Properties

Label 4017.2.a.d.1.2
Level 4017
Weight 2
Character 4017.1
Self dual yes
Analytic conductor 32.076
Analytic rank 1
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 4017 = 3 \cdot 13 \cdot 103 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 4017.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(32.0759064919\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\)
Character \(\chi\) = 4017.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214 q^{2} -1.00000 q^{3} -1.82843 q^{4} -0.585786 q^{5} -0.414214 q^{6} -2.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.414214 q^{2} -1.00000 q^{3} -1.82843 q^{4} -0.585786 q^{5} -0.414214 q^{6} -2.82843 q^{7} -1.58579 q^{8} +1.00000 q^{9} -0.242641 q^{10} -1.41421 q^{11} +1.82843 q^{12} +1.00000 q^{13} -1.17157 q^{14} +0.585786 q^{15} +3.00000 q^{16} +6.82843 q^{17} +0.414214 q^{18} -5.65685 q^{19} +1.07107 q^{20} +2.82843 q^{21} -0.585786 q^{22} +7.65685 q^{23} +1.58579 q^{24} -4.65685 q^{25} +0.414214 q^{26} -1.00000 q^{27} +5.17157 q^{28} +3.65685 q^{29} +0.242641 q^{30} -4.58579 q^{31} +4.41421 q^{32} +1.41421 q^{33} +2.82843 q^{34} +1.65685 q^{35} -1.82843 q^{36} +11.0711 q^{37} -2.34315 q^{38} -1.00000 q^{39} +0.928932 q^{40} +1.17157 q^{41} +1.17157 q^{42} -2.82843 q^{43} +2.58579 q^{44} -0.585786 q^{45} +3.17157 q^{46} +7.07107 q^{47} -3.00000 q^{48} +1.00000 q^{49} -1.92893 q^{50} -6.82843 q^{51} -1.82843 q^{52} -3.65685 q^{53} -0.414214 q^{54} +0.828427 q^{55} +4.48528 q^{56} +5.65685 q^{57} +1.51472 q^{58} -6.00000 q^{59} -1.07107 q^{60} +4.00000 q^{61} -1.89949 q^{62} -2.82843 q^{63} -4.17157 q^{64} -0.585786 q^{65} +0.585786 q^{66} -4.58579 q^{67} -12.4853 q^{68} -7.65685 q^{69} +0.686292 q^{70} +4.24264 q^{71} -1.58579 q^{72} -7.75736 q^{73} +4.58579 q^{74} +4.65685 q^{75} +10.3431 q^{76} +4.00000 q^{77} -0.414214 q^{78} -6.48528 q^{79} -1.75736 q^{80} +1.00000 q^{81} +0.485281 q^{82} -0.828427 q^{83} -5.17157 q^{84} -4.00000 q^{85} -1.17157 q^{86} -3.65685 q^{87} +2.24264 q^{88} +5.75736 q^{89} -0.242641 q^{90} -2.82843 q^{91} -14.0000 q^{92} +4.58579 q^{93} +2.92893 q^{94} +3.31371 q^{95} -4.41421 q^{96} +0.343146 q^{97} +0.414214 q^{98} -1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} + 2q^{6} - 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} + 2q^{6} - 6q^{8} + 2q^{9} + 8q^{10} - 2q^{12} + 2q^{13} - 8q^{14} + 4q^{15} + 6q^{16} + 8q^{17} - 2q^{18} - 12q^{20} - 4q^{22} + 4q^{23} + 6q^{24} + 2q^{25} - 2q^{26} - 2q^{27} + 16q^{28} - 4q^{29} - 8q^{30} - 12q^{31} + 6q^{32} - 8q^{35} + 2q^{36} + 8q^{37} - 16q^{38} - 2q^{39} + 16q^{40} + 8q^{41} + 8q^{42} + 8q^{44} - 4q^{45} + 12q^{46} - 6q^{48} + 2q^{49} - 18q^{50} - 8q^{51} + 2q^{52} + 4q^{53} + 2q^{54} - 4q^{55} - 8q^{56} + 20q^{58} - 12q^{59} + 12q^{60} + 8q^{61} + 16q^{62} - 14q^{64} - 4q^{65} + 4q^{66} - 12q^{67} - 8q^{68} - 4q^{69} + 24q^{70} - 6q^{72} - 24q^{73} + 12q^{74} - 2q^{75} + 32q^{76} + 8q^{77} + 2q^{78} + 4q^{79} - 12q^{80} + 2q^{81} - 16q^{82} + 4q^{83} - 16q^{84} - 8q^{85} - 8q^{86} + 4q^{87} - 4q^{88} + 20q^{89} + 8q^{90} - 28q^{92} + 12q^{93} + 20q^{94} - 16q^{95} - 6q^{96} + 12q^{97} - 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214 0.292893 0.146447 0.989219i \(-0.453216\pi\)
0.146447 + 0.989219i \(0.453216\pi\)
\(3\) −1.00000 −0.577350
\(4\) −1.82843 −0.914214
\(5\) −0.585786 −0.261972 −0.130986 0.991384i \(-0.541814\pi\)
−0.130986 + 0.991384i \(0.541814\pi\)
\(6\) −0.414214 −0.169102
\(7\) −2.82843 −1.06904 −0.534522 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) −1.58579 −0.560660
\(9\) 1.00000 0.333333
\(10\) −0.242641 −0.0767297
\(11\) −1.41421 −0.426401 −0.213201 0.977008i \(-0.568389\pi\)
−0.213201 + 0.977008i \(0.568389\pi\)
\(12\) 1.82843 0.527821
\(13\) 1.00000 0.277350
\(14\) −1.17157 −0.313116
\(15\) 0.585786 0.151249
\(16\) 3.00000 0.750000
\(17\) 6.82843 1.65614 0.828068 0.560627i \(-0.189440\pi\)
0.828068 + 0.560627i \(0.189440\pi\)
\(18\) 0.414214 0.0976311
\(19\) −5.65685 −1.29777 −0.648886 0.760886i \(-0.724765\pi\)
−0.648886 + 0.760886i \(0.724765\pi\)
\(20\) 1.07107 0.239498
\(21\) 2.82843 0.617213
\(22\) −0.585786 −0.124890
\(23\) 7.65685 1.59656 0.798282 0.602284i \(-0.205742\pi\)
0.798282 + 0.602284i \(0.205742\pi\)
\(24\) 1.58579 0.323697
\(25\) −4.65685 −0.931371
\(26\) 0.414214 0.0812340
\(27\) −1.00000 −0.192450
\(28\) 5.17157 0.977335
\(29\) 3.65685 0.679061 0.339530 0.940595i \(-0.389732\pi\)
0.339530 + 0.940595i \(0.389732\pi\)
\(30\) 0.242641 0.0442999
\(31\) −4.58579 −0.823632 −0.411816 0.911267i \(-0.635105\pi\)
−0.411816 + 0.911267i \(0.635105\pi\)
\(32\) 4.41421 0.780330
\(33\) 1.41421 0.246183
\(34\) 2.82843 0.485071
\(35\) 1.65685 0.280059
\(36\) −1.82843 −0.304738
\(37\) 11.0711 1.82007 0.910036 0.414529i \(-0.136054\pi\)
0.910036 + 0.414529i \(0.136054\pi\)
\(38\) −2.34315 −0.380108
\(39\) −1.00000 −0.160128
\(40\) 0.928932 0.146877
\(41\) 1.17157 0.182969 0.0914845 0.995807i \(-0.470839\pi\)
0.0914845 + 0.995807i \(0.470839\pi\)
\(42\) 1.17157 0.180778
\(43\) −2.82843 −0.431331 −0.215666 0.976467i \(-0.569192\pi\)
−0.215666 + 0.976467i \(0.569192\pi\)
\(44\) 2.58579 0.389822
\(45\) −0.585786 −0.0873239
