Properties

Label 4017.2.a.d.1.1
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 1.1
Root \(-1.41421\)
Character \(\chi\) = 4017.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -3.41421 q^{5} +2.41421 q^{6} +2.82843 q^{7} -4.41421 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.41421 q^{2} -1.00000 q^{3} +3.82843 q^{4} -3.41421 q^{5} +2.41421 q^{6} +2.82843 q^{7} -4.41421 q^{8} +1.00000 q^{9} +8.24264 q^{10} +1.41421 q^{11} -3.82843 q^{12} +1.00000 q^{13} -6.82843 q^{14} +3.41421 q^{15} +3.00000 q^{16} +1.17157 q^{17} -2.41421 q^{18} +5.65685 q^{19} -13.0711 q^{20} -2.82843 q^{21} -3.41421 q^{22} -3.65685 q^{23} +4.41421 q^{24} +6.65685 q^{25} -2.41421 q^{26} -1.00000 q^{27} +10.8284 q^{28} -7.65685 q^{29} -8.24264 q^{30} -7.41421 q^{31} +1.58579 q^{32} -1.41421 q^{33} -2.82843 q^{34} -9.65685 q^{35} +3.82843 q^{36} -3.07107 q^{37} -13.6569 q^{38} -1.00000 q^{39} +15.0711 q^{40} +6.82843 q^{41} +6.82843 q^{42} +2.82843 q^{43} +5.41421 q^{44} -3.41421 q^{45} +8.82843 q^{46} -7.07107 q^{47} -3.00000 q^{48} +1.00000 q^{49} -16.0711 q^{50} -1.17157 q^{51} +3.82843 q^{52} +7.65685 q^{53} +2.41421 q^{54} -4.82843 q^{55} -12.4853 q^{56} -5.65685 q^{57} +18.4853 q^{58} -6.00000 q^{59} +13.0711 q^{60} +4.00000 q^{61} +17.8995 q^{62} +2.82843 q^{63} -9.82843 q^{64} -3.41421 q^{65} +3.41421 q^{66} -7.41421 q^{67} +4.48528 q^{68} +3.65685 q^{69} +23.3137 q^{70} -4.24264 q^{71} -4.41421 q^{72} -16.2426 q^{73} +7.41421 q^{74} -6.65685 q^{75} +21.6569 q^{76} +4.00000 q^{77} +2.41421 q^{78} +10.4853 q^{79} -10.2426 q^{80} +1.00000 q^{81} -16.4853 q^{82} +4.82843 q^{83} -10.8284 q^{84} -4.00000 q^{85} -6.82843 q^{86} +7.65685 q^{87} -6.24264 q^{88} +14.2426 q^{89} +8.24264 q^{90} +2.82843 q^{91} -14.0000 q^{92} +7.41421 q^{93} +17.0711 q^{94} -19.3137 q^{95} -1.58579 q^{96} +11.6569 q^{97} -2.41421 q^{98} +1.41421 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} + 2q^{6} - 6q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{2} - 2q^{3} + 2q^{4} - 4q^{5} + 2q^{6} - 6q^{8} + 2q^{9} + 8q^{10} - 2q^{12} + 2q^{13} - 8q^{14} + 4q^{15} + 6q^{16} + 8q^{17} - 2q^{18} - 12q^{20} - 4q^{22} + 4q^{23} + 6q^{24} + 2q^{25} - 2q^{26} - 2q^{27} + 16q^{28} - 4q^{29} - 8q^{30} - 12q^{31} + 6q^{32} - 8q^{35} + 2q^{36} + 8q^{37} - 16q^{38} - 2q^{39} + 16q^{40} + 8q^{41} + 8q^{42} + 8q^{44} - 4q^{45} + 12q^{46} - 6q^{48} + 2q^{49} - 18q^{50} - 8q^{51} + 2q^{52} + 4q^{53} + 2q^{54} - 4q^{55} - 8q^{56} + 20q^{58} - 12q^{59} + 12q^{60} + 8q^{61} + 16q^{62} - 14q^{64} - 4q^{65} + 4q^{66} - 12q^{67} - 8q^{68} - 4q^{69} + 24q^{70} - 6q^{72} - 24q^{73} + 12q^{74} - 2q^{75} + 32q^{76} + 8q^{77} + 2q^{78} + 4q^{79} - 12q^{80} + 2q^{81} - 16q^{82} + 4q^{83} - 16q^{84} - 8q^{85} - 8q^{86} + 4q^{87} - 4q^{88} + 20q^{89} + 8q^{90} - 28q^{92} + 12q^{93} + 20q^{94} - 16q^{95} - 6q^{96} + 12q^{97} - 2q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.41421 −1.70711 −0.853553 0.521005i \(-0.825557\pi\)
−0.853553 + 0.521005i \(0.825557\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.82843 1.91421
\(5\) −3.41421 −1.52688 −0.763441 0.645877i \(-0.776492\pi\)
−0.763441 + 0.645877i \(0.776492\pi\)
\(6\) 2.41421 0.985599
\(7\) 2.82843 1.06904 0.534522 0.845154i \(-0.320491\pi\)
0.534522 + 0.845154i \(0.320491\pi\)
\(8\) −4.41421 −1.56066
\(9\) 1.00000 0.333333
\(10\) 8.24264 2.60655
\(11\) 1.41421 0.426401 0.213201 0.977008i \(-0.431611\pi\)
0.213201 + 0.977008i \(0.431611\pi\)
\(12\) −3.82843 −1.10517
\(13\) 1.00000 0.277350
\(14\) −6.82843 −1.82497
\(15\) 3.41421 0.881546
\(16\) 3.00000 0.750000
\(17\) 1.17157 0.284148 0.142074 0.989856i \(-0.454623\pi\)
0.142074 + 0.989856i \(0.454623\pi\)
\(18\) −2.41421 −0.569036
\(19\) 5.65685 1.29777 0.648886 0.760886i \(-0.275235\pi\)
0.648886 + 0.760886i \(0.275235\pi\)
\(20\) −13.0711 −2.92278
\(21\) −2.82843 −0.617213
\(22\) −3.41421 −0.727913
\(23\) −3.65685 −0.762507 −0.381253 0.924471i \(-0.624507\pi\)
−0.381253 + 0.924471i \(0.624507\pi\)
\(24\) 4.41421 0.901048
\(25\) 6.65685 1.33137
\(26\) −2.41421 −0.473466
\(27\) −1.00000 −0.192450
\(28\) 10.8284 2.04638
\(29\) −7.65685 −1.42184 −0.710921 0.703272i \(-0.751722\pi\)
−0.710921 + 0.703272i \(0.751722\pi\)
\(30\) −8.24264 −1.50489
\(31\) −7.41421 −1.33163 −0.665816 0.746116i \(-0.731916\pi\)
−0.665816 + 0.746116i \(0.731916\pi\)
\(32\) 1.58579 0.280330
\(33\) −1.41421 −0.246183
\(34\) −2.82843 −0.485071
\(35\) −9.65685 −1.63231
\(36\) 3.82843 0.638071
\(37\) −3.07107 −0.504880 −0.252440 0.967612i \(-0.581233\pi\)
−0.252440 + 0.967612i \(0.581233\pi\)
\(38\) −13.6569 −2.21543
\(39\) −1.00000 −0.160128
\(40\) 15.0711 2.38295
\(41\) 6.82843 1.06642 0.533211 0.845983i \(-0.320985\pi\)
0.533211 + 0.845983i \(0.320985\pi\)
\(42\) 6.82843 1.05365
\(43\) 2.82843 0.431331 0.215666 0.976467i \(-0.430808\pi\)
0.215666 + 0.976467i \(0.430808\pi\)
\(44\) 5.41421 0.816223
\(45\) −3.41421 −0.508961
\(46\) 8.82843 1.30168
\(47\) −7.07107 −1.03142 −0.515711 0.856763i \(-0.672472\pi\)
−0.515711 + 0.856763i \(0.672472\pi\)
\(48\) −3.00000 −0.433013
\(49\) 1.00000 0.142857
\(50\) −16.0711 −2.27279
