Properties

Label 177.10.a.b.1.9
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 1.9
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.43662 q^{2} -81.0000 q^{3} -492.316 q^{4} +339.476 q^{5} +359.366 q^{6} -12342.0 q^{7} +4455.77 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q-4.43662 q^{2} -81.0000 q^{3} -492.316 q^{4} +339.476 q^{5} +359.366 q^{6} -12342.0 q^{7} +4455.77 q^{8} +6561.00 q^{9} -1506.13 q^{10} +52983.7 q^{11} +39877.6 q^{12} -128050. q^{13} +54756.7 q^{14} -27497.6 q^{15} +232297. q^{16} -563735. q^{17} -29108.6 q^{18} +833020. q^{19} -167130. q^{20} +999702. q^{21} -235068. q^{22} +2.12601e6 q^{23} -360917. q^{24} -1.83788e6 q^{25} +568108. q^{26} -531441. q^{27} +6.07617e6 q^{28} +5.73663e6 q^{29} +121996. q^{30} +1.18319e6 q^{31} -3.31197e6 q^{32} -4.29168e6 q^{33} +2.50108e6 q^{34} -4.18981e6 q^{35} -3.23009e6 q^{36} +5.87411e6 q^{37} -3.69579e6 q^{38} +1.03720e7 q^{39} +1.51263e6 q^{40} -2.15771e7 q^{41} -4.43529e6 q^{42} +2.71725e7 q^{43} -2.60847e7 q^{44} +2.22730e6 q^{45} -9.43231e6 q^{46} -2.75285e7 q^{47} -1.88161e7 q^{48} +1.11971e8 q^{49} +8.15398e6 q^{50} +4.56626e7 q^{51} +6.30410e7 q^{52} +2.71958e7 q^{53} +2.35780e6 q^{54} +1.79867e7 q^{55} -5.49931e7 q^{56} -6.74746e7 q^{57} -2.54512e7 q^{58} -1.21174e7 q^{59} +1.35375e7 q^{60} +9.59563e7 q^{61} -5.24937e6 q^{62} -8.09758e7 q^{63} -1.04242e8 q^{64} -4.34699e7 q^{65} +1.90405e7 q^{66} +1.74709e8 q^{67} +2.77536e8 q^{68} -1.72207e8 q^{69} +1.85886e7 q^{70} +1.39282e8 q^{71} +2.92343e7 q^{72} -1.56504e8 q^{73} -2.60612e7 q^{74} +1.48868e8 q^{75} -4.10110e8 q^{76} -6.53924e8 q^{77} -4.60168e7 q^{78} -5.55442e8 q^{79} +7.88595e7 q^{80} +4.30467e7 q^{81} +9.57292e7 q^{82} -2.03005e8 q^{83} -4.92169e8 q^{84} -1.91375e8 q^{85} -1.20554e8 q^{86} -4.64667e8 q^{87} +2.36083e8 q^{88} +1.86479e7 q^{89} -9.88169e6 q^{90} +1.58039e9 q^{91} -1.04667e9 q^{92} -9.58386e7 q^{93} +1.22133e8 q^{94} +2.82791e8 q^{95} +2.68269e8 q^{96} -6.59088e8 q^{97} -4.96774e8 q^{98} +3.47626e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21q + 20q^{2} - 1701q^{3} + 4950q^{4} + 2058q^{5} - 1620q^{6} - 17167q^{7} - 2853q^{8} + 137781q^{9} + O(q^{10}) \) \( 21q + 20q^{2} - 1701q^{3} + 4950q^{4} + 2058q^{5} - 1620q^{6} - 17167q^{7} - 2853q^{8} + 137781q^{9} - 31559q^{10} - 38751q^{11} - 400950q^{12} - 58915q^{13} + 3453q^{14} - 166698q^{15} + 1655714q^{16} - 64233q^{17} + 131220q^{18} - 1937236q^{19} - 1065507q^{20} + 1390527q^{21} - 5386882q^{22} - 1838574q^{23} + 231093q^{24} + 4565755q^{25} - 839702q^{26} - 11160261q^{27} - 4471034q^{28} + 15658544q^{29} + 2556279q^{30} - 14282802q^{31} - 2205286q^{32} + 3138831q^{33} + 19005532q^{34} - 8633300q^{35} + 32476950q^{36} + 7531195q^{37} + 26649773q^{38} + 4772115q^{39} + 17775672q^{40} + 18338245q^{41} - 279693q^{42} - 22480305q^{43} - 80230922q^{44} + 13502538q^{45} - 83894107q^{46} - 110397260q^{47} - 134112834q^{48} + 130653638q^{49} + 65575693q^{50} + 5202873q^{51} + 177908014q^{52} + 145498338q^{53} - 10628820q^{54} + 86448944q^{55} + 354387888q^{56} + 156916116q^{57} + 115508368q^{58} - 254464581q^{59} + 86306067q^{60} + 287595506q^{61} + 819899030q^{62} - 112632687q^{63} + 822446413q^{64} + 77238206q^{65} + 436337442q^{66} - 392860610q^{67} + 167325073q^{68} + 148924494q^{69} - 424902116q^{70} - 248960491q^{71} - 18718533q^{72} - 758406074q^{73} - 923266846q^{74} - 369826155q^{75} - 2312747568q^{76} - 878126795q^{77} + 68015862q^{78} - 1925801029q^{79} - 1898919861q^{80} + 903981141q^{81} - 3249102191q^{82} - 1650336307q^{83} + 362153754q^{84} - 2342480762q^{85} - 3609864952q^{86} - 1268342064q^{87} - 5987792887q^{88} - 574997526q^{89} - 207058599q^{90} - 4481387117q^{91} - 5317166770q^{92} + 1156906962q^{93} - 5360726568q^{94} - 2789231462q^{95} + 178628166q^{96} - 4651540898q^{97} - 5566652976q^{98} - 254245311q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.43662 −0.196073 −0.0980363 0.995183i \(-0.531256\pi\)
−0.0980363 + 0.995183i \(0.531256\pi\)
\(3\) −81.0000 −0.577350
\(4\) −492.316 −0.961556
\(5\) 339.476 0.242909 0.121455 0.992597i \(-0.461244\pi\)
0.121455 + 0.992597i \(0.461244\pi\)
\(6\) 359.366 0.113203
\(7\) −12342.0 −1.94287 −0.971436 0.237302i \(-0.923737\pi\)
−0.971436 + 0.237302i \(0.923737\pi\)
\(8\) 4455.77 0.384607
\(9\) 6561.00 0.333333
\(10\) −1506.13 −0.0476279
\(11\) 52983.7 1.09113 0.545563 0.838070i \(-0.316316\pi\)
0.545563 + 0.838070i \(0.316316\pi\)
\(12\) 39877.6 0.555154
\(13\) −128050. −1.24347 −0.621733 0.783229i \(-0.713571\pi\)
−0.621733 + 0.783229i \(0.713571\pi\)
\(14\) 54756.7 0.380944
\(15\) −27497.6 −0.140244
\(16\) 232297. 0.886145
\(17\) −563735. −1.63702 −0.818512 0.574489i \(-0.805201\pi\)
−0.818512 + 0.574489i \(0.805201\pi\)
\(18\) −29108.6 −0.0653576
\(19\) 833020. 1.46644 0.733220 0.679991i \(-0.238016\pi\)
0.733220 + 0.679991i \(0.238016\pi\)
\(20\) −167130. −0.233571
\(21\) 999702. 1.12172
\(22\) −235068. −0.213940
\(23\) 2.12601e6 1.58413 0.792065 0.610437i \(-0.209006\pi\)
0.792065 + 0.610437i \(0.209006\pi\)
\(24\) −360917. −0.222053
\(25\) −1.83788e6 −0.940995
\(26\) 568108. 0.243810
\(27\) −531441. −0.192450
\(28\) 6.07617e6 1.86818
\(29\) 5.73663e6 1.50614 0.753071 0.657939i \(-0.228572\pi\)
0.753071 + 0.657939i \(0.228572\pi\)
\(30\) 121996. 0.0274980
\(31\) 1.18319e6 0.230106 0.115053 0.993359i \(-0.463296\pi\)
0.115053 + 0.993359i \(0.463296\pi\)
\(32\) −3.31197e6 −0.558356
\(33\) −4.29168e6 −0.629962
\(34\) 2.50108e6 0.320976
\(35\) −4.18981e6 −0.471942
\(36\) −3.23009e6 −0.320519
\(37\) 5.87411e6 0.515270 0.257635 0.966242i \(-0.417057\pi\)
0.257635 + 0.966242i \(0.417057\pi\)
\(38\) −3.69579e6 −0.287529
