Properties

Label 4012.2.b.b.237.16
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 237.16
Character \(\chi\) = 4012.237
Dual form 4012.2.b.b.237.31

$q$-expansion

\(f(q)\) \(=\) \(q-1.43806i q^{3} +2.66437i q^{5} +3.93282i q^{7} +0.931993 q^{9} +O(q^{10})\) \(q-1.43806i q^{3} +2.66437i q^{5} +3.93282i q^{7} +0.931993 q^{9} -4.93836i q^{11} +5.59880 q^{13} +3.83152 q^{15} +(-2.22849 - 3.46898i) q^{17} -6.57098 q^{19} +5.65561 q^{21} -6.13081i q^{23} -2.09887 q^{25} -5.65443i q^{27} +1.64457i q^{29} -9.54938i q^{31} -7.10165 q^{33} -10.4785 q^{35} -6.71951i q^{37} -8.05138i q^{39} +9.66660i q^{41} +9.17449 q^{43} +2.48318i q^{45} -8.93156 q^{47} -8.46706 q^{49} +(-4.98859 + 3.20470i) q^{51} +5.74193 q^{53} +13.1576 q^{55} +9.44944i q^{57} +1.00000 q^{59} -11.2509i q^{61} +3.66536i q^{63} +14.9173i q^{65} -0.829919 q^{67} -8.81645 q^{69} +5.67113i q^{71} -5.67456i q^{73} +3.01829i q^{75} +19.4217 q^{77} -2.84110i q^{79} -5.33541 q^{81} +4.48971 q^{83} +(9.24265 - 5.93753i) q^{85} +2.36499 q^{87} +14.0734 q^{89} +22.0190i q^{91} -13.7326 q^{93} -17.5075i q^{95} -11.1433i q^{97} -4.60252i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 46q - 54q^{9} + O(q^{10}) \) \( 46q - 54q^{9} + 8q^{13} - 10q^{15} + q^{17} - 20q^{19} - 24q^{21} - 54q^{25} + 2q^{33} + 26q^{35} - 38q^{43} + 6q^{47} - 66q^{49} + 26q^{51} + 18q^{53} - 20q^{55} + 46q^{59} + 48q^{67} + 28q^{69} + 22q^{77} + 70q^{81} - 52q^{83} - 2q^{85} + 44q^{87} - 76q^{89} - 26q^{93} + O(q^{100}) \)

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
\(3\) 1.43806i 0.830262i −0.909762 0.415131i \(-0.863736\pi\)
0.909762 0.415131i \(-0.136264\pi\)
\(4\) 0 0
\(5\) 2.66437i 1.19154i 0.803154 + 0.595771i \(0.203154\pi\)
−0.803154 + 0.595771i \(0.796846\pi\)
\(6\) 0 0
\(7\) 3.93282i 1.48647i 0.669033 + 0.743233i \(0.266709\pi\)
−0.669033 + 0.743233i \(0.733291\pi\)
\(8\) 0 0
\(9\) 0.931993 0.310664
\(10\) 0 0
\(11\) 4.93836i 1.48897i −0.667638 0.744486i \(-0.732694\pi\)
0.667638 0.744486i \(-0.267306\pi\)
\(12\) 0 0
\(13\) 5.59880 1.55283 0.776413 0.630224i \(-0.217037\pi\)
0.776413 + 0.630224i \(0.217037\pi\)
\(14\) 0 0
\(15\) 3.83152 0.989293
\(16\) 0 0
\(17\) −2.22849 3.46898i −0.540489 0.841351i
\(18\) 0 0
\(19\) −6.57098 −1.50749 −0.753743 0.657169i \(-0.771754\pi\)
−0.753743 + 0.657169i \(0.771754\pi\)
\(20\) 0 0
\(21\) 5.65561 1.23416
\(22\) 0 0
\(23\) 6.13081i 1.27836i −0.769056 0.639181i \(-0.779273\pi\)
0.769056 0.639181i \(-0.220727\pi\)
\(24\) 0 0
\(25\) −2.09887 −0.419774
\(26\) 0 0
\(27\) 5.65443i 1.08820i
\(28\) 0 0
\(29\) 1.64457i 0.305389i 0.988273 + 0.152695i \(0.0487951\pi\)
−0.988273 + 0.152695i \(0.951205\pi\)
\(30\) 0 0
\(31\) 9.54938i 1.71512i −0.514384 0.857560i \(-0.671980\pi\)
0.514384 0.857560i \(-0.328020\pi\)
\(32\) 0 0
\(33\) −7.10165 −1.23624
\(34\) 0 0
\(35\) −10.4785 −1.77119
\(36\) 0 0
\(37\) 6.71951i 1.10468i −0.833619 0.552341i \(-0.813735\pi\)
0.833619 0.552341i \(-0.186265\pi\)
\(38\) 0 0
\(39\) 8.05138i 1.28925i
\(40\) 0 0
\(41\) 9.66660i 1.50967i 0.655915 + 0.754835i \(0.272283\pi\)
−0.655915 + 0.754835i \(0.727717\pi\)
\(42\) 0 0
\(43\) 9.17449 1.39910 0.699548 0.714585i \(-0.253385\pi\)
0.699548 + 0.714585i \(0.253385\pi\)
\(44\) 0 0
\(45\) 2.48318i 0.370170i
\(46\) 0 0
\(47\) −8.93156 −1.30280 −0.651401 0.758734i \(-0.725818\pi\)
−0.651401 + 0.758734i \(0.725818\pi\)
\(48\) 0 0
\(49\) −8.46706 −1.20958
\(50\) 0 0
