Properties

Label 4032.2.v.e.1583.18
Level 4032
Weight 2
Character 4032.1583
Analytic conductor 32.196
Analytic rank 0
Dimension 40
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4032.v (of order \(4\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(40\)
Relative dimension: \(20\) over \(\Q(i)\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 1008)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 1583.18
Character \(\chi\) = 4032.1583
Dual form 4032.2.v.e.3599.18

$q$-expansion

\(f(q)\) \(=\) \(q+(2.51504 + 2.51504i) q^{5} -1.00000 q^{7} +O(q^{10})\) \(q+(2.51504 + 2.51504i) q^{5} -1.00000 q^{7} +(-0.984548 + 0.984548i) q^{11} +(-3.26086 - 3.26086i) q^{13} -5.50927i q^{17} +(1.21444 - 1.21444i) q^{19} -8.62021i q^{23} +7.65081i q^{25} +(2.04663 - 2.04663i) q^{29} -0.164437i q^{31} +(-2.51504 - 2.51504i) q^{35} +(-8.31105 + 8.31105i) q^{37} -9.56578 q^{41} +(-5.01867 - 5.01867i) q^{43} +3.37837 q^{47} +1.00000 q^{49} +(-3.72717 - 3.72717i) q^{53} -4.95235 q^{55} +(-10.0781 + 10.0781i) q^{59} +(-1.07042 - 1.07042i) q^{61} -16.4023i q^{65} +(3.12073 - 3.12073i) q^{67} +7.66257i q^{71} -8.40588i q^{73} +(0.984548 - 0.984548i) q^{77} -13.8712i q^{79} +(-7.31214 - 7.31214i) q^{83} +(13.8560 - 13.8560i) q^{85} -7.49158 q^{89} +(3.26086 + 3.26086i) q^{91} +6.10874 q^{95} -7.84085 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40q - 40q^{7} + O(q^{10}) \) \( 40q - 40q^{7} - 24q^{13} + 32q^{19} - 8q^{37} - 32q^{43} + 40q^{49} - 48q^{55} - 24q^{61} + 64q^{85} + 24q^{91} + 64q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 2.51504 + 2.51504i 1.12476 + 1.12476i 0.991016 + 0.133742i \(0.0426994\pi\)
0.133742 + 0.991016i \(0.457301\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −0.984548 + 0.984548i −0.296852 + 0.296852i −0.839780 0.542927i \(-0.817316\pi\)
0.542927 + 0.839780i \(0.317316\pi\)
\(12\) 0 0
\(13\) −3.26086 3.26086i −0.904399 0.904399i 0.0914142 0.995813i \(-0.470861\pi\)
−0.995813 + 0.0914142i \(0.970861\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 5.50927i 1.33619i −0.744074 0.668097i \(-0.767109\pi\)
0.744074 0.668097i \(-0.232891\pi\)
\(18\) 0 0
\(19\) 1.21444 1.21444i 0.278613 0.278613i −0.553942 0.832555i \(-0.686877\pi\)
0.832555 + 0.553942i \(0.186877\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 8.62021i 1.79744i −0.438526 0.898719i \(-0.644499\pi\)
0.438526 0.898719i \(-0.355501\pi\)
\(24\) 0 0
\(25\) 7.65081i 1.53016i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.04663 2.04663i 0.380050 0.380050i −0.491070 0.871120i \(-0.663394\pi\)
0.871120 + 0.491070i \(0.163394\pi\)
\(30\) 0 0
\(31\) 0.164437i 0.0295337i −0.999891 0.0147669i \(-0.995299\pi\)
0.999891 0.0147669i \(-0.00470061\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2.51504 2.51504i −0.425119 0.425119i
\(36\) 0 0
\(37\) −8.31105 + 8.31105i −1.36633 + 1.36633i −0.500717 + 0.865611i \(0.666930\pi\)
−0.865611 + 0.500717i \(0.833070\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −9.56578 −1.49392 −0.746962 0.664867i \(-0.768488\pi\)
−0.746962 + 0.664867i \(0.768488\pi\)
\(42\) 0 0
\(43\) −5.01867 5.01867i −0.765340 0.765340i 0.211942 0.977282i \(-0.432021\pi\)
−0.977282 + 0.211942i \(0.932021\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.37837 0.492786 0.246393 0.969170i \(-0.420755\pi\)
0.246393 + 0.969170i \(0.420755\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.72717 3.72717i −0.511966 0.511966i 0.403162 0.915129i \(-0.367911\pi\)
−0.915129 + 0.403162i \(0.867911\pi\)
\(54\) 0 0
\(55\) −4.95235 −0.667774