\(39\) 1.03720e7 0.717916
\(40\) 1.51263e6 0.0934248
\(41\) −2.15771e7 −1.19252 −0.596259 0.802792i \(-0.703347\pi\)
−0.596259 + 0.802792i \(0.703347\pi\)
\(42\) −4.43529e6 −0.219938
\(43\) 2.71725e7 1.21205 0.606025 0.795445i \(-0.292763\pi\)
0.606025 + 0.795445i \(0.292763\pi\)
\(44\) −2.60847e7 −1.04918
\(45\) 2.22730e6 0.0809698
\(46\) −9.43231e6 −0.310604
\(47\) −2.75285e7 −0.822889 −0.411445 0.911435i \(-0.634976\pi\)
−0.411445 + 0.911435i \(0.634976\pi\)
\(48\) −1.88161e7 −0.511616
\(49\) 1.11971e8 2.77475
\(50\) 8.15398e6 0.184503
\(51\) 4.56626e7 0.945136
\(52\) 6.30410e7 1.19566
\(53\) 2.71958e7 0.473435 0.236717 0.971579i \(-0.423928\pi\)
0.236717 + 0.971579i \(0.423928\pi\)
\(54\) 2.35780e6 0.0377342
\(55\) 1.79867e7 0.265045
\(56\) −5.49931e7 −0.747243
\(57\) −6.74746e7 −0.846650
\(58\) −2.54512e7 −0.295313
\(59\) −1.21174e7 −0.130189
\(60\) 1.35375e7 0.134852
\(61\) 9.59563e7 0.887338 0.443669 0.896191i \(-0.353677\pi\)
0.443669 + 0.896191i \(0.353677\pi\)
\(62\) −5.24937e6 −0.0451175
\(63\) −8.09758e7 −0.647624
\(64\) −1.04242e8 −0.776666
\(65\) −4.34699e7 −0.302050
\(66\) 1.90405e7 0.123518
\(67\) 1.74709e8 1.05920 0.529602 0.848246i \(-0.322341\pi\)
0.529602 + 0.848246i \(0.322341\pi\)
\(68\) 2.77536e8 1.57409
\(69\) −1.72207e8 −0.914597
\(70\) 1.85886e7 0.0925349
\(71\) 1.39282e8 0.650480 0.325240 0.945632i \(-0.394555\pi\)
0.325240 + 0.945632i \(0.394555\pi\)
\(72\) 2.92343e7 0.128202
\(73\) −1.56504e8 −0.645020 −0.322510 0.946566i \(-0.604527\pi\)
−0.322510 + 0.946566i \(0.604527\pi\)
\(74\) −2.60612e7 −0.101030
\(75\) 1.48868e8 0.543284
\(76\) −4.10110e8 −1.41006
\(77\) −6.53924e8 −2.11992
\(78\) −4.60168e7 −0.140764
\(79\) −5.55442e8 −1.60442 −0.802208 0.597044i \(-0.796342\pi\)
−0.802208 + 0.597044i \(0.796342\pi\)
\(80\) 7.88595e7 0.215253
\(81\) 4.30467e7 0.111111
\(82\) 9.57292e7 0.233820
\(83\) −2.03005e8 −0.469521 −0.234761 0.972053i \(-0.575431\pi\)
−0.234761 + 0.972053i \(0.575431\pi\)
\(84\) −4.92169e8 −1.07859
\(85\) −1.91375e8 −0.397649
\(86\) −1.20554e8 −0.237650
\(87\) −4.64667e8 −0.869571
\(88\) 2.36083e8 0.419655
\(89\) 1.86479e7 0.0315046 0.0157523 0.999876i \(-0.494986\pi\)
0.0157523 + 0.999876i \(0.494986\pi\)
\(90\) −9.88169e6 −0.0158760
\(91\) 1.58039e9 2.41590
\(92\) −1.04667e9 −1.52323
\(93\) −9.58386e7 −0.132852
\(94\) 1.22133e8 0.161346
\(95\) 2.82791e8 0.356212
\(96\) 2.68269e8 0.322367
\(97\) −6.59088e8 −0.755911 −0.377955 0.925824i \(-0.623373\pi\)
−0.377955 + 0.925824i \(0.623373\pi\)
\(98\) −4.96774e8 −0.544053
\(99\) 3.47626e8 0.363709
\(100\) 9.04819e8 0.904819
\(101\) −1.82533e8 −0.174540 −0.0872699 0.996185i \(-0.527814\pi\)
−0.0872699 + 0.996185i \(0.527814\pi\)
\(102\) −2.02587e8 −0.185315
\(103\) 1.33454e9 1.16832 0.584162 0.811637i \(-0.301423\pi\)
0.584162 + 0.811637i \(0.301423\pi\)
\(104\) −5.70560e8 −0.478246
\(105\) 3.39375e8 0.272476
\(106\) −1.20657e8 −0.0928276
\(107\) −1.46868e9 −1.08318 −0.541590 0.840643i \(-0.682177\pi\)
−0.541590 + 0.840643i \(0.682177\pi\)
\(108\) 2.61637e8 0.185051
\(109\) −5.59949e8 −0.379952 −0.189976 0.981789i \(-0.560841\pi\)
−0.189976 + 0.981789i \(0.560841\pi\)
\(110\) −7.98001e7 −0.0519681
\(111\) −4.75803e8 −0.297491
\(112\) −2.86701e9 −1.72167
\(113\) 9.22036e8 0.531979 0.265990 0.963976i \(-0.414301\pi\)
0.265990 + 0.963976i \(0.414301\pi\)
\(114\) 2.99359e8 0.166005
\(115\) 7.21731e8 0.384800
\(116\) −2.82424e9 −1.44824
\(117\) −8.40135e8 −0.414489
\(118\) 5.37601e7 0.0255265
\(119\) 6.95762e9 3.18053
\(120\) −1.22523e8 −0.0539388
\(121\) 4.49324e8 0.190557
\(122\) −4.25721e8 −0.173983
\(123\) 1.74774e9 0.688501
\(124\) −5.82505e8 −0.221260
\(125\) −1.28696e9 −0.471486
\(126\) 3.59259e8 0.126981
\(127\) −2.91682e9 −0.994932 −0.497466 0.867483i \(-0.665736\pi\)
−0.497466 + 0.867483i \(0.665736\pi\)
\(128\) 2.15821e9 0.710639
\(129\) −2.20097e9 −0.699778
\(130\) 1.92859e8 0.0592237
\(131\) −4.78537e9 −1.41969 −0.709846 0.704356i \(-0.751236\pi\)
−0.709846 + 0.704356i \(0.751236\pi\)
\(132\) 2.11286e9 0.605744
\(133\) −1.02811e10 −2.84911
\(134\) −7.75119e8 −0.207681
\(135\) −1.80412e8 −0.0467479
\(136\) −2.51187e9 −0.629612
\(137\) 3.90657e9 0.947443 0.473721 0.880675i \(-0.342910\pi\)
0.473721 + 0.880675i \(0.342910\pi\)
\(138\) 7.64017e8 0.179328
\(139\) −2.16547e9 −0.492024 −0.246012 0.969267i \(-0.579120\pi\)
−0.246012 + 0.969267i \(0.579120\pi\)
\(140\) 2.06271e9 0.453798
\(141\) 2.22980e9 0.475095
\(142\) −6.17943e8 −0.127541
\(143\) −6.78455e9 −1.35678
\(144\) 1.52410e9 0.295382
\(145\) 1.94745e9 0.365856
\(146\) 6.94349e8 0.126471
\(147\) −9.06967e9 −1.60200
\(148\) −2.89192e9 −0.495460
\(149\) −9.94536e9 −1.65304 −0.826518 0.562911i \(-0.809681\pi\)
−0.826518 + 0.562911i \(0.809681\pi\)
\(150\) −6.60472e8 −0.106523
\(151\) 6.31624e8 0.0988695 0.0494347 0.998777i \(-0.484258\pi\)
0.0494347 + 0.998777i \(0.484258\pi\)
\(152\) 3.71175e9 0.564004
\(153\) −3.69867e9 −0.545675
\(154\) 2.90121e9 0.415658
\(155\) 4.01666e8 0.0558949
\(156\) −5.10632e9 −0.690316
\(157\) −1.31608e10 −1.72875 −0.864375 0.502847i \(-0.832286\pi\)
−0.864375 + 0.502847i \(0.832286\pi\)
\(158\) 2.46429e9 0.314582
\(159\) −2.20286e9 −0.273338
\(160\) −1.12433e9 −0.135630
\(161\) −2.62392e10 −3.07776
\(162\) −1.90982e8 −0.0217859
\(163\) 1.18425e10 1.31402 0.657008 0.753884i \(-0.271822\pi\)
0.657008 + 0.753884i \(0.271822\pi\)
\(164\) 1.06227e10 1.14667
\(165\) −1.45692e9 −0.153024
\(166\) 9.00655e8 0.0920603
\(167\) −5.43925e9 −0.541147 −0.270573 0.962699i \(-0.587213\pi\)
−0.270573 + 0.962699i \(0.587213\pi\)
\(168\) 4.45444e9 0.431421
\(169\) 5.79226e9 0.546208
\(170\) 8.49057e8 0.0779680
\(171\) 5.46545e9 0.488813
\(172\) −1.33774e10 −1.16545
