Properties

Label 8016.2.a.be.1.5
Level 8016
Weight 2
Character 8016.1
Self dual yes
Analytic conductor 64.008
Analytic rank 0
Dimension 11
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: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - x^{10} - 33 x^{9} + 22 x^{8} + 417 x^{7} - 151 x^{6} - 2470 x^{5} + 272 x^{4} + 6584 x^{3} + 292 x^{2} - 5687 x + 242\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 4008)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(1.09328\) of defining polynomial
Character \(\chi\) \(=\) 8016.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -0.0932775 q^{5} -3.86231 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -0.0932775 q^{5} -3.86231 q^{7} +1.00000 q^{9} -4.61303 q^{11} -0.601864 q^{13} -0.0932775 q^{15} -0.832355 q^{17} -3.67672 q^{19} -3.86231 q^{21} -6.30257 q^{23} -4.99130 q^{25} +1.00000 q^{27} +4.62585 q^{29} -4.74106 q^{31} -4.61303 q^{33} +0.360267 q^{35} +5.70292 q^{37} -0.601864 q^{39} -4.02105 q^{41} +3.85227 q^{43} -0.0932775 q^{45} -2.09137 q^{47} +7.91745 q^{49} -0.832355 q^{51} +7.76489 q^{53} +0.430292 q^{55} -3.67672 q^{57} +4.38419 q^{59} +1.61504 q^{61} -3.86231 q^{63} +0.0561404 q^{65} +12.7662 q^{67} -6.30257 q^{69} +13.1673 q^{71} +2.17998 q^{73} -4.99130 q^{75} +17.8170 q^{77} +8.12808 q^{79} +1.00000 q^{81} +2.46750 q^{83} +0.0776400 q^{85} +4.62585 q^{87} +4.23125 q^{89} +2.32459 q^{91} -4.74106 q^{93} +0.342955 q^{95} +3.07885 q^{97} -4.61303 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q + 11q^{3} + 10q^{5} + q^{7} + 11q^{9} + O(q^{10}) \) \( 11q + 11q^{3} + 10q^{5} + q^{7} + 11q^{9} + q^{11} + 10q^{13} + 10q^{15} + 17q^{17} - 2q^{19} + q^{21} + 3q^{23} + 21q^{25} + 11q^{27} + 17q^{29} + 15q^{31} + q^{33} - 11q^{35} + 4q^{37} + 10q^{39} + 16q^{41} - 10q^{43} + 10q^{45} + 16q^{47} + 22q^{49} + 17q^{51} + 42q^{53} + 5q^{55} - 2q^{57} + 2q^{59} + 12q^{61} + q^{63} + 10q^{65} + q^{67} + 3q^{69} + 9q^{71} + 24q^{73} + 21q^{75} + 22q^{77} + 30q^{79} + 11q^{81} - 16q^{83} + 25q^{85} + 17q^{87} + 37q^{89} - q^{91} + 15q^{93} - 5q^{95} + 4q^{97} + q^{99} + O(q^{100}) \)

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
\(5\) −0.0932775 −0.0417150 −0.0208575 0.999782i \(-0.506640\pi\)
−0.0208575 + 0.999782i \(0.506640\pi\)
\(6\) 0 0
\(7\) −3.86231 −1.45982 −0.729908 0.683545i \(-0.760437\pi\)
−0.729908 + 0.683545i \(0.760437\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −4.61303 −1.39088 −0.695441 0.718584i \(-0.744791\pi\)
−0.695441 + 0.718584i \(0.744791\pi\)
\(12\) 0 0
\(13\) −0.601864 −0.166927 −0.0834636 0.996511i \(-0.526598\pi\)
−0.0834636 + 0.996511i \(0.526598\pi\)
\(14\) 0 0
\(15\) −0.0932775 −0.0240842
\(16\) 0 0
\(17\) −0.832355 −0.201876 −0.100938 0.994893i \(-0.532184\pi\)
−0.100938 + 0.994893i \(0.532184\pi\)
\(18\) 0 0
\(19\) −3.67672 −0.843497 −0.421748 0.906713i \(-0.638583\pi\)
−0.421748 + 0.906713i \(0.638583\pi\)
\(20\) 0 0
\(21\) −3.86231 −0.842825
\(22\) 0 0
\(23\) −6.30257 −1.31418 −0.657088 0.753814i \(-0.728212\pi\)
−0.657088 + 0.753814i \(0.728212\pi\)
\(24\) 0 0
\(25\) −4.99130 −0.998260
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.62585 0.858999 0.429500 0.903067i \(-0.358690\pi\)
0.429500 + 0.903067i \(0.358690\pi\)
\(30\) 0 0
\(31\) −4.74106 −0.851519 −0.425760 0.904836i \(-0.639993\pi\)
−0.425760 + 0.904836i \(0.639993\pi\)
\(32\) 0 0
\(33\) −4.61303 −0.803026
\(34\) 0 0
\(35\) 0.360267 0.0608962
\(36\) 0 0
\(37\) 5.70292 0.937555 0.468777 0.883316i \(-0.344695\pi\)
0.468777 + 0.883316i \(0.344695\pi\)
\(38\) 0 0
\(39\) −0.601864 −0.0963754
\(40\) 0 0
\(41\) −4.02105 −0.627982 −0.313991 0.949426i \(-0.601666\pi\)
−0.313991 + 0.949426i \(0.601666\pi\)
\(42\) 0 0
\(43\) 3.85227 0.587466 0.293733 0.955887i \(-0.405102\pi\)
0.293733 + 0.955887i \(0.405102\pi\)
\(44\) 0 0
\(45\) −0.0932775 −0.0139050
\(46\) 0 0
\(47\) −2.09137 −0.305057 −0.152529 0.988299i \(-0.548742\pi\)
−0.152529 + 0.988299i \(0.548742\pi\)
\(48\) 0 0
\(49\) 7.91745 1.13106
\(50\) 0 0
\(51\) −0.832355 −0.116553
\(52\) 0 0
\(53\) 7.76489 1.06659 0.533295 0.845930i \(-0.320954\pi\)
0.533295 + 0.845930i \(0.320954\pi\)
\(54\) 0 0
\(55\) 0.430292 0.0580206
\(56\) 0 0
\(57\) −3.67672 −0.486993
\(58\) 0 0
