Properties

Label 177.12.a.a.1.3
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 1.3
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-70.0902 q^{2} -243.000 q^{3} +2864.64 q^{4} +3425.74 q^{5} +17031.9 q^{6} -19416.7 q^{7} -57238.6 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-70.0902 q^{2} -243.000 q^{3} +2864.64 q^{4} +3425.74 q^{5} +17031.9 q^{6} -19416.7 q^{7} -57238.6 q^{8} +59049.0 q^{9} -240111. q^{10} -662510. q^{11} -696108. q^{12} +1.45490e6 q^{13} +1.36092e6 q^{14} -832455. q^{15} -1.85492e6 q^{16} +1.01042e7 q^{17} -4.13876e6 q^{18} +4.12705e6 q^{19} +9.81352e6 q^{20} +4.71827e6 q^{21} +4.64355e7 q^{22} -8.18826e6 q^{23} +1.39090e7 q^{24} -3.70924e7 q^{25} -1.01974e8 q^{26} -1.43489e7 q^{27} -5.56220e7 q^{28} -1.12718e8 q^{29} +5.83470e7 q^{30} -6.52662e7 q^{31} +2.47236e8 q^{32} +1.60990e8 q^{33} -7.08205e8 q^{34} -6.65167e7 q^{35} +1.69154e8 q^{36} -6.08937e6 q^{37} -2.89266e8 q^{38} -3.53541e8 q^{39} -1.96085e8 q^{40} -6.32776e8 q^{41} -3.30704e8 q^{42} +9.00775e8 q^{43} -1.89786e9 q^{44} +2.02287e8 q^{45} +5.73917e8 q^{46} -1.96782e9 q^{47} +4.50745e8 q^{48} -1.60032e9 q^{49} +2.59982e9 q^{50} -2.45532e9 q^{51} +4.16777e9 q^{52} +5.33676e9 q^{53} +1.00572e9 q^{54} -2.26959e9 q^{55} +1.11139e9 q^{56} -1.00287e9 q^{57} +7.90045e9 q^{58} +7.14924e8 q^{59} -2.38469e9 q^{60} -6.94170e9 q^{61} +4.57452e9 q^{62} -1.14654e9 q^{63} -1.35300e10 q^{64} +4.98411e9 q^{65} -1.12838e10 q^{66} -4.15991e9 q^{67} +2.89449e10 q^{68} +1.98975e9 q^{69} +4.66217e9 q^{70} +1.26410e10 q^{71} -3.37988e9 q^{72} +1.33068e10 q^{73} +4.26805e8 q^{74} +9.01346e9 q^{75} +1.18225e10 q^{76} +1.28638e10 q^{77} +2.47798e10 q^{78} +3.13876e10 q^{79} -6.35447e9 q^{80} +3.48678e9 q^{81} +4.43514e10 q^{82} +3.96569e9 q^{83} +1.35161e10 q^{84} +3.46144e10 q^{85} -6.31356e10 q^{86} +2.73905e10 q^{87} +3.79212e10 q^{88} +3.45441e10 q^{89} -1.41783e10 q^{90} -2.82494e10 q^{91} -2.34564e10 q^{92} +1.58597e10 q^{93} +1.37925e11 q^{94} +1.41382e10 q^{95} -6.00784e10 q^{96} +2.67913e10 q^{97} +1.12167e11 q^{98} -3.91206e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 26q - 78q^{2} - 6318q^{3} + 23070q^{4} + 3808q^{5} + 18954q^{6} - 98819q^{7} - 117645q^{8} + 1535274q^{9} + O(q^{10}) \) \( 26q - 78q^{2} - 6318q^{3} + 23070q^{4} + 3808q^{5} + 18954q^{6} - 98819q^{7} - 117645q^{8} + 1535274q^{9} - 859751q^{10} + 579094q^{11} - 5606010q^{12} - 2018538q^{13} + 4157413q^{14} - 925344q^{15} + 20190274q^{16} - 13084493q^{17} - 4605822q^{18} + 9917231q^{19} + 10165633q^{20} + 24013017q^{21} - 89820518q^{22} - 63513223q^{23} + 28587735q^{24} + 218986852q^{25} - 77999532q^{26} - 373071582q^{27} - 444601862q^{28} + 81530981q^{29} + 208919493q^{30} - 408861231q^{31} - 26253128q^{32} - 140719842q^{33} - 508910076q^{34} - 75731421q^{35} + 1362260430q^{36} - 802381301q^{37} + 732704675q^{38} + 490504734q^{39} - 646130800q^{40} - 1354472849q^{41} - 1010251359q^{42} + 282952194q^{43} + 1846047996q^{44} + 224858592q^{45} + 9629305849q^{46} - 1196794197q^{47} - 4906236582q^{48} + 10889725683q^{49} - 6236232091q^{50} + 3179531799q^{51} - 1968200812q^{52} - 8276044236q^{53} + 1119214746q^{54} - 6672895076q^{55} + 2579741342q^{56} - 2409887133q^{57} - 9401656060q^{58} + 18588031774q^{59} - 2470248819q^{60} - 21181559029q^{61} - 6117706514q^{62} - 5835163131q^{63} + 42975855037q^{64} + 25680681860q^{65} + 21826385874q^{66} + 26234163394q^{67} + 19707344091q^{68} + 15433713189q^{69} + 129203099090q^{70} + 52088830406q^{71} - 6946819605q^{72} + 20943384867q^{73} + 41969200146q^{74} - 53213805036q^{75} + 223987219368q^{76} + 94604773153q^{77} + 18953886276q^{78} + 68965662774q^{79} + 218947784293q^{80} + 90656394426q^{81} + 11938614923q^{82} + 17947446393q^{83} + 108038252466q^{84} - 52849386709q^{85} + 384986147852q^{86} - 19812028383q^{87} - 49061112607q^{88} + 38570593981q^{89} - 50767436799q^{90} - 226268806999q^{91} - 79559686310q^{92} + 99353279133q^{93} - 16709400108q^{94} - 252795831501q^{95} + 6379510104q^{96} - 186894587836q^{97} - 252443311612q^{98} + 34194921606q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −70.0902 −1.54879 −0.774395 0.632702i \(-0.781946\pi\)
−0.774395 + 0.632702i \(0.781946\pi\)
\(3\) −243.000 −0.577350
\(4\) 2864.64 1.39875
\(5\) 3425.74 0.490252 0.245126 0.969491i \(-0.421171\pi\)
0.245126 + 0.969491i \(0.421171\pi\)
\(6\) 17031.9 0.894194
\(7\) −19416.7 −0.436654 −0.218327 0.975876i \(-0.570060\pi\)
−0.218327 + 0.975876i \(0.570060\pi\)
\(8\) −57238.6 −0.617582
\(9\) 59049.0 0.333333
\(10\) −240111. −0.759298
\(11\) −662510. −1.24032 −0.620159 0.784476i \(-0.712932\pi\)
−0.620159 + 0.784476i \(0.712932\pi\)
\(12\) −696108. −0.807569
\(13\) 1.45490e6 1.08679 0.543394 0.839478i \(-0.317139\pi\)
0.543394 + 0.839478i \(0.317139\pi\)
\(14\) 1.36092e6 0.676285
\(15\) −832455. −0.283047
\(16\) −1.85492e6 −0.442247
\(17\) 1.01042e7 1.72597 0.862984 0.505232i \(-0.168593\pi\)
0.862984 + 0.505232i \(0.168593\pi\)
\(18\) −4.13876e6 −0.516263
\(19\) 4.12705e6 0.382380 0.191190 0.981553i \(-0.438765\pi\)
0.191190 + 0.981553i \(0.438765\pi\)
\(20\) 9.81352e6 0.685741
\(21\) 4.71827e6 0.252102
\(22\) 4.64355e7 1.92099
\(23\) −8.18826e6 −0.265270 −0.132635 0.991165i \(-0.542344\pi\)
−0.132635 + 0.991165i \(0.542344\pi\)
\(24\) 1.39090e7 0.356561
\(25\) −3.70924e7 −0.759653
\(26\) −1.01974e8 −1.68321
\(27\) −1.43489e7 −0.192450
\(28\) −5.56220e7 −0.610770
\(29\) −1.12718e8 −1.02048 −0.510241 0.860032i \(-0.670444\pi\)
−0.510241 + 0.860032i \(0.670444\pi\)
\(30\) 5.83470e7 0.438381
\(31\) −6.52662e7 −0.409448 −0.204724 0.978820i \(-0.565630\pi\)
−0.204724 + 0.978820i \(0.565630\pi\)
\(32\) 2.47236e8 1.30253
\(33\) 1.60990e8 0.716098
\(34\) −7.08205e8 −2.67316
