Properties

Label 177.12.a.d.1.5
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $0$
Dimension $28$
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: \(0\)
Dimension: \(28\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-67.3482 q^{2} +243.000 q^{3} +2487.78 q^{4} +9961.02 q^{5} -16365.6 q^{6} -37608.5 q^{7} -29618.6 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-67.3482 q^{2} +243.000 q^{3} +2487.78 q^{4} +9961.02 q^{5} -16365.6 q^{6} -37608.5 q^{7} -29618.6 q^{8} +59049.0 q^{9} -670857. q^{10} +181051. q^{11} +604531. q^{12} -10898.1 q^{13} +2.53287e6 q^{14} +2.42053e6 q^{15} -3.10022e6 q^{16} -8.94272e6 q^{17} -3.97684e6 q^{18} -483641. q^{19} +2.47809e7 q^{20} -9.13887e6 q^{21} -1.21934e7 q^{22} -5.76947e7 q^{23} -7.19731e6 q^{24} +5.03939e7 q^{25} +733965. q^{26} +1.43489e7 q^{27} -9.35619e7 q^{28} -7.44117e7 q^{29} -1.63018e8 q^{30} -3.78363e6 q^{31} +2.69453e8 q^{32} +4.39953e7 q^{33} +6.02276e8 q^{34} -3.74620e8 q^{35} +1.46901e8 q^{36} +2.66928e8 q^{37} +3.25723e7 q^{38} -2.64823e6 q^{39} -2.95031e8 q^{40} +9.21400e8 q^{41} +6.15487e8 q^{42} +1.72347e9 q^{43} +4.50414e8 q^{44} +5.88188e8 q^{45} +3.88563e9 q^{46} +5.56453e8 q^{47} -7.53354e8 q^{48} -5.62925e8 q^{49} -3.39394e9 q^{50} -2.17308e9 q^{51} -2.71120e7 q^{52} -5.10556e8 q^{53} -9.66373e8 q^{54} +1.80345e9 q^{55} +1.11391e9 q^{56} -1.17525e8 q^{57} +5.01149e9 q^{58} +7.14924e8 q^{59} +6.02175e9 q^{60} +4.70006e9 q^{61} +2.54821e8 q^{62} -2.22075e9 q^{63} -1.17979e10 q^{64} -1.08556e8 q^{65} -2.96300e9 q^{66} +6.76322e8 q^{67} -2.22475e10 q^{68} -1.40198e10 q^{69} +2.52300e10 q^{70} +2.31176e10 q^{71} -1.74895e9 q^{72} +1.24600e10 q^{73} -1.79771e10 q^{74} +1.22457e10 q^{75} -1.20319e9 q^{76} -6.80905e9 q^{77} +1.78353e8 q^{78} -1.81125e10 q^{79} -3.08814e10 q^{80} +3.48678e9 q^{81} -6.20547e10 q^{82} +2.88119e10 q^{83} -2.27355e10 q^{84} -8.90786e10 q^{85} -1.16073e11 q^{86} -1.80820e10 q^{87} -5.36246e9 q^{88} +1.17986e10 q^{89} -3.96134e10 q^{90} +4.09860e8 q^{91} -1.43532e11 q^{92} -9.19423e8 q^{93} -3.74761e10 q^{94} -4.81756e9 q^{95} +6.54771e10 q^{96} -6.35855e9 q^{97} +3.79120e10 q^{98} +1.06909e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + O(q^{10}) \) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + 616281q^{10} + 1082362q^{11} + 7099002q^{12} + 503712q^{13} + 1321669q^{14} + 6454566q^{15} + 34870338q^{16} + 13513579q^{17} + 5668704q^{18} + 35971687q^{19} + 96105997q^{20} + 34586919q^{21} - 47598882q^{22} + 61380539q^{23} + 80756433q^{24} + 294744746q^{25} + 62820734q^{26} + 401769396q^{27} + 148068294q^{28} + 322339307q^{29} + 149756283q^{30} + 151247077q^{31} + 466383494q^{32} + 263013966q^{33} + 684479860q^{34} + 960297361q^{35} + 1725057486q^{36} + 863508437q^{37} + 992640509q^{38} + 122402016q^{39} + 3067680252q^{40} + 3081170377q^{41} + 321165567q^{42} + 2554238300q^{43} + 4350123570q^{44} + 1568459538q^{45} - 1987059155q^{46} + 6203398333q^{47} + 8473492134q^{48} + 10327857997q^{49} + 17577682253q^{50} + 3283799697q^{51} + 32137181618q^{52} + 14571770754q^{53} + 1377495072q^{54} + 18251419334q^{55} + 33498842836q^{56} + 8741119941q^{57} + 11860778276q^{58} + 20017880372q^{59} + 23353757271q^{60} + 2761613771q^{61} + 13785829526q^{62} + 8404621317q^{63} + 86547545293q^{64} + 32034985256q^{65} - 11566528326q^{66} + 39381333296q^{67} + 38995496621q^{68} + 14915470977q^{69} + 8551800364q^{70} + 26130020296q^{71} + 19623813219q^{72} + 41382402799q^{73} + 23815315058q^{74} + 71622973278q^{75} + 10611720128q^{76} - 8426124313q^{77} + 15265438362q^{78} + 59825111206q^{79} + 4009687655q^{80} + 97629963228q^{81} - 39592715115q^{82} + 35433122727q^{83} + 35980595442q^{84} - 8950496085q^{85} - 182032360688q^{86} + 78328451601q^{87} - 220003602335q^{88} + 102303043039q^{89} + 36390776769q^{90} - 111146323655q^{91} - 163000203526q^{92} + 36753039711q^{93} - 81314346008q^{94} + 208102168887q^{95} + 113331189042q^{96} - 171891031490q^{97} + 72304707792q^{98} + 63912393738q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −67.3482 −1.48820 −0.744100 0.668069i \(-0.767121\pi\)
−0.744100 + 0.668069i \(0.767121\pi\)
\(3\) 243.000 0.577350
\(4\) 2487.78 1.21474
\(5\) 9961.02 1.42551 0.712753 0.701415i \(-0.247448\pi\)
0.712753 + 0.701415i \(0.247448\pi\)
\(6\) −16365.6 −0.859212
\(7\) −37608.5 −0.845760 −0.422880 0.906186i \(-0.638981\pi\)
−0.422880 + 0.906186i \(0.638981\pi\)
\(8\) −29618.6 −0.319572
\(9\) 59049.0 0.333333
\(10\) −670857. −2.12144
\(11\) 181051. 0.338954 0.169477 0.985534i \(-0.445792\pi\)
0.169477 + 0.985534i \(0.445792\pi\)
\(12\) 604531. 0.701329
\(13\) −10898.1 −0.00814068 −0.00407034 0.999992i \(-0.501296\pi\)
−0.00407034 + 0.999992i \(0.501296\pi\)
\(14\) 2.53287e6 1.25866
\(15\) 2.42053e6 0.823016
\(16\) −3.10022e6 −0.739150
\(17\) −8.94272e6 −1.52757 −0.763784 0.645472i \(-0.776661\pi\)
−0.763784 + 0.645472i \(0.776661\pi\)
\(18\) −3.97684e6 −0.496066
\(19\) −483641. −0.0448103 −0.0224052 0.999749i \(-0.507132\pi\)
−0.0224052 + 0.999749i \(0.507132\pi\)
\(20\) 2.47809e7 1.73162
\(21\) −9.13887e6 −0.488300
\(22\) −1.21934e7 −0.504430
\(23\) −5.76947e7 −1.86910 −0.934550 0.355831i \(-0.884198\pi\)
−0.934550 + 0.355831i \(0.884198\pi\)
\(24\) −7.19731e6 −0.184505
\(25\) 5.03939e7 1.03207
\(26\) 733965. 0.0121150
\(27\) 1.43489e7 0.192450
\(28\) −9.35619e7 −1.02738
\(29\) −7.44117e7 −0.673678 −0.336839 0.941562i \(-0.609358\pi\)
−0.336839 + 0.941562i \(0.609358\pi\)
\(30\) −1.63018e8 −1.22481
\(31\) −3.78363e6 −0.0237367 −0.0118683 0.999930i \(-0.503778\pi\)
−0.0118683 + 0.999930i \(0.503778\pi\)
\(32\) 2.69453e8 1.41958
\(33\) 4.39953e7 0.195695
\(34\) 6.02276e8 2.27332
\(35\) −3.74620e8 −1.20564
\(36\) 1.46901e8 0.404912