\(173\) −1.31101e10 −1.11275 −0.556376 0.830930i \(-0.687809\pi\)
−0.556376 + 0.830930i \(0.687809\pi\)
\(174\) 2.06155e9 0.170499
\(175\) 2.26831e10 1.82823
\(176\) 1.23080e10 0.966896
\(177\) 9.81506e8 0.0751646
\(178\) −8.27335e7 −0.00617719
\(179\) 1.76458e10 1.28470 0.642351 0.766411i \(-0.277959\pi\)
0.642351 + 0.766411i \(0.277959\pi\)
\(180\) −1.09654e9 −0.0778570
\(181\) 2.64661e10 1.83289 0.916444 0.400163i \(-0.131047\pi\)
0.916444 + 0.400163i \(0.131047\pi\)
\(182\) −7.01159e9 −0.473691
\(183\) −7.77246e9 −0.512305
\(184\) 9.47302e9 0.609268
\(185\) 1.99412e9 0.125164
\(186\) 4.25199e8 0.0260486
\(187\) −2.98688e10 −1.78620
\(188\) 1.35527e10 0.791254
\(189\) 6.55904e9 0.373906
\(190\) −1.25463e9 −0.0698435
\(191\) 2.91012e10 1.58220 0.791099 0.611688i \(-0.209509\pi\)
0.791099 + 0.611688i \(0.209509\pi\)
\(192\) 8.44363e9 0.448408
\(193\) 1.79428e10 0.930856 0.465428 0.885086i \(-0.345900\pi\)
0.465428 + 0.885086i \(0.345900\pi\)
\(194\) 2.92412e9 0.148213
\(195\) 3.52106e9 0.174388
\(196\) −5.51253e10 −2.66808
\(197\) −1.30845e10 −0.618957 −0.309478 0.950906i \(-0.600154\pi\)
−0.309478 + 0.950906i \(0.600154\pi\)
\(198\) −1.54228e9 −0.0713134
\(199\) −1.01257e10 −0.457705 −0.228852 0.973461i \(-0.573497\pi\)
−0.228852 + 0.973461i \(0.573497\pi\)
\(200\) −8.18917e9 −0.361914
\(201\) −1.41515e10 −0.611532
\(202\) 8.09827e8 0.0342225
\(203\) −7.08015e10 −2.92624
\(204\) −2.24804e10 −0.908801
\(205\) −7.32490e9 −0.289674
\(206\) −5.92083e9 −0.229076
\(207\) 1.39488e10 0.528043
\(208\) −2.97457e10 −1.10189
\(209\) 4.41365e10 1.60007
\(210\) −1.50568e9 −0.0534250
\(211\) 1.18601e10 0.411923 0.205962 0.978560i \(-0.433968\pi\)
0.205962 + 0.978560i \(0.433968\pi\)
\(212\) −1.33889e10 −0.455234
\(213\) −1.12819e10 −0.375555
\(214\) 6.51597e9 0.212382
\(215\) 9.22440e9 0.294419
\(216\) −2.36798e9 −0.0740177
\(217\) −1.46030e10 −0.447066
\(218\) 2.48428e9 0.0744983
\(219\) 1.26768e10 0.372402
\(220\) −8.85515e9 −0.254855
\(221\) 7.21862e10 2.03558
\(222\) 2.11096e9 0.0583299
\(223\) −4.02908e10 −1.09102 −0.545512 0.838103i \(-0.683665\pi\)
−0.545512 + 0.838103i \(0.683665\pi\)
\(224\) 4.08763e10 1.08481
\(225\) −1.20583e10 −0.313665
\(226\) −4.09072e9 −0.104307
\(227\) −6.59239e9 −0.164788 −0.0823941 0.996600i \(-0.526257\pi\)
−0.0823941 + 0.996600i \(0.526257\pi\)
\(228\) 3.32189e10 0.814101
\(229\) 3.98522e10 0.957619 0.478810 0.877919i \(-0.341068\pi\)
0.478810 + 0.877919i \(0.341068\pi\)
\(230\) −3.20204e9 −0.0754487
\(231\) 5.29679e10 1.22394
\(232\) 2.55611e10 0.579273
\(233\) −9.10647e9 −0.202418 −0.101209 0.994865i \(-0.532271\pi\)
−0.101209 + 0.994865i \(0.532271\pi\)
\(234\) 3.72736e9 0.0812699
\(235\) −9.34526e9 −0.199888
\(236\) 5.96558e9 0.125184
\(237\) 4.49908e10 0.926310
\(238\) −3.08683e10 −0.623615
\(239\) 2.75413e10 0.546001 0.273001 0.962014i \(-0.411984\pi\)
0.273001 + 0.962014i \(0.411984\pi\)
\(240\) −6.38762e9 −0.124276
\(241\) 2.12666e10 0.406089 0.203045 0.979169i \(-0.434916\pi\)
0.203045 + 0.979169i \(0.434916\pi\)
\(242\) −1.99348e9 −0.0373631
\(243\) −3.48678e9 −0.0641500
\(244\) −4.72409e10 −0.853225
\(245\) 3.80116e10 0.674013
\(246\) −7.75407e9 −0.134996
\(247\) −1.06668e11 −1.82347
\(248\) 5.27203e9 0.0885005
\(249\) 1.64434e10 0.271078
\(250\) 5.70973e9 0.0924455
\(251\) −3.66142e10 −0.582261 −0.291131 0.956683i \(-0.594031\pi\)
−0.291131 + 0.956683i \(0.594031\pi\)
\(252\) 3.98657e10 0.622726
\(253\) 1.12644e11 1.72849
\(254\) 1.29408e10 0.195079
\(255\) 1.55014e10 0.229583
\(256\) 4.37969e10 0.637329
\(257\) 3.46810e10 0.495898 0.247949 0.968773i \(-0.420243\pi\)
0.247949 + 0.968773i \(0.420243\pi\)
\(258\) 9.76486e9 0.137207
\(259\) −7.24983e10 −1.00110
\(260\) 2.14009e10 0.290437
\(261\) 3.76380e10 0.502047
\(262\) 2.12308e10 0.278363
\(263\) 1.18027e11 1.52119 0.760593 0.649229i \(-0.224908\pi\)
0.760593 + 0.649229i \(0.224908\pi\)
\(264\) −1.91227e10 −0.242288
\(265\) 9.23232e9 0.115002
\(266\) 4.56135e10 0.558632
\(267\) −1.51048e9 −0.0181892
\(268\) −8.60123e10 −1.01848
\(269\) 1.47003e11 1.71175 0.855873 0.517185i \(-0.173020\pi\)
0.855873 + 0.517185i \(0.173020\pi\)
\(270\) 8.00417e8 0.00916599
\(271\) −2.83854e10 −0.319693 −0.159846 0.987142i \(-0.551100\pi\)
−0.159846 + 0.987142i \(0.551100\pi\)
\(272\) −1.30954e11 −1.45064
\(273\) −1.28012e11 −1.39482
\(274\) −1.73320e10 −0.185768
\(275\) −9.73777e10 −1.02674
\(276\) 8.47803e10 0.879436
\(277\) 4.37749e10 0.446751 0.223376 0.974732i \(-0.428292\pi\)
0.223376 + 0.974732i \(0.428292\pi\)
\(278\) 9.60737e9 0.0964724
\(279\) 7.76293e9 0.0767020
\(280\) −1.86688e10 −0.181512
\(281\) 9.03302e10 0.864280 0.432140 0.901806i \(-0.357759\pi\)
0.432140 + 0.901806i \(0.357759\pi\)
\(282\) −9.89279e9 −0.0931532
\(283\) 1.27299e9 0.0117973 0.00589867 0.999983i \(-0.498122\pi\)
0.00589867 + 0.999983i \(0.498122\pi\)
\(284\) −6.85711e10 −0.625473
\(285\) −2.29060e10 −0.205659
\(286\) 3.01005e10 0.266027
\(287\) 2.66304e11 2.31691
\(288\) −2.17298e10 −0.186119
\(289\) 1.99210e11 1.67985
\(290\) −8.64009e9 −0.0717344
\(291\) 5.33861e10 0.436425
\(292\) 7.70496e10 0.620222
\(293\) −1.74934e11 −1.38666 −0.693330 0.720620i \(-0.743857\pi\)
−0.693330 + 0.720620i \(0.743857\pi\)
\(294\) 4.02387e10 0.314109
\(295\) −4.11356e9 −0.0316241
\(296\) 2.61737e10 0.198176
\(297\) −2.81577e10 −0.209987
\(298\) 4.41237e10 0.324115
\(299\) −2.72236e11 −1.96981
\(300\) −7.32903e10 −0.522397
\(301\) −3.35362e11 −2.35486
\(302\) −2.80227e9 −0.0193856
\(303\) 1.47851e10 0.100771
\(304\) 1.93509e11 1.29948
\(305\) 3.25749e10 0.215543
\(306\) 1.64096e10 0.106992
\(307\) 4.06289e10 0.261043 0.130522 0.991445i \(-0.458335\pi\)
