Properties

Label 177.10.a.b.1.12
Level $177$
Weight $10$
Character 177.1
Self dual yes
Analytic conductor $91.161$
Analytic rank $1$
Dimension $21$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(91.1613430010\)
Analytic rank: \(1\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+7.87274 q^{2} -81.0000 q^{3} -450.020 q^{4} -1394.70 q^{5} -637.692 q^{6} -8519.00 q^{7} -7573.74 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q+7.87274 q^{2} -81.0000 q^{3} -450.020 q^{4} -1394.70 q^{5} -637.692 q^{6} -8519.00 q^{7} -7573.74 q^{8} +6561.00 q^{9} -10980.1 q^{10} -32784.5 q^{11} +36451.6 q^{12} +173232. q^{13} -67067.9 q^{14} +112971. q^{15} +170784. q^{16} +557614. q^{17} +51653.1 q^{18} +329203. q^{19} +627644. q^{20} +690039. q^{21} -258104. q^{22} +121336. q^{23} +613473. q^{24} -7930.40 q^{25} +1.36381e6 q^{26} -531441. q^{27} +3.83372e6 q^{28} -5991.02 q^{29} +889391. q^{30} -7.64818e6 q^{31} +5.22229e6 q^{32} +2.65554e6 q^{33} +4.38995e6 q^{34} +1.18815e7 q^{35} -2.95258e6 q^{36} -7.81561e6 q^{37} +2.59173e6 q^{38} -1.40318e7 q^{39} +1.05631e7 q^{40} -1.42595e7 q^{41} +5.43250e6 q^{42} +1.08394e7 q^{43} +1.47537e7 q^{44} -9.15064e6 q^{45} +955249. q^{46} +1.53262e7 q^{47} -1.38335e7 q^{48} +3.22198e7 q^{49} -62434.0 q^{50} -4.51668e7 q^{51} -7.79577e7 q^{52} -2.75622e6 q^{53} -4.18390e6 q^{54} +4.57246e7 q^{55} +6.45207e7 q^{56} -2.66654e7 q^{57} -47165.8 q^{58} -1.21174e7 q^{59} -5.08391e7 q^{60} +1.86412e8 q^{61} -6.02122e7 q^{62} -5.58932e7 q^{63} -4.63277e7 q^{64} -2.41607e8 q^{65} +2.09064e7 q^{66} -3.01770e7 q^{67} -2.50938e8 q^{68} -9.82824e6 q^{69} +9.35398e7 q^{70} -9.51388e6 q^{71} -4.96913e7 q^{72} +2.10691e8 q^{73} -6.15303e7 q^{74} +642363. q^{75} -1.48148e8 q^{76} +2.79291e8 q^{77} -1.10468e8 q^{78} +3.67936e7 q^{79} -2.38193e8 q^{80} +4.30467e7 q^{81} -1.12261e8 q^{82} +8.97616e7 q^{83} -3.10531e8 q^{84} -7.77706e8 q^{85} +8.53361e7 q^{86} +485273. q^{87} +2.48301e8 q^{88} -8.83063e7 q^{89} -7.20407e7 q^{90} -1.47576e9 q^{91} -5.46037e7 q^{92} +6.19503e8 q^{93} +1.20660e8 q^{94} -4.59140e8 q^{95} -4.23006e8 q^{96} -1.72684e8 q^{97} +2.53658e8 q^{98} -2.15099e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21q + 20q^{2} - 1701q^{3} + 4950q^{4} + 2058q^{5} - 1620q^{6} - 17167q^{7} - 2853q^{8} + 137781q^{9} + O(q^{10}) \) \( 21q + 20q^{2} - 1701q^{3} + 4950q^{4} + 2058q^{5} - 1620q^{6} - 17167q^{7} - 2853q^{8} + 137781q^{9} - 31559q^{10} - 38751q^{11} - 400950q^{12} - 58915q^{13} + 3453q^{14} - 166698q^{15} + 1655714q^{16} - 64233q^{17} + 131220q^{18} - 1937236q^{19} - 1065507q^{20} + 1390527q^{21} - 5386882q^{22} - 1838574q^{23} + 231093q^{24} + 4565755q^{25} - 839702q^{26} - 11160261q^{27} - 4471034q^{28} + 15658544q^{29} + 2556279q^{30} - 14282802q^{31} - 2205286q^{32} + 3138831q^{33} + 19005532q^{34} - 8633300q^{35} + 32476950q^{36} + 7531195q^{37} + 26649773q^{38} + 4772115q^{39} + 17775672q^{40} + 18338245q^{41} - 279693q^{42} - 22480305q^{43} - 80230922q^{44} + 13502538q^{45} - 83894107q^{46} - 110397260q^{47} - 134112834q^{48} + 130653638q^{49} + 65575693q^{50} + 5202873q^{51} + 177908014q^{52} + 145498338q^{53} - 10628820q^{54} + 86448944q^{55} + 354387888q^{56} + 156916116q^{57} + 115508368q^{58} - 254464581q^{59} + 86306067q^{60} + 287595506q^{61} + 819899030q^{62} - 112632687q^{63} + 822446413q^{64} + 77238206q^{65} + 436337442q^{66} - 392860610q^{67} + 167325073q^{68} + 148924494q^{69} - 424902116q^{70} - 248960491q^{71} - 18718533q^{72} - 758406074q^{73} - 923266846q^{74} - 369826155q^{75} - 2312747568q^{76} - 878126795q^{77} + 68015862q^{78} - 1925801029q^{79} - 1898919861q^{80} + 903981141q^{81} - 3249102191q^{82} - 1650336307q^{83} + 362153754q^{84} - 2342480762q^{85} - 3609864952q^{86} - 1268342064q^{87} - 5987792887q^{88} - 574997526q^{89} - 207058599q^{90} - 4481387117q^{91} - 5317166770q^{92} + 1156906962q^{93} - 5360726568q^{94} - 2789231462q^{95} + 178628166q^{96} - 4651540898q^{97} - 5566652976q^{98} - 254245311q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.87274 0.347929 0.173965 0.984752i \(-0.444342\pi\)
0.173965 + 0.984752i \(0.444342\pi\)
\(3\) −81.0000 −0.577350
\(4\) −450.020 −0.878945
\(5\) −1394.70 −0.997968 −0.498984 0.866611i \(-0.666293\pi\)
−0.498984 + 0.866611i \(0.666293\pi\)
\(6\) −637.692 −0.200877
\(7\) −8519.00 −1.34106 −0.670529 0.741883i \(-0.733933\pi\)
−0.670529 + 0.741883i \(0.733933\pi\)
\(8\) −7573.74 −0.653740
\(9\) 6561.00 0.333333
\(10\) −10980.1 −0.347222
\(11\) −32784.5 −0.675151 −0.337576 0.941298i \(-0.609607\pi\)
−0.337576 + 0.941298i \(0.609607\pi\)
\(12\) 36451.6 0.507459
\(13\) 173232. 1.68222 0.841109 0.540866i \(-0.181903\pi\)
0.841109 + 0.540866i \(0.181903\pi\)
\(14\) −67067.9 −0.466593
\(15\) 112971. 0.576177
\(16\) 170784. 0.651490
\(17\) 557614. 1.61925 0.809625 0.586948i \(-0.199671\pi\)
0.809625 + 0.586948i \(0.199671\pi\)
\(18\) 51653.1 0.115976
\(19\) 329203. 0.579525 0.289763 0.957099i \(-0.406424\pi\)
0.289763 + 0.957099i \(0.406424\pi\)
\(20\) 627644. 0.877159
\(21\) 690039. 0.774260
\(22\) −258104. −0.234905
\(23\) 121336. 0.0904098 0.0452049 0.998978i \(-0.485606\pi\)
0.0452049 + 0.998978i \(0.485606\pi\)
\(24\) 613473. 0.377437
\(25\) −7930.40 −0.00406037
\(26\) 1.36381e6 0.585293
\(27\) −531441. −0.192450
\(28\) 3.83372e6 1.17872
\(29\) −5991.02 −0.00157293 −0.000786466 1.00000i \(-0.500250\pi\)
−0.000786466 1.00000i \(0.500250\pi\)
\(30\) 889391. 0.200469
\(31\) −7.64818e6 −1.48741 −0.743705 0.668508i \(-0.766933\pi\)
−0.743705 + 0.668508i \(0.766933\pi\)
\(32\) 5.22229e6 0.880413
\(33\) 2.65554e6 0.389799
\(34\) 4.38995e6 0.563384
\(35\) 1.18815e7 1.33833
\(36\) −2.95258e6 −0.292982
\(37\) −7.81561e6 −0.685575 −0.342788 0.939413i \(-0.611371\pi\)
