Properties

Label 289.10.a.i.1.17
Level $289$
Weight $10$
Character 289.1
Self dual yes
Analytic conductor $148.845$
Analytic rank $0$
Dimension $52$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 289 = 17^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 289.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(148.845356651\)
Analytic rank: \(0\)
Dimension: \(52\)
Twist minimal: no (minimal twist has level 17)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.17
Character \(\chi\) \(=\) 289.1

$q$-expansion

\(f(q)\) \(=\) \(q-14.8865 q^{2} -140.613 q^{3} -290.392 q^{4} +1537.70 q^{5} +2093.24 q^{6} +2138.55 q^{7} +11944.8 q^{8} +89.0110 q^{9} +O(q^{10})\) \(q-14.8865 q^{2} -140.613 q^{3} -290.392 q^{4} +1537.70 q^{5} +2093.24 q^{6} +2138.55 q^{7} +11944.8 q^{8} +89.0110 q^{9} -22891.0 q^{10} -84061.4 q^{11} +40832.8 q^{12} +12765.7 q^{13} -31835.6 q^{14} -216220. q^{15} -29136.1 q^{16} -1325.06 q^{18} +781004. q^{19} -446535. q^{20} -300708. q^{21} +1.25138e6 q^{22} -1.32503e6 q^{23} -1.67960e6 q^{24} +411393. q^{25} -190037. q^{26} +2.75517e6 q^{27} -621018. q^{28} +2.18670e6 q^{29} +3.21877e6 q^{30} +2.91345e6 q^{31} -5.68201e6 q^{32} +1.18201e7 q^{33} +3.28845e6 q^{35} -25848.0 q^{36} +962092. q^{37} -1.16264e7 q^{38} -1.79502e6 q^{39} +1.83675e7 q^{40} +2.92396e7 q^{41} +4.47650e6 q^{42} -9.73937e6 q^{43} +2.44107e7 q^{44} +136872. q^{45} +1.97251e7 q^{46} -1.22741e7 q^{47} +4.09692e6 q^{48} -3.57802e7 q^{49} -6.12420e6 q^{50} -3.70706e6 q^{52} -5.42576e7 q^{53} -4.10149e7 q^{54} -1.29261e8 q^{55} +2.55446e7 q^{56} -1.09819e8 q^{57} -3.25523e7 q^{58} +1.10830e8 q^{59} +6.27886e7 q^{60} -8.49900e7 q^{61} -4.33711e7 q^{62} +190355. q^{63} +9.95030e7 q^{64} +1.96298e7 q^{65} -1.75960e8 q^{66} -1.27439e8 q^{67} +1.86317e8 q^{69} -4.89536e7 q^{70} -3.50617e8 q^{71} +1.06322e6 q^{72} -2.51809e8 q^{73} -1.43222e7 q^{74} -5.78472e7 q^{75} -2.26797e8 q^{76} -1.79770e8 q^{77} +2.67217e7 q^{78} -1.63817e8 q^{79} -4.48026e7 q^{80} -3.89165e8 q^{81} -4.35276e8 q^{82} +7.86881e8 q^{83} +8.73232e7 q^{84} +1.44985e8 q^{86} -3.07478e8 q^{87} -1.00410e9 q^{88} -2.83737e8 q^{89} -2.03755e6 q^{90} +2.73002e7 q^{91} +3.84778e8 q^{92} -4.09669e8 q^{93} +1.82719e8 q^{94} +1.20095e9 q^{95} +7.98964e8 q^{96} -2.62185e8 q^{97} +5.32642e8 q^{98} -7.48239e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 52q + 64q^{2} + 13312q^{4} + 49152q^{8} + 341172q^{9} + O(q^{10}) \) \( 52q + 64q^{2} + 13312q^{4} + 49152q^{8} + 341172q^{9} + 156200q^{13} + 1207872q^{15} + 3407880q^{16} + 2193336q^{18} + 1185568q^{19} + 5198336q^{21} + 13827692q^{25} + 3618944q^{26} - 9167544q^{30} + 61884888q^{32} + 1635208q^{33} + 46992776q^{35} + 156027320q^{36} + 84813952q^{38} - 4635776q^{42} + 125448912q^{43} + 164193176q^{47} + 270850284q^{49} - 226223888q^{50} + 103553016q^{52} + 426167208q^{53} + 677761520q^{55} + 375214512q^{59} + 336918024q^{60} + 190014416q^{64} + 1377178928q^{66} + 311910088q^{67} + 533688136q^{69} + 1477690280q^{70} + 2757942680q^{72} + 4047975520q^{76} + 3440336432q^{77} + 3266558756q^{81} + 2072890608q^{83} + 2630025952q^{84} + 1538547296q^{86} - 1010436256q^{87} + 1873849184q^{89} - 1998451624q^{93} - 6880776704q^{94} - 4667454128q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −14.8865 −0.657897 −0.328949 0.944348i \(-0.606694\pi\)
−0.328949 + 0.944348i \(0.606694\pi\)
\(3\) −140.613 −1.00226 −0.501129 0.865372i \(-0.667082\pi\)
−0.501129 + 0.865372i \(0.667082\pi\)
\(4\) −290.392 −0.567171
\(5\) 1537.70 1.10029 0.550144 0.835070i \(-0.314573\pi\)
0.550144 + 0.835070i \(0.314573\pi\)
\(6\) 2093.24 0.659383
\(7\) 2138.55 0.336650 0.168325 0.985732i \(-0.446164\pi\)
0.168325 + 0.985732i \(0.446164\pi\)
\(8\) 11944.8 1.03104
\(9\) 89.0110 0.00452223
\(10\) −22891.0 −0.723876
\(11\) −84061.4 −1.73113 −0.865565 0.500797i \(-0.833040\pi\)
−0.865565 + 0.500797i \(0.833040\pi\)
\(12\) 40832.8 0.568452
\(13\) 12765.7 0.123965 0.0619826 0.998077i \(-0.480258\pi\)
0.0619826 + 0.998077i \(0.480258\pi\)
\(14\) −31835.6 −0.221481
\(15\) −216220. −1.10277
\(16\) −29136.1 −0.111146
\(17\) 0 0
\(18\) −1325.06 −0.00297516
\(19\) 781004. 1.37487 0.687436 0.726245i \(-0.258736\pi\)
0.687436 + 0.726245i \(0.258736\pi\)
\(20\) −446535. −0.624052
\(21\) −300708. −0.337411
\(22\) 1.25138e6 1.13891
\(23\) −1.32503e6 −0.987303 −0.493652 0.869660i \(-0.664338\pi\)
−0.493652 + 0.869660i \(0.664338\pi\)
\(24\) −1.67960e6 −1.03337
\(25\) 411393. 0.210633
\(26\) −190037. −0.0815564
\(27\) 2.75517e6 0.997726
\(28\) −621018. −0.190938
\(29\) 2.18670e6 0.574114 0.287057 0.957914i \(-0.407323\pi\)
0.287057 + 0.957914i \(0.407323\pi\)
\(30\) 3.21877e6 0.725511
\(31\) 2.91345e6 0.566605 0.283302 0.959031i \(-0.408570\pi\)
0.283302 + 0.959031i \(0.408570\pi\)
\(32\) −5.68201e6 −0.957915
\(33\) 1.18201e7 1.73504
\(34\) 0 0
\(35\) 3.28845e6 0.370412
\(36\) −25848.0 −0.00256488
\(37\) 962092. 0.0843935 0.0421968 0.999109i \(-0.486564\pi\)
0.0421968 + 0.999109i \(0.486564\pi\)
\(38\) −1.16264e7 −0.904524
\(39\) −1.79502e6 −0.124245
\(40\) 1.83675e7 1.13444
\(41\) 2.92396e7 1.61601 0.808005 0.589176i \(-0.200547\pi\)
0.808005 + 0.589176i \(0.200547\pi\)
\(42\) 4.47650e6 0.221982
\(43\) −9.73937e6 −0.434433 −0.217216 0.976123i \(-0.569698\pi\)
−0.217216 + 0.976123i \(0.569698\pi\)
\(44\) 2.44107e7 0.981847
\(45\) 136872. 0.00497575
\(46\) 1.97251e7 0.649544
\(47\) −1.22741e7 −0.366902 −0.183451 0.983029i \(-0.558727\pi\)
−0.183451 + 0.983029i \(0.558727\pi\)
\(48\) 4.09692e6 0.111397
