Properties

Label 6012.2.h.a.3005.14
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 3005.14
Character \(\chi\) \(=\) 6012.3005
Dual form 6012.2.h.a.3005.13

$q$-expansion

\(f(q)\) \(=\) \(q-2.21423 q^{5} -1.10661 q^{7} +O(q^{10})\) \(q-2.21423 q^{5} -1.10661 q^{7} -1.52623i q^{11} +3.26430i q^{13} -5.68069 q^{17} +6.25231 q^{19} +4.86296 q^{23} -0.0972042 q^{25} -8.93164i q^{29} -1.89423 q^{31} +2.45029 q^{35} +4.41435i q^{37} -2.75307 q^{41} +0.818459i q^{43} +3.00307i q^{47} -5.77541 q^{49} -9.36175 q^{53} +3.37942i q^{55} -4.10719 q^{59} -12.1085 q^{61} -7.22790i q^{65} -9.17056i q^{67} +10.9769 q^{71} -5.25379i q^{73} +1.68895i q^{77} +11.5472i q^{79} +5.75501 q^{83} +12.5783 q^{85} -8.72974i q^{89} -3.61231i q^{91} -13.8440 q^{95} +16.1024 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56q + O(q^{10}) \) \( 56q + 8q^{19} + 64q^{25} - 8q^{31} + 56q^{49} - 8q^{61} + 32q^{85} - 48q^{97} + O(q^{100}) \)

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
\(5\) −2.21423 −0.990232 −0.495116 0.868827i \(-0.664874\pi\)
−0.495116 + 0.868827i \(0.664874\pi\)
\(6\) 0 0
\(7\) −1.10661 −0.418260 −0.209130 0.977888i \(-0.567063\pi\)
−0.209130 + 0.977888i \(0.567063\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.52623i 0.460176i −0.973170 0.230088i \(-0.926099\pi\)
0.973170 0.230088i \(-0.0739015\pi\)
\(12\) 0 0
\(13\) 3.26430i 0.905354i 0.891675 + 0.452677i \(0.149531\pi\)
−0.891675 + 0.452677i \(0.850469\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5.68069 −1.37777 −0.688885 0.724870i \(-0.741900\pi\)
−0.688885 + 0.724870i \(0.741900\pi\)
\(18\) 0 0
\(19\) 6.25231 1.43438 0.717189 0.696879i \(-0.245428\pi\)
0.717189 + 0.696879i \(0.245428\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.86296 1.01400 0.506998 0.861947i \(-0.330755\pi\)
0.506998 + 0.861947i \(0.330755\pi\)
\(24\) 0 0
\(25\) −0.0972042 −0.0194408
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.93164i 1.65856i −0.558830 0.829282i \(-0.688750\pi\)
0.558830 0.829282i \(-0.311250\pi\)
\(30\) 0 0
\(31\) −1.89423 −0.340214 −0.170107 0.985426i \(-0.554411\pi\)
−0.170107 + 0.985426i \(0.554411\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 2.45029 0.414174
\(36\) 0 0
\(37\) 4.41435i 0.725714i 0.931845 + 0.362857i \(0.118199\pi\)
−0.931845 + 0.362857i \(0.881801\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.75307 −0.429957 −0.214979 0.976619i \(-0.568968\pi\)
−0.214979 + 0.976619i \(0.568968\pi\)
\(42\) 0 0
\(43\) 0.818459i 0.124814i 0.998051 + 0.0624069i \(0.0198777\pi\)
−0.998051 + 0.0624069i \(0.980122\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.00307i 0.438042i 0.975720 + 0.219021i \(0.0702864\pi\)
−0.975720 + 0.219021i \(0.929714\pi\)
\(48\) 0 0
\(49\) −5.77541 −0.825059
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −9.36175 −1.28594 −0.642968 0.765893i \(-0.722297\pi\)
−0.642968 + 0.765893i \(0.722297\pi\)
\(54\) 0 0
\(55\) 3.37942i 0.455681i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −4.10719 −0.534710 −0.267355 0.963598i \(-0.586150\pi\)
−0.267355 + 0.963598i \(0.586150\pi\)
\(60\) 0 0
\(61\) −12.1085 −1.55033 −0.775166 0.631757i \(-0.782334\pi\)
−0.775166 + 0.631757i \(0.782334\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7.22790i 0.896510i
\(66\) 0 0
\(67\) 9.17056i 1.12036i −0.828370 0.560181i \(-0.810732\pi\)
0.828370 0.560181i \(-0.189268\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.9769 1.30271 0.651357 0.758771i \(-0.274200\pi\)
0.651357 + 0.758771i \(0.274200\pi\)
\(72\) 0 0
\(73\) 5.25379i 0.614910i −0.951563 0.307455i \(-0.900523\pi\)
