Properties

Label 10.10.a.c.1.1
Level 10
Weight 10
Character 10.1
Self dual yes
Analytic conductor 5.150
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 10 = 2 \cdot 5 \)
Weight: \( k \) = \( 10 \)
Character orbit: \([\chi]\) = 10.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(5.15035836164\)
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
Root \(0\)
Character \(\chi\) = 10.1

$q$-expansion

\(f(q)\) \(=\) \(q+16.0000 q^{2} +174.000 q^{3} +256.000 q^{4} -625.000 q^{5} +2784.00 q^{6} +4658.00 q^{7} +4096.00 q^{8} +10593.0 q^{9} +O(q^{10})\) \(q+16.0000 q^{2} +174.000 q^{3} +256.000 q^{4} -625.000 q^{5} +2784.00 q^{6} +4658.00 q^{7} +4096.00 q^{8} +10593.0 q^{9} -10000.0 q^{10} +28992.0 q^{11} +44544.0 q^{12} -164446. q^{13} +74528.0 q^{14} -108750. q^{15} +65536.0 q^{16} -594822. q^{17} +169488. q^{18} -295780. q^{19} -160000. q^{20} +810492. q^{21} +463872. q^{22} +2.54453e6 q^{23} +712704. q^{24} +390625. q^{25} -2.63114e6 q^{26} -1.58166e6 q^{27} +1.19245e6 q^{28} -3.72297e6 q^{29} -1.74000e6 q^{30} +2.33577e6 q^{31} +1.04858e6 q^{32} +5.04461e6 q^{33} -9.51715e6 q^{34} -2.91125e6 q^{35} +2.71181e6 q^{36} +1.08404e7 q^{37} -4.73248e6 q^{38} -2.86136e7 q^{39} -2.56000e6 q^{40} +2.15939e7 q^{41} +1.29679e7 q^{42} +1.08323e7 q^{43} +7.42195e6 q^{44} -6.62062e6 q^{45} +4.07125e7 q^{46} +5.17214e6 q^{47} +1.14033e7 q^{48} -1.86566e7 q^{49} +6.25000e6 q^{50} -1.03499e8 q^{51} -4.20982e7 q^{52} +9.81797e7 q^{53} -2.53066e7 q^{54} -1.81200e7 q^{55} +1.90792e7 q^{56} -5.14657e7 q^{57} -5.95675e7 q^{58} +1.61629e7 q^{59} -2.78400e7 q^{60} -4.39282e7 q^{61} +3.73724e7 q^{62} +4.93422e7 q^{63} +1.67772e7 q^{64} +1.02779e8 q^{65} +8.07137e7 q^{66} -8.15574e7 q^{67} -1.52274e8 q^{68} +4.42749e8 q^{69} -4.65800e7 q^{70} +1.61308e8 q^{71} +4.33889e7 q^{72} -2.47148e8 q^{73} +1.73447e8 q^{74} +6.79688e7 q^{75} -7.57197e7 q^{76} +1.35045e8 q^{77} -4.57818e8 q^{78} -5.83346e8 q^{79} -4.09600e7 q^{80} -4.83711e8 q^{81} +3.45502e8 q^{82} -1.45718e7 q^{83} +2.07486e8 q^{84} +3.71764e8 q^{85} +1.73317e8 q^{86} -6.47797e8 q^{87} +1.18751e8 q^{88} +4.70134e8 q^{89} -1.05930e8 q^{90} -7.65989e8 q^{91} +6.51401e8 q^{92} +4.06424e8 q^{93} +8.27542e7 q^{94} +1.84862e8 q^{95} +1.82452e8 q^{96} -1.17838e8 q^{97} -2.98506e8 q^{98} +3.07112e8 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0000 0.707107
\(3\) 174.000 1.24023 0.620117 0.784509i \(-0.287085\pi\)
0.620117 + 0.784509i \(0.287085\pi\)
\(4\) 256.000 0.500000
\(5\) −625.000 −0.447214
\(6\) 2784.00 0.876978
\(7\) 4658.00 0.733261 0.366630 0.930367i \(-0.380511\pi\)
0.366630 + 0.930367i \(0.380511\pi\)
\(8\) 4096.00 0.353553
\(9\) 10593.0 0.538180
\(10\) −10000.0 −0.316228
\(11\) 28992.0 0.597051 0.298525 0.954402i \(-0.403505\pi\)
0.298525 + 0.954402i \(0.403505\pi\)
\(12\) 44544.0 0.620117
\(13\) −164446. −1.59690 −0.798451 0.602060i \(-0.794347\pi\)
−0.798451 + 0.602060i \(0.794347\pi\)
\(14\) 74528.0 0.518493
\(15\) −108750. −0.554649
\(16\) 65536.0 0.250000
\(17\) −594822. −1.72730 −0.863648 0.504095i \(-0.831826\pi\)
−0.863648 + 0.504095i \(0.831826\pi\)
\(18\) 169488. 0.380551
\(19\) −295780. −0.520688 −0.260344 0.965516i \(-0.583836\pi\)
−0.260344 + 0.965516i \(0.583836\pi\)
\(20\) −160000. −0.223607
\(21\) 810492. 0.909415
\(22\) 463872. 0.422178
\(23\) 2.54453e6 1.89598 0.947988 0.318305i \(-0.103114\pi\)
0.947988 + 0.318305i \(0.103114\pi\)
\(24\) 712704. 0.438489
\(25\) 390625. 0.200000
\(26\) −2.63114e6 −1.12918
\(27\) −1.58166e6 −0.572765
\(28\) 1.19245e6 0.366630
\(29\) −3.72297e6 −0.977459 −0.488729 0.872435i \(-0.662539\pi\)
−0.488729 + 0.872435i \(0.662539\pi\)
\(30\) −1.74000e6 −0.392196
\(31\) 2.33577e6 0.454258 0.227129 0.973865i \(-0.427066\pi\)
0.227129 + 0.973865i \(0.427066\pi\)
\(32\) 1.04858e6 0.176777
\(33\) 5.04461e6 0.740482
\(34\) −9.51715e6 −1.22138
\(35\) −2.91125e6 −0.327924
\(36\) 2.71181e6 0.269090
\(37\) 1.08404e7 0.950907 0.475454 0.879741i \(-0.342284\pi\)
0.475454 + 0.879741i \(0.342284\pi\)
\(38\) −4.73248e6 −0.368182
\(39\) −2.86136e7 −1.98053
\(40\) −2.56000e6 −0.158114
\(41\) 2.15939e7 1.19345 0.596723 0.802447i \(-0.296469\pi\)
0.596723 + 0.802447i \(0.296469\pi\)
\(42\) 1.29679e7 0.643053
\(43\) 1.08323e7 0.483184 0.241592 0.970378i \(-0.422330\pi\)
0.241592 + 0.970378i \(0.422330\pi\)
\(44\) 7.42195e6 0.298525
\(45\) −6.62062e6 −0.240681
\(46\) 4.07125e7 1.34066
\(47\) 5.17214e6 0.154607 0.0773036 0.997008i \(-0.475369\pi\)
0.0773036 + 0.997008i \(0.475369\pi\)
\(48\) 1.14033e7 0.310058
\(49\) −1.86566e7 −0.462329
\(50\) 6.25000e6 0.141421
\(51\) −1.03499e8 −2.14225
\(52\) −4.20982e7 −0.798451
\(53\) 9.81797e7 1.70915 0.854575 0.519328i \(-0.173818\pi\)
0.854575 + 0.519328i \(0.173818\pi\)
\(54\) −2.53066e7 −0.405006
\(55\) −1.81200e7 −0.267009
\(56\) 1.90792e7 0.259247
\(57\) −5.14657e7 −0.645775
\(58\) −5.95675e7 −0.691168
\(59\) 1.61629e7 0.173654 0.0868269 0.996223i \(-0.472327\pi\)
0.0868269 + 0.996223i \(0.472327\pi\)
\(60\) −2.78400e7 −0.277325
\(61\) −4.39282e7 −0.406218 −0.203109 0.979156i \(-0.565105\pi\)
−0.203109 + 0.979156i \(0.565105\pi\)
\(62\) 3.73724e7 0.321209
\(63\) 4.93422e7 0.394626
\(64\) 1.67772e7 0.125000
\(65\) 1.02779e8 0.714156
\(66\) 8.07137e7 0.523600
\(67\) −8.15574e7 −0.494455 −0.247228 0.968957i \(-0.579520\pi\)
−0.247228 + 0.968957i \(0.579520\pi\)
\(68\) −1.52274e8 −0.863648
\(69\) 4.42749e8 2.35145
\(70\) −4.65800e7 −0.231877
\(71\) 1.61308e8 0.753343 0.376671 0.926347i \(-0.377069\pi\)
0.376671 + 0.926347i \(0.377069\pi\)
\(72\) 4.33889e7 0.190275
