Properties

Label 4032.2.p.k.1567.7
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 1567.7
Root \(-2.23871 + 0.500000i\)
Character \(\chi\) = 4032.1567
Dual form 4032.2.p.k.1567.8

$q$-expansion

\(f(q)\) \(=\) \(q\)\(+1.11575 q^{5}\) \(+(1.37268 - 2.26180i) q^{7}\) \(+O(q^{10})\) \(q\)\(+1.11575 q^{5}\) \(+(1.37268 - 2.26180i) q^{7}\) \(+0.812826 q^{11}\) \(+2.00000 q^{13}\) \(+3.55819i q^{17}\) \(+4.52360i q^{19}\) \(+3.63935i q^{23}\) \(-3.75510 q^{25}\) \(+8.95482i q^{29}\) \(+5.08975 q^{31}\) \(+(1.53157 - 2.52360i) q^{35}\) \(+0.718741i q^{37}\) \(+10.4864i q^{41}\) \(-1.11971 q^{43}\) \(-8.55385 q^{47}\) \(+(-3.23150 - 6.20946i) q^{49}\) \(+5.08975i q^{53}\) \(+0.906910 q^{55}\) \(-2.23150i q^{59}\) \(+5.27871 q^{61}\) \(+2.23150 q^{65}\) \(+15.1643 q^{67}\) \(+5.87085i q^{71}\) \(+3.86507i q^{73}\) \(+(1.11575 - 1.83845i) q^{77}\) \(-1.70789i q^{79}\) \(-7.27871i q^{83}\) \(+3.97004i q^{85}\) \(-3.37002i q^{89}\) \(+(2.74536 - 4.52360i) q^{91}\) \(+5.04721i q^{95}\) \(-8.55385i q^{97}\) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(12q \) \(\mathstrut +\mathstrut O(q^{10}) \) \(12q \) \(\mathstrut +\mathstrut 24q^{13} \) \(\mathstrut +\mathstrut 36q^{25} \) \(\mathstrut -\mathstrut 12q^{49} \) \(\mathstrut -\mathstrut 72q^{61} \) \(\mathstrut +\mathstrut O(q^{100}) \)

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
\(5\) 1.11575 0.498978 0.249489 0.968378i \(-0.419737\pi\)
0.249489 + 0.968378i \(0.419737\pi\)
\(6\) 0 0
\(7\) 1.37268 2.26180i 0.518824 0.854881i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.812826 0.245076 0.122538 0.992464i \(-0.460897\pi\)
0.122538 + 0.992464i \(0.460897\pi\)
\(12\) 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
\(17\) 3.55819i 0.862987i 0.902116 + 0.431493i \(0.142013\pi\)
−0.902116 + 0.431493i \(0.857987\pi\)
\(18\) 0 0
\(19\) 4.52360i 1.03779i 0.854839 + 0.518893i \(0.173656\pi\)
−0.854839 + 0.518893i \(0.826344\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.63935i 0.758858i 0.925221 + 0.379429i \(0.123880\pi\)
−0.925221 + 0.379429i \(0.876120\pi\)
\(24\) 0 0
\(25\) −3.75510 −0.751021
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.95482i 1.66287i 0.555623 + 0.831434i \(0.312480\pi\)
−0.555623 + 0.831434i \(0.687520\pi\)
\(30\) 0 0
\(31\) 5.08975 0.914147 0.457073 0.889429i \(-0.348898\pi\)
0.457073 + 0.889429i \(0.348898\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.53157 2.52360i 0.258882 0.426567i
\(36\) 0 0
\(37\) 0.718741i 0.118160i 0.998253 + 0.0590802i \(0.0188168\pi\)
−0.998253 + 0.0590802i \(0.981183\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 10.4864i 1.63770i 0.574008 + 0.818849i \(0.305388\pi\)
−0.574008 + 0.818849i \(0.694612\pi\)
\(42\) 0 0
\(43\) −1.11971 −0.170754 −0.0853770 0.996349i \(-0.527209\pi\)
−0.0853770 + 0.996349i \(0.527209\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.55385 −1.24771 −0.623854 0.781541i \(-0.714434\pi\)
−0.623854 + 0.781541i \(0.714434\pi\)
\(48\) 0 0
\(49\) −3.23150 6.20946i −0.461643 0.887066i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.08975i 0.699131i 0.936912 + 0.349566i \(0.113671\pi\)
−0.936912 + 0.349566i \(0.886329\pi\)
\(54\) 0 0
\(55\) 0.906910 0.122288
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.23150i 0.290516i −0.989394 0.145258i \(-0.953599\pi\)
