Properties

Label 531.8.a.b.1.9
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 1.9
Root \(1.97136\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.97136 q^{2} -124.114 q^{4} +339.775 q^{5} +364.700 q^{7} -497.007 q^{8} +O(q^{10})\) \(q+1.97136 q^{2} -124.114 q^{4} +339.775 q^{5} +364.700 q^{7} -497.007 q^{8} +669.820 q^{10} +3588.22 q^{11} +857.316 q^{13} +718.956 q^{14} +14906.8 q^{16} -3649.35 q^{17} -9650.25 q^{19} -42170.8 q^{20} +7073.68 q^{22} -116133. q^{23} +37322.4 q^{25} +1690.08 q^{26} -45264.3 q^{28} -247675. q^{29} +114054. q^{31} +93003.6 q^{32} -7194.19 q^{34} +123916. q^{35} -194589. q^{37} -19024.1 q^{38} -168871. q^{40} +594925. q^{41} +247788. q^{43} -445348. q^{44} -228939. q^{46} +1.30718e6 q^{47} -690537. q^{49} +73575.9 q^{50} -106405. q^{52} -1.20394e6 q^{53} +1.21919e6 q^{55} -181259. q^{56} -488257. q^{58} -205379. q^{59} +428862. q^{61} +224843. q^{62} -1.72472e6 q^{64} +291295. q^{65} -2.01026e6 q^{67} +452934. q^{68} +244284. q^{70} -1.87858e6 q^{71} +2.35049e6 q^{73} -383605. q^{74} +1.19773e6 q^{76} +1.30863e6 q^{77} -3.74874e6 q^{79} +5.06496e6 q^{80} +1.17281e6 q^{82} -3.21289e6 q^{83} -1.23996e6 q^{85} +488480. q^{86} -1.78337e6 q^{88} +544547. q^{89} +312663. q^{91} +1.44136e7 q^{92} +2.57692e6 q^{94} -3.27892e6 q^{95} -2.91764e6 q^{97} -1.36130e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} + O(q^{10}) \) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} - 3479q^{10} - 898q^{11} - 8172q^{13} + 13315q^{14} + 3138q^{16} + 44985q^{17} - 40137q^{19} - 130657q^{20} + 109394q^{22} + 2833q^{23} + 285746q^{25} + 129420q^{26} + 112890q^{28} - 144375q^{29} - 141759q^{31} + 36224q^{32} - 341332q^{34} + 78859q^{35} - 297971q^{37} - 329075q^{38} - 203048q^{40} - 659077q^{41} - 1431608q^{43} - 254916q^{44} + 873113q^{46} + 1574073q^{47} + 1893545q^{49} - 302533q^{50} - 4972548q^{52} - 587736q^{53} - 4624036q^{55} + 5798506q^{56} - 6991380q^{58} - 3286064q^{59} - 6117131q^{61} + 11570258q^{62} - 19063011q^{64} + 5335514q^{65} - 16518710q^{67} + 17284669q^{68} - 39189486q^{70} + 10882582q^{71} - 21097441q^{73} + 16717030q^{74} - 40864952q^{76} + 3404601q^{77} - 3784458q^{79} + 27466195q^{80} - 24990117q^{82} + 1951425q^{83} - 23238675q^{85} + 35910572q^{86} - 27843055q^{88} - 10499443q^{89} + 699217q^{91} + 20062766q^{92} - 59358988q^{94} + 29236333q^{95} - 25158976q^{97} - 2120460q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.97136 0.174245 0.0871227 0.996198i \(-0.472233\pi\)
0.0871227 + 0.996198i \(0.472233\pi\)
\(3\) 0 0
\(4\) −124.114 −0.969639
\(5\) 339.775 1.21562 0.607809 0.794083i \(-0.292049\pi\)
0.607809 + 0.794083i \(0.292049\pi\)
\(6\) 0 0
\(7\) 364.700 0.401877 0.200938 0.979604i \(-0.435601\pi\)
0.200938 + 0.979604i \(0.435601\pi\)
\(8\) −497.007 −0.343200
\(9\) 0 0
\(10\) 669.820 0.211816
\(11\) 3588.22 0.812840 0.406420 0.913686i \(-0.366777\pi\)
0.406420 + 0.913686i \(0.366777\pi\)
\(12\) 0 0
\(13\) 857.316 0.108228 0.0541140 0.998535i \(-0.482767\pi\)
0.0541140 + 0.998535i \(0.482767\pi\)
\(14\) 718.956 0.0700252
\(15\) 0 0
\(16\) 14906.8 0.909837
\(17\) −3649.35 −0.180154 −0.0900771 0.995935i \(-0.528711\pi\)
−0.0900771 + 0.995935i \(0.528711\pi\)
\(18\) 0 0
\(19\) −9650.25 −0.322776 −0.161388 0.986891i \(-0.551597\pi\)
−0.161388 + 0.986891i \(0.551597\pi\)
\(20\) −42170.8 −1.17871
\(21\) 0 0
\(22\) 7073.68 0.141634
\(23\) −116133. −1.99024 −0.995122 0.0986484i \(-0.968548\pi\)
−0.995122 + 0.0986484i \(0.968548\pi\)
\(24\) 0 0
\(25\) 37322.4 0.477727
\(26\) 1690.08 0.0188582
\(27\) 0 0
\(28\) −45264.3 −0.389675
\(29\) −247675. −1.88577 −0.942886 0.333116i \(-0.891900\pi\)
−0.942886 + 0.333116i \(0.891900\pi\)
\(30\) 0 0
\(31\) 114054. 0.687617 0.343808 0.939040i \(-0.388283\pi\)
0.343808 + 0.939040i \(0.388283\pi\)
\(32\) 93003.6 0.501735
\(33\) 0 0
\(34\) −7194.19 −0.0313910
\(35\) 123916. 0.488529
\(36\) 0 0
\(37\) −194589. −0.631556 −0.315778 0.948833i \(-0.602266\pi\)
−0.315778 + 0.948833i \(0.602266\pi\)
\(38\) −19024.1 −0.0562422
\(39\) 0 0
\(40\) −168871. −0.417201
\(41\) 594925. 1.34809 0.674045 0.738691i \(-0.264556\pi\)
0.674045 + 0.738691i \(0.264556\pi\)
\(42\) 0 0
\(43\) 247788. 0.475270 0.237635 0.971354i \(-0.423628\pi\)
0.237635 + 0.971354i \(0.423628\pi\)
\(44\) −445348. −0.788161
\(45\) 0 0
\(46\) −228939. −0.346791
\(47\) 1.30718e6 1.83651 0.918254 0.395993i \(-0.129600\pi\)
0.918254 + 0.395993i \(0.129600\pi\)
\(48\) 0 0
\(49\) −690537. −0.838495
\(50\) 73575.9 0.0832416
\(51\) 0 0
\(52\) −106405. −0.104942
\(53\) −1.20394e6 −1.11081 −0.555403 0.831581i \(-0.687436\pi\)
−0.555403 + 0.831581i \(0.687436\pi\)
\(54\) 0 0
\(55\) 1.21919e6 0.988102
\(56\) −181259. −0.137924
\(57\) 0 0
\(58\) −488257. −0.328587