0.130522 + 0.991445i \(0.458335\pi\)
\(308\) 3.21938e11 2.03842
\(309\) −1.08097e11 −0.674532
\(310\) −1.78204e9 −0.0109595
\(311\) −4.16431e10 −0.252419 −0.126209 0.992004i \(-0.540281\pi\)
−0.126209 + 0.992004i \(0.540281\pi\)
\(312\) 4.62154e10 0.276116
\(313\) −2.10609e11 −1.24030 −0.620152 0.784482i \(-0.712929\pi\)
−0.620152 + 0.784482i \(0.712929\pi\)
\(314\) 5.83892e10 0.338961
\(315\) −2.74894e10 −0.157314
\(316\) 2.73453e11 1.54274
\(317\) 6.47864e10 0.360344 0.180172 0.983635i \(-0.442335\pi\)
0.180172 + 0.983635i \(0.442335\pi\)
\(318\) 9.77324e9 0.0535941
\(319\) 3.03948e11 1.64339
\(320\) −3.53878e10 −0.188660
\(321\) 1.18963e11 0.625374
\(322\) 1.16413e11 0.603465
\(323\) −4.69603e11 −2.40060
\(324\) −2.11926e10 −0.106840
\(325\) 2.35340e11 1.17010
\(326\) −5.25408e10 −0.257643
\(327\) 4.53559e10 0.219366
\(328\) −9.61424e10 −0.458651
\(329\) 3.39756e11 1.59877
\(330\) 6.46381e9 0.0300038
\(331\) −2.47766e11 −1.13453 −0.567265 0.823535i \(-0.691999\pi\)
−0.567265 + 0.823535i \(0.691999\pi\)
\(332\) 9.99427e10 0.451471
\(333\) 3.85401e10 0.171757
\(334\) 2.41319e10 0.106104
\(335\) 5.93097e10 0.257291
\(336\) 2.32228e11 0.994004
\(337\) −2.17120e11 −0.916991 −0.458495 0.888697i \(-0.651611\pi\)
−0.458495 + 0.888697i \(0.651611\pi\)
\(338\) −2.56981e10 −0.107096
\(339\) −7.46849e10 −0.307138
\(340\) 9.42169e10 0.382361
\(341\) 6.26899e10 0.251075
\(342\) −2.42481e10 −0.0958430
\(343\) −8.83904e11 −3.44811
\(344\) 1.21074e11 0.466164
\(345\) −5.84602e10 −0.222164
\(346\) 5.81645e10 0.218180
\(347\) −1.52482e11 −0.564594 −0.282297 0.959327i \(-0.591096\pi\)
−0.282297 + 0.959327i \(0.591096\pi\)
\(348\) 2.28763e11 0.836141
\(349\) 3.33989e11 1.20508 0.602542 0.798087i \(-0.294154\pi\)
0.602542 + 0.798087i \(0.294154\pi\)
\(350\) −1.00636e11 −0.358466
\(351\) 6.80509e10 0.239305
\(352\) −1.75480e11 −0.609237
\(353\) 2.18994e11 0.750666 0.375333 0.926890i \(-0.377528\pi\)
0.375333 + 0.926890i \(0.377528\pi\)
\(354\) −4.35457e9 −0.0147377
\(355\) 4.72831e10 0.158008
\(356\) −9.18065e9 −0.0302934
\(357\) −5.63567e11 −1.83628
\(358\) −7.82876e10 −0.251895
\(359\) −3.71576e11 −1.18066 −0.590328 0.807164i \(-0.701001\pi\)
−0.590328 + 0.807164i \(0.701001\pi\)
\(360\) 9.92435e9 0.0311416
\(361\) 3.71235e11 1.15045
\(362\) −1.17420e11 −0.359379
\(363\) −3.63952e10 −0.110018
\(364\) −7.78052e11 −2.32302
\(365\) −5.31295e10 −0.156681
\(366\) 3.44834e10 0.100449
\(367\) 1.98110e11 0.570045 0.285022 0.958521i \(-0.407999\pi\)
0.285022 + 0.958521i \(0.407999\pi\)
\(368\) 4.93867e11 1.40377
\(369\) −1.41567e11 −0.397506
\(370\) −8.84715e9 −0.0245412
\(371\) −3.35650e11 −0.919823
\(372\) 4.71829e10 0.127744
\(373\) 4.59095e11 1.22804 0.614021 0.789290i \(-0.289551\pi\)
0.614021 + 0.789290i \(0.289551\pi\)
\(374\) 1.32516e11 0.350225
\(375\) 1.04243e11 0.272213
\(376\) −1.22660e11 −0.316489
\(377\) −7.34575e11 −1.87284
\(378\) −2.91000e10 −0.0733127
\(379\) −3.04151e11 −0.757205 −0.378602 0.925559i \(-0.623595\pi\)
−0.378602 + 0.925559i \(0.623595\pi\)
\(380\) −1.39222e11 −0.342518
\(381\) 2.36263e11 0.574424
\(382\) −1.29111e11 −0.310226
\(383\) 1.53059e11 0.363465 0.181733 0.983348i \(-0.441829\pi\)
0.181733 + 0.983348i \(0.441829\pi\)
\(384\) −1.74815e11 −0.410288
\(385\) −2.21992e11 −0.514948
\(386\) −7.96054e10 −0.182515
\(387\) 1.78278e11 0.404017
\(388\) 3.24480e11 0.726850
\(389\) 4.19117e11 0.928030 0.464015 0.885827i \(-0.346408\pi\)
0.464015 + 0.885827i \(0.346408\pi\)
\(390\) −1.56216e10 −0.0341928
\(391\) −1.19851e12 −2.59326
\(392\) 4.98918e11 1.06719
\(393\) 3.87615e11 0.819660
\(394\) 5.80511e10 0.121361
\(395\) −1.88560e11 −0.389728
\(396\) −1.71142e11 −0.349726
\(397\) −4.01994e11 −0.812198 −0.406099 0.913829i \(-0.633111\pi\)
−0.406099 + 0.913829i \(0.633111\pi\)
\(398\) 4.49238e10 0.0897434
\(399\) 8.32772e11 1.64493
\(400\) −4.26935e11 −0.833858
\(401\) −5.23348e11 −1.01074 −0.505372 0.862902i \(-0.668645\pi\)
−0.505372 + 0.862902i \(0.668645\pi\)
\(402\) 6.27846e10 0.119905
\(403\) −1.51508e11 −0.286129
\(404\) 8.98638e10 0.167830
\(405\) 1.46133e10 0.0269899
\(406\) 3.14119e11 0.573756
\(407\) 3.11232e11 0.562224
\(408\) 2.03462e11 0.363506
\(409\) −5.56139e11 −0.982716 −0.491358 0.870958i \(-0.663499\pi\)
−0.491358 + 0.870958i \(0.663499\pi\)
\(410\) 3.24978e10 0.0567971
\(411\) −3.16432e11 −0.547006
\(412\) −6.57014e11 −1.12341
\(413\) 1.49552e11 0.252940
\(414\) −6.18854e10 −0.103535
\(415\) −6.89154e10 −0.114051
\(416\) 4.24097e11 0.694297
\(417\) 1.75403e11 0.284070
\(418\) −1.95817e11 −0.313730
\(419\) −2.40013e10 −0.0380427 −0.0190214 0.999819i \(-0.506055\pi\)
−0.0190214 + 0.999819i \(0.506055\pi\)
\(420\) −1.67080e11 −0.262001
\(421\) −1.44975e11 −0.224917 −0.112459 0.993656i \(-0.535873\pi\)
−0.112459 + 0.993656i \(0.535873\pi\)
\(422\) −5.26186e10 −0.0807669
\(423\) −1.80614e11 −0.274296
\(424\) 1.21178e11 0.182087
\(425\) 1.03608e12 1.54043
\(426\) 5.00534e10 0.0736360
\(427\) −1.18429e12 −1.72398
\(428\) 7.23055e11 1.04154
\(429\) 5.49549e11 0.783337
\(430\) −4.09251e10 −0.0577274
\(431\) −9.40468e11 −1.31279 −0.656397 0.754416i \(-0.727920\pi\)
−0.656397 + 0.754416i \(0.727920\pi\)
\(432\) −1.23452e11 −0.170539
\(433\) −8.32077e11 −1.13754 −0.568772 0.822495i \(-0.692581\pi\)
−0.568772 + 0.822495i \(0.692581\pi\)
\(434\) 6.47878e10 0.0876575
\(435\) −1.57743e11 −0.211227
\(436\) 2.75672e11 0.365345
\(437\) 1.77101e12 2.32303
\(438\) −5.62423e10 −0.0730179
\(439\) 9.47325e10 0.121733 0.0608666 0.998146i \(-0.480614\pi\)
0.0608666 + 0.998146i \(0.480614\pi\)
\(440\) 8.01446e10 0.101938
\(441\) 7.34643e11 0.924917
\(442\) −3.20263e11 −0.399122
\(443\) −5.67930e11 −0.700612 −0.350306 0.936635i \(-0.613922\pi\)
