Properties

Label 177.8.a.a.1.12
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + 1488374176 x^{9} + 42885295136 x^{8} - 226132003872 x^{7} - 3353576629440 x^{6} + 16796366777600 x^{5} + 99470801612800 x^{4} - 494039551757568 x^{3} - 493048066650624 x^{2} + 3193975642099712 x - 2385018853548032\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(-13.0039\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+13.0039 q^{2} -27.0000 q^{3} +41.1009 q^{4} +167.303 q^{5} -351.105 q^{6} +887.373 q^{7} -1130.03 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+13.0039 q^{2} -27.0000 q^{3} +41.1009 q^{4} +167.303 q^{5} -351.105 q^{6} +887.373 q^{7} -1130.03 q^{8} +729.000 q^{9} +2175.59 q^{10} -7784.37 q^{11} -1109.72 q^{12} +4609.99 q^{13} +11539.3 q^{14} -4517.19 q^{15} -19955.6 q^{16} -6055.74 q^{17} +9479.83 q^{18} +13552.7 q^{19} +6876.32 q^{20} -23959.1 q^{21} -101227. q^{22} +35274.4 q^{23} +30510.7 q^{24} -50134.6 q^{25} +59947.7 q^{26} -19683.0 q^{27} +36471.8 q^{28} +78335.1 q^{29} -58741.0 q^{30} -170575. q^{31} -114857. q^{32} +210178. q^{33} -78748.2 q^{34} +148460. q^{35} +29962.6 q^{36} -263573. q^{37} +176238. q^{38} -124470. q^{39} -189057. q^{40} -198905. q^{41} -311561. q^{42} -444435. q^{43} -319945. q^{44} +121964. q^{45} +458704. q^{46} -531622. q^{47} +538802. q^{48} -36112.2 q^{49} -651945. q^{50} +163505. q^{51} +189475. q^{52} -1.00107e6 q^{53} -255955. q^{54} -1.30235e6 q^{55} -1.00275e6 q^{56} -365924. q^{57} +1.01866e6 q^{58} +205379. q^{59} -185661. q^{60} -1.77312e6 q^{61} -2.21813e6 q^{62} +646895. q^{63} +1.06073e6 q^{64} +771266. q^{65} +2.73313e6 q^{66} -2.37225e6 q^{67} -248897. q^{68} -952409. q^{69} +1.93056e6 q^{70} -5.87275e6 q^{71} -823789. q^{72} -2.90906e6 q^{73} -3.42748e6 q^{74} +1.35363e6 q^{75} +557030. q^{76} -6.90764e6 q^{77} -1.61859e6 q^{78} -1.66441e6 q^{79} -3.33864e6 q^{80} +531441. q^{81} -2.58653e6 q^{82} +8.18425e6 q^{83} -984739. q^{84} -1.01315e6 q^{85} -5.77938e6 q^{86} -2.11505e6 q^{87} +8.79653e6 q^{88} +1.13356e7 q^{89} +1.58601e6 q^{90} +4.09078e6 q^{91} +1.44981e6 q^{92} +4.60552e6 q^{93} -6.91315e6 q^{94} +2.26742e6 q^{95} +3.10115e6 q^{96} +243947. q^{97} -469599. q^{98} -5.67480e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} + O(q^{10}) \) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} - 3479q^{10} + 898q^{11} - 26298q^{12} - 8172q^{13} - 13315q^{14} + 1836q^{15} + 3138q^{16} - 44985q^{17} - 4374q^{18} - 40137q^{19} + 130657q^{20} + 63261q^{21} + 109394q^{22} - 2833q^{23} - 22113q^{24} + 285746q^{25} - 129420q^{26} - 314928q^{27} + 112890q^{28} + 144375q^{29} + 93933q^{30} - 141759q^{31} - 36224q^{32} - 24246q^{33} - 341332q^{34} - 78859q^{35} + 710046q^{36} - 297971q^{37} + 329075q^{38} + 220644q^{39} - 203048q^{40} + 659077q^{41} + 359505q^{42} - 1431608q^{43} + 254916q^{44} - 49572q^{45} + 873113q^{46} - 1574073q^{47} - 84726q^{48} + 1893545q^{49} + 302533q^{50} + 1214595q^{51} - 4972548q^{52} + 587736q^{53} + 118098q^{54} - 4624036q^{55} - 5798506q^{56} + 1083699q^{57} - 6991380q^{58} + 3286064q^{59} - 3527739q^{60} - 6117131q^{61} - 11570258q^{62} - 1708047q^{63} - 19063011q^{64} - 5335514q^{65} - 2953638q^{66} - 16518710q^{67} - 17284669q^{68} + 76491q^{69} - 39189486q^{70} - 10882582q^{71} + 597051q^{72} - 21097441q^{73} - 16717030q^{74} - 7715142q^{75} - 40864952q^{76} - 3404601q^{77} + 3494340q^{78} - 3784458q^{79} - 27466195q^{80} + 8503056q^{81} - 24990117q^{82} - 1951425q^{83} - 3048030q^{84} - 23238675q^{85} - 35910572q^{86} - 3898125q^{87} - 27843055q^{88} + 10499443q^{89} - 2536191q^{90} + 699217q^{91} - 20062766q^{92} + 3827493q^{93} - 59358988q^{94} - 29236333q^{95} + 978048q^{96} - 25158976q^{97} + 2120460q^{98} + 654642q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 13.0039 1.14939 0.574696 0.818367i \(-0.305120\pi\)
0.574696 + 0.818367i \(0.305120\pi\)
\(3\) −27.0000 −0.577350
\(4\) 41.1009 0.321101
\(5\) 167.303 0.598562 0.299281 0.954165i \(-0.403253\pi\)
0.299281 + 0.954165i \(0.403253\pi\)
\(6\) −351.105 −0.663601
\(7\) 887.373 0.977829 0.488915 0.872332i \(-0.337393\pi\)
0.488915 + 0.872332i \(0.337393\pi\)
\(8\) −1130.03 −0.780321
\(9\) 729.000 0.333333
\(10\) 2175.59 0.687982
\(11\) −7784.37 −1.76339 −0.881696 0.471818i \(-0.843598\pi\)
−0.881696 + 0.471818i \(0.843598\pi\)
\(12\) −1109.72 −0.185388
\(13\) 4609.99 0.581967 0.290983 0.956728i \(-0.406018\pi\)
0.290983 + 0.956728i \(0.406018\pi\)
\(14\) 11539.3 1.12391
\(15\) −4517.19 −0.345580
\(16\) −19955.6 −1.21800
\(17\) −6055.74 −0.298948 −0.149474 0.988766i \(-0.547758\pi\)
−0.149474 + 0.988766i \(0.547758\pi\)
\(18\) 9479.83 0.383131
\(19\) 13552.7 0.453304 0.226652 0.973976i \(-0.427222\pi\)
0.226652 + 0.973976i \(0.427222\pi\)
\(20\) 6876.32 0.192199
\(21\) −23959.1 −0.564550
\(22\) −101227. −2.02683
\(23\) 35274.4 0.604523 0.302261 0.953225i \(-0.402258\pi\)
0.302261 + 0.953225i \(0.402258\pi\)
\(24\) 30510.7 0.450519
\(25\) −50134.6 −0.641723
\(26\) 59947.7 0.668907
\(27\) −19683.0 −0.192450
\(28\) 36471.8 0.313982
\(29\) 78335.1 0.596435 0.298218 0.954498i \(-0.403608\pi\)
0.298218 + 0.954498i \(0.403608\pi\)
\(30\) −58741.0 −0.397207
\(31\) −170575. −1.02837 −0.514184 0.857680i \(-0.671905\pi\)
−0.514184 + 0.857680i \(0.671905\pi\)
\(32\) −114857. −0.619632
\(33\) 210178. 1.01809
\(34\) −78748.2 −0.343609
\(35\) 148460. 0.585292
\(36\) 29962.6 0.107034
\(37\) −263573. −0.855451 −0.427726 0.903909i \(-0.640685\pi\)
−0.427726 + 0.903909i \(0.640685\pi\)
\(38\) 176238. 0.521024
\(39\) −124470. −0.335999
\(40\) −189057. −0.467071
