Properties

Label 531.8.a.b.1.7
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $1$
Dimension $16$
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: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + 1488374176 x^{9} + 42885295136 x^{8} - 226132003872 x^{7} - 3353576629440 x^{6} + 16796366777600 x^{5} + 99470801612800 x^{4} - 494039551757568 x^{3} - 493048066650624 x^{2} + 3193975642099712 x - 2385018853548032\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-3.09726\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.09726 q^{2} -118.407 q^{4} +156.435 q^{5} +11.3597 q^{7} +763.187 q^{8} +O(q^{10})\) \(q-3.09726 q^{2} -118.407 q^{4} +156.435 q^{5} +11.3597 q^{7} +763.187 q^{8} -484.520 q^{10} -4296.19 q^{11} +1741.86 q^{13} -35.1841 q^{14} +12792.3 q^{16} +4764.38 q^{17} +18278.8 q^{19} -18523.0 q^{20} +13306.4 q^{22} -82004.5 q^{23} -53653.2 q^{25} -5395.00 q^{26} -1345.07 q^{28} +157425. q^{29} +37778.8 q^{31} -137309. q^{32} -14756.5 q^{34} +1777.05 q^{35} +450442. q^{37} -56614.3 q^{38} +119389. q^{40} +437505. q^{41} -738856. q^{43} +508699. q^{44} +253989. q^{46} -869122. q^{47} -823414. q^{49} +166178. q^{50} -206248. q^{52} +991560. q^{53} -672074. q^{55} +8669.60 q^{56} -487588. q^{58} -205379. q^{59} -1.30896e6 q^{61} -117011. q^{62} -1.21213e6 q^{64} +272487. q^{65} +2.85617e6 q^{67} -564135. q^{68} -5504.01 q^{70} +4.56945e6 q^{71} +937053. q^{73} -1.39514e6 q^{74} -2.16434e6 q^{76} -48803.6 q^{77} +6.61566e6 q^{79} +2.00116e6 q^{80} -1.35507e6 q^{82} +3.65210e6 q^{83} +745314. q^{85} +2.28843e6 q^{86} -3.27880e6 q^{88} +5.71107e6 q^{89} +19787.0 q^{91} +9.70990e6 q^{92} +2.69190e6 q^{94} +2.85944e6 q^{95} -1.38687e7 q^{97} +2.55033e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} + O(q^{10}) \) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} - 3479q^{10} - 898q^{11} - 8172q^{13} + 13315q^{14} + 3138q^{16} + 44985q^{17} - 40137q^{19} - 130657q^{20} + 109394q^{22} + 2833q^{23} + 285746q^{25} + 129420q^{26} + 112890q^{28} - 144375q^{29} - 141759q^{31} + 36224q^{32} - 341332q^{34} + 78859q^{35} - 297971q^{37} - 329075q^{38} - 203048q^{40} - 659077q^{41} - 1431608q^{43} - 254916q^{44} + 873113q^{46} + 1574073q^{47} + 1893545q^{49} - 302533q^{50} - 4972548q^{52} - 587736q^{53} - 4624036q^{55} + 5798506q^{56} - 6991380q^{58} - 3286064q^{59} - 6117131q^{61} + 11570258q^{62} - 19063011q^{64} + 5335514q^{65} - 16518710q^{67} + 17284669q^{68} - 39189486q^{70} + 10882582q^{71} - 21097441q^{73} + 16717030q^{74} - 40864952q^{76} + 3404601q^{77} - 3784458q^{79} + 27466195q^{80} - 24990117q^{82} + 1951425q^{83} - 23238675q^{85} + 35910572q^{86} - 27843055q^{88} - 10499443q^{89} + 699217q^{91} + 20062766q^{92} - 59358988q^{94} + 29236333q^{95} - 25158976q^{97} - 2120460q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.09726 −0.273762 −0.136881 0.990587i \(-0.543708\pi\)
−0.136881 + 0.990587i \(0.543708\pi\)
\(3\) 0 0
\(4\) −118.407 −0.925054
\(5\) 156.435 0.559678 0.279839 0.960047i \(-0.409719\pi\)
0.279839 + 0.960047i \(0.409719\pi\)
\(6\) 0 0
\(7\) 11.3597 0.0125177 0.00625885 0.999980i \(-0.498008\pi\)
0.00625885 + 0.999980i \(0.498008\pi\)
\(8\) 763.187 0.527007
\(9\) 0 0
\(10\) −484.520 −0.153219
\(11\) −4296.19 −0.973217 −0.486608 0.873620i \(-0.661766\pi\)
−0.486608 + 0.873620i \(0.661766\pi\)
\(12\) 0 0
\(13\) 1741.86 0.219893 0.109947 0.993938i \(-0.464932\pi\)
0.109947 + 0.993938i \(0.464932\pi\)
\(14\) −35.1841 −0.00342687
\(15\) 0 0
\(16\) 12792.3 0.780780
\(17\) 4764.38 0.235199 0.117599 0.993061i \(-0.462480\pi\)
0.117599 + 0.993061i \(0.462480\pi\)
\(18\) 0 0
\(19\) 18278.8 0.611379 0.305689 0.952131i \(-0.401113\pi\)
0.305689 + 0.952131i \(0.401113\pi\)
\(20\) −18523.0 −0.517732
\(21\) 0 0
\(22\) 13306.4 0.266430
\(23\) −82004.5 −1.40537 −0.702684 0.711502i \(-0.748015\pi\)
−0.702684 + 0.711502i \(0.748015\pi\)
\(24\) 0 0
\(25\) −53653.2 −0.686761
\(26\) −5395.00 −0.0601984
\(27\) 0 0
\(28\) −1345.07 −0.0115796
\(29\) 157425. 1.19862 0.599311 0.800517i \(-0.295441\pi\)
0.599311 + 0.800517i \(0.295441\pi\)
\(30\) 0 0
\(31\) 37778.8 0.227762 0.113881 0.993494i \(-0.463672\pi\)
0.113881 + 0.993494i \(0.463672\pi\)
\(32\) −137309. −0.740755
\(33\) 0 0
\(34\) −14756.5 −0.0643885
\(35\) 1777.05 0.00700588
\(36\) 0 0
\(37\) 450442. 1.46195 0.730975 0.682404i \(-0.239066\pi\)
0.730975 + 0.682404i \(0.239066\pi\)
\(38\) −56614.3 −0.167372
\(39\) 0 0
\(40\) 119389. 0.294954
\(41\) 437505. 0.991378 0.495689 0.868500i \(-0.334916\pi\)
0.495689 + 0.868500i \(0.334916\pi\)
\(42\) 0 0
\(43\) −738856. −1.41716 −0.708582 0.705629i \(-0.750665\pi\)
−0.708582 + 0.705629i \(0.750665\pi\)
\(44\) 508699. 0.900278
\(45\) 0 0
\(46\) 253989. 0.384736
\(47\) −869122. −1.22106 −0.610532 0.791992i \(-0.709044\pi\)
