Properties

Label 3150.2.d.f.3149.1
Level 3150
Weight 2
Character 3150.3149
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.d (of order \(2\), degree \(1\), not 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: \( 2^{4}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 630)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3149.1
Root \(-2.73923i\)
Character \(\chi\) = 3150.3149
Dual form 3150.2.d.f.3149.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} +(-1.93693 - 1.80230i) q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} +(-1.93693 - 1.80230i) q^{7} +1.00000 q^{8} +3.87386i q^{11} -1.60461 q^{13} +(-1.93693 - 1.80230i) q^{14} +1.00000 q^{16} +8.11650i q^{17} -2.63803i q^{19} +3.87386i q^{22} +5.47847 q^{23} -1.60461 q^{26} +(-1.93693 - 1.80230i) q^{28} -5.47847i q^{29} -3.73074i q^{31} +1.00000 q^{32} +8.11650i q^{34} +4.51190i q^{37} -2.63803i q^{38} -1.60461 q^{41} +10.1165i q^{43} +3.87386i q^{44} +5.47847 q^{46} +11.1097i q^{47} +(0.503406 + 6.98188i) q^{49} -1.60461 q^{52} -2.26926 q^{53} +(-1.93693 - 1.80230i) q^{56} -5.47847i q^{58} -4.61142 q^{59} +11.8472i q^{61} -3.73074i q^{62} +1.00000 q^{64} +6.90729i q^{67} +8.11650i q^{68} +2.63803i q^{71} -13.7477 q^{73} +4.51190i q^{74} -2.63803i q^{76} +(6.98188 - 7.50341i) q^{77} +8.01698 q^{79} -1.60461 q^{82} -3.20921i q^{83} +10.1165i q^{86} +3.87386i q^{88} +17.8376 q^{89} +(3.10801 + 2.89199i) q^{91} +5.47847 q^{92} +11.1097i q^{94} -8.68768 q^{97} +(0.503406 + 6.98188i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 8q^{2} + 8q^{4} + 8q^{8} + O(q^{10}) \) \( 8q + 8q^{2} + 8q^{4} + 8q^{8} + 8q^{13} + 8q^{16} - 8q^{23} + 8q^{26} + 8q^{32} + 8q^{41} - 8q^{46} - 4q^{49} + 8q^{52} - 8q^{53} + 8q^{64} - 48q^{73} - 4q^{77} - 8q^{79} + 8q^{82} - 8q^{89} - 4q^{91} - 8q^{92} + 24q^{97} - 4q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −1.93693 1.80230i −0.732091 0.681207i
\(8\) 1.00000 0.353553
\(9\) 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
\(13\) −1.60461 −0.445038 −0.222519 0.974928i \(-0.571428\pi\)
−0.222519 + 0.974928i \(0.571428\pi\)
\(14\) −1.93693 1.80230i −0.517667 0.481686i
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 8.11650i 1.96854i 0.176667 + 0.984271i \(0.443468\pi\)
−0.176667 + 0.984271i \(0.556532\pi\)
\(18\) 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
\(22\) 3.87386i 0.825910i
\(23\) 5.47847 1.14234 0.571170 0.820832i \(-0.306490\pi\)
0.571170 + 0.820832i \(0.306490\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −1.60461 −0.314689
\(27\) 0 0
\(28\) −1.93693 1.80230i −0.366046 0.340603i
\(29\) 5.47847i 1.01733i −0.860966 0.508663i \(-0.830140\pi\)
0.860966 0.508663i \(-0.169860\pi\)
\(30\) 0 0
\(31\) 3.73074i 0.670061i −0.942207 0.335031i \(-0.891253\pi\)
0.942207 0.335031i \(-0.108747\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) 8.11650i 1.39197i
\(35\) 0 0
\(36\) 0 0
\(37\) 4.51190i 0.741751i 0.928683 + 0.370876i \(0.120942\pi\)
−0.928683 + 0.370876i \(0.879058\pi\)
\(38\) 2.63803i 0.427946i
\(39\) 0 0
\(40\) 0 0
\(41\) −1.60461 −0.250597 −0.125299 0.992119i \(-0.539989\pi\)
−0.125299 + 0.992119i \(0.539989\pi\)
\(42\) 0 0
\(43\) 10.1165i 1.54275i 0.636379 + 0.771376i \(0.280431\pi\)
−0.636379 + 0.771376i \(0.719569\pi\)
\(44\) 3.87386i 0.584007i
\(45\) 0 0
\(46\) 5.47847 0.807756
\(47\) 11.1097i 1.62052i 0.586074 + 0.810258i \(0.300673\pi\)
