Properties

Label 177.8.a.d.1.8
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 1.8
Root \(-6.28523\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.28523 q^{2} +27.0000 q^{3} -100.066 q^{4} +498.961 q^{5} -142.701 q^{6} +341.176 q^{7} +1205.38 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-5.28523 q^{2} +27.0000 q^{3} -100.066 q^{4} +498.961 q^{5} -142.701 q^{6} +341.176 q^{7} +1205.38 q^{8} +729.000 q^{9} -2637.13 q^{10} +2091.72 q^{11} -2701.79 q^{12} +4481.72 q^{13} -1803.19 q^{14} +13472.0 q^{15} +6437.77 q^{16} +22387.5 q^{17} -3852.93 q^{18} +6302.72 q^{19} -49929.3 q^{20} +9211.75 q^{21} -11055.2 q^{22} +8949.56 q^{23} +32545.3 q^{24} +170838. q^{25} -23686.9 q^{26} +19683.0 q^{27} -34140.2 q^{28} -232539. q^{29} -71202.4 q^{30} -109526. q^{31} -188314. q^{32} +56476.3 q^{33} -118323. q^{34} +170234. q^{35} -72948.4 q^{36} +123858. q^{37} -33311.3 q^{38} +121006. q^{39} +601440. q^{40} +133392. q^{41} -48686.2 q^{42} +326147. q^{43} -209310. q^{44} +363743. q^{45} -47300.5 q^{46} -479181. q^{47} +173820. q^{48} -707142. q^{49} -902916. q^{50} +604464. q^{51} -448469. q^{52} -906342. q^{53} -104029. q^{54} +1.04369e6 q^{55} +411248. q^{56} +170173. q^{57} +1.22902e6 q^{58} +205379. q^{59} -1.34809e6 q^{60} +3.23653e6 q^{61} +578868. q^{62} +248717. q^{63} +171249. q^{64} +2.23620e6 q^{65} -298490. q^{66} +1.18481e6 q^{67} -2.24024e6 q^{68} +241638. q^{69} -899724. q^{70} +1.35464e6 q^{71} +878724. q^{72} -5.84208e6 q^{73} -654616. q^{74} +4.61261e6 q^{75} -630690. q^{76} +713643. q^{77} -639546. q^{78} +4.69904e6 q^{79} +3.21220e6 q^{80} +531441. q^{81} -705006. q^{82} +1.26796e6 q^{83} -921786. q^{84} +1.11705e7 q^{85} -1.72376e6 q^{86} -6.27856e6 q^{87} +2.52132e6 q^{88} +2.48381e6 q^{89} -1.92246e6 q^{90} +1.52905e6 q^{91} -895550. q^{92} -2.95719e6 q^{93} +2.53258e6 q^{94} +3.14481e6 q^{95} -5.08448e6 q^{96} +4.51095e6 q^{97} +3.73741e6 q^{98} +1.52486e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + O(q^{10}) \) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + 3609q^{10} + 15070q^{11} + 36666q^{12} + 13662q^{13} + 20861q^{14} + 18306q^{15} + 60482q^{16} + 71919q^{17} + 17496q^{18} + 56231q^{19} + 143053q^{20} + 83187q^{21} + 274198q^{22} + 150029q^{23} + 110889q^{24} + 399672q^{25} + 182846q^{26} + 354294q^{27} + 434150q^{28} + 591285q^{29} + 97443q^{30} + 426733q^{31} + 1205630q^{32} + 406890q^{33} + 403548q^{34} + 912879q^{35} + 989982q^{36} + 7703q^{37} - 417859q^{38} + 368874q^{39} + 618020q^{40} + 770959q^{41} + 563247q^{42} + 793050q^{43} + 2591274q^{44} + 494262q^{45} - 4068019q^{46} + 1410373q^{47} + 1633014q^{48} + 1637427q^{49} + 1021549q^{50} + 1941813q^{51} - 3749190q^{52} + 1037934q^{53} + 472392q^{54} + 331974q^{55} - 391748q^{56} + 1518237q^{57} + 653724q^{58} + 3696822q^{59} + 3862431q^{60} - 1374623q^{61} + 5251718q^{62} + 2246049q^{63} + 5077197q^{64} + 3257170q^{65} + 7403346q^{66} - 2436904q^{67} + 14119909q^{68} + 4050783q^{69} + 5185580q^{70} + 14289172q^{71} + 2994003q^{72} + 5482515q^{73} + 14934154q^{74} + 10791144q^{75} + 3822912q^{76} + 23157109q^{77} + 4936842q^{78} + 19786414q^{79} + 31978143q^{80} + 9565938q^{81} + 9749509q^{82} + 30227337q^{83} + 11722050q^{84} + 9946981q^{85} + 44295864q^{86} + 15964695q^{87} + 39970897q^{88} + 31061677q^{89} + 2630961q^{90} + 26377785q^{91} + 4719698q^{92} + 11521791q^{93} + 44488296q^{94} + 15534599q^{95} + 32552010q^{96} + 12084118q^{97} + 42274744q^{98} + 10986030q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.28523 −0.467153 −0.233576 0.972338i \(-0.575043\pi\)
−0.233576 + 0.972338i \(0.575043\pi\)
\(3\) 27.0000 0.577350
\(4\) −100.066 −0.781768
\(5\) 498.961 1.78514 0.892569 0.450910i \(-0.148900\pi\)
0.892569 + 0.450910i \(0.148900\pi\)
\(6\) −142.701 −0.269711
\(7\) 341.176 0.375954 0.187977 0.982173i \(-0.439807\pi\)
0.187977 + 0.982173i \(0.439807\pi\)
\(8\) 1205.38 0.832358
\(9\) 729.000 0.333333
\(10\) −2637.13 −0.833932
\(11\) 2091.72 0.473836 0.236918 0.971530i \(-0.423863\pi\)
0.236918 + 0.971530i \(0.423863\pi\)
\(12\) −2701.79 −0.451354
\(13\) 4481.72 0.565774 0.282887 0.959153i \(-0.408708\pi\)
0.282887 + 0.959153i \(0.408708\pi\)
\(14\) −1803.19 −0.175628
\(15\) 13472.0 1.03065
\(16\) 6437.77 0.392930
\(17\) 22387.5 1.10519 0.552593 0.833451i \(-0.313639\pi\)
0.552593 + 0.833451i \(0.313639\pi\)
\(18\) −3852.93 −0.155718
\(19\) 6302.72 0.210809 0.105405 0.994429i \(-0.466386\pi\)
0.105405 + 0.994429i \(0.466386\pi\)
\(20\) −49929.3 −1.39557
\(21\) 9211.75 0.217057
\(22\) −11055.2 −0.221354
\(23\) 8949.56 0.153375 0.0766875 0.997055i \(-0.475566\pi\)
0.0766875 + 0.997055i \(0.475566\pi\)
\(24\) 32545.3 0.480562
\(25\) 170838. 2.18672
\(26\) −23686.9 −0.264303
\(27\) 19683.0 0.192450
\(28\) −34140.2 −0.293909
\(29\) −232539. −1.77053 −0.885265 0.465087i \(-0.846023\pi\)
−0.885265 + 0.465087i \(0.846023\pi\)
\(30\) −71202.4 −0.481471
\(31\) −109526. −0.660312 −0.330156 0.943926i \(-0.607101\pi\)
−0.330156 + 0.943926i \(0.607101\pi\)
\(32\) −188314. −1.01592
\(33\) 56476.3 0.273569
\(34\) −118323. −0.516290
\(35\) 170234. 0.671131
\(36\) −72948.4 −0.260589
\(37\) 123858. 0.401992 0.200996 0.979592i \(-0.435582\pi\)
0.200996 + 0.979592i \(0.435582\pi\)
\(38\) −33311.3 −0.0984802
\(39\) 121006. 0.326650
\(40\) 601440. 1.48587
\(41\) 133392. 0.302263 0.151132 0.988514i \(-0.451708\pi\)
0.151132 + 0.988514i \(0.451708\pi\)
\(42\) −48686.2 −0.101399
\(43\) 326147. 0.625567 0.312784 0.949824i \(-0.398739\pi\)
0.312784 + 0.949824i \(0.398739\pi\)
