Properties

Label 4002.2.a.x.1.1
Level 4002
Weight 2
Character 4002.1
Self dual yes
Analytic conductor 31.956
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 4002 = 2 \cdot 3 \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4002.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.9561308889\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Defining polynomial: \(x^{2} - 6\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.44949\) of defining polynomial
Character \(\chi\) \(=\) 4002.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} -0.449490 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.00000 q^{5} +1.00000 q^{6} -0.449490 q^{7} +1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{10} -4.89898 q^{11} +1.00000 q^{12} +2.89898 q^{13} -0.449490 q^{14} +4.00000 q^{15} +1.00000 q^{16} +4.89898 q^{17} +1.00000 q^{18} +0.449490 q^{19} +4.00000 q^{20} -0.449490 q^{21} -4.89898 q^{22} -1.00000 q^{23} +1.00000 q^{24} +11.0000 q^{25} +2.89898 q^{26} +1.00000 q^{27} -0.449490 q^{28} +1.00000 q^{29} +4.00000 q^{30} -6.00000 q^{31} +1.00000 q^{32} -4.89898 q^{33} +4.89898 q^{34} -1.79796 q^{35} +1.00000 q^{36} -2.00000 q^{37} +0.449490 q^{38} +2.89898 q^{39} +4.00000 q^{40} +0.898979 q^{41} -0.449490 q^{42} +7.55051 q^{43} -4.89898 q^{44} +4.00000 q^{45} -1.00000 q^{46} +4.00000 q^{47} +1.00000 q^{48} -6.79796 q^{49} +11.0000 q^{50} +4.89898 q^{51} +2.89898 q^{52} +0.898979 q^{53} +1.00000 q^{54} -19.5959 q^{55} -0.449490 q^{56} +0.449490 q^{57} +1.00000 q^{58} +4.89898 q^{59} +4.00000 q^{60} -6.89898 q^{61} -6.00000 q^{62} -0.449490 q^{63} +1.00000 q^{64} +11.5959 q^{65} -4.89898 q^{66} +2.00000 q^{67} +4.89898 q^{68} -1.00000 q^{69} -1.79796 q^{70} -1.10102 q^{71} +1.00000 q^{72} +2.89898 q^{73} -2.00000 q^{74} +11.0000 q^{75} +0.449490 q^{76} +2.20204 q^{77} +2.89898 q^{78} +10.0000 q^{79} +4.00000 q^{80} +1.00000 q^{81} +0.898979 q^{82} -6.44949 q^{83} -0.449490 q^{84} +19.5959 q^{85} +7.55051 q^{86} +1.00000 q^{87} -4.89898 q^{88} +3.10102 q^{89} +4.00000 q^{90} -1.30306 q^{91} -1.00000 q^{92} -6.00000 q^{93} +4.00000 q^{94} +1.79796 q^{95} +1.00000 q^{96} +15.1464 q^{97} -6.79796 q^{98} -4.89898 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 8q^{5} + 2q^{6} + 4q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{3} + 2q^{4} + 8q^{5} + 2q^{6} + 4q^{7} + 2q^{8} + 2q^{9} + 8q^{10} + 2q^{12} - 4q^{13} + 4q^{14} + 8q^{15} + 2q^{16} + 2q^{18} - 4q^{19} + 8q^{20} + 4q^{21} - 2q^{23} + 2q^{24} + 22q^{25} - 4q^{26} + 2q^{27} + 4q^{28} + 2q^{29} + 8q^{30} - 12q^{31} + 2q^{32} + 16q^{35} + 2q^{36} - 4q^{37} - 4q^{38} - 4q^{39} + 8q^{40} - 8q^{41} + 4q^{42} + 20q^{43} + 8q^{45} - 2q^{46} + 8q^{47} + 2q^{48} + 6q^{49} + 22q^{50} - 4q^{52} - 8q^{53} + 2q^{54} + 4q^{56} - 4q^{57} + 2q^{58} + 8q^{60} - 4q^{61} - 12q^{62} + 4q^{63} + 2q^{64} - 16q^{65} + 4q^{67} - 2q^{69} + 16q^{70} - 12q^{71} + 2q^{72} - 4q^{73} - 4q^{74} + 22q^{75} - 4q^{76} + 24q^{77} - 4q^{78} + 20q^{79} + 8q^{80} + 2q^{81} - 8q^{82} - 8q^{83} + 4q^{84} + 20q^{86} + 2q^{87} + 16q^{89} + 8q^{90} - 32q^{91} - 2q^{92} - 12q^{93} + 8q^{94} - 16q^{95} + 2q^{96} - 4q^{97} + 6q^{98} + O(q^{100}) \)

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
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 4.00000 1.78885 0.894427 0.447214i \(-0.147584\pi\)
0.894427 + 0.447214i \(0.147584\pi\)
\(6\) 1.00000 0.408248
\(7\) −0.449490 −0.169891 −0.0849456 0.996386i \(-0.527072\pi\)
−0.0849456 + 0.996386i \(0.527072\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 4.00000 1.26491
\(11\) −4.89898 −1.47710 −0.738549 0.674200i \(-0.764489\pi\)
−0.738549 + 0.674200i \(0.764489\pi\)
\(12\) 1.00000 0.288675
\(13\) 2.89898 0.804032 0.402016 0.915633i \(-0.368310\pi\)
0.402016 + 0.915633i \(0.368310\pi\)
\(14\) −0.449490 −0.120131
\(15\) 4.00000 1.03280
\(16\) 1.00000 0.250000
\(17\) 4.89898 1.18818 0.594089 0.804400i \(-0.297513\pi\)
0.594089 + 0.804400i \(0.297513\pi\)
\(18\) 1.00000 0.235702
\(19\) 0.449490 0.103120 0.0515600 0.998670i \(-0.483581\pi\)
0.0515600 + 0.998670i \(0.483581\pi\)
\(20\) 4.00000 0.894427
\(21\) −0.449490 −0.0980867
\(22\) −4.89898 −1.04447
\(23\) −1.00000 −0.208514
\(24\) 1.00000 0.204124
\(25\) 11.0000 2.20000
\(26\) 2.89898 0.568537
\(27\) 1.00000 0.192450
\(28\) −0.449490 −0.0849456
\(29\) 1.00000 0.185695
\(30\) 4.00000 0.730297
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 1.00000 0.176777
\(33\) −4.89898 −0.852803
\(34\) 4.89898 0.840168
\(35\) −1.79796 −0.303911
\(36\) 1.00000 0.166667
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 0.449490 0.0729169
\(39\) 2.89898 0.464208
\(40\) 4.00000 0.632456
\(41\) 0.898979 0.140397 0.0701985 0.997533i \(-0.477637\pi\)
0.0701985 + 0.997533i \(0.477637\pi\)
\(42\) −0.449490 −0.0693578
\(43\) 7.55051 1.15144 0.575721 0.817646i \(-0.304721\pi\)
0.575721 + 0.817646i \(0.304721\pi\)
\(44\) −4.89898 −0.738549
\(45\) 4.00000 0.596285
\(46\) −1.00000 −0.147442
\(47\) 4.00000 0.583460 0.291730 0.956501i \(-0.405769\pi\)
0.291730 + 0.956501i \(0.405769\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.79796 −0.971137
\(50\) 11.0000 1.55563
\(51\) 4.89898 0.685994
\(52\) 2.89898 0.402016