−0.610532 + 0.791992i \(0.709044\pi\)
\(48\) 0 0
\(49\) −823414. −0.999843
\(50\) 166178. 0.188009
\(51\) 0 0
\(52\) −206248. −0.203413
\(53\) 991560. 0.914858 0.457429 0.889246i \(-0.348770\pi\)
0.457429 + 0.889246i \(0.348770\pi\)
\(54\) 0 0
\(55\) −672074. −0.544688
\(56\) 8669.60 0.00659691
\(57\) 0 0
\(58\) −487588. −0.328137
\(59\) −205379. −0.130189
\(60\) 0 0
\(61\) −1.30896e6 −0.738368 −0.369184 0.929356i \(-0.620363\pi\)
−0.369184 + 0.929356i \(0.620363\pi\)
\(62\) −117011. −0.0623527
\(63\) 0 0
\(64\) −1.21213e6 −0.577989
\(65\) 272487. 0.123069
\(66\) 0 0
\(67\) 2.85617e6 1.16017 0.580085 0.814556i \(-0.303019\pi\)
0.580085 + 0.814556i \(0.303019\pi\)
\(68\) −564135. −0.217572
\(69\) 0 0
\(70\) −5504.01 −0.00191794
\(71\) 4.56945e6 1.51516 0.757582 0.652740i \(-0.226381\pi\)
0.757582 + 0.652740i \(0.226381\pi\)
\(72\) 0 0
\(73\) 937053. 0.281925 0.140963 0.990015i \(-0.454980\pi\)
0.140963 + 0.990015i \(0.454980\pi\)
\(74\) −1.39514e6 −0.400226
\(75\) 0 0
\(76\) −2.16434e6 −0.565559
\(77\) −48803.6 −0.0121824
\(78\) 0 0
\(79\) 6.61566e6 1.50966 0.754829 0.655922i \(-0.227720\pi\)
0.754829 + 0.655922i \(0.227720\pi\)
\(80\) 2.00116e6 0.436985
\(81\) 0 0
\(82\) −1.35507e6 −0.271402
\(83\) 3.65210e6 0.701082 0.350541 0.936547i \(-0.385998\pi\)
0.350541 + 0.936547i \(0.385998\pi\)
\(84\) 0 0
\(85\) 745314. 0.131635
\(86\) 2.28843e6 0.387966
\(87\) 0 0
\(88\) −3.27880e6 −0.512892
\(89\) 5.71107e6 0.858722 0.429361 0.903133i \(-0.358739\pi\)
0.429361 + 0.903133i \(0.358739\pi\)
\(90\) 0 0
\(91\) 19787.0 0.00275256
\(92\) 9.70990e6 1.30004
\(93\) 0 0
\(94\) 2.69190e6 0.334281
\(95\) 2.85944e6 0.342175
\(96\) 0 0
\(97\) −1.38687e7 −1.54289 −0.771446 0.636295i \(-0.780466\pi\)
−0.771446 + 0.636295i \(0.780466\pi\)
\(98\) 2.55033e6 0.273719
\(99\) 0 0
\(100\) 6.35291e6 0.635291
\(101\) −1.27095e7 −1.22745 −0.613727 0.789518i \(-0.710330\pi\)
−0.613727 + 0.789518i \(0.710330\pi\)
\(102\) 0 0
\(103\) 1.41595e7 1.27679 0.638394 0.769710i \(-0.279599\pi\)
0.638394 + 0.769710i \(0.279599\pi\)
\(104\) 1.32937e6 0.115885
\(105\) 0 0
\(106\) −3.07112e6 −0.250453
\(107\) 7.44830e6 0.587779 0.293889 0.955839i \(-0.405050\pi\)
0.293889 + 0.955839i \(0.405050\pi\)
\(108\) 0 0
\(109\) −2.45904e7 −1.81875 −0.909374 0.415980i \(-0.863439\pi\)
−0.909374 + 0.415980i \(0.863439\pi\)
\(110\) 2.08159e6 0.149115
\(111\) 0 0
\(112\) 145317. 0.00977357
\(113\) −5.72325e6 −0.373137 −0.186569 0.982442i \(-0.559737\pi\)
−0.186569 + 0.982442i \(0.559737\pi\)
\(114\) 0 0
\(115\) −1.28283e7 −0.786553
\(116\) −1.86403e7 −1.10879
\(117\) 0 0
\(118\) 636113. 0.0356408
\(119\) 54122.0 0.00294415
\(120\) 0 0
\(121\) −1.02989e6 −0.0528495
\(122\) 4.05420e6 0.202137
\(123\) 0 0
\(124\) −4.47327e6 −0.210693
\(125\) −2.06147e7 −0.944043
\(126\) 0 0
\(127\) −1.29403e7 −0.560574 −0.280287 0.959916i \(-0.590430\pi\)
−0.280287 + 0.959916i \(0.590430\pi\)
\(128\) 2.13299e7 0.898986
\(129\) 0 0
\(130\) −843965. −0.0336917
\(131\) 4.66242e7 1.81201 0.906007 0.423263i \(-0.139115\pi\)
0.906007 + 0.423263i \(0.139115\pi\)
\(132\) 0 0
\(133\) 207642. 0.00765306
\(134\) −8.84630e6 −0.317611
\(135\) 0 0
\(136\) 3.63611e6 0.123951
\(137\) 2.07610e7 0.689803 0.344902 0.938639i \(-0.387912\pi\)
0.344902 + 0.938639i \(0.387912\pi\)
\(138\) 0 0
\(139\) −1.80615e6 −0.0570430 −0.0285215 0.999593i \(-0.509080\pi\)
−0.0285215 + 0.999593i \(0.509080\pi\)
\(140\) −210416. −0.00648082
\(141\) 0 0
\(142\) −1.41528e7 −0.414794
\(143\) −7.48337e6 −0.214004
\(144\) 0 0
\(145\) 2.46268e7 0.670842
\(146\) −2.90230e6 −0.0771805
\(147\) 0 0
\(148\) −5.33354e7 −1.35238
\(149\) −7.35804e7 −1.82226 −0.911130 0.412119i \(-0.864789\pi\)
−0.911130 + 0.412119i \(0.864789\pi\)
\(150\) 0 0
\(151\) 2.57500e6 0.0608636 0.0304318 0.999537i \(-0.490312\pi\)
0.0304318 + 0.999537i \(0.490312\pi\)
\(152\) 1.39502e7 0.322201
\(153\) 0 0
\(154\) 151158. 0.00333509
\(155\) 5.90991e6 0.127474
\(156\) 0 0
\(157\) −2.53317e7 −0.522414 −0.261207 0.965283i \(-0.584121\pi\)
−0.261207 + 0.965283i \(0.584121\pi\)
\(158\) −2.04904e7 −0.413287
\(159\) 0 0
\(160\) −2.14799e7 −0.414584
\(161\) −931548. −0.0175920
\(162\) 0 0
\(163\) −1.12501e6 −0.0203470 −0.0101735 0.999948i \(-0.503238\pi\)
−0.0101735 + 0.999948i \(0.503238\pi\)
\(164\) −5.18036e7 −0.917078
\(165\) 0 0
\(166\) −1.13115e7 −0.191930
\(167\) 5.17400e7 0.859645 0.429823 0.902913i \(-0.358576\pi\)
0.429823 + 0.902913i \(0.358576\pi\)
\(168\) 0 0
\(169\) −5.97144e7 −0.951647
\(170\) −2.30843e6 −0.0360368
\(171\) 0 0
\(172\) 8.74857e7 1.31095
\(173\) 5.17329e7 0.759636 0.379818 0.925061i \(-0.375987\pi\)
0.379818 + 0.925061i \(0.375987\pi\)
\(174\) 0 0
\(175\) −609485. −0.00859667
\(176\) −5.49582e7 −0.759868
\(177\) 0 0
\(178\) −1.76887e7 −0.235086
\(179\) −4.40517e7 −0.574086 −0.287043 0.957918i \(-0.592672\pi\)