−0.350306 + 0.936635i \(0.613922\pi\)
\(444\) 2.34246e11 0.286054
\(445\) 6.33051e9 0.00765277
\(446\) 1.78755e11 0.213920
\(447\) 8.05574e11 0.954380
\(448\) 1.28656e12 1.50896
\(449\) 6.19806e11 0.719693 0.359847 0.933011i \(-0.382829\pi\)
0.359847 + 0.933011i \(0.382829\pi\)
\(450\) 5.34982e10 0.0615011
\(451\) −1.14323e12 −1.30119
\(452\) −4.53933e11 −0.511528
\(453\) −5.11615e10 −0.0570823
\(454\) 2.92479e10 0.0323105
\(455\) 5.36505e11 0.586844
\(456\) −3.00651e11 −0.325628
\(457\) −3.07940e11 −0.330251 −0.165125 0.986273i \(-0.552803\pi\)
−0.165125 + 0.986273i \(0.552803\pi\)
\(458\) −1.76809e11 −0.187763
\(459\) 2.99592e11 0.315045
\(460\) −3.55320e11 −0.370006
\(461\) −1.40714e12 −1.45105 −0.725526 0.688195i \(-0.758403\pi\)
−0.725526 + 0.688195i \(0.758403\pi\)
\(462\) −2.34998e11 −0.239980
\(463\) 1.36244e12 1.37786 0.688929 0.724829i \(-0.258081\pi\)
0.688929 + 0.724829i \(0.258081\pi\)
\(464\) 1.33260e12 1.33466
\(465\) −3.25349e10 −0.0322709
\(466\) 4.04019e10 0.0396886
\(467\) −9.45768e11 −0.920150 −0.460075 0.887880i \(-0.652178\pi\)
−0.460075 + 0.887880i \(0.652178\pi\)
\(468\) 4.13612e11 0.398554
\(469\) −2.15626e12 −2.05790
\(470\) 4.14613e10 0.0391925
\(471\) 1.06602e12 0.998095
\(472\) −5.39922e10 −0.0500716
\(473\) 1.43970e12 1.32250
\(474\) −1.99607e11 −0.181624
\(475\) −1.53099e12 −1.37991
\(476\) −3.42535e12 −3.05825
\(477\) 1.78432e11 0.157812
\(478\) −1.22190e11 −0.107056
\(479\) −1.03116e12 −0.894989 −0.447495 0.894287i \(-0.647684\pi\)
−0.447495 + 0.894287i \(0.647684\pi\)
\(480\) 9.10711e10 0.0783060
\(481\) −7.52179e11 −0.640720
\(482\) −9.43518e10 −0.0796230
\(483\) 2.12538e12 1.77695
\(484\) −2.21210e11 −0.183231
\(485\) −2.23745e11 −0.183618
\(486\) 1.54695e10 0.0125781
\(487\) −7.67349e11 −0.618177 −0.309088 0.951033i \(-0.600024\pi\)
−0.309088 + 0.951033i \(0.600024\pi\)
\(488\) 4.27559e11 0.341277
\(489\) −9.59245e11 −0.758647
\(490\) −1.68643e11 −0.132156
\(491\) −1.35773e12 −1.05426 −0.527129 0.849785i \(-0.676731\pi\)
−0.527129 + 0.849785i \(0.676731\pi\)
\(492\) −8.60442e11 −0.662032
\(493\) −3.23394e12 −2.46559
\(494\) 4.73246e11 0.357532
\(495\) 1.18011e11 0.0883483
\(496\) 2.74853e11 0.203907
\(497\) −1.71902e12 −1.26380
\(498\) −7.29531e10 −0.0531510
\(499\) 6.93814e11 0.500945 0.250473 0.968124i \(-0.419414\pi\)
0.250473 + 0.968124i \(0.419414\pi\)
\(500\) 6.33590e11 0.453360
\(501\) 4.40579e11 0.312431
\(502\) 1.62443e11 0.114166
\(503\) −2.15196e12 −1.49892 −0.749458 0.662052i \(-0.769686\pi\)
−0.749458 + 0.662052i \(0.769686\pi\)
\(504\) −3.60810e11 −0.249081
\(505\) −6.19655e10 −0.0423973
\(506\) −4.99758e11 −0.338909
\(507\) −4.69173e11 −0.315353
\(508\) 1.43600e12 0.956683
\(509\) −7.20802e11 −0.475977 −0.237989 0.971268i \(-0.576488\pi\)
−0.237989 + 0.971268i \(0.576488\pi\)
\(510\) −6.87736e10 −0.0450149
\(511\) 1.93157e12 1.25319
\(512\) −1.29931e12 −0.835602
\(513\) −4.42701e11 −0.282217
\(514\) −1.53866e11 −0.0972320
\(515\) 4.53044e11 0.283797
\(516\) 1.08357e12 0.672875
\(517\) −1.45856e12 −0.897876
\(518\) 3.21647e11 0.196289
\(519\) 1.06192e12 0.642448
\(520\) −1.93692e11 −0.116171
\(521\) 1.79148e10 0.0106523 0.00532615 0.999986i \(-0.498305\pi\)
0.00532615 + 0.999986i \(0.498305\pi\)
\(522\) −1.66986e11 −0.0984377
\(523\) 1.03328e12 0.603896 0.301948 0.953324i \(-0.402363\pi\)
0.301948 + 0.953324i \(0.402363\pi\)
\(524\) 2.35591e12 1.36511
\(525\) −1.83733e12 −1.05553
\(526\) −5.23643e11 −0.298263
\(527\) −6.67008e11 −0.376689
\(528\) −9.96946e11 −0.558238
\(529\) 2.71878e12 1.50947
\(530\) −4.09603e10 −0.0225487
\(531\) −7.95020e10 −0.0433963
\(532\) 5.06157e12 2.73957
\(533\) 2.76294e12 1.48286
\(534\) 6.70141e9 0.00356640
\(535\) −4.98582e11 −0.263114
\(536\) 7.78465e11 0.407378
\(537\) −1.42931e12 −0.741723
\(538\) −6.52194e11 −0.335627
\(539\) 5.93265e12 3.02760
\(540\) 8.88196e10 0.0449507
\(541\) −3.38071e12 −1.69676 −0.848380 0.529388i \(-0.822422\pi\)
−0.848380 + 0.529388i \(0.822422\pi\)
\(542\) 1.25935e11 0.0626830
\(543\) −2.14375e12 −1.05822
\(544\) 1.86707e12 0.914043
\(545\) −1.90089e11 −0.0922940
\(546\) 5.67939e11 0.273486
\(547\) 2.74815e11 0.131249 0.0656246 0.997844i \(-0.479096\pi\)
0.0656246 + 0.997844i \(0.479096\pi\)
\(548\) −1.92327e12 −0.911019
\(549\) 6.29569e11 0.295779
\(550\) 4.32028e11 0.201317
\(551\) 4.77873e12 2.20867
\(552\) −7.67315e11 −0.351761
\(553\) 6.85527e12 3.11718
\(554\) −1.94212e11 −0.0875957
\(555\) −1.61524e11 −0.0722634
\(556\) 1.06610e12 0.473108
\(557\) 1.71495e11 0.0754925 0.0377463 0.999287i \(-0.487982\pi\)
0.0377463 + 0.999287i \(0.487982\pi\)
\(558\) −3.44411e10 −0.0150392
\(559\) −3.47943e12 −1.50714
\(560\) −9.73283e11 −0.418209
\(561\) 2.41937e12 1.03126
\(562\) −4.00761e11 −0.169462
\(563\) 1.49060e12 0.625277 0.312639 0.949872i \(-0.398787\pi\)
0.312639 + 0.949872i \(0.398787\pi\)
\(564\) −1.09777e12 −0.456831
\(565\) 3.13009e11 0.129223
\(566\) −5.64775e9 −0.00231314
\(567\) −5.31282e11 −0.215875
\(568\) 6.20610e11 0.250179
\(569\) 2.46114e12 0.984308 0.492154 0.870508i \(-0.336210\pi\)
0.492154 + 0.870508i \(0.336210\pi\)
\(570\) 1.01625e11 0.0403241
\(571\) 5.91742e11 0.232954 0.116477 0.993193i \(-0.462840\pi\)
0.116477 + 0.993193i \(0.462840\pi\)
\(572\) 3.34015e12 1.30462
\(573\) −2.35720e12 −0.913483
\(574\) −1.18149e12 −0.454283
\(575\) −3.90736e12 −1.49066
\(576\) −6.83934e11 −0.258889
\(577\) −2.01620e12 −0.757255 −0.378627 0.925549i \(-0.623604\pi\)
−0.378627 + 0.925549i \(0.623604\pi\)
\(578\) −8.83817e11 −0.329372
\(579\) −1.45337e12 −0.537430
\(580\) −9.58761e11 −0.351791
\(581\) 2.50549e12 0.912220
\(582\) −2.36854e11 −0.0855711
\(583\) 1.44093e12 0.516577