\(35\) −6.65167e7 −0.214070
\(36\) 1.69154e8 0.466250
\(37\) −6.08937e6 −0.0144365 −0.00721826 0.999974i \(-0.502298\pi\)
−0.00721826 + 0.999974i \(0.502298\pi\)
\(38\) −2.89266e8 −0.592226
\(39\) −3.53541e8 −0.627457
\(40\) −1.96085e8 −0.302771
\(41\) −6.32776e8 −0.852980 −0.426490 0.904492i \(-0.640250\pi\)
−0.426490 + 0.904492i \(0.640250\pi\)
\(42\) −3.30704e8 −0.390453
\(43\) 9.00775e8 0.934415 0.467208 0.884148i \(-0.345260\pi\)
0.467208 + 0.884148i \(0.345260\pi\)
\(44\) −1.89786e9 −1.73490
\(45\) 2.02287e8 0.163417
\(46\) 5.73917e8 0.410848
\(47\) −1.96782e9 −1.25154 −0.625772 0.780006i \(-0.715216\pi\)
−0.625772 + 0.780006i \(0.715216\pi\)
\(48\) 4.50745e8 0.255331
\(49\) −1.60032e9 −0.809334
\(50\) 2.59982e9 1.17654
\(51\) −2.45532e9 −0.996488
\(52\) 4.16777e9 1.52015
\(53\) 5.33676e9 1.75291 0.876456 0.481482i \(-0.159901\pi\)
0.876456 + 0.481482i \(0.159901\pi\)
\(54\) 1.00572e9 0.298065
\(55\) −2.26959e9 −0.608069
\(56\) 1.11139e9 0.269669
\(57\) −1.00287e9 −0.220767
\(58\) 7.90045e9 1.58051
\(59\) 7.14924e8 0.130189
\(60\) −2.38469e9 −0.395913
\(61\) −6.94170e9 −1.05233 −0.526164 0.850383i \(-0.676370\pi\)
−0.526164 + 0.850383i \(0.676370\pi\)
\(62\) 4.57452e9 0.634149
\(63\) −1.14654e9 −0.145551
\(64\) −1.35300e10 −1.57510
\(65\) 4.98411e9 0.532800
\(66\) −1.12838e10 −1.10909
\(67\) −4.15991e9 −0.376420 −0.188210 0.982129i \(-0.560268\pi\)
−0.188210 + 0.982129i \(0.560268\pi\)
\(68\) 2.89449e10 2.41420
\(69\) 1.98975e9 0.153154
\(70\) 4.66217e9 0.331550
\(71\) 1.26410e10 0.831498 0.415749 0.909479i \(-0.363519\pi\)
0.415749 + 0.909479i \(0.363519\pi\)
\(72\) −3.37988e9 −0.205861
\(73\) 1.33068e10 0.751274 0.375637 0.926767i \(-0.377424\pi\)
0.375637 + 0.926767i \(0.377424\pi\)
\(74\) 4.26805e8 0.0223592
\(75\) 9.01346e9 0.438586
\(76\) 1.18225e10 0.534854
\(77\) 1.28638e10 0.541589
\(78\) 2.47798e10 0.971800
\(79\) 3.13876e10 1.14765 0.573824 0.818979i \(-0.305459\pi\)
0.573824 + 0.818979i \(0.305459\pi\)
\(80\) −6.35447e9 −0.216812
\(81\) 3.48678e9 0.111111
\(82\) 4.43514e10 1.32109
\(83\) 3.96569e9 0.110507 0.0552535 0.998472i \(-0.482403\pi\)
0.0552535 + 0.998472i \(0.482403\pi\)
\(84\) 1.35161e10 0.352628
\(85\) 3.46144e10 0.846160
\(86\) −6.31356e10 −1.44721
\(87\) 2.73905e10 0.589175
\(88\) 3.79212e10 0.765997
\(89\) 3.45441e10 0.655736 0.327868 0.944724i \(-0.393670\pi\)
0.327868 + 0.944724i \(0.393670\pi\)
\(90\) −1.41783e10 −0.253099
\(91\) −2.82494e10 −0.474550
\(92\) −2.34564e10 −0.371047
\(93\) 1.58597e10 0.236395
\(94\) 1.37925e11 1.93838
\(95\) 1.41382e10 0.187462
\(96\) −6.00784e10 −0.752015
\(97\) 2.67913e10 0.316774 0.158387 0.987377i \(-0.449371\pi\)
0.158387 + 0.987377i \(0.449371\pi\)
\(98\) 1.12167e11 1.25349
\(99\) −3.91206e10 −0.413439
\(100\) −1.06256e11 −1.06256
\(101\) 4.49865e9 0.0425907 0.0212954 0.999773i \(-0.493221\pi\)
0.0212954 + 0.999773i \(0.493221\pi\)
\(102\) 1.72094e11 1.54335
\(103\) 9.34612e10 0.794377 0.397188 0.917737i \(-0.369986\pi\)
0.397188 + 0.917737i \(0.369986\pi\)
\(104\) −8.32765e10 −0.671180
\(105\) 1.61636e10 0.123594
\(106\) −3.74055e11 −2.71489
\(107\) 9.02495e10 0.622063 0.311031 0.950400i \(-0.399326\pi\)
0.311031 + 0.950400i \(0.399326\pi\)
\(108\) −4.11045e10 −0.269190
\(109\) 3.59947e10 0.224074 0.112037 0.993704i \(-0.464262\pi\)
0.112037 + 0.993704i \(0.464262\pi\)
\(110\) 1.59076e11 0.941771
\(111\) 1.47972e9 0.00833493
\(112\) 3.60164e10 0.193109
\(113\) −1.23124e11 −0.628653 −0.314326 0.949315i \(-0.601779\pi\)
−0.314326 + 0.949315i \(0.601779\pi\)
\(114\) 7.02916e10 0.341922
\(115\) −2.80509e10 −0.130049
\(116\) −3.22897e11 −1.42740
\(117\) 8.59104e10 0.362263
\(118\) −5.01092e10 −0.201635
\(119\) −1.96191e11 −0.753650
\(120\) 4.76486e10 0.174805
\(121\) 1.53608e11 0.538388
\(122\) 4.86545e11 1.62984
\(123\) 1.53765e11 0.492468
\(124\) −1.86964e11 −0.572716
\(125\) −2.94342e11 −0.862674
\(126\) 8.03612e10 0.225428
\(127\) 1.28688e11 0.345636 0.172818 0.984954i \(-0.444713\pi\)
0.172818 + 0.984954i \(0.444713\pi\)
\(128\) 4.41980e11 1.13697
\(129\) −2.18888e11 −0.539485
\(130\) −3.49338e11 −0.825196
\(131\) 6.02975e11 1.36555 0.682774 0.730630i \(-0.260773\pi\)
0.682774 + 0.730630i \(0.260773\pi\)
\(132\) 4.61179e11 1.00164
\(133\) −8.01338e10 −0.166967
\(134\) 2.91569e11 0.582995
\(135\) −4.91557e10 −0.0943491
\(136\) −5.78350e11 −1.06593
\(137\) 9.53644e10 0.168820 0.0844099 0.996431i \(-0.473099\pi\)
0.0844099 + 0.996431i \(0.473099\pi\)
\(138\) −1.39462e11 −0.237203
\(139\) −1.03821e12 −1.69709 −0.848543 0.529127i \(-0.822519\pi\)
−0.848543 + 0.529127i \(0.822519\pi\)
\(140\) −1.90547e11 −0.299431
\(141\) 4.78179e11 0.722580
\(142\) −8.86012e11 −1.28782
\(143\) −9.63887e11 −1.34796
\(144\) −1.09531e11 −0.147416
\(145\) −3.86144e11 −0.500293
\(146\) −9.32678e11 −1.16357
\(147\) 3.88877e11 0.467269
\(148\) −1.74439e10 −0.0201931
\(149\) 8.53346e11 0.951920 0.475960 0.879467i \(-0.342101\pi\)
0.475960 + 0.879467i \(0.342101\pi\)
\(150\) −6.31755e11 −0.679277
\(151\) −7.65158e11 −0.793191 −0.396595 0.917994i \(-0.629808\pi\)
−0.396595 + 0.917994i \(0.629808\pi\)
\(152\) −2.36227e11 −0.236151
\(153\) 5.96643e11 0.575323
\(154\) −9.01626e11 −0.838808
\(155\) −2.23585e11 −0.200733
\(156\) −1.01277e12 −0.877657
\(157\) −4.07737e11 −0.341139 −0.170570 0.985346i \(-0.554561\pi\)
−0.170570 + 0.985346i \(0.554561\pi\)
\(158\) −2.19996e12 −1.77747
\(159\) −1.29683e12 −1.01204
\(160\) 8.46968e11 0.638568
\(161\) 1.58989e11 0.115831
\(162\) −2.44390e11 −0.172088
\(163\) −2.63213e12 −1.79174 −0.895871 0.444314i \(-0.853448\pi\)
−0.895871 + 0.444314i \(0.853448\pi\)
\(164\) −1.81268e12 −1.19311
\(165\) 5.51510e11 0.351069
\(166\) −2.77956e11 −0.171152
