Properties

Label 177.12.a.d.1.19
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 1.19
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+31.7299 q^{2} +243.000 q^{3} -1041.21 q^{4} +1823.69 q^{5} +7710.37 q^{6} -52038.7 q^{7} -98020.5 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q+31.7299 q^{2} +243.000 q^{3} -1041.21 q^{4} +1823.69 q^{5} +7710.37 q^{6} -52038.7 q^{7} -98020.5 q^{8} +59049.0 q^{9} +57865.4 q^{10} -443322. q^{11} -253015. q^{12} +780277. q^{13} -1.65118e6 q^{14} +443156. q^{15} -977779. q^{16} +6.30507e6 q^{17} +1.87362e6 q^{18} -9.72491e6 q^{19} -1.89884e6 q^{20} -1.26454e7 q^{21} -1.40666e7 q^{22} -1.54842e7 q^{23} -2.38190e7 q^{24} -4.55023e7 q^{25} +2.47581e7 q^{26} +1.43489e7 q^{27} +5.41833e7 q^{28} -1.80080e8 q^{29} +1.40613e7 q^{30} +2.28440e8 q^{31} +1.69721e8 q^{32} -1.07727e8 q^{33} +2.00059e8 q^{34} -9.49022e7 q^{35} -6.14825e7 q^{36} -3.90774e8 q^{37} -3.08571e8 q^{38} +1.89607e8 q^{39} -1.78759e8 q^{40} +7.69638e8 q^{41} -4.01237e8 q^{42} +2.12524e8 q^{43} +4.61593e8 q^{44} +1.07687e8 q^{45} -4.91312e8 q^{46} +1.63456e9 q^{47} -2.37600e8 q^{48} +7.30697e8 q^{49} -1.44378e9 q^{50} +1.53213e9 q^{51} -8.12434e8 q^{52} +4.66931e9 q^{53} +4.55290e8 q^{54} -8.08481e8 q^{55} +5.10085e9 q^{56} -2.36315e9 q^{57} -5.71393e9 q^{58} +7.14924e8 q^{59} -4.61419e8 q^{60} +1.49764e9 q^{61} +7.24839e9 q^{62} -3.07283e9 q^{63} +7.38773e9 q^{64} +1.42298e9 q^{65} -3.41818e9 q^{66} +1.66508e8 q^{67} -6.56491e9 q^{68} -3.76266e9 q^{69} -3.01124e9 q^{70} -1.80803e10 q^{71} -5.78801e9 q^{72} +1.81596e10 q^{73} -1.23992e10 q^{74} -1.10571e10 q^{75} +1.01257e10 q^{76} +2.30699e10 q^{77} +6.01623e9 q^{78} +1.92745e10 q^{79} -1.78316e9 q^{80} +3.48678e9 q^{81} +2.44206e10 q^{82} +3.05360e10 q^{83} +1.31665e10 q^{84} +1.14985e10 q^{85} +6.74336e9 q^{86} -4.37595e10 q^{87} +4.34547e10 q^{88} -8.33362e9 q^{89} +3.41689e9 q^{90} -4.06046e10 q^{91} +1.61223e10 q^{92} +5.55109e10 q^{93} +5.18643e10 q^{94} -1.77352e10 q^{95} +4.12422e10 q^{96} +8.42053e10 q^{97} +2.31849e10 q^{98} -2.61777e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + O(q^{10}) \) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + 616281q^{10} + 1082362q^{11} + 7099002q^{12} + 503712q^{13} + 1321669q^{14} + 6454566q^{15} + 34870338q^{16} + 13513579q^{17} + 5668704q^{18} + 35971687q^{19} + 96105997q^{20} + 34586919q^{21} - 47598882q^{22} + 61380539q^{23} + 80756433q^{24} + 294744746q^{25} + 62820734q^{26} + 401769396q^{27} + 148068294q^{28} + 322339307q^{29} + 149756283q^{30} + 151247077q^{31} + 466383494q^{32} + 263013966q^{33} + 684479860q^{34} + 960297361q^{35} + 1725057486q^{36} + 863508437q^{37} + 992640509q^{38} + 122402016q^{39} + 3067680252q^{40} + 3081170377q^{41} + 321165567q^{42} + 2554238300q^{43} + 4350123570q^{44} + 1568459538q^{45} - 1987059155q^{46} + 6203398333q^{47} + 8473492134q^{48} + 10327857997q^{49} + 17577682253q^{50} + 3283799697q^{51} + 32137181618q^{52} + 14571770754q^{53} + 1377495072q^{54} + 18251419334q^{55} + 33498842836q^{56} + 8741119941q^{57} + 11860778276q^{58} + 20017880372q^{59} + 23353757271q^{60} + 2761613771q^{61} + 13785829526q^{62} + 8404621317q^{63} + 86547545293q^{64} + 32034985256q^{65} - 11566528326q^{66} + 39381333296q^{67} + 38995496621q^{68} + 14915470977q^{69} + 8551800364q^{70} + 26130020296q^{71} + 19623813219q^{72} + 41382402799q^{73} + 23815315058q^{74} + 71622973278q^{75} + 10611720128q^{76} - 8426124313q^{77} + 15265438362q^{78} + 59825111206q^{79} + 4009687655q^{80} + 97629963228q^{81} - 39592715115q^{82} + 35433122727q^{83} + 35980595442q^{84} - 8950496085q^{85} - 182032360688q^{86} + 78328451601q^{87} - 220003602335q^{88} + 102303043039q^{89} + 36390776769q^{90} - 111146323655q^{91} - 163000203526q^{92} + 36753039711q^{93} - 81314346008q^{94} + 208102168887q^{95} + 113331189042q^{96} - 171891031490q^{97} + 72304707792q^{98} + 63912393738q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 31.7299 0.701139 0.350569 0.936537i \(-0.385988\pi\)
0.350569 + 0.936537i \(0.385988\pi\)
\(3\) 243.000 0.577350
\(4\) −1041.21 −0.508404
\(5\) 1823.69 0.260985 0.130492 0.991449i \(-0.458344\pi\)
0.130492 + 0.991449i \(0.458344\pi\)
\(6\) 7710.37 0.404803
\(7\) −52038.7 −1.17027 −0.585136 0.810935i \(-0.698959\pi\)
−0.585136 + 0.810935i \(0.698959\pi\)
\(8\) −98020.5 −1.05760
\(9\) 59049.0 0.333333
\(10\) 57865.4 0.182986
\(11\) −443322. −0.829965 −0.414983 0.909829i \(-0.636212\pi\)
−0.414983 + 0.909829i \(0.636212\pi\)
\(12\) −253015. −0.293527
\(13\) 780277. 0.582855 0.291427 0.956593i \(-0.405870\pi\)
0.291427 + 0.956593i \(0.405870\pi\)
\(14\) −1.65118e6 −0.820523
\(15\) 443156. 0.150680
\(16\) −977779. −0.233121
\(17\) 6.30507e6 1.07701 0.538506 0.842621i \(-0.318989\pi\)
0.538506 + 0.842621i \(0.318989\pi\)
\(18\) 1.87362e6 0.233713
\(19\) −9.72491e6 −0.901034 −0.450517 0.892768i \(-0.648760\pi\)
−0.450517 + 0.892768i \(0.648760\pi\)
\(20\) −1.89884e6 −0.132686
\(21\) −1.26454e7 −0.675657
\(22\) −1.40666e7 −0.581921
\(23\) −1.54842e7 −0.501632 −0.250816 0.968035i \(-0.580699\pi\)
−0.250816 + 0.968035i \(0.580699\pi\)
\(24\) −2.38190e7 −0.610606
\(25\) −4.55023e7 −0.931887
\(26\) 2.47581e7 0.408662
\(27\) 1.43489e7 0.192450
\(28\) 5.41833e7 0.594972
\(29\) −1.80080e8 −1.63034 −0.815168 0.579224i \(-0.803356\pi\)
−0.815168 + 0.579224i \(0.803356\pi\)
\(30\) 1.40613e7 0.105647
\(31\) 2.28440e8 1.43312 0.716561 0.697525i \(-0.245715\pi\)
0.716561 + 0.697525i \(0.245715\pi\)
\(32\) 1.69721e8 0.894151
\(33\) −1.07727e8 −0.479181
\(34\) 2.00059e8 0.755135
\(35\) −9.49022e7 −0.305423
\(36\) −6.14825e7 −0.169468
\(37\) −3.90774e8 −0.926437 −0.463219 0.886244i \(-0.653306\pi\)