\(584\) −6.97346e11 −0.248079
\(585\) −2.85206e11 −0.100683
\(586\) 7.76116e11 0.271886
\(587\) −1.31299e12 −0.456448 −0.228224 0.973609i \(-0.573292\pi\)
−0.228224 + 0.973609i \(0.573292\pi\)
\(588\) 4.46515e12 1.54042
\(589\) 9.85624e11 0.337437
\(590\) 1.82503e10 0.00620062
\(591\) 1.05985e12 0.357355
\(592\) 1.36454e12 0.456603
\(593\) −4.87910e11 −0.162029 −0.0810147 0.996713i \(-0.525816\pi\)
−0.0810147 + 0.996713i \(0.525816\pi\)
\(594\) 1.24925e11 0.0411728
\(595\) 2.36195e12 0.772580
\(596\) 4.89626e12 1.58949
\(597\) 8.20180e11 0.264256
\(598\) 1.20781e12 0.386226
\(599\) 1.61047e12 0.511130 0.255565 0.966792i \(-0.417739\pi\)
0.255565 + 0.966792i \(0.417739\pi\)
\(600\) 6.63323e11 0.208951
\(601\) 2.28175e12 0.713399 0.356699 0.934219i \(-0.383902\pi\)
0.356699 + 0.934219i \(0.383902\pi\)
\(602\) 1.48787e12 0.461724
\(603\) 1.14627e12 0.353068
\(604\) −3.10959e11 −0.0950685
\(605\) 1.52535e11 0.0462882
\(606\) −6.55960e10 −0.0197584
\(607\) −5.34157e12 −1.59706 −0.798528 0.601958i \(-0.794387\pi\)
−0.798528 + 0.601958i \(0.794387\pi\)
\(608\) −2.75894e12 −0.818796
\(609\) 5.73492e12 1.68947
\(610\) −1.44522e11 −0.0422620
\(611\) 3.52501e12 1.02324
\(612\) 1.82091e12 0.524697
\(613\) −1.89520e12 −0.542103 −0.271052 0.962565i \(-0.587371\pi\)
−0.271052 + 0.962565i \(0.587371\pi\)
\(614\) −1.80255e11 −0.0511835
\(615\) 5.93317e11 0.167243
\(616\) −2.91374e12 −0.815337
\(617\) −4.56332e11 −0.126765 −0.0633823 0.997989i \(-0.520189\pi\)
−0.0633823 + 0.997989i \(0.520189\pi\)
\(618\) 4.79587e11 0.132257
\(619\) −3.43083e11 −0.0939272 −0.0469636 0.998897i \(-0.514954\pi\)
−0.0469636 + 0.998897i \(0.514954\pi\)
\(620\) −1.97747e11 −0.0537461
\(621\) −1.12985e12 −0.304866
\(622\) 1.84755e11 0.0494924
\(623\) −2.30152e11 −0.0612094
\(624\) 2.40940e12 0.636177
\(625\) 3.15272e12 0.826467
\(626\) 9.34392e11 0.243190
\(627\) −3.57506e12 −0.923802
\(628\) 6.47926e12 1.66229
\(629\) −3.31145e12 −0.843509
\(630\) 1.21960e11 0.0308450
\(631\) −2.43342e12 −0.611062 −0.305531 0.952182i \(-0.598834\pi\)
−0.305531 + 0.952182i \(0.598834\pi\)
\(632\) −2.47492e12 −0.617071
\(633\) −9.60666e11 −0.237824
\(634\) −2.87433e11 −0.0706536
\(635\) −9.90193e11 −0.241678
\(636\) 1.08450e12 0.262829
\(637\) −1.43379e13 −3.45031
\(638\) −1.34850e12 −0.322224
\(639\) 9.13832e11 0.216827
\(640\) 7.32661e11 0.172621
\(641\) −6.11253e12 −1.43008 −0.715039 0.699085i \(-0.753591\pi\)
−0.715039 + 0.699085i \(0.753591\pi\)
\(642\) −5.27794e11 −0.122619
\(643\) 1.07901e12 0.248929 0.124465 0.992224i \(-0.460279\pi\)
0.124465 + 0.992224i \(0.460279\pi\)
\(644\) 1.29180e13 2.95944
\(645\) −7.47177e11 −0.169983
\(646\) 2.08345e12 0.470692
\(647\) −2.21917e12 −0.497877 −0.248939 0.968519i \(-0.580082\pi\)
−0.248939 + 0.968519i \(0.580082\pi\)
\(648\) 1.91806e11 0.0427342
\(649\) −6.42023e11 −0.142053
\(650\) −1.04412e12 −0.229424
\(651\) 1.18284e12 0.258114
\(652\) −5.83027e12 −1.26350
\(653\) −4.76854e12 −1.02630 −0.513152 0.858298i \(-0.671522\pi\)
−0.513152 + 0.858298i \(0.671522\pi\)
\(654\) −2.01227e11 −0.0430116
\(655\) −1.62452e12 −0.344857
\(656\) −5.01230e12 −1.05674
\(657\) −1.02682e12 −0.215007
\(658\) −1.50737e12 −0.313475
\(659\) 2.36550e12 0.488584 0.244292 0.969702i \(-0.421444\pi\)
0.244292 + 0.969702i \(0.421444\pi\)
\(660\) 7.17267e11 0.147141
\(661\) −3.11734e11 −0.0635152 −0.0317576 0.999496i \(-0.510110\pi\)
−0.0317576 + 0.999496i \(0.510110\pi\)
\(662\) 1.09924e12 0.222451
\(663\) −5.84708e12 −1.17525
\(664\) −9.04543e11 −0.180581
\(665\) −3.49020e12 −0.692075
\(666\) −1.70987e11 −0.0336768
\(667\) 1.21961e13 2.38592
\(668\) 2.67783e12 0.520343
\(669\) 3.26356e12 0.629903
\(670\) −2.63134e11 −0.0504477
\(671\) 5.08412e12 0.968198
\(672\) −3.31098e12 −0.626318
\(673\) 5.18257e12 0.973817 0.486908 0.873453i \(-0.338125\pi\)
0.486908 + 0.873453i \(0.338125\pi\)
\(674\) 9.63278e11 0.179797
\(675\) 9.76725e11 0.181095
\(676\) −2.85163e12 −0.525209
\(677\) 3.26977e12 0.598229 0.299115 0.954217i \(-0.403309\pi\)
0.299115 + 0.954217i \(0.403309\pi\)
\(678\) 3.31348e11 0.0602214
\(679\) 8.13446e12 1.46864
\(680\) −8.52722e11 −0.152939
\(681\) 5.33983e11 0.0951406
\(682\) −2.78131e11 −0.0492289
\(683\) −4.71601e12 −0.829242 −0.414621 0.909994i \(-0.636086\pi\)
−0.414621 + 0.909994i \(0.636086\pi\)
\(684\) −2.69073e12 −0.470021
\(685\) 1.32619e12 0.230143
\(686\) 3.92155e12 0.676081
\(687\) −3.22803e12 −0.552882
\(688\) 6.31209e12 1.07405
\(689\) −3.48242e12 −0.588700
\(690\) 2.59366e11 0.0435603
\(691\) 8.20731e12 1.36946 0.684730 0.728797i \(-0.259920\pi\)
0.684730 + 0.728797i \(0.259920\pi\)
\(692\) 6.45432e12 1.06997
\(693\) −4.29040e12 −0.706640
\(694\) 6.76505e11 0.110701
\(695\) −7.35126e11 −0.119517
\(696\) −2.07045e12 −0.334444
\(697\) 1.21638e13 1.95218
\(698\) −1.48178e12 −0.236284
\(699\) 7.37624e11 0.116866
\(700\) −1.11673e13 −1.75795
\(701\) −4.18348e12 −0.654345 −0.327172 0.944965i \(-0.606096\pi\)
−0.327172 + 0.944965i \(0.606096\pi\)
\(702\) −3.01916e11 −0.0469212
\(703\) 4.89326e12 0.755612
\(704\) −5.52315e12 −0.847441
\(705\) 7.56966e11 0.115405
\(706\) −9.71594e11 −0.147185
\(707\) 2.25282e12 0.339108
\(708\) −4.83212e11 −0.0722749
\(709\) −5.97767e12 −0.888431 −0.444216 0.895920i \(-0.646518\pi\)
−0.444216 + 0.895920i \(0.646518\pi\)
\(710\) −2.09777e11 −0.0309810
\(711\) −3.64426e12 −0.534806
\(712\) 8.30906e10 0.0121169
\(713\) 2.51548e12 0.364518
\(714\) 2.50033e12 0.360044
\(715\) −2.30319e12 −0.329574
\(716\) −8.68731e12 −1.23531
\(717\) −2.23084e12 −0.315234
\(718\) 1.64854e12 0.231494
\(719\) 1.85467e11 0.0258813 0.0129407 0.999916i \(-0.495881\pi\)
0.0129407 + 0.999916i \(0.495881\pi\)
