Properties

Label 531.8.a.c.1.16
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(165.876448532\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + 12724944 x^{10} + 71331230512 x^{9} + 50741131904 x^{8} - 7480805165760 x^{7} - 10751966150272 x^{6} + 413177144536320 x^{5} + 886760582981376 x^{4} - 10454479722123264 x^{3} - 29180140031461376 x^{2} + 79787300207378432 x + 248723246810300416\)
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Root \(-20.9857\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+20.9857 q^{2} +312.401 q^{4} +465.631 q^{5} +286.579 q^{7} +3869.79 q^{8} +O(q^{10})\) \(q+20.9857 q^{2} +312.401 q^{4} +465.631 q^{5} +286.579 q^{7} +3869.79 q^{8} +9771.60 q^{10} +3497.66 q^{11} -4677.13 q^{13} +6014.07 q^{14} +41223.0 q^{16} +1899.70 q^{17} +6396.17 q^{19} +145463. q^{20} +73400.9 q^{22} -9375.09 q^{23} +138687. q^{25} -98153.0 q^{26} +89527.5 q^{28} +160070. q^{29} -272299. q^{31} +369763. q^{32} +39866.7 q^{34} +133440. q^{35} -35341.8 q^{37} +134228. q^{38} +1.80189e6 q^{40} +727812. q^{41} -712974. q^{43} +1.09267e6 q^{44} -196743. q^{46} +476612. q^{47} -741416. q^{49} +2.91045e6 q^{50} -1.46114e6 q^{52} +1.49764e6 q^{53} +1.62862e6 q^{55} +1.10900e6 q^{56} +3.35919e6 q^{58} +205379. q^{59} -1.04757e6 q^{61} -5.71439e6 q^{62} +2.48319e6 q^{64} -2.17782e6 q^{65} -2.56238e6 q^{67} +593470. q^{68} +2.80033e6 q^{70} -3.16847e6 q^{71} +1.62960e6 q^{73} -741674. q^{74} +1.99817e6 q^{76} +1.00235e6 q^{77} -2.11177e6 q^{79} +1.91947e7 q^{80} +1.52737e7 q^{82} -5.43782e6 q^{83} +884561. q^{85} -1.49623e7 q^{86} +1.35352e7 q^{88} +2.27504e6 q^{89} -1.34037e6 q^{91} -2.92879e6 q^{92} +1.00021e7 q^{94} +2.97825e6 q^{95} +1.00026e7 q^{97} -1.55591e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 2q^{2} + 1166q^{4} + 318q^{5} + 3145q^{7} - 2355q^{8} + O(q^{10}) \) \( 17q - 2q^{2} + 1166q^{4} + 318q^{5} + 3145q^{7} - 2355q^{8} + 6521q^{10} + 1764q^{11} + 18192q^{13} + 7827q^{14} + 139226q^{16} + 15507q^{17} + 52083q^{19} - 721q^{20} - 234434q^{22} - 63823q^{23} + 202153q^{25} + 367956q^{26} + 182306q^{28} + 502955q^{29} + 347531q^{31} + 243908q^{32} - 330872q^{34} - 92641q^{35} + 447615q^{37} - 775669q^{38} + 2203270q^{40} - 940335q^{41} + 478562q^{43} + 596924q^{44} - 3078663q^{46} - 703121q^{47} + 1895082q^{49} + 876967q^{50} + 6278296q^{52} + 1005974q^{53} + 5212846q^{55} - 3425294q^{56} + 6710166q^{58} + 3491443q^{59} + 11510749q^{61} - 5996234q^{62} + 29496941q^{64} - 11094180q^{65} + 14007144q^{67} - 19688159q^{68} + 30909708q^{70} - 5229074q^{71} + 5452211q^{73} - 12819662q^{74} + 41929340q^{76} - 9930777q^{77} + 15275654q^{79} - 36576105q^{80} + 32025935q^{82} - 7826609q^{83} + 11836945q^{85} - 51649136q^{86} + 30223741q^{88} + 6436185q^{89} + 11633535q^{91} - 43357972q^{92} - 4494252q^{94} - 23741055q^{95} + 26377540q^{97} - 26517816q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 20.9857 1.85489 0.927447 0.373954i \(-0.121998\pi\)
0.927447 + 0.373954i \(0.121998\pi\)
\(3\) 0 0
\(4\) 312.401 2.44063
\(5\) 465.631 1.66589 0.832945 0.553355i \(-0.186653\pi\)
0.832945 + 0.553355i \(0.186653\pi\)
\(6\) 0 0
\(7\) 286.579 0.315792 0.157896 0.987456i \(-0.449529\pi\)
0.157896 + 0.987456i \(0.449529\pi\)
\(8\) 3869.79 2.67222
\(9\) 0 0
\(10\) 9771.60 3.09005
\(11\) 3497.66 0.792324 0.396162 0.918181i \(-0.370342\pi\)
0.396162 + 0.918181i \(0.370342\pi\)
\(12\) 0 0
\(13\) −4677.13 −0.590443 −0.295221 0.955429i \(-0.595393\pi\)
−0.295221 + 0.955429i \(0.595393\pi\)
\(14\) 6014.07 0.585761
\(15\) 0 0
\(16\) 41223.0 2.51605
\(17\) 1899.70 0.0937810 0.0468905 0.998900i \(-0.485069\pi\)
0.0468905 + 0.998900i \(0.485069\pi\)
\(18\) 0 0
\(19\) 6396.17 0.213935 0.106968 0.994263i \(-0.465886\pi\)
0.106968 + 0.994263i \(0.465886\pi\)
\(20\) 145463. 4.06583
\(21\) 0 0
\(22\) 73400.9 1.46968
\(23\) −9375.09 −0.160668 −0.0803338 0.996768i \(-0.525599\pi\)
−0.0803338 + 0.996768i \(0.525599\pi\)
\(24\) 0 0
\(25\) 138687. 1.77519
\(26\) −98153.0 −1.09521
\(27\) 0 0
\(28\) 89527.5 0.770732
\(29\) 160070. 1.21876 0.609379 0.792879i \(-0.291419\pi\)
0.609379 + 0.792879i \(0.291419\pi\)
\(30\) 0 0
\(31\) −272299. −1.64165 −0.820823 0.571183i \(-0.806485\pi\)
−0.820823 + 0.571183i \(0.806485\pi\)
\(32\) 369763. 1.99479
\(33\) 0 0
\(34\) 39866.7 0.173954
\(35\) 133440. 0.526075
\(36\) 0 0
\(37\) −35341.8 −0.114705 −0.0573525 0.998354i \(-0.518266\pi\)
−0.0573525 + 0.998354i \(0.518266\pi\)
\(38\) 134228. 0.396827
\(39\) 0 0
\(40\) 1.80189e6 4.45163
\(41\) 727812. 1.64921 0.824605 0.565709i \(-0.191397\pi\)
0.824605 + 0.565709i \(0.191397\pi\)
\(42\) 0 0
\(43\) −712974. −1.36752 −0.683761 0.729706i \(-0.739657\pi\)
−0.683761 + 0.729706i \(0.739657\pi\)
\(44\) 1.09267e6 1.93377
\(45\) 0 0
\(46\) −196743. −0.298021