−0.287043 + 0.957918i \(0.592672\pi\)
\(180\) 0 0
\(181\) 8.76285e7 1.09843 0.549213 0.835683i \(-0.314928\pi\)
0.549213 + 0.835683i \(0.314928\pi\)
\(182\) −61285.7 −0.000753545 0
\(183\) 0 0
\(184\) −6.25848e7 −0.740638
\(185\) 7.04647e7 0.818221
\(186\) 0 0
\(187\) −2.04687e7 −0.228899
\(188\) 1.02910e8 1.12955
\(189\) 0 0
\(190\) −8.85644e6 −0.0936746
\(191\) −1.82917e8 −1.89949 −0.949744 0.313028i \(-0.898656\pi\)
−0.949744 + 0.313028i \(0.898656\pi\)
\(192\) 0 0
\(193\) −8.82731e7 −0.883848 −0.441924 0.897053i \(-0.645704\pi\)
−0.441924 + 0.897053i \(0.645704\pi\)
\(194\) 4.29551e7 0.422385
\(195\) 0 0
\(196\) 9.74979e7 0.924909
\(197\) −1.70359e8 −1.58757 −0.793785 0.608198i \(-0.791893\pi\)
−0.793785 + 0.608198i \(0.791893\pi\)
\(198\) 0 0
\(199\) 9.62750e7 0.866020 0.433010 0.901389i \(-0.357451\pi\)
0.433010 + 0.901389i \(0.357451\pi\)
\(200\) −4.09474e7 −0.361928
\(201\) 0 0
\(202\) 3.93648e7 0.336030
\(203\) 1.78831e6 0.0150040
\(204\) 0 0
\(205\) 6.84409e7 0.554852
\(206\) −4.38558e7 −0.349536
\(207\) 0 0
\(208\) 2.22824e7 0.171688
\(209\) −7.85293e7 −0.595004
\(210\) 0 0
\(211\) −2.06758e8 −1.51521 −0.757605 0.652713i \(-0.773631\pi\)
−0.757605 + 0.652713i \(0.773631\pi\)
\(212\) −1.17408e8 −0.846293
\(213\) 0 0
\(214\) −2.30693e7 −0.160912
\(215\) −1.15583e8 −0.793155
\(216\) 0 0
\(217\) 429156. 0.00285106
\(218\) 7.61629e7 0.497904
\(219\) 0 0
\(220\) 7.95782e7 0.503866
\(221\) 8.29888e6 0.0517186
\(222\) 0 0
\(223\) −4.61080e7 −0.278426 −0.139213 0.990262i \(-0.544457\pi\)
−0.139213 + 0.990262i \(0.544457\pi\)
\(224\) −1.55979e6 −0.00927255
\(225\) 0 0
\(226\) 1.77264e7 0.102151
\(227\) −1.52563e8 −0.865683 −0.432841 0.901470i \(-0.642489\pi\)
−0.432841 + 0.901470i \(0.642489\pi\)
\(228\) 0 0
\(229\) −2.88554e8 −1.58782 −0.793912 0.608033i \(-0.791959\pi\)
−0.793912 + 0.608033i \(0.791959\pi\)
\(230\) 3.97328e7 0.215328
\(231\) 0 0
\(232\) 1.20145e8 0.631682
\(233\) −3.41845e7 −0.177045 −0.0885224 0.996074i \(-0.528214\pi\)
−0.0885224 + 0.996074i \(0.528214\pi\)
\(234\) 0 0
\(235\) −1.35961e8 −0.683402
\(236\) 2.43183e7 0.120432
\(237\) 0 0
\(238\) −167630. −0.000805996 0
\(239\) −1.22378e8 −0.579843 −0.289922 0.957050i \(-0.593629\pi\)
−0.289922 + 0.957050i \(0.593629\pi\)
\(240\) 0 0
\(241\) −1.04747e8 −0.482040 −0.241020 0.970520i \(-0.577482\pi\)
−0.241020 + 0.970520i \(0.577482\pi\)
\(242\) 3.18983e6 0.0144682
\(243\) 0 0
\(244\) 1.54990e8 0.683030
\(245\) −1.28811e8 −0.559590
\(246\) 0 0
\(247\) 3.18391e7 0.134438
\(248\) 2.88323e7 0.120032
\(249\) 0 0
\(250\) 6.38491e7 0.258443
\(251\) −1.01531e8 −0.405268 −0.202634 0.979255i \(-0.564950\pi\)
−0.202634 + 0.979255i \(0.564950\pi\)
\(252\) 0 0
\(253\) 3.52307e8 1.36773
\(254\) 4.00797e7 0.153464
\(255\) 0 0
\(256\) 8.90886e7 0.331881
\(257\) −2.43506e8 −0.894836 −0.447418 0.894325i \(-0.647656\pi\)
−0.447418 + 0.894325i \(0.647656\pi\)
\(258\) 0 0
\(259\) 5.11689e6 0.0183003
\(260\) −3.22644e7 −0.113846
\(261\) 0 0
\(262\) −1.44407e8 −0.496061
\(263\) 5.30485e8 1.79816 0.899079 0.437787i \(-0.144238\pi\)
0.899079 + 0.437787i \(0.144238\pi\)
\(264\) 0 0
\(265\) 1.55114e8 0.512026
\(266\) −643123. −0.00209512
\(267\) 0 0
\(268\) −3.38190e8 −1.07322
\(269\) 1.66365e8 0.521110 0.260555 0.965459i \(-0.416094\pi\)
0.260555 + 0.965459i \(0.416094\pi\)
\(270\) 0 0
\(271\) −5.31604e8 −1.62254 −0.811271 0.584670i \(-0.801224\pi\)
−0.811271 + 0.584670i \(0.801224\pi\)
\(272\) 6.09473e7 0.183638
\(273\) 0 0
\(274\) −6.43022e7 −0.188842
\(275\) 2.30504e8 0.668367
\(276\) 0 0
\(277\) −3.03386e8 −0.857661 −0.428831 0.903385i \(-0.641074\pi\)
−0.428831 + 0.903385i \(0.641074\pi\)
\(278\) 5.59412e6 0.0156162
\(279\) 0 0
\(280\) 1.35623e6 0.00369215
\(281\) −6.31916e7 −0.169898 −0.0849489 0.996385i \(-0.527073\pi\)
−0.0849489 + 0.996385i \(0.527073\pi\)
\(282\) 0 0
\(283\) −2.67112e8 −0.700552 −0.350276 0.936647i \(-0.613912\pi\)
−0.350276 + 0.936647i \(0.613912\pi\)
\(284\) −5.41054e8 −1.40161
\(285\) 0 0
\(286\) 2.31780e7 0.0585861
\(287\) 4.96993e6 0.0124098
\(288\) 0 0
\(289\) −3.87639e8 −0.944682
\(290\) −7.62757e7 −0.183651
\(291\) 0 0
\(292\) −1.10954e8 −0.260796
\(293\) 4.43690e8 1.03049 0.515244 0.857043i \(-0.327701\pi\)
0.515244 + 0.857043i \(0.327701\pi\)
\(294\) 0 0
\(295\) −3.21284e7 −0.0728638
\(296\) 3.43771e8 0.770458
\(297\) 0 0
\(298\) 2.27898e8 0.498866
\(299\) −1.42840e8 −0.309031
\(300\) 0 0
\(301\) −8.39320e6 −0.0177396
\(302\) −7.97544e6 −0.0166621
\(303\) 0 0
\(304\) 2.33828e8 0.477352
\(305\) −2.04767e8 −0.413248
\(306\) 0 0
\(307\) −4.86871e8 −0.960351 −0.480175 0.877173i \(-0.659427\pi\)
−0.480175 + 0.877173i \(0.659427\pi\)
\(308\) 5.77868e6 0.0112694
\(309\) 0 0
\(310\) −1.83046e7 −0.0348974
\(311\) 5.91749e8 1.11552 0.557758 0.830003i \(-0.311662\pi\)
