Properties

Label 177.8.a.a.1.3
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(16.2952\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-16.2952 q^{2} -27.0000 q^{3} +137.534 q^{4} +495.569 q^{5} +439.971 q^{6} +565.301 q^{7} -155.363 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-16.2952 q^{2} -27.0000 q^{3} +137.534 q^{4} +495.569 q^{5} +439.971 q^{6} +565.301 q^{7} -155.363 q^{8} +729.000 q^{9} -8075.41 q^{10} -826.834 q^{11} -3713.43 q^{12} +11479.1 q^{13} -9211.71 q^{14} -13380.4 q^{15} -15072.7 q^{16} -23564.6 q^{17} -11879.2 q^{18} -53546.6 q^{19} +68157.8 q^{20} -15263.1 q^{21} +13473.4 q^{22} -87155.1 q^{23} +4194.81 q^{24} +167464. q^{25} -187055. q^{26} -19683.0 q^{27} +77748.3 q^{28} +26745.6 q^{29} +218036. q^{30} -231681. q^{31} +265500. q^{32} +22324.5 q^{33} +383990. q^{34} +280146. q^{35} +100263. q^{36} +409893. q^{37} +872554. q^{38} -309936. q^{39} -76993.4 q^{40} +189635. q^{41} +248716. q^{42} -849460. q^{43} -113718. q^{44} +361270. q^{45} +1.42021e6 q^{46} -902990. q^{47} +406963. q^{48} -503977. q^{49} -2.72886e6 q^{50} +636244. q^{51} +1.57877e6 q^{52} -324949. q^{53} +320739. q^{54} -409753. q^{55} -87827.2 q^{56} +1.44576e6 q^{57} -435826. q^{58} +205379. q^{59} -1.84026e6 q^{60} -1.58592e6 q^{61} +3.77529e6 q^{62} +412105. q^{63} -2.39707e6 q^{64} +5.68870e6 q^{65} -363783. q^{66} -2.57819e6 q^{67} -3.24094e6 q^{68} +2.35319e6 q^{69} -4.56504e6 q^{70} -678037. q^{71} -113260. q^{72} +2.50036e6 q^{73} -6.67930e6 q^{74} -4.52153e6 q^{75} -7.36449e6 q^{76} -467410. q^{77} +5.05048e6 q^{78} -7.33035e6 q^{79} -7.46957e6 q^{80} +531441. q^{81} -3.09015e6 q^{82} +3.91602e6 q^{83} -2.09920e6 q^{84} -1.16779e7 q^{85} +1.38421e7 q^{86} -722131. q^{87} +128460. q^{88} +3.80382e6 q^{89} -5.88698e6 q^{90} +6.48916e6 q^{91} -1.19868e7 q^{92} +6.25539e6 q^{93} +1.47144e7 q^{94} -2.65361e7 q^{95} -7.16849e6 q^{96} -7.68854e6 q^{97} +8.21243e6 q^{98} -602762. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} + O(q^{10}) \) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} - 3479q^{10} + 898q^{11} - 26298q^{12} - 8172q^{13} - 13315q^{14} + 1836q^{15} + 3138q^{16} - 44985q^{17} - 4374q^{18} - 40137q^{19} + 130657q^{20} + 63261q^{21} + 109394q^{22} - 2833q^{23} - 22113q^{24} + 285746q^{25} - 129420q^{26} - 314928q^{27} + 112890q^{28} + 144375q^{29} + 93933q^{30} - 141759q^{31} - 36224q^{32} - 24246q^{33} - 341332q^{34} - 78859q^{35} + 710046q^{36} - 297971q^{37} + 329075q^{38} + 220644q^{39} - 203048q^{40} + 659077q^{41} + 359505q^{42} - 1431608q^{43} + 254916q^{44} - 49572q^{45} + 873113q^{46} - 1574073q^{47} - 84726q^{48} + 1893545q^{49} + 302533q^{50} + 1214595q^{51} - 4972548q^{52} + 587736q^{53} + 118098q^{54} - 4624036q^{55} - 5798506q^{56} + 1083699q^{57} - 6991380q^{58} + 3286064q^{59} - 3527739q^{60} - 6117131q^{61} - 11570258q^{62} - 1708047q^{63} - 19063011q^{64} - 5335514q^{65} - 2953638q^{66} - 16518710q^{67} - 17284669q^{68} + 76491q^{69} - 39189486q^{70} - 10882582q^{71} + 597051q^{72} - 21097441q^{73} - 16717030q^{74} - 7715142q^{75} - 40864952q^{76} - 3404601q^{77} + 3494340q^{78} - 3784458q^{79} - 27466195q^{80} + 8503056q^{81} - 24990117q^{82} - 1951425q^{83} - 3048030q^{84} - 23238675q^{85} - 35910572q^{86} - 3898125q^{87} - 27843055q^{88} + 10499443q^{89} - 2536191q^{90} + 699217q^{91} - 20062766q^{92} + 3827493q^{93} - 59358988q^{94} - 29236333q^{95} + 978048q^{96} - 25158976q^{97} + 2120460q^{98} + 654642q^{99} + O(q^{100}) \)

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 \(-0.755929\pi\)
−0.720154 + 0.693814i \(0.755929\pi\)
\(3\) −27.0000 −0.577350
\(4\) 137.534 1.07449
\(5\) 495.569 1.77300 0.886502 0.462726i \(-0.153128\pi\)
0.886502 + 0.462726i \(0.153128\pi\)
\(6\) 439.971 0.831562
\(7\) 565.301 0.622927 0.311463 0.950258i \(-0.399181\pi\)
0.311463 + 0.950258i \(0.399181\pi\)
\(8\) −155.363 −0.107284
\(9\) 729.000 0.333333
\(10\) −8075.41 −2.55367
\(11\) −826.834 −0.187303 −0.0936513 0.995605i \(-0.529854\pi\)
−0.0936513 + 0.995605i \(0.529854\pi\)
\(12\) −3713.43 −0.620355
\(13\) 11479.1 1.44913 0.724564 0.689207i \(-0.242041\pi\)
0.724564 + 0.689207i \(0.242041\pi\)
\(14\) −9211.71 −0.897206
\(15\) −13380.4 −1.02364
\(16\) −15072.7 −0.919965
\(17\) −23564.6 −1.16329 −0.581646 0.813442i \(-0.697591\pi\)
−0.581646 + 0.813442i \(0.697591\pi\)
\(18\) −11879.2 −0.480103
\(19\) −53546.6 −1.79099 −0.895497 0.445067i \(-0.853180\pi\)
−0.895497 + 0.445067i \(0.853180\pi\)
\(20\) 68157.8 1.90507
\(21\) −15263.1 −0.359647
\(22\) 13473.4 0.269773
\(23\) −87155.1 −1.49364 −0.746819 0.665028i \(-0.768420\pi\)
−0.746819 + 0.665028i \(0.768420\pi\)
\(24\) 4194.81 0.0619403
\(25\) 167464. 2.14354
\(26\) −187055. −2.08719
\(27\) −19683.0 −0.192450
\(28\) 77748.3 0.669326
\(29\) 26745.6 0.203638 0.101819 0.994803i \(-0.467534\pi\)
0.101819 + 0.994803i \(0.467534\pi\)
\(30\) 218036. 1.47436
\(31\) −231681. −1.39677 −0.698384 0.715723i \(-0.746097\pi\)
−0.698384 + 0.715723i \(0.746097\pi\)
\(32\) 265500. 1.43232
\(33\) 22324.5 0.108139
\(34\) 383990. 1.67550
\(35\) 280146. 1.10445
\(36\) 100263. 0.358162
\(37\) 409893. 1.33035 0.665173 0.746689i \(-0.268358\pi\)
0.665173 + 0.746689i \(0.268358\pi\)
\(38\) 872554. 2.57958
\(39\) −309936. −0.836655
\(40\) −76993.4 −0.190214
\(41\) 189635. 0.429710 0.214855 0.976646i \(-0.431072\pi\)
0.214855 + 0.976646i \(0.431072\pi\)
\(42\) 248716. 0.518002
\(43\) −849460. −1.62931 −0.814654 0.579947i \(-0.803073\pi\)
−0.814654 + 0.579947i \(0.803073\pi\)
