Properties

Label 4016.2.a.m.1.17
Level 4016
Weight 2
Character 4016.1
Self dual yes
Analytic conductor 32.068
Analytic rank 0
Dimension 23
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4016 = 2^{4} \cdot 251 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4016.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0679214517\)
Analytic rank: \(0\)
Dimension: \(23\)
Coefficient ring index: multiple of None
Twist minimal: no (minimal twist has level 2008)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.17
Character \(\chi\) = 4016.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.28390 q^{3} +3.72143 q^{5} +0.978625 q^{7} -1.35160 q^{9} +O(q^{10})\) \(q+1.28390 q^{3} +3.72143 q^{5} +0.978625 q^{7} -1.35160 q^{9} -0.158851 q^{11} +0.124371 q^{13} +4.77795 q^{15} +0.487197 q^{17} -4.09613 q^{19} +1.25646 q^{21} +5.58572 q^{23} +8.84905 q^{25} -5.58702 q^{27} +9.50175 q^{29} +4.53723 q^{31} -0.203949 q^{33} +3.64189 q^{35} -4.70240 q^{37} +0.159680 q^{39} +8.31648 q^{41} -12.3553 q^{43} -5.02988 q^{45} -4.37024 q^{47} -6.04229 q^{49} +0.625513 q^{51} +10.3814 q^{53} -0.591152 q^{55} -5.25902 q^{57} +6.42238 q^{59} +6.11573 q^{61} -1.32271 q^{63} +0.462838 q^{65} +1.25850 q^{67} +7.17151 q^{69} +15.1543 q^{71} +15.8278 q^{73} +11.3613 q^{75} -0.155455 q^{77} +5.29643 q^{79} -3.11838 q^{81} +14.8743 q^{83} +1.81307 q^{85} +12.1993 q^{87} -5.47610 q^{89} +0.121713 q^{91} +5.82535 q^{93} -15.2435 q^{95} +5.60112 q^{97} +0.214703 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 23q - 2q^{3} + 8q^{5} - 2q^{7} + 45q^{9} + O(q^{10}) \) \( 23q - 2q^{3} + 8q^{5} - 2q^{7} + 45q^{9} - 8q^{11} + 8q^{13} - 7q^{15} + 19q^{17} + 9q^{19} + 9q^{21} - 21q^{23} + 65q^{25} - 5q^{27} + 10q^{29} + 9q^{31} + 34q^{33} - 12q^{35} + 11q^{37} + 9q^{39} + 35q^{41} + 9q^{43} + 29q^{45} - 37q^{47} + 77q^{49} + 17q^{51} + 38q^{53} + 20q^{55} + 51q^{57} - 17q^{59} - 22q^{63} + 41q^{65} - 9q^{67} + 8q^{69} - 13q^{71} + 41q^{73} - 25q^{75} + 36q^{77} + 36q^{79} + 127q^{81} - 29q^{83} + 34q^{85} - 10q^{87} + 36q^{89} + 6q^{91} + 36q^{93} - 25q^{95} + 40q^{97} - 19q^{99} + O(q^{100}) \)

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
\(3\) 1.28390 0.741260 0.370630 0.928781i \(-0.379142\pi\)
0.370630 + 0.928781i \(0.379142\pi\)
\(4\) 0 0
\(5\) 3.72143 1.66427 0.832137 0.554570i \(-0.187117\pi\)
0.832137 + 0.554570i \(0.187117\pi\)
\(6\) 0 0
\(7\) 0.978625 0.369886 0.184943 0.982749i \(-0.440790\pi\)
0.184943 + 0.982749i \(0.440790\pi\)
\(8\) 0 0
\(9\) −1.35160 −0.450533
\(10\) 0 0
\(11\) −0.158851 −0.0478953 −0.0239477 0.999713i \(-0.507624\pi\)
−0.0239477 + 0.999713i \(0.507624\pi\)
\(12\) 0 0
\(13\) 0.124371 0.0344943 0.0172471 0.999851i \(-0.494510\pi\)
0.0172471 + 0.999851i \(0.494510\pi\)
\(14\) 0 0
\(15\) 4.77795 1.23366
\(16\) 0 0
\(17\) 0.487197 0.118163 0.0590813 0.998253i \(-0.481183\pi\)
0.0590813 + 0.998253i \(0.481183\pi\)
\(18\) 0 0
\(19\) −4.09613 −0.939716 −0.469858 0.882742i \(-0.655695\pi\)
−0.469858 + 0.882742i \(0.655695\pi\)
\(20\) 0 0
\(21\) 1.25646 0.274182
\(22\) 0 0
\(23\) 5.58572 1.16470 0.582351 0.812937i \(-0.302133\pi\)
0.582351 + 0.812937i \(0.302133\pi\)
\(24\) 0 0
\(25\) 8.84905 1.76981
\(26\) 0 0
\(27\) −5.58702 −1.07522
\(28\) 0 0
\(29\) 9.50175 1.76443 0.882215 0.470847i \(-0.156052\pi\)
0.882215 + 0.470847i \(0.156052\pi\)
\(30\) 0 0
\(31\) 4.53723 0.814911 0.407455 0.913225i \(-0.366416\pi\)
0.407455 + 0.913225i \(0.366416\pi\)
\(32\) 0 0
\(33\) −0.203949 −0.0355029
\(34\) 0 0
\(35\) 3.64189 0.615591
\(36\) 0 0
\(37\) −4.70240 −0.773069 −0.386535 0.922275i \(-0.626328\pi\)
−0.386535 + 0.922275i \(0.626328\pi\)
\(38\) 0 0
\(39\) 0.159680 0.0255692
\(40\) 0 0
\(41\) 8.31648 1.29882 0.649408 0.760440i \(-0.275017\pi\)
0.649408 + 0.760440i \(0.275017\pi\)
\(42\) 0 0
\(43\) −12.3553 −1.88417 −0.942084 0.335376i \(-0.891136\pi\)
−0.942084 + 0.335376i \(0.891136\pi\)
\(44\) 0 0
\(45\) −5.02988 −0.749811
\(46\) 0 0
\(47\) −4.37024 −0.637465 −0.318732 0.947845i \(-0.603257\pi\)
−0.318732 + 0.947845i \(0.603257\pi\)
\(48\) 0 0
\(49\) −6.04229 −0.863185
\(50\) 0 0
\(51\) 0.625513 0.0875893
\(52\) 0 0
\(53\) 10.3814 1.42600 0.713000 0.701164i \(-0.247336\pi\)
0.713000 + 0.701164i \(0.247336\pi\)
\(54\) 0 0
