Properties

Label 8016.2.a.w.1.6
Level $8016$
Weight $2$
Character 8016.1
Self dual yes
Analytic conductor $64.008$
Analytic rank $1$
Dimension $7$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 8016 = 2^{4} \cdot 3 \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(64.0080822603\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
Defining polynomial: \(x^{7} - 2 x^{6} - 7 x^{5} + 11 x^{4} + 12 x^{3} - 14 x^{2} - 6 x + 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 4008)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-1.20126\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} +1.82640 q^{5} -2.26944 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} +1.82640 q^{5} -2.26944 q^{7} +1.00000 q^{9} +0.576121 q^{11} -1.28343 q^{13} +1.82640 q^{15} -1.88405 q^{17} -5.66549 q^{19} -2.26944 q^{21} +2.15348 q^{23} -1.66425 q^{25} +1.00000 q^{27} +6.70504 q^{29} -2.07544 q^{31} +0.576121 q^{33} -4.14491 q^{35} -5.63675 q^{37} -1.28343 q^{39} +8.23751 q^{41} -2.54507 q^{43} +1.82640 q^{45} -9.37328 q^{47} -1.84966 q^{49} -1.88405 q^{51} +6.17753 q^{53} +1.05223 q^{55} -5.66549 q^{57} +3.25370 q^{59} +1.95266 q^{61} -2.26944 q^{63} -2.34406 q^{65} -8.37621 q^{67} +2.15348 q^{69} +2.61175 q^{71} -5.82577 q^{73} -1.66425 q^{75} -1.30747 q^{77} +1.10559 q^{79} +1.00000 q^{81} -14.4804 q^{83} -3.44103 q^{85} +6.70504 q^{87} -3.01816 q^{89} +2.91266 q^{91} -2.07544 q^{93} -10.3475 q^{95} -5.58912 q^{97} +0.576121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7q + 7q^{3} - 3q^{5} - 8q^{7} + 7q^{9} + O(q^{10}) \) \( 7q + 7q^{3} - 3q^{5} - 8q^{7} + 7q^{9} - q^{11} - 2q^{13} - 3q^{15} + 11q^{17} - 2q^{19} - 8q^{21} - 17q^{23} + 4q^{25} + 7q^{27} - 7q^{29} - 10q^{31} - q^{33} - 10q^{35} - 21q^{37} - 2q^{39} + 8q^{41} + 12q^{43} - 3q^{45} - 25q^{47} - 7q^{49} + 11q^{51} - 7q^{53} - 15q^{55} - 2q^{57} - 3q^{59} - 14q^{61} - 8q^{63} + 4q^{65} - 4q^{67} - 17q^{69} - 27q^{71} - 12q^{73} + 4q^{75} + 16q^{77} - 8q^{79} + 7q^{81} - 15q^{83} - 3q^{85} - 7q^{87} + 14q^{89} + 3q^{91} - 10q^{93} - 37q^{95} + 3q^{97} - q^{99} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) 1.82640 0.816793 0.408396 0.912805i \(-0.366088\pi\)
0.408396 + 0.912805i \(0.366088\pi\)
\(6\) 0 0
\(7\) −2.26944 −0.857766 −0.428883 0.903360i \(-0.641093\pi\)
−0.428883 + 0.903360i \(0.641093\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0.576121 0.173707 0.0868535 0.996221i \(-0.472319\pi\)
0.0868535 + 0.996221i \(0.472319\pi\)
\(12\) 0 0
\(13\) −1.28343 −0.355959 −0.177979 0.984034i \(-0.556956\pi\)
−0.177979 + 0.984034i \(0.556956\pi\)
\(14\) 0 0
\(15\) 1.82640 0.471576
\(16\) 0 0
\(17\) −1.88405 −0.456949 −0.228474 0.973550i \(-0.573374\pi\)
−0.228474 + 0.973550i \(0.573374\pi\)
\(18\) 0 0
\(19\) −5.66549 −1.29975 −0.649876 0.760040i \(-0.725179\pi\)
−0.649876 + 0.760040i \(0.725179\pi\)
\(20\) 0 0
\(21\) −2.26944 −0.495231
\(22\) 0 0
\(23\) 2.15348 0.449032 0.224516 0.974470i \(-0.427920\pi\)
0.224516 + 0.974470i \(0.427920\pi\)
\(24\) 0 0
\(25\) −1.66425 −0.332850
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 6.70504 1.24509 0.622547 0.782582i \(-0.286098\pi\)
0.622547 + 0.782582i \(0.286098\pi\)
\(30\) 0 0
\(31\) −2.07544 −0.372761 −0.186380 0.982478i \(-0.559676\pi\)
−0.186380 + 0.982478i \(0.559676\pi\)
\(32\) 0 0
\(33\) 0.576121 0.100290
\(34\) 0 0
\(35\) −4.14491 −0.700617
\(36\) 0 0
\(37\) −5.63675 −0.926676 −0.463338 0.886182i \(-0.653348\pi\)
−0.463338 + 0.886182i \(0.653348\pi\)
\(38\) 0 0
\(39\) −1.28343 −0.205513
\(40\) 0 0
\(41\) 8.23751 1.28648 0.643241 0.765664i \(-0.277589\pi\)
0.643241 + 0.765664i \(0.277589\pi\)
\(42\) 0 0
\(43\) −2.54507 −0.388120 −0.194060 0.980990i \(-0.562166\pi\)
−0.194060 + 0.980990i \(0.562166\pi\)
\(44\) 0 0
\(45\) 1.82640 0.272264
\(46\) 0 0
\(47\) −9.37328 −1.36723 −0.683617 0.729841i \(-0.739594\pi\)
−0.683617 + 0.729841i \(0.739594\pi\)
\(48\) 0 0
\(49\) −1.84966 −0.264237
\(50\) 0 0
\(51\) −1.88405 −0.263820
\(52\) 0 0
\(53\) 6.17753 0.848548 0.424274 0.905534i \(-0.360529\pi\)
0.424274 + 0.905534i \(0.360529\pi\)
\(54\) 0 0
\(55\) 1.05223 0.141883
\(56\) 0 0
\(57\) −5.66549 −0.750413
\(58\) 0 0
\(59\) 3.25370 0.423596 0.211798 0.977313i \(-0.432068\pi\)
