Properties

Label 1338.2.a.h.1.5
Level 1338
Weight 2
Character 1338.1
Self dual yes
Analytic conductor 10.684
Analytic rank 0
Dimension 5
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 1338 = 2 \cdot 3 \cdot 223 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1338.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(10.6839837904\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.356173.1
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(-1.38363\)
Character \(\chi\) = 1338.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.35484 q^{5} -1.00000 q^{6} +3.05676 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +4.35484 q^{5} -1.00000 q^{6} +3.05676 q^{7} -1.00000 q^{8} +1.00000 q^{9} -4.35484 q^{10} -0.734963 q^{11} +1.00000 q^{12} -2.43689 q^{13} -3.05676 q^{14} +4.35484 q^{15} +1.00000 q^{16} -0.681711 q^{17} -1.00000 q^{18} -1.87811 q^{19} +4.35484 q^{20} +3.05676 q^{21} +0.734963 q^{22} +6.91115 q^{23} -1.00000 q^{24} +13.9646 q^{25} +2.43689 q^{26} +1.00000 q^{27} +3.05676 q^{28} +0.113306 q^{29} -4.35484 q^{30} +1.56736 q^{31} -1.00000 q^{32} -0.734963 q^{33} +0.681711 q^{34} +13.3117 q^{35} +1.00000 q^{36} -3.71884 q^{37} +1.87811 q^{38} -2.43689 q^{39} -4.35484 q^{40} -11.1102 q^{41} -3.05676 q^{42} -10.9512 q^{43} -0.734963 q^{44} +4.35484 q^{45} -6.91115 q^{46} -6.22966 q^{47} +1.00000 q^{48} +2.34379 q^{49} -13.9646 q^{50} -0.681711 q^{51} -2.43689 q^{52} +6.47426 q^{53} -1.00000 q^{54} -3.20064 q^{55} -3.05676 q^{56} -1.87811 q^{57} -0.113306 q^{58} +7.94099 q^{59} +4.35484 q^{60} -4.54537 q^{61} -1.56736 q^{62} +3.05676 q^{63} +1.00000 q^{64} -10.6122 q^{65} +0.734963 q^{66} -2.23117 q^{67} -0.681711 q^{68} +6.91115 q^{69} -13.3117 q^{70} +10.5641 q^{71} -1.00000 q^{72} -3.30232 q^{73} +3.71884 q^{74} +13.9646 q^{75} -1.87811 q^{76} -2.24661 q^{77} +2.43689 q^{78} -3.98486 q^{79} +4.35484 q^{80} +1.00000 q^{81} +11.1102 q^{82} -9.77231 q^{83} +3.05676 q^{84} -2.96874 q^{85} +10.9512 q^{86} +0.113306 q^{87} +0.734963 q^{88} -3.23802 q^{89} -4.35484 q^{90} -7.44898 q^{91} +6.91115 q^{92} +1.56736 q^{93} +6.22966 q^{94} -8.17887 q^{95} -1.00000 q^{96} -11.1019 q^{97} -2.34379 q^{98} -0.734963 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 5q^{2} + 5q^{3} + 5q^{4} + 5q^{5} - 5q^{6} - q^{7} - 5q^{8} + 5q^{9} + O(q^{10}) \) \( 5q - 5q^{2} + 5q^{3} + 5q^{4} + 5q^{5} - 5q^{6} - q^{7} - 5q^{8} + 5q^{9} - 5q^{10} + 9q^{11} + 5q^{12} + q^{14} + 5q^{15} + 5q^{16} + 6q^{17} - 5q^{18} - 4q^{19} + 5q^{20} - q^{21} - 9q^{22} + 16q^{23} - 5q^{24} + 8q^{25} + 5q^{27} - q^{28} + 8q^{29} - 5q^{30} - q^{31} - 5q^{32} + 9q^{33} - 6q^{34} + 22q^{35} + 5q^{36} - 2q^{37} + 4q^{38} - 5q^{40} + 4q^{41} + q^{42} + 3q^{43} + 9q^{44} + 5q^{45} - 16q^{46} + 18q^{47} + 5q^{48} + 2q^{49} - 8q^{50} + 6q^{51} + 26q^{53} - 5q^{54} + q^{55} + q^{56} - 4q^{57} - 8q^{58} + 21q^{59} + 5q^{60} - 20q^{61} + q^{62} - q^{63} + 5q^{64} - 3q^{65} - 9q^{66} - 5q^{67} + 6q^{68} + 16q^{69} - 22q^{70} + 17q^{71} - 5q^{72} + 5q^{73} + 2q^{74} + 8q^{75} - 4q^{76} + 2q^{77} - 21q^{79} + 5q^{80} + 5q^{81} - 4q^{82} + 11q^{83} - q^{84} - 12q^{85} - 3q^{86} + 8q^{87} - 9q^{88} - 5q^{89} - 5q^{90} - 10q^{91} + 16q^{92} - q^{93} - 18q^{94} + 10q^{95} - 5q^{96} - 11q^{97} - 2q^{98} + 9q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 1.00000 0.577350
\(4\) 1.00000 0.500000
\(5\) 4.35484 1.94754 0.973772 0.227528i \(-0.0730644\pi\)
0.973772 + 0.227528i \(0.0730644\pi\)
\(6\) −1.00000 −0.408248
\(7\) 3.05676 1.15535 0.577674 0.816268i \(-0.303961\pi\)
0.577674 + 0.816268i \(0.303961\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −4.35484 −1.37712
\(11\) −0.734963 −0.221600 −0.110800 0.993843i \(-0.535341\pi\)
−0.110800 + 0.993843i \(0.535341\pi\)
\(12\) 1.00000 0.288675
\(13\) −2.43689 −0.675871 −0.337935 0.941169i \(-0.609728\pi\)
−0.337935 + 0.941169i \(0.609728\pi\)
\(14\) −3.05676 −0.816954
\(15\) 4.35484 1.12441
\(16\) 1.00000 0.250000
\(17\) −0.681711 −0.165339 −0.0826696 0.996577i \(-0.526345\pi\)
−0.0826696 + 0.996577i \(0.526345\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.87811 −0.430868 −0.215434 0.976518i \(-0.569117\pi\)
−0.215434 + 0.976518i \(0.569117\pi\)
\(20\) 4.35484 0.973772
\(21\) 3.05676 0.667040
\(22\) 0.734963 0.156695
\(23\) 6.91115 1.44107 0.720537 0.693416i \(-0.243895\pi\)
0.720537 + 0.693416i \(0.243895\pi\)
\(24\) −1.00000 −0.204124
\(25\) 13.9646 2.79292
\(26\) 2.43689 0.477913
\(27\) 1.00000 0.192450
\(28\) 3.05676 0.577674
\(29\) 0.113306 0.0210404 0.0105202 0.999945i \(-0.496651\pi\)
0.0105202 + 0.999945i \(0.496651\pi\)
\(30\) −4.35484 −0.795081
\(31\) 1.56736 0.281506 0.140753 0.990045i \(-0.455048\pi\)
0.140753 + 0.990045i \(0.455048\pi\)
\(32\) −1.00000 −0.176777
\(33\) −0.734963 −0.127941
\(34\) 0.681711 0.116912
\(35\) 13.3117 2.25009
\(36\) 1.00000 0.166667
\(37\) −3.71884 −0.611374 −0.305687 0.952132i \(-0.598886\pi\)
−0.305687 + 0.952132i \(0.598886\pi\)
\(38\) 1.87811 0.304670
\(39\) −2.43689 −0.390214
\(40\) −4.35484 −0.688560
\(41\) −11.1102 −1.73513 −0.867563 0.497327i \(-0.834315\pi\)
−0.867563 + 0.497327i \(0.834315\pi\)
\(42\) −3.05676 −0.471669
\(43\) −10.9512 −1.67004 −0.835022 0.550217i \(-0.814545\pi\)
−0.835022 + 0.550217i \(0.814545\pi\)
\(44\) −0.734963 −0.110800
