Properties

Label 4012.2.b.b.237.10
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 237.10
Character \(\chi\) = 4012.237
Dual form 4012.2.b.b.237.37

$q$-expansion

\(f(q)\) \(=\) \(q-1.82354i q^{3} -0.284793i q^{5} -0.645277i q^{7} -0.325298 q^{9} +O(q^{10})\) \(q-1.82354i q^{3} -0.284793i q^{5} -0.645277i q^{7} -0.325298 q^{9} +1.56411i q^{11} +1.12492 q^{13} -0.519331 q^{15} +(1.35405 - 3.89443i) q^{17} -0.845911 q^{19} -1.17669 q^{21} +2.49622i q^{23} +4.91889 q^{25} -4.87743i q^{27} +9.28760i q^{29} -5.94750i q^{31} +2.85222 q^{33} -0.183770 q^{35} -6.09474i q^{37} -2.05134i q^{39} -1.30564i q^{41} +3.23300 q^{43} +0.0926424i q^{45} +0.319650 q^{47} +6.58362 q^{49} +(-7.10164 - 2.46916i) q^{51} -3.07538 q^{53} +0.445448 q^{55} +1.54255i q^{57} +1.00000 q^{59} +4.55451i q^{61} +0.209907i q^{63} -0.320370i q^{65} +4.98126 q^{67} +4.55196 q^{69} -13.3681i q^{71} -13.2696i q^{73} -8.96980i q^{75} +1.00929 q^{77} +1.65938i q^{79} -9.87007 q^{81} +4.62138 q^{83} +(-1.10910 - 0.385622i) q^{85} +16.9363 q^{87} +1.32179 q^{89} -0.725887i q^{91} -10.8455 q^{93} +0.240909i q^{95} -6.36118i q^{97} -0.508802i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46q - 54q^{9} + O(q^{10}) \) \( 46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100}) \)

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
\(3\) 1.82354i 1.05282i −0.850230 0.526411i \(-0.823537\pi\)
0.850230 0.526411i \(-0.176463\pi\)
\(4\) 0 0
\(5\) 0.284793i 0.127363i −0.997970 0.0636816i \(-0.979716\pi\)
0.997970 0.0636816i \(-0.0202842\pi\)
\(6\) 0 0
\(7\) 0.645277i 0.243892i −0.992537 0.121946i \(-0.961087\pi\)
0.992537 0.121946i \(-0.0389135\pi\)
\(8\) 0 0
\(9\) −0.325298 −0.108433
\(10\) 0 0
\(11\) 1.56411i 0.471598i 0.971802 + 0.235799i \(0.0757707\pi\)
−0.971802 + 0.235799i \(0.924229\pi\)
\(12\) 0 0
\(13\) 1.12492 0.311997 0.155999 0.987757i \(-0.450140\pi\)
0.155999 + 0.987757i \(0.450140\pi\)
\(14\) 0 0
\(15\) −0.519331 −0.134091
\(16\) 0 0
\(17\) 1.35405 3.89443i 0.328404 0.944537i
\(18\) 0 0
\(19\) −0.845911 −0.194065 −0.0970327 0.995281i \(-0.530935\pi\)
−0.0970327 + 0.995281i \(0.530935\pi\)
\(20\) 0 0
\(21\) −1.17669 −0.256775
\(22\) 0 0
\(23\) 2.49622i 0.520498i 0.965542 + 0.260249i \(0.0838047\pi\)
−0.965542 + 0.260249i \(0.916195\pi\)
\(24\) 0 0
\(25\) 4.91889 0.983779
\(26\) 0 0
\(27\) 4.87743i 0.938661i
\(28\) 0 0
\(29\) 9.28760i 1.72466i 0.506344 + 0.862332i \(0.330997\pi\)
−0.506344 + 0.862332i \(0.669003\pi\)
\(30\) 0 0
\(31\) 5.94750i 1.06820i −0.845420 0.534102i \(-0.820650\pi\)
0.845420 0.534102i \(-0.179350\pi\)
\(32\) 0 0
\(33\) 2.85222 0.496508
\(34\) 0 0
\(35\) −0.183770 −0.0310629
\(36\) 0 0
\(37\) 6.09474i 1.00197i −0.865456 0.500985i \(-0.832971\pi\)
0.865456 0.500985i \(-0.167029\pi\)
\(38\) 0 0
\(39\) 2.05134i 0.328477i
\(40\) 0 0
\(41\) 1.30564i 0.203907i −0.994789 0.101954i \(-0.967491\pi\)
0.994789 0.101954i \(-0.0325094\pi\)
\(42\) 0 0
\(43\) 3.23300 0.493028 0.246514 0.969139i \(-0.420715\pi\)
0.246514 + 0.969139i \(0.420715\pi\)
\(44\) 0 0
\(45\) 0.0926424i 0.0138103i
\(46\) 0 0
\(47\) 0.319650 0.0466257 0.0233128 0.999728i \(-0.492579\pi\)
0.0233128 + 0.999728i \(0.492579\pi\)
\(48\) 0 0
\(49\) 6.58362 0.940517
\(50\) 0 0
\(51\) −7.10164 2.46916i −0.994429 0.345751i
\(52\) 0 0
\(53\) −3.07538 −0.422436 −0.211218 0.977439i \(-0.567743\pi\)
−0.211218 + 0.977439i \(0.567743\pi\)
\(54\) 0 0
\(55\) 0.445448 0.0600642
