Properties

Label 177.8.a.b.1.7
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $17$
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: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.7
Root \(-4.85375\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-6.85375 q^{2} +27.0000 q^{3} -81.0262 q^{4} -190.727 q^{5} -185.051 q^{6} -799.288 q^{7} +1432.61 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-6.85375 q^{2} +27.0000 q^{3} -81.0262 q^{4} -190.727 q^{5} -185.051 q^{6} -799.288 q^{7} +1432.61 q^{8} +729.000 q^{9} +1307.20 q^{10} +972.348 q^{11} -2187.71 q^{12} -1278.77 q^{13} +5478.12 q^{14} -5149.64 q^{15} +552.586 q^{16} +36467.5 q^{17} -4996.38 q^{18} +1900.72 q^{19} +15453.9 q^{20} -21580.8 q^{21} -6664.23 q^{22} +80181.7 q^{23} +38680.5 q^{24} -41748.1 q^{25} +8764.35 q^{26} +19683.0 q^{27} +64763.2 q^{28} -201337. q^{29} +35294.3 q^{30} +106900. q^{31} -187162. q^{32} +26253.4 q^{33} -249939. q^{34} +152446. q^{35} -59068.1 q^{36} -246438. q^{37} -13027.1 q^{38} -34526.7 q^{39} -273238. q^{40} -348348. q^{41} +147909. q^{42} +303791. q^{43} -78785.6 q^{44} -139040. q^{45} -549545. q^{46} +1.12223e6 q^{47} +14919.8 q^{48} -184682. q^{49} +286131. q^{50} +984622. q^{51} +103614. q^{52} +534787. q^{53} -134902. q^{54} -185453. q^{55} -1.14507e6 q^{56} +51319.5 q^{57} +1.37991e6 q^{58} -205379. q^{59} +417255. q^{60} -2.14348e6 q^{61} -732666. q^{62} -582681. q^{63} +1.21203e6 q^{64} +243896. q^{65} -179934. q^{66} -158432. q^{67} -2.95482e6 q^{68} +2.16491e6 q^{69} -1.04483e6 q^{70} -3.94945e6 q^{71} +1.04437e6 q^{72} +3.25842e6 q^{73} +1.68903e6 q^{74} -1.12720e6 q^{75} -154008. q^{76} -777186. q^{77} +236637. q^{78} -5.28930e6 q^{79} -105393. q^{80} +531441. q^{81} +2.38749e6 q^{82} -3.63337e6 q^{83} +1.74861e6 q^{84} -6.95534e6 q^{85} -2.08211e6 q^{86} -5.43609e6 q^{87} +1.39300e6 q^{88} -1.34439e6 q^{89} +952946. q^{90} +1.02210e6 q^{91} -6.49682e6 q^{92} +2.88630e6 q^{93} -7.69146e6 q^{94} -362519. q^{95} -5.05336e6 q^{96} +2.79048e6 q^{97} +1.26576e6 q^{98} +708842. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} - 6391q^{10} - 8888q^{11} + 31482q^{12} - 12702q^{13} - 17555q^{14} - 28944q^{15} + 139226q^{16} - 36167q^{17} - 23328q^{18} - 71037q^{19} - 274883q^{20} - 64989q^{21} - 325182q^{22} - 269995q^{23} - 179415q^{24} + 97329q^{25} - 336906q^{26} + 334611q^{27} - 901362q^{28} - 543825q^{29} - 172557q^{30} - 633109q^{31} - 837062q^{32} - 239976q^{33} - 529288q^{34} - 287621q^{35} + 850014q^{36} - 867607q^{37} - 1727169q^{38} - 342954q^{39} - 815662q^{40} - 1428939q^{41} - 473985q^{42} - 477060q^{43} - 1667926q^{44} - 781488q^{45} + 5305549q^{46} - 1217849q^{47} + 3759102q^{48} + 4350738q^{49} + 4561369q^{50} - 976509q^{51} + 4175994q^{52} - 3487068q^{53} - 629856q^{54} - 960484q^{55} - 5363196q^{56} - 1917999q^{57} - 3082906q^{58} - 3491443q^{59} - 7421841q^{60} + 998917q^{61} - 5742614q^{62} - 1754703q^{63} + 17531621q^{64} - 6075816q^{65} - 8779914q^{66} - 356026q^{67} - 16149231q^{68} - 7289865q^{69} - 548798q^{70} - 12879428q^{71} - 4844205q^{72} - 6176157q^{73} - 5971906q^{74} + 2627883q^{75} - 17624580q^{76} + 239687q^{77} - 9096462q^{78} - 18886490q^{79} - 70463349q^{80} + 9034497q^{81} - 19351611q^{82} - 22824893q^{83} - 24336774q^{84} - 7973079q^{85} - 27502196q^{86} - 14683275q^{87} - 62527651q^{88} - 30609647q^{89} - 4659039q^{90} - 36301521q^{91} - 41388548q^{92} - 17093943q^{93} + 1010176q^{94} - 29303629q^{95} - 22600674q^{96} - 26249806q^{97} - 93110852q^{98} - 6479352q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.85375 −0.605791 −0.302896 0.953024i \(-0.597953\pi\)
−0.302896 + 0.953024i \(0.597953\pi\)
\(3\) 27.0000 0.577350
\(4\) −81.0262 −0.633017
\(5\) −190.727 −0.682367 −0.341183 0.939997i \(-0.610828\pi\)
−0.341183 + 0.939997i \(0.610828\pi\)
\(6\) −185.051 −0.349754
\(7\) −799.288 −0.880765 −0.440383 0.897810i \(-0.645157\pi\)
−0.440383 + 0.897810i \(0.645157\pi\)
\(8\) 1432.61 0.989267
\(9\) 729.000 0.333333
\(10\) 1307.20 0.413372
\(11\) 972.348 0.220266 0.110133 0.993917i \(-0.464872\pi\)
0.110133 + 0.993917i \(0.464872\pi\)
\(12\) −2187.71 −0.365472
\(13\) −1278.77 −0.161432 −0.0807161 0.996737i \(-0.525721\pi\)
−0.0807161 + 0.996737i \(0.525721\pi\)
\(14\) 5478.12 0.533560
\(15\) −5149.64 −0.393965
\(16\) 552.586 0.0337272
\(17\) 36467.5 1.80026 0.900128 0.435625i \(-0.143473\pi\)
0.900128 + 0.435625i \(0.143473\pi\)
\(18\) −4996.38 −0.201930
\(19\) 1900.72 0.0635742 0.0317871 0.999495i \(-0.489880\pi\)
0.0317871 + 0.999495i \(0.489880\pi\)
\(20\) 15453.9 0.431950
\(21\) −21580.8 −0.508510
\(22\) −6664.23 −0.133435
\(23\) 80181.7 1.37413 0.687065 0.726596i \(-0.258899\pi\)
0.687065 + 0.726596i \(0.258899\pi\)
\(24\) 38680.5 0.571154
\(25\) −41748.1 −0.534376
\(26\) 8764.35 0.0977942
\(27\) 19683.0 0.192450
\(28\) 64763.2 0.557539
\(29\) −201337. −1.53296 −0.766479 0.642270i \(-0.777993\pi\)
−0.766479 + 0.642270i \(0.777993\pi\)
\(30\) 35294.3 0.238660
\(31\) 106900. 0.644484 0.322242 0.946657i \(-0.395564\pi\)
0.322242 + 0.946657i \(0.395564\pi\)
\(32\) −187162. −1.00970
\(33\) 26253.4 0.127171
\(34\) −249939. −1.09058
\(35\) 152446. 0.601005
\(36\) −59068.1 −0.211006
\(37\) −246438. −0.799838 −0.399919 0.916550i \(-0.630962\pi\)
−0.399919 + 0.916550i \(0.630962\pi\)
\(38\) −13027.1 −0.0385127
\(39\) −34526.7 −0.0932029
\(40\) −273238. −0.675043
\(41\) −348348. −0.789350 −0.394675 0.918821i \(-0.629143\pi\)
−0.394675 + 0.918821i \(0.629143\pi\)
\(42\) 147909. 0.308051
\(43\) 303791. 0.582687 0.291343 0.956619i \(-0.405898\pi\)