\(49\) −3.57802e7 −0.886667
\(50\) −6.12420e6 −0.138575
\(51\) 0 0
\(52\) −3.70706e6 −0.0703095
\(53\) −5.42576e7 −0.944537 −0.472268 0.881455i \(-0.656565\pi\)
−0.472268 + 0.881455i \(0.656565\pi\)
\(54\) −4.10149e7 −0.656401
\(55\) −1.29261e8 −1.90474
\(56\) 2.55446e7 0.347099
\(57\) −1.09819e8 −1.37798
\(58\) −3.25523e7 −0.377708
\(59\) 1.10830e8 1.19076 0.595380 0.803444i \(-0.297001\pi\)
0.595380 + 0.803444i \(0.297001\pi\)
\(60\) 6.27886e7 0.625461
\(61\) −8.49900e7 −0.785930 −0.392965 0.919553i \(-0.628551\pi\)
−0.392965 + 0.919553i \(0.628551\pi\)
\(62\) −4.33711e7 −0.372768
\(63\) 190355. 0.00152241
\(64\) 9.95030e7 0.741355
\(65\) 1.96298e7 0.136397
\(66\) −1.75960e8 −1.14148
\(67\) −1.27439e8 −0.772619 −0.386310 0.922369i \(-0.626250\pi\)
−0.386310 + 0.922369i \(0.626250\pi\)
\(68\) 0 0
\(69\) 1.86317e8 0.989533
\(70\) −4.89536e7 −0.243693
\(71\) −3.50617e8 −1.63746 −0.818730 0.574179i \(-0.805321\pi\)
−0.818730 + 0.574179i \(0.805321\pi\)
\(72\) 1.06322e6 0.00466259
\(73\) −2.51809e8 −1.03781 −0.518905 0.854832i \(-0.673660\pi\)
−0.518905 + 0.854832i \(0.673660\pi\)
\(74\) −1.43222e7 −0.0555223
\(75\) −5.78472e7 −0.211109
\(76\) −2.26797e8 −0.779787
\(77\) −1.79770e8 −0.582785
\(78\) 2.67217e7 0.0817406
\(79\) −1.63817e8 −0.473192 −0.236596 0.971608i \(-0.576032\pi\)
−0.236596 + 0.971608i \(0.576032\pi\)
\(80\) −4.48026e7 −0.122292
\(81\) −3.89165e8 −1.00450
\(82\) −4.35276e8 −1.06317
\(83\) 7.86881e8 1.81994 0.909972 0.414671i \(-0.136103\pi\)
0.909972 + 0.414671i \(0.136103\pi\)
\(84\) 8.73232e7 0.191370
\(85\) 0 0
\(86\) 1.44985e8 0.285812
\(87\) −3.07478e8 −0.575411
\(88\) −1.00410e9 −1.78486
\(89\) −2.83737e8 −0.479360 −0.239680 0.970852i \(-0.577042\pi\)
−0.239680 + 0.970852i \(0.577042\pi\)
\(90\) −2.03755e6 −0.00327353
\(91\) 2.73002e7 0.0417329
\(92\) 3.84778e8 0.559970
\(93\) −4.09669e8 −0.567884
\(94\) 1.82719e8 0.241384
\(95\) 1.20095e9 1.51275
\(96\) 7.98964e8 0.960079
\(97\) −2.62185e8 −0.300701 −0.150350 0.988633i \(-0.548040\pi\)
−0.150350 + 0.988633i \(0.548040\pi\)
\(98\) 5.32642e8 0.583335
\(99\) −7.48239e6 −0.00782856
\(100\) −1.19465e8 −0.119465
\(101\) −1.01701e9 −0.972477 −0.486238 0.873826i \(-0.661631\pi\)
−0.486238 + 0.873826i \(0.661631\pi\)
\(102\) 0 0
\(103\) 1.11236e9 0.973814 0.486907 0.873454i \(-0.338125\pi\)
0.486907 + 0.873454i \(0.338125\pi\)
\(104\) 1.52484e8 0.127813
\(105\) −4.62399e8 −0.371249
\(106\) 8.07706e8 0.621408
\(107\) −1.17459e9 −0.866284 −0.433142 0.901326i \(-0.642595\pi\)
−0.433142 + 0.901326i \(0.642595\pi\)
\(108\) −8.00078e8 −0.565882
\(109\) −1.13422e9 −0.769625 −0.384813 0.922995i \(-0.625734\pi\)
−0.384813 + 0.922995i \(0.625734\pi\)
\(110\) 1.92425e9 1.25312
\(111\) −1.35283e8 −0.0845841
\(112\) −6.23092e7 −0.0374172
\(113\) 1.37503e9 0.793337 0.396668 0.917962i \(-0.370166\pi\)
0.396668 + 0.917962i \(0.370166\pi\)
\(114\) 1.63483e9 0.906567
\(115\) −2.03750e9 −1.08632
\(116\) −6.34999e8 −0.325621
\(117\) 1.13629e6 0.000560599 0
\(118\) −1.64988e9 −0.783398
\(119\) 0 0
\(120\) −2.58271e9 −1.13700
\(121\) 4.70837e9 1.99681
\(122\) 1.26521e9 0.517061
\(123\) −4.11147e9 −1.61966
\(124\) −8.46042e8 −0.321362
\(125\) −2.37072e9 −0.868531
\(126\) −2.83372e6 −0.00100159
\(127\) −1.63757e9 −0.558577 −0.279289 0.960207i \(-0.590099\pi\)
−0.279289 + 0.960207i \(0.590099\pi\)
\(128\) 1.42794e9 0.470180
\(129\) 1.36948e9 0.435414
\(130\) −2.92220e8 −0.0897355
\(131\) 4.89940e9 1.45352 0.726762 0.686890i \(-0.241024\pi\)
0.726762 + 0.686890i \(0.241024\pi\)
\(132\) −3.43247e9 −0.984064
\(133\) 1.67022e9 0.462851
\(134\) 1.89712e9 0.508304
\(135\) 4.23662e9 1.09779
\(136\) 0 0
\(137\) −3.10441e9 −0.752899 −0.376449 0.926437i \(-0.622855\pi\)
−0.376449 + 0.926437i \(0.622855\pi\)
\(138\) −2.77360e9 −0.651011
\(139\) 2.82102e8 0.0640972 0.0320486 0.999486i \(-0.489797\pi\)
0.0320486 + 0.999486i \(0.489797\pi\)
\(140\) −9.54939e8 −0.210087
\(141\) 1.72590e9 0.367730
\(142\) 5.21947e9 1.07728
\(143\) −1.07310e9 −0.214600
\(144\) −2.59344e6 −0.000502625 0
\(145\) 3.36249e9 0.631691
\(146\) 3.74855e9 0.682772
\(147\) 5.03116e9 0.888669
\(148\) −2.79384e8 −0.0478656
\(149\) −1.55842e9 −0.259028 −0.129514 0.991578i \(-0.541342\pi\)
−0.129514 + 0.991578i \(0.541342\pi\)
\(150\) 8.61142e8 0.138888
\(151\) 8.38462e9 1.31246 0.656231 0.754560i \(-0.272150\pi\)
0.656231 + 0.754560i \(0.272150\pi\)
\(152\) 9.32895e9 1.41754
\(153\) 0 0
\(154\) 2.67615e9 0.383413
\(155\) 4.48001e9 0.623428
\(156\) 5.21260e8 0.0704683
\(157\) 8.18273e9 1.07486 0.537428 0.843310i \(-0.319396\pi\)
0.537428 + 0.843310i \(0.319396\pi\)
\(158\) 2.43867e9 0.311312
\(159\) 7.62932e9 0.946670
\(160\) −8.73722e9 −1.05398
\(161\) −2.83365e9 −0.332376
\(162\) 5.79330e9 0.660859
\(163\) −9.23855e9 −1.02508 −0.512542 0.858662i \(-0.671296\pi\)
−0.512542 + 0.858662i \(0.671296\pi\)
\(164\) −8.49093e9 −0.916554
\(165\) 1.81758e10 1.90904
\(166\) −1.17139e10 −1.19734
\(167\) 1.07606e10 1.07056 0.535279 0.844675i \(-0.320206\pi\)
0.535279 + 0.844675i \(0.320206\pi\)
\(168\) −3.59191e9 −0.347883
\(169\) −1.04415e10 −0.984633
\(170\) 0 0
\(171\) 6.95179e7 0.00621748
\(172\) 2.82823e9 0.246398
\(173\) −1.65396e10 −1.40384 −0.701921 0.712255i \(-0.747674\pi\)
−0.701921 + 0.712255i \(0.747674\pi\)
\(174\) 4.57728e9 0.378561
\(175\) 8.79785e8 0.0709097
\(176\) 2.44922e9 0.192407
\(177\) −1.55842e10 −1.19345
\(178\) 4.22386e9 0.315369
\(179\) 1.20555e10 0.877699 0.438849 0.898561i \(-0.355386\pi\)
0.438849 + 0.898561i \(0.355386\pi\)
\(180\) −3.97465e7 −0.00282210
