Properties

Label 588.4.a.k.1.1
Level $588$
Weight $4$
Character 588.1
Self dual yes
Analytic conductor $34.693$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 588.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(34.6931230834\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.136768.1
Defining polynomial: \(x^{4} - 2 x^{3} - 23 x^{2} + 18 x + 119\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-3.31012\) of defining polynomial
Character \(\chi\) \(=\) 588.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} -16.6547 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -16.6547 q^{5} +9.00000 q^{9} -71.7600 q^{11} -65.3878 q^{13} -49.9641 q^{15} +90.4316 q^{17} +163.818 q^{19} +79.2288 q^{23} +152.379 q^{25} +27.0000 q^{27} -43.2995 q^{29} +135.636 q^{31} -215.280 q^{33} +270.740 q^{37} -196.163 q^{39} +152.241 q^{41} -177.641 q^{43} -149.892 q^{45} -45.6331 q^{47} +271.295 q^{51} -158.438 q^{53} +1195.14 q^{55} +491.455 q^{57} -391.769 q^{59} +551.295 q^{61} +1089.01 q^{65} +458.630 q^{67} +237.687 q^{69} -486.786 q^{71} +574.892 q^{73} +457.136 q^{75} -668.319 q^{79} +81.0000 q^{81} +76.2450 q^{83} -1506.11 q^{85} -129.899 q^{87} +1366.80 q^{89} +406.907 q^{93} -2728.34 q^{95} +242.655 q^{97} -645.840 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 12q^{3} + 36q^{9} + O(q^{10}) \) \( 4q + 12q^{3} + 36q^{9} + 48q^{17} + 192q^{19} + 192q^{23} + 324q^{25} + 108q^{27} + 96q^{29} + 48q^{31} + 256q^{37} + 1008q^{41} - 112q^{43} + 864q^{47} + 144q^{51} - 648q^{53} + 2352q^{55} + 576q^{57} + 336q^{59} + 960q^{61} - 360q^{65} + 720q^{67} + 576q^{69} - 1344q^{71} + 672q^{73} + 972q^{75} - 1984q^{79} + 324q^{81} + 3120q^{83} + 680q^{85} + 288q^{87} + 2160q^{89} + 144q^{93} - 3744q^{95} + 2016q^{97} + O(q^{100}) \)

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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −16.6547 −1.48964 −0.744820 0.667265i \(-0.767465\pi\)
−0.744820 + 0.667265i \(0.767465\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −71.7600 −1.96695 −0.983475 0.181044i \(-0.942052\pi\)
−0.983475 + 0.181044i \(0.942052\pi\)
\(12\) 0 0
\(13\) −65.3878 −1.39502 −0.697512 0.716573i \(-0.745709\pi\)
−0.697512 + 0.716573i \(0.745709\pi\)
\(14\) 0 0
\(15\) −49.9641 −0.860044
\(16\) 0 0
\(17\) 90.4316 1.29017 0.645085 0.764111i \(-0.276822\pi\)
0.645085 + 0.764111i \(0.276822\pi\)
\(18\) 0 0
\(19\) 163.818 1.97803 0.989014 0.147824i \(-0.0472268\pi\)
0.989014 + 0.147824i \(0.0472268\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 79.2288 0.718276 0.359138 0.933284i \(-0.383071\pi\)
0.359138 + 0.933284i \(0.383071\pi\)
\(24\) 0 0
\(25\) 152.379 1.21903
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −43.2995 −0.277259 −0.138630 0.990344i \(-0.544270\pi\)
−0.138630 + 0.990344i \(0.544270\pi\)
\(30\) 0 0
\(31\) 135.636 0.785836 0.392918 0.919574i \(-0.371466\pi\)
0.392918 + 0.919574i \(0.371466\pi\)
\(32\) 0 0
\(33\) −215.280 −1.13562
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 270.740 1.20296 0.601479 0.798889i \(-0.294578\pi\)
0.601479 + 0.798889i \(0.294578\pi\)
\(38\) 0 0
\(39\) −196.163 −0.805417
\(40\) 0 0
\(41\) 152.241 0.579902 0.289951 0.957041i \(-0.406361\pi\)
0.289951 + 0.957041i \(0.406361\pi\)
\(42\) 0 0
\(43\) −177.641 −0.630001 −0.315001 0.949091i \(-0.602005\pi\)
−0.315001 + 0.949091i \(0.602005\pi\)
\(44\) 0 0
\(45\) −149.892 −0.496547
\(46\) 0 0
\(47\) −45.6331 −0.141623 −0.0708114 0.997490i \(-0.522559\pi\)
−0.0708114 + 0.997490i \(0.522559\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 271.295 0.744880
\(52\) 0 0
\(53\) −158.438 −0.410624 −0.205312 0.978697i \(-0.565821\pi\)
−0.205312 + 0.978697i \(0.565821\pi\)
\(54\) 0 0
\(55\) 1195.14 2.93005
\(56\) 0 0
\(57\) 491.455 1.14201
\(58\) 0 0
\(59\) −391.769 −0.864474 −0.432237 0.901760i \(-0.642276\pi\)
−0.432237 + 0.901760i \(0.642276\pi\)
\(60\) 0 0
\(61\) 551.295 1.15715 0.578574 0.815630i \(-0.303609\pi\)
0.578574 + 0.815630i \(0.303609\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1089.01 2.07808
\(66\) 0 0
\(67\) 458.630 0.836277 0.418139 0.908383i \(-0.362683\pi\)
0.418139 + 0.908383i \(0.362683\pi\)
\(68\) 0 0
\(69\) 237.687 0.414697
