Properties

Label 531.8.a.b.1.14
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 1.14
Root \(16.2952\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+16.2952 q^{2} +137.534 q^{4} -495.569 q^{5} +565.301 q^{7} +155.363 q^{8} +O(q^{10})\) \(q+16.2952 q^{2} +137.534 q^{4} -495.569 q^{5} +565.301 q^{7} +155.363 q^{8} -8075.41 q^{10} +826.834 q^{11} +11479.1 q^{13} +9211.71 q^{14} -15072.7 q^{16} +23564.6 q^{17} -53546.6 q^{19} -68157.8 q^{20} +13473.4 q^{22} +87155.1 q^{23} +167464. q^{25} +187055. q^{26} +77748.3 q^{28} -26745.6 q^{29} -231681. q^{31} -265500. q^{32} +383990. q^{34} -280146. q^{35} +409893. q^{37} -872554. q^{38} -76993.4 q^{40} -189635. q^{41} -849460. q^{43} +113718. q^{44} +1.42021e6 q^{46} +902990. q^{47} -503977. q^{49} +2.72886e6 q^{50} +1.57877e6 q^{52} +324949. q^{53} -409753. q^{55} +87827.2 q^{56} -435826. q^{58} -205379. q^{59} -1.58592e6 q^{61} -3.77529e6 q^{62} -2.39707e6 q^{64} -5.68870e6 q^{65} -2.57819e6 q^{67} +3.24094e6 q^{68} -4.56504e6 q^{70} +678037. q^{71} +2.50036e6 q^{73} +6.67930e6 q^{74} -7.36449e6 q^{76} +467410. q^{77} -7.33035e6 q^{79} +7.46957e6 q^{80} -3.09015e6 q^{82} -3.91602e6 q^{83} -1.16779e7 q^{85} -1.38421e7 q^{86} +128460. q^{88} -3.80382e6 q^{89} +6.48916e6 q^{91} +1.19868e7 q^{92} +1.47144e7 q^{94} +2.65361e7 q^{95} -7.68854e6 q^{97} -8.21243e6 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} + O(q^{10}) \) \( 16q + 6q^{2} + 974q^{4} + 68q^{5} - 2343q^{7} - 819q^{8} - 3479q^{10} - 898q^{11} - 8172q^{13} + 13315q^{14} + 3138q^{16} + 44985q^{17} - 40137q^{19} - 130657q^{20} + 109394q^{22} + 2833q^{23} + 285746q^{25} + 129420q^{26} + 112890q^{28} - 144375q^{29} - 141759q^{31} + 36224q^{32} - 341332q^{34} + 78859q^{35} - 297971q^{37} - 329075q^{38} - 203048q^{40} - 659077q^{41} - 1431608q^{43} - 254916q^{44} + 873113q^{46} + 1574073q^{47} + 1893545q^{49} - 302533q^{50} - 4972548q^{52} - 587736q^{53} - 4624036q^{55} + 5798506q^{56} - 6991380q^{58} - 3286064q^{59} - 6117131q^{61} + 11570258q^{62} - 19063011q^{64} + 5335514q^{65} - 16518710q^{67} + 17284669q^{68} - 39189486q^{70} + 10882582q^{71} - 21097441q^{73} + 16717030q^{74} - 40864952q^{76} + 3404601q^{77} - 3784458q^{79} + 27466195q^{80} - 24990117q^{82} + 1951425q^{83} - 23238675q^{85} + 35910572q^{86} - 27843055q^{88} - 10499443q^{89} + 699217q^{91} + 20062766q^{92} - 59358988q^{94} + 29236333q^{95} - 25158976q^{97} - 2120460q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.2952 1.44031 0.720154 0.693814i \(-0.244071\pi\)
0.720154 + 0.693814i \(0.244071\pi\)
\(3\) 0 0
\(4\) 137.534 1.07449
\(5\) −495.569 −1.77300 −0.886502 0.462726i \(-0.846872\pi\)
−0.886502 + 0.462726i \(0.846872\pi\)
\(6\) 0 0
\(7\) 565.301 0.622927 0.311463 0.950258i \(-0.399181\pi\)
0.311463 + 0.950258i \(0.399181\pi\)
\(8\) 155.363 0.107284
\(9\) 0 0
\(10\) −8075.41 −2.55367
\(11\) 826.834 0.187303 0.0936513 0.995605i \(-0.470146\pi\)
0.0936513 + 0.995605i \(0.470146\pi\)
\(12\) 0 0
\(13\) 11479.1 1.44913 0.724564 0.689207i \(-0.242041\pi\)
0.724564 + 0.689207i \(0.242041\pi\)
\(14\) 9211.71 0.897206
\(15\) 0 0
\(16\) −15072.7 −0.919965
\(17\) 23564.6 1.16329 0.581646 0.813442i \(-0.302409\pi\)
0.581646 + 0.813442i \(0.302409\pi\)
\(18\) 0 0
\(19\) −53546.6 −1.79099 −0.895497 0.445067i \(-0.853180\pi\)
−0.895497 + 0.445067i \(0.853180\pi\)
\(20\) −68157.8 −1.90507
\(21\) 0 0
\(22\) 13473.4 0.269773
\(23\) 87155.1 1.49364 0.746819 0.665028i \(-0.231580\pi\)
0.746819 + 0.665028i \(0.231580\pi\)
\(24\) 0 0
\(25\) 167464. 2.14354
\(26\) 187055. 2.08719
\(27\) 0 0
\(28\) 77748.3 0.669326
\(29\) −26745.6 −0.203638 −0.101819 0.994803i \(-0.532466\pi\)
−0.101819 + 0.994803i \(0.532466\pi\)
\(30\) 0 0
\(31\) −231681. −1.39677 −0.698384 0.715723i \(-0.746097\pi\)
−0.698384 + 0.715723i \(0.746097\pi\)
\(32\) −265500. −1.43232
\(33\) 0 0
\(34\) 383990. 1.67550
\(35\) −280146. −1.10445
\(36\) 0 0
\(37\) 409893. 1.33035 0.665173 0.746689i \(-0.268358\pi\)
0.665173 + 0.746689i \(0.268358\pi\)
\(38\) −872554. −2.57958
\(39\) 0 0
\(40\) −76993.4 −0.190214
\(41\) −189635. −0.429710 −0.214855 0.976646i \(-0.568928\pi\)
−0.214855 + 0.976646i \(0.568928\pi\)
\(42\) 0 0
\(43\) −849460. −1.62931 −0.814654 0.579947i \(-0.803073\pi\)
−0.814654 + 0.579947i \(0.803073\pi\)
\(44\) 113718. 0.201254
\(45\) 0 0
\(46\) 1.42021e6 2.15130
\(47\) 902990. 1.26865 0.634323 0.773068i \(-0.281279\pi\)
0.634323 + 0.773068i \(0.281279\pi\)
\(48\) 0 0
\(49\) −503977. −0.611963
\(50\) 2.72886e6 3.08736
\(51\) 0 0
\(52\) 1.57877e6 1.55707
\(53\) 324949. 0.299812 0.149906 0.988700i \(-0.452103\pi\)
0.149906 + 0.988700i \(0.452103\pi\)
\(54\) 0 0
\(55\) −409753. −0.332088
\(56\) 87827.2 0.0668299
\(57\) 0 0
\(58\) −435826. −0.293302
