Properties

Label 177.12.a.a.1.1
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $1$
Dimension $26$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(135.996742959\)
Analytic rank: \(1\)
Dimension: \(26\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-87.0020 q^{2} -243.000 q^{3} +5521.36 q^{4} -497.667 q^{5} +21141.5 q^{6} +10880.7 q^{7} -302189. q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-87.0020 q^{2} -243.000 q^{3} +5521.36 q^{4} -497.667 q^{5} +21141.5 q^{6} +10880.7 q^{7} -302189. q^{8} +59049.0 q^{9} +43298.0 q^{10} +310893. q^{11} -1.34169e6 q^{12} -1.09899e6 q^{13} -946641. q^{14} +120933. q^{15} +1.49833e7 q^{16} +7.19696e6 q^{17} -5.13738e6 q^{18} +6.26538e6 q^{19} -2.74780e6 q^{20} -2.64400e6 q^{21} -2.70483e7 q^{22} +1.54620e7 q^{23} +7.34319e7 q^{24} -4.85805e7 q^{25} +9.56141e7 q^{26} -1.43489e7 q^{27} +6.00761e7 q^{28} -7.10429e6 q^{29} -1.05214e7 q^{30} +4.03274e7 q^{31} -6.84697e8 q^{32} -7.55470e7 q^{33} -6.26150e8 q^{34} -5.41495e6 q^{35} +3.26031e8 q^{36} -7.18924e8 q^{37} -5.45101e8 q^{38} +2.67054e8 q^{39} +1.50390e8 q^{40} +3.85411e8 q^{41} +2.30034e8 q^{42} -6.53643e8 q^{43} +1.71655e9 q^{44} -2.93867e7 q^{45} -1.34522e9 q^{46} +5.83771e8 q^{47} -3.64095e9 q^{48} -1.85894e9 q^{49} +4.22660e9 q^{50} -1.74886e9 q^{51} -6.06790e9 q^{52} -4.97322e9 q^{53} +1.24838e9 q^{54} -1.54721e8 q^{55} -3.28802e9 q^{56} -1.52249e9 q^{57} +6.18088e8 q^{58} +7.14924e8 q^{59} +6.67715e8 q^{60} +4.77228e8 q^{61} -3.50857e9 q^{62} +6.42493e8 q^{63} +2.88842e10 q^{64} +5.46930e8 q^{65} +6.57274e9 q^{66} -2.18049e9 q^{67} +3.97370e10 q^{68} -3.75726e9 q^{69} +4.71112e8 q^{70} +1.54140e10 q^{71} -1.78440e10 q^{72} +2.19704e10 q^{73} +6.25478e10 q^{74} +1.18050e10 q^{75} +3.45934e10 q^{76} +3.38273e9 q^{77} -2.32342e10 q^{78} -4.35217e9 q^{79} -7.45671e9 q^{80} +3.48678e9 q^{81} -3.35315e10 q^{82} +6.40974e10 q^{83} -1.45985e10 q^{84} -3.58169e9 q^{85} +5.68683e10 q^{86} +1.72634e9 q^{87} -9.39485e10 q^{88} -4.53166e10 q^{89} +2.55671e9 q^{90} -1.19577e10 q^{91} +8.53710e10 q^{92} -9.79957e9 q^{93} -5.07892e10 q^{94} -3.11807e9 q^{95} +1.66381e11 q^{96} -1.41445e11 q^{97} +1.61731e11 q^{98} +1.83579e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 26q - 78q^{2} - 6318q^{3} + 23070q^{4} + 3808q^{5} + 18954q^{6} - 98819q^{7} - 117645q^{8} + 1535274q^{9} + O(q^{10}) \) \( 26q - 78q^{2} - 6318q^{3} + 23070q^{4} + 3808q^{5} + 18954q^{6} - 98819q^{7} - 117645q^{8} + 1535274q^{9} - 859751q^{10} + 579094q^{11} - 5606010q^{12} - 2018538q^{13} + 4157413q^{14} - 925344q^{15} + 20190274q^{16} - 13084493q^{17} - 4605822q^{18} + 9917231q^{19} + 10165633q^{20} + 24013017q^{21} - 89820518q^{22} - 63513223q^{23} + 28587735q^{24} + 218986852q^{25} - 77999532q^{26} - 373071582q^{27} - 444601862q^{28} + 81530981q^{29} + 208919493q^{30} - 408861231q^{31} - 26253128q^{32} - 140719842q^{33} - 508910076q^{34} - 75731421q^{35} + 1362260430q^{36} - 802381301q^{37} + 732704675q^{38} + 490504734q^{39} - 646130800q^{40} - 1354472849q^{41} - 1010251359q^{42} + 282952194q^{43} + 1846047996q^{44} + 224858592q^{45} + 9629305849q^{46} - 1196794197q^{47} - 4906236582q^{48} + 10889725683q^{49} - 6236232091q^{50} + 3179531799q^{51} - 1968200812q^{52} - 8276044236q^{53} + 1119214746q^{54} - 6672895076q^{55} + 2579741342q^{56} - 2409887133q^{57} - 9401656060q^{58} + 18588031774q^{59} - 2470248819q^{60} - 21181559029q^{61} - 6117706514q^{62} - 5835163131q^{63} + 42975855037q^{64} + 25680681860q^{65} + 21826385874q^{66} + 26234163394q^{67} + 19707344091q^{68} + 15433713189q^{69} + 129203099090q^{70} + 52088830406q^{71} - 6946819605q^{72} + 20943384867q^{73} + 41969200146q^{74} - 53213805036q^{75} + 223987219368q^{76} + 94604773153q^{77} + 18953886276q^{78} + 68965662774q^{79} + 218947784293q^{80} + 90656394426q^{81} + 11938614923q^{82} + 17947446393q^{83} + 108038252466q^{84} - 52849386709q^{85} + 384986147852q^{86} - 19812028383q^{87} - 49061112607q^{88} + 38570593981q^{89} - 50767436799q^{90} - 226268806999q^{91} - 79559686310q^{92} + 99353279133q^{93} - 16709400108q^{94} - 252795831501q^{95} + 6379510104q^{96} - 186894587836q^{97} - 252443311612q^{98} + 34194921606q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −87.0020 −1.92249 −0.961246 0.275693i \(-0.911093\pi\)
−0.961246 + 0.275693i \(0.911093\pi\)
\(3\) −243.000 −0.577350
\(4\) 5521.36 2.69597
\(5\) −497.667 −0.0712203 −0.0356102 0.999366i \(-0.511337\pi\)
−0.0356102 + 0.999366i \(0.511337\pi\)
\(6\) 21141.5 1.10995
\(7\) 10880.7 0.244690 0.122345 0.992488i \(-0.460959\pi\)
0.122345 + 0.992488i \(0.460959\pi\)
\(8\) −302189. −3.26050
\(9\) 59049.0 0.333333
\(10\) 43298.0 0.136920
\(11\) 310893. 0.582038 0.291019 0.956717i \(-0.406006\pi\)
0.291019 + 0.956717i \(0.406006\pi\)
\(12\) −1.34169e6 −1.55652
\(13\) −1.09899e6 −0.820926 −0.410463 0.911877i \(-0.634633\pi\)
−0.410463 + 0.911877i \(0.634633\pi\)
\(14\) −946641. −0.470415
\(15\) 120933. 0.0411191
\(16\) 1.49833e7 3.57230
\(17\) 7.19696e6 1.22936 0.614681 0.788776i \(-0.289285\pi\)
0.614681 + 0.788776i \(0.289285\pi\)
\(18\) −5.13738e6 −0.640831
\(19\) 6.26538e6 0.580501 0.290250 0.956951i \(-0.406261\pi\)
0.290250 + 0.956951i \(0.406261\pi\)
\(20\) −2.74780e6 −0.192008
\(21\) −2.64400e6 −0.141272
\(22\) −2.70483e7 −1.11896
\(23\) 1.54620e7 0.500912 0.250456 0.968128i \(-0.419419\pi\)
0.250456 + 0.968128i \(0.419419\pi\)
\(24\) 7.34319e7 1.88245
\(25\) −4.85805e7 −0.994928
\(26\) 9.56141e7 1.57822
\(27\) −1.43489e7 −0.192450
\(28\) 6.00761e7 0.659679
\(29\) −7.10429e6 −0.0643179 −0.0321589 0.999483i \(-0.510238\pi\)
−0.0321589 + 0.999483i \(0.510238\pi\)
\(30\) −1.05214e7 −0.0790511
\(31\) 4.03274e7 0.252995 0.126497 0.991967i \(-0.459626\pi\)