\(70\) 0 0
\(71\) −486.786 −0.813674 −0.406837 0.913501i \(-0.633368\pi\)
−0.406837 + 0.913501i \(0.633368\pi\)
\(72\) 0 0
\(73\) 574.892 0.921726 0.460863 0.887471i \(-0.347540\pi\)
0.460863 + 0.887471i \(0.347540\pi\)
\(74\) 0 0
\(75\) 457.136 0.703806
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −668.319 −0.951795 −0.475897 0.879501i \(-0.657877\pi\)
−0.475897 + 0.879501i \(0.657877\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 76.2450 0.100831 0.0504155 0.998728i \(-0.483945\pi\)
0.0504155 + 0.998728i \(0.483945\pi\)
\(84\) 0 0
\(85\) −1506.11 −1.92189
\(86\) 0 0
\(87\) −129.899 −0.160076
\(88\) 0 0
\(89\) 1366.80 1.62787 0.813937 0.580953i \(-0.197320\pi\)
0.813937 + 0.580953i \(0.197320\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 406.907 0.453702
\(94\) 0 0
\(95\) −2728.34 −2.94655
\(96\) 0 0
\(97\) 242.655 0.253999 0.127000 0.991903i \(-0.459465\pi\)
0.127000 + 0.991903i \(0.459465\pi\)
\(98\) 0 0
\(99\) −645.840 −0.655650
\(100\) 0 0
\(101\) 694.743 0.684450 0.342225 0.939618i \(-0.388819\pi\)
0.342225 + 0.939618i \(0.388819\pi\)
\(102\) 0 0
\(103\) −174.218 −0.166662 −0.0833310 0.996522i \(-0.526556\pi\)
−0.0833310 + 0.996522i \(0.526556\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 47.6051 0.0430108 0.0215054 0.999769i \(-0.493154\pi\)
0.0215054 + 0.999769i \(0.493154\pi\)
\(108\) 0 0
\(109\) −1633.10 −1.43507 −0.717534 0.696523i \(-0.754729\pi\)
−0.717534 + 0.696523i \(0.754729\pi\)
\(110\) 0 0
\(111\) 812.221 0.694528
\(112\) 0 0
\(113\) −1127.29 −0.938464 −0.469232 0.883075i \(-0.655469\pi\)
−0.469232 + 0.883075i \(0.655469\pi\)
\(114\) 0 0
\(115\) −1319.53 −1.06997
\(116\) 0 0
\(117\) −588.490 −0.465008
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 3818.50 2.86889
\(122\) 0 0
\(123\) 456.722 0.334807
\(124\) 0 0
\(125\) −455.982 −0.326274
\(126\) 0 0
\(127\) 2398.66 1.67596 0.837979 0.545702i \(-0.183737\pi\)
0.837979 + 0.545702i \(0.183737\pi\)
\(128\) 0 0
\(129\) −532.924 −0.363731
\(130\) 0 0
\(131\) 1548.96 1.03308 0.516541 0.856263i \(-0.327219\pi\)
0.516541 + 0.856263i \(0.327219\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −449.677 −0.286681
\(136\) 0 0
\(137\) 1523.27 0.949939 0.474969 0.880002i \(-0.342459\pi\)
0.474969 + 0.880002i \(0.342459\pi\)
\(138\) 0 0
\(139\) 551.445 0.336496 0.168248 0.985745i \(-0.446189\pi\)
0.168248 + 0.985745i \(0.446189\pi\)
\(140\) 0 0
\(141\) −136.899 −0.0817660
\(142\) 0 0
\(143\) 4692.23 2.74394
\(144\) 0 0
\(145\) 721.140 0.413017
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2400.24 −1.31970 −0.659851 0.751397i \(-0.729381\pi\)
−0.659851 + 0.751397i \(0.729381\pi\)
\(150\) 0 0
\(151\) 621.560 0.334979 0.167490 0.985874i \(-0.446434\pi\)
0.167490 + 0.985874i \(0.446434\pi\)
\(152\) 0 0
\(153\) 813.884 0.430056
\(154\) 0 0
\(155\) −2258.97 −1.17061
\(156\) 0 0
\(157\) −2691.58 −1.36822 −0.684112 0.729377i \(-0.739810\pi\)
−0.684112 + 0.729377i \(0.739810\pi\)
\(158\) 0 0
\(159\) −475.313 −0.237074
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1057.31 0.508065 0.254032 0.967196i \(-0.418243\pi\)
0.254032 + 0.967196i \(0.418243\pi\)
\(164\) 0 0
\(165\) 3585.42 1.69166
\(166\) 0 0
\(167\) −243.840 −0.112987 −0.0564937 0.998403i \(-0.517992\pi\)
−0.0564937 + 0.998403i \(0.517992\pi\)
\(168\) 0 0
\(169\) 2078.56 0.946090
\(170\) 0 0
\(171\) 1474.37 0.659342
\(172\) 0 0
\(173\) −1687.26 −0.741505 −0.370753 0.928732i \(-0.620900\pi\)
−0.370753 + 0.928732i \(0.620900\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1175.31 −0.499104
\(178\) 0 0
\(179\) −359.139 −0.149963 −0.0749814 0.997185i \(-0.523890\pi\)
−0.0749814 + 0.997185i \(0.523890\pi\)
\(180\) 0 0
\(181\) 2982.58 1.22483 0.612413 0.790538i \(-0.290199\pi\)
0.612413 + 0.790538i \(0.290199\pi\)
\(182\) 0 0
\(183\) 1653.88 0.668080
\(184\) 0 0
\(185\) −4509.10 −1.79198
\(186\) 0 0
\(187\) −6489.37 −2.53770
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3490.77 1.32243 0.661213 0.750198i \(-0.270042\pi\)
0.661213 + 0.750198i \(0.270042\pi\)
\(192\) 0 0
\(193\) 1604.23 0.598315 0.299158 0.954204i \(-0.403294\pi\)
0.299158 + 0.954204i \(0.403294\pi\)
\(194\) 0 0
\(195\) 3267.04 1.19978
\(196\) 0 0
