Properties

Label 387.8.a.d.1.2
Level $387$
Weight $8$
Character 387.1
Self dual yes
Analytic conductor $120.893$
Analytic rank $1$
Dimension $13$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 387 = 3^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 387.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(120.893004862\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Defining polynomial: \(x^{13} - 3 x^{12} - 1279 x^{11} + 3765 x^{10} + 598742 x^{9} - 1518614 x^{8} - 124677082 x^{7} + 193428526 x^{6} + 11160446785 x^{5} + 1754605765 x^{4} - 349352939351 x^{3} - 481872751923 x^{2} + 2098464001560 x + 1551032970660\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 43)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(17.8435\) of defining polynomial
Character \(\chi\) \(=\) 387.1

$q$-expansion

\(f(q)\) \(=\) \(q-18.8435 q^{2} +227.079 q^{4} -70.1157 q^{5} +1065.82 q^{7} -1867.00 q^{8} +O(q^{10})\) \(q-18.8435 q^{2} +227.079 q^{4} -70.1157 q^{5} +1065.82 q^{7} -1867.00 q^{8} +1321.23 q^{10} +1455.37 q^{11} +9741.20 q^{13} -20083.8 q^{14} +6114.77 q^{16} -21354.5 q^{17} +45230.6 q^{19} -15921.8 q^{20} -27424.2 q^{22} +39774.0 q^{23} -73208.8 q^{25} -183559. q^{26} +242025. q^{28} -106837. q^{29} -195235. q^{31} +123752. q^{32} +402394. q^{34} -74730.8 q^{35} -34774.3 q^{37} -852305. q^{38} +130906. q^{40} -217638. q^{41} -79507.0 q^{43} +330483. q^{44} -749483. q^{46} -914679. q^{47} +312430. q^{49} +1.37951e6 q^{50} +2.21202e6 q^{52} -1.04811e6 q^{53} -102044. q^{55} -1.98989e6 q^{56} +2.01318e6 q^{58} -2.62505e6 q^{59} +305720. q^{61} +3.67892e6 q^{62} -3.11462e6 q^{64} -683011. q^{65} -1.51055e6 q^{67} -4.84916e6 q^{68} +1.40819e6 q^{70} +3.03204e6 q^{71} -4.88668e6 q^{73} +655271. q^{74} +1.02709e7 q^{76} +1.55116e6 q^{77} -2.20917e6 q^{79} -428742. q^{80} +4.10107e6 q^{82} +8.61841e6 q^{83} +1.49729e6 q^{85} +1.49819e6 q^{86} -2.71717e6 q^{88} +4.49534e6 q^{89} +1.03824e7 q^{91} +9.03184e6 q^{92} +1.72358e7 q^{94} -3.17138e6 q^{95} +1.11102e7 q^{97} -5.88729e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q - 16q^{2} + 922q^{4} - 998q^{5} + 1360q^{7} - 3870q^{8} + O(q^{10}) \) \( 13q - 16q^{2} + 922q^{4} - 998q^{5} + 1360q^{7} - 3870q^{8} + 4667q^{10} - 1620q^{11} + 13550q^{13} - 44160q^{14} + 114026q^{16} - 110880q^{17} + 105058q^{19} - 167251q^{20} + 201504q^{22} - 160184q^{23} + 270149q^{25} - 272104q^{26} + 208172q^{28} - 285546q^{29} - 99616q^{31} - 200126q^{32} - 80941q^{34} + 187104q^{35} + 176038q^{37} - 652165q^{38} - 895387q^{40} + 410260q^{41} - 1033591q^{43} + 2177076q^{44} - 3975765q^{46} + 424556q^{47} - 1561359q^{49} + 4063801q^{50} - 4172312q^{52} - 3992458q^{53} + 406960q^{55} - 1559556q^{56} - 4052005q^{58} - 2248836q^{59} + 6210394q^{61} - 885317q^{62} - 3096318q^{64} - 5600420q^{65} - 1993648q^{67} - 9327135q^{68} - 1105098q^{70} - 4978064q^{71} + 8224814q^{73} + 3613563q^{74} + 10687121q^{76} - 17261892q^{77} + 6945708q^{79} - 15822799q^{80} - 508449q^{82} - 22937328q^{83} - 575532q^{85} + 1272112q^{86} + 11202656q^{88} - 9291302q^{89} + 25581108q^{91} + 14388137q^{92} - 24645805q^{94} - 30750464q^{95} + 10001852q^{97} + 32304856q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −18.8435 −1.66555 −0.832775 0.553612i \(-0.813249\pi\)
−0.832775 + 0.553612i \(0.813249\pi\)
\(3\) 0 0
\(4\) 227.079 1.77405
\(5\) −70.1157 −0.250854 −0.125427 0.992103i \(-0.540030\pi\)
−0.125427 + 0.992103i \(0.540030\pi\)
\(6\) 0 0
\(7\) 1065.82 1.17447 0.587234 0.809418i \(-0.300217\pi\)
0.587234 + 0.809418i \(0.300217\pi\)
\(8\) −1867.00 −1.28923
\(9\) 0 0
\(10\) 1321.23 0.417809
\(11\) 1455.37 0.329684 0.164842 0.986320i \(-0.447289\pi\)
0.164842 + 0.986320i \(0.447289\pi\)
\(12\) 0 0
\(13\) 9741.20 1.22973 0.614866 0.788631i \(-0.289210\pi\)
0.614866 + 0.788631i \(0.289210\pi\)
\(14\) −20083.8 −1.95613
\(15\) 0 0
\(16\) 6114.77 0.373216
\(17\) −21354.5 −1.05419 −0.527094 0.849807i \(-0.676719\pi\)
−0.527094 + 0.849807i \(0.676719\pi\)
\(18\) 0 0
\(19\) 45230.6 1.51285 0.756424 0.654082i \(-0.226945\pi\)
0.756424 + 0.654082i \(0.226945\pi\)
\(20\) −15921.8 −0.445028
\(21\) 0 0
\(22\) −27424.2 −0.549105
\(23\) 39774.0 0.681635 0.340817 0.940130i \(-0.389296\pi\)
0.340817 + 0.940130i \(0.389296\pi\)
\(24\) 0 0
\(25\) −73208.8 −0.937072
\(26\) −183559. −2.04818
\(27\) 0 0
\(28\) 242025. 2.08357
\(29\) −106837. −0.813443 −0.406721 0.913552i \(-0.633328\pi\)
−0.406721 + 0.913552i \(0.633328\pi\)
\(30\) 0 0
\(31\) −195235. −1.17704 −0.588521 0.808482i \(-0.700290\pi\)
−0.588521 + 0.808482i \(0.700290\pi\)
\(32\) 123752. 0.667617
\(33\) 0 0
\(34\) 402394. 1.75580
\(35\) −74730.8 −0.294619
\(36\) 0 0
\(37\) −34774.3 −0.112863 −0.0564316 0.998406i \(-0.517972\pi\)
−0.0564316 + 0.998406i \(0.517972\pi\)
\(38\) −852305. −2.51972
\(39\) 0 0
