Properties

Label 624.4.c.c.337.2
Level $624$
Weight $4$
Character 624.337
Analytic conductor $36.817$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 624.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.8171918436\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.1362828.1
Defining polynomial: \( x^{4} + 23x^{2} + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 39)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 337.2
Root \(-1.52356i\) of defining polynomial
Character \(\chi\) \(=\) 624.337
Dual form 624.4.c.c.337.3

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -9.65841i q^{5} -22.3639i q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -9.65841i q^{5} -22.3639i q^{7} +9.00000 q^{9} +50.3050i q^{11} +(-39.7151 + 24.8940i) q^{13} +28.9752i q^{15} -86.1454 q^{17} +116.880i q^{19} +67.0918i q^{21} +72.0000 q^{23} +31.7151 q^{25} -27.0000 q^{27} +14.1454 q^{29} -196.215i q^{31} -150.915i q^{33} -216.000 q^{35} -154.424i q^{37} +(119.145 - 74.6819i) q^{39} +265.726i q^{41} +211.855 q^{43} -86.9257i q^{45} -67.5535i q^{47} -157.145 q^{49} +258.436 q^{51} +686.581 q^{53} +485.866 q^{55} -350.639i q^{57} +91.9304i q^{59} +329.006 q^{61} -201.275i q^{63} +(240.436 + 383.585i) q^{65} +768.370i q^{67} -216.000 q^{69} +264.969i q^{71} +771.306i q^{73} -95.1454 q^{75} +1125.02 q^{77} -1226.86 q^{79} +81.0000 q^{81} -514.019i q^{83} +832.027i q^{85} -42.4361 q^{87} +527.889i q^{89} +(556.727 + 888.186i) q^{91} +588.646i q^{93} +1128.87 q^{95} +74.2755i q^{97} +452.745i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{3} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{3} + 36 q^{9} - 12 q^{13} + 96 q^{17} + 288 q^{23} - 20 q^{25} - 108 q^{27} - 384 q^{29} - 864 q^{35} + 36 q^{39} + 1288 q^{43} - 188 q^{49} - 288 q^{51} + 984 q^{53} + 328 q^{55} + 288 q^{61} - 360 q^{65} - 864 q^{69} + 60 q^{75} + 1416 q^{77} - 4320 q^{79} + 324 q^{81} + 1152 q^{87} + 24 q^{91} + 1872 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(3\) −3.00000 −0.577350
\(4\) 0 0
\(5\) 9.65841i 0.863874i −0.901904 0.431937i \(-0.857830\pi\)
0.901904 0.431937i \(-0.142170\pi\)
\(6\) 0 0
\(7\) 22.3639i 1.20754i −0.797159 0.603769i \(-0.793665\pi\)
0.797159 0.603769i \(-0.206335\pi\)
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 50.3050i 1.37887i 0.724349 + 0.689433i \(0.242140\pi\)
−0.724349 + 0.689433i \(0.757860\pi\)
\(12\) 0 0
\(13\) −39.7151 + 24.8940i −0.847307 + 0.531103i
\(14\) 0 0
\(15\) 28.9752i 0.498758i
\(16\) 0 0
\(17\) −86.1454 −1.22902 −0.614509 0.788910i \(-0.710646\pi\)
−0.614509 + 0.788910i \(0.710646\pi\)
\(18\) 0 0
\(19\) 116.880i 1.41127i 0.708578 + 0.705633i \(0.249337\pi\)
−0.708578 + 0.705633i \(0.750663\pi\)
\(20\) 0 0
\(21\) 67.0918i 0.697173i
\(22\) 0 0
\(23\) 72.0000 0.652741 0.326370 0.945242i \(-0.394174\pi\)
0.326370 + 0.945242i \(0.394174\pi\)
\(24\) 0 0
\(25\) 31.7151 0.253721
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) 0 0
\(29\) 14.1454 0.0905768 0.0452884 0.998974i \(-0.485579\pi\)
0.0452884 + 0.998974i \(0.485579\pi\)
\(30\) 0 0
\(31\) 196.215i 1.13682i −0.822747 0.568408i \(-0.807559\pi\)
0.822747 0.568408i \(-0.192441\pi\)
\(32\) 0 0
\(33\) 150.915i 0.796089i
\(34\) 0 0
\(35\) −216.000 −1.04316
\(36\) 0 0
\(37\) 154.424i 0.686138i −0.939310 0.343069i \(-0.888533\pi\)
0.939310 0.343069i \(-0.111467\pi\)
\(38\) 0 0
\(39\) 119.145 74.6819i 0.489193 0.306633i
\(40\) 0 0
\(41\) 265.726i 1.01218i 0.862480 + 0.506091i \(0.168910\pi\)
−0.862480 + 0.506091i \(0.831090\pi\)
\(42\) 0 0