\(59\) −205379. −0.130189
\(60\) 0 0
\(61\) 428862. 0.241915 0.120958 0.992658i \(-0.461404\pi\)
0.120958 + 0.992658i \(0.461404\pi\)
\(62\) 224843. 0.119814
\(63\) 0 0
\(64\) −1.72472e6 −0.822412
\(65\) 291295. 0.131564
\(66\) 0 0
\(67\) −2.01026e6 −0.816565 −0.408283 0.912856i \(-0.633872\pi\)
−0.408283 + 0.912856i \(0.633872\pi\)
\(68\) 452934. 0.174684
\(69\) 0 0
\(70\) 244284. 0.0851238
\(71\) −1.87858e6 −0.622909 −0.311454 0.950261i \(-0.600816\pi\)
−0.311454 + 0.950261i \(0.600816\pi\)
\(72\) 0 0
\(73\) 2.35049e6 0.707178 0.353589 0.935401i \(-0.384961\pi\)
0.353589 + 0.935401i \(0.384961\pi\)
\(74\) −383605. −0.110046
\(75\) 0 0
\(76\) 1.19773e6 0.312976
\(77\) 1.30863e6 0.326661
\(78\) 0 0
\(79\) −3.74874e6 −0.855442 −0.427721 0.903911i \(-0.640683\pi\)
−0.427721 + 0.903911i \(0.640683\pi\)
\(80\) 5.06496e6 1.10601
\(81\) 0 0
\(82\) 1.17281e6 0.234898
\(83\) −3.21289e6 −0.616768 −0.308384 0.951262i \(-0.599788\pi\)
−0.308384 + 0.951262i \(0.599788\pi\)
\(84\) 0 0
\(85\) −1.23996e6 −0.218999
\(86\) 488480. 0.0828137
\(87\) 0 0
\(88\) −1.78337e6 −0.278967
\(89\) 544547. 0.0818786 0.0409393 0.999162i \(-0.486965\pi\)
0.0409393 + 0.999162i \(0.486965\pi\)
\(90\) 0 0
\(91\) 312663. 0.0434943
\(92\) 1.44136e7 1.92982
\(93\) 0 0
\(94\) 2.57692e6 0.320003
\(95\) −3.27892e6 −0.392372
\(96\) 0 0
\(97\) −2.91764e6 −0.324586 −0.162293 0.986743i \(-0.551889\pi\)
−0.162293 + 0.986743i \(0.551889\pi\)
\(98\) −1.36130e6 −0.146104
\(99\) 0 0
\(100\) −4.63222e6 −0.463222
\(101\) 1.99790e7 1.92952 0.964761 0.263129i \(-0.0847544\pi\)
0.964761 + 0.263129i \(0.0847544\pi\)
\(102\) 0 0
\(103\) −1.14245e7 −1.03016 −0.515082 0.857141i \(-0.672239\pi\)
−0.515082 + 0.857141i \(0.672239\pi\)
\(104\) −426092. −0.0371439
\(105\) 0 0
\(106\) −2.37340e6 −0.193553
\(107\) −2.29986e7 −1.81492 −0.907460 0.420138i \(-0.861982\pi\)
−0.907460 + 0.420138i \(0.861982\pi\)
\(108\) 0 0
\(109\) 2.02074e7 1.49457 0.747287 0.664502i \(-0.231356\pi\)
0.747287 + 0.664502i \(0.231356\pi\)
\(110\) 2.40346e6 0.172172
\(111\) 0 0
\(112\) 5.43650e6 0.365643
\(113\) 1.26444e7 0.824374 0.412187 0.911099i \(-0.364765\pi\)
0.412187 + 0.911099i \(0.364765\pi\)
\(114\) 0 0
\(115\) −3.94590e7 −2.41938
\(116\) 3.07399e7 1.82852
\(117\) 0 0
\(118\) −404876. −0.0226848
\(119\) −1.33092e6 −0.0723998
\(120\) 0 0
\(121\) −6.61184e6 −0.339292
\(122\) 845441. 0.0421526
\(123\) 0 0
\(124\) −1.41557e7 −0.666740
\(125\) −1.38637e7 −0.634885
\(126\) 0 0
\(127\) 3.20643e7 1.38902 0.694509 0.719484i \(-0.255621\pi\)
0.694509 + 0.719484i \(0.255621\pi\)
\(128\) −1.53045e7 −0.645037
\(129\) 0 0
\(130\) 574248. 0.0229244
\(131\) 2.38103e7 0.925369 0.462684 0.886523i \(-0.346886\pi\)
0.462684 + 0.886523i \(0.346886\pi\)
\(132\) 0 0
\(133\) −3.51945e6 −0.129716
\(134\) −3.96295e6 −0.142283
\(135\) 0 0
\(136\) 1.81375e6 0.0618290
\(137\) −2.13799e7 −0.710370 −0.355185 0.934796i \(-0.615582\pi\)
−0.355185 + 0.934796i \(0.615582\pi\)
\(138\) 0 0
\(139\) −3.64257e7 −1.15042 −0.575209 0.818006i \(-0.695079\pi\)
−0.575209 + 0.818006i \(0.695079\pi\)
\(140\) −1.53797e7 −0.473696
\(141\) 0 0
\(142\) −3.70335e6 −0.108539
\(143\) 3.07624e6 0.0879719
\(144\) 0 0
\(145\) −8.41539e7 −2.29238
\(146\) 4.63367e6 0.123222
\(147\) 0 0
\(148\) 2.41512e7 0.612381
\(149\) −95229.9 −0.00235842 −0.00117921 0.999999i \(-0.500375\pi\)
−0.00117921 + 0.999999i \(0.500375\pi\)
\(150\) 0 0
\(151\) 4.71653e7 1.11482 0.557408 0.830238i \(-0.311796\pi\)
0.557408 + 0.830238i \(0.311796\pi\)
\(152\) 4.79624e6 0.110777
\(153\) 0 0
\(154\) 2.57977e6 0.0569192
\(155\) 3.87529e7 0.835879
\(156\) 0 0
\(157\) 4.44571e7 0.916838 0.458419 0.888736i \(-0.348416\pi\)
0.458419 + 0.888736i \(0.348416\pi\)
\(158\) −7.39012e6 −0.149057
\(159\) 0 0
\(160\) 3.16003e7 0.609918
\(161\) −4.23536e7 −0.799833
\(162\) 0 0
\(163\) −9.85994e7 −1.78327 −0.891636 0.452753i \(-0.850442\pi\)
−0.891636 + 0.452753i \(0.850442\pi\)
\(164\) −7.38384e7 −1.30716
\(165\) 0 0
\(166\) −6.33376e6 −0.107469
\(167\) −1.60639e7 −0.266897 −0.133449 0.991056i \(-0.542605\pi\)
−0.133449 + 0.991056i \(0.542605\pi\)
\(168\) 0 0
\(169\) −6.20135e7 −0.988287
\(170\) −2.44441e6 −0.0381595
\(171\) 0 0
\(172\) −3.07539e7 −0.460841
\(173\) −9.63596e7 −1.41493 −0.707463 0.706751i \(-0.750160\pi\)
−0.707463 + 0.706751i \(0.750160\pi\)
\(174\) 0 0
\(175\) 1.36115e7 0.191987
\(176\) 5.34888e7 0.739552
\(177\) 0 0
\(178\) 1.07350e6 0.0142670
\(179\) −6.96623e7 −0.907846 −0.453923 0.891041i \(-0.649976\pi\)
−0.453923 + 0.891041i \(0.649976\pi\)
\(180\) 0 0
\(181\) −1.04761e8 −1.31319 −0.656593 0.754245i \(-0.728003\pi\)
−0.656593 + 0.754245i \(0.728003\pi\)
\(182\) 616372. 0.00757868
\(183\) 0 0
\(184\) 5.77187e7 0.683053
\(185\) −6.61165e7 −0.767731
\(186\) 0 0
