Properties

Label 4014.2.a.r.1.2
Level 4014
Weight 2
Character 4014.1
Self dual yes
Analytic conductor 32.052
Analytic rank 1
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4014 = 2 \cdot 3^{2} \cdot 223 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4014.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0519513713\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.356173.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1338)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.75496\)
Character \(\chi\) = 4014.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.54501 q^{5} -1.28984 q^{7} +1.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{4} -1.54501 q^{5} -1.28984 q^{7} +1.00000 q^{8} -1.54501 q^{10} +2.49412 q^{11} -2.65927 q^{13} -1.28984 q^{14} +1.00000 q^{16} -1.92011 q^{17} +4.69562 q^{19} -1.54501 q^{20} +2.49412 q^{22} -3.63334 q^{23} -2.61296 q^{25} -2.65927 q^{26} -1.28984 q^{28} -9.08085 q^{29} +5.96966 q^{31} +1.00000 q^{32} -1.92011 q^{34} +1.99281 q^{35} +5.70604 q^{37} +4.69562 q^{38} -1.54501 q^{40} -6.53614 q^{41} +6.44584 q^{43} +2.49412 q^{44} -3.63334 q^{46} -12.1071 q^{47} -5.33631 q^{49} -2.61296 q^{50} -2.65927 q^{52} -2.97407 q^{53} -3.85342 q^{55} -1.28984 q^{56} -9.08085 q^{58} +12.6372 q^{59} -10.5220 q^{61} +5.96966 q^{62} +1.00000 q^{64} +4.10859 q^{65} +4.51299 q^{67} -1.92011 q^{68} +1.99281 q^{70} -6.01319 q^{71} -9.46377 q^{73} +5.70604 q^{74} +4.69562 q^{76} -3.21701 q^{77} -9.23357 q^{79} -1.54501 q^{80} -6.53614 q^{82} -12.9523 q^{83} +2.96659 q^{85} +6.44584 q^{86} +2.49412 q^{88} +0.168223 q^{89} +3.43004 q^{91} -3.63334 q^{92} -12.1071 q^{94} -7.25476 q^{95} +1.05018 q^{97} -5.33631 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q + 5q^{2} + 5q^{4} - 5q^{5} - q^{7} + 5q^{8} + O(q^{10}) \) \( 5q + 5q^{2} + 5q^{4} - 5q^{5} - q^{7} + 5q^{8} - 5q^{10} - 9q^{11} - q^{14} + 5q^{16} - 6q^{17} - 4q^{19} - 5q^{20} - 9q^{22} - 16q^{23} + 8q^{25} - q^{28} - 8q^{29} - q^{31} + 5q^{32} - 6q^{34} - 22q^{35} - 2q^{37} - 4q^{38} - 5q^{40} - 4q^{41} + 3q^{43} - 9q^{44} - 16q^{46} - 18q^{47} + 2q^{49} + 8q^{50} - 26q^{53} + q^{55} - q^{56} - 8q^{58} - 21q^{59} - 20q^{61} - q^{62} + 5q^{64} + 3q^{65} - 5q^{67} - 6q^{68} - 22q^{70} - 17q^{71} + 5q^{73} - 2q^{74} - 4q^{76} - 2q^{77} - 21q^{79} - 5q^{80} - 4q^{82} - 11q^{83} - 12q^{85} + 3q^{86} - 9q^{88} + 5q^{89} - 10q^{91} - 16q^{92} - 18q^{94} - 10q^{95} - 11q^{97} + 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.54501 −0.690947 −0.345474 0.938428i \(-0.612282\pi\)
−0.345474 + 0.938428i \(0.612282\pi\)
\(6\) 0 0
\(7\) −1.28984 −0.487514 −0.243757 0.969836i \(-0.578380\pi\)
−0.243757 + 0.969836i \(0.578380\pi\)
\(8\) 1.00000 0.353553
\(9\) 0 0
\(10\) −1.54501 −0.488574
\(11\) 2.49412 0.752005 0.376002 0.926619i \(-0.377298\pi\)
0.376002 + 0.926619i \(0.377298\pi\)
\(12\) 0 0
\(13\) −2.65927 −0.737549 −0.368775 0.929519i \(-0.620223\pi\)
−0.368775 + 0.929519i \(0.620223\pi\)
\(14\) −1.28984 −0.344724
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.92011 −0.465696 −0.232848 0.972513i \(-0.574805\pi\)
−0.232848 + 0.972513i \(0.574805\pi\)
\(18\) 0 0
\(19\) 4.69562 1.07725 0.538624 0.842546i \(-0.318944\pi\)
0.538624 + 0.842546i \(0.318944\pi\)
\(20\) −1.54501 −0.345474
\(21\) 0 0
\(22\) 2.49412 0.531747
\(23\) −3.63334 −0.757604 −0.378802 0.925478i \(-0.623664\pi\)
−0.378802 + 0.925478i \(0.623664\pi\)
\(24\) 0 0
\(25\) −2.61296 −0.522592
\(26\) −2.65927 −0.521526
\(27\) 0 0
\(28\) −1.28984 −0.243757
\(29\) −9.08085 −1.68627 −0.843136 0.537700i \(-0.819293\pi\)
−0.843136 + 0.537700i \(0.819293\pi\)
\(30\) 0 0
\(31\) 5.96966 1.07218 0.536091 0.844160i \(-0.319900\pi\)
0.536091 + 0.844160i \(0.319900\pi\)
\(32\) 1.00000 0.176777
\(33\) 0 0
\(34\) −1.92011 −0.329297
\(35\) 1.99281 0.336846
\(36\) 0 0
\(37\) 5.70604 0.938067 0.469033 0.883180i \(-0.344602\pi\)
0.469033 + 0.883180i \(0.344602\pi\)
\(38\) 4.69562 0.761730
\(39\) 0 0
\(40\) −1.54501 −0.244287
\(41\) −6.53614 −1.02077 −0.510387 0.859945i \(-0.670498\pi\)
−0.510387 + 0.859945i \(0.670498\pi\)
\(42\) 0 0
\(43\) 6.44584 0.982981 0.491491 0.870883i \(-0.336452\pi\)
0.491491 + 0.870883i \(0.336452\pi\)
\(44\) 2.49412 0.376002
\(45\) 0 0
\(46\) −3.63334 −0.535707
\(47\) −12.1071 −1.76600 −0.882999 0.469374i \(-0.844479\pi\)
−0.882999 + 0.469374i \(0.844479\pi\)
\(48\) 0 0
\(49\) −5.33631 −0.762330
\(50\) −2.61296 −0.369528
\(51\) 0 0
\(52\) −2.65927 −0.368775
\(53\) −2.97407 −0.408520 −0.204260 0.978917i \(-0.565479\pi\)
−0.204260 + 0.978917i \(0.565479\pi\)
