Properties

Label 1.16.a.a.1.1
Level $1$
Weight $16$
Character 1.1
Self dual yes
Analytic conductor $1.427$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.42693505100\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

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

$q$-expansion

\(f(q)\) \(=\) \(q+216.000 q^{2} -3348.00 q^{3} +13888.0 q^{4} +52110.0 q^{5} -723168. q^{6} +2.82246e6 q^{7} -4.07808e6 q^{8} -3.13980e6 q^{9} +O(q^{10})\) \(q+216.000 q^{2} -3348.00 q^{3} +13888.0 q^{4} +52110.0 q^{5} -723168. q^{6} +2.82246e6 q^{7} -4.07808e6 q^{8} -3.13980e6 q^{9} +1.12558e7 q^{10} +2.05869e7 q^{11} -4.64970e7 q^{12} -1.90073e8 q^{13} +6.09650e8 q^{14} -1.74464e8 q^{15} -1.33595e9 q^{16} +1.64653e9 q^{17} -6.78197e8 q^{18} +1.56326e9 q^{19} +7.23704e8 q^{20} -9.44958e9 q^{21} +4.44676e9 q^{22} +9.45112e9 q^{23} +1.36534e10 q^{24} -2.78021e10 q^{25} -4.10558e10 q^{26} +5.85522e10 q^{27} +3.91983e10 q^{28} -3.69026e10 q^{29} -3.76843e10 q^{30} +7.15885e10 q^{31} -1.54934e11 q^{32} -6.89248e10 q^{33} +3.55650e11 q^{34} +1.47078e11 q^{35} -4.36056e10 q^{36} -1.03365e12 q^{37} +3.37664e11 q^{38} +6.36366e11 q^{39} -2.12509e11 q^{40} +1.64197e12 q^{41} -2.04111e12 q^{42} -4.92403e11 q^{43} +2.85910e11 q^{44} -1.63615e11 q^{45} +2.04144e12 q^{46} -3.41068e12 q^{47} +4.47275e12 q^{48} +3.21870e12 q^{49} -6.00526e12 q^{50} -5.51258e12 q^{51} -2.63974e12 q^{52} +6.79715e12 q^{53} +1.26473e13 q^{54} +1.07278e12 q^{55} -1.15102e13 q^{56} -5.23379e12 q^{57} -7.97095e12 q^{58} +9.85886e12 q^{59} -2.42296e12 q^{60} +4.93184e12 q^{61} +1.54631e13 q^{62} -8.86196e12 q^{63} +1.03106e13 q^{64} -9.90472e12 q^{65} -1.48878e13 q^{66} -2.88378e13 q^{67} +2.28670e13 q^{68} -3.16423e13 q^{69} +3.17689e13 q^{70} +1.25050e14 q^{71} +1.28044e13 q^{72} -8.21715e13 q^{73} -2.23269e14 q^{74} +9.30815e13 q^{75} +2.17105e13 q^{76} +5.81055e13 q^{77} +1.37455e14 q^{78} -2.54131e13 q^{79} -6.96162e13 q^{80} -1.50980e14 q^{81} +3.54666e14 q^{82} -2.81737e14 q^{83} -1.31236e14 q^{84} +8.58006e13 q^{85} -1.06359e14 q^{86} +1.23550e14 q^{87} -8.39548e13 q^{88} +7.15619e14 q^{89} -3.53409e13 q^{90} -5.36474e14 q^{91} +1.31257e14 q^{92} -2.39678e14 q^{93} -7.36708e14 q^{94} +8.14613e13 q^{95} +5.18719e14 q^{96} +6.12786e14 q^{97} +6.95238e14 q^{98} -6.46387e13 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 216.000 1.19324 0.596621 0.802523i \(-0.296509\pi\)
0.596621 + 0.802523i \(0.296509\pi\)
\(3\) −3348.00 −0.883845 −0.441922 0.897053i \(-0.645703\pi\)
−0.441922 + 0.897053i \(0.645703\pi\)
\(4\) 13888.0 0.423828
\(5\) 52110.0 0.298295 0.149148 0.988815i \(-0.452347\pi\)
0.149148 + 0.988815i \(0.452347\pi\)
\(6\) −723168. −1.05464
\(7\) 2.82246e6 1.29536 0.647682 0.761911i \(-0.275739\pi\)
0.647682 + 0.761911i \(0.275739\pi\)
\(8\) −4.07808e6 −0.687513
\(9\) −3.13980e6 −0.218818
\(10\) 1.12558e7 0.355938
\(11\) 2.05869e7 0.318526 0.159263 0.987236i \(-0.449088\pi\)
0.159263 + 0.987236i \(0.449088\pi\)
\(12\) −4.64970e7 −0.374598
\(13\) −1.90073e8 −0.840129 −0.420065 0.907494i \(-0.637993\pi\)
−0.420065 + 0.907494i \(0.637993\pi\)
\(14\) 6.09650e8 1.54568
\(15\) −1.74464e8 −0.263647
\(16\) −1.33595e9 −1.24420
\(17\) 1.64653e9 0.973200 0.486600 0.873625i \(-0.338237\pi\)
0.486600 + 0.873625i \(0.338237\pi\)
\(18\) −6.78197e8 −0.261103
\(19\) 1.56326e9 0.401216 0.200608 0.979672i \(-0.435708\pi\)
0.200608 + 0.979672i \(0.435708\pi\)
\(20\) 7.23704e8 0.126426
\(21\) −9.44958e9 −1.14490
\(22\) 4.44676e9 0.380079
\(23\) 9.45112e9 0.578794 0.289397 0.957209i \(-0.406545\pi\)
0.289397 + 0.957209i \(0.406545\pi\)
\(24\) 1.36534e10 0.607655
\(25\) −2.78021e10 −0.911020
\(26\) −4.10558e10 −1.00248
\(27\) 5.85522e10 1.07725
\(28\) 3.91983e10 0.549012
\(29\) −3.69026e10 −0.397257 −0.198629 0.980075i \(-0.563649\pi\)
−0.198629 + 0.980075i \(0.563649\pi\)
\(30\) −3.76843e10 −0.314594
\(31\) 7.15885e10 0.467337 0.233669 0.972316i \(-0.424927\pi\)
0.233669 + 0.972316i \(0.424927\pi\)
\(32\) −1.54934e11 −0.797117
\(33\) −6.89248e10 −0.281528
\(34\) 3.55650e11 1.16126
\(35\) 1.47078e11 0.386401
\(36\) −4.36056e10 −0.0927413
\(37\) −1.03365e12 −1.79003 −0.895017 0.446031i \(-0.852837\pi\)
−0.895017 + 0.446031i \(0.852837\pi\)
\(38\) 3.37664e11 0.478748
\(39\) 6.36366e11 0.742544
\(40\) −2.12509e11 −0.205082
\(41\) 1.64197e12 1.31670 0.658351 0.752711i \(-0.271254\pi\)
0.658351 + 0.752711i \(0.271254\pi\)
\(42\) −2.04111e12 −1.36614
\(43\) −4.92403e11 −0.276253 −0.138127 0.990415i \(-0.544108\pi\)
−0.138127 + 0.990415i \(0.544108\pi\)
\(44\) 2.85910e11 0.135000
\(45\) −1.63615e11 −0.0652724
\(46\) 2.04144e12 0.690642
\(47\) −3.41068e12 −0.981991 −0.490996 0.871162i \(-0.663367\pi\)
−0.490996 + 0.871162i \(0.663367\pi\)
\(48\) 4.47275e12 1.09968
\(49\) 3.21870e12 0.677968
\(50\) −6.00526e12 −1.08707
\(51\) −5.51258e12 −0.860158
\(52\) −2.63974e12 −0.356070
\(53\) 6.79715e12 0.794800 0.397400 0.917645i \(-0.369913\pi\)
0.397400 + 0.917645i \(0.369913\pi\)
\(54\) 1.26473e13 1.28542
\(55\) 1.07278e12 0.0950147
\(56\) −1.15102e13 −0.890580
\(57\) −5.23379e12 −0.354613
\(58\) −7.97095e12 −0.474024
\(59\) 9.85886e12 0.515747 0.257873 0.966179i \(-0.416978\pi\)
0.257873 + 0.966179i \(0.416978\pi\)
\(60\) −2.42296e12 −0.111741
\(61\) 4.93184e12 0.200926 0.100463 0.994941i \(-0.467968\pi\)
0.100463 + 0.994941i \(0.467968\pi\)
\(62\) 1.54631e13 0.557647
\(63\) −8.86196e12 −0.283449
\(64\) 1.03106e13 0.293044
\(65\) −9.90472e12 −0.250606
\(66\) −1.48878e13 −0.335931
\(67\) −2.88378e13 −0.581302 −0.290651 0.956829i \(-0.593872\pi\)
−0.290651 + 0.956829i \(0.593872\pi\)
\(68\) 2.28670e13 0.412470