\(44\) −209310. −0.370430
\(45\) 363743. 0.595046
\(46\) −47300.5 −0.0716495
\(47\) −479181. −0.673221 −0.336610 0.941644i \(-0.609280\pi\)
−0.336610 + 0.941644i \(0.609280\pi\)
\(48\) 173820. 0.226858
\(49\) −707142. −0.858658
\(50\) −902916. −1.02153
\(51\) 604464. 0.638079
\(52\) −448469. −0.442304
\(53\) −906342. −0.836231 −0.418116 0.908394i \(-0.637309\pi\)
−0.418116 + 0.908394i \(0.637309\pi\)
\(54\) −104029. −0.0899036
\(55\) 1.04369e6 0.845863
\(56\) 411248. 0.312929
\(57\) 170173. 0.121711
\(58\) 1.22902e6 0.827108
\(59\) 205379. 0.130189
\(60\) −1.34809e6 −0.805730
\(61\) 3.23653e6 1.82568 0.912842 0.408312i \(-0.133883\pi\)
0.912842 + 0.408312i \(0.133883\pi\)
\(62\) 578868. 0.308467
\(63\) 248717. 0.125318
\(64\) 171249. 0.0816578
\(65\) 2.23620e6 1.00998
\(66\) −298490. −0.127799
\(67\) 1.18481e6 0.481267 0.240633 0.970616i \(-0.422645\pi\)
0.240633 + 0.970616i \(0.422645\pi\)
\(68\) −2.24024e6 −0.863999
\(69\) 241638. 0.0885511
\(70\) −899724. −0.313521
\(71\) 1.35464e6 0.449181 0.224590 0.974453i \(-0.427896\pi\)
0.224590 + 0.974453i \(0.427896\pi\)
\(72\) 878724. 0.277453
\(73\) −5.84208e6 −1.75767 −0.878836 0.477125i \(-0.841679\pi\)
−0.878836 + 0.477125i \(0.841679\pi\)
\(74\) −654616. −0.187791
\(75\) 4.61261e6 1.26250
\(76\) −630690. −0.164804
\(77\) 713643. 0.178141
\(78\) −639546. −0.152595
\(79\) 4.69904e6 1.07230 0.536148 0.844124i \(-0.319879\pi\)
0.536148 + 0.844124i \(0.319879\pi\)
\(80\) 3.21220e6 0.701435
\(81\) 531441. 0.111111
\(82\) −705006. −0.141203
\(83\) 1.26796e6 0.243406 0.121703 0.992567i \(-0.461165\pi\)
0.121703 + 0.992567i \(0.461165\pi\)
\(84\) −921786. −0.169689
\(85\) 1.11705e7 1.97291
\(86\) −1.72376e6 −0.292235
\(87\) −6.27856e6 −1.02222
\(88\) 2.52132e6 0.394401
\(89\) 2.48381e6 0.373468 0.186734 0.982411i \(-0.440210\pi\)
0.186734 + 0.982411i \(0.440210\pi\)
\(90\) −1.92246e6 −0.277977
\(91\) 1.52905e6 0.212705
\(92\) −895550. −0.119904
\(93\) −2.95719e6 −0.381232
\(94\) 2.53258e6 0.314497
\(95\) 3.14481e6 0.376324
\(96\) −5.08448e6 −0.586540
\(97\) 4.51095e6 0.501841 0.250921 0.968008i \(-0.419267\pi\)
0.250921 + 0.968008i \(0.419267\pi\)
\(98\) 3.73741e6 0.401125
\(99\) 1.52486e6 0.157945
\(100\) −1.70951e7 −1.70951
\(101\) −812528. −0.0784718 −0.0392359 0.999230i \(-0.512492\pi\)
−0.0392359 + 0.999230i \(0.512492\pi\)
\(102\) −3.19473e6 −0.298080
\(103\) 1.18581e7 1.06926 0.534630 0.845086i \(-0.320451\pi\)
0.534630 + 0.845086i \(0.320451\pi\)
\(104\) 5.40219e6 0.470926
\(105\) 4.59631e6 0.387478
\(106\) 4.79022e6 0.390648
\(107\) 1.53725e7 1.21311 0.606555 0.795041i \(-0.292551\pi\)
0.606555 + 0.795041i \(0.292551\pi\)
\(108\) −1.96961e6 −0.150451
\(109\) 68968.8 0.00510106 0.00255053 0.999997i \(-0.499188\pi\)
0.00255053 + 0.999997i \(0.499188\pi\)
\(110\) −5.51612e6 −0.395147
\(111\) 3.34416e6 0.232090
\(112\) 2.19641e6 0.147724
\(113\) 2.00851e7 1.30948 0.654740 0.755854i \(-0.272778\pi\)
0.654740 + 0.755854i \(0.272778\pi\)
\(114\) −899405. −0.0568576
\(115\) 4.46549e6 0.273796
\(116\) 2.32694e7 1.38414
\(117\) 3.26717e6 0.188591
\(118\) −1.08548e6 −0.0608181
\(119\) 7.63809e6 0.415499
\(120\) 1.62389e7 0.857870
\(121\) −1.51119e7 −0.775479
\(122\) −1.71058e7 −0.852873
\(123\) 3.60158e6 0.174512
\(124\) 1.09598e7 0.516211
\(125\) 4.62600e7 2.11846
\(126\) −1.31453e6 −0.0585427
\(127\) 2.70641e6 0.117241 0.0586207 0.998280i \(-0.481330\pi\)
0.0586207 + 0.998280i \(0.481330\pi\)
\(128\) 2.31991e7 0.977770
\(129\) 8.80597e6 0.361171
\(130\) −1.18189e7 −0.471817
\(131\) 2.11862e6 0.0823386 0.0411693 0.999152i \(-0.486892\pi\)
0.0411693 + 0.999152i \(0.486892\pi\)
\(132\) −5.65138e6 −0.213868
\(133\) 2.15033e6 0.0792547
\(134\) −6.26198e6 −0.224825
\(135\) 9.82106e6 0.343550
\(136\) 2.69856e7 0.919910
\(137\) 3.48905e7 1.15927 0.579636 0.814875i \(-0.303195\pi\)
0.579636 + 0.814875i \(0.303195\pi\)
\(138\) −1.27711e6 −0.0413669
\(139\) −3.13281e7 −0.989422 −0.494711 0.869058i \(-0.664726\pi\)
−0.494711 + 0.869058i \(0.664726\pi\)
\(140\) −1.70347e7 −0.524669
\(141\) −1.29379e7 −0.388684
\(142\) −7.15960e6 −0.209836
\(143\) 9.37448e6 0.268084
\(144\) 4.69313e6 0.130977
\(145\) −1.16028e8 −3.16064
\(146\) 3.08767e7 0.821101
\(147\) −1.90928e7 −0.495747
\(148\) −1.23940e7 −0.314264
\(149\) −1.74202e6 −0.0431421 −0.0215710 0.999767i \(-0.506867\pi\)
−0.0215710 + 0.999767i \(0.506867\pi\)
\(150\) −2.43787e7 −0.589782
\(151\) −2.12520e7 −0.502321 −0.251160 0.967945i \(-0.580812\pi\)
−0.251160 + 0.967945i \(0.580812\pi\)
\(152\) 7.59719e6 0.175469
\(153\) 1.63205e7 0.368395
\(154\) −3.77177e6 −0.0832189
\(155\) −5.46490e7 −1.17875
\(156\) −1.21087e7 −0.255364
\(157\) −2.17178e7 −0.447886 −0.223943 0.974602i \(-0.571893\pi\)
−0.223943 + 0.974602i \(0.571893\pi\)
\(158\) −2.48355e7 −0.500926
\(159\) −2.44712e7 −0.482798
\(160\) −9.39615e7 −1.81355
\(161\) 3.05338e6 0.0576620
\(162\) −2.80879e6 −0.0519059
\(163\) −5.00693e7 −0.905556 −0.452778 0.891623i \(-0.649567\pi\)
−0.452778 + 0.891623i \(0.649567\pi\)
\(164\) −1.33480e7 −0.236300
\(165\) 2.81795e7 0.488359
\(166\) −6.70143e6 −0.113708
\(167\) 3.77408e7 0.627052 0.313526 0.949580i \(-0.398490\pi\)
0.313526 + 0.949580i \(0.398490\pi\)
\(168\) 1.11037e7 0.180669
\(169\) −4.26627e7 −0.679900
\(170\) −5.90388e7 −0.921650
\(171\) 4.59468e6 0.0702698
\(172\) −3.26364e7 −0.489049
\(173\) −1.26205e8 −1.85317 −0.926584 0.376087i \(-0.877269\pi\)
−0.926584 + 0.376087i \(0.877269\pi\)
\(174\) 3.31836e7 0.477531
\(175\) 5.82857e7 0.822107
\(176\) 1.34660e7 0.186184
\(177\) 5.54523e6 0.0751646
