Properties

Label 177.8.a.d.1.17
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
Dimension $18$
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: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + 3375594921 x^{11} + 115310342333 x^{10} - 774932111214 x^{9} - 14600047830166 x^{8} + 102185027148481 x^{7} + 988557475638619 x^{6} - 7379206238519716 x^{5} - 30152342836849520 x^{4} + 260578770749067175 x^{3} + 182609347488069978 x^{2} - 3481290425710753600 x + 5164646074739714048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.17
Root \(19.9328\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+20.9328 q^{2} +27.0000 q^{3} +310.183 q^{4} -149.882 q^{5} +565.186 q^{6} +1201.09 q^{7} +3813.60 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+20.9328 q^{2} +27.0000 q^{3} +310.183 q^{4} -149.882 q^{5} +565.186 q^{6} +1201.09 q^{7} +3813.60 q^{8} +729.000 q^{9} -3137.45 q^{10} +3090.18 q^{11} +8374.94 q^{12} -9762.44 q^{13} +25142.2 q^{14} -4046.82 q^{15} +40126.1 q^{16} +774.830 q^{17} +15260.0 q^{18} +42295.3 q^{19} -46490.9 q^{20} +32429.5 q^{21} +64686.2 q^{22} +19299.9 q^{23} +102967. q^{24} -55660.4 q^{25} -204355. q^{26} +19683.0 q^{27} +372558. q^{28} -148022. q^{29} -84711.3 q^{30} -77522.7 q^{31} +351811. q^{32} +83434.9 q^{33} +16219.4 q^{34} -180022. q^{35} +226123. q^{36} +381041. q^{37} +885360. q^{38} -263586. q^{39} -571591. q^{40} +313892. q^{41} +678840. q^{42} +86822.9 q^{43} +958522. q^{44} -109264. q^{45} +404002. q^{46} +1.09797e6 q^{47} +1.08340e6 q^{48} +619078. q^{49} -1.16513e6 q^{50} +20920.4 q^{51} -3.02814e6 q^{52} -1.35146e6 q^{53} +412021. q^{54} -463163. q^{55} +4.58049e6 q^{56} +1.14197e6 q^{57} -3.09852e6 q^{58} +205379. q^{59} -1.25525e6 q^{60} -1.17823e6 q^{61} -1.62277e6 q^{62} +875596. q^{63} +2.22825e6 q^{64} +1.46321e6 q^{65} +1.74653e6 q^{66} -2.20281e6 q^{67} +240339. q^{68} +521098. q^{69} -3.76837e6 q^{70} +64342.9 q^{71} +2.78012e6 q^{72} -1.88910e6 q^{73} +7.97626e6 q^{74} -1.50283e6 q^{75} +1.31193e7 q^{76} +3.71159e6 q^{77} -5.51760e6 q^{78} -267273. q^{79} -6.01418e6 q^{80} +531441. q^{81} +6.57065e6 q^{82} -9.61801e6 q^{83} +1.00591e7 q^{84} -116133. q^{85} +1.81745e6 q^{86} -3.99660e6 q^{87} +1.17847e7 q^{88} +1.26033e7 q^{89} -2.28720e6 q^{90} -1.17256e7 q^{91} +5.98651e6 q^{92} -2.09311e6 q^{93} +2.29835e7 q^{94} -6.33931e6 q^{95} +9.49889e6 q^{96} -7.21193e6 q^{97} +1.29590e7 q^{98} +2.25274e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + O(q^{10}) \) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + 3609q^{10} + 15070q^{11} + 36666q^{12} + 13662q^{13} + 20861q^{14} + 18306q^{15} + 60482q^{16} + 71919q^{17} + 17496q^{18} + 56231q^{19} + 143053q^{20} + 83187q^{21} + 274198q^{22} + 150029q^{23} + 110889q^{24} + 399672q^{25} + 182846q^{26} + 354294q^{27} + 434150q^{28} + 591285q^{29} + 97443q^{30} + 426733q^{31} + 1205630q^{32} + 406890q^{33} + 403548q^{34} + 912879q^{35} + 989982q^{36} + 7703q^{37} - 417859q^{38} + 368874q^{39} + 618020q^{40} + 770959q^{41} + 563247q^{42} + 793050q^{43} + 2591274q^{44} + 494262q^{45} - 4068019q^{46} + 1410373q^{47} + 1633014q^{48} + 1637427q^{49} + 1021549q^{50} + 1941813q^{51} - 3749190q^{52} + 1037934q^{53} + 472392q^{54} + 331974q^{55} - 391748q^{56} + 1518237q^{57} + 653724q^{58} + 3696822q^{59} + 3862431q^{60} - 1374623q^{61} + 5251718q^{62} + 2246049q^{63} + 5077197q^{64} + 3257170q^{65} + 7403346q^{66} - 2436904q^{67} + 14119909q^{68} + 4050783q^{69} + 5185580q^{70} + 14289172q^{71} + 2994003q^{72} + 5482515q^{73} + 14934154q^{74} + 10791144q^{75} + 3822912q^{76} + 23157109q^{77} + 4936842q^{78} + 19786414q^{79} + 31978143q^{80} + 9565938q^{81} + 9749509q^{82} + 30227337q^{83} + 11722050q^{84} + 9946981q^{85} + 44295864q^{86} + 15964695q^{87} + 39970897q^{88} + 31061677q^{89} + 2630961q^{90} + 26377785q^{91} + 4719698q^{92} + 11521791q^{93} + 44488296q^{94} + 15534599q^{95} + 32552010q^{96} + 12084118q^{97} + 42274744q^{98} + 10986030q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 20.9328 1.85022 0.925109 0.379702i \(-0.123974\pi\)
0.925109 + 0.379702i \(0.123974\pi\)
\(3\) 27.0000 0.577350
\(4\) 310.183 2.42330
\(5\) −149.882 −0.536234 −0.268117 0.963386i \(-0.586401\pi\)
−0.268117 + 0.963386i \(0.586401\pi\)
\(6\) 565.186 1.06822
\(7\) 1201.09 1.32353 0.661764 0.749712i \(-0.269808\pi\)
0.661764 + 0.749712i \(0.269808\pi\)
\(8\) 3813.60 2.63342
\(9\) 729.000 0.333333
\(10\) −3137.45 −0.992150
\(11\) 3090.18 0.700019 0.350009 0.936746i \(-0.386178\pi\)
0.350009 + 0.936746i \(0.386178\pi\)
\(12\) 8374.94 1.39910
\(13\) −9762.44 −1.23241 −0.616207 0.787584i \(-0.711332\pi\)
−0.616207 + 0.787584i \(0.711332\pi\)
\(14\) 25142.2 2.44881
\(15\) −4046.82 −0.309595
\(16\) 40126.1 2.44910
\(17\) 774.830 0.0382503 0.0191252 0.999817i \(-0.493912\pi\)
0.0191252 + 0.999817i \(0.493912\pi\)
\(18\) 15260.0 0.616739
\(19\) 42295.3 1.41467 0.707334 0.706879i \(-0.249898\pi\)
0.707334 + 0.706879i \(0.249898\pi\)
\(20\) −46490.9 −1.29946
\(21\) 32429.5 0.764139
\(22\) 64686.2 1.29519
\(23\) 19299.9 0.330756 0.165378 0.986230i \(-0.447116\pi\)
0.165378 + 0.986230i \(0.447116\pi\)
\(24\) 102967. 1.52041
\(25\) −55660.4 −0.712453
\(26\) −204355. −2.28023
\(27\) 19683.0 0.192450
\(28\) 372558. 3.20731
\(29\) −148022. −1.12703 −0.563513 0.826107i \(-0.690551\pi\)
−0.563513 + 0.826107i \(0.690551\pi\)
\(30\) −84711.3 −0.572818
\(31\) −77522.7 −0.467372 −0.233686 0.972312i \(-0.575079\pi\)
−0.233686 + 0.972312i \(0.575079\pi\)
\(32\) 351811. 1.89795
\(33\) 83434.9 0.404156
\(34\) 16219.4 0.0707714
