Properties

Label 6.16.a.b.1.1
Level $6$
Weight $16$
Character 6.1
Self dual yes
Analytic conductor $8.562$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6 = 2 \cdot 3 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 6.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.56161030600\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 6.1

$q$-expansion

\(f(q)\) \(=\) \(q+128.000 q^{2} -2187.00 q^{3} +16384.0 q^{4} -114810. q^{5} -279936. q^{6} -3.03453e6 q^{7} +2.09715e6 q^{8} +4.78297e6 q^{9} +O(q^{10})\) \(q+128.000 q^{2} -2187.00 q^{3} +16384.0 q^{4} -114810. q^{5} -279936. q^{6} -3.03453e6 q^{7} +2.09715e6 q^{8} +4.78297e6 q^{9} -1.46957e7 q^{10} -1.03452e8 q^{11} -3.58318e7 q^{12} -1.04366e8 q^{13} -3.88420e8 q^{14} +2.51089e8 q^{15} +2.68435e8 q^{16} +9.97690e8 q^{17} +6.12220e8 q^{18} +4.93402e9 q^{19} -1.88105e9 q^{20} +6.63651e9 q^{21} -1.32418e10 q^{22} +8.32492e9 q^{23} -4.58647e9 q^{24} -1.73362e10 q^{25} -1.33588e10 q^{26} -1.04604e10 q^{27} -4.97177e10 q^{28} +1.04128e11 q^{29} +3.21395e10 q^{30} -2.96697e11 q^{31} +3.43597e10 q^{32} +2.26249e11 q^{33} +1.27704e11 q^{34} +3.48394e11 q^{35} +7.83642e10 q^{36} -1.78337e11 q^{37} +6.31554e11 q^{38} +2.28248e11 q^{39} -2.40774e11 q^{40} -1.79088e12 q^{41} +8.49474e11 q^{42} -2.86346e12 q^{43} -1.69495e12 q^{44} -5.49133e11 q^{45} +1.06559e12 q^{46} +4.33291e12 q^{47} -5.87068e11 q^{48} +4.46080e12 q^{49} -2.21904e12 q^{50} -2.18195e12 q^{51} -1.70993e12 q^{52} +9.73232e12 q^{53} -1.33893e12 q^{54} +1.18773e13 q^{55} -6.36387e12 q^{56} -1.07907e13 q^{57} +1.33284e13 q^{58} -1.35148e13 q^{59} +4.11385e12 q^{60} +5.35266e12 q^{61} -3.79772e13 q^{62} -1.45141e13 q^{63} +4.39805e12 q^{64} +1.19822e13 q^{65} +2.89599e13 q^{66} -5.32339e13 q^{67} +1.63461e13 q^{68} -1.82066e13 q^{69} +4.45945e13 q^{70} -2.02297e13 q^{71} +1.00306e13 q^{72} +2.62642e13 q^{73} -2.28272e13 q^{74} +3.79144e13 q^{75} +8.08389e13 q^{76} +3.13927e14 q^{77} +2.92158e13 q^{78} -3.39031e14 q^{79} -3.08191e13 q^{80} +2.28768e13 q^{81} -2.29233e14 q^{82} +1.31685e14 q^{83} +1.08733e14 q^{84} -1.14545e14 q^{85} -3.66523e14 q^{86} -2.27728e14 q^{87} -2.16954e14 q^{88} -3.93521e13 q^{89} -7.02890e13 q^{90} +3.16701e14 q^{91} +1.36395e14 q^{92} +6.48876e14 q^{93} +5.54612e14 q^{94} -5.66474e14 q^{95} -7.51447e13 q^{96} +1.12875e15 q^{97} +5.70982e14 q^{98} -4.94806e14 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 128.000 0.707107
\(3\) −2187.00 −0.577350
\(4\) 16384.0 0.500000
\(5\) −114810. −0.657211 −0.328605 0.944467i \(-0.606579\pi\)
−0.328605 + 0.944467i \(0.606579\pi\)
\(6\) −279936. −0.408248
\(7\) −3.03453e6 −1.39269 −0.696347 0.717705i \(-0.745193\pi\)
−0.696347 + 0.717705i \(0.745193\pi\)
\(8\) 2.09715e6 0.353553
\(9\) 4.78297e6 0.333333
\(10\) −1.46957e7 −0.464718
\(11\) −1.03452e8 −1.60064 −0.800318 0.599576i \(-0.795336\pi\)
−0.800318 + 0.599576i \(0.795336\pi\)
\(12\) −3.58318e7 −0.288675
\(13\) −1.04366e8 −0.461300 −0.230650 0.973037i \(-0.574085\pi\)
−0.230650 + 0.973037i \(0.574085\pi\)
\(14\) −3.88420e8 −0.984784
\(15\) 2.51089e8 0.379441
\(16\) 2.68435e8 0.250000
\(17\) 9.97690e8 0.589697 0.294848 0.955544i \(-0.404731\pi\)
0.294848 + 0.955544i \(0.404731\pi\)
\(18\) 6.12220e8 0.235702
\(19\) 4.93402e9 1.26633 0.633167 0.774015i \(-0.281754\pi\)
0.633167 + 0.774015i \(0.281754\pi\)
\(20\) −1.88105e9 −0.328605
\(21\) 6.63651e9 0.804073
\(22\) −1.32418e10 −1.13182
\(23\) 8.32492e9 0.509825 0.254913 0.966964i \(-0.417953\pi\)
0.254913 + 0.966964i \(0.417953\pi\)
\(24\) −4.58647e9 −0.204124
\(25\) −1.73362e10 −0.568074
\(26\) −1.33588e10 −0.326188
\(27\) −1.04604e10 −0.192450
\(28\) −4.97177e10 −0.696347
\(29\) 1.04128e11 1.12094 0.560472 0.828174i \(-0.310620\pi\)
0.560472 + 0.828174i \(0.310620\pi\)
\(30\) 3.21395e10 0.268305
\(31\) −2.96697e11 −1.93687 −0.968434 0.249270i \(-0.919809\pi\)
−0.968434 + 0.249270i \(0.919809\pi\)
\(32\) 3.43597e10 0.176777
\(33\) 2.26249e11 0.924127
\(34\) 1.27704e11 0.416978
\(35\) 3.48394e11 0.915294
\(36\) 7.83642e10 0.166667
\(37\) −1.78337e11 −0.308837 −0.154419 0.988006i \(-0.549350\pi\)
−0.154419 + 0.988006i \(0.549350\pi\)
\(38\) 6.31554e11 0.895434
\(39\) 2.28248e11 0.266332
\(40\) −2.40774e11 −0.232359
\(41\) −1.79088e12 −1.43611 −0.718056 0.695986i \(-0.754968\pi\)
−0.718056 + 0.695986i \(0.754968\pi\)
\(42\) 8.49474e11 0.568565
\(43\) −2.86346e12 −1.60649 −0.803244 0.595650i \(-0.796895\pi\)
−0.803244 + 0.595650i \(0.796895\pi\)
\(44\) −1.69495e12 −0.800318
\(45\) −5.49133e11 −0.219070
\(46\) 1.06559e12 0.360501
\(47\) 4.33291e12 1.24751 0.623757 0.781618i \(-0.285605\pi\)
0.623757 + 0.781618i \(0.285605\pi\)
\(48\) −5.87068e11 −0.144338
\(49\) 4.46080e12 0.939598
\(50\) −2.21904e12 −0.401689
\(51\) −2.18195e12 −0.340461
\(52\) −1.70993e12 −0.230650
\(53\) 9.73232e12 1.13801 0.569007 0.822333i \(-0.307328\pi\)
0.569007 + 0.822333i \(0.307328\pi\)
\(54\) −1.33893e12 −0.136083
\(55\) 1.18773e13 1.05196
\(56\) −6.36387e12 −0.492392
\(57\) −1.07907e13 −0.731119
\(58\) 1.33284e13 0.792626
\(59\) −1.35148e13 −0.707002 −0.353501 0.935434i \(-0.615009\pi\)
−0.353501 + 0.935434i \(0.615009\pi\)
\(60\) 4.11385e12 0.189720
\(61\) 5.35266e12 0.218070 0.109035 0.994038i \(-0.465224\pi\)
0.109035 + 0.994038i \(0.465224\pi\)
\(62\) −3.79772e13 −1.36957
\(63\) −1.45141e13 −0.464231
\(64\) 4.39805e12 0.125000
\(65\) 1.19822e13 0.303171
\(66\) 2.89599e13 0.653457
\(67\) −5.32339e13 −1.07307 −0.536534 0.843879i \(-0.680267\pi\)
−0.536534 + 0.843879i \(0.680267\pi\)
\(68\) 1.63461e13 0.294848
\(69\) −1.82066e13 −0.294348