0.291343 + 0.956619i \(0.405898\pi\)
\(44\) −78785.6 −0.139432
\(45\) −139040. −0.227456
\(46\) −549545. −0.832436
\(47\) 1.12223e6 1.57666 0.788330 0.615252i \(-0.210946\pi\)
0.788330 + 0.615252i \(0.210946\pi\)
\(48\) 14919.8 0.0194724
\(49\) −184682. −0.224253
\(50\) 286131. 0.323720
\(51\) 984622. 1.03938
\(52\) 103614. 0.102189
\(53\) 534787. 0.493418 0.246709 0.969090i \(-0.420651\pi\)
0.246709 + 0.969090i \(0.420651\pi\)
\(54\) −134902. −0.116585
\(55\) −185453. −0.150302
\(56\) −1.14507e6 −0.871312
\(57\) 51319.5 0.0367046
\(58\) 1.37991e6 0.928652
\(59\) −205379. −0.130189
\(60\) 417255. 0.249386
\(61\) −2.14348e6 −1.20911 −0.604555 0.796564i \(-0.706649\pi\)
−0.604555 + 0.796564i \(0.706649\pi\)
\(62\) −732666. −0.390423
\(63\) −582681. −0.293588
\(64\) 1.21203e6 0.577940
\(65\) 243896. 0.110156
\(66\) −179934. −0.0770389
\(67\) −158432. −0.0643548 −0.0321774 0.999482i \(-0.510244\pi\)
−0.0321774 + 0.999482i \(0.510244\pi\)
\(68\) −2.95482e6 −1.13959
\(69\) 2.16491e6 0.793355
\(70\) −1.04483e6 −0.364084
\(71\) −3.94945e6 −1.30958 −0.654790 0.755811i \(-0.727243\pi\)
−0.654790 + 0.755811i \(0.727243\pi\)
\(72\) 1.04437e6 0.329756
\(73\) 3.25842e6 0.980342 0.490171 0.871626i \(-0.336934\pi\)
0.490171 + 0.871626i \(0.336934\pi\)
\(74\) 1.68903e6 0.484535
\(75\) −1.12720e6 −0.308522
\(76\) −154008. −0.0402435
\(77\) −777186. −0.194003
\(78\) 236637. 0.0564615
\(79\) −5.28930e6 −1.20699 −0.603494 0.797367i \(-0.706225\pi\)
−0.603494 + 0.797367i \(0.706225\pi\)
\(80\) −105393. −0.0230143
\(81\) 531441. 0.111111
\(82\) 2.38749e6 0.478181
\(83\) −3.63337e6 −0.697487 −0.348744 0.937218i \(-0.613392\pi\)
−0.348744 + 0.937218i \(0.613392\pi\)
\(84\) 1.74861e6 0.321895
\(85\) −6.95534e6 −1.22844
\(86\) −2.08211e6 −0.352987
\(87\) −5.43609e6 −0.885053
\(88\) 1.39300e6 0.217902
\(89\) −1.34439e6 −0.202143 −0.101072 0.994879i \(-0.532227\pi\)
−0.101072 + 0.994879i \(0.532227\pi\)
\(90\) 952946. 0.137791
\(91\) 1.02210e6 0.142184
\(92\) −6.49682e6 −0.869848
\(93\) 2.88630e6 0.372093
\(94\) −7.69146e6 −0.955128
\(95\) −362519. −0.0433809
\(96\) −5.05336e6 −0.582950
\(97\) 2.79048e6 0.310441 0.155220 0.987880i \(-0.450391\pi\)
0.155220 + 0.987880i \(0.450391\pi\)
\(98\) 1.26576e6 0.135850
\(99\) 708842. 0.0734220
\(100\) 3.38269e6 0.338269
\(101\) −1.43415e7 −1.38506 −0.692532 0.721387i \(-0.743505\pi\)
−0.692532 + 0.721387i \(0.743505\pi\)
\(102\) −6.74835e6 −0.629647
\(103\) 1.62771e6 0.146773 0.0733867 0.997304i \(-0.476619\pi\)
0.0733867 + 0.997304i \(0.476619\pi\)
\(104\) −1.83198e6 −0.159700
\(105\) 4.11604e6 0.346990
\(106\) −3.66529e6 −0.298908
\(107\) −7.86386e6 −0.620572 −0.310286 0.950643i \(-0.600425\pi\)
−0.310286 + 0.950643i \(0.600425\pi\)
\(108\) −1.59484e6 −0.121824
\(109\) 2.63958e6 0.195228 0.0976141 0.995224i \(-0.468879\pi\)
0.0976141 + 0.995224i \(0.468879\pi\)
\(110\) 1.27105e6 0.0910518
\(111\) −6.65383e6 −0.461787
\(112\) −441675. −0.0297057
\(113\) −2.91448e7 −1.90015 −0.950074 0.312026i \(-0.898992\pi\)
−0.950074 + 0.312026i \(0.898992\pi\)
\(114\) −351731. −0.0222353
\(115\) −1.52928e7 −0.937661
\(116\) 1.63135e7 0.970388
\(117\) −932222. −0.0538107
\(118\) 1.40762e6 0.0788673
\(119\) −2.91480e7 −1.58560
\(120\) −7.37743e6 −0.389736
\(121\) −1.85417e7 −0.951483
\(122\) 1.46909e7 0.732468
\(123\) −9.40539e6 −0.455731
\(124\) −8.66170e6 −0.407969
\(125\) 2.28631e7 1.04701
\(126\) 3.99355e6 0.177853
\(127\) −3.71615e7 −1.60983 −0.804915 0.593390i \(-0.797789\pi\)
−0.804915 + 0.593390i \(0.797789\pi\)
\(128\) 1.56498e7 0.659588
\(129\) 8.20235e6 0.336414
\(130\) −1.67160e6 −0.0667315
\(131\) 1.50104e7 0.583369 0.291684 0.956515i \(-0.405784\pi\)
0.291684 + 0.956515i \(0.405784\pi\)
\(132\) −2.12721e6 −0.0805011
\(133\) −1.51922e6 −0.0559939
\(134\) 1.08585e6 0.0389856
\(135\) −3.75409e6 −0.131322
\(136\) 5.22437e7 1.78094
\(137\) 1.75306e7 0.582470 0.291235 0.956652i \(-0.405934\pi\)
0.291235 + 0.956652i \(0.405934\pi\)
\(138\) −1.48377e7 −0.480607
\(139\) −1.16871e7 −0.369110 −0.184555 0.982822i \(-0.559084\pi\)
−0.184555 + 0.982822i \(0.559084\pi\)
\(140\) −1.23521e7 −0.380446
\(141\) 3.03001e7 0.910286
\(142\) 2.70685e7 0.793332
\(143\) −1.24341e6 −0.0355580
\(144\) 402835. 0.0112424
\(145\) 3.84004e7 1.04604
\(146\) −2.23324e7 −0.593883
\(147\) −4.98641e6 −0.129472
\(148\) 1.99679e7 0.506311
\(149\) −6.76391e7 −1.67512 −0.837559 0.546346i \(-0.816018\pi\)
−0.837559 + 0.546346i \(0.816018\pi\)
\(150\) 7.72553e6 0.186900
\(151\) 4.31830e7 1.02069 0.510345 0.859970i \(-0.329518\pi\)
0.510345 + 0.859970i \(0.329518\pi\)
\(152\) 2.72300e6 0.0628919
\(153\) 2.65848e7 0.600085
\(154\) 5.32664e6 0.117525
\(155\) −2.03888e7 −0.439775
\(156\) 2.79757e6 0.0589990
\(157\) 4.62920e7 0.954679 0.477340 0.878719i \(-0.341601\pi\)
0.477340 + 0.878719i \(0.341601\pi\)
\(158\) 3.62515e7 0.731183
\(159\) 1.44392e7 0.284875
\(160\) 3.56968e7 0.688985
\(161\) −6.40883e7 −1.21029
\(162\) −3.64236e6 −0.0673101
\(163\) 7.22501e7 1.30672 0.653359 0.757048i \(-0.273359\pi\)
0.653359 + 0.757048i \(0.273359\pi\)
\(164\) 2.82253e7 0.499672
\(165\) −5.00724e6 −0.0867770
\(166\) 2.49022e7 0.422532
\(167\) 3.24667e7 0.539425 0.269713 0.962941i \(-0.413071\pi\)
0.269713 + 0.962941i \(0.413071\pi\)
\(168\) −3.09169e7 −0.503052
\(169\) −6.11133e7 −0.973940
\(170\) 4.76702e7 0.744175
\(171\) 1.38563e6 0.0211914
\(172\) −2.46150e7 −0.368851
\(173\) 1.31381e7 0.192918 0.0964589 0.995337i \(-0.469248\pi\)
0.0964589 + 0.995337i \(0.469248\pi\)
\(174\) 3.72576e7 0.536158
\(175\) 3.33687e7 0.470659