\(187\) −1.30947e7 −0.146436
\(188\) −1.62239e8 −1.78075
\(189\) 0 0
\(190\) −6.46393e6 −0.0683690
\(191\) −3.38716e7 −0.351738 −0.175869 0.984414i \(-0.556273\pi\)
−0.175869 + 0.984414i \(0.556273\pi\)
\(192\) 0 0
\(193\) −5.74536e7 −0.575264 −0.287632 0.957741i \(-0.592868\pi\)
−0.287632 + 0.957741i \(0.592868\pi\)
\(194\) −5.75172e6 −0.0565576
\(195\) 0 0
\(196\) 8.57051e7 0.813037
\(197\) −7.82632e7 −0.729333 −0.364667 0.931138i \(-0.618817\pi\)
−0.364667 + 0.931138i \(0.618817\pi\)
\(198\) 0 0
\(199\) −1.18386e8 −1.06491 −0.532455 0.846458i \(-0.678731\pi\)
−0.532455 + 0.846458i \(0.678731\pi\)
\(200\) −1.85495e7 −0.163956
\(201\) 0 0
\(202\) 3.93859e7 0.336210
\(203\) −9.03271e7 −0.757848
\(204\) 0 0
\(205\) 2.02141e8 1.63876
\(206\) −2.25218e7 −0.179501
\(207\) 0 0
\(208\) 1.27798e7 0.0984698
\(209\) −3.46272e7 −0.262365
\(210\) 0 0
\(211\) 4.78501e7 0.350666 0.175333 0.984509i \(-0.443900\pi\)
0.175333 + 0.984509i \(0.443900\pi\)
\(212\) 1.49425e8 1.07708
\(213\) 0 0
\(214\) −4.53385e7 −0.316241
\(215\) 8.41923e7 0.577747
\(216\) 0 0
\(217\) 4.15957e7 0.276337
\(218\) 3.98361e7 0.260423
\(219\) 0 0
\(220\) −1.51318e8 −0.958102
\(221\) −3.12865e6 −0.0194977
\(222\) 0 0
\(223\) 1.24666e8 0.752800 0.376400 0.926457i \(-0.377162\pi\)
0.376400 + 0.926457i \(0.377162\pi\)
\(224\) 3.39184e7 0.201636
\(225\) 0 0
\(226\) 2.49267e7 0.143643
\(227\) −7.13606e7 −0.404919 −0.202460 0.979291i \(-0.564893\pi\)
−0.202460 + 0.979291i \(0.564893\pi\)
\(228\) 0 0
\(229\) −3.39682e8 −1.86917 −0.934584 0.355743i \(-0.884228\pi\)
−0.934584 + 0.355743i \(0.884228\pi\)
\(230\) −7.77879e7 −0.421565
\(231\) 0 0
\(232\) 1.23096e8 0.647198
\(233\) 3.19085e8 1.65257 0.826286 0.563250i \(-0.190449\pi\)
0.826286 + 0.563250i \(0.190449\pi\)
\(234\) 0 0
\(235\) 4.44147e8 2.23249
\(236\) 2.54904e7 0.126236
\(237\) 0 0
\(238\) −2.62372e6 −0.0126153
\(239\) −2.48784e8 −1.17877 −0.589387 0.807851i \(-0.700631\pi\)
−0.589387 + 0.807851i \(0.700631\pi\)
\(240\) 0 0
\(241\) −1.02085e8 −0.469788 −0.234894 0.972021i \(-0.575474\pi\)
−0.234894 + 0.972021i \(0.575474\pi\)
\(242\) −1.30343e7 −0.0591200
\(243\) 0 0
\(244\) −5.32276e7 −0.234570
\(245\) −2.34627e8 −1.01929
\(246\) 0 0
\(247\) −8.27331e6 −0.0349333
\(248\) −5.66859e7 −0.235990
\(249\) 0 0
\(250\) −2.73304e7 −0.110626
\(251\) −1.69390e8 −0.676131 −0.338066 0.941123i \(-0.609773\pi\)
−0.338066 + 0.941123i \(0.609773\pi\)
\(252\) 0 0
\(253\) −4.16709e8 −1.61775
\(254\) 6.32102e7 0.242030
\(255\) 0 0
\(256\) 1.90594e8 0.710018
\(257\) 4.15160e8 1.52563 0.762815 0.646617i \(-0.223817\pi\)
0.762815 + 0.646617i \(0.223817\pi\)
\(258\) 0 0
\(259\) −7.09666e7 −0.253808
\(260\) −3.61537e7 −0.127569
\(261\) 0 0
\(262\) 4.69387e7 0.161241
\(263\) 2.46530e8 0.835650 0.417825 0.908527i \(-0.362792\pi\)
0.417825 + 0.908527i \(0.362792\pi\)
\(264\) 0 0
\(265\) −4.09068e8 −1.35032
\(266\) −6.93810e6 −0.0226024
\(267\) 0 0
\(268\) 2.49501e8 0.791773
\(269\) −3.18928e8 −0.998987 −0.499494 0.866318i \(-0.666481\pi\)
−0.499494 + 0.866318i \(0.666481\pi\)
\(270\) 0 0
\(271\) −3.87904e8 −1.18394 −0.591972 0.805958i \(-0.701651\pi\)
−0.591972 + 0.805958i \(0.701651\pi\)
\(272\) −5.44000e7 −0.163911
\(273\) 0 0
\(274\) −4.21476e7 −0.123779
\(275\) 1.33921e8 0.388315
\(276\) 0 0
\(277\) −7.42290e7 −0.209843 −0.104921 0.994481i \(-0.533459\pi\)
−0.104921 + 0.994481i \(0.533459\pi\)
\(278\) −7.18082e7 −0.200455
\(279\) 0 0
\(280\) −6.15873e7 −0.167663
\(281\) 6.16298e8 1.65698 0.828492 0.560001i \(-0.189199\pi\)
0.828492 + 0.560001i \(0.189199\pi\)
\(282\) 0 0
\(283\) 7.96612e7 0.208927 0.104463 0.994529i \(-0.466687\pi\)
0.104463 + 0.994529i \(0.466687\pi\)
\(284\) 2.33157e8 0.603996
\(285\) 0 0
\(286\) 6.06438e6 0.0153287
\(287\) 2.16969e8 0.541766
\(288\) 0 0
\(289\) −3.97021e8 −0.967544
\(290\) −1.65898e8 −0.399436
\(291\) 0 0
\(292\) −2.91728e8 −0.685707
\(293\) −2.50390e8 −0.581542 −0.290771 0.956793i \(-0.593912\pi\)
−0.290771 + 0.956793i \(0.593912\pi\)
\(294\) 0 0
\(295\) −6.97828e7 −0.158260
\(296\) 9.67121e7 0.216750
\(297\) 0 0
\(298\) −187732. −0.000410944 0
\(299\) −9.95623e7 −0.215400
\(300\) 0 0
\(301\) 9.03684e7 0.191000
\(302\) 9.29799e7 0.194252
\(303\) 0 0
\(304\) −1.43854e8 −0.293673
\(305\) 1.45717e8 0.294076
\(306\) 0 0
\(307\) −3.31396e8 −0.653677 −0.326839 0.945080i \(-0.605983\pi\)
−0.326839 + 0.945080i \(0.605983\pi\)
\(308\) −1.62418e8 −0.316743
\(309\) 0 0
\(310\) 7.63960e7 0.145648
\(311\) 9.41919e7 0.177563 0.0887815 0.996051i \(-0.471703\pi\)
0.0887815 + 0.996051i \(0.471703\pi\)
\(312\) 0 0
\(313\) −5.39491e8 −0.994441 −0.497220 0.867624i \(-0.665646\pi\)
−0.497220 + 0.867624i \(0.665646\pi\)
\(314\) 8.76411e7 0.159755
\(315\) 0 0
\(316\) 4.65270e8 0.829470