\(55\) −0.591152 −0.0797110
\(56\) 0 0
\(57\) −5.25902 −0.696574
\(58\) 0 0
\(59\) 6.42238 0.836123 0.418062 0.908419i \(-0.362710\pi\)
0.418062 + 0.908419i \(0.362710\pi\)
\(60\) 0 0
\(61\) 6.11573 0.783039 0.391520 0.920170i \(-0.371950\pi\)
0.391520 + 0.920170i \(0.371950\pi\)
\(62\) 0 0
\(63\) −1.32271 −0.166646
\(64\) 0 0
\(65\) 0.462838 0.0574080
\(66\) 0 0
\(67\) 1.25850 0.153750 0.0768749 0.997041i \(-0.475506\pi\)
0.0768749 + 0.997041i \(0.475506\pi\)
\(68\) 0 0
\(69\) 7.17151 0.863348
\(70\) 0 0
\(71\) 15.1543 1.79849 0.899244 0.437447i \(-0.144117\pi\)
0.899244 + 0.437447i \(0.144117\pi\)
\(72\) 0 0
\(73\) 15.8278 1.85250 0.926250 0.376909i \(-0.123013\pi\)
0.926250 + 0.376909i \(0.123013\pi\)
\(74\) 0 0
\(75\) 11.3613 1.31189
\(76\) 0 0
\(77\) −0.155455 −0.0177158
\(78\) 0 0
\(79\) 5.29643 0.595895 0.297948 0.954582i \(-0.403698\pi\)
0.297948 + 0.954582i \(0.403698\pi\)
\(80\) 0 0
\(81\) −3.11838 −0.346487
\(82\) 0 0
\(83\) 14.8743 1.63267 0.816334 0.577580i \(-0.196003\pi\)
0.816334 + 0.577580i \(0.196003\pi\)
\(84\) 0 0
\(85\) 1.81307 0.196655
\(86\) 0 0
\(87\) 12.1993 1.30790
\(88\) 0 0
\(89\) −5.47610 −0.580465 −0.290233 0.956956i \(-0.593733\pi\)
−0.290233 + 0.956956i \(0.593733\pi\)
\(90\) 0 0
\(91\) 0.121713 0.0127589
\(92\) 0 0
\(93\) 5.82535 0.604061
\(94\) 0 0
\(95\) −15.2435 −1.56395
\(96\) 0 0
\(97\) 5.60112 0.568708 0.284354 0.958719i \(-0.408221\pi\)
0.284354 + 0.958719i \(0.408221\pi\)
\(98\) 0 0
\(99\) 0.214703 0.0215784
\(100\) 0 0
\(101\) 13.3827 1.33163 0.665815 0.746117i \(-0.268084\pi\)
0.665815 + 0.746117i \(0.268084\pi\)
\(102\) 0 0
\(103\) −12.1928 −1.20139 −0.600695 0.799478i \(-0.705109\pi\)
−0.600695 + 0.799478i \(0.705109\pi\)
\(104\) 0 0
\(105\) 4.67582 0.456313
\(106\) 0 0
\(107\) −18.0707 −1.74696 −0.873481 0.486859i \(-0.838143\pi\)
−0.873481 + 0.486859i \(0.838143\pi\)
\(108\) 0 0
\(109\) −0.754731 −0.0722901 −0.0361451 0.999347i \(-0.511508\pi\)
−0.0361451 + 0.999347i \(0.511508\pi\)
\(110\) 0 0
\(111\) −6.03741 −0.573045
\(112\) 0 0
\(113\) −14.2548 −1.34097 −0.670487 0.741921i \(-0.733915\pi\)
−0.670487 + 0.741921i \(0.733915\pi\)
\(114\) 0 0
\(115\) 20.7869 1.93838
\(116\) 0 0
\(117\) −0.168100 −0.0155408
\(118\) 0 0
\(119\) 0.476783 0.0437067
\(120\) 0 0
\(121\) −10.9748 −0.997706
\(122\) 0 0
\(123\) 10.6775 0.962761
\(124\) 0 0
\(125\) 14.3240 1.28118
\(126\) 0 0
\(127\) −16.6350 −1.47612 −0.738061 0.674735i \(-0.764258\pi\)
−0.738061 + 0.674735i \(0.764258\pi\)
\(128\) 0 0
\(129\) −15.8630 −1.39666
\(130\) 0 0
\(131\) −15.1564 −1.32422 −0.662110 0.749407i \(-0.730339\pi\)
−0.662110 + 0.749407i \(0.730339\pi\)
\(132\) 0 0
\(133\) −4.00857 −0.347587
\(134\) 0 0
\(135\) −20.7917 −1.78947
\(136\) 0 0
\(137\) 3.64085 0.311058 0.155529 0.987831i \(-0.450292\pi\)
0.155529 + 0.987831i \(0.450292\pi\)
\(138\) 0 0
\(139\) −8.66321 −0.734804 −0.367402 0.930062i \(-0.619753\pi\)
−0.367402 + 0.930062i \(0.619753\pi\)
\(140\) 0 0
\(141\) −5.61095 −0.472527
\(142\) 0 0
\(143\) −0.0197564 −0.00165211
\(144\) 0 0
\(145\) 35.3601 2.93650
\(146\) 0 0
\(147\) −7.75770 −0.639845
\(148\) 0 0
\(149\) 11.5090 0.942855 0.471428 0.881905i \(-0.343739\pi\)
0.471428 + 0.881905i \(0.343739\pi\)
\(150\) 0 0
\(151\) 4.44178 0.361467 0.180734 0.983532i \(-0.442153\pi\)
0.180734 + 0.983532i \(0.442153\pi\)
\(152\) 0 0
\(153\) −0.658495 −0.0532362
\(154\) 0 0
\(155\) 16.8850 1.35624
\(156\) 0 0
\(157\) −6.60217 −0.526910 −0.263455 0.964672i \(-0.584862\pi\)
−0.263455 + 0.964672i \(0.584862\pi\)
\(158\) 0 0
\(159\) 13.3287 1.05704
\(160\) 0 0
\(161\) 5.46632 0.430807
\(162\) 0 0
\(163\) −12.2503 −0.959520 −0.479760 0.877400i \(-0.659276\pi\)
−0.479760 + 0.877400i \(0.659276\pi\)
\(164\) 0 0
\(165\) −0.758981 −0.0590866
\(166\) 0 0
\(167\) −20.1260 −1.55740 −0.778698 0.627399i \(-0.784120\pi\)
−0.778698 + 0.627399i \(0.784120\pi\)
\(168\) 0 0
\(169\) −12.9845 −0.998810
\(170\) 0 0
\(171\) 5.53632 0.423373
\(172\) 0 0
\(173\) −20.9139 −1.59006 −0.795029 0.606572i \(-0.792544\pi\)
−0.795029 + 0.606572i \(0.792544\pi\)
\(174\) 0 0
\(175\) 8.65990 0.654627
\(176\) 0 0
\(177\) 8.24570 0.619785
\(178\) 0 0
\(179\) 9.33184 0.697495 0.348747 0.937217i \(-0.386607\pi\)
0.348747 + 0.937217i \(0.386607\pi\)
\(180\) 0 0