\(44\) −113718. −0.201254
\(45\) 361270. 0.591001
\(46\) 1.42021e6 2.15130
\(47\) −902990. −1.26865 −0.634323 0.773068i \(-0.718721\pi\)
−0.634323 + 0.773068i \(0.718721\pi\)
\(48\) 406963. 0.531142
\(49\) −503977. −0.611963
\(50\) −2.72886e6 −3.08736
\(51\) 636244. 0.671627
\(52\) 1.57877e6 1.55707
\(53\) −324949. −0.299812 −0.149906 0.988700i \(-0.547897\pi\)
−0.149906 + 0.988700i \(0.547897\pi\)
\(54\) 320739. 0.277187
\(55\) −409753. −0.332088
\(56\) −87827.2 −0.0668299
\(57\) 1.44576e6 1.03403
\(58\) −435826. −0.293302
\(59\) 205379. 0.130189
\(60\) −1.84026e6 −1.09989
\(61\) −1.58592e6 −0.894598 −0.447299 0.894384i \(-0.647614\pi\)
−0.447299 + 0.894384i \(0.647614\pi\)
\(62\) 3.77529e6 2.01178
\(63\) 412105. 0.207642
\(64\) −2.39707e6 −1.14301
\(65\) 5.68870e6 2.56931
\(66\) −363783. −0.155754
\(67\) −2.57819e6 −1.04726 −0.523629 0.851947i \(-0.675422\pi\)
−0.523629 + 0.851947i \(0.675422\pi\)
\(68\) −3.24094e6 −1.24994
\(69\) 2.35319e6 0.862352
\(70\) −4.56504e6 −1.59075
\(71\) −678037. −0.224827 −0.112414 0.993661i \(-0.535858\pi\)
−0.112414 + 0.993661i \(0.535858\pi\)
\(72\) −113260. −0.0357613
\(73\) 2.50036e6 0.752269 0.376135 0.926565i \(-0.377253\pi\)
0.376135 + 0.926565i \(0.377253\pi\)
\(74\) −6.67930e6 −1.91611
\(75\) −4.52153e6 −1.23757
\(76\) −7.36449e6 −1.92440
\(77\) −467410. −0.116676
\(78\) 5.05048e6 1.20504
\(79\) −7.33035e6 −1.67275 −0.836373 0.548161i \(-0.815328\pi\)
−0.836373 + 0.548161i \(0.815328\pi\)
\(80\) −7.46957e6 −1.63110
\(81\) 531441. 0.111111
\(82\) −3.09015e6 −0.618914
\(83\) 3.91602e6 0.751747 0.375874 0.926671i \(-0.377343\pi\)
0.375874 + 0.926671i \(0.377343\pi\)
\(84\) −2.09920e6 −0.386436
\(85\) −1.16779e7 −2.06252
\(86\) 1.38421e7 2.34670
\(87\) −722131. −0.117571
\(88\) 128460. 0.0200945
\(89\) 3.80382e6 0.571946 0.285973 0.958238i \(-0.407683\pi\)
0.285973 + 0.958238i \(0.407683\pi\)
\(90\) −5.88698e6 −0.851223
\(91\) 6.48916e6 0.902700
\(92\) −1.19868e7 −1.60489
\(93\) 6.25539e6 0.806425
\(94\) 1.47144e7 1.82724
\(95\) −2.65361e7 −3.17544
\(96\) −7.16849e6 −0.826948
\(97\) −7.68854e6 −0.855348 −0.427674 0.903933i \(-0.640667\pi\)
−0.427674 + 0.903933i \(0.640667\pi\)
\(98\) 8.21243e6 0.881414
\(99\) −602762. −0.0624342
\(100\) 2.30321e7 2.30321
\(101\) −1.11320e7 −1.07510 −0.537551 0.843231i \(-0.680650\pi\)
−0.537551 + 0.843231i \(0.680650\pi\)
\(102\) −1.03677e7 −0.967349
\(103\) 9.10836e6 0.821315 0.410658 0.911790i \(-0.365299\pi\)
0.410658 + 0.911790i \(0.365299\pi\)
\(104\) −1.78344e6 −0.155468
\(105\) −7.56394e6 −0.637655
\(106\) 5.29511e6 0.431822
\(107\) 2.43881e7 1.92458 0.962289 0.272029i \(-0.0876945\pi\)
0.962289 + 0.272029i \(0.0876945\pi\)
\(108\) −2.70709e6 −0.206785
\(109\) −1.92554e7 −1.42416 −0.712080 0.702098i \(-0.752247\pi\)
−0.712080 + 0.702098i \(0.752247\pi\)
\(110\) 6.67702e6 0.478309
\(111\) −1.10671e7 −0.768076
\(112\) −8.52062e6 −0.573071
\(113\) 9.40627e6 0.613258 0.306629 0.951829i \(-0.400799\pi\)
0.306629 + 0.951829i \(0.400799\pi\)
\(114\) −2.35590e7 −1.48932
\(115\) −4.31914e7 −2.64822
\(116\) 3.67844e6 0.218807
\(117\) 8.36828e6 0.483043
\(118\) −3.34670e6 −0.187512
\(119\) −1.33211e7 −0.724645
\(120\) 2.07882e6 0.109820
\(121\) −1.88035e7 −0.964918
\(122\) 2.58430e7 1.28850
\(123\) −5.12015e6 −0.248093
\(124\) −3.18641e7 −1.50081
\(125\) 4.42737e7 2.02750
\(126\) −6.71534e6 −0.299069
\(127\) 2.55998e7 1.10898 0.554490 0.832190i \(-0.312913\pi\)
0.554490 + 0.832190i \(0.312913\pi\)
\(128\) 5.07683e6 0.213972
\(129\) 2.29354e7 0.940681
\(130\) −9.26986e7 −3.70060
\(131\) 2.04746e7 0.795729 0.397865 0.917444i \(-0.369751\pi\)
0.397865 + 0.917444i \(0.369751\pi\)
\(132\) 3.07039e6 0.116194
\(133\) −3.02700e7 −1.11566
\(134\) 4.20122e7 1.50837
\(135\) −9.75429e6 −0.341215
\(136\) 3.66108e6 0.124802
\(137\) 1.13505e6 0.0377132 0.0188566 0.999822i \(-0.493997\pi\)
0.0188566 + 0.999822i \(0.493997\pi\)
\(138\) −3.83457e7 −1.24205
\(139\) −1.44843e7 −0.457453 −0.228726 0.973491i \(-0.573456\pi\)
−0.228726 + 0.973491i \(0.573456\pi\)
\(140\) 3.85297e7 1.18672
\(141\) 2.43807e7 0.732453
\(142\) 1.10488e7 0.323821
\(143\) −9.49132e6 −0.271425
\(144\) −1.09880e7 −0.306655
\(145\) 1.32543e7 0.361051
\(146\) −4.07440e7 −1.08350
\(147\) 1.36074e7 0.353317
\(148\) 5.63744e7 1.42944
\(149\) 6.91223e7 1.71185 0.855926 0.517098i \(-0.172988\pi\)
0.855926 + 0.517098i \(0.172988\pi\)
\(150\) 7.36793e7 1.78249
\(151\) 5.36019e7 1.26695 0.633477 0.773762i \(-0.281627\pi\)
0.633477 + 0.773762i \(0.281627\pi\)
\(152\) 8.31919e6 0.192145
\(153\) −1.71786e7 −0.387764
\(154\) 7.61655e6 0.168049
\(155\) −1.14814e8 −2.47648
\(156\) −4.26269e7 −0.898974
\(157\) −8.33413e7 −1.71874 −0.859372 0.511350i \(-0.829145\pi\)
−0.859372 + 0.511350i \(0.829145\pi\)
\(158\) 1.19450e8 2.40927
\(159\) 8.77361e6 0.173097
\(160\) 1.31574e8 2.53950
\(161\) −4.92689e7 −0.930426
\(162\) −8.65995e6 −0.160034
\(163\) −5.74632e6 −0.103928 −0.0519641 0.998649i \(-0.516548\pi\)
−0.0519641 + 0.998649i \(0.516548\pi\)
\(164\) 2.60813e7 0.461717
\(165\) 1.10633e7 0.191731
\(166\) −6.38125e7 −1.08275
\(167\) −5.42463e7 −0.901286 −0.450643 0.892704i \(-0.648805\pi\)
−0.450643 + 0.892704i \(0.648805\pi\)
\(168\) 2.37133e6 0.0385843
\(169\) 6.90217e7 1.09997
\(170\) 1.90294e8 2.97066
\(171\) −3.90355e7 −0.596998
\(172\) −1.16830e8 −1.75067
\(173\) −1.02709e8 −1.50816 −0.754081 0.656781i \(-0.771918\pi\)
−0.754081 + 0.656781i \(0.771918\pi\)
\(174\) 1.17673e7 0.169338
\(175\) 9.46676e7 1.33527
\(176\) 1.24626e7 0.172312
\(177\) −5.54523e6 −0.0751646
