Properties

Label 4012.2.b.b.237.35
Level 4012
Weight 2
Character 4012.237
Analytic conductor 32.036
Analytic rank 0
Dimension 46
CM no
Inner twists 2

Related objects

Downloads

Learn more about

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 237.35
Character \(\chi\) = 4012.237
Dual form 4012.2.b.b.237.12

$q$-expansion

\(f(q)\) \(=\) \(q+1.77972i q^{3} -1.91545i q^{5} -2.45686i q^{7} -0.167415 q^{9} +O(q^{10})\) \(q+1.77972i q^{3} -1.91545i q^{5} -2.45686i q^{7} -0.167415 q^{9} +1.63017i q^{11} +2.53859 q^{13} +3.40896 q^{15} +(-3.23137 - 2.56091i) q^{17} +2.17325 q^{19} +4.37253 q^{21} +2.41510i q^{23} +1.33107 q^{25} +5.04122i q^{27} -4.33492i q^{29} -5.11391i q^{31} -2.90125 q^{33} -4.70599 q^{35} -4.50693i q^{37} +4.51800i q^{39} +4.47471i q^{41} -4.13699 q^{43} +0.320674i q^{45} +11.7078 q^{47} +0.963830 q^{49} +(4.55771 - 5.75094i) q^{51} -14.4131 q^{53} +3.12250 q^{55} +3.86779i q^{57} +1.00000 q^{59} +3.02775i q^{61} +0.411315i q^{63} -4.86254i q^{65} +6.34987 q^{67} -4.29820 q^{69} -6.97310i q^{71} -9.10446i q^{73} +2.36893i q^{75} +4.00511 q^{77} +4.05009i q^{79} -9.47422 q^{81} -14.8812 q^{83} +(-4.90529 + 6.18951i) q^{85} +7.71496 q^{87} +5.92912 q^{89} -6.23698i q^{91} +9.10134 q^{93} -4.16275i q^{95} -11.7962i q^{97} -0.272914i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46q - 54q^{9} + O(q^{10}) \) \( 46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100}) \)

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
\(3\) 1.77972i 1.02752i 0.857933 + 0.513762i \(0.171748\pi\)
−0.857933 + 0.513762i \(0.828252\pi\)
\(4\) 0 0
\(5\) 1.91545i 0.856614i −0.903633 0.428307i \(-0.859110\pi\)
0.903633 0.428307i \(-0.140890\pi\)
\(6\) 0 0
\(7\) 2.45686i 0.928607i −0.885676 0.464303i \(-0.846305\pi\)
0.885676 0.464303i \(-0.153695\pi\)
\(8\) 0 0
\(9\) −0.167415 −0.0558049
\(10\) 0 0
\(11\) 1.63017i 0.491515i 0.969331 + 0.245758i \(0.0790367\pi\)
−0.969331 + 0.245758i \(0.920963\pi\)
\(12\) 0 0
\(13\) 2.53859 0.704079 0.352040 0.935985i \(-0.385488\pi\)
0.352040 + 0.935985i \(0.385488\pi\)
\(14\) 0 0
\(15\) 3.40896 0.880191
\(16\) 0 0
\(17\) −3.23137 2.56091i −0.783722 0.621112i
\(18\) 0 0
\(19\) 2.17325 0.498579 0.249289 0.968429i \(-0.419803\pi\)
0.249289 + 0.968429i \(0.419803\pi\)
\(20\) 0 0
\(21\) 4.37253 0.954165
\(22\) 0 0
\(23\) 2.41510i 0.503582i 0.967782 + 0.251791i \(0.0810196\pi\)
−0.967782 + 0.251791i \(0.918980\pi\)
\(24\) 0 0
\(25\) 1.33107 0.266213
\(26\) 0 0
\(27\) 5.04122i 0.970183i
\(28\) 0 0
\(29\) 4.33492i 0.804974i −0.915426 0.402487i \(-0.868146\pi\)
0.915426 0.402487i \(-0.131854\pi\)
\(30\) 0 0
\(31\) 5.11391i 0.918485i −0.888311 0.459242i \(-0.848121\pi\)
0.888311 0.459242i \(-0.151879\pi\)
\(32\) 0 0
\(33\) −2.90125 −0.505043
\(34\) 0 0
\(35\) −4.70599 −0.795457
\(36\) 0 0
\(37\) 4.50693i 0.740935i −0.928845 0.370467i \(-0.879198\pi\)
0.928845 0.370467i \(-0.120802\pi\)
\(38\) 0 0
\(39\) 4.51800i 0.723458i
\(40\) 0 0
\(41\) 4.47471i 0.698833i 0.936968 + 0.349416i \(0.113620\pi\)
−0.936968 + 0.349416i \(0.886380\pi\)
\(42\) 0 0
\(43\) −4.13699 −0.630884 −0.315442 0.948945i \(-0.602153\pi\)
−0.315442 + 0.948945i \(0.602153\pi\)
\(44\) 0 0
\(45\) 0.320674i 0.0478032i
\(46\) 0 0
\(47\) 11.7078 1.70775 0.853876 0.520476i \(-0.174246\pi\)
0.853876 + 0.520476i \(0.174246\pi\)
\(48\) 0 0
\(49\) 0.963830 0.137690
\(50\) 0 0
\(51\) 4.55771 5.75094i 0.638208 0.805293i
\(52\) 0 0