\(35\) −180022. −0.709721
\(36\) 226123. 0.807768
\(37\) 381041. 1.23670 0.618352 0.785901i \(-0.287801\pi\)
0.618352 + 0.785901i \(0.287801\pi\)
\(38\) 885360. 2.61744
\(39\) −263586. −0.711535
\(40\) −571591. −1.41213
\(41\) 313892. 0.711274 0.355637 0.934624i \(-0.384264\pi\)
0.355637 + 0.934624i \(0.384264\pi\)
\(42\) 678840. 1.41382
\(43\) 86822.9 0.166531 0.0832654 0.996527i \(-0.473465\pi\)
0.0832654 + 0.996527i \(0.473465\pi\)
\(44\) 958522. 1.69636
\(45\) −109264. −0.178745
\(46\) 404002. 0.611971
\(47\) 1.09797e6 1.54258 0.771288 0.636486i \(-0.219613\pi\)
0.771288 + 0.636486i \(0.219613\pi\)
\(48\) 1.08340e6 1.41399
\(49\) 619078. 0.751725
\(50\) −1.16513e6 −1.31819
\(51\) 20920.4 0.0220838
\(52\) −3.02814e6 −2.98652
\(53\) −1.35146e6 −1.24692 −0.623458 0.781857i \(-0.714273\pi\)
−0.623458 + 0.781857i \(0.714273\pi\)
\(54\) 412021. 0.356075
\(55\) −463163. −0.375374
\(56\) 4.58049e6 3.48541
\(57\) 1.14197e6 0.816759
\(58\) −3.09852e6 −2.08524
\(59\) 205379. 0.130189
\(60\) −1.25525e6 −0.750243
\(61\) −1.17823e6 −0.664622 −0.332311 0.943170i \(-0.607828\pi\)
−0.332311 + 0.943170i \(0.607828\pi\)
\(62\) −1.62277e6 −0.864741
\(63\) 875596. 0.441176
\(64\) 2.22825e6 1.06251
\(65\) 1.46321e6 0.660863
\(66\) 1.74653e6 0.747776
\(67\) −2.20281e6 −0.894777 −0.447389 0.894340i \(-0.647646\pi\)
−0.447389 + 0.894340i \(0.647646\pi\)
\(68\) 240339. 0.0926922
\(69\) 521098. 0.190962
\(70\) −3.76837e6 −1.31314
\(71\) 64342.9 0.0213352 0.0106676 0.999943i \(-0.496604\pi\)
0.0106676 + 0.999943i \(0.496604\pi\)
\(72\) 2.78012e6 0.877808
\(73\) −1.88910e6 −0.568363 −0.284181 0.958771i \(-0.591722\pi\)
−0.284181 + 0.958771i \(0.591722\pi\)
\(74\) 7.97626e6 2.28817
\(75\) −1.50283e6 −0.411335
\(76\) 1.31193e7 3.42817
\(77\) 3.71159e6 0.926494
\(78\) −5.51760e6 −1.31649
\(79\) −267273. −0.0609902 −0.0304951 0.999535i \(-0.509708\pi\)
−0.0304951 + 0.999535i \(0.509708\pi\)
\(80\) −6.01418e6 −1.31329
\(81\) 531441. 0.111111
\(82\) 6.57065e6 1.31601
\(83\) −9.61801e6 −1.84634 −0.923170 0.384391i \(-0.874411\pi\)
−0.923170 + 0.384391i \(0.874411\pi\)
\(84\) 1.00591e7 1.85174
\(85\) −116133. −0.0205111
\(86\) 1.81745e6 0.308118
\(87\) −3.99660e6 −0.650689
\(88\) 1.17847e7 1.84345
\(89\) 1.26033e7 1.89505 0.947523 0.319687i \(-0.103578\pi\)
0.947523 + 0.319687i \(0.103578\pi\)
\(90\) −2.28720e6 −0.330717
\(91\) −1.17256e7 −1.63113
\(92\) 5.98651e6 0.801523
\(93\) −2.09311e6 −0.269838
\(94\) 2.29835e7 2.85410
\(95\) −6.33931e6 −0.758594
\(96\) 9.49889e6 1.09578
\(97\) −7.21193e6 −0.802325 −0.401162 0.916007i \(-0.631394\pi\)
−0.401162 + 0.916007i \(0.631394\pi\)
\(98\) 1.29590e7 1.39085
\(99\) 2.25274e6 0.233340
\(100\) −1.72649e7 −1.72649
\(101\) −3.44100e6 −0.332322 −0.166161 0.986099i \(-0.553137\pi\)
−0.166161 + 0.986099i \(0.553137\pi\)
\(102\) 437923. 0.0408599
\(103\) −8.00277e6 −0.721623 −0.360811 0.932639i \(-0.617500\pi\)
−0.360811 + 0.932639i \(0.617500\pi\)
\(104\) −3.72301e7 −3.24547
\(105\) −4.86060e6 −0.409758
\(106\) −2.82899e7 −2.30707
\(107\) −1.89058e7 −1.49194 −0.745972 0.665978i \(-0.768015\pi\)
−0.745972 + 0.665978i \(0.768015\pi\)
\(108\) 6.10533e6 0.466365
\(109\) −1.48351e7 −1.09723 −0.548613 0.836076i \(-0.684844\pi\)
−0.548613 + 0.836076i \(0.684844\pi\)
\(110\) −9.69530e6 −0.694524
\(111\) 1.02881e7 0.714011
\(112\) 4.81951e7 3.24145
\(113\) 1.51244e7 0.986061 0.493030 0.870012i \(-0.335889\pi\)
0.493030 + 0.870012i \(0.335889\pi\)
\(114\) 2.39047e7 1.51118
\(115\) −2.89271e6 −0.177363
\(116\) −4.59140e7 −2.73113
\(117\) −7.11682e6 −0.410805
\(118\) 4.29916e6 0.240878
\(119\) 930642. 0.0506254
\(120\) −1.54330e7 −0.815295
\(121\) −9.93795e6 −0.509974
\(122\) −2.46636e7 −1.22970
\(123\) 8.47509e6 0.410654
\(124\) −2.40462e7 −1.13259
\(125\) 2.00520e7 0.918276
\(126\) 1.83287e7 0.816271
\(127\) 3.60355e7 1.56105 0.780527 0.625122i \(-0.214951\pi\)
0.780527 + 0.625122i \(0.214951\pi\)
\(128\) 1.61184e6 0.0679339
\(129\) 2.34422e6 0.0961466
\(130\) 3.06292e7 1.22274
\(131\) 1.15356e7 0.448321 0.224161 0.974552i \(-0.428036\pi\)
0.224161 + 0.974552i \(0.428036\pi\)
\(132\) 2.58801e7 0.979393
\(133\) 5.08005e7 1.87235
\(134\) −4.61110e7 −1.65553
\(135\) −2.95013e6 −0.103198
\(136\) 2.95490e6 0.100729
\(137\) −3.77725e7 −1.25503 −0.627515 0.778605i \(-0.715928\pi\)
−0.627515 + 0.778605i \(0.715928\pi\)
\(138\) 1.09080e7 0.353322
\(139\) −3.71316e6 −0.117271 −0.0586357 0.998279i \(-0.518675\pi\)
−0.0586357 + 0.998279i \(0.518675\pi\)
\(140\) −5.58398e7 −1.71987
\(141\) 2.96451e7 0.890607
\(142\) 1.34688e6 0.0394747
\(143\) −3.01677e7 −0.862713
\(144\) 2.92519e7 0.816367
\(145\) 2.21859e7 0.604350
\(146\) −3.95442e7 −1.05159
\(147\) 1.67151e7 0.434009
\(148\) 1.18192e8 2.99691
\(149\) −6.47154e7 −1.60271 −0.801357 0.598187i \(-0.795888\pi\)
−0.801357 + 0.598187i \(0.795888\pi\)
\(150\) −3.14585e7 −0.761059
\(151\) 2.42807e7 0.573907 0.286953 0.957945i \(-0.407358\pi\)
0.286953 + 0.957945i \(0.407358\pi\)
\(152\) 1.61298e8 3.72542
\(153\) 564851. 0.0127501
\(154\) 7.76941e7 1.71422
\(155\) 1.16193e7 0.250621
\(156\) −8.17599e7 −1.72427
\(157\) 4.14110e7 0.854018 0.427009 0.904247i \(-0.359567\pi\)
0.427009 + 0.904247i \(0.359567\pi\)
\(158\) −5.59477e6 −0.112845
\(159\) −3.64894e7 −0.719908
\(160\) −5.27301e7 −1.01774
\(161\) 2.31810e7 0.437765
\(162\) 1.11246e7 0.205580
\(163\) −8.91039e7 −1.61154 −0.805769 0.592231i \(-0.798247\pi\)
−0.805769 + 0.592231i \(0.798247\pi\)
\(164\) 9.73640e7 1.72363
\(165\) −1.25054e7 −0.216722
\(166\) −2.01332e8 −3.41613
\(167\) 5.76977e7 0.958630 0.479315 0.877643i \(-0.340885\pi\)