\(46\) 3.17157 0.467623
\(47\) 7.07107 1.03142 0.515711 0.856763i \(-0.327528\pi\)
0.515711 + 0.856763i \(0.327528\pi\)
\(48\) −3.00000 −0.433013
\(49\) 1.00000 0.142857
\(50\) −1.92893 −0.272792
\(51\) −6.82843 −0.956171
\(52\) −1.82843 −0.253557
\(53\) −3.65685 −0.502308 −0.251154 0.967947i \(-0.580810\pi\)
−0.251154 + 0.967947i \(0.580810\pi\)
\(54\) −0.414214 −0.0563673
\(55\) 0.828427 0.111705
\(56\) 4.48528 0.599371
\(57\) 5.65685 0.749269
\(58\) 1.51472 0.198892
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) −1.07107 −0.138274
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) −1.89949 −0.241236
\(63\) −2.82843 −0.356348
\(64\) −4.17157 −0.521447
\(65\) −0.585786 −0.0726579
\(66\) 0.585786 0.0721053
\(67\) −4.58579 −0.560243 −0.280121 0.959965i \(-0.590375\pi\)
−0.280121 + 0.959965i \(0.590375\pi\)
\(68\) −12.4853 −1.51406
\(69\) −7.65685 −0.921777
\(70\) 0.686292 0.0820275
\(71\) 4.24264 0.503509 0.251754 0.967791i \(-0.418992\pi\)
0.251754 + 0.967791i \(0.418992\pi\)
\(72\) −1.58579 −0.186887
\(73\) −7.75736 −0.907930 −0.453965 0.891019i \(-0.649991\pi\)
−0.453965 + 0.891019i \(0.649991\pi\)
\(74\) 4.58579 0.533087
\(75\) 4.65685 0.537727
\(76\) 10.3431 1.18644
\(77\) 4.00000 0.455842
\(78\) −0.414214 −0.0469005
\(79\) −6.48528 −0.729651 −0.364826 0.931076i \(-0.618871\pi\)
−0.364826 + 0.931076i \(0.618871\pi\)
\(80\) −1.75736 −0.196479
\(81\) 1.00000 0.111111
\(82\) 0.485281 0.0535904
\(83\) −0.828427 −0.0909317 −0.0454658 0.998966i \(-0.514477\pi\)
−0.0454658 + 0.998966i \(0.514477\pi\)
\(84\) −5.17157 −0.564265
\(85\) −4.00000 −0.433861
\(86\) −1.17157 −0.126334
\(87\) −3.65685 −0.392056
\(88\) 2.24264 0.239066
\(89\) 5.75736 0.610279 0.305139 0.952308i \(-0.401297\pi\)
0.305139 + 0.952308i \(0.401297\pi\)
\(90\) −0.242641 −0.0255766
\(91\) −2.82843 −0.296500
\(92\) −14.0000 −1.45960
\(93\) 4.58579 0.475524
\(94\) 2.92893 0.302096
\(95\) 3.31371 0.339979
\(96\) −4.41421 −0.450524
\(97\) 0.343146 0.0348412 0.0174206 0.999848i \(-0.494455\pi\)
0.0174206 + 0.999848i \(0.494455\pi\)
\(98\) 0.414214 0.0418419
\(99\) −1.41421 −0.142134
\(100\) 8.51472 0.851472
\(101\) −3.65685 −0.363871 −0.181935 0.983311i \(-0.558236\pi\)
−0.181935 + 0.983311i \(0.558236\pi\)
\(102\) −2.82843 −0.280056
\(103\) 1.00000 0.0985329
\(104\) −1.58579 −0.155499
\(105\) −1.65685 −0.161692
\(106\) −1.51472 −0.147122
\(107\) 2.34315 0.226520 0.113260 0.993565i \(-0.463871\pi\)
0.113260 + 0.993565i \(0.463871\pi\)
\(108\) 1.82843 0.175940
\(109\) 8.72792 0.835983 0.417992 0.908451i \(-0.362734\pi\)
0.417992 + 0.908451i \(0.362734\pi\)
\(110\) 0.343146 0.0327177
\(111\) −11.0711 −1.05082
\(112\) −8.48528 −0.801784
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 2.34315 0.219456
\(115\) −4.48528 −0.418255
\(116\) −6.68629 −0.620807
\(117\) 1.00000 0.0924500
\(118\) −2.48528 −0.228789
\(119\) −19.3137 −1.77048
\(120\) −0.928932 −0.0847995
\(121\) −9.00000 −0.818182
\(122\) 1.65685 0.150005
\(123\) −1.17157 −0.105637
\(124\) 8.38478 0.752975
\(125\) 5.65685 0.505964
\(126\) −1.17157 −0.104372
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −10.5563 −0.933058
\(129\) 2.82843 0.249029
\(130\) −0.242641 −0.0212810
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) −2.58579 −0.225064
\(133\) 16.0000 1.38738
\(134\) −1.89949 −0.164091
\(135\) 0.585786 0.0504165
\(136\) −10.8284 −0.928530
\(137\) −8.00000 −0.683486 −0.341743 0.939793i \(-0.611017\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) −3.17157 −0.269982
\(139\) −8.14214 −0.690607 −0.345303 0.938491i \(-0.612224\pi\)
−0.345303 + 0.938491i \(0.612224\pi\)
\(140\) −3.02944 −0.256034
\(141\) −7.07107 −0.595491
\(142\) 1.75736 0.147474
\(143\) −1.41421 −0.118262
\(144\) 3.00000 0.250000
\(145\) −2.14214 −0.177895
\(146\) −3.21320 −0.265927
\(147\) −1.00000 −0.0824786
\(148\) −20.2426 −1.66393
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 1.92893 0.157497
\(151\) −22.7279 −1.84957 −0.924786 0.380488i \(-0.875756\pi\)
−0.924786 + 0.380488i \(0.875756\pi\)
\(152\) 8.97056 0.727609
\(153\) 6.82843 0.552046
\(154\) 1.65685 0.133513
\(155\) 2.68629 0.215768
\(156\) 1.82843 0.146391
\(157\) 12.1421 0.969048 0.484524 0.874778i \(-0.338993\pi\)
0.484524 + 0.874778i \(0.338993\pi\)
\(158\) −2.68629 −0.213710
\(159\) 3.65685 0.290007
\(160\) −2.58579 −0.204424
\(161\) −21.6569 −1.70680
\(162\) 0.414214 0.0325437
\(163\) −5.65685 −0.443079 −0.221540 0.975151i \(-0.571108\pi\)
−0.221540 + 0.975151i \(0.571108\pi\)
\(164\) −2.14214 −0.167273
\(165\) −0.828427 −0.0644930
\(166\) −0.343146 −0.0266333
\(167\) 4.14214 0.320528 0.160264 0.987074i \(-0.448765\pi\)
0.160264 + 0.987074i \(0.448765\pi\)
\(168\) −4.48528 −0.346047
\(169\) 1.00000 0.0769231
\(170\) −1.65685 −0.127075
\(171\) −5.65685 −0.432590
\(172\) 5.17157 0.394329
\(173\) 13.7990 1.04912 0.524559 0.851374i \(-0.324230\pi\)
0.524559 + 0.851374i \(0.324230\pi\)
\(174\) −1.51472 −0.114831
\(175\) 13.1716 0.995677
\(176\) −4.24264 −0.319801
\(177\) 6.00000 0.450988
\(178\) 2.38478 0.178747
\(179\) −9.31371 −0.696139 −0.348070 0.937469i \(-0.613163\pi\)
−0.348070 + 0.937469i \(0.613163\pi\)
\(180\) 1.07107 0.0798327