0.211798 + 0.977313i \(0.432068\pi\)
\(60\) 0 0
\(61\) 1.95266 0.250013 0.125006 0.992156i \(-0.460105\pi\)
0.125006 + 0.992156i \(0.460105\pi\)
\(62\) 0 0
\(63\) −2.26944 −0.285922
\(64\) 0 0
\(65\) −2.34406 −0.290745
\(66\) 0 0
\(67\) −8.37621 −1.02332 −0.511659 0.859189i \(-0.670969\pi\)
−0.511659 + 0.859189i \(0.670969\pi\)
\(68\) 0 0
\(69\) 2.15348 0.259249
\(70\) 0 0
\(71\) 2.61175 0.309958 0.154979 0.987918i \(-0.450469\pi\)
0.154979 + 0.987918i \(0.450469\pi\)
\(72\) 0 0
\(73\) −5.82577 −0.681855 −0.340928 0.940090i \(-0.610741\pi\)
−0.340928 + 0.940090i \(0.610741\pi\)
\(74\) 0 0
\(75\) −1.66425 −0.192171
\(76\) 0 0
\(77\) −1.30747 −0.149000
\(78\) 0 0
\(79\) 1.10559 0.124389 0.0621944 0.998064i \(-0.480190\pi\)
0.0621944 + 0.998064i \(0.480190\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −14.4804 −1.58943 −0.794717 0.606980i \(-0.792381\pi\)
−0.794717 + 0.606980i \(0.792381\pi\)
\(84\) 0 0
\(85\) −3.44103 −0.373233
\(86\) 0 0
\(87\) 6.70504 0.718856
\(88\) 0 0
\(89\) −3.01816 −0.319925 −0.159962 0.987123i \(-0.551137\pi\)
−0.159962 + 0.987123i \(0.551137\pi\)
\(90\) 0 0
\(91\) 2.91266 0.305329
\(92\) 0 0
\(93\) −2.07544 −0.215214
\(94\) 0 0
\(95\) −10.3475 −1.06163
\(96\) 0 0
\(97\) −5.58912 −0.567490 −0.283745 0.958900i \(-0.591577\pi\)
−0.283745 + 0.958900i \(0.591577\pi\)
\(98\) 0 0
\(99\) 0.576121 0.0579023
\(100\) 0 0
\(101\) −10.9923 −1.09377 −0.546887 0.837206i \(-0.684187\pi\)
−0.546887 + 0.837206i \(0.684187\pi\)
\(102\) 0 0
\(103\) −12.3818 −1.22002 −0.610009 0.792395i \(-0.708834\pi\)
−0.610009 + 0.792395i \(0.708834\pi\)
\(104\) 0 0
\(105\) −4.14491 −0.404501
\(106\) 0 0
\(107\) 11.3065 1.09304 0.546519 0.837447i \(-0.315953\pi\)
0.546519 + 0.837447i \(0.315953\pi\)
\(108\) 0 0
\(109\) −4.32015 −0.413795 −0.206898 0.978363i \(-0.566337\pi\)
−0.206898 + 0.978363i \(0.566337\pi\)
\(110\) 0 0
\(111\) −5.63675 −0.535017
\(112\) 0 0
\(113\) 11.4319 1.07542 0.537712 0.843128i \(-0.319289\pi\)
0.537712 + 0.843128i \(0.319289\pi\)
\(114\) 0 0
\(115\) 3.93313 0.366766
\(116\) 0 0
\(117\) −1.28343 −0.118653
\(118\) 0 0
\(119\) 4.27573 0.391955
\(120\) 0 0
\(121\) −10.6681 −0.969826
\(122\) 0 0
\(123\) 8.23751 0.742751
\(124\) 0 0
\(125\) −12.1716 −1.08866
\(126\) 0 0
\(127\) −13.9407 −1.23704 −0.618519 0.785770i \(-0.712267\pi\)
−0.618519 + 0.785770i \(0.712267\pi\)
\(128\) 0 0
\(129\) −2.54507 −0.224081
\(130\) 0 0
\(131\) −10.1443 −0.886308 −0.443154 0.896445i \(-0.646141\pi\)
−0.443154 + 0.896445i \(0.646141\pi\)
\(132\) 0 0
\(133\) 12.8575 1.11488
\(134\) 0 0
\(135\) 1.82640 0.157192
\(136\) 0 0
\(137\) 23.0801 1.97187 0.985933 0.167144i \(-0.0534544\pi\)
0.985933 + 0.167144i \(0.0534544\pi\)
\(138\) 0 0
\(139\) 11.7976 1.00066 0.500331 0.865834i \(-0.333212\pi\)
0.500331 + 0.865834i \(0.333212\pi\)
\(140\) 0 0
\(141\) −9.37328 −0.789373
\(142\) 0 0
\(143\) −0.739410 −0.0618325
\(144\) 0 0
\(145\) 12.2461 1.01698
\(146\) 0 0
\(147\) −1.84966 −0.152557
\(148\) 0 0
\(149\) −17.6449 −1.44553 −0.722764 0.691095i \(-0.757129\pi\)
−0.722764 + 0.691095i \(0.757129\pi\)
\(150\) 0 0
\(151\) −18.2039 −1.48141 −0.740707 0.671828i \(-0.765510\pi\)
−0.740707 + 0.671828i \(0.765510\pi\)
\(152\) 0 0
\(153\) −1.88405 −0.152316
\(154\) 0 0
\(155\) −3.79060 −0.304468
\(156\) 0 0
\(157\) 15.6419 1.24836 0.624180 0.781280i \(-0.285433\pi\)
0.624180 + 0.781280i \(0.285433\pi\)
\(158\) 0 0
\(159\) 6.17753 0.489910
\(160\) 0 0
\(161\) −4.88719 −0.385165
\(162\) 0 0
\(163\) −8.11923 −0.635947 −0.317974 0.948100i \(-0.603002\pi\)
−0.317974 + 0.948100i \(0.603002\pi\)
\(164\) 0 0
\(165\) 1.05223 0.0819160
\(166\) 0 0
\(167\) 1.00000 0.0773823
\(168\) 0 0
\(169\) −11.3528 −0.873293
\(170\) 0 0
\(171\) −5.66549 −0.433251
\(172\) 0 0
\(173\) −20.4282 −1.55312 −0.776562 0.630041i \(-0.783038\pi\)
−0.776562 + 0.630041i \(0.783038\pi\)
\(174\) 0 0
\(175\) 3.77690 0.285507
\(176\) 0 0
\(177\) 3.25370 0.244563
\(178\) 0 0
\(179\) −5.48020 −0.409609 −0.204805 0.978803i \(-0.565656\pi\)
−0.204805 + 0.978803i \(0.565656\pi\)
\(180\) 0 0
\(181\) −22.3021 −1.65770 −0.828851 0.559470i \(-0.811005\pi\)
−0.828851 + 0.559470i \(0.811005\pi\)
\(182\) 0 0
\(183\) 1.95266 0.144345
\(184\) 0 0
\(185\) −10.2950 −0.756902