0.989394 0.145258i \(-0.0464013\pi\)
\(60\) 0 0
\(61\) 5.27871 0.675869 0.337935 0.941170i \(-0.390272\pi\)
0.337935 + 0.941170i \(0.390272\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.23150 0.276783
\(66\) 0 0
\(67\) 15.1643 1.85261 0.926306 0.376772i \(-0.122966\pi\)
0.926306 + 0.376772i \(0.122966\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.87085i 0.696742i 0.937357 + 0.348371i \(0.113265\pi\)
−0.937357 + 0.348371i \(0.886735\pi\)
\(72\) 0 0
\(73\) 3.86507i 0.452372i 0.974084 + 0.226186i \(0.0726257\pi\)
−0.974084 + 0.226186i \(0.927374\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.11575 1.83845i 0.127151 0.209511i
\(78\) 0 0
\(79\) 1.70789i 0.192153i −0.995374 0.0960766i \(-0.969371\pi\)
0.995374 0.0960766i \(-0.0306294\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 7.27871i 0.798942i −0.916746 0.399471i \(-0.869194\pi\)
0.916746 0.399471i \(-0.130806\pi\)
\(84\) 0 0
\(85\) 3.97004i 0.430612i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 3.37002i 0.357221i −0.983920 0.178611i \(-0.942840\pi\)
0.983920 0.178611i \(-0.0571602\pi\)
\(90\) 0 0
\(91\) 2.74536 4.52360i 0.287792 0.474203i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.04721i 0.517833i
\(96\) 0 0
\(97\) 8.55385i 0.868512i −0.900789 0.434256i \(-0.857011\pi\)
0.900789 0.434256i \(-0.142989\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.11575 0.111021 0.0555106 0.998458i \(-0.482321\pi\)
0.0555106 + 0.998458i \(0.482321\pi\)
\(102\) 0 0
\(103\) 14.2574 1.40482 0.702410 0.711772i \(-0.252107\pi\)
0.702410 + 0.711772i \(0.252107\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.2317 1.27916 0.639581 0.768724i \(-0.279108\pi\)
0.639581 + 0.768724i \(0.279108\pi\)
\(108\) 0 0
\(109\) 1.43748i 0.137686i −0.997628 0.0688429i \(-0.978069\pi\)
0.997628 0.0688429i \(-0.0219307\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −8.23150 −0.774354 −0.387177 0.922005i \(-0.626550\pi\)
−0.387177 + 0.922005i \(0.626550\pi\)
\(114\) 0 0
\(115\) 4.06061i 0.378654i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 8.04791 + 4.88425i 0.737751 + 0.447739i
\(120\) 0 0
\(121\) −10.3393 −0.939938
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −9.76850 −0.873721
\(126\) 0 0
\(127\) 2.75510i 0.244476i −0.992501 0.122238i \(-0.960993\pi\)
0.992501 0.122238i \(-0.0390071\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 19.2787i 1.68439i −0.539174 0.842194i \(-0.681264\pi\)
0.539174 0.842194i \(-0.318736\pi\)
\(132\) 0 0
\(133\) 10.2315 + 6.20946i 0.887183 + 0.538429i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.58421 −0.562527 −0.281264 0.959631i \(-0.590754\pi\)
−0.281264 + 0.959631i \(0.590754\pi\)
\(138\) 0 0
\(139\) 17.5102i 1.48520i −0.669737 0.742598i \(-0.733593\pi\)
0.669737 0.742598i \(-0.266407\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.62565 0.135944
\(144\) 0 0
\(145\) 9.99134i 0.829735i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 10.5805i 0.866786i 0.901205 + 0.433393i \(0.142684\pi\)
−0.901205 + 0.433393i \(0.857316\pi\)
\(150\) 0 0
\(151\) 12.5236i 1.01916i 0.860424 + 0.509578i \(0.170199\pi\)
−0.860424 + 0.509578i \(0.829801\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 5.67889 0.456139
\(156\) 0 0