\(181\) 4.34315 0.322823 0.161412 0.986887i \(-0.448395\pi\)
0.161412 + 0.986887i \(0.448395\pi\)
\(182\) −1.17157 −0.0868428
\(183\) −4.00000 −0.295689
\(184\) −12.1421 −0.895130
\(185\) −6.48528 −0.476807
\(186\) 1.89949 0.139278
\(187\) −9.65685 −0.706179
\(188\) −12.9289 −0.942939
\(189\) 2.82843 0.205738
\(190\) 1.37258 0.0995776
\(191\) −8.97056 −0.649087 −0.324544 0.945871i \(-0.605211\pi\)
−0.324544 + 0.945871i \(0.605211\pi\)
\(192\) 4.17157 0.301057
\(193\) −16.7279 −1.20410 −0.602051 0.798458i \(-0.705650\pi\)
−0.602051 + 0.798458i \(0.705650\pi\)
\(194\) 0.142136 0.0102047
\(195\) 0.585786 0.0419490
\(196\) −1.82843 −0.130602
\(197\) −11.8995 −0.847804 −0.423902 0.905708i \(-0.639340\pi\)
−0.423902 + 0.905708i \(0.639340\pi\)
\(198\) −0.585786 −0.0416300
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) 7.38478 0.522183
\(201\) 4.58579 0.323456
\(202\) −1.51472 −0.106575
\(203\) −10.3431 −0.725947
\(204\) 12.4853 0.874145
\(205\) −0.686292 −0.0479327
\(206\) 0.414214 0.0288596
\(207\) 7.65685 0.532188
\(208\) 3.00000 0.208013
\(209\) 8.00000 0.553372
\(210\) −0.686292 −0.0473586
\(211\) 5.17157 0.356026 0.178013 0.984028i \(-0.443033\pi\)
0.178013 + 0.984028i \(0.443033\pi\)
\(212\) 6.68629 0.459216
\(213\) −4.24264 −0.290701
\(214\) 0.970563 0.0663463
\(215\) 1.65685 0.112997
\(216\) 1.58579 0.107899
\(217\) 12.9706 0.880499
\(218\) 3.61522 0.244854
\(219\) 7.75736 0.524194
\(220\) −1.51472 −0.102122
\(221\) 6.82843 0.459330
\(222\) −4.58579 −0.307778
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) −12.4853 −0.834208
\(225\) −4.65685 −0.310457
\(226\) −4.14214 −0.275531
\(227\) −7.75736 −0.514874 −0.257437 0.966295i \(-0.582878\pi\)
−0.257437 + 0.966295i \(0.582878\pi\)
\(228\) −10.3431 −0.684992
\(229\) −29.3137 −1.93710 −0.968552 0.248810i \(-0.919960\pi\)
−0.968552 + 0.248810i \(0.919960\pi\)
\(230\) −1.85786 −0.122504
\(231\) −4.00000 −0.263181
\(232\) −5.79899 −0.380722
\(233\) 16.6274 1.08930 0.544649 0.838664i \(-0.316663\pi\)
0.544649 + 0.838664i \(0.316663\pi\)
\(234\) 0.414214 0.0270780
\(235\) −4.14214 −0.270203
\(236\) 10.9706 0.714123
\(237\) 6.48528 0.421264
\(238\) −8.00000 −0.518563
\(239\) 23.6569 1.53023 0.765117 0.643891i \(-0.222681\pi\)
0.765117 + 0.643891i \(0.222681\pi\)
\(240\) 1.75736 0.113437
\(241\) 4.72792 0.304552 0.152276 0.988338i \(-0.451340\pi\)
0.152276 + 0.988338i \(0.451340\pi\)
\(242\) −3.72792 −0.239640
\(243\) −1.00000 −0.0641500
\(244\) −7.31371 −0.468212
\(245\) −0.585786 −0.0374245
\(246\) −0.485281 −0.0309404
\(247\) −5.65685 −0.359937
\(248\) 7.27208 0.461777
\(249\) 0.828427 0.0524994
\(250\) 2.34315 0.148194
\(251\) 7.51472 0.474325 0.237162 0.971470i \(-0.423783\pi\)
0.237162 + 0.971470i \(0.423783\pi\)
\(252\) 5.17157 0.325778
\(253\) −10.8284 −0.680777
\(254\) 0 0
\(255\) 4.00000 0.250490
\(256\) 3.97056 0.248160
\(257\) −2.97056 −0.185299 −0.0926493 0.995699i \(-0.529534\pi\)
−0.0926493 + 0.995699i \(0.529534\pi\)
\(258\) 1.17157 0.0729389
\(259\) −31.3137 −1.94574
\(260\) 1.07107 0.0664248
\(261\) 3.65685 0.226354
\(262\) −2.48528 −0.153541
\(263\) −6.82843 −0.421059 −0.210529 0.977588i \(-0.567519\pi\)
−0.210529 + 0.977588i \(0.567519\pi\)
\(264\) −2.24264 −0.138025
\(265\) 2.14214 0.131590
\(266\) 6.62742 0.406353
\(267\) −5.75736 −0.352345
\(268\) 8.38478 0.512182
\(269\) −2.82843 −0.172452 −0.0862261 0.996276i \(-0.527481\pi\)
−0.0862261 + 0.996276i \(0.527481\pi\)
\(270\) 0.242641 0.0147666
\(271\) −23.4142 −1.42231 −0.711156 0.703034i \(-0.751828\pi\)
−0.711156 + 0.703034i \(0.751828\pi\)
\(272\) 20.4853 1.24210
\(273\) 2.82843 0.171184
\(274\) −3.31371 −0.200188
\(275\) 6.58579 0.397138
\(276\) 14.0000 0.842701
\(277\) 8.82843 0.530449 0.265224 0.964187i \(-0.414554\pi\)
0.265224 + 0.964187i \(0.414554\pi\)
\(278\) −3.37258 −0.202274
\(279\) −4.58579 −0.274544
\(280\) −2.62742 −0.157018
\(281\) −19.8995 −1.18710 −0.593552 0.804796i \(-0.702275\pi\)
−0.593552 + 0.804796i \(0.702275\pi\)
\(282\) −2.92893 −0.174415
\(283\) 18.1421 1.07844 0.539219 0.842166i \(-0.318720\pi\)
0.539219 + 0.842166i \(0.318720\pi\)
\(284\) −7.75736 −0.460315
\(285\) −3.31371 −0.196287
\(286\) −0.585786 −0.0346383
\(287\) −3.31371 −0.195602
\(288\) 4.41421 0.260110
\(289\) 29.6274 1.74279
\(290\) −0.887302 −0.0521041
\(291\) −0.343146 −0.0201156
\(292\) 14.1838 0.830042
\(293\) 21.5563 1.25934 0.629668 0.776865i \(-0.283191\pi\)
0.629668 + 0.776865i \(0.283191\pi\)
\(294\) −0.414214 −0.0241574
\(295\) 3.51472 0.204635
\(296\) −17.5563 −1.02044
\(297\) 1.41421 0.0820610
\(298\) −1.65685 −0.0959790
\(299\) 7.65685 0.442807
\(300\) −8.51472 −0.491598
\(301\) 8.00000 0.461112
\(302\) −9.41421 −0.541727
\(303\) 3.65685 0.210081
\(304\) −16.9706 −0.973329
\(305\) −2.34315 −0.134168
\(306\) 2.82843 0.161690
\(307\) −6.92893 −0.395455 −0.197728 0.980257i \(-0.563356\pi\)
−0.197728 + 0.980257i \(0.563356\pi\)
\(308\) −7.31371 −0.416737
\(309\) −1.00000 −0.0568880
\(310\) 1.11270 0.0631970
\(311\) −33.6569 −1.90851 −0.954253 0.299002i \(-0.903346\pi\)
−0.954253 + 0.299002i \(0.903346\pi\)
\(312\) 1.58579 0.0897775
\(313\) 29.3137 1.65691 0.828454 0.560057i \(-0.189221\pi\)
