Properties

Label 1.18.a.a.1.1
Level $1$
Weight $18$
Character 1.1
Self dual yes
Analytic conductor $1.832$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1 \)
Weight: \( k \) \(=\) \( 18 \)
Character orbit: \([\chi]\) \(=\) 1.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.83222087345\)
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 \(\chi\) \(=\) 1.1

$q$-expansion

\(f(q)\) \(=\) \(q-528.000 q^{2} -4284.00 q^{3} +147712. q^{4} -1.02585e6 q^{5} +2.26195e6 q^{6} +3.22599e6 q^{7} -8.78592e6 q^{8} -1.10788e8 q^{9} +O(q^{10})\) \(q-528.000 q^{2} -4284.00 q^{3} +147712. q^{4} -1.02585e6 q^{5} +2.26195e6 q^{6} +3.22599e6 q^{7} -8.78592e6 q^{8} -1.10788e8 q^{9} +5.41649e8 q^{10} -7.53618e8 q^{11} -6.32798e8 q^{12} +2.54106e9 q^{13} -1.70332e9 q^{14} +4.39474e9 q^{15} -1.47219e10 q^{16} -5.42974e9 q^{17} +5.84958e10 q^{18} +1.48750e9 q^{19} -1.51530e11 q^{20} -1.38201e10 q^{21} +3.97910e11 q^{22} -3.17092e11 q^{23} +3.76389e10 q^{24} +2.89429e11 q^{25} -1.34168e12 q^{26} +1.02785e12 q^{27} +4.76518e11 q^{28} +2.43341e12 q^{29} -2.32042e12 q^{30} -8.84972e12 q^{31} +8.92477e12 q^{32} +3.22850e12 q^{33} +2.86690e12 q^{34} -3.30938e12 q^{35} -1.63646e13 q^{36} +1.26917e13 q^{37} -7.85400e11 q^{38} -1.08859e13 q^{39} +9.01304e12 q^{40} +4.88642e13 q^{41} +7.29704e12 q^{42} -9.10200e13 q^{43} -1.11318e14 q^{44} +1.13651e14 q^{45} +1.67424e14 q^{46} -4.93050e13 q^{47} +6.30688e13 q^{48} -2.22223e14 q^{49} -1.52818e14 q^{50} +2.32610e13 q^{51} +3.75346e14 q^{52} +2.29405e13 q^{53} -5.42705e14 q^{54} +7.73099e14 q^{55} -2.83433e13 q^{56} -6.37245e12 q^{57} -1.28484e15 q^{58} +3.26951e13 q^{59} +6.49156e14 q^{60} -1.30829e15 q^{61} +4.67265e15 q^{62} -3.57400e14 q^{63} -2.78265e15 q^{64} -2.60675e15 q^{65} -1.70465e15 q^{66} +5.19614e15 q^{67} -8.02038e14 q^{68} +1.35842e15 q^{69} +1.74735e15 q^{70} -3.70949e15 q^{71} +9.73370e14 q^{72} +3.40237e15 q^{73} -6.70119e15 q^{74} -1.23991e15 q^{75} +2.19722e14 q^{76} -2.43117e15 q^{77} +5.74777e15 q^{78} +2.36653e15 q^{79} +1.51025e16 q^{80} +9.90381e15 q^{81} -2.58003e16 q^{82} -2.97668e16 q^{83} -2.04140e15 q^{84} +5.57010e15 q^{85} +4.80585e16 q^{86} -1.04247e16 q^{87} +6.62123e15 q^{88} +2.91672e16 q^{89} -6.00079e16 q^{90} +8.19745e15 q^{91} -4.68383e16 q^{92} +3.79122e16 q^{93} +2.60330e16 q^{94} -1.52595e15 q^{95} -3.82337e16 q^{96} -6.37699e16 q^{97} +1.17334e17 q^{98} +8.34915e16 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −528.000 −1.45841 −0.729204 0.684297i \(-0.760109\pi\)
−0.729204 + 0.684297i \(0.760109\pi\)
\(3\) −4284.00 −0.376980 −0.188490 0.982075i \(-0.560359\pi\)
−0.188490 + 0.982075i \(0.560359\pi\)
\(4\) 147712. 1.12695
\(5\) −1.02585e6 −1.17446 −0.587231 0.809420i \(-0.699782\pi\)
−0.587231 + 0.809420i \(0.699782\pi\)
\(6\) 2.26195e6 0.549791
\(7\) 3.22599e6 0.211510 0.105755 0.994392i \(-0.466274\pi\)
0.105755 + 0.994392i \(0.466274\pi\)
\(8\) −8.78592e6 −0.185149
\(9\) −1.10788e8 −0.857886
\(10\) 5.41649e8 1.71284
\(11\) −7.53618e8 −1.06002 −0.530009 0.847992i \(-0.677812\pi\)
−0.530009 + 0.847992i \(0.677812\pi\)
\(12\) −6.32798e8 −0.424839
\(13\) 2.54106e9 0.863967 0.431984 0.901881i \(-0.357814\pi\)
0.431984 + 0.901881i \(0.357814\pi\)
\(14\) −1.70332e9 −0.308467
\(15\) 4.39474e9 0.442749
\(16\) −1.47219e10 −0.856930
\(17\) −5.42974e9 −0.188783 −0.0943916 0.995535i \(-0.530091\pi\)
−0.0943916 + 0.995535i \(0.530091\pi\)
\(18\) 5.84958e10 1.25115
\(19\) 1.48750e9 0.0200933 0.0100467 0.999950i \(-0.496802\pi\)
0.0100467 + 0.999950i \(0.496802\pi\)
\(20\) −1.51530e11 −1.32356
\(21\) −1.38201e10 −0.0797350
\(22\) 3.97910e11 1.54594
\(23\) −3.17092e11 −0.844303 −0.422152 0.906525i \(-0.638725\pi\)
−0.422152 + 0.906525i \(0.638725\pi\)
\(24\) 3.76389e10 0.0697977
\(25\) 2.89429e11 0.379360
\(26\) −1.34168e12 −1.26002
\(27\) 1.02785e12 0.700387
\(28\) 4.76518e11 0.238361
\(29\) 2.43341e12 0.903301 0.451650 0.892195i \(-0.350835\pi\)
0.451650 + 0.892195i \(0.350835\pi\)
\(30\) −2.32042e12 −0.645709
\(31\) −8.84972e12 −1.86361 −0.931805 0.362958i \(-0.881767\pi\)
−0.931805 + 0.362958i \(0.881767\pi\)
\(32\) 8.92477e12 1.43490
\(33\) 3.22850e12 0.399606
\(34\) 2.86690e12 0.275323
\(35\) −3.30938e12 −0.248410
\(36\) −1.63646e13 −0.966797
\(37\) 1.26917e13 0.594023 0.297012 0.954874i \(-0.404010\pi\)
0.297012 + 0.954874i \(0.404010\pi\)
\(38\) −7.85400e11 −0.0293042
\(39\) −1.08859e13 −0.325699
\(40\) 9.01304e12 0.217451
\(41\) 4.88642e13 0.955713 0.477857 0.878438i \(-0.341414\pi\)
0.477857 + 0.878438i \(0.341414\pi\)
\(42\) 7.29704e12 0.116286
\(43\) −9.10200e13 −1.18756 −0.593779 0.804628i \(-0.702365\pi\)
−0.593779 + 0.804628i \(0.702365\pi\)
\(44\) −1.11318e14 −1.19459
\(45\) 1.13651e14 1.00755
\(46\) 1.67424e14 1.23134
\(47\) −4.93050e13 −0.302036 −0.151018 0.988531i \(-0.548255\pi\)
−0.151018 + 0.988531i \(0.548255\pi\)
\(48\) 6.30688e13 0.323046
\(49\) −2.22223e14 −0.955264
\(50\) −1.52818e14 −0.553262
\(51\) 2.32610e13 0.0711676
\(52\) 3.75346e14 0.973650
\(53\) 2.29405e13 0.0506124 0.0253062 0.999680i \(-0.491944\pi\)
0.0253062 + 0.999680i \(0.491944\pi\)
\(54\) −5.42705e14 −1.02145
\(55\) 7.73099e14 1.24495
\(56\) −2.83433e13 −0.0391609
\(57\) −6.37245e12 −0.00757478
\(58\) −1.28484e15 −1.31738
\(59\) 3.26951e13 0.0289895 0.0144947 0.999895i \(-0.495386\pi\)
0.0144947 + 0.999895i \(0.495386\pi\)
\(60\) 6.49156e14 0.498957
\(61\) −1.30829e15 −0.873774 −0.436887 0.899516i \(-0.643919\pi\)
−0.436887 + 0.899516i \(0.643919\pi\)
\(62\) 4.67265e15 2.71790
\(63\) −3.57400e14 −0.181451
\(64\) −2.78265e15 −1.23574
\(65\) −2.60675e15 −1.01470
\(66\) −1.70465e15 −0.582789