0.951563 0.307455i \(-0.0994773\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.68895i 0.192473i
\(78\) 0 0
\(79\) 11.5472i 1.29916i 0.760294 + 0.649579i \(0.225055\pi\)
−0.760294 + 0.649579i \(0.774945\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 5.75501 0.631694 0.315847 0.948810i \(-0.397711\pi\)
0.315847 + 0.948810i \(0.397711\pi\)
\(84\) 0 0
\(85\) 12.5783 1.36431
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8.72974i 0.925350i −0.886528 0.462675i \(-0.846890\pi\)
0.886528 0.462675i \(-0.153110\pi\)
\(90\) 0 0
\(91\) 3.61231i 0.378673i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −13.8440 −1.42037
\(96\) 0 0
\(97\) 16.1024 1.63495 0.817473 0.575966i \(-0.195374\pi\)
0.817473 + 0.575966i \(0.195374\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.4647 1.33978 0.669892 0.742458i \(-0.266340\pi\)
0.669892 + 0.742458i \(0.266340\pi\)
\(102\) 0 0
\(103\) 9.52408i 0.938435i 0.883083 + 0.469218i \(0.155464\pi\)
−0.883083 + 0.469218i \(0.844536\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 15.5883i 1.50698i 0.657459 + 0.753490i \(0.271631\pi\)
−0.657459 + 0.753490i \(0.728369\pi\)
\(108\) 0 0
\(109\) 16.7828i 1.60750i 0.594969 + 0.803748i \(0.297164\pi\)
−0.594969 + 0.803748i \(0.702836\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.82248 0.359589 0.179794 0.983704i \(-0.442457\pi\)
0.179794 + 0.983704i \(0.442457\pi\)
\(114\) 0 0
\(115\) −10.7677 −1.00409
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 6.28632 0.576266
\(120\) 0 0
\(121\) 8.67061 0.788238
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.2864 1.00948
\(126\) 0 0
\(127\) 18.2667 1.62091 0.810454 0.585802i \(-0.199220\pi\)
0.810454 + 0.585802i \(0.199220\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0.659035 0.0575801 0.0287901 0.999585i \(-0.490835\pi\)
0.0287901 + 0.999585i \(0.490835\pi\)
\(132\) 0 0
\(133\) −6.91888 −0.599943
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.04200i 0.345331i −0.984980 0.172666i \(-0.944762\pi\)
0.984980 0.172666i \(-0.0552380\pi\)
\(138\) 0 0
\(139\) 7.65339i 0.649152i −0.945860 0.324576i \(-0.894778\pi\)
0.945860 0.324576i \(-0.105222\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.98208 0.416623
\(144\) 0 0
\(145\) 19.7767i 1.64236i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.28994 0.597215 0.298608 0.954376i \(-0.403478\pi\)
0.298608 + 0.954376i \(0.403478\pi\)
\(150\) 0 0
\(151\) 3.47956i 0.283163i 0.989927 + 0.141581i \(0.0452187\pi\)
−0.989927 + 0.141581i \(0.954781\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.19425 0.336890
\(156\) 0 0
\(157\) 3.65246 0.291498 0.145749 0.989322i \(-0.453441\pi\)
0.145749 + 0.989322i \(0.453441\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −5.38140 −0.424114
\(162\) 0 0
\(163\) 0.688646i 0.0539389i 0.999636 + 0.0269695i \(0.00858569\pi\)
−0.999636 + 0.0269695i \(0.991414\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.71933 9.53799i −0.674722 0.738072i
\(168\) 0 0
\(169\) 2.34434 0.180334
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 22.9194i 1.74253i −0.490817 0.871263i \(-0.663301\pi\)
0.490817 0.871263i \(-0.336699\pi\)
\(174\) 0 0
\(175\) 0.107567 0.00813132
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.0216i 0.898538i −0.893396 0.449269i \(-0.851684\pi\)
0.893396 0.449269i \(-0.148316\pi\)
\(180\) 0 0
\(181\) 3.52740 0.262190 0.131095 0.991370i \(-0.458151\pi\)
0.131095 + 0.991370i \(0.458151\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 9.77436i 0.718625i
\(186\) 0 0
\(187\) 8.67006i 0.634018i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 12.9390i 0.936233i −0.883667 0.468116i \(-0.844933\pi\)