\(56\) 0 0
\(57\) 1.54255i 0.204316i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 4.55451i 0.583145i 0.956549 + 0.291572i \(0.0941785\pi\)
−0.956549 + 0.291572i \(0.905822\pi\)
\(62\) 0 0
\(63\) 0.209907i 0.0264458i
\(64\) 0 0
\(65\) 0.320370i 0.0397370i
\(66\) 0 0
\(67\) 4.98126 0.608558 0.304279 0.952583i \(-0.401585\pi\)
0.304279 + 0.952583i \(0.401585\pi\)
\(68\) 0 0
\(69\) 4.55196 0.547991
\(70\) 0 0
\(71\) 13.3681i 1.58650i −0.608896 0.793250i \(-0.708387\pi\)
0.608896 0.793250i \(-0.291613\pi\)
\(72\) 0 0
\(73\) 13.2696i 1.55309i −0.630065 0.776543i \(-0.716972\pi\)
0.630065 0.776543i \(-0.283028\pi\)
\(74\) 0 0
\(75\) 8.96980i 1.03574i
\(76\) 0 0
\(77\) 1.00929 0.115019
\(78\) 0 0
\(79\) 1.65938i 0.186695i 0.995634 + 0.0933476i \(0.0297568\pi\)
−0.995634 + 0.0933476i \(0.970243\pi\)
\(80\) 0 0
\(81\) −9.87007 −1.09667
\(82\) 0 0
\(83\) 4.62138 0.507262 0.253631 0.967301i \(-0.418375\pi\)
0.253631 + 0.967301i \(0.418375\pi\)
\(84\) 0 0
\(85\) −1.10910 0.385622i −0.120299 0.0418266i
\(86\) 0 0
\(87\) 16.9363 1.81576
\(88\) 0 0
\(89\) 1.32179 0.140110 0.0700550 0.997543i \(-0.477683\pi\)
0.0700550 + 0.997543i \(0.477683\pi\)
\(90\) 0 0
\(91\) 0.725887i 0.0760936i
\(92\) 0 0
\(93\) −10.8455 −1.12463
\(94\) 0 0
\(95\) 0.240909i 0.0247168i
\(96\) 0 0
\(97\) 6.36118i 0.645880i −0.946419 0.322940i \(-0.895329\pi\)
0.946419 0.322940i \(-0.104671\pi\)
\(98\) 0 0
\(99\) 0.508802i 0.0511365i
\(100\) 0 0
\(101\) 1.83767 0.182855 0.0914275 0.995812i \(-0.470857\pi\)
0.0914275 + 0.995812i \(0.470857\pi\)
\(102\) 0 0
\(103\) −9.66717 −0.952535 −0.476267 0.879301i \(-0.658011\pi\)
−0.476267 + 0.879301i \(0.658011\pi\)
\(104\) 0 0
\(105\) 0.335113i 0.0327036i
\(106\) 0 0
\(107\) 15.8474i 1.53202i 0.642828 + 0.766011i \(0.277761\pi\)
−0.642828 + 0.766011i \(0.722239\pi\)
\(108\) 0 0
\(109\) 3.38945i 0.324650i 0.986737 + 0.162325i \(0.0518994\pi\)
−0.986737 + 0.162325i \(0.948101\pi\)
\(110\) 0 0
\(111\) −11.1140 −1.05489
\(112\) 0 0
\(113\) 15.2698i 1.43647i −0.695803 0.718233i \(-0.744951\pi\)
0.695803 0.718233i \(-0.255049\pi\)
\(114\) 0 0
\(115\) 0.710906 0.0662923
\(116\) 0 0
\(117\) −0.365934 −0.0338306
\(118\) 0 0
\(119\) −2.51299 0.873735i −0.230365 0.0800951i
\(120\) 0 0
\(121\) 8.55355 0.777595
\(122\) 0 0
\(123\) −2.38089 −0.214678
\(124\) 0 0
\(125\) 2.82483i 0.252660i
\(126\) 0 0
\(127\) 1.76464 0.156586 0.0782931 0.996930i \(-0.475053\pi\)
0.0782931 + 0.996930i \(0.475053\pi\)
\(128\) 0 0
\(129\) 5.89550i 0.519070i
\(130\) 0 0
\(131\) 9.57745i 0.836785i 0.908266 + 0.418393i \(0.137406\pi\)
−0.908266 + 0.418393i \(0.862594\pi\)
\(132\) 0 0
\(133\) 0.545848i 0.0473310i
\(134\) 0 0
\(135\) −1.38906 −0.119551
\(136\) 0 0
\(137\) −6.59426 −0.563385 −0.281693 0.959505i \(-0.590896\pi\)
−0.281693 + 0.959505i \(0.590896\pi\)
\(138\) 0 0
\(139\) 22.3298i 1.89399i −0.321247 0.946995i \(-0.604102\pi\)
0.321247 0.946995i \(-0.395898\pi\)
\(140\) 0 0
\(141\) 0.582894i 0.0490885i
\(142\) 0 0
\(143\) 1.75951i 0.147137i
\(144\) 0 0
\(145\) 2.64504 0.219659
\(146\) 0 0
\(147\) 12.0055i 0.990196i
\(148\) 0 0
\(149\) 18.9103 1.54919 0.774595 0.632458i \(-0.217954\pi\)
0.774595 + 0.632458i \(0.217954\pi\)
\(150\) 0 0
\(151\) −9.21004 −0.749502 −0.374751 0.927125i \(-0.622272\pi\)
−0.374751 + 0.927125i \(0.622272\pi\)
\(152\) 0 0
\(153\) −0.440467 + 1.26685i −0.0356097 + 0.102419i
\(154\) 0 0
\(155\) −1.69381 −0.136050
\(156\) 0 0