\(186\) 0 0
\(187\) −1.08544 −0.0793752
\(188\) 0 0
\(189\) −2.26944 −0.165077
\(190\) 0 0
\(191\) −2.13733 −0.154652 −0.0773261 0.997006i \(-0.524638\pi\)
−0.0773261 + 0.997006i \(0.524638\pi\)
\(192\) 0 0
\(193\) 17.7899 1.28054 0.640272 0.768148i \(-0.278822\pi\)
0.640272 + 0.768148i \(0.278822\pi\)
\(194\) 0 0
\(195\) −2.34406 −0.167861
\(196\) 0 0
\(197\) 23.9537 1.70663 0.853317 0.521393i \(-0.174587\pi\)
0.853317 + 0.521393i \(0.174587\pi\)
\(198\) 0 0
\(199\) −9.64250 −0.683538 −0.341769 0.939784i \(-0.611026\pi\)
−0.341769 + 0.939784i \(0.611026\pi\)
\(200\) 0 0
\(201\) −8.37621 −0.590813
\(202\) 0 0
\(203\) −15.2167 −1.06800
\(204\) 0 0
\(205\) 15.0450 1.05079
\(206\) 0 0
\(207\) 2.15348 0.149677
\(208\) 0 0
\(209\) −3.26401 −0.225776
\(210\) 0 0
\(211\) −0.232543 −0.0160089 −0.00800446 0.999968i \(-0.502548\pi\)
−0.00800446 + 0.999968i \(0.502548\pi\)
\(212\) 0 0
\(213\) 2.61175 0.178954
\(214\) 0 0
\(215\) −4.64833 −0.317013
\(216\) 0 0
\(217\) 4.71009 0.319742
\(218\) 0 0
\(219\) −5.82577 −0.393669
\(220\) 0 0
\(221\) 2.41804 0.162655
\(222\) 0 0
\(223\) 21.2086 1.42023 0.710117 0.704084i \(-0.248642\pi\)
0.710117 + 0.704084i \(0.248642\pi\)
\(224\) 0 0
\(225\) −1.66425 −0.110950
\(226\) 0 0
\(227\) 9.72014 0.645148 0.322574 0.946544i \(-0.395452\pi\)
0.322574 + 0.946544i \(0.395452\pi\)
\(228\) 0 0
\(229\) 0.371580 0.0245547 0.0122773 0.999925i \(-0.496092\pi\)
0.0122773 + 0.999925i \(0.496092\pi\)
\(230\) 0 0
\(231\) −1.30747 −0.0860252
\(232\) 0 0
\(233\) 14.0679 0.921619 0.460810 0.887499i \(-0.347559\pi\)
0.460810 + 0.887499i \(0.347559\pi\)
\(234\) 0 0
\(235\) −17.1194 −1.11675
\(236\) 0 0
\(237\) 1.10559 0.0718159
\(238\) 0 0
\(239\) −4.93902 −0.319479 −0.159739 0.987159i \(-0.551065\pi\)
−0.159739 + 0.987159i \(0.551065\pi\)
\(240\) 0 0
\(241\) −11.4721 −0.738985 −0.369493 0.929234i \(-0.620469\pi\)
−0.369493 + 0.929234i \(0.620469\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −3.37823 −0.215827
\(246\) 0 0
\(247\) 7.27125 0.462658
\(248\) 0 0
\(249\) −14.4804 −0.917661
\(250\) 0 0
\(251\) −20.6662 −1.30444 −0.652219 0.758031i \(-0.726162\pi\)
−0.652219 + 0.758031i \(0.726162\pi\)
\(252\) 0 0
\(253\) 1.24067 0.0780001
\(254\) 0 0
\(255\) −3.44103 −0.215486
\(256\) 0 0
\(257\) 24.4158 1.52302 0.761509 0.648155i \(-0.224459\pi\)
0.761509 + 0.648155i \(0.224459\pi\)
\(258\) 0 0
\(259\) 12.7922 0.794871
\(260\) 0 0
\(261\) 6.70504 0.415031
\(262\) 0 0
\(263\) −7.72951 −0.476622 −0.238311 0.971189i \(-0.576594\pi\)
−0.238311 + 0.971189i \(0.576594\pi\)
\(264\) 0 0
\(265\) 11.2827 0.693088
\(266\) 0 0
\(267\) −3.01816 −0.184709
\(268\) 0 0
\(269\) 11.1431 0.679409 0.339705 0.940532i \(-0.389673\pi\)
0.339705 + 0.940532i \(0.389673\pi\)
\(270\) 0 0
\(271\) −10.4906 −0.637259 −0.318630 0.947879i \(-0.603223\pi\)
−0.318630 + 0.947879i \(0.603223\pi\)
\(272\) 0 0
\(273\) 2.91266 0.176282
\(274\) 0 0
\(275\) −0.958808 −0.0578183
\(276\) 0 0
\(277\) 7.71364 0.463468 0.231734 0.972779i \(-0.425560\pi\)
0.231734 + 0.972779i \(0.425560\pi\)
\(278\) 0 0
\(279\) −2.07544 −0.124254
\(280\) 0 0
\(281\) 26.5013 1.58093 0.790467 0.612504i \(-0.209838\pi\)
0.790467 + 0.612504i \(0.209838\pi\)
\(282\) 0 0
\(283\) −9.42532 −0.560277 −0.280139 0.959960i \(-0.590380\pi\)
−0.280139 + 0.959960i \(0.590380\pi\)
\(284\) 0 0
\(285\) −10.3475 −0.612932
\(286\) 0 0
\(287\) −18.6945 −1.10350
\(288\) 0 0
\(289\) −13.4504 −0.791198
\(290\) 0 0
\(291\) −5.58912 −0.327640
\(292\) 0 0
\(293\) −8.72067 −0.509467 −0.254734 0.967011i \(-0.581988\pi\)
−0.254734 + 0.967011i \(0.581988\pi\)
\(294\) 0 0
\(295\) 5.94257 0.345990
\(296\) 0 0
\(297\) 0.576121 0.0334299
\(298\) 0 0
\(299\) −2.76384 −0.159837
\(300\) 0 0
\(301\) 5.77587 0.332916
\(302\) 0 0
\(303\) −10.9923 −0.631491
\(304\) 0 0
\(305\) 3.56635 0.204209
\(306\) 0 0
\(307\) 8.58502 0.489973 0.244986 0.969527i \(-0.421217\pi\)
0.244986 + 0.969527i \(0.421217\pi\)
\(308\) 0 0
\(309\) −12.3818 −0.704377
\(310\) 0 0
\(311\) −14.4072 −0.816955 −0.408478 0.912768i \(-0.633940\pi\)
−0.408478 + 0.912768i \(0.633940\pi\)
\(312\) 0 0
\(313\) −25.1446 −1.42126 −0.710628 0.703568i \(-0.751589\pi\)
−0.710628 + 0.703568i \(0.751589\pi\)
\(314\) 0 0
\(315\) −4.14491 −0.233539