\(59\) −205379. −0.130189
\(60\) 0 0
\(61\) −1.58592e6 −0.894598 −0.447299 0.894384i \(-0.647614\pi\)
−0.447299 + 0.894384i \(0.647614\pi\)
\(62\) −3.77529e6 −2.01178
\(63\) 0 0
\(64\) −2.39707e6 −1.14301
\(65\) −5.68870e6 −2.56931
\(66\) 0 0
\(67\) −2.57819e6 −1.04726 −0.523629 0.851947i \(-0.675422\pi\)
−0.523629 + 0.851947i \(0.675422\pi\)
\(68\) 3.24094e6 1.24994
\(69\) 0 0
\(70\) −4.56504e6 −1.59075
\(71\) 678037. 0.224827 0.112414 0.993661i \(-0.464142\pi\)
0.112414 + 0.993661i \(0.464142\pi\)
\(72\) 0 0
\(73\) 2.50036e6 0.752269 0.376135 0.926565i \(-0.377253\pi\)
0.376135 + 0.926565i \(0.377253\pi\)
\(74\) 6.67930e6 1.91611
\(75\) 0 0
\(76\) −7.36449e6 −1.92440
\(77\) 467410. 0.116676
\(78\) 0 0
\(79\) −7.33035e6 −1.67275 −0.836373 0.548161i \(-0.815328\pi\)
−0.836373 + 0.548161i \(0.815328\pi\)
\(80\) 7.46957e6 1.63110
\(81\) 0 0
\(82\) −3.09015e6 −0.618914
\(83\) −3.91602e6 −0.751747 −0.375874 0.926671i \(-0.622657\pi\)
−0.375874 + 0.926671i \(0.622657\pi\)
\(84\) 0 0
\(85\) −1.16779e7 −2.06252
\(86\) −1.38421e7 −2.34670
\(87\) 0 0
\(88\) 128460. 0.0200945
\(89\) −3.80382e6 −0.571946 −0.285973 0.958238i \(-0.592317\pi\)
−0.285973 + 0.958238i \(0.592317\pi\)
\(90\) 0 0
\(91\) 6.48916e6 0.902700
\(92\) 1.19868e7 1.60489
\(93\) 0 0
\(94\) 1.47144e7 1.82724
\(95\) 2.65361e7 3.17544
\(96\) 0 0
\(97\) −7.68854e6 −0.855348 −0.427674 0.903933i \(-0.640667\pi\)
−0.427674 + 0.903933i \(0.640667\pi\)
\(98\) −8.21243e6 −0.881414
\(99\) 0 0
\(100\) 2.30321e7 2.30321
\(101\) 1.11320e7 1.07510 0.537551 0.843231i \(-0.319350\pi\)
0.537551 + 0.843231i \(0.319350\pi\)
\(102\) 0 0
\(103\) 9.10836e6 0.821315 0.410658 0.911790i \(-0.365299\pi\)
0.410658 + 0.911790i \(0.365299\pi\)
\(104\) 1.78344e6 0.155468
\(105\) 0 0
\(106\) 5.29511e6 0.431822
\(107\) −2.43881e7 −1.92458 −0.962289 0.272029i \(-0.912305\pi\)
−0.962289 + 0.272029i \(0.912305\pi\)
\(108\) 0 0
\(109\) −1.92554e7 −1.42416 −0.712080 0.702098i \(-0.752247\pi\)
−0.712080 + 0.702098i \(0.752247\pi\)
\(110\) −6.67702e6 −0.478309
\(111\) 0 0
\(112\) −8.52062e6 −0.573071
\(113\) −9.40627e6 −0.613258 −0.306629 0.951829i \(-0.599201\pi\)
−0.306629 + 0.951829i \(0.599201\pi\)
\(114\) 0 0
\(115\) −4.31914e7 −2.64822
\(116\) −3.67844e6 −0.218807
\(117\) 0 0
\(118\) −3.34670e6 −0.187512
\(119\) 1.33211e7 0.724645
\(120\) 0 0
\(121\) −1.88035e7 −0.964918
\(122\) −2.58430e7 −1.28850
\(123\) 0 0
\(124\) −3.18641e7 −1.50081
\(125\) −4.42737e7 −2.02750
\(126\) 0 0
\(127\) 2.55998e7 1.10898 0.554490 0.832190i \(-0.312913\pi\)
0.554490 + 0.832190i \(0.312913\pi\)
\(128\) −5.07683e6 −0.213972
\(129\) 0 0
\(130\) −9.26986e7 −3.70060
\(131\) −2.04746e7 −0.795729 −0.397865 0.917444i \(-0.630249\pi\)
−0.397865 + 0.917444i \(0.630249\pi\)
\(132\) 0 0
\(133\) −3.02700e7 −1.11566
\(134\) −4.20122e7 −1.50837
\(135\) 0 0
\(136\) 3.66108e6 0.124802
\(137\) −1.13505e6 −0.0377132 −0.0188566 0.999822i \(-0.506003\pi\)
−0.0188566 + 0.999822i \(0.506003\pi\)
\(138\) 0 0
\(139\) −1.44843e7 −0.457453 −0.228726 0.973491i \(-0.573456\pi\)
−0.228726 + 0.973491i \(0.573456\pi\)
\(140\) −3.85297e7 −1.18672
\(141\) 0 0
\(142\) 1.10488e7 0.323821
\(143\) 9.49132e6 0.271425
\(144\) 0 0
\(145\) 1.32543e7 0.361051
\(146\) 4.07440e7 1.08350
\(147\) 0 0
\(148\) 5.63744e7 1.42944
\(149\) −6.91223e7 −1.71185 −0.855926 0.517098i \(-0.827012\pi\)
−0.855926 + 0.517098i \(0.827012\pi\)
\(150\) 0 0
\(151\) 5.36019e7 1.26695 0.633477 0.773762i \(-0.281627\pi\)
0.633477 + 0.773762i \(0.281627\pi\)
\(152\) −8.31919e6 −0.192145
\(153\) 0 0
\(154\) 7.61655e6 0.168049
\(155\) 1.14814e8 2.47648
\(156\) 0 0
\(157\) −8.33413e7 −1.71874 −0.859372 0.511350i \(-0.829145\pi\)
−0.859372 + 0.511350i \(0.829145\pi\)
\(158\) −1.19450e8 −2.40927
\(159\) 0 0
\(160\) 1.31574e8 2.53950
\(161\) 4.92689e7 0.930426
\(162\) 0 0
\(163\) −5.74632e6 −0.103928 −0.0519641 0.998649i \(-0.516548\pi\)
−0.0519641 + 0.998649i \(0.516548\pi\)
\(164\) −2.60813e7 −0.461717
\(165\) 0 0
\(166\) −6.38125e7 −1.08275
\(167\) 5.42463e7 0.901286 0.450643 0.892704i \(-0.351195\pi\)
0.450643 + 0.892704i \(0.351195\pi\)
\(168\) 0 0
\(169\) 6.90217e7 1.09997
\(170\) −1.90294e8 −2.97066
\(171\) 0 0
\(172\) −1.16830e8 −1.75067
\(173\) 1.02709e8 1.50816 0.754081 0.656781i \(-0.228082\pi\)
0.754081 + 0.656781i \(0.228082\pi\)
\(174\) 0 0
\(175\) 9.46676e7 1.33527
\(176\) −1.24626e7 −0.172312
\(177\) 0 0
\(178\) −6.19841e7 −0.823779
\(179\) −1.18297e8 −1.54166 −0.770828 0.637043i \(-0.780157\pi\)
−0.770828 + 0.637043i \(0.780157\pi\)
\(180\) 0 0
\(181\) −7.54155e7 −0.945334 −0.472667 0.881241i \(-0.656709\pi\)
−0.472667 + 0.881241i \(0.656709\pi\)
\(182\) 1.05742e8 1.30017
\(183\) 0 0
\(184\) 1.35407e7 0.160243
\(185\) −2.03131e8 −2.35871