\(40\) 130906. 0.323407
\(41\) −217638. −0.493164 −0.246582 0.969122i \(-0.579308\pi\)
−0.246582 + 0.969122i \(0.579308\pi\)
\(42\) 0 0
\(43\) −79507.0 −0.152499
\(44\) 330483. 0.584877
\(45\) 0 0
\(46\) −749483. −1.13530
\(47\) −914679. −1.28507 −0.642534 0.766257i \(-0.722117\pi\)
−0.642534 + 0.766257i \(0.722117\pi\)
\(48\) 0 0
\(49\) 312430. 0.379373
\(50\) 1.37951e6 1.56074
\(51\) 0 0
\(52\) 2.21202e6 2.18161
\(53\) −1.04811e6 −0.967031 −0.483515 0.875336i \(-0.660640\pi\)
−0.483515 + 0.875336i \(0.660640\pi\)
\(54\) 0 0
\(55\) −102044. −0.0827024
\(56\) −1.98989e6 −1.51415
\(57\) 0 0
\(58\) 2.01318e6 1.35483
\(59\) −2.62505e6 −1.66401 −0.832005 0.554768i \(-0.812807\pi\)
−0.832005 + 0.554768i \(0.812807\pi\)
\(60\) 0 0
\(61\) 305720. 0.172452 0.0862262 0.996276i \(-0.472519\pi\)
0.0862262 + 0.996276i \(0.472519\pi\)
\(62\) 3.67892e6 1.96042
\(63\) 0 0
\(64\) −3.11462e6 −1.48517
\(65\) −683011. −0.308483
\(66\) 0 0
\(67\) −1.51055e6 −0.613583 −0.306791 0.951777i \(-0.599255\pi\)
−0.306791 + 0.951777i \(0.599255\pi\)
\(68\) −4.84916e6 −1.87019
\(69\) 0 0
\(70\) 1.40819e6 0.490703
\(71\) 3.03204e6 1.00538 0.502690 0.864467i \(-0.332344\pi\)
0.502690 + 0.864467i \(0.332344\pi\)
\(72\) 0 0
\(73\) −4.88668e6 −1.47023 −0.735113 0.677945i \(-0.762871\pi\)
−0.735113 + 0.677945i \(0.762871\pi\)
\(74\) 655271. 0.187979
\(75\) 0 0
\(76\) 1.02709e7 2.68387
\(77\) 1.55116e6 0.387203
\(78\) 0 0
\(79\) −2.20917e6 −0.504120 −0.252060 0.967712i \(-0.581108\pi\)
−0.252060 + 0.967712i \(0.581108\pi\)
\(80\) −428742. −0.0936226
\(81\) 0 0
\(82\) 4.10107e6 0.821389
\(83\) 8.61841e6 1.65445 0.827225 0.561871i \(-0.189918\pi\)
0.827225 + 0.561871i \(0.189918\pi\)
\(84\) 0 0
\(85\) 1.49729e6 0.264447
\(86\) 1.49819e6 0.253994
\(87\) 0 0
\(88\) −2.71717e6 −0.425037
\(89\) 4.49534e6 0.675923 0.337962 0.941160i \(-0.390263\pi\)
0.337962 + 0.941160i \(0.390263\pi\)
\(90\) 0 0
\(91\) 1.03824e7 1.44428
\(92\) 9.03184e6 1.20926
\(93\) 0 0
\(94\) 1.72358e7 2.14034
\(95\) −3.17138e6 −0.379503
\(96\) 0 0
\(97\) 1.11102e7 1.23601 0.618003 0.786175i \(-0.287942\pi\)
0.618003 + 0.786175i \(0.287942\pi\)
\(98\) −5.88729e6 −0.631865
\(99\) 0 0
\(100\) −1.66242e7 −1.66242
\(101\) 4.64538e6 0.448638 0.224319 0.974516i \(-0.427984\pi\)
0.224319 + 0.974516i \(0.427984\pi\)
\(102\) 0 0
\(103\) −1.71624e6 −0.154756 −0.0773782 0.997002i \(-0.524655\pi\)
−0.0773782 + 0.997002i \(0.524655\pi\)
\(104\) −1.81868e7 −1.58540
\(105\) 0 0
\(106\) 1.97501e7 1.61064
\(107\) 7.01098e6 0.553268 0.276634 0.960975i \(-0.410781\pi\)
0.276634 + 0.960975i \(0.410781\pi\)
\(108\) 0 0
\(109\) 9.93632e6 0.734908 0.367454 0.930042i \(-0.380230\pi\)
0.367454 + 0.930042i \(0.380230\pi\)
\(110\) 1.92287e6 0.137745
\(111\) 0 0
\(112\) 6.51725e6 0.438330
\(113\) 1.02922e7 0.671017 0.335508 0.942037i \(-0.391092\pi\)
0.335508 + 0.942037i \(0.391092\pi\)
\(114\) 0 0
\(115\) −2.78878e6 −0.170990
\(116\) −2.42604e7 −1.44309
\(117\) 0 0
\(118\) 4.94653e7 2.77149
\(119\) −2.27601e7 −1.23811
\(120\) 0 0
\(121\) −1.73691e7 −0.891308
\(122\) −5.76085e6 −0.287228
\(123\) 0 0
\(124\) −4.43338e7 −2.08814
\(125\) 1.06109e7 0.485922
\(126\) 0 0
\(127\) −2.57945e7 −1.11741 −0.558706 0.829365i \(-0.688702\pi\)
−0.558706 + 0.829365i \(0.688702\pi\)
\(128\) 4.28502e7 1.80600
\(129\) 0 0
\(130\) 1.28703e7 0.513793
\(131\) 218805. 0.00850371 0.00425185 0.999991i \(-0.498647\pi\)
0.00425185 + 0.999991i \(0.498647\pi\)
\(132\) 0 0
\(133\) 4.82077e7 1.77679
\(134\) 2.84641e7 1.02195
\(135\) 0 0
\(136\) 3.98689e7 1.35909
\(137\) −1.69620e7 −0.563578 −0.281789 0.959476i \(-0.590928\pi\)
−0.281789 + 0.959476i \(0.590928\pi\)
\(138\) 0 0
\(139\) −4.70595e7 −1.48626 −0.743131 0.669146i \(-0.766660\pi\)
−0.743131 + 0.669146i \(0.766660\pi\)
\(140\) −1.69698e7 −0.522671
\(141\) 0 0
\(142\) −5.71343e7 −1.67451
\(143\) 1.41770e7 0.405423
\(144\) 0 0
\(145\) 7.49092e6 0.204055
\(146\) 9.20824e7 2.44873
\(147\) 0 0
\(148\) −7.89652e6 −0.200226
\(149\) −3.38648e7 −0.838681 −0.419341 0.907829i \(-0.637739\pi\)
−0.419341 + 0.907829i \(0.637739\pi\)
\(150\) 0 0
\(151\) −5.68824e7 −1.34449 −0.672247 0.740327i \(-0.734671\pi\)
−0.672247 + 0.740327i \(0.734671\pi\)
\(152\) −8.44456e7 −1.95040
\(153\) 0 0
\(154\) −2.92293e7 −0.644906
\(155\) 1.36891e7 0.295265
\(156\) 0 0
\(157\) 9.38899e7 1.93629 0.968144 0.250392i \(-0.0805596\pi\)
0.968144 + 0.250392i \(0.0805596\pi\)
\(158\) 4.16285e7 0.839636
\(159\) 0 0
\(160\) −8.67696e6 −0.167474
\(161\) 4.23919e7 0.800557
\(162\) 0 0
\(163\) 4.02174e7 0.727373 0.363686 0.931521i \(-0.381518\pi\)
0.363686 + 0.931521i \(0.381518\pi\)
\(164\) −4.94211e7 −0.874900
\(165\) 0 0
\(166\) −1.62401e8 −2.75557
\(167\) −1.22868e7 −0.204141 −0.102071 0.994777i \(-0.532547\pi\)
−0.102071 + 0.994777i \(0.532547\pi\)
\(168\) 0 0
\(169\) 3.21424e7 0.512242