\(37\) 2.66928e8 0.632827 0.316413 0.948621i \(-0.397521\pi\)
0.316413 + 0.948621i \(0.397521\pi\)
\(38\) 3.25723e7 0.0666867
\(39\) −2.64823e6 −0.00470002
\(40\) −2.95031e8 −0.455552
\(41\) 9.21400e8 1.24204 0.621022 0.783793i \(-0.286718\pi\)
0.621022 + 0.783793i \(0.286718\pi\)
\(42\) 6.15487e8 0.726688
\(43\) 1.72347e9 1.78784 0.893919 0.448229i \(-0.147945\pi\)
0.893919 + 0.448229i \(0.147945\pi\)
\(44\) 4.50414e8 0.411740
\(45\) 5.88188e8 0.475169
\(46\) 3.88563e9 2.78159
\(47\) 5.56453e8 0.353908 0.176954 0.984219i \(-0.443376\pi\)
0.176954 + 0.984219i \(0.443376\pi\)
\(48\) −7.53354e8 −0.426749
\(49\) −5.62925e8 −0.284690
\(50\) −3.39394e9 −1.53592
\(51\) −2.17308e9 −0.881941
\(52\) −2.71120e7 −0.00988879
\(53\) −5.10556e8 −0.167697 −0.0838487 0.996478i \(-0.526721\pi\)
−0.0838487 + 0.996478i \(0.526721\pi\)
\(54\) −9.66373e8 −0.286404
\(55\) 1.80345e9 0.483180
\(56\) 1.11391e9 0.270281
\(57\) −1.17525e8 −0.0258713
\(58\) 5.01149e9 1.00257
\(59\) 7.14924e8 0.130189
\(60\) 6.02175e9 0.999748
\(61\) 4.70006e9 0.712507 0.356253 0.934389i \(-0.384054\pi\)
0.356253 + 0.934389i \(0.384054\pi\)
\(62\) 2.54821e8 0.0353249
\(63\) −2.22075e9 −0.281920
\(64\) −1.17979e10 −1.37346
\(65\) −1.08556e8 −0.0116046
\(66\) −2.96300e9 −0.291233
\(67\) 6.76322e8 0.0611987 0.0305994 0.999532i \(-0.490258\pi\)
0.0305994 + 0.999532i \(0.490258\pi\)
\(68\) −2.22475e10 −1.85559
\(69\) −1.40198e10 −1.07913
\(70\) 2.52300e10 1.79423
\(71\) 2.31176e10 1.52063 0.760313 0.649556i \(-0.225045\pi\)
0.760313 + 0.649556i \(0.225045\pi\)
\(72\) −1.74895e9 −0.106524
\(73\) 1.24600e10 0.703467 0.351733 0.936100i \(-0.385592\pi\)
0.351733 + 0.936100i \(0.385592\pi\)
\(74\) −1.79771e10 −0.941772
\(75\) 1.22457e10 0.595864
\(76\) −1.20319e9 −0.0544328
\(77\) −6.80905e9 −0.286673
\(78\) 1.78353e8 0.00699457
\(79\) −1.81125e10 −0.662262 −0.331131 0.943585i \(-0.607430\pi\)
−0.331131 + 0.943585i \(0.607430\pi\)
\(80\) −3.08814e10 −1.05366
\(81\) 3.48678e9 0.111111
\(82\) −6.20547e10 −1.84841
\(83\) 2.88119e10 0.802864 0.401432 0.915889i \(-0.368513\pi\)
0.401432 + 0.915889i \(0.368513\pi\)
\(84\) −2.27355e10 −0.593156
\(85\) −8.90786e10 −2.17756
\(86\) −1.16073e11 −2.66066
\(87\) −1.80820e10 −0.388948
\(88\) −5.36246e9 −0.108320
\(89\) 1.17986e10 0.223967 0.111984 0.993710i \(-0.464280\pi\)
0.111984 + 0.993710i \(0.464280\pi\)
\(90\) −3.96134e10 −0.707146
\(91\) 4.09860e8 0.00688506
\(92\) −1.43532e11 −2.27047
\(93\) −9.19423e8 −0.0137044
\(94\) −3.74761e10 −0.526686
\(95\) −4.81756e9 −0.0638774
\(96\) 6.54771e10 0.819592
\(97\) −6.35855e9 −0.0751819 −0.0375909 0.999293i \(-0.511968\pi\)
−0.0375909 + 0.999293i \(0.511968\pi\)
\(98\) 3.79120e10 0.423675
\(99\) 1.06909e10 0.112985
\(100\) 1.25369e11 1.25369
\(101\) 8.42346e10 0.797486 0.398743 0.917063i \(-0.369447\pi\)
0.398743 + 0.917063i \(0.369447\pi\)
\(102\) 1.46353e11 1.31250
\(103\) −1.34168e9 −0.0114036 −0.00570181 0.999984i \(-0.501815\pi\)
−0.00570181 + 0.999984i \(0.501815\pi\)
\(104\) 3.22785e8 0.00260153
\(105\) −9.10325e10 −0.696074
\(106\) 3.43851e10 0.249567
\(107\) 3.85074e10 0.265420 0.132710 0.991155i \(-0.457632\pi\)
0.132710 + 0.991155i \(0.457632\pi\)
\(108\) 3.56970e10 0.233776
\(109\) −9.99136e10 −0.621983 −0.310992 0.950413i \(-0.600661\pi\)
−0.310992 + 0.950413i \(0.600661\pi\)
\(110\) −1.21459e11 −0.719068
\(111\) 6.48635e10 0.365363
\(112\) 1.16595e11 0.625144
\(113\) 1.27390e11 0.650437 0.325219 0.945639i \(-0.394562\pi\)
0.325219 + 0.945639i \(0.394562\pi\)
\(114\) 7.91508e9 0.0385016
\(115\) −5.74698e11 −2.66441
\(116\) −1.85120e11 −0.818341
\(117\) −6.43519e8 −0.00271356
\(118\) −4.81489e10 −0.193747
\(119\) 3.36322e11 1.29196
\(120\) −7.16926e10 −0.263013
\(121\) −2.52532e11 −0.885110
\(122\) −3.16540e11 −1.06035
\(123\) 2.23900e11 0.717095
\(124\) −9.41285e9 −0.0288338
\(125\) 1.55964e10 0.0457108
\(126\) 1.49563e11 0.419553
\(127\) 3.52951e11 0.947968 0.473984 0.880533i \(-0.342815\pi\)
0.473984 + 0.880533i \(0.342815\pi\)
\(128\) 2.42730e11 0.624408
\(129\) 4.18804e11 1.03221
\(130\) 7.31104e9 0.0172699
\(131\) −3.81317e11 −0.863563 −0.431782 0.901978i \(-0.642115\pi\)
−0.431782 + 0.901978i \(0.642115\pi\)
\(132\) 1.09451e11 0.237718
\(133\) 1.81890e10 0.0378988
\(134\) −4.55491e10 −0.0910759
\(135\) 1.42930e11 0.274339
\(136\) 2.64870e11 0.488168
\(137\) 6.74048e11 1.19324 0.596620 0.802524i \(-0.296510\pi\)
0.596620 + 0.802524i \(0.296510\pi\)
\(138\) 9.44209e11 1.60595
\(139\) 7.18276e11 1.17411 0.587057 0.809546i \(-0.300287\pi\)
0.587057 + 0.809546i \(0.300287\pi\)
\(140\) −9.31972e11 −1.46453
\(141\) 1.35218e11 0.204329
\(142\) −1.55693e12 −2.26300
\(143\) −1.97310e9 −0.00275931
\(144\) −1.83065e11 −0.246383
\(145\) −7.41216e11 −0.960331
\(146\) −8.39161e11 −1.04690
\(147\) −1.36791e11 −0.164366
\(148\) 6.64059e11 0.768718
\(149\) −1.16882e12 −1.30384 −0.651920 0.758287i \(-0.726036\pi\)
−0.651920 + 0.758287i \(0.726036\pi\)
\(150\) −8.24727e11 −0.886764
\(151\) 1.56463e11 0.162196 0.0810979 0.996706i \(-0.474157\pi\)
0.0810979 + 0.996706i \(0.474157\pi\)
\(152\) 1.43247e10 0.0143201
\(153\) −5.28058e11 −0.509189
\(154\) 4.58577e11 0.426627
\(155\) −3.76889e10 −0.0338367
\(156\) −6.58821e9 −0.00570929
\(157\) 2.01386e12 1.68492 0.842462 0.538756i \(-0.181106\pi\)
0.842462 + 0.538756i \(0.181106\pi\)
\(158\) 1.21985e12 0.985578
\(159\) −1.24065e11 −0.0968202
\(160\) 2.68403e12 2.02361
\(161\) 2.16981e12 1.58081
\(162\) −2.34829e11 −0.165355
\(163\) −2.10323e12 −1.43171 −0.715855 0.698249i \(-0.753963\pi\)
−0.715855 + 0.698249i \(0.753963\pi\)
\(164\) 2.29224e12 1.50876
\(165\) 4.38238e11 0.278964
\(166\) −1.94043e12 −1.19482
\(167\) 1.48637e12 0.885494 0.442747 0.896647i \(-0.354004\pi\)