\(53\) −14.4131 −1.97979 −0.989897 0.141791i \(-0.954714\pi\)
−0.989897 + 0.141791i \(0.954714\pi\)
\(54\) 0 0
\(55\) 3.12250 0.421038
\(56\) 0 0
\(57\) 3.86779i 0.512301i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 3.02775i 0.387664i 0.981035 + 0.193832i \(0.0620917\pi\)
−0.981035 + 0.193832i \(0.937908\pi\)
\(62\) 0 0
\(63\) 0.411315i 0.0518208i
\(64\) 0 0
\(65\) 4.86254i 0.603124i
\(66\) 0 0
\(67\) 6.34987 0.775760 0.387880 0.921710i \(-0.373208\pi\)
0.387880 + 0.921710i \(0.373208\pi\)
\(68\) 0 0
\(69\) −4.29820 −0.517443
\(70\) 0 0
\(71\) 6.97310i 0.827555i −0.910378 0.413778i \(-0.864209\pi\)
0.910378 0.413778i \(-0.135791\pi\)
\(72\) 0 0
\(73\) 9.10446i 1.06560i −0.846242 0.532798i \(-0.821140\pi\)
0.846242 0.532798i \(-0.178860\pi\)
\(74\) 0 0
\(75\) 2.36893i 0.273540i
\(76\) 0 0
\(77\) 4.00511 0.456424
\(78\) 0 0
\(79\) 4.05009i 0.455670i 0.973700 + 0.227835i \(0.0731647\pi\)
−0.973700 + 0.227835i \(0.926835\pi\)
\(80\) 0 0
\(81\) −9.47422 −1.05269
\(82\) 0 0
\(83\) −14.8812 −1.63342 −0.816712 0.577045i \(-0.804206\pi\)
−0.816712 + 0.577045i \(0.804206\pi\)
\(84\) 0 0
\(85\) −4.90529 + 6.18951i −0.532053 + 0.671347i
\(86\) 0 0
\(87\) 7.71496 0.827130
\(88\) 0 0
\(89\) 5.92912 0.628485 0.314243 0.949343i \(-0.398249\pi\)
0.314243 + 0.949343i \(0.398249\pi\)
\(90\) 0 0
\(91\) 6.23698i 0.653813i
\(92\) 0 0
\(93\) 9.10134 0.943765
\(94\) 0 0
\(95\) 4.16275i 0.427089i
\(96\) 0 0
\(97\) 11.7962i 1.19773i −0.800851 0.598863i \(-0.795619\pi\)
0.800851 0.598863i \(-0.204381\pi\)
\(98\) 0 0
\(99\) 0.272914i 0.0274289i
\(100\) 0 0
\(101\) 8.17839 0.813780 0.406890 0.913477i \(-0.366613\pi\)
0.406890 + 0.913477i \(0.366613\pi\)
\(102\) 0 0
\(103\) 9.16265 0.902822 0.451411 0.892316i \(-0.350921\pi\)
0.451411 + 0.892316i \(0.350921\pi\)
\(104\) 0 0
\(105\) 8.37535i 0.817351i
\(106\) 0 0
\(107\) 10.0224i 0.968904i −0.874818 0.484452i \(-0.839019\pi\)
0.874818 0.484452i \(-0.160981\pi\)
\(108\) 0 0
\(109\) 10.4574i 1.00164i −0.865553 0.500818i \(-0.833033\pi\)
0.865553 0.500818i \(-0.166967\pi\)
\(110\) 0 0
\(111\) 8.02109 0.761328
\(112\) 0 0
\(113\) 0.100302i 0.00943565i 0.999989 + 0.00471783i \(0.00150174\pi\)
−0.999989 + 0.00471783i \(0.998498\pi\)
\(114\) 0 0
\(115\) 4.62599 0.431375
\(116\) 0 0
\(117\) −0.424998 −0.0392911
\(118\) 0 0
\(119\) −6.29181 + 7.93902i −0.576769 + 0.727769i
\(120\) 0 0
\(121\) 8.34254 0.758413
\(122\) 0 0
\(123\) −7.96375 −0.718067
\(124\) 0 0
\(125\) 12.1268i 1.08466i
\(126\) 0 0
\(127\) 7.30255 0.647997 0.323998 0.946058i \(-0.394973\pi\)
0.323998 + 0.946058i \(0.394973\pi\)
\(128\) 0 0
\(129\) 7.36269i 0.648249i
\(130\) 0 0
\(131\) 18.0951i 1.58098i −0.612475 0.790490i \(-0.709826\pi\)
0.612475 0.790490i \(-0.290174\pi\)
\(132\) 0 0
\(133\) 5.33938i 0.462983i
\(134\) 0 0
\(135\) 9.65618 0.831072
\(136\) 0 0
\(137\) −16.0065 −1.36752 −0.683762 0.729705i \(-0.739657\pi\)
−0.683762 + 0.729705i \(0.739657\pi\)
\(138\) 0 0
\(139\) 15.4708i 1.31222i −0.754667 0.656108i \(-0.772201\pi\)
0.754667 0.656108i \(-0.227799\pi\)
\(140\) 0 0
\(141\) 20.8366i 1.75476i
\(142\) 0 0
\(143\) 4.13834i 0.346066i
\(144\) 0 0
\(145\) −8.30330 −0.689552
\(146\) 0 0
\(147\) 1.71535i 0.141480i
\(148\) 0 0
\(149\) 9.70814 0.795322 0.397661 0.917532i \(-0.369822\pi\)
0.397661 + 0.917532i \(0.369822\pi\)
\(150\) 0 0
\(151\) −14.5573 −1.18466 −0.592330 0.805696i \(-0.701792\pi\)
−0.592330 + 0.805696i \(0.701792\pi\)