\(186\) 0 0
\(187\) 1.94840e7 0.217888
\(188\) 1.24192e8 1.36314
\(189\) 0 0
\(190\) 4.32411e8 4.57361
\(191\) −5.67288e7 −0.589097 −0.294548 0.955637i \(-0.595169\pi\)
−0.294548 + 0.955637i \(0.595169\pi\)
\(192\) 0 0
\(193\) 1.56586e7 0.156784 0.0783920 0.996923i \(-0.475021\pi\)
0.0783920 + 0.996923i \(0.475021\pi\)
\(194\) −1.25286e8 −1.23196
\(195\) 0 0
\(196\) −6.93142e7 −0.657546
\(197\) −1.95180e6 −0.0181888 −0.00909440 0.999959i \(-0.502895\pi\)
−0.00909440 + 0.999959i \(0.502895\pi\)
\(198\) 0 0
\(199\) 4.03128e7 0.362624 0.181312 0.983426i \(-0.441966\pi\)
0.181312 + 0.983426i \(0.441966\pi\)
\(200\) 2.60178e7 0.229967
\(201\) 0 0
\(202\) 1.81399e8 1.54848
\(203\) −1.51193e7 −0.126852
\(204\) 0 0
\(205\) 9.39774e7 0.761877
\(206\) 1.48423e8 1.18295
\(207\) 0 0
\(208\) −1.73021e8 −1.33315
\(209\) −4.42741e7 −0.335458
\(210\) 0 0
\(211\) 1.16617e8 0.854620 0.427310 0.904105i \(-0.359461\pi\)
0.427310 + 0.904105i \(0.359461\pi\)
\(212\) 4.46916e7 0.322144
\(213\) 0 0
\(214\) −3.97410e8 −2.77199
\(215\) 4.20966e8 2.88877
\(216\) 0 0
\(217\) −1.30970e8 −0.870084
\(218\) −3.13770e8 −2.05123
\(219\) 0 0
\(220\) −5.63552e7 −0.356824
\(221\) 2.70501e8 1.68576
\(222\) 0 0
\(223\) 1.52043e8 0.918121 0.459060 0.888405i \(-0.348186\pi\)
0.459060 + 0.888405i \(0.348186\pi\)
\(224\) −1.50087e8 −0.892228
\(225\) 0 0
\(226\) −1.53277e8 −0.883280
\(227\) 2.19156e8 1.24355 0.621774 0.783197i \(-0.286412\pi\)
0.621774 + 0.783197i \(0.286412\pi\)
\(228\) 0 0
\(229\) −1.50483e8 −0.828061 −0.414030 0.910263i \(-0.635879\pi\)
−0.414030 + 0.910263i \(0.635879\pi\)
\(230\) −7.03813e8 −3.81426
\(231\) 0 0
\(232\) −4.15529e6 −0.0218471
\(233\) −1.05923e8 −0.548586 −0.274293 0.961646i \(-0.588444\pi\)
−0.274293 + 0.961646i \(0.588444\pi\)
\(234\) 0 0
\(235\) −4.47494e8 −2.24931
\(236\) −2.82467e7 −0.139886
\(237\) 0 0
\(238\) 2.17070e8 1.04371
\(239\) 2.39069e8 1.13274 0.566370 0.824151i \(-0.308347\pi\)
0.566370 + 0.824151i \(0.308347\pi\)
\(240\) 0 0
\(241\) −6.39780e7 −0.294422 −0.147211 0.989105i \(-0.547030\pi\)
−0.147211 + 0.989105i \(0.547030\pi\)
\(242\) −3.06408e8 −1.38978
\(243\) 0 0
\(244\) −2.18119e8 −0.961234
\(245\) 2.49756e8 1.08501
\(246\) 0 0
\(247\) −6.14668e8 −2.59538
\(248\) −3.59948e7 −0.149851
\(249\) 0 0
\(250\) −7.21450e8 −2.92022
\(251\) −1.43897e8 −0.574373 −0.287186 0.957875i \(-0.592720\pi\)
−0.287186 + 0.957875i \(0.592720\pi\)
\(252\) 0 0
\(253\) 7.20627e7 0.279762
\(254\) 4.17155e8 1.59727
\(255\) 0 0
\(256\) 2.24097e8 0.834826
\(257\) 1.50715e8 0.553849 0.276925 0.960892i \(-0.410685\pi\)
0.276925 + 0.960892i \(0.410685\pi\)
\(258\) 0 0
\(259\) 2.31713e8 0.828708
\(260\) −7.82391e8 −2.76069
\(261\) 0 0
\(262\) −3.33638e8 −1.14609
\(263\) 5.02309e8 1.70265 0.851325 0.524638i \(-0.175799\pi\)
0.851325 + 0.524638i \(0.175799\pi\)
\(264\) 0 0
\(265\) −1.61035e8 −0.531568
\(266\) −4.93256e8 −1.60689
\(267\) 0 0
\(268\) −3.54590e8 −1.12526
\(269\) 2.76426e8 0.865858 0.432929 0.901428i \(-0.357480\pi\)
0.432929 + 0.901428i \(0.357480\pi\)
\(270\) 0 0
\(271\) 1.17995e8 0.360139 0.180069 0.983654i \(-0.442368\pi\)
0.180069 + 0.983654i \(0.442368\pi\)
\(272\) −3.55182e8 −1.07019
\(273\) 0 0
\(274\) −1.84959e7 −0.0543186
\(275\) 1.38465e8 0.401491
\(276\) 0 0
\(277\) −6.34905e7 −0.179485 −0.0897427 0.995965i \(-0.528604\pi\)
−0.0897427 + 0.995965i \(0.528604\pi\)
\(278\) −2.36025e8 −0.658872
\(279\) 0 0
\(280\) −4.35245e7 −0.118490
\(281\) −3.92031e8 −1.05402 −0.527009 0.849860i \(-0.676687\pi\)
−0.527009 + 0.849860i \(0.676687\pi\)
\(282\) 0 0
\(283\) 3.91102e8 1.02574 0.512871 0.858466i \(-0.328582\pi\)
0.512871 + 0.858466i \(0.328582\pi\)
\(284\) 9.32533e7 0.241574
\(285\) 0 0
\(286\) 1.54663e8 0.390936
\(287\) −1.07201e8 −0.267678
\(288\) 0 0
\(289\) 1.44951e8 0.353248
\(290\) 2.15982e8 0.520025
\(291\) 0 0
\(292\) 3.43886e8 0.808303
\(293\) 3.95594e8 0.918783 0.459392 0.888234i \(-0.348067\pi\)
0.459392 + 0.888234i \(0.348067\pi\)
\(294\) 0 0
\(295\) 1.01780e8 0.230825
\(296\) 6.36824e7 0.142725
\(297\) 0 0
\(298\) −1.12636e9 −2.46559
\(299\) 1.00046e9 2.16447
\(300\) 0 0
\(301\) −4.80201e8 −1.01494
\(302\) 8.73455e8 1.82480
\(303\) 0 0
\(304\) 8.07092e8 1.64765
\(305\) 7.85935e8 1.58612
\(306\) 0 0
\(307\) −3.74974e8 −0.739634 −0.369817 0.929105i \(-0.620580\pi\)
−0.369817 + 0.929105i \(0.620580\pi\)
\(308\) 6.42849e7 0.125367
\(309\) 0 0
\(310\) 1.87092e9 3.56689
\(311\) −5.67572e8 −1.06994 −0.534970 0.844871i \(-0.679677\pi\)
−0.534970 + 0.844871i \(0.679677\pi\)
\(312\) 0 0
\(313\) −9.61191e8 −1.77176 −0.885879 0.463916i \(-0.846444\pi\)
−0.885879 + 0.463916i \(0.846444\pi\)
\(314\) −1.35806e9 −2.47552
\(315\) 0 0
\(316\) −1.00817e9 −1.79734