\(181\) 4.39638 0.326780 0.163390 0.986562i \(-0.447757\pi\)
0.163390 + 0.986562i \(0.447757\pi\)
\(182\) 0 0
\(183\) 7.85199 0.580436
\(184\) 0 0
\(185\) −17.4996 −1.28660
\(186\) 0 0
\(187\) −0.0773917 −0.00565944
\(188\) 0 0
\(189\) −5.46760 −0.397709
\(190\) 0 0
\(191\) 4.38279 0.317127 0.158564 0.987349i \(-0.449314\pi\)
0.158564 + 0.987349i \(0.449314\pi\)
\(192\) 0 0
\(193\) 25.1081 1.80732 0.903660 0.428251i \(-0.140870\pi\)
0.903660 + 0.428251i \(0.140870\pi\)
\(194\) 0 0
\(195\) 0.594238 0.0425543
\(196\) 0 0
\(197\) 8.19301 0.583728 0.291864 0.956460i \(-0.405725\pi\)
0.291864 + 0.956460i \(0.405725\pi\)
\(198\) 0 0
\(199\) −17.3067 −1.22684 −0.613420 0.789757i \(-0.710207\pi\)
−0.613420 + 0.789757i \(0.710207\pi\)
\(200\) 0 0
\(201\) 1.61578 0.113969
\(202\) 0 0
\(203\) 9.29865 0.652637
\(204\) 0 0
\(205\) 30.9492 2.16159
\(206\) 0 0
\(207\) −7.54965 −0.524737
\(208\) 0 0
\(209\) 0.650673 0.0450080
\(210\) 0 0
\(211\) −7.72391 −0.531736 −0.265868 0.964009i \(-0.585659\pi\)
−0.265868 + 0.964009i \(0.585659\pi\)
\(212\) 0 0
\(213\) 19.4567 1.33315
\(214\) 0 0
\(215\) −45.9795 −3.13577
\(216\) 0 0
\(217\) 4.44025 0.301424
\(218\) 0 0
\(219\) 20.3213 1.37319
\(220\) 0 0
\(221\) 0.0605932 0.00407594
\(222\) 0 0
\(223\) −8.79546 −0.588987 −0.294494 0.955653i \(-0.595151\pi\)
−0.294494 + 0.955653i \(0.595151\pi\)
\(224\) 0 0
\(225\) −11.9604 −0.797358
\(226\) 0 0
\(227\) 2.29104 0.152061 0.0760307 0.997105i \(-0.475775\pi\)
0.0760307 + 0.997105i \(0.475775\pi\)
\(228\) 0 0
\(229\) 6.59267 0.435656 0.217828 0.975987i \(-0.430103\pi\)
0.217828 + 0.975987i \(0.430103\pi\)
\(230\) 0 0
\(231\) −0.199589 −0.0131320
\(232\) 0 0
\(233\) −28.8484 −1.88992 −0.944962 0.327180i \(-0.893902\pi\)
−0.944962 + 0.327180i \(0.893902\pi\)
\(234\) 0 0
\(235\) −16.2635 −1.06092
\(236\) 0 0
\(237\) 6.80010 0.441714
\(238\) 0 0
\(239\) 6.03122 0.390127 0.195064 0.980791i \(-0.437509\pi\)
0.195064 + 0.980791i \(0.437509\pi\)
\(240\) 0 0
\(241\) −18.5017 −1.19180 −0.595899 0.803059i \(-0.703204\pi\)
−0.595899 + 0.803059i \(0.703204\pi\)
\(242\) 0 0
\(243\) 12.7574 0.818386
\(244\) 0 0
\(245\) −22.4860 −1.43658
\(246\) 0 0
\(247\) −0.509439 −0.0324148
\(248\) 0 0
\(249\) 19.0971 1.21023
\(250\) 0 0
\(251\) 1.00000 0.0631194
\(252\) 0 0
\(253\) −0.887296 −0.0557838
\(254\) 0 0
\(255\) 2.32780 0.145773
\(256\) 0 0
\(257\) 18.5580 1.15761 0.578807 0.815465i \(-0.303519\pi\)
0.578807 + 0.815465i \(0.303519\pi\)
\(258\) 0 0
\(259\) −4.60188 −0.285947
\(260\) 0 0
\(261\) −12.8426 −0.794934
\(262\) 0 0
\(263\) 13.4324 0.828276 0.414138 0.910214i \(-0.364083\pi\)
0.414138 + 0.910214i \(0.364083\pi\)
\(264\) 0 0
\(265\) 38.6338 2.37326
\(266\) 0 0
\(267\) −7.03077 −0.430276
\(268\) 0 0
\(269\) −19.5121 −1.18968 −0.594838 0.803846i \(-0.702784\pi\)
−0.594838 + 0.803846i \(0.702784\pi\)
\(270\) 0 0
\(271\) 19.8337 1.20481 0.602405 0.798191i \(-0.294209\pi\)
0.602405 + 0.798191i \(0.294209\pi\)
\(272\) 0 0
\(273\) 0.156267 0.00945770
\(274\) 0 0
\(275\) −1.40568 −0.0847656
\(276\) 0 0
\(277\) −24.3978 −1.46592 −0.732960 0.680271i \(-0.761862\pi\)
−0.732960 + 0.680271i \(0.761862\pi\)
\(278\) 0 0
\(279\) −6.13252 −0.367144
\(280\) 0 0
\(281\) 16.6184 0.991370 0.495685 0.868502i \(-0.334917\pi\)
0.495685 + 0.868502i \(0.334917\pi\)
\(282\) 0 0
\(283\) −1.11551 −0.0663101 −0.0331550 0.999450i \(-0.510556\pi\)
−0.0331550 + 0.999450i \(0.510556\pi\)
\(284\) 0 0
\(285\) −19.5711 −1.15929
\(286\) 0 0
\(287\) 8.13872 0.480413
\(288\) 0 0
\(289\) −16.7626 −0.986038
\(290\) 0 0
\(291\) 7.19128 0.421560
\(292\) 0 0
\(293\) −3.13480 −0.183137 −0.0915685 0.995799i \(-0.529188\pi\)
−0.0915685 + 0.995799i \(0.529188\pi\)
\(294\) 0 0
\(295\) 23.9005 1.39154
\(296\) 0 0
\(297\) 0.887503 0.0514981
\(298\) 0 0
\(299\) 0.694701 0.0401756
\(300\) 0 0
\(301\) −12.0912 −0.696927
\(302\) 0 0
\(303\) 17.1821 0.987084
\(304\) 0 0
\(305\) 22.7593 1.30319
\(306\) 0 0
\(307\) −7.08635 −0.404439 −0.202220 0.979340i \(-0.564816\pi\)
−0.202220 + 0.979340i \(0.564816\pi\)
\(308\) 0 0
\(309\) −15.6543 −0.890543
\(310\) 0 0
\(311\) −6.79534 −0.385328 −0.192664 0.981265i \(-0.561713\pi\)
−0.192664 + 0.981265i \(0.561713\pi\)
\(312\) 0 0
\(313\) −21.1021 −1.19276 −0.596381 0.802701i \(-0.703395\pi\)