\(54\) 0 0
\(55\) −3.85342 −0.519595
\(56\) −1.28984 −0.172362
\(57\) 0 0
\(58\) −9.08085 −1.19237
\(59\) 12.6372 1.64523 0.822615 0.568599i \(-0.192514\pi\)
0.822615 + 0.568599i \(0.192514\pi\)
\(60\) 0 0
\(61\) −10.5220 −1.34721 −0.673603 0.739094i \(-0.735254\pi\)
−0.673603 + 0.739094i \(0.735254\pi\)
\(62\) 5.96966 0.758147
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.10859 0.509608
\(66\) 0 0
\(67\) 4.51299 0.551349 0.275675 0.961251i \(-0.411099\pi\)
0.275675 + 0.961251i \(0.411099\pi\)
\(68\) −1.92011 −0.232848
\(69\) 0 0
\(70\) 1.99281 0.238186
\(71\) −6.01319 −0.713635 −0.356817 0.934174i \(-0.616138\pi\)
−0.356817 + 0.934174i \(0.616138\pi\)
\(72\) 0 0
\(73\) −9.46377 −1.10765 −0.553825 0.832633i \(-0.686832\pi\)
−0.553825 + 0.832633i \(0.686832\pi\)
\(74\) 5.70604 0.663313
\(75\) 0 0
\(76\) 4.69562 0.538624
\(77\) −3.21701 −0.366613
\(78\) 0 0
\(79\) −9.23357 −1.03886 −0.519429 0.854514i \(-0.673855\pi\)
−0.519429 + 0.854514i \(0.673855\pi\)
\(80\) −1.54501 −0.172737
\(81\) 0 0
\(82\) −6.53614 −0.721796
\(83\) −12.9523 −1.42170 −0.710852 0.703342i \(-0.751690\pi\)
−0.710852 + 0.703342i \(0.751690\pi\)
\(84\) 0 0
\(85\) 2.96659 0.321772
\(86\) 6.44584 0.695073
\(87\) 0 0
\(88\) 2.49412 0.265874
\(89\) 0.168223 0.0178316 0.00891580 0.999960i \(-0.497162\pi\)
0.00891580 + 0.999960i \(0.497162\pi\)
\(90\) 0 0
\(91\) 3.43004 0.359565
\(92\) −3.63334 −0.378802
\(93\) 0 0
\(94\) −12.1071 −1.24875
\(95\) −7.25476 −0.744322
\(96\) 0 0
\(97\) 1.05018 0.106630 0.0533151 0.998578i \(-0.483021\pi\)
0.0533151 + 0.998578i \(0.483021\pi\)
\(98\) −5.33631 −0.539049
\(99\) 0 0
\(100\) −2.61296 −0.261296
\(101\) 4.99849 0.497368 0.248684 0.968585i \(-0.420002\pi\)
0.248684 + 0.968585i \(0.420002\pi\)
\(102\) 0 0
\(103\) 15.2450 1.50213 0.751065 0.660228i \(-0.229540\pi\)
0.751065 + 0.660228i \(0.229540\pi\)
\(104\) −2.65927 −0.260763
\(105\) 0 0
\(106\) −2.97407 −0.288867
\(107\) 11.7155 1.13258 0.566288 0.824207i \(-0.308379\pi\)
0.566288 + 0.824207i \(0.308379\pi\)
\(108\) 0 0
\(109\) −16.5487 −1.58508 −0.792541 0.609818i \(-0.791242\pi\)
−0.792541 + 0.609818i \(0.791242\pi\)
\(110\) −3.85342 −0.367409
\(111\) 0 0
\(112\) −1.28984 −0.121878
\(113\) −2.38720 −0.224569 −0.112284 0.993676i \(-0.535817\pi\)
−0.112284 + 0.993676i \(0.535817\pi\)
\(114\) 0 0
\(115\) 5.61353 0.523465
\(116\) −9.08085 −0.843136
\(117\) 0 0
\(118\) 12.6372 1.16335
\(119\) 2.47664 0.227033
\(120\) 0 0
\(121\) −4.77938 −0.434489
\(122\) −10.5220 −0.952618
\(123\) 0 0
\(124\) 5.96966 0.536091
\(125\) 11.7621 1.05203
\(126\) 0 0
\(127\) −14.2746 −1.26667 −0.633333 0.773880i \(-0.718314\pi\)
−0.633333 + 0.773880i \(0.718314\pi\)
\(128\) 1.00000 0.0883883
\(129\) 0 0
\(130\) 4.10859 0.360347
\(131\) −19.9949 −1.74696 −0.873480 0.486859i \(-0.838142\pi\)
−0.873480 + 0.486859i \(0.838142\pi\)
\(132\) 0 0
\(133\) −6.05660 −0.525174
\(134\) 4.51299 0.389863
\(135\) 0 0
\(136\) −1.92011 −0.164648
\(137\) −10.4458 −0.892448 −0.446224 0.894921i \(-0.647232\pi\)
−0.446224 + 0.894921i \(0.647232\pi\)
\(138\) 0 0
\(139\) 12.4043 1.05212 0.526058 0.850449i \(-0.323669\pi\)
0.526058 + 0.850449i \(0.323669\pi\)
\(140\) 1.99281 0.168423
\(141\) 0 0
\(142\) −6.01319 −0.504616
\(143\) −6.63253 −0.554640
\(144\) 0 0
\(145\) 14.0300 1.16513
\(146\) −9.46377 −0.783227
\(147\) 0 0
\(148\) 5.70604 0.469033
\(149\) −2.70297 −0.221436 −0.110718 0.993852i \(-0.535315\pi\)
−0.110718 + 0.993852i \(0.535315\pi\)
\(150\) 0 0
\(151\) −22.0553 −1.79483 −0.897415 0.441187i \(-0.854558\pi\)
−0.897415 + 0.441187i \(0.854558\pi\)
\(152\) 4.69562 0.380865
\(153\) 0 0
\(154\) −3.21701 −0.259234
\(155\) −9.22315 −0.740821
\(156\) 0 0
\(157\) 7.38327 0.589249 0.294625 0.955613i \(-0.404805\pi\)
0.294625 + 0.955613i \(0.404805\pi\)
\(158\) −9.23357 −0.734583
\(159\) 0 0
\(160\) −1.54501 −0.122143
\(161\) 4.68643 0.369343
\(162\) 0 0
\(163\) −4.26962 −0.334423 −0.167211 0.985921i \(-0.553476\pi\)
−0.167211 + 0.985921i \(0.553476\pi\)
\(164\) −6.53614 −0.510387
\(165\) 0 0
\(166\) −12.9523 −1.00530
\(167\) 8.38192 0.648613 0.324306 0.945952i \(-0.394869\pi\)
0.324306 + 0.945952i \(0.394869\pi\)
\(168\) 0 0
\(169\) −5.92827 −0.456021
\(170\) 2.96659 0.227527
\(171\) 0 0
\(172\) 6.44584 0.491491
\(173\) 6.01306 0.457164 0.228582 0.973525i \(-0.426591\pi\)
0.228582 + 0.973525i \(0.426591\pi\)
\(174\) 0 0
\(175\) 3.37030 0.254771
\(176\) 2.49412 0.188001
\(177\) 0 0
\(178\) 0.168223 0.0126089
\(179\) −23.1043 −1.72690 −0.863449 0.504435i \(-0.831701\pi\)
−0.863449 + 0.504435i \(0.831701\pi\)