−0.342788 + 0.939413i \(0.611371\pi\)
\(38\) 2.59173e6 0.201634
\(39\) −1.40318e7 −0.971229
\(40\) 1.05631e7 0.652412
\(41\) −1.42595e7 −0.788092 −0.394046 0.919091i \(-0.628925\pi\)
−0.394046 + 0.919091i \(0.628925\pi\)
\(42\) 5.43250e6 0.269388
\(43\) 1.08394e7 0.483502 0.241751 0.970338i \(-0.422278\pi\)
0.241751 + 0.970338i \(0.422278\pi\)
\(44\) 1.47537e7 0.593421
\(45\) −9.15064e6 −0.332656
\(46\) 955249. 0.0314562
\(47\) 1.53262e7 0.458137 0.229069 0.973410i \(-0.426432\pi\)
0.229069 + 0.973410i \(0.426432\pi\)
\(48\) −1.38335e7 −0.376138
\(49\) 3.22198e7 0.798437
\(50\) −62434.0 −0.00141272
\(51\) −4.51668e7 −0.934874
\(52\) −7.79577e7 −1.47858
\(53\) −2.75622e6 −0.0479813 −0.0239907 0.999712i \(-0.507637\pi\)
−0.0239907 + 0.999712i \(0.507637\pi\)
\(54\) −4.18390e6 −0.0669590
\(55\) 4.57246e7 0.673779
\(56\) 6.45207e7 0.876704
\(57\) −2.66654e7 −0.334589
\(58\) −47165.8 −0.000547269 0
\(59\) −1.21174e7 −0.130189
\(60\) −5.08391e7 −0.506428
\(61\) 1.86412e8 1.72381 0.861906 0.507067i \(-0.169270\pi\)
0.861906 + 0.507067i \(0.169270\pi\)
\(62\) −6.02122e7 −0.517514
\(63\) −5.58932e7 −0.447019
\(64\) −4.63277e7 −0.345168
\(65\) −2.41607e8 −1.67880
\(66\) 2.09064e7 0.135622
\(67\) −3.01770e7 −0.182953 −0.0914765 0.995807i \(-0.529159\pi\)
−0.0914765 + 0.995807i \(0.529159\pi\)
\(68\) −2.50938e8 −1.42323
\(69\) −9.82824e6 −0.0521981
\(70\) 9.35398e7 0.465645
\(71\) −9.51388e6 −0.0444319 −0.0222160 0.999753i \(-0.507072\pi\)
−0.0222160 + 0.999753i \(0.507072\pi\)
\(72\) −4.96913e7 −0.217913
\(73\) 2.10691e8 0.868348 0.434174 0.900829i \(-0.357040\pi\)
0.434174 + 0.900829i \(0.357040\pi\)
\(74\) −6.15303e7 −0.238532
\(75\) 642363. 0.00234425
\(76\) −1.48148e8 −0.509371
\(77\) 2.79291e8 0.905417
\(78\) −1.10468e8 −0.337919
\(79\) 3.67936e7 0.106280 0.0531398 0.998587i \(-0.483077\pi\)
0.0531398 + 0.998587i \(0.483077\pi\)
\(80\) −2.38193e8 −0.650166
\(81\) 4.30467e7 0.111111
\(82\) −1.12261e8 −0.274200
\(83\) 8.97616e7 0.207606 0.103803 0.994598i \(-0.466899\pi\)
0.103803 + 0.994598i \(0.466899\pi\)
\(84\) −3.10531e8 −0.680532
\(85\) −7.77706e8 −1.61596
\(86\) 8.53361e7 0.168225
\(87\) 485273. 0.000908133 0
\(88\) 2.48301e8 0.441374
\(89\) −8.83063e7 −0.149189 −0.0745944 0.997214i \(-0.523766\pi\)
−0.0745944 + 0.997214i \(0.523766\pi\)
\(90\) −7.20407e7 −0.115741
\(91\) −1.47576e9 −2.25595
\(92\) −5.46037e7 −0.0794652
\(93\) 6.19503e8 0.858756
\(94\) 1.20660e8 0.159399
\(95\) −4.59140e8 −0.578347
\(96\) −4.23006e8 −0.508306
\(97\) −1.72684e8 −0.198053 −0.0990263 0.995085i \(-0.531573\pi\)
−0.0990263 + 0.995085i \(0.531573\pi\)
\(98\) 2.53658e8 0.277800
\(99\) −2.15099e8 −0.225050
\(100\) 3.56884e6 0.00356884
\(101\) 1.82690e9 1.74690 0.873452 0.486911i \(-0.161876\pi\)
0.873452 + 0.486911i \(0.161876\pi\)
\(102\) −3.55586e8 −0.325270
\(103\) 1.49179e9 1.30599 0.652994 0.757363i \(-0.273513\pi\)
0.652994 + 0.757363i \(0.273513\pi\)
\(104\) −1.31201e9 −1.09973
\(105\) −9.62399e8 −0.772687
\(106\) −2.16990e7 −0.0166941
\(107\) −1.25467e9 −0.925344 −0.462672 0.886529i \(-0.653109\pi\)
−0.462672 + 0.886529i \(0.653109\pi\)
\(108\) 2.39159e8 0.169153
\(109\) −6.65149e7 −0.0451335 −0.0225668 0.999745i \(-0.507184\pi\)
−0.0225668 + 0.999745i \(0.507184\pi\)
\(110\) 3.59978e8 0.234428
\(111\) 6.33064e8 0.395817
\(112\) −1.45491e9 −0.873685
\(113\) 2.19833e9 1.26835 0.634177 0.773188i \(-0.281339\pi\)
0.634177 + 0.773188i \(0.281339\pi\)
\(114\) −2.09930e8 −0.116413
\(115\) −1.69228e8 −0.0902260
\(116\) 2.69608e6 0.00138252
\(117\) 1.13657e9 0.560739
\(118\) −9.53969e7 −0.0452965
\(119\) −4.75032e9 −2.17151
\(120\) −8.55612e8 −0.376670
\(121\) −1.28313e9 −0.544171
\(122\) 1.46758e9 0.599765
\(123\) 1.15502e9 0.455005
\(124\) 3.44183e9 1.30735
\(125\) 2.73509e9 1.00202
\(126\) −4.40033e8 −0.155531
\(127\) 2.05903e9 0.702337 0.351169 0.936312i \(-0.385784\pi\)
0.351169 + 0.936312i \(0.385784\pi\)
\(128\) −3.03854e9 −1.00051
\(129\) −8.77994e8 −0.279150
\(130\) −1.90211e9 −0.584104
\(131\) −5.98798e9 −1.77648 −0.888239 0.459381i \(-0.848071\pi\)
−0.888239 + 0.459381i \(0.848071\pi\)
\(132\) −1.19505e9 −0.342612
\(133\) −2.80448e9 −0.777177
\(134\) −2.37576e8 −0.0636547
\(135\) 7.41202e8 0.192059
\(136\) −4.22322e9 −1.05857
\(137\) 2.48377e9 0.602377 0.301189 0.953565i \(-0.402617\pi\)
0.301189 + 0.953565i \(0.402617\pi\)
\(138\) −7.73752e7 −0.0181613
\(139\) −3.99988e9 −0.908825 −0.454413 0.890791i \(-0.650151\pi\)
−0.454413 + 0.890791i \(0.650151\pi\)
\(140\) −5.34690e9 −1.17632
\(141\) −1.24143e9 −0.264506
\(142\) −7.49003e7 −0.0154592
\(143\) −5.67931e9 −1.13575
\(144\) 1.12051e9 0.217163
\(145\) 8.35569e6 0.00156974
\(146\) 1.65872e9 0.302124
\(147\) −2.60980e9 −0.460978
\(148\) 3.51718e9 0.602583
\(149\) −1.05074e10 −1.74645 −0.873226 0.487316i \(-0.837976\pi\)
−0.873226 + 0.487316i \(0.837976\pi\)
\(150\) 5.05716e6 0.000815635 0
\(151\) 4.68248e9 0.732960 0.366480 0.930426i \(-0.380563\pi\)
0.366480 + 0.930426i \(0.380563\pi\)
\(152\) −2.49329e9 −0.378859
\(153\) 3.65851e9 0.539750
\(154\) 2.19879e9 0.315021
\(155\) 1.06669e10 1.48439
\(156\) 6.31457e9 0.853657
\(157\) 1.03498e10 1.35951 0.679756 0.733439i \(-0.262086\pi\)
0.679756 + 0.733439i \(0.262086\pi\)
\(158\) 2.89666e8 0.0369778
\(159\) 2.23254e8 0.0277020
\(160\) −7.28354e9 −0.878623
\(161\) −1.03366e9 −0.121245
\(162\) 3.38896e8 0.0386588
\(163\) −1.12940e10 −1.25315 −0.626577 0.779359i \(-0.715545\pi\)
−0.626577 + 0.779359i \(0.715545\pi\)
\(164\) 6.41706e9 0.692690
\(165\) −3.70369e9 −0.389007
\(166\) 7.06670e8 0.0722321
\(167\) −4.72352e9 −0.469940 −0.234970 0.972003i \(-0.575499\pi\)
−0.234970 + 0.972003i \(0.575499\pi\)
\(168\) −5.22617e9 −0.506165