\(53\) 0.898979 0.123484 0.0617422 0.998092i \(-0.480334\pi\)
0.0617422 + 0.998092i \(0.480334\pi\)
\(54\) 1.00000 0.136083
\(55\) −19.5959 −2.64231
\(56\) −0.449490 −0.0600656
\(57\) 0.449490 0.0595364
\(58\) 1.00000 0.131306
\(59\) 4.89898 0.637793 0.318896 0.947790i \(-0.396688\pi\)
0.318896 + 0.947790i \(0.396688\pi\)
\(60\) 4.00000 0.516398
\(61\) −6.89898 −0.883324 −0.441662 0.897182i \(-0.645611\pi\)
−0.441662 + 0.897182i \(0.645611\pi\)
\(62\) −6.00000 −0.762001
\(63\) −0.449490 −0.0566304
\(64\) 1.00000 0.125000
\(65\) 11.5959 1.43830
\(66\) −4.89898 −0.603023
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) 4.89898 0.594089
\(69\) −1.00000 −0.120386
\(70\) −1.79796 −0.214897
\(71\) −1.10102 −0.130667 −0.0653335 0.997863i \(-0.520811\pi\)
−0.0653335 + 0.997863i \(0.520811\pi\)
\(72\) 1.00000 0.117851
\(73\) 2.89898 0.339300 0.169650 0.985504i \(-0.445736\pi\)
0.169650 + 0.985504i \(0.445736\pi\)
\(74\) −2.00000 −0.232495
\(75\) 11.0000 1.27017
\(76\) 0.449490 0.0515600
\(77\) 2.20204 0.250946
\(78\) 2.89898 0.328245
\(79\) 10.0000 1.12509 0.562544 0.826767i \(-0.309823\pi\)
0.562544 + 0.826767i \(0.309823\pi\)
\(80\) 4.00000 0.447214
\(81\) 1.00000 0.111111
\(82\) 0.898979 0.0992757
\(83\) −6.44949 −0.707923 −0.353962 0.935260i \(-0.615166\pi\)
−0.353962 + 0.935260i \(0.615166\pi\)
\(84\) −0.449490 −0.0490434
\(85\) 19.5959 2.12548
\(86\) 7.55051 0.814192
\(87\) 1.00000 0.107211
\(88\) −4.89898 −0.522233
\(89\) 3.10102 0.328708 0.164354 0.986401i \(-0.447446\pi\)
0.164354 + 0.986401i \(0.447446\pi\)
\(90\) 4.00000 0.421637
\(91\) −1.30306 −0.136598
\(92\) −1.00000 −0.104257
\(93\) −6.00000 −0.622171
\(94\) 4.00000 0.412568
\(95\) 1.79796 0.184467
\(96\) 1.00000 0.102062
\(97\) 15.1464 1.53789 0.768943 0.639317i \(-0.220783\pi\)
0.768943 + 0.639317i \(0.220783\pi\)
\(98\) −6.79796 −0.686698
\(99\) −4.89898 −0.492366
\(100\) 11.0000 1.10000
\(101\) −14.8990 −1.48250 −0.741252 0.671227i \(-0.765768\pi\)
−0.741252 + 0.671227i \(0.765768\pi\)
\(102\) 4.89898 0.485071
\(103\) −18.2474 −1.79797 −0.898987 0.437975i \(-0.855696\pi\)
−0.898987 + 0.437975i \(0.855696\pi\)
\(104\) 2.89898 0.284268
\(105\) −1.79796 −0.175463
\(106\) 0.898979 0.0873166
\(107\) −14.4495 −1.39688 −0.698442 0.715666i \(-0.746123\pi\)
−0.698442 + 0.715666i \(0.746123\pi\)
\(108\) 1.00000 0.0962250
\(109\) 0.449490 0.0430533 0.0215267 0.999768i \(-0.493147\pi\)
0.0215267 + 0.999768i \(0.493147\pi\)
\(110\) −19.5959 −1.86840
\(111\) −2.00000 −0.189832
\(112\) −0.449490 −0.0424728
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0.449490 0.0420986
\(115\) −4.00000 −0.373002
\(116\) 1.00000 0.0928477
\(117\) 2.89898 0.268011
\(118\) 4.89898 0.450988
\(119\) −2.20204 −0.201861
\(120\) 4.00000 0.365148
\(121\) 13.0000 1.18182
\(122\) −6.89898 −0.624604
\(123\) 0.898979 0.0810583
\(124\) −6.00000 −0.538816
\(125\) 24.0000 2.14663
\(126\) −0.449490 −0.0400437
\(127\) −13.7980 −1.22437 −0.612185 0.790714i \(-0.709709\pi\)
−0.612185 + 0.790714i \(0.709709\pi\)
\(128\) 1.00000 0.0883883
\(129\) 7.55051 0.664785
\(130\) 11.5959 1.01703
\(131\) 9.10102 0.795160 0.397580 0.917568i \(-0.369850\pi\)
0.397580 + 0.917568i \(0.369850\pi\)
\(132\) −4.89898 −0.426401
\(133\) −0.202041 −0.0175192
\(134\) 2.00000 0.172774
\(135\) 4.00000 0.344265
\(136\) 4.89898 0.420084
\(137\) 13.7980 1.17884 0.589420 0.807827i \(-0.299356\pi\)
0.589420 + 0.807827i \(0.299356\pi\)
\(138\) −1.00000 −0.0851257
\(139\) 6.00000 0.508913 0.254457 0.967084i \(-0.418103\pi\)
0.254457 + 0.967084i \(0.418103\pi\)
\(140\) −1.79796 −0.151955
\(141\) 4.00000 0.336861
\(142\) −1.10102 −0.0923956
\(143\) −14.2020 −1.18763
\(144\) 1.00000 0.0833333
\(145\) 4.00000 0.332182
\(146\) 2.89898 0.239921
\(147\) −6.79796 −0.560686
\(148\) −2.00000 −0.164399
\(149\) 10.6969 0.876327 0.438164 0.898895i \(-0.355629\pi\)
0.438164 + 0.898895i \(0.355629\pi\)
\(150\) 11.0000 0.898146
\(151\) 5.79796 0.471831 0.235916 0.971774i \(-0.424191\pi\)
0.235916 + 0.971774i \(0.424191\pi\)
\(152\) 0.449490 0.0364584
\(153\) 4.89898 0.396059
\(154\) 2.20204 0.177446
\(155\) −24.0000 −1.92773
\(156\) 2.89898 0.232104
\(157\) 3.79796 0.303110 0.151555 0.988449i \(-0.451572\pi\)
0.151555 + 0.988449i \(0.451572\pi\)
\(158\) 10.0000 0.795557
\(159\) 0.898979 0.0712937
\(160\) 4.00000 0.316228
\(161\) 0.449490 0.0354248
\(162\) 1.00000 0.0785674
\(163\) −8.89898 −0.697022 −0.348511 0.937305i \(-0.613313\pi\)
−0.348511 + 0.937305i \(0.613313\pi\)
\(164\) 0.898979 0.0701985
\(165\) −19.5959 −1.52554
\(166\) −6.44949 −0.500577
\(167\) −9.79796 −0.758189 −0.379094 0.925358i \(-0.623764\pi\)
−0.379094 + 0.925358i \(0.623764\pi\)
\(168\) −0.449490 −0.0346789
\(169\) −4.59592 −0.353532
\(170\) 19.5959 1.50294
\(171\) 0.449490 0.0343733
\(172\) 7.55051 0.575721
\(173\) −12.8990 −0.980691 −0.490346 0.871528i \(-0.663129\pi\)
−0.490346 + 0.871528i \(0.663129\pi\)
\(174\) 1.00000 0.0758098
\(175\) −4.94439 −0.373761
\(176\) −4.89898 −0.369274
\(177\) 4.89898 0.368230
\(178\) 3.10102 0.232431
\(179\) −14.6969 −1.09850 −0.549250 0.835658i \(-0.685087\pi\)
−0.549250 + 0.835658i \(0.685087\pi\)
\(180\) 4.00000 0.298142