−0.463219 + 0.886244i \(0.653306\pi\)
\(38\) −3.08571e8 −0.631750
\(39\) 1.89607e8 0.336511
\(40\) −1.78759e8 −0.276018
\(41\) 7.69638e8 1.03747 0.518735 0.854935i \(-0.326403\pi\)
0.518735 + 0.854935i \(0.326403\pi\)
\(42\) −4.01237e8 −0.473729
\(43\) 2.12524e8 0.220460 0.110230 0.993906i \(-0.464841\pi\)
0.110230 + 0.993906i \(0.464841\pi\)
\(44\) 4.61593e8 0.421958
\(45\) 1.07687e8 0.0869949
\(46\) −4.91312e8 −0.351714
\(47\) 1.63456e9 1.03959 0.519794 0.854291i \(-0.326009\pi\)
0.519794 + 0.854291i \(0.326009\pi\)
\(48\) −2.37600e8 −0.134592
\(49\) 7.30697e8 0.369538
\(50\) −1.44378e9 −0.653382
\(51\) 1.53213e9 0.621814
\(52\) −8.12434e8 −0.296326
\(53\) 4.66931e9 1.53368 0.766842 0.641836i \(-0.221827\pi\)
0.766842 + 0.641836i \(0.221827\pi\)
\(54\) 4.55290e8 0.134934
\(55\) −8.08481e8 −0.216608
\(56\) 5.10085e9 1.23768
\(57\) −2.36315e9 −0.520212
\(58\) −5.71393e9 −1.14309
\(59\) 7.14924e8 0.130189
\(60\) −4.61419e8 −0.0766061
\(61\) 1.49764e9 0.227035 0.113517 0.993536i \(-0.463788\pi\)
0.113517 + 0.993536i \(0.463788\pi\)
\(62\) 7.24839e9 1.00482
\(63\) −3.07283e9 −0.390091
\(64\) 7.38773e9 0.860045
\(65\) 1.42298e9 0.152116
\(66\) −3.41818e9 −0.335972
\(67\) 1.66508e8 0.0150669 0.00753345 0.999972i \(-0.497602\pi\)
0.00753345 + 0.999972i \(0.497602\pi\)
\(68\) −6.56491e9 −0.547558
\(69\) −3.76266e9 −0.289618
\(70\) −3.01124e9 −0.214144
\(71\) −1.80803e10 −1.18928 −0.594641 0.803991i \(-0.702706\pi\)
−0.594641 + 0.803991i \(0.702706\pi\)
\(72\) −5.78801e9 −0.352534
\(73\) 1.81596e10 1.02525 0.512625 0.858613i \(-0.328673\pi\)
0.512625 + 0.858613i \(0.328673\pi\)
\(74\) −1.23992e10 −0.649561
\(75\) −1.10571e10 −0.538025
\(76\) 1.01257e10 0.458089
\(77\) 2.30699e10 0.971285
\(78\) 6.01623e9 0.235941
\(79\) 1.92745e10 0.704749 0.352375 0.935859i \(-0.385374\pi\)
0.352375 + 0.935859i \(0.385374\pi\)
\(80\) −1.78316e9 −0.0608409
\(81\) 3.48678e9 0.111111
\(82\) 2.44206e10 0.727410
\(83\) 3.05360e10 0.850907 0.425453 0.904980i \(-0.360115\pi\)
0.425453 + 0.904980i \(0.360115\pi\)
\(84\) 1.31665e10 0.343507
\(85\) 1.14985e10 0.281084
\(86\) 6.74336e9 0.154573
\(87\) −4.37595e10 −0.941275
\(88\) 4.34547e10 0.877772
\(89\) −8.33362e9 −0.158193 −0.0790967 0.996867i \(-0.525204\pi\)
−0.0790967 + 0.996867i \(0.525204\pi\)
\(90\) 3.41689e9 0.0609955
\(91\) −4.06046e10 −0.682099
\(92\) 1.61223e10 0.255032
\(93\) 5.55109e10 0.827413
\(94\) 5.18643e10 0.728896
\(95\) −1.77352e10 −0.235156
\(96\) 4.12422e10 0.516238
\(97\) 8.42053e10 0.995623 0.497811 0.867285i \(-0.334137\pi\)
0.497811 + 0.867285i \(0.334137\pi\)
\(98\) 2.31849e10 0.259097
\(99\) −2.61777e10 −0.276655
\(100\) 4.73775e10 0.473775
\(101\) 5.16619e10 0.489106 0.244553 0.969636i \(-0.421359\pi\)
0.244553 + 0.969636i \(0.421359\pi\)
\(102\) 4.86144e10 0.435978
\(103\) 9.70573e10 0.824942 0.412471 0.910971i \(-0.364666\pi\)
0.412471 + 0.910971i \(0.364666\pi\)
\(104\) −7.64831e10 −0.616428
\(105\) −2.30612e10 −0.176336
\(106\) 1.48157e11 1.07533
\(107\) 4.76546e10 0.328469 0.164234 0.986421i \(-0.447485\pi\)
0.164234 + 0.986421i \(0.447485\pi\)
\(108\) −1.49403e10 −0.0978425
\(109\) 7.42731e10 0.462366 0.231183 0.972910i \(-0.425740\pi\)
0.231183 + 0.972910i \(0.425740\pi\)
\(110\) −2.56530e10 −0.151872
\(111\) −9.49580e10 −0.534879
\(112\) 5.08823e10 0.272815
\(113\) 3.03166e10 0.154792 0.0773961 0.997000i \(-0.475339\pi\)
0.0773961 + 0.997000i \(0.475339\pi\)
\(114\) −7.49827e10 −0.364741
\(115\) −2.82383e10 −0.130918
\(116\) 1.87502e11 0.828870
\(117\) 4.60746e10 0.194285
\(118\) 2.26845e10 0.0912805
\(119\) −3.28107e11 −1.26040
\(120\) −4.34383e10 −0.159359
\(121\) −8.87769e10 −0.311158
\(122\) 4.75199e10 0.159183
\(123\) 1.87022e11 0.598983
\(124\) −2.37855e11 −0.728605
\(125\) −1.72029e11 −0.504193
\(126\) −9.75007e10 −0.273508
\(127\) −2.86207e11 −0.768704 −0.384352 0.923187i \(-0.625575\pi\)
−0.384352 + 0.923187i \(0.625575\pi\)
\(128\) −1.13177e11 −0.291140
\(129\) 5.16432e10 0.127283
\(130\) 4.51510e10 0.106655
\(131\) 2.74019e11 0.620566 0.310283 0.950644i \(-0.399576\pi\)
0.310283 + 0.950644i \(0.399576\pi\)
\(132\) 1.12167e11 0.243618
\(133\) 5.06072e11 1.05445
\(134\) 5.28329e9 0.0105640
\(135\) 2.61679e10 0.0502265
\(136\) −6.18026e11 −1.13905
\(137\) −6.66596e11 −1.18005 −0.590024 0.807386i \(-0.700882\pi\)
−0.590024 + 0.807386i \(0.700882\pi\)
\(138\) −1.19389e11 −0.203062
\(139\) 1.81154e11 0.296119 0.148059 0.988978i \(-0.452697\pi\)
0.148059 + 0.988978i \(0.452697\pi\)
\(140\) 9.88133e10 0.155278
\(141\) 3.97197e11 0.600207
\(142\) −5.73687e11 −0.833852
\(143\) −3.45914e11 −0.483749
\(144\) −5.77369e10 −0.0777069
\(145\) −3.28410e11 −0.425493
\(146\) 5.76201e11 0.718842
\(147\) 1.77559e11 0.213353
\(148\) 4.06878e11 0.471005
\(149\) 4.69746e10 0.0524009 0.0262005 0.999657i \(-0.491659\pi\)
0.0262005 + 0.999657i \(0.491659\pi\)
\(150\) −3.50840e11 −0.377230
\(151\) 1.57217e12 1.62977 0.814885 0.579623i \(-0.196800\pi\)
0.814885 + 0.579623i \(0.196800\pi\)
\(152\) 9.53241e11 0.952934
\(153\) 3.72308e11 0.359004
\(154\) 7.32006e11 0.681006
\(155\) 4.16603e11 0.374023
\(156\) −1.97421e11 −0.171084
\(157\) 1.25470e12 1.04977 0.524883 0.851175i \(-0.324109\pi\)
0.524883 + 0.851175i \(0.324109\pi\)
\(158\) 6.11579e11 0.494127
\(159\) 1.13464e12 0.885473
\(160\) 3.09518e11 0.233360
\(161\) 8.05777e11 0.587047
\(162\) 1.10635e11 0.0779043
\(163\) 5.24687e11 0.357165 0.178582 0.983925i \(-0.442849\pi\)
0.178582 + 0.983925i \(0.442849\pi\)
\(164\) −8.01356e11 −0.527454
\(165\) −1.96461e11 −0.125059
\(166\) 9.68904e11 0.596604
\(167\) 2.89074e11 0.172214 0.0861071 0.996286i \(-0.472557\pi\)
0.0861071 + 0.996286i \(0.472557\pi\)
\(168\) 1.23951e12 0.714576
\(169\) −1.18333e12 −0.660280