\(152\) 0 0
\(153\) 0.540978 + 0.428734i 0.0437355 + 0.0346611i
\(154\) 0 0
\(155\) −9.79541 −0.786787
\(156\) 0 0
\(157\) 18.0953 1.44416 0.722080 0.691810i \(-0.243186\pi\)
0.722080 + 0.691810i \(0.243186\pi\)
\(158\) 0 0
\(159\) 25.6514i 2.03428i
\(160\) 0 0
\(161\) 5.93356 0.467630
\(162\) 0 0
\(163\) 9.15635i 0.717180i 0.933495 + 0.358590i \(0.116742\pi\)
−0.933495 + 0.358590i \(0.883258\pi\)
\(164\) 0 0
\(165\) 5.55719i 0.432627i
\(166\) 0 0
\(167\) 22.3032i 1.72587i 0.505314 + 0.862935i \(0.331377\pi\)
−0.505314 + 0.862935i \(0.668623\pi\)
\(168\) 0 0
\(169\) −6.55554 −0.504272
\(170\) 0 0
\(171\) −0.363834 −0.0278231
\(172\) 0 0
\(173\) 6.25558i 0.475603i −0.971314 0.237802i \(-0.923573\pi\)
0.971314 0.237802i \(-0.0764268\pi\)
\(174\) 0 0
\(175\) 3.27025i 0.247207i
\(176\) 0 0
\(177\) 1.77972i 0.133772i
\(178\) 0 0
\(179\) 24.4193 1.82518 0.912590 0.408875i \(-0.134079\pi\)
0.912590 + 0.408875i \(0.134079\pi\)
\(180\) 0 0
\(181\) 13.2410i 0.984194i −0.870540 0.492097i \(-0.836230\pi\)
0.870540 0.492097i \(-0.163770\pi\)
\(182\) 0 0
\(183\) −5.38856 −0.398334
\(184\) 0 0
\(185\) −8.63278 −0.634695
\(186\) 0 0
\(187\) 4.17472 5.26768i 0.305286 0.385211i
\(188\) 0 0
\(189\) 12.3856 0.900918
\(190\) 0 0
\(191\) −8.22406 −0.595072 −0.297536 0.954711i \(-0.596165\pi\)
−0.297536 + 0.954711i \(0.596165\pi\)
\(192\) 0 0
\(193\) 4.01119i 0.288732i −0.989524 0.144366i \(-0.953886\pi\)
0.989524 0.144366i \(-0.0461142\pi\)
\(194\) 0 0
\(195\) 8.65398 0.619724
\(196\) 0 0
\(197\) 3.07376i 0.218996i −0.993987 0.109498i \(-0.965076\pi\)
0.993987 0.109498i \(-0.0349244\pi\)
\(198\) 0 0
\(199\) 9.20338i 0.652411i 0.945299 + 0.326205i \(0.105770\pi\)
−0.945299 + 0.326205i \(0.894230\pi\)
\(200\) 0 0
\(201\) 11.3010i 0.797112i
\(202\) 0 0
\(203\) −10.6503 −0.747504
\(204\) 0 0
\(205\) 8.57107 0.598629
\(206\) 0 0
\(207\) 0.404322i 0.0281023i
\(208\) 0 0
\(209\) 3.54278i 0.245059i
\(210\) 0 0
\(211\) 18.7010i 1.28743i −0.765265 0.643715i \(-0.777392\pi\)
0.765265 0.643715i \(-0.222608\pi\)
\(212\) 0 0
\(213\) 12.4102 0.850333
\(214\) 0 0
\(215\) 7.92417i 0.540424i
\(216\) 0 0
\(217\) −12.5642 −0.852911
\(218\) 0 0
\(219\) 16.2034 1.09493
\(220\) 0 0
\(221\) −8.20313 6.50112i −0.551802 0.437312i
\(222\) 0 0
\(223\) 17.6784 1.18383 0.591917 0.805999i \(-0.298371\pi\)
0.591917 + 0.805999i \(0.298371\pi\)
\(224\) 0 0
\(225\) −0.222840 −0.0148560
\(226\) 0 0
\(227\) 19.7808i 1.31290i 0.754369 + 0.656450i \(0.227943\pi\)
−0.754369 + 0.656450i \(0.772057\pi\)
\(228\) 0 0
\(229\) −18.4331 −1.21809 −0.609046 0.793135i \(-0.708448\pi\)
−0.609046 + 0.793135i \(0.708448\pi\)
\(230\) 0 0
\(231\) 7.12798i 0.468987i
\(232\) 0 0
\(233\) 13.8125i 0.904884i −0.891794 0.452442i \(-0.850553\pi\)
0.891794 0.452442i \(-0.149447\pi\)
\(234\) 0 0
\(235\) 22.4256i 1.46288i
\(236\) 0 0
\(237\) −7.20803 −0.468212
\(238\) 0 0
\(239\) 18.4220 1.19162 0.595810 0.803125i \(-0.296831\pi\)
0.595810 + 0.803125i \(0.296831\pi\)
\(240\) 0 0
\(241\) 24.5213i 1.57955i 0.613394 + 0.789777i \(0.289804\pi\)
−0.613394 + 0.789777i \(0.710196\pi\)
\(242\) 0 0
\(243\) 1.73783i 0.111482i
\(244\) 0 0
\(245\) 1.84616i 0.117947i
\(246\) 0 0
\(247\) 5.51701 0.351039
\(248\) 0 0
\(249\) 26.4844i 1.67838i
\(250\) 0 0
\(251\) 17.5505 1.10777 0.553887 0.832592i \(-0.313144\pi\)
0.553887 + 0.832592i \(0.313144\pi\)
\(252\) 0 0
\(253\) −3.93702 −0.247518
\(254\) 0 0
\(255\) −11.0156 8.73005i −0.689824 0.546697i
