Properties

Label 6012.2.h.a.3005.18
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 3005.18
Character \(\chi\) \(=\) 6012.3005
Dual form 6012.2.h.a.3005.17

$q$-expansion

\(f(q)\) \(=\) \(q-1.39469 q^{5} -4.57475 q^{7} +O(q^{10})\) \(q-1.39469 q^{5} -4.57475 q^{7} +3.16668i q^{11} -0.678448i q^{13} +2.04770 q^{17} -3.83917 q^{19} +1.63709 q^{23} -3.05483 q^{25} -2.49162i q^{29} -4.99578 q^{31} +6.38038 q^{35} +10.1215i q^{37} -12.1835 q^{41} +4.71460i q^{43} -3.25120i q^{47} +13.9284 q^{49} -7.08207 q^{53} -4.41655i q^{55} +5.99637 q^{59} +9.50097 q^{61} +0.946227i q^{65} +9.54542i q^{67} -9.35726 q^{71} -10.4904i q^{73} -14.4868i q^{77} -17.2275i q^{79} -15.9720 q^{83} -2.85592 q^{85} +15.3957i q^{89} +3.10373i q^{91} +5.35446 q^{95} +7.44085 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 56q + O(q^{10}) \) \( 56q + 8q^{19} + 64q^{25} - 8q^{31} + 56q^{49} - 8q^{61} + 32q^{85} - 48q^{97} + O(q^{100}) \)

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
\(5\) −1.39469 −0.623726 −0.311863 0.950127i \(-0.600953\pi\)
−0.311863 + 0.950127i \(0.600953\pi\)
\(6\) 0 0
\(7\) −4.57475 −1.72909 −0.864547 0.502552i \(-0.832394\pi\)
−0.864547 + 0.502552i \(0.832394\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.16668i 0.954791i 0.878689 + 0.477395i \(0.158419\pi\)
−0.878689 + 0.477395i \(0.841581\pi\)
\(12\) 0 0
\(13\) 0.678448i 0.188168i −0.995564 0.0940839i \(-0.970008\pi\)
0.995564 0.0940839i \(-0.0299922\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.04770 0.496641 0.248320 0.968678i \(-0.420121\pi\)
0.248320 + 0.968678i \(0.420121\pi\)
\(18\) 0 0
\(19\) −3.83917 −0.880765 −0.440383 0.897810i \(-0.645157\pi\)
−0.440383 + 0.897810i \(0.645157\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.63709 0.341356 0.170678 0.985327i \(-0.445404\pi\)
0.170678 + 0.985327i \(0.445404\pi\)
\(24\) 0 0
\(25\) −3.05483 −0.610966
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.49162i 0.462681i −0.972873 0.231341i \(-0.925689\pi\)
0.972873 0.231341i \(-0.0743112\pi\)
\(30\) 0 0
\(31\) −4.99578 −0.897269 −0.448635 0.893715i \(-0.648090\pi\)
−0.448635 + 0.893715i \(0.648090\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.38038 1.07848
\(36\) 0 0
\(37\) 10.1215i 1.66396i 0.554802 + 0.831982i \(0.312794\pi\)
−0.554802 + 0.831982i \(0.687206\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −12.1835 −1.90275 −0.951374 0.308037i \(-0.900328\pi\)
−0.951374 + 0.308037i \(0.900328\pi\)
\(42\) 0 0
\(43\) 4.71460i 0.718970i 0.933151 + 0.359485i \(0.117048\pi\)
−0.933151 + 0.359485i \(0.882952\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3.25120i 0.474237i −0.971481 0.237118i \(-0.923797\pi\)
0.971481 0.237118i \(-0.0762029\pi\)
\(48\) 0 0
\(49\) 13.9284 1.98977
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −7.08207 −0.972797 −0.486398 0.873737i \(-0.661690\pi\)
−0.486398 + 0.873737i \(0.661690\pi\)
\(54\) 0 0
\(55\) 4.41655i 0.595528i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.99637 0.780661 0.390330 0.920675i \(-0.372361\pi\)
0.390330 + 0.920675i \(0.372361\pi\)
\(60\) 0 0
\(61\) 9.50097 1.21647 0.608237 0.793755i \(-0.291877\pi\)
0.608237 + 0.793755i \(0.291877\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.946227i 0.117365i
\(66\) 0 0
\(67\) 9.54542i 1.16616i 0.812415 + 0.583079i \(0.198152\pi\)
−0.812415 + 0.583079i \(0.801848\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.35726 −1.11050 −0.555251 0.831683i \(-0.687378\pi\)
−0.555251 + 0.831683i \(0.687378\pi\)
\(72\) 0 0
\(73\) 10.4904i 1.22781i −0.789381 0.613903i \(-0.789598\pi\)