\(157\) 13.0359 1.04038 0.520190 0.854051i \(-0.325861\pi\)
0.520190 + 0.854051i \(0.325861\pi\)
\(158\) 0 0
\(159\) 5.60808i 0.444750i
\(160\) 0 0
\(161\) 1.61076 0.126945
\(162\) 0 0
\(163\) 9.99477i 0.782851i −0.920210 0.391425i \(-0.871982\pi\)
0.920210 0.391425i \(-0.128018\pi\)
\(164\) 0 0
\(165\) 0.812292i 0.0632369i
\(166\) 0 0
\(167\) 14.8575i 1.14971i −0.818257 0.574853i \(-0.805059\pi\)
0.818257 0.574853i \(-0.194941\pi\)
\(168\) 0 0
\(169\) −11.7346 −0.902658
\(170\) 0 0
\(171\) 0.275173 0.0210430
\(172\) 0 0
\(173\) 6.62290i 0.503530i −0.967788 0.251765i \(-0.918989\pi\)
0.967788 0.251765i \(-0.0810110\pi\)
\(174\) 0 0
\(175\) 3.17405i 0.239936i
\(176\) 0 0
\(177\) 1.82354i 0.137066i
\(178\) 0 0
\(179\) −25.6065 −1.91392 −0.956960 0.290219i \(-0.906272\pi\)
−0.956960 + 0.290219i \(0.906272\pi\)
\(180\) 0 0
\(181\) 15.4252i 1.14655i 0.819363 + 0.573275i \(0.194327\pi\)
−0.819363 + 0.573275i \(0.805673\pi\)
\(182\) 0 0
\(183\) 8.30533 0.613947
\(184\) 0 0
\(185\) −1.73574 −0.127614
\(186\) 0 0
\(187\) 6.09133 + 2.11788i 0.445442 + 0.154875i
\(188\) 0 0
\(189\) −3.14729 −0.228932
\(190\) 0 0
\(191\) −20.2292 −1.46373 −0.731867 0.681447i \(-0.761351\pi\)
−0.731867 + 0.681447i \(0.761351\pi\)
\(192\) 0 0
\(193\) 14.2173i 1.02338i 0.859169 + 0.511692i \(0.170981\pi\)
−0.859169 + 0.511692i \(0.829019\pi\)
\(194\) 0 0
\(195\) −0.584207 −0.0418359
\(196\) 0 0
\(197\) 9.98969i 0.711736i −0.934536 0.355868i \(-0.884185\pi\)
0.934536 0.355868i \(-0.115815\pi\)
\(198\) 0 0
\(199\) 1.47073i 0.104257i 0.998640 + 0.0521286i \(0.0166006\pi\)
−0.998640 + 0.0521286i \(0.983399\pi\)
\(200\) 0 0
\(201\) 9.08353i 0.640703i
\(202\) 0 0
\(203\) 5.99308 0.420631
\(204\) 0 0
\(205\) −0.371838 −0.0259703
\(206\) 0 0
\(207\) 0.812015i 0.0564389i
\(208\) 0 0
\(209\) 1.32310i 0.0915208i
\(210\) 0 0
\(211\) 5.98113i 0.411758i 0.978577 + 0.205879i \(0.0660053\pi\)
−0.978577 + 0.205879i \(0.933995\pi\)
\(212\) 0 0
\(213\) −24.3772 −1.67030
\(214\) 0 0
\(215\) 0.920735i 0.0627936i
\(216\) 0 0
\(217\) −3.83779 −0.260526
\(218\) 0 0
\(219\) −24.1976 −1.63512
\(220\) 0 0
\(221\) 1.52320 4.38093i 0.102461 0.294693i
\(222\) 0 0
\(223\) −27.7405 −1.85764 −0.928822 0.370527i \(-0.879177\pi\)
−0.928822 + 0.370527i \(0.879177\pi\)
\(224\) 0 0
\(225\) −1.60010 −0.106674
\(226\) 0 0
\(227\) 11.6734i 0.774793i −0.921913 0.387396i \(-0.873375\pi\)
0.921913 0.387396i \(-0.126625\pi\)
\(228\) 0 0
\(229\) −7.25514 −0.479433 −0.239717 0.970843i \(-0.577055\pi\)
−0.239717 + 0.970843i \(0.577055\pi\)
\(230\) 0 0
\(231\) 1.84048i 0.121094i
\(232\) 0 0
\(233\) 13.5248i 0.886038i 0.896512 + 0.443019i \(0.146092\pi\)
−0.896512 + 0.443019i \(0.853908\pi\)
\(234\) 0 0
\(235\) 0.0910339i 0.00593840i
\(236\) 0 0
\(237\) 3.02595 0.196557
\(238\) 0 0
\(239\) 2.89559 0.187300 0.0936501 0.995605i \(-0.470146\pi\)
0.0936501 + 0.995605i \(0.470146\pi\)
\(240\) 0 0
\(241\) 10.2515i 0.660356i 0.943919 + 0.330178i \(0.107109\pi\)
−0.943919 + 0.330178i \(0.892891\pi\)
\(242\) 0 0
\(243\) 3.36619i 0.215941i
\(244\) 0 0
\(245\) 1.87497i 0.119787i
\(246\) 0 0
\(247\) −0.951584 −0.0605479
\(248\) 0 0
\(249\) 8.42727i 0.534057i
\(250\) 0 0
\(251\) 13.3824 0.844691 0.422345 0.906435i \(-0.361207\pi\)
0.422345 + 0.906435i \(0.361207\pi\)
\(252\) 0 0
\(253\) −3.90437 −0.245466
\(254\) 0 0
\(255\) −0.703197 + 2.02250i −0.0440359 + 0.126654i
\(256\) 0 0
\(257\) 22.1868 1.38397 0.691987 0.721910i \(-0.256736\pi\)
