Properties

Label 6012.2.h.a.3005.20
Level 6012
Weight 2
Character 6012.3005
Analytic conductor 48.006
Analytic rank 0
Dimension 56
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) \(=\) \( 6012 = 2^{2} \cdot 3^{2} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6012.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.0060616952\)
Analytic rank: \(0\)
Dimension: \(56\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3005.20
Character \(\chi\) \(=\) 6012.3005
Dual form 6012.2.h.a.3005.19

$q$-expansion

\(f(q)\) \(=\) \(q-1.15018 q^{5} -0.123775 q^{7} +O(q^{10})\) \(q-1.15018 q^{5} -0.123775 q^{7} -0.794122i q^{11} -0.814543i q^{13} -4.10883 q^{17} +2.35843 q^{19} -4.07130 q^{23} -3.67709 q^{25} -4.97503i q^{29} +8.23579 q^{31} +0.142364 q^{35} -4.60513i q^{37} -1.20278 q^{41} +10.0697i q^{43} +8.74772i q^{47} -6.98468 q^{49} +6.14215 q^{53} +0.913381i q^{55} +5.67351 q^{59} +5.76977 q^{61} +0.936870i q^{65} -1.63628i q^{67} +6.18282 q^{71} +10.9470i q^{73} +0.0982927i q^{77} -7.10481i q^{79} -11.9780 q^{83} +4.72589 q^{85} +2.78947i q^{89} +0.100820i q^{91} -2.71262 q^{95} +13.6099 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56q + O(q^{10}) \) \( 56q + 8q^{19} + 64q^{25} - 8q^{31} + 56q^{49} - 8q^{61} + 32q^{85} - 48q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6012\mathbb{Z}\right)^\times\).

\(n\) \(3007\) \(3341\) \(4681\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −1.15018 −0.514375 −0.257188 0.966361i \(-0.582796\pi\)
−0.257188 + 0.966361i \(0.582796\pi\)
\(6\) 0 0
\(7\) −0.123775 −0.0467827 −0.0233914 0.999726i \(-0.507446\pi\)
−0.0233914 + 0.999726i \(0.507446\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.794122i 0.239437i −0.992808 0.119718i \(-0.961801\pi\)
0.992808 0.119718i \(-0.0381991\pi\)
\(12\) 0 0
\(13\) 0.814543i 0.225914i −0.993600 0.112957i \(-0.963968\pi\)
0.993600 0.112957i \(-0.0360322\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.10883 −0.996538 −0.498269 0.867022i \(-0.666031\pi\)
−0.498269 + 0.867022i \(0.666031\pi\)
\(18\) 0 0
\(19\) 2.35843 0.541062 0.270531 0.962711i \(-0.412801\pi\)
0.270531 + 0.962711i \(0.412801\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.07130 −0.848925 −0.424463 0.905445i \(-0.639537\pi\)
−0.424463 + 0.905445i \(0.639537\pi\)
\(24\) 0 0
\(25\) −3.67709 −0.735418
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 4.97503i 0.923840i −0.886922 0.461920i \(-0.847161\pi\)
0.886922 0.461920i \(-0.152839\pi\)
\(30\) 0 0
\(31\) 8.23579 1.47919 0.739596 0.673051i \(-0.235017\pi\)
0.739596 + 0.673051i \(0.235017\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.142364 0.0240639
\(36\) 0 0
\(37\) 4.60513i 0.757079i −0.925585 0.378539i \(-0.876426\pi\)
0.925585 0.378539i \(-0.123574\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.20278 −0.187842 −0.0939210 0.995580i \(-0.529940\pi\)
−0.0939210 + 0.995580i \(0.529940\pi\)
\(42\) 0 0
\(43\) 10.0697i 1.53561i 0.640681 + 0.767807i \(0.278652\pi\)
−0.640681 + 0.767807i \(0.721348\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 8.74772i 1.27599i 0.770042 + 0.637993i \(0.220235\pi\)
−0.770042 + 0.637993i \(0.779765\pi\)
\(48\) 0 0
\(49\) −6.98468 −0.997811
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 6.14215 0.843690 0.421845 0.906668i \(-0.361383\pi\)
0.421845 + 0.906668i \(0.361383\pi\)
\(54\) 0 0
\(55\) 0.913381i 0.123160i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.67351 0.738628 0.369314 0.929305i \(-0.379593\pi\)
0.369314 + 0.929305i \(0.379593\pi\)
\(60\) 0 0
\(61\) 5.76977 0.738743 0.369371 0.929282i \(-0.379573\pi\)
0.369371 + 0.929282i \(0.379573\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.936870i 0.116204i
\(66\) 0 0