\(169\) 1.94047e10 1.82986
\(170\) −6.12268e9 −0.562240
\(171\) 2.15990e9 0.193175
\(172\) −4.87796e9 −0.424972
\(173\) 7.29035e9 0.618787 0.309393 0.950934i \(-0.399874\pi\)
0.309393 + 0.950934i \(0.399874\pi\)
\(174\) 3.82043e6 0.000315966 0
\(175\) 6.75591e7 0.00544519
\(176\) −5.59907e9 −0.439854
\(177\) 9.81506e8 0.0751646
\(178\) −6.95213e8 −0.0519072
\(179\) 1.12010e10 0.815491 0.407745 0.913096i \(-0.366315\pi\)
0.407745 + 0.913096i \(0.366315\pi\)
\(180\) 4.11797e9 0.292386
\(181\) −2.04979e10 −1.41957 −0.709783 0.704421i \(-0.751207\pi\)
−0.709783 + 0.704421i \(0.751207\pi\)
\(182\) −1.16183e10 −0.784912
\(183\) −1.50994e10 −0.995244
\(184\) −9.18969e8 −0.0591045
\(185\) 1.09004e10 0.684182
\(186\) 4.87719e9 0.298787
\(187\) −1.82811e10 −1.09324
\(188\) −6.89712e9 −0.402677
\(189\) 4.52735e9 0.258087
\(190\) −3.61469e9 −0.201224
\(191\) −3.24595e10 −1.76479 −0.882393 0.470514i \(-0.844069\pi\)
−0.882393 + 0.470514i \(0.844069\pi\)
\(192\) 3.75254e9 0.199283
\(193\) −2.97274e10 −1.54223 −0.771114 0.636697i \(-0.780300\pi\)
−0.771114 + 0.636697i \(0.780300\pi\)
\(194\) −1.35950e9 −0.0689083
\(195\) 1.95701e10 0.969255
\(196\) −1.44995e10 −0.701782
\(197\) −3.17107e10 −1.50006 −0.750029 0.661405i \(-0.769960\pi\)
−0.750029 + 0.661405i \(0.769960\pi\)
\(198\) −1.69342e9 −0.0783016
\(199\) −2.20685e9 −0.0997548 −0.0498774 0.998755i \(-0.515883\pi\)
−0.0498774 + 0.998755i \(0.515883\pi\)
\(200\) 6.00628e7 0.00265442
\(201\) 2.44434e9 0.105628
\(202\) 1.43827e10 0.607799
\(203\) 5.10375e7 0.00210939
\(204\) 2.03259e10 0.821703
\(205\) 1.98878e10 0.786491
\(206\) 1.17445e10 0.454392
\(207\) 7.96087e8 0.0301366
\(208\) 2.95852e10 1.09595
\(209\) −1.07927e10 −0.391267
\(210\) −7.57672e9 −0.268840
\(211\) −4.43141e10 −1.53911 −0.769556 0.638579i \(-0.779523\pi\)
−0.769556 + 0.638579i \(0.779523\pi\)
\(212\) 1.24035e9 0.0421730
\(213\) 7.70624e8 0.0256528
\(214\) −9.87771e9 −0.321954
\(215\) −1.51178e10 −0.482520
\(216\) 4.02499e9 0.125812
\(217\) 6.51549e10 1.99470
\(218\) −5.23654e8 −0.0157033
\(219\) −1.70660e10 −0.501341
\(220\) −2.05770e10 −0.592215
\(221\) 9.65965e10 2.72393
\(222\) 4.98395e9 0.137716
\(223\) 3.13863e10 0.849900 0.424950 0.905217i \(-0.360292\pi\)
0.424950 + 0.905217i \(0.360292\pi\)
\(224\) −4.44887e10 −1.18068
\(225\) −5.20314e7 −0.00135346
\(226\) 1.73069e10 0.441298
\(227\) −1.35755e10 −0.339342 −0.169671 0.985501i \(-0.554271\pi\)
−0.169671 + 0.985501i \(0.554271\pi\)
\(228\) 1.20000e10 0.294085
\(229\) −3.87818e10 −0.931898 −0.465949 0.884812i \(-0.654287\pi\)
−0.465949 + 0.884812i \(0.654287\pi\)
\(230\) −1.33229e9 −0.0313923
\(231\) −2.26226e10 −0.522743
\(232\) 4.53744e7 0.00102829
\(233\) 4.13830e10 0.919856 0.459928 0.887956i \(-0.347875\pi\)
0.459928 + 0.887956i \(0.347875\pi\)
\(234\) 8.94795e9 0.195098
\(235\) −2.13755e10 −0.457206
\(236\) 5.45305e9 0.114429
\(237\) −2.98028e9 −0.0613606
\(238\) −3.73980e10 −0.755531
\(239\) 1.41322e10 0.280168 0.140084 0.990140i \(-0.455263\pi\)
0.140084 + 0.990140i \(0.455263\pi\)
\(240\) 1.92936e10 0.375373
\(241\) 6.77959e9 0.129457 0.0647287 0.997903i \(-0.479382\pi\)
0.0647287 + 0.997903i \(0.479382\pi\)
\(242\) −1.01017e10 −0.189333
\(243\) −3.48678e9 −0.0641500
\(244\) −8.38892e10 −1.51514
\(245\) −4.49370e10 −0.796814
\(246\) 9.09318e9 0.158310
\(247\) 5.70283e10 0.974887
\(248\) 5.79253e10 0.972380
\(249\) −7.27069e9 −0.119861
\(250\) 2.15327e10 0.348632
\(251\) 5.23978e10 0.833262 0.416631 0.909076i \(-0.363211\pi\)
0.416631 + 0.909076i \(0.363211\pi\)
\(252\) 2.51530e10 0.392905
\(253\) −3.97795e9 −0.0610403
\(254\) 1.62102e10 0.244364
\(255\) 6.29942e10 0.932974
\(256\) −2.01862e8 −0.00293748
\(257\) 1.29728e11 1.85497 0.927483 0.373866i \(-0.121968\pi\)
0.927483 + 0.373866i \(0.121968\pi\)
\(258\) −6.91223e9 −0.0971246
\(259\) 6.65812e10 0.919396
\(260\) 1.08728e11 1.47557
\(261\) −3.93071e7 −0.000524311 0
\(262\) −4.71419e10 −0.618089
\(263\) −3.39579e10 −0.437663 −0.218831 0.975763i \(-0.570224\pi\)
−0.218831 + 0.975763i \(0.570224\pi\)
\(264\) −2.01124e10 −0.254827
\(265\) 3.84410e9 0.0478838
\(266\) −2.20789e10 −0.270403
\(267\) 7.15281e9 0.0861342
\(268\) 1.35803e10 0.160806
\(269\) 2.99218e10 0.348419 0.174210 0.984709i \(-0.444263\pi\)
0.174210 + 0.984709i \(0.444263\pi\)
\(270\) 5.83529e9 0.0668230
\(271\) −1.27158e11 −1.43212 −0.716061 0.698037i \(-0.754057\pi\)
−0.716061 + 0.698037i \(0.754057\pi\)
\(272\) 9.52317e10 1.05492
\(273\) 1.19537e11 1.30247
\(274\) 1.95541e10 0.209585
\(275\) 2.59994e8 0.00274136
\(276\) 4.42290e9 0.0458793
\(277\) −6.25830e10 −0.638700 −0.319350 0.947637i \(-0.603465\pi\)
−0.319350 + 0.947637i \(0.603465\pi\)
\(278\) −3.14900e10 −0.316207
\(279\) −5.01797e10 −0.495803
\(280\) −8.99871e10 −0.874922
\(281\) −5.33771e10 −0.510712 −0.255356 0.966847i \(-0.582193\pi\)
−0.255356 + 0.966847i \(0.582193\pi\)
\(282\) −9.77343e9 −0.0920293
\(283\) 1.22464e11 1.13493 0.567465 0.823397i \(-0.307924\pi\)
0.567465 + 0.823397i \(0.307924\pi\)
\(284\) 4.28144e9 0.0390532
\(285\) 3.71903e10 0.333909
\(286\) −4.47117e10 −0.395161
\(287\) 1.21477e11 1.05688
\(288\) 3.42635e10 0.293471
\(289\) 1.92346e11 1.62197
\(290\) 6.57822e7 0.000546157 0
\(291\) 1.39874e10 0.114346
\(292\) −9.48153e10 −0.763231
\(293\) 1.36171e11 1.07939 0.539697 0.841859i \(-0.318539\pi\)
0.539697 + 0.841859i \(0.318539\pi\)
\(294\) −2.05463e10 −0.160388
\(295\) 1.69001e10 0.129924
\(296\) 5.91934e10 0.448188
\(297\) 1.74230e10 0.129933
\(298\) −8.27220e10 −0.607642
\(299\) 2.10193e10 0.152089
\(300\) −2.89076e8 −0.00206047
\(301\) −9.23412e10 −0.648405
\(302\) 3.68640e10 0.255018
\(303\) −1.47979e11 −1.00858
\(304\) 5.62226e10 0.377555
\(305\) −2.59990e11 −1.72031