\(167\) 2.43531e12 1.45082 0.725411 0.688316i \(-0.241650\pi\)
0.725411 + 0.688316i \(0.241650\pi\)
\(168\) −2.70067e11 −0.155694
\(169\) 3.24575e11 0.181108
\(170\) −2.42613e12 −1.31052
\(171\) 2.43698e11 0.127460
\(172\) 2.58040e12 1.30701
\(173\) 2.68448e12 1.31706 0.658531 0.752554i \(-0.271178\pi\)
0.658531 + 0.752554i \(0.271178\pi\)
\(174\) −1.91981e12 −0.912509
\(175\) 7.20214e11 0.331705
\(176\) 1.22890e12 0.548526
\(177\) −1.73727e11 −0.0751646
\(178\) −2.42120e12 −1.01560
\(179\) 2.10047e12 0.854330 0.427165 0.904174i \(-0.359512\pi\)
0.427165 + 0.904174i \(0.359512\pi\)
\(180\) 5.79479e11 0.228580
\(181\) 1.55084e12 0.593383 0.296691 0.954973i \(-0.404117\pi\)
0.296691 + 0.954973i \(0.404117\pi\)
\(182\) 1.98001e12 0.734978
\(183\) 1.68683e12 0.607562
\(184\) 4.68685e11 0.163826
\(185\) −2.08606e10 −0.00707754
\(186\) −1.11161e12 −0.366126
\(187\) −6.69413e12 −2.14075
\(188\) −5.63709e12 −1.75060
\(189\) 2.78609e11 0.0840340
\(190\) −9.90950e11 −0.290340
\(191\) −1.58977e12 −0.452532 −0.226266 0.974065i \(-0.572652\pi\)
−0.226266 + 0.974065i \(0.572652\pi\)
\(192\) 3.28779e12 0.909383
\(193\) 6.85823e10 0.0184351 0.00921757 0.999958i \(-0.497066\pi\)
0.00921757 + 0.999958i \(0.497066\pi\)
\(194\) −1.87781e12 −0.490616
\(195\) −1.21114e12 −0.307612
\(196\) −4.58434e12 −1.13206
\(197\) −8.56601e11 −0.205691 −0.102845 0.994697i \(-0.532795\pi\)
−0.102845 + 0.994697i \(0.532795\pi\)
\(198\) 2.74197e12 0.640331
\(199\) −7.42003e12 −1.68544 −0.842721 0.538350i \(-0.819048\pi\)
−0.842721 + 0.538350i \(0.819048\pi\)
\(200\) 2.12312e12 0.469148
\(201\) 1.01086e12 0.217326
\(202\) −3.15312e11 −0.0659641
\(203\) 2.18862e12 0.445597
\(204\) −7.03361e12 −1.39384
\(205\) −2.16773e12 −0.418175
\(206\) −6.55072e12 −1.23032
\(207\) −4.83509e11 −0.0884235
\(208\) −2.69872e12 −0.480628
\(209\) −2.73421e12 −0.474272
\(210\) −1.13291e12 −0.191421
\(211\) −3.72312e12 −0.612850 −0.306425 0.951895i \(-0.599133\pi\)
−0.306425 + 0.951895i \(0.599133\pi\)
\(212\) 1.52879e13 2.45189
\(213\) −3.07177e12 −0.480066
\(214\) −6.32561e12 −0.963445
\(215\) 3.08582e12 0.458099
\(216\) 8.21312e11 0.118854
\(217\) 1.26726e12 0.178787
\(218\) −2.52287e12 −0.347044
\(219\) −3.23356e12 −0.433748
\(220\) −6.50156e12 −0.850537
\(221\) 1.47006e13 1.87576
\(222\) −1.03714e11 −0.0129091
\(223\) −1.17804e13 −1.43049 −0.715243 0.698875i \(-0.753684\pi\)
−0.715243 + 0.698875i \(0.753684\pi\)
\(224\) −4.80052e12 −0.568754
\(225\) −2.19027e12 −0.253218
\(226\) 8.62978e12 0.973651
\(227\) −1.59234e13 −1.75345 −0.876724 0.480994i \(-0.840276\pi\)
−0.876724 + 0.480994i \(0.840276\pi\)
\(228\) −2.87287e12 −0.308798
\(229\) −1.55582e13 −1.63254 −0.816272 0.577668i \(-0.803963\pi\)
−0.816272 + 0.577668i \(0.803963\pi\)
\(230\) 1.96609e12 0.201419
\(231\) −3.12590e12 −0.312687
\(232\) 6.45184e12 0.630231
\(233\) 3.92685e12 0.374616 0.187308 0.982301i \(-0.440024\pi\)
0.187308 + 0.982301i \(0.440024\pi\)
\(234\) −6.02148e12 −0.561069
\(235\) −6.74123e12 −0.613573
\(236\) 2.04800e12 0.182102
\(237\) −7.62718e12 −0.662595
\(238\) 1.37510e13 1.16725
\(239\) −2.94919e12 −0.244632 −0.122316 0.992491i \(-0.539032\pi\)
−0.122316 + 0.992491i \(0.539032\pi\)
\(240\) 1.54414e12 0.125177
\(241\) 1.62453e13 1.28716 0.643580 0.765378i \(-0.277448\pi\)
0.643580 + 0.765378i \(0.277448\pi\)
\(242\) −1.07664e13 −0.833850
\(243\) −8.47289e11 −0.0641500
\(244\) −1.98855e13 −1.47195
\(245\) −5.48227e12 −0.396778
\(246\) −1.07774e13 −0.762730
\(247\) 6.00444e12 0.415565
\(248\) 3.73575e12 0.252868
\(249\) −9.63664e11 −0.0638012
\(250\) 2.06305e13 1.33610
\(251\) 4.69952e12 0.297747 0.148874 0.988856i \(-0.452435\pi\)
0.148874 + 0.988856i \(0.452435\pi\)
\(252\) −3.28442e12 −0.203590
\(253\) 5.42481e12 0.329020
\(254\) −9.01980e12 −0.535318
\(255\) −8.41129e12 −0.488530
\(256\) −3.26907e12 −0.185825
\(257\) 3.31576e13 1.84480 0.922402 0.386231i \(-0.126223\pi\)
0.922402 + 0.386231i \(0.126223\pi\)
\(258\) 1.53419e13 0.835549
\(259\) 1.18236e11 0.00630376
\(260\) 1.42777e13 0.745255
\(261\) −6.65590e12 −0.340160
\(262\) −4.22626e13 −2.11495
\(263\) −1.73861e13 −0.852012 −0.426006 0.904720i \(-0.640080\pi\)
−0.426006 + 0.904720i \(0.640080\pi\)
\(264\) −9.21485e12 −0.442249
\(265\) 1.82824e13 0.859369
\(266\) 5.61660e12 0.258597
\(267\) −8.39422e12 −0.378589
\(268\) −1.19166e13 −0.526517
\(269\) 4.37856e12 0.189537 0.0947684 0.995499i \(-0.469789\pi\)
0.0947684 + 0.995499i \(0.469789\pi\)
\(270\) 3.44533e12 0.146127
\(271\) 3.90100e13 1.62123 0.810615 0.585579i \(-0.199133\pi\)
0.810615 + 0.585579i \(0.199133\pi\)
\(272\) −1.87424e13 −0.763303
\(273\) 6.86461e12 0.273981
\(274\) −6.68412e12 −0.261466
\(275\) 2.45741e13 0.942211
\(276\) 5.69992e12 0.214224
\(277\) 4.37063e13 1.61030 0.805148 0.593074i \(-0.202086\pi\)
0.805148 + 0.593074i \(0.202086\pi\)
\(278\) 7.27684e13 2.62843
\(279\) −3.85390e12 −0.136483
\(280\) 3.80733e12 0.132206
\(281\) 3.37117e13 1.14788 0.573940 0.818898i \(-0.305414\pi\)
0.573940 + 0.818898i \(0.305414\pi\)
\(282\) −3.35157e13 −1.11912
\(283\) −5.53312e13 −1.81194 −0.905972 0.423338i \(-0.860858\pi\)
−0.905972 + 0.423338i \(0.860858\pi\)
\(284\) 3.62120e13 1.16306
\(285\) −3.43558e12 −0.108231
\(286\) 6.75591e13 2.08771
\(287\) 1.22864e13 0.372457
\(288\) 1.45991e13 0.434176
\(289\) 6.78229e13 1.97896
\(290\) 2.70649e13 0.774849
\(291\) −6.51029e12 −0.182889
\(292\) 3.81193e13 1.05085
\(293\) −3.18979e13 −0.862959 −0.431479 0.902123i \(-0.642008\pi\)
−0.431479 + 0.902123i \(0.642008\pi\)
\(294\) −2.72565e13 −0.723702
\(295\) 2.44915e12 0.0638254
\(296\) 3.48547e11 0.00891573
\(297\) 9.50630e12 0.238699
\(298\) −5.98112e13 −1.47432
\(299\) −1.19131e13 −0.288293
\(300\) 2.58203e13 0.613472