0.557758 + 0.830003i \(0.311662\pi\)
\(312\) 0 0
\(313\) −6.83534e8 −1.25995 −0.629977 0.776614i \(-0.716936\pi\)
−0.629977 + 0.776614i \(0.716936\pi\)
\(314\) 7.84589e7 0.143017
\(315\) 0 0
\(316\) −7.83340e8 −1.39651
\(317\) 3.17450e8 0.559716 0.279858 0.960041i \(-0.409713\pi\)
0.279858 + 0.960041i \(0.409713\pi\)
\(318\) 0 0
\(319\) −6.76330e8 −1.16652
\(320\) −1.89619e8 −0.323488
\(321\) 0 0
\(322\) 2.88525e6 0.00481602
\(323\) 8.70872e7 0.143796
\(324\) 0 0
\(325\) −9.34563e7 −0.151014
\(326\) 3.48446e6 0.00557024
\(327\) 0 0
\(328\) 3.33898e8 0.522463
\(329\) −9.87298e6 −0.0152849
\(330\) 0 0
\(331\) 1.88196e8 0.285242 0.142621 0.989777i \(-0.454447\pi\)
0.142621 + 0.989777i \(0.454447\pi\)
\(332\) −4.32434e8 −0.648539
\(333\) 0 0
\(334\) −1.60253e8 −0.235338
\(335\) 4.46804e8 0.649321
\(336\) 0 0
\(337\) −7.47824e8 −1.06438 −0.532188 0.846626i \(-0.678630\pi\)
−0.532188 + 0.846626i \(0.678630\pi\)
\(338\) 1.84951e8 0.260525
\(339\) 0 0
\(340\) −8.82504e7 −0.121770
\(341\) −1.62305e8 −0.221662
\(342\) 0 0
\(343\) −1.87090e7 −0.0250334
\(344\) −5.63885e8 −0.746855
\(345\) 0 0
\(346\) −1.60230e8 −0.207959
\(347\) 1.23827e8 0.159097 0.0795483 0.996831i \(-0.474652\pi\)
0.0795483 + 0.996831i \(0.474652\pi\)
\(348\) 0 0
\(349\) −1.33403e9 −1.67987 −0.839935 0.542688i \(-0.817407\pi\)
−0.839935 + 0.542688i \(0.817407\pi\)
\(350\) 1.88774e6 0.00235344
\(351\) 0 0
\(352\) 5.89907e8 0.720915
\(353\) −1.33244e9 −1.61226 −0.806132 0.591735i \(-0.798443\pi\)
−0.806132 + 0.591735i \(0.798443\pi\)
\(354\) 0 0
\(355\) 7.14820e8 0.848003
\(356\) −6.76231e8 −0.794365
\(357\) 0 0
\(358\) 1.36440e8 0.157163
\(359\) −7.56062e8 −0.862435 −0.431218 0.902248i \(-0.641916\pi\)
−0.431218 + 0.902248i \(0.641916\pi\)
\(360\) 0 0
\(361\) −5.59757e8 −0.626216
\(362\) −2.71409e8 −0.300707
\(363\) 0 0
\(364\) −2.34292e6 −0.00254626
\(365\) 1.46588e8 0.157787
\(366\) 0 0
\(367\) −1.10105e9 −1.16272 −0.581358 0.813648i \(-0.697479\pi\)
−0.581358 + 0.813648i \(0.697479\pi\)
\(368\) −1.04903e9 −1.09728
\(369\) 0 0
\(370\) −2.18248e8 −0.223998
\(371\) 1.12639e7 0.0114519
\(372\) 0 0
\(373\) −1.72328e9 −1.71940 −0.859698 0.510803i \(-0.829348\pi\)
−0.859698 + 0.510803i \(0.829348\pi\)
\(374\) 6.33969e7 0.0626639
\(375\) 0 0
\(376\) −6.63303e8 −0.643509
\(377\) 2.74213e8 0.263569
\(378\) 0 0
\(379\) 3.29484e8 0.310883 0.155441 0.987845i \(-0.450320\pi\)
0.155441 + 0.987845i \(0.450320\pi\)
\(380\) −3.38578e8 −0.316531
\(381\) 0 0
\(382\) 5.66541e8 0.520008
\(383\) 6.36350e8 0.578762 0.289381 0.957214i \(-0.406551\pi\)
0.289381 + 0.957214i \(0.406551\pi\)
\(384\) 0 0
\(385\) −7.63457e6 −0.00681824
\(386\) 2.73405e8 0.241964
\(387\) 0 0
\(388\) 1.64215e9 1.42726
\(389\) 9.45771e8 0.814634 0.407317 0.913287i \(-0.366464\pi\)
0.407317 + 0.913287i \(0.366464\pi\)
\(390\) 0 0
\(391\) −3.90700e8 −0.330541
\(392\) −6.28419e8 −0.526924
\(393\) 0 0
\(394\) 5.27647e8 0.434617
\(395\) 1.03492e9 0.844922
\(396\) 0 0
\(397\) 1.20371e9 0.965506 0.482753 0.875756i \(-0.339637\pi\)
0.482753 + 0.875756i \(0.339637\pi\)
\(398\) −2.98189e8 −0.237083
\(399\) 0 0
\(400\) −6.86348e8 −0.536209
\(401\) −1.24141e9 −0.961409 −0.480705 0.876883i \(-0.659619\pi\)
−0.480705 + 0.876883i \(0.659619\pi\)
\(402\) 0 0
\(403\) 6.58053e7 0.0500834
\(404\) 1.50490e9 1.13546
\(405\) 0 0
\(406\) −5.53887e6 −0.00410752
\(407\) −1.93518e9 −1.42279
\(408\) 0 0
\(409\) −1.31447e9 −0.949990 −0.474995 0.879988i \(-0.657550\pi\)
−0.474995 + 0.879988i \(0.657550\pi\)
\(410\) −2.11980e8 −0.151897
\(411\) 0 0
\(412\) −1.67659e9 −1.18110
\(413\) −2.33305e6 −0.00162967
\(414\) 0 0
\(415\) 5.71315e8 0.392380
\(416\) −2.39173e8 −0.162887
\(417\) 0 0
\(418\) 2.43226e8 0.162890
\(419\) 1.30216e9 0.864796 0.432398 0.901683i \(-0.357667\pi\)
0.432398 + 0.901683i \(0.357667\pi\)
\(420\) 0 0
\(421\) 2.34962e9 1.53465 0.767327 0.641256i \(-0.221587\pi\)
0.767327 + 0.641256i \(0.221587\pi\)
\(422\) 6.40383e8 0.414807
\(423\) 0 0
\(424\) 7.56746e8 0.482136
\(425\) −2.55624e8 −0.161525
\(426\) 0 0
\(427\) −1.48694e7 −0.00924267
\(428\) −8.81930e8 −0.543727
\(429\) 0 0
\(430\) 3.57990e8 0.217136
\(431\) 1.37817e9 0.829147 0.414573 0.910016i \(-0.363931\pi\)
0.414573 + 0.910016i \(0.363931\pi\)
\(432\) 0 0
\(433\) −1.93157e9 −1.14341 −0.571705 0.820459i \(-0.693718\pi\)
−0.571705 + 0.820459i \(0.693718\pi\)
\(434\) −1.32921e6 −0.000780512 0
\(435\) 0 0
\(436\) 2.91167e9 1.68244
\(437\) −1.49894e9 −0.859212
\(438\) 0 0
\(439\) 2.84522e9 1.60505 0.802526 0.596617i \(-0.203489\pi\)
0.802526 + 0.596617i \(0.203489\pi\)
\(440\) −5.12918e8 −0.287054
\(441\) 0 0
\(442\) −2.57038e7 −0.0141586
\(443\) −3.69126e8 −0.201726 −0.100863 0.994900i \(-0.532160\pi\)
−0.100863 + 0.994900i \(0.532160\pi\)
\(444\) 0 0
\(445\) 8.93410e8 0.480608
\(446\) 1.42809e8 0.0762224