\(69\) −3.16423e13 −0.511564
\(70\) 3.17689e13 0.461070
\(71\) 1.25050e14 1.63172 0.815862 0.578247i \(-0.196263\pi\)
0.815862 + 0.578247i \(0.196263\pi\)
\(72\) 1.28044e13 0.150440
\(73\) −8.21715e13 −0.870562 −0.435281 0.900295i \(-0.643351\pi\)
−0.435281 + 0.900295i \(0.643351\pi\)
\(74\) −2.23269e14 −2.13595
\(75\) 9.30815e13 0.805200
\(76\) 2.17105e13 0.170047
\(77\) 5.81055e13 0.412607
\(78\) 1.37455e14 0.886035
\(79\) −2.54131e13 −0.148886 −0.0744430 0.997225i \(-0.523718\pi\)
−0.0744430 + 0.997225i \(0.523718\pi\)
\(80\) −6.96162e13 −0.371138
\(81\) −1.50980e14 −0.733300
\(82\) 3.54666e14 1.57114
\(83\) −2.81737e14 −1.13961 −0.569807 0.821779i \(-0.692982\pi\)
−0.569807 + 0.821779i \(0.692982\pi\)
\(84\) −1.31236e14 −0.485241
\(85\) 8.58006e13 0.290301
\(86\) −1.06359e14 −0.329637
\(87\) 1.23550e14 0.351114
\(88\) −8.39548e13 −0.218991
\(89\) 7.15619e14 1.71497 0.857485 0.514509i \(-0.172026\pi\)
0.857485 + 0.514509i \(0.172026\pi\)
\(90\) −3.53409e13 −0.0778858
\(91\) −5.36474e14 −1.08827
\(92\) 1.31257e14 0.245309
\(93\) −2.39678e14 −0.413054
\(94\) −7.36708e14 −1.17175
\(95\) 8.14613e13 0.119681
\(96\) 5.18719e14 0.704528
\(97\) 6.12786e14 0.770054 0.385027 0.922905i \(-0.374192\pi\)
0.385027 + 0.922905i \(0.374192\pi\)
\(98\) 6.95238e14 0.808981
\(99\) −6.46387e13 −0.0696993
\(100\) −3.86116e14 −0.386116
\(101\) −8.17642e14 −0.758844 −0.379422 0.925224i \(-0.623877\pi\)
−0.379422 + 0.925224i \(0.623877\pi\)
\(102\) −1.19072e15 −1.02638
\(103\) 7.41115e14 0.593753 0.296877 0.954916i \(-0.404055\pi\)
0.296877 + 0.954916i \(0.404055\pi\)
\(104\) 7.75134e14 0.577600
\(105\) −4.92418e14 −0.341518
\(106\) 1.46818e15 0.948389
\(107\) −2.51430e15 −1.51370 −0.756849 0.653590i \(-0.773262\pi\)
−0.756849 + 0.653590i \(0.773262\pi\)
\(108\) 8.13173e14 0.456567
\(109\) 1.26835e15 0.664572 0.332286 0.943179i \(-0.392180\pi\)
0.332286 + 0.943179i \(0.392180\pi\)
\(110\) 2.31721e14 0.113376
\(111\) 3.46067e15 1.58211
\(112\) −3.77065e15 −1.61169
\(113\) −2.05416e15 −0.821385 −0.410692 0.911774i \(-0.634713\pi\)
−0.410692 + 0.911774i \(0.634713\pi\)
\(114\) −1.13050e15 −0.423139
\(115\) 4.92498e14 0.172652
\(116\) −5.12503e14 −0.168369
\(117\) 5.96793e14 0.183836
\(118\) 2.12951e15 0.615411
\(119\) 4.64725e15 1.26065
\(120\) 7.11479e14 0.181260
\(121\) −3.75343e15 −0.898541
\(122\) 1.06528e15 0.239753
\(123\) −5.49733e15 −1.16376
\(124\) 9.94221e14 0.198071
\(125\) −3.03904e15 −0.570048
\(126\) −1.91418e15 −0.338224
\(127\) 2.99068e15 0.498014 0.249007 0.968502i \(-0.419896\pi\)
0.249007 + 0.968502i \(0.419896\pi\)
\(128\) 7.30396e15 1.14679
\(129\) 1.64857e15 0.244165
\(130\) −2.13942e15 −0.299034
\(131\) −1.62623e15 −0.214608 −0.107304 0.994226i \(-0.534222\pi\)
−0.107304 + 0.994226i \(0.534222\pi\)
\(132\) −9.57227e14 −0.119319
\(133\) 4.41222e15 0.519721
\(134\) −6.22897e15 −0.693634
\(135\) 3.05116e15 0.321337
\(136\) −6.71467e15 −0.669088
\(137\) 1.05922e16 0.999038 0.499519 0.866303i \(-0.333510\pi\)
0.499519 + 0.866303i \(0.333510\pi\)
\(138\) −6.83474e15 −0.610421
\(139\) −1.86709e16 −1.57963 −0.789813 0.613347i \(-0.789823\pi\)
−0.789813 + 0.613347i \(0.789823\pi\)
\(140\) 2.04262e15 0.163767
\(141\) 1.14190e16 0.867928
\(142\) 2.70108e16 1.94704
\(143\) −3.91301e15 −0.267603
\(144\) 4.19461e15 0.272253
\(145\) −1.92299e15 −0.118500
\(146\) −1.77490e16 −1.03879
\(147\) −1.07762e16 −0.599219
\(148\) −1.43554e16 −0.758667
\(149\) −1.25560e16 −0.630889 −0.315444 0.948944i \(-0.602154\pi\)
−0.315444 + 0.948944i \(0.602154\pi\)
\(150\) 2.01056e16 0.960799
\(151\) 2.87588e16 1.30751 0.653753 0.756708i \(-0.273194\pi\)
0.653753 + 0.756708i \(0.273194\pi\)
\(152\) −6.37509e15 −0.275841
\(153\) −5.16977e15 −0.212954
\(154\) 1.25508e16 0.492340
\(155\) 3.73048e15 0.139404
\(156\) 8.83784e15 0.314711
\(157\) −1.45276e16 −0.493114 −0.246557 0.969128i \(-0.579299\pi\)
−0.246557 + 0.969128i \(0.579299\pi\)
\(158\) −5.48922e15 −0.177657
\(159\) −2.27569e16 −0.702480
\(160\) −8.07362e15 −0.237776
\(161\) 2.66754e16 0.749750
\(162\) −3.26117e16 −0.875005
\(163\) 1.67741e16 0.429767 0.214884 0.976640i \(-0.431063\pi\)
0.214884 + 0.976640i \(0.431063\pi\)
\(164\) 2.28037e16 0.558055
\(165\) −3.59167e15 −0.0839783
\(166\) −6.08551e16 −1.35984
\(167\) 6.41999e16 1.37139 0.685695 0.727889i \(-0.259498\pi\)
0.685695 + 0.727889i \(0.259498\pi\)
\(168\) 3.85362e16 0.787134
\(169\) −1.50580e16 −0.294183
\(170\) 1.85329e16 0.346399
\(171\) −4.90832e15 −0.0877934
\(172\) −6.83849e15 −0.117084
\(173\) −7.59860e16 −1.24563 −0.622814 0.782370i \(-0.714010\pi\)
−0.622814 + 0.782370i \(0.714010\pi\)
\(174\) 2.66868e16 0.418964
\(175\) −7.84703e16 −1.18010
\(176\) −2.75029e16 −0.396309
\(177\) −3.30075e16 −0.455840
\(178\) 1.54574e17 2.04638
\(179\) 9.33749e16 1.18531 0.592655 0.805456i \(-0.298080\pi\)
0.592655 + 0.805456i \(0.298080\pi\)
\(180\) −2.27229e15 −0.0276643
\(181\) 7.43177e16 0.867966 0.433983 0.900921i \(-0.357108\pi\)
0.433983 + 0.900921i \(0.357108\pi\)
\(182\) −1.15878e17 −1.29857
\(183\) −1.65118e16 −0.177587
\(184\) −3.85424e16 −0.397929
\(185\) −5.38636e16 −0.533958
\(186\) −5.17705e16 −0.492873
\(187\) 3.38968e16 0.309990
\(188\) −4.73676e16 −0.416196
\(189\) 1.65261e17 1.39543
\(190\) 1.75956e16 0.142808
\(191\) −9.86224e16 −0.769529 −0.384765 0.923015i \(-0.625717\pi\)
−0.384765 + 0.923015i \(0.625717\pi\)
\(192\) −3.45197e16 −0.259005
\(193\) −8.91178e15 −0.0643109 −0.0321554 0.999483i \(-0.510237\pi\)
−0.0321554 + 0.999483i \(0.510237\pi\)
\(194\) 1.32362e17 0.918861
\(195\) 3.31610e16 0.221497
\(196\) 4.47013e16 0.287342