0.691987 + 0.721910i \(0.256736\pi\)
\(258\) 0 0
\(259\) −3.93280 −0.244372
\(260\) 0 0
\(261\) 3.02123i 0.187010i
\(262\) 0 0
\(263\) −2.12333 −0.130930 −0.0654651 0.997855i \(-0.520853\pi\)
−0.0654651 + 0.997855i \(0.520853\pi\)
\(264\) 0 0
\(265\) 0.875847i 0.0538028i
\(266\) 0 0
\(267\) 2.41034i 0.147511i
\(268\) 0 0
\(269\) 12.8565i 0.783877i −0.919991 0.391939i \(-0.871805\pi\)
0.919991 0.391939i \(-0.128195\pi\)
\(270\) 0 0
\(271\) −25.5328 −1.55101 −0.775504 0.631342i \(-0.782504\pi\)
−0.775504 + 0.631342i \(0.782504\pi\)
\(272\) 0 0
\(273\) −1.32368 −0.0801130
\(274\) 0 0
\(275\) 7.69371i 0.463948i
\(276\) 0 0
\(277\) 15.8770i 0.953954i −0.878916 0.476977i \(-0.841733\pi\)
0.878916 0.476977i \(-0.158267\pi\)
\(278\) 0 0
\(279\) 1.93471i 0.115828i
\(280\) 0 0
\(281\) 27.1720 1.62095 0.810474 0.585775i \(-0.199210\pi\)
0.810474 + 0.585775i \(0.199210\pi\)
\(282\) 0 0
\(283\) 4.03060i 0.239594i −0.992798 0.119797i \(-0.961776\pi\)
0.992798 0.119797i \(-0.0382244\pi\)
\(284\) 0 0
\(285\) 0.439308 0.0260224
\(286\) 0 0
\(287\) −0.842503 −0.0497314
\(288\) 0 0
\(289\) −13.3331 10.5465i −0.784301 0.620380i
\(290\) 0 0
\(291\) −11.5999 −0.679996
\(292\) 0 0
\(293\) −1.89615 −0.110774 −0.0553870 0.998465i \(-0.517639\pi\)
−0.0553870 + 0.998465i \(0.517639\pi\)
\(294\) 0 0
\(295\) 0.284793i 0.0165813i
\(296\) 0 0
\(297\) 7.62885 0.442671
\(298\) 0 0
\(299\) 2.80805i 0.162394i
\(300\) 0 0
\(301\) 2.08618i 0.120245i
\(302\) 0 0
\(303\) 3.35107i 0.192514i
\(304\) 0 0
\(305\) 1.29709 0.0742712
\(306\) 0 0
\(307\) 4.01367 0.229072 0.114536 0.993419i \(-0.463462\pi\)
0.114536 + 0.993419i \(0.463462\pi\)
\(308\) 0 0
\(309\) 17.6285i 1.00285i
\(310\) 0 0
\(311\) 13.3227i 0.755464i 0.925915 + 0.377732i \(0.123296\pi\)
−0.925915 + 0.377732i \(0.876704\pi\)
\(312\) 0 0
\(313\) 22.7398i 1.28533i −0.766147 0.642665i \(-0.777829\pi\)
0.766147 0.642665i \(-0.222171\pi\)
\(314\) 0 0
\(315\) 0.0597800 0.00336822
\(316\) 0 0
\(317\) 23.0591i 1.29513i 0.762010 + 0.647565i \(0.224213\pi\)
−0.762010 + 0.647565i \(0.775787\pi\)
\(318\) 0 0
\(319\) −14.5269 −0.813348
\(320\) 0 0
\(321\) 28.8983 1.61294
\(322\) 0 0
\(323\) −1.14540 + 3.29434i −0.0637319 + 0.183302i
\(324\) 0 0
\(325\) 5.53337 0.306936
\(326\) 0 0
\(327\) 6.18080 0.341799
\(328\) 0 0
\(329\) 0.206263i 0.0113716i
\(330\) 0 0
\(331\) −5.41739 −0.297767 −0.148883 0.988855i \(-0.547568\pi\)
−0.148883 + 0.988855i \(0.547568\pi\)
\(332\) 0 0
\(333\) 1.98261i 0.108646i
\(334\) 0 0
\(335\) 1.41863i 0.0775079i
\(336\) 0 0
\(337\) 0.00668662i 0.000364243i −1.00000 0.000182122i \(-0.999942\pi\)
1.00000 0.000182122i \(-5.79711e-5\pi\)
\(338\) 0 0
\(339\) −27.8452 −1.51234
\(340\) 0 0
\(341\) 9.30257 0.503763
\(342\) 0 0
\(343\) 8.76520i 0.473276i
\(344\) 0 0
\(345\) 1.29636i 0.0697939i
\(346\) 0 0
\(347\) 14.1057i 0.757236i −0.925553 0.378618i \(-0.876399\pi\)
0.925553 0.378618i \(-0.123601\pi\)
\(348\) 0 0
\(349\) 23.9796 1.28360 0.641800 0.766872i \(-0.278188\pi\)
0.641800 + 0.766872i \(0.278188\pi\)
\(350\) 0 0
\(351\) 5.48672i 0.292860i
\(352\) 0 0
\(353\) −14.8194 −0.788755 −0.394378 0.918948i \(-0.629040\pi\)
−0.394378 + 0.918948i \(0.629040\pi\)
\(354\) 0 0
\(355\) −3.80713 −0.202062
\(356\) 0 0
\(357\) −1.59329 + 4.58253i −0.0843259 + 0.242533i
\(358\) 0 0
\(359\) −14.1611 −0.747393 −0.373697 0.927551i \(-0.621910\pi\)
−0.373697 + 0.927551i \(0.621910\pi\)
