Properties

Label 2.34.a.a.1.1
Level 2
Weight 34
Character 2.1
Self dual yes
Analytic conductor 13.797
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 2 \)
Weight: \( k \) \(=\) \( 34 \)
Character orbit: \([\chi]\) \(=\) 2.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.7965657762\)
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 \(\chi\) \(=\) 2.1

$q$-expansion

\(f(q)\) \(=\) \(q-65536.0 q^{2} -1.33006e8 q^{3} +4.29497e9 q^{4} +5.38799e11 q^{5} +8.71665e12 q^{6} -3.33473e13 q^{7} -2.81475e14 q^{8} +1.21314e16 q^{9} +O(q^{10})\) \(q-65536.0 q^{2} -1.33006e8 q^{3} +4.29497e9 q^{4} +5.38799e11 q^{5} +8.71665e12 q^{6} -3.33473e13 q^{7} -2.81475e14 q^{8} +1.21314e16 q^{9} -3.53107e16 q^{10} -8.58823e16 q^{11} -5.71255e17 q^{12} +1.14405e18 q^{13} +2.18545e18 q^{14} -7.16633e19 q^{15} +1.84467e19 q^{16} -1.39114e20 q^{17} -7.95045e20 q^{18} +8.06950e19 q^{19} +2.31412e21 q^{20} +4.43538e21 q^{21} +5.62838e21 q^{22} -1.41204e22 q^{23} +3.74377e22 q^{24} +1.73889e23 q^{25} -7.49767e22 q^{26} -8.74160e23 q^{27} -1.43226e23 q^{28} -1.63269e24 q^{29} +4.69652e24 q^{30} -1.89408e24 q^{31} -1.20893e24 q^{32} +1.14228e25 q^{33} +9.11695e24 q^{34} -1.79675e25 q^{35} +5.21040e25 q^{36} -9.64442e25 q^{37} -5.28843e24 q^{38} -1.52166e26 q^{39} -1.51658e26 q^{40} +6.41768e26 q^{41} -2.90677e26 q^{42} -8.17976e26 q^{43} -3.68862e26 q^{44} +6.53640e27 q^{45} +9.25393e26 q^{46} -6.22925e27 q^{47} -2.45352e27 q^{48} -6.61895e27 q^{49} -1.13960e28 q^{50} +1.85029e28 q^{51} +4.91367e27 q^{52} -2.13221e28 q^{53} +5.72890e28 q^{54} -4.62733e28 q^{55} +9.38643e27 q^{56} -1.07329e28 q^{57} +1.07000e29 q^{58} +2.98988e29 q^{59} -3.07791e29 q^{60} -4.55882e29 q^{61} +1.24130e29 q^{62} -4.04550e29 q^{63} +7.92282e28 q^{64} +6.16415e29 q^{65} -7.48606e29 q^{66} +1.17233e30 q^{67} -5.97489e29 q^{68} +1.87809e30 q^{69} +1.17752e30 q^{70} +2.59152e30 q^{71} -3.41469e30 q^{72} -2.82517e30 q^{73} +6.32057e30 q^{74} -2.31282e31 q^{75} +3.46582e29 q^{76} +2.86394e30 q^{77} +9.97232e30 q^{78} +9.20688e29 q^{79} +9.93909e30 q^{80} +4.88289e31 q^{81} -4.20589e31 q^{82} -1.61992e31 q^{83} +1.90498e31 q^{84} -7.49543e31 q^{85} +5.36068e31 q^{86} +2.17156e32 q^{87} +2.41737e31 q^{88} -2.03492e32 q^{89} -4.28369e32 q^{90} -3.81511e31 q^{91} -6.06465e31 q^{92} +2.51923e32 q^{93} +4.08240e32 q^{94} +4.34784e31 q^{95} +1.60794e32 q^{96} +2.26807e32 q^{97} +4.33780e32 q^{98} -1.04187e33 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −65536.0 −0.707107
\(3\) −1.33006e8 −1.78389 −0.891947 0.452140i \(-0.850661\pi\)
−0.891947 + 0.452140i \(0.850661\pi\)
\(4\) 4.29497e9 0.500000
\(5\) 5.38799e11 1.57914 0.789572 0.613658i \(-0.210303\pi\)
0.789572 + 0.613658i \(0.210303\pi\)
\(6\) 8.71665e12 1.26140
\(7\) −3.33473e13 −0.379265 −0.189633 0.981855i \(-0.560730\pi\)
−0.189633 + 0.981855i \(0.560730\pi\)
\(8\) −2.81475e14 −0.353553
\(9\) 1.21314e16 2.18228
\(10\) −3.53107e16 −1.11662
\(11\) −8.58823e16 −0.563539 −0.281770 0.959482i \(-0.590921\pi\)
−0.281770 + 0.959482i \(0.590921\pi\)
\(12\) −5.71255e17 −0.891947
\(13\) 1.14405e18 0.476849 0.238425 0.971161i \(-0.423369\pi\)
0.238425 + 0.971161i \(0.423369\pi\)
\(14\) 2.18545e18 0.268181
\(15\) −7.16633e19 −2.81703
\(16\) 1.84467e19 0.250000
\(17\) −1.39114e20 −0.693366 −0.346683 0.937982i \(-0.612692\pi\)
−0.346683 + 0.937982i \(0.612692\pi\)
\(18\) −7.95045e20 −1.54310
\(19\) 8.06950e19 0.0641818 0.0320909 0.999485i \(-0.489783\pi\)
0.0320909 + 0.999485i \(0.489783\pi\)
\(20\) 2.31412e21 0.789572
\(21\) 4.43538e21 0.676569
\(22\) 5.62838e21 0.398482
\(23\) −1.41204e22 −0.480106 −0.240053 0.970760i \(-0.577165\pi\)
−0.240053 + 0.970760i \(0.577165\pi\)
\(24\) 3.74377e22 0.630702
\(25\) 1.73889e23 1.49370
\(26\) −7.49767e22 −0.337183
\(27\) −8.74160e23 −2.10906
\(28\) −1.43226e23 −0.189633
\(29\) −1.63269e24 −1.21153 −0.605767 0.795642i \(-0.707134\pi\)
−0.605767 + 0.795642i \(0.707134\pi\)
\(30\) 4.69652e24 1.99194
\(31\) −1.89408e24 −0.467660 −0.233830 0.972278i \(-0.575126\pi\)
−0.233830 + 0.972278i \(0.575126\pi\)
\(32\) −1.20893e24 −0.176777
\(33\) 1.14228e25 1.00529
\(34\) 9.11695e24 0.490284
\(35\) −1.79675e25 −0.598915
\(36\) 5.21040e25 1.09114
\(37\) −9.64442e25 −1.28513 −0.642566 0.766230i \(-0.722130\pi\)
−0.642566 + 0.766230i \(0.722130\pi\)
\(38\) −5.28843e24 −0.0453834
\(39\) −1.52166e26 −0.850649
\(40\) −1.51658e26 −0.558312
\(41\) 6.41768e26 1.57197 0.785986 0.618245i \(-0.212156\pi\)
0.785986 + 0.618245i \(0.212156\pi\)
\(42\) −2.90677e26 −0.478407
\(43\) −8.17976e26 −0.913084 −0.456542 0.889702i \(-0.650912\pi\)
−0.456542 + 0.889702i \(0.650912\pi\)
\(44\) −3.68862e26 −0.281770
\(45\) 6.53640e27 3.44613
\(46\) 9.25393e26 0.339486
\(47\) −6.22925e27 −1.60258 −0.801292 0.598273i \(-0.795854\pi\)
−0.801292 + 0.598273i \(0.795854\pi\)
\(48\) −2.45352e27 −0.445974
\(49\) −6.61895e27 −0.856158
\(50\) −1.13960e28 −1.05620
\(51\) 1.85029e28 1.23689
\(52\) 4.91367e27 0.238425
\(53\) −2.13221e28 −0.755577 −0.377788 0.925892i \(-0.623315\pi\)
−0.377788 + 0.925892i \(0.623315\pi\)
\(54\) 5.72890e28 1.49133
\(55\) −4.62733e28 −0.889910
\(56\) 9.38643e27 0.134091
\(57\) −1.07329e28 −0.114494
\(58\) 1.07000e29 0.856685
\(59\) 2.98988e29 1.80549 0.902746 0.430175i \(-0.141548\pi\)
0.902746 + 0.430175i \(0.141548\pi\)
\(60\) −3.07791e29 −1.40851
\(61\) −4.55882e29 −1.58822 −0.794109 0.607775i \(-0.792062\pi\)
−0.794109 + 0.607775i \(0.792062\pi\)
\(62\) 1.24130e29 0.330685
\(63\) −4.04550e29 −0.827663
\(64\) 7.92282e28 0.125000
\(65\) 6.16415e29 0.753014