\(51\) −1.17157 −0.164053
\(52\) 3.82843 0.530907
\(53\) 7.65685 1.05175 0.525875 0.850562i \(-0.323738\pi\)
0.525875 + 0.850562i \(0.323738\pi\)
\(54\) 2.41421 0.328533
\(55\) −4.82843 −0.651065
\(56\) −12.4853 −1.66842
\(57\) −5.65685 −0.749269
\(58\) 18.4853 2.42724
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 13.0711 1.68747
\(61\) 4.00000 0.512148 0.256074 0.966657i \(-0.417571\pi\)
0.256074 + 0.966657i \(0.417571\pi\)
\(62\) 17.8995 2.27324
\(63\) 2.82843 0.356348
\(64\) −9.82843 −1.22855
\(65\) −3.41421 −0.423481
\(66\) 3.41421 0.420261
\(67\) −7.41421 −0.905790 −0.452895 0.891564i \(-0.649609\pi\)
−0.452895 + 0.891564i \(0.649609\pi\)
\(68\) 4.48528 0.543920
\(69\) 3.65685 0.440234
\(70\) 23.3137 2.78652
\(71\) −4.24264 −0.503509 −0.251754 0.967791i \(-0.581008\pi\)
−0.251754 + 0.967791i \(0.581008\pi\)
\(72\) −4.41421 −0.520220
\(73\) −16.2426 −1.90106 −0.950529 0.310637i \(-0.899458\pi\)
−0.950529 + 0.310637i \(0.899458\pi\)
\(74\) 7.41421 0.861885
\(75\) −6.65685 −0.768667
\(76\) 21.6569 2.48421
\(77\) 4.00000 0.455842
\(78\) 2.41421 0.273356
\(79\) 10.4853 1.17969 0.589843 0.807518i \(-0.299190\pi\)
0.589843 + 0.807518i \(0.299190\pi\)
\(80\) −10.2426 −1.14516
\(81\) 1.00000 0.111111
\(82\) −16.4853 −1.82049
\(83\) 4.82843 0.529989 0.264994 0.964250i \(-0.414630\pi\)
0.264994 + 0.964250i \(0.414630\pi\)
\(84\) −10.8284 −1.18148
\(85\) −4.00000 −0.433861
\(86\) −6.82843 −0.736328
\(87\) 7.65685 0.820901
\(88\) −6.24264 −0.665468
\(89\) 14.2426 1.50972 0.754858 0.655888i \(-0.227706\pi\)
0.754858 + 0.655888i \(0.227706\pi\)
\(90\) 8.24264 0.868851
\(91\) 2.82843 0.296500
\(92\) −14.0000 −1.45960
\(93\) 7.41421 0.768818
\(94\) 17.0711 1.76075
\(95\) −19.3137 −1.98154
\(96\) −1.58579 −0.161849
\(97\) 11.6569 1.18357 0.591787 0.806094i \(-0.298423\pi\)
0.591787 + 0.806094i \(0.298423\pi\)
\(98\) −2.41421 −0.243872
\(99\) 1.41421 0.142134
\(100\) 25.4853 2.54853
\(101\) 7.65685 0.761885 0.380943 0.924599i \(-0.375599\pi\)
0.380943 + 0.924599i \(0.375599\pi\)
\(102\) 2.82843 0.280056
\(103\) 1.00000 0.0985329
\(104\) −4.41421 −0.432849
\(105\) 9.65685 0.942412
\(106\) −18.4853 −1.79545
\(107\) 13.6569 1.32026 0.660129 0.751152i \(-0.270502\pi\)
0.660129 + 0.751152i \(0.270502\pi\)
\(108\) −3.82843 −0.368391
\(109\) −16.7279 −1.60224 −0.801122 0.598501i \(-0.795763\pi\)
−0.801122 + 0.598501i \(0.795763\pi\)
\(110\) 11.6569 1.11144
\(111\) 3.07107 0.291493
\(112\) 8.48528 0.801784
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 13.6569 1.27908
\(115\) 12.4853 1.16426
\(116\) −29.3137 −2.72171
\(117\) 1.00000 0.0924500
\(118\) 14.4853 1.33348
\(119\) 3.31371 0.303767
\(120\) −15.0711 −1.37579
\(121\) −9.00000 −0.818182
\(122\) −9.65685 −0.874291
\(123\) −6.82843 −0.615699
\(124\) −28.3848 −2.54903
\(125\) −5.65685 −0.505964
\(126\) −6.82843 −0.608325
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 20.5563 1.81694
\(129\) −2.82843 −0.249029
\(130\) 8.24264 0.722927
\(131\) −6.00000 −0.524222 −0.262111 0.965038i \(-0.584419\pi\)
−0.262111 + 0.965038i \(0.584419\pi\)
\(132\) −5.41421 −0.471247
\(133\) 16.0000 1.38738
\(134\) 17.8995 1.54628
\(135\) 3.41421 0.293849
\(136\) −5.17157 −0.443459
\(137\) −8.00000 −0.683486 −0.341743 0.939793i \(-0.611017\pi\)
−0.341743 + 0.939793i \(0.611017\pi\)
\(138\) −8.82843 −0.751526
\(139\) 20.1421 1.70843 0.854217 0.519917i \(-0.174037\pi\)
0.854217 + 0.519917i \(0.174037\pi\)
\(140\) −36.9706 −3.12458
\(141\) 7.07107 0.595491
\(142\) 10.2426 0.859543
\(143\) 1.41421 0.118262
\(144\) 3.00000 0.250000
\(145\) 26.1421 2.17099
\(146\) 39.2132 3.24531
\(147\) −1.00000 −0.0824786
\(148\) −11.7574 −0.966449
\(149\) −4.00000 −0.327693 −0.163846 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 16.0711 1.31220
\(151\) 2.72792 0.221995 0.110998 0.993821i \(-0.464595\pi\)
0.110998 + 0.993821i \(0.464595\pi\)
\(152\) −24.9706 −2.02538
\(153\) 1.17157 0.0947161
\(154\) −9.65685 −0.778171
\(155\) 25.3137 2.03325
\(156\) −3.82843 −0.306519
\(157\) −16.1421 −1.28828 −0.644141 0.764906i \(-0.722785\pi\)
−0.644141 + 0.764906i \(0.722785\pi\)
\(158\) −25.3137 −2.01385
\(159\) −7.65685 −0.607228
\(160\) −5.41421 −0.428031
\(161\) −10.3431 −0.815154
\(162\) −2.41421 −0.189679
\(163\) 5.65685 0.443079 0.221540 0.975151i \(-0.428892\pi\)
0.221540 + 0.975151i \(0.428892\pi\)
\(164\) 26.1421 2.04136
\(165\) 4.82843 0.375893
\(166\) −11.6569 −0.904747
\(167\) −24.1421 −1.86817 −0.934087 0.357045i \(-0.883784\pi\)
−0.934087 + 0.357045i \(0.883784\pi\)
\(168\) 12.4853 0.963260
\(169\) 1.00000 0.0769231
\(170\) 9.65685 0.740647
\(171\) 5.65685 0.432590
\(172\) 10.8284 0.825660
\(173\) −25.7990 −1.96146 −0.980730 0.195366i \(-0.937411\pi\)
−0.980730 + 0.195366i \(0.937411\pi\)
\(174\) −18.4853 −1.40137
\(175\) 18.8284 1.42330
\(176\) 4.24264 0.319801
\(177\) 6.00000 0.450988
\(178\) −34.3848 −2.57725
\(179\) 13.3137 0.995113 0.497557 0.867431i \(-0.334231\pi\)
0.497557 + 0.867431i \(0.334231\pi\)
\(180\) −13.0711 −0.974260
\(181\) 15.6569 1.16376 0.581882 0.813273i \(-0.302316\pi\)
0.581882 + 0.813273i \(0.302316\pi\)
