Properties

Label 3150.2.b.e.251.7
Level 3150
Weight 2
Character 3150.251
Analytic conductor 25.153
Analytic rank 0
Dimension 8
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 3150.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(25.1528766367\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.7442857984.4
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 251.7
Root \(-2.73923i\)
Character \(\chi\) = 3150.251
Dual form 3150.2.b.e.251.3

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000i q^{2} -1.00000 q^{4} +(1.80230 + 1.93693i) q^{7} -1.00000i q^{8} +O(q^{10})\) \(q+1.00000i q^{2} -1.00000 q^{4} +(1.80230 + 1.93693i) q^{7} -1.00000i q^{8} +3.87386i q^{11} -1.60461i q^{13} +(-1.93693 + 1.80230i) q^{14} +1.00000 q^{16} +8.11650 q^{17} -2.63803i q^{19} -3.87386 q^{22} -5.47847i q^{23} +1.60461 q^{26} +(-1.80230 - 1.93693i) q^{28} +5.47847i q^{29} +3.73074i q^{31} +1.00000i q^{32} +8.11650i q^{34} -4.51190 q^{37} +2.63803 q^{38} +1.60461 q^{41} +10.1165 q^{43} -3.87386i q^{44} +5.47847 q^{46} +11.1097 q^{47} +(-0.503406 + 6.98188i) q^{49} +1.60461i q^{52} +2.26926i q^{53} +(1.93693 - 1.80230i) q^{56} -5.47847 q^{58} -4.61142 q^{59} -11.8472i q^{61} -3.73074 q^{62} -1.00000 q^{64} -6.90729 q^{67} -8.11650 q^{68} +2.63803i q^{71} -13.7477i q^{73} -4.51190i q^{74} +2.63803i q^{76} +(-7.50341 + 6.98188i) q^{77} -8.01698 q^{79} +1.60461i q^{82} +3.20921 q^{83} +10.1165i q^{86} +3.87386 q^{88} +17.8376 q^{89} +(3.10801 - 2.89199i) q^{91} +5.47847i q^{92} +11.1097i q^{94} +8.68768i q^{97} +(-6.98188 - 0.503406i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 8q^{4} + 4q^{7} + O(q^{10}) \) \( 8q - 8q^{4} + 4q^{7} + 8q^{16} - 8q^{26} - 4q^{28} + 8q^{37} + 8q^{38} - 8q^{41} + 16q^{43} - 8q^{46} + 40q^{47} + 4q^{49} + 8q^{58} - 40q^{62} - 8q^{64} - 32q^{67} - 52q^{77} + 8q^{79} - 16q^{83} - 8q^{89} - 4q^{91} + 4q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) 1.80230 + 1.93693i 0.681207 + 0.732091i
\(8\) 1.00000i 0.353553i
\(9\) 0 0
\(10\) 0 0
\(11\) 3.87386i 1.16801i 0.811749 + 0.584007i \(0.198516\pi\)
−0.811749 + 0.584007i \(0.801484\pi\)
\(12\) 0 0
\(13\) 1.60461i 0.445038i −0.974928 0.222519i \(-0.928572\pi\)
0.974928 0.222519i \(-0.0714279\pi\)
\(14\) −1.93693 + 1.80230i −0.517667 + 0.481686i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 8.11650 1.96854 0.984271 0.176667i \(-0.0565317\pi\)
0.984271 + 0.176667i \(0.0565317\pi\)
\(18\) 0 0
\(19\) 2.63803i 0.605207i −0.953117 0.302603i \(-0.902144\pi\)
0.953117 0.302603i \(-0.0978557\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −3.87386 −0.825910
\(23\) 5.47847i 1.14234i −0.820832 0.571170i \(-0.806490\pi\)
0.820832 0.571170i \(-0.193510\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.60461 0.314689
\(27\) 0 0
\(28\) −1.80230 1.93693i −0.340603 0.366046i
\(29\) 5.47847i 1.01733i 0.860966 + 0.508663i \(0.169860\pi\)
−0.860966 + 0.508663i \(0.830140\pi\)
\(30\) 0 0
\(31\) 3.73074i 0.670061i 0.942207 + 0.335031i \(0.108747\pi\)
−0.942207 + 0.335031i \(0.891253\pi\)
\(32\) 1.00000i 0.176777i
\(33\) 0 0
\(34\) 8.11650i 1.39197i
\(35\) 0 0
\(36\) 0 0
\(37\) −4.51190 −0.741751 −0.370876 0.928683i \(-0.620942\pi\)
−0.370876 + 0.928683i \(0.620942\pi\)
\(38\) 2.63803 0.427946
\(39\) 0 0
\(40\) 0 0
\(41\) 1.60461 0.250597 0.125299 0.992119i \(-0.460011\pi\)
0.125299 + 0.992119i \(0.460011\pi\)
\(42\) 0 0
\(43\) 10.1165 1.54275 0.771376 0.636379i \(-0.219569\pi\)
0.771376 + 0.636379i \(0.219569\pi\)
\(44\) 3.87386i 0.584007i
\(45\) 0 0
\(46\) 5.47847 0.807756
\(47\) 11.1097 1.62052 0.810258 0.586074i \(-0.199327\pi\)
0.810258 + 0.586074i \(0.199327\pi\)
\(48\) 0 0
\(49\) −0.503406 + 6.98188i −0.0719152 + 0.997411i
