Properties

Label 4032.2.p.k.1567.2
Level 4032
Weight 2
Character 4032.1567
Analytic conductor 32.196
Analytic rank 0
Dimension 12
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.p (of order \(2\) and degree \(1\))

Newform invariants

Self dual: No
Analytic conductor: \(32.195682095\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{14} \)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.2
Root \(1.75780 - 0.500000i\)
Character \(\chi\) = 4032.1567
Dual form 4032.2.p.k.1567.1

$q$-expansion

\(f(q)\) \(=\) \(q\)\(-3.88448 q^{5}\) \(+(-2.62383 + 0.339877i) q^{7}\) \(+O(q^{10})\) \(q\)\(-3.88448 q^{5}\) \(+(-2.62383 + 0.339877i) q^{7}\) \(+1.48046 q^{11}\) \(+2.00000 q^{13}\) \(+3.76720i q^{17}\) \(-0.679754i q^{19}\) \(+5.20473i q^{23}\) \(+10.0892 q^{25}\) \(+7.03122i q^{29}\) \(+6.42503 q^{31}\) \(+(10.1922 - 1.32025i) q^{35}\) \(-8.71176i q^{37}\) \(-3.16101i q^{41}\) \(+8.20859 q^{43}\) \(-9.88913 q^{47}\) \(+(6.76897 - 1.78356i) q^{49}\) \(-6.42503i q^{53}\) \(-5.75084 q^{55}\) \(-7.76897i q^{59}\) \(-12.4095 q^{61}\) \(-7.76897 q^{65}\) \(-8.81478 q^{67}\) \(+12.9737i q^{71}\) \(+13.4562i q^{73}\) \(+(-3.88448 + 0.503175i) q^{77}\) \(-4.44872i q^{79}\) \(-10.4095i q^{83}\) \(-14.6336i q^{85}\) \(+10.6954i q^{89}\) \(+(-5.24766 + 0.679754i) q^{91}\) \(+2.64049i q^{95}\) \(+9.88913i q^{97}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut +\mathstrut 24q^{13} \) \(\mathstrut +\mathstrut 36q^{25} \) \(\mathstrut -\mathstrut 12q^{49} \) \(\mathstrut -\mathstrut 72q^{61} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Character Values

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

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −3.88448 −1.73719 −0.868597 0.495519i \(-0.834978\pi\)
−0.868597 + 0.495519i \(0.834978\pi\)
\(6\) 0 0
\(7\) −2.62383 + 0.339877i −0.991715 + 0.128461i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.48046 0.446376 0.223188 0.974775i \(-0.428354\pi\)
0.223188 + 0.974775i \(0.428354\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.76720i 0.913679i 0.889549 + 0.456840i \(0.151019\pi\)
−0.889549 + 0.456840i \(0.848981\pi\)
\(18\) 0 0
\(19\) 0.679754i 0.155946i −0.996955 0.0779731i \(-0.975155\pi\)
0.996955 0.0779731i \(-0.0248448\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.20473i 1.08526i 0.839971 + 0.542631i \(0.182572\pi\)
−0.839971 + 0.542631i \(0.817428\pi\)
\(24\) 0 0
\(25\) 10.0892 2.01784
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.03122i 1.30566i 0.757503 + 0.652832i \(0.226419\pi\)
−0.757503 + 0.652832i \(0.773581\pi\)
\(30\) 0 0
\(31\) 6.42503 1.15397 0.576985 0.816755i \(-0.304229\pi\)
0.576985 + 0.816755i \(0.304229\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 10.1922 1.32025i 1.72280 0.223162i
\(36\) 0 0
\(37\) 8.71176i 1.43220i −0.697995 0.716102i \(-0.745924\pi\)
0.697995 0.716102i \(-0.254076\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.16101i 0.493666i −0.969058 0.246833i \(-0.920610\pi\)
0.969058 0.246833i \(-0.0793900\pi\)
\(42\) 0 0
\(43\) 8.20859 1.25180 0.625899 0.779904i \(-0.284732\pi\)
0.625899 + 0.779904i \(0.284732\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −9.88913 −1.44248 −0.721239 0.692686i \(-0.756427\pi\)
−0.721239 + 0.692686i \(0.756427\pi\)
\(48\) 0 0