\(73\) −2.47148e8 −1.01860 −0.509301 0.860589i \(-0.670096\pi\)
−0.509301 + 0.860589i \(0.670096\pi\)
\(74\) 1.73447e8 0.672393
\(75\) 6.79688e7 0.248047
\(76\) −7.57197e7 −0.260344
\(77\) 1.35045e8 0.437794
\(78\) −4.57818e8 −1.40045
\(79\) −5.83346e8 −1.68502 −0.842508 0.538684i \(-0.818922\pi\)
−0.842508 + 0.538684i \(0.818922\pi\)
\(80\) −4.09600e7 −0.111803
\(81\) −4.83711e8 −1.24854
\(82\) 3.45502e8 0.843894
\(83\) −1.45718e7 −0.0337024 −0.0168512 0.999858i \(-0.505364\pi\)
−0.0168512 + 0.999858i \(0.505364\pi\)
\(84\) 2.07486e8 0.454707
\(85\) 3.71764e8 0.772470
\(86\) 1.73317e8 0.341663
\(87\) −6.47797e8 −1.21228
\(88\) 1.18751e8 0.211089
\(89\) 4.70134e8 0.794267 0.397133 0.917761i \(-0.370005\pi\)
0.397133 + 0.917761i \(0.370005\pi\)
\(90\) −1.05930e8 −0.170188
\(91\) −7.65989e8 −1.17095
\(92\) 6.51401e8 0.947988
\(93\) 4.06424e8 0.563386
\(94\) 8.27542e7 0.109324
\(95\) 1.84862e8 0.232859
\(96\) 1.82452e8 0.219244
\(97\) −1.17838e8 −0.135149 −0.0675747 0.997714i \(-0.521526\pi\)
−0.0675747 + 0.997714i \(0.521526\pi\)
\(98\) −2.98506e8 −0.326916
\(99\) 3.07112e8 0.321321
\(100\) 1.00000e8 0.100000
\(101\) −8.60927e7 −0.0823228 −0.0411614 0.999153i \(-0.513106\pi\)
−0.0411614 + 0.999153i \(0.513106\pi\)
\(102\) −1.65598e9 −1.51480
\(103\) 1.92872e9 1.68850 0.844252 0.535947i \(-0.180045\pi\)
0.844252 + 0.535947i \(0.180045\pi\)
\(104\) −6.73571e8 −0.564590
\(105\) −5.06557e8 −0.406703
\(106\) 1.57087e9 1.20855
\(107\) 1.39685e9 1.03020 0.515100 0.857130i \(-0.327755\pi\)
0.515100 + 0.857130i \(0.327755\pi\)
\(108\) −4.04905e8 −0.286382
\(109\) −6.04327e8 −0.410065 −0.205033 0.978755i \(-0.565730\pi\)
−0.205033 + 0.978755i \(0.565730\pi\)
\(110\) −2.89920e8 −0.188804
\(111\) 1.88623e9 1.17935
\(112\) 3.05267e8 0.183315
\(113\) −1.68580e9 −0.972643 −0.486322 0.873780i \(-0.661662\pi\)
−0.486322 + 0.873780i \(0.661662\pi\)
\(114\) −8.23452e8 −0.456632
\(115\) −1.59033e9 −0.847907
\(116\) −9.53080e8 −0.488729
\(117\) −1.74198e9 −0.859421
\(118\) 2.58606e8 0.122792
\(119\) −2.77068e9 −1.26656
\(120\) −4.45440e8 −0.196098
\(121\) −1.51741e9 −0.643531
\(122\) −7.02851e8 −0.287239
\(123\) 3.75733e9 1.48015
\(124\) 5.97958e8 0.227129
\(125\) −2.44141e8 −0.0894427
\(126\) 7.89475e8 0.279043
\(127\) 3.70716e9 1.26452 0.632258 0.774758i \(-0.282128\pi\)
0.632258 + 0.774758i \(0.282128\pi\)
\(128\) 2.68435e8 0.0883883
\(129\) 1.88482e9 0.599261
\(130\) 1.64446e9 0.504985
\(131\) 2.54080e9 0.753789 0.376895 0.926256i \(-0.376992\pi\)
0.376895 + 0.926256i \(0.376992\pi\)
\(132\) 1.29142e9 0.370241
\(133\) −1.37774e9 −0.381800
\(134\) −1.30492e9 −0.349633
\(135\) 9.88538e8 0.256148
\(136\) −2.43639e9 −0.610692
\(137\) −1.15390e9 −0.279849 −0.139925 0.990162i \(-0.544686\pi\)
−0.139925 + 0.990162i \(0.544686\pi\)
\(138\) 7.08398e9 1.66273
\(139\) −5.62721e9 −1.27858 −0.639288 0.768968i \(-0.720771\pi\)
−0.639288 + 0.768968i \(0.720771\pi\)
\(140\) −7.45280e8 −0.163962
\(141\) 8.99952e8 0.191749
\(142\) 2.58092e9 0.532694
\(143\) −4.76762e9 −0.953431
\(144\) 6.94223e8 0.134545
\(145\) 2.32686e9 0.437133
\(146\) −3.95437e9 −0.720260
\(147\) −3.24626e9 −0.573396
\(148\) 2.77515e9 0.475454
\(149\) −2.13688e9 −0.355174 −0.177587 0.984105i \(-0.556829\pi\)
−0.177587 + 0.984105i \(0.556829\pi\)
\(150\) 1.08750e9 0.175396
\(151\) −9.67515e6 −0.00151447 −0.000757236 1.00000i \(-0.500241\pi\)
−0.000757236 1.00000i \(0.500241\pi\)
\(152\) −1.21151e9 −0.184091
\(153\) −6.30095e9 −0.929597
\(154\) 2.16072e9 0.309567
\(155\) −1.45986e9 −0.203150
\(156\) −7.32508e9 −0.990266
\(157\) −6.88488e8 −0.0904373 −0.0452187 0.998977i \(-0.514398\pi\)
−0.0452187 + 0.998977i \(0.514398\pi\)
\(158\) −9.33353e9 −1.19149
\(159\) 1.70833e10 2.11975
\(160\) −6.55360e8 −0.0790569
\(161\) 1.18524e10 1.39024
\(162\) −7.73937e9 −0.882853
\(163\) −1.43082e10 −1.58759 −0.793797 0.608182i \(-0.791899\pi\)
−0.793797 + 0.608182i \(0.791899\pi\)
\(164\) 5.52803e9 0.596723
\(165\) −3.15288e9 −0.331154
\(166\) −2.33149e8 −0.0238312
\(167\) 9.98735e9 0.993633 0.496817 0.867856i \(-0.334502\pi\)
0.496817 + 0.867856i \(0.334502\pi\)
\(168\) 3.31978e9 0.321527
\(169\) 1.64380e10 1.55010
\(170\) 5.94822e9 0.546219
\(171\) −3.13320e9 −0.280224
\(172\) 2.77307e9 0.241592
\(173\) 3.51396e9 0.298256 0.149128 0.988818i \(-0.452353\pi\)
0.149128 + 0.988818i \(0.452353\pi\)
\(174\) −1.03647e10 −0.857210
\(175\) 1.81953e9 0.146652
\(176\) 1.90002e9 0.149263
\(177\) 2.81234e9 0.215371
\(178\) 7.52214e9 0.561631
\(179\) 1.19502e9 0.0870038 0.0435019 0.999053i \(-0.486149\pi\)
0.0435019 + 0.999053i \(0.486149\pi\)
\(180\) −1.69488e9 −0.120341
\(181\) −9.12053e9 −0.631635 −0.315818 0.948820i \(-0.602279\pi\)
−0.315818 + 0.948820i \(0.602279\pi\)
\(182\) −1.22558e10 −0.827983
\(183\) −7.64350e9 −0.503805
\(184\) 1.04224e10 0.670329
\(185\) −6.77526e9 −0.425259
\(186\) 6.50279e9 0.398374
\(187\) −1.72451e10 −1.03128
\(188\) 1.32407e9 0.0773036
\(189\) −7.36737e9 −0.419986
\(190\) 2.95780e9 0.164656
\(191\) −9.37431e9 −0.509670 −0.254835 0.966985i \(-0.582021\pi\)
−0.254835 + 0.966985i \(0.582021\pi\)
\(192\) 2.91924e9 0.155029
\(193\) 2.40000e10 1.24510 0.622550 0.782580i \(-0.286097\pi\)
0.622550 + 0.782580i \(0.286097\pi\)
\(194\) −1.88542e9 −0.0955651
\(195\) 1.78835e10 0.885721
\(196\) −4.77610e9 −0.231165
\(197\) −5.56124e8 −0.0263071 −0.0131536 0.999913i \(-0.504187\pi\)
−0.0131536 + 0.999913i \(0.504187\pi\)
\(198\) 4.91380e9 0.227208
\(199\) −2.51255e10 −1.13573 −0.567866 0.823121i \(-0.692231\pi\)
−0.567866 + 0.823121i \(0.692231\pi\)