\(67\) 1.63628i 0.199903i −0.994992 0.0999515i \(-0.968131\pi\)
0.994992 0.0999515i \(-0.0318688\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 6.18282 0.733766 0.366883 0.930267i \(-0.380425\pi\)
0.366883 + 0.930267i \(0.380425\pi\)
\(72\) 0 0
\(73\) 10.9470i 1.28124i 0.767856 + 0.640622i \(0.221324\pi\)
−0.767856 + 0.640622i \(0.778676\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.0982927i 0.0112015i
\(78\) 0 0
\(79\) 7.10481i 0.799354i −0.916656 0.399677i \(-0.869122\pi\)
0.916656 0.399677i \(-0.130878\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −11.9780 −1.31476 −0.657378 0.753561i \(-0.728335\pi\)
−0.657378 + 0.753561i \(0.728335\pi\)
\(84\) 0 0
\(85\) 4.72589 0.512595
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.78947i 0.295683i 0.989011 + 0.147841i \(0.0472325\pi\)
−0.989011 + 0.147841i \(0.952767\pi\)
\(90\) 0 0
\(91\) 0.100820i 0.0105689i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −2.71262 −0.278309
\(96\) 0 0
\(97\) 13.6099 1.38188 0.690938 0.722914i \(-0.257198\pi\)
0.690938 + 0.722914i \(0.257198\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −16.8665 −1.67828 −0.839141 0.543913i \(-0.816942\pi\)
−0.839141 + 0.543913i \(0.816942\pi\)
\(102\) 0 0
\(103\) 7.67764i 0.756500i 0.925703 + 0.378250i \(0.123474\pi\)
−0.925703 + 0.378250i \(0.876526\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 5.59639i 0.541023i 0.962717 + 0.270512i \(0.0871928\pi\)
−0.962717 + 0.270512i \(0.912807\pi\)
\(108\) 0 0
\(109\) 7.09482i 0.679561i 0.940505 + 0.339780i \(0.110353\pi\)
−0.940505 + 0.339780i \(0.889647\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −11.5035 −1.08216 −0.541080 0.840971i \(-0.681984\pi\)
−0.541080 + 0.840971i \(0.681984\pi\)
\(114\) 0 0
\(115\) 4.68272 0.436666
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0.508573 0.0466208
\(120\) 0 0
\(121\) 10.3694 0.942670
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 9.98020 0.892656
\(126\) 0 0
\(127\) −20.7013 −1.83694 −0.918472 0.395485i \(-0.870577\pi\)
−0.918472 + 0.395485i \(0.870577\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −0.790714 −0.0690850 −0.0345425 0.999403i \(-0.510997\pi\)
−0.0345425 + 0.999403i \(0.510997\pi\)
\(132\) 0 0
\(133\) −0.291916 −0.0253123
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.47688i 0.724229i 0.932134 + 0.362115i \(0.117945\pi\)
−0.932134 + 0.362115i \(0.882055\pi\)
\(138\) 0 0
\(139\) 4.17420i 0.354051i 0.984206 + 0.177026i \(0.0566476\pi\)
−0.984206 + 0.177026i \(0.943352\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −0.646846 −0.0540920
\(144\) 0 0
\(145\) 5.72217i 0.475200i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.51667 −0.124250 −0.0621252 0.998068i \(-0.519788\pi\)
−0.0621252 + 0.998068i \(0.519788\pi\)
\(150\) 0 0
\(151\) 4.54670i 0.370005i −0.982738 0.185002i \(-0.940771\pi\)
0.982738 0.185002i \(-0.0592293\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −9.47263 −0.760860
\(156\) 0 0
\(157\) 14.9852 1.19595 0.597974 0.801515i \(-0.295972\pi\)
0.597974 + 0.801515i \(0.295972\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.503927 0.0397150
\(162\) 0 0
\(163\) 5.16688i 0.404701i 0.979313 + 0.202351i \(0.0648580\pi\)
−0.979313 + 0.202351i \(0.935142\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.57244 + 8.68150i 0.740738 + 0.671794i
\(168\) 0 0
\(169\) 12.3365 0.948963
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.57331i 0.195645i −0.995204 0.0978226i \(-0.968812\pi\)
0.995204 0.0978226i \(-0.0311878\pi\)
\(174\) 0 0
\(175\) 0.455133 0.0344049
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 20.5854i 1.53863i −0.638872 0.769313i \(-0.720599\pi\)
0.638872 0.769313i \(-0.279401\pi\)