\(49\) 6.76897 1.78356i 0.966995 0.254794i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.42503i 0.882545i −0.897373 0.441273i \(-0.854527\pi\)
0.897373 0.441273i \(-0.145473\pi\)
\(54\) 0 0
\(55\) −5.75084 −0.775442
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.76897i 1.01143i −0.862700 0.505717i \(-0.831228\pi\)
0.862700 0.505717i \(-0.168772\pi\)
\(60\) 0 0
\(61\) −12.4095 −1.58887 −0.794434 0.607350i \(-0.792233\pi\)
−0.794434 + 0.607350i \(0.792233\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −7.76897 −0.963622
\(66\) 0 0
\(67\) −8.81478 −1.07690 −0.538448 0.842659i \(-0.680989\pi\)
−0.538448 + 0.842659i \(0.680989\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.9737i 1.53969i 0.638228 + 0.769847i \(0.279668\pi\)
−0.638228 + 0.769847i \(0.720332\pi\)
\(72\) 0 0
\(73\) 13.4562i 1.57493i 0.616356 + 0.787467i \(0.288608\pi\)
−0.616356 + 0.787467i \(0.711392\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −3.88448 + 0.503175i −0.442678 + 0.0573421i
\(78\) 0 0
\(79\) 4.44872i 0.500520i −0.968179 0.250260i \(-0.919484\pi\)
0.968179 0.250260i \(-0.0805161\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 10.4095i 1.14259i −0.820746 0.571293i \(-0.806442\pi\)
0.820746 0.571293i \(-0.193558\pi\)
\(84\) 0 0
\(85\) 14.6336i 1.58724i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.6954i 1.13371i 0.823818 + 0.566855i \(0.191840\pi\)
−0.823818 + 0.566855i \(0.808160\pi\)
\(90\) 0 0
\(91\) −5.24766 + 0.679754i −0.550104 + 0.0712576i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.64049i 0.270909i
\(96\) 0 0
\(97\) 9.88913i 1.00409i 0.864842 + 0.502044i \(0.167419\pi\)
−0.864842 + 0.502044i \(0.832581\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.88448 −0.386521 −0.193260 0.981148i \(-0.561906\pi\)
−0.193260 + 0.981148i \(0.561906\pi\)
\(102\) 0 0
\(103\) −3.06394 −0.301899 −0.150950 0.988541i \(-0.548233\pi\)
−0.150950 + 0.988541i \(0.548233\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.08665 −0.201724 −0.100862 0.994900i \(-0.532160\pi\)
−0.100862 + 0.994900i \(0.532160\pi\)
\(108\) 0 0
\(109\) 17.4235i 1.66887i 0.551106 + 0.834435i \(0.314206\pi\)
−0.551106 + 0.834435i \(0.685794\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.76897 0.166410 0.0832052 0.996532i \(-0.473484\pi\)
0.0832052 + 0.996532i \(0.473484\pi\)
\(114\) 0 0
\(115\) 20.2177i 1.88531i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.28038 9.88448i −0.117373 0.906109i
\(120\) 0 0
\(121\) −8.80823 −0.800748
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −19.7690 −1.76819
\(126\) 0 0
\(127\) 11.0892i 0.984009i −0.870593 0.492004i \(-0.836264\pi\)
0.870593 0.492004i \(-0.163736\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.59054i 0.138966i 0.997583 + 0.0694831i \(0.0221350\pi\)
−0.997583 + 0.0694831i \(0.977865\pi\)
\(132\) 0 0
\(133\) 0.231033 + 1.78356i 0.0200331 + 0.154654i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −18.8974 −1.61452 −0.807259 0.590198i \(-0.799050\pi\)
−0.807259 + 0.590198i \(0.799050\pi\)
\(138\) 0 0
\(139\) 10.1784i 0.863323i −0.902036 0.431661i \(-0.857928\pi\)
0.902036 0.431661i \(-0.142072\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.96093 0.247605
\(144\) 0 0
\(145\) 27.3127i 2.26819i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 4.07029i 0.333451i 0.986003 + 0.166726i \(0.0533194\pi\)