0.828454 + 0.560057i \(0.189221\pi\)
\(314\) 5.02944 0.283828
\(315\) 1.65685 0.0933532
\(316\) 11.8579 0.667057
\(317\) 25.4558 1.42974 0.714871 0.699256i \(-0.246485\pi\)
0.714871 + 0.699256i \(0.246485\pi\)
\(318\) 1.51472 0.0849412
\(319\) −5.17157 −0.289552
\(320\) 2.44365 0.136604
\(321\) −2.34315 −0.130782
\(322\) −8.97056 −0.499910
\(323\) −38.6274 −2.14929
\(324\) −1.82843 −0.101579
\(325\) −4.65685 −0.258316
\(326\) −2.34315 −0.129775
\(327\) −8.72792 −0.482655
\(328\) −1.85786 −0.102583
\(329\) −20.0000 −1.10264
\(330\) −0.343146 −0.0188896
\(331\) −9.07107 −0.498591 −0.249295 0.968427i \(-0.580199\pi\)
−0.249295 + 0.968427i \(0.580199\pi\)
\(332\) 1.51472 0.0831310
\(333\) 11.0711 0.606691
\(334\) 1.71573 0.0938805
\(335\) 2.68629 0.146768
\(336\) 8.48528 0.462910
\(337\) −32.6274 −1.77733 −0.888664 0.458558i \(-0.848366\pi\)
−0.888664 + 0.458558i \(0.848366\pi\)
\(338\) 0.414214 0.0225302
\(339\) 10.0000 0.543125
\(340\) 7.31371 0.396642
\(341\) 6.48528 0.351198
\(342\) −2.34315 −0.126703
\(343\) 16.9706 0.916324
\(344\) 4.48528 0.241830
\(345\) 4.48528 0.241479
\(346\) 5.71573 0.307279
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 6.68629 0.358423
\(349\) −22.8701 −1.22421 −0.612103 0.790778i \(-0.709676\pi\)
−0.612103 + 0.790778i \(0.709676\pi\)
\(350\) 5.45584 0.291627
\(351\) −1.00000 −0.0533761
\(352\) −6.24264 −0.332734
\(353\) −29.0711 −1.54730 −0.773648 0.633615i \(-0.781570\pi\)
−0.773648 + 0.633615i \(0.781570\pi\)
\(354\) 2.48528 0.132091
\(355\) −2.48528 −0.131905
\(356\) −10.5269 −0.557925
\(357\) 19.3137 1.02219
\(358\) −3.85786 −0.203894
\(359\) 5.51472 0.291056 0.145528 0.989354i \(-0.453512\pi\)
0.145528 + 0.989354i \(0.453512\pi\)
\(360\) 0.928932 0.0489590
\(361\) 13.0000 0.684211
\(362\) 1.79899 0.0945528
\(363\) 9.00000 0.472377
\(364\) 5.17157 0.271064
\(365\) 4.54416 0.237852
\(366\) −1.65685 −0.0866052
\(367\) −20.8284 −1.08724 −0.543618 0.839333i \(-0.682946\pi\)
−0.543618 + 0.839333i \(0.682946\pi\)
\(368\) 22.9706 1.19742
\(369\) 1.17157 0.0609896
\(370\) −2.68629 −0.139654
\(371\) 10.3431 0.536989
\(372\) −8.38478 −0.434730
\(373\) −9.65685 −0.500013 −0.250006 0.968244i \(-0.580433\pi\)
−0.250006 + 0.968244i \(0.580433\pi\)
\(374\) −4.00000 −0.206835
\(375\) −5.65685 −0.292119
\(376\) −11.2132 −0.578277
\(377\) 3.65685 0.188338
\(378\) 1.17157 0.0602592
\(379\) −7.41421 −0.380843 −0.190421 0.981702i \(-0.560985\pi\)
−0.190421 + 0.981702i \(0.560985\pi\)
\(380\) −6.05887 −0.310814
\(381\) 0 0
\(382\) −3.71573 −0.190113
\(383\) −5.41421 −0.276653 −0.138327 0.990387i \(-0.544172\pi\)
−0.138327 + 0.990387i \(0.544172\pi\)
\(384\) 10.5563 0.538701
\(385\) −2.34315 −0.119418
\(386\) −6.92893 −0.352673
\(387\) −2.82843 −0.143777
\(388\) −0.627417 −0.0318523
\(389\) −6.48528 −0.328817 −0.164408 0.986392i \(-0.552571\pi\)
−0.164408 + 0.986392i \(0.552571\pi\)
\(390\) 0.242641 0.0122866
\(391\) 52.2843 2.64413
\(392\) −1.58579 −0.0800943
\(393\) 6.00000 0.302660
\(394\) −4.92893 −0.248316
\(395\) 3.79899 0.191148
\(396\) 2.58579 0.129941
\(397\) 26.8701 1.34857 0.674285 0.738471i \(-0.264452\pi\)
0.674285 + 0.738471i \(0.264452\pi\)
\(398\) −1.65685 −0.0830506
\(399\) −16.0000 −0.801002
\(400\) −13.9706 −0.698528
\(401\) −30.8284 −1.53950 −0.769749 0.638347i \(-0.779619\pi\)
−0.769749 + 0.638347i \(0.779619\pi\)
\(402\) 1.89949 0.0947382
\(403\) −4.58579 −0.228434
\(404\) 6.68629 0.332655
\(405\) −0.585786 −0.0291080
\(406\) −4.28427 −0.212625
\(407\) −15.6569 −0.776081
\(408\) 10.8284 0.536087
\(409\) 20.8284 1.02990 0.514950 0.857220i \(-0.327811\pi\)
0.514950 + 0.857220i \(0.327811\pi\)
\(410\) −0.284271 −0.0140392
\(411\) 8.00000 0.394611
\(412\) −1.82843 −0.0900801
\(413\) 16.9706 0.835067
\(414\) 3.17157 0.155874
\(415\) 0.485281 0.0238215
\(416\) 4.41421 0.216425
\(417\) 8.14214 0.398722
\(418\) 3.31371 0.162079
\(419\) −18.3431 −0.896121 −0.448061 0.894003i \(-0.647885\pi\)
−0.448061 + 0.894003i \(0.647885\pi\)
\(420\) 3.02944 0.147821
\(421\) 6.48528 0.316073 0.158037 0.987433i \(-0.449484\pi\)
0.158037 + 0.987433i \(0.449484\pi\)
\(422\) 2.14214 0.104278
\(423\) 7.07107 0.343807
\(424\) 5.79899 0.281624
\(425\) −31.7990 −1.54248
\(426\) −1.75736 −0.0851443
\(427\) −11.3137 −0.547509
\(428\) −4.28427 −0.207088
\(429\) 1.41421 0.0682789
\(430\) 0.686292 0.0330959
\(431\) −13.5147 −0.650981 −0.325491 0.945545i \(-0.605529\pi\)
−0.325491 + 0.945545i \(0.605529\pi\)
\(432\) −3.00000 −0.144338
\(433\) 7.65685 0.367965 0.183982 0.982930i \(-0.441101\pi\)
0.183982 + 0.982930i \(0.441101\pi\)
\(434\) 5.37258 0.257892
\(435\) 2.14214 0.102708
\(436\) −15.9584 −0.764267
\(437\) −43.3137 −2.07198
\(438\) 3.21320 0.153533
\(439\) 25.4558 1.21494 0.607471 0.794342i \(-0.292184\pi\)
0.607471 + 0.794342i \(0.292184\pi\)
\(440\) −1.31371 −0.0626286
\(441\) 1.00000 0.0476190
\(442\) 2.82843 0.134535
\(443\) 2.82843 0.134383 0.0671913 0.997740i \(-0.478596\pi\)
0.0671913 + 0.997740i \(0.478596\pi\)
\(444\) 20.2426 0.960673
\(445\) −3.37258 −0.159876
\(446\) 0 0
\(447\) 4.00000 0.189194
\(448\) 11.7990 0.557450
\(449\) −31.8995 −1.50543 −0.752715 0.658346i \(-0.771256\pi\)