\(178\) −6.19841e7 −0.823779
\(179\) 1.18297e8 1.54166 0.770828 0.637043i \(-0.219843\pi\)
0.770828 + 0.637043i \(0.219843\pi\)
\(180\) 4.96870e7 0.635023
\(181\) −7.54155e7 −0.945334 −0.472667 0.881241i \(-0.656709\pi\)
−0.472667 + 0.881241i \(0.656709\pi\)
\(182\) −1.05742e8 −1.30017
\(183\) 4.28199e7 0.516496
\(184\) 1.35407e7 0.160243
\(185\) 2.03131e8 2.35871
\(186\) −1.01933e8 −1.16150
\(187\) 1.94840e7 0.217888
\(188\) −1.24192e8 −1.36314
\(189\) −1.11268e7 −0.119882
\(190\) 4.32411e8 4.57361
\(191\) 5.67288e7 0.589097 0.294548 0.955637i \(-0.404831\pi\)
0.294548 + 0.955637i \(0.404831\pi\)
\(192\) 6.47209e7 0.659918
\(193\) 1.56586e7 0.156784 0.0783920 0.996923i \(-0.475021\pi\)
0.0783920 + 0.996923i \(0.475021\pi\)
\(194\) 1.25286e8 1.23196
\(195\) −1.53595e8 −1.48339
\(196\) −6.93142e7 −0.657546
\(197\) 1.95180e6 0.0181888 0.00909440 0.999959i \(-0.497105\pi\)
0.00909440 + 0.999959i \(0.497105\pi\)
\(198\) 9.82214e6 0.0899244
\(199\) 4.03128e7 0.362624 0.181312 0.983426i \(-0.441966\pi\)
0.181312 + 0.983426i \(0.441966\pi\)
\(200\) −2.60178e7 −0.229967
\(201\) 6.96112e7 0.604634
\(202\) 1.81399e8 1.54848
\(203\) 1.51193e7 0.126852
\(204\) 8.75054e7 0.721654
\(205\) 9.39774e7 0.761877
\(206\) −1.48423e8 −1.18295
\(207\) −6.35360e7 −0.497879
\(208\) −1.73021e8 −1.33315
\(209\) 4.42741e7 0.335458
\(210\) 1.23256e8 0.918419
\(211\) 1.16617e8 0.854620 0.427310 0.904105i \(-0.359461\pi\)
0.427310 + 0.904105i \(0.359461\pi\)
\(212\) −4.46916e7 −0.322144
\(213\) 1.83070e7 0.129804
\(214\) −3.97410e8 −2.77199
\(215\) −4.20966e8 −2.88877
\(216\) 3.05802e6 0.0206468
\(217\) −1.30970e8 −0.870084
\(218\) 3.13770e8 2.05123
\(219\) −6.75098e7 −0.434323
\(220\) −5.63552e7 −0.356824
\(221\) −2.70501e8 −1.68576
\(222\) 1.80341e8 1.10627
\(223\) 1.52043e8 0.918121 0.459060 0.888405i \(-0.348186\pi\)
0.459060 + 0.888405i \(0.348186\pi\)
\(224\) 1.50087e8 0.892228
\(225\) 1.22081e8 0.714513
\(226\) −1.53277e8 −0.883280
\(227\) −2.19156e8 −1.24355 −0.621774 0.783197i \(-0.713588\pi\)
−0.621774 + 0.783197i \(0.713588\pi\)
\(228\) 1.98841e8 1.11105
\(229\) −1.50483e8 −0.828061 −0.414030 0.910263i \(-0.635879\pi\)
−0.414030 + 0.910263i \(0.635879\pi\)
\(230\) 7.03813e8 3.81426
\(231\) 1.26201e7 0.0673628
\(232\) −4.15529e6 −0.0218471
\(233\) 1.05923e8 0.548586 0.274293 0.961646i \(-0.411556\pi\)
0.274293 + 0.961646i \(0.411556\pi\)
\(234\) −1.36363e8 −0.695730
\(235\) −4.47494e8 −2.24931
\(236\) 2.82467e7 0.139886
\(237\) 1.97919e8 0.965760
\(238\) 2.17070e8 1.04371
\(239\) −2.39069e8 −1.13274 −0.566370 0.824151i \(-0.691653\pi\)
−0.566370 + 0.824151i \(0.691653\pi\)
\(240\) 2.01678e8 0.941717
\(241\) −6.39780e7 −0.294422 −0.147211 0.989105i \(-0.547030\pi\)
−0.147211 + 0.989105i \(0.547030\pi\)
\(242\) 3.06408e8 1.38978
\(243\) −1.43489e7 −0.0641500
\(244\) −2.18119e8 −0.961234
\(245\) −2.49756e8 −1.08501
\(246\) 8.34340e7 0.357330
\(247\) −6.14668e8 −2.59538
\(248\) 3.59948e7 0.149851
\(249\) −1.05733e8 −0.434022
\(250\) −7.21450e8 −2.92022
\(251\) 1.43897e8 0.574373 0.287186 0.957875i \(-0.407280\pi\)
0.287186 + 0.957875i \(0.407280\pi\)
\(252\) 5.66785e7 0.223109
\(253\) 7.20627e7 0.279762
\(254\) −4.17155e8 −1.59727
\(255\) 3.15303e8 1.19080
\(256\) 2.24097e8 0.834826
\(257\) −1.50715e8 −0.553849 −0.276925 0.960892i \(-0.589315\pi\)
−0.276925 + 0.960892i \(0.589315\pi\)
\(258\) −3.73738e8 −1.35487
\(259\) 2.31713e8 0.828708
\(260\) 7.82391e8 2.76069
\(261\) 1.94975e7 0.0678794
\(262\) −3.33638e8 −1.14609
\(263\) −5.02309e8 −1.70265 −0.851325 0.524638i \(-0.824201\pi\)
−0.851325 + 0.524638i \(0.824201\pi\)
\(264\) −3.46841e6 −0.0116016
\(265\) −1.61035e8 −0.531568
\(266\) 4.93256e8 1.60689
\(267\) −1.02703e8 −0.330213
\(268\) −3.54590e8 −1.12526
\(269\) −2.76426e8 −0.865858 −0.432929 0.901428i \(-0.642520\pi\)
−0.432929 + 0.901428i \(0.642520\pi\)
\(270\) 1.58948e8 0.491454
\(271\) 1.17995e8 0.360139 0.180069 0.983654i \(-0.442368\pi\)
0.180069 + 0.983654i \(0.442368\pi\)
\(272\) 3.55182e8 1.07019
\(273\) −1.75207e8 −0.521174
\(274\) −1.84959e7 −0.0543186
\(275\) −1.38465e8 −0.401491
\(276\) 3.23644e8 0.926586
\(277\) −6.34905e7 −0.179485 −0.0897427 0.995965i \(-0.528604\pi\)
−0.0897427 + 0.995965i \(0.528604\pi\)
\(278\) 2.36025e8 0.658872
\(279\) −1.68895e8 −0.465590
\(280\) −4.35245e7 −0.118490
\(281\) 3.92031e8 1.05402 0.527009 0.849860i \(-0.323313\pi\)
0.527009 + 0.849860i \(0.323313\pi\)
\(282\) −3.97290e8 −1.05496
\(283\) 3.91102e8 1.02574 0.512871 0.858466i \(-0.328582\pi\)
0.512871 + 0.858466i \(0.328582\pi\)
\(284\) −9.32533e7 −0.241574
\(285\) 7.16474e8 1.83334
\(286\) 1.54663e8 0.390936
\(287\) 1.07201e8 0.267678
\(288\) 1.93549e8 0.477439
\(289\) 1.44951e8 0.353248
\(290\) −2.15982e8 −0.520025
\(291\) 2.07590e8 0.493835
\(292\) 3.43886e8 0.808303
\(293\) −3.95594e8 −0.918783 −0.459392 0.888234i \(-0.651933\pi\)
−0.459392 + 0.888234i \(0.651933\pi\)
\(294\) −2.21735e8 −0.508885
\(295\) 1.01780e8 0.230825
\(296\) −6.36824e7 −0.142725
\(297\) 1.62746e7 0.0360464
\(298\) −1.12636e9 −2.46559
\(299\) −1.00046e9 −2.16447
\(300\) −6.21865e8 −1.32976
\(301\) −4.80201e8 −1.01494
\(302\) −8.73455e8 −1.82480
\(303\) 3.00565e8 0.620710
\(304\) 8.07092e8 1.64765
\(305\) −7.85935e8 −1.58612
\(306\) 2.79929e8 0.558499
\(307\) −3.74974e8 −0.739634 −0.369817 0.929105i \(-0.620580\pi\)
−0.369817 + 0.929105i \(0.620580\pi\)
\(308\) −6.42849e7 −0.125367
\(309\) −2.45926e8 −0.474187
\(310\) 1.87092e9 3.56689
\(311\) 5.67572e8 1.06994 0.534970 0.844871i \(-0.320323\pi\)