\(47\) 476612. 0.669611 0.334805 0.942287i \(-0.391329\pi\)
0.334805 + 0.942287i \(0.391329\pi\)
\(48\) 0 0
\(49\) −741416. −0.900275
\(50\) 2.91045e6 3.29279
\(51\) 0 0
\(52\) −1.46114e6 −1.44105
\(53\) 1.49764e6 1.38179 0.690894 0.722956i \(-0.257217\pi\)
0.690894 + 0.722956i \(0.257217\pi\)
\(54\) 0 0
\(55\) 1.62862e6 1.31993
\(56\) 1.10900e6 0.843866
\(57\) 0 0
\(58\) 3.35919e6 2.26067
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) −1.04757e6 −0.590919 −0.295460 0.955355i \(-0.595473\pi\)
−0.295460 + 0.955355i \(0.595473\pi\)
\(62\) −5.71439e6 −3.04508
\(63\) 0 0
\(64\) 2.48319e6 1.18408
\(65\) −2.17782e6 −0.983613
\(66\) 0 0
\(67\) −2.56238e6 −1.04083 −0.520416 0.853913i \(-0.674223\pi\)
−0.520416 + 0.853913i \(0.674223\pi\)
\(68\) 593470. 0.228885
\(69\) 0 0
\(70\) 2.80033e6 0.975813
\(71\) −3.16847e6 −1.05062 −0.525309 0.850912i \(-0.676050\pi\)
−0.525309 + 0.850912i \(0.676050\pi\)
\(72\) 0 0
\(73\) 1.62960e6 0.490287 0.245144 0.969487i \(-0.421165\pi\)
0.245144 + 0.969487i \(0.421165\pi\)
\(74\) −741674. −0.212766
\(75\) 0 0
\(76\) 1.99817e6 0.522138
\(77\) 1.00235e6 0.250210
\(78\) 0 0
\(79\) −2.11177e6 −0.481893 −0.240947 0.970538i \(-0.577458\pi\)
−0.240947 + 0.970538i \(0.577458\pi\)
\(80\) 1.91947e7 4.19147
\(81\) 0 0
\(82\) 1.52737e7 3.05911
\(83\) −5.43782e6 −1.04388 −0.521941 0.852982i \(-0.674792\pi\)
−0.521941 + 0.852982i \(0.674792\pi\)
\(84\) 0 0
\(85\) 884561. 0.156229
\(86\) −1.49623e7 −2.53661
\(87\) 0 0
\(88\) 1.35352e7 2.11727
\(89\) 2.27504e6 0.342078 0.171039 0.985264i \(-0.445288\pi\)
0.171039 + 0.985264i \(0.445288\pi\)
\(90\) 0 0
\(91\) −1.34037e6 −0.186457
\(92\) −2.92879e6 −0.392130
\(93\) 0 0
\(94\) 1.00021e7 1.24206
\(95\) 2.97825e6 0.356393
\(96\) 0 0
\(97\) 1.00026e7 1.11279 0.556395 0.830918i \(-0.312184\pi\)
0.556395 + 0.830918i \(0.312184\pi\)
\(98\) −1.55591e7 −1.66992
\(99\) 0 0
\(100\) 4.33259e7 4.33259
\(101\) 4.08392e6 0.394414 0.197207 0.980362i \(-0.436813\pi\)
0.197207 + 0.980362i \(0.436813\pi\)
\(102\) 0 0
\(103\) −1.63687e7 −1.47599 −0.737996 0.674805i \(-0.764228\pi\)
−0.737996 + 0.674805i \(0.764228\pi\)
\(104\) −1.80995e7 −1.57779
\(105\) 0 0
\(106\) 3.14290e7 2.56307
\(107\) 1.43241e7 1.13038 0.565189 0.824961i \(-0.308803\pi\)
0.565189 + 0.824961i \(0.308803\pi\)
\(108\) 0 0
\(109\) −666192. −0.0492728 −0.0246364 0.999696i \(-0.507843\pi\)
−0.0246364 + 0.999696i \(0.507843\pi\)
\(110\) 3.41777e7 2.44832
\(111\) 0 0
\(112\) 1.18137e7 0.794550
\(113\) −2.37986e7 −1.55159 −0.775795 0.630986i \(-0.782651\pi\)
−0.775795 + 0.630986i \(0.782651\pi\)
\(114\) 0 0
\(115\) −4.36533e6 −0.267655
\(116\) 5.00060e7 2.97454
\(117\) 0 0
\(118\) 4.31003e6 0.241487
\(119\) 544415. 0.0296153
\(120\) 0 0
\(121\) −7.25356e6 −0.372222
\(122\) −2.19840e7 −1.09609
\(123\) 0 0
\(124\) −8.50663e7 −4.00665
\(125\) 2.81995e7 1.29139
\(126\) 0 0
\(127\) 7.78497e6 0.337244 0.168622 0.985681i \(-0.446068\pi\)
0.168622 + 0.985681i \(0.446068\pi\)
\(128\) 4.78197e6 0.201545
\(129\) 0 0
\(130\) −4.57031e7 −1.82450
\(131\) 2.68679e7 1.04420 0.522101 0.852884i \(-0.325148\pi\)
0.522101 + 0.852884i \(0.325148\pi\)
\(132\) 0 0
\(133\) 1.83301e6 0.0675590
\(134\) −5.37733e7 −1.93063
\(135\) 0 0
\(136\) 7.35146e6 0.250604
\(137\) 5.02442e7 1.66941 0.834707 0.550695i \(-0.185637\pi\)
0.834707 + 0.550695i \(0.185637\pi\)
\(138\) 0 0
\(139\) −5.46930e7 −1.72735 −0.863675 0.504049i \(-0.831843\pi\)
−0.863675 + 0.504049i \(0.831843\pi\)
\(140\) 4.16868e7 1.28396
\(141\) 0 0
\(142\) −6.64926e7 −1.94879
\(143\) −1.63590e7 −0.467822
\(144\) 0 0
\(145\) 7.45335e7 2.03032
\(146\) 3.41983e7 0.909431
\(147\) 0 0
\(148\) −1.10408e7 −0.279953
\(149\) −3.71390e7 −0.919769 −0.459884 0.887979i \(-0.652109\pi\)
−0.459884 + 0.887979i \(0.652109\pi\)
\(150\) 0 0
\(151\) 1.36501e7 0.322639 0.161319 0.986902i \(-0.448425\pi\)
0.161319 + 0.986902i \(0.448425\pi\)
\(152\) 2.47518e7 0.571682
\(153\) 0 0
\(154\) 2.10352e7 0.464112
\(155\) −1.26791e8 −2.73480
\(156\) 0 0
\(157\) 3.77615e7 0.778755 0.389378 0.921078i \(-0.372690\pi\)
0.389378 + 0.921078i \(0.372690\pi\)
\(158\) −4.43169e7 −0.893861
\(159\) 0 0
\(160\) 1.72173e8 3.32311
\(161\) −2.68670e6 −0.0507375
\(162\) 0 0
\(163\) 4.92529e7 0.890790 0.445395 0.895334i \(-0.353063\pi\)
0.445395 + 0.895334i \(0.353063\pi\)
\(164\) 2.27369e8 4.02512
\(165\) 0 0
\(166\) −1.14117e8 −1.93629
\(167\) −1.04186e8 −1.73102 −0.865508 0.500895i \(-0.833004\pi\)
−0.865508 + 0.500895i \(0.833004\pi\)
\(168\) 0 0
\(169\) −4.08730e7 −0.651377
\(170\) 1.85632e7 0.289788
\(171\) 0 0
\(172\) −2.22734e8 −3.33762
\(173\) −9.60679e7 −1.41064 −0.705321 0.708888i \(-0.749197\pi\)
−0.705321 + 0.708888i \(0.749197\pi\)
\(174\) 0 0
\(175\) 3.97447e7 0.560591
\(176\) 1.44184e8 1.99353
\(177\) 0 0