\(178\) −1.31275e7 −0.174467
\(179\) −8.64515e7 −1.12664 −0.563322 0.826237i \(-0.690477\pi\)
−0.563322 + 0.826237i \(0.690477\pi\)
\(180\) −3.63984e7 −0.465188
\(181\) 1.43002e7 0.179254 0.0896268 0.995975i \(-0.471433\pi\)
0.0896268 + 0.995975i \(0.471433\pi\)
\(182\) −8.08140e6 −0.0993658
\(183\) 8.73864e7 1.05406
\(184\) 1.07877e7 0.127663
\(185\) 6.18002e7 0.717611
\(186\) 1.56294e7 0.178093
\(187\) 4.68284e7 0.523677
\(188\) 4.79499e7 0.526303
\(189\) 6.71536e6 0.0723525
\(190\) −1.66211e7 −0.175801
\(191\) −1.45821e8 −1.51427 −0.757136 0.653257i \(-0.773402\pi\)
−0.757136 + 0.653257i \(0.773402\pi\)
\(192\) 4.62372e6 0.0471452
\(193\) −1.60189e8 −1.60391 −0.801957 0.597381i \(-0.796208\pi\)
−0.801957 + 0.597381i \(0.796208\pi\)
\(194\) −2.38414e7 −0.234437
\(195\) 6.03775e7 0.583115
\(196\) 7.07611e7 0.671272
\(197\) −1.81066e7 −0.168735 −0.0843677 0.996435i \(-0.526887\pi\)
−0.0843677 + 0.996435i \(0.526887\pi\)
\(198\) −8.05924e6 −0.0737846
\(199\) 2.17069e8 1.95260 0.976298 0.216430i \(-0.0694412\pi\)
0.976298 + 0.216430i \(0.0694412\pi\)
\(200\) 2.05925e8 1.82013
\(201\) 3.19898e7 0.277859
\(202\) 4.29440e6 0.0366583
\(203\) −7.93368e7 −0.665638
\(204\) −6.04865e7 −0.498830
\(205\) 6.65573e7 0.539582
\(206\) −6.26726e7 −0.499508
\(207\) 6.52423e6 0.0511250
\(208\) 2.88523e7 0.222310
\(209\) 1.31835e7 0.0998891
\(210\) −2.42925e7 −0.181011
\(211\) −1.96840e7 −0.144253 −0.0721266 0.997395i \(-0.522979\pi\)
−0.0721266 + 0.997395i \(0.522979\pi\)
\(212\) 9.06943e7 0.653739
\(213\) 3.65754e7 0.259335
\(214\) −8.12470e7 −0.566708
\(215\) 1.62735e8 1.11672
\(216\) 2.37256e7 0.160187
\(217\) −3.73675e7 −0.248247
\(218\) −364516. −0.00238297
\(219\) −1.57736e8 −1.01479
\(220\) −1.04438e8 −0.661269
\(221\) 1.00335e8 0.625285
\(222\) −1.76746e7 −0.108421
\(223\) 1.80353e8 1.08907 0.544536 0.838738i \(-0.316706\pi\)
0.544536 + 0.838738i \(0.316706\pi\)
\(224\) −6.42482e7 −0.381938
\(225\) 1.24541e8 0.728907
\(226\) −1.06154e8 −0.611727
\(227\) −1.17899e8 −0.668993 −0.334496 0.942397i \(-0.608566\pi\)
−0.334496 + 0.942397i \(0.608566\pi\)
\(228\) −1.70286e7 −0.0951497
\(229\) 7.05269e7 0.388089 0.194044 0.980993i \(-0.437839\pi\)
0.194044 + 0.980993i \(0.437839\pi\)
\(230\) −2.36011e7 −0.127904
\(231\) 1.92684e7 0.102850
\(232\) −2.80299e8 −1.47371
\(233\) −2.32980e8 −1.20663 −0.603313 0.797505i \(-0.706153\pi\)
−0.603313 + 0.797505i \(0.706153\pi\)
\(234\) −1.72678e7 −0.0881009
\(235\) −2.39093e8 −1.20179
\(236\) −2.05515e7 −0.101778
\(237\) 1.26874e8 0.619090
\(238\) −4.03690e7 −0.194102
\(239\) 4.21184e8 1.99562 0.997812 0.0661120i \(-0.0210595\pi\)
0.997812 + 0.0661120i \(0.0210595\pi\)
\(240\) 8.67294e7 0.404974
\(241\) 1.65428e8 0.761291 0.380645 0.924721i \(-0.375702\pi\)
0.380645 + 0.924721i \(0.375702\pi\)
\(242\) 7.98699e7 0.362267
\(243\) 1.43489e7 0.0641500
\(244\) −3.23868e8 −1.42726
\(245\) −3.52837e8 −1.53282
\(246\) −1.90352e7 −0.0815237
\(247\) 2.82470e7 0.119270
\(248\) −1.32020e8 −0.549616
\(249\) 3.42348e7 0.140530
\(250\) −2.44495e8 −0.989645
\(251\) 4.33192e8 1.72911 0.864554 0.502540i \(-0.167601\pi\)
0.864554 + 0.502540i \(0.167601\pi\)
\(252\) −2.48882e7 −0.0979698
\(253\) 1.87199e7 0.0726746
\(254\) −1.43040e7 −0.0547696
\(255\) 3.01604e8 1.13906
\(256\) −1.44533e8 −0.538426
\(257\) −1.31387e8 −0.482821 −0.241411 0.970423i \(-0.577610\pi\)
−0.241411 + 0.970423i \(0.577610\pi\)
\(258\) −4.65416e7 −0.168722
\(259\) 4.22573e7 0.151131
\(260\) −2.23769e8 −0.789574
\(261\) −1.69521e8 −0.590177
\(262\) −1.11974e7 −0.0384647
\(263\) −2.55280e8 −0.865311 −0.432655 0.901559i \(-0.642423\pi\)
−0.432655 + 0.901559i \(0.642423\pi\)
\(264\) 6.80756e7 0.227708
\(265\) −4.52230e8 −1.49279
\(266\) −1.13650e7 −0.0370241
\(267\) 6.70629e7 0.215622
\(268\) −1.18559e8 −0.376239
\(269\) −1.16484e8 −0.364867 −0.182434 0.983218i \(-0.558397\pi\)
−0.182434 + 0.983218i \(0.558397\pi\)
\(270\) −5.19065e7 −0.160490
\(271\) 2.19711e8 0.670593 0.335297 0.942113i \(-0.391164\pi\)
0.335297 + 0.942113i \(0.391164\pi\)
\(272\) 1.44126e8 0.434261
\(273\) 4.12845e7 0.122805
\(274\) −1.84404e8 −0.541557
\(275\) 3.57344e8 1.03615
\(276\) −2.41799e7 −0.0692264
\(277\) −2.25909e8 −0.638638 −0.319319 0.947647i \(-0.603454\pi\)
−0.319319 + 0.947647i \(0.603454\pi\)
\(278\) 1.65576e8 0.462211
\(279\) −7.98441e7 −0.220104
\(280\) 2.05197e8 0.558621
\(281\) 2.43072e8 0.653527 0.326763 0.945106i \(-0.394042\pi\)
0.326763 + 0.945106i \(0.394042\pi\)
\(282\) 6.83797e7 0.181575
\(283\) 2.39190e6 0.00627323 0.00313662 0.999995i \(-0.499002\pi\)
0.00313662 + 0.999995i \(0.499002\pi\)
\(284\) −1.35554e8 −0.351155
\(285\) 8.49099e7 0.217271
\(286\) −4.95463e7 −0.125236
\(287\) 4.55100e7 0.113637
\(288\) −1.37281e8 −0.338639
\(289\) 9.08632e7 0.221435
\(290\) 6.13235e8 1.47650
\(291\) 1.21796e8 0.289738
\(292\) 5.84596e8 1.37409
\(293\) −5.43356e8 −1.26197 −0.630984 0.775796i \(-0.717349\pi\)
−0.630984 + 0.775796i \(0.717349\pi\)
\(294\) 1.00910e8 0.231589
\(295\) 1.02476e8 0.232405
\(296\) 1.49296e8 0.334601
\(297\) 4.11712e7 0.0911898
\(298\) 9.20697e6 0.0201539
\(299\) 4.01094e7 0.0867755
\(300\) −4.61568e8 −0.986986
\(301\) 1.11274e8 0.235185
\(302\) 1.12322e8 0.234661
\(303\) −2.19383e7 −0.0453057
\(304\) 4.05754e7 0.0828334
\(305\) 1.61491e9 3.25910
\(306\) −8.62577e7 −0.172097
\(307\) −2.04302e8 −0.402985 −0.201492 0.979490i \(-0.564579\pi\)
−0.201492 + 0.979490i \(0.564579\pi\)
\(308\) −7.14116e7 −0.139265
\(309\) 3.20168e8 0.617338
\(310\) 2.88833e8 0.550656