\(45\) 4.35484 0.649181
\(46\) −6.91115 −1.01899
\(47\) −6.22966 −0.908689 −0.454344 0.890826i \(-0.650126\pi\)
−0.454344 + 0.890826i \(0.650126\pi\)
\(48\) 1.00000 0.144338
\(49\) 2.34379 0.334827
\(50\) −13.9646 −1.97490
\(51\) −0.681711 −0.0954586
\(52\) −2.43689 −0.337935
\(53\) 6.47426 0.889309 0.444654 0.895702i \(-0.353327\pi\)
0.444654 + 0.895702i \(0.353327\pi\)
\(54\) −1.00000 −0.136083
\(55\) −3.20064 −0.431575
\(56\) −3.05676 −0.408477
\(57\) −1.87811 −0.248762
\(58\) −0.113306 −0.0148778
\(59\) 7.94099 1.03383 0.516915 0.856037i \(-0.327080\pi\)
0.516915 + 0.856037i \(0.327080\pi\)
\(60\) 4.35484 0.562207
\(61\) −4.54537 −0.581974 −0.290987 0.956727i \(-0.593984\pi\)
−0.290987 + 0.956727i \(0.593984\pi\)
\(62\) −1.56736 −0.199055
\(63\) 3.05676 0.385116
\(64\) 1.00000 0.125000
\(65\) −10.6122 −1.31629
\(66\) 0.734963 0.0904677
\(67\) −2.23117 −0.272581 −0.136290 0.990669i \(-0.543518\pi\)
−0.136290 + 0.990669i \(0.543518\pi\)
\(68\) −0.681711 −0.0826696
\(69\) 6.91115 0.832005
\(70\) −13.3117 −1.59105
\(71\) 10.5641 1.25372 0.626862 0.779130i \(-0.284339\pi\)
0.626862 + 0.779130i \(0.284339\pi\)
\(72\) −1.00000 −0.117851
\(73\) −3.30232 −0.386507 −0.193254 0.981149i \(-0.561904\pi\)
−0.193254 + 0.981149i \(0.561904\pi\)
\(74\) 3.71884 0.432307
\(75\) 13.9646 1.61250
\(76\) −1.87811 −0.215434
\(77\) −2.24661 −0.256024
\(78\) 2.43689 0.275923
\(79\) −3.98486 −0.448332 −0.224166 0.974551i \(-0.571966\pi\)
−0.224166 + 0.974551i \(0.571966\pi\)
\(80\) 4.35484 0.486886
\(81\) 1.00000 0.111111
\(82\) 11.1102 1.22692
\(83\) −9.77231 −1.07265 −0.536325 0.844011i \(-0.680188\pi\)
−0.536325 + 0.844011i \(0.680188\pi\)
\(84\) 3.05676 0.333520
\(85\) −2.96874 −0.322005
\(86\) 10.9512 1.18090
\(87\) 0.113306 0.0121477
\(88\) 0.734963 0.0783473
\(89\) −3.23802 −0.343230 −0.171615 0.985164i \(-0.554898\pi\)
−0.171615 + 0.985164i \(0.554898\pi\)
\(90\) −4.35484 −0.459040
\(91\) −7.44898 −0.780865
\(92\) 6.91115 0.720537
\(93\) 1.56736 0.162528
\(94\) 6.22966 0.642540
\(95\) −8.17887 −0.839134
\(96\) −1.00000 −0.102062
\(97\) −11.1019 −1.12723 −0.563613 0.826039i \(-0.690589\pi\)
−0.563613 + 0.826039i \(0.690589\pi\)
\(98\) −2.34379 −0.236759
\(99\) −0.734963 −0.0738665
\(100\) 13.9646 1.39646
\(101\) 19.8419 1.97434 0.987171 0.159664i \(-0.0510410\pi\)
0.987171 + 0.159664i \(0.0510410\pi\)
\(102\) 0.681711 0.0674994
\(103\) −4.42428 −0.435937 −0.217968 0.975956i \(-0.569943\pi\)
−0.217968 + 0.975956i \(0.569943\pi\)
\(104\) 2.43689 0.238956
\(105\) 13.3117 1.29909
\(106\) −6.47426 −0.628836
\(107\) 11.4126 1.10330 0.551651 0.834075i \(-0.313998\pi\)
0.551651 + 0.834075i \(0.313998\pi\)
\(108\) 1.00000 0.0962250
\(109\) −15.1912 −1.45506 −0.727528 0.686078i \(-0.759331\pi\)
−0.727528 + 0.686078i \(0.759331\pi\)
\(110\) 3.20064 0.305169
\(111\) −3.71884 −0.352977
\(112\) 3.05676 0.288837
\(113\) −0.723916 −0.0681003 −0.0340501 0.999420i \(-0.510841\pi\)
−0.0340501 + 0.999420i \(0.510841\pi\)
\(114\) 1.87811 0.175901
\(115\) 30.0969 2.80655
\(116\) 0.113306 0.0105202
\(117\) −2.43689 −0.225290
\(118\) −7.94099 −0.731028
\(119\) −2.08383 −0.191024
\(120\) −4.35484 −0.397541
\(121\) −10.4598 −0.950894
\(122\) 4.54537 0.411518
\(123\) −11.1102 −1.00178
\(124\) 1.56736 0.140753
\(125\) 39.0395 3.49180
\(126\) −3.05676 −0.272318
\(127\) 17.7134 1.57181 0.785905 0.618347i \(-0.212197\pi\)
0.785905 + 0.618347i \(0.212197\pi\)
\(128\) −1.00000 −0.0883883
\(129\) −10.9512 −0.964200
\(130\) 10.6122 0.930755
\(131\) −4.62868 −0.404409 −0.202205 0.979343i \(-0.564811\pi\)
−0.202205 + 0.979343i \(0.564811\pi\)
\(132\) −0.734963 −0.0639703
\(133\) −5.74094 −0.497802
\(134\) 2.23117 0.192744
\(135\) 4.35484 0.374805
\(136\) 0.681711 0.0584562
\(137\) −6.95121 −0.593882 −0.296941 0.954896i \(-0.595966\pi\)
−0.296941 + 0.954896i \(0.595966\pi\)
\(138\) −6.91115 −0.588316
\(139\) 23.4964 1.99294 0.996468 0.0839681i \(-0.0267594\pi\)
0.996468 + 0.0839681i \(0.0267594\pi\)
\(140\) 13.3117 1.12504
\(141\) −6.22966 −0.524632
\(142\) −10.5641 −0.886517
\(143\) 1.79102 0.149773
\(144\) 1.00000 0.0833333
\(145\) 0.493430 0.0409772
\(146\) 3.30232 0.273302
\(147\) 2.34379 0.193313
\(148\) −3.71884 −0.305687
\(149\) −8.25494 −0.676271 −0.338136 0.941097i \(-0.609796\pi\)
−0.338136 + 0.941097i \(0.609796\pi\)
\(150\) −13.9646 −1.14021
\(151\) −15.3701 −1.25080 −0.625402 0.780303i \(-0.715065\pi\)
−0.625402 + 0.780303i \(0.715065\pi\)
\(152\) 1.87811 0.152335
\(153\) −0.681711 −0.0551131
\(154\) 2.24661 0.181037
\(155\) 6.82559 0.548245
\(156\) −2.43689 −0.195107
\(157\) 5.70675 0.455448 0.227724 0.973726i \(-0.426872\pi\)
0.227724 + 0.973726i \(0.426872\pi\)
\(158\) 3.98486 0.317019
\(159\) 6.47426 0.513443
\(160\) −4.35484 −0.344280
\(161\) 21.1257 1.66494
\(162\) −1.00000 −0.0785674
\(163\) 1.46144 0.114469 0.0572343 0.998361i \(-0.481772\pi\)
0.0572343 + 0.998361i \(0.481772\pi\)
\(164\) −11.1102 −0.867563
\(165\) −3.20064 −0.249170
\(166\) 9.77231 0.758478
\(167\) −0.335943 −0.0259960 −0.0129980 0.999916i \(-0.504138\pi\)
−0.0129980 + 0.999916i \(0.504138\pi\)
\(168\) −3.05676 −0.235834
\(169\) −7.06159 −0.543199
\(170\) 2.96874 0.227692
\(171\) −1.87811 −0.143623
\(172\) −10.9512 −0.835022
\(173\) 3.25577 0.247532 0.123766 0.992311i \(-0.460503\pi\)