\(178\) 4.77435e7 0.634518
\(179\) 8.54665e7 1.11381 0.556904 0.830577i \(-0.311989\pi\)
0.556904 + 0.830577i \(0.311989\pi\)
\(180\) 0 0
\(181\) 4.14050e7 0.519012 0.259506 0.965741i \(-0.416440\pi\)
0.259506 + 0.965741i \(0.416440\pi\)
\(182\) −2.81286e7 −0.345858
\(183\) 0 0
\(184\) −3.62796e7 −0.429339
\(185\) −1.64562e7 −0.191086
\(186\) 0 0
\(187\) 6.64452e6 0.0743050
\(188\) 1.48894e8 1.63427
\(189\) 0 0
\(190\) 6.25008e7 0.661071
\(191\) 8.00346e7 0.831114 0.415557 0.909567i \(-0.363587\pi\)
0.415557 + 0.909567i \(0.363587\pi\)
\(192\) 0 0
\(193\) 1.36187e8 1.36360 0.681798 0.731540i \(-0.261198\pi\)
0.681798 + 0.731540i \(0.261198\pi\)
\(194\) 2.09913e8 2.06411
\(195\) 0 0
\(196\) −2.31619e8 −2.19724
\(197\) 1.36747e8 1.27434 0.637171 0.770723i \(-0.280105\pi\)
0.637171 + 0.770723i \(0.280105\pi\)
\(198\) 0 0
\(199\) −4.28301e7 −0.385268 −0.192634 0.981271i \(-0.561703\pi\)
−0.192634 + 0.981271i \(0.561703\pi\)
\(200\) 5.36689e8 4.74371
\(201\) 0 0
\(202\) 8.57040e7 0.731596
\(203\) 4.58727e7 0.384874
\(204\) 0 0
\(205\) 3.38892e8 2.74740
\(206\) −3.43509e8 −2.73781
\(207\) 0 0
\(208\) −1.92806e8 −1.48559
\(209\) 2.23716e7 0.169506
\(210\) 0 0
\(211\) −4.75756e7 −0.348655 −0.174327 0.984688i \(-0.555775\pi\)
−0.174327 + 0.984688i \(0.555775\pi\)
\(212\) 4.67864e8 3.37244
\(213\) 0 0
\(214\) 3.00602e8 2.09673
\(215\) −3.31982e8 −2.27814
\(216\) 0 0
\(217\) −7.80350e7 −0.518419
\(218\) −1.39805e7 −0.0913958
\(219\) 0 0
\(220\) 5.08781e8 3.22145
\(221\) −8.88517e6 −0.0553723
\(222\) 0 0
\(223\) 1.14043e8 0.688652 0.344326 0.938850i \(-0.388107\pi\)
0.344326 + 0.938850i \(0.388107\pi\)
\(224\) 1.05966e8 0.629940
\(225\) 0 0
\(226\) −4.99431e8 −2.87803
\(227\) −5.49344e7 −0.311712 −0.155856 0.987780i \(-0.549814\pi\)
−0.155856 + 0.987780i \(0.549814\pi\)
\(228\) 0 0
\(229\) −5.39847e7 −0.297062 −0.148531 0.988908i \(-0.547454\pi\)
−0.148531 + 0.988908i \(0.547454\pi\)
\(230\) −9.16096e7 −0.496471
\(231\) 0 0
\(232\) 6.19437e8 3.25679
\(233\) 2.85964e8 1.48104 0.740518 0.672036i \(-0.234580\pi\)
0.740518 + 0.672036i \(0.234580\pi\)
\(234\) 0 0
\(235\) 2.21925e8 1.11550
\(236\) 6.41606e7 0.317743
\(237\) 0 0
\(238\) 1.14250e7 0.0549332
\(239\) 8.27309e7 0.391990 0.195995 0.980605i \(-0.437206\pi\)
0.195995 + 0.980605i \(0.437206\pi\)
\(240\) 0 0
\(241\) −2.15524e8 −0.991825 −0.495912 0.868373i \(-0.665166\pi\)
−0.495912 + 0.868373i \(0.665166\pi\)
\(242\) −1.52221e8 −0.690433
\(243\) 0 0
\(244\) −3.27262e8 −1.44222
\(245\) −3.45226e8 −1.49976
\(246\) 0 0
\(247\) −2.99157e7 −0.126317
\(248\) −1.05374e9 −4.38684
\(249\) 0 0
\(250\) 5.91786e8 2.39538
\(251\) 3.78187e8 1.50955 0.754776 0.655982i \(-0.227745\pi\)
0.754776 + 0.655982i \(0.227745\pi\)
\(252\) 0 0
\(253\) −3.27909e7 −0.127301
\(254\) 1.63373e8 0.625551
\(255\) 0 0
\(256\) −2.17495e8 −0.810233
\(257\) −2.36018e8 −0.867320 −0.433660 0.901076i \(-0.642778\pi\)
−0.433660 + 0.901076i \(0.642778\pi\)
\(258\) 0 0
\(259\) −1.01282e7 −0.0362229
\(260\) −6.80352e8 −2.40064
\(261\) 0 0
\(262\) 5.63843e8 1.93688
\(263\) 3.99714e8 1.35489 0.677445 0.735573i \(-0.263087\pi\)
0.677445 + 0.735573i \(0.263087\pi\)
\(264\) 0 0
\(265\) 6.97346e8 2.30191
\(266\) 3.84670e7 0.125315
\(267\) 0 0
\(268\) −8.00489e8 −2.54029
\(269\) −3.97589e8 −1.24538 −0.622689 0.782470i \(-0.713960\pi\)
−0.622689 + 0.782470i \(0.713960\pi\)
\(270\) 0 0
\(271\) 5.14541e8 1.57046 0.785232 0.619202i \(-0.212544\pi\)
0.785232 + 0.619202i \(0.212544\pi\)
\(272\) 7.83116e7 0.235958
\(273\) 0 0
\(274\) 1.05441e9 3.09658
\(275\) 4.85079e8 1.40653
\(276\) 0 0
\(277\) −2.94631e8 −0.832911 −0.416455 0.909156i \(-0.636728\pi\)
−0.416455 + 0.909156i \(0.636728\pi\)
\(278\) −1.14777e9 −3.20405
\(279\) 0 0
\(280\) 5.16384e8 1.40579
\(281\) −6.15936e8 −1.65601 −0.828006 0.560719i \(-0.810525\pi\)
−0.828006 + 0.560719i \(0.810525\pi\)
\(282\) 0 0
\(283\) −4.48859e6 −0.0117722 −0.00588609 0.999983i \(-0.501874\pi\)
−0.00588609 + 0.999983i \(0.501874\pi\)
\(284\) −9.89832e8 −2.56417
\(285\) 0 0
\(286\) −3.43306e8 −0.867761
\(287\) 2.08576e8 0.520807
\(288\) 0 0
\(289\) −4.06730e8 −0.991205
\(290\) 1.56414e9 3.76602
\(291\) 0 0
\(292\) 5.09088e8 1.19661
\(293\) −3.57368e8 −0.830002 −0.415001 0.909821i \(-0.636219\pi\)
−0.415001 + 0.909821i \(0.636219\pi\)
\(294\) 0 0
\(295\) 9.56307e7 0.216880
\(296\) −1.36765e8 −0.306517
\(297\) 0 0
\(298\) −7.79390e8 −1.70607
\(299\) 4.38485e7 0.0948650
\(300\) 0 0
\(301\) −2.04323e8 −0.431852
\(302\) 2.86457e8 0.598461
\(303\) 0 0
\(304\) 2.63670e8 0.538273
\(305\) −4.87780e8 −0.984407
\(306\) 0 0
\(307\) −4.17291e8 −0.823105 −0.411552 0.911386i \(-0.635013\pi\)
−0.411552 + 0.911386i \(0.635013\pi\)
\(308\) 3.13137e8 0.610670
\(309\) 0 0
\(310\) −2.66079e9 −5.07277