\(67\) 5.19614e15 1.56331 0.781655 0.623711i \(-0.214376\pi\)
0.781655 + 0.623711i \(0.214376\pi\)
\(68\) −8.02038e14 −0.212750
\(69\) 1.35842e15 0.318286
\(70\) 1.74735e15 0.362283
\(71\) −3.70949e15 −0.681739 −0.340870 0.940111i \(-0.610721\pi\)
−0.340870 + 0.940111i \(0.610721\pi\)
\(72\) 9.73370e14 0.158837
\(73\) 3.40237e15 0.493785 0.246892 0.969043i \(-0.420591\pi\)
0.246892 + 0.969043i \(0.420591\pi\)
\(74\) −6.70119e15 −0.866328
\(75\) −1.23991e15 −0.143011
\(76\) 2.19722e14 0.0226442
\(77\) −2.43117e15 −0.224204
\(78\) 5.74777e15 0.475001
\(79\) 2.36653e15 0.175502 0.0877511 0.996142i \(-0.472032\pi\)
0.0877511 + 0.996142i \(0.472032\pi\)
\(80\) 1.51025e16 1.00643
\(81\) 9.90381e15 0.593854
\(82\) −2.58003e16 −1.39382
\(83\) −2.97668e16 −1.45067 −0.725333 0.688398i \(-0.758314\pi\)
−0.725333 + 0.688398i \(0.758314\pi\)
\(84\) −2.04140e15 −0.0898576
\(85\) 5.57010e15 0.221719
\(86\) 4.80585e16 1.73194
\(87\) −1.04247e16 −0.340527
\(88\) 6.62123e15 0.196262
\(89\) 2.91672e16 0.785379 0.392690 0.919671i \(-0.371545\pi\)
0.392690 + 0.919671i \(0.371545\pi\)
\(90\) −6.00079e16 −1.46942
\(91\) 8.19745e15 0.182737
\(92\) −4.68383e16 −0.951490
\(93\) 3.79122e16 0.702545
\(94\) 2.60330e16 0.440492
\(95\) −1.52595e15 −0.0235988
\(96\) −3.82337e16 −0.540930
\(97\) −6.37699e16 −0.826144 −0.413072 0.910698i \(-0.635544\pi\)
−0.413072 + 0.910698i \(0.635544\pi\)
\(98\) 1.17334e17 1.39316
\(99\) 8.34915e16 0.909375
\(100\) 4.27521e16 0.427521
\(101\) −1.60611e17 −1.47586 −0.737929 0.674878i \(-0.764196\pi\)
−0.737929 + 0.674878i \(0.764196\pi\)
\(102\) −1.22818e16 −0.103791
\(103\) −9.07136e16 −0.705596 −0.352798 0.935700i \(-0.614770\pi\)
−0.352798 + 0.935700i \(0.614770\pi\)
\(104\) −2.23256e16 −0.159963
\(105\) 1.41774e16 0.0936456
\(106\) −1.21126e16 −0.0738135
\(107\) 1.95549e17 1.10026 0.550129 0.835080i \(-0.314579\pi\)
0.550129 + 0.835080i \(0.314579\pi\)
\(108\) 1.51826e17 0.789303
\(109\) 2.13756e17 1.02753 0.513763 0.857932i \(-0.328251\pi\)
0.513763 + 0.857932i \(0.328251\pi\)
\(110\) −4.08196e17 −1.81565
\(111\) −5.43710e16 −0.223935
\(112\) −4.74929e16 −0.181249
\(113\) −2.81383e17 −0.995706 −0.497853 0.867262i \(-0.665878\pi\)
−0.497853 + 0.867262i \(0.665878\pi\)
\(114\) 3.36465e15 0.0110471
\(115\) 3.25289e17 0.991602
\(116\) 3.59444e17 1.01798
\(117\) −2.81518e17 −0.741185
\(118\) −1.72630e16 −0.0422785
\(119\) −1.75163e16 −0.0399295
\(120\) −3.86118e16 −0.0819747
\(121\) 6.24934e16 0.123640
\(122\) 6.90775e17 1.27432
\(123\) −2.09334e17 −0.360285
\(124\) −1.30721e18 −2.10020
\(125\) 4.85751e17 0.728918
\(126\) 1.88707e17 0.264630
\(127\) −8.70305e17 −1.14114 −0.570571 0.821248i \(-0.693278\pi\)
−0.570571 + 0.821248i \(0.693278\pi\)
\(128\) 2.99449e17 0.367315
\(129\) 3.89930e17 0.447686
\(130\) 1.37636e18 1.47984
\(131\) 1.31783e17 0.132756 0.0663781 0.997795i \(-0.478856\pi\)
0.0663781 + 0.997795i \(0.478856\pi\)
\(132\) 4.76888e17 0.450338
\(133\) 4.79866e15 0.00424993
\(134\) −2.74356e18 −2.27994
\(135\) −1.05442e18 −0.822577
\(136\) 4.77053e16 0.0349531
\(137\) 1.87128e18 1.28829 0.644147 0.764902i \(-0.277213\pi\)
0.644147 + 0.764902i \(0.277213\pi\)
\(138\) −7.17246e17 −0.464190
\(139\) 1.58706e18 0.965980 0.482990 0.875626i \(-0.339551\pi\)
0.482990 + 0.875626i \(0.339551\pi\)
\(140\) −4.88836e17 −0.279946
\(141\) 2.11223e17 0.113862
\(142\) 1.95861e18 0.994254
\(143\) −1.91499e18 −0.915821
\(144\) 1.63101e18 0.735148
\(145\) −2.49631e18 −1.06089
\(146\) −1.79645e18 −0.720139
\(147\) 9.52005e17 0.360116
\(148\) 1.87471e18 0.669436
\(149\) −4.77240e17 −0.160936 −0.0804680 0.996757i \(-0.525641\pi\)
−0.0804680 + 0.996757i \(0.525641\pi\)
\(150\) 6.54674e17 0.208569
\(151\) −3.92964e18 −1.18317 −0.591587 0.806241i \(-0.701499\pi\)
−0.591587 + 0.806241i \(0.701499\pi\)
\(152\) −1.30691e16 −0.00372026
\(153\) 6.01548e17 0.161954
\(154\) 1.28366e18 0.326981
\(155\) 9.07849e18 2.18874
\(156\) −1.60798e18 −0.367047
\(157\) −2.29453e18 −0.496075 −0.248038 0.968750i \(-0.579786\pi\)
−0.248038 + 0.968750i \(0.579786\pi\)
\(158\) −1.24953e18 −0.255954
\(159\) −9.82769e16 −0.0190799
\(160\) −9.15548e18 −1.68524
\(161\) −1.02294e18 −0.178578
\(162\) −5.22921e18 −0.866081
\(163\) 8.01044e18 1.25910 0.629552 0.776958i \(-0.283238\pi\)
0.629552 + 0.776958i \(0.283238\pi\)
\(164\) 7.21782e18 1.07704
\(165\) −3.31196e18 −0.469322
\(166\) 1.57168e19 2.11566
\(167\) −8.61477e18 −1.10193 −0.550965 0.834528i \(-0.685740\pi\)
−0.550965 + 0.834528i \(0.685740\pi\)
\(168\) 1.21423e17 0.0147629
\(169\) −2.19341e18 −0.253561
\(170\) −2.94101e18 −0.323356
\(171\) −1.64796e17 −0.0172378
\(172\) −1.34447e19 −1.33832
\(173\) 2.31430e18 0.219294 0.109647 0.993971i \(-0.465028\pi\)
0.109647 + 0.993971i \(0.465028\pi\)
\(174\) 5.50426e18 0.496627
\(175\) 9.33695e17 0.0802383
\(176\) 1.10947e19 0.908362
\(177\) −1.40066e17 −0.0109285
\(178\) −1.54003e19 −1.14540
\(179\) −7.90307e18 −0.560461 −0.280230 0.959933i \(-0.590411\pi\)
−0.280230 + 0.959933i \(0.590411\pi\)
\(180\) 1.67877e19 1.13547
\(181\) 1.40729e19 0.908060 0.454030 0.890986i \(-0.349986\pi\)
0.454030 + 0.890986i \(0.349986\pi\)
\(182\) −4.32826e18 −0.266505
\(183\) 5.60470e18 0.329396
\(184\) 2.78594e18 0.156322
\(185\) −1.30197e19 −0.697658
\(186\) −2.00176e19 −1.02460
\(187\) 4.09195e18 0.200114
\(188\) −7.28294e18 −0.340381
\(189\) 3.31584e18 0.148138
\(190\) 8.05703e17 0.0344167
\(191\) −2.82501e19 −1.15408 −0.577040 0.816716i \(-0.695792\pi\)
−0.577040 + 0.816716i \(0.695792\pi\)
\(192\) 1.19209e19 0.465851
\(193\) 4.91755e19 1.83870 0.919351 0.393437i \(-0.128714\pi\)