0.479315 + 0.877643i \(0.340885\pi\)
\(168\) 1.23673e8 2.01230
\(169\) 3.25568e7 0.518845
\(170\) −2.43099e6 −0.0379501
\(171\) 3.08333e7 0.471556
\(172\) 2.69310e7 0.403555
\(173\) 1.70091e7 0.249759 0.124880 0.992172i \(-0.460146\pi\)
0.124880 + 0.992172i \(0.460146\pi\)
\(174\) −8.36601e7 −1.20392
\(175\) −6.68532e7 −0.942951
\(176\) 1.23997e8 1.71442
\(177\) 5.54523e6 0.0751646
\(178\) 2.63823e8 3.50625
\(179\) −3.67383e7 −0.478777 −0.239388 0.970924i \(-0.576947\pi\)
−0.239388 + 0.970924i \(0.576947\pi\)
\(180\) −3.38918e7 −0.433153
\(181\) 1.07531e7 0.134791 0.0673954 0.997726i \(-0.478531\pi\)
0.0673954 + 0.997726i \(0.478531\pi\)
\(182\) −2.45450e8 −3.01795
\(183\) −3.18121e7 −0.383720
\(184\) 7.36023e7 0.871021
\(185\) −5.71112e7 −0.663163
\(186\) −4.38148e7 −0.499258
\(187\) 2.39437e6 0.0267759
\(188\) 3.40571e8 3.73813
\(189\) 2.36411e7 0.254713
\(190\) −1.32700e8 −1.40356
\(191\) −4.65850e7 −0.483759 −0.241880 0.970306i \(-0.577764\pi\)
−0.241880 + 0.970306i \(0.577764\pi\)
\(192\) 6.01628e7 0.613442
\(193\) −6.10866e7 −0.611639 −0.305820 0.952089i \(-0.598930\pi\)
−0.305820 + 0.952089i \(0.598930\pi\)
\(194\) −1.50966e8 −1.48448
\(195\) 3.95068e7 0.381549
\(196\) 1.92027e8 1.82166
\(197\) 7.42960e7 0.692363 0.346181 0.938168i \(-0.387478\pi\)
0.346181 + 0.938168i \(0.387478\pi\)
\(198\) 4.71563e7 0.431729
\(199\) −1.79787e8 −1.61723 −0.808617 0.588335i \(-0.799784\pi\)
−0.808617 + 0.588335i \(0.799784\pi\)
\(200\) −2.12267e8 −1.87619
\(201\) −5.94758e7 −0.516600
\(202\) −7.20298e7 −0.614869
\(203\) −1.77788e8 −1.49165
\(204\) 6.48916e6 0.0535159
\(205\) −4.70468e7 −0.381410
\(206\) −1.67521e8 −1.33516
\(207\) 1.40696e7 0.110252
\(208\) −3.91729e8 −3.01831
\(209\) 1.30700e8 0.990294
\(210\) −1.01746e8 −0.758141
\(211\) −1.70096e8 −1.24654 −0.623269 0.782008i \(-0.714196\pi\)
−0.623269 + 0.782008i \(0.714196\pi\)
\(212\) −4.19200e8 −3.02166
\(213\) 1.73726e6 0.0123179
\(214\) −3.95752e8 −2.76042
\(215\) −1.30132e7 −0.0892996
\(216\) 7.50632e7 0.506803
\(217\) −9.31119e7 −0.618580
\(218\) −3.10540e8 −2.03011
\(219\) −5.10058e7 −0.328144
\(220\) −1.43665e8 −0.909646
\(221\) −7.56423e6 −0.0471402
\(222\) 2.15359e8 1.32108
\(223\) −1.26142e8 −0.761713 −0.380857 0.924634i \(-0.624371\pi\)
−0.380857 + 0.924634i \(0.624371\pi\)
\(224\) 4.22557e8 2.51199
\(225\) −4.05764e7 −0.237484
\(226\) 3.16596e8 1.82443
\(227\) 1.79611e8 1.01916 0.509579 0.860424i \(-0.329801\pi\)
0.509579 + 0.860424i \(0.329801\pi\)
\(228\) 3.54221e8 1.97926
\(229\) 3.37131e8 1.85513 0.927564 0.373664i \(-0.121899\pi\)
0.927564 + 0.373664i \(0.121899\pi\)
\(230\) −6.05526e7 −0.328160
\(231\) 1.00213e8 0.534912
\(232\) −5.64498e8 −2.96794
\(233\) 2.03401e8 1.05343 0.526716 0.850041i \(-0.323423\pi\)
0.526716 + 0.850041i \(0.323423\pi\)
\(234\) −1.48975e8 −0.760078
\(235\) −1.64565e8 −0.827182
\(236\) 6.37051e7 0.315487
\(237\) −7.21636e6 −0.0352127
\(238\) 1.94810e7 0.0936679
\(239\) 2.40718e8 1.14056 0.570278 0.821452i \(-0.306836\pi\)
0.570278 + 0.821452i \(0.306836\pi\)
\(240\) −1.62383e8 −0.758230
\(241\) 6.28354e7 0.289164 0.144582 0.989493i \(-0.453816\pi\)
0.144582 + 0.989493i \(0.453816\pi\)
\(242\) −2.08029e8 −0.943563
\(243\) 1.43489e7 0.0641500
\(244\) −3.65466e8 −1.61058
\(245\) −9.27887e7 −0.403101
\(246\) 1.77408e8 0.759800
\(247\) −4.12905e8 −1.74346
\(248\) −2.95641e8 −1.23079
\(249\) −2.59686e8 −1.06599
\(250\) 4.19746e8 1.69901
\(251\) 1.10247e8 0.440058 0.220029 0.975493i \(-0.429385\pi\)
0.220029 + 0.975493i \(0.429385\pi\)
\(252\) 2.71595e8 1.06910
\(253\) 5.96403e7 0.231536
\(254\) 7.54326e8 2.88829
\(255\) −3.13559e6 −0.0118421
\(256\) −2.51476e8 −0.936821
\(257\) −8.00040e7 −0.293999 −0.146999 0.989137i \(-0.546962\pi\)
−0.146999 + 0.989137i \(0.546962\pi\)
\(258\) 4.90711e7 0.177892
\(259\) 4.57665e8 1.63681
\(260\) 4.53864e8 1.60147
\(261\) −1.07908e8 −0.375675
\(262\) 2.41472e8 0.829492
\(263\) −1.42844e8 −0.484190 −0.242095 0.970253i \(-0.577835\pi\)
−0.242095 + 0.970253i \(0.577835\pi\)
\(264\) 3.18188e8 1.06431
\(265\) 2.02559e8 0.668639
\(266\) 1.06340e9 3.46426
\(267\) 3.40290e8 1.09411
\(268\) −6.83274e8 −2.16832
\(269\) −1.90832e8 −0.597747 −0.298873 0.954293i \(-0.596611\pi\)
−0.298873 + 0.954293i \(0.596611\pi\)
\(270\) −6.17545e7 −0.190939
\(271\) 1.37623e8 0.420048 0.210024 0.977696i \(-0.432646\pi\)
0.210024 + 0.977696i \(0.432646\pi\)
\(272\) 3.10909e7 0.0936789
\(273\) −3.16591e8 −0.941736
\(274\) −7.90685e8 −2.32208
\(275\) −1.72001e8 −0.498730
\(276\) 1.61636e8 0.462760
\(277\) 1.18249e8 0.334286 0.167143 0.985933i \(-0.446546\pi\)
0.167143 + 0.985933i \(0.446546\pi\)
\(278\) −7.77269e7 −0.216978
\(279\) −5.65141e7 −0.155791
\(280\) −6.86533e8 −1.86900
\(281\) −1.62690e8 −0.437411 −0.218705 0.975791i \(-0.570183\pi\)
−0.218705 + 0.975791i \(0.570183\pi\)
\(282\) 6.20555e8 1.64782
\(283\) −4.15031e8 −1.08850 −0.544249 0.838924i \(-0.683185\pi\)
−0.544249 + 0.838924i \(0.683185\pi\)
\(284\) 1.99581e7 0.0517016
\(285\) −1.71161e8 −0.437974
\(286\) −6.31495e8 −1.59621
\(287\) 3.77013e8 0.941391
\(288\) 2.56470e8 0.632649
\(289\) −4.09738e8 −0.998537
\(290\) 4.64413e8 1.11818
\(291\) −1.94722e8 −0.463222
\(292\) −5.85968e8 −1.37732
\(293\) 4.61387e8 1.07159 0.535796 0.844348i \(-0.320012\pi\)
0.535796 + 0.844348i \(0.320012\pi\)
\(294\) 3.49894e8 0.803010
\(295\) −3.07826e7 −0.0698118
\(296\) 1.45314e9 3.25676
\(297\) 6.08240e7 0.134719
\(298\) −1.35468e9 −2.96537
\(299\) −1.88414e8 −0.407629
\(300\) −4.66152e8 −0.996790
\(301\) 1.04282e8 0.220408
\(302\) 5.08263e8 1.06185
\(303\) −9.29070e7 −0.191866