0.789381 0.613903i \(-0.210402\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 14.4868i 1.65092i
\(78\) 0 0
\(79\) 17.2275i 1.93824i −0.246590 0.969120i \(-0.579310\pi\)
0.246590 0.969120i \(-0.420690\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −15.9720 −1.75316 −0.876580 0.481257i \(-0.840180\pi\)
−0.876580 + 0.481257i \(0.840180\pi\)
\(84\) 0 0
\(85\) −2.85592 −0.309768
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 15.3957i 1.63194i 0.578094 + 0.815970i \(0.303797\pi\)
−0.578094 + 0.815970i \(0.696203\pi\)
\(90\) 0 0
\(91\) 3.10373i 0.325360i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 5.35446 0.549356
\(96\) 0 0
\(97\) 7.44085 0.755503 0.377752 0.925907i \(-0.376697\pi\)
0.377752 + 0.925907i \(0.376697\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 6.07700 0.604684 0.302342 0.953200i \(-0.402232\pi\)
0.302342 + 0.953200i \(0.402232\pi\)
\(102\) 0 0
\(103\) 14.5730i 1.43592i 0.696083 + 0.717961i \(0.254924\pi\)
−0.696083 + 0.717961i \(0.745076\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.24282i 0.410169i 0.978744 + 0.205084i \(0.0657469\pi\)
−0.978744 + 0.205084i \(0.934253\pi\)
\(108\) 0 0
\(109\) 5.51313i 0.528062i −0.964514 0.264031i \(-0.914948\pi\)
0.964514 0.264031i \(-0.0850521\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 10.6859 1.00524 0.502620 0.864507i \(-0.332369\pi\)
0.502620 + 0.864507i \(0.332369\pi\)
\(114\) 0 0
\(115\) −2.28323 −0.212913
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −9.36773 −0.858738
\(120\) 0 0
\(121\) 0.972122 0.0883747
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.2340 1.00480
\(126\) 0 0
\(127\) 14.3908 1.27698 0.638490 0.769630i \(-0.279559\pi\)
0.638490 + 0.769630i \(0.279559\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.21806 −0.368534 −0.184267 0.982876i \(-0.558991\pi\)
−0.184267 + 0.982876i \(0.558991\pi\)
\(132\) 0 0
\(133\) 17.5632 1.52293
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 15.7664i 1.34701i −0.739182 0.673505i \(-0.764788\pi\)
0.739182 0.673505i \(-0.235212\pi\)
\(138\) 0 0
\(139\) 1.32836i 0.112670i 0.998412 + 0.0563350i \(0.0179415\pi\)
−0.998412 + 0.0563350i \(0.982059\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.14843 0.179661
\(144\) 0 0
\(145\) 3.47504i 0.288586i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 11.7715 0.964362 0.482181 0.876072i \(-0.339845\pi\)
0.482181 + 0.876072i \(0.339845\pi\)
\(150\) 0 0
\(151\) 10.5889i 0.861709i −0.902421 0.430854i \(-0.858212\pi\)
0.902421 0.430854i \(-0.141788\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 6.96759 0.559650
\(156\) 0 0
\(157\) 15.4110 1.22993 0.614965 0.788555i \(-0.289170\pi\)
0.614965 + 0.788555i \(0.289170\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −7.48927 −0.590237
\(162\) 0 0
\(163\) 12.6578i 0.991436i −0.868484 0.495718i \(-0.834905\pi\)
0.868484 0.495718i \(-0.165095\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.96454 12.7726i 0.152021 0.988377i
\(168\) 0 0
\(169\) 12.5397 0.964593
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 20.4501i 1.55479i −0.629013 0.777395i \(-0.716541\pi\)
0.629013 0.777395i \(-0.283459\pi\)
\(174\) 0 0
\(175\) 13.9751 1.05642
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0.305626i 0.0228436i −0.999935 0.0114218i \(-0.996364\pi\)
0.999935 0.0114218i \(-0.00363574\pi\)
\(180\) 0 0
\(181\) 10.4692 0.778168 0.389084 0.921202i \(-0.372792\pi\)
0.389084 + 0.921202i \(0.372792\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 14.1164i 1.03786i
\(186\) 0 0
\(187\) 6.48442i 0.474188i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 14.0013i 1.01310i −0.862212 0.506548i \(-0.830921\pi\)
