −0.558725 0.829353i \(-0.688709\pi\)
0.558725 0.829353i \(-0.311291\pi\)
\(110\) 0 0
\(111\) −354.454 −0.303093
\(112\) 0 0
\(113\) 1439.91 1.19872 0.599361 0.800479i \(-0.295421\pi\)
0.599361 + 0.800479i \(0.295421\pi\)
\(114\) 0 0
\(115\) 1033.93i 0.838388i
\(116\) 0 0
\(117\) −502.740 −0.397251
\(118\) 0 0
\(119\) −7.34264 50.7923i −0.00565629 0.0391271i
\(120\) 0 0
\(121\) −139.328 −0.104679
\(122\) 0 0
\(123\) 840.360 0.616038
\(124\) 0 0
\(125\) −1264.48 −0.904789
\(126\) 0 0
\(127\) 627.432i 0.438390i 0.975681 + 0.219195i \(0.0703432\pi\)
−0.975681 + 0.219195i \(0.929657\pi\)
\(128\) 0 0
\(129\) 45.4589i 0.0310266i
\(130\) 0 0
\(131\) 2258.12i 1.50605i −0.657991 0.753026i \(-0.728593\pi\)
0.657991 0.753026i \(-0.271407\pi\)
\(132\) 0 0
\(133\) 179.855 + 1244.14i 0.117259 + 0.811133i
\(134\) 0 0
\(135\) 158.351i 0.100953i
\(136\) 0 0
\(137\) 2568.10 1.60151 0.800757 0.598990i \(-0.204431\pi\)
0.800757 + 0.598990i \(0.204431\pi\)
\(138\) 0 0
\(139\) 1203.02i 0.734095i −0.930202 0.367047i \(-0.880369\pi\)
0.930202 0.367047i \(-0.119631\pi\)
\(140\) 0 0
\(141\) 19.0295i 0.0113657i
\(142\) 0 0
\(143\) −1928.32 −1.12765
\(144\) 0 0
\(145\) 680.613i 0.389806i
\(146\) 0 0
\(147\) −291.418 986.872i −0.163509 0.553713i
\(148\) 0 0
\(149\) 1629.70i 0.896044i −0.894022 0.448022i \(-0.852129\pi\)
0.894022 0.448022i \(-0.147871\pi\)
\(150\) 0 0
\(151\) 3458.49i 1.86389i 0.362597 + 0.931946i \(0.381890\pi\)
−0.362597 + 0.931946i \(0.618110\pi\)
\(152\) 0 0
\(153\) 24.9393i 0.0131779i
\(154\) 0 0
\(155\) 1831.73 0.949213
\(156\) 0 0
\(157\) 1673.84 0.850875 0.425437 0.904988i \(-0.360120\pi\)
0.425437 + 0.904988i \(0.360120\pi\)
\(158\) 0 0
\(159\) −69.6872 −0.0347582
\(160\) 0 0
\(161\) 467.138 + 3231.40i 0.228669 + 1.58180i
\(162\) 0 0
\(163\) 2843.38 1.36632 0.683162 0.730267i \(-0.260604\pi\)
0.683162 + 0.730267i \(0.260604\pi\)
\(164\) 0 0
\(165\) 607.374i 0.286570i
\(166\) 0 0
\(167\) 3912.38 1.81287 0.906434 0.422348i \(-0.138794\pi\)
0.906434 + 0.422348i \(0.138794\pi\)
\(168\) 0 0
\(169\) 923.343 0.420274
\(170\) 0 0
\(171\) 610.880i 0.273188i
\(172\) 0 0
\(173\) 746.440 0.328039 0.164019 0.986457i \(-0.447554\pi\)
0.164019 + 0.986457i \(0.447554\pi\)
\(174\) 0 0
\(175\) −1660.74 + 240.080i −0.717372 + 0.103705i
\(176\) 0 0
\(177\) 866.653 0.368032
\(178\) 0 0
\(179\) 3708.08 1.54835 0.774176 0.632970i \(-0.218164\pi\)
0.774176 + 0.632970i \(0.218164\pi\)
\(180\) 0 0
\(181\) 2146.34 0.881415 0.440708 0.897651i \(-0.354728\pi\)
0.440708 + 0.897651i \(0.354728\pi\)
\(182\) 0 0
\(183\) 1544.43i 0.623867i
\(184\) 0 0
\(185\) 692.940i 0.275384i
\(186\) 0 0
\(187\) 95.6579i 0.0374075i
\(188\) 0 0
\(189\) 71.5441 + 494.902i 0.0275347 + 0.190470i
\(190\) 0 0
\(191\) 3770.63i 1.42845i 0.699918 + 0.714224i \(0.253220\pi\)
−0.699918 + 0.714224i \(0.746780\pi\)
\(192\) 0 0