\(41\) −198905. −0.450714 −0.225357 0.974276i \(-0.572355\pi\)
−0.225357 + 0.974276i \(0.572355\pi\)
\(42\) −311561. −0.648889
\(43\) −444435. −0.852449 −0.426225 0.904617i \(-0.640157\pi\)
−0.426225 + 0.904617i \(0.640157\pi\)
\(44\) −319945. −0.566227
\(45\) 121964. 0.199521
\(46\) 458704. 0.694833
\(47\) −531622. −0.746896 −0.373448 0.927651i \(-0.621825\pi\)
−0.373448 + 0.927651i \(0.621825\pi\)
\(48\) 538802. 0.703210
\(49\) −36112.2 −0.0438499
\(50\) −651945. −0.737591
\(51\) 163505. 0.172598
\(52\) 189475. 0.186870
\(53\) −1.00107e6 −0.923631 −0.461816 0.886976i \(-0.652802\pi\)
−0.461816 + 0.886976i \(0.652802\pi\)
\(54\) −255955. −0.221200
\(55\) −1.30235e6 −1.05550
\(56\) −1.00275e6 −0.763021
\(57\) −365924. −0.261715
\(58\) 1.01866e6 0.685538
\(59\) 205379. 0.130189
\(60\) −185661. −0.110966
\(61\) −1.77312e6 −1.00019 −0.500097 0.865969i \(-0.666702\pi\)
−0.500097 + 0.865969i \(0.666702\pi\)
\(62\) −2.21813e6 −1.18200
\(63\) 646895. 0.325943
\(64\) 1.06073e6 0.505795
\(65\) 771266. 0.348343
\(66\) 2.73313e6 1.17019
\(67\) −2.37225e6 −0.963606 −0.481803 0.876280i \(-0.660018\pi\)
−0.481803 + 0.876280i \(0.660018\pi\)
\(68\) −248897. −0.0959926
\(69\) −952409. −0.349021
\(70\) 1.93056e6 0.672729
\(71\) −5.87275e6 −1.94732 −0.973659 0.228008i \(-0.926779\pi\)
−0.973659 + 0.228008i \(0.926779\pi\)
\(72\) −823789. −0.260107
\(73\) −2.90906e6 −0.875232 −0.437616 0.899162i \(-0.644177\pi\)
−0.437616 + 0.899162i \(0.644177\pi\)
\(74\) −3.42748e6 −0.983249
\(75\) 1.35363e6 0.370499
\(76\) 557030. 0.145556
\(77\) −6.90764e6 −1.72430
\(78\) −1.61859e6 −0.386194
\(79\) −1.66441e6 −0.379808 −0.189904 0.981803i \(-0.560818\pi\)
−0.189904 + 0.981803i \(0.560818\pi\)
\(80\) −3.33864e6 −0.729046
\(81\) 531441. 0.111111
\(82\) −2.58653e6 −0.518047
\(83\) 8.18425e6 1.57111 0.785553 0.618794i \(-0.212379\pi\)
0.785553 + 0.618794i \(0.212379\pi\)
\(84\) −984739. −0.181277
\(85\) −1.01315e6 −0.178939
\(86\) −5.77938e6 −0.979798
\(87\) −2.11505e6 −0.344352
\(88\) 8.79653e6 1.37601
\(89\) 1.13356e7 1.70444 0.852219 0.523185i \(-0.175256\pi\)
0.852219 + 0.523185i \(0.175256\pi\)
\(90\) 1.58601e6 0.229327
\(91\) 4.09078e6 0.569064
\(92\) 1.44981e6 0.194113
\(93\) 4.60552e6 0.593729
\(94\) −6.91315e6 −0.858476
\(95\) 2.26742e6 0.271331
\(96\) 3.10115e6 0.357745
\(97\) 243947. 0.0271390 0.0135695 0.999908i \(-0.495681\pi\)
0.0135695 + 0.999908i \(0.495681\pi\)
\(98\) −469599. −0.0504006
\(99\) −5.67480e6 −0.587797
\(100\) −2.06058e6 −0.206058
\(101\) 1.91745e7 1.85183 0.925913 0.377736i \(-0.123297\pi\)
0.925913 + 0.377736i \(0.123297\pi\)
\(102\) 2.12620e6 0.198383
\(103\) 8.63345e6 0.778492 0.389246 0.921134i \(-0.372736\pi\)
0.389246 + 0.921134i \(0.372736\pi\)
\(104\) −5.20940e6 −0.454121
\(105\) −4.00843e6 −0.337918
\(106\) −1.30178e7 −1.06161
\(107\) −1.22715e7 −0.968397 −0.484199 0.874958i \(-0.660889\pi\)
−0.484199 + 0.874958i \(0.660889\pi\)
\(108\) −808989. −0.0617959
\(109\) −6.90858e6 −0.510971 −0.255485 0.966813i \(-0.582235\pi\)
−0.255485 + 0.966813i \(0.582235\pi\)
\(110\) −1.69356e7 −1.21318
\(111\) 7.11648e6 0.493895
\(112\) −1.77081e7 −1.19099
\(113\) 9.23590e6 0.602150 0.301075 0.953601i \(-0.402655\pi\)
0.301075 + 0.953601i \(0.402655\pi\)
\(114\) −4.75843e6 −0.300813
\(115\) 5.90153e6 0.361844
\(116\) 3.21964e6 0.191516
\(117\) 3.36068e6 0.193989
\(118\) 2.67072e6 0.149638
\(119\) −5.37370e6 −0.292320
\(120\) 5.10454e6 0.269663
\(121\) 4.11092e7 2.10955
\(122\) −2.30575e7 −1.14961
\(123\) 5.37042e6 0.260220
\(124\) −7.01078e6 −0.330210
\(125\) −2.14583e7 −0.982674
\(126\) 8.41214e6 0.374636
\(127\) 3.43987e6 0.149015 0.0745073 0.997220i \(-0.476262\pi\)
0.0745073 + 0.997220i \(0.476262\pi\)
\(128\) 2.84953e7 1.20099
\(129\) 1.19997e7 0.492162
\(130\) 1.00295e7 0.400383
\(131\) −2.36186e6 −0.0917919 −0.0458960 0.998946i \(-0.514614\pi\)
−0.0458960 + 0.998946i \(0.514614\pi\)
\(132\) 8.63850e6 0.326911
\(133\) 1.20263e7 0.443254
\(134\) −3.08485e7 −1.10756
\(135\) −3.29303e6 −0.115193
\(136\) 6.84314e6 0.233276
\(137\) 3.14805e6 0.104597 0.0522985 0.998631i \(-0.483345\pi\)
0.0522985 + 0.998631i \(0.483345\pi\)
\(138\) −1.23850e7 −0.401162
\(139\) 1.60628e7 0.507305 0.253652 0.967295i \(-0.418368\pi\)
0.253652 + 0.967295i \(0.418368\pi\)
\(140\) 6.10186e6 0.187938
\(141\) 1.43538e7 0.431221
\(142\) −7.63685e7 −2.23823
\(143\) −3.58858e7 −1.02624
\(144\) −1.45477e7 −0.405998
\(145\) 1.31057e7 0.357004
\(146\) −3.78291e7 −1.00598
\(147\) 975030. 0.0253167
\(148\) −1.08331e7 −0.274686
\(149\) −3.30906e7 −0.819506 −0.409753 0.912196i \(-0.634385\pi\)
−0.409753 + 0.912196i \(0.634385\pi\)
\(150\) 1.76025e7 0.425848
\(151\) −8.23311e7 −1.94601 −0.973003 0.230791i \(-0.925869\pi\)
−0.973003 + 0.230791i \(0.925869\pi\)
\(152\) −1.53149e7 −0.353722
\(153\) −4.41464e6 −0.0996495
\(154\) −8.98261e7 −1.98189
\(155\) −2.85377e7 −0.615543
\(156\) −5.11582e6 −0.107889
\(157\) 7.06660e7 1.45734 0.728671 0.684863i \(-0.240138\pi\)
0.728671 + 0.684863i \(0.240138\pi\)
\(158\) −2.16437e7 −0.436548
\(159\) 2.70289e7 0.533259
\(160\) −1.92160e7 −0.370889
\(161\) 3.13016e7 0.591120
\(162\) 6.91080e6 0.127710
\(163\) −5.37840e7 −0.972740 −0.486370 0.873753i \(-0.661679\pi\)
−0.486370 + 0.873753i \(0.661679\pi\)
\(164\) −8.17516e6 −0.144725
\(165\) 3.51634e7 0.609393
\(166\) 1.06427e8 1.80582
\(167\) 4.12989e7 0.686169 0.343084 0.939305i \(-0.388528\pi\)
0.343084 + 0.939305i \(0.388528\pi\)
\(168\) 2.70744e7 0.440530
\(169\) −4.14965e7 −0.661315
\(170\) −1.31748e7 −0.205671
\(171\) 9.87994e6 0.151101
\(172\) −1.82667e7 −0.273722