−0.752715 + 0.658346i \(0.771256\pi\)
\(450\) −1.92893 −0.0909307
\(451\) −1.65685 −0.0780182
\(452\) 18.2843 0.860020
\(453\) 22.7279 1.06785
\(454\) −3.21320 −0.150803
\(455\) 1.65685 0.0776745
\(456\) −8.97056 −0.420085
\(457\) −26.5858 −1.24363 −0.621815 0.783164i \(-0.713605\pi\)
−0.621815 + 0.783164i \(0.713605\pi\)
\(458\) −12.1421 −0.567365
\(459\) −6.82843 −0.318724
\(460\) 8.20101 0.382374
\(461\) 29.9411 1.39450 0.697249 0.716829i \(-0.254407\pi\)
0.697249 + 0.716829i \(0.254407\pi\)
\(462\) −1.65685 −0.0770838
\(463\) −9.75736 −0.453463 −0.226731 0.973957i \(-0.572804\pi\)
−0.226731 + 0.973957i \(0.572804\pi\)
\(464\) 10.9706 0.509296
\(465\) −2.68629 −0.124574
\(466\) 6.88730 0.319048
\(467\) 35.3137 1.63412 0.817062 0.576550i \(-0.195601\pi\)
0.817062 + 0.576550i \(0.195601\pi\)
\(468\) −1.82843 −0.0845191
\(469\) 12.9706 0.598925
\(470\) −1.71573 −0.0791407
\(471\) −12.1421 −0.559480
\(472\) 9.51472 0.437950
\(473\) 4.00000 0.183920
\(474\) 2.68629 0.123385
\(475\) 26.3431 1.20871
\(476\) 35.3137 1.61860
\(477\) −3.65685 −0.167436
\(478\) 9.79899 0.448195
\(479\) −2.58579 −0.118148 −0.0590738 0.998254i \(-0.518815\pi\)
−0.0590738 + 0.998254i \(0.518815\pi\)
\(480\) 2.58579 0.118024
\(481\) 11.0711 0.504797
\(482\) 1.95837 0.0892013
\(483\) 21.6569 0.985421
\(484\) 16.4558 0.747993
\(485\) −0.201010 −0.00912740
\(486\) −0.414214 −0.0187891
\(487\) 29.5563 1.33933 0.669663 0.742665i \(-0.266439\pi\)
0.669663 + 0.742665i \(0.266439\pi\)
\(488\) −6.34315 −0.287141
\(489\) 5.65685 0.255812
\(490\) −0.242641 −0.0109614
\(491\) −18.3431 −0.827815 −0.413907 0.910319i \(-0.635836\pi\)
−0.413907 + 0.910319i \(0.635836\pi\)
\(492\) 2.14214 0.0965749
\(493\) 24.9706 1.12462
\(494\) −2.34315 −0.105423
\(495\) 0.828427 0.0372350
\(496\) −13.7574 −0.617724
\(497\) −12.0000 −0.538274
\(498\) 0.343146 0.0153767
\(499\) −3.41421 −0.152841 −0.0764206 0.997076i \(-0.524349\pi\)
−0.0764206 + 0.997076i \(0.524349\pi\)
\(500\) −10.3431 −0.462560
\(501\) −4.14214 −0.185057
\(502\) 3.11270 0.138927
\(503\) −38.2843 −1.70701 −0.853506 0.521084i \(-0.825528\pi\)
−0.853506 + 0.521084i \(0.825528\pi\)
\(504\) 4.48528 0.199790
\(505\) 2.14214 0.0953238
\(506\) −4.48528 −0.199395
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) −29.4558 −1.30561 −0.652804 0.757527i \(-0.726407\pi\)
−0.652804 + 0.757527i \(0.726407\pi\)
\(510\) 1.65685 0.0733667
\(511\) 21.9411 0.970618
\(512\) 22.7574 1.00574
\(513\) 5.65685 0.249756
\(514\) −1.23045 −0.0542727
\(515\) −0.585786 −0.0258128
\(516\) −5.17157 −0.227666
\(517\) −10.0000 −0.439799
\(518\) −12.9706 −0.569894
\(519\) −13.7990 −0.605708
\(520\) 0.928932 0.0407364
\(521\) −25.7990 −1.13027 −0.565137 0.824997i \(-0.691177\pi\)
−0.565137 + 0.824997i \(0.691177\pi\)
\(522\) 1.51472 0.0662974
\(523\) 0.142136 0.00621516 0.00310758 0.999995i \(-0.499011\pi\)
0.00310758 + 0.999995i \(0.499011\pi\)
\(524\) 10.9706 0.479251
\(525\) −13.1716 −0.574855
\(526\) −2.82843 −0.123325
\(527\) −31.3137 −1.36405
\(528\) 4.24264 0.184637
\(529\) 35.6274 1.54902
\(530\) 0.887302 0.0385419
\(531\) −6.00000 −0.260378
\(532\) −29.2548 −1.26836
\(533\) 1.17157 0.0507465
\(534\) −2.38478 −0.103199
\(535\) −1.37258 −0.0593419
\(536\) 7.27208 0.314106
\(537\) 9.31371 0.401916
\(538\) −1.17157 −0.0505101
\(539\) −1.41421 −0.0609145
\(540\) −1.07107 −0.0460914
\(541\) −25.1127 −1.07968 −0.539840 0.841768i \(-0.681515\pi\)
−0.539840 + 0.841768i \(0.681515\pi\)
\(542\) −9.69848 −0.416586
\(543\) −4.34315 −0.186382
\(544\) 30.1421 1.29233
\(545\) −5.11270 −0.219004
\(546\) 1.17157 0.0501387
\(547\) −12.8284 −0.548504 −0.274252 0.961658i \(-0.588430\pi\)
−0.274252 + 0.961658i \(0.588430\pi\)
\(548\) 14.6274 0.624852
\(549\) 4.00000 0.170716
\(550\) 2.72792 0.116319
\(551\) −20.6863 −0.881266
\(552\) 12.1421 0.516804
\(553\) 18.3431 0.780030
\(554\) 3.65685 0.155365
\(555\) 6.48528 0.275285
\(556\) 14.8873 0.631362
\(557\) −33.0711 −1.40127 −0.700633 0.713522i \(-0.747099\pi\)
−0.700633 + 0.713522i \(0.747099\pi\)
\(558\) −1.89949 −0.0804120
\(559\) −2.82843 −0.119630
\(560\) 4.97056 0.210045
\(561\) 9.65685 0.407713
\(562\) −8.24264 −0.347695
\(563\) −19.3137 −0.813976 −0.406988 0.913434i \(-0.633421\pi\)
−0.406988 + 0.913434i \(0.633421\pi\)
\(564\) 12.9289 0.544406
\(565\) 5.85786 0.246442
\(566\) 7.51472 0.315867
\(567\) −2.82843 −0.118783
\(568\) −6.72792 −0.282297
\(569\) 8.14214 0.341336 0.170668 0.985329i \(-0.445407\pi\)
0.170668 + 0.985329i \(0.445407\pi\)
\(570\) −1.37258 −0.0574912
\(571\) 34.6274 1.44911 0.724556 0.689216i \(-0.242045\pi\)
0.724556 + 0.689216i \(0.242045\pi\)
\(572\) 2.58579 0.108117
\(573\) 8.97056 0.374751
\(574\) −1.37258 −0.0572905
\(575\) −35.6569 −1.48699
\(576\) −4.17157 −0.173816
\(577\) −18.3848 −0.765368 −0.382684 0.923879i \(-0.625000\pi\)
−0.382684 + 0.923879i \(0.625000\pi\)
\(578\) 12.2721 0.510451
\(579\) 16.7279 0.695189
\(580\) 3.91674 0.162634
\(581\) 2.34315 0.0972101
\(582\) −0.142136 −0.00589171
\(583\) 5.17157 0.214185
\(584\) 12.3015 0.509040
\(585\) −0.585786 −0.0242193
\(586\) 8.92893 0.368851
\(587\) 11.1716 0.461100 0.230550 0.973060i \(-0.425947\pi\)
0.230550 + 0.973060i \(0.425947\pi\)