\(256\) 0 0
\(257\) −28.3471 −1.76825 −0.884123 0.467255i \(-0.845243\pi\)
−0.884123 + 0.467255i \(0.845243\pi\)
\(258\) 0 0
\(259\) −11.0729 −0.688037
\(260\) 0 0
\(261\) 0.725729i 0.0449215i
\(262\) 0 0
\(263\) 27.9897 1.72592 0.862961 0.505271i \(-0.168607\pi\)
0.862961 + 0.505271i \(0.168607\pi\)
\(264\) 0 0
\(265\) 27.6075i 1.69592i
\(266\) 0 0
\(267\) 10.5522i 0.645784i
\(268\) 0 0
\(269\) 6.44275i 0.392821i −0.980522 0.196411i \(-0.937071\pi\)
0.980522 0.196411i \(-0.0629285\pi\)
\(270\) 0 0
\(271\) 27.9996 1.70085 0.850426 0.526094i \(-0.176344\pi\)
0.850426 + 0.526094i \(0.176344\pi\)
\(272\) 0 0
\(273\) 11.1001 0.671808
\(274\) 0 0
\(275\) 2.16987i 0.130848i
\(276\) 0 0
\(277\) 7.48484i 0.449720i −0.974391 0.224860i \(-0.927807\pi\)
0.974391 0.224860i \(-0.0721925\pi\)
\(278\) 0 0
\(279\) 0.856143i 0.0512559i
\(280\) 0 0
\(281\) 0.510434 0.0304499 0.0152250 0.999884i \(-0.495154\pi\)
0.0152250 + 0.999884i \(0.495154\pi\)
\(282\) 0 0
\(283\) 31.2509i 1.85767i −0.370489 0.928837i \(-0.620810\pi\)
0.370489 0.928837i \(-0.379190\pi\)
\(284\) 0 0
\(285\) 7.40854 0.438844
\(286\) 0 0
\(287\) 10.9937 0.648940
\(288\) 0 0
\(289\) 3.88347 + 16.5505i 0.228439 + 0.973558i
\(290\) 0 0
\(291\) 20.9940 1.23069
\(292\) 0 0
\(293\) −11.2697 −0.658383 −0.329192 0.944263i \(-0.606776\pi\)
−0.329192 + 0.944263i \(0.606776\pi\)
\(294\) 0 0
\(295\) 1.91545i 0.111522i
\(296\) 0 0
\(297\) −8.21805 −0.476859
\(298\) 0 0
\(299\) 6.13095i 0.354562i
\(300\) 0 0
\(301\) 10.1640i 0.585843i
\(302\) 0 0
\(303\) 14.5553i 0.836178i
\(304\) 0 0
\(305\) 5.79950 0.332078
\(306\) 0 0
\(307\) −2.78761 −0.159097 −0.0795486 0.996831i \(-0.525348\pi\)
−0.0795486 + 0.996831i \(0.525348\pi\)
\(308\) 0 0
\(309\) 16.3070i 0.927671i
\(310\) 0 0
\(311\) 27.7573i 1.57397i 0.616969 + 0.786987i \(0.288360\pi\)
−0.616969 + 0.786987i \(0.711640\pi\)
\(312\) 0 0
\(313\) 5.64200i 0.318905i 0.987206 + 0.159452i \(0.0509728\pi\)
−0.987206 + 0.159452i \(0.949027\pi\)
\(314\) 0 0
\(315\) 0.787851 0.0443904
\(316\) 0 0
\(317\) 5.43933i 0.305503i 0.988265 + 0.152752i \(0.0488134\pi\)
−0.988265 + 0.152752i \(0.951187\pi\)
\(318\) 0 0
\(319\) 7.06666 0.395657
\(320\) 0 0
\(321\) 17.8371 0.995572
\(322\) 0 0
\(323\) −7.02258 5.56551i −0.390747 0.309673i
\(324\) 0 0
\(325\) 3.37904 0.187435
\(326\) 0 0
\(327\) 18.6113 1.02920
\(328\) 0 0
\(329\) 28.7644i 1.58583i
\(330\) 0 0
\(331\) 7.87894 0.433066 0.216533 0.976275i \(-0.430525\pi\)
0.216533 + 0.976275i \(0.430525\pi\)
\(332\) 0 0
\(333\) 0.754526i 0.0413478i
\(334\) 0 0
\(335\) 12.1628i 0.664526i
\(336\) 0 0
\(337\) 33.0543i 1.80058i 0.435288 + 0.900291i \(0.356647\pi\)
−0.435288 + 0.900291i \(0.643353\pi\)
\(338\) 0 0
\(339\) −0.178510 −0.00969536
\(340\) 0 0
\(341\) 8.33654 0.451449
\(342\) 0 0
\(343\) 19.5660i 1.05647i
\(344\) 0 0
\(345\) 8.23297i 0.443248i
\(346\) 0 0
\(347\) 36.4162i 1.95492i 0.211110 + 0.977462i \(0.432292\pi\)
−0.211110 + 0.977462i \(0.567708\pi\)
\(348\) 0 0
\(349\) −13.1050 −0.701496 −0.350748 0.936470i \(-0.614073\pi\)
−0.350748 + 0.936470i \(0.614073\pi\)
\(350\) 0 0
\(351\) 12.7976i 0.683086i
\(352\) 0 0
\(353\) −4.53985 −0.241632 −0.120816 0.992675i \(-0.538551\pi\)
−0.120816 + 0.992675i \(0.538551\pi\)
\(354\) 0 0
\(355\) −13.3566 −0.708895
\(356\) 0 0
\(357\) −14.1293 11.1977i −0.747800 0.592644i
\(358\) 0 0
\(359\) −10.2595 −0.541475 −0.270738 0.962653i \(-0.587268\pi\)