−0.586074 + 0.810258i \(0.699327\pi\)
\(48\) 0 0
\(49\) 0.503406 + 6.98188i 0.0719152 + 0.997411i
\(50\) 0 0
\(51\) 0 0
\(52\) −1.60461 −0.222519
\(53\) −2.26926 −0.311706 −0.155853 0.987780i \(-0.549813\pi\)
−0.155853 + 0.987780i \(0.549813\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.93693 1.80230i −0.258833 0.240843i
\(57\) 0 0
\(58\) 5.47847i 0.719358i
\(59\) −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 \(0.274039\pi\)
−0.651740 + 0.758442i \(0.725961\pi\)
\(62\) 3.73074i 0.473805i
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) 6.90729i 0.843860i 0.906628 + 0.421930i \(0.138647\pi\)
−0.906628 + 0.421930i \(0.861353\pi\)
\(68\) 8.11650i 0.984271i
\(69\) 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
\(73\) −13.7477 −1.60905 −0.804525 0.593919i \(-0.797580\pi\)
−0.804525 + 0.593919i \(0.797580\pi\)
\(74\) 4.51190i 0.524497i
\(75\) 0 0
\(76\) 2.63803i 0.302603i
\(77\) 6.98188 7.50341i 0.795659 0.855092i
\(78\) 0 0
\(79\) 8.01698 0.901981 0.450990 0.892529i \(-0.351071\pi\)
0.450990 + 0.892529i \(0.351071\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −1.60461 −0.177199
\(83\) 3.20921i 0.352257i −0.984367 0.176128i \(-0.943643\pi\)
0.984367 0.176128i \(-0.0563574\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 10.1165i 1.09089i
\(87\) 0 0
\(88\) 3.87386i 0.412955i
\(89\) 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
\(92\) 5.47847 0.571170
\(93\) 0 0
\(94\) 11.1097i 1.14588i
\(95\) 0 0
\(96\) 0 0
\(97\) −8.68768 −0.882100 −0.441050 0.897482i \(-0.645394\pi\)
−0.441050 + 0.897482i \(0.645394\pi\)
\(98\) 0.503406 + 6.98188i 0.0508517 + 0.705276i
\(99\) 0 0
\(100\) 0 0
\(101\) 6.21603 0.618518 0.309259 0.950978i \(-0.399919\pi\)
0.309259 + 0.950978i \(0.399919\pi\)
\(102\) 0 0
\(103\) 12.8807 1.26917 0.634585 0.772853i \(-0.281171\pi\)
0.634585 + 0.772853i \(0.281171\pi\)
\(104\) −1.60461 −0.157345
\(105\) 0 0
\(106\) −2.26926 −0.220410
\(107\) 19.9638 1.92997 0.964984 0.262308i \(-0.0844835\pi\)
0.964984 + 0.262308i \(0.0844835\pi\)
\(108\) 0 0
\(109\) 15.0238 1.43902 0.719509 0.694483i \(-0.244367\pi\)
0.719509 + 0.694483i \(0.244367\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.93693 1.80230i −0.183023 0.170302i
\(113\) −10.2330 −0.962640 −0.481320 0.876545i \(-0.659843\pi\)
−0.481320 + 0.876545i \(0.659843\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 5.47847i 0.508663i
\(117\) 0 0
\(118\) −4.61142 −0.424515
\(119\) 14.6284 15.7211i 1.34098 1.44115i
\(120\) 0 0
\(121\) −4.00681 −0.364256
\(122\) 11.8472i 1.07260i
\(123\) 0 0
\(124\) 3.73074i 0.335031i
\(125\) 0 0
\(126\) 0 0
\(127\) 2.81382i 0.249686i −0.992177 0.124843i \(-0.960157\pi\)
0.992177 0.124843i \(-0.0398427\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −8.62840 −0.753867 −0.376933 0.926240i \(-0.623021\pi\)
−0.376933 + 0.926240i \(0.623021\pi\)
\(132\) 0 0
\(133\) −4.75454 + 5.10969i −0.412271 + 0.443066i
\(134\) 6.90729i 0.596699i
\(135\) 0 0
\(136\) 8.11650i 0.695984i
\(137\) 0.723932 0.0618496 0.0309248 0.999522i \(-0.490155\pi\)
0.0309248 + 0.999522i \(0.490155\pi\)
\(138\) 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
\(142\) 2.63803i 0.221379i
\(143\) 6.21603i 0.519810i
\(144\) 0 0
\(145\) 0 0
\(146\) −13.7477 −1.13777
\(147\) 0 0
\(148\) 4.51190i 0.370876i
\(149\) 3.00681i 0.246328i −0.992386 0.123164i \(-0.960696\pi\)
0.992386 0.123164i \(-0.0393041\pi\)