\(301\) −1.74901e13 −0.408016
\(302\) 5.36301e13 1.22849
\(303\) −1.09317e12 −0.0245898
\(304\) −7.65533e12 −0.169106
\(305\) −2.37805e13 −0.515907
\(306\) −4.18188e13 −0.891054
\(307\) −2.27684e13 −0.476509 −0.238255 0.971203i \(-0.576575\pi\)
−0.238255 + 0.971203i \(0.576575\pi\)
\(308\) 3.68502e13 0.757548
\(309\) −2.27111e13 −0.458634
\(310\) 1.56711e13 0.310893
\(311\) −9.71409e13 −1.89330 −0.946651 0.322261i \(-0.895557\pi\)
−0.946651 + 0.322261i \(0.895557\pi\)
\(312\) 2.02362e13 0.387506
\(313\) −5.13532e13 −0.966214 −0.483107 0.875561i \(-0.660492\pi\)
−0.483107 + 0.875561i \(0.660492\pi\)
\(314\) 2.85784e13 0.528353
\(315\) −3.92775e12 −0.0713568
\(316\) 8.99141e13 1.60527
\(317\) −1.29652e13 −0.227486 −0.113743 0.993510i \(-0.536284\pi\)
−0.113743 + 0.993510i \(0.536284\pi\)
\(318\) 9.08953e13 1.56744
\(319\) 7.46770e13 1.26572
\(320\) −4.63502e13 −0.772195
\(321\) −2.19306e13 −0.359148
\(322\) −1.11436e13 −0.179398
\(323\) 4.17005e13 0.659975
\(324\) 9.98839e12 0.155417
\(325\) −5.39658e13 −0.825581
\(326\) 1.84487e14 2.77503
\(327\) −8.74670e12 −0.129369
\(328\) 3.62192e13 0.526785
\(329\) 3.82086e13 0.546491
\(330\) −3.86555e13 −0.543732
\(331\) −6.87540e13 −0.951140 −0.475570 0.879678i \(-0.657758\pi\)
−0.475570 + 0.879678i \(0.657758\pi\)
\(332\) 1.13603e13 0.154572
\(333\) −3.59571e11 −0.00481218
\(334\) −1.70692e14 −2.24702
\(335\) −1.42508e13 −0.184541
\(336\) −8.75199e12 −0.111491
\(337\) −8.03327e13 −1.00676 −0.503382 0.864064i \(-0.667911\pi\)
−0.503382 + 0.864064i \(0.667911\pi\)
\(338\) −2.27495e13 −0.280498
\(339\) 2.99191e13 0.362953
\(340\) 9.91578e13 1.18357
\(341\) 4.32395e13 0.507846
\(342\) −1.70809e13 −0.197409
\(343\) 6.94662e13 0.790052
\(344\) −5.15592e13 −0.577078
\(345\) 6.81636e12 0.0750841
\(346\) −1.88156e14 −2.03985
\(347\) 6.13215e13 0.654336 0.327168 0.944966i \(-0.393906\pi\)
0.327168 + 0.944966i \(0.393906\pi\)
\(348\) 7.84641e13 0.824109
\(349\) −2.53450e12 −0.0262031 −0.0131015 0.999914i \(-0.504170\pi\)
−0.0131015 + 0.999914i \(0.504170\pi\)
\(350\) −5.04800e13 −0.513742
\(351\) −2.08762e13 −0.209152
\(352\) −1.63797e14 −1.61555
\(353\) 1.10956e14 1.07743 0.538714 0.842489i \(-0.318910\pi\)
0.538714 + 0.842489i \(0.318910\pi\)
\(354\) 1.21765e13 0.116414
\(355\) 4.33049e13 0.407644
\(356\) 9.89565e13 0.917211
\(357\) 4.76743e13 0.435120
\(358\) −1.47223e14 −1.32318
\(359\) 3.70842e13 0.328223 0.164112 0.986442i \(-0.447524\pi\)
0.164112 + 0.986442i \(0.447524\pi\)
\(360\) −1.15786e13 −0.100924
\(361\) −9.94577e13 −0.853786
\(362\) −1.08699e14 −0.919026
\(363\) −3.73268e13 −0.310838
\(364\) −8.09245e13 −0.663777
\(365\) 4.55857e13 0.368314
\(366\) −1.18230e14 −0.940987
\(367\) −9.94812e13 −0.779970 −0.389985 0.920821i \(-0.627520\pi\)
−0.389985 + 0.920821i \(0.627520\pi\)
\(368\) 1.51886e13 0.117315
\(369\) −3.73648e13 −0.284327
\(370\) 1.46212e12 0.0109616
\(371\) −1.03622e14 −0.765415
\(372\) 4.54323e13 0.330658
\(373\) −3.99225e13 −0.286299 −0.143149 0.989701i \(-0.545723\pi\)
−0.143149 + 0.989701i \(0.545723\pi\)
\(374\) 4.69194e14 3.31557
\(375\) 7.15250e13 0.498065
\(376\) 1.12635e14 0.772931
\(377\) −1.63994e14 −1.10905
\(378\) −1.95278e13 −0.130151
\(379\) −1.52327e14 −1.00060 −0.500299 0.865853i \(-0.666777\pi\)
−0.500299 + 0.865853i \(0.666777\pi\)
\(380\) 4.05009e13 0.262213
\(381\) −3.12713e13 −0.199553
\(382\) 1.11427e14 0.700878
\(383\) −1.88218e14 −1.16699 −0.583496 0.812116i \(-0.698316\pi\)
−0.583496 + 0.812116i \(0.698316\pi\)
\(384\) −1.07401e14 −0.656428
\(385\) 4.40680e13 0.265515
\(386\) −4.80695e12 −0.0285522
\(387\) 5.31899e13 0.311472
\(388\) 7.67475e13 0.443088
\(389\) −2.80258e14 −1.59527 −0.797636 0.603139i \(-0.793916\pi\)
−0.797636 + 0.603139i \(0.793916\pi\)
\(390\) 8.48891e13 0.476427
\(391\) −8.27358e13 −0.457848
\(392\) 9.16000e13 0.499830
\(393\) −1.46523e14 −0.788399
\(394\) 6.00394e13 0.318571
\(395\) 1.07526e14 0.562637
\(396\) −1.12066e14 −0.578299
\(397\) −2.00577e14 −1.02078 −0.510392 0.859942i \(-0.670500\pi\)
−0.510392 + 0.859942i \(0.670500\pi\)
\(398\) 5.20072e14 2.61040
\(399\) 1.94725e13 0.0963987
\(400\) 6.88034e13 0.335954
\(401\) 8.95389e13 0.431239 0.215619 0.976477i \(-0.430823\pi\)
0.215619 + 0.976477i \(0.430823\pi\)
\(402\) −7.08512e13 −0.336592
\(403\) −9.49557e13 −0.444983
\(404\) 1.28870e13 0.0595738
\(405\) 1.19448e13 0.0544725
\(406\) −1.53401e14 −0.690136
\(407\) 4.03427e12 0.0179059
\(408\) 1.40539e14 0.615413
\(409\) −5.10887e12 −0.0220723 −0.0110361 0.999939i \(-0.503513\pi\)
−0.0110361 + 0.999939i \(0.503513\pi\)
\(410\) 1.51936e14 0.647666
\(411\) −2.31736e13 −0.0974681
\(412\) 2.67733e14 1.11114
\(413\) −1.38815e13 −0.0568475
\(414\) 3.38893e13 0.136949
\(415\) 1.35854e13 0.0541763
\(416\) 3.59704e14 1.41557
\(417\) 2.52285e14 0.979813
\(418\) 1.91642e14 0.734548
\(419\) −2.25293e13 −0.0852257 −0.0426129 0.999092i \(-0.513568\pi\)
−0.0426129 + 0.999092i \(0.513568\pi\)
\(420\) 4.63028e13 0.172877
\(421\) 1.05584e14 0.389087 0.194544 0.980894i \(-0.437677\pi\)
0.194544 + 0.980894i \(0.437677\pi\)
\(422\) 2.60955e14 0.949176
\(423\) −1.16198e14 −0.417181
\(424\) −3.05469e14 −1.08257
\(425\) −3.74789e14 −1.31114
\(426\) 2.15301e14 0.743521
\(427\) 1.34785e14 0.459503
\(428\) 2.58533e14 0.870111
\(429\) 2.34224e14 0.778246
\(430\) −2.16286e14 −0.709500
\(431\) −2.21242e14 −0.716544 −0.358272 0.933617i \(-0.616634\pi\)
−0.358272 + 0.933617i \(0.616634\pi\)
\(432\) 2.66160e13 0.0851104
\(433\) 2.77234e14 0.875311 0.437656 0.899143i \(-0.355809\pi\)
0.437656 + 0.899143i \(0.355809\pi\)
\(434\) −8.88223e13 −0.276903
\(435\) 9.38329e13 0.288844
\(436\) 1.03112e14 0.313424
\(437\) −3.37934e13 −0.101434
\(438\) 2.26641e14 0.671785