0.919351 + 0.393437i \(0.128714\pi\)
\(194\) 3.36705e19 1.20486
\(195\) 1.11673e19 0.382521
\(196\) −3.28251e19 −1.07654
\(197\) 1.29458e19 0.406598 0.203299 0.979117i \(-0.434834\pi\)
0.203299 + 0.979117i \(0.434834\pi\)
\(198\) −4.40835e19 −1.32624
\(199\) −5.51755e19 −1.59036 −0.795179 0.606375i \(-0.792623\pi\)
−0.795179 + 0.606375i \(0.792623\pi\)
\(200\) −2.54290e18 −0.0702383
\(201\) −2.22603e19 −0.589338
\(202\) 8.48028e19 2.15240
\(203\) 7.85016e18 0.191057
\(204\) 3.43593e18 0.0802025
\(205\) −5.01273e19 −1.12245
\(206\) 4.78968e19 1.02905
\(207\) 3.51298e19 0.724316
\(208\) −3.74094e19 −0.740359
\(209\) −1.12101e18 −0.0212993
\(210\) −7.48567e18 −0.136574
\(211\) 1.73510e19 0.304035 0.152017 0.988378i \(-0.451423\pi\)
0.152017 + 0.988378i \(0.451423\pi\)
\(212\) 3.38858e18 0.0570378
\(213\) 1.58915e19 0.257002
\(214\) −1.03250e20 −1.60462
\(215\) 9.33728e19 1.39474
\(216\) −9.03061e18 −0.129676
\(217\) −2.85491e19 −0.394171
\(218\) −1.12863e20 −1.49855
\(219\) −1.45758e19 −0.186147
\(220\) 1.14196e20 1.40300
\(221\) −1.37973e19 −0.163102
\(222\) 2.87079e19 0.326589
\(223\) 9.48415e19 1.03850 0.519251 0.854622i \(-0.326211\pi\)
0.519251 + 0.854622i \(0.326211\pi\)
\(224\) 2.87912e19 0.303496
\(225\) −3.20651e19 −0.325448
\(226\) 1.48570e20 1.45214
\(227\) −1.83782e20 −1.73015 −0.865073 0.501647i \(-0.832728\pi\)
−0.865073 + 0.501647i \(0.832728\pi\)
\(228\) −9.41287e17 −0.00853642
\(229\) −9.14908e19 −0.799422 −0.399711 0.916641i \(-0.630889\pi\)
−0.399711 + 0.916641i \(0.630889\pi\)
\(230\) −1.71752e20 −1.44616
\(231\) 1.04151e19 0.0845205
\(232\) −2.13798e19 −0.167246
\(233\) 9.87497e19 0.744750 0.372375 0.928082i \(-0.378544\pi\)
0.372375 + 0.928082i \(0.378544\pi\)
\(234\) 1.48642e20 1.08095
\(235\) 5.05795e19 0.354730
\(236\) 4.82946e18 0.0326698
\(237\) −1.01382e19 −0.0661609
\(238\) 9.24861e18 0.0582334
\(239\) 1.89337e20 1.15041 0.575206 0.818009i \(-0.304922\pi\)
0.575206 + 0.818009i \(0.304922\pi\)
\(240\) −6.46991e19 −0.379405
\(241\) −1.38762e20 −0.785463 −0.392731 0.919653i \(-0.628470\pi\)
−0.392731 + 0.919653i \(0.628470\pi\)
\(242\) −3.29965e19 −0.180317
\(243\) −1.75165e20 −0.924258
\(244\) −1.93250e20 −0.984702
\(245\) 2.27968e20 1.12192
\(246\) 1.10528e20 0.525443
\(247\) 3.77983e18 0.0173600
\(248\) 7.77529e19 0.345046
\(249\) 1.27521e20 0.546873
\(250\) −2.56476e20 −1.06306
\(251\) −3.34508e20 −1.34023 −0.670115 0.742257i \(-0.733755\pi\)
−0.670115 + 0.742257i \(0.733755\pi\)
\(252\) −5.27922e19 −0.204487
\(253\) 2.38966e20 0.894977
\(254\) 4.59521e20 1.66425
\(255\) −2.38623e19 −0.0835836
\(256\) 2.06618e20 0.700048
\(257\) 4.91382e19 0.161060 0.0805300 0.996752i \(-0.474339\pi\)
0.0805300 + 0.996752i \(0.474339\pi\)
\(258\) −2.05883e20 −0.652909
\(259\) 4.09432e19 0.125642
\(260\) −3.85048e20 −1.14352
\(261\) −2.69591e20 −0.774929
\(262\) −6.95816e19 −0.193613
\(263\) 1.78845e19 0.0481784 0.0240892 0.999710i \(-0.492331\pi\)
0.0240892 + 0.999710i \(0.492331\pi\)
\(264\) −2.83653e19 −0.0739869
\(265\) −2.35335e19 −0.0594423
\(266\) −2.53369e18 −0.00619812
\(267\) −1.24952e20 −0.296073
\(268\) 7.67533e20 1.76178
\(269\) 1.15237e20 0.256270 0.128135 0.991757i \(-0.459101\pi\)
0.128135 + 0.991757i \(0.459101\pi\)
\(270\) 5.56734e20 1.19965
\(271\) −1.47067e20 −0.307098 −0.153549 0.988141i \(-0.549070\pi\)
−0.153549 + 0.988141i \(0.549070\pi\)
\(272\) 7.99363e19 0.161774
\(273\) −3.51179e19 −0.0688884
\(274\) −9.88038e20 −1.87886
\(275\) −2.18119e20 −0.402129
\(276\) 2.00655e20 0.358693
\(277\) −1.78744e20 −0.309852 −0.154926 0.987926i \(-0.549514\pi\)
−0.154926 + 0.987926i \(0.549514\pi\)
\(278\) −8.37969e20 −1.40879
\(279\) 9.80439e20 1.59877
\(280\) 2.90760e19 0.0459929
\(281\) 2.80546e20 0.430527 0.215263 0.976556i \(-0.430939\pi\)
0.215263 + 0.976556i \(0.430939\pi\)
\(282\) −1.11526e20 −0.166057
\(283\) 3.39877e20 0.491062 0.245531 0.969389i \(-0.421038\pi\)
0.245531 + 0.969389i \(0.421038\pi\)
\(284\) −5.47936e20 −0.768288
\(285\) 6.53718e18 0.00889629
\(286\) 1.01112e21 1.33564
\(287\) 1.57635e20 0.202142
\(288\) −9.88753e20 −1.23098
\(289\) −7.97758e20 −0.964361
\(290\) 1.31805e21 1.54721
\(291\) 2.73190e20 0.311440
\(292\) 5.02571e20 0.556472
\(293\) −7.64887e20 −0.822664 −0.411332 0.911486i \(-0.634936\pi\)
−0.411332 + 0.911486i \(0.634936\pi\)
\(294\) −5.02659e20 −0.525196
\(295\) −3.35403e19 −0.0340470
\(296\) −1.11508e20 −0.109983
\(297\) −7.74607e20 −0.742423
\(298\) 2.51983e20 0.234710
\(299\) −8.05751e20 −0.729450
\(300\) −1.83150e20 −0.161167
\(301\) −2.93630e20 −0.251180
\(302\) 2.07485e21 1.72555
\(303\) 6.88059e20 0.556370
\(304\) −2.18989e19 −0.0172185
\(305\) 1.34211e21 1.02621
\(306\) −3.17617e20 −0.236196
\(307\) 1.38472e21 1.00158 0.500789 0.865569i \(-0.333043\pi\)
0.500789 + 0.865569i \(0.333043\pi\)
\(308\) −3.59112e20 −0.252667
\(309\) 3.88617e20 0.265996
\(310\) −4.79344e21 −3.19207
\(311\) −2.34733e21 −1.52094 −0.760469 0.649374i \(-0.775031\pi\)
−0.760469 + 0.649374i \(0.775031\pi\)
\(312\) 9.56428e19 0.0603029
\(313\) 1.58424e21 0.972065 0.486033 0.873941i \(-0.338444\pi\)
0.486033 + 0.873941i \(0.338444\pi\)
\(314\) 1.21151e21 0.723480
\(315\) 3.66638e20 0.213107
\(316\) 3.49565e20 0.197783
\(317\) 8.98378e20 0.494829 0.247415 0.968910i \(-0.420419\pi\)
0.247415 + 0.968910i \(0.420419\pi\)
\(318\) 5.18902e19 0.0278263
\(319\) −1.83386e21 −0.957516
\(320\) 2.85458e21 1.45133
\(321\) −8.37734e20 −0.414776
\(322\) 5.40110e20 0.260440
\(323\) −8.07674e18 −0.00379328
\(324\) 1.46291e21 0.669245
\(325\) 7.35457e20 0.327755
\(326\) −4.22951e21 −1.83629
\(327\) −9.15730e20 −0.387357
