Properties

Label 4012.2.b.b.237.41
Level 4012
Weight 2
Character 4012.237
Analytic conductor 32.036
Analytic rank 0
Dimension 46
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4012.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.0359812909\)
Analytic rank: \(0\)
Dimension: \(46\)
Coefficient ring index: multiple of None
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 237.41
Character \(\chi\) = 4012.237
Dual form 4012.2.b.b.237.6

$q$-expansion

\(f(q)\) \(=\) \(q+2.74502i q^{3} -1.20235i q^{5} +3.93843i q^{7} -4.53516 q^{9} +O(q^{10})\) \(q+2.74502i q^{3} -1.20235i q^{5} +3.93843i q^{7} -4.53516 q^{9} +1.10522i q^{11} -0.0851767 q^{13} +3.30049 q^{15} +(3.92673 + 1.25729i) q^{17} +7.38549 q^{19} -10.8111 q^{21} -5.67905i q^{23} +3.55435 q^{25} -4.21405i q^{27} +5.59463i q^{29} +5.38593i q^{31} -3.03385 q^{33} +4.73538 q^{35} +7.96819i q^{37} -0.233812i q^{39} +7.91312i q^{41} +4.18238 q^{43} +5.45286i q^{45} +4.24249 q^{47} -8.51122 q^{49} +(-3.45129 + 10.7790i) q^{51} +3.82436 q^{53} +1.32886 q^{55} +20.2733i q^{57} +1.00000 q^{59} +2.00065i q^{61} -17.8614i q^{63} +0.102412i q^{65} -8.91072 q^{67} +15.5891 q^{69} -8.59825i q^{71} -7.38541i q^{73} +9.75677i q^{75} -4.35283 q^{77} -13.5159i q^{79} -2.03780 q^{81} -14.8577 q^{83} +(1.51170 - 4.72132i) q^{85} -15.3574 q^{87} -10.9415 q^{89} -0.335462i q^{91} -14.7845 q^{93} -8.87996i q^{95} +15.6089i q^{97} -5.01235i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46q - 54q^{9} + O(q^{10}) \) \( 46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4012\mathbb{Z}\right)^\times\).

\(n\) \(2007\) \(3129\) \(3777\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.74502i 1.58484i 0.609975 + 0.792420i \(0.291179\pi\)
−0.609975 + 0.792420i \(0.708821\pi\)
\(4\) 0 0
\(5\) 1.20235i 0.537708i −0.963181 0.268854i \(-0.913355\pi\)
0.963181 0.268854i \(-0.0866450\pi\)
\(6\) 0 0
\(7\) 3.93843i 1.48859i 0.667853 + 0.744293i \(0.267213\pi\)
−0.667853 + 0.744293i \(0.732787\pi\)
\(8\) 0 0
\(9\) −4.53516 −1.51172
\(10\) 0 0
\(11\) 1.10522i 0.333236i 0.986022 + 0.166618i \(0.0532847\pi\)
−0.986022 + 0.166618i \(0.946715\pi\)
\(12\) 0 0
\(13\) −0.0851767 −0.0236238 −0.0118119 0.999930i \(-0.503760\pi\)
−0.0118119 + 0.999930i \(0.503760\pi\)
\(14\) 0 0
\(15\) 3.30049 0.852182
\(16\) 0 0
\(17\) 3.92673 + 1.25729i 0.952372 + 0.304937i
\(18\) 0 0
\(19\) 7.38549 1.69435 0.847173 0.531317i \(-0.178303\pi\)
0.847173 + 0.531317i \(0.178303\pi\)
\(20\) 0 0
\(21\) −10.8111 −2.35917
\(22\) 0 0
\(23\) 5.67905i 1.18416i −0.805878 0.592081i \(-0.798306\pi\)
0.805878 0.592081i \(-0.201694\pi\)
\(24\) 0 0
\(25\) 3.55435 0.710870
\(26\) 0 0
\(27\) 4.21405i 0.810995i
\(28\) 0 0
\(29\) 5.59463i 1.03890i 0.854502 + 0.519449i \(0.173863\pi\)
−0.854502 + 0.519449i \(0.826137\pi\)
\(30\) 0 0
\(31\) 5.38593i 0.967342i 0.875250 + 0.483671i \(0.160697\pi\)
−0.875250 + 0.483671i \(0.839303\pi\)
\(32\) 0 0
\(33\) −3.03385 −0.528126
\(34\) 0 0
\(35\) 4.73538 0.800425
\(36\) 0 0
\(37\) 7.96819i 1.30996i 0.755646 + 0.654981i \(0.227323\pi\)
−0.755646 + 0.654981i \(0.772677\pi\)
\(38\) 0 0
\(39\) 0.233812i 0.0374399i
\(40\) 0 0
\(41\) 7.91312i 1.23582i 0.786248 + 0.617911i \(0.212021\pi\)
−0.786248 + 0.617911i \(0.787979\pi\)
\(42\) 0 0
\(43\) 4.18238 0.637808 0.318904 0.947787i \(-0.396685\pi\)
0.318904 + 0.947787i \(0.396685\pi\)
\(44\) 0 0
\(45\) 5.45286i 0.812865i
\(46\) 0 0
\(47\) 4.24249 0.618830 0.309415 0.950927i \(-0.399867\pi\)