\(180\) 0 0
\(181\) −20.2695 −1.50662 −0.753310 0.657666i \(-0.771544\pi\)
−0.753310 + 0.657666i \(0.771544\pi\)
\(182\) 3.43004 0.254251
\(183\) 0 0
\(184\) −3.63334 −0.267854
\(185\) −8.81586 −0.648155
\(186\) 0 0
\(187\) −4.78899 −0.350206
\(188\) −12.1071 −0.882999
\(189\) 0 0
\(190\) −7.25476 −0.526315
\(191\) −5.79518 −0.419325 −0.209662 0.977774i \(-0.567236\pi\)
−0.209662 + 0.977774i \(0.567236\pi\)
\(192\) 0 0
\(193\) 12.3097 0.886070 0.443035 0.896504i \(-0.353902\pi\)
0.443035 + 0.896504i \(0.353902\pi\)
\(194\) 1.05018 0.0753989
\(195\) 0 0
\(196\) −5.33631 −0.381165
\(197\) −18.7257 −1.33415 −0.667077 0.744989i \(-0.732455\pi\)
−0.667077 + 0.744989i \(0.732455\pi\)
\(198\) 0 0
\(199\) 3.65510 0.259103 0.129552 0.991573i \(-0.458646\pi\)
0.129552 + 0.991573i \(0.458646\pi\)
\(200\) −2.61296 −0.184764
\(201\) 0 0
\(202\) 4.99849 0.351692
\(203\) 11.7128 0.822081
\(204\) 0 0
\(205\) 10.0984 0.705301
\(206\) 15.2450 1.06217
\(207\) 0 0
\(208\) −2.65927 −0.184387
\(209\) 11.7114 0.810096
\(210\) 0 0
\(211\) 26.2300 1.80575 0.902873 0.429907i \(-0.141454\pi\)
0.902873 + 0.429907i \(0.141454\pi\)
\(212\) −2.97407 −0.204260
\(213\) 0 0
\(214\) 11.7155 0.800852
\(215\) −9.95885 −0.679188
\(216\) 0 0
\(217\) −7.69990 −0.522703
\(218\) −16.5487 −1.12082
\(219\) 0 0
\(220\) −3.85342 −0.259798
\(221\) 5.10611 0.343474
\(222\) 0 0
\(223\) −1.00000 −0.0669650
\(224\) −1.28984 −0.0861811
\(225\) 0 0
\(226\) −2.38720 −0.158794
\(227\) 27.2280 1.80718 0.903592 0.428395i \(-0.140921\pi\)
0.903592 + 0.428395i \(0.140921\pi\)
\(228\) 0 0
\(229\) −2.43949 −0.161206 −0.0806029 0.996746i \(-0.525685\pi\)
−0.0806029 + 0.996746i \(0.525685\pi\)
\(230\) 5.61353 0.370145
\(231\) 0 0
\(232\) −9.08085 −0.596187
\(233\) 11.3011 0.740358 0.370179 0.928961i \(-0.379296\pi\)
0.370179 + 0.928961i \(0.379296\pi\)
\(234\) 0 0
\(235\) 18.7055 1.22021
\(236\) 12.6372 0.822615
\(237\) 0 0
\(238\) 2.47664 0.160537
\(239\) 8.55819 0.553584 0.276792 0.960930i \(-0.410729\pi\)
0.276792 + 0.960930i \(0.410729\pi\)
\(240\) 0 0
\(241\) 17.4393 1.12336 0.561681 0.827354i \(-0.310155\pi\)
0.561681 + 0.827354i \(0.310155\pi\)
\(242\) −4.77938 −0.307230
\(243\) 0 0
\(244\) −10.5220 −0.673603
\(245\) 8.24463 0.526730
\(246\) 0 0
\(247\) −12.4869 −0.794524
\(248\) 5.96966 0.379074
\(249\) 0 0
\(250\) 11.7621 0.743898
\(251\) −0.239068 −0.0150898 −0.00754492 0.999972i \(-0.502402\pi\)
−0.00754492 + 0.999972i \(0.502402\pi\)
\(252\) 0 0
\(253\) −9.06198 −0.569722
\(254\) −14.2746 −0.895668
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −0.145523 −0.00907746 −0.00453873 0.999990i \(-0.501445\pi\)
−0.00453873 + 0.999990i \(0.501445\pi\)
\(258\) 0 0
\(259\) −7.35988 −0.457320
\(260\) 4.10859 0.254804
\(261\) 0 0
\(262\) −19.9949 −1.23529
\(263\) 8.17409 0.504036 0.252018 0.967723i \(-0.418906\pi\)
0.252018 + 0.967723i \(0.418906\pi\)
\(264\) 0 0
\(265\) 4.59496 0.282266
\(266\) −6.05660 −0.371354
\(267\) 0 0
\(268\) 4.51299 0.275675
\(269\) 21.6988 1.32300 0.661501 0.749944i \(-0.269920\pi\)
0.661501 + 0.749944i \(0.269920\pi\)
\(270\) 0 0
\(271\) −31.1848 −1.89434 −0.947170 0.320732i \(-0.896071\pi\)
−0.947170 + 0.320732i \(0.896071\pi\)
\(272\) −1.92011 −0.116424
\(273\) 0 0
\(274\) −10.4458 −0.631056
\(275\) −6.51703 −0.392991
\(276\) 0 0
\(277\) 19.0999 1.14760 0.573802 0.818994i \(-0.305468\pi\)
0.573802 + 0.818994i \(0.305468\pi\)
\(278\) 12.4043 0.743959
\(279\) 0 0
\(280\) 1.99281 0.119093
\(281\) −7.56711 −0.451416 −0.225708 0.974195i \(-0.572469\pi\)
−0.225708 + 0.974195i \(0.572469\pi\)
\(282\) 0 0
\(283\) −5.87036 −0.348956 −0.174478 0.984661i \(-0.555824\pi\)
−0.174478 + 0.984661i \(0.555824\pi\)
\(284\) −6.01319 −0.356817
\(285\) 0 0
\(286\) −6.63253 −0.392190
\(287\) 8.43058 0.497641
\(288\) 0 0
\(289\) −13.3132 −0.783127
\(290\) 14.0300 0.823868
\(291\) 0 0
\(292\) −9.46377 −0.553825
\(293\) 16.2606 0.949954 0.474977 0.879998i \(-0.342456\pi\)
0.474977 + 0.879998i \(0.342456\pi\)
\(294\) 0 0
\(295\) −19.5246 −1.13677
\(296\) 5.70604 0.331657
\(297\) 0 0
\(298\) −2.70297 −0.156579
\(299\) 9.66205 0.558771
\(300\) 0 0
\(301\) −8.31410 −0.479217
\(302\) −22.0553 −1.26914
\(303\) 0 0
\(304\) 4.69562 0.269312
\(305\) 16.2566 0.930848
\(306\) 0 0
\(307\) −7.70583 −0.439795 −0.219897 0.975523i \(-0.570572\pi\)
−0.219897 + 0.975523i \(0.570572\pi\)
\(308\) −3.21701 −0.183306
\(309\) 0 0
\(310\) −9.22315 −0.523840
\(311\) −8.35322 −0.473668 −0.236834 0.971550i \(-0.576110\pi\)
−0.236834 + 0.971550i \(0.576110\pi\)
\(312\) 0 0