0.883667 0.468116i \(-0.155067\pi\)
\(192\) 0 0
\(193\) 10.2433i 0.737327i 0.929563 + 0.368664i \(0.120185\pi\)
−0.929563 + 0.368664i \(0.879815\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 18.8008 1.33950 0.669750 0.742587i \(-0.266401\pi\)
0.669750 + 0.742587i \(0.266401\pi\)
\(198\) 0 0
\(199\) 15.3128 1.08549 0.542747 0.839896i \(-0.317384\pi\)
0.542747 + 0.839896i \(0.317384\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 9.88386i 0.693711i
\(204\) 0 0
\(205\) 6.09592 0.425757
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 9.54248i 0.660067i
\(210\) 0 0
\(211\) 5.41188 0.372569 0.186285 0.982496i \(-0.440355\pi\)
0.186285 + 0.982496i \(0.440355\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.81225i 0.123595i
\(216\) 0 0
\(217\) 2.09618 0.142298
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 18.5435i 1.24737i
\(222\) 0 0
\(223\) 20.0075 1.33980 0.669902 0.742450i \(-0.266336\pi\)
0.669902 + 0.742450i \(0.266336\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 15.2335 1.01108 0.505541 0.862802i \(-0.331293\pi\)
0.505541 + 0.862802i \(0.331293\pi\)
\(228\) 0 0
\(229\) 5.19317 0.343175 0.171587 0.985169i \(-0.445110\pi\)
0.171587 + 0.985169i \(0.445110\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 17.8919i 1.17213i 0.810262 + 0.586067i \(0.199325\pi\)
−0.810262 + 0.586067i \(0.800675\pi\)
\(234\) 0 0
\(235\) 6.64947i 0.433763i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 9.92922i 0.642268i 0.947034 + 0.321134i \(0.104064\pi\)
−0.947034 + 0.321134i \(0.895936\pi\)
\(240\) 0 0
\(241\) 1.09323i 0.0704210i 0.999380 + 0.0352105i \(0.0112102\pi\)
−0.999380 + 0.0352105i \(0.988790\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 12.7881 0.816999
\(246\) 0 0
\(247\) 20.4094i 1.29862i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.1099i 0.764368i 0.924086 + 0.382184i \(0.124828\pi\)
−0.924086 + 0.382184i \(0.875172\pi\)
\(252\) 0 0
\(253\) 7.42200i 0.466617i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.89139 0.117982 0.0589908 0.998259i \(-0.481212\pi\)
0.0589908 + 0.998259i \(0.481212\pi\)
\(258\) 0 0
\(259\) 4.88497i 0.303537i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0.0895652i 0.00552282i −0.999996 0.00276141i \(-0.999121\pi\)
0.999996 0.00276141i \(-0.000878986\pi\)
\(264\) 0 0
\(265\) 20.7290 1.27337
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −25.1217 −1.53170 −0.765849 0.643020i \(-0.777681\pi\)
−0.765849 + 0.643020i \(0.777681\pi\)
\(270\) 0 0
\(271\) 32.9056i 1.99887i −0.0335917 0.999436i \(-0.510695\pi\)
0.0335917 0.999436i \(-0.489305\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0.148356i 0.00894621i
\(276\) 0 0
\(277\) 3.38325i 0.203280i 0.994821 + 0.101640i \(0.0324090\pi\)
−0.994821 + 0.101640i \(0.967591\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.9847i 1.01322i 0.862175 + 0.506611i \(0.169102\pi\)
−0.862175 + 0.506611i \(0.830898\pi\)
\(282\) 0 0
\(283\) −15.2318 −0.905436 −0.452718 0.891654i \(-0.649546\pi\)
−0.452718 + 0.891654i \(0.649546\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.04658 0.179834
\(288\) 0 0
\(289\) 15.2703 0.898252
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5.64421i 0.329738i 0.986315 + 0.164869i \(0.0527202\pi\)
−0.986315 + 0.164869i \(0.947280\pi\)
\(294\) 0 0
\(295\) 9.09424 0.529487
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 15.8742i 0.918026i
\(300\) 0 0
\(301\) 0.905717i 0.0522046i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 26.8109 1.53519
\(306\) 0 0
\(307\) 23.8883i 1.36338i 0.731643 + 0.681688i \(0.238754\pi\)