\(197\) −455.935 −0.164893 −0.0824467 0.996595i \(-0.526273\pi\)
−0.0824467 + 0.996595i \(0.526273\pi\)
\(198\) 0 0
\(199\) 2084.16 0.742423 0.371212 0.928548i \(-0.378942\pi\)
0.371212 + 0.928548i \(0.378942\pi\)
\(200\) 0 0
\(201\) 1375.89 0.482825
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −2535.52 −0.863846
\(206\) 0 0
\(207\) 713.060 0.239425
\(208\) 0 0
\(209\) −11755.6 −3.89068
\(210\) 0 0
\(211\) 2568.06 0.837880 0.418940 0.908014i \(-0.362402\pi\)
0.418940 + 0.908014i \(0.362402\pi\)
\(212\) 0 0
\(213\) −1460.36 −0.469775
\(214\) 0 0
\(215\) 2958.56 0.938475
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 1724.68 0.532159
\(220\) 0 0
\(221\) −5913.12 −1.79982
\(222\) 0 0
\(223\) −4659.55 −1.39922 −0.699612 0.714523i \(-0.746644\pi\)
−0.699612 + 0.714523i \(0.746644\pi\)
\(224\) 0 0
\(225\) 1371.41 0.406343
\(226\) 0 0
\(227\) 4828.77 1.41188 0.705940 0.708271i \(-0.250525\pi\)
0.705940 + 0.708271i \(0.250525\pi\)
\(228\) 0 0
\(229\) −3222.59 −0.929932 −0.464966 0.885328i \(-0.653934\pi\)
−0.464966 + 0.885328i \(0.653934\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −5702.42 −1.60334 −0.801670 0.597767i \(-0.796055\pi\)
−0.801670 + 0.597767i \(0.796055\pi\)
\(234\) 0 0
\(235\) 760.005 0.210967
\(236\) 0 0
\(237\) −2004.96 −0.549519
\(238\) 0 0
\(239\) 1994.47 0.539796 0.269898 0.962889i \(-0.413010\pi\)
0.269898 + 0.962889i \(0.413010\pi\)
\(240\) 0 0
\(241\) −1420.89 −0.379781 −0.189891 0.981805i \(-0.560813\pi\)
−0.189891 + 0.981805i \(0.560813\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −10711.7 −2.75939
\(248\) 0 0
\(249\) 228.735 0.0582148
\(250\) 0 0
\(251\) −3855.44 −0.969534 −0.484767 0.874643i \(-0.661096\pi\)
−0.484767 + 0.874643i \(0.661096\pi\)
\(252\) 0 0
\(253\) −5685.46 −1.41281
\(254\) 0 0
\(255\) −4518.33 −1.10960
\(256\) 0 0
\(257\) 4053.28 0.983800 0.491900 0.870652i \(-0.336303\pi\)
0.491900 + 0.870652i \(0.336303\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −389.696 −0.0924198
\(262\) 0 0
\(263\) −500.787 −0.117414 −0.0587069 0.998275i \(-0.518698\pi\)
−0.0587069 + 0.998275i \(0.518698\pi\)
\(264\) 0 0
\(265\) 2638.73 0.611683
\(266\) 0 0
\(267\) 4100.41 0.939853
\(268\) 0 0
\(269\) 3281.74 0.743834 0.371917 0.928266i \(-0.378701\pi\)
0.371917 + 0.928266i \(0.378701\pi\)
\(270\) 0 0
\(271\) 3324.36 0.745168 0.372584 0.927999i \(-0.378472\pi\)
0.372584 + 0.927999i \(0.378472\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −10934.7 −2.39777
\(276\) 0 0
\(277\) 8619.75 1.86971 0.934856 0.355026i \(-0.115528\pi\)
0.934856 + 0.355026i \(0.115528\pi\)
\(278\) 0 0
\(279\) 1220.72 0.261945
\(280\) 0 0
\(281\) −7710.80 −1.63697 −0.818484 0.574529i \(-0.805185\pi\)
−0.818484 + 0.574529i \(0.805185\pi\)
\(282\) 0 0
\(283\) −139.271 −0.0292537 −0.0146268 0.999893i \(-0.504656\pi\)
−0.0146268 + 0.999893i \(0.504656\pi\)
\(284\) 0 0
\(285\) −8185.03 −1.70119
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 3264.87 0.664537
\(290\) 0 0
\(291\) 727.966 0.146646
\(292\) 0 0
\(293\) −4437.09 −0.884702 −0.442351 0.896842i \(-0.645855\pi\)
−0.442351 + 0.896842i \(0.645855\pi\)
\(294\) 0 0
\(295\) 6524.79 1.28776
\(296\) 0 0
\(297\) −1937.52 −0.378540
\(298\) 0 0
\(299\) −5180.60 −1.00201
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 2084.23 0.395168
\(304\) 0 0
\(305\) −9181.64 −1.72373
\(306\) 0 0
\(307\) −299.480 −0.0556749 −0.0278375 0.999612i \(-0.508862\pi\)
−0.0278375 + 0.999612i \(0.508862\pi\)
\(308\) 0 0
\(309\) −522.654 −0.0962224
\(310\) 0 0
\(311\) 913.944 0.166640 0.0833199 0.996523i \(-0.473448\pi\)
0.0833199 + 0.996523i \(0.473448\pi\)
\(312\) 0 0
\(313\) 3887.38 0.702005 0.351002 0.936375i \(-0.385841\pi\)
0.351002 + 0.936375i \(0.385841\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2879.30 0.510151 0.255075 0.966921i \(-0.417900\pi\)
0.255075 + 0.966921i \(0.417900\pi\)
\(318\) 0 0
\(319\) 3107.17 0.545355
\(320\) 0 0
\(321\) 142.815 0.0248323
\(322\) 0 0
\(323\) 14814.4 2.55199
\(324\) 0 0
\(325\) −9963.70 −1.70057
\(326\) 0 0
\(327\) −4899.29 −0.828537
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2810.79 −0.466752 −0.233376 0.972387i \(-0.574977\pi\)