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −10.0781 + 10.0781i −1.31206 + 1.31206i −0.392162 + 0.919896i \(0.628273\pi\)
−0.919896 + 0.392162i \(0.871727\pi\)
\(60\) 0 0
\(61\) −1.07042 1.07042i −0.137054 0.137054i 0.635252 0.772305i \(-0.280896\pi\)
−0.772305 + 0.635252i \(0.780896\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 16.4023i 2.03446i
\(66\) 0 0
\(67\) 3.12073 3.12073i 0.381258 0.381258i −0.490297 0.871555i \(-0.663112\pi\)
0.871555 + 0.490297i \(0.163112\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.66257i 0.909380i 0.890650 + 0.454690i \(0.150250\pi\)
−0.890650 + 0.454690i \(0.849750\pi\)
\(72\) 0 0
\(73\) 8.40588i 0.983834i −0.870642 0.491917i \(-0.836296\pi\)
0.870642 0.491917i \(-0.163704\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.984548 0.984548i 0.112200 0.112200i
\(78\) 0 0
\(79\) 13.8712i 1.56063i −0.625387 0.780315i \(-0.715059\pi\)
0.625387 0.780315i \(-0.284941\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −7.31214 7.31214i −0.802611 0.802611i 0.180892 0.983503i \(-0.442102\pi\)
−0.983503 + 0.180892i \(0.942102\pi\)
\(84\) 0 0
\(85\) 13.8560 13.8560i 1.50290 1.50290i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.49158 −0.794106 −0.397053 0.917796i \(-0.629967\pi\)
−0.397053 + 0.917796i \(0.629967\pi\)
\(90\) 0 0
\(91\) 3.26086 + 3.26086i 0.341831 + 0.341831i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 6.10874 0.626744
\(96\) 0 0
\(97\) −7.84085 −0.796118 −0.398059 0.917360i \(-0.630316\pi\)
−0.398059 + 0.917360i \(0.630316\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −0.748257 0.748257i −0.0744543 0.0744543i 0.668899 0.743353i \(-0.266766\pi\)
−0.743353 + 0.668899i \(0.766766\pi\)
\(102\) 0 0
\(103\) 11.8161 1.16428 0.582140 0.813089i \(-0.302216\pi\)
0.582140 + 0.813089i \(0.302216\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −4.41189 + 4.41189i −0.426514 + 0.426514i −0.887439 0.460925i \(-0.847518\pi\)
0.460925 + 0.887439i \(0.347518\pi\)
\(108\) 0 0
\(109\) −0.128743 0.128743i −0.0123314 0.0123314i 0.700914 0.713246i \(-0.252776\pi\)
−0.713246 + 0.700914i \(0.752776\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 14.3042i 1.34563i −0.739811 0.672815i \(-0.765085\pi\)
0.739811 0.672815i \(-0.234915\pi\)
\(114\) 0 0
\(115\) 21.6801 21.6801i 2.02168 2.02168i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 5.50927i 0.505034i
\(120\) 0 0
\(121\) 9.06133i 0.823757i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −6.66689 + 6.66689i −0.596304 + 0.596304i
\(126\) 0 0
\(127\) 1.31914i 0.117055i −0.998286 0.0585274i \(-0.981359\pi\)
0.998286 0.0585274i \(-0.0186405\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.84141 3.84141i −0.335626 0.335626i 0.519093 0.854718i \(-0.326270\pi\)
−0.854718 + 0.519093i \(0.826270\pi\)
\(132\) 0 0
\(133\) −1.21444 + 1.21444i −0.105306 + 0.105306i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 11.8262 1.01038 0.505191 0.863008i \(-0.331422\pi\)
0.505191 + 0.863008i \(0.331422\pi\)
\(138\) 0 0
\(139\) 10.6322 + 10.6322i 0.901814 + 0.901814i 0.995593 0.0937788i \(-0.0298947\pi\)
−0.0937788 + 0.995593i \(0.529895\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.42094 0.536946
\(144\) 0 0
\(145\) 10.2947 0.854930
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −9.66703 9.66703i −0.791954 0.791954i 0.189858 0.981812i \(-0.439197\pi\)
−0.981812 + 0.189858i \(0.939197\pi\)
\(150\) 0 0
\(151\) 5.05132 0.411070 0.205535 0.978650i \(-0.434107\pi\)
0.205535 + 0.978650i \(0.434107\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0.413565 0.413565i 0.0332183 0.0332183i