\(200\) 1.60000e9 0.0707107
\(201\) −1.41910e10 −0.613240
\(202\) −1.37748e9 −0.0582110
\(203\) −1.73416e10 −0.716732
\(204\) −2.64958e10 −1.07113
\(205\) −1.34962e10 −0.533726
\(206\) 3.08595e10 1.19395
\(207\) 2.69542e10 1.02038
\(208\) −1.07771e10 −0.399225
\(209\) −8.57525e9 −0.310877
\(210\) −8.10492e9 −0.287582
\(211\) −1.63915e10 −0.569309 −0.284654 0.958630i \(-0.591879\pi\)
−0.284654 + 0.958630i \(0.591879\pi\)
\(212\) 2.51340e10 0.854575
\(213\) 2.80675e10 0.934321
\(214\) 2.23495e10 0.728461
\(215\) −6.77018e9 −0.216086
\(216\) −6.47848e9 −0.202503
\(217\) 1.08800e10 0.333090
\(218\) −9.66924e9 −0.289960
\(219\) −4.30037e10 −1.26330
\(220\) −4.63872e9 −0.133505
\(221\) 9.78161e10 2.75832
\(222\) 3.01797e10 0.833925
\(223\) 4.65257e10 1.25986 0.629928 0.776654i \(-0.283084\pi\)
0.629928 + 0.776654i \(0.283084\pi\)
\(224\) 4.88427e9 0.129623
\(225\) 4.13789e9 0.107636
\(226\) −2.69728e10 −0.687763
\(227\) −1.91415e9 −0.0478475 −0.0239237 0.999714i \(-0.507616\pi\)
−0.0239237 + 0.999714i \(0.507616\pi\)
\(228\) −1.31752e10 −0.322887
\(229\) −1.45825e10 −0.350406 −0.175203 0.984532i \(-0.556058\pi\)
−0.175203 + 0.984532i \(0.556058\pi\)
\(230\) −2.54453e10 −0.599560
\(231\) 2.34978e10 0.542966
\(232\) −1.52493e10 −0.345584
\(233\) 4.12790e10 0.917545 0.458773 0.888554i \(-0.348289\pi\)
0.458773 + 0.888554i \(0.348289\pi\)
\(234\) −2.78716e10 −0.607702
\(235\) −3.23259e9 −0.0691424
\(236\) 4.13769e9 0.0868269
\(237\) −1.01502e11 −2.08981
\(238\) −4.43309e10 −0.895592
\(239\) 3.65502e10 0.724602 0.362301 0.932061i \(-0.381991\pi\)
0.362301 + 0.932061i \(0.381991\pi\)
\(240\) −7.12704e9 −0.138662
\(241\) −8.66070e10 −1.65377 −0.826887 0.562368i \(-0.809890\pi\)
−0.826887 + 0.562368i \(0.809890\pi\)
\(242\) −2.42786e10 −0.455045
\(243\) −5.30339e10 −0.975720
\(244\) −1.12456e10 −0.203109
\(245\) 1.16604e10 0.206760
\(246\) 6.01173e10 1.04663
\(247\) 4.86398e10 0.831488
\(248\) 9.56732e9 0.160605
\(249\) −2.53549e9 −0.0417989
\(250\) −3.90625e9 −0.0632456
\(251\) −7.20769e10 −1.14621 −0.573105 0.819482i \(-0.694261\pi\)
−0.573105 + 0.819482i \(0.694261\pi\)
\(252\) 1.26316e10 0.197313
\(253\) 7.37711e10 1.13199
\(254\) 5.93145e10 0.894148
\(255\) 6.46869e10 0.958044
\(256\) 4.29497e9 0.0625000
\(257\) −1.12729e11 −1.61190 −0.805948 0.591986i \(-0.798344\pi\)
−0.805948 + 0.591986i \(0.798344\pi\)
\(258\) 3.01571e10 0.423741
\(259\) 5.04947e10 0.697263
\(260\) 2.63114e10 0.357078
\(261\) −3.94374e10 −0.526049
\(262\) 4.06528e10 0.533009
\(263\) 5.55225e10 0.715596 0.357798 0.933799i \(-0.383528\pi\)
0.357798 + 0.933799i \(0.383528\pi\)
\(264\) 2.06627e10 0.261800
\(265\) −6.13623e10 −0.764355
\(266\) −2.20439e10 −0.269973
\(267\) 8.18033e10 0.985076
\(268\) −2.08787e10 −0.247228
\(269\) 2.79726e10 0.325722 0.162861 0.986649i \(-0.447928\pi\)
0.162861 + 0.986649i \(0.447928\pi\)
\(270\) 1.58166e10 0.181124
\(271\) 3.32884e10 0.374914 0.187457 0.982273i \(-0.439975\pi\)
0.187457 + 0.982273i \(0.439975\pi\)
\(272\) −3.89823e10 −0.431824
\(273\) −1.33282e11 −1.45225
\(274\) −1.84623e10 −0.197883
\(275\) 1.13250e10 0.119410
\(276\) 1.13344e11 1.17573
\(277\) 5.46240e10 0.557474 0.278737 0.960367i \(-0.410084\pi\)
0.278737 + 0.960367i \(0.410084\pi\)
\(278\) −9.00353e10 −0.904090
\(279\) 2.47428e10 0.244473
\(280\) −1.19245e10 −0.115939
\(281\) −8.37818e10 −0.801625 −0.400813 0.916160i \(-0.631272\pi\)
−0.400813 + 0.916160i \(0.631272\pi\)
\(282\) 1.43992e10 0.135587
\(283\) −8.36086e10 −0.774840 −0.387420 0.921903i \(-0.626634\pi\)
−0.387420 + 0.921903i \(0.626634\pi\)
\(284\) 4.12948e10 0.376671
\(285\) 3.21661e10 0.288799
\(286\) −7.62819e10 −0.674178
\(287\) 1.00584e11 0.875107
\(288\) 1.11076e10 0.0951377
\(289\) 2.35225e11 1.98355
\(290\) 3.72297e10 0.309100
\(291\) −2.05039e10 −0.167617
\(292\) −6.32699e10 −0.509301
\(293\) −2.28547e11 −1.81164 −0.905819 0.423666i \(-0.860743\pi\)
−0.905819 + 0.423666i \(0.860743\pi\)
\(294\) −5.19401e10 −0.405452
\(295\) −1.01018e10 −0.0776603
\(296\) 4.44024e10 0.336197
\(297\) −4.58555e10 −0.341969
\(298\) −3.41900e10 −0.251146
\(299\) −4.18438e11 −3.02769
\(300\) 1.74000e10 0.124023
\(301\) 5.04568e10 0.354300
\(302\) −1.54802e8 −0.00107089
\(303\) −1.49801e10 −0.102100
\(304\) −1.93842e10 −0.130172
\(305\) 2.74551e10 0.181666
\(306\) −1.00815e11 −0.657324
\(307\) 1.95064e11 1.25330 0.626648 0.779302i \(-0.284426\pi\)
0.626648 + 0.779302i \(0.284426\pi\)
\(308\) 3.45715e10 0.218897
\(309\) 3.35598e11 2.09414
\(310\) −2.33577e10 −0.143649
\(311\) 2.15637e11 1.30708 0.653540 0.756892i \(-0.273283\pi\)
0.653540 + 0.756892i \(0.273283\pi\)
\(312\) −1.17201e11 −0.700224
\(313\) 1.91755e11 1.12927 0.564635 0.825341i \(-0.309017\pi\)
0.564635 + 0.825341i \(0.309017\pi\)
\(314\) −1.10158e10 −0.0639488
\(315\) −3.08389e10 −0.176482
\(316\) −1.49337e11 −0.842508
\(317\) −3.38886e11 −1.88489 −0.942446 0.334358i \(-0.891481\pi\)
−0.942446 + 0.334358i \(0.891481\pi\)
\(318\) 2.73332e11 1.49889
\(319\) −1.07936e11 −0.583592
\(320\) −1.04858e10 −0.0559017
\(321\) 2.43051e11 1.27769
\(322\) 1.89639e11 0.983052
\(323\) 1.75936e11 0.899383
\(324\) −1.23830e11 −0.624271
\(325\) −6.42367e10 −0.319380
\(326\) −2.28931e11 −1.12260
\(327\) −1.05153e11 −0.508577
\(328\) 8.84485e10 0.421947
\(329\) 2.40918e10 0.113367
\(330\) −5.04461e10 −0.234161
\(331\) −1.78427e11 −0.817022 −0.408511 0.912753i \(-0.633952\pi\)
−0.408511 + 0.912753i \(0.633952\pi\)
\(332\) −3.73038e9 −0.0168512
\(333\) 1.14833e11 0.511760
\(334\) 1.59798e11 0.702605
\(335\) 5.09734e10 0.221127
\(336\) 5.31164e10 0.227354