0.862212 0.506548i \(-0.169079\pi\)
\(192\) 0 0
\(193\) 11.7565i 0.846255i −0.906070 0.423127i \(-0.860932\pi\)
0.906070 0.423127i \(-0.139068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −21.3280 −1.51956 −0.759780 0.650181i \(-0.774693\pi\)
−0.759780 + 0.650181i \(0.774693\pi\)
\(198\) 0 0
\(199\) −10.0439 −0.711992 −0.355996 0.934488i \(-0.615858\pi\)
−0.355996 + 0.934488i \(0.615858\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 11.3985i 0.800020i
\(204\) 0 0
\(205\) 16.9923 1.18679
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 12.1574i 0.840947i
\(210\) 0 0
\(211\) −2.36608 −0.162888 −0.0814438 0.996678i \(-0.525953\pi\)
−0.0814438 + 0.996678i \(0.525953\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.57542i 0.448440i
\(216\) 0 0
\(217\) 22.8545 1.55146
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.38926i 0.0934517i
\(222\) 0 0
\(223\) 5.63935 0.377639 0.188819 0.982012i \(-0.439534\pi\)
0.188819 + 0.982012i \(0.439534\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 14.2355 0.944844 0.472422 0.881372i \(-0.343380\pi\)
0.472422 + 0.881372i \(0.343380\pi\)
\(228\) 0 0
\(229\) 5.42754 0.358662 0.179331 0.983789i \(-0.442607\pi\)
0.179331 + 0.983789i \(0.442607\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 7.63054i 0.499894i −0.968260 0.249947i \(-0.919587\pi\)
0.968260 0.249947i \(-0.0804131\pi\)
\(234\) 0 0
\(235\) 4.53443i 0.295794i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 6.38304i 0.412884i −0.978459 0.206442i \(-0.933811\pi\)
0.978459 0.206442i \(-0.0661885\pi\)
\(240\) 0 0
\(241\) 17.0010i 1.09513i 0.836764 + 0.547564i \(0.184445\pi\)
−0.836764 + 0.547564i \(0.815555\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −19.4258 −1.24107
\(246\) 0 0
\(247\) 2.60468i 0.165732i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 15.5787i 0.983316i 0.870788 + 0.491658i \(0.163609\pi\)
−0.870788 + 0.491658i \(0.836391\pi\)
\(252\) 0 0
\(253\) 5.18413i 0.325924i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −22.5886 −1.40904 −0.704520 0.709684i \(-0.748838\pi\)
−0.704520 + 0.709684i \(0.748838\pi\)
\(258\) 0 0
\(259\) 46.3034i 2.87715i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.71783i 0.290914i −0.989365 0.145457i \(-0.953535\pi\)
0.989365 0.145457i \(-0.0464652\pi\)
\(264\) 0 0
\(265\) 9.87731 0.606758
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 22.3567 1.36311 0.681556 0.731766i \(-0.261303\pi\)
0.681556 + 0.731766i \(0.261303\pi\)
\(270\) 0 0
\(271\) 6.60680i 0.401335i 0.979659 + 0.200667i \(0.0643110\pi\)
−0.979659 + 0.200667i \(0.935689\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 9.67368i 0.583345i
\(276\) 0 0
\(277\) 26.6210i 1.59950i −0.600334 0.799749i \(-0.704966\pi\)
0.600334 0.799749i \(-0.295034\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 32.8224i 1.95802i −0.203805 0.979011i \(-0.565331\pi\)
0.203805 0.979011i \(-0.434669\pi\)
\(282\) 0 0
\(283\) 1.71459 0.101922 0.0509609 0.998701i \(-0.483772\pi\)
0.0509609 + 0.998701i \(0.483772\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 55.7367 3.29003
\(288\) 0 0
\(289\) −12.8069 −0.753348
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.20416i 0.479292i −0.970860 0.239646i \(-0.922969\pi\)
0.970860 0.239646i \(-0.0770314\pi\)
\(294\) 0 0
\(295\) −8.36310 −0.486918
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.11068i 0.0642322i
\(300\) 0 0
\(301\) 21.5681i 1.24317i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −13.2509 −0.758747
\(306\) 0 0
\(307\) 4.84943i 0.276772i 0.990378 + 0.138386i \(0.0441914\pi\)