\(180\) 0 0
\(181\) 14.6593 1.08962 0.544810 0.838559i \(-0.316602\pi\)
0.544810 + 0.838559i \(0.316602\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 5.29672i 0.389423i
\(186\) 0 0
\(187\) 3.26291i 0.238608i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.56029i 0.619401i 0.950834 + 0.309700i \(0.100229\pi\)
−0.950834 + 0.309700i \(0.899771\pi\)
\(192\) 0 0
\(193\) 15.4273i 1.11048i 0.831690 + 0.555241i \(0.187374\pi\)
−0.831690 + 0.555241i \(0.812626\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −11.5232 −0.820992 −0.410496 0.911862i \(-0.634644\pi\)
−0.410496 + 0.911862i \(0.634644\pi\)
\(198\) 0 0
\(199\) −2.53256 −0.179529 −0.0897643 0.995963i \(-0.528611\pi\)
−0.0897643 + 0.995963i \(0.528611\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0.615786i 0.0432197i
\(204\) 0 0
\(205\) 1.38341 0.0966213
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.87288i 0.129550i
\(210\) 0 0
\(211\) −0.916720 −0.0631096 −0.0315548 0.999502i \(-0.510046\pi\)
−0.0315548 + 0.999502i \(0.510046\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 11.5819i 0.789882i
\(216\) 0 0
\(217\) −1.01939 −0.0692006
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.34682i 0.225132i
\(222\) 0 0
\(223\) 8.01407 0.536662 0.268331 0.963327i \(-0.413528\pi\)
0.268331 + 0.963327i \(0.413528\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.99466 0.397879 0.198940 0.980012i \(-0.436250\pi\)
0.198940 + 0.980012i \(0.436250\pi\)
\(228\) 0 0
\(229\) −0.823194 −0.0543982 −0.0271991 0.999630i \(-0.508659\pi\)
−0.0271991 + 0.999630i \(0.508659\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 12.4559i 0.816015i 0.912979 + 0.408008i \(0.133776\pi\)
−0.912979 + 0.408008i \(0.866224\pi\)
\(234\) 0 0
\(235\) 10.0614i 0.656336i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.60828i 0.298085i 0.988831 + 0.149042i \(0.0476190\pi\)
−0.988831 + 0.149042i \(0.952381\pi\)
\(240\) 0 0
\(241\) 30.8197i 1.98527i 0.121137 + 0.992636i \(0.461346\pi\)
−0.121137 + 0.992636i \(0.538654\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 8.03363 0.513250
\(246\) 0 0
\(247\) 1.92105i 0.122233i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 17.6086i 1.11145i 0.831367 + 0.555724i \(0.187559\pi\)
−0.831367 + 0.555724i \(0.812441\pi\)
\(252\) 0 0
\(253\) 3.23311i 0.203264i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −25.2238 −1.57342 −0.786710 0.617323i \(-0.788217\pi\)
−0.786710 + 0.617323i \(0.788217\pi\)
\(258\) 0 0
\(259\) 0.570002i 0.0354182i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 22.8911i 1.41153i 0.708448 + 0.705763i \(0.249396\pi\)
−0.708448 + 0.705763i \(0.750604\pi\)
\(264\) 0 0
\(265\) −7.06457 −0.433973
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −17.1408 −1.04509 −0.522545 0.852612i \(-0.675017\pi\)
−0.522545 + 0.852612i \(0.675017\pi\)
\(270\) 0 0
\(271\) 15.2752i 0.927900i 0.885861 + 0.463950i \(0.153568\pi\)
−0.885861 + 0.463950i \(0.846432\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.92006i 0.176086i
\(276\) 0 0
\(277\) 27.2829i 1.63927i 0.572888 + 0.819634i \(0.305823\pi\)
−0.572888 + 0.819634i \(0.694177\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 9.76299i 0.582411i 0.956660 + 0.291206i \(0.0940564\pi\)
−0.956660 + 0.291206i \(0.905944\pi\)
\(282\) 0 0
\(283\) 18.7180 1.11267 0.556334 0.830959i \(-0.312207\pi\)
0.556334 + 0.830959i \(0.312207\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0.148874 0.00878776
\(288\) 0 0
\(289\) −0.117498 −0.00691166
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 11.3088i 0.660667i 0.943864 + 0.330334i \(0.107161\pi\)
−0.943864 + 0.330334i \(0.892839\pi\)
\(294\) 0 0
\(295\) −6.52555 −0.379932