\(317\) −1.03762e8 −0.182950 −0.0914748 0.995807i \(-0.529158\pi\)
−0.0914748 + 0.995807i \(0.529158\pi\)
\(318\) 0 0
\(319\) −2.21142e7 −0.0381420
\(320\) 1.18791e9 2.02656
\(321\) 0 0
\(322\) 8.02847e8 1.34010
\(323\) −1.26180e9 −2.08345
\(324\) 0 0
\(325\) 1.92234e9 3.10626
\(326\) −9.36376e7 −0.149689
\(327\) 0 0
\(328\) −2.94624e7 −0.0461009
\(329\) 5.10462e8 0.790273
\(330\) 0 0
\(331\) −4.86499e8 −0.737367 −0.368684 0.929555i \(-0.620191\pi\)
−0.368684 + 0.929555i \(0.620191\pi\)
\(332\) −5.38587e8 −0.807743
\(333\) 0 0
\(334\) 8.83956e8 1.29813
\(335\) 1.27767e9 1.85679
\(336\) 0 0
\(337\) −2.32747e8 −0.331269 −0.165634 0.986187i \(-0.552967\pi\)
−0.165634 + 0.986187i \(0.552967\pi\)
\(338\) 1.12472e9 1.58430
\(339\) 0 0
\(340\) −1.60611e9 −2.21615
\(341\) −1.91562e8 −0.261618
\(342\) 0 0
\(343\) −7.50449e8 −1.00413
\(344\) −1.31975e8 −0.174798
\(345\) 0 0
\(346\) 1.67367e9 2.17222
\(347\) 1.12320e8 0.144312 0.0721560 0.997393i \(-0.477012\pi\)
0.0721560 + 0.997393i \(0.477012\pi\)
\(348\) 0 0
\(349\) 1.23358e9 1.55338 0.776688 0.629885i \(-0.216898\pi\)
0.776688 + 0.629885i \(0.216898\pi\)
\(350\) 1.54263e9 1.92320
\(351\) 0 0
\(352\) −2.19524e8 −0.268277
\(353\) −1.25683e9 −1.52078 −0.760390 0.649466i \(-0.774992\pi\)
−0.760390 + 0.649466i \(0.774992\pi\)
\(354\) 0 0
\(355\) −3.36014e8 −0.398619
\(356\) −5.23156e8 −0.614549
\(357\) 0 0
\(358\) −1.92767e9 −2.22046
\(359\) −1.53369e8 −0.174947 −0.0874735 0.996167i \(-0.527879\pi\)
−0.0874735 + 0.996167i \(0.527879\pi\)
\(360\) 0 0
\(361\) 1.97337e9 2.20766
\(362\) −1.22891e9 −1.36157
\(363\) 0 0
\(364\) 8.92482e8 0.969940
\(365\) −1.23910e9 −1.33378
\(366\) 0 0
\(367\) −9.10830e8 −0.961847 −0.480924 0.876762i \(-0.659699\pi\)
−0.480924 + 0.876762i \(0.659699\pi\)
\(368\) −1.31366e9 −1.37409
\(369\) 0 0
\(370\) −3.31006e9 −3.39727
\(371\) 1.83694e8 0.186761
\(372\) 0 0
\(373\) −8.48680e8 −0.846766 −0.423383 0.905951i \(-0.639157\pi\)
−0.423383 + 0.905951i \(0.639157\pi\)
\(374\) 3.17496e8 0.313825
\(375\) 0 0
\(376\) 1.40292e8 0.136105
\(377\) −3.07016e8 −0.295098
\(378\) 0 0
\(379\) 1.11369e9 1.05082 0.525410 0.850849i \(-0.323912\pi\)
0.525410 + 0.850849i \(0.323912\pi\)
\(380\) 3.64962e9 3.41197
\(381\) 0 0
\(382\) −9.24408e8 −0.848481
\(383\) 1.24305e9 1.13056 0.565279 0.824900i \(-0.308768\pi\)
0.565279 + 0.824900i \(0.308768\pi\)
\(384\) 0 0
\(385\) −2.31634e8 −0.206866
\(386\) 2.55160e8 0.225817
\(387\) 0 0
\(388\) −1.05744e9 −0.919060
\(389\) −3.31934e8 −0.285909 −0.142955 0.989729i \(-0.545660\pi\)
−0.142955 + 0.989729i \(0.545660\pi\)
\(390\) 0 0
\(391\) 2.05377e9 1.73754
\(392\) −7.82997e7 −0.0656536
\(393\) 0 0
\(394\) −3.18051e7 −0.0261975
\(395\) 3.63270e9 2.96578
\(396\) 0 0
\(397\) −2.50412e8 −0.200858 −0.100429 0.994944i \(-0.532021\pi\)
−0.100429 + 0.994944i \(0.532021\pi\)
\(398\) 6.56906e8 0.522291
\(399\) 0 0
\(400\) −2.52414e9 −1.97198
\(401\) −1.57100e9 −1.21666 −0.608332 0.793683i \(-0.708161\pi\)
−0.608332 + 0.793683i \(0.708161\pi\)
\(402\) 0 0
\(403\) −2.65949e9 −2.02410
\(404\) 1.53104e9 1.15518
\(405\) 0 0
\(406\) −2.46373e8 −0.182705
\(407\) 3.38914e8 0.249177
\(408\) 0 0
\(409\) −2.37344e8 −0.171532 −0.0857661 0.996315i \(-0.527334\pi\)
−0.0857661 + 0.996315i \(0.527334\pi\)
\(410\) 1.53138e9 1.09734
\(411\) 0 0
\(412\) 1.25271e9 0.882492
\(413\) −1.16101e8 −0.0810981
\(414\) 0 0
\(415\) 1.94066e9 1.33285
\(416\) −3.04770e9 −2.07561
\(417\) 0 0
\(418\) −7.21457e8 −0.483163
\(419\) 1.61768e9 1.07435 0.537173 0.843472i \(-0.319492\pi\)
0.537173 + 0.843472i \(0.319492\pi\)
\(420\) 0 0
\(421\) −1.17270e9 −0.765946 −0.382973 0.923759i \(-0.625100\pi\)
−0.382973 + 0.923759i \(0.625100\pi\)
\(422\) 1.90030e9 1.23092
\(423\) 0 0
\(424\) 5.04852e7 0.0321650
\(425\) 3.94622e9 2.49356
\(426\) 0 0
\(427\) −8.96525e8 −0.557269
\(428\) −3.35421e9 −2.06793
\(429\) 0 0
\(430\) 6.85974e9 4.16071
\(431\) −2.74618e9 −1.65219 −0.826094 0.563533i \(-0.809442\pi\)
−0.826094 + 0.563533i \(0.809442\pi\)
\(432\) 0 0
\(433\) 1.44724e8 0.0856709 0.0428355 0.999082i \(-0.486361\pi\)
0.0428355 + 0.999082i \(0.486361\pi\)
\(434\) −2.13418e9 −1.25319
\(435\) 0 0
\(436\) −2.64827e9 −1.53024
\(437\) −4.66686e9 −2.67510
\(438\) 0 0
\(439\) −2.83060e8 −0.159681 −0.0798403 0.996808i \(-0.525441\pi\)
−0.0798403 + 0.996808i \(0.525441\pi\)
\(440\) −6.36607e7 −0.0356276
\(441\) 0 0
\(442\) 4.40787e9 2.42801
\(443\) 6.58269e8 0.359741 0.179871 0.983690i \(-0.442432\pi\)
0.179871 + 0.983690i \(0.442432\pi\)
\(444\) 0 0
\(445\) 1.88506e9 1.01406
\(446\) 2.47758e9 1.32238
\(447\) 0 0
\(448\) −1.35507e9 −0.712012
\(449\) −2.96392e9 −1.54527 −0.772636 0.634850i \(-0.781062\pi\)
−0.772636 + 0.634850i \(0.781062\pi\)
\(450\) 0 0
\(451\) −1.56797e8 −0.0804857