\(70\) 4.45945e13 0.647210
\(71\) −2.02297e13 −0.263968 −0.131984 0.991252i \(-0.542135\pi\)
−0.131984 + 0.991252i \(0.542135\pi\)
\(72\) 1.00306e13 0.117851
\(73\) 2.62642e13 0.278254 0.139127 0.990275i \(-0.455570\pi\)
0.139127 + 0.990275i \(0.455570\pi\)
\(74\) −2.28272e13 −0.218381
\(75\) 3.79144e13 0.327978
\(76\) 8.08389e13 0.633167
\(77\) 3.13927e14 2.22920
\(78\) 2.92158e13 0.188325
\(79\) −3.39031e14 −1.98626 −0.993131 0.117005i \(-0.962671\pi\)
−0.993131 + 0.117005i \(0.962671\pi\)
\(80\) −3.08191e13 −0.164303
\(81\) 2.28768e13 0.111111
\(82\) −2.29233e14 −1.01548
\(83\) 1.31685e14 0.532660 0.266330 0.963882i \(-0.414189\pi\)
0.266330 + 0.963882i \(0.414189\pi\)
\(84\) 1.08733e14 0.402036
\(85\) −1.14545e14 −0.387555
\(86\) −3.66523e14 −1.13596
\(87\) −2.27728e14 −0.647177
\(88\) −2.16954e14 −0.565910
\(89\) −3.93521e13 −0.0943069 −0.0471534 0.998888i \(-0.515015\pi\)
−0.0471534 + 0.998888i \(0.515015\pi\)
\(90\) −7.02890e13 −0.154906
\(91\) 3.16701e14 0.642450
\(92\) 1.36395e14 0.254913
\(93\) 6.48876e14 1.11825
\(94\) 5.54612e14 0.882126
\(95\) −5.66474e14 −0.832249
\(96\) −7.51447e13 −0.102062
\(97\) 1.12875e15 1.41844 0.709219 0.704989i \(-0.249048\pi\)
0.709219 + 0.704989i \(0.249048\pi\)
\(98\) 5.70982e14 0.664396
\(99\) −4.94806e14 −0.533545
\(100\) −2.84037e14 −0.284037
\(101\) 3.79528e13 0.0352236 0.0176118 0.999845i \(-0.494394\pi\)
0.0176118 + 0.999845i \(0.494394\pi\)
\(102\) −2.79289e14 −0.240743
\(103\) −2.07297e14 −0.166079 −0.0830393 0.996546i \(-0.526463\pi\)
−0.0830393 + 0.996546i \(0.526463\pi\)
\(104\) −2.18871e14 −0.163094
\(105\) −7.61938e14 −0.528445
\(106\) 1.24574e15 0.804697
\(107\) −1.99692e15 −1.20221 −0.601107 0.799169i \(-0.705273\pi\)
−0.601107 + 0.799169i \(0.705273\pi\)
\(108\) −1.71382e14 −0.0962250
\(109\) 1.35603e13 0.00710510 0.00355255 0.999994i \(-0.498869\pi\)
0.00355255 + 0.999994i \(0.498869\pi\)
\(110\) 1.52029e15 0.743845
\(111\) 3.90024e14 0.178307
\(112\) −8.14575e14 −0.348174
\(113\) −7.08794e14 −0.283421 −0.141710 0.989908i \(-0.545260\pi\)
−0.141710 + 0.989908i \(0.545260\pi\)
\(114\) −1.38121e15 −0.516979
\(115\) −9.55784e14 −0.335063
\(116\) 1.70604e15 0.560472
\(117\) −4.99179e14 −0.153767
\(118\) −1.72990e15 −0.499926
\(119\) −3.02752e15 −0.821267
\(120\) 5.26573e14 0.134153
\(121\) 6.52501e15 1.56203
\(122\) 6.85141e14 0.154199
\(123\) 3.91666e15 0.829139
\(124\) −4.86108e15 −0.968434
\(125\) 5.49410e15 1.03056
\(126\) −1.85780e15 −0.328261
\(127\) −1.23021e15 −0.204858 −0.102429 0.994740i \(-0.532661\pi\)
−0.102429 + 0.994740i \(0.532661\pi\)
\(128\) 5.62950e14 0.0883883
\(129\) 6.26239e15 0.927507
\(130\) 1.53373e15 0.214374
\(131\) 7.94836e15 1.04892 0.524460 0.851435i \(-0.324267\pi\)
0.524460 + 0.851435i \(0.324267\pi\)
\(132\) 3.70686e15 0.462064
\(133\) −1.49724e16 −1.76362
\(134\) −6.81394e15 −0.758774
\(135\) 1.20095e15 0.126480
\(136\) 2.09231e15 0.208489
\(137\) −1.78131e16 −1.68010 −0.840050 0.542508i \(-0.817475\pi\)
−0.840050 + 0.542508i \(0.817475\pi\)
\(138\) −2.33044e15 −0.208135
\(139\) 3.89941e15 0.329904 0.164952 0.986302i \(-0.447253\pi\)
0.164952 + 0.986302i \(0.447253\pi\)
\(140\) 5.70809e15 0.457647
\(141\) −9.47607e15 −0.720253
\(142\) −2.58940e15 −0.186653
\(143\) 1.07968e16 0.738373
\(144\) 1.28392e15 0.0833333
\(145\) −1.19550e16 −0.736696
\(146\) 3.36181e15 0.196756
\(147\) −9.75577e15 −0.542477
\(148\) −2.92188e15 −0.154419
\(149\) −3.48726e15 −0.175222 −0.0876108 0.996155i \(-0.527923\pi\)
−0.0876108 + 0.996155i \(0.527923\pi\)
\(150\) 4.85304e15 0.231915
\(151\) 6.85712e15 0.311756 0.155878 0.987776i \(-0.450179\pi\)
0.155878 + 0.987776i \(0.450179\pi\)
\(152\) 1.03474e16 0.447717
\(153\) 4.77192e15 0.196566
\(154\) 4.01827e16 1.57628
\(155\) 3.40637e16 1.27293
\(156\) 3.73962e15 0.133166
\(157\) 3.69836e16 1.25534 0.627670 0.778480i \(-0.284009\pi\)
0.627670 + 0.778480i \(0.284009\pi\)
\(158\) −4.33960e16 −1.40450
\(159\) −2.12846e16 −0.657032
\(160\) −3.94484e15 −0.116180
\(161\) −2.52622e16 −0.710031
\(162\) 2.92823e15 0.0785674
\(163\) 7.42535e15 0.190244 0.0951218 0.995466i \(-0.469676\pi\)
0.0951218 + 0.995466i \(0.469676\pi\)
\(164\) −2.93418e16 −0.718056
\(165\) −2.59756e16 −0.607347
\(166\) 1.68557e16 0.376647
\(167\) −1.47365e16 −0.314789 −0.157395 0.987536i \(-0.550309\pi\)
−0.157395 + 0.987536i \(0.550309\pi\)
\(168\) 1.39178e16 0.284283
\(169\) −4.02937e16 −0.787203
\(170\) −1.46617e16 −0.274043
\(171\) 2.35992e16 0.422112
\(172\) −4.69149e16 −0.803244
\(173\) 3.40039e16 0.557421 0.278710 0.960375i \(-0.410093\pi\)
0.278710 + 0.960375i \(0.410093\pi\)
\(174\) −2.91492e16 −0.457623
\(175\) 5.26073e16 0.791153
\(176\) −2.77701e16 −0.400159
\(177\) 2.95569e16 0.408188
\(178\) −5.03707e15 −0.0666850
\(179\) 3.81276e16 0.483996 0.241998 0.970277i \(-0.422197\pi\)
0.241998 + 0.970277i \(0.422197\pi\)
\(180\) −8.99699e15 −0.109535
\(181\) −5.14124e16 −0.600452 −0.300226 0.953868i \(-0.597062\pi\)
−0.300226 + 0.953868i \(0.597062\pi\)
\(182\) 4.05377e16 0.454280
\(183\) −1.17063e16 −0.125903
\(184\) 1.74586e16 0.180250
\(185\) 2.04749e16 0.202971
\(186\) 8.30561e16 0.790723
\(187\) −1.03213e17 −0.943889
\(188\) 7.09904e16 0.623757
\(189\) 3.17422e16 0.268024
\(190\) −7.25087e16 −0.588489
\(191\) 6.75568e16 0.527131 0.263566 0.964641i \(-0.415101\pi\)
0.263566 + 0.964641i \(0.415101\pi\)
\(192\) −9.61853e15 −0.0721688
\(193\) −2.07235e16 −0.149549 −0.0747746 0.997200i \(-0.523824\pi\)
−0.0747746 + 0.997200i \(0.523824\pi\)
\(194\) 1.44480e17 1.00299
\(195\) −2.62052e16 −0.175036
\(196\) 7.30857e16 0.469799