0.123766 + 0.992311i \(0.460503\pi\)
\(174\) −0.113306 −0.00858972
\(175\) 42.6865 3.22680
\(176\) −0.734963 −0.0553999
\(177\) 7.94099 0.596881
\(178\) 3.23802 0.242700
\(179\) 2.01680 0.150743 0.0753715 0.997156i \(-0.475986\pi\)
0.0753715 + 0.997156i \(0.475986\pi\)
\(180\) 4.35484 0.324591
\(181\) −0.718726 −0.0534225 −0.0267113 0.999643i \(-0.508503\pi\)
−0.0267113 + 0.999643i \(0.508503\pi\)
\(182\) 7.44898 0.552155
\(183\) −4.54537 −0.336003
\(184\) −6.91115 −0.509497
\(185\) −16.1950 −1.19068
\(186\) −1.56736 −0.114924
\(187\) 0.501032 0.0366391
\(188\) −6.22966 −0.454344
\(189\) 3.05676 0.222347
\(190\) 8.17887 0.593357
\(191\) 4.95760 0.358719 0.179360 0.983784i \(-0.442597\pi\)
0.179360 + 0.983784i \(0.442597\pi\)
\(192\) 1.00000 0.0721688
\(193\) −8.59130 −0.618415 −0.309208 0.950995i \(-0.600064\pi\)
−0.309208 + 0.950995i \(0.600064\pi\)
\(194\) 11.1019 0.797070
\(195\) −10.6122 −0.759959
\(196\) 2.34379 0.167414
\(197\) 17.0864 1.21736 0.608678 0.793417i \(-0.291700\pi\)
0.608678 + 0.793417i \(0.291700\pi\)
\(198\) 0.734963 0.0522315
\(199\) −0.350646 −0.0248566 −0.0124283 0.999923i \(-0.503956\pi\)
−0.0124283 + 0.999923i \(0.503956\pi\)
\(200\) −13.9646 −0.987448
\(201\) −2.23117 −0.157374
\(202\) −19.8419 −1.39607
\(203\) 0.346350 0.0243090
\(204\) −0.681711 −0.0477293
\(205\) −48.3833 −3.37923
\(206\) 4.42428 0.308254
\(207\) 6.91115 0.480358
\(208\) −2.43689 −0.168968
\(209\) 1.38034 0.0954802
\(210\) −13.3117 −0.918595
\(211\) −12.3103 −0.847475 −0.423738 0.905785i \(-0.639282\pi\)
−0.423738 + 0.905785i \(0.639282\pi\)
\(212\) 6.47426 0.444654
\(213\) 10.5641 0.723838
\(214\) −11.4126 −0.780152
\(215\) −47.6907 −3.25248
\(216\) −1.00000 −0.0680414
\(217\) 4.79104 0.325237
\(218\) 15.1912 1.02888
\(219\) −3.30232 −0.223150
\(220\) −3.20064 −0.215787
\(221\) 1.66125 0.111748
\(222\) 3.71884 0.249592
\(223\) −1.00000 −0.0669650
\(224\) −3.05676 −0.204238
\(225\) 13.9646 0.930975
\(226\) 0.723916 0.0481542
\(227\) 11.9814 0.795233 0.397616 0.917552i \(-0.369838\pi\)
0.397616 + 0.917552i \(0.369838\pi\)
\(228\) −1.87811 −0.124381
\(229\) −0.278694 −0.0184166 −0.00920831 0.999958i \(-0.502931\pi\)
−0.00920831 + 0.999958i \(0.502931\pi\)
\(230\) −30.0969 −1.98453
\(231\) −2.24661 −0.147816
\(232\) −0.113306 −0.00743892
\(233\) −12.2226 −0.800732 −0.400366 0.916355i \(-0.631117\pi\)
−0.400366 + 0.916355i \(0.631117\pi\)
\(234\) 2.43689 0.159304
\(235\) −27.1291 −1.76971
\(236\) 7.94099 0.516915
\(237\) −3.98486 −0.258845
\(238\) 2.08383 0.135074
\(239\) 22.2407 1.43863 0.719316 0.694683i \(-0.244455\pi\)
0.719316 + 0.694683i \(0.244455\pi\)
\(240\) 4.35484 0.281104
\(241\) −22.3491 −1.43963 −0.719814 0.694166i \(-0.755773\pi\)
−0.719814 + 0.694166i \(0.755773\pi\)
\(242\) 10.4598 0.672383
\(243\) 1.00000 0.0641500
\(244\) −4.54537 −0.290987
\(245\) 10.2068 0.652091
\(246\) 11.1102 0.708362
\(247\) 4.57674 0.291211
\(248\) −1.56736 −0.0995274
\(249\) −9.77231 −0.619295
\(250\) −39.0395 −2.46907
\(251\) 19.7825 1.24866 0.624331 0.781160i \(-0.285372\pi\)
0.624331 + 0.781160i \(0.285372\pi\)
\(252\) 3.05676 0.192558
\(253\) −5.07944 −0.319341
\(254\) −17.7134 −1.11144
\(255\) −2.96874 −0.185910
\(256\) 1.00000 0.0625000
\(257\) −11.4401 −0.713617 −0.356808 0.934178i \(-0.616135\pi\)
−0.356808 + 0.934178i \(0.616135\pi\)
\(258\) 10.9512 0.681793
\(259\) −11.3676 −0.706349
\(260\) −10.6122 −0.658143
\(261\) 0.113306 0.00701348
\(262\) 4.62868 0.285961
\(263\) 13.3295 0.821929 0.410965 0.911651i \(-0.365192\pi\)
0.410965 + 0.911651i \(0.365192\pi\)
\(264\) 0.734963 0.0452338
\(265\) 28.1944 1.73197
\(266\) 5.74094 0.351999
\(267\) −3.23802 −0.198164
\(268\) −2.23117 −0.136290
\(269\) 3.03055 0.184776 0.0923878 0.995723i \(-0.470550\pi\)
0.0923878 + 0.995723i \(0.470550\pi\)
\(270\) −4.35484 −0.265027
\(271\) 21.4604 1.30362 0.651812 0.758381i \(-0.274009\pi\)
0.651812 + 0.758381i \(0.274009\pi\)
\(272\) −0.681711 −0.0413348
\(273\) −7.44898 −0.450833
\(274\) 6.95121 0.419938
\(275\) −10.2635 −0.618911
\(276\) 6.91115 0.416002
\(277\) 28.2695 1.69855 0.849276 0.527950i \(-0.177039\pi\)
0.849276 + 0.527950i \(0.177039\pi\)
\(278\) −23.4964 −1.40922
\(279\) 1.56736 0.0938353
\(280\) −13.3117 −0.795526
\(281\) −12.7637 −0.761420 −0.380710 0.924694i \(-0.624320\pi\)
−0.380710 + 0.924694i \(0.624320\pi\)
\(282\) 6.22966 0.370971
\(283\) 32.5585 1.93540 0.967702 0.252098i \(-0.0811205\pi\)
0.967702 + 0.252098i \(0.0811205\pi\)
\(284\) 10.5641 0.626862
\(285\) −8.17887 −0.484474
\(286\) −1.79102 −0.105905
\(287\) −33.9613 −2.00467
\(288\) −1.00000 −0.0589256
\(289\) −16.5353 −0.972663
\(290\) −0.493430 −0.0289752
\(291\) −11.1019 −0.650805
\(292\) −3.30232 −0.193254
\(293\) 28.3976 1.65901 0.829503 0.558503i \(-0.188624\pi\)
0.829503 + 0.558503i \(0.188624\pi\)
\(294\) −2.34379 −0.136693
\(295\) 34.5817 2.01343
\(296\) 3.71884 0.216153
\(297\) −0.734963 −0.0426469
\(298\) 8.25494 0.478196
\(299\) −16.8417 −0.973980
\(300\) 13.9646 0.806248
\(301\) −33.4752 −1.92948
\(302\) 15.3701 0.884452
\(303\) 19.8419 1.13989
\(304\) −1.87811 −0.107717
\(305\) −19.7943 −1.13342
\(306\) 0.681711 0.0389708
\(307\) −27.9578 −1.59563 −0.797817 0.602900i \(-0.794012\pi\)
−0.797817 + 0.602900i \(0.794012\pi\)
\(308\) −2.24661 −0.128012