\(173\) 5.36462e7 0.787731 0.393865 0.919168i \(-0.371138\pi\)
0.393865 + 0.919168i \(0.371138\pi\)
\(174\) −2.75038e7 −0.395795
\(175\) −4.44881e7 −0.627496
\(176\) 1.55342e8 2.14780
\(177\) −5.54523e6 −0.0751646
\(178\) 1.47407e8 1.95907
\(179\) −3.62629e7 −0.472581 −0.236291 0.971682i \(-0.575932\pi\)
−0.236291 + 0.971682i \(0.575932\pi\)
\(180\) 5.01283e6 0.0640663
\(181\) −1.64625e7 −0.206358 −0.103179 0.994663i \(-0.532901\pi\)
−0.103179 + 0.994663i \(0.532901\pi\)
\(182\) 5.31960e7 0.654077
\(183\) 4.78743e7 0.577462
\(184\) −3.98610e7 −0.471722
\(185\) −4.40967e7 −0.512041
\(186\) 5.98896e7 0.682427
\(187\) 4.71401e7 0.527163
\(188\) −2.18501e7 −0.239829
\(189\) −1.74662e7 −0.188183
\(190\) 2.94852e7 0.311865
\(191\) −7.01561e7 −0.728532 −0.364266 0.931295i \(-0.618680\pi\)
−0.364266 + 0.931295i \(0.618680\pi\)
\(192\) −2.86397e7 −0.292021
\(193\) 9.06871e7 0.908019 0.454010 0.890997i \(-0.349993\pi\)
0.454010 + 0.890997i \(0.349993\pi\)
\(194\) 3.17225e6 0.0311933
\(195\) −2.08242e7 −0.201116
\(196\) −1.48425e6 −0.0140802
\(197\) −3.02563e7 −0.281958 −0.140979 0.990013i \(-0.545025\pi\)
−0.140979 + 0.990013i \(0.545025\pi\)
\(198\) −7.37945e7 −0.675609
\(199\) 1.78612e8 1.60666 0.803332 0.595532i \(-0.203059\pi\)
0.803332 + 0.595532i \(0.203059\pi\)
\(200\) 5.66534e7 0.500750
\(201\) 6.40509e7 0.556338
\(202\) 2.49343e8 2.12847
\(203\) 6.95125e7 0.583212
\(204\) 6.72021e6 0.0554213
\(205\) −3.32774e7 −0.269781
\(206\) 1.12268e8 0.894792
\(207\) 2.57151e7 0.201508
\(208\) −9.19952e7 −0.708832
\(209\) −1.05499e8 −0.799352
\(210\) −5.21252e7 −0.388401
\(211\) −1.01493e8 −0.743786 −0.371893 0.928276i \(-0.621291\pi\)
−0.371893 + 0.928276i \(0.621291\pi\)
\(212\) −4.11449e7 −0.296579
\(213\) 1.58564e8 1.12428
\(214\) −1.59577e8 −1.11307
\(215\) −7.43554e7 −0.510244
\(216\) 2.22423e7 0.150173
\(217\) −1.51363e8 −1.00557
\(218\) −8.98383e7 −0.587305
\(219\) 7.85447e7 0.505315
\(220\) −5.35278e7 −0.338922
\(221\) −2.79169e7 −0.173978
\(222\) 9.25418e7 0.567679
\(223\) 1.17677e8 0.710600 0.355300 0.934752i \(-0.384379\pi\)
0.355300 + 0.934752i \(0.384379\pi\)
\(224\) −1.01921e8 −0.605895
\(225\) −3.65481e7 −0.213908
\(226\) 1.20102e8 0.692106
\(227\) 2.17121e8 1.23200 0.616000 0.787746i \(-0.288752\pi\)
0.616000 + 0.787746i \(0.288752\pi\)
\(228\) −1.50398e7 −0.0840369
\(229\) 6.89428e7 0.379372 0.189686 0.981845i \(-0.439253\pi\)
0.189686 + 0.981845i \(0.439253\pi\)
\(230\) 7.67427e7 0.415901
\(231\) 1.86506e8 0.995523
\(232\) −8.85207e7 −0.465411
\(233\) 2.42127e8 1.25400 0.627000 0.779019i \(-0.284282\pi\)
0.627000 + 0.779019i \(0.284282\pi\)
\(234\) 4.37019e7 0.222969
\(235\) −8.89421e7 −0.447064
\(236\) 8.44126e6 0.0418038
\(237\) 4.49389e7 0.219282
\(238\) −6.98790e7 −0.335991
\(239\) 1.66165e8 0.787312 0.393656 0.919258i \(-0.371210\pi\)
0.393656 + 0.919258i \(0.371210\pi\)
\(240\) 9.01433e7 0.420915
\(241\) 2.91718e7 0.134247 0.0671233 0.997745i \(-0.478618\pi\)
0.0671233 + 0.997745i \(0.478618\pi\)
\(242\) 5.34579e8 2.42470
\(243\) −1.43489e7 −0.0641500
\(244\) −7.28769e7 −0.321163
\(245\) −6.04170e6 −0.0262469
\(246\) 6.98363e7 0.299095
\(247\) 6.24779e7 0.263808
\(248\) 1.92754e8 0.802458
\(249\) −2.20975e8 −0.907078
\(250\) −2.79041e8 −1.12948
\(251\) 2.96094e8 1.18188 0.590938 0.806717i \(-0.298758\pi\)
0.590938 + 0.806717i \(0.298758\pi\)
\(252\) 2.65880e7 0.104661
\(253\) −2.74589e8 −1.06601
\(254\) 4.47317e7 0.171276
\(255\) 2.73549e7 0.103311
\(256\) 2.34777e8 0.874611
\(257\) 2.67505e8 0.983027 0.491514 0.870870i \(-0.336444\pi\)
0.491514 + 0.870870i \(0.336444\pi\)
\(258\) 1.56043e8 0.565686
\(259\) −2.33888e8 −0.836485
\(260\) 3.16997e7 0.111853
\(261\) 5.71063e7 0.198812
\(262\) −3.07133e7 −0.105505
\(263\) 2.10348e8 0.713006 0.356503 0.934294i \(-0.383969\pi\)
0.356503 + 0.934294i \(0.383969\pi\)
\(264\) −2.37506e8 −0.794441
\(265\) −1.67482e8 −0.552851
\(266\) 1.56389e8 0.509472
\(267\) −3.06062e8 −0.984058
\(268\) −9.75018e7 −0.309415
\(269\) −4.76854e8 −1.49366 −0.746831 0.665014i \(-0.768426\pi\)
−0.746831 + 0.665014i \(0.768426\pi\)
\(270\) −4.28222e7 −0.132402
\(271\) −2.49889e8 −0.762701 −0.381350 0.924431i \(-0.624541\pi\)
−0.381350 + 0.924431i \(0.624541\pi\)
\(272\) 1.20846e8 0.364118
\(273\) −1.10451e8 −0.328549
\(274\) 4.09368e7 0.120223
\(275\) 3.90266e8 1.13161
\(276\) −3.91449e7 −0.112071
\(277\) −2.02351e8 −0.572039 −0.286020 0.958224i \(-0.592332\pi\)
−0.286020 + 0.958224i \(0.592332\pi\)
\(278\) 2.08879e8 0.583092
\(279\) −1.24349e8 −0.342790
\(280\) −1.67764e8 −0.456715
\(281\) −2.77424e8 −0.745885 −0.372942 0.927855i \(-0.621651\pi\)
−0.372942 + 0.927855i \(0.621651\pi\)
\(282\) 1.86655e8 0.495641
\(283\) −3.44360e8 −0.903151 −0.451575 0.892233i \(-0.649138\pi\)
−0.451575 + 0.892233i \(0.649138\pi\)
\(284\) −2.41375e8 −0.625286
\(285\) −6.12203e7 −0.156653
\(286\) −4.66655e8 −1.17955
\(287\) −1.76503e8 −0.440722
\(288\) −8.37310e7 −0.206544
\(289\) −3.73667e8 −0.910630
\(290\) 1.70425e8 0.410337
\(291\) −6.58656e6 −0.0156687
\(292\) −1.19565e8 −0.281038
\(293\) −3.56077e8 −0.827004 −0.413502 0.910503i \(-0.635694\pi\)
−0.413502 + 0.910503i \(0.635694\pi\)
\(294\) 1.26792e7 0.0290988
\(295\) 3.43606e7 0.0779262
\(296\) 2.97845e8 0.667527
\(297\) 1.53220e8 0.339365
\(298\) −4.30306e8 −0.941934
\(299\) 1.62615e8 0.351812
\(300\) 5.56356e7 0.118968
\(301\) −3.94379e8 −0.833550
\(302\) −1.07062e9 −2.23672
\(303\) −5.17713e8 −1.06915
\(304\) −2.70453e8 −0.552122
\(305\) −2.96649e8 −0.598679
\(306\) −5.74074e7 −0.114536
\(307\) −1.96195e8 −0.386993 −0.193496 0.981101i \(-0.561983\pi\)