\(157\) 0.815710 0.0651008 0.0325504 0.999470i \(-0.489637\pi\)
0.0325504 + 0.999470i \(0.489637\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.23150 + 4.99567i 0.648733 + 0.393714i
\(162\) 0 0
\(163\) −4.18284 −0.327626 −0.163813 0.986491i \(-0.552379\pi\)
−0.163813 + 0.986491i \(0.552379\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.4189 0.961005 0.480503 0.876993i \(-0.340454\pi\)
0.480503 + 0.876993i \(0.340454\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 25.4417 1.93429 0.967147 0.254218i \(-0.0818180\pi\)
0.967147 + 0.254218i \(0.0818180\pi\)
\(174\) 0 0
\(175\) −5.15456 + 8.49330i −0.389648 + 0.642033i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 9.36668 0.700099 0.350049 0.936731i \(-0.386165\pi\)
0.350049 + 0.936731i \(0.386165\pi\)
\(180\) 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
\(185\) 0.801935i 0.0589595i
\(186\) 0 0
\(187\) 2.89218i 0.211497i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 11.1764i 0.808693i 0.914606 + 0.404346i \(0.132501\pi\)
−0.914606 + 0.404346i \(0.867499\pi\)
\(192\) 0 0
\(193\) 13.2181 0.951460 0.475730 0.879591i \(-0.342184\pi\)
0.475730 + 0.879591i \(0.342184\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.5718i 1.46568i 0.680401 + 0.732840i \(0.261806\pi\)
−0.680401 + 0.732840i \(0.738194\pi\)
\(198\) 0 0
\(199\) −13.7268 −0.973067 −0.486534 0.873662i \(-0.661739\pi\)
−0.486534 + 0.873662i \(0.661739\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 20.2540 + 12.2921i 1.42155 + 0.862737i
\(204\) 0 0
\(205\) 11.7002i 0.817176i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.67690i 0.254337i
\(210\) 0 0
\(211\) 9.86173 0.678910 0.339455 0.940622i \(-0.389757\pi\)
0.339455 + 0.940622i \(0.389757\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.24931 −0.0852026
\(216\) 0 0
\(217\) 6.98660 11.5120i 0.474281 0.781486i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 7.11637i 0.478699i
\(222\) 0 0
\(223\) −25.3438 −1.69715 −0.848573 0.529079i \(-0.822538\pi\)
−0.848573 + 0.529079i \(0.822538\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.5574i 0.966210i −0.875563 0.483105i \(-0.839509\pi\)
0.875563 0.483105i \(-0.160491\pi\)
\(228\) 0 0
\(229\) −14.5842 −0.963752 −0.481876 0.876239i \(-0.660044\pi\)
−0.481876 + 0.876239i \(0.660044\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −27.8361 −1.82361 −0.911803 0.410629i \(-0.865309\pi\)
−0.911803 + 0.410629i \(0.865309\pi\)
\(234\) 0 0
\(235\) −9.54396 −0.622579
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 20.1024i 1.30031i −0.759800 0.650157i \(-0.774703\pi\)
0.759800 0.650157i \(-0.225297\pi\)
\(240\) 0 0
\(241\) 16.2840i 1.04894i 0.851428 + 0.524472i \(0.175737\pi\)
−0.851428 + 0.524472i \(0.824263\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −3.60554 6.92820i −0.230350 0.442627i
\(246\) 0 0
\(247\) 9.04721i 0.575660i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.5842i 0.794308i 0.917752 + 0.397154i \(0.130002\pi\)
−0.917752 + 0.397154i \(0.869998\pi\)
\(252\) 0 0
\(253\) 2.95816i 0.185978i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 28.2079i 1.75956i −0.475383 0.879779i \(-0.657690\pi\)
0.475383 0.879779i \(-0.342310\pi\)
\(258\) 0 0
\(259\) 1.62565 + 0.986602i 0.101013 + 0.0613045i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.12915i 0.377939i 0.981983 + 0.188970i \(0.0605148\pi\)