\(311\) −1.57129e8 −0.296207 −0.148103 0.988972i \(-0.547317\pi\)
−0.148103 + 0.988972i \(0.547317\pi\)
\(312\) 0 0
\(313\) 8.52228e8 1.57091 0.785454 0.618920i \(-0.212430\pi\)
0.785454 + 0.618920i \(0.212430\pi\)
\(314\) 7.92454e8 1.44451
\(315\) 0 0
\(316\) −6.59718e8 −1.17612
\(317\) 7.40371e8 1.30539 0.652697 0.757619i \(-0.273637\pi\)
0.652697 + 0.757619i \(0.273637\pi\)
\(318\) 0 0
\(319\) 5.59870e8 0.965651
\(320\) 1.15625e9 1.97254
\(321\) 0 0
\(322\) −5.63824e7 −0.0941127
\(323\) 1.21508e7 0.0200631
\(324\) 0 0
\(325\) −6.48657e8 −1.04815
\(326\) 1.03361e9 1.65232
\(327\) 0 0
\(328\) 2.81648e9 4.40705
\(329\) 1.36587e8 0.211458
\(330\) 0 0
\(331\) −2.16470e8 −0.328095 −0.164047 0.986452i \(-0.552455\pi\)
−0.164047 + 0.986452i \(0.552455\pi\)
\(332\) −1.69878e9 −2.54773
\(333\) 0 0
\(334\) −2.18642e9 −3.21085
\(335\) −1.19312e9 −1.73391
\(336\) 0 0
\(337\) −1.73965e8 −0.247604 −0.123802 0.992307i \(-0.539509\pi\)
−0.123802 + 0.992307i \(0.539509\pi\)
\(338\) −8.57749e8 −1.20824
\(339\) 0 0
\(340\) 2.76338e8 0.381297
\(341\) −9.52407e8 −1.30072
\(342\) 0 0
\(343\) −4.48484e8 −0.600092
\(344\) −2.75906e9 −3.65432
\(345\) 0 0
\(346\) −2.01606e9 −2.61659
\(347\) 7.38996e7 0.0949486 0.0474743 0.998872i \(-0.484883\pi\)
0.0474743 + 0.998872i \(0.484883\pi\)
\(348\) 0 0
\(349\) 5.44309e8 0.685419 0.342710 0.939441i \(-0.388655\pi\)
0.342710 + 0.939441i \(0.388655\pi\)
\(350\) 8.34072e8 1.03984
\(351\) 0 0
\(352\) 1.29330e9 1.58052
\(353\) −1.58939e9 −1.92317 −0.961587 0.274499i \(-0.911488\pi\)
−0.961587 + 0.274499i \(0.911488\pi\)
\(354\) 0 0
\(355\) −1.47533e9 −1.75021
\(356\) 7.10726e8 0.834886
\(357\) 0 0
\(358\) 1.79358e9 2.06599
\(359\) −1.69409e9 −1.93244 −0.966221 0.257713i \(-0.917031\pi\)
−0.966221 + 0.257713i \(0.917031\pi\)
\(360\) 0 0
\(361\) −8.52961e8 −0.954232
\(362\) 8.68914e8 0.962713
\(363\) 0 0
\(364\) −4.18732e8 −0.455073
\(365\) 7.58791e8 0.816765
\(366\) 0 0
\(367\) −1.50784e9 −1.59230 −0.796151 0.605098i \(-0.793134\pi\)
−0.796151 + 0.605098i \(0.793134\pi\)
\(368\) −3.86470e8 −0.404248
\(369\) 0 0
\(370\) −3.45346e8 −0.354445
\(371\) 4.29192e8 0.436358
\(372\) 0 0
\(373\) −9.09574e8 −0.907522 −0.453761 0.891123i \(-0.649918\pi\)
−0.453761 + 0.891123i \(0.649918\pi\)
\(374\) 1.39440e8 0.137828
\(375\) 0 0
\(376\) 1.84439e9 1.78935
\(377\) −7.48669e8 −0.719606
\(378\) 0 0
\(379\) 1.38013e9 1.30221 0.651106 0.758987i \(-0.274305\pi\)
0.651106 + 0.758987i \(0.274305\pi\)
\(380\) 9.30409e8 0.869824
\(381\) 0 0
\(382\) 1.67958e9 1.54163
\(383\) 1.20160e9 1.09286 0.546430 0.837505i \(-0.315987\pi\)
0.546430 + 0.837505i \(0.315987\pi\)
\(384\) 0 0
\(385\) 4.66727e8 0.416822
\(386\) 2.85799e9 2.52933
\(387\) 0 0
\(388\) 3.12483e9 2.71591
\(389\) −1.11547e9 −0.960802 −0.480401 0.877049i \(-0.659509\pi\)
−0.480401 + 0.877049i \(0.659509\pi\)
\(390\) 0 0
\(391\) −1.78099e7 −0.0150676
\(392\) −2.86912e9 −2.40574
\(393\) 0 0
\(394\) 2.86973e9 2.36377
\(395\) −9.83303e8 −0.802782
\(396\) 0 0
\(397\) −1.21715e9 −0.976286 −0.488143 0.872764i \(-0.662325\pi\)
−0.488143 + 0.872764i \(0.662325\pi\)
\(398\) −8.98820e8 −0.714631
\(399\) 0 0
\(400\) 5.71709e9 4.46648
\(401\) 9.38458e8 0.726791 0.363396 0.931635i \(-0.381617\pi\)
0.363396 + 0.931635i \(0.381617\pi\)
\(402\) 0 0
\(403\) 1.27358e9 0.969298
\(404\) 1.27582e9 0.962620
\(405\) 0 0
\(406\) 9.62672e8 0.713900
\(407\) −1.23614e8 −0.0908836
\(408\) 0 0
\(409\) −2.00800e9 −1.45122 −0.725608 0.688108i \(-0.758441\pi\)
−0.725608 + 0.688108i \(0.758441\pi\)
\(410\) 7.11189e9 5.09614
\(411\) 0 0
\(412\) −5.11360e9 −3.60235
\(413\) 5.88573e7 0.0411126
\(414\) 0 0
\(415\) −2.53201e9 −1.73899
\(416\) −1.72943e9 −1.17781
\(417\) 0 0
\(418\) 4.69485e8 0.314416
\(419\) 2.40847e9 1.59953 0.799765 0.600313i \(-0.204957\pi\)
0.799765 + 0.600313i \(0.204957\pi\)
\(420\) 0 0
\(421\) −5.54976e8 −0.362482 −0.181241 0.983439i \(-0.558011\pi\)
−0.181241 + 0.983439i \(0.558011\pi\)
\(422\) −9.98409e8 −0.646718
\(423\) 0 0
\(424\) 5.79555e9 3.69244
\(425\) 2.63464e8 0.166479
\(426\) 0 0
\(427\) −3.00211e8 −0.186608
\(428\) 4.47486e9 2.75884
\(429\) 0 0
\(430\) −6.96690e9 −4.22571
\(431\) 2.19055e9 1.31790 0.658951 0.752186i \(-0.271001\pi\)
0.658951 + 0.752186i \(0.271001\pi\)
\(432\) 0 0
\(433\) 2.56663e8 0.151934 0.0759672 0.997110i \(-0.475796\pi\)
0.0759672 + 0.997110i \(0.475796\pi\)
\(434\) −1.63762e9 −0.961611
\(435\) 0 0
\(436\) −2.08119e8 −0.120257
\(437\) −5.99647e7 −0.0343725
\(438\) 0 0
\(439\) 2.12343e9 1.19787 0.598937 0.800796i \(-0.295590\pi\)
0.598937 + 0.800796i \(0.295590\pi\)
\(440\) 6.30240e9 3.52713
\(441\) 0 0
\(442\) −1.86462e8 −0.102710
\(443\) −3.50552e9 −1.91575 −0.957875 0.287185i \(-0.907281\pi\)
−0.957875 + 0.287185i \(0.907281\pi\)
\(444\) 0 0
\(445\) 1.05933e9 0.569864
\(446\) 2.39327e9 1.27738