\(50\) 0 0
\(51\) 0 0
\(52\) 1.60461i 0.222519i
\(53\) 2.26926i 0.311706i 0.987780 + 0.155853i \(0.0498127\pi\)
−0.987780 + 0.155853i \(0.950187\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.93693 1.80230i 0.258833 0.240843i
\(57\) 0 0
\(58\) −5.47847 −0.719358
\(59\) −4.61142 −0.600356 −0.300178 0.953883i \(-0.597046\pi\)
−0.300178 + 0.953883i \(0.597046\pi\)
\(60\) 0 0
\(61\) 11.8472i 1.51688i −0.651740 0.758442i \(-0.725961\pi\)
0.651740 0.758442i \(-0.274039\pi\)
\(62\) −3.73074 −0.473805
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −6.90729 −0.843860 −0.421930 0.906628i \(-0.638647\pi\)
−0.421930 + 0.906628i \(0.638647\pi\)
\(68\) −8.11650 −0.984271
\(69\) 0 0
\(70\) 0 0
\(71\) 2.63803i 0.313077i 0.987672 + 0.156539i \(0.0500335\pi\)
−0.987672 + 0.156539i \(0.949966\pi\)
\(72\) 0 0
\(73\) 13.7477i 1.60905i −0.593919 0.804525i \(-0.702420\pi\)
0.593919 0.804525i \(-0.297580\pi\)
\(74\) 4.51190i 0.524497i
\(75\) 0 0
\(76\) 2.63803i 0.302603i
\(77\) −7.50341 + 6.98188i −0.855092 + 0.795659i
\(78\) 0 0
\(79\) −8.01698 −0.901981 −0.450990 0.892529i \(-0.648929\pi\)
−0.450990 + 0.892529i \(0.648929\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 1.60461i 0.177199i
\(83\) 3.20921 0.352257 0.176128 0.984367i \(-0.443643\pi\)
0.176128 + 0.984367i \(0.443643\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 10.1165i 1.09089i
\(87\) 0 0
\(88\) 3.87386 0.412955
\(89\) 17.8376 1.89078 0.945392 0.325937i \(-0.105680\pi\)
0.945392 + 0.325937i \(0.105680\pi\)
\(90\) 0 0
\(91\) 3.10801 2.89199i 0.325808 0.303163i
\(92\) 5.47847i 0.571170i
\(93\) 0 0
\(94\) 11.1097i 1.14588i
\(95\) 0 0
\(96\) 0 0
\(97\) 8.68768i 0.882100i 0.897482 + 0.441050i \(0.145394\pi\)
−0.897482 + 0.441050i \(0.854606\pi\)
\(98\) −6.98188 0.503406i −0.705276 0.0508517i
\(99\) 0 0
\(100\) 0 0
\(101\) −6.21603 −0.618518 −0.309259 0.950978i \(-0.600081\pi\)
−0.309259 + 0.950978i \(0.600081\pi\)
\(102\) 0 0
\(103\) 12.8807i 1.26917i 0.772853 + 0.634585i \(0.218829\pi\)
−0.772853 + 0.634585i \(0.781171\pi\)
\(104\) −1.60461 −0.157345
\(105\) 0 0
\(106\) −2.26926 −0.220410
\(107\) 19.9638i 1.92997i 0.262308 + 0.964984i \(0.415516\pi\)
−0.262308 + 0.964984i \(0.584484\pi\)
\(108\) 0 0
\(109\) −15.0238 −1.43902 −0.719509 0.694483i \(-0.755633\pi\)
−0.719509 + 0.694483i \(0.755633\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.80230 + 1.93693i 0.170302 + 0.183023i
\(113\) 10.2330i 0.962640i 0.876545 + 0.481320i \(0.159843\pi\)
−0.876545 + 0.481320i \(0.840157\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 5.47847i 0.508663i
\(117\) 0 0
\(118\) 4.61142i 0.424515i
\(119\) 14.6284 + 15.7211i 1.34098 + 1.44115i
\(120\) 0 0
\(121\) −4.00681 −0.364256
\(122\) 11.8472 1.07260
\(123\) 0 0
\(124\) 3.73074i 0.335031i
\(125\) 0 0
\(126\) 0 0
\(127\) 2.81382 0.249686 0.124843 0.992177i \(-0.460157\pi\)
0.124843 + 0.992177i \(0.460157\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) 0 0
\(130\) 0 0
\(131\) 8.62840 0.753867 0.376933 0.926240i \(-0.376979\pi\)
0.376933 + 0.926240i \(0.376979\pi\)
\(132\) 0 0
\(133\) 5.10969 4.75454i 0.443066 0.412271i
\(134\) 6.90729i 0.596699i
\(135\) 0 0
\(136\) 8.11650i 0.695984i
\(137\) 0.723932i 0.0618496i 0.999522 + 0.0309248i \(0.00984525\pi\)
−0.999522 + 0.0309248i \(0.990155\pi\)
\(138\) 0 0
\(139\) 2.84043i 0.240923i 0.992718 + 0.120461i \(0.0384374\pi\)
−0.992718 + 0.120461i \(0.961563\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −2.63803 −0.221379
\(143\) 6.21603 0.519810
\(144\) 0 0
\(145\) 0 0
\(146\) 13.7477 1.13777
\(147\) 0 0
\(148\) 4.51190 0.370876
\(149\) 3.00681i 0.246328i 0.992386 + 0.123164i \(0.0393041\pi\)
−0.992386 + 0.123164i \(0.960696\pi\)