\(181\) −10.6515 −0.791722 −0.395861 0.918310i \(-0.629554\pi\)
−0.395861 + 0.918310i \(0.629554\pi\)
\(182\) −1.30306 −0.0965893
\(183\) −6.89898 −0.509987
\(184\) −1.00000 −0.0737210
\(185\) −8.00000 −0.588172
\(186\) −6.00000 −0.439941
\(187\) −24.0000 −1.75505
\(188\) 4.00000 0.291730
\(189\) −0.449490 −0.0326956
\(190\) 1.79796 0.130438
\(191\) −7.34847 −0.531717 −0.265858 0.964012i \(-0.585655\pi\)
−0.265858 + 0.964012i \(0.585655\pi\)
\(192\) 1.00000 0.0721688
\(193\) −17.5959 −1.26658 −0.633291 0.773914i \(-0.718296\pi\)
−0.633291 + 0.773914i \(0.718296\pi\)
\(194\) 15.1464 1.08745
\(195\) 11.5959 0.830401
\(196\) −6.79796 −0.485568
\(197\) −15.7980 −1.12556 −0.562779 0.826607i \(-0.690268\pi\)
−0.562779 + 0.826607i \(0.690268\pi\)
\(198\) −4.89898 −0.348155
\(199\) −1.75255 −0.124235 −0.0621175 0.998069i \(-0.519785\pi\)
−0.0621175 + 0.998069i \(0.519785\pi\)
\(200\) 11.0000 0.777817
\(201\) 2.00000 0.141069
\(202\) −14.8990 −1.04829
\(203\) −0.449490 −0.0315480
\(204\) 4.89898 0.342997
\(205\) 3.59592 0.251150
\(206\) −18.2474 −1.27136
\(207\) −1.00000 −0.0695048
\(208\) 2.89898 0.201008
\(209\) −2.20204 −0.152318
\(210\) −1.79796 −0.124071
\(211\) 15.1010 1.03960 0.519799 0.854289i \(-0.326007\pi\)
0.519799 + 0.854289i \(0.326007\pi\)
\(212\) 0.898979 0.0617422
\(213\) −1.10102 −0.0754407
\(214\) −14.4495 −0.987747
\(215\) 30.2020 2.05976
\(216\) 1.00000 0.0680414
\(217\) 2.69694 0.183080
\(218\) 0.449490 0.0304433
\(219\) 2.89898 0.195895
\(220\) −19.5959 −1.32116
\(221\) 14.2020 0.955333
\(222\) −2.00000 −0.134231
\(223\) −5.79796 −0.388260 −0.194130 0.980976i \(-0.562188\pi\)
−0.194130 + 0.980976i \(0.562188\pi\)
\(224\) −0.449490 −0.0300328
\(225\) 11.0000 0.733333
\(226\) 0 0
\(227\) −7.34847 −0.487735 −0.243868 0.969809i \(-0.578416\pi\)
−0.243868 + 0.969809i \(0.578416\pi\)
\(228\) 0.449490 0.0297682
\(229\) 16.6969 1.10336 0.551682 0.834054i \(-0.313986\pi\)
0.551682 + 0.834054i \(0.313986\pi\)
\(230\) −4.00000 −0.263752
\(231\) 2.20204 0.144884
\(232\) 1.00000 0.0656532
\(233\) −12.6969 −0.831804 −0.415902 0.909409i \(-0.636534\pi\)
−0.415902 + 0.909409i \(0.636534\pi\)
\(234\) 2.89898 0.189512
\(235\) 16.0000 1.04372
\(236\) 4.89898 0.318896
\(237\) 10.0000 0.649570
\(238\) −2.20204 −0.142737
\(239\) −10.8990 −0.704996 −0.352498 0.935812i \(-0.614668\pi\)
−0.352498 + 0.935812i \(0.614668\pi\)
\(240\) 4.00000 0.258199
\(241\) −8.69694 −0.560219 −0.280110 0.959968i \(-0.590371\pi\)
−0.280110 + 0.959968i \(0.590371\pi\)
\(242\) 13.0000 0.835672
\(243\) 1.00000 0.0641500
\(244\) −6.89898 −0.441662
\(245\) −27.1918 −1.73722
\(246\) 0.898979 0.0573168
\(247\) 1.30306 0.0829118
\(248\) −6.00000 −0.381000
\(249\) −6.44949 −0.408720
\(250\) 24.0000 1.51789
\(251\) 12.8990 0.814176 0.407088 0.913389i \(-0.366544\pi\)
0.407088 + 0.913389i \(0.366544\pi\)
\(252\) −0.449490 −0.0283152
\(253\) 4.89898 0.307996
\(254\) −13.7980 −0.865761
\(255\) 19.5959 1.22714
\(256\) 1.00000 0.0625000
\(257\) −10.8990 −0.679860 −0.339930 0.940451i \(-0.610403\pi\)
−0.339930 + 0.940451i \(0.610403\pi\)
\(258\) 7.55051 0.470074
\(259\) 0.898979 0.0558599
\(260\) 11.5959 0.719148
\(261\) 1.00000 0.0618984
\(262\) 9.10102 0.562263
\(263\) 3.75255 0.231392 0.115696 0.993285i \(-0.463090\pi\)
0.115696 + 0.993285i \(0.463090\pi\)
\(264\) −4.89898 −0.301511
\(265\) 3.59592 0.220895
\(266\) −0.202041 −0.0123879
\(267\) 3.10102 0.189779
\(268\) 2.00000 0.122169
\(269\) 10.8990 0.664523 0.332261 0.943187i \(-0.392188\pi\)
0.332261 + 0.943187i \(0.392188\pi\)
\(270\) 4.00000 0.243432
\(271\) −13.7980 −0.838166 −0.419083 0.907948i \(-0.637648\pi\)
−0.419083 + 0.907948i \(0.637648\pi\)
\(272\) 4.89898 0.297044
\(273\) −1.30306 −0.0788649
\(274\) 13.7980 0.833565
\(275\) −53.8888 −3.24962
\(276\) −1.00000 −0.0601929
\(277\) 24.6969 1.48390 0.741948 0.670458i \(-0.233902\pi\)
0.741948 + 0.670458i \(0.233902\pi\)
\(278\) 6.00000 0.359856
\(279\) −6.00000 −0.359211
\(280\) −1.79796 −0.107449
\(281\) −26.4495 −1.57784 −0.788922 0.614493i \(-0.789361\pi\)
−0.788922 + 0.614493i \(0.789361\pi\)
\(282\) 4.00000 0.238197
\(283\) −21.5959 −1.28374 −0.641872 0.766812i \(-0.721842\pi\)
−0.641872 + 0.766812i \(0.721842\pi\)
\(284\) −1.10102 −0.0653335
\(285\) 1.79796 0.106502
\(286\) −14.2020 −0.839784
\(287\) −0.404082 −0.0238522
\(288\) 1.00000 0.0589256
\(289\) 7.00000 0.411765
\(290\) 4.00000 0.234888
\(291\) 15.1464 0.887899
\(292\) 2.89898 0.169650
\(293\) −9.14643 −0.534340 −0.267170 0.963649i \(-0.586089\pi\)
−0.267170 + 0.963649i \(0.586089\pi\)
\(294\) −6.79796 −0.396465
\(295\) 19.5959 1.14092
\(296\) −2.00000 −0.116248
\(297\) −4.89898 −0.284268
\(298\) 10.6969 0.619657
\(299\) −2.89898 −0.167652
\(300\) 11.0000 0.635085
\(301\) −3.39388 −0.195620
\(302\) 5.79796 0.333635
\(303\) −14.8990 −0.855924
\(304\) 0.449490 0.0257800
\(305\) −27.5959 −1.58014
\(306\) 4.89898 0.280056
\(307\) 6.69694 0.382214 0.191107 0.981569i \(-0.438792\pi\)
0.191107 + 0.981569i \(0.438792\pi\)
\(308\) 2.20204 0.125473
\(309\) −18.2474 −1.03806
\(310\) −24.0000 −1.36311
\(311\) 26.6969 1.51384 0.756922 0.653505i \(-0.226702\pi\)
0.756922 + 0.653505i \(0.226702\pi\)
\(312\) 2.89898 0.164122