\(588\) 1.82843 0.0754031
\(589\) 25.9411 1.06889
\(590\) 1.45584 0.0599362
\(591\) 11.8995 0.489480
\(592\) 33.2132 1.36505
\(593\) −33.0711 −1.35807 −0.679033 0.734108i \(-0.737601\pi\)
−0.679033 + 0.734108i \(0.737601\pi\)
\(594\) 0.585786 0.0240351
\(595\) 11.3137 0.463817
\(596\) 7.31371 0.299581
\(597\) 4.00000 0.163709
\(598\) 3.17157 0.129695
\(599\) 27.1127 1.10779 0.553897 0.832585i \(-0.313140\pi\)
0.553897 + 0.832585i \(0.313140\pi\)
\(600\) −7.38478 −0.301482
\(601\) −17.3137 −0.706241 −0.353120 0.935578i \(-0.614879\pi\)
−0.353120 + 0.935578i \(0.614879\pi\)
\(602\) 3.31371 0.135057
\(603\) −4.58579 −0.186748
\(604\) 41.5563 1.69090
\(605\) 5.27208 0.214340
\(606\) 1.51472 0.0615312
\(607\) 3.31371 0.134499 0.0672496 0.997736i \(-0.478578\pi\)
0.0672496 + 0.997736i \(0.478578\pi\)
\(608\) −24.9706 −1.01269
\(609\) 10.3431 0.419125
\(610\) −0.970563 −0.0392969
\(611\) 7.07107 0.286065
\(612\) −12.4853 −0.504688
\(613\) 29.7990 1.20357 0.601785 0.798658i \(-0.294456\pi\)
0.601785 + 0.798658i \(0.294456\pi\)
\(614\) −2.87006 −0.115826
\(615\) 0.686292 0.0276739
\(616\) −6.34315 −0.255573
\(617\) 38.0416 1.53150 0.765749 0.643139i \(-0.222368\pi\)
0.765749 + 0.643139i \(0.222368\pi\)
\(618\) −0.414214 −0.0166621
\(619\) −38.8284 −1.56065 −0.780323 0.625377i \(-0.784945\pi\)
−0.780323 + 0.625377i \(0.784945\pi\)
\(620\) −4.91169 −0.197258
\(621\) −7.65685 −0.307259
\(622\) −13.9411 −0.558988
\(623\) −16.2843 −0.652416
\(624\) −3.00000 −0.120096
\(625\) 19.9706 0.798823
\(626\) 12.1421 0.485297
\(627\) −8.00000 −0.319489
\(628\) −22.2010 −0.885917
\(629\) 75.5980 3.01429
\(630\) 0.686292 0.0273425
\(631\) −11.5147 −0.458394 −0.229197 0.973380i \(-0.573610\pi\)
−0.229197 + 0.973380i \(0.573610\pi\)
\(632\) 10.2843 0.409086
\(633\) −5.17157 −0.205552
\(634\) 10.5442 0.418762
\(635\) 0 0
\(636\) −6.68629 −0.265129
\(637\) 1.00000 0.0396214
\(638\) −2.14214 −0.0848080
\(639\) 4.24264 0.167836
\(640\) 6.18377 0.244435
\(641\) −41.3137 −1.63179 −0.815897 0.578198i \(-0.803756\pi\)
−0.815897 + 0.578198i \(0.803756\pi\)
\(642\) −0.970563 −0.0383051
\(643\) 33.4558 1.31937 0.659685 0.751542i \(-0.270690\pi\)
0.659685 + 0.751542i \(0.270690\pi\)
\(644\) 39.5980 1.56038
\(645\) −1.65685 −0.0652386
\(646\) −16.0000 −0.629512
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −1.58579 −0.0622956
\(649\) 8.48528 0.333076
\(650\) −1.92893 −0.0756589
\(651\) −12.9706 −0.508356
\(652\) 10.3431 0.405069
\(653\) −11.1716 −0.437177 −0.218589 0.975817i \(-0.570145\pi\)
−0.218589 + 0.975817i \(0.570145\pi\)
\(654\) −3.61522 −0.141366
\(655\) 3.51472 0.137331
\(656\) 3.51472 0.137227
\(657\) −7.75736 −0.302643
\(658\) −8.28427 −0.322955
\(659\) −8.62742 −0.336076 −0.168038 0.985780i \(-0.553743\pi\)
−0.168038 + 0.985780i \(0.553743\pi\)
\(660\) 1.51472 0.0589603
\(661\) 23.0711 0.897361 0.448680 0.893692i \(-0.351894\pi\)
0.448680 + 0.893692i \(0.351894\pi\)
\(662\) −3.75736 −0.146034
\(663\) −6.82843 −0.265194
\(664\) 1.31371 0.0509818
\(665\) −9.37258 −0.363453
\(666\) 4.58579 0.177696
\(667\) 28.0000 1.08416
\(668\) −7.57359 −0.293031
\(669\) 0 0
\(670\) 1.11270 0.0429873
\(671\) −5.65685 −0.218380
\(672\) 12.4853 0.481630
\(673\) 41.3137 1.59253 0.796263 0.604950i \(-0.206807\pi\)
0.796263 + 0.604950i \(0.206807\pi\)
\(674\) −13.5147 −0.520568
\(675\) 4.65685 0.179242
\(676\) −1.82843 −0.0703241
\(677\) 2.68629 0.103243 0.0516213 0.998667i \(-0.483561\pi\)
0.0516213 + 0.998667i \(0.483561\pi\)
\(678\) 4.14214 0.159078
\(679\) −0.970563 −0.0372468
\(680\) 6.34315 0.243249
\(681\) 7.75736 0.297263
\(682\) 2.68629 0.102863
\(683\) −17.6985 −0.677214 −0.338607 0.940928i \(-0.609956\pi\)
−0.338607 + 0.940928i \(0.609956\pi\)
\(684\) 10.3431 0.395480
\(685\) 4.68629 0.179054
\(686\) 7.02944 0.268385
\(687\) 29.3137 1.11839
\(688\) −8.48528 −0.323498
\(689\) −3.65685 −0.139315
\(690\) 1.85786 0.0707277
\(691\) −8.87006 −0.337433 −0.168716 0.985665i \(-0.553962\pi\)
−0.168716 + 0.985665i \(0.553962\pi\)
\(692\) −25.2304 −0.959118
\(693\) 4.00000 0.151947
\(694\) 11.5980 0.440253
\(695\) 4.76955 0.180919
\(696\) 5.79899 0.219810
\(697\) 8.00000 0.303022
\(698\) −9.47309 −0.358562
\(699\) −16.6274 −0.628907
\(700\) −24.0833 −0.910262
\(701\) 35.1127 1.32619 0.663094 0.748536i \(-0.269243\pi\)
0.663094 + 0.748536i \(0.269243\pi\)
\(702\) −0.414214 −0.0156335
\(703\) −62.6274 −2.36204
\(704\) 5.89949 0.222346
\(705\) 4.14214 0.156002
\(706\) −12.0416 −0.453193
\(707\) 10.3431 0.388994
\(708\) −10.9706 −0.412299
\(709\) 30.0000 1.12667 0.563337 0.826227i \(-0.309517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) −1.02944 −0.0386341
\(711\) −6.48528 −0.243217
\(712\) −9.12994 −0.342159
\(713\) −35.1127 −1.31498
\(714\) 8.00000 0.299392
\(715\) 0.828427 0.0309814
\(716\) 17.0294 0.636420
\(717\) −23.6569 −0.883481
\(718\) 2.28427 0.0852482
\(719\) 43.7990 1.63343 0.816713 0.577044i \(-0.195794\pi\)
0.816713 + 0.577044i \(0.195794\pi\)
\(720\) −1.75736 −0.0654929
\(721\) −2.82843 −0.105336
\(722\) 5.38478 0.200401
\(723\) −4.72792 −0.175833
\(724\) −7.94113 −0.295130
\(725\) −17.0294 −0.632457
\(726\) 3.72792 0.138356