\(360\) 0 0
\(361\) −18.2844 −0.962339
\(362\) 0 0
\(363\) 15.5977i 0.818669i
\(364\) 0 0
\(365\) −3.77908 −0.197806
\(366\) 0 0
\(367\) 13.8046i 0.720596i 0.932837 + 0.360298i \(0.117325\pi\)
−0.932837 + 0.360298i \(0.882675\pi\)
\(368\) 0 0
\(369\) 0.424723i 0.0221102i
\(370\) 0 0
\(371\) 1.98448i 0.103029i
\(372\) 0 0
\(373\) 34.7891 1.80131 0.900655 0.434534i \(-0.143087\pi\)
0.900655 + 0.434534i \(0.143087\pi\)
\(374\) 0 0
\(375\) −5.15119 −0.266006
\(376\) 0 0
\(377\) 10.4478i 0.538090i
\(378\) 0 0
\(379\) 9.91747i 0.509426i −0.967017 0.254713i \(-0.918019\pi\)
0.967017 0.254713i \(-0.0819810\pi\)
\(380\) 0 0
\(381\) 3.21788i 0.164857i
\(382\) 0 0
\(383\) 28.4705 1.45477 0.727386 0.686228i \(-0.240735\pi\)
0.727386 + 0.686228i \(0.240735\pi\)
\(384\) 0 0
\(385\) 0.287438i 0.0146492i
\(386\) 0 0
\(387\) −1.05169 −0.0534602
\(388\) 0 0
\(389\) −33.3610 −1.69147 −0.845734 0.533604i \(-0.820837\pi\)
−0.845734 + 0.533604i \(0.820837\pi\)
\(390\) 0 0
\(391\) 9.72135 + 3.38000i 0.491630 + 0.170934i
\(392\) 0 0
\(393\) 17.4649 0.880985
\(394\) 0 0
\(395\) 0.472580 0.0237781
\(396\) 0 0
\(397\) 13.4188i 0.673472i 0.941599 + 0.336736i \(0.109323\pi\)
−0.941599 + 0.336736i \(0.890677\pi\)
\(398\) 0 0
\(399\) 0.995375 0.0498311
\(400\) 0 0
\(401\) 35.3351i 1.76455i 0.470734 + 0.882275i \(0.343989\pi\)
−0.470734 + 0.882275i \(0.656011\pi\)
\(402\) 0 0
\(403\) 6.69048i 0.333277i
\(404\) 0 0
\(405\) 2.81093i 0.139676i
\(406\) 0 0
\(407\) 9.53287 0.472527
\(408\) 0 0
\(409\) 25.4471 1.25828 0.629139 0.777293i \(-0.283407\pi\)
0.629139 + 0.777293i \(0.283407\pi\)
\(410\) 0 0
\(411\) 12.0249i 0.593144i
\(412\) 0 0
\(413\) 0.645277i 0.0317520i
\(414\) 0 0
\(415\) 1.31614i 0.0646066i
\(416\) 0 0
\(417\) −40.7193 −1.99403
\(418\) 0 0
\(419\) 25.0810i 1.22529i −0.790360 0.612643i \(-0.790106\pi\)
0.790360 0.612643i \(-0.209894\pi\)
\(420\) 0 0
\(421\) 3.09469 0.150826 0.0754131 0.997152i \(-0.475972\pi\)
0.0754131 + 0.997152i \(0.475972\pi\)
\(422\) 0 0
\(423\) −0.103981 −0.00505574
\(424\) 0 0
\(425\) 6.66040 19.1563i 0.323077 0.929216i
\(426\) 0 0
\(427\) 2.93892 0.142224
\(428\) 0 0
\(429\) 3.20853 0.154909
\(430\) 0 0
\(431\) 6.19055i 0.298189i −0.988823 0.149094i \(-0.952364\pi\)
0.988823 0.149094i \(-0.0476358\pi\)
\(432\) 0 0
\(433\) −2.71795 −0.130616 −0.0653081 0.997865i \(-0.520803\pi\)
−0.0653081 + 0.997865i \(0.520803\pi\)
\(434\) 0 0
\(435\) 4.82334i 0.231261i
\(436\) 0 0
\(437\) 2.11158i 0.101011i
\(438\) 0 0
\(439\) 19.8137i 0.945658i −0.881154 0.472829i \(-0.843233\pi\)
0.881154 0.472829i \(-0.156767\pi\)
\(440\) 0 0
\(441\) −2.14163 −0.101983
\(442\) 0 0
\(443\) 28.6289 1.36020 0.680101 0.733118i \(-0.261936\pi\)
0.680101 + 0.733118i \(0.261936\pi\)
\(444\) 0 0
\(445\) 0.376437i 0.0178448i
\(446\) 0 0
\(447\) 34.4836i 1.63102i
\(448\) 0 0
\(449\) 22.7766i 1.07489i 0.843298 + 0.537447i \(0.180611\pi\)
−0.843298 + 0.537447i \(0.819389\pi\)
\(450\) 0 0
\(451\) 2.04218 0.0961623
\(452\) 0 0
\(453\) 16.7949i 0.789092i
\(454\) 0 0
\(455\) −0.206727 −0.00969153
\(456\) 0 0
\(457\) −26.1381 −1.22269 −0.611345 0.791364i \(-0.709371\pi\)
−0.611345 + 0.791364i \(0.709371\pi\)
\(458\) 0 0
\(459\) −18.9948 6.60426i −0.886600 0.308260i
\(460\) 0 0
\(461\) −26.1242 −1.21672 −0.608362 0.793660i \(-0.708173\pi\)
−0.608362 + 0.793660i \(0.708173\pi\)
\(462\) 0 0
\(463\) −20.0320 −0.930964 −0.465482 0.885057i \(-0.654119\pi\)