\(150\) 0 0
\(151\) −15.2262 −1.23909 −0.619545 0.784961i \(-0.712683\pi\)
−0.619545 + 0.784961i \(0.712683\pi\)
\(152\) −2.63803 −0.213973
\(153\) 0 0
\(154\) −6.98188 7.50341i −0.562616 0.604642i
\(155\) 0 0
\(156\) 0 0
\(157\) 2.64767i 0.211307i −0.994403 0.105653i \(-0.966307\pi\)
0.994403 0.105653i \(-0.0336934\pi\)
\(158\) 8.01698i 0.637797i
\(159\) 0 0
\(160\) 0 0
\(161\) 10.6114 9.87386i 0.836297 0.778169i
\(162\) 0 0
\(163\) 9.32572 0.730446 0.365223 0.930920i \(-0.380993\pi\)
0.365223 + 0.930920i \(0.380993\pi\)
\(164\) −1.60461 −0.125299
\(165\) 0 0
\(166\) 3.20921i 0.249083i
\(167\) 17.3257 1.34070 0.670352 0.742043i \(-0.266143\pi\)
0.670352 + 0.742043i \(0.266143\pi\)
\(168\) 0 0
\(169\) 10.4252 0.801941
\(170\) 0 0
\(171\) 0 0
\(172\) −10.1165 −0.771376
\(173\) 2.99319 0.227568 0.113784 0.993506i \(-0.463703\pi\)
0.113784 + 0.993506i \(0.463703\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.87386i 0.292003i
\(177\) 0 0
\(178\) 17.8376i 1.33699i
\(179\) 7.38858i 0.552248i 0.961122 + 0.276124i \(0.0890501\pi\)
−0.961122 + 0.276124i \(0.910950\pi\)
\(180\) 0 0
\(181\) 11.1097i 0.825777i 0.910782 + 0.412888i \(0.135480\pi\)
−0.910782 + 0.412888i \(0.864520\pi\)
\(182\) 2.89199 + 3.10801i 0.214368 + 0.230381i
\(183\) 0 0
\(184\) −5.47847 −0.403878
\(185\) 0 0
\(186\) 0 0
\(187\) 31.4422i 2.29928i
\(188\) −11.1097 −0.810258
\(189\) 0 0
\(190\) 0 0
\(191\) 9.36197i 0.677408i 0.940893 + 0.338704i \(0.109989\pi\)
−0.940893 + 0.338704i \(0.890011\pi\)
\(192\) 0 0
\(193\) −17.4422 −1.25552 −0.627759 0.778408i \(-0.716028\pi\)
−0.627759 + 0.778408i \(0.716028\pi\)
\(194\) −8.68768 −0.623739
\(195\) 0 0
\(196\) 0.503406 6.98188i 0.0359576 0.498705i
\(197\) 5.27607i 0.375904i −0.982178 0.187952i \(-0.939815\pi\)
0.982178 0.187952i \(-0.0601850\pi\)
\(198\) 0 0
\(199\) 12.2160i 0.865971i 0.901401 + 0.432986i \(0.142540\pi\)
−0.901401 + 0.432986i \(0.857460\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 6.21603i 0.437358i
\(203\) −10.6114 + 9.87386i −0.744776 + 0.693009i
\(204\) 0 0
\(205\) 0 0
\(206\) −12.8807 −0.897439
\(207\) 0 0
\(208\) 1.60461i 0.111259i
\(209\) 10.2194 0.706889
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) 2.26926i 0.155853i
\(213\) 0 0
\(214\) −19.9638 −1.36469
\(215\) 0 0
\(216\) 0 0
\(217\) −7.22619 + 6.72393i −0.490546 + 0.456450i
\(218\) 15.0238i 1.01754i
\(219\) 0 0
\(220\) 0 0
\(221\) 13.0238i 0.876075i
\(222\) 0 0
\(223\) 3.65784i 0.244947i −0.992472 0.122473i \(-0.960917\pi\)
0.992472 0.122473i \(-0.0390826\pi\)
\(224\) −1.93693 + 1.80230i −0.129417 + 0.120421i
\(225\) 0 0
\(226\) −10.2330 −0.680689
\(227\) 4.23301 0.280955 0.140477 0.990084i \(-0.455136\pi\)
0.140477 + 0.990084i \(0.455136\pi\)
\(228\) 0 0
\(229\) 7.59497i 0.501890i 0.968001 + 0.250945i \(0.0807413\pi\)
−0.968001 + 0.250945i \(0.919259\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 5.47847 0.359679
\(233\) 9.94677i 0.651634i −0.945433 0.325817i \(-0.894361\pi\)
0.945433 0.325817i \(-0.105639\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 4.61142 0.300178
\(237\) 0 0
\(238\) −15.7211 + 14.6284i −1.01905 + 0.948218i
\(239\) 9.36197i 0.605575i −0.953058 0.302788i \(-0.902083\pi\)
0.953058 0.302788i \(-0.0979173\pi\)
\(240\) 0 0
\(241\) 23.0408i 1.48419i 0.670296 + 0.742093i \(0.266167\pi\)
−0.670296 + 0.742093i \(0.733833\pi\)
\(242\) 4.00681i 0.257568i
\(243\) 0 0
\(244\) 11.8472i 0.758442i
\(245\) 0 0
\(246\) 0 0
\(247\) −4.23301 −0.269340
\(248\) 3.73074 0.236902
\(249\) 0 0
\(250\) 0 0
\(251\) −23.8376 −1.50462 −0.752308 0.658811i \(-0.771060\pi\)
−0.752308 + 0.658811i \(0.771060\pi\)
\(252\) 0 0
\(253\) 21.2228 1.33427