\(197\) 1.71244e16 0.105954 0.0529772 0.998596i \(-0.483129\pi\)
0.0529772 + 0.998596i \(0.483129\pi\)
\(198\) −6.33352e16 −0.377273
\(199\) −1.05743e17 −0.606533 −0.303267 0.952906i \(-0.598077\pi\)
−0.303267 + 0.952906i \(0.598077\pi\)
\(200\) −3.63567e16 −0.200844
\(201\) 1.16423e17 0.619536
\(202\) 4.85796e15 0.0249069
\(203\) −3.15980e17 −1.56113
\(204\) −3.57490e16 −0.170231
\(205\) 2.05611e17 0.943828
\(206\) −2.65340e16 −0.117435
\(207\) 3.98178e16 0.169942
\(208\) −2.80155e16 −0.115325
\(209\) −5.10432e17 −2.02694
\(210\) −9.75281e16 −0.373667
\(211\) −3.13093e17 −1.15759 −0.578795 0.815473i \(-0.696477\pi\)
−0.578795 + 0.815473i \(0.696477\pi\)
\(212\) 1.59454e17 0.569007
\(213\) 4.42423e16 0.152402
\(214\) −2.55605e17 −0.850093
\(215\) 3.28754e17 1.05580
\(216\) −2.19370e16 −0.0680414
\(217\) 9.00334e17 2.69747
\(218\) 1.73572e15 0.00502406
\(219\) −5.74397e16 −0.160650
\(220\) 1.94598e17 0.525978
\(221\) −1.04125e17 −0.272027
\(222\) 4.99231e16 0.126082
\(223\) 3.35017e17 0.818052 0.409026 0.912523i \(-0.365869\pi\)
0.409026 + 0.912523i \(0.365869\pi\)
\(224\) −1.04266e17 −0.246196
\(225\) −8.29187e16 −0.189358
\(226\) −9.07256e16 −0.200409
\(227\) 4.67560e17 0.999180 0.499590 0.866262i \(-0.333484\pi\)
0.499590 + 0.866262i \(0.333484\pi\)
\(228\) −1.76795e17 −0.365559
\(229\) 3.61186e17 0.722711 0.361355 0.932428i \(-0.382314\pi\)
0.361355 + 0.932428i \(0.382314\pi\)
\(230\) −1.22340e17 −0.236925
\(231\) −6.86559e17 −1.28703
\(232\) 2.18373e17 0.396313
\(233\) 3.61787e17 0.635747 0.317873 0.948133i \(-0.397031\pi\)
0.317873 + 0.948133i \(0.397031\pi\)
\(234\) −6.38949e16 −0.108729
\(235\) −4.97461e17 −0.819880
\(236\) −2.21427e17 −0.353501
\(237\) 7.41462e17 1.14677
\(238\) −3.87522e17 −0.580724
\(239\) −1.06842e18 −1.55152 −0.775762 0.631026i \(-0.782634\pi\)
−0.775762 + 0.631026i \(0.782634\pi\)
\(240\) 6.74013e16 0.0948602
\(241\) −1.00588e18 −1.37220 −0.686102 0.727505i \(-0.740680\pi\)
−0.686102 + 0.727505i \(0.740680\pi\)
\(242\) 8.35201e17 1.10453
\(243\) −5.00315e16 −0.0641500
\(244\) 8.76980e16 0.109035
\(245\) −5.12144e17 −0.617514
\(246\) 5.01332e17 0.586290
\(247\) −5.14943e17 −0.584160
\(248\) −6.22218e17 −0.684786
\(249\) −2.87995e17 −0.307531
\(250\) 7.03244e17 0.728713
\(251\) 1.74766e18 1.75754 0.878768 0.477249i \(-0.158366\pi\)
0.878768 + 0.477249i \(0.158366\pi\)
\(252\) −2.37798e17 −0.232116
\(253\) −8.61227e17 −0.816044
\(254\) −1.57467e17 −0.144856
\(255\) 2.50509e17 0.223755
\(256\) 7.20576e16 0.0625000
\(257\) −2.77264e17 −0.233558 −0.116779 0.993158i \(-0.537257\pi\)
−0.116779 + 0.993158i \(0.537257\pi\)
\(258\) 8.01585e17 0.655846
\(259\) 5.41170e17 0.430116
\(260\) 1.96317e17 0.151586
\(261\) 4.98042e17 0.373648
\(262\) 1.01739e18 0.741699
\(263\) 1.73303e18 1.22783 0.613914 0.789373i \(-0.289594\pi\)
0.613914 + 0.789373i \(0.289594\pi\)
\(264\) 4.74478e17 0.326728
\(265\) −1.11737e18 −0.747914
\(266\) −1.91647e18 −1.24707
\(267\) 8.60631e16 0.0544481
\(268\) −8.72184e17 −0.536534
\(269\) −5.11271e16 −0.0305850 −0.0152925 0.999883i \(-0.504868\pi\)
−0.0152925 + 0.999883i \(0.504868\pi\)
\(270\) 1.53722e17 0.0894351
\(271\) 2.08455e17 0.117962 0.0589812 0.998259i \(-0.481215\pi\)
0.0589812 + 0.998259i \(0.481215\pi\)
\(272\) 2.67815e17 0.147424
\(273\) −6.92625e17 −0.370918
\(274\) −2.28008e18 −1.18801
\(275\) 1.79346e18 0.909279
\(276\) −2.98297e17 −0.147174
\(277\) 3.29723e18 1.58326 0.791628 0.611003i \(-0.209234\pi\)
0.791628 + 0.611003i \(0.209234\pi\)
\(278\) 4.99124e17 0.233277
\(279\) −1.41909e18 −0.645623
\(280\) 7.30636e17 0.323605
\(281\) −3.98328e18 −1.71768 −0.858841 0.512242i \(-0.828815\pi\)
−0.858841 + 0.512242i \(0.828815\pi\)
\(282\) −1.21294e18 −0.509296
\(283\) −2.19051e18 −0.895668 −0.447834 0.894117i \(-0.647804\pi\)
−0.447834 + 0.894117i \(0.647804\pi\)
\(284\) −3.31443e17 −0.131984
\(285\) 1.23888e18 0.480499
\(286\) 1.38199e18 0.522108
\(287\) 5.43448e18 2.00006
\(288\) 1.64342e17 0.0589256
\(289\) −1.86704e18 −0.652258
\(290\) −1.53024e18 −0.520923
\(291\) −2.46858e18 −0.818935
\(292\) 4.30312e17 0.139127
\(293\) −5.27183e18 −1.66132 −0.830662 0.556778i \(-0.812038\pi\)
−0.830662 + 0.556778i \(0.812038\pi\)
\(294\) −1.24874e18 −0.383589
\(295\) 1.55164e18 0.464649
\(296\) −3.74001e17 −0.109190
\(297\) 1.08214e18 0.308042
\(298\) −4.46369e17 −0.123900
\(299\) −8.68837e17 −0.235182
\(300\) 6.21189e17 0.163989
\(301\) 8.68925e18 2.23735
\(302\) 8.77712e17 0.220445
\(303\) −8.30029e16 −0.0203364
\(304\) 1.32446e18 0.316584
\(305\) −6.14539e17 −0.143318
\(306\) 6.10806e17 0.138993
\(307\) 4.77067e18 1.05935 0.529677 0.848199i \(-0.322313\pi\)
0.529677 + 0.848199i \(0.322313\pi\)
\(308\) 5.14338e18 1.11460
\(309\) 4.53359e17 0.0958856
\(310\) 4.36016e18 0.900098
\(311\) −6.46666e18 −1.30310 −0.651549 0.758607i \(-0.725880\pi\)
−0.651549 + 0.758607i \(0.725880\pi\)
\(312\) 4.78671e17 0.0941624
\(313\) −6.00129e18 −1.15256 −0.576278 0.817254i \(-0.695496\pi\)
−0.576278 + 0.817254i \(0.695496\pi\)
\(314\) 4.73389e18 0.887659
\(315\) 1.66636e18 0.305098
\(316\) −5.55469e18 −0.993131
\(317\) −9.53943e17 −0.166563 −0.0832814 0.996526i \(-0.526540\pi\)
−0.0832814 + 0.996526i \(0.526540\pi\)
\(318\) −2.72443e18 −0.464592
\(319\) −1.07722e19 −1.79422
\(320\) −5.04940e17 −0.0821513
\(321\) 4.36726e18 0.694098
\(322\) −3.23356e18 −0.502068
\(323\) 4.92262e18 0.746753
\(324\) 3.74813e17 0.0555556
\(325\) 1.80931e18 0.262052
\(326\) 9.50445e17 0.134523
\(327\) −2.96563e16 −0.00410213
\(328\) −3.75575e18 −0.507742
\(329\) −1.31483e19 −1.73741