−0.981983 + 0.188970i \(0.939485\pi\)
\(264\) 0 0
\(265\) 5.67889i 0.348851i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 23.2102 1.41515 0.707574 0.706639i \(-0.249789\pi\)
0.707574 + 0.706639i \(0.249789\pi\)
\(270\) 0 0
\(271\) 21.1856 1.28693 0.643466 0.765475i \(-0.277496\pi\)
0.643466 + 0.765475i \(0.277496\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −3.05224 −0.184057
\(276\) 0 0
\(277\) 6.20946i 0.373090i −0.982446 0.186545i \(-0.940271\pi\)
0.982446 0.186545i \(-0.0597291\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −25.8629 −1.54285 −0.771426 0.636319i \(-0.780456\pi\)
−0.771426 + 0.636319i \(0.780456\pi\)
\(282\) 0 0
\(283\) 27.0810i 1.60980i −0.593411 0.804900i \(-0.702219\pi\)
0.593411 0.804900i \(-0.297781\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 23.7181 + 14.3945i 1.40004 + 0.849678i
\(288\) 0 0
\(289\) 4.33931 0.255254
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 25.4417 1.48632 0.743159 0.669115i \(-0.233327\pi\)
0.743159 + 0.669115i \(0.233327\pi\)
\(294\) 0 0
\(295\) 2.48979i 0.144961i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7.27871i 0.420939i
\(300\) 0 0
\(301\) −1.53700 + 2.53256i −0.0885913 + 0.145974i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.88972 0.337244
\(306\) 0 0
\(307\) 10.0338i 0.572660i 0.958131 + 0.286330i \(0.0924353\pi\)
−0.958131 + 0.286330i \(0.907565\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −10.7933 −0.612030 −0.306015 0.952027i \(-0.598996\pi\)
−0.306015 + 0.952027i \(0.598996\pi\)
\(312\) 0 0
\(313\) 30.9641i 1.75020i 0.483946 + 0.875098i \(0.339203\pi\)
−0.483946 + 0.875098i \(0.660797\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 10.5805i 0.594259i 0.954837 + 0.297129i \(0.0960292\pi\)
−0.954837 + 0.297129i \(0.903971\pi\)
\(318\) 0 0
\(319\) 7.27871i 0.407529i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −16.0958 −0.895596
\(324\) 0 0
\(325\) −7.51021 −0.416591
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −11.7417 + 19.3471i −0.647341 + 1.06664i
\(330\) 0 0
\(331\) 5.80849 0.319264 0.159632 0.987177i \(-0.448969\pi\)
0.159632 + 0.987177i \(0.448969\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 16.9195 0.924413
\(336\) 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
\(341\) 4.13708 0.224035
\(342\) 0 0
\(343\) −18.4804 1.21459i −0.997847 0.0655819i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 24.0250 1.28973 0.644865 0.764296i \(-0.276913\pi\)
0.644865 + 0.764296i \(0.276913\pi\)
\(348\) 0 0
\(349\) 33.2787 1.78137 0.890684 0.454623i \(-0.150226\pi\)
0.890684 + 0.454623i \(0.150226\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 24.7191i 1.31567i −0.753164 0.657833i \(-0.771473\pi\)
0.753164 0.657833i \(-0.228527\pi\)
\(354\) 0 0
\(355\) 6.55040i 0.347659i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 29.8709i 1.57652i −0.615340 0.788262i \(-0.710981\pi\)
0.615340 0.788262i \(-0.289019\pi\)
\(360\) 0 0
\(361\) −1.46300 −0.0769999
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 4.31245i 0.225724i
\(366\) 0 0
\(367\) 0.505942 0.0264100 0.0132050 0.999913i \(-0.495797\pi\)
0.0132050 + 0.999913i \(0.495797\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 11.5120 + 6.98660i 0.597674 + 0.362726i