\(316\) 0 0
\(317\) −5.08844 −0.285795 −0.142898 0.989737i \(-0.545642\pi\)
−0.142898 + 0.989737i \(0.545642\pi\)
\(318\) 0 0
\(319\) 3.86291 0.216282
\(320\) 0 0
\(321\) 11.3065 0.631065
\(322\) 0 0
\(323\) 10.6741 0.593920
\(324\) 0 0
\(325\) 2.13594 0.118481
\(326\) 0 0
\(327\) −4.32015 −0.238905
\(328\) 0 0
\(329\) 21.2721 1.17277
\(330\) 0 0
\(331\) −21.6459 −1.18976 −0.594882 0.803813i \(-0.702801\pi\)
−0.594882 + 0.803813i \(0.702801\pi\)
\(332\) 0 0
\(333\) −5.63675 −0.308892
\(334\) 0 0
\(335\) −15.2984 −0.835838
\(336\) 0 0
\(337\) 16.5621 0.902194 0.451097 0.892475i \(-0.351033\pi\)
0.451097 + 0.892475i \(0.351033\pi\)
\(338\) 0 0
\(339\) 11.4319 0.620897
\(340\) 0 0
\(341\) −1.19571 −0.0647512
\(342\) 0 0
\(343\) 20.0837 1.08442
\(344\) 0 0
\(345\) 3.93313 0.211753
\(346\) 0 0
\(347\) 9.57361 0.513938 0.256969 0.966420i \(-0.417276\pi\)
0.256969 + 0.966420i \(0.417276\pi\)
\(348\) 0 0
\(349\) −19.9918 −1.07014 −0.535068 0.844809i \(-0.679714\pi\)
−0.535068 + 0.844809i \(0.679714\pi\)
\(350\) 0 0
\(351\) −1.28343 −0.0685043
\(352\) 0 0
\(353\) 21.3481 1.13624 0.568121 0.822945i \(-0.307671\pi\)
0.568121 + 0.822945i \(0.307671\pi\)
\(354\) 0 0
\(355\) 4.77011 0.253171
\(356\) 0 0
\(357\) 4.27573 0.226295
\(358\) 0 0
\(359\) −7.71901 −0.407394 −0.203697 0.979034i \(-0.565296\pi\)
−0.203697 + 0.979034i \(0.565296\pi\)
\(360\) 0 0
\(361\) 13.0978 0.689357
\(362\) 0 0
\(363\) −10.6681 −0.559929
\(364\) 0 0
\(365\) −10.6402 −0.556934
\(366\) 0 0
\(367\) 0.0577004 0.00301194 0.00150597 0.999999i \(-0.499521\pi\)
0.00150597 + 0.999999i \(0.499521\pi\)
\(368\) 0 0
\(369\) 8.23751 0.428827
\(370\) 0 0
\(371\) −14.0195 −0.727856
\(372\) 0 0
\(373\) −9.25746 −0.479333 −0.239666 0.970855i \(-0.577038\pi\)
−0.239666 + 0.970855i \(0.577038\pi\)
\(374\) 0 0
\(375\) −12.1716 −0.628539
\(376\) 0 0
\(377\) −8.60543 −0.443202
\(378\) 0 0
\(379\) −5.75854 −0.295796 −0.147898 0.989003i \(-0.547251\pi\)
−0.147898 + 0.989003i \(0.547251\pi\)
\(380\) 0 0
\(381\) −13.9407 −0.714204
\(382\) 0 0
\(383\) −15.2551 −0.779498 −0.389749 0.920921i \(-0.627438\pi\)
−0.389749 + 0.920921i \(0.627438\pi\)
\(384\) 0 0
\(385\) −2.38797 −0.121702
\(386\) 0 0
\(387\) −2.54507 −0.129373
\(388\) 0 0
\(389\) 3.29042 0.166831 0.0834156 0.996515i \(-0.473417\pi\)
0.0834156 + 0.996515i \(0.473417\pi\)
\(390\) 0 0
\(391\) −4.05727 −0.205185
\(392\) 0 0
\(393\) −10.1443 −0.511710
\(394\) 0 0
\(395\) 2.01926 0.101600
\(396\) 0 0
\(397\) −4.37992 −0.219822 −0.109911 0.993941i \(-0.535057\pi\)
−0.109911 + 0.993941i \(0.535057\pi\)
\(398\) 0 0
\(399\) 12.8575 0.643678
\(400\) 0 0
\(401\) −8.73266 −0.436088 −0.218044 0.975939i \(-0.569968\pi\)
−0.218044 + 0.975939i \(0.569968\pi\)
\(402\) 0 0
\(403\) 2.66368 0.132688
\(404\) 0 0
\(405\) 1.82640 0.0907548
\(406\) 0 0
\(407\) −3.24745 −0.160970
\(408\) 0 0
\(409\) 36.6405 1.81176 0.905878 0.423538i \(-0.139212\pi\)
0.905878 + 0.423538i \(0.139212\pi\)
\(410\) 0 0
\(411\) 23.0801 1.13846
\(412\) 0 0
\(413\) −7.38406 −0.363346
\(414\) 0 0
\(415\) −26.4471 −1.29824
\(416\) 0 0
\(417\) 11.7976 0.577733
\(418\) 0 0
\(419\) −7.95948 −0.388846 −0.194423 0.980918i \(-0.562283\pi\)
−0.194423 + 0.980918i \(0.562283\pi\)
\(420\) 0 0
\(421\) −35.9794 −1.75353 −0.876765 0.480918i \(-0.840303\pi\)
−0.876765 + 0.480918i \(0.840303\pi\)
\(422\) 0 0
\(423\) −9.37328 −0.455745
\(424\) 0 0
\(425\) 3.13552 0.152095
\(426\) 0 0
\(427\) −4.43144 −0.214453
\(428\) 0 0
\(429\) −0.739410 −0.0356990
\(430\) 0 0
\(431\) −26.4280 −1.27299 −0.636496 0.771280i \(-0.719617\pi\)
−0.636496 + 0.771280i \(0.719617\pi\)
\(432\) 0 0
\(433\) 2.50877 0.120564 0.0602818 0.998181i \(-0.480800\pi\)
0.0602818 + 0.998181i \(0.480800\pi\)
\(434\) 0 0
\(435\) 12.2461 0.587156
\(436\) 0 0
\(437\) −12.2005 −0.583631
\(438\) 0 0
\(439\) 14.1782 0.676691 0.338345 0.941022i \(-0.390133\pi\)
0.338345 + 0.941022i \(0.390133\pi\)
\(440\) 0 0
\(441\) −1.84966 −0.0880791
\(442\) 0 0
\(443\) −5.51322 −0.261941 −0.130971 0.991386i \(-0.541809\pi\)
−0.130971 + 0.991386i \(0.541809\pi\)
\(444\) 0 0
\(445\) −5.51239 −0.261312
\(446\) 0 0
\(447\) −17.6449 −0.834576
\(448\) 0 0
\(449\) −33.5392 −1.58281 −0.791406 0.611290i \(-0.790651\pi\)