−0.270738 + 0.962653i \(0.587268\pi\)
\(360\) 0 0
\(361\) −14.2770 −0.751419
\(362\) 0 0
\(363\) 14.8474i 0.779287i
\(364\) 0 0
\(365\) −17.4391 −0.912804
\(366\) 0 0
\(367\) 15.5045i 0.809331i 0.914465 + 0.404665i \(0.132612\pi\)
−0.914465 + 0.404665i \(0.867388\pi\)
\(368\) 0 0
\(369\) 0.749132i 0.0389983i
\(370\) 0 0
\(371\) 35.4110i 1.83845i
\(372\) 0 0
\(373\) 11.6689 0.604192 0.302096 0.953277i \(-0.402314\pi\)
0.302096 + 0.953277i \(0.402314\pi\)
\(374\) 0 0
\(375\) 21.5824 1.11451
\(376\) 0 0
\(377\) 11.0046i 0.566766i
\(378\) 0 0
\(379\) 11.1980i 0.575203i 0.957750 + 0.287601i \(0.0928579\pi\)
−0.957750 + 0.287601i \(0.907142\pi\)
\(380\) 0 0
\(381\) 12.9965i 0.665832i
\(382\) 0 0
\(383\) −1.55995 −0.0797095 −0.0398548 0.999205i \(-0.512690\pi\)
−0.0398548 + 0.999205i \(0.512690\pi\)
\(384\) 0 0
\(385\) 7.67156i 0.390979i
\(386\) 0 0
\(387\) 0.692592 0.0352064
\(388\) 0 0
\(389\) 21.5522 1.09274 0.546369 0.837545i \(-0.316010\pi\)
0.546369 + 0.837545i \(0.316010\pi\)
\(390\) 0 0
\(391\) 6.18485 7.80406i 0.312781 0.394668i
\(392\) 0 0
\(393\) 32.2043 1.62449
\(394\) 0 0
\(395\) 7.75772 0.390333
\(396\) 0 0
\(397\) 3.56484i 0.178914i −0.995991 0.0894570i \(-0.971487\pi\)
0.995991 0.0894570i \(-0.0285132\pi\)
\(398\) 0 0
\(399\) 9.50263 0.475726
\(400\) 0 0
\(401\) 37.3067i 1.86301i 0.363729 + 0.931505i \(0.381503\pi\)
−0.363729 + 0.931505i \(0.618497\pi\)
\(402\) 0 0
\(403\) 12.9821i 0.646686i
\(404\) 0 0
\(405\) 18.1474i 0.901749i
\(406\) 0 0
\(407\) 7.34707 0.364181
\(408\) 0 0
\(409\) 19.3836 0.958457 0.479228 0.877690i \(-0.340917\pi\)
0.479228 + 0.877690i \(0.340917\pi\)
\(410\) 0 0
\(411\) 28.4871i 1.40516i
\(412\) 0 0
\(413\) 2.45686i 0.120894i
\(414\) 0 0
\(415\) 28.5041i 1.39921i
\(416\) 0 0
\(417\) 27.5338 1.34833
\(418\) 0 0
\(419\) 5.51292i 0.269324i −0.990892 0.134662i \(-0.957005\pi\)
0.990892 0.134662i \(-0.0429948\pi\)
\(420\) 0 0
\(421\) −13.7120 −0.668282 −0.334141 0.942523i \(-0.608446\pi\)
−0.334141 + 0.942523i \(0.608446\pi\)
\(422\) 0 0
\(423\) −1.96005 −0.0953009
\(424\) 0 0
\(425\) −4.30116 3.40874i −0.208637 0.165348i
\(426\) 0 0
\(427\) 7.43877 0.359988
\(428\) 0 0
\(429\) −7.36511 −0.355591
\(430\) 0 0
\(431\) 10.4744i 0.504533i −0.967658 0.252267i \(-0.918824\pi\)
0.967658 0.252267i \(-0.0811760\pi\)
\(432\) 0 0
\(433\) −17.6141 −0.846481 −0.423240 0.906017i \(-0.639107\pi\)
−0.423240 + 0.906017i \(0.639107\pi\)
\(434\) 0 0
\(435\) 14.7776i 0.708531i
\(436\) 0 0
\(437\) 5.24862i 0.251075i
\(438\) 0 0
\(439\) 21.4576i 1.02412i 0.858951 + 0.512059i \(0.171117\pi\)
−0.858951 + 0.512059i \(0.828883\pi\)
\(440\) 0 0
\(441\) −0.161359 −0.00768377
\(442\) 0 0
\(443\) 23.5184 1.11739 0.558696 0.829372i \(-0.311302\pi\)
0.558696 + 0.829372i \(0.311302\pi\)
\(444\) 0 0
\(445\) 11.3569i 0.538369i
\(446\) 0 0
\(447\) 17.2778i 0.817212i
\(448\) 0 0
\(449\) 37.1905i 1.75513i −0.479461 0.877563i \(-0.659168\pi\)
0.479461 0.877563i \(-0.340832\pi\)
\(450\) 0 0
\(451\) −7.29455 −0.343487
\(452\) 0 0
\(453\) 25.9080i 1.21727i
\(454\) 0 0
\(455\) −11.9466 −0.560065
\(456\) 0 0
\(457\) 14.0684 0.658094 0.329047 0.944314i \(-0.393273\pi\)
0.329047 + 0.944314i \(0.393273\pi\)
\(458\) 0 0
\(459\) 12.9101 16.2900i 0.602592 0.760353i
\(460\) 0 0
\(461\) 25.6177 1.19313 0.596567 0.802564i \(-0.296531\pi\)
0.596567 + 0.802564i \(0.296531\pi\)
\(462\) 0 0
\(463\) −14.8828 −0.691663 −0.345832 0.938297i \(-0.612403\pi\)