−0.465482 + 0.885057i \(0.654119\pi\)
\(464\) 0 0
\(465\) 3.08872i 0.143236i
\(466\) 0 0
\(467\) −21.7634 −1.00709 −0.503544 0.863970i \(-0.667971\pi\)
−0.503544 + 0.863970i \(0.667971\pi\)
\(468\) 0 0
\(469\) 3.21430i 0.148422i
\(470\) 0 0
\(471\) 23.7715i 1.09533i
\(472\) 0 0
\(473\) 5.05678i 0.232511i
\(474\) 0 0
\(475\) −4.16095 −0.190917
\(476\) 0 0
\(477\) 1.00041 0.0458058
\(478\) 0 0
\(479\) 7.90053i 0.360984i 0.983576 + 0.180492i \(0.0577691\pi\)
−0.983576 + 0.180492i \(0.942231\pi\)
\(480\) 0 0
\(481\) 6.85611i 0.312612i
\(482\) 0 0
\(483\) 2.93728i 0.133651i
\(484\) 0 0
\(485\) −1.81162 −0.0822613
\(486\) 0 0
\(487\) 12.6887i 0.574979i −0.957784 0.287489i \(-0.907179\pi\)
0.957784 0.287489i \(-0.0928206\pi\)
\(488\) 0 0
\(489\) −18.2259 −0.824202
\(490\) 0 0
\(491\) 27.8725 1.25787 0.628934 0.777459i \(-0.283492\pi\)
0.628934 + 0.777459i \(0.283492\pi\)
\(492\) 0 0
\(493\) 36.1699 + 12.5758i 1.62901 + 0.566387i
\(494\) 0 0
\(495\) −0.144903 −0.00651291
\(496\) 0 0
\(497\) −8.62613 −0.386935
\(498\) 0 0
\(499\) 18.1286i 0.811546i 0.913974 + 0.405773i \(0.132998\pi\)
−0.913974 + 0.405773i \(0.867002\pi\)
\(500\) 0 0
\(501\) −27.0932 −1.21043
\(502\) 0 0
\(503\) 37.0824i 1.65342i 0.562628 + 0.826710i \(0.309790\pi\)
−0.562628 + 0.826710i \(0.690210\pi\)
\(504\) 0 0
\(505\) 0.523355i 0.0232890i
\(506\) 0 0
\(507\) 21.3984i 0.950337i
\(508\) 0 0
\(509\) 17.6485 0.782257 0.391129 0.920336i \(-0.372085\pi\)
0.391129 + 0.920336i \(0.372085\pi\)
\(510\) 0 0
\(511\) −8.56255 −0.378785
\(512\) 0 0
\(513\) 4.12587i 0.182162i
\(514\) 0 0
\(515\) 2.75314i 0.121318i
\(516\) 0 0
\(517\) 0.499968i 0.0219886i
\(518\) 0 0
\(519\) −12.0771 −0.530127
\(520\) 0 0
\(521\) 29.6418i 1.29863i 0.760520 + 0.649315i \(0.224944\pi\)
−0.760520 + 0.649315i \(0.775056\pi\)
\(522\) 0 0
\(523\) 31.1883 1.36377 0.681884 0.731460i \(-0.261161\pi\)
0.681884 + 0.731460i \(0.261161\pi\)
\(524\) 0 0
\(525\) −5.78801 −0.252609
\(526\) 0 0
\(527\) −23.1621 8.05319i −1.00896 0.350802i
\(528\) 0 0
\(529\) 16.7689 0.729082
\(530\) 0 0
\(531\) −0.325298 −0.0141167
\(532\) 0 0
\(533\) 1.46875i 0.0636185i
\(534\) 0 0
\(535\) 4.51321 0.195123
\(536\) 0 0
\(537\) 46.6945i 2.01502i
\(538\) 0 0
\(539\) 10.2975i 0.443546i
\(540\) 0 0
\(541\) 40.3276i 1.73382i −0.498466 0.866909i \(-0.666103\pi\)
0.498466 0.866909i \(-0.333897\pi\)
\(542\) 0 0
\(543\) 28.1286 1.20711
\(544\) 0 0
\(545\) 0.965291 0.0413485
\(546\) 0 0
\(547\) 33.7138i 1.44150i −0.693196 0.720749i \(-0.743798\pi\)
0.693196 0.720749i \(-0.256202\pi\)
\(548\) 0 0
\(549\) 1.48157i 0.0632319i
\(550\) 0 0
\(551\) 7.85648i 0.334697i
\(552\) 0 0
\(553\) 1.07076 0.0455334
\(554\) 0 0
\(555\) 3.16519i 0.134355i
\(556\) 0 0
\(557\) −2.16460 −0.0917171 −0.0458586 0.998948i \(-0.514602\pi\)
−0.0458586 + 0.998948i \(0.514602\pi\)
\(558\) 0 0
\(559\) 3.63687 0.153823
\(560\) 0 0
\(561\) 3.86204 11.1078i 0.163055 0.468971i
\(562\) 0 0
\(563\) 28.2368 1.19004 0.595020 0.803711i \(-0.297144\pi\)
0.595020 + 0.803711i \(0.297144\pi\)
\(564\) 0 0
\(565\) −4.34874 −0.182953
\(566\) 0 0
\(567\) 6.36894i 0.267470i
\(568\) 0 0
\(569\) −1.10346 −0.0462596 −0.0231298 0.999732i \(-0.507363\pi\)
−0.0231298 + 0.999732i \(0.507363\pi\)
\(570\) 0 0
\(571\) 25.9942i 1.08782i 0.839143 + 0.543911i \(0.183057\pi\)
−0.839143 + 0.543911i \(0.816943\pi\)
\(572\) 0 0
\(573\) 36.8888i 1.54105i
\(574\) 0 0
\(575\) 12.2786i 0.512055i
\(576\) 0 0