\(66\) −7.48606e29 −0.710851
\(67\) 1.17233e30 0.868593 0.434297 0.900770i \(-0.356997\pi\)
0.434297 + 0.900770i \(0.356997\pi\)
\(68\) −5.97489e29 −0.346683
\(69\) 1.87809e30 0.856458
\(70\) 1.17752e30 0.423497
\(71\) 2.59152e30 0.737552 0.368776 0.929518i \(-0.379777\pi\)
0.368776 + 0.929518i \(0.379777\pi\)
\(72\) −3.41469e30 −0.771552
\(73\) −2.82517e30 −0.508415 −0.254207 0.967150i \(-0.581815\pi\)
−0.254207 + 0.967150i \(0.581815\pi\)
\(74\) 6.32057e30 0.908726
\(75\) −2.31282e31 −2.66460
\(76\) 3.46582e29 0.0320909
\(77\) 2.86394e30 0.213731
\(78\) 9.97232e30 0.601500
\(79\) 9.20688e29 0.0450055 0.0225028 0.999747i \(-0.492837\pi\)
0.0225028 + 0.999747i \(0.492837\pi\)
\(80\) 9.93909e30 0.394786
\(81\) 4.88289e31 1.58006
\(82\) −4.20589e31 −1.11155
\(83\) −1.61992e31 −0.350514 −0.175257 0.984523i \(-0.556076\pi\)
−0.175257 + 0.984523i \(0.556076\pi\)
\(84\) 1.90498e31 0.338285
\(85\) −7.49543e31 −1.09493
\(86\) 5.36068e31 0.645648
\(87\) 2.17156e32 2.16125
\(88\) 2.41737e31 0.199241
\(89\) −2.03492e32 −1.39191 −0.695955 0.718085i \(-0.745019\pi\)
−0.695955 + 0.718085i \(0.745019\pi\)
\(90\) −4.28369e32 −2.43678
\(91\) −3.81511e31 −0.180852
\(92\) −6.06465e31 −0.240053
\(93\) 2.51923e32 0.834256
\(94\) 4.08240e32 1.13320
\(95\) 4.34784e31 0.101352
\(96\) 1.60794e32 0.315351
\(97\) 2.26807e32 0.374906 0.187453 0.982274i \(-0.439977\pi\)
0.187453 + 0.982274i \(0.439977\pi\)
\(98\) 4.33780e32 0.605395
\(99\) −1.04187e33 −1.22980
\(100\) 7.46848e32 0.746848
\(101\) 7.19854e32 0.610860 0.305430 0.952215i \(-0.401200\pi\)
0.305430 + 0.952215i \(0.401200\pi\)
\(102\) −1.21261e33 −0.874615
\(103\) −2.42535e33 −1.48922 −0.744611 0.667499i \(-0.767365\pi\)
−0.744611 + 0.667499i \(0.767365\pi\)
\(104\) −3.22023e32 −0.168592
\(105\) 2.38978e33 1.06840
\(106\) 1.39737e33 0.534273
\(107\) −4.14367e33 −1.35692 −0.678458 0.734639i \(-0.737352\pi\)
−0.678458 + 0.734639i \(0.737352\pi\)
\(108\) −3.75449e33 −1.05453
\(109\) 9.64390e32 0.232657 0.116328 0.993211i \(-0.462888\pi\)
0.116328 + 0.993211i \(0.462888\pi\)
\(110\) 3.03257e33 0.629261
\(111\) 1.28276e34 2.29254
\(112\) −6.15149e32 −0.0948163
\(113\) −1.19165e33 −0.158619 −0.0793095 0.996850i \(-0.525272\pi\)
−0.0793095 + 0.996850i \(0.525272\pi\)
\(114\) 7.03390e32 0.0809592
\(115\) −7.60804e33 −0.758156
\(116\) −7.01234e33 −0.605767
\(117\) 1.38790e34 1.04062
\(118\) −1.95945e34 −1.27668
\(119\) 4.63907e33 0.262970
\(120\) 2.01714e34 0.995969
\(121\) −1.58494e34 −0.682424
\(122\) 2.98767e34 1.12304
\(123\) −8.53587e34 −2.80423
\(124\) −8.13501e33 −0.233830
\(125\) 3.09669e34 0.779618
\(126\) 2.65126e34 0.585246
\(127\) −2.18147e34 −0.422656 −0.211328 0.977415i \(-0.567779\pi\)
−0.211328 + 0.977415i \(0.567779\pi\)
\(128\) −5.19230e33 −0.0883883
\(129\) 1.08795e35 1.62885
\(130\) −4.03974e34 −0.532461
\(131\) 1.54600e35 1.79569 0.897847 0.440309i \(-0.145131\pi\)
0.897847 + 0.440309i \(0.145131\pi\)
\(132\) 4.90606e34 0.502647
\(133\) −2.69096e33 −0.0243420
\(134\) −7.68300e34 −0.614188
\(135\) −4.70997e35 −3.33051
\(136\) 3.91570e34 0.245142
\(137\) 1.66110e35 0.921522 0.460761 0.887524i \(-0.347577\pi\)
0.460761 + 0.887524i \(0.347577\pi\)
\(138\) −1.23082e35 −0.605607
\(139\) −1.53290e35 −0.669531 −0.334766 0.942301i \(-0.608657\pi\)
−0.334766 + 0.942301i \(0.608657\pi\)
\(140\) −7.71698e34 −0.299457
\(141\) 8.28524e35 2.85884
\(142\) −1.69838e35 −0.521528
\(143\) −9.82539e34 −0.268723
\(144\) 2.23785e35 0.545570
\(145\) −8.79690e35 −1.91319
\(146\) 1.85151e35 0.359503
\(147\) 8.80357e35 1.52729
\(148\) −4.14225e35 −0.642566
\(149\) 3.23613e35 0.449213 0.224606 0.974450i \(-0.427890\pi\)
0.224606 + 0.974450i \(0.427890\pi\)
\(150\) 1.51573e36 1.88415
\(151\) −7.40848e35 −0.825295 −0.412648 0.910891i \(-0.635396\pi\)
−0.412648 + 0.910891i \(0.635396\pi\)
\(152\) −2.27136e34 −0.0226917
\(153\) −1.68765e36 −1.51312
\(154\) −1.87691e35 −0.151131
\(155\) −1.02053e36 −0.738502
\(156\) −6.53546e35 −0.425324
\(157\) 5.54604e35 0.324817 0.162408 0.986724i \(-0.448074\pi\)
0.162408 + 0.986724i \(0.448074\pi\)
\(158\) −6.03382e34 −0.0318237
\(159\) 2.83596e36 1.34787
\(160\) −6.51368e35 −0.279156
\(161\) 4.70876e35 0.182087
\(162\) −3.20005e36 −1.11727
\(163\) −2.15178e36 −0.678737 −0.339369 0.940653i \(-0.610213\pi\)
−0.339369 + 0.940653i \(0.610213\pi\)
\(164\) 2.75637e36 0.785986
\(165\) 6.15460e36 1.58750
\(166\) 1.06163e36 0.247851
\(167\) 9.54101e35 0.201731 0.100866 0.994900i \(-0.467839\pi\)
0.100866 + 0.994900i \(0.467839\pi\)
\(168\) −1.24845e36 −0.239203
\(169\) −4.44727e36 −0.772615
\(170\) 4.91221e36 0.774229
\(171\) 9.78945e35 0.140063
\(172\) −3.51318e36 −0.456542
\(173\) 8.72939e36 1.03092 0.515458 0.856915i \(-0.327622\pi\)
0.515458 + 0.856915i \(0.327622\pi\)
\(174\) −1.42316e37 −1.52823
\(175\) −5.79874e36 −0.566507
\(176\) −1.58425e36 −0.140885
\(177\) −3.97671e37 −3.22081
\(178\) 1.33360e37 0.984230
\(179\) 1.44217e37 0.970377 0.485189 0.874409i \(-0.338751\pi\)
0.485189 + 0.874409i \(0.338751\pi\)
\(180\) 2.80736e37 1.72307
\(181\) −6.19356e36 −0.346932 −0.173466 0.984840i \(-0.555497\pi\)
−0.173466 + 0.984840i \(0.555497\pi\)
\(182\) 2.50027e36 0.127882
\(183\) 6.06348e37 2.83321
\(184\) 3.97453e36 0.169743
\(185\) −5.19641e37 −2.02941
\(186\) −1.65100e37 −0.589908
\(187\) 1.19474e37 0.390739
\(188\) −2.67544e37 −0.801292
\(189\) 2.91509e37 0.799893
\(190\) −2.84940e36 −0.0716670
\(191\) −2.37825e37 −0.548538 −0.274269 0.961653i \(-0.588436\pi\)
−0.274269 + 0.961653i \(0.588436\pi\)