\(727\) −52.9706 −1.96457 −0.982285 0.187395i \(-0.939996\pi\)
−0.982285 + 0.187395i \(0.939996\pi\)
\(728\) 4.48528 0.166236
\(729\) 1.00000 0.0370370
\(730\) 1.88225 0.0696652
\(731\) −19.3137 −0.714343
\(732\) 7.31371 0.270322
\(733\) −53.2132 −1.96547 −0.982737 0.185007i \(-0.940769\pi\)
−0.982737 + 0.185007i \(0.940769\pi\)
\(734\) −8.62742 −0.318444
\(735\) 0.585786 0.0216071
\(736\) 33.7990 1.24585
\(737\) 6.48528 0.238888
\(738\) 0.485281 0.0178635
\(739\) 34.6274 1.27379 0.636895 0.770951i \(-0.280218\pi\)
0.636895 + 0.770951i \(0.280218\pi\)
\(740\) 11.8579 0.435904
\(741\) 5.65685 0.207810
\(742\) 4.28427 0.157281
\(743\) −29.8995 −1.09691 −0.548453 0.836181i \(-0.684783\pi\)
−0.548453 + 0.836181i \(0.684783\pi\)
\(744\) −7.27208 −0.266607
\(745\) 2.34315 0.0858462
\(746\) −4.00000 −0.146450
\(747\) −0.828427 −0.0303106
\(748\) 17.6569 0.645599
\(749\) −6.62742 −0.242161
\(750\) −2.34315 −0.0855596
\(751\) 3.17157 0.115732 0.0578662 0.998324i \(-0.481570\pi\)
0.0578662 + 0.998324i \(0.481570\pi\)
\(752\) 21.2132 0.773566
\(753\) −7.51472 −0.273852
\(754\) 1.51472 0.0551628
\(755\) 13.3137 0.484535
\(756\) −5.17157 −0.188088
\(757\) −23.6569 −0.859823 −0.429911 0.902871i \(-0.641455\pi\)
−0.429911 + 0.902871i \(0.641455\pi\)
\(758\) −3.07107 −0.111546
\(759\) 10.8284 0.393047
\(760\) −5.25483 −0.190613
\(761\) −8.58579 −0.311235 −0.155617 0.987817i \(-0.549737\pi\)
−0.155617 + 0.987817i \(0.549737\pi\)
\(762\) 0 0
\(763\) −24.6863 −0.893704
\(764\) 16.4020 0.593404
\(765\) −4.00000 −0.144620
\(766\) −2.24264 −0.0810299
\(767\) −6.00000 −0.216647
\(768\) −3.97056 −0.143275
\(769\) 2.58579 0.0932458 0.0466229 0.998913i \(-0.485154\pi\)
0.0466229 + 0.998913i \(0.485154\pi\)
\(770\) −0.970563 −0.0349767
\(771\) 2.97056 0.106982
\(772\) 30.5858 1.10081
\(773\) −12.0000 −0.431610 −0.215805 0.976436i \(-0.569238\pi\)
−0.215805 + 0.976436i \(0.569238\pi\)
\(774\) −1.17157 −0.0421113
\(775\) 21.3553 0.767106
\(776\) −0.544156 −0.0195341
\(777\) 31.3137 1.12337
\(778\) −2.68629 −0.0963082
\(779\) −6.62742 −0.237452
\(780\) −1.07107 −0.0383504
\(781\) −6.00000 −0.214697
\(782\) 21.6569 0.774448
\(783\) −3.65685 −0.130685
\(784\) 3.00000 0.107143
\(785\) −7.11270 −0.253863
\(786\) 2.48528 0.0886471
\(787\) −21.6569 −0.771983 −0.385992 0.922502i \(-0.626141\pi\)
−0.385992 + 0.922502i \(0.626141\pi\)
\(788\) 21.7574 0.775074
\(789\) 6.82843 0.243098
\(790\) 1.57359 0.0559859
\(791\) 28.2843 1.00567
\(792\) 2.24264 0.0796888
\(793\) 4.00000 0.142044
\(794\) 11.1299 0.394987
\(795\) −2.14214 −0.0759737
\(796\) 7.31371 0.259228
\(797\) 41.3137 1.46341 0.731703 0.681623i \(-0.238726\pi\)
0.731703 + 0.681623i \(0.238726\pi\)
\(798\) −6.62742 −0.234608
\(799\) 48.2843 1.70817
\(800\) −20.5563 −0.726777
\(801\) 5.75736 0.203426
\(802\) −12.7696 −0.450909
\(803\) 10.9706 0.387143
\(804\) −8.38478 −0.295708
\(805\) 12.6863 0.447133
\(806\) −1.89949 −0.0669069
\(807\) 2.82843 0.0995654
\(808\) 5.79899 0.204008
\(809\) 36.1421 1.27069 0.635345 0.772228i \(-0.280858\pi\)
0.635345 + 0.772228i \(0.280858\pi\)
\(810\) −0.242641 −0.00852552
\(811\) 18.9289 0.664685 0.332342 0.943159i \(-0.392161\pi\)
0.332342 + 0.943159i \(0.392161\pi\)
\(812\) 18.9117 0.663670
\(813\) 23.4142 0.821172
\(814\) −6.48528 −0.227309
\(815\) 3.31371 0.116074
\(816\) −20.4853 −0.717128
\(817\) 16.0000 0.559769
\(818\) 8.62742 0.301651
\(819\) −2.82843 −0.0988332
\(820\) 1.25483 0.0438207
\(821\) 15.3137 0.534452 0.267226 0.963634i \(-0.413893\pi\)
0.267226 + 0.963634i \(0.413893\pi\)
\(822\) 3.31371 0.115579
\(823\) −27.5147 −0.959103 −0.479551 0.877514i \(-0.659201\pi\)
−0.479551 + 0.877514i \(0.659201\pi\)
\(824\) −1.58579 −0.0552435
\(825\) −6.58579 −0.229288
\(826\) 7.02944 0.244585
\(827\) −32.5269 −1.13107 −0.565536 0.824724i \(-0.691331\pi\)
−0.565536 + 0.824724i \(0.691331\pi\)
\(828\) −14.0000 −0.486534
\(829\) 24.6274 0.855346 0.427673 0.903934i \(-0.359334\pi\)
0.427673 + 0.903934i \(0.359334\pi\)
\(830\) 0.201010 0.00697716
\(831\) −8.82843 −0.306255
\(832\) −4.17157 −0.144623
\(833\) 6.82843 0.236591
\(834\) 3.37258 0.116783
\(835\) −2.42641 −0.0839693
\(836\) −14.6274 −0.505900
\(837\) 4.58579 0.158508
\(838\) −7.59798 −0.262468
\(839\) 0.343146 0.0118467 0.00592335 0.999982i \(-0.498115\pi\)
0.00592335 + 0.999982i \(0.498115\pi\)
\(840\) 2.62742 0.0906545
\(841\) −15.6274 −0.538876
\(842\) 2.68629 0.0925757
\(843\) 19.8995 0.685375
\(844\) −9.45584 −0.325484
\(845\) −0.585786 −0.0201517
\(846\) 2.92893 0.100699
\(847\) 25.4558 0.874673
\(848\) −10.9706 −0.376731
\(849\) −18.1421 −0.622636
\(850\) −13.1716 −0.451781
\(851\) 84.7696 2.90586
\(852\) 7.75736 0.265763
\(853\) 9.02944 0.309162 0.154581 0.987980i \(-0.450597\pi\)
0.154581 + 0.987980i \(0.450597\pi\)
\(854\) −4.68629 −0.160362
\(855\) 3.31371 0.113326
\(856\) −3.71573 −0.127001
\(857\) −16.4853 −0.563126 −0.281563 0.959543i \(-0.590853\pi\)
−0.281563 + 0.959543i \(0.590853\pi\)
\(858\) 0.585786 0.0199984
\(859\) −12.4853 −0.425992 −0.212996 0.977053i \(-0.568322\pi\)
−0.212996 + 0.977053i \(0.568322\pi\)
\(860\) −3.02944 −0.103303
\(861\) 3.31371 0.112931
\(862\) −5.59798 −0.190668