\(176\) 537306. 0.00742895
\(177\) −5.54523e6 −0.0751646
\(178\) 9.21409e6 0.122457
\(179\) −1.99708e7 −0.260261 −0.130131 0.991497i \(-0.541540\pi\)
−0.130131 + 0.991497i \(0.541540\pi\)
\(180\) 1.12659e7 0.143983
\(181\) 9.38499e7 1.17641 0.588205 0.808712i \(-0.299835\pi\)
0.588205 + 0.808712i \(0.299835\pi\)
\(182\) −7.00524e6 −0.0861337
\(183\) −5.78740e7 −0.698080
\(184\) 1.14869e8 1.35938
\(185\) 4.70025e7 0.545783
\(186\) −1.97820e7 −0.225411
\(187\) 3.54591e7 0.396535
\(188\) −9.09298e7 −0.998053
\(189\) −1.57324e7 −0.169503
\(190\) 2.48462e6 0.0262798
\(191\) −7.50591e7 −0.779447 −0.389724 0.920932i \(-0.627429\pi\)
−0.389724 + 0.920932i \(0.627429\pi\)
\(192\) 3.27247e7 0.333674
\(193\) −7.13691e7 −0.714594 −0.357297 0.933991i \(-0.616302\pi\)
−0.357297 + 0.933991i \(0.616302\pi\)
\(194\) −1.91253e7 −0.188062
\(195\) 6.58519e6 0.0635985
\(196\) 1.49641e7 0.141956
\(197\) −1.40656e8 −1.31077 −0.655384 0.755296i \(-0.727493\pi\)
−0.655384 + 0.755296i \(0.727493\pi\)
\(198\) −4.85822e6 −0.0444784
\(199\) 2.79140e7 0.251094 0.125547 0.992088i \(-0.459931\pi\)
0.125547 + 0.992088i \(0.459931\pi\)
\(200\) −5.98088e7 −0.528640
\(201\) −4.27767e6 −0.0371553
\(202\) 9.82931e7 0.839060
\(203\) 1.60926e8 1.35018
\(204\) −7.97801e7 −0.657944
\(205\) 6.64394e7 0.538626
\(206\) −1.11559e7 −0.0889140
\(207\) 5.84525e7 0.458043
\(208\) −706629. −0.00544465
\(209\) 1.84816e6 0.0140032
\(210\) −2.82103e7 −0.210204
\(211\) 1.66531e8 1.22041 0.610207 0.792242i \(-0.291086\pi\)
0.610207 + 0.792242i \(0.291086\pi\)
\(212\) −4.33317e7 −0.312342
\(213\) −1.06635e8 −0.756087
\(214\) 5.38969e7 0.375937
\(215\) −5.79412e7 −0.397606
\(216\) 2.81981e7 0.190385
\(217\) −8.54439e7 −0.567639
\(218\) −1.80910e7 −0.118268
\(219\) 8.79775e7 0.566001
\(220\) 1.50266e7 0.0951438
\(221\) −4.66334e7 −0.290619
\(222\) 4.56037e7 0.279746
\(223\) 8.94297e6 0.0540026 0.0270013 0.999635i \(-0.491404\pi\)
0.0270013 + 0.999635i \(0.491404\pi\)
\(224\) 1.49596e8 0.889308
\(225\) −3.04344e7 −0.178125
\(226\) 1.99751e8 1.15109
\(227\) −6.88172e7 −0.390487 −0.195243 0.980755i \(-0.562550\pi\)
−0.195243 + 0.980755i \(0.562550\pi\)
\(228\) −4.15822e6 −0.0232346
\(229\) 1.97507e8 1.08682 0.543411 0.839467i \(-0.317132\pi\)
0.543411 + 0.839467i \(0.317132\pi\)
\(230\) 1.04813e8 0.568027
\(231\) −2.09840e7 −0.112007
\(232\) −2.88438e8 −1.51651
\(233\) 2.46365e8 1.27595 0.637973 0.770058i \(-0.279773\pi\)
0.637973 + 0.770058i \(0.279773\pi\)
\(234\) 6.38921e6 0.0325981
\(235\) −2.14039e8 −1.07586
\(236\) 1.66411e7 0.0824118
\(237\) −1.42811e8 −0.696855
\(238\) 1.99773e8 0.960545
\(239\) −3.78259e8 −1.79224 −0.896120 0.443812i \(-0.853626\pi\)
−0.896120 + 0.443812i \(0.853626\pi\)
\(240\) −2.84562e6 −0.0132873
\(241\) −2.71983e8 −1.25165 −0.625824 0.779964i \(-0.715237\pi\)
−0.625824 + 0.779964i \(0.715237\pi\)
\(242\) 1.27080e8 0.576400
\(243\) 1.43489e7 0.0641500
\(244\) 1.73678e8 0.765387
\(245\) 3.52239e7 0.153023
\(246\) 6.44622e7 0.276078
\(247\) −2.43058e6 −0.0102629
\(248\) 1.53146e8 0.637567
\(249\) −9.81010e7 −0.402695
\(250\) −1.56698e8 −0.634268
\(251\) 1.67816e8 0.669847 0.334923 0.942245i \(-0.391290\pi\)
0.334923 + 0.942245i \(0.391290\pi\)
\(252\) 4.72124e7 0.185846
\(253\) 7.79645e7 0.302674
\(254\) 2.54695e8 0.975221
\(255\) −1.87794e8 −0.709237
\(256\) −2.62399e8 −0.977513
\(257\) −3.77370e8 −1.38676 −0.693381 0.720571i \(-0.743880\pi\)
−0.693381 + 0.720571i \(0.743880\pi\)
\(258\) −5.62169e7 −0.203797
\(259\) 1.96975e8 0.704470
\(260\) −1.97619e7 −0.0697305
\(261\) −1.46775e8 −0.510986
\(262\) −1.02878e8 −0.353400
\(263\) −2.37395e8 −0.804686 −0.402343 0.915489i \(-0.631804\pi\)
−0.402343 + 0.915489i \(0.631804\pi\)
\(264\) 3.76109e7 0.125806
\(265\) −1.01998e8 −0.336692
\(266\) 1.04124e7 0.0339206
\(267\) −3.62985e7 −0.116707
\(268\) 1.28371e7 0.0407377
\(269\) 2.56851e8 0.804542 0.402271 0.915521i \(-0.368221\pi\)
0.402271 + 0.915521i \(0.368221\pi\)
\(270\) 2.57296e7 0.0795535
\(271\) −2.99436e8 −0.913927 −0.456964 0.889485i \(-0.651063\pi\)
−0.456964 + 0.889485i \(0.651063\pi\)
\(272\) 2.01514e7 0.0607176
\(273\) 2.75968e7 0.0820898
\(274\) −1.20150e8 −0.352855
\(275\) −4.05937e7 −0.117705
\(276\) −1.75414e8 −0.502207
\(277\) −1.07342e8 −0.303453 −0.151727 0.988422i \(-0.548483\pi\)
−0.151727 + 0.988422i \(0.548483\pi\)
\(278\) 8.01006e7 0.223604
\(279\) 7.79302e7 0.214828
\(280\) 2.18396e8 0.594555
\(281\) 1.26999e8 0.341451 0.170726 0.985319i \(-0.445389\pi\)
0.170726 + 0.985319i \(0.445389\pi\)
\(282\) −2.07669e8 −0.551443
\(283\) 6.79420e8 1.78191 0.890955 0.454091i \(-0.150036\pi\)
0.890955 + 0.454091i \(0.150036\pi\)
\(284\) 3.20009e8 0.828986
\(285\) −9.78802e6 −0.0250460
\(286\) 8.52200e6 0.0215407
\(287\) 2.78430e8 0.695232
\(288\) −1.36441e8 −0.336566
\(289\) 9.19537e8 2.24092
\(290\) −2.63187e8 −0.633682
\(291\) 7.53431e7 0.179233
\(292\) −2.64018e8 −0.620573
\(293\) 3.54246e8 0.822752 0.411376 0.911466i \(-0.365048\pi\)
0.411376 + 0.911466i \(0.365048\pi\)
\(294\) 3.41756e7 0.0784333
\(295\) 3.91714e7 0.0888366
\(296\) −3.53051e8 −0.791254
\(297\) 1.91387e7 0.0423902
\(298\) 4.63581e8 1.01477
\(299\) −1.02534e8 −0.221829
\(300\) 9.13325e7 0.195300
\(301\) −2.42816e8 −0.513210
\(302\) −2.95966e8 −0.618325
\(303\) −3.87221e8 −0.799667
\(304\) 1.05031e6 0.00214418
\(305\) 4.08821e8 0.825056
\(306\) −1.82205e8 −0.363527
\(307\) −4.47178e8 −0.882056 −0.441028 0.897493i \(-0.645386\pi\)
−0.441028 + 0.897493i \(0.645386\pi\)
\(308\) 6.29724e7 0.122807
\(309\) 4.39482e7 0.0847396
\(310\) 1.39739e8 0.266412