\(150\) 0 0
\(151\) −15.2262 −1.23909 −0.619545 0.784961i \(-0.712683\pi\)
−0.619545 + 0.784961i \(0.712683\pi\)
\(152\) 2.63803i 0.213973i
\(153\) 0 0
\(154\) 6.98188 7.50341i 0.562616 0.604642i
\(155\) 0 0
\(156\) 0 0
\(157\) 2.64767 0.211307 0.105653 0.994403i \(-0.466307\pi\)
0.105653 + 0.994403i \(0.466307\pi\)
\(158\) 8.01698 0.637797
\(159\) 0 0
\(160\) 0 0
\(161\) −10.6114 9.87386i −0.836297 0.778169i
\(162\) 0 0
\(163\) 9.32572i 0.730446i 0.930920 + 0.365223i \(0.119007\pi\)
−0.930920 + 0.365223i \(0.880993\pi\)
\(164\) −1.60461 −0.125299
\(165\) 0 0
\(166\) 3.20921i 0.249083i
\(167\) 17.3257i 1.34070i 0.742043 + 0.670352i \(0.233857\pi\)
−0.742043 + 0.670352i \(0.766143\pi\)
\(168\) 0 0
\(169\) −10.4252 −0.801941
\(170\) 0 0
\(171\) 0 0
\(172\) 10.1165i 0.771376i
\(173\) 2.99319i 0.227568i −0.993506 0.113784i \(-0.963703\pi\)
0.993506 0.113784i \(-0.0362972\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.87386i 0.292003i
\(177\) 0 0
\(178\) 17.8376 1.33699
\(179\) 7.38858i 0.552248i −0.961122 0.276124i \(-0.910950\pi\)
0.961122 0.276124i \(-0.0890501\pi\)
\(180\) 0 0
\(181\) 11.1097i 0.825777i −0.910782 0.412888i \(-0.864520\pi\)
0.910782 0.412888i \(-0.135480\pi\)
\(182\) 3.10801 + 2.89199i 0.230381 + 0.214368i
\(183\) 0 0
\(184\) 5.47847 0.403878
\(185\) 0 0
\(186\) 0 0
\(187\) −31.4422 −2.29928
\(188\) 11.1097i 0.810258i
\(189\) 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
\(193\) 17.4422i 1.25552i −0.778408 0.627759i \(-0.783972\pi\)
0.778408 0.627759i \(-0.216028\pi\)
\(194\) −8.68768 −0.623739
\(195\) 0 0
\(196\) 0.503406 + 6.98188i 0.0359576 + 0.498705i
\(197\) −5.27607 −0.375904 −0.187952 0.982178i \(-0.560185\pi\)
−0.187952 + 0.982178i \(0.560185\pi\)
\(198\) 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
\(202\) 6.21603 0.437358
\(203\) −9.87386 + 10.6114i −0.693009 + 0.744776i
\(204\) 0 0
\(205\) 0 0
\(206\) 12.8807 0.897439
\(207\) 0 0
\(208\) −1.60461 −0.111259
\(209\) 10.2194 0.706889
\(210\) 0 0
\(211\) 4.00000 0.275371 0.137686 0.990476i \(-0.456034\pi\)
0.137686 + 0.990476i \(0.456034\pi\)
\(212\) −2.26926 −0.155853
\(213\) 0 0
\(214\) 19.9638 1.36469
\(215\) 0 0
\(216\) 0 0
\(217\) −6.72393 + 7.22619i −0.456450 + 0.490546i
\(218\) 15.0238 1.01754
\(219\) 0 0
\(220\) 0 0
\(221\) 13.0238i 0.876075i
\(222\) 0 0
\(223\) −3.65784 −0.244947 −0.122473 0.992472i \(-0.539083\pi\)
−0.122473 + 0.992472i \(0.539083\pi\)
\(224\) −1.93693 1.80230i −0.129417 0.120421i
\(225\) 0 0
\(226\) −10.2330 −0.680689
\(227\) 4.23301i 0.280955i 0.990084 + 0.140477i \(0.0448637\pi\)
−0.990084 + 0.140477i \(0.955136\pi\)
\(228\) 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
\(232\) 5.47847i 0.359679i
\(233\) 9.94677 0.651634 0.325817 0.945433i \(-0.394361\pi\)
0.325817 + 0.945433i \(0.394361\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −4.61142 −0.300178
\(237\) 0 0
\(238\) 14.6284 15.7211i 0.948218 1.01905i
\(239\) 9.36197i 0.605575i 0.953058 + 0.302788i \(0.0979173\pi\)
−0.953058 + 0.302788i \(0.902083\pi\)
\(240\) 0 0
\(241\) 23.0408i 1.48419i −0.670296 0.742093i \(-0.733833\pi\)
0.670296 0.742093i \(-0.266167\pi\)
\(242\) −4.00681 −0.257568
\(243\) 0 0
\(244\) 11.8472i 0.758442i
\(245\) 0 0
\(246\) 0 0
\(247\) 4.23301i 0.269340i
\(248\) 3.73074i 0.236902i
\(249\) 0 0
\(250\) 0 0
\(251\) 23.8376 1.50462 0.752308 0.658811i \(-0.228940\pi\)