\(170\) 3.64845e11 0.197079
\(171\) −5.74246e11 −0.300345
\(172\) −2.21282e11 −0.112083
\(173\) −3.71003e12 −1.82022 −0.910109 0.414369i \(-0.864002\pi\)
−0.910109 + 0.414369i \(0.864002\pi\)
\(174\) −1.38849e12 −0.659965
\(175\) 2.36788e12 1.09056
\(176\) 4.33471e11 0.193482
\(177\) 1.73727e11 0.0751646
\(178\) −2.64425e11 −0.110916
\(179\) 2.04322e12 0.831043 0.415522 0.909583i \(-0.363599\pi\)
0.415522 + 0.909583i \(0.363599\pi\)
\(180\) −1.12125e11 −0.0442286
\(181\) −7.73103e11 −0.295805 −0.147902 0.989002i \(-0.547252\pi\)
−0.147902 + 0.989002i \(0.547252\pi\)
\(182\) −1.28838e12 −0.478246
\(183\) 3.63926e11 0.131079
\(184\) 1.51777e12 0.530527
\(185\) −7.12649e11 −0.241786
\(186\) 1.76136e12 0.580131
\(187\) −2.79518e12 −0.893883
\(188\) −1.70192e12 −0.528531
\(189\) −7.46698e11 −0.225219
\(190\) −5.62736e11 −0.164877
\(191\) 5.26945e12 1.49997 0.749984 0.661457i \(-0.230061\pi\)
0.749984 + 0.661457i \(0.230061\pi\)
\(192\) 1.79522e12 0.496547
\(193\) 3.18046e12 0.854918 0.427459 0.904035i \(-0.359409\pi\)
0.427459 + 0.904035i \(0.359409\pi\)
\(194\) 2.67183e12 0.698070
\(195\) 3.45784e11 0.0878243
\(196\) −7.60810e11 −0.187875
\(197\) −5.18962e12 −1.24615 −0.623077 0.782161i \(-0.714117\pi\)
−0.623077 + 0.782161i \(0.714117\pi\)
\(198\) −8.30618e11 −0.193974
\(199\) −5.44048e11 −0.123579 −0.0617895 0.998089i \(-0.519681\pi\)
−0.0617895 + 0.998089i \(0.519681\pi\)
\(200\) 4.46016e12 0.985564
\(201\) 4.04615e10 0.00869888
\(202\) 1.63923e12 0.342931
\(203\) 9.37114e12 1.90794
\(204\) −1.59527e12 −0.316133
\(205\) 1.40358e12 0.270764
\(206\) 3.07962e12 0.578399
\(207\) −9.14326e11 −0.167211
\(208\) −7.62939e11 −0.135876
\(209\) 4.31127e12 0.747827
\(210\) −7.31731e11 −0.123636
\(211\) 4.24001e12 0.697933 0.348966 0.937135i \(-0.386533\pi\)
0.348966 + 0.937135i \(0.386533\pi\)
\(212\) −4.86175e12 −0.779731
\(213\) −4.39352e12 −0.686633
\(214\) 1.51208e12 0.230302
\(215\) 3.87576e11 0.0575368
\(216\) −1.40649e12 −0.203535
\(217\) −1.18877e13 −1.67714
\(218\) 2.35668e12 0.324183
\(219\) 4.41277e12 0.591928
\(220\) 8.41800e11 0.110125
\(221\) 4.91970e12 0.627742
\(222\) −3.01301e12 −0.375024
\(223\) 1.47233e13 1.78784 0.893921 0.448225i \(-0.147944\pi\)
0.893921 + 0.448225i \(0.147944\pi\)
\(224\) −8.83206e12 −1.04640
\(225\) −2.68687e12 −0.310629
\(226\) 9.61943e11 0.108531
\(227\) −3.07095e12 −0.338167 −0.169083 0.985602i \(-0.554081\pi\)
−0.169083 + 0.985602i \(0.554081\pi\)
\(228\) 2.46054e12 0.264478
\(229\) 1.73696e12 0.182261 0.0911306 0.995839i \(-0.470952\pi\)
0.0911306 + 0.995839i \(0.470952\pi\)
\(230\) −8.95999e11 −0.0917920
\(231\) 5.60599e12 0.560772
\(232\) 1.76516e13 1.72425
\(233\) −8.97649e12 −0.856345 −0.428173 0.903697i \(-0.640842\pi\)
−0.428173 + 0.903697i \(0.640842\pi\)
\(234\) 1.46194e12 0.136221
\(235\) 2.98092e12 0.271317
\(236\) −7.44388e11 −0.0661886
\(237\) 4.68371e12 0.406887
\(238\) −1.04108e13 −0.883714
\(239\) −1.45199e12 −0.120441 −0.0602205 0.998185i \(-0.519180\pi\)
−0.0602205 + 0.998185i \(0.519180\pi\)
\(240\) −4.33308e11 −0.0351265
\(241\) 1.32929e13 1.05323 0.526617 0.850103i \(-0.323460\pi\)
0.526617 + 0.850103i \(0.323460\pi\)
\(242\) −2.81688e12 −0.218165
\(243\) 8.47289e11 0.0641500
\(244\) −1.55936e12 −0.115426
\(245\) 1.33256e12 0.0964436
\(246\) 5.93420e12 0.419970
\(247\) −7.58813e12 −0.525172
\(248\) −2.23918e13 −1.51567
\(249\) 7.42024e12 0.491271
\(250\) −5.45847e12 −0.353509
\(251\) 1.27985e13 0.810876 0.405438 0.914123i \(-0.367119\pi\)
0.405438 + 0.914123i \(0.367119\pi\)
\(252\) 3.19947e12 0.198324
\(253\) 6.86449e12 0.416338
\(254\) −9.08131e12 −0.538968
\(255\) 2.79413e12 0.162284
\(256\) −1.87212e13 −1.06417
\(257\) 1.91661e12 0.106635 0.0533176 0.998578i \(-0.483020\pi\)
0.0533176 + 0.998578i \(0.483020\pi\)
\(258\) 1.63864e12 0.0892430
\(259\) 2.03353e13 1.08418
\(260\) −1.48162e12 −0.0773365
\(261\) −1.06336e13 −0.543445
\(262\) 8.69460e12 0.435103
\(263\) 6.84308e12 0.335348 0.167674 0.985843i \(-0.446374\pi\)
0.167674 + 0.985843i \(0.446374\pi\)
\(264\) 1.05595e13 0.506782
\(265\) 8.51536e12 0.400268
\(266\) 1.60576e13 0.739319
\(267\) −2.02507e12 −0.0913330
\(268\) −1.73370e11 −0.00766008
\(269\) 7.41245e12 0.320866 0.160433 0.987047i \(-0.448711\pi\)
0.160433 + 0.987047i \(0.448711\pi\)
\(270\) 8.30305e11 0.0352158
\(271\) 7.82419e12 0.325168 0.162584 0.986695i \(-0.448017\pi\)
0.162584 + 0.986695i \(0.448017\pi\)
\(272\) −6.16497e12 −0.251074
\(273\) −9.86691e12 −0.393810
\(274\) −2.11511e13 −0.827378
\(275\) 2.01722e13 0.773434
\(276\) 3.91773e12 0.147243
\(277\) 3.55026e13 1.30804 0.654021 0.756476i \(-0.273081\pi\)
0.654021 + 0.756476i \(0.273081\pi\)
\(278\) 5.74799e12 0.207620
\(279\) 1.34892e13 0.477707
\(280\) 9.30236e12 0.323016
\(281\) 4.78222e12 0.162834 0.0814170 0.996680i \(-0.474055\pi\)
0.0814170 + 0.996680i \(0.474055\pi\)
\(282\) 1.26030e13 0.420828
\(283\) −3.81288e13 −1.24861 −0.624307 0.781179i \(-0.714619\pi\)
−0.624307 + 0.781179i \(0.714619\pi\)
\(284\) 1.88254e13 0.604636
\(285\) −4.30965e12 −0.135767
\(286\) −1.09758e13 −0.339175
\(287\) −4.00509e13 −1.21412
\(288\) 1.00219e13 0.298050
\(289\) 5.48200e12 0.159956
\(290\) −1.04204e13 −0.298330
\(291\) 2.04619e13 0.574823
\(292\) −1.89079e13 −0.521241
\(293\) −5.77895e13 −1.56343 −0.781713 0.623639i \(-0.785654\pi\)
−0.781713 + 0.623639i \(0.785654\pi\)
\(294\) 5.63394e12 0.149590
\(295\) 1.30380e12 0.0339773
\(296\) 3.83038e13 0.979801
\(297\) −6.36119e12 −0.159727
\(298\) 1.49050e12 0.0367403
\(299\) −1.20820e13 −0.292379
\(300\) 1.15127e13 0.273534
\(301\) −1.10594e13 −0.257999
\(302\) 4.98848e13 1.14269
\(303\) 1.25538e13 0.282385
\(304\) 9.50882e12 0.210050
\(305\) 2.73122e12 0.0592526