\(337\) 2.64693e11 1.11791 0.558957 0.829196i \(-0.311202\pi\)
0.558957 + 0.829196i \(0.311202\pi\)
\(338\) 2.63008e11 1.09608
\(339\) −2.93330e11 −1.20631
\(340\) 9.51715e10 0.386235
\(341\) 6.77187e10 0.271215
\(342\) −5.01312e10 −0.198148
\(343\) −2.74870e11 −1.07227
\(344\) 4.43691e10 0.170831
\(345\) −2.76718e11 −1.05160
\(346\) 5.62233e10 0.210899
\(347\) −8.11912e10 −0.300626 −0.150313 0.988638i \(-0.548028\pi\)
−0.150313 + 0.988638i \(0.548028\pi\)
\(348\) −1.65836e11 −0.606139
\(349\) −2.26688e11 −0.817926 −0.408963 0.912551i \(-0.634110\pi\)
−0.408963 + 0.912551i \(0.634110\pi\)
\(350\) 2.91125e10 0.103699
\(351\) 2.60098e11 0.914649
\(352\) 3.04003e10 0.105545
\(353\) 3.28733e10 0.112683 0.0563413 0.998412i \(-0.482056\pi\)
0.0563413 + 0.998412i \(0.482056\pi\)
\(354\) 4.49974e10 0.152290
\(355\) −1.00817e11 −0.336905
\(356\) 1.20354e11 0.397133
\(357\) −4.82098e11 −1.57083
\(358\) 1.91204e10 0.0615210
\(359\) −3.12096e11 −0.991661 −0.495831 0.868419i \(-0.665136\pi\)
−0.495831 + 0.868419i \(0.665136\pi\)
\(360\) −2.71181e10 −0.0850938
\(361\) −2.35202e11 −0.728884
\(362\) −1.45928e11 −0.446634
\(363\) −2.64030e11 −0.798129
\(364\) −1.96093e11 −0.585473
\(365\) 1.54467e11 0.455532
\(366\) −1.22296e11 −0.356244
\(367\) 6.28209e11 1.80762 0.903810 0.427934i \(-0.140759\pi\)
0.903810 + 0.427934i \(0.140759\pi\)
\(368\) 1.66759e11 0.473994
\(369\) 2.28744e11 0.642289
\(370\) −1.08404e11 −0.300703
\(371\) 4.57321e11 1.25325
\(372\) 1.04045e11 0.281693
\(373\) −9.84770e10 −0.263418 −0.131709 0.991288i \(-0.542046\pi\)
−0.131709 + 0.991288i \(0.542046\pi\)
\(374\) −2.75921e11 −0.729227
\(375\) −4.24805e10 −0.110930
\(376\) 2.11851e10 0.0546619
\(377\) 6.12228e11 1.56091
\(378\) −1.17878e11 −0.296975
\(379\) 2.89024e11 0.719544 0.359772 0.933040i \(-0.382855\pi\)
0.359772 + 0.933040i \(0.382855\pi\)
\(380\) 4.73248e10 0.116429
\(381\) 6.45046e11 1.56830
\(382\) −1.49989e11 −0.360391
\(383\) 2.01350e11 0.478141 0.239071 0.971002i \(-0.423157\pi\)
0.239071 + 0.971002i \(0.423157\pi\)
\(384\) 4.67078e10 0.109622
\(385\) −8.44030e10 −0.195787
\(386\) 3.84001e11 0.880418
\(387\) 1.14746e11 0.260040
\(388\) −3.01666e10 −0.0675747
\(389\) 3.93769e11 0.871904 0.435952 0.899970i \(-0.356412\pi\)
0.435952 + 0.899970i \(0.356412\pi\)
\(390\) 2.86136e11 0.626299
\(391\) −1.51354e12 −3.27491
\(392\) −7.64176e10 −0.163458
\(393\) 4.42099e11 0.934875
\(394\) −8.89798e9 −0.0186019
\(395\) 3.64591e11 0.753562
\(396\) 7.86207e10 0.160660
\(397\) −5.15625e11 −1.04178 −0.520891 0.853623i \(-0.674400\pi\)
−0.520891 + 0.853623i \(0.674400\pi\)
\(398\) −4.02008e11 −0.803084
\(399\) −2.39727e11 −0.473521
\(400\) 2.56000e10 0.0500000
\(401\) −8.43309e11 −1.62869 −0.814343 0.580384i \(-0.802902\pi\)
−0.814343 + 0.580384i \(0.802902\pi\)
\(402\) −2.27056e11 −0.433626
\(403\) −3.84108e11 −0.725406
\(404\) −2.20397e10 −0.0411614
\(405\) 3.02319e11 0.558365
\(406\) −2.77466e11 −0.506806
\(407\) 3.14285e11 0.567740
\(408\) −4.23932e11 −0.757400
\(409\) 5.64549e11 0.997577 0.498789 0.866724i \(-0.333778\pi\)
0.498789 + 0.866724i \(0.333778\pi\)
\(410\) −2.15939e11 −0.377401
\(411\) −2.00778e11 −0.347078
\(412\) 4.93753e11 0.844252
\(413\) 7.52866e10 0.127333
\(414\) 4.31268e11 0.721516
\(415\) 9.10737e9 0.0150722
\(416\) −1.72434e11 −0.282295
\(417\) −9.79134e11 −1.58573
\(418\) −1.37204e11 −0.219823
\(419\) 6.92475e11 1.09759 0.548796 0.835956i \(-0.315086\pi\)
0.548796 + 0.835956i \(0.315086\pi\)
\(420\) −1.29679e11 −0.203351
\(421\) 4.08125e11 0.633176 0.316588 0.948563i \(-0.397463\pi\)
0.316588 + 0.948563i \(0.397463\pi\)
\(422\) −2.62264e11 −0.402562
\(423\) 5.47885e10 0.0832065
\(424\) 4.02144e11 0.604276
\(425\) −2.32352e11 −0.345459
\(426\) 4.49081e11 0.660665
\(427\) −2.04617e11 −0.297863
\(428\) 3.57592e11 0.515100
\(429\) −8.29566e11 −1.18248
\(430\) −1.08323e11 −0.152796
\(431\) 4.41055e11 0.615665 0.307833 0.951441i \(-0.400396\pi\)
0.307833 + 0.951441i \(0.400396\pi\)
\(432\) −1.03656e11 −0.143191
\(433\) −7.15390e10 −0.0978019 −0.0489009 0.998804i \(-0.515572\pi\)
−0.0489009 + 0.998804i \(0.515572\pi\)
\(434\) 1.74080e11 0.235530
\(435\) 4.04873e11 0.542147
\(436\) −1.54708e11 −0.205033
\(437\) −7.52622e11 −0.987212
\(438\) −6.88060e11 −0.893291
\(439\) 4.74967e11 0.610342 0.305171 0.952298i \(-0.401286\pi\)
0.305171 + 0.952298i \(0.401286\pi\)
\(440\) −7.42195e10 −0.0944020
\(441\) −1.97630e11 −0.248816
\(442\) 1.56506e12 1.95043
\(443\) −1.97072e11 −0.243113 −0.121556 0.992585i \(-0.538789\pi\)
−0.121556 + 0.992585i \(0.538789\pi\)
\(444\) 4.82876e11 0.589674
\(445\) −2.93834e11 −0.355207
\(446\) 7.44411e11 0.890853
\(447\) −3.71817e11 −0.440499
\(448\) 7.81483e10 0.0916576
\(449\) −6.20156e11 −0.720100 −0.360050 0.932933i \(-0.617240\pi\)
−0.360050 + 0.932933i \(0.617240\pi\)
\(450\) 6.62063e10 0.0761102
\(451\) 6.26049e11 0.712548
\(452\) −4.31565e11 −0.486322
\(453\) −1.68348e9 −0.00187830
\(454\) −3.06264e10 −0.0338333
\(455\) 4.78743e11 0.523663
\(456\) −2.10804e11 −0.228316
\(457\) 7.88006e11 0.845097 0.422549 0.906340i \(-0.361136\pi\)
0.422549 + 0.906340i \(0.361136\pi\)
\(458\) −2.33319e11 −0.247774
\(459\) 9.40806e11 0.989334
\(460\) −4.07125e11 −0.423953
\(461\) 1.53496e10 0.0158286 0.00791429 0.999969i \(-0.497481\pi\)
0.00791429 + 0.999969i \(0.497481\pi\)
\(462\) 3.75965e11 0.383935
\(463\) 1.94494e11 0.196694 0.0983471 0.995152i \(-0.468644\pi\)
0.0983471 + 0.995152i \(0.468644\pi\)
\(464\) −2.43989e11 −0.244365
\(465\) −2.54015e11 −0.251954
\(466\) 6.60464e11 0.648802