0.534970 + 0.844871i \(0.320323\pi\)
\(312\) 4.81528e7 0.0897594
\(313\) −9.61191e8 −1.77176 −0.885879 0.463916i \(-0.846444\pi\)
−0.885879 + 0.463916i \(0.846444\pi\)
\(314\) 1.35806e9 2.47552
\(315\) 2.04226e8 0.368150
\(316\) −1.00817e9 −1.79734
\(317\) 1.03762e8 0.182950 0.0914748 0.995807i \(-0.470842\pi\)
0.0914748 + 0.995807i \(0.470842\pi\)
\(318\) −1.42968e8 −0.249312
\(319\) −2.21142e7 −0.0381420
\(320\) −1.18791e9 −2.02656
\(321\) −6.58480e8 −1.11116
\(322\) 8.02847e8 1.34010
\(323\) 1.26180e9 2.08345
\(324\) 7.30914e7 0.119387
\(325\) 1.92234e9 3.10626
\(326\) 9.36376e7 0.149689
\(327\) 5.19895e8 0.822240
\(328\) −2.94624e7 −0.0461009
\(329\) −5.10462e8 −0.790273
\(330\) −1.80280e8 −0.276152
\(331\) −4.86499e8 −0.737367 −0.368684 0.929555i \(-0.620191\pi\)
−0.368684 + 0.929555i \(0.620191\pi\)
\(332\) 5.38587e8 0.807743
\(333\) 2.98812e8 0.443449
\(334\) 8.83956e8 1.29813
\(335\) −1.27767e9 −1.85679
\(336\) 2.30057e8 0.330862
\(337\) −2.32747e8 −0.331269 −0.165634 0.986187i \(-0.552967\pi\)
−0.165634 + 0.986187i \(0.552967\pi\)
\(338\) −1.12472e9 −1.58430
\(339\) −2.53969e8 −0.354064
\(340\) −1.60611e9 −2.21615
\(341\) 1.91562e8 0.261618
\(342\) 6.36092e8 0.859861
\(343\) −7.50449e8 −1.00413
\(344\) 1.31975e8 0.174798
\(345\) 1.16617e9 1.52895
\(346\) 1.67367e9 2.17222
\(347\) −1.12320e8 −0.144312 −0.0721560 0.997393i \(-0.522988\pi\)
−0.0721560 + 0.997393i \(0.522988\pi\)
\(348\) −9.93178e7 −0.126328
\(349\) 1.23358e9 1.55338 0.776688 0.629885i \(-0.216898\pi\)
0.776688 + 0.629885i \(0.216898\pi\)
\(350\) −1.54263e9 −1.92320
\(351\) −2.25943e8 −0.278885
\(352\) −2.19524e8 −0.268277
\(353\) 1.25683e9 1.52078 0.760390 0.649466i \(-0.225008\pi\)
0.760390 + 0.649466i \(0.225008\pi\)
\(354\) 9.03608e7 0.108260
\(355\) −3.36014e8 −0.398619
\(356\) 5.23156e8 0.614549
\(357\) 3.59669e8 0.418374
\(358\) −1.92767e9 −2.22046
\(359\) 1.53369e8 0.174947 0.0874735 0.996167i \(-0.472121\pi\)
0.0874735 + 0.996167i \(0.472121\pi\)
\(360\) −5.61282e7 −0.0634048
\(361\) 1.97337e9 2.20766
\(362\) 1.22891e9 1.36157
\(363\) 5.07695e8 0.557096
\(364\) 8.92482e8 0.969940
\(365\) 1.23910e9 1.33378
\(366\) −6.97761e8 −0.743914
\(367\) −9.10830e8 −0.961847 −0.480924 0.876762i \(-0.659699\pi\)
−0.480924 + 0.876762i \(0.659699\pi\)
\(368\) 1.31366e9 1.37409
\(369\) 1.38244e8 0.143237
\(370\) −3.31006e9 −3.39727
\(371\) −1.83694e8 −0.186761
\(372\) 8.60330e8 0.866493
\(373\) −8.48680e8 −0.846766 −0.423383 0.905951i \(-0.639157\pi\)
−0.423383 + 0.905951i \(0.639157\pi\)
\(374\) −3.17496e8 −0.313825
\(375\) −1.19539e9 −1.17058
\(376\) 1.40292e8 0.136105
\(377\) 3.07016e8 0.295098
\(378\) 1.81314e8 0.172667
\(379\) 1.11369e9 1.05082 0.525410 0.850849i \(-0.323912\pi\)
0.525410 + 0.850849i \(0.323912\pi\)
\(380\) −3.64962e9 −3.41197
\(381\) −6.91195e8 −0.640270
\(382\) −9.24408e8 −0.848481
\(383\) −1.24305e9 −1.13056 −0.565279 0.824900i \(-0.691232\pi\)
−0.565279 + 0.824900i \(0.691232\pi\)
\(384\) −1.37074e8 −0.123537
\(385\) −2.31634e8 −0.206866
\(386\) −2.55160e8 −0.225817
\(387\) −6.19256e8 −0.543103
\(388\) −1.05744e9 −0.919060
\(389\) 3.31934e8 0.285909 0.142955 0.989729i \(-0.454340\pi\)
0.142955 + 0.989729i \(0.454340\pi\)
\(390\) 2.50286e9 2.13654
\(391\) 2.05377e9 1.73754
\(392\) 7.82997e7 0.0656536
\(393\) −5.52813e8 −0.459414
\(394\) −3.18051e7 −0.0261975
\(395\) −3.63270e9 −2.96578
\(396\) −8.29004e7 −0.0670847
\(397\) −2.50412e8 −0.200858 −0.100429 0.994944i \(-0.532021\pi\)
−0.100429 + 0.994944i \(0.532021\pi\)
\(398\) −6.56906e8 −0.522291
\(399\) 8.17289e8 0.644126
\(400\) −2.52414e9 −1.97198
\(401\) 1.57100e9 1.21666 0.608332 0.793683i \(-0.291839\pi\)
0.608332 + 0.793683i \(0.291839\pi\)
\(402\) −1.13433e9 −0.870859
\(403\) −2.65949e9 −2.02410
\(404\) −1.53104e9 −1.15518
\(405\) 2.63366e8 0.197000
\(406\) −2.46373e8 −0.182705
\(407\) −3.38914e8 −0.249177
\(408\) −9.88491e7 −0.0720546
\(409\) −2.37344e8 −0.171532 −0.0857661 0.996315i \(-0.527334\pi\)
−0.0857661 + 0.996315i \(0.527334\pi\)
\(410\) −1.53138e9 −1.09734
\(411\) −3.06464e7 −0.0217737
\(412\) 1.25271e9 0.882492
\(413\) 1.16101e8 0.0810981
\(414\) 1.03533e9 0.717099
\(415\) 1.94066e9 1.33285
\(416\) 3.04770e9 2.07561
\(417\) 3.91076e8 0.264110
\(418\) −7.21457e8 −0.483163
\(419\) −1.61768e9 −1.07435 −0.537173 0.843472i \(-0.680508\pi\)
−0.537173 + 0.843472i \(0.680508\pi\)
\(420\) −1.04030e9 −0.685152
\(421\) −1.17270e9 −0.765946 −0.382973 0.923759i \(-0.625100\pi\)
−0.382973 + 0.923759i \(0.625100\pi\)
\(422\) −1.90030e9 −1.23092
\(423\) −6.58280e8 −0.422882
\(424\) 5.04852e7 0.0321650
\(425\) −3.94622e9 −2.49356
\(426\) −2.98317e8 −0.186958
\(427\) −8.96525e8 −0.557269
\(428\) 3.35421e9 2.06793
\(429\) 2.56266e8 0.156708
\(430\) 6.85974e9 4.16071
\(431\) 2.74618e9 1.65219 0.826094 0.563533i \(-0.190558\pi\)
0.826094 + 0.563533i \(0.190558\pi\)
\(432\) 2.96676e8 0.177047
\(433\) 1.44724e8 0.0856709 0.0428355 0.999082i \(-0.486361\pi\)
0.0428355 + 0.999082i \(0.486361\pi\)
\(434\) 2.13418e9 1.25319
\(435\) −3.57866e8 −0.208453
\(436\) −2.64827e9 −1.53024
\(437\) 4.66686e9 2.67510
\(438\) 1.10009e9 0.625559
\(439\) −2.83060e8 −0.159681 −0.0798403 0.996808i \(-0.525441\pi\)
−0.0798403 + 0.996808i \(0.525441\pi\)
\(440\) 6.36607e7 0.0356276
\(441\) −3.67400e8 −0.203988
\(442\) 4.40787e9 2.42801
\(443\) −6.58269e8 −0.359741 −0.179871 0.983690i \(-0.557568\pi\)
−0.179871 + 0.983690i \(0.557568\pi\)
\(444\) −1.52211e9 −0.825287
\(445\) 1.88506e9 1.01406
\(446\) −2.47758e9 −1.32238