0.126497 + 0.991967i \(0.459626\pi\)
\(32\) −6.84697e8 −3.60723
\(33\) −7.55470e7 −0.336040
\(34\) −6.26150e8 −2.36344
\(35\) −5.41495e6 −0.0174269
\(36\) 3.26031e8 0.898658
\(37\) −7.18924e8 −1.70441 −0.852204 0.523210i \(-0.824734\pi\)
−0.852204 + 0.523210i \(0.824734\pi\)
\(38\) −5.45101e8 −1.11601
\(39\) 2.67054e8 0.473962
\(40\) 1.50390e8 0.232214
\(41\) 3.85411e8 0.519532 0.259766 0.965672i \(-0.416355\pi\)
0.259766 + 0.965672i \(0.416355\pi\)
\(42\) 2.30034e8 0.271594
\(43\) −6.53643e8 −0.678054 −0.339027 0.940777i \(-0.610098\pi\)
−0.339027 + 0.940777i \(0.610098\pi\)
\(44\) 1.71655e9 1.56916
\(45\) −2.93867e7 −0.0237401
\(46\) −1.34522e9 −0.962999
\(47\) 5.83771e8 0.371282 0.185641 0.982618i \(-0.440564\pi\)
0.185641 + 0.982618i \(0.440564\pi\)
\(48\) −3.64095e9 −2.06247
\(49\) −1.85894e9 −0.940127
\(50\) 4.22660e9 1.91274
\(51\) −1.74886e9 −0.709773
\(52\) −6.06790e9 −2.21320
\(53\) −4.97322e9 −1.63351 −0.816753 0.576987i \(-0.804228\pi\)
−0.816753 + 0.576987i \(0.804228\pi\)
\(54\) 1.24838e9 0.369984
\(55\) −1.54721e8 −0.0414529
\(56\) −3.28802e9 −0.797812
\(57\) −1.52249e9 −0.335152
\(58\) 6.18088e8 0.123651
\(59\) 7.14924e8 0.130189
\(60\) 6.67715e8 0.110856
\(61\) 4.77228e8 0.0723456 0.0361728 0.999346i \(-0.488483\pi\)
0.0361728 + 0.999346i \(0.488483\pi\)
\(62\) −3.50857e9 −0.486380
\(63\) 6.42493e8 0.0815634
\(64\) 2.88842e10 3.36256
\(65\) 5.46930e8 0.0584666
\(66\) 6.57274e9 0.646033
\(67\) −2.18049e9 −0.197307 −0.0986537 0.995122i \(-0.531454\pi\)
−0.0986537 + 0.995122i \(0.531454\pi\)
\(68\) 3.97370e10 3.31433
\(69\) −3.75726e9 −0.289202
\(70\) 4.71112e8 0.0335031
\(71\) 1.54140e10 1.01390 0.506950 0.861976i \(-0.330773\pi\)
0.506950 + 0.861976i \(0.330773\pi\)
\(72\) −1.78440e10 −1.08683
\(73\) 2.19704e10 1.24040 0.620200 0.784444i \(-0.287052\pi\)
0.620200 + 0.784444i \(0.287052\pi\)
\(74\) 6.25478e10 3.27671
\(75\) 1.18050e10 0.574422
\(76\) 3.45934e10 1.56502
\(77\) 3.38273e9 0.142419
\(78\) −2.32342e10 −0.911188
\(79\) −4.35217e9 −0.159132 −0.0795659 0.996830i \(-0.525353\pi\)
−0.0795659 + 0.996830i \(0.525353\pi\)
\(80\) −7.45671e9 −0.254421
\(81\) 3.48678e9 0.111111
\(82\) −3.35315e10 −0.998797
\(83\) 6.40974e10 1.78612 0.893060 0.449938i \(-0.148554\pi\)
0.893060 + 0.449938i \(0.148554\pi\)
\(84\) −1.45985e10 −0.380866
\(85\) −3.58169e9 −0.0875556
\(86\) 5.68683e10 1.30355
\(87\) 1.72634e9 0.0371339
\(88\) −9.39485e10 −1.89773
\(89\) −4.53166e10 −0.860226 −0.430113 0.902775i \(-0.641526\pi\)
−0.430113 + 0.902775i \(0.641526\pi\)
\(90\) 2.55671e9 0.0456402
\(91\) −1.19577e10 −0.200873
\(92\) 8.53710e10 1.35045
\(93\) −9.79957e9 −0.146067
\(94\) −5.07892e10 −0.713787
\(95\) −3.11807e9 −0.0413434
\(96\) 1.66381e11 2.08263
\(97\) −1.41445e11 −1.67241 −0.836204 0.548418i \(-0.815230\pi\)
−0.836204 + 0.548418i \(0.815230\pi\)
\(98\) 1.61731e11 1.80739
\(99\) 1.83579e10 0.194013
\(100\) −2.68230e11 −2.68230
\(101\) 1.59623e11 1.51122 0.755609 0.655023i \(-0.227341\pi\)
0.755609 + 0.655023i \(0.227341\pi\)
\(102\) 1.52154e11 1.36453
\(103\) 1.16182e11 0.987495 0.493748 0.869605i \(-0.335627\pi\)
0.493748 + 0.869605i \(0.335627\pi\)
\(104\) 3.32102e11 2.67663
\(105\) 1.31583e9 0.0100614
\(106\) 4.32681e11 3.14040
\(107\) −1.39522e11 −0.961685 −0.480842 0.876807i \(-0.659669\pi\)
−0.480842 + 0.876807i \(0.659669\pi\)
\(108\) −7.92254e10 −0.518841
\(109\) 4.90855e10 0.305568 0.152784 0.988260i \(-0.451176\pi\)
0.152784 + 0.988260i \(0.451176\pi\)
\(110\) 1.34611e10 0.0796929
\(111\) 1.74698e11 0.984040
\(112\) 1.63029e11 0.874108
\(113\) −5.90391e9 −0.0301445 −0.0150722 0.999886i \(-0.504798\pi\)
−0.0150722 + 0.999886i \(0.504798\pi\)
\(114\) 1.32460e11 0.644328
\(115\) −7.69491e9 −0.0356751
\(116\) −3.92253e10 −0.173399
\(117\) −6.48941e10 −0.273642
\(118\) −6.21999e10 −0.250287
\(119\) 7.83078e10 0.300813
\(120\) −3.65447e10 −0.134069
\(121\) −1.88657e11 −0.661232
\(122\) −4.15199e10 −0.139084
\(123\) −9.36548e10 −0.299952
\(124\) 2.22662e11 0.682067
\(125\) 4.84770e10 0.142079
\(126\) −5.58982e10 −0.156805
\(127\) 1.07144e11 0.287771 0.143886 0.989594i \(-0.454040\pi\)
0.143886 + 0.989594i \(0.454040\pi\)
\(128\) −1.11072e12 −2.85727
\(129\) 1.58835e11 0.391475
\(130\) −4.75840e10 −0.112402
\(131\) −1.95333e11 −0.442367 −0.221183 0.975232i \(-0.570992\pi\)
−0.221183 + 0.975232i \(0.570992\pi\)
\(132\) −4.17122e11 −0.905954
\(133\) 6.81716e10 0.142043
\(134\) 1.89707e11 0.379322
\(135\) 7.14098e9 0.0137064
\(136\) −2.17484e12 −4.00833
\(137\) 7.38374e10 0.130711 0.0653557 0.997862i \(-0.479182\pi\)
0.0653557 + 0.997862i \(0.479182\pi\)
\(138\) 3.26889e11 0.555988
\(139\) 3.14483e11 0.514062 0.257031 0.966403i \(-0.417256\pi\)
0.257031 + 0.966403i \(0.417256\pi\)
\(140\) −2.98979e10 −0.0469825
\(141\) −1.41856e11 −0.214360
\(142\) −1.34105e12 −1.94921
\(143\) −3.41667e11 −0.477810
\(144\) 8.84751e11 1.19077
\(145\) 3.53557e9 0.00458074
\(146\) −1.91147e12 −2.38466
\(147\) 4.51722e11 0.542782
\(148\) −3.96943e12 −4.59504
\(149\) 2.10310e11 0.234604 0.117302 0.993096i \(-0.462575\pi\)
0.117302 + 0.993096i \(0.462575\pi\)
\(150\) −1.02706e12 −1.10432
\(151\) 1.38807e12 1.43893 0.719463 0.694531i \(-0.244388\pi\)
0.719463 + 0.694531i \(0.244388\pi\)
\(152\) −1.89333e12 −1.89272
\(153\) 4.24973e11 0.409787
\(154\) −2.94304e11 −0.273799
\(155\) −2.00696e10 −0.0180184
\(156\) 1.47450e12 1.27779
\(157\) 2.74934e11 0.230028 0.115014 0.993364i \(-0.463309\pi\)
0.115014 + 0.993364i \(0.463309\pi\)
\(158\) 3.78648e11 0.305930
\(159\) 1.20849e12 0.943105
\(160\) 3.40751e11 0.256908
\(161\) 1.68237e11 0.122568
\(162\) −3.03357e11 −0.213610
\(163\) −3.64398e11 −0.248053 −0.124027 0.992279i \(-0.539581\pi\)
−0.124027 + 0.992279i \(0.539581\pi\)
\(164\) 2.12799e12 1.40065