−0.791406 + 0.611290i \(0.790651\pi\)
\(450\) 0 0
\(451\) 4.74580 0.223471
\(452\) 0 0
\(453\) −18.2039 −0.855295
\(454\) 0 0
\(455\) 5.31969 0.249391
\(456\) 0 0
\(457\) 19.1715 0.896806 0.448403 0.893831i \(-0.351993\pi\)
0.448403 + 0.893831i \(0.351993\pi\)
\(458\) 0 0
\(459\) −1.88405 −0.0879399
\(460\) 0 0
\(461\) 20.4315 0.951588 0.475794 0.879557i \(-0.342161\pi\)
0.475794 + 0.879557i \(0.342161\pi\)
\(462\) 0 0
\(463\) −3.77055 −0.175232 −0.0876161 0.996154i \(-0.527925\pi\)
−0.0876161 + 0.996154i \(0.527925\pi\)
\(464\) 0 0
\(465\) −3.79060 −0.175785
\(466\) 0 0
\(467\) −3.99456 −0.184846 −0.0924230 0.995720i \(-0.529461\pi\)
−0.0924230 + 0.995720i \(0.529461\pi\)
\(468\) 0 0
\(469\) 19.0093 0.877767
\(470\) 0 0
\(471\) 15.6419 0.720741
\(472\) 0 0
\(473\) −1.46627 −0.0674191
\(474\) 0 0
\(475\) 9.42878 0.432622
\(476\) 0 0
\(477\) 6.17753 0.282849
\(478\) 0 0
\(479\) 11.3905 0.520445 0.260222 0.965549i \(-0.416204\pi\)
0.260222 + 0.965549i \(0.416204\pi\)
\(480\) 0 0
\(481\) 7.23436 0.329859
\(482\) 0 0
\(483\) −4.88719 −0.222375
\(484\) 0 0
\(485\) −10.2080 −0.463521
\(486\) 0 0
\(487\) 30.4636 1.38044 0.690218 0.723601i \(-0.257515\pi\)
0.690218 + 0.723601i \(0.257515\pi\)
\(488\) 0 0
\(489\) −8.11923 −0.367164
\(490\) 0 0
\(491\) 43.2815 1.95327 0.976633 0.214915i \(-0.0689476\pi\)
0.976633 + 0.214915i \(0.0689476\pi\)
\(492\) 0 0
\(493\) −12.6326 −0.568944
\(494\) 0 0
\(495\) 1.05223 0.0472942
\(496\) 0 0
\(497\) −5.92720 −0.265871
\(498\) 0 0
\(499\) 5.84147 0.261500 0.130750 0.991415i \(-0.458261\pi\)
0.130750 + 0.991415i \(0.458261\pi\)
\(500\) 0 0
\(501\) 1.00000 0.0446767
\(502\) 0 0
\(503\) 22.5856 1.00704 0.503520 0.863983i \(-0.332038\pi\)
0.503520 + 0.863983i \(0.332038\pi\)
\(504\) 0 0
\(505\) −20.0764 −0.893387
\(506\) 0 0
\(507\) −11.3528 −0.504196
\(508\) 0 0
\(509\) −5.82494 −0.258186 −0.129093 0.991633i \(-0.541207\pi\)
−0.129093 + 0.991633i \(0.541207\pi\)
\(510\) 0 0
\(511\) 13.2212 0.584872
\(512\) 0 0
\(513\) −5.66549 −0.250138
\(514\) 0 0
\(515\) −22.6142 −0.996501
\(516\) 0 0
\(517\) −5.40015 −0.237498
\(518\) 0 0
\(519\) −20.4282 −0.896697
\(520\) 0 0
\(521\) 2.08275 0.0912470 0.0456235 0.998959i \(-0.485473\pi\)
0.0456235 + 0.998959i \(0.485473\pi\)
\(522\) 0 0
\(523\) 1.94456 0.0850296 0.0425148 0.999096i \(-0.486463\pi\)
0.0425148 + 0.999096i \(0.486463\pi\)
\(524\) 0 0
\(525\) 3.77690 0.164838
\(526\) 0 0
\(527\) 3.91024 0.170333
\(528\) 0 0
\(529\) −18.3625 −0.798370
\(530\) 0 0
\(531\) 3.25370 0.141199
\(532\) 0 0
\(533\) −10.5722 −0.457935
\(534\) 0 0
\(535\) 20.6502 0.892785
\(536\) 0 0
\(537\) −5.48020 −0.236488
\(538\) 0 0
\(539\) −1.06563 −0.0458999
\(540\) 0 0
\(541\) −26.7186 −1.14872 −0.574362 0.818601i \(-0.694750\pi\)
−0.574362 + 0.818601i \(0.694750\pi\)
\(542\) 0 0
\(543\) −22.3021 −0.957074
\(544\) 0 0
\(545\) −7.89034 −0.337985
\(546\) 0 0
\(547\) 29.9878 1.28218 0.641092 0.767464i \(-0.278482\pi\)
0.641092 + 0.767464i \(0.278482\pi\)
\(548\) 0 0
\(549\) 1.95266 0.0833376
\(550\) 0 0
\(551\) −37.9873 −1.61831
\(552\) 0 0
\(553\) −2.50907 −0.106697
\(554\) 0 0
\(555\) −10.2950 −0.436998
\(556\) 0 0
\(557\) −0.764104 −0.0323761 −0.0161881 0.999869i \(-0.505153\pi\)
−0.0161881 + 0.999869i \(0.505153\pi\)
\(558\) 0 0
\(559\) 3.26641 0.138155
\(560\) 0 0
\(561\) −1.08544 −0.0458273
\(562\) 0 0
\(563\) 28.6526 1.20756 0.603781 0.797150i \(-0.293660\pi\)
0.603781 + 0.797150i \(0.293660\pi\)
\(564\) 0 0
\(565\) 20.8793 0.878399
\(566\) 0 0
\(567\) −2.26944 −0.0953073
\(568\) 0 0
\(569\) 39.4889 1.65546 0.827729 0.561127i \(-0.189632\pi\)
0.827729 + 0.561127i \(0.189632\pi\)
\(570\) 0 0
\(571\) 39.5948 1.65699 0.828496 0.559995i \(-0.189197\pi\)
0.828496 + 0.559995i \(0.189197\pi\)
\(572\) 0 0
\(573\) −2.13733 −0.0892884
\(574\) 0 0
\(575\) −3.58393 −0.149460
\(576\) 0 0
\(577\) −16.3008 −0.678610 −0.339305 0.940676i \(-0.610192\pi\)
−0.339305 + 0.940676i \(0.610192\pi\)
\(578\) 0 0
\(579\) 17.7899 0.739322
\(580\) 0 0
\(581\) 32.8624 1.36336
\(582\) 0 0
\(583\) 3.55900 0.147399
\(584\) 0 0
\(585\) −2.34406 −0.0969149
\(586\) 0 0
\(587\) −13.1281 −0.541855 −0.270927 0.962600i \(-0.587330\pi\)
−0.270927 + 0.962600i \(0.587330\pi\)
\(588\) 0 0
\(589\) 11.7584 0.484497