\(309\) −4.42428 −0.251688
\(310\) −6.82559 −0.387668
\(311\) −3.59459 −0.203831 −0.101915 0.994793i \(-0.532497\pi\)
−0.101915 + 0.994793i \(0.532497\pi\)
\(312\) 2.43689 0.137961
\(313\) 11.4643 0.647998 0.323999 0.946057i \(-0.394972\pi\)
0.323999 + 0.946057i \(0.394972\pi\)
\(314\) −5.70675 −0.322050
\(315\) 13.3117 0.750030
\(316\) −3.98486 −0.224166
\(317\) −33.3632 −1.87386 −0.936931 0.349514i \(-0.886347\pi\)
−0.936931 + 0.349514i \(0.886347\pi\)
\(318\) −6.47426 −0.363059
\(319\) −0.0832759 −0.00466255
\(320\) 4.35484 0.243443
\(321\) 11.4126 0.636992
\(322\) −21.1257 −1.17729
\(323\) 1.28033 0.0712394
\(324\) 1.00000 0.0555556
\(325\) −34.0302 −1.88765
\(326\) −1.46144 −0.0809415
\(327\) −15.1912 −0.840077
\(328\) 11.1102 0.613460
\(329\) −19.0426 −1.04985
\(330\) 3.20064 0.176190
\(331\) −11.6186 −0.638615 −0.319308 0.947651i \(-0.603450\pi\)
−0.319308 + 0.947651i \(0.603450\pi\)
\(332\) −9.77231 −0.536325
\(333\) −3.71884 −0.203791
\(334\) 0.335943 0.0183820
\(335\) −9.71638 −0.530862
\(336\) 3.05676 0.166760
\(337\) −7.91263 −0.431029 −0.215514 0.976501i \(-0.569143\pi\)
−0.215514 + 0.976501i \(0.569143\pi\)
\(338\) 7.06159 0.384100
\(339\) −0.723916 −0.0393177
\(340\) −2.96874 −0.161003
\(341\) −1.15195 −0.0623816
\(342\) 1.87811 0.101557
\(343\) −14.2329 −0.768505
\(344\) 10.9512 0.590450
\(345\) 30.0969 1.62036
\(346\) −3.25577 −0.175031
\(347\) −21.1718 −1.13656 −0.568280 0.822835i \(-0.692391\pi\)
−0.568280 + 0.822835i \(0.692391\pi\)
\(348\) 0.113306 0.00607385
\(349\) −11.3838 −0.609360 −0.304680 0.952455i \(-0.598550\pi\)
−0.304680 + 0.952455i \(0.598550\pi\)
\(350\) −42.6865 −2.28169
\(351\) −2.43689 −0.130071
\(352\) 0.734963 0.0391736
\(353\) −20.6961 −1.10154 −0.550772 0.834656i \(-0.685667\pi\)
−0.550772 + 0.834656i \(0.685667\pi\)
\(354\) −7.94099 −0.422059
\(355\) 46.0048 2.44168
\(356\) −3.23802 −0.171615
\(357\) −2.08383 −0.110288
\(358\) −2.01680 −0.106591
\(359\) −13.4560 −0.710183 −0.355092 0.934832i \(-0.615550\pi\)
−0.355092 + 0.934832i \(0.615550\pi\)
\(360\) −4.35484 −0.229520
\(361\) −15.4727 −0.814353
\(362\) 0.718726 0.0377754
\(363\) −10.4598 −0.548999
\(364\) −7.44898 −0.390433
\(365\) −14.3811 −0.752740
\(366\) 4.54537 0.237590
\(367\) 8.80612 0.459675 0.229838 0.973229i \(-0.426180\pi\)
0.229838 + 0.973229i \(0.426180\pi\)
\(368\) 6.91115 0.360269
\(369\) −11.1102 −0.578375
\(370\) 16.1950 0.841936
\(371\) 19.7903 1.02746
\(372\) 1.56736 0.0812638
\(373\) 5.86346 0.303598 0.151799 0.988411i \(-0.451493\pi\)
0.151799 + 0.988411i \(0.451493\pi\)
\(374\) −0.501032 −0.0259078
\(375\) 39.0395 2.01599
\(376\) 6.22966 0.321270
\(377\) −0.276114 −0.0142206
\(378\) −3.05676 −0.157223
\(379\) −20.7423 −1.06546 −0.532729 0.846286i \(-0.678833\pi\)
−0.532729 + 0.846286i \(0.678833\pi\)
\(380\) −8.17887 −0.419567
\(381\) 17.7134 0.907485
\(382\) −4.95760 −0.253653
\(383\) 25.8637 1.32158 0.660788 0.750573i \(-0.270222\pi\)
0.660788 + 0.750573i \(0.270222\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −9.78361 −0.498619
\(386\) 8.59130 0.437286
\(387\) −10.9512 −0.556681
\(388\) −11.1019 −0.563613
\(389\) 2.44929 0.124184 0.0620920 0.998070i \(-0.480223\pi\)
0.0620920 + 0.998070i \(0.480223\pi\)
\(390\) 10.6122 0.537372
\(391\) −4.71141 −0.238266
\(392\) −2.34379 −0.118379
\(393\) −4.62868 −0.233486
\(394\) −17.0864 −0.860801
\(395\) −17.3534 −0.873146
\(396\) −0.734963 −0.0369333
\(397\) 19.7131 0.989370 0.494685 0.869072i \(-0.335283\pi\)
0.494685 + 0.869072i \(0.335283\pi\)
\(398\) 0.350646 0.0175763
\(399\) −5.74094 −0.287406
\(400\) 13.9646 0.698231
\(401\) 11.7643 0.587483 0.293742 0.955885i \(-0.405100\pi\)
0.293742 + 0.955885i \(0.405100\pi\)
\(402\) 2.23117 0.111281
\(403\) −3.81947 −0.190262
\(404\) 19.8419 0.987171
\(405\) 4.35484 0.216394
\(406\) −0.346350 −0.0171891
\(407\) 2.73321 0.135480
\(408\) 0.681711 0.0337497
\(409\) 29.9196 1.47943 0.739715 0.672920i \(-0.234960\pi\)
0.739715 + 0.672920i \(0.234960\pi\)
\(410\) 48.3833 2.38948
\(411\) −6.95121 −0.342878
\(412\) −4.42428 −0.217968
\(413\) 24.2737 1.19443
\(414\) −6.91115 −0.339664
\(415\) −42.5568 −2.08903
\(416\) 2.43689 0.119478
\(417\) 23.4964 1.15062
\(418\) −1.38034 −0.0675147
\(419\) −36.6787 −1.79187 −0.895935 0.444185i \(-0.853493\pi\)
−0.895935 + 0.444185i \(0.853493\pi\)
\(420\) 13.3117 0.649545
\(421\) 30.9616 1.50898 0.754488 0.656313i \(-0.227885\pi\)
0.754488 + 0.656313i \(0.227885\pi\)
\(422\) 12.3103 0.599255
\(423\) −6.22966 −0.302896
\(424\) −6.47426 −0.314418
\(425\) −9.51983 −0.461780
\(426\) −10.5641 −0.511831
\(427\) −13.8941 −0.672383
\(428\) 11.4126 0.551651
\(429\) 1.79102 0.0864713
\(430\) 47.6907 2.29985
\(431\) −32.4852 −1.56476 −0.782378 0.622804i \(-0.785994\pi\)
−0.782378 + 0.622804i \(0.785994\pi\)
\(432\) 1.00000 0.0481125
\(433\) −24.9126 −1.19722 −0.598612 0.801039i \(-0.704281\pi\)
−0.598612 + 0.801039i \(0.704281\pi\)
\(434\) −4.79104 −0.229977
\(435\) 0.493430 0.0236582
\(436\) −15.1912 −0.727528
\(437\) −12.9799 −0.620913
\(438\) 3.30232 0.157791
\(439\) 31.8189 1.51863 0.759317 0.650721i \(-0.225533\pi\)
0.759317 + 0.650721i \(0.225533\pi\)
\(440\) 3.20064 0.152585
\(441\) 2.34379 0.111609
\(442\) −1.66125 −0.0790177
\(443\) −22.4064 −1.06456 −0.532281 0.846568i \(-0.678665\pi\)