\(313\) 17.5959 0.994580 0.497290 0.867584i \(-0.334328\pi\)
0.497290 + 0.867584i \(0.334328\pi\)
\(314\) 3.79796 0.214331
\(315\) −1.79796 −0.101304
\(316\) 10.0000 0.562544
\(317\) 1.10102 0.0618395 0.0309197 0.999522i \(-0.490156\pi\)
0.0309197 + 0.999522i \(0.490156\pi\)
\(318\) 0.898979 0.0504123
\(319\) −4.89898 −0.274290
\(320\) 4.00000 0.223607
\(321\) −14.4495 −0.806492
\(322\) 0.449490 0.0250491
\(323\) 2.20204 0.122525
\(324\) 1.00000 0.0555556
\(325\) 31.8888 1.76887
\(326\) −8.89898 −0.492869
\(327\) 0.449490 0.0248568
\(328\) 0.898979 0.0496378
\(329\) −1.79796 −0.0991247
\(330\) −19.5959 −1.07872
\(331\) −20.8990 −1.14871 −0.574356 0.818606i \(-0.694747\pi\)
−0.574356 + 0.818606i \(0.694747\pi\)
\(332\) −6.44949 −0.353962
\(333\) −2.00000 −0.109599
\(334\) −9.79796 −0.536120
\(335\) 8.00000 0.437087
\(336\) −0.449490 −0.0245217
\(337\) 15.5505 0.847090 0.423545 0.905875i \(-0.360786\pi\)
0.423545 + 0.905875i \(0.360786\pi\)
\(338\) −4.59592 −0.249985
\(339\) 0 0
\(340\) 19.5959 1.06274
\(341\) 29.3939 1.59177
\(342\) 0.449490 0.0243056
\(343\) 6.20204 0.334879
\(344\) 7.55051 0.407096
\(345\) −4.00000 −0.215353
\(346\) −12.8990 −0.693453
\(347\) 19.5959 1.05196 0.525982 0.850496i \(-0.323698\pi\)
0.525982 + 0.850496i \(0.323698\pi\)
\(348\) 1.00000 0.0536056
\(349\) 14.0000 0.749403 0.374701 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) −4.94439 −0.264289
\(351\) 2.89898 0.154736
\(352\) −4.89898 −0.261116
\(353\) 15.7980 0.840841 0.420420 0.907329i \(-0.361883\pi\)
0.420420 + 0.907329i \(0.361883\pi\)
\(354\) 4.89898 0.260378
\(355\) −4.40408 −0.233744
\(356\) 3.10102 0.164354
\(357\) −2.20204 −0.116544
\(358\) −14.6969 −0.776757
\(359\) −5.55051 −0.292945 −0.146472 0.989215i \(-0.546792\pi\)
−0.146472 + 0.989215i \(0.546792\pi\)
\(360\) 4.00000 0.210819
\(361\) −18.7980 −0.989366
\(362\) −10.6515 −0.559832
\(363\) 13.0000 0.682323
\(364\) −1.30306 −0.0682990
\(365\) 11.5959 0.606958
\(366\) −6.89898 −0.360615
\(367\) −30.0000 −1.56599 −0.782994 0.622030i \(-0.786308\pi\)
−0.782994 + 0.622030i \(0.786308\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 0.898979 0.0467990
\(370\) −8.00000 −0.415900
\(371\) −0.404082 −0.0209789
\(372\) −6.00000 −0.311086
\(373\) −20.0454 −1.03791 −0.518956 0.854801i \(-0.673679\pi\)
−0.518956 + 0.854801i \(0.673679\pi\)
\(374\) −24.0000 −1.24101
\(375\) 24.0000 1.23935
\(376\) 4.00000 0.206284
\(377\) 2.89898 0.149305
\(378\) −0.449490 −0.0231193
\(379\) 17.3485 0.891131 0.445566 0.895249i \(-0.353003\pi\)
0.445566 + 0.895249i \(0.353003\pi\)
\(380\) 1.79796 0.0922333
\(381\) −13.7980 −0.706891
\(382\) −7.34847 −0.375980
\(383\) −9.79796 −0.500652 −0.250326 0.968162i \(-0.580538\pi\)
−0.250326 + 0.968162i \(0.580538\pi\)
\(384\) 1.00000 0.0510310
\(385\) 8.80816 0.448906
\(386\) −17.5959 −0.895609
\(387\) 7.55051 0.383814
\(388\) 15.1464 0.768943
\(389\) 31.3485 1.58943 0.794715 0.606982i \(-0.207620\pi\)
0.794715 + 0.606982i \(0.207620\pi\)
\(390\) 11.5959 0.587182
\(391\) −4.89898 −0.247752
\(392\) −6.79796 −0.343349
\(393\) 9.10102 0.459086
\(394\) −15.7980 −0.795890
\(395\) 40.0000 2.01262
\(396\) −4.89898 −0.246183
\(397\) −13.5959 −0.682360 −0.341180 0.939998i \(-0.610826\pi\)
−0.341180 + 0.939998i \(0.610826\pi\)
\(398\) −1.75255 −0.0878475
\(399\) −0.202041 −0.0101147
\(400\) 11.0000 0.550000
\(401\) 32.2474 1.61036 0.805180 0.593030i \(-0.202068\pi\)
0.805180 + 0.593030i \(0.202068\pi\)
\(402\) 2.00000 0.0997509
\(403\) −17.3939 −0.866451
\(404\) −14.8990 −0.741252
\(405\) 4.00000 0.198762
\(406\) −0.449490 −0.0223078
\(407\) 9.79796 0.485667
\(408\) 4.89898 0.242536
\(409\) −6.00000 −0.296681 −0.148340 0.988936i \(-0.547393\pi\)
−0.148340 + 0.988936i \(0.547393\pi\)
\(410\) 3.59592 0.177590
\(411\) 13.7980 0.680603
\(412\) −18.2474 −0.898987
\(413\) −2.20204 −0.108355
\(414\) −1.00000 −0.0491473
\(415\) −25.7980 −1.26637
\(416\) 2.89898 0.142134
\(417\) 6.00000 0.293821
\(418\) −2.20204 −0.107705
\(419\) −32.2474 −1.57539 −0.787695 0.616065i \(-0.788726\pi\)
−0.787695 + 0.616065i \(0.788726\pi\)
\(420\) −1.79796 −0.0877314
\(421\) −14.4949 −0.706438 −0.353219 0.935541i \(-0.614913\pi\)
−0.353219 + 0.935541i \(0.614913\pi\)
\(422\) 15.1010 0.735106
\(423\) 4.00000 0.194487
\(424\) 0.898979 0.0436583
\(425\) 53.8888 2.61399
\(426\) −1.10102 −0.0533446
\(427\) 3.10102 0.150069
\(428\) −14.4495 −0.698442
\(429\) −14.2020 −0.685681
\(430\) 30.2020 1.45647
\(431\) 36.4949 1.75790 0.878949 0.476916i \(-0.158246\pi\)
0.878949 + 0.476916i \(0.158246\pi\)
\(432\) 1.00000 0.0481125
\(433\) 26.2474 1.26137 0.630686 0.776038i \(-0.282774\pi\)
0.630686 + 0.776038i \(0.282774\pi\)
\(434\) 2.69694 0.129457
\(435\) 4.00000 0.191785
\(436\) 0.449490 0.0215267
\(437\) −0.449490 −0.0215020
\(438\) 2.89898 0.138519
\(439\) 21.7980 1.04036 0.520180 0.854057i \(-0.325865\pi\)
0.520180 + 0.854057i \(0.325865\pi\)
\(440\) −19.5959 −0.934199
\(441\) −6.79796 −0.323712
\(442\) 14.2020 0.675522
\(443\) 18.8990 0.897918 0.448959 0.893552i \(-0.351795\pi\)
0.448959 + 0.893552i \(0.351795\pi\)
\(444\) −2.00000 −0.0949158
\(445\) 12.4041 0.588010
\(446\) −5.79796 −0.274541
\(447\) 10.6969 0.505948
\(448\) −0.449490 −0.0212364