−0.986003 + 0.166726i \(0.946681\pi\)
\(150\) 0 0
\(151\) 8.67975i 0.706348i −0.935558 0.353174i \(-0.885102\pi\)
0.935558 0.353174i \(-0.114898\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −24.9579 −2.00467
\(156\) 0 0
\(157\) 3.12847 0.249679 0.124840 0.992177i \(-0.460158\pi\)
0.124840 + 0.992177i \(0.460158\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1.76897 13.6563i −0.139414 1.07627i
\(162\) 0 0
\(163\) −12.1759 −0.953687 −0.476844 0.878988i \(-0.658219\pi\)
−0.476844 + 0.878988i \(0.658219\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3.56712 −0.276032 −0.138016 0.990430i \(-0.544073\pi\)
−0.138016 + 0.990430i \(0.544073\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −4.93444 −0.375158 −0.187579 0.982249i \(-0.560064\pi\)
−0.187579 + 0.982249i \(0.560064\pi\)
\(174\) 0 0
\(175\) −26.4724 + 3.42909i −2.00112 + 0.259215i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 11.3696 0.849803 0.424902 0.905240i \(-0.360309\pi\)
0.424902 + 0.905240i \(0.360309\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 33.8407i 2.48802i
\(186\) 0 0
\(187\) 5.57720i 0.407845i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 22.3332i 1.61597i −0.589200 0.807987i \(-0.700557\pi\)
0.589200 0.807987i \(-0.299443\pi\)
\(192\) 0 0
\(193\) −20.6271 −1.48477 −0.742387 0.669971i \(-0.766307\pi\)
−0.742387 + 0.669971i \(0.766307\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 23.2424i 1.65595i −0.560765 0.827975i \(-0.689493\pi\)
0.560765 0.827975i \(-0.310507\pi\)
\(198\) 0 0
\(199\) 26.2383 1.85998 0.929992 0.367580i \(-0.119814\pi\)
0.929992 + 0.367580i \(0.119814\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.38975 18.4487i −0.167727 1.29485i
\(204\) 0 0
\(205\) 12.2789i 0.857594i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.00635i 0.0696107i
\(210\) 0 0
\(211\) −12.7821 −0.879953 −0.439976 0.898009i \(-0.645013\pi\)
−0.439976 + 0.898009i \(0.645013\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −31.8861 −2.17462
\(216\) 0 0
\(217\) −16.8582 + 2.18372i −1.14441 + 0.148240i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 7.53439i 0.506818i
\(222\) 0 0
\(223\) −4.03528 −0.270222 −0.135111 0.990830i \(-0.543139\pi\)
−0.135111 + 0.990830i \(0.543139\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 20.8189i 1.38180i −0.722950 0.690900i \(-0.757214\pi\)
0.722950 0.690900i \(-0.242786\pi\)
\(228\) 0 0
\(229\) −26.8974 −1.77743 −0.888717 0.458457i \(-0.848402\pi\)
−0.888717 + 0.458457i \(0.848402\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 25.2284 1.65277 0.826383 0.563108i \(-0.190395\pi\)
0.826383 + 0.563108i \(0.190395\pi\)
\(234\) 0 0
\(235\) 38.4142 2.50586
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.74266i 0.565516i −0.959191 0.282758i \(-0.908751\pi\)
0.959191 0.282758i \(-0.0912493\pi\)
\(240\) 0 0
\(241\) 17.0234i 1.09657i 0.836291 + 0.548286i \(0.184719\pi\)
−0.836291 + 0.548286i \(0.815281\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −26.2939 + 6.92820i −1.67986 + 0.442627i
\(246\) 0 0
\(247\) 1.35951i 0.0865034i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.8974i 1.57151i −0.618536 0.785756i \(-0.712274\pi\)
0.618536 0.785756i \(-0.287726\pi\)
\(252\) 0 0
\(253\) 7.70541i 0.484435i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.56117i 0.222139i 0.993813 + 0.111070i \(0.0354277\pi\)