\(165\) 3.75972e10 0.0239328
\(166\) −5.57660e12 −3.43380
\(167\) −2.21384e12 −1.31888 −0.659442 0.751756i \(-0.729207\pi\)
−0.659442 + 0.751756i \(0.729207\pi\)
\(168\) 7.98989e11 0.460617
\(169\) −5.84388e11 −0.326080
\(170\) 3.11614e11 0.168325
\(171\) 3.69965e11 0.193500
\(172\) −3.60900e12 −1.82802
\(173\) −8.02401e11 −0.393675 −0.196838 0.980436i \(-0.563067\pi\)
−0.196838 + 0.980436i \(0.563067\pi\)
\(174\) −1.50195e11 −0.0713897
\(175\) −5.28588e11 −0.243449
\(176\) 4.65821e12 2.07922
\(177\) −1.73727e11 −0.0751646
\(178\) 3.94264e12 1.65378
\(179\) 3.13610e12 1.27555 0.637776 0.770222i \(-0.279855\pi\)
0.637776 + 0.770222i \(0.279855\pi\)
\(180\) −1.62255e11 −0.0640027
\(181\) 5.10220e11 0.195221 0.0976103 0.995225i \(-0.468880\pi\)
0.0976103 + 0.995225i \(0.468880\pi\)
\(182\) 1.04035e12 0.386176
\(183\) −1.15967e11 −0.0417688
\(184\) −4.67243e12 −1.63322
\(185\) 3.57785e11 0.121388
\(186\) 8.52582e11 0.280812
\(187\) 2.23748e12 0.715535
\(188\) 3.22321e12 1.00097
\(189\) −1.56126e11 −0.0470907
\(190\) 2.71279e11 0.0794824
\(191\) −8.22711e11 −0.234187 −0.117094 0.993121i \(-0.537358\pi\)
−0.117094 + 0.993121i \(0.537358\pi\)
\(192\) −7.01886e12 −1.94138
\(193\) 1.70154e12 0.457379 0.228690 0.973499i \(-0.426556\pi\)
0.228690 + 0.973499i \(0.426556\pi\)
\(194\) 1.23060e13 3.21519
\(195\) −1.32904e11 −0.0337557
\(196\) −1.02639e13 −2.53456
\(197\) 1.95873e12 0.470338 0.235169 0.971955i \(-0.424436\pi\)
0.235169 + 0.971955i \(0.424436\pi\)
\(198\) −1.59718e12 −0.372988
\(199\) −4.45516e12 −1.01198 −0.505989 0.862540i \(-0.668872\pi\)
−0.505989 + 0.862540i \(0.668872\pi\)
\(200\) 1.46805e13 3.24396
\(201\) 5.29860e11 0.113916
\(202\) −1.38875e13 −2.90530
\(203\) −7.72995e10 −0.0157380
\(204\) −9.65608e12 −1.91353
\(205\) −1.91806e11 −0.0370013
\(206\) −1.01081e13 −1.89845
\(207\) 9.13013e11 0.166971
\(208\) −1.64665e13 −2.93260
\(209\) 1.94786e12 0.337873
\(210\) −1.14480e11 −0.0193430
\(211\) −2.33831e12 −0.384900 −0.192450 0.981307i \(-0.561643\pi\)
−0.192450 + 0.981307i \(0.561643\pi\)
\(212\) −2.74589e13 −4.40389
\(213\) −3.74561e12 −0.585375
\(214\) 1.21387e13 1.84883
\(215\) 3.25297e11 0.0482912
\(216\) 4.33608e12 0.627483
\(217\) 4.38790e11 0.0619053
\(218\) −4.27054e12 −0.587452
\(219\) −5.33880e12 −0.716145
\(220\) −8.54271e11 −0.111756
\(221\) −7.90936e12 −1.00922
\(222\) −1.51991e13 −1.89181
\(223\) −4.82525e12 −0.585926 −0.292963 0.956124i \(-0.594641\pi\)
−0.292963 + 0.956124i \(0.594641\pi\)
\(224\) −7.44997e12 −0.882653
\(225\) −2.86863e12 −0.331643
\(226\) 5.13652e11 0.0579525
\(227\) 1.20927e13 1.33163 0.665813 0.746119i \(-0.268085\pi\)
0.665813 + 0.746119i \(0.268085\pi\)
\(228\) −8.40620e12 −0.903562
\(229\) 1.31247e13 1.37719 0.688593 0.725148i \(-0.258228\pi\)
0.688593 + 0.725148i \(0.258228\pi\)
\(230\) 6.69473e11 0.0685851
\(231\) −8.22002e11 −0.0822256
\(232\) 2.14684e12 0.209708
\(233\) −1.03669e12 −0.0988990 −0.0494495 0.998777i \(-0.515747\pi\)
−0.0494495 + 0.998777i \(0.515747\pi\)
\(234\) 5.64592e12 0.526075
\(235\) −2.90523e11 −0.0264428
\(236\) 3.94735e12 0.350986
\(237\) 1.05758e12 0.0918748
\(238\) −6.81294e12 −0.578310
\(239\) −1.32504e13 −1.09911 −0.549553 0.835459i \(-0.685202\pi\)
−0.549553 + 0.835459i \(0.685202\pi\)
\(240\) 1.81198e12 0.146890
\(241\) −4.03205e12 −0.319471 −0.159736 0.987160i \(-0.551064\pi\)
−0.159736 + 0.987160i \(0.551064\pi\)
\(242\) 1.64136e13 1.27121
\(243\) −8.47289e11 −0.0641500
\(244\) 2.63495e12 0.195042
\(245\) 9.25132e11 0.0669561
\(246\) 8.14816e12 0.576656
\(247\) −6.88558e12 −0.476548
\(248\) −1.21865e13 −0.824888
\(249\) −1.55757e13 −1.03122
\(250\) −4.21760e12 −0.273146
\(251\) 2.46265e13 1.56026 0.780129 0.625619i \(-0.215153\pi\)
0.780129 + 0.625619i \(0.215153\pi\)
\(252\) 3.54743e12 0.219893
\(253\) 4.80701e12 0.291550
\(254\) −9.32175e12 −0.553238
\(255\) 8.70350e11 0.0505502
\(256\) 3.74804e13 2.13052
\(257\) −2.39787e13 −1.33411 −0.667057 0.745007i \(-0.732446\pi\)
−0.667057 + 0.745007i \(0.732446\pi\)
\(258\) −1.38190e13 −0.752607
\(259\) −7.82238e12 −0.417052
\(260\) 3.01979e12 0.157625
\(261\) −4.19501e11 −0.0214393
\(262\) 1.69943e13 0.850447
\(263\) 5.97421e12 0.292768 0.146384 0.989228i \(-0.453236\pi\)
0.146384 + 0.989228i \(0.453236\pi\)
\(264\) 2.28295e13 1.09566
\(265\) 2.47501e12 0.116339
\(266\) −5.93107e12 −0.273076
\(267\) 1.10119e13 0.496652
\(268\) −1.20393e13 −0.531936
\(269\) 3.10192e12 0.134274 0.0671372 0.997744i \(-0.478613\pi\)
0.0671372 + 0.997744i \(0.478613\pi\)
\(270\) −6.21280e11 −0.0263504
\(271\) 4.47567e12 0.186006 0.0930030 0.995666i \(-0.470353\pi\)
0.0930030 + 0.995666i \(0.470353\pi\)
\(272\) 1.07834e14 4.39166
\(273\) 2.90573e12 0.115974
\(274\) −6.42401e12 −0.251292
\(275\) −1.51033e13 −0.579085
\(276\) −2.07451e13 −0.779680
\(277\) 2.20233e13 0.811417 0.405708 0.914003i \(-0.367025\pi\)
0.405708 + 0.914003i \(0.367025\pi\)
\(278\) −2.73606e13 −0.988279
\(279\) 2.38129e12 0.0843315
\(280\) 1.63634e12 0.0568204
\(281\) −5.04649e13 −1.71832 −0.859162 0.511704i \(-0.829014\pi\)
−0.859162 + 0.511704i \(0.829014\pi\)
\(282\) 1.23418e13 0.412105
\(283\) 1.78252e13 0.583725 0.291863 0.956460i \(-0.405725\pi\)
0.291863 + 0.956460i \(0.405725\pi\)
\(284\) 8.51063e13 2.73345
\(285\) 7.57692e11 0.0238696
\(286\) 2.97258e13 0.918586
\(287\) 4.19353e12 0.127125
\(288\) −4.04307e13 −1.20241
\(289\) 1.75243e13 0.511332
\(290\) −3.07602e11 −0.00880643
\(291\) 3.43711e13 0.965565
\(292\) 1.21306e14 3.34409
\(293\) 4.57165e12 0.123680 0.0618402 0.998086i \(-0.480303\pi\)
0.0618402 + 0.998086i \(0.480303\pi\)
\(294\) −3.93007e13 −1.04349
\(295\) −3.55794e11 −0.00927209
\(296\) 2.17251e14 5.55722
\(297\) −4.46097e12 −0.112013
\(298\) −1.82974e13 −0.451024