−0.532281 + 0.846568i \(0.678665\pi\)
\(444\) −3.71884 −0.176488
\(445\) −14.1011 −0.668455
\(446\) 1.00000 0.0473514
\(447\) −8.25494 −0.390445
\(448\) 3.05676 0.144418
\(449\) 27.5968 1.30237 0.651187 0.758917i \(-0.274271\pi\)
0.651187 + 0.758917i \(0.274271\pi\)
\(450\) −13.9646 −0.658298
\(451\) 8.16560 0.384503
\(452\) −0.723916 −0.0340501
\(453\) −15.3701 −0.722152
\(454\) −11.9814 −0.562314
\(455\) −32.4391 −1.52077
\(456\) 1.87811 0.0879506
\(457\) 6.26270 0.292957 0.146478 0.989214i \(-0.453206\pi\)
0.146478 + 0.989214i \(0.453206\pi\)
\(458\) 0.278694 0.0130225
\(459\) −0.681711 −0.0318195
\(460\) 30.0969 1.40328
\(461\) −24.4624 −1.13933 −0.569664 0.821878i \(-0.692926\pi\)
−0.569664 + 0.821878i \(0.692926\pi\)
\(462\) 2.24661 0.104522
\(463\) −10.5467 −0.490146 −0.245073 0.969505i \(-0.578812\pi\)
−0.245073 + 0.969505i \(0.578812\pi\)
\(464\) 0.113306 0.00526011
\(465\) 6.82559 0.316529
\(466\) 12.2226 0.566203
\(467\) −27.0318 −1.25088 −0.625442 0.780271i \(-0.715081\pi\)
−0.625442 + 0.780271i \(0.715081\pi\)
\(468\) −2.43689 −0.112645
\(469\) −6.82015 −0.314925
\(470\) 27.1291 1.25137
\(471\) 5.70675 0.262953
\(472\) −7.94099 −0.365514
\(473\) 8.04873 0.370081
\(474\) 3.98486 0.183031
\(475\) −26.2271 −1.20338
\(476\) −2.08383 −0.0955121
\(477\) 6.47426 0.296436
\(478\) −22.2407 −1.01727
\(479\) 23.9419 1.09393 0.546967 0.837154i \(-0.315782\pi\)
0.546967 + 0.837154i \(0.315782\pi\)
\(480\) −4.35484 −0.198770
\(481\) 9.06239 0.413210
\(482\) 22.3491 1.01797
\(483\) 21.1257 0.961254
\(484\) −10.4598 −0.475447
\(485\) −48.3470 −2.19532
\(486\) −1.00000 −0.0453609
\(487\) −2.04679 −0.0927490 −0.0463745 0.998924i \(-0.514767\pi\)
−0.0463745 + 0.998924i \(0.514767\pi\)
\(488\) 4.54537 0.205759
\(489\) 1.46144 0.0660885
\(490\) −10.2068 −0.461098
\(491\) −30.6746 −1.38433 −0.692163 0.721741i \(-0.743342\pi\)
−0.692163 + 0.721741i \(0.743342\pi\)
\(492\) −11.1102 −0.500888
\(493\) −0.0772421 −0.00347881
\(494\) −4.57674 −0.205917
\(495\) −3.20064 −0.143858
\(496\) 1.56736 0.0703765
\(497\) 32.2918 1.44849
\(498\) 9.77231 0.437908
\(499\) 29.0813 1.30186 0.650929 0.759138i \(-0.274379\pi\)
0.650929 + 0.759138i \(0.274379\pi\)
\(500\) 39.0395 1.74590
\(501\) −0.335943 −0.0150088
\(502\) −19.7825 −0.882937
\(503\) 32.4064 1.44493 0.722466 0.691407i \(-0.243009\pi\)
0.722466 + 0.691407i \(0.243009\pi\)
\(504\) −3.05676 −0.136159
\(505\) 86.4083 3.84512
\(506\) 5.07944 0.225809
\(507\) −7.06159 −0.313616
\(508\) 17.7134 0.785905
\(509\) −20.3134 −0.900376 −0.450188 0.892934i \(-0.648643\pi\)
−0.450188 + 0.892934i \(0.648643\pi\)
\(510\) 2.96874 0.131458
\(511\) −10.0944 −0.446550
\(512\) −1.00000 −0.0441942
\(513\) −1.87811 −0.0829206
\(514\) 11.4401 0.504603
\(515\) −19.2670 −0.849006
\(516\) −10.9512 −0.482100
\(517\) 4.57856 0.201365
\(518\) 11.3676 0.499464
\(519\) 3.25577 0.142912
\(520\) 10.6122 0.465378
\(521\) −23.8912 −1.04669 −0.523346 0.852120i \(-0.675317\pi\)
−0.523346 + 0.852120i \(0.675317\pi\)
\(522\) −0.113306 −0.00495928
\(523\) 26.0276 1.13811 0.569054 0.822300i \(-0.307309\pi\)
0.569054 + 0.822300i \(0.307309\pi\)
\(524\) −4.62868 −0.202205
\(525\) 42.6865 1.86299
\(526\) −13.3295 −0.581192
\(527\) −1.06849 −0.0465440
\(528\) −0.734963 −0.0319851
\(529\) 24.7640 1.07670
\(530\) −28.1944 −1.22469
\(531\) 7.94099 0.344610
\(532\) −5.74094 −0.248901
\(533\) 27.0744 1.17272
\(534\) 3.23802 0.140123
\(535\) 49.7002 2.14873
\(536\) 2.23117 0.0963718
\(537\) 2.01680 0.0870315
\(538\) −3.03055 −0.130656
\(539\) −1.72260 −0.0741976
\(540\) 4.35484 0.187402
\(541\) −5.46873 −0.235119 −0.117559 0.993066i \(-0.537507\pi\)
−0.117559 + 0.993066i \(0.537507\pi\)
\(542\) −21.4604 −0.921801
\(543\) −0.718726 −0.0308435
\(544\) 0.681711 0.0292281
\(545\) −66.1554 −2.83378
\(546\) 7.44898 0.318787
\(547\) 0.600980 0.0256961 0.0128480 0.999917i \(-0.495910\pi\)
0.0128480 + 0.999917i \(0.495910\pi\)
\(548\) −6.95121 −0.296941
\(549\) −4.54537 −0.193991
\(550\) 10.2635 0.437636
\(551\) −0.212802 −0.00906565
\(552\) −6.91115 −0.294158
\(553\) −12.1808 −0.517979
\(554\) −28.2695 −1.20106
\(555\) −16.1950 −0.687438
\(556\) 23.4964 0.996468
\(557\) −33.5046 −1.41963 −0.709817 0.704386i \(-0.751222\pi\)
−0.709817 + 0.704386i \(0.751222\pi\)
\(558\) −1.56736 −0.0663516
\(559\) 26.6868 1.12873
\(560\) 13.3117 0.562522
\(561\) 0.501032 0.0211536
\(562\) 12.7637 0.538405
\(563\) −0.0190630 −0.000803411 0 −0.000401705 1.00000i \(-0.500128\pi\)
−0.000401705 1.00000i \(0.500128\pi\)
\(564\) −6.22966 −0.262316
\(565\) −3.15254 −0.132628
\(566\) −32.5585 −1.36854
\(567\) 3.05676 0.128372
\(568\) −10.5641 −0.443258
\(569\) 17.4587 0.731908 0.365954 0.930633i \(-0.380743\pi\)
0.365954 + 0.930633i \(0.380743\pi\)
\(570\) 8.17887 0.342575
\(571\) 18.0879 0.756955 0.378477 0.925611i \(-0.376448\pi\)
0.378477 + 0.925611i \(0.376448\pi\)
\(572\) 1.79102 0.0748863
\(573\) 4.95760 0.207107
\(574\) 33.9613 1.41752
\(575\) 96.5116 4.02481
\(576\) 1.00000 0.0416667
\(577\) 35.8487 1.49240 0.746200 0.665722i \(-0.231876\pi\)
0.746200 + 0.665722i \(0.231876\pi\)
\(578\) 16.5353 0.687777
\(579\) −8.59130 −0.357042
\(580\) 0.493430 0.0204886
\(581\) −29.8716 −1.23928
\(582\) 11.1019 0.460188
\(583\) −4.75834 −0.197070
\(584\) 3.30232 0.136651
\(585\) −10.6122 −0.438762