\(197\) 3.54176e16 0.219140 0.109570 0.993979i \(-0.465053\pi\)
0.109570 + 0.993979i \(0.465053\pi\)
\(198\) −1.39620e16 −0.0831682
\(199\) −2.86461e17 −1.64311 −0.821556 0.570127i \(-0.806894\pi\)
−0.821556 + 0.570127i \(0.806894\pi\)
\(200\) 1.13379e17 0.626338
\(201\) 9.65490e16 0.513780
\(202\) −1.76611e17 −0.905485
\(203\) −1.04156e17 −0.514593
\(204\) −7.65587e16 −0.364559
\(205\) 8.55633e16 0.392766
\(206\) 1.60081e17 0.708492
\(207\) −2.96746e16 −0.126651
\(208\) 2.53928e17 1.04529
\(209\) 3.21825e16 0.127798
\(210\) −1.06362e17 −0.407514
\(211\) 3.75834e17 1.38956 0.694780 0.719222i \(-0.255502\pi\)
0.694780 + 0.719222i \(0.255502\pi\)
\(212\) 9.43988e16 0.336859
\(213\) −4.18668e17 −1.44219
\(214\) −5.43089e17 −1.80621
\(215\) −2.56591e16 −0.0824050
\(216\) −2.38781e17 −0.740621
\(217\) 2.02055e17 0.605372
\(218\) 2.73964e17 0.792995
\(219\) 2.75110e17 0.769441
\(220\) 1.48988e16 0.0402699
\(221\) −3.12961e17 −0.817614
\(222\) 7.47504e17 1.88784
\(223\) −2.53078e16 −0.0617970 −0.0308985 0.999523i \(-0.509837\pi\)
−0.0308985 + 0.999523i \(0.509837\pi\)
\(224\) −4.37295e17 −1.03256
\(225\) 8.72932e16 0.199348
\(226\) −4.43699e17 −0.980111
\(227\) 3.03692e17 0.648992 0.324496 0.945887i \(-0.394805\pi\)
0.324496 + 0.945887i \(0.394805\pi\)
\(228\) −7.26868e16 −0.150295
\(229\) 1.07992e17 0.216085 0.108042 0.994146i \(-0.465542\pi\)
0.108042 + 0.994146i \(0.465542\pi\)
\(230\) 1.06379e17 0.206015
\(231\) −1.94537e17 −0.364681
\(232\) 1.50492e17 0.273119
\(233\) −7.90506e17 −1.38911 −0.694554 0.719441i \(-0.744398\pi\)
−0.694554 + 0.719441i \(0.744398\pi\)
\(234\) 1.28907e17 0.219360
\(235\) −1.77731e17 −0.292923
\(236\) 1.36920e17 0.218588
\(237\) 8.50830e16 0.131592
\(238\) 1.00381e18 1.50426
\(239\) 3.52956e17 0.512551 0.256275 0.966604i \(-0.417505\pi\)
0.256275 + 0.966604i \(0.417505\pi\)
\(240\) 2.33075e17 0.328028
\(241\) 6.85690e16 0.0935405 0.0467703 0.998906i \(-0.485107\pi\)
0.0467703 + 0.998906i \(0.485107\pi\)
\(242\) −8.10741e17 −1.07218
\(243\) −3.34679e17 −0.429123
\(244\) 6.84934e16 0.0851580
\(245\) 1.67726e17 0.202235
\(246\) −1.18742e18 −1.38865
\(247\) −2.97134e17 −0.337073
\(248\) −2.91944e17 −0.321300
\(249\) 9.43255e17 1.00724
\(250\) −6.56433e17 −0.680205
\(251\) 1.58806e18 1.59703 0.798515 0.601975i \(-0.205619\pi\)
0.798515 + 0.601975i \(0.205619\pi\)
\(252\) −1.23075e17 −0.120134
\(253\) 1.94569e17 0.184361
\(254\) 6.45986e17 0.594251
\(255\) −2.87260e17 −0.256581
\(256\) 1.23980e18 1.07535
\(257\) −8.28562e17 −0.697954 −0.348977 0.937131i \(-0.613471\pi\)
−0.348977 + 0.937131i \(0.613471\pi\)
\(258\) 3.56090e17 0.291348
\(259\) −2.91744e18 −2.31875
\(260\) −1.37557e17 −0.106214
\(261\) 1.15867e17 0.0869271
\(262\) −3.51265e17 −0.256080
\(263\) 1.40445e18 0.995038 0.497519 0.867453i \(-0.334244\pi\)
0.497519 + 0.867453i \(0.334244\pi\)
\(264\) 2.81081e17 0.193554
\(265\) 3.54200e17 0.237085
\(266\) 9.53041e17 0.620153
\(267\) −2.39589e18 −1.51577
\(268\) −4.00500e17 −0.246372
\(269\) 1.43582e18 0.858930 0.429465 0.903083i \(-0.358702\pi\)
0.429465 + 0.903083i \(0.358702\pi\)
\(270\) 6.59050e17 0.383433
\(271\) 5.09160e17 0.288127 0.144064 0.989568i \(-0.453983\pi\)
0.144064 + 0.989568i \(0.453983\pi\)
\(272\) −2.19967e18 −1.21085
\(273\) 1.79611e18 0.961865
\(274\) 2.28792e18 1.19209
\(275\) −5.72358e17 −0.290184
\(276\) −4.39449e17 −0.216815
\(277\) 5.68946e17 0.273195 0.136598 0.990627i \(-0.456383\pi\)
0.136598 + 0.990627i \(0.456383\pi\)
\(278\) −4.03292e18 −1.88488
\(279\) −2.24774e17 −0.102262
\(280\) −5.99797e17 −0.265655
\(281\) −4.06184e18 −1.75156 −0.875780 0.482710i \(-0.839652\pi\)
−0.875780 + 0.482710i \(0.839652\pi\)
\(282\) 2.46650e18 1.03565
\(283\) 2.78506e18 1.13877 0.569385 0.822071i \(-0.307181\pi\)
0.569385 + 0.822071i \(0.307181\pi\)
\(284\) 1.73670e18 0.691570
\(285\) −2.72733e17 −0.105779
\(286\) −8.45211e17 −0.319315
\(287\) 4.63440e18 1.70561
\(288\) 4.86463e17 0.174424
\(289\) −1.51369e17 −0.0528813
\(290\) −4.15366e17 −0.141399
\(291\) −2.05161e18 −0.680608
\(292\) −1.14120e18 −0.368969
\(293\) −3.63803e18 −1.14646 −0.573230 0.819395i \(-0.694310\pi\)
−0.573230 + 0.819395i \(0.694310\pi\)
\(294\) −2.32766e18 −0.715013
\(295\) 5.13745e17 0.153845
\(296\) 4.21532e18 1.23067
\(297\) 1.20541e18 0.343131
\(298\) −2.71209e18 −0.752803
\(299\) −1.79641e18 −0.486262
\(300\) 1.29272e18 0.341267
\(301\) −1.38979e18 −0.357849
\(302\) 6.21190e18 1.56017
\(303\) 2.73746e18 0.670701
\(304\) −2.08843e18 −0.499192
\(305\) 2.56998e17 0.0599351
\(306\) −1.11667e18 −0.254106
\(307\) −9.75296e17 −0.216570 −0.108285 0.994120i \(-0.534536\pi\)
−0.108285 + 0.994120i \(0.534536\pi\)
\(308\) 8.06969e17 0.174874
\(309\) −2.48125e18 −0.524786
\(310\) 8.05783e17 0.166343
\(311\) 3.36692e17 0.0678468 0.0339234 0.999424i \(-0.489200\pi\)
0.0339234 + 0.999424i \(0.489200\pi\)
\(312\) −2.59515e18 −0.510508
\(313\) 3.65551e18 0.702046 0.351023 0.936367i \(-0.385834\pi\)
0.351023 + 0.936367i \(0.385834\pi\)
\(314\) −3.13797e18 −0.588405
\(315\) −4.61797e17 −0.0845515
\(316\) −3.52937e17 −0.0631021
\(317\) −7.97380e17 −0.139226 −0.0696131 0.997574i \(-0.522176\pi\)
−0.0696131 + 0.997574i \(0.522176\pi\)
\(318\) −4.91548e18 −0.838229
\(319\) −7.59708e17 −0.126537
\(320\) 5.37283e17 0.0874135
\(321\) 8.41788e18 1.33787
\(322\) 5.76188e18 0.894633
\(323\) 2.57395e18 0.390464
\(324\) −2.09681e18 −0.310793
\(325\) 5.28444e18 0.765375
\(326\) 3.62321e18 0.512816
\(327\) −4.24645e18 −0.587378
\(328\) −6.69610e18 −0.905249
\(329\) −9.62651e18 −1.27204
\(330\) −7.75801e17 −0.100206
\(331\) −1.01585e19 −1.28269 −0.641343 0.767255i \(-0.721622\pi\)