\(447\) 0 0
\(448\) −1.37695e7 −0.00723510
\(449\) 3.21750e8 0.167748 0.0838739 0.996476i \(-0.473271\pi\)
0.0838739 + 0.996476i \(0.473271\pi\)
\(450\) 0 0
\(451\) −1.87961e9 −0.964825
\(452\) 6.77673e8 0.345172
\(453\) 0 0
\(454\) 4.72528e8 0.236991
\(455\) 3.09538e6 0.00154054
\(456\) 0 0
\(457\) 3.67556e9 1.80143 0.900714 0.434413i \(-0.143044\pi\)
0.900714 + 0.434413i \(0.143044\pi\)
\(458\) 8.93727e8 0.434686
\(459\) 0 0
\(460\) 1.51897e9 0.727604
\(461\) 8.20164e8 0.389895 0.194947 0.980814i \(-0.437546\pi\)
0.194947 + 0.980814i \(0.437546\pi\)
\(462\) 0 0
\(463\) −1.84327e9 −0.863089 −0.431544 0.902092i \(-0.642031\pi\)
−0.431544 + 0.902092i \(0.642031\pi\)
\(464\) 2.01383e9 0.935859
\(465\) 0 0
\(466\) 1.05878e8 0.0484682
\(467\) −7.89693e8 −0.358797 −0.179399 0.983776i \(-0.557415\pi\)
−0.179399 + 0.983776i \(0.557415\pi\)
\(468\) 0 0
\(469\) 3.24453e7 0.0145227
\(470\) 4.21107e8 0.187090
\(471\) 0 0
\(472\) −1.56743e8 −0.0686104
\(473\) 3.17427e9 1.37921
\(474\) 0 0
\(475\) −9.80717e8 −0.419871
\(476\) −6.40842e6 −0.00272350
\(477\) 0 0
\(478\) 3.79037e8 0.158739
\(479\) −3.05692e9 −1.27089 −0.635447 0.772144i \(-0.719184\pi\)
−0.635447 + 0.772144i \(0.719184\pi\)
\(480\) 0 0
\(481\) 7.84606e8 0.321473
\(482\) 3.24430e8 0.131964
\(483\) 0 0
\(484\) 1.21946e8 0.0488887
\(485\) −2.16955e9 −0.863522
\(486\) 0 0
\(487\) 2.29199e9 0.899209 0.449605 0.893228i \(-0.351565\pi\)
0.449605 + 0.893228i \(0.351565\pi\)
\(488\) −9.98983e8 −0.389125
\(489\) 0 0
\(490\) 3.98960e8 0.153195
\(491\) 2.91988e9 1.11322 0.556608 0.830775i \(-0.312103\pi\)
0.556608 + 0.830775i \(0.312103\pi\)
\(492\) 0 0
\(493\) 7.50034e8 0.281914
\(494\) −9.86142e7 −0.0368040
\(495\) 0 0
\(496\) 4.83277e8 0.177832
\(497\) 5.19077e7 0.0189664
\(498\) 0 0
\(499\) 2.09323e9 0.754163 0.377082 0.926180i \(-0.376928\pi\)
0.377082 + 0.926180i \(0.376928\pi\)
\(500\) 2.44092e9 0.873291
\(501\) 0 0
\(502\) 3.14469e8 0.110947
\(503\) −2.06891e9 −0.724860 −0.362430 0.932011i \(-0.618053\pi\)
−0.362430 + 0.932011i \(0.618053\pi\)
\(504\) 0 0
\(505\) −1.98821e9 −0.686979
\(506\) −1.09119e9 −0.374432
\(507\) 0 0
\(508\) 1.53223e9 0.518561
\(509\) 3.44598e9 1.15824 0.579122 0.815241i \(-0.303395\pi\)
0.579122 + 0.815241i \(0.303395\pi\)
\(510\) 0 0
\(511\) 1.06447e7 0.00352906
\(512\) −3.00615e9 −0.989843
\(513\) 0 0
\(514\) 7.54202e8 0.244972
\(515\) 2.21504e9 0.714589
\(516\) 0 0
\(517\) 3.73392e9 1.18836
\(518\) −1.58484e7 −0.00500992
\(519\) 0 0
\(520\) 2.07959e8 0.0648583
\(521\) −3.18767e9 −0.987509 −0.493755 0.869601i \(-0.664376\pi\)
−0.493755 + 0.869601i \(0.664376\pi\)
\(522\) 0 0
\(523\) 4.05354e9 1.23902 0.619510 0.784988i \(-0.287331\pi\)
0.619510 + 0.784988i \(0.287331\pi\)
\(524\) −5.52062e9 −1.67621
\(525\) 0 0
\(526\) −1.64305e9 −0.492267
\(527\) 1.79992e8 0.0535694
\(528\) 0 0
\(529\) 3.31991e9 0.975059
\(530\) −4.80430e8 −0.140173
\(531\) 0 0
\(532\) −2.45863e7 −0.00707950
\(533\) 7.62072e8 0.217997
\(534\) 0 0
\(535\) 1.16517e9 0.328967
\(536\) 2.17979e9 0.611418
\(537\) 0 0
\(538\) −5.15277e8 −0.142660
\(539\) 3.53755e9 0.973064
\(540\) 0 0
\(541\) −1.94039e9 −0.526865 −0.263432 0.964678i \(-0.584855\pi\)
−0.263432 + 0.964678i \(0.584855\pi\)
\(542\) 1.64652e9 0.444190
\(543\) 0 0
\(544\) −6.54192e8 −0.174225
\(545\) −3.84679e9 −1.01791
\(546\) 0 0
\(547\) 3.90843e9 1.02105 0.510525 0.859863i \(-0.329451\pi\)
0.510525 + 0.859863i \(0.329451\pi\)
\(548\) −2.45824e9 −0.638106
\(549\) 0 0
\(550\) −7.13933e8 −0.182974
\(551\) 2.87755e9 0.732812
\(552\) 0 0
\(553\) 7.51521e7 0.0188974
\(554\) 9.39665e8 0.234795
\(555\) 0 0
\(556\) 2.13861e8 0.0527678
\(557\) 1.37471e9 0.337069 0.168534 0.985696i \(-0.446097\pi\)
0.168534 + 0.985696i \(0.446097\pi\)
\(558\) 0 0
\(559\) −1.28698e9 −0.311625
\(560\) 2.27326e7 0.00547005
\(561\) 0 0
\(562\) 1.95721e8 0.0465115
\(563\) 7.49339e9 1.76970 0.884849 0.465879i \(-0.154262\pi\)
0.884849 + 0.465879i \(0.154262\pi\)
\(564\) 0 0
\(565\) −8.95316e8 −0.208837
\(566\) 8.27315e8 0.191785
\(567\) 0 0
\(568\) 3.48734e9 0.798501
\(569\) −4.64326e9 −1.05665 −0.528324 0.849043i \(-0.677179\pi\)
−0.528324 + 0.849043i \(0.677179\pi\)
\(570\) 0 0
\(571\) −7.07238e9 −1.58979 −0.794894 0.606748i \(-0.792474\pi\)
−0.794894 + 0.606748i \(0.792474\pi\)
\(572\) 8.86083e8 0.197965
\(573\) 0 0
\(574\) −1.53932e7 −0.00339733
\(575\) 4.39980e9 0.965152
\(576\) 0 0
\(577\) 7.54152e8 0.163434 0.0817172 0.996656i \(-0.473960\pi\)
0.0817172 + 0.996656i \(0.473960\pi\)
\(578\) 1.20062e9 0.258618
\(579\) 0 0
\(580\) −2.91599e9 −0.620565
\(581\) 4.14868e7 0.00877594
\(582\) 0 0
\(583\) −4.25994e9 −0.890355
\(584\) 7.15147e8 0.148577
\(585\) 0 0
\(586\) −1.37423e9 −0.282109
\(587\) −4.19667e9 −0.856389 −0.428195 0.903687i \(-0.640850\pi\)
−0.428195 + 0.903687i \(0.640850\pi\)