\(863\) −16.4437 −0.559748 −0.279874 0.960037i \(-0.590293\pi\)
−0.279874 + 0.960037i \(0.590293\pi\)
\(864\) −4.41421 −0.150175
\(865\) −8.08326 −0.274839
\(866\) 3.17157 0.107774
\(867\) −29.6274 −1.00620
\(868\) −23.7157 −0.804964
\(869\) 9.17157 0.311124
\(870\) 0.887302 0.0300823
\(871\) −4.58579 −0.155383
\(872\) −13.8406 −0.468703
\(873\) 0.343146 0.0116137
\(874\) −17.9411 −0.606868
\(875\) −16.0000 −0.540899
\(876\) −14.1838 −0.479225
\(877\) −17.2132 −0.581249 −0.290624 0.956837i \(-0.593863\pi\)
−0.290624 + 0.956837i \(0.593863\pi\)
\(878\) 10.5442 0.355848
\(879\) −21.5563 −0.727078
\(880\) 2.48528 0.0837788
\(881\) 17.5147 0.590086 0.295043 0.955484i \(-0.404666\pi\)
0.295043 + 0.955484i \(0.404666\pi\)
\(882\) 0.414214 0.0139473
\(883\) −53.1127 −1.78738 −0.893692 0.448680i \(-0.851894\pi\)
−0.893692 + 0.448680i \(0.851894\pi\)
\(884\) −12.4853 −0.419925
\(885\) −3.51472 −0.118146
\(886\) 1.17157 0.0393598
\(887\) 44.2843 1.48692 0.743460 0.668780i \(-0.233183\pi\)
0.743460 + 0.668780i \(0.233183\pi\)
\(888\) 17.5563 0.589153
\(889\) 0 0
\(890\) −1.39697 −0.0468265
\(891\) −1.41421 −0.0473779
\(892\) 0 0
\(893\) −40.0000 −1.33855
\(894\) 1.65685 0.0554135
\(895\) 5.45584 0.182369
\(896\) 29.8579 0.997481
\(897\) −7.65685 −0.255655
\(898\) −13.2132 −0.440930
\(899\) −16.7696 −0.559296
\(900\) 8.51472 0.283824
\(901\) −24.9706 −0.831890
\(902\) −0.686292 −0.0228510
\(903\) −8.00000 −0.266223
\(904\) 15.8579 0.527425
\(905\) −2.54416 −0.0845706
\(906\) 9.41421 0.312766
\(907\) 47.3137 1.57103 0.785513 0.618845i \(-0.212399\pi\)
0.785513 + 0.618845i \(0.212399\pi\)
\(908\) 14.1838 0.470705
\(909\) −3.65685 −0.121290
\(910\) 0.686292 0.0227503
\(911\) −2.62742 −0.0870502 −0.0435251 0.999052i \(-0.513859\pi\)
−0.0435251 + 0.999052i \(0.513859\pi\)
\(912\) 16.9706 0.561951
\(913\) 1.17157 0.0387734
\(914\) −11.0122 −0.364251
\(915\) 2.34315 0.0774620
\(916\) 53.5980 1.77093
\(917\) 16.9706 0.560417
\(918\) −2.82843 −0.0933520
\(919\) 46.8284 1.54473 0.772364 0.635181i \(-0.219074\pi\)
0.772364 + 0.635181i \(0.219074\pi\)
\(920\) 7.11270 0.234499
\(921\) 6.92893 0.228316
\(922\) 12.4020 0.408439
\(923\) 4.24264 0.139648
\(924\) 7.31371 0.240603
\(925\) −51.5563 −1.69516
\(926\) −4.04163 −0.132816
\(927\) 1.00000 0.0328443
\(928\) 16.1421 0.529892
\(929\) −16.2843 −0.534270 −0.267135 0.963659i \(-0.586077\pi\)
−0.267135 + 0.963659i \(0.586077\pi\)
\(930\) −1.11270 −0.0364868
\(931\) −5.65685 −0.185396
\(932\) −30.4020 −0.995851
\(933\) 33.6569 1.10188
\(934\) 14.6274 0.478624
\(935\) 5.65685 0.184999
\(936\) −1.58579 −0.0518331
\(937\) 36.1421 1.18071 0.590356 0.807143i \(-0.298987\pi\)
0.590356 + 0.807143i \(0.298987\pi\)
\(938\) 5.37258 0.175421
\(939\) −29.3137 −0.956617
\(940\) 7.57359 0.247023
\(941\) −23.3137 −0.760005 −0.380003 0.924985i \(-0.624077\pi\)
−0.380003 + 0.924985i \(0.624077\pi\)
\(942\) −5.02944 −0.163868
\(943\) 8.97056 0.292122
\(944\) −18.0000 −0.585850
\(945\) −1.65685 −0.0538975
\(946\) 1.65685 0.0538690
\(947\) −1.61522 −0.0524877 −0.0262439 0.999656i \(-0.508355\pi\)
−0.0262439 + 0.999656i \(0.508355\pi\)
\(948\) −11.8579 −0.385126
\(949\) −7.75736 −0.251815
\(950\) 10.9117 0.354022
\(951\) −25.4558 −0.825462
\(952\) 30.6274 0.992640
\(953\) −30.1421 −0.976400 −0.488200 0.872732i \(-0.662346\pi\)
−0.488200 + 0.872732i \(0.662346\pi\)
\(954\) −1.51472 −0.0490408
\(955\) 5.25483 0.170042
\(956\) −43.2548 −1.39896
\(957\) 5.17157 0.167173
\(958\) −1.07107 −0.0346046
\(959\) 22.6274 0.730677
\(960\) −2.44365 −0.0788685
\(961\) −9.97056 −0.321631
\(962\) 4.58579 0.147852
\(963\) 2.34315 0.0755068
\(964\) −8.64466 −0.278426
\(965\) 9.79899 0.315441
\(966\) 8.97056 0.288623
\(967\) −17.2721 −0.555433 −0.277716 0.960663i \(-0.589578\pi\)
−0.277716 + 0.960663i \(0.589578\pi\)
\(968\) 14.2721 0.458722
\(969\) 38.6274 1.24089
\(970\) −0.0832611 −0.00267335
\(971\) −32.2843 −1.03605 −0.518026 0.855365i \(-0.673333\pi\)
−0.518026 + 0.855365i \(0.673333\pi\)
\(972\) 1.82843 0.0586468
\(973\) 23.0294 0.738290
\(974\) 12.2426 0.392280
\(975\) 4.65685 0.149139
\(976\) 12.0000 0.384111
\(977\) −13.8579 −0.443352 −0.221676 0.975120i \(-0.571153\pi\)
−0.221676 + 0.975120i \(0.571153\pi\)
\(978\) 2.34315 0.0749255
\(979\) −8.14214 −0.260224
\(980\) 1.07107 0.0342140
\(981\) 8.72792 0.278661
\(982\) −7.59798 −0.242461
\(983\) −47.4558 −1.51361 −0.756803 0.653643i \(-0.773240\pi\)
−0.756803 + 0.653643i \(0.773240\pi\)
\(984\) 1.85786 0.0592266
\(985\) 6.97056 0.222101
\(986\) 10.3431 0.329393
\(987\) 20.0000 0.636607
\(988\) 10.3431 0.329059
\(989\) −21.6569 −0.688648
\(990\) 0.343146 0.0109059
\(991\) 3.17157 0.100748 0.0503742 0.998730i \(-0.483959\pi\)
0.0503742 + 0.998730i \(0.483959\pi\)
\(992\) −20.2426 −0.642704
\(993\) 9.07107 0.287862
\(994\) −4.97056 −0.157657
\(995\) 2.34315 0.0742827
\(996\) −1.51472 −0.0479957
\(997\) −24.3431 −0.770955 −0.385478 0.922717i \(-0.625963\pi\)
−0.385478 + 0.922717i \(0.625963\pi\)
\(998\) −1.41421 −0.0447661
\(999\) −11.0711 −0.350273
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 4017.2.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4017.2.a.d.1.2 2 1.1 even 1 trivial