\(43\) 211.855 0.751338 0.375669 0.926754i \(-0.377413\pi\)
0.375669 + 0.926754i \(0.377413\pi\)
\(44\) 0 0
\(45\) 86.9257i 0.287958i
\(46\) 0 0
\(47\) 67.5535i 0.209653i −0.994491 0.104827i \(-0.966571\pi\)
0.994491 0.104827i \(-0.0334287\pi\)
\(48\) 0 0
\(49\) −157.145 −0.458150
\(50\) 0 0
\(51\) 258.436 0.709574
\(52\) 0 0
\(53\) 686.581 1.77942 0.889710 0.456527i \(-0.150907\pi\)
0.889710 + 0.456527i \(0.150907\pi\)
\(54\) 0 0
\(55\) 485.866 1.19117
\(56\) 0 0
\(57\) 350.639i 0.814795i
\(58\) 0 0
\(59\) 91.9304i 0.202853i 0.994843 + 0.101426i \(0.0323406\pi\)
−0.994843 + 0.101426i \(0.967659\pi\)
\(60\) 0 0
\(61\) 329.006 0.690572 0.345286 0.938498i \(-0.387782\pi\)
0.345286 + 0.938498i \(0.387782\pi\)
\(62\) 0 0
\(63\) 201.275i 0.402513i
\(64\) 0 0
\(65\) 240.436 + 383.585i 0.458807 + 0.731967i
\(66\) 0 0
\(67\) 768.370i 1.40106i 0.713621 + 0.700532i \(0.247054\pi\)
−0.713621 + 0.700532i \(0.752946\pi\)
\(68\) 0 0
\(69\) −216.000 −0.376860
\(70\) 0 0
\(71\) 264.969i 0.442902i 0.975171 + 0.221451i \(0.0710793\pi\)
−0.975171 + 0.221451i \(0.928921\pi\)
\(72\) 0 0
\(73\) 771.306i 1.23664i 0.785927 + 0.618319i \(0.212186\pi\)
−0.785927 + 0.618319i \(0.787814\pi\)
\(74\) 0 0
\(75\) −95.1454 −0.146486
\(76\) 0 0
\(77\) 1125.02 1.66503
\(78\) 0 0
\(79\) −1226.86 −1.74725 −0.873624 0.486602i \(-0.838236\pi\)
−0.873624 + 0.486602i \(0.838236\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 514.019i 0.679771i −0.940467 0.339885i \(-0.889612\pi\)
0.940467 0.339885i \(-0.110388\pi\)
\(84\) 0 0
\(85\) 832.027i 1.06172i
\(86\) 0 0
\(87\) −42.4361 −0.0522945
\(88\) 0 0
\(89\) 527.889i 0.628720i 0.949304 + 0.314360i \(0.101790\pi\)
−0.949304 + 0.314360i \(0.898210\pi\)
\(90\) 0 0
\(91\) 556.727 + 888.186i 0.641328 + 1.02316i
\(92\) 0 0
\(93\) 588.646i 0.656341i
\(94\) 0 0
\(95\) 1128.87 1.21916
\(96\) 0 0
\(97\) 74.2755i 0.0777478i 0.999244 + 0.0388739i \(0.0123771\pi\)
−0.999244 + 0.0388739i \(0.987623\pi\)
\(98\) 0 0
\(99\) 452.745i 0.459622i
\(100\) 0 0
\(101\) −609.419 −0.600390 −0.300195 0.953878i \(-0.597052\pi\)
−0.300195 + 0.953878i \(0.597052\pi\)
\(102\) 0 0
\(103\) −32.4361 −0.0310293 −0.0155147 0.999880i \(-0.504939\pi\)
−0.0155147 + 0.999880i \(0.504939\pi\)
\(104\) 0 0
\(105\) 648.000 0.602270
\(106\) 0 0
\(107\) 1725.45 1.55893 0.779467 0.626444i \(-0.215490\pi\)
0.779467 + 0.626444i \(0.215490\pi\)
\(108\) 0 0
\(109\) 1273.43i 1.11902i −0.828825 0.559508i \(-0.810990\pi\)
0.828825 0.559508i \(-0.189010\pi\)
\(110\) 0 0
\(111\) 463.271i 0.396142i
\(112\) 0 0
\(113\) 1114.73 0.928006 0.464003 0.885834i \(-0.346413\pi\)
0.464003 + 0.885834i \(0.346413\pi\)
\(114\) 0 0
\(115\) 695.406i 0.563886i
\(116\) 0 0
\(117\) −357.436 + 224.046i −0.282436 + 0.177034i
\(118\) 0 0
\(119\) 1926.55i 1.48409i
\(120\) 0 0
\(121\) −1199.59 −0.901272
\(122\) 0 0
\(123\) 797.179i 0.584384i
\(124\) 0 0
\(125\) 1513.62i 1.08306i
\(126\) 0 0
\(127\) 1174.01 0.820289 0.410144 0.912021i \(-0.365478\pi\)
0.410144 + 0.912021i \(0.365478\pi\)
\(128\) 0 0
\(129\) −635.564 −0.433785
\(130\) 0 0
\(131\) 1445.16 0.963851 0.481925 0.876212i \(-0.339938\pi\)
0.481925 + 0.876212i \(0.339938\pi\)
\(132\) 0 0
\(133\) 2613.89 1.70416
\(134\) 0 0
\(135\) 260.777i 0.166253i
\(136\) 0 0
\(137\) 508.793i 0.317293i 0.987335 + 0.158646i \(0.0507130\pi\)
−0.987335 + 0.158646i \(0.949287\pi\)
\(138\) 0 0
\(139\) 757.018 0.461938 0.230969 0.972961i \(-0.425810\pi\)
0.230969 + 0.972961i \(0.425810\pi\)
\(140\) 0 0
\(141\) 202.661i 0.121043i
\(142\) 0 0