0.752308 + 0.658811i \(0.228940\pi\)
\(252\) 0 0
\(253\) 21.2228i 1.33427i
\(254\) 2.81382i 0.176555i
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 29.7441i 1.85539i −0.373341 0.927694i \(-0.621788\pi\)
0.373341 0.927694i \(-0.378212\pi\)
\(258\) 0 0
\(259\) 8.13181 8.73923i 0.505286 0.543030i
\(260\) 0 0
\(261\) 0 0
\(262\) −8.62840 −0.533064
\(263\) −22.7545 −1.40310 −0.701552 0.712618i \(-0.747509\pi\)
−0.701552 + 0.712618i \(0.747509\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.75454 + 5.10969i −0.291519 + 0.313295i
\(267\) 0 0
\(268\) 6.90729i 0.421930i
\(269\) 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 \(-0.886646\pi\)
0.937259 0.348634i \(-0.113354\pi\)
\(272\) 8.11650i 0.492135i
\(273\) 0 0
\(274\) 0.723932 0.0437343
\(275\) 0 0
\(276\) 0 0
\(277\) 23.7211i 1.42526i 0.701538 + 0.712632i \(0.252497\pi\)
−0.701538 + 0.712632i \(0.747503\pi\)
\(278\) 2.84043i 0.170358i
\(279\) 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
\(283\) −32.1798 −1.91289 −0.956445 0.291914i \(-0.905708\pi\)
−0.956445 + 0.291914i \(0.905708\pi\)
\(284\) 2.63803i 0.156539i
\(285\) 0 0
\(286\) 6.21603i 0.367561i
\(287\) 3.10801 + 2.89199i 0.183460 + 0.170709i
\(288\) 0 0
\(289\) −48.8776 −2.87515
\(290\) 0 0
\(291\) 0 0
\(292\) −13.7477 −0.804525
\(293\) 15.8146i 0.923898i −0.886907 0.461949i \(-0.847150\pi\)
0.886907 0.461949i \(-0.152850\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 4.51190i 0.262249i
\(297\) 0 0
\(298\) 3.00681i 0.174180i
\(299\) −8.79079 −0.508384
\(300\) 0 0
\(301\) 18.2330 19.5950i 1.05093 1.12944i
\(302\) −15.2262 −0.876169
\(303\) 0 0
\(304\) 2.63803i 0.151302i
\(305\) 0 0
\(306\) 0 0
\(307\) 6.72393 0.383755 0.191878 0.981419i \(-0.438542\pi\)
0.191878 + 0.981419i \(0.438542\pi\)
\(308\) 6.98188 7.50341i 0.397829 0.427546i
\(309\) 0 0
\(310\) 0 0
\(311\) 1.02379 0.0580540 0.0290270 0.999579i \(-0.490759\pi\)
0.0290270 + 0.999579i \(0.490759\pi\)
\(312\) 0 0
\(313\) −23.2568 −1.31455 −0.657276 0.753650i \(-0.728291\pi\)
−0.657276 + 0.753650i \(0.728291\pi\)
\(314\) 2.64767 0.149417
\(315\) 0 0
\(316\) 8.01698 0.450990
\(317\) 4.46830 0.250965 0.125482 0.992096i \(-0.459952\pi\)
0.125482 + 0.992096i \(0.459952\pi\)
\(318\) 0 0
\(319\) 21.2228 1.18825
\(320\) 0 0
\(321\) 0 0
\(322\) −10.6114 9.87386i −0.591351 0.550249i
\(323\) 21.4116 1.19137
\(324\) 0 0
\(325\) 0 0
\(326\) 9.32572i 0.516504i
\(327\) 0 0
\(328\) −1.60461 −0.0885996
\(329\) 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 \(0.627764\pi\)
\(332\) 3.20921i 0.176128i
\(333\) 0 0
\(334\) 17.3257i 0.948021i
\(335\) 0 0
\(336\) 0 0
\(337\) 23.4286i 1.27624i 0.769938 + 0.638118i \(0.220287\pi\)
−0.769938 + 0.638118i \(0.779713\pi\)
\(338\) −10.4252 −0.567058
\(339\) 0 0
\(340\) 0 0
\(341\) 14.4524 0.782641
\(342\) 0 0
\(343\) 11.6084 14.4307i 0.626794 0.779185i
\(344\) 10.1165i 0.545445i
\(345\) 0 0
\(346\) 2.99319i 0.160915i
\(347\) −20.1798 −1.08331 −0.541654 0.840602i \(-0.682202\pi\)
−0.541654 + 0.840602i \(0.682202\pi\)
\(348\) 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
\(352\) 3.87386i 0.206478i
\(353\) 12.3495i 0.657298i −0.944452 0.328649i \(-0.893407\pi\)
0.944452 0.328649i \(-0.106593\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 17.8376 0.945392
\(357\) 0 0
\(358\) 7.38858i 0.390499i
\(359\) 25.5757i 1.34983i −0.737894 0.674917i \(-0.764179\pi\)