\(304\) 1.69714e9 3.46467
\(305\) 1.76595e8 0.356393
\(306\) 1.18239e7 0.0235905
\(307\) 7.27720e8 1.43542 0.717712 0.696340i \(-0.245189\pi\)
0.717712 + 0.696340i \(0.245189\pi\)
\(308\) 1.15127e9 2.24518
\(309\) −2.16075e8 −0.416629
\(310\) 2.43224e8 0.463704
\(311\) −1.57441e8 −0.296794 −0.148397 0.988928i \(-0.547411\pi\)
−0.148397 + 0.988928i \(0.547411\pi\)
\(312\) −1.00521e9 −1.87377
\(313\) −1.78754e8 −0.329497 −0.164748 0.986336i \(-0.552681\pi\)
−0.164748 + 0.986336i \(0.552681\pi\)
\(314\) 8.66850e8 1.58012
\(315\) −1.31236e8 −0.236574
\(316\) −8.29035e7 −0.147798
\(317\) 5.72794e8 1.00993 0.504965 0.863140i \(-0.331505\pi\)
0.504965 + 0.863140i \(0.331505\pi\)
\(318\) −7.63826e8 −1.33199
\(319\) −4.57416e8 −0.788939
\(320\) −3.33975e8 −0.569756
\(321\) −5.10457e8 −0.861374
\(322\) 4.85243e8 0.809961
\(323\) 3.27717e7 0.0541115
\(324\) 1.64844e8 0.269256
\(325\) 5.43381e8 0.878037
\(326\) −1.86520e9 −2.98169
\(327\) −4.00547e8 −0.633484
\(328\) 1.19706e9 1.87309
\(329\) 1.31876e9 2.04164
\(330\) −2.61773e8 −0.400983
\(331\) 2.90899e8 0.440905 0.220452 0.975398i \(-0.429247\pi\)
0.220452 + 0.975398i \(0.429247\pi\)
\(332\) −2.98334e9 −4.47425
\(333\) 2.77779e8 0.412235
\(334\) 1.20778e9 1.77367
\(335\) 3.30162e8 0.479810
\(336\) 1.30127e9 1.87145
\(337\) 1.34379e9 1.91261 0.956304 0.292374i \(-0.0944453\pi\)
0.956304 + 0.292374i \(0.0944453\pi\)
\(338\) 6.81505e8 0.959976
\(339\) 4.08359e8 0.569302
\(340\) −3.60225e7 −0.0497047
\(341\) −2.39559e8 −0.327169
\(342\) 6.45428e8 0.872481
\(343\) −2.45581e8 −0.328599
\(344\) 3.31108e8 0.438546
\(345\) −7.81032e7 −0.102401
\(346\) 3.56049e8 0.462109
\(347\) 2.97139e8 0.381774 0.190887 0.981612i \(-0.438864\pi\)
0.190887 + 0.981612i \(0.438864\pi\)
\(348\) −1.23968e9 −1.57682
\(349\) 1.52383e9 1.91888 0.959440 0.281912i \(-0.0909687\pi\)
0.959440 + 0.281912i \(0.0909687\pi\)
\(350\) −1.39943e9 −1.74466
\(351\) −1.92154e8 −0.237178
\(352\) 1.08716e9 1.32860
\(353\) −5.87541e8 −0.710930 −0.355465 0.934690i \(-0.615677\pi\)
−0.355465 + 0.934690i \(0.615677\pi\)
\(354\) 1.16077e8 0.139071
\(355\) −9.64384e6 −0.0114407
\(356\) 3.90934e9 4.59228
\(357\) 2.51273e7 0.0292286
\(358\) −7.69036e8 −0.885841
\(359\) 1.34525e9 1.53452 0.767262 0.641334i \(-0.221619\pi\)
0.767262 + 0.641334i \(0.221619\pi\)
\(360\) −4.16690e8 −0.470711
\(361\) 8.95021e8 1.00129
\(362\) 2.25094e8 0.249392
\(363\) −2.68325e8 −0.294434
\(364\) −3.63708e9 −3.95274
\(365\) 2.83143e8 0.304776
\(366\) −6.65918e8 −0.709965
\(367\) 1.37516e9 1.45218 0.726090 0.687600i \(-0.241336\pi\)
0.726090 + 0.687600i \(0.241336\pi\)
\(368\) 7.74430e8 0.810056
\(369\) 2.28827e8 0.237091
\(370\) −1.19550e9 −1.22700
\(371\) −1.62323e9 −1.65033
\(372\) −6.49248e8 −0.653899
\(373\) −1.95269e8 −0.194829 −0.0974143 0.995244i \(-0.531057\pi\)
−0.0974143 + 0.995244i \(0.531057\pi\)
\(374\) 5.01208e7 0.0495413
\(375\) 5.41405e8 0.530167
\(376\) 4.18721e9 4.06226
\(377\) 1.44506e9 1.38896
\(378\) 4.94875e8 0.471274
\(379\) −1.78491e9 −1.68414 −0.842071 0.539367i \(-0.818664\pi\)
−0.842071 + 0.539367i \(0.818664\pi\)
\(380\) −1.96635e9 −1.83830
\(381\) 9.72960e8 0.901275
\(382\) −9.75155e8 −0.895059
\(383\) 1.02371e9 0.931072 0.465536 0.885029i \(-0.345862\pi\)
0.465536 + 0.885029i \(0.345862\pi\)
\(384\) 4.35196e7 0.0392216
\(385\) −5.56301e8 −0.496818
\(386\) −1.27872e9 −1.13167
\(387\) 6.32939e7 0.0555103
\(388\) −2.23702e9 −1.94428
\(389\) 5.45849e8 0.470163 0.235082 0.971976i \(-0.424464\pi\)
0.235082 + 0.971976i \(0.424464\pi\)
\(390\) 8.26989e8 0.705949
\(391\) 1.49542e7 0.0126515
\(392\) 2.36092e9 1.97961
\(393\) 3.11460e8 0.258838
\(394\) 1.55522e9 1.28102
\(395\) 4.00594e7 0.0327050
\(396\) 6.98762e8 0.565453
\(397\) 1.54111e9 1.23614 0.618070 0.786123i \(-0.287915\pi\)
0.618070 + 0.786123i \(0.287915\pi\)
\(398\) −3.76345e9 −2.99224
\(399\) 1.37161e9 1.08100
\(400\) −2.23343e9 −1.74487
\(401\) −1.87893e9 −1.45514 −0.727570 0.686034i \(-0.759350\pi\)
−0.727570 + 0.686034i \(0.759350\pi\)
\(402\) −1.24500e9 −0.955822
\(403\) 7.56811e8 0.575996
\(404\) −1.06734e9 −0.805319
\(405\) −7.96535e7 −0.0595816
\(406\) −3.72161e9 −2.75988
\(407\) 1.17749e9 0.865716
\(408\) 7.97822e7 0.0581561
\(409\) −1.97782e9 −1.42941 −0.714703 0.699428i \(-0.753438\pi\)
−0.714703 + 0.699428i \(0.753438\pi\)
\(410\) −9.84822e8 −0.705691
\(411\) −1.01986e9 −0.724592
\(412\) −2.48232e9 −1.74871
\(413\) 2.46679e8 0.172309
\(414\) 2.94517e8 0.203990
\(415\) 1.44157e9 0.990071
\(416\) −3.43453e9 −2.33906
\(417\) −1.00255e8 −0.0677066
\(418\) 2.73592e9 1.83226
\(419\) 1.09100e9 0.724561 0.362280 0.932069i \(-0.381998\pi\)
0.362280 + 0.932069i \(0.381998\pi\)
\(420\) −1.50767e9 −0.992968
\(421\) 1.23434e9 0.806211 0.403105 0.915154i \(-0.367931\pi\)
0.403105 + 0.915154i \(0.367931\pi\)
\(422\) −3.56059e9 −2.30637
\(423\) 8.00417e8 0.514192
\(424\) −5.15393e9 −3.28366
\(425\) −4.31273e7 −0.0272515
\(426\) 3.63657e7 0.0227907
\(427\) −1.41516e9 −0.879645
\(428\) −5.86426e9 −3.61543
\(429\) −8.14528e8 −0.498088
\(430\) −2.72403e8 −0.165224
\(431\) 2.37469e8 0.142869 0.0714343 0.997445i \(-0.477242\pi\)
0.0714343 + 0.997445i \(0.477242\pi\)
\(432\) 7.89802e8 0.471330
\(433\) 2.89301e9 1.71255 0.856274 0.516522i \(-0.172774\pi\)
0.856274 + 0.516522i \(0.172774\pi\)
\(434\) −1.94909e9 −1.14451
\(435\) 5.99019e8 0.348922
\(436\) −4.60158e9 −2.65892
\(437\) 8.16296e8 0.467910
\(438\) −1.06769e9 −0.607138
\(439\) −1.35570e9 −0.764781 −0.382391 0.924001i \(-0.624899\pi\)
−0.382391 + 0.924001i \(0.624899\pi\)
\(440\) −1.76632e9 −0.988519