−0.233376 + 0.972387i \(0.574977\pi\)
\(332\) 0 0
\(333\) 2436.66 0.400986
\(334\) 0 0
\(335\) −7638.34 −1.24575
\(336\) 0 0
\(337\) 2960.90 0.478607 0.239303 0.970945i \(-0.423081\pi\)
0.239303 + 0.970945i \(0.423081\pi\)
\(338\) 0 0
\(339\) −3381.87 −0.541823
\(340\) 0 0
\(341\) −9733.22 −1.54570
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −3958.59 −0.617749
\(346\) 0 0
\(347\) −4284.51 −0.662838 −0.331419 0.943484i \(-0.607527\pi\)
−0.331419 + 0.943484i \(0.607527\pi\)
\(348\) 0 0
\(349\) 3295.36 0.505434 0.252717 0.967540i \(-0.418676\pi\)
0.252717 + 0.967540i \(0.418676\pi\)
\(350\) 0 0
\(351\) −1765.47 −0.268472
\(352\) 0 0
\(353\) −10629.7 −1.60272 −0.801360 0.598183i \(-0.795890\pi\)
−0.801360 + 0.598183i \(0.795890\pi\)
\(354\) 0 0
\(355\) 8107.26 1.21208
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3185.01 0.468241 0.234121 0.972208i \(-0.424779\pi\)
0.234121 + 0.972208i \(0.424779\pi\)
\(360\) 0 0
\(361\) 19977.5 2.91259
\(362\) 0 0
\(363\) 11455.5 1.65636
\(364\) 0 0
\(365\) −9574.65 −1.37304
\(366\) 0 0
\(367\) 7603.03 1.08140 0.540702 0.841214i \(-0.318159\pi\)
0.540702 + 0.841214i \(0.318159\pi\)
\(368\) 0 0
\(369\) 1370.17 0.193301
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 1511.96 0.209882 0.104941 0.994478i \(-0.466535\pi\)
0.104941 + 0.994478i \(0.466535\pi\)
\(374\) 0 0
\(375\) −1367.94 −0.188374
\(376\) 0 0
\(377\) 2831.26 0.386783
\(378\) 0 0
\(379\) 8848.36 1.19923 0.599617 0.800287i \(-0.295320\pi\)
0.599617 + 0.800287i \(0.295320\pi\)
\(380\) 0 0
\(381\) 7195.98 0.967615
\(382\) 0 0
\(383\) 4134.10 0.551548 0.275774 0.961223i \(-0.411066\pi\)
0.275774 + 0.961223i \(0.411066\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1598.77 −0.210000
\(388\) 0 0
\(389\) 7630.61 0.994569 0.497284 0.867588i \(-0.334331\pi\)
0.497284 + 0.867588i \(0.334331\pi\)
\(390\) 0 0
\(391\) 7164.79 0.926698
\(392\) 0 0
\(393\) 4646.89 0.596450
\(394\) 0 0
\(395\) 11130.6 1.41783
\(396\) 0 0
\(397\) −11618.5 −1.46881 −0.734403 0.678714i \(-0.762538\pi\)
−0.734403 + 0.678714i \(0.762538\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3047.01 0.379453 0.189726 0.981837i \(-0.439240\pi\)
0.189726 + 0.981837i \(0.439240\pi\)
\(402\) 0 0
\(403\) −8868.92 −1.09626
\(404\) 0 0
\(405\) −1349.03 −0.165516
\(406\) 0 0
\(407\) −19428.3 −2.36616
\(408\) 0 0
\(409\) 11582.8 1.40033 0.700164 0.713982i \(-0.253110\pi\)
0.700164 + 0.713982i \(0.253110\pi\)
\(410\) 0 0
\(411\) 4569.80 0.548447
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −1269.84 −0.150202
\(416\) 0 0
\(417\) 1654.33 0.194276
\(418\) 0 0
\(419\) −1318.83 −0.153768 −0.0768841 0.997040i \(-0.524497\pi\)
−0.0768841 + 0.997040i \(0.524497\pi\)
\(420\) 0 0
\(421\) −9733.56 −1.12680 −0.563402 0.826183i \(-0.690508\pi\)
−0.563402 + 0.826183i \(0.690508\pi\)
\(422\) 0 0
\(423\) −410.698 −0.0472076
\(424\) 0 0
\(425\) 13779.8 1.57275
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 14076.7 1.58422
\(430\) 0 0
\(431\) −14428.9 −1.61257 −0.806284 0.591528i \(-0.798525\pi\)
−0.806284 + 0.591528i \(0.798525\pi\)
\(432\) 0 0
\(433\) −16620.3 −1.84463 −0.922313 0.386444i \(-0.873703\pi\)
−0.922313 + 0.386444i \(0.873703\pi\)
\(434\) 0 0
\(435\) 2163.42 0.238455
\(436\) 0 0
\(437\) 12979.1 1.42077
\(438\) 0 0
\(439\) −515.565 −0.0560515 −0.0280257 0.999607i \(-0.508922\pi\)
−0.0280257 + 0.999607i \(0.508922\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 8976.09 0.962679 0.481340 0.876534i \(-0.340150\pi\)
0.481340 + 0.876534i \(0.340150\pi\)
\(444\) 0 0
\(445\) −22763.7 −2.42495
\(446\) 0 0
\(447\) −7200.73 −0.761930
\(448\) 0 0
\(449\) 2106.83 0.221442 0.110721 0.993852i \(-0.464684\pi\)
0.110721 + 0.993852i \(0.464684\pi\)
\(450\) 0 0
\(451\) −10924.8 −1.14064
\(452\) 0 0
\(453\) 1864.68 0.193400
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 15726.2 1.60971 0.804857 0.593469i \(-0.202242\pi\)
0.804857 + 0.593469i \(0.202242\pi\)
\(458\) 0 0
\(459\) 2441.65 0.248293
\(460\) 0 0
\(461\) −7921.16 −0.800272 −0.400136 0.916456i \(-0.631037\pi\)
−0.400136 + 0.916456i \(0.631037\pi\)
\(462\) 0 0
\(463\) −5821.20 −0.584306 −0.292153 0.956372i \(-0.594372\pi\)
−0.292153 + 0.956372i \(0.594372\pi\)