\(439\) −5.73176e14 −1.67777 −0.838887 0.544306i \(-0.816793\pi\)
−0.838887 + 0.544306i \(0.816793\pi\)
\(440\) 1.29908e14 0.375532
\(441\) −9.44971e13 −0.269778
\(442\) −1.03037e15 −2.90516
\(443\) −3.67632e14 −1.02375 −0.511874 0.859061i \(-0.671049\pi\)
−0.511874 + 0.859061i \(0.671049\pi\)
\(444\) 4.23886e12 0.0116585
\(445\) 1.18339e14 0.321476
\(446\) 8.25692e14 2.21552
\(447\) −2.07363e14 −0.549591
\(448\) 2.62708e14 0.687772
\(449\) −1.71819e14 −0.444341 −0.222171 0.975008i \(-0.571314\pi\)
−0.222171 + 0.975008i \(0.571314\pi\)
\(450\) 1.53517e14 0.392181
\(451\) 4.19221e14 1.05797
\(452\) −3.52706e14 −0.879328
\(453\) 1.85933e14 0.457949
\(454\) 1.11607e15 2.71572
\(455\) −9.67752e13 −0.232649
\(456\) 5.74031e13 0.136342
\(457\) 4.34071e14 1.01864 0.509322 0.860576i \(-0.329896\pi\)
0.509322 + 0.860576i \(0.329896\pi\)
\(458\) 1.09048e15 2.52847
\(459\) −1.44984e14 −0.332163
\(460\) −8.03557e13 −0.181907
\(461\) −3.43568e14 −0.768524 −0.384262 0.923224i \(-0.625544\pi\)
−0.384262 + 0.923224i \(0.625544\pi\)
\(462\) 2.19095e14 0.484286
\(463\) −6.49507e14 −1.41869 −0.709347 0.704860i \(-0.751010\pi\)
−0.709347 + 0.704860i \(0.751010\pi\)
\(464\) 2.09083e14 0.451305
\(465\) 5.43312e13 0.115893
\(466\) −2.75234e14 −0.580202
\(467\) −4.46394e14 −0.929983 −0.464992 0.885315i \(-0.653943\pi\)
−0.464992 + 0.885315i \(0.653943\pi\)
\(468\) 2.46103e14 0.506715
\(469\) 8.07718e13 0.164365
\(470\) 4.72494e14 0.950295
\(471\) 9.90800e13 0.196957
\(472\) −4.09213e13 −0.0804023
\(473\) −5.96773e14 −1.15897
\(474\) 5.34591e14 1.02622
\(475\) −1.53082e14 −0.290476
\(476\) −5.62016e14 −1.05417
\(477\) 3.15130e14 0.584304
\(478\) 2.06709e14 0.378884
\(479\) 1.92556e13 0.0348910 0.0174455 0.999848i \(-0.494447\pi\)
0.0174455 + 0.999848i \(0.494447\pi\)
\(480\) −2.05813e14 −0.368677
\(481\) −8.85942e12 −0.0156894
\(482\) −1.13863e15 −1.99354
\(483\) −3.86344e13 −0.0668752
\(484\) 4.40033e14 0.753071
\(485\) 9.17801e13 0.155299
\(486\) 5.93867e13 0.0993549
\(487\) −6.15329e14 −1.01788 −0.508942 0.860801i \(-0.669963\pi\)
−0.508942 + 0.860801i \(0.669963\pi\)
\(488\) 3.97333e14 0.649899
\(489\) 6.39608e14 1.03446
\(490\) 3.84254e14 0.614525
\(491\) 9.75978e14 1.54345 0.771724 0.635958i \(-0.219395\pi\)
0.771724 + 0.635958i \(0.219395\pi\)
\(492\) 4.40480e14 0.688840
\(493\) −1.13893e15 −1.76132
\(494\) −4.20853e14 −0.643624
\(495\) −1.34017e14 −0.202690
\(496\) 1.21063e14 0.181077
\(497\) −2.45447e14 −0.363077
\(498\) 6.75434e13 0.0988147
\(499\) −7.05491e14 −1.02079 −0.510397 0.859939i \(-0.670502\pi\)
−0.510397 + 0.859939i \(0.670502\pi\)
\(500\) −8.43183e14 −1.20667
\(501\) −5.91781e14 −0.837633
\(502\) −3.29391e14 −0.461148
\(503\) 1.04627e15 1.44883 0.724417 0.689362i \(-0.242109\pi\)
0.724417 + 0.689362i \(0.242109\pi\)
\(504\) 6.56263e13 0.0898897
\(505\) 1.54112e13 0.0208802
\(506\) −3.80226e14 −0.509582
\(507\) −7.88717e13 −0.104563
\(508\) 3.68646e14 0.483459
\(509\) −7.28565e13 −0.0945194 −0.0472597 0.998883i \(-0.515049\pi\)
−0.0472597 + 0.998883i \(0.515049\pi\)
\(510\) 5.89549e14 0.756631
\(511\) −2.58375e14 −0.328047
\(512\) −6.76045e14 −0.849162
\(513\) −5.92186e13 −0.0735890
\(514\) −2.32402e15 −2.85722
\(515\) 3.20174e14 0.389445
\(516\) −6.27037e14 −0.754605
\(517\) 1.30370e15 1.55231
\(518\) −8.28717e12 −0.00976320
\(519\) −6.52328e14 −0.760406
\(520\) −2.85284e14 −0.329048
\(521\) −9.60568e14 −1.09628 −0.548139 0.836387i \(-0.684663\pi\)
−0.548139 + 0.836387i \(0.684663\pi\)
\(522\) 4.66513e14 0.526837
\(523\) 1.46279e15 1.63465 0.817323 0.576180i \(-0.195457\pi\)
0.817323 + 0.576180i \(0.195457\pi\)
\(524\) 1.72731e15 1.91006
\(525\) −1.75012e14 −0.191510
\(526\) 1.21860e15 1.31959
\(527\) −6.59462e14 −0.706694
\(528\) −2.98623e14 −0.316692
\(529\) −8.85762e14 −0.929632
\(530\) −1.28141e15 −1.33098
\(531\) 4.22156e13 0.0433963
\(532\) −2.29555e14 −0.233546
\(533\) −9.20626e14 −0.927008
\(534\) 5.88353e14 0.586355
\(535\) 3.09172e14 0.304968
\(536\) 2.38107e14 0.232470
\(537\) −5.10415e14 −0.493248
\(538\) −3.06894e14 −0.293553
\(539\) 1.06023e15 1.00383
\(540\) −1.40813e14 −0.131971
\(541\) 6.17096e14 0.572490 0.286245 0.958157i \(-0.407593\pi\)
0.286245 + 0.958157i \(0.407593\pi\)
\(542\) −2.73422e15 −2.51095
\(543\) −3.76854e14 −0.342590
\(544\) 2.49812e15 2.24812
\(545\) 1.23308e14 0.109853
\(546\) −4.81142e14 −0.424340
\(547\) −6.62825e14 −0.578720 −0.289360 0.957220i \(-0.593443\pi\)
−0.289360 + 0.957220i \(0.593443\pi\)
\(548\) 2.73185e14 0.236137
\(549\) −4.09900e14 −0.350776
\(550\) −1.72241e15 −1.45929
\(551\) −4.65193e14 −0.390211
\(552\) −1.13890e14 −0.0945850
\(553\) −6.09444e14 −0.501124
\(554\) −3.06339e15 −2.49401
\(555\) 5.06913e12 0.00408622
\(556\) −2.97410e15 −2.37380
\(557\) 2.08314e14 0.164632 0.0823161 0.996606i \(-0.473768\pi\)
0.0823161 + 0.996606i \(0.473768\pi\)
\(558\) 2.70121e14 0.211383
\(559\) 1.31054e15 1.01551
\(560\) 1.23383e14 0.0946719
\(561\) 1.62667e15 1.23596
\(562\) −2.36286e15 −1.77782
\(563\) 1.17930e15 0.878674 0.439337 0.898322i \(-0.355213\pi\)
0.439337 + 0.898322i \(0.355213\pi\)
\(564\) 1.36981e15 1.01071
\(565\) −4.21791e14 −0.308198
\(566\) 3.87818e15 2.80632
\(567\) −6.77020e13 −0.0485171
\(568\) −7.23555e14 −0.513518
\(569\) −2.24886e15 −1.58069 −0.790343 0.612665i \(-0.790098\pi\)
−0.790343 + 0.612665i \(0.790098\pi\)
\(570\) 2.40801e14 0.167628
\(571\) −4.46356e14 −0.307739 −0.153870 0.988091i \(-0.549174\pi\)
−0.153870 + 0.988091i \(0.549174\pi\)
\(572\) −2.76119e15 −1.88546
\(573\) 3.86313e14 0.261270
\(574\) −8.61160e14 −0.576857
\(575\) 3.03723e14 0.201513
\(576\) −7.98932e14 −0.525032
\(577\) 2.81221e14 0.183055 0.0915273 0.995803i \(-0.470825\pi\)
0.0915273 + 0.995803i \(0.470825\pi\)