\(299\) −1.69925e13 −0.411212
\(300\) 6.51799e13 1.54863
\(301\) −7.11208e12 −0.165913
\(302\) −1.20765e14 −2.76632
\(303\) −3.87883e13 −0.872502
\(304\) 9.38763e13 2.07373
\(305\) −2.37501e11 −0.00515248
\(306\) −3.69735e13 −0.787813
\(307\) −2.93239e13 −0.613706 −0.306853 0.951757i \(-0.599276\pi\)
−0.306853 + 0.951757i \(0.599276\pi\)
\(308\) 1.86772e13 0.383958
\(309\) −2.82323e13 −0.570131
\(310\) 1.74610e12 0.0346401
\(311\) 2.26927e13 0.442286 0.221143 0.975241i \(-0.429021\pi\)
0.221143 + 0.975241i \(0.429021\pi\)
\(312\) −8.07008e13 −1.54535
\(313\) −6.25481e13 −1.17685 −0.588424 0.808552i \(-0.700251\pi\)
−0.588424 + 0.808552i \(0.700251\pi\)
\(314\) −2.39199e13 −0.442227
\(315\) −3.19748e11 −0.00580897
\(316\) −2.40299e13 −0.429015
\(317\) 3.29402e13 0.577963 0.288981 0.957335i \(-0.406683\pi\)
0.288981 + 0.957335i \(0.406683\pi\)
\(318\) −1.05141e14 −1.81311
\(319\) −2.20867e12 −0.0374354
\(320\) −1.43747e13 −0.239483
\(321\) 3.39039e13 0.555229
\(322\) −1.46369e13 −0.235636
\(323\) 4.50917e13 0.713646
\(324\) 1.92518e13 0.299553
\(325\) 5.33893e13 0.816762
\(326\) 3.17034e13 0.476880
\(327\) −1.19278e13 −0.176420
\(328\) −1.16467e14 −1.69393
\(329\) 6.35182e12 0.0908491
\(330\) −3.27104e12 −0.0460107
\(331\) −1.05376e14 −1.45777 −0.728885 0.684636i \(-0.759961\pi\)
−0.728885 + 0.684636i \(0.759961\pi\)
\(332\) 3.53904e14 4.81533
\(333\) −4.24517e13 −0.568136
\(334\) 1.92609e14 2.53554
\(335\) 1.08516e12 0.0140523
\(336\) −3.96160e13 −0.504666
\(337\) −1.28291e14 −1.60780 −0.803900 0.594764i \(-0.797245\pi\)
−0.803900 + 0.594764i \(0.797245\pi\)
\(338\) 5.08429e13 0.626886
\(339\) 1.43465e12 0.0174039
\(340\) −1.97758e13 −0.236048
\(341\) 1.25375e13 0.147252
\(342\) −3.21877e13 −0.372003
\(343\) −4.17411e13 −0.474730
\(344\) 1.97524e14 2.21079
\(345\) 1.86986e12 0.0205970
\(346\) 6.98105e13 0.756837
\(347\) −1.02566e14 −1.09444 −0.547219 0.836989i \(-0.684314\pi\)
−0.547219 + 0.836989i \(0.684314\pi\)
\(348\) 9.53175e12 0.100112
\(349\) −7.67112e12 −0.0793083 −0.0396542 0.999213i \(-0.512626\pi\)
−0.0396542 + 0.999213i \(0.512626\pi\)
\(350\) 4.59882e13 0.468029
\(351\) 1.57693e13 0.157987
\(352\) −2.12868e14 −2.09954
\(353\) −8.27561e13 −0.803598 −0.401799 0.915728i \(-0.631615\pi\)
−0.401799 + 0.915728i \(0.631615\pi\)
\(354\) 1.51146e13 0.144503
\(355\) −7.67105e12 −0.0722102
\(356\) −2.50209e14 −2.31915
\(357\) −1.90288e13 −0.173674
\(358\) −2.72847e14 −2.45224
\(359\) 5.25379e13 0.465000 0.232500 0.972596i \(-0.425309\pi\)
0.232500 + 0.972596i \(0.425309\pi\)
\(360\) 8.88035e12 0.0774045
\(361\) −7.72352e13 −0.663019
\(362\) −4.43902e13 −0.375310
\(363\) 4.58437e13 0.381762
\(364\) −6.60228e13 −0.541548
\(365\) −1.09339e13 −0.0883416
\(366\) 1.00893e13 0.0803001
\(367\) −8.76316e13 −0.687064 −0.343532 0.939141i \(-0.611623\pi\)
−0.343532 + 0.939141i \(0.611623\pi\)
\(368\) 2.31672e14 1.78941
\(369\) 2.27581e13 0.173177
\(370\) −3.11280e13 −0.233368
\(371\) −5.41120e13 −0.399703
\(372\) −5.41069e13 −0.393792
\(373\) −2.72590e14 −1.95484 −0.977419 0.211310i \(-0.932227\pi\)
−0.977419 + 0.211310i \(0.932227\pi\)
\(374\) −1.94666e14 −1.37561
\(375\) −1.17799e13 −0.0820296
\(376\) −1.76409e14 −1.21056
\(377\) 7.80752e12 0.0528002
\(378\) 1.35833e13 0.0905314
\(379\) −6.18379e13 −0.406199 −0.203100 0.979158i \(-0.565101\pi\)
−0.203100 + 0.979158i \(0.565101\pi\)
\(380\) −1.72160e13 −0.111461
\(381\) −2.60360e13 −0.166145
\(382\) 7.15775e13 0.450223
\(383\) 1.20075e14 0.744491 0.372245 0.928134i \(-0.378588\pi\)
0.372245 + 0.928134i \(0.378588\pi\)
\(384\) 2.69906e14 1.64965
\(385\) −1.68347e12 −0.0101431
\(386\) −1.48037e14 −0.879308
\(387\) −3.85970e13 −0.226018
\(388\) −7.80967e14 −4.50877
\(389\) −1.68458e14 −0.958888 −0.479444 0.877573i \(-0.659162\pi\)
−0.479444 + 0.877573i \(0.659162\pi\)
\(390\) 1.15629e13 0.0648951
\(391\) 1.11279e14 0.615802
\(392\) 5.61751e14 3.06528
\(393\) 4.74658e13 0.255401
\(394\) −1.70413e14 −0.904220
\(395\) 2.16593e12 0.0113334
\(396\) 1.01361e14 0.523053
\(397\) 5.51988e13 0.280919 0.140460 0.990086i \(-0.455142\pi\)
0.140460 + 0.990086i \(0.455142\pi\)
\(398\) 3.87608e14 1.94552
\(399\) −1.65657e13 −0.0820085
\(400\) −7.27897e14 −3.55418
\(401\) −7.47170e13 −0.359853 −0.179927 0.983680i \(-0.557586\pi\)
−0.179927 + 0.983680i \(0.557586\pi\)
\(402\) −4.60989e13 −0.219002
\(403\) −4.43193e13 −0.207690
\(404\) 8.81334e14 4.07421
\(405\) −1.73526e12 −0.00791337
\(406\) 6.72521e12 0.0302561
\(407\) −2.23508e14 −0.992029
\(408\) 5.28487e14 2.31421
\(409\) 3.86108e14 1.66813 0.834066 0.551664i \(-0.186007\pi\)
0.834066 + 0.551664i \(0.186007\pi\)
\(410\) 1.66875e13 0.0711346
\(411\) −1.79425e13 −0.0754663
\(412\) 6.41484e14 2.66226
\(413\) 7.77886e12 0.0318560
\(414\) −7.94340e13 −0.321000
\(415\) −3.18991e13 −0.127208
\(416\) 7.52473e14 2.96127
\(417\) −7.64193e13 −0.296794
\(418\) −1.69468e14 −0.649559
\(419\) 2.76663e14 1.04658 0.523291 0.852154i \(-0.324704\pi\)
0.523291 + 0.852154i \(0.324704\pi\)
\(420\) 7.26519e12 0.0271254
\(421\) 5.22220e13 0.192443 0.0962214 0.995360i \(-0.469324\pi\)
0.0962214 + 0.995360i \(0.469324\pi\)
\(422\) 2.03437e14 0.739967
\(423\) 3.44711e13 0.123761
\(424\) 1.50285e15 5.32604
\(425\) −3.49631e14 −1.22313
\(426\) 3.25875e14 1.12538
\(427\) 5.19257e12 0.0177023
\(428\) −7.70352e14 −2.59268
\(429\) 8.30252e13 0.275864
\(430\) −2.83015e13 −0.0928395
\(431\) −4.64077e14 −1.50302 −0.751511 0.659720i \(-0.770675\pi\)
−0.751511 + 0.659720i \(0.770675\pi\)
\(432\) −2.14994e14 −0.687490
\(433\) 7.89752e13 0.249349 0.124674 0.992198i \(-0.460211\pi\)
0.124674 + 0.992198i \(0.460211\pi\)
\(434\) −3.81756e13 −0.119012
\(435\) −8.59144e11 −0.00264469
\(436\) 2.71019e14 0.823803
\(437\) 9.68751e13 0.290780