\(317\) −6.91838e8 −1.21982 −0.609912 0.792469i \(-0.708795\pi\)
−0.609912 + 0.792469i \(0.708795\pi\)
\(318\) 0 0
\(319\) −8.88713e8 −1.53283
\(320\) −5.86019e8 −0.999739
\(321\) 0 0
\(322\) −8.34942e7 −0.139367
\(323\) 3.52171e7 0.0581494
\(324\) 0 0
\(325\) 3.19971e7 0.0517033
\(326\) −1.94375e8 −0.310727
\(327\) 0 0
\(328\) −2.95682e8 −0.462665
\(329\) 4.76729e8 0.738050
\(330\) 0 0
\(331\) −8.54742e8 −1.29550 −0.647750 0.761853i \(-0.724290\pi\)
−0.647750 + 0.761853i \(0.724290\pi\)
\(332\) 3.98763e8 0.598042
\(333\) 0 0
\(334\) −3.16678e7 −0.0465056
\(335\) −6.83038e8 −0.992631
\(336\) 0 0
\(337\) 1.27131e8 0.180945 0.0904723 0.995899i \(-0.471162\pi\)
0.0904723 + 0.995899i \(0.471162\pi\)
\(338\) −1.22251e8 −0.172204
\(339\) 0 0
\(340\) 1.53896e8 0.212349
\(341\) 4.09253e8 0.558922
\(342\) 0 0
\(343\) −5.52185e8 −0.738848
\(344\) −1.23152e8 −0.163113
\(345\) 0 0
\(346\) −1.89960e8 −0.246544
\(347\) 6.17139e7 0.0792920 0.0396460 0.999214i \(-0.487377\pi\)
0.0396460 + 0.999214i \(0.487377\pi\)
\(348\) 0 0
\(349\) 3.92987e8 0.494868 0.247434 0.968905i \(-0.420413\pi\)
0.247434 + 0.968905i \(0.420413\pi\)
\(350\) 2.68332e7 0.0334529
\(351\) 0 0
\(352\) 3.33717e8 0.407830
\(353\) 1.45872e9 1.76506 0.882531 0.470254i \(-0.155838\pi\)
0.882531 + 0.470254i \(0.155838\pi\)
\(354\) 0 0
\(355\) −6.38294e8 −0.757219
\(356\) −6.75857e7 −0.0793926
\(357\) 0 0
\(358\) −1.37330e8 −0.158188
\(359\) 1.00957e9 1.15162 0.575808 0.817585i \(-0.304687\pi\)
0.575808 + 0.817585i \(0.304687\pi\)
\(360\) 0 0
\(361\) −8.00744e8 −0.895816
\(362\) −2.06523e8 −0.228817
\(363\) 0 0
\(364\) −3.88058e7 −0.0421737
\(365\) 7.98639e8 0.859658
\(366\) 0 0
\(367\) 6.89693e8 0.728324 0.364162 0.931336i \(-0.381355\pi\)
0.364162 + 0.931336i \(0.381355\pi\)
\(368\) −1.73116e9 −1.81080
\(369\) 0 0
\(370\) −1.30340e8 −0.133774
\(371\) −4.39076e8 −0.446407
\(372\) 0 0
\(373\) 1.52953e7 0.0152608 0.00763041 0.999971i \(-0.497571\pi\)
0.00763041 + 0.999971i \(0.497571\pi\)
\(374\) −2.58143e7 −0.0255159
\(375\) 0 0
\(376\) −6.49678e8 −0.630290
\(377\) −2.12336e8 −0.204093
\(378\) 0 0
\(379\) −1.31066e9 −1.23667 −0.618335 0.785915i \(-0.712192\pi\)
−0.618335 + 0.785915i \(0.712192\pi\)
\(380\) 4.06959e8 0.380459
\(381\) 0 0
\(382\) −6.67731e7 −0.0612886
\(383\) −6.67359e8 −0.606966 −0.303483 0.952837i \(-0.598149\pi\)
−0.303483 + 0.952837i \(0.598149\pi\)
\(384\) 0 0
\(385\) 4.44639e8 0.397095
\(386\) −1.13262e8 −0.100237
\(387\) 0 0
\(388\) 3.62119e8 0.314731
\(389\) −2.25715e9 −1.94418 −0.972089 0.234611i \(-0.924618\pi\)
−0.972089 + 0.234611i \(0.924618\pi\)
\(390\) 0 0
\(391\) 4.23808e8 0.358551
\(392\) 3.43202e8 0.287772
\(393\) 0 0
\(394\) −1.54285e8 −0.127083
\(395\) −1.27373e9 −1.03989
\(396\) 0 0
\(397\) −9.81086e8 −0.786938 −0.393469 0.919338i \(-0.628725\pi\)
−0.393469 + 0.919338i \(0.628725\pi\)
\(398\) −2.33381e8 −0.185556
\(399\) 0 0
\(400\) 5.56357e8 0.434654
\(401\) 5.90695e7 0.0457465 0.0228732 0.999738i \(-0.492719\pi\)
0.0228732 + 0.999738i \(0.492719\pi\)
\(402\) 0 0
\(403\) 9.77807e7 0.0744193
\(404\) −2.47967e9 −1.87094
\(405\) 0 0
\(406\) −1.78067e8 −0.132051
\(407\) −6.98228e8 −0.513354
\(408\) 0 0
\(409\) −7.74847e8 −0.559995 −0.279998 0.960001i \(-0.590334\pi\)
−0.279998 + 0.960001i \(0.590334\pi\)
\(410\) 3.98493e8 0.285547
\(411\) 0 0
\(412\) 1.41794e9 0.998887
\(413\) −7.49018e7 −0.0523199
\(414\) 0 0
\(415\) −1.09166e9 −0.749754
\(416\) 7.97335e7 0.0543018
\(417\) 0 0
\(418\) −6.82628e7 −0.0457159
\(419\) −2.71966e8 −0.180620 −0.0903098 0.995914i \(-0.528786\pi\)
−0.0903098 + 0.995914i \(0.528786\pi\)
\(420\) 0 0
\(421\) −8.59819e8 −0.561591 −0.280795 0.959768i \(-0.590598\pi\)
−0.280795 + 0.959768i \(0.590598\pi\)
\(422\) 9.43298e7 0.0611020
\(423\) 0 0
\(424\) 5.98366e8 0.381229
\(425\) −1.36202e8 −0.0860644
\(426\) 0 0
\(427\) 1.56406e8 0.0972200
\(428\) 2.85444e9 1.75982
\(429\) 0 0
\(430\) 1.65973e8 0.100670
\(431\) 6.62428e8 0.398536 0.199268 0.979945i \(-0.436144\pi\)
0.199268 + 0.979945i \(0.436144\pi\)
\(432\) 0 0
\(433\) 2.01470e9 1.19262 0.596312 0.802753i \(-0.296632\pi\)
0.596312 + 0.802753i \(0.296632\pi\)
\(434\) 8.20001e7 0.0481505
\(435\) 0 0
\(436\) −2.50801e9 −1.44920
\(437\) 1.12071e9 0.642403
\(438\) 0 0
\(439\) 2.41587e9 1.36285 0.681425 0.731888i \(-0.261360\pi\)
0.681425 + 0.731888i \(0.261360\pi\)
\(440\) −6.05946e8 −0.339117
\(441\) 0 0
\(442\) −6.16769e6 −0.00339739
\(443\) 2.74104e9 1.49796 0.748982 0.662590i \(-0.230543\pi\)
0.748982 + 0.662590i \(0.230543\pi\)
\(444\) 0 0
\(445\) 1.85024e8 0.0995330
\(446\) 2.45761e8 0.131172
\(447\) 0 0
\(448\) −6.29007e8 −0.330508
\(449\) 1.37537e9 0.717063 0.358532 0.933518i \(-0.383277\pi\)
0.358532 + 0.933518i \(0.383277\pi\)
\(450\) 0 0
\(451\) 2.13472e9 1.09578