0.309415 + 0.950927i \(0.399867\pi\)
\(48\) 0 0
\(49\) −8.51122 −1.21589
\(50\) 0 0
\(51\) −3.45129 + 10.7790i −0.483277 + 1.50936i
\(52\) 0 0
\(53\) 3.82436 0.525317 0.262658 0.964889i \(-0.415401\pi\)
0.262658 + 0.964889i \(0.415401\pi\)
\(54\) 0 0
\(55\) 1.32886 0.179184
\(56\) 0 0
\(57\) 20.2733i 2.68527i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 2.00065i 0.256156i 0.991764 + 0.128078i \(0.0408809\pi\)
−0.991764 + 0.128078i \(0.959119\pi\)
\(62\) 0 0
\(63\) 17.8614i 2.25033i
\(64\) 0 0
\(65\) 0.102412i 0.0127027i
\(66\) 0 0
\(67\) −8.91072 −1.08862 −0.544309 0.838885i \(-0.683208\pi\)
−0.544309 + 0.838885i \(0.683208\pi\)
\(68\) 0 0
\(69\) 15.5891 1.87671
\(70\) 0 0
\(71\) 8.59825i 1.02042i −0.860048 0.510212i \(-0.829567\pi\)
0.860048 0.510212i \(-0.170433\pi\)
\(72\) 0 0
\(73\) 7.38541i 0.864397i −0.901778 0.432199i \(-0.857738\pi\)
0.901778 0.432199i \(-0.142262\pi\)
\(74\) 0 0
\(75\) 9.75677i 1.12662i
\(76\) 0 0
\(77\) −4.35283 −0.496051
\(78\) 0 0
\(79\) 13.5159i 1.52066i −0.649536 0.760331i \(-0.725037\pi\)
0.649536 0.760331i \(-0.274963\pi\)
\(80\) 0 0
\(81\) −2.03780 −0.226422
\(82\) 0 0
\(83\) −14.8577 −1.63085 −0.815424 0.578865i \(-0.803496\pi\)
−0.815424 + 0.578865i \(0.803496\pi\)
\(84\) 0 0
\(85\) 1.51170 4.72132i 0.163967 0.512099i
\(86\) 0 0
\(87\) −15.3574 −1.64649
\(88\) 0 0
\(89\) −10.9415 −1.15980 −0.579900 0.814687i \(-0.696909\pi\)
−0.579900 + 0.814687i \(0.696909\pi\)
\(90\) 0 0
\(91\) 0.335462i 0.0351660i
\(92\) 0 0
\(93\) −14.7845 −1.53308
\(94\) 0 0
\(95\) 8.87996i 0.911064i
\(96\) 0 0
\(97\) 15.6089i 1.58484i 0.609975 + 0.792421i \(0.291179\pi\)
−0.609975 + 0.792421i \(0.708821\pi\)
\(98\) 0 0
\(99\) 5.01235i 0.503760i
\(100\) 0 0
\(101\) 14.4865 1.44146 0.720729 0.693217i \(-0.243807\pi\)
0.720729 + 0.693217i \(0.243807\pi\)
\(102\) 0 0
\(103\) −2.27955 −0.224611 −0.112306 0.993674i \(-0.535824\pi\)
−0.112306 + 0.993674i \(0.535824\pi\)
\(104\) 0 0
\(105\) 12.9987i 1.26855i
\(106\) 0 0
\(107\) 2.26402i 0.218871i 0.993994 + 0.109435i \(0.0349043\pi\)
−0.993994 + 0.109435i \(0.965096\pi\)
\(108\) 0 0
\(109\) 8.30760i 0.795724i 0.917445 + 0.397862i \(0.130248\pi\)
−0.917445 + 0.397862i \(0.869752\pi\)
\(110\) 0 0
\(111\) −21.8729 −2.07608
\(112\) 0 0
\(113\) 13.7598i 1.29441i −0.762315 0.647206i \(-0.775937\pi\)
0.762315 0.647206i \(-0.224063\pi\)
\(114\) 0 0
\(115\) −6.82821 −0.636734
\(116\) 0 0
\(117\) 0.386290 0.0357125
\(118\) 0 0
\(119\) −4.95174 + 15.4652i −0.453926 + 1.41769i
\(120\) 0 0
\(121\) 9.77849 0.888954
\(122\) 0 0
\(123\) −21.7217 −1.95858
\(124\) 0 0
\(125\) 10.2853i 0.919949i
\(126\) 0 0
\(127\) 17.3837 1.54256 0.771278 0.636499i \(-0.219618\pi\)
0.771278 + 0.636499i \(0.219618\pi\)
\(128\) 0 0
\(129\) 11.4807i 1.01082i
\(130\) 0 0
\(131\) 15.6197i 1.36470i −0.731025 0.682350i \(-0.760958\pi\)
0.731025 0.682350i \(-0.239042\pi\)
\(132\) 0 0
\(133\) 29.0872i 2.52218i
\(134\) 0 0
\(135\) −5.06678 −0.436079
\(136\) 0 0
\(137\) −2.32537 −0.198670 −0.0993349 0.995054i \(-0.531672\pi\)
−0.0993349 + 0.995054i \(0.531672\pi\)
\(138\) 0 0
\(139\) 6.36572i 0.539933i −0.962870 0.269967i \(-0.912987\pi\)
0.962870 0.269967i \(-0.0870126\pi\)
\(140\) 0 0
\(141\) 11.6457i 0.980747i
\(142\) 0 0
\(143\) 0.0941389i 0.00787229i
\(144\) 0 0
\(145\) 6.72672 0.558624
\(146\) 0 0
\(147\) 23.3635i 1.92699i
\(148\) 0 0
\(149\) −6.98869 −0.572536 −0.286268 0.958150i \(-0.592415\pi\)