\(182\) −6.82843 −0.506157
\(183\) −4.00000 −0.295689
\(184\) 16.1421 1.19001
\(185\) 10.4853 0.770893
\(186\) −17.8995 −1.31245
\(187\) 1.65685 0.121161
\(188\) −27.0711 −1.97436
\(189\) −2.82843 −0.205738
\(190\) 46.6274 3.38271
\(191\) 24.9706 1.80681 0.903403 0.428792i \(-0.141061\pi\)
0.903403 + 0.428792i \(0.141061\pi\)
\(192\) 9.82843 0.709306
\(193\) 8.72792 0.628250 0.314125 0.949382i \(-0.398289\pi\)
0.314125 + 0.949382i \(0.398289\pi\)
\(194\) −28.1421 −2.02049
\(195\) 3.41421 0.244497
\(196\) 3.82843 0.273459
\(197\) 7.89949 0.562816 0.281408 0.959588i \(-0.409199\pi\)
0.281408 + 0.959588i \(0.409199\pi\)
\(198\) −3.41421 −0.242638
\(199\) −4.00000 −0.283552 −0.141776 0.989899i \(-0.545281\pi\)
−0.141776 + 0.989899i \(0.545281\pi\)
\(200\) −29.3848 −2.07782
\(201\) 7.41421 0.522958
\(202\) −18.4853 −1.30062
\(203\) −21.6569 −1.52001
\(204\) −4.48528 −0.314033
\(205\) −23.3137 −1.62830
\(206\) −2.41421 −0.168206
\(207\) −3.65685 −0.254169
\(208\) 3.00000 0.208013
\(209\) 8.00000 0.553372
\(210\) −23.3137 −1.60880
\(211\) 10.8284 0.745460 0.372730 0.927940i \(-0.378422\pi\)
0.372730 + 0.927940i \(0.378422\pi\)
\(212\) 29.3137 2.01327
\(213\) 4.24264 0.290701
\(214\) −32.9706 −2.25382
\(215\) −9.65685 −0.658592
\(216\) 4.41421 0.300349
\(217\) −20.9706 −1.42357
\(218\) 40.3848 2.73520
\(219\) 16.2426 1.09758
\(220\) −18.4853 −1.24628
\(221\) 1.17157 0.0788085
\(222\) −7.41421 −0.497609
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 4.48528 0.299685
\(225\) 6.65685 0.443790
\(226\) 24.1421 1.60591
\(227\) −16.2426 −1.07806 −0.539031 0.842286i \(-0.681209\pi\)
−0.539031 + 0.842286i \(0.681209\pi\)
\(228\) −21.6569 −1.43426
\(229\) −6.68629 −0.441843 −0.220921 0.975292i \(-0.570906\pi\)
−0.220921 + 0.975292i \(0.570906\pi\)
\(230\) −30.1421 −1.98751
\(231\) −4.00000 −0.263181
\(232\) 33.7990 2.21901
\(233\) −28.6274 −1.87544 −0.937722 0.347386i \(-0.887069\pi\)
−0.937722 + 0.347386i \(0.887069\pi\)
\(234\) −2.41421 −0.157822
\(235\) 24.1421 1.57486
\(236\) −22.9706 −1.49526
\(237\) −10.4853 −0.681092
\(238\) −8.00000 −0.518563
\(239\) 12.3431 0.798412 0.399206 0.916861i \(-0.369286\pi\)
0.399206 + 0.916861i \(0.369286\pi\)
\(240\) 10.2426 0.661160
\(241\) −20.7279 −1.33520 −0.667601 0.744519i \(-0.732679\pi\)
−0.667601 + 0.744519i \(0.732679\pi\)
\(242\) 21.7279 1.39672
\(243\) −1.00000 −0.0641500
\(244\) 15.3137 0.980360
\(245\) −3.41421 −0.218126
\(246\) 16.4853 1.05106
\(247\) 5.65685 0.359937
\(248\) 32.7279 2.07823
\(249\) −4.82843 −0.305989
\(250\) 13.6569 0.863735
\(251\) 24.4853 1.54550 0.772749 0.634712i \(-0.218881\pi\)
0.772749 + 0.634712i \(0.218881\pi\)
\(252\) 10.8284 0.682127
\(253\) −5.17157 −0.325134
\(254\) 0 0
\(255\) 4.00000 0.250490
\(256\) −29.9706 −1.87316
\(257\) 30.9706 1.93189 0.965945 0.258746i \(-0.0833094\pi\)
0.965945 + 0.258746i \(0.0833094\pi\)
\(258\) 6.82843 0.425119
\(259\) −8.68629 −0.539740
\(260\) −13.0711 −0.810633
\(261\) −7.65685 −0.473947
\(262\) 14.4853 0.894904
\(263\) −1.17157 −0.0722423 −0.0361211 0.999347i \(-0.511500\pi\)
−0.0361211 + 0.999347i \(0.511500\pi\)
\(264\) 6.24264 0.384208
\(265\) −26.1421 −1.60590
\(266\) −38.6274 −2.36840
\(267\) −14.2426 −0.871635
\(268\) −28.3848 −1.73388
\(269\) 2.82843 0.172452 0.0862261 0.996276i \(-0.472519\pi\)
0.0862261 + 0.996276i \(0.472519\pi\)
\(270\) −8.24264 −0.501631
\(271\) −20.5858 −1.25050 −0.625249 0.780426i \(-0.715002\pi\)
−0.625249 + 0.780426i \(0.715002\pi\)
\(272\) 3.51472 0.213111
\(273\) −2.82843 −0.171184
\(274\) 19.3137 1.16678
\(275\) 9.41421 0.567698
\(276\) 14.0000 0.842701
\(277\) 3.17157 0.190561 0.0952807 0.995450i \(-0.469625\pi\)
0.0952807 + 0.995450i \(0.469625\pi\)
\(278\) −48.6274 −2.91648
\(279\) −7.41421 −0.443877
\(280\) 42.6274 2.54748
\(281\) −0.100505 −0.00599563 −0.00299781 0.999996i \(-0.500954\pi\)
−0.00299781 + 0.999996i \(0.500954\pi\)
\(282\) −17.0711 −1.01657
\(283\) −10.1421 −0.602887 −0.301444 0.953484i \(-0.597469\pi\)
−0.301444 + 0.953484i \(0.597469\pi\)
\(284\) −16.2426 −0.963823
\(285\) 19.3137 1.14405
\(286\) −3.41421 −0.201887
\(287\) 19.3137 1.14005
\(288\) 1.58579 0.0934434
\(289\) −15.6274 −0.919260
\(290\) −63.1127 −3.70611
\(291\) −11.6569 −0.683337
\(292\) −62.1838 −3.63903
\(293\) −9.55635 −0.558288 −0.279144 0.960249i \(-0.590051\pi\)
−0.279144 + 0.960249i \(0.590051\pi\)
\(294\) 2.41421 0.140800
\(295\) 20.4853 1.19270
\(296\) 13.5563 0.787947
\(297\) −1.41421 −0.0820610
\(298\) 9.65685 0.559407
\(299\) −3.65685 −0.211481
\(300\) −25.4853 −1.47139
\(301\) 8.00000 0.461112
\(302\) −6.58579 −0.378969
\(303\) −7.65685 −0.439875
\(304\) 16.9706 0.973329
\(305\) −13.6569 −0.781989
\(306\) −2.82843 −0.161690
\(307\) −21.0711 −1.20259 −0.601295 0.799027i \(-0.705348\pi\)
−0.601295 + 0.799027i \(0.705348\pi\)
\(308\) 15.3137 0.872580
\(309\) −1.00000 −0.0568880
\(310\) −61.1127 −3.47097
\(311\) −22.3431 −1.26696 −0.633482 0.773758i \(-0.718375\pi\)
−0.633482 + 0.773758i \(0.718375\pi\)
\(312\) 4.41421 0.249906
\(313\) 6.68629 0.377932 0.188966 0.981984i \(-0.439486\pi\)
0.188966 + 0.981984i \(0.439486\pi\)