\(449\) −20.4949 −0.967214 −0.483607 0.875285i \(-0.660674\pi\)
−0.483607 + 0.875285i \(0.660674\pi\)
\(450\) 11.0000 0.518545
\(451\) −4.40408 −0.207380
\(452\) 0 0
\(453\) 5.79796 0.272412
\(454\) −7.34847 −0.344881
\(455\) −5.21225 −0.244354
\(456\) 0.449490 0.0210493
\(457\) −22.0000 −1.02912 −0.514558 0.857455i \(-0.672044\pi\)
−0.514558 + 0.857455i \(0.672044\pi\)
\(458\) 16.6969 0.780197
\(459\) 4.89898 0.228665
\(460\) −4.00000 −0.186501
\(461\) 8.69694 0.405057 0.202528 0.979276i \(-0.435084\pi\)
0.202528 + 0.979276i \(0.435084\pi\)
\(462\) 2.20204 0.102448
\(463\) −29.7980 −1.38483 −0.692414 0.721500i \(-0.743453\pi\)
−0.692414 + 0.721500i \(0.743453\pi\)
\(464\) 1.00000 0.0464238
\(465\) −24.0000 −1.11297
\(466\) −12.6969 −0.588174
\(467\) 17.7980 0.823591 0.411796 0.911276i \(-0.364902\pi\)
0.411796 + 0.911276i \(0.364902\pi\)
\(468\) 2.89898 0.134005
\(469\) −0.898979 −0.0415110
\(470\) 16.0000 0.738025
\(471\) 3.79796 0.175001
\(472\) 4.89898 0.225494
\(473\) −36.9898 −1.70079
\(474\) 10.0000 0.459315
\(475\) 4.94439 0.226864
\(476\) −2.20204 −0.100930
\(477\) 0.898979 0.0411614
\(478\) −10.8990 −0.498508
\(479\) 24.2474 1.10789 0.553947 0.832552i \(-0.313121\pi\)
0.553947 + 0.832552i \(0.313121\pi\)
\(480\) 4.00000 0.182574
\(481\) −5.79796 −0.264364
\(482\) −8.69694 −0.396135
\(483\) 0.449490 0.0204525
\(484\) 13.0000 0.590909
\(485\) 60.5857 2.75106
\(486\) 1.00000 0.0453609
\(487\) −32.0000 −1.45006 −0.725029 0.688718i \(-0.758174\pi\)
−0.725029 + 0.688718i \(0.758174\pi\)
\(488\) −6.89898 −0.312302
\(489\) −8.89898 −0.402426
\(490\) −27.1918 −1.22840
\(491\) −5.10102 −0.230206 −0.115103 0.993354i \(-0.536720\pi\)
−0.115103 + 0.993354i \(0.536720\pi\)
\(492\) 0.898979 0.0405291
\(493\) 4.89898 0.220639
\(494\) 1.30306 0.0586275
\(495\) −19.5959 −0.880771
\(496\) −6.00000 −0.269408
\(497\) 0.494897 0.0221992
\(498\) −6.44949 −0.289009
\(499\) 3.79796 0.170020 0.0850100 0.996380i \(-0.472908\pi\)
0.0850100 + 0.996380i \(0.472908\pi\)
\(500\) 24.0000 1.07331
\(501\) −9.79796 −0.437741
\(502\) 12.8990 0.575710
\(503\) 35.3485 1.57611 0.788055 0.615605i \(-0.211088\pi\)
0.788055 + 0.615605i \(0.211088\pi\)
\(504\) −0.449490 −0.0200219
\(505\) −59.5959 −2.65198
\(506\) 4.89898 0.217786
\(507\) −4.59592 −0.204112
\(508\) −13.7980 −0.612185
\(509\) 12.8990 0.571737 0.285869 0.958269i \(-0.407718\pi\)
0.285869 + 0.958269i \(0.407718\pi\)
\(510\) 19.5959 0.867722
\(511\) −1.30306 −0.0576440
\(512\) 1.00000 0.0441942
\(513\) 0.449490 0.0198455
\(514\) −10.8990 −0.480733
\(515\) −72.9898 −3.21631
\(516\) 7.55051 0.332393
\(517\) −19.5959 −0.861827
\(518\) 0.898979 0.0394989
\(519\) −12.8990 −0.566202
\(520\) 11.5959 0.508515
\(521\) 18.4495 0.808287 0.404143 0.914696i \(-0.367570\pi\)
0.404143 + 0.914696i \(0.367570\pi\)
\(522\) 1.00000 0.0437688
\(523\) 19.7980 0.865704 0.432852 0.901465i \(-0.357507\pi\)
0.432852 + 0.901465i \(0.357507\pi\)
\(524\) 9.10102 0.397580
\(525\) −4.94439 −0.215791
\(526\) 3.75255 0.163619
\(527\) −29.3939 −1.28042
\(528\) −4.89898 −0.213201
\(529\) 1.00000 0.0434783
\(530\) 3.59592 0.156197
\(531\) 4.89898 0.212598
\(532\) −0.202041 −0.00875959
\(533\) 2.60612 0.112884
\(534\) 3.10102 0.134194
\(535\) −57.7980 −2.49882
\(536\) 2.00000 0.0863868
\(537\) −14.6969 −0.634220
\(538\) 10.8990 0.469888
\(539\) 33.3031 1.43446
\(540\) 4.00000 0.172133
\(541\) 2.20204 0.0946731 0.0473366 0.998879i \(-0.484927\pi\)
0.0473366 + 0.998879i \(0.484927\pi\)
\(542\) −13.7980 −0.592673
\(543\) −10.6515 −0.457101
\(544\) 4.89898 0.210042
\(545\) 1.79796 0.0770161
\(546\) −1.30306 −0.0557659
\(547\) 27.7980 1.18855 0.594277 0.804260i \(-0.297438\pi\)
0.594277 + 0.804260i \(0.297438\pi\)
\(548\) 13.7980 0.589420
\(549\) −6.89898 −0.294441
\(550\) −53.8888 −2.29783
\(551\) 0.449490 0.0191489
\(552\) −1.00000 −0.0425628
\(553\) −4.49490 −0.191142
\(554\) 24.6969 1.04927
\(555\) −8.00000 −0.339581
\(556\) 6.00000 0.254457
\(557\) −21.3939 −0.906488 −0.453244 0.891387i \(-0.649733\pi\)
−0.453244 + 0.891387i \(0.649733\pi\)
\(558\) −6.00000 −0.254000
\(559\) 21.8888 0.925797
\(560\) −1.79796 −0.0759776
\(561\) −24.0000 −1.01328
\(562\) −26.4495 −1.11570
\(563\) 7.10102 0.299272 0.149636 0.988741i \(-0.452190\pi\)
0.149636 + 0.988741i \(0.452190\pi\)
\(564\) 4.00000 0.168430
\(565\) 0 0
\(566\) −21.5959 −0.907744
\(567\) −0.449490 −0.0188768
\(568\) −1.10102 −0.0461978
\(569\) 1.79796 0.0753744 0.0376872 0.999290i \(-0.488001\pi\)
0.0376872 + 0.999290i \(0.488001\pi\)
\(570\) 1.79796 0.0753082
\(571\) −34.8990 −1.46048 −0.730238 0.683192i \(-0.760591\pi\)
−0.730238 + 0.683192i \(0.760591\pi\)
\(572\) −14.2020 −0.593817
\(573\) −7.34847 −0.306987
\(574\) −0.404082 −0.0168661
\(575\) −11.0000 −0.458732
\(576\) 1.00000 0.0416667
\(577\) −6.00000 −0.249783 −0.124892 0.992170i \(-0.539858\pi\)
−0.124892 + 0.992170i \(0.539858\pi\)
\(578\) 7.00000 0.291162
\(579\) −17.5959 −0.731261
\(580\) 4.00000 0.166091
\(581\) 2.89898 0.120270
\(582\) 15.1464 0.627840
\(583\) −4.40408 −0.182398
\(584\) 2.89898 0.119961
\(585\) 11.5959 0.479432
\(586\) −9.14643 −0.377835
\(587\) 47.1918 1.94782 0.973908 0.226944i \(-0.0728735\pi\)