−0.990378 + 0.138386i \(0.955809\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 7.14797i 0.405324i −0.979249 0.202662i \(-0.935041\pi\)
0.979249 0.202662i \(-0.0649593\pi\)
\(312\) 0 0
\(313\) 11.0218i 0.622988i 0.950248 + 0.311494i \(0.100829\pi\)
−0.950248 + 0.311494i \(0.899171\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.26539i 0.295734i 0.989007 + 0.147867i \(0.0472408\pi\)
−0.989007 + 0.147867i \(0.952759\pi\)
\(318\) 0 0
\(319\) 7.89016 0.441764
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −7.86147 −0.437424
\(324\) 0 0
\(325\) 2.07254i 0.114964i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 14.8734i 0.820000i
\(330\) 0 0
\(331\) 8.27343i 0.454749i 0.973807 + 0.227374i \(0.0730141\pi\)
−0.973807 + 0.227374i \(0.926986\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 13.3129i 0.727363i
\(336\) 0 0
\(337\) 34.1026 1.85769 0.928844 0.370471i \(-0.120804\pi\)
0.928844 + 0.370471i \(0.120804\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 15.8201i 0.856704i
\(342\) 0 0
\(343\) −31.6955 −1.71140
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −17.7405 −0.952359 −0.476179 0.879348i \(-0.657979\pi\)
−0.476179 + 0.879348i \(0.657979\pi\)
\(348\) 0 0
\(349\) 32.9216i 1.76226i 0.472879 + 0.881128i \(0.343215\pi\)
−0.472879 + 0.881128i \(0.656785\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 23.4839i 1.24992i −0.780656 0.624961i \(-0.785115\pi\)
0.780656 0.624961i \(-0.214885\pi\)
\(354\) 0 0
\(355\) 13.0505 0.692649
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 19.7233i 1.04096i 0.853875 + 0.520479i \(0.174246\pi\)
−0.853875 + 0.520479i \(0.825754\pi\)
\(360\) 0 0
\(361\) −4.26079 −0.224252
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 14.6309i 0.765815i
\(366\) 0 0
\(367\) 24.6349 1.28593 0.642967 0.765894i \(-0.277703\pi\)
0.642967 + 0.765894i \(0.277703\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 32.3987 1.68206
\(372\) 0 0
\(373\) 5.09861i 0.263996i 0.991250 + 0.131998i \(0.0421393\pi\)
−0.991250 + 0.131998i \(0.957861\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.69043 −0.0870617
\(378\) 0 0
\(379\) 2.07234i 0.106449i 0.998583 + 0.0532244i \(0.0169498\pi\)
−0.998583 + 0.0532244i \(0.983050\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 21.5206i 1.09965i 0.835280 + 0.549825i \(0.185305\pi\)
−0.835280 + 0.549825i \(0.814695\pi\)
\(384\) 0 0
\(385\) 20.2046i 1.02972i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 11.7818 0.597359 0.298679 0.954354i \(-0.403454\pi\)
0.298679 + 0.954354i \(0.403454\pi\)
\(390\) 0 0
\(391\) 3.35227 0.169531
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 24.0270i 1.20893i
\(396\) 0 0
\(397\) −1.66291 −0.0834590 −0.0417295 0.999129i \(-0.513287\pi\)
−0.0417295 + 0.999129i \(0.513287\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.873382 0.0436146 0.0218073 0.999762i \(-0.493058\pi\)
0.0218073 + 0.999762i \(0.493058\pi\)
\(402\) 0 0
\(403\) 3.38938i 0.168837i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −32.0516 −1.58874
\(408\) 0 0
\(409\) −22.8880 −1.13174 −0.565869 0.824495i \(-0.691459\pi\)
−0.565869 + 0.824495i \(0.691459\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −27.4319 −1.34984
\(414\) 0 0
\(415\) 22.2761 1.09349
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0.989747i 0.0483523i −0.999708 0.0241762i \(-0.992304\pi\)
0.999708 0.0241762i \(-0.00769626\pi\)
\(420\) 0 0
\(421\) −13.0798 −0.637468 −0.318734 0.947844i \(-0.603258\pi\)
−0.318734 + 0.947844i \(0.603258\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −6.25538 −0.303431
\(426\) 0 0
\(427\) −43.4646 −2.10340
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 25.2048i 1.21407i −0.794674 0.607037i \(-0.792358\pi\)