\(170\) −2.82142e7 −0.440450
\(171\) 0 0
\(172\) −1.80544e7 −0.270541
\(173\) 5.25459e6 0.0771574 0.0385787 0.999256i \(-0.487717\pi\)
0.0385787 + 0.999256i \(0.487717\pi\)
\(174\) 0 0
\(175\) −7.80274e7 −1.10056
\(176\) 8.89923e6 0.123043
\(177\) 0 0
\(178\) −8.47081e7 −1.12578
\(179\) −5.70630e7 −0.743651 −0.371826 0.928303i \(-0.621268\pi\)
−0.371826 + 0.928303i \(0.621268\pi\)
\(180\) 0 0
\(181\) −6.26864e7 −0.785775 −0.392888 0.919586i \(-0.628524\pi\)
−0.392888 + 0.919586i \(0.628524\pi\)
\(182\) −1.95641e8 −2.40552
\(183\) 0 0
\(184\) −7.42580e7 −0.878782
\(185\) 2.43823e6 0.0283121
\(186\) 0 0
\(187\) −3.10786e7 −0.347549
\(188\) −2.07704e8 −2.27978
\(189\) 0 0
\(190\) 5.97600e7 0.632081
\(191\) 7.15929e7 0.743453 0.371726 0.928342i \(-0.378766\pi\)
0.371726 + 0.928342i \(0.378766\pi\)
\(192\) 0 0
\(193\) −6.80335e7 −0.681196 −0.340598 0.940209i \(-0.610630\pi\)
−0.340598 + 0.940209i \(0.610630\pi\)
\(194\) −2.09356e8 −2.05863
\(195\) 0 0
\(196\) 7.09463e7 0.673029
\(197\) −1.26363e8 −1.17758 −0.588788 0.808288i \(-0.700395\pi\)
−0.588788 + 0.808288i \(0.700395\pi\)
\(198\) 0 0
\(199\) −2.86067e7 −0.257325 −0.128662 0.991688i \(-0.541068\pi\)
−0.128662 + 0.991688i \(0.541068\pi\)
\(200\) 1.36681e8 1.20810
\(201\) 0 0
\(202\) −8.75354e7 −0.747229
\(203\) −1.13869e8 −0.955362
\(204\) 0 0
\(205\) 1.52599e7 0.123712
\(206\) 3.23401e7 0.257754
\(207\) 0 0
\(208\) 5.95652e7 0.458956
\(209\) 6.58271e7 0.498761
\(210\) 0 0
\(211\) −1.14904e8 −0.842066 −0.421033 0.907045i \(-0.638332\pi\)
−0.421033 + 0.907045i \(0.638332\pi\)
\(212\) −2.38003e8 −1.71557
\(213\) 0 0
\(214\) −1.32112e8 −0.921495
\(215\) 5.57469e6 0.0382548
\(216\) 0 0
\(217\) −2.08086e8 −1.38240
\(218\) −1.87235e8 −1.22402
\(219\) 0 0
\(220\) −2.31721e7 −0.146719
\(221\) −2.08018e8 −1.29637
\(222\) 0 0
\(223\) −8.16498e7 −0.493047 −0.246523 0.969137i \(-0.579288\pi\)
−0.246523 + 0.969137i \(0.579288\pi\)
\(224\) 1.31897e8 0.784094
\(225\) 0 0
\(226\) −1.93941e8 −1.11761
\(227\) −3.41280e8 −1.93651 −0.968257 0.249958i \(-0.919583\pi\)
−0.968257 + 0.249958i \(0.919583\pi\)
\(228\) 0 0
\(229\) −1.15657e8 −0.636427 −0.318213 0.948019i \(-0.603083\pi\)
−0.318213 + 0.948019i \(0.603083\pi\)
\(230\) 5.25505e7 0.284793
\(231\) 0 0
\(232\) 1.99464e8 1.04871
\(233\) 1.82084e8 0.943029 0.471514 0.881858i \(-0.343708\pi\)
0.471514 + 0.881858i \(0.343708\pi\)
\(234\) 0 0
\(235\) 6.41334e7 0.322364
\(236\) −5.96094e8 −2.95204
\(237\) 0 0
\(238\) 4.28880e8 2.06213
\(239\) 3.58475e6 0.0169850 0.00849252 0.999964i \(-0.497297\pi\)
0.00849252 + 0.999964i \(0.497297\pi\)
\(240\) 0 0
\(241\) 1.50219e8 0.691299 0.345650 0.938364i \(-0.387659\pi\)
0.345650 + 0.938364i \(0.387659\pi\)
\(242\) 3.27295e8 1.48452
\(243\) 0 0
\(244\) 6.94226e7 0.305940
\(245\) −2.19063e7 −0.0951671
\(246\) 0 0
\(247\) 4.40601e8 1.86040
\(248\) 3.64504e8 1.51748
\(249\) 0 0
\(250\) −1.99946e8 −0.809327
\(251\) 2.63281e8 1.05090 0.525450 0.850824i \(-0.323897\pi\)
0.525450 + 0.850824i \(0.323897\pi\)
\(252\) 0 0
\(253\) 5.78857e7 0.224724
\(254\) 4.86059e8 1.86111
\(255\) 0 0
\(256\) −4.08778e8 −1.52282
\(257\) −2.87010e8 −1.05470 −0.527352 0.849647i \(-0.676815\pi\)
−0.527352 + 0.849647i \(0.676815\pi\)
\(258\) 0 0
\(259\) −3.70632e7 −0.132554
\(260\) −1.55098e8 −0.547266
\(261\) 0 0
\(262\) −4.12307e6 −0.0141633
\(263\) −4.14761e8 −1.40589 −0.702947 0.711242i \(-0.748133\pi\)
−0.702947 + 0.711242i \(0.748133\pi\)
\(264\) 0 0
\(265\) 7.34888e7 0.242583
\(266\) −9.08404e8 −2.95933
\(267\) 0 0
\(268\) −3.43014e8 −1.08853
\(269\) 1.11250e8 0.348470 0.174235 0.984704i \(-0.444255\pi\)
0.174235 + 0.984704i \(0.444255\pi\)
\(270\) 0 0
\(271\) 1.01854e8 0.310874 0.155437 0.987846i \(-0.450321\pi\)
0.155437 + 0.987846i \(0.450321\pi\)
\(272\) −1.30578e8 −0.393440
\(273\) 0 0
\(274\) 3.19623e8 0.938667
\(275\) −1.06546e8 −0.308938
\(276\) 0 0
\(277\) 3.01221e8 0.851543 0.425772 0.904831i \(-0.360003\pi\)
0.425772 + 0.904831i \(0.360003\pi\)
\(278\) 8.86767e8 2.47544
\(279\) 0 0
\(280\) 1.39522e8 0.379831
\(281\) −539783. −0.00145127 −0.000725633 1.00000i \(-0.500231\pi\)
−0.000725633 1.00000i \(0.500231\pi\)
\(282\) 0 0
\(283\) 7.75243e7 0.203322 0.101661 0.994819i \(-0.467584\pi\)
0.101661 + 0.994819i \(0.467584\pi\)
\(284\) 6.88512e8 1.78360
\(285\) 0 0
\(286\) −2.67145e8 −0.675252
\(287\) −2.31963e8 −0.579205
\(288\) 0 0
\(289\) 4.56762e7 0.111313
\(290\) −1.41156e8 −0.339864
\(291\) 0 0
\(292\) −1.10966e9 −2.60826
\(293\) 3.93510e8 0.913943 0.456971 0.889481i \(-0.348934\pi\)
0.456971 + 0.889481i \(0.348934\pi\)
\(294\) 0 0
\(295\) 1.84057e8 0.417423
\(296\) 6.49236e7 0.145506
\(297\) 0 0
\(298\) 6.38133e8 1.39687
\(299\) 3.87446e8 0.838228
\(300\) 0 0
\(301\) −8.47402e7 −0.179105
\(302\) 1.07187e9 2.23932
\(303\) 0 0
\(304\) 2.76575e8 0.564619