\(578\) −4.75372e15 −3.06500
\(579\) −1.66655e13 −0.0106435
\(580\) −1.10616e15 −0.699786
\(581\) −7.70008e13 −0.0482533
\(582\) 4.56308e14 0.283257
\(583\) −3.53566e15 −2.17417
\(584\) −7.61664e14 −0.463973
\(585\) 2.94307e14 0.177600
\(586\) 2.23573e15 1.33654
\(587\) −2.71962e15 −1.61064 −0.805320 0.592840i \(-0.798007\pi\)
−0.805320 + 0.592840i \(0.798007\pi\)
\(588\) 1.11399e15 0.653593
\(589\) −2.69356e14 −0.156564
\(590\) −1.71661e14 −0.0988522
\(591\) 2.08154e14 0.118755
\(592\) 1.12953e13 0.00638451
\(593\) −1.67708e15 −0.939187 −0.469594 0.882883i \(-0.655600\pi\)
−0.469594 + 0.882883i \(0.655600\pi\)
\(594\) −6.66299e14 −0.369695
\(595\) −6.72098e14 −0.369479
\(596\) 2.44453e15 1.33150
\(597\) 1.80307e15 0.973091
\(598\) 8.34993e14 0.446505
\(599\) −1.44572e14 −0.0766015 −0.0383008 0.999266i \(-0.512194\pi\)
−0.0383008 + 0.999266i \(0.512194\pi\)
\(600\) −5.15918e14 −0.270862
\(601\) 2.98266e15 1.55165 0.775827 0.630946i \(-0.217333\pi\)
0.775827 + 0.630946i \(0.217333\pi\)
\(602\) 1.22589e15 0.631931
\(603\) −2.45638e14 −0.125473
\(604\) −2.19190e15 −1.10948
\(605\) 5.26223e14 0.263946
\(606\) 7.66207e13 0.0380844
\(607\) −1.60781e14 −0.0791949 −0.0395974 0.999216i \(-0.512608\pi\)
−0.0395974 + 0.999216i \(0.512608\pi\)
\(608\) 1.02036e15 0.498060
\(609\) −5.31835e14 −0.257265
\(610\) 1.66678e15 0.799031
\(611\) −2.86298e15 −1.36016
\(612\) 1.70917e15 0.804733
\(613\) 8.60595e14 0.401575 0.200787 0.979635i \(-0.435650\pi\)
0.200787 + 0.979635i \(0.435650\pi\)
\(614\) 1.59584e15 0.738013
\(615\) 5.26758e14 0.241434
\(616\) −7.36306e14 −0.334476
\(617\) 3.61157e15 1.62603 0.813014 0.582244i \(-0.197825\pi\)
0.813014 + 0.582244i \(0.197825\pi\)
\(618\) 1.59182e15 0.710327
\(619\) 5.25634e14 0.232479 0.116240 0.993221i \(-0.462916\pi\)
0.116240 + 0.993221i \(0.462916\pi\)
\(620\) −6.40491e14 −0.280775
\(621\) 1.17493e14 0.0510513
\(622\) 6.80863e15 2.93233
\(623\) −6.70734e14 −0.286329
\(624\) 6.55789e14 0.277491
\(625\) 8.02815e14 0.336725
\(626\) 3.59936e15 1.49646
\(627\) 6.64413e14 0.273821
\(628\) −1.16802e15 −0.477169
\(629\) −6.15282e13 −0.0249170
\(630\) 2.75297e14 0.110517
\(631\) 1.03394e15 0.411465 0.205733 0.978608i \(-0.434042\pi\)
0.205733 + 0.978608i \(0.434042\pi\)
\(632\) −1.79658e15 −0.708766
\(633\) 9.04719e14 0.353829
\(634\) 9.08736e14 0.352328
\(635\) 4.40853e14 0.169449
\(636\) −3.71496e15 −1.41560
\(637\) −2.32830e15 −0.879574
\(638\) −5.23413e15 −1.96034
\(639\) 7.46440e14 0.277166
\(640\) 1.51411e15 0.557400
\(641\) −4.90083e15 −1.78875 −0.894377 0.447313i \(-0.852381\pi\)
−0.894377 + 0.447313i \(0.852381\pi\)
\(642\) 1.53712e15 0.556245
\(643\) −4.26885e15 −1.53162 −0.765810 0.643067i \(-0.777662\pi\)
−0.765810 + 0.643067i \(0.777662\pi\)
\(644\) 4.55448e14 0.162019
\(645\) −7.49855e14 −0.264484
\(646\) −2.92280e15 −1.02216
\(647\) −9.88713e14 −0.342844 −0.171422 0.985198i \(-0.554836\pi\)
−0.171422 + 0.985198i \(0.554836\pi\)
\(648\) −1.99579e14 −0.0686202
\(649\) −4.73645e14 −0.161476
\(650\) 3.78247e15 1.27865
\(651\) −3.07943e14 −0.103223
\(652\) −7.54011e15 −2.50620
\(653\) −3.20209e15 −1.05538 −0.527692 0.849436i \(-0.676943\pi\)
−0.527692 + 0.849436i \(0.676943\pi\)
\(654\) 6.13058e14 0.200366
\(655\) 2.06564e15 0.669463
\(656\) 1.17375e15 0.377227
\(657\) 7.85754e14 0.250425
\(658\) −2.67805e15 −0.846400
\(659\) 6.20183e14 0.194379 0.0971896 0.995266i \(-0.469015\pi\)
0.0971896 + 0.995266i \(0.469015\pi\)
\(660\) 1.57988e15 0.491058
\(661\) −3.73583e14 −0.115154 −0.0575771 0.998341i \(-0.518337\pi\)
−0.0575771 + 0.998341i \(0.518337\pi\)
\(662\) 4.81899e15 1.47312
\(663\) −3.57225e15 −1.08297
\(664\) −2.26991e14 −0.0682471
\(665\) −2.74518e14 −0.0818561
\(666\) 2.52024e13 0.00745305
\(667\) 9.22967e14 0.270704
\(668\) 6.97630e15 2.02934
\(669\) 2.86264e15 0.825892
\(670\) 9.98840e14 0.285815
\(671\) 4.59895e15 1.30522
\(672\) 1.16653e15 0.328370
\(673\) 2.48508e15 0.693837 0.346918 0.937895i \(-0.387228\pi\)
0.346918 + 0.937895i \(0.387228\pi\)
\(674\) 5.63054e15 1.55927
\(675\) 5.32236e14 0.146195
\(676\) 9.29790e14 0.253325
\(677\) 1.35648e15 0.366586 0.183293 0.983058i \(-0.441324\pi\)
0.183293 + 0.983058i \(0.441324\pi\)
\(678\) −2.09704e15 −0.562138
\(679\) −5.20200e14 −0.138320
\(680\) −1.98128e15 −0.522573
\(681\) 3.86938e15 1.01235
\(682\) −3.03067e15 −0.786546
\(683\) −5.28071e14 −0.135950 −0.0679749 0.997687i \(-0.521654\pi\)
−0.0679749 + 0.997687i \(0.521654\pi\)
\(684\) 6.98108e14 0.178285
\(685\) 3.26694e14 0.0827643
\(686\) −4.86890e15 −1.22362
\(687\) 3.78065e15 0.942549
\(688\) −1.67086e15 −0.413242
\(689\) 7.76445e15 1.90504
\(690\) −4.77761e14 −0.116289
\(691\) 1.72140e15 0.415674 0.207837 0.978163i \(-0.433358\pi\)
0.207837 + 0.978163i \(0.433358\pi\)
\(692\) 7.69007e15 1.84224
\(693\) 7.59594e14 0.180530
\(694\) −4.29804e15 −1.01343
\(695\) −3.55664e15 −0.832000
\(696\) −1.56780e15 −0.363864
\(697\) −6.39369e15 −1.47222
\(698\) 1.77643e14 0.0405830
\(699\) −9.54224e14 −0.216285
\(700\) 2.06315e15 0.463973
\(701\) 5.47245e15 1.22105 0.610524 0.791998i \(-0.290959\pi\)
0.610524 + 0.791998i \(0.290959\pi\)
\(702\) 1.46322e15 0.323933
\(703\) −2.51311e13 −0.00552023
\(704\) 8.96375e15 1.95362
\(705\) 1.63812e15 0.354246
\(706\) −7.77690e15 −1.66871
\(707\) −8.73492e13 −0.0185974
\(708\) −4.97665e14 −0.105137
\(709\) 3.41159e14 0.0715158 0.0357579 0.999360i \(-0.488615\pi\)
0.0357579 + 0.999360i \(0.488615\pi\)
\(710\) −3.03525e15 −0.631355
\(711\) 1.85340e15 0.382549
\(712\) −1.97726e15 −0.404970
\(713\) 5.34416e14 0.108614
\(714\) −3.34150e15 −0.673910
\(715\) −3.30203e15 −0.660842
\(716\) 6.01711e15 1.19500
\(717\) 7.16653e14 0.141239
\(718\) −2.59924e15 −0.508349