\(254\) 2.81382i 0.176555i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −29.7441 −1.85539 −0.927694 0.373341i \(-0.878212\pi\)
−0.927694 + 0.373341i \(0.878212\pi\)
\(258\) 0 0
\(259\) −8.13181 8.73923i −0.505286 0.543030i
\(260\) 0 0
\(261\) 0 0
\(262\) 8.62840i 0.533064i
\(263\) 22.7545i 1.40310i 0.712618 + 0.701552i \(0.247509\pi\)
−0.712618 + 0.701552i \(0.752491\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 4.75454 + 5.10969i 0.291519 + 0.313295i
\(267\) 0 0
\(268\) 6.90729 0.421930
\(269\) 0.202401 0.0123406 0.00617030 0.999981i \(-0.498036\pi\)
0.00617030 + 0.999981i \(0.498036\pi\)
\(270\) 0 0
\(271\) 11.4785i 0.697267i 0.937259 + 0.348634i \(0.113354\pi\)
−0.937259 + 0.348634i \(0.886646\pi\)
\(272\) 8.11650 0.492135
\(273\) 0 0
\(274\) −0.723932 −0.0437343
\(275\) 0 0
\(276\) 0 0
\(277\) −23.7211 −1.42526 −0.712632 0.701538i \(-0.752497\pi\)
−0.712632 + 0.701538i \(0.752497\pi\)
\(278\) −2.84043 −0.170358
\(279\) 0 0
\(280\) 0 0
\(281\) 5.57194i 0.332394i −0.986093 0.166197i \(-0.946851\pi\)
0.986093 0.166197i \(-0.0531488\pi\)
\(282\) 0 0
\(283\) 32.1798i 1.91289i −0.291914 0.956445i \(-0.594292\pi\)
0.291914 0.956445i \(-0.405708\pi\)
\(284\) 2.63803i 0.156539i
\(285\) 0 0
\(286\) 6.21603i 0.367561i
\(287\) 2.89199 + 3.10801i 0.170709 + 0.183460i
\(288\) 0 0
\(289\) 48.8776 2.87515
\(290\) 0 0
\(291\) 0 0
\(292\) 13.7477i 0.804525i
\(293\) 15.8146 0.923898 0.461949 0.886907i \(-0.347150\pi\)
0.461949 + 0.886907i \(0.347150\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 4.51190i 0.262249i
\(297\) 0 0
\(298\) −3.00681 −0.174180
\(299\) −8.79079 −0.508384
\(300\) 0 0
\(301\) 18.2330 + 19.5950i 1.05093 + 1.12944i
\(302\) 15.2262i 0.876169i
\(303\) 0 0
\(304\) 2.63803i 0.151302i
\(305\) 0 0
\(306\) 0 0
\(307\) 6.72393i 0.383755i −0.981419 0.191878i \(-0.938542\pi\)
0.981419 0.191878i \(-0.0614576\pi\)
\(308\) 7.50341 6.98188i 0.427546 0.397829i
\(309\) 0 0
\(310\) 0 0
\(311\) −1.02379 −0.0580540 −0.0290270 0.999579i \(-0.509241\pi\)
−0.0290270 + 0.999579i \(0.509241\pi\)
\(312\) 0 0
\(313\) 23.2568i 1.31455i −0.753650 0.657276i \(-0.771709\pi\)
0.753650 0.657276i \(-0.228291\pi\)
\(314\) 2.64767 0.149417
\(315\) 0 0
\(316\) 8.01698 0.450990
\(317\) 4.46830i 0.250965i 0.992096 + 0.125482i \(0.0400478\pi\)
−0.992096 + 0.125482i \(0.959952\pi\)
\(318\) 0 0
\(319\) −21.2228 −1.18825
\(320\) 0 0
\(321\) 0 0
\(322\) 9.87386 + 10.6114i 0.550249 + 0.591351i
\(323\) 21.4116i 1.19137i
\(324\) 0 0
\(325\) 0 0
\(326\) 9.32572i 0.516504i
\(327\) 0 0
\(328\) 1.60461i 0.0885996i
\(329\) 20.0230 + 21.5187i 1.10391 + 1.18636i
\(330\) 0 0
\(331\) −14.2160 −0.781383 −0.390692 0.920522i \(-0.627764\pi\)
−0.390692 + 0.920522i \(0.627764\pi\)
\(332\) −3.20921 −0.176128
\(333\) 0 0
\(334\) 17.3257i 0.948021i
\(335\) 0 0
\(336\) 0 0
\(337\) −23.4286 −1.27624 −0.638118 0.769938i \(-0.720287\pi\)
−0.638118 + 0.769938i \(0.720287\pi\)
\(338\) 10.4252i 0.567058i
\(339\) 0 0
\(340\) 0 0
\(341\) −14.4524 −0.782641
\(342\) 0 0
\(343\) −14.4307 + 11.6084i −0.779185 + 0.626794i
\(344\) 10.1165i 0.545445i
\(345\) 0 0
\(346\) 2.99319i 0.160915i
\(347\) 20.1798i 1.08331i −0.840602 0.541654i \(-0.817798\pi\)
0.840602 0.541654i \(-0.182202\pi\)
\(348\) 0 0
\(349\) 27.0372i 1.44727i −0.690184 0.723634i \(-0.742470\pi\)
0.690184 0.723634i \(-0.257530\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −3.87386 −0.206478
\(353\) 12.3495 0.657298 0.328649 0.944452i \(-0.393407\pi\)
0.328649 + 0.944452i \(0.393407\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −17.8376 −0.945392
\(357\) 0 0
\(358\) −7.38858 −0.390499
\(359\) 25.5757i 1.34983i 0.737894 + 0.674917i \(0.235821\pi\)
−0.737894 + 0.674917i \(0.764179\pi\)
\(360\) 0 0
\(361\) 12.0408 0.633725