−0.596381 + 0.802701i \(0.703395\pi\)
\(314\) 0 0
\(315\) −4.92237 −0.277344
\(316\) 0 0
\(317\) −17.3791 −0.976106 −0.488053 0.872814i \(-0.662293\pi\)
−0.488053 + 0.872814i \(0.662293\pi\)
\(318\) 0 0
\(319\) −1.50936 −0.0845079
\(320\) 0 0
\(321\) −23.2010 −1.29495
\(322\) 0 0
\(323\) −1.99562 −0.111039
\(324\) 0 0
\(325\) 1.10056 0.0610483
\(326\) 0 0
\(327\) −0.968999 −0.0535858
\(328\) 0 0
\(329\) −4.27682 −0.235789
\(330\) 0 0
\(331\) 9.36302 0.514638 0.257319 0.966326i \(-0.417161\pi\)
0.257319 + 0.966326i \(0.417161\pi\)
\(332\) 0 0
\(333\) 6.35576 0.348293
\(334\) 0 0
\(335\) 4.68341 0.255882
\(336\) 0 0
\(337\) −6.97394 −0.379895 −0.189947 0.981794i \(-0.560832\pi\)
−0.189947 + 0.981794i \(0.560832\pi\)
\(338\) 0 0
\(339\) −18.3017 −0.994011
\(340\) 0 0
\(341\) −0.720743 −0.0390304
\(342\) 0 0
\(343\) −12.7635 −0.689165
\(344\) 0 0
\(345\) 26.6883 1.43685
\(346\) 0 0
\(347\) 20.8163 1.11748 0.558740 0.829343i \(-0.311285\pi\)
0.558740 + 0.829343i \(0.311285\pi\)
\(348\) 0 0
\(349\) −6.73006 −0.360252 −0.180126 0.983644i \(-0.557651\pi\)
−0.180126 + 0.983644i \(0.557651\pi\)
\(350\) 0 0
\(351\) −0.694863 −0.0370890
\(352\) 0 0
\(353\) 22.9404 1.22099 0.610496 0.792019i \(-0.290970\pi\)
0.610496 + 0.792019i \(0.290970\pi\)
\(354\) 0 0
\(355\) 56.3958 2.99318
\(356\) 0 0
\(357\) 0.612143 0.0323980
\(358\) 0 0
\(359\) 9.60134 0.506739 0.253370 0.967370i \(-0.418461\pi\)
0.253370 + 0.967370i \(0.418461\pi\)
\(360\) 0 0
\(361\) −2.22175 −0.116934
\(362\) 0 0
\(363\) −14.0905 −0.739560
\(364\) 0 0
\(365\) 58.9020 3.08307
\(366\) 0 0
\(367\) 20.2307 1.05604 0.528018 0.849233i \(-0.322935\pi\)
0.528018 + 0.849233i \(0.322935\pi\)
\(368\) 0 0
\(369\) −11.2406 −0.585160
\(370\) 0 0
\(371\) 10.1595 0.527457
\(372\) 0 0
\(373\) −22.1957 −1.14925 −0.574625 0.818417i \(-0.694852\pi\)
−0.574625 + 0.818417i \(0.694852\pi\)
\(374\) 0 0
\(375\) 18.3906 0.949684
\(376\) 0 0
\(377\) 1.18174 0.0608628
\(378\) 0 0
\(379\) 6.31846 0.324557 0.162279 0.986745i \(-0.448116\pi\)
0.162279 + 0.986745i \(0.448116\pi\)
\(380\) 0 0
\(381\) −21.3577 −1.09419
\(382\) 0 0
\(383\) −3.99897 −0.204338 −0.102169 0.994767i \(-0.532578\pi\)
−0.102169 + 0.994767i \(0.532578\pi\)
\(384\) 0 0
\(385\) −0.578517 −0.0294839
\(386\) 0 0
\(387\) 16.6994 0.848880
\(388\) 0 0
\(389\) 36.6251 1.85697 0.928484 0.371372i \(-0.121113\pi\)
0.928484 + 0.371372i \(0.121113\pi\)
\(390\) 0 0
\(391\) 2.72135 0.137624
\(392\) 0 0
\(393\) −19.4593 −0.981592
\(394\) 0 0
\(395\) 19.7103 0.991734
\(396\) 0 0
\(397\) −21.1127 −1.05962 −0.529808 0.848118i \(-0.677736\pi\)
−0.529808 + 0.848118i \(0.677736\pi\)
\(398\) 0 0
\(399\) −5.14661 −0.257653
\(400\) 0 0
\(401\) −31.0701 −1.55156 −0.775782 0.631001i \(-0.782645\pi\)
−0.775782 + 0.631001i \(0.782645\pi\)
\(402\) 0 0
\(403\) 0.564300 0.0281098
\(404\) 0 0
\(405\) −11.6048 −0.576649
\(406\) 0 0
\(407\) 0.746979 0.0370264
\(408\) 0 0
\(409\) −1.73077 −0.0855809 −0.0427905 0.999084i \(-0.513625\pi\)
−0.0427905 + 0.999084i \(0.513625\pi\)
\(410\) 0 0
\(411\) 4.67448 0.230575
\(412\) 0 0
\(413\) 6.28511 0.309270
\(414\) 0 0
\(415\) 55.3538 2.71721
\(416\) 0 0
\(417\) −11.1227 −0.544681
\(418\) 0 0
\(419\) −11.4727 −0.560480 −0.280240 0.959930i \(-0.590414\pi\)
−0.280240 + 0.959930i \(0.590414\pi\)
\(420\) 0 0
\(421\) −30.9162 −1.50676 −0.753382 0.657583i \(-0.771579\pi\)
−0.753382 + 0.657583i \(0.771579\pi\)
\(422\) 0 0
\(423\) 5.90681 0.287199
\(424\) 0 0
\(425\) 4.31123 0.209125
\(426\) 0 0
\(427\) 5.98501 0.289635
\(428\) 0 0
\(429\) −0.0253653 −0.00122465
\(430\) 0 0
\(431\) 2.84348 0.136966 0.0684829 0.997652i \(-0.478184\pi\)
0.0684829 + 0.997652i \(0.478184\pi\)
\(432\) 0 0
\(433\) −9.87776 −0.474695 −0.237347 0.971425i \(-0.576278\pi\)
−0.237347 + 0.971425i \(0.576278\pi\)
\(434\) 0 0
\(435\) 45.3988 2.17671
\(436\) 0 0
\(437\) −22.8798 −1.09449
\(438\) 0 0
\(439\) −21.7174 −1.03652 −0.518259 0.855224i \(-0.673420\pi\)
−0.518259 + 0.855224i \(0.673420\pi\)
\(440\) 0 0
\(441\) 8.16676 0.388893
\(442\) 0 0
\(443\) 5.12886 0.243679 0.121840 0.992550i \(-0.461121\pi\)
0.121840 + 0.992550i \(0.461121\pi\)
\(444\) 0 0
\(445\) −20.3789 −0.966054
\(446\) 0 0
\(447\) 14.7764 0.698901
\(448\) 0 0