\(438\) 4.64486e14 1.37678
\(439\) −1.27763e14 −0.373982 −0.186991 0.982362i \(-0.559873\pi\)
−0.186991 + 0.982362i \(0.559873\pi\)
\(440\) 4.67551e13 0.135157
\(441\) −1.09768e14 −0.313376
\(442\) 6.88131e14 1.94021
\(443\) 3.11261e14 0.866772 0.433386 0.901208i \(-0.357319\pi\)
0.433386 + 0.901208i \(0.357319\pi\)
\(444\) 9.64573e14 2.65295
\(445\) 2.25526e13 0.0612656
\(446\) 4.19806e14 1.12644
\(447\) −5.11053e13 −0.135449
\(448\) 3.14279e14 0.822786
\(449\) −7.05942e13 −0.182563 −0.0912817 0.995825i \(-0.529096\pi\)
−0.0912817 + 0.995825i \(0.529096\pi\)
\(450\) 2.49576e14 0.637580
\(451\) 1.19822e14 0.302387
\(452\) −3.25976e13 −0.0812688
\(453\) −3.37301e14 −0.830765
\(454\) −1.05209e15 −2.56004
\(455\) 5.95096e12 0.0143062
\(456\) 4.60079e14 1.09276
\(457\) −3.60614e14 −0.846259 −0.423130 0.906069i \(-0.639069\pi\)
−0.423130 + 0.906069i \(0.639069\pi\)
\(458\) −1.14187e15 −2.64763
\(459\) −1.03268e14 −0.236591
\(460\) −4.24863e13 −0.0961792
\(461\) −7.46336e14 −1.66947 −0.834736 0.550650i \(-0.814380\pi\)
−0.834736 + 0.550650i \(0.814380\pi\)
\(462\) 7.15159e13 0.158078
\(463\) −7.54481e14 −1.64798 −0.823991 0.566602i \(-0.808258\pi\)
−0.823991 + 0.566602i \(0.808258\pi\)
\(464\) −1.06446e14 −0.229763
\(465\) 4.87692e12 0.0104029
\(466\) 9.01942e13 0.190132
\(467\) −2.58798e14 −0.539161 −0.269581 0.962978i \(-0.586885\pi\)
−0.269581 + 0.962978i \(0.586885\pi\)
\(468\) −3.58303e14 −0.737732
\(469\) −2.37253e13 −0.0482792
\(470\) 2.52761e13 0.0508361
\(471\) −6.68091e13 −0.132807
\(472\) −2.16042e14 −0.424481
\(473\) −2.03213e14 −0.394653
\(474\) −9.20114e13 −0.176629
\(475\) −3.04375e14 −0.577556
\(476\) 4.32365e14 0.810984
\(477\) −2.93664e14 −0.544502
\(478\) 1.15281e15 2.11302
\(479\) 9.95737e13 0.180426 0.0902130 0.995922i \(-0.471245\pi\)
0.0902130 + 0.995922i \(0.471245\pi\)
\(480\) −8.28025e13 −0.148326
\(481\) 7.90088e14 1.39919
\(482\) 3.50796e14 0.614181
\(483\) −4.08815e13 −0.0707648
\(484\) −1.04164e15 −1.78266
\(485\) 7.03924e13 0.119109
\(486\) 7.37158e13 0.123328
\(487\) −4.77882e14 −0.790517 −0.395258 0.918570i \(-0.629345\pi\)
−0.395258 + 0.918570i \(0.629345\pi\)
\(488\) −1.44213e14 −0.235883
\(489\) 8.85488e13 0.143214
\(490\) −8.04884e13 −0.128723
\(491\) −1.03256e15 −1.63293 −0.816465 0.577395i \(-0.804069\pi\)
−0.816465 + 0.577395i \(0.804069\pi\)
\(492\) −5.17102e14 −0.808663
\(493\) −5.11293e13 −0.0790700
\(494\) 5.99059e14 0.916160
\(495\) −9.13613e12 −0.0138176
\(496\) 6.04239e14 0.903774
\(497\) 1.67715e14 0.248091
\(498\) 1.35511e15 1.98251
\(499\) −4.81188e13 −0.0696245 −0.0348122 0.999394i \(-0.511083\pi\)
−0.0348122 + 0.999394i \(0.511083\pi\)
\(500\) 2.67659e14 0.383042
\(501\) 5.37964e14 0.761458
\(502\) −2.14255e15 −2.99958
\(503\) −1.16003e15 −1.60637 −0.803184 0.595731i \(-0.796862\pi\)
−0.803184 + 0.595731i \(0.796862\pi\)
\(504\) −1.94154e14 −0.265937
\(505\) −7.94390e13 −0.107629
\(506\) −4.18220e14 −0.560502
\(507\) 1.42006e14 0.188262
\(508\) 5.91580e14 0.775824
\(509\) 3.15698e14 0.409566 0.204783 0.978807i \(-0.434351\pi\)
0.204783 + 0.978807i \(0.434351\pi\)
\(510\) −7.57223e13 −0.0971824
\(511\) 2.39052e14 0.303514
\(512\) −9.86111e14 −1.23863
\(513\) −8.99014e13 −0.111717
\(514\) 2.08619e15 2.56482
\(515\) −5.78201e13 −0.0703297
\(516\) 8.76987e14 1.05541
\(517\) 1.81490e14 0.216100
\(518\) 6.80563e14 0.801779
\(519\) 1.94983e14 0.227288
\(520\) −1.65276e14 −0.190630
\(521\) 1.24006e15 1.41525 0.707626 0.706587i \(-0.249766\pi\)
0.707626 + 0.706587i \(0.249766\pi\)
\(522\) 3.64975e13 0.0412169
\(523\) −1.64034e15 −1.83305 −0.916526 0.399975i \(-0.869019\pi\)
−0.916526 + 0.399975i \(0.869019\pi\)
\(524\) −1.07850e15 −1.19261
\(525\) 1.28447e14 0.140555
\(526\) −5.19769e14 −0.562845
\(527\) 2.90235e14 0.311022
\(528\) −1.13195e15 −1.20044
\(529\) −7.13738e14 −0.749087
\(530\) −2.15331e14 −0.223660
\(531\) 4.22156e13 0.0433963
\(532\) 3.76400e14 0.382944
\(533\) −4.23561e14 −0.426498
\(534\) −9.58062e14 −0.954809
\(535\) 6.94356e13 0.0684915
\(536\) 6.58922e14 0.643320
\(537\) −7.62072e14 −0.736440
\(538\) −2.69874e14 −0.258142
\(539\) −5.77931e14 −0.547189
\(540\) 3.94279e13 0.0369520
\(541\) 1.20176e15 1.11489 0.557444 0.830215i \(-0.311782\pi\)
0.557444 + 0.830215i \(0.311782\pi\)
\(542\) −3.89392e14 −0.357595
\(543\) −1.23984e14 −0.112711
\(544\) −4.92774e15 −4.43459
\(545\) −2.44282e13 −0.0217626
\(546\) −2.52804e14 −0.222959
\(547\) 6.94610e14 0.606472 0.303236 0.952915i \(-0.401933\pi\)
0.303236 + 0.952915i \(0.401933\pi\)
\(548\) 4.07683e14 0.352395
\(549\) 2.81799e13 0.0241152
\(550\) 1.31402e15 1.11329
\(551\) −4.45111e13 −0.0373366
\(552\) 1.13540e15 0.942941
\(553\) −4.73546e13 −0.0389380
\(554\) −1.91607e15 −1.55994
\(555\) −8.69417e13 −0.0700836
\(556\) 1.73637e15 1.38590
\(557\) −4.88417e14 −0.386000 −0.193000 0.981199i \(-0.561822\pi\)
−0.193000 + 0.981199i \(0.561822\pi\)
\(558\) −2.07178e14 −0.162127
\(559\) 7.18346e14 0.556633
\(560\) −8.11340e13 −0.0622542
\(561\) −5.43709e14 −0.413114
\(562\) 4.39055e15 3.30346
\(563\) −2.37293e14 −0.176803 −0.0884013 0.996085i \(-0.528176\pi\)
−0.0884013 + 0.996085i \(0.528176\pi\)
\(564\) −7.83239e14 −0.577909
\(565\) 2.93818e12 0.00214690
\(566\) −1.55083e15 −1.12221
\(567\) 3.79386e13 0.0271878
\(568\) −4.65795e15 −3.30582
\(569\) 1.01923e15 0.716399 0.358199 0.933645i \(-0.383391\pi\)
0.358199 + 0.933645i \(0.383391\pi\)
\(570\) −6.59208e13 −0.0458892
\(571\) 2.20057e15 1.51718 0.758589 0.651569i \(-0.225889\pi\)
0.758589 + 0.651569i \(0.225889\pi\)
\(572\) −1.88647e15 −1.28816
\(573\) 1.99919e14 0.135208
\(574\) −3.64846e14 −0.244396
\(575\) −7.51149e14 −0.498371
\(576\) 1.70558e15 1.12085
\(577\) 1.03069e15 0.670908 0.335454 0.942057i \(-0.391110\pi\)