\(447\) 0 0
\(448\) 7.11630e8 0.373922
\(449\) −1.44417e9 −0.752933 −0.376467 0.926430i \(-0.622861\pi\)
−0.376467 + 0.926430i \(0.622861\pi\)
\(450\) 0 0
\(451\) 2.54564e9 1.30671
\(452\) −7.43470e9 −3.78686
\(453\) 0 0
\(454\) −1.15284e9 −0.578193
\(455\) −6.24116e8 −0.310617
\(456\) 0 0
\(457\) −3.55204e7 −0.0174089 −0.00870445 0.999962i \(-0.502771\pi\)
−0.00870445 + 0.999962i \(0.502771\pi\)
\(458\) −1.13291e9 −0.551018
\(459\) 0 0
\(460\) −1.36373e9 −0.653246
\(461\) 1.47773e9 0.702491 0.351245 0.936283i \(-0.385758\pi\)
0.351245 + 0.936283i \(0.385758\pi\)
\(462\) 0 0
\(463\) −1.25958e9 −0.589782 −0.294891 0.955531i \(-0.595283\pi\)
−0.294891 + 0.955531i \(0.595283\pi\)
\(464\) 6.59858e9 3.06646
\(465\) 0 0
\(466\) 6.00117e9 2.74717
\(467\) −1.25254e9 −0.569090 −0.284545 0.958663i \(-0.591843\pi\)
−0.284545 + 0.958663i \(0.591843\pi\)
\(468\) 0 0
\(469\) −7.34323e8 −0.328687
\(470\) 4.65726e9 2.06913
\(471\) 0 0
\(472\) 7.94773e8 0.347894
\(473\) −2.49374e9 −1.08352
\(474\) 0 0
\(475\) 8.87065e8 0.379776
\(476\) 1.70076e8 0.0722800
\(477\) 0 0
\(478\) 1.73617e9 0.727100
\(479\) −3.28427e9 −1.36541 −0.682707 0.730693i \(-0.739197\pi\)
−0.682707 + 0.730693i \(0.739197\pi\)
\(480\) 0 0
\(481\) 1.65298e8 0.0677268
\(482\) −4.52292e9 −1.83973
\(483\) 0 0
\(484\) −2.26602e9 −0.908458
\(485\) 4.65753e9 1.85379
\(486\) 0 0
\(487\) 1.74837e8 0.0685935 0.0342968 0.999412i \(-0.489081\pi\)
0.0342968 + 0.999412i \(0.489081\pi\)
\(488\) −4.05387e9 −1.57907
\(489\) 0 0
\(490\) −7.24482e9 −2.78190
\(491\) 1.05796e9 0.403353 0.201676 0.979452i \(-0.435361\pi\)
0.201676 + 0.979452i \(0.435361\pi\)
\(492\) 0 0
\(493\) 3.04086e8 0.114296
\(494\) −6.27804e8 −0.234304
\(495\) 0 0
\(496\) −1.12250e10 −4.13047
\(497\) −9.08015e8 −0.331777
\(498\) 0 0
\(499\) −7.20840e8 −0.259709 −0.129854 0.991533i \(-0.541451\pi\)
−0.129854 + 0.991533i \(0.541451\pi\)
\(500\) 8.80954e9 3.15180
\(501\) 0 0
\(502\) 7.93653e9 2.80006
\(503\) −5.82049e8 −0.203925 −0.101963 0.994788i \(-0.532512\pi\)
−0.101963 + 0.994788i \(0.532512\pi\)
\(504\) 0 0
\(505\) 1.90160e9 0.657051
\(506\) −6.88140e8 −0.236129
\(507\) 0 0
\(508\) 2.43203e9 0.823088
\(509\) 5.21497e9 1.75283 0.876414 0.481558i \(-0.159929\pi\)
0.876414 + 0.481558i \(0.159929\pi\)
\(510\) 0 0
\(511\) 4.67009e8 0.154829
\(512\) −5.17639e9 −1.70444
\(513\) 0 0
\(514\) −4.95301e9 −1.60879
\(515\) −7.62177e9 −2.45884
\(516\) 0 0
\(517\) 1.66703e9 0.530549
\(518\) −2.12548e8 −0.0671897
\(519\) 0 0
\(520\) −8.42769e9 −2.62843
\(521\) −4.72585e8 −0.146402 −0.0732011 0.997317i \(-0.523321\pi\)
−0.0732011 + 0.997317i \(0.523321\pi\)
\(522\) 0 0
\(523\) 8.79553e8 0.268847 0.134424 0.990924i \(-0.457082\pi\)
0.134424 + 0.990924i \(0.457082\pi\)
\(524\) 8.39357e9 2.54851
\(525\) 0 0
\(526\) 8.38829e9 2.51318
\(527\) −5.17287e8 −0.153955
\(528\) 0 0
\(529\) −3.31693e9 −0.974186
\(530\) 1.46343e10 4.26980
\(531\) 0 0
\(532\) 5.72633e8 0.164887
\(533\) −3.40407e9 −0.973764
\(534\) 0 0
\(535\) 6.66974e9 1.88309
\(536\) −9.91585e9 −2.78134
\(537\) 0 0
\(538\) −8.34369e9 −2.31004
\(539\) −2.59322e9 −0.713310
\(540\) 0 0
\(541\) −1.83033e8 −0.0496981 −0.0248491 0.999691i \(-0.507911\pi\)
−0.0248491 + 0.999691i \(0.507911\pi\)
\(542\) 1.07980e10 2.91304
\(543\) 0 0
\(544\) 7.02440e8 0.187074
\(545\) −3.10200e8 −0.0820830
\(546\) 0 0
\(547\) −2.64278e9 −0.690408 −0.345204 0.938528i \(-0.612190\pi\)
−0.345204 + 0.938528i \(0.612190\pi\)
\(548\) 1.56963e10 4.07442
\(549\) 0 0
\(550\) 1.01797e10 2.60896
\(551\) 1.02384e9 0.260735
\(552\) 0 0
\(553\) −6.05187e8 −0.152178
\(554\) −6.18304e9 −1.54496
\(555\) 0 0
\(556\) −1.70862e10 −4.21583
\(557\) −2.28927e9 −0.561313 −0.280656 0.959808i \(-0.590552\pi\)
−0.280656 + 0.959808i \(0.590552\pi\)
\(558\) 0 0
\(559\) 3.33467e9 0.807443
\(560\) 5.50080e9 1.32363
\(561\) 0 0
\(562\) −1.29259e10 −3.07173
\(563\) −1.07781e9 −0.254543 −0.127271 0.991868i \(-0.540622\pi\)
−0.127271 + 0.991868i \(0.540622\pi\)
\(564\) 0 0
\(565\) −1.10814e10 −2.58478
\(566\) −9.41963e7 −0.0218362
\(567\) 0 0
\(568\) −1.22613e10 −2.80748
\(569\) 1.08728e9 0.247429 0.123714 0.992318i \(-0.460519\pi\)
0.123714 + 0.992318i \(0.460519\pi\)
\(570\) 0 0
\(571\) 2.15499e8 0.0484416 0.0242208 0.999707i \(-0.492290\pi\)
0.0242208 + 0.999707i \(0.492290\pi\)
\(572\) −5.11057e9 −1.14178
\(573\) 0 0
\(574\) 4.37711e9 0.966042
\(575\) −1.30020e9 −0.285216
\(576\) 0 0
\(577\) 7.86457e9 1.70435 0.852176 0.523255i \(-0.175282\pi\)
0.852176 + 0.523255i \(0.175282\pi\)
\(578\) −8.53552e9 −1.83858
\(579\) 0 0
\(580\) 2.32843e10 4.95526
\(581\) −1.55836e9 −0.329649
\(582\) 0 0
\(583\) 5.23823e9 1.09482
\(584\) 6.30620e9 1.31016
\(585\) 0 0
\(586\) −7.49963e9 −1.53957
\(587\) −7.86924e9 −1.60583 −0.802914 0.596094i \(-0.796718\pi\)