0.737894 0.674917i \(-0.235821\pi\)
\(360\) 0 0
\(361\) 12.0408 0.633725
\(362\) 11.1097i 0.583912i
\(363\) 0 0
\(364\) 3.10801 + 2.89199i 0.162904 + 0.151581i
\(365\) 0 0
\(366\) 0 0
\(367\) 20.8444 1.08807 0.544035 0.839062i \(-0.316896\pi\)
0.544035 + 0.839062i \(0.316896\pi\)
\(368\) 5.47847 0.285585
\(369\) 0 0
\(370\) 0 0
\(371\) 4.39539 + 4.08989i 0.228197 + 0.212336i
\(372\) 0 0
\(373\) 7.48810i 0.387719i −0.981029 0.193860i \(-0.937899\pi\)
0.981029 0.193860i \(-0.0621006\pi\)
\(374\) −31.4422 −1.62584
\(375\) 0 0
\(376\) 11.1097i 0.572939i
\(377\) 8.79079i 0.452749i
\(378\) 0 0
\(379\) −32.6651 −1.67789 −0.838946 0.544215i \(-0.816827\pi\)
−0.838946 + 0.544215i \(0.816827\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 9.36197i 0.479000i
\(383\) 0.890309i 0.0454927i 0.999741 + 0.0227463i \(0.00724101\pi\)
−0.999741 + 0.0227463i \(0.992759\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 17.4422i 0.887786i
\(387\) 0 0
\(388\) −8.68768 −0.441050
\(389\) 13.5317i 0.686084i −0.939320 0.343042i \(-0.888543\pi\)
0.939320 0.343042i \(-0.111457\pi\)
\(390\) 0 0
\(391\) 44.4660i 2.24874i
\(392\) 0.503406 + 6.98188i 0.0254258 + 0.352638i
\(393\) 0 0
\(394\) −5.27607 −0.265804
\(395\) 0 0
\(396\) 0 0
\(397\) 25.2991 1.26973 0.634863 0.772625i \(-0.281057\pi\)
0.634863 + 0.772625i \(0.281057\pi\)
\(398\) 12.2160i 0.612334i
\(399\) 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
\(403\) 5.98638i 0.298203i
\(404\) 6.21603 0.309259
\(405\) 0 0
\(406\) −9.87386 + 10.6114i −0.490032 + 0.526636i
\(407\) −17.4785 −0.866376
\(408\) 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
\(412\) 12.8807 0.634585
\(413\) 8.93200 + 8.31117i 0.439515 + 0.408966i
\(414\) 0 0
\(415\) 0 0
\(416\) −1.60461 −0.0786723
\(417\) 0 0
\(418\) 10.2194 0.499846
\(419\) −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 \(0.792879\pi\)
\(422\) 4.00000 0.194717
\(423\) 0 0
\(424\) −2.26926 −0.110205
\(425\) 0 0
\(426\) 0 0
\(427\) 21.3523 22.9473i 1.03331 1.11050i
\(428\) 19.9638 0.964984
\(429\) 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
\(433\) 6.95358 0.334168 0.167084 0.985943i \(-0.446565\pi\)
0.167084 + 0.985943i \(0.446565\pi\)
\(434\) −6.72393 + 7.22619i −0.322759 + 0.346868i
\(435\) 0 0
\(436\) 15.0238 0.719509
\(437\) 14.4524i 0.691352i
\(438\) 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
\(442\) 13.0238i 0.619479i
\(443\) 2.80441 0.133242 0.0666208 0.997778i \(-0.478778\pi\)
0.0666208 + 0.997778i \(0.478778\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −3.65784 −0.173204
\(447\) 0 0
\(448\) −1.93693 1.80230i −0.0915114 0.0851508i
\(449\) 30.1226i 1.42157i 0.703409 + 0.710786i \(0.251660\pi\)
−0.703409 + 0.710786i \(0.748340\pi\)
\(450\) 0 0
\(451\) 6.21603i 0.292701i
\(452\) −10.2330 −0.481320
\(453\) 0 0
\(454\) 4.23301i 0.198665i
\(455\) 0 0
\(456\) 0 0
\(457\) 22.0340i 1.03071i −0.856978 0.515353i \(-0.827661\pi\)
0.856978 0.515353i \(-0.172339\pi\)
\(458\) 7.59497i 0.354890i
\(459\) 0 0
\(460\) 0 0
\(461\) 15.4116 0.717790 0.358895 0.933378i \(-0.383154\pi\)
0.358895 + 0.933378i \(0.383154\pi\)
\(462\) 0 0
\(463\) 25.0605i 1.16466i 0.812953 + 0.582329i \(0.197858\pi\)
−0.812953 + 0.582329i \(0.802142\pi\)
\(464\) 5.47847i 0.254332i
\(465\) 0 0
\(466\) 9.94677 0.460775
\(467\) 9.62764i 0.445514i 0.974874 + 0.222757i \(0.0715056\pi\)
−0.974874 + 0.222757i \(0.928494\pi\)
\(468\) 0 0