\(467\) 1.06503e11 0.103618 0.0518089 0.998657i \(-0.483501\pi\)
0.0518089 + 0.998657i \(0.483501\pi\)
\(468\) −4.45946e11 −0.429710
\(469\) −3.79894e11 −0.362564
\(470\) −5.17214e10 −0.0488911
\(471\) −1.19797e11 −0.112163
\(472\) 6.62031e10 0.0613959
\(473\) 3.14050e11 0.288485
\(474\) −1.62403e12 −1.47772
\(475\) −1.15539e11 −0.104138
\(476\) −7.09294e11 −0.633279
\(477\) 1.04002e12 0.919831
\(478\) 5.84803e11 0.512371
\(479\) −8.31146e11 −0.721386 −0.360693 0.932685i \(-0.617460\pi\)
−0.360693 + 0.932685i \(0.617460\pi\)
\(480\) −1.14033e11 −0.0980491
\(481\) −1.78266e12 −1.51851
\(482\) −1.38571e12 −1.16939
\(483\) 2.06232e12 1.72423
\(484\) −3.88457e11 −0.321765
\(485\) 7.36490e10 0.0604407
\(486\) −8.48542e11 −0.689938
\(487\) −1.38566e12 −1.11629 −0.558146 0.829743i \(-0.688487\pi\)
−0.558146 + 0.829743i \(0.688487\pi\)
\(488\) −1.79930e11 −0.143620
\(489\) −2.48962e12 −1.96899
\(490\) 1.86566e11 0.146201
\(491\) 3.92804e11 0.305007 0.152503 0.988303i \(-0.451266\pi\)
0.152503 + 0.988303i \(0.451266\pi\)
\(492\) 9.61877e11 0.740076
\(493\) 2.21450e12 1.68836
\(494\) 7.78237e11 0.587951
\(495\) −1.91945e11 −0.143699
\(496\) 1.53077e11 0.113565
\(497\) 7.51371e11 0.552397
\(498\) −4.05679e10 −0.0295563
\(499\) −7.52029e11 −0.542978 −0.271489 0.962442i \(-0.587516\pi\)
−0.271489 + 0.962442i \(0.587516\pi\)
\(500\) −6.25000e10 −0.0447214
\(501\) 1.73780e12 1.23234
\(502\) −1.15323e12 −0.810493
\(503\) −1.83429e12 −1.27765 −0.638826 0.769351i \(-0.720580\pi\)
−0.638826 + 0.769351i \(0.720580\pi\)
\(504\) 2.02106e11 0.139521
\(505\) 5.38079e10 0.0368159
\(506\) 1.18034e12 0.800441
\(507\) 2.86021e12 1.92248
\(508\) 9.49033e11 0.632258
\(509\) −6.19864e11 −0.409323 −0.204662 0.978833i \(-0.565609\pi\)
−0.204662 + 0.978833i \(0.565609\pi\)
\(510\) 1.03499e12 0.677439
\(511\) −1.15122e12 −0.746900
\(512\) 6.87195e10 0.0441942
\(513\) 4.67823e11 0.298232
\(514\) −1.80367e12 −1.13978
\(515\) −1.20545e12 −0.755122
\(516\) 4.82514e11 0.299630
\(517\) 1.49951e11 0.0923083
\(518\) 8.07915e11 0.493039
\(519\) 6.11428e11 0.369907
\(520\) 4.20982e11 0.252492
\(521\) 5.25683e11 0.312575 0.156287 0.987712i \(-0.450047\pi\)
0.156287 + 0.987712i \(0.450047\pi\)
\(522\) −6.30999e11 −0.371973
\(523\) 1.68426e11 0.0984353 0.0492177 0.998788i \(-0.484327\pi\)
0.0492177 + 0.998788i \(0.484327\pi\)
\(524\) 6.50445e11 0.376895
\(525\) 3.16598e11 0.181883
\(526\) 8.88360e11 0.506003
\(527\) −1.38937e12 −0.784639
\(528\) 3.30603e11 0.185121
\(529\) 4.67350e12 2.59473
\(530\) −9.81797e11 −0.540481
\(531\) 1.71213e11 0.0934570
\(532\) −3.52702e11 −0.190900
\(533\) −3.55102e12 −1.90582
\(534\) 1.30885e12 0.696554
\(535\) −8.73028e11 −0.460719
\(536\) −3.34059e11 −0.174816
\(537\) 2.07934e11 0.107905
\(538\) 4.47561e11 0.230320
\(539\) −5.40893e11 −0.276034
\(540\) 2.53066e11 0.128074
\(541\) 2.20612e12 1.10724 0.553620 0.832769i \(-0.313246\pi\)
0.553620 + 0.832769i \(0.313246\pi\)
\(542\) 5.32615e11 0.265104
\(543\) −1.58697e12 −0.783375
\(544\) −6.23716e11 −0.305346
\(545\) 3.77705e11 0.183387
\(546\) −2.13251e12 −1.02689
\(547\) −3.51639e12 −1.67940 −0.839701 0.543049i \(-0.817270\pi\)
−0.839701 + 0.543049i \(0.817270\pi\)
\(548\) −2.95397e11 −0.139925
\(549\) −4.65331e11 −0.218618
\(550\) 1.81200e11 0.0844357
\(551\) 1.10118e12 0.508951
\(552\) 1.81350e12 0.831365
\(553\) −2.71722e12 −1.23556
\(554\) 8.73985e11 0.394194
\(555\) −1.17890e12 −0.527420
\(556\) −1.44057e12 −0.639288
\(557\) 6.29996e11 0.277325 0.138663 0.990340i \(-0.455720\pi\)
0.138663 + 0.990340i \(0.455720\pi\)
\(558\) 3.95885e11 0.172868
\(559\) −1.78133e12 −0.771597
\(560\) −1.90792e11 −0.0819810
\(561\) −3.00064e12 −1.27903
\(562\) −1.34051e12 −0.566835
\(563\) 4.02619e11 0.168891 0.0844455 0.996428i \(-0.473088\pi\)
0.0844455 + 0.996428i \(0.473088\pi\)
\(564\) 2.30388e11 0.0958746
\(565\) 1.05363e12 0.434979
\(566\) −1.33774e12 −0.547894
\(567\) −2.25313e12 −0.915507
\(568\) 6.60716e11 0.266347
\(569\) 2.55482e12 1.02177 0.510887 0.859648i \(-0.329317\pi\)
0.510887 + 0.859648i \(0.329317\pi\)
\(570\) 5.14657e11 0.204212
\(571\) 1.16243e12 0.457620 0.228810 0.973471i \(-0.426516\pi\)
0.228810 + 0.973471i \(0.426516\pi\)
\(572\) −1.22051e12 −0.476716
\(573\) −1.63113e12 −0.632110
\(574\) 1.60935e12 0.618794
\(575\) 9.93959e11 0.379195
\(576\) 1.77721e11 0.0672725
\(577\) 4.66332e12 1.75148 0.875738 0.482787i \(-0.160375\pi\)
0.875738 + 0.482787i \(0.160375\pi\)
\(578\) 3.76361e12 1.40258
\(579\) 4.17601e12 1.54421
\(580\) 5.95675e11 0.218566
\(581\) −6.78754e10 −0.0247127
\(582\) −3.28062e11 −0.118523
\(583\) 2.84643e12 1.02045
\(584\) −1.01232e12 −0.360130
\(585\) 1.08874e12 0.384345
\(586\) −3.65675e12 −1.28102
\(587\) 3.88422e12 1.35030 0.675152 0.737678i \(-0.264078\pi\)
0.675152 + 0.737678i \(0.264078\pi\)
\(588\) −8.31042e11 −0.286698
\(589\) −6.90875e11 −0.236527
\(590\) −1.61629e11 −0.0549141
\(591\) −9.67655e10 −0.0326270
\(592\) 7.10438e11 0.237727
\(593\) −2.01341e12 −0.668631 −0.334315 0.942461i \(-0.608505\pi\)
−0.334315 + 0.942461i \(0.608505\pi\)
\(594\) −7.33688e11 −0.241809
\(595\) 1.73168e12 0.566422
\(596\) −5.47041e11 −0.177587
\(597\) −4.37184e12 −1.40857
\(598\) −6.69502e12 −2.14090
\(599\) 2.10578e12 0.668333 0.334166 0.942514i \(-0.391545\pi\)
0.334166 + 0.942514i \(0.391545\pi\)
\(600\) 2.78400e11 0.0876978
\(601\) 4.74058e12 1.48216 0.741082 0.671415i \(-0.234313\pi\)
0.741082 + 0.671415i \(0.234313\pi\)
\(602\) 8.07309e11 0.250528
\(603\) −8.63938e11 −0.266106
\(604\) −2.47684e9 −0.000757236 0
\(605\) 9.48382e11 0.287796