\(51\) −4.98859 + 3.20470i −0.698542 + 0.448748i
\(52\) 0 0
\(53\) 5.74193 0.788715 0.394357 0.918957i \(-0.370967\pi\)
0.394357 + 0.918957i \(0.370967\pi\)
\(54\) 0 0
\(55\) 13.1576 1.77417
\(56\) 0 0
\(57\) 9.44944i 1.25161i
\(58\) 0 0
\(59\) 1.00000 0.130189
\(60\) 0 0
\(61\) 11.2509i 1.44054i −0.693696 0.720268i \(-0.744019\pi\)
0.693696 0.720268i \(-0.255981\pi\)
\(62\) 0 0
\(63\) 3.66536i 0.461792i
\(64\) 0 0
\(65\) 14.9173i 1.85026i
\(66\) 0 0
\(67\) −0.829919 −0.101391 −0.0506953 0.998714i \(-0.516144\pi\)
−0.0506953 + 0.998714i \(0.516144\pi\)
\(68\) 0 0
\(69\) −8.81645 −1.06138
\(70\) 0 0
\(71\) 5.67113i 0.673039i 0.941677 + 0.336519i \(0.109250\pi\)
−0.941677 + 0.336519i \(0.890750\pi\)
\(72\) 0 0
\(73\) 5.67456i 0.664157i −0.943252 0.332079i \(-0.892250\pi\)
0.943252 0.332079i \(-0.107750\pi\)
\(74\) 0 0
\(75\) 3.01829i 0.348523i
\(76\) 0 0
\(77\) 19.4217 2.21331
\(78\) 0 0
\(79\) 2.84110i 0.319649i −0.987145 0.159824i \(-0.948907\pi\)
0.987145 0.159824i \(-0.0510928\pi\)
\(80\) 0 0
\(81\) −5.33541 −0.592823
\(82\) 0 0
\(83\) 4.48971 0.492810 0.246405 0.969167i \(-0.420751\pi\)
0.246405 + 0.969167i \(0.420751\pi\)
\(84\) 0 0
\(85\) 9.24265 5.93753i 1.00251 0.644016i
\(86\) 0 0
\(87\) 2.36499 0.253553
\(88\) 0 0
\(89\) 14.0734 1.49177 0.745887 0.666073i \(-0.232026\pi\)
0.745887 + 0.666073i \(0.232026\pi\)
\(90\) 0 0
\(91\) 22.0190i 2.30822i
\(92\) 0 0
\(93\) −13.7326 −1.42400
\(94\) 0 0
\(95\) 17.5075i 1.79623i
\(96\) 0 0
\(97\) 11.1433i 1.13143i −0.824601 0.565715i \(-0.808600\pi\)
0.824601 0.565715i \(-0.191400\pi\)
\(98\) 0 0
\(99\) 4.60252i 0.462571i
\(100\) 0 0
\(101\) −3.64143 −0.362336 −0.181168 0.983452i \(-0.557988\pi\)
−0.181168 + 0.983452i \(0.557988\pi\)
\(102\) 0 0
\(103\) 4.72995 0.466056 0.233028 0.972470i \(-0.425137\pi\)
0.233028 + 0.972470i \(0.425137\pi\)
\(104\) 0 0
\(105\) 15.0687i 1.47055i
\(106\) 0 0
\(107\) 18.5925i 1.79740i −0.438560 0.898702i \(-0.644511\pi\)
0.438560 0.898702i \(-0.355489\pi\)
\(108\) 0 0
\(109\) 0.790027i 0.0756708i −0.999284 0.0378354i \(-0.987954\pi\)
0.999284 0.0378354i \(-0.0120463\pi\)
\(110\) 0 0
\(111\) −9.66304 −0.917175
\(112\) 0 0
\(113\) 11.5730i 1.08870i 0.838858 + 0.544350i \(0.183224\pi\)
−0.838858 + 0.544350i \(0.816776\pi\)
\(114\) 0 0
\(115\) 16.3347 1.52322
\(116\) 0 0
\(117\) 5.21804 0.482408
\(118\) 0 0
\(119\) 13.6429 8.76426i 1.25064 0.803418i
\(120\) 0 0
\(121\) −13.3874 −1.21704
\(122\) 0 0
\(123\) 13.9011 1.25342
\(124\) 0 0
\(125\) 7.72968i 0.691364i
\(126\) 0 0
\(127\) 5.60393 0.497268 0.248634 0.968598i \(-0.420018\pi\)
0.248634 + 0.968598i \(0.420018\pi\)
\(128\) 0 0
\(129\) 13.1934i 1.16162i
\(130\) 0 0
\(131\) 20.2437i 1.76870i 0.466825 + 0.884350i \(0.345398\pi\)
−0.466825 + 0.884350i \(0.654602\pi\)
\(132\) 0 0
\(133\) 25.8425i 2.24083i
\(134\) 0 0
\(135\) 15.0655 1.29663
\(136\) 0 0
\(137\) 6.73196 0.575150 0.287575 0.957758i \(-0.407151\pi\)
0.287575 + 0.957758i \(0.407151\pi\)
\(138\) 0 0
\(139\) 0.142436i 0.0120812i 0.999982 + 0.00604061i \(0.00192280\pi\)
−0.999982 + 0.00604061i \(0.998077\pi\)
\(140\) 0 0
\(141\) 12.8441i 1.08167i
\(142\) 0 0
\(143\) 27.6489i 2.31212i
\(144\) 0 0
\(145\) −4.38175 −0.363884
\(146\) 0 0
\(147\) 12.1761i 1.00427i
\(148\) 0 0
\(149\) 19.5104 1.59836 0.799179 0.601093i \(-0.205268\pi\)
0.799179 + 0.601093i \(0.205268\pi\)
\(150\) 0 0
\(151\) −19.0832 −1.55297 −0.776485 0.630135i \(-0.782999\pi\)