−0.641343 + 0.767255i \(0.721622\pi\)
\(332\) −3.91276e18 −0.483000
\(333\) 3.24546e18 0.391692
\(334\) 1.38672e19 1.63640
\(335\) −1.50274e18 −0.173399
\(336\) 1.26241e19 1.42448
\(337\) −4.81465e18 −0.531301 −0.265651 0.964069i \(-0.585587\pi\)
−0.265651 + 0.964069i \(0.585587\pi\)
\(338\) −3.25253e18 −0.351032
\(339\) 6.87734e18 0.725977
\(340\) 1.19160e18 0.123038
\(341\) 1.47378e18 0.148859
\(342\) −1.06020e18 −0.104759
\(343\) −4.31515e18 −0.417148
\(344\) 2.00806e18 0.189928
\(345\) −1.64888e18 −0.152597
\(346\) −1.64130e19 −1.48634
\(347\) 4.50275e18 0.399031 0.199516 0.979895i \(-0.436063\pi\)
0.199516 + 0.979895i \(0.436063\pi\)
\(348\) 1.71586e18 0.148812
\(349\) 2.24323e19 1.90407 0.952036 0.305986i \(-0.0989860\pi\)
0.952036 + 0.305986i \(0.0989860\pi\)
\(350\) −1.69496e19 −1.40815
\(351\) −1.11292e19 −0.905026
\(352\) −3.18961e18 −0.253902
\(353\) 8.02510e18 0.625374 0.312687 0.949856i \(-0.398771\pi\)
0.312687 + 0.949856i \(0.398771\pi\)
\(354\) −7.12961e18 −0.543928
\(355\) 6.51636e18 0.486735
\(356\) 9.93851e18 0.726853
\(357\) −1.55590e19 −1.11422
\(358\) 2.01690e19 1.41436
\(359\) 1.61507e18 0.110913 0.0554567 0.998461i \(-0.482339\pi\)
0.0554567 + 0.998461i \(0.482339\pi\)
\(360\) 6.67236e17 0.0448756
\(361\) −1.27374e19 −0.839026
\(362\) 1.60526e19 1.03569
\(363\) 1.25665e19 0.794171
\(364\) −7.45055e18 −0.461241
\(365\) −4.28195e18 −0.259684
\(366\) −3.56655e18 −0.211905
\(367\) −9.97799e18 −0.580828 −0.290414 0.956901i \(-0.593793\pi\)
−0.290414 + 0.956901i \(0.593793\pi\)
\(368\) −1.26262e19 −0.720135
\(369\) −5.15547e18 −0.288118
\(370\) −1.16345e19 −0.637142
\(371\) 1.91847e19 1.02956
\(372\) −3.32865e18 −0.175064
\(373\) −2.36866e19 −1.22092 −0.610459 0.792048i \(-0.709015\pi\)
−0.610459 + 0.792048i \(0.709015\pi\)
\(374\) 7.32171e18 0.369893
\(375\) 1.01747e19 0.503834
\(376\) 1.39090e19 0.675132
\(377\) 7.01419e18 0.333747
\(378\) 3.56964e19 1.66508
\(379\) 1.86851e19 0.854480 0.427240 0.904138i \(-0.359486\pi\)
0.427240 + 0.904138i \(0.359486\pi\)
\(380\) 1.13133e18 0.0507241
\(381\) −1.00128e19 −0.440167
\(382\) −2.13024e19 −0.918235
\(383\) −3.02521e19 −1.27869 −0.639343 0.768921i \(-0.720794\pi\)
−0.639343 + 0.768921i \(0.720794\pi\)
\(384\) −2.44537e19 −1.01358
\(385\) 3.02788e18 0.123079
\(386\) −1.92494e18 −0.0767385
\(387\) 1.54605e18 0.0604493
\(388\) 8.51037e18 0.326370
\(389\) −1.00714e18 −0.0378852 −0.0189426 0.999821i \(-0.506030\pi\)
−0.0189426 + 0.999821i \(0.506030\pi\)
\(390\) 7.16278e18 0.264300
\(391\) 1.55615e19 0.563283
\(392\) −1.31261e19 −0.466112
\(393\) 5.44461e18 0.189680
\(394\) 7.65020e18 0.261487
\(395\) −1.32428e18 −0.0444120
\(396\) −8.97702e17 −0.0295405
\(397\) 3.56324e19 1.15058 0.575290 0.817950i \(-0.304889\pi\)
0.575290 + 0.817950i \(0.304889\pi\)
\(398\) −6.18756e19 −1.96063
\(399\) −1.47721e19 −0.459353
\(400\) 3.71422e19 1.13349
\(401\) 3.94327e19 1.18106 0.590532 0.807014i \(-0.298918\pi\)
0.590532 + 0.807014i \(0.298918\pi\)
\(402\) 2.08546e19 0.613065
\(403\) −1.36071e19 −0.392624
\(404\) −1.13554e19 −0.321620
\(405\) −7.86757e18 −0.218740
\(406\) −2.24977e19 −0.614034
\(407\) −2.12796e19 −0.570172
\(408\) 2.24807e19 0.591370
\(409\) −5.27823e19 −1.36321 −0.681607 0.731719i \(-0.738718\pi\)
−0.681607 + 0.731719i \(0.738718\pi\)
\(410\) 1.84817e19 0.468665
\(411\) −3.54627e19 −0.882995
\(412\) 1.02926e19 0.251649
\(413\) 2.78262e19 0.668080
\(414\) −6.40972e18 −0.151125
\(415\) −1.46813e19 −0.339941
\(416\) 2.94488e19 0.669681
\(417\) 6.25102e19 1.39615
\(418\) 6.95143e18 0.152494
\(419\) 8.62630e18 0.185874 0.0929372 0.995672i \(-0.470374\pi\)
0.0929372 + 0.995672i \(0.470374\pi\)
\(420\) −6.83870e18 −0.144745
\(421\) −4.29249e19 −0.892469 −0.446235 0.894916i \(-0.647235\pi\)
−0.446235 + 0.894916i \(0.647235\pi\)
\(422\) 8.11801e19 1.65808
\(423\) 1.07089e19 0.214878
\(424\) −2.77193e19 −0.546435
\(425\) −4.57770e19 −0.886605
\(426\) −9.04322e19 −1.72088
\(427\) 1.39199e19 0.260272
\(428\) −3.49186e19 −0.641547
\(429\) 1.31008e19 0.236519
\(430\) −5.54237e18 −0.0983291
\(431\) 5.04764e19 0.880053 0.440026 0.897985i \(-0.354969\pi\)
0.440026 + 0.897985i \(0.354969\pi\)
\(432\) −7.82227e19 −1.34031
\(433\) 5.05734e19 0.851653 0.425827 0.904805i \(-0.359983\pi\)
0.425827 + 0.904805i \(0.359983\pi\)
\(434\) 4.36440e19 0.722356
\(435\) 6.43818e18 0.104735
\(436\) 1.76149e19 0.281664
\(437\) 1.47745e19 0.232222
\(438\) 5.94238e19 0.918130
\(439\) 2.47946e19 0.376594 0.188297 0.982112i \(-0.439703\pi\)
0.188297 + 0.982112i \(0.439703\pi\)
\(440\) −4.37489e18 −0.0653238
\(441\) −1.01061e19 −0.148352
\(442\) −6.75996e19 −0.975612
\(443\) −1.30654e20 −1.85394 −0.926970 0.375135i \(-0.877596\pi\)
−0.926970 + 0.375135i \(0.877596\pi\)
\(444\) 4.80617e19 0.670544
\(445\) 3.72909e19 0.511567
\(446\) −5.46648e18 −0.0737389
\(447\) 4.20373e19 0.557608
\(448\) 2.91011e19 0.379598
\(449\) −7.78280e19 −0.998363 −0.499181 0.866498i \(-0.666366\pi\)
−0.499181 + 0.866498i \(0.666366\pi\)
\(450\) 1.88553e19 0.237870
\(451\) 3.38031e19 0.419404
\(452\) −2.85282e19 −0.348126
\(453\) −9.62844e19 −1.15563
\(454\) 6.55975e19 0.774405
\(455\) −2.79556e19 −0.324627
\(456\) 2.13438e19 0.243801
\(457\) −1.18451e20 −1.33096 −0.665482 0.746414i \(-0.731774\pi\)
−0.665482 + 0.746414i \(0.731774\pi\)
\(458\) 2.33262e19 0.257841
\(459\) 9.64078e19 1.04838
\(460\) 6.83981e18 0.0731746
\(461\) 1.38643e20 1.45929 0.729644 0.683827i \(-0.239686\pi\)
0.729644 + 0.683827i \(0.239686\pi\)
\(462\) −4.20200e19 −0.435153
\(463\) 1.75645e20 1.78969 0.894846 0.446375i \(-0.147285\pi\)
0.894846 + 0.446375i \(0.147285\pi\)