−0.286268 + 0.958150i \(0.592415\pi\)
\(150\) 0 0
\(151\) −10.7987 −0.878790 −0.439395 0.898294i \(-0.644807\pi\)
−0.439395 + 0.898294i \(0.644807\pi\)
\(152\) 0 0
\(153\) −17.8084 5.70201i −1.43972 0.460980i
\(154\) 0 0
\(155\) 6.47579 0.520148
\(156\) 0 0
\(157\) −18.8562 −1.50489 −0.752444 0.658656i \(-0.771125\pi\)
−0.752444 + 0.658656i \(0.771125\pi\)
\(158\) 0 0
\(159\) 10.4980i 0.832544i
\(160\) 0 0
\(161\) 22.3665 1.76273
\(162\) 0 0
\(163\) 3.55300i 0.278292i −0.990272 0.139146i \(-0.955564\pi\)
0.990272 0.139146i \(-0.0444358\pi\)
\(164\) 0 0
\(165\) 3.64776i 0.283978i
\(166\) 0 0
\(167\) 7.67778i 0.594124i 0.954858 + 0.297062i \(0.0960069\pi\)
−0.954858 + 0.297062i \(0.903993\pi\)
\(168\) 0 0
\(169\) −12.9927 −0.999442
\(170\) 0 0
\(171\) −33.4944 −2.56138
\(172\) 0 0
\(173\) 21.8061i 1.65788i −0.559335 0.828942i \(-0.688943\pi\)
0.559335 0.828942i \(-0.311057\pi\)
\(174\) 0 0
\(175\) 13.9985i 1.05819i
\(176\) 0 0
\(177\) 2.74502i 0.206329i
\(178\) 0 0
\(179\) 12.2648 0.916715 0.458357 0.888768i \(-0.348438\pi\)
0.458357 + 0.888768i \(0.348438\pi\)
\(180\) 0 0
\(181\) 19.1303i 1.42194i 0.703220 + 0.710972i \(0.251745\pi\)
−0.703220 + 0.710972i \(0.748255\pi\)
\(182\) 0 0
\(183\) −5.49182 −0.405967
\(184\) 0 0
\(185\) 9.58057 0.704377
\(186\) 0 0
\(187\) −1.38958 + 4.33990i −0.101616 + 0.317365i
\(188\) 0 0
\(189\) 16.5967 1.20724
\(190\) 0 0
\(191\) 8.10126 0.586187 0.293093 0.956084i \(-0.405315\pi\)
0.293093 + 0.956084i \(0.405315\pi\)
\(192\) 0 0
\(193\) 15.9862i 1.15071i 0.817904 + 0.575355i \(0.195136\pi\)
−0.817904 + 0.575355i \(0.804864\pi\)
\(194\) 0 0
\(195\) −0.281125 −0.0201318
\(196\) 0 0
\(197\) 13.6949i 0.975723i −0.872921 0.487862i \(-0.837777\pi\)
0.872921 0.487862i \(-0.162223\pi\)
\(198\) 0 0
\(199\) 26.4946i 1.87815i −0.343710 0.939076i \(-0.611684\pi\)
0.343710 0.939076i \(-0.388316\pi\)
\(200\) 0 0
\(201\) 24.4602i 1.72529i
\(202\) 0 0
\(203\) −22.0341 −1.54649
\(204\) 0 0
\(205\) 9.51436 0.664512
\(206\) 0 0
\(207\) 25.7554i 1.79012i
\(208\) 0 0
\(209\) 8.16258i 0.564618i
\(210\) 0 0
\(211\) 22.4564i 1.54596i −0.634428 0.772982i \(-0.718764\pi\)
0.634428 0.772982i \(-0.281236\pi\)
\(212\) 0 0
\(213\) 23.6024 1.61721
\(214\) 0 0
\(215\) 5.02870i 0.342955i
\(216\) 0 0
\(217\) −21.2121 −1.43997
\(218\) 0 0
\(219\) 20.2731 1.36993
\(220\) 0 0
\(221\) −0.334466 0.107092i −0.0224986 0.00720377i
\(222\) 0 0
\(223\) −13.9282 −0.932699 −0.466349 0.884601i \(-0.654431\pi\)
−0.466349 + 0.884601i \(0.654431\pi\)
\(224\) 0 0
\(225\) −16.1195 −1.07464
\(226\) 0 0
\(227\) 12.4714i 0.827757i 0.910332 + 0.413879i \(0.135826\pi\)
−0.910332 + 0.413879i \(0.864174\pi\)
\(228\) 0 0
\(229\) 24.1944 1.59881 0.799404 0.600794i \(-0.205149\pi\)
0.799404 + 0.600794i \(0.205149\pi\)
\(230\) 0 0
\(231\) 11.9486i 0.786161i
\(232\) 0 0
\(233\) 14.8016i 0.969683i 0.874602 + 0.484842i \(0.161123\pi\)
−0.874602 + 0.484842i \(0.838877\pi\)
\(234\) 0 0
\(235\) 5.10096i 0.332750i
\(236\) 0 0
\(237\) 37.1016 2.41001
\(238\) 0 0
\(239\) 22.6552 1.46544 0.732721 0.680529i \(-0.238250\pi\)
0.732721 + 0.680529i \(0.238250\pi\)
\(240\) 0 0
\(241\) 21.8877i 1.40991i 0.709252 + 0.704955i \(0.249033\pi\)
−0.709252 + 0.704955i \(0.750967\pi\)
\(242\) 0 0
\(243\) 18.2360i 1.16984i
\(244\) 0 0
\(245\) 10.2335i 0.653794i
\(246\) 0 0
\(247\) −0.629071 −0.0400269
\(248\) 0 0
\(249\) 40.7848i 2.58463i
\(250\) 0 0
\(251\) −3.71941 −0.234767 −0.117384 0.993087i \(-0.537451\pi\)