\(193\) 361.180 0.134706 0.0673531 0.997729i \(-0.478545\pi\)
0.0673531 + 0.997729i \(0.478545\pi\)
\(194\) 0 0
\(195\) 982.831i 0.360933i
\(196\) 0 0
\(197\) 4532.77i 1.63932i −0.572848 0.819662i \(-0.694161\pi\)
0.572848 0.819662i \(-0.305839\pi\)
\(198\) 0 0
\(199\) −318.530 −0.113467 −0.0567337 0.998389i \(-0.518069\pi\)
−0.0567337 + 0.998389i \(0.518069\pi\)
\(200\) 0 0
\(201\) 885.772i 0.310833i
\(202\) 0 0
\(203\) 307.506 + 2127.16i 0.106319 + 0.735454i
\(204\) 0 0
\(205\) 1642.86i 0.559719i
\(206\) 0 0
\(207\) 1586.64i 0.532749i
\(208\) 0 0
\(209\) 2343.11i 0.775484i
\(210\) 0 0
\(211\) −791.205 −0.258146 −0.129073 0.991635i \(-0.541200\pi\)
−0.129073 + 0.991635i \(0.541200\pi\)
\(212\) 0 0
\(213\) 1427.64 0.459252
\(214\) 0 0
\(215\) −88.8697 −0.0281901
\(216\) 0 0
\(217\) 5724.80 827.589i 1.79090 0.258896i
\(218\) 0 0
\(219\) −1421.49 −0.438608
\(220\) 0 0
\(221\) 154.790i 0.0471146i
\(222\) 0 0
\(223\) −1013.41 −0.304317 −0.152158 0.988356i \(-0.548622\pi\)
−0.152158 + 0.988356i \(0.548622\pi\)
\(224\) 0 0
\(225\) 815.432 0.241610
\(226\) 0 0
\(227\) 6000.78i 1.75456i 0.479976 + 0.877282i \(0.340645\pi\)
−0.479976 + 0.877282i \(0.659355\pi\)
\(228\) 0 0
\(229\) 1222.77 0.352851 0.176426 0.984314i \(-0.443546\pi\)
0.176426 + 0.984314i \(0.443546\pi\)
\(230\) 0 0
\(231\) 274.416 + 1898.26i 0.0781613 + 0.540676i
\(232\) 0 0
\(233\) 4159.85 1.16962 0.584808 0.811172i \(-0.301170\pi\)
0.584808 + 0.811172i \(0.301170\pi\)
\(234\) 0 0
\(235\) 37.2016 0.0103267
\(236\) 0 0
\(237\) 2388.05 0.654516
\(238\) 0 0
\(239\) 3144.05i 0.850929i 0.904975 + 0.425464i \(0.139889\pi\)
−0.904975 + 0.425464i \(0.860111\pi\)
\(240\) 0 0
\(241\) 578.042i 0.154502i −0.997012 0.0772509i \(-0.975386\pi\)
0.997012 0.0772509i \(-0.0246143\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) 1929.28 569.708i 0.503091 0.148560i
\(246\) 0 0
\(247\) 3791.53i 0.976719i
\(248\) 0 0
\(249\) 2633.30 0.670195
\(250\) 0 0
\(251\) 1928.17i 0.484882i −0.970166 0.242441i \(-0.922052\pi\)
0.970166 0.242441i \(-0.0779480\pi\)
\(252\) 0 0
\(253\) 6085.75i 1.51228i
\(254\) 0 0
\(255\) −48.7551 −0.0119732
\(256\) 0 0
\(257\) 1012.74i 0.245810i 0.992418 + 0.122905i \(0.0392210\pi\)
−0.992418 + 0.122905i \(0.960779\pi\)
\(258\) 0 0
\(259\) −313.076 2165.68i −0.0751103 0.519572i
\(260\) 0 0
\(261\) 1044.45i 0.247700i
\(262\) 0 0
\(263\) 1647.19i 0.386198i −0.981179 0.193099i \(-0.938146\pi\)
0.981179 0.193099i \(-0.0618539\pi\)
\(264\) 0 0
\(265\) 136.235i 0.0315805i
\(266\) 0 0
\(267\) −101.573 −0.0232816
\(268\) 0 0
\(269\) −2289.73 −0.518985 −0.259493 0.965745i \(-0.583555\pi\)
−0.259493 + 0.965745i \(0.583555\pi\)
\(270\) 0 0
\(271\) −3502.19 −0.785030 −0.392515 0.919746i \(-0.628395\pi\)
−0.392515 + 0.919746i \(0.628395\pi\)
\(272\) 0 0
\(273\) −444.051 3071.70i −0.0984438 0.680980i
\(274\) 0 0
\(275\) 3127.69 0.685843