\(181\) 2.05478e10 1.42302 0.711511 0.702675i \(-0.248011\pi\)
0.711511 + 0.702675i \(0.248011\pi\)
\(182\) −4.06404e8 −0.0274560
\(183\) 1.19507e10 0.787705
\(184\) −1.58272e10 −1.01795
\(185\) 1.47941e9 0.0928571
\(186\) 6.09854e9 0.373610
\(187\) 0 0
\(188\) 3.56430e9 0.208096
\(189\) 5.89208e9 0.335885
\(190\) −1.78779e10 −0.995236
\(191\) −3.26529e10 −1.77530 −0.887651 0.460517i \(-0.847664\pi\)
−0.887651 + 0.460517i \(0.847664\pi\)
\(192\) −1.39914e10 −0.743030
\(193\) 2.78185e10 1.44320 0.721600 0.692310i \(-0.243407\pi\)
0.721600 + 0.692310i \(0.243407\pi\)
\(194\) 3.90302e9 0.197830
\(195\) −2.76021e9 −0.136705
\(196\) 1.03903e10 0.502892
\(197\) −2.76208e10 −1.30659 −0.653293 0.757105i \(-0.726613\pi\)
−0.653293 + 0.757105i \(0.726613\pi\)
\(198\) 1.11387e8 0.00515039
\(199\) −3.75699e10 −1.69825 −0.849123 0.528195i \(-0.822869\pi\)
−0.849123 + 0.528195i \(0.822869\pi\)
\(200\) 4.91401e9 0.217171
\(201\) 1.79196e10 0.774364
\(202\) 1.51397e10 0.639790
\(203\) 4.67638e9 0.193276
\(204\) 0 0
\(205\) 4.49617e10 1.77808
\(206\) −1.65591e10 −0.640670
\(207\) −1.17942e8 −0.00446481
\(208\) −3.71943e8 −0.0137782
\(209\) −6.56523e10 −2.38008
\(210\) 6.88351e9 0.244244
\(211\) −3.68658e9 −0.128042 −0.0640211 0.997949i \(-0.520392\pi\)
−0.0640211 + 0.997949i \(0.520392\pi\)
\(212\) 1.57559e10 0.535714
\(213\) 4.93013e10 1.64116
\(214\) 1.74856e10 0.569926
\(215\) −1.49762e10 −0.478001
\(216\) 3.29100e10 1.02869
\(217\) 6.23057e9 0.190748
\(218\) 1.68846e10 0.506334
\(219\) 3.54076e10 1.04015
\(220\) 3.75363e10 1.08031
\(221\) 0 0
\(222\) 2.01389e9 0.0556477
\(223\) −1.26499e10 −0.342544 −0.171272 0.985224i \(-0.554788\pi\)
−0.171272 + 0.985224i \(0.554788\pi\)
\(224\) −1.21513e10 −0.322483
\(225\) 3.66185e7 0.000952530 0
\(226\) −2.04693e10 −0.521934
\(227\) 4.42724e10 1.10667 0.553333 0.832960i \(-0.313356\pi\)
0.553333 + 0.832960i \(0.313356\pi\)
\(228\) 3.18906e10 0.781549
\(229\) 6.67619e10 1.60424 0.802119 0.597164i \(-0.203706\pi\)
0.802119 + 0.597164i \(0.203706\pi\)
\(230\) 3.03312e10 0.714685
\(231\) 2.52780e10 0.584101
\(232\) 2.61197e10 0.591933
\(233\) 6.22710e10 1.38415 0.692076 0.721824i \(-0.256696\pi\)
0.692076 + 0.721824i \(0.256696\pi\)
\(234\) −1.69154e7 −0.000368816 0
\(235\) −1.88739e10 −0.403697
\(236\) −3.21842e10 −0.675365
\(237\) 2.30348e10 0.474261
\(238\) 0 0
\(239\) 6.82772e10 1.35358 0.676792 0.736174i \(-0.263370\pi\)
0.676792 + 0.736174i \(0.263370\pi\)
\(240\) 6.29983e9 0.122568
\(241\) −6.20183e10 −1.18425 −0.592125 0.805846i \(-0.701711\pi\)
−0.592125 + 0.805846i \(0.701711\pi\)
\(242\) −7.00912e10 −1.31369
\(243\) 4.91595e8 0.00904438
\(244\) 2.46804e10 0.445757
\(245\) −5.50192e10 −0.975588
\(246\) 6.12054e10 1.06557
\(247\) 9.97007e9 0.170436
\(248\) 3.48006e10 0.584191
\(249\) −1.10646e11 −1.82405
\(250\) 3.52918e10 0.571404
\(251\) 8.72683e10 1.38779 0.693897 0.720074i \(-0.255892\pi\)
0.693897 + 0.720074i \(0.255892\pi\)
\(252\) −5.52774e7 −0.000863467 0
\(253\) 1.11384e11 1.70915
\(254\) 2.43777e10 0.367486
\(255\) 0 0
\(256\) −7.22025e10 −1.05069
\(257\) −2.32433e10 −0.332353 −0.166176 0.986096i \(-0.553142\pi\)
−0.166176 + 0.986096i \(0.553142\pi\)
\(258\) −2.03868e10 −0.286458
\(259\) 2.05749e9 0.0284111
\(260\) −5.70033e9 −0.0773607
\(261\) 1.94640e8 0.00259627
\(262\) −7.29350e10 −0.956269
\(263\) 2.54583e10 0.328117 0.164058 0.986451i \(-0.447541\pi\)
0.164058 + 0.986451i \(0.447541\pi\)
\(264\) 1.41189e11 1.78889
\(265\) −8.34318e10 −1.03926
\(266\) −2.48637e10 −0.304508
\(267\) 3.98972e10 0.480442
\(268\) 3.70072e10 0.438207
\(269\) −1.66536e10 −0.193921 −0.0969603 0.995288i \(-0.530912\pi\)
−0.0969603 + 0.995288i \(0.530912\pi\)
\(270\) −6.30685e10 −0.722230
\(271\) 7.55383e10 0.850756 0.425378 0.905016i \(-0.360141\pi\)
0.425378 + 0.905016i \(0.360141\pi\)
\(272\) 0 0
\(273\) −3.83876e9 −0.0418272
\(274\) 4.62138e10 0.495330
\(275\) −3.45822e10 −0.364633
\(276\) −5.41048e10 −0.561235
\(277\) −3.08726e10 −0.315076 −0.157538 0.987513i \(-0.550356\pi\)
−0.157538 + 0.987513i \(0.550356\pi\)
\(278\) −4.19951e9 −0.0421694
\(279\) 2.59329e8 0.00256231
\(280\) 3.92800e10 0.381909
\(281\) 5.90092e10 0.564601 0.282300 0.959326i \(-0.408903\pi\)
0.282300 + 0.959326i \(0.408903\pi\)
\(282\) −2.56926e10 −0.241929
\(283\) 1.83598e11 1.70149 0.850745 0.525579i \(-0.176151\pi\)
0.850745 + 0.525579i \(0.176151\pi\)
\(284\) 1.01816e11 0.928720
\(285\) −1.68869e11 −1.51617
\(286\) 1.59748e10 0.141185
\(287\) 6.25304e10 0.544030
\(288\) −5.05761e8 −0.00433191
\(289\) 0 0
\(290\) −5.00557e10 −0.415587
\(291\) 3.68666e10 0.301380
\(292\) 7.31231e10 0.588616
\(293\) 8.80276e10 0.697774 0.348887 0.937165i \(-0.386560\pi\)
0.348887 + 0.937165i \(0.386560\pi\)
\(294\) −7.48964e10 −0.584653
\(295\) 1.70424e11 1.31018
\(296\) 1.14920e10 0.0870129
\(297\) −2.31603e11 −1.72719
\(298\) 2.31995e10 0.170414
\(299\) −1.69150e10 −0.122391
\(300\) 1.67983e10 0.119735
\(301\) −2.08282e10 −0.146252
\(302\) −1.24818e11 −0.863465
\(303\) 1.43005e11 0.974673
\(304\) −2.27554e10 −0.152811
\(305\) −1.30689e11 −0.864749
\(306\) 0 0
\(307\) 1.25066e11 0.803558 0.401779 0.915737i \(-0.368392\pi\)
0.401779 + 0.915737i \(0.368392\pi\)
\(308\) 5.22037e10 0.330539
\(309\) −1.56412e11 −0.976014
\(310\) −6.66917e10 −0.410152
\(311\) −1.23644e11 −0.749466 −0.374733 0.927133i \(-0.622266\pi\)
−0.374733 + 0.927133i \(0.622266\pi\)
\(312\) −2.14412e10 −0.128101
\(313\) 4.92496e10 0.290037 0.145018 0.989429i \(-0.453676\pi\)
0.145018 + 0.989429i \(0.453676\pi\)