\(372\) 0 0
\(373\) 21.1609i 1.09567i 0.836586 + 0.547836i \(0.184548\pi\)
−0.836586 + 0.547836i \(0.815452\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 17.9096i 0.922394i
\(378\) 0 0
\(379\) −21.4787 −1.10329 −0.551644 0.834080i \(-0.685999\pi\)
−0.551644 + 0.834080i \(0.685999\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1.24931 −0.0638370 −0.0319185 0.999490i \(-0.510162\pi\)
−0.0319185 + 0.999490i \(0.510162\pi\)
\(384\) 0 0
\(385\) 1.24490 2.05125i 0.0634458 0.104541i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 20.5718i 1.04303i −0.853241 0.521516i \(-0.825367\pi\)
0.853241 0.521516i \(-0.174633\pi\)
\(390\) 0 0
\(391\) −12.9495 −0.654884
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.90558i 0.0958802i
\(396\) 0 0
\(397\) −22.3259 −1.12051 −0.560253 0.828322i \(-0.689296\pi\)
−0.560253 + 0.828322i \(0.689296\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 25.6046 1.27863 0.639317 0.768943i \(-0.279217\pi\)
0.639317 + 0.768943i \(0.279217\pi\)
\(402\) 0 0
\(403\) 10.1795 0.507077
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0.584211i 0.0289583i
\(408\) 0 0
\(409\) 16.2840i 0.805192i 0.915378 + 0.402596i \(0.131892\pi\)
−0.915378 + 0.402596i \(0.868108\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −5.04721 3.06313i −0.248357 0.150727i
\(414\) 0 0
\(415\) 8.12121i 0.398655i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 18.6945i 0.913286i 0.889650 + 0.456643i \(0.150948\pi\)
−0.889650 + 0.456643i \(0.849052\pi\)
\(420\) 0 0
\(421\) 31.0473i 1.51315i 0.653905 + 0.756577i \(0.273130\pi\)
−0.653905 + 0.756577i \(0.726870\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 13.3614i 0.648121i
\(426\) 0 0
\(427\) 7.24598 11.9394i 0.350657 0.577788i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.99207i 0.0959545i −0.998848 0.0479772i \(-0.984723\pi\)
0.998848 0.0479772i \(-0.0152775\pi\)
\(432\) 0 0
\(433\) 10.6051i 0.509649i −0.966987 0.254824i \(-0.917982\pi\)
0.966987 0.254824i \(-0.0820177\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −16.4630 −0.787532
\(438\) 0 0
\(439\) 22.0925 1.05442 0.527208 0.849736i \(-0.323239\pi\)
0.527208 + 0.849736i \(0.323239\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7.55286 −0.358847 −0.179424 0.983772i \(-0.557423\pi\)
−0.179424 + 0.983772i \(0.557423\pi\)
\(444\) 0 0
\(445\) 3.76010i 0.178246i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.3259 0.864853 0.432427 0.901669i \(-0.357657\pi\)
0.432427 + 0.901669i \(0.357657\pi\)
\(450\) 0 0
\(451\) 8.52360i 0.401361i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.06313 5.04721i 0.143602 0.236617i
\(456\) 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
\(461\) −29.5787 −1.37762 −0.688810 0.724942i \(-0.741866\pi\)
−0.688810 + 0.724942i \(0.741866\pi\)
\(462\) 0 0
\(463\) 16.2653i 0.755913i 0.925823 + 0.377957i \(0.123373\pi\)
−0.925823 + 0.377957i \(0.876627\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 21.5102i 0.995374i −0.867357 0.497687i \(-0.834183\pi\)
0.867357 0.497687i \(-0.165817\pi\)
\(468\) 0 0
\(469\) 20.8157 34.2986i 0.961180 1.58376i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −0.910128 −0.0418477
\(474\) 0 0