\(362\) −11.1097 −0.583912
\(363\) 0 0
\(364\) −3.10801 + 2.89199i −0.162904 + 0.151581i
\(365\) 0 0
\(366\) 0 0
\(367\) 20.8444i 1.08807i −0.839062 0.544035i \(-0.816896\pi\)
0.839062 0.544035i \(-0.183104\pi\)
\(368\) 5.47847i 0.285585i
\(369\) 0 0
\(370\) 0 0
\(371\) −4.39539 + 4.08989i −0.228197 + 0.212336i
\(372\) 0 0
\(373\) −7.48810 −0.387719 −0.193860 0.981029i \(-0.562101\pi\)
−0.193860 + 0.981029i \(0.562101\pi\)
\(374\) −31.4422 −1.62584
\(375\) 0 0
\(376\) 11.1097i 0.572939i
\(377\) 8.79079 0.452749
\(378\) 0 0
\(379\) 32.6651 1.67789 0.838946 0.544215i \(-0.183173\pi\)
0.838946 + 0.544215i \(0.183173\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −9.36197 −0.479000
\(383\) −0.890309 −0.0454927 −0.0227463 0.999741i \(-0.507241\pi\)
−0.0227463 + 0.999741i \(0.507241\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 17.4422i 0.887786i
\(387\) 0 0
\(388\) 8.68768i 0.441050i
\(389\) 13.5317i 0.686084i 0.939320 + 0.343042i \(0.111457\pi\)
−0.939320 + 0.343042i \(0.888543\pi\)
\(390\) 0 0
\(391\) 44.4660i 2.24874i
\(392\) 6.98188 + 0.503406i 0.352638 + 0.0254258i
\(393\) 0 0
\(394\) 5.27607 0.265804
\(395\) 0 0
\(396\) 0 0
\(397\) 25.2991i 1.26973i −0.772625 0.634863i \(-0.781057\pi\)
0.772625 0.634863i \(-0.218943\pi\)
\(398\) −12.2160 −0.612334
\(399\) 0 0
\(400\) 0 0
\(401\) 32.7619i 1.63605i −0.575182 0.818025i \(-0.695069\pi\)
0.575182 0.818025i \(-0.304931\pi\)
\(402\) 0 0
\(403\) 5.98638 0.298203
\(404\) 6.21603 0.309259
\(405\) 0 0
\(406\) −9.87386 10.6114i −0.490032 0.526636i
\(407\) 17.4785i 0.866376i
\(408\) 0 0
\(409\) 12.0000i 0.593362i 0.954977 + 0.296681i \(0.0958798\pi\)
−0.954977 + 0.296681i \(0.904120\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 12.8807i 0.634585i
\(413\) −8.31117 8.93200i −0.408966 0.439515i
\(414\) 0 0
\(415\) 0 0
\(416\) 1.60461 0.0786723
\(417\) 0 0
\(418\) 10.2194i 0.499846i
\(419\) −4.17937 −0.204176 −0.102088 0.994775i \(-0.532552\pi\)
−0.102088 + 0.994775i \(0.532552\pi\)
\(420\) 0 0
\(421\) −32.6514 −1.59133 −0.795667 0.605735i \(-0.792879\pi\)
−0.795667 + 0.605735i \(0.792879\pi\)
\(422\) 4.00000i 0.194717i
\(423\) 0 0
\(424\) 2.26926 0.110205
\(425\) 0 0
\(426\) 0 0
\(427\) 22.9473 21.3523i 1.11050 1.03331i
\(428\) 19.9638i 0.964984i
\(429\) 0 0
\(430\) 0 0
\(431\) 13.3087i 0.641059i −0.947239 0.320530i \(-0.896139\pi\)
0.947239 0.320530i \(-0.103861\pi\)
\(432\) 0 0
\(433\) 6.95358i 0.334168i 0.985943 + 0.167084i \(0.0534351\pi\)
−0.985943 + 0.167084i \(0.946565\pi\)
\(434\) −6.72393 7.22619i −0.322759 0.346868i
\(435\) 0 0
\(436\) 15.0238 0.719509
\(437\) −14.4524 −0.691352
\(438\) 0 0
\(439\) 26.9739i 1.28739i −0.765280 0.643697i \(-0.777400\pi\)
0.765280 0.643697i \(-0.222600\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 13.0238 0.619479
\(443\) 2.80441i 0.133242i −0.997778 0.0666208i \(-0.978778\pi\)
0.997778 0.0666208i \(-0.0212218\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 3.65784 0.173204
\(447\) 0 0
\(448\) −1.80230 1.93693i −0.0851508 0.0915114i
\(449\) 30.1226i 1.42157i −0.703409 0.710786i \(-0.748340\pi\)
0.703409 0.710786i \(-0.251660\pi\)
\(450\) 0 0
\(451\) 6.21603i 0.292701i
\(452\) 10.2330i 0.481320i
\(453\) 0 0
\(454\) 4.23301i 0.198665i
\(455\) 0 0
\(456\) 0 0
\(457\) 22.0340 1.03071 0.515353 0.856978i \(-0.327661\pi\)
0.515353 + 0.856978i \(0.327661\pi\)
\(458\) −7.59497 −0.354890
\(459\) 0 0
\(460\) 0 0
\(461\) −15.4116 −0.717790 −0.358895 0.933378i \(-0.616846\pi\)
−0.358895 + 0.933378i \(0.616846\pi\)
\(462\) 0 0
\(463\) 25.0605 1.16466 0.582329 0.812953i \(-0.302142\pi\)
0.582329 + 0.812953i \(0.302142\pi\)
\(464\) 5.47847i 0.254332i
\(465\) 0 0
\(466\) 9.94677 0.460775