0.794674 0.607037i \(-0.207642\pi\)
\(432\) 0 0
\(433\) −7.31288 −0.351435 −0.175717 0.984441i \(-0.556224\pi\)
−0.175717 + 0.984441i \(0.556224\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −6.28505 −0.300655
\(438\) 0 0
\(439\) 32.2996i 1.54158i 0.637090 + 0.770789i \(0.280138\pi\)
−0.637090 + 0.770789i \(0.719862\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 18.0186 0.856092 0.428046 0.903757i \(-0.359202\pi\)
0.428046 + 0.903757i \(0.359202\pi\)
\(444\) 0 0
\(445\) 21.4723i 1.01788i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 40.4917i 1.91092i 0.295119 + 0.955460i \(0.404641\pi\)
−0.295119 + 0.955460i \(0.595359\pi\)
\(450\) 0 0
\(451\) 38.5814i 1.81673i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 4.32876i 0.202935i
\(456\) 0 0
\(457\) 29.5031i 1.38010i −0.723763 0.690049i \(-0.757589\pi\)
0.723763 0.690049i \(-0.242411\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 18.5480i 0.863865i 0.901906 + 0.431932i \(0.142168\pi\)
−0.901906 + 0.431932i \(0.857832\pi\)
\(462\) 0 0
\(463\) 34.3790i 1.59773i −0.601513 0.798863i \(-0.705435\pi\)
0.601513 0.798863i \(-0.294565\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.5733i 0.859470i 0.902955 + 0.429735i \(0.141393\pi\)
−0.902955 + 0.429735i \(0.858607\pi\)
\(468\) 0 0
\(469\) 43.6679i 2.01640i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −14.9296 −0.686466
\(474\) 0 0
\(475\) 11.7280 0.538118
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −20.6138 −0.941868 −0.470934 0.882169i \(-0.656083\pi\)
−0.470934 + 0.882169i \(0.656083\pi\)
\(480\) 0 0
\(481\) 6.86691 0.313104
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −10.3777 −0.471227
\(486\) 0 0
\(487\) 29.2837i 1.32697i 0.748190 + 0.663485i \(0.230923\pi\)
−0.748190 + 0.663485i \(0.769077\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 32.2853i 1.45701i −0.685038 0.728507i \(-0.740214\pi\)
0.685038 0.728507i \(-0.259786\pi\)
\(492\) 0 0
\(493\) 5.10209i 0.229786i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 42.8071 1.92016
\(498\) 0 0
\(499\) 14.7516i 0.660373i 0.943916 + 0.330187i \(0.107112\pi\)
−0.943916 + 0.330187i \(0.892888\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 18.1022i 0.807137i 0.914949 + 0.403569i \(0.132230\pi\)
−0.914949 + 0.403569i \(0.867770\pi\)
\(504\) 0 0
\(505\) −8.47555 −0.377157
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 38.7177i 1.71613i −0.513541 0.858065i \(-0.671666\pi\)
0.513541 0.858065i \(-0.328334\pi\)
\(510\) 0 0
\(511\) 47.9909i 2.12299i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 20.3249i 0.895621i
\(516\) 0 0
\(517\) 10.2955 0.452797
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 25.1043 1.09984 0.549919 0.835218i \(-0.314659\pi\)
0.549919 + 0.835218i \(0.314659\pi\)
\(522\) 0 0
\(523\) −30.4808 −1.33283 −0.666416 0.745580i \(-0.732173\pi\)
−0.666416 + 0.745580i \(0.732173\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10.2299 −0.445620
\(528\) 0 0
\(529\) −20.3199 −0.883476
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.26590i 0.358036i
\(534\) 0 0
\(535\) 5.91743i 0.255833i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 44.1067i 1.89981i
\(540\) 0 0
\(541\) 9.04366i 0.388817i −0.980921 0.194409i \(-0.937721\pi\)
0.980921 0.194409i \(-0.0622788\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.68912i 0.329366i
\(546\) 0 0
\(547\) 39.1689i 1.67474i −0.546635 0.837371i \(-0.684092\pi\)
0.546635 0.837371i \(-0.315908\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 9.56573i 0.407514i
\(552\) 0 0