−0.776485 + 0.630135i \(0.782999\pi\)
\(152\) 0 0
\(153\) −2.07694 3.23307i −0.167911 0.261378i
\(154\) 0 0
\(155\) 25.4431 2.04364
\(156\) 0 0
\(157\) −10.6262 −0.848063 −0.424032 0.905647i \(-0.639385\pi\)
−0.424032 + 0.905647i \(0.639385\pi\)
\(158\) 0 0
\(159\) 8.25722i 0.654840i
\(160\) 0 0
\(161\) 24.1114 1.90024
\(162\) 0 0
\(163\) 4.91575i 0.385031i 0.981294 + 0.192516i \(0.0616646\pi\)
−0.981294 + 0.192516i \(0.938335\pi\)
\(164\) 0 0
\(165\) 18.9214i 1.47303i
\(166\) 0 0
\(167\) 16.4227i 1.27083i 0.772171 + 0.635414i \(0.219171\pi\)
−0.772171 + 0.635414i \(0.780829\pi\)
\(168\) 0 0
\(169\) 18.3465 1.41127
\(170\) 0 0
\(171\) −6.12411 −0.468322
\(172\) 0 0
\(173\) 17.5017i 1.33063i 0.746564 + 0.665313i \(0.231702\pi\)
−0.746564 + 0.665313i \(0.768298\pi\)
\(174\) 0 0
\(175\) 8.25447i 0.623980i
\(176\) 0 0
\(177\) 1.43806i 0.108091i
\(178\) 0 0
\(179\) −3.84481 −0.287375 −0.143687 0.989623i \(-0.545896\pi\)
−0.143687 + 0.989623i \(0.545896\pi\)
\(180\) 0 0
\(181\) 4.67637i 0.347592i −0.984782 0.173796i \(-0.944397\pi\)
0.984782 0.173796i \(-0.0556033\pi\)
\(182\) 0 0
\(183\) −16.1795 −1.19602
\(184\) 0 0
\(185\) 17.9033 1.31627
\(186\) 0 0
\(187\) −17.1311 + 11.0051i −1.25275 + 0.804773i
\(188\) 0 0
\(189\) 22.2378 1.61756
\(190\) 0 0
\(191\) 5.08518 0.367951 0.183975 0.982931i \(-0.441103\pi\)
0.183975 + 0.982931i \(0.441103\pi\)
\(192\) 0 0
\(193\) 0.285281i 0.0205349i −0.999947 0.0102675i \(-0.996732\pi\)
0.999947 0.0102675i \(-0.00326830\pi\)
\(194\) 0 0
\(195\) 21.4519 1.53620
\(196\) 0 0
\(197\) 11.9224i 0.849434i −0.905326 0.424717i \(-0.860374\pi\)
0.905326 0.424717i \(-0.139626\pi\)
\(198\) 0 0
\(199\) 12.3728i 0.877082i −0.898711 0.438541i \(-0.855495\pi\)
0.898711 0.438541i \(-0.144505\pi\)
\(200\) 0 0
\(201\) 1.19347i 0.0841809i
\(202\) 0 0
\(203\) −6.46780 −0.453950
\(204\) 0 0
\(205\) −25.7554 −1.79884
\(206\) 0 0
\(207\) 5.71387i 0.397142i
\(208\) 0 0
\(209\) 32.4499i 2.24461i
\(210\) 0 0
\(211\) 23.1918i 1.59659i −0.602268 0.798294i \(-0.705736\pi\)
0.602268 0.798294i \(-0.294264\pi\)
\(212\) 0 0
\(213\) 8.15540 0.558799
\(214\) 0 0
\(215\) 24.4442i 1.66708i
\(216\) 0 0
\(217\) 37.5560 2.54947
\(218\) 0 0
\(219\) −8.16034 −0.551425
\(220\) 0 0
\(221\) −12.4769 19.4221i −0.839285 1.30647i
\(222\) 0 0
\(223\) −1.71690 −0.114972 −0.0574862 0.998346i \(-0.518309\pi\)
−0.0574862 + 0.998346i \(0.518309\pi\)
\(224\) 0 0
\(225\) −1.95613 −0.130409
\(226\) 0 0
\(227\) 17.3219i 1.14970i 0.818260 + 0.574848i \(0.194939\pi\)
−0.818260 + 0.574848i \(0.805061\pi\)
\(228\) 0 0
\(229\) 27.1142 1.79175 0.895877 0.444302i \(-0.146548\pi\)
0.895877 + 0.444302i \(0.146548\pi\)
\(230\) 0 0
\(231\) 27.9295i 1.83762i
\(232\) 0 0
\(233\) 8.90548i 0.583418i −0.956507 0.291709i \(-0.905776\pi\)
0.956507 0.291709i \(-0.0942238\pi\)
\(234\) 0 0
\(235\) 23.7970i 1.55234i
\(236\) 0 0
\(237\) −4.08566 −0.265392
\(238\) 0 0
\(239\) 15.3486 0.992820 0.496410 0.868088i \(-0.334651\pi\)
0.496410 + 0.868088i \(0.334651\pi\)
\(240\) 0 0
\(241\) 9.83383i 0.633452i 0.948517 + 0.316726i \(0.102584\pi\)
−0.948517 + 0.316726i \(0.897416\pi\)
\(242\) 0 0
\(243\) 9.29067i 0.595997i
\(244\) 0 0
\(245\) 22.5594i 1.44127i
\(246\) 0 0
\(247\) −36.7896 −2.34086
\(248\) 0 0
\(249\) 6.45646i 0.409161i
\(250\) 0 0
\(251\) −16.2017 −1.02264 −0.511320 0.859391i \(-0.670843\pi\)
−0.511320 + 0.859391i \(0.670843\pi\)
\(252\) 0 0
\(253\) −30.2762 −1.90345