\(590\) 0 0
\(591\) 23.9537 0.985325
\(592\) 0 0
\(593\) 4.17646 0.171507 0.0857533 0.996316i \(-0.472670\pi\)
0.0857533 + 0.996316i \(0.472670\pi\)
\(594\) 0 0
\(595\) 7.80921 0.320146
\(596\) 0 0
\(597\) −9.64250 −0.394641
\(598\) 0 0
\(599\) 39.9237 1.63124 0.815618 0.578590i \(-0.196397\pi\)
0.815618 + 0.578590i \(0.196397\pi\)
\(600\) 0 0
\(601\) −0.528562 −0.0215605 −0.0107803 0.999942i \(-0.503432\pi\)
−0.0107803 + 0.999942i \(0.503432\pi\)
\(602\) 0 0
\(603\) −8.37621 −0.341106
\(604\) 0 0
\(605\) −19.4842 −0.792147
\(606\) 0 0
\(607\) 11.2201 0.455411 0.227706 0.973730i \(-0.426878\pi\)
0.227706 + 0.973730i \(0.426878\pi\)
\(608\) 0 0
\(609\) −15.2167 −0.616610
\(610\) 0 0
\(611\) 12.0299 0.486679
\(612\) 0 0
\(613\) 24.9044 1.00588 0.502940 0.864321i \(-0.332252\pi\)
0.502940 + 0.864321i \(0.332252\pi\)
\(614\) 0 0
\(615\) 15.0450 0.606674
\(616\) 0 0
\(617\) −6.39995 −0.257652 −0.128826 0.991667i \(-0.541121\pi\)
−0.128826 + 0.991667i \(0.541121\pi\)
\(618\) 0 0
\(619\) 10.9438 0.439869 0.219935 0.975515i \(-0.429416\pi\)
0.219935 + 0.975515i \(0.429416\pi\)
\(620\) 0 0
\(621\) 2.15348 0.0864163
\(622\) 0 0
\(623\) 6.84953 0.274421
\(624\) 0 0
\(625\) −13.9090 −0.556362
\(626\) 0 0
\(627\) −3.26401 −0.130352
\(628\) 0 0
\(629\) 10.6199 0.423444
\(630\) 0 0
\(631\) −15.6786 −0.624155 −0.312078 0.950057i \(-0.601025\pi\)
−0.312078 + 0.950057i \(0.601025\pi\)
\(632\) 0 0
\(633\) −0.232543 −0.00924275
\(634\) 0 0
\(635\) −25.4614 −1.01040
\(636\) 0 0
\(637\) 2.37391 0.0940576
\(638\) 0 0
\(639\) 2.61175 0.103319
\(640\) 0 0
\(641\) −0.388862 −0.0153591 −0.00767957 0.999971i \(-0.502445\pi\)
−0.00767957 + 0.999971i \(0.502445\pi\)
\(642\) 0 0
\(643\) −22.2838 −0.878787 −0.439393 0.898295i \(-0.644807\pi\)
−0.439393 + 0.898295i \(0.644807\pi\)
\(644\) 0 0
\(645\) −4.64833 −0.183028
\(646\) 0 0
\(647\) −43.4363 −1.70766 −0.853829 0.520554i \(-0.825726\pi\)
−0.853829 + 0.520554i \(0.825726\pi\)
\(648\) 0 0
\(649\) 1.87452 0.0735815
\(650\) 0 0
\(651\) 4.71009 0.184603
\(652\) 0 0
\(653\) 17.1896 0.672680 0.336340 0.941741i \(-0.390811\pi\)
0.336340 + 0.941741i \(0.390811\pi\)
\(654\) 0 0
\(655\) −18.5275 −0.723930
\(656\) 0 0
\(657\) −5.82577 −0.227285
\(658\) 0 0
\(659\) 4.43602 0.172803 0.0864015 0.996260i \(-0.472463\pi\)
0.0864015 + 0.996260i \(0.472463\pi\)
\(660\) 0 0
\(661\) 46.9317 1.82543 0.912715 0.408597i \(-0.133982\pi\)
0.912715 + 0.408597i \(0.133982\pi\)
\(662\) 0 0
\(663\) 2.41804 0.0939089
\(664\) 0 0
\(665\) 23.4829 0.910629
\(666\) 0 0
\(667\) 14.4392 0.559088
\(668\) 0 0
\(669\) 21.2086 0.819973
\(670\) 0 0
\(671\) 1.12497 0.0434290
\(672\) 0 0
\(673\) −41.9815 −1.61827 −0.809134 0.587624i \(-0.800063\pi\)
−0.809134 + 0.587624i \(0.800063\pi\)
\(674\) 0 0
\(675\) −1.66425 −0.0640569
\(676\) 0 0
\(677\) −15.9875 −0.614449 −0.307224 0.951637i \(-0.599400\pi\)
−0.307224 + 0.951637i \(0.599400\pi\)
\(678\) 0 0
\(679\) 12.6842 0.486773
\(680\) 0 0
\(681\) 9.72014 0.372476
\(682\) 0 0
\(683\) 1.80502 0.0690673 0.0345337 0.999404i \(-0.489005\pi\)
0.0345337 + 0.999404i \(0.489005\pi\)
\(684\) 0 0
\(685\) 42.1536 1.61061
\(686\) 0 0
\(687\) 0.371580 0.0141767
\(688\) 0 0
\(689\) −7.92841 −0.302048
\(690\) 0 0
\(691\) −16.6566 −0.633645 −0.316823 0.948485i \(-0.602616\pi\)
−0.316823 + 0.948485i \(0.602616\pi\)
\(692\) 0 0
\(693\) −1.30747 −0.0496667
\(694\) 0 0
\(695\) 21.5472 0.817334
\(696\) 0 0
\(697\) −15.5199 −0.587857
\(698\) 0 0
\(699\) 14.0679 0.532097
\(700\) 0 0
\(701\) −42.6642 −1.61140 −0.805702 0.592321i \(-0.798212\pi\)
−0.805702 + 0.592321i \(0.798212\pi\)
\(702\) 0 0
\(703\) 31.9350 1.20445
\(704\) 0 0
\(705\) −17.1194 −0.644754
\(706\) 0 0
\(707\) 24.9463 0.938203
\(708\) 0 0
\(709\) 4.94574 0.185741 0.0928706 0.995678i \(-0.470396\pi\)
0.0928706 + 0.995678i \(0.470396\pi\)
\(710\) 0 0
\(711\) 1.10559 0.0414630
\(712\) 0 0
\(713\) −4.46944 −0.167382
\(714\) 0 0
\(715\) −1.35046 −0.0505044
\(716\) 0 0
\(717\) −4.93902 −0.184451
\(718\) 0 0
\(719\) −14.4179 −0.537696 −0.268848 0.963183i \(-0.586643\pi\)
−0.268848 + 0.963183i \(0.586643\pi\)
\(720\) 0 0
\(721\) 28.0998 1.04649
\(722\) 0 0
\(723\) −11.4721 −0.426653
\(724\) 0 0
\(725\) −11.1588 −0.414429
\(726\) 0 0