\(143\) −1252.29 1997.87i −0.732320 1.16832i
\(144\) 0 0
\(145\) 136.622i 0.0782470i
\(146\) 0 0
\(147\) 471.436 0.264513
\(148\) 0 0
\(149\) 3247.79i 1.78570i 0.450352 + 0.892851i \(0.351298\pi\)
−0.450352 + 0.892851i \(0.648702\pi\)
\(150\) 0 0
\(151\) 795.296i 0.428611i −0.976767 0.214305i \(-0.931251\pi\)
0.976767 0.214305i \(-0.0687488\pi\)
\(152\) 0 0
\(153\) −775.308 −0.409673
\(154\) 0 0
\(155\) −1895.13 −0.982067
\(156\) 0 0
\(157\) −65.2732 −0.0331807 −0.0165903 0.999862i \(-0.505281\pi\)
−0.0165903 + 0.999862i \(0.505281\pi\)
\(158\) 0 0
\(159\) −2059.74 −1.02735
\(160\) 0 0
\(161\) 1610.20i 0.788210i
\(162\) 0 0
\(163\) 1855.39i 0.891568i 0.895140 + 0.445784i \(0.147075\pi\)
−0.895140 + 0.445784i \(0.852925\pi\)
\(164\) 0 0
\(165\) −1457.60 −0.687721
\(166\) 0 0
\(167\) 3532.54i 1.63686i 0.574604 + 0.818432i \(0.305156\pi\)
−0.574604 + 0.818432i \(0.694844\pi\)
\(168\) 0 0
\(169\) 957.581 1977.33i 0.435859 0.900015i
\(170\) 0 0
\(171\) 1051.92i 0.470422i
\(172\) 0 0
\(173\) 3178.36 1.39680 0.698400 0.715708i \(-0.253896\pi\)
0.698400 + 0.715708i \(0.253896\pi\)
\(174\) 0 0
\(175\) 709.275i 0.306378i
\(176\) 0 0
\(177\) 275.791i 0.117117i
\(178\) 0 0
\(179\) 2741.16 1.14460 0.572302 0.820043i \(-0.306050\pi\)
0.572302 + 0.820043i \(0.306050\pi\)
\(180\) 0 0
\(181\) −3871.09 −1.58970 −0.794850 0.606806i \(-0.792450\pi\)
−0.794850 + 0.606806i \(0.792450\pi\)
\(182\) 0 0
\(183\) −987.018 −0.398702
\(184\) 0 0
\(185\) −1491.49 −0.592737
\(186\) 0 0
\(187\) 4333.54i 1.69465i
\(188\) 0 0
\(189\) 603.826i 0.232391i
\(190\) 0 0
\(191\) 928.291 0.351669 0.175834 0.984420i \(-0.443738\pi\)
0.175834 + 0.984420i \(0.443738\pi\)
\(192\) 0 0
\(193\) 2261.51i 0.843456i −0.906722 0.421728i \(-0.861424\pi\)
0.906722 0.421728i \(-0.138576\pi\)
\(194\) 0 0
\(195\) −721.308 1150.75i −0.264892 0.422601i
\(196\) 0 0
\(197\) 2265.59i 0.819374i 0.912226 + 0.409687i \(0.134362\pi\)
−0.912226 + 0.409687i \(0.865638\pi\)
\(198\) 0 0
\(199\) −260.895 −0.0929366 −0.0464683 0.998920i \(-0.514797\pi\)
−0.0464683 + 0.998920i \(0.514797\pi\)
\(200\) 0 0
\(201\) 2305.11i 0.808905i
\(202\) 0 0
\(203\) 316.346i 0.109375i
\(204\) 0 0
\(205\) 2566.49 0.874399
\(206\) 0 0
\(207\) 648.000 0.217580
\(208\) 0 0
\(209\) −5879.63 −1.94595
\(210\) 0 0
\(211\) −5851.22 −1.90907 −0.954537 0.298092i \(-0.903650\pi\)
−0.954537 + 0.298092i \(0.903650\pi\)
\(212\) 0 0
\(213\) 794.907i 0.255710i
\(214\) 0 0
\(215\) 2046.18i 0.649062i
\(216\) 0 0
\(217\) −4388.15 −1.37275
\(218\) 0 0
\(219\) 2313.92i 0.713973i
\(220\) 0 0
\(221\) 3421.27 2144.50i 1.04136 0.652736i
\(222\) 0 0
\(223\) 3463.60i 1.04009i −0.854139 0.520045i \(-0.825915\pi\)
0.854139 0.520045i \(-0.174085\pi\)
\(224\) 0 0
\(225\) 285.436 0.0845737
\(226\) 0 0
\(227\) 5329.15i 1.55819i 0.626908 + 0.779093i \(0.284320\pi\)
−0.626908 + 0.779093i \(0.715680\pi\)
\(228\) 0 0
\(229\) 4773.95i 1.37761i 0.724949 + 0.688803i \(0.241863\pi\)
−0.724949 + 0.688803i \(0.758137\pi\)
\(230\) 0 0
\(231\) −3375.05 −0.961308
\(232\) 0 0
\(233\) 4813.78 1.35348 0.676741 0.736221i \(-0.263392\pi\)
0.676741 + 0.736221i \(0.263392\pi\)
\(234\) 0 0
\(235\) −652.459 −0.181114
\(236\) 0 0
\(237\) 3680.58 1.00877
\(238\) 0 0
\(239\) 1683.19i 0.455549i 0.973714 + 0.227775i \(0.0731449\pi\)
−0.973714 + 0.227775i \(0.926855\pi\)
\(240\) 0 0
\(241\) 664.861i 0.177707i −0.996045 0.0888537i \(-0.971680\pi\)
0.996045 0.0888537i \(-0.0283204\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 0 0
\(245\) 1517.77i 0.395784i
\(246\) 0 0