\(475\) 16.9866i 0.779399i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −9.99134 −0.456516 −0.228258 0.973601i \(-0.573303\pi\)
−0.228258 + 0.973601i \(0.573303\pi\)
\(480\) 0 0
\(481\) 1.43748i 0.0655436i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.54396i 0.433369i
\(486\) 0 0
\(487\) 19.0810i 0.864644i −0.901719 0.432322i \(-0.857694\pi\)
0.901719 0.432322i \(-0.142306\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −38.0696 −1.71806 −0.859028 0.511928i \(-0.828931\pi\)
−0.859028 + 0.511928i \(0.828931\pi\)
\(492\) 0 0
\(493\) −31.8629 −1.43503
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 13.2787 + 8.05880i 0.595631 + 0.361487i
\(498\) 0 0
\(499\) 11.2992 0.505822 0.252911 0.967490i \(-0.418612\pi\)
0.252911 + 0.967490i \(0.418612\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 22.4103 0.999224 0.499612 0.866249i \(-0.333476\pi\)
0.499612 + 0.866249i \(0.333476\pi\)
\(504\) 0 0
\(505\) 1.24490 0.0553972
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −11.2102 −0.496882 −0.248441 0.968647i \(-0.579918\pi\)
−0.248441 + 0.968647i \(0.579918\pi\)
\(510\) 0 0
\(511\) 8.74202 + 5.30550i 0.386724 + 0.234702i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 15.9077 0.700975
\(516\) 0 0
\(517\) −6.95279 −0.305783
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 17.7909i 0.779435i 0.920935 + 0.389717i \(0.127427\pi\)
−0.920935 + 0.389717i \(0.872573\pi\)
\(522\) 0 0
\(523\) 26.4362i 1.15597i 0.816046 + 0.577987i \(0.196162\pi\)
−0.816046 + 0.577987i \(0.803838\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 18.1103i 0.788896i
\(528\) 0 0
\(529\) 9.75510 0.424135
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 20.9728i 0.908432i
\(534\) 0 0
\(535\) 14.7633 0.638274
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −2.62664 5.04721i −0.113138 0.217399i
\(540\) 0 0
\(541\) 31.4236i 1.35101i −0.737356 0.675504i \(-0.763926\pi\)
0.737356 0.675504i \(-0.236074\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1.60387i 0.0687023i
\(546\) 0 0
\(547\) −21.2906 −0.910318 −0.455159 0.890410i \(-0.650418\pi\)
−0.455159 + 0.890410i \(0.650418\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −40.5081 −1.72570
\(552\) 0 0
\(553\) −3.86292 2.34439i −0.164268 0.0996937i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 35.0420i 1.48478i −0.669970 0.742388i \(-0.733693\pi\)
0.669970 0.742388i \(-0.266307\pi\)
\(558\) 0 0
\(559\) −2.23942 −0.0947173
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 2.81571i 0.118668i 0.998238 + 0.0593340i \(0.0188977\pi\)
−0.998238 + 0.0593340i \(0.981102\pi\)
\(564\) 0 0
\(565\) −9.18429 −0.386386
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −10.7213 −0.449460 −0.224730 0.974421i \(-0.572150\pi\)
−0.224730 + 0.974421i \(0.572150\pi\)
\(570\) 0 0
\(571\) 22.4688 0.940291 0.470145 0.882589i \(-0.344201\pi\)
0.470145 + 0.882589i \(0.344201\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 13.6661i 0.569918i
\(576\) 0 0
\(577\) 25.2142i 1.04968i −0.851201 0.524840i \(-0.824125\pi\)
0.851201 0.524840i \(-0.175875\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −16.4630 9.99134i −0.683000 0.414511i
\(582\) 0 0
\(583\) 4.13708i 0.171340i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 9.51021i 0.392528i −0.980551 0.196264i \(-0.937119\pi\)