−0.993813 + 0.111070i \(0.964572\pi\)
\(258\) 0 0
\(259\) 2.96093 + 22.8582i 0.183983 + 1.42034i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 24.9737i 1.53994i −0.638078 0.769972i \(-0.720270\pi\)
0.638078 0.769972i \(-0.279730\pi\)
\(264\) 0 0
\(265\) 24.9579i 1.53315i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 2.83453 0.172824 0.0864122 0.996259i \(-0.472460\pi\)
0.0864122 + 0.996259i \(0.472460\pi\)
\(270\) 0 0
\(271\) 3.86426 0.234737 0.117369 0.993088i \(-0.462554\pi\)
0.117369 + 0.993088i \(0.462554\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 14.9367 0.900717
\(276\) 0 0
\(277\) 1.78356i 0.107164i −0.998563 0.0535818i \(-0.982936\pi\)
0.998563 0.0535818i \(-0.0170638\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −20.4880 −1.22221 −0.611105 0.791549i \(-0.709275\pi\)
−0.611105 + 0.791549i \(0.709275\pi\)
\(282\) 0 0
\(283\) 12.1392i 0.721599i −0.932643 0.360799i \(-0.882504\pi\)
0.932643 0.360799i \(-0.117496\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.07435 + 8.29394i 0.0634171 + 0.489576i
\(288\) 0 0
\(289\) 2.80823 0.165190
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −4.93444 −0.288273 −0.144136 0.989558i \(-0.546040\pi\)
−0.144136 + 0.989558i \(0.546040\pi\)
\(294\) 0 0
\(295\) 30.1784i 1.75706i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 10.4095i 0.601995i
\(300\) 0 0
\(301\) −21.5379 + 2.78991i −1.24143 + 0.160808i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 48.2043 2.76017
\(306\) 0 0
\(307\) 21.4987i 1.22699i 0.789697 + 0.613497i \(0.210238\pi\)
−0.789697 + 0.613497i \(0.789762\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 6.52804 0.370171 0.185086 0.982722i \(-0.440744\pi\)
0.185086 + 0.982722i \(0.440744\pi\)
\(312\) 0 0
\(313\) 33.6347i 1.90114i −0.310505 0.950572i \(-0.600498\pi\)
0.310505 0.950572i \(-0.399502\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 4.07029i 0.228610i 0.993446 + 0.114305i \(0.0364642\pi\)
−0.993446 + 0.114305i \(0.963536\pi\)
\(318\) 0 0
\(319\) 10.4095i 0.582818i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.56077 0.142485
\(324\) 0 0
\(325\) 20.1784 1.11930
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 25.9474 3.36109i 1.43053 0.185303i
\(330\) 0 0
\(331\) 15.1368 0.831993 0.415997 0.909366i \(-0.363433\pi\)
0.415997 + 0.909366i \(0.363433\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 34.2409 1.87078
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 9.51202 0.515105
\(342\) 0 0
\(343\) −17.1544 + 6.98037i −0.926252 + 0.376905i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −8.61470 −0.462461 −0.231231 0.972899i \(-0.574275\pi\)
−0.231231 + 0.972899i \(0.574275\pi\)
\(348\) 0 0
\(349\) 15.5905 0.834542 0.417271 0.908782i \(-0.362987\pi\)
0.417271 + 0.908782i \(0.362987\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 11.9078i 0.633787i −0.948461 0.316894i \(-0.897360\pi\)
0.948461 0.316894i \(-0.102640\pi\)
\(354\) 0 0
\(355\) 50.3961i 2.67475i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 11.0263i 0.581946i 0.956731 + 0.290973i \(0.0939790\pi\)
−0.956731 + 0.290973i \(0.906021\pi\)
\(360\) 0 0
\(361\) 18.5379 0.975681
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 52.2706i 2.73597i
\(366\) 0 0