\(441\) 4.51308e8 0.250575
\(442\) −1.58341e8 −0.0872197
\(443\) 1.36994e9 0.748664 0.374332 0.927295i \(-0.377872\pi\)
0.374332 + 0.927295i \(0.377872\pi\)
\(444\) 3.19120e9 1.73027
\(445\) −1.88901e9 −1.01619
\(446\) −2.64050e9 −1.40934
\(447\) −1.74732e9 −0.925327
\(448\) 2.67633e9 1.40627
\(449\) 3.56894e9 1.86070 0.930352 0.366669i \(-0.119502\pi\)
0.930352 + 0.366669i \(0.119502\pi\)
\(450\) −8.49379e8 −0.439397
\(451\) 9.69984e8 0.497905
\(452\) 4.69133e9 2.38953
\(453\) 6.55578e8 0.331345
\(454\) 3.75976e9 1.88566
\(455\) 1.75746e9 0.874670
\(456\) 4.35503e9 2.15087
\(457\) −1.58612e9 −0.777374 −0.388687 0.921370i \(-0.627071\pi\)
−0.388687 + 0.921370i \(0.627071\pi\)
\(458\) 7.05709e9 3.43239
\(459\) 1.52510e7 0.00736128
\(460\) −8.97270e8 −0.429804
\(461\) −2.42902e9 −1.15472 −0.577361 0.816489i \(-0.695917\pi\)
−0.577361 + 0.816489i \(0.695917\pi\)
\(462\) 2.09774e9 0.989703
\(463\) −2.06490e9 −0.966865 −0.483432 0.875382i \(-0.660610\pi\)
−0.483432 + 0.875382i \(0.660610\pi\)
\(464\) −5.93955e9 −2.76020
\(465\) 3.13720e8 0.144696
\(466\) 4.25775e9 1.94908
\(467\) −2.28871e9 −1.03988 −0.519938 0.854204i \(-0.674045\pi\)
−0.519938 + 0.854204i \(0.674045\pi\)
\(468\) −2.20752e9 −0.995505
\(469\) −2.64578e9 −1.18426
\(470\) −3.44482e9 −1.53047
\(471\) 1.11810e9 0.493068
\(472\) 7.83234e8 0.342843
\(473\) 2.68299e8 0.116575
\(474\) −1.51059e8 −0.0651511
\(475\) −2.35417e9 −1.00788
\(476\) 2.88669e8 0.122681
\(477\) −9.85214e8 −0.415639
\(478\) 5.03891e9 2.11028
\(479\) −9.59752e8 −0.399011 −0.199505 0.979897i \(-0.563934\pi\)
−0.199505 + 0.979897i \(0.563934\pi\)
\(480\) −1.42371e9 −0.587595
\(481\) −3.71989e9 −1.52413
\(482\) 1.31532e9 0.535017
\(483\) 6.25886e8 0.252744
\(484\) −3.08258e9 −1.23582
\(485\) 1.08094e9 0.430234
\(486\) 3.00363e8 0.118692
\(487\) 8.78521e8 0.344668 0.172334 0.985039i \(-0.444869\pi\)
0.172334 + 0.985039i \(0.444869\pi\)
\(488\) −4.49329e9 −1.75023
\(489\) −2.40581e9 −0.930421
\(490\) −1.94233e9 −0.745824
\(491\) 1.17143e9 0.446612 0.223306 0.974748i \(-0.428315\pi\)
0.223306 + 0.974748i \(0.428315\pi\)
\(492\) 2.62883e9 0.995140
\(493\) −1.14692e8 −0.0431091
\(494\) −8.64328e9 −3.22578
\(495\) −3.37646e8 −0.125125
\(496\) −3.11068e9 −1.14464
\(497\) 7.72817e7 0.0282377
\(498\) −5.43597e9 −1.97230
\(499\) 3.72675e9 1.34270 0.671349 0.741142i \(-0.265715\pi\)
0.671349 + 0.741142i \(0.265715\pi\)
\(500\) 6.21980e9 2.22526
\(501\) 1.55784e9 0.553465
\(502\) 2.30778e9 0.814202
\(503\) 5.09523e9 1.78516 0.892578 0.450894i \(-0.148895\pi\)
0.892578 + 0.450894i \(0.148895\pi\)
\(504\) 3.33918e9 1.16180
\(505\) 5.15744e8 0.178203
\(506\) 1.24844e9 0.428391
\(507\) 8.79032e8 0.299555
\(508\) 1.11776e10 3.78291
\(509\) 3.59219e9 1.20739 0.603693 0.797217i \(-0.293695\pi\)
0.603693 + 0.797217i \(0.293695\pi\)
\(510\) −6.56368e7 −0.0219105
\(511\) −2.26899e9 −0.752244
\(512\) −5.47042e9 −1.80126
\(513\) 8.32499e8 0.272253
\(514\) −1.67471e9 −0.543962
\(515\) 1.19947e9 0.386959
\(516\) 7.27137e8 0.232993
\(517\) 3.39292e9 1.07983
\(518\) 9.58022e9 3.02846
\(519\) 4.59247e8 0.144198
\(520\) 5.58012e9 1.74033
\(521\) 5.07000e9 1.57064 0.785318 0.619092i \(-0.212499\pi\)
0.785318 + 0.619092i \(0.212499\pi\)
\(522\) −2.25882e9 −0.695081
\(523\) 4.99855e8 0.152788 0.0763938 0.997078i \(-0.475659\pi\)
0.0763938 + 0.997078i \(0.475659\pi\)
\(524\) 3.57814e9 1.08642
\(525\) −1.80504e9 −0.544413
\(526\) −2.99012e9 −0.895856
\(527\) −6.00669e7 −0.0178771
\(528\) 3.34792e9 0.989819
\(529\) −3.03234e9 −0.890600
\(530\) 4.24014e9 1.23713
\(531\) 1.49721e8 0.0433963
\(532\) 1.57575e10 4.53728
\(533\) −3.06435e9 −0.876584
\(534\) 7.12322e9 2.02433
\(535\) 2.83364e9 0.800031
\(536\) −8.40064e9 −2.35633
\(537\) −9.91933e8 −0.276422
\(538\) −3.99464e9 −1.10596
\(539\) 1.91306e9 0.526222
\(540\) −9.15080e8 −0.250081
\(541\) 5.11313e9 1.38834 0.694171 0.719810i \(-0.255771\pi\)
0.694171 + 0.719810i \(0.255771\pi\)
\(542\) 2.88084e9 0.777181
\(543\) 2.90335e8 0.0778215
\(544\) 2.72593e8 0.0725971
\(545\) 2.22351e9 0.588371
\(546\) −6.62714e9 −1.74242
\(547\) −3.84269e8 −0.100387 −0.0501937 0.998739i \(-0.515984\pi\)
−0.0501937 + 0.998739i \(0.515984\pi\)
\(548\) −1.17164e10 −3.04132
\(549\) −8.58928e8 −0.221541
\(550\) −3.60046e9 −0.922759
\(551\) −6.26065e9 −1.59437
\(552\) 1.98726e9 0.502884
\(553\) −3.21019e8 −0.0807222
\(554\) 2.47528e9 0.618501
\(555\) −1.54200e9 −0.382877
\(556\) −1.15176e9 −0.284184
\(557\) 5.07247e9 1.24373 0.621866 0.783124i \(-0.286375\pi\)
0.621866 + 0.783124i \(0.286375\pi\)
\(558\) −1.18300e9 −0.288247
\(559\) −8.47604e8 −0.205235
\(560\) −7.22358e9 −1.73818
\(561\) 6.46479e7 0.0154591
\(562\) −3.40557e9 −0.809305
\(563\) −8.25453e8 −0.194945 −0.0974727 0.995238i \(-0.531076\pi\)
−0.0974727 + 0.995238i \(0.531076\pi\)
\(564\) 9.19540e9 2.15821
\(565\) −2.26688e9 −0.528760
\(566\) −8.68776e9 −2.01396
\(567\) 6.38309e8 0.147059
\(568\) 2.45378e8 0.0561845
\(569\) 4.84625e9 1.10284 0.551420 0.834228i \(-0.314086\pi\)
0.551420 + 0.834228i \(0.314086\pi\)
\(570\) −3.58289e9 −0.810348
\(571\) 2.06740e9 0.464728 0.232364 0.972629i \(-0.425354\pi\)
0.232364 + 0.972629i \(0.425354\pi\)
\(572\) −9.35751e9 −2.09062
\(573\) −1.25779e9 −0.279298
\(574\) 7.89195e9 1.74178
\(575\) −1.07424e9 −0.235648
\(576\) 1.62440e9 0.354171
\(577\) 5.23872e9 1.13530 0.567649 0.823270i \(-0.307853\pi\)
0.567649 + 0.823270i \(0.307853\pi\)
\(578\) −8.57698e9 −1.84751
\(579\) −1.64934e9 −0.353130
\(580\) 6.88169e9 1.46453
\(581\) −1.15521e10 −2.44368
\(582\) −4.07608e9 −0.857062
\(583\) −4.17625e9 −0.872865