\(156\) 0 0
\(157\) 11.1102 + 11.1102i 0.886692 + 0.886692i 0.994204 0.107512i \(-0.0342884\pi\)
−0.107512 + 0.994204i \(0.534288\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.62021i 0.679367i
\(162\) 0 0
\(163\) −0.936756 + 0.936756i −0.0733724 + 0.0733724i −0.742841 0.669468i \(-0.766522\pi\)
0.669468 + 0.742841i \(0.266522\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 17.9147i 1.38628i −0.720804 0.693139i \(-0.756227\pi\)
0.720804 0.693139i \(-0.243773\pi\)
\(168\) 0 0
\(169\) 8.26636i 0.635874i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.95689 1.95689i 0.148780 0.148780i −0.628793 0.777573i \(-0.716451\pi\)
0.777573 + 0.628793i \(0.216451\pi\)
\(174\) 0 0
\(175\) 7.65081i 0.578347i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 16.4536 + 16.4536i 1.22980 + 1.22980i 0.964039 + 0.265760i \(0.0856230\pi\)
0.265760 + 0.964039i \(0.414377\pi\)
\(180\) 0 0
\(181\) 2.01639 2.01639i 0.149877 0.149877i −0.628186 0.778063i \(-0.716202\pi\)
0.778063 + 0.628186i \(0.216202\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −41.8052 −3.07358
\(186\) 0 0
\(187\) 5.42414 + 5.42414i 0.396653 + 0.396653i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −6.72330 −0.486481 −0.243240 0.969966i \(-0.578210\pi\)
−0.243240 + 0.969966i \(0.578210\pi\)
\(192\) 0 0
\(193\) 1.37543 0.0990058 0.0495029 0.998774i \(-0.484236\pi\)
0.0495029 + 0.998774i \(0.484236\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 10.2035 + 10.2035i 0.726967 + 0.726967i 0.970014 0.243048i \(-0.0781472\pi\)
−0.243048 + 0.970014i \(0.578147\pi\)
\(198\) 0 0
\(199\) 14.9391 1.05900 0.529501 0.848309i \(-0.322379\pi\)
0.529501 + 0.848309i \(0.322379\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.04663 + 2.04663i −0.143646 + 0.143646i
\(204\) 0 0
\(205\) −24.0583 24.0583i −1.68030 1.68030i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 2.39136i 0.165414i
\(210\) 0 0
\(211\) −4.93531 + 4.93531i −0.339761 + 0.339761i −0.856277 0.516516i \(-0.827228\pi\)
0.516516 + 0.856277i \(0.327228\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 25.2443i 1.72164i
\(216\) 0 0
\(217\) 0.164437i 0.0111627i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −17.9649 + 17.9649i −1.20845 + 1.20845i
\(222\) 0 0
\(223\) 5.25747i 0.352066i 0.984384 + 0.176033i \(0.0563266\pi\)
−0.984384 + 0.176033i \(0.943673\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −6.30931 6.30931i −0.418763 0.418763i 0.466014 0.884777i \(-0.345690\pi\)
−0.884777 + 0.466014i \(0.845690\pi\)
\(228\) 0 0
\(229\) 19.1785 19.1785i 1.26735 1.26735i 0.319899 0.947452i \(-0.396351\pi\)
0.947452 0.319899i \(-0.103649\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.555029 0.0363611 0.0181806 0.999835i \(-0.494213\pi\)
0.0181806 + 0.999835i \(0.494213\pi\)
\(234\) 0 0
\(235\) 8.49672 + 8.49672i 0.554265 + 0.554265i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −23.0408 −1.49039 −0.745194 0.666848i \(-0.767643\pi\)
−0.745194 + 0.666848i \(0.767643\pi\)
\(240\) 0 0
\(241\) −7.47827 −0.481718 −0.240859 0.970560i \(-0.577429\pi\)
−0.240859 + 0.970560i \(0.577429\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.51504 + 2.51504i 0.160680 + 0.160680i
\(246\) 0 0
\(247\) −7.92026 −0.503954
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.7049 18.7049i 1.18064 1.18064i 0.201066 0.979578i \(-0.435559\pi\)
0.979578 0.201066i \(-0.0644407\pi\)
\(252\) 0 0
\(253\) 8.48701 + 8.48701i 0.533574 + 0.533574i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 18.3076i 1.14200i 0.820951 + 0.570998i \(0.193444\pi\)