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.31625i 0.191784i
\(300\) 0 0
\(301\) 1.24638i 0.0718402i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −6.63626 −0.379991
\(306\) 0 0
\(307\) 22.2817i 1.27169i 0.771818 + 0.635843i \(0.219348\pi\)
−0.771818 + 0.635843i \(0.780652\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 22.5078i 1.27630i −0.769912 0.638150i \(-0.779700\pi\)
0.769912 0.638150i \(-0.220300\pi\)
\(312\) 0 0
\(313\) 28.3219i 1.60085i −0.599434 0.800424i \(-0.704608\pi\)
0.599434 0.800424i \(-0.295392\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 31.3547i 1.76106i 0.473994 + 0.880528i \(0.342812\pi\)
−0.473994 + 0.880528i \(0.657188\pi\)
\(318\) 0 0
\(319\) −3.95078 −0.221201
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −9.69041 −0.539189
\(324\) 0 0
\(325\) 2.99515i 0.166141i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.08275i 0.0596941i
\(330\) 0 0
\(331\) 0.436333i 0.0239830i 0.999928 + 0.0119915i \(0.00381711\pi\)
−0.999928 + 0.0119915i \(0.996183\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.88201i 0.102825i
\(336\) 0 0
\(337\) −6.04105 −0.329077 −0.164539 0.986371i \(-0.552614\pi\)
−0.164539 + 0.986371i \(0.552614\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.54022i 0.354173i
\(342\) 0 0
\(343\) 1.73096 0.0934631
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 36.3162 1.94956 0.974778 0.223175i \(-0.0716421\pi\)
0.974778 + 0.223175i \(0.0716421\pi\)
\(348\) 0 0
\(349\) 24.4596i 1.30929i 0.755936 + 0.654646i \(0.227182\pi\)
−0.755936 + 0.654646i \(0.772818\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 15.6082i 0.830742i −0.909652 0.415371i \(-0.863652\pi\)
0.909652 0.415371i \(-0.136348\pi\)
\(354\) 0 0
\(355\) −7.11135 −0.377431
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 21.7241i 1.14655i 0.819362 + 0.573277i \(0.194328\pi\)
−0.819362 + 0.573277i \(0.805672\pi\)
\(360\) 0 0
\(361\) −13.4378 −0.707252
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 12.5910i 0.659041i
\(366\) 0 0
\(367\) 3.18168 0.166082 0.0830411 0.996546i \(-0.473537\pi\)
0.0830411 + 0.996546i \(0.473537\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −0.760248 −0.0394701
\(372\) 0 0
\(373\) 4.76949i 0.246955i 0.992347 + 0.123477i \(0.0394047\pi\)
−0.992347 + 0.123477i \(0.960595\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −4.05238 −0.208708
\(378\) 0 0
\(379\) 37.7989i 1.94160i −0.239889 0.970800i \(-0.577111\pi\)
0.239889 0.970800i \(-0.422889\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 10.2238i 0.522410i 0.965283 + 0.261205i \(0.0841198\pi\)
−0.965283 + 0.261205i \(0.915880\pi\)
\(384\) 0 0
\(385\) 0.113054i 0.00576177i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −16.2159 −0.822181 −0.411090 0.911595i \(-0.634852\pi\)
−0.411090 + 0.911595i \(0.634852\pi\)
\(390\) 0 0
\(391\) 16.7283 0.845987
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 8.17180i 0.411168i
\(396\) 0 0
\(397\) 16.4032 0.823253 0.411626 0.911353i \(-0.364961\pi\)
0.411626 + 0.911353i \(0.364961\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 26.7573 1.33620 0.668098 0.744073i \(-0.267109\pi\)
0.668098 + 0.744073i \(0.267109\pi\)
\(402\) 0 0
\(403\) 6.70841i 0.334170i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.65703 −0.181272
\(408\) 0 0
\(409\) 8.02809 0.396964 0.198482 0.980105i \(-0.436399\pi\)
0.198482 + 0.980105i \(0.436399\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −0.702241 −0.0345550
\(414\) 0 0
\(415\) 13.7768 0.676278
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 37.5741i 1.83561i −0.397028 0.917806i \(-0.629959\pi\)
0.397028 0.917806i \(-0.370041\pi\)
\(420\) 0 0
\(421\) 0.251731 0.0122686 0.00613432 0.999981i \(-0.498047\pi\)