\(330\) −3.32488e18 −0.429459
\(331\) −5.29071e18 −0.668042 −0.334021 0.942566i \(-0.608406\pi\)
−0.334021 + 0.942566i \(0.608406\pi\)
\(332\) 2.15752e18 0.266330
\(333\) −8.52983e17 −0.102946
\(334\) −1.88627e18 −0.222590
\(335\) 6.11179e18 0.705232
\(336\) 1.78148e18 0.201018
\(337\) 6.22409e18 0.686834 0.343417 0.939183i \(-0.388416\pi\)
0.343417 + 0.939183i \(0.388416\pi\)
\(338\) −5.15759e18 −0.556636
\(339\) 1.55013e18 0.163633
\(340\) −1.87670e18 −0.193777
\(341\) 3.06938e19 3.10022
\(342\) 3.02070e18 0.298478
\(343\) 8.70190e17 0.0841217
\(344\) −6.00511e18 −0.567979
\(345\) 2.09030e18 0.193448
\(346\) 4.35250e18 0.394156
\(347\) −3.22035e18 −0.285386 −0.142693 0.989767i \(-0.545576\pi\)
−0.142693 + 0.989767i \(0.545576\pi\)
\(348\) −3.73110e18 −0.323588
\(349\) −6.17407e18 −0.524060 −0.262030 0.965060i \(-0.584392\pi\)
−0.262030 + 0.965060i \(0.584392\pi\)
\(350\) 6.73374e18 0.559430
\(351\) 1.09170e18 0.0887772
\(352\) −3.55457e18 −0.282955
\(353\) −6.14267e18 −0.478681 −0.239341 0.970936i \(-0.576931\pi\)
−0.239341 + 0.970936i \(0.576931\pi\)
\(354\) 3.78329e18 0.288632
\(355\) 2.32257e18 0.173483
\(356\) −6.44746e17 −0.0471534
\(357\) 6.62118e18 0.474159
\(358\) 4.88033e18 0.342237
\(359\) −5.81103e18 −0.399066 −0.199533 0.979891i \(-0.563943\pi\)
−0.199533 + 0.979891i \(0.563943\pi\)
\(360\) −1.15161e18 −0.0774530
\(361\) 9.16338e18 0.603603
\(362\) −6.58079e18 −0.424584
\(363\) −1.42702e19 −0.901841
\(364\) 5.18883e18 0.321225
\(365\) −3.01539e18 −0.182872
\(366\) −1.49840e18 −0.0890268
\(367\) −2.77147e17 −0.0161330 −0.00806650 0.999967i \(-0.502568\pi\)
−0.00806650 + 0.999967i \(0.502568\pi\)
\(368\) 2.23470e18 0.127456
\(369\) −8.56574e18 −0.478704
\(370\) 2.62079e18 0.143522
\(371\) −2.95330e19 −1.58490
\(372\) 1.06312e19 0.559126
\(373\) 2.30498e19 1.18809 0.594047 0.804430i \(-0.297529\pi\)
0.594047 + 0.804430i \(0.297529\pi\)
\(374\) −1.32112e19 −0.667431
\(375\) −1.20156e19 −0.594991
\(376\) 9.08677e18 0.441063
\(377\) −1.08674e19 −0.517091
\(378\) 4.06301e18 0.189522
\(379\) −1.34397e19 −0.614604 −0.307302 0.951612i \(-0.599426\pi\)
−0.307302 + 0.951612i \(0.599426\pi\)
\(380\) −9.28112e18 −0.416124
\(381\) 2.69048e18 0.118275
\(382\) 8.64727e18 0.372738
\(383\) 2.64151e19 1.11651 0.558254 0.829670i \(-0.311471\pi\)
0.558254 + 0.829670i \(0.311471\pi\)
\(384\) −1.23117e18 −0.0510310
\(385\) −3.60420e19 −1.46505
\(386\) −2.65261e18 −0.105747
\(387\) −1.36958e19 −0.535496
\(388\) 1.84935e19 0.709219
\(389\) 3.95275e19 1.48688 0.743442 0.668800i \(-0.233192\pi\)
0.743442 + 0.668800i \(0.233192\pi\)
\(390\) −3.35426e18 −0.123769
\(391\) 8.30569e18 0.300642
\(392\) 9.35497e18 0.332198
\(393\) −1.73831e19 −0.605595
\(394\) 2.19192e18 0.0749210
\(395\) 3.89242e19 1.30539
\(396\) −8.10691e18 −0.266773
\(397\) 2.82890e18 0.0913457 0.0456729 0.998956i \(-0.485457\pi\)
0.0456729 + 0.998956i \(0.485457\pi\)
\(398\) −1.35351e19 −0.428884
\(399\) 3.27447e19 1.01822
\(400\) −4.65366e18 −0.142018
\(401\) −4.63681e19 −1.38879 −0.694395 0.719594i \(-0.744328\pi\)
−0.694395 + 0.719594i \(0.744328\pi\)
\(402\) 1.49021e19 0.438078
\(403\) 3.09650e19 0.893477
\(404\) 6.21819e17 0.0176118
\(405\) −2.62648e18 −0.0730234
\(406\) −4.04454e19 −1.10389
\(407\) 1.84493e19 0.494336
\(408\) −4.57588e18 −0.120371
\(409\) −1.17092e19 −0.302415 −0.151207 0.988502i \(-0.548316\pi\)
−0.151207 + 0.988502i \(0.548316\pi\)
\(410\) 2.63182e19 0.667387
\(411\) 3.89573e19 0.970007
\(412\) −3.39635e18 −0.0830393
\(413\) 4.10112e19 0.984638
\(414\) 5.09668e18 0.120167
\(415\) −1.51187e19 −0.350070
\(416\) −3.58598e18 −0.0815470
\(417\) −8.52801e18 −0.190470
\(418\) −6.53353e19 −1.43326
\(419\) −4.20102e19 −0.905210 −0.452605 0.891711i \(-0.649505\pi\)
−0.452605 + 0.891711i \(0.649505\pi\)
\(420\) −1.24836e19 −0.264223
\(421\) −1.59718e19 −0.332077 −0.166039 0.986119i \(-0.553098\pi\)
−0.166039 + 0.986119i \(0.553098\pi\)
\(422\) −4.00759e19 −0.818540
\(423\) 2.07242e19 0.415838
\(424\) 2.04101e19 0.402348
\(425\) −1.72962e19 −0.334991
\(426\) 5.66301e18 0.107764
\(427\) −1.62428e19 −0.303705
\(428\) −3.27175e19 −0.601107
\(429\) −2.36127e19 −0.426300
\(430\) 4.20805e19 0.746564
\(431\) −9.76365e19 −1.70229 −0.851143 0.524934i \(-0.824090\pi\)
−0.851143 + 0.524934i \(0.824090\pi\)
\(432\) −2.80793e18 −0.0481125
\(433\) −6.91455e19 −1.16441 −0.582204 0.813043i \(-0.697809\pi\)
−0.582204 + 0.813043i \(0.697809\pi\)
\(434\) 1.15243e20 1.90740
\(435\) 2.61455e19 0.425332
\(436\) 2.22172e17 0.00355255
\(437\) 4.10753e19 0.645609
\(438\) −7.35229e18 −0.113597
\(439\) 5.48990e19 0.833836 0.416918 0.908944i \(-0.363110\pi\)
0.416918 + 0.908944i \(0.363110\pi\)
\(440\) 2.49085e19 0.371922
\(441\) 2.13359e19 0.313199
\(442\) −1.33280e19 −0.192352
\(443\) 3.80504e19 0.539923 0.269961 0.962871i \(-0.412989\pi\)
0.269961 + 0.962871i \(0.412989\pi\)
\(444\) 6.39015e18 0.0891536
\(445\) 4.51802e18 0.0619795
\(446\) 4.28822e19 0.578450
\(447\) 7.62664e18 0.101164
\(448\) −1.33460e19 −0.174087
\(449\) 1.11994e18 0.0143663 0.00718317 0.999974i \(-0.497714\pi\)
0.00718317 + 0.999974i \(0.497714\pi\)
\(450\) −1.06136e19 −0.133896
\(451\) 1.85270e20 2.29869
\(452\) −1.16129e19 −0.141710
\(453\) −1.49965e19 −0.179992
\(454\) 5.98477e19 0.706527
\(455\) −3.63604e19 −0.422225
\(456\) −2.26297e19 −0.258489
\(457\) −9.70734e19 −1.09076 −0.545379 0.838189i \(-0.683614\pi\)
−0.545379 + 0.838189i \(0.683614\pi\)
\(458\) 4.62318e19 0.511034
\(459\) −1.04362e19 −0.113487
\(460\) −1.56596e19 −0.167531
\(461\) −9.78165e19 −1.02957 −0.514784 0.857320i \(-0.672128\pi\)