\(577\) 13.6580 0.568588 0.284294 0.958737i \(-0.408241\pi\)
0.284294 + 0.958737i \(0.408241\pi\)
\(578\) 0 0
\(579\) 25.9258 1.07744
\(580\) 0 0
\(581\) 2.98207i 0.123717i
\(582\) 0 0
\(583\) 4.81025i 0.199220i
\(584\) 0 0
\(585\) 0.104215i 0.00430878i
\(586\) 0 0
\(587\) 23.2472 0.959514 0.479757 0.877401i \(-0.340725\pi\)
0.479757 + 0.877401i \(0.340725\pi\)
\(588\) 0 0
\(589\) 5.03106i 0.207301i
\(590\) 0 0
\(591\) −18.2166 −0.749330
\(592\) 0 0
\(593\) −39.9531 −1.64068 −0.820338 0.571879i \(-0.806215\pi\)
−0.820338 + 0.571879i \(0.806215\pi\)
\(594\) 0 0
\(595\) −0.248833 + 0.715680i −0.0102012 + 0.0293400i
\(596\) 0 0
\(597\) 2.68193 0.109764
\(598\) 0 0
\(599\) 46.7706 1.91100 0.955498 0.294998i \(-0.0953189\pi\)
0.955498 + 0.294998i \(0.0953189\pi\)
\(600\) 0 0
\(601\) 12.0432i 0.491252i 0.969365 + 0.245626i \(0.0789935\pi\)
−0.969365 + 0.245626i \(0.921007\pi\)
\(602\) 0 0
\(603\) −1.62039 −0.0659875
\(604\) 0 0
\(605\) 2.43599i 0.0990370i
\(606\) 0 0
\(607\) 10.1246i 0.410945i 0.978663 + 0.205472i \(0.0658730\pi\)
−0.978663 + 0.205472i \(0.934127\pi\)
\(608\) 0 0
\(609\) 10.9286i 0.442850i
\(610\) 0 0
\(611\) 0.359581 0.0145471
\(612\) 0 0
\(613\) 3.63880 0.146970 0.0734848 0.997296i \(-0.476588\pi\)
0.0734848 + 0.997296i \(0.476588\pi\)
\(614\) 0 0
\(615\) 0.678061i 0.0273421i
\(616\) 0 0
\(617\) 17.4331i 0.701829i −0.936407 0.350915i \(-0.885871\pi\)
0.936407 0.350915i \(-0.114129\pi\)
\(618\) 0 0
\(619\) 12.8289i 0.515638i −0.966193 0.257819i \(-0.916996\pi\)
0.966193 0.257819i \(-0.0830038\pi\)
\(620\) 0 0
\(621\) 12.1751 0.488571
\(622\) 0 0
\(623\) 0.852924i 0.0341717i
\(624\) 0 0
\(625\) 23.7900 0.951599
\(626\) 0 0
\(627\) −2.41273 −0.0963551
\(628\) 0 0
\(629\) −23.7355 8.25256i −0.946398 0.329051i
\(630\) 0 0
\(631\) 22.7859 0.907091 0.453546 0.891233i \(-0.350159\pi\)
0.453546 + 0.891233i \(0.350159\pi\)
\(632\) 0 0
\(633\) 10.9068 0.433507
\(634\) 0 0
\(635\) 0.502556i 0.0199433i
\(636\) 0 0
\(637\) 7.40606 0.293439
\(638\) 0 0
\(639\) 4.34861i 0.172028i
\(640\) 0 0
\(641\) 28.0217i 1.10679i 0.832919 + 0.553396i \(0.186668\pi\)
−0.832919 + 0.553396i \(0.813332\pi\)
\(642\) 0 0
\(643\) 19.2975i 0.761018i −0.924777 0.380509i \(-0.875749\pi\)
0.924777 0.380509i \(-0.124251\pi\)
\(644\) 0 0
\(645\) −1.67900 −0.0661104
\(646\) 0 0
\(647\) −19.3721 −0.761597 −0.380798 0.924658i \(-0.624351\pi\)
−0.380798 + 0.924658i \(0.624351\pi\)
\(648\) 0 0
\(649\) 1.56411i 0.0613968i
\(650\) 0 0
\(651\) 6.99836i 0.274288i
\(652\) 0 0
\(653\) 11.8464i 0.463587i −0.972765 0.231794i \(-0.925541\pi\)
0.972765 0.231794i \(-0.0744594\pi\)
\(654\) 0 0
\(655\) 2.72759 0.106576
\(656\) 0 0
\(657\) 4.31656i 0.168405i
\(658\) 0 0
\(659\) −11.9559 −0.465737 −0.232869 0.972508i \(-0.574811\pi\)
−0.232869 + 0.972508i \(0.574811\pi\)
\(660\) 0 0
\(661\) 18.9669 0.737727 0.368863 0.929484i \(-0.379747\pi\)
0.368863 + 0.929484i \(0.379747\pi\)
\(662\) 0 0
\(663\) −7.98880 2.77761i −0.310259 0.107873i
\(664\) 0 0
\(665\) 0.155453 0.00602823
\(666\) 0 0
\(667\) −23.1839 −0.897684
\(668\) 0 0
\(669\) 50.5860i 1.95577i
\(670\) 0 0
\(671\) −7.12377 −0.275010
\(672\) 0 0
\(673\) 25.6384i 0.988287i 0.869380 + 0.494144i \(0.164518\pi\)
−0.869380 + 0.494144i \(0.835482\pi\)
\(674\) 0 0
\(675\) 23.9915i 0.923435i
\(676\) 0 0
\(677\) 5.12360i 0.196916i 0.995141 + 0.0984580i \(0.0313910\pi\)
−0.995141 + 0.0984580i \(0.968609\pi\)
\(678\) 0 0