−0.117384 + 0.993087i \(0.537451\pi\)
\(252\) 0 0
\(253\) 6.27659 0.394606
\(254\) 0 0
\(255\) 12.9601 + 4.14967i 0.811595 + 0.259862i
\(256\) 0 0
\(257\) −10.4585 −0.652382 −0.326191 0.945304i \(-0.605765\pi\)
−0.326191 + 0.945304i \(0.605765\pi\)
\(258\) 0 0
\(259\) −31.3821 −1.94999
\(260\) 0 0
\(261\) 25.3726i 1.57052i
\(262\) 0 0
\(263\) 4.33686 0.267422 0.133711 0.991020i \(-0.457311\pi\)
0.133711 + 0.991020i \(0.457311\pi\)
\(264\) 0 0
\(265\) 4.59823i 0.282467i
\(266\) 0 0
\(267\) 30.0348i 1.83810i
\(268\) 0 0
\(269\) 8.89174i 0.542139i 0.962560 + 0.271069i \(0.0873773\pi\)
−0.962560 + 0.271069i \(0.912623\pi\)
\(270\) 0 0
\(271\) 11.0797 0.673043 0.336521 0.941676i \(-0.390750\pi\)
0.336521 + 0.941676i \(0.390750\pi\)
\(272\) 0 0
\(273\) 0.920853 0.0557325
\(274\) 0 0
\(275\) 3.92833i 0.236887i
\(276\) 0 0
\(277\) 16.8618i 1.01313i −0.862202 0.506564i \(-0.830915\pi\)
0.862202 0.506564i \(-0.169085\pi\)
\(278\) 0 0
\(279\) 24.4261i 1.46235i
\(280\) 0 0
\(281\) −23.0983 −1.37793 −0.688966 0.724794i \(-0.741935\pi\)
−0.688966 + 0.724794i \(0.741935\pi\)
\(282\) 0 0
\(283\) 10.3712i 0.616504i 0.951305 + 0.308252i \(0.0997439\pi\)
−0.951305 + 0.308252i \(0.900256\pi\)
\(284\) 0 0
\(285\) 24.3757 1.44389
\(286\) 0 0
\(287\) −31.1653 −1.83963
\(288\) 0 0
\(289\) 13.8384 + 9.87408i 0.814026 + 0.580828i
\(290\) 0 0
\(291\) −42.8468 −2.51172
\(292\) 0 0
\(293\) −23.4558 −1.37030 −0.685152 0.728400i \(-0.740264\pi\)
−0.685152 + 0.728400i \(0.740264\pi\)
\(294\) 0 0
\(295\) 1.20235i 0.0700037i
\(296\) 0 0
\(297\) 4.65745 0.270253
\(298\) 0 0
\(299\) 0.483722i 0.0279744i
\(300\) 0 0
\(301\) 16.4720i 0.949432i
\(302\) 0 0
\(303\) 39.7657i 2.28448i
\(304\) 0 0
\(305\) 2.40548 0.137737
\(306\) 0 0
\(307\) −10.6235 −0.606315 −0.303157 0.952941i \(-0.598041\pi\)
−0.303157 + 0.952941i \(0.598041\pi\)
\(308\) 0 0
\(309\) 6.25743i 0.355973i
\(310\) 0 0
\(311\) 9.27315i 0.525832i −0.964819 0.262916i \(-0.915316\pi\)
0.964819 0.262916i \(-0.0846842\pi\)
\(312\) 0 0
\(313\) 2.10767i 0.119133i 0.998224 + 0.0595663i \(0.0189718\pi\)
−0.998224 + 0.0595663i \(0.981028\pi\)
\(314\) 0 0
\(315\) −21.4757 −1.21002
\(316\) 0 0
\(317\) 10.9507i 0.615054i −0.951539 0.307527i \(-0.900499\pi\)
0.951539 0.307527i \(-0.0995014\pi\)
\(318\) 0 0
\(319\) −6.18330 −0.346198
\(320\) 0 0
\(321\) −6.21479 −0.346875
\(322\) 0 0
\(323\) 29.0008 + 9.28569i 1.61365 + 0.516670i
\(324\) 0 0
\(325\) −0.302748 −0.0167934
\(326\) 0 0
\(327\) −22.8046 −1.26110
\(328\) 0 0
\(329\) 16.7087i 0.921182i
\(330\) 0 0
\(331\) 29.7366 1.63447 0.817235 0.576304i \(-0.195506\pi\)
0.817235 + 0.576304i \(0.195506\pi\)
\(332\) 0 0
\(333\) 36.1370i 1.98030i
\(334\) 0 0
\(335\) 10.7138i 0.585359i
\(336\) 0 0
\(337\) 27.4455i 1.49505i 0.664232 + 0.747527i \(0.268759\pi\)
−0.664232 + 0.747527i \(0.731241\pi\)
\(338\) 0 0
\(339\) 37.7710 2.05144
\(340\) 0 0
\(341\) −5.95264 −0.322353
\(342\) 0 0
\(343\) 5.95185i 0.321370i
\(344\) 0 0
\(345\) 18.7436i 1.00912i
\(346\) 0 0
\(347\) 10.1963i 0.547366i −0.961820 0.273683i \(-0.911758\pi\)
0.961820 0.273683i \(-0.0882419\pi\)
\(348\) 0 0
\(349\) −31.5314 −1.68784 −0.843918 0.536472i \(-0.819757\pi\)
−0.843918 + 0.536472i \(0.819757\pi\)
\(350\) 0 0
\(351\) 0.358939i 0.0191588i
\(352\) 0 0
\(353\) 14.8932 0.792683 0.396342 0.918103i \(-0.370280\pi\)
0.396342 + 0.918103i \(0.370280\pi\)
\(354\) 0 0
\(355\) −10.3381 −0.548691
\(356\) 0 0