\(313\) −12.8779 −0.727900 −0.363950 0.931418i \(-0.618572\pi\)
−0.363950 + 0.931418i \(0.618572\pi\)
\(314\) 7.38327 0.416662
\(315\) 0 0
\(316\) −9.23357 −0.519429
\(317\) −17.0655 −0.958493 −0.479246 0.877680i \(-0.659090\pi\)
−0.479246 + 0.877680i \(0.659090\pi\)
\(318\) 0 0
\(319\) −22.6487 −1.26808
\(320\) −1.54501 −0.0863684
\(321\) 0 0
\(322\) 4.68643 0.261165
\(323\) −9.01613 −0.501671
\(324\) 0 0
\(325\) 6.94857 0.385437
\(326\) −4.26962 −0.236472
\(327\) 0 0
\(328\) −6.53614 −0.360898
\(329\) 15.6162 0.860949
\(330\) 0 0
\(331\) −19.9208 −1.09495 −0.547473 0.836823i \(-0.684410\pi\)
−0.547473 + 0.836823i \(0.684410\pi\)
\(332\) −12.9523 −0.710852
\(333\) 0 0
\(334\) 8.38192 0.458638
\(335\) −6.97259 −0.380953
\(336\) 0 0
\(337\) 4.53383 0.246973 0.123487 0.992346i \(-0.460592\pi\)
0.123487 + 0.992346i \(0.460592\pi\)
\(338\) −5.92827 −0.322456
\(339\) 0 0
\(340\) 2.96659 0.160886
\(341\) 14.8890 0.806286
\(342\) 0 0
\(343\) 15.9119 0.859160
\(344\) 6.44584 0.347536
\(345\) 0 0
\(346\) 6.01306 0.323264
\(347\) 34.7965 1.86797 0.933986 0.357309i \(-0.116306\pi\)
0.933986 + 0.357309i \(0.116306\pi\)
\(348\) 0 0
\(349\) −21.9889 −1.17704 −0.588518 0.808484i \(-0.700288\pi\)
−0.588518 + 0.808484i \(0.700288\pi\)
\(350\) 3.37030 0.180150
\(351\) 0 0
\(352\) 2.49412 0.132937
\(353\) 5.13896 0.273519 0.136760 0.990604i \(-0.456331\pi\)
0.136760 + 0.990604i \(0.456331\pi\)
\(354\) 0 0
\(355\) 9.29041 0.493084
\(356\) 0.168223 0.00891580
\(357\) 0 0
\(358\) −23.1043 −1.22110
\(359\) −11.4094 −0.602163 −0.301081 0.953598i \(-0.597348\pi\)
−0.301081 + 0.953598i \(0.597348\pi\)
\(360\) 0 0
\(361\) 3.04884 0.160465
\(362\) −20.2695 −1.06534
\(363\) 0 0
\(364\) 3.43004 0.179783
\(365\) 14.6216 0.765328
\(366\) 0 0
\(367\) 23.4709 1.22517 0.612585 0.790405i \(-0.290130\pi\)
0.612585 + 0.790405i \(0.290130\pi\)
\(368\) −3.63334 −0.189401
\(369\) 0 0
\(370\) −8.81586 −0.458315
\(371\) 3.83608 0.199159
\(372\) 0 0
\(373\) 14.7702 0.764773 0.382387 0.924002i \(-0.375102\pi\)
0.382387 + 0.924002i \(0.375102\pi\)
\(374\) −4.78899 −0.247633
\(375\) 0 0
\(376\) −12.1071 −0.624375
\(377\) 24.1485 1.24371
\(378\) 0 0
\(379\) 9.14574 0.469785 0.234893 0.972021i \(-0.424526\pi\)
0.234893 + 0.972021i \(0.424526\pi\)
\(380\) −7.25476 −0.372161
\(381\) 0 0
\(382\) −5.79518 −0.296507
\(383\) 15.2947 0.781520 0.390760 0.920493i \(-0.372212\pi\)
0.390760 + 0.920493i \(0.372212\pi\)
\(384\) 0 0
\(385\) 4.97030 0.253310
\(386\) 12.3097 0.626546
\(387\) 0 0
\(388\) 1.05018 0.0533151
\(389\) 37.7337 1.91317 0.956586 0.291449i \(-0.0941375\pi\)
0.956586 + 0.291449i \(0.0941375\pi\)
\(390\) 0 0
\(391\) 6.97644 0.352814
\(392\) −5.33631 −0.269524
\(393\) 0 0
\(394\) −18.7257 −0.943390
\(395\) 14.2659 0.717796
\(396\) 0 0
\(397\) −26.6903 −1.33955 −0.669774 0.742565i \(-0.733609\pi\)
−0.669774 + 0.742565i \(0.733609\pi\)
\(398\) 3.65510 0.183214
\(399\) 0 0
\(400\) −2.61296 −0.130648
\(401\) 31.3412 1.56510 0.782552 0.622585i \(-0.213918\pi\)
0.782552 + 0.622585i \(0.213918\pi\)
\(402\) 0 0
\(403\) −15.8749 −0.790787
\(404\) 4.99849 0.248684
\(405\) 0 0
\(406\) 11.7128 0.581299
\(407\) 14.2315 0.705430
\(408\) 0 0
\(409\) 38.0880 1.88333 0.941666 0.336550i \(-0.109260\pi\)
0.941666 + 0.336550i \(0.109260\pi\)
\(410\) 10.0984 0.498723
\(411\) 0 0
\(412\) 15.2450 0.751065
\(413\) −16.3000 −0.802072
\(414\) 0 0
\(415\) 20.0114 0.982323
\(416\) −2.65927 −0.130382
\(417\) 0 0
\(418\) 11.7114 0.572824
\(419\) −22.7446 −1.11115 −0.555574 0.831467i \(-0.687501\pi\)
−0.555574 + 0.831467i \(0.687501\pi\)
\(420\) 0 0
\(421\) −21.6762 −1.05643 −0.528217 0.849109i \(-0.677139\pi\)
−0.528217 + 0.849109i \(0.677139\pi\)
\(422\) 26.2300 1.27686
\(423\) 0 0
\(424\) −2.97407 −0.144434
\(425\) 5.01718 0.243369
\(426\) 0 0
\(427\) 13.5717 0.656781
\(428\) 11.7155 0.566288
\(429\) 0 0
\(430\) −9.95885 −0.480259
\(431\) 24.7372 1.19155 0.595775 0.803151i \(-0.296845\pi\)
0.595775 + 0.803151i \(0.296845\pi\)
\(432\) 0 0
\(433\) 29.6583 1.42529 0.712645 0.701525i \(-0.247497\pi\)
0.712645 + 0.701525i \(0.247497\pi\)
\(434\) −7.69990 −0.369607
\(435\) 0 0
\(436\) −16.5487 −0.792541
\(437\) −17.0608 −0.816128
\(438\) 0 0
\(439\) 27.9533 1.33414 0.667069 0.744996i \(-0.267548\pi\)
0.667069 + 0.744996i \(0.267548\pi\)
\(440\) −3.85342 −0.183705
\(441\) 0 0
\(442\) 5.10611 0.242873
\(443\) −21.5451 −1.02364 −0.511819 0.859093i \(-0.671028\pi\)
−0.511819 + 0.859093i \(0.671028\pi\)
\(444\) 0 0
\(445\) −0.259905 −0.0123207
\(446\) −1.00000 −0.0473514
\(447\) 0 0
\(448\) −1.28984 −0.0609392