\(59\) 4.38419 0.570772 0.285386 0.958413i \(-0.407878\pi\)
0.285386 + 0.958413i \(0.407878\pi\)
\(60\) 0 0
\(61\) 1.61504 0.206785 0.103393 0.994641i \(-0.467030\pi\)
0.103393 + 0.994641i \(0.467030\pi\)
\(62\) 0 0
\(63\) −3.86231 −0.486606
\(64\) 0 0
\(65\) 0.0561404 0.00696336
\(66\) 0 0
\(67\) 12.7662 1.55964 0.779821 0.626002i \(-0.215310\pi\)
0.779821 + 0.626002i \(0.215310\pi\)
\(68\) 0 0
\(69\) −6.30257 −0.758740
\(70\) 0 0
\(71\) 13.1673 1.56267 0.781337 0.624109i \(-0.214538\pi\)
0.781337 + 0.624109i \(0.214538\pi\)
\(72\) 0 0
\(73\) 2.17998 0.255147 0.127574 0.991829i \(-0.459281\pi\)
0.127574 + 0.991829i \(0.459281\pi\)
\(74\) 0 0
\(75\) −4.99130 −0.576346
\(76\) 0 0
\(77\) 17.8170 2.03043
\(78\) 0 0
\(79\) 8.12808 0.914481 0.457240 0.889343i \(-0.348838\pi\)
0.457240 + 0.889343i \(0.348838\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 2.46750 0.270844 0.135422 0.990788i \(-0.456761\pi\)
0.135422 + 0.990788i \(0.456761\pi\)
\(84\) 0 0
\(85\) 0.0776400 0.00842124
\(86\) 0 0
\(87\) 4.62585 0.495944
\(88\) 0 0
\(89\) 4.23125 0.448512 0.224256 0.974530i \(-0.428005\pi\)
0.224256 + 0.974530i \(0.428005\pi\)
\(90\) 0 0
\(91\) 2.32459 0.243683
\(92\) 0 0
\(93\) −4.74106 −0.491625
\(94\) 0 0
\(95\) 0.342955 0.0351864
\(96\) 0 0
\(97\) 3.07885 0.312610 0.156305 0.987709i \(-0.450042\pi\)
0.156305 + 0.987709i \(0.450042\pi\)
\(98\) 0 0
\(99\) −4.61303 −0.463627
\(100\) 0 0
\(101\) 7.37707 0.734046 0.367023 0.930212i \(-0.380377\pi\)
0.367023 + 0.930212i \(0.380377\pi\)
\(102\) 0 0
\(103\) 5.13078 0.505551 0.252775 0.967525i \(-0.418657\pi\)
0.252775 + 0.967525i \(0.418657\pi\)
\(104\) 0 0
\(105\) 0.360267 0.0351584
\(106\) 0 0
\(107\) −4.13448 −0.399696 −0.199848 0.979827i \(-0.564045\pi\)
−0.199848 + 0.979827i \(0.564045\pi\)
\(108\) 0 0
\(109\) −16.1790 −1.54967 −0.774833 0.632166i \(-0.782166\pi\)
−0.774833 + 0.632166i \(0.782166\pi\)
\(110\) 0 0
\(111\) 5.70292 0.541298
\(112\) 0 0
\(113\) −15.0108 −1.41210 −0.706050 0.708162i \(-0.749524\pi\)
−0.706050 + 0.708162i \(0.749524\pi\)
\(114\) 0 0
\(115\) 0.587888 0.0548208
\(116\) 0 0
\(117\) −0.601864 −0.0556424
\(118\) 0 0
\(119\) 3.21482 0.294702
\(120\) 0 0
\(121\) 10.2801 0.934551
\(122\) 0 0
\(123\) −4.02105 −0.362565
\(124\) 0 0
\(125\) 0.931964 0.0833574
\(126\) 0 0
\(127\) −0.742937 −0.0659250 −0.0329625 0.999457i \(-0.510494\pi\)
−0.0329625 + 0.999457i \(0.510494\pi\)
\(128\) 0 0
\(129\) 3.85227 0.339174
\(130\) 0 0
\(131\) −21.4308 −1.87242 −0.936209 0.351444i \(-0.885691\pi\)
−0.936209 + 0.351444i \(0.885691\pi\)
\(132\) 0 0
\(133\) 14.2006 1.23135
\(134\) 0 0
\(135\) −0.0932775 −0.00802805
\(136\) 0 0
\(137\) 14.6892 1.25498 0.627492 0.778623i \(-0.284082\pi\)
0.627492 + 0.778623i \(0.284082\pi\)
\(138\) 0 0
\(139\) 17.1246 1.45249 0.726247 0.687434i \(-0.241263\pi\)
0.726247 + 0.687434i \(0.241263\pi\)
\(140\) 0 0
\(141\) −2.09137 −0.176125
\(142\) 0 0
\(143\) 2.77642 0.232176
\(144\) 0 0
\(145\) −0.431488 −0.0358331
\(146\) 0 0
\(147\) 7.91745 0.653020
\(148\) 0 0
\(149\) 16.3475 1.33924 0.669620 0.742704i \(-0.266457\pi\)
0.669620 + 0.742704i \(0.266457\pi\)
\(150\) 0 0
\(151\) 5.37955 0.437782 0.218891 0.975749i \(-0.429756\pi\)
0.218891 + 0.975749i \(0.429756\pi\)
\(152\) 0 0
\(153\) −0.832355 −0.0672919
\(154\) 0 0
\(155\) 0.442234 0.0355211
\(156\) 0 0
\(157\) −2.59090 −0.206776 −0.103388 0.994641i \(-0.532968\pi\)
−0.103388 + 0.994641i \(0.532968\pi\)
\(158\) 0 0
\(159\) 7.76489 0.615796
\(160\) 0 0
\(161\) 24.3425 1.91846
\(162\) 0 0
\(163\) −14.8771 −1.16526 −0.582631 0.812737i \(-0.697977\pi\)
−0.582631 + 0.812737i \(0.697977\pi\)
\(164\) 0 0
\(165\) 0.430292 0.0334982
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) −12.6378 −0.972135
\(170\) 0 0
\(171\) −3.67672 −0.281166
\(172\) 0 0
\(173\) 2.39765 0.182290 0.0911448 0.995838i \(-0.470947\pi\)
0.0911448 + 0.995838i \(0.470947\pi\)
\(174\) 0 0
\(175\) 19.2780 1.45728
\(176\) 0 0
\(177\) 4.38419 0.329536
\(178\) 0 0
\(179\) 9.24300 0.690854 0.345427 0.938446i \(-0.387734\pi\)
0.345427 + 0.938446i \(0.387734\pi\)
\(180\) 0 0
\(181\) 7.04367 0.523552 0.261776 0.965129i \(-0.415692\pi\)
0.261776 + 0.965129i \(0.415692\pi\)
\(182\) 0 0
\(183\) 1.61504 0.119387
\(184\) 0 0