\(584\) −7.20429e9 −1.49674
\(585\) 1.06668e9 0.220288
\(586\) 9.65814e9 1.98268
\(587\) −5.79577e9 −1.18271 −0.591355 0.806411i \(-0.701407\pi\)
−0.591355 + 0.806411i \(0.701407\pi\)
\(588\) 5.18474e9 1.05174
\(589\) −3.27885e9 −0.661177
\(590\) −6.44367e8 −0.129167
\(591\) 2.00599e9 0.399736
\(592\) 1.52897e10 3.02881
\(593\) 6.07221e8 0.119579 0.0597896 0.998211i \(-0.480957\pi\)
0.0597896 + 0.998211i \(0.480957\pi\)
\(594\) 1.27322e9 0.249259
\(595\) −1.39486e8 −0.0271471
\(596\) −2.00736e10 −3.88386
\(597\) −4.85426e9 −0.933711
\(598\) −3.94404e9 −0.754202
\(599\) 7.83137e9 1.48883 0.744413 0.667720i \(-0.232730\pi\)
0.744413 + 0.667720i \(0.232730\pi\)
\(600\) −5.73120e9 −1.08322
\(601\) −9.69564e9 −1.82186 −0.910932 0.412556i \(-0.864636\pi\)
−0.910932 + 0.412556i \(0.864636\pi\)
\(602\) 2.18292e9 0.407803
\(603\) −1.60585e9 −0.298259
\(604\) 7.53145e9 1.39075
\(605\) 1.48952e9 0.273466
\(606\) −1.94480e9 −0.354995
\(607\) 2.75108e9 0.499278 0.249639 0.968339i \(-0.419688\pi\)
0.249639 + 0.968339i \(0.419688\pi\)
\(608\) 1.48799e10 2.68497
\(609\) −4.80028e9 −0.861205
\(610\) 3.69663e9 0.659405
\(611\) −1.07188e10 −1.90109
\(612\) 1.75207e8 0.0308974
\(613\) −1.06940e10 −1.87511 −0.937557 0.347832i \(-0.886918\pi\)
−0.937557 + 0.347832i \(0.886918\pi\)
\(614\) 1.52332e10 2.65585
\(615\) −1.27026e9 −0.220207
\(616\) 1.41545e10 2.43985
\(617\) 3.67080e9 0.629162 0.314581 0.949231i \(-0.398136\pi\)
0.314581 + 0.949231i \(0.398136\pi\)
\(618\) −4.52306e9 −0.770854
\(619\) −7.09568e9 −1.20248 −0.601238 0.799070i \(-0.705326\pi\)
−0.601238 + 0.799070i \(0.705326\pi\)
\(620\) 3.60410e9 0.607331
\(621\) 3.79880e8 0.0636541
\(622\) −3.29568e9 −0.549134
\(623\) 1.51377e10 2.50815
\(624\) −1.05767e10 −1.74262
\(625\) 1.34303e9 0.220042
\(626\) −3.74183e9 −0.609640
\(627\) 3.52890e9 0.571747
\(628\) 1.28450e10 2.06955
\(629\) 2.95242e8 0.0473043
\(630\) −2.74714e9 −0.437713
\(631\) −1.14862e10 −1.82000 −0.910001 0.414606i \(-0.863919\pi\)
−0.910001 + 0.414606i \(0.863919\pi\)
\(632\) −1.01927e9 −0.160613
\(633\) −4.59259e9 −0.719689
\(634\) 1.19902e10 1.86859
\(635\) −5.40108e9 −0.837091
\(636\) −1.13184e10 −1.74456
\(637\) −6.04371e9 −0.926437
\(638\) −9.57500e9 −1.45971
\(639\) 4.69059e7 0.00711172
\(640\) −2.41586e8 −0.0364285
\(641\) −7.81739e9 −1.17235 −0.586177 0.810183i \(-0.699367\pi\)
−0.586177 + 0.810183i \(0.699367\pi\)
\(642\) −1.06853e10 −1.59373
\(643\) 2.34946e9 0.348522 0.174261 0.984699i \(-0.444246\pi\)
0.174261 + 0.984699i \(0.444246\pi\)
\(644\) 7.19034e9 1.06084
\(645\) −3.51356e8 −0.0515571
\(646\) 6.86003e8 0.100118
\(647\) 7.89807e9 1.14645 0.573226 0.819397i \(-0.305692\pi\)
0.573226 + 0.819397i \(0.305692\pi\)
\(648\) 2.02671e9 0.292603
\(649\) 6.34658e8 0.0911347
\(650\) 1.13745e10 1.62456
\(651\) −2.51402e9 −0.357137
\(652\) −2.76385e10 −3.90525
\(653\) 6.38444e8 0.0897277 0.0448639 0.998993i \(-0.485715\pi\)
0.0448639 + 0.998993i \(0.485715\pi\)
\(654\) −8.38457e9 −1.17208
\(655\) −1.72897e9 −0.240405
\(656\) 1.25953e10 1.74198
\(657\) −1.37716e9 −0.189454
\(658\) 2.76053e10 3.77748
\(659\) 8.08275e9 1.10017 0.550085 0.835108i \(-0.314595\pi\)
0.550085 + 0.835108i \(0.314595\pi\)
\(660\) −3.87896e9 −0.525184
\(661\) 9.43857e9 1.27116 0.635581 0.772034i \(-0.280761\pi\)
0.635581 + 0.772034i \(0.280761\pi\)
\(662\) 6.08934e9 0.815770
\(663\) −2.04234e8 −0.0272164
\(664\) −3.66793e10 −4.86220
\(665\) −7.61409e9 −1.00402
\(666\) 5.81470e9 0.762724
\(667\) −2.85682e9 −0.372771
\(668\) 1.78969e10 2.32305
\(669\) −3.40583e9 −0.439775
\(670\) 6.91121e9 0.887754
\(671\) −3.64094e9 −0.465248
\(672\) 1.14090e10 1.45030
\(673\) −1.08913e10 −1.37730 −0.688649 0.725095i \(-0.741796\pi\)
−0.688649 + 0.725095i \(0.741796\pi\)
\(674\) 2.81293e10 3.53874
\(675\) −1.09556e9 −0.137112
\(676\) 1.00986e10 1.25732
\(677\) −1.25627e10 −1.55604 −0.778021 0.628238i \(-0.783777\pi\)
−0.778021 + 0.628238i \(0.783777\pi\)
\(678\) 8.54810e9 1.05333
\(679\) −8.66218e9 −1.06190
\(680\) −4.42886e8 −0.0540145
\(681\) 4.84949e9 0.588411
\(682\) −5.01465e9 −0.605335
\(683\) −1.74046e9 −0.209022 −0.104511 0.994524i \(-0.533328\pi\)
−0.104511 + 0.994524i \(0.533328\pi\)
\(684\) 9.56396e9 1.14272
\(685\) 5.66142e9 0.672990
\(686\) −5.14071e9 −0.607979
\(687\) 9.10253e9 1.07106
\(688\) 3.48386e9 0.407851
\(689\) 1.31935e10 1.53672
\(690\) −1.63492e9 −0.189463
\(691\) −1.58954e10 −1.83272 −0.916362 0.400351i \(-0.868888\pi\)
−0.916362 + 0.400351i \(0.868888\pi\)
\(692\) 5.27595e9 0.605242
\(693\) 2.70575e9 0.308831
\(694\) 6.21995e9 0.706364
\(695\) 5.56536e8 0.0628849
\(696\) −1.52415e10 −1.71354
\(697\) 2.43213e8 0.0272065
\(698\) 3.18981e10 3.55035
\(699\) 5.49182e9 0.608200
\(700\) −2.07367e10 −2.28506
\(701\) 1.64503e10 1.80369 0.901844 0.432062i \(-0.142214\pi\)
0.901844 + 0.432062i \(0.142214\pi\)
\(702\) −4.02233e9 −0.438831
\(703\) 1.61162e10 1.74953
\(704\) 6.88570e9 0.743779
\(705\) −4.44327e9 −0.477574
\(706\) −1.22989e10 −1.31538
\(707\) −4.13295e9 −0.439838
\(708\) 1.72004e9 0.182147
\(709\) −8.21342e9 −0.865491 −0.432745 0.901516i \(-0.642455\pi\)
−0.432745 + 0.901516i \(0.642455\pi\)
\(710\) −2.01873e8 −0.0211677
\(711\) −1.94842e8 −0.0203301
\(712\) 4.80641e10 4.99046
\(713\) −1.49618e9 −0.154586
\(714\) 5.25986e8 0.0540792
\(715\) 4.52160e9 0.462616
\(716\) −1.13956e10 −1.16022
\(717\) 6.49939e9 0.658500
\(718\) 2.81600e10 2.83920
\(719\) −9.47920e9 −0.951088 −0.475544 0.879692i \(-0.657749\pi\)
−0.475544 + 0.879692i \(0.657749\pi\)
\(720\) −4.38434e9 −0.437764
\(721\) −9.61206e9 −0.955087
\(722\) 1.87353e10 1.85260