−0.802914 + 0.596094i \(0.796718\pi\)
\(588\) 0 0
\(589\) −1.74167e9 −0.351206
\(590\) 2.00688e9 0.402290
\(591\) 0 0
\(592\) −1.45690e9 −0.288604
\(593\) 9.64856e9 1.90008 0.950038 0.312134i \(-0.101044\pi\)
0.950038 + 0.312134i \(0.101044\pi\)
\(594\) 0 0
\(595\) 2.53496e8 0.0493358
\(596\) −1.16023e10 −2.24482
\(597\) 0 0
\(598\) 9.20194e8 0.175965
\(599\) −5.43654e9 −1.03354 −0.516772 0.856123i \(-0.672866\pi\)
−0.516772 + 0.856123i \(0.672866\pi\)
\(600\) 0 0
\(601\) −2.28199e9 −0.428798 −0.214399 0.976746i \(-0.568779\pi\)
−0.214399 + 0.976746i \(0.568779\pi\)
\(602\) −4.28787e9 −0.801040
\(603\) 0 0
\(604\) 4.26431e9 0.787443
\(605\) −3.37748e9 −0.620082
\(606\) 0 0
\(607\) −2.22820e9 −0.404383 −0.202191 0.979346i \(-0.564806\pi\)
−0.202191 + 0.979346i \(0.564806\pi\)
\(608\) 2.36507e9 0.426757
\(609\) 0 0
\(610\) −1.02364e10 −1.82597
\(611\) −2.22918e9 −0.395367
\(612\) 0 0
\(613\) 9.10899e8 0.159720 0.0798599 0.996806i \(-0.474553\pi\)
0.0798599 + 0.996806i \(0.474553\pi\)
\(614\) −8.75716e9 −1.52677
\(615\) 0 0
\(616\) 3.87890e9 0.668615
\(617\) −6.17621e9 −1.05858 −0.529290 0.848441i \(-0.677542\pi\)
−0.529290 + 0.848441i \(0.677542\pi\)
\(618\) 0 0
\(619\) −9.92262e9 −1.68155 −0.840774 0.541387i \(-0.817900\pi\)
−0.840774 + 0.541387i \(0.817900\pi\)
\(620\) −3.96095e10 −6.67465
\(621\) 0 0
\(622\) −3.29747e9 −0.549432
\(623\) 6.51980e8 0.108025
\(624\) 0 0
\(625\) 2.29562e9 0.376114
\(626\) 1.78846e10 2.91387
\(627\) 0 0
\(628\) 1.17967e10 1.90066
\(629\) −6.71390e7 −0.0107572
\(630\) 0 0
\(631\) 3.99548e9 0.633091 0.316546 0.948577i \(-0.397477\pi\)
0.316546 + 0.948577i \(0.397477\pi\)
\(632\) −8.17209e9 −1.28773
\(633\) 0 0
\(634\) 1.55372e10 2.42137
\(635\) 3.62492e9 0.561811
\(636\) 0 0
\(637\) 3.46770e9 0.531561
\(638\) 1.17493e10 1.79118
\(639\) 0 0
\(640\) 2.22663e9 0.335752
\(641\) 4.03744e9 0.605485 0.302742 0.953072i \(-0.402098\pi\)
0.302742 + 0.953072i \(0.402098\pi\)
\(642\) 0 0
\(643\) −8.64628e9 −1.28260 −0.641299 0.767291i \(-0.721604\pi\)
−0.641299 + 0.767291i \(0.721604\pi\)
\(644\) −8.39329e8 −0.123832
\(645\) 0 0
\(646\) 2.54994e8 0.0372149
\(647\) −7.61025e9 −1.10467 −0.552337 0.833621i \(-0.686264\pi\)
−0.552337 + 0.833621i \(0.686264\pi\)
\(648\) 0 0
\(649\) 7.18346e8 0.103152
\(650\) −1.36125e10 −1.94421
\(651\) 0 0
\(652\) 1.53867e10 2.17409
\(653\) 1.04681e10 1.47120 0.735599 0.677417i \(-0.236901\pi\)
0.735599 + 0.677417i \(0.236901\pi\)
\(654\) 0 0
\(655\) 1.25105e10 1.73953
\(656\) 3.00026e10 4.14950
\(657\) 0 0
\(658\) 2.86638e9 0.392232
\(659\) −3.31882e9 −0.451737 −0.225868 0.974158i \(-0.572522\pi\)
−0.225868 + 0.974158i \(0.572522\pi\)
\(660\) 0 0
\(661\) 7.95737e9 1.07168 0.535839 0.844320i \(-0.319995\pi\)
0.535839 + 0.844320i \(0.319995\pi\)
\(662\) −4.54278e9 −0.608581
\(663\) 0 0
\(664\) −2.10432e10 −2.78948
\(665\) 8.53504e8 0.112546
\(666\) 0 0
\(667\) −1.50067e9 −0.195815
\(668\) −3.25478e10 −4.22477
\(669\) 0 0
\(670\) −2.50385e10 −3.21623
\(671\) −3.66404e9 −0.468200
\(672\) 0 0
\(673\) −7.14650e9 −0.903734 −0.451867 0.892085i \(-0.649242\pi\)
−0.451867 + 0.892085i \(0.649242\pi\)
\(674\) −3.65079e9 −0.459279
\(675\) 0 0
\(676\) −1.27687e10 −1.58977
\(677\) 7.35419e9 0.910908 0.455454 0.890259i \(-0.349477\pi\)
0.455454 + 0.890259i \(0.349477\pi\)
\(678\) 0 0
\(679\) 2.86654e9 0.351410
\(680\) 3.42306e9 0.417478
\(681\) 0 0
\(682\) −1.99870e10 −2.41269
\(683\) −4.04823e9 −0.486175 −0.243087 0.970004i \(-0.578160\pi\)
−0.243087 + 0.970004i \(0.578160\pi\)
\(684\) 0 0
\(685\) 2.33952e10 2.78106
\(686\) −9.41177e9 −1.11311
\(687\) 0 0
\(688\) −2.93910e10 −3.44076
\(689\) −7.00465e9 −0.815867
\(690\) 0 0
\(691\) −1.70309e9 −0.196365 −0.0981823 0.995168i \(-0.531303\pi\)
−0.0981823 + 0.995168i \(0.531303\pi\)
\(692\) −3.00117e10 −3.44286
\(693\) 0 0
\(694\) 1.55084e9 0.176120
\(695\) −2.54668e10 −2.87758
\(696\) 0 0
\(697\) 1.38263e9 0.154665
\(698\) 1.14227e10 1.27138
\(699\) 0 0
\(700\) 1.24163e10 1.36820
\(701\) −7.38709e9 −0.809954 −0.404977 0.914327i \(-0.632721\pi\)
−0.404977 + 0.914327i \(0.632721\pi\)
\(702\) 0 0
\(703\) −2.26052e8 −0.0245395
\(704\) 8.68536e9 0.938174
\(705\) 0 0
\(706\) −3.33545e10 −3.56729
\(707\) 1.17037e9 0.124553
\(708\) 0 0
\(709\) −3.00194e9 −0.316330 −0.158165 0.987413i \(-0.550558\pi\)
−0.158165 + 0.987413i \(0.550558\pi\)
\(710\) −3.09610e10 −3.24646
\(711\) 0 0
\(712\) 8.80394e9 0.914108
\(713\) 2.55282e9 0.263759
\(714\) 0 0
\(715\) −7.61726e9 −0.779341
\(716\) 2.66998e10 2.71839
\(717\) 0 0
\(718\) −3.55518e10 −3.58448
\(719\) −3.86807e9 −0.388100 −0.194050 0.980992i \(-0.562162\pi\)
−0.194050 + 0.980992i \(0.562162\pi\)
\(720\) 0 0
\(721\) −4.69092e9 −0.466106
\(722\) −1.79000e10 −1.77000
\(723\) 0 0
\(724\) 1.29350e10 1.26672