\(185\) −0.531955 −0.0391101
\(186\) 0 0
\(187\) 3.83968 0.280785
\(188\) 0 0
\(189\) −3.86231 −0.280942
\(190\) 0 0
\(191\) −11.2775 −0.816009 −0.408004 0.912980i \(-0.633775\pi\)
−0.408004 + 0.912980i \(0.633775\pi\)
\(192\) 0 0
\(193\) −0.525080 −0.0377961 −0.0188980 0.999821i \(-0.506016\pi\)
−0.0188980 + 0.999821i \(0.506016\pi\)
\(194\) 0 0
\(195\) 0.0561404 0.00402030
\(196\) 0 0
\(197\) −17.7353 −1.26359 −0.631795 0.775136i \(-0.717681\pi\)
−0.631795 + 0.775136i \(0.717681\pi\)
\(198\) 0 0
\(199\) −14.8380 −1.05184 −0.525920 0.850534i \(-0.676279\pi\)
−0.525920 + 0.850534i \(0.676279\pi\)
\(200\) 0 0
\(201\) 12.7662 0.900460
\(202\) 0 0
\(203\) −17.8665 −1.25398
\(204\) 0 0
\(205\) 0.375073 0.0261962
\(206\) 0 0
\(207\) −6.30257 −0.438059
\(208\) 0 0
\(209\) 16.9608 1.17320
\(210\) 0 0
\(211\) −25.0538 −1.72478 −0.862388 0.506247i \(-0.831032\pi\)
−0.862388 + 0.506247i \(0.831032\pi\)
\(212\) 0 0
\(213\) 13.1673 0.902210
\(214\) 0 0
\(215\) −0.359331 −0.0245061
\(216\) 0 0
\(217\) 18.3114 1.24306
\(218\) 0 0
\(219\) 2.17998 0.147309
\(220\) 0 0
\(221\) 0.500965 0.0336986
\(222\) 0 0
\(223\) −6.95620 −0.465822 −0.232911 0.972498i \(-0.574825\pi\)
−0.232911 + 0.972498i \(0.574825\pi\)
\(224\) 0 0
\(225\) −4.99130 −0.332753
\(226\) 0 0
\(227\) −10.5654 −0.701250 −0.350625 0.936516i \(-0.614031\pi\)
−0.350625 + 0.936516i \(0.614031\pi\)
\(228\) 0 0
\(229\) 17.6950 1.16932 0.584660 0.811279i \(-0.301228\pi\)
0.584660 + 0.811279i \(0.301228\pi\)
\(230\) 0 0
\(231\) 17.8170 1.17227
\(232\) 0 0
\(233\) 3.03342 0.198726 0.0993628 0.995051i \(-0.468320\pi\)
0.0993628 + 0.995051i \(0.468320\pi\)
\(234\) 0 0
\(235\) 0.195078 0.0127255
\(236\) 0 0
\(237\) 8.12808 0.527976
\(238\) 0 0
\(239\) 5.47139 0.353915 0.176957 0.984219i \(-0.443375\pi\)
0.176957 + 0.984219i \(0.443375\pi\)
\(240\) 0 0
\(241\) −20.1326 −1.29685 −0.648427 0.761277i \(-0.724573\pi\)
−0.648427 + 0.761277i \(0.724573\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −0.738520 −0.0471823
\(246\) 0 0
\(247\) 2.21288 0.140802
\(248\) 0 0
\(249\) 2.46750 0.156372
\(250\) 0 0
\(251\) −11.8256 −0.746424 −0.373212 0.927746i \(-0.621744\pi\)
−0.373212 + 0.927746i \(0.621744\pi\)
\(252\) 0 0
\(253\) 29.0739 1.82786
\(254\) 0 0
\(255\) 0.0776400 0.00486201
\(256\) 0 0
\(257\) 27.9960 1.74634 0.873172 0.487412i \(-0.162059\pi\)
0.873172 + 0.487412i \(0.162059\pi\)
\(258\) 0 0
\(259\) −22.0265 −1.36866
\(260\) 0 0
\(261\) 4.62585 0.286333
\(262\) 0 0
\(263\) −1.13973 −0.0702790 −0.0351395 0.999382i \(-0.511188\pi\)
−0.0351395 + 0.999382i \(0.511188\pi\)
\(264\) 0 0
\(265\) −0.724289 −0.0444927
\(266\) 0 0
\(267\) 4.23125 0.258948
\(268\) 0 0
\(269\) 3.10035 0.189031 0.0945157 0.995523i \(-0.469870\pi\)
0.0945157 + 0.995523i \(0.469870\pi\)
\(270\) 0 0
\(271\) 26.9980 1.64001 0.820007 0.572354i \(-0.193970\pi\)
0.820007 + 0.572354i \(0.193970\pi\)
\(272\) 0 0
\(273\) 2.32459 0.140690
\(274\) 0 0
\(275\) 23.0250 1.38846
\(276\) 0 0
\(277\) 9.24414 0.555427 0.277713 0.960664i \(-0.410423\pi\)
0.277713 + 0.960664i \(0.410423\pi\)
\(278\) 0 0
\(279\) −4.74106 −0.283840
\(280\) 0 0
\(281\) −1.83309 −0.109353 −0.0546765 0.998504i \(-0.517413\pi\)
−0.0546765 + 0.998504i \(0.517413\pi\)
\(282\) 0 0
\(283\) −2.27493 −0.135230 −0.0676152 0.997711i \(-0.521539\pi\)
−0.0676152 + 0.997711i \(0.521539\pi\)
\(284\) 0 0
\(285\) 0.342955 0.0203149
\(286\) 0 0
\(287\) 15.5305 0.916738
\(288\) 0 0
\(289\) −16.3072 −0.959246
\(290\) 0 0
\(291\) 3.07885 0.180485
\(292\) 0 0
\(293\) 33.0304 1.92966 0.964830 0.262876i \(-0.0846711\pi\)
0.964830 + 0.262876i \(0.0846711\pi\)
\(294\) 0 0
\(295\) −0.408946 −0.0238098
\(296\) 0 0
\(297\) −4.61303 −0.267675
\(298\) 0 0
\(299\) 3.79329 0.219372
\(300\) 0 0
\(301\) −14.8787 −0.857593
\(302\) 0 0
\(303\) 7.37707 0.423801
\(304\) 0 0
\(305\) −0.150647 −0.00862603
\(306\) 0 0
\(307\) −3.54712 −0.202445 −0.101222 0.994864i \(-0.532275\pi\)
−0.101222 + 0.994864i \(0.532275\pi\)
\(308\) 0 0
\(309\) 5.13078 0.291880
\(310\) 0 0
\(311\) 9.22270 0.522971 0.261486 0.965207i \(-0.415788\pi\)
0.261486 + 0.965207i \(0.415788\pi\)
\(312\) 0 0
\(313\) −13.6463 −0.771335 −0.385668 0.922638i \(-0.626029\pi\)
−0.385668 + 0.922638i \(0.626029\pi\)
\(314\) 0 0