\(447\) −1.86630e9 −0.988339
\(448\) −1.35507e9 −0.712012
\(449\) 2.96392e9 1.54527 0.772636 0.634850i \(-0.218938\pi\)
0.772636 + 0.634850i \(0.218938\pi\)
\(450\) −1.98934e9 −1.02912
\(451\) −1.56797e8 −0.0804857
\(452\) 1.29368e9 0.658937
\(453\) −1.44725e9 −0.731476
\(454\) 3.57119e9 1.79109
\(455\) 3.21583e9 1.60049
\(456\) −2.24618e8 −0.110935
\(457\) −2.62985e8 −0.128891 −0.0644456 0.997921i \(-0.520528\pi\)
−0.0644456 + 0.997921i \(0.520528\pi\)
\(458\) 2.45215e9 1.19266
\(459\) 4.63822e8 0.223876
\(460\) −5.94030e9 −2.84548
\(461\) 7.39703e8 0.351645 0.175822 0.984422i \(-0.443742\pi\)
0.175822 + 0.984422i \(0.443742\pi\)
\(462\) −2.05647e8 −0.0970231
\(463\) −8.00108e8 −0.374641 −0.187320 0.982299i \(-0.559980\pi\)
−0.187320 + 0.982299i \(0.559980\pi\)
\(464\) −4.03129e8 −0.187340
\(465\) 3.09998e9 1.42979
\(466\) −1.72604e9 −0.790132
\(467\) 3.44400e9 1.56478 0.782392 0.622786i \(-0.213999\pi\)
0.782392 + 0.622786i \(0.213999\pi\)
\(468\) 1.15093e9 0.519023
\(469\) −1.45745e9 −0.652364
\(470\) 7.29202e9 3.23970
\(471\) 2.25021e9 0.992318
\(472\) −3.19084e7 −0.0139672
\(473\) 7.02362e8 0.305174
\(474\) −3.22514e9 −1.39099
\(475\) −8.96713e9 −3.83907
\(476\) −1.83211e9 −0.778622
\(477\) −2.36888e8 −0.0999374
\(478\) 3.89568e9 1.63149
\(479\) −2.17648e9 −0.904857 −0.452429 0.891801i \(-0.649442\pi\)
−0.452429 + 0.891801i \(0.649442\pi\)
\(480\) −3.55248e9 −1.46618
\(481\) 4.70521e9 1.92784
\(482\) 1.04254e9 0.424059
\(483\) 1.33026e9 0.537182
\(484\) −2.58613e9 −1.03679
\(485\) −3.81020e9 −1.51653
\(486\) 2.33819e8 0.0923958
\(487\) 1.73782e9 0.681796 0.340898 0.940100i \(-0.389269\pi\)
0.340898 + 0.940100i \(0.389269\pi\)
\(488\) 2.46395e8 0.0959758
\(489\) 1.55151e8 0.0600030
\(490\) 4.06983e9 1.56275
\(491\) −7.21828e8 −0.275200 −0.137600 0.990488i \(-0.543939\pi\)
−0.137600 + 0.990488i \(0.543939\pi\)
\(492\) −7.04196e8 −0.266573
\(493\) −6.30249e8 −0.236891
\(494\) 1.00161e10 3.73815
\(495\) −2.98710e8 −0.110696
\(496\) 3.49206e9 1.28498
\(497\) −3.83295e8 −0.140051
\(498\) 1.72294e9 0.625125
\(499\) −1.78127e9 −0.641769 −0.320885 0.947118i \(-0.603980\pi\)
−0.320885 + 0.947118i \(0.603980\pi\)
\(500\) 6.08915e9 2.17852
\(501\) 1.46465e9 0.520358
\(502\) −2.34483e9 −0.827273
\(503\) 1.72676e9 0.604986 0.302493 0.953152i \(-0.402181\pi\)
0.302493 + 0.953152i \(0.402181\pi\)
\(504\) −6.40260e7 −0.0222766
\(505\) −5.51669e9 −1.90616
\(506\) −1.17428e9 −0.402944
\(507\) −1.86358e9 −0.635069
\(508\) 3.52085e9 1.19158
\(509\) 1.96038e8 0.0658912 0.0329456 0.999457i \(-0.489511\pi\)
0.0329456 + 0.999457i \(0.489511\pi\)
\(510\) −5.13793e9 −1.71511
\(511\) 1.41346e9 0.468608
\(512\) −4.30154e9 −1.41638
\(513\) 1.05396e9 0.344677
\(514\) 2.45594e9 0.797713
\(515\) 4.51383e9 1.45619
\(516\) 3.15441e9 1.01075
\(517\) 7.46623e8 0.237621
\(518\) −3.77582e9 −1.19359
\(519\) 2.77315e9 0.870738
\(520\) −8.83816e8 −0.275645
\(521\) −2.84883e9 −0.882540 −0.441270 0.897374i \(-0.645472\pi\)
−0.441270 + 0.897374i \(0.645472\pi\)
\(522\) −3.17717e8 −0.0977673
\(523\) −3.08000e9 −0.941445 −0.470722 0.882281i \(-0.656007\pi\)
−0.470722 + 0.882281i \(0.656007\pi\)
\(524\) 2.81596e9 0.855000
\(525\) −2.55603e9 −0.770917
\(526\) 8.18523e9 2.45234
\(527\) 5.45947e9 1.62485
\(528\) −3.36491e8 −0.0994843
\(529\) 4.19118e9 1.23095
\(530\) 2.62410e9 0.765621
\(531\) 1.49721e8 0.0433963
\(532\) −4.16316e9 −1.19876
\(533\) 2.17684e9 0.622705
\(534\) 1.67357e9 0.475609
\(535\) 1.20860e10 3.41228
\(536\) 4.00557e8 0.112354
\(537\) −3.19402e9 −0.890076
\(538\) 4.50443e9 1.24710
\(539\) 4.16706e8 0.114622
\(540\) −1.34155e9 −0.366631
\(541\) 3.69807e9 1.00412 0.502060 0.864833i \(-0.332576\pi\)
0.502060 + 0.864833i \(0.332576\pi\)
\(542\) −1.92275e9 −0.518711
\(543\) 2.03622e9 0.545789
\(544\) −6.25639e9 −1.66620
\(545\) −9.54237e9 −2.52504
\(546\) 2.85504e9 0.750651
\(547\) −1.21521e9 −0.317464 −0.158732 0.987322i \(-0.550741\pi\)
−0.158732 + 0.987322i \(0.550741\pi\)
\(548\) 1.56108e8 0.0405223
\(549\) −1.15614e9 −0.298199
\(550\) 2.25632e9 0.578270
\(551\) −1.43214e9 −0.364715
\(552\) −3.65599e8 −0.0925163
\(553\) −4.14386e9 −1.04200
\(554\) 1.03459e9 0.258514
\(555\) −5.48453e9 −1.36180
\(556\) −1.99209e9 −0.491527
\(557\) −6.35497e9 −1.55819 −0.779095 0.626906i \(-0.784321\pi\)
−0.779095 + 0.626906i \(0.784321\pi\)
\(558\) 2.75219e9 0.670592
\(559\) −9.75105e9 −2.36108
\(560\) −4.22256e9 −1.01606
\(561\) −5.26068e8 −0.125797
\(562\) −6.38823e9 −1.51811
\(563\) 2.41689e9 0.570791 0.285395 0.958410i \(-0.407875\pi\)
0.285395 + 0.958410i \(0.407875\pi\)
\(564\) 3.35319e9 0.787011
\(565\) 4.66146e9 1.08731
\(566\) −6.37310e9 −1.47738
\(567\) 3.00424e8 0.0692141
\(568\) 1.05342e8 0.0241203
\(569\) 2.02038e9 0.459769 0.229884 0.973218i \(-0.426165\pi\)
0.229884 + 0.973218i \(0.426165\pi\)
\(570\) −1.16751e10 −2.64057
\(571\) −5.09913e9 −1.14622 −0.573112 0.819477i \(-0.694264\pi\)
−0.573112 + 0.819477i \(0.694264\pi\)
\(572\) −1.30538e9 −0.291643
\(573\) −1.53168e9 −0.340115
\(574\) −1.74686e9 −0.385538
\(575\) −1.45953e10 −3.20167
\(576\) −1.74746e9 −0.381004
\(577\) −5.65588e9 −1.22570 −0.612851 0.790199i \(-0.709977\pi\)
−0.612851 + 0.790199i \(0.709977\pi\)
\(578\) −2.36201e9 −0.508785
\(579\) −4.22782e8 −0.0905193
\(580\) 1.82292e9 0.387945
\(581\) 2.21373e9 0.468283
\(582\) −3.38273e9 −0.711275
\(583\) 2.68678e8 0.0561556
\(584\) −3.88465e8 −0.0807063
\(585\) 4.14706e9 0.856436
\(586\) 6.44629e9 1.32333
\(587\) 8.74088e9 1.78370 0.891850 0.452331i \(-0.149407\pi\)