−0.731643 + 0.681688i \(0.761246\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 14.1909i 0.804690i −0.915488 0.402345i \(-0.868195\pi\)
0.915488 0.402345i \(-0.131805\pi\)
\(312\) 0 0
\(313\) 9.69765i 0.548143i 0.961709 + 0.274072i \(0.0883706\pi\)
−0.961709 + 0.274072i \(0.911629\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 8.30637i 0.466532i −0.972413 0.233266i \(-0.925059\pi\)
0.972413 0.233266i \(-0.0749413\pi\)
\(318\) 0 0
\(319\) −13.6318 −0.763232
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −35.5175 −1.97624
\(324\) 0 0
\(325\) 0.317304i 0.0176008i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.32323i 0.183216i
\(330\) 0 0
\(331\) 14.1080i 0.775447i −0.921776 0.387723i \(-0.873262\pi\)
0.921776 0.387723i \(-0.126738\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 20.3057i 1.10942i
\(336\) 0 0
\(337\) 29.0529 1.58261 0.791307 0.611419i \(-0.209401\pi\)
0.791307 + 0.611419i \(0.209401\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.89103i 0.156558i
\(342\) 0 0
\(343\) 14.1374 0.763349
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 16.1997 0.869645 0.434822 0.900516i \(-0.356811\pi\)
0.434822 + 0.900516i \(0.356811\pi\)
\(348\) 0 0
\(349\) 16.0230i 0.857694i 0.903377 + 0.428847i \(0.141080\pi\)
−0.903377 + 0.428847i \(0.858920\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 11.7462i 0.625185i 0.949887 + 0.312593i \(0.101197\pi\)
−0.949887 + 0.312593i \(0.898803\pi\)
\(354\) 0 0
\(355\) −24.3053 −1.28999
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 9.78407i 0.516383i 0.966094 + 0.258192i \(0.0831266\pi\)
−0.966094 + 0.258192i \(0.916873\pi\)
\(360\) 0 0
\(361\) 20.0914 1.05744
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 11.6331i 0.608903i
\(366\) 0 0
\(367\) 8.46441 0.441838 0.220919 0.975292i \(-0.429094\pi\)
0.220919 + 0.975292i \(0.429094\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 10.3598 0.537855
\(372\) 0 0
\(373\) 36.5181i 1.89084i 0.325859 + 0.945418i \(0.394346\pi\)
−0.325859 + 0.945418i \(0.605654\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 29.1556 1.50159
\(378\) 0 0
\(379\) 35.7780i 1.83779i −0.394497 0.918897i \(-0.629081\pi\)
0.394497 0.918897i \(-0.370919\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 31.2989i 1.59930i −0.600465 0.799651i \(-0.705018\pi\)
0.600465 0.799651i \(-0.294982\pi\)
\(384\) 0 0
\(385\) 3.73971i 0.190593i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 35.3084 1.79021 0.895104 0.445857i \(-0.147101\pi\)
0.895104 + 0.445857i \(0.147101\pi\)
\(390\) 0 0
\(391\) −27.6250 −1.39705
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 25.5680i 1.28647i
\(396\) 0 0
\(397\) 16.7323 0.839773 0.419886 0.907577i \(-0.362070\pi\)
0.419886 + 0.907577i \(0.362070\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −17.1969 −0.858772 −0.429386 0.903121i \(-0.641270\pi\)
−0.429386 + 0.903121i \(0.641270\pi\)
\(402\) 0 0
\(403\) 6.18333i 0.308014i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 6.73732 0.333957
\(408\) 0 0
\(409\) −16.6768 −0.824614 −0.412307 0.911045i \(-0.635277\pi\)
−0.412307 + 0.911045i \(0.635277\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4.54506 0.223648
\(414\) 0 0
\(415\) −12.7429 −0.625523
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 5.19272i 0.253681i 0.991923 + 0.126841i \(0.0404836\pi\)
−0.991923 + 0.126841i \(0.959516\pi\)
\(420\) 0 0
\(421\) 38.6414 1.88327 0.941633 0.336641i \(-0.109291\pi\)
0.941633 + 0.336641i \(0.109291\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0.552187 0.0267850
\(426\) 0 0
\(427\) 13.3994 0.648442
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 29.8086i 1.43583i 0.696130 + 0.717916i \(0.254904\pi\)