\(311\) −1.03177e9 −1.94501 −0.972506 0.232879i \(-0.925185\pi\)
−0.972506 + 0.232879i \(0.925185\pi\)
\(312\) −4.94634e7 −0.0922026
\(313\) −1.47667e8 −0.272194 −0.136097 0.990695i \(-0.543456\pi\)
−0.136097 + 0.990695i \(0.543456\pi\)
\(314\) −3.17274e8 −0.578336
\(315\) 1.11133e8 0.200335
\(316\) 4.28571e8 0.764044
\(317\) −8.08129e8 −1.42486 −0.712431 0.701742i \(-0.752406\pi\)
−0.712431 + 0.701742i \(0.752406\pi\)
\(318\) −9.89629e7 −0.172575
\(319\) −1.95769e8 −0.337658
\(320\) −2.31167e8 −0.394367
\(321\) −2.12324e8 −0.358288
\(322\) 4.39245e8 0.733181
\(323\) 6.93145e7 0.114450
\(324\) −4.30606e7 −0.0703352
\(325\) 5.33861e7 0.0862654
\(326\) −4.95184e8 −0.791599
\(327\) 7.12687e7 0.112715
\(328\) −4.99047e8 −0.780878
\(329\) −8.96983e8 −1.38867
\(330\) 3.43184e7 0.0525688
\(331\) −1.71934e8 −0.260594 −0.130297 0.991475i \(-0.541593\pi\)
−0.130297 + 0.991475i \(0.541593\pi\)
\(332\) 2.94398e8 0.441521
\(333\) −1.79654e8 −0.266613
\(334\) −2.22519e8 −0.326779
\(335\) 3.02173e7 0.0439136
\(336\) −1.19252e7 −0.0171506
\(337\) 2.64986e8 0.377154 0.188577 0.982058i \(-0.439613\pi\)
0.188577 + 0.982058i \(0.439613\pi\)
\(338\) 4.18855e8 0.590004
\(339\) −7.86911e8 −1.09705
\(340\) 5.63565e8 0.777620
\(341\) 1.03944e8 0.141958
\(342\) −9.49673e6 −0.0128376
\(343\) 8.05862e8 1.07828
\(344\) 4.35215e8 0.576433
\(345\) −4.12907e8 −0.541359
\(346\) −9.00455e7 −0.116868
\(347\) 5.14421e8 0.660945 0.330473 0.943816i \(-0.392792\pi\)
0.330473 + 0.943816i \(0.392792\pi\)
\(348\) 4.40466e8 0.560254
\(349\) −6.09218e7 −0.0767157 −0.0383578 0.999264i \(-0.512213\pi\)
−0.0383578 + 0.999264i \(0.512213\pi\)
\(350\) −2.28701e8 −0.285121
\(351\) −2.51700e7 −0.0310676
\(352\) −1.81986e8 −0.222402
\(353\) −1.90611e8 −0.230641 −0.115320 0.993328i \(-0.536789\pi\)
−0.115320 + 0.993328i \(0.536789\pi\)
\(354\) 3.80056e7 0.0455341
\(355\) 7.53268e8 0.893614
\(356\) 1.08931e8 0.127960
\(357\) −7.86996e8 −0.915448
\(358\) 1.36875e8 0.157664
\(359\) −1.39955e9 −1.59646 −0.798229 0.602354i \(-0.794230\pi\)
−0.798229 + 0.602354i \(0.794230\pi\)
\(360\) −1.99191e8 −0.225014
\(361\) −8.90259e8 −0.995958
\(362\) −6.43224e8 −0.712659
\(363\) −5.00626e8 −0.549339
\(364\) −8.28171e7 −0.0900047
\(365\) −6.21471e8 −0.668953
\(366\) 3.96654e8 0.422891
\(367\) −1.50266e9 −1.58683 −0.793413 0.608684i \(-0.791698\pi\)
−0.793413 + 0.608684i \(0.791698\pi\)
\(368\) 4.43073e7 0.0463455
\(369\) −2.53946e8 −0.263117
\(370\) −3.22143e8 −0.330631
\(371\) −4.27449e8 −0.434585
\(372\) −2.33866e8 −0.235541
\(373\) 8.89561e7 0.0887554 0.0443777 0.999015i \(-0.485869\pi\)
0.0443777 + 0.999015i \(0.485869\pi\)
\(374\) −2.43027e8 −0.240218
\(375\) 6.17303e8 0.604490
\(376\) 1.60772e9 1.55974
\(377\) 2.57463e8 0.247469
\(378\) 1.07826e8 0.102684
\(379\) −7.27075e8 −0.686028 −0.343014 0.939330i \(-0.611448\pi\)
−0.343014 + 0.939330i \(0.611448\pi\)
\(380\) 2.93736e7 0.0274608
\(381\) −1.00336e9 −0.929436
\(382\) 5.14436e8 0.472182
\(383\) 2.06745e9 1.88036 0.940178 0.340683i \(-0.110658\pi\)
0.940178 + 0.340683i \(0.110658\pi\)
\(384\) 4.22544e8 0.380813
\(385\) 1.48231e8 0.132381
\(386\) 4.89145e8 0.432895
\(387\) 2.21464e8 0.194229
\(388\) −2.26102e8 −0.196514
\(389\) −8.24531e7 −0.0710205 −0.0355102 0.999369i \(-0.511306\pi\)
−0.0355102 + 0.999369i \(0.511306\pi\)
\(390\) −4.51332e7 −0.0385274
\(391\) 2.92402e9 2.47379
\(392\) −2.64578e8 −0.221846
\(393\) 4.05281e8 0.336808
\(394\) 9.64019e8 0.794051
\(395\) 1.00881e9 0.823609
\(396\) −5.74347e7 −0.0464774
\(397\) 2.60792e8 0.209183 0.104592 0.994515i \(-0.466646\pi\)
0.104592 + 0.994515i \(0.466646\pi\)
\(398\) −1.91316e8 −0.152111
\(399\) −4.10190e7 −0.0323281
\(400\) −2.30694e7 −0.0180230
\(401\) −5.57165e8 −0.431498 −0.215749 0.976449i \(-0.569219\pi\)
−0.215749 + 0.976449i \(0.569219\pi\)
\(402\) 2.93180e7 0.0225084
\(403\) −1.36700e8 −0.104040
\(404\) 1.16204e9 0.876769
\(405\) −1.01360e8 −0.0758185
\(406\) −1.10295e9 −0.817925
\(407\) −2.39624e8 −0.176177
\(408\) 1.41058e9 1.02822
\(409\) −1.59001e9 −1.14913 −0.574564 0.818460i \(-0.694828\pi\)
−0.574564 + 0.818460i \(0.694828\pi\)
\(410\) −4.55359e8 −0.326295
\(411\) 4.73325e8 0.336289
\(412\) −1.31887e8 −0.0929100
\(413\) 1.64157e8 0.114666
\(414\) −4.00618e8 −0.277479
\(415\) 6.92983e8 0.475942
\(416\) 2.39336e8 0.162998
\(417\) −3.15552e8 −0.213106
\(418\) −1.26668e7 −0.00848303
\(419\) 5.88021e8 0.390520 0.195260 0.980751i \(-0.437445\pi\)
0.195260 + 0.980751i \(0.437445\pi\)
\(420\) −3.33507e8 −0.219651
\(421\) −7.50924e8 −0.490466 −0.245233 0.969464i \(-0.578864\pi\)
−0.245233 + 0.969464i \(0.578864\pi\)
\(422\) −1.14136e9 −0.739316
\(423\) 8.18104e8 0.525554
\(424\) 7.66142e8 0.488123
\(425\) −1.52245e9 −0.962013
\(426\) 7.30850e8 0.458031
\(427\) 1.71326e9 1.06494
\(428\) 6.37178e8 0.392833
\(429\) −3.35720e7 −0.0205294
\(430\) 3.97114e8 0.240866
\(431\) −2.36435e9 −1.42247 −0.711233 0.702957i \(-0.751863\pi\)
−0.711233 + 0.702957i \(0.751863\pi\)
\(432\) 1.08766e7 0.00649080
\(433\) 6.97398e7 0.0412832 0.0206416 0.999787i \(-0.493429\pi\)
0.0206416 + 0.999787i \(0.493429\pi\)
\(434\) 5.85611e8 0.343871
\(435\) 1.03681e9 0.603931
\(436\) −2.13875e8 −0.123583
\(437\) 1.52403e8 0.0873592
\(438\) −6.02975e8 −0.342878
\(439\) −2.54774e9 −1.43724 −0.718620 0.695403i \(-0.755226\pi\)
−0.718620 + 0.695403i \(0.755226\pi\)
\(440\) −2.65683e8 −0.148689
\(441\) −1.34633e8 −0.0747510
\(442\) 3.19614e8 0.176055
\(443\) 1.15431e9 0.630825 0.315412 0.948955i \(-0.397857\pi\)
0.315412 + 0.948955i \(0.397857\pi\)
\(444\) 5.39135e8 0.292319