\(314\) 38.9706 2.19924
\(315\) −9.65685 −0.544102
\(316\) 40.1421 2.25817
\(317\) −25.4558 −1.42974 −0.714871 0.699256i \(-0.753515\pi\)
−0.714871 + 0.699256i \(0.753515\pi\)
\(318\) 18.4853 1.03660
\(319\) −10.8284 −0.606276
\(320\) 33.5563 1.87586
\(321\) −13.6569 −0.762251
\(322\) 24.9706 1.39156
\(323\) 6.62742 0.368759
\(324\) 3.82843 0.212690
\(325\) 6.65685 0.369256
\(326\) −13.6569 −0.756383
\(327\) 16.7279 0.925056
\(328\) −30.1421 −1.66432
\(329\) −20.0000 −1.10264
\(330\) −11.6569 −0.641689
\(331\) 5.07107 0.278731 0.139366 0.990241i \(-0.455494\pi\)
0.139366 + 0.990241i \(0.455494\pi\)
\(332\) 18.4853 1.01451
\(333\) −3.07107 −0.168293
\(334\) 58.2843 3.18917
\(335\) 25.3137 1.38304
\(336\) −8.48528 −0.462910
\(337\) 12.6274 0.687859 0.343930 0.938995i \(-0.388242\pi\)
0.343930 + 0.938995i \(0.388242\pi\)
\(338\) −2.41421 −0.131316
\(339\) 10.0000 0.543125
\(340\) −15.3137 −0.830502
\(341\) −10.4853 −0.567810
\(342\) −13.6569 −0.738478
\(343\) −16.9706 −0.916324
\(344\) −12.4853 −0.673161
\(345\) −12.4853 −0.672185
\(346\) 62.2843 3.34842
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 29.3137 1.57138
\(349\) 30.8701 1.65244 0.826218 0.563350i \(-0.190488\pi\)
0.826218 + 0.563350i \(0.190488\pi\)
\(350\) −45.4558 −2.42972
\(351\) −1.00000 −0.0533761
\(352\) 2.24264 0.119533
\(353\) −14.9289 −0.794587 −0.397293 0.917692i \(-0.630050\pi\)
−0.397293 + 0.917692i \(0.630050\pi\)
\(354\) −14.4853 −0.769884
\(355\) 14.4853 0.768799
\(356\) 54.5269 2.88992
\(357\) −3.31371 −0.175380
\(358\) −32.1421 −1.69876
\(359\) 22.4853 1.18673 0.593364 0.804934i \(-0.297800\pi\)
0.593364 + 0.804934i \(0.297800\pi\)
\(360\) 15.0711 0.794315
\(361\) 13.0000 0.684211
\(362\) −37.7990 −1.98667
\(363\) 9.00000 0.472377
\(364\) 10.8284 0.567564
\(365\) 55.4558 2.90269
\(366\) 9.65685 0.504772
\(367\) −15.1716 −0.791950 −0.395975 0.918261i \(-0.629593\pi\)
−0.395975 + 0.918261i \(0.629593\pi\)
\(368\) −10.9706 −0.571880
\(369\) 6.82843 0.355474
\(370\) −25.3137 −1.31600
\(371\) 21.6569 1.12437
\(372\) 28.3848 1.47168
\(373\) 1.65685 0.0857887 0.0428943 0.999080i \(-0.486342\pi\)
0.0428943 + 0.999080i \(0.486342\pi\)
\(374\) −4.00000 −0.206835
\(375\) 5.65685 0.292119
\(376\) 31.2132 1.60970
\(377\) −7.65685 −0.394348
\(378\) 6.82843 0.351216
\(379\) −4.58579 −0.235556 −0.117778 0.993040i \(-0.537577\pi\)
−0.117778 + 0.993040i \(0.537577\pi\)
\(380\) −73.9411 −3.79310
\(381\) 0 0
\(382\) −60.2843 −3.08441
\(383\) −2.58579 −0.132128 −0.0660638 0.997815i \(-0.521044\pi\)
−0.0660638 + 0.997815i \(0.521044\pi\)
\(384\) −20.5563 −1.04901
\(385\) −13.6569 −0.696018
\(386\) −21.0711 −1.07249
\(387\) 2.82843 0.143777
\(388\) 44.6274 2.26561
\(389\) 10.4853 0.531625 0.265812 0.964025i \(-0.414360\pi\)
0.265812 + 0.964025i \(0.414360\pi\)
\(390\) −8.24264 −0.417382
\(391\) −4.28427 −0.216665
\(392\) −4.41421 −0.222951
\(393\) 6.00000 0.302660
\(394\) −19.0711 −0.960787
\(395\) −35.7990 −1.80124
\(396\) 5.41421 0.272074
\(397\) −26.8701 −1.34857 −0.674285 0.738471i \(-0.735548\pi\)
−0.674285 + 0.738471i \(0.735548\pi\)
\(398\) 9.65685 0.484054
\(399\) −16.0000 −0.801002
\(400\) 19.9706 0.998528
\(401\) −25.1716 −1.25701 −0.628504 0.777806i \(-0.716332\pi\)
−0.628504 + 0.777806i \(0.716332\pi\)
\(402\) −17.8995 −0.892746
\(403\) −7.41421 −0.369328
\(404\) 29.3137 1.45841
\(405\) −3.41421 −0.169654
\(406\) 52.2843 2.59482
\(407\) −4.34315 −0.215282
\(408\) 5.17157 0.256031
\(409\) 15.1716 0.750186 0.375093 0.926987i \(-0.377611\pi\)
0.375093 + 0.926987i \(0.377611\pi\)
\(410\) 56.2843 2.77968
\(411\) 8.00000 0.394611
\(412\) 3.82843 0.188613
\(413\) −16.9706 −0.835067
\(414\) 8.82843 0.433894
\(415\) −16.4853 −0.809231
\(416\) 1.58579 0.0777496
\(417\) −20.1421 −0.986365
\(418\) −19.3137 −0.944664
\(419\) −29.6569 −1.44883 −0.724416 0.689363i \(-0.757891\pi\)
−0.724416 + 0.689363i \(0.757891\pi\)
\(420\) 36.9706 1.80398
\(421\) −10.4853 −0.511021 −0.255511 0.966806i \(-0.582244\pi\)
−0.255511 + 0.966806i \(0.582244\pi\)
\(422\) −26.1421 −1.27258
\(423\) −7.07107 −0.343807
\(424\) −33.7990 −1.64142
\(425\) 7.79899 0.378307
\(426\) −10.2426 −0.496258
\(427\) 11.3137 0.547509
\(428\) 52.2843 2.52726
\(429\) −1.41421 −0.0682789
\(430\) 23.3137 1.12429
\(431\) −30.4853 −1.46842 −0.734212 0.678920i \(-0.762448\pi\)
−0.734212 + 0.678920i \(0.762448\pi\)
\(432\) −3.00000 −0.144338
\(433\) −3.65685 −0.175737 −0.0878686 0.996132i \(-0.528006\pi\)
−0.0878686 + 0.996132i \(0.528006\pi\)
\(434\) 50.6274 2.43019
\(435\) −26.1421 −1.25342
\(436\) −64.0416 −3.06704
\(437\) −20.6863 −0.989560
\(438\) −39.2132 −1.87368
\(439\) −25.4558 −1.21494 −0.607471 0.794342i \(-0.707816\pi\)
−0.607471 + 0.794342i \(0.707816\pi\)
\(440\) 21.3137 1.01609
\(441\) 1.00000 0.0476190
\(442\) −2.82843 −0.134535
\(443\) −2.82843 −0.134383 −0.0671913 0.997740i \(-0.521404\pi\)
−0.0671913 + 0.997740i \(0.521404\pi\)
\(444\) 11.7574 0.557980
\(445\) −48.6274 −2.30516
\(446\) 0 0
\(447\) 4.00000 0.189194
\(448\) −27.7990 −1.31338
\(449\) −12.1005 −0.571058 −0.285529 0.958370i \(-0.592169\pi\)