\(464\) 4.92999e19 0.494266
\(465\) −1.24896e19 −0.123212
\(466\) −1.70749e20 −1.65754
\(467\) −1.36631e20 −1.30519 −0.652593 0.757708i \(-0.726319\pi\)
−0.652593 + 0.757708i \(0.726319\pi\)
\(468\) 8.28826e18 0.0779147
\(469\) −8.13935e19 −0.752997
\(470\) −3.83899e19 −0.349528
\(471\) 4.86385e19 0.435837
\(472\) −4.02052e19 −0.354583
\(473\) −1.01370e19 −0.0879938
\(474\) 1.83779e19 0.157021
\(475\) −4.34619e19 −0.365516
\(476\) 6.45410e19 0.534298
\(477\) −2.13417e19 −0.173917
\(478\) 7.62386e19 0.611597
\(479\) 6.41058e19 0.506269 0.253134 0.967431i \(-0.418539\pi\)
0.253134 + 0.967431i \(0.418539\pi\)
\(480\) 2.70305e19 0.210157
\(481\) 1.96470e20 1.50386
\(482\) 1.48109e19 0.111617
\(483\) −8.93091e19 −0.662662
\(484\) −5.21276e19 −0.380827
\(485\) 3.19323e19 0.229703
\(486\) −7.22907e19 −0.512047
\(487\) −2.41343e19 −0.168332 −0.0841662 0.996452i \(-0.526823\pi\)
−0.0841662 + 0.996452i \(0.526823\pi\)
\(488\) −2.01124e19 −0.138139
\(489\) −5.61598e19 −0.379847
\(490\) 3.62289e19 0.241315
\(491\) −2.80908e19 −0.184269 −0.0921346 0.995747i \(-0.529369\pi\)
−0.0921346 + 0.995747i \(0.529369\pi\)
\(492\) −7.63469e19 −0.493234
\(493\) −6.07611e19 −0.386611
\(494\) −6.41808e19 −0.402210
\(495\) −3.36832e18 −0.0207910
\(496\) −9.56384e19 −0.581460
\(497\) 3.52948e20 2.11368
\(498\) 2.03743e20 1.20188
\(499\) −1.71994e20 −0.999443 −0.499722 0.866186i \(-0.666564\pi\)
−0.499722 + 0.866186i \(0.666564\pi\)
\(500\) −4.22062e19 −0.241602
\(501\) −2.14941e20 −1.21210
\(502\) 3.43020e20 1.90564
\(503\) −1.83497e20 −1.00431 −0.502155 0.864778i \(-0.667459\pi\)
−0.502155 + 0.864778i \(0.667459\pi\)
\(504\) 3.61398e19 0.194875
\(505\) −4.26073e19 −0.226359
\(506\) 4.20268e19 0.219987
\(507\) 5.04142e19 0.260012
\(508\) 4.15345e19 0.211072
\(509\) 2.67204e20 1.33801 0.669004 0.743258i \(-0.266721\pi\)
0.669004 + 0.743258i \(0.266721\pi\)
\(510\) −6.20482e19 −0.306163
\(511\) −2.31925e20 −1.12769
\(512\) 2.84604e19 0.136369
\(513\) 9.15321e19 0.432209
\(514\) −1.78969e20 −0.832828
\(515\) 3.86195e19 0.177114
\(516\) 2.28953e19 0.103484
\(517\) −7.02153e19 −0.312790
\(518\) −6.30167e20 −2.76683
\(519\) 2.54401e20 1.10094
\(520\) 4.03922e19 0.172295
\(521\) −2.01468e20 −0.847076 −0.423538 0.905878i \(-0.639212\pi\)
−0.423538 + 0.905878i \(0.639212\pi\)
\(522\) 2.50272e19 0.103725
\(523\) 3.58989e20 1.46662 0.733311 0.679894i \(-0.237974\pi\)
0.733311 + 0.679894i \(0.237974\pi\)
\(524\) −2.25850e19 −0.0909570
\(525\) 2.62718e20 1.04303
\(526\) 3.03362e20 1.18732
\(527\) 1.17872e20 0.454813
\(528\) 9.20799e19 0.350276
\(529\) −1.77312e20 −0.664997
\(530\) 7.65071e19 0.282900
\(531\) −3.09549e19 −0.112855
\(532\) 6.12770e19 0.220272
\(533\) −3.12095e20 −1.10620
\(534\) −5.17512e20 −1.80868
\(535\) −1.31020e20 −0.451528
\(536\) 1.17603e20 0.399652
\(537\) −3.12619e20 −1.04763
\(538\) 3.10137e20 1.02491
\(539\) 6.62628e19 0.215950
\(540\) 4.23744e19 0.136192
\(541\) 2.02328e20 0.641323 0.320662 0.947194i \(-0.396095\pi\)
0.320662 + 0.947194i \(0.396095\pi\)
\(542\) 1.09979e20 0.343806
\(543\) −2.48816e20 −0.767147
\(544\) −2.55103e20 −0.775755
\(545\) 6.60939e19 0.198238
\(546\) 3.87961e20 1.14774
\(547\) 7.40963e19 0.216218 0.108109 0.994139i \(-0.465520\pi\)
0.108109 + 0.994139i \(0.465520\pi\)
\(548\) 1.47104e20 0.423420
\(549\) −1.54850e19 −0.0439662
\(550\) −1.23629e20 −0.346259
\(551\) −5.76882e19 −0.159386
\(552\) 1.29040e20 0.351707
\(553\) −7.17273e19 −0.192862
\(554\) 1.22892e20 0.325988
\(555\) 1.80335e20 0.471936
\(556\) −2.59302e20 −0.669490
\(557\) 2.09626e18 0.00533987 0.00266994 0.999996i \(-0.499150\pi\)
0.00266994 + 0.999996i \(0.499150\pi\)
\(558\) −4.85511e19 −0.122023
\(559\) 9.35927e19 0.232088
\(560\) −1.96489e20 −0.480759
\(561\) −1.13487e20 −0.273983
\(562\) −8.77357e20 −2.09004
\(563\) 6.87353e20 1.61572 0.807861 0.589373i \(-0.200625\pi\)
0.807861 + 0.589373i \(0.200625\pi\)
\(564\) 1.58587e20 0.367852
\(565\) −1.07042e20 −0.245015
\(566\) 6.01573e20 1.35883
\(567\) −4.26134e20 −0.949891
\(568\) −5.09964e20 −1.12183
\(569\) −9.05218e19 −0.196522 −0.0982610 0.995161i \(-0.531328\pi\)
−0.0982610 + 0.995161i \(0.531328\pi\)
\(570\) −5.89102e19 −0.126220
\(571\) 2.05774e20 0.435130 0.217565 0.976046i \(-0.430189\pi\)
0.217565 + 0.976046i \(0.430189\pi\)
\(572\) −5.43439e19 −0.113418
\(573\) 3.30188e20 0.680145
\(574\) 1.00103e21 2.03520
\(575\) −2.62761e20 −0.527293
\(576\) −3.23731e19 −0.0641233
\(577\) 5.70778e20 1.11596 0.557980 0.829854i \(-0.311576\pi\)
0.557980 + 0.829854i \(0.311576\pi\)
\(578\) −3.26956e19 −0.0631002
\(579\) 2.98366e19 0.0568408
\(580\) −2.67065e19 −0.0502236
\(581\) −7.95190e20 −1.47622
\(582\) −4.43147e20 −0.812130
\(583\) 1.39932e20 0.253164
\(584\) 3.35102e20 0.598522
\(585\) 3.10989e19 0.0548373
\(586\) −7.85814e20 −1.36800
\(587\) −9.30363e20 −1.59907 −0.799534 0.600621i \(-0.794920\pi\)
−0.799534 + 0.600621i \(0.794920\pi\)
\(588\) −1.49660e20 −0.253966
\(589\) 1.11911e20 0.187503
\(590\) 1.10969e20 0.183574
\(591\) −1.18578e20 −0.193686
\(592\) 1.38090e21 2.22716
\(593\) 3.54225e20 0.564116 0.282058 0.959397i \(-0.408983\pi\)
0.282058 + 0.959397i \(0.408983\pi\)
\(594\) 2.60368e20 0.409438
\(595\) 2.42168e20 0.376045
\(596\) −1.74377e20 −0.267388
\(597\) 9.59071e20 1.45226
\(598\) −3.88024e20 −0.580229
\(599\) −3.30045e20 −0.487385 −0.243693 0.969853i \(-0.578359\pi\)
−0.243693 + 0.969853i \(0.578359\pi\)
\(600\) −3.79594e20 −0.553586
\(601\) −3.35884e20 −0.483761 −0.241880 0.970306i \(-0.577764\pi\)
−0.241880 + 0.970306i \(0.577764\pi\)
\(602\) −3.00194e20 −0.427000
\(603\) 9.05451e19 0.127199
\(604\) 3.99402e20 0.554158