\(306\) 1.18133e13 0.251712
\(307\) −6.85156e13 −1.43393 −0.716966 0.697108i \(-0.754470\pi\)
−0.716966 + 0.697108i \(0.754470\pi\)
\(308\) −2.40207e13 −0.493806
\(309\) 2.35849e13 0.476280
\(310\) 1.32188e13 0.262242
\(311\) −1.67459e13 −0.326382 −0.163191 0.986594i \(-0.552179\pi\)
−0.163191 + 0.986594i \(0.552179\pi\)
\(312\) −1.85854e13 −0.355895
\(313\) 2.32665e13 0.437761 0.218881 0.975752i \(-0.429759\pi\)
0.218881 + 0.975752i \(0.429759\pi\)
\(314\) 3.98116e13 0.736031
\(315\) −5.60388e12 −0.101808
\(316\) −2.00689e13 −0.358298
\(317\) −2.53473e13 −0.444740 −0.222370 0.974962i \(-0.571379\pi\)
−0.222370 + 0.974962i \(0.571379\pi\)
\(318\) 3.60021e13 0.620839
\(319\) 7.98336e13 1.35312
\(320\) 1.34729e13 0.224458
\(321\) 1.15801e13 0.189642
\(322\) 2.55672e13 0.411601
\(323\) −6.13163e13 −0.970425
\(324\) −3.63048e12 −0.0564894
\(325\) −3.55044e13 −0.543155
\(326\) 1.66483e13 0.250422
\(327\) 1.80484e13 0.266947
\(328\) −7.54403e13 −1.09723
\(329\) −8.50601e13 −1.21660
\(330\) −6.23369e12 −0.0876836
\(331\) −9.52814e13 −1.31812 −0.659059 0.752092i \(-0.729045\pi\)
−0.659059 + 0.752092i \(0.729045\pi\)
\(332\) −3.17944e13 −0.432605
\(333\) −2.30748e13 −0.308812
\(334\) 9.17231e12 0.120746
\(335\) 3.03659e11 0.00393223
\(336\) 1.23644e13 0.157510
\(337\) 1.35144e13 0.169369 0.0846844 0.996408i \(-0.473012\pi\)
0.0846844 + 0.996408i \(0.473012\pi\)
\(338\) −3.75469e13 −0.462948
\(339\) 7.36693e12 0.0893693
\(340\) −1.19723e13 −0.142904
\(341\) −1.01273e14 −1.18944
\(342\) −1.82208e13 −0.210583
\(343\) 6.48730e13 0.737813
\(344\) −2.08317e13 −0.233159
\(345\) −6.86191e12 −0.0755858
\(346\) −1.17719e14 −1.27623
\(347\) 6.87579e13 0.733686 0.366843 0.930283i \(-0.380439\pi\)
0.366843 + 0.930283i \(0.380439\pi\)
\(348\) 4.55629e13 0.478548
\(349\) −1.03182e14 −1.06676 −0.533378 0.845877i \(-0.679078\pi\)
−0.533378 + 0.845877i \(0.679078\pi\)
\(350\) 7.51326e13 0.764635
\(351\) 1.11961e13 0.112170
\(352\) −7.52411e13 −0.742114
\(353\) 1.08150e13 0.105018 0.0525092 0.998620i \(-0.483278\pi\)
0.0525092 + 0.998620i \(0.483278\pi\)
\(354\) 5.51233e12 0.0527008
\(355\) −3.29728e13 −0.310384
\(356\) 8.67706e12 0.0804262
\(357\) −7.97301e13 −0.727691
\(358\) 6.48312e13 0.582677
\(359\) −2.15264e13 −0.190525 −0.0952627 0.995452i \(-0.530369\pi\)
−0.0952627 + 0.995452i \(0.530369\pi\)
\(360\) −1.05555e13 −0.0920059
\(361\) −2.19163e13 −0.188138
\(362\) −2.45305e13 −0.207400
\(363\) −2.15728e13 −0.179647
\(364\) 4.22780e13 0.346782
\(365\) 3.31173e13 0.267574
\(366\) 1.15473e13 0.0919044
\(367\) 1.54251e13 0.120939 0.0604693 0.998170i \(-0.480740\pi\)
0.0604693 + 0.998170i \(0.480740\pi\)
\(368\) 1.51401e13 0.116941
\(369\) 4.54464e13 0.345823
\(370\) −2.26123e13 −0.169525
\(371\) −2.42985e14 −1.79483
\(372\) −5.77987e13 −0.420660
\(373\) −2.20761e14 −1.58315 −0.791577 0.611069i \(-0.790740\pi\)
−0.791577 + 0.611069i \(0.790740\pi\)
\(374\) −8.86908e13 −0.626736
\(375\) −4.18031e13 −0.291096
\(376\) −1.60220e14 −1.09947
\(377\) −1.40513e14 −0.950249
\(378\) −2.36927e13 −0.157910
\(379\) −3.21113e13 −0.210932 −0.105466 0.994423i \(-0.533633\pi\)
−0.105466 + 0.994423i \(0.533633\pi\)
\(380\) 1.84661e13 0.119554
\(381\) −6.95482e13 −0.443812
\(382\) 1.67199e14 1.05169
\(383\) −1.99474e14 −1.23678 −0.618392 0.785870i \(-0.712215\pi\)
−0.618392 + 0.785870i \(0.712215\pi\)
\(384\) −2.75019e13 −0.168090
\(385\) 4.20723e13 0.253491
\(386\) 1.00916e14 0.599416
\(387\) 1.25493e13 0.0734868
\(388\) −8.76756e13 −0.506179
\(389\) −2.66409e14 −1.51644 −0.758222 0.651996i \(-0.773932\pi\)
−0.758222 + 0.651996i \(0.773932\pi\)
\(390\) 1.09717e13 0.0615770
\(391\) −9.76289e13 −0.540264
\(392\) −7.16232e13 −0.390823
\(393\) 6.65866e13 0.358284
\(394\) −1.64666e14 −0.873726
\(395\) 3.51507e13 0.183929
\(396\) 2.72566e13 0.140653
\(397\) −1.13244e14 −0.576327 −0.288164 0.957581i \(-0.593045\pi\)
−0.288164 + 0.957581i \(0.593045\pi\)
\(398\) −1.72626e13 −0.0866461
\(399\) 1.22975e14 0.608790
\(400\) 4.44912e13 0.217242
\(401\) −2.36341e13 −0.113827 −0.0569136 0.998379i \(-0.518126\pi\)
−0.0569136 + 0.998379i \(0.518126\pi\)
\(402\) 1.28384e12 0.00609912
\(403\) 1.78247e14 0.835302
\(404\) −5.37910e13 −0.248664
\(405\) 6.35880e12 0.0289983
\(406\) 2.97346e14 1.33773
\(407\) 1.73239e14 0.768911
\(408\) −1.50180e14 −0.657631
\(409\) 3.20954e14 1.38664 0.693321 0.720629i \(-0.256147\pi\)
0.693321 + 0.720629i \(0.256147\pi\)
\(410\) 4.45354e13 0.189843
\(411\) −1.61983e14 −0.681301
\(412\) −1.01057e14 −0.419404
\(413\) −3.72037e13 −0.152356
\(414\) −2.90115e13 −0.117238
\(415\) 5.56880e13 0.222074
\(416\) 1.32429e14 0.521160
\(417\) 4.40203e13 0.170964
\(418\) 1.36796e14 0.524330
\(419\) 3.70709e14 1.40235 0.701174 0.712990i \(-0.252659\pi\)
0.701174 + 0.712990i \(0.252659\pi\)
\(420\) 2.40116e13 0.0896501
\(421\) 5.33573e12 0.0196627 0.00983133 0.999952i \(-0.496871\pi\)
0.00983133 + 0.999952i \(0.496871\pi\)
\(422\) 1.34535e14 0.489348
\(423\) 9.65189e13 0.346530
\(424\) −4.57688e14 −1.62203
\(425\) −2.86895e14 −1.00365
\(426\) −1.39406e14 −0.481425
\(427\) −7.79351e13 −0.265693
\(428\) −4.96185e13 −0.166995
\(429\) −8.40572e13 −0.279293
\(430\) 1.22978e13 0.0403413
\(431\) 4.38169e14 1.41911 0.709557 0.704648i \(-0.248895\pi\)
0.709557 + 0.704648i \(0.248895\pi\)
\(432\) −1.40301e13 −0.0448641
\(433\) −1.97721e14 −0.624267 −0.312133 0.950038i \(-0.601044\pi\)
−0.312133 + 0.950038i \(0.601044\pi\)
\(434\) −3.77196e14 −1.17591
\(435\) −7.98036e13 −0.245658
\(436\) −7.73340e13 −0.235069
\(437\) 1.50583e14 0.451988
\(438\) 1.40017e14 0.415024
\(439\) 2.87713e14 0.842180 0.421090 0.907019i \(-0.361648\pi\)
0.421090 + 0.907019i \(0.361648\pi\)