−0.696130 + 0.717916i \(0.745096\pi\)
\(432\) 0 0
\(433\) −23.5768 −1.13303 −0.566515 0.824052i \(-0.691709\pi\)
−0.566515 + 0.824052i \(0.691709\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 30.4047 1.45445
\(438\) 0 0
\(439\) 33.7776i 1.61212i −0.591836 0.806058i \(-0.701597\pi\)
0.591836 0.806058i \(-0.298403\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 5.82132 0.276579 0.138290 0.990392i \(-0.455839\pi\)
0.138290 + 0.990392i \(0.455839\pi\)
\(444\) 0 0
\(445\) 19.3296i 0.916311i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 31.1572i 1.47040i −0.677850 0.735201i \(-0.737088\pi\)
0.677850 0.735201i \(-0.262912\pi\)
\(450\) 0 0
\(451\) 4.20183i 0.197856i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 7.99848i 0.374974i
\(456\) 0 0
\(457\) 9.38086i 0.438818i 0.975633 + 0.219409i \(0.0704129\pi\)
−0.975633 + 0.219409i \(0.929587\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 42.4425i 1.97674i −0.152053 0.988372i \(-0.548588\pi\)
0.152053 0.988372i \(-0.451412\pi\)
\(462\) 0 0
\(463\) 9.06979i 0.421509i 0.977539 + 0.210754i \(0.0675921\pi\)
−0.977539 + 0.210754i \(0.932408\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.77793i 0.221096i −0.993871 0.110548i \(-0.964739\pi\)
0.993871 0.110548i \(-0.0352606\pi\)
\(468\) 0 0
\(469\) 10.1482i 0.468603i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.24916 0.0574364
\(474\) 0 0
\(475\) −0.607750 −0.0278855
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −0.0102115 −0.000466577 −0.000233289 1.00000i \(-0.500074\pi\)
−0.000233289 1.00000i \(0.500074\pi\)
\(480\) 0 0
\(481\) −14.4098 −0.657028
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −35.6543 −1.61898
\(486\) 0 0
\(487\) 33.4280i 1.51477i 0.652971 + 0.757383i \(0.273522\pi\)
−0.652971 + 0.757383i \(0.726478\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 22.1905i 1.00144i 0.865608 + 0.500722i \(0.166932\pi\)
−0.865608 + 0.500722i \(0.833068\pi\)
\(492\) 0 0
\(493\) 50.7379i 2.28512i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −12.1471 −0.544873
\(498\) 0 0
\(499\) 41.2840i 1.84813i 0.382238 + 0.924064i \(0.375153\pi\)
−0.382238 + 0.924064i \(0.624847\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 32.2498i 1.43795i −0.695037 0.718974i \(-0.744612\pi\)
0.695037 0.718974i \(-0.255388\pi\)
\(504\) 0 0
\(505\) −29.8138 −1.32670
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 15.7838i 0.699603i 0.936824 + 0.349802i \(0.113751\pi\)
−0.936824 + 0.349802i \(0.886249\pi\)
\(510\) 0 0
\(511\) 5.81391i 0.257192i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 21.0885i 0.929269i
\(516\) 0 0
\(517\) 4.58338 0.201577
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −38.8058 −1.70011 −0.850057 0.526691i \(-0.823432\pi\)
−0.850057 + 0.526691i \(0.823432\pi\)
\(522\) 0 0
\(523\) 19.8033 0.865940 0.432970 0.901408i \(-0.357466\pi\)
0.432970 + 0.901408i \(0.357466\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 10.7605 0.468736
\(528\) 0 0
\(529\) 0.648346 0.0281890
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.98685i 0.389264i
\(534\) 0 0
\(535\) 34.5161i 1.49226i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 8.81462i 0.379673i
\(540\) 0 0
\(541\) 29.3779i 1.26305i −0.775354 0.631527i \(-0.782429\pi\)
0.775354 0.631527i \(-0.217571\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 37.1608i 1.59179i
\(546\) 0 0
\(547\) 43.1267i 1.84397i −0.387231 0.921983i \(-0.626568\pi\)
0.387231 0.921983i \(-0.373432\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 55.8434i 2.37901i
\(552\) 0 0