\(311\) 4.81659e8 0.907984 0.453992 0.891006i \(-0.349999\pi\)
0.453992 + 0.891006i \(0.349999\pi\)
\(312\) 1.45859e8 0.271889
\(313\) −1.77600e8 −0.327369 −0.163684 0.986513i \(-0.552338\pi\)
−0.163684 + 0.986513i \(0.552338\pi\)
\(314\) 1.14784e8 0.209231
\(315\) 1.24100e8 0.223710
\(316\) −4.70216e8 −0.838287
\(317\) 4.19257e8 0.739218 0.369609 0.929187i \(-0.379492\pi\)
0.369609 + 0.929187i \(0.379492\pi\)
\(318\) 1.29336e8 0.225541
\(319\) −4.86406e8 −0.838941
\(320\) 8.54466e7 0.145771
\(321\) 4.15057e8 0.700390
\(322\) −1.61378e7 −0.0269370
\(323\) 1.41102e8 0.232984
\(324\) −5.31794e7 −0.0868632
\(325\) 7.65646e8 1.23719
\(326\) 2.64628e8 0.423033
\(327\) 1.86216e6 0.00294510
\(328\) 1.60788e8 0.251591
\(329\) −1.63485e8 −0.253100
\(330\) −1.48935e8 −0.228138
\(331\) −6.53330e8 −0.990226 −0.495113 0.868829i \(-0.664873\pi\)
−0.495113 + 0.868829i \(0.664873\pi\)
\(332\) −1.26880e8 −0.190287
\(333\) 9.02923e7 0.133997
\(334\) −1.99469e8 −0.292929
\(335\) 5.91173e8 0.859128
\(336\) 5.93031e7 0.0852884
\(337\) 2.22015e8 0.315993 0.157997 0.987440i \(-0.449497\pi\)
0.157997 + 0.987440i \(0.449497\pi\)
\(338\) 2.25482e8 0.317617
\(339\) 5.42297e8 0.756029
\(340\) −1.11779e9 −1.54236
\(341\) −2.29096e8 −0.312880
\(342\) −2.42839e7 −0.0328267
\(343\) −5.22233e8 −0.698771
\(344\) 3.93132e8 0.520696
\(345\) 1.20568e8 0.158076
\(346\) 6.67022e8 0.865713
\(347\) −7.82082e8 −1.00485 −0.502423 0.864622i \(-0.667558\pi\)
−0.502423 + 0.864622i \(0.667558\pi\)
\(348\) 6.28273e8 0.799136
\(349\) 2.24931e8 0.283243 0.141622 0.989921i \(-0.454768\pi\)
0.141622 + 0.989921i \(0.454768\pi\)
\(350\) −3.08053e8 −0.384050
\(351\) 8.82136e7 0.108883
\(352\) −3.93899e8 −0.481378
\(353\) −9.53452e8 −1.15368 −0.576842 0.816855i \(-0.695715\pi\)
−0.576842 + 0.816855i \(0.695715\pi\)
\(354\) −2.93078e7 −0.0351133
\(355\) 6.75915e8 0.801850
\(356\) −2.48546e8 −0.291965
\(357\) 2.06228e8 0.239889
\(358\) 4.56916e8 0.526315
\(359\) 8.80036e8 1.00385 0.501926 0.864910i \(-0.332625\pi\)
0.501926 + 0.864910i \(0.332625\pi\)
\(360\) 4.38450e8 0.495291
\(361\) −8.54148e8 −0.955559
\(362\) −7.55800e7 −0.0837388
\(363\) −4.08021e8 −0.447723
\(364\) −1.53007e8 −0.166286
\(365\) −2.91497e9 −3.13769
\(366\) −4.61857e8 −0.492407
\(367\) −4.42919e8 −0.467727 −0.233864 0.972269i \(-0.575137\pi\)
−0.233864 + 0.972269i \(0.575137\pi\)
\(368\) 5.76152e7 0.0602656
\(369\) 9.72426e7 0.100754
\(370\) −3.26628e8 −0.335234
\(371\) −3.09222e8 −0.314385
\(372\) 2.95915e8 0.298035
\(373\) 6.56422e8 0.654942 0.327471 0.944861i \(-0.393804\pi\)
0.327471 + 0.944861i \(0.393804\pi\)
\(374\) −2.47499e8 −0.244637
\(375\) 1.24902e9 1.22309
\(376\) −5.77597e8 −0.560360
\(377\) −1.04218e9 −1.00172
\(378\) −3.54922e7 −0.0337996
\(379\) 4.00683e7 0.0378062 0.0189031 0.999821i \(-0.493983\pi\)
0.0189031 + 0.999821i \(0.493983\pi\)
\(380\) −3.14690e8 −0.294198
\(381\) 7.30731e7 0.0676893
\(382\) 7.70699e8 0.707397
\(383\) 4.34731e8 0.395389 0.197694 0.980264i \(-0.436655\pi\)
0.197694 + 0.980264i \(0.436655\pi\)
\(384\) 6.26376e8 0.564516
\(385\) 3.56080e8 0.318006
\(386\) 8.46634e8 0.749273
\(387\) 2.37761e8 0.208522
\(388\) −4.51394e8 −0.392324
\(389\) −7.97933e8 −0.687294 −0.343647 0.939099i \(-0.611662\pi\)
−0.343647 + 0.939099i \(0.611662\pi\)
\(390\) −3.19109e8 −0.272404
\(391\) 2.00359e8 0.169508
\(392\) −8.52377e8 −0.714711
\(393\) 5.72027e7 0.0475382
\(394\) 9.56977e7 0.0788252
\(395\) 2.34464e9 1.91420
\(396\) −1.52587e8 −0.123477
\(397\) 2.94069e8 0.235875 0.117938 0.993021i \(-0.462372\pi\)
0.117938 + 0.993021i \(0.462372\pi\)
\(398\) −1.14726e9 −0.912161
\(399\) 5.80590e7 0.0457577
\(400\) 1.09981e9 0.859229
\(401\) 1.98521e9 1.53745 0.768724 0.639580i \(-0.220892\pi\)
0.768724 + 0.639580i \(0.220892\pi\)
\(402\) −1.69073e8 −0.129803
\(403\) −4.90863e8 −0.373587
\(404\) 8.13067e7 0.0613468
\(405\) 2.65169e8 0.198349
\(406\) 4.19313e8 0.310955
\(407\) 2.59075e8 0.190478
\(408\) 7.28610e8 0.531110
\(409\) 8.07703e8 0.583741 0.291871 0.956458i \(-0.405722\pi\)
0.291871 + 0.956458i \(0.405722\pi\)
\(410\) −3.51771e8 −0.252067
\(411\) 9.42044e8 0.669306
\(412\) −1.18659e9 −0.835914
\(413\) 7.00704e7 0.0489451
\(414\) −3.44821e7 −0.0238832
\(415\) 6.32661e8 0.434513
\(416\) −8.43971e8 −0.574779
\(417\) −8.45857e8 −0.571243
\(418\) −6.96777e7 −0.0466635
\(419\) 1.07006e9 0.710653 0.355326 0.934742i \(-0.384370\pi\)
0.355326 + 0.934742i \(0.384370\pi\)
\(420\) −4.59936e8 −0.302918
\(421\) −1.95522e9 −1.27705 −0.638526 0.769600i \(-0.720456\pi\)
−0.638526 + 0.769600i \(0.720456\pi\)
\(422\) 1.04035e8 0.0673882
\(423\) −3.49323e8 −0.224407
\(424\) −1.09249e9 −0.696044
\(425\) 3.82463e9 2.41673
\(426\) −1.93309e8 −0.121149
\(427\) 1.10423e9 0.686374
\(428\) −1.53827e9 −0.948371
\(429\) 2.53111e8 0.154778
\(430\) −8.60091e8 −0.521681
\(431\) −2.40194e9 −1.44508 −0.722540 0.691329i \(-0.757025\pi\)
−0.722540 + 0.691329i \(0.757025\pi\)
\(432\) 1.26715e8 0.0756194
\(433\) −1.31134e9 −0.776259 −0.388129 0.921605i \(-0.626879\pi\)
−0.388129 + 0.921605i \(0.626879\pi\)
\(434\) 1.97496e8 0.115969
\(435\) −3.13276e9 −1.82480
\(436\) −6.90146e6 −0.00398785
\(437\) 5.64066e7 0.0323329
\(438\) 8.33672e8 0.474063
\(439\) −2.63455e8 −0.148621 −0.0743105 0.997235i \(-0.523676\pi\)
−0.0743105 + 0.997235i \(0.523676\pi\)
\(440\) 1.25804e9 0.704061
\(441\) −5.15507e8 −0.286219
\(442\) −5.30291e8 −0.292103
\(443\) 1.46341e9 0.799748 0.399874 0.916570i \(-0.369054\pi\)
0.399874 + 0.916570i \(0.369054\pi\)
\(444\) −3.34638e8 −0.181441
\(445\) 1.23933e9 0.666692