0.973908 + 0.226944i \(0.0728735\pi\)
\(588\) −6.79796 −0.280343
\(589\) −2.69694 −0.111125
\(590\) 19.5959 0.806751
\(591\) −15.7980 −0.649841
\(592\) −2.00000 −0.0821995
\(593\) −17.5959 −0.722578 −0.361289 0.932454i \(-0.617663\pi\)
−0.361289 + 0.932454i \(0.617663\pi\)
\(594\) −4.89898 −0.201008
\(595\) −8.80816 −0.361100
\(596\) 10.6969 0.438164
\(597\) −1.75255 −0.0717271
\(598\) −2.89898 −0.118548
\(599\) −6.69694 −0.273629 −0.136815 0.990597i \(-0.543687\pi\)
−0.136815 + 0.990597i \(0.543687\pi\)
\(600\) 11.0000 0.449073
\(601\) −19.3939 −0.791093 −0.395546 0.918446i \(-0.629445\pi\)
−0.395546 + 0.918446i \(0.629445\pi\)
\(602\) −3.39388 −0.138324
\(603\) 2.00000 0.0814463
\(604\) 5.79796 0.235916
\(605\) 52.0000 2.11410
\(606\) −14.8990 −0.605230
\(607\) −3.79796 −0.154154 −0.0770772 0.997025i \(-0.524559\pi\)
−0.0770772 + 0.997025i \(0.524559\pi\)
\(608\) 0.449490 0.0182292
\(609\) −0.449490 −0.0182142
\(610\) −27.5959 −1.11733
\(611\) 11.5959 0.469121
\(612\) 4.89898 0.198030
\(613\) 36.0454 1.45586 0.727930 0.685651i \(-0.240482\pi\)
0.727930 + 0.685651i \(0.240482\pi\)
\(614\) 6.69694 0.270266
\(615\) 3.59592 0.145001
\(616\) 2.20204 0.0887228
\(617\) −19.5959 −0.788902 −0.394451 0.918917i \(-0.629065\pi\)
−0.394451 + 0.918917i \(0.629065\pi\)
\(618\) −18.2474 −0.734020
\(619\) −24.0454 −0.966467 −0.483233 0.875492i \(-0.660538\pi\)
−0.483233 + 0.875492i \(0.660538\pi\)
\(620\) −24.0000 −0.963863
\(621\) −1.00000 −0.0401286
\(622\) 26.6969 1.07045
\(623\) −1.39388 −0.0558445
\(624\) 2.89898 0.116052
\(625\) 41.0000 1.64000
\(626\) 17.5959 0.703274
\(627\) −2.20204 −0.0879410
\(628\) 3.79796 0.151555
\(629\) −9.79796 −0.390670
\(630\) −1.79796 −0.0716324
\(631\) −8.44949 −0.336369 −0.168184 0.985756i \(-0.553790\pi\)
−0.168184 + 0.985756i \(0.553790\pi\)
\(632\) 10.0000 0.397779
\(633\) 15.1010 0.600212
\(634\) 1.10102 0.0437271
\(635\) −55.1918 −2.19022
\(636\) 0.898979 0.0356469
\(637\) −19.7071 −0.780825
\(638\) −4.89898 −0.193952
\(639\) −1.10102 −0.0435557
\(640\) 4.00000 0.158114
\(641\) −0.404082 −0.0159603 −0.00798014 0.999968i \(-0.502540\pi\)
−0.00798014 + 0.999968i \(0.502540\pi\)
\(642\) −14.4495 −0.570276
\(643\) −47.7980 −1.88497 −0.942484 0.334252i \(-0.891516\pi\)
−0.942484 + 0.334252i \(0.891516\pi\)
\(644\) 0.449490 0.0177124
\(645\) 30.2020 1.18920
\(646\) 2.20204 0.0866381
\(647\) −14.8990 −0.585739 −0.292870 0.956152i \(-0.594610\pi\)
−0.292870 + 0.956152i \(0.594610\pi\)
\(648\) 1.00000 0.0392837
\(649\) −24.0000 −0.942082
\(650\) 31.8888 1.25078
\(651\) 2.69694 0.105701
\(652\) −8.89898 −0.348511
\(653\) −30.0000 −1.17399 −0.586995 0.809590i \(-0.699689\pi\)
−0.586995 + 0.809590i \(0.699689\pi\)
\(654\) 0.449490 0.0175764
\(655\) 36.4041 1.42243
\(656\) 0.898979 0.0350993
\(657\) 2.89898 0.113100
\(658\) −1.79796 −0.0700917
\(659\) −43.5959 −1.69826 −0.849128 0.528187i \(-0.822872\pi\)
−0.849128 + 0.528187i \(0.822872\pi\)
\(660\) −19.5959 −0.762770
\(661\) −33.3485 −1.29711 −0.648553 0.761170i \(-0.724625\pi\)
−0.648553 + 0.761170i \(0.724625\pi\)
\(662\) −20.8990 −0.812262
\(663\) 14.2020 0.551562
\(664\) −6.44949 −0.250289
\(665\) −0.808164 −0.0313393
\(666\) −2.00000 −0.0774984
\(667\) −1.00000 −0.0387202
\(668\) −9.79796 −0.379094
\(669\) −5.79796 −0.224162
\(670\) 8.00000 0.309067
\(671\) 33.7980 1.30476
\(672\) −0.449490 −0.0173394
\(673\) −41.5959 −1.60340 −0.801702 0.597723i \(-0.796072\pi\)
−0.801702 + 0.597723i \(0.796072\pi\)
\(674\) 15.5505 0.598983
\(675\) 11.0000 0.423390
\(676\) −4.59592 −0.176766
\(677\) −4.65153 −0.178773 −0.0893864 0.995997i \(-0.528491\pi\)
−0.0893864 + 0.995997i \(0.528491\pi\)
\(678\) 0 0
\(679\) −6.80816 −0.261273
\(680\) 19.5959 0.751469
\(681\) −7.34847 −0.281594
\(682\) 29.3939 1.12555
\(683\) −19.5959 −0.749817 −0.374908 0.927062i \(-0.622326\pi\)
−0.374908 + 0.927062i \(0.622326\pi\)
\(684\) 0.449490 0.0171867
\(685\) 55.1918 2.10877
\(686\) 6.20204 0.236795
\(687\) 16.6969 0.637028
\(688\) 7.55051 0.287861
\(689\) 2.60612 0.0992854
\(690\) −4.00000 −0.152277
\(691\) 29.3939 1.11820 0.559098 0.829102i \(-0.311148\pi\)
0.559098 + 0.829102i \(0.311148\pi\)
\(692\) −12.8990 −0.490346
\(693\) 2.20204 0.0836486
\(694\) 19.5959 0.743851
\(695\) 24.0000 0.910372
\(696\) 1.00000 0.0379049
\(697\) 4.40408 0.166817
\(698\) 14.0000 0.529908
\(699\) −12.6969 −0.480242
\(700\) −4.94439 −0.186880
\(701\) 6.20204 0.234248 0.117124 0.993117i \(-0.462633\pi\)
0.117124 + 0.993117i \(0.462633\pi\)
\(702\) 2.89898 0.109415
\(703\) −0.898979 −0.0339057
\(704\) −4.89898 −0.184637
\(705\) 16.0000 0.602595
\(706\) 15.7980 0.594564
\(707\) 6.69694 0.251864
\(708\) 4.89898 0.184115
\(709\) −18.2474 −0.685297 −0.342649 0.939464i \(-0.611324\pi\)
−0.342649 + 0.939464i \(0.611324\pi\)
\(710\) −4.40408 −0.165282
\(711\) 10.0000 0.375029
\(712\) 3.10102 0.116216
\(713\) 6.00000 0.224702
\(714\) −2.20204 −0.0824093
\(715\) −56.8082 −2.12450
\(716\) −14.6969 −0.549250
\(717\) −10.8990 −0.407030
\(718\) −5.55051 −0.207143
\(719\) 37.3939 1.39456 0.697278 0.716801i \(-0.254394\pi\)
0.697278 + 0.716801i \(0.254394\pi\)
\(720\) 4.00000 0.149071
\(721\) 8.20204 0.305460
\(722\) −18.7980 −0.699588