\(315\) 0.360267 0.0202987
\(316\) 0 0
\(317\) 13.8395 0.777304 0.388652 0.921385i \(-0.372941\pi\)
0.388652 + 0.921385i \(0.372941\pi\)
\(318\) 0 0
\(319\) −21.3392 −1.19477
\(320\) 0 0
\(321\) −4.13448 −0.230764
\(322\) 0 0
\(323\) 3.06033 0.170282
\(324\) 0 0
\(325\) 3.00409 0.166637
\(326\) 0 0
\(327\) −16.1790 −0.894701
\(328\) 0 0
\(329\) 8.07752 0.445328
\(330\) 0 0
\(331\) 33.2468 1.82741 0.913705 0.406377i \(-0.133208\pi\)
0.913705 + 0.406377i \(0.133208\pi\)
\(332\) 0 0
\(333\) 5.70292 0.312518
\(334\) 0 0
\(335\) −1.19080 −0.0650604
\(336\) 0 0
\(337\) 32.7473 1.78386 0.891930 0.452174i \(-0.149351\pi\)
0.891930 + 0.452174i \(0.149351\pi\)
\(338\) 0 0
\(339\) −15.0108 −0.815276
\(340\) 0 0
\(341\) 21.8707 1.18436
\(342\) 0 0
\(343\) −3.54348 −0.191330
\(344\) 0 0
\(345\) 0.587888 0.0316508
\(346\) 0 0
\(347\) −0.842952 −0.0452521 −0.0226260 0.999744i \(-0.507203\pi\)
−0.0226260 + 0.999744i \(0.507203\pi\)
\(348\) 0 0
\(349\) 0.514223 0.0275257 0.0137629 0.999905i \(-0.495619\pi\)
0.0137629 + 0.999905i \(0.495619\pi\)
\(350\) 0 0
\(351\) −0.601864 −0.0321251
\(352\) 0 0
\(353\) −14.5789 −0.775958 −0.387979 0.921668i \(-0.626827\pi\)
−0.387979 + 0.921668i \(0.626827\pi\)
\(354\) 0 0
\(355\) −1.22822 −0.0651869
\(356\) 0 0
\(357\) 3.21482 0.170146
\(358\) 0 0
\(359\) 3.12726 0.165050 0.0825252 0.996589i \(-0.473702\pi\)
0.0825252 + 0.996589i \(0.473702\pi\)
\(360\) 0 0
\(361\) −5.48176 −0.288514
\(362\) 0 0
\(363\) 10.2801 0.539563
\(364\) 0 0
\(365\) −0.203343 −0.0106435
\(366\) 0 0
\(367\) −20.4895 −1.06955 −0.534773 0.844996i \(-0.679603\pi\)
−0.534773 + 0.844996i \(0.679603\pi\)
\(368\) 0 0
\(369\) −4.02105 −0.209327
\(370\) 0 0
\(371\) −29.9904 −1.55702
\(372\) 0 0
\(373\) −17.4296 −0.902470 −0.451235 0.892405i \(-0.649016\pi\)
−0.451235 + 0.892405i \(0.649016\pi\)
\(374\) 0 0
\(375\) 0.931964 0.0481264
\(376\) 0 0
\(377\) −2.78414 −0.143390
\(378\) 0 0
\(379\) 14.3457 0.736890 0.368445 0.929650i \(-0.379890\pi\)
0.368445 + 0.929650i \(0.379890\pi\)
\(380\) 0 0
\(381\) −0.742937 −0.0380618
\(382\) 0 0
\(383\) 28.3329 1.44774 0.723872 0.689934i \(-0.242361\pi\)
0.723872 + 0.689934i \(0.242361\pi\)
\(384\) 0 0
\(385\) −1.66192 −0.0846994
\(386\) 0 0
\(387\) 3.85227 0.195822
\(388\) 0 0
\(389\) −16.8631 −0.854991 −0.427496 0.904017i \(-0.640604\pi\)
−0.427496 + 0.904017i \(0.640604\pi\)
\(390\) 0 0
\(391\) 5.24598 0.265300
\(392\) 0 0
\(393\) −21.4308 −1.08104
\(394\) 0 0
\(395\) −0.758167 −0.0381475
\(396\) 0 0
\(397\) 31.3038 1.57109 0.785546 0.618803i \(-0.212382\pi\)
0.785546 + 0.618803i \(0.212382\pi\)
\(398\) 0 0
\(399\) 14.2006 0.710920
\(400\) 0 0
\(401\) −0.0449210 −0.00224325 −0.00112162 0.999999i \(-0.500357\pi\)
−0.00112162 + 0.999999i \(0.500357\pi\)
\(402\) 0 0
\(403\) 2.85348 0.142142
\(404\) 0 0
\(405\) −0.0932775 −0.00463500
\(406\) 0 0
\(407\) −26.3078 −1.30403
\(408\) 0 0
\(409\) 19.4314 0.960821 0.480411 0.877044i \(-0.340488\pi\)
0.480411 + 0.877044i \(0.340488\pi\)
\(410\) 0 0
\(411\) 14.6892 0.724565
\(412\) 0 0
\(413\) −16.9331 −0.833223
\(414\) 0 0
\(415\) −0.230163 −0.0112982
\(416\) 0 0
\(417\) 17.1246 0.838597
\(418\) 0 0
\(419\) −34.5480 −1.68778 −0.843889 0.536517i \(-0.819740\pi\)
−0.843889 + 0.536517i \(0.819740\pi\)
\(420\) 0 0
\(421\) 1.22382 0.0596455 0.0298227 0.999555i \(-0.490506\pi\)
0.0298227 + 0.999555i \(0.490506\pi\)
\(422\) 0 0
\(423\) −2.09137 −0.101686
\(424\) 0 0
\(425\) 4.15453 0.201525
\(426\) 0 0
\(427\) −6.23780 −0.301868
\(428\) 0 0
\(429\) 2.77642 0.134047
\(430\) 0 0
\(431\) 12.4784 0.601063 0.300531 0.953772i \(-0.402836\pi\)
0.300531 + 0.953772i \(0.402836\pi\)
\(432\) 0 0
\(433\) −16.8043 −0.807565 −0.403782 0.914855i \(-0.632305\pi\)
−0.403782 + 0.914855i \(0.632305\pi\)
\(434\) 0 0
\(435\) −0.431488 −0.0206883
\(436\) 0 0
\(437\) 23.1728 1.10850
\(438\) 0 0
\(439\) 13.2911 0.634350 0.317175 0.948367i \(-0.397266\pi\)
0.317175 + 0.948367i \(0.397266\pi\)
\(440\) 0 0
\(441\) 7.91745 0.377021
\(442\) 0 0
\(443\) −23.5380 −1.11832 −0.559162 0.829059i \(-0.688877\pi\)
−0.559162 + 0.829059i \(0.688877\pi\)
\(444\) 0 0
\(445\) −0.394681 −0.0187097
\(446\) 0 0
\(447\) 16.3475 0.773211
\(448\) 0 0
\(449\) −31.0913 −1.46729 −0.733644 0.679534i \(-0.762182\pi\)