\(605\) −1.95591e20 −0.268030
\(606\) 5.91292e20 0.800309
\(607\) −1.33438e21 −1.78387 −0.891934 0.452165i \(-0.850652\pi\)
−0.891934 + 0.452165i \(0.850652\pi\)
\(608\) −2.42202e20 −0.319816
\(609\) 3.48714e20 0.454820
\(610\) 5.55116e19 0.0715172
\(611\) 6.48280e20 0.825000
\(612\) −7.17978e19 −0.0902559
\(613\) 5.68844e18 0.00706381 0.00353191 0.999994i \(-0.498876\pi\)
0.00353191 + 0.999994i \(0.498876\pi\)
\(614\) −2.10664e20 −0.258421
\(615\) −2.86466e20 −0.347144
\(616\) −2.36959e20 −0.283673
\(617\) 3.98915e20 0.471783 0.235891 0.971779i \(-0.424199\pi\)
0.235891 + 0.971779i \(0.424199\pi\)
\(618\) −5.35950e20 −0.626197
\(619\) −5.40017e20 −0.623343 −0.311672 0.950190i \(-0.600889\pi\)
−0.311672 + 0.950190i \(0.600889\pi\)
\(620\) 5.18088e19 0.0590835
\(621\) 5.53384e20 0.623504
\(622\) 7.27254e19 0.0809577
\(623\) 2.01980e21 2.22151
\(624\) −8.50151e20 −0.923871
\(625\) 6.90089e20 0.740978
\(626\) 7.89591e20 0.837711
\(627\) −1.07747e20 −0.112953
\(628\) −2.01760e20 −0.208996
\(629\) −1.70194e21 −1.74206
\(630\) −9.97480e19 −0.100891
\(631\) 9.59111e20 0.958625 0.479312 0.877644i \(-0.340886\pi\)
0.479312 + 0.877644i \(0.340886\pi\)
\(632\) 1.03637e20 0.102361
\(633\) −1.25829e21 −1.22816
\(634\) −1.72234e20 −0.166131
\(635\) 1.55844e20 0.148555
\(636\) −3.16047e20 −0.297731
\(637\) −6.11788e20 −0.569581
\(638\) −1.64097e20 −0.150989
\(639\) −3.92633e20 −0.357051
\(640\) 3.80609e20 0.342082
\(641\) −9.25925e20 −0.822509 −0.411255 0.911521i \(-0.634909\pi\)
−0.411255 + 0.911521i \(0.634909\pi\)
\(642\) 1.81826e21 1.59641
\(643\) −7.65928e20 −0.664669 −0.332335 0.943162i \(-0.607836\pi\)
−0.332335 + 0.943162i \(0.607836\pi\)
\(644\) 3.70467e20 0.317765
\(645\) 8.59068e19 0.0728332
\(646\) 5.55972e20 0.465918
\(647\) 1.36075e21 1.12719 0.563596 0.826051i \(-0.309418\pi\)
0.563596 + 0.826051i \(0.309418\pi\)
\(648\) 6.15709e20 0.504153
\(649\) 2.02963e20 0.164279
\(650\) 1.14144e21 0.913278
\(651\) −6.76481e20 −0.535055
\(652\) 2.32959e20 0.182147
\(653\) −2.71809e20 −0.210094 −0.105047 0.994467i \(-0.533499\pi\)
−0.105047 + 0.994467i \(0.533499\pi\)
\(654\) −9.17233e20 −0.700885
\(655\) −8.47427e19 −0.0640166
\(656\) −2.19359e21 −1.63824
\(657\) 2.58002e20 0.190495
\(658\) −2.07933e21 −1.51785
\(659\) 6.74316e20 0.486657 0.243329 0.969944i \(-0.421761\pi\)
0.243329 + 0.969944i \(0.421761\pi\)
\(660\) −4.98811e19 −0.0355923
\(661\) 1.26727e21 0.894042 0.447021 0.894524i \(-0.352485\pi\)
0.447021 + 0.894524i \(0.352485\pi\)
\(662\) −2.19424e21 −1.53056
\(663\) 1.04779e21 0.722644
\(664\) 1.14894e21 0.783499
\(665\) 2.29921e20 0.155030
\(666\) 7.01020e20 0.467384
\(667\) −3.48770e20 −0.229930
\(668\) 8.91609e20 0.581234
\(669\) 8.47305e19 0.0546190
\(670\) −3.24592e20 −0.206908
\(671\) 1.01531e20 0.0640000
\(672\) 1.46406e21 0.912620
\(673\) −1.13945e21 −0.702394 −0.351197 0.936302i \(-0.614225\pi\)
−0.351197 + 0.936302i \(0.614225\pi\)
\(674\) −1.03997e21 −0.633971
\(675\) −1.62788e21 −0.981393
\(676\) −2.09126e20 −0.124683
\(677\) 1.74431e21 1.02851 0.514256 0.857637i \(-0.328068\pi\)
0.514256 + 0.857637i \(0.328068\pi\)
\(678\) 1.48550e21 0.866266
\(679\) 1.72956e21 0.997500
\(680\) −3.49902e20 −0.199586
\(681\) −1.01676e21 −0.573608
\(682\) 3.18337e20 0.177625
\(683\) −1.43739e21 −0.793267 −0.396634 0.917977i \(-0.629822\pi\)
−0.396634 + 0.917977i \(0.629822\pi\)
\(684\) −6.81667e19 −0.0372093
\(685\) 5.51960e20 0.298008
\(686\) −9.32073e20 −0.497759
\(687\) −3.61556e20 −0.190985
\(688\) 6.57825e20 0.343714
\(689\) −1.29196e21 −0.667735
\(690\) −3.56159e20 −0.182085
\(691\) −1.77548e21 −0.897903 −0.448951 0.893556i \(-0.648202\pi\)
−0.448951 + 0.893556i \(0.648202\pi\)
\(692\) −1.05529e21 −0.527932
\(693\) −1.82440e20 −0.0902860
\(694\) 9.72594e20 0.476141
\(695\) −9.72941e20 −0.471195
\(696\) −5.03846e20 −0.241395
\(697\) 2.70356e21 1.28141
\(698\) 4.84538e21 2.27202
\(699\) 2.64661e21 1.22776
\(700\) −1.08980e21 −0.500161
\(701\) 1.43100e21 0.649764 0.324882 0.945755i \(-0.394675\pi\)
0.324882 + 0.945755i \(0.394675\pi\)
\(702\) −2.40391e21 −1.07992
\(703\) −1.61586e21 −0.718191
\(704\) 2.12262e20 0.0933420
\(705\) 5.95043e20 0.258899
\(706\) 1.73342e21 0.746223
\(707\) −2.30776e21 −0.982980
\(708\) −4.58408e20 −0.193198
\(709\) −2.41840e21 −1.00851 −0.504257 0.863554i \(-0.668234\pi\)
−0.504257 + 0.863554i \(0.668234\pi\)
\(710\) 1.40753e21 0.580793
\(711\) 7.97921e19 0.0325790
\(712\) −2.91835e21 −1.17906
\(713\) 6.76591e20 0.270492
\(714\) −3.36074e21 −1.32953
\(715\) −2.03907e20 −0.0798246
\(716\) 1.29679e21 0.502368
\(717\) −1.18170e21 −0.453015
\(718\) 3.48856e20 0.132347
\(719\) 4.74444e21 1.78122 0.890611 0.454766i \(-0.150277\pi\)
0.890611 + 0.454766i \(0.150277\pi\)
\(720\) 2.18581e20 0.0812118
\(721\) 2.09176e21 0.769127
\(722\) −2.75127e21 −1.00116
\(723\) −2.29569e20 −0.0826753
\(724\) 1.03212e21 0.367868
\(725\) 1.02597e21 0.361909
\(726\) 2.71436e21 0.947639
\(727\) −3.59265e21 −1.24138 −0.620692 0.784054i \(-0.713148\pi\)
−0.620692 + 0.784054i \(0.713148\pi\)
\(728\) 2.18778e21 0.748202
\(729\) 3.28690e21 1.11258
\(730\) −9.24902e20 −0.309866
\(731\) −8.10755e20 −0.268850
\(732\) −2.29316e20 −0.0752664
\(733\) −2.76824e21 −0.899339 −0.449669 0.893195i \(-0.648458\pi\)
−0.449669 + 0.893195i \(0.648458\pi\)
\(734\) −2.15525e21 −0.693069
\(735\) −5.61548e20 −0.178744
\(736\) −1.46430e21 −0.461367
\(737\) −5.93680e20 −0.185160
\(738\) −1.11358e21 −0.343795
\(739\) 3.55824e21 1.08743 0.543716 0.839269i \(-0.317017\pi\)
0.543716 + 0.839269i \(0.317017\pi\)
\(740\) −7.48058e20 −0.226307
\(741\) 9.94803e20 0.297921