\(306\) 2.88025e10 0.187795
\(307\) 1.35328e11 0.869494 0.434747 0.900553i \(-0.356838\pi\)
0.434747 + 0.900553i \(0.356838\pi\)
\(308\) −1.25686e11 −0.795812
\(309\) −1.20835e11 −0.754012
\(310\) 8.39781e10 0.516462
\(311\) 1.26392e11 0.766122 0.383061 0.923723i \(-0.374870\pi\)
0.383061 + 0.923723i \(0.374870\pi\)
\(312\) 1.06273e11 0.634931
\(313\) 3.20969e11 1.89022 0.945111 0.326748i \(-0.105953\pi\)
0.945111 + 0.326748i \(0.105953\pi\)
\(314\) 8.14812e10 0.473014
\(315\) 7.79543e10 0.446111
\(316\) −1.65578e10 −0.0934139
\(317\) −2.55439e11 −1.42076 −0.710381 0.703818i \(-0.751477\pi\)
−0.710381 + 0.703818i \(0.751477\pi\)
\(318\) 1.75762e9 0.00963835
\(319\) 1.96412e8 0.00106197
\(320\) 6.46133e10 0.344467
\(321\) 1.01628e11 0.534248
\(322\) −8.13777e9 −0.0421846
\(323\) 1.83568e11 0.938396
\(324\) −1.93719e10 −0.0976606
\(325\) −1.37380e9 −0.00683042
\(326\) −8.89150e10 −0.436009
\(327\) 5.38770e9 0.0260579
\(328\) 1.07998e11 0.515208
\(329\) −1.30564e11 −0.614388
\(330\) −2.91582e10 −0.135347
\(331\) −1.73240e11 −0.793272 −0.396636 0.917976i \(-0.629822\pi\)
−0.396636 + 0.917976i \(0.629822\pi\)
\(332\) −4.03945e10 −0.182474
\(333\) −5.12782e10 −0.228525
\(334\) −3.71871e10 −0.163506
\(335\) 4.20879e10 0.182581
\(336\) 1.17848e11 0.504423
\(337\) −2.70295e11 −1.14157 −0.570785 0.821099i \(-0.693361\pi\)
−0.570785 + 0.821099i \(0.693361\pi\)
\(338\) 1.52768e11 0.636661
\(339\) −1.78065e11 −0.732284
\(340\) 3.49983e11 1.42034
\(341\) 2.50742e11 1.00423
\(342\) 1.70043e10 0.0672113
\(343\) 6.92920e10 0.270308
\(344\) −8.20950e10 −0.316085
\(345\) 1.37075e10 0.0520920
\(346\) 5.73951e10 0.215294
\(347\) 1.51065e9 0.00559345 0.00279673 0.999996i \(-0.499110\pi\)
0.00279673 + 0.999996i \(0.499110\pi\)
\(348\) −2.18382e8 −0.000798199 0
\(349\) −2.59300e11 −0.935595 −0.467798 0.883836i \(-0.654952\pi\)
−0.467798 + 0.883836i \(0.654952\pi\)
\(350\) 5.31876e8 0.00189454
\(351\) −9.20624e10 −0.323743
\(352\) −1.71210e11 −0.594412
\(353\) −7.36010e10 −0.252289 −0.126144 0.992012i \(-0.540260\pi\)
−0.126144 + 0.992012i \(0.540260\pi\)
\(354\) 7.72715e9 0.0261520
\(355\) 1.32690e10 0.0443416
\(356\) 3.97396e10 0.131129
\(357\) 3.84776e11 1.25372
\(358\) 8.81828e10 0.283733
\(359\) −3.72647e11 −1.18406 −0.592028 0.805917i \(-0.701673\pi\)
−0.592028 + 0.805917i \(0.701673\pi\)
\(360\) 6.93045e10 0.217471
\(361\) −2.14313e11 −0.664151
\(362\) −1.61375e11 −0.493909
\(363\) 1.03933e11 0.314177
\(364\) 6.64122e11 1.98286
\(365\) −2.93852e11 −0.866584
\(366\) −1.18874e11 −0.346275
\(367\) −3.22104e11 −0.926827 −0.463414 0.886142i \(-0.653376\pi\)
−0.463414 + 0.886142i \(0.653376\pi\)
\(368\) 2.07223e10 0.0589010
\(369\) −9.35566e10 −0.262697
\(370\) 8.58164e10 0.238047
\(371\) 2.34802e10 0.0643457
\(372\) −2.78789e11 −0.754800
\(373\) −3.92320e11 −1.04942 −0.524711 0.851280i \(-0.675827\pi\)
−0.524711 + 0.851280i \(0.675827\pi\)
\(374\) −1.43922e11 −0.380370
\(375\) −2.21542e11 −0.578516
\(376\) −1.16077e11 −0.299503
\(377\) −1.03783e9 −0.00264601
\(378\) 3.56426e10 0.0897960
\(379\) −4.57826e11 −1.13979 −0.569894 0.821718i \(-0.693016\pi\)
−0.569894 + 0.821718i \(0.693016\pi\)
\(380\) 2.06622e11 0.508336
\(381\) −1.66781e11 −0.405495
\(382\) −2.55545e11 −0.614021
\(383\) 1.14027e11 0.270777 0.135389 0.990793i \(-0.456772\pi\)
0.135389 + 0.990793i \(0.456772\pi\)
\(384\) 2.46122e11 0.577643
\(385\) −3.89528e11 −0.903577
\(386\) −2.34036e11 −0.536586
\(387\) 7.11176e10 0.161167
\(388\) 7.77115e10 0.174077
\(389\) 2.66221e11 0.589480 0.294740 0.955577i \(-0.404767\pi\)
0.294740 + 0.955577i \(0.404767\pi\)
\(390\) 1.54071e11 0.337232
\(391\) 6.76588e10 0.146396
\(392\) −2.44024e11 −0.521970
\(393\) 4.85027e11 1.02565
\(394\) −2.49650e11 −0.521914
\(395\) −5.13161e10 −0.106064
\(396\) 9.67988e10 0.197807
\(397\) −7.04699e11 −1.42379 −0.711896 0.702285i \(-0.752163\pi\)
−0.711896 + 0.702285i \(0.752163\pi\)
\(398\) −1.73740e10 −0.0347076
\(399\) 2.27163e11 0.448703
\(400\) −1.35439e9 −0.00264529
\(401\) 3.31737e11 0.640685 0.320343 0.947302i \(-0.396202\pi\)
0.320343 + 0.947302i \(0.396202\pi\)
\(402\) 1.92436e10 0.0367511
\(403\) −1.32491e12 −2.50215
\(404\) −8.22142e11 −1.53543
\(405\) −6.00374e10 −0.110885
\(406\) 4.01805e8 0.000733920 0
\(407\) 2.56231e11 0.462867
\(408\) 3.42081e11 0.611165
\(409\) 4.79405e11 0.847125 0.423563 0.905867i \(-0.360779\pi\)
0.423563 + 0.905867i \(0.360779\pi\)
\(410\) 1.56571e11 0.273643
\(411\) −2.01185e11 −0.347783
\(412\) −6.71334e11 −1.14789
\(413\) 1.03228e11 0.174591
\(414\) 6.26739e9 0.0104854
\(415\) −1.25191e11 −0.207184
\(416\) 9.04666e11 1.48105
\(417\) 3.23990e11 0.524710
\(418\) −8.49685e10 −0.136133
\(419\) 8.85242e11 1.40313 0.701566 0.712604i \(-0.252484\pi\)
0.701566 + 0.712604i \(0.252484\pi\)
\(420\) 4.33099e11 0.679149
\(421\) −1.09924e12 −1.70539 −0.852697 0.522405i \(-0.825035\pi\)
−0.852697 + 0.522405i \(0.825035\pi\)
\(422\) −3.48873e11 −0.535503
\(423\) 1.00555e11 0.152712
\(424\) 2.08749e10 0.0313673
\(425\) −4.42211e9 −0.00657475
\(426\) 6.06693e9 0.00892536
\(427\) −1.58805e12 −2.31173
\(428\) 5.64628e11 0.813327
\(429\) 4.60024e11 0.655726
\(430\) −1.19018e11 −0.167883
\(431\) 5.15254e10 0.0719240 0.0359620 0.999353i \(-0.488550\pi\)
0.0359620 + 0.999353i \(0.488550\pi\)
\(432\) −9.07617e10 −0.125379
\(433\) −7.35694e11 −1.00578 −0.502888 0.864351i \(-0.667729\pi\)
−0.502888 + 0.864351i \(0.667729\pi\)
\(434\) 5.12948e11 0.694016
\(435\) −6.76811e8 −0.000906287 0
\(436\) 2.99330e10 0.0396699
\(437\) 3.99442e10 0.0523947
\(438\) −1.34356e11 −0.174431
\(439\) −4.53463e11 −0.582708 −0.291354 0.956615i \(-0.594106\pi\)
−0.291354 + 0.956615i \(0.594106\pi\)
\(440\) −3.46306e11 −0.440477
\(441\) 2.11394e11 0.266146