\(254\) 0 0
\(255\) −8.53850 13.2914i −0.534702 0.832343i
\(256\) 0 0
\(257\) −4.30176 −0.268336 −0.134168 0.990959i \(-0.542836\pi\)
−0.134168 + 0.990959i \(0.542836\pi\)
\(258\) 0 0
\(259\) 26.4266 1.64207
\(260\) 0 0
\(261\) 1.53273i 0.0948736i
\(262\) 0 0
\(263\) −11.4217 −0.704295 −0.352148 0.935944i \(-0.614548\pi\)
−0.352148 + 0.935944i \(0.614548\pi\)
\(264\) 0 0
\(265\) 15.2986i 0.939787i
\(266\) 0 0
\(267\) 20.2383i 1.23856i
\(268\) 0 0
\(269\) 1.59411i 0.0971945i 0.998818 + 0.0485972i \(0.0154751\pi\)
−0.998818 + 0.0485972i \(0.984525\pi\)
\(270\) 0 0
\(271\) 14.8878 0.904371 0.452185 0.891924i \(-0.350645\pi\)
0.452185 + 0.891924i \(0.350645\pi\)
\(272\) 0 0
\(273\) 31.6646 1.91643
\(274\) 0 0
\(275\) 10.3650i 0.625032i
\(276\) 0 0
\(277\) 3.21746i 0.193318i 0.995318 + 0.0966592i \(0.0308157\pi\)
−0.995318 + 0.0966592i \(0.969184\pi\)
\(278\) 0 0
\(279\) 8.89996i 0.532827i
\(280\) 0 0
\(281\) −20.0117 −1.19380 −0.596898 0.802317i \(-0.703600\pi\)
−0.596898 + 0.802317i \(0.703600\pi\)
\(282\) 0 0
\(283\) 1.03613i 0.0615915i −0.999526 0.0307958i \(-0.990196\pi\)
0.999526 0.0307958i \(-0.00980414\pi\)
\(284\) 0 0
\(285\) −25.1768 −1.49135
\(286\) 0 0
\(287\) −38.0170 −2.24407
\(288\) 0 0
\(289\) −7.06764 + 15.4612i −0.415744 + 0.909482i
\(290\) 0 0
\(291\) −16.0247 −0.939383
\(292\) 0 0
\(293\) −19.0273 −1.11159 −0.555793 0.831321i \(-0.687585\pi\)
−0.555793 + 0.831321i \(0.687585\pi\)
\(294\) 0 0
\(295\) 2.66437i 0.155126i
\(296\) 0 0
\(297\) −27.9236 −1.62029
\(298\) 0 0
\(299\) 34.3251i 1.98507i
\(300\) 0 0
\(301\) 36.0816i 2.07971i
\(302\) 0 0
\(303\) 5.23659i 0.300834i
\(304\) 0 0
\(305\) 29.9767 1.71646
\(306\) 0 0
\(307\) 12.5768 0.717794 0.358897 0.933377i \(-0.383153\pi\)
0.358897 + 0.933377i \(0.383153\pi\)
\(308\) 0 0
\(309\) 6.80194i 0.386949i
\(310\) 0 0
\(311\) 25.2938i 1.43428i −0.696930 0.717139i \(-0.745451\pi\)
0.696930 0.717139i \(-0.254549\pi\)
\(312\) 0 0
\(313\) 15.6164i 0.882688i 0.897338 + 0.441344i \(0.145498\pi\)
−0.897338 + 0.441344i \(0.854502\pi\)
\(314\) 0 0
\(315\) −9.76588 −0.550245
\(316\) 0 0
\(317\) 15.5970i 0.876017i 0.898971 + 0.438008i \(0.144316\pi\)
−0.898971 + 0.438008i \(0.855684\pi\)
\(318\) 0 0
\(319\) 8.12149 0.454716
\(320\) 0 0
\(321\) −26.7371 −1.49232
\(322\) 0 0
\(323\) 14.6434 + 22.7946i 0.814780 + 1.26833i
\(324\) 0 0
\(325\) −11.7511 −0.651836
\(326\) 0 0
\(327\) −1.13610 −0.0628266
\(328\) 0 0
\(329\) 35.1262i 1.93657i
\(330\) 0 0
\(331\) −13.6759 −0.751693 −0.375846 0.926682i \(-0.622648\pi\)
−0.375846 + 0.926682i \(0.622648\pi\)
\(332\) 0 0
\(333\) 6.26254i 0.343185i
\(334\) 0 0
\(335\) 2.21121i 0.120811i
\(336\) 0 0
\(337\) 22.7871i 1.24129i −0.784091 0.620646i \(-0.786870\pi\)
0.784091 0.620646i \(-0.213130\pi\)
\(338\) 0 0
\(339\) 16.6427 0.903907
\(340\) 0 0
\(341\) −47.1583 −2.55377
\(342\) 0 0
\(343\) 5.76967i 0.311533i
\(344\) 0 0
\(345\) 23.4903i 1.26467i
\(346\) 0 0
\(347\) 26.7914i 1.43824i 0.694887 + 0.719119i \(0.255454\pi\)
−0.694887 + 0.719119i \(0.744546\pi\)
\(348\) 0 0
\(349\) −5.34385 −0.286050 −0.143025 0.989719i \(-0.545683\pi\)
−0.143025 + 0.989719i \(0.545683\pi\)
\(350\) 0 0
\(351\) 31.6580i 1.68978i
\(352\) 0 0
\(353\) −10.5447 −0.561236 −0.280618 0.959820i \(-0.590539\pi\)
−0.280618 + 0.959820i \(0.590539\pi\)
\(354\) 0 0
\(355\) −15.1100 −0.801954
\(356\) 0 0
\(357\) −12.6035 19.6192i −0.667048 1.03836i
\(358\) 0 0