0.891850 + 0.452331i \(0.149407\pi\)
\(588\) 1.87148e9 0.379634
\(589\) 1.24057e10 2.50161
\(590\) −1.65852e9 −0.332460
\(591\) −5.26987e7 −0.0105013
\(592\) −6.17820e9 −1.22387
\(593\) −7.16535e9 −1.41106 −0.705531 0.708679i \(-0.749291\pi\)
−0.705531 + 0.708679i \(0.749291\pi\)
\(594\) −2.65198e8 −0.0519179
\(595\) −6.60153e9 −1.28480
\(596\) 9.50669e9 1.83936
\(597\) −1.08845e9 −0.209361
\(598\) 1.63028e10 3.11751
\(599\) −1.12486e8 −0.0213847 −0.0106923 0.999943i \(-0.503404\pi\)
−0.0106923 + 0.999943i \(0.503404\pi\)
\(600\) 7.02481e8 0.132772
\(601\) 2.43804e9 0.458121 0.229061 0.973412i \(-0.426435\pi\)
0.229061 + 0.973412i \(0.426435\pi\)
\(602\) 7.82498e9 1.46182
\(603\) −1.87950e9 −0.349086
\(604\) 7.37210e9 1.36132
\(605\) −9.31845e9 −1.71080
\(606\) −4.89777e9 −0.894014
\(607\) 3.37659e9 0.612800 0.306400 0.951903i \(-0.400876\pi\)
0.306400 + 0.951903i \(0.400876\pi\)
\(608\) −1.42166e10 −2.56527
\(609\) −4.08222e8 −0.0732379
\(610\) 1.28070e10 2.28451
\(611\) −1.03655e10 −1.83843
\(612\) −2.36264e9 −0.416647
\(613\) −3.64627e9 −0.639348 −0.319674 0.947528i \(-0.603573\pi\)
−0.319674 + 0.947528i \(0.603573\pi\)
\(614\) 6.11029e9 1.06530
\(615\) −2.53739e9 −0.439870
\(616\) 7.26184e7 0.0125174
\(617\) −2.27884e8 −0.0390585 −0.0195293 0.999809i \(-0.506217\pi\)
−0.0195293 + 0.999809i \(0.506217\pi\)
\(618\) 4.00741e9 0.682975
\(619\) 2.30512e9 0.390640 0.195320 0.980740i \(-0.437426\pi\)
0.195320 + 0.980740i \(0.437426\pi\)
\(620\) −1.57909e10 −2.66094
\(621\) 1.71547e9 0.287451
\(622\) −9.24870e9 −1.54104
\(623\) 2.15031e9 0.356280
\(624\) 4.67158e9 0.769693
\(625\) 8.85757e9 1.45122
\(626\) 1.56628e10 2.55188
\(627\) −1.19540e9 −0.193677
\(628\) −1.14623e10 −1.84677
\(629\) −9.65897e9 −1.54758
\(630\) −3.32792e9 −0.530250
\(631\) 5.67126e9 0.898622 0.449311 0.893375i \(-0.351670\pi\)
0.449311 + 0.893375i \(0.351670\pi\)
\(632\) 1.13887e9 0.179458
\(633\) −3.14866e9 −0.493415
\(634\) −1.69083e9 −0.263504
\(635\) 1.26865e10 1.96623
\(636\) 1.20667e9 0.185990
\(637\) −5.78522e9 −0.886812
\(638\) 3.60355e8 0.0549362
\(639\) −4.94289e8 −0.0749424
\(640\) 2.51592e9 0.379373
\(641\) −8.04557e9 −1.20657 −0.603287 0.797524i \(-0.706142\pi\)
−0.603287 + 0.797524i \(0.706142\pi\)
\(642\) 1.07301e10 1.60041
\(643\) 6.95787e9 1.03214 0.516069 0.856547i \(-0.327395\pi\)
0.516069 + 0.856547i \(0.327395\pi\)
\(644\) −6.77616e9 −0.999731
\(645\) 1.13661e10 1.66783
\(646\) −2.05614e10 −3.00081
\(647\) −1.50113e9 −0.217898 −0.108949 0.994047i \(-0.534748\pi\)
−0.108949 + 0.994047i \(0.534748\pi\)
\(648\) −8.25665e7 −0.0119204
\(649\) −1.69814e8 −0.0243847
\(650\) −3.13250e10 −4.47398
\(651\) 3.53618e9 0.502343
\(652\) −7.90316e8 −0.111669
\(653\) −1.05745e9 −0.148616 −0.0743078 0.997235i \(-0.523675\pi\)
−0.0743078 + 0.997235i \(0.523675\pi\)
\(654\) −8.47180e9 −1.18428
\(655\) 1.01466e10 1.41083
\(656\) −2.85831e9 −0.395318
\(657\) 1.82277e9 0.250756
\(658\) 8.31808e9 1.13824
\(659\) −1.27158e10 −1.73079 −0.865396 0.501088i \(-0.832933\pi\)
−0.865396 + 0.501088i \(0.832933\pi\)
\(660\) 1.52159e9 0.206013
\(661\) −1.38832e10 −1.86976 −0.934878 0.354968i \(-0.884492\pi\)
−0.934878 + 0.354968i \(0.884492\pi\)
\(662\) 7.92761e9 1.06204
\(663\) 7.30352e9 0.973273
\(664\) −6.08407e8 −0.0806503
\(665\) −1.50009e10 −1.97807
\(666\) −4.86921e9 −0.638703
\(667\) −2.33101e9 −0.304162
\(668\) −7.46073e9 −0.968420
\(669\) −4.10516e9 −0.530077
\(670\) 2.08200e10 2.67435
\(671\) 1.31130e9 0.167560
\(672\) −4.05236e9 −0.515128
\(673\) 8.66625e9 1.09592 0.547960 0.836505i \(-0.315405\pi\)
0.547960 + 0.836505i \(0.315405\pi\)
\(674\) 3.79267e9 0.477129
\(675\) −3.29620e9 −0.412524
\(676\) 9.49284e9 1.18191
\(677\) −1.30940e10 −1.62186 −0.810929 0.585144i \(-0.801038\pi\)
−0.810929 + 0.585144i \(0.801038\pi\)
\(678\) 4.13849e9 0.509962
\(679\) −4.34634e9 −0.532819
\(680\) 1.81432e9 0.221275
\(681\) 5.91721e9 0.717963
\(682\) −3.12154e9 −0.376811
\(683\) 5.07430e9 0.609402 0.304701 0.952448i \(-0.401443\pi\)
0.304701 + 0.952448i \(0.401443\pi\)
\(684\) −5.36872e9 −0.641467
\(685\) 5.62496e8 0.0668656
\(686\) 1.22287e10 1.44626
\(687\) 4.06303e9 0.478081
\(688\) 1.28037e10 1.49891
\(689\) −3.73012e9 −0.434466
\(690\) −1.90030e10 −2.20216
\(691\) 7.09480e9 0.818026 0.409013 0.912529i \(-0.365873\pi\)
0.409013 + 0.912529i \(0.365873\pi\)
\(692\) −1.41260e10 −1.62050
\(693\) −3.40742e8 −0.0388919
\(694\) 1.83027e9 0.207854
\(695\) −7.17798e9 −0.811065
\(696\) 1.12193e8 0.0126134
\(697\) −4.46867e9 −0.499878
\(698\) −2.01014e10 −2.23734
\(699\) −2.85992e9 −0.316726
\(700\) 1.30200e10 1.43473
\(701\) 5.74769e9 0.630203 0.315101 0.949058i \(-0.397961\pi\)
0.315101 + 0.949058i \(0.397961\pi\)
\(702\) 3.68180e9 0.401680
\(703\) −2.19484e10 −2.38264
\(704\) 1.98198e9 0.214089
\(705\) 1.20823e10 1.29864
\(706\) −2.04804e10 −2.19039
\(707\) −6.29295e9 −0.669709
\(708\) −7.62660e8 −0.0807634
\(709\) −5.47895e9 −0.577345 −0.288672 0.957428i \(-0.593214\pi\)
−0.288672 + 0.957428i \(0.593214\pi\)
\(710\) 5.47543e9 0.574135
\(711\) −5.34383e9 −0.557582
\(712\) −5.90975e8 −0.0613605
\(713\) 2.01922e10 2.08627
\(714\) −5.86089e9 −0.602588
\(715\) −4.70361e9 −0.481238
\(716\) 1.62699e10 1.65649
\(717\) 6.45486e9 0.653988
\(718\) −2.49918e9 −0.251978
\(719\) 1.62636e10 1.63179 0.815895 0.578200i \(-0.196245\pi\)
0.815895 + 0.578200i \(0.196245\pi\)
\(720\) −5.44532e9 −0.543700
\(721\) 5.14897e9 0.511619
\(722\) −3.21565e10 −3.17971
\(723\) 1.72741e9 0.169985
\(724\) −1.03722e10 −1.01575
\(725\) 4.47893e9 0.436507