\(442\) 7.60479e11 0.947735
\(443\) 3.20088e11 0.394869 0.197434 0.980316i \(-0.436739\pi\)
0.197434 + 0.980316i \(0.436739\pi\)
\(444\) −2.84892e11 −0.347901
\(445\) 1.23161e11 0.148886
\(446\) 2.47096e11 0.295705
\(447\) 8.51098e11 1.00831
\(448\) 3.94666e11 0.462891
\(449\) −1.45474e12 −1.68918 −0.844590 0.535413i \(-0.820156\pi\)
−0.844590 + 0.535413i \(0.820156\pi\)
\(450\) −4.09630e8 −0.000470907 0
\(451\) 4.67490e11 0.532081
\(452\) −9.89293e11 −1.11481
\(453\) −3.79281e11 −0.423174
\(454\) −1.06876e11 −0.118067
\(455\) 2.05825e12 2.25137
\(456\) 2.01957e11 0.218734
\(457\) −6.52938e11 −0.700243 −0.350122 0.936704i \(-0.613860\pi\)
−0.350122 + 0.936704i \(0.613860\pi\)
\(458\) −3.05319e11 −0.324235
\(459\) −2.96339e11 −0.311625
\(460\) 7.61560e10 0.0793037
\(461\) 1.56956e12 1.61854 0.809272 0.587435i \(-0.199862\pi\)
0.809272 + 0.587435i \(0.199862\pi\)
\(462\) −1.78102e11 −0.181878
\(463\) −1.86374e12 −1.88482 −0.942412 0.334454i \(-0.891448\pi\)
−0.942412 + 0.334454i \(0.891448\pi\)
\(464\) −1.02317e9 −0.00102475
\(465\) −8.64022e11 −0.857011
\(466\) 3.25798e11 0.320045
\(467\) −3.21350e11 −0.312646 −0.156323 0.987706i \(-0.549964\pi\)
−0.156323 + 0.987706i \(0.549964\pi\)
\(468\) −5.11480e11 −0.492859
\(469\) 2.57078e11 0.245351
\(470\) −1.68284e11 −0.159075
\(471\) −8.38333e11 −0.784914
\(472\) 9.17737e10 0.0851097
\(473\) −3.55365e11 −0.326437
\(474\) −2.34630e10 −0.0213491
\(475\) −2.61071e9 −0.00235308
\(476\) 2.13774e12 1.90864
\(477\) −1.80836e10 −0.0159938
\(478\) 1.11259e11 0.0974788
\(479\) 1.47615e12 1.28121 0.640604 0.767871i \(-0.278684\pi\)
0.640604 + 0.767871i \(0.278684\pi\)
\(480\) 5.89967e11 0.507273
\(481\) −1.35391e12 −1.15329
\(482\) 5.33740e10 0.0450420
\(483\) 8.37268e10 0.0700007
\(484\) 5.77432e11 0.478296
\(485\) 2.40843e11 0.197650
\(486\) −2.74506e10 −0.0223197
\(487\) 4.49035e11 0.361742 0.180871 0.983507i \(-0.442108\pi\)
0.180871 + 0.983507i \(0.442108\pi\)
\(488\) −1.41184e12 −1.12693
\(489\) 9.14816e11 0.723509
\(490\) −3.53778e11 −0.277235
\(491\) 1.06546e12 0.827314 0.413657 0.910433i \(-0.364251\pi\)
0.413657 + 0.910433i \(0.364251\pi\)
\(492\) −5.19782e11 −0.399925
\(493\) −3.34068e9 −0.00254697
\(494\) 4.48970e11 0.339192
\(495\) 2.99999e11 0.224593
\(496\) −1.30619e12 −0.969032
\(497\) 8.10488e10 0.0595858
\(498\) −5.72403e10 −0.0417032
\(499\) 2.45188e11 0.177030 0.0885152 0.996075i \(-0.471788\pi\)
0.0885152 + 0.996075i \(0.471788\pi\)
\(500\) −1.23084e12 −0.880720
\(501\) 3.82605e11 0.271320
\(502\) 4.12514e11 0.289916
\(503\) 9.03179e11 0.629097 0.314549 0.949241i \(-0.398147\pi\)
0.314549 + 0.949241i \(0.398147\pi\)
\(504\) 4.23320e11 0.292235
\(505\) −2.54798e12 −1.74335
\(506\) −3.13173e10 −0.0212377
\(507\) −1.57178e12 −1.05647
\(508\) −9.26604e11 −0.617316
\(509\) 1.50017e12 0.990625 0.495312 0.868715i \(-0.335054\pi\)
0.495312 + 0.868715i \(0.335054\pi\)
\(510\) 4.95937e11 0.324609
\(511\) −1.79488e12 −1.16451
\(512\) 1.55414e12 0.999485
\(513\) −1.74952e11 −0.111530
\(514\) 1.02132e12 0.645397
\(515\) −2.08060e12 −1.30333
\(516\) 3.95115e11 0.245358
\(517\) −5.02463e11 −0.309312
\(518\) 5.24177e11 0.319885
\(519\) −5.90518e11 −0.357257
\(520\) 1.82986e12 1.09750
\(521\) −2.75051e12 −1.63547 −0.817737 0.575592i \(-0.804772\pi\)
−0.817737 + 0.575592i \(0.804772\pi\)
\(522\) −3.09455e8 −0.000182423 0
\(523\) −5.31073e11 −0.310382 −0.155191 0.987884i \(-0.549599\pi\)
−0.155191 + 0.987884i \(0.549599\pi\)
\(524\) 2.69471e12 1.56143
\(525\) −5.47229e9 −0.00314378
\(526\) −2.67341e11 −0.152276
\(527\) −4.26474e12 −2.40849
\(528\) 4.53524e11 0.253950
\(529\) −1.78643e12 −0.991826
\(530\) 3.02636e10 0.0166602
\(531\) −7.95020e10 −0.0433963
\(532\) 1.26207e12 0.683096
\(533\) −2.47020e12 −1.32574
\(534\) 5.63122e10 0.0299686
\(535\) 1.74989e12 0.923464
\(536\) 2.28553e11 0.119604
\(537\) −9.07283e11 −0.470824
\(538\) 2.35567e11 0.121225
\(539\) −1.05631e12 −0.539065
\(540\) −3.33556e11 −0.168809
\(541\) 2.81202e12 1.41134 0.705668 0.708543i \(-0.250647\pi\)
0.705668 + 0.708543i \(0.250647\pi\)
\(542\) −1.00108e12 −0.498278
\(543\) 1.66033e12 0.819587
\(544\) 2.91202e12 1.42561
\(545\) 9.27684e10 0.0450418
\(546\) 9.41081e11 0.453169
\(547\) 2.80003e12 1.33727 0.668637 0.743589i \(-0.266878\pi\)
0.668637 + 0.743589i \(0.266878\pi\)
\(548\) −1.11775e12 −0.529456
\(549\) 1.22305e12 0.574604
\(550\) 2.04687e9 0.000953800 0
\(551\) −1.97226e9 −0.000911554 0
\(552\) 7.44365e10 0.0341240
\(553\) −3.13444e11 −0.142527
\(554\) −4.92700e11 −0.222223
\(555\) −8.82936e11 −0.395013
\(556\) 1.80003e12 0.798807
\(557\) 1.98817e12 0.875195 0.437597 0.899171i \(-0.355830\pi\)
0.437597 + 0.899171i \(0.355830\pi\)
\(558\) −3.95052e11 −0.172505
\(559\) 1.87773e12 0.813356
\(560\) 2.02917e12 0.871910
\(561\) 1.48077e12 0.631181
\(562\) −4.20224e11 −0.177692
\(563\) −2.61155e12 −1.09549 −0.547747 0.836644i \(-0.684514\pi\)
−0.547747 + 0.836644i \(0.684514\pi\)
\(564\) 5.58666e11 0.232486
\(565\) −3.06602e12 −1.26578
\(566\) 9.64127e11 0.394876
\(567\) −3.66715e11 −0.149006
\(568\) 7.20556e10 0.0290469
\(569\) 6.12579e11 0.244995 0.122497 0.992469i \(-0.460910\pi\)
0.122497 + 0.992469i \(0.460910\pi\)
\(570\) 2.92790e11 0.116177
\(571\) 1.92887e12 0.759347 0.379673 0.925121i \(-0.376036\pi\)
0.379673 + 0.925121i \(0.376036\pi\)
\(572\) 2.55580e12 0.998263
\(573\) 2.62922e12 1.01890
\(574\) 9.56355e11 0.367719
\(575\) −9.62246e8 −0.000367097 0
\(576\) −3.03956e11 −0.115056
\(577\) 1.13524e12 0.426381 0.213190 0.977011i \(-0.431615\pi\)
0.213190 + 0.977011i \(0.431615\pi\)
\(578\) 1.51429e12 0.564331
\(579\) 2.40792e12 0.890405
\(580\) −3.76023e9 −0.00137971
\(581\) −7.64679e11 −0.278411
\(582\) 1.10120e11 0.0397842