\(247\) −2909.60 4641.89i −0.749528 1.19578i
\(248\) 0 0
\(249\) 1542.06i 0.392466i
\(250\) 0 0
\(251\) 2142.04 0.538662 0.269331 0.963048i \(-0.413198\pi\)
0.269331 + 0.963048i \(0.413198\pi\)
\(252\) 0 0
\(253\) 3621.96i 0.900042i
\(254\) 0 0
\(255\) 2496.08i 0.612983i
\(256\) 0 0
\(257\) −3152.62 −0.765194 −0.382597 0.923915i \(-0.624970\pi\)
−0.382597 + 0.923915i \(0.624970\pi\)
\(258\) 0 0
\(259\) −3453.52 −0.828539
\(260\) 0 0
\(261\) 127.308 0.0301923
\(262\) 0 0
\(263\) 2167.71 0.508238 0.254119 0.967173i \(-0.418214\pi\)
0.254119 + 0.967173i \(0.418214\pi\)
\(264\) 0 0
\(265\) 6631.28i 1.53719i
\(266\) 0 0
\(267\) 1583.67i 0.362992i
\(268\) 0 0
\(269\) −3248.98 −0.736409 −0.368204 0.929745i \(-0.620027\pi\)
−0.368204 + 0.929745i \(0.620027\pi\)
\(270\) 0 0
\(271\) 4897.70i 1.09784i 0.835876 + 0.548919i \(0.184960\pi\)
−0.835876 + 0.548919i \(0.815040\pi\)
\(272\) 0 0
\(273\) −1670.18 2664.56i −0.370271 0.590719i
\(274\) 0 0
\(275\) 1595.43i 0.349847i
\(276\) 0 0
\(277\) −2900.33 −0.629111 −0.314555 0.949239i \(-0.601855\pi\)
−0.314555 + 0.949239i \(0.601855\pi\)
\(278\) 0 0
\(279\) 1765.94i 0.378939i
\(280\) 0 0
\(281\) 4396.53i 0.933364i 0.884425 + 0.466682i \(0.154551\pi\)
−0.884425 + 0.466682i \(0.845449\pi\)
\(282\) 0 0
\(283\) −559.151 −0.117449 −0.0587245 0.998274i \(-0.518703\pi\)
−0.0587245 + 0.998274i \(0.518703\pi\)
\(284\) 0 0
\(285\) −3386.62 −0.703880
\(286\) 0 0
\(287\) 5942.69 1.22225
\(288\) 0 0
\(289\) 2508.02 0.510487
\(290\) 0 0
\(291\) 222.827i 0.0448877i
\(292\) 0 0
\(293\) 6918.65i 1.37950i −0.724050 0.689748i \(-0.757722\pi\)
0.724050 0.689748i \(-0.242278\pi\)
\(294\) 0 0
\(295\) 887.901 0.175239
\(296\) 0 0
\(297\) 1358.24i 0.265363i
\(298\) 0 0
\(299\) −2859.49 + 1792.37i −0.553072 + 0.346673i
\(300\) 0 0
\(301\) 4737.90i 0.907270i
\(302\) 0 0
\(303\) 1828.26 0.346635
\(304\) 0 0
\(305\) 3177.67i 0.596567i
\(306\) 0 0
\(307\) 8980.94i 1.66961i 0.550548 + 0.834803i \(0.314419\pi\)
−0.550548 + 0.834803i \(0.685581\pi\)
\(308\) 0 0
\(309\) 97.3082 0.0179148
\(310\) 0 0
\(311\) −7943.13 −1.44827 −0.724137 0.689656i \(-0.757762\pi\)
−0.724137 + 0.689656i \(0.757762\pi\)
\(312\) 0 0
\(313\) −5059.57 −0.913686 −0.456843 0.889547i \(-0.651020\pi\)
−0.456843 + 0.889547i \(0.651020\pi\)
\(314\) 0 0
\(315\) −1944.00 −0.347721
\(316\) 0 0
\(317\) 8702.12i 1.54183i 0.636939 + 0.770914i \(0.280200\pi\)
−0.636939 + 0.770914i \(0.719800\pi\)
\(318\) 0 0
\(319\) 711.582i 0.124893i
\(320\) 0 0
\(321\) −5176.36 −0.900051
\(322\) 0 0
\(323\) 10068.6i 1.73447i
\(324\) 0 0
\(325\) −1259.57 + 789.515i −0.214980 + 0.134752i
\(326\) 0 0
\(327\) 3820.30i 0.646064i
\(328\) 0 0
\(329\) −1510.76 −0.253164
\(330\) 0 0
\(331\) 1737.65i 0.288549i −0.989538 0.144275i \(-0.953915\pi\)
0.989538 0.144275i \(-0.0460848\pi\)
\(332\) 0 0
\(333\) 1389.81i 0.228713i
\(334\) 0 0
\(335\) 7421.23 1.21034
\(336\) 0 0
\(337\) −2917.47 −0.471586 −0.235793 0.971803i \(-0.575769\pi\)
−0.235793 + 0.971803i \(0.575769\pi\)
\(338\) 0 0
\(339\) −3344.18 −0.535785
\(340\) 0 0
\(341\) 9870.61 1.56752
\(342\) 0 0
\(343\) 4156.44i 0.654305i
\(344\) 0 0
\(345\) 2086.22i 0.325560i
\(346\) 0 0
\(347\) −5081.23 −0.786095 −0.393047 0.919518i \(-0.628579\pi\)
−0.393047 + 0.919518i \(0.628579\pi\)
\(348\) 0 0
\(349\) 4266.14i 0.654330i 0.944967 + 0.327165i \(0.106093\pi\)
−0.944967 + 0.327165i \(0.893907\pi\)
\(350\) 0 0
\(351\) 1072.31 672.137i 0.163064 0.102211i
\(352\) 0 0
\(353\) 3264.49i 0.492213i −0.969243 0.246106i \(-0.920849\pi\)
0.969243 0.246106i \(-0.0791513\pi\)