0.335454 + 0.942057i \(0.391110\pi\)
\(578\) −1.52465e15 −0.983031
\(579\) −4.13474e14 −0.264068
\(580\) 1.95211e13 0.0123496
\(581\) 6.97422e14 0.437046
\(582\) −2.99035e15 −1.85629
\(583\) −1.54614e15 −0.950762
\(584\) −6.63920e15 −4.04432
\(585\) 3.22957e13 0.0194889
\(586\) −3.97743e14 −0.237775
\(587\) 9.95367e14 0.589486 0.294743 0.955577i \(-0.404766\pi\)
0.294743 + 0.955577i \(0.404766\pi\)
\(588\) 2.49412e15 1.46333
\(589\) 2.52667e14 0.146864
\(590\) 3.09548e13 0.0178255
\(591\) −4.75971e14 −0.271550
\(592\) −1.07719e16 −6.08866
\(593\) −3.14138e15 −1.75922 −0.879610 0.475695i \(-0.842197\pi\)
−0.879610 + 0.475695i \(0.842197\pi\)
\(594\) 3.88114e14 0.215344
\(595\) −3.89712e13 −0.0214240
\(596\) 1.16120e15 0.632486
\(597\) 1.08260e15 0.584266
\(598\) 1.47838e15 0.790551
\(599\) 1.13720e15 0.602547 0.301274 0.953538i \(-0.402588\pi\)
0.301274 + 0.953538i \(0.402588\pi\)
\(600\) −3.56736e15 −1.87290
\(601\) −1.10569e15 −0.575208 −0.287604 0.957749i \(-0.592859\pi\)
−0.287604 + 0.957749i \(0.592859\pi\)
\(602\) 6.18766e14 0.318967
\(603\) −1.28756e14 −0.0657692
\(604\) 7.66404e15 3.87931
\(605\) 9.38885e13 0.0470931
\(606\) 3.37466e15 1.67738
\(607\) 3.39894e15 1.67420 0.837098 0.547053i \(-0.184250\pi\)
0.837098 + 0.547053i \(0.184250\pi\)
\(608\) −4.28989e15 −2.09400
\(609\) 1.87838e13 0.00908631
\(610\) 2.06631e13 0.00990560
\(611\) −6.41557e14 −0.304795
\(612\) 2.34643e15 1.10478
\(613\) −2.72226e15 −1.27027 −0.635137 0.772400i \(-0.719056\pi\)
−0.635137 + 0.772400i \(0.719056\pi\)
\(614\) 2.55124e15 1.17984
\(615\) 4.66089e13 0.0213627
\(616\) −1.02222e15 −0.464357
\(617\) −3.28617e15 −1.47952 −0.739761 0.672870i \(-0.765061\pi\)
−0.739761 + 0.672870i \(0.765061\pi\)
\(618\) 2.45627e15 1.09607
\(619\) −2.92121e15 −1.29200 −0.646002 0.763336i \(-0.723560\pi\)
−0.646002 + 0.763336i \(0.723560\pi\)
\(620\) −1.10812e14 −0.0485770
\(621\) −2.21862e14 −0.0964005
\(622\) −1.97431e15 −0.850292
\(623\) −4.93076e14 −0.210489
\(624\) 4.00136e15 1.69314
\(625\) 2.34797e15 0.984809
\(626\) 5.44182e15 2.26248
\(627\) −4.73331e14 −0.195071
\(628\) 1.51801e15 0.620150
\(629\) −5.17406e15 −2.09533
\(630\) 2.78187e13 0.0111677
\(631\) 2.34220e15 0.932098 0.466049 0.884759i \(-0.345677\pi\)
0.466049 + 0.884759i \(0.345677\pi\)
\(632\) 1.31518e15 0.518849
\(633\) 5.68208e14 0.222222
\(634\) −2.86586e15 −1.11113
\(635\) −5.33221e13 −0.0204952
\(636\) 6.67252e15 2.54259
\(637\) 2.04295e15 0.771775
\(638\) 1.92159e14 0.0719693
\(639\) 9.10182e14 0.337966
\(640\) 5.52771e14 0.203496
\(641\) −2.51161e14 −0.0916712 −0.0458356 0.998949i \(-0.514595\pi\)
−0.0458356 + 0.998949i \(0.514595\pi\)
\(642\) −2.94971e15 −1.06742
\(643\) 2.37034e15 0.850452 0.425226 0.905087i \(-0.360195\pi\)
0.425226 + 0.905087i \(0.360195\pi\)
\(644\) 9.28894e14 0.330441
\(645\) −7.90471e13 −0.0278810
\(646\) −3.92307e15 −1.37198
\(647\) 3.65947e15 1.26895 0.634476 0.772943i \(-0.281216\pi\)
0.634476 + 0.772943i \(0.281216\pi\)
\(648\) −1.05367e15 −0.362277
\(649\) 2.22265e14 0.0757749
\(650\) −4.64498e15 −1.57022
\(651\) −1.06626e14 −0.0357410
\(652\) −2.01197e15 −0.668745
\(653\) 3.28323e15 1.08213 0.541063 0.840982i \(-0.318022\pi\)
0.541063 + 0.840982i \(0.318022\pi\)
\(654\) 1.03774e15 0.339165
\(655\) 9.72106e13 0.0315055
\(656\) 5.77474e15 1.85593
\(657\) 1.29733e15 0.413466
\(658\) −5.52621e14 −0.174657
\(659\) 5.06314e15 1.58690 0.793451 0.608634i \(-0.208282\pi\)
0.793451 + 0.608634i \(0.208282\pi\)
\(660\) 2.07588e14 0.0645223
\(661\) 1.00841e15 0.310834 0.155417 0.987849i \(-0.450328\pi\)
0.155417 + 0.987849i \(0.450328\pi\)
\(662\) 9.16795e15 2.80255
\(663\) 1.92198e15 0.582671
\(664\) −1.93695e16 −5.82364
\(665\) −3.39268e13 −0.0101163
\(666\) 3.69339e15 1.09224
\(667\) −1.09846e14 −0.0322176
\(668\) −1.22234e16 −3.55568
\(669\) 1.17253e15 0.338284
\(670\) −9.44112e13 −0.0270154
\(671\) 1.48367e14 0.0421079
\(672\) 1.81034e15 0.509600
\(673\) 3.76567e15 1.05138 0.525690 0.850676i \(-0.323807\pi\)
0.525690 + 0.850676i \(0.323807\pi\)
\(674\) 1.11616e16 3.09098
\(675\) 6.97076e14 0.191474
\(676\) −3.22661e15 −0.879103
\(677\) −6.19641e15 −1.67457 −0.837283 0.546769i \(-0.815858\pi\)
−0.837283 + 0.546769i \(0.815858\pi\)
\(678\) −1.24817e14 −0.0334589
\(679\) −1.53901e15 −0.409222
\(680\) 1.08235e15 0.285475
\(681\) −2.93853e15 −0.768815
\(682\) −1.09079e15 −0.283092
\(683\) 3.26788e15 0.841301 0.420651 0.907223i \(-0.361802\pi\)
0.420651 + 0.907223i \(0.361802\pi\)
\(684\) 2.04271e15 0.521672
\(685\) −3.67465e13 −0.00930931
\(686\) 3.63157e15 0.912665
\(687\) −3.18929e15 −0.795119
\(688\) −9.79376e15 −2.42222
\(689\) 5.46551e15 1.34099
\(690\) −1.62682e14 −0.0395976
\(691\) −2.65936e15 −0.642167 −0.321084 0.947051i \(-0.604047\pi\)
−0.321084 + 0.947051i \(0.604047\pi\)
\(692\) −4.43034e15 −1.06134
\(693\) 1.99747e14 0.0474730
\(694\) 8.92345e15 2.10405
\(695\) −1.56508e14 −0.0366116
\(696\) −5.21682e14 −0.121075
\(697\) 2.77379e15 0.638694
\(698\) 6.67403e14 0.152470
\(699\) 2.51916e14 0.0570993
\(700\) −2.91852e15 −0.656333
\(701\) −8.34269e15 −1.86147 −0.930737 0.365689i \(-0.880833\pi\)
−0.930737 + 0.365689i \(0.880833\pi\)
\(702\) −1.37196e15 −0.303729
\(703\) −4.50433e15 −0.989410
\(704\) 8.97989e15 1.95714
\(705\) 7.05972e13 0.0152668
\(706\) 7.19995e15 1.54491
\(707\) 1.73680e15 0.369780
\(708\) −9.59206e14 −0.202642
\(709\) −5.12010e15 −1.07331 −0.536653 0.843803i \(-0.680312\pi\)
−0.536653 + 0.843803i \(0.680312\pi\)
\(710\) 6.67397e14 0.138824
\(711\) −2.56991e14 −0.0530439
\(712\) 1.36942e16 2.80477
\(713\) 6.23541e14 0.126728
\(714\) 1.65554e15 0.333888
\(715\) 1.70037e14 0.0340298
\(716\) 1.73155e16 3.43885
\(717\) 3.21984e15 0.634569
\(718\) −4.57090e15 −0.893959