\(725\) 2.21996e10 2.16353
\(726\) 0 0
\(727\) 1.88812e10 1.82247 0.911235 0.411887i \(-0.135130\pi\)
0.911235 + 0.411887i \(0.135130\pi\)
\(728\) −5.18694e9 −0.498255
\(729\) 0 0
\(730\) 1.59238e10 1.51501
\(731\) −1.35444e9 −0.128247
\(732\) 0 0
\(733\) −2.89464e9 −0.271476 −0.135738 0.990745i \(-0.543340\pi\)
−0.135738 + 0.990745i \(0.543340\pi\)
\(734\) −3.16432e10 −2.95355
\(735\) 0 0
\(736\) −3.46656e9 −0.320499
\(737\) −8.96231e9 −0.824677
\(738\) 0 0
\(739\) −4.83541e9 −0.440735 −0.220368 0.975417i \(-0.570726\pi\)
−0.220368 + 0.975417i \(0.570726\pi\)
\(740\) −5.14094e9 −0.466371
\(741\) 0 0
\(742\) 9.00690e9 0.809397
\(743\) 1.65827e10 1.48318 0.741591 0.670852i \(-0.234071\pi\)
0.741591 + 0.670852i \(0.234071\pi\)
\(744\) 0 0
\(745\) −1.72931e10 −1.53223
\(746\) −1.90881e10 −1.68336
\(747\) 0 0
\(748\) 2.07575e9 0.181351
\(749\) 4.10498e9 0.356965
\(750\) 0 0
\(751\) 1.94126e10 1.67242 0.836208 0.548413i \(-0.184768\pi\)
0.836208 + 0.548413i \(0.184768\pi\)
\(752\) 1.96474e10 1.68478
\(753\) 0 0
\(754\) −1.57114e10 −1.33479
\(755\) 6.35591e9 0.537481
\(756\) 0 0
\(757\) 1.23613e10 1.03569 0.517843 0.855475i \(-0.326735\pi\)
0.517843 + 0.855475i \(0.326735\pi\)
\(758\) 2.89630e10 2.41547
\(759\) 0 0
\(760\) 1.15252e10 0.952361
\(761\) 1.27885e10 1.05189 0.525947 0.850517i \(-0.323711\pi\)
0.525947 + 0.850517i \(0.323711\pi\)
\(762\) 0 0
\(763\) −1.90917e8 −0.0155599
\(764\) 2.50029e10 2.02845
\(765\) 0 0
\(766\) 2.52165e10 2.02714
\(767\) −9.60585e8 −0.0768691
\(768\) 0 0
\(769\) −6.74352e9 −0.534742 −0.267371 0.963594i \(-0.586155\pi\)
−0.267371 + 0.963594i \(0.586155\pi\)
\(770\) 9.79461e9 0.773160
\(771\) 0 0
\(772\) 4.25450e10 3.32804
\(773\) −3.73822e9 −0.291096 −0.145548 0.989351i \(-0.546495\pi\)
−0.145548 + 0.989351i \(0.546495\pi\)
\(774\) 0 0
\(775\) −3.77642e10 −2.91424
\(776\) 3.87081e10 2.97362
\(777\) 0 0
\(778\) −2.34089e10 −1.78219
\(779\) 4.65521e9 0.352824
\(780\) 0 0
\(781\) −1.10822e10 −0.832430
\(782\) −3.73754e8 −0.0279487
\(783\) 0 0
\(784\) −3.05634e10 −2.26514
\(785\) 1.75829e10 1.29732
\(786\) 0 0
\(787\) −1.47464e10 −1.07839 −0.539193 0.842182i \(-0.681271\pi\)
−0.539193 + 0.842182i \(0.681271\pi\)
\(788\) 4.27199e10 3.11020
\(789\) 0 0
\(790\) −2.06353e10 −1.48907
\(791\) −6.82017e9 −0.489979
\(792\) 0 0
\(793\) 4.89962e9 0.348904
\(794\) −2.55428e10 −1.81091
\(795\) 0 0
\(796\) −1.33802e10 −0.940298
\(797\) 1.86115e10 1.30220 0.651099 0.758993i \(-0.274308\pi\)
0.651099 + 0.758993i \(0.274308\pi\)
\(798\) 0 0
\(799\) 9.05422e8 0.0627968
\(800\) 5.12812e10 3.54114
\(801\) 0 0
\(802\) 1.96942e10 1.34812
\(803\) 5.69978e9 0.388466
\(804\) 0 0
\(805\) −1.25101e9 −0.0845231
\(806\) 2.67269e10 1.79795
\(807\) 0 0
\(808\) 1.58039e10 1.05396
\(809\) −4.53515e9 −0.301143 −0.150571 0.988599i \(-0.548111\pi\)
−0.150571 + 0.988599i \(0.548111\pi\)
\(810\) 0 0
\(811\) 7.48809e9 0.492945 0.246472 0.969150i \(-0.420729\pi\)
0.246472 + 0.969150i \(0.420729\pi\)
\(812\) 1.43307e10 0.939335
\(813\) 0 0
\(814\) −2.59412e9 −0.168579
\(815\) 2.29337e10 1.48396
\(816\) 0 0
\(817\) −4.56030e9 −0.292561
\(818\) −4.21393e10 −2.69185
\(819\) 0 0
\(820\) 1.05870e11 6.70540
\(821\) 1.66654e10 1.05103 0.525514 0.850785i \(-0.323873\pi\)
0.525514 + 0.850785i \(0.323873\pi\)
\(822\) 0 0
\(823\) 1.72880e10 1.08105 0.540524 0.841328i \(-0.318226\pi\)
0.540524 + 0.841328i \(0.318226\pi\)
\(824\) −6.33434e10 −3.94418
\(825\) 0 0
\(826\) 1.23516e9 0.0762595
\(827\) 1.68089e10 1.03341 0.516703 0.856165i \(-0.327159\pi\)
0.516703 + 0.856165i \(0.327159\pi\)
\(828\) 0 0
\(829\) 3.99422e9 0.243495 0.121748 0.992561i \(-0.461150\pi\)
0.121748 + 0.992561i \(0.461150\pi\)
\(830\) −5.31362e10 −3.22565
\(831\) 0 0
\(832\) −1.16142e10 −0.699130
\(833\) −1.40847e9 −0.0844287
\(834\) 0 0
\(835\) −4.85121e10 −2.88368
\(836\) 6.98892e9 0.413702
\(837\) 0 0
\(838\) 5.05435e10 2.96696
\(839\) 1.32657e10 0.775467 0.387734 0.921771i \(-0.373258\pi\)
0.387734 + 0.921771i \(0.373258\pi\)
\(840\) 0 0
\(841\) 8.37255e9 0.485369
\(842\) −1.16466e10 −0.672367
\(843\) 0 0
\(844\) −1.48627e10 −0.850938
\(845\) −1.90317e10 −1.08512
\(846\) 0 0
\(847\) −2.07872e9 −0.117545
\(848\) 6.17372e10 3.47665
\(849\) 0 0
\(850\) 5.52899e9 0.308801
\(851\) 3.31333e8 0.0184294
\(852\) 0 0
\(853\) 3.06500e10 1.69087 0.845433 0.534082i \(-0.179343\pi\)
0.845433 + 0.534082i \(0.179343\pi\)
\(854\) −6.30015e9 −0.346137
\(855\) 0 0
\(856\) 5.54312e10 3.02062
\(857\) 5.05197e9 0.274175 0.137087 0.990559i \(-0.456226\pi\)
0.137087 + 0.990559i \(0.456226\pi\)
\(858\) 0 0
\(859\) −6.74004e7 −0.00362816 −0.00181408 0.999998i \(-0.500577\pi\)
−0.00181408 + 0.999998i \(0.500577\pi\)
\(860\) −1.03712e11 −5.56010
\(861\) 0 0
\(862\) 4.59703e10 2.44457