\(354\) 0 0
\(355\) 2559.18 0.382612
\(356\) 0 0
\(357\) 5779.65i 0.856838i
\(358\) 0 0
\(359\) 4416.98i 0.649357i 0.945824 + 0.324679i \(0.105256\pi\)
−0.945824 + 0.324679i \(0.894744\pi\)
\(360\) 0 0
\(361\) −6801.87 −0.991670
\(362\) 0 0
\(363\) 3598.78 0.520350
\(364\) 0 0
\(365\) 7449.59 1.06830
\(366\) 0 0
\(367\) 1740.22 0.247516 0.123758 0.992312i \(-0.460505\pi\)
0.123758 + 0.992312i \(0.460505\pi\)
\(368\) 0 0
\(369\) 2391.54i 0.337394i
\(370\) 0 0
\(371\) 15354.7i 2.14872i
\(372\) 0 0
\(373\) 1176.28 0.163285 0.0816427 0.996662i \(-0.473983\pi\)
0.0816427 + 0.996662i \(0.473983\pi\)
\(374\) 0 0
\(375\) 4540.86i 0.625304i
\(376\) 0 0
\(377\) −561.785 + 352.134i −0.0767464 + 0.0481056i
\(378\) 0 0
\(379\) 7135.54i 0.967092i −0.875319 0.483546i \(-0.839349\pi\)
0.875319 0.483546i \(-0.160651\pi\)
\(380\) 0 0
\(381\) −3522.04 −0.473594
\(382\) 0 0
\(383\) 1942.87i 0.259207i 0.991566 + 0.129603i \(0.0413704\pi\)
−0.991566 + 0.129603i \(0.958630\pi\)
\(384\) 0 0
\(385\) 10865.9i 1.43838i
\(386\) 0 0
\(387\) 1906.69 0.250446
\(388\) 0 0
\(389\) 7545.92 0.983531 0.491766 0.870728i \(-0.336352\pi\)
0.491766 + 0.870728i \(0.336352\pi\)
\(390\) 0 0
\(391\) −6202.47 −0.802231
\(392\) 0 0
\(393\) −4335.49 −0.556480
\(394\) 0 0
\(395\) 11849.5i 1.50940i
\(396\) 0 0
\(397\) 415.922i 0.0525806i −0.999654 0.0262903i \(-0.991631\pi\)
0.999654 0.0262903i \(-0.00836943\pi\)
\(398\) 0 0
\(399\) −7841.67 −0.983896
\(400\) 0 0
\(401\) 958.178i 0.119324i 0.998219 + 0.0596622i \(0.0190024\pi\)
−0.998219 + 0.0596622i \(0.980998\pi\)
\(402\) 0 0
\(403\) 4884.58 + 7792.71i 0.603767 + 0.963233i
\(404\) 0 0
\(405\) 782.331i 0.0959860i
\(406\) 0 0
\(407\) 7768.29 0.946093
\(408\) 0 0
\(409\) 4284.83i 0.518022i 0.965874 + 0.259011i \(0.0833966\pi\)
−0.965874 + 0.259011i \(0.916603\pi\)
\(410\) 0 0
\(411\) 1526.38i 0.183189i
\(412\) 0 0
\(413\) 2055.92 0.244953
\(414\) 0 0
\(415\) −4964.61 −0.587236
\(416\) 0 0
\(417\) −2271.05 −0.266700
\(418\) 0 0
\(419\) 9949.01 1.16000 0.580001 0.814616i \(-0.303052\pi\)
0.580001 + 0.814616i \(0.303052\pi\)
\(420\) 0 0
\(421\) 377.250i 0.0436724i −0.999762 0.0218362i \(-0.993049\pi\)
0.999762 0.0218362i \(-0.00695122\pi\)
\(422\) 0 0
\(423\) 607.982i 0.0698843i
\(424\) 0 0
\(425\) −2732.11 −0.311828
\(426\) 0 0
\(427\) 7357.86i 0.833892i
\(428\) 0 0
\(429\) 3756.87 + 5993.61i 0.422805 + 0.674532i
\(430\) 0 0
\(431\) 2437.13i 0.272373i −0.990683 0.136186i \(-0.956515\pi\)
0.990683 0.136186i \(-0.0434846\pi\)
\(432\) 0 0
\(433\) −11215.1 −1.24471 −0.622357 0.782733i \(-0.713825\pi\)
−0.622357 + 0.782733i \(0.713825\pi\)
\(434\) 0 0
\(435\) 409.865i 0.0451759i
\(436\) 0 0
\(437\) 8415.34i 0.921191i
\(438\) 0 0
\(439\) −1835.19 −0.199519 −0.0997596 0.995012i \(-0.531807\pi\)
−0.0997596 + 0.995012i \(0.531807\pi\)
\(440\) 0 0
\(441\) −1414.31 −0.152717
\(442\) 0 0
\(443\) −11610.1 −1.24518 −0.622588 0.782550i \(-0.713919\pi\)
−0.622588 + 0.782550i \(0.713919\pi\)
\(444\) 0 0
\(445\) 5098.56 0.543135
\(446\) 0 0
\(447\) 9743.38i 1.03098i
\(448\) 0 0
\(449\) 14087.0i 1.48064i −0.672255 0.740319i \(-0.734674\pi\)
0.672255 0.740319i \(-0.265326\pi\)
\(450\) 0 0
\(451\) −13367.4 −1.39566
\(452\) 0 0
\(453\) 2385.89i 0.247459i
\(454\) 0 0
\(455\) 8578.47 5377.10i 0.883878 0.554027i
\(456\) 0 0
\(457\) 2375.01i 0.243103i −0.992585 0.121552i \(-0.961213\pi\)
0.992585 0.121552i \(-0.0387870\pi\)
\(458\) 0 0
\(459\) 2325.92 0.236525
\(460\) 0 0
\(461\) 6372.06i 0.643766i 0.946779 + 0.321883i \(0.104316\pi\)
−0.946779 + 0.321883i \(0.895684\pi\)