\(276\) 0 0
\(277\) 1410.70i 0.305996i −0.988227 0.152998i \(-0.951107\pi\)
0.988227 0.152998i \(-0.0488928\pi\)
\(278\) 0 0
\(279\) −2810.91 −0.603172
\(280\) 0 0
\(281\) −5875.05 −1.24725 −0.623623 0.781725i \(-0.714340\pi\)
−0.623623 + 0.781725i \(0.714340\pi\)
\(282\) 0 0
\(283\) 7588.94i 1.59405i −0.603947 0.797024i \(-0.706406\pi\)
0.603947 0.797024i \(-0.293594\pi\)
\(284\) 0 0
\(285\) 1194.24 0.248213
\(286\) 0 0
\(287\) 742.257 + 5134.52i 0.152662 + 1.05603i
\(288\) 0 0
\(289\) 4905.32 0.998437
\(290\) 0 0
\(291\) 2101.13 0.423266
\(292\) 0 0
\(293\) −4140.21 −0.825507 −0.412754 0.910843i \(-0.635433\pi\)
−0.412754 + 0.910843i \(0.635433\pi\)
\(294\) 0 0
\(295\) 1694.26i 0.334385i
\(296\) 0 0
\(297\) 932.056i 0.182099i
\(298\) 0 0
\(299\) 9847.74i 1.90471i
\(300\) 0 0
\(301\) −277.749 + 40.1520i −0.0531867 + 0.00768878i
\(302\) 0 0
\(303\) 877.109i 0.166299i
\(304\) 0 0
\(305\) −3019.28 −0.566832
\(306\) 0 0
\(307\) 2083.37i 0.387310i 0.981070 + 0.193655i \(0.0620342\pi\)
−0.981070 + 0.193655i \(0.937966\pi\)
\(308\) 0 0
\(309\) 1474.33i 0.271429i
\(310\) 0 0
\(311\) −3078.67 −0.561336 −0.280668 0.959805i \(-0.590556\pi\)
−0.280668 + 0.959805i \(0.590556\pi\)
\(312\) 0 0
\(313\) 4280.52i 0.773001i −0.922289 0.386500i \(-0.873684\pi\)
0.922289 0.386500i \(-0.126316\pi\)
\(314\) 0 0
\(315\) −967.509 + 139.865i −0.173057 + 0.0250175i
\(316\) 0 0
\(317\) 800.596i 0.141848i 0.997482 + 0.0709242i \(0.0225949\pi\)
−0.997482 + 0.0709242i \(0.977405\pi\)
\(318\) 0 0
\(319\) 4006.10i 0.703131i
\(320\) 0 0
\(321\) 5861.11i 1.01911i
\(322\) 0 0
\(323\) 188.086 0.0324005
\(324\) 0 0
\(325\) −5061.12 −0.863817
\(326\) 0 0
\(327\) −5662.79 −0.957654
\(328\) 0 0
\(329\) 116.268 16.8080i 0.0194835 0.00281657i
\(330\) 0 0
\(331\) 1068.57 0.177444 0.0887218 0.996056i \(-0.471722\pi\)
0.0887218 + 0.996056i \(0.471722\pi\)
\(332\) 0 0
\(333\) 1063.36i 0.174991i
\(334\) 0 0
\(335\) 1731.64 0.282416
\(336\) 0 0
\(337\) 4758.07 0.769105 0.384553 0.923103i \(-0.374356\pi\)
0.384553 + 0.923103i \(0.374356\pi\)
\(338\) 0 0
\(339\) 4319.74i 0.692082i
\(340\) 0 0
\(341\) −10781.6 −1.71219
\(342\) 0 0
\(343\) 5772.30 2652.20i 0.908673 0.417509i
\(344\) 0 0
\(345\) 3101.80 0.484044
\(346\) 0 0
\(347\) 2519.35 0.389757 0.194878 0.980827i \(-0.437569\pi\)
0.194878 + 0.980827i \(0.437569\pi\)
\(348\) 0 0
\(349\) 4415.55 0.677246 0.338623 0.940922i \(-0.390039\pi\)
0.338623 + 0.940922i \(0.390039\pi\)
\(350\) 0 0
\(351\) 1508.22i 0.229353i
\(352\) 0 0
\(353\) 11413.0i 1.72083i 0.509598 + 0.860413i \(0.329794\pi\)
−0.509598 + 0.860413i \(0.670206\pi\)
\(354\) 0 0
\(355\) 2790.97i 0.417266i
\(356\) 0 0
\(357\) −152.377 + 22.0279i −0.0225900 + 0.00326566i
\(358\) 0 0
\(359\) 6476.75i 0.952173i −0.879399 0.476086i \(-0.842055\pi\)
0.879399 0.476086i \(-0.157945\pi\)
\(360\) 0 0
\(361\) 2251.90 0.328313