−0.193496 + 0.981101i \(0.561983\pi\)
\(308\) −2.83910e8 −0.553673
\(309\) −2.33103e8 −0.449463
\(310\) −3.71101e8 −0.707500
\(311\) 5.23320e7 0.0986519 0.0493260 0.998783i \(-0.484293\pi\)
0.0493260 + 0.998783i \(0.484293\pi\)
\(312\) 1.40654e8 0.262187
\(313\) −3.12015e8 −0.575136 −0.287568 0.957760i \(-0.592847\pi\)
−0.287568 + 0.957760i \(0.592847\pi\)
\(314\) 9.18932e8 1.67506
\(315\) 1.08228e8 0.195097
\(316\) −6.84086e7 −0.121957
\(317\) 5.38220e8 0.948970 0.474485 0.880264i \(-0.342634\pi\)
0.474485 + 0.880264i \(0.342634\pi\)
\(318\) 3.51480e8 0.612923
\(319\) −6.09789e8 −1.05175
\(320\) 1.77463e8 0.302750
\(321\) 3.31330e8 0.559104
\(322\) 4.07042e8 0.679428
\(323\) −8.20719e7 −0.135514
\(324\) 2.18427e7 0.0356779
\(325\) −2.31120e8 −0.373461
\(326\) −6.99401e8 −1.11806
\(327\) 1.86532e8 0.295009
\(328\) 2.24767e8 0.351702
\(329\) −4.71747e8 −0.730337
\(330\) 4.57261e8 0.700431
\(331\) −7.08848e8 −1.07437 −0.537187 0.843463i \(-0.680513\pi\)
−0.537187 + 0.843463i \(0.680513\pi\)
\(332\) 3.36380e8 0.504483
\(333\) −1.92145e8 −0.285150
\(334\) 5.37046e8 0.788676
\(335\) −3.96886e8 −0.576778
\(336\) 4.78118e8 0.687619
\(337\) 1.28631e8 0.183079 0.0915397 0.995801i \(-0.470821\pi\)
0.0915397 + 0.995801i \(0.470821\pi\)
\(338\) −5.39616e8 −0.760110
\(339\) −2.49369e8 −0.347651
\(340\) −4.16412e7 −0.0574575
\(341\) 1.32782e9 1.81342
\(342\) 1.28478e8 0.173675
\(343\) −7.62835e8 −1.02071
\(344\) 5.02223e8 0.665184
\(345\) −1.59341e8 −0.208911
\(346\) 6.97609e8 0.905411
\(347\) −6.00582e7 −0.0771648 −0.0385824 0.999255i \(-0.512284\pi\)
−0.0385824 + 0.999255i \(0.512284\pi\)
\(348\) −8.69304e7 −0.110572
\(349\) 1.11674e9 1.40625 0.703124 0.711067i \(-0.251788\pi\)
0.703124 + 0.711067i \(0.251788\pi\)
\(350\) −5.78518e8 −0.721238
\(351\) −9.07384e7 −0.112000
\(352\) 8.94092e8 1.09265
\(353\) 3.99147e8 0.482971 0.241486 0.970404i \(-0.422365\pi\)
0.241486 + 0.970404i \(0.422365\pi\)
\(354\) −7.21095e7 −0.0863936
\(355\) −9.82530e8 −1.16559
\(356\) 4.65905e8 0.547296
\(357\) 1.45090e8 0.168771
\(358\) −4.71558e8 −0.543181
\(359\) −6.35935e8 −0.725408 −0.362704 0.931904i \(-0.618146\pi\)
−0.362704 + 0.931904i \(0.618146\pi\)
\(360\) −1.37823e8 −0.155690
\(361\) −7.10195e8 −0.794516
\(362\) −2.14077e8 −0.237186
\(363\) −1.10995e9 −1.21795
\(364\) 1.68135e8 0.182727
\(365\) −4.86696e8 −0.523881
\(366\) 6.22552e8 0.663730
\(367\) −2.61675e8 −0.276332 −0.138166 0.990409i \(-0.544121\pi\)
−0.138166 + 0.990409i \(0.544121\pi\)
\(368\) −7.03923e8 −0.736305
\(369\) −1.45001e8 −0.150238
\(370\) −5.73428e8 −0.588536
\(371\) −8.88322e8 −0.903154
\(372\) 1.89291e8 0.190647
\(373\) 1.69724e9 1.69341 0.846706 0.532060i \(-0.178582\pi\)
0.846706 + 0.532060i \(0.178582\pi\)
\(374\) 6.13004e8 0.605917
\(375\) 5.79373e8 0.567347
\(376\) 6.00746e8 0.582819
\(377\) 3.61124e8 0.347106
\(378\) −2.27128e8 −0.216296
\(379\) 1.10612e9 1.04367 0.521837 0.853045i \(-0.325247\pi\)
0.521837 + 0.853045i \(0.325247\pi\)
\(380\) 9.31929e7 0.0871245
\(381\) −9.28765e7 −0.0860337
\(382\) −9.12301e8 −0.837368
\(383\) −6.71833e8 −0.611034 −0.305517 0.952187i \(-0.598829\pi\)
−0.305517 + 0.952187i \(0.598829\pi\)
\(384\) −7.69374e8 −0.693391
\(385\) −1.15567e9 −1.03210
\(386\) 1.17928e9 1.04367
\(387\) −3.23993e8 −0.284150
\(388\) 1.00264e7 0.00871435
\(389\) 1.16525e9 1.00368 0.501839 0.864961i \(-0.332657\pi\)
0.501839 + 0.864961i \(0.332657\pi\)
\(390\) −2.70795e8 −0.231161
\(391\) −2.13613e8 −0.180721
\(392\) 4.08077e7 0.0342170
\(393\) 6.37702e7 0.0529961
\(394\) −3.93450e8 −0.324080
\(395\) −2.78460e8 −0.227339
\(396\) −2.33240e8 −0.188742
\(397\) 6.93011e8 0.555870 0.277935 0.960600i \(-0.410350\pi\)
0.277935 + 0.960600i \(0.410350\pi\)
\(398\) 2.32265e9 1.84669
\(399\) −3.24711e8 −0.255913
\(400\) 1.00047e9 0.781616
\(401\) −2.25957e9 −1.74993 −0.874965 0.484186i \(-0.839116\pi\)
−0.874965 + 0.484186i \(0.839116\pi\)
\(402\) 8.32910e8 0.639450
\(403\) −7.86348e8 −0.598476
\(404\) 7.88091e8 0.594623
\(405\) 8.89118e7 0.0665069
\(406\) 9.03932e8 0.670339
\(407\) 2.05175e9 1.50850
\(408\) −1.84765e8 −0.134682
\(409\) 8.72327e7 0.0630446 0.0315223 0.999503i \(-0.489964\pi\)
0.0315223 + 0.999503i \(0.489964\pi\)
\(410\) −4.32735e8 −0.310084
\(411\) −8.49973e7 −0.0603891
\(412\) 3.54843e8 0.249974
\(413\) 1.82248e8 0.127303
\(414\) 3.34395e8 0.231611
\(415\) 1.36925e9 0.940405
\(416\) −5.29491e8 −0.360605
\(417\) −4.33695e8 −0.292893
\(418\) −1.37190e9 −0.918769
\(419\) −8.24265e8 −0.547416 −0.273708 0.961813i \(-0.588250\pi\)
−0.273708 + 0.961813i \(0.588250\pi\)
\(420\) −1.64750e8 −0.108506
\(421\) −1.82215e9 −1.19014 −0.595068 0.803675i \(-0.702875\pi\)
−0.595068 + 0.803675i \(0.702875\pi\)
\(422\) −1.31980e9 −0.854901
\(423\) −3.87552e8 −0.248965
\(424\) 1.13123e9 0.720729
\(425\) 3.03602e8 0.191842
\(426\) 2.06195e9 1.29224
\(427\) −1.57342e9 −0.978019
\(428\) −5.04369e8 −0.310953
\(429\) 9.68918e8 0.592497
\(430\) −9.66909e8 −0.586470
\(431\) 3.24182e9 1.95038 0.975189 0.221375i \(-0.0710544\pi\)
0.975189 + 0.221375i \(0.0710544\pi\)
\(432\) 3.92787e8 0.234403
\(433\) 1.71326e9 1.01418 0.507090 0.861893i \(-0.330721\pi\)
0.507090 + 0.861893i \(0.330721\pi\)
\(434\) −1.96831e9 −1.15579
\(435\) −3.53854e8 −0.206116
\(436\) −2.83949e8 −0.164073
\(437\) 4.78065e8 0.274032
\(438\) 1.02139e9 0.580805
\(439\) −1.70403e9 −0.961283 −0.480642 0.876917i \(-0.659596\pi\)
−0.480642 + 0.876917i \(0.659596\pi\)
\(440\) 1.47169e9 0.823629
\(441\) −2.63258e7 −0.0146166
\(442\) −3.63028e8 −0.199969