\(553\) 12.7782i 0.543386i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 39.1934i 1.66068i 0.557259 + 0.830339i \(0.311853\pi\)
−0.557259 + 0.830339i \(0.688147\pi\)
\(558\) 0 0
\(559\) −2.67170 −0.113001
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 32.6676i 1.37678i −0.725342 0.688388i \(-0.758319\pi\)
0.725342 0.688388i \(-0.241681\pi\)
\(564\) 0 0
\(565\) −8.46384 −0.356076
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −27.4263 −1.14977 −0.574885 0.818234i \(-0.694953\pi\)
−0.574885 + 0.818234i \(0.694953\pi\)
\(570\) 0 0
\(571\) 9.53874i 0.399184i 0.979879 + 0.199592i \(0.0639616\pi\)
−0.979879 + 0.199592i \(0.936038\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −0.472700 −0.0197129
\(576\) 0 0
\(577\) −8.45388 −0.351940 −0.175970 0.984396i \(-0.556306\pi\)
−0.175970 + 0.984396i \(0.556306\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −6.36856 −0.264212
\(582\) 0 0
\(583\) 14.2882i 0.591757i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −16.7100 −0.689696 −0.344848 0.938659i \(-0.612070\pi\)
−0.344848 + 0.938659i \(0.612070\pi\)
\(588\) 0 0
\(589\) −11.8433 −0.487995
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5.86014 0.240647 0.120324 0.992735i \(-0.461607\pi\)
0.120324 + 0.992735i \(0.461607\pi\)
\(594\) 0 0
\(595\) −13.9193 −0.570637
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 46.3364i 1.89326i 0.322329 + 0.946628i \(0.395534\pi\)
−0.322329 + 0.946628i \(0.604466\pi\)
\(600\) 0 0
\(601\) 14.8468 0.605615 0.302807 0.953052i \(-0.402076\pi\)
0.302807 + 0.953052i \(0.402076\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −19.1987 −0.780538
\(606\) 0 0
\(607\) 16.2655i 0.660196i 0.943947 + 0.330098i \(0.107082\pi\)
−0.943947 + 0.330098i \(0.892918\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −9.80291 −0.396583
\(612\) 0 0
\(613\) 0.116515 0.00470602 0.00235301 0.999997i \(-0.499251\pi\)
0.00235301 + 0.999997i \(0.499251\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.36390i 0.0549084i 0.999623 + 0.0274542i \(0.00874004\pi\)
−0.999623 + 0.0274542i \(0.991260\pi\)
\(618\) 0 0
\(619\) 18.6289i 0.748759i 0.927276 + 0.374379i \(0.122144\pi\)
−0.927276 + 0.374379i \(0.877856\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 9.66043i 0.387037i
\(624\) 0 0
\(625\) −24.5045 −0.980181
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 25.0766i 0.999868i
\(630\) 0 0
\(631\) 8.81751 0.351019 0.175510 0.984478i \(-0.443843\pi\)
0.175510 + 0.984478i \(0.443843\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −40.4466 −1.60508
\(636\) 0 0
\(637\) 18.8527i 0.746970i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 14.4116 0.569224 0.284612 0.958643i \(-0.408135\pi\)
0.284612 + 0.958643i \(0.408135\pi\)
\(642\) 0 0
\(643\) 36.3292i 1.43268i 0.697749 + 0.716342i \(0.254185\pi\)
−0.697749 + 0.716342i \(0.745815\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 13.6166 0.535322 0.267661 0.963513i \(-0.413749\pi\)
0.267661 + 0.963513i \(0.413749\pi\)
\(648\) 0 0
\(649\) 6.26853i 0.246061i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 22.5752i 0.883437i 0.897154 + 0.441718i \(0.145631\pi\)
−0.897154 + 0.441718i \(0.854369\pi\)
\(654\) 0 0
\(655\) −1.45925 −0.0570177
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 2.04987 0.0798518 0.0399259 0.999203i \(-0.487288\pi\)
0.0399259 + 0.999203i \(0.487288\pi\)
\(660\) 0 0
\(661\) 24.1084i 0.937708i 0.883276 + 0.468854i \(0.155333\pi\)
−0.883276 + 0.468854i \(0.844667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 15.3200 0.594082
\(666\) 0 0
\(667\) 43.4342i 1.68178i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 18.4804i 0.713426i
\(672\) 0 0