\(362\) 0 0
\(363\) 417.985i 0.0604367i
\(364\) 0 0
\(365\) 2778.93i 0.398509i
\(366\) 0 0
\(367\) 6158.99 0.876013 0.438006 0.898972i \(-0.355685\pi\)
0.438006 + 0.898972i \(0.355685\pi\)
\(368\) 0 0
\(369\) 2521.08i 0.355670i
\(370\) 0 0
\(371\) −61.5519 425.782i −0.00861352 0.0595836i
\(372\) 0 0
\(373\) 9414.04i 1.30681i 0.757008 + 0.653406i \(0.226660\pi\)
−0.757008 + 0.653406i \(0.773340\pi\)
\(374\) 0 0
\(375\) 3793.44i 0.522380i
\(376\) 0 0
\(377\) 6482.54i 0.885590i
\(378\) 0 0
\(379\) −9747.56 −1.32110 −0.660552 0.750780i \(-0.729678\pi\)
−0.660552 + 0.750780i \(0.729678\pi\)
\(380\) 0 0
\(381\) 1882.30 0.253105
\(382\) 0 0
\(383\) 11501.0 1.53440 0.767200 0.641408i \(-0.221650\pi\)
0.767200 + 0.641408i \(0.221650\pi\)
\(384\) 0 0
\(385\) −3711.00 + 536.469i −0.491247 + 0.0710156i
\(386\) 0 0
\(387\) 136.377 0.0179132
\(388\) 0 0
\(389\) 5810.06i 0.757280i −0.925544 0.378640i \(-0.876392\pi\)
0.925544 0.378640i \(-0.123608\pi\)
\(390\) 0 0
\(391\) 488.515 0.0631848
\(392\) 0 0
\(393\) −6774.36 −0.869519
\(394\) 0 0
\(395\) 4668.51i 0.594679i
\(396\) 0 0
\(397\) 6292.07 0.795441 0.397720 0.917507i \(-0.369801\pi\)
0.397720 + 0.917507i \(0.369801\pi\)
\(398\) 0 0
\(399\) 3732.42 539.566i 0.468308 0.0676995i
\(400\) 0 0
\(401\) 2540.23 0.316342 0.158171 0.987412i \(-0.449440\pi\)
0.158171 + 0.987412i \(0.449440\pi\)
\(402\) 0 0
\(403\) 17446.4 2.15649
\(404\) 0 0
\(405\) 475.052 0.0582853
\(406\) 0 0
\(407\) 4078.66i 0.496736i
\(408\) 0 0
\(409\) 6379.47i 0.771258i −0.922654 0.385629i \(-0.873984\pi\)
0.922654 0.385629i \(-0.126016\pi\)
\(410\) 0 0
\(411\) 7704.29i 0.924634i
\(412\) 0 0
\(413\) 765.480 + 5295.17i 0.0912029 + 0.630892i
\(414\) 0 0
\(415\) 5147.96i 0.608924i
\(416\) 0 0
\(417\) −3609.07 −0.423830
\(418\) 0 0
\(419\) 3727.78i 0.434639i 0.976101 + 0.217320i \(0.0697314\pi\)
−0.976101 + 0.217320i \(0.930269\pi\)
\(420\) 0 0
\(421\) 5998.64i 0.694432i −0.937785 0.347216i \(-0.887127\pi\)
0.937785 0.347216i \(-0.112873\pi\)
\(422\) 0 0
\(423\) −57.0884 −0.00656201
\(424\) 0 0
\(425\) 251.066i 0.0286553i
\(426\) 0 0
\(427\) −9436.33 + 1364.13i −1.06945 + 0.154602i
\(428\) 0 0
\(429\) 5784.97i 0.651051i
\(430\) 0 0
\(431\) 430.232i 0.0480824i −0.999711 0.0240412i \(-0.992347\pi\)
0.999711 0.0240412i \(-0.00765329\pi\)
\(432\) 0 0
\(433\) 10945.0i 1.21474i 0.794419 + 0.607370i \(0.207775\pi\)
−0.794419 + 0.607370i \(0.792225\pi\)
\(434\) 0 0
\(435\) 2041.84 0.225054
\(436\) 0 0
\(437\) −11966.0 −1.30987
\(438\) 0 0
\(439\) −14202.7 −1.54410 −0.772048 0.635564i \(-0.780767\pi\)
−0.772048 + 0.635564i \(0.780767\pi\)
\(440\) 0 0
\(441\) −2960.62 + 874.255i −0.319686 + 0.0944018i
\(442\) 0 0
\(443\) −11866.0 −1.27262 −0.636309 0.771434i \(-0.719540\pi\)
−0.636309 + 0.771434i \(0.719540\pi\)
\(444\) 0 0
\(445\) 198.570i 0.0211531i
\(446\) 0 0
\(447\) −4889.11 −0.517331