\(727\) −7.72774 −0.286606 −0.143303 0.989679i \(-0.545772\pi\)
−0.143303 + 0.989679i \(0.545772\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 4.79504 0.177351
\(732\) 0 0
\(733\) −2.86167 −0.105698 −0.0528491 0.998603i \(-0.516830\pi\)
−0.0528491 + 0.998603i \(0.516830\pi\)
\(734\) 0 0
\(735\) −3.37823 −0.124608
\(736\) 0 0
\(737\) −4.82571 −0.177757
\(738\) 0 0
\(739\) 44.8796 1.65092 0.825461 0.564459i \(-0.190915\pi\)
0.825461 + 0.564459i \(0.190915\pi\)
\(740\) 0 0
\(741\) 7.27125 0.267116
\(742\) 0 0
\(743\) 32.0693 1.17651 0.588255 0.808675i \(-0.299815\pi\)
0.588255 + 0.808675i \(0.299815\pi\)
\(744\) 0 0
\(745\) −32.2268 −1.18070
\(746\) 0 0
\(747\) −14.4804 −0.529812
\(748\) 0 0
\(749\) −25.6593 −0.937570
\(750\) 0 0
\(751\) 30.4610 1.11154 0.555769 0.831337i \(-0.312424\pi\)
0.555769 + 0.831337i \(0.312424\pi\)
\(752\) 0 0
\(753\) −20.6662 −0.753117
\(754\) 0 0
\(755\) −33.2477 −1.21001
\(756\) 0 0
\(757\) −9.07724 −0.329918 −0.164959 0.986300i \(-0.552749\pi\)
−0.164959 + 0.986300i \(0.552749\pi\)
\(758\) 0 0
\(759\) 1.24067 0.0450334
\(760\) 0 0
\(761\) −45.4211 −1.64651 −0.823256 0.567670i \(-0.807845\pi\)
−0.823256 + 0.567670i \(0.807845\pi\)
\(762\) 0 0
\(763\) 9.80431 0.354940
\(764\) 0 0
\(765\) −3.44103 −0.124411
\(766\) 0 0
\(767\) −4.17589 −0.150783
\(768\) 0 0
\(769\) −37.9043 −1.36686 −0.683432 0.730014i \(-0.739514\pi\)
−0.683432 + 0.730014i \(0.739514\pi\)
\(770\) 0 0
\(771\) 24.4158 0.879315
\(772\) 0 0
\(773\) −10.3435 −0.372030 −0.186015 0.982547i \(-0.559557\pi\)
−0.186015 + 0.982547i \(0.559557\pi\)
\(774\) 0 0
\(775\) 3.45406 0.124073
\(776\) 0 0
\(777\) 12.7922 0.458919
\(778\) 0 0
\(779\) −46.6695 −1.67211
\(780\) 0 0
\(781\) 1.50468 0.0538418
\(782\) 0 0
\(783\) 6.70504 0.239619
\(784\) 0 0
\(785\) 28.5684 1.01965
\(786\) 0 0
\(787\) 30.9881 1.10461 0.552304 0.833643i \(-0.313749\pi\)
0.552304 + 0.833643i \(0.313749\pi\)
\(788\) 0 0
\(789\) −7.72951 −0.275178
\(790\) 0 0
\(791\) −25.9440 −0.922462
\(792\) 0 0
\(793\) −2.50610 −0.0889943
\(794\) 0 0
\(795\) 11.2827 0.400155
\(796\) 0 0
\(797\) −43.3862 −1.53682 −0.768409 0.639960i \(-0.778951\pi\)
−0.768409 + 0.639960i \(0.778951\pi\)
\(798\) 0 0
\(799\) 17.6597 0.624756
\(800\) 0 0
\(801\) −3.01816 −0.106642
\(802\) 0 0
\(803\) −3.35635 −0.118443
\(804\) 0 0
\(805\) −8.92599 −0.314600
\(806\) 0 0
\(807\) 11.1431 0.392257
\(808\) 0 0
\(809\) 31.5421 1.10896 0.554480 0.832197i \(-0.312917\pi\)
0.554480 + 0.832197i \(0.312917\pi\)
\(810\) 0 0
\(811\) 0.0655369 0.00230131 0.00115066 0.999999i \(-0.499634\pi\)
0.00115066 + 0.999999i \(0.499634\pi\)
\(812\) 0 0
\(813\) −10.4906 −0.367922
\(814\) 0 0
\(815\) −14.8290 −0.519437
\(816\) 0 0
\(817\) 14.4191 0.504459
\(818\) 0 0
\(819\) 2.91266 0.101776
\(820\) 0 0
\(821\) −6.79246 −0.237058 −0.118529 0.992951i \(-0.537818\pi\)
−0.118529 + 0.992951i \(0.537818\pi\)
\(822\) 0 0
\(823\) 4.87356 0.169882 0.0849408 0.996386i \(-0.472930\pi\)
0.0849408 + 0.996386i \(0.472930\pi\)
\(824\) 0 0
\(825\) −0.958808 −0.0333814
\(826\) 0 0
\(827\) −31.0388 −1.07933 −0.539663 0.841881i \(-0.681448\pi\)
−0.539663 + 0.841881i \(0.681448\pi\)
\(828\) 0 0
\(829\) 23.2306 0.806834 0.403417 0.915016i \(-0.367822\pi\)
0.403417 + 0.915016i \(0.367822\pi\)
\(830\) 0 0
\(831\) 7.71364 0.267583
\(832\) 0 0
\(833\) 3.48485 0.120743
\(834\) 0 0
\(835\) 1.82640 0.0632053
\(836\) 0 0
\(837\) −2.07544 −0.0717379
\(838\) 0 0
\(839\) 2.35242 0.0812146 0.0406073 0.999175i \(-0.487071\pi\)
0.0406073 + 0.999175i \(0.487071\pi\)
\(840\) 0 0
\(841\) 15.9575 0.550260
\(842\) 0 0
\(843\) 26.5013 0.912753
\(844\) 0 0
\(845\) −20.7348 −0.713300
\(846\) 0 0
\(847\) 24.2105 0.831884
\(848\) 0 0
\(849\) −9.42532 −0.323476
\(850\) 0 0
\(851\) −12.1387 −0.416108
\(852\) 0 0
\(853\) −51.6691 −1.76912 −0.884558 0.466430i \(-0.845540\pi\)
−0.884558 + 0.466430i \(0.845540\pi\)
\(854\) 0 0
\(855\) −10.3475 −0.353876
\(856\) 0 0
\(857\) −6.17501 −0.210934 −0.105467 0.994423i \(-0.533634\pi\)
−0.105467 + 0.994423i \(0.533634\pi\)
\(858\) 0 0
\(859\) 28.9287 0.987034 0.493517 0.869736i \(-0.335711\pi\)
0.493517 + 0.869736i \(0.335711\pi\)
\(860\) 0 0
\(861\) −18.6945 −0.637107
\(862\) 0 0
\(863\) −9.28425 −0.316039 −0.158020 0.987436i \(-0.550511\pi\)