0.00613432 + 0.999981i \(0.498047\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 15.1085 0.732872
\(426\) 0 0
\(427\) −0.714155 −0.0345604
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 36.2347i 1.74537i −0.488287 0.872683i \(-0.662378\pi\)
0.488287 0.872683i \(-0.337622\pi\)
\(432\) 0 0
\(433\) 34.5240 1.65912 0.829560 0.558418i \(-0.188591\pi\)
0.829560 + 0.558418i \(0.188591\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −9.60190 −0.459321
\(438\) 0 0
\(439\) 4.48008i 0.213823i −0.994269 0.106911i \(-0.965904\pi\)
0.994269 0.106911i \(-0.0340961\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −19.2513 −0.914659 −0.457330 0.889297i \(-0.651194\pi\)
−0.457330 + 0.889297i \(0.651194\pi\)
\(444\) 0 0
\(445\) 3.20838i 0.152092i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 14.5088i 0.684712i 0.939570 + 0.342356i \(0.111225\pi\)
−0.939570 + 0.342356i \(0.888775\pi\)
\(450\) 0 0
\(451\) 0.955150i 0.0449762i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.115962i 0.00543636i
\(456\) 0 0
\(457\) 23.3890i 1.09409i −0.837103 0.547045i \(-0.815753\pi\)
0.837103 0.547045i \(-0.184247\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 36.1915i 1.68560i 0.538224 + 0.842802i \(0.319096\pi\)
−0.538224 + 0.842802i \(0.680904\pi\)
\(462\) 0 0
\(463\) 35.9351i 1.67005i 0.550214 + 0.835023i \(0.314546\pi\)
−0.550214 + 0.835023i \(0.685454\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 24.9986i 1.15680i 0.815755 + 0.578398i \(0.196322\pi\)
−0.815755 + 0.578398i \(0.803678\pi\)
\(468\) 0 0
\(469\) 0.202531i 0.00935200i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7.99656 0.367682
\(474\) 0 0
\(475\) −8.67217 −0.397906
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −14.4680 −0.661059 −0.330529 0.943796i \(-0.607227\pi\)
−0.330529 + 0.943796i \(0.607227\pi\)
\(480\) 0 0
\(481\) −3.75108 −0.171034
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −15.6538 −0.710803
\(486\) 0 0
\(487\) 32.4111i 1.46869i 0.678777 + 0.734344i \(0.262510\pi\)
−0.678777 + 0.734344i \(0.737490\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 11.1482i 0.503112i −0.967843 0.251556i \(-0.919058\pi\)
0.967843 0.251556i \(-0.0809423\pi\)
\(492\) 0 0
\(493\) 20.4416i 0.920641i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −0.765282 −0.0343276
\(498\) 0 0
\(499\) 4.33292i 0.193968i 0.995286 + 0.0969840i \(0.0309196\pi\)
−0.995286 + 0.0969840i \(0.969080\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 19.1302i 0.852974i −0.904494 0.426487i \(-0.859751\pi\)
0.904494 0.426487i \(-0.140249\pi\)
\(504\) 0 0
\(505\) 19.3995 0.863267
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 4.13258i 0.183173i −0.995797 0.0915866i \(-0.970806\pi\)
0.995797 0.0915866i \(-0.0291938\pi\)
\(510\) 0 0
\(511\) 1.35496i 0.0599401i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 8.83065i 0.389125i
\(516\) 0 0
\(517\) 6.94675 0.305518
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −32.0773 −1.40533 −0.702666 0.711520i \(-0.748007\pi\)
−0.702666 + 0.711520i \(0.748007\pi\)
\(522\) 0 0
\(523\) −20.6678 −0.903741 −0.451871 0.892083i \(-0.649243\pi\)
−0.451871 + 0.892083i \(0.649243\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −33.8395 −1.47407
\(528\) 0 0
\(529\) −6.42449 −0.279326
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.979713i 0.0424361i
\(534\) 0 0
\(535\) 6.43684i 0.278289i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 5.54668i 0.238913i
\(540\) 0 0
\(541\) 29.8851i 1.28486i 0.766345 + 0.642430i \(0.222074\pi\)
−0.766345 + 0.642430i \(0.777926\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8.16031i 0.349549i
\(546\) 0 0
\(547\) 2.10073i 0.0898207i −0.998991 0.0449103i \(-0.985700\pi\)