\(553\) 78.8113i 3.35140i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.61535i 0.322673i −0.986899 0.161336i \(-0.948420\pi\)
0.986899 0.161336i \(-0.0515804\pi\)
\(558\) 0 0
\(559\) 3.19861 0.135287
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 41.6553i 1.75556i 0.479062 + 0.877781i \(0.340977\pi\)
−0.479062 + 0.877781i \(0.659023\pi\)
\(564\) 0 0
\(565\) −14.9035 −0.626995
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −24.2979 −1.01862 −0.509311 0.860582i \(-0.670100\pi\)
−0.509311 + 0.860582i \(0.670100\pi\)
\(570\) 0 0
\(571\) 39.5904i 1.65681i 0.560133 + 0.828403i \(0.310750\pi\)
−0.560133 + 0.828403i \(0.689250\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5.00102 −0.208557
\(576\) 0 0
\(577\) 13.4711 0.560807 0.280404 0.959882i \(-0.409532\pi\)
0.280404 + 0.959882i \(0.409532\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 73.0681 3.03138
\(582\) 0 0
\(583\) 22.4267i 0.928817i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.30680 −0.0539376 −0.0269688 0.999636i \(-0.508585\pi\)
−0.0269688 + 0.999636i \(0.508585\pi\)
\(588\) 0 0
\(589\) 19.1797 0.790284
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 13.0491 0.535862 0.267931 0.963438i \(-0.413660\pi\)
0.267931 + 0.963438i \(0.413660\pi\)
\(594\) 0 0
\(595\) 13.0651 0.535617
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 15.3900i 0.628820i 0.949287 + 0.314410i \(0.101807\pi\)
−0.949287 + 0.314410i \(0.898193\pi\)
\(600\) 0 0
\(601\) 44.7420 1.82506 0.912532 0.409006i \(-0.134124\pi\)
0.912532 + 0.409006i \(0.134124\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1.35581 −0.0551216
\(606\) 0 0
\(607\) 7.47030i 0.303210i −0.988441 0.151605i \(-0.951556\pi\)
0.988441 0.151605i \(-0.0484442\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.20577 −0.0892360
\(612\) 0 0
\(613\) 10.4294 0.421238 0.210619 0.977568i \(-0.432452\pi\)
0.210619 + 0.977568i \(0.432452\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 19.7728i 0.796021i 0.917381 + 0.398011i \(0.130299\pi\)
−0.917381 + 0.398011i \(0.869701\pi\)
\(618\) 0 0
\(619\) 13.2880i 0.534090i −0.963684 0.267045i \(-0.913953\pi\)
0.963684 0.267045i \(-0.0860473\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 70.4315i 2.82178i
\(624\) 0 0
\(625\) −0.393856 −0.0157543
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 20.7258i 0.826392i
\(630\) 0 0
\(631\) 36.7439 1.46275 0.731375 0.681975i \(-0.238879\pi\)
0.731375 + 0.681975i \(0.238879\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −20.0708 −0.796486
\(636\) 0 0
\(637\) 9.44967i 0.374410i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −31.1486 −1.23029 −0.615147 0.788412i \(-0.710904\pi\)
−0.615147 + 0.788412i \(0.710904\pi\)
\(642\) 0 0
\(643\) 27.0760i 1.06777i 0.845556 + 0.533887i \(0.179269\pi\)
−0.845556 + 0.533887i \(0.820731\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −25.5380 −1.00400 −0.502000 0.864867i \(-0.667402\pi\)
−0.502000 + 0.864867i \(0.667402\pi\)
\(648\) 0 0
\(649\) 18.9886i 0.745368i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 26.2589i 1.02759i −0.857913 0.513794i \(-0.828239\pi\)
0.857913 0.513794i \(-0.171761\pi\)
\(654\) 0 0
\(655\) 5.88291 0.229864
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −29.7145 −1.15751 −0.578756 0.815501i \(-0.696462\pi\)
−0.578756 + 0.815501i \(0.696462\pi\)
\(660\) 0 0
\(661\) 2.40757i 0.0936436i −0.998903 0.0468218i \(-0.985091\pi\)
0.998903 0.0468218i \(-0.0149093\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −24.4953 −0.949888
\(666\) 0 0
\(667\) 4.07899i 0.157939i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 30.0866i 1.16148i
\(672\) 0 0