−0.733644 + 0.679534i \(0.762182\pi\)
\(450\) 0 0
\(451\) 18.5492 0.873448
\(452\) 0 0
\(453\) 5.37955 0.252753
\(454\) 0 0
\(455\) −0.216832 −0.0101652
\(456\) 0 0
\(457\) −23.8267 −1.11457 −0.557283 0.830323i \(-0.688156\pi\)
−0.557283 + 0.830323i \(0.688156\pi\)
\(458\) 0 0
\(459\) −0.832355 −0.0388510
\(460\) 0 0
\(461\) 8.24371 0.383948 0.191974 0.981400i \(-0.438511\pi\)
0.191974 + 0.981400i \(0.438511\pi\)
\(462\) 0 0
\(463\) 5.22478 0.242816 0.121408 0.992603i \(-0.461259\pi\)
0.121408 + 0.992603i \(0.461259\pi\)
\(464\) 0 0
\(465\) 0.442234 0.0205081
\(466\) 0 0
\(467\) 16.6902 0.772328 0.386164 0.922430i \(-0.373800\pi\)
0.386164 + 0.922430i \(0.373800\pi\)
\(468\) 0 0
\(469\) −49.3071 −2.27679
\(470\) 0 0
\(471\) −2.59090 −0.119382
\(472\) 0 0
\(473\) −17.7707 −0.817096
\(474\) 0 0
\(475\) 18.3516 0.842029
\(476\) 0 0
\(477\) 7.76489 0.355530
\(478\) 0 0
\(479\) 9.87991 0.451425 0.225712 0.974194i \(-0.427529\pi\)
0.225712 + 0.974194i \(0.427529\pi\)
\(480\) 0 0
\(481\) −3.43239 −0.156503
\(482\) 0 0
\(483\) 24.3425 1.10762
\(484\) 0 0
\(485\) −0.287187 −0.0130405
\(486\) 0 0
\(487\) 3.04928 0.138176 0.0690881 0.997611i \(-0.477991\pi\)
0.0690881 + 0.997611i \(0.477991\pi\)
\(488\) 0 0
\(489\) −14.8771 −0.672764
\(490\) 0 0
\(491\) 31.2489 1.41024 0.705121 0.709087i \(-0.250893\pi\)
0.705121 + 0.709087i \(0.250893\pi\)
\(492\) 0 0
\(493\) −3.85035 −0.173411
\(494\) 0 0
\(495\) 0.430292 0.0193402
\(496\) 0 0
\(497\) −50.8563 −2.28122
\(498\) 0 0
\(499\) −25.0818 −1.12282 −0.561409 0.827539i \(-0.689740\pi\)
−0.561409 + 0.827539i \(0.689740\pi\)
\(500\) 0 0
\(501\) −1.00000 −0.0446767
\(502\) 0 0
\(503\) −18.7306 −0.835155 −0.417578 0.908641i \(-0.637121\pi\)
−0.417578 + 0.908641i \(0.637121\pi\)
\(504\) 0 0
\(505\) −0.688115 −0.0306207
\(506\) 0 0
\(507\) −12.6378 −0.561263
\(508\) 0 0
\(509\) 15.6596 0.694098 0.347049 0.937847i \(-0.387184\pi\)
0.347049 + 0.937847i \(0.387184\pi\)
\(510\) 0 0
\(511\) −8.41977 −0.372468
\(512\) 0 0
\(513\) −3.67672 −0.162331
\(514\) 0 0
\(515\) −0.478586 −0.0210890
\(516\) 0 0
\(517\) 9.64755 0.424299
\(518\) 0 0
\(519\) 2.39765 0.105245
\(520\) 0 0
\(521\) 23.7109 1.03879 0.519397 0.854533i \(-0.326156\pi\)
0.519397 + 0.854533i \(0.326156\pi\)
\(522\) 0 0
\(523\) 27.3744 1.19700 0.598500 0.801123i \(-0.295764\pi\)
0.598500 + 0.801123i \(0.295764\pi\)
\(524\) 0 0
\(525\) 19.2780 0.841359
\(526\) 0 0
\(527\) 3.94625 0.171901
\(528\) 0 0
\(529\) 16.7224 0.727060
\(530\) 0 0
\(531\) 4.38419 0.190257
\(532\) 0 0
\(533\) 2.42012 0.104827
\(534\) 0 0
\(535\) 0.385654 0.0166733
\(536\) 0 0
\(537\) 9.24300 0.398865
\(538\) 0 0
\(539\) −36.5234 −1.57318
\(540\) 0 0
\(541\) −27.8135 −1.19580 −0.597899 0.801572i \(-0.703998\pi\)
−0.597899 + 0.801572i \(0.703998\pi\)
\(542\) 0 0
\(543\) 7.04367 0.302273
\(544\) 0 0
\(545\) 1.50914 0.0646443
\(546\) 0 0
\(547\) −13.5049 −0.577427 −0.288714 0.957415i \(-0.593228\pi\)
−0.288714 + 0.957415i \(0.593228\pi\)
\(548\) 0 0
\(549\) 1.61504 0.0689283
\(550\) 0 0
\(551\) −17.0079 −0.724563
\(552\) 0 0
\(553\) −31.3932 −1.33497
\(554\) 0 0
\(555\) −0.531955 −0.0225802
\(556\) 0 0
\(557\) 9.31791 0.394812 0.197406 0.980322i \(-0.436748\pi\)
0.197406 + 0.980322i \(0.436748\pi\)
\(558\) 0 0
\(559\) −2.31855 −0.0980641
\(560\) 0 0
\(561\) 3.83968 0.162111
\(562\) 0 0
\(563\) −31.6760 −1.33498 −0.667492 0.744617i \(-0.732632\pi\)
−0.667492 + 0.744617i \(0.732632\pi\)
\(564\) 0 0
\(565\) 1.40017 0.0589057
\(566\) 0 0
\(567\) −3.86231 −0.162202
\(568\) 0 0
\(569\) −16.3201 −0.684175 −0.342087 0.939668i \(-0.611134\pi\)
−0.342087 + 0.939668i \(0.611134\pi\)
\(570\) 0 0
\(571\) −13.0316 −0.545357 −0.272679 0.962105i \(-0.587910\pi\)
−0.272679 + 0.962105i \(0.587910\pi\)
\(572\) 0 0
\(573\) −11.2775 −0.471123
\(574\) 0 0
\(575\) 31.4580 1.31189
\(576\) 0 0
\(577\) −11.6936 −0.486810 −0.243405 0.969925i \(-0.578264\pi\)
−0.243405 + 0.969925i \(0.578264\pi\)
\(578\) 0 0
\(579\) −0.525080 −0.0218216
\(580\) 0 0
\(581\) −9.53027 −0.395382
\(582\) 0 0
\(583\) −35.8197 −1.48350
\(584\) 0 0
\(585\) 0.0561404 0.00232112
\(586\) 0 0
\(587\) 33.9904 1.40294 0.701468 0.712701i \(-0.252528\pi\)
0.701468 + 0.712701i \(0.252528\pi\)
\(588\) 0 0
\(589\) 17.4315 0.718254