\(192\) −1.05378e37 −0.222987
\(193\) 7.58741e37 1.47366 0.736832 0.676076i \(-0.236321\pi\)
0.736832 + 0.676076i \(0.236321\pi\)
\(194\) −1.48640e37 −0.265098
\(195\) −8.19867e37 −1.34330
\(196\) −2.84282e37 −0.428079
\(197\) −1.46746e37 −0.203177 −0.101589 0.994827i \(-0.532393\pi\)
−0.101589 + 0.994827i \(0.532393\pi\)
\(198\) 6.82802e37 0.869600
\(199\) 5.46042e37 0.639956 0.319978 0.947425i \(-0.396324\pi\)
0.319978 + 0.947425i \(0.396324\pi\)
\(200\) −4.89455e37 −0.528102
\(201\) −1.55927e38 −1.54948
\(202\) −4.71764e37 −0.431943
\(203\) 5.44457e37 0.459493
\(204\) 7.94693e37 0.618446
\(205\) 3.45784e38 2.48237
\(206\) 1.58947e38 1.05304
\(207\) −1.71300e38 −1.04772
\(208\) 2.11041e37 0.119212
\(209\) −6.93027e36 −0.0361690
\(210\) −1.56616e38 −0.755473
\(211\) 2.20449e37 0.0983213 0.0491607 0.998791i \(-0.484345\pi\)
0.0491607 + 0.998791i \(0.484345\pi\)
\(212\) −9.15778e37 −0.377788
\(213\) −3.44687e38 −1.31571
\(214\) 2.71560e38 0.959485
\(215\) −4.40725e38 −1.44189
\(216\) 2.46054e38 0.745665
\(217\) 6.31624e37 0.177367
\(218\) −6.32023e37 −0.164513
\(219\) 3.75764e38 0.906958
\(220\) −1.98742e38 −0.444955
\(221\) −1.59154e38 −0.330631
\(222\) −8.40671e38 −1.62107
\(223\) −3.00366e38 −0.537799 −0.268900 0.963168i \(-0.586660\pi\)
−0.268900 + 0.963168i \(0.586660\pi\)
\(224\) 4.03144e37 0.0670453
\(225\) 2.10952e39 3.25966
\(226\) 7.80961e37 0.112161
\(227\) 7.55693e38 1.00906 0.504532 0.863393i \(-0.331665\pi\)
0.504532 + 0.863393i \(0.331665\pi\)
\(228\) −4.60974e37 −0.0572468
\(229\) −4.47790e38 −0.517355 −0.258678 0.965964i \(-0.583287\pi\)
−0.258678 + 0.965964i \(0.583287\pi\)
\(230\) 4.98601e38 0.536097
\(231\) −3.80920e38 −0.381273
\(232\) 4.59561e38 0.428342
\(233\) −8.40517e38 −0.729750 −0.364875 0.931057i \(-0.618888\pi\)
−0.364875 + 0.931057i \(0.618888\pi\)
\(234\) −9.09574e38 −0.735828
\(235\) −3.35631e39 −2.53071
\(236\) 1.28414e39 0.902746
\(237\) −1.22457e38 −0.0802851
\(238\) −3.04026e38 −0.185948
\(239\) 1.59672e39 0.911306 0.455653 0.890158i \(-0.349406\pi\)
0.455653 + 0.890158i \(0.349406\pi\)
\(240\) −1.32195e39 −0.704257
\(241\) 3.46992e39 1.72599 0.862996 0.505212i \(-0.168586\pi\)
0.862996 + 0.505212i \(0.168586\pi\)
\(242\) 1.03871e39 0.482546
\(243\) −1.63500e39 −0.709603
\(244\) −1.95800e39 −0.794109
\(245\) −3.56628e39 −1.35200
\(246\) 5.59407e39 1.98289
\(247\) 9.23194e37 0.0306051
\(248\) 5.33136e38 0.165343
\(249\) 2.15459e39 0.625280
\(250\) −2.02944e39 −0.551273
\(251\) −1.48419e39 −0.377461 −0.188731 0.982029i \(-0.560437\pi\)
−0.188731 + 0.982029i \(0.560437\pi\)
\(252\) −1.73753e39 −0.413831
\(253\) 1.21269e39 0.270558
\(254\) 1.42965e39 0.298863
\(255\) 9.96934e39 1.95323
\(256\) 3.40282e38 0.0625000
\(257\) −1.36871e39 −0.235730 −0.117865 0.993030i \(-0.537605\pi\)
−0.117865 + 0.993030i \(0.537605\pi\)
\(258\) −7.13001e39 −1.15177
\(259\) 3.21616e39 0.487406
\(260\) 2.64748e39 0.376507
\(261\) −1.98068e40 −2.64391
\(262\) −1.01318e40 −1.26975
\(263\) 6.59217e39 0.775816 0.387908 0.921698i \(-0.373198\pi\)
0.387908 + 0.921698i \(0.373198\pi\)
\(264\) −3.21524e39 −0.355425
\(265\) −1.14883e40 −1.19316
\(266\) 1.76355e38 0.0172124
\(267\) 2.70655e40 2.48302
\(268\) 5.03513e39 0.434297
\(269\) 5.00218e39 0.405739 0.202870 0.979206i \(-0.434973\pi\)
0.202870 + 0.979206i \(0.434973\pi\)
\(270\) 3.08672e40 2.35503
\(271\) −2.49179e40 −1.78862 −0.894308 0.447451i \(-0.852332\pi\)
−0.894308 + 0.447451i \(0.852332\pi\)
\(272\) −2.56619e39 −0.173342
\(273\) 5.07431e39 0.322622
\(274\) −1.08862e40 −0.651614
\(275\) −1.49340e40 −0.841757
\(276\) 8.06633e39 0.428229
\(277\) 3.71250e40 1.85673 0.928367 0.371665i \(-0.121213\pi\)
0.928367 + 0.371665i \(0.121213\pi\)
\(278\) 1.00460e40 0.473430
\(279\) −2.29779e40 −1.02056
\(280\) 5.05740e39 0.211748
\(281\) −2.42056e40 −0.955570 −0.477785 0.878477i \(-0.658560\pi\)
−0.477785 + 0.878477i \(0.658560\pi\)
\(282\) −5.42982e40 −2.02151
\(283\) 2.77995e40 0.976249 0.488125 0.872774i \(-0.337681\pi\)
0.488125 + 0.872774i \(0.337681\pi\)
\(284\) 1.11305e40 0.368776
\(285\) −5.78287e39 −0.180802
\(286\) 6.43917e39 0.190016
\(287\) −2.14012e40 −0.596194
\(288\) −1.46660e40 −0.385776
\(289\) −2.09019e40 −0.519243
\(290\) 5.76514e40 1.35283
\(291\) −3.01666e40 −0.668792
\(292\) −1.21340e40 −0.254207
\(293\) 1.14215e40 0.226155 0.113078 0.993586i \(-0.463929\pi\)
0.113078 + 0.993586i \(0.463929\pi\)
\(294\) −5.76951e40 −1.07996
\(295\) 1.61094e41 2.85113
\(296\) 2.71466e40 0.454363
\(297\) 7.50749e40 1.18854
\(298\) −2.12083e40 −0.317642
\(299\) −1.61545e40 −0.228938
\(300\) −9.93350e40 −1.33230
\(301\) 2.72773e40 0.346301
\(302\) 4.85522e40 0.583572
\(303\) −9.57446e40 −1.08971
\(304\) 1.48856e39 0.0160455
\(305\) −2.45629e41 −2.50803
\(306\) 1.10602e41 1.06994
\(307\) 1.67395e40 0.153447 0.0767236 0.997052i \(-0.475554\pi\)
0.0767236 + 0.997052i \(0.475554\pi\)
\(308\) 1.23005e40 0.106865
\(309\) 3.22584e41 2.65661
\(310\) 6.68813e40 0.522200
\(311\) −2.59200e41 −1.91906 −0.959528 0.281612i \(-0.909131\pi\)
−0.959528 + 0.281612i \(0.909131\pi\)
\(312\) 4.28308e40 0.300750
\(313\) −1.65212e41 −1.10042 −0.550212 0.835025i \(-0.685453\pi\)
−0.550212 + 0.835025i \(0.685453\pi\)
\(314\) −3.63465e40 −0.229680
\(315\) −2.17971e41 −1.30700
\(316\) 3.95433e39 0.0225028
\(317\) 5.16289e40 0.278879 0.139439 0.990231i \(-0.455470\pi\)
0.139439 + 0.990231i \(0.455470\pi\)
\(318\) −1.85858e41 −0.953087
\(319\) 1.40219e41 0.682748
\(320\) 4.26881e40 0.197393
\(321\) 5.51131e41 2.42060
\(322\) −3.08594e40 −0.128755