\(464\) 0 0
\(465\) −6776.92 −0.675854
\(466\) 0 0
\(467\) 11787.9 1.16805 0.584024 0.811737i \(-0.301477\pi\)
0.584024 + 0.811737i \(0.301477\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −8074.73 −0.789944
\(472\) 0 0
\(473\) 12747.5 1.23918
\(474\) 0 0
\(475\) 24962.4 2.41127
\(476\) 0 0
\(477\) −1425.94 −0.136875
\(478\) 0 0
\(479\) −5314.43 −0.506936 −0.253468 0.967344i \(-0.581571\pi\)
−0.253468 + 0.967344i \(0.581571\pi\)
\(480\) 0 0
\(481\) −17703.1 −1.67816
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −4041.35 −0.378367
\(486\) 0 0
\(487\) −18662.7 −1.73652 −0.868261 0.496107i \(-0.834762\pi\)
−0.868261 + 0.496107i \(0.834762\pi\)
\(488\) 0 0
\(489\) 3171.92 0.293331
\(490\) 0 0
\(491\) 13519.1 1.24259 0.621293 0.783578i \(-0.286608\pi\)
0.621293 + 0.783578i \(0.286608\pi\)
\(492\) 0 0
\(493\) −3915.64 −0.357711
\(494\) 0 0
\(495\) 10756.3 0.976683
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 13832.3 1.24092 0.620462 0.784237i \(-0.286945\pi\)
0.620462 + 0.784237i \(0.286945\pi\)
\(500\) 0 0
\(501\) −731.520 −0.0652334
\(502\) 0 0
\(503\) 13469.0 1.19394 0.596970 0.802264i \(-0.296371\pi\)
0.596970 + 0.802264i \(0.296371\pi\)
\(504\) 0 0
\(505\) −11570.7 −1.01958
\(506\) 0 0
\(507\) 6235.68 0.546226
\(508\) 0 0
\(509\) 8775.25 0.764157 0.382079 0.924130i \(-0.375208\pi\)
0.382079 + 0.924130i \(0.375208\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 4423.10 0.380672
\(514\) 0 0
\(515\) 2901.54 0.248267
\(516\) 0 0
\(517\) 3274.63 0.278565
\(518\) 0 0
\(519\) −5061.79 −0.428108
\(520\) 0 0
\(521\) −1660.76 −0.139653 −0.0698267 0.997559i \(-0.522245\pi\)
−0.0698267 + 0.997559i \(0.522245\pi\)
\(522\) 0 0
\(523\) 6324.82 0.528805 0.264402 0.964412i \(-0.414825\pi\)
0.264402 + 0.964412i \(0.414825\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12265.8 1.01386
\(528\) 0 0
\(529\) −5889.79 −0.484079
\(530\) 0 0
\(531\) −3525.92 −0.288158
\(532\) 0 0
\(533\) −9954.68 −0.808977
\(534\) 0 0
\(535\) −792.848 −0.0640707
\(536\) 0 0
\(537\) −1077.42 −0.0865810
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 4467.88 0.355063 0.177532 0.984115i \(-0.443189\pi\)
0.177532 + 0.984115i \(0.443189\pi\)
\(542\) 0 0
\(543\) 8947.74 0.707153
\(544\) 0 0
\(545\) 27198.7 2.13774
\(546\) 0 0
\(547\) −10939.0 −0.855063 −0.427531 0.904000i \(-0.640617\pi\)
−0.427531 + 0.904000i \(0.640617\pi\)
\(548\) 0 0
\(549\) 4961.65 0.385716
\(550\) 0 0
\(551\) −7093.26 −0.548427
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −13527.3 −1.03460
\(556\) 0 0
\(557\) −14325.1 −1.08972 −0.544862 0.838526i \(-0.683418\pi\)
−0.544862 + 0.838526i \(0.683418\pi\)
\(558\) 0 0
\(559\) 11615.6 0.878867
\(560\) 0 0
\(561\) −19468.1 −1.46514
\(562\) 0 0
\(563\) 20402.8 1.52731 0.763656 0.645624i \(-0.223402\pi\)
0.763656 + 0.645624i \(0.223402\pi\)
\(564\) 0 0
\(565\) 18774.6 1.39797
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −7250.43 −0.534189 −0.267095 0.963670i \(-0.586064\pi\)
−0.267095 + 0.963670i \(0.586064\pi\)
\(570\) 0 0
\(571\) 24894.8 1.82454 0.912272 0.409586i \(-0.134327\pi\)
0.912272 + 0.409586i \(0.134327\pi\)
\(572\) 0 0
\(573\) 10472.3 0.763503
\(574\) 0 0
\(575\) 12072.8 0.875599
\(576\) 0 0
\(577\) 1363.45 0.0983726 0.0491863 0.998790i \(-0.484337\pi\)
0.0491863 + 0.998790i \(0.484337\pi\)
\(578\) 0 0
\(579\) 4812.68 0.345437
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 11369.5 0.807677
\(584\) 0 0
\(585\) 9801.12 0.692694
\(586\) 0 0
\(587\) 23854.8 1.67733 0.838666 0.544647i \(-0.183336\pi\)
0.838666 + 0.544647i \(0.183336\pi\)
\(588\) 0 0
\(589\) 22219.6 1.55440
\(590\) 0 0
\(591\) −1367.80 −0.0952013
\(592\) 0 0
\(593\) 3437.20 0.238025 0.119013 0.992893i \(-0.462027\pi\)
0.119013 + 0.992893i \(0.462027\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 6252.48 0.428638
\(598\) 0 0
\(599\) 16511.7 1.12630 0.563148 0.826356i \(-0.309590\pi\)
0.563148 + 0.826356i \(0.309590\pi\)
\(600\) 0 0
\(601\) −11148.3 −0.756656 −0.378328 0.925672i \(-0.623501\pi\)
−0.378328 + 0.925672i \(0.623501\pi\)
\(602\) 0 0
\(603\) 4127.67 0.278759
\(604\) 0 0
\(605\) −63595.8 −4.27362
\(606\) 0 0
\(607\) −391.583 −0.0261843 −0.0130921 0.999914i \(-0.504167\pi\)