\(305\) −2.14358e7 −0.0432603
\(306\) 0 0
\(307\) 5.00982e8 0.988184 0.494092 0.869409i \(-0.335501\pi\)
0.494092 + 0.869409i \(0.335501\pi\)
\(308\) 3.52236e8 0.686919
\(309\) 0 0
\(310\) −2.57950e8 −0.491779
\(311\) 4.05811e8 0.765002 0.382501 0.923955i \(-0.375063\pi\)
0.382501 + 0.923955i \(0.375063\pi\)
\(312\) 0 0
\(313\) −8.16758e7 −0.150553 −0.0752763 0.997163i \(-0.523984\pi\)
−0.0752763 + 0.997163i \(0.523984\pi\)
\(314\) −1.76922e9 −3.22499
\(315\) 0 0
\(316\) −5.01655e8 −0.894336
\(317\) 3.74278e8 0.659913 0.329956 0.943996i \(-0.392966\pi\)
0.329956 + 0.943996i \(0.392966\pi\)
\(318\) 0 0
\(319\) −1.55486e8 −0.268179
\(320\) 2.18384e8 0.372559
\(321\) 0 0
\(322\) −7.98814e8 −1.33337
\(323\) −9.65878e8 −1.59483
\(324\) 0 0
\(325\) −7.13141e8 −1.15235
\(326\) −7.57838e8 −1.21148
\(327\) 0 0
\(328\) 4.06330e8 0.635801
\(329\) −9.74883e8 −1.50927
\(330\) 0 0
\(331\) 8.80018e7 0.133381 0.0666904 0.997774i \(-0.478756\pi\)
0.0666904 + 0.997774i \(0.478756\pi\)
\(332\) 1.95706e9 2.93509
\(333\) 0 0
\(334\) 2.31527e8 0.340007
\(335\) 1.05913e8 0.153919
\(336\) 0 0
\(337\) −2.56628e8 −0.365257 −0.182629 0.983182i \(-0.558461\pi\)
−0.182629 + 0.983182i \(0.558461\pi\)
\(338\) −6.05677e8 −0.853165
\(339\) 0 0
\(340\) 3.40002e8 0.469144
\(341\) −2.84139e8 −0.388052
\(342\) 0 0
\(343\) −5.44755e8 −0.728906
\(344\) 1.48440e8 0.196605
\(345\) 0 0
\(346\) −9.90151e7 −0.128510
\(347\) 3.07698e8 0.395341 0.197670 0.980269i \(-0.436662\pi\)
0.197670 + 0.980269i \(0.436662\pi\)
\(348\) 0 0
\(349\) −1.28877e9 −1.62288 −0.811439 0.584437i \(-0.801316\pi\)
−0.811439 + 0.584437i \(0.801316\pi\)
\(350\) 1.47031e9 1.83304
\(351\) 0 0
\(352\) 1.80104e8 0.220103
\(353\) −1.44420e9 −1.74749 −0.873747 0.486380i \(-0.838317\pi\)
−0.873747 + 0.486380i \(0.838317\pi\)
\(354\) 0 0
\(355\) −2.12593e8 −0.252203
\(356\) 1.02080e9 1.19912
\(357\) 0 0
\(358\) 1.07527e9 1.23859
\(359\) −8.83030e8 −1.00727 −0.503634 0.863917i \(-0.668004\pi\)
−0.503634 + 0.863917i \(0.668004\pi\)
\(360\) 0 0
\(361\) 1.15194e9 1.28871
\(362\) 1.18123e9 1.30875
\(363\) 0 0
\(364\) 2.35762e9 2.56223
\(365\) 3.42633e8 0.368811
\(366\) 0 0
\(367\) −1.60796e8 −0.169803 −0.0849015 0.996389i \(-0.527058\pi\)
−0.0849015 + 0.996389i \(0.527058\pi\)
\(368\) 2.43209e8 0.254397
\(369\) 0 0
\(370\) −4.59448e7 −0.0471553
\(371\) −1.11709e9 −1.13575
\(372\) 0 0
\(373\) 1.15384e8 0.115124 0.0575620 0.998342i \(-0.481667\pi\)
0.0575620 + 0.998342i \(0.481667\pi\)
\(374\) 5.85631e8 0.578860
\(375\) 0 0
\(376\) 1.70770e9 1.65674
\(377\) −1.04072e9 −1.00032
\(378\) 0 0
\(379\) 1.76476e9 1.66513 0.832565 0.553928i \(-0.186872\pi\)
0.832565 + 0.553928i \(0.186872\pi\)
\(380\) −7.20153e8 −0.673259
\(381\) 0 0
\(382\) −1.34906e9 −1.23826
\(383\) −2.52128e8 −0.229311 −0.114656 0.993405i \(-0.536576\pi\)
−0.114656 + 0.993405i \(0.536576\pi\)
\(384\) 0 0
\(385\) −1.08761e8 −0.0971313
\(386\) 1.28199e9 1.13457
\(387\) 0 0
\(388\) 2.52289e9 2.19274
\(389\) 1.17424e9 1.01142 0.505710 0.862703i \(-0.331230\pi\)
0.505710 + 0.862703i \(0.331230\pi\)
\(390\) 0 0
\(391\) −8.49354e8 −0.718571
\(392\) −5.83307e8 −0.489098
\(393\) 0 0
\(394\) 2.38113e9 1.96131
\(395\) 1.54897e8 0.126460
\(396\) 0 0
\(397\) −1.20026e9 −0.962738 −0.481369 0.876518i \(-0.659860\pi\)
−0.481369 + 0.876518i \(0.659860\pi\)
\(398\) 5.39051e8 0.428587
\(399\) 0 0
\(400\) −4.47655e8 −0.349730
\(401\) 1.92289e8 0.148919 0.0744593 0.997224i \(-0.476277\pi\)
0.0744593 + 0.997224i \(0.476277\pi\)
\(402\) 0 0
\(403\) −1.90183e9 −1.44745
\(404\) 1.05487e9 0.795909
\(405\) 0 0
\(406\) 2.14569e9 1.59120
\(407\) −5.06093e7 −0.0372092
\(408\) 0 0
\(409\) 4.65392e8 0.336347 0.168173 0.985757i \(-0.446213\pi\)
0.168173 + 0.985757i \(0.446213\pi\)
\(410\) −2.87550e8 −0.206049
\(411\) 0 0
\(412\) −3.89723e8 −0.274546
\(413\) −2.79783e9 −1.95432
\(414\) 0 0
\(415\) −6.04286e8 −0.415025
\(416\) 1.20549e9 0.820990
\(417\) 0 0
\(418\) −1.24042e9 −0.830712
\(419\) 2.24682e9 1.49217 0.746086 0.665850i \(-0.231931\pi\)
0.746086 + 0.665850i \(0.231931\pi\)
\(420\) 0 0
\(421\) 6.62749e8 0.432874 0.216437 0.976297i \(-0.430556\pi\)
0.216437 + 0.976297i \(0.430556\pi\)
\(422\) 2.16520e9 1.40250
\(423\) 0 0
\(424\) 1.95682e9 1.24672
\(425\) 1.56334e9 0.987851
\(426\) 0 0
\(427\) 3.25843e8 0.202540
\(428\) 1.59205e9 0.981528
\(429\) 0 0
\(430\) −1.05047e8 −0.0637153
\(431\) 4.38748e8 0.263964 0.131982 0.991252i \(-0.457866\pi\)
0.131982 + 0.991252i \(0.457866\pi\)
\(432\) 0 0
\(433\) 1.08042e8 0.0639567 0.0319784 0.999489i \(-0.489819\pi\)
0.0319784 + 0.999489i \(0.489819\pi\)
\(434\) 3.92107e9 2.30245
\(435\) 0 0
\(436\) 2.25633e9 1.30377
\(437\) 1.79900e9 1.03121
\(438\) 0 0
\(439\) −2.59454e9 −1.46364 −0.731820 0.681498i \(-0.761329\pi\)
−0.731820 + 0.681498i \(0.761329\pi\)
\(440\) 1.90516e8 0.106622
\(441\) 0 0