\(445\) 2.56411e8 0.137936
\(446\) −6.12929e7 −0.0327143
\(447\) −1.82625e9 −0.967130
\(448\) −9.68759e8 −0.509029
\(449\) 5.55994e8 0.289873 0.144937 0.989441i \(-0.453702\pi\)
0.144937 + 0.989441i \(0.453702\pi\)
\(450\) 2.08589e8 0.107907
\(451\) −3.38715e8 −0.173867
\(452\) 2.36150e9 1.20283
\(453\) 1.16594e9 0.589296
\(454\) 4.71655e8 0.236554
\(455\) −1.94943e8 −0.0970215
\(456\) 7.35209e7 0.0363106
\(457\) 3.93559e8 0.192887 0.0964435 0.995338i \(-0.469253\pi\)
0.0964435 + 0.995338i \(0.469253\pi\)
\(458\) −1.35366e9 −0.658388
\(459\) 7.17789e8 0.346460
\(460\) 1.23912e9 0.593555
\(461\) −2.99382e9 −1.42322 −0.711611 0.702574i \(-0.752034\pi\)
−0.711611 + 0.702574i \(0.752034\pi\)
\(462\) 1.43819e8 0.0678531
\(463\) −1.18369e9 −0.554248 −0.277124 0.960834i \(-0.589381\pi\)
−0.277124 + 0.960834i \(0.589381\pi\)
\(464\) −1.11256e8 −0.0517023
\(465\) −5.50497e8 −0.253904
\(466\) −1.68852e9 −0.772957
\(467\) 3.15650e9 1.43416 0.717079 0.696992i \(-0.245479\pi\)
0.717079 + 0.696992i \(0.245479\pi\)
\(468\) 7.55343e7 0.0340631
\(469\) 1.26633e8 0.0566815
\(470\) 1.46697e9 0.651747
\(471\) 1.24988e9 0.551184
\(472\) −2.94228e8 −0.128792
\(473\) 2.95390e8 0.128346
\(474\) 9.78791e8 0.422149
\(475\) −7.93515e7 −0.0339725
\(476\) 2.36175e9 1.00371
\(477\) 3.89860e8 0.164473
\(478\) 2.59249e9 1.08572
\(479\) 2.27050e9 0.943945 0.471972 0.881613i \(-0.343542\pi\)
0.471972 + 0.881613i \(0.343542\pi\)
\(480\) 9.63815e8 0.397786
\(481\) 3.15137e8 0.129120
\(482\) 1.86410e9 0.758238
\(483\) −1.73038e9 −0.698759
\(484\) 1.50236e9 0.602305
\(485\) −5.32222e8 −0.211834
\(486\) −9.83438e7 −0.0388615
\(487\) 2.04995e8 0.0804252 0.0402126 0.999191i \(-0.487196\pi\)
0.0402126 + 0.999191i \(0.487196\pi\)
\(488\) −3.07078e9 −1.19613
\(489\) 1.95075e9 0.754434
\(490\) −2.41416e8 −0.0926999
\(491\) 1.08333e9 0.413023 0.206512 0.978444i \(-0.433789\pi\)
0.206512 + 0.978444i \(0.433789\pi\)
\(492\) 7.62083e8 0.288486
\(493\) −7.34224e9 −2.75972
\(494\) 1.66586e7 0.00621718
\(495\) −1.35195e8 −0.0501007
\(496\) 5.90715e7 0.0217366
\(497\) 3.15675e9 1.15343
\(498\) 6.72360e8 0.243949
\(499\) 1.23774e9 0.445942 0.222971 0.974825i \(-0.428425\pi\)
0.222971 + 0.974825i \(0.428425\pi\)
\(500\) −1.85251e9 −0.662773
\(501\) 8.76602e8 0.311437
\(502\) −1.15017e9 −0.405787
\(503\) 3.42069e9 1.19847 0.599234 0.800574i \(-0.295472\pi\)
0.599234 + 0.800574i \(0.295472\pi\)
\(504\) −8.34756e8 −0.290437
\(505\) 2.73532e9 0.945122
\(506\) −5.34349e8 −0.183357
\(507\) −1.65006e9 −0.562304
\(508\) 3.01105e9 1.01905
\(509\) −3.62806e9 −1.21944 −0.609722 0.792616i \(-0.708719\pi\)
−0.609722 + 0.792616i \(0.708719\pi\)
\(510\) 1.28709e9 0.429650
\(511\) −2.60442e9 −0.863451
\(512\) −2.04753e8 −0.0674195
\(513\) 3.74119e7 0.0122349
\(514\) 2.58640e9 0.840088
\(515\) −3.10449e8 −0.100153
\(516\) −6.64605e8 −0.212956
\(517\) 1.09120e9 0.347285
\(518\) −1.35002e9 −0.426762
\(519\) 3.54730e8 0.111381
\(520\) 3.49408e8 0.108974
\(521\) −1.36162e9 −0.421817 −0.210909 0.977506i \(-0.567642\pi\)
−0.210909 + 0.977506i \(0.567642\pi\)
\(522\) 1.00596e9 0.309551
\(523\) 1.07415e9 0.328329 0.164165 0.986433i \(-0.447507\pi\)
0.164165 + 0.986433i \(0.447507\pi\)
\(524\) −1.21624e9 −0.369282
\(525\) 9.00956e8 0.271735
\(526\) 1.62705e9 0.487472
\(527\) 3.89838e9 1.16024
\(528\) 1.45073e7 0.00428911
\(529\) 3.02428e9 0.888234
\(530\) 6.99072e8 0.203965
\(531\) −1.49721e8 −0.0433963
\(532\) 1.23097e8 0.0354451
\(533\) 4.45456e8 0.127426
\(534\) 2.48780e8 0.0707004
\(535\) 1.49985e9 0.423458
\(536\) −2.26972e8 −0.0636642
\(537\) −5.39211e8 −0.150262
\(538\) −1.76039e9 −0.487385
\(539\) −1.79575e8 −0.0493953
\(540\) 3.04179e8 0.0831288
\(541\) −6.13348e9 −1.66539 −0.832696 0.553730i \(-0.813204\pi\)
−0.832696 + 0.553730i \(0.813204\pi\)
\(542\) 2.05226e9 0.553649
\(543\) 2.53395e9 0.679201
\(544\) −6.82531e9 −1.81772
\(545\) −5.03440e8 −0.133217
\(546\) −1.89141e8 −0.0497293
\(547\) −6.39180e9 −1.66981 −0.834906 0.550392i \(-0.814478\pi\)
−0.834906 + 0.550392i \(0.814478\pi\)
\(548\) −1.42043e9 −0.368713
\(549\) −1.56260e9 −0.403036
\(550\) 2.78219e8 0.0713045
\(551\) −3.82685e8 −0.0974565
\(552\) 3.10147e9 0.784840
\(553\) 4.22767e9 1.06307
\(554\) 7.35697e8 0.183829
\(555\) 1.26907e9 0.315108
\(556\) 9.46963e8 0.233653
\(557\) 5.20568e9 1.27639 0.638197 0.769873i \(-0.279681\pi\)
0.638197 + 0.769873i \(0.279681\pi\)
\(558\) −5.34114e8 −0.130141
\(559\) −3.88478e8 −0.0940643
\(560\) 8.42396e7 0.0202702
\(561\) 9.57395e8 0.228940
\(562\) −8.70420e8 −0.206848
\(563\) 3.32429e9 0.785089 0.392545 0.919733i \(-0.371595\pi\)
0.392545 + 0.919733i \(0.371595\pi\)
\(564\) −2.45510e9 −0.576226
\(565\) 5.55872e9 1.29660
\(566\) −4.65657e9 −1.07947
\(567\) −4.24774e8 −0.0978628
\(568\) −5.65803e9 −1.29553
\(569\) 1.61858e9 0.368333 0.184166 0.982895i \(-0.441041\pi\)
0.184166 + 0.982895i \(0.441041\pi\)
\(570\) 6.70846e7 0.0151726
\(571\) −6.26853e8 −0.140909 −0.0704546 0.997515i \(-0.522445\pi\)
−0.0704546 + 0.997515i \(0.522445\pi\)
\(572\) 1.00749e8 0.0225088
\(573\) −2.02660e9 −0.450014
\(574\) −1.90829e9 −0.421165
\(575\) −3.34743e9 −0.734302
\(576\) 8.83568e8 0.192647
\(577\) 2.33845e9 0.506772 0.253386 0.967365i \(-0.418456\pi\)
0.253386 + 0.967365i \(0.418456\pi\)
\(578\) −6.30228e9 −1.35753
\(579\) −1.92696e9 −0.412571
\(580\) −3.11144e9 −0.662161
\(581\) 2.90411e9 0.614323
\(582\) −5.16382e8 −0.108578
\(583\) 5.19999e8 0.108683
\(584\) 4.66806e9 0.969820
\(585\) 1.77800e8 0.0367186
\(586\) −2.42791e9 −0.498416
\(587\) −5.68734e9 −1.16058 −0.580291 0.814409i \(-0.697061\pi\)