\(323\) −1.12258e40 −0.0445015
\(324\) 2.09718e41 0.790031
\(325\) 1.98939e41 0.712268
\(326\) 1.41019e41 0.479940
\(327\) −1.28269e41 −0.415035
\(328\) −1.80642e41 −0.555776
\(329\) 2.07729e41 0.607805
\(330\) −4.03348e41 −1.12254
\(331\) −2.93097e40 −0.0775977 −0.0387989 0.999247i \(-0.512353\pi\)
−0.0387989 + 0.999247i \(0.512353\pi\)
\(332\) −6.95751e40 −0.175257
\(333\) −1.17001e42 −2.80452
\(334\) −6.25280e40 −0.142645
\(335\) 6.31652e41 1.37163
\(336\) 8.18183e40 0.169142
\(337\) −5.29809e41 −1.04286 −0.521429 0.853294i \(-0.674601\pi\)
−0.521429 + 0.853294i \(0.674601\pi\)
\(338\) 2.91456e41 0.546321
\(339\) 1.58496e41 0.282960
\(340\) −3.21926e41 −0.547463
\(341\) 1.62668e41 0.263545
\(342\) −6.41561e40 −0.0990393
\(343\) 4.78532e41 0.703976
\(344\) 2.30240e41 0.322824
\(345\) 1.01191e42 1.35247
\(346\) −5.72089e41 −0.728967
\(347\) 2.52378e40 0.0306630 0.0153315 0.999882i \(-0.495120\pi\)
0.0153315 + 0.999882i \(0.495120\pi\)
\(348\) 9.32680e41 1.08063
\(349\) −1.27613e42 −1.41018 −0.705092 0.709116i \(-0.749094\pi\)
−0.705092 + 0.709116i \(0.749094\pi\)
\(350\) 3.80026e41 0.400581
\(351\) −1.00009e42 −1.00570
\(352\) 1.03825e41 0.0996206
\(353\) −6.97275e41 −0.638442 −0.319221 0.947680i \(-0.603421\pi\)
−0.319221 + 0.947680i \(0.603421\pi\)
\(354\) 2.60617e42 2.27745
\(355\) 1.39631e42 1.16470
\(356\) −8.73990e41 −0.695955
\(357\) −6.17022e41 −0.469110
\(358\) −9.45139e41 −0.686160
\(359\) 8.87202e41 0.615126 0.307563 0.951528i \(-0.400487\pi\)
0.307563 + 0.951528i \(0.400487\pi\)
\(360\) −1.83983e42 −1.21839
\(361\) −1.57426e42 −0.995881
\(362\) 4.05901e41 0.245318
\(363\) 2.10806e42 1.21737
\(364\) −1.63858e41 −0.0904262
\(365\) −1.52220e42 −0.802860
\(366\) −3.97376e42 −2.00338
\(367\) −3.14846e42 −1.51743 −0.758714 0.651424i \(-0.774172\pi\)
−0.758714 + 0.651424i \(0.774172\pi\)
\(368\) −2.60475e41 −0.120026
\(369\) 7.78556e42 3.43048
\(370\) 3.40552e42 1.43501
\(371\) 7.11035e41 0.286564
\(372\) 1.08200e42 0.417128
\(373\) −9.37335e41 −0.345700 −0.172850 0.984948i \(-0.555298\pi\)
−0.172850 + 0.984948i \(0.555298\pi\)
\(374\) −7.82985e41 −0.276294
\(375\) −4.11877e42 −1.39076
\(376\) 1.75338e42 0.566599
\(377\) −1.86788e42 −0.577720
\(378\) −1.91043e42 −0.565610
\(379\) 3.51502e42 0.996278 0.498139 0.867097i \(-0.334017\pi\)
0.498139 + 0.867097i \(0.334017\pi\)
\(380\) 1.86738e41 0.0506762
\(381\) 2.90147e42 0.753974
\(382\) 1.55861e42 0.387875
\(383\) 4.14787e42 0.988657 0.494329 0.869275i \(-0.335414\pi\)
0.494329 + 0.869275i \(0.335414\pi\)
\(384\) 6.90604e41 0.157675
\(385\) 1.54309e42 0.337512
\(386\) −4.97249e42 −1.04204
\(387\) −9.92321e42 −1.99260
\(388\) 9.74127e41 0.187453
\(389\) −1.63044e42 −0.300703 −0.150351 0.988633i \(-0.548041\pi\)
−0.150351 + 0.988633i \(0.548041\pi\)
\(390\) 5.37308e42 0.949855
\(391\) 1.96434e42 0.332889
\(392\) 1.86307e42 0.302697
\(393\) −2.05626e43 −3.20333
\(394\) 9.61717e41 0.143668
\(395\) 4.96066e41 0.0710702
\(396\) −4.47481e42 −0.614900
\(397\) 7.05205e42 0.929548 0.464774 0.885429i \(-0.346136\pi\)
0.464774 + 0.885429i \(0.346136\pi\)
\(398\) −3.57854e42 −0.452517
\(399\) 3.57913e41 0.0434235
\(400\) 3.20769e42 0.373424
\(401\) 9.06730e42 1.01297 0.506484 0.862249i \(-0.330945\pi\)
0.506484 + 0.862249i \(0.330945\pi\)
\(402\) 1.02188e43 1.09565
\(403\) −2.16693e42 −0.223003
\(404\) 3.09175e42 0.305430
\(405\) 2.63090e43 2.49515
\(406\) −3.56815e42 −0.324911
\(407\) 8.28285e42 0.724222
\(408\) −5.20810e42 −0.437307
\(409\) 1.80287e42 0.145388 0.0726942 0.997354i \(-0.476840\pi\)
0.0726942 + 0.997354i \(0.476840\pi\)
\(410\) −2.26613e43 −1.75530
\(411\) −2.20935e43 −1.64390
\(412\) −1.04168e43 −0.744611
\(413\) −9.97044e42 −0.684760
\(414\) 1.12263e43 0.740853
\(415\) −8.72813e42 −0.553512
\(416\) −1.38308e42 −0.0842959
\(417\) 2.03885e43 1.19437
\(418\) 4.54182e41 0.0255753
\(419\) 2.18879e43 1.18488 0.592438 0.805616i \(-0.298166\pi\)
0.592438 + 0.805616i \(0.298166\pi\)
\(420\) 1.02640e43 0.534200
\(421\) 1.25344e43 0.627263 0.313632 0.949545i \(-0.398454\pi\)
0.313632 + 0.949545i \(0.398454\pi\)
\(422\) −1.44473e42 −0.0695237
\(423\) −7.55696e43 −3.49729
\(424\) 6.00164e42 0.267137
\(425\) −2.41904e43 −1.03568
\(426\) 2.25894e43 0.930351
\(427\) 1.52024e43 0.602356
\(428\) −1.77969e43 −0.678458
\(429\) 1.30683e43 0.479374
\(430\) 2.88833e43 1.01957
\(431\) 1.38369e43 0.470072 0.235036 0.971987i \(-0.424479\pi\)
0.235036 + 0.971987i \(0.424479\pi\)
\(432\) −1.61254e43 −0.527265
\(433\) −4.86806e43 −1.53216 −0.766082 0.642743i \(-0.777796\pi\)
−0.766082 + 0.642743i \(0.777796\pi\)
\(434\) −4.13941e42 −0.125418
\(435\) 1.17004e44 3.41293
\(436\) 4.14203e42 0.116328
\(437\) −1.13944e42 −0.0308141
\(438\) −2.46261e43 −0.641316
\(439\) 2.44480e43 0.613167 0.306584 0.951844i \(-0.400814\pi\)
0.306584 + 0.951844i \(0.400814\pi\)
\(440\) 1.30248e43 0.314631
\(441\) −8.02973e43 −1.86837
\(442\) 1.04303e43 0.233792
\(443\) 3.59045e43 0.775331 0.387666 0.921800i \(-0.373282\pi\)
0.387666 + 0.921800i \(0.373282\pi\)
\(444\) 5.50942e43 1.14627
\(445\) −1.09641e44 −2.19803
\(446\) 1.96848e43 0.380282
\(447\) −4.30423e43 −0.801348
\(448\) −2.64205e42 −0.0474082
\(449\) 8.52869e43 1.47509 0.737544 0.675299i \(-0.235986\pi\)
0.737544 + 0.675299i \(0.235986\pi\)
\(450\) −1.38250e44 −2.30493
\(451\) −5.51165e43 −0.885867
\(452\) −5.11811e42 −0.0793095
\(453\) 9.85369e43 1.47224
\(454\) −4.95251e43 −0.713516
\(455\) −2.05558e43 −0.285592
\(456\) 3.02104e42 0.0404796
\(457\) 6.07893e43 0.785615 0.392807 0.919621i \(-0.371504\pi\)