−0.514784 + 0.857320i \(0.672128\pi\)
\(462\) −8.78795e19 −0.910066
\(463\) −2.17159e19 −0.221269 −0.110634 0.993861i \(-0.535288\pi\)
−0.110634 + 0.993861i \(0.535288\pi\)
\(464\) 2.79517e19 0.280236
\(465\) −7.44974e19 −0.734927
\(466\) 4.63088e19 0.449541
\(467\) −1.98443e20 −1.89565 −0.947826 0.318788i \(-0.896724\pi\)
−0.947826 + 0.318788i \(0.896724\pi\)
\(468\) −8.17854e18 −0.0768833
\(469\) 1.61540e20 1.49446
\(470\) −6.36750e19 −0.579743
\(471\) −8.08830e19 −0.724771
\(472\) −2.83427e19 −0.249963
\(473\) 2.96230e20 2.57140
\(474\) 9.49071e19 0.810888
\(475\) −8.55373e19 −0.719372
\(476\) −4.96028e19 −0.410634
\(477\) 4.65494e19 0.379338
\(478\) −1.36758e20 −1.09709
\(479\) −3.78661e19 −0.299043 −0.149522 0.988758i \(-0.547773\pi\)
−0.149522 + 0.988758i \(0.547773\pi\)
\(480\) 8.62737e18 0.0670763
\(481\) 1.86123e19 0.142467
\(482\) −1.28753e20 −0.970295
\(483\) 5.52484e19 0.409936
\(484\) 1.06906e20 0.781017
\(485\) −1.29592e20 −0.932212
\(486\) −6.40404e18 −0.0453609
\(487\) 3.89265e19 0.271505 0.135752 0.990743i \(-0.456655\pi\)
0.135752 + 0.990743i \(0.456655\pi\)
\(488\) 1.12253e19 0.0770994
\(489\) −1.62392e19 −0.109837
\(490\) −6.55545e19 −0.436648
\(491\) 1.40516e20 0.921753 0.460876 0.887464i \(-0.347535\pi\)
0.460876 + 0.887464i \(0.347535\pi\)
\(492\) 6.41706e19 0.414570
\(493\) 1.03888e20 0.661016
\(494\) −6.59127e19 −0.413063
\(495\) 5.68087e19 0.350652
\(496\) −7.96439e19 −0.484217
\(497\) 6.13875e19 0.367627
\(498\) −3.68633e19 −0.217457
\(499\) −1.53703e20 −0.893157 −0.446578 0.894745i \(-0.647358\pi\)
−0.446578 + 0.894745i \(0.647358\pi\)
\(500\) 9.00153e19 0.515278
\(501\) 3.22287e19 0.181744
\(502\) 2.23701e20 1.24277
\(503\) 5.62265e18 0.0307738 0.0153869 0.999882i \(-0.495102\pi\)
0.0153869 + 0.999882i \(0.495102\pi\)
\(504\) −3.04382e19 −0.164131
\(505\) −4.35737e18 −0.0231493
\(506\) −1.10237e20 −0.577031
\(507\) 8.81222e19 0.454492
\(508\) −2.01558e19 −0.102429
\(509\) 3.29307e20 1.64899 0.824495 0.565869i \(-0.191459\pi\)
0.824495 + 0.565869i \(0.191459\pi\)
\(510\) 3.20652e19 0.158219
\(511\) −7.96993e19 −0.387523
\(512\) 9.22337e18 0.0441942
\(513\) −5.16115e19 −0.243706
\(514\) −3.54898e19 −0.165151
\(515\) 2.37998e19 0.109149
\(516\) 1.02603e20 0.463753
\(517\) −4.48247e20 −1.99682
\(518\) 6.92698e19 0.304138
\(519\) −7.43666e19 −0.321827
\(520\) 2.51286e19 0.107187
\(521\) 1.71775e20 0.722231 0.361116 0.932521i \(-0.382396\pi\)
0.361116 + 0.932521i \(0.382396\pi\)
\(522\) 6.37494e19 0.264209
\(523\) −1.59414e20 −0.651275 −0.325637 0.945495i \(-0.605579\pi\)
−0.325637 + 0.945495i \(0.605579\pi\)
\(524\) 1.30226e20 0.524460
\(525\) −1.15052e20 −0.456773
\(526\) 2.21828e20 0.868205
\(527\) −2.96011e20 −1.14216
\(528\) 6.07332e19 0.231032
\(529\) −1.97331e20 −0.740078
\(530\) −1.43023e20 −0.528855
\(531\) −6.46410e19 −0.235667
\(532\) −2.45308e20 −0.881809
\(533\) 1.86907e20 0.662478
\(534\) 1.10161e19 0.0385006
\(535\) 2.29266e20 0.790108
\(536\) −1.11640e20 −0.379387
\(537\) −8.33851e19 −0.279435
\(538\) −6.54427e18 −0.0216269
\(539\) −4.61477e20 −1.50395
\(540\) 1.96764e19 0.0632401
\(541\) 4.74121e19 0.150283 0.0751415 0.997173i \(-0.476059\pi\)
0.0751415 + 0.997173i \(0.476059\pi\)
\(542\) 2.66823e19 0.0834120
\(543\) 1.12439e20 0.346671
\(544\) 3.42804e19 0.104245
\(545\) −1.55686e18 −0.00466955
\(546\) −8.86560e19 −0.262279
\(547\) −2.80986e20 −0.819934 −0.409967 0.912100i \(-0.634460\pi\)
−0.409967 + 0.912100i \(0.634460\pi\)
\(548\) −2.91850e20 −0.840050
\(549\) 2.56016e19 0.0726900
\(550\) 2.29563e20 0.642958
\(551\) 5.13770e20 1.41949
\(552\) −3.81820e19 −0.104068
\(553\) 1.02880e21 2.76626
\(554\) 4.22046e20 1.11953
\(555\) −4.47787e19 −0.117185
\(556\) 6.38879e19 0.164952
\(557\) −3.49642e20 −0.890654 −0.445327 0.895368i \(-0.646913\pi\)
−0.445327 + 0.895368i \(0.646913\pi\)
\(558\) −1.81644e20 −0.456524
\(559\) 2.98847e20 0.741073
\(560\) 9.35213e19 0.228823
\(561\) 2.25726e20 0.544955
\(562\) −5.09859e20 −1.21459
\(563\) 2.68215e20 0.630479 0.315240 0.949012i \(-0.397915\pi\)
0.315240 + 0.949012i \(0.397915\pi\)
\(564\) −1.55256e20 −0.360126
\(565\) 8.13766e19 0.186267
\(566\) −2.80385e20 −0.633333
\(567\) −6.94203e19 −0.154744
\(568\) −4.24247e19 −0.0933267
\(569\) −7.95691e20 −1.72744 −0.863719 0.503974i \(-0.831871\pi\)
−0.863719 + 0.503974i \(0.831871\pi\)
\(570\) 1.58577e20 0.339764
\(571\) 8.84470e20 1.87030 0.935152 0.354246i \(-0.115262\pi\)
0.935152 + 0.354246i \(0.115262\pi\)
\(572\) 1.76895e20 0.369186
\(573\) −1.47747e20 −0.304339
\(574\) 6.95614e20 1.41426
\(575\) −1.44323e20 −0.289618
\(576\) 2.10357e19 0.0416667
\(577\) 3.15722e20 0.617286 0.308643 0.951178i \(-0.400125\pi\)
0.308643 + 0.951178i \(0.400125\pi\)
\(578\) −2.38981e20 −0.461216
\(579\) 4.53224e19 0.0863423
\(580\) −1.95870e20 −0.368348
\(581\) −3.99601e20 −0.741832
\(582\) −3.15978e20 −0.579075
\(583\) −1.00682e21 −1.82154
\(584\) 5.50799e19 0.0983778
\(585\) 5.73107e19 0.101057
\(586\) −6.74794e20 −1.17473
\(587\) 7.33397e20 1.26053 0.630265 0.776380i \(-0.282946\pi\)
0.630265 + 0.776380i \(0.282946\pi\)
\(588\) −1.59838e20 −0.271239
\(589\) −1.46391e21 −2.45272
\(590\) 1.98610e20 0.328557
\(591\) −3.74511e19 −0.0611728
\(592\) −4.78721e19 −0.0772093
\(593\) −3.25571e20 −0.518485 −0.259242 0.965812i \(-0.583473\pi\)
−0.259242 + 0.965812i \(0.583473\pi\)
\(594\) 1.38514e20 0.217819
\(595\) 3.47589e20 0.539746
\(596\) −5.71353e19 −0.0876108
\(597\) 2.31261e20 0.350182
\(598\) −1.11211e20 −0.166299
\(599\) 6.61173e20 0.976368 0.488184 0.872741i \(-0.337659\pi\)