\(440\) 7.92477e13 0.229085
\(441\) 4.31469e13 0.123179
\(442\) 1.56102e14 0.440134
\(443\) 5.49354e14 1.52979 0.764895 0.644155i \(-0.222791\pi\)
0.764895 + 0.644155i \(0.222791\pi\)
\(444\) 9.88714e13 0.271935
\(445\) −1.51979e13 −0.0412861
\(446\) 4.67170e14 1.25352
\(447\) 1.14148e13 0.0302537
\(448\) −3.84448e14 −1.00649
\(449\) −3.18321e14 −0.823210 −0.411605 0.911362i \(-0.635032\pi\)
−0.411605 + 0.911362i \(0.635032\pi\)
\(450\) −8.52540e13 −0.217794
\(451\) −3.41198e14 −0.861064
\(452\) −3.15660e13 −0.0786970
\(453\) 3.82037e14 0.940948
\(454\) −9.74411e13 −0.237102
\(455\) −7.40500e13 −0.178017
\(456\) 2.31637e14 0.550177
\(457\) 7.42812e13 0.174317 0.0871585 0.996194i \(-0.472221\pi\)
0.0871585 + 0.996194i \(0.472221\pi\)
\(458\) 5.51136e13 0.127790
\(459\) 9.04709e13 0.207271
\(460\) 2.94021e13 0.0665595
\(461\) 4.18115e14 0.935279 0.467639 0.883919i \(-0.345105\pi\)
0.467639 + 0.883919i \(0.345105\pi\)
\(462\) 1.77878e14 0.393179
\(463\) 3.81881e14 0.834128 0.417064 0.908877i \(-0.363059\pi\)
0.417064 + 0.908877i \(0.363059\pi\)
\(464\) 1.76079e14 0.380065
\(465\) 1.01235e14 0.215942
\(466\) −2.84823e14 −0.600417
\(467\) −8.74922e14 −1.82275 −0.911374 0.411580i \(-0.864977\pi\)
−0.911374 + 0.411580i \(0.864977\pi\)
\(468\) −4.79734e13 −0.0987753
\(469\) −8.66486e12 −0.0176324
\(470\) 9.45842e13 0.190231
\(471\) 3.04892e14 0.606082
\(472\) −7.00772e13 −0.137688
\(473\) −9.42165e13 −0.182974
\(474\) 1.48614e14 0.285284
\(475\) 4.42506e14 0.839661
\(476\) 3.41629e14 0.640792
\(477\) 2.75718e14 0.511228
\(478\) −4.60714e13 −0.0844458
\(479\) 1.37469e14 0.249092 0.124546 0.992214i \(-0.460253\pi\)
0.124546 + 0.992214i \(0.460253\pi\)
\(480\) 7.52128e13 0.134730
\(481\) −3.04912e14 −0.539978
\(482\) 4.21782e14 0.738463
\(483\) 1.95804e14 0.338932
\(484\) 9.24356e13 0.158194
\(485\) 1.53564e14 0.259842
\(486\) 2.68844e13 0.0449781
\(487\) 6.78733e14 1.12277 0.561384 0.827556i \(-0.310269\pi\)
0.561384 + 0.827556i \(0.310269\pi\)
\(488\) −1.46799e14 −0.240112
\(489\) 1.27499e14 0.206209
\(490\) 4.22821e13 0.0676204
\(491\) 2.33040e14 0.368538 0.184269 0.982876i \(-0.441008\pi\)
0.184269 + 0.982876i \(0.441008\pi\)
\(492\) −1.94730e14 −0.304526
\(493\) −1.13542e15 −1.75589
\(494\) −2.40771e14 −0.368218
\(495\) −4.77400e13 −0.0722027
\(496\) −2.23364e14 −0.334090
\(497\) 9.40875e14 1.39178
\(498\) 2.35444e14 0.344449
\(499\) −2.66776e14 −0.386006 −0.193003 0.981198i \(-0.561823\pi\)
−0.193003 + 0.981198i \(0.561823\pi\)
\(500\) 1.79119e14 0.256334
\(501\) 7.02451e13 0.0994280
\(502\) 4.06096e14 0.568536
\(503\) −2.58819e14 −0.358403 −0.179202 0.983812i \(-0.557351\pi\)
−0.179202 + 0.983812i \(0.557351\pi\)
\(504\) 3.01200e14 0.412560
\(505\) 9.42151e13 0.127649
\(506\) 2.17810e14 0.291910
\(507\) −2.87549e14 −0.381213
\(508\) 2.98002e14 0.390813
\(509\) 1.20045e15 1.55738 0.778691 0.627407i \(-0.215884\pi\)
0.778691 + 0.627407i \(0.215884\pi\)
\(510\) 8.86574e13 0.113783
\(511\) −9.44999e14 −1.19982
\(512\) −3.62235e14 −0.454994
\(513\) −1.39542e14 −0.173404
\(514\) 6.08137e13 0.0747661
\(515\) 1.77002e14 0.215297
\(516\) −5.37716e13 −0.0647112
\(517\) −7.24635e14 −0.862822
\(518\) 6.45239e14 0.760163
\(519\) −9.01536e14 −1.05090
\(520\) −1.39481e14 −0.160878
\(521\) 5.94863e14 0.678906 0.339453 0.940623i \(-0.389758\pi\)
0.339453 + 0.940623i \(0.389758\pi\)
\(522\) −3.37402e14 −0.381031
\(523\) −1.47836e15 −1.65204 −0.826019 0.563643i \(-0.809399\pi\)
−0.826019 + 0.563643i \(0.809399\pi\)
\(524\) −2.85312e14 −0.315499
\(525\) 5.75395e14 0.629636
\(526\) 2.17130e14 0.235125
\(527\) 1.44033e15 1.54349
\(528\) 1.05334e14 0.111707
\(529\) −7.13049e14 −0.748365
\(530\) 2.70192e14 0.280643
\(531\) 4.22156e13 0.0433963
\(532\) −5.26928e14 −0.536089
\(533\) 6.00531e14 0.604694
\(534\) −6.42553e13 −0.0640371
\(535\) 8.69070e13 0.0857253
\(536\) −1.63212e13 −0.0159348
\(537\) 4.96503e14 0.479803
\(538\) 2.35196e14 0.224972
\(539\) −3.23934e14 −0.306703
\(540\) −2.72463e13 −0.0255354
\(541\) −1.80702e15 −1.67640 −0.838201 0.545361i \(-0.816393\pi\)
−0.838201 + 0.545361i \(0.816393\pi\)
\(542\) 2.48261e14 0.227988
\(543\) −1.87864e14 −0.170783
\(544\) 1.07010e15 0.963012
\(545\) 1.35451e14 0.120670
\(546\) −3.13076e14 −0.276115
\(547\) −2.61646e14 −0.228446 −0.114223 0.993455i \(-0.536438\pi\)
−0.114223 + 0.993455i \(0.536438\pi\)
\(548\) 6.94068e14 0.599942
\(549\) 8.84340e13 0.0756783
\(550\) 6.40062e14 0.542285
\(551\) 1.75127e15 1.46899
\(552\) 3.68818e14 0.306300
\(553\) −1.00302e15 −0.824749
\(554\) 1.12650e15 0.917119
\(555\) −1.73174e14 −0.139595
\(556\) −1.88619e14 −0.150548
\(557\) −1.71485e13 −0.0135526 −0.00677629 0.999977i \(-0.502157\pi\)
−0.00677629 + 0.999977i \(0.502157\pi\)
\(558\) 4.28010e14 0.334939
\(559\) 1.65827e14 0.128496
\(560\) 9.27934e13 0.0712005
\(561\) −6.79228e14 −0.516084
\(562\) 1.51740e14 0.114169
\(563\) 2.23152e15 1.66266 0.831331 0.555778i \(-0.187580\pi\)
0.831331 + 0.555778i \(0.187580\pi\)
\(564\) −4.13566e14 −0.305148
\(565\) 5.52880e13 0.0403984
\(566\) −1.20983e15 −0.875452
\(567\) −1.81448e14 −0.130030
\(568\) 1.77224e15 1.25779
\(569\) 1.21180e15 0.851755 0.425878 0.904781i \(-0.359965\pi\)
0.425878 + 0.904781i \(0.359965\pi\)
\(570\) −1.36745e14 −0.0951918
\(571\) −4.26898e14 −0.294324 −0.147162 0.989112i \(-0.547014\pi\)
−0.147162 + 0.989112i \(0.547014\pi\)
\(572\) 3.60170e14 0.245940
\(573\) 1.28048e15 0.866006
\(574\) −1.27081e15 −0.851268
\(575\) 7.04567e14 0.467465
\(576\) 4.36238e14 0.286682
\(577\) 2.37727e15 1.54743 0.773717 0.633531i \(-0.218395\pi\)
0.773717 + 0.633531i \(0.218395\pi\)
\(578\) 1.73944e14 0.112152
\(579\) 7.72852e14 0.493587
\(580\) 3.41944e14 0.216322
\(581\) −1.58905e15 −0.995793