\(679\) −4.10473 −0.157525
\(680\) 0 0
\(681\) −21.2870 −0.815718
\(682\) 0 0
\(683\) 23.8909i 0.914162i 0.889425 + 0.457081i \(0.151105\pi\)
−0.889425 + 0.457081i \(0.848895\pi\)
\(684\) 0 0
\(685\) 1.87800i 0.0717546i
\(686\) 0 0
\(687\) 13.2300i 0.504758i
\(688\) 0 0
\(689\) −3.45957 −0.131799
\(690\) 0 0
\(691\) 4.91390i 0.186934i −0.995622 0.0934668i \(-0.970205\pi\)
0.995622 0.0934668i \(-0.0297949\pi\)
\(692\) 0 0
\(693\) −0.328319 −0.0124718
\(694\) 0 0
\(695\) −6.35937 −0.241225
\(696\) 0 0
\(697\) −5.08474 1.76790i −0.192598 0.0669640i
\(698\) 0 0
\(699\) 24.6630 0.932839
\(700\) 0 0
\(701\) 35.7046 1.34855 0.674273 0.738482i \(-0.264457\pi\)
0.674273 + 0.738482i \(0.264457\pi\)
\(702\) 0 0
\(703\) 5.15561i 0.194448i
\(704\) 0 0
\(705\) −0.166004 −0.00625207
\(706\) 0 0
\(707\) 1.18581i 0.0445969i
\(708\) 0 0
\(709\) 6.96839i 0.261703i −0.991402 0.130852i \(-0.958229\pi\)
0.991402 0.130852i \(-0.0417712\pi\)
\(710\) 0 0
\(711\) 0.539793i 0.0202438i
\(712\) 0 0
\(713\) 14.8463 0.555998
\(714\) 0 0
\(715\) 0.501094 0.0187399
\(716\) 0 0
\(717\) 5.28023i 0.197194i
\(718\) 0 0
\(719\) 21.3544i 0.796383i −0.917302 0.398192i \(-0.869638\pi\)
0.917302 0.398192i \(-0.130362\pi\)
\(720\) 0 0
\(721\) 6.23801i 0.232316i
\(722\) 0 0
\(723\) 18.6940 0.695237
\(724\) 0 0
\(725\) 45.6847i 1.69669i
\(726\) 0 0
\(727\) 17.0508 0.632378 0.316189 0.948696i \(-0.397597\pi\)
0.316189 + 0.948696i \(0.397597\pi\)
\(728\) 0 0
\(729\) −23.4718 −0.869327
\(730\) 0 0
\(731\) 4.37763 12.5907i 0.161912 0.465683i
\(732\) 0 0
\(733\) 32.8717 1.21414 0.607072 0.794647i \(-0.292344\pi\)
0.607072 + 0.794647i \(0.292344\pi\)
\(734\) 0 0
\(735\) −3.41908 −0.126115
\(736\) 0 0
\(737\) 7.79126i 0.286995i
\(738\) 0 0
\(739\) 9.33927 0.343551 0.171775 0.985136i \(-0.445050\pi\)
0.171775 + 0.985136i \(0.445050\pi\)
\(740\) 0 0
\(741\) 1.73525i 0.0637461i
\(742\) 0 0
\(743\) 13.7678i 0.505092i 0.967585 + 0.252546i \(0.0812679\pi\)
−0.967585 + 0.252546i \(0.918732\pi\)
\(744\) 0 0
\(745\) 5.38551i 0.197310i
\(746\) 0 0
\(747\) −1.50332 −0.0550037
\(748\) 0 0
\(749\) 10.2259 0.373648
\(750\) 0 0
\(751\) 17.6064i 0.642467i 0.947000 + 0.321233i \(0.104097\pi\)
−0.947000 + 0.321233i \(0.895903\pi\)
\(752\) 0 0
\(753\) 24.4034i 0.889308i
\(754\) 0 0
\(755\) 2.62295i 0.0954590i
\(756\) 0 0
\(757\) 22.4997 0.817764 0.408882 0.912587i \(-0.365919\pi\)
0.408882 + 0.912587i \(0.365919\pi\)
\(758\) 0 0
\(759\) 7.11978i 0.258432i
\(760\) 0 0
\(761\) 24.3751 0.883597 0.441798 0.897114i \(-0.354341\pi\)
0.441798 + 0.897114i \(0.354341\pi\)
\(762\) 0 0
\(763\) 2.18714 0.0791796
\(764\) 0 0
\(765\) 0.360789 + 0.125442i 0.0130444 + 0.00453536i
\(766\) 0 0
\(767\) 1.12492 0.0406186
\(768\) 0 0
\(769\) 29.6146 1.06793 0.533965 0.845507i \(-0.320702\pi\)
0.533965 + 0.845507i \(0.320702\pi\)
\(770\) 0 0
\(771\) 40.4585i 1.45708i
\(772\) 0 0
\(773\) 48.6428 1.74956 0.874781 0.484519i \(-0.161005\pi\)
0.874781 + 0.484519i \(0.161005\pi\)
\(774\) 0 0
\(775\) 29.2551i 1.05088i
\(776\) 0 0
\(777\) 7.17162i 0.257280i
\(778\) 0 0
\(779\) 1.10446i 0.0395714i
\(780\) 0 0
\(781\) 20.9092 0.748190
\(782\) 0 0
\(783\) 45.2996 1.61887
\(784\) 0 0
\(785\) 3.71253i 0.132506i
\(786\) 0 0
\(787\) 13.3538i 0.476012i −0.971264 0.238006i \(-0.923506\pi\)
0.971264 0.238006i \(-0.0764938\pi\)
\(788\) 0 0
\(789\) 3.87198i 0.137846i
\(790\) 0 0
\(791\) −9.85328 −0.350342
\(792\) 0 0
\(793\) 5.12347i 0.181940i