\(452\) −1.56935e9 −0.799344
\(453\) 0 0
\(454\) −1.40678e8 −0.0705553
\(455\) 1.06235e8 0.0528724
\(456\) 0 0
\(457\) 1.14427e9 0.560816 0.280408 0.959881i \(-0.409530\pi\)
0.280408 + 0.959881i \(0.409530\pi\)
\(458\) −6.69636e8 −0.325694
\(459\) 0 0
\(460\) 4.89740e9 2.34592
\(461\) −7.69708e8 −0.365909 −0.182954 0.983121i \(-0.558566\pi\)
−0.182954 + 0.983121i \(0.558566\pi\)
\(462\) 0 0
\(463\) −2.23996e8 −0.104884 −0.0524418 0.998624i \(-0.516700\pi\)
−0.0524418 + 0.998624i \(0.516700\pi\)
\(464\) −3.69204e9 −1.71575
\(465\) 0 0
\(466\) 6.29032e8 0.287953
\(467\) −3.61971e8 −0.164462 −0.0822308 0.996613i \(-0.526204\pi\)
−0.0822308 + 0.996613i \(0.526204\pi\)
\(468\) 0 0
\(469\) −7.33143e8 −0.328159
\(470\) 8.75575e8 0.389001
\(471\) 0 0
\(472\) 1.02075e8 0.0446809
\(473\) 8.89119e8 0.386319
\(474\) 0 0
\(475\) −3.60170e8 −0.154199
\(476\) 1.65185e8 0.0702016
\(477\) 0 0
\(478\) −4.90444e8 −0.205396
\(479\) 1.79221e9 0.745100 0.372550 0.928012i \(-0.378484\pi\)
0.372550 + 0.928012i \(0.378484\pi\)
\(480\) 0 0
\(481\) −1.66824e8 −0.0683520
\(482\) −2.01246e8 −0.0818583
\(483\) 0 0
\(484\) 8.20620e8 0.328990
\(485\) −9.91341e8 −0.394573
\(486\) 0 0
\(487\) 1.52055e9 0.596554 0.298277 0.954479i \(-0.403588\pi\)
0.298277 + 0.954479i \(0.403588\pi\)
\(488\) −2.13147e8 −0.0830253
\(489\) 0 0
\(490\) −4.62536e8 −0.177606
\(491\) 3.45739e9 1.31815 0.659073 0.752079i \(-0.270949\pi\)
0.659073 + 0.752079i \(0.270949\pi\)
\(492\) 0 0
\(493\) 9.03853e8 0.339730
\(494\) −1.63097e7 −0.00608697
\(495\) 0 0
\(496\) 1.70018e9 0.625619
\(497\) −6.85117e8 −0.250333
\(498\) 0 0
\(499\) −5.24675e8 −0.189033 −0.0945167 0.995523i \(-0.530131\pi\)
−0.0945167 + 0.995523i \(0.530131\pi\)
\(500\) 1.72068e9 0.615609
\(501\) 0 0
\(502\) −3.33930e8 −0.117813
\(503\) 2.08499e9 0.730492 0.365246 0.930911i \(-0.380985\pi\)
0.365246 + 0.930911i \(0.380985\pi\)
\(504\) 0 0
\(505\) 6.78839e9 2.34556
\(506\) −8.21485e8 −0.281885
\(507\) 0 0
\(508\) −3.97961e9 −1.34685
\(509\) −3.13440e9 −1.05352 −0.526760 0.850014i \(-0.676593\pi\)
−0.526760 + 0.850014i \(0.676593\pi\)
\(510\) 0 0
\(511\) 8.57224e8 0.284198
\(512\) 2.33471e9 0.768754
\(513\) 0 0
\(514\) 8.18430e8 0.265834
\(515\) −3.88176e9 −1.25229
\(516\) 0 0
\(517\) 4.69045e9 1.49279
\(518\) −1.39901e8 −0.0442248
\(519\) 0 0
\(520\) −1.44776e8 −0.0451527
\(521\) −9.76015e8 −0.302360 −0.151180 0.988506i \(-0.548307\pi\)
−0.151180 + 0.988506i \(0.548307\pi\)
\(522\) 0 0
\(523\) −1.99309e9 −0.609216 −0.304608 0.952478i \(-0.598526\pi\)
−0.304608 + 0.952478i \(0.598526\pi\)
\(524\) −2.95518e9 −0.897273
\(525\) 0 0
\(526\) 4.86000e8 0.145608
\(527\) −4.16225e8 −0.123877
\(528\) 0 0
\(529\) 1.00819e10 2.96107
\(530\) −8.06422e8 −0.235286
\(531\) 0 0
\(532\) 4.36812e8 0.125778
\(533\) 5.10039e8 0.145901
\(534\) 0 0
\(535\) −7.81435e9 −2.20625
\(536\) 9.99115e8 0.280246
\(537\) 0 0
\(538\) −6.28723e8 −0.174069
\(539\) −2.47780e9 −0.681562
\(540\) 0 0
\(541\) 3.17125e9 0.861072 0.430536 0.902573i \(-0.358325\pi\)
0.430536 + 0.902573i \(0.358325\pi\)
\(542\) −7.64698e8 −0.206297
\(543\) 0 0
\(544\) −3.39403e8 −0.0903897
\(545\) 6.86597e9 1.81683
\(546\) 0 0
\(547\) 2.77827e9 0.725803 0.362901 0.931828i \(-0.381786\pi\)
0.362901 + 0.931828i \(0.381786\pi\)
\(548\) 2.65354e9 0.688802
\(549\) 0 0
\(550\) 2.64007e8 0.0676621
\(551\) 2.39012e9 0.608681
\(552\) 0 0
\(553\) −1.36717e9 −0.343782
\(554\) −1.46332e8 −0.0365641
\(555\) 0 0
\(556\) 4.52093e9 1.11549
\(557\) 7.99807e8 0.196107 0.0980533 0.995181i \(-0.468738\pi\)
0.0980533 + 0.995181i \(0.468738\pi\)
\(558\) 0 0
\(559\) 2.12433e8 0.0514375
\(560\) 1.84719e9 0.444482
\(561\) 0 0
\(562\) 1.21495e9 0.288722
\(563\) 1.37529e9 0.324800 0.162400 0.986725i \(-0.448077\pi\)
0.162400 + 0.986725i \(0.448077\pi\)
\(564\) 0 0
\(565\) 4.29626e9 1.00212
\(566\) 1.57041e8 0.0364045
\(567\) 0 0
\(568\) 9.33666e8 0.213783
\(569\) −4.45989e8 −0.101492 −0.0507459 0.998712i \(-0.516160\pi\)
−0.0507459 + 0.998712i \(0.516160\pi\)
\(570\) 0 0
\(571\) −7.16360e9 −1.61029 −0.805147 0.593076i \(-0.797914\pi\)
−0.805147 + 0.593076i \(0.797914\pi\)
\(572\) −3.81804e8 −0.0853010
\(573\) 0 0
\(574\) 4.27725e8 0.0944002
\(575\) −4.33434e9 −0.950793
\(576\) 0 0
\(577\) −5.75941e9 −1.24814 −0.624069 0.781369i \(-0.714522\pi\)
−0.624069 + 0.781369i \(0.714522\pi\)
\(578\) −7.82672e8 −0.168590
\(579\) 0 0
\(580\) 1.04447e10 2.22278
\(581\) −1.17174e9 −0.247865
\(582\) 0 0
\(583\) −4.31999e9 −0.902908
\(584\) −1.16821e9 −0.242704
\(585\) 0 0
\(586\) −4.93610e8 −0.101331
\(587\) 2.89213e9 0.590180 0.295090 0.955469i \(-0.404650\pi\)
0.295090 + 0.955469i \(0.404650\pi\)
\(588\) 0 0
\(589\) −1.10065e9 −0.221946
\(590\) −1.37567e8 −0.0275761
\(591\) 0 0