−0.285529 + 0.958370i \(0.592169\pi\)
\(450\) −16.0711 −0.757597
\(451\) 9.65685 0.454724
\(452\) −38.2843 −1.80074
\(453\) −2.72792 −0.128169
\(454\) 39.2132 1.84037
\(455\) −9.65685 −0.452720
\(456\) 24.9706 1.16935
\(457\) −29.4142 −1.37594 −0.687969 0.725740i \(-0.741498\pi\)
−0.687969 + 0.725740i \(0.741498\pi\)
\(458\) 16.1421 0.754272
\(459\) −1.17157 −0.0546843
\(460\) 47.7990 2.22864
\(461\) −37.9411 −1.76709 −0.883547 0.468342i \(-0.844851\pi\)
−0.883547 + 0.468342i \(0.844851\pi\)
\(462\) 9.65685 0.449278
\(463\) −18.2426 −0.847807 −0.423904 0.905707i \(-0.639341\pi\)
−0.423904 + 0.905707i \(0.639341\pi\)
\(464\) −22.9706 −1.06638
\(465\) −25.3137 −1.17390
\(466\) 69.1127 3.20158
\(467\) 12.6863 0.587052 0.293526 0.955951i \(-0.405171\pi\)
0.293526 + 0.955951i \(0.405171\pi\)
\(468\) 3.82843 0.176969
\(469\) −20.9706 −0.968331
\(470\) −58.2843 −2.68845
\(471\) 16.1421 0.743790
\(472\) 26.4853 1.21908
\(473\) 4.00000 0.183920
\(474\) 25.3137 1.16270
\(475\) 37.6569 1.72781
\(476\) 12.6863 0.581475
\(477\) 7.65685 0.350583
\(478\) −29.7990 −1.36297
\(479\) −5.41421 −0.247382 −0.123691 0.992321i \(-0.539473\pi\)
−0.123691 + 0.992321i \(0.539473\pi\)
\(480\) 5.41421 0.247124
\(481\) −3.07107 −0.140029
\(482\) 50.0416 2.27933
\(483\) 10.3431 0.470629
\(484\) −34.4558 −1.56617
\(485\) −39.7990 −1.80718
\(486\) 2.41421 0.109511
\(487\) −1.55635 −0.0705249 −0.0352625 0.999378i \(-0.511227\pi\)
−0.0352625 + 0.999378i \(0.511227\pi\)
\(488\) −17.6569 −0.799288
\(489\) −5.65685 −0.255812
\(490\) 8.24264 0.372365
\(491\) −29.6569 −1.33840 −0.669198 0.743085i \(-0.733362\pi\)
−0.669198 + 0.743085i \(0.733362\pi\)
\(492\) −26.1421 −1.17858
\(493\) −8.97056 −0.404014
\(494\) −13.6569 −0.614451
\(495\) −4.82843 −0.217022
\(496\) −22.2426 −0.998724
\(497\) −12.0000 −0.538274
\(498\) 11.6569 0.522356
\(499\) −0.585786 −0.0262234 −0.0131117 0.999914i \(-0.504174\pi\)
−0.0131117 + 0.999914i \(0.504174\pi\)
\(500\) −21.6569 −0.968524
\(501\) 24.1421 1.07859
\(502\) −59.1127 −2.63833
\(503\) 18.2843 0.815255 0.407628 0.913148i \(-0.366356\pi\)
0.407628 + 0.913148i \(0.366356\pi\)
\(504\) −12.4853 −0.556139
\(505\) −26.1421 −1.16331
\(506\) 12.4853 0.555038
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) 21.4558 0.951014 0.475507 0.879712i \(-0.342265\pi\)
0.475507 + 0.879712i \(0.342265\pi\)
\(510\) −9.65685 −0.427613
\(511\) −45.9411 −2.03232
\(512\) 31.2426 1.38074
\(513\) −5.65685 −0.249756
\(514\) −74.7696 −3.29794
\(515\) −3.41421 −0.150448
\(516\) −10.8284 −0.476695
\(517\) −10.0000 −0.439799
\(518\) 20.9706 0.921394
\(519\) 25.7990 1.13245
\(520\) 15.0711 0.660910
\(521\) 13.7990 0.604545 0.302272 0.953222i \(-0.402255\pi\)
0.302272 + 0.953222i \(0.402255\pi\)
\(522\) 18.4853 0.809079
\(523\) −28.1421 −1.23057 −0.615285 0.788305i \(-0.710959\pi\)
−0.615285 + 0.788305i \(0.710959\pi\)
\(524\) −22.9706 −1.00347
\(525\) −18.8284 −0.821740
\(526\) 2.82843 0.123325
\(527\) −8.68629 −0.378381
\(528\) −4.24264 −0.184637
\(529\) −9.62742 −0.418583
\(530\) 63.1127 2.74144
\(531\) −6.00000 −0.260378
\(532\) 61.2548 2.65573
\(533\) 6.82843 0.295772
\(534\) 34.3848 1.48797
\(535\) −46.6274 −2.01588
\(536\) 32.7279 1.41363
\(537\) −13.3137 −0.574529
\(538\) −6.82843 −0.294394
\(539\) 1.41421 0.0609145
\(540\) 13.0711 0.562489
\(541\) 37.1127 1.59560 0.797800 0.602922i \(-0.205997\pi\)
0.797800 + 0.602922i \(0.205997\pi\)
\(542\) 49.6985 2.13473
\(543\) −15.6569 −0.671900
\(544\) 1.85786 0.0796553
\(545\) 57.1127 2.44644
\(546\) 6.82843 0.292230
\(547\) −7.17157 −0.306634 −0.153317 0.988177i \(-0.548996\pi\)
−0.153317 + 0.988177i \(0.548996\pi\)
\(548\) −30.6274 −1.30834
\(549\) 4.00000 0.170716
\(550\) −22.7279 −0.969122
\(551\) −43.3137 −1.84523
\(552\) −16.1421 −0.687055
\(553\) 29.6569 1.26114
\(554\) −7.65685 −0.325309
\(555\) −10.4853 −0.445075
\(556\) 77.1127 3.27031
\(557\) −18.9289 −0.802045 −0.401022 0.916068i \(-0.631345\pi\)
−0.401022 + 0.916068i \(0.631345\pi\)
\(558\) 17.8995 0.757746
\(559\) 2.82843 0.119630
\(560\) −28.9706 −1.22423
\(561\) −1.65685 −0.0699524
\(562\) 0.242641 0.0102352
\(563\) 3.31371 0.139656 0.0698281 0.997559i \(-0.477755\pi\)
0.0698281 + 0.997559i \(0.477755\pi\)
\(564\) 27.0711 1.13990
\(565\) 34.1421 1.43637
\(566\) 24.4853 1.02919
\(567\) 2.82843 0.118783
\(568\) 18.7279 0.785806
\(569\) −20.1421 −0.844402 −0.422201 0.906502i \(-0.638742\pi\)
−0.422201 + 0.906502i \(0.638742\pi\)
\(570\) −46.6274 −1.95301
\(571\) −10.6274 −0.444744 −0.222372 0.974962i \(-0.571380\pi\)
−0.222372 + 0.974962i \(0.571380\pi\)
\(572\) 5.41421 0.226380
\(573\) −24.9706 −1.04316
\(574\) −46.6274 −1.94619
\(575\) −24.3431 −1.01518
\(576\) −9.82843 −0.409518
\(577\) 18.3848 0.765368 0.382684 0.923879i \(-0.375000\pi\)
0.382684 + 0.923879i \(0.375000\pi\)
\(578\) 37.7279 1.56927
\(579\) −8.72792 −0.362720
\(580\) 100.083 4.15573
\(581\) 13.6569 0.566582
\(582\) 28.1421 1.16653
\(583\) 10.8284 0.448468
\(584\) 71.6985 2.96690
\(585\) −3.41421 −0.141160
\(586\) 23.0711 0.953057
\(587\) 16.8284 0.694584 0.347292 0.937757i \(-0.387101\pi\)
0.347292 + 0.937757i \(0.387101\pi\)