\(314\) −1.21812e11 −0.707144
\(315\) 2.92708e8 0.00167509
\(316\) 4.75712e10 0.268381
\(317\) 2.94046e11 1.63549 0.817745 0.575580i \(-0.195224\pi\)
0.817745 + 0.575580i \(0.195224\pi\)
\(318\) −1.13574e11 −0.622812
\(319\) −1.83817e11 −0.993865
\(320\) 1.53006e11 0.815704
\(321\) 1.65163e11 0.868240
\(322\) 4.21832e10 0.218669
\(323\) 0 0
\(324\) 1.13010e11 0.569725
\(325\) 5.25172e9 0.0261112
\(326\) 1.37530e11 0.674400
\(327\) 1.59487e11 0.771363
\(328\) 3.49262e11 1.66617
\(329\) −2.62488e10 −0.123518
\(330\) −2.70574e11 −1.25595
\(331\) 3.07114e10 0.140629 0.0703143 0.997525i \(-0.477600\pi\)
0.0703143 + 0.997525i \(0.477600\pi\)
\(332\) −2.28504e11 −1.03222
\(333\) 8.56368e7 0.000381646 0
\(334\) −1.60187e11 −0.704318
\(335\) −1.95963e11 −0.850104
\(336\) 8.76148e9 0.0375017
\(337\) −4.27513e11 −1.80557 −0.902787 0.430088i \(-0.858482\pi\)
−0.902787 + 0.430088i \(0.858482\pi\)
\(338\) 1.55438e11 0.647787
\(339\) −1.93346e11 −0.795129
\(340\) 0 0
\(341\) −2.44909e11 −0.980866
\(342\) −1.03488e9 −0.00409046
\(343\) −1.62816e11 −0.635147
\(344\) −1.16335e11 −0.447917
\(345\) 2.86499e11 1.08877
\(346\) 2.46217e11 0.923584
\(347\) −1.31346e11 −0.486335 −0.243167 0.969984i \(-0.578186\pi\)
−0.243167 + 0.969984i \(0.578186\pi\)
\(348\) 8.92892e10 0.326356
\(349\) −1.90494e11 −0.687333 −0.343666 0.939092i \(-0.611669\pi\)
−0.343666 + 0.939092i \(0.611669\pi\)
\(350\) −1.30969e10 −0.0466513
\(351\) 3.51717e10 0.123683
\(352\) 4.77638e11 1.65828
\(353\) −5.20746e10 −0.178501 −0.0892503 0.996009i \(-0.528447\pi\)
−0.0892503 + 0.996009i \(0.528447\pi\)
\(354\) 2.31994e11 0.785167
\(355\) −5.39144e11 −1.80168
\(356\) 8.23950e10 0.271879
\(357\) 0 0
\(358\) −1.79464e11 −0.577435
\(359\) 6.15238e11 1.95487 0.977436 0.211232i \(-0.0677474\pi\)
0.977436 + 0.211232i \(0.0677474\pi\)
\(360\) 1.63491e9 0.00513019
\(361\) 2.87279e11 0.890271
\(362\) −3.05885e11 −0.936202
\(363\) −6.62058e11 −2.00132
\(364\) −7.92774e9 −0.0236697
\(365\) −3.87206e11 −1.14189
\(366\) −1.77904e11 −0.518229
\(367\) 5.60031e11 1.61144 0.805721 0.592295i \(-0.201778\pi\)
0.805721 + 0.592295i \(0.201778\pi\)
\(368\) 3.86063e10 0.109734
\(369\) 2.60264e9 0.00730796
\(370\) −2.20232e10 −0.0610904
\(371\) −1.16033e11 −0.317979
\(372\) 1.18964e11 0.322088
\(373\) −2.00817e10 −0.0537169 −0.0268585 0.999639i \(-0.508550\pi\)
−0.0268585 + 0.999639i \(0.508550\pi\)
\(374\) 0 0
\(375\) 3.33354e11 0.870492
\(376\) −1.46612e11 −0.378289
\(377\) 2.79148e10 0.0711702
\(378\) −8.77125e10 −0.220978
\(379\) −2.22361e10 −0.0553581 −0.0276791 0.999617i \(-0.508812\pi\)
−0.0276791 + 0.999617i \(0.508812\pi\)
\(380\) −3.48746e11 −0.857990
\(381\) 2.30264e11 0.559839
\(382\) 4.86089e11 1.16797
\(383\) 1.05889e11 0.251453 0.125726 0.992065i \(-0.459874\pi\)
0.125726 + 0.992065i \(0.459874\pi\)
\(384\) −2.00786e11 −0.471242
\(385\) −2.76432e11 −0.641231
\(386\) −4.14121e11 −0.949477
\(387\) −8.66911e8 −0.00196460
\(388\) 7.61363e10 0.170549
\(389\) 6.41740e10 0.142097 0.0710486 0.997473i \(-0.477365\pi\)
0.0710486 + 0.997473i \(0.477365\pi\)
\(390\) 4.10899e10 0.0899381
\(391\) 0 0
\(392\) −4.27388e11 −0.914187
\(393\) −6.88919e11 −1.45681
\(394\) 4.11177e11 0.859600
\(395\) −2.51902e11 −0.520648
\(396\) 2.17282e9 0.00444013
\(397\) −1.37059e11 −0.276917 −0.138458 0.990368i \(-0.544215\pi\)
−0.138458 + 0.990368i \(0.544215\pi\)
\(398\) 5.59284e11 1.11727
\(399\) −2.34854e11 −0.463896
\(400\) −1.19864e10 −0.0234109
\(401\) −4.83309e11 −0.933415 −0.466708 0.884412i \(-0.654560\pi\)
−0.466708 + 0.884412i \(0.654560\pi\)
\(402\) −2.66760e11 −0.509452
\(403\) 3.71923e10 0.0702393
\(404\) 2.95331e11 0.551561
\(405\) −5.98418e11 −1.10524
\(406\) −6.96149e10 −0.127156
\(407\) −8.08748e10 −0.146096
\(408\) 0 0
\(409\) −2.08768e11 −0.368900 −0.184450 0.982842i \(-0.559050\pi\)
−0.184450 + 0.982842i \(0.559050\pi\)
\(410\) −6.69323e11 −1.16979
\(411\) 4.36520e11 0.754599
\(412\) −3.23019e11 −0.552319
\(413\) 2.37016e11 0.400870
\(414\) 1.75575e9 0.00293739
\(415\) 1.20999e12 2.00246
\(416\) −7.25349e10 −0.118748
\(417\) −3.96672e10 −0.0642420
\(418\) 9.77333e11 1.56585
\(419\) −1.79984e11 −0.285280 −0.142640 0.989775i \(-0.545559\pi\)
−0.142640 + 0.989775i \(0.545559\pi\)
\(420\) 1.34277e11 0.210562
\(421\) 1.24378e12 1.92963 0.964813 0.262937i \(-0.0846912\pi\)
0.964813 + 0.262937i \(0.0846912\pi\)
\(422\) 5.48804e10 0.0842386
\(423\) −1.09253e9 −0.00165921
\(424\) −6.48097e11 −0.973853
\(425\) 0 0
\(426\) −7.33925e11 −1.07971
\(427\) −1.81756e11 −0.264584
\(428\) 3.41092e11 0.491331
\(429\) 1.50892e11 0.215084
\(430\) 2.22944e11 0.314476
\(431\) −8.50306e11 −1.18694 −0.593468 0.804857i \(-0.702242\pi\)
−0.593468 + 0.804857i \(0.702242\pi\)
\(432\) −8.02750e10 −0.110893
\(433\) 6.91161e11 0.944895 0.472448 0.881359i \(-0.343371\pi\)
0.472448 + 0.881359i \(0.343371\pi\)
\(434\) −9.27515e10 −0.125492
\(435\) −4.72809e11 −0.633117
\(436\) 3.29369e11 0.436509
\(437\) −1.03485e12 −1.35741
\(438\) −5.27095e11 −0.684314
\(439\) 1.98550e11 0.255140 0.127570 0.991830i \(-0.459282\pi\)
0.127570 + 0.991830i \(0.459282\pi\)
\(440\) −1.54400e12 −1.96386
\(441\) −3.18483e9 −0.00400971
\(442\) 0 0
\(443\) 1.17901e12 1.45445 0.727227 0.686397i \(-0.240809\pi\)
0.727227 + 0.686397i \(0.240809\pi\)
\(444\) 3.92850e10 0.0479737
\(445\) −4.36303e11 −0.527434
\(446\) 1.88313e11 0.225359
\(447\) 2.19134e11 0.259613
\(448\) 2.12793e11 0.249578
\(449\) −7.72876e10 −0.0897432 −0.0448716 0.998993i \(-0.514288\pi\)
−0.0448716 + 0.998993i \(0.514288\pi\)