\(606\) −2.39682e11 −0.0721953
\(607\) −4.01476e12 −1.20036 −0.600179 0.799866i \(-0.704904\pi\)
−0.600179 + 0.799866i \(0.704904\pi\)
\(608\) −3.10148e11 −0.0920455
\(609\) −3.01744e12 −0.888915
\(610\) 4.39282e11 0.128457
\(611\) −8.50537e11 −0.246893
\(612\) −1.61304e12 −0.464798
\(613\) 1.30399e12 0.372993 0.186496 0.982456i \(-0.440287\pi\)
0.186496 + 0.982456i \(0.440287\pi\)
\(614\) 3.12102e12 0.886214
\(615\) −2.34833e12 −0.661944
\(616\) 5.53143e11 0.154783
\(617\) 5.10164e12 1.41719 0.708593 0.705618i \(-0.249330\pi\)
0.708593 + 0.705618i \(0.249330\pi\)
\(618\) 5.36956e12 1.48078
\(619\) −4.50630e11 −0.123371 −0.0616854 0.998096i \(-0.519648\pi\)
−0.0616854 + 0.998096i \(0.519648\pi\)
\(620\) −3.73724e11 −0.101575
\(621\) −4.02459e12 −1.08595
\(622\) 3.45020e12 0.924246
\(623\) 2.18988e12 0.582404
\(624\) −1.87522e12 −0.495133
\(625\) 1.52588e11 0.0400000
\(626\) 3.06808e12 0.798514
\(627\) −1.49209e12 −0.385560
\(628\) −1.76253e11 −0.0452187
\(629\) −6.44812e12 −1.64250
\(630\) −4.93422e11 −0.124792
\(631\) 7.97105e10 0.0200163 0.0100081 0.999950i \(-0.496814\pi\)
0.0100081 + 0.999950i \(0.496814\pi\)
\(632\) −2.38938e12 −0.595743
\(633\) −2.85212e12 −0.706076
\(634\) −5.42217e12 −1.33282
\(635\) −2.31697e12 −0.565509
\(636\) 4.37332e12 1.05987
\(637\) 3.06801e12 0.738294
\(638\) −1.72698e12 −0.412662
\(639\) 1.70873e12 0.405434
\(640\) −1.67772e11 −0.0395285
\(641\) −3.12648e12 −0.731468 −0.365734 0.930719i \(-0.619182\pi\)
−0.365734 + 0.930719i \(0.619182\pi\)
\(642\) 3.88882e12 0.903462
\(643\) −5.94982e12 −1.37263 −0.686316 0.727303i \(-0.740773\pi\)
−0.686316 + 0.727303i \(0.740773\pi\)
\(644\) 3.03422e12 0.695122
\(645\) −1.17801e12 −0.267998
\(646\) 2.81498e12 0.635960
\(647\) 3.56187e12 0.799115 0.399557 0.916708i \(-0.369164\pi\)
0.399557 + 0.916708i \(0.369164\pi\)
\(648\) −1.98128e12 −0.441426
\(649\) 4.68594e11 0.103680
\(650\) −1.02779e12 −0.225836
\(651\) 1.89312e12 0.413109
\(652\) −3.66289e12 −0.793797
\(653\) −1.37883e11 −0.0296757 −0.0148379 0.999890i \(-0.504723\pi\)
−0.0148379 + 0.999890i \(0.504723\pi\)
\(654\) −1.68245e12 −0.359618
\(655\) −1.58800e12 −0.337105
\(656\) 1.41518e12 0.298362
\(657\) −2.61804e12 −0.548191
\(658\) 3.85469e11 0.0801628
\(659\) −4.16654e12 −0.860581 −0.430290 0.902691i \(-0.641589\pi\)
−0.430290 + 0.902691i \(0.641589\pi\)
\(660\) −8.07137e11 −0.165577
\(661\) −3.29083e12 −0.670500 −0.335250 0.942129i \(-0.608821\pi\)
−0.335250 + 0.942129i \(0.608821\pi\)
\(662\) −2.85483e12 −0.577722
\(663\) 1.70200e13 3.42097
\(664\) −5.96860e10 −0.0119156
\(665\) 8.61090e11 0.170746
\(666\) 1.83732e12 0.361869
\(667\) −9.47322e12 −1.85324
\(668\) 2.55676e12 0.496817
\(669\) 8.09547e12 1.56252
\(670\) 8.15574e11 0.156360
\(671\) −1.27357e12 −0.242532
\(672\) 8.49862e11 0.160763
\(673\) 5.74732e12 1.07993 0.539967 0.841686i \(-0.318437\pi\)
0.539967 + 0.841686i \(0.318437\pi\)
\(674\) 4.23510e12 0.790485
\(675\) −6.17836e11 −0.114553
\(676\) 4.20812e12 0.775048
\(677\) 9.31245e12 1.70379 0.851893 0.523716i \(-0.175455\pi\)
0.851893 + 0.523716i \(0.175455\pi\)
\(678\) −4.69327e12 −0.852987
\(679\) −5.48892e11 −0.0990998
\(680\) 1.52274e12 0.273110
\(681\) −3.33062e11 −0.0593421
\(682\) 1.08350e12 0.191778
\(683\) 9.19967e12 1.61763 0.808815 0.588063i \(-0.200109\pi\)
0.808815 + 0.588063i \(0.200109\pi\)
\(684\) −8.02099e11 −0.140112
\(685\) 7.21185e11 0.125152
\(686\) −4.39792e12 −0.758208
\(687\) −2.53735e12 −0.434585
\(688\) 7.09905e11 0.120796
\(689\) −1.61453e13 −2.72934
\(690\) −4.42749e12 −0.743595
\(691\) 1.03125e13 1.72074 0.860369 0.509672i \(-0.170233\pi\)
0.860369 + 0.509672i \(0.170233\pi\)
\(692\) 8.99573e11 0.149128
\(693\) 1.43053e12 0.235612
\(694\) −1.29906e12 −0.212574
\(695\) 3.51701e12 0.571796
\(696\) −2.65338e12 −0.428605
\(697\) −1.28445e13 −2.06144
\(698\) −3.62701e12 −0.578361
\(699\) 7.18255e12 1.13797
\(700\) 4.65800e11 0.0733261
\(701\) −9.52566e12 −1.48992 −0.744961 0.667108i \(-0.767532\pi\)
−0.744961 + 0.667108i \(0.767532\pi\)
\(702\) 4.16156e12 0.646754
\(703\) −3.20638e12 −0.495126
\(704\) 4.86405e11 0.0746313
\(705\) −5.62470e11 −0.0857528
\(706\) 5.25973e11 0.0796787
\(707\) −4.01020e11 −0.0603641
\(708\) 7.19958e11 0.107686
\(709\) −8.38537e12 −1.24628 −0.623138 0.782112i \(-0.714142\pi\)
−0.623138 + 0.782112i \(0.714142\pi\)
\(710\) −1.61308e12 −0.238228
\(711\) −6.17938e12 −0.906842
\(712\) 1.92567e12 0.280816
\(713\) 5.94345e12 0.861263
\(714\) −7.71358e12 −1.11074
\(715\) 2.97976e12 0.426387
\(716\) 3.05926e11 0.0435019
\(717\) 6.35974e12 0.898676
\(718\) −4.99354e12 −0.701211
\(719\) 6.94013e12 0.968473 0.484236 0.874937i \(-0.339097\pi\)
0.484236 + 0.874937i \(0.339097\pi\)
\(720\) −4.33889e11 −0.0601704
\(721\) 8.98398e12 1.23811
\(722\) −3.76323e12 −0.515399
\(723\) −1.50696e13 −2.05107
\(724\) −2.33485e12 −0.315818
\(725\) −1.45429e12 −0.195492
\(726\) −4.22447e12 −0.564362
\(727\) 7.74984e12 1.02893 0.514467 0.857510i \(-0.327990\pi\)
0.514467 + 0.857510i \(0.327990\pi\)
\(728\) −3.13749e12 −0.413992
\(729\) 2.92986e11 0.0384214
\(730\) 2.47148e12 0.322110
\(731\) −6.44329e12 −0.834602
\(732\) −1.95674e12 −0.251902
\(733\) 4.12555e12 0.527854 0.263927 0.964543i \(-0.414982\pi\)
0.263927 + 0.964543i \(0.414982\pi\)
\(734\) 1.00513e13 1.27818
\(735\) 2.02891e12 0.256431
\(736\) 2.66814e12 0.335164
\(737\) −2.36451e12 −0.295215
\(738\) 3.65990e12 0.454167
\(739\) 3.33931e12 0.411867 0.205933 0.978566i \(-0.433977\pi\)
0.205933 + 0.978566i \(0.433977\pi\)
\(740\) −1.73447e12 −0.212629