−0.820951 + 0.570998i \(0.806556\pi\)
\(258\) 0 0
\(259\) 8.31105 8.31105i 0.516423 0.516423i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 20.1306i 1.24130i −0.784086 0.620652i \(-0.786868\pi\)
0.784086 0.620652i \(-0.213132\pi\)
\(264\) 0 0
\(265\) 18.7479i 1.15168i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 8.69779 8.69779i 0.530313 0.530313i −0.390352 0.920666i \(-0.627647\pi\)
0.920666 + 0.390352i \(0.127647\pi\)
\(270\) 0 0
\(271\) 11.6299i 0.706466i −0.935535 0.353233i \(-0.885082\pi\)
0.935535 0.353233i \(-0.114918\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −7.53259 7.53259i −0.454232 0.454232i
\(276\) 0 0
\(277\) 22.9876 22.9876i 1.38119 1.38119i 0.538679 0.842511i \(-0.318923\pi\)
0.842511 0.538679i \(-0.181077\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −11.5187 −0.687148 −0.343574 0.939126i \(-0.611638\pi\)
−0.343574 + 0.939126i \(0.611638\pi\)
\(282\) 0 0
\(283\) 15.6255 + 15.6255i 0.928839 + 0.928839i 0.997631 0.0687924i \(-0.0219146\pi\)
−0.0687924 + 0.997631i \(0.521915\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 9.56578 0.564650
\(288\) 0 0
\(289\) −13.3521 −0.785417
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3.77816 + 3.77816i 0.220722 + 0.220722i 0.808803 0.588080i \(-0.200116\pi\)
−0.588080 + 0.808803i \(0.700116\pi\)
\(294\) 0 0
\(295\) −50.6936 −2.95150
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −28.1093 + 28.1093i −1.62560 + 1.62560i
\(300\) 0 0
\(301\) 5.01867 + 5.01867i 0.289271 + 0.289271i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.38430i 0.308304i
\(306\) 0 0
\(307\) −0.708546 + 0.708546i −0.0404389 + 0.0404389i −0.727037 0.686598i \(-0.759103\pi\)
0.686598 + 0.727037i \(0.259103\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 21.6074i 1.22525i 0.790376 + 0.612623i \(0.209885\pi\)
−0.790376 + 0.612623i \(0.790115\pi\)
\(312\) 0 0
\(313\) 5.11637i 0.289194i 0.989491 + 0.144597i \(0.0461886\pi\)
−0.989491 + 0.144597i \(0.953811\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 7.88622 7.88622i 0.442934 0.442934i −0.450063 0.892997i \(-0.648598\pi\)
0.892997 + 0.450063i \(0.148598\pi\)
\(318\) 0 0
\(319\) 4.03002i 0.225638i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −6.69070 6.69070i −0.372281 0.372281i
\(324\) 0 0
\(325\) 24.9482 24.9482i 1.38388 1.38388i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −3.37837 −0.186256
\(330\) 0 0
\(331\) −0.144453 0.144453i −0.00793985 0.00793985i 0.703126 0.711066i \(-0.251787\pi\)
−0.711066 + 0.703126i \(0.751787\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 15.6975 0.857646
\(336\) 0 0
\(337\) −18.3255 −0.998256 −0.499128 0.866528i \(-0.666346\pi\)
−0.499128 + 0.866528i \(0.666346\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0.161896 + 0.161896i 0.00876716 + 0.00876716i
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.70010 + 7.70010i −0.413363 + 0.413363i −0.882908 0.469545i \(-0.844418\pi\)
0.469545 + 0.882908i \(0.344418\pi\)
\(348\) 0 0
\(349\) 9.23902 + 9.23902i 0.494553 + 0.494553i 0.909737 0.415184i \(-0.136283\pi\)
−0.415184 + 0.909737i \(0.636283\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 13.9051i 0.740092i −0.929013 0.370046i \(-0.879342\pi\)
0.929013 0.370046i \(-0.120658\pi\)
\(354\) 0 0
\(355\) −19.2716 + 19.2716i −1.02283 + 1.02283i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 20.3979i 1.07656i −0.842766 0.538281i \(-0.819074\pi\)
0.842766 0.538281i \(-0.180926\pi\)
\(360\) 0 0