\(442\) 3.91980e9 2.15917
\(443\) −1.32379e9 −0.723446 −0.361723 0.932286i \(-0.617811\pi\)
−0.361723 + 0.932286i \(0.617811\pi\)
\(444\) 0 0
\(445\) −3.15194e8 −0.169558
\(446\) 1.53857e9 0.821194
\(447\) 0 0
\(448\) −3.31962e9 −1.74428
\(449\) −1.73729e9 −0.905753 −0.452876 0.891573i \(-0.649602\pi\)
−0.452876 + 0.891573i \(0.649602\pi\)
\(450\) 0 0
\(451\) −3.16743e8 −0.162588
\(452\) 2.33714e9 1.19042
\(453\) 0 0
\(454\) 6.43093e9 3.22536
\(455\) −7.27967e8 −0.362303
\(456\) 0 0
\(457\) −1.37475e9 −0.673776 −0.336888 0.941545i \(-0.609374\pi\)
−0.336888 + 0.941545i \(0.609374\pi\)
\(458\) 2.17939e9 1.06000
\(459\) 0 0
\(460\) −6.33274e8 −0.303347
\(461\) 1.34864e9 0.641124 0.320562 0.947228i \(-0.396128\pi\)
0.320562 + 0.947228i \(0.396128\pi\)
\(462\) 0 0
\(463\) −1.82505e9 −0.854557 −0.427279 0.904120i \(-0.640528\pi\)
−0.427279 + 0.904120i \(0.640528\pi\)
\(464\) −6.53281e8 −0.303590
\(465\) 0 0
\(466\) −3.43110e9 −1.57066
\(467\) −1.36048e9 −0.618135 −0.309067 0.951040i \(-0.600017\pi\)
−0.309067 + 0.951040i \(0.600017\pi\)
\(468\) 0 0
\(469\) −1.60997e9 −0.720633
\(470\) −1.20850e9 −0.536913
\(471\) 0 0
\(472\) 4.90097e9 2.14529
\(473\) −1.15712e8 −0.0502763
\(474\) 0 0
\(475\) −3.31128e9 −1.41765
\(476\) −5.16833e9 −2.19647
\(477\) 0 0
\(478\) −6.75494e7 −0.0282894
\(479\) −3.05531e9 −1.27023 −0.635113 0.772419i \(-0.719046\pi\)
−0.635113 + 0.772419i \(0.719046\pi\)
\(480\) 0 0
\(481\) −3.38743e8 −0.138792
\(482\) −2.83066e9 −1.15139
\(483\) 0 0
\(484\) −3.94415e9 −1.58123
\(485\) −7.79000e8 −0.310057
\(486\) 0 0
\(487\) −3.84229e9 −1.50743 −0.753717 0.657199i \(-0.771741\pi\)
−0.753717 + 0.657199i \(0.771741\pi\)
\(488\) −5.70779e8 −0.222330
\(489\) 0 0
\(490\) 4.12792e8 0.158506
\(491\) 9.60708e7 0.0366274 0.0183137 0.999832i \(-0.494170\pi\)
0.0183137 + 0.999832i \(0.494170\pi\)
\(492\) 0 0
\(493\) 2.28144e9 0.857522
\(494\) −8.30247e9 −3.09858
\(495\) 0 0
\(496\) −1.19382e9 −0.439291
\(497\) 3.23160e9 1.18079
\(498\) 0 0
\(499\) 4.64015e8 0.167178 0.0835892 0.996500i \(-0.473362\pi\)
0.0835892 + 0.996500i \(0.473362\pi\)
\(500\) 2.40951e9 0.862052
\(501\) 0 0
\(502\) −4.96115e9 −1.75033
\(503\) −3.05552e9 −1.07052 −0.535262 0.844686i \(-0.679787\pi\)
−0.535262 + 0.844686i \(0.679787\pi\)
\(504\) 0 0
\(505\) −3.25714e8 −0.112543
\(506\) −1.09077e9 −0.374289
\(507\) 0 0
\(508\) −5.85738e9 −1.98235
\(509\) 2.74667e9 0.923198 0.461599 0.887089i \(-0.347276\pi\)
0.461599 + 0.887089i \(0.347276\pi\)
\(510\) 0 0
\(511\) −5.20832e9 −1.72673
\(512\) 2.21800e9 0.730326
\(513\) 0 0
\(514\) 5.40828e9 1.75666
\(515\) 1.20336e8 0.0388212
\(516\) 0 0
\(517\) −1.33119e9 −0.423666
\(518\) 6.98401e8 0.220775
\(519\) 0 0
\(520\) 1.27518e9 0.397704
\(521\) 3.04048e9 0.941912 0.470956 0.882157i \(-0.343909\pi\)
0.470956 + 0.882157i \(0.343909\pi\)
\(522\) 0 0
\(523\) 3.17164e9 0.969457 0.484729 0.874665i \(-0.338918\pi\)
0.484729 + 0.874665i \(0.338918\pi\)
\(524\) 4.96861e7 0.0150860
\(525\) 0 0
\(526\) 7.81557e9 2.34159
\(527\) 4.16915e9 1.24082
\(528\) 0 0
\(529\) −1.82286e9 −0.535374
\(530\) −1.38479e9 −0.404034
\(531\) 0 0
\(532\) 1.09470e10 3.15212
\(533\) −2.12006e9 −0.606460
\(534\) 0 0
\(535\) −4.91580e8 −0.138789
\(536\) 2.82020e9 0.791047
\(537\) 0 0
\(538\) −2.09634e9 −0.580394
\(539\) 4.54700e8 0.125073
\(540\) 0 0
\(541\) 4.49965e9 1.22177 0.610884 0.791720i \(-0.290814\pi\)
0.610884 + 0.791720i \(0.290814\pi\)
\(542\) −1.91929e9 −0.517777
\(543\) 0 0
\(544\) −2.64266e9 −0.703794
\(545\) −6.96692e8 −0.184354
\(546\) 0 0
\(547\) 4.37317e8 0.114246 0.0571229 0.998367i \(-0.481807\pi\)
0.0571229 + 0.998367i \(0.481807\pi\)
\(548\) −3.85170e9 −0.999818
\(549\) 0 0
\(550\) 2.00770e9 0.514551
\(551\) −4.83229e9 −1.23061
\(552\) 0 0
\(553\) −2.35457e9 −0.592072
\(554\) −5.67608e9 −1.41829
\(555\) 0 0
\(556\) −1.06862e10 −2.63671
\(557\) 7.67789e9 1.88256 0.941280 0.337627i \(-0.109624\pi\)
0.941280 + 0.337627i \(0.109624\pi\)
\(558\) 0 0
\(559\) −7.74493e8 −0.187532
\(560\) −4.56962e8 −0.109957
\(561\) 0 0
\(562\) 1.01714e7 0.00241716
\(563\) −7.00323e8 −0.165394 −0.0826968 0.996575i \(-0.526353\pi\)
−0.0826968 + 0.996575i \(0.526353\pi\)
\(564\) 0 0
\(565\) −7.21644e8 −0.168327
\(566\) −1.46083e9 −0.338643
\(567\) 0 0
\(568\) −5.66081e9 −1.29616
\(569\) −1.86146e9 −0.423605 −0.211803 0.977312i \(-0.567933\pi\)
−0.211803 + 0.977312i \(0.567933\pi\)
\(570\) 0 0
\(571\) 6.78752e9 1.52575 0.762877 0.646544i \(-0.223786\pi\)
0.762877 + 0.646544i \(0.223786\pi\)
\(572\) 3.21930e9 0.719243
\(573\) 0 0
\(574\) 4.37101e9 0.964695
\(575\) −2.91180e9 −0.638741
\(576\) 0 0
\(577\) −4.99855e8 −0.108325 −0.0541625 0.998532i \(-0.517249\pi\)
−0.0541625 + 0.998532i \(0.517249\pi\)
\(578\) −8.60702e8 −0.185398
\(579\) 0 0
\(580\) 1.70103e9 0.362005
\(581\) 9.18567e9 1.94310
\(582\) 0 0