0.980551 0.196264i \(-0.0628810\pi\)
\(588\) 0 0
\(589\) 23.0240i 0.948689i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 28.3960i 1.16609i −0.812442 0.583043i \(-0.801862\pi\)
0.812442 0.583043i \(-0.198138\pi\)
\(594\) 0 0
\(595\) 8.97945 + 5.44960i 0.368122 + 0.223412i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 39.9653i 1.63294i 0.577390 + 0.816468i \(0.304071\pi\)
−0.577390 + 0.816468i \(0.695929\pi\)
\(600\) 0 0
\(601\) 46.6343i 1.90225i −0.308799 0.951127i \(-0.599927\pi\)
0.308799 0.951127i \(-0.400073\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −11.5361 −0.469009
\(606\) 0 0
\(607\) 30.4582 1.23626 0.618130 0.786076i \(-0.287891\pi\)
0.618130 + 0.786076i \(0.287891\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −17.1077 −0.692104
\(612\) 0 0
\(613\) 14.8683i 0.600525i −0.953857 0.300262i \(-0.902926\pi\)
0.953857 0.300262i \(-0.0970742\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −18.5842 −0.748172 −0.374086 0.927394i \(-0.622044\pi\)
−0.374086 + 0.927394i \(0.622044\pi\)
\(618\) 0 0
\(619\) 35.6046i 1.43107i 0.698577 + 0.715535i \(0.253817\pi\)
−0.698577 + 0.715535i \(0.746183\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −7.62231 4.62596i −0.305382 0.185335i
\(624\) 0 0
\(625\) 7.87632 0.315053
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.55742 −0.101971
\(630\) 0 0
\(631\) 24.2653i 0.965987i −0.875624 0.482993i \(-0.839550\pi\)
0.875624 0.482993i \(-0.160450\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3.07400i 0.121988i
\(636\) 0 0
\(637\) −6.46300 12.4189i −0.256073 0.492056i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 42.6518 1.68465 0.842323 0.538973i \(-0.181188\pi\)
0.842323 + 0.538973i \(0.181188\pi\)
\(642\) 0 0
\(643\) 8.06061i 0.317879i −0.987288 0.158940i \(-0.949192\pi\)
0.987288 0.158940i \(-0.0508075\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −21.1609 −0.831923 −0.415961 0.909382i \(-0.636555\pi\)
−0.415961 + 0.909382i \(0.636555\pi\)
\(648\) 0 0
\(649\) 1.81382i 0.0711986i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4.71342i 0.184450i 0.995738 + 0.0922251i \(0.0293979\pi\)
−0.995738 + 0.0922251i \(0.970602\pi\)
\(654\) 0 0
\(655\) 21.5102i 0.840473i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −44.1959 −1.72163 −0.860813 0.508921i \(-0.830045\pi\)
−0.860813 + 0.508921i \(0.830045\pi\)
\(660\) 0 0
\(661\) 43.3731 1.68702 0.843510 0.537114i \(-0.180486\pi\)
0.843510 + 0.537114i \(0.180486\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 11.4158 + 6.92820i 0.442685 + 0.268664i
\(666\) 0 0
\(667\) −32.5898 −1.26188
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.29067 0.165639
\(672\) 0 0
\(673\) −11.2449 −0.433459 −0.216729 0.976232i \(-0.569539\pi\)
−0.216729 + 0.976232i \(0.569539\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 16.5157 0.634749 0.317374 0.948300i \(-0.397199\pi\)
0.317374 + 0.948300i \(0.397199\pi\)
\(678\) 0 0
\(679\) −19.3471 11.7417i −0.742475 0.450605i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 23.0349 0.881407 0.440703 0.897653i \(-0.354729\pi\)
0.440703 + 0.897653i \(0.354729\pi\)
\(684\) 0 0
\(685\) −7.34633 −0.280689
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 10.1795i 0.387808i
\(690\) 0 0
\(691\) 11.8788i 0.451890i −0.974140 0.225945i \(-0.927453\pi\)