\(449\) 20.9927 0.990707 0.495354 0.868691i \(-0.335039\pi\)
0.495354 + 0.868691i \(0.335039\pi\)
\(450\) 0 0
\(451\) −1.32108 −0.0622072
\(452\) 0 0
\(453\) 5.70281 0.267941
\(454\) 0 0
\(455\) 0.452945 0.0212344
\(456\) 0 0
\(457\) 16.0836 0.752357 0.376178 0.926547i \(-0.377238\pi\)
0.376178 + 0.926547i \(0.377238\pi\)
\(458\) 0 0
\(459\) −2.72198 −0.127051
\(460\) 0 0
\(461\) −15.4988 −0.721850 −0.360925 0.932595i \(-0.617539\pi\)
−0.360925 + 0.932595i \(0.617539\pi\)
\(462\) 0 0
\(463\) 39.1923 1.82142 0.910710 0.413046i \(-0.135535\pi\)
0.910710 + 0.413046i \(0.135535\pi\)
\(464\) 0 0
\(465\) 21.6787 1.00532
\(466\) 0 0
\(467\) 12.4339 0.575372 0.287686 0.957725i \(-0.407114\pi\)
0.287686 + 0.957725i \(0.407114\pi\)
\(468\) 0 0
\(469\) 1.23160 0.0568698
\(470\) 0 0
\(471\) −8.47653 −0.390578
\(472\) 0 0
\(473\) 1.96265 0.0902428
\(474\) 0 0
\(475\) −36.2468 −1.66312
\(476\) 0 0
\(477\) −14.0315 −0.642460
\(478\) 0 0
\(479\) −1.77090 −0.0809143 −0.0404571 0.999181i \(-0.512881\pi\)
−0.0404571 + 0.999181i \(0.512881\pi\)
\(480\) 0 0
\(481\) −0.584841 −0.0266665
\(482\) 0 0
\(483\) 7.01822 0.319340
\(484\) 0 0
\(485\) 20.8442 0.946486
\(486\) 0 0
\(487\) 6.05297 0.274286 0.137143 0.990551i \(-0.456208\pi\)
0.137143 + 0.990551i \(0.456208\pi\)
\(488\) 0 0
\(489\) −15.7282 −0.711254
\(490\) 0 0
\(491\) 0.330789 0.0149283 0.00746415 0.999972i \(-0.497624\pi\)
0.00746415 + 0.999972i \(0.497624\pi\)
\(492\) 0 0
\(493\) 4.62922 0.208490
\(494\) 0 0
\(495\) 0.799001 0.0359124
\(496\) 0 0
\(497\) 14.8304 0.665235
\(498\) 0 0
\(499\) −9.18444 −0.411152 −0.205576 0.978641i \(-0.565907\pi\)
−0.205576 + 0.978641i \(0.565907\pi\)
\(500\) 0 0
\(501\) −25.8398 −1.15444
\(502\) 0 0
\(503\) −13.1410 −0.585930 −0.292965 0.956123i \(-0.594642\pi\)
−0.292965 + 0.956123i \(0.594642\pi\)
\(504\) 0 0
\(505\) 49.8028 2.21620
\(506\) 0 0
\(507\) −16.6708 −0.740378
\(508\) 0 0
\(509\) 12.2484 0.542902 0.271451 0.962452i \(-0.412497\pi\)
0.271451 + 0.962452i \(0.412497\pi\)
\(510\) 0 0
\(511\) 15.4895 0.685213
\(512\) 0 0
\(513\) 22.8851 1.01040
\(514\) 0 0
\(515\) −45.3746 −1.99944
\(516\) 0 0
\(517\) 0.694216 0.0305316
\(518\) 0 0
\(519\) −26.8514 −1.17865
\(520\) 0 0
\(521\) −2.33003 −0.102080 −0.0510402 0.998697i \(-0.516254\pi\)
−0.0510402 + 0.998697i \(0.516254\pi\)
\(522\) 0 0
\(523\) 43.3538 1.89573 0.947864 0.318675i \(-0.103238\pi\)
0.947864 + 0.318675i \(0.103238\pi\)
\(524\) 0 0
\(525\) 11.1185 0.485249
\(526\) 0 0
\(527\) 2.21053 0.0962920
\(528\) 0 0
\(529\) 8.20024 0.356532
\(530\) 0 0
\(531\) −8.68049 −0.376701
\(532\) 0 0
\(533\) 1.03433 0.0448017
\(534\) 0 0
\(535\) −67.2489 −2.90742
\(536\) 0 0
\(537\) 11.9812 0.517025
\(538\) 0 0
\(539\) 0.959823 0.0413425
\(540\) 0 0
\(541\) −11.7559 −0.505427 −0.252713 0.967541i \(-0.581323\pi\)
−0.252713 + 0.967541i \(0.581323\pi\)
\(542\) 0 0
\(543\) 5.64451 0.242229
\(544\) 0 0
\(545\) −2.80868 −0.120311
\(546\) 0 0
\(547\) −20.7563 −0.887477 −0.443738 0.896156i \(-0.646348\pi\)
−0.443738 + 0.896156i \(0.646348\pi\)
\(548\) 0 0
\(549\) −8.26602 −0.352785
\(550\) 0 0
\(551\) −38.9203 −1.65806
\(552\) 0 0
\(553\) 5.18323 0.220413
\(554\) 0 0
\(555\) −22.4678 −0.953705
\(556\) 0 0
\(557\) 25.0496 1.06139 0.530694 0.847564i \(-0.321931\pi\)
0.530694 + 0.847564i \(0.321931\pi\)
\(558\) 0 0
\(559\) −1.53664 −0.0649930
\(560\) 0 0
\(561\) −0.0993632 −0.00419512
\(562\) 0 0
\(563\) −25.3564 −1.06865 −0.534323 0.845280i \(-0.679433\pi\)
−0.534323 + 0.845280i \(0.679433\pi\)
\(564\) 0 0
\(565\) −53.0481 −2.23175
\(566\) 0 0
\(567\) −3.05173 −0.128160
\(568\) 0 0
\(569\) −17.7495 −0.744100 −0.372050 0.928213i \(-0.621345\pi\)
−0.372050 + 0.928213i \(0.621345\pi\)
\(570\) 0 0
\(571\) 42.6572 1.78515 0.892574 0.450900i \(-0.148897\pi\)
0.892574 + 0.450900i \(0.148897\pi\)
\(572\) 0 0
\(573\) 5.62706 0.235074
\(574\) 0 0
\(575\) 49.4283 2.06130
\(576\) 0 0
\(577\) 35.3567 1.47192 0.735960 0.677025i \(-0.236731\pi\)
0.735960 + 0.677025i \(0.236731\pi\)
\(578\) 0 0
\(579\) 32.2363 1.33969
\(580\) 0 0
\(581\) 14.5564 0.603901
\(582\) 0 0
\(583\) −1.64910 −0.0682987
\(584\) 0 0
\(585\) −0.625571 −0.0258642
\(586\) 0 0
\(587\) −38.7462 −1.59923 −0.799614 0.600514i \(-0.794963\pi\)
−0.799614 + 0.600514i \(0.794963\pi\)
\(588\) 0 0