\(462\) 0 0
\(463\) 63.4732i 0.00637117i 0.999995 + 0.00318558i \(0.00101400\pi\)
−0.999995 + 0.00318558i \(0.998986\pi\)
\(464\) 0 0
\(465\) 5685.38 0.566996
\(466\) 0 0
\(467\) −7855.78 −0.778420 −0.389210 0.921149i \(-0.627252\pi\)
−0.389210 + 0.921149i \(0.627252\pi\)
\(468\) 0 0
\(469\) 17183.8 1.69184
\(470\) 0 0
\(471\) 195.820 0.0191569
\(472\) 0 0
\(473\) 10657.3i 1.03599i
\(474\) 0 0
\(475\) 3706.85i 0.358068i
\(476\) 0 0
\(477\) 6179.23 0.593140
\(478\) 0 0
\(479\) 13033.3i 1.24323i 0.783324 + 0.621613i \(0.213523\pi\)
−0.783324 + 0.621613i \(0.786477\pi\)
\(480\) 0 0
\(481\) 3844.22 + 6132.96i 0.364410 + 0.581370i
\(482\) 0 0
\(483\) 4830.61i 0.455073i
\(484\) 0 0
\(485\) 717.384 0.0671643
\(486\) 0 0
\(487\) 69.8976i 0.00650383i −0.999995 0.00325191i \(-0.998965\pi\)
0.999995 0.00325191i \(-0.00103512\pi\)
\(488\) 0 0
\(489\) 5566.18i 0.514747i
\(490\) 0 0
\(491\) −2625.66 −0.241333 −0.120667 0.992693i \(-0.538503\pi\)
−0.120667 + 0.992693i \(0.538503\pi\)
\(492\) 0 0
\(493\) −1218.56 −0.111321
\(494\) 0 0
\(495\) 4372.80 0.397056
\(496\) 0 0
\(497\) 5925.75 0.534821
\(498\) 0 0
\(499\) 7631.34i 0.684621i 0.939587 + 0.342310i \(0.111209\pi\)
−0.939587 + 0.342310i \(0.888791\pi\)
\(500\) 0 0
\(501\) 10597.6i 0.945044i
\(502\) 0 0
\(503\) 4320.14 0.382953 0.191477 0.981497i \(-0.438672\pi\)
0.191477 + 0.981497i \(0.438672\pi\)
\(504\) 0 0
\(505\) 5886.01i 0.518662i
\(506\) 0 0
\(507\) −2872.74 + 5932.00i −0.251643 + 0.519624i
\(508\) 0 0
\(509\) 12450.7i 1.08422i 0.840308 + 0.542109i \(0.182374\pi\)
−0.840308 + 0.542109i \(0.817626\pi\)
\(510\) 0 0
\(511\) 17249.4 1.49329
\(512\) 0 0
\(513\) 3155.75i 0.271598i
\(514\) 0 0
\(515\) 313.281i 0.0268054i
\(516\) 0 0
\(517\) 3398.28 0.289083
\(518\) 0 0
\(519\) −9535.08 −0.806443
\(520\) 0 0
\(521\) 14373.1 1.20863 0.604314 0.796746i \(-0.293447\pi\)
0.604314 + 0.796746i \(0.293447\pi\)
\(522\) 0 0
\(523\) 16946.7 1.41688 0.708439 0.705772i \(-0.249400\pi\)
0.708439 + 0.705772i \(0.249400\pi\)
\(524\) 0 0
\(525\) 2127.82i 0.176887i
\(526\) 0 0
\(527\) 16903.0i 1.39717i
\(528\) 0 0
\(529\) −6983.00 −0.573929
\(530\) 0 0
\(531\) 827.373i 0.0676176i
\(532\) 0 0
\(533\) −6614.98 10553.4i −0.537574 0.857630i
\(534\) 0 0
\(535\) 16665.1i 1.34672i
\(536\) 0 0
\(537\) −8223.49 −0.660837
\(538\) 0 0
\(539\) 7905.20i 0.631727i
\(540\) 0 0
\(541\) 815.667i 0.0648212i 0.999475 + 0.0324106i \(0.0103184\pi\)
−0.999475 + 0.0324106i \(0.989682\pi\)
\(542\) 0 0
\(543\) 11613.3 0.917814
\(544\) 0 0
\(545\) −12299.3 −0.966689
\(546\) 0 0
\(547\) 17971.4 1.40476 0.702378 0.711804i \(-0.252122\pi\)
0.702378 + 0.711804i \(0.252122\pi\)
\(548\) 0 0
\(549\) 2961.05 0.230191
\(550\) 0 0
\(551\) 1653.31i 0.127828i
\(552\) 0 0
\(553\) 27437.4i 2.10987i
\(554\) 0 0
\(555\) 4474.47 0.342217
\(556\) 0 0
\(557\) 6760.83i 0.514301i 0.966371 + 0.257151i \(0.0827836\pi\)
−0.966371 + 0.257151i \(0.917216\pi\)
\(558\) 0 0
\(559\) −8413.83 + 5273.90i −0.636614 + 0.399038i
\(560\) 0 0
\(561\) 13000.6i 0.978408i
\(562\) 0 0
\(563\) −12962.7 −0.970359 −0.485179 0.874415i \(-0.661246\pi\)
−0.485179 + 0.874415i \(0.661246\pi\)
\(564\) 0 0
\(565\) 10766.5i 0.801681i
\(566\) 0 0
\(567\) 1811.48i 0.134171i
\(568\) 0 0
\(569\) 164.757 0.0121388 0.00606938 0.999982i \(-0.498068\pi\)
0.00606938 + 0.999982i \(0.498068\pi\)
\(570\) 0 0
\(571\) 5216.55 0.382322 0.191161 0.981559i \(-0.438775\pi\)
0.191161 + 0.981559i \(0.438775\pi\)
\(572\) 0 0
\(573\) −2784.87 −0.203036
\(574\) 0 0
\(575\) 2283.49 0.165614
