Properties

Label 177.10.a.b.1.8
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 1.8
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-12.5454 q^{2} -81.0000 q^{3} -354.612 q^{4} +2378.79 q^{5} +1016.18 q^{6} +4679.29 q^{7} +10872.0 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q-12.5454 q^{2} -81.0000 q^{3} -354.612 q^{4} +2378.79 q^{5} +1016.18 q^{6} +4679.29 q^{7} +10872.0 q^{8} +6561.00 q^{9} -29843.0 q^{10} +27377.3 q^{11} +28723.6 q^{12} -94968.3 q^{13} -58703.7 q^{14} -192682. q^{15} +45167.3 q^{16} -363572. q^{17} -82310.6 q^{18} -703094. q^{19} -843549. q^{20} -379022. q^{21} -343460. q^{22} +1.27907e6 q^{23} -880634. q^{24} +3.70553e6 q^{25} +1.19142e6 q^{26} -531441. q^{27} -1.65933e6 q^{28} -6.03733e6 q^{29} +2.41728e6 q^{30} -2.91796e6 q^{31} -6.13312e6 q^{32} -2.21756e6 q^{33} +4.56116e6 q^{34} +1.11311e7 q^{35} -2.32661e6 q^{36} +8.60765e6 q^{37} +8.82061e6 q^{38} +7.69243e6 q^{39} +2.58623e7 q^{40} -1.66287e7 q^{41} +4.75500e6 q^{42} +3.19196e7 q^{43} -9.70831e6 q^{44} +1.56073e7 q^{45} -1.60465e7 q^{46} +4.06112e7 q^{47} -3.65855e6 q^{48} -1.84579e7 q^{49} -4.64875e7 q^{50} +2.94493e7 q^{51} +3.36769e7 q^{52} -6.97055e7 q^{53} +6.66716e6 q^{54} +6.51248e7 q^{55} +5.08733e7 q^{56} +5.69506e7 q^{57} +7.57409e7 q^{58} -1.21174e7 q^{59} +6.83275e7 q^{60} -6.34803e7 q^{61} +3.66070e7 q^{62} +3.07008e7 q^{63} +5.38170e7 q^{64} -2.25910e8 q^{65} +2.78202e7 q^{66} -1.72165e8 q^{67} +1.28927e8 q^{68} -1.03605e8 q^{69} -1.39644e8 q^{70} -1.91685e8 q^{71} +7.13313e7 q^{72} +1.49098e8 q^{73} -1.07987e8 q^{74} -3.00148e8 q^{75} +2.49326e8 q^{76} +1.28106e8 q^{77} -9.65048e7 q^{78} -7.22461e7 q^{79} +1.07444e8 q^{80} +4.30467e7 q^{81} +2.08614e8 q^{82} -1.57839e8 q^{83} +1.34406e8 q^{84} -8.64861e8 q^{85} -4.00445e8 q^{86} +4.89024e8 q^{87} +2.97646e8 q^{88} +5.53997e8 q^{89} -1.95800e8 q^{90} -4.44384e8 q^{91} -4.53574e8 q^{92} +2.36355e8 q^{93} -5.09485e8 q^{94} -1.67251e9 q^{95} +4.96783e8 q^{96} +6.86877e8 q^{97} +2.31562e8 q^{98} +1.79622e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21q + 20q^{2} - 1701q^{3} + 4950q^{4} + 2058q^{5} - 1620q^{6} - 17167q^{7} - 2853q^{8} + 137781q^{9} + O(q^{10}) \) \( 21q + 20q^{2} - 1701q^{3} + 4950q^{4} + 2058q^{5} - 1620q^{6} - 17167q^{7} - 2853q^{8} + 137781q^{9} - 31559q^{10} - 38751q^{11} - 400950q^{12} - 58915q^{13} + 3453q^{14} - 166698q^{15} + 1655714q^{16} - 64233q^{17} + 131220q^{18} - 1937236q^{19} - 1065507q^{20} + 1390527q^{21} - 5386882q^{22} - 1838574q^{23} + 231093q^{24} + 4565755q^{25} - 839702q^{26} - 11160261q^{27} - 4471034q^{28} + 15658544q^{29} + 2556279q^{30} - 14282802q^{31} - 2205286q^{32} + 3138831q^{33} + 19005532q^{34} - 8633300q^{35} + 32476950q^{36} + 7531195q^{37} + 