0.488184 + 0.872741i \(0.337659\pi\)
\(600\) 7.95122e19 0.115958
\(601\) −2.49467e20 −0.359297 −0.179649 0.983731i \(-0.557496\pi\)
−0.179649 + 0.983731i \(0.557496\pi\)
\(602\) 1.11222e21 1.58204
\(603\) −2.54616e20 −0.357689
\(604\) 1.12347e20 0.155878
\(605\) −7.49136e20 −1.02659
\(606\) −1.06244e19 −0.0143800
\(607\) 1.08917e21 1.45606 0.728031 0.685544i \(-0.240436\pi\)
0.728031 + 0.685544i \(0.240436\pi\)
\(608\) 1.69531e20 0.223858
\(609\) 6.91048e20 0.901320
\(610\) −7.86610e19 −0.101341
\(611\) −4.52208e20 −0.575478
\(612\) 7.81831e19 0.0982828
\(613\) 1.99450e20 0.247674 0.123837 0.992303i \(-0.460480\pi\)
0.123837 + 0.992303i \(0.460480\pi\)
\(614\) 6.10645e20 0.749076
\(615\) −4.49672e20 −0.544919
\(616\) 6.58353e20 0.788140
\(617\) 4.21127e20 0.498052 0.249026 0.968497i \(-0.419890\pi\)
0.249026 + 0.968497i \(0.419890\pi\)
\(618\) 5.80299e19 0.0678013
\(619\) −1.27272e21 −1.46910 −0.734552 0.678552i \(-0.762608\pi\)
−0.734552 + 0.678552i \(0.762608\pi\)
\(620\) 5.58100e20 0.636465
\(621\) −8.70816e19 −0.0981159
\(622\) −8.27732e20 −0.921429
\(623\) 1.19415e20 0.131341
\(624\) 6.12699e19 0.0665829
\(625\) −1.01717e20 −0.109218
\(626\) −7.68165e20 −0.814980
\(627\) 1.11632e21 1.17025
\(628\) 6.05938e20 0.627670
\(629\) −1.77925e20 −0.182120
\(630\) 2.13294e20 0.215737
\(631\) −3.42884e20 −0.342711 −0.171355 0.985209i \(-0.554815\pi\)
−0.171355 + 0.985209i \(0.554815\pi\)
\(632\) −7.11000e20 −0.702250
\(633\) 6.84735e20 0.668335
\(634\) −1.22105e20 −0.117778
\(635\) 1.41241e20 0.134635
\(636\) −3.48727e20 −0.328516
\(637\) −4.65555e20 −0.433436
\(638\) −1.37885e21 −1.26871
\(639\) −9.67578e19 −0.0879893
\(640\) −6.46323e19 −0.0580898
\(641\) 1.00075e21 0.888980 0.444490 0.895784i \(-0.353385\pi\)
0.444490 + 0.895784i \(0.353385\pi\)
\(642\) 5.59009e20 0.490802
\(643\) −1.35729e20 −0.117785 −0.0588927 0.998264i \(-0.518757\pi\)
−0.0588927 + 0.998264i \(0.518757\pi\)
\(644\) −4.13896e20 −0.355015
\(645\) −7.18985e20 −0.609567
\(646\) 6.30095e20 0.528034
\(647\) 2.15462e21 1.78479 0.892397 0.451251i \(-0.149022\pi\)
0.892397 + 0.451251i \(0.149022\pi\)
\(648\) 4.79761e19 0.0392837
\(649\) 1.39813e21 1.13165
\(650\) 2.31592e20 0.185299
\(651\) −1.96903e21 −1.55738
\(652\) 1.21657e20 0.0951218
\(653\) −2.06101e21 −1.59306 −0.796529 0.604601i \(-0.793333\pi\)
−0.796529 + 0.604601i \(0.793333\pi\)
\(654\) −3.79601e18 −0.00290064
\(655\) −9.12551e20 −0.689362
\(656\) −4.80736e20 −0.359028
\(657\) 1.25621e20 0.0927515
\(658\) −1.68299e21 −1.22853
\(659\) −1.61309e21 −1.16418 −0.582089 0.813125i \(-0.697764\pi\)
−0.582089 + 0.813125i \(0.697764\pi\)
\(660\) −4.25585e20 −0.303673
\(661\) 1.60888e21 1.13504 0.567521 0.823359i \(-0.307903\pi\)
0.567521 + 0.823359i \(0.307903\pi\)
\(662\) −6.77211e20 −0.472377
\(663\) 2.27721e20 0.157055
\(664\) 2.76163e20 0.188324
\(665\) 1.71898e21 1.15907
\(666\) −1.09182e20 −0.0727936
\(667\) 8.66859e20 0.571485
\(668\) −2.41442e20 −0.157395
\(669\) −7.32683e20 −0.472303
\(670\) 7.82309e20 0.498674
\(671\) −5.53742e20 −0.349051
\(672\) 2.28029e20 0.142141
\(673\) 1.67294e21 1.03126 0.515628 0.856812i \(-0.327559\pi\)
0.515628 + 0.856812i \(0.327559\pi\)
\(674\) 7.96684e20 0.485665
\(675\) 1.81343e20 0.109326
\(676\) −6.60171e20 −0.393601
\(677\) 7.27000e20 0.428666 0.214333 0.976761i \(-0.431242\pi\)
0.214333 + 0.976761i \(0.431242\pi\)
\(678\) 1.98417e20 0.115706
\(679\) −3.42523e21 −1.97545
\(680\) −2.40218e20 −0.137021
\(681\) −1.02255e21 −0.576877
\(682\) 3.92880e21 2.19219
\(683\) 2.85909e21 1.57787 0.788936 0.614475i \(-0.210632\pi\)
0.788936 + 0.614475i \(0.210632\pi\)
\(684\) 3.86650e20 0.211056
\(685\) 2.04512e21 1.10418
\(686\) 1.11384e20 0.0594830
\(687\) −7.89913e20 −0.417257
\(688\) −7.68654e20 −0.401622
\(689\) −1.01572e21 −0.524965
\(690\) 2.67558e20 0.136789
\(691\) 1.71274e21 0.866178 0.433089 0.901351i \(-0.357424\pi\)
0.433089 + 0.901351i \(0.357424\pi\)
\(692\) 5.57120e20 0.278710
\(693\) 1.50150e21 0.743065
\(694\) −4.12205e20 −0.201798
\(695\) −4.47691e20 −0.216817
\(696\) −4.77581e20 −0.228812
\(697\) −1.78675e21 −0.846870
\(698\) −7.90281e20 −0.370566
\(699\) −7.91229e20 −0.367049
\(700\) 8.61918e20 0.395577
\(701\) −8.95519e20 −0.406621 −0.203311 0.979114i \(-0.565170\pi\)
−0.203311 + 0.979114i \(0.565170\pi\)
\(702\) 1.39738e20 0.0627749
\(703\) −8.79920e20 −0.391091
\(704\) −4.54985e20 −0.200079
\(705\) 1.08795e21 0.473358
\(706\) −7.86261e20 −0.338479
\(707\) −1.15169e20 −0.0490557
\(708\) 4.84261e20 0.204094
\(709\) −3.99642e21 −1.66657 −0.833286 0.552842i \(-0.813543\pi\)
−0.833286 + 0.552842i \(0.813543\pi\)
\(710\) 2.97289e20 0.122671
\(711\) −1.62158e21 −0.662088
\(712\) −8.25274e19 −0.0333425
\(713\) −2.46998e21 −0.987464
\(714\) 8.47511e20 0.335281
\(715\) −1.23958e21 −0.485267
\(716\) 6.24683e20 0.241998
\(717\) 2.33664e21 0.895772
\(718\) −7.43812e20 −0.282183
\(719\) −2.66669e20 −0.100117 −0.0500583 0.998746i \(-0.515941\pi\)
−0.0500583 + 0.998746i \(0.515941\pi\)
\(720\) −1.47407e20 −0.0547676
\(721\) 6.29049e20 0.231297
\(722\) 1.17291e21 0.426812
\(723\) 2.19986e21 0.792242
\(724\) −8.42341e20 −0.300226
\(725\) −1.80519e21 −0.636779
\(726\) −1.82658e21 −0.637698
\(727\) −1.69447e21 −0.585499 −0.292750 0.956189i \(-0.594570\pi\)
−0.292750 + 0.956189i \(0.594570\pi\)
\(728\) 6.64170e20 0.227140
\(729\) 1.09419e20 0.0370370
\(730\) −3.85970e20 −0.129310
\(731\) −2.85684e21 −0.947341
\(732\) −1.91796e20 −0.0629514
\(733\) 3.16239e21 1.02739 0.513695 0.857973i \(-0.328276\pi\)
0.513695 + 0.857973i \(0.328276\pi\)