\(448\) 0 0
\(449\) −37.3423 −0.00392493 −0.00196246 0.999998i \(-0.500625\pi\)
−0.00196246 + 0.999998i \(0.500625\pi\)
\(450\) 0 0
\(451\) 9669.91i 1.00962i
\(452\) 0 0
\(453\) 10375.5 1.07612
\(454\) 0 0
\(455\) 6005.01 868.096i 0.618723 0.0894439i
\(456\) 0 0
\(457\) −18183.5 −1.86124 −0.930621 0.365984i \(-0.880732\pi\)
−0.930621 + 0.365984i \(0.880732\pi\)
\(458\) 0 0
\(459\) 74.8180 0.00760829
\(460\) 0 0
\(461\) −9339.91 −0.943607 −0.471803 0.881704i \(-0.656397\pi\)
−0.471803 + 0.881704i \(0.656397\pi\)
\(462\) 0 0
\(463\) 8483.34i 0.851521i −0.904836 0.425761i \(-0.860007\pi\)
0.904836 0.425761i \(-0.139993\pi\)
\(464\) 0 0
\(465\) 5495.19i 0.548028i
\(466\) 0 0
\(467\) 15257.9i 1.51188i 0.654638 + 0.755942i \(0.272821\pi\)
−0.654638 + 0.755942i \(0.727179\pi\)
\(468\) 0 0
\(469\) 5411.98 782.367i 0.532840 0.0770284i
\(470\) 0 0
\(471\) 5021.53i 0.491253i
\(472\) 0 0
\(473\) 523.089 0.0508492
\(474\) 0 0
\(475\) 6149.77i 0.594044i
\(476\) 0 0
\(477\) 209.061i 0.0200676i
\(478\) 0 0
\(479\) 651.587 0.0621540 0.0310770 0.999517i \(-0.490106\pi\)
0.0310770 + 0.999517i \(0.490106\pi\)
\(480\) 0 0
\(481\) 6599.95i 0.625637i
\(482\) 0 0
\(483\) 9694.21 1401.41i 0.913255 0.132022i
\(484\) 0 0
\(485\) 4107.60i 0.384570i
\(486\) 0 0
\(487\) 12697.4i 1.18147i −0.806866 0.590735i \(-0.798838\pi\)
0.806866 0.590735i \(-0.201162\pi\)
\(488\) 0 0
\(489\) 8530.14i 0.788847i
\(490\) 0 0
\(491\) −17950.7 −1.64990 −0.824951 0.565205i \(-0.808797\pi\)
−0.824951 + 0.565205i \(0.808797\pi\)
\(492\) 0 0
\(493\) 321.578 0.0293776
\(494\) 0 0
\(495\) 1822.12 0.165451
\(496\) 0 0
\(497\) 1260.98 + 8722.77i 0.113808 + 0.787263i
\(498\) 0 0
\(499\) −11302.1 −1.01394 −0.506968 0.861965i \(-0.669234\pi\)
−0.506968 + 0.861965i \(0.669234\pi\)
\(500\) 0 0
\(501\) 11737.1i 1.04666i
\(502\) 0 0
\(503\) 16281.7 1.44327 0.721636 0.692272i \(-0.243390\pi\)
0.721636 + 0.692272i \(0.243390\pi\)
\(504\) 0 0
\(505\) 1714.70 0.151096
\(506\) 0 0
\(507\) 2770.03i 0.242646i
\(508\) 0 0
\(509\) −21438.3 −1.86687 −0.933436 0.358743i \(-0.883205\pi\)
−0.933436 + 0.358743i \(0.883205\pi\)
\(510\) 0 0
\(511\) −1255.54 8685.15i −0.108693 0.751875i
\(512\) 0 0
\(513\) −1832.64 −0.157725
\(514\) 0 0
\(515\) −2882.23 −0.246614
\(516\) 0 0
\(517\) −218.969 −0.0186272
\(518\) 0 0
\(519\) 2239.32i 0.189393i
\(520\) 0 0
\(521\) 10394.4i 0.874067i 0.899445 + 0.437034i \(0.143971\pi\)
−0.899445 + 0.437034i \(0.856029\pi\)
\(522\) 0 0
\(523\) 4872.79i 0.407404i −0.979033 0.203702i \(-0.934703\pi\)
0.979033 0.203702i \(-0.0652974\pi\)
\(524\) 0 0
\(525\) 720.239 + 4982.22i 0.0598739 + 0.414175i
\(526\) 0 0
\(527\) 865.460i 0.0715371i
\(528\) 0 0
\(529\) −18912.3 −1.55439
\(530\) 0 0
\(531\) 2599.96i 0.212483i
\(532\) 0 0
\(533\) 15647.5i 1.27161i
\(534\) 0 0
\(535\) −11458.2 −0.925943
\(536\) 0 0
\(537\) 11124.2i 0.893942i