\(443\) 8.96154e7 0.0489745 0.0244872 0.999700i \(-0.492205\pi\)
0.0244872 + 0.999700i \(0.492205\pi\)
\(444\) 2.92494e8 0.158590
\(445\) 1.89649e9 1.02021
\(446\) 1.53026e9 0.816757
\(447\) 8.93445e8 0.473142
\(448\) 9.41262e8 0.494581
\(449\) −2.38975e9 −1.24592 −0.622959 0.782255i \(-0.714070\pi\)
−0.622959 + 0.782255i \(0.714070\pi\)
\(450\) −4.75268e8 −0.245864
\(451\) 1.54835e9 0.794786
\(452\) 3.79604e8 0.193351
\(453\) 2.22294e9 1.12353
\(454\) 2.82341e9 1.41605
\(455\) 6.84401e8 0.340620
\(456\) 4.13503e8 0.204222
\(457\) 3.10251e9 1.52057 0.760286 0.649588i \(-0.225059\pi\)
0.760286 + 0.649588i \(0.225059\pi\)
\(458\) 8.96525e8 0.436047
\(459\) 1.19195e8 0.0575326
\(460\) 2.42558e8 0.116189
\(461\) −9.42015e8 −0.447821 −0.223911 0.974610i \(-0.571882\pi\)
−0.223911 + 0.974610i \(0.571882\pi\)
\(462\) 2.42530e9 1.14425
\(463\) −2.76277e9 −1.29363 −0.646817 0.762645i \(-0.723900\pi\)
−0.646817 + 0.762645i \(0.723900\pi\)
\(464\) −1.56323e9 −0.726455
\(465\) 7.70518e8 0.355384
\(466\) 3.14859e9 1.44134
\(467\) 602426. 0.000273712 0 0.000136856 1.00000i \(-0.499956\pi\)
0.000136856 1.00000i \(0.499956\pi\)
\(468\) 1.38127e8 0.0622900
\(469\) −2.10507e9 −0.942242
\(470\) −1.15659e9 −0.513852
\(471\) −1.90798e9 −0.841397
\(472\) −2.32083e8 −0.101589
\(473\) 3.45964e9 1.50320
\(474\) 5.84381e8 0.252041
\(475\) −6.79461e8 −0.290896
\(476\) −2.20864e8 −0.0938643
\(477\) −7.29780e8 −0.307877
\(478\) 2.16079e9 0.904930
\(479\) −2.45402e9 −1.02024 −0.510121 0.860103i \(-0.670399\pi\)
−0.510121 + 0.860103i \(0.670399\pi\)
\(480\) 5.18832e8 0.214133
\(481\) −1.21507e9 −0.497844
\(482\) 3.79346e8 0.154302
\(483\) −8.45142e8 −0.341283
\(484\) 1.68963e9 0.677379
\(485\) 4.08131e7 0.0162444
\(486\) −1.86591e8 −0.0737335
\(487\) −1.78601e9 −0.700701 −0.350350 0.936619i \(-0.613937\pi\)
−0.350350 + 0.936619i \(0.613937\pi\)
\(488\) 2.00367e9 0.780472
\(489\) 1.45217e9 0.561612
\(490\) −7.85655e7 −0.0301679
\(491\) −2.06965e9 −0.789063 −0.394532 0.918882i \(-0.629093\pi\)
−0.394532 + 0.918882i \(0.629093\pi\)
\(492\) 2.20729e8 0.0835569
\(493\) −4.74377e8 −0.178303
\(494\) 8.12456e8 0.303218
\(495\) −9.49413e8 −0.351833
\(496\) 3.40393e9 1.25255
\(497\) −5.21132e9 −1.90415
\(498\) −2.87353e9 −1.04259
\(499\) −1.63444e9 −0.588866 −0.294433 0.955672i \(-0.595131\pi\)
−0.294433 + 0.955672i \(0.595131\pi\)
\(500\) −8.81954e8 −0.315537
\(501\) −1.11507e9 −0.396160
\(502\) 3.85037e9 1.35844
\(503\) −1.96230e9 −0.687507 −0.343754 0.939060i \(-0.611699\pi\)
−0.343754 + 0.939060i \(0.611699\pi\)
\(504\) −7.31008e8 −0.254340
\(505\) 3.20796e9 1.10843
\(506\) −3.57072e9 −1.22526
\(507\) 1.12041e9 0.381810
\(508\) 1.41382e8 0.0478487
\(509\) −1.74020e8 −0.0584906 −0.0292453 0.999572i \(-0.509310\pi\)
−0.0292453 + 0.999572i \(0.509310\pi\)
\(510\) 3.55720e8 0.118744
\(511\) −2.58142e9 −0.855827
\(512\) −5.94396e8 −0.195718
\(513\) −2.66759e8 −0.0872384
\(514\) 3.47860e9 1.12988
\(515\) 1.44440e9 0.465976
\(516\) 4.93200e8 0.158034
\(517\) 4.13834e9 1.31707
\(518\) −3.04145e9 −0.961449
\(519\) −1.44845e9 −0.454797
\(520\) −8.71550e8 −0.271820
\(521\) −1.04646e9 −0.324184 −0.162092 0.986776i \(-0.551824\pi\)
−0.162092 + 0.986776i \(0.551824\pi\)
\(522\) 7.42603e8 0.228513
\(523\) −3.71160e9 −1.13450 −0.567252 0.823545i \(-0.691993\pi\)
−0.567252 + 0.823545i \(0.691993\pi\)
\(524\) −9.70745e7 −0.0294745
\(525\) 1.20118e9 0.362285
\(526\) 2.73534e9 0.819523
\(527\) 1.03296e9 0.307429
\(528\) −4.19423e9 −1.24003
\(529\) −2.16054e9 −0.634553
\(530\) −2.17792e9 −0.635442
\(531\) 1.49721e8 0.0433963
\(532\) 4.94293e8 0.142329
\(533\) −9.16948e8 −0.262301
\(534\) −3.98000e9 −1.13107
\(535\) −2.05306e9 −0.579646
\(536\) 2.68071e9 0.751922
\(537\) 9.79097e8 0.272845
\(538\) −6.20095e9 −1.71680
\(539\) 2.81111e8 0.0773245
\(540\) −1.35347e8 −0.0369887
\(541\) 3.26372e9 0.886180 0.443090 0.896477i \(-0.353882\pi\)
0.443090 + 0.896477i \(0.353882\pi\)
\(542\) −3.24952e9 −0.876642
\(543\) 4.44488e8 0.119141
\(544\) 6.95547e8 0.185238
\(545\) −1.15583e9 −0.305848
\(546\) −1.43629e9 −0.377632
\(547\) 7.46387e8 0.194988 0.0974941 0.995236i \(-0.468917\pi\)
0.0974941 + 0.995236i \(0.468917\pi\)
\(548\) 1.29388e8 0.0335862
\(549\) −1.29261e9 −0.333398
\(550\) 5.07498e9 1.30066
\(551\) 1.06166e9 0.270366
\(552\) 1.07625e9 0.272349
\(553\) −1.47695e9 −0.371388
\(554\) −2.63135e9 −0.657497
\(555\) 1.19061e9 0.295627
\(556\) 6.60195e8 0.162896
\(557\) 5.33461e9 1.30801 0.654003 0.756492i \(-0.273088\pi\)
0.654003 + 0.756492i \(0.273088\pi\)
\(558\) −1.61702e9 −0.393999
\(559\) −2.04884e9 −0.496097
\(560\) −2.96262e9 −0.712883
\(561\) −1.27278e9 −0.304358
\(562\) −3.60759e9 −0.857313
\(563\) 7.90248e9 1.86631 0.933156 0.359472i \(-0.117043\pi\)
0.933156 + 0.359472i \(0.117043\pi\)
\(564\) 5.89954e8 0.138465
\(565\) 1.54520e9 0.360424
\(566\) −4.47802e9 −1.03807
\(567\) 4.71586e8 0.108648
\(568\) 6.63635e9 1.51953
\(569\) 5.54526e9 1.26191 0.630956 0.775819i \(-0.282663\pi\)
0.630956 + 0.775819i \(0.282663\pi\)
\(570\) −7.96101e8 −0.180055
\(571\) 5.09241e9 1.14471 0.572357 0.820004i \(-0.306029\pi\)
0.572357 + 0.820004i \(0.306029\pi\)
\(572\) −1.47494e9 −0.329525
\(573\) 1.89421e9 0.420618
\(574\) −2.29522e9 −0.506562
\(575\) −1.76847e9 −0.387936
\(576\) 7.73271e8 0.168598
\(577\) 1.69420e9 0.367154 0.183577 0.983005i \(-0.441232\pi\)
0.183577 + 0.983005i \(0.441232\pi\)
\(578\) −4.85912e9 −1.04667
\(579\) −2.44855e9 −0.524245
\(580\) 5.38657e8 0.114634
\(581\) 7.26248e9 1.53627
\(582\) −8.56508e7 −0.0180095
\(583\) 7.79269e9 1.62872
\(584\) 3.28732e9 0.682962
\(585\) 5.62253e8 0.116114