\(723\) −8.69694 −0.323443
\(724\) −10.6515 −0.395861
\(725\) 11.0000 0.408530
\(726\) 13.0000 0.482475
\(727\) 0.202041 0.00749329 0.00374664 0.999993i \(-0.498807\pi\)
0.00374664 + 0.999993i \(0.498807\pi\)
\(728\) −1.30306 −0.0482947
\(729\) 1.00000 0.0370370
\(730\) 11.5959 0.429184
\(731\) 36.9898 1.36812
\(732\) −6.89898 −0.254994
\(733\) −0.202041 −0.00746256 −0.00373128 0.999993i \(-0.501188\pi\)
−0.00373128 + 0.999993i \(0.501188\pi\)
\(734\) −30.0000 −1.10732
\(735\) −27.1918 −1.00299
\(736\) −1.00000 −0.0368605
\(737\) −9.79796 −0.360912
\(738\) 0.898979 0.0330919
\(739\) −14.2020 −0.522431 −0.261215 0.965281i \(-0.584123\pi\)
−0.261215 + 0.965281i \(0.584123\pi\)
\(740\) −8.00000 −0.294086
\(741\) 1.30306 0.0478692
\(742\) −0.404082 −0.0148343
\(743\) −1.14643 −0.0420584 −0.0210292 0.999779i \(-0.506694\pi\)
−0.0210292 + 0.999779i \(0.506694\pi\)
\(744\) −6.00000 −0.219971
\(745\) 42.7878 1.56762
\(746\) −20.0454 −0.733915
\(747\) −6.44949 −0.235974
\(748\) −24.0000 −0.877527
\(749\) 6.49490 0.237318
\(750\) 24.0000 0.876356
\(751\) −11.7980 −0.430514 −0.215257 0.976557i \(-0.569059\pi\)
−0.215257 + 0.976557i \(0.569059\pi\)
\(752\) 4.00000 0.145865
\(753\) 12.8990 0.470065
\(754\) 2.89898 0.105575
\(755\) 23.1918 0.844037
\(756\) −0.449490 −0.0163478
\(757\) −42.4949 −1.54450 −0.772252 0.635317i \(-0.780870\pi\)
−0.772252 + 0.635317i \(0.780870\pi\)
\(758\) 17.3485 0.630125
\(759\) 4.89898 0.177822
\(760\) 1.79796 0.0652188
\(761\) −47.3939 −1.71803 −0.859013 0.511953i \(-0.828922\pi\)
−0.859013 + 0.511953i \(0.828922\pi\)
\(762\) −13.7980 −0.499847
\(763\) −0.202041 −0.00731438
\(764\) −7.34847 −0.265858
\(765\) 19.5959 0.708492
\(766\) −9.79796 −0.354015
\(767\) 14.2020 0.512806
\(768\) 1.00000 0.0360844
\(769\) 34.2474 1.23499 0.617497 0.786573i \(-0.288147\pi\)
0.617497 + 0.786573i \(0.288147\pi\)
\(770\) 8.80816 0.317424
\(771\) −10.8990 −0.392517
\(772\) −17.5959 −0.633291
\(773\) 54.4495 1.95841 0.979206 0.202868i \(-0.0650264\pi\)
0.979206 + 0.202868i \(0.0650264\pi\)
\(774\) 7.55051 0.271397
\(775\) −66.0000 −2.37079
\(776\) 15.1464 0.543725
\(777\) 0.898979 0.0322507
\(778\) 31.3485 1.12390
\(779\) 0.404082 0.0144777
\(780\) 11.5959 0.415200
\(781\) 5.39388 0.193008
\(782\) −4.89898 −0.175187
\(783\) 1.00000 0.0357371
\(784\) −6.79796 −0.242784
\(785\) 15.1918 0.542220
\(786\) 9.10102 0.324623
\(787\) 34.0000 1.21197 0.605985 0.795476i \(-0.292779\pi\)
0.605985 + 0.795476i \(0.292779\pi\)
\(788\) −15.7980 −0.562779
\(789\) 3.75255 0.133594
\(790\) 40.0000 1.42314
\(791\) 0 0
\(792\) −4.89898 −0.174078
\(793\) −20.0000 −0.710221
\(794\) −13.5959 −0.482501
\(795\) 3.59592 0.127534
\(796\) −1.75255 −0.0621175
\(797\) −39.8434 −1.41132 −0.705662 0.708548i \(-0.749350\pi\)
−0.705662 + 0.708548i \(0.749350\pi\)
\(798\) −0.202041 −0.00715217
\(799\) 19.5959 0.693254
\(800\) 11.0000 0.388909
\(801\) 3.10102 0.109569
\(802\) 32.2474 1.13870
\(803\) −14.2020 −0.501179
\(804\) 2.00000 0.0705346
\(805\) 1.79796 0.0633697
\(806\) −17.3939 −0.612673
\(807\) 10.8990 0.383662
\(808\) −14.8990 −0.524144
\(809\) 20.4949 0.720562 0.360281 0.932844i \(-0.382681\pi\)
0.360281 + 0.932844i \(0.382681\pi\)
\(810\) 4.00000 0.140546
\(811\) −45.5959 −1.60109 −0.800545 0.599273i \(-0.795456\pi\)
−0.800545 + 0.599273i \(0.795456\pi\)
\(812\) −0.449490 −0.0157740
\(813\) −13.7980 −0.483916
\(814\) 9.79796 0.343418
\(815\) −35.5959 −1.24687
\(816\) 4.89898 0.171499
\(817\) 3.39388 0.118737
\(818\) −6.00000 −0.209785
\(819\) −1.30306 −0.0455327
\(820\) 3.59592 0.125575
\(821\) 32.8990 1.14818 0.574091 0.818791i \(-0.305356\pi\)
0.574091 + 0.818791i \(0.305356\pi\)
\(822\) 13.7980 0.481259
\(823\) −26.2020 −0.913346 −0.456673 0.889635i \(-0.650959\pi\)
−0.456673 + 0.889635i \(0.650959\pi\)
\(824\) −18.2474 −0.635680
\(825\) −53.8888 −1.87617
\(826\) −2.20204 −0.0766188
\(827\) 7.59592 0.264136 0.132068 0.991241i \(-0.457838\pi\)
0.132068 + 0.991241i \(0.457838\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) −25.7980 −0.895460
\(831\) 24.6969 0.856727
\(832\) 2.89898 0.100504
\(833\) −33.3031 −1.15388
\(834\) 6.00000 0.207763
\(835\) −39.1918 −1.35629
\(836\) −2.20204 −0.0761592
\(837\) −6.00000 −0.207390
\(838\) −32.2474 −1.11397
\(839\) 52.7423 1.82087 0.910434 0.413654i \(-0.135748\pi\)
0.910434 + 0.413654i \(0.135748\pi\)
\(840\) −1.79796 −0.0620355
\(841\) 1.00000 0.0344828
\(842\) −14.4949 −0.499527
\(843\) −26.4495 −0.910969
\(844\) 15.1010 0.519799
\(845\) −18.3837 −0.632418
\(846\) 4.00000 0.137523
\(847\) −5.84337 −0.200780
\(848\) 0.898979 0.0308711
\(849\) −21.5959 −0.741170
\(850\) 53.8888 1.84837
\(851\) 2.00000 0.0685591
\(852\) −1.10102 −0.0377203
\(853\) −25.1918 −0.862552 −0.431276 0.902220i \(-0.641936\pi\)
−0.431276 + 0.902220i \(0.641936\pi\)
\(854\) 3.10102 0.106115
\(855\) 1.79796 0.0614889
\(856\) −14.4495 −0.493873
\(857\) 34.8990 1.19213 0.596063 0.802938i \(-0.296731\pi\)
0.596063 + 0.802938i \(0.296731\pi\)
\(858\) −14.2020 −0.484850
\(859\) 3.59592 0.122691 0.0613456 0.998117i \(-0.480461\pi\)
0.0613456 + 0.998117i \(0.480461\pi\)
\(860\) 30.2020 1.02988
\(861\) −0.404082 −0.0137711
\(862\) 36.4949 1.24302