\(673\) 12.5706i 0.484561i 0.970206 + 0.242280i \(0.0778954\pi\)
−0.970206 + 0.242280i \(0.922105\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 51.1838i 1.96715i −0.180495 0.983576i \(-0.557770\pi\)
0.180495 0.983576i \(-0.442230\pi\)
\(678\) 0 0
\(679\) −34.0400 −1.30634
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 32.4862 1.24305 0.621524 0.783395i \(-0.286514\pi\)
0.621524 + 0.783395i \(0.286514\pi\)
\(684\) 0 0
\(685\) 21.9892i 0.840165i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4.80482i 0.183049i
\(690\) 0 0
\(691\) 14.5875i 0.554936i −0.960735 0.277468i \(-0.910505\pi\)
0.960735 0.277468i \(-0.0894953\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.85265i 0.0702752i
\(696\) 0 0
\(697\) −24.9482 −0.944982
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 34.6612i 1.30913i −0.756004 0.654567i \(-0.772851\pi\)
0.756004 0.654567i \(-0.227149\pi\)
\(702\) 0 0
\(703\) 38.8581i 1.46556i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −27.8008 −1.04555
\(708\) 0 0
\(709\) 15.6451i 0.587563i 0.955873 + 0.293781i \(0.0949138\pi\)
−0.955873 + 0.293781i \(0.905086\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −8.17853 −0.306288
\(714\) 0 0
\(715\) −2.99640 −0.112059
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 18.1948 0.678550 0.339275 0.940687i \(-0.389818\pi\)
0.339275 + 0.940687i \(0.389818\pi\)
\(720\) 0 0
\(721\) 66.6679i 2.48284i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 7.61146i 0.282683i
\(726\) 0 0
\(727\) 2.38903i 0.0886041i 0.999018 + 0.0443021i \(0.0141064\pi\)
−0.999018 + 0.0443021i \(0.985894\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 9.65410i 0.357070i
\(732\) 0 0
\(733\) 10.8454 0.400585 0.200292 0.979736i \(-0.435811\pi\)
0.200292 + 0.979736i \(0.435811\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −30.2273 −1.11344
\(738\) 0 0
\(739\) 48.9673i 1.80129i 0.434555 + 0.900645i \(0.356906\pi\)
−0.434555 + 0.900645i \(0.643094\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.09351i 0.113490i −0.998389 0.0567448i \(-0.981928\pi\)
0.998389 0.0567448i \(-0.0180722\pi\)
\(744\) 0 0
\(745\) −16.4177 −0.601497
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 19.4098i 0.709220i
\(750\) 0 0
\(751\) 1.99295i 0.0727238i −0.999339 0.0363619i \(-0.988423\pi\)
0.999339 0.0363619i \(-0.0115769\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 14.7682i 0.537470i
\(756\) 0 0
\(757\) 14.4304 0.524483 0.262241 0.965002i \(-0.415538\pi\)
0.262241 + 0.965002i \(0.415538\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 31.3336i 1.13584i −0.823083 0.567921i \(-0.807748\pi\)
0.823083 0.567921i \(-0.192252\pi\)
\(762\) 0 0
\(763\) 25.2212i 0.913068i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.06823i 0.146895i
\(768\) 0 0
\(769\) 41.6037i 1.50027i 0.661287 + 0.750133i \(0.270011\pi\)
−0.661287 + 0.750133i \(0.729989\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −7.95828 −0.286240 −0.143120 0.989705i \(-0.545713\pi\)
−0.143120 + 0.989705i \(0.545713\pi\)
\(774\) 0 0
\(775\) 15.2613 0.548201
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 46.7746 1.67588
\(780\) 0 0
\(781\) 29.6315i 1.06030i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −21.4936 −0.767139
\(786\) 0 0
\(787\) 53.0474i 1.89093i 0.325718 + 0.945467i \(0.394394\pi\)
−0.325718 + 0.945467i \(0.605606\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −48.8851 −1.73816
\(792\) 0 0
\(793\) 6.44592i 0.228901i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −10.5774 −0.374671 −0.187336 0.982296i \(-0.559985\pi\)
−0.187336 + 0.982296i \(0.559985\pi\)
\(798\) 0 0 <