−0.345832 + 0.938297i \(0.612403\pi\)
\(464\) 0 0
\(465\) 17.4331i 0.808442i
\(466\) 0 0
\(467\) −16.1674 −0.748139 −0.374070 0.927401i \(-0.622038\pi\)
−0.374070 + 0.927401i \(0.622038\pi\)
\(468\) 0 0
\(469\) 15.6008i 0.720376i
\(470\) 0 0
\(471\) 32.2046i 1.48391i
\(472\) 0 0
\(473\) 6.74399i 0.310089i
\(474\) 0 0
\(475\) 2.89274 0.132728
\(476\) 0 0
\(477\) 2.41297 0.110482
\(478\) 0 0
\(479\) 31.9120i 1.45809i −0.684463 0.729047i \(-0.739963\pi\)
0.684463 0.729047i \(-0.260037\pi\)
\(480\) 0 0
\(481\) 11.4413i 0.521677i
\(482\) 0 0
\(483\) 10.5601i 0.480501i
\(484\) 0 0
\(485\) −22.5951 −1.02599
\(486\) 0 0
\(487\) 8.32478i 0.377232i −0.982051 0.188616i \(-0.939600\pi\)
0.982051 0.188616i \(-0.0604001\pi\)
\(488\) 0 0
\(489\) −16.2958 −0.736920
\(490\) 0 0
\(491\) −41.6313 −1.87880 −0.939398 0.342830i \(-0.888615\pi\)
−0.939398 + 0.342830i \(0.888615\pi\)
\(492\) 0 0
\(493\) −11.1013 + 14.0077i −0.499979 + 0.630876i
\(494\) 0 0
\(495\) −0.522753 −0.0234960
\(496\) 0 0
\(497\) −17.1320 −0.768473
\(498\) 0 0
\(499\) 44.1304i 1.97555i −0.155899 0.987773i \(-0.549827\pi\)
0.155899 0.987773i \(-0.450173\pi\)
\(500\) 0 0
\(501\) −39.6935 −1.77337
\(502\) 0 0
\(503\) 12.3377i 0.550113i −0.961428 0.275056i \(-0.911303\pi\)
0.961428 0.275056i \(-0.0886966\pi\)
\(504\) 0 0
\(505\) 15.6653i 0.697095i
\(506\) 0 0
\(507\) 11.6670i 0.518151i
\(508\) 0 0
\(509\) −33.3709 −1.47914 −0.739570 0.673080i \(-0.764971\pi\)
−0.739570 + 0.673080i \(0.764971\pi\)
\(510\) 0 0
\(511\) −22.3684 −0.989520
\(512\) 0 0
\(513\) 10.9558i 0.483712i
\(514\) 0 0
\(515\) 17.5506i 0.773370i
\(516\) 0 0
\(517\) 19.0857i 0.839386i
\(518\) 0 0
\(519\) 11.1332 0.488694
\(520\) 0 0
\(521\) 34.9617i 1.53170i −0.643019 0.765850i \(-0.722318\pi\)
0.643019 0.765850i \(-0.277682\pi\)
\(522\) 0 0
\(523\) −8.28760 −0.362391 −0.181196 0.983447i \(-0.557997\pi\)
−0.181196 + 0.983447i \(0.557997\pi\)
\(524\) 0 0
\(525\) 5.82013 0.254011
\(526\) 0 0
\(527\) −13.0963 + 16.5249i −0.570482 + 0.719836i
\(528\) 0 0
\(529\) 17.1673 0.746405
\(530\) 0 0
\(531\) −0.167415 −0.00726518
\(532\) 0 0
\(533\) 11.3595i 0.492034i
\(534\) 0 0
\(535\) −19.1974 −0.829977
\(536\) 0 0
\(537\) 43.4595i 1.87542i
\(538\) 0 0
\(539\) 1.57121i 0.0676767i
\(540\) 0 0
\(541\) 37.1610i 1.59768i 0.601547 + 0.798838i \(0.294551\pi\)
−0.601547 + 0.798838i \(0.705449\pi\)
\(542\) 0 0
\(543\) 23.5653 1.01128
\(544\) 0 0
\(545\) −20.0306 −0.858015
\(546\) 0 0
\(547\) 36.8004i 1.57347i −0.617290 0.786736i \(-0.711769\pi\)
0.617290 0.786736i \(-0.288231\pi\)
\(548\) 0 0
\(549\) 0.506890i 0.0216336i
\(550\) 0 0
\(551\) 9.42088i 0.401343i
\(552\) 0 0
\(553\) 9.95050 0.423138
\(554\) 0 0
\(555\) 15.3640i 0.652164i
\(556\) 0 0
\(557\) 39.5920 1.67757 0.838783 0.544466i \(-0.183268\pi\)
0.838783 + 0.544466i \(0.183268\pi\)
\(558\) 0 0
\(559\) −10.5021 −0.444193
\(560\) 0 0
\(561\) 9.37501 + 7.42985i 0.395813 + 0.313689i
\(562\) 0 0
\(563\) −22.7253 −0.957757 −0.478879 0.877881i \(-0.658957\pi\)
−0.478879 + 0.877881i \(0.658957\pi\)
\(564\) 0 0
\(565\) 0.192124 0.00808271
\(566\) 0 0
\(567\) 23.2768i 0.977535i
\(568\) 0 0
\(569\) −5.98876 −0.251062 −0.125531 0.992090i \(-0.540063\pi\)
−0.125531 + 0.992090i \(0.540063\pi\)
\(570\) 0 0
\(571\) 4.11101i 0.172041i −0.996293 0.0860203i \(-0.972585\pi\)
0.996293 0.0860203i \(-0.0274150\pi\)
\(572\) 0 0
\(573\) 14.6365i 0.611450i
\(574\) 0 0
\(575\) 3.21465i 0.134060i
\(576\) 0 0