\(359\) −4.03950 −0.213197 −0.106598 0.994302i \(-0.533996\pi\)
−0.106598 + 0.994302i \(0.533996\pi\)
\(360\) 0 0
\(361\) 24.1778 1.27251
\(362\) 0 0
\(363\) 19.2519i 1.01046i
\(364\) 0 0
\(365\) 15.1191 0.791372
\(366\) 0 0
\(367\) 34.8079i 1.81696i −0.417933 0.908478i \(-0.637245\pi\)
0.417933 0.908478i \(-0.362755\pi\)
\(368\) 0 0
\(369\) 9.00921i 0.469001i
\(370\) 0 0
\(371\) 22.5820i 1.17240i
\(372\) 0 0
\(373\) 16.4544 0.851978 0.425989 0.904728i \(-0.359926\pi\)
0.425989 + 0.904728i \(0.359926\pi\)
\(374\) 0 0
\(375\) 11.1157 0.574014
\(376\) 0 0
\(377\) 9.20762i 0.474216i
\(378\) 0 0
\(379\) 24.3735i 1.25198i 0.779831 + 0.625990i \(0.215305\pi\)
−0.779831 + 0.625990i \(0.784695\pi\)
\(380\) 0 0
\(381\) 8.05876i 0.412863i
\(382\) 0 0
\(383\) −29.3363 −1.49901 −0.749507 0.661997i \(-0.769709\pi\)
−0.749507 + 0.661997i \(0.769709\pi\)
\(384\) 0 0
\(385\) 51.7466i 2.63725i
\(386\) 0 0
\(387\) 8.55057 0.434650
\(388\) 0 0
\(389\) −7.76624 −0.393764 −0.196882 0.980427i \(-0.563082\pi\)
−0.196882 + 0.980427i \(0.563082\pi\)
\(390\) 0 0
\(391\) −21.2677 + 13.6625i −1.07555 + 0.690940i
\(392\) 0 0
\(393\) 29.1116 1.46848
\(394\) 0 0
\(395\) 7.56974 0.380875
\(396\) 0 0
\(397\) 38.7854i 1.94658i −0.229575 0.973291i \(-0.573734\pi\)
0.229575 0.973291i \(-0.426266\pi\)
\(398\) 0 0
\(399\) −37.1629 −1.86047
\(400\) 0 0
\(401\) 3.17880i 0.158742i 0.996845 + 0.0793708i \(0.0252911\pi\)
−0.996845 + 0.0793708i \(0.974709\pi\)
\(402\) 0 0
\(403\) 53.4650i 2.66328i
\(404\) 0 0
\(405\) 14.2155i 0.706374i
\(406\) 0 0
\(407\) −33.1834 −1.64484
\(408\) 0 0
\(409\) −0.994863 −0.0491928 −0.0245964 0.999697i \(-0.507830\pi\)
−0.0245964 + 0.999697i \(0.507830\pi\)
\(410\) 0 0
\(411\) 9.68093i 0.477525i
\(412\) 0 0
\(413\) 3.93282i 0.193521i
\(414\) 0 0
\(415\) 11.9623i 0.587204i
\(416\) 0 0
\(417\) 0.204830 0.0100306
\(418\) 0 0
\(419\) 1.88003i 0.0918454i 0.998945 + 0.0459227i \(0.0146228\pi\)
−0.998945 + 0.0459227i \(0.985377\pi\)
\(420\) 0 0
\(421\) 25.5582 1.24563 0.622816 0.782368i \(-0.285988\pi\)
0.622816 + 0.782368i \(0.285988\pi\)
\(422\) 0 0
\(423\) −8.32415 −0.404734
\(424\) 0 0
\(425\) 4.67732 + 7.28094i 0.226883 + 0.353177i
\(426\) 0 0
\(427\) 44.2479 2.14131
\(428\) 0 0
\(429\) −39.7607 −1.91966
\(430\) 0 0
\(431\) 23.6517i 1.13926i −0.821901 0.569630i \(-0.807087\pi\)
0.821901 0.569630i \(-0.192913\pi\)
\(432\) 0 0
\(433\) −23.1584 −1.11292 −0.556462 0.830873i \(-0.687841\pi\)
−0.556462 + 0.830873i \(0.687841\pi\)
\(434\) 0 0
\(435\) 6.30120i 0.302119i
\(436\) 0 0
\(437\) 40.2854i 1.92711i
\(438\) 0 0
\(439\) 10.0893i 0.481535i −0.970583 0.240768i \(-0.922601\pi\)
0.970583 0.240768i \(-0.0773992\pi\)
\(440\) 0 0
\(441\) −7.89124 −0.375773
\(442\) 0 0
\(443\) 33.6961 1.60095 0.800475 0.599366i \(-0.204580\pi\)
0.800475 + 0.599366i \(0.204580\pi\)
\(444\) 0 0
\(445\) 37.4967i 1.77751i
\(446\) 0 0
\(447\) 28.0571i 1.32706i
\(448\) 0 0
\(449\) 18.4832i 0.872278i −0.899879 0.436139i \(-0.856346\pi\)
0.899879 0.436139i \(-0.143654\pi\)
\(450\) 0 0
\(451\) 47.7372 2.24786
\(452\) 0 0
\(453\) 27.4428i 1.28937i
\(454\) 0 0
\(455\) −58.6669 −2.75035
\(456\) 0 0
\(457\) 14.2318 0.665735 0.332868 0.942974i \(-0.391984\pi\)
0.332868 + 0.942974i \(0.391984\pi\)
\(458\) 0 0
\(459\) −19.6151 + 12.6009i −0.915554 + 0.588157i
\(460\) 0 0
\(461\) −11.4183 −0.531804 −0.265902 0.964000i \(-0.585670\pi\)
−0.265902 + 0.964000i \(0.585670\pi\)
\(462\) 0 0