−0.580291 + 0.814409i \(0.697061\pi\)
\(588\) 4.04030e8 0.0819583
\(589\) 2.03187e8 0.0409725
\(590\) −2.68471e8 −0.0538164
\(591\) −3.79770e9 −0.756772
\(592\) −1.36178e8 −0.0269763
\(593\) −6.92748e9 −1.36422 −0.682109 0.731250i \(-0.738937\pi\)
−0.682109 + 0.731250i \(0.738937\pi\)
\(594\) −1.31172e8 −0.0256796
\(595\) 5.55932e9 1.08196
\(596\) 5.48053e9 1.06038
\(597\) 7.53679e8 0.144969
\(598\) 7.02741e8 0.134382
\(599\) 1.04577e7 0.00198812 0.000994062 1.00000i \(-0.499684\pi\)
0.000994062 1.00000i \(0.499684\pi\)
\(600\) −1.61484e9 −0.305211
\(601\) 5.50968e9 1.03530 0.517649 0.855593i \(-0.326807\pi\)
0.517649 + 0.855593i \(0.326807\pi\)
\(602\) 1.66420e9 0.310898
\(603\) −1.15497e8 −0.0214516
\(604\) −3.49896e9 −0.646114
\(605\) 3.53641e9 0.649260
\(606\) 2.65391e9 0.484432
\(607\) −1.08515e10 −1.96939 −0.984693 0.174298i \(-0.944234\pi\)
−0.984693 + 0.174298i \(0.944234\pi\)
\(608\) −3.55742e8 −0.0641908
\(609\) 4.34500e9 0.779524
\(610\) −2.80195e9 −0.499812
\(611\) −1.43507e9 −0.254524
\(612\) −2.15406e9 −0.379864
\(613\) −5.00263e9 −0.877177 −0.438588 0.898688i \(-0.644521\pi\)
−0.438588 + 0.898688i \(0.644521\pi\)
\(614\) 3.06485e9 0.534342
\(615\) 1.79386e9 0.310976
\(616\) −1.11341e9 −0.191920
\(617\) 1.47593e9 0.252970 0.126485 0.991969i \(-0.459630\pi\)
0.126485 + 0.991969i \(0.459630\pi\)
\(618\) −3.01210e8 −0.0513345
\(619\) −9.36559e9 −1.58715 −0.793575 0.608472i \(-0.791783\pi\)
−0.793575 + 0.608472i \(0.791783\pi\)
\(620\) 1.65202e9 0.278385
\(621\) 1.57822e9 0.264452
\(622\) 7.07150e9 1.17827
\(623\) 1.07455e9 0.178041
\(624\) −1.90790e7 −0.00314347
\(625\) −1.09904e9 −0.180067
\(626\) 1.01207e9 0.164893
\(627\) 4.99004e7 0.00808477
\(628\) −3.75087e9 −0.604328
\(629\) −8.98698e9 −1.43991
\(630\) −7.61679e8 −0.121361
\(631\) −1.19966e10 −1.90088 −0.950438 0.310914i \(-0.899365\pi\)
−0.950438 + 0.310914i \(0.899365\pi\)
\(632\) −7.57751e9 −1.19403
\(633\) 4.49634e9 0.704606
\(634\) 5.53871e9 0.863170
\(635\) 7.08771e9 1.09849
\(636\) −1.16996e9 −0.180331
\(637\) 2.36165e8 0.0362016
\(638\) 1.34175e9 0.204551
\(639\) −2.87915e9 −0.436527
\(640\) −2.98484e9 −0.450081
\(641\) −1.25626e10 −1.88398 −0.941989 0.335644i \(-0.891046\pi\)
−0.941989 + 0.335644i \(0.891046\pi\)
\(642\) 1.45522e9 0.217047
\(643\) −4.11211e9 −0.609995 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(644\) 5.19283e9 0.766131
\(645\) −1.56441e9 −0.229558
\(646\) −4.75064e8 −0.0693327
\(647\) −8.36038e9 −1.21356 −0.606780 0.794870i \(-0.707539\pi\)
−0.606780 + 0.794870i \(0.707539\pi\)
\(648\) 7.61349e8 0.109919
\(649\) −1.99700e8 −0.0286762
\(650\) −3.65895e8 −0.0522588
\(651\) −2.30699e9 −0.327727
\(652\) −5.85415e9 −0.827175
\(653\) 5.92578e8 0.0832817 0.0416409 0.999133i \(-0.486741\pi\)
0.0416409 + 0.999133i \(0.486741\pi\)
\(654\) −4.88458e8 −0.0682818
\(655\) −2.86290e9 −0.398072
\(656\) −1.92492e8 −0.0266225
\(657\) 2.37539e9 0.326781
\(658\) 6.14769e9 0.841243
\(659\) 3.61916e9 0.492617 0.246308 0.969192i \(-0.420782\pi\)
0.246308 + 0.969192i \(0.420782\pi\)
\(660\) 4.05717e8 0.0549313
\(661\) 1.49686e9 0.201593 0.100797 0.994907i \(-0.467861\pi\)
0.100797 + 0.994907i \(0.467861\pi\)
\(662\) 1.17839e9 0.157866
\(663\) −1.25910e9 −0.167789
\(664\) −5.20521e9 −0.690002
\(665\) 2.89757e8 0.0382084
\(666\) 1.23130e9 0.161512
\(667\) −1.61435e10 −2.10648
\(668\) −2.63066e9 −0.341465
\(669\) 2.41460e8 0.0311784
\(670\) −2.07102e8 −0.0266025
\(671\) −2.08421e9 −0.266326
\(672\) 4.03909e9 0.513442
\(673\) 1.55009e10 1.96022 0.980109 0.198461i \(-0.0635943\pi\)
0.980109 + 0.198461i \(0.0635943\pi\)
\(674\) −1.81615e9 −0.228476
\(675\) −8.21728e8 −0.102841
\(676\) 4.95177e9 0.616520
\(677\) −7.97429e9 −0.987715 −0.493857 0.869543i \(-0.664414\pi\)
−0.493857 + 0.869543i \(0.664414\pi\)
\(678\) 5.39329e9 0.664584
\(679\) −2.23040e9 −0.273425
\(680\) −9.96431e9 −1.21525
\(681\) −1.85806e9 −0.225448
\(682\) −7.12407e8 −0.0859969
\(683\) 7.37364e9 0.885542 0.442771 0.896635i \(-0.353995\pi\)
0.442771 + 0.896635i \(0.353995\pi\)
\(684\) −1.12272e8 −0.0134145
\(685\) −3.34356e9 −0.397458
\(686\) −5.52317e9 −0.653212
\(687\) 5.33269e9 0.627477
\(688\) 1.67871e8 0.0196524
\(689\) −6.83868e8 −0.0796535
\(690\) 2.82996e9 0.327951
\(691\) 5.70889e9 0.658232 0.329116 0.944290i \(-0.393249\pi\)
0.329116 + 0.944290i \(0.393249\pi\)
\(692\) −1.06453e9 −0.122120
\(693\) −5.66569e8 −0.0646675
\(694\) −3.52571e9 −0.400395
\(695\) 2.22905e9 0.251868
\(696\) −7.78781e9 −0.875555
\(697\) −1.27034e10 −1.42103
\(698\) 4.17543e8 0.0464737
\(699\) 6.65184e9 0.736668
\(700\) −2.70374e9 −0.297935
\(701\) 1.23928e10 1.35880 0.679402 0.733767i \(-0.262239\pi\)
0.679402 + 0.733767i \(0.262239\pi\)
\(702\) 1.72509e8 0.0188205
\(703\) −4.68410e8 −0.0508491
\(704\) 1.17851e9 0.127300
\(705\) −5.77906e9 −0.621149
\(706\) 1.30640e9 0.139720
\(707\) 1.14630e10 1.21992
\(708\) 4.49309e8 0.0475805
\(709\) 1.39979e10 1.47504 0.737518 0.675328i \(-0.235998\pi\)
0.737518 + 0.675328i \(0.235998\pi\)
\(710\) −5.16271e9 −0.541344
\(711\) −3.85590e9 −0.402330
\(712\) −1.92599e9 −0.199974
\(713\) 8.57143e9 0.885605
\(714\) 5.39387e9 0.554571
\(715\) 2.37152e8 0.0242636
\(716\) 1.61815e9 0.164750
\(717\) −1.02130e10 −1.03475
\(718\) 9.59215e9 0.967120
\(719\) 7.66012e9 0.768572 0.384286 0.923214i \(-0.374448\pi\)
0.384286 + 0.923214i \(0.374448\pi\)
\(720\) −7.68317e7 −0.00767144
\(721\) −1.30101e9 −0.129273
\(722\) 6.10161e9 0.603343
\(723\) −7.34354e9 −0.722639
\(724\) −7.60430e9 −0.744688
\(725\) 8.40543e9 0.819175