\(588\) −3.82843 −0.157882
\(589\) −41.9411 −1.72815
\(590\) −49.4558 −2.03606
\(591\) −7.89949 −0.324942
\(592\) −9.21320 −0.378660
\(593\) −18.9289 −0.777318 −0.388659 0.921382i \(-0.627062\pi\)
−0.388659 + 0.921382i \(0.627062\pi\)
\(594\) 3.41421 0.140087
\(595\) −11.3137 −0.463817
\(596\) −15.3137 −0.627274
\(597\) 4.00000 0.163709
\(598\) 8.82843 0.361021
\(599\) −35.1127 −1.43467 −0.717333 0.696731i \(-0.754637\pi\)
−0.717333 + 0.696731i \(0.754637\pi\)
\(600\) 29.3848 1.19963
\(601\) 5.31371 0.216751 0.108375 0.994110i \(-0.465435\pi\)
0.108375 + 0.994110i \(0.465435\pi\)
\(602\) −19.3137 −0.787168
\(603\) −7.41421 −0.301930
\(604\) 10.4437 0.424946
\(605\) 30.7279 1.24927
\(606\) 18.4853 0.750913
\(607\) −19.3137 −0.783919 −0.391960 0.919982i \(-0.628203\pi\)
−0.391960 + 0.919982i \(0.628203\pi\)
\(608\) 8.97056 0.363804
\(609\) 21.6569 0.877580
\(610\) 32.9706 1.33494
\(611\) −7.07107 −0.286065
\(612\) 4.48528 0.181307
\(613\) −9.79899 −0.395777 −0.197889 0.980224i \(-0.563408\pi\)
−0.197889 + 0.980224i \(0.563408\pi\)
\(614\) 50.8701 2.05295
\(615\) 23.3137 0.940099
\(616\) −17.6569 −0.711415
\(617\) −10.0416 −0.404261 −0.202130 0.979359i \(-0.564786\pi\)
−0.202130 + 0.979359i \(0.564786\pi\)
\(618\) 2.41421 0.0971139
\(619\) −33.1716 −1.33328 −0.666639 0.745381i \(-0.732268\pi\)
−0.666639 + 0.745381i \(0.732268\pi\)
\(620\) 96.9117 3.89207
\(621\) 3.65685 0.146745
\(622\) 53.9411 2.16284
\(623\) 40.2843 1.61396
\(624\) −3.00000 −0.120096
\(625\) −13.9706 −0.558823
\(626\) −16.1421 −0.645169
\(627\) −8.00000 −0.319489
\(628\) −61.7990 −2.46605
\(629\) −3.59798 −0.143461
\(630\) 23.3137 0.928840
\(631\) −28.4853 −1.13398 −0.566991 0.823724i \(-0.691892\pi\)
−0.566991 + 0.823724i \(0.691892\pi\)
\(632\) −46.2843 −1.84109
\(633\) −10.8284 −0.430391
\(634\) 61.4558 2.44072
\(635\) 0 0
\(636\) −29.3137 −1.16236
\(637\) 1.00000 0.0396214
\(638\) 26.1421 1.03498
\(639\) −4.24264 −0.167836
\(640\) −70.1838 −2.77426
\(641\) −18.6863 −0.738064 −0.369032 0.929417i \(-0.620311\pi\)
−0.369032 + 0.929417i \(0.620311\pi\)
\(642\) 32.9706 1.30124
\(643\) −17.4558 −0.688391 −0.344196 0.938898i \(-0.611848\pi\)
−0.344196 + 0.938898i \(0.611848\pi\)
\(644\) −39.5980 −1.56038
\(645\) 9.65685 0.380238
\(646\) −16.0000 −0.629512
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) −4.41421 −0.173407
\(649\) −8.48528 −0.333076
\(650\) −16.0711 −0.630359
\(651\) 20.9706 0.821901
\(652\) 21.6569 0.848148
\(653\) −16.8284 −0.658547 −0.329274 0.944235i \(-0.606804\pi\)
−0.329274 + 0.944235i \(0.606804\pi\)
\(654\) −40.3848 −1.57917
\(655\) 20.4853 0.800426
\(656\) 20.4853 0.799816
\(657\) −16.2426 −0.633686
\(658\) 48.2843 1.88232
\(659\) 36.6274 1.42680 0.713401 0.700756i \(-0.247154\pi\)
0.713401 + 0.700756i \(0.247154\pi\)
\(660\) 18.4853 0.719539
\(661\) 8.92893 0.347295 0.173648 0.984808i \(-0.444445\pi\)
0.173648 + 0.984808i \(0.444445\pi\)
\(662\) −12.2426 −0.475824
\(663\) −1.17157 −0.0455001
\(664\) −21.3137 −0.827132
\(665\) −54.6274 −2.11836
\(666\) 7.41421 0.287295
\(667\) 28.0000 1.08416
\(668\) −92.4264 −3.57609
\(669\) 0 0
\(670\) −61.1127 −2.36099
\(671\) 5.65685 0.218380
\(672\) −4.48528 −0.173023
\(673\) 18.6863 0.720304 0.360152 0.932894i \(-0.382725\pi\)
0.360152 + 0.932894i \(0.382725\pi\)
\(674\) −30.4853 −1.17425
\(675\) −6.65685 −0.256222
\(676\) 3.82843 0.147247
\(677\) 25.3137 0.972885 0.486442 0.873713i \(-0.338294\pi\)
0.486442 + 0.873713i \(0.338294\pi\)
\(678\) −24.1421 −0.927173
\(679\) 32.9706 1.26529
\(680\) 17.6569 0.677109
\(681\) 16.2426 0.622419
\(682\) 25.3137 0.969312
\(683\) 41.6985 1.59555 0.797774 0.602956i \(-0.206011\pi\)
0.797774 + 0.602956i \(0.206011\pi\)
\(684\) 21.6569 0.828071
\(685\) 27.3137 1.04360
\(686\) 40.9706 1.56426
\(687\) 6.68629 0.255098
\(688\) 8.48528 0.323498
\(689\) 7.65685 0.291703
\(690\) 30.1421 1.14749
\(691\) 44.8701 1.70694 0.853469 0.521144i \(-0.174495\pi\)
0.853469 + 0.521144i \(0.174495\pi\)
\(692\) −98.7696 −3.75466
\(693\) 4.00000 0.151947
\(694\) −67.5980 −2.56598
\(695\) −68.7696 −2.60858
\(696\) −33.7990 −1.28115
\(697\) 8.00000 0.303022
\(698\) −74.5269 −2.82089
\(699\) 28.6274 1.08279
\(700\) 72.0833 2.72449
\(701\) −27.1127 −1.02403 −0.512016 0.858976i \(-0.671101\pi\)
−0.512016 + 0.858976i \(0.671101\pi\)
\(702\) 2.41421 0.0911186
\(703\) −17.3726 −0.655219
\(704\) −13.8995 −0.523857
\(705\) −24.1421 −0.909245
\(706\) 36.0416 1.35644
\(707\) 21.6569 0.814490
\(708\) 22.9706 0.863287
\(709\) 30.0000 1.12667 0.563337 0.826227i \(-0.309517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) −34.9706 −1.31242
\(711\) 10.4853 0.393229
\(712\) −62.8701 −2.35616
\(713\) 27.1127 1.01538
\(714\) 8.00000 0.299392
\(715\) −4.82843 −0.180573
\(716\) 50.9706 1.90486
\(717\) −12.3431 −0.460963
\(718\) −54.2843 −2.02587
\(719\) 4.20101 0.156671 0.0783356 0.996927i \(-0.475039\pi\)
0.0783356 + 0.996927i \(0.475039\pi\)
\(720\) −10.2426 −0.381721
\(721\) 2.82843 0.105336
\(722\) −31.3848 −1.16802
\(723\) 20.7279 0.770880
\(724\) 59.9411 2.22769
\(725\) −50.9706 −1.89300
\(726\) −21.7279 −0.806399