\(446\) −9.53207e8 −0.508763
\(447\) −4.70345e7 −0.0249081
\(448\) 5.84260e7 0.0306996
\(449\) −2.17952e9 −1.13631 −0.568157 0.822920i \(-0.692343\pi\)
−0.568157 + 0.822920i \(0.692343\pi\)
\(450\) −6.58226e8 −0.340511
\(451\) 2.79018e8 0.143223
\(452\) −2.00984e9 −1.02371
\(453\) −5.73805e8 −0.290015
\(454\) 6.23126e8 0.312522
\(455\) 7.62939e8 0.379708
\(456\) 2.05124e8 0.101307
\(457\) 2.47142e9 1.21127 0.605634 0.795743i \(-0.292920\pi\)
0.605634 + 0.795743i \(0.292920\pi\)
\(458\) −3.72751e8 −0.181297
\(459\) 4.40654e8 0.212693
\(460\) −4.46845e8 −0.214045
\(461\) 2.72822e9 1.29696 0.648479 0.761232i \(-0.275405\pi\)
0.648479 + 0.761232i \(0.275405\pi\)
\(462\) −1.01838e8 −0.0480465
\(463\) −2.36027e9 −1.10517 −0.552583 0.833458i \(-0.686358\pi\)
−0.552583 + 0.833458i \(0.686358\pi\)
\(464\) −1.49703e9 −0.695695
\(465\) −1.47552e9 −0.680551
\(466\) 1.23135e9 0.563678
\(467\) −3.71806e9 −1.68930 −0.844650 0.535318i \(-0.820192\pi\)
−0.844650 + 0.535318i \(0.820192\pi\)
\(468\) −3.26934e8 −0.147435
\(469\) 4.04228e8 0.180934
\(470\) 1.26366e9 0.561420
\(471\) −5.86381e8 −0.258587
\(472\) 2.47560e8 0.108364
\(473\) 6.82207e8 0.296416
\(474\) −6.70559e8 −0.289210
\(475\) 1.07674e9 0.460981
\(476\) −7.64316e8 −0.324824
\(477\) −6.60723e8 −0.278744
\(478\) −2.22605e9 −0.932261
\(479\) −3.64251e9 −1.51435 −0.757175 0.653212i \(-0.773421\pi\)
−0.757175 + 0.653212i \(0.773421\pi\)
\(480\) −2.53696e9 −1.04705
\(481\) 5.55095e8 0.227436
\(482\) −8.74327e8 −0.355639
\(483\) 8.24411e7 0.0332912
\(484\) 1.51219e9 0.606245
\(485\) 2.25079e9 0.895857
\(486\) −7.58373e7 −0.0299679
\(487\) −4.68073e9 −1.83638 −0.918190 0.396140i \(-0.870350\pi\)
−0.918190 + 0.396140i \(0.870350\pi\)
\(488\) 3.90126e9 1.51962
\(489\) −1.35187e9 −0.522823
\(490\) 1.86482e9 0.716063
\(491\) 4.42989e9 1.68892 0.844458 0.535622i \(-0.179923\pi\)
0.844458 + 0.535622i \(0.179923\pi\)
\(492\) −3.60397e8 −0.136428
\(493\) −5.20598e9 −1.95676
\(494\) −1.49292e8 −0.0557175
\(495\) 7.60847e8 0.281954
\(496\) −7.05100e8 −0.259457
\(497\) 4.62172e8 0.168871
\(498\) −1.80939e8 −0.0656491
\(499\) 2.99262e9 1.07820 0.539100 0.842242i \(-0.318765\pi\)
0.539100 + 0.842242i \(0.318765\pi\)
\(500\) −4.62907e9 −1.65615
\(501\) 1.01900e9 0.362029
\(502\) −2.28952e9 −0.807757
\(503\) −6.91728e8 −0.242353 −0.121176 0.992631i \(-0.538667\pi\)
−0.121176 + 0.992631i \(0.538667\pi\)
\(504\) 2.99799e8 0.104310
\(505\) −4.05420e8 −0.140083
\(506\) −9.89392e7 −0.0339501
\(507\) −1.15189e9 −0.392541
\(508\) −2.70821e8 −0.0916556
\(509\) 1.02248e9 0.343671 0.171835 0.985126i \(-0.445030\pi\)
0.171835 + 0.985126i \(0.445030\pi\)
\(510\) −1.59405e9 −0.532115
\(511\) −1.99318e9 −0.660804
\(512\) −2.20560e9 −0.726243
\(513\) 1.24056e8 0.0405703
\(514\) 6.94411e8 0.225551
\(515\) 5.91671e9 1.90878
\(516\) −8.81182e8 −0.282352
\(517\) −1.00231e9 −0.318996
\(518\) −2.23339e8 −0.0706010
\(519\) −3.40753e9 −1.06993
\(520\) 2.69548e9 0.840669
\(521\) −1.33952e9 −0.414970 −0.207485 0.978238i \(-0.566528\pi\)
−0.207485 + 0.978238i \(0.566528\pi\)
\(522\) 8.95958e8 0.275703
\(523\) 2.03575e9 0.622255 0.311127 0.950368i \(-0.399293\pi\)
0.311127 + 0.950368i \(0.399293\pi\)
\(524\) −2.12002e8 −0.0643697
\(525\) 1.57371e9 0.474644
\(526\) 1.34921e9 0.404232
\(527\) −2.45201e9 −0.729768
\(528\) 3.63581e8 0.107494
\(529\) −3.32473e9 −0.976476
\(530\) 2.39014e9 0.697361
\(531\) 1.49721e8 0.0433963
\(532\) −2.15176e8 −0.0619589
\(533\) 5.97824e8 0.171013
\(534\) −3.54443e8 −0.100728
\(535\) 7.67027e9 2.16557
\(536\) 1.42815e9 0.400586
\(537\) −2.33419e9 −0.650468
\(538\) 6.15647e8 0.170449
\(539\) −1.47914e9 −0.406863
\(540\) −9.82758e8 −0.268577
\(541\) −3.66614e9 −0.995449 −0.497724 0.867335i \(-0.665831\pi\)
−0.497724 + 0.867335i \(0.665831\pi\)
\(542\) −1.16122e9 −0.313269
\(543\) 3.86106e8 0.103492
\(544\) −4.21589e9 −1.12278
\(545\) 3.44128e7 0.00910610
\(546\) −2.18198e8 −0.0573688
\(547\) −1.78158e9 −0.465426 −0.232713 0.972545i \(-0.574760\pi\)
−0.232713 + 0.972545i \(0.574760\pi\)
\(548\) −3.49137e9 −0.906282
\(549\) 2.35943e9 0.608561
\(550\) −1.88864e9 −0.484039
\(551\) −1.46563e9 −0.373244
\(552\) 2.91267e8 0.0737062
\(553\) 1.60320e9 0.403134
\(554\) 1.19398e9 0.298341
\(555\) 1.66861e9 0.414313
\(556\) 3.13488e9 0.773499
\(557\) 5.09999e9 1.25048 0.625239 0.780434i \(-0.285002\pi\)
0.625239 + 0.780434i \(0.285002\pi\)
\(558\) 4.21994e8 0.102822
\(559\) 1.46170e9 0.353930
\(560\) 1.09592e9 0.263708
\(561\) 1.26437e9 0.302345
\(562\) −1.28469e9 −0.305297
\(563\) −5.89137e9 −1.39135 −0.695676 0.718356i \(-0.744895\pi\)
−0.695676 + 0.718356i \(0.744895\pi\)
\(564\) 1.29465e9 0.303861
\(565\) 1.00217e10 2.33760
\(566\) −1.26418e7 −0.00293056
\(567\) 1.81315e8 0.0417727
\(568\) 1.63286e9 0.373879
\(569\) 8.17341e9 1.85999 0.929994 0.367574i \(-0.119812\pi\)
0.929994 + 0.367574i \(0.119812\pi\)
\(570\) −4.48768e8 −0.101499
\(571\) 5.67551e9 1.27579 0.637894 0.770124i \(-0.279806\pi\)
0.637894 + 0.770124i \(0.279806\pi\)
\(572\) −9.38070e8 −0.209580
\(573\) −3.93717e9 −0.874266
\(574\) −2.40531e8 −0.0530859
\(575\) 1.52892e9 0.335388
\(576\) 1.24840e8 0.0272193
\(577\) 7.16256e9 1.55222 0.776109 0.630598i \(-0.217190\pi\)
0.776109 + 0.630598i \(0.217190\pi\)
\(578\) −4.80233e8 −0.103444
\(579\) −4.32509e9 −0.926021
\(580\) 1.16105e10 2.47089
\(581\) 4.32596e8 0.0915094
\(582\) −6.43717e8 −0.135352
\(583\) −1.89581e9 −0.396237
\(584\) −7.04195e9 −1.46301
\(585\) 1.63019e9 0.336662
\(586\) 2.87176e9 0.589531
\(587\) −2.83241e9 −0.577993 −0.288997 0.957330i \(-0.593322\pi\)