\(463\) 35.3216 1.64154 0.820768 0.571262i \(-0.193546\pi\)
0.820768 + 0.571262i \(0.193546\pi\)
\(464\) 0 0
\(465\) 36.5886i 1.69676i
\(466\) 0 0
\(467\) 3.77136 0.174518 0.0872589 0.996186i \(-0.472189\pi\)
0.0872589 + 0.996186i \(0.472189\pi\)
\(468\) 0 0
\(469\) 3.26392i 0.150714i
\(470\) 0 0
\(471\) 15.2811i 0.704115i
\(472\) 0 0
\(473\) 45.3070i 2.08322i
\(474\) 0 0
\(475\) 13.7916 0.632804
\(476\) 0 0
\(477\) 5.35144 0.245026
\(478\) 0 0
\(479\) 16.0809i 0.734755i 0.930072 + 0.367377i \(0.119744\pi\)
−0.930072 + 0.367377i \(0.880256\pi\)
\(480\) 0 0
\(481\) 37.6212i 1.71538i
\(482\) 0 0
\(483\) 34.6735i 1.57770i
\(484\) 0 0
\(485\) 29.6898 1.34815
\(486\) 0 0
\(487\) 27.0721i 1.22675i −0.789790 0.613377i \(-0.789811\pi\)
0.789790 0.613377i \(-0.210189\pi\)
\(488\) 0 0
\(489\) 7.06913 0.319677
\(490\) 0 0
\(491\) 20.3412 0.917985 0.458993 0.888440i \(-0.348210\pi\)
0.458993 + 0.888440i \(0.348210\pi\)
\(492\) 0 0
\(493\) 5.70498 3.66491i 0.256940 0.165059i
\(494\) 0 0
\(495\) 12.2628 0.551173
\(496\) 0 0
\(497\) −22.3035 −1.00045
\(498\) 0 0
\(499\) 13.1752i 0.589803i −0.955528 0.294901i \(-0.904713\pi\)
0.955528 0.294901i \(-0.0952868\pi\)
\(500\) 0 0
\(501\) 23.6168 1.05512
\(502\) 0 0
\(503\) 3.05368i 0.136157i −0.997680 0.0680785i \(-0.978313\pi\)
0.997680 0.0680785i \(-0.0216868\pi\)
\(504\) 0 0
\(505\) 9.70213i 0.431739i
\(506\) 0 0
\(507\) 26.3833i 1.17172i
\(508\) 0 0
\(509\) −0.477036 −0.0211443 −0.0105721 0.999944i \(-0.503365\pi\)
−0.0105721 + 0.999944i \(0.503365\pi\)
\(510\) 0 0
\(511\) 22.3170 0.987247
\(512\) 0 0
\(513\) 37.1551i 1.64044i
\(514\) 0 0
\(515\) 12.6023i 0.555326i
\(516\) 0 0
\(517\) 44.1073i 1.93984i
\(518\) 0 0
\(519\) 25.1684 1.10477
\(520\) 0 0
\(521\) 7.29393i 0.319553i −0.987153 0.159776i \(-0.948923\pi\)
0.987153 0.159776i \(-0.0510773\pi\)
\(522\) 0 0
\(523\) 15.0795 0.659382 0.329691 0.944089i \(-0.393055\pi\)
0.329691 + 0.944089i \(0.393055\pi\)
\(524\) 0 0
\(525\) −11.8704 −0.518067
\(526\) 0 0
\(527\) −33.1266 + 21.2807i −1.44302 + 0.927003i
\(528\) 0 0
\(529\) −14.5868 −0.634209
\(530\) 0 0
\(531\) 0.931993 0.0404451
\(532\) 0 0
\(533\) 54.1213i 2.34426i
\(534\) 0 0
\(535\) 49.5373 2.14168
\(536\) 0 0
\(537\) 5.52906i 0.238596i
\(538\) 0 0
\(539\) 41.8134i 1.80103i
\(540\) 0 0
\(541\) 9.43840i 0.405789i −0.979201 0.202894i \(-0.934965\pi\)
0.979201 0.202894i \(-0.0650348\pi\)
\(542\) 0 0
\(543\) −6.72489 −0.288593
\(544\) 0 0
\(545\) 2.10492 0.0901650
\(546\) 0 0
\(547\) 37.4448i 1.60102i 0.599318 + 0.800511i \(0.295438\pi\)
−0.599318 + 0.800511i \(0.704562\pi\)
\(548\) 0 0
\(549\) 10.4858i 0.447523i
\(550\) 0 0
\(551\) 10.8064i 0.460370i
\(552\) 0 0
\(553\) 11.1735 0.475147
\(554\) 0 0
\(555\) 25.7459i 1.09285i
\(556\) 0 0
\(557\) 18.3004 0.775415 0.387707 0.921783i \(-0.373267\pi\)
0.387707 + 0.921783i \(0.373267\pi\)
\(558\) 0 0
\(559\) 51.3661 2.17255
\(560\) 0 0
\(561\) 15.8260 + 24.6355i 0.668173 + 1.04011i
\(562\) 0 0
\(563\) 14.9555 0.630298 0.315149 0.949042i \(-0.397946\pi\)
0.315149 + 0.949042i \(0.397946\pi\)
\(564\) 0 0
\(565\) −30.8349 −1.29723
\(566\) 0 0
\(567\) 20.9832i 0.881211i
\(568\) 0 0
\(569\) 25.0101 1.04848 0.524239 0.851571i \(-0.324350\pi\)
0.524239 + 0.851571i \(0.324350\pi\)
\(570\) 0 0
\(571\) 34.4005i 1.43961i 0.694174 + 0.719807i \(0.255770\pi\)
−0.694174 + 0.719807i \(0.744230\pi\)
\(572\) 0 0
\(573\) 7.31278i 0.305496i
\(574\) 0 0
\(575\) 12.8678i 0.536623i