0.392807 + 0.919621i \(0.371504\pi\)
\(458\) 2.93464e43 0.365825
\(459\) 1.21608e44 1.46235
\(460\) −3.26763e43 −0.379078
\(461\) −1.43000e44 −1.60055 −0.800277 0.599631i \(-0.795314\pi\)
−0.800277 + 0.599631i \(0.795314\pi\)
\(462\) 2.49640e43 0.269601
\(463\) 1.40454e43 0.146369 0.0731844 0.997318i \(-0.476684\pi\)
0.0731844 + 0.997318i \(0.476684\pi\)
\(464\) −3.01178e43 −0.302884
\(465\) 1.35736e44 1.31741
\(466\) 5.50841e43 0.516011
\(467\) −5.13667e43 −0.464465 −0.232233 0.972660i \(-0.574603\pi\)
−0.232233 + 0.972660i \(0.574603\pi\)
\(468\) 5.96098e43 0.520309
\(469\) −3.90941e43 −0.329427
\(470\) 2.19959e44 1.78948
\(471\) −7.37654e43 −0.579439
\(472\) −8.41576e43 −0.638338
\(473\) 7.02496e43 0.514559
\(474\) 8.02532e42 0.0567701
\(475\) 1.40320e43 0.0958682
\(476\) 1.99246e43 0.131485
\(477\) −2.58668e44 −1.64888
\(478\) −1.04643e44 −0.644390
\(479\) −1.67218e44 −0.994823 −0.497412 0.867515i \(-0.665716\pi\)
−0.497412 + 0.867515i \(0.665716\pi\)
\(480\) 8.66356e43 0.497985
\(481\) −1.10337e44 −0.612814
\(482\) −2.27405e44 −1.22046
\(483\) −6.26292e43 −0.324825
\(484\) −6.80726e43 −0.341212
\(485\) 1.22203e44 0.592030
\(486\) 1.07152e44 0.501765
\(487\) 1.67480e44 0.758117 0.379058 0.925373i \(-0.376248\pi\)
0.379058 + 0.925373i \(0.376248\pi\)
\(488\) 1.28319e44 0.561520
\(489\) 2.86198e44 1.21080
\(490\) 2.33720e44 0.956006
\(491\) 2.77223e44 1.09644 0.548220 0.836334i \(-0.315306\pi\)
0.548220 + 0.836334i \(0.315306\pi\)
\(492\) −3.66613e44 −1.40212
\(493\) 2.27129e44 0.840037
\(494\) −6.05025e42 −0.0216411
\(495\) −5.61361e44 −1.94203
\(496\) −3.49396e43 −0.116915
\(497\) −8.64204e43 −0.279728
\(498\) −1.41203e44 −0.442140
\(499\) 4.56174e44 1.38188 0.690942 0.722910i \(-0.257196\pi\)
0.690942 + 0.722910i \(0.257196\pi\)
\(500\) 1.33002e44 0.389809
\(501\) −1.26901e44 −0.359867
\(502\) 9.72677e43 0.266905
\(503\) −5.38096e44 −1.42885 −0.714427 0.699710i \(-0.753312\pi\)
−0.714427 + 0.699710i \(0.753312\pi\)
\(504\) 1.13871e44 0.292623
\(505\) 3.87857e44 0.964636
\(506\) −7.94748e43 −0.191314
\(507\) 5.91512e44 1.37826
\(508\) −9.36933e43 −0.211328
\(509\) 6.21555e43 0.135718 0.0678588 0.997695i \(-0.478383\pi\)
0.0678588 + 0.997695i \(0.478383\pi\)
\(510\) −6.53351e44 −1.38114
\(511\) 9.42120e43 0.192824
\(512\) −2.23007e43 −0.0441942
\(513\) −7.05404e43 −0.135363
\(514\) 8.96998e43 0.166686
\(515\) −1.30677e45 −2.35170
\(516\) 4.67272e44 0.814423
\(517\) 5.34982e44 0.903119
\(518\) −2.10774e44 −0.344648
\(519\) −1.16106e45 −1.83904
\(520\) −1.73505e44 −0.266231
\(521\) −7.97253e44 −1.18515 −0.592576 0.805515i \(-0.701889\pi\)
−0.592576 + 0.805515i \(0.701889\pi\)
\(522\) 1.29806e45 1.86952
\(523\) 1.35653e44 0.189301 0.0946504 0.995511i \(-0.469827\pi\)
0.0946504 + 0.995511i \(0.469827\pi\)
\(524\) 6.64000e44 0.897847
\(525\) 7.71264e44 1.01059
\(526\) −4.32025e44 −0.548585
\(527\) 2.63492e44 0.324260
\(528\) 2.10714e44 0.251324
\(529\) −6.65620e44 −0.769499
\(530\) 7.52900e44 0.843695
\(531\) 3.62715e45 3.94009
\(532\) −1.15576e43 −0.0121710
\(533\) 7.34218e44 0.749593
\(534\) −1.77377e45 −1.75576
\(535\) −2.23261e45 −2.14277
\(536\) −3.29982e44 −0.307094
\(537\) −1.91816e45 −1.73105
\(538\) −3.27823e44 −0.286901
\(539\) 5.68450e44 0.482479
\(540\) −2.02292e45 −1.66526
\(541\) 1.73198e45 1.38289 0.691445 0.722429i \(-0.256974\pi\)
0.691445 + 0.722429i \(0.256974\pi\)
\(542\) 1.63302e45 1.26474
\(543\) 8.23778e44 0.618889
\(544\) 1.68178e44 0.122571
\(545\) 5.19613e44 0.367399
\(546\) −3.32550e44 −0.228128
\(547\) −1.10462e45 −0.735227 −0.367614 0.929979i \(-0.619825\pi\)
−0.367614 + 0.929979i \(0.619825\pi\)
\(548\) 7.13436e44 0.460761
\(549\) −5.53049e45 −3.46593
\(550\) 9.78714e44 0.595212
\(551\) −1.31750e44 −0.0777586
\(552\) −5.28635e44 −0.302804
\(553\) −3.07025e43 −0.0170690
\(554\) −2.43302e45 −1.31291
\(555\) 6.91151e45 3.62025
\(556\) −6.58378e44 −0.334766
\(557\) 1.13068e45 0.558120 0.279060 0.960274i \(-0.409977\pi\)
0.279060 + 0.960274i \(0.409977\pi\)
\(558\) 1.50588e45 0.721648
\(559\) −9.35808e44 −0.435404
\(560\) −3.31442e44 −0.149729
\(561\) −1.58907e45 −0.697037
\(562\) 1.58634e45 0.675690
\(563\) 2.85343e45 1.18026 0.590131 0.807307i \(-0.299076\pi\)
0.590131 + 0.807307i \(0.299076\pi\)
\(564\) 3.55849e45 1.42942
\(565\) −6.42061e44 −0.250482
\(566\) −1.82187e45 −0.690312
\(567\) −1.62831e45 −0.599263
\(568\) −7.29449e44 −0.260764
\(569\) 4.66376e44 0.161951 0.0809756 0.996716i \(-0.474196\pi\)
0.0809756 + 0.996716i \(0.474196\pi\)
\(570\) 3.78986e44 0.127846
\(571\) −5.04939e45 −1.65479 −0.827396 0.561619i \(-0.810179\pi\)
−0.827396 + 0.561619i \(0.810179\pi\)
\(572\) −4.21997e44 −0.134362
\(573\) 3.16320e45 0.978533
\(574\) 1.40255e45 0.421573
\(575\) −2.45538e45 −0.717132
\(576\) 9.61150e44 0.272785
\(577\) 2.74486e44 0.0757040 0.0378520 0.999283i \(-0.487948\pi\)
0.0378520 + 0.999283i \(0.487948\pi\)
\(578\) 1.36983e45 0.367161
\(579\) −1.00917e46 −2.62886
\(580\) −3.77824e45 −0.956594
\(581\) 5.40200e44 0.132938
\(582\) 1.97700e45 0.472908
\(583\) 1.83119e45 0.425797
\(584\) 7.95216e44 0.179752
\(585\) 7.47799e45 1.64329
\(586\) −7.48518e44 −0.159916
\(587\) 3.25926e45 0.677002 0.338501 0.940966i \(-0.390080\pi\)
0.338501 + 0.940966i \(0.390080\pi\)
\(588\) 3.78111e45 0.763647
\(589\) −1.52843e44 −0.0300153
\(590\) −1.05575e46 −2.01605
\(591\) 1.95181e45 0.362446
\(592\) −1.77908e45 −0.321283
\(593\) 1.56542e45 0.274933 0.137467 0.990506i \(-0.456104\pi\)
0.137467 + 0.990506i \(0.456104\pi\)
\(594\) −4.92011e45 −0.840423