\(450\) −5.45121e8 −0.000626667 0
\(451\) −2.45792e12 −2.79752
\(452\) −3.99296e11 −0.449958
\(453\) −1.17899e12 −1.31543
\(454\) −6.59062e11 −0.728073
\(455\) 4.19794e10 0.0459182
\(456\) −1.31177e12 −1.42075
\(457\) −2.62124e11 −0.281115 −0.140558 0.990073i \(-0.544890\pi\)
−0.140558 + 0.990073i \(0.544890\pi\)
\(458\) −9.93851e11 −1.05542
\(459\) 0 0
\(460\) 5.91672e11 0.616128
\(461\) 7.85754e11 0.810275 0.405138 0.914256i \(-0.367224\pi\)
0.405138 + 0.914256i \(0.367224\pi\)
\(462\) −3.76301e11 −0.384279
\(463\) 1.23202e12 1.24596 0.622981 0.782237i \(-0.285921\pi\)
0.622981 + 0.782237i \(0.285921\pi\)
\(464\) −6.37120e10 −0.0638102
\(465\) −6.29948e11 −0.624836
\(466\) −9.26998e11 −0.910630
\(467\) 5.56830e10 0.0541747 0.0270874 0.999633i \(-0.491377\pi\)
0.0270874 + 0.999633i \(0.491377\pi\)
\(468\) −3.29969e8 −0.000317955 0
\(469\) −2.72535e11 −0.260103
\(470\) 2.80966e11 0.265591
\(471\) −1.15060e12 −1.07728
\(472\) 1.32385e12 1.22772
\(473\) 8.18705e11 0.752059
\(474\) −3.42908e11 −0.312015
\(475\) 3.21299e11 0.289593
\(476\) 0 0
\(477\) −4.82952e9 −0.00427141
\(478\) −1.01641e12 −0.890519
\(479\) 1.63312e12 1.41745 0.708726 0.705484i \(-0.249270\pi\)
0.708726 + 0.705484i \(0.249270\pi\)
\(480\) 1.22857e12 1.05636
\(481\) 1.22818e10 0.0104619
\(482\) 9.23237e11 0.779115
\(483\) 3.98448e11 0.333127
\(484\) −1.36727e12 −1.13253
\(485\) −4.03161e11 −0.330857
\(486\) −7.31813e9 −0.00595027
\(487\) −6.96278e11 −0.560922 −0.280461 0.959865i \(-0.590487\pi\)
−0.280461 + 0.959865i \(0.590487\pi\)
\(488\) −1.01519e12 −0.810323
\(489\) 1.29906e12 1.02740
\(490\) 8.19044e11 0.641837
\(491\) 7.38660e10 0.0573559 0.0286779 0.999589i \(-0.490870\pi\)
0.0286779 + 0.999589i \(0.490870\pi\)
\(492\) 1.19394e12 0.918624
\(493\) 0 0
\(494\) −1.48420e11 −0.112129
\(495\) −1.15057e10 −0.00861367
\(496\) −8.48867e10 −0.0629756
\(497\) −7.49814e11 −0.551251
\(498\) 1.64713e12 1.20004
\(499\) 4.42678e11 0.319621 0.159811 0.987148i \(-0.448912\pi\)
0.159811 + 0.987148i \(0.448912\pi\)
\(500\) 6.88437e11 0.492606
\(501\) −1.51307e12 −1.07298
\(502\) −1.29912e12 −0.913026
\(503\) −7.67499e11 −0.534591 −0.267296 0.963615i \(-0.586130\pi\)
−0.267296 + 0.963615i \(0.586130\pi\)
\(504\) 2.27375e9 0.00156966
\(505\) −1.56386e12 −1.07000
\(506\) −1.65812e12 −1.12444
\(507\) 1.46822e12 0.986856
\(508\) 4.75537e11 0.316809
\(509\) 1.23640e12 0.816452 0.408226 0.912881i \(-0.366148\pi\)
0.408226 + 0.912881i \(0.366148\pi\)
\(510\) 0 0
\(511\) −5.38506e11 −0.349379
\(512\) 3.43741e11 0.221063
\(513\) 2.15180e12 1.37174
\(514\) 3.46012e11 0.218654
\(515\) 1.71047e12 1.07148
\(516\) −3.97686e11 −0.246954
\(517\) 1.03178e12 0.635154
\(518\) −3.06288e10 −0.0186916
\(519\) 2.32569e12 1.40701
\(520\) 2.34474e11 0.140631
\(521\) 6.77564e11 0.402885 0.201442 0.979500i \(-0.435437\pi\)
0.201442 + 0.979500i \(0.435437\pi\)
\(522\) −2.89751e9 −0.00170808
\(523\) −1.65170e12 −0.965324 −0.482662 0.875807i \(-0.660330\pi\)
−0.482662 + 0.875807i \(0.660330\pi\)
\(524\) −1.42274e12 −0.824397
\(525\) −1.23709e11 −0.0710698
\(526\) −3.78985e11 −0.215867
\(527\) 0 0
\(528\) −3.44393e11 −0.192842
\(529\) −4.54466e10 −0.0252320
\(530\) 1.24201e12 0.683728
\(531\) 9.86511e9 0.00538489
\(532\) −4.85018e11 −0.262516
\(533\) 3.73264e11 0.200329
\(534\) −5.93930e11 −0.316082
\(535\) −1.80617e12 −0.953161
\(536\) −1.52223e12 −0.796600
\(537\) −1.69516e12 −0.879681
\(538\) 2.47915e11 0.127580
\(539\) 3.00773e12 1.53493
\(540\) −1.23028e12 −0.622633
\(541\) 9.85219e10 0.0494476 0.0247238 0.999694i \(-0.492129\pi\)
0.0247238 + 0.999694i \(0.492129\pi\)
\(542\) −1.12450e12 −0.559710
\(543\) −2.88929e12 −1.42624
\(544\) 0 0
\(545\) −1.74409e12 −0.846809
\(546\) 5.71457e10 0.0275180
\(547\) −1.58944e11 −0.0759102 −0.0379551 0.999279i \(-0.512084\pi\)
−0.0379551 + 0.999279i \(0.512084\pi\)
\(548\) 9.01495e11 0.427022
\(549\) −7.56505e9 −0.00355415
\(550\) 5.14809e11 0.239891
\(551\) 1.70782e12 0.789333
\(552\) 2.22552e12 1.02025
\(553\) −3.50332e11 −0.159300
\(554\) 4.59586e11 0.207287
\(555\) −2.08024e11 −0.0930669
\(556\) −8.19200e10 −0.0363541
\(557\) 2.80868e12 1.23639 0.618194 0.786026i \(-0.287865\pi\)
0.618194 + 0.786026i \(0.287865\pi\)
\(558\) −3.86051e9 −0.00168574
\(559\) −1.24330e11 −0.0538546
\(560\) −9.58128e10 −0.0411697
\(561\) 0 0
\(562\) −8.78442e11 −0.371449
\(563\) 3.00188e11 0.125923 0.0629616 0.998016i \(-0.479945\pi\)
0.0629616 + 0.998016i \(0.479945\pi\)
\(564\) −5.01187e11 −0.208566
\(565\) 2.11437e12 0.872899
\(566\) −2.73314e12 −1.11940
\(567\) −8.32249e11 −0.338166
\(568\) −4.18806e12 −1.68828
\(569\) 4.28791e12 1.71491 0.857454 0.514561i \(-0.172045\pi\)
0.857454 + 0.514561i \(0.172045\pi\)
\(570\) 2.51387e12 0.997484
\(571\) 1.67794e11 0.0660562 0.0330281 0.999454i \(-0.489485\pi\)
0.0330281 + 0.999454i \(0.489485\pi\)
\(572\) 3.11620e11 0.121715
\(573\) 4.59143e12 1.77931
\(574\) −9.30860e11 −0.357916
\(575\) −5.45108e11 −0.207959
\(576\) 8.85686e9 0.00335258
\(577\) 7.33555e11 0.275513 0.137756 0.990466i \(-0.456011\pi\)
0.137756 + 0.990466i \(0.456011\pi\)
\(578\) 0 0
\(579\) −3.91165e12 −1.44646
\(580\) −9.76438e11 −0.358277
\(581\) 1.68279e12 0.612684
\(582\) −5.48815e11 −0.198277
\(583\) 4.56097e12 1.63512
\(584\) −3.00781e12 −1.07002
\(585\) 1.74727e9 0.000616820 0
\(586\) −1.31042e12 −0.459064
\(587\) 3.71592e12 1.29180 0.645899 0.763423i \(-0.276483\pi\)
0.645899 + 0.763423i \(0.276483\pi\)
\(588\) −1.46101e12 −0.504028
\(589\) 2.27542e12 0.779008
\(590\) −2.53701e12 −0.861963