\(357\) −42.4522 13.5927i −2.24681 0.719400i
\(358\) 0 0
\(359\) −16.3543 −0.863145 −0.431572 0.902078i \(-0.642041\pi\)
−0.431572 + 0.902078i \(0.642041\pi\)
\(360\) 0 0
\(361\) 35.5454 1.87081
\(362\) 0 0
\(363\) 26.8422i 1.40885i
\(364\) 0 0
\(365\) −8.87987 −0.464794
\(366\) 0 0
\(367\) 6.28407i 0.328026i 0.986458 + 0.164013i \(0.0524439\pi\)
−0.986458 + 0.164013i \(0.947556\pi\)
\(368\) 0 0
\(369\) 35.8873i 1.86822i
\(370\) 0 0
\(371\) 15.0620i 0.781980i
\(372\) 0 0
\(373\) 12.5989 0.652348 0.326174 0.945310i \(-0.394240\pi\)
0.326174 + 0.945310i \(0.394240\pi\)
\(374\) 0 0
\(375\) 28.2335 1.45797
\(376\) 0 0
\(377\) 0.476532i 0.0245427i
\(378\) 0 0
\(379\) 32.3527i 1.66184i 0.556389 + 0.830922i \(0.312187\pi\)
−0.556389 + 0.830922i \(0.687813\pi\)
\(380\) 0 0
\(381\) 47.7187i 2.44470i
\(382\) 0 0
\(383\) −33.7881 −1.72649 −0.863246 0.504783i \(-0.831572\pi\)
−0.863246 + 0.504783i \(0.831572\pi\)
\(384\) 0 0
\(385\) 5.23363i 0.266731i
\(386\) 0 0
\(387\) −18.9678 −0.964187
\(388\) 0 0
\(389\) −12.2883 −0.623041 −0.311520 0.950239i \(-0.600838\pi\)
−0.311520 + 0.950239i \(0.600838\pi\)
\(390\) 0 0
\(391\) 7.14020 22.3001i 0.361096 1.12776i
\(392\) 0 0
\(393\) 42.8765 2.16283
\(394\) 0 0
\(395\) −16.2509 −0.817672
\(396\) 0 0
\(397\) 12.6943i 0.637107i −0.947905 0.318554i \(-0.896803\pi\)
0.947905 0.318554i \(-0.103197\pi\)
\(398\) 0 0
\(399\) −79.8451 −3.99726
\(400\) 0 0
\(401\) 11.2890i 0.563746i −0.959452 0.281873i \(-0.909044\pi\)
0.959452 0.281873i \(-0.0909556\pi\)
\(402\) 0 0
\(403\) 0.458756i 0.0228523i
\(404\) 0 0
\(405\) 2.45016i 0.121749i
\(406\) 0 0
\(407\) −8.80659 −0.436527
\(408\) 0 0
\(409\) −19.2092 −0.949835 −0.474918 0.880030i \(-0.657522\pi\)
−0.474918 + 0.880030i \(0.657522\pi\)
\(410\) 0 0
\(411\) 6.38320i 0.314860i
\(412\) 0 0
\(413\) 3.93843i 0.193797i
\(414\) 0 0
\(415\) 17.8642i 0.876920i
\(416\) 0 0
\(417\) 17.4740 0.855708
\(418\) 0 0
\(419\) 33.3995i 1.63167i 0.578282 + 0.815837i \(0.303724\pi\)
−0.578282 + 0.815837i \(0.696276\pi\)
\(420\) 0 0
\(421\) −22.4111 −1.09225 −0.546125 0.837704i \(-0.683898\pi\)
−0.546125 + 0.837704i \(0.683898\pi\)
\(422\) 0 0
\(423\) −19.2404 −0.935498
\(424\) 0 0
\(425\) 13.9570 + 4.46884i 0.677013 + 0.216771i
\(426\) 0 0
\(427\) −7.87940 −0.381311
\(428\) 0 0
\(429\) 0.258414 0.0124763
\(430\) 0 0
\(431\) 39.1271i 1.88468i −0.334651 0.942342i \(-0.608618\pi\)
0.334651 0.942342i \(-0.391382\pi\)
\(432\) 0 0
\(433\) −12.4682 −0.599182 −0.299591 0.954068i \(-0.596850\pi\)
−0.299591 + 0.954068i \(0.596850\pi\)
\(434\) 0 0
\(435\) 18.4650i 0.885330i
\(436\) 0 0
\(437\) 41.9425i 2.00638i
\(438\) 0 0
\(439\) 9.33016i 0.445304i −0.974898 0.222652i \(-0.928529\pi\)
0.974898 0.222652i \(-0.0714714\pi\)
\(440\) 0 0
\(441\) 38.5998 1.83808
\(442\) 0 0
\(443\) 18.4886 0.878420 0.439210 0.898384i \(-0.355258\pi\)
0.439210 + 0.898384i \(0.355258\pi\)
\(444\) 0 0
\(445\) 13.1556i 0.623635i
\(446\) 0 0
\(447\) 19.1841i 0.907378i
\(448\) 0 0
\(449\) 0.117579i 0.00554888i 0.999996 + 0.00277444i \(0.000883133\pi\)
−0.999996 + 0.00277444i \(0.999117\pi\)
\(450\) 0 0
\(451\) −8.74574 −0.411821
\(452\) 0 0
\(453\) 29.6428i 1.39274i
\(454\) 0 0
\(455\) −0.403344 −0.0189091
\(456\) 0 0
\(457\) −38.3980 −1.79618 −0.898091 0.439809i \(-0.855046\pi\)
−0.898091 + 0.439809i \(0.855046\pi\)
\(458\) 0 0
\(459\) 5.29828 16.5475i 0.247303 0.772369i
\(460\) 0 0
\(461\) −34.9206 −1.62642 −0.813208 0.581973i \(-0.802281\pi\)