\(586\) −28.3976 −1.17309
\(587\) 2.69524 0.111244 0.0556222 0.998452i \(-0.482286\pi\)
0.0556222 + 0.998452i \(0.482286\pi\)
\(588\) 2.34379 0.0966563
\(589\) −2.94367 −0.121292
\(590\) −34.5817 −1.42371
\(591\) 17.0864 0.702841
\(592\) −3.71884 −0.152843
\(593\) −31.1847 −1.28060 −0.640302 0.768124i \(-0.721191\pi\)
−0.640302 + 0.768124i \(0.721191\pi\)
\(594\) 0.734963 0.0301559
\(595\) −9.07473 −0.372028
\(596\) −8.25494 −0.338136
\(597\) −0.350646 −0.0143510
\(598\) 16.8417 0.688708
\(599\) −5.49303 −0.224439 −0.112220 0.993683i \(-0.535796\pi\)
−0.112220 + 0.993683i \(0.535796\pi\)
\(600\) −13.9646 −0.570103
\(601\) −25.3653 −1.03467 −0.517335 0.855783i \(-0.673076\pi\)
−0.517335 + 0.855783i \(0.673076\pi\)
\(602\) 33.4752 1.36435
\(603\) −2.23117 −0.0908602
\(604\) −15.3701 −0.625402
\(605\) −45.5509 −1.85191
\(606\) −19.8419 −0.806022
\(607\) −12.3964 −0.503154 −0.251577 0.967837i \(-0.580949\pi\)
−0.251577 + 0.967837i \(0.580949\pi\)
\(608\) 1.87811 0.0761674
\(609\) 0.346350 0.0140348
\(610\) 19.7943 0.801449
\(611\) 15.1810 0.614156
\(612\) −0.681711 −0.0275565
\(613\) 11.0514 0.446363 0.223181 0.974777i \(-0.428356\pi\)
0.223181 + 0.974777i \(0.428356\pi\)
\(614\) 27.9578 1.12828
\(615\) −48.3833 −1.95100
\(616\) 2.24661 0.0905183
\(617\) 24.7403 0.996006 0.498003 0.867175i \(-0.334067\pi\)
0.498003 + 0.867175i \(0.334067\pi\)
\(618\) 4.42428 0.177970
\(619\) −4.37667 −0.175913 −0.0879566 0.996124i \(-0.528034\pi\)
−0.0879566 + 0.996124i \(0.528034\pi\)
\(620\) 6.82559 0.274122
\(621\) 6.91115 0.277335
\(622\) 3.59459 0.144130
\(623\) −9.89786 −0.396550
\(624\) −2.43689 −0.0975535
\(625\) 100.187 4.00750
\(626\) −11.4643 −0.458204
\(627\) 1.38034 0.0551255
\(628\) 5.70675 0.227724
\(629\) 2.53518 0.101084
\(630\) −13.3117 −0.530351
\(631\) −39.8325 −1.58571 −0.792854 0.609411i \(-0.791406\pi\)
−0.792854 + 0.609411i \(0.791406\pi\)
\(632\) 3.98486 0.158509
\(633\) −12.3103 −0.489290
\(634\) 33.3632 1.32502
\(635\) 77.1390 3.06117
\(636\) 6.47426 0.256721
\(637\) −5.71155 −0.226300
\(638\) 0.0832759 0.00329692
\(639\) 10.5641 0.417908
\(640\) −4.35484 −0.172140
\(641\) −44.5598 −1.76000 −0.880002 0.474969i \(-0.842459\pi\)
−0.880002 + 0.474969i \(0.842459\pi\)
\(642\) −11.4126 −0.450421
\(643\) 33.6455 1.32685 0.663425 0.748243i \(-0.269102\pi\)
0.663425 + 0.748243i \(0.269102\pi\)
\(644\) 21.1257 0.832471
\(645\) −47.6907 −1.87782
\(646\) −1.28033 −0.0503738
\(647\) −7.69538 −0.302537 −0.151268 0.988493i \(-0.548336\pi\)
−0.151268 + 0.988493i \(0.548336\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −5.83633 −0.229096
\(650\) 34.0302 1.33477
\(651\) 4.79104 0.187776
\(652\) 1.46144 0.0572343
\(653\) 11.4671 0.448742 0.224371 0.974504i \(-0.427967\pi\)
0.224371 + 0.974504i \(0.427967\pi\)
\(654\) 15.1912 0.594024
\(655\) −20.1571 −0.787605
\(656\) −11.1102 −0.433782
\(657\) −3.30232 −0.128836
\(658\) 19.0426 0.742357
\(659\) 35.9351 1.39983 0.699916 0.714225i \(-0.253221\pi\)
0.699916 + 0.714225i \(0.253221\pi\)
\(660\) −3.20064 −0.124585
\(661\) −45.1850 −1.75749 −0.878747 0.477288i \(-0.841620\pi\)
−0.878747 + 0.477288i \(0.841620\pi\)
\(662\) 11.6186 0.451569
\(663\) 1.66125 0.0645177
\(664\) 9.77231 0.379239
\(665\) −25.0009 −0.969491
\(666\) 3.71884 0.144102
\(667\) 0.783076 0.0303208
\(668\) −0.335943 −0.0129980
\(669\) −1.00000 −0.0386622
\(670\) 9.71638 0.375376
\(671\) 3.34067 0.128965
\(672\) −3.05676 −0.117917
\(673\) 1.54949 0.0597284 0.0298642 0.999554i \(-0.490493\pi\)
0.0298642 + 0.999554i \(0.490493\pi\)
\(674\) 7.91263 0.304783
\(675\) 13.9646 0.537498
\(676\) −7.06159 −0.271600
\(677\) −18.2028 −0.699590 −0.349795 0.936826i \(-0.613749\pi\)
−0.349795 + 0.936826i \(0.613749\pi\)
\(678\) 0.723916 0.0278018
\(679\) −33.9358 −1.30234
\(680\) 2.96874 0.113846
\(681\) 11.9814 0.459128
\(682\) 1.15195 0.0441104
\(683\) 32.0454 1.22618 0.613092 0.790012i \(-0.289926\pi\)
0.613092 + 0.790012i \(0.289926\pi\)
\(684\) −1.87811 −0.0718114
\(685\) −30.2714 −1.15661
\(686\) 14.2329 0.543415
\(687\) −0.278694 −0.0106328
\(688\) −10.9512 −0.417511
\(689\) −15.7770 −0.601057
\(690\) −30.0969 −1.14577
\(691\) 8.17206 0.310880 0.155440 0.987845i \(-0.450321\pi\)
0.155440 + 0.987845i \(0.450321\pi\)
\(692\) 3.25577 0.123766
\(693\) −2.24661 −0.0853415
\(694\) 21.1718 0.803669
\(695\) 102.323 3.88133
\(696\) −0.113306 −0.00429486
\(697\) 7.57397 0.286884
\(698\) 11.3838 0.430883
\(699\) −12.2226 −0.462303
\(700\) 42.6865 1.61340
\(701\) −25.3718 −0.958281 −0.479140 0.877738i \(-0.659052\pi\)
−0.479140 + 0.877738i \(0.659052\pi\)
\(702\) 2.43689 0.0919743
\(703\) 6.98440 0.263422
\(704\) −0.734963 −0.0276999
\(705\) −27.1291 −1.02174
\(706\) 20.6961 0.778910
\(707\) 60.6520 2.28105
\(708\) 7.94099 0.298441
\(709\) 7.95024 0.298578 0.149289 0.988794i \(-0.452302\pi\)
0.149289 + 0.988794i \(0.452302\pi\)
\(710\) −46.0048 −1.72653
\(711\) −3.98486 −0.149444
\(712\) 3.23802 0.121350
\(713\) 10.8322 0.405671
\(714\) 2.08383 0.0779853
\(715\) 7.79960 0.291689
\(716\) 2.01680 0.0753715
\(717\) 22.2407 0.830595
\(718\) 13.4560 0.502175
\(719\) −31.5123 −1.17521 −0.587606 0.809147i \(-0.699929\pi\)
−0.587606 + 0.809147i \(0.699929\pi\)
\(720\) 4.35484 0.162295
\(721\) −13.5240 −0.503658
\(722\) 15.4727 0.575834