\(741\) 8.46333e12 1.03124
\(742\) 7.31713e12 0.886183
\(743\) −1.26471e13 −1.52245 −0.761224 0.648489i \(-0.775401\pi\)
−0.761224 + 0.648489i \(0.775401\pi\)
\(744\) 1.66471e12 0.199187
\(745\) 1.33555e12 0.158839
\(746\) −1.57563e12 −0.186265
\(747\) −1.54359e11 −0.0181380
\(748\) −4.41474e12 −0.515642
\(749\) 6.50651e12 0.755405
\(750\) −6.79687e11 −0.0784393
\(751\) 1.56552e12 0.179588 0.0897941 0.995960i \(-0.471379\pi\)
0.0897941 + 0.995960i \(0.471379\pi\)
\(752\) 3.38961e11 0.0386518
\(753\) −1.25414e13 −1.42157
\(754\) 9.79564e12 1.10373
\(755\) 6.04697e9 0.000677293 0
\(756\) −1.88605e12 −0.209993
\(757\) 9.54781e11 0.105675 0.0528375 0.998603i \(-0.483173\pi\)
0.0528375 + 0.998603i \(0.483173\pi\)
\(758\) 4.62438e12 0.508794
\(759\) 1.28362e13 1.40394
\(760\) 7.57197e11 0.0823280
\(761\) 5.27330e12 0.569969 0.284985 0.958532i \(-0.408012\pi\)
0.284985 + 0.958532i \(0.408012\pi\)
\(762\) 1.03207e13 1.10895
\(763\) −2.81496e12 −0.300685
\(764\) −2.39982e12 −0.254835
\(765\) 3.93809e12 0.415728
\(766\) 3.22159e12 0.338097
\(767\) −2.65792e12 −0.277308
\(768\) 7.47324e11 0.0775146
\(769\) −6.96923e12 −0.718648 −0.359324 0.933213i \(-0.616993\pi\)
−0.359324 + 0.933213i \(0.616993\pi\)
\(770\) −1.35045e12 −0.138442
\(771\) −1.96149e13 −1.99913
\(772\) 6.14401e12 0.622550
\(773\) −3.79383e12 −0.382182 −0.191091 0.981572i \(-0.561203\pi\)
−0.191091 + 0.981572i \(0.561203\pi\)
\(774\) 1.83594e12 0.183876
\(775\) 9.12411e11 0.0908517
\(776\) −4.82666e11 −0.0477825
\(777\) 8.78607e12 0.864769
\(778\) 6.30031e12 0.616530
\(779\) −6.38703e12 −0.621413
\(780\) 4.57818e12 0.442860
\(781\) 4.67663e12 0.449784
\(782\) −2.42167e13 −2.31571
\(783\) 5.88847e12 0.559854
\(784\) −1.22268e12 −0.115582
\(785\) 4.30305e11 0.0404448
\(786\) 7.07359e12 0.661056
\(787\) −2.05939e13 −1.91361 −0.956803 0.290736i \(-0.906100\pi\)
−0.956803 + 0.290736i \(0.906100\pi\)
\(788\) −1.42368e11 −0.0131536
\(789\) 9.66092e12 0.887507
\(790\) 5.83346e12 0.532849
\(791\) −7.85247e12 −0.713201
\(792\) 1.25793e12 0.113604
\(793\) 7.22381e12 0.648690
\(794\) −8.25000e12 −0.736651
\(795\) −1.06770e13 −0.947979
\(796\) −6.43213e12 −0.567866
\(797\) 1.45507e12 0.127738 0.0638690 0.997958i \(-0.479656\pi\)
0.0638690 + 0.997958i \(0.479656\pi\)
\(798\) −3.83564e12 −0.334830
\(799\) −3.07650e12 −0.267052
\(800\) 4.09600e11 0.0353553
\(801\) 4.98013e12 0.427459
\(802\) −1.34929e13 −1.15165
\(803\) −7.16531e12 −0.608156
\(804\) −3.63289e12 −0.306620
\(805\) −7.40777e12 −0.621736
\(806\) −6.14573e12 −0.512939
\(807\) 4.86722e12 0.403971
\(808\) −3.52636e11 −0.0291055
\(809\) 1.31749e13 1.08138 0.540690 0.841222i \(-0.318163\pi\)
0.540690 + 0.841222i \(0.318163\pi\)
\(810\) 4.83711e12 0.394824
\(811\) 9.97833e12 0.809961 0.404980 0.914325i \(-0.367278\pi\)
0.404980 + 0.914325i \(0.367278\pi\)
\(812\) −4.43945e12 −0.358366
\(813\) 5.79219e12 0.464981
\(814\) 5.02857e12 0.401453
\(815\) 8.94260e12 0.709994
\(816\) −6.78291e12 −0.535563
\(817\) −3.20398e12 −0.251588
\(818\) 9.03278e12 0.705394
\(819\) −8.11413e12 −0.630179
\(820\) −3.45502e12 −0.266863
\(821\) −1.80913e13 −1.38972 −0.694859 0.719146i \(-0.744533\pi\)
−0.694859 + 0.719146i \(0.744533\pi\)
\(822\) −3.21245e12 −0.245422
\(823\) 1.39801e12 0.106221 0.0531107 0.998589i \(-0.483086\pi\)
0.0531107 + 0.998589i \(0.483086\pi\)
\(824\) 7.90004e12 0.596976
\(825\) 1.97055e12 0.148096
\(826\) 1.20459e12 0.0900383
\(827\) 2.26201e13 1.68159 0.840794 0.541354i \(-0.182088\pi\)
0.840794 + 0.541354i \(0.182088\pi\)
\(828\) 6.90029e12 0.510189
\(829\) 1.72646e13 1.26959 0.634793 0.772682i \(-0.281085\pi\)
0.634793 + 0.772682i \(0.281085\pi\)
\(830\) 1.45718e11 0.0106576
\(831\) 9.50458e12 0.691398
\(832\) −2.75895e12 −0.199613
\(833\) 1.10974e13 0.798579
\(834\) −1.56662e13 −1.12128
\(835\) −6.24209e12 −0.444366
\(836\) −2.19526e12 −0.155439
\(837\) −3.69440e12 −0.260183
\(838\) 1.10796e13 0.776115
\(839\) −1.76607e12 −0.123049 −0.0615245 0.998106i \(-0.519596\pi\)
−0.0615245 + 0.998106i \(0.519596\pi\)
\(840\) −2.07486e12 −0.143791
\(841\) −6.46640e11 −0.0445739
\(842\) 6.53001e12 0.447723
\(843\) −1.45780e13 −0.994203
\(844\) −4.19622e12 −0.284654
\(845\) −1.02737e13 −0.693224
\(846\) 8.76615e11 0.0588359
\(847\) −7.06810e12 −0.471876
\(848\) 6.43430e12 0.427287
\(849\) −1.45479e13 −0.960983
\(850\) −3.71764e12 −0.244277
\(851\) 2.75838e13 1.80290
\(852\) 7.18529e12 0.467161
\(853\) −2.37844e13 −1.53823 −0.769116 0.639109i \(-0.779303\pi\)
−0.769116 + 0.639109i \(0.779303\pi\)
\(854\) −3.27388e12 −0.210621
\(855\) 1.95825e12 0.125320
\(856\) 5.72148e12 0.364230
\(857\) 7.61941e12 0.482511 0.241256 0.970462i \(-0.422441\pi\)
0.241256 + 0.970462i \(0.422441\pi\)
\(858\) −1.32730e13 −0.836138
\(859\) −1.27695e13 −0.800212 −0.400106 0.916469i \(-0.631027\pi\)
−0.400106 + 0.916469i \(0.631027\pi\)
\(860\) −1.73317e12 −0.108043
\(861\) 1.75017e13 1.08534
\(862\) 7.05687e12 0.435341
\(863\) 2.63038e13 1.61425 0.807124 0.590382i \(-0.201023\pi\)
0.807124 + 0.590382i \(0.201023\pi\)
\(864\) −1.65849e12 −0.101251
\(865\) −2.19622e12 −0.133384
\(866\) −1.14462e12 −0.0691564
\(867\) 4.09292e13 2.46007
\(868\) 2.78529e12 0.166545
\(869\) −1.69124e13 −1.00604
\(870\) 6.47797e12 0.383356
\(871\) 1.34118e13 0.789596
\(872\) −2.47532e12 −0.144980
\(873\) −1.24826e12 −0.0727347
\(874\) −1.20420e13 −0.698065
\(875\) −1.13721e12 −0.0655848
\(876\) −1.10090e13 −0.631652
\(877\) 4.76832e11 0.0272187 0.0136093 0.999907i \(-0.495668\pi\)
0.0136093 + 0.999907i \(0.495668\pi\)
\(878\) 7.59947e12 0.431577