\(467\) 9.62764 0.445514 0.222757 0.974874i \(-0.428494\pi\)
0.222757 + 0.974874i \(0.428494\pi\)
\(468\) 0 0
\(469\) −12.4490 13.3789i −0.574843 0.617782i
\(470\) 0 0
\(471\) 0 0
\(472\) 4.61142i 0.212258i
\(473\) 39.1899i 1.80196i
\(474\) 0 0
\(475\) 0 0
\(476\) −14.6284 15.7211i −0.670492 0.720576i
\(477\) 0 0
\(478\) 9.36197 0.428206
\(479\) 37.0238 1.69166 0.845830 0.533452i \(-0.179106\pi\)
0.845830 + 0.533452i \(0.179106\pi\)
\(480\) 0 0
\(481\) 7.23982i 0.330107i
\(482\) −23.0408 −1.04948
\(483\) 0 0
\(484\) 4.00681 0.182128
\(485\) 0 0
\(486\) 0 0
\(487\) 11.6386 0.527394 0.263697 0.964606i \(-0.415058\pi\)
0.263697 + 0.964606i \(0.415058\pi\)
\(488\) −11.8472 −0.536300
\(489\) 0 0
\(490\) 0 0
\(491\) 32.1069i 1.44896i 0.689294 + 0.724481i \(0.257921\pi\)
−0.689294 + 0.724481i \(0.742079\pi\)
\(492\) 0 0
\(493\) 44.4660i 2.00265i
\(494\) 4.23301i 0.190452i
\(495\) 0 0
\(496\) 3.73074i 0.167515i
\(497\) −5.10969 + 4.75454i −0.229201 + 0.213270i
\(498\) 0 0
\(499\) 13.6122 0.609365 0.304682 0.952454i \(-0.401450\pi\)
0.304682 + 0.952454i \(0.401450\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 23.8376i 1.06392i
\(503\) 3.07573 0.137140 0.0685700 0.997646i \(-0.478156\pi\)
0.0685700 + 0.997646i \(0.478156\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 21.2228i 0.943470i
\(507\) 0 0
\(508\) −2.81382 −0.124843
\(509\) 14.7908 0.655590 0.327795 0.944749i \(-0.393694\pi\)
0.327795 + 0.944749i \(0.393694\pi\)
\(510\) 0 0
\(511\) 26.6284 24.7776i 1.17797 1.09610i
\(512\) 1.00000i 0.0441942i
\(513\) 0 0
\(514\) 29.7441i 1.31196i
\(515\) 0 0
\(516\) 0 0
\(517\) 43.0374i 1.89278i
\(518\) 8.73923 8.13181i 0.383980 0.357291i
\(519\) 0 0
\(520\) 0 0
\(521\) −27.0332 −1.18435 −0.592173 0.805811i \(-0.701730\pi\)
−0.592173 + 0.805811i \(0.701730\pi\)
\(522\) 0 0
\(523\) 3.51472i 0.153688i −0.997043 0.0768440i \(-0.975516\pi\)
0.997043 0.0768440i \(-0.0244843\pi\)
\(524\) −8.62840 −0.376933
\(525\) 0 0
\(526\) −22.7545 −0.992145
\(527\) 30.2806i 1.31904i
\(528\) 0 0
\(529\) −7.01362 −0.304940
\(530\) 0 0
\(531\) 0 0
\(532\) −5.10969 + 4.75454i −0.221533 + 0.206135i
\(533\) 2.57476i 0.111525i
\(534\) 0 0
\(535\) 0 0
\(536\) 6.90729i 0.298350i
\(537\) 0 0
\(538\) 0.202401i 0.00872612i
\(539\) −27.0468 1.95013i −1.16499 0.0839979i
\(540\) 0 0
\(541\) −39.6616 −1.70519 −0.852593 0.522576i \(-0.824971\pi\)
−0.852593 + 0.522576i \(0.824971\pi\)
\(542\) −11.4785 −0.493042
\(543\) 0 0
\(544\) 8.11650i 0.347992i
\(545\) 0 0
\(546\) 0 0
\(547\) 34.3495 1.46868 0.734339 0.678782i \(-0.237492\pi\)
0.734339 + 0.678782i \(0.237492\pi\)
\(548\) 0.723932i 0.0309248i
\(549\) 0 0
\(550\) 0 0
\(551\) 14.4524 0.615692
\(552\) 0 0
\(553\) −14.4490 15.5283i −0.614435 0.660332i
\(554\) 23.7211i 1.00781i
\(555\) 0 0
\(556\) 2.84043i 0.120461i
\(557\) 38.4660i 1.62986i −0.579561 0.814929i \(-0.696776\pi\)
0.579561 0.814929i \(-0.303224\pi\)
\(558\) 0 0
\(559\) 16.2330i 0.686583i
\(560\) 0 0
\(561\) 0 0
\(562\) 5.57194 0.235038
\(563\) 10.2194 0.430696 0.215348 0.976537i \(-0.430911\pi\)
0.215348 + 0.976537i \(0.430911\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 32.1798 1.35262
\(567\) 0 0
\(568\) 2.63803 0.110689
\(569\) 2.48129i 0.104021i −0.998647 0.0520106i \(-0.983437\pi\)
0.998647 0.0520106i \(-0.0165629\pi\)
\(570\) 0 0
\(571\) 20.6345 0.863525 0.431762 0.901987i \(-0.357892\pi\)
0.431762 + 0.901987i \(0.357892\pi\)
\(572\) −6.21603 −0.259905
\(573\) 0 0
\(574\) −3.10801 + 2.89199i −0.129726 + 0.120709i
\(575\) 0 0
\(576\) 0 0
\(577\) 2.50455i 0.104266i 0.998640 + 0.0521329i \(0.0166019\pi\)
−0.998640 + 0.0521329i \(0.983398\pi\)
\(578\) 48.8776i 2.03304i
\(579\) 0 0
\(580\) 0 0
\(581\) 5.78397 + 6.21603i 0.239960 + 0.257884i