\(592\) −2.90069e9 −0.574613
\(593\) 2.52669e9 0.497577 0.248789 0.968558i \(-0.419968\pi\)
0.248789 + 0.968558i \(0.419968\pi\)
\(594\) 0 0
\(595\) −4.52214e8 −0.0880104
\(596\) 1.18193e7 0.00228682
\(597\) 0 0
\(598\) −1.96273e8 −0.0375325
\(599\) 9.87541e9 1.87742 0.938709 0.344709i \(-0.112023\pi\)
0.938709 + 0.344709i \(0.112023\pi\)
\(600\) 0 0
\(601\) −1.69668e9 −0.318815 −0.159408 0.987213i \(-0.550958\pi\)
−0.159408 + 0.987213i \(0.550958\pi\)
\(602\) 1.78149e8 0.0332809
\(603\) 0 0
\(604\) −5.85386e9 −1.08097
\(605\) −2.24654e9 −0.412449
\(606\) 0 0
\(607\) −9.97951e9 −1.81113 −0.905564 0.424211i \(-0.860552\pi\)
−0.905564 + 0.424211i \(0.860552\pi\)
\(608\) −8.97507e8 −0.161948
\(609\) 0 0
\(610\) 2.87260e8 0.0512414
\(611\) 1.12067e9 0.198761
\(612\) 0 0
\(613\) −6.06527e9 −1.06350 −0.531751 0.846901i \(-0.678466\pi\)
−0.531751 + 0.846901i \(0.678466\pi\)
\(614\) −6.53302e8 −0.113900
\(615\) 0 0
\(616\) −6.50396e8 −0.112110
\(617\) 1.40806e9 0.241336 0.120668 0.992693i \(-0.461496\pi\)
0.120668 + 0.992693i \(0.461496\pi\)
\(618\) 0 0
\(619\) −2.17547e9 −0.368668 −0.184334 0.982864i \(-0.559013\pi\)
−0.184334 + 0.982864i \(0.559013\pi\)
\(620\) −4.80977e9 −0.810501
\(621\) 0 0
\(622\) 1.85686e8 0.0309395
\(623\) 1.98596e8 0.0329051
\(624\) 0 0
\(625\) −7.62637e9 −1.24950
\(626\) −1.06353e9 −0.173277
\(627\) 0 0
\(628\) −5.51774e9 −0.889001
\(629\) 7.10123e8 0.113777
\(630\) 0 0
\(631\) −3.80261e9 −0.602531 −0.301266 0.953540i \(-0.597409\pi\)
−0.301266 + 0.953540i \(0.597409\pi\)
\(632\) 1.86315e9 0.293588
\(633\) 0 0
\(634\) −1.36386e9 −0.212549
\(635\) 1.08946e10 1.68852
\(636\) 0 0
\(637\) −5.92008e8 −0.0907486
\(638\) −1.75197e9 −0.267089
\(639\) 0 0
\(640\) −5.20010e9 −0.784118
\(641\) −3.59687e9 −0.539413 −0.269706 0.962943i \(-0.586927\pi\)
−0.269706 + 0.962943i \(0.586927\pi\)
\(642\) 0 0
\(643\) 4.76649e9 0.707066 0.353533 0.935422i \(-0.384980\pi\)
0.353533 + 0.935422i \(0.384980\pi\)
\(644\) 5.25666e9 0.775549
\(645\) 0 0
\(646\) 6.94257e7 0.0101323
\(647\) −1.17944e9 −0.171203 −0.0856016 0.996329i \(-0.527281\pi\)
−0.0856016 + 0.996329i \(0.527281\pi\)
\(648\) 0 0
\(649\) −7.36945e8 −0.105823
\(650\) 6.30778e7 0.00900907
\(651\) 0 0
\(652\) 1.22375e10 1.72913
\(653\) −8.76325e9 −1.23160 −0.615799 0.787903i \(-0.711167\pi\)
−0.615799 + 0.787903i \(0.711167\pi\)
\(654\) 0 0
\(655\) 8.09015e9 1.12489
\(656\) 8.86841e9 1.22654
\(657\) 0 0
\(658\) 9.39804e8 0.128602
\(659\) 1.34224e10 1.82697 0.913487 0.406868i \(-0.133379\pi\)
0.913487 + 0.406868i \(0.133379\pi\)
\(660\) 0 0
\(661\) −5.67841e9 −0.764754 −0.382377 0.924006i \(-0.624894\pi\)
−0.382377 + 0.924006i \(0.624894\pi\)
\(662\) −1.68501e9 −0.225735
\(663\) 0 0
\(664\) 1.59683e9 0.211675
\(665\) −1.19582e9 −0.157685
\(666\) 0 0
\(667\) 2.87631e10 3.75315
\(668\) 1.99375e9 0.258794
\(669\) 0 0
\(670\) −1.34651e9 −0.172961
\(671\) 1.53885e9 0.196638
\(672\) 0 0
\(673\) 2.49548e9 0.315574 0.157787 0.987473i \(-0.449564\pi\)
0.157787 + 0.987473i \(0.449564\pi\)
\(674\) 2.50621e8 0.0315288
\(675\) 0 0
\(676\) 7.69673e9 0.958281
\(677\) −5.54895e9 −0.687307 −0.343654 0.939097i \(-0.611665\pi\)
−0.343654 + 0.939097i \(0.611665\pi\)
\(678\) 0 0
\(679\) −1.06406e9 −0.130444
\(680\) 6.16269e8 0.0751604
\(681\) 0 0
\(682\) 8.06785e8 0.0973896
\(683\) −4.93572e9 −0.592759 −0.296379 0.955070i \(-0.595779\pi\)
−0.296379 + 0.955070i \(0.595779\pi\)
\(684\) 0 0
\(685\) −7.26438e9 −0.863538
\(686\) −1.08856e9 −0.128741
\(687\) 0 0
\(688\) 3.69372e9 0.432419
\(689\) −1.03215e9 −0.120220
\(690\) 0 0
\(691\) −8.63015e9 −0.995050 −0.497525 0.867450i \(-0.665758\pi\)
−0.497525 + 0.867450i \(0.665758\pi\)
\(692\) 1.19595e10 1.37197
\(693\) 0 0
\(694\) 1.21660e8 0.0138163
\(695\) −1.23766e10 −1.39847
\(696\) 0 0
\(697\) −2.17109e9 −0.242864
\(698\) 7.74720e8 0.0862285
\(699\) 0 0
\(700\) −1.68937e9 −0.186158
\(701\) −8.06267e9 −0.884028 −0.442014 0.897008i \(-0.645736\pi\)
−0.442014 + 0.897008i \(0.645736\pi\)
\(702\) 0 0
\(703\) 1.87783e9 0.203851
\(704\) −6.18869e9 −0.668489
\(705\) 0 0
\(706\) 2.87566e9 0.307554
\(707\) 7.28636e9 0.775430
\(708\) 0 0
\(709\) 1.13085e10 1.19163 0.595817 0.803120i \(-0.296828\pi\)
0.595817 + 0.803120i \(0.296828\pi\)
\(710\) −1.25831e9 −0.131942
\(711\) 0 0
\(712\) −2.70644e8 −0.0281008
\(713\) −1.32454e10 −1.36853
\(714\) 0 0
\(715\) 1.04523e9 0.106940
\(716\) 8.64605e9 0.880283
\(717\) 0 0
\(718\) 1.99024e9 0.200664
\(719\) 1.36390e10 1.36846 0.684231 0.729266i \(-0.260138\pi\)
0.684231 + 0.729266i \(0.260138\pi\)
\(720\) 0 0
\(721\) −4.16652e9 −0.413999
\(722\) −1.57856e9 −0.156092
\(723\) 0 0
\(724\) 1.30023e10 1.27332
\(725\) −9.24382e9 −0.900883
\(726\) 0 0
\(727\) 3.69976e9 0.357111 0.178556 0.983930i \(-0.442858\pi\)