\(583\) 9.03612e10 0.0323947
\(584\) −1.59572e12 −0.567674
\(585\) −1.58518e12 −0.559600
\(586\) 1.07204e12 0.375553
\(587\) 1.79042e12 0.622420 0.311210 0.950341i \(-0.399266\pi\)
0.311210 + 0.950341i \(0.399266\pi\)
\(588\) 1.17446e12 0.405174
\(589\) −2.51780e12 −0.861991
\(590\) 1.33050e11 0.0452045
\(591\) 2.56857e12 0.866058
\(592\) −1.33478e12 −0.446645
\(593\) −8.61080e11 −0.285955 −0.142977 0.989726i \(-0.545668\pi\)
−0.142977 + 0.989726i \(0.545668\pi\)
\(594\) 1.37167e11 0.0452075
\(595\) 6.62528e12 2.16709
\(596\) 4.72853e12 1.53504
\(597\) 1.78755e11 0.0575935
\(598\) 1.65479e11 0.0529162
\(599\) −3.61522e12 −1.14740 −0.573699 0.819066i \(-0.694492\pi\)
−0.573699 + 0.819066i \(0.694492\pi\)
\(600\) −4.86509e9 −0.00153253
\(601\) 6.50374e11 0.203343 0.101671 0.994818i \(-0.467581\pi\)
0.101671 + 0.994818i \(0.467581\pi\)
\(602\) −7.26979e11 −0.225599
\(603\) −1.97991e11 −0.0609843
\(604\) −2.10721e12 −0.644231
\(605\) 1.78958e12 0.543065
\(606\) −1.16500e12 −0.350913
\(607\) −1.14782e12 −0.343181 −0.171590 0.985168i \(-0.554891\pi\)
−0.171590 + 0.985168i \(0.554891\pi\)
\(608\) 1.71919e12 0.510221
\(609\) −4.13404e9 −0.00121786
\(610\) −2.04683e12 −0.598546
\(611\) 2.65499e12 0.770686
\(612\) −1.64640e12 −0.474410
\(613\) 3.57940e12 1.02385 0.511927 0.859029i \(-0.328932\pi\)
0.511927 + 0.859029i \(0.328932\pi\)
\(614\) 1.06541e12 0.302522
\(615\) −1.61091e12 −0.454081
\(616\) −2.11528e12 −0.591907
\(617\) 2.70222e12 0.750650 0.375325 0.926893i \(-0.377531\pi\)
0.375325 + 0.926893i \(0.377531\pi\)
\(618\) −9.51301e11 −0.262343
\(619\) 5.10159e12 1.39668 0.698341 0.715765i \(-0.253922\pi\)
0.698341 + 0.715765i \(0.253922\pi\)
\(620\) −4.80033e12 −1.30469
\(621\) −6.44831e10 −0.0173994
\(622\) 9.95052e11 0.266556
\(623\) 7.52281e11 0.200071
\(624\) −2.39640e12 −0.632746
\(625\) −3.79915e12 −0.995923
\(626\) 2.52690e12 0.657664
\(627\) 8.74212e11 0.225898
\(628\) −4.65761e12 −1.19494
\(629\) −4.35809e12 −1.11012
\(630\) 6.13715e11 0.155215
\(631\) −5.04642e12 −1.26722 −0.633609 0.773654i \(-0.718427\pi\)
−0.633609 + 0.773654i \(0.718427\pi\)
\(632\) −2.78665e11 −0.0694792
\(633\) 3.58944e12 0.888607
\(634\) −2.01101e12 −0.494325
\(635\) −2.87173e12 −0.700910
\(636\) −1.00469e11 −0.0243486
\(637\) 5.58149e12 1.34314
\(638\) 1.54631e9 0.000369490 0
\(639\) −6.24206e10 −0.0148106
\(640\) 4.23786e12 0.998474
\(641\) 3.76398e12 0.880614 0.440307 0.897847i \(-0.354870\pi\)
0.440307 + 0.897847i \(0.354870\pi\)
\(642\) 8.00095e11 0.185881
\(643\) −3.84154e12 −0.886250 −0.443125 0.896460i \(-0.646130\pi\)
−0.443125 + 0.896460i \(0.646130\pi\)
\(644\) 4.65169e11 0.106567
\(645\) 1.22454e12 0.278583
\(646\) 1.44519e12 0.326495
\(647\) −2.67586e12 −0.600336 −0.300168 0.953886i \(-0.597043\pi\)
−0.300168 + 0.953886i \(0.597043\pi\)
\(648\) −3.26024e11 −0.0726378
\(649\) 3.97261e11 0.0878972
\(650\) −1.08156e10 −0.00237650
\(651\) −5.27755e12 −1.15164
\(652\) 5.08254e12 1.10145
\(653\) 1.56783e12 0.337434 0.168717 0.985665i \(-0.446038\pi\)
0.168717 + 0.985665i \(0.446038\pi\)
\(654\) 4.24160e10 0.00906630
\(655\) 8.35146e12 1.77287
\(656\) −2.43530e12 −0.513434
\(657\) 1.38235e12 0.289449
\(658\) −1.02790e12 −0.213764
\(659\) 1.34203e12 0.277191 0.138596 0.990349i \(-0.455741\pi\)
0.138596 + 0.990349i \(0.455741\pi\)
\(660\) 1.66673e12 0.341915
\(661\) −2.80128e12 −0.570755 −0.285378 0.958415i \(-0.592119\pi\)
−0.285378 + 0.958415i \(0.592119\pi\)
\(662\) −1.36387e12 −0.276003
\(663\) −7.82431e12 −1.57266
\(664\) −6.79831e11 −0.135720
\(665\) 3.91141e12 0.775597
\(666\) −4.03700e11 −0.0795106
\(667\) −7.26928e8 −0.000142208 0
\(668\) 2.12568e12 0.413051
\(669\) −2.54229e12 −0.490690
\(670\) 3.31348e11 0.0635254
\(671\) −6.11143e12 −1.16383
\(672\) 3.60359e12 0.681668
\(673\) 3.95540e12 0.743229 0.371614 0.928387i \(-0.378804\pi\)
0.371614 + 0.928387i \(0.378804\pi\)
\(674\) −2.12796e12 −0.397186
\(675\) 4.21454e9 0.000781418 0
\(676\) −8.73251e12 −1.60834
\(677\) −4.10835e12 −0.751655 −0.375828 0.926690i \(-0.622641\pi\)
−0.375828 + 0.926690i \(0.622641\pi\)
\(678\) −1.40186e12 −0.254783
\(679\) 1.47110e12 0.265600
\(680\) 5.89014e12 1.05642
\(681\) 1.09961e12 0.195919
\(682\) 1.97402e12 0.349400
\(683\) −2.63864e12 −0.463966 −0.231983 0.972720i \(-0.574521\pi\)
−0.231983 + 0.972720i \(0.574521\pi\)
\(684\) −9.71998e11 −0.169790
\(685\) −3.46412e12 −0.601153
\(686\) 5.45518e11 0.0940482
\(687\) 3.14133e12 0.538032
\(688\) 1.85120e12 0.314997
\(689\) −4.77464e11 −0.0807150
\(690\) 1.07915e11 0.0181243
\(691\) −4.12989e12 −0.689109 −0.344554 0.938766i \(-0.611970\pi\)
−0.344554 + 0.938766i \(0.611970\pi\)
\(692\) −3.28080e12 −0.543879
\(693\) 1.83243e12 0.301806
\(694\) 1.18929e10 0.00194613
\(695\) 5.57864e12 0.906978
\(696\) −3.67533e9 −0.000593683 0
\(697\) −7.95130e12 −1.27612
\(698\) −2.04140e12 −0.325521
\(699\) −3.35202e12 −0.531079
\(700\) −3.04029e10 −0.00478602
\(701\) −3.39242e12 −0.530613 −0.265307 0.964164i \(-0.585473\pi\)
−0.265307 + 0.964164i \(0.585473\pi\)
\(702\) −7.24784e11 −0.112640
\(703\) −2.57292e12 −0.397308
\(704\) 1.51883e12 0.233041
\(705\) 1.73142e12 0.263968
\(706\) −5.79442e11 −0.0877786
\(707\) −1.55634e13 −2.34270
\(708\) −4.41697e11 −0.0660656
\(709\) −9.50357e12 −1.41247 −0.706234 0.707979i \(-0.749607\pi\)
−0.706234 + 0.707979i \(0.749607\pi\)
\(710\) 1.04464e11 0.0154278
\(711\) 2.41403e11 0.0354265
\(712\) 6.68808e11 0.0975308
\(713\) −9.28002e11 −0.134476
\(714\) 3.02924e12 0.436206
\(715\) 7.92094e12 1.13344
\(716\) −5.04068e12 −0.716772
\(717\) −1.14471e12 −0.161755
\(718\) −2.93375e12 −0.411968
\(719\) 6.87302e12 0.959108 0.479554 0.877512i \(-0.340799\pi\)
0.479554 + 0.877512i \(0.340799\pi\)