\(591\) 3.88384e12 1.30954
\(592\) −2.80316e10 −0.00937996
\(593\) 2.82729e11 0.0938910 0.0469455 0.998897i \(-0.485051\pi\)
0.0469455 + 0.998897i \(0.485051\pi\)
\(594\) 3.44777e12 1.13632
\(595\) 0 0
\(596\) 4.52553e11 0.146913
\(597\) 5.28281e12 1.70208
\(598\) 2.51805e11 0.0805209
\(599\) −5.50565e12 −1.74738 −0.873691 0.486482i \(-0.838280\pi\)
−0.873691 + 0.486482i \(0.838280\pi\)
\(600\) −6.90974e11 −0.217661
\(601\) 1.54272e12 0.482338 0.241169 0.970483i \(-0.422469\pi\)
0.241169 + 0.970483i \(0.422469\pi\)
\(602\) 3.10059e11 0.0962188
\(603\) −1.13435e10 −0.00349396
\(604\) −2.43482e12 −0.744391
\(605\) 7.24005e12 2.19706
\(606\) −2.12884e12 −0.641235
\(607\) 3.59153e12 1.07382 0.536909 0.843640i \(-0.319592\pi\)
0.536909 + 0.843640i \(0.319592\pi\)
\(608\) −4.43767e12 −1.31701
\(609\) −6.57559e11 −0.193712
\(610\) 1.94551e12 0.568916
\(611\) −1.56688e11 −0.0454830
\(612\) 0 0
\(613\) 5.79969e12 1.65895 0.829474 0.558546i \(-0.188641\pi\)
0.829474 + 0.558546i \(0.188641\pi\)
\(614\) −1.86180e12 −0.528658
\(615\) −6.32220e12 −1.78209
\(616\) −2.14732e12 −0.600873
\(617\) −5.14222e12 −1.42846 −0.714229 0.699912i \(-0.753222\pi\)
−0.714229 + 0.699912i \(0.753222\pi\)
\(618\) 2.32842e12 0.642117
\(619\) 1.66025e11 0.0454534 0.0227267 0.999742i \(-0.492765\pi\)
0.0227267 + 0.999742i \(0.492765\pi\)
\(620\) −1.30096e12 −0.353591
\(621\) −3.65068e12 −0.985058
\(622\) 1.84063e12 0.493071
\(623\) −6.06788e11 −0.161377
\(624\) 5.23001e10 0.0138093
\(625\) −4.44895e12 −1.16627
\(626\) −7.33155e11 −0.190815
\(627\) 9.23156e12 2.38546
\(628\) −2.37620e12 −0.609627
\(629\) 0 0
\(630\) −4.35741e9 −0.00110204
\(631\) 2.14564e12 0.538796 0.269398 0.963029i \(-0.413175\pi\)
0.269398 + 0.963029i \(0.413175\pi\)
\(632\) −1.95677e12 −0.487879
\(633\) 5.18382e11 0.128331
\(634\) −4.37731e12 −1.07598
\(635\) −2.51809e12 −0.614595
\(636\) −2.21549e12 −0.536924
\(637\) −4.56759e11 −0.109916
\(638\) 2.73639e12 0.653861
\(639\) −3.12088e10 −0.00740496
\(640\) 2.19574e12 0.517333
\(641\) −6.35807e11 −0.148752 −0.0743762 0.997230i \(-0.523697\pi\)
−0.0743762 + 0.997230i \(0.523697\pi\)
\(642\) −2.45870e12 −0.571213
\(643\) 4.36286e12 1.00652 0.503259 0.864135i \(-0.332134\pi\)
0.503259 + 0.864135i \(0.332134\pi\)
\(644\) 8.22868e11 0.188514
\(645\) 2.10585e12 0.479081
\(646\) 0 0
\(647\) 5.37030e12 1.20484 0.602419 0.798180i \(-0.294204\pi\)
0.602419 + 0.798180i \(0.294204\pi\)
\(648\) −4.64850e12 −1.03568
\(649\) −9.31655e12 −2.06136
\(650\) −7.81798e10 −0.0171785
\(651\) −8.76099e11 −0.191178
\(652\) 2.68280e12 0.581399
\(653\) 1.91038e12 0.411160 0.205580 0.978640i \(-0.434092\pi\)
0.205580 + 0.978640i \(0.434092\pi\)
\(654\) −2.37420e12 −0.507478
\(655\) 7.53380e12 1.59929
\(656\) −8.51928e11 −0.179612
\(657\) −2.24137e10 −0.00469321
\(658\) 3.90754e11 0.0812619
\(659\) −4.74561e11 −0.0980184 −0.0490092 0.998798i \(-0.515606\pi\)
−0.0490092 + 0.998798i \(0.515606\pi\)
\(660\) −5.27810e12 −1.08275
\(661\) −6.60336e12 −1.34542 −0.672710 0.739906i \(-0.734870\pi\)
−0.672710 + 0.739906i \(0.734870\pi\)
\(662\) −4.57186e11 −0.0925192
\(663\) 0 0
\(664\) 9.39915e12 1.87643
\(665\) 2.56829e12 0.509269
\(666\) −1.27483e9 −0.000251084 0
\(667\) −2.89744e12 −0.566825
\(668\) −3.12478e12 −0.607190
\(669\) 1.77874e12 0.343317
\(670\) 2.91720e12 0.559281
\(671\) 7.14438e12 1.36055
\(672\) 1.70863e12 0.323211
\(673\) 2.74504e12 0.515800 0.257900 0.966172i \(-0.416969\pi\)
0.257900 + 0.966172i \(0.416969\pi\)
\(674\) 6.36418e12 1.18788
\(675\) 1.13346e12 0.210154
\(676\) 3.03214e12 0.558455
\(677\) 1.15812e12 0.211887 0.105943 0.994372i \(-0.466214\pi\)
0.105943 + 0.994372i \(0.466214\pi\)
\(678\) 2.87825e12 0.523113
\(679\) −5.60696e11 −0.101231
\(680\) 0 0
\(681\) −6.22527e12 −1.10917
\(682\) 3.64584e12 0.645309
\(683\) −6.11786e12 −1.07574 −0.537869 0.843029i \(-0.680770\pi\)
−0.537869 + 0.843029i \(0.680770\pi\)
\(684\) −2.01874e10 −0.00352637
\(685\) −4.77365e12 −0.828405
\(686\) 2.42377e12 0.417861
\(687\) −9.38758e12 −1.60786
\(688\) 2.83767e11 0.0482853
\(689\) −6.92636e11 −0.117090
\(690\) −4.26497e12 −0.716300
\(691\) −2.38401e12 −0.397793 −0.198897 0.980020i \(-0.563736\pi\)
−0.198897 + 0.980020i \(0.563736\pi\)
\(692\) 4.80297e12 0.796219
\(693\) −1.60015e10 −0.00263549
\(694\) 1.95529e12 0.319958
\(695\) 4.33787e11 0.0705253
\(696\) −3.67277e12 −0.593270
\(697\) 0 0
\(698\) 2.83579e12 0.452194
\(699\) −8.75611e12 −1.38728
\(700\) −2.55482e11 −0.0402179
\(701\) −3.30170e12 −0.516424 −0.258212 0.966088i \(-0.583133\pi\)
−0.258212 + 0.966088i \(0.583133\pi\)
\(702\) −5.23584e11 −0.0813709
\(703\) 7.51398e11 0.116030
\(704\) −8.36436e12 −1.28338
\(705\) 2.65391e12 0.404609
\(706\) 7.75209e11 0.117435
\(707\) −2.17493e12 −0.327385
\(708\) 4.52551e12 0.676890
\(709\) −6.50115e12 −0.966234 −0.483117 0.875556i \(-0.660495\pi\)
−0.483117 + 0.875556i \(0.660495\pi\)
\(710\) 8.02597e12 1.18532
\(711\) −1.45815e10 −0.00213988
\(712\) −3.38919e12 −0.494238
\(713\) −3.86041e12 −0.559411
\(714\) 0 0
\(715\) −1.65011e12 −0.236122
\(716\) −3.50081e12 −0.497805
\(717\) −9.60066e12 −1.35664
\(718\) −9.15875e12 −1.28610
\(719\) 2.37012e12 0.330742 0.165371 0.986231i \(-0.447118\pi\)
0.165371 + 0.986231i \(0.447118\pi\)
\(720\) −3.98792e9 −0.000553032 0
\(721\) 2.37883e12 0.327835
\(722\) −4.27659e12 −0.585706
\(723\) 8.72058e12 1.18692
\(724\) −5.96691e12 −0.807097
\(725\) 8.99592e11 0.120927
\(726\) 9.85573e12 1.31666
\(727\) −6.84668e12 −0.909023 −0.454512 0.890741i \(-0.650186\pi\)