−0.288997 + 0.957330i \(0.593322\pi\)
\(588\) 1.91055e9 0.387559
\(589\) −6.90308e8 −0.139200
\(590\) −5.41610e8 −0.108569
\(591\) −4.88879e8 −0.0974194
\(592\) 7.97367e8 0.157955
\(593\) −3.85485e9 −0.759130 −0.379565 0.925165i \(-0.623926\pi\)
−0.379565 + 0.925165i \(0.623926\pi\)
\(594\) −2.17599e8 −0.0425996
\(595\) 3.81111e9 0.741724
\(596\) 1.74318e8 0.0337271
\(597\) 5.86087e9 1.12733
\(598\) −2.11987e8 −0.0405374
\(599\) −9.88088e9 −1.87846 −0.939229 0.343290i \(-0.888459\pi\)
−0.939229 + 0.343290i \(0.888459\pi\)
\(600\) 5.55997e9 1.05086
\(601\) 7.33104e9 1.37754 0.688771 0.724979i \(-0.258150\pi\)
0.688771 + 0.724979i \(0.258150\pi\)
\(602\) −5.88106e8 −0.109867
\(603\) 8.63724e8 0.160422
\(604\) 2.12661e9 0.392699
\(605\) −7.54026e9 −1.38434
\(606\) 1.15949e8 0.0211647
\(607\) −6.93033e9 −1.25775 −0.628873 0.777508i \(-0.716484\pi\)
−0.628873 + 0.777508i \(0.716484\pi\)
\(608\) −1.18689e9 −0.214165
\(609\) −2.14209e9 −0.384307
\(610\) −8.53515e9 −1.52250
\(611\) −2.14756e9 −0.380890
\(612\) −1.63313e9 −0.288000
\(613\) −5.61223e9 −0.984065 −0.492033 0.870577i \(-0.663746\pi\)
−0.492033 + 0.870577i \(0.663746\pi\)
\(614\) 1.07978e9 0.188255
\(615\) 1.79705e9 0.311528
\(616\) 8.60213e8 0.148277
\(617\) −5.35454e9 −0.917750 −0.458875 0.888501i \(-0.651747\pi\)
−0.458875 + 0.888501i \(0.651747\pi\)
\(618\) −1.69216e9 −0.288391
\(619\) −1.02326e10 −1.73408 −0.867038 0.498242i \(-0.833979\pi\)
−0.867038 + 0.498242i \(0.833979\pi\)
\(620\) 5.46853e9 0.921509
\(621\) 1.76154e8 0.0295170
\(622\) −2.54568e9 −0.424167
\(623\) 8.47416e8 0.140407
\(624\) 7.79011e8 0.128350
\(625\) 9.73528e9 1.59503
\(626\) 9.38656e8 0.152931
\(627\) 3.55954e8 0.0576710
\(628\) 2.17322e9 0.350143
\(629\) 2.77287e9 0.444275
\(630\) −6.55899e8 −0.104507
\(631\) −2.22654e9 −0.352800 −0.176400 0.984319i \(-0.556445\pi\)
−0.176400 + 0.984319i \(0.556445\pi\)
\(632\) 5.66415e9 0.892534
\(633\) −5.31469e8 −0.0832846
\(634\) −2.21587e9 −0.345328
\(635\) 1.35040e9 0.209292
\(636\) 2.44875e9 0.377437
\(637\) −3.16921e9 −0.485806
\(638\) 2.57077e9 0.391913
\(639\) 9.87536e8 0.149727
\(640\) 1.15755e10 1.74545
\(641\) −4.46510e9 −0.669620 −0.334810 0.942286i \(-0.608672\pi\)
−0.334810 + 0.942286i \(0.608672\pi\)
\(642\) −2.19367e9 −0.327189
\(643\) 8.21135e9 1.21808 0.609041 0.793139i \(-0.291555\pi\)
0.609041 + 0.793139i \(0.291555\pi\)
\(644\) −3.05540e8 −0.0450783
\(645\) 4.39384e9 0.644741
\(646\) −7.45758e8 −0.108839
\(647\) 1.31007e10 1.90164 0.950820 0.309745i \(-0.100244\pi\)
0.950820 + 0.309745i \(0.100244\pi\)
\(648\) 6.40590e8 0.0924842
\(649\) 4.29594e8 0.0616882
\(650\) −4.04661e9 −0.577956
\(651\) −1.00892e9 −0.143326
\(652\) 5.01025e9 0.707935
\(653\) −4.31273e9 −0.606116 −0.303058 0.952972i \(-0.598008\pi\)
−0.303058 + 0.952972i \(0.598008\pi\)
\(654\) −9.84194e6 −0.00137581
\(655\) 1.05711e9 0.146986
\(656\) 8.58745e8 0.118768
\(657\) −4.25888e9 −0.585890
\(658\) 8.64056e8 0.118236
\(659\) 3.34441e9 0.455219 0.227610 0.973752i \(-0.426909\pi\)
0.227610 + 0.973752i \(0.426909\pi\)
\(660\) −2.81982e9 −0.381784
\(661\) 7.28830e9 0.981570 0.490785 0.871281i \(-0.336710\pi\)
0.490785 + 0.871281i \(0.336710\pi\)
\(662\) 3.45300e9 0.462587
\(663\) 2.70903e9 0.361008
\(664\) 1.52837e9 0.202601
\(665\) 1.07293e9 0.141481
\(666\) −4.77215e8 −0.0625971
\(667\) −2.08112e9 −0.271555
\(668\) −3.77659e9 −0.490210
\(669\) 4.86953e9 0.628776
\(670\) −3.12449e9 −0.401344
\(671\) 6.76991e9 0.865075
\(672\) −1.73470e9 −0.220512
\(673\) −2.74970e9 −0.347723 −0.173861 0.984770i \(-0.555624\pi\)
−0.173861 + 0.984770i \(0.555624\pi\)
\(674\) −1.17340e9 −0.147617
\(675\) 3.36260e9 0.420835
\(676\) 4.26910e9 0.531524
\(677\) 7.33717e9 0.908800 0.454400 0.890798i \(-0.349854\pi\)
0.454400 + 0.890798i \(0.349854\pi\)
\(678\) −2.86616e9 −0.353181
\(679\) 1.53903e9 0.188670
\(680\) 1.34648e10 1.64217
\(681\) −3.18329e9 −0.386243
\(682\) 1.21083e9 0.146163
\(683\) 5.10390e9 0.612956 0.306478 0.951878i \(-0.400849\pi\)
0.306478 + 0.951878i \(0.400849\pi\)
\(684\) −4.59773e8 −0.0549347
\(685\) 1.74090e10 2.06946
\(686\) 2.76012e9 0.326433
\(687\) 1.90423e9 0.224063
\(688\) 2.09966e9 0.245804
\(689\) −4.06197e9 −0.473118
\(690\) −6.37230e8 −0.0738456
\(691\) −1.60504e10 −1.85061 −0.925303 0.379229i \(-0.876189\pi\)
−0.925303 + 0.379229i \(0.876189\pi\)
\(692\) 1.26289e10 1.44875
\(693\) 5.20246e8 0.0593803
\(694\) 4.13348e9 0.469416
\(695\) −1.56315e10 −1.76626
\(696\) −7.56807e9 −0.850849
\(697\) 2.98631e9 0.334057
\(698\) −1.18881e9 −0.132318
\(699\) −6.29046e9 −0.696646
\(700\) −5.83243e9 −0.642698
\(701\) −1.74438e10 −1.91261 −0.956307 0.292366i \(-0.905558\pi\)
−0.956307 + 0.292366i \(0.905558\pi\)
\(702\) −4.66229e8 −0.0508651
\(703\) 7.80640e8 0.0847436
\(704\) 3.58204e8 0.0386924
\(705\) −6.45551e9 −0.693855
\(706\) 5.03921e9 0.538947
\(707\) −2.77215e8 −0.0295018
\(708\) −5.54891e8 −0.0587613
\(709\) 1.51506e9 0.159650 0.0798250 0.996809i \(-0.474564\pi\)
0.0798250 + 0.996809i \(0.474564\pi\)
\(710\) −3.57237e9 −0.374586
\(711\) 3.42560e9 0.357432
\(712\) 2.99394e9 0.310859
\(713\) −9.80206e8 −0.101275
\(714\) −1.08996e9 −0.112065
\(715\) 4.67750e9 0.478567
\(716\) 8.65089e9 0.880775
\(717\) 1.13720e10 1.15217
\(718\) −4.65119e9 −0.468952
\(719\) −7.46523e9 −0.749018 −0.374509 0.927223i \(-0.622189\pi\)
−0.374509 + 0.927223i \(0.622189\pi\)
\(720\) 2.34169e9 0.233812
\(721\) 4.04568e9 0.401993
\(722\) 4.51437e9 0.446392
\(723\) 4.46657e9 0.439531
\(724\) −1.43097e9 −0.140135
\(725\) −3.97264e10 −3.87165