\(720\) −1.56278e12 −0.216722
\(721\) −1.27085e13 −1.75141
\(722\) −1.68723e12 −0.231078
\(723\) −5.49147e11 −0.0747423
\(724\) 9.22446e12 1.24772
\(725\) 4.75112e7 6.38668e−6 0
\(726\) 8.18240e11 0.109311
\(727\) 1.23689e13 1.64220 0.821099 0.570786i \(-0.193362\pi\)
0.821099 + 0.570786i \(0.193362\pi\)
\(728\) 1.11770e13 1.47481
\(729\) 2.82430e11 0.0370370
\(730\) −2.31342e12 −0.301510
\(731\) 6.04423e12 0.782911
\(732\) 6.79503e12 0.874765
\(733\) −1.42337e13 −1.82116 −0.910581 0.413330i \(-0.864366\pi\)
−0.910581 + 0.413330i \(0.864366\pi\)
\(734\) −2.53584e12 −0.322470
\(735\) 3.63990e12 0.460041
\(736\) 6.33653e11 0.0795979
\(737\) 9.89337e11 0.123521
\(738\) −7.36547e11 −0.0914002
\(739\) 4.57007e12 0.563668 0.281834 0.959463i \(-0.409057\pi\)
0.281834 + 0.959463i \(0.409057\pi\)
\(740\) −4.90542e12 −0.601358
\(741\) −4.61930e12 −0.562852
\(742\) 1.84854e11 0.0223878
\(743\) −4.31982e12 −0.520015 −0.260008 0.965607i \(-0.583725\pi\)
−0.260008 + 0.965607i \(0.583725\pi\)
\(744\) −4.69195e12 −0.561404
\(745\) 1.46547e13 1.74290
\(746\) −3.08863e12 −0.365125
\(747\) 5.88926e11 0.0692019
\(748\) 8.22685e12 0.960896
\(749\) 1.06886e13 1.24094
\(750\) −1.74414e12 −0.201283
\(751\) −1.64985e13 −1.89262 −0.946311 0.323258i \(-0.895222\pi\)
−0.946311 + 0.323258i \(0.895222\pi\)
\(752\) 2.61748e12 0.298472
\(753\) −4.24422e12 −0.481084
\(754\) −8.17061e9 −0.000920626 0
\(755\) −6.53067e12 −0.731470
\(756\) −2.03740e12 −0.226844
\(757\) 9.88795e12 1.09440 0.547199 0.837003i \(-0.315694\pi\)
0.547199 + 0.837003i \(0.315694\pi\)
\(758\) −3.60435e12 −0.396566
\(759\) 3.22214e11 0.0352416
\(760\) 3.47740e12 0.378089
\(761\) −6.61292e11 −0.0714763 −0.0357382 0.999361i \(-0.511378\pi\)
−0.0357382 + 0.999361i \(0.511378\pi\)
\(762\) −1.31303e12 −0.141083
\(763\) 5.66640e11 0.0605267
\(764\) 1.46074e13 1.55115
\(765\) −5.10253e12 −0.538653
\(766\) 8.97703e11 0.0942114
\(767\) −2.09911e12 −0.219006
\(768\) 1.63508e10 0.00169595
\(769\) −1.35353e12 −0.139572 −0.0697861 0.997562i \(-0.522232\pi\)
−0.0697861 + 0.997562i \(0.522232\pi\)
\(770\) −3.06665e12 −0.314381
\(771\) −1.05080e13 −1.07096
\(772\) 1.33779e13 1.35553
\(773\) −4.96259e12 −0.499920 −0.249960 0.968256i \(-0.580417\pi\)
−0.249960 + 0.968256i \(0.580417\pi\)
\(774\) 5.59890e11 0.0560749
\(775\) 6.06532e10 0.00603943
\(776\) 1.30787e12 0.129475
\(777\) −5.39308e12 −0.530813
\(778\) 2.09589e12 0.205098
\(779\) −4.69427e12 −0.456719
\(780\) −8.80695e12 −0.851922
\(781\) 3.11908e11 0.0299983
\(782\) 5.32661e11 0.0509355
\(783\) 3.18388e9 0.000302711 0
\(784\) 5.50263e12 0.520173
\(785\) −1.44349e13 −1.35675
\(786\) 3.81849e12 0.356854
\(787\) −5.22654e12 −0.485655 −0.242827 0.970069i \(-0.578075\pi\)
−0.242827 + 0.970069i \(0.578075\pi\)
\(788\) 1.42704e13 1.31847
\(789\) 2.75059e12 0.252685
\(790\) −4.03998e11 −0.0369026
\(791\) −1.87276e13 −1.70094
\(792\) 1.62910e12 0.147125
\(793\) 3.22925e13 2.89983
\(794\) −5.54791e12 −0.495379
\(795\) −3.11372e11 −0.0276457
\(796\) 9.93126e11 0.0876790
\(797\) −5.16094e12 −0.453071 −0.226536 0.974003i \(-0.572740\pi\)
−0.226536 + 0.974003i \(0.572740\pi\)
\(798\) 1.78839e12 0.156117
\(799\) 8.54613e12 0.741838
\(800\) −4.14149e10 −0.00357480
\(801\) −5.79377e11 −0.0497296
\(802\) 2.61168e12 0.222913
\(803\) −6.90741e12 −0.586266
\(804\) −1.10000e12 −0.0928412
\(805\) 1.44165e12 0.120998
\(806\) −1.04307e13 −0.870571
\(807\) −2.42367e12 −0.201160
\(808\) −1.38365e13 −1.14202
\(809\) 3.07165e12 0.252118 0.126059 0.992023i \(-0.459767\pi\)
0.126059 + 0.992023i \(0.459767\pi\)
\(810\) −4.72659e11 −0.0385803
\(811\) −1.64233e13 −1.33311 −0.666556 0.745455i \(-0.732232\pi\)
−0.666556 + 0.745455i \(0.732232\pi\)
\(812\) −2.29679e10 −0.00185404
\(813\) 1.02998e13 0.826837
\(814\) 2.01724e12 0.161045
\(815\) 1.57518e13 1.25061
\(816\) −7.71376e12 −0.609061
\(817\) 3.56837e12 0.280202
\(818\) 3.77423e12 0.294740
\(819\) −9.68247e12 −0.751984
\(820\) −8.94989e12 −0.691282
\(821\) 2.30152e13 1.76795 0.883976 0.467532i \(-0.154857\pi\)
0.883976 + 0.467532i \(0.154857\pi\)
\(822\) −1.58388e12 −0.121004
\(823\) −8.71598e11 −0.0662242 −0.0331121 0.999452i \(-0.510542\pi\)
−0.0331121 + 0.999452i \(0.510542\pi\)
\(824\) −1.12984e13 −0.853777
\(825\) −2.10595e10 −0.00158273
\(826\) 8.12686e11 0.0607453
\(827\) 3.51025e12 0.260954 0.130477 0.991451i \(-0.458349\pi\)
0.130477 + 0.991451i \(0.458349\pi\)
\(828\) −3.58255e11 −0.0264884
\(829\) −1.62965e13 −1.19839 −0.599195 0.800603i \(-0.704513\pi\)
−0.599195 + 0.800603i \(0.704513\pi\)
\(830\) −9.85595e11 −0.0720853
\(831\) 5.06922e12 0.368754
\(832\) −8.02542e12 −0.580648
\(833\) 1.79662e13 1.29287
\(834\) 2.55069e12 0.182562
\(835\) 6.58791e12 0.468984
\(836\) 4.85695e12 0.343902
\(837\) 4.06456e12 0.286252
\(838\) 6.96928e12 0.488191
\(839\) 2.35510e12 0.164089 0.0820446 0.996629i \(-0.473855\pi\)
0.0820446 + 0.996629i \(0.473855\pi\)
\(840\) 7.28896e12 0.505136
\(841\) −1.45071e13 −0.999998
\(842\) −8.65407e12 −0.593357
\(843\) 4.32354e12 0.294860
\(844\) 1.99422e13 1.35280
\(845\) −2.70638e13 −1.82614
\(846\) 7.91648e11 0.0531331
\(847\) 1.09310e13 0.729765
\(848\) −4.70718e11 −0.0312593
\(849\) −9.91958e12 −0.655252
\(850\) −3.48141e10 −0.00228755
\(851\) −9.48317e11 −0.0619827
\(852\) −3.46796e11 −0.0225474
\(853\) 8.18557e12 0.529393 0.264696 0.964332i \(-0.414728\pi\)
0.264696 + 0.964332i \(0.414728\pi\)
\(854\) −1.25023e13 −0.804320
\(855\) −3.01242e12 −0.192782
\(856\) 9.50256e12 0.604935
\(857\) −4.09656e12 −0.259421 −0.129711 0.991552i \(-0.541405\pi\)
−0.129711 + 0.991552i \(0.541405\pi\)
\(858\) 3.62165e12 0.228146
\(859\) 1.63156e13 1.02243 0.511214 0.859453i \(-0.329196\pi\)
0.511214 + 0.859453i \(0.329196\pi\)