\(720\) 5.17397e11 0.0717509
\(721\) −1.64708e13 −2.26990
\(722\) −1.64703e12 −0.225571
\(723\) −1.72259e12 −0.234456
\(724\) −1.30297e13 −1.76242
\(725\) −1.05432e13 −1.41727
\(726\) 1.61472e11 0.0215716
\(727\) −6.25007e12 −0.829812 −0.414906 0.909864i \(-0.636186\pi\)
−0.414906 + 0.909864i \(0.636186\pi\)
\(728\) 7.04185e12 0.929171
\(729\) 2.82430e11 0.0370370
\(730\) 2.35715e11 0.0307209
\(731\) −1.53181e13 −1.98416
\(732\) 3.82651e12 0.492610
\(733\) −5.09676e12 −0.652118 −0.326059 0.945349i \(-0.605721\pi\)
−0.326059 + 0.945349i \(0.605721\pi\)
\(734\) −8.78938e11 −0.111770
\(735\) −3.07894e12 −0.389142
\(736\) −7.04129e12 −0.884508
\(737\) 9.25675e12 1.15573
\(738\) 6.28079e11 0.0779401
\(739\) −1.10301e13 −1.36044 −0.680220 0.733008i \(-0.738116\pi\)
−0.680220 + 0.733008i \(0.738116\pi\)
\(740\) −9.81739e11 −0.120352
\(741\) 8.64012e12 1.05278
\(742\) 1.48915e12 0.180352
\(743\) −4.17587e12 −0.502687 −0.251343 0.967898i \(-0.580872\pi\)
−0.251343 + 0.967898i \(0.580872\pi\)
\(744\) −4.27035e11 −0.0510958
\(745\) −3.37621e12 −0.401538
\(746\) −2.03683e12 −0.240785
\(747\) −1.33192e12 −0.156507
\(748\) 1.47049e13 1.71753
\(749\) 1.81264e13 2.10448
\(750\) −4.62488e11 −0.0533734
\(751\) 6.90534e12 0.792146 0.396073 0.918219i \(-0.370373\pi\)
0.396073 + 0.918219i \(0.370373\pi\)
\(752\) −6.39479e12 −0.729199
\(753\) 2.96575e12 0.336169
\(754\) 3.25903e12 0.367212
\(755\) 2.14421e11 0.0240163
\(756\) −3.22912e12 −0.359531
\(757\) 1.53101e13 1.69451 0.847257 0.531183i \(-0.178252\pi\)
0.847257 + 0.531183i \(0.178252\pi\)
\(758\) 1.34940e12 0.148467
\(759\) −9.12416e12 −0.997942
\(760\) 1.26005e12 0.137002
\(761\) −1.27908e12 −0.138251 −0.0691253 0.997608i \(-0.522021\pi\)
−0.0691253 + 0.997608i \(0.522021\pi\)
\(762\) −1.04821e12 −0.112629
\(763\) 6.91089e12 0.738199
\(764\) −1.43270e13 −1.52137
\(765\) −1.25561e12 −0.132550
\(766\) −6.79062e11 −0.0712656
\(767\) 1.55163e12 0.161886
\(768\) −3.54755e12 −0.367962
\(769\) −3.09767e12 −0.319423 −0.159712 0.987164i \(-0.551056\pi\)
−0.159712 + 0.987164i \(0.551056\pi\)
\(770\) 9.84893e11 0.100967
\(771\) −2.80916e12 −0.286307
\(772\) −8.83354e12 −0.895070
\(773\) −6.48475e12 −0.653259 −0.326630 0.945152i \(-0.605913\pi\)
−0.326630 + 0.945152i \(0.605913\pi\)
\(774\) −7.90953e11 −0.0792167
\(775\) −2.17457e12 −0.216529
\(776\) −2.93674e12 −0.290729
\(777\) 5.87236e12 0.577987
\(778\) −1.85946e12 −0.181961
\(779\) −1.79741e13 −1.74876
\(780\) −1.73348e12 −0.167684
\(781\) 7.37970e12 0.709756
\(782\) 5.31732e12 0.508467
\(783\) −3.04868e12 −0.289857
\(784\) 2.60106e13 2.45883
\(785\) −4.46776e12 −0.419930
\(786\) −1.71970e12 −0.160713
\(787\) 8.41991e12 0.782386 0.391193 0.920309i \(-0.372062\pi\)
0.391193 + 0.920309i \(0.372062\pi\)
\(788\) 6.44173e12 0.595161
\(789\) −9.56022e12 −0.878257
\(790\) 8.36567e11 0.0764150
\(791\) −1.13798e13 −1.03357
\(792\) 1.54894e12 0.139885
\(793\) −1.22872e13 −1.10337
\(794\) 1.78349e12 0.159250
\(795\) −7.47818e11 −0.0663963
\(796\) 4.98504e12 0.440109
\(797\) 2.85116e12 0.250299 0.125150 0.992138i \(-0.460059\pi\)
0.125150 + 0.992138i \(0.460059\pi\)
\(798\) −3.69469e12 −0.322526
\(799\) 1.55188e13 1.34709
\(800\) 6.08700e12 0.525410
\(801\) 1.22349e11 0.0105015
\(802\) 2.32190e12 0.198179
\(803\) −8.29217e12 −0.703798
\(804\) 6.96700e12 0.588022
\(805\) −8.90760e12 −0.747617
\(806\) 6.72182e11 0.0561021
\(807\) −1.19072e13 −0.988277
\(808\) −8.13323e11 −0.0671293
\(809\) 1.86626e13 1.53180 0.765902 0.642958i \(-0.222293\pi\)
0.765902 + 0.642958i \(0.222293\pi\)
\(810\) −6.48338e10 −0.00529199
\(811\) 8.25962e12 0.670450 0.335225 0.942138i \(-0.391188\pi\)
0.335225 + 0.942138i \(0.391188\pi\)
\(812\) 3.48567e13 2.81374
\(813\) 2.29922e12 0.184575
\(814\) −1.38082e12 −0.110237
\(815\) 4.02026e12 0.319187
\(816\) 1.06073e13 0.837528
\(817\) 2.26352e13 1.77740
\(818\) 2.46737e12 0.192684
\(819\) 1.03689e13 0.805298
\(820\) 3.60617e12 0.278538
\(821\) 1.10007e13 0.845041 0.422521 0.906353i \(-0.361145\pi\)
0.422521 + 0.906353i \(0.361145\pi\)
\(822\) 1.40389e12 0.107253
\(823\) −1.30547e13 −0.991901 −0.495950 0.868351i \(-0.665180\pi\)
−0.495950 + 0.868351i \(0.665180\pi\)
\(824\) 5.94639e12 0.449346
\(825\) 7.88760e12 0.592791
\(826\) −6.63507e11 −0.0495947
\(827\) −1.98955e13 −1.47904 −0.739519 0.673136i \(-0.764947\pi\)
−0.739519 + 0.673136i \(0.764947\pi\)
\(828\) −6.86721e12 −0.507743
\(829\) −2.50037e13 −1.83869 −0.919346 0.393450i \(-0.871282\pi\)
−0.919346 + 0.393450i \(0.871282\pi\)
\(830\) 3.05751e11 0.0223623
\(831\) −3.54576e12 −0.257932
\(832\) 1.33482e13 0.965758
\(833\) −6.31221e13 −4.54234
\(834\) −7.78197e11 −0.0556984
\(835\) −1.84650e12 −0.131450
\(836\) −2.17291e13 −1.53856
\(837\) −6.28797e11 −0.0442839
\(838\) 1.06485e11 0.00745914
\(839\) −2.47580e13 −1.72499 −0.862494 0.506067i \(-0.831099\pi\)
−0.862494 + 0.506067i \(0.831099\pi\)
\(840\) 1.51218e12 0.104796
\(841\) 1.84018e13 1.26846
\(842\) 6.43197e11 0.0441001
\(843\) −7.31675e12 −0.498992
\(844\) −5.83891e12 −0.396087
\(845\) 1.96634e12 0.132679
\(846\) 8.01316e11 0.0537820
\(847\) −5.54556e12 −0.370228
\(848\) 6.31751e12 0.419532
\(849\) −1.03112e11 −0.00681120
\(850\) −4.59668e12 −0.302037
\(851\) 1.24884e13 0.816254
\(852\) 5.55426e12 0.361117
\(853\) −1.01946e13 −0.659327 −0.329664 0.944098i \(-0.606935\pi\)
−0.329664 + 0.944098i \(0.606935\pi\)
\(854\) 5.25425e12 0.338026
\(855\) 1.85539e12 0.118737
\(856\) −6.54410e12 −0.416599
\(857\) 2.82216e13 1.78718 0.893589 0.448885i \(-0.148179\pi\)
0.893589 + 0.448885i \(0.148179\pi\)
\(858\) −2.43814e12 −0.153591
\(859\) −2.63641e13 −1.65213 −0.826064 0.563577i \(-0.809425\pi\)
−0.826064 + 0.563577i \(0.809425\pi\)