\(590\) 0 0
\(591\) −17.7353 −0.729534
\(592\) 0 0
\(593\) 27.0891 1.11242 0.556208 0.831043i \(-0.312256\pi\)
0.556208 + 0.831043i \(0.312256\pi\)
\(594\) 0 0
\(595\) −0.299870 −0.0122935
\(596\) 0 0
\(597\) −14.8380 −0.607280
\(598\) 0 0
\(599\) 43.6427 1.78319 0.891597 0.452830i \(-0.149585\pi\)
0.891597 + 0.452830i \(0.149585\pi\)
\(600\) 0 0
\(601\) 15.2968 0.623969 0.311984 0.950087i \(-0.399006\pi\)
0.311984 + 0.950087i \(0.399006\pi\)
\(602\) 0 0
\(603\) 12.7662 0.519881
\(604\) 0 0
\(605\) −0.958898 −0.0389848
\(606\) 0 0
\(607\) −28.1380 −1.14208 −0.571042 0.820921i \(-0.693461\pi\)
−0.571042 + 0.820921i \(0.693461\pi\)
\(608\) 0 0
\(609\) −17.8665 −0.723987
\(610\) 0 0
\(611\) 1.25872 0.0509224
\(612\) 0 0
\(613\) −45.5990 −1.84173 −0.920864 0.389884i \(-0.872515\pi\)
−0.920864 + 0.389884i \(0.872515\pi\)
\(614\) 0 0
\(615\) 0.375073 0.0151244
\(616\) 0 0
\(617\) −9.68654 −0.389965 −0.194983 0.980807i \(-0.562465\pi\)
−0.194983 + 0.980807i \(0.562465\pi\)
\(618\) 0 0
\(619\) 3.87505 0.155752 0.0778758 0.996963i \(-0.475186\pi\)
0.0778758 + 0.996963i \(0.475186\pi\)
\(620\) 0 0
\(621\) −6.30257 −0.252913
\(622\) 0 0
\(623\) −16.3424 −0.654745
\(624\) 0 0
\(625\) 24.8696 0.994783
\(626\) 0 0
\(627\) 16.9608 0.677349
\(628\) 0 0
\(629\) −4.74686 −0.189270
\(630\) 0 0
\(631\) 22.6390 0.901244 0.450622 0.892715i \(-0.351202\pi\)
0.450622 + 0.892715i \(0.351202\pi\)
\(632\) 0 0
\(633\) −25.0538 −0.995800
\(634\) 0 0
\(635\) 0.0692993 0.00275006
\(636\) 0 0
\(637\) −4.76523 −0.188805
\(638\) 0 0
\(639\) 13.1673 0.520891
\(640\) 0 0
\(641\) 5.55587 0.219444 0.109722 0.993962i \(-0.465004\pi\)
0.109722 + 0.993962i \(0.465004\pi\)
\(642\) 0 0
\(643\) 19.1357 0.754639 0.377320 0.926083i \(-0.376846\pi\)
0.377320 + 0.926083i \(0.376846\pi\)
\(644\) 0 0
\(645\) −0.359331 −0.0141486
\(646\) 0 0
\(647\) 10.6418 0.418372 0.209186 0.977876i \(-0.432919\pi\)
0.209186 + 0.977876i \(0.432919\pi\)
\(648\) 0 0
\(649\) −20.2244 −0.793876
\(650\) 0 0
\(651\) 18.3114 0.717682
\(652\) 0 0
\(653\) 6.42895 0.251584 0.125792 0.992057i \(-0.459853\pi\)
0.125792 + 0.992057i \(0.459853\pi\)
\(654\) 0 0
\(655\) 1.99901 0.0781079
\(656\) 0 0
\(657\) 2.17998 0.0850492
\(658\) 0 0
\(659\) 0.838749 0.0326730 0.0163365 0.999867i \(-0.494800\pi\)
0.0163365 + 0.999867i \(0.494800\pi\)
\(660\) 0 0
\(661\) 15.2337 0.592521 0.296261 0.955107i \(-0.404260\pi\)
0.296261 + 0.955107i \(0.404260\pi\)
\(662\) 0 0
\(663\) 0.500965 0.0194559
\(664\) 0 0
\(665\) −1.32460 −0.0513657
\(666\) 0 0
\(667\) −29.1548 −1.12888
\(668\) 0 0
\(669\) −6.95620 −0.268942
\(670\) 0 0
\(671\) −7.45024 −0.287613
\(672\) 0 0
\(673\) −35.9761 −1.38678 −0.693388 0.720565i \(-0.743883\pi\)
−0.693388 + 0.720565i \(0.743883\pi\)
\(674\) 0 0
\(675\) −4.99130 −0.192115
\(676\) 0 0
\(677\) 5.27845 0.202867 0.101434 0.994842i \(-0.467657\pi\)
0.101434 + 0.994842i \(0.467657\pi\)
\(678\) 0 0
\(679\) −11.8915 −0.456353
\(680\) 0 0
\(681\) −10.5654 −0.404867
\(682\) 0 0
\(683\) 37.0882 1.41914 0.709570 0.704635i \(-0.248889\pi\)
0.709570 + 0.704635i \(0.248889\pi\)
\(684\) 0 0
\(685\) −1.37017 −0.0523516
\(686\) 0 0
\(687\) 17.6950 0.675107
\(688\) 0 0
\(689\) −4.67341 −0.178043
\(690\) 0 0
\(691\) −10.1700 −0.386883 −0.193442 0.981112i \(-0.561965\pi\)
−0.193442 + 0.981112i \(0.561965\pi\)
\(692\) 0 0
\(693\) 17.8170 0.676810
\(694\) 0 0
\(695\) −1.59734 −0.0605907
\(696\) 0 0
\(697\) 3.34694 0.126774
\(698\) 0 0
\(699\) 3.03342 0.114734
\(700\) 0 0
\(701\) 12.3197 0.465310 0.232655 0.972559i \(-0.425259\pi\)
0.232655 + 0.972559i \(0.425259\pi\)
\(702\) 0 0
\(703\) −20.9680 −0.790824
\(704\) 0 0
\(705\) 0.195078 0.00734705
\(706\) 0 0
\(707\) −28.4925 −1.07157
\(708\) 0 0
\(709\) 14.4565 0.542926 0.271463 0.962449i \(-0.412493\pi\)
0.271463 + 0.962449i \(0.412493\pi\)
\(710\) 0 0
\(711\) 8.12808 0.304827
\(712\) 0 0
\(713\) 29.8809 1.11905
\(714\) 0 0
\(715\) −0.258978 −0.00968521
\(716\) 0 0
\(717\) 5.47139 0.204333
\(718\) 0 0
\(719\) −1.61087 −0.0600752 −0.0300376 0.999549i \(-0.509563\pi\)
−0.0300376 + 0.999549i \(0.509563\pi\)
\(720\) 0 0
\(721\) −19.8167 −0.738011
\(722\) 0 0
\(723\) −20.1326 −0.748739
\(724\) 0 0
\(725\) −23.0890 −0.857505
\(726\) 0 0