\(734\) −3.54749e19 −0.0114078
\(735\) 1.12006e21 0.356522
\(736\) 2.86042e20 0.0901252
\(737\) 5.50714e21 1.71759
\(738\) −1.09641e21 −0.338495
\(739\) 2.01381e21 0.615440 0.307720 0.951477i \(-0.400434\pi\)
0.307720 + 0.951477i \(0.400434\pi\)
\(740\) 3.35461e20 0.101486
\(741\) 1.12618e21 0.337265
\(742\) −3.78022e21 −1.12070
\(743\) 2.11411e21 0.620458 0.310229 0.950662i \(-0.399594\pi\)
0.310229 + 0.950662i \(0.399594\pi\)
\(744\) 1.36079e21 0.395361
\(745\) 4.00372e20 0.115157
\(746\) 2.95038e21 0.840110
\(747\) 6.29844e20 0.177553
\(748\) −1.69104e21 −0.471945
\(749\) 6.05970e21 1.67432
\(750\) −1.53800e21 −0.420722
\(751\) 4.95087e21 1.34086 0.670428 0.741974i \(-0.266110\pi\)
0.670428 + 0.741974i \(0.266110\pi\)
\(752\) 1.16311e21 0.311879
\(753\) −3.82213e21 −1.01471
\(754\) −1.39103e21 −0.365638
\(755\) −7.87266e20 −0.204889
\(756\) 5.20065e20 0.134012
\(757\) −4.85879e21 −1.23968 −0.619839 0.784729i \(-0.712802\pi\)
−0.619839 + 0.784729i \(0.712802\pi\)
\(758\) −1.72028e21 −0.434590
\(759\) 1.88350e21 0.471143
\(760\) −1.18798e21 −0.294244
\(761\) −3.69092e21 −0.905211 −0.452605 0.891711i \(-0.649505\pi\)
−0.452605 + 0.891711i \(0.649505\pi\)
\(762\) 3.44381e20 0.0836329
\(763\) −4.11491e19 −0.00989523
\(764\) 1.10685e21 0.263566
\(765\) −5.47864e20 −0.129185
\(766\) 3.38114e21 0.789491
\(767\) 1.41049e21 0.326140
\(768\) −1.57590e20 −0.0360844
\(769\) −2.33815e21 −0.530181 −0.265091 0.964224i \(-0.585402\pi\)
−0.265091 + 0.964224i \(0.585402\pi\)
\(770\) −4.61337e21 −1.03595
\(771\) 6.06376e20 0.134845
\(772\) −3.39534e20 −0.0747746
\(773\) −5.20682e21 −1.13560 −0.567801 0.823166i \(-0.692206\pi\)
−0.567801 + 0.823166i \(0.692206\pi\)
\(774\) −1.75307e21 −0.378653
\(775\) 5.14361e21 1.10028
\(776\) 2.36716e21 0.501493
\(777\) −1.18354e21 −0.248328
\(778\) 5.05952e21 1.05139
\(779\) −8.83624e21 −1.81860
\(780\) −4.29345e20 −0.0875180
\(781\) 2.09279e21 0.422516
\(782\) 1.06313e21 0.212586
\(783\) −1.08922e21 −0.215726
\(784\) 1.19744e21 0.234899
\(785\) −4.24608e21 −0.825023
\(786\) −2.22503e21 −0.428220
\(787\) −1.92196e21 −0.366381 −0.183191 0.983077i \(-0.558643\pi\)
−0.183191 + 0.983077i \(0.558643\pi\)
\(788\) 2.80566e20 0.0529772
\(789\) −3.79013e21 −0.708887
\(790\) 4.98230e21 0.923052
\(791\) 2.15085e21 0.394718
\(792\) −1.03768e21 −0.188637
\(793\) −5.58635e20 −0.100596
\(794\) 3.62099e20 0.0645912
\(795\) 2.44368e21 0.431809
\(796\) −1.73250e21 −0.303267
\(797\) −6.23880e20 −0.108184 −0.0540921 0.998536i \(-0.517226\pi\)
−0.0540921 + 0.998536i \(0.517226\pi\)
\(798\) 4.19132e21 0.719994
\(799\) 4.32290e21 0.735655
\(800\) −5.95669e20 −0.100422
\(801\) −1.88220e20 −0.0314356
\(802\) −5.93511e21 −0.982022
\(803\) −2.71707e21 −0.445384
\(804\) 1.90747e21 0.309768
\(805\) 2.90035e21 0.466640
\(806\) 3.96352e21 0.631783
\(807\) 1.11815e20 0.0176583
\(808\) 7.95929e19 0.0124534
\(809\) 5.72858e21 0.888042 0.444021 0.896016i \(-0.353552\pi\)
0.444021 + 0.896016i \(0.353552\pi\)
\(810\) −3.36190e20 −0.0516354
\(811\) 4.80014e21 0.730462 0.365231 0.930917i \(-0.380990\pi\)
0.365231 + 0.930917i \(0.380990\pi\)
\(812\) −5.17702e21 −0.780566
\(813\) −4.55892e20 −0.0681056
\(814\) 2.36151e21 0.349548
\(815\) −8.52505e20 −0.125030
\(816\) −5.85712e20 −0.0851154
\(817\) −1.41284e22 −2.03435
\(818\) −1.49878e21 −0.213840
\(819\) 1.51477e21 0.214150
\(820\) 3.36873e21 0.471914
\(821\) 8.89775e21 1.23511 0.617556 0.786527i \(-0.288123\pi\)
0.617556 + 0.786527i \(0.288123\pi\)
\(822\) 4.98653e21 0.685898
\(823\) −1.39946e21 −0.190749 −0.0953744 0.995441i \(-0.530405\pi\)
−0.0953744 + 0.995441i \(0.530405\pi\)
\(824\) −4.34733e20 −0.0587177
\(825\) −3.92231e21 −0.524973
\(826\) 5.24943e21 0.696244
\(827\) 6.61343e21 0.869232 0.434616 0.900616i \(-0.356884\pi\)
0.434616 + 0.900616i \(0.356884\pi\)
\(828\) 6.52375e20 0.0849709
\(829\) 4.14735e21 0.535318 0.267659 0.963514i \(-0.413750\pi\)
0.267659 + 0.963514i \(0.413750\pi\)
\(830\) −1.93520e21 −0.247537
\(831\) −7.21105e21 −0.914093
\(832\) −4.59006e20 −0.0576625
\(833\) 4.45049e21 0.554078
\(834\) −1.09158e21 −0.134683
\(835\) 1.69190e21 0.206883
\(836\) −8.36292e21 −1.01347
\(837\) 3.10355e21 0.372750
\(838\) −5.37730e21 −0.640080
\(839\) 9.57811e21 1.12997 0.564983 0.825103i \(-0.308883\pi\)
0.564983 + 0.825103i \(0.308883\pi\)
\(840\) −1.59790e21 −0.186834
\(841\) 2.21350e21 0.256513
\(842\) −2.04440e21 −0.234814
\(843\) 8.71142e21 0.991705
\(844\) −5.12972e21 −0.578795
\(845\) 4.62612e21 0.517358
\(846\) 2.65269e21 0.294042
\(847\) −1.98003e22 −2.17544
\(848\) 2.61250e21 0.284503
\(849\) 4.79065e21 0.517114
\(850\) −2.21391e21 −0.236875
\(851\) −1.48465e21 −0.157453
\(852\) 7.24865e20 0.0762009
\(853\) −8.15361e21 −0.849634 −0.424817 0.905279i \(-0.639662\pi\)
−0.424817 + 0.905279i \(0.639662\pi\)
\(854\) −2.07908e21 −0.214752
\(855\) −2.70943e21 −0.277416
\(856\) −4.18784e21 −0.425047
\(857\) −1.53552e22 −1.54490 −0.772449 0.635076i \(-0.780969\pi\)
−0.772449 + 0.635076i \(0.780969\pi\)
\(858\) −3.02242e21 −0.301439
\(859\) −4.87617e21 −0.482092 −0.241046 0.970514i \(-0.577490\pi\)
−0.241046 + 0.970514i \(0.577490\pi\)
\(860\) 5.38630e21 0.527901
\(861\) −1.18852e22 −1.15474
\(862\) −1.24975e22 −1.20370
\(863\) −4.76667e21 −0.455128 −0.227564 0.973763i \(-0.573076\pi\)
−0.227564 + 0.973763i \(0.573076\pi\)
\(864\) −3.59415e20 −0.0340207
\(865\) −3.90399e21 −0.366343
\(866\) −8.85063e21 −0.823360
\(867\) 4.08321e21 0.376581
\(868\) 1.47511e22 1.34873
\(869\) 3.50734e22 3.17928
\(870\) 3.34662e21 0.300755
\(871\) 5.55580e21 0.495006
\(872\) 2.84380e19 0.00251203