\(727\) −19.0294 −0.705763 −0.352881 0.935668i \(-0.614798\pi\)
−0.352881 + 0.935668i \(0.614798\pi\)
\(728\) −12.4853 −0.462735
\(729\) 1.00000 0.0370370
\(730\) −133.882 −4.95520
\(731\) 3.31371 0.122562
\(732\) −15.3137 −0.566011
\(733\) −10.7868 −0.398419 −0.199210 0.979957i \(-0.563837\pi\)
−0.199210 + 0.979957i \(0.563837\pi\)
\(734\) 36.6274 1.35194
\(735\) 3.41421 0.125935
\(736\) −5.79899 −0.213754
\(737\) −10.4853 −0.386230
\(738\) −16.4853 −0.606832
\(739\) −10.6274 −0.390936 −0.195468 0.980710i \(-0.562623\pi\)
−0.195468 + 0.980710i \(0.562623\pi\)
\(740\) 40.1421 1.47565
\(741\) −5.65685 −0.207810
\(742\) −52.2843 −1.91942
\(743\) −10.1005 −0.370552 −0.185276 0.982687i \(-0.559318\pi\)
−0.185276 + 0.982687i \(0.559318\pi\)
\(744\) −32.7279 −1.19986
\(745\) 13.6569 0.500348
\(746\) −4.00000 −0.146450
\(747\) 4.82843 0.176663
\(748\) 6.34315 0.231928
\(749\) 38.6274 1.41142
\(750\) −13.6569 −0.498678
\(751\) 8.82843 0.322154 0.161077 0.986942i \(-0.448503\pi\)
0.161077 + 0.986942i \(0.448503\pi\)
\(752\) −21.2132 −0.773566
\(753\) −24.4853 −0.892293
\(754\) 18.4853 0.673194
\(755\) −9.31371 −0.338961
\(756\) −10.8284 −0.393826
\(757\) −12.3431 −0.448619 −0.224310 0.974518i \(-0.572013\pi\)
−0.224310 + 0.974518i \(0.572013\pi\)
\(758\) 11.0711 0.402119
\(759\) 5.17157 0.187716
\(760\) 85.2548 3.09252
\(761\) −11.4142 −0.413765 −0.206882 0.978366i \(-0.566332\pi\)
−0.206882 + 0.978366i \(0.566332\pi\)
\(762\) 0 0
\(763\) −47.3137 −1.71287
\(764\) 95.5980 3.45861
\(765\) −4.00000 −0.144620
\(766\) 6.24264 0.225556
\(767\) −6.00000 −0.216647
\(768\) 29.9706 1.08147
\(769\) 5.41421 0.195242 0.0976208 0.995224i \(-0.468877\pi\)
0.0976208 + 0.995224i \(0.468877\pi\)
\(770\) 32.9706 1.18818
\(771\) −30.9706 −1.11538
\(772\) 33.4142 1.20260
\(773\) −12.0000 −0.431610 −0.215805 0.976436i \(-0.569238\pi\)
−0.215805 + 0.976436i \(0.569238\pi\)
\(774\) −6.82843 −0.245443
\(775\) −49.3553 −1.77290
\(776\) −51.4558 −1.84716
\(777\) 8.68629 0.311619
\(778\) −25.3137 −0.907540
\(779\) 38.6274 1.38397
\(780\) 13.0711 0.468019
\(781\) −6.00000 −0.214697
\(782\) 10.3431 0.369870
\(783\) 7.65685 0.273634
\(784\) 3.00000 0.107143
\(785\) 55.1127 1.96706
\(786\) −14.4853 −0.516673
\(787\) −10.3431 −0.368693 −0.184347 0.982861i \(-0.559017\pi\)
−0.184347 + 0.982861i \(0.559017\pi\)
\(788\) 30.2426 1.07735
\(789\) 1.17157 0.0417091
\(790\) 86.4264 3.07491
\(791\) −28.2843 −1.00567
\(792\) −6.24264 −0.221823
\(793\) 4.00000 0.142044
\(794\) 64.8701 2.30215
\(795\) 26.1421 0.927166
\(796\) −15.3137 −0.542780
\(797\) 18.6863 0.661902 0.330951 0.943648i \(-0.392630\pi\)
0.330951 + 0.943648i \(0.392630\pi\)
\(798\) 38.6274 1.36740
\(799\) −8.28427 −0.293076
\(800\) 10.5563 0.373223
\(801\) 14.2426 0.503239
\(802\) 60.7696 2.14585
\(803\) −22.9706 −0.810614
\(804\) 28.3848 1.00105
\(805\) 35.3137 1.24464
\(806\) 17.8995 0.630483
\(807\) −2.82843 −0.0995654
\(808\) −33.7990 −1.18904
\(809\) 7.85786 0.276268 0.138134 0.990414i \(-0.455890\pi\)
0.138134 + 0.990414i \(0.455890\pi\)
\(810\) 8.24264 0.289617
\(811\) 33.0711 1.16128 0.580641 0.814160i \(-0.302802\pi\)
0.580641 + 0.814160i \(0.302802\pi\)
\(812\) −82.9117 −2.90963
\(813\) 20.5858 0.721975
\(814\) 10.4853 0.367509
\(815\) −19.3137 −0.676530
\(816\) −3.51472 −0.123040
\(817\) 16.0000 0.559769
\(818\) −36.6274 −1.28065
\(819\) 2.82843 0.0988332
\(820\) −89.2548 −3.11691
\(821\) −7.31371 −0.255250 −0.127625 0.991822i \(-0.540735\pi\)
−0.127625 + 0.991822i \(0.540735\pi\)
\(822\) −19.3137 −0.673643
\(823\) −44.4853 −1.55066 −0.775330 0.631557i \(-0.782416\pi\)
−0.775330 + 0.631557i \(0.782416\pi\)
\(824\) −4.41421 −0.153776
\(825\) −9.41421 −0.327761
\(826\) 40.9706 1.42555
\(827\) 32.5269 1.13107 0.565536 0.824724i \(-0.308669\pi\)
0.565536 + 0.824724i \(0.308669\pi\)
\(828\) −14.0000 −0.486534
\(829\) −20.6274 −0.716420 −0.358210 0.933641i \(-0.616613\pi\)
−0.358210 + 0.933641i \(0.616613\pi\)
\(830\) 39.7990 1.38144
\(831\) −3.17157 −0.110021
\(832\) −9.82843 −0.340739
\(833\) 1.17157 0.0405926
\(834\) 48.6274 1.68383
\(835\) 82.4264 2.85248
\(836\) 30.6274 1.05927
\(837\) 7.41421 0.256273
\(838\) 71.5980 2.47331
\(839\) 11.6569 0.402439 0.201220 0.979546i \(-0.435510\pi\)
0.201220 + 0.979546i \(0.435510\pi\)
\(840\) −42.6274 −1.47079
\(841\) 29.6274 1.02164
\(842\) 25.3137 0.872368
\(843\) 0.100505 0.00346158
\(844\) 41.4558 1.42697
\(845\) −3.41421 −0.117453
\(846\) 17.0711 0.586915
\(847\) −25.4558 −0.874673
\(848\) 22.9706 0.788812
\(849\) 10.1421 0.348077
\(850\) −18.8284 −0.645810
\(851\) 11.2304 0.384975
\(852\) 16.2426 0.556464
\(853\) 42.9706 1.47128 0.735642 0.677371i \(-0.236881\pi\)
0.735642 + 0.677371i \(0.236881\pi\)
\(854\) −27.3137 −0.934656
\(855\) −19.3137 −0.660515
\(856\) −60.2843 −2.06047
\(857\) 0.485281 0.0165769 0.00828845 0.999966i \(-0.497362\pi\)
0.00828845 + 0.999966i \(0.497362\pi\)
\(858\) 3.41421 0.116559
\(859\) 4.48528 0.153036 0.0765179 0.997068i \(-0.475620\pi\)
0.0765179 + 0.997068i \(0.475620\pi\)
\(860\) −36.9706 −1.26069
\(861\) −19.3137 −0.658209
\(862\) 73.5980 2.50676