\(742\) 4.14389e21 1.22851
\(743\) −1.94092e21 −0.569628 −0.284814 0.958583i \(-0.591932\pi\)
−0.284814 + 0.958583i \(0.591932\pi\)
\(744\) 9.77427e20 0.283980
\(745\) −6.54291e20 −0.188191
\(746\) −5.11630e21 −1.45685
\(747\) 8.84598e20 0.249368
\(748\) 4.70759e20 0.131382
\(749\) −7.09651e21 −1.96079
\(750\) 2.19774e21 0.601196
\(751\) 4.75565e21 1.28798 0.643992 0.765032i \(-0.277277\pi\)
0.643992 + 0.765032i \(0.277277\pi\)
\(752\) 4.55650e21 1.22179
\(753\) −5.31681e21 −1.41153
\(754\) 1.51507e21 0.398241
\(755\) 1.49862e21 0.390022
\(756\) 2.29514e21 0.591421
\(757\) 3.62137e21 0.923960 0.461980 0.886890i \(-0.347139\pi\)
0.461980 + 0.886890i \(0.347139\pi\)
\(758\) 4.03599e21 1.01960
\(759\) −6.51416e20 −0.162947
\(760\) −3.32206e20 −0.0822821
\(761\) −3.86361e21 −0.947564 −0.473782 0.880642i \(-0.657112\pi\)
−0.473782 + 0.880642i \(0.657112\pi\)
\(762\) −2.16276e21 −0.525226
\(763\) 3.57987e21 0.860862
\(764\) −1.36967e21 −0.326148
\(765\) −2.69397e20 −0.0635231
\(766\) −6.53445e21 −1.52578
\(767\) −1.87391e21 −0.433294
\(768\) −4.15085e21 −0.950446
\(769\) −5.39327e21 −1.22294 −0.611469 0.791268i \(-0.709421\pi\)
−0.611469 + 0.791268i \(0.709421\pi\)
\(770\) 6.54021e20 0.146863
\(771\) 2.77403e21 0.616883
\(772\) −1.23767e20 −0.0272568
\(773\) 6.57037e21 1.43299 0.716496 0.697591i \(-0.245745\pi\)
0.716496 + 0.697591i \(0.245745\pi\)
\(774\) 3.33947e20 0.0721306
\(775\) −1.99031e21 −0.425754
\(776\) −2.49899e21 −0.529422
\(777\) 9.76758e21 2.04941
\(778\) −2.17543e20 −0.0452062
\(779\) 2.56683e21 0.528282
\(780\) 4.60540e20 0.0938767
\(781\) 2.57439e21 0.519746
\(782\) 3.36129e21 0.672133
\(783\) −2.16073e21 −0.427944
\(784\) −4.30001e21 −0.843527
\(785\) −7.57035e20 −0.147094
\(786\) 1.17604e21 0.226335
\(787\) −3.72074e20 −0.0709281 −0.0354641 0.999371i \(-0.511291\pi\)
−0.0354641 + 0.999371i \(0.511291\pi\)
\(788\) 4.91879e20 0.0928778
\(789\) −4.70211e21 −0.879459
\(790\) −2.86044e20 −0.0529943
\(791\) −5.79778e21 −1.06399
\(792\) 2.63602e20 0.0479192
\(793\) −9.37412e20 −0.168804
\(794\) 7.69660e21 1.37292
\(795\) −1.18586e21 −0.209546
\(796\) −3.97837e21 −0.696397
\(797\) 2.61511e21 0.453474 0.226737 0.973956i \(-0.427194\pi\)
0.226737 + 0.973956i \(0.427194\pi\)
\(798\) −3.19078e21 −0.548119
\(799\) −5.61579e21 −0.955674
\(800\) 4.30750e21 0.726190
\(801\) −2.24690e21 −0.375267
\(802\) 8.51746e21 1.40930
\(803\) −1.69165e21 −0.277296
\(804\) 1.34087e21 0.217755
\(805\) 1.39005e21 0.223647
\(806\) −2.93913e21 −0.468495
\(807\) −4.80713e21 −0.759161
\(808\) 3.33441e21 0.521715
\(809\) 5.34899e21 0.829198 0.414599 0.910004i \(-0.363922\pi\)
0.414599 + 0.910004i \(0.363922\pi\)
\(810\) −1.69939e21 −0.261010
\(811\) 8.46492e21 1.28815 0.644075 0.764962i \(-0.277242\pi\)
0.644075 + 0.764962i \(0.277242\pi\)
\(812\) −1.44652e21 −0.218099
\(813\) −1.70467e21 −0.254660
\(814\) −4.59640e21 −0.680354
\(815\) 8.74100e20 0.128197
\(816\) 7.36451e21 1.07021
\(817\) −7.69753e20 −0.110837
\(818\) −1.14010e22 −1.62664
\(819\) 1.68442e21 0.238134
\(820\) 1.18830e21 0.166465
\(821\) 7.99397e21 1.10966 0.554829 0.831965i \(-0.312784\pi\)
0.554829 + 0.831965i \(0.312784\pi\)
\(822\) −7.65994e21 −1.05363
\(823\) 1.96841e21 0.268297 0.134148 0.990961i \(-0.457170\pi\)
0.134148 + 0.990961i \(0.457170\pi\)
\(824\) −3.02232e21 −0.408213
\(825\) 1.91626e21 0.256477
\(826\) 6.01046e21 0.797181
\(827\) −1.43539e22 −1.88659 −0.943296 0.331954i \(-0.892292\pi\)
−0.943296 + 0.331954i \(0.892292\pi\)
\(828\) −4.12121e20 −0.0536782
\(829\) −8.83327e21 −1.14015 −0.570076 0.821592i \(-0.693086\pi\)
−0.570076 + 0.821592i \(0.693086\pi\)
\(830\) −3.17116e21 −0.405632
\(831\) −1.90483e21 −0.241462
\(832\) −1.95976e21 −0.246195
\(833\) 5.29967e21 0.659799
\(834\) 1.35022e22 1.66594
\(835\) 3.34546e21 0.409079
\(836\) 4.46951e20 0.0541643
\(837\) 4.19166e21 0.503437
\(838\) 1.86328e21 0.221793
\(839\) 1.26696e22 1.49469 0.747343 0.664439i \(-0.231329\pi\)
0.747343 + 0.664439i \(0.231329\pi\)
\(840\) 2.00812e21 0.234798
\(841\) −7.26739e21 −0.842187
\(842\) −9.27177e21 −1.06493
\(843\) 1.35990e22 1.54811
\(844\) 5.21958e21 0.588935
\(845\) −7.84673e20 −0.0877533
\(846\) 2.31312e21 0.256401
\(847\) −1.05939e22 −1.16394
\(848\) −9.08064e21 −0.988889
\(849\) −9.32438e21 −1.00650
\(850\) −9.88783e21 −1.05793
\(851\) −9.76917e21 −1.03606
\(852\) −5.81446e21 −0.611241
\(853\) 6.00532e21 0.625776 0.312888 0.949790i \(-0.398704\pi\)
0.312888 + 0.949790i \(0.398704\pi\)
\(854\) 3.00670e21 0.310568
\(855\) −2.55773e20 −0.0261883
\(856\) 1.02535e22 1.04069
\(857\) 1.47589e22 1.48491 0.742453 0.669898i \(-0.233662\pi\)
0.742453 + 0.669898i \(0.233662\pi\)
\(858\) 2.82976e21 0.282225
\(859\) 9.64956e20 0.0954023 0.0477012 0.998862i \(-0.484810\pi\)
0.0477012 + 0.998862i \(0.484810\pi\)
\(860\) −3.56354e20 −0.0349255
\(861\) −1.55160e22 −1.50749
\(862\) 1.09029e22 1.05012
\(863\) 3.44391e21 0.328829 0.164415 0.986391i \(-0.447426\pi\)
0.164415 + 0.986391i \(0.447426\pi\)
\(864\) −9.07173e21 −0.858691
\(865\) −3.95963e21 −0.371564
\(866\) 1.09239e22 1.01623
\(867\) 5.06782e20 0.0467389
\(868\) 2.80614e21 0.256574
\(869\) −5.23175e20 −0.0474241
\(870\) 1.39065e21 0.124975
\(871\) 5.48130e21 0.488368
\(872\) −5.17245e21 −0.456901
\(873\) −1.92403e21 −0.168502
\(874\) 3.19130e21 0.277097
\(875\) −8.57756e21 −0.738420
\(876\) 3.82073e21 0.326111
\(877\) −1.09850e22 −0.929617 −0.464808 0.885411i \(-0.653877\pi\)
−0.464808 + 0.885411i \(0.653877\pi\)
\(878\) 5.35564e21 0.449368
\(879\) 1.21801e22 1.01329
\(880\) −1.43318e21 −0.118217
\(881\) −7.98462e21 −0.653033 −0.326516 0.945192i \(-0.605875\pi\)