\(726\) 3.43116e9 0.332785
\(727\) 1.02618e10 0.990493 0.495247 0.868752i \(-0.335078\pi\)
0.495247 + 0.868752i \(0.335078\pi\)
\(728\) 1.46428e9 0.140658
\(729\) 3.87420e8 0.0370370
\(730\) 4.25940e9 0.405246
\(731\) 1.10785e10 1.04899
\(732\) 4.68931e9 0.441896
\(733\) −1.67714e10 −1.57292 −0.786459 0.617643i \(-0.788088\pi\)
−0.786459 + 0.617643i \(0.788088\pi\)
\(734\) 1.02988e10 0.961285
\(735\) 9.51045e8 0.0883477
\(736\) −1.50069e10 −1.38746
\(737\) −1.54051e8 −0.0141752
\(738\) 1.74048e9 0.159394
\(739\) 5.49527e9 0.500880 0.250440 0.968132i \(-0.419425\pi\)
0.250440 + 0.968132i \(0.419425\pi\)
\(740\) −3.80843e9 −0.345490
\(741\) −6.56257e7 −0.00592530
\(742\) 2.92962e9 0.263268
\(743\) −1.77986e9 −0.159193 −0.0795966 0.996827i \(-0.525363\pi\)
−0.0795966 + 0.996827i \(0.525363\pi\)
\(744\) 4.13495e9 0.368100
\(745\) 1.29006e10 1.14305
\(746\) −6.09683e8 −0.0537673
\(747\) −2.64873e9 −0.232496
\(748\) −2.87311e9 −0.251013
\(749\) 6.28548e9 0.546578
\(750\) −4.23084e9 −0.366195
\(751\) −7.12062e9 −0.613448 −0.306724 0.951798i \(-0.599233\pi\)
−0.306724 + 0.951798i \(0.599233\pi\)
\(752\) 6.20127e8 0.0531763
\(753\) 4.53103e9 0.386736
\(754\) −1.76459e9 −0.149914
\(755\) −8.23619e9 −0.696485
\(756\) 1.27473e9 0.107298
\(757\) 6.43143e9 0.538855 0.269428 0.963021i \(-0.413166\pi\)
0.269428 + 0.963021i \(0.413166\pi\)
\(758\) 4.98319e9 0.415590
\(759\) 2.10504e9 0.174749
\(760\) −5.19350e8 −0.0429153
\(761\) 1.46912e10 1.20840 0.604202 0.796831i \(-0.293492\pi\)
0.604202 + 0.796831i \(0.293492\pi\)
\(762\) 6.87678e9 0.563044
\(763\) −2.10979e9 −0.171950
\(764\) 6.08175e9 0.493403
\(765\) −5.07044e9 −0.409478
\(766\) −1.41698e10 −1.13910
\(767\) 2.62632e8 0.0210167
\(768\) −7.08477e9 −0.564367
\(769\) 1.24049e10 0.983677 0.491838 0.870687i \(-0.336325\pi\)
0.491838 + 0.870687i \(0.336325\pi\)
\(770\) −1.01594e9 −0.0801952
\(771\) −1.01890e10 −0.800647
\(772\) 5.78276e9 0.452350
\(773\) −2.65186e9 −0.206501 −0.103250 0.994655i \(-0.532924\pi\)
−0.103250 + 0.994655i \(0.532924\pi\)
\(774\) −1.51786e9 −0.117662
\(775\) −4.46287e9 −0.344396
\(776\) 3.99768e9 0.307109
\(777\) 5.31833e9 0.406726
\(778\) 5.65113e8 0.0430236
\(779\) −6.62112e8 −0.0501823
\(780\) −5.33573e8 −0.0402590
\(781\) −3.84024e9 −0.288456
\(782\) −2.00405e10 −1.49860
\(783\) −3.96291e9 −0.295018
\(784\) −1.02053e8 −0.00756342
\(785\) −8.82916e9 −0.651441
\(786\) −2.77770e9 −0.204036
\(787\) −4.05957e9 −0.296871 −0.148436 0.988922i \(-0.547424\pi\)
−0.148436 + 0.988922i \(0.547424\pi\)
\(788\) 1.13968e10 0.829738
\(789\) −6.40967e9 −0.464586
\(790\) −6.91415e9 −0.498935
\(791\) 2.32951e10 1.67358
\(792\) 1.01550e9 0.0726340
\(793\) 2.74102e9 0.195189
\(794\) −1.78740e9 −0.126721
\(795\) −2.75396e9 −0.194389
\(796\) −2.26177e9 −0.158947
\(797\) 2.36833e10 1.65706 0.828530 0.559944i \(-0.189177\pi\)
0.828530 + 0.559944i \(0.189177\pi\)
\(798\) 2.81134e8 0.0195841
\(799\) 4.09248e10 2.83839
\(800\) 7.81364e9 0.539558
\(801\) −9.80058e8 −0.0673811
\(802\) 3.81867e9 0.261398
\(803\) 3.16832e9 0.215936
\(804\) 3.46603e8 0.0235199
\(805\) 1.22234e10 0.825859
\(806\) 9.36910e8 0.0630268
\(807\) 6.93499e9 0.464503
\(808\) −2.05458e10 −1.37020
\(809\) −9.66394e9 −0.641704 −0.320852 0.947129i \(-0.603969\pi\)
−0.320852 + 0.947129i \(0.603969\pi\)
\(810\) 6.94698e8 0.0459302
\(811\) 1.27359e10 0.838413 0.419207 0.907891i \(-0.362308\pi\)
0.419207 + 0.907891i \(0.362308\pi\)
\(812\) −1.30392e10 −0.854684
\(813\) −8.08477e9 −0.527656
\(814\) 1.64232e9 0.106727
\(815\) −1.37801e10 −0.891661
\(816\) 5.44088e8 0.0350553
\(817\) 5.77422e8 0.0370438
\(818\) 1.08975e10 0.696132
\(819\) 7.45113e8 0.0473946
\(820\) −5.38333e9 −0.340959
\(821\) 1.18429e10 0.746893 0.373446 0.927652i \(-0.378176\pi\)
0.373446 + 0.927652i \(0.378176\pi\)
\(822\) −3.24405e9 −0.203721
\(823\) 1.12076e10 0.700829 0.350415 0.936595i \(-0.386041\pi\)
0.350415 + 0.936595i \(0.386041\pi\)
\(824\) 2.33188e9 0.145198
\(825\) −1.09603e9 −0.0679569
\(826\) −1.12509e9 −0.0694636
\(827\) −1.69838e10 −1.04416 −0.522078 0.852898i \(-0.674843\pi\)
−0.522078 + 0.852898i \(0.674843\pi\)
\(828\) −4.73618e9 −0.289949
\(829\) −4.59293e9 −0.279994 −0.139997 0.990152i \(-0.544709\pi\)
−0.139997 + 0.990152i \(0.544709\pi\)
\(830\) −4.74953e9 −0.288322
\(831\) −2.89824e9 −0.175199
\(832\) −1.54990e9 −0.0932980
\(833\) −6.73488e9 −0.403713
\(834\) 2.16272e9 0.129098
\(835\) −6.19230e9 −0.368086
\(836\) −1.49749e8 −0.00886428
\(837\) 2.10411e9 0.124031
\(838\) −4.03015e9 −0.236574
\(839\) −2.30945e10 −1.35002 −0.675012 0.737806i \(-0.735862\pi\)
−0.675012 + 0.737806i \(0.735862\pi\)
\(840\) 5.89669e9 0.343266
\(841\) 2.32866e10 1.34996
\(842\) 5.14664e9 0.297120
\(843\) 3.42898e9 0.197137
\(844\) −1.34934e10 −0.772542
\(845\) 1.16560e10 0.664584
\(846\) −5.60708e9 −0.318376
\(847\) 1.48202e10 0.838033
\(848\) 2.95516e8 0.0166416
\(849\) 1.83443e10 1.02879
\(850\) 1.04345e10 0.582779
\(851\) −1.97598e10 −1.09908
\(852\) 8.64023e9 0.478616
\(853\) 2.22496e10 1.22744 0.613721 0.789523i \(-0.289672\pi\)
0.613721 + 0.789523i \(0.289672\pi\)
\(854\) −1.17422e10 −0.645132
\(855\) −2.64277e8 −0.0144603
\(856\) −1.12659e10 −0.613912
\(857\) 1.07105e10 0.581270 0.290635 0.956834i \(-0.406133\pi\)
0.290635 + 0.956834i \(0.406133\pi\)
\(858\) 2.30094e8 0.0124365
\(859\) 3.19391e10 1.71928 0.859639 0.510902i \(-0.170688\pi\)
0.859639 + 0.510902i \(0.170688\pi\)
\(860\) 4.69475e9 0.251691
\(861\) 7.51761e9 0.401392
\(862\) 1.62047e10 0.861717
\(863\) 2.90663e10 1.53940 0.769700 0.638406i \(-0.220406\pi\)