\(873\) 5.39878e21 0.472812
\(874\) 5.25764e21 0.456515
\(875\) −1.66720e22 −1.43525
\(876\) −9.41093e20 −0.0803252
\(877\) 1.57944e22 1.33661 0.668306 0.743887i \(-0.267020\pi\)
0.668306 + 0.743887i \(0.267020\pi\)
\(878\) 7.02707e21 0.589611
\(879\) 1.15295e22 0.959165
\(880\) 3.18829e21 0.262989
\(881\) −1.12789e22 −0.922461 −0.461230 0.887280i \(-0.652592\pi\)
−0.461230 + 0.887280i \(0.652592\pi\)
\(882\) 2.73099e21 0.221465
\(883\) 4.48307e21 0.360471 0.180236 0.983623i \(-0.442314\pi\)
0.180236 + 0.983623i \(0.442314\pi\)
\(884\) −1.70598e21 −0.136013
\(885\) −3.39343e21 −0.268265
\(886\) 4.87045e21 0.381783
\(887\) −8.56246e21 −0.665536 −0.332768 0.943009i \(-0.607983\pi\)
−0.332768 + 0.943009i \(0.607983\pi\)
\(888\) 8.17940e20 0.0630411
\(889\) 3.73312e21 0.285304
\(890\) 5.78307e20 0.0438261
\(891\) −2.36664e21 −0.177848
\(892\) 5.48893e21 0.409026
\(893\) 2.13786e22 1.57977
\(894\) 9.76210e20 0.0715339
\(895\) −4.37743e21 −0.318087
\(896\) −1.70829e21 −0.123098
\(897\) 1.90015e21 0.135783
\(898\) 1.43352e20 0.0101585
\(899\) −3.08945e22 −2.17112
\(900\) −1.35854e21 −0.0946790
\(901\) 9.70983e21 0.671082
\(902\) 2.37145e22 1.62542
\(903\) −1.90034e22 −1.29173
\(904\) −1.48645e21 −0.100204
\(905\) 5.90266e21 0.394624
\(906\) −1.91956e21 −0.127274
\(907\) 7.34638e21 0.483079 0.241540 0.970391i \(-0.422348\pi\)
0.241540 + 0.970391i \(0.422348\pi\)
\(908\) 7.66051e21 0.499590
\(909\) 1.81527e20 0.0117412
\(910\) −4.65414e21 −0.298558
\(911\) −1.18999e22 −0.757106 −0.378553 0.925580i \(-0.623578\pi\)
−0.378553 + 0.925580i \(0.623578\pi\)
\(912\) −2.89660e21 −0.182780
\(913\) −1.36230e22 −0.852594
\(914\) −1.24254e22 −0.771283
\(915\) 1.34400e21 0.0827447
\(916\) 5.91767e21 0.361355
\(917\) −2.41195e22 −1.46083
\(918\) −1.33583e21 −0.0802475
\(919\) 1.92629e22 1.14777 0.573886 0.818935i \(-0.305435\pi\)
0.573886 + 0.818935i \(0.305435\pi\)
\(920\) −2.00442e21 −0.118463
\(921\) −1.04334e22 −0.611618
\(922\) −1.25205e22 −0.728015
\(923\) 2.11129e21 0.121768
\(924\) −1.12486e22 −0.643514
\(925\) 3.09170e21 0.175442
\(926\) −2.77964e21 −0.156461
\(927\) −9.91495e20 −0.0553596
\(928\) 3.57782e21 0.198157
\(929\) 8.78109e21 0.482426 0.241213 0.970472i \(-0.422455\pi\)
0.241213 + 0.970472i \(0.422455\pi\)
\(930\) −9.53567e21 −0.519672
\(931\) 2.20096e22 1.18985
\(932\) 5.92753e21 0.317873
\(933\) 1.41426e22 0.752343
\(934\) −2.54007e22 −1.34043
\(935\) 1.18499e22 0.620334
\(936\) −1.04685e21 −0.0543647
\(937\) −2.35424e22 −1.21284 −0.606421 0.795144i \(-0.707395\pi\)
−0.606421 + 0.795144i \(0.707395\pi\)
\(938\) 2.06771e22 1.05674
\(939\) 1.31248e22 0.665429
\(940\) −8.15040e21 −0.409940
\(941\) −2.31660e22 −1.15592 −0.577961 0.816064i \(-0.696152\pi\)
−0.577961 + 0.816064i \(0.696152\pi\)
\(942\) −1.03530e22 −0.512490
\(943\) −1.49090e22 −0.732166
\(944\) −3.62786e21 −0.176751
\(945\) −3.64433e21 −0.176148
\(946\) 3.79174e22 1.81826
\(947\) 2.02578e22 0.963755 0.481877 0.876239i \(-0.339955\pi\)
0.481877 + 0.876239i \(0.339955\pi\)
\(948\) 1.21481e22 0.573385
\(949\) −2.74108e21 −0.128359
\(950\) −1.09488e22 −0.508673
\(951\) 2.08627e21 0.0961651
\(952\) −6.34916e21 −0.290362
\(953\) −1.66351e22 −0.754794 −0.377397 0.926052i \(-0.623181\pi\)
−0.377397 + 0.926052i \(0.623181\pi\)
\(954\) 5.95832e21 0.268232
\(955\) −7.75620e21 −0.346436
\(956\) −1.75050e22 −0.775762
\(957\) 2.35589e22 1.03589
\(958\) −4.84686e21 −0.211456
\(959\) 5.40544e22 2.33987
\(960\) 1.10430e21 0.0474301
\(961\) 6.45637e22 2.75146
\(962\) 2.38238e21 0.100739
\(963\) −9.55119e21 −0.400738
\(964\) −1.64804e22 −0.686102
\(965\) 2.37927e21 0.0982853
\(966\) 7.07180e21 0.289869
\(967\) −2.10794e22 −0.857355 −0.428677 0.903458i \(-0.641020\pi\)
−0.428677 + 0.903458i \(0.641020\pi\)
\(968\) 1.36839e22 0.552263
\(969\) −1.07658e22 −0.431138
\(970\) −1.65878e22 −0.659174
\(971\) −3.09261e22 −1.21950 −0.609749 0.792594i \(-0.708730\pi\)
−0.609749 + 0.792594i \(0.708730\pi\)
\(972\) −8.19717e20 −0.0320750
\(973\) −1.18329e22 −0.459456
\(974\) 4.98259e21 0.191983
\(975\) −3.95696e21 −0.151296
\(976\) 1.43684e21 0.0545175
\(977\) 7.85725e21 0.295843 0.147922 0.988999i \(-0.452742\pi\)
0.147922 + 0.988999i \(0.452742\pi\)
\(978\) −2.07862e21 −0.0776666
\(979\) 4.07105e21 0.150951
\(980\) −8.39097e21 −0.308757
\(981\) 6.48584e19 0.00236837
\(982\) 1.79860e22 0.651778
\(983\) 1.27981e22 0.460251 0.230125 0.973161i \(-0.426086\pi\)
0.230125 + 0.973161i \(0.426086\pi\)
\(984\) 8.21383e21 0.293145
\(985\) −1.96605e21 −0.0696343
\(986\) 1.32976e22 0.467409
\(987\) 2.87554e22 1.00309
\(988\) −8.43682e21 −0.292080
\(989\) −2.38381e22 −0.819028
\(990\) 7.27151e21 0.247948
\(991\) −5.39385e22 −1.82535 −0.912675 0.408686i \(-0.865987\pi\)
−0.912675 + 0.408686i \(0.865987\pi\)
\(992\) −1.01944e22 −0.342393
\(993\) 1.15708e22 0.385694
\(994\) 7.85760e21 0.259951
\(995\) 1.21404e22 0.398620
\(996\) −4.71850e21 −0.153766
\(997\) −1.99678e22 −0.645828 −0.322914 0.946428i \(-0.604663\pi\)
−0.322914 + 0.946428i \(0.604663\pi\)
\(998\) −1.96740e22 −0.631557
\(999\) 1.86547e21 0.0594358
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 6.16.a.b.1.1 1
3.2 odd 2 18.16.a.b.1.1 1
4.3 odd 2 48.16.a.d.1.1 1
5.2 odd 4 150.16.c.a.49.2 2
5.3 odd 4 150.16.c.a.49.1 2
5.4 even 2 150.16.a.f.1.1 1
12.11 even 2 144.16.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6.16.a.b.1.1 1 1.1 even 1 trivial
18.16.a.b.1.1 1 3.2 odd 2
48.16.a.d.1.1 1 4.3 odd 2
144.16.a.j.1.1 1 12.11 even 2
150.16.a.f.1.1 1 5.4 even 2
150.16.c.a.49.1 2 5.3 odd 4
150.16.c.a.49.2 2 5.2 odd 4