−0.454512 + 0.890741i \(0.650186\pi\)
\(728\) 3.26095e11 0.0430282
\(729\) 7.59080e12 0.995437
\(730\) 5.76414e12 0.751246
\(731\) 0 0
\(732\) −3.47039e12 −0.446764
\(733\) 1.22017e13 1.56118 0.780592 0.625041i \(-0.214918\pi\)
0.780592 + 0.625041i \(0.214918\pi\)
\(734\) −8.33691e12 −1.06016
\(735\) 7.73641e12 0.977792
\(736\) 7.52884e12 0.945753
\(737\) 1.07127e13 1.33750
\(738\) −3.87443e10 −0.00480789
\(739\) 6.41729e12 0.791501 0.395751 0.918358i \(-0.370484\pi\)
0.395751 + 0.918358i \(0.370484\pi\)
\(740\) −4.29608e11 −0.0526659
\(741\) −1.40192e12 −0.170821
\(742\) 1.72732e12 0.209197
\(743\) 1.87182e11 0.0225327 0.0112664 0.999937i \(-0.496414\pi\)
0.0112664 + 0.999937i \(0.496414\pi\)
\(744\) −4.89342e12 −0.585510
\(745\) −2.39638e12 −0.285005
\(746\) 2.98947e11 0.0353402
\(747\) 7.00411e10 0.00823019
\(748\) 0 0
\(749\) −2.51193e12 −0.291635
\(750\) −4.96248e12 −0.572695
\(751\) 1.47873e13 1.69633 0.848163 0.529735i \(-0.177709\pi\)
0.848163 + 0.529735i \(0.177709\pi\)
\(752\) 3.57620e11 0.0407795
\(753\) −1.22711e13 −1.39093
\(754\) −4.15554e11 −0.0468226
\(755\) 1.28930e13 1.44409
\(756\) −1.71101e12 −0.190504
\(757\) 1.41642e13 1.56770 0.783848 0.620952i \(-0.213254\pi\)
0.783848 + 0.620952i \(0.213254\pi\)
\(758\) 3.31017e11 0.0364200
\(759\) −1.56620e13 −1.71301
\(760\) 1.43451e13 1.55971
\(761\) −1.94664e12 −0.210404 −0.105202 0.994451i \(-0.533549\pi\)
−0.105202 + 0.994451i \(0.533549\pi\)
\(762\) −3.42782e12 −0.368316
\(763\) −2.42560e12 −0.259095
\(764\) 9.48215e12 1.00690
\(765\) 0 0
\(766\) −1.57632e12 −0.165430
\(767\) 1.41483e12 0.147613
\(768\) 1.01526e13 1.05306
\(769\) −9.54772e12 −0.984535 −0.492267 0.870444i \(-0.663832\pi\)
−0.492267 + 0.870444i \(0.663832\pi\)
\(770\) 4.11511e12 0.421864
\(771\) 3.26831e12 0.333103
\(772\) −8.07827e12 −0.818542
\(773\) −6.14763e12 −0.619298 −0.309649 0.950851i \(-0.600212\pi\)
−0.309649 + 0.950851i \(0.600212\pi\)
\(774\) 1.29053e10 0.00129251
\(775\) 1.19857e12 0.119346
\(776\) −3.13175e12 −0.310034
\(777\) −2.89309e11 −0.0284753
\(778\) −9.55327e11 −0.0934854
\(779\) 2.28362e13 2.22180
\(780\) 8.01541e11 0.0775354
\(781\) 2.94734e13 2.83465
\(782\) 0 0
\(783\) 6.02473e12 0.572808
\(784\) 1.04250e12 0.0985490
\(785\) 1.25826e13 1.18265
\(786\) 1.02556e13 0.958429
\(787\) 9.50792e12 0.883485 0.441742 0.897142i \(-0.354361\pi\)
0.441742 + 0.897142i \(0.354361\pi\)
\(788\) 8.02085e12 0.741058
\(789\) −3.57977e12 −0.328858
\(790\) 3.74994e12 0.342533
\(791\) 2.94057e12 0.267077
\(792\) −8.93757e10 −0.00807154
\(793\) −1.08496e12 −0.0974280
\(794\) 2.04033e12 0.182183
\(795\) 1.17316e13 1.04161
\(796\) 1.09100e13 0.963196
\(797\) 4.57017e12 0.401208 0.200604 0.979672i \(-0.435710\pi\)
0.200604 + 0.979672i \(0.435710\pi\)
\(798\) 3.49616e12 0.305196
\(799\) 0 0
\(800\) −2.33754e12 −0.201769
\(801\) −2.52557e10 −0.00216777
\(802\) 7.19478e12 0.614091
\(803\) 2.11674e13 1.79658
\(804\) −5.20369e12 −0.439197
\(805\) −4.35730e12 −0.365709
\(806\) −5.53663e11 −0.0462102
\(807\) 2.34172e12 0.194359
\(808\) −1.21480e13 −1.00266
\(809\) 1.08135e13 0.887560 0.443780 0.896136i \(-0.353637\pi\)
0.443780 + 0.896136i \(0.353637\pi\)
\(810\) 8.90836e12 0.727135
\(811\) −1.66684e13 −1.35301 −0.676504 0.736439i \(-0.736506\pi\)
−0.676504 + 0.736439i \(0.736506\pi\)
\(812\) −1.35798e12 −0.109620
\(813\) −1.06217e13 −0.852678
\(814\) 1.20394e12 0.0961162
\(815\) −1.42061e13 −1.12789
\(816\) 0 0
\(817\) −7.60648e12 −0.597289
\(818\) 3.10782e12 0.242698
\(819\) 2.43001e9 0.000188726 0
\(820\) −1.30565e13 −1.00847
\(821\) 7.04533e12 0.541199 0.270600 0.962692i \(-0.412778\pi\)
0.270600 + 0.962692i \(0.412778\pi\)
\(822\) −6.49827e12 −0.496449
\(823\) −4.13230e12 −0.313973 −0.156986 0.987601i \(-0.550178\pi\)
−0.156986 + 0.987601i \(0.550178\pi\)
\(824\) 1.32869e13 1.00404
\(825\) 4.86271e12 0.365457
\(826\) −3.52835e12 −0.263731
\(827\) 3.87796e12 0.288289 0.144145 0.989557i \(-0.453957\pi\)
0.144145 + 0.989557i \(0.453957\pi\)
\(828\) 3.42495e10 0.00253231
\(829\) 2.77616e12 0.204150 0.102075 0.994777i \(-0.467452\pi\)
0.102075 + 0.994777i \(0.467452\pi\)
\(830\) −1.80125e13 −1.31741
\(831\) 4.34109e12 0.315787
\(832\) 1.27023e12 0.0919023
\(833\) 0 0
\(834\) 5.90506e11 0.0422646
\(835\) 1.65465e13 1.17792
\(836\) 1.90649e13 1.34991
\(837\) 8.02705e12 0.565316
\(838\) 2.67934e12 0.187685
\(839\) 1.10624e13 0.770760 0.385380 0.922758i \(-0.374070\pi\)
0.385380 + 0.922758i \(0.374070\pi\)
\(840\) −5.52327e12 −0.382771
\(841\) −9.72549e12 −0.670393
\(842\) −1.85155e13 −1.26950
\(843\) −8.29746e12 −0.565876
\(844\) 1.07055e12 0.0726218
\(845\) −1.60559e13 −1.08338
\(846\) 1.62640e10 0.00109159
\(847\) 1.00691e13 0.672226
\(848\) 1.58086e12 0.104981
\(849\) −2.58163e13 −1.70533
\(850\) 0 0
\(851\) −1.27480e12 −0.0833220
\(852\) −1.43167e13 −0.930818
\(853\) −2.50474e13 −1.61992 −0.809958 0.586488i \(-0.800510\pi\)
−0.809958 + 0.586488i \(0.800510\pi\)
\(854\) 2.70571e12 0.174069
\(855\) 1.06898e11 0.00684101
\(856\) −1.40303e13 −0.893171
\(857\) 2.13270e12 0.135057 0.0675284 0.997717i \(-0.478489\pi\)
0.0675284 + 0.997717i \(0.478489\pi\)
\(858\) −2.24626e12 −0.141503
\(859\) 4.78118e11 0.0299617 0.0149808 0.999888i \(-0.495231\pi\)
0.0149808 + 0.999888i \(0.495231\pi\)
\(860\) 4.34897e12 0.271109
\(861\) −8.79259e12 −0.545259
\(862\) 1.26581e13 0.780882
\(863\) 8.51279e12 0.522424 0.261212 0.965281i \(-0.415878\pi\)
0.261212 + 0.965281i \(0.415878\pi\)
\(864\) −1.56549e13 −0.955737
\(865\) −2.54330e13 −1.54463
\(866\) −1.02890e13 −0.621644