\(449\) −38.3133 −1.80812 −0.904059 0.427407i \(-0.859427\pi\)
−0.904059 + 0.427407i \(0.859427\pi\)
\(450\) 0 0
\(451\) −16.3019 −0.767627
\(452\) −2.38720 −0.112284
\(453\) 0 0
\(454\) 27.2280 1.27787
\(455\) −5.29942 −0.248441
\(456\) 0 0
\(457\) −8.77811 −0.410623 −0.205311 0.978697i \(-0.565821\pi\)
−0.205311 + 0.978697i \(0.565821\pi\)
\(458\) −2.43949 −0.113990
\(459\) 0 0
\(460\) 5.61353 0.261732
\(461\) 14.4028 0.670807 0.335404 0.942075i \(-0.391127\pi\)
0.335404 + 0.942075i \(0.391127\pi\)
\(462\) 0 0
\(463\) 13.6803 0.635779 0.317889 0.948128i \(-0.397026\pi\)
0.317889 + 0.948128i \(0.397026\pi\)
\(464\) −9.08085 −0.421568
\(465\) 0 0
\(466\) 11.3011 0.523512
\(467\) −8.37193 −0.387407 −0.193703 0.981060i \(-0.562050\pi\)
−0.193703 + 0.981060i \(0.562050\pi\)
\(468\) 0 0
\(469\) −5.82103 −0.268790
\(470\) 18.7055 0.862820
\(471\) 0 0
\(472\) 12.6372 0.581676
\(473\) 16.0767 0.739206
\(474\) 0 0
\(475\) −12.2695 −0.562961
\(476\) 2.47664 0.113517
\(477\) 0 0
\(478\) 8.55819 0.391443
\(479\) 21.7260 0.992688 0.496344 0.868126i \(-0.334675\pi\)
0.496344 + 0.868126i \(0.334675\pi\)
\(480\) 0 0
\(481\) −15.1739 −0.691870
\(482\) 17.4393 0.794337
\(483\) 0 0
\(484\) −4.77938 −0.217245
\(485\) −1.62254 −0.0736758
\(486\) 0 0
\(487\) 2.38053 0.107872 0.0539360 0.998544i \(-0.482823\pi\)
0.0539360 + 0.998544i \(0.482823\pi\)
\(488\) −10.5220 −0.476309
\(489\) 0 0
\(490\) 8.24463 0.372454
\(491\) −20.8266 −0.939890 −0.469945 0.882696i \(-0.655726\pi\)
−0.469945 + 0.882696i \(0.655726\pi\)
\(492\) 0 0
\(493\) 17.4363 0.785291
\(494\) −12.4869 −0.561813
\(495\) 0 0
\(496\) 5.96966 0.268045
\(497\) 7.75606 0.347907
\(498\) 0 0
\(499\) 18.7813 0.840765 0.420382 0.907347i \(-0.361896\pi\)
0.420382 + 0.907347i \(0.361896\pi\)
\(500\) 11.7621 0.526015
\(501\) 0 0
\(502\) −0.239068 −0.0106701
\(503\) 11.5451 0.514771 0.257385 0.966309i \(-0.417139\pi\)
0.257385 + 0.966309i \(0.417139\pi\)
\(504\) 0 0
\(505\) −7.72269 −0.343655
\(506\) −9.06198 −0.402854
\(507\) 0 0
\(508\) −14.2746 −0.633333
\(509\) −21.8205 −0.967177 −0.483589 0.875295i \(-0.660667\pi\)
−0.483589 + 0.875295i \(0.660667\pi\)
\(510\) 0 0
\(511\) 12.2068 0.539995
\(512\) 1.00000 0.0441942
\(513\) 0 0
\(514\) −0.145523 −0.00641873
\(515\) −23.5535 −1.03789
\(516\) 0 0
\(517\) −30.1965 −1.32804
\(518\) −7.35988 −0.323374
\(519\) 0 0
\(520\) 4.10859 0.180174
\(521\) 5.31636 0.232914 0.116457 0.993196i \(-0.462846\pi\)
0.116457 + 0.993196i \(0.462846\pi\)
\(522\) 0 0
\(523\) 31.5525 1.37969 0.689847 0.723955i \(-0.257678\pi\)
0.689847 + 0.723955i \(0.257678\pi\)
\(524\) −19.9949 −0.873480
\(525\) 0 0
\(526\) 8.17409 0.356407
\(527\) −11.4624 −0.499311
\(528\) 0 0
\(529\) −9.79882 −0.426036
\(530\) 4.59496 0.199592
\(531\) 0 0
\(532\) −6.05660 −0.262587
\(533\) 17.3814 0.752871
\(534\) 0 0
\(535\) −18.1004 −0.782550
\(536\) 4.51299 0.194931
\(537\) 0 0
\(538\) 21.6988 0.935504
\(539\) −13.3094 −0.573276
\(540\) 0 0
\(541\) −10.2150 −0.439179 −0.219590 0.975592i \(-0.570472\pi\)
−0.219590 + 0.975592i \(0.570472\pi\)
\(542\) −31.1848 −1.33950
\(543\) 0 0
\(544\) −1.92011 −0.0823242
\(545\) 25.5679 1.09521
\(546\) 0 0
\(547\) 6.88780 0.294501 0.147251 0.989099i \(-0.452958\pi\)
0.147251 + 0.989099i \(0.452958\pi\)
\(548\) −10.4458 −0.446224
\(549\) 0 0
\(550\) −6.51703 −0.277887
\(551\) −42.6402 −1.81653
\(552\) 0 0
\(553\) 11.9098 0.506457
\(554\) 19.0999 0.811478
\(555\) 0 0
\(556\) 12.4043 0.526058
\(557\) 25.1326 1.06490 0.532451 0.846461i \(-0.321271\pi\)
0.532451 + 0.846461i \(0.321271\pi\)
\(558\) 0 0
\(559\) −17.1412 −0.724997
\(560\) 1.99281 0.0842116
\(561\) 0 0
\(562\) −7.56711 −0.319199
\(563\) −36.3836 −1.53339 −0.766693 0.642013i \(-0.778099\pi\)
−0.766693 + 0.642013i \(0.778099\pi\)
\(564\) 0 0
\(565\) 3.68824 0.155165
\(566\) −5.87036 −0.246749
\(567\) 0 0
\(568\) −6.01319 −0.252308
\(569\) 40.7614 1.70881 0.854404 0.519610i \(-0.173923\pi\)
0.854404 + 0.519610i \(0.173923\pi\)
\(570\) 0 0
\(571\) −12.3995 −0.518903 −0.259451 0.965756i \(-0.583542\pi\)
−0.259451 + 0.965756i \(0.583542\pi\)
\(572\) −6.63253 −0.277320
\(573\) 0 0
\(574\) 8.43058 0.351886
\(575\) 9.49378 0.395918
\(576\) 0 0
\(577\) −9.50735 −0.395796 −0.197898 0.980223i \(-0.563412\pi\)
−0.197898 + 0.980223i \(0.563412\pi\)
\(578\) −13.3132 −0.553754
\(579\) 0 0
\(580\) 14.0300 0.582563
\(581\) 16.7064 0.693100
\(582\) 0 0
\(583\) −7.41768 −0.307209
\(584\) −9.46377 −0.391614
\(585\) 0 0
\(586\) 16.2606 0.671719
\(587\) 28.0628 1.15827 0.579137 0.815230i \(-0.303390\pi\)
0.579137 + 0.815230i \(0.303390\pi\)