\(582\) 0 0
\(583\) −8.79079 −0.364077
\(584\) −13.7477 −0.568885
\(585\) 0 0
\(586\) 15.8146i 0.653294i
\(587\) −31.4422 −1.29776 −0.648880 0.760891i \(-0.724762\pi\)
−0.648880 + 0.760891i \(0.724762\pi\)
\(588\) 0 0
\(589\) 9.84183 0.405526
\(590\) 0 0
\(591\) 0 0
\(592\) −4.51190 −0.185438
\(593\) −27.1267 −1.11396 −0.556979 0.830526i \(-0.688040\pi\)
−0.556979 + 0.830526i \(0.688040\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 3.00681i 0.123164i
\(597\) 0 0
\(598\) 8.79079i 0.359482i
\(599\) 33.7747i 1.38000i −0.723810 0.689999i \(-0.757611\pi\)
0.723810 0.689999i \(-0.242389\pi\)
\(600\) 0 0
\(601\) 24.4886i 0.998912i 0.866339 + 0.499456i \(0.166467\pi\)
−0.866339 + 0.499456i \(0.833533\pi\)
\(602\) −19.5950 + 18.2330i −0.798631 + 0.743122i
\(603\) 0 0
\(604\) 15.2262 0.619545
\(605\) 0 0
\(606\) 0 0
\(607\) 27.3331i 1.10941i −0.832045 0.554707i \(-0.812830\pi\)
0.832045 0.554707i \(-0.187170\pi\)
\(608\) 2.63803 0.106986
\(609\) 0 0
\(610\) 0 0
\(611\) 17.8267i 0.721190i
\(612\) 0 0
\(613\) −11.3163 −0.457061 −0.228531 0.973537i \(-0.573392\pi\)
−0.228531 + 0.973537i \(0.573392\pi\)
\(614\) 6.72393 0.271356
\(615\) 0 0
\(616\) 6.98188 + 7.50341i 0.281308 + 0.302321i
\(617\) 10.6843i 0.430135i −0.976599 0.215067i \(-0.931003\pi\)
0.976599 0.215067i \(-0.0689971\pi\)
\(618\) 0 0
\(619\) 9.56437i 0.384424i −0.981353 0.192212i \(-0.938434\pi\)
0.981353 0.192212i \(-0.0615662\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 1.02379i 0.0410504i
\(623\) 32.1488 + 34.5502i 1.28801 + 1.38423i
\(624\) 0 0
\(625\) 0 0
\(626\) 23.2568 0.929529
\(627\) 0 0
\(628\) 2.64767i 0.105653i
\(629\) −36.6208 −1.46017
\(630\) 0 0
\(631\) 17.1956 0.684546 0.342273 0.939601i \(-0.388803\pi\)
0.342273 + 0.939601i \(0.388803\pi\)
\(632\) 8.01698i 0.318898i
\(633\) 0 0
\(634\) −4.46830 −0.177459
\(635\) 0 0
\(636\) 0 0
\(637\) 11.2032 + 0.807769i 0.443885 + 0.0320050i
\(638\) 21.2228i 0.840220i
\(639\) 0 0
\(640\) 0 0
\(641\) 16.5289i 0.652851i −0.945223 0.326426i \(-0.894156\pi\)
0.945223 0.326426i \(-0.105844\pi\)
\(642\) 0 0
\(643\) 37.4286i 1.47604i 0.674779 + 0.738020i \(0.264239\pi\)
−0.674779 + 0.738020i \(0.735761\pi\)
\(644\) −10.6114 + 9.87386i −0.418148 + 0.389085i
\(645\) 0 0
\(646\) 21.4116 0.842429
\(647\) 5.52812 0.217333 0.108666 0.994078i \(-0.465342\pi\)
0.108666 + 0.994078i \(0.465342\pi\)
\(648\) 0 0
\(649\) 17.8640i 0.701223i
\(650\) 0 0
\(651\) 0 0
\(652\) −9.32572 −0.365223
\(653\) 18.5159i 0.724583i −0.932065 0.362291i \(-0.881995\pi\)
0.932065 0.362291i \(-0.118005\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.60461 0.0626494
\(657\) 0 0
\(658\) −21.5187 + 20.0230i −0.838887 + 0.780579i
\(659\) 43.3693i 1.68943i −0.535217 0.844714i \(-0.679770\pi\)
0.535217 0.844714i \(-0.320230\pi\)
\(660\) 0 0
\(661\) 36.1142i 1.40468i −0.711842 0.702340i \(-0.752139\pi\)
0.711842 0.702340i \(-0.247861\pi\)
\(662\) 14.2160i 0.552522i
\(663\) 0 0
\(664\) 3.20921i 0.124542i
\(665\) 0 0
\(666\) 0 0
\(667\) 30.0136 1.16213
\(668\) −17.3257 −0.670352
\(669\) 0 0
\(670\) 0 0
\(671\) 45.8946 1.77174
\(672\) 0 0
\(673\) 33.8743 1.30576 0.652879 0.757463i \(-0.273561\pi\)
0.652879 + 0.757463i \(0.273561\pi\)
\(674\) 23.4286i 0.902436i
\(675\) 0 0
\(676\) −10.4252 −0.400971
\(677\) 4.38446 0.168509 0.0842543 0.996444i \(-0.473149\pi\)
0.0842543 + 0.996444i \(0.473149\pi\)
\(678\) 0 0
\(679\) −16.8274 + 15.6578i −0.645778 + 0.600893i
\(680\) 0 0
\(681\) 0 0
\(682\) 14.4524i 0.553411i
\(683\) 1.95013i 0.0746195i −0.999304 0.0373098i \(-0.988121\pi\)
0.999304 0.0373098i \(-0.0118788\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −11.6084 14.4307i −0.443211 0.550967i
\(687\) 0 0