\(726\) −8.27300e9 −0.802389
\(727\) −2.55116e9 −0.246245 −0.123122 0.992391i \(-0.539291\pi\)
−0.123122 + 0.992391i \(0.539291\pi\)
\(728\) −1.00818e9 −0.0968451
\(729\) 3.87420e8 0.0370370
\(730\) −2.01915e10 −1.92105
\(731\) 2.00172e10 1.89536
\(732\) 5.88921e9 0.554968
\(733\) 1.41305e10 1.32524 0.662619 0.748957i \(-0.269445\pi\)
0.662619 + 0.748957i \(0.269445\pi\)
\(734\) 1.48422e10 1.38536
\(735\) 6.74341e9 0.626432
\(736\) −2.31396e10 −2.13936
\(737\) 2.13174e9 0.196154
\(738\) −2.25272e9 −0.206305
\(739\) 1.92246e10 1.75227 0.876136 0.482064i \(-0.160113\pi\)
0.876136 + 0.482064i \(0.160113\pi\)
\(740\) 2.79374e10 2.53440
\(741\) 1.65960e10 1.49844
\(742\) 2.99333e9 0.268993
\(743\) 5.89705e9 0.527441 0.263721 0.964599i \(-0.415050\pi\)
0.263721 + 0.964599i \(0.415050\pi\)
\(744\) −9.71859e8 −0.0865163
\(745\) 3.42549e10 3.03512
\(746\) 1.38294e10 1.21960
\(747\) 2.85478e9 0.250582
\(748\) 2.67972e9 0.234117
\(749\) 1.37867e10 1.19887
\(750\) 1.94791e10 1.68599
\(751\) −3.11911e9 −0.268715 −0.134357 0.990933i \(-0.542897\pi\)
−0.134357 + 0.990933i \(0.542897\pi\)
\(752\) 1.36105e10 1.16711
\(753\) −3.88522e9 −0.331614
\(754\) −5.00289e9 −0.425032
\(755\) 2.65635e10 2.24631
\(756\) −1.53032e9 −0.128812
\(757\) −1.03709e10 −0.868921 −0.434461 0.900691i \(-0.643061\pi\)
−0.434461 + 0.900691i \(0.643061\pi\)
\(758\) −1.81479e10 −1.51350
\(759\) −1.94569e9 −0.161521
\(760\) 4.12273e9 0.340673
\(761\) 1.48892e9 0.122468 0.0612342 0.998123i \(-0.480496\pi\)
0.0612342 + 0.998123i \(0.480496\pi\)
\(762\) 1.12632e10 0.922186
\(763\) −1.08851e10 −0.887147
\(764\) 7.80215e9 0.632977
\(765\) −8.51318e9 −0.687507
\(766\) 2.02558e10 1.62835
\(767\) 2.35757e9 0.188660
\(768\) −6.05062e9 −0.481987
\(769\) 1.76503e10 1.39962 0.699809 0.714330i \(-0.253268\pi\)
0.699809 + 0.714330i \(0.253268\pi\)
\(770\) 3.77453e9 0.297951
\(771\) 4.06931e9 0.319765
\(772\) 2.15359e9 0.168462
\(773\) −6.95196e8 −0.0541351 −0.0270676 0.999634i \(-0.508617\pi\)
−0.0270676 + 0.999634i \(0.508617\pi\)
\(774\) 1.00909e10 0.782235
\(775\) −3.87982e10 −2.99403
\(776\) 1.19452e9 0.0917649
\(777\) −6.25626e9 −0.478455
\(778\) −5.40894e9 −0.411798
\(779\) −1.01543e10 −0.769608
\(780\) −2.11246e10 −1.59388
\(781\) 5.60624e8 0.0421107
\(782\) −3.34667e10 −2.50259
\(783\) −5.26434e8 −0.0391902
\(784\) 7.59631e9 0.562984
\(785\) −4.13014e10 −3.04734
\(786\) 9.00822e9 0.661698
\(787\) −7.19358e8 −0.0526058 −0.0263029 0.999654i \(-0.508373\pi\)
−0.0263029 + 0.999654i \(0.508373\pi\)
\(788\) 2.68440e8 0.0195436
\(789\) 1.35623e10 0.983026
\(790\) 5.91956e10 4.27164
\(791\) 5.31738e9 0.382014
\(792\) 9.36471e7 0.00669817
\(793\) −1.82050e10 −1.29639
\(794\) 4.08052e9 0.289297
\(795\) 4.34793e9 0.306901
\(796\) 5.54439e9 0.389635
\(797\) 4.07407e9 0.285052 0.142526 0.989791i \(-0.454477\pi\)
0.142526 + 0.989791i \(0.454477\pi\)
\(798\) −1.33179e10 −0.927739
\(799\) 2.12786e10 1.47581
\(800\) 4.44616e10 3.07023
\(801\) 2.77299e9 0.190649
\(802\) −2.55998e10 −1.75237
\(803\) −2.06739e9 −0.140902
\(804\) 9.57392e9 0.649671
\(805\) −2.44161e10 −1.64965
\(806\) 4.33371e10 2.91532
\(807\) 7.46351e9 0.499903
\(808\) 1.72951e9 0.115341
\(809\) −1.12391e10 −0.746296 −0.373148 0.927772i \(-0.621722\pi\)
−0.373148 + 0.927772i \(0.621722\pi\)
\(810\) −4.29161e9 −0.283741
\(811\) 1.57592e10 1.03744 0.518719 0.854945i \(-0.326409\pi\)
0.518719 + 0.854945i \(0.326409\pi\)
\(812\) 2.07943e9 0.136300
\(813\) −3.18586e9 −0.207926
\(814\) 5.52267e9 0.358892
\(815\) −2.84770e9 −0.184265
\(816\) −9.58992e9 −0.617873
\(817\) 4.54857e10 2.91808
\(818\) 3.86757e9 0.247059
\(819\) 4.73060e9 0.300900
\(820\) 1.29251e10 0.818627
\(821\) −1.83331e10 −1.15621 −0.578103 0.815964i \(-0.696207\pi\)
−0.578103 + 0.815964i \(0.696207\pi\)
\(822\) 4.99389e8 0.0313608
\(823\) 3.71121e9 0.232069 0.116034 0.993245i \(-0.462982\pi\)
0.116034 + 0.993245i \(0.462982\pi\)
\(824\) −1.41511e9 −0.0881138
\(825\) 3.73855e9 0.231801
\(826\) −1.89189e9 −0.116806
\(827\) 3.00461e9 0.184722 0.0923610 0.995726i \(-0.470559\pi\)
0.0923610 + 0.995726i \(0.470559\pi\)
\(828\) −8.73838e9 −0.534964
\(829\) 1.97799e10 1.20582 0.602912 0.797808i \(-0.294007\pi\)
0.602912 + 0.797808i \(0.294007\pi\)
\(830\) −3.16235e10 −1.91971
\(831\) 1.71424e9 0.103626
\(832\) −2.75162e10 −1.65637
\(833\) 1.18760e10 0.711891
\(834\) −6.37268e9 −0.380400
\(835\) −2.68828e10 −1.59798
\(836\) 6.08921e9 0.360445
\(837\) 4.56018e9 0.268808
\(838\) 2.63605e10 1.54739
\(839\) −7.17247e9 −0.419278 −0.209639 0.977779i \(-0.567229\pi\)
−0.209639 + 0.977779i \(0.567229\pi\)
\(840\) 1.17516e9 0.0684100
\(841\) −1.65345e10 −0.958531
\(842\) 1.91094e10 1.10320
\(843\) −1.05848e10 −0.608538
\(844\) 1.60388e10 0.918278
\(845\) 3.42050e10 1.95025
\(846\) 1.07268e10 0.609080
\(847\) −1.06297e10 −0.601073
\(848\) 4.89786e9 0.275817
\(849\) −1.05598e10 −0.592212
\(850\) 6.43046e10 3.59150
\(851\) −3.57243e10 −1.98706
\(852\) 2.51784e9 0.139473
\(853\) −2.70770e9 −0.149375 −0.0746877 0.997207i \(-0.523796\pi\)
−0.0746877 + 0.997207i \(0.523796\pi\)
\(854\) 1.46091e10 0.802639
\(855\) −1.93448e10 −1.05848
\(856\) −3.78903e9 −0.206476
\(857\) −1.93553e10 −1.05043 −0.525215 0.850969i \(-0.676015\pi\)
−0.525215 + 0.850969i \(0.676015\pi\)
\(858\) −4.17591e9 −0.225707
\(859\) −3.02146e10 −1.62645 −0.813224 0.581951i \(-0.802290\pi\)
−0.813224 + 0.581951i \(0.802290\pi\)
\(860\) −5.78973e10 −3.10394
\(861\) −2.89443e9 −0.154544
\(862\) −4.47497e10 −2.37966
\(863\) 4.20793e9 0.222859 0.111430 0.993772i \(-0.464457\pi\)