\(586\) −4.63039e9 −0.950551
\(587\) −8.65701e9 −1.76659 −0.883293 0.468821i \(-0.844679\pi\)
−0.883293 + 0.468821i \(0.844679\pi\)
\(588\) 4.00746e7 0.00812922
\(589\) −2.31175e9 −0.466163
\(590\) 4.46821e8 0.0895677
\(591\) 8.16921e8 0.162789
\(592\) 5.25977e9 1.04194
\(593\) −7.39803e9 −1.45688 −0.728441 0.685108i \(-0.759755\pi\)
−0.728441 + 0.685108i \(0.759755\pi\)
\(594\) 1.99245e9 0.390063
\(595\) −8.99038e8 −0.174972
\(596\) −1.36005e9 −0.263144
\(597\) −4.82252e9 −0.927607
\(598\) 2.11462e9 0.404370
\(599\) 4.39303e9 0.835162 0.417581 0.908640i \(-0.362878\pi\)
0.417581 + 0.908640i \(0.362878\pi\)
\(600\) −1.52964e9 −0.289108
\(601\) −1.08818e9 −0.204475 −0.102238 0.994760i \(-0.532600\pi\)
−0.102238 + 0.994760i \(0.532600\pi\)
\(602\) −5.12846e9 −0.958075
\(603\) −1.72937e9 −0.321202
\(604\) −3.38388e9 −0.624864
\(605\) 6.87770e9 1.26270
\(606\) −6.73227e9 −1.22887
\(607\) −5.98074e9 −1.08541 −0.542706 0.839923i \(-0.682600\pi\)
−0.542706 + 0.839923i \(0.682600\pi\)
\(608\) −1.55663e9 −0.280882
\(609\) −1.87684e9 −0.336718
\(610\) −3.85759e9 −0.688116
\(611\) −2.45077e9 −0.434669
\(612\) −1.81446e8 −0.0319975
\(613\) −5.60656e9 −0.983070 −0.491535 0.870858i \(-0.663564\pi\)
−0.491535 + 0.870858i \(0.663564\pi\)
\(614\) −2.55129e9 −0.444806
\(615\) 8.98489e8 0.155758
\(616\) 7.80580e9 1.34550
\(617\) −9.07967e9 −1.55622 −0.778112 0.628126i \(-0.783822\pi\)
−0.778112 + 0.628126i \(0.783822\pi\)
\(618\) −3.03125e9 −0.516609
\(619\) −3.61181e9 −0.612079 −0.306040 0.952019i \(-0.599004\pi\)
−0.306040 + 0.952019i \(0.599004\pi\)
\(620\) −1.17293e9 −0.197651
\(621\) −6.94306e8 −0.116340
\(622\) 6.80518e8 0.113390
\(623\) 1.00589e10 1.66665
\(624\) 2.48387e9 0.409245
\(625\) 3.26731e8 0.0535316
\(626\) −4.05741e9 −0.661056
\(627\) 2.84849e9 0.461506
\(628\) 2.90444e9 0.467954
\(629\) 1.59613e9 0.255736
\(630\) 1.40738e9 0.224243
\(631\) 2.73746e9 0.433756 0.216878 0.976199i \(-0.430413\pi\)
0.216878 + 0.976199i \(0.430413\pi\)
\(632\) 1.88082e9 0.296372
\(633\) 2.74031e9 0.429425
\(634\) 6.99895e9 1.09074
\(635\) 5.75502e8 0.0891946
\(636\) 1.11091e9 0.171230
\(637\) −1.66477e8 −0.0255191
\(638\) −7.92963e9 −1.20887
\(639\) −4.28123e9 −0.649106
\(640\) 4.76736e9 0.718867
\(641\) −7.16076e9 −1.07388 −0.536940 0.843620i \(-0.680420\pi\)
−0.536940 + 0.843620i \(0.680420\pi\)
\(642\) 4.30857e9 0.642630
\(643\) −1.59933e9 −0.237247 −0.118623 0.992939i \(-0.537848\pi\)
−0.118623 + 0.992939i \(0.537848\pi\)
\(644\) 1.28652e9 0.189809
\(645\) 2.00760e9 0.294589
\(646\) −1.06725e9 −0.155759
\(647\) −1.56410e9 −0.227038 −0.113519 0.993536i \(-0.536212\pi\)
−0.113519 + 0.993536i \(0.536212\pi\)
\(648\) −6.00542e8 −0.0867023
\(649\) −1.59875e9 −0.229574
\(650\) −3.00546e9 −0.429253
\(651\) 4.08681e9 0.580566
\(652\) −2.21057e9 −0.312348
\(653\) 1.10948e10 1.55928 0.779639 0.626229i \(-0.215402\pi\)
0.779639 + 0.626229i \(0.215402\pi\)
\(654\) 2.42563e9 0.339081
\(655\) −3.95147e8 −0.0549432
\(656\) 3.96927e9 0.548968
\(657\) −2.12071e9 −0.291744
\(658\) −6.13454e9 −0.839443
\(659\) 8.31935e9 1.13238 0.566188 0.824276i \(-0.308418\pi\)
0.566188 + 0.824276i \(0.308418\pi\)
\(660\) 1.44525e9 0.195677
\(661\) 5.78095e9 0.778563 0.389282 0.921119i \(-0.372723\pi\)
0.389282 + 0.921119i \(0.372723\pi\)
\(662\) −9.21778e9 −1.23488
\(663\) 7.53756e8 0.100446
\(664\) −9.24841e9 −1.22597
\(665\) 2.01204e9 0.265315
\(666\) −2.49863e9 −0.327750
\(667\) 2.76323e9 0.360559
\(668\) 1.69742e9 0.220329
\(669\) −3.17728e9 −0.410265
\(670\) −5.16106e9 −0.662944
\(671\) 1.38026e10 1.76373
\(672\) 2.75188e9 0.349813
\(673\) −6.94500e9 −0.878252 −0.439126 0.898425i \(-0.644712\pi\)
−0.439126 + 0.898425i \(0.644712\pi\)
\(674\) 1.67270e9 0.210430
\(675\) 9.86800e8 0.123500
\(676\) −1.70554e9 −0.212349
\(677\) 2.90331e9 0.359611 0.179806 0.983702i \(-0.442453\pi\)
0.179806 + 0.983702i \(0.442453\pi\)
\(678\) −3.24277e9 −0.399587
\(679\) 2.16472e8 0.0265373
\(680\) 1.14488e9 0.139630
\(681\) −5.86226e9 −0.711296
\(682\) 1.72668e10 2.08433
\(683\) −4.42751e9 −0.531726 −0.265863 0.964011i \(-0.585657\pi\)
−0.265863 + 0.964011i \(0.585657\pi\)
\(684\) 4.06075e8 0.0485187
\(685\) 5.26679e8 0.0626078
\(686\) −9.91981e9 −1.17319
\(687\) −1.86146e9 −0.219030
\(688\) 8.86898e9 1.03828
\(689\) −4.61492e9 −0.537523
\(690\) −2.07205e9 −0.240120
\(691\) 3.87979e9 0.447337 0.223669 0.974665i \(-0.428197\pi\)
0.223669 + 0.974665i \(0.428197\pi\)
\(692\) 2.20491e9 0.252941
\(693\) −5.03567e9 −0.574765
\(694\) −7.80989e8 −0.0886925
\(695\) 2.68736e9 0.303654
\(696\) 2.39006e9 0.268705
\(697\) 1.20451e9 0.134740
\(698\) 1.45219e10 1.61633
\(699\) −6.53743e9 −0.723998
\(700\) −1.82850e9 −0.201489
\(701\) −1.56016e10 −1.71063 −0.855315 0.518108i \(-0.826636\pi\)
−0.855315 + 0.518108i \(0.826636\pi\)
\(702\) −1.17995e9 −0.128731
\(703\) −3.57214e9 −0.387779
\(704\) −8.25710e9 −0.891915
\(705\) 2.40144e9 0.258113
\(706\) 5.19046e9 0.555123
\(707\) 1.70150e10 1.81077
\(708\) −2.27914e8 −0.0241354
\(709\) 1.35775e10 1.43073 0.715364 0.698752i \(-0.246261\pi\)
0.715364 + 0.698752i \(0.246261\pi\)
\(710\) −1.27767e10 −1.33972
\(711\) −1.21335e9 −0.126603
\(712\) −1.28096e10 −1.33001
\(713\) −6.01693e9 −0.621672
\(714\) 1.88673e9 0.193984
\(715\) −6.00382e9 −0.614266
\(716\) −1.49044e9 −0.151746
\(717\) −4.48646e9 −0.454555
\(718\) −8.26963e9 −0.833778
\(719\) 1.51786e10 1.52293 0.761466 0.648205i \(-0.224480\pi\)
0.761466 + 0.648205i \(0.224480\pi\)
\(720\) −2.43387e9 −0.243015
\(721\) 7.66109e9 0.761232
\(722\) −9.23529e9 −0.913210
\(723\) −7.87638e8 −0.0775073