\(588\) 0 0
\(589\) 28.0312 1.15501
\(590\) −19.5246 −0.803815
\(591\) 0 0
\(592\) 5.70604 0.234517
\(593\) 38.8259 1.59439 0.797194 0.603723i \(-0.206317\pi\)
0.797194 + 0.603723i \(0.206317\pi\)
\(594\) 0 0
\(595\) −3.82642 −0.156868
\(596\) −2.70297 −0.110718
\(597\) 0 0
\(598\) 9.66205 0.395110
\(599\) −40.9391 −1.67273 −0.836363 0.548176i \(-0.815322\pi\)
−0.836363 + 0.548176i \(0.815322\pi\)
\(600\) 0 0
\(601\) −20.9510 −0.854608 −0.427304 0.904108i \(-0.640537\pi\)
−0.427304 + 0.904108i \(0.640537\pi\)
\(602\) −8.31410 −0.338857
\(603\) 0 0
\(604\) −22.0553 −0.897415
\(605\) 7.38417 0.300209
\(606\) 0 0
\(607\) 18.1587 0.737039 0.368519 0.929620i \(-0.379865\pi\)
0.368519 + 0.929620i \(0.379865\pi\)
\(608\) 4.69562 0.190432
\(609\) 0 0
\(610\) 16.2566 0.658209
\(611\) 32.1960 1.30251
\(612\) 0 0
\(613\) 33.1238 1.33786 0.668928 0.743327i \(-0.266753\pi\)
0.668928 + 0.743327i \(0.266753\pi\)
\(614\) −7.70583 −0.310982
\(615\) 0 0
\(616\) −3.21701 −0.129617
\(617\) −20.7781 −0.836493 −0.418247 0.908333i \(-0.637355\pi\)
−0.418247 + 0.908333i \(0.637355\pi\)
\(618\) 0 0
\(619\) −44.0401 −1.77012 −0.885060 0.465476i \(-0.845883\pi\)
−0.885060 + 0.465476i \(0.845883\pi\)
\(620\) −9.22315 −0.370411
\(621\) 0 0
\(622\) −8.35322 −0.334934
\(623\) −0.216981 −0.00869315
\(624\) 0 0
\(625\) −5.10765 −0.204306
\(626\) −12.8779 −0.514703
\(627\) 0 0
\(628\) 7.38327 0.294625
\(629\) −10.9562 −0.436854
\(630\) 0 0
\(631\) −42.0837 −1.67533 −0.837664 0.546186i \(-0.816079\pi\)
−0.837664 + 0.546186i \(0.816079\pi\)
\(632\) −9.23357 −0.367292
\(633\) 0 0
\(634\) −17.0655 −0.677757
\(635\) 22.0543 0.875199
\(636\) 0 0
\(637\) 14.1907 0.562256
\(638\) −22.6487 −0.896671
\(639\) 0 0
\(640\) −1.54501 −0.0610717
\(641\) −25.3047 −0.999475 −0.499738 0.866177i \(-0.666570\pi\)
−0.499738 + 0.866177i \(0.666570\pi\)
\(642\) 0 0
\(643\) 33.5383 1.32262 0.661310 0.750113i \(-0.270001\pi\)
0.661310 + 0.750113i \(0.270001\pi\)
\(644\) 4.68643 0.184671
\(645\) 0 0
\(646\) −9.01613 −0.354735
\(647\) −26.8197 −1.05439 −0.527195 0.849744i \(-0.676756\pi\)
−0.527195 + 0.849744i \(0.676756\pi\)
\(648\) 0 0
\(649\) 31.5188 1.23722
\(650\) 6.94857 0.272545
\(651\) 0 0
\(652\) −4.26962 −0.167211
\(653\) −33.5965 −1.31473 −0.657366 0.753571i \(-0.728329\pi\)
−0.657366 + 0.753571i \(0.728329\pi\)
\(654\) 0 0
\(655\) 30.8922 1.20706
\(656\) −6.53614 −0.255193
\(657\) 0 0
\(658\) 15.6162 0.608783
\(659\) −7.35401 −0.286471 −0.143236 0.989689i \(-0.545751\pi\)
−0.143236 + 0.989689i \(0.545751\pi\)
\(660\) 0 0
\(661\) −25.7191 −1.00036 −0.500179 0.865922i \(-0.666732\pi\)
−0.500179 + 0.865922i \(0.666732\pi\)
\(662\) −19.9208 −0.774243
\(663\) 0 0
\(664\) −12.9523 −0.502648
\(665\) 9.35747 0.362867
\(666\) 0 0
\(667\) 32.9939 1.27753
\(668\) 8.38192 0.324306
\(669\) 0 0
\(670\) −6.97259 −0.269375
\(671\) −26.2431 −1.01310
\(672\) 0 0
\(673\) 18.1684 0.700340 0.350170 0.936686i \(-0.386124\pi\)
0.350170 + 0.936686i \(0.386124\pi\)
\(674\) 4.53383 0.174636
\(675\) 0 0
\(676\) −5.92827 −0.228011
\(677\) 13.2991 0.511125 0.255563 0.966793i \(-0.417739\pi\)
0.255563 + 0.966793i \(0.417739\pi\)
\(678\) 0 0
\(679\) −1.35457 −0.0519836
\(680\) 2.96659 0.113763
\(681\) 0 0
\(682\) 14.8890 0.570130
\(683\) 25.8252 0.988175 0.494087 0.869412i \(-0.335502\pi\)
0.494087 + 0.869412i \(0.335502\pi\)
\(684\) 0 0
\(685\) 16.1389 0.616635
\(686\) 15.9119 0.607518
\(687\) 0 0
\(688\) 6.44584 0.245745
\(689\) 7.90886 0.301304
\(690\) 0 0
\(691\) −30.4652 −1.15895 −0.579476 0.814989i \(-0.696743\pi\)
−0.579476 + 0.814989i \(0.696743\pi\)
\(692\) 6.01306 0.228582
\(693\) 0 0
\(694\) 34.7965 1.32086
\(695\) −19.1647 −0.726957
\(696\) 0 0
\(697\) 12.5501 0.475371
\(698\) −21.9889 −0.832290
\(699\) 0 0
\(700\) 3.37030 0.127385
\(701\) 5.22755 0.197442 0.0987209 0.995115i \(-0.468525\pi\)
0.0987209 + 0.995115i \(0.468525\pi\)
\(702\) 0 0
\(703\) 26.7934 1.01053
\(704\) 2.49412 0.0940006
\(705\) 0 0
\(706\) 5.13896 0.193407
\(707\) −6.44725 −0.242474
\(708\) 0 0
\(709\) 30.7149 1.15352 0.576760 0.816913i \(-0.304317\pi\)
0.576760 + 0.816913i \(0.304317\pi\)
\(710\) 9.29041 0.348663
\(711\) 0 0
\(712\) 0.168223 0.00630443
\(713\) −21.6898 −0.812290
\(714\) 0 0
\(715\) 10.2473 0.383227
\(716\) −23.1043 −0.863449
\(717\) 0 0
\(718\) −11.4094 −0.425793
\(719\) 19.0574 0.710722 0.355361 0.934729i \(-0.384358\pi\)
0.355361 + 0.934729i \(0.384358\pi\)
\(720\) 0 0
\(721\) −19.6636 −0.732309
\(722\) 3.04884 0.113466
\(723\) 0 0
\(724\) −20.2695 −0.753310
\(725\) 23.7279 0.881232