\(576\) 0 0
\(577\) 12753.3i 0.920151i −0.887880 0.460076i \(-0.847822\pi\)
0.887880 0.460076i \(-0.152178\pi\)
\(578\) 0 0
\(579\) 6784.53i 0.486970i
\(580\) 0 0
\(581\) −11495.5 −0.820849
\(582\) 0 0
\(583\) 34538.5i 2.45358i
\(584\) 0 0
\(585\) 2163.92 + 3452.26i 0.152936 + 0.243989i
\(586\) 0 0
\(587\) 1575.38i 0.110771i 0.998465 + 0.0553857i \(0.0176388\pi\)
−0.998465 + 0.0553857i \(0.982361\pi\)
\(588\) 0 0
\(589\) 22933.6 1.60435
\(590\) 0 0
\(591\) 6796.77i 0.473066i
\(592\) 0 0
\(593\) 3845.95i 0.266331i 0.991094 + 0.133165i \(0.0425142\pi\)
−0.991094 + 0.133165i \(0.957486\pi\)
\(594\) 0 0
\(595\) 18607.4 1.28207
\(596\) 0 0
\(597\) 782.686 0.0536570
\(598\) 0 0
\(599\) −6107.20 −0.416583 −0.208292 0.978067i \(-0.566790\pi\)
−0.208292 + 0.978067i \(0.566790\pi\)
\(600\) 0 0
\(601\) 9638.90 0.654208 0.327104 0.944988i \(-0.393927\pi\)
0.327104 + 0.944988i \(0.393927\pi\)
\(602\) 0 0
\(603\) 6915.33i 0.467022i
\(604\) 0 0
\(605\) 11586.2i 0.778586i
\(606\) 0 0
\(607\) −11821.7 −0.790489 −0.395244 0.918576i \(-0.629340\pi\)
−0.395244 + 0.918576i \(0.629340\pi\)
\(608\) 0 0
\(609\) 949.037i 0.0631477i
\(610\) 0 0
\(611\) 1681.67 + 2682.90i 0.111347 + 0.177640i
\(612\) 0 0
\(613\) 8107.86i 0.534214i −0.963667 0.267107i \(-0.913932\pi\)
0.963667 0.267107i \(-0.0860677\pi\)
\(614\) 0 0
\(615\) −7699.48 −0.504834
\(616\) 0 0
\(617\) 27647.6i 1.80397i −0.431768 0.901985i \(-0.642110\pi\)
0.431768 0.901985i \(-0.357890\pi\)
\(618\) 0 0
\(619\) 29181.9i 1.89486i −0.319956 0.947432i \(-0.603668\pi\)
0.319956 0.947432i \(-0.396332\pi\)
\(620\) 0 0
\(621\) −1944.00 −0.125620
\(622\) 0 0
\(623\) 11805.7 0.759204
\(624\) 0 0
\(625\) −10654.8 −0.681905
\(626\) 0 0
\(627\) 17638.9 1.12349
\(628\) 0 0
\(629\) 13302.9i 0.843277i
\(630\) 0 0
\(631\) 2209.34i 0.139386i 0.997569 + 0.0696928i \(0.0222019\pi\)
−0.997569 + 0.0696928i \(0.977798\pi\)
\(632\) 0 0
\(633\) 17553.7 1.10220
\(634\) 0 0
\(635\) 11339.1i 0.708627i
\(636\) 0 0
\(637\) 6241.05 3911.97i 0.388194 0.243325i
\(638\) 0 0
\(639\) 2384.72i 0.147634i
\(640\) 0 0
\(641\) 18256.0 1.12491 0.562455 0.826828i \(-0.309857\pi\)
0.562455 + 0.826828i \(0.309857\pi\)
\(642\) 0 0
\(643\) 1281.61i 0.0786033i 0.999227 + 0.0393016i \(0.0125133\pi\)
−0.999227 + 0.0393016i \(0.987487\pi\)
\(644\) 0 0
\(645\) 6138.54i 0.374736i
\(646\) 0 0
\(647\) 16393.2 0.996107 0.498054 0.867146i \(-0.334048\pi\)
0.498054 + 0.867146i \(0.334048\pi\)
\(648\) 0 0
\(649\) −4624.56 −0.279707
\(650\) 0 0
\(651\) 13164.4 0.792557
\(652\) 0 0
\(653\) 16759.2 1.00434 0.502172 0.864768i \(-0.332534\pi\)
0.502172 + 0.864768i \(0.332534\pi\)
\(654\) 0 0
\(655\) 13958.0i 0.832646i
\(656\) 0 0
\(657\) 6941.76i 0.412213i
\(658\) 0 0
\(659\) 29659.3 1.75320 0.876601 0.481217i \(-0.159805\pi\)
0.876601 + 0.481217i \(0.159805\pi\)
\(660\) 0 0
\(661\) 10386.0i 0.611147i −0.952169 0.305573i \(-0.901152\pi\)
0.952169 0.305573i \(-0.0988481\pi\)
\(662\) 0 0
\(663\) −10263.8 + 6433.50i −0.601227 + 0.376857i
\(664\) 0 0
\(665\) 25246.0i 1.47218i
\(666\) 0 0
\(667\) 1018.47 0.0591232
\(668\) 0 0
\(669\) 10390.8i 0.600496i
\(670\) 0 0
\(671\) 16550.6i 0.952206i
\(672\) 0 0
\(673\) 19449.6 1.11400 0.557002 0.830511i \(-0.311951\pi\)
0.557002 + 0.830511i \(0.311951\pi\)
\(674\) 0 0
\(675\) −856.308 −0.0488286
\(676\) 0 0
\(677\) −6629.48 −0.376354 −0.188177 0.982135i \(-0.560258\pi\)
−0.188177 + 0.982135i \(0.560258\pi\)
\(678\) 0 0
\(679\) 1661.09 0.0938835
\(680\) 0 0
\(681\) 15987.5i 0.899619i
\(682\) 0 0
\(683\) 2526.40i 0.141537i 0.997493 + 0.0707687i \(0.0225452\pi\)