\(582\) 6.49254e14 0.403031
\(583\) −2.07001e15 −1.27290
\(584\) −1.78001e15 −1.08430
\(585\) 8.40256e13 0.0507054
\(586\) −1.83366e15 −1.09618
\(587\) 3.05800e15 1.81104 0.905521 0.424302i \(-0.139480\pi\)
0.905521 + 0.424302i \(0.139480\pi\)
\(588\) −1.84877e14 −0.108469
\(589\) −2.22156e15 −1.29129
\(590\) 4.13694e13 0.0238228
\(591\) −1.26108e15 −0.719467
\(592\) 3.82090e14 0.215972
\(593\) −1.64773e15 −0.922754 −0.461377 0.887204i \(-0.652645\pi\)
−0.461377 + 0.887204i \(0.652645\pi\)
\(594\) −2.01840e14 −0.111991
\(595\) −5.98365e14 −0.328945
\(596\) −4.89106e13 −0.0266409
\(597\) −1.32204e14 −0.0713484
\(598\) −3.83360e14 −0.204998
\(599\) −2.26430e15 −1.19974 −0.599869 0.800098i \(-0.704781\pi\)
−0.599869 + 0.800098i \(0.704781\pi\)
\(600\) 1.08382e15 0.569016
\(601\) −3.57425e15 −1.85941 −0.929704 0.368307i \(-0.879937\pi\)
−0.929704 + 0.368307i \(0.879937\pi\)
\(602\) −3.50915e14 −0.180893
\(603\) 9.83214e12 0.00502230
\(604\) −1.63696e15 −0.828582
\(605\) −1.61901e14 −0.0812074
\(606\) 3.98333e14 0.197991
\(607\) 3.14155e15 1.54741 0.773706 0.633544i \(-0.218401\pi\)
0.773706 + 0.633544i \(0.218401\pi\)
\(608\) −1.65052e15 −0.805660
\(609\) 2.27719e15 1.10155
\(610\) 8.66614e13 0.0415443
\(611\) 1.27541e15 0.605929
\(612\) −3.87652e14 −0.182519
\(613\) 3.80047e15 1.77339 0.886695 0.462354i \(-0.152995\pi\)
0.886695 + 0.462354i \(0.152995\pi\)
\(614\) −2.17400e15 −1.00539
\(615\) 3.41069e14 0.156325
\(616\) −2.26132e15 −1.02723
\(617\) −1.04144e15 −0.468883 −0.234442 0.972130i \(-0.575326\pi\)
−0.234442 + 0.972130i \(0.575326\pi\)
\(618\) 7.48348e14 0.333939
\(619\) 1.05252e15 0.465514 0.232757 0.972535i \(-0.425225\pi\)
0.232757 + 0.972535i \(0.425225\pi\)
\(620\) −4.33772e14 −0.190155
\(621\) −2.22181e14 −0.0965392
\(622\) −5.31346e14 −0.228839
\(623\) 4.33670e14 0.185129
\(624\) −1.85394e14 −0.0784478
\(625\) 1.90806e15 0.800300
\(626\) 7.38245e14 0.306932
\(627\) 1.04764e15 0.431758
\(628\) −1.30641e15 −0.533705
\(629\) −2.46386e15 −0.997784
\(630\) −1.77811e14 −0.0713813
\(631\) −1.55786e15 −0.619966 −0.309983 0.950742i \(-0.600323\pi\)
−0.309983 + 0.950742i \(0.600323\pi\)
\(632\) −1.88930e15 −0.745343
\(633\) 1.03032e15 0.402952
\(634\) −8.04269e14 −0.311824
\(635\) −5.21951e14 −0.200620
\(636\) −1.18140e15 −0.450178
\(637\) 5.70146e14 0.215387
\(638\) 2.53312e15 0.948727
\(639\) −1.06762e15 −0.396427
\(640\) −2.06399e14 −0.0759831
\(641\) 1.92660e15 0.703189 0.351595 0.936152i \(-0.385640\pi\)
0.351595 + 0.936152i \(0.385640\pi\)
\(642\) 3.67435e14 0.132965
\(643\) −6.88029e14 −0.246857 −0.123429 0.992353i \(-0.539389\pi\)
−0.123429 + 0.992353i \(0.539389\pi\)
\(644\) −8.38985e14 −0.298457
\(645\) 9.41810e13 0.0332189
\(646\) −1.94556e15 −0.680402
\(647\) 1.09245e15 0.378815 0.189408 0.981899i \(-0.439343\pi\)
0.189408 + 0.981899i \(0.439343\pi\)
\(648\) −3.41776e14 −0.117511
\(649\) −3.16942e14 −0.108052
\(650\) −1.12655e15 −0.380827
\(651\) −2.88872e15 −0.968299
\(652\) −5.46311e14 −0.181584
\(653\) 3.74060e15 1.23287 0.616437 0.787404i \(-0.288575\pi\)
0.616437 + 0.787404i \(0.288575\pi\)
\(654\) 5.72673e14 0.187167
\(655\) 4.99724e14 0.161958
\(656\) −7.52536e14 −0.241856
\(657\) 1.07230e15 0.341750
\(658\) −2.69895e15 −0.853007
\(659\) 3.53097e15 1.10668 0.553342 0.832954i \(-0.313352\pi\)
0.553342 + 0.832954i \(0.313352\pi\)
\(660\) 2.04557e14 0.0635804
\(661\) 3.00196e15 0.925332 0.462666 0.886533i \(-0.346893\pi\)
0.462666 + 0.886533i \(0.346893\pi\)
\(662\) −3.02327e15 −0.924183
\(663\) 1.19549e15 0.362427
\(664\) −2.99315e15 −0.899920
\(665\) 9.22916e14 0.275197
\(666\) −7.32162e14 −0.216520
\(667\) 2.78840e15 0.817830
\(668\) −3.00988e14 −0.0875545
\(669\) 3.57777e15 1.03221
\(670\) 9.63506e12 0.00275704
\(671\) −6.63936e14 −0.188431
\(672\) −2.14619e15 −0.604139
\(673\) 6.86134e13 0.0191569 0.00957847 0.999954i \(-0.496951\pi\)
0.00957847 + 0.999954i \(0.496951\pi\)
\(674\) 4.28812e14 0.118751
\(675\) −6.52908e14 −0.179342
\(676\) 1.23210e15 0.335689
\(677\) 3.47540e15 0.939220 0.469610 0.882874i \(-0.344395\pi\)
0.469610 + 0.882874i \(0.344395\pi\)
\(678\) 2.33752e14 0.0626603
\(679\) −4.38193e15 −1.16515
\(680\) −1.12708e15 −0.297274
\(681\) −7.46241e14 −0.195241
\(682\) −3.21337e15 −0.833963
\(683\) −3.88839e15 −1.00105 −0.500525 0.865722i \(-0.666860\pi\)
−0.500525 + 0.865722i \(0.666860\pi\)
\(684\) 5.97912e14 0.152696
\(685\) −1.21566e15 −0.307975
\(686\) 2.05841e15 0.517309
\(687\) 4.22081e14 0.105229
\(688\) −2.07801e14 −0.0513939
\(689\) 3.64336e15 0.893915
\(690\) −2.17728e14 −0.0529961
\(691\) 2.20209e15 0.531748 0.265874 0.964008i \(-0.414340\pi\)
0.265874 + 0.964008i \(0.414340\pi\)
\(692\) 3.86292e15 0.925407
\(693\) 1.36226e15 0.323762
\(694\) 2.18168e15 0.514416
\(695\) 3.30367e14 0.0772824
\(696\) 4.28933e15 0.995493
\(697\) 4.85262e15 1.11737
\(698\) −3.27397e15 −0.747945
\(699\) −2.18129e15 −0.494411
\(700\) −2.46546e15 −0.554446
\(701\) −2.64860e15 −0.590974 −0.295487 0.955347i \(-0.595482\pi\)
−0.295487 + 0.955347i \(0.595482\pi\)
\(702\) 3.55252e14 0.0786471
\(703\) 3.80024e15 0.834751
\(704\) −3.27514e15 −0.713807
\(705\) 7.24363e14 0.156645
\(706\) 3.43159e14 0.0736325
\(707\) −2.68842e15 −0.572387
\(708\) −1.80886e14 −0.0382140
\(709\) −6.31638e15 −1.32408 −0.662040 0.749469i \(-0.730309\pi\)
−0.662040 + 0.749469i \(0.730309\pi\)
\(710\) −1.04622e15 −0.217623
\(711\) 1.13814e15 0.234916
\(712\) 8.16865e14 0.167306
\(713\) −3.53721e15 −0.718900
\(714\) −2.52983e15 −0.510213
\(715\) −6.30839e14 −0.126251
\(716\) −2.12743e15 −0.422506
\(717\) −3.52833e14 −0.0695366
\(718\) −6.83033e14 −0.133585
\(719\) 8.99398e15 1.74559 0.872797 0.488083i \(-0.162304\pi\)