\(688\) 10.1165 0.385688
\(689\) 3.64126 0.138721
\(690\) 0 0
\(691\) 12.1471i 0.462098i −0.972942 0.231049i \(-0.925784\pi\)
0.972942 0.231049i \(-0.0742157\pi\)
\(692\) −2.99319 −0.113784
\(693\) 0 0
\(694\) 20.1798 0.766014
\(695\) 0 0
\(696\) 0 0
\(697\) 13.0238 0.493311
\(698\) 27.0372 1.02337
\(699\) 0 0
\(700\) 0 0
\(701\) 11.0600i 0.417732i 0.977944 + 0.208866i \(0.0669773\pi\)
−0.977944 + 0.208866i \(0.933023\pi\)
\(702\) 0 0
\(703\) 11.9025i 0.448913i
\(704\) 3.87386i 0.146002i
\(705\) 0 0
\(706\) 12.3495i 0.464780i
\(707\) −11.2032 12.0400i −0.421338 0.452811i
\(708\) 0 0
\(709\) 14.2330 0.534532 0.267266 0.963623i \(-0.413880\pi\)
0.267266 + 0.963623i \(0.413880\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 17.8376i 0.668493i
\(713\) 20.4388 0.765438
\(714\) 0 0
\(715\) 0 0
\(716\) 7.38858i 0.276124i
\(717\) 0 0
\(718\) −25.5757 −0.954477
\(719\) −20.4660 −0.763254 −0.381627 0.924317i \(-0.624636\pi\)
−0.381627 + 0.924317i \(0.624636\pi\)
\(720\) 0 0
\(721\) −24.9490 + 23.2149i −0.929149 + 0.864567i
\(722\) 12.0408i 0.448111i
\(723\) 0 0
\(724\) 11.1097i 0.412888i
\(725\) 0 0
\(726\) 0 0
\(727\) 25.5717i 0.948402i −0.880417 0.474201i \(-0.842737\pi\)
0.880417 0.474201i \(-0.157263\pi\)
\(728\) −2.89199 3.10801i −0.107184 0.115191i
\(729\) 0 0
\(730\) 0 0
\(731\) 82.1106 3.03697
\(732\) 0 0
\(733\) 22.6624i 0.837053i 0.908204 + 0.418527i \(0.137453\pi\)
−0.908204 + 0.418527i \(0.862547\pi\)
\(734\) 20.8444 0.769382
\(735\) 0 0
\(736\) 5.47847 0.201939
\(737\) 26.7579i 0.985640i
\(738\) 0 0
\(739\) −44.8675 −1.65048 −0.825238 0.564785i \(-0.808959\pi\)
−0.825238 + 0.564785i \(0.808959\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −4.08989 4.39539i −0.150145 0.161360i
\(743\) 5.37536i 0.197203i 0.995127 + 0.0986015i \(0.0314369\pi\)
−0.995127 + 0.0986015i \(0.968563\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 7.48810i 0.274159i
\(747\) 0 0
\(748\) 31.4422i 1.14964i
\(749\) −38.6684 + 35.9807i −1.41291 + 1.31471i
\(750\) 0 0
\(751\) 13.6276 0.497280 0.248640 0.968596i \(-0.420016\pi\)
0.248640 + 0.968596i \(0.420016\pi\)
\(752\) 11.1097 0.405129
\(753\) 0 0
\(754\) 8.79079i 0.320142i
\(755\) 0 0
\(756\) 0 0
\(757\) 4.54586 0.165222 0.0826110 0.996582i \(-0.473674\pi\)
0.0826110 + 0.996582i \(0.473674\pi\)
\(758\) 32.6651i 1.18645i
\(759\) 0 0
\(760\) 0 0
\(761\) −15.6522 −0.567392 −0.283696 0.958914i \(-0.591561\pi\)
−0.283696 + 0.958914i \(0.591561\pi\)
\(762\) 0 0
\(763\) −27.0774 29.1001i −0.980269 1.05349i
\(764\) 9.36197i 0.338704i
\(765\) 0 0
\(766\) 0.890309i 0.0321682i
\(767\) 7.39951i 0.267181i
\(768\) 0 0
\(769\) 17.1922i 0.619968i −0.950742 0.309984i \(-0.899676\pi\)
0.950742 0.309984i \(-0.100324\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 17.4422 0.627759
\(773\) −42.6990 −1.53578 −0.767889 0.640584i \(-0.778693\pi\)
−0.767889 + 0.640584i \(0.778693\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 8.68768 0.311870
\(777\) 0 0
\(778\) −13.5317 −0.485135
\(779\) 4.23301i 0.151663i
\(780\) 0 0
\(781\) −10.2194 −0.365678
\(782\) 44.4660 1.59010
\(783\) 0 0
\(784\) −0.503406 + 6.98188i −0.0179788 + 0.249353i
\(785\) 0 0
\(786\) 0 0
\(787\) 30.3992i 1.08361i −0.840503 0.541806i \(-0.817741\pi\)
0.840503 0.541806i \(-0.182259\pi\)
\(788\) 5.27607i 0.187952i
\(789\) 0 0
\(790\) 0 0
\(791\) −19.8206 + 18.4430i −0.704741 + 0.655757i
\(792\) 0 0
\(793\) −19.0102 −0.675071
\(794\) 25.2991 0.897831
\(795\) 0 0
\(796\) 12.2160i 0.432986i
\(797\) −2.99319 −0.106024 −0.0530121 0.998594i \(-0.516882\pi\)
−0.0530121 + 0.998594i \(0.516882\pi\)
\(798\) 0 0
\(799\) 90.1718 3.19005
\(800\) 0 0
\(801\) 0 0
\(802\) 32.7619 1.15686
\(803\) 53.2568 1.87939