\(588\) 0 0
\(589\) 6.90551e8 0.139249
\(590\) 9.95101e7 0.0199474
\(591\) 0 0
\(592\) 5.76218e9 1.14146
\(593\) 9.07288e8 0.178671 0.0893355 0.996002i \(-0.471526\pi\)
0.0893355 + 0.996002i \(0.471526\pi\)
\(594\) 0 0
\(595\) 8.46656e6 0.00164777
\(596\) 8.71244e9 1.68569
\(597\) 0 0
\(598\) 4.42414e8 0.0846009
\(599\) 7.00263e9 1.33127 0.665637 0.746276i \(-0.268160\pi\)
0.665637 + 0.746276i \(0.268160\pi\)
\(600\) 0 0
\(601\) 3.01689e7 0.00566890 0.00283445 0.999996i \(-0.499098\pi\)
0.00283445 + 0.999996i \(0.499098\pi\)
\(602\) 2.59959e7 0.00485644
\(603\) 0 0
\(604\) −3.04898e8 −0.0563021
\(605\) −1.61110e8 −0.0295787
\(606\) 0 0
\(607\) −1.46131e9 −0.265205 −0.132602 0.991169i \(-0.542333\pi\)
−0.132602 + 0.991169i \(0.542333\pi\)
\(608\) −2.50985e9 −0.452882
\(609\) 0 0
\(610\) 6.34218e8 0.113132
\(611\) −1.51389e9 −0.268503
\(612\) 0 0
\(613\) −5.73275e9 −1.00520 −0.502599 0.864520i \(-0.667623\pi\)
−0.502599 + 0.864520i \(0.667623\pi\)
\(614\) 1.50797e9 0.262908
\(615\) 0 0
\(616\) −3.72463e7 −0.00642023
\(617\) −3.48905e9 −0.598010 −0.299005 0.954251i \(-0.596655\pi\)
−0.299005 + 0.954251i \(0.596655\pi\)
\(618\) 0 0
\(619\) −5.02284e9 −0.851200 −0.425600 0.904911i \(-0.639937\pi\)
−0.425600 + 0.904911i \(0.639937\pi\)
\(620\) −6.99775e8 −0.117920
\(621\) 0 0
\(622\) −1.83280e9 −0.305386
\(623\) 6.48762e7 0.0107492
\(624\) 0 0
\(625\) 9.66804e8 0.158401
\(626\) 2.11708e9 0.344928
\(627\) 0 0
\(628\) 2.99945e9 0.483262
\(629\) 2.14607e9 0.343849
\(630\) 0 0
\(631\) −7.21168e9 −1.14270 −0.571352 0.820705i \(-0.693581\pi\)
−0.571352 + 0.820705i \(0.693581\pi\)
\(632\) 5.04899e9 0.795600
\(633\) 0 0
\(634\) −9.83226e8 −0.153229
\(635\) −2.02432e9 −0.313741
\(636\) 0 0
\(637\) −1.43427e9 −0.219859
\(638\) 2.09477e9 0.319348
\(639\) 0 0
\(640\) 3.33673e9 0.503143
\(641\) −1.16522e10 −1.74746 −0.873728 0.486415i \(-0.838304\pi\)
−0.873728 + 0.486415i \(0.838304\pi\)
\(642\) 0 0
\(643\) −1.68105e9 −0.249369 −0.124684 0.992196i \(-0.539792\pi\)
−0.124684 + 0.992196i \(0.539792\pi\)
\(644\) 1.10302e8 0.0162735
\(645\) 0 0
\(646\) −2.69732e8 −0.0393658
\(647\) −3.46626e9 −0.503148 −0.251574 0.967838i \(-0.580948\pi\)
−0.251574 + 0.967838i \(0.580948\pi\)
\(648\) 0 0
\(649\) 8.82348e8 0.126702
\(650\) 2.89459e8 0.0413419
\(651\) 0 0
\(652\) 1.33209e8 0.0188221
\(653\) −8.92600e9 −1.25447 −0.627236 0.778830i \(-0.715814\pi\)
−0.627236 + 0.778830i \(0.715814\pi\)
\(654\) 0 0
\(655\) 7.29364e9 1.01414
\(656\) 5.59669e9 0.774048
\(657\) 0 0
\(658\) 3.05792e7 0.00418443
\(659\) −1.71451e8 −0.0233368 −0.0116684 0.999932i \(-0.503714\pi\)
−0.0116684 + 0.999932i \(0.503714\pi\)
\(660\) 0 0
\(661\) 1.10119e10 1.48305 0.741526 0.670924i \(-0.234102\pi\)
0.741526 + 0.670924i \(0.234102\pi\)
\(662\) −5.82894e8 −0.0780884
\(663\) 0 0
\(664\) 2.78723e9 0.369475
\(665\) 3.24825e7 0.00428325
\(666\) 0 0
\(667\) −1.29096e10 −1.68450
\(668\) −6.12638e9 −0.795218
\(669\) 0 0
\(670\) −1.38387e9 −0.177760
\(671\) 5.62356e9 0.718592
\(672\) 0 0
\(673\) −5.18178e9 −0.655279 −0.327640 0.944803i \(-0.606253\pi\)
−0.327640 + 0.944803i \(0.606253\pi\)
\(674\) 2.31621e9 0.291386
\(675\) 0 0
\(676\) 7.07061e9 0.880325
\(677\) 8.09065e9 1.00213 0.501064 0.865410i \(-0.332942\pi\)
0.501064 + 0.865410i \(0.332942\pi\)
\(678\) 0 0
\(679\) −1.57545e8 −0.0193135
\(680\) 5.68814e8 0.0693728
\(681\) 0 0
\(682\) 5.02701e8 0.0606827
\(683\) −1.31475e9 −0.157896 −0.0789478 0.996879i \(-0.525156\pi\)
−0.0789478 + 0.996879i \(0.525156\pi\)
\(684\) 0 0
\(685\) 3.24773e9 0.386068
\(686\) 5.79466e7 0.00685321
\(687\) 0 0
\(688\) −9.45166e9 −1.10649
\(689\) 1.72716e9 0.201171
\(690\) 0 0
\(691\) 9.51511e9 1.09709 0.548543 0.836123i \(-0.315183\pi\)
0.548543 + 0.836123i \(0.315183\pi\)
\(692\) −6.12553e9 −0.702704
\(693\) 0 0
\(694\) −3.83524e8 −0.0435546
\(695\) −2.82545e8 −0.0319257
\(696\) 0 0
\(697\) 2.08444e9 0.233171
\(698\) 4.13183e9 0.459884
\(699\) 0 0
\(700\) 7.21673e7 0.00795238
\(701\) 1.01482e10 1.11270 0.556349 0.830949i \(-0.312202\pi\)
0.556349 + 0.830949i \(0.312202\pi\)
\(702\) 0 0
\(703\) 8.23354e9 0.893805
\(704\) 5.20755e9 0.562509
\(705\) 0 0
\(706\) 4.12692e9 0.441377
\(707\) −1.44377e8 −0.0153649
\(708\) 0 0
\(709\) 1.71054e10 1.80248 0.901241 0.433319i \(-0.142658\pi\)
0.901241 + 0.433319i \(0.142658\pi\)
\(710\) −2.21399e9 −0.232151
\(711\) 0 0
\(712\) 4.35862e9 0.452553
\(713\) −3.09803e9 −0.320090
\(714\) 0 0
\(715\) −1.17066e9 −0.119773
\(716\) 5.21602e9 0.531060
\(717\) 0 0
\(718\) 2.34172e9 0.236102
\(719\) −4.37544e9 −0.439006 −0.219503 0.975612i \(-0.570444\pi\)
−0.219503 + 0.975612i \(0.570444\pi\)
\(720\) 0 0
\(721\) 1.60848e8 0.0159824
\(722\) 1.73371e9 0.171434
\(723\) 0 0
\(724\) −1.03758e10 −1.01610
\(725\) −8.44638e9 −0.823166
\(726\) 0 0