\(727\) 2.46431 0.0913961 0.0456981 0.998955i \(-0.485449\pi\)
0.0456981 + 0.998955i \(0.485449\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −3.20646 −0.118595
\(732\) 0 0
\(733\) −23.5980 −0.871611 −0.435806 0.900041i \(-0.643537\pi\)
−0.435806 + 0.900041i \(0.643537\pi\)
\(734\) 0 0
\(735\) −0.738520 −0.0272407
\(736\) 0 0
\(737\) −58.8910 −2.16928
\(738\) 0 0
\(739\) −52.2196 −1.92093 −0.960464 0.278405i \(-0.910194\pi\)
−0.960464 + 0.278405i \(0.910194\pi\)
\(740\) 0 0
\(741\) 2.21288 0.0812924
\(742\) 0 0
\(743\) 35.0041 1.28417 0.642087 0.766632i \(-0.278069\pi\)
0.642087 + 0.766632i \(0.278069\pi\)
\(744\) 0 0
\(745\) −1.52486 −0.0558664
\(746\) 0 0
\(747\) 2.46750 0.0902813
\(748\) 0 0
\(749\) 15.9687 0.583482
\(750\) 0 0
\(751\) 35.6060 1.29928 0.649641 0.760241i \(-0.274919\pi\)
0.649641 + 0.760241i \(0.274919\pi\)
\(752\) 0 0
\(753\) −11.8256 −0.430948
\(754\) 0 0
\(755\) −0.501791 −0.0182620
\(756\) 0 0
\(757\) 26.4806 0.962453 0.481227 0.876596i \(-0.340191\pi\)
0.481227 + 0.876596i \(0.340191\pi\)
\(758\) 0 0
\(759\) 29.0739 1.05532
\(760\) 0 0
\(761\) 34.9119 1.26555 0.632777 0.774334i \(-0.281915\pi\)
0.632777 + 0.774334i \(0.281915\pi\)
\(762\) 0 0
\(763\) 62.4883 2.26223
\(764\) 0 0
\(765\) 0.0776400 0.00280708
\(766\) 0 0
\(767\) −2.63869 −0.0952774
\(768\) 0 0
\(769\) 19.5078 0.703468 0.351734 0.936100i \(-0.385592\pi\)
0.351734 + 0.936100i \(0.385592\pi\)
\(770\) 0 0
\(771\) 27.9960 1.00825
\(772\) 0 0
\(773\) −15.7496 −0.566475 −0.283237 0.959050i \(-0.591408\pi\)
−0.283237 + 0.959050i \(0.591408\pi\)
\(774\) 0 0
\(775\) 23.6640 0.850038
\(776\) 0 0
\(777\) −22.0265 −0.790195
\(778\) 0 0
\(779\) 14.7842 0.529700
\(780\) 0 0
\(781\) −60.7413 −2.17349
\(782\) 0 0
\(783\) 4.62585 0.165315
\(784\) 0 0
\(785\) 0.241672 0.00862565
\(786\) 0 0
\(787\) −3.68308 −0.131288 −0.0656438 0.997843i \(-0.520910\pi\)
−0.0656438 + 0.997843i \(0.520910\pi\)
\(788\) 0 0
\(789\) −1.13973 −0.0405756
\(790\) 0 0
\(791\) 57.9765 2.06141
\(792\) 0 0
\(793\) −0.972037 −0.0345180
\(794\) 0 0
\(795\) −0.724289 −0.0256879
\(796\) 0 0
\(797\) 49.7204 1.76119 0.880593 0.473873i \(-0.157144\pi\)
0.880593 + 0.473873i \(0.157144\pi\)
\(798\) 0 0
\(799\) 1.74076 0.0615837
\(800\) 0 0
\(801\) 4.23125 0.149504
\(802\) 0 0
\(803\) −10.0563 −0.354880
\(804\) 0 0
\(805\) −2.27061 −0.0800284
\(806\) 0 0
\(807\) 3.10035 0.109137
\(808\) 0 0
\(809\) 50.2283 1.76593 0.882967 0.469436i \(-0.155543\pi\)
0.882967 + 0.469436i \(0.155543\pi\)
\(810\) 0 0
\(811\) 45.2989 1.59066 0.795329 0.606177i \(-0.207298\pi\)
0.795329 + 0.606177i \(0.207298\pi\)
\(812\) 0 0
\(813\) 26.9980 0.946862
\(814\) 0 0
\(815\) 1.38770 0.0486089
\(816\) 0 0
\(817\) −14.1637 −0.495526
\(818\) 0 0
\(819\) 2.32459 0.0812277
\(820\) 0 0
\(821\) 3.61746 0.126250 0.0631252 0.998006i \(-0.479893\pi\)
0.0631252 + 0.998006i \(0.479893\pi\)
\(822\) 0 0
\(823\) −32.4501 −1.13114 −0.565570 0.824700i \(-0.691344\pi\)
−0.565570 + 0.824700i \(0.691344\pi\)
\(824\) 0 0
\(825\) 23.0250 0.801628
\(826\) 0 0
\(827\) −33.1508 −1.15277 −0.576384 0.817179i \(-0.695537\pi\)
−0.576384 + 0.817179i \(0.695537\pi\)
\(828\) 0 0
\(829\) −32.9423 −1.14413 −0.572066 0.820208i \(-0.693858\pi\)
−0.572066 + 0.820208i \(0.693858\pi\)
\(830\) 0 0
\(831\) 9.24414 0.320676
\(832\) 0 0
\(833\) −6.59013 −0.228335
\(834\) 0 0
\(835\) 0.0932775 0.00322800
\(836\) 0 0
\(837\) −4.74106 −0.163875
\(838\) 0 0
\(839\) 45.4962 1.57070 0.785352 0.619050i \(-0.212482\pi\)
0.785352 + 0.619050i \(0.212482\pi\)
\(840\) 0 0
\(841\) −7.60148 −0.262120
\(842\) 0 0
\(843\) −1.83309 −0.0631350
\(844\) 0 0
\(845\) 1.17882 0.0405526
\(846\) 0 0
\(847\) −39.7048 −1.36427
\(848\) 0 0
\(849\) −2.27493 −0.0780753
\(850\) 0 0
\(851\) −35.9431 −1.23211
\(852\) 0 0
\(853\) −22.1984 −0.760059 −0.380029 0.924974i \(-0.624086\pi\)
−0.380029 + 0.924974i \(0.624086\pi\)
\(854\) 0 0
\(855\) 0.342955 0.0117288
\(856\) 0 0
\(857\) −0.277939 −0.00949421 −0.00474710 0.999989i \(-0.501511\pi\)
−0.00474710 + 0.999989i \(0.501511\pi\)
\(858\) 0 0
\(859\) −29.8460 −1.01833 −0.509165 0.860669i \(-0.670046\pi\)
−0.509165 + 0.860669i \(0.670046\pi\)
\(860\) 0 0
\(861\) 15.5305 0.529279
\(862\) 0 0
\(863\) 37.6018 1.27998 0.639990 0.768383i \(-0.278938\pi\)