\(723\) 1.69655e9 0.166949
\(724\) 3.33544e9 0.326639
\(725\) 8.23897e9 0.802953
\(726\) −5.61679e9 −0.544766
\(727\) −1.39173e10 −1.34333 −0.671667 0.740853i \(-0.734421\pi\)
−0.671667 + 0.740853i \(0.734421\pi\)
\(728\) −4.47168e10 −4.29547
\(729\) 3.87420e8 0.0370370
\(730\) 5.92697e9 0.563901
\(731\) 6.72730e7 0.00636986
\(732\) −9.86759e9 −0.929870
\(733\) −1.17895e10 −1.10569 −0.552843 0.833286i \(-0.686457\pi\)
−0.552843 + 0.833286i \(0.686457\pi\)
\(734\) 2.87859e10 2.68685
\(735\) −2.50529e9 −0.232730
\(736\) 6.78992e9 0.627758
\(737\) −6.80708e9 −0.626361
\(738\) 4.79000e9 0.438671
\(739\) −4.71766e9 −0.430003 −0.215001 0.976614i \(-0.568976\pi\)
−0.215001 + 0.976614i \(0.568976\pi\)
\(740\) −1.77149e10 −1.60705
\(741\) −1.11484e10 −1.00659
\(742\) −3.39787e10 −3.05347
\(743\) −8.49980e8 −0.0760235 −0.0380118 0.999277i \(-0.512102\pi\)
−0.0380118 + 0.999277i \(0.512102\pi\)
\(744\) −7.98231e9 −0.710597
\(745\) 9.69968e9 0.859430
\(746\) −4.08753e9 −0.360475
\(747\) −7.01153e9 −0.615447
\(748\) 7.42691e8 0.0648863
\(749\) −2.27076e10 −1.97463
\(750\) 1.13331e10 0.980924
\(751\) 1.12483e9 0.0969050 0.0484525 0.998825i \(-0.484571\pi\)
0.0484525 + 0.998825i \(0.484571\pi\)
\(752\) 4.40571e10 3.77793
\(753\) 2.97667e9 0.254067
\(754\) 3.02492e10 2.56988
\(755\) −3.63924e9 −0.307748
\(756\) 7.33306e9 0.617247
\(757\) −2.61362e9 −0.218981 −0.109491 0.993988i \(-0.534922\pi\)
−0.109491 + 0.993988i \(0.534922\pi\)
\(758\) −3.73632e10 −3.11603
\(759\) 1.61029e9 0.133677
\(760\) −2.41756e10 −1.99770
\(761\) 1.20945e10 0.994817 0.497408 0.867516i \(-0.334285\pi\)
0.497408 + 0.867516i \(0.334285\pi\)
\(762\) 2.03668e10 1.66756
\(763\) −1.78183e10 −1.45221
\(764\) −1.44499e10 −1.17230
\(765\) −8.46610e7 −0.00683705
\(766\) 2.14292e10 1.72269
\(767\) −2.00500e9 −0.160447
\(768\) −6.78985e9 −0.540874
\(769\) 3.67326e9 0.291279 0.145639 0.989338i \(-0.453476\pi\)
0.145639 + 0.989338i \(0.453476\pi\)
\(770\) −1.16449e10 −0.919221
\(771\) −2.16011e9 −0.169740
\(772\) −1.89480e10 −1.48219
\(773\) −2.06426e10 −1.60744 −0.803721 0.595007i \(-0.797149\pi\)
−0.803721 + 0.595007i \(0.797149\pi\)
\(774\) 1.32492e9 0.102706
\(775\) 4.31494e9 0.332981
\(776\) −2.75034e10 −2.11286
\(777\) 1.23570e10 0.945013
\(778\) 1.14262e10 0.869904
\(779\) 1.32762e10 1.00622
\(780\) 1.22543e10 0.924610
\(781\) 1.98831e8 0.0149350
\(782\) 3.13033e8 0.0234081
\(783\) −2.91352e9 −0.216896
\(784\) 2.48412e10 1.84105
\(785\) −6.20677e9 −0.457954
\(786\) 6.51974e9 0.478907
\(787\) 8.84372e7 0.00646731 0.00323365 0.999995i \(-0.498971\pi\)
0.00323365 + 0.999995i \(0.498971\pi\)
\(788\) 2.30454e10 1.67781
\(789\) −3.85678e9 −0.279547
\(790\) 8.38556e8 0.0605114
\(791\) 1.81658e10 1.30508
\(792\) 8.59107e9 0.614482
\(793\) 1.15024e10 0.819090
\(794\) 3.22598e10 2.28713
\(795\) 5.46911e9 0.386039
\(796\) −5.57670e10 −3.91905
\(797\) 1.12406e10 0.786478 0.393239 0.919436i \(-0.371354\pi\)
0.393239 + 0.919436i \(0.371354\pi\)
\(798\) 2.87118e10 2.00009
\(799\) 8.50737e8 0.0590040
\(800\) −1.95819e10 −1.35220
\(801\) 9.18782e9 0.631682
\(802\) −3.93312e10 −2.69232
\(803\) −5.83767e9 −0.397864
\(804\) −1.84484e10 −1.25188
\(805\) −3.47441e9 −0.234745
\(806\) 1.58422e10 1.06572
\(807\) −5.15245e9 −0.345109
\(808\) −1.31226e10 −0.875146
\(809\) 2.81091e10 1.86650 0.933249 0.359230i \(-0.116961\pi\)
0.933249 + 0.359230i \(0.116961\pi\)
\(810\) −1.66737e9 −0.110239
\(811\) 2.83599e8 0.0186695 0.00933474 0.999956i \(-0.497029\pi\)
0.00933474 + 0.999956i \(0.497029\pi\)
\(812\) −5.51469e10 −3.61472
\(813\) 3.71583e9 0.242515
\(814\) 2.46481e10 1.60176
\(815\) 1.33551e10 0.864162
\(816\) 8.39454e8 0.0540856
\(817\) 3.67220e9 0.235586
\(818\) −4.14014e10 −2.64471
\(819\) −8.54795e9 −0.543711
\(820\) −1.45931e10 −0.924272
\(821\) −1.71158e10 −1.07943 −0.539717 0.841846i \(-0.681469\pi\)
−0.539717 + 0.841846i \(0.681469\pi\)
\(822\) −2.13485e10 −1.34065
\(823\) −1.08622e10 −0.679235 −0.339618 0.940564i \(-0.610298\pi\)
−0.339618 + 0.940564i \(0.610298\pi\)
\(824\) −3.05194e10 −1.90034
\(825\) −4.64402e9 −0.287942
\(826\) 5.16369e9 0.318808
\(827\) 1.34081e10 0.824321 0.412161 0.911111i \(-0.364774\pi\)
0.412161 + 0.911111i \(0.364774\pi\)
\(828\) 4.36416e9 0.267174
\(829\) 1.85211e10 1.12908 0.564542 0.825404i \(-0.309053\pi\)
0.564542 + 0.825404i \(0.309053\pi\)
\(830\) 3.01761e10 1.83185
\(831\) 3.19272e9 0.193000
\(832\) −2.17532e10 −1.30946
\(833\) 4.79680e8 0.0287537
\(834\) −2.09863e9 −0.125272
\(835\) −8.64785e9 −0.514050
\(836\) 4.05410e10 2.39978
\(837\) −1.52588e9 −0.0899459
\(838\) 2.28377e10 1.34059
\(839\) −3.61962e9 −0.211591 −0.105795 0.994388i \(-0.533739\pi\)
−0.105795 + 0.994388i \(0.533739\pi\)
\(840\) −1.85364e10 −1.07907
\(841\) 4.66072e9 0.270189
\(842\) 2.58383e10 1.49167
\(843\) −4.39264e9 −0.252539
\(844\) −5.27609e10 −3.02074
\(845\) −4.87967e9 −0.278223
\(846\) 1.67550e10 0.951367
\(847\) −1.19364e10 −0.674965
\(848\) −5.42288e10 −3.05383
\(849\) −1.12058e10 −0.628444
\(850\) −9.02776e8 −0.0504213
\(851\) 7.35406e9 0.409047
\(852\) 5.38868e8 0.0298499
\(853\) −1.71198e10 −0.944448 −0.472224 0.881478i \(-0.656549\pi\)
−0.472224 + 0.881478i \(0.656549\pi\)
\(854\) −2.96233e10 −1.62754
\(855\) −4.62136e9 −0.252865
\(856\) −7.20993e10 −3.92892
\(857\) −1.15451e10 −0.626565 −0.313282 0.949660i \(-0.601429\pi\)
−0.313282 + 0.949660i \(0.601429\pi\)
\(858\) −1.70504e10 −0.921570
\(859\) 4.99390e9 0.268821 0.134411 0.990926i \(-0.457086\pi\)
0.134411 + 0.990926i \(0.457086\pi\)
\(860\) −4.03647e9 −0.216400
\(861\) 1.01794e10 0.543512
\(862\) 4.97090e9 0.264338