\(583\) −1.52538e9 −0.318814
\(584\) 9.12343e9 1.89545
\(585\) 0 0
\(586\) −7.41512e9 −1.52222
\(587\) −7.26067e9 −1.48164 −0.740821 0.671703i \(-0.765563\pi\)
−0.740821 + 0.671703i \(0.765563\pi\)
\(588\) 0 0
\(589\) −8.83061e9 −1.78069
\(590\) −3.46829e9 −0.695238
\(591\) 0 0
\(592\) −2.12637e8 −0.0421224
\(593\) −6.06525e9 −1.19442 −0.597210 0.802085i \(-0.703724\pi\)
−0.597210 + 0.802085i \(0.703724\pi\)
\(594\) 0 0
\(595\) 1.59584e9 0.310584
\(596\) −7.68999e9 −1.48787
\(597\) 0 0
\(598\) −7.30086e9 −1.39611
\(599\) 5.97141e8 0.113523 0.0567614 0.998388i \(-0.481923\pi\)
0.0567614 + 0.998388i \(0.481923\pi\)
\(600\) 0 0
\(601\) 2.66025e8 0.0499876 0.0249938 0.999688i \(-0.492043\pi\)
0.0249938 + 0.999688i \(0.492043\pi\)
\(602\) 1.59681e9 0.298308
\(603\) 0 0
\(604\) −1.29168e10 −2.38521
\(605\) 1.21785e9 0.223588
\(606\) 0 0
\(607\) 5.53315e9 1.00418 0.502090 0.864815i \(-0.332564\pi\)
0.502090 + 0.864815i \(0.332564\pi\)
\(608\) 5.59738e9 1.01000
\(609\) 0 0
\(610\) 4.03926e8 0.0720522
\(611\) −8.91007e9 −1.58029
\(612\) 0 0
\(613\) 3.58911e9 0.629325 0.314662 0.949204i \(-0.398109\pi\)
0.314662 + 0.949204i \(0.398109\pi\)
\(614\) −9.44028e9 −1.64587
\(615\) 0 0
\(616\) −2.89601e9 −0.499192
\(617\) 3.27338e9 0.561046 0.280523 0.959847i \(-0.409492\pi\)
0.280523 + 0.959847i \(0.409492\pi\)
\(618\) 0 0
\(619\) −9.05095e9 −1.53383 −0.766914 0.641749i \(-0.778209\pi\)
−0.766914 + 0.641749i \(0.778209\pi\)
\(620\) 3.10850e9 0.523817
\(621\) 0 0
\(622\) −7.64691e9 −1.27415
\(623\) 4.79122e9 0.793850
\(624\) 0 0
\(625\) 4.97545e9 0.815177
\(626\) 1.53906e9 0.250753
\(627\) 0 0
\(628\) 2.13204e10 3.43508
\(629\) 7.42588e8 0.118979
\(630\) 0 0
\(631\) −4.70226e9 −0.745082 −0.372541 0.928016i \(-0.621513\pi\)
−0.372541 + 0.928016i \(0.621513\pi\)
\(632\) 4.12451e9 0.649925
\(633\) 0 0
\(634\) −7.05271e9 −1.09912
\(635\) 1.80860e9 0.280307
\(636\) 0 0
\(637\) 3.04344e9 0.466528
\(638\) 2.92991e9 0.446665
\(639\) 0 0
\(640\) −3.00447e9 −0.453041
\(641\) −8.63472e9 −1.29493 −0.647463 0.762097i \(-0.724170\pi\)
−0.647463 + 0.762097i \(0.724170\pi\)
\(642\) 0 0
\(643\) 1.85229e9 0.274770 0.137385 0.990518i \(-0.456130\pi\)
0.137385 + 0.990518i \(0.456130\pi\)
\(644\) 9.62632e9 1.42023
\(645\) 0 0
\(646\) 1.82006e10 2.65626
\(647\) −4.30999e9 −0.625622 −0.312811 0.949815i \(-0.601271\pi\)
−0.312811 + 0.949815i \(0.601271\pi\)
\(648\) 0 0
\(649\) −3.82041e9 −0.548597
\(650\) 1.34381e10 1.91929
\(651\) 0 0
\(652\) 9.13252e9 1.29040
\(653\) 5.51577e9 0.775193 0.387597 0.921829i \(-0.373305\pi\)
0.387597 + 0.921829i \(0.373305\pi\)
\(654\) 0 0
\(655\) −1.53417e7 −0.00213319
\(656\) −1.33081e9 −0.184057
\(657\) 0 0
\(658\) 1.83703e10 2.51376
\(659\) −1.13199e10 −1.54079 −0.770397 0.637564i \(-0.779942\pi\)
−0.770397 + 0.637564i \(0.779942\pi\)
\(660\) 0 0
\(661\) 5.19183e9 0.699222 0.349611 0.936895i \(-0.386314\pi\)
0.349611 + 0.936895i \(0.386314\pi\)
\(662\) −1.65826e9 −0.222152
\(663\) 0 0
\(664\) −1.60906e10 −2.13296
\(665\) −3.38012e9 −0.445714
\(666\) 0 0
\(667\) −4.24932e9 −0.554471
\(668\) −2.79007e9 −0.362158
\(669\) 0 0
\(670\) −1.99578e9 −0.256360
\(671\) 4.44934e8 0.0568548
\(672\) 0 0
\(673\) −7.58270e9 −0.958895 −0.479447 0.877571i \(-0.659163\pi\)
−0.479447 + 0.877571i \(0.659163\pi\)
\(674\) 4.83577e9 0.608354
\(675\) 0 0
\(676\) 7.29887e9 0.908746
\(677\) −4.96130e9 −0.614519 −0.307259 0.951626i \(-0.599412\pi\)
−0.307259 + 0.951626i \(0.599412\pi\)
\(678\) 0 0
\(679\) 1.18415e10 1.45165
\(680\) −2.79543e9 −0.340932
\(681\) 0 0
\(682\) 5.35418e9 0.646320
\(683\) 1.05886e10 1.27165 0.635823 0.771835i \(-0.280661\pi\)
0.635823 + 0.771835i \(0.280661\pi\)
\(684\) 0 0
\(685\) 1.18930e9 0.141376
\(686\) 1.02651e10 1.21403
\(687\) 0 0
\(688\) −4.86167e8 −0.0569149
\(689\) −1.02098e10 −1.18919
\(690\) 0 0
\(691\) −1.26917e10 −1.46335 −0.731674 0.681655i \(-0.761261\pi\)
−0.731674 + 0.681655i \(0.761261\pi\)
\(692\) 1.19321e9 0.136882
\(693\) 0 0
\(694\) −5.79812e9 −0.658460
\(695\) 3.29961e9 0.372834
\(696\) 0 0
\(697\) 4.64756e9 0.519888
\(698\) 2.42850e10 2.70298
\(699\) 0 0
\(700\) −1.77184e10 −1.95246
\(701\) 1.15011e10 1.26104 0.630519 0.776174i \(-0.282842\pi\)
0.630519 + 0.776174i \(0.282842\pi\)
\(702\) 0 0
\(703\) −1.57286e9 −0.170745
\(704\) −4.53291e9 −0.489635
\(705\) 0 0
\(706\) 2.72138e10 2.91054
\(707\) 4.95114e9 0.526911
\(708\) 0 0
\(709\) −1.09458e9 −0.115342 −0.0576709 0.998336i \(-0.518367\pi\)
−0.0576709 + 0.998336i \(0.518367\pi\)
\(710\) 4.00601e9 0.420057
\(711\) 0 0
\(712\) −8.39280e9 −0.871418
\(713\) −7.76528e9 −0.802313
\(714\) 0 0
\(715\) −9.94031e8 −0.101702
\(716\) −1.29578e10 −1.31928
\(717\) 0 0
\(718\) 1.66394e10 1.67765
\(719\) 6.22726e9 0.624807 0.312403 0.949950i \(-0.398866\pi\)
0.312403 + 0.949950i \(0.398866\pi\)
\(720\) 0 0