\(469\) 12.4490 13.3789i 0.574843 0.617782i
\(470\) 0 0
\(471\) 0 0
\(472\) −4.61142 −0.212258
\(473\) −39.1899 −1.80196
\(474\) 0 0
\(475\) 0 0
\(476\) 14.6284 15.7211i 0.670492 0.720576i
\(477\) 0 0
\(478\) 9.36197i 0.428206i
\(479\) 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
\(482\) 23.0408i 1.04948i
\(483\) 0 0
\(484\) −4.00681 −0.182128
\(485\) 0 0
\(486\) 0 0
\(487\) 11.6386i 0.527394i −0.964606 0.263697i \(-0.915058\pi\)
0.964606 0.263697i \(-0.0849419\pi\)
\(488\) 11.8472i 0.536300i
\(489\) 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
\(493\) 44.4660 2.00265
\(494\) 4.23301i 0.190452i
\(495\) 0 0
\(496\) 3.73074i 0.167515i
\(497\) 4.75454 5.10969i 0.213270 0.229201i
\(498\) 0 0
\(499\) −13.6122 −0.609365 −0.304682 0.952454i \(-0.598550\pi\)
−0.304682 + 0.952454i \(0.598550\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 23.8376 1.06392
\(503\) 3.07573i 0.137140i −0.997646 0.0685700i \(-0.978156\pi\)
0.997646 0.0685700i \(-0.0218436\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 21.2228i 0.943470i
\(507\) 0 0
\(508\) 2.81382i 0.124843i
\(509\) 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
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) 29.7441i 1.31196i
\(515\) 0 0
\(516\) 0 0
\(517\) −43.0374 −1.89278
\(518\) 8.13181 8.73923i 0.357291 0.383980i
\(519\) 0 0
\(520\) 0 0
\(521\) 27.0332 1.18435 0.592173 0.805811i \(-0.298270\pi\)
0.592173 + 0.805811i \(0.298270\pi\)
\(522\) 0 0
\(523\) −3.51472 −0.153688 −0.0768440 0.997043i \(-0.524484\pi\)
−0.0768440 + 0.997043i \(0.524484\pi\)
\(524\) −8.62840 −0.376933
\(525\) 0 0
\(526\) −22.7545 −0.992145
\(527\) 30.2806 1.31904
\(528\) 0 0
\(529\) 7.01362 0.304940
\(530\) 0 0
\(531\) 0 0
\(532\) −4.75454 + 5.10969i −0.206135 + 0.221533i
\(533\) 2.57476 0.111525
\(534\) 0 0
\(535\) 0 0
\(536\) 6.90729i 0.298350i
\(537\) 0 0
\(538\) 0.202401 0.00872612
\(539\) −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 \(0.824971\pi\)
\(542\) 11.4785i 0.493042i
\(543\) 0 0
\(544\) 8.11650i 0.347992i
\(545\) 0 0
\(546\) 0 0
\(547\) 34.3495i 1.46868i −0.678782 0.734339i \(-0.737492\pi\)
0.678782 0.734339i \(-0.262508\pi\)
\(548\) 0.723932 0.0309248
\(549\) 0 0
\(550\) 0 0
\(551\) −14.4524 −0.615692
\(552\) 0 0
\(553\) −15.5283 14.4490i −0.660332 0.614435i
\(554\) 23.7211i 1.00781i
\(555\) 0 0
\(556\) 2.84043i 0.120461i
\(557\) −38.4660 −1.62986 −0.814929 0.579561i \(-0.803224\pi\)
−0.814929 + 0.579561i \(0.803224\pi\)
\(558\) 0 0
\(559\) 16.2330i 0.686583i
\(560\) 0 0
\(561\) 0 0
\(562\) 5.57194i 0.235038i
\(563\) 10.2194i 0.430696i −0.976537 0.215348i \(-0.930911\pi\)
0.976537 0.215348i \(-0.0690885\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −32.1798 −1.35262
\(567\) 0 0
\(568\) 2.63803i 0.110689i
\(569\) 2.48129i 0.104021i 0.998647 + 0.0520106i \(0.0165629\pi\)
−0.998647 + 0.0520106i \(0.983437\pi\)
\(570\) 0 0
\(571\) 20.6345 0.863525 0.431762 0.901987i \(-0.357892\pi\)
0.431762 + 0.901987i \(0.357892\pi\)
\(572\) 6.21603i 0.259905i
\(573\) 0 0
\(574\) 3.10801 + 2.89199i 0.129726 + 0.120709i
\(575\) 0 0
\(576\) 0 0
\(577\) −2.50455 −0.104266 −0.0521329 0.998640i \(-0.516602\pi\)
−0.0521329 + 0.998640i \(0.516602\pi\)
\(578\) −48.8776 −2.03304
\(579\) 0 0
\(580\) 0 0
\(581\) −5.78397 + 6.21603i −0.239960 + 0.257884i
\(582\) 0 0
\(583\) 8.79079i 0.364077i
\(584\) −13.7477 −0.568885
\(585\) 0 0