\(328\) −4.29317e20 −0.176950
\(329\) −1.59058e20 −0.0638836
\(330\) 1.74871e21 0.684463
\(331\) 4.00469e21 1.52767 0.763837 0.645409i \(-0.223313\pi\)
0.763837 + 0.645409i \(0.223313\pi\)
\(332\) −4.39691e21 −1.63483
\(333\) −1.40608e21 −0.509604
\(334\) 4.54860e21 1.60706
\(335\) −5.33046e21 −1.83605
\(336\) 2.03459e20 0.0683273
\(337\) −1.06753e21 −0.349564 −0.174782 0.984607i \(-0.555922\pi\)
−0.174782 + 0.984607i \(0.555922\pi\)
\(338\) 1.15812e21 0.369795
\(339\) 1.20544e21 0.375362
\(340\) 8.22771e20 0.249867
\(341\) 6.66931e21 1.97546
\(342\) 8.70125e19 0.0251397
\(343\) −1.46736e21 −0.413557
\(344\) 7.99694e20 0.219876
\(345\) −1.39354e21 −0.373814
\(346\) −1.22195e21 −0.319821
\(347\) −1.61841e21 −0.413321 −0.206660 0.978413i \(-0.566260\pi\)
−0.206660 + 0.978413i \(0.566260\pi\)
\(348\) −1.53986e21 −0.383758
\(349\) −5.60078e21 −1.36217 −0.681086 0.732203i \(-0.738492\pi\)
−0.681086 + 0.732203i \(0.738492\pi\)
\(350\) −4.92991e20 −0.117020
\(351\) 2.61183e21 0.605111
\(352\) −6.72587e21 −1.52102
\(353\) 5.10774e21 1.12757 0.563785 0.825921i \(-0.309344\pi\)
0.563785 + 0.825921i \(0.309344\pi\)
\(354\) 7.39547e19 0.0159382
\(355\) 3.80538e21 0.800676
\(356\) 4.30834e21 0.885086
\(357\) 7.50399e19 0.0150526
\(358\) 4.17282e21 0.817380
\(359\) 5.84965e21 1.11899 0.559496 0.828833i \(-0.310995\pi\)
0.559496 + 0.828833i \(0.310995\pi\)
\(360\) −9.98532e20 −0.186548
\(361\) −5.47817e21 −0.999596
\(362\) −7.43047e21 −1.32432
\(363\) −2.67722e20 −0.0466098
\(364\) 1.21086e21 0.205936
\(365\) −3.49032e21 −0.579931
\(366\) −2.95928e21 −0.480393
\(367\) 4.29535e20 0.0681297 0.0340649 0.999420i \(-0.489155\pi\)
0.0340649 + 0.999420i \(0.489155\pi\)
\(368\) 4.66821e21 0.723509
\(369\) −5.41354e21 −0.819893
\(370\) 6.87442e21 1.01747
\(371\) 7.40057e19 0.0107050
\(372\) 5.60009e21 0.791735
\(373\) −6.54734e21 −0.904774 −0.452387 0.891822i \(-0.649427\pi\)
−0.452387 + 0.891822i \(0.649427\pi\)
\(374\) −2.16055e21 −0.291847
\(375\) −2.08096e21 −0.274788
\(376\) 4.33190e20 0.0559218
\(377\) 6.18345e21 0.780422
\(378\) −1.75076e21 −0.216046
\(379\) 1.51850e21 0.183223 0.0916117 0.995795i \(-0.470798\pi\)
0.0916117 + 0.995795i \(0.470798\pi\)
\(380\) −2.25401e20 −0.0265948
\(381\) 3.72839e21 0.430188
\(382\) 1.49160e22 1.68312
\(383\) −8.04597e21 −0.887951 −0.443975 0.896039i \(-0.646432\pi\)
−0.443975 + 0.896039i \(0.646432\pi\)
\(384\) −1.28284e21 −0.138470
\(385\) 2.49401e21 0.263319
\(386\) −2.59647e22 −2.68158
\(387\) 1.00839e22 1.01879
\(388\) −9.41958e21 −0.931026
\(389\) 6.84040e21 0.661470 0.330735 0.943724i \(-0.392703\pi\)
0.330735 + 0.943724i \(0.392703\pi\)
\(390\) −5.89635e21 −0.557871
\(391\) 1.72173e21 0.159390
\(392\) 1.95244e21 0.176867
\(393\) −5.64560e20 −0.0500465
\(394\) −6.83536e21 −0.592985
\(395\) −2.42771e21 −0.206121
\(396\) 1.23327e22 1.02482
\(397\) 6.23458e21 0.507093 0.253547 0.967323i \(-0.418403\pi\)
0.253547 + 0.967323i \(0.418403\pi\)
\(398\) 2.91327e22 2.31939
\(399\) −2.05575e19 −0.00160214
\(400\) −4.26095e21 −0.325085
\(401\) −2.32048e21 −0.173320 −0.0866602 0.996238i \(-0.527619\pi\)
−0.0866602 + 0.996238i \(0.527619\pi\)
\(402\) 1.17534e22 0.859494
\(403\) −2.24877e22 −1.61010
\(404\) −2.37242e22 −1.66322
\(405\) −1.01598e22 −0.697458
\(406\) −4.14489e21 −0.278639
\(407\) −9.56466e21 −0.629676
\(408\) −2.04369e20 −0.0131766
\(409\) 2.38590e22 1.50662 0.753312 0.657663i \(-0.228455\pi\)
0.753312 + 0.657663i \(0.228455\pi\)
\(410\) 2.64672e22 1.63699
\(411\) −8.01658e21 −0.485661
\(412\) −1.33995e22 −0.795173
\(413\) 1.05474e20 0.00613155
\(414\) −1.85485e22 −1.05635
\(415\) 3.05362e22 1.70375
\(416\) 2.26784e22 1.23971
\(417\) −6.79898e21 −0.364156
\(418\) 5.91892e20 0.0310630
\(419\) −2.61839e22 −1.34653 −0.673263 0.739403i \(-0.735108\pi\)
−0.673263 + 0.739403i \(0.735108\pi\)
\(420\) 2.09417e21 0.105534
\(421\) 1.11903e22 0.552642 0.276321 0.961065i \(-0.410885\pi\)
0.276321 + 0.961065i \(0.410885\pi\)
\(422\) −9.16132e21 −0.443407
\(423\) 5.46238e21 0.259113
\(424\) −2.01553e20 −0.00937086
\(425\) −1.57152e21 −0.0716168
\(426\) −8.39069e21 −0.374814
\(427\) −4.22052e21 −0.184812
\(428\) 2.88850e22 1.23994
\(429\) 8.20383e21 0.345247
\(430\) −4.93009e22 −2.03410
\(431\) 1.74296e22 0.705068 0.352534 0.935799i \(-0.385320\pi\)
0.352534 + 0.935799i \(0.385320\pi\)
\(432\) −1.51319e22 −0.600182
\(433\) −4.51413e22 −1.75560 −0.877802 0.479024i \(-0.840991\pi\)
−0.877802 + 0.479024i \(0.840991\pi\)
\(434\) 1.50739e22 0.574863
\(435\) 1.06942e22 0.399936
\(436\) 3.15743e22 1.15797
\(437\) −4.71674e20 −0.0169648
\(438\) 7.69600e21 0.271478
\(439\) 3.85266e22 1.33294 0.666472 0.745530i \(-0.267804\pi\)
0.666472 + 0.745530i \(0.267804\pi\)
\(440\) −6.79239e21 −0.230502
\(441\) 2.46196e22 0.819507
\(442\) 7.28499e21 0.237870
\(443\) −3.84966e21 −0.123308 −0.0616539 0.998098i \(-0.519637\pi\)
−0.0616539 + 0.998098i \(0.519637\pi\)
\(444\) −8.03126e21 −0.252364
\(445\) −2.99212e22 −0.922398
\(446\) −5.00763e22 −1.51456
\(447\) 2.04449e21 0.0606697
\(448\) −8.97679e21 −0.261371
\(449\) −2.77614e21 −0.0793136 −0.0396568 0.999213i \(-0.512626\pi\)
−0.0396568 + 0.999213i \(0.512626\pi\)
\(450\) 1.69304e22 0.474635
\(451\) −3.68249e22 −1.01307
\(452\) −4.15636e22 −1.12211
\(453\) 1.68346e22 0.446034
\(454\) 9.70368e22 2.52326
\(455\) −8.40936e21 −0.214618
\(456\) 5.59878e19 0.00140247
\(457\) −1.34139e22 −0.329814 −0.164907 0.986309i \(-0.552732\pi\)
−0.164907 + 0.986309i \(0.552732\pi\)
\(458\) 4.83071e22 1.16588
\(459\) −5.58096e21 −0.132221
\(460\) 4.80490e22 1.11749
\(461\) −4.32728e22 −0.988000 −0.494000 0.869462i \(-0.664466\pi\)