\(719\) −2.63545e14 −0.0511500 −0.0255750 0.999673i \(-0.508142\pi\)
−0.0255750 + 0.999673i \(0.508142\pi\)
\(720\) −4.40311e14 −0.0848069
\(721\) 1.26414e15 0.241630
\(722\) 6.71962e15 1.27465
\(723\) 9.79787e14 0.184447
\(724\) 2.81711e15 0.526310
\(725\) 3.45130e14 0.0639916
\(726\) −3.98850e15 −0.733935
\(727\) 2.19833e15 0.401471 0.200735 0.979646i \(-0.435667\pi\)
0.200735 + 0.979646i \(0.435667\pi\)
\(728\) 3.61349e15 0.654945
\(729\) 2.05891e14 0.0370370
\(730\) 9.51274e14 0.169836
\(731\) −4.70424e15 −0.833574
\(732\) −6.40292e14 −0.112608
\(733\) −2.95864e15 −0.516440 −0.258220 0.966086i \(-0.583136\pi\)
−0.258220 + 0.966086i \(0.583136\pi\)
\(734\) 7.62413e15 1.32088
\(735\) −2.24807e14 −0.0386571
\(736\) −1.05868e16 −1.80690
\(737\) −6.77900e14 −0.114840
\(738\) −1.98000e15 −0.332932
\(739\) 2.61427e15 0.436321 0.218161 0.975913i \(-0.429994\pi\)
0.218161 + 0.975913i \(0.429994\pi\)
\(740\) 1.97546e15 0.327260
\(741\) 1.67319e15 0.275135
\(742\) 4.70786e15 0.768426
\(743\) −7.87753e15 −1.27630 −0.638148 0.769913i \(-0.720299\pi\)
−0.638148 + 0.769913i \(0.720299\pi\)
\(744\) 2.96132e15 0.476249
\(745\) −1.04664e14 −0.0167086
\(746\) 2.37159e16 3.75816
\(747\) 3.78488e15 0.595373
\(748\) 1.23539e16 1.92906
\(749\) −1.51810e15 −0.235315
\(750\) 1.02488e15 0.157701
\(751\) −9.92797e15 −1.51649 −0.758247 0.651967i \(-0.773944\pi\)
−0.758247 + 0.651967i \(0.773944\pi\)
\(752\) 8.74683e15 1.32633
\(753\) −5.98423e15 −0.900815
\(754\) −6.79270e14 −0.101508
\(755\) −6.90797e14 −0.102481
\(756\) −8.62026e14 −0.126955
\(757\) 8.71053e15 1.27355 0.636777 0.771048i \(-0.280267\pi\)
0.636777 + 0.771048i \(0.280267\pi\)
\(758\) 5.38002e15 0.780914
\(759\) −1.16810e15 −0.168326
\(760\) 9.42248e14 0.134800
\(761\) 1.30399e16 1.85207 0.926037 0.377432i \(-0.123193\pi\)
0.926037 + 0.377432i \(0.123193\pi\)
\(762\) 2.26519e15 0.319412
\(763\) 5.34084e14 0.0747695
\(764\) −4.54248e15 −0.631363
\(765\) −2.11495e14 −0.0291852
\(766\) −1.04468e16 −1.43128
\(767\) −7.85693e14 −0.106876
\(768\) −9.10774e15 −1.23005
\(769\) −6.96474e15 −0.933921 −0.466961 0.884278i \(-0.654651\pi\)
−0.466961 + 0.884278i \(0.654651\pi\)
\(770\) 1.46465e14 0.0195001
\(771\) 5.82682e15 0.770251
\(772\) 9.39479e15 1.23308
\(773\) 3.66907e15 0.478155 0.239077 0.971001i \(-0.423155\pi\)
0.239077 + 0.971001i \(0.423155\pi\)
\(774\) 3.35802e15 0.434518
\(775\) −1.95913e15 −0.251711
\(776\) 4.27431e16 5.45288
\(777\) 1.90084e15 0.240785
\(778\) 1.46562e16 1.84345
\(779\) 2.41475e15 0.301589
\(780\) −7.33810e14 −0.0910046
\(781\) 4.79211e15 0.590128
\(782\) −9.68151e15 −1.18387
\(783\) 1.01939e14 0.0123780
\(784\) −2.78531e16 −3.35842
\(785\) −1.36826e14 −0.0163827
\(786\) −4.12962e15 −0.491006
\(787\) 1.29771e15 0.153221 0.0766105 0.997061i \(-0.475590\pi\)
0.0766105 + 0.997061i \(0.475590\pi\)
\(788\) 1.08148e16 1.26802
\(789\) −1.45173e15 −0.169030
\(790\) −1.88441e14 −0.0217884
\(791\) −6.42385e13 −0.00737606
\(792\) −5.54756e15 −0.632577
\(793\) −5.24468e14 −0.0593904
\(794\) −4.80241e15 −0.540065
\(795\) −6.01427e14 −0.0671682
\(796\) −2.45985e16 −2.72827
\(797\) 2.62688e15 0.289347 0.144674 0.989479i \(-0.453787\pi\)
0.144674 + 0.989479i \(0.453787\pi\)
\(798\) 1.44125e15 0.157661
\(799\) 4.20137e15 0.456440
\(800\) 3.32629e16 3.58893
\(801\) −2.67590e15 −0.286742
\(802\) 6.50053e15 0.691815
\(803\) 6.83043e15 0.721959
\(804\) 2.92555e15 0.307113
\(805\) −8.37258e13 −0.00872935
\(806\) 3.85587e15 0.399282
\(807\) −7.53767e14 −0.0775234
\(808\) −4.82362e16 −4.92732
\(809\) 7.36382e15 0.747113 0.373556 0.927607i \(-0.378138\pi\)
0.373556 + 0.927607i \(0.378138\pi\)
\(810\) 1.50971e14 0.0152134
\(811\) 1.11764e16 1.11863 0.559314 0.828956i \(-0.311065\pi\)
0.559314 + 0.828956i \(0.311065\pi\)
\(812\) −4.26798e14 −0.0424291
\(813\) −1.08759e15 −0.107391
\(814\) 1.94457e16 1.90717
\(815\) 1.81349e14 0.0176664
\(816\) −2.62038e16 −2.53552
\(817\) −4.09533e15 −0.393611
\(818\) −3.35922e16 −3.20697
\(819\) −7.06092e14 −0.0669575
\(820\) −1.05903e15 −0.0997544
\(821\) −8.65540e15 −0.809841 −0.404921 0.914352i \(-0.632701\pi\)
−0.404921 + 0.914352i \(0.632701\pi\)
\(822\) 1.56103e15 0.145083
\(823\) −1.76556e16 −1.62999 −0.814994 0.579469i \(-0.803260\pi\)
−0.814994 + 0.579469i \(0.803260\pi\)
\(824\) −3.51090e16 −3.21973
\(825\) 3.67011e15 0.334335
\(826\) −6.76777e14 −0.0612428
\(827\) −1.25112e16 −1.12465 −0.562325 0.826916i \(-0.690093\pi\)
−0.562325 + 0.826916i \(0.690093\pi\)
\(828\) 5.04107e15 0.450149
\(829\) 1.85690e16 1.64717 0.823586 0.567192i \(-0.191970\pi\)
0.823586 + 0.567192i \(0.191970\pi\)
\(830\) 2.77529e15 0.244556
\(831\) −5.35166e15 −0.468472
\(832\) −3.17434e16 −2.76042
\(833\) −1.33787e16 −1.15576
\(834\) 6.64863e15 0.570583
\(835\) 1.10176e15 0.0939313
\(836\) 1.07548e16 0.910898
\(837\) −5.78655e14 −0.0486888
\(838\) −2.40702e16 −2.01205
\(839\) −1.68722e16 −1.40114 −0.700569 0.713584i \(-0.747071\pi\)
−0.700569 + 0.713584i \(0.747071\pi\)
\(840\) −3.97631e14 −0.0328053
\(841\) −1.21500e16 −0.995863
\(842\) −4.54342e15 −0.369970
\(843\) 1.22630e16 0.992075
\(844\) −1.29106e16 −1.03768
\(845\) 2.90830e14 0.0232235
\(846\) −2.99905e15 −0.237929
\(847\) −2.05272e15 −0.161797
\(848\) −7.45155e16 −5.83538
\(849\) −4.33152e15 −0.337014
\(850\) 3.04187e16 2.35145
\(851\) −1.11160e16 −0.853758
\(852\) −2.06808e16 −1.57816
\(853\) 3.17535e15 0.240753 0.120377 0.992728i \(-0.461590\pi\)
0.120377 + 0.992728i \(0.461590\pi\)
\(854\) −4.51764e14 −0.0340325
\(855\) −1.84119e14 −0.0137811
\(856\) 4.21621e16 3.13557
\(857\) −3.21870e15 −0.237841 −0.118920 0.992904i \(-0.537943\pi\)
−0.118920 + 0.992904i \(0.537943\pi\)
\(858\) −7.22336e15 −0.530346
\(859\) 1.77583e16 1.29550 0.647751 0.761852i \(-0.275710\pi\)