\(673\) 5.80060i 0.223597i 0.993731 + 0.111798i \(0.0356611\pi\)
−0.993731 + 0.111798i \(0.964339\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.60083i 0.0615248i −0.999527 0.0307624i \(-0.990206\pi\)
0.999527 0.0307624i \(-0.00979352\pi\)
\(678\) 0 0
\(679\) −17.8191 −0.683833
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 26.7110 1.02207 0.511034 0.859560i \(-0.329263\pi\)
0.511034 + 0.859560i \(0.329263\pi\)
\(684\) 0 0
\(685\) 8.94990i 0.341958i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 30.5596i 1.16423i
\(690\) 0 0
\(691\) 14.0321i 0.533808i −0.963723 0.266904i \(-0.913999\pi\)
0.963723 0.266904i \(-0.0860006\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 16.9463i 0.642811i
\(696\) 0 0
\(697\) 15.6393 0.592382
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 50.1976i 1.89594i 0.318364 + 0.947969i \(0.396867\pi\)
−0.318364 + 0.947969i \(0.603133\pi\)
\(702\) 0 0
\(703\) 27.5999i 1.04095i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −14.9002 −0.560378
\(708\) 0 0
\(709\) 23.4320i 0.880009i −0.897996 0.440004i \(-0.854977\pi\)
0.897996 0.440004i \(-0.145023\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −9.21155 −0.344975
\(714\) 0 0
\(715\) −11.0315 −0.412553
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −36.5966 −1.36482 −0.682411 0.730969i \(-0.739069\pi\)
−0.682411 + 0.730969i \(0.739069\pi\)
\(720\) 0 0
\(721\) 10.5395i 0.392510i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0.868193i 0.0322439i
\(726\) 0 0
\(727\) 24.8889i 0.923080i −0.887119 0.461540i \(-0.847297\pi\)
0.887119 0.461540i \(-0.152703\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 4.64942i 0.171965i
\(732\) 0 0
\(733\) 11.4373 0.422446 0.211223 0.977438i \(-0.432255\pi\)
0.211223 + 0.977438i \(0.432255\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −13.9964 −0.515564
\(738\) 0 0
\(739\) 36.0468i 1.32600i −0.748618 0.663001i \(-0.769282\pi\)
0.748618 0.663001i \(-0.230718\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 12.2477i 0.449324i 0.974437 + 0.224662i \(0.0721277\pi\)
−0.974437 + 0.224662i \(0.927872\pi\)
\(744\) 0 0
\(745\) −16.1416 −0.591382
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 17.2502i 0.630310i
\(750\) 0 0
\(751\) 8.39744i 0.306427i 0.988193 + 0.153213i \(0.0489622\pi\)
−0.988193 + 0.153213i \(0.951038\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 7.70453i 0.280397i
\(756\) 0 0
\(757\) 39.7794 1.44581 0.722904 0.690949i \(-0.242807\pi\)
0.722904 + 0.690949i \(0.242807\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 16.9492i 0.614409i 0.951644 + 0.307205i \(0.0993936\pi\)
−0.951644 + 0.307205i \(0.900606\pi\)
\(762\) 0 0
\(763\) 18.5720i 0.672351i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 13.4071i 0.484102i
\(768\) 0 0
\(769\) 0.808888i 0.0291692i −0.999894 0.0145846i \(-0.995357\pi\)
0.999894 0.0145846i \(-0.00464259\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 21.1557 0.760919 0.380459 0.924798i \(-0.375766\pi\)
0.380459 + 0.924798i \(0.375766\pi\)
\(774\) 0 0
\(775\) 0.184127 0.00661404
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −17.2130 −0.616721
\(780\) 0 0
\(781\) 16.7533i 0.599479i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −8.08738 −0.288651
\(786\) 0 0
\(787\) 20.5075i 0.731015i −0.930808 0.365508i \(-0.880895\pi\)
0.930808 0.365508i \(-0.119105\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4.23000 −0.150402
\(792\) 0 0
\(793\) 39.5257i 1.40360i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.49394 0.0883398 0.0441699 0.999024i \(-0.485936\pi\)
0.0441699 + 0.999024i \(0.485936\pi\)
\(798\) 0 0