\(577\) −8.01381 −0.333619 −0.166810 0.985989i \(-0.553347\pi\)
−0.166810 + 0.985989i \(0.553347\pi\)
\(578\) 0 0
\(579\) 7.13881 0.296679
\(580\) 0 0
\(581\) 36.5611i 1.51681i
\(582\) 0 0
\(583\) 23.4958i 0.973098i
\(584\) 0 0
\(585\) 0.814060i 0.0336573i
\(586\) 0 0
\(587\) −38.3009 −1.58085 −0.790423 0.612562i \(-0.790139\pi\)
−0.790423 + 0.612562i \(0.790139\pi\)
\(588\) 0 0
\(589\) 11.1138i 0.457937i
\(590\) 0 0
\(591\) 5.47044 0.225024
\(592\) 0 0
\(593\) −17.0545 −0.700345 −0.350172 0.936685i \(-0.613877\pi\)
−0.350172 + 0.936685i \(0.613877\pi\)
\(594\) 0 0
\(595\) 15.2068 + 12.0516i 0.623417 + 0.494068i
\(596\) 0 0
\(597\) −16.3795 −0.670367
\(598\) 0 0
\(599\) 35.7691 1.46149 0.730743 0.682653i \(-0.239174\pi\)
0.730743 + 0.682653i \(0.239174\pi\)
\(600\) 0 0
\(601\) 4.22983i 0.172538i 0.996272 + 0.0862692i \(0.0274945\pi\)
−0.996272 + 0.0862692i \(0.972505\pi\)
\(602\) 0 0
\(603\) −1.06306 −0.0432912
\(604\) 0 0
\(605\) 15.9797i 0.649667i
\(606\) 0 0
\(607\) 23.5365i 0.955316i 0.878546 + 0.477658i \(0.158514\pi\)
−0.878546 + 0.477658i \(0.841486\pi\)
\(608\) 0 0
\(609\) 18.9546i 0.768078i
\(610\) 0 0
\(611\) 29.7213 1.20239
\(612\) 0 0
\(613\) −37.4083 −1.51091 −0.755453 0.655202i \(-0.772583\pi\)
−0.755453 + 0.655202i \(0.772583\pi\)
\(614\) 0 0
\(615\) 15.2541i 0.615106i
\(616\) 0 0
\(617\) 7.09868i 0.285782i 0.989738 + 0.142891i \(0.0456398\pi\)
−0.989738 + 0.142891i \(0.954360\pi\)
\(618\) 0 0
\(619\) 9.14215i 0.367454i 0.982977 + 0.183727i \(0.0588162\pi\)
−0.982977 + 0.183727i \(0.941184\pi\)
\(620\) 0 0
\(621\) −12.1750 −0.488567
\(622\) 0 0
\(623\) 14.5670i 0.583616i
\(624\) 0 0
\(625\) −16.5729 −0.662917
\(626\) 0 0
\(627\) −6.30516 −0.251804
\(628\) 0 0
\(629\) −11.5419 + 14.5635i −0.460204 + 0.580687i
\(630\) 0 0
\(631\) −19.0906 −0.759985 −0.379992 0.924990i \(-0.624073\pi\)
−0.379992 + 0.924990i \(0.624073\pi\)
\(632\) 0 0
\(633\) 33.2826 1.32287
\(634\) 0 0
\(635\) 13.9876i 0.555083i
\(636\) 0 0
\(637\) 2.44677 0.0969447
\(638\) 0 0
\(639\) 1.16740i 0.0461816i
\(640\) 0 0
\(641\) 14.0504i 0.554956i −0.960732 0.277478i \(-0.910501\pi\)
0.960732 0.277478i \(-0.0894985\pi\)
\(642\) 0 0
\(643\) 26.8748i 1.05984i 0.848048 + 0.529919i \(0.177778\pi\)
−0.848048 + 0.529919i \(0.822222\pi\)
\(644\) 0 0
\(645\) −14.1028 −0.555299
\(646\) 0 0
\(647\) 28.7167 1.12897 0.564485 0.825443i \(-0.309075\pi\)
0.564485 + 0.825443i \(0.309075\pi\)
\(648\) 0 0
\(649\) 1.63017i 0.0639898i
\(650\) 0 0
\(651\) 22.3607i 0.876386i
\(652\) 0 0
\(653\) 17.4582i 0.683192i −0.939847 0.341596i \(-0.889032\pi\)
0.939847 0.341596i \(-0.110968\pi\)
\(654\) 0 0
\(655\) −34.6603 −1.35429
\(656\) 0 0
\(657\) 1.52422i 0.0594655i
\(658\) 0 0
\(659\) −19.5925 −0.763214 −0.381607 0.924325i \(-0.624629\pi\)
−0.381607 + 0.924325i \(0.624629\pi\)
\(660\) 0 0
\(661\) −9.43303 −0.366902 −0.183451 0.983029i \(-0.558727\pi\)
−0.183451 + 0.983029i \(0.558727\pi\)
\(662\) 0 0
\(663\) 11.5702 14.5993i 0.449349 0.566990i
\(664\) 0 0
\(665\) −10.2273 −0.396598
\(666\) 0 0
\(667\) 10.4692 0.405371
\(668\) 0 0
\(669\) 31.4627i 1.21642i
\(670\) 0 0
\(671\) −4.93576 −0.190543
\(672\) 0 0
\(673\) 44.4240i 1.71242i −0.516627 0.856211i \(-0.672813\pi\)
0.516627 0.856211i \(-0.327187\pi\)
\(674\) 0 0
\(675\) 6.71019i 0.258275i
\(676\) 0 0
\(677\) 32.6181i 1.25361i 0.779175 + 0.626807i \(0.215639\pi\)
−0.779175 + 0.626807i \(0.784361\pi\)
\(678\) 0 0