\(721\) −1.82921e9 −0.181756
\(722\) −2.17066e10 −2.14640
\(723\) 0 0
\(724\) −1.42348e10 −1.39401
\(725\) 7.82138e9 0.762255
\(726\) 0 0
\(727\) 1.91377e10 1.84722 0.923611 0.383332i \(-0.125223\pi\)
0.923611 + 0.383332i \(0.125223\pi\)
\(728\) −1.93839e10 −1.86201
\(729\) 0 0
\(730\) −6.45642e9 −0.614273
\(731\) 1.69783e9 0.160762
\(732\) 0 0
\(733\) 9.03821e9 0.847654 0.423827 0.905743i \(-0.360686\pi\)
0.423827 + 0.905743i \(0.360686\pi\)
\(734\) 3.02997e9 0.282815
\(735\) 0 0
\(736\) 4.92211e9 0.455071
\(737\) −2.19840e9 −0.202288
\(738\) 0 0
\(739\) −1.24567e9 −0.113540 −0.0567698 0.998387i \(-0.518080\pi\)
−0.0567698 + 0.998387i \(0.518080\pi\)
\(740\) 5.53670e8 0.0502273
\(741\) 0 0
\(742\) 2.10500e10 1.89164
\(743\) −1.95885e10 −1.75203 −0.876013 0.482287i \(-0.839806\pi\)
−0.876013 + 0.482287i \(0.839806\pi\)
\(744\) 0 0
\(745\) 2.37446e9 0.210386
\(746\) −2.17425e9 −0.191745
\(747\) 0 0
\(748\) −7.05730e9 −0.616571
\(749\) 7.47244e9 0.649795
\(750\) 0 0
\(751\) 1.10559e10 0.952473 0.476236 0.879317i \(-0.342001\pi\)
0.476236 + 0.879317i \(0.342001\pi\)
\(752\) −5.59305e9 −0.479608
\(753\) 0 0
\(754\) 1.96108e10 1.66608
\(755\) 3.98835e9 0.337271
\(756\) 0 0
\(757\) −1.11431e10 −0.933617 −0.466808 0.884358i \(-0.654596\pi\)
−0.466808 + 0.884358i \(0.654596\pi\)
\(758\) −3.32543e10 −2.77335
\(759\) 0 0
\(760\) 5.92096e9 0.489266
\(761\) 1.64914e10 1.35647 0.678236 0.734844i \(-0.262745\pi\)
0.678236 + 0.734844i \(0.262745\pi\)
\(762\) 0 0
\(763\) 1.05903e10 0.863125
\(764\) 1.62573e10 1.31893
\(765\) 0 0
\(766\) 4.75099e9 0.381929
\(767\) −2.55712e10 −2.04629
\(768\) 0 0
\(769\) −2.19764e10 −1.74266 −0.871332 0.490694i \(-0.836743\pi\)
−0.871332 + 0.490694i \(0.836743\pi\)
\(770\) 2.04943e9 0.161777
\(771\) 0 0
\(772\) −1.54490e10 −1.20848
\(773\) −1.65022e10 −1.28503 −0.642517 0.766271i \(-0.722110\pi\)
−0.642517 + 0.766271i \(0.722110\pi\)
\(774\) 0 0
\(775\) 1.42929e10 1.10297
\(776\) −2.07427e10 −1.59349
\(777\) 0 0
\(778\) −2.21268e10 −1.68457
\(779\) −9.84391e9 −0.746082
\(780\) 0 0
\(781\) 4.41272e9 0.331457
\(782\) 1.60048e10 1.19682
\(783\) 0 0
\(784\) 1.91044e9 0.141588
\(785\) −6.58316e9 −0.485725
\(786\) 0 0
\(787\) 1.33119e10 0.973483 0.486741 0.873546i \(-0.338185\pi\)
0.486741 + 0.873546i \(0.338185\pi\)
\(788\) −2.86944e10 −2.08908
\(789\) 0 0
\(790\) −2.91881e9 −0.210626
\(791\) 1.09696e10 0.788087
\(792\) 0 0
\(793\) 2.97808e9 0.212070
\(794\) 2.26171e10 1.60349
\(795\) 0 0
\(796\) −6.49597e9 −0.456508
\(797\) −6.86024e9 −0.479993 −0.239997 0.970774i \(-0.577146\pi\)
−0.239997 + 0.970774i \(0.577146\pi\)
\(798\) 0 0
\(799\) 1.95325e10 1.35470
\(800\) −9.05974e9 −0.625606
\(801\) 0 0
\(802\) −3.62340e9 −0.248031
\(803\) −7.11191e9 −0.484710
\(804\) 0 0
\(805\) −2.97234e9 −0.200823
\(806\) 3.58371e10 2.41080
\(807\) 0 0
\(808\) −8.67292e9 −0.578396
\(809\) −7.51249e9 −0.498843 −0.249422 0.968395i \(-0.580240\pi\)
−0.249422 + 0.968395i \(0.580240\pi\)
\(810\) 0 0
\(811\) 1.36176e10 0.896453 0.448226 0.893920i \(-0.352056\pi\)
0.448226 + 0.893920i \(0.352056\pi\)
\(812\) −2.58572e10 −1.69486
\(813\) 0 0
\(814\) 9.53659e8 0.0619737
\(815\) −2.81987e9 −0.182464
\(816\) 0 0
\(817\) −3.59615e9 −0.230707
\(818\) −8.76964e9 −0.560202
\(819\) 0 0
\(820\) 3.46519e9 0.219472
\(821\) 8.07643e8 0.0509353 0.0254676 0.999676i \(-0.491893\pi\)
0.0254676 + 0.999676i \(0.491893\pi\)
\(822\) 0 0
\(823\) −2.23299e10 −1.39633 −0.698163 0.715938i \(-0.745999\pi\)
−0.698163 + 0.715938i \(0.745999\pi\)
\(824\) 3.20423e9 0.199516
\(825\) 0 0
\(826\) 5.27211e10 3.25502
\(827\) −1.07538e9 −0.0661136 −0.0330568 0.999453i \(-0.510524\pi\)
−0.0330568 + 0.999453i \(0.510524\pi\)
\(828\) 0 0
\(829\) 1.30676e10 0.796628 0.398314 0.917249i \(-0.369595\pi\)
0.398314 + 0.917249i \(0.369595\pi\)
\(830\) 1.13869e10 0.691244
\(831\) 0 0
\(832\) −3.03401e10 −1.82636
\(833\) −6.67179e9 −0.399931
\(834\) 0 0
\(835\) 8.61497e8 0.0512096
\(836\) 1.49480e10 0.884830
\(837\) 0 0
\(838\) −4.23380e10 −2.48529
\(839\) 7.30739e9 0.427165 0.213582 0.976925i \(-0.431487\pi\)
0.213582 + 0.976925i \(0.431487\pi\)
\(840\) 0 0
\(841\) −5.83582e9 −0.338311
\(842\) −1.24885e10 −0.720974
\(843\) 0 0
\(844\) −2.60923e10 −1.49387
\(845\) −2.25369e9 −0.128498
\(846\) 0 0
\(847\) −1.85123e10 −1.04681
\(848\) −6.40894e9 −0.360911
\(849\) 0 0
\(850\) −2.94588e10 −1.64531
\(851\) −1.38311e9 −0.0769315
\(852\) 0 0
\(853\) −2.19746e10 −1.21227 −0.606134 0.795363i \(-0.707280\pi\)
−0.606134 + 0.795363i \(0.707280\pi\)
\(854\) −6.14003e9 −0.337340
\(855\) 0 0
\(856\) −1.30895e10 −0.713288
\(857\) 4.09566e9 0.222275 0.111138 0.993805i \(-0.464551\pi\)
0.111138 + 0.993805i \(0.464551\pi\)
\(858\) 0 0
\(859\) −2.48200e10 −1.33606 −0.668031 0.744134i \(-0.732863\pi\)
−0.668031 + 0.744134i \(0.732863\pi\)
\(860\) 1.26590e9 0.0678661