−0.494000 + 0.869462i \(0.664466\pi\)
\(462\) −5.49918e21 −0.123265
\(463\) 3.85721e22 0.848859 0.424429 0.905461i \(-0.360475\pi\)
0.424429 + 0.905461i \(0.360475\pi\)
\(464\) −3.58245e22 −0.774065
\(465\) −3.88922e22 −0.825112
\(466\) −5.21398e22 −1.08615
\(467\) 4.96724e22 1.01607 0.508033 0.861338i \(-0.330373\pi\)
0.508033 + 0.861338i \(0.330373\pi\)
\(468\) −4.15836e22 −0.835281
\(469\) 1.67627e22 0.330655
\(470\) −2.67060e22 −0.517341
\(471\) 9.82978e21 0.187011
\(472\) −2.87256e20 −0.00536739
\(473\) 6.85943e22 1.25883
\(474\) 5.35299e21 0.0964895
\(475\) 4.30525e20 0.00762260
\(476\) −2.58737e21 −0.0449986
\(477\) −2.54152e21 −0.0434197
\(478\) −9.99699e22 −1.67777
\(479\) −4.26513e22 −0.703202 −0.351601 0.936150i \(-0.614363\pi\)
−0.351601 + 0.936150i \(0.614363\pi\)
\(480\) 3.92221e22 0.635302
\(481\) 3.22503e22 0.513217
\(482\) 7.32663e22 1.14553
\(483\) 4.38226e21 0.0673205
\(484\) 9.23103e21 0.139336
\(485\) 6.54183e22 0.970275
\(486\) 9.24869e22 1.34794
\(487\) −9.28126e22 −1.32926 −0.664631 0.747172i \(-0.731411\pi\)
−0.664631 + 0.747172i \(0.731411\pi\)
\(488\) 1.14945e22 0.161779
\(489\) −3.43167e22 −0.474658
\(490\) −1.20367e23 −1.63622
\(491\) 1.29877e21 0.0173516 0.00867579 0.999962i \(-0.497238\pi\)
0.00867579 + 0.999962i \(0.497238\pi\)
\(492\) −3.09211e22 −0.406024
\(493\) −1.32128e22 −0.170528
\(494\) −1.99575e21 −0.0253179
\(495\) −8.56497e22 −1.06803
\(496\) 1.30285e23 1.59698
\(497\) −1.19668e22 −0.144194
\(498\) −6.73310e22 −0.797564
\(499\) −7.66788e22 −0.892937 −0.446468 0.894799i \(-0.647318\pi\)
−0.446468 + 0.894799i \(0.647318\pi\)
\(500\) 7.17512e22 0.821456
\(501\) 3.69057e22 0.415406
\(502\) 1.76620e23 1.95460
\(503\) 6.67712e22 0.726543 0.363272 0.931683i \(-0.381660\pi\)
0.363272 + 0.931683i \(0.381660\pi\)
\(504\) 3.14008e21 0.0335956
\(505\) 1.64763e23 1.73334
\(506\) −1.26174e23 −1.30524
\(507\) 9.39656e21 0.0955875
\(508\) −1.28555e23 −1.28601
\(509\) −1.15659e23 −1.13783 −0.568916 0.822395i \(-0.692637\pi\)
−0.568916 + 0.822395i \(0.692637\pi\)
\(510\) 1.25993e22 0.121899
\(511\) 1.09760e22 0.104440
\(512\) −1.48344e23 −1.38827
\(513\) 1.52893e21 0.0140731
\(514\) −2.59450e22 −0.234891
\(515\) 9.30585e22 0.828695
\(516\) 5.75973e22 0.504521
\(517\) 3.71571e22 0.320164
\(518\) −2.16180e22 −0.183237
\(519\) −9.91446e21 −0.0826697
\(520\) 2.29027e22 0.187870
\(521\) −1.79855e23 −1.45145 −0.725725 0.687985i \(-0.758496\pi\)
−0.725725 + 0.687985i \(0.758496\pi\)
\(522\) 1.42344e23 1.13016
\(523\) 9.37211e22 0.732104 0.366052 0.930594i \(-0.380709\pi\)
0.366052 + 0.930594i \(0.380709\pi\)
\(524\) 1.94660e22 0.149610
\(525\) −3.99995e21 −0.0302483
\(526\) −9.44299e21 −0.0702637
\(527\) 4.80517e22 0.351818
\(528\) −4.75298e22 −0.342435
\(529\) −4.05028e22 −0.287152
\(530\) 1.24257e22 0.0866912
\(531\) −3.62221e21 −0.0248697
\(532\) 7.08820e20 0.00478947
\(533\) 1.24167e23 0.825705
\(534\) 6.59748e22 0.431795
\(535\) −2.00604e23 −1.29221
\(536\) −4.56529e22 −0.289446
\(537\) 3.38568e22 0.211283
\(538\) −6.08452e22 −0.373746
\(539\) 1.67472e23 1.01260
\(540\) −1.55750e23 −0.927006
\(541\) −1.52396e23 −0.892889 −0.446444 0.894811i \(-0.647310\pi\)
−0.446444 + 0.894811i \(0.647310\pi\)
\(542\) 7.76514e22 0.447874
\(543\) −6.02881e22 −0.342321
\(544\) −4.84592e22 −0.270886
\(545\) −2.19281e23 −1.20679
\(546\) 1.85422e22 0.100467
\(547\) 2.04979e23 1.09349 0.546747 0.837298i \(-0.315866\pi\)
0.546747 + 0.837298i \(0.315866\pi\)
\(548\) 2.76411e23 1.45185
\(549\) 1.44942e23 0.749598
\(550\) 1.15167e23 0.586468
\(551\) 3.61970e21 0.0181503
\(552\) −1.19350e22 −0.0589304
\(553\) 7.63442e21 0.0371204
\(554\) 9.43770e22 0.451890
\(555\) 5.57765e22 0.263003
\(556\) 2.34428e23 1.08861
\(557\) −4.07359e23 −1.86298 −0.931490 0.363767i \(-0.881491\pi\)
−0.931490 + 0.363767i \(0.881491\pi\)
\(558\) −5.17672e23 −2.33165
\(559\) −2.31288e23 −1.02601
\(560\) 4.87206e22 0.212870
\(561\) −1.75299e22 −0.0754390
\(562\) −1.48128e23 −0.627884
\(563\) 2.21297e23 0.923962 0.461981 0.886890i \(-0.347139\pi\)
0.461981 + 0.886890i \(0.347139\pi\)
\(564\) 3.12001e22 0.128317
\(565\) 2.88657e23 1.16942
\(566\) −1.79455e23 −0.716169
\(567\) 3.19496e22 0.125606
\(568\) 3.25913e22 0.126224
\(569\) 4.88686e23 1.86456 0.932278 0.361743i \(-0.117818\pi\)
0.932278 + 0.361743i \(0.117818\pi\)
\(570\) −3.45163e21 −0.0129744
\(571\) 3.42218e20 0.00126735 0.000633673 1.00000i \(-0.499798\pi\)
0.000633673 1.00000i \(0.499798\pi\)
\(572\) −2.82867e23 −1.03209
\(573\) 1.21023e23 0.435065
\(574\) −8.32315e22 −0.294806
\(575\) −9.17755e22 −0.320295
\(576\) 3.08282e23 1.06013
\(577\) −1.95911e23 −0.663840 −0.331920 0.943308i \(-0.607696\pi\)
−0.331920 + 0.943308i \(0.607696\pi\)
\(578\) 4.21216e23 1.40643
\(579\) −2.10668e23 −0.693155
\(580\) −3.68736e23 −1.19558
\(581\) −9.60273e22 −0.306830
\(582\) −1.44244e23 −0.454207
\(583\) −1.72883e22 −0.0536501
\(584\) −2.98930e22 −0.0914239
\(585\) 2.88795e23 0.870493
\(586\) 4.03860e23 1.19978
\(587\) −4.80676e23 −1.40744 −0.703719 0.710479i \(-0.748478\pi\)
−0.703719 + 0.710479i \(0.748478\pi\)
\(588\) 1.40623e23 0.405834
\(589\) −1.31640e22 −0.0374461
\(590\) 1.77093e22 0.0496545
\(591\) −5.54596e22 −0.153279
\(592\) −1.86846e23 −0.509036
\(593\) −6.40804e23 −1.72092 −0.860459 0.509519i \(-0.829823\pi\)
−0.860459 + 0.509519i \(0.829823\pi\)
\(594\) 4.08992e23 1.08276
\(595\) 1.79691e22 0.0468956
\(596\) −7.04940e22 −0.181367
\(597\) 2.36372e23 0.599534
\(598\) 4.25436e23 1.06384
\(599\) 3.60950e23 0.889854 0.444927 0.895567i \(-0.353230\pi\)
0.444927 + 0.895567i \(0.353230\pi\)