\(538\) 0 0
\(539\) −11355.8 + 3353.31i −0.907475 + 0.267973i
\(540\) 0 0
\(541\) 3456.08i 0.274656i −0.990526 0.137328i \(-0.956149\pi\)
0.990526 0.137328i \(-0.0438514\pi\)
\(542\) 0 0
\(543\) 6439.02i 0.508885i
\(544\) 0 0
\(545\) 11070.5i 0.870103i
\(546\) 0 0
\(547\) 20798.4 1.62573 0.812867 0.582449i \(-0.197905\pi\)
0.812867 + 0.582449i \(0.197905\pi\)
\(548\) 0 0
\(549\) 4633.29 0.360190
\(550\) 0 0
\(551\) −7876.94 −0.609018
\(552\) 0 0
\(553\) 2109.27 + 14590.7i 0.162197 + 1.12199i
\(554\) 0 0
\(555\) −2078.82 −0.158993
\(556\) 0 0
\(557\) 2239.72i 0.170377i −0.996365 0.0851885i \(-0.972851\pi\)
0.996365 0.0851885i \(-0.0271493\pi\)
\(558\) 0 0
\(559\) −846.445 −0.0640443
\(560\) 0 0
\(561\) 286.974 0.0215972
\(562\) 0 0
\(563\) 14008.9i 1.04868i 0.851510 + 0.524339i \(0.175688\pi\)
−0.851510 + 0.524339i \(0.824312\pi\)
\(564\) 0 0
\(565\) 8444.86 0.628811
\(566\) 0 0
\(567\) 1484.71 214.632i 0.109968 0.0158972i
\(568\) 0 0
\(569\) −11092.1 −0.817231 −0.408615 0.912707i \(-0.633988\pi\)
−0.408615 + 0.912707i \(0.633988\pi\)
\(570\) 0 0
\(571\) −8828.17 −0.647018 −0.323509 0.946225i \(-0.604863\pi\)
−0.323509 + 0.946225i \(0.604863\pi\)
\(572\) 0 0
\(573\) 11311.9 0.824714
\(574\) 0 0
\(575\) 15972.8i 1.15846i
\(576\) 0 0
\(577\) 18544.3i 1.33797i −0.743274 0.668987i \(-0.766728\pi\)
0.743274 0.668987i \(-0.233272\pi\)
\(578\) 0 0
\(579\) 1083.54i 0.0777727i
\(580\) 0 0
\(581\) 2325.89 + 16089.2i 0.166083 + 1.14887i
\(582\) 0 0
\(583\) 801.881i 0.0569649i
\(584\) 0 0
\(585\) −2948.49 −0.208385
\(586\) 0 0
\(587\) 2437.71i 0.171406i 0.996321 + 0.0857029i \(0.0273136\pi\)
−0.996321 + 0.0857029i \(0.972686\pi\)
\(588\) 0 0
\(589\) 21199.1i 1.48301i
\(590\) 0 0
\(591\) −13598.3 −0.946464
\(592\) 0 0
\(593\) 19750.9i 1.36774i −0.729602 0.683872i \(-0.760295\pi\)
0.729602 0.683872i \(-0.239705\pi\)
\(594\) 0 0
\(595\) −43.0634 297.889i −0.00296711 0.0205248i
\(596\) 0 0
\(597\) 955.591i 0.0655104i
\(598\) 0 0
\(599\) 12894.2i 0.879538i −0.898111 0.439769i \(-0.855060\pi\)
0.898111 0.439769i \(-0.144940\pi\)
\(600\) 0 0
\(601\) 16930.0i 1.14907i −0.818482 0.574533i \(-0.805184\pi\)
0.818482 0.574533i \(-0.194816\pi\)
\(602\) 0 0
\(603\) −2657.31 −0.179460
\(604\) 0 0
\(605\) −817.139 −0.0549114
\(606\) 0 0
\(607\) −3134.16 −0.209574 −0.104787 0.994495i \(-0.533416\pi\)
−0.104787 + 0.994495i \(0.533416\pi\)
\(608\) 0 0
\(609\) 6381.47 922.518i 0.424615 0.0613831i
\(610\) 0 0
\(611\) 354.329 0.0234609
\(612\) 0 0
\(613\) 19788.1i 1.30381i 0.758301 + 0.651905i \(0.226030\pi\)
−0.758301 + 0.651905i \(0.773970\pi\)
\(614\) 0 0
\(615\) 4928.58 0.323154
\(616\) 0 0
\(617\) −19220.2 −1.25410 −0.627048 0.778981i \(-0.715737\pi\)
−0.627048 + 0.778981i \(0.715737\pi\)
\(618\) 0 0
\(619\) 16075.2i 1.04381i 0.853005 + 0.521903i \(0.174778\pi\)
−0.853005 + 0.521903i \(0.825222\pi\)