0.111430 + 0.993772i \(0.464457\pi\)
\(864\) −5.22583e9 −0.275649
\(865\) −5.08996e10 −2.67398
\(866\) −2.35831e9 −0.123393
\(867\) −3.91368e9 −0.203948
\(868\) −1.80128e10 −0.934894
\(869\) 6.06098e9 0.313310
\(870\) 5.83151e9 0.300237
\(871\) −2.95954e10 −1.51761
\(872\) 2.99158e9 0.152789
\(873\) −5.60494e9 −0.285116
\(874\) −7.60475e10 −3.85296
\(875\) 2.50280e10 1.26298
\(876\) −9.28492e9 −0.466674
\(877\) −1.97999e10 −0.991206 −0.495603 0.868549i \(-0.665053\pi\)
−0.495603 + 0.868549i \(0.665053\pi\)
\(878\) 4.61252e9 0.229989
\(879\) 1.06810e10 0.530460
\(880\) 6.17609e9 0.305509
\(881\) 5.47108e9 0.269561 0.134781 0.990875i \(-0.456967\pi\)
0.134781 + 0.990875i \(0.456967\pi\)
\(882\) 5.98686e9 0.293805
\(883\) 1.99701e10 0.976153 0.488076 0.872801i \(-0.337699\pi\)
0.488076 + 0.872801i \(0.337699\pi\)
\(884\) −3.72031e10 −1.81133
\(885\) −2.74805e9 −0.133267
\(886\) 1.07266e10 0.518138
\(887\) −6.76780e9 −0.325623 −0.162811 0.986657i \(-0.552056\pi\)
−0.162811 + 0.986657i \(0.552056\pi\)
\(888\) 1.71943e9 0.0824021
\(889\) 1.44716e10 0.690813
\(890\) −3.07174e10 −1.46056
\(891\) −4.39413e8 −0.0208114
\(892\) 2.09111e10 0.986508
\(893\) 4.83521e10 2.27214
\(894\) 3.04118e10 1.42351
\(895\) 5.86243e10 2.73336
\(896\) 2.86994e9 0.133289
\(897\) 2.70125e10 1.24966
\(898\) −4.82978e10 −2.22567
\(899\) −6.19645e9 −0.284436
\(900\) 1.67904e10 0.767735
\(901\) 7.65728e9 0.348769
\(902\) 2.55504e9 0.115924
\(903\) 1.29654e10 0.585975
\(904\) −1.46139e9 −0.0657926
\(905\) −3.73736e10 −1.67608
\(906\) 2.35833e10 1.05355
\(907\) −2.47127e10 −1.09975 −0.549877 0.835246i \(-0.685325\pi\)
−0.549877 + 0.835246i \(0.685325\pi\)
\(908\) −3.01414e10 −1.33618
\(909\) −8.11525e9 −0.358367
\(910\) −5.24027e10 −2.30520
\(911\) −3.74402e9 −0.164068 −0.0820340 0.996630i \(-0.526142\pi\)
−0.0820340 + 0.996630i \(0.526142\pi\)
\(912\) −2.17915e10 −0.951273
\(913\) −3.23790e9 −0.140804
\(914\) 4.28539e9 0.185643
\(915\) 2.12203e10 0.915750
\(916\) −2.06965e10 −0.889740
\(917\) 1.15743e10 0.495681
\(918\) −7.55808e9 −0.322450
\(919\) −3.76618e9 −0.160065 −0.0800326 0.996792i \(-0.525502\pi\)
−0.0800326 + 0.996792i \(0.525502\pi\)
\(920\) 6.71036e9 0.284111
\(921\) 1.01243e10 0.427028
\(922\) −1.20536e10 −0.506477
\(923\) −7.78326e9 −0.325804
\(924\) 1.73569e9 0.0723804
\(925\) 6.86424e10 2.85165
\(926\) 1.30379e10 0.539598
\(927\) 6.63999e9 0.273772
\(928\) 7.10095e9 0.291675
\(929\) 3.28415e10 1.34390 0.671952 0.740595i \(-0.265456\pi\)
0.671952 + 0.740595i \(0.265456\pi\)
\(930\) −5.05149e10 −2.05934
\(931\) 2.69863e10 1.09602
\(932\) 1.45680e10 0.589448
\(933\) −1.53244e10 −0.617730
\(934\) −5.61208e10 −2.25377
\(935\) 9.65567e9 0.386315
\(936\) −1.30012e9 −0.0518226
\(937\) −3.67023e10 −1.45749 −0.728744 0.684787i \(-0.759895\pi\)
−0.728744 + 0.684787i \(0.759895\pi\)
\(938\) 2.37496e10 0.939605
\(939\) 2.59522e10 1.02293
\(940\) −6.15458e10 −2.41686
\(941\) 2.52107e10 0.986329 0.493165 0.869936i \(-0.335840\pi\)
0.493165 + 0.869936i \(0.335840\pi\)
\(942\) −3.66677e10 −1.42924
\(943\) −1.65277e10 −0.641831
\(944\) −3.09562e9 −0.119769
\(945\) −5.51411e9 −0.212552
\(946\) −1.14451e10 −0.439544
\(947\) 8.87956e9 0.339755 0.169878 0.985465i \(-0.445663\pi\)
0.169878 + 0.985465i \(0.445663\pi\)
\(948\) 2.72207e10 1.03770
\(949\) 2.87020e10 1.09013
\(950\) 1.46121e11 5.52944
\(951\) −2.80158e9 −0.105626
\(952\) 2.06961e9 0.0777427
\(953\) 1.58158e10 0.591925 0.295962 0.955200i \(-0.404360\pi\)
0.295962 + 0.955200i \(0.404360\pi\)
\(954\) 3.86014e9 0.143941
\(955\) 2.81131e10 1.04447
\(956\) −3.28802e10 −1.21711
\(957\) 5.97082e8 0.0220213
\(958\) 3.54662e10 1.30327
\(959\) 6.41645e8 0.0234925
\(960\) 3.20737e10 1.17004
\(961\) 2.61635e10 0.950963
\(962\) −7.66725e10 −2.77669
\(963\) 1.77790e10 0.641526
\(964\) −8.79917e9 −0.316353
\(965\) 7.75992e9 0.277979
\(966\) −2.16769e10 −0.773707
\(967\) 5.18915e10 1.84546 0.922728 0.385451i \(-0.125954\pi\)
0.922728 + 0.385451i \(0.125954\pi\)
\(968\) 2.92138e9 0.103520
\(969\) −3.40687e10 −1.20288
\(970\) 6.20881e10 2.18428
\(971\) −1.69159e10 −0.592963 −0.296481 0.955039i \(-0.595813\pi\)
−0.296481 + 0.955039i \(0.595813\pi\)
\(972\) −1.97347e9 −0.0689284
\(973\) −8.18800e9 −0.284959
\(974\) −2.83182e10 −0.981996
\(975\) −5.19032e10 −1.79340
\(976\) 2.39042e10 0.822999
\(977\) 8.24427e9 0.282827 0.141414 0.989951i \(-0.454835\pi\)
0.141414 + 0.989951i \(0.454835\pi\)
\(978\) −2.52821e9 −0.0864227
\(979\) −3.14513e9 −0.107127
\(980\) −3.43500e10 −1.16583
\(981\) −1.40372e10 −0.474720
\(982\) 1.17623e10 0.396373
\(983\) −4.10669e9 −0.137897 −0.0689484 0.997620i \(-0.521964\pi\)
−0.0689484 + 0.997620i \(0.521964\pi\)
\(984\) 7.95484e8 0.0266164
\(985\) 9.67254e8 0.0322488
\(986\) 1.02701e10 0.341196
\(987\) 1.37825e10 0.456265
\(988\) −8.45379e10 −2.78870
\(989\) 7.40347e10 2.43359
\(990\) 4.86755e9 0.159436
\(991\) −1.53943e9 −0.0502460 −0.0251230 0.999684i \(-0.507998\pi\)
−0.0251230 + 0.999684i \(0.507998\pi\)
\(992\) −6.15112e10 −2.00062
\(993\) 1.31355e10 0.425719
\(994\) 6.24588e9 0.201716
\(995\) 1.99778e10 0.642934
\(996\) −1.45419e10 −0.466350
\(997\) −2.17704e10 −0.695717 −0.347858 0.937547i \(-0.613091\pi\)
−0.347858 + 0.937547i \(0.613091\pi\)
\(998\) 2.90263e10 0.924345
\(999\) −8.06793e9 −0.256025
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 177.8.a.a.1.3 16
3.2 odd 2 531.8.a.b.1.14 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.3 16 1.1 even 1 trivial
531.8.a.b.1.14 16 3.2 odd 2