0.974140 0.225945i \(-0.0725470\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 19.5370i 0.741081i
\(696\) 0 0
\(697\) −37.3125 −1.41331
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 28.9157i 1.09213i 0.837742 + 0.546066i \(0.183875\pi\)
−0.837742 + 0.546066i \(0.816125\pi\)
\(702\) 0 0
\(703\) −3.25130 −0.122625
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.53157 2.52360i 0.0576005 0.0949099i
\(708\) 0 0
\(709\) 38.6943i 1.45319i −0.687064 0.726597i \(-0.741101\pi\)
0.687064 0.726597i \(-0.258899\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 18.5234i 0.693707i
\(714\) 0 0
\(715\) 1.81382 0.0678330
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −31.9542 −1.19169 −0.595846 0.803099i \(-0.703183\pi\)
−0.595846 + 0.803099i \(0.703183\pi\)
\(720\) 0 0
\(721\) 19.5708 32.2474i 0.728855 1.20095i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 33.6263i 1.24885i
\(726\) 0 0
\(727\) 7.32917 0.271824 0.135912 0.990721i \(-0.456604\pi\)
0.135912 + 0.990721i \(0.456604\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 3.98413i 0.147358i
\(732\) 0 0
\(733\) 33.6046 1.24122 0.620608 0.784121i \(-0.286886\pi\)
0.620608 + 0.784121i \(0.286886\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.3259 0.454031
\(738\) 0 0
\(739\) −44.2653 −1.62833 −0.814163 0.580636i \(-0.802804\pi\)
−0.814163 + 0.580636i \(0.802804\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 25.7338i 0.944081i −0.881577 0.472040i \(-0.843518\pi\)
0.881577 0.472040i \(-0.156482\pi\)
\(744\) 0 0
\(745\) 11.8052i 0.432507i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 18.1630 29.9276i 0.663660 1.09353i
\(750\) 0 0
\(751\) 18.9598i 0.691853i −0.938262 0.345927i \(-0.887565\pi\)
0.938262 0.345927i \(-0.112435\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 13.9732i 0.508537i
\(756\) 0 0
\(757\) 39.0706i 1.42004i −0.704179 0.710022i \(-0.748685\pi\)
0.704179 0.710022i \(-0.251315\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 10.6746i 0.386952i 0.981105 + 0.193476i \(0.0619762\pi\)
−0.981105 + 0.193476i \(0.938024\pi\)
\(762\) 0 0
\(763\) −3.25130 1.97320i −0.117705 0.0714348i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.46300i 0.161150i
\(768\) 0 0
\(769\) 12.7953i 0.461409i −0.973024 0.230704i \(-0.925897\pi\)
0.973024 0.230704i \(-0.0741030\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −20.6528 −0.742828 −0.371414 0.928467i \(-0.621127\pi\)
−0.371414 + 0.928467i \(0.621127\pi\)
\(774\) 0 0
\(775\) −19.1125 −0.686543
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −47.4363 −1.69958
\(780\) 0 0
\(781\) 4.77198i 0.170755i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0.910128 0.0324839
\(786\) 0 0
\(787\) 4.00000i 0.142585i 0.997455 + 0.0712923i \(0.0227123\pi\)
−0.997455 + 0.0712923i \(0.977288\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −11.2992 + 18.6180i −0.401754 + 0.661981i
\(792\) 0 0
\(793\) 10.5574 0.374905
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −50.6101 −1.79270 −0.896351 0.443346i \(-0.853791\pi\)
−0.896351 + 0.443346i \(0.853791\pi\)
\(798\) 0 0
\(799\) 30.4362i 1.07676i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.14163i 0.110866i
\(804\) 0 0
\(805\) 9.18429 + 5.57391i 0.323704 + 0.196455i