\(576\) 0 0
\(577\) −36.2824 −1.51045 −0.755227 0.655463i \(-0.772474\pi\)
−0.755227 + 0.655463i \(0.772474\pi\)
\(578\) 0 0
\(579\) −0.410250 −0.0170494
\(580\) 0 0
\(581\) 17.6572i 0.732545i
\(582\) 0 0
\(583\) 28.3557i 1.17437i
\(584\) 0 0
\(585\) 13.9028i 0.574810i
\(586\) 0 0
\(587\) −17.4887 −0.721836 −0.360918 0.932598i \(-0.617537\pi\)
−0.360918 + 0.932598i \(0.617537\pi\)
\(588\) 0 0
\(589\) 62.7488i 2.58552i
\(590\) 0 0
\(591\) −17.1450 −0.705253
\(592\) 0 0
\(593\) 37.2797 1.53090 0.765448 0.643498i \(-0.222518\pi\)
0.765448 + 0.643498i \(0.222518\pi\)
\(594\) 0 0
\(595\) 23.3512 + 36.3496i 0.957307 + 1.49019i
\(596\) 0 0
\(597\) −17.7927 −0.728208
\(598\) 0 0
\(599\) −9.61792 −0.392978 −0.196489 0.980506i \(-0.562954\pi\)
−0.196489 + 0.980506i \(0.562954\pi\)
\(600\) 0 0
\(601\) 38.0506i 1.55212i 0.630661 + 0.776059i \(0.282784\pi\)
−0.630661 + 0.776059i \(0.717216\pi\)
\(602\) 0 0
\(603\) −0.773479 −0.0314985
\(604\) 0 0
\(605\) 35.6691i 1.45015i
\(606\) 0 0
\(607\) 25.9507i 1.05331i 0.850080 + 0.526654i \(0.176554\pi\)
−0.850080 + 0.526654i \(0.823446\pi\)
\(608\) 0 0
\(609\) 9.30106i 0.376898i
\(610\) 0 0
\(611\) −50.0060 −2.02302
\(612\) 0 0
\(613\) −36.9504 −1.49241 −0.746206 0.665715i \(-0.768127\pi\)
−0.746206 + 0.665715i \(0.768127\pi\)
\(614\) 0 0
\(615\) 37.0377i 1.49351i
\(616\) 0 0
\(617\) 25.6827i 1.03395i 0.856001 + 0.516974i \(0.172942\pi\)
−0.856001 + 0.516974i \(0.827058\pi\)
\(618\) 0 0
\(619\) 3.17128i 0.127464i 0.997967 + 0.0637322i \(0.0203004\pi\)
−0.997967 + 0.0637322i \(0.979700\pi\)
\(620\) 0 0
\(621\) −34.6662 −1.39111
\(622\) 0 0
\(623\) 55.3480i 2.21747i
\(624\) 0 0
\(625\) −31.0891 −1.24356
\(626\) 0 0
\(627\) 46.6648 1.86361
\(628\) 0 0
\(629\) −23.3099 + 14.9744i −0.929425 + 0.597068i
\(630\) 0 0
\(631\) −10.2218 −0.406925 −0.203463 0.979083i \(-0.565220\pi\)
−0.203463 + 0.979083i \(0.565220\pi\)
\(632\) 0 0
\(633\) −33.3511 −1.32559
\(634\) 0 0
\(635\) 14.9309i 0.592516i
\(636\) 0 0
\(637\) −47.4053 −1.87827
\(638\) 0 0
\(639\) 5.28545i 0.209089i
\(640\) 0 0
\(641\) 20.8179i 0.822256i 0.911578 + 0.411128i \(0.134865\pi\)
−0.911578 + 0.411128i \(0.865135\pi\)
\(642\) 0 0
\(643\) 26.7086i 1.05329i 0.850087 + 0.526643i \(0.176549\pi\)
−0.850087 + 0.526643i \(0.823451\pi\)
\(644\) 0 0
\(645\) 35.1522 1.38412
\(646\) 0 0
\(647\) 42.0526 1.65326 0.826630 0.562746i \(-0.190255\pi\)
0.826630 + 0.562746i \(0.190255\pi\)
\(648\) 0 0
\(649\) 4.93836i 0.193848i
\(650\) 0 0
\(651\) 54.0076i 2.11673i
\(652\) 0 0
\(653\) 1.46051i 0.0571543i 0.999592 + 0.0285772i \(0.00909763\pi\)
−0.999592 + 0.0285772i \(0.990902\pi\)
\(654\) 0 0
\(655\) −53.9367 −2.10748
\(656\) 0 0
\(657\) 5.28865i 0.206330i
\(658\) 0 0
\(659\) 35.2978 1.37501 0.687504 0.726181i \(-0.258707\pi\)
0.687504 + 0.726181i \(0.258707\pi\)
\(660\) 0 0
\(661\) 30.3176 1.17922 0.589608 0.807689i \(-0.299282\pi\)
0.589608 + 0.807689i \(0.299282\pi\)
\(662\) 0 0
\(663\) −27.9301 + 17.9425i −1.08471 + 0.696827i
\(664\) 0 0
\(665\) 68.8539 2.67004
\(666\) 0 0
\(667\) 10.0826 0.390398
\(668\) 0 0
\(669\) 2.46901i 0.0954573i
\(670\) 0 0
\(671\) −55.5612 −2.14492
\(672\) 0 0
\(673\) 12.7565i 0.491727i −0.969305 0.245863i \(-0.920929\pi\)
0.969305 0.245863i \(-0.0790714\pi\)
\(674\) 0 0
\(675\) 11.8679i 0.456796i
\(676\) 0 0
\(677\) 26.4740i 1.01748i −0.860921 0.508739i \(-0.830112\pi\)
0.860921 0.508739i \(-0.169888\pi\)
\(678\) 0 0