\(860\) 6.80331e12 0.424109
\(861\) −9.83962e12 −0.610188
\(862\) 4.05646e11 0.0250245
\(863\) −1.09372e13 −0.671211 −0.335606 0.942003i \(-0.608941\pi\)
−0.335606 + 0.942003i \(0.608941\pi\)
\(864\) −2.77534e12 −0.169435
\(865\) −1.01679e13 −0.617529
\(866\) −5.79193e12 −0.349939
\(867\) −1.55800e13 −0.936444
\(868\) −2.93210e13 −1.75323
\(869\) −1.20626e12 −0.0717548
\(870\) −5.32836e9 −0.000315324 0
\(871\) −5.22761e12 −0.307767
\(872\) 5.03766e11 0.0295056
\(873\) −1.13298e12 −0.0660175
\(874\) 3.14471e11 0.0182297
\(875\) −2.33002e13 −1.34377
\(876\) 7.68004e12 0.440651
\(877\) 6.57105e12 0.375091 0.187546 0.982256i \(-0.439947\pi\)
0.187546 + 0.982256i \(0.439947\pi\)
\(878\) −3.57000e12 −0.202741
\(879\) −1.10298e13 −0.623189
\(880\) 7.80903e12 0.438960
\(881\) 2.58625e11 0.0144637 0.00723184 0.999974i \(-0.497698\pi\)
0.00723184 + 0.999974i \(0.497698\pi\)
\(882\) 1.66425e12 0.0925998
\(883\) −3.12545e13 −1.73017 −0.865086 0.501623i \(-0.832736\pi\)
−0.865086 + 0.501623i \(0.832736\pi\)
\(884\) −4.34703e13 −2.39419
\(885\) −1.36891e12 −0.0750118
\(886\) 2.51997e12 0.137386
\(887\) −2.78715e13 −1.51183 −0.755916 0.654668i \(-0.772808\pi\)
−0.755916 + 0.654668i \(0.772808\pi\)
\(888\) −4.79466e12 −0.258761
\(889\) −1.75409e13 −0.941875
\(890\) 9.69615e11 0.0518017
\(891\) −1.41126e12 −0.0750168
\(892\) −1.41244e13 −0.747015
\(893\) 5.04544e12 0.265502
\(894\) 6.70048e12 0.350822
\(895\) −1.56221e13 −0.813833
\(896\) 2.58853e13 1.34174
\(897\) −1.70256e12 −0.0878086
\(898\) −1.14528e13 −0.587716
\(899\) 4.58204e10 0.00233959
\(900\) 2.34152e10 0.00118961
\(901\) −1.53691e12 −0.0776937
\(902\) 3.68043e12 0.185127
\(903\) 7.47964e12 0.374357
\(904\) −1.66496e13 −0.829174
\(905\) 2.85885e13 1.41668
\(906\) −2.98598e12 −0.147235
\(907\) 9.71090e12 0.476460 0.238230 0.971209i \(-0.423433\pi\)
0.238230 + 0.971209i \(0.423433\pi\)
\(908\) 6.10922e12 0.298263
\(909\) 1.19863e13 0.582301
\(910\) 1.62041e13 0.783317
\(911\) −2.34403e13 −1.12754 −0.563768 0.825933i \(-0.690649\pi\)
−0.563768 + 0.825933i \(0.690649\pi\)
\(912\) −4.55403e12 −0.217981
\(913\) −2.94279e12 −0.140165
\(914\) −5.14041e12 −0.243635
\(915\) 2.10592e13 0.993221
\(916\) 1.74526e13 0.819087
\(917\) 5.10117e13 2.38236
\(918\) −2.33300e12 −0.108423
\(919\) 1.71926e13 0.795099 0.397550 0.917581i \(-0.369861\pi\)
0.397550 + 0.917581i \(0.369861\pi\)
\(920\) 1.28169e12 0.0589844
\(921\) −1.09616e13 −0.502002
\(922\) 1.23568e13 0.563139
\(923\) −1.64811e12 −0.0747442
\(924\) 1.01806e13 0.459462
\(925\) 6.19809e10 0.00278369
\(926\) −1.46727e13 −0.655786
\(927\) 9.78761e12 0.435329
\(928\) −3.12869e10 −0.00138483
\(929\) 2.39639e13 1.05557 0.527785 0.849378i \(-0.323023\pi\)
0.527785 + 0.849378i \(0.323023\pi\)
\(930\) −6.80222e12 −0.298179
\(931\) 1.06068e13 0.462714
\(932\) −1.86232e13 −0.808503
\(933\) −1.02377e13 −0.442321
\(934\) −2.52991e12 −0.108779
\(935\) 2.54967e13 1.09102
\(936\) −8.60810e12 −0.366578
\(937\) 3.32009e13 1.40709 0.703544 0.710652i \(-0.251600\pi\)
0.703544 + 0.710652i \(0.251600\pi\)
\(938\) 2.02391e12 0.0853647
\(939\) −2.59985e13 −1.09132
\(940\) 9.61942e12 0.401859
\(941\) −1.23066e13 −0.511665 −0.255832 0.966721i \(-0.582350\pi\)
−0.255832 + 0.966721i \(0.582350\pi\)
\(942\) −6.59998e12 −0.273095
\(943\) −1.73020e12 −0.0712512
\(944\) −2.06945e12 −0.0848167
\(945\) −6.31430e12 −0.257562
\(946\) −2.79770e12 −0.113577
\(947\) 2.39621e13 0.968167 0.484084 0.875022i \(-0.339153\pi\)
0.484084 + 0.875022i \(0.339153\pi\)
\(948\) 1.34118e12 0.0539326
\(949\) 3.64984e13 1.46075
\(950\) −2.05535e10 −0.000818707 0
\(951\) 2.06906e13 0.820277
\(952\) 3.59777e13 1.41960
\(953\) 1.41217e13 0.554584 0.277292 0.960786i \(-0.410563\pi\)
0.277292 + 0.960786i \(0.410563\pi\)
\(954\) −1.42367e11 −0.00556470
\(955\) 4.52714e13 1.76120
\(956\) −6.35977e12 −0.246253
\(957\) −1.59094e10 −0.000613127 0
\(958\) 1.16213e13 0.445770
\(959\) −2.11592e13 −0.807823
\(960\) −5.23368e12 −0.198878
\(961\) 3.20551e13 1.21239
\(962\) −1.06590e13 −0.401262
\(963\) −8.23191e12 −0.308448
\(964\) −3.05095e12 −0.113786
\(965\) 4.14608e13 1.53909
\(966\) 6.59159e11 0.0243553
\(967\) 2.92844e12 0.107700 0.0538502 0.998549i \(-0.482851\pi\)
0.0538502 + 0.998549i \(0.482851\pi\)
\(968\) 9.71806e12 0.355746
\(969\) −1.48690e13 −0.541783
\(970\) 1.89610e12 0.0687683
\(971\) 5.04989e12 0.182304 0.0911518 0.995837i \(-0.470945\pi\)
0.0911518 + 0.995837i \(0.470945\pi\)
\(972\) 1.56912e12 0.0563844
\(973\) 3.40750e13 1.21879
\(974\) 3.53513e12 0.125861
\(975\) 1.11278e11 0.00394354
\(976\) 3.18362e13 1.12305
\(977\) 4.83238e13 1.69682 0.848410 0.529339i \(-0.177560\pi\)
0.848410 + 0.529339i \(0.177560\pi\)
\(978\) 7.20211e12 0.251730
\(979\) 2.89507e12 0.100725
\(980\) 2.02226e13 0.700356
\(981\) −4.36404e11 −0.0150445
\(982\) 8.38810e12 0.287847
\(983\) 2.76295e11 0.00943804 0.00471902 0.999989i \(-0.498498\pi\)
0.00471902 + 0.999989i \(0.498498\pi\)
\(984\) −8.74782e12 −0.297455
\(985\) 4.42270e13 1.49701
\(986\) −2.63003e10 −0.000886166 0
\(987\) 1.05757e13 0.354717
\(988\) −2.56639e13 −0.856873
\(989\) 1.31522e12 0.0437134
\(990\) 2.36181e12 0.0781425
\(991\) −4.44286e13 −1.46329 −0.731646 0.681685i \(-0.761248\pi\)
−0.731646 + 0.681685i \(0.761248\pi\)
\(992\) −3.99410e13 −1.30953
\(993\) 1.40324e13 0.457996
\(994\) 6.38076e11 0.0207316
\(995\) 3.07790e12 0.0995521
\(996\) 3.27196e12 0.105351
\(997\) 4.47458e13 1.43425 0.717124 0.696946i \(-0.245458\pi\)
0.717124 + 0.696946i \(0.245458\pi\)
\(998\) 1.93031e12 0.0615940
\(999\) 4.15353e12 0.131939
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 177.10.a.b.1.12 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.10.a.b.1.12 21 1.1 even 1 trivial