\(367\) 11.1695 0.583044 0.291522 0.956564i \(-0.405838\pi\)
0.291522 + 0.956564i \(0.405838\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.18372 + 16.8582i 0.113373 + 0.875233i
\(372\) 0 0
\(373\) 8.14058i 0.421503i 0.977540 + 0.210752i \(0.0675912\pi\)
−0.977540 + 0.210752i \(0.932409\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 14.0624i 0.724252i
\(378\) 0 0
\(379\) −17.4915 −0.898479 −0.449240 0.893411i \(-0.648305\pi\)
−0.449240 + 0.893411i \(0.648305\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −31.8861 −1.62930 −0.814652 0.579950i \(-0.803072\pi\)
−0.814652 + 0.579950i \(0.803072\pi\)
\(384\) 0 0
\(385\) 15.0892 1.95458i 0.769018 0.0996144i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 23.2424i 1.17843i 0.807975 + 0.589217i \(0.200564\pi\)
−0.807975 + 0.589217i \(0.799436\pi\)
\(390\) 0 0
\(391\) −19.6072 −0.991581
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 17.2810i 0.869501i
\(396\) 0 0
\(397\) 3.04995 0.153073 0.0765364 0.997067i \(-0.475614\pi\)
0.0765364 + 0.997067i \(0.475614\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −17.4594 −0.871881 −0.435941 0.899975i \(-0.643584\pi\)
−0.435941 + 0.899975i \(0.643584\pi\)
\(402\) 0 0
\(403\) 12.8501 0.640107
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 12.8974i 0.639302i
\(408\) 0 0
\(409\) 17.0234i 0.841751i 0.907118 + 0.420876i \(0.138277\pi\)
−0.907118 + 0.420876i \(0.861723\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 2.64049 + 20.3844i 0.129930 + 1.00305i
\(414\) 0 0
\(415\) 40.4354i 1.98489i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 11.3069i 0.552378i 0.961103 + 0.276189i \(0.0890716\pi\)
−0.961103 + 0.276189i \(0.910928\pi\)
\(420\) 0 0
\(421\) 8.91779i 0.434627i 0.976102 + 0.217313i \(0.0697293\pi\)
−0.976102 + 0.217313i \(0.930271\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 38.0081i 1.84366i
\(426\) 0 0
\(427\) 32.5603 4.21769i 1.57570 0.204108i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 15.4617i 0.744763i 0.928080 + 0.372381i \(0.121459\pi\)
−0.928080 + 0.372381i \(0.878541\pi\)
\(432\) 0 0
\(433\) 7.93455i 0.381310i 0.981657 + 0.190655i \(0.0610612\pi\)
−0.981657 + 0.190655i \(0.938939\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.53793 0.169242
\(438\) 0 0
\(439\) −1.88657 −0.0900412 −0.0450206 0.998986i \(-0.514335\pi\)
−0.0450206 + 0.998986i \(0.514335\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −22.8713 −1.08665 −0.543323 0.839524i \(-0.682834\pi\)
−0.543323 + 0.839524i \(0.682834\pi\)
\(444\) 0 0
\(445\) 41.5461i 1.96947i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −7.04995 −0.332708 −0.166354 0.986066i \(-0.553199\pi\)
−0.166354 + 0.986066i \(0.553199\pi\)
\(450\) 0 0
\(451\) 4.67975i 0.220361i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 20.3844 2.64049i 0.955638 0.123788i
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −4.57758 −0.213199 −0.106600 0.994302i \(-0.533996\pi\)
−0.106600 + 0.994302i \(0.533996\pi\)
\(462\) 0 0
\(463\) 25.2676i 1.17429i 0.809483 + 0.587143i \(0.199748\pi\)
−0.809483 + 0.587143i \(0.800252\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.17843i 0.285904i −0.989730 0.142952i \(-0.954341\pi\)
0.989730 0.142952i \(-0.0456594\pi\)
\(468\) 0 0