\(595\) 2.49953e45 0.415267
\(596\) 1.38991e45 0.224606
\(597\) −7.26266e45 −1.14161
\(598\) 1.05870e45 0.161884
\(599\) 2.66724e45 0.396753 0.198376 0.980126i \(-0.436433\pi\)
0.198376 + 0.980126i \(0.436433\pi\)
\(600\) 6.51002e45 0.942077
\(601\) 1.01736e46 1.43234 0.716170 0.697926i \(-0.245893\pi\)
0.716170 + 0.697926i \(0.245893\pi\)
\(602\) −1.78764e45 −0.244872
\(603\) 1.42221e46 1.89551
\(604\) −3.18192e45 −0.412648
\(605\) −8.53964e45 −1.07765
\(606\) 6.27472e45 0.770541
\(607\) −1.91910e45 −0.229343 −0.114671 0.993403i \(-0.536581\pi\)
−0.114671 + 0.993403i \(0.536581\pi\)
\(608\) −9.75543e43 −0.0113459
\(609\) −7.24158e45 −0.819687
\(610\) 1.60975e46 1.77344
\(611\) −7.12659e45 −0.764191
\(612\) −7.24839e45 −0.756559
\(613\) −9.21265e45 −0.936023 −0.468012 0.883722i \(-0.655029\pi\)
−0.468012 + 0.883722i \(0.655029\pi\)
\(614\) −1.09704e45 −0.108504
\(615\) −4.59912e46 −4.42828
\(616\) −8.06128e44 −0.0755653
\(617\) 1.85344e46 1.69150 0.845752 0.533577i \(-0.179152\pi\)
0.845752 + 0.533577i \(0.179152\pi\)
\(618\) −2.11409e46 −1.87851
\(619\) 1.79796e46 1.55555 0.777777 0.628540i \(-0.216347\pi\)
0.777777 + 0.628540i \(0.216347\pi\)
\(620\) −4.38313e45 −0.369251
\(621\) 1.23435e46 1.01257
\(622\) 1.69869e46 1.35698
\(623\) 6.78590e45 0.527904
\(624\) −2.80696e45 −0.212662
\(625\) −3.55844e45 −0.262567
\(626\) 1.08273e46 0.778117
\(627\) 9.21764e44 0.0645217
\(628\) 2.38200e45 0.162408
\(629\) 1.34167e46 0.891067
\(630\) 1.42850e46 0.924188
\(631\) −1.17228e46 −0.738831 −0.369416 0.929264i \(-0.620442\pi\)
−0.369416 + 0.929264i \(0.620442\pi\)
\(632\) −2.59151e44 −0.0159119
\(633\) −2.93209e45 −0.175395
\(634\) −3.38355e45 −0.197197
\(635\) −1.17537e46 −0.667435
\(636\) 1.21804e46 0.673934
\(637\) −7.57244e45 −0.408258
\(638\) −9.18938e45 −0.482775
\(639\) 3.14389e46 1.60954
\(640\) −2.79761e45 −0.139578
\(641\) −1.41216e46 −0.686638 −0.343319 0.939219i \(-0.611551\pi\)
−0.343319 + 0.939219i \(0.611551\pi\)
\(642\) −3.61190e46 −1.71162
\(643\) −4.07405e45 −0.188168 −0.0940838 0.995564i \(-0.529992\pi\)
−0.0940838 + 0.995564i \(0.529992\pi\)
\(644\) 2.02240e45 0.0910437
\(645\) 5.86188e46 2.57218
\(646\) 7.35693e44 0.0314673
\(647\) −3.65004e45 −0.152187 −0.0760933 0.997101i \(-0.524245\pi\)
−0.0760933 + 0.997101i \(0.524245\pi\)
\(648\) −1.37441e46 −0.558636
\(649\) −2.56778e46 −1.01747
\(650\) −1.30376e46 −0.503650
\(651\) −8.40096e45 −0.316404
\(652\) −9.24181e45 −0.339369
\(653\) −5.21698e46 −1.86789 −0.933946 0.357414i \(-0.883659\pi\)
−0.933946 + 0.357414i \(0.883659\pi\)
\(654\) 8.40626e45 0.293474
\(655\) 8.32982e46 2.83566
\(656\) 1.18385e46 0.392993
\(657\) −3.42734e46 −1.10950
\(658\) −1.36137e46 −0.429783
\(659\) −5.05202e45 −0.155545 −0.0777724 0.996971i \(-0.524781\pi\)
−0.0777724 + 0.996971i \(0.524781\pi\)
\(660\) 2.64338e46 0.793752
\(661\) −5.49372e46 −1.60895 −0.804475 0.593987i \(-0.797553\pi\)
−0.804475 + 0.593987i \(0.797553\pi\)
\(662\) 1.92084e45 0.0548699
\(663\) 2.11683e46 0.589811
\(664\) 4.55968e45 0.123925
\(665\) −1.44989e45 −0.0384395
\(666\) 7.66775e46 1.98309
\(667\) 2.30541e46 0.581665
\(668\) 4.09783e45 0.100866
\(669\) 3.99503e46 0.959377
\(670\) −4.13959e46 −0.969892
\(671\) 3.91522e46 0.895023
\(672\) −5.36204e45 −0.119602
\(673\) 3.75226e45 0.0816666 0.0408333 0.999166i \(-0.486999\pi\)
0.0408333 + 0.999166i \(0.486999\pi\)
\(674\) 3.47215e46 0.737413
\(675\) −1.52007e47 −3.15030
\(676\) −1.91009e46 −0.386307
\(677\) 4.64616e46 0.917025 0.458513 0.888688i \(-0.348382\pi\)
0.458513 + 0.888688i \(0.348382\pi\)
\(678\) −1.03872e46 −0.200083
\(679\) −7.56339e45 −0.142189
\(680\) 2.10978e46 0.387115
\(681\) −1.00511e47 −1.80006
\(682\) −1.06606e46 −0.186354
\(683\) 8.92876e46 1.52352 0.761762 0.647857i \(-0.224334\pi\)
0.761762 + 0.647857i \(0.224334\pi\)
\(684\) 4.20454e45 0.0700313
\(685\) 8.94997e46 1.45522
\(686\) −3.13611e46 −0.497786
\(687\) 5.95585e46 0.922907
\(688\) −1.50890e46 −0.228271
\(689\) −2.43937e46 −0.360296
\(690\) −6.63167e46 −0.956341
\(691\) −1.40554e47 −1.97904 −0.989522 0.144382i \(-0.953881\pi\)
−0.989522 + 0.144382i \(0.953881\pi\)
\(692\) 3.74924e46 0.515458
\(693\) 3.47437e46 0.466420
\(694\) −1.65398e45 −0.0216820
\(695\) −8.25928e46 −1.05729
\(696\) −6.11241e46 −0.764117
\(697\) −8.92787e46 −1.08995
\(698\) 8.36325e46 0.997150
\(699\) 1.11793e47 1.30180
\(700\) −2.49054e46 −0.283254
\(701\) −5.92277e46 −0.657926 −0.328963 0.944343i \(-0.606699\pi\)
−0.328963 + 0.944343i \(0.606699\pi\)
\(702\) 6.55417e46 0.711140
\(703\) −7.78257e45 −0.0824821
\(704\) −6.80429e45 −0.0704424
\(705\) 4.46408e47 4.51452
\(706\) 4.56966e46 0.451447
\(707\) −2.40052e46 −0.231678
\(708\) −1.70798e47 −1.61040
\(709\) 6.19740e46 0.570882 0.285441 0.958396i \(-0.407860\pi\)
0.285441 + 0.958396i \(0.407860\pi\)
\(710\) −9.15086e46 −0.823568
\(711\) 1.11693e46 0.0982146
\(712\) 5.72778e46 0.492115
\(713\) 2.67451e46 0.224526
\(714\) 4.04371e46 0.331711
\(715\) −5.29391e46 −0.424353
\(716\) 6.19406e46 0.485189
\(717\) −2.12373e47 −1.62567
\(718\) −5.81436e46 −0.434960
\(719\) −2.33935e46 −0.171028 −0.0855142 0.996337i \(-0.527253\pi\)
−0.0855142 + 0.996337i \(0.527253\pi\)
\(720\) 1.20575e47 0.861533
\(721\) 8.08787e46 0.564810
\(722\) 1.03171e47 0.704194
\(723\) −4.61519e47 −3.07899
\(724\) −2.66011e46 −0.173466
\(725\) −2.83907e47 −1.80967
\(726\) −1.38154e47 −0.860812
\(727\) 3.68313e46 0.224336 0.112168 0.993689i \(-0.464220\pi\)
0.112168 + 0.993689i \(0.464220\pi\)
\(728\) 1.07386e46 0.0639410
\(729\) −5.39781e46 −0.314205