\(600\) 1.08938e22 0.0264785
\(601\) 6.53955e23 1.56717 0.783583 0.621287i \(-0.213390\pi\)
0.783583 + 0.621287i \(0.213390\pi\)
\(602\) 1.55036e23 0.366323
\(603\) −5.75668e23 −1.34114
\(604\) −5.80455e23 −1.33338
\(605\) −6.41089e22 −0.145210
\(606\) −3.63295e23 −0.811414
\(607\) 4.08715e23 0.900154 0.450077 0.892990i \(-0.351397\pi\)
0.450077 + 0.892990i \(0.351397\pi\)
\(608\) 1.32756e22 0.0288319
\(609\) −3.36301e22 −0.0720247
\(610\) −7.08631e23 −1.49664
\(611\) −1.25287e23 −0.260949
\(612\) 8.88558e22 0.182515
\(613\) 1.80349e23 0.365343 0.182671 0.983174i \(-0.441526\pi\)
0.182671 + 0.983174i \(0.441526\pi\)
\(614\) −7.31131e23 −1.46071
\(615\) 2.14745e23 0.423141
\(616\) 2.13600e22 0.0415113
\(617\) 4.42851e23 0.848856 0.424428 0.905462i \(-0.360475\pi\)
0.424428 + 0.905462i \(0.360475\pi\)
\(618\) −2.05190e23 −0.387930
\(619\) 1.71098e22 0.0319062 0.0159531 0.999873i \(-0.494922\pi\)
0.0159531 + 0.999873i \(0.494922\pi\)
\(620\) 1.34100e24 2.46661
\(621\) −3.25923e23 −0.591339
\(622\) 1.23939e24 2.21815
\(623\) 9.40931e22 0.166115
\(624\) 1.60262e23 0.279101
\(625\) −7.19124e23 −1.23545
\(626\) −8.36481e23 −1.41767
\(627\) 4.80239e21 0.00802941
\(628\) −3.38930e23 −0.559053
\(629\) −6.89124e22 −0.112142
\(630\) −1.93585e23 −0.310797
\(631\) −2.82293e23 −0.447147 −0.223574 0.974687i \(-0.571772\pi\)
−0.223574 + 0.974687i \(0.571772\pi\)
\(632\) −2.07922e22 −0.0324941
\(633\) −7.43316e22 −0.114615
\(634\) −4.74344e23 −0.721663
\(635\) 8.92802e23 1.34023
\(636\) −1.45167e22 −0.0215021
\(637\) −5.64684e23 −0.825316
\(638\) 9.68279e23 1.39645
\(639\) 4.10965e23 0.584854
\(640\) −3.07190e23 −0.431397
\(641\) −3.77613e23 −0.523304 −0.261652 0.965162i \(-0.584267\pi\)
−0.261652 + 0.965162i \(0.584267\pi\)
\(642\) 4.42323e23 0.604912
\(643\) −4.65536e23 −0.628289 −0.314145 0.949375i \(-0.601718\pi\)
−0.314145 + 0.949375i \(0.601718\pi\)
\(644\) −1.51100e23 −0.201249
\(645\) −4.00009e23 −0.525790
\(646\) 4.26452e21 0.00553215
\(647\) 1.00412e24 1.28558 0.642788 0.766045i \(-0.277778\pi\)
0.642788 + 0.766045i \(0.277778\pi\)
\(648\) −8.70141e22 −0.109952
\(649\) −2.46396e22 −0.0307294
\(650\) −3.88321e23 −0.478000
\(651\) 1.22304e23 0.148595
\(652\) 1.18324e24 1.41895
\(653\) 1.26098e23 0.149261 0.0746304 0.997211i \(-0.476222\pi\)
0.0746304 + 0.997211i \(0.476222\pi\)
\(654\) 4.83505e23 0.564924
\(655\) −1.35190e23 −0.155917
\(656\) −7.19375e23 −0.818979
\(657\) −3.76940e23 −0.423611
\(658\) 8.39824e22 0.0931683
\(659\) −1.06105e24 −1.16201 −0.581005 0.813900i \(-0.697340\pi\)
−0.581005 + 0.813900i \(0.697340\pi\)
\(660\) −4.89216e23 −0.528904
\(661\) 1.57511e24 1.68112 0.840560 0.541719i \(-0.182226\pi\)
0.840560 + 0.541719i \(0.182226\pi\)
\(662\) −2.11448e24 −2.22797
\(663\) 5.91077e22 0.0614864
\(664\) 2.61528e23 0.268590
\(665\) −4.92271e21 −0.00499137
\(666\) 7.42408e23 0.743211
\(667\) −7.71615e23 −0.762660
\(668\) −1.27251e24 −1.24182
\(669\) −4.06301e23 −0.391495
\(670\) 2.81449e24 2.67771
\(671\) 9.85948e23 0.926217
\(672\) −1.23342e23 −0.114412
\(673\) 2.91330e23 0.266844 0.133422 0.991059i \(-0.457403\pi\)
0.133422 + 0.991059i \(0.457403\pi\)
\(674\) 5.63657e23 0.509807
\(675\) 2.97489e23 0.265699
\(676\) −3.23993e23 −0.285751
\(677\) 1.33861e24 1.16587 0.582937 0.812517i \(-0.301903\pi\)
0.582937 + 0.812517i \(0.301903\pi\)
\(678\) −6.36475e23 −0.547430
\(679\) −2.05721e23 −0.174737
\(680\) −4.89385e22 −0.0410511
\(681\) 7.87321e23 0.652231
\(682\) −3.52140e24 −2.88103
\(683\) −1.05295e24 −0.850812 −0.425406 0.905003i \(-0.639869\pi\)
−0.425406 + 0.905003i \(0.639869\pi\)
\(684\) −2.43424e22 −0.0194261
\(685\) −1.91966e24 −1.51305
\(686\) 7.74764e23 0.603135
\(687\) 3.91947e23 0.301366
\(688\) 1.33999e24 1.01765
\(689\) 5.82932e22 0.0437275
\(690\) 7.35787e23 0.545174
\(691\) −5.71311e22 −0.0418128 −0.0209064 0.999781i \(-0.506655\pi\)
−0.0209064 + 0.999781i \(0.506655\pi\)
\(692\) 3.41850e23 0.247134
\(693\) 2.69343e23 0.192341
\(694\) 8.54519e23 0.602790
\(695\) −1.62809e24 −1.13451
\(696\) 9.15909e22 0.0630483
\(697\) −2.65320e23 −0.180423
\(698\) 2.95721e24 1.98660
\(699\) −4.23044e23 −0.280756
\(700\) 1.37918e23 0.0904248
\(701\) −3.18403e23 −0.206241 −0.103120 0.994669i \(-0.532883\pi\)
−0.103120 + 0.994669i \(0.532883\pi\)
\(702\) −1.37905e24 −0.882498
\(703\) 1.88788e22 0.0119359
\(704\) 2.09705e24 1.30991
\(705\) −2.16683e23 −0.133726
\(706\) −2.69689e24 −1.64446
\(707\) −5.18131e23 −0.312158
\(708\) −2.06894e22 −0.0123159
\(709\) 2.12997e24 1.25279 0.626397 0.779504i \(-0.284529\pi\)
0.626397 + 0.779504i \(0.284529\pi\)
\(710\) −2.00924e24 −1.16771
\(711\) −2.62182e23 −0.150561
\(712\) −2.56261e23 −0.145413
\(713\) 2.80617e24 1.57345
\(714\) −3.96210e22 −0.0219529
\(715\) 1.96450e24 1.07560
\(716\) −1.16738e24 −0.631613
\(717\) −8.11119e23 −0.433683
\(718\) −3.08861e24 −1.63195
\(719\) −2.34234e24 −1.22308 −0.611540 0.791213i \(-0.709450\pi\)
−0.611540 + 0.791213i \(0.709450\pi\)
\(720\) −1.67317e24 −0.863403
\(721\) −2.92641e23 −0.149240
\(722\) 2.89248e24 1.45782
\(723\) 5.94456e23 0.296104
\(724\) 2.07873e24 1.02334
\(725\) 7.04299e23 0.342676
\(726\) 1.41357e23 0.0679761
\(727\) 3.58185e24 1.70241 0.851207 0.524830i \(-0.175871\pi\)
0.851207 + 0.524830i \(0.175871\pi\)
\(728\) −7.20222e22 −0.0338337
\(729\) −5.28574e23 −0.245427
\(730\) 1.84289e24 0.845776
\(731\) 4.94215e23 0.224191
\(732\) 8.27881e23 0.371213
\(733\) −3.29013e24 −1.45824 −0.729119 0.684387i \(-0.760070\pi\)
−0.729119 + 0.684387i \(0.760070\pi\)
\(734\) −2.26794e23 −0.0993609
\(735\) −9.76615e23 −0.422942
\(736\) −2.82997e24 −1.21149