\(469\) 23.1285 2.99594i 1.06797 0.138340i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 12.1525 0.558773
\(474\) 0 0
\(475\) 6.85818i 0.314675i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −27.3127 −1.24795 −0.623973 0.781446i \(-0.714483\pi\)
−0.623973 + 0.781446i \(0.714483\pi\)
\(480\) 0 0
\(481\) 17.4235i 0.794444i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 38.4142i 1.74430i
\(486\) 0 0
\(487\) 20.1392i 0.912593i −0.889828 0.456296i \(-0.849176\pi\)
0.889828 0.456296i \(-0.150824\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 9.22089 0.416133 0.208066 0.978115i \(-0.433283\pi\)
0.208066 + 0.978115i \(0.433283\pi\)
\(492\) 0 0
\(493\) −26.4880 −1.19296
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4.40946 34.0408i −0.197791 1.52694i
\(498\) 0 0
\(499\) 4.64147 0.207781 0.103890 0.994589i \(-0.466871\pi\)
0.103890 + 0.994589i \(0.466871\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 23.7455 1.05876 0.529381 0.848384i \(-0.322424\pi\)
0.529381 + 0.848384i \(0.322424\pi\)
\(504\) 0 0
\(505\) 15.0892 0.671461
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 9.16547 0.406252 0.203126 0.979153i \(-0.434890\pi\)
0.203126 + 0.979153i \(0.434890\pi\)
\(510\) 0 0
\(511\) −4.57347 35.3069i −0.202318 1.56189i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 11.9018 0.524457
\(516\) 0 0
\(517\) −14.6405 −0.643888
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 18.8360i 0.825219i 0.910908 + 0.412610i \(0.135383\pi\)
−0.910908 + 0.412610i \(0.864617\pi\)
\(522\) 0 0
\(523\) 41.2543i 1.80392i 0.431815 + 0.901962i \(0.357873\pi\)
−0.431815 + 0.901962i \(0.642127\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 24.2043i 1.05436i
\(528\) 0 0
\(529\) −4.08921 −0.177792
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 6.32201i 0.273837i
\(534\) 0 0
\(535\) 8.10557 0.350434
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 10.0212 2.64049i 0.431644 0.113734i
\(540\) 0 0
\(541\) 37.8430i 1.62700i −0.581567 0.813499i \(-0.697560\pi\)
0.581567 0.813499i \(-0.302440\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 67.6814i 2.89915i
\(546\) 0 0
\(547\) −31.9541 −1.36626 −0.683130 0.730297i \(-0.739382\pi\)
−0.683130 + 0.730297i \(0.739382\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 4.77950 0.203613
\(552\) 0 0
\(553\) 1.51202 + 11.6727i 0.0642975 + 0.496373i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 17.7207i 0.750849i 0.926853 + 0.375424i \(0.122503\pi\)
−0.926853 + 0.375424i \(0.877497\pi\)
\(558\) 0 0
\(559\) 16.4172 0.694372
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5.12847i 0.216139i −0.994143 0.108070i \(-0.965533\pi\)
0.994143 0.108070i \(-0.0344670\pi\)
\(564\) 0 0
\(565\) −6.87153 −0.289087
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −28.4095 −1.19099 −0.595493 0.803360i \(-0.703043\pi\)
−0.595493 + 0.803360i \(0.703043\pi\)
\(570\) 0 0
\(571\) −30.8118 −1.28943 −0.644716 0.764422i \(-0.723024\pi\)
−0.644716 + 0.764422i \(0.723024\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 52.5116i 2.18989i
\(576\) 0 0
\(577\) 36.0594i 1.50117i −0.660772 0.750587i \(-0.729771\pi\)
0.660772 0.750587i \(-0.270229\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 3.53793 + 27.3127i 0.146778 + 1.13312i
\(582\) 0 0
\(583\) 9.51202i 0.393948i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 18.1784i 0.750304i −0.926963 0.375152i \(-0.877591\pi\)