\(863\) −1.41398e9 −0.0748867 −0.0374434 0.999299i \(-0.511921\pi\)
−0.0374434 + 0.999299i \(0.511921\pi\)
\(864\) 6.92469e9 0.365260
\(865\) −2.54937e9 −0.133929
\(866\) 6.05589e10 3.16859
\(867\) −1.10629e10 −0.576506
\(868\) −2.88817e10 −1.49901
\(869\) −8.25921e8 −0.0426943
\(870\) 1.25392e10 0.645581
\(871\) 2.15048e10 1.10274
\(872\) −5.65750e10 −2.88946
\(873\) −5.25749e9 −0.267442
\(874\) 1.70874e10 0.865736
\(875\) 2.40843e10 1.21536
\(876\) −1.58211e10 −0.795194
\(877\) −4.65935e9 −0.233253 −0.116626 0.993176i \(-0.537208\pi\)
−0.116626 + 0.993176i \(0.537208\pi\)
\(878\) −2.83786e10 −1.41501
\(879\) 1.24575e10 0.618683
\(880\) −1.85849e10 −0.919329
\(881\) 3.44643e10 1.69806 0.849031 0.528342i \(-0.177186\pi\)
0.849031 + 0.528342i \(0.177186\pi\)
\(882\) 9.44715e9 0.463618
\(883\) −1.49854e10 −0.732498 −0.366249 0.930517i \(-0.619358\pi\)
−0.366249 + 0.930517i \(0.619358\pi\)
\(884\) −2.34630e9 −0.114235
\(885\) −8.31131e8 −0.0403058
\(886\) 2.86766e10 1.38519
\(887\) 2.02649e10 0.975015 0.487508 0.873119i \(-0.337906\pi\)
0.487508 + 0.873119i \(0.337906\pi\)
\(888\) 3.92348e10 1.88029
\(889\) 4.32820e10 2.06610
\(890\) −3.95423e10 −1.88017
\(891\) 1.64225e9 0.0777798
\(892\) −3.91270e10 −1.84586
\(893\) 4.64388e10 2.18223
\(894\) −3.65763e10 −1.71206
\(895\) 5.50641e9 0.256737
\(896\) 1.93596e9 0.0899123
\(897\) −5.08719e9 −0.235345
\(898\) 7.47080e10 3.44271
\(899\) 1.14751e10 0.526741
\(900\) −1.25861e10 −0.575497
\(901\) −1.04715e9 −0.0476950
\(902\) 2.03045e10 0.921233
\(903\) 2.81562e9 0.127253
\(904\) 5.76785e10 2.59672
\(905\) −1.61170e9 −0.0722795
\(906\) 1.37231e10 0.613061
\(907\) 2.55043e10 1.13498 0.567491 0.823380i \(-0.307914\pi\)
0.567491 + 0.823380i \(0.307914\pi\)
\(908\) 5.57122e10 2.46973
\(909\) −2.50849e9 −0.110774
\(910\) 3.67885e10 1.61833
\(911\) −3.56300e10 −1.56135 −0.780677 0.624935i \(-0.785126\pi\)
−0.780677 + 0.624935i \(0.785126\pi\)
\(912\) 4.58229e10 2.00033
\(913\) −2.97214e10 −1.29247
\(914\) −3.32020e10 −1.43831
\(915\) 4.76807e9 0.205764
\(916\) 1.04572e11 4.49554
\(917\) 1.38553e10 0.593366
\(918\) 3.19246e8 0.0136200
\(919\) −2.41602e10 −1.02682 −0.513412 0.858143i \(-0.671619\pi\)
−0.513412 + 0.858143i \(0.671619\pi\)
\(920\) −1.10317e10 −0.467072
\(921\) 1.96485e10 0.828742
\(922\) −5.08462e10 −2.13649
\(923\) −6.28143e8 −0.0262938
\(924\) 3.10844e10 1.29625
\(925\) −2.12089e10 −0.881093
\(926\) −4.32242e10 −1.78891
\(927\) −5.83402e9 −0.240541
\(928\) −5.20758e10 −2.13904
\(929\) 1.41353e9 0.0578427 0.0289214 0.999582i \(-0.490793\pi\)
0.0289214 + 0.999582i \(0.490793\pi\)
\(930\) 6.56705e9 0.267719
\(931\) 2.61841e10 1.06344
\(932\) 6.30915e10 2.55279
\(933\) −4.25090e9 −0.171354
\(934\) −4.79091e10 −1.92400
\(935\) −3.58872e8 −0.0143582
\(936\) −2.71407e10 −1.08182
\(937\) 3.20821e10 1.27402 0.637008 0.770857i \(-0.280172\pi\)
0.637008 + 0.770857i \(0.280172\pi\)
\(938\) −5.53835e10 −2.19114
\(939\) −4.82636e9 −0.190235
\(940\) −5.10454e10 −2.00451
\(941\) −4.82625e10 −1.88819 −0.944096 0.329671i \(-0.893062\pi\)
−0.944096 + 0.329671i \(0.893062\pi\)
\(942\) 2.34049e10 0.912282
\(943\) 6.05809e9 0.235258
\(944\) 8.24105e9 0.318846
\(945\) −3.54337e9 −0.136586
\(946\) 5.61625e9 0.215689
\(947\) 3.71988e9 0.142333 0.0711663 0.997464i \(-0.477328\pi\)
0.0711663 + 0.997464i \(0.477328\pi\)
\(948\) −2.23839e9 −0.0853311
\(949\) 1.84423e10 0.700458
\(950\) −4.92795e10 −1.86480
\(951\) 1.54654e10 0.583083
\(952\) 3.54910e9 0.133318
\(953\) −3.84659e10 −1.43963 −0.719814 0.694167i \(-0.755773\pi\)
−0.719814 + 0.694167i \(0.755773\pi\)
\(954\) −2.06233e10 −0.769022
\(955\) 6.98225e9 0.259408
\(956\) 7.46667e10 2.76391
\(957\) −1.23502e10 −0.455494
\(958\) −2.00903e10 −0.738257
\(959\) −4.53682e10 −1.66107
\(960\) −9.01733e9 −0.328949
\(961\) −2.15028e10 −0.781563
\(962\) −7.78678e10 −2.81997
\(963\) −1.37823e10 −0.497314
\(964\) 1.94905e10 0.700733
\(965\) 9.15579e9 0.327982
\(966\) 1.31016e10 0.467631
\(967\) −1.01210e10 −0.359942 −0.179971 0.983672i \(-0.557600\pi\)
−0.179971 + 0.983672i \(0.557600\pi\)
\(968\) −3.78994e10 −1.34298
\(969\) 8.84835e8 0.0312413
\(970\) 2.26271e10 0.796027
\(971\) 3.40438e10 1.19336 0.596679 0.802480i \(-0.296487\pi\)
0.596679 + 0.802480i \(0.296487\pi\)
\(972\) 4.45079e9 0.155455
\(973\) −4.45985e9 −0.155212
\(974\) 1.83899e10 0.637710
\(975\) 1.46713e10 0.506935
\(976\) −4.72776e10 −1.62773
\(977\) −1.56624e10 −0.537314 −0.268657 0.963236i \(-0.586580\pi\)
−0.268657 + 0.963236i \(0.586580\pi\)
\(978\) −5.03603e10 −1.72148
\(979\) 3.89465e10 1.32657
\(980\) −2.87815e10 −0.976836
\(981\) −1.08148e10 −0.365742
\(982\) 2.45213e10 0.826330
\(983\) −3.86584e10 −1.29810 −0.649048 0.760747i \(-0.724833\pi\)
−0.649048 + 0.760747i \(0.724833\pi\)
\(984\) 3.23206e10 1.08143
\(985\) −1.11356e10 −0.371269
\(986\) −2.40083e9 −0.0797613
\(987\) 3.56065e10 1.17874
\(988\) −1.28076e11 −4.22493
\(989\) 1.67568e9 0.0550811
\(990\) −7.06788e9 −0.231508
\(991\) 1.35643e10 0.442731 0.221365 0.975191i \(-0.428949\pi\)
0.221365 + 0.975191i \(0.428949\pi\)
\(992\) −2.72733e10 −0.887048
\(993\) 7.85428e9 0.254556
\(994\) 1.61772e9 0.0522459
\(995\) 2.69469e10 0.867217
\(996\) −8.05503e10 −2.58321
\(997\) −4.26350e10 −1.36249 −0.681245 0.732055i \(-0.738561\pi\)
−0.681245 + 0.732055i \(0.738561\pi\)
\(998\) 7.80113e10 2.48428
\(999\) 7.50003e9 0.238004
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.d.1.17 18
3.2 odd 2 531.8.a.e.1.2 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.17 18 1.1 even 1 trivial
531.8.a.e.1.2 18 3.2 odd 2