\(863\) 17.7980 0.605850 0.302925 0.953014i \(-0.402037\pi\)
0.302925 + 0.953014i \(0.402037\pi\)
\(864\) 1.00000 0.0340207
\(865\) −51.5959 −1.75431
\(866\) 26.2474 0.891925
\(867\) 7.00000 0.237732
\(868\) 2.69694 0.0915401
\(869\) −48.9898 −1.66186
\(870\) 4.00000 0.135613
\(871\) 5.79796 0.196456
\(872\) 0.449490 0.0152216
\(873\) 15.1464 0.512629
\(874\) −0.449490 −0.0152042
\(875\) −10.7878 −0.364693
\(876\) 2.89898 0.0979474
\(877\) 12.6969 0.428745 0.214373 0.976752i \(-0.431229\pi\)
0.214373 + 0.976752i \(0.431229\pi\)
\(878\) 21.7980 0.735645
\(879\) −9.14643 −0.308501
\(880\) −19.5959 −0.660578
\(881\) −11.1010 −0.374003 −0.187001 0.982360i \(-0.559877\pi\)
−0.187001 + 0.982360i \(0.559877\pi\)
\(882\) −6.79796 −0.228899
\(883\) −17.7980 −0.598949 −0.299475 0.954104i \(-0.596811\pi\)
−0.299475 + 0.954104i \(0.596811\pi\)
\(884\) 14.2020 0.477666
\(885\) 19.5959 0.658710
\(886\) 18.8990 0.634924
\(887\) −4.89898 −0.164492 −0.0822458 0.996612i \(-0.526209\pi\)
−0.0822458 + 0.996612i \(0.526209\pi\)
\(888\) −2.00000 −0.0671156
\(889\) 6.20204 0.208010
\(890\) 12.4041 0.415786
\(891\) −4.89898 −0.164122
\(892\) −5.79796 −0.194130
\(893\) 1.79796 0.0601664
\(894\) 10.6969 0.357759
\(895\) −58.7878 −1.96506
\(896\) −0.449490 −0.0150164
\(897\) −2.89898 −0.0967941
\(898\) −20.4949 −0.683924
\(899\) −6.00000 −0.200111
\(900\) 11.0000 0.366667
\(901\) 4.40408 0.146721
\(902\) −4.40408 −0.146640
\(903\) −3.39388 −0.112941
\(904\) 0 0
\(905\) −42.6061 −1.41628
\(906\) 5.79796 0.192624
\(907\) 25.3485 0.841682 0.420841 0.907134i \(-0.361735\pi\)
0.420841 + 0.907134i \(0.361735\pi\)
\(908\) −7.34847 −0.243868
\(909\) −14.8990 −0.494168
\(910\) −5.21225 −0.172784
\(911\) −27.8434 −0.922492 −0.461246 0.887272i \(-0.652597\pi\)
−0.461246 + 0.887272i \(0.652597\pi\)
\(912\) 0.449490 0.0148841
\(913\) 31.5959 1.04567
\(914\) −22.0000 −0.727695
\(915\) −27.5959 −0.912293
\(916\) 16.6969 0.551682
\(917\) −4.09082 −0.135091
\(918\) 4.89898 0.161690
\(919\) −26.2474 −0.865823 −0.432912 0.901436i \(-0.642514\pi\)
−0.432912 + 0.901436i \(0.642514\pi\)
\(920\) −4.00000 −0.131876
\(921\) 6.69694 0.220672
\(922\) 8.69694 0.286418
\(923\) −3.19184 −0.105061
\(924\) 2.20204 0.0724418
\(925\) −22.0000 −0.723356
\(926\) −29.7980 −0.979222
\(927\) −18.2474 −0.599325
\(928\) 1.00000 0.0328266
\(929\) 26.0000 0.853032 0.426516 0.904480i \(-0.359741\pi\)
0.426516 + 0.904480i \(0.359741\pi\)
\(930\) −24.0000 −0.786991
\(931\) −3.05561 −0.100144
\(932\) −12.6969 −0.415902
\(933\) 26.6969 0.874019
\(934\) 17.7980 0.582367
\(935\) −96.0000 −3.13954
\(936\) 2.89898 0.0947561
\(937\) 7.79796 0.254748 0.127374 0.991855i \(-0.459345\pi\)
0.127374 + 0.991855i \(0.459345\pi\)
\(938\) −0.898979 −0.0293527
\(939\) 17.5959 0.574221
\(940\) 16.0000 0.521862
\(941\) 35.1918 1.14722 0.573611 0.819128i \(-0.305542\pi\)
0.573611 + 0.819128i \(0.305542\pi\)
\(942\) 3.79796 0.123744
\(943\) −0.898979 −0.0292748
\(944\) 4.89898 0.159448
\(945\) −1.79796 −0.0584876
\(946\) −36.9898 −1.20264
\(947\) 9.10102 0.295743 0.147872 0.989007i \(-0.452758\pi\)
0.147872 + 0.989007i \(0.452758\pi\)
\(948\) 10.0000 0.324785
\(949\) 8.40408 0.272808
\(950\) 4.94439 0.160417
\(951\) 1.10102 0.0357030
\(952\) −2.20204 −0.0713686
\(953\) −54.9444 −1.77982 −0.889912 0.456133i \(-0.849234\pi\)
−0.889912 + 0.456133i \(0.849234\pi\)
\(954\) 0.898979 0.0291055
\(955\) −29.3939 −0.951164
\(956\) −10.8990 −0.352498
\(957\) −4.89898 −0.158362
\(958\) 24.2474 0.783400
\(959\) −6.20204 −0.200274
\(960\) 4.00000 0.129099
\(961\) 5.00000 0.161290
\(962\) −5.79796 −0.186934
\(963\) −14.4495 −0.465628
\(964\) −8.69694 −0.280110
\(965\) −70.3837 −2.26573
\(966\) 0.449490 0.0144621
\(967\) 57.3939 1.84566 0.922831 0.385204i \(-0.125869\pi\)
0.922831 + 0.385204i \(0.125869\pi\)
\(968\) 13.0000 0.417836
\(969\) 2.20204 0.0707397
\(970\) 60.5857 1.94529
\(971\) 38.2929 1.22888 0.614438 0.788965i \(-0.289383\pi\)
0.614438 + 0.788965i \(0.289383\pi\)
\(972\) 1.00000 0.0320750
\(973\) −2.69694 −0.0864599
\(974\) −32.0000 −1.02535
\(975\) 31.8888 1.02126
\(976\) −6.89898 −0.220831
\(977\) −44.7423 −1.43143 −0.715717 0.698390i \(-0.753900\pi\)
−0.715717 + 0.698390i \(0.753900\pi\)
\(978\) −8.89898 −0.284558
\(979\) −15.1918 −0.485533
\(980\) −27.1918 −0.868611
\(981\) 0.449490 0.0143511
\(982\) −5.10102 −0.162780
\(983\) −14.9444 −0.476652 −0.238326 0.971185i \(-0.576599\pi\)
−0.238326 + 0.971185i \(0.576599\pi\)
\(984\) 0.898979 0.0286584
\(985\) −63.1918 −2.01346
\(986\) 4.89898 0.156015
\(987\) −1.79796 −0.0572297
\(988\) 1.30306 0.0414559
\(989\) −7.55051 −0.240092
\(990\) −19.5959 −0.622799
\(991\) −43.1010 −1.36915 −0.684575 0.728943i \(-0.740012\pi\)
−0.684575 + 0.728943i \(0.740012\pi\)
\(992\) −6.00000 −0.190500
\(993\) −20.8990 −0.663209
\(994\) 0.494897 0.0156972
\(995\) −7.01021 −0.222238
\(996\) −6.44949 −0.204360
\(997\) −1.59592 −0.0505432 −0.0252716 0.999681i \(-0.508045\pi\)
−0.0252716 + 0.999681i \(0.508045\pi\)
\(998\) 3.79796 0.120222
\(999\) −2.00000 −0.0632772
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 4002.2.a.x.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4002.2.a.x.1.1 2 1.1 even 1 trivial