\(724\) −6.76625e8 −0.0662618
\(725\) −3.92730e9 −0.382746
\(726\) −1.44336e10 −1.39990
\(727\) −1.49034e10 −1.43852 −0.719260 0.694741i \(-0.755519\pi\)
−0.719260 + 0.694741i \(0.755519\pi\)
\(728\) −4.62268e9 −0.444053
\(729\) 3.87420e8 0.0370370
\(730\) −6.32893e9 −0.602144
\(731\) 2.69138e9 0.254838
\(732\) 1.96768e9 0.185424
\(733\) 1.05230e10 0.986905 0.493453 0.869773i \(-0.335735\pi\)
0.493453 + 0.869773i \(0.335735\pi\)
\(734\) −3.40279e9 −0.317614
\(735\) 1.63126e8 0.0151536
\(736\) −4.05153e9 −0.374582
\(737\) 1.84665e10 1.69921
\(738\) −1.88558e9 −0.172682
\(739\) 1.82290e10 1.66153 0.830764 0.556624i \(-0.187904\pi\)
0.830764 + 0.556624i \(0.187904\pi\)
\(740\) −1.81241e9 −0.164417
\(741\) −1.68690e9 −0.152309
\(742\) −1.15516e10 −1.03808
\(743\) −1.41694e9 −0.126733 −0.0633667 0.997990i \(-0.520184\pi\)
−0.0633667 + 0.997990i \(0.520184\pi\)
\(744\) −5.20435e9 −0.463299
\(745\) −5.53616e9 −0.490526
\(746\) 2.20707e10 1.94639
\(747\) 5.96632e9 0.523702
\(748\) 1.93750e9 0.169273
\(749\) −1.08894e10 −0.946927
\(750\) 7.53409e9 0.652104
\(751\) −1.51027e10 −1.30111 −0.650554 0.759460i \(-0.725463\pi\)
−0.650554 + 0.759460i \(0.725463\pi\)
\(752\) 1.06088e10 0.909716
\(753\) −7.99454e9 −0.682356
\(754\) 4.69601e9 0.398960
\(755\) −1.37743e10 −1.16481
\(756\) −7.17875e8 −0.0604258
\(757\) −8.01056e9 −0.671162 −0.335581 0.942011i \(-0.608933\pi\)
−0.335581 + 0.942011i \(0.608933\pi\)
\(758\) 1.43838e10 1.19959
\(759\) 7.41390e9 0.615461
\(760\) −2.56224e9 −0.211725
\(761\) 9.09051e9 0.747725 0.373863 0.927484i \(-0.378033\pi\)
0.373863 + 0.927484i \(0.378033\pi\)
\(762\) −1.20775e9 −0.0988864
\(763\) −6.13049e9 −0.499642
\(764\) −2.88348e9 −0.233932
\(765\) −7.38583e8 −0.0596464
\(766\) −8.73643e9 −0.702317
\(767\) 9.46795e8 0.0757656
\(768\) −6.33897e9 −0.504957
\(769\) −1.56881e10 −1.24402 −0.622009 0.783010i \(-0.713683\pi\)
−0.622009 + 0.783010i \(0.713683\pi\)
\(770\) −1.50282e10 −1.18629
\(771\) −7.22263e9 −0.567551
\(772\) 3.72732e9 0.291566
\(773\) 2.02623e10 1.57783 0.788916 0.614502i \(-0.210643\pi\)
0.788916 + 0.614502i \(0.210643\pi\)
\(774\) −4.21317e9 −0.326599
\(775\) 8.55170e9 0.659928
\(776\) −2.75666e8 −0.0211771
\(777\) 6.31497e9 0.482945
\(778\) 1.51527e10 1.15362
\(779\) −2.69570e9 −0.204310
\(780\) −8.55893e8 −0.0645785
\(781\) 4.57156e10 3.43389
\(782\) −2.77780e9 −0.207719
\(783\) −1.54187e9 −0.114784
\(784\) 7.20643e8 0.0534089
\(785\) 1.18227e10 0.872311
\(786\) 8.29260e8 0.0609133
\(787\) 2.40184e9 0.175644 0.0878219 0.996136i \(-0.472009\pi\)
0.0878219 + 0.996136i \(0.472009\pi\)
\(788\) −1.24356e9 −0.0905370
\(789\) −5.67939e9 −0.411654
\(790\) −3.62107e9 −0.261301
\(791\) 8.19568e9 0.588800
\(792\) 6.41267e9 0.458671
\(793\) −8.17407e9 −0.582080
\(794\) 9.01184e9 0.638913
\(795\) 4.52202e9 0.319189
\(796\) 7.34111e9 0.515901
\(797\) 2.37890e9 0.166446 0.0832228 0.996531i \(-0.473479\pi\)
0.0832228 + 0.996531i \(0.473479\pi\)
\(798\) −4.22250e9 −0.294144
\(799\) 3.21936e9 0.223283
\(800\) 5.75833e9 0.397632
\(801\) 8.26368e9 0.568146
\(802\) −2.93832e10 −2.01136
\(803\) 2.26452e10 1.54338
\(804\) 2.63255e9 0.178641
\(805\) 5.23685e9 0.353822
\(806\) −1.02256e10 −0.687884
\(807\) 1.28751e10 0.862366
\(808\) −2.16677e10 −1.44502
\(809\) 1.39600e10 0.926967 0.463483 0.886106i \(-0.346599\pi\)
0.463483 + 0.886106i \(0.346599\pi\)
\(810\) 1.15620e9 0.0764425
\(811\) −1.88244e10 −1.23922 −0.619611 0.784909i \(-0.712710\pi\)
−0.619611 + 0.784909i \(0.712710\pi\)
\(812\) 2.85703e9 0.187270
\(813\) 6.74700e9 0.440346
\(814\) 2.66807e10 1.73385
\(815\) −8.99825e9 −0.582246
\(816\) −3.26285e9 −0.210223
\(817\) −6.02331e9 −0.386418
\(818\) 1.13436e9 0.0724629
\(819\) 2.98218e9 0.189688
\(820\) −1.36773e9 −0.0866268
\(821\) 7.81215e9 0.492685 0.246343 0.969183i \(-0.420771\pi\)
0.246343 + 0.969183i \(0.420771\pi\)
\(822\) −1.10529e9 −0.0694107
\(823\) −1.87867e10 −1.17476 −0.587381 0.809310i \(-0.699841\pi\)
−0.587381 + 0.809310i \(0.699841\pi\)
\(824\) −9.75602e9 −0.607474
\(825\) −1.05372e10 −0.653335
\(826\) 2.36993e9 0.146320
\(827\) −1.42506e10 −0.876119 −0.438059 0.898946i \(-0.644334\pi\)
−0.438059 + 0.898946i \(0.644334\pi\)
\(828\) 1.05691e9 0.0647042
\(829\) 7.49011e8 0.0456612 0.0228306 0.999739i \(-0.492732\pi\)
0.0228306 + 0.999739i \(0.492732\pi\)
\(830\) 1.78056e10 1.08089
\(831\) 5.46347e9 0.330267
\(832\) 4.88995e9 0.294356
\(833\) 2.18686e8 0.0131088
\(834\) −5.63972e9 −0.336648
\(835\) 6.90944e9 0.410715
\(836\) −4.33612e9 −0.256673
\(837\) 3.35742e9 0.197910
\(838\) −1.07186e10 −0.629196
\(839\) −2.65131e10 −1.54987 −0.774933 0.632043i \(-0.782217\pi\)
−0.774933 + 0.632043i \(0.782217\pi\)
\(840\) 4.52963e9 0.263685
\(841\) −1.11135e10 −0.644265
\(842\) −2.36950e10 −1.36793
\(843\) 7.49044e9 0.430637
\(844\) −4.17146e9 −0.238830
\(845\) −6.94250e9 −0.395838
\(846\) −5.03968e9 −0.286159
\(847\) 3.64792e10 2.06278
\(848\) 1.99770e10 1.12498
\(849\) 9.29772e9 0.521434
\(850\) 3.94801e9 0.220502
\(851\) −9.29739e9 −0.517140
\(852\) 6.51713e9 0.361009
\(853\) −1.34893e10 −0.744163 −0.372081 0.928200i \(-0.621356\pi\)
−0.372081 + 0.928200i \(0.621356\pi\)
\(854\) −2.04606e10 −1.12413
\(855\) 1.65295e9 0.0904435
\(856\) 1.38671e10 0.755660
\(857\) 2.26740e10 1.23054 0.615269 0.788317i \(-0.289047\pi\)
0.615269 + 0.788317i \(0.289047\pi\)
\(858\) 1.25997e10 0.681011
\(859\) −1.97914e10 −1.06537 −0.532685 0.846314i \(-0.678817\pi\)
−0.532685 + 0.846314i \(0.678817\pi\)
\(860\) −3.05607e9 −0.163840
\(861\) 4.76557e9 0.254451
\(862\) 4.21563e10 2.24175