\(723\) −22.3491 −0.831170
\(724\) −0.718726 −0.0267113
\(725\) 1.58228 0.0587643
\(726\) 10.4598 0.388201
\(727\) −41.5993 −1.54283 −0.771416 0.636332i \(-0.780451\pi\)
−0.771416 + 0.636332i \(0.780451\pi\)
\(728\) 7.44898 0.276078
\(729\) 1.00000 0.0370370
\(730\) 14.3811 0.532267
\(731\) 7.46556 0.276124
\(732\) −4.54537 −0.168002
\(733\) −0.280816 −0.0103722 −0.00518608 0.999987i \(-0.501651\pi\)
−0.00518608 + 0.999987i \(0.501651\pi\)
\(734\) −8.80612 −0.325040
\(735\) 10.2068 0.376485
\(736\) −6.91115 −0.254748
\(737\) 1.63983 0.0604038
\(738\) 11.1102 0.408973
\(739\) 30.0973 1.10715 0.553574 0.832800i \(-0.313264\pi\)
0.553574 + 0.832800i \(0.313264\pi\)
\(740\) −16.1950 −0.595339
\(741\) 4.57674 0.168131
\(742\) −19.7903 −0.726524
\(743\) 1.49835 0.0549692 0.0274846 0.999622i \(-0.491250\pi\)
0.0274846 + 0.999622i \(0.491250\pi\)
\(744\) −1.56736 −0.0574621
\(745\) −35.9489 −1.31707
\(746\) −5.86346 −0.214676
\(747\) −9.77231 −0.357550
\(748\) 0.501032 0.0183195
\(749\) 34.8857 1.27470
\(750\) −39.0395 −1.42552
\(751\) 19.0975 0.696878 0.348439 0.937331i \(-0.386712\pi\)
0.348439 + 0.937331i \(0.386712\pi\)
\(752\) −6.22966 −0.227172
\(753\) 19.7825 0.720915
\(754\) 0.276114 0.0100555
\(755\) −66.9344 −2.43599
\(756\) 3.05676 0.111173
\(757\) −29.6151 −1.07638 −0.538189 0.842824i \(-0.680891\pi\)
−0.538189 + 0.842824i \(0.680891\pi\)
\(758\) 20.7423 0.753392
\(759\) −5.07944 −0.184372
\(760\) 8.17887 0.296679
\(761\) 14.8382 0.537886 0.268943 0.963156i \(-0.413326\pi\)
0.268943 + 0.963156i \(0.413326\pi\)
\(762\) −17.7134 −0.641689
\(763\) −46.4360 −1.68110
\(764\) 4.95760 0.179360
\(765\) −2.96874 −0.107335
\(766\) −25.8637 −0.934495
\(767\) −19.3513 −0.698735
\(768\) 1.00000 0.0360844
\(769\) −23.8346 −0.859498 −0.429749 0.902948i \(-0.641398\pi\)
−0.429749 + 0.902948i \(0.641398\pi\)
\(770\) 9.78361 0.352577
\(771\) −11.4401 −0.412007
\(772\) −8.59130 −0.309208
\(773\) 55.4521 1.99447 0.997237 0.0742897i \(-0.0236690\pi\)
0.997237 + 0.0742897i \(0.0236690\pi\)
\(774\) 10.9512 0.393633
\(775\) 21.8876 0.786224
\(776\) 11.1019 0.398535
\(777\) −11.3676 −0.407811
\(778\) −2.44929 −0.0878113
\(779\) 20.8662 0.747611
\(780\) −10.6122 −0.379979
\(781\) −7.76419 −0.277825
\(782\) 4.71141 0.168480
\(783\) 0.113306 0.00404923
\(784\) 2.34379 0.0837068
\(785\) 24.8520 0.887005
\(786\) 4.62868 0.165099
\(787\) 33.1602 1.18203 0.591017 0.806659i \(-0.298726\pi\)
0.591017 + 0.806659i \(0.298726\pi\)
\(788\) 17.0864 0.608678
\(789\) 13.3295 0.474541
\(790\) 17.3534 0.617407
\(791\) −2.21284 −0.0786795
\(792\) 0.734963 0.0261158
\(793\) 11.0765 0.393339
\(794\) −19.7131 −0.699590
\(795\) 28.1944 0.999951
\(796\) −0.350646 −0.0124283
\(797\) 15.2713 0.540936 0.270468 0.962729i \(-0.412821\pi\)
0.270468 + 0.962729i \(0.412821\pi\)
\(798\) 5.74094 0.203227
\(799\) 4.24682 0.150242
\(800\) −13.9646 −0.493724
\(801\) −3.23802 −0.114410
\(802\) −11.7643 −0.415413
\(803\) 2.42708 0.0856499
\(804\) −2.23117 −0.0786872
\(805\) 91.9992 3.24255
\(806\) 3.81947 0.134535
\(807\) 3.03055 0.106680
\(808\) −19.8419 −0.698036
\(809\) 39.8830 1.40221 0.701106 0.713057i \(-0.252690\pi\)
0.701106 + 0.713057i \(0.252690\pi\)
\(810\) −4.35484 −0.153013
\(811\) −21.4838 −0.754400 −0.377200 0.926132i \(-0.623113\pi\)
−0.377200 + 0.926132i \(0.623113\pi\)
\(812\) 0.346350 0.0121545
\(813\) 21.4604 0.752648
\(814\) −2.73321 −0.0957990
\(815\) 6.36432 0.222932
\(816\) −0.681711 −0.0238647
\(817\) 20.5676 0.719569
\(818\) −29.9196 −1.04611
\(819\) −7.44898 −0.260288
\(820\) −48.3833 −1.68962
\(821\) 14.5566 0.508029 0.254014 0.967200i \(-0.418249\pi\)
0.254014 + 0.967200i \(0.418249\pi\)
\(822\) 6.95121 0.242451
\(823\) 3.79452 0.132269 0.0661344 0.997811i \(-0.478933\pi\)
0.0661344 + 0.997811i \(0.478933\pi\)
\(824\) 4.42428 0.154127
\(825\) −10.2635 −0.357328
\(826\) −24.2737 −0.844591
\(827\) 56.9287 1.97960 0.989802 0.142452i \(-0.0454986\pi\)
0.989802 + 0.142452i \(0.0454986\pi\)
\(828\) 6.91115 0.240179
\(829\) −44.0562 −1.53013 −0.765067 0.643951i \(-0.777294\pi\)
−0.765067 + 0.643951i \(0.777294\pi\)
\(830\) 42.5568 1.47717
\(831\) 28.2695 0.980659
\(832\) −2.43689 −0.0844838
\(833\) −1.59779 −0.0553601
\(834\) −23.4964 −0.813613
\(835\) −1.46298 −0.0506284
\(836\) 1.38034 0.0477401
\(837\) 1.56736 0.0541758
\(838\) 36.6787 1.26704
\(839\) 2.03615 0.0702957 0.0351479 0.999382i \(-0.488810\pi\)
0.0351479 + 0.999382i \(0.488810\pi\)
\(840\) −13.3117 −0.459297
\(841\) −28.9872 −0.999557
\(842\) −30.9616 −1.06701
\(843\) −12.7637 −0.439606
\(844\) −12.3103 −0.423738
\(845\) −30.7521 −1.05790
\(846\) 6.22966 0.214180
\(847\) −31.9732 −1.09861
\(848\) 6.47426 0.222327
\(849\) 32.5585 1.11741
\(850\) 9.51983 0.326528
\(851\) −25.7015 −0.881035
\(852\) 10.5641 0.361919
\(853\) 45.1825 1.54702 0.773510 0.633785i \(-0.218499\pi\)
0.773510 + 0.633785i \(0.218499\pi\)
\(854\) 13.8941 0.475446
\(855\) −8.17887 −0.279711
\(856\) −11.4126 −0.390076
\(857\) −13.9020 −0.474883 −0.237442 0.971402i \(-0.576309\pi\)
−0.237442 + 0.971402i \(0.576309\pi\)
\(858\) −1.79102 −0.0611444
\(859\) 25.9715 0.886134 0.443067 0.896488i \(-0.353890\pi\)
0.443067 + 0.896488i \(0.353890\pi\)
\(860\) −47.6907 −1.62624
\(861\) −33.9613 −1.15740
\(862\) 32.4852 1.10645