\(589\) −18.5851 −0.765785
\(590\) 0 0
\(591\) 10.5190 0.432694
\(592\) 0 0
\(593\) 35.0722 1.44024 0.720121 0.693849i \(-0.244086\pi\)
0.720121 + 0.693849i \(0.244086\pi\)
\(594\) 0 0
\(595\) 1.77432 0.0727399
\(596\) 0 0
\(597\) −22.2201 −0.909408
\(598\) 0 0
\(599\) −18.9351 −0.773667 −0.386834 0.922149i \(-0.626431\pi\)
−0.386834 + 0.922149i \(0.626431\pi\)
\(600\) 0 0
\(601\) 20.2368 0.825476 0.412738 0.910850i \(-0.364572\pi\)
0.412738 + 0.910850i \(0.364572\pi\)
\(602\) 0 0
\(603\) −1.70098 −0.0692694
\(604\) 0 0
\(605\) −40.8418 −1.66046
\(606\) 0 0
\(607\) 11.4054 0.462931 0.231466 0.972843i \(-0.425648\pi\)
0.231466 + 0.972843i \(0.425648\pi\)
\(608\) 0 0
\(609\) 11.9385 0.483774
\(610\) 0 0
\(611\) −0.543530 −0.0219889
\(612\) 0 0
\(613\) 10.5342 0.425473 0.212737 0.977110i \(-0.431762\pi\)
0.212737 + 0.977110i \(0.431762\pi\)
\(614\) 0 0
\(615\) 39.7357 1.60230
\(616\) 0 0
\(617\) 8.84836 0.356222 0.178111 0.984010i \(-0.443001\pi\)
0.178111 + 0.984010i \(0.443001\pi\)
\(618\) 0 0
\(619\) −27.6532 −1.11148 −0.555738 0.831358i \(-0.687564\pi\)
−0.555738 + 0.831358i \(0.687564\pi\)
\(620\) 0 0
\(621\) −31.2075 −1.25231
\(622\) 0 0
\(623\) −5.35905 −0.214706
\(624\) 0 0
\(625\) 9.06044 0.362417
\(626\) 0 0
\(627\) 0.835399 0.0333626
\(628\) 0 0
\(629\) −2.29099 −0.0913479
\(630\) 0 0
\(631\) −30.2853 −1.20564 −0.602820 0.797877i \(-0.705956\pi\)
−0.602820 + 0.797877i \(0.705956\pi\)
\(632\) 0 0
\(633\) −9.91673 −0.394155
\(634\) 0 0
\(635\) −61.9062 −2.45667
\(636\) 0 0
\(637\) −0.751486 −0.0297749
\(638\) 0 0
\(639\) −20.4826 −0.810279
\(640\) 0 0
\(641\) 6.48028 0.255956 0.127978 0.991777i \(-0.459151\pi\)
0.127978 + 0.991777i \(0.459151\pi\)
\(642\) 0 0
\(643\) 31.9557 1.26021 0.630105 0.776510i \(-0.283012\pi\)
0.630105 + 0.776510i \(0.283012\pi\)
\(644\) 0 0
\(645\) −59.0331 −2.32442
\(646\) 0 0
\(647\) −28.7890 −1.13181 −0.565907 0.824469i \(-0.691474\pi\)
−0.565907 + 0.824469i \(0.691474\pi\)
\(648\) 0 0
\(649\) −1.02020 −0.0400464
\(650\) 0 0
\(651\) 5.70084 0.223434
\(652\) 0 0
\(653\) −35.7344 −1.39840 −0.699198 0.714928i \(-0.746459\pi\)
−0.699198 + 0.714928i \(0.746459\pi\)
\(654\) 0 0
\(655\) −56.4035 −2.20387
\(656\) 0 0
\(657\) −21.3928 −0.834613
\(658\) 0 0
\(659\) −8.06551 −0.314188 −0.157094 0.987584i \(-0.550213\pi\)
−0.157094 + 0.987584i \(0.550213\pi\)
\(660\) 0 0
\(661\) −42.6595 −1.65926 −0.829632 0.558311i \(-0.811450\pi\)
−0.829632 + 0.558311i \(0.811450\pi\)
\(662\) 0 0
\(663\) 0.0777956 0.00302133
\(664\) 0 0
\(665\) −14.9176 −0.578481
\(666\) 0 0
\(667\) 53.0741 2.05504
\(668\) 0 0
\(669\) −11.2925 −0.436593
\(670\) 0 0
\(671\) −0.971489 −0.0375039
\(672\) 0 0
\(673\) −21.0702 −0.812196 −0.406098 0.913830i \(-0.633111\pi\)
−0.406098 + 0.913830i \(0.633111\pi\)
\(674\) 0 0
\(675\) −49.4398 −1.90294
\(676\) 0 0
\(677\) 40.8903 1.57154 0.785770 0.618519i \(-0.212267\pi\)
0.785770 + 0.618519i \(0.212267\pi\)
\(678\) 0 0
\(679\) 5.48140 0.210357
\(680\) 0 0
\(681\) 2.94146 0.112717
\(682\) 0 0
\(683\) −2.36369 −0.0904442 −0.0452221 0.998977i \(-0.514400\pi\)
−0.0452221 + 0.998977i \(0.514400\pi\)
\(684\) 0 0
\(685\) 13.5492 0.517687
\(686\) 0 0
\(687\) 8.46433 0.322934
\(688\) 0 0
\(689\) 1.29115 0.0491889
\(690\) 0 0
\(691\) −9.94822 −0.378448 −0.189224 0.981934i \(-0.560597\pi\)
−0.189224 + 0.981934i \(0.560597\pi\)
\(692\) 0 0
\(693\) 0.210113 0.00798155
\(694\) 0 0
\(695\) −32.2395 −1.22292
\(696\) 0 0
\(697\) 4.05177 0.153472
\(698\) 0 0
\(699\) −37.0385 −1.40093
\(700\) 0 0
\(701\) −20.3292 −0.767825 −0.383912 0.923370i \(-0.625424\pi\)
−0.383912 + 0.923370i \(0.625424\pi\)
\(702\) 0 0
\(703\) 19.2616 0.726465
\(704\) 0 0
\(705\) −20.8808 −0.786415
\(706\) 0 0
\(707\) 13.0967 0.492551
\(708\) 0 0
\(709\) −15.5399 −0.583612 −0.291806 0.956478i \(-0.594256\pi\)
−0.291806 + 0.956478i \(0.594256\pi\)
\(710\) 0 0
\(711\) −7.15866 −0.268471
\(712\) 0 0
\(713\) 25.3437 0.949129
\(714\) 0 0
\(715\) −0.0735222 −0.00274957
\(716\) 0 0
\(717\) 7.74349 0.289186
\(718\) 0 0
\(719\) 40.0743 1.49452 0.747259 0.664532i \(-0.231369\pi\)
0.747259 + 0.664532i \(0.231369\pi\)
\(720\) 0 0
\(721\) −11.9322 −0.444377
\(722\) 0 0
\(723\) −23.7543 −0.883433
\(724\) 0 0
\(725\) 84.0814 3.12271
\(726\) 0 0