−0.813208 + 0.581973i \(0.802281\pi\)
\(462\) 0 0
\(463\) 13.6149 0.632736 0.316368 0.948636i \(-0.397536\pi\)
0.316368 + 0.948636i \(0.397536\pi\)
\(464\) 0 0
\(465\) 17.7762i 0.824352i
\(466\) 0 0
\(467\) −2.25425 −0.104314 −0.0521571 0.998639i \(-0.516610\pi\)
−0.0521571 + 0.998639i \(0.516610\pi\)
\(468\) 0 0
\(469\) 35.0942i 1.62050i
\(470\) 0 0
\(471\) 51.7607i 2.38501i
\(472\) 0 0
\(473\) 4.62245i 0.212541i
\(474\) 0 0
\(475\) 26.2506 1.20446
\(476\) 0 0
\(477\) −17.3441 −0.794132
\(478\) 0 0
\(479\) 41.2559i 1.88503i −0.334162 0.942516i \(-0.608453\pi\)
0.334162 0.942516i \(-0.391547\pi\)
\(480\) 0 0
\(481\) 0.678704i 0.0309462i
\(482\) 0 0
\(483\) 61.3966i 2.79364i
\(484\) 0 0
\(485\) 18.7674 0.852182
\(486\) 0 0
\(487\) 23.8489i 1.08070i 0.841441 + 0.540348i \(0.181708\pi\)
−0.841441 + 0.540348i \(0.818292\pi\)
\(488\) 0 0
\(489\) 9.75307 0.441049
\(490\) 0 0
\(491\) 33.6879 1.52031 0.760156 0.649741i \(-0.225123\pi\)
0.760156 + 0.649741i \(0.225123\pi\)
\(492\) 0 0
\(493\) −7.03407 + 21.9686i −0.316799 + 0.989417i
\(494\) 0 0
\(495\) −6.02661 −0.270876
\(496\) 0 0
\(497\) 33.8636 1.51899
\(498\) 0 0
\(499\) 17.4764i 0.782350i −0.920316 0.391175i \(-0.872069\pi\)
0.920316 0.391175i \(-0.127931\pi\)
\(500\) 0 0
\(501\) −21.0757 −0.941592
\(502\) 0 0
\(503\) 4.58844i 0.204588i −0.994754 0.102294i \(-0.967382\pi\)
0.994754 0.102294i \(-0.0326183\pi\)
\(504\) 0 0
\(505\) 17.4179i 0.775084i
\(506\) 0 0
\(507\) 35.6654i 1.58396i
\(508\) 0 0
\(509\) 28.4657 1.26172 0.630861 0.775896i \(-0.282702\pi\)
0.630861 + 0.775896i \(0.282702\pi\)
\(510\) 0 0
\(511\) 29.0869 1.28673
\(512\) 0 0
\(513\) 31.1228i 1.37411i
\(514\) 0 0
\(515\) 2.74083i 0.120775i
\(516\) 0 0
\(517\) 4.68888i 0.206217i
\(518\) 0 0
\(519\) 59.8582 2.62748
\(520\) 0 0
\(521\) 2.63818i 0.115581i 0.998329 + 0.0577905i \(0.0184055\pi\)
−0.998329 + 0.0577905i \(0.981594\pi\)
\(522\) 0 0
\(523\) 31.7380 1.38781 0.693903 0.720068i \(-0.255890\pi\)
0.693903 + 0.720068i \(0.255890\pi\)
\(524\) 0 0
\(525\) −38.4264 −1.67706
\(526\) 0 0
\(527\) −6.77168 + 21.1491i −0.294979 + 0.921270i
\(528\) 0 0
\(529\) −9.25156 −0.402242
\(530\) 0 0
\(531\) −4.53516 −0.196809
\(532\) 0 0
\(533\) 0.674014i 0.0291948i
\(534\) 0 0
\(535\) 2.72215 0.117689
\(536\) 0 0
\(537\) 33.6672i 1.45285i
\(538\) 0 0
\(539\) 9.40677i 0.405178i
\(540\) 0 0
\(541\) 18.6626i 0.802368i −0.915998 0.401184i \(-0.868599\pi\)
0.915998 0.401184i \(-0.131401\pi\)
\(542\) 0 0
\(543\) −52.5131 −2.25355
\(544\) 0 0
\(545\) 9.98867 0.427868
\(546\) 0 0
\(547\) 25.0800i 1.07234i 0.844109 + 0.536172i \(0.180130\pi\)
−0.844109 + 0.536172i \(0.819870\pi\)
\(548\) 0 0
\(549\) 9.07325i 0.387237i
\(550\) 0 0
\(551\) 41.3191i 1.76025i
\(552\) 0 0
\(553\) 53.2315 2.26364
\(554\) 0 0
\(555\) 26.2989i 1.11633i
\(556\) 0 0
\(557\) 45.0667 1.90954 0.954768 0.297352i \(-0.0961034\pi\)
0.954768 + 0.297352i \(0.0961034\pi\)
\(558\) 0 0
\(559\) −0.356242 −0.0150674
\(560\) 0 0
\(561\) −11.9131 3.81443i −0.502973 0.161045i
\(562\) 0 0
\(563\) 0.964781 0.0406607 0.0203303 0.999793i \(-0.493528\pi\)
0.0203303 + 0.999793i \(0.493528\pi\)
\(564\) 0 0
\(565\) −16.5441 −0.696016
\(566\) 0 0
\(567\) 8.02574i 0.337049i
\(568\) 0 0
\(569\) −43.1527 −1.80906 −0.904528 0.426415i \(-0.859776\pi\)
−0.904528 + 0.426415i \(0.859776\pi\)
\(570\) 0 0
\(571\) 10.9017i 0.456222i 0.973635 + 0.228111i \(0.0732549\pi\)
−0.973635 + 0.228111i \(0.926745\pi\)