\(860\) −4.54132e12 −0.283100
\(861\) −2.15706e13 −1.33767
\(862\) 4.17250e12 0.257403
\(863\) 1.78000e13 1.09237 0.546186 0.837664i \(-0.316079\pi\)
0.546186 + 0.837664i \(0.316079\pi\)
\(864\) 1.76012e12 0.107456
\(865\) −4.45057e12 −0.270298
\(866\) 3.69161e12 0.223041
\(867\) −1.61360e13 −0.969861
\(868\) 7.18928e12 0.429879
\(869\) −2.94294e13 −1.75062
\(870\) 6.99847e11 0.0414159
\(871\) −2.23715e13 −1.31708
\(872\) −2.49500e12 −0.146133
\(873\) −4.32428e12 −0.251970
\(874\) −7.85730e12 −0.455483
\(875\) 1.58836e13 0.916037
\(876\) −6.24102e12 −0.358086
\(877\) 2.80232e13 1.59963 0.799815 0.600246i \(-0.204931\pi\)
0.799815 + 0.600246i \(0.204931\pi\)
\(878\) −4.20292e11 −0.0238685
\(879\) 1.41697e13 0.800589
\(880\) 4.17827e12 0.234868
\(881\) 5.48469e12 0.306733 0.153366 0.988169i \(-0.450989\pi\)
0.153366 + 0.988169i \(0.450989\pi\)
\(882\) −3.25933e12 −0.181351
\(883\) −1.68563e13 −0.933122 −0.466561 0.884489i \(-0.654507\pi\)
−0.466561 + 0.884489i \(0.654507\pi\)
\(884\) −3.55385e13 −1.95733
\(885\) 3.33198e11 0.0182582
\(886\) 2.51969e12 0.137371
\(887\) −1.17075e13 −0.635052 −0.317526 0.948250i \(-0.602852\pi\)
−0.317526 + 0.948250i \(0.602852\pi\)
\(888\) −2.12007e12 −0.114417
\(889\) 3.59994e13 1.93303
\(890\) −2.80860e10 −0.00150050
\(891\) 2.28077e12 0.121236
\(892\) 1.98358e13 1.04908
\(893\) −2.29318e13 −1.20672
\(894\) −3.57402e12 −0.187128
\(895\) 5.99032e12 0.312066
\(896\) −2.66366e13 −1.38068
\(897\) 2.20511e13 1.13727
\(898\) −2.74984e12 −0.141112
\(899\) 6.78754e12 0.346572
\(900\) 5.93652e12 0.301606
\(901\) −1.53312e13 −0.775024
\(902\) 5.07209e12 0.255127
\(903\) 2.71643e13 1.35958
\(904\) 4.10838e12 0.204603
\(905\) 8.98460e12 0.445226
\(906\) 2.26984e11 0.0111923
\(907\) 1.59339e13 0.781791 0.390895 0.920435i \(-0.372165\pi\)
0.390895 + 0.920435i \(0.372165\pi\)
\(908\) 3.24554e12 0.158453
\(909\) −1.19760e12 −0.0581799
\(910\) −2.38027e12 −0.115064
\(911\) −1.50810e13 −0.725433 −0.362716 0.931900i \(-0.618151\pi\)
−0.362716 + 0.931900i \(0.618151\pi\)
\(912\) −1.56742e13 −0.750254
\(913\) −1.07560e13 −0.512307
\(914\) 1.36621e12 0.0647531
\(915\) −2.63857e12 −0.124444
\(916\) −1.96199e13 −0.920804
\(917\) 5.90610e13 2.75828
\(918\) −1.32918e12 −0.0617718
\(919\) 3.13312e13 1.44896 0.724481 0.689295i \(-0.242079\pi\)
0.724481 + 0.689295i \(0.242079\pi\)
\(920\) 3.21587e12 0.147997
\(921\) −3.29094e12 −0.150713
\(922\) 6.24294e12 0.284512
\(923\) −1.78351e13 −0.808850
\(924\) −2.60770e13 −1.17688
\(925\) −1.07959e13 −0.484866
\(926\) −6.04465e12 −0.270160
\(927\) 8.75590e12 0.389441
\(928\) −1.89995e13 −0.840964
\(929\) 3.82774e13 1.68605 0.843027 0.537871i \(-0.180771\pi\)
0.843027 + 0.537871i \(0.180771\pi\)
\(930\) 1.44345e11 0.00632745
\(931\) 9.32743e13 4.06901
\(932\) 4.48327e12 0.194636
\(933\) 3.37309e12 0.145734
\(934\) 4.19601e12 0.180416
\(935\) −1.01397e13 −0.433885
\(936\) −3.74345e12 −0.159415
\(937\) 2.32522e13 0.985454 0.492727 0.870184i \(-0.336000\pi\)
0.492727 + 0.870184i \(0.336000\pi\)
\(938\) 9.56651e12 0.403498
\(939\) 1.70593e13 0.716089
\(940\) 4.60082e12 0.192203
\(941\) 3.51809e13 1.46270 0.731348 0.682004i \(-0.238891\pi\)
0.731348 + 0.682004i \(0.238891\pi\)
\(942\) −4.72953e12 −0.195699
\(943\) −4.58731e13 −1.88910
\(944\) −2.81483e12 −0.115366
\(945\) 2.22664e12 0.0908253
\(946\) −6.38739e12 −0.259306
\(947\) −3.64229e13 −1.47163 −0.735816 0.677181i \(-0.763201\pi\)
−0.735816 + 0.677181i \(0.763201\pi\)
\(948\) −2.21497e13 −0.890699
\(949\) 2.00403e13 0.802060
\(950\) 6.79243e12 0.270563
\(951\) −5.24770e12 −0.208045
\(952\) 3.10015e13 1.22325
\(953\) −2.17649e13 −0.854750 −0.427375 0.904074i \(-0.640562\pi\)
−0.427375 + 0.904074i \(0.640562\pi\)
\(954\) −7.91633e11 −0.0309425
\(955\) 9.87917e12 0.384331
\(956\) −1.35590e13 −0.525011
\(957\) −2.46198e13 −0.948813
\(958\) 4.57488e12 0.175483
\(959\) −4.82148e13 −1.84076
\(960\) 2.86641e12 0.108923
\(961\) −2.50397e13 −0.947051
\(962\) 3.33713e12 0.125628
\(963\) −9.63601e12 −0.361060
\(964\) −1.04699e13 −0.390477
\(965\) 6.09116e12 0.226114
\(966\) −9.42949e12 −0.348410
\(967\) 2.07696e13 0.763851 0.381925 0.924193i \(-0.375261\pi\)
0.381925 + 0.924193i \(0.375261\pi\)
\(968\) 2.00208e12 0.0732897
\(969\) 3.80378e13 1.38599
\(970\) 9.92670e11 0.0360024
\(971\) 2.81746e13 1.01712 0.508559 0.861027i \(-0.330179\pi\)
0.508559 + 0.861027i \(0.330179\pi\)
\(972\) 1.71660e12 0.0616838
\(973\) 2.67262e13 0.955939
\(974\) 3.40444e12 0.121208
\(975\) −1.90626e13 −0.675555
\(976\) 2.22904e13 0.786310
\(977\) 3.55629e13 1.24874 0.624369 0.781130i \(-0.285356\pi\)
0.624369 + 0.781130i \(0.285356\pi\)
\(978\) 4.25581e12 0.148750
\(979\) 9.88033e11 0.0343755
\(980\) −1.87137e13 −0.648101
\(981\) −3.67382e12 −0.126651
\(982\) 6.02373e12 0.206711
\(983\) 3.19675e12 0.109199 0.0545994 0.998508i \(-0.482612\pi\)
0.0545994 + 0.998508i \(0.482612\pi\)
\(984\) 7.78753e12 0.264802
\(985\) −4.44189e12 −0.150350
\(986\) 1.43478e13 0.483435
\(987\) −2.75202e13 −0.923049
\(988\) 5.25145e13 1.75337
\(989\) 5.77690e13 1.92005
\(990\) −5.23569e11 −0.0173227
\(991\) −2.46648e12 −0.0812356 −0.0406178 0.999175i \(-0.512933\pi\)
−0.0406178 + 0.999175i \(0.512933\pi\)
\(992\) −3.91870e12 −0.128481
\(993\) 2.00691e13 0.655022
\(994\) 7.62665e12 0.247796
\(995\) −3.43743e12 −0.111181
\(996\) −8.09536e12 −0.260657
\(997\) 5.14255e13 1.64835 0.824177 0.566333i \(-0.191638\pi\)
0.824177 + 0.566333i \(0.191638\pi\)
\(998\) −3.07819e12 −0.0982217
\(999\) −3.12174e12 −0.0991637
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.1.9 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.10.a.b.1.9 21 1.1 even 1 trivial