\(679\) −28.9817 −1.11222
\(680\) 0 0
\(681\) −35.2044 −1.34904
\(682\) 0 0
\(683\) 17.6943i 0.677055i 0.940956 + 0.338528i \(0.109929\pi\)
−0.940956 + 0.338528i \(0.890071\pi\)
\(684\) 0 0
\(685\) 30.6595i 1.17144i
\(686\) 0 0
\(687\) 32.8058i 1.25162i
\(688\) 0 0
\(689\) −36.5891 −1.39393
\(690\) 0 0
\(691\) 20.6096i 0.784027i 0.919959 + 0.392013i \(0.128221\pi\)
−0.919959 + 0.392013i \(0.871779\pi\)
\(692\) 0 0
\(693\) −0.670513 −0.0254707
\(694\) 0 0
\(695\) −29.6335 −1.12406
\(696\) 0 0
\(697\) 11.4593 14.4594i 0.434053 0.547690i
\(698\) 0 0
\(699\) 24.5824 0.929790
\(700\) 0 0
\(701\) −48.1164 −1.81733 −0.908665 0.417525i \(-0.862898\pi\)
−0.908665 + 0.417525i \(0.862898\pi\)
\(702\) 0 0
\(703\) 9.79471i 0.369414i
\(704\) 0 0
\(705\) 39.9113 1.50315
\(706\) 0 0
\(707\) 20.0932i 0.755681i
\(708\) 0 0
\(709\) 2.20794i 0.0829209i −0.999140 0.0414605i \(-0.986799\pi\)
0.999140 0.0414605i \(-0.0132011\pi\)
\(710\) 0 0
\(711\) 0.678043i 0.0254286i
\(712\) 0 0
\(713\) 12.3506 0.462533
\(714\) 0 0
\(715\) 7.92677 0.296445
\(716\) 0 0
\(717\) 32.7861i 1.22442i
\(718\) 0 0
\(719\) 38.7086i 1.44359i 0.692109 + 0.721793i \(0.256682\pi\)
−0.692109 + 0.721793i \(0.743318\pi\)
\(720\) 0 0
\(721\) 22.5114i 0.838367i
\(722\) 0 0
\(723\) −43.6411 −1.62303
\(724\) 0 0
\(725\) 5.77006i 0.214295i
\(726\) 0 0
\(727\) 36.8702 1.36744 0.683719 0.729745i \(-0.260361\pi\)
0.683719 + 0.729745i \(0.260361\pi\)
\(728\) 0 0
\(729\) −25.3298 −0.938141
\(730\) 0 0
\(731\) 13.3681 + 10.5945i 0.494438 + 0.391850i
\(732\) 0 0
\(733\) −8.96503 −0.331131 −0.165565 0.986199i \(-0.552945\pi\)
−0.165565 + 0.986199i \(0.552945\pi\)
\(734\) 0 0
\(735\) 3.28566 0.121193
\(736\) 0 0
\(737\) 10.3514i 0.381298i
\(738\) 0 0
\(739\) −19.4158 −0.714221 −0.357111 0.934062i \(-0.616238\pi\)
−0.357111 + 0.934062i \(0.616238\pi\)
\(740\) 0 0
\(741\) 9.81875i 0.360701i
\(742\) 0 0
\(743\) 4.08704i 0.149939i −0.997186 0.0749695i \(-0.976114\pi\)
0.997186 0.0749695i \(-0.0238859\pi\)
\(744\) 0 0
\(745\) 18.5954i 0.681284i
\(746\) 0 0
\(747\) 2.49133 0.0911530
\(748\) 0 0
\(749\) −24.6237 −0.899731
\(750\) 0 0
\(751\) 38.1845i 1.39337i −0.717377 0.696685i \(-0.754657\pi\)
0.717377 0.696685i \(-0.245343\pi\)
\(752\) 0 0
\(753\) 31.2350i 1.13826i
\(754\) 0 0
\(755\) 27.8838i 1.01480i
\(756\) 0 0
\(757\) −20.0495 −0.728710 −0.364355 0.931260i \(-0.618710\pi\)
−0.364355 + 0.931260i \(0.618710\pi\)
\(758\) 0 0
\(759\) 7.00680i 0.254331i
\(760\) 0 0
\(761\) −0.504268 −0.0182797 −0.00913985 0.999958i \(-0.502909\pi\)
−0.00913985 + 0.999958i \(0.502909\pi\)
\(762\) 0 0
\(763\) −25.6924 −0.930126
\(764\) 0 0
\(765\) 0.821217 1.03621i 0.0296912 0.0374644i
\(766\) 0 0
\(767\) 2.53859 0.0916633
\(768\) 0 0
\(769\) −36.3548 −1.31099 −0.655494 0.755200i \(-0.727540\pi\)
−0.655494 + 0.755200i \(0.727540\pi\)
\(770\) 0 0
\(771\) 50.4500i 1.81691i
\(772\) 0 0
\(773\) −25.4730 −0.916202 −0.458101 0.888900i \(-0.651470\pi\)
−0.458101 + 0.888900i \(0.651470\pi\)
\(774\) 0 0
\(775\) 6.80695i 0.244513i
\(776\) 0 0
\(777\) 19.7067i 0.706974i
\(778\) 0 0
\(779\) 9.72468i 0.348423i
\(780\) 0 0
\(781\) 11.3674 0.406756
\(782\) 0 0
\(783\) 21.8533 0.780972
\(784\) 0 0
\(785\) 34.6605i 1.23709i
\(786\) 0 0
\(787\) 11.0753i 0.394791i −0.980324 0.197396i \(-0.936752\pi\)
0.980324 0.197396i \(-0.0632484\pi\)
\(788\) 0 0
\(789\) 49.8140i 1.77343i
\(790\) 0 0
\(791\) 0.246429 0.00876201
\(792\) 0 0
\(793\) 7.68624i