\(726\) 2.15649e9 0.209155
\(727\) −9.95066e9 −0.960465 −0.480233 0.877141i \(-0.659448\pi\)
−0.480233 + 0.877141i \(0.659448\pi\)
\(728\) 1.84310e9 0.177047
\(729\) 3.87420e8 0.0370370
\(730\) 1.54063e10 1.46578
\(731\) 7.30163e9 0.691368
\(732\) −8.74444e9 −0.824030
\(733\) −1.32840e10 −1.24585 −0.622926 0.782281i \(-0.714056\pi\)
−0.622926 + 0.782281i \(0.714056\pi\)
\(734\) 2.34093e9 0.218500
\(735\) −9.52659e9 −0.884977
\(736\) −1.68533e9 −0.155816
\(737\) 2.47828e9 0.228042
\(738\) −5.13949e8 −0.0470677
\(739\) −1.90326e10 −1.73477 −0.867386 0.497637i \(-0.834201\pi\)
−0.867386 + 0.497637i \(0.834201\pi\)
\(740\) −6.18412e9 −0.561005
\(741\) 7.62669e8 0.0688608
\(742\) 1.63431e9 0.146866
\(743\) 1.49051e10 1.33313 0.666567 0.745445i \(-0.267763\pi\)
0.666567 + 0.745445i \(0.267763\pi\)
\(744\) −3.56455e9 −0.317321
\(745\) −8.69201e8 −0.0770146
\(746\) −3.46934e9 −0.305958
\(747\) 9.24339e8 0.0811352
\(748\) −4.68594e9 −0.409394
\(749\) 5.24472e9 0.456074
\(750\) −6.60136e9 −0.571372
\(751\) 1.38756e10 1.19540 0.597699 0.801721i \(-0.296082\pi\)
0.597699 + 0.801721i \(0.296082\pi\)
\(752\) −3.08486e9 −0.264529
\(753\) 1.16962e10 0.998301
\(754\) 5.50813e9 0.467956
\(755\) −1.06039e10 −0.896713
\(756\) −6.71982e8 −0.0565629
\(757\) 2.56500e9 0.214907 0.107454 0.994210i \(-0.465730\pi\)
0.107454 + 0.994210i \(0.465730\pi\)
\(758\) −2.11770e8 −0.0176613
\(759\) 5.05438e8 0.0419587
\(760\) 3.79070e9 0.313236
\(761\) 6.51304e8 0.0535720 0.0267860 0.999641i \(-0.491473\pi\)
0.0267860 + 0.999641i \(0.491473\pi\)
\(762\) −3.86208e8 −0.0316213
\(763\) 2.35305e7 0.00191776
\(764\) 1.45918e10 1.18381
\(765\) 8.14331e9 0.657636
\(766\) −2.29765e9 −0.184707
\(767\) 9.20451e8 0.0736575
\(768\) −3.90238e9 −0.310860
\(769\) −2.24597e10 −1.78099 −0.890496 0.454991i \(-0.849643\pi\)
−0.890496 + 0.454991i \(0.849643\pi\)
\(770\) −1.88197e9 −0.148557
\(771\) −3.54745e9 −0.278757
\(772\) 1.60295e10 1.25389
\(773\) −1.23734e9 −0.0963518 −0.0481759 0.998839i \(-0.515341\pi\)
−0.0481759 + 0.998839i \(0.515341\pi\)
\(774\) −1.25662e9 −0.0974118
\(775\) −1.87111e10 −1.44392
\(776\) 5.43742e9 0.417712
\(777\) 1.14095e9 0.0872552
\(778\) 4.21726e9 0.321071
\(779\) 8.40730e8 0.0637200
\(780\) −6.04176e9 −0.455861
\(781\) 2.83353e9 0.212838
\(782\) −1.05894e9 −0.0791860
\(783\) −4.57707e9 −0.340739
\(784\) −4.55242e9 −0.337393
\(785\) −1.08364e10 −0.799538
\(786\) −3.02329e8 −0.0222076
\(787\) −2.95191e9 −0.215870 −0.107935 0.994158i \(-0.534424\pi\)
−0.107935 + 0.994158i \(0.534424\pi\)
\(788\) 1.81187e9 0.131912
\(789\) −6.89257e9 −0.499587
\(790\) −1.23920e10 −0.894222
\(791\) 6.85254e9 0.492305
\(792\) 1.83804e9 0.131467
\(793\) 1.45052e10 1.03292
\(794\) −1.55422e9 −0.110190
\(795\) −1.22102e10 −0.861862
\(796\) −2.17213e10 −1.52648
\(797\) 1.12703e10 0.788553 0.394276 0.918992i \(-0.370995\pi\)
0.394276 + 0.918992i \(0.370995\pi\)
\(798\) −3.06855e8 −0.0213759
\(799\) −1.07277e10 −0.744034
\(800\) −3.21711e10 −2.22153
\(801\) 1.81070e9 0.124489
\(802\) −1.04923e10 −0.718223
\(803\) −1.22200e10 −0.832848
\(804\) −3.20110e9 −0.217222
\(805\) 1.52352e9 0.102935
\(806\) 2.59432e9 0.174522
\(807\) −3.14508e9 −0.210656
\(808\) −9.79407e8 −0.0653166
\(809\) −2.44387e10 −1.62277 −0.811386 0.584511i \(-0.801286\pi\)
−0.811386 + 0.584511i \(0.801286\pi\)
\(810\) −1.40148e9 −0.0926592
\(811\) 1.77574e10 1.16898 0.584488 0.811402i \(-0.301295\pi\)
0.584488 + 0.811402i \(0.301295\pi\)
\(812\) 7.93894e9 0.520375
\(813\) 5.93219e9 0.387167
\(814\) −1.36927e9 −0.0889824
\(815\) −2.49827e10 −1.61654
\(816\) 3.89140e9 0.250721
\(817\) 2.05561e9 0.131876
\(818\) −4.26890e9 −0.272696
\(819\) 1.11468e9 0.0709017
\(820\) −6.66015e9 −0.421828
\(821\) 1.19253e10 0.752090 0.376045 0.926601i \(-0.377284\pi\)
0.376045 + 0.926601i \(0.377284\pi\)
\(822\) −4.97892e9 −0.312668
\(823\) −2.32580e10 −1.45436 −0.727182 0.686445i \(-0.759170\pi\)
−0.727182 + 0.686445i \(0.759170\pi\)
\(824\) 1.42935e10 0.890007
\(825\) 9.64828e9 0.598220
\(826\) −3.70338e8 −0.0228648
\(827\) 6.50185e9 0.399731 0.199866 0.979823i \(-0.435949\pi\)
0.199866 + 0.979823i \(0.435949\pi\)
\(828\) −6.52856e8 −0.0399679
\(829\) 1.11632e10 0.680529 0.340265 0.940330i \(-0.389483\pi\)
0.340265 + 0.940330i \(0.389483\pi\)
\(830\) −3.34376e9 −0.202984
\(831\) −6.09955e9 −0.368718
\(832\) 7.67489e8 0.0461999
\(833\) −1.58312e10 −0.948977
\(834\) 4.47055e9 0.266858
\(835\) 1.88312e10 1.11938
\(836\) −1.31922e9 −0.0780902
\(837\) −2.15579e9 −0.127077
\(838\) −5.65549e9 −0.331983
\(839\) 2.99442e10 1.75043 0.875217 0.483731i \(-0.160719\pi\)
0.875217 + 0.483731i \(0.160719\pi\)
\(840\) 5.54031e9 0.322520
\(841\) 3.68246e10 2.13478
\(842\) 1.03338e10 0.596578
\(843\) 6.56295e9 0.377314
\(844\) 1.96971e9 0.112773
\(845\) −2.12871e10 −1.21372
\(846\) 1.84625e9 0.104832
\(847\) −5.15582e9 −0.291545
\(848\) −5.83482e9 −0.328581
\(849\) 6.45814e7 0.00362185
\(850\) −2.02141e10 −1.12898
\(851\) 1.10847e9 0.0616554
\(852\) −3.65997e9 −0.202740
\(853\) −2.44170e10 −1.34701 −0.673504 0.739184i \(-0.735212\pi\)
−0.673504 + 0.739184i \(0.735212\pi\)
\(854\) −5.83609e9 −0.320641
\(855\) 2.29257e9 0.125441
\(856\) 1.85297e10 1.00974
\(857\) 6.74647e9 0.366137 0.183069 0.983100i \(-0.441397\pi\)
0.183069 + 0.983100i \(0.441397\pi\)
\(858\) −1.33775e9 −0.0723051
\(859\) −1.73978e10 −0.936520 −0.468260 0.883591i \(-0.655119\pi\)
−0.468260 + 0.883591i \(0.655119\pi\)
\(860\) −1.62843e10 −0.873020
\(861\) 1.22877e9 0.0656085
\(862\) 1.26948e10 0.675073