0.872797 + 0.488083i \(0.162304\pi\)
\(720\) −1.05294e14 −0.0202803
\(721\) −5.05073e15 −0.965407
\(722\) −6.95402e14 −0.131911
\(723\) 3.23017e15 0.608085
\(724\) 8.04964e14 0.150388
\(725\) 8.19407e15 1.51929
\(726\) −6.84503e14 −0.125957
\(727\) −9.72835e15 −1.77664 −0.888321 0.459223i \(-0.848128\pi\)
−0.888321 + 0.459223i \(0.848128\pi\)
\(728\) 3.98008e15 0.721388
\(729\) 2.05891e14 0.0370370
\(730\) 1.05081e15 0.187607
\(731\) 1.33998e15 0.237439
\(732\) −3.78924e14 −0.0666410
\(733\) −1.55093e15 −0.270719 −0.135360 0.990797i \(-0.543219\pi\)
−0.135360 + 0.990797i \(0.543219\pi\)
\(734\) 4.89438e14 0.0847948
\(735\) 3.23812e14 0.0556818
\(736\) −2.62799e15 −0.448535
\(737\) −7.38168e13 −0.0125050
\(738\) 1.44201e15 0.242470
\(739\) 4.95578e15 0.827118 0.413559 0.910477i \(-0.364285\pi\)
0.413559 + 0.910477i \(0.364285\pi\)
\(740\) 7.42018e14 0.122925
\(741\) −1.84391e15 −0.303208
\(742\) −7.70989e15 −1.25842
\(743\) 4.31572e15 0.699222 0.349611 0.936895i \(-0.386314\pi\)
0.349611 + 0.936895i \(0.386314\pi\)
\(744\) −5.44121e15 −0.875073
\(745\) 8.56670e13 0.0136758
\(746\) −7.00472e15 −1.11001
\(747\) 1.80312e15 0.283636
\(748\) 2.91037e15 0.454454
\(749\) −2.47988e15 −0.384398
\(750\) −1.32641e15 −0.204099
\(751\) 5.80093e14 0.0886090 0.0443045 0.999018i \(-0.485893\pi\)
0.0443045 + 0.999018i \(0.485893\pi\)
\(752\) −1.59823e15 −0.242350
\(753\) 3.11004e15 0.468159
\(754\) −4.45845e15 −0.666257
\(755\) 2.86714e15 0.425345
\(756\) 7.77471e14 0.114502
\(757\) −2.22175e15 −0.324838 −0.162419 0.986722i \(-0.551930\pi\)
−0.162419 + 0.986722i \(0.551930\pi\)
\(758\) −1.01889e15 −0.147893
\(759\) 1.66807e15 0.240373
\(760\) 1.73841e15 0.248701
\(761\) 4.37184e15 0.620939 0.310469 0.950583i \(-0.399514\pi\)
0.310469 + 0.950583i \(0.399514\pi\)
\(762\) −2.20676e15 −0.311174
\(763\) −3.86507e15 −0.541094
\(764\) −5.48661e15 −0.762590
\(765\) 6.78973e14 0.0936946
\(766\) −6.32930e15 −0.867157
\(767\) 5.57839e14 0.0758812
\(768\) −4.54924e15 −0.614401
\(769\) −1.74235e15 −0.233636 −0.116818 0.993153i \(-0.537269\pi\)
−0.116818 + 0.993153i \(0.537269\pi\)
\(770\) 1.33495e15 0.177732
\(771\) 4.65735e14 0.0615659
\(772\) −3.31153e15 −0.434644
\(773\) −2.41985e15 −0.315356 −0.157678 0.987491i \(-0.550401\pi\)
−0.157678 + 0.987491i \(0.550401\pi\)
\(774\) 3.98188e14 0.0515245
\(775\) −1.03945e16 −1.33551
\(776\) −8.25384e15 −1.05297
\(777\) 4.94149e15 0.625954
\(778\) −8.45314e15 −1.06324
\(779\) −7.48466e15 −0.934795
\(780\) −3.60035e14 −0.0446503
\(781\) 8.01541e15 0.987063
\(782\) −3.09776e15 −0.378800
\(783\) −2.58396e15 −0.313758
\(784\) −7.14460e14 −0.0861469
\(785\) 2.28818e15 0.273973
\(786\) 2.11279e15 0.251207
\(787\) −1.07655e16 −1.27108 −0.635538 0.772069i \(-0.719222\pi\)
−0.635538 + 0.772069i \(0.719222\pi\)
\(788\) 5.40350e15 0.633550
\(789\) 1.66287e15 0.193613
\(790\) 1.11533e15 0.128960
\(791\) −1.57764e15 −0.181149
\(792\) 2.56595e15 0.292591
\(793\) 1.16857e15 0.132328
\(794\) −3.59324e15 −0.404086
\(795\) 2.06923e15 0.231095
\(796\) 5.66469e14 0.0628281
\(797\) −2.58738e15 −0.284997 −0.142498 0.989795i \(-0.545514\pi\)
−0.142498 + 0.989795i \(0.545514\pi\)
\(798\) 3.90200e15 0.426846
\(799\) 1.03060e16 1.11965
\(800\) −7.72270e15 −0.833248
\(801\) −4.92092e14 −0.0527312
\(802\) −7.49910e14 −0.0798086
\(803\) −8.05054e15 −0.850921
\(804\) −4.21290e13 −0.00442255
\(805\) 1.46948e15 0.153210
\(806\) 5.65575e15 0.585663
\(807\) 1.80122e15 0.185252
\(808\) −5.06393e15 −0.517279
\(809\) −1.13803e16 −1.15461 −0.577307 0.816527i \(-0.695896\pi\)
−0.577307 + 0.816527i \(0.695896\pi\)
\(810\) 2.01764e14 0.0203318
\(811\) 8.98215e15 0.899012 0.449506 0.893277i \(-0.351600\pi\)
0.449506 + 0.893277i \(0.351600\pi\)
\(812\) −9.75734e15 −0.970004
\(813\) 1.90128e15 0.187736
\(814\) 5.49685e15 0.539113
\(815\) 9.56865e14 0.0932146
\(816\) −1.49809e15 −0.144958
\(817\) −2.06677e15 −0.198642
\(818\) 1.01838e16 0.972228
\(819\) −2.39766e15 −0.227366
\(820\) −1.46142e15 −0.137657
\(821\) 6.30672e15 0.590087 0.295043 0.955484i \(-0.404666\pi\)
0.295043 + 0.955484i \(0.404666\pi\)
\(822\) −5.13971e15 −0.477687
\(823\) 2.13866e16 1.97443 0.987215 0.159396i \(-0.0509546\pi\)
0.987215 + 0.159396i \(0.0509546\pi\)
\(824\) −9.51360e15 −0.872459
\(825\) 4.90184e15 0.446542
\(826\) −1.18047e15 −0.106823
\(827\) −8.24854e15 −0.741476 −0.370738 0.928738i \(-0.620895\pi\)
−0.370738 + 0.928738i \(0.620895\pi\)
\(828\) 9.52008e14 0.0850107
\(829\) 8.52629e15 0.756328 0.378164 0.925739i \(-0.376556\pi\)
0.378164 + 0.925739i \(0.376556\pi\)
\(830\) 1.76698e15 0.155704
\(831\) 8.62714e15 0.755199
\(832\) 5.76447e15 0.501281
\(833\) 4.60709e15 0.397997
\(834\) 1.39676e15 0.119870
\(835\) 5.27181e14 0.0449453
\(836\) −4.48895e15 −0.380198
\(837\) 3.27787e15 0.275804
\(838\) 1.17626e16 0.983241
\(839\) −2.94730e15 −0.244756 −0.122378 0.992484i \(-0.539052\pi\)
−0.122378 + 0.992484i \(0.539052\pi\)
\(840\) 2.26047e15 0.186493
\(841\) 2.02284e16 1.65800
\(842\) 1.69302e14 0.0137863
\(843\) 1.16208e15 0.0940122
\(844\) −4.41475e15 −0.354832
\(845\) −2.15802e15 −0.172323
\(846\) 3.06254e15 0.242965
\(847\) 4.61983e15 0.364139
\(848\) −4.56556e15 −0.357533
\(849\) −9.26531e15 −0.720888
\(850\) −9.10316e15 −0.703701
\(851\) 6.05082e15 0.464731
\(852\) 4.57458e15 0.349087
\(853\) −1.10689e16 −0.839237 −0.419618 0.907701i \(-0.637836\pi\)
−0.419618 + 0.907701i \(0.637836\pi\)
\(854\) −2.47287e15 −0.186288
\(855\) −1.04725e15 −0.0783853
\(856\) −4.67113e15 −0.347389
\(857\) 1.86672e16 1.37939 0.689693 0.724102i \(-0.257746\pi\)
0.689693 + 0.724102i \(0.257746\pi\)
\(858\) −2.66713e15 −0.195823
\(859\) −1.61594e16 −1.17886 −0.589432 0.807818i \(-0.700649\pi\)