\(730\) 9.97590e46 0.567708
\(731\) 1.13792e47 0.633102
\(732\) 2.60425e47 1.41661
\(733\) 1.06749e47 0.567740 0.283870 0.958863i \(-0.408382\pi\)
0.283870 + 0.958863i \(0.408382\pi\)
\(734\) 2.06337e47 1.07298
\(735\) 4.74336e47 2.41182
\(736\) 1.70705e46 0.0848715
\(737\) −1.00683e47 −0.489486
\(738\) −5.10234e47 −2.42572
\(739\) 1.86383e47 0.866509 0.433254 0.901272i \(-0.357365\pi\)
0.433254 + 0.901272i \(0.357365\pi\)
\(740\) −2.23184e47 −1.01470
\(741\) −1.22790e46 −0.0545962
\(742\) −4.65984e46 −0.202631
\(743\) −1.41560e47 −0.602040 −0.301020 0.953618i \(-0.597327\pi\)
−0.301020 + 0.953618i \(0.597327\pi\)
\(744\) −7.09100e46 −0.294954
\(745\) 1.74362e47 0.709372
\(746\) 6.14292e46 0.244447
\(747\) −1.96520e47 −0.764920
\(748\) 5.13137e46 0.195370
\(749\) 1.38180e47 0.514631
\(750\) 2.69927e47 0.983413
\(751\) 2.61886e47 0.933368 0.466684 0.884424i \(-0.345449\pi\)
0.466684 + 0.884424i \(0.345449\pi\)
\(752\) −1.14909e47 −0.400646
\(753\) 1.97405e47 0.673351
\(754\) 1.22414e47 0.408509
\(755\) −3.99168e47 −1.30326
\(756\) 1.25202e47 0.399947
\(757\) 2.21676e47 0.692846 0.346423 0.938078i \(-0.387396\pi\)
0.346423 + 0.938078i \(0.387396\pi\)
\(758\) −2.30360e47 −0.704475
\(759\) −1.61294e47 −0.482648
\(760\) −1.22381e46 −0.0358335
\(761\) 1.15570e47 0.331131 0.165565 0.986199i \(-0.447055\pi\)
0.165565 + 0.986199i \(0.447055\pi\)
\(762\) −1.90151e47 −0.533140
\(763\) −3.21598e46 −0.0882386
\(764\) −1.02145e47 −0.274269
\(765\) −9.09302e47 −2.38943
\(766\) −2.71835e47 −0.699086
\(767\) 3.42058e47 0.860947
\(768\) −4.52594e46 −0.111493
\(769\) −3.82649e47 −0.922605 −0.461302 0.887243i \(-0.652618\pi\)
−0.461302 + 0.887243i \(0.652618\pi\)
\(770\) −1.01128e47 −0.238657
\(771\) 1.82046e47 0.420518
\(772\) 3.25877e47 0.736832
\(773\) 7.89533e47 1.74746 0.873732 0.486407i \(-0.161693\pi\)
0.873732 + 0.486407i \(0.161693\pi\)
\(774\) 6.50327e47 1.40898
\(775\) −3.29360e47 −0.698542
\(776\) −6.38404e46 −0.132549
\(777\) −4.27767e47 −0.869481
\(778\) 1.06853e47 0.212629
\(779\) 5.17875e46 0.100892
\(780\) −3.52130e47 −0.671649
\(781\) −2.22566e47 −0.415639
\(782\) −1.28735e47 −0.235388
\(783\) 1.42723e48 2.55520
\(784\) −1.22098e47 −0.214039
\(785\) 2.98820e47 0.512933
\(786\) 1.34759e48 2.26509
\(787\) −1.15500e48 −1.90107 −0.950534 0.310620i \(-0.899463\pi\)
−0.950534 + 0.310620i \(0.899463\pi\)
\(788\) −6.30271e46 −0.101589
\(789\) −8.76795e47 −1.38397
\(790\) −3.25102e46 −0.0502542
\(791\) 3.97384e46 0.0601587
\(792\) 2.93261e47 0.434800
\(793\) −5.21554e47 −0.757341
\(794\) −4.62163e47 −0.657290
\(795\) 1.52801e48 2.12848
\(796\) 2.34523e47 0.319978
\(797\) 4.08413e47 0.545805 0.272902 0.962042i \(-0.412016\pi\)
0.272902 + 0.962042i \(0.412016\pi\)
\(798\) −2.34562e46 −0.0307050
\(799\) 8.66573e47 1.11118
\(800\) −2.10219e47 −0.264051
\(801\) −2.46864e48 −3.03754
\(802\) −5.94235e47 −0.716277
\(803\) 2.42632e47 0.286512
\(804\) −6.69700e47 −0.774739
\(805\) 2.53708e47 0.287542
\(806\) 1.42012e47 0.157687
\(807\) −6.65318e47 −0.723796
\(808\) −2.02621e47 −0.215972
\(809\) 7.61932e47 0.795730 0.397865 0.917444i \(-0.369751\pi\)
0.397865 + 0.917444i \(0.369751\pi\)
\(810\) −1.72418e48 −1.76433
\(811\) −1.03654e48 −1.03930 −0.519651 0.854378i \(-0.673938\pi\)
−0.519651 + 0.854378i \(0.673938\pi\)
\(812\) 2.33843e47 0.229747
\(813\) 3.31421e48 3.19070
\(814\) −5.42825e47 −0.512103
\(815\) −1.15938e48 −1.07182
\(816\) 3.41318e47 0.309223
\(817\) −6.60065e46 −0.0586034
\(818\) −1.18153e47 −0.102805
\(819\) −4.62827e47 −0.394670
\(820\) 1.48513e48 1.24118
\(821\) 8.66860e47 0.710047 0.355023 0.934857i \(-0.384473\pi\)
0.355023 + 0.934857i \(0.384473\pi\)
\(822\) 1.44792e48 1.16241
\(823\) 6.30492e47 0.496115 0.248058 0.968745i \(-0.420208\pi\)
0.248058 + 0.968745i \(0.420208\pi\)
\(824\) 6.82674e47 0.526519
\(825\) 1.98630e48 1.50161
\(826\) 6.53423e47 0.484199
\(827\) 2.94601e47 0.213990 0.106995 0.994260i \(-0.465877\pi\)
0.106995 + 0.994260i \(0.465877\pi\)
\(828\) −7.35729e47 −0.523862
\(829\) −1.63460e48 −1.14094 −0.570470 0.821318i \(-0.693239\pi\)
−0.570470 + 0.821318i \(0.693239\pi\)
\(830\) 5.72006e47 0.391392
\(831\) −4.93782e48 −3.31222
\(832\) 9.06413e46 0.0596062
\(833\) 9.20787e47 0.593631
\(834\) −1.33618e48 −0.844549
\(835\) 5.14069e47 0.318562
\(836\) −2.97653e46 −0.0180845
\(837\) 1.65573e48 0.986323
\(838\) −1.43445e48 −0.837833
\(839\) −2.42256e48 −1.38740 −0.693702 0.720262i \(-0.744021\pi\)
−0.693702 + 0.720262i \(0.744021\pi\)
\(840\) −6.72663e47 −0.377737
\(841\) 8.49591e47 0.467817
\(842\) −8.21456e47 −0.443542
\(843\) 3.21949e48 1.70464
\(844\) 9.46821e46 0.0491607
\(845\) −2.39619e48 −1.22007
\(846\) 4.95253e48 2.47295
\(847\) 5.28535e47 0.258820
\(848\) −3.93324e47 −0.188894
\(849\) −3.69749e48 −1.74153
\(850\) 1.58534e48 0.732335
\(851\) 1.36183e48 0.616999
\(852\) −1.48042e48 −0.657857
\(853\) −2.74926e48 −1.19827 −0.599137 0.800647i \(-0.704489\pi\)
−0.599137 + 0.800647i \(0.704489\pi\)
\(854\) −9.96307e47 −0.425930
\(855\) 5.27455e47 0.221179
\(856\) 1.16634e48 0.479742
\(857\) −7.23564e47 −0.291940 −0.145970 0.989289i \(-0.546630\pi\)
−0.145970 + 0.989289i \(0.546630\pi\)
\(858\) −8.56446e47 −0.338969
\(859\) −8.35479e47 −0.324376 −0.162188 0.986760i \(-0.551855\pi\)
−0.162188 + 0.986760i \(0.551855\pi\)
\(860\) −1.89290e48 −0.720946
\(861\) 2.84648e48 1.06355
\(862\) −9.06816e47 −0.332391
\(863\) 6.91469e46 0.0248653 0.0124327 0.999923i \(-0.496042\pi\)
0.0124327 + 0.999923i \(0.496042\pi\)
\(864\) 1.05679e48 0.372833