−0.326516 + 0.945192i \(0.605875\pi\)
\(882\) −2.18291e21 −0.177020
\(883\) 5.45236e21 0.438409 0.219204 0.975679i \(-0.429654\pi\)
0.219204 + 0.975679i \(0.429654\pi\)
\(884\) −4.34640e21 −0.346528
\(885\) −1.72002e21 −0.135975
\(886\) −2.82213e22 −2.21220
\(887\) 1.67127e22 1.29903 0.649517 0.760347i \(-0.274971\pi\)
0.649517 + 0.760347i \(0.274971\pi\)
\(888\) −1.41129e22 −1.08772
\(889\) 8.44105e21 0.645109
\(890\) 8.05483e21 0.610424
\(891\) −3.10820e21 −0.233575
\(892\) −3.51475e20 −0.0261913
\(893\) −5.33178e21 −0.393991
\(894\) 9.08007e21 0.665361
\(895\) 4.86576e21 0.353572
\(896\) 2.06151e22 1.48551
\(897\) 6.01436e21 0.429780
\(898\) −1.68108e22 −1.19129
\(899\) −2.64180e21 −0.185653
\(900\) 1.21233e21 0.0844892
\(901\) 1.11917e22 0.773500
\(902\) 7.30146e21 0.500450
\(903\) 4.65300e21 0.316283
\(904\) 8.37704e21 0.564712
\(905\) 3.87269e21 0.258910
\(906\) −2.07974e22 −1.37895
\(907\) −1.53384e22 −1.00862 −0.504309 0.863523i \(-0.668253\pi\)
−0.504309 + 0.863523i \(0.668253\pi\)
\(908\) 4.21767e21 0.275061
\(909\) 2.56723e21 0.166049
\(910\) −6.03842e21 −0.387358
\(911\) −1.49134e22 −0.948829 −0.474415 0.880302i \(-0.657340\pi\)
−0.474415 + 0.880302i \(0.657340\pi\)
\(912\) 6.99206e21 0.441209
\(913\) −5.80007e21 −0.362997
\(914\) −2.55854e22 −1.58816
\(915\) −8.60430e20 −0.0529734
\(916\) 1.49979e21 0.0915827
\(917\) −4.58995e21 −0.277996
\(918\) 2.08241e22 1.25097
\(919\) −5.86667e21 −0.349563 −0.174782 0.984607i \(-0.555922\pi\)
−0.174782 + 0.984607i \(0.555922\pi\)
\(920\) −2.00844e21 −0.118700
\(921\) 3.26529e21 0.191414
\(922\) 2.99469e22 1.74129
\(923\) −2.37687e22 −1.37086
\(924\) −2.70173e21 −0.154562
\(925\) 2.87377e22 1.63076
\(926\) 3.79393e22 2.13554
\(927\) −2.32695e21 −0.129924
\(928\) 5.71747e21 0.316660
\(929\) −1.67946e22 −0.922684 −0.461342 0.887222i \(-0.652632\pi\)
−0.461342 + 0.887222i \(0.652632\pi\)
\(930\) −2.69776e21 −0.147022
\(931\) 5.03165e21 0.272012
\(932\) −1.09786e22 −0.588743
\(933\) −1.12724e21 −0.0599661
\(934\) −2.95123e22 −1.55740
\(935\) 1.76636e21 0.0924683
\(936\) −2.43377e21 −0.126389
\(937\) 5.04466e21 0.259887 0.129944 0.991521i \(-0.458520\pi\)
0.129944 + 0.991521i \(0.458520\pi\)
\(938\) −1.75810e22 −0.898508
\(939\) −1.22387e22 −0.620500
\(940\) −2.46833e21 −0.124149
\(941\) 1.65425e22 0.825430 0.412715 0.910860i \(-0.364581\pi\)
0.412715 + 0.910860i \(0.364581\pi\)
\(942\) 1.05059e22 0.520059
\(943\) 1.55185e22 0.762100
\(944\) −1.31709e22 −0.641691
\(945\) 8.61175e21 0.416249
\(946\) −2.18960e21 −0.104998
\(947\) −9.81583e21 −0.466984 −0.233492 0.972359i \(-0.575015\pi\)
−0.233492 + 0.972359i \(0.575015\pi\)
\(948\) 1.18163e21 0.0557725
\(949\) 1.56186e22 0.731384
\(950\) −9.38776e21 −0.436149
\(951\) 2.66963e21 0.123054
\(952\) −1.89519e22 −0.866712
\(953\) −5.97914e21 −0.271295 −0.135648 0.990757i \(-0.543311\pi\)
−0.135648 + 0.990757i \(0.543311\pi\)
\(954\) −4.60981e21 −0.207525
\(955\) −5.13921e21 −0.229547
\(956\) 4.90186e21 0.217233
\(957\) 2.54350e21 0.111839
\(958\) 1.38469e22 0.604101
\(959\) 2.98960e22 1.29412
\(960\) −1.79882e21 −0.0772600
\(961\) −1.83404e22 −0.781596
\(962\) 4.24375e22 1.79447
\(963\) 7.89441e21 0.331225
\(964\) 9.52286e20 0.0396451
\(965\) −4.64393e20 −0.0191836
\(966\) −1.92908e22 −0.790717
\(967\) −1.44757e22 −0.588764 −0.294382 0.955688i \(-0.595114\pi\)
−0.294382 + 0.955688i \(0.595114\pi\)
\(968\) 1.53068e22 0.617759
\(969\) −8.61757e21 −0.345109
\(970\) 6.89737e21 0.274092
\(971\) 1.77921e21 0.0701590 0.0350795 0.999385i \(-0.488832\pi\)
0.0350795 + 0.999385i \(0.488832\pi\)
\(972\) −4.64802e21 −0.181874
\(973\) −5.26978e22 −2.04619
\(974\) −5.21301e21 −0.200861
\(975\) −1.76923e22 −0.676472
\(976\) −6.58868e21 −0.249991
\(977\) 1.04088e22 0.391913 0.195957 0.980613i \(-0.437219\pi\)
0.195957 + 0.980613i \(0.437219\pi\)
\(978\) −1.21305e22 −0.453250
\(979\) 1.47323e22 0.546262
\(980\) 2.32938e21 0.0857127
\(981\) −3.98238e21 −0.145420
\(982\) −6.06761e21 −0.219878
\(983\) 3.26461e22 1.17403 0.587017 0.809575i \(-0.300302\pi\)
0.587017 + 0.809575i \(0.300302\pi\)
\(984\) 2.24185e22 0.800100
\(985\) 1.84561e21 0.0653684
\(986\) −1.31244e22 −0.461320
\(987\) 3.22295e22 1.12428
\(988\) −4.12659e21 −0.142861
\(989\) −4.65376e21 −0.159894
\(990\) −7.27557e20 −0.0248087
\(991\) −7.47327e21 −0.252906 −0.126453 0.991973i \(-0.540359\pi\)
−0.126453 + 0.991973i \(0.540359\pi\)
\(992\) −1.10915e22 −0.372523
\(993\) 3.40107e22 1.13369
\(994\) 7.62369e22 2.52213
\(995\) −1.49275e22 −0.490132
\(996\) 1.30999e22 0.426897
\(997\) −3.10809e22 −1.00526 −0.502632 0.864500i \(-0.667635\pi\)
−0.502632 + 0.864500i \(0.667635\pi\)
\(998\) −3.71506e22 −1.19258
\(999\) −6.05226e22 −1.92831
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 1.16.a.a.1.1 1
3.2 odd 2 9.16.a.a.1.1 1
4.3 odd 2 16.16.a.d.1.1 1
5.2 odd 4 25.16.b.a.24.2 2
5.3 odd 4 25.16.b.a.24.1 2
5.4 even 2 25.16.a.a.1.1 1
7.2 even 3 49.16.c.c.18.1 2
7.3 odd 6 49.16.c.b.30.1 2
7.4 even 3 49.16.c.c.30.1 2
7.5 odd 6 49.16.c.b.18.1 2
7.6 odd 2 49.16.a.a.1.1 1
8.3 odd 2 64.16.a.c.1.1 1
8.5 even 2 64.16.a.i.1.1 1
11.10 odd 2 121.16.a.a.1.1 1
12.11 even 2 144.16.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.16.a.a.1.1 1 1.1 even 1 trivial
9.16.a.a.1.1 1 3.2 odd 2
16.16.a.d.1.1 1 4.3 odd 2
25.16.a.a.1.1 1 5.4 even 2
25.16.b.a.24.1 2 5.3 odd 4
25.16.b.a.24.2 2 5.2 odd 4
49.16.a.a.1.1 1 7.6 odd 2
49.16.c.b.18.1 2 7.5 odd 6
49.16.c.b.30.1 2 7.3 odd 6
49.16.c.c.18.1 2 7.2 even 3
49.16.c.c.30.1 2 7.4 even 3
64.16.a.c.1.1 1 8.3 odd 2
64.16.a.i.1.1 1 8.5 even 2
121.16.a.a.1.1 1 11.10 odd 2
144.16.a.f.1.1 1 12.11 even 2