0.926963 0.375152i \(-0.122409\pi\)
\(588\) 0 0
\(589\) 4.36744i 0.179957i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 10.9014i 0.447668i −0.974627 0.223834i \(-0.928143\pi\)
0.974627 0.223834i \(-0.0718574\pi\)
\(594\) 0 0
\(595\) 4.97363 + 38.3961i 0.203899 + 1.57409i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 5.74532i 0.234747i −0.993088 0.117374i \(-0.962552\pi\)
0.993088 0.117374i \(-0.0374475\pi\)
\(600\) 0 0
\(601\) 35.9894i 1.46804i 0.679129 + 0.734019i \(0.262358\pi\)
−0.679129 + 0.734019i \(0.737642\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 34.2154 1.39105
\(606\) 0 0
\(607\) 22.4652 0.911832 0.455916 0.890023i \(-0.349312\pi\)
0.455916 + 0.890023i \(0.349312\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −19.7783 −0.800143
\(612\) 0 0
\(613\) 36.1954i 1.46192i 0.682420 + 0.730960i \(0.260927\pi\)
−0.682420 + 0.730960i \(0.739073\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −30.8974 −1.24388 −0.621942 0.783063i \(-0.713656\pi\)
−0.621942 + 0.783063i \(0.713656\pi\)
\(618\) 0 0
\(619\) 7.45941i 0.299819i 0.988700 + 0.149910i \(0.0478982\pi\)
−0.988700 + 0.149910i \(0.952102\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.63512 28.0629i −0.145638 1.12432i
\(624\) 0 0
\(625\) 26.3462 1.05385
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 32.8189 1.30858
\(630\) 0 0
\(631\) 17.2676i 0.687414i −0.939077 0.343707i \(-0.888317\pi\)
0.939077 0.343707i \(-0.111683\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 43.0759i 1.70941i
\(636\) 0 0
\(637\) 13.5379 3.56712i 0.536393 0.141334i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −8.09990 −0.319927 −0.159963 0.987123i \(-0.551138\pi\)
−0.159963 + 0.987123i \(0.551138\pi\)
\(642\) 0 0
\(643\) 24.2177i 0.955052i 0.878618 + 0.477526i \(0.158466\pi\)
−0.878618 + 0.477526i \(0.841534\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 8.14058 0.320039 0.160020 0.987114i \(-0.448844\pi\)
0.160020 + 0.987114i \(0.448844\pi\)
\(648\) 0 0
\(649\) 11.5017i 0.451480i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 35.3502i 1.38336i −0.722204 0.691681i \(-0.756871\pi\)
0.722204 0.691681i \(-0.243129\pi\)
\(654\) 0 0
\(655\) 6.17843i 0.241411i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −31.5480 −1.22894 −0.614468 0.788942i \(-0.710629\pi\)
−0.614468 + 0.788942i \(0.710629\pi\)
\(660\) 0 0
\(661\) 10.3096 0.400995 0.200498 0.979694i \(-0.435744\pi\)
0.200498 + 0.979694i \(0.435744\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −0.897442 6.92820i −0.0348013 0.268664i
\(666\) 0 0
\(667\) −36.5956 −1.41699
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −18.3717 −0.709233
\(672\) 0 0
\(673\) −25.0892 −0.967118 −0.483559 0.875312i \(-0.660656\pi\)
−0.483559 + 0.875312i \(0.660656\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 26.1414 1.00470 0.502348 0.864665i \(-0.332469\pi\)
0.502348 + 0.864665i \(0.332469\pi\)
\(678\) 0 0
\(679\) −3.36109 25.9474i −0.128987 0.995770i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 39.6886 1.51864 0.759321 0.650716i \(-0.225531\pi\)
0.759321 + 0.650716i \(0.225531\pi\)
\(684\) 0 0
\(685\) 73.4068 2.80473
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 12.8501i 0.489548i
\(690\) 0 0