\(452\) −1.29368e9 −0.658937
\(453\) 0 0
\(454\) 3.57119e9 1.79109
\(455\) −3.21583e9 −1.60049
\(456\) 0 0
\(457\) −2.62985e8 −0.128891 −0.0644456 0.997921i \(-0.520528\pi\)
−0.0644456 + 0.997921i \(0.520528\pi\)
\(458\) −2.45215e9 −1.19266
\(459\) 0 0
\(460\) −5.94030e9 −2.84548
\(461\) −7.39703e8 −0.351645 −0.175822 0.984422i \(-0.556258\pi\)
−0.175822 + 0.984422i \(0.556258\pi\)
\(462\) 0 0
\(463\) −8.00108e8 −0.374641 −0.187320 0.982299i \(-0.559980\pi\)
−0.187320 + 0.982299i \(0.559980\pi\)
\(464\) 4.03129e8 0.187340
\(465\) 0 0
\(466\) −1.72604e9 −0.790132
\(467\) −3.44400e9 −1.56478 −0.782392 0.622786i \(-0.786001\pi\)
−0.782392 + 0.622786i \(0.786001\pi\)
\(468\) 0 0
\(469\) −1.45745e9 −0.652364
\(470\) −7.29202e9 −3.23970
\(471\) 0 0
\(472\) −3.19084e7 −0.0139672
\(473\) −7.02362e8 −0.305174
\(474\) 0 0
\(475\) −8.96713e9 −3.83907
\(476\) 1.83211e9 0.778622
\(477\) 0 0
\(478\) 3.89568e9 1.63149
\(479\) 2.17648e9 0.904857 0.452429 0.891801i \(-0.350558\pi\)
0.452429 + 0.891801i \(0.350558\pi\)
\(480\) 0 0
\(481\) 4.70521e9 1.92784
\(482\) −1.04254e9 −0.424059
\(483\) 0 0
\(484\) −2.58613e9 −1.03679
\(485\) 3.81020e9 1.51653
\(486\) 0 0
\(487\) 1.73782e9 0.681796 0.340898 0.940100i \(-0.389269\pi\)
0.340898 + 0.940100i \(0.389269\pi\)
\(488\) −2.46395e8 −0.0959758
\(489\) 0 0
\(490\) 4.06983e9 1.56275
\(491\) 7.21828e8 0.275200 0.137600 0.990488i \(-0.456061\pi\)
0.137600 + 0.990488i \(0.456061\pi\)
\(492\) 0 0
\(493\) −6.30249e8 −0.236891
\(494\) −1.00161e10 −3.73815
\(495\) 0 0
\(496\) 3.49206e9 1.28498
\(497\) 3.83295e8 0.140051
\(498\) 0 0
\(499\) −1.78127e9 −0.641769 −0.320885 0.947118i \(-0.603980\pi\)
−0.320885 + 0.947118i \(0.603980\pi\)
\(500\) −6.08915e9 −2.17852
\(501\) 0 0
\(502\) −2.34483e9 −0.827273
\(503\) −1.72676e9 −0.604986 −0.302493 0.953152i \(-0.597819\pi\)
−0.302493 + 0.953152i \(0.597819\pi\)
\(504\) 0 0
\(505\) −5.51669e9 −1.90616
\(506\) 1.17428e9 0.402944
\(507\) 0 0
\(508\) 3.52085e9 1.19158
\(509\) −1.96038e8 −0.0658912 −0.0329456 0.999457i \(-0.510489\pi\)
−0.0329456 + 0.999457i \(0.510489\pi\)
\(510\) 0 0
\(511\) 1.41346e9 0.468608
\(512\) 4.30154e9 1.41638
\(513\) 0 0
\(514\) 2.45594e9 0.797713
\(515\) −4.51383e9 −1.45619
\(516\) 0 0
\(517\) 7.46623e8 0.237621
\(518\) 3.77582e9 1.19359
\(519\) 0 0
\(520\) −8.83816e8 −0.275645
\(521\) 2.84883e9 0.882540 0.441270 0.897374i \(-0.354528\pi\)
0.441270 + 0.897374i \(0.354528\pi\)
\(522\) 0 0
\(523\) −3.08000e9 −0.941445 −0.470722 0.882281i \(-0.656007\pi\)
−0.470722 + 0.882281i \(0.656007\pi\)
\(524\) −2.81596e9 −0.855000
\(525\) 0 0
\(526\) 8.18523e9 2.45234
\(527\) −5.45947e9 −1.62485
\(528\) 0 0
\(529\) 4.19118e9 1.23095
\(530\) −2.62410e9 −0.765621
\(531\) 0 0
\(532\) −4.16316e9 −1.19876
\(533\) −2.17684e9 −0.622705
\(534\) 0 0
\(535\) 1.20860e10 3.41228
\(536\) −4.00557e8 −0.112354
\(537\) 0 0
\(538\) 4.50443e9 1.24710
\(539\) −4.16706e8 −0.114622
\(540\) 0 0
\(541\) 3.69807e9 1.00412 0.502060 0.864833i \(-0.332576\pi\)
0.502060 + 0.864833i \(0.332576\pi\)
\(542\) 1.92275e9 0.518711
\(543\) 0 0
\(544\) −6.25639e9 −1.66620
\(545\) 9.54237e9 2.52504
\(546\) 0 0
\(547\) −1.21521e9 −0.317464 −0.158732 0.987322i \(-0.550741\pi\)
−0.158732 + 0.987322i \(0.550741\pi\)
\(548\) −1.56108e8 −0.0405223
\(549\) 0 0
\(550\) 2.25632e9 0.578270
\(551\) 1.43214e9 0.364715
\(552\) 0 0
\(553\) −4.14386e9 −1.04200
\(554\) −1.03459e9 −0.258514
\(555\) 0 0
\(556\) −1.99209e9 −0.491527
\(557\) 6.35497e9 1.55819 0.779095 0.626906i \(-0.215679\pi\)
0.779095 + 0.626906i \(0.215679\pi\)
\(558\) 0 0
\(559\) −9.75105e9 −2.36108
\(560\) 4.22256e9 1.01606
\(561\) 0 0
\(562\) −6.38823e9 −1.51811
\(563\) −2.41689e9 −0.570791 −0.285395 0.958410i \(-0.592125\pi\)
−0.285395 + 0.958410i \(0.592125\pi\)
\(564\) 0 0
\(565\) 4.66146e9 1.08731
\(566\) 6.37310e9 1.47738
\(567\) 0 0
\(568\) 1.05342e8 0.0241203
\(569\) −2.02038e9 −0.459769 −0.229884 0.973218i \(-0.573835\pi\)
−0.229884 + 0.973218i \(0.573835\pi\)
\(570\) 0 0
\(571\) −5.09913e9 −1.14622 −0.573112 0.819477i \(-0.694264\pi\)
−0.573112 + 0.819477i \(0.694264\pi\)
\(572\) 1.30538e9 0.291643
\(573\) 0 0
\(574\) −1.74686e9 −0.385538
\(575\) 1.45953e10 3.20167
\(576\) 0 0
\(577\) −5.65588e9 −1.22570 −0.612851 0.790199i \(-0.709977\pi\)
−0.612851 + 0.790199i \(0.709977\pi\)
\(578\) 2.36201e9 0.508785
\(579\) 0 0
\(580\) 1.82292e9 0.387945
\(581\) −2.21373e9 −0.468283
\(582\) 0 0
\(583\) 2.68678e8 0.0561556
\(584\) 3.88465e8 0.0807063
\(585\) 0 0
\(586\) 6.44629e9 1.32333
\(587\) −8.74088e9 −1.78370 −0.891850 0.452331i \(-0.850593\pi\)
−0.891850 + 0.452331i \(0.850593\pi\)
\(588\) 0 0
\(589\) 1.24057e10 2.50161
\(590\) 1.65852e9 0.332460
\(591\) 0 0
\(592\) −6.17820e9 −1.22387