\(737\) −3.91591e24 −1.65714
\(738\) 2.85835e24 1.19574
\(739\) 1.74130e24 0.720103 0.360052 0.932932i \(-0.382759\pi\)
0.360052 + 0.932932i \(0.382759\pi\)
\(740\) −1.92317e24 −0.786227
\(741\) −1.61928e22 −0.00654436
\(742\) −3.90750e22 −0.0156123
\(743\) −4.61544e24 −1.82309 −0.911545 0.411199i \(-0.865110\pi\)
−0.911545 + 0.411199i \(0.865110\pi\)
\(744\) −3.33094e23 −0.130076
\(745\) 4.89576e23 0.189013
\(746\) 3.45700e24 1.31953
\(747\) 3.29778e24 1.24451
\(748\) 6.04431e23 0.225519
\(749\) 6.30841e23 0.232715
\(750\) 1.09875e24 0.400753
\(751\) −8.19607e22 −0.0295574 −0.0147787 0.999891i \(-0.504704\pi\)
−0.0147787 + 0.999891i \(0.504704\pi\)
\(752\) 7.25865e23 0.258824
\(753\) 1.43303e24 0.505241
\(754\) −3.26486e24 −1.13817
\(755\) 4.03122e24 1.38959
\(756\) 4.89789e23 0.166945
\(757\) −2.51608e24 −0.848025 −0.424013 0.905656i \(-0.639379\pi\)
−0.424013 + 0.905656i \(0.639379\pi\)
\(758\) −8.01767e23 −0.267215
\(759\) −1.02373e24 −0.337389
\(760\) 1.34069e22 0.00436931
\(761\) −5.27745e23 −0.170081 −0.0850403 0.996378i \(-0.527102\pi\)
−0.0850403 + 0.996378i \(0.527102\pi\)
\(762\) −1.96859e24 −0.627390
\(763\) 6.89574e23 0.217331
\(764\) −4.17287e24 −1.30059
\(765\) −6.17098e23 −0.190209
\(766\) 4.24827e24 1.29499
\(767\) 8.30803e22 0.0250460
\(768\) −8.85151e23 −0.263905
\(769\) −7.51719e23 −0.221657 −0.110828 0.993840i \(-0.535350\pi\)
−0.110828 + 0.993840i \(0.535350\pi\)
\(770\) −1.31684e24 −0.384027
\(771\) −2.10508e23 −0.0607165
\(772\) 7.26381e24 2.07213
\(773\) 2.81914e24 0.795409 0.397704 0.917514i \(-0.369807\pi\)
0.397704 + 0.917514i \(0.369807\pi\)
\(774\) −5.32429e24 −1.48581
\(775\) −2.56136e24 −0.706980
\(776\) 5.60277e23 0.152960
\(777\) −1.75401e23 −0.0473644
\(778\) −3.61173e24 −0.964693
\(779\) 7.26854e22 0.0192034
\(780\) 1.64955e24 0.431083
\(781\) 2.79554e24 0.722656
\(782\) −9.09072e23 −0.232456
\(783\) 2.50118e24 0.632660
\(784\) 3.27156e24 0.818594
\(785\) 2.35385e24 0.582621
\(786\) 2.98088e23 0.0729882
\(787\) −3.20732e24 −0.776887 −0.388443 0.921473i \(-0.626987\pi\)
−0.388443 + 0.921473i \(0.626987\pi\)
\(788\) 1.91224e24 0.458216
\(789\) −7.66170e22 −0.0181623
\(790\) 1.28183e24 0.300608
\(791\) −9.07739e23 −0.210601
\(792\) −7.33550e23 −0.168370
\(793\) −3.32444e24 −0.754912
\(794\) −3.29186e24 −0.739549
\(795\) 1.00817e23 0.0224086
\(796\) −8.15008e24 −1.79226
\(797\) 2.36328e24 0.514186 0.257093 0.966387i \(-0.417235\pi\)
0.257093 + 0.966387i \(0.417235\pi\)
\(798\) 1.08543e22 0.00233657
\(799\) 2.67713e23 0.0570194
\(800\) 2.58309e24 0.544345
\(801\) −3.23136e24 −0.673766
\(802\) 1.22521e24 0.252772
\(803\) −2.56409e24 −0.523421
\(804\) −3.28811e24 −0.664156
\(805\) 1.04938e24 0.209733
\(806\) 1.18735e25 2.34818
\(807\) −4.93676e23 −0.0966088
\(808\) 1.41112e24 0.273254
\(809\) 9.60721e24 1.84092 0.920460 0.390836i \(-0.127814\pi\)
0.920460 + 0.390836i \(0.127814\pi\)
\(810\) 5.36439e24 1.01718
\(811\) −6.64616e24 −1.24708 −0.623539 0.781792i \(-0.714306\pi\)
−0.623539 + 0.781792i \(0.714306\pi\)
\(812\) 1.15956e24 0.215312
\(813\) 6.30035e23 0.115770
\(814\) 5.05014e24 0.918324
\(815\) −8.21751e24 −1.47877
\(816\) −3.42447e23 −0.0609856
\(817\) −1.35392e23 −0.0238620
\(818\) −1.25976e25 −2.19727
\(819\) −9.08175e23 −0.156768
\(820\) −7.40440e24 −1.26495
\(821\) −5.09458e24 −0.861373 −0.430687 0.902502i \(-0.641729\pi\)
−0.430687 + 0.902502i \(0.641729\pi\)
\(822\) 4.23275e24 0.708292
\(823\) 5.78198e24 0.957586 0.478793 0.877928i \(-0.341074\pi\)
0.478793 + 0.877928i \(0.341074\pi\)
\(824\) 7.97002e23 0.130641
\(825\) 9.34421e23 0.151595
\(826\) −5.56903e22 −0.00894230
\(827\) 8.38907e24 1.33327 0.666633 0.745386i \(-0.267735\pi\)
0.666633 + 0.745386i \(0.267735\pi\)
\(828\) 5.18909e24 0.816270
\(829\) 5.02769e24 0.782808 0.391404 0.920219i \(-0.371989\pi\)
0.391404 + 0.920219i \(0.371989\pi\)
\(830\) −1.61231e25 −2.48477
\(831\) 7.65741e23 0.116808
\(832\) −7.07088e24 −1.06764
\(833\) 1.20662e24 0.180338
\(834\) 3.58986e24 0.531087
\(835\) 8.83747e24 1.29417
\(836\) −1.65586e23 −0.0240033
\(837\) −9.09619e24 −1.30525
\(838\) 1.38251e25 1.96378
\(839\) 9.38328e23 0.131940 0.0659702 0.997822i \(-0.478986\pi\)
0.0659702 + 0.997822i \(0.478986\pi\)
\(840\) −1.24562e23 −0.0173384
\(841\) −1.33566e24 −0.184048
\(842\) −5.90847e24 −0.805977
\(843\) −1.20186e24 −0.162300
\(844\) 2.56295e24 0.342633
\(845\) 2.25011e24 0.297798
\(846\) −2.88414e24 −0.377892
\(847\) 2.01603e23 0.0261510
\(848\) −3.37728e23 −0.0433713
\(849\) −1.45603e24 −0.185121
\(850\) 8.29764e23 0.104447
\(851\) −4.02442e24 −0.501536
\(852\) 2.34736e24 0.289630
\(853\) 2.33188e24 0.284865 0.142433 0.989805i \(-0.454508\pi\)
0.142433 + 0.989805i \(0.454508\pi\)
\(854\) 2.22843e24 0.269531
\(855\) 1.69056e23 0.0202451
\(856\) −1.71808e24 −0.203712
\(857\) −1.09008e25 −1.27974 −0.639872 0.768482i \(-0.721012\pi\)
−0.639872 + 0.768482i \(0.721012\pi\)
\(858\) −4.33162e24 −0.503510
\(859\) −1.47838e25 −1.70155 −0.850773 0.525533i \(-0.823866\pi\)
−0.850773 + 0.525533i \(0.823866\pi\)
\(860\) 1.37923e25 1.57181
\(861\) −6.75310e23 −0.0762037
\(862\) −9.20284e24 −1.02828
\(863\) 5.86087e24 0.648441 0.324220 0.945982i \(-0.394898\pi\)
0.324220 + 0.945982i \(0.394898\pi\)
\(864\) 9.17333e24 1.00499
\(865\) −2.37412e24 −0.257553
\(866\) 2.38346e25 2.56039
\(867\) 3.41760e24 0.363545
\(868\) −4.21705e24 −0.444213
\(869\) −1.78346e24 −0.186036
\(870\) −5.64654e24 −0.583269
\(871\) 1.32037e25 1.35065
\(872\) −1.87804e24 −0.190246
\(873\) 7.06491e24 0.708737
\(874\) 2.49044e23 0.0247417
\(875\) 1.56703e24 0.154173
\(876\) −2.15302e24 −0.209779