\(586\) 15.8146i 0.653294i
\(587\) 31.4422i 1.29776i −0.760891 0.648880i \(-0.775238\pi\)
0.760891 0.648880i \(-0.224762\pi\)
\(588\) 0 0
\(589\) −9.84183 −0.405526
\(590\) 0 0
\(591\) 0 0
\(592\) 4.51190i 0.185438i
\(593\) 27.1267i 1.11396i 0.830526 + 0.556979i \(0.188040\pi\)
−0.830526 + 0.556979i \(0.811960\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 3.00681i 0.123164i
\(597\) 0 0
\(598\) −8.79079 −0.359482
\(599\) 33.7747i 1.38000i 0.723810 + 0.689999i \(0.242389\pi\)
−0.723810 + 0.689999i \(0.757611\pi\)
\(600\) 0 0
\(601\) 24.4886i 0.998912i −0.866339 0.499456i \(-0.833533\pi\)
0.866339 0.499456i \(-0.166467\pi\)
\(602\) 18.2330 19.5950i 0.743122 0.798631i
\(603\) 0 0
\(604\) −15.2262 −0.619545
\(605\) 0 0
\(606\) 0 0
\(607\) 27.3331 1.10941 0.554707 0.832045i \(-0.312830\pi\)
0.554707 + 0.832045i \(0.312830\pi\)
\(608\) 2.63803i 0.106986i
\(609\) 0 0
\(610\) 0 0
\(611\) 17.8267i 0.721190i
\(612\) 0 0
\(613\) 11.3163i 0.457061i −0.973537 0.228531i \(-0.926608\pi\)
0.973537 0.228531i \(-0.0733921\pi\)
\(614\) 6.72393 0.271356
\(615\) 0 0
\(616\) 6.98188 7.50341i 0.281308 0.302321i
\(617\) −10.6843 −0.430135 −0.215067 0.976599i \(-0.568997\pi\)
−0.215067 + 0.976599i \(0.568997\pi\)
\(618\) 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
\(622\) 1.02379 0.0410504
\(623\) −34.5502 32.1488i −1.38423 1.28801i
\(624\) 0 0
\(625\) 0 0
\(626\) −23.2568 −0.929529
\(627\) 0 0
\(628\) 2.64767 0.105653
\(629\) −36.6208 −1.46017
\(630\) 0 0
\(631\) 17.1956 0.684546 0.342273 0.939601i \(-0.388803\pi\)
0.342273 + 0.939601i \(0.388803\pi\)
\(632\) 8.01698 0.318898
\(633\) 0 0
\(634\) 4.46830 0.177459
\(635\) 0 0
\(636\) 0 0
\(637\) −0.807769 11.2032i −0.0320050 0.443885i
\(638\) 21.2228 0.840220
\(639\) 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
\(643\) 37.4286 1.47604 0.738020 0.674779i \(-0.235761\pi\)
0.738020 + 0.674779i \(0.235761\pi\)
\(644\) −10.6114 9.87386i −0.418148 0.389085i
\(645\) 0 0
\(646\) 21.4116 0.842429
\(647\) 5.52812i 0.217333i 0.994078 + 0.108666i \(0.0346580\pi\)
−0.994078 + 0.108666i \(0.965342\pi\)
\(648\) 0 0
\(649\) 17.8640i 0.701223i
\(650\) 0 0
\(651\) 0 0
\(652\) 9.32572i 0.365223i
\(653\) 18.5159 0.724583 0.362291 0.932065i \(-0.381995\pi\)
0.362291 + 0.932065i \(0.381995\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −1.60461 −0.0626494
\(657\) 0 0
\(658\) 20.0230 21.5187i 0.780579 0.838887i
\(659\) 43.3693i 1.68943i 0.535217 + 0.844714i \(0.320230\pi\)
−0.535217 + 0.844714i \(0.679770\pi\)
\(660\) 0 0
\(661\) 36.1142i 1.40468i 0.711842 + 0.702340i \(0.247861\pi\)
−0.711842 + 0.702340i \(0.752139\pi\)
\(662\) −14.2160 −0.552522
\(663\) 0 0
\(664\) 3.20921i 0.124542i
\(665\) 0 0
\(666\) 0 0
\(667\) 30.0136i 1.16213i
\(668\) 17.3257i 0.670352i
\(669\) 0 0
\(670\) 0 0
\(671\) −45.8946 −1.77174
\(672\) 0 0
\(673\) 33.8743i 1.30576i 0.757463 + 0.652879i \(0.226439\pi\)
−0.757463 + 0.652879i \(0.773561\pi\)
\(674\) 23.4286i 0.902436i
\(675\) 0 0
\(676\) −10.4252 −0.400971
\(677\) 4.38446i 0.168509i 0.996444 + 0.0842543i \(0.0268508\pi\)
−0.996444 + 0.0842543i \(0.973149\pi\)
\(678\) 0 0
\(679\) 16.8274 + 15.6578i 0.645778 + 0.600893i
\(680\) 0 0
\(681\) 0 0
\(682\) 14.4524 0.553411
\(683\) 1.95013 0.0746195 0.0373098 0.999304i \(-0.488121\pi\)
0.0373098 + 0.999304i \(0.488121\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 11.6084 14.4307i 0.443211 0.550967i