\(719\) −9.73487e14 −0.188939 −0.0944694 0.995528i \(-0.530115\pi\)
−0.0944694 + 0.995528i \(0.530115\pi\)
\(720\) −3.75225e14 −0.0722708
\(721\) −1.81471e15 −0.346868
\(722\) 6.97102e15 1.32234
\(723\) −3.94760e15 −0.743143
\(724\) 4.44260e15 0.829995
\(725\) 4.18099e15 0.775211
\(726\) 2.61625e15 0.481424
\(727\) −7.93057e14 −0.144832 −0.0724161 0.997375i \(-0.523071\pi\)
−0.0724161 + 0.997375i \(0.523071\pi\)
\(728\) 1.61696e15 0.293073
\(729\) 2.05891e14 0.0370370
\(730\) −3.19511e15 −0.570441
\(731\) 9.10161e15 1.61277
\(732\) 4.83217e15 0.849828
\(733\) 1.18996e14 0.0207711 0.0103855 0.999946i \(-0.496694\pi\)
0.0103855 + 0.999946i \(0.496694\pi\)
\(734\) 6.97266e15 1.20801
\(735\) 1.33219e15 0.229080
\(736\) −2.02444e15 −0.345522
\(737\) 2.75598e15 0.466880
\(738\) 2.61891e15 0.440362
\(739\) 3.87846e15 0.647314 0.323657 0.946175i \(-0.395088\pi\)
0.323657 + 0.946175i \(0.395088\pi\)
\(740\) −5.97582e13 −0.00989972
\(741\) −1.45908e15 −0.239927
\(742\) 7.26292e15 1.18547
\(743\) 8.40639e15 1.36198 0.680991 0.732292i \(-0.261549\pi\)
0.680991 + 0.732292i \(0.261549\pi\)
\(744\) −9.07786e14 −0.145993
\(745\) 2.92334e15 0.466681
\(746\) 2.79818e15 0.443416
\(747\) 2.34170e14 0.0368357
\(748\) −1.91763e16 −2.99437
\(749\) −1.75235e15 −0.271626
\(750\) −5.01321e15 −0.771398
\(751\) 1.22233e16 1.86710 0.933549 0.358449i \(-0.116694\pi\)
0.933549 + 0.358449i \(0.116694\pi\)
\(752\) 3.65014e15 0.553491
\(753\) −1.14198e15 −0.171905
\(754\) 1.14944e16 1.71768
\(755\) −2.62123e15 −0.388864
\(756\) 7.98115e14 0.117543
\(757\) 1.63077e14 0.0238433 0.0119216 0.999929i \(-0.496205\pi\)
0.0119216 + 0.999929i \(0.496205\pi\)
\(758\) 1.06766e16 1.54972
\(759\) −1.31823e15 −0.189960
\(760\) −8.09251e14 −0.115773
\(761\) −7.28896e15 −1.03526 −0.517631 0.855604i \(-0.673186\pi\)
−0.517631 + 0.855604i \(0.673186\pi\)
\(762\) 2.19181e15 0.309066
\(763\) −6.98899e14 −0.0978429
\(764\) −4.55411e15 −0.632980
\(765\) 2.04394e15 0.282053
\(766\) 1.31923e16 1.80743
\(767\) 1.04014e15 0.141488
\(768\) 7.94383e14 0.107286
\(769\) 9.15102e15 1.22708 0.613542 0.789662i \(-0.289744\pi\)
0.613542 + 0.789662i \(0.289744\pi\)
\(770\) −3.08874e15 −0.411228
\(771\) −8.05729e15 −1.06510
\(772\) 1.96464e14 0.0257862
\(773\) 1.29011e15 0.168128 0.0840641 0.996460i \(-0.473210\pi\)
0.0840641 + 0.996460i \(0.473210\pi\)
\(774\) −3.72809e15 −0.482404
\(775\) 2.42088e15 0.311038
\(776\) −1.53350e15 −0.195634
\(777\) −2.87313e13 −0.00363948
\(778\) 1.96433e16 2.47074
\(779\) −2.61150e15 −0.326162
\(780\) −3.46948e15 −0.430273
\(781\) −8.37481e15 −1.03132
\(782\) 5.79897e15 0.709111
\(783\) 1.61738e15 0.196392
\(784\) 2.96846e15 0.357925
\(785\) −1.39680e15 −0.167244
\(786\) 1.02698e16 1.22106
\(787\) 1.65538e15 0.195450 0.0977252 0.995213i \(-0.468843\pi\)
0.0977252 + 0.995213i \(0.468843\pi\)
\(788\) −2.45385e15 −0.287710
\(789\) 4.22483e15 0.491910
\(790\) −7.53650e15 −0.871406
\(791\) 2.39066e15 0.274503
\(792\) 2.23921e15 0.255332
\(793\) −1.00995e16 −1.14366
\(794\) 1.40585e16 1.58098
\(795\) −4.44261e15 −0.496157
\(796\) −2.12557e16 −2.35751
\(797\) −2.41106e13 −0.00265575 −0.00132788 0.999999i \(-0.500423\pi\)
−0.00132788 + 0.999999i \(0.500423\pi\)
\(798\) −1.36483e15 −0.149301
\(799\) −1.98832e16 −2.16013
\(800\) −9.17059e15 −0.989470
\(801\) 2.03979e15 0.218579
\(802\) −6.27580e15 −0.667898
\(803\) −8.81590e15 −0.931819
\(804\) 2.89574e15 0.303985
\(805\) 5.44657e14 0.0567865
\(806\) 6.65547e15 0.689185
\(807\) −1.06399e15 −0.109429
\(808\) −2.57497e14 −0.0263032
\(809\) 1.21304e16 1.23072 0.615361 0.788245i \(-0.289010\pi\)
0.615361 + 0.788245i \(0.289010\pi\)
\(810\) −8.37216e14 −0.0843664
\(811\) 8.99516e15 0.900315 0.450157 0.892949i \(-0.351368\pi\)
0.450157 + 0.892949i \(0.351368\pi\)
\(812\) 6.26961e15 0.623279
\(813\) −9.47943e15 −0.936018
\(814\) −2.82763e14 −0.0277325
\(815\) −9.01700e15 −0.878406
\(816\) 4.55441e15 0.440693
\(817\) 3.71754e15 0.357301
\(818\) 3.58082e14 0.0341853
\(819\) −1.66810e15 −0.158183
\(820\) −6.20976e15 −0.584923
\(821\) 9.08905e15 0.850416 0.425208 0.905096i \(-0.360201\pi\)
0.425208 + 0.905096i \(0.360201\pi\)
\(822\) 1.62424e15 0.150958
\(823\) −1.46245e16 −1.35015 −0.675077 0.737748i \(-0.735889\pi\)
−0.675077 + 0.737748i \(0.735889\pi\)
\(824\) −5.34959e15 −0.490593
\(825\) −5.97151e15 −0.543986
\(826\) 9.72957e14 0.0880448
\(827\) −9.76403e15 −0.877705 −0.438853 0.898559i \(-0.644615\pi\)
−0.438853 + 0.898559i \(0.644615\pi\)
\(828\) −1.38508e15 −0.123682
\(829\) −6.18201e15 −0.548378 −0.274189 0.961676i \(-0.588409\pi\)
−0.274189 + 0.961676i \(0.588409\pi\)
\(830\) −9.52207e14 −0.0839077
\(831\) −1.06206e16 −0.929705
\(832\) −1.96848e16 −1.71180
\(833\) −1.61699e16 −1.39688
\(834\) −1.76827e16 −1.51752
\(835\) 8.34275e15 0.711269
\(836\) −7.83254e15 −0.663389
\(837\) 9.36498e14 0.0787983
\(838\) 1.57908e15 0.131997
\(839\) 4.06156e15 0.337289 0.168645 0.985677i \(-0.446061\pi\)
0.168645 + 0.985677i \(0.446061\pi\)
\(840\) −9.25180e14 −0.0763291
\(841\) 5.04886e14 0.0413824
\(842\) −7.40041e15 −0.602614
\(843\) −8.19195e15 −0.662728
\(844\) −1.06654e16 −0.857224
\(845\) 1.11191e15 0.0887886
\(846\) 8.14432e15 0.646127
\(847\) −2.98257e15 −0.235089
\(848\) −9.89924e15 −0.775220
\(849\) 1.34455e16 1.04613
\(850\) 2.62691e16 2.03067
\(851\) 4.98614e13 0.00382958
\(852\) −8.79952e15 −0.671492
\(853\) −7.72221e15 −0.585493 −0.292746 0.956190i \(-0.594569\pi\)
−0.292746 + 0.956190i \(0.594569\pi\)
\(854\) −9.44712e15 −0.711674
\(855\) 8.34846e14 0.0624875
\(856\) −5.16576e15 −0.384174
\(857\) −1.63335e16 −1.20694 −0.603468 0.797387i \(-0.706215\pi\)
−0.603468 + 0.797387i \(0.706215\pi\)
\(858\) −1.64168e16 −1.20534
\(859\) −2.01926e16 −1.47310 −0.736548 0.676386i \(-0.763545\pi\)