\(863\) 3.06467e10 1.62310 0.811551 0.584282i \(-0.198624\pi\)
0.811551 + 0.584282i \(0.198624\pi\)
\(864\) −3.70659e9 −0.195513
\(865\) −6.29714e10 −3.30816
\(866\) 6.93072e9 0.362631
\(867\) 2.45331e9 0.127845
\(868\) 3.73923e9 0.194072
\(869\) 9.82906e9 0.508092
\(870\) 1.65574e10 0.852459
\(871\) 5.30997e9 0.272288
\(872\) 8.31339e7 0.00424590
\(873\) 3.28848e9 0.167280
\(874\) −2.98122e8 −0.0151044
\(875\) 1.57828e10 0.796445
\(876\) 1.57841e10 0.793332
\(877\) 1.20979e10 0.605634 0.302817 0.953049i \(-0.402073\pi\)
0.302817 + 0.953049i \(0.402073\pi\)
\(878\) 1.39242e9 0.0694287
\(879\) −1.46706e10 −0.728597
\(880\) 6.71900e9 0.332365
\(881\) −2.44049e10 −1.20243 −0.601217 0.799086i \(-0.705317\pi\)
−0.601217 + 0.799086i \(0.705317\pi\)
\(882\) 2.72457e9 0.133708
\(883\) −4.56189e9 −0.222989 −0.111494 0.993765i \(-0.535564\pi\)
−0.111494 + 0.993765i \(0.535564\pi\)
\(884\) −1.00401e10 −0.488828
\(885\) 2.76686e9 0.134179
\(886\) −7.73446e9 −0.373604
\(887\) 2.10603e9 0.101328 0.0506642 0.998716i \(-0.483866\pi\)
0.0506642 + 0.998716i \(0.483866\pi\)
\(888\) 4.03099e9 0.193182
\(889\) 9.23363e8 0.0440774
\(890\) −6.55012e9 −0.311447
\(891\) 1.11162e9 0.0526485
\(892\) −1.80473e10 −0.851402
\(893\) −3.02014e9 −0.141921
\(894\) 2.48588e8 0.0116359
\(895\) −4.31360e10 −2.01122
\(896\) 7.91498e9 0.367597
\(897\) 1.08295e9 0.0500999
\(898\) 1.15193e10 0.530832
\(899\) 2.54690e10 1.16910
\(900\) −1.24623e10 −0.569836
\(901\) −2.02908e10 −0.924191
\(902\) −1.47467e9 −0.0669071
\(903\) 3.00439e9 0.135784
\(904\) 2.42102e10 1.08996
\(905\) 7.13526e9 0.319992
\(906\) 3.03269e9 0.135481
\(907\) 1.54356e10 0.686906 0.343453 0.939170i \(-0.388403\pi\)
0.343453 + 0.939170i \(0.388403\pi\)
\(908\) 1.17978e10 0.522997
\(909\) −5.92333e8 −0.0261573
\(910\) −4.03231e9 −0.177382
\(911\) 1.76826e10 0.774874 0.387437 0.921896i \(-0.373360\pi\)
0.387437 + 0.921896i \(0.373360\pi\)
\(912\) 1.09554e9 0.0478239
\(913\) 2.65220e9 0.115334
\(914\) −1.30620e10 −0.565847
\(915\) 4.36025e10 1.88164
\(916\) −7.05737e9 −0.303395
\(917\) 7.22822e8 0.0309555
\(918\) −2.32896e9 −0.0993601
\(919\) 7.03662e9 0.299061 0.149531 0.988757i \(-0.452224\pi\)
0.149531 + 0.988757i \(0.452224\pi\)
\(920\) 5.38262e9 0.227896
\(921\) −5.51616e9 −0.232663
\(922\) −1.44193e10 −0.605878
\(923\) 6.07113e9 0.254135
\(924\) −1.92811e9 −0.0804046
\(925\) 2.11595e10 0.879043
\(926\) 1.24745e10 0.516281
\(927\) 8.64453e9 0.356420
\(928\) 4.37904e10 1.79871
\(929\) 1.13152e10 0.463028 0.231514 0.972832i \(-0.425632\pi\)
0.231514 + 0.972832i \(0.425632\pi\)
\(930\) 7.79848e9 0.317921
\(931\) −4.45692e9 −0.181013
\(932\) 2.33134e10 0.943302
\(933\) 1.30048e10 0.524225
\(934\) 1.96508e10 0.789161
\(935\) 2.33655e10 0.934836
\(936\) 3.93819e9 0.156975
\(937\) −4.20315e10 −1.66911 −0.834557 0.550921i \(-0.814276\pi\)
−0.834557 + 0.550921i \(0.814276\pi\)
\(938\) −2.13644e9 −0.0845240
\(939\) −4.79520e9 −0.189007
\(940\) 2.39252e10 0.939523
\(941\) −1.96707e10 −0.769585 −0.384792 0.923003i \(-0.625727\pi\)
−0.384792 + 0.923003i \(0.625727\pi\)
\(942\) 3.09916e9 0.120800
\(943\) 1.19380e9 0.0463596
\(944\) 1.32218e9 0.0511551
\(945\) 3.35071e9 0.129159
\(946\) −3.60562e9 −0.138472
\(947\) 1.69371e10 0.648058 0.324029 0.946047i \(-0.394962\pi\)
0.324029 + 0.946047i \(0.394962\pi\)
\(948\) −1.26958e10 −0.483985
\(949\) −2.61826e10 −0.994444
\(950\) −5.69082e9 −0.215349
\(951\) 1.13199e10 0.426788
\(952\) 9.20682e9 0.345844
\(953\) 1.91845e10 0.718002 0.359001 0.933337i \(-0.383117\pi\)
0.359001 + 0.933337i \(0.383117\pi\)
\(954\) 3.49207e9 0.130216
\(955\) −7.27592e10 −2.70319
\(956\) −4.21463e10 −1.56012
\(957\) −1.31330e10 −0.484363
\(958\) 1.92515e10 0.707433
\(959\) 1.19038e10 0.435833
\(960\) 2.30706e9 0.0841607
\(961\) −1.55168e10 −0.563987
\(962\) −2.93381e9 −0.106247
\(963\) 1.12065e10 0.404370
\(964\) −1.65538e10 −0.595153
\(965\) −7.99280e10 −2.86321
\(966\) −4.35720e8 −0.0155521
\(967\) −8.35028e9 −0.296967 −0.148484 0.988915i \(-0.547439\pi\)
−0.148484 + 0.988915i \(0.547439\pi\)
\(968\) −1.82156e10 −0.645476
\(969\) 3.80976e9 0.134513
\(970\) −1.18959e10 −0.418502
\(971\) 1.35240e10 0.474064 0.237032 0.971502i \(-0.423825\pi\)
0.237032 + 0.971502i \(0.423825\pi\)
\(972\) −1.43584e9 −0.0501505
\(973\) −1.06884e10 −0.371978
\(974\) 2.47388e10 0.857870
\(975\) 2.06724e10 0.714291
\(976\) 2.08361e10 0.717366
\(977\) −4.40527e10 −1.51127 −0.755633 0.654995i \(-0.772671\pi\)
−0.755633 + 0.654995i \(0.772671\pi\)
\(978\) 7.14495e9 0.244238
\(979\) 5.19542e9 0.176963
\(980\) 3.53071e10 1.19831
\(981\) 5.02783e7 0.00170035
\(982\) −2.34130e10 −0.788981
\(983\) −9.32750e9 −0.313204 −0.156602 0.987662i \(-0.550054\pi\)
−0.156602 + 0.987662i \(0.550054\pi\)
\(984\) 4.34128e9 0.145256
\(985\) −9.03452e9 −0.301216
\(986\) 2.75148e10 0.914107
\(987\) −4.41410e9 −0.146127
\(988\) −2.82657e9 −0.0932419
\(989\) 2.91888e9 0.0959464
\(990\) −4.02125e9 −0.131716
\(991\) 4.28591e10 1.39890 0.699448 0.714683i \(-0.253429\pi\)
0.699448 + 0.714683i \(0.253429\pi\)
\(992\) 2.06252e10 0.670822
\(993\) −1.76399e10 −0.571707
\(994\) −2.44268e9 −0.0788887
\(995\) 1.08309e11 3.48566
\(996\) −3.42575e9 −0.109862
\(997\) −1.75144e10 −0.559709 −0.279855 0.960042i \(-0.590286\pi\)
−0.279855 + 0.960042i \(0.590286\pi\)
\(998\) −1.58167e10 −0.503684
\(999\) 2.43789e9 0.0773633
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.1.8 18
3.2 odd 2 531.8.a.e.1.11 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.8 18 1.1 even 1 trivial
531.8.a.e.1.11 18 3.2 odd 2