\(865\) 4.70339e48 1.62796
\(866\) 3.19033e48 1.08340
\(867\) 2.78007e48 0.926275
\(868\) 2.71281e47 0.0886836
\(869\) −7.90708e46 −0.0253624
\(870\) −7.66795e48 −2.41330
\(871\) 1.34121e48 0.414188
\(872\) −2.71452e47 −0.0822566
\(873\) 2.75149e48 0.818149
\(874\) 7.46746e46 0.0217888
\(875\) −1.03266e48 −0.295682
\(876\) 1.61389e48 0.453479
\(877\) 4.76181e48 1.31304 0.656521 0.754308i \(-0.272027\pi\)
0.656521 + 0.754308i \(0.272027\pi\)
\(878\) −1.60222e48 −0.433575
\(879\) −1.51912e48 −0.403437
\(880\) −8.53592e47 −0.222477
\(881\) 1.74378e48 0.446055 0.223028 0.974812i \(-0.428406\pi\)
0.223028 + 0.974812i \(0.428406\pi\)
\(882\) 5.26236e48 1.32114
\(883\) −7.49080e47 −0.184577 −0.0922883 0.995732i \(-0.529418\pi\)
−0.0922883 + 0.995732i \(0.529418\pi\)
\(884\) −6.83559e47 −0.165316
\(885\) −2.14265e49 −5.08612
\(886\) −2.35304e48 −0.548242
\(887\) 1.36136e47 0.0311338 0.0155669 0.999879i \(-0.495045\pi\)
0.0155669 + 0.999879i \(0.495045\pi\)
\(888\) −3.61065e48 −0.810535
\(889\) 7.27461e47 0.160299
\(890\) 7.18544e48 1.55424
\(891\) −4.19354e48 −0.890427
\(892\) −1.29006e48 −0.268900
\(893\) −5.02669e47 −0.102857
\(894\) 2.82082e48 0.566639
\(895\) 7.77039e48 1.53237
\(896\) 1.73149e47 0.0335226
\(897\) 2.14863e48 0.408401
\(898\) −5.58936e48 −1.04304
\(899\) 3.09244e48 0.566586
\(900\) 9.06033e48 1.62983
\(901\) 2.96620e48 0.523891
\(902\) 3.61212e48 0.626403
\(903\) −3.62803e48 −0.617765
\(904\) 3.35420e47 0.0560803
\(905\) −3.33708e48 −0.547855
\(906\) −6.45771e48 −1.04103
\(907\) −6.26172e48 −0.991228 −0.495614 0.868543i \(-0.665057\pi\)
−0.495614 + 0.868543i \(0.665057\pi\)
\(908\) 3.24568e48 0.504532
\(909\) 8.73285e48 1.33307
\(910\) 1.34714e48 0.201944
\(911\) −1.10745e49 −1.63031 −0.815155 0.579243i \(-0.803348\pi\)
−0.815155 + 0.579243i \(0.803348\pi\)
\(912\) −1.97987e47 −0.0286234
\(913\) 1.39123e48 0.197528
\(914\) −3.98389e48 −0.555514
\(915\) 3.26700e49 4.47405
\(916\) −1.92324e48 −0.258678
\(917\) −5.15548e48 −0.681044
\(918\) −7.96968e48 −1.03404
\(919\) 6.88858e48 0.877857 0.438929 0.898522i \(-0.355358\pi\)
0.438929 + 0.898522i \(0.355358\pi\)
\(920\) 2.14147e48 0.268049
\(921\) −2.22644e48 −0.273733
\(922\) 9.37164e48 1.13176
\(923\) 2.96484e48 0.351701
\(924\) −1.63604e48 −0.190637
\(925\) −1.67706e49 −1.91960
\(926\) −9.20481e47 −0.103498
\(927\) −2.94229e49 −3.24990
\(928\) 1.97380e48 0.214171
\(929\) −1.62943e49 −1.73691 −0.868456 0.495767i \(-0.834887\pi\)
−0.868456 + 0.495767i \(0.834887\pi\)
\(930\) −8.89559e48 −0.931550
\(931\) −5.34116e47 −0.0549498
\(932\) −3.60999e48 −0.364875
\(933\) 3.44750e49 3.42339
\(934\) 3.36637e48 0.328426
\(935\) 6.43725e48 0.617033
\(936\) −3.90659e48 −0.367914
\(937\) −1.49919e49 −1.38724 −0.693621 0.720340i \(-0.743986\pi\)
−0.693621 + 0.720340i \(0.743986\pi\)
\(938\) 2.56207e48 0.232940
\(939\) 2.19741e49 1.96304
\(940\) −1.44153e49 −1.26536
\(941\) 8.73939e48 0.753794 0.376897 0.926255i \(-0.376991\pi\)
0.376897 + 0.926255i \(0.376991\pi\)
\(942\) 4.83429e48 0.409725
\(943\) −9.06201e48 −0.754712
\(944\) 5.51535e48 0.451373
\(945\) 1.57065e49 1.26315
\(946\) −4.60388e48 −0.363848
\(947\) 7.24316e48 0.562540 0.281270 0.959629i \(-0.409244\pi\)
0.281270 + 0.959629i \(0.409244\pi\)
\(948\) −5.25947e47 −0.0401425
\(949\) −3.23215e48 −0.242437
\(950\) −9.19600e47 −0.0677891
\(951\) −6.86694e48 −0.497490
\(952\) −1.30578e48 −0.0929738
\(953\) 8.48504e48 0.593774 0.296887 0.954913i \(-0.404052\pi\)
0.296887 + 0.954913i \(0.404052\pi\)
\(954\) 1.69520e49 1.16593
\(955\) −1.28140e49 −0.866220
\(956\) 6.85787e48 0.455653
\(957\) −1.86499e49 −1.21795
\(958\) 1.09588e49 0.703446
\(959\) −5.53931e48 −0.349501
\(960\) −5.67775e48 −0.352128
\(961\) −1.28159e49 −0.781294
\(962\) 7.23107e48 0.433325
\(963\) −5.02686e49 −2.96117
\(964\) 1.49032e49 0.862996
\(965\) 4.08809e49 2.32713
\(966\) 4.10447e48 0.229686
\(967\) 3.55845e49 1.95760 0.978800 0.204820i \(-0.0656608\pi\)
0.978800 + 0.204820i \(0.0656608\pi\)
\(968\) 4.46121e48 0.241273
\(969\) 1.49309e48 0.0793860
\(970\) −8.00871e48 −0.418629
\(971\) −1.83055e49 −0.940728 −0.470364 0.882473i \(-0.655877\pi\)
−0.470364 + 0.882473i \(0.655877\pi\)
\(972\) −7.02229e48 −0.354801
\(973\) 5.11183e48 0.253930
\(974\) −1.09760e49 −0.536069
\(975\) −2.64599e49 −1.27061
\(976\) −8.40954e48 −0.397054
\(977\) 9.99310e48 0.463916 0.231958 0.972726i \(-0.425487\pi\)
0.231958 + 0.972726i \(0.425487\pi\)
\(978\) −1.87563e49 −0.856162
\(979\) 1.74763e49 0.784396
\(980\) −1.53171e49 −0.675998
\(981\) 1.16994e49 0.507722
\(982\) −1.81681e49 −0.775300
\(983\) −2.70444e49 −1.13487 −0.567434 0.823419i \(-0.692064\pi\)
−0.567434 + 0.823419i \(0.692064\pi\)
\(984\) 2.40264e49 0.991445
\(985\) −7.90668e48 −0.320846
\(986\) −1.48851e49 −0.593996
\(987\) −2.76291e49 −1.08426
\(988\) 3.96509e47 0.0153025
\(989\) 1.15501e49 0.438377
\(990\) 3.67893e49 1.37322
\(991\) 2.95819e49 1.08595 0.542977 0.839748i \(-0.317297\pi\)
0.542977 + 0.839748i \(0.317297\pi\)
\(992\) 2.28980e48 0.0826714
\(993\) 3.89835e48 0.138426
\(994\) 5.66364e48 0.197797
\(995\) 2.94207e49 1.01058
\(996\) 9.25388e48 0.312640
\(997\) 4.00799e49 1.33185 0.665927 0.746017i \(-0.268036\pi\)
0.665927 + 0.746017i \(0.268036\pi\)
\(998\) −2.98958e49 −0.977140
\(999\) 8.43077e49 2.71042
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 2.34.a.a.1.1 1
3.2 odd 2 18.34.a.c.1.1 1
4.3 odd 2 16.34.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2.34.a.a.1.1 1 1.1 even 1 trivial
16.34.a.a.1.1 1 4.3 odd 2
18.34.a.c.1.1 1 3.2 odd 2