\(572\) 0 0
\(573\) 22.2382i 0.929012i
\(574\) 0 0
\(575\) 20.1853i 0.841785i
\(576\) 0 0
\(577\) −10.0756 −0.419454 −0.209727 0.977760i \(-0.567258\pi\)
−0.209727 + 0.977760i \(0.567258\pi\)
\(578\) 0 0
\(579\) −43.8825 −1.82369
\(580\) 0 0
\(581\) 58.5161i 2.42766i
\(582\) 0 0
\(583\) 4.22676i 0.175055i
\(584\) 0 0
\(585\) 0.464457i 0.0192029i
\(586\) 0 0
\(587\) 30.9781 1.27860 0.639302 0.768956i \(-0.279223\pi\)
0.639302 + 0.768956i \(0.279223\pi\)
\(588\) 0 0
\(589\) 39.7777i 1.63901i
\(590\) 0 0
\(591\) 37.5929 1.54637
\(592\) 0 0
\(593\) 27.3220 1.12198 0.560989 0.827823i \(-0.310421\pi\)
0.560989 + 0.827823i \(0.310421\pi\)
\(594\) 0 0
\(595\) 18.5946 + 5.95374i 0.762303 + 0.244080i
\(596\) 0 0
\(597\) 72.7283 2.97657
\(598\) 0 0
\(599\) 9.83617 0.401895 0.200947 0.979602i \(-0.435598\pi\)
0.200947 + 0.979602i \(0.435598\pi\)
\(600\) 0 0
\(601\) 8.88222i 0.362313i −0.983454 0.181157i \(-0.942016\pi\)
0.983454 0.181157i \(-0.0579841\pi\)
\(602\) 0 0
\(603\) 40.4116 1.64569
\(604\) 0 0
\(605\) 11.7572i 0.477998i
\(606\) 0 0
\(607\) 14.9458i 0.606633i −0.952890 0.303317i \(-0.901906\pi\)
0.952890 0.303317i \(-0.0980940\pi\)
\(608\) 0 0
\(609\) 60.4841i 2.45094i
\(610\) 0 0
\(611\) −0.361361 −0.0146191
\(612\) 0 0
\(613\) 29.1577 1.17767 0.588835 0.808253i \(-0.299587\pi\)
0.588835 + 0.808253i \(0.299587\pi\)
\(614\) 0 0
\(615\) 26.1172i 1.05315i
\(616\) 0 0
\(617\) 13.9156i 0.560219i 0.959968 + 0.280110i \(0.0903709\pi\)
−0.959968 + 0.280110i \(0.909629\pi\)
\(618\) 0 0
\(619\) 1.51547i 0.0609120i 0.999536 + 0.0304560i \(0.00969594\pi\)
−0.999536 + 0.0304560i \(0.990304\pi\)
\(620\) 0 0
\(621\) −23.9318 −0.960350
\(622\) 0 0
\(623\) 43.0925i 1.72646i
\(624\) 0 0
\(625\) 5.40514 0.216205
\(626\) 0 0
\(627\) −22.4065 −0.894829
\(628\) 0 0
\(629\) −10.0183 + 31.2889i −0.399456 + 1.24757i
\(630\) 0 0
\(631\) −25.3766 −1.01023 −0.505113 0.863053i \(-0.668549\pi\)
−0.505113 + 0.863053i \(0.668549\pi\)
\(632\) 0 0
\(633\) 61.6434 2.45011
\(634\) 0 0
\(635\) 20.9014i 0.829445i
\(636\) 0 0
\(637\) 0.724958 0.0287239
\(638\) 0 0
\(639\) 38.9945i 1.54260i
\(640\) 0 0
\(641\) 40.4994i 1.59963i 0.600248 + 0.799814i \(0.295069\pi\)
−0.600248 + 0.799814i \(0.704931\pi\)
\(642\) 0 0
\(643\) 26.3431i 1.03887i 0.854510 + 0.519435i \(0.173858\pi\)
−0.854510 + 0.519435i \(0.826142\pi\)
\(644\) 0 0
\(645\) 13.8039 0.543528
\(646\) 0 0
\(647\) 17.4076 0.684363 0.342181 0.939634i \(-0.388834\pi\)
0.342181 + 0.939634i \(0.388834\pi\)
\(648\) 0 0
\(649\) 1.10522i 0.0433837i
\(650\) 0 0
\(651\) 58.2278i 2.28213i
\(652\) 0 0
\(653\) 9.26968i 0.362750i 0.983414 + 0.181375i \(0.0580548\pi\)
−0.983414 + 0.181375i \(0.941945\pi\)
\(654\) 0 0
\(655\) −18.7804 −0.733811
\(656\) 0 0
\(657\) 33.4940i 1.30673i
\(658\) 0 0
\(659\) 37.0162 1.44195 0.720973 0.692963i \(-0.243695\pi\)
0.720973 + 0.692963i \(0.243695\pi\)
\(660\) 0 0
\(661\) 16.1524 0.628255 0.314128 0.949381i \(-0.398288\pi\)
0.314128 + 0.949381i \(0.398288\pi\)
\(662\) 0 0
\(663\) 0.293970 0.918118i 0.0114168 0.0356567i
\(664\) 0 0
\(665\) 34.9731 1.35620
\(666\) 0 0
\(667\) 31.7722 1.23022
\(668\) 0 0
\(669\) 38.2332i 1.47818i
\(670\) 0 0
\(671\) −2.21115 −0.0853606
\(672\) 0 0
\(673\) 5.06734i 0.195331i 0.995219 + 0.0976657i \(0.0311376\pi\)
−0.995219 + 0.0976657i \(0.968862\pi\)
\(674\) 0 0
\(675\) 14.9782i 0.576512i
\(676\) 0 0
\(677\) 3.68162i 0.141496i −0.997494 0.0707480i \(-0.977461\pi\)