−0.736548 + 0.676386i \(0.763545\pi\)
\(860\) 8.83978e15 0.640767
\(861\) −2.98561e15 −0.215038
\(862\) 1.55069e16 1.10978
\(863\) −1.02354e16 −0.727858 −0.363929 0.931427i \(-0.618565\pi\)
−0.363929 + 0.931427i \(0.618565\pi\)
\(864\) −3.54757e15 −0.250672
\(865\) 9.19633e15 0.645693
\(866\) −1.94314e16 −1.35567
\(867\) −1.64810e16 −1.14256
\(868\) 3.63023e15 0.250078
\(869\) −2.07946e16 −1.42345
\(870\) −6.57677e15 −0.447360
\(871\) −6.05225e15 −0.409088
\(872\) −2.06029e15 −0.138384
\(873\) 1.58200e15 0.105591
\(874\) 2.36858e15 0.157100
\(875\) 5.71515e15 0.376690
\(876\) −9.26298e15 −0.606706
\(877\) −2.94855e16 −1.91916 −0.959578 0.281442i \(-0.909187\pi\)
−0.959578 + 0.281442i \(0.909187\pi\)
\(878\) 4.01741e16 2.59852
\(879\) 7.75119e15 0.498230
\(880\) 4.20990e15 0.268916
\(881\) 4.35657e15 0.276552 0.138276 0.990394i \(-0.455844\pi\)
0.138276 + 0.990394i \(0.455844\pi\)
\(882\) 6.62333e15 0.417829
\(883\) 1.91271e16 1.19912 0.599562 0.800328i \(-0.295341\pi\)
0.599562 + 0.800328i \(0.295341\pi\)
\(884\) 4.21119e16 2.62372
\(885\) −5.95143e14 −0.0368496
\(886\) 2.57674e16 1.58557
\(887\) −2.54821e16 −1.55831 −0.779157 0.626829i \(-0.784353\pi\)
−0.779157 + 0.626829i \(0.784353\pi\)
\(888\) −8.46970e13 −0.00514750
\(889\) −2.49871e15 −0.150923
\(890\) −8.29442e15 −0.497899
\(891\) −2.31003e15 −0.137813
\(892\) −3.37467e16 −2.00089
\(893\) −8.12127e15 −0.478565
\(894\) 1.45341e16 0.851202
\(895\) 7.19568e15 0.418837
\(896\) −8.58181e15 −0.496460
\(897\) 2.89489e15 0.166446
\(898\) 1.20428e16 0.688191
\(899\) 7.35668e15 0.417834
\(900\) −6.27434e15 −0.354188
\(901\) 5.39236e16 3.02547
\(902\) −2.93833e16 −1.63857
\(903\) 4.25010e15 0.235568
\(904\) 7.04744e15 0.388244
\(905\) 5.31278e15 0.290907
\(906\) −1.30321e16 −0.709267
\(907\) 1.89738e16 1.02639 0.513197 0.858271i \(-0.328461\pi\)
0.513197 + 0.858271i \(0.328461\pi\)
\(908\) −4.56148e16 −2.45264
\(909\) 2.65641e14 0.0141969
\(910\) 6.78300e15 0.360325
\(911\) 1.73615e15 0.0916719 0.0458359 0.998949i \(-0.485405\pi\)
0.0458359 + 0.998949i \(0.485405\pi\)
\(912\) 1.86025e15 0.0976334
\(913\) −2.62731e15 −0.137064
\(914\) −3.04242e16 −1.57766
\(915\) 5.77865e15 0.297859
\(916\) −4.45687e16 −2.28352
\(917\) −1.17078e16 −0.596271
\(918\) 1.01620e16 0.514450
\(919\) 6.66355e15 0.335328 0.167664 0.985844i \(-0.446378\pi\)
0.167664 + 0.985844i \(0.446378\pi\)
\(920\) 1.60559e15 0.0803161
\(921\) 5.53272e15 0.275113
\(922\) 2.40808e16 1.19028
\(923\) 1.83914e16 0.903662
\(924\) −8.95459e15 −0.437371
\(925\) 2.25869e14 0.0109667
\(926\) 4.55241e16 2.19726
\(927\) 5.51879e15 0.264792
\(928\) −2.78680e16 −1.32921
\(929\) −3.02861e16 −1.43601 −0.718004 0.696039i \(-0.754944\pi\)
−0.718004 + 0.696039i \(0.754944\pi\)
\(930\) −3.80808e15 −0.179494
\(931\) −6.60458e15 −0.309473
\(932\) 1.12490e16 0.523995
\(933\) 2.36052e16 1.09310
\(934\) 3.12878e16 1.44035
\(935\) −2.29324e16 −1.04951
\(936\) −4.91739e15 −0.223727
\(937\) −1.38014e16 −0.624247 −0.312123 0.950042i \(-0.601040\pi\)
−0.312123 + 0.950042i \(0.601040\pi\)
\(938\) −5.66132e15 −0.254567
\(939\) 1.24788e16 0.557844
\(940\) −1.93112e16 −0.858235
\(941\) 3.48870e16 1.54142 0.770709 0.637187i \(-0.219902\pi\)
0.770709 + 0.637187i \(0.219902\pi\)
\(942\) −6.94454e15 −0.305045
\(943\) 5.18134e15 0.226270
\(944\) −1.32613e15 −0.0575756
\(945\) 9.54442e14 0.0411979
\(946\) 4.18280e16 1.79500
\(947\) 5.17522e15 0.220802 0.110401 0.993887i \(-0.464786\pi\)
0.110401 + 0.993887i \(0.464786\pi\)
\(948\) −2.18491e16 −0.926805
\(949\) 1.93601e16 0.816476
\(950\) 1.07296e16 0.449886
\(951\) 3.15055e15 0.131339
\(952\) 1.12297e16 0.465440
\(953\) 2.81904e16 1.16169 0.580846 0.814013i \(-0.302722\pi\)
0.580846 + 0.814013i \(0.302722\pi\)
\(954\) −2.20876e16 −0.904964
\(955\) −5.44613e15 −0.221855
\(956\) −8.44837e15 −0.342180
\(957\) −1.81465e16 −0.730764
\(958\) −1.34963e15 −0.0540388
\(959\) −1.85167e15 −0.0737158
\(960\) 1.12631e16 0.445827
\(961\) −2.11488e16 −0.832352
\(962\) 6.20959e14 0.0242997
\(963\) 5.32914e15 0.207354
\(964\) 4.65368e16 1.80042
\(965\) 2.34945e14 0.00903787
\(966\) 2.70790e15 0.103576
\(967\) −1.11969e16 −0.425844 −0.212922 0.977069i \(-0.568298\pi\)
−0.212922 + 0.977069i \(0.568298\pi\)
\(968\) −8.79234e15 −0.332499
\(969\) −1.01332e16 −0.381037
\(970\) −6.43289e15 −0.240526
\(971\) 1.55940e16 0.579764 0.289882 0.957062i \(-0.406384\pi\)
0.289882 + 0.957062i \(0.406384\pi\)
\(972\) −2.42718e15 −0.0897299
\(973\) 2.01586e16 0.741038
\(974\) 4.31286e16 1.57649
\(975\) 1.31137e16 0.476650
\(976\) 1.28763e16 0.465389
\(977\) −2.06463e16 −0.742032 −0.371016 0.928626i \(-0.620991\pi\)
−0.371016 + 0.928626i \(0.620991\pi\)
\(978\) −4.48303e16 −1.60217
\(979\) −2.28858e16 −0.813321
\(980\) −1.57048e16 −0.554993
\(981\) 2.12545e15 0.0746915
\(982\) −6.84066e16 −2.39048
\(983\) −3.86690e16 −1.34375 −0.671875 0.740665i \(-0.734511\pi\)
−0.671875 + 0.740665i \(0.734511\pi\)
\(984\) −8.80127e15 −0.304139
\(985\) −2.93449e15 −0.100840
\(986\) 7.98277e16 2.72791
\(987\) −9.28468e15 −0.315517
\(988\) 1.72006e16 0.581273
\(989\) −7.37579e15 −0.247873
\(990\) 9.39328e15 0.313924
\(991\) −4.43033e16 −1.47242 −0.736208 0.676755i \(-0.763386\pi\)
−0.736208 + 0.676755i \(0.763386\pi\)
\(992\) −1.61362e16 −0.533318
\(993\) 1.67072e16 0.549141
\(994\) 1.72035e16 0.562330
\(995\) −2.54191e16 −0.826292
\(996\) −2.76055e15 −0.0892420
\(997\) 2.59343e16 0.833780 0.416890 0.908957i \(-0.363120\pi\)
0.416890 + 0.908957i \(0.363120\pi\)
\(998\) 4.94480e16 1.58100
\(999\) 8.73758e13 0.00277831
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.1.3 26
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.a.1.3 26 1.1 even 1 trivial