−0.0130921 + 0.999914i \(0.504167\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2983.85 0.197567
\(612\) 0 0
\(613\) −11740.5 −0.773563 −0.386782 0.922171i \(-0.626413\pi\)
−0.386782 + 0.922171i \(0.626413\pi\)
\(614\) 0 0
\(615\) −7606.56 −0.498742
\(616\) 0 0
\(617\) −8818.95 −0.575426 −0.287713 0.957717i \(-0.592895\pi\)
−0.287713 + 0.957717i \(0.592895\pi\)
\(618\) 0 0
\(619\) −15924.7 −1.03404 −0.517018 0.855975i \(-0.672958\pi\)
−0.517018 + 0.855975i \(0.672958\pi\)
\(620\) 0 0
\(621\) 2139.18 0.138232
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −11453.1 −0.732998
\(626\) 0 0
\(627\) −35266.8 −2.24629
\(628\) 0 0
\(629\) 24483.5 1.55202
\(630\) 0 0
\(631\) 29206.4 1.84261 0.921307 0.388836i \(-0.127123\pi\)
0.921307 + 0.388836i \(0.127123\pi\)
\(632\) 0 0
\(633\) 7704.18 0.483750
\(634\) 0 0
\(635\) −39948.9 −2.49658
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −4381.07 −0.271225
\(640\) 0 0
\(641\) 5613.01 0.345867 0.172933 0.984934i \(-0.444675\pi\)
0.172933 + 0.984934i \(0.444675\pi\)
\(642\) 0 0
\(643\) −18995.0 −1.16499 −0.582496 0.812834i \(-0.697924\pi\)
−0.582496 + 0.812834i \(0.697924\pi\)
\(644\) 0 0
\(645\) 8875.68 0.541829
\(646\) 0 0
\(647\) 22665.0 1.37721 0.688604 0.725138i \(-0.258224\pi\)
0.688604 + 0.725138i \(0.258224\pi\)
\(648\) 0 0
\(649\) 28113.3 1.70038
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 26527.8 1.58976 0.794879 0.606768i \(-0.207534\pi\)
0.794879 + 0.606768i \(0.207534\pi\)
\(654\) 0 0
\(655\) −25797.5 −1.53892
\(656\) 0 0
\(657\) 5174.03 0.307242
\(658\) 0 0
\(659\) −17114.8 −1.01168 −0.505839 0.862628i \(-0.668817\pi\)
−0.505839 + 0.862628i \(0.668817\pi\)
\(660\) 0 0
\(661\) −1191.82 −0.0701305 −0.0350653 0.999385i \(-0.511164\pi\)
−0.0350653 + 0.999385i \(0.511164\pi\)
\(662\) 0 0
\(663\) −17739.4 −1.03912
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −3430.57 −0.199149
\(668\) 0 0
\(669\) −13978.7 −0.807842
\(670\) 0 0
\(671\) −39560.9 −2.27605
\(672\) 0 0
\(673\) −9415.20 −0.539271 −0.269636 0.962962i \(-0.586903\pi\)
−0.269636 + 0.962962i \(0.586903\pi\)
\(674\) 0 0
\(675\) 4114.22 0.234602
\(676\) 0 0
\(677\) 7917.13 0.449453 0.224727 0.974422i \(-0.427851\pi\)
0.224727 + 0.974422i \(0.427851\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 14486.3 0.815150
\(682\) 0 0
\(683\) 29234.4 1.63781 0.818904 0.573930i \(-0.194582\pi\)
0.818904 + 0.573930i \(0.194582\pi\)
\(684\) 0 0
\(685\) −25369.6 −1.41507
\(686\) 0 0
\(687\) −9667.76 −0.536897
\(688\) 0 0
\(689\) 10359.9 0.572831
\(690\) 0 0
\(691\) −1222.46 −0.0673001 −0.0336501 0.999434i \(-0.510713\pi\)
−0.0336501 + 0.999434i \(0.510713\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −9184.14 −0.501258
\(696\) 0 0
\(697\) 13767.4 0.748172
\(698\) 0 0
\(699\) −17107.3 −0.925688
\(700\) 0 0
\(701\) 13054.6 0.703372 0.351686 0.936118i \(-0.385608\pi\)
0.351686 + 0.936118i \(0.385608\pi\)
\(702\) 0 0
\(703\) 44352.3 2.37948
\(704\) 0 0
\(705\) 2280.02 0.121802
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −19975.7 −1.05812 −0.529058 0.848585i \(-0.677455\pi\)
−0.529058 + 0.848585i \(0.677455\pi\)
\(710\) 0 0
\(711\) −6014.87 −0.317265
\(712\) 0 0
\(713\) 10746.3 0.564447
\(714\) 0 0
\(715\) −78147.5 −4.08749
\(716\) 0 0
\(717\) 5983.40 0.311651
\(718\) 0 0
\(719\) 6512.26 0.337783 0.168892 0.985635i \(-0.445981\pi\)
0.168892 + 0.985635i \(0.445981\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −4262.66 −0.219267
\(724\) 0 0
\(725\) −6597.92 −0.337987
\(726\) 0 0
\(727\) 11437.1 0.583467 0.291733 0.956500i \(-0.405768\pi\)
0.291733 + 0.956500i \(0.405768\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −16064.4 −0.812808
\(732\) 0 0
\(733\) −12276.0 −0.618589 −0.309295 0.950966i \(-0.600093\pi\)
−0.309295 + 0.950966i \(0.600093\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −32911.3 −1.64492
\(738\) 0 0
\(739\) 12093.8 0.602000 0.301000 0.953624i \(-0.402680\pi\)
0.301000 + 0.953624i \(0.402680\pi\)
\(740\) 0 0
\(741\) −32135.2 −1.59314
\(742\) 0 0
\(743\) 34331.6 1.69516 0.847580 0.530668i \(-0.178059\pi\)
0.847580 + 0.530668i \(0.178059\pi\)
\(744\) 0 0
\(745\) 39975.3 1.96588
\(746\) 0 0