\(863\) −47.5563 −1.61884 −0.809418 0.587232i \(-0.800218\pi\)
−0.809418 + 0.587232i \(0.800218\pi\)
\(864\) −1.58579 −0.0539496
\(865\) 88.0833 2.99492
\(866\) 8.82843 0.300002
\(867\) 15.6274 0.530735
\(868\) −80.2843 −2.72503
\(869\) 14.8284 0.503020
\(870\) 63.1127 2.13972
\(871\) −7.41421 −0.251221
\(872\) 73.8406 2.50056
\(873\) 11.6569 0.394525
\(874\) 49.9411 1.68928
\(875\) −16.0000 −0.540899
\(876\) 62.1838 2.10099
\(877\) 25.2132 0.851389 0.425695 0.904867i \(-0.360030\pi\)
0.425695 + 0.904867i \(0.360030\pi\)
\(878\) 61.4558 2.07403
\(879\) 9.55635 0.322328
\(880\) −14.4853 −0.488299
\(881\) 34.4853 1.16184 0.580919 0.813961i \(-0.302693\pi\)
0.580919 + 0.813961i \(0.302693\pi\)
\(882\) −2.41421 −0.0812908
\(883\) 9.11270 0.306667 0.153333 0.988175i \(-0.450999\pi\)
0.153333 + 0.988175i \(0.450999\pi\)
\(884\) 4.48528 0.150856
\(885\) −20.4853 −0.688605
\(886\) 6.82843 0.229405
\(887\) −12.2843 −0.412465 −0.206233 0.978503i \(-0.566120\pi\)
−0.206233 + 0.978503i \(0.566120\pi\)
\(888\) −13.5563 −0.454921
\(889\) 0 0
\(890\) 117.397 3.93516
\(891\) 1.41421 0.0473779
\(892\) 0 0
\(893\) −40.0000 −1.33855
\(894\) −9.65685 −0.322974
\(895\) −45.4558 −1.51942
\(896\) 58.1421 1.94239
\(897\) 3.65685 0.122099
\(898\) 29.2132 0.974857
\(899\) 56.7696 1.89337
\(900\) 25.4853 0.849509
\(901\) 8.97056 0.298853
\(902\) −23.3137 −0.776262
\(903\) −8.00000 −0.266223
\(904\) 44.1421 1.46815
\(905\) −53.4558 −1.77693
\(906\) 6.58579 0.218798
\(907\) 24.6863 0.819695 0.409847 0.912154i \(-0.365582\pi\)
0.409847 + 0.912154i \(0.365582\pi\)
\(908\) −62.1838 −2.06364
\(909\) 7.65685 0.253962
\(910\) 23.3137 0.772842
\(911\) 42.6274 1.41231 0.706155 0.708058i \(-0.250428\pi\)
0.706155 + 0.708058i \(0.250428\pi\)
\(912\) −16.9706 −0.561951
\(913\) 6.82843 0.225988
\(914\) 71.0122 2.34887
\(915\) 13.6569 0.451482
\(916\) −25.5980 −0.845781
\(917\) −16.9706 −0.560417
\(918\) 2.82843 0.0933520
\(919\) 41.1716 1.35812 0.679062 0.734081i \(-0.262387\pi\)
0.679062 + 0.734081i \(0.262387\pi\)
\(920\) −55.1127 −1.81701
\(921\) 21.0711 0.694315
\(922\) 91.5980 3.01662
\(923\) −4.24264 −0.139648
\(924\) −15.3137 −0.503784
\(925\) −20.4437 −0.672183
\(926\) 44.0416 1.44730
\(927\) 1.00000 0.0328443
\(928\) −12.1421 −0.398585
\(929\) 40.2843 1.32168 0.660842 0.750525i \(-0.270199\pi\)
0.660842 + 0.750525i \(0.270199\pi\)
\(930\) 61.1127 2.00396
\(931\) 5.65685 0.185396
\(932\) −109.598 −3.59000
\(933\) 22.3431 0.731482
\(934\) −30.6274 −1.00216
\(935\) −5.65685 −0.184999
\(936\) −4.41421 −0.144283
\(937\) 7.85786 0.256705 0.128353 0.991729i \(-0.459031\pi\)
0.128353 + 0.991729i \(0.459031\pi\)
\(938\) 50.6274 1.65304
\(939\) −6.68629 −0.218199
\(940\) 92.4264 3.01462
\(941\) −0.686292 −0.0223725 −0.0111862 0.999937i \(-0.503561\pi\)
−0.0111862 + 0.999937i \(0.503561\pi\)
\(942\) −38.9706 −1.26973
\(943\) −24.9706 −0.813153
\(944\) −18.0000 −0.585850
\(945\) 9.65685 0.314137
\(946\) −9.65685 −0.313971
\(947\) −38.3848 −1.24734 −0.623669 0.781689i \(-0.714359\pi\)
−0.623669 + 0.781689i \(0.714359\pi\)
\(948\) −40.1421 −1.30376
\(949\) −16.2426 −0.527258
\(950\) −90.9117 −2.94956
\(951\) 25.4558 0.825462
\(952\) −14.6274 −0.474077
\(953\) −1.85786 −0.0601821 −0.0300911 0.999547i \(-0.509580\pi\)
−0.0300911 + 0.999547i \(0.509580\pi\)
\(954\) −18.4853 −0.598483
\(955\) −85.2548 −2.75878
\(956\) 47.2548 1.52833
\(957\) 10.8284 0.350033
\(958\) 13.0711 0.422307
\(959\) −22.6274 −0.730677
\(960\) −33.5563 −1.08303
\(961\) 23.9706 0.773244
\(962\) 7.41421 0.239044
\(963\) 13.6569 0.440086
\(964\) −79.3553 −2.55586
\(965\) −29.7990 −0.959263
\(966\) −24.9706 −0.803415
\(967\) −42.7279 −1.37404 −0.687019 0.726640i \(-0.741081\pi\)
−0.687019 + 0.726640i \(0.741081\pi\)
\(968\) 39.7279 1.27690
\(969\) −6.62742 −0.212903
\(970\) 96.0833 3.08505
\(971\) 24.2843 0.779319 0.389660 0.920959i \(-0.372593\pi\)
0.389660 + 0.920959i \(0.372593\pi\)
\(972\) −3.82843 −0.122797
\(973\) 56.9706 1.82639
\(974\) 3.75736 0.120394
\(975\) −6.65685 −0.213190
\(976\) 12.0000 0.384111
\(977\) −42.1421 −1.34825 −0.674123 0.738619i \(-0.735478\pi\)
−0.674123 + 0.738619i \(0.735478\pi\)
\(978\) 13.6569 0.436698
\(979\) 20.1421 0.643745
\(980\) −13.0711 −0.417540
\(981\) −16.7279 −0.534081
\(982\) 71.5980 2.28478
\(983\) 3.45584 0.110224 0.0551122 0.998480i \(-0.482448\pi\)
0.0551122 + 0.998480i \(0.482448\pi\)
\(984\) 30.1421 0.960896
\(985\) −26.9706 −0.859354
\(986\) 21.6569 0.689695
\(987\) 20.0000 0.636607
\(988\) 21.6569 0.688996
\(989\) −10.3431 −0.328893
\(990\) 11.6569 0.370479
\(991\) 8.82843 0.280444 0.140222 0.990120i \(-0.455218\pi\)
0.140222 + 0.990120i \(0.455218\pi\)
\(992\) −11.7574 −0.373297
\(993\) −5.07107 −0.160925
\(994\) 28.9706 0.918890
\(995\) 13.6569 0.432951
\(996\) −18.4853 −0.585729
\(997\) −35.6569 −1.12926 −0.564632 0.825343i \(-0.690982\pi\)
−0.564632 + 0.825343i \(0.690982\pi\)
\(998\) 1.41421 0.0447661
\(999\) 3.07107 0.0971643
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.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4017.2.a.d.1.1 2 1.1 even 1 trivial