\(620\) 0 0
\(621\) −4759.92 −0.307583
\(622\) 0 0
\(623\) −89.7156 620.603i −0.00576947 0.0399100i
\(624\) 0 0
\(625\) 3909.46 0.250206
\(626\) 0 0
\(627\) −7029.32 −0.447726
\(628\) 0 0
\(629\) −327.402 −0.0207542
\(630\) 0 0
\(631\) 19087.7i 1.20423i −0.798408 0.602116i \(-0.794324\pi\)
0.798408 0.602116i \(-0.205676\pi\)
\(632\) 0 0
\(633\) 2373.61i 0.149041i
\(634\) 0 0
\(635\) 3679.79i 0.229965i
\(636\) 0 0
\(637\) 18375.6 5426.21i 1.14296 0.337511i
\(638\) 0 0
\(639\) 4282.93i 0.265149i
\(640\) 0 0
\(641\) −26888.1 −1.65681 −0.828407 0.560126i \(-0.810753\pi\)
−0.828407 + 0.560126i \(0.810753\pi\)
\(642\) 0 0
\(643\) 23591.9i 1.44692i −0.690364 0.723462i \(-0.742550\pi\)
0.690364 0.723462i \(-0.257450\pi\)
\(644\) 0 0
\(645\) 266.609i 0.0162755i
\(646\) 0 0
\(647\) 14156.1 0.860178 0.430089 0.902787i \(-0.358482\pi\)
0.430089 + 0.902787i \(0.358482\pi\)
\(648\) 0 0
\(649\) 9972.46i 0.603164i
\(650\) 0 0
\(651\) −2482.77 17174.4i −0.149474 1.03398i
\(652\) 0 0
\(653\) 1229.44i 0.0736779i −0.999321 0.0368389i \(-0.988271\pi\)
0.999321 0.0368389i \(-0.0117289\pi\)
\(654\) 0 0
\(655\) 13243.5i 0.790026i
\(656\) 0 0
\(657\) 4264.46i 0.253230i
\(658\) 0 0
\(659\) 27180.6 1.60668 0.803341 0.595519i \(-0.203053\pi\)
0.803341 + 0.595519i \(0.203053\pi\)
\(660\) 0 0
\(661\) 20449.2 1.20330 0.601649 0.798761i \(-0.294511\pi\)
0.601649 + 0.798761i \(0.294511\pi\)
\(662\) 0 0
\(663\) −464.371 −0.0272016
\(664\) 0 0
\(665\) 1054.82 + 7296.69i 0.0615103 + 0.425494i
\(666\) 0 0
\(667\) −20458.8 −1.18766
\(668\) 0 0
\(669\) 3040.22i 0.175697i
\(670\) 0 0
\(671\) 17771.6 1.02245
\(672\) 0 0
\(673\) −15959.2 −0.914087 −0.457044 0.889444i \(-0.651092\pi\)
−0.457044 + 0.889444i \(0.651092\pi\)
\(674\) 0 0
\(675\) 2446.30i 0.139493i
\(676\) 0 0
\(677\) −15758.8 −0.894621 −0.447311 0.894379i \(-0.647618\pi\)
−0.447311 + 0.894379i \(0.647618\pi\)
\(678\) 0 0
\(679\) 1855.84 + 12837.7i 0.104891 + 0.725576i
\(680\) 0 0
\(681\) 18002.3 1.01300
\(682\) 0 0
\(683\) −27607.3 −1.54665 −0.773325 0.634009i \(-0.781408\pi\)
−0.773325 + 0.634009i \(0.781408\pi\)
\(684\) 0 0
\(685\) 15061.5 0.840102
\(686\) 0 0
\(687\) 3668.31i 0.203719i
\(688\) 0 0
\(689\) 1297.58i 0.0717470i
\(690\) 0 0
\(691\) 106.769i 0.00587798i −0.999996 0.00293899i \(-0.999064\pi\)
0.999996 0.00293899i \(-0.000935511\pi\)
\(692\) 0 0
\(693\) 5694.78 823.248i 0.312160 0.0451264i
\(694\) 0 0
\(695\) 7055.55i 0.385082i
\(696\) 0 0
\(697\) 776.223 0.0421830
\(698\) 0 0
\(699\) 12479.5i 0.675278i
\(700\) 0 0
\(701\) 10439.6i 0.562478i −0.959638 0.281239i \(-0.909255\pi\)
0.959638 0.281239i \(-0.0907453\pi\)
\(702\) 0 0
\(703\) 8019.60 0.430249
\(704\) 0 0
\(705\) 111.605i 0.00596210i
\(706\) 0 0
\(707\) 5359.06 774.716i 0.285075 0.0412110i
\(708\) 0 0
\(709\) 3868.07i 0.204892i −0.994739 0.102446i \(-0.967333\pi\)