\(747\) 686.205 0.0336103
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −14557.9 −0.707358 −0.353679 0.935367i \(-0.615070\pi\)
−0.353679 + 0.935367i \(0.615070\pi\)
\(752\) 0 0
\(753\) −11566.3 −0.559761
\(754\) 0 0
\(755\) −10351.9 −0.498998
\(756\) 0 0
\(757\) 27115.5 1.30189 0.650944 0.759126i \(-0.274373\pi\)
0.650944 + 0.759126i \(0.274373\pi\)
\(758\) 0 0
\(759\) −17056.4 −0.815688
\(760\) 0 0
\(761\) −10964.0 −0.522264 −0.261132 0.965303i \(-0.584096\pi\)
−0.261132 + 0.965303i \(0.584096\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −13555.0 −0.640629
\(766\) 0 0
\(767\) 25616.9 1.20596
\(768\) 0 0
\(769\) −7835.53 −0.367434 −0.183717 0.982979i \(-0.558813\pi\)
−0.183717 + 0.982979i \(0.558813\pi\)
\(770\) 0 0
\(771\) 12159.8 0.567997
\(772\) 0 0
\(773\) −28767.7 −1.33856 −0.669278 0.743012i \(-0.733396\pi\)
−0.669278 + 0.743012i \(0.733396\pi\)
\(774\) 0 0
\(775\) 20668.0 0.957956
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 24939.8 1.14706
\(780\) 0 0
\(781\) 34931.7 1.60046
\(782\) 0 0
\(783\) −1169.09 −0.0533586
\(784\) 0 0
\(785\) 44827.3 2.03816
\(786\) 0 0
\(787\) −2695.83 −0.122104 −0.0610520 0.998135i \(-0.519446\pi\)
−0.0610520 + 0.998135i \(0.519446\pi\)
\(788\) 0 0
\(789\) −1502.36 −0.0677889
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −36047.9 −1.61425
\(794\) 0 0
\(795\) 7916.19 0.353155
\(796\) 0 0
\(797\) 12592.1 0.559643 0.279822 0.960052i \(-0.409725\pi\)
0.279822 + 0.960052i \(0.409725\pi\)
\(798\) 0 0
\(799\) −4126.67 −0.182717
\(800\) 0 0
\(801\) 12301.2 0.542625
\(802\) 0 0
\(803\) −41254.2 −1.81299
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 9845.23 0.429453
\(808\) 0 0
\(809\) −7799.49 −0.338956 −0.169478 0.985534i \(-0.554208\pi\)
−0.169478 + 0.985534i \(0.554208\pi\)
\(810\) 0 0
\(811\) −19323.5 −0.836669 −0.418335 0.908293i \(-0.637386\pi\)
−0.418335 + 0.908293i \(0.637386\pi\)
\(812\) 0 0
\(813\) 9973.08 0.430223
\(814\) 0 0
\(815\) −17609.1 −0.756834
\(816\) 0 0
\(817\) −29100.9 −1.24616
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1085.30 0.0461354 0.0230677 0.999734i \(-0.492657\pi\)
0.0230677 + 0.999734i \(0.492657\pi\)
\(822\) 0 0
\(823\) −24350.8 −1.03137 −0.515683 0.856779i \(-0.672462\pi\)
−0.515683 + 0.856779i \(0.672462\pi\)
\(824\) 0 0
\(825\) −32804.1 −1.38435
\(826\) 0 0
\(827\) 31664.9 1.33143 0.665716 0.746205i \(-0.268126\pi\)
0.665716 + 0.746205i \(0.268126\pi\)
\(828\) 0 0
\(829\) −3418.02 −0.143200 −0.0716000 0.997433i \(-0.522810\pi\)
−0.0716000 + 0.997433i \(0.522810\pi\)
\(830\) 0 0
\(831\) 25859.2 1.07948
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 4061.08 0.168311
\(836\) 0 0
\(837\) 3662.17 0.151234
\(838\) 0 0
\(839\) 9114.25 0.375040 0.187520 0.982261i \(-0.439955\pi\)
0.187520 + 0.982261i \(0.439955\pi\)
\(840\) 0 0
\(841\) −22514.2 −0.923127
\(842\) 0 0
\(843\) −23132.4 −0.945104
\(844\) 0 0
\(845\) −34617.8 −1.40933
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −417.813 −0.0168896
\(850\) 0 0
\(851\) 21450.5 0.864056
\(852\) 0 0
\(853\) −36229.5 −1.45425 −0.727124 0.686506i \(-0.759143\pi\)
−0.727124 + 0.686506i \(0.759143\pi\)
\(854\) 0 0
\(855\) −24555.1 −0.982183
\(856\) 0 0
\(857\) 8645.18 0.344590 0.172295 0.985045i \(-0.444882\pi\)
0.172295 + 0.985045i \(0.444882\pi\)
\(858\) 0 0
\(859\) −8212.58 −0.326205 −0.163102 0.986609i \(-0.552150\pi\)
−0.163102 + 0.986609i \(0.552150\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −2611.61 −0.103013 −0.0515065 0.998673i \(-0.516402\pi\)
−0.0515065 + 0.998673i \(0.516402\pi\)
\(864\) 0 0
\(865\) 28100.9 1.10458
\(866\) 0 0
\(867\) 9794.61 0.383670
\(868\) 0 0
\(869\) 47958.6 1.87213
\(870\) 0 0
\(871\) −29988.8 −1.16663
\(872\) 0 0
\(873\) 2183.90 0.0846664
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −6446.58 −0.248216 −0.124108 0.992269i \(-0.539607\pi\)
−0.124108 + 0.992269i \(0.539607\pi\)
\(878\) 0 0
\(879\) −13311.3 −0.510783
\(880\) 0 0
\(881\) 37166.3 1.42130 0.710649 0.703547i \(-0.248401\pi\)
0.710649 + 0.703547i \(0.248401\pi\)
\(882\) 0 0
\(883\) 24377.3 0.929060 0.464530 0.885557i \(-0.346223\pi\)
0.464530 + 0.885557i \(0.346223\pi\)
\(884\) 0 0
\(885\) 19574.4 0.743486
\(886\) 0 0