\(727\) −1.22749e10 −1.18481 −0.592405 0.805641i \(-0.701821\pi\)
−0.592405 + 0.805641i \(0.701821\pi\)
\(728\) 1.51012e7 0.00145062
\(729\) 0 0
\(730\) −4.54020e8 −0.0431962
\(731\) −3.52019e9 −0.333315
\(732\) 0 0
\(733\) −7.53477e9 −0.706653 −0.353327 0.935500i \(-0.614950\pi\)
−0.353327 + 0.935500i \(0.614950\pi\)
\(734\) 3.41023e9 0.318308
\(735\) 0 0
\(736\) 1.12600e10 1.04103
\(737\) −1.22706e10 −1.12910
\(738\) 0 0
\(739\) 1.70023e10 1.54972 0.774858 0.632136i \(-0.217821\pi\)
0.774858 + 0.632136i \(0.217821\pi\)
\(740\) −8.34351e9 −0.756899
\(741\) 0 0
\(742\) −3.48871e7 −0.00313510
\(743\) 1.25008e10 1.11809 0.559045 0.829137i \(-0.311168\pi\)
0.559045 + 0.829137i \(0.311168\pi\)
\(744\) 0 0
\(745\) −1.15105e10 −1.01988
\(746\) 5.33747e9 0.470705
\(747\) 0 0
\(748\) 2.42364e9 0.211744
\(749\) 8.46106e7 0.00735764
\(750\) 0 0
\(751\) −6.82394e9 −0.587889 −0.293945 0.955822i \(-0.594968\pi\)
−0.293945 + 0.955822i \(0.594968\pi\)
\(752\) −1.11181e10 −0.953382
\(753\) 0 0
\(754\) −8.49311e8 −0.0721551
\(755\) 4.02819e8 0.0340640
\(756\) 0 0
\(757\) 1.76261e10 1.47680 0.738400 0.674363i \(-0.235582\pi\)
0.738400 + 0.674363i \(0.235582\pi\)
\(758\) −1.02050e9 −0.0851079
\(759\) 0 0
\(760\) 2.18229e9 0.180329
\(761\) 8.48847e9 0.698205 0.349103 0.937084i \(-0.386486\pi\)
0.349103 + 0.937084i \(0.386486\pi\)
\(762\) 0 0
\(763\) −2.79340e8 −0.0227665
\(764\) 2.16586e10 1.75713
\(765\) 0 0
\(766\) −1.97094e9 −0.158443
\(767\) −3.57741e8 −0.0286276
\(768\) 0 0
\(769\) 1.91757e10 1.52058 0.760288 0.649586i \(-0.225058\pi\)
0.760288 + 0.649586i \(0.225058\pi\)
\(770\) 2.36463e7 0.00186657
\(771\) 0 0
\(772\) 1.04521e10 0.817607
\(773\) 7.92188e8 0.0616879 0.0308440 0.999524i \(-0.490181\pi\)
0.0308440 + 0.999524i \(0.490181\pi\)
\(774\) 0 0
\(775\) −2.02695e9 −0.156418
\(776\) −1.05844e10 −0.813114
\(777\) 0 0
\(778\) −2.92930e9 −0.223016
\(779\) 7.99707e9 0.606108
\(780\) 0 0
\(781\) −1.96312e10 −1.47458
\(782\) 1.21010e9 0.0904895
\(783\) 0 0
\(784\) −1.05334e10 −0.780658
\(785\) −3.96275e9 −0.292384
\(786\) 0 0
\(787\) 2.00728e10 1.46790 0.733950 0.679203i \(-0.237675\pi\)
0.733950 + 0.679203i \(0.237675\pi\)
\(788\) 2.01717e10 1.46859
\(789\) 0 0
\(790\) −3.20542e9 −0.231307
\(791\) −6.50146e7 −0.00467082
\(792\) 0 0
\(793\) −2.28003e9 −0.162362
\(794\) −3.72821e9 −0.264319
\(795\) 0 0
\(796\) −1.13996e10 −0.801115
\(797\) −1.65749e10 −1.15971 −0.579853 0.814721i \(-0.696890\pi\)
−0.579853 + 0.814721i \(0.696890\pi\)
\(798\) 0 0
\(799\) −4.14082e9 −0.287192
\(800\) 7.36707e9 0.508721
\(801\) 0 0
\(802\) 3.84496e9 0.263197
\(803\) −4.02576e9 −0.274374
\(804\) 0 0
\(805\) −1.45726e8 −0.00984584
\(806\) −2.03817e8 −0.0137109
\(807\) 0 0
\(808\) −9.69977e9 −0.646877
\(809\) 7.11284e9 0.472305 0.236153 0.971716i \(-0.424113\pi\)
0.236153 + 0.971716i \(0.424113\pi\)
\(810\) 0 0
\(811\) 1.80923e9 0.119102 0.0595511 0.998225i \(-0.481033\pi\)
0.0595511 + 0.998225i \(0.481033\pi\)
\(812\) −2.11748e8 −0.0138795
\(813\) 0 0
\(814\) 5.99378e9 0.389507
\(815\) −1.75991e8 −0.0113878
\(816\) 0 0
\(817\) −1.35054e10 −0.866424
\(818\) 4.07126e9 0.260071
\(819\) 0 0
\(820\) −8.10388e9 −0.513268
\(821\) −1.91315e10 −1.20655 −0.603277 0.797531i \(-0.706139\pi\)
−0.603277 + 0.797531i \(0.706139\pi\)
\(822\) 0 0
\(823\) −1.55173e10 −0.970325 −0.485163 0.874424i \(-0.661240\pi\)
−0.485163 + 0.874424i \(0.661240\pi\)
\(824\) 1.08064e10 0.672876
\(825\) 0 0
\(826\) 7.22607e6 0.000446141 0
\(827\) −2.67848e10 −1.64672 −0.823360 0.567519i \(-0.807903\pi\)
−0.823360 + 0.567519i \(0.807903\pi\)
\(828\) 0 0
\(829\) 1.50178e10 0.915515 0.457757 0.889077i \(-0.348653\pi\)
0.457757 + 0.889077i \(0.348653\pi\)
\(830\) −1.76951e9 −0.107419
\(831\) 0 0
\(832\) −2.11136e9 −0.127096
\(833\) −3.92305e9 −0.235162
\(834\) 0 0
\(835\) 8.09394e9 0.481124
\(836\) 9.29842e9 0.550411
\(837\) 0 0
\(838\) −4.03312e9 −0.236748
\(839\) −1.78661e10 −1.04439 −0.522197 0.852825i \(-0.674887\pi\)
−0.522197 + 0.852825i \(0.674887\pi\)
\(840\) 0 0
\(841\) 7.53290e9 0.436693
\(842\) −7.27739e9 −0.420130
\(843\) 0 0
\(844\) 2.44815e10 1.40165
\(845\) −9.34141e9 −0.532616
\(846\) 0 0
\(847\) −1.16992e7 −0.000661555 0
\(848\) 1.26843e10 0.714303
\(849\) 0 0
\(850\) 7.91735e8 0.0442195
\(851\) −3.69382e10 −2.05458
\(852\) 0 0
\(853\) −1.05414e10 −0.581538 −0.290769 0.956793i \(-0.593911\pi\)
−0.290769 + 0.956793i \(0.593911\pi\)
\(854\) 4.60546e7 0.00253029
\(855\) 0 0
\(856\) 5.68445e9 0.309763
\(857\) 1.68053e10 0.912037 0.456018 0.889970i \(-0.349275\pi\)
0.456018 + 0.889970i \(0.349275\pi\)
\(858\) 0 0
\(859\) −3.46312e9 −0.186420 −0.0932099 0.995646i \(-0.529713\pi\)
−0.0932099 + 0.995646i \(0.529713\pi\)
\(860\) 1.36858e10 0.733712
\(861\) 0 0
\(862\) −4.26855e9 −0.226989
\(863\) −1.26642e10 −0.670716 −0.335358 0.942091i \(-0.608857\pi\)