0.994739 0.102446i \(-0.0326669\pi\)
\(710\) 0 0
\(711\) 7164.14i 0.377885i
\(712\) 0 0
\(713\) 55060.5i 2.89205i
\(714\) 0 0
\(715\) −11309.3 −0.591530
\(716\) 0 0
\(717\) 9432.16 0.491284
\(718\) 0 0
\(719\) −20194.2 −1.04745 −0.523724 0.851888i \(-0.675458\pi\)
−0.523724 + 0.851888i \(0.675458\pi\)
\(720\) 0 0
\(721\) −9008.00 + 1302.21i −0.465292 + 0.0672635i
\(722\) 0 0
\(723\) −1734.13 −0.0892017
\(724\) 0 0
\(725\) 10514.5i 0.538620i
\(726\) 0 0
\(727\) 13941.0 0.711202 0.355601 0.934638i \(-0.384276\pi\)
0.355601 + 0.934638i \(0.384276\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 41.9894i 0.00212453i
\(732\) 0 0
\(733\) −32974.3 −1.66158 −0.830788 0.556589i \(-0.812110\pi\)
−0.830788 + 0.556589i \(0.812110\pi\)
\(734\) 0 0
\(735\) −1709.12 5787.85i −0.0857714 0.290460i
\(736\) 0 0
\(737\) −10192.5 −0.509422
\(738\) 0 0
\(739\) 15559.4 0.774508 0.387254 0.921973i \(-0.373424\pi\)
0.387254 + 0.921973i \(0.373424\pi\)
\(740\) 0 0
\(741\) 11374.6 0.563909
\(742\) 0 0
\(743\) 5200.10i 0.256761i −0.991725 0.128380i \(-0.959022\pi\)
0.991725 0.128380i \(-0.0409779\pi\)
\(744\) 0 0
\(745\) 9557.96i 0.470036i
\(746\) 0 0
\(747\) 7899.90i 0.386937i
\(748\) 0 0
\(749\) −35810.8 + 5176.89i −1.74700 + 0.252549i
\(750\) 0 0
\(751\) 10449.1i 0.507714i 0.967242 + 0.253857i \(0.0816993\pi\)
−0.967242 + 0.253857i \(0.918301\pi\)
\(752\) 0 0
\(753\) −5784.52 −0.279947
\(754\) 0 0
\(755\) 20283.5i 0.977737i
\(756\) 0 0
\(757\) 11354.8i 0.545175i 0.962131 + 0.272588i \(0.0878794\pi\)
−0.962131 + 0.272588i \(0.912121\pi\)
\(758\) 0 0
\(759\) −18257.2 −0.873117
\(760\) 0 0
\(761\) 1559.85i 0.0743028i −0.999310 0.0371514i \(-0.988172\pi\)
0.999310 0.0371514i \(-0.0118284\pi\)
\(762\) 0 0
\(763\) −5001.72 34599.1i −0.237319 1.64164i
\(764\) 0 0
\(765\) 146.265i 0.00691272i
\(766\) 0 0
\(767\) 16137.1i 0.759682i
\(768\) 0 0
\(769\) 4818.32i 0.225947i 0.993598 + 0.112973i \(0.0360375\pi\)
−0.993598 + 0.112973i \(0.963962\pi\)
\(770\) 0 0
\(771\) 3038.23 0.141918
\(772\) 0 0
\(773\) −4434.61 −0.206341 −0.103171 0.994664i \(-0.532899\pi\)
−0.103171 + 0.994664i \(0.532899\pi\)
\(774\) 0 0
\(775\) −28297.6 −1.31159
\(776\) 0 0
\(777\) −6497.05 + 939.227i −0.299975 + 0.0433650i
\(778\) 0 0
\(779\) −19013.3 −0.874484
\(780\) 0 0
\(781\) 16427.7i 0.752663i
\(782\) 0 0
\(783\) −3133.34 −0.143010
\(784\) 0 0
\(785\) 9816.83 0.446341
\(786\) 0 0
\(787\) 28432.5i 1.28781i −0.765104 0.643907i \(-0.777312\pi\)
0.765104 0.643907i \(-0.222688\pi\)
\(788\) 0 0
\(789\) −4941.57 −0.222972
\(790\) 0 0
\(791\) 26393.2 3815.45i 1.18639 0.171507i
\(792\) 0 0
\(793\) −28757.3 −1.28777
\(794\) 0 0
\(795\) −408.704 −0.0182330
\(796\) 0 0
\(797\) 4011.19 0.178273 0.0891365 0.996019i \(-0.471589\pi\)
0.0891365 + 0.996019i \(0.471589\pi\)
\(798\) 0 0
\(799\) 17.5771i 0.000778264i
\(800\) 0 0
\(801\) 304.720i 0.0134416i
\(802\) 0