\(593\) 7.16535e9 1.41106 0.705531 0.708679i \(-0.250709\pi\)
0.705531 + 0.708679i \(0.250709\pi\)
\(594\) 0 0
\(595\) −6.60153e9 −1.28480
\(596\) −9.50669e9 −1.83936
\(597\) 0 0
\(598\) 1.63028e10 3.11751
\(599\) 1.12486e8 0.0213847 0.0106923 0.999943i \(-0.496596\pi\)
0.0106923 + 0.999943i \(0.496596\pi\)
\(600\) 0 0
\(601\) 2.43804e9 0.458121 0.229061 0.973412i \(-0.426435\pi\)
0.229061 + 0.973412i \(0.426435\pi\)
\(602\) −7.82498e9 −1.46182
\(603\) 0 0
\(604\) 7.37210e9 1.36132
\(605\) 9.31845e9 1.71080
\(606\) 0 0
\(607\) 3.37659e9 0.612800 0.306400 0.951903i \(-0.400876\pi\)
0.306400 + 0.951903i \(0.400876\pi\)
\(608\) 1.42166e10 2.56527
\(609\) 0 0
\(610\) 1.28070e10 2.28451
\(611\) 1.03655e10 1.83843
\(612\) 0 0
\(613\) −3.64627e9 −0.639348 −0.319674 0.947528i \(-0.603573\pi\)
−0.319674 + 0.947528i \(0.603573\pi\)
\(614\) −6.11029e9 −1.06530
\(615\) 0 0
\(616\) 7.26184e7 0.0125174
\(617\) 2.27884e8 0.0390585 0.0195293 0.999809i \(-0.493783\pi\)
0.0195293 + 0.999809i \(0.493783\pi\)
\(618\) 0 0
\(619\) 2.30512e9 0.390640 0.195320 0.980740i \(-0.437426\pi\)
0.195320 + 0.980740i \(0.437426\pi\)
\(620\) 1.57909e10 2.66094
\(621\) 0 0
\(622\) −9.24870e9 −1.54104
\(623\) −2.15031e9 −0.356280
\(624\) 0 0
\(625\) 8.85757e9 1.45122
\(626\) −1.56628e10 −2.55188
\(627\) 0 0
\(628\) −1.14623e10 −1.84677
\(629\) 9.65897e9 1.54758
\(630\) 0 0
\(631\) 5.67126e9 0.898622 0.449311 0.893375i \(-0.351670\pi\)
0.449311 + 0.893375i \(0.351670\pi\)
\(632\) −1.13887e9 −0.179458
\(633\) 0 0
\(634\) −1.69083e9 −0.263504
\(635\) −1.26865e10 −1.96623
\(636\) 0 0
\(637\) −5.78522e9 −0.886812
\(638\) −3.60355e8 −0.0549362
\(639\) 0 0
\(640\) 2.51592e9 0.379373
\(641\) 8.04557e9 1.20657 0.603287 0.797524i \(-0.293858\pi\)
0.603287 + 0.797524i \(0.293858\pi\)
\(642\) 0 0
\(643\) 6.95787e9 1.03214 0.516069 0.856547i \(-0.327395\pi\)
0.516069 + 0.856547i \(0.327395\pi\)
\(644\) 6.77616e9 0.999731
\(645\) 0 0
\(646\) −2.05614e10 −3.00081
\(647\) 1.50113e9 0.217898 0.108949 0.994047i \(-0.465252\pi\)
0.108949 + 0.994047i \(0.465252\pi\)
\(648\) 0 0
\(649\) −1.69814e8 −0.0243847
\(650\) 3.13250e10 4.47398
\(651\) 0 0
\(652\) −7.90316e8 −0.111669
\(653\) 1.05745e9 0.148616 0.0743078 0.997235i \(-0.476325\pi\)
0.0743078 + 0.997235i \(0.476325\pi\)
\(654\) 0 0
\(655\) 1.01466e10 1.41083
\(656\) 2.85831e9 0.395318
\(657\) 0 0
\(658\) 8.31808e9 1.13824
\(659\) 1.27158e10 1.73079 0.865396 0.501088i \(-0.167067\pi\)
0.865396 + 0.501088i \(0.167067\pi\)
\(660\) 0 0
\(661\) −1.38832e10 −1.86976 −0.934878 0.354968i \(-0.884492\pi\)
−0.934878 + 0.354968i \(0.884492\pi\)
\(662\) −7.92761e9 −1.06204
\(663\) 0 0
\(664\) −6.08407e8 −0.0806503
\(665\) 1.50009e10 1.97807
\(666\) 0 0
\(667\) −2.33101e9 −0.304162
\(668\) 7.46073e9 0.968420
\(669\) 0 0
\(670\) 2.08200e10 2.67435
\(671\) −1.31130e9 −0.167560
\(672\) 0 0
\(673\) 8.66625e9 1.09592 0.547960 0.836505i \(-0.315405\pi\)
0.547960 + 0.836505i \(0.315405\pi\)
\(674\) −3.79267e9 −0.477129
\(675\) 0 0
\(676\) 9.49284e9 1.18191
\(677\) 1.30940e10 1.62186 0.810929 0.585144i \(-0.198962\pi\)
0.810929 + 0.585144i \(0.198962\pi\)
\(678\) 0 0
\(679\) −4.34634e9 −0.532819
\(680\) −1.81432e9 −0.221275
\(681\) 0 0
\(682\) −3.12154e9 −0.376811
\(683\) −5.07430e9 −0.609402 −0.304701 0.952448i \(-0.598557\pi\)
−0.304701 + 0.952448i \(0.598557\pi\)
\(684\) 0 0
\(685\) 5.62496e8 0.0668656
\(686\) −1.22287e10 −1.44626
\(687\) 0 0
\(688\) 1.28037e10 1.49891
\(689\) 3.73012e9 0.434466
\(690\) 0 0
\(691\) 7.09480e9 0.818026 0.409013 0.912529i \(-0.365873\pi\)
0.409013 + 0.912529i \(0.365873\pi\)
\(692\) 1.41260e10 1.62050
\(693\) 0 0
\(694\) 1.83027e9 0.207854
\(695\) 7.17798e9 0.811065
\(696\) 0 0
\(697\) −4.46867e9 −0.499878
\(698\) 2.01014e10 2.23734
\(699\) 0 0
\(700\) 1.30200e10 1.43473
\(701\) −5.74769e9 −0.630203 −0.315101 0.949058i \(-0.602039\pi\)
−0.315101 + 0.949058i \(0.602039\pi\)
\(702\) 0 0
\(703\) −2.19484e10 −2.38264
\(704\) −1.98198e9 −0.214089
\(705\) 0 0
\(706\) −2.04804e10 −2.19039
\(707\) 6.29295e9 0.669709
\(708\) 0 0
\(709\) −5.47895e9 −0.577345 −0.288672 0.957428i \(-0.593214\pi\)
−0.288672 + 0.957428i \(0.593214\pi\)
\(710\) −5.47543e9 −0.574135
\(711\) 0 0
\(712\) −5.90975e8 −0.0613605
\(713\) −2.01922e10 −2.08627
\(714\) 0 0
\(715\) −4.70361e9 −0.481238
\(716\) −1.62699e10 −1.65649
\(717\) 0 0
\(718\) −2.49918e9 −0.251978
\(719\) −1.62636e10 −1.63179 −0.815895 0.578200i \(-0.803755\pi\)
−0.815895 + 0.578200i \(0.803755\pi\)
\(720\) 0 0
\(721\) 5.14897e9 0.511619
\(722\) 3.21565e10 3.17971
\(723\) 0 0
\(724\) −1.03722e10 −1.01575
\(725\) −4.47893e9 −0.436507
\(726\) 0 0
\(727\) −2.55116e9 −0.246245 −0.123122 0.992391i \(-0.539291\pi\)
−0.123122 + 0.992391i \(0.539291\pi\)