0.647751 + 0.761852i \(0.275710\pi\)
\(860\) 1.79608e15 0.130192
\(861\) −1.01903e15 −0.0733954
\(862\) 4.03757e16 2.88955
\(863\) −1.16475e16 −0.828274 −0.414137 0.910215i \(-0.635917\pi\)
−0.414137 + 0.910215i \(0.635917\pi\)
\(864\) 9.82465e15 0.694211
\(865\) 3.99329e14 0.0280377
\(866\) −6.87100e15 −0.479371
\(867\) −4.25841e15 −0.295217
\(868\) 2.42271e15 0.166895
\(869\) −1.35306e15 −0.0926207
\(870\) 7.47472e13 0.00508440
\(871\) 2.39634e15 0.161975
\(872\) −1.48331e16 −0.996303
\(873\) −8.35217e15 −0.557469
\(874\) −8.42833e15 −0.559022
\(875\) 5.27463e14 0.0347654
\(876\) −2.94774e16 −1.93071
\(877\) 3.25080e15 0.211589 0.105795 0.994388i \(-0.466261\pi\)
0.105795 + 0.994388i \(0.466261\pi\)
\(878\) 1.11156e16 0.718976
\(879\) −1.11091e15 −0.0714069
\(880\) −2.31824e15 −0.148082
\(881\) 6.06655e15 0.385101 0.192550 0.981287i \(-0.438324\pi\)
0.192550 + 0.981287i \(0.438324\pi\)
\(882\) 9.55008e15 0.602462
\(883\) 2.27045e15 0.142340 0.0711701 0.997464i \(-0.477327\pi\)
0.0711701 + 0.997464i \(0.477327\pi\)
\(884\) −4.36704e16 −2.72082
\(885\) 8.64580e13 0.00535325
\(886\) −2.70804e16 −1.66636
\(887\) −2.68480e16 −1.64185 −0.820924 0.571038i \(-0.806541\pi\)
−0.820924 + 0.571038i \(0.806541\pi\)
\(888\) −5.27920e16 −3.20846
\(889\) 1.16580e15 0.0704148
\(890\) −1.96212e15 −0.117783
\(891\) 1.08402e15 0.0646709
\(892\) −2.66419e16 −1.57964
\(893\) 3.65755e15 0.215530
\(894\) 4.44627e15 0.260399
\(895\) −1.56073e15 −0.0908451
\(896\) −1.20854e16 −0.699146
\(897\) 4.12918e15 0.237413
\(898\) 6.14184e15 0.350977
\(899\) −2.86498e14 −0.0162721
\(900\) −1.58387e16 −0.894100
\(901\) −3.57921e16 −2.00817
\(902\) −1.04247e16 −0.581337
\(903\) 1.72824e15 0.0957901
\(904\) 1.78410e15 0.0982860
\(905\) −2.53920e14 −0.0139037
\(906\) 2.93459e16 1.59714
\(907\) 5.46793e14 0.0295789 0.0147895 0.999891i \(-0.495292\pi\)
0.0147895 + 0.999891i \(0.495292\pi\)
\(908\) 6.67683e16 3.59003
\(909\) 9.42556e15 0.503739
\(910\) −5.17746e14 −0.0275036
\(911\) 8.84768e15 0.467174 0.233587 0.972336i \(-0.424954\pi\)
0.233587 + 0.972336i \(0.424954\pi\)
\(912\) −2.28119e16 −1.19727
\(913\) 1.99274e16 1.03959
\(914\) 3.13742e16 1.62693
\(915\) 5.77127e13 0.00297478
\(916\) 7.24659e16 3.71286
\(917\) −2.12535e15 −0.108243
\(918\) 8.98457e15 0.454844
\(919\) −2.50273e16 −1.25944 −0.629722 0.776821i \(-0.716831\pi\)
−0.629722 + 0.776821i \(0.716831\pi\)
\(920\) 2.32532e15 0.116319
\(921\) 7.12570e15 0.354323
\(922\) 6.49327e16 3.20955
\(923\) −1.69398e16 −0.832337
\(924\) −4.53857e15 −0.221678
\(925\) 3.49256e16 1.69576
\(926\) 6.56414e16 3.16823
\(927\) 6.86045e15 0.329165
\(928\) 4.86429e15 0.232009
\(929\) 3.42467e16 1.62380 0.811901 0.583796i \(-0.198433\pi\)
0.811901 + 0.583796i \(0.198433\pi\)
\(930\) −4.24302e14 −0.0199995
\(931\) −1.16470e16 −0.545744
\(932\) −5.72394e15 −0.266629
\(933\) −5.51432e15 −0.255354
\(934\) 2.25160e16 1.03653
\(935\) −1.11352e15 −0.0509606
\(936\) 1.96103e16 0.892209
\(937\) 3.20191e16 1.44824 0.724121 0.689673i \(-0.242246\pi\)
0.724121 + 0.689673i \(0.242246\pi\)
\(938\) 2.06415e15 0.0928164
\(939\) 1.51992e16 0.679454
\(940\) −1.60408e15 −0.0712892
\(941\) −2.01964e16 −0.892343 −0.446171 0.894948i \(-0.647213\pi\)
−0.446171 + 0.894948i \(0.647213\pi\)
\(942\) 5.81253e15 0.255320
\(943\) 5.95920e15 0.260240
\(944\) 1.07119e16 0.465074
\(945\) 7.76987e13 0.00335381
\(946\) 1.76800e16 0.758717
\(947\) −2.12489e16 −0.906592 −0.453296 0.891360i \(-0.649752\pi\)
−0.453296 + 0.891360i \(0.649752\pi\)
\(948\) 5.83926e15 0.247692
\(949\) −2.41452e16 −1.01828
\(950\) 2.64813e16 1.11035
\(951\) −8.00446e15 −0.333687
\(952\) −2.36637e16 −0.980800
\(953\) 2.57898e16 1.06276 0.531382 0.847132i \(-0.321673\pi\)
0.531382 + 0.847132i \(0.321673\pi\)
\(954\) 2.55494e16 1.04680
\(955\) 4.09436e14 0.0166789
\(956\) −7.31600e16 −2.96316
\(957\) 5.36708e14 0.0216134
\(958\) −8.66312e15 −0.346868
\(959\) 8.03401e14 0.0319838
\(960\) 3.49305e15 0.138265
\(961\) −2.37822e16 −0.935994
\(962\) −6.87393e16 −2.68994
\(963\) −8.23865e15 −0.320562
\(964\) −2.22624e16 −0.861286
\(965\) −8.46799e14 −0.0325747
\(966\) 3.55677e15 0.136045
\(967\) −6.39376e15 −0.243170 −0.121585 0.992581i \(-0.538798\pi\)
−0.121585 + 0.992581i \(0.538798\pi\)
\(968\) 5.70101e16 2.15595
\(969\) −1.09573e16 −0.412024
\(970\) −6.12428e15 −0.228987
\(971\) 1.52562e16 0.567205 0.283602 0.958942i \(-0.408470\pi\)
0.283602 + 0.958942i \(0.408470\pi\)
\(972\) −4.67818e15 −0.172947
\(973\) 3.42178e15 0.125786
\(974\) 4.15767e16 1.51976
\(975\) −1.29736e16 −0.471558
\(976\) 7.15047e15 0.258441
\(977\) −3.95039e16 −1.41978 −0.709889 0.704314i \(-0.751255\pi\)
−0.709889 + 0.704314i \(0.751255\pi\)
\(978\) −7.70393e15 −0.275327
\(979\) −1.40886e16 −0.500684
\(980\) 5.10798e15 0.180512
\(981\) 2.89845e15 0.101856
\(982\) 8.98349e16 3.13929
\(983\) −3.17929e16 −1.10480 −0.552402 0.833578i \(-0.686289\pi\)
−0.552402 + 0.833578i \(0.686289\pi\)
\(984\) 2.83015e16 0.977993
\(985\) −9.74794e14 −0.0334976
\(986\) 4.44835e15 0.152011
\(987\) −1.54349e15 −0.0524518
\(988\) −3.80177e16 −1.28476
\(989\) −1.01066e16 −0.339645
\(990\) 7.94862e14 0.0265643
\(991\) −3.94441e16 −1.31092 −0.655461 0.755229i \(-0.727525\pi\)
−0.655461 + 0.755229i \(0.727525\pi\)
\(992\) −2.76121e16 −0.912609
\(993\) 2.56064e16 0.841644
\(994\) −1.45915e16 −0.476953
\(995\) 2.21718e15 0.0720734
\(996\) −8.59987e16 −2.78013
\(997\) −4.52291e16 −1.45410 −0.727050 0.686584i \(-0.759109\pi\)
−0.727050 + 0.686584i \(0.759109\pi\)
\(998\) 4.18644e15 0.133853
\(999\) 1.03158e16 0.328013
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 177.12.a.a.1.1 26
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.a.1.1 26 1.1 even 1 trivial