\(887\) −36344.5 −1.37579 −0.687896 0.725809i \(-0.741466\pi\)
−0.687896 + 0.725809i \(0.741466\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −5812.56 −0.218550
\(892\) 0 0
\(893\) −7475.55 −0.280134
\(894\) 0 0
\(895\) 5981.35 0.223391
\(896\) 0 0
\(897\) −15541.8 −0.578512
\(898\) 0 0
\(899\) −5872.97 −0.217880
\(900\) 0 0
\(901\) −14327.8 −0.529775
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −49673.9 −1.82455
\(906\) 0 0
\(907\) 326.875 0.0119666 0.00598329 0.999982i \(-0.498095\pi\)
0.00598329 + 0.999982i \(0.498095\pi\)
\(908\) 0 0
\(909\) 6252.68 0.228150
\(910\) 0 0
\(911\) −38230.7 −1.39038 −0.695191 0.718825i \(-0.744680\pi\)
−0.695191 + 0.718825i \(0.744680\pi\)
\(912\) 0 0
\(913\) −5471.34 −0.198330
\(914\) 0 0
\(915\) −27544.9 −0.995199
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −20049.3 −0.719659 −0.359829 0.933018i \(-0.617165\pi\)
−0.359829 + 0.933018i \(0.617165\pi\)
\(920\) 0 0
\(921\) −898.439 −0.0321439
\(922\) 0 0
\(923\) 31829.8 1.13509
\(924\) 0 0
\(925\) 41255.0 1.46644
\(926\) 0 0
\(927\) −1567.96 −0.0555540
\(928\) 0 0
\(929\) 43341.5 1.53066 0.765332 0.643635i \(-0.222575\pi\)
0.765332 + 0.643635i \(0.222575\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 2741.83 0.0962095
\(934\) 0 0
\(935\) 108078. 3.78026
\(936\) 0 0
\(937\) 17530.0 0.611185 0.305592 0.952162i \(-0.401146\pi\)
0.305592 + 0.952162i \(0.401146\pi\)
\(938\) 0 0
\(939\) 11662.1 0.405303
\(940\) 0 0
\(941\) −32625.0 −1.13023 −0.565115 0.825012i \(-0.691168\pi\)
−0.565115 + 0.825012i \(0.691168\pi\)
\(942\) 0 0
\(943\) 12061.9 0.416530
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −19394.2 −0.665497 −0.332749 0.943016i \(-0.607976\pi\)
−0.332749 + 0.943016i \(0.607976\pi\)
\(948\) 0 0
\(949\) −37590.9 −1.28583
\(950\) 0 0
\(951\) 8637.91 0.294536
\(952\) 0 0
\(953\) −49618.5 −1.68657 −0.843285 0.537467i \(-0.819381\pi\)
−0.843285 + 0.537467i \(0.819381\pi\)
\(954\) 0 0
\(955\) −58137.7 −1.96994
\(956\) 0 0
\(957\) 9321.52 0.314861
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −11393.9 −0.382462
\(962\) 0 0
\(963\) 428.446 0.0143369
\(964\) 0 0
\(965\) −26717.9 −0.891274
\(966\) 0 0
\(967\) 28432.8 0.945538 0.472769 0.881186i \(-0.343254\pi\)
0.472769 + 0.881186i \(0.343254\pi\)
\(968\) 0 0
\(969\) 44443.1 1.47339
\(970\) 0 0
\(971\) 42690.2 1.41091 0.705455 0.708755i \(-0.250743\pi\)
0.705455 + 0.708755i \(0.250743\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −29891.1 −0.981827
\(976\) 0 0
\(977\) 22735.1 0.744482 0.372241 0.928136i \(-0.378589\pi\)
0.372241 + 0.928136i \(0.378589\pi\)
\(978\) 0 0
\(979\) −98081.7 −3.20195
\(980\) 0 0
\(981\) −14697.9 −0.478356
\(982\) 0 0
\(983\) 5643.83 0.183123 0.0915616 0.995799i \(-0.470814\pi\)
0.0915616 + 0.995799i \(0.470814\pi\)
\(984\) 0 0
\(985\) 7593.45 0.245632
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −14074.3 −0.452515
\(990\) 0 0
\(991\) −54844.6 −1.75802 −0.879009 0.476806i \(-0.841794\pi\)
−0.879009 + 0.476806i \(0.841794\pi\)
\(992\) 0 0
\(993\) −8432.36 −0.269479
\(994\) 0 0
\(995\) −34711.0 −1.10594
\(996\) 0 0
\(997\) 44881.1 1.42568 0.712838 0.701328i \(-0.247409\pi\)
0.712838 + 0.701328i \(0.247409\pi\)
\(998\) 0 0
\(999\) 7309.99 0.231509
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 588.4.a.k.1.1 yes 4
3.2 odd 2 1764.4.a.ba.1.4 4
4.3 odd 2 2352.4.a.cl.1.1 4
7.2 even 3 588.4.i.k.361.4 8
7.3 odd 6 588.4.i.l.373.1 8
7.4 even 3 588.4.i.k.373.4 8
7.5 odd 6 588.4.i.l.361.1 8
7.6 odd 2 588.4.a.j.1.4 4
21.2 odd 6 1764.4.k.bd.361.1 8
21.5 even 6 1764.4.k.bb.361.4 8
21.11 odd 6 1764.4.k.bd.1549.1 8
21.17 even 6 1764.4.k.bb.1549.4 8
21.20 even 2 1764.4.a.bc.1.1 4
28.27 even 2 2352.4.a.cq.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
588.4.a.j.1.4 4 7.6 odd 2
588.4.a.k.1.1 yes 4 1.1 even 1 trivial
588.4.i.k.361.4 8 7.2 even 3
588.4.i.k.373.4 8 7.4 even 3
588.4.i.l.361.1 8 7.5 odd 6
588.4.i.l.373.1 8 7.3 odd 6
1764.4.a.ba.1.4 4 3.2 odd 2
1764.4.a.bc.1.1 4 21.20 even 2
1764.4.k.bb.361.4 8 21.5 even 6
1764.4.k.bb.1549.4 8 21.17 even 6
1764.4.k.bd.361.1 8 21.2 odd 6
1764.4.k.bd.1549.1 8 21.11 odd 6
2352.4.a.cl.1.1 4 4.3 odd 2
2352.4.a.cq.1.4 4 28.27 even 2