0.442747 + 0.896647i \(0.354004\pi\)
\(168\) 2.70680e11 0.156047
\(169\) −1.79204e12 −0.999934
\(170\) 5.99929e12 3.24064
\(171\) −2.85585e10 −0.0149368
\(172\) 4.28763e12 2.17175
\(173\) 3.24076e12 1.58999 0.794994 0.606618i \(-0.207474\pi\)
0.794994 + 0.606618i \(0.207474\pi\)
\(174\) 1.21779e12 0.578832
\(175\) −1.89524e12 −0.872881
\(176\) −5.61297e11 −0.250538
\(177\) 1.73727e11 0.0751646
\(178\) −7.94613e11 −0.333308
\(179\) 4.64086e12 1.88759 0.943793 0.330536i \(-0.107230\pi\)
0.943793 + 0.330536i \(0.107230\pi\)
\(180\) 1.46328e12 0.577205
\(181\) 9.82599e11 0.375962 0.187981 0.982173i \(-0.439806\pi\)
0.187981 + 0.982173i \(0.439806\pi\)
\(182\) −2.76033e10 −0.0102463
\(183\) 1.14211e12 0.411366
\(184\) 1.70883e12 0.597312
\(185\) 2.65888e12 0.902098
\(186\) 6.19215e10 0.0203948
\(187\) −1.61908e12 −0.517774
\(188\) 1.38433e12 0.429905
\(189\) −5.39641e11 −0.162767
\(190\) 3.24454e11 0.0950623
\(191\) −2.35067e12 −0.669126 −0.334563 0.942373i \(-0.608589\pi\)
−0.334563 + 0.942373i \(0.608589\pi\)
\(192\) −2.86690e12 −0.792968
\(193\) 5.53479e12 1.48777 0.743886 0.668307i \(-0.232981\pi\)
0.743886 + 0.668307i \(0.232981\pi\)
\(194\) 4.28237e11 0.111886
\(195\) −2.63791e10 −0.00669991
\(196\) −1.40043e12 −0.345823
\(197\) 2.76354e12 0.663594 0.331797 0.943351i \(-0.392345\pi\)
0.331797 + 0.943351i \(0.392345\pi\)
\(198\) −7.20010e11 −0.168143
\(199\) 7.05978e12 1.60361 0.801806 0.597584i \(-0.203872\pi\)
0.801806 + 0.597584i \(0.203872\pi\)
\(200\) −1.49259e12 −0.329820
\(201\) 1.64346e11 0.0353331
\(202\) −5.67305e12 −1.18682
\(203\) 2.79851e12 0.569770
\(204\) −5.40615e12 −1.07133
\(205\) 9.17809e12 1.77054
\(206\) 9.03594e10 0.0169709
\(207\) −3.40681e12 −0.623033
\(208\) 3.37864e10 0.00601719
\(209\) −8.75634e10 −0.0151886
\(210\) 6.13088e12 1.03590
\(211\) −7.30005e12 −1.20163 −0.600817 0.799386i \(-0.705158\pi\)
−0.600817 + 0.799386i \(0.705158\pi\)
\(212\) −1.27015e12 −0.203708
\(213\) 5.61759e12 0.877934
\(214\) −2.59341e12 −0.394998
\(215\) 1.71676e13 2.54857
\(216\) −4.24994e11 −0.0615017
\(217\) 1.42297e11 0.0200755
\(218\) 6.72900e12 0.925635
\(219\) 3.02779e12 0.406147
\(220\) 4.48659e12 0.586937
\(221\) 9.74582e10 0.0124354
\(222\) −4.36844e12 −0.543733
\(223\) −5.75506e12 −0.698832 −0.349416 0.936968i \(-0.613620\pi\)
−0.349416 + 0.936968i \(0.613620\pi\)
\(224\) −1.01337e13 −1.20062
\(225\) 2.97571e12 0.344022
\(226\) −8.57952e12 −0.967980
\(227\) −5.66844e12 −0.624196 −0.312098 0.950050i \(-0.601032\pi\)
−0.312098 + 0.950050i \(0.601032\pi\)
\(228\) −2.92376e11 −0.0314268
\(229\) 1.33296e12 0.139869 0.0699345 0.997552i \(-0.477721\pi\)
0.0699345 + 0.997552i \(0.477721\pi\)
\(230\) 3.87049e13 3.96518
\(231\) −1.65460e12 −0.165511
\(232\) 2.20397e12 0.215289
\(233\) 1.11677e13 1.06538 0.532692 0.846309i \(-0.321181\pi\)
0.532692 + 0.846309i \(0.321181\pi\)
\(234\) 4.33399e10 0.00403832
\(235\) 5.54284e12 0.504498
\(236\) 1.77858e12 0.158145
\(237\) −4.40134e12 −0.382357
\(238\) −2.26507e13 −1.92269
\(239\) −7.64841e12 −0.634429 −0.317214 0.948354i \(-0.602747\pi\)
−0.317214 + 0.948354i \(0.602747\pi\)
\(240\) −7.50417e12 −0.608333
\(241\) −9.77672e12 −0.774639 −0.387319 0.921946i \(-0.626599\pi\)
−0.387319 + 0.921946i \(0.626599\pi\)
\(242\) 1.70076e13 1.31722
\(243\) 8.47289e11 0.0641500
\(244\) 1.16927e13 0.865508
\(245\) −5.60730e12 −0.405827
\(246\) −1.50793e13 −1.06718
\(247\) 5.27074e9 0.000364786 0
\(248\) 1.12066e11 0.00758558
\(249\) 7.00128e12 0.463534
\(250\) −1.05039e12 −0.0680267
\(251\) 8.03236e12 0.508906 0.254453 0.967085i \(-0.418105\pi\)
0.254453 + 0.967085i \(0.418105\pi\)
\(252\) −5.52473e12 −0.342459
\(253\) −1.04457e13 −0.633538
\(254\) −2.37706e13 −1.41077
\(255\) −2.16461e13 −1.25721
\(256\) 7.81474e12 0.444217
\(257\) −6.58689e11 −0.0366478 −0.0183239 0.999832i \(-0.505833\pi\)
−0.0183239 + 0.999832i \(0.505833\pi\)
\(258\) −2.82057e13 −1.53613
\(259\) −1.00388e13 −0.535220
\(260\) −2.70063e11 −0.0140965
\(261\) −4.39394e12 −0.224559
\(262\) 2.56810e13 1.28515
\(263\) 1.51824e13 0.744017 0.372008 0.928229i \(-0.378669\pi\)
0.372008 + 0.928229i \(0.378669\pi\)
\(264\) −1.30308e12 −0.0625387
\(265\) −5.08566e12 −0.239054
\(266\) −1.22500e12 −0.0564010
\(267\) 2.86705e12 0.129308
\(268\) 1.68254e12 0.0743404
\(269\) −2.72108e13 −1.17789 −0.588943 0.808175i \(-0.700456\pi\)
−0.588943 + 0.808175i \(0.700456\pi\)
\(270\) −9.62607e12 −0.408271
\(271\) −7.63387e12 −0.317259 −0.158629 0.987338i \(-0.550707\pi\)
−0.158629 + 0.987338i \(0.550707\pi\)
\(272\) 2.77244e13 1.12910
\(273\) 9.95960e10 0.00397509
\(274\) −4.53959e13 −1.77578
\(275\) 9.12384e12 0.349823
\(276\) −3.48782e13 −1.31085
\(277\) 4.00591e13 1.47592 0.737959 0.674845i \(-0.235790\pi\)
0.737959 + 0.674845i \(0.235790\pi\)
\(278\) −4.83746e13 −1.74731
\(279\) −2.23420e11 −0.00791222
\(280\) 1.10957e13 0.385288
\(281\) −2.49514e13 −0.849593 −0.424797 0.905289i \(-0.639654\pi\)
−0.424797 + 0.905289i \(0.639654\pi\)
\(282\) −9.10670e12 −0.304082
\(283\) −3.60074e13 −1.17914 −0.589572 0.807716i \(-0.700704\pi\)
−0.589572 + 0.807716i \(0.700704\pi\)
\(284\) 5.75117e13 1.84716
\(285\) −1.17067e12 −0.0368796
\(286\) 1.32885e11 0.00410641
\(287\) −3.46525e13 −1.05047
\(288\) 1.59109e13 0.473192
\(289\) 4.57003e13 1.33346
\(290\) 4.99196e13 1.42916
\(291\) −1.54513e12 −0.0434063
\(292\) 3.09979e13 0.854527
\(293\) −5.02048e13 −1.35823 −0.679115 0.734032i \(-0.737636\pi\)
−0.679115 + 0.734032i \(0.737636\pi\)
\(294\) 9.21261e12 0.244609
\(295\) 7.12138e12 0.185585
\(296\) −7.90603e12 −0.202234
\(297\) 2.59788e12 0.0652316
\(298\) 7.87182e13 1.94037
\(299\) 6.28760e11 0.0152157
\(300\) 3.04647e13 0.723818
\(301\) −6.48173e13 −1.51208
\(302\) −1.05375e13 −0.241380