−0.158020 + 0.987436i \(0.550511\pi\)
\(864\) 0 0
\(865\) −37.3101 −1.26858
\(866\) 0 0
\(867\) −13.4504 −0.456798
\(868\) 0 0
\(869\) 0.636955 0.0216072
\(870\) 0 0
\(871\) 10.7503 0.364259
\(872\) 0 0
\(873\) −5.58912 −0.189163
\(874\) 0 0
\(875\) 27.6227 0.933817
\(876\) 0 0
\(877\) 14.8580 0.501718 0.250859 0.968024i \(-0.419287\pi\)
0.250859 + 0.968024i \(0.419287\pi\)
\(878\) 0 0
\(879\) −8.72067 −0.294141
\(880\) 0 0
\(881\) −23.8922 −0.804947 −0.402474 0.915432i \(-0.631850\pi\)
−0.402474 + 0.915432i \(0.631850\pi\)
\(882\) 0 0
\(883\) 56.7643 1.91027 0.955135 0.296170i \(-0.0957094\pi\)
0.955135 + 0.296170i \(0.0957094\pi\)
\(884\) 0 0
\(885\) 5.94257 0.199757
\(886\) 0 0
\(887\) −42.4315 −1.42471 −0.712355 0.701820i \(-0.752371\pi\)
−0.712355 + 0.701820i \(0.752371\pi\)
\(888\) 0 0
\(889\) 31.6375 1.06109
\(890\) 0 0
\(891\) 0.576121 0.0193008
\(892\) 0 0
\(893\) 53.1043 1.77707
\(894\) 0 0
\(895\) −10.0091 −0.334566
\(896\) 0 0
\(897\) −2.76384 −0.0922820
\(898\) 0 0
\(899\) −13.9159 −0.464122
\(900\) 0 0
\(901\) −11.6388 −0.387743
\(902\) 0 0
\(903\) 5.77587 0.192209
\(904\) 0 0
\(905\) −40.7326 −1.35400
\(906\) 0 0
\(907\) −42.7520 −1.41956 −0.709779 0.704424i \(-0.751205\pi\)
−0.709779 + 0.704424i \(0.751205\pi\)
\(908\) 0 0
\(909\) −10.9923 −0.364592
\(910\) 0 0
\(911\) 4.12245 0.136583 0.0682914 0.997665i \(-0.478245\pi\)
0.0682914 + 0.997665i \(0.478245\pi\)
\(912\) 0 0
\(913\) −8.34248 −0.276096
\(914\) 0 0
\(915\) 3.56635 0.117900
\(916\) 0 0
\(917\) 23.0217 0.760245
\(918\) 0 0
\(919\) 15.5909 0.514295 0.257148 0.966372i \(-0.417217\pi\)
0.257148 + 0.966372i \(0.417217\pi\)
\(920\) 0 0
\(921\) 8.58502 0.282886
\(922\) 0 0
\(923\) −3.35199 −0.110332
\(924\) 0 0
\(925\) 9.38095 0.308444
\(926\) 0 0
\(927\) −12.3818 −0.406672
\(928\) 0 0
\(929\) −35.6152 −1.16850 −0.584248 0.811575i \(-0.698611\pi\)
−0.584248 + 0.811575i \(0.698611\pi\)
\(930\) 0 0
\(931\) 10.4792 0.343443
\(932\) 0 0
\(933\) −14.4072 −0.471669
\(934\) 0 0
\(935\) −1.98245 −0.0648331
\(936\) 0 0
\(937\) −5.39850 −0.176361 −0.0881806 0.996104i \(-0.528105\pi\)
−0.0881806 + 0.996104i \(0.528105\pi\)
\(938\) 0 0
\(939\) −25.1446 −0.820563
\(940\) 0 0
\(941\) −19.2566 −0.627747 −0.313873 0.949465i \(-0.601627\pi\)
−0.313873 + 0.949465i \(0.601627\pi\)
\(942\) 0 0
\(943\) 17.7393 0.577672
\(944\) 0 0
\(945\) −4.14491 −0.134834
\(946\) 0 0
\(947\) 12.0455 0.391425 0.195712 0.980661i \(-0.437298\pi\)
0.195712 + 0.980661i \(0.437298\pi\)
\(948\) 0 0
\(949\) 7.47696 0.242712
\(950\) 0 0
\(951\) −5.08844 −0.165004
\(952\) 0 0
\(953\) −24.9635 −0.808647 −0.404324 0.914616i \(-0.632493\pi\)
−0.404324 + 0.914616i \(0.632493\pi\)
\(954\) 0 0
\(955\) −3.90364 −0.126319
\(956\) 0 0
\(957\) 3.86291 0.124870
\(958\) 0 0
\(959\) −52.3788 −1.69140
\(960\) 0 0
\(961\) −26.6925 −0.861049
\(962\) 0 0
\(963\) 11.3065 0.364346
\(964\) 0 0
\(965\) 32.4915 1.04594
\(966\) 0 0
\(967\) −18.5543 −0.596667 −0.298333 0.954462i \(-0.596431\pi\)
−0.298333 + 0.954462i \(0.596431\pi\)
\(968\) 0 0
\(969\) 10.6741 0.342900
\(970\) 0 0
\(971\) 12.8450 0.412214 0.206107 0.978529i \(-0.433920\pi\)
0.206107 + 0.978529i \(0.433920\pi\)
\(972\) 0 0
\(973\) −26.7740 −0.858334
\(974\) 0 0
\(975\) 2.13594 0.0684049
\(976\) 0 0
\(977\) 46.2296 1.47902 0.739508 0.673148i \(-0.235058\pi\)
0.739508 + 0.673148i \(0.235058\pi\)
\(978\) 0 0
\(979\) −1.73883 −0.0555732
\(980\) 0 0
\(981\) −4.32015 −0.137932
\(982\) 0 0
\(983\) 8.20446 0.261682 0.130841 0.991403i \(-0.458232\pi\)
0.130841 + 0.991403i \(0.458232\pi\)
\(984\) 0 0
\(985\) 43.7492 1.39397
\(986\) 0 0
\(987\) 21.2721 0.677097
\(988\) 0 0
\(989\) −5.48077 −0.174278
\(990\) 0 0
\(991\) −19.6139 −0.623056 −0.311528 0.950237i \(-0.600841\pi\)
−0.311528 + 0.950237i \(0.600841\pi\)
\(992\) 0 0
\(993\) −21.6459 −0.686911
\(994\) 0 0
\(995\) −17.6111 −0.558309
\(996\) 0 0
\(997\) 3.59543 0.113868 0.0569342 0.998378i \(-0.481867\pi\)
0.0569342 + 0.998378i \(0.481867\pi\)
\(998\) 0 0
\(999\) −5.63675 −0.178339
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 8016.2.a.w.1.6 7
4.3 odd 2 4008.2.a.g.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.g.1.6 7 4.3 odd 2
8016.2.a.w.1.6 7 1.1 even 1 trivial