\(727\) −27.3048 −1.01268 −0.506340 0.862334i \(-0.669002\pi\)
−0.506340 + 0.862334i \(0.669002\pi\)
\(728\) 0 0
\(729\) 25.7343 0.953124
\(730\) 0 0
\(731\) −6.01948 −0.222638
\(732\) 0 0
\(733\) −11.4085 −0.421382 −0.210691 0.977553i \(-0.567571\pi\)
−0.210691 + 0.977553i \(0.567571\pi\)
\(734\) 0 0
\(735\) −28.8698 −1.06488
\(736\) 0 0
\(737\) −0.199913 −0.00736389
\(738\) 0 0
\(739\) 44.5501 1.63880 0.819402 0.573220i \(-0.194306\pi\)
0.819402 + 0.573220i \(0.194306\pi\)
\(740\) 0 0
\(741\) −0.654069 −0.0240278
\(742\) 0 0
\(743\) 42.3103 1.55221 0.776106 0.630602i \(-0.217192\pi\)
0.776106 + 0.630602i \(0.217192\pi\)
\(744\) 0 0
\(745\) 42.8300 1.56917
\(746\) 0 0
\(747\) −20.1041 −0.735571
\(748\) 0 0
\(749\) −17.6845 −0.646176
\(750\) 0 0
\(751\) 25.2451 0.921206 0.460603 0.887606i \(-0.347633\pi\)
0.460603 + 0.887606i \(0.347633\pi\)
\(752\) 0 0
\(753\) 1.28390 0.0467879
\(754\) 0 0
\(755\) 16.5298 0.601581
\(756\) 0 0
\(757\) 8.42631 0.306260 0.153130 0.988206i \(-0.451065\pi\)
0.153130 + 0.988206i \(0.451065\pi\)
\(758\) 0 0
\(759\) −1.13920 −0.0413503
\(760\) 0 0
\(761\) −18.7686 −0.680361 −0.340180 0.940360i \(-0.610488\pi\)
−0.340180 + 0.940360i \(0.610488\pi\)
\(762\) 0 0
\(763\) −0.738599 −0.0267391
\(764\) 0 0
\(765\) −2.45055 −0.0885997
\(766\) 0 0
\(767\) 0.798758 0.0288415
\(768\) 0 0
\(769\) −38.5958 −1.39180 −0.695900 0.718138i \(-0.744995\pi\)
−0.695900 + 0.718138i \(0.744995\pi\)
\(770\) 0 0
\(771\) 23.8266 0.858093
\(772\) 0 0
\(773\) −7.12987 −0.256444 −0.128222 0.991746i \(-0.540927\pi\)
−0.128222 + 0.991746i \(0.540927\pi\)
\(774\) 0 0
\(775\) 40.1502 1.44224
\(776\) 0 0
\(777\) −5.90836 −0.211961
\(778\) 0 0
\(779\) −34.0654 −1.22052
\(780\) 0 0
\(781\) −2.40728 −0.0861392
\(782\) 0 0
\(783\) −53.0865 −1.89715
\(784\) 0 0
\(785\) −24.5695 −0.876923
\(786\) 0 0
\(787\) 7.16095 0.255260 0.127630 0.991822i \(-0.459263\pi\)
0.127630 + 0.991822i \(0.459263\pi\)
\(788\) 0 0
\(789\) 17.2458 0.613968
\(790\) 0 0
\(791\) −13.9501 −0.496007
\(792\) 0 0
\(793\) 0.760619 0.0270104
\(794\) 0 0
\(795\) 49.6020 1.75920
\(796\) 0 0
\(797\) −7.57184 −0.268208 −0.134104 0.990967i \(-0.542816\pi\)
−0.134104 + 0.990967i \(0.542816\pi\)
\(798\) 0 0
\(799\) −2.12917 −0.0753245
\(800\) 0 0
\(801\) 7.40149 0.261519
\(802\) 0 0
\(803\) −2.51425 −0.0887261
\(804\) 0 0
\(805\) 20.3426 0.716981
\(806\) 0 0
\(807\) −25.0516 −0.881859
\(808\) 0 0
\(809\) 41.3021 1.45211 0.726053 0.687639i \(-0.241353\pi\)
0.726053 + 0.687639i \(0.241353\pi\)
\(810\) 0 0
\(811\) 51.5573 1.81042 0.905211 0.424962i \(-0.139713\pi\)
0.905211 + 0.424962i \(0.139713\pi\)
\(812\) 0 0
\(813\) 25.4645 0.893077
\(814\) 0 0
\(815\) −45.5888 −1.59690
\(816\) 0 0
\(817\) 50.6089 1.77058
\(818\) 0 0
\(819\) −0.164507 −0.00574833
\(820\) 0 0
\(821\) −37.2073 −1.29854 −0.649271 0.760557i \(-0.724926\pi\)
−0.649271 + 0.760557i \(0.724926\pi\)
\(822\) 0 0
\(823\) −11.7945 −0.411130 −0.205565 0.978643i \(-0.565903\pi\)
−0.205565 + 0.978643i \(0.565903\pi\)
\(824\) 0 0
\(825\) −1.80475 −0.0628334
\(826\) 0 0
\(827\) −54.3940 −1.89147 −0.945733 0.324946i \(-0.894654\pi\)
−0.945733 + 0.324946i \(0.894654\pi\)
\(828\) 0 0
\(829\) −19.4462 −0.675394 −0.337697 0.941255i \(-0.609648\pi\)
−0.337697 + 0.941255i \(0.609648\pi\)
\(830\) 0 0
\(831\) −31.3243 −1.08663
\(832\) 0 0
\(833\) −2.94379 −0.101996
\(834\) 0 0
\(835\) −74.8975 −2.59193
\(836\) 0 0
\(837\) −25.3496 −0.876211
\(838\) 0 0
\(839\) 40.1624 1.38656 0.693280 0.720669i \(-0.256165\pi\)
0.693280 + 0.720669i \(0.256165\pi\)
\(840\) 0 0
\(841\) 61.2832 2.11321
\(842\) 0 0
\(843\) 21.3364 0.734863
\(844\) 0 0
\(845\) −48.3210 −1.66229
\(846\) 0 0
\(847\) −10.7402 −0.369037
\(848\) 0 0
\(849\) −1.43220 −0.0491530
\(850\) 0 0
\(851\) −26.2663 −0.900396
\(852\) 0 0
\(853\) 28.9463 0.991102 0.495551 0.868579i \(-0.334966\pi\)
0.495551 + 0.868579i \(0.334966\pi\)
\(854\) 0 0
\(855\) 20.6030 0.704609
\(856\) 0 0
\(857\) −0.535180 −0.0182814 −0.00914071 0.999958i \(-0.502910\pi\)
−0.00914071 + 0.999958i \(0.502910\pi\)
\(858\) 0 0
\(859\) 15.3975 0.525357 0.262678 0.964883i \(-0.415394\pi\)
0.262678 + 0.964883i \(0.415394\pi\)
\(860\) 0 0
\(861\) 10.4493 0.356111
\(862\) 0 0
\(863\) 29.1111 0.990955 0.495477 0.868621i \(-0.334993\pi\)