\(691\) 20.4354i 0.777398i −0.921365 0.388699i \(-0.872925\pi\)
0.921365 0.388699i \(-0.127075\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 39.5379i 1.49976i
\(696\) 0 0
\(697\) 11.9081 0.451053
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 23.0482i 0.870520i 0.900305 + 0.435260i \(0.143343\pi\)
−0.900305 + 0.435260i \(0.856657\pi\)
\(702\) 0 0
\(703\) −5.92185 −0.223347
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 10.1922 1.32025i 0.383318 0.0496530i
\(708\) 0 0
\(709\) 6.72217i 0.252457i 0.992001 + 0.126228i \(0.0402872\pi\)
−0.992001 + 0.126228i \(0.959713\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 33.4405i 1.25236i
\(714\) 0 0
\(715\) −11.5017 −0.430138
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 14.6686 0.547047 0.273524 0.961865i \(-0.411811\pi\)
0.273524 + 0.961865i \(0.411811\pi\)
\(720\) 0 0
\(721\) 8.03926 1.04136i 0.299398 0.0387824i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 70.9395i 2.63463i
\(726\) 0 0
\(727\) −9.99214 −0.370588 −0.185294 0.982683i \(-0.559324\pi\)
−0.185294 + 0.982683i \(0.559324\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 30.9234i 1.14374i
\(732\) 0 0
\(733\) −9.45941 −0.349391 −0.174696 0.984622i \(-0.555894\pi\)
−0.174696 + 0.984622i \(0.555894\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −13.0500 −0.480701
\(738\) 0 0
\(739\) −12.3119 −0.452899 −0.226450 0.974023i \(-0.572712\pi\)
−0.226450 + 0.974023i \(0.572712\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.51429i 0.0555537i 0.999614 + 0.0277769i \(0.00884279\pi\)
−0.999614 + 0.0277769i \(0.991157\pi\)
\(744\) 0 0
\(745\) 15.8110i 0.579270i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 5.47502 0.709205i 0.200053 0.0259138i
\(750\) 0 0
\(751\) 52.5745i 1.91847i −0.282606 0.959236i \(-0.591199\pi\)
0.282606 0.959236i \(-0.408801\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 33.7164i 1.22706i
\(756\) 0 0
\(757\) 22.2030i 0.806982i −0.914984 0.403491i \(-0.867797\pi\)
0.914984 0.403491i \(-0.132203\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 11.3016i 0.409682i 0.978795 + 0.204841i \(0.0656678\pi\)
−0.978795 + 0.204841i \(0.934332\pi\)
\(762\) 0 0
\(763\) −5.92185 45.7164i −0.214385 1.65504i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 15.5379i 0.561042i
\(768\) 0 0
\(769\) 32.4923i 1.17170i −0.810419 0.585851i \(-0.800760\pi\)
0.810419 0.585851i \(-0.199240\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −35.6535 −1.28237 −0.641183 0.767388i \(-0.721556\pi\)
−0.641183 + 0.767388i \(0.721556\pi\)
\(774\) 0 0
\(775\) 64.8235 2.32853
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.14871 −0.0769854
\(780\) 0 0
\(781\) 19.2071i 0.687283i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −12.1525 −0.433742
\(786\) 0 0
\(787\) 4.00000i 0.142585i −0.997455 0.0712923i \(-0.977288\pi\)
0.997455 0.0712923i \(-0.0227123\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4.64147 + 0.601231i −0.165032 + 0.0213773i
\(792\) 0 0
\(793\) −24.8189 −0.881346
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −44.8604 −1.58904 −0.794519 0.607239i \(-0.792277\pi\)
−0.794519 + 0.607239i \(0.792277\pi\)
\(798\) 0 0
\(799\) 37.2543i 1.31796i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 19.9215i 0.703014i
\(804\) 0 0
\(805\) 6.87153 + 53.0478i 0.242189