\(361\) 16.0503i 0.844750i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 21.1411 21.1411i 1.10658 1.10658i
\(366\) 0 0
\(367\) 21.2755i 1.11057i −0.831660 0.555285i \(-0.812609\pi\)
0.831660 0.555285i \(-0.187391\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.72717 + 3.72717i 0.193505 + 0.193505i
\(372\) 0 0
\(373\) 14.6527 14.6527i 0.758688 0.758688i −0.217396 0.976084i \(-0.569756\pi\)
0.976084 + 0.217396i \(0.0697562\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −13.3476 −0.687434
\(378\) 0 0
\(379\) 8.85565 + 8.85565i 0.454884 + 0.454884i 0.896972 0.442088i \(-0.145762\pi\)
−0.442088 + 0.896972i \(0.645762\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −33.3573 −1.70448 −0.852238 0.523154i \(-0.824755\pi\)
−0.852238 + 0.523154i \(0.824755\pi\)
\(384\) 0 0
\(385\) 4.95235 0.252395
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −2.00738 2.00738i −0.101778 0.101778i 0.654384 0.756162i \(-0.272928\pi\)
−0.756162 + 0.654384i \(0.772928\pi\)
\(390\) 0 0
\(391\) −47.4911 −2.40173
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 34.8865 34.8865i 1.75533 1.75533i
\(396\) 0 0
\(397\) −2.43985 2.43985i −0.122453 0.122453i 0.643225 0.765677i \(-0.277596\pi\)
−0.765677 + 0.643225i \(0.777596\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 11.6798i 0.583263i −0.956531 0.291632i \(-0.905802\pi\)
0.956531 0.291632i \(-0.0941981\pi\)
\(402\) 0 0
\(403\) −0.536205 + 0.536205i −0.0267103 + 0.0267103i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 16.3653i 0.811195i
\(408\) 0 0
\(409\) 15.2951i 0.756292i 0.925746 + 0.378146i \(0.123438\pi\)
−0.925746 + 0.378146i \(0.876562\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 10.0781 10.0781i 0.495911 0.495911i
\(414\) 0 0
\(415\) 36.7806i 1.80549i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 13.1833 + 13.1833i 0.644045 + 0.644045i 0.951547 0.307502i \(-0.0994932\pi\)
−0.307502 + 0.951547i \(0.599493\pi\)
\(420\) 0 0
\(421\) −22.2574 + 22.2574i −1.08476 + 1.08476i −0.0887017 + 0.996058i \(0.528272\pi\)
−0.996058 + 0.0887017i \(0.971728\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 42.1504 2.04459
\(426\) 0 0
\(427\) 1.07042 + 1.07042i 0.0518014 + 0.0518014i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −19.0326 −0.916769 −0.458385 0.888754i \(-0.651572\pi\)
−0.458385 + 0.888754i \(0.651572\pi\)
\(432\) 0 0
\(433\) −31.7120 −1.52398 −0.761992 0.647587i \(-0.775778\pi\)
−0.761992 + 0.647587i \(0.775778\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −10.4688 10.4688i −0.500789 0.500789i
\(438\) 0 0
\(439\) −32.9521 −1.57272 −0.786359 0.617770i \(-0.788036\pi\)
−0.786359 + 0.617770i \(0.788036\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −4.94776 + 4.94776i −0.235075 + 0.235075i −0.814807 0.579732i \(-0.803157\pi\)
0.579732 + 0.814807i \(0.303157\pi\)
\(444\) 0 0
\(445\) −18.8416 18.8416i −0.893177 0.893177i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 31.5258i 1.48779i 0.668294 + 0.743897i \(0.267025\pi\)
−0.668294 + 0.743897i \(0.732975\pi\)
\(450\) 0 0
\(451\) 9.41796 9.41796i 0.443475 0.443475i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 16.4023i 0.768954i
\(456\) 0 0
\(457\) 35.2517i 1.64901i 0.565857 + 0.824503i \(0.308545\pi\)
−0.565857 + 0.824503i \(0.691455\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 15.8769 15.8769i 0.739462 0.739462i −0.233012 0.972474i \(-0.574858\pi\)
0.972474 + 0.233012i \(0.0748582\pi\)
\(462\) 0 0
\(463\) 15.9217i 0.739946i 0.929043 + 0.369973i \(0.120633\pi\)