\(303\) 2.04690e13 0.460429
\(304\) 1.49939e12 0.0331216
\(305\) 4.68174e13 1.01568
\(306\) 3.55638e13 0.757775
\(307\) 1.79977e13 0.376666 0.188333 0.982105i \(-0.439692\pi\)
0.188333 + 0.982105i \(0.439692\pi\)
\(308\) −1.69394e13 −0.348233
\(309\) −3.26027e11 −0.00658388
\(310\) 2.53828e12 0.0503558
\(311\) 3.13168e13 0.610374 0.305187 0.952293i \(-0.401281\pi\)
0.305187 + 0.952293i \(0.401281\pi\)
\(312\) 7.84367e10 0.00150200
\(313\) −3.78048e13 −0.711301 −0.355650 0.934619i \(-0.615741\pi\)
−0.355650 + 0.934619i \(0.615741\pi\)
\(314\) −1.35630e14 −2.50750
\(315\) −2.21209e13 −0.401879
\(316\) −4.50600e13 −0.804475
\(317\) 2.71569e13 0.476491 0.238246 0.971205i \(-0.423428\pi\)
0.238246 + 0.971205i \(0.423428\pi\)
\(318\) 8.35557e12 0.144088
\(319\) −1.34723e13 −0.228345
\(320\) −1.17520e14 −1.95788
\(321\) 9.35730e12 0.153240
\(322\) −1.46133e14 −2.35256
\(323\) 4.32506e12 0.0684508
\(324\) 8.67436e12 0.134971
\(325\) −5.49195e11 −0.00840172
\(326\) 1.41649e14 2.13067
\(327\) −2.42790e13 −0.359102
\(328\) −2.72905e13 −0.396923
\(329\) −2.09274e13 −0.299321
\(330\) −2.95146e13 −0.415154
\(331\) 5.48421e13 0.758683 0.379341 0.925257i \(-0.376151\pi\)
0.379341 + 0.925257i \(0.376151\pi\)
\(332\) 7.16776e13 0.975269
\(333\) 1.57618e13 0.210942
\(334\) −1.00104e14 −1.31779
\(335\) 6.73686e12 0.0872391
\(336\) 2.83325e13 0.360927
\(337\) −2.15246e12 −0.0269756 −0.0134878 0.999909i \(-0.504293\pi\)
−0.0134878 + 0.999909i \(0.504293\pi\)
\(338\) 1.20691e14 1.48810
\(339\) 3.09559e13 0.375530
\(340\) −2.21608e14 −2.64516
\(341\) −6.85029e11 −0.00804563
\(342\) 1.92336e12 0.0222289
\(343\) 9.55351e13 1.08654
\(344\) −5.10468e13 −0.571343
\(345\) −1.39652e14 −1.53830
\(346\) −2.18260e14 −2.36622
\(347\) 1.26013e14 1.34463 0.672317 0.740264i \(-0.265299\pi\)
0.672317 + 0.740264i \(0.265299\pi\)
\(348\) −4.49842e13 −0.472470
\(349\) 2.68537e13 0.277629 0.138814 0.990318i \(-0.455671\pi\)
0.138814 + 0.990318i \(0.455671\pi\)
\(350\) 1.27641e14 1.29902
\(351\) −1.56375e11 −0.00156667
\(352\) 4.87847e13 0.481170
\(353\) 4.36781e13 0.424134 0.212067 0.977255i \(-0.431980\pi\)
0.212067 + 0.977255i \(0.431980\pi\)
\(354\) −1.17002e13 −0.111860
\(355\) 2.30275e14 2.16766
\(356\) 2.93523e13 0.272061
\(357\) 8.17264e13 0.745911
\(358\) −3.12554e14 −2.80911
\(359\) 1.69035e14 1.49609 0.748046 0.663647i \(-0.230992\pi\)
0.748046 + 0.663647i \(0.230992\pi\)
\(360\) −1.74213e13 −0.151851
\(361\) −1.16256e14 −0.997992
\(362\) −6.61763e13 −0.559507
\(363\) −6.13654e13 −0.511019
\(364\) 1.01964e12 0.00836354
\(365\) 1.24115e14 1.00280
\(366\) −7.69193e13 −0.612194
\(367\) 5.93070e13 0.464989 0.232494 0.972598i \(-0.425311\pi\)
0.232494 + 0.972598i \(0.425311\pi\)
\(368\) 1.78866e14 1.38155
\(369\) 5.44078e13 0.414015
\(370\) −1.79071e14 −1.34250
\(371\) 1.92013e13 0.141832
\(372\) −2.28732e12 −0.0166472
\(373\) 5.12519e13 0.367546 0.183773 0.982969i \(-0.441169\pi\)
0.183773 + 0.982969i \(0.441169\pi\)
\(374\) 1.09042e14 0.770552
\(375\) 3.78992e12 0.0263911
\(376\) −1.64813e13 −0.113099
\(377\) 8.10943e11 0.00548419
\(378\) 3.63439e13 0.242229
\(379\) 1.27387e14 0.836775 0.418388 0.908269i \(-0.362595\pi\)
0.418388 + 0.908269i \(0.362595\pi\)
\(380\) −1.19850e13 −0.0775942
\(381\) 8.57670e13 0.547310
\(382\) 1.58313e14 0.995792
\(383\) 1.87412e14 1.16199 0.580996 0.813906i \(-0.302663\pi\)
0.580996 + 0.813906i \(0.302663\pi\)
\(384\) 5.89834e13 0.360502
\(385\) −6.78251e13 −0.408655
\(386\) −3.72758e14 −2.21410
\(387\) 1.01769e14 0.595946
\(388\) −1.58187e13 −0.0913263
\(389\) 1.98704e14 1.13106 0.565529 0.824729i \(-0.308672\pi\)
0.565529 + 0.824729i \(0.308672\pi\)
\(390\) 1.77658e12 0.00997080
\(391\) 5.15947e14 2.85518
\(392\) 1.66730e13 0.0909789
\(393\) −9.26600e13 −0.498578
\(394\) −1.86120e14 −0.987559
\(395\) −1.80419e14 −0.944059
\(396\) 2.65965e13 0.137247
\(397\) −1.60357e14 −0.816092 −0.408046 0.912961i \(-0.633790\pi\)
−0.408046 + 0.912961i \(0.633790\pi\)
\(398\) −4.75464e14 −2.38650
\(399\) 4.41993e12 0.0218809
\(400\) −1.56232e14 −0.762852
\(401\) −2.18004e14 −1.04995 −0.524977 0.851116i \(-0.675926\pi\)
−0.524977 + 0.851116i \(0.675926\pi\)
\(402\) −1.10684e13 −0.0525827
\(403\) 4.12342e10 0.000193233 0
\(404\) 2.09557e14 0.968736
\(405\) 3.47319e13 0.158390
\(406\) −1.88475e14 −0.847931
\(407\) 4.83275e13 0.214499
\(408\) 6.43635e13 0.281844
\(409\) −1.26493e13 −0.0546498 −0.0273249 0.999627i \(-0.508699\pi\)
−0.0273249 + 0.999627i \(0.508699\pi\)
\(410\) −6.18128e14 −2.63492
\(411\) 1.63794e14 0.688917
\(412\) −3.33780e12 −0.0138524
\(413\) −2.68873e13 −0.110109
\(414\) 2.29443e14 0.927198
\(415\) 2.86996e14 1.14449
\(416\) −2.93652e12 −0.0115563
\(417\) 1.74541e14 0.677875
\(418\) 5.89724e12 0.0226037
\(419\) 3.16015e14 1.19545 0.597725 0.801702i \(-0.296072\pi\)
0.597725 + 0.801702i \(0.296072\pi\)
\(420\) −2.26469e14 −0.845547
\(421\) −4.96360e14 −1.82913 −0.914567 0.404435i \(-0.867468\pi\)
−0.914567 + 0.404435i \(0.867468\pi\)
\(422\) 4.91645e14 1.78827
\(423\) 3.28580e13 0.117969
\(424\) 1.51219e13 0.0535914
\(425\) −4.50658e14 −1.57655
\(426\) −3.78334e14 −1.30654
\(427\) −1.76762e14 −0.602610
\(428\) 9.57981e13 0.322416
\(429\) −4.79463e11 −0.00159309
\(430\) −1.15620e15 −3.79278
\(431\) 3.66809e14 1.18800 0.593998 0.804467i \(-0.297549\pi\)
0.593998 + 0.804467i \(0.297549\pi\)
\(432\) −4.44848e13 −0.142250
\(433\) −6.77200e13 −0.213813 −0.106906 0.994269i \(-0.534095\pi\)
−0.106906 + 0.994269i \(0.534095\pi\)
\(434\) −9.58344e12 −0.0298764
\(435\) −1.80116e14 −0.554447
\(436\) −2.48563e14 −0.755546
\(437\) 2.79035e13 0.0837550
\(438\) −2.03916e14 −0.604427
\(439\) −7.84505e13 −0.229636 −0.114818 0.993387i \(-0.536629\pi\)