\(863\) −1.08103e10 −0.572534 −0.286267 0.958150i \(-0.592415\pi\)
−0.286267 + 0.958150i \(0.592415\pi\)
\(864\) 2.26074e9 0.119248
\(865\) 8.97519e9 0.471506
\(866\) 2.22790e10 1.16569
\(867\) 1.00890e10 0.525752
\(868\) −6.22117e9 −0.322889
\(869\) 1.29563e10 0.669751
\(870\) −4.60148e9 −0.236908
\(871\) −1.09361e10 −0.560786
\(872\) 7.80687e9 0.398721
\(873\) 1.77837e8 0.00904633
\(874\) 6.21670e9 0.314970
\(875\) −1.90415e10 −0.960887
\(876\) 3.22826e9 0.162257
\(877\) −8.46440e7 −0.00423738 −0.00211869 0.999998i \(-0.500674\pi\)
−0.00211869 + 0.999998i \(0.500674\pi\)
\(878\) −2.21590e10 −1.10489
\(879\) 9.61408e9 0.477471
\(880\) 2.59892e10 1.28559
\(881\) −3.78092e10 −1.86286 −0.931432 0.363915i \(-0.881440\pi\)
−0.931432 + 0.363915i \(0.881440\pi\)
\(882\) −3.42338e8 −0.0168002
\(883\) 3.77215e10 1.84385 0.921927 0.387364i \(-0.126614\pi\)
0.921927 + 0.387364i \(0.126614\pi\)
\(884\) −1.14741e9 −0.0558645
\(885\) −9.27736e8 −0.0449907
\(886\) 1.16535e9 0.0562908
\(887\) 1.24757e10 0.600250 0.300125 0.953900i \(-0.402972\pi\)
0.300125 + 0.953900i \(0.402972\pi\)
\(888\) −8.04180e9 −0.385397
\(889\) 3.05245e9 0.145711
\(890\) 2.46617e10 1.17262
\(891\) −4.13693e9 −0.195932
\(892\) 4.83664e9 0.228174
\(893\) −7.20493e9 −0.338571
\(894\) 1.16183e10 0.543826
\(895\) −6.06689e9 −0.282869
\(896\) 2.52860e10 1.17436
\(897\) −4.39060e9 −0.203119
\(898\) −3.10760e10 −1.43205
\(899\) −1.33620e10 −0.613356
\(900\) −1.50216e9 −0.0686859
\(901\) 6.06222e9 0.276118
\(902\) 2.01345e10 0.913520
\(903\) 1.06482e10 0.481250
\(904\) −1.04368e10 −0.469870
\(905\) −2.75424e9 −0.123518
\(906\) 2.89068e10 1.29137
\(907\) 1.59259e8 0.00708725 0.00354363 0.999994i \(-0.498872\pi\)
0.00354363 + 0.999994i \(0.498872\pi\)
\(908\) 8.92386e9 0.395596
\(909\) 1.39782e10 0.617275
\(910\) 8.89986e9 0.391506
\(911\) 2.28968e10 1.00337 0.501683 0.865051i \(-0.332714\pi\)
0.501683 + 0.865051i \(0.332714\pi\)
\(912\) 7.30224e9 0.318768
\(913\) −6.37092e10 −2.77048
\(914\) 4.03447e10 1.74773
\(915\) 8.00953e9 0.345647
\(916\) 2.83361e9 0.121817
\(917\) −2.09585e9 −0.0897568
\(918\) 1.55000e9 0.0661275
\(919\) −1.19936e10 −0.509734 −0.254867 0.966976i \(-0.582032\pi\)
−0.254867 + 0.966976i \(0.582032\pi\)
\(920\) −6.66887e9 −0.282355
\(921\) 5.29726e9 0.223430
\(922\) −1.22498e10 −0.514722
\(923\) −2.70733e10 −1.13327
\(924\) 7.66557e9 0.319663
\(925\) 1.32141e10 0.548963
\(926\) −3.59268e10 −1.48689
\(927\) 6.29379e9 0.259497
\(928\) −8.99737e9 −0.369571
\(929\) −5.52148e9 −0.225944 −0.112972 0.993598i \(-0.536037\pi\)
−0.112972 + 0.993598i \(0.536037\pi\)
\(930\) 1.00197e10 0.408475
\(931\) −4.89420e8 −0.0198773
\(932\) 9.95165e9 0.402661
\(933\) −1.41296e9 −0.0569567
\(934\) 7.83387e6 0.000314603 0
\(935\) 7.88670e9 0.315540
\(936\) −3.79766e9 −0.151374
\(937\) 4.28153e10 1.70024 0.850120 0.526589i \(-0.176529\pi\)
0.850120 + 0.526589i \(0.176529\pi\)
\(938\) −2.73741e10 −1.08300
\(939\) 8.42441e9 0.332055
\(940\) −3.65560e9 −0.143553
\(941\) 4.62683e10 1.81017 0.905087 0.425227i \(-0.139806\pi\)
0.905087 + 0.425227i \(0.139806\pi\)
\(942\) −2.48112e10 −0.967095
\(943\) −7.01624e9 −0.272467
\(944\) −4.09847e9 −0.158569
\(945\) −2.92215e9 −0.112639
\(946\) 4.49888e10 1.72777
\(947\) 1.84946e10 0.707651 0.353826 0.935311i \(-0.384881\pi\)
0.353826 + 0.935311i \(0.384881\pi\)
\(948\) 1.84703e9 0.0704117
\(949\) −1.34107e10 −0.509356
\(950\) −8.83563e9 −0.334353
\(951\) −1.45319e10 −0.547888
\(952\) 6.07242e9 0.228104
\(953\) −1.08055e10 −0.404409 −0.202205 0.979343i \(-0.564811\pi\)
−0.202205 + 0.979343i \(0.564811\pi\)
\(954\) −9.48997e9 −0.353871
\(955\) −1.17373e10 −0.436072
\(956\) 6.82953e9 0.252807
\(957\) 1.64643e10 0.607228
\(958\) −3.19117e10 −1.17266
\(959\) 2.79349e9 0.102278
\(960\) −4.79151e9 −0.174793
\(961\) 1.58314e9 0.0575423
\(962\) −1.58006e10 −0.572218
\(963\) −8.94590e9 −0.322799
\(964\) 1.19899e9 0.0431067
\(965\) 1.51723e10 0.543506
\(966\) −1.09901e10 −0.392268
\(967\) 2.41210e10 0.857834 0.428917 0.903344i \(-0.358895\pi\)
0.428917 + 0.903344i \(0.358895\pi\)
\(968\) −4.64544e10 −1.64613
\(969\) 2.21594e9 0.0782393
\(970\) 5.30728e8 0.0186712
\(971\) −3.65187e10 −1.28011 −0.640057 0.768328i \(-0.721089\pi\)
−0.640057 + 0.768328i \(0.721089\pi\)
\(972\) −5.89753e8 −0.0205986
\(973\) 1.42537e10 0.496058
\(974\) −2.32251e10 −0.805379
\(975\) 6.24024e9 0.215618
\(976\) 3.53838e10 1.21823
\(977\) 1.82046e10 0.624524 0.312262 0.949996i \(-0.398913\pi\)
0.312262 + 0.949996i \(0.398913\pi\)
\(978\) 1.88838e10 0.645512
\(979\) −8.82408e10 −3.00559
\(980\) −2.48319e8 −0.00842789
\(981\) −5.03635e9 −0.170324
\(982\) −2.69135e10 −0.906942
\(983\) −4.38690e10 −1.47306 −0.736531 0.676404i \(-0.763537\pi\)
−0.736531 + 0.676404i \(0.763537\pi\)
\(984\) −6.06872e9 −0.203055
\(985\) −5.06199e9 −0.168770
\(986\) −6.16875e9 −0.204940
\(987\) 1.27372e10 0.421660
\(988\) 2.56790e9 0.0847089
\(989\) −1.56772e10 −0.515325
\(990\) −1.23461e10 −0.404394
\(991\) −2.39803e10 −0.782702 −0.391351 0.920242i \(-0.627992\pi\)
−0.391351 + 0.920242i \(0.627992\pi\)
\(992\) 1.95918e10 0.637210
\(993\) 1.91389e10 0.620290
\(994\) −6.77673e10 −2.18861
\(995\) 2.98824e10 0.961688
\(996\) −9.08226e9 −0.291264
\(997\) 5.54581e10 1.77228 0.886139 0.463419i \(-0.153378\pi\)
0.886139 + 0.463419i \(0.153378\pi\)
\(998\) −2.12540e10 −0.676837
\(999\) 5.18791e9 0.164632
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 177.8.a.a.1.12 16
3.2 odd 2 531.8.a.b.1.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.12 16 1.1 even 1 trivial
531.8.a.b.1.5 16 3.2 odd 2