−0.929043 + 0.369973i \(0.879367\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6.52420 6.52420i −0.301904 0.301904i 0.539854 0.841758i \(-0.318479\pi\)
−0.841758 + 0.539854i \(0.818479\pi\)
\(468\) 0 0
\(469\) −3.12073 + 3.12073i −0.144102 + 0.144102i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9.88224 0.454386
\(474\) 0 0
\(475\) 9.29148 + 9.29148i 0.426323 + 0.426323i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −31.5891 −1.44334 −0.721671 0.692237i \(-0.756625\pi\)
−0.721671 + 0.692237i \(0.756625\pi\)
\(480\) 0 0
\(481\) 54.2023 2.47141
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −19.7200 19.7200i −0.895440 0.895440i
\(486\) 0 0
\(487\) 29.4810 1.33591 0.667955 0.744202i \(-0.267170\pi\)
0.667955 + 0.744202i \(0.267170\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −6.06862 + 6.06862i −0.273873 + 0.273873i −0.830657 0.556784i \(-0.812035\pi\)
0.556784 + 0.830657i \(0.312035\pi\)
\(492\) 0 0
\(493\) −11.2755 11.2755i −0.507821 0.507821i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 7.66257i 0.343713i
\(498\) 0 0
\(499\) −15.8968 + 15.8968i −0.711638 + 0.711638i −0.966878 0.255240i \(-0.917846\pi\)
0.255240 + 0.966878i \(0.417846\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 17.1978i 0.766813i 0.923580 + 0.383406i \(0.125249\pi\)
−0.923580 + 0.383406i \(0.874751\pi\)
\(504\) 0 0
\(505\) 3.76379i 0.167486i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −0.959431 + 0.959431i −0.0425260 + 0.0425260i −0.728050 0.685524i \(-0.759573\pi\)
0.685524 + 0.728050i \(0.259573\pi\)
\(510\) 0 0
\(511\) 8.40588i 0.371854i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 29.7180 + 29.7180i 1.30953 + 1.30953i
\(516\) 0 0
\(517\) −3.32617 + 3.32617i −0.146285 + 0.146285i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 9.04026 0.396061 0.198030 0.980196i \(-0.436546\pi\)
0.198030 + 0.980196i \(0.436546\pi\)
\(522\) 0 0
\(523\) 5.32615 + 5.32615i 0.232896 + 0.232896i 0.813901 0.581004i \(-0.197340\pi\)
−0.581004 + 0.813901i \(0.697340\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −0.905927 −0.0394628
\(528\) 0 0
\(529\) −51.3080 −2.23078
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 31.1926 + 31.1926i 1.35110 + 1.35110i
\(534\) 0 0
\(535\) −22.1921 −0.959449
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −0.984548 + 0.984548i −0.0424075 + 0.0424075i
\(540\) 0 0
\(541\) 12.7082 + 12.7082i 0.546367 + 0.546367i 0.925388 0.379021i \(-0.123739\pi\)
−0.379021 + 0.925388i \(0.623739\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.647588i 0.0277396i
\(546\) 0 0
\(547\) −19.7217 + 19.7217i −0.843239 + 0.843239i −0.989279 0.146040i \(-0.953347\pi\)
0.146040 + 0.989279i \(0.453347\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 4.97105i 0.211774i
\(552\) 0 0
\(553\) 13.8712i 0.589863i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −32.2571 + 32.2571i −1.36678 + 1.36678i −0.501789 + 0.864990i \(0.667325\pi\)
−0.864990 + 0.501789i \(0.832675\pi\)
\(558\) 0 0
\(559\) 32.7303i 1.38434i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −27.0814 27.0814i −1.14134 1.14134i −0.988205 0.153138i \(-0.951062\pi\)
−0.153138 0.988205i \(1.45106\pi\)
\(564\) 0 0
\(565\) 35.9757 35.9757i 1.51351 1.51351i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −28.4823 −1.19404 −0.597020 0.802226i \(-0.703649\pi\)
−0.597020 + 0.802226i \(0.703649\pi\)
\(570\) 0 0
\(571\) −15.8844 15.8844i −0.664743 0.664743i 0.291751 0.956494i \(-0.405762\pi\)
−0.956494 + 0.291751i \(0.905762\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 65.9516 2.75037
\(576\) 0 0
\(577\) 4.65713 0.193879 0.0969395 0.995290i \(-0.469095\pi\)