\(687\) 0 0
\(688\) 10.1165i 0.385688i
\(689\) 3.64126 0.138721
\(690\) 0 0
\(691\) 12.1471i 0.462098i 0.972942 + 0.231049i \(0.0742157\pi\)
−0.972942 + 0.231049i \(0.925784\pi\)
\(692\) 2.99319i 0.113784i
\(693\) 0 0
\(694\) −20.1798 −0.766014
\(695\) 0 0
\(696\) 0 0
\(697\) 13.0238i 0.493311i
\(698\) 27.0372i 1.02337i
\(699\) 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
\(703\) 11.9025 0.448913
\(704\) 3.87386i 0.146002i
\(705\) 0 0
\(706\) 12.3495i 0.464780i
\(707\) −12.0400 11.2032i −0.452811 0.421338i
\(708\) 0 0
\(709\) −14.2330 −0.534532 −0.267266 0.963623i \(-0.586120\pi\)
−0.267266 + 0.963623i \(0.586120\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 17.8376 0.668493
\(713\) 20.4388i 0.765438i
\(714\) 0 0
\(715\) 0 0
\(716\) 7.38858i 0.276124i
\(717\) 0 0
\(718\) 25.5757i 0.954477i
\(719\) −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
\(722\) 12.0408 0.448111
\(723\) 0 0
\(724\) 11.1097i 0.412888i
\(725\) 0 0
\(726\) 0 0
\(727\) 25.5717 0.948402 0.474201 0.880417i \(-0.342737\pi\)
0.474201 + 0.880417i \(0.342737\pi\)
\(728\) 3.10801 + 2.89199i 0.115191 + 0.107184i
\(729\) 0 0
\(730\) 0 0
\(731\) −82.1106 −3.03697
\(732\) 0 0
\(733\) 22.6624 0.837053 0.418527 0.908204i \(-0.362547\pi\)
0.418527 + 0.908204i \(0.362547\pi\)
\(734\) 20.8444 0.769382
\(735\) 0 0
\(736\) 5.47847 0.201939
\(737\) −26.7579 −0.985640
\(738\) 0 0
\(739\) 44.8675 1.65048 0.825238 0.564785i \(-0.191041\pi\)
0.825238 + 0.564785i \(0.191041\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 4.39539 + 4.08989i 0.161360 + 0.150145i
\(743\) −5.37536 −0.197203 −0.0986015 0.995127i \(-0.531437\pi\)
−0.0986015 + 0.995127i \(0.531437\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 7.48810i 0.274159i
\(747\) 0 0
\(748\) −31.4422 −1.14964
\(749\) −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 \(0.420016\pi\)
\(752\) 11.1097i 0.405129i
\(753\) 0 0
\(754\) 8.79079i 0.320142i
\(755\) 0 0
\(756\) 0 0
\(757\) 4.54586i 0.165222i −0.996582 0.0826110i \(-0.973674\pi\)
0.996582 0.0826110i \(-0.0263259\pi\)
\(758\) −32.6651 −1.18645
\(759\) 0 0
\(760\) 0 0
\(761\) 15.6522 0.567392 0.283696 0.958914i \(-0.408439\pi\)
0.283696 + 0.958914i \(0.408439\pi\)
\(762\) 0 0
\(763\) −29.1001 27.0774i −1.05349 0.980269i
\(764\) 9.36197i 0.338704i
\(765\) 0 0
\(766\) 0.890309i 0.0321682i
\(767\) 7.39951 0.267181
\(768\) 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
\(772\) 17.4422i 0.627759i
\(773\) 42.6990i 1.53578i 0.640584 + 0.767889i \(0.278693\pi\)
−0.640584 + 0.767889i \(0.721307\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −8.68768 −0.311870
\(777\) 0 0
\(778\) 13.5317i 0.485135i
\(779\) 4.23301i 0.151663i
\(780\) 0 0
\(781\) −10.2194 −0.365678
\(782\) 44.4660i 1.59010i
\(783\) 0 0
\(784\) 0.503406 + 6.98188i 0.0179788 + 0.249353i
\(785\) 0 0
\(786\) 0 0
\(787\) 30.3992 1.08361 0.541806 0.840503i \(-0.317741\pi\)
0.541806 + 0.840503i \(0.317741\pi\)
\(788\) −5.27607 −0.187952
\(789\) 0 0
\(790\) 0 0
\(791\) 19.8206 + 18.4430i 0.704741 + 0.655757i
\(792\) 0 0
\(793\) 19.0102i 0.675071i
\(794\) 25.2991 0.897831
\(795\) 0 0
\(796\) 12.2160i 0.432986i
\(797\) 2.99319i 0.106024i −0.998594 0.0530121i \(-0.983118\pi\)
0.998594 0.0530121i \(-0.0168822\pi\)
\(798\) 0 0
\(799\) −90.1718 −3.19005
\(800\) 0 0
\(801\) 0 0
\(802\) 32.7619i 1.15686i
\(803\) 53.2568i 1.87939i
\(804\) 0 0
\(805\)