0.178556 + 0.983930i \(0.442858\pi\)
\(728\) −1.55396e8 −0.0149273
\(729\) 0 0
\(730\) 1.57441e9 0.149791
\(731\) −9.04265e8 −0.0856219
\(732\) 0 0
\(733\) 1.64513e10 1.54289 0.771447 0.636294i \(-0.219533\pi\)
0.771447 + 0.636294i \(0.219533\pi\)
\(734\) 1.35963e9 0.126907
\(735\) 0 0
\(736\) −1.08007e10 −0.998576
\(737\) −7.21327e9 −0.663737
\(738\) 0 0
\(739\) −4.40695e9 −0.401682 −0.200841 0.979624i \(-0.564367\pi\)
−0.200841 + 0.979624i \(0.564367\pi\)
\(740\) 8.20597e9 0.744421
\(741\) 0 0
\(742\) −8.65578e8 −0.0777844
\(743\) −2.97820e9 −0.266374 −0.133187 0.991091i \(-0.542521\pi\)
−0.133187 + 0.991091i \(0.542521\pi\)
\(744\) 0 0
\(745\) −3.23568e7 −0.00286694
\(746\) 3.01526e7 0.00265913
\(747\) 0 0
\(748\) 1.62523e9 0.141990
\(749\) −8.38758e9 −0.729374
\(750\) 0 0
\(751\) 6.84373e9 0.589594 0.294797 0.955560i \(-0.404748\pi\)
0.294797 + 0.955560i \(0.404748\pi\)
\(752\) 1.94858e10 1.67092
\(753\) 0 0
\(754\) −4.18590e8 −0.0355623
\(755\) 1.60256e10 1.35519
\(756\) 0 0
\(757\) −1.28064e10 −1.07298 −0.536491 0.843906i \(-0.680250\pi\)
−0.536491 + 0.843906i \(0.680250\pi\)
\(758\) −2.58379e9 −0.215484
\(759\) 0 0
\(760\) 1.62965e9 0.134662
\(761\) 1.33708e9 0.109979 0.0549897 0.998487i \(-0.482487\pi\)
0.0549897 + 0.998487i \(0.482487\pi\)
\(762\) 0 0
\(763\) 7.36964e9 0.600634
\(764\) 4.20393e9 0.341058
\(765\) 0 0
\(766\) −1.31561e9 −0.105761
\(767\) −1.76075e8 −0.0140901
\(768\) 0 0
\(769\) 1.10898e10 0.879390 0.439695 0.898147i \(-0.355086\pi\)
0.439695 + 0.898147i \(0.355086\pi\)
\(770\) 8.76544e8 0.0691920
\(771\) 0 0
\(772\) 7.13078e9 0.557798
\(773\) 1.78044e10 1.38644 0.693218 0.720728i \(-0.256192\pi\)
0.693218 + 0.720728i \(0.256192\pi\)
\(774\) 0 0
\(775\) 4.25679e9 0.328493
\(776\) 1.45009e9 0.111398
\(777\) 0 0
\(778\) −4.44965e9 −0.338764
\(779\) −5.74117e9 −0.435130
\(780\) 0 0
\(781\) −6.74075e9 −0.506325
\(782\) 8.35479e8 0.0624758
\(783\) 0 0
\(784\) −1.02937e10 −0.762894
\(785\) 1.51054e10 1.11452
\(786\) 0 0
\(787\) 1.97486e10 1.44419 0.722095 0.691794i \(-0.243179\pi\)
0.722095 + 0.691794i \(0.243179\pi\)
\(788\) 9.71354e9 0.707190
\(789\) 0 0
\(790\) −2.51098e9 −0.181196
\(791\) 4.61142e9 0.331297
\(792\) 0 0
\(793\) 3.67670e8 0.0261820
\(794\) −1.93408e9 −0.137120
\(795\) 0 0
\(796\) 1.46933e10 1.03258
\(797\) −1.84228e10 −1.28900 −0.644499 0.764605i \(-0.722934\pi\)
−0.644499 + 0.764605i \(0.722934\pi\)
\(798\) 0 0
\(799\) −4.77035e9 −0.330854
\(800\) 3.47112e9 0.239692
\(801\) 0 0
\(802\) 1.16447e8 0.00797111
\(803\) 8.43408e9 0.574822
\(804\) 0 0
\(805\) −1.43907e10 −0.972291
\(806\) 1.92761e8 0.0129672
\(807\) 0 0
\(808\) −9.92972e9 −0.662213
\(809\) −1.12171e10 −0.744838 −0.372419 0.928065i \(-0.621472\pi\)
−0.372419 + 0.928065i \(0.621472\pi\)
\(810\) 0 0
\(811\) 1.91381e10 1.25987 0.629934 0.776649i \(-0.283082\pi\)
0.629934 + 0.776649i \(0.283082\pi\)
\(812\) 1.12108e10 0.734838
\(813\) 0 0
\(814\) −1.37646e9 −0.0894495
\(815\) −3.35017e10 −2.16778
\(816\) 0 0
\(817\) −2.39122e9 −0.153406
\(818\) −1.52750e9 −0.0975766
\(819\) 0 0
\(820\) −2.50885e10 −1.58901
\(821\) −1.26324e10 −0.796685 −0.398342 0.917237i \(-0.630414\pi\)
−0.398342 + 0.917237i \(0.630414\pi\)
\(822\) 0 0
\(823\) −5.60518e9 −0.350502 −0.175251 0.984524i \(-0.556074\pi\)
−0.175251 + 0.984524i \(0.556074\pi\)
\(824\) 5.67806e9 0.353553
\(825\) 0 0
\(826\) −1.47658e8 −0.00911650
\(827\) 9.05725e9 0.556836 0.278418 0.960460i \(-0.410190\pi\)
0.278418 + 0.960460i \(0.410190\pi\)
\(828\) 0 0
\(829\) 1.80271e10 1.09897 0.549483 0.835505i \(-0.314825\pi\)
0.549483 + 0.835505i \(0.314825\pi\)
\(830\) −2.15206e9 −0.130641
\(831\) 0 0
\(832\) −1.47863e9 −0.0890080
\(833\) 2.52001e9 0.151058
\(834\) 0 0
\(835\) −5.45813e9 −0.324445
\(836\) 4.29771e9 0.254399
\(837\) 0 0
\(838\) −5.36143e8 −0.0314721
\(839\) −2.52101e10 −1.47370 −0.736848 0.676059i \(-0.763687\pi\)
−0.736848 + 0.676059i \(0.763687\pi\)
\(840\) 0 0
\(841\) 4.40930e10 2.55613
\(842\) −1.69501e9 −0.0978546
\(843\) 0 0
\(844\) −5.93885e9 −0.340020
\(845\) −2.10707e10 −1.20138
\(846\) 0 0
\(847\) −2.41134e9 −0.136353
\(848\) −1.79468e10 −1.01065
\(849\) 0 0
\(850\) −2.68504e8 −0.0149963
\(851\) 2.25981e10 1.25695
\(852\) 0 0
\(853\) −4.60485e9 −0.254035 −0.127017 0.991900i \(-0.540540\pi\)
−0.127017 + 0.991900i \(0.540540\pi\)
\(854\) 3.08333e8 0.0169401
\(855\) 0 0
\(856\) 1.14305e10 0.622881
\(857\) −2.64820e10 −1.43720 −0.718600 0.695423i \(-0.755217\pi\)
−0.718600 + 0.695423i \(0.755217\pi\)
\(858\) 0 0
\(859\) −3.30189e10 −1.77740 −0.888702 0.458485i \(-0.848392\pi\)
−0.888702 + 0.458485i \(0.848392\pi\)
\(860\) −1.04494e10 −0.560206
\(861\) 0 0
\(862\) 1.30588e9 0.0694431
\(863\) 1.44548e10 0.765552 0.382776 0.923841i \(-0.374968\pi\)