\(863\) 24.8717 0.846642 0.423321 0.905980i \(-0.360864\pi\)
0.423321 + 0.905980i \(0.360864\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 14.1784 0.482079
\(866\) 24.9126 0.846566
\(867\) −16.5353 −0.561567
\(868\) 4.79104 0.162619
\(869\) 2.92872 0.0993502
\(870\) −0.493430 −0.0167289
\(871\) 5.43710 0.184229
\(872\) 15.1912 0.514440
\(873\) −11.1019 −0.375742
\(874\) 12.9799 0.439052
\(875\) 119.334 4.03424
\(876\) −3.30232 −0.111575
\(877\) −41.9092 −1.41517 −0.707586 0.706627i \(-0.750216\pi\)
−0.707586 + 0.706627i \(0.750216\pi\)
\(878\) −31.8189 −1.07384
\(879\) 28.3976 0.957827
\(880\) −3.20064 −0.107894
\(881\) 0.208232 0.00701552 0.00350776 0.999994i \(-0.498883\pi\)
0.00350776 + 0.999994i \(0.498883\pi\)
\(882\) −2.34379 −0.0789196
\(883\) −4.62099 −0.155509 −0.0777544 0.996973i \(-0.524775\pi\)
−0.0777544 + 0.996973i \(0.524775\pi\)
\(884\) 1.66125 0.0558739
\(885\) 34.5817 1.16245
\(886\) 22.4064 0.752759
\(887\) −11.1644 −0.374865 −0.187433 0.982277i \(-0.560017\pi\)
−0.187433 + 0.982277i \(0.560017\pi\)
\(888\) 3.71884 0.124796
\(889\) 54.1457 1.81599
\(890\) 14.1011 0.472669
\(891\) −0.734963 −0.0246222
\(892\) −1.00000 −0.0334825
\(893\) 11.7000 0.391525
\(894\) 8.25494 0.276087
\(895\) 8.78285 0.293578
\(896\) −3.05676 −0.102119
\(897\) −16.8417 −0.562327
\(898\) −27.5968 −0.920918
\(899\) 0.177591 0.00592301
\(900\) 13.9646 0.465487
\(901\) −4.41358 −0.147038
\(902\) −8.16560 −0.271885
\(903\) −33.4752 −1.11399
\(904\) 0.723916 0.0240771
\(905\) −3.12994 −0.104043
\(906\) 15.3701 0.510638
\(907\) 0.749208 0.0248770 0.0124385 0.999923i \(-0.496041\pi\)
0.0124385 + 0.999923i \(0.496041\pi\)
\(908\) 11.9814 0.397616
\(909\) 19.8419 0.658114
\(910\) 32.4391 1.07535
\(911\) −45.7170 −1.51467 −0.757336 0.653026i \(-0.773499\pi\)
−0.757336 + 0.653026i \(0.773499\pi\)
\(912\) −1.87811 −0.0621905
\(913\) 7.18228 0.237699
\(914\) −6.26270 −0.207152
\(915\) −19.7943 −0.654380
\(916\) −0.278694 −0.00920831
\(917\) −14.1488 −0.467233
\(918\) 0.681711 0.0224998
\(919\) −4.43703 −0.146364 −0.0731821 0.997319i \(-0.523315\pi\)
−0.0731821 + 0.997319i \(0.523315\pi\)
\(920\) −30.0969 −0.992267
\(921\) −27.9578 −0.921240
\(922\) 24.4624 0.805626
\(923\) −25.7434 −0.847355
\(924\) −2.24661 −0.0739079
\(925\) −51.9322 −1.70752
\(926\) 10.5467 0.346585
\(927\) −4.42428 −0.145312
\(928\) −0.113306 −0.00371946
\(929\) 0.789965 0.0259179 0.0129590 0.999916i \(-0.495875\pi\)
0.0129590 + 0.999916i \(0.495875\pi\)
\(930\) −6.82559 −0.223820
\(931\) −4.40190 −0.144266
\(932\) −12.2226 −0.400366
\(933\) −3.59459 −0.117682
\(934\) 27.0318 0.884508
\(935\) 2.18191 0.0713562
\(936\) 2.43689 0.0796521
\(937\) 11.0887 0.362252 0.181126 0.983460i \(-0.442026\pi\)
0.181126 + 0.983460i \(0.442026\pi\)
\(938\) 6.82015 0.222686
\(939\) 11.4643 0.374122
\(940\) −27.1291 −0.884855
\(941\) −49.8346 −1.62456 −0.812281 0.583267i \(-0.801774\pi\)
−0.812281 + 0.583267i \(0.801774\pi\)
\(942\) −5.70675 −0.185936
\(943\) −76.7845 −2.50045
\(944\) 7.94099 0.258457
\(945\) 13.3117 0.433030
\(946\) −8.04873 −0.261687
\(947\) 3.22300 0.104733 0.0523667 0.998628i \(-0.483324\pi\)
0.0523667 + 0.998628i \(0.483324\pi\)
\(948\) −3.98486 −0.129422
\(949\) 8.04738 0.261229
\(950\) 26.2271 0.850919
\(951\) −33.3632 −1.08188
\(952\) 2.08383 0.0675372
\(953\) 35.4866 1.14952 0.574762 0.818320i \(-0.305095\pi\)
0.574762 + 0.818320i \(0.305095\pi\)
\(954\) −6.47426 −0.209612
\(955\) 21.5895 0.698621
\(956\) 22.2407 0.719316
\(957\) −0.0832759 −0.00269193
\(958\) −23.9419 −0.773528
\(959\) −21.2482 −0.686140
\(960\) 4.35484 0.140552
\(961\) −28.5434 −0.920754
\(962\) −9.06239 −0.292183
\(963\) 11.4126 0.367767
\(964\) −22.3491 −0.719814
\(965\) −37.4137 −1.20439
\(966\) −21.1257 −0.679709
\(967\) −4.41806 −0.142075 −0.0710376 0.997474i \(-0.522631\pi\)
−0.0710376 + 0.997474i \(0.522631\pi\)
\(968\) 10.4598 0.336192
\(969\) 1.28033 0.0411301
\(970\) 48.3470 1.55233
\(971\) 12.8435 0.412169 0.206084 0.978534i \(-0.433928\pi\)
0.206084 + 0.978534i \(0.433928\pi\)
\(972\) 1.00000 0.0320750
\(973\) 71.8228 2.30253
\(974\) 2.04679 0.0655834
\(975\) −34.0302 −1.08984
\(976\) −4.54537 −0.145494
\(977\) 14.9739 0.479057 0.239528 0.970889i \(-0.423007\pi\)
0.239528 + 0.970889i \(0.423007\pi\)
\(978\) −1.46144 −0.0467316
\(979\) 2.37983 0.0760596
\(980\) 10.2068 0.326045
\(981\) −15.1912 −0.485019
\(982\) 30.6746 0.978866
\(983\) −22.0615 −0.703652 −0.351826 0.936065i \(-0.614439\pi\)
−0.351826 + 0.936065i \(0.614439\pi\)
\(984\) 11.1102 0.354181
\(985\) 74.4086 2.37085
\(986\) 0.0772421 0.00245989
\(987\) −19.0426 −0.606132
\(988\) 4.57674 0.145606
\(989\) −75.6855 −2.40666
\(990\) 3.20064 0.101723
\(991\) −8.96101 −0.284656 −0.142328 0.989820i \(-0.545459\pi\)
−0.142328 + 0.989820i \(0.545459\pi\)
\(992\) −1.56736 −0.0497637
\(993\) −11.6186 −0.368705
\(994\) −32.2918 −1.02423
\(995\) −1.52701 −0.0484093
\(996\) −9.77231 −0.309648
\(997\) 51.5423 1.63236 0.816180 0.577798i \(-0.196088\pi\)
0.816180 + 0.577798i \(0.196088\pi\)
\(998\) −29.0813 −0.920553
\(999\) −3.71884 −0.117659
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 1338.2.a.h.1.5 5
3.2 odd 2 4014.2.a.r.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1338.2.a.h.1.5 5 1.1 even 1 trivial
4014.2.a.r.1.1 5 3.2 odd 2