\(679\) 43.8245 1.68183
\(680\) 0 0
\(681\) 24.9099 0.954550
\(682\) 0 0
\(683\) 4.49043i 0.171821i −0.996303 0.0859107i \(-0.972620\pi\)
0.996303 0.0859107i \(-0.0273800\pi\)
\(684\) 0 0
\(685\) 17.9364i 0.685316i
\(686\) 0 0
\(687\) 38.9917i 1.48763i
\(688\) 0 0
\(689\) 32.1479 1.22474
\(690\) 0 0
\(691\) 0.816424i 0.0310582i −0.999879 0.0155291i \(-0.995057\pi\)
0.999879 0.0155291i \(-0.00494327\pi\)
\(692\) 0 0
\(693\) 18.1009 0.687596
\(694\) 0 0
\(695\) −0.379501 −0.0143953
\(696\) 0 0
\(697\) 33.5333 21.5420i 1.27016 0.815960i
\(698\) 0 0
\(699\) −12.8066 −0.484390
\(700\) 0 0
\(701\) −12.5635 −0.474518 −0.237259 0.971446i \(-0.576249\pi\)
−0.237259 + 0.971446i \(0.576249\pi\)
\(702\) 0 0
\(703\) 44.1538i 1.66529i
\(704\) 0 0
\(705\) −34.2214 −1.28885
\(706\) 0 0
\(707\) 14.3211i 0.538600i
\(708\) 0 0
\(709\) 44.5609i 1.67352i 0.547571 + 0.836759i \(0.315552\pi\)
−0.547571 + 0.836759i \(0.684448\pi\)
\(710\) 0 0
\(711\) 2.64788i 0.0993034i
\(712\) 0 0
\(713\) −58.5454 −2.19254
\(714\) 0 0
\(715\) 73.6669 2.75498
\(716\) 0 0
\(717\) 22.0722i 0.824301i
\(718\) 0 0
\(719\) 3.38195i 0.126126i −0.998010 0.0630628i \(-0.979913\pi\)
0.998010 0.0630628i \(-0.0200868\pi\)
\(720\) 0 0
\(721\) 18.6020i 0.692776i
\(722\) 0 0
\(723\) 14.1416 0.525932
\(724\) 0 0
\(725\) 3.45174i 0.128194i
\(726\) 0 0
\(727\) −27.2106 −1.00919 −0.504593 0.863357i \(-0.668358\pi\)
−0.504593 + 0.863357i \(0.668358\pi\)
\(728\) 0 0
\(729\) −29.3667 −1.08766
\(730\) 0 0
\(731\) −20.4453 31.8261i −0.756196 1.17713i
\(732\) 0 0
\(733\) −2.20268 −0.0813578 −0.0406789 0.999172i \(-0.512952\pi\)
−0.0406789 + 0.999172i \(0.512952\pi\)
\(734\) 0 0
\(735\) −32.4417 −1.19663
\(736\) 0 0
\(737\) 4.09844i 0.150968i
\(738\) 0 0
\(739\) −12.9774 −0.477381 −0.238690 0.971096i \(-0.576718\pi\)
−0.238690 + 0.971096i \(0.576718\pi\)
\(740\) 0 0
\(741\) 52.9055i 1.94353i
\(742\) 0 0
\(743\) 20.5515i 0.753962i −0.926221 0.376981i \(-0.876962\pi\)
0.926221 0.376981i \(-0.123038\pi\)
\(744\) 0 0
\(745\) 51.9830i 1.90451i
\(746\) 0 0
\(747\) 4.18438 0.153099
\(748\) 0 0
\(749\) 73.1209 2.67178
\(750\) 0 0
\(751\) 45.4481i 1.65843i −0.558933 0.829213i \(-0.688789\pi\)
0.558933 0.829213i \(-0.311211\pi\)
\(752\) 0 0
\(753\) 23.2989i 0.849059i
\(754\) 0 0
\(755\) 50.8448i 1.85043i
\(756\) 0 0
\(757\) 21.3682 0.776640 0.388320 0.921525i \(-0.373056\pi\)
0.388320 + 0.921525i \(0.373056\pi\)
\(758\) 0 0
\(759\) 43.5388i 1.58036i
\(760\) 0 0
\(761\) 13.0180 0.471902 0.235951 0.971765i \(-0.424180\pi\)
0.235951 + 0.971765i \(0.424180\pi\)
\(762\) 0 0
\(763\) 3.10703 0.112482
\(764\) 0 0
\(765\) 8.61409 5.53374i 0.311443 0.200073i
\(766\) 0 0
\(767\) 5.59880 0.202161
\(768\) 0 0
\(769\) 2.63006 0.0948425 0.0474212 0.998875i \(-0.484900\pi\)
0.0474212 + 0.998875i \(0.484900\pi\)
\(770\) 0 0
\(771\) 6.18618i 0.222790i
\(772\) 0 0
\(773\) −29.6444 −1.06624 −0.533118 0.846041i \(-0.678980\pi\)
−0.533118 + 0.846041i \(0.678980\pi\)
\(774\) 0 0
\(775\) 20.0429i 0.719963i
\(776\) 0 0
\(777\) 38.0030i 1.36335i
\(778\) 0 0
\(779\) 63.5191i 2.27581i
\(780\) 0 0
\(781\) 28.0061 1.00214
\(782\) 0 0
\(783\) 9.29911 0.332323
\(784\) 0 0
\(785\) 28.3121i 1.01050i
\(786\) 0 0
\(787\) 32.9315i 1.17388i 0.809631 + 0.586940i \(0.199667\pi\)
−0.809631 + 0.586940i \(0.800333\pi\)
\(788\) 0 0
\(789\) 16.4251i 0.584750i
\(790\) 0 0
\(791\) −45.5147 −1.61831
\(792\) 0 0
\(793\) 62.9917i 2.23690i