0.998991 0.0449103i \(-0.0143002\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 11.7333i 0.499854i
\(552\) 0 0
\(553\) 0.879401i 0.0373959i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 11.2694i 0.477500i 0.971081 + 0.238750i \(0.0767376\pi\)
−0.971081 + 0.238750i \(0.923262\pi\)
\(558\) 0 0
\(559\) 8.20220 0.346916
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 25.6691i 1.08183i 0.841079 + 0.540913i \(0.181921\pi\)
−0.841079 + 0.540913i \(0.818079\pi\)
\(564\) 0 0
\(565\) 13.2311 0.556636
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 28.5577 1.19720 0.598601 0.801047i \(-0.295724\pi\)
0.598601 + 0.801047i \(0.295724\pi\)
\(570\) 0 0
\(571\) 24.9317i 1.04336i −0.853142 0.521679i \(-0.825306\pi\)
0.853142 0.521679i \(-0.174694\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 14.9705 0.624315
\(576\) 0 0
\(577\) 14.3062 0.595575 0.297787 0.954632i \(-0.403751\pi\)
0.297787 + 0.954632i \(0.403751\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.48258 0.0615079
\(582\) 0 0
\(583\) 4.87762i 0.202010i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −37.9790 −1.56756 −0.783780 0.621039i \(-0.786711\pi\)
−0.783780 + 0.621039i \(0.786711\pi\)
\(588\) 0 0
\(589\) 19.4236 0.800334
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 21.9951 0.903231 0.451615 0.892213i \(-0.350848\pi\)
0.451615 + 0.892213i \(0.350848\pi\)
\(594\) 0 0
\(595\) −0.584949 −0.0239806
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 10.0813i 0.411912i −0.978561 0.205956i \(-0.933970\pi\)
0.978561 0.205956i \(-0.0660304\pi\)
\(600\) 0 0
\(601\) −37.3683 −1.52428 −0.762142 0.647410i \(-0.775852\pi\)
−0.762142 + 0.647410i \(0.775852\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −11.9266 −0.484886
\(606\) 0 0
\(607\) 10.1385i 0.411511i −0.978603 0.205755i \(-0.934035\pi\)
0.978603 0.205755i \(-0.0659651\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7.12539 0.288263
\(612\) 0 0
\(613\) 18.2559 0.737347 0.368674 0.929559i \(-0.379812\pi\)
0.368674 + 0.929559i \(0.379812\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 41.1896i 1.65823i −0.559078 0.829115i \(-0.688845\pi\)
0.559078 0.829115i \(-0.311155\pi\)
\(618\) 0 0
\(619\) 0.482020i 0.0193740i −0.999953 0.00968701i \(-0.996916\pi\)
0.999953 0.00968701i \(-0.00308352\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0.345268i 0.0138329i
\(624\) 0 0
\(625\) 6.90644 0.276258
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 18.9217i 0.754458i
\(630\) 0 0
\(631\) −32.9041 −1.30989 −0.654946 0.755676i \(-0.727309\pi\)
−0.654946 + 0.755676i \(0.727309\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 23.8102 0.944879
\(636\) 0 0
\(637\) 5.68932i 0.225419i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −49.4849 −1.95454 −0.977268 0.212008i \(-0.932000\pi\)
−0.977268 + 0.212008i \(0.932000\pi\)
\(642\) 0 0
\(643\) 34.0548i 1.34299i −0.741009 0.671495i \(-0.765652\pi\)
0.741009 0.671495i \(-0.234348\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 31.4190 1.23521 0.617603 0.786490i \(-0.288104\pi\)
0.617603 + 0.786490i \(0.288104\pi\)
\(648\) 0 0
\(649\) 4.50546i 0.176855i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 43.7737i 1.71300i 0.516148 + 0.856499i \(0.327365\pi\)
−0.516148 + 0.856499i \(0.672635\pi\)
\(654\) 0 0
\(655\) 0.909462 0.0355356
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 41.8148 1.62887 0.814437 0.580253i \(-0.197046\pi\)
0.814437 + 0.580253i \(0.197046\pi\)
\(660\) 0 0
\(661\) 30.3568i 1.18074i −0.807131 0.590372i \(-0.798981\pi\)
0.807131 0.590372i \(-0.201019\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.335756 0.0130200
\(666\) 0 0