\(728\) 1.00818e9 0.0968451
\(729\) 0 0
\(730\) −2.01915e10 −1.92105
\(731\) −2.00172e10 −1.89536
\(732\) 0 0
\(733\) 1.41305e10 1.32524 0.662619 0.748957i \(-0.269445\pi\)
0.662619 + 0.748957i \(0.269445\pi\)
\(734\) −1.48422e10 −1.38536
\(735\) 0 0
\(736\) −2.31396e10 −2.13936
\(737\) −2.13174e9 −0.196154
\(738\) 0 0
\(739\) 1.92246e10 1.75227 0.876136 0.482064i \(-0.160113\pi\)
0.876136 + 0.482064i \(0.160113\pi\)
\(740\) −2.79374e10 −2.53440
\(741\) 0 0
\(742\) 2.99333e9 0.268993
\(743\) −5.89705e9 −0.527441 −0.263721 0.964599i \(-0.584950\pi\)
−0.263721 + 0.964599i \(0.584950\pi\)
\(744\) 0 0
\(745\) 3.42549e10 3.03512
\(746\) −1.38294e10 −1.21960
\(747\) 0 0
\(748\) 2.67972e9 0.234117
\(749\) −1.37867e10 −1.19887
\(750\) 0 0
\(751\) −3.11911e9 −0.268715 −0.134357 0.990933i \(-0.542897\pi\)
−0.134357 + 0.990933i \(0.542897\pi\)
\(752\) −1.36105e10 −1.16711
\(753\) 0 0
\(754\) −5.00289e9 −0.425032
\(755\) −2.65635e10 −2.24631
\(756\) 0 0
\(757\) −1.03709e10 −0.868921 −0.434461 0.900691i \(-0.643061\pi\)
−0.434461 + 0.900691i \(0.643061\pi\)
\(758\) 1.81479e10 1.51350
\(759\) 0 0
\(760\) 4.12273e9 0.340673
\(761\) −1.48892e9 −0.122468 −0.0612342 0.998123i \(-0.519504\pi\)
−0.0612342 + 0.998123i \(0.519504\pi\)
\(762\) 0 0
\(763\) −1.08851e10 −0.887147
\(764\) −7.80215e9 −0.632977
\(765\) 0 0
\(766\) 2.02558e10 1.62835
\(767\) −2.35757e9 −0.188660
\(768\) 0 0
\(769\) 1.76503e10 1.39962 0.699809 0.714330i \(-0.253268\pi\)
0.699809 + 0.714330i \(0.253268\pi\)
\(770\) −3.77453e9 −0.297951
\(771\) 0 0
\(772\) 2.15359e9 0.168462
\(773\) 6.95196e8 0.0541351 0.0270676 0.999634i \(-0.491383\pi\)
0.0270676 + 0.999634i \(0.491383\pi\)
\(774\) 0 0
\(775\) −3.87982e10 −2.99403
\(776\) −1.19452e9 −0.0917649
\(777\) 0 0
\(778\) −5.40894e9 −0.411798
\(779\) 1.01543e10 0.769608
\(780\) 0 0
\(781\) 5.60624e8 0.0421107
\(782\) 3.34667e10 2.50259
\(783\) 0 0
\(784\) 7.59631e9 0.562984
\(785\) 4.13014e10 3.04734
\(786\) 0 0
\(787\) −7.19358e8 −0.0526058 −0.0263029 0.999654i \(-0.508373\pi\)
−0.0263029 + 0.999654i \(0.508373\pi\)
\(788\) −2.68440e8 −0.0195436
\(789\) 0 0
\(790\) 5.91956e10 4.27164
\(791\) −5.31738e9 −0.382014
\(792\) 0 0
\(793\) −1.82050e10 −1.29639
\(794\) −4.08052e9 −0.289297
\(795\) 0 0
\(796\) 5.54439e9 0.389635
\(797\) −4.07407e9 −0.285052 −0.142526 0.989791i \(-0.545523\pi\)
−0.142526 + 0.989791i \(0.545523\pi\)
\(798\) 0 0
\(799\) 2.12786e10 1.47581
\(800\) −4.44616e10 −3.07023
\(801\) 0 0
\(802\) −2.55998e10 −1.75237
\(803\) 2.06739e9 0.140902
\(804\) 0 0
\(805\) −2.44161e10 −1.64965
\(806\) −4.33371e10 −2.91532
\(807\) 0 0
\(808\) 1.72951e9 0.115341
\(809\) 1.12391e10 0.746296 0.373148 0.927772i \(-0.378278\pi\)
0.373148 + 0.927772i \(0.378278\pi\)
\(810\) 0 0
\(811\) 1.57592e10 1.03744 0.518719 0.854945i \(-0.326409\pi\)
0.518719 + 0.854945i \(0.326409\pi\)
\(812\) −2.07943e9 −0.136300
\(813\) 0 0
\(814\) 5.52267e9 0.358892
\(815\) 2.84770e9 0.184265
\(816\) 0 0
\(817\) 4.54857e10 2.91808
\(818\) −3.86757e9 −0.247059
\(819\) 0 0
\(820\) 1.29251e10 0.818627
\(821\) 1.83331e10 1.15621 0.578103 0.815964i \(-0.303793\pi\)
0.578103 + 0.815964i \(0.303793\pi\)
\(822\) 0 0
\(823\) 3.71121e9 0.232069 0.116034 0.993245i \(-0.462982\pi\)
0.116034 + 0.993245i \(0.462982\pi\)
\(824\) 1.41511e9 0.0881138
\(825\) 0 0
\(826\) −1.89189e9 −0.116806
\(827\) −3.00461e9 −0.184722 −0.0923610 0.995726i \(-0.529441\pi\)
−0.0923610 + 0.995726i \(0.529441\pi\)
\(828\) 0 0
\(829\) 1.97799e10 1.20582 0.602912 0.797808i \(-0.294007\pi\)
0.602912 + 0.797808i \(0.294007\pi\)
\(830\) 3.16235e10 1.91971
\(831\) 0 0
\(832\) −2.75162e10 −1.65637
\(833\) −1.18760e10 −0.711891
\(834\) 0 0
\(835\) −2.68828e10 −1.59798
\(836\) −6.08921e9 −0.360445
\(837\) 0 0
\(838\) 2.63605e10 1.54739
\(839\) 7.17247e9 0.419278 0.209639 0.977779i \(-0.432771\pi\)
0.209639 + 0.977779i \(0.432771\pi\)
\(840\) 0 0
\(841\) −1.65345e10 −0.958531
\(842\) −1.91094e10 −1.10320
\(843\) 0 0
\(844\) 1.60388e10 0.918278
\(845\) −3.42050e10 −1.95025
\(846\) 0 0
\(847\) −1.06297e10 −0.601073
\(848\) −4.89786e9 −0.275817
\(849\) 0 0
\(850\) 6.43046e10 3.59150
\(851\) 3.57243e10 1.98706
\(852\) 0 0
\(853\) −2.70770e9 −0.149375 −0.0746877 0.997207i \(-0.523796\pi\)
−0.0746877 + 0.997207i \(0.523796\pi\)
\(854\) −1.46091e10 −0.802639
\(855\) 0 0
\(856\) −3.78903e9 −0.206476
\(857\) 1.93553e10 1.05043 0.525215 0.850969i \(-0.323985\pi\)
0.525215 + 0.850969i \(0.323985\pi\)
\(858\) 0 0
\(859\) −3.02146e10 −1.62645 −0.813224 0.581951i \(-0.802290\pi\)
−0.813224 + 0.581951i \(0.802290\pi\)
\(860\) 5.78973e10 3.10394
\(861\) 0 0
\(862\) −4.47497e10 −2.37966
\(863\) −4.20793e9 −0.222859 −0.111430 0.993772i \(-0.535543\pi\)
−0.111430 + 0.993772i \(0.535543\pi\)