\(726\) 0 0
\(727\) 39.1391 1.45159 0.725794 0.687913i \(-0.241473\pi\)
0.725794 + 0.687913i \(0.241473\pi\)
\(728\) 3.43004 0.127126
\(729\) 0 0
\(730\) 14.6216 0.541169
\(731\) −12.3768 −0.457771
\(732\) 0 0
\(733\) −18.2484 −0.674020 −0.337010 0.941501i \(-0.609416\pi\)
−0.337010 + 0.941501i \(0.609416\pi\)
\(734\) 23.4709 0.866326
\(735\) 0 0
\(736\) −3.63334 −0.133927
\(737\) 11.2559 0.414617
\(738\) 0 0
\(739\) −20.2613 −0.745323 −0.372661 0.927967i \(-0.621555\pi\)
−0.372661 + 0.927967i \(0.621555\pi\)
\(740\) −8.81586 −0.324077
\(741\) 0 0
\(742\) 3.83608 0.140827
\(743\) −6.60499 −0.242314 −0.121157 0.992633i \(-0.538660\pi\)
−0.121157 + 0.992633i \(0.538660\pi\)
\(744\) 0 0
\(745\) 4.17610 0.153001
\(746\) 14.7702 0.540776
\(747\) 0 0
\(748\) −4.78899 −0.175103
\(749\) −15.1111 −0.552146
\(750\) 0 0
\(751\) 29.0359 1.05953 0.529767 0.848143i \(-0.322279\pi\)
0.529767 + 0.848143i \(0.322279\pi\)
\(752\) −12.1071 −0.441500
\(753\) 0 0
\(754\) 24.1485 0.879435
\(755\) 34.0755 1.24013
\(756\) 0 0
\(757\) −27.4584 −0.997993 −0.498997 0.866604i \(-0.666298\pi\)
−0.498997 + 0.866604i \(0.666298\pi\)
\(758\) 9.14574 0.332188
\(759\) 0 0
\(760\) −7.25476 −0.263158
\(761\) −6.91985 −0.250844 −0.125422 0.992103i \(-0.540029\pi\)
−0.125422 + 0.992103i \(0.540029\pi\)
\(762\) 0 0
\(763\) 21.3452 0.772750
\(764\) −5.79518 −0.209662
\(765\) 0 0
\(766\) 15.2947 0.552618
\(767\) −33.6059 −1.21344
\(768\) 0 0
\(769\) −11.4749 −0.413796 −0.206898 0.978362i \(-0.566337\pi\)
−0.206898 + 0.978362i \(0.566337\pi\)
\(770\) 4.97030 0.179117
\(771\) 0 0
\(772\) 12.3097 0.443035
\(773\) −21.8130 −0.784559 −0.392279 0.919846i \(-0.628313\pi\)
−0.392279 + 0.919846i \(0.628313\pi\)
\(774\) 0 0
\(775\) −15.5985 −0.560313
\(776\) 1.05018 0.0376994
\(777\) 0 0
\(778\) 37.7337 1.35282
\(779\) −30.6912 −1.09963
\(780\) 0 0
\(781\) −14.9976 −0.536657
\(782\) 6.97644 0.249477
\(783\) 0 0
\(784\) −5.33631 −0.190583
\(785\) −11.4072 −0.407140
\(786\) 0 0
\(787\) 14.3504 0.511538 0.255769 0.966738i \(-0.417671\pi\)
0.255769 + 0.966738i \(0.417671\pi\)
\(788\) −18.7257 −0.667077
\(789\) 0 0
\(790\) 14.2659 0.507558
\(791\) 3.07911 0.109480
\(792\) 0 0
\(793\) 27.9809 0.993630
\(794\) −26.6903 −0.947204
\(795\) 0 0
\(796\) 3.65510 0.129552
\(797\) 2.16335 0.0766298 0.0383149 0.999266i \(-0.487801\pi\)
0.0383149 + 0.999266i \(0.487801\pi\)
\(798\) 0 0
\(799\) 23.2470 0.822419
\(800\) −2.61296 −0.0923821
\(801\) 0 0
\(802\) 31.3412 1.10670
\(803\) −23.6038 −0.832958
\(804\) 0 0
\(805\) −7.24056 −0.255196
\(806\) −15.8749 −0.559171
\(807\) 0 0
\(808\) 4.99849 0.175846
\(809\) −7.90979 −0.278093 −0.139047 0.990286i \(-0.544404\pi\)
−0.139047 + 0.990286i \(0.544404\pi\)
\(810\) 0 0
\(811\) −23.1783 −0.813900 −0.406950 0.913450i \(-0.633408\pi\)
−0.406950 + 0.913450i \(0.633408\pi\)
\(812\) 11.7128 0.411040
\(813\) 0 0
\(814\) 14.2315 0.498815
\(815\) 6.59659 0.231068
\(816\) 0 0
\(817\) 30.2672 1.05892
\(818\) 38.0880 1.33172
\(819\) 0 0
\(820\) 10.0984 0.352650
\(821\) 13.0185 0.454347 0.227174 0.973854i \(-0.427051\pi\)
0.227174 + 0.973854i \(0.427051\pi\)
\(822\) 0 0
\(823\) −23.0296 −0.802762 −0.401381 0.915911i \(-0.631470\pi\)
−0.401381 + 0.915911i \(0.631470\pi\)
\(824\) 15.2450 0.531083
\(825\) 0 0
\(826\) −16.3000 −0.567150
\(827\) 48.7420 1.69492 0.847462 0.530856i \(-0.178129\pi\)
0.847462 + 0.530856i \(0.178129\pi\)
\(828\) 0 0
\(829\) −3.02728 −0.105142 −0.0525709 0.998617i \(-0.516742\pi\)
−0.0525709 + 0.998617i \(0.516742\pi\)
\(830\) 20.0114 0.694607
\(831\) 0 0
\(832\) −2.65927 −0.0921937
\(833\) 10.2463 0.355014
\(834\) 0 0
\(835\) −12.9501 −0.448157
\(836\) 11.7114 0.405048
\(837\) 0 0
\(838\) −22.7446 −0.785700
\(839\) 21.2196 0.732583 0.366292 0.930500i \(-0.380627\pi\)
0.366292 + 0.930500i \(0.380627\pi\)
\(840\) 0 0
\(841\) 53.4619 1.84351
\(842\) −21.6762 −0.747012
\(843\) 0 0
\(844\) 26.2300 0.902873
\(845\) 9.15921 0.315087
\(846\) 0 0
\(847\) 6.16464 0.211819
\(848\) −2.97407 −0.102130
\(849\) 0 0
\(850\) 5.01718 0.172088
\(851\) −20.7320 −0.710684
\(852\) 0 0
\(853\) −23.0700 −0.789901 −0.394950 0.918703i \(-0.629238\pi\)
−0.394950 + 0.918703i \(0.629238\pi\)
\(854\) 13.5717 0.464414
\(855\) 0 0
\(856\) 11.7155 0.400426
\(857\) −2.90304 −0.0991661 −0.0495831 0.998770i \(-0.515789\pi\)
−0.0495831 + 0.998770i \(0.515789\pi\)
\(858\) 0 0
\(859\) 30.4007 1.03726 0.518629 0.854999i \(-0.326443\pi\)
0.518629 + 0.854999i \(0.326443\pi\)
\(860\) −9.95885 −0.339594
\(861\) 0 0
\(862\) 24.7372 0.842553
\(863\) −38.0933 −1.29671 −0.648356 0.761337i \(-0.724543\pi\)