0.769700 + 0.638406i \(0.220406\pi\)
\(864\) −3.68390e9 −0.194317
\(865\) −2.50580e9 −0.131641
\(866\) −4.77979e8 −0.0250090
\(867\) 2.48275e10 1.29380
\(868\) 6.92319e9 0.359325
\(869\) −5.14304e9 −0.265858
\(870\) −7.10604e9 −0.365856
\(871\) 2.02598e8 0.0103889
\(872\) 3.78150e9 0.193133
\(873\) 2.03426e9 0.103480
\(874\) −1.04453e9 −0.0529215
\(875\) −1.82742e10 −0.922167
\(876\) −7.12848e9 −0.358288
\(877\) −1.87443e9 −0.0938364 −0.0469182 0.998899i \(-0.514940\pi\)
−0.0469182 + 0.998899i \(0.514940\pi\)
\(878\) 1.74616e10 0.870668
\(879\) 9.56465e9 0.475016
\(880\) −1.02479e8 −0.00506927
\(881\) −3.84330e10 −1.89360 −0.946802 0.321818i \(-0.895706\pi\)
−0.946802 + 0.321818i \(0.895706\pi\)
\(882\) 9.22741e8 0.0452835
\(883\) 1.17045e10 0.572125 0.286063 0.958211i \(-0.407653\pi\)
0.286063 + 0.958211i \(0.407653\pi\)
\(884\) 3.77853e9 0.183967
\(885\) 1.05763e9 0.0512898
\(886\) −7.91134e9 −0.382148
\(887\) 3.09354e9 0.148841 0.0744205 0.997227i \(-0.476289\pi\)
0.0744205 + 0.997227i \(0.476289\pi\)
\(888\) −9.53236e9 −0.456831
\(889\) 2.97027e10 1.41788
\(890\) −1.75738e9 −0.0835604
\(891\) 5.16746e8 0.0244740
\(892\) −7.24615e8 −0.0341846
\(893\) 2.13304e9 0.100235
\(894\) 1.25167e10 0.585879
\(895\) 3.80897e9 0.177594
\(896\) −1.25087e10 −0.580942
\(897\) −2.76841e9 −0.128073
\(898\) −3.81064e9 −0.175603
\(899\) −2.15229e10 −0.987967
\(900\) 2.46598e9 0.112756
\(901\) 1.95023e10 0.888279
\(902\) 2.32147e9 0.105327
\(903\) −6.55604e9 −0.296302
\(904\) −4.17533e10 −1.87975
\(905\) −1.78997e10 −0.802744
\(906\) −7.99107e9 −0.356990
\(907\) −5.95597e9 −0.265050 −0.132525 0.991180i \(-0.542308\pi\)
−0.132525 + 0.991180i \(0.542308\pi\)
\(908\) 5.57599e9 0.247185
\(909\) −1.04550e10 −0.461688
\(910\) 1.33609e9 0.0587748
\(911\) −4.08258e10 −1.78904 −0.894521 0.447026i \(-0.852483\pi\)
−0.894521 + 0.447026i \(0.852483\pi\)
\(912\) 2.83584e7 0.00123794
\(913\) −3.53290e9 −0.153633
\(914\) −2.69735e9 −0.116849
\(915\) 1.10382e10 0.476346
\(916\) −1.60032e10 −0.687977
\(917\) −1.19976e10 −0.513811
\(918\) −4.91954e9 −0.209882
\(919\) −1.12519e10 −0.478214 −0.239107 0.970993i \(-0.576855\pi\)
−0.239107 + 0.970993i \(0.576855\pi\)
\(920\) −2.19087e10 −0.927598
\(921\) −1.20738e10 −0.509255
\(922\) 2.05189e10 0.862176
\(923\) 5.05043e9 0.211408
\(924\) 1.70025e9 0.0709026
\(925\) 1.02883e10 0.427414
\(926\) 8.11271e9 0.335759
\(927\) 1.18660e9 0.0489244
\(928\) 3.76825e10 1.54783
\(929\) 1.44429e10 0.591018 0.295509 0.955340i \(-0.404511\pi\)
0.295509 + 0.955340i \(0.404511\pi\)
\(930\) 3.77297e9 0.153813
\(931\) −3.51029e8 −0.0142567
\(932\) −1.99620e10 −0.807696
\(933\) −2.78578e10 −1.12295
\(934\) −2.16339e10 −0.868801
\(935\) −6.76301e9 −0.270582
\(936\) −1.33551e9 −0.0532332
\(937\) 2.43718e10 0.967829 0.483915 0.875115i \(-0.339215\pi\)
0.483915 + 0.875115i \(0.339215\pi\)
\(938\) −8.67909e8 −0.0343372
\(939\) −3.98702e9 −0.157151
\(940\) 1.73428e10 0.681038
\(941\) −1.04613e10 −0.409280 −0.204640 0.978837i \(-0.565602\pi\)
−0.204640 + 0.978837i \(0.565602\pi\)
\(942\) −8.56639e9 −0.333903
\(943\) −2.79311e10 −1.08467
\(944\) −1.13490e8 −0.00439090
\(945\) 3.00060e9 0.115663
\(946\) −2.02453e9 −0.0777509
\(947\) −6.80097e9 −0.260223 −0.130112 0.991499i \(-0.541534\pi\)
−0.130112 + 0.991499i \(0.541534\pi\)
\(948\) 1.15714e10 0.441121
\(949\) −4.16677e9 −0.158259
\(950\) 5.43855e8 0.0205802
\(951\) −2.18195e10 −0.822645
\(952\) −4.17578e10 −1.56859
\(953\) −9.17306e8 −0.0343312 −0.0171656 0.999853i \(-0.505464\pi\)
−0.0171656 + 0.999853i \(0.505464\pi\)
\(954\) −2.67200e9 −0.0996362
\(955\) 1.43158e10 0.531869
\(956\) 3.06488e10 1.13452
\(957\) −5.28577e9 −0.194947
\(958\) −1.55614e10 −0.571833
\(959\) −1.40120e10 −0.513019
\(960\) −6.24150e9 −0.227688
\(961\) −1.60850e10 −0.584640
\(962\) −2.15987e9 −0.0782195
\(963\) −5.73275e9 −0.206857
\(964\) 2.20377e10 0.792314
\(965\) 1.36120e10 0.487615
\(966\) 1.18596e10 0.423302
\(967\) −3.69970e10 −1.31575 −0.657875 0.753127i \(-0.728544\pi\)
−0.657875 + 0.753127i \(0.728544\pi\)
\(968\) −2.65631e10 −0.941271
\(969\) 1.87149e9 0.0660776
\(970\) 3.64771e9 0.128327
\(971\) −1.31210e10 −0.459937 −0.229968 0.973198i \(-0.573862\pi\)
−0.229968 + 0.973198i \(0.573862\pi\)
\(972\) −1.16264e9 −0.0406080
\(973\) 9.34138e9 0.325099
\(974\) −1.40498e9 −0.0487209
\(975\) 1.44142e9 0.0498053
\(976\) −1.18446e9 −0.0407798
\(977\) −3.90962e10 −1.34123 −0.670615 0.741806i \(-0.733970\pi\)
−0.670615 + 0.741806i \(0.733970\pi\)
\(978\) −1.33700e10 −0.457030
\(979\) −1.30721e9 −0.0445253
\(980\) −2.85406e9 −0.0968660
\(981\) 1.92426e9 0.0650760
\(982\) −7.42485e9 −0.250206
\(983\) −3.75059e10 −1.25940 −0.629699 0.776840i \(-0.716822\pi\)
−0.629699 + 0.776840i \(0.716822\pi\)
\(984\) −1.34743e10 −0.450840
\(985\) 2.68269e10 0.894424
\(986\) 5.03219e10 1.67181
\(987\) −2.42185e10 −0.801748
\(988\) 1.96941e8 0.00649660
\(989\) 2.43585e10 0.800688
\(990\) 9.26596e8 0.0303506
\(991\) 5.39247e10 1.76007 0.880036 0.474907i \(-0.157518\pi\)
0.880036 + 0.474907i \(0.157518\pi\)
\(992\) −2.00076e10 −0.650735
\(993\) −4.64223e9 −0.150454
\(994\) −2.16355e10 −0.698739
\(995\) −5.32397e9 −0.171338
\(996\) 7.94875e9 0.254912
\(997\) 2.01581e10 0.644192 0.322096 0.946707i \(-0.395613\pi\)
0.322096 + 0.946707i \(0.395613\pi\)
\(998\) −8.48317e9 −0.270148
\(999\) −4.85064e9 −0.153929
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.b.1.7 17
3.2 odd 2 531.8.a.d.1.11 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.7 17 1.1 even 1 trivial
531.8.a.d.1.11 17 3.2 odd 2