\(861\) 0 0
\(862\) −8.26757e9 −0.439645
\(863\) 9.32141e9 0.493678 0.246839 0.969056i \(-0.420608\pi\)
0.246839 + 0.969056i \(0.420608\pi\)
\(864\) 0 0
\(865\) −3.68430e8 −0.0193552
\(866\) −2.03590e9 −0.106523
\(867\) 0 0
\(868\) −4.72519e10 −2.45245
\(869\) −3.21515e9 −0.166200
\(870\) 0 0
\(871\) −1.47146e10 −0.754543
\(872\) −1.85511e10 −0.947463
\(873\) 0 0
\(874\) −3.38996e10 −1.71753
\(875\) 1.13093e10 0.570699
\(876\) 0 0
\(877\) −1.06318e10 −0.532239 −0.266120 0.963940i \(-0.585742\pi\)
−0.266120 + 0.963940i \(0.585742\pi\)
\(878\) 4.88903e10 2.43777
\(879\) 0 0
\(880\) −6.23976e8 −0.0308659
\(881\) −8.99181e9 −0.443028 −0.221514 0.975157i \(-0.571100\pi\)
−0.221514 + 0.975157i \(0.571100\pi\)
\(882\) 0 0
\(883\) 7.28785e9 0.356235 0.178118 0.984009i \(-0.442999\pi\)
0.178118 + 0.984009i \(0.442999\pi\)
\(884\) −4.72366e10 −2.29983
\(885\) 0 0
\(886\) 2.49449e10 1.20494
\(887\) −2.86600e10 −1.37893 −0.689467 0.724317i \(-0.742155\pi\)
−0.689467 + 0.724317i \(0.742155\pi\)
\(888\) 0 0
\(889\) −2.74923e10 −1.31236
\(890\) 5.93937e9 0.282407
\(891\) 0 0
\(892\) −1.85410e10 −0.874692
\(893\) −4.13715e10 −1.94411
\(894\) 0 0
\(895\) 4.00102e9 0.186548
\(896\) 4.56706e10 2.12109
\(897\) 0 0
\(898\) 3.27367e10 1.50858
\(899\) 2.08583e10 0.957457
\(900\) 0 0
\(901\) 2.23818e10 1.01943
\(902\) 5.96856e9 0.270799
\(903\) 0 0
\(904\) −1.92155e10 −0.865092
\(905\) 4.39530e9 0.197115
\(906\) 0 0
\(907\) 2.38033e9 0.105928 0.0529642 0.998596i \(-0.483133\pi\)
0.0529642 + 0.998596i \(0.483133\pi\)
\(908\) −7.74976e10 −3.43548
\(909\) 0 0
\(910\) 1.37175e10 0.603434
\(911\) 3.27857e10 1.43671 0.718356 0.695676i \(-0.244895\pi\)
0.718356 + 0.695676i \(0.244895\pi\)
\(912\) 0 0
\(913\) 1.25429e10 0.545446
\(914\) 2.59051e10 1.12221
\(915\) 0 0
\(916\) −2.62633e10 −1.12906
\(917\) 2.33207e8 0.00998732
\(918\) 0 0
\(919\) 1.71738e10 0.729899 0.364950 0.931027i \(-0.381086\pi\)
0.364950 + 0.931027i \(0.381086\pi\)
\(920\) 5.20665e9 0.220446
\(921\) 0 0
\(922\) −2.54131e10 −1.06782
\(923\) 2.95357e10 1.23635
\(924\) 0 0
\(925\) 2.54578e9 0.105761
\(926\) 3.43904e10 1.42331
\(927\) 0 0
\(928\) −1.32212e10 −0.543068
\(929\) 4.09833e9 0.167707 0.0838536 0.996478i \(-0.473277\pi\)
0.0838536 + 0.996478i \(0.473277\pi\)
\(930\) 0 0
\(931\) 1.41314e10 0.573934
\(932\) 4.13474e10 1.67299
\(933\) 0 0
\(934\) 2.56363e10 1.02953
\(935\) 2.17910e9 0.0871839
\(936\) 0 0
\(937\) 3.03563e10 1.20548 0.602740 0.797938i \(-0.294076\pi\)
0.602740 + 0.797938i \(0.294076\pi\)
\(938\) 3.03376e10 1.20025
\(939\) 0 0
\(940\) 1.45633e10 0.571891
\(941\) 3.15855e10 1.23573 0.617867 0.786283i \(-0.287997\pi\)
0.617867 + 0.786283i \(0.287997\pi\)
\(942\) 0 0
\(943\) −8.65634e9 −0.336158
\(944\) −1.60516e10 −0.621035
\(945\) 0 0
\(946\) 2.18042e9 0.0837377
\(947\) 1.44552e10 0.553095 0.276548 0.961000i \(-0.410810\pi\)
0.276548 + 0.961000i \(0.410810\pi\)
\(948\) 0 0
\(949\) −4.76021e10 −1.80798
\(950\) 6.23962e10 2.36116
\(951\) 0 0
\(952\) 4.24930e10 1.59620
\(953\) 8.95669e9 0.335214 0.167607 0.985854i \(-0.446396\pi\)
0.167607 + 0.985854i \(0.446396\pi\)
\(954\) 0 0
\(955\) −5.01979e9 −0.186498
\(956\) 8.14022e8 0.0301324
\(957\) 0 0
\(958\) 5.75729e10 2.11562
\(959\) −1.80784e10 −0.661904
\(960\) 0 0
\(961\) 1.06042e10 0.385430
\(962\) 6.38313e9 0.231164
\(963\) 0 0
\(964\) 3.41117e10 1.22640
\(965\) 4.77022e9 0.170881
\(966\) 0 0
\(967\) 3.78882e10 1.34745 0.673723 0.738984i \(-0.264694\pi\)
0.673723 + 0.738984i \(0.264694\pi\)
\(968\) 3.24281e10 1.14910
\(969\) 0 0
\(970\) 1.46791e10 0.516415
\(971\) −3.53468e10 −1.23903 −0.619516 0.784984i \(-0.712671\pi\)
−0.619516 + 0.784984i \(0.712671\pi\)
\(972\) 0 0
\(973\) −5.01570e10 −1.74557
\(974\) 7.24023e10 2.51071
\(975\) 0 0
\(976\) 1.86941e9 0.0643620
\(977\) −4.28246e10 −1.46914 −0.734568 0.678535i \(-0.762615\pi\)
−0.734568 + 0.678535i \(0.762615\pi\)
\(978\) 0 0
\(979\) 6.54236e9 0.222841
\(980\) −4.97445e9 −0.168832
\(981\) 0 0
\(982\) −1.81031e9 −0.0610048
\(983\) −1.30981e10 −0.439815 −0.219908 0.975521i \(-0.570576\pi\)
−0.219908 + 0.975521i \(0.570576\pi\)
\(984\) 0 0
\(985\) 8.86005e9 0.295399
\(986\) −4.29905e10 −1.42825
\(987\) 0 0
\(988\) 1.00051e11 3.30045
\(989\) −3.16231e9 −0.103948
\(990\) 0 0
\(991\) −2.48042e10 −0.809593 −0.404796 0.914407i \(-0.632658\pi\)
−0.404796 + 0.914407i \(0.632658\pi\)
\(992\) −2.41608e10 −0.785814
\(993\) 0 0
\(994\) −6.08949e10 −1.96666
\(995\) 2.00578e9 0.0645508
\(996\) 0 0
\(997\) 4.36888e10 1.39617 0.698083 0.716017i \(-0.254037\pi\)
0.698083 + 0.716017i \(0.254037\pi\)
\(998\) −8.74368e9 −0.278444
\(999\) 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 387.8.a.d.1.2 13
3.2 odd 2 43.8.a.b.1.12 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.12 13 3.2 odd 2
387.8.a.d.1.2 13 1.1 even 1 trivial