0.639990 + 0.768383i \(0.278938\pi\)
\(864\) 0 0
\(865\) −0.223646 −0.00760421
\(866\) 0 0
\(867\) −16.3072 −0.553821
\(868\) 0 0
\(869\) −37.4951 −1.27193
\(870\) 0 0
\(871\) −7.68354 −0.260347
\(872\) 0 0
\(873\) 3.07885 0.104203
\(874\) 0 0
\(875\) −3.59953 −0.121686
\(876\) 0 0
\(877\) −23.8596 −0.805681 −0.402840 0.915270i \(-0.631977\pi\)
−0.402840 + 0.915270i \(0.631977\pi\)
\(878\) 0 0
\(879\) 33.0304 1.11409
\(880\) 0 0
\(881\) −9.20697 −0.310191 −0.155095 0.987900i \(-0.549568\pi\)
−0.155095 + 0.987900i \(0.549568\pi\)
\(882\) 0 0
\(883\) 19.8744 0.668827 0.334414 0.942426i \(-0.391462\pi\)
0.334414 + 0.942426i \(0.391462\pi\)
\(884\) 0 0
\(885\) −0.408946 −0.0137466
\(886\) 0 0
\(887\) 6.76687 0.227209 0.113605 0.993526i \(-0.463760\pi\)
0.113605 + 0.993526i \(0.463760\pi\)
\(888\) 0 0
\(889\) 2.86945 0.0962384
\(890\) 0 0
\(891\) −4.61303 −0.154542
\(892\) 0 0
\(893\) 7.68937 0.257315
\(894\) 0 0
\(895\) −0.862164 −0.0288190
\(896\) 0 0
\(897\) 3.79329 0.126654
\(898\) 0 0
\(899\) −21.9314 −0.731455
\(900\) 0 0
\(901\) −6.46314 −0.215319
\(902\) 0 0
\(903\) −14.8787 −0.495132
\(904\) 0 0
\(905\) −0.657016 −0.0218400
\(906\) 0 0
\(907\) −4.25694 −0.141349 −0.0706747 0.997499i \(-0.522515\pi\)
−0.0706747 + 0.997499i \(0.522515\pi\)
\(908\) 0 0
\(909\) 7.37707 0.244682
\(910\) 0 0
\(911\) −13.7840 −0.456686 −0.228343 0.973581i \(-0.573331\pi\)
−0.228343 + 0.973581i \(0.573331\pi\)
\(912\) 0 0
\(913\) −11.3827 −0.376712
\(914\) 0 0
\(915\) −0.150647 −0.00498024
\(916\) 0 0
\(917\) 82.7724 2.73339
\(918\) 0 0
\(919\) 11.3999 0.376046 0.188023 0.982165i \(-0.439792\pi\)
0.188023 + 0.982165i \(0.439792\pi\)
\(920\) 0 0
\(921\) −3.54712 −0.116881
\(922\) 0 0
\(923\) −7.92495 −0.260853
\(924\) 0 0
\(925\) −28.4650 −0.935923
\(926\) 0 0
\(927\) 5.13078 0.168517
\(928\) 0 0
\(929\) 20.6773 0.678399 0.339199 0.940715i \(-0.389844\pi\)
0.339199 + 0.940715i \(0.389844\pi\)
\(930\) 0 0
\(931\) −29.1102 −0.954049
\(932\) 0 0
\(933\) 9.22270 0.301938
\(934\) 0 0
\(935\) −0.358156 −0.0117129
\(936\) 0 0
\(937\) −45.4516 −1.48484 −0.742419 0.669936i \(-0.766322\pi\)
−0.742419 + 0.669936i \(0.766322\pi\)
\(938\) 0 0
\(939\) −13.6463 −0.445331
\(940\) 0 0
\(941\) 3.11181 0.101442 0.0507211 0.998713i \(-0.483848\pi\)
0.0507211 + 0.998713i \(0.483848\pi\)
\(942\) 0 0
\(943\) 25.3429 0.825279
\(944\) 0 0
\(945\) 0.360267 0.0117195
\(946\) 0 0
\(947\) 24.8799 0.808490 0.404245 0.914651i \(-0.367534\pi\)
0.404245 + 0.914651i \(0.367534\pi\)
\(948\) 0 0
\(949\) −1.31205 −0.0425910
\(950\) 0 0
\(951\) 13.8395 0.448776
\(952\) 0 0
\(953\) 35.2552 1.14203 0.571013 0.820941i \(-0.306551\pi\)
0.571013 + 0.820941i \(0.306551\pi\)
\(954\) 0 0
\(955\) 1.05193 0.0340398
\(956\) 0 0
\(957\) −21.3392 −0.689799
\(958\) 0 0
\(959\) −56.7343 −1.83205
\(960\) 0 0
\(961\) −8.52236 −0.274915
\(962\) 0 0
\(963\) −4.13448 −0.133232
\(964\) 0 0
\(965\) 0.0489782 0.00157666
\(966\) 0 0
\(967\) 43.5362 1.40003 0.700016 0.714128i \(-0.253176\pi\)
0.700016 + 0.714128i \(0.253176\pi\)
\(968\) 0 0
\(969\) 3.06033 0.0983121
\(970\) 0 0
\(971\) 8.29834 0.266306 0.133153 0.991095i \(-0.457490\pi\)
0.133153 + 0.991095i \(0.457490\pi\)
\(972\) 0 0
\(973\) −66.1407 −2.12037
\(974\) 0 0
\(975\) 3.00409 0.0962077
\(976\) 0 0
\(977\) 19.0429 0.609235 0.304618 0.952475i \(-0.401471\pi\)
0.304618 + 0.952475i \(0.401471\pi\)
\(978\) 0 0
\(979\) −19.5189 −0.623827
\(980\) 0 0
\(981\) −16.1790 −0.516556
\(982\) 0 0
\(983\) 6.70203 0.213761 0.106881 0.994272i \(-0.465914\pi\)
0.106881 + 0.994272i \(0.465914\pi\)
\(984\) 0 0
\(985\) 1.65431 0.0527106
\(986\) 0 0
\(987\) 8.07752 0.257110
\(988\) 0 0
\(989\) −24.2792 −0.772034
\(990\) 0 0
\(991\) −26.3822 −0.838057 −0.419029 0.907973i \(-0.637629\pi\)
−0.419029 + 0.907973i \(0.637629\pi\)
\(992\) 0 0
\(993\) 33.2468 1.05506
\(994\) 0 0
\(995\) 1.38405 0.0438775
\(996\) 0 0
\(997\) −46.3122 −1.46672 −0.733361 0.679839i \(-0.762050\pi\)
−0.733361 + 0.679839i \(0.762050\pi\)
\(998\) 0 0
\(999\) 5.70292 0.180433
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.be.1.5 11
4.3 odd 2 4008.2.a.k.1.5 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4008.2.a.k.1.5 11 4.3 odd 2
8016.2.a.be.1.5 11 1.1 even 1 trivial