0.382776 + 0.923841i \(0.374968\pi\)
\(864\) 0 0
\(865\) −3.27406e10 −1.72001
\(866\) 3.97171e9 0.207809
\(867\) 0 0
\(868\) −5.16260e9 −0.267947
\(869\) −1.34513e10 −0.695337
\(870\) 0 0
\(871\) −1.72343e9 −0.0883751
\(872\) −1.00432e10 −0.512938
\(873\) 0 0
\(874\) 2.20932e9 0.111936
\(875\) −5.05610e9 −0.255145
\(876\) 0 0
\(877\) 1.61500e9 0.0808491 0.0404245 0.999183i \(-0.487129\pi\)
0.0404245 + 0.999183i \(0.487129\pi\)
\(878\) 4.76256e9 0.237470
\(879\) 0 0
\(880\) 1.81742e10 0.899013
\(881\) −3.62460e10 −1.78585 −0.892925 0.450206i \(-0.851351\pi\)
−0.892925 + 0.450206i \(0.851351\pi\)
\(882\) 0 0
\(883\) 1.90627e10 0.931796 0.465898 0.884838i \(-0.345731\pi\)
0.465898 + 0.884838i \(0.345731\pi\)
\(884\) 3.88308e8 0.0189057
\(885\) 0 0
\(886\) 5.40357e9 0.261013
\(887\) 1.75867e9 0.0846157 0.0423079 0.999105i \(-0.486529\pi\)
0.0423079 + 0.999105i \(0.486529\pi\)
\(888\) 0 0
\(889\) 1.16938e10 0.558214
\(890\) 3.64748e8 0.0173432
\(891\) 0 0
\(892\) −1.54727e10 −0.729944
\(893\) −1.26146e10 −0.592780
\(894\) 0 0
\(895\) −2.36695e10 −1.10359
\(896\) −5.58156e9 −0.259225
\(897\) 0 0
\(898\) 2.71135e9 0.124945
\(899\) −2.82484e10 −1.29669
\(900\) 0 0
\(901\) 4.39359e9 0.200116
\(902\) 4.20831e9 0.190935
\(903\) 0 0
\(904\) −6.28436e9 −0.282925
\(905\) −3.55954e10 −1.59633
\(906\) 0 0
\(907\) −3.74603e10 −1.66704 −0.833520 0.552489i \(-0.813678\pi\)
−0.833520 + 0.552489i \(0.813678\pi\)
\(908\) 8.85684e9 0.392625
\(909\) 0 0
\(910\) 2.09428e8 0.00921277
\(911\) −8.20611e9 −0.359603 −0.179801 0.983703i \(-0.557545\pi\)
−0.179801 + 0.983703i \(0.557545\pi\)
\(912\) 0 0
\(913\) −1.15285e10 −0.501334
\(914\) 2.25576e9 0.0977196
\(915\) 0 0
\(916\) 4.21592e10 1.81242
\(917\) 8.68361e9 0.371884
\(918\) 0 0
\(919\) −1.35638e9 −0.0576472 −0.0288236 0.999585i \(-0.509176\pi\)
−0.0288236 + 0.999585i \(0.509176\pi\)
\(920\) 1.96114e10 0.830331
\(921\) 0 0
\(922\) −1.51737e9 −0.0637579
\(923\) −1.61053e9 −0.0674161
\(924\) 0 0
\(925\) −7.26252e9 −0.301711
\(926\) −4.41578e8 −0.0182755
\(927\) 0 0
\(928\) −2.30347e10 −0.946158
\(929\) −6.23174e9 −0.255008 −0.127504 0.991838i \(-0.540697\pi\)
−0.127504 + 0.991838i \(0.540697\pi\)
\(930\) 0 0
\(931\) 6.66385e9 0.270646
\(932\) −3.96028e10 −1.60240
\(933\) 0 0
\(934\) −7.13575e8 −0.0286567
\(935\) −4.44925e9 −0.178011
\(936\) 0 0
\(937\) −3.18495e10 −1.26478 −0.632389 0.774651i \(-0.717926\pi\)
−0.632389 + 0.774651i \(0.717926\pi\)
\(938\) −1.44529e9 −0.0571801
\(939\) 0 0
\(940\) −5.51248e10 −2.16471
\(941\) 4.56004e10 1.78404 0.892021 0.451995i \(-0.149287\pi\)
0.892021 + 0.451995i \(0.149287\pi\)
\(942\) 0 0
\(943\) −6.90901e10 −2.68303
\(944\) −3.06154e9 −0.118451
\(945\) 0 0
\(946\) 1.75277e9 0.0673142
\(947\) −1.39598e10 −0.534138 −0.267069 0.963677i \(-0.586055\pi\)
−0.267069 + 0.963677i \(0.586055\pi\)
\(948\) 0 0
\(949\) 2.01511e9 0.0765363
\(950\) −7.10026e8 −0.0268684
\(951\) 0 0
\(952\) 6.61476e8 0.0248476
\(953\) 1.88757e10 0.706444 0.353222 0.935540i \(-0.385086\pi\)
0.353222 + 0.935540i \(0.385086\pi\)
\(954\) 0 0
\(955\) −1.15087e10 −0.427578
\(956\) 3.08776e10 1.14298
\(957\) 0 0
\(958\) 3.53309e9 0.129830
\(959\) −7.79727e9 −0.285481
\(960\) 0 0
\(961\) −1.45042e10 −0.527183
\(962\) −3.28871e8 −0.0119100
\(963\) 0 0
\(964\) 1.26701e10 0.455524
\(965\) −1.95213e10 −0.699301
\(966\) 0 0
\(967\) −4.38222e10 −1.55848 −0.779241 0.626724i \(-0.784395\pi\)
−0.779241 + 0.626724i \(0.784395\pi\)
\(968\) 3.28613e9 0.116445
\(969\) 0 0
\(970\) −1.95429e9 −0.0687525
\(971\) −7.68907e8 −0.0269529 −0.0134765 0.999909i \(-0.504290\pi\)
−0.0134765 + 0.999909i \(0.504290\pi\)
\(972\) 0 0
\(973\) −1.32845e10 −0.462326
\(974\) 2.99756e9 0.103947
\(975\) 0 0
\(976\) 6.39295e9 0.220103
\(977\) 3.25858e10 1.11788 0.558942 0.829206i \(-0.311207\pi\)
0.558942 + 0.829206i \(0.311207\pi\)
\(978\) 0 0
\(979\) 1.95395e9 0.0665541
\(980\) 2.91205e10 0.988342
\(981\) 0 0
\(982\) 6.81577e9 0.229681
\(983\) 3.29965e10 1.10798 0.553988 0.832525i \(-0.313106\pi\)
0.553988 + 0.832525i \(0.313106\pi\)
\(984\) 0 0
\(985\) −2.65919e10 −0.886590
\(986\) 1.78182e9 0.0591963
\(987\) 0 0
\(988\) 1.02683e9 0.0338727
\(989\) −2.87763e10 −0.945904
\(990\) 0 0
\(991\) 2.11125e10 0.689100 0.344550 0.938768i \(-0.388031\pi\)
0.344550 + 0.938768i \(0.388031\pi\)
\(992\) 1.06075e10 0.345002
\(993\) 0 0
\(994\) −1.35061e9 −0.0436193
\(995\) −4.02245e10 −1.29452
\(996\) 0 0
\(997\) −2.81284e10 −0.898902 −0.449451 0.893305i \(-0.648380\pi\)
−0.449451 + 0.893305i \(0.648380\pi\)
\(998\) −1.03432e9 −0.0329382
\(999\) 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.1.9 16
3.2 odd 2 177.8.a.a.1.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.8 16 3.2 odd 2
531.8.a.b.1.9 16 1.1 even 1 trivial