−0.114818 + 0.993387i \(0.536629\pi\)
\(440\) −5.34156e13 −0.154411
\(441\) −3.32401e13 −0.0948966
\(442\) −6.56364e12 −0.0185064
\(443\) −6.76883e14 −1.88492 −0.942461 0.334317i \(-0.891495\pi\)
−0.942461 + 0.334317i \(0.891495\pi\)
\(444\) 1.61366e14 0.443820
\(445\) 1.17526e14 0.319267
\(446\) 3.87593e14 1.04000
\(447\) −2.84024e14 −0.752773
\(448\) 4.43703e14 1.16162
\(449\) 2.48132e14 0.641694 0.320847 0.947131i \(-0.396032\pi\)
0.320847 + 0.947131i \(0.396032\pi\)
\(450\) −2.00409e14 −0.511974
\(451\) 1.66820e14 0.420995
\(452\) 3.16920e14 0.790111
\(453\) 3.80206e13 0.0936438
\(454\) 3.81759e14 0.928928
\(455\) 4.08262e12 0.00981469
\(456\) 3.48091e12 0.00826773
\(457\) 7.18385e13 0.168585 0.0842924 0.996441i \(-0.473137\pi\)
0.0842924 + 0.996441i \(0.473137\pi\)
\(458\) −8.97724e13 −0.208153
\(459\) −1.28318e14 −0.293980
\(460\) −1.42972e15 −3.23656
\(461\) 1.29741e14 0.290217 0.145108 0.989416i \(-0.453647\pi\)
0.145108 + 0.989416i \(0.453647\pi\)
\(462\) 1.11434e14 0.246313
\(463\) 5.56814e14 1.21623 0.608114 0.793850i \(-0.291926\pi\)
0.608114 + 0.793850i \(0.291926\pi\)
\(464\) 2.30693e14 0.497949
\(465\) −9.15839e12 −0.0195357
\(466\) −7.52124e14 −1.58550
\(467\) −7.28759e14 −1.51824 −0.759121 0.650950i \(-0.774371\pi\)
−0.759121 + 0.650950i \(0.774371\pi\)
\(468\) −1.60094e12 −0.00329626
\(469\) −2.54355e13 −0.0517594
\(470\) −3.73301e14 −0.750794
\(471\) 4.89367e14 0.972791
\(472\) −2.11750e13 −0.0416048
\(473\) 3.12036e14 0.605994
\(474\) 2.96423e14 0.569024
\(475\) −2.43725e13 −0.0462472
\(476\) 8.36697e14 1.56939
\(477\) −3.01478e13 −0.0558992
\(478\) 5.15107e14 0.944157
\(479\) −8.15098e14 −1.47695 −0.738473 0.674283i \(-0.764453\pi\)
−0.738473 + 0.674283i \(0.764453\pi\)
\(480\) 6.52219e14 1.16833
\(481\) −2.90900e12 −0.00515164
\(482\) 6.58444e14 1.15282
\(483\) 5.27264e14 0.912681
\(484\) −6.28246e14 −1.07518
\(485\) −6.33376e13 −0.107172
\(486\) −5.70634e13 −0.0954680
\(487\) −7.69461e14 −1.27285 −0.636425 0.771338i \(-0.719588\pi\)
−0.636425 + 0.771338i \(0.719588\pi\)
\(488\) −1.39209e14 −0.227697
\(489\) −5.11085e14 −0.826598
\(490\) 3.77642e14 0.603951
\(491\) 4.61449e14 0.729753 0.364877 0.931056i \(-0.381111\pi\)
0.364877 + 0.931056i \(0.381111\pi\)
\(492\) 5.57015e14 0.871082
\(493\) 6.65443e14 1.02909
\(494\) −3.54975e11 −0.000542875 0
\(495\) 1.06492e14 0.161060
\(496\) 1.17301e13 0.0175450
\(497\) −8.69421e14 −1.28609
\(498\) −4.71524e14 −0.689830
\(499\) 8.66882e13 0.125432 0.0627158 0.998031i \(-0.480024\pi\)
0.0627158 + 0.998031i \(0.480024\pi\)
\(500\) 3.88004e13 0.0555266
\(501\) 3.61187e14 0.511240
\(502\) −5.40965e14 −0.757354
\(503\) 9.55447e14 1.32307 0.661535 0.749914i \(-0.269905\pi\)
0.661535 + 0.749914i \(0.269905\pi\)
\(504\) 6.57753e13 0.0900938
\(505\) 8.39063e14 1.13682
\(506\) 7.03496e14 0.942831
\(507\) −4.35466e14 −0.577312
\(508\) 8.78065e14 1.15153
\(509\) 1.23788e14 0.160595 0.0802973 0.996771i \(-0.474413\pi\)
0.0802973 + 0.996771i \(0.474413\pi\)
\(510\) 1.45783e15 1.87098
\(511\) −4.68604e14 −0.594964
\(512\) −1.02342e15 −1.28549
\(513\) −6.93972e12 −0.00862375
\(514\) 4.43615e13 0.0545393
\(515\) −1.33645e13 −0.0162559
\(516\) 1.04189e15 1.25386
\(517\) 1.00746e14 0.119958
\(518\) 6.76094e14 0.796514
\(519\) 7.87505e14 0.917980
\(520\) 3.21527e12 0.00370850
\(521\) −8.29006e14 −0.946128 −0.473064 0.881028i \(-0.656852\pi\)
−0.473064 + 0.881028i \(0.656852\pi\)
\(522\) 2.95924e14 0.334189
\(523\) −6.50445e14 −0.726860 −0.363430 0.931621i \(-0.618394\pi\)
−0.363430 + 0.931621i \(0.618394\pi\)
\(524\) −9.48634e14 −1.04900
\(525\) −4.60543e14 −0.503958
\(526\) −1.02250e15 −1.10725
\(527\) 3.38359e13 0.0362593
\(528\) −1.36395e14 −0.144648
\(529\) 2.37587e15 2.49354
\(530\) 3.42510e14 0.355760
\(531\) 4.22156e13 0.0433963
\(532\) 4.52503e13 0.0460371
\(533\) −1.00415e13 −0.0101111
\(534\) −1.93091e14 −0.192435
\(535\) 3.83573e14 0.378358
\(536\) −2.00317e13 −0.0195574
\(537\) 1.12773e15 1.08980
\(538\) 1.83260e15 1.75293
\(539\) −1.01918e14 −0.0964966
\(540\) 3.55578e14 0.333249
\(541\) −1.05175e15 −0.975725 −0.487862 0.872921i \(-0.662223\pi\)
−0.487862 + 0.872921i \(0.662223\pi\)
\(542\) 5.14127e14 0.472144
\(543\) 2.38772e14 0.217062
\(544\) −2.40964e15 −2.16850
\(545\) −9.95241e14 −0.886641
\(546\) −6.70761e12 −0.00591573
\(547\) 1.20682e15 1.05369 0.526846 0.849961i \(-0.323375\pi\)
0.526846 + 0.849961i \(0.323375\pi\)
\(548\) 1.67688e15 1.44947
\(549\) 2.77534e14 0.237502
\(550\) −6.14474e14 −0.520606
\(551\) 3.59885e13 0.0301877
\(552\) 4.15246e14 0.344859
\(553\) 6.81186e14 0.560115
\(554\) −2.69791e15 −2.19646
\(555\) 6.46107e14 0.520827
\(556\) 1.78691e15 1.42624
\(557\) 4.93551e14 0.390057 0.195028 0.980798i \(-0.437520\pi\)
0.195028 + 0.980798i \(0.437520\pi\)
\(558\) 1.50469e13 0.0117750
\(559\) −1.87825e13 −0.0145542
\(560\) 1.16140e15 0.891146
\(561\) −3.93437e14 −0.298937
\(562\) 1.68044e15 1.26436
\(563\) −1.00373e14 −0.0747857 −0.0373929 0.999301i \(-0.511905\pi\)
−0.0373929 + 0.999301i \(0.511905\pi\)
\(564\) 3.36393e14 0.248206
\(565\) 1.26894e15 0.927202
\(566\) 2.42504e15 1.75480
\(567\) −1.31133e14 −0.0939734
\(568\) −6.84711e14 −0.485950
\(569\) 1.45512e15 1.02278 0.511388 0.859350i \(-0.329132\pi\)
0.511388 + 0.859350i \(0.329132\pi\)
\(570\) 7.88423e13 0.0548842
\(571\) 1.64681e15 1.13539 0.567694 0.823240i \(-0.307836\pi\)
0.567694 + 0.823240i \(0.307836\pi\)
\(572\) −4.90864e12 −0.00335184
\(573\) −5.71212e14 −0.386320
\(574\) 2.33379e15 1.56331
\(575\) −2.90746e15 −1.92904
\(576\) −6.96656e14 −0.457820
\(577\) −2.39991e15 −1.56217 −0.781084 0.624426i \(-0.785333\pi\)
−0.781084 + 0.624426i \(0.785333\pi\)
\(578\) −3.07783e15 −1.98446
\(579\) 1.34495e15 0.858965
\(580\) −1.84399e15 −1.16655