−0.997493 + 0.0707687i \(0.977455\pi\)
\(684\) 0 0
\(685\) 4914.13 0.274101
\(686\) 0 0
\(687\) 14321.9i 0.795361i
\(688\) 0 0
\(689\) −27267.7 + 17091.7i −1.50771 + 0.945055i
\(690\) 0 0
\(691\) 3808.76i 0.209685i 0.994489 + 0.104842i \(0.0334338\pi\)
−0.994489 + 0.104842i \(0.966566\pi\)
\(692\) 0 0
\(693\) 10125.2 0.555011
\(694\) 0 0
\(695\) 7311.59i 0.399056i
\(696\) 0 0
\(697\) 22891.1i 1.24399i
\(698\) 0 0
\(699\) −14441.3 −0.781433
\(700\) 0 0
\(701\) −33617.8 −1.81131 −0.905655 0.424015i \(-0.860620\pi\)
−0.905655 + 0.424015i \(0.860620\pi\)
\(702\) 0 0
\(703\) 18049.0 0.968323
\(704\) 0 0
\(705\) 1957.38 0.104566
\(706\) 0 0
\(707\) 13629.0i 0.724994i
\(708\) 0 0
\(709\) 26606.5i 1.40935i 0.709530 + 0.704675i \(0.248907\pi\)
−0.709530 + 0.704675i \(0.751093\pi\)
\(710\) 0 0
\(711\) −11041.7 −0.582416
\(712\) 0 0
\(713\) 14127.5i 0.742046i
\(714\) 0 0
\(715\) −19296.2 + 12095.1i −1.00928 + 0.632633i
\(716\) 0 0
\(717\) 5049.56i 0.263011i
\(718\) 0 0
\(719\) −16539.3 −0.857877 −0.428939 0.903334i \(-0.641112\pi\)
−0.428939 + 0.903334i \(0.641112\pi\)
\(720\) 0 0
\(721\) 725.398i 0.0374691i
\(722\) 0 0
\(723\) 1994.58i 0.102599i
\(724\) 0 0
\(725\) 448.622 0.0229812
\(726\) 0 0
\(727\) −12757.5 −0.650823 −0.325411 0.945573i \(-0.605503\pi\)
−0.325411 + 0.945573i \(0.605503\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −18250.3 −0.923408
\(732\) 0 0
\(733\) 20523.4i 1.03417i 0.855933 + 0.517087i \(0.172984\pi\)
−0.855933 + 0.517087i \(0.827016\pi\)
\(734\) 0 0
\(735\) 4553.32i 0.228506i
\(736\) 0 0
\(737\) −38652.9 −1.93188
\(738\) 0 0
\(739\) 7462.66i 0.371473i −0.982600 0.185736i \(-0.940533\pi\)
0.982600 0.185736i \(-0.0594670\pi\)
\(740\) 0 0
\(741\) 8728.80 + 13925.7i 0.432740 + 0.690381i
\(742\) 0 0
\(743\) 18847.8i 0.930632i −0.885145 0.465316i \(-0.845941\pi\)
0.885145 0.465316i \(-0.154059\pi\)
\(744\) 0 0
\(745\) 31368.5 1.54262
\(746\) 0 0
\(747\) 4626.17i 0.226590i
\(748\) 0 0
\(749\) 38587.9i 1.88247i
\(750\) 0 0
\(751\) −20832.6 −1.01224 −0.506119 0.862464i \(-0.668920\pi\)
−0.506119 + 0.862464i \(0.668920\pi\)
\(752\) 0 0
\(753\) −6426.11 −0.310996
\(754\) 0 0
\(755\) −7681.29 −0.370266
\(756\) 0 0
\(757\) −20860.9 −1.00159 −0.500795 0.865566i \(-0.666959\pi\)
−0.500795 + 0.865566i \(0.666959\pi\)
\(758\) 0 0
\(759\) 10865.9i 0.519640i
\(760\) 0 0
\(761\) 4464.86i 0.212682i 0.994330 + 0.106341i \(0.0339135\pi\)
−0.994330 + 0.106341i \(0.966086\pi\)
\(762\) 0 0
\(763\) −28479.0 −1.35126
\(764\) 0 0
\(765\) 7488.24i 0.353906i
\(766\) 0 0
\(767\) −2288.51 3651.03i −0.107736 0.171879i
\(768\) 0 0
\(769\) 23797.7i 1.11595i 0.829857 + 0.557977i \(0.188422\pi\)
−0.829857 + 0.557977i \(0.811578\pi\)
\(770\) 0 0
\(771\) 9457.85 0.441785
\(772\) 0 0
\(773\) 29418.7i 1.36885i 0.729085 + 0.684423i \(0.239946\pi\)
−0.729085 + 0.684423i \(0.760054\pi\)
\(774\) 0 0
\(775\) 6222.99i 0.288434i
\(776\) 0 0
\(777\) 10360.6 0.478357
\(778\) 0 0
\(779\) −31058.0 −1.42846
\(780\) 0 0
\(781\) −13329.3 −0.610703
\(782\) 0 0
\(783\) −381.925 −0.0174315
\(784\) 0 0
\(785\) 630.435i 0.0286640i
\(786\) 0 0
\(787\) 23896.0i 1.08234i 0.840913 + 0.541170i \(0.182018\pi\)
−0.840913 + 0.541170i \(0.817982\pi\)
\(788\) 0 0
\(789\) −6503.13 −0.293432
\(790\) 0 0
\(791\) 24929.7i 1.12060i
\(792\) 0 0
\(793\) −13066.5 + 8190.26i −0.585126 + 0.366765i
\(794\) 0 0
\(795\) 19893.9i 0.887500i
\(796\) 0 0
\(797\) 1034.67 0.0459847 0.0229923 0.999736i \(-0.492681\pi\)
0.0229923 + 0.999736i \(0.492681\pi\)
\(798\)