\(879\) −3.97672e13 −2.24685
\(880\) −1.18751e12 −0.0667523
\(881\) 1.74839e13 0.977791 0.488896 0.872342i \(-0.337400\pi\)
0.488896 + 0.872342i \(0.337400\pi\)
\(882\) −3.16208e12 −0.175940
\(883\) −6.48321e12 −0.358894 −0.179447 0.983768i \(-0.557431\pi\)
−0.179447 + 0.983768i \(0.557431\pi\)
\(884\) 2.50409e13 1.37916
\(885\) −1.75771e12 −0.0963170
\(886\) −3.15315e12 −0.171907
\(887\) −9.92450e12 −0.538335 −0.269167 0.963093i \(-0.586748\pi\)
−0.269167 + 0.963093i \(0.586748\pi\)
\(888\) 7.72601e12 0.416962
\(889\) 1.72679e13 0.927220
\(890\) −4.70134e12 −0.251169
\(891\) −1.40237e13 −0.745443
\(892\) 1.19106e13 0.629928
\(893\) −1.52981e12 −0.0805021
\(894\) −5.94907e12 −0.311480
\(895\) −7.46891e11 −0.0389093
\(896\) 1.25037e12 0.0648117
\(897\) −7.28083e13 −3.75504
\(898\) −9.92250e12 −0.509187
\(899\) −8.69601e12 −0.444019
\(900\) 1.05930e12 0.0538180
\(901\) −5.83994e13 −2.95221
\(902\) 1.00168e13 0.503847
\(903\) 8.77949e12 0.439414
\(904\) −6.90505e12 −0.343881
\(905\) 5.70033e12 0.282476
\(906\) −2.69356e10 −0.00132816
\(907\) 1.40768e12 0.0690671 0.0345335 0.999404i \(-0.489005\pi\)
0.0345335 + 0.999404i \(0.489005\pi\)
\(908\) −4.90022e11 −0.0239237
\(909\) −9.11980e11 −0.0443045
\(910\) 7.65989e12 0.370285
\(911\) 6.91343e12 0.332553 0.166277 0.986079i \(-0.446826\pi\)
0.166277 + 0.986079i \(0.446826\pi\)
\(912\) −3.37286e12 −0.161444
\(913\) −4.22465e11 −0.0201221
\(914\) 1.26081e13 0.597574
\(915\) 4.77719e12 0.225308
\(916\) −3.73311e12 −0.175203
\(917\) 1.18351e13 0.552724
\(918\) 1.50529e13 0.699565
\(919\) −8.50189e12 −0.393184 −0.196592 0.980485i \(-0.562987\pi\)
−0.196592 + 0.980485i \(0.562987\pi\)
\(920\) −6.51401e12 −0.299780
\(921\) 3.39411e13 1.55438
\(922\) 2.45593e11 0.0111925
\(923\) −2.65264e13 −1.20301
\(924\) 6.01543e12 0.271483
\(925\) 4.23454e12 0.190181
\(926\) 3.11190e12 0.139084
\(927\) 2.04309e13 0.908719
\(928\) −3.90382e12 −0.172792
\(929\) 8.90639e12 0.392311 0.196156 0.980573i \(-0.437154\pi\)
0.196156 + 0.980573i \(0.437154\pi\)
\(930\) −4.06424e12 −0.178158
\(931\) 5.51826e12 0.240729
\(932\) 1.05674e13 0.458773
\(933\) 3.75209e13 1.62109
\(934\) 1.70404e12 0.0732689
\(935\) 1.07782e13 0.461204
\(936\) −7.13514e12 −0.303851
\(937\) 7.40064e12 0.313647 0.156824 0.987627i \(-0.449875\pi\)
0.156824 + 0.987627i \(0.449875\pi\)
\(938\) −6.07831e12 −0.256372
\(939\) 3.33654e13 1.40056
\(940\) −8.27542e11 −0.0345712
\(941\) −8.13867e12 −0.338377 −0.169188 0.985584i \(-0.554115\pi\)
−0.169188 + 0.985584i \(0.554115\pi\)
\(942\) −1.91675e12 −0.0793115
\(943\) 5.49463e13 2.26275
\(944\) 1.05925e12 0.0434134
\(945\) 4.60461e12 0.187823
\(946\) 5.02480e12 0.203990
\(947\) −1.16975e13 −0.472627 −0.236313 0.971677i \(-0.575939\pi\)
−0.236313 + 0.971677i \(0.575939\pi\)
\(948\) −2.59846e13 −1.04491
\(949\) 4.06425e13 1.62661
\(950\) −1.84863e12 −0.0736364
\(951\) −5.89661e13 −2.33771
\(952\) −1.13487e13 −0.447796
\(953\) −2.50410e13 −0.983406 −0.491703 0.870763i \(-0.663625\pi\)
−0.491703 + 0.870763i \(0.663625\pi\)
\(954\) 1.66403e13 0.650418
\(955\) 5.85894e12 0.227931
\(956\) 9.35686e12 0.362301
\(957\) −1.87809e13 −0.723791
\(958\) −1.32983e13 −0.510097
\(959\) −5.37485e12 −0.205202
\(960\) −1.82452e12 −0.0693312
\(961\) −2.09838e13 −0.793649
\(962\) −2.85226e13 −1.07375
\(963\) 1.47968e13 0.554433
\(964\) −2.21714e13 −0.826887
\(965\) −1.50000e13 −0.556825
\(966\) 3.29972e13 1.21921
\(967\) 2.68904e13 0.988960 0.494480 0.869189i \(-0.335359\pi\)
0.494480 + 0.869189i \(0.335359\pi\)
\(968\) −6.21532e12 −0.227522
\(969\) 3.06129e13 1.11544
\(970\) 1.17838e12 0.0427380
\(971\) 3.28780e13 1.18691 0.593456 0.804866i \(-0.297763\pi\)
0.593456 + 0.804866i \(0.297763\pi\)
\(972\) −1.35767e13 −0.487860
\(973\) −2.62115e13 −0.937529
\(974\) −2.21706e13 −0.789338
\(975\) −1.11772e13 −0.396106
\(976\) −2.87888e12 −0.101554
\(977\) −8.13505e11 −0.0285650 −0.0142825 0.999898i \(-0.504546\pi\)
−0.0142825 + 0.999898i \(0.504546\pi\)
\(978\) −3.98339e13 −1.39229
\(979\) 1.36301e13 0.474217
\(980\) 2.98506e12 0.103380
\(981\) −6.40164e12 −0.220689
\(982\) 6.28487e12 0.215672
\(983\) −4.13240e13 −1.41160 −0.705800 0.708412i \(-0.749412\pi\)
−0.705800 + 0.708412i \(0.749412\pi\)
\(984\) 1.53900e13 0.523313
\(985\) 3.47577e11 0.0117649
\(986\) 3.54321e13 1.19385
\(987\) 4.19198e12 0.140602
\(988\) 1.24518e13 0.415744
\(989\) 2.75631e13 0.916105
\(990\) −3.07112e12 −0.101611
\(991\) −3.61227e13 −1.18973 −0.594866 0.803825i \(-0.702795\pi\)
−0.594866 + 0.803825i \(0.702795\pi\)
\(992\) 2.44923e12 0.0803023
\(993\) −3.10462e13 −1.01330
\(994\) 1.20219e13 0.390603
\(995\) 1.57034e13 0.507915
\(996\) −6.49086e11 −0.0208995
\(997\) −3.07166e13 −0.984565 −0.492283 0.870435i \(-0.663837\pi\)
−0.492283 + 0.870435i \(0.663837\pi\)
\(998\) −1.20325e13 −0.383944
\(999\) −1.71459e13 −0.544646
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 10.10.a.c.1.1 1
3.2 odd 2 90.10.a.e.1.1 1
4.3 odd 2 80.10.a.a.1.1 1
5.2 odd 4 50.10.b.d.49.2 2
5.3 odd 4 50.10.b.d.49.1 2
5.4 even 2 50.10.a.a.1.1 1
8.3 odd 2 320.10.a.i.1.1 1
8.5 even 2 320.10.a.b.1.1 1
20.3 even 4 400.10.c.c.49.1 2
20.7 even 4 400.10.c.c.49.2 2
20.19 odd 2 400.10.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.10.a.c.1.1 1 1.1 even 1 trivial
50.10.a.a.1.1 1 5.4 even 2
50.10.b.d.49.1 2 5.3 odd 4
50.10.b.d.49.2 2 5.2 odd 4
80.10.a.a.1.1 1 4.3 odd 2
90.10.a.e.1.1 1 3.2 odd 2
320.10.a.b.1.1 1 8.5 even 2
320.10.a.i.1.1 1 8.3 odd 2
400.10.a.j.1.1 1 20.19 odd 2
400.10.c.c.49.1 2 20.3 even 4
400.10.c.c.49.2 2 20.7 even 4