\(667\) 20.2549i 0.784271i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.58190i 0.176882i
\(672\) 0 0
\(673\) 19.8641i 0.765707i −0.923809 0.382853i \(-0.874941\pi\)
0.923809 0.382853i \(-0.125059\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 42.0025i 1.61429i −0.590356 0.807143i \(-0.701013\pi\)
0.590356 0.807143i \(-0.298987\pi\)
\(678\) 0 0
\(679\) −1.68457 −0.0646479
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 19.1373 0.732269 0.366135 0.930562i \(-0.380681\pi\)
0.366135 + 0.930562i \(0.380681\pi\)
\(684\) 0 0
\(685\) 9.74993i 0.372526i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 5.00305i 0.190601i
\(690\) 0 0
\(691\) 33.3900i 1.27022i 0.772423 + 0.635109i \(0.219045\pi\)
−0.772423 + 0.635109i \(0.780955\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 4.80108i 0.182115i
\(696\) 0 0
\(697\) 4.94200 0.187192
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 5.30339i 0.200306i 0.994972 + 0.100153i \(0.0319333\pi\)
−0.994972 + 0.100153i \(0.968067\pi\)
\(702\) 0 0
\(703\) 10.8609i 0.409626i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.08766 0.0785146
\(708\) 0 0
\(709\) 25.2444i 0.948074i 0.880505 + 0.474037i \(0.157204\pi\)
−0.880505 + 0.474037i \(0.842796\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −33.5304 −1.25572
\(714\) 0 0
\(715\) 0.743989 0.0278236
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −19.1715 −0.714975 −0.357487 0.933918i \(-0.616366\pi\)
−0.357487 + 0.933918i \(0.616366\pi\)
\(720\) 0 0
\(721\) 0.950303i 0.0353911i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 18.2936i 0.679408i
\(726\) 0 0
\(727\) 9.95814i 0.369327i −0.982802 0.184663i \(-0.940881\pi\)
0.982802 0.184663i \(-0.0591195\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 41.3747i 1.53030i
\(732\) 0 0
\(733\) −7.86615 −0.290543 −0.145271 0.989392i \(-0.546406\pi\)
−0.145271 + 0.989392i \(0.546406\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.29940 −0.0478641
\(738\) 0 0
\(739\) 47.6692i 1.75354i −0.480910 0.876770i \(-0.659694\pi\)
0.480910 0.876770i \(-0.340306\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 7.00382i 0.256945i 0.991713 + 0.128473i \(0.0410074\pi\)
−0.991713 + 0.128473i \(0.958993\pi\)
\(744\) 0 0
\(745\) 1.74444 0.0639113
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0.692695i 0.0253105i
\(750\) 0 0
\(751\) 42.7064i 1.55838i −0.626789 0.779189i \(-0.715631\pi\)
0.626789 0.779189i \(-0.284369\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 5.22951i 0.190321i
\(756\) 0 0
\(757\) −36.1458 −1.31374 −0.656870 0.754004i \(-0.728120\pi\)
−0.656870 + 0.754004i \(0.728120\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 21.1228i 0.765702i −0.923810 0.382851i \(-0.874942\pi\)
0.923810 0.382851i \(-0.125058\pi\)
\(762\) 0 0
\(763\) 0.878165i 0.0317917i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.62132i 0.166866i
\(768\) 0 0
\(769\) 44.7684i 1.61439i 0.590284 + 0.807195i \(0.299016\pi\)
−0.590284 + 0.807195i \(0.700984\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 16.3112 0.586673 0.293337 0.956009i \(-0.405234\pi\)
0.293337 + 0.956009i \(0.405234\pi\)
\(774\) 0 0
\(775\) −30.2837 −1.08782
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −2.83666 −0.101634
\(780\) 0 0
\(781\) 4.90991i 0.175691i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −17.2356 −0.615166
\(786\) 0 0
\(787\) 26.6793i 0.951016i 0.879711 + 0.475508i \(0.157736\pi\)
−0.879711 + 0.475508i \(0.842264\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.42385 0.0506264
\(792\) 0 0
\(793\) 4.69973i 0.166892i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 36.7400 1.30140 0.650698 0.759337i \(-0.274476\pi\)
0.650698 + 0.759337i \(0.274476\pi\)
\(798\) 0 0