\(863\) −3.32493e10 −1.76094 −0.880471 0.474101i \(-0.842773\pi\)
−0.880471 + 0.474101i \(0.842773\pi\)
\(864\) 0 0
\(865\) −4.47322e10 −2.34998
\(866\) 5.38627e9 0.281822
\(867\) 0 0
\(868\) −2.43782e10 −1.26527
\(869\) −7.38623e9 −0.381816
\(870\) 0 0
\(871\) 1.19846e10 0.614552
\(872\) −2.57802e9 −0.131668
\(873\) 0 0
\(874\) −1.25840e9 −0.0637573
\(875\) 8.08137e9 0.407809
\(876\) 0 0
\(877\) −3.04224e10 −1.52298 −0.761491 0.648176i \(-0.775532\pi\)
−0.761491 + 0.648176i \(0.775532\pi\)
\(878\) 4.45616e10 2.22193
\(879\) 0 0
\(880\) 6.71365e10 3.32101
\(881\) −1.67047e10 −0.823044 −0.411522 0.911400i \(-0.635003\pi\)
−0.411522 + 0.911400i \(0.635003\pi\)
\(882\) 0 0
\(883\) 4.43529e9 0.216800 0.108400 0.994107i \(-0.465427\pi\)
0.108400 + 0.994107i \(0.465427\pi\)
\(884\) −2.77574e9 −0.135144
\(885\) 0 0
\(886\) −7.35658e10 −3.55351
\(887\) −3.65741e10 −1.75971 −0.879855 0.475242i \(-0.842360\pi\)
−0.879855 + 0.475242i \(0.842360\pi\)
\(888\) 0 0
\(889\) 2.23101e9 0.106499
\(890\) 2.22308e10 1.05704
\(891\) 0 0
\(892\) 3.56270e10 1.68075
\(893\) 3.04849e9 0.143253
\(894\) 0 0
\(895\) 3.97958e10 1.85548
\(896\) 1.37041e9 0.0636463
\(897\) 0 0
\(898\) −3.03070e10 −1.39661
\(899\) −4.35869e10 −2.00077
\(900\) 0 0
\(901\) 2.84507e9 0.129585
\(902\) 5.34221e10 2.42381
\(903\) 0 0
\(904\) −9.20955e10 −4.14619
\(905\) 1.92794e10 0.864618
\(906\) 0 0
\(907\) 1.91299e10 0.851307 0.425654 0.904886i \(-0.360044\pi\)
0.425654 + 0.904886i \(0.360044\pi\)
\(908\) −1.71616e10 −0.760775
\(909\) 0 0
\(910\) −1.30975e10 −0.576162
\(911\) 3.39426e10 1.48741 0.743704 0.668509i \(-0.233067\pi\)
0.743704 + 0.668509i \(0.233067\pi\)
\(912\) 0 0
\(913\) −1.90196e10 −0.827093
\(914\) −7.45422e8 −0.0322917
\(915\) 0 0
\(916\) −1.68649e10 −0.725018
\(917\) 7.69978e9 0.329751
\(918\) 0 0
\(919\) −3.53345e10 −1.50174 −0.750870 0.660450i \(-0.770366\pi\)
−0.750870 + 0.660450i \(0.770366\pi\)
\(920\) −1.68929e10 −0.715232
\(921\) 0 0
\(922\) 3.10111e10 1.30305
\(923\) 1.48193e10 0.620330
\(924\) 0 0
\(925\) −4.90144e9 −0.203624
\(926\) −2.64332e10 −1.09398
\(927\) 0 0
\(928\) 5.91879e10 2.43117
\(929\) 3.23304e10 1.32299 0.661493 0.749951i \(-0.269923\pi\)
0.661493 + 0.749951i \(0.269923\pi\)
\(930\) 0 0
\(931\) −4.74222e9 −0.192601
\(932\) 8.93354e10 3.61467
\(933\) 0 0
\(934\) −2.62854e10 −1.05560
\(935\) 3.09389e9 0.123784
\(936\) 0 0
\(937\) 9.18986e9 0.364939 0.182469 0.983212i \(-0.441591\pi\)
0.182469 + 0.983212i \(0.441591\pi\)
\(938\) −1.54103e10 −0.609679
\(939\) 0 0
\(940\) 6.93296e10 2.72252
\(941\) 1.09465e10 0.428265 0.214133 0.976805i \(-0.431307\pi\)
0.214133 + 0.976805i \(0.431307\pi\)
\(942\) 0 0
\(943\) −6.82331e9 −0.264974
\(944\) 8.46635e9 0.327562
\(945\) 0 0
\(946\) −5.23329e10 −2.00982
\(947\) 2.72786e10 1.04375 0.521875 0.853022i \(-0.325233\pi\)
0.521875 + 0.853022i \(0.325233\pi\)
\(948\) 0 0
\(949\) −7.62185e9 −0.289487
\(950\) 1.86157e10 0.704445
\(951\) 0 0
\(952\) 2.10677e9 0.0791386
\(953\) 2.27671e10 0.852085 0.426043 0.904703i \(-0.359907\pi\)
0.426043 + 0.904703i \(0.359907\pi\)
\(954\) 0 0
\(955\) 3.72665e10 1.38455
\(956\) 2.58452e10 0.956704
\(957\) 0 0
\(958\) −6.89227e10 −2.53270
\(959\) 1.43989e10 0.527187
\(960\) 0 0
\(961\) 4.66339e10 1.69500
\(962\) 3.46891e9 0.125626
\(963\) 0 0
\(964\) −6.73298e10 −2.42068
\(965\) 6.34129e10 2.27160
\(966\) 0 0
\(967\) −1.38243e10 −0.491645 −0.245822 0.969315i \(-0.579058\pi\)
−0.245822 + 0.969315i \(0.579058\pi\)
\(968\) −2.80697e10 −0.994660
\(969\) 0 0
\(970\) 9.77418e10 3.43858
\(971\) −1.62886e10 −0.570973 −0.285487 0.958383i \(-0.592155\pi\)
−0.285487 + 0.958383i \(0.592155\pi\)
\(972\) 0 0
\(973\) −1.56739e10 −0.545483
\(974\) 3.66909e9 0.127234
\(975\) 0 0
\(976\) −4.31840e10 −1.48679
\(977\) −1.93224e10 −0.662874 −0.331437 0.943477i \(-0.607533\pi\)
−0.331437 + 0.943477i \(0.607533\pi\)
\(978\) 0 0
\(979\) 7.95733e9 0.271037
\(980\) −1.07849e11 −3.66036
\(981\) 0 0
\(982\) 2.22021e10 0.748177
\(983\) −3.13862e10 −1.05391 −0.526953 0.849895i \(-0.676666\pi\)
−0.526953 + 0.849895i \(0.676666\pi\)
\(984\) 0 0
\(985\) 6.36735e10 2.12291
\(986\) 6.38146e9 0.212007
\(987\) 0 0
\(988\) −9.34571e9 −0.308292
\(989\) 6.68420e9 0.219716
\(990\) 0 0
\(991\) −2.26508e9 −0.0739308 −0.0369654 0.999317i \(-0.511769\pi\)
−0.0369654 + 0.999317i \(0.511769\pi\)
\(992\) −1.00686e11 −3.27475
\(993\) 0 0
\(994\) −1.90554e10 −0.615411
\(995\) −1.99430e10 −0.641814
\(996\) 0 0
\(997\) 4.99508e10 1.59628 0.798140 0.602472i \(-0.205817\pi\)
0.798140 + 0.602472i \(0.205817\pi\)
\(998\) −1.51273e10 −0.481732
\(999\) 0 0
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 531.8.a.c.1.16 17
3.2 odd 2 177.8.a.c.1.2 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.2 17 3.2 odd 2
531.8.a.c.1.16 17 1.1 even 1 trivial