\(864\) 0 0
\(865\) −5.08996e10 −2.67398
\(866\) 2.35831e9 0.123393
\(867\) 0 0
\(868\) −1.80128e10 −0.934894
\(869\) −6.06098e9 −0.313310
\(870\) 0 0
\(871\) −2.95954e10 −1.51761
\(872\) −2.99158e9 −0.152789
\(873\) 0 0
\(874\) −7.60475e10 −3.85296
\(875\) −2.50280e10 −1.26298
\(876\) 0 0
\(877\) −1.97999e10 −0.991206 −0.495603 0.868549i \(-0.665053\pi\)
−0.495603 + 0.868549i \(0.665053\pi\)
\(878\) −4.61252e9 −0.229989
\(879\) 0 0
\(880\) 6.17609e9 0.305509
\(881\) −5.47108e9 −0.269561 −0.134781 0.990875i \(-0.543033\pi\)
−0.134781 + 0.990875i \(0.543033\pi\)
\(882\) 0 0
\(883\) 1.99701e10 0.976153 0.488076 0.872801i \(-0.337699\pi\)
0.488076 + 0.872801i \(0.337699\pi\)
\(884\) 3.72031e10 1.81133
\(885\) 0 0
\(886\) 1.07266e10 0.518138
\(887\) 6.76780e9 0.325623 0.162811 0.986657i \(-0.447944\pi\)
0.162811 + 0.986657i \(0.447944\pi\)
\(888\) 0 0
\(889\) 1.44716e10 0.690813
\(890\) 3.07174e10 1.46056
\(891\) 0 0
\(892\) 2.09111e10 0.986508
\(893\) −4.83521e10 −2.27214
\(894\) 0 0
\(895\) 5.86243e10 2.73336
\(896\) −2.86994e9 −0.133289
\(897\) 0 0
\(898\) −4.82978e10 −2.22567
\(899\) 6.19645e9 0.284436
\(900\) 0 0
\(901\) 7.65728e9 0.348769
\(902\) −2.55504e9 −0.115924
\(903\) 0 0
\(904\) −1.46139e9 −0.0657926
\(905\) 3.73736e10 1.67608
\(906\) 0 0
\(907\) −2.47127e10 −1.09975 −0.549877 0.835246i \(-0.685325\pi\)
−0.549877 + 0.835246i \(0.685325\pi\)
\(908\) 3.01414e10 1.33618
\(909\) 0 0
\(910\) −5.24027e10 −2.30520
\(911\) 3.74402e9 0.164068 0.0820340 0.996630i \(-0.473858\pi\)
0.0820340 + 0.996630i \(0.473858\pi\)
\(912\) 0 0
\(913\) −3.23790e9 −0.140804
\(914\) −4.28539e9 −0.185643
\(915\) 0 0
\(916\) −2.06965e10 −0.889740
\(917\) −1.15743e10 −0.495681
\(918\) 0 0
\(919\) −3.76618e9 −0.160065 −0.0800326 0.996792i \(-0.525502\pi\)
−0.0800326 + 0.996792i \(0.525502\pi\)
\(920\) −6.71036e9 −0.284111
\(921\) 0 0
\(922\) −1.20536e10 −0.506477
\(923\) 7.78326e9 0.325804
\(924\) 0 0
\(925\) 6.86424e10 2.85165
\(926\) −1.30379e10 −0.539598
\(927\) 0 0
\(928\) 7.10095e9 0.291675
\(929\) −3.28415e10 −1.34390 −0.671952 0.740595i \(-0.734544\pi\)
−0.671952 + 0.740595i \(0.734544\pi\)
\(930\) 0 0
\(931\) 2.69863e10 1.09602
\(932\) −1.45680e10 −0.589448
\(933\) 0 0
\(934\) −5.61208e10 −2.25377
\(935\) −9.65567e9 −0.386315
\(936\) 0 0
\(937\) −3.67023e10 −1.45749 −0.728744 0.684787i \(-0.759895\pi\)
−0.728744 + 0.684787i \(0.759895\pi\)
\(938\) −2.37496e10 −0.939605
\(939\) 0 0
\(940\) −6.15458e10 −2.41686
\(941\) −2.52107e10 −0.986329 −0.493165 0.869936i \(-0.664160\pi\)
−0.493165 + 0.869936i \(0.664160\pi\)
\(942\) 0 0
\(943\) −1.65277e10 −0.641831
\(944\) 3.09562e9 0.119769
\(945\) 0 0
\(946\) −1.14451e10 −0.439544
\(947\) −8.87956e9 −0.339755 −0.169878 0.985465i \(-0.554337\pi\)
−0.169878 + 0.985465i \(0.554337\pi\)
\(948\) 0 0
\(949\) 2.87020e10 1.09013
\(950\) −1.46121e11 −5.52944
\(951\) 0 0
\(952\) 2.06961e9 0.0777427
\(953\) −1.58158e10 −0.591925 −0.295962 0.955200i \(-0.595640\pi\)
−0.295962 + 0.955200i \(0.595640\pi\)
\(954\) 0 0
\(955\) 2.81131e10 1.04447
\(956\) 3.28802e10 1.21711
\(957\) 0 0
\(958\) 3.54662e10 1.30327
\(959\) −6.41645e8 −0.0234925
\(960\) 0 0
\(961\) 2.61635e10 0.950963
\(962\) 7.66725e10 2.77669
\(963\) 0 0
\(964\) −8.79917e9 −0.316353
\(965\) −7.75992e9 −0.277979
\(966\) 0 0
\(967\) 5.18915e10 1.84546 0.922728 0.385451i \(-0.125954\pi\)
0.922728 + 0.385451i \(0.125954\pi\)
\(968\) −2.92138e9 −0.103520
\(969\) 0 0
\(970\) 6.20881e10 2.18428
\(971\) 1.69159e10 0.592963 0.296481 0.955039i \(-0.404187\pi\)
0.296481 + 0.955039i \(0.404187\pi\)
\(972\) 0 0
\(973\) −8.18800e9 −0.284959
\(974\) 2.83182e10 0.981996
\(975\) 0 0
\(976\) 2.39042e10 0.822999
\(977\) −8.24427e9 −0.282827 −0.141414 0.989951i \(-0.545165\pi\)
−0.141414 + 0.989951i \(0.545165\pi\)
\(978\) 0 0
\(979\) −3.14513e9 −0.107127
\(980\) 3.43500e10 1.16583
\(981\) 0 0
\(982\) 1.17623e10 0.396373
\(983\) 4.10669e9 0.137897 0.0689484 0.997620i \(-0.478036\pi\)
0.0689484 + 0.997620i \(0.478036\pi\)
\(984\) 0 0
\(985\) 9.67254e8 0.0322488
\(986\) −1.02701e10 −0.341196
\(987\) 0 0
\(988\) −8.45379e10 −2.78870
\(989\) −7.40347e10 −2.43359
\(990\) 0 0
\(991\) −1.53943e9 −0.0502460 −0.0251230 0.999684i \(-0.507998\pi\)
−0.0251230 + 0.999684i \(0.507998\pi\)
\(992\) 6.15112e10 2.00062
\(993\) 0 0
\(994\) 6.24588e9 0.201716
\(995\) −1.99778e10 −0.642934
\(996\) 0 0
\(997\) −2.17704e10 −0.695717 −0.347858 0.937547i \(-0.613091\pi\)
−0.347858 + 0.937547i \(0.613091\pi\)
\(998\) −2.90263e10 −0.924345
\(999\) 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.1.14 16
3.2 odd 2 177.8.a.a.1.3 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.3 16 3.2 odd 2
531.8.a.b.1.14 16 1.1 even 1 trivial