−0.648356 + 0.761337i \(0.724543\pi\)
\(864\) 0 0
\(865\) −9.29021 −0.315877
\(866\) 29.6583 1.00783
\(867\) 0 0
\(868\) −7.69990 −0.261352
\(869\) −23.0296 −0.781226
\(870\) 0 0
\(871\) −12.0013 −0.406647
\(872\) −16.5487 −0.560411
\(873\) 0 0
\(874\) −17.0608 −0.577090
\(875\) −15.1712 −0.512879
\(876\) 0 0
\(877\) −45.3907 −1.53274 −0.766368 0.642402i \(-0.777938\pi\)
−0.766368 + 0.642402i \(0.777938\pi\)
\(878\) 27.9533 0.943379
\(879\) 0 0
\(880\) −3.85342 −0.129899
\(881\) −17.8676 −0.601976 −0.300988 0.953628i \(-0.597316\pi\)
−0.300988 + 0.953628i \(0.597316\pi\)
\(882\) 0 0
\(883\) 25.3660 0.853634 0.426817 0.904338i \(-0.359635\pi\)
0.426817 + 0.904338i \(0.359635\pi\)
\(884\) 5.10611 0.171737
\(885\) 0 0
\(886\) −21.5451 −0.723822
\(887\) 31.6442 1.06251 0.531255 0.847212i \(-0.321721\pi\)
0.531255 + 0.847212i \(0.321721\pi\)
\(888\) 0 0
\(889\) 18.4119 0.617517
\(890\) −0.259905 −0.00871205
\(891\) 0 0
\(892\) −1.00000 −0.0334825
\(893\) −56.8502 −1.90242
\(894\) 0 0
\(895\) 35.6963 1.19320
\(896\) −1.28984 −0.0430905
\(897\) 0 0
\(898\) −38.3133 −1.27853
\(899\) −54.2096 −1.80799
\(900\) 0 0
\(901\) 5.71056 0.190246
\(902\) −16.3019 −0.542794
\(903\) 0 0
\(904\) −2.38720 −0.0793971
\(905\) 31.3165 1.04099
\(906\) 0 0
\(907\) 27.9750 0.928894 0.464447 0.885601i \(-0.346253\pi\)
0.464447 + 0.885601i \(0.346253\pi\)
\(908\) 27.2280 0.903592
\(909\) 0 0
\(910\) −5.29942 −0.175674
\(911\) −8.88228 −0.294283 −0.147142 0.989115i \(-0.547007\pi\)
−0.147142 + 0.989115i \(0.547007\pi\)
\(912\) 0 0
\(913\) −32.3046 −1.06913
\(914\) −8.77811 −0.290354
\(915\) 0 0
\(916\) −2.43949 −0.0806029
\(917\) 25.7902 0.851667
\(918\) 0 0
\(919\) −38.5695 −1.27229 −0.636146 0.771569i \(-0.719472\pi\)
−0.636146 + 0.771569i \(0.719472\pi\)
\(920\) 5.61353 0.185073
\(921\) 0 0
\(922\) 14.4028 0.474332
\(923\) 15.9907 0.526341
\(924\) 0 0
\(925\) −14.9096 −0.490226
\(926\) 13.6803 0.449563
\(927\) 0 0
\(928\) −9.08085 −0.298094
\(929\) 16.4905 0.541037 0.270518 0.962715i \(-0.412805\pi\)
0.270518 + 0.962715i \(0.412805\pi\)
\(930\) 0 0
\(931\) −25.0573 −0.821220
\(932\) 11.3011 0.370179
\(933\) 0 0
\(934\) −8.37193 −0.273938
\(935\) 7.39902 0.241974
\(936\) 0 0
\(937\) −19.5436 −0.638462 −0.319231 0.947677i \(-0.603425\pi\)
−0.319231 + 0.947677i \(0.603425\pi\)
\(938\) −5.82103 −0.190063
\(939\) 0 0
\(940\) 18.7055 0.610106
\(941\) 58.8382 1.91807 0.959035 0.283287i \(-0.0914250\pi\)
0.959035 + 0.283287i \(0.0914250\pi\)
\(942\) 0 0
\(943\) 23.7480 0.773343
\(944\) 12.6372 0.411307
\(945\) 0 0
\(946\) 16.0767 0.522698
\(947\) 19.1164 0.621199 0.310600 0.950541i \(-0.399470\pi\)
0.310600 + 0.950541i \(0.399470\pi\)
\(948\) 0 0
\(949\) 25.1667 0.816947
\(950\) −12.2695 −0.398074
\(951\) 0 0
\(952\) 2.47664 0.0802684
\(953\) 8.48256 0.274777 0.137389 0.990517i \(-0.456129\pi\)
0.137389 + 0.990517i \(0.456129\pi\)
\(954\) 0 0
\(955\) 8.95359 0.289731
\(956\) 8.55819 0.276792
\(957\) 0 0
\(958\) 21.7260 0.701936
\(959\) 13.4735 0.435081
\(960\) 0 0
\(961\) 4.63679 0.149574
\(962\) −15.1739 −0.489226
\(963\) 0 0
\(964\) 17.4393 0.561681
\(965\) −19.0185 −0.612228
\(966\) 0 0
\(967\) −6.04992 −0.194552 −0.0972762 0.995257i \(-0.531013\pi\)
−0.0972762 + 0.995257i \(0.531013\pi\)
\(968\) −4.77938 −0.153615
\(969\) 0 0
\(970\) −1.62254 −0.0520967
\(971\) −8.71377 −0.279638 −0.139819 0.990177i \(-0.544652\pi\)
−0.139819 + 0.990177i \(0.544652\pi\)
\(972\) 0 0
\(973\) −15.9995 −0.512921
\(974\) 2.38053 0.0762770
\(975\) 0 0
\(976\) −10.5220 −0.336801
\(977\) 59.1766 1.89323 0.946614 0.322370i \(-0.104479\pi\)
0.946614 + 0.322370i \(0.104479\pi\)
\(978\) 0 0
\(979\) 0.419568 0.0134095
\(980\) 8.24463 0.263365
\(981\) 0 0
\(982\) −20.8266 −0.664603
\(983\) −28.0539 −0.894780 −0.447390 0.894339i \(-0.647646\pi\)
−0.447390 + 0.894339i \(0.647646\pi\)
\(984\) 0 0
\(985\) 28.9314 0.921830
\(986\) 17.4363 0.555284
\(987\) 0 0
\(988\) −12.4869 −0.397262
\(989\) −23.4199 −0.744711
\(990\) 0 0
\(991\) 5.76653 0.183180 0.0915899 0.995797i \(-0.470805\pi\)
0.0915899 + 0.995797i \(0.470805\pi\)
\(992\) 5.96966 0.189537
\(993\) 0 0
\(994\) 7.75606 0.246007
\(995\) −5.64715 −0.179027
\(996\) 0 0
\(997\) −27.9499 −0.885183 −0.442592 0.896723i \(-0.645941\pi\)
−0.442592 + 0.896723i \(0.645941\pi\)
\(998\) 18.7813 0.594511
\(999\) 0 0
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 4014.2.a.r.1.2 5
3.2 odd 2 1338.2.a.h.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1338.2.a.h.1.4 5 3.2 odd 2
4014.2.a.r.1.2 5 1.1 even 1 trivial