0.997494 0.0707480i \(-0.0225386\pi\)
\(678\) 0 0
\(679\) −61.4745 −2.35917
\(680\) 0 0
\(681\) −34.2344 −1.31186
\(682\) 0 0
\(683\) 39.8989i 1.52669i 0.645991 + 0.763345i \(0.276444\pi\)
−0.645991 + 0.763345i \(0.723556\pi\)
\(684\) 0 0
\(685\) 2.79592i 0.106826i
\(686\) 0 0
\(687\) 66.4141i 2.53386i
\(688\) 0 0
\(689\) −0.325747 −0.0124100
\(690\) 0 0
\(691\) 37.1353i 1.41269i −0.707866 0.706346i \(-0.750342\pi\)
0.707866 0.706346i \(-0.249658\pi\)
\(692\) 0 0
\(693\) 19.7408 0.749890
\(694\) 0 0
\(695\) −7.65384 −0.290327
\(696\) 0 0
\(697\) −9.94908 + 31.0727i −0.376848 + 1.17696i
\(698\) 0 0
\(699\) −40.6307 −1.53679
\(700\) 0 0
\(701\) 7.57706 0.286182 0.143091 0.989710i \(-0.454296\pi\)
0.143091 + 0.989710i \(0.454296\pi\)
\(702\) 0 0
\(703\) 58.8489i 2.21953i
\(704\) 0 0
\(705\) 14.0023 0.527356
\(706\) 0 0
\(707\) 57.0540i 2.14574i
\(708\) 0 0
\(709\) 30.0160i 1.12727i −0.826022 0.563637i \(-0.809402\pi\)
0.826022 0.563637i \(-0.190598\pi\)
\(710\) 0 0
\(711\) 61.2969i 2.29881i
\(712\) 0 0
\(713\) 30.5870 1.14549
\(714\) 0 0
\(715\) −0.113188 −0.00423300
\(716\) 0 0
\(717\) 62.1890i 2.32249i
\(718\) 0 0
\(719\) 34.1138i 1.27223i −0.771594 0.636115i \(-0.780540\pi\)
0.771594 0.636115i \(-0.219460\pi\)
\(720\) 0 0
\(721\) 8.97786i 0.334353i
\(722\) 0 0
\(723\) −60.0822 −2.23448
\(724\) 0 0
\(725\) 19.8853i 0.738521i
\(726\) 0 0
\(727\) 14.4116 0.534497 0.267249 0.963628i \(-0.413885\pi\)
0.267249 + 0.963628i \(0.413885\pi\)
\(728\) 0 0
\(729\) 43.9448 1.62759
\(730\) 0 0
\(731\) 16.4231 + 5.25847i 0.607430 + 0.194491i
\(732\) 0 0
\(733\) −0.588587 −0.0217400 −0.0108700 0.999941i \(-0.503460\pi\)
−0.0108700 + 0.999941i \(0.503460\pi\)
\(734\) 0 0
\(735\) −28.0912 −1.03616
\(736\) 0 0
\(737\) 9.84830i 0.362767i
\(738\) 0 0
\(739\) −43.4714 −1.59912 −0.799561 0.600584i \(-0.794935\pi\)
−0.799561 + 0.600584i \(0.794935\pi\)
\(740\) 0 0
\(741\) 1.72682i 0.0634362i
\(742\) 0 0
\(743\) 4.80932i 0.176437i −0.996101 0.0882185i \(-0.971883\pi\)
0.996101 0.0882185i \(-0.0281174\pi\)
\(744\) 0 0
\(745\) 8.40287i 0.307857i
\(746\) 0 0
\(747\) 67.3822 2.46538
\(748\) 0 0
\(749\) −8.91667 −0.325808
\(750\) 0 0
\(751\) 28.9960i 1.05808i −0.848598 0.529039i \(-0.822553\pi\)
0.848598 0.529039i \(-0.177447\pi\)
\(752\) 0 0
\(753\) 10.2099i 0.372069i
\(754\) 0 0
\(755\) 12.9839i 0.472533i
\(756\) 0 0
\(757\) −2.51423 −0.0913813 −0.0456906 0.998956i \(-0.514549\pi\)
−0.0456906 + 0.998956i \(0.514549\pi\)
\(758\) 0 0
\(759\) 17.2294i 0.625387i
\(760\) 0 0
\(761\) 27.5862 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(762\) 0 0
\(763\) −32.7189 −1.18450
\(764\) 0 0
\(765\) −6.85582 + 21.4119i −0.247873 + 0.774150i
\(766\) 0 0
\(767\) −0.0851767 −0.00307555
\(768\) 0 0
\(769\) −20.9417 −0.755175 −0.377588 0.925974i \(-0.623246\pi\)
−0.377588 + 0.925974i \(0.623246\pi\)
\(770\) 0 0
\(771\) 28.7088i 1.03392i
\(772\) 0 0
\(773\) −24.6969 −0.888284 −0.444142 0.895956i \(-0.646491\pi\)
−0.444142 + 0.895956i \(0.646491\pi\)
\(774\) 0 0
\(775\) 19.1435i 0.687654i
\(776\) 0 0
\(777\) 86.1447i 3.09042i
\(778\) 0 0
\(779\) 58.4423i 2.09391i
\(780\) 0 0
\(781\) 9.50295 0.340042
\(782\) 0 0
\(783\) 23.5761 0.842540
\(784\) 0 0
\(785\) 22.6718i 0.809191i
\(786\) 0 0
\(787\) 43.3112i 1.54388i 0.635696 + 0.771939i \(0.280713\pi\)
−0.635696 + 0.771939i \(0.719287\pi\)
\(788\) 0 0
\(789\) 11.9048i 0.423822i
\(790\) 0 0
\(791\) 54.1920 1.92684
\(792\) 0 0
\(793\) 0.170408i 0.00605138i