−0.589432 + 0.807818i \(0.700649\pi\)
\(860\) −4.03549e14 −0.0292520
\(861\) −9.73238e15 −0.700974
\(862\) 1.39031e16 0.994996
\(863\) 8.57698e15 0.609923 0.304961 0.952365i \(-0.401356\pi\)
0.304961 + 0.952365i \(0.401356\pi\)
\(864\) 2.43531e15 0.172079
\(865\) −6.76592e15 −0.475049
\(866\) −6.27368e15 −0.437698
\(867\) 1.33213e15 0.0923508
\(868\) 1.23776e16 0.852667
\(869\) −8.54483e15 −0.584917
\(870\) −2.53216e15 −0.172241
\(871\) 1.29922e14 0.00878182
\(872\) −7.28028e15 −0.488998
\(873\) 4.97224e15 0.331874
\(874\) 4.77797e15 0.316906
\(875\) 8.95216e15 0.590043
\(876\) −4.59463e15 −0.300939
\(877\) −1.34626e16 −0.876255 −0.438128 0.898913i \(-0.644358\pi\)
−0.438128 + 0.898913i \(0.644358\pi\)
\(878\) 9.12912e15 0.590485
\(879\) −1.40429e16 −0.902644
\(880\) 7.90516e14 0.0504959
\(881\) 8.04558e15 0.510728 0.255364 0.966845i \(-0.417805\pi\)
0.255364 + 0.966845i \(0.417805\pi\)
\(882\) 1.36905e15 0.0863657
\(883\) −1.38620e15 −0.0869047 −0.0434523 0.999056i \(-0.513836\pi\)
−0.0434523 + 0.999056i \(0.513836\pi\)
\(884\) −5.12245e15 −0.319147
\(885\) 3.16823e14 0.0196168
\(886\) 1.74310e16 1.07260
\(887\) −1.93804e16 −1.18517 −0.592587 0.805507i \(-0.701893\pi\)
−0.592587 + 0.805507i \(0.701893\pi\)
\(888\) 9.30783e15 0.565688
\(889\) 1.48938e16 0.899593
\(890\) −4.82228e14 −0.0289473
\(891\) −1.54577e15 −0.0922184
\(892\) −1.53301e16 −0.908946
\(893\) −1.58959e16 −0.936704
\(894\) 3.62192e14 0.0212120
\(895\) 3.72619e15 0.216890
\(896\) 5.88956e15 0.340713
\(897\) −2.93592e15 −0.168805
\(898\) −1.01003e16 −0.577184
\(899\) −4.11376e16 −2.33647
\(900\) 2.79760e15 0.157925
\(901\) 2.94404e16 1.65180
\(902\) −1.08262e16 −0.603725
\(903\) −2.68744e15 −0.148956
\(904\) −2.97165e15 −0.163708
\(905\) −1.40990e15 −0.0772005
\(906\) 1.21220e16 0.659735
\(907\) 2.31034e16 1.24979 0.624893 0.780710i \(-0.285143\pi\)
0.624893 + 0.780710i \(0.285143\pi\)
\(908\) 3.19751e15 0.171925
\(909\) 3.05058e15 0.163035
\(910\) −2.34960e15 −0.124815
\(911\) 3.61189e16 1.90715 0.953573 0.301161i \(-0.0973743\pi\)
0.953573 + 0.301161i \(0.0973743\pi\)
\(912\) 2.31064e15 0.121272
\(913\) −1.35373e16 −0.706223
\(914\) 2.35694e15 0.122220
\(915\) 6.63687e14 0.0342095
\(916\) −1.80854e15 −0.0926624
\(917\) −1.42596e16 −0.726232
\(918\) 2.87063e15 0.145326
\(919\) 5.85313e15 0.294546 0.147273 0.989096i \(-0.452950\pi\)
0.147273 + 0.989096i \(0.452950\pi\)
\(920\) 2.76793e15 0.138459
\(921\) −1.66493e16 −0.827881
\(922\) 1.32668e16 0.655760
\(923\) −1.41077e16 −0.693179
\(924\) −5.83702e15 −0.285099
\(925\) 1.77811e16 0.863335
\(926\) 1.21171e16 0.584840
\(927\) 5.73113e15 0.274981
\(928\) −3.05634e16 −1.45777
\(929\) 2.08034e16 0.986390 0.493195 0.869919i \(-0.335829\pi\)
0.493195 + 0.869919i \(0.335829\pi\)
\(930\) 3.21216e15 0.151405
\(931\) −7.10596e15 −0.332966
\(932\) 9.34643e15 0.435370
\(933\) −4.06926e15 −0.188437
\(934\) −2.77612e16 −1.27800
\(935\) −5.09753e15 −0.233290
\(936\) −4.51625e15 −0.205476
\(937\) 2.51133e16 1.13589 0.567945 0.823067i \(-0.307739\pi\)
0.567945 + 0.823067i \(0.307739\pi\)
\(938\) −2.74935e14 −0.0123628
\(939\) 5.65376e15 0.252742
\(940\) −3.10377e15 −0.137939
\(941\) 1.42223e16 0.628386 0.314193 0.949359i \(-0.398266\pi\)
0.314193 + 0.949359i \(0.398266\pi\)
\(942\) 9.67421e15 0.424948
\(943\) −1.19172e16 −0.520428
\(944\) −6.99038e14 −0.0303497
\(945\) −1.36174e15 −0.0587787
\(946\) −2.98948e15 −0.128291
\(947\) 3.67105e16 1.56627 0.783133 0.621855i \(-0.213620\pi\)
0.783133 + 0.621855i \(0.213620\pi\)
\(948\) −4.87673e15 −0.206863
\(949\) 1.41695e16 0.597571
\(950\) 1.40407e16 0.588719
\(951\) −6.15940e15 −0.256771
\(952\) 3.21612e16 1.33300
\(953\) 3.72037e16 1.53312 0.766559 0.642174i \(-0.221967\pi\)
0.766559 + 0.642174i \(0.221967\pi\)
\(954\) 8.74852e15 0.358442
\(955\) 9.60982e15 0.391468
\(956\) 1.51183e15 0.0612327
\(957\) 1.93996e16 0.781226
\(958\) 4.36188e15 0.174648
\(959\) 3.46888e16 1.38098
\(960\) 3.27391e15 0.129591
\(961\) 2.67764e16 1.05384
\(962\) −9.67483e15 −0.378600
\(963\) 2.81396e15 0.109490
\(964\) −1.38407e16 −0.535469
\(965\) 5.80016e15 0.223121
\(966\) 6.21284e15 0.237638
\(967\) 3.66743e16 1.39481 0.697406 0.716676i \(-0.254337\pi\)
0.697406 + 0.716676i \(0.254337\pi\)
\(968\) 8.70195e15 0.329081
\(969\) −1.48999e16 −0.560275
\(970\) 4.87257e15 0.182186
\(971\) 1.90896e16 0.709725 0.354863 0.934918i \(-0.384528\pi\)
0.354863 + 0.934918i \(0.384528\pi\)
\(972\) −8.82207e14 −0.0326142
\(973\) −9.42700e15 −0.346539
\(974\) 2.15362e16 0.787216
\(975\) −8.62757e15 −0.313591
\(976\) −1.46436e15 −0.0529266
\(977\) 3.80992e16 1.36929 0.684646 0.728875i \(-0.259957\pi\)
0.684646 + 0.728875i \(0.259957\pi\)
\(978\) 4.04554e15 0.144581
\(979\) 3.69448e15 0.131295
\(980\) −1.38748e15 −0.0490324
\(981\) 4.38575e15 0.154122
\(982\) 7.39433e15 0.258396
\(983\) −2.80815e16 −0.975835 −0.487917 0.872890i \(-0.662243\pi\)
−0.487917 + 0.872890i \(0.662243\pi\)
\(984\) −1.83320e16 −0.633485
\(985\) −9.46424e15 −0.325227
\(986\) −3.60268e16 −1.23112
\(987\) −2.06696e16 −0.702405
\(988\) 7.90085e15 0.267000
\(989\) −3.29076e15 −0.110590
\(990\) −1.51479e15 −0.0506241
\(991\) −2.53187e16 −0.841464 −0.420732 0.907185i \(-0.638227\pi\)
−0.420732 + 0.907185i \(0.638227\pi\)
\(992\) 3.87711e16 1.28143
\(993\) −2.31534e16 −0.761015
\(994\) 2.98539e16 0.975834
\(995\) −9.92172e14 −0.0322522
\(996\) −7.72604e15 −0.249764
\(997\) −3.10539e16 −0.998374 −0.499187 0.866494i \(-0.666368\pi\)
−0.499187 + 0.866494i \(0.666368\pi\)
\(998\) −8.46479e15 −0.270644
\(999\) −5.60718e15 −0.178293
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.1.19 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.d.1.19 28 1.1 even 1 trivial