0.495477 + 0.868621i \(0.334993\pi\)
\(864\) 0 0
\(865\) −77.8298 −2.64629
\(866\) 0 0
\(867\) −21.5216 −0.730911
\(868\) 0 0
\(869\) −0.841343 −0.0285406
\(870\) 0 0
\(871\) 0.156520 0.00530349
\(872\) 0 0
\(873\) −7.57047 −0.256222
\(874\) 0 0
\(875\) 14.0178 0.473888
\(876\) 0 0
\(877\) −4.25776 −0.143774 −0.0718871 0.997413i \(-0.522902\pi\)
−0.0718871 + 0.997413i \(0.522902\pi\)
\(878\) 0 0
\(879\) −4.02477 −0.135752
\(880\) 0 0
\(881\) −9.60023 −0.323440 −0.161720 0.986837i \(-0.551704\pi\)
−0.161720 + 0.986837i \(0.551704\pi\)
\(882\) 0 0
\(883\) −20.1278 −0.677355 −0.338678 0.940902i \(-0.609980\pi\)
−0.338678 + 0.940902i \(0.609980\pi\)
\(884\) 0 0
\(885\) 30.6858 1.03149
\(886\) 0 0
\(887\) 30.4028 1.02082 0.510412 0.859930i \(-0.329493\pi\)
0.510412 + 0.859930i \(0.329493\pi\)
\(888\) 0 0
\(889\) −16.2795 −0.545996
\(890\) 0 0
\(891\) 0.495357 0.0165951
\(892\) 0 0
\(893\) 17.9010 0.599036
\(894\) 0 0
\(895\) 34.7278 1.16082
\(896\) 0 0
\(897\) 0.891927 0.0297806
\(898\) 0 0
\(899\) 43.1116 1.43785
\(900\) 0 0
\(901\) 5.05781 0.168500
\(902\) 0 0
\(903\) −15.5239 −0.516604
\(904\) 0 0
\(905\) 16.3608 0.543852
\(906\) 0 0
\(907\) −24.9774 −0.829360 −0.414680 0.909967i \(-0.636107\pi\)
−0.414680 + 0.909967i \(0.636107\pi\)
\(908\) 0 0
\(909\) −18.0881 −0.599943
\(910\) 0 0
\(911\) −31.4566 −1.04221 −0.521103 0.853494i \(-0.674479\pi\)
−0.521103 + 0.853494i \(0.674479\pi\)
\(912\) 0 0
\(913\) −2.36280 −0.0781972
\(914\) 0 0
\(915\) 29.2206 0.966005
\(916\) 0 0
\(917\) −14.8324 −0.489810
\(918\) 0 0
\(919\) 23.1207 0.762681 0.381341 0.924435i \(-0.375463\pi\)
0.381341 + 0.924435i \(0.375463\pi\)
\(920\) 0 0
\(921\) −9.09817 −0.299795
\(922\) 0 0
\(923\) 1.88476 0.0620376
\(924\) 0 0
\(925\) −41.6117 −1.36819
\(926\) 0 0
\(927\) 16.4797 0.541266
\(928\) 0 0
\(929\) −25.6481 −0.841486 −0.420743 0.907180i \(-0.638231\pi\)
−0.420743 + 0.907180i \(0.638231\pi\)
\(930\) 0 0
\(931\) 24.7500 0.811148
\(932\) 0 0
\(933\) −8.72454 −0.285629
\(934\) 0 0
\(935\) −0.288008 −0.00941886
\(936\) 0 0
\(937\) −19.4551 −0.635572 −0.317786 0.948163i \(-0.602939\pi\)
−0.317786 + 0.948163i \(0.602939\pi\)
\(938\) 0 0
\(939\) −27.0930 −0.884148
\(940\) 0 0
\(941\) −10.6946 −0.348634 −0.174317 0.984690i \(-0.555772\pi\)
−0.174317 + 0.984690i \(0.555772\pi\)
\(942\) 0 0
\(943\) 46.4535 1.51273
\(944\) 0 0
\(945\) −20.3473 −0.661898
\(946\) 0 0
\(947\) −27.6893 −0.899781 −0.449890 0.893084i \(-0.648537\pi\)
−0.449890 + 0.893084i \(0.648537\pi\)
\(948\) 0 0
\(949\) 1.96852 0.0639007
\(950\) 0 0
\(951\) −22.3130 −0.723549
\(952\) 0 0
\(953\) −40.6787 −1.31771 −0.658856 0.752269i \(-0.728959\pi\)
−0.658856 + 0.752269i \(0.728959\pi\)
\(954\) 0 0
\(955\) 16.3102 0.527787
\(956\) 0 0
\(957\) −1.93787 −0.0626424
\(958\) 0 0
\(959\) 3.56302 0.115056
\(960\) 0 0
\(961\) −10.4135 −0.335920
\(962\) 0 0
\(963\) 24.4244 0.787064
\(964\) 0 0
\(965\) 93.4380 3.00788
\(966\) 0 0
\(967\) −36.7809 −1.18279 −0.591397 0.806381i \(-0.701423\pi\)
−0.591397 + 0.806381i \(0.701423\pi\)
\(968\) 0 0
\(969\) −2.56218 −0.0823090
\(970\) 0 0
\(971\) 6.66929 0.214028 0.107014 0.994258i \(-0.465871\pi\)
0.107014 + 0.994258i \(0.465871\pi\)
\(972\) 0 0
\(973\) −8.47804 −0.271793
\(974\) 0 0
\(975\) 1.41302 0.0452527
\(976\) 0 0
\(977\) −41.6789 −1.33343 −0.666713 0.745315i \(-0.732299\pi\)
−0.666713 + 0.745315i \(0.732299\pi\)
\(978\) 0 0
\(979\) 0.869883 0.0278016
\(980\) 0 0
\(981\) 1.02009 0.0325691
\(982\) 0 0
\(983\) −57.4121 −1.83116 −0.915580 0.402135i \(-0.868268\pi\)
−0.915580 + 0.402135i \(0.868268\pi\)
\(984\) 0 0
\(985\) 30.4897 0.971484
\(986\) 0 0
\(987\) −5.49102 −0.174781
\(988\) 0 0
\(989\) −69.0133 −2.19450
\(990\) 0 0
\(991\) 32.2071 1.02309 0.511546 0.859256i \(-0.329073\pi\)
0.511546 + 0.859256i \(0.329073\pi\)
\(992\) 0 0
\(993\) 12.0212 0.381481
\(994\) 0 0
\(995\) −64.4058 −2.04180
\(996\) 0 0
\(997\) −31.6186 −1.00137 −0.500686 0.865629i \(-0.666919\pi\)
−0.500686 + 0.865629i \(0.666919\pi\)
\(998\) 0 0
\(999\) 26.2724 0.831221
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 4016.2.a.m.1.17 23
4.3 odd 2 2008.2.a.d.1.7 23
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2008.2.a.d.1.7 23 4.3 odd 2
4016.2.a.m.1.17 23 1.1 even 1 trivial