0.0969395 + 0.995290i \(0.469095\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 7.31214 + 7.31214i 0.303359 + 0.303359i
\(582\) 0 0
\(583\) 7.33916 0.303957
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −13.2034 + 13.2034i −0.544963 + 0.544963i −0.924980 0.380017i \(-0.875918\pi\)
0.380017 + 0.924980i \(0.375918\pi\)
\(588\) 0 0
\(589\) −0.199699 0.199699i −0.00822847 0.00822847i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 8.00745i 0.328827i −0.986392 0.164413i \(-0.947427\pi\)
0.986392 0.164413i \(-0.0525731\pi\)
\(594\) 0 0
\(595\) −13.8560 + 13.8560i −0.568041 + 0.568041i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 29.8782i 1.22079i −0.792096 0.610396i \(-0.791010\pi\)
0.792096 0.610396i \(-0.208990\pi\)
\(600\) 0 0
\(601\) 28.4594i 1.16088i −0.814302 0.580441i \(-0.802880\pi\)
0.814302 0.580441i \(-0.197120\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −22.7896 + 22.7896i −0.926528 + 0.926528i
\(606\) 0 0
\(607\) 27.1606i 1.10242i −0.834368 0.551208i \(-0.814167\pi\)
0.834368 0.551208i \(-0.185833\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −11.0164 11.0164i −0.445675 0.445675i
\(612\) 0 0
\(613\) −25.1239 + 25.1239i −1.01474 + 1.01474i −0.0148539 + 0.999890i \(0.504728\pi\)
−0.999890 + 0.0148539i \(0.995272\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 20.2712 0.816089 0.408044 0.912962i \(-0.366211\pi\)
0.408044 + 0.912962i \(0.366211\pi\)
\(618\) 0 0
\(619\) 13.2344 + 13.2344i 0.531937 + 0.531937i 0.921148 0.389211i \(-0.127252\pi\)
−0.389211 + 0.921148i \(0.627252\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 7.49158 0.300144
\(624\) 0 0
\(625\) 4.71914 0.188766
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 45.7878 + 45.7878i 1.82568 + 1.82568i
\(630\) 0 0
\(631\) 47.8911 1.90651 0.953257 0.302161i \(-0.0977080\pi\)
0.953257 + 0.302161i \(0.0977080\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3.31769 3.31769i 0.131658 0.131658i
\(636\) 0 0
\(637\) −3.26086 3.26086i −0.129200 0.129200i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 23.5861i 0.931595i −0.884891 0.465797i \(-0.845768\pi\)
0.884891 0.465797i \(-0.154232\pi\)
\(642\) 0 0
\(643\) 29.9073 29.9073i 1.17943 1.17943i 0.199537 0.979890i \(-0.436056\pi\)
0.979890 0.199537i \(-0.0639437\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 35.8559i 1.40964i −0.709385 0.704821i \(-0.751027\pi\)
0.709385 0.704821i \(-0.248973\pi\)
\(648\) 0 0
\(649\) 19.8448i 0.778975i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 26.7966 26.7966i 1.04863 1.04863i 0.0498765 0.998755i \(-0.484117\pi\)
0.998755 0.0498765i \(-0.0158828\pi\)
\(654\) 0 0
\(655\) 19.3226i 0.754995i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −19.9220 19.9220i −0.776050 0.776050i 0.203107 0.979157i \(-0.434896\pi\)
−0.979157 + 0.203107i \(0.934896\pi\)
\(660\) 0 0
\(661\) 3.02969 3.02969i 0.117841 0.117841i −0.645727 0.763568i \(-0.723446\pi\)
0.763568 + 0.645727i \(0.223446\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −6.10874 −0.236887
\(666\) 0 0
\(667\) −17.6424 17.6424i −0.683117 0.683117i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2.10776 0.0813693
\(672\) 0 0
\(673\) 41.8102 1.61166 0.805832 0.592144i \(-0.201718\pi\)
0.805832 + 0.592144i \(0.201718\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 11.0622 + 11.0622i 0.425154 + 0.425154i 0.886974 0.461820i \(-0.152803\pi\)
−0.461820 + 0.886974i \(0.652803\pi\)
\(678\) 0 0
\(679\) 7.84085 0.300904