−0.335358 + 0.942091i \(0.608857\pi\)
\(864\) 0 0
\(865\) 8.09282e9 0.425151
\(866\) 5.98257e9 0.313022
\(867\) 0 0
\(868\) −5.08151e7 −0.00263739
\(869\) −2.84222e10 −1.46922
\(870\) 0 0
\(871\) 4.97504e9 0.255113
\(872\) −1.87671e10 −0.958492
\(873\) 0 0
\(874\) 4.64263e9 0.235220
\(875\) −2.34177e8 −0.0118172
\(876\) 0 0
\(877\) −1.35643e10 −0.679044 −0.339522 0.940598i \(-0.610265\pi\)
−0.339522 + 0.940598i \(0.610265\pi\)
\(878\) −8.81238e9 −0.439403
\(879\) 0 0
\(880\) −8.59737e9 −0.425281
\(881\) −3.52328e10 −1.73593 −0.867963 0.496630i \(-0.834571\pi\)
−0.867963 + 0.496630i \(0.834571\pi\)
\(882\) 0 0
\(883\) 1.37573e10 0.672469 0.336234 0.941778i \(-0.390847\pi\)
0.336234 + 0.941778i \(0.390847\pi\)
\(884\) −9.82645e8 −0.0478425
\(885\) 0 0
\(886\) 1.14328e9 0.0552248
\(887\) 2.09679e10 1.00884 0.504420 0.863459i \(-0.331706\pi\)
0.504420 + 0.863459i \(0.331706\pi\)
\(888\) 0 0
\(889\) −1.46999e8 −0.00701709
\(890\) −2.76713e9 −0.131572
\(891\) 0 0
\(892\) 5.45951e9 0.257559
\(893\) −1.58865e10 −0.746532
\(894\) 0 0
\(895\) −6.89121e9 −0.321303
\(896\) 2.42301e8 0.0112532
\(897\) 0 0
\(898\) −9.96545e8 −0.0459230
\(899\) 5.94734e9 0.273001
\(900\) 0 0
\(901\) 4.72417e9 0.215173
\(902\) 5.82163e9 0.264133
\(903\) 0 0
\(904\) −4.36792e9 −0.196646
\(905\) 1.37081e10 0.614764
\(906\) 0 0
\(907\) −6.58180e9 −0.292900 −0.146450 0.989218i \(-0.546785\pi\)
−0.146450 + 0.989218i \(0.546785\pi\)
\(908\) 1.80645e10 0.800803
\(909\) 0 0
\(910\) −9.58721e6 −0.000421743 0
\(911\) −2.97498e10 −1.30368 −0.651839 0.758357i \(-0.726002\pi\)
−0.651839 + 0.758357i \(0.726002\pi\)
\(912\) 0 0
\(913\) −1.56901e10 −0.682305
\(914\) −1.13842e10 −0.493163
\(915\) 0 0
\(916\) 3.41668e10 1.46882
\(917\) 5.29638e8 0.0226822
\(918\) 0 0
\(919\) −9.95637e9 −0.423152 −0.211576 0.977361i \(-0.567860\pi\)
−0.211576 + 0.977361i \(0.567860\pi\)
\(920\) −9.79043e9 −0.414519
\(921\) 0 0
\(922\) −2.54027e9 −0.106738
\(923\) 7.95934e9 0.333174
\(924\) 0 0
\(925\) −2.41676e10 −1.00401
\(926\) 5.70909e9 0.236281
\(927\) 0 0
\(928\) −2.16160e10 −0.887884
\(929\) 6.90890e9 0.282718 0.141359 0.989958i \(-0.454853\pi\)
0.141359 + 0.989958i \(0.454853\pi\)
\(930\) 0 0
\(931\) −1.50510e10 −0.611283
\(932\) 4.04768e9 0.163776
\(933\) 0 0
\(934\) 2.44589e9 0.0982251
\(935\) −3.20201e9 −0.128110
\(936\) 0 0
\(937\) −6.04161e9 −0.239919 −0.119959 0.992779i \(-0.538276\pi\)
−0.119959 + 0.992779i \(0.538276\pi\)
\(938\) −1.00492e8 −0.00397575
\(939\) 0 0
\(940\) 1.60987e10 0.632184
\(941\) 1.95376e10 0.764376 0.382188 0.924085i \(-0.375171\pi\)
0.382188 + 0.924085i \(0.375171\pi\)
\(942\) 0 0
\(943\) −3.58773e10 −1.39325
\(944\) −2.62727e9 −0.101649
\(945\) 0 0
\(946\) −9.83154e9 −0.377575
\(947\) 2.08163e10 0.796486 0.398243 0.917280i \(-0.369620\pi\)
0.398243 + 0.917280i \(0.369620\pi\)
\(948\) 0 0
\(949\) 1.63222e9 0.0619934
\(950\) 3.03754e9 0.114945
\(951\) 0 0
\(952\) 4.13052e7 0.00155159
\(953\) 2.45125e10 0.917409 0.458705 0.888589i \(-0.348314\pi\)
0.458705 + 0.888589i \(0.348314\pi\)
\(954\) 0 0
\(955\) −2.86145e10 −1.06310
\(956\) 1.44904e10 0.536387
\(957\) 0 0
\(958\) 9.46808e9 0.347923
\(959\) 2.35839e8 0.00863475
\(960\) 0 0
\(961\) −2.60854e10 −0.948124
\(962\) −2.43013e9 −0.0880070
\(963\) 0 0
\(964\) 1.24028e10 0.445913
\(965\) −1.38090e10 −0.494670
\(966\) 0 0
\(967\) −9.86699e9 −0.350907 −0.175454 0.984488i \(-0.556139\pi\)
−0.175454 + 0.984488i \(0.556139\pi\)
\(968\) −7.85997e8 −0.0278521
\(969\) 0 0
\(970\) 6.71967e9 0.236400
\(971\) −1.19828e10 −0.420039 −0.210020 0.977697i \(-0.567353\pi\)
−0.210020 + 0.977697i \(0.567353\pi\)
\(972\) 0 0
\(973\) −2.05174e7 −0.000714047 0
\(974\) −7.09889e9 −0.246169
\(975\) 0 0
\(976\) −1.67446e10 −0.576503
\(977\) −7.54352e9 −0.258787 −0.129394 0.991593i \(-0.541303\pi\)
−0.129394 + 0.991593i \(0.541303\pi\)
\(978\) 0 0
\(979\) −2.45359e10 −0.835723
\(980\) 1.52521e10 0.517651
\(981\) 0 0
\(982\) −9.04363e9 −0.304756
\(983\) −2.03133e10 −0.682091 −0.341045 0.940047i \(-0.610781\pi\)
−0.341045 + 0.940047i \(0.610781\pi\)
\(984\) 0 0
\(985\) −2.66500e10 −0.888528
\(986\) −2.32305e9 −0.0771774
\(987\) 0 0
\(988\) −3.76998e9 −0.124362
\(989\) 6.05895e10 1.99164
\(990\) 0 0
\(991\) −3.47760e10 −1.13507 −0.567534 0.823350i \(-0.692103\pi\)
−0.567534 + 0.823350i \(0.692103\pi\)
\(992\) −5.18737e9 −0.168716
\(993\) 0 0
\(994\) −1.60772e8 −0.00519227
\(995\) 1.50608e10 0.484692
\(996\) 0 0
\(997\) −1.31923e10 −0.421586 −0.210793 0.977531i \(-0.567605\pi\)
−0.210793 + 0.977531i \(0.567605\pi\)
\(998\) −6.48329e9 −0.206461
\(999\) 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.b.1.7 16
3.2 odd 2 177.8.a.a.1.10 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.10 16 3.2 odd 2
531.8.a.b.1.7 16 1.1 even 1 trivial