\(581\) −1.08357e15 −0.679030
\(582\) 1.04062e14 0.0645972
\(583\) −9.24365e13 −0.0568416
\(584\) −3.69048e14 −0.224808
\(585\) −6.41011e12 −0.00386819
\(586\) 3.38120e15 2.02132
\(587\) −1.14328e15 −0.677085 −0.338543 0.940951i \(-0.609934\pi\)
−0.338543 + 0.940951i \(0.609934\pi\)
\(588\) −3.40305e14 −0.199661
\(589\) 1.82992e12 0.00106365
\(590\) −4.79612e14 −0.276188
\(591\) 6.71541e14 0.383126
\(592\) −8.27536e14 −0.467754
\(593\) −1.90189e14 −0.106508 −0.0532542 0.998581i \(-0.516959\pi\)
−0.0532542 + 0.998581i \(0.516959\pi\)
\(594\) −1.74962e14 −0.0970777
\(595\) 3.35012e15 1.84169
\(596\) −2.90778e15 −1.58382
\(597\) 1.71553e15 0.925846
\(598\) −4.23459e13 −0.0226441
\(599\) −1.12882e15 −0.598104 −0.299052 0.954237i \(-0.596670\pi\)
−0.299052 + 0.954237i \(0.596670\pi\)
\(600\) −3.62700e14 −0.190421
\(601\) −3.10749e15 −1.61659 −0.808295 0.588778i \(-0.799609\pi\)
−0.808295 + 0.588778i \(0.799609\pi\)
\(602\) 4.36533e15 2.25028
\(603\) 3.99362e13 0.0203996
\(604\) 3.89247e14 0.197025
\(605\) −2.51548e15 −1.26173
\(606\) −1.37855e15 −0.685210
\(607\) 1.59490e15 0.785587 0.392794 0.919627i \(-0.371509\pi\)
0.392794 + 0.919627i \(0.371509\pi\)
\(608\) −1.30319e14 −0.0636116
\(609\) 6.80039e14 0.328957
\(610\) −3.15307e15 −1.51154
\(611\) −6.06426e12 −0.00288105
\(612\) −1.31369e15 −0.618531
\(613\) −2.80189e15 −1.30743 −0.653716 0.756740i \(-0.726791\pi\)
−0.653716 + 0.756740i \(0.726791\pi\)
\(614\) −1.21211e15 −0.560554
\(615\) 2.23028e15 1.02222
\(616\) 2.01674e14 0.0916129
\(617\) 1.29479e15 0.582949 0.291474 0.956579i \(-0.405854\pi\)
0.291474 + 0.956579i \(0.405854\pi\)
\(618\) 2.19573e13 0.00979813
\(619\) 2.21313e15 0.978831 0.489415 0.872051i \(-0.337210\pi\)
0.489415 + 0.872051i \(0.337210\pi\)
\(620\) −9.37617e13 −0.0411028
\(621\) −8.27855e14 −0.359709
\(622\) −2.10913e15 −0.908358
\(623\) −4.43727e14 −0.189423
\(624\) 8.21009e12 0.00347402
\(625\) −2.30528e15 −0.966905
\(626\) 2.54609e15 1.05856
\(627\) −2.12779e13 −0.00876915
\(628\) 5.01003e15 2.04674
\(629\) −2.38706e15 −0.966686
\(630\) 1.48980e15 0.598076
\(631\) 1.83838e14 0.0731600 0.0365800 0.999331i \(-0.488354\pi\)
0.0365800 + 0.999331i \(0.488354\pi\)
\(632\) 5.36467e14 0.211641
\(633\) −1.77391e15 −0.693764
\(634\) −1.82897e15 −0.709114
\(635\) 3.51575e15 1.35133
\(636\) −3.08647e14 −0.117611
\(637\) 6.13478e12 0.00231757
\(638\) 9.07334e14 0.339824
\(639\) 1.36507e15 0.506876
\(640\) 2.41784e15 0.890097
\(641\) −1.34809e15 −0.492040 −0.246020 0.969265i \(-0.579123\pi\)
−0.246020 + 0.969265i \(0.579123\pi\)
\(642\) −6.30198e14 −0.228052
\(643\) −4.86516e15 −1.74557 −0.872784 0.488106i \(-0.837688\pi\)
−0.872784 + 0.488106i \(0.837688\pi\)
\(644\) 5.39802e15 1.92027
\(645\) 4.17172e15 1.47142
\(646\) −2.91285e14 −0.101868
\(647\) 3.74698e15 1.29929 0.649647 0.760236i \(-0.274917\pi\)
0.649647 + 0.760236i \(0.274917\pi\)
\(648\) −1.03274e14 −0.0355080
\(649\) 1.29437e14 0.0441280
\(650\) 3.69873e13 0.0125034
\(651\) 3.45781e13 0.0115906
\(652\) −5.23238e15 −1.73915
\(653\) 1.46018e15 0.481264 0.240632 0.970616i \(-0.422645\pi\)
0.240632 + 0.970616i \(0.422645\pi\)
\(654\) 1.63515e15 0.534416
\(655\) −3.79831e15 −1.23101
\(656\) −2.85654e15 −0.918057
\(657\) 7.35753e14 0.234489
\(658\) 1.40942e15 0.445450
\(659\) −4.59347e14 −0.143970 −0.0719849 0.997406i \(-0.522933\pi\)
−0.0719849 + 0.997406i \(0.522933\pi\)
\(660\) 1.09024e15 0.338868
\(661\) 3.53993e15 1.09115 0.545577 0.838061i \(-0.316310\pi\)
0.545577 + 0.838061i \(0.316310\pi\)
\(662\) −3.69352e15 −1.12907
\(663\) 2.36823e13 0.00717960
\(664\) −8.53366e14 −0.256573
\(665\) 1.81181e14 0.0540249
\(666\) −1.06153e15 −0.313924
\(667\) 4.29316e15 1.25917
\(668\) 3.69776e15 1.07564
\(669\) −1.39848e15 −0.403471
\(670\) −4.53716e14 −0.129829
\(671\) 8.50948e14 0.241507
\(672\) −2.46250e15 −0.693178
\(673\) −4.34346e15 −1.21270 −0.606349 0.795198i \(-0.707367\pi\)
−0.606349 + 0.795198i \(0.707367\pi\)
\(674\) 1.44964e14 0.0401451
\(675\) 7.23097e14 0.198621
\(676\) −4.45821e15 −1.21466
\(677\) 6.18104e15 1.67041 0.835207 0.549936i \(-0.185348\pi\)
0.835207 + 0.549936i \(0.185348\pi\)
\(678\) −2.08482e15 −0.558864
\(679\) 2.39136e14 0.0635858
\(680\) 2.63838e15 0.695886
\(681\) −1.37743e15 −0.360380
\(682\) 4.61355e13 0.0119735
\(683\) 7.29907e15 1.87912 0.939558 0.342391i \(-0.111237\pi\)
0.939558 + 0.342391i \(0.111237\pi\)
\(684\) −7.10473e13 −0.0181443
\(685\) 6.71421e15 1.70097
\(686\) −6.43412e15 −1.61699
\(687\) 3.23909e14 0.0807535
\(688\) −5.34315e15 −1.32148
\(689\) 5.56407e12 0.00136517
\(690\) 9.40529e15 2.28930
\(691\) −1.15596e15 −0.279135 −0.139567 0.990213i \(-0.544571\pi\)
−0.139567 + 0.990213i \(0.544571\pi\)
\(692\) 8.06231e15 1.93142
\(693\) −4.02067e14 −0.0955578
\(694\) −8.48676e15 −2.00108
\(695\) 7.15476e15 1.67371
\(696\) 5.35564e14 0.124297
\(697\) −8.23982e15 −1.89731
\(698\) −1.80855e15 −0.413167
\(699\) 2.71375e15 0.615099
\(700\) −4.71494e15 −1.06032
\(701\) −2.84226e15 −0.634183 −0.317092 0.948395i \(-0.602706\pi\)
−0.317092 + 0.948395i \(0.602706\pi\)
\(702\) 1.05316e13 0.00233152
\(703\) −1.29097e14 −0.0283572
\(704\) −2.13602e15 −0.465539
\(705\) 1.34691e15 0.291272
\(706\) −2.94164e15 −0.631196
\(707\) −3.16794e15 −0.674482
\(708\) 4.32194e14 0.0913053
\(709\) −4.22203e15 −0.885049 −0.442524 0.896756i \(-0.645917\pi\)
−0.442524 + 0.896756i \(0.645917\pi\)
\(710\) −1.55086e16 −3.22591
\(711\) −1.06953e15 −0.220754
\(712\) −3.49457e14 −0.0715737
\(713\) 2.18295e14 0.0443662
\(714\) −5.50413e15 −1.11006
\(715\) −1.96541e13 −0.00393341
\(716\) 1.15455e16 2.29292
\(717\) −1.85856e15 −0.366288
\(718\) −1.13842e16 −2.22648
\(719\) 6.37112e15 1.23654 0.618268 0.785968i \(-0.287835\pi\)