\(867\) 0 0
\(868\) −1.80931e12 −0.108187
\(869\) 1.37707e13 0.819157
\(870\) 7.03848e12 0.416526
\(871\) −1.62685e12 −0.0957779
\(872\) −1.35481e13 −0.793512
\(873\) −2.33373e10 −0.00135984
\(874\) 1.54054e13 0.893039
\(875\) −5.06991e12 −0.292391
\(876\) −1.02821e13 −0.589945
\(877\) −1.88647e13 −1.07684 −0.538421 0.842676i \(-0.680979\pi\)
−0.538421 + 0.842676i \(0.680979\pi\)
\(878\) −2.95571e12 −0.167856
\(879\) −1.23778e13 −0.699350
\(880\) 3.76617e12 0.211703
\(881\) −2.99720e12 −0.167619 −0.0838095 0.996482i \(-0.526709\pi\)
−0.0838095 + 0.996482i \(0.526709\pi\)
\(882\) 4.74110e10 0.00263797
\(883\) −9.35302e12 −0.517760 −0.258880 0.965909i \(-0.583353\pi\)
−0.258880 + 0.965909i \(0.583353\pi\)
\(884\) 0 0
\(885\) −2.39638e13 −1.31314
\(886\) −1.75513e13 −0.956881
\(887\) 1.00237e13 0.543716 0.271858 0.962337i \(-0.412362\pi\)
0.271858 + 0.962337i \(0.412362\pi\)
\(888\) −1.61593e12 −0.0872094
\(889\) −3.50203e12 −0.188045
\(890\) 6.49502e12 0.346997
\(891\) 3.27137e13 1.73892
\(892\) 3.67343e12 0.194281
\(893\) −9.58613e12 −0.504442
\(894\) −3.26215e12 −0.170799
\(895\) 1.85377e13 0.965721
\(896\) 3.05372e12 0.158286
\(897\) 2.37846e12 0.122668
\(898\) 1.15054e12 0.0590418
\(899\) 6.37084e12 0.325296
\(900\) −1.06337e10 −0.000540248 0
\(901\) 0 0
\(902\) 3.65899e13 1.84048
\(903\) 2.92871e12 0.146582
\(904\) 1.64244e13 0.817960
\(905\) 3.15963e13 1.56573
\(906\) 1.75510e13 0.865416
\(907\) −9.97808e12 −0.489569 −0.244785 0.969577i \(-0.578717\pi\)
−0.244785 + 0.969577i \(0.578717\pi\)
\(908\) −1.28563e13 −0.627669
\(909\) −9.05251e10 −0.00439776
\(910\) −6.24927e11 −0.0302095
\(911\) 2.06967e13 0.995562 0.497781 0.867303i \(-0.334148\pi\)
0.497781 + 0.867303i \(0.334148\pi\)
\(912\) 3.19971e12 0.153156
\(913\) −6.61463e13 −3.15056
\(914\) 3.90212e12 0.184945
\(915\) 1.83766e13 0.866702
\(916\) −1.93871e13 −0.909878
\(917\) 1.04776e13 0.489329
\(918\) 0 0
\(919\) 1.75832e13 0.813164 0.406582 0.913614i \(-0.366721\pi\)
0.406582 + 0.913614i \(0.366721\pi\)
\(920\) −2.43375e13 −1.12003
\(921\) −1.75859e13 −0.805373
\(922\) −1.16971e13 −0.533078
\(923\) −4.47588e12 −0.202988
\(924\) −7.34051e12 −0.331286
\(925\) 3.95798e11 0.0177761
\(926\) −1.83405e13 −0.819715
\(927\) 9.90119e10 0.00440381
\(928\) −1.24248e13 −0.549953
\(929\) −2.50697e13 −1.10428 −0.552138 0.833753i \(-0.686188\pi\)
−0.552138 + 0.833753i \(0.686188\pi\)
\(930\) 9.37772e12 0.411078
\(931\) −2.79445e13 −1.21905
\(932\) −1.80830e13 −0.785052
\(933\) 1.73860e13 0.751158
\(934\) −8.28926e11 −0.0356414
\(935\) 0 0
\(936\) 1.35728e10 0.000577998 0
\(937\) −3.59638e13 −1.52418 −0.762091 0.647470i \(-0.775827\pi\)
−0.762091 + 0.647470i \(0.775827\pi\)
\(938\) 4.05710e12 0.171121
\(939\) −6.92514e12 −0.290692
\(940\) 5.48082e12 0.228966
\(941\) −4.41801e13 −1.83685 −0.918425 0.395596i \(-0.870538\pi\)
−0.918425 + 0.395596i \(0.870538\pi\)
\(942\) 1.71284e13 0.708741
\(943\) −3.87434e13 −1.59549
\(944\) −3.22916e12 −0.132348
\(945\) 9.06024e12 0.369570
\(946\) −1.21877e13 −0.494778
\(947\) 1.28335e13 0.518527 0.259263 0.965807i \(-0.416520\pi\)
0.259263 + 0.965807i \(0.416520\pi\)
\(948\) −6.68912e12 −0.268987
\(949\) −3.21451e12 −0.128652
\(950\) −4.78303e12 −0.190523
\(951\) −4.13466e13 −1.63918
\(952\) 0 0
\(953\) −3.92880e13 −1.54291 −0.771457 0.636281i \(-0.780472\pi\)
−0.771457 + 0.636281i \(0.780472\pi\)
\(954\) 7.18947e10 0.00281015
\(955\) −5.02104e13 −1.95334
\(956\) −1.98271e13 −0.767714
\(957\) 2.58471e13 0.996110
\(958\) −2.43115e13 −0.932537
\(959\) −6.63895e12 −0.253464
\(960\) −2.15146e13 −0.817546
\(961\) −1.79514e13 −0.678959
\(962\) −1.82833e11 −0.00688283
\(963\) −1.04552e11 −0.00391753
\(964\) 1.80096e13 0.671673
\(965\) 4.27765e13 1.58794
\(966\) −5.93150e12 −0.219163
\(967\) −1.46165e13 −0.537556 −0.268778 0.963202i \(-0.586620\pi\)
−0.268778 + 0.963202i \(0.586620\pi\)
\(968\) 5.62406e13 2.05878
\(969\) 0 0
\(970\) 6.00167e12 0.217670
\(971\) 2.39472e13 0.864508 0.432254 0.901752i \(-0.357718\pi\)
0.432254 + 0.901752i \(0.357718\pi\)
\(972\) −1.42755e11 −0.00512971
\(973\) 6.03290e11 0.0215783
\(974\) 1.03652e13 0.369029
\(975\) −7.38460e11 −0.0261701
\(976\) 2.47628e12 0.0873526
\(977\) 4.54828e13 1.59706 0.798530 0.601954i \(-0.205611\pi\)
0.798530 + 0.601954i \(0.205611\pi\)
\(978\) −1.93385e13 −0.675923
\(979\) 2.38514e13 0.829833
\(980\) 1.59771e13 0.553326
\(981\) −1.00958e11 −0.00348042
\(982\) −1.09961e12 −0.0377343
\(983\) 3.89910e12 0.133191 0.0665954 0.997780i \(-0.478786\pi\)
0.0665954 + 0.997780i \(0.478786\pi\)
\(984\) −4.91107e13 −1.66993
\(985\) −4.24725e13 −1.43762
\(986\) 0 0
\(987\) 3.69093e12 0.123797
\(988\) −2.89522e12 −0.0966665
\(989\) 1.29050e13 0.428917
\(990\) 1.71279e11 0.00566691
\(991\) 1.86771e13 0.615147 0.307574 0.951524i \(-0.400483\pi\)
0.307574 + 0.951524i \(0.400483\pi\)
\(992\) −1.65543e13 −0.542759
\(993\) −4.31842e12 −0.140946
\(994\) 1.11621e13 0.362667
\(995\) −5.77711e13 −1.86856
\(996\) 3.21306e13 1.03455
\(997\) −4.71408e13 −1.51101 −0.755507 0.655140i \(-0.772610\pi\)
−0.755507 + 0.655140i \(0.772610\pi\)
\(998\) −6.58993e12 −0.210278
\(999\) 2.65073e12 0.0842016
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 289.10.a.i.1.17 52
17.3 odd 16 17.10.d.a.9.5 yes 52
17.6 odd 16 17.10.d.a.2.5 52
17.16 even 2 inner 289.10.a.i.1.18 52
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
17.10.d.a.2.5 52 17.6 odd 16
17.10.d.a.9.5 yes 52 17.3 odd 16
289.10.a.i.1.17 52 1.1 even 1 trivial
289.10.a.i.1.18 52 17.16 even 2 inner