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −25.6844 + 25.6844i −0.982786 + 0.982786i −0.999854 0.0170687i \(-0.994567\pi\)
0.0170687 + 0.999854i \(0.494567\pi\)
\(684\) 0 0
\(685\) 29.7434 + 29.7434i 1.13644 + 1.13644i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 24.3075i 0.926044i
\(690\) 0 0
\(691\) −15.7276 + 15.7276i −0.598306 + 0.598306i −0.939862 0.341556i \(-0.889046\pi\)
0.341556 + 0.939862i \(0.389046\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 53.4809i 2.02865i
\(696\) 0 0
\(697\) 52.7005i 1.99617i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 4.35804 4.35804i 0.164601 0.164601i −0.620000 0.784601i \(-0.712868\pi\)
0.784601 + 0.620000i \(0.212868\pi\)
\(702\) 0 0
\(703\) 20.1866i 0.761352i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0.748257 + 0.748257i 0.0281411 + 0.0281411i
\(708\) 0 0
\(709\) −2.68056 + 2.68056i −0.100670 + 0.100670i −0.755648 0.654978i \(-0.772678\pi\)
0.654978 + 0.755648i \(0.272678\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.41748 −0.0530850
\(714\) 0 0
\(715\) 16.1489 + 16.1489i 0.603934 + 0.603934i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 19.9117 0.742581 0.371290 0.928517i \(-0.378915\pi\)
0.371290 + 0.928517i \(0.378915\pi\)
\(720\) 0 0
\(721\) −11.8161 −0.440056
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 15.6584 + 15.6584i 0.581539 + 0.581539i
\(726\) 0 0
\(727\) −24.0642 −0.892491 −0.446246 0.894911i \(-0.647239\pi\)
−0.446246 + 0.894911i \(0.647239\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −27.6492 + 27.6492i −1.02264 + 1.02264i
\(732\) 0 0
\(733\) −3.18890 3.18890i −0.117785 0.117785i 0.645758 0.763542i \(-0.276542\pi\)
−0.763542 + 0.645758i \(0.776542\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.14502i 0.226355i
\(738\) 0 0
\(739\) −20.8311 + 20.8311i −0.766285 + 0.766285i −0.977450 0.211165i \(-0.932274\pi\)
0.211165 + 0.977450i \(0.432274\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 28.6291i 1.05030i 0.851010 + 0.525150i \(0.175991\pi\)
−0.851010 + 0.525150i \(0.824009\pi\)
\(744\) 0 0
\(745\) 48.6258i 1.78151i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 4.41189 4.41189i 0.161207 0.161207i
\(750\) 0 0
\(751\) 28.1706i 1.02796i 0.857802 + 0.513980i \(0.171830\pi\)
−0.857802 + 0.513980i \(0.828170\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 12.7042 + 12.7042i 0.462355 + 0.462355i
\(756\) 0 0
\(757\) −4.64056 + 4.64056i −0.168664 + 0.168664i −0.786392 0.617728i \(-0.788053\pi\)
0.617728 + 0.786392i \(0.288053\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −6.29768 −0.228291 −0.114145 0.993464i \(-0.536413\pi\)
−0.114145 + 0.993464i \(0.536413\pi\)
\(762\) 0 0
\(763\) 0.128743 + 0.128743i 0.00466082 + 0.00466082i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 65.7265 2.37325
\(768\) 0 0
\(769\) 4.73382 0.170706 0.0853529 0.996351i \(-0.472798\pi\)
0.0853529 + 0.996351i \(0.472798\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 5.44275 + 5.44275i 0.195762 + 0.195762i 0.798180 0.602418i \(-0.205796\pi\)
−0.602418 + 0.798180i \(0.705796\pi\)
\(774\) 0 0
\(775\) 1.25808 0.0451914
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −11.6171 + 11.6171i −0.416226 + 0.416226i
\(780\) 0 0
\(781\) −7.54417 7.54417i −0.269952 0.269952i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 55.8852i 1.99463i
\(786\) 0 0
\(787\) 27.1082 27.1082i 0.966302 0.966302i −0.0331485 0.999450i \(-0.510553\pi\)
0.999450 + 0.0331485i \(0.0105534\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 14.3042i 0.508600i
\(792\) 0 0
\(793\) 6.98099i 0.247902i
\(794\) 0 0