0.618268 + 0.785968i \(0.287835\pi\)
\(720\) −1.82351e15 −0.351221
\(721\) 5.04584e13 0.00964472
\(722\) 7.82966e15 1.48521
\(723\) −2.37574e15 −0.447238
\(724\) 2.44449e15 0.456695
\(725\) −3.74989e15 −0.695280
\(726\) 4.13285e15 0.760498
\(727\) −4.81292e15 −0.878961 −0.439481 0.898252i \(-0.644837\pi\)
−0.439481 + 0.898252i \(0.644837\pi\)
\(728\) −1.21395e13 −0.00220027
\(729\) 2.05891e14 0.0370370
\(730\) −8.35890e15 −1.49236
\(731\) −1.54125e16 −2.73104
\(732\) 2.84133e15 0.499702
\(733\) −8.66668e15 −1.51280 −0.756399 0.654110i \(-0.773043\pi\)
−0.756399 + 0.654110i \(0.773043\pi\)
\(734\) −3.99422e15 −0.691996
\(735\) −1.36258e15 −0.234304
\(736\) −1.55460e16 −2.65333
\(737\) 1.22449e14 0.0207435
\(738\) −3.66427e15 −0.616137
\(739\) −6.34588e15 −1.05913 −0.529563 0.848271i \(-0.677644\pi\)
−0.529563 + 0.848271i \(0.677644\pi\)
\(740\) 6.61471e15 1.09581
\(741\) 1.28079e12 0.000210610 0
\(742\) −1.29317e15 −0.211074
\(743\) −1.45746e15 −0.236134 −0.118067 0.993006i \(-0.537670\pi\)
−0.118067 + 0.993006i \(0.537670\pi\)
\(744\) 2.72320e13 0.00437953
\(745\) −1.16427e16 −1.85863
\(746\) −3.45172e15 −0.546981
\(747\) 1.70131e15 0.267621
\(748\) −4.02793e15 −0.628960
\(749\) −1.44821e15 −0.224482
\(750\) −2.55244e14 −0.0392752
\(751\) −1.26410e16 −1.93092 −0.965458 0.260558i \(-0.916093\pi\)
−0.965458 + 0.260558i \(0.916093\pi\)
\(752\) −1.72513e15 −0.261591
\(753\) 1.95186e15 0.293817
\(754\) −5.46155e13 −0.00816157
\(755\) 1.55854e15 0.231211
\(756\) −1.34251e15 −0.197719
\(757\) 5.70005e15 0.833396 0.416698 0.909045i \(-0.363187\pi\)
0.416698 + 0.909045i \(0.363187\pi\)
\(758\) −8.57927e15 −1.24529
\(759\) −2.53829e15 −0.365773
\(760\) 1.42689e14 0.0204134
\(761\) 1.12577e16 1.59895 0.799476 0.600698i \(-0.205111\pi\)
0.799476 + 0.600698i \(0.205111\pi\)
\(762\) −5.77626e15 −0.814506
\(763\) 3.75760e15 0.526049
\(764\) −5.84795e15 −0.812812
\(765\) −5.26000e15 −0.725852
\(766\) −1.26218e16 −1.72928
\(767\) −7.79129e12 −0.00105983
\(768\) 1.89898e15 0.256469
\(769\) 8.29147e15 1.11183 0.555913 0.831241i \(-0.312369\pi\)
0.555913 + 0.831241i \(0.312369\pi\)
\(770\) 4.56790e15 0.608160
\(771\) −1.60061e14 −0.0211586
\(772\) 1.37694e16 1.80725
\(773\) 2.75697e15 0.359289 0.179645 0.983732i \(-0.442505\pi\)
0.179645 + 0.983732i \(0.442505\pi\)
\(774\) −6.85399e15 −0.886886
\(775\) −1.90672e14 −0.0244978
\(776\) 1.88331e14 0.0240260
\(777\) −2.43942e15 −0.309009
\(778\) −1.33824e16 −1.68324
\(779\) −4.45627e14 −0.0556564
\(780\) −6.56254e13 −0.00813863
\(781\) 4.18546e15 0.515422
\(782\) −3.47481e16 −4.24907
\(783\) −1.06773e15 −0.129649
\(784\) 1.74519e15 0.210428
\(785\) 2.00601e16 2.40187
\(786\) 6.24049e15 0.741984
\(787\) 6.89600e15 0.814210 0.407105 0.913381i \(-0.366538\pi\)
0.407105 + 0.913381i \(0.366538\pi\)
\(788\) 6.87510e15 0.806092
\(789\) 3.68931e15 0.429558
\(790\) 1.21509e16 1.40495
\(791\) −4.79097e15 −0.550114
\(792\) −3.16648e14 −0.0361067
\(793\) −5.12215e13 −0.00580029
\(794\) 1.07997e16 1.21451
\(795\) −1.23582e15 −0.138018
\(796\) 1.75632e16 1.94797
\(797\) 9.13323e15 1.00601 0.503007 0.864283i \(-0.332227\pi\)
0.503007 + 0.864283i \(0.332227\pi\)
\(798\) −2.97675e14 −0.0325631
\(799\) −4.97620e15 −0.540618
\(800\) 1.35788e16 1.46510
\(801\) 6.96694e14 0.0746558
\(802\) 1.46822e16 1.56254
\(803\) 2.25590e15 0.238443
\(804\) 4.08858e14 0.0429204
\(805\) 2.16136e16 2.25345
\(806\) −2.77705e12 −0.000287569 0
\(807\) −6.61222e15 −0.680053
\(808\) −2.49491e15 −0.254854
\(809\) 1.06543e16 1.08096 0.540479 0.841357i \(-0.318243\pi\)
0.540479 + 0.841357i \(0.318243\pi\)
\(810\) −2.33913e15 −0.235715
\(811\) 9.11486e15 0.912295 0.456147 0.889904i \(-0.349229\pi\)
0.456147 + 0.889904i \(0.349229\pi\)
\(812\) 6.96210e15 0.692121
\(813\) −1.85503e15 −0.183169
\(814\) −3.25477e15 −0.319217
\(815\) −2.09503e16 −2.04091
\(816\) 6.73703e15 0.651887
\(817\) −8.33542e14 −0.0801136
\(818\) 8.51908e14 0.0813298
\(819\) 2.42018e13 0.00229502
\(820\) 2.28331e16 2.15074
\(821\) −2.30932e15 −0.216071 −0.108035 0.994147i \(-0.534456\pi\)
−0.108035 + 0.994147i \(0.534456\pi\)
\(822\) −1.10312e16 −1.02525
\(823\) −2.26195e15 −0.208825 −0.104413 0.994534i \(-0.533296\pi\)
−0.104413 + 0.994534i \(0.533296\pi\)
\(824\) 3.97385e13 0.00364428
\(825\) 2.21709e15 0.201970
\(826\) 1.81081e15 0.163864
\(827\) 6.19451e15 0.556835 0.278418 0.960460i \(-0.410190\pi\)
0.278418 + 0.960460i \(0.410190\pi\)
\(828\) −8.47541e15 −0.756822
\(829\) −5.84390e15 −0.518385 −0.259193 0.965826i \(-0.583456\pi\)
−0.259193 + 0.965826i \(0.583456\pi\)
\(830\) −1.93286e16 −1.70322
\(831\) 9.73436e15 0.852122
\(832\) 1.28575e14 0.0111809
\(833\) 5.03407e15 0.434883
\(834\) −1.17550e16 −1.00881
\(835\) 1.48057e16 1.26228
\(836\) −2.17839e14 −0.0184502
\(837\) −5.42910e13 −0.00456812
\(838\) −2.12831e16 −1.77907
\(839\) 4.69763e15 0.390111 0.195056 0.980792i \(-0.437511\pi\)
0.195056 + 0.980792i \(0.437511\pi\)
\(840\) 2.69625e15 0.222446
\(841\) −6.66341e15 −0.546158
\(842\) 3.34290e16 2.72212
\(843\) −6.06320e15 −0.490513
\(844\) −1.81609e16 −1.45967
\(845\) −1.78506e16 −1.42541
\(846\) −2.21293e15 −0.175562
\(847\) 9.49737e15 0.748591
\(848\) 1.58284e15 0.123954
\(849\) −8.74981e15 −0.680779
\(850\) 3.03510e16 2.34622
\(851\) −1.54003e16 −1.18282
\(852\) 1.39753e16 1.06646
\(853\) −5.10348e15 −0.386943 −0.193471 0.981106i \(-0.561975\pi\)
−0.193471 + 0.981106i \(0.561975\pi\)
\(854\) 1.19046e16 0.896803
\(855\) −2.84472e14 −0.0212925
\(856\) −1.14053e15 −0.0848208
\(857\) 9.17857e15 0.678235 0.339118 0.940744i \(-0.389872\pi\)
0.339118 + 0.940744i \(0.389872\pi\)
\(858\) 3.22910e13 0.00237083
\(859\) 2.19842e16 1.60379 0.801896 0.597464i \(-0.203825\pi\)