\(877\) −1.08364e25 −1.04565 −0.522827 0.852439i \(-0.675122\pi\)
−0.522827 + 0.852439i \(0.675122\pi\)
\(878\) −2.03420e25 −1.94398
\(879\) 3.27678e24 0.310128
\(880\) −1.13815e25 −1.06684
\(881\) 5.00054e24 0.464217 0.232109 0.972690i \(-0.425437\pi\)
0.232109 + 0.972690i \(0.425437\pi\)
\(882\) −1.29991e25 −1.19518
\(883\) −1.04108e24 −0.0948025 −0.0474012 0.998876i \(-0.515094\pi\)
−0.0474012 + 0.998876i \(0.515094\pi\)
\(884\) −2.03803e24 −0.183809
\(885\) 1.43686e23 0.0128351
\(886\) 2.03262e24 0.179833
\(887\) 1.47651e25 1.29386 0.646929 0.762550i \(-0.276053\pi\)
0.646929 + 0.762550i \(0.276053\pi\)
\(888\) 4.77700e23 0.0414615
\(889\) −2.80760e24 −0.241363
\(890\) 1.57984e25 1.34523
\(891\) −7.46369e24 −0.629496
\(892\) 1.40092e25 1.17034
\(893\) −7.33412e22 −0.00606891
\(894\) −1.07949e24 −0.0884812
\(895\) 8.10737e24 0.658240
\(896\) 9.66021e23 0.0776906
\(897\) 3.45184e24 0.274988
\(898\) 1.46580e24 0.115672
\(899\) −2.15350e25 −1.68340
\(900\) −4.73640e24 −0.366764
\(901\) −1.24561e23 −0.00955478
\(902\) 1.94436e25 1.47747
\(903\) 1.25791e24 0.0946899
\(904\) 2.47221e24 0.184354
\(905\) −1.44366e25 −1.06648
\(906\) −8.88866e24 −0.650499
\(907\) −2.99747e23 −0.0217317 −0.0108658 0.999941i \(-0.503459\pi\)
−0.0108658 + 0.999941i \(0.503459\pi\)
\(908\) −2.71468e25 −1.94979
\(909\) 1.77937e25 1.26612
\(910\) 4.44014e24 0.313000
\(911\) −1.71431e24 −0.119724 −0.0598621 0.998207i \(-0.519066\pi\)
−0.0598621 + 0.998207i \(0.519066\pi\)
\(912\) 9.38148e22 0.00649106
\(913\) 2.24328e25 1.53773
\(914\) 7.08256e24 0.481003
\(915\) −5.74958e24 −0.386863
\(916\) −1.35143e25 −0.900911
\(917\) 4.25132e23 0.0280792
\(918\) 2.94675e24 0.192832
\(919\) −2.42539e25 −1.57253 −0.786265 0.617889i \(-0.787988\pi\)
−0.786265 + 0.617889i \(0.787988\pi\)
\(920\) −2.85796e24 −0.183594
\(921\) −5.93213e24 −0.377575
\(922\) 2.28480e25 1.44091
\(923\) −9.42605e24 −0.589000
\(924\) 1.53844e24 0.0952507
\(925\) 3.67333e24 0.225349
\(926\) −2.03661e25 −1.23798
\(927\) 1.00499e25 0.605320
\(928\) 2.17176e25 1.29615
\(929\) 1.40138e25 0.828745 0.414373 0.910107i \(-0.364001\pi\)
0.414373 + 0.910107i \(0.364001\pi\)
\(930\) 2.05351e25 1.20335
\(931\) −3.30557e23 −0.0191944
\(932\) 1.45865e25 0.839298
\(933\) 1.00560e25 0.573364
\(934\) −2.62270e25 −1.48184
\(935\) −4.19773e24 −0.235026
\(936\) 2.47340e24 0.137230
\(937\) −2.43217e25 −1.33724 −0.668618 0.743606i \(-0.733114\pi\)
−0.668618 + 0.743606i \(0.733114\pi\)
\(938\) −8.85072e24 −0.482230
\(939\) −6.78690e24 −0.366449
\(940\) 7.47120e24 0.399764
\(941\) 5.98394e23 0.0317304 0.0158652 0.999874i \(-0.494950\pi\)
0.0158652 + 0.999874i \(0.494950\pi\)
\(942\) −5.19012e24 −0.272738
\(943\) −1.54944e25 −0.806912
\(944\) −4.81335e23 −0.0248419
\(945\) −3.40155e24 −0.173983
\(946\) −3.62178e25 −1.83589
\(947\) 3.62477e25 1.82098 0.910491 0.413530i \(-0.135704\pi\)
0.910491 + 0.413530i \(0.135704\pi\)
\(948\) −1.49754e24 −0.0745602
\(949\) 8.64565e24 0.426614
\(950\) −2.27317e23 −0.0111169
\(951\) −3.84865e24 −0.186541
\(952\) 1.53897e23 0.00739291
\(953\) −1.88157e25 −0.895839 −0.447919 0.894074i \(-0.647835\pi\)
−0.447919 + 0.894074i \(0.647835\pi\)
\(954\) 1.34192e24 0.0633236
\(955\) 2.89803e25 1.35542
\(956\) 2.79673e25 1.29646
\(957\) 7.85627e24 0.360965
\(958\) 2.25199e25 1.02556
\(959\) 6.03674e24 0.272486
\(960\) −1.22290e25 −0.547124
\(961\) 5.57675e25 2.47305
\(962\) −1.70282e25 −0.748479
\(963\) −2.16644e25 −0.943896
\(964\) −2.04968e25 −0.885180
\(965\) −5.04467e25 −2.15949
\(966\) −2.31383e24 −0.0981807
\(967\) −1.81212e24 −0.0762186 −0.0381093 0.999274i \(-0.512134\pi\)
−0.0381093 + 0.999274i \(0.512134\pi\)
\(968\) −5.49062e23 −0.0228919
\(969\) 3.46008e22 0.00142999
\(970\) −3.45409e25 −1.41506
\(971\) 2.03685e25 0.827170 0.413585 0.910465i \(-0.364276\pi\)
0.413585 + 0.910465i \(0.364276\pi\)
\(972\) −2.58739e25 −1.04160
\(973\) 5.11985e24 0.204314
\(974\) 4.90050e25 1.93861
\(975\) −3.15070e24 −0.123557
\(976\) 1.92605e25 0.748763
\(977\) −2.52590e25 −0.973449 −0.486724 0.873556i \(-0.661808\pi\)
−0.486724 + 0.873556i \(0.661808\pi\)
\(978\) 1.81192e25 0.692244
\(979\) −2.19809e25 −0.832517
\(980\) 3.36736e25 1.26435
\(981\) −2.36815e25 −0.881499
\(982\) −6.85750e23 −0.0253057
\(983\) −1.51840e24 −0.0555495 −0.0277747 0.999614i \(-0.508842\pi\)
−0.0277747 + 0.999614i \(0.508842\pi\)
\(984\) 1.83919e24 0.0667066
\(985\) −1.32804e25 −0.477533
\(986\) 6.97635e24 0.248699
\(987\) 6.81402e23 0.0240828
\(988\) 5.58327e23 0.0195639
\(989\) 2.88617e25 1.00266
\(990\) 4.52231e25 1.55762
\(991\) −6.69650e24 −0.228677 −0.114338 0.993442i \(-0.536475\pi\)
−0.114338 + 0.993442i \(0.536475\pi\)
\(992\) −7.89818e25 −2.67410
\(993\) −1.71561e25 −0.575903
\(994\) 6.31846e24 0.210294
\(995\) 5.66018e25 1.86781
\(996\) 1.88363e25 0.616300
\(997\) −3.45326e25 −1.12026 −0.560132 0.828404i \(-0.689249\pi\)
−0.560132 + 0.828404i \(0.689249\pi\)
\(998\) 4.04864e25 1.30227
\(999\) 1.30451e25 0.416046
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1.18.a.a.1.1 1
3.2 odd 2 9.18.a.b.1.1 1
4.3 odd 2 16.18.a.b.1.1 1
5.2 odd 4 25.18.b.a.24.1 2
5.3 odd 4 25.18.b.a.24.2 2
5.4 even 2 25.18.a.a.1.1 1
7.6 odd 2 49.18.a.a.1.1 1
8.3 odd 2 64.18.a.b.1.1 1
8.5 even 2 64.18.a.d.1.1 1
11.10 odd 2 121.18.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.18.a.a.1.1 1 1.1 even 1 trivial
9.18.a.b.1.1 1 3.2 odd 2
16.18.a.b.1.1 1 4.3 odd 2
25.18.a.a.1.1 1 5.4 even 2
25.18.b.a.24.1 2 5.2 odd 4
25.18.b.a.24.2 2 5.3 odd 4
49.18.a.a.1.1 1 7.6 odd 2
64.18.a.b.1.1 1 8.3 odd 2
64.18.a.d.1.1 1 8.5 even 2
121.18.a.b.1.1 1 11.10 odd 2