0.801896 + 0.597464i \(0.203825\pi\)
\(860\) 4.27092e16 3.09585
\(861\) −8.42056e15 −0.606490
\(862\) −2.47039e16 −1.76797
\(863\) 1.88282e16 1.33891 0.669454 0.742854i \(-0.266528\pi\)
0.669454 + 0.742854i \(0.266528\pi\)
\(864\) 3.86636e15 0.273197
\(865\) 3.22813e16 2.26654
\(866\) 4.56082e15 0.318196
\(867\) 1.11052e16 0.769875
\(868\) 3.54004e14 0.0243865
\(869\) −3.27928e15 −0.224476
\(870\) 1.21305e16 0.825128
\(871\) −7.37060e12 −0.000498199 0
\(872\) 2.95930e15 0.198769
\(873\) −3.75466e14 −0.0250606
\(874\) −1.87925e15 −0.124644
\(875\) −5.86557e14 −0.0386603
\(876\) 7.53248e15 0.493362
\(877\) 7.00820e15 0.456151 0.228075 0.973643i \(-0.426757\pi\)
0.228075 + 0.973643i \(0.426757\pi\)
\(878\) 5.28350e15 0.341745
\(879\) −1.21998e16 −0.784174
\(880\) −5.59109e15 −0.357143
\(881\) −2.67563e16 −1.69847 −0.849237 0.528011i \(-0.822938\pi\)
−0.849237 + 0.528011i \(0.822938\pi\)
\(882\) 2.23866e15 0.141225
\(883\) −7.83053e15 −0.490916 −0.245458 0.969407i \(-0.578938\pi\)
−0.245458 + 0.969407i \(0.578938\pi\)
\(884\) 2.42455e14 0.0151058
\(885\) 1.73049e15 0.107148
\(886\) 4.55869e16 2.80514
\(887\) 4.92294e15 0.301054 0.150527 0.988606i \(-0.451903\pi\)
0.150527 + 0.988606i \(0.451903\pi\)
\(888\) −1.92116e15 −0.116760
\(889\) −1.32740e16 −0.801754
\(890\) −7.91516e15 −0.475132
\(891\) 6.31284e14 0.0376615
\(892\) −1.43173e16 −0.848898
\(893\) −2.69124e14 −0.0158587
\(894\) 1.91285e16 1.12028
\(895\) 4.62277e16 2.69077
\(896\) −9.12872e15 −0.528100
\(897\) 1.52789e14 0.00878481
\(898\) −1.67113e16 −0.954969
\(899\) 2.81546e14 0.0159909
\(900\) 7.40291e15 0.417897
\(901\) 4.56576e15 0.256169
\(902\) −1.12350e16 −0.626525
\(903\) −1.57506e16 −0.873001
\(904\) −3.77312e15 −0.207862
\(905\) 9.78769e15 0.535936
\(906\) −2.56062e15 −0.139361
\(907\) −3.37947e14 −0.0182814 −0.00914069 0.999958i \(-0.502910\pi\)
−0.00914069 + 0.999958i \(0.502910\pi\)
\(908\) −1.41018e16 −0.758234
\(909\) 4.97397e15 0.265829
\(910\) −2.74958e14 −0.0146062
\(911\) −3.12034e16 −1.64760 −0.823799 0.566882i \(-0.808150\pi\)
−0.823799 + 0.566882i \(0.808150\pi\)
\(912\) 3.64353e14 0.0191227
\(913\) 5.21640e15 0.272133
\(914\) −4.83820e15 −0.250888
\(915\) 1.13766e16 0.586404
\(916\) 3.31611e15 0.169904
\(917\) 1.43408e16 0.730367
\(918\) 8.64200e15 0.437502
\(919\) 1.51270e16 0.761232 0.380616 0.924733i \(-0.375712\pi\)
0.380616 + 0.924733i \(0.375712\pi\)
\(920\) 1.70217e16 0.851472
\(921\) 4.37345e15 0.217468
\(922\) −8.73783e15 −0.431900
\(923\) −2.51937e14 −0.0123789
\(924\) −4.11628e15 −0.201052
\(925\) 1.34515e16 0.653119
\(926\) −3.75004e16 −1.80999
\(927\) −7.92246e13 −0.00380121
\(928\) −2.00505e16 −0.956336
\(929\) −4.80904e15 −0.228020 −0.114010 0.993480i \(-0.536370\pi\)
−0.114010 + 0.993480i \(0.536370\pi\)
\(930\) 6.16801e14 0.0290729
\(931\) 2.72253e14 0.0127570
\(932\) 2.77828e16 1.29416
\(933\) 7.60999e15 0.352399
\(934\) 4.90806e16 2.25945
\(935\) −1.61277e16 −0.738090
\(936\) 1.90601e13 0.000867178 0
\(937\) 2.84930e16 1.28876 0.644378 0.764708i \(-0.277117\pi\)
0.644378 + 0.764708i \(0.277117\pi\)
\(938\) 1.71304e15 0.0770284
\(939\) −9.18657e15 −0.410670
\(940\) 1.37894e16 0.612833
\(941\) 2.63741e16 1.16529 0.582646 0.812726i \(-0.302017\pi\)
0.582646 + 0.812726i \(0.302017\pi\)
\(942\) −3.29580e16 −1.44771
\(943\) −5.31599e16 −2.32151
\(944\) −2.21642e15 −0.0962292
\(945\) −5.37538e15 −0.232025
\(946\) −2.10151e16 −0.901840
\(947\) 2.20792e16 0.942018 0.471009 0.882128i \(-0.343890\pi\)
0.471009 + 0.882128i \(0.343890\pi\)
\(948\) −1.09496e16 −0.464464
\(949\) −1.35790e14 −0.00572670
\(950\) 1.64145e15 0.0688251
\(951\) 6.59913e15 0.275102
\(952\) −9.96139e15 −0.412873
\(953\) 1.76709e15 0.0728196 0.0364098 0.999337i \(-0.488408\pi\)
0.0364098 + 0.999337i \(0.488408\pi\)
\(954\) 2.03040e15 0.0831891
\(955\) −2.34151e16 −0.953842
\(956\) −1.90276e16 −0.770665
\(957\) −3.27376e15 −0.131835
\(958\) 5.48954e16 2.19799
\(959\) −2.53500e16 −1.00919
\(960\) −2.85572e16 −1.13038
\(961\) −2.53942e16 −0.999437
\(962\) 1.95916e14 0.00766667
\(963\) 2.27382e15 0.0884733
\(964\) −2.43223e16 −0.940983
\(965\) 5.51322e16 2.12083
\(966\) −3.55103e16 −1.35825
\(967\) 2.32514e16 0.884307 0.442153 0.896939i \(-0.354215\pi\)
0.442153 + 0.896939i \(0.354215\pi\)
\(968\) 7.47964e15 0.282857
\(969\) 1.05099e15 0.0395201
\(970\) 4.26568e15 0.159494
\(971\) −2.77595e16 −1.03206 −0.516032 0.856569i \(-0.672591\pi\)
−0.516032 + 0.856569i \(0.672591\pi\)
\(972\) 2.10787e15 0.0779254
\(973\) −2.70133e16 −0.993018
\(974\) 5.18218e16 1.89426
\(975\) −1.33454e14 −0.00485074
\(976\) −1.45712e16 −0.526649
\(977\) −3.62673e16 −1.30345 −0.651727 0.758454i \(-0.725955\pi\)
−0.651727 + 0.758454i \(0.725955\pi\)
\(978\) 3.44206e16 1.23014
\(979\) 2.13614e15 0.0759145
\(980\) −1.39498e16 −0.492973
\(981\) −5.89980e15 −0.207328
\(982\) −3.10778e16 −1.08602
\(983\) −2.32610e16 −0.808321 −0.404160 0.914688i \(-0.632436\pi\)
−0.404160 + 0.914688i \(0.632436\pi\)
\(984\) −6.63160e15 −0.229163
\(985\) 2.75277e16 0.945956
\(986\) −4.48164e16 −1.53149
\(987\) −5.08536e15 −0.172813
\(988\) 1.31125e13 0.000443120 0
\(989\) −9.94352e16 −3.34165
\(990\) −7.17204e15 −0.239689
\(991\) −5.06166e16 −1.68224 −0.841120 0.540849i \(-0.818103\pi\)
−0.841120 + 0.540849i \(0.818103\pi\)
\(992\) −1.01951e15 −0.0336960
\(993\) 1.33266e16 0.438026
\(994\) 5.85539e16 1.91395
\(995\) 7.03227e16 2.28596
\(996\) 1.74177e16 0.563072
\(997\) −3.16497e16 −1.01753 −0.508764 0.860906i \(-0.669897\pi\)
−0.508764 + 0.860906i \(0.669897\pi\)
\(998\) −5.83830e15 −0.186667
\(999\) 3.83013e15 0.121788
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.d.1.5 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.d.1.5 28 1.1 even 1 trivial