26649773q^{38} + 4772115q^{39} + 17775672q^{40} + 18338245q^{41} - 279693q^{42} - 22480305q^{43} - 80230922q^{44} + 13502538q^{45} - 83894107q^{46} - 110397260q^{47} - 134112834q^{48} + 130653638q^{49} + 65575693q^{50} + 5202873q^{51} + 177908014q^{52} + 145498338q^{53} - 10628820q^{54} + 86448944q^{55} + 354387888q^{56} + 156916116q^{57} + 115508368q^{58} - 254464581q^{59} + 86306067q^{60} + 287595506q^{61} + 819899030q^{62} - 112632687q^{63} + 822446413q^{64} + 77238206q^{65} + 436337442q^{66} - 392860610q^{67} + 167325073q^{68} + 148924494q^{69} - 424902116q^{70} - 248960491q^{71} - 18718533q^{72} - 758406074q^{73} - 923266846q^{74} - 369826155q^{75} - 2312747568q^{76} - 878126795q^{77} + 68015862q^{78} - 1925801029q^{79} - 1898919861q^{80} + 903981141q^{81} - 3249102191q^{82} - 1650336307q^{83} + 362153754q^{84} - 2342480762q^{85} - 3609864952q^{86} - 1268342064q^{87} - 5987792887q^{88} - 574997526q^{89} - 207058599q^{90} - 4481387117q^{91} - 5317166770q^{92} + 1156906962q^{93} - 5360726568q^{94} - 2789231462q^{95} + 178628166q^{96} - 4651540898q^{97} - 5566652976q^{98} - 254245311q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −12.5454 −0.554435 −0.277217 0.960807i \(-0.589412\pi\)
−0.277217 + 0.960807i \(0.589412\pi\)
\(3\) −81.0000 −0.577350
\(4\) −354.612 −0.692602
\(5\) 2378.79 1.70213 0.851063 0.525064i \(-0.175959\pi\)
0.851063 + 0.525064i \(0.175959\pi\)
\(6\) 1016.18 0.320103
\(7\) 4679.29 0.736612 0.368306 0.929705i \(-0.379938\pi\)
0.368306 + 0.929705i \(0.379938\pi\)
\(8\) 10872.0 0.938438
\(9\) 6561.00 0.333333
\(10\) −29843.0 −0.943718
\(11\) 27377.3 0.563797 0.281899 0.959444i \(-0.409036\pi\)
0.281899 + 0.959444i \(0.409036\pi\)
\(12\) 28723.6 0.399874
\(13\) −94968.3 −0.922218 −0.461109 0.887344i \(-0.652548\pi\)
−0.461109 + 0.887344i \(0.652548\pi\)
\(14\) −58703.7 −0.408403
\(15\) −192682. −0.982723
\(16\) 45167.3 0.172299
\(17\) −363572. −1.05577 −0.527886 0.849315i \(-0.677015\pi\)
−0.527886 + 0.849315i \(0.677015\pi\)
\(18\) −82310.6 −0.184812
\(19\) −703094. −1.23772 −0.618859 0.785502i \(-0.712405\pi\)
−0.618859 + 0.785502i \(0.712405\pi\)
\(20\) −843549. −1.17890
\(21\) −379022. −0.425283
\(22\) −343460. −0.312589
\(23\) 1.27907e6 0.953057 0.476529 0.879159i \(-0.341895\pi\)
0.476529 + 0.879159i \(0.341895\pi\)
\(24\) −880634. −0.541807
\(25\) 3.70553e6 1.89723
\(26\) 1.19142e6 0.511310
\(27\) −531441. −0.192450
\(28\) −1.65933e6 −0.510179
\(29\) −6.03733e6 −1.58509 −0.792545 0.609813i \(-0.791245\pi\)
−0.792545 + 0.609813i \(0.791245\pi\)
\(30\) 2.41728e6 0.544856
\(31\) −2.91796e6 −0.567481 −0.283740 0.958901i \(-0.591575\pi\)
−0.283740 + 0.958901i \(0.591575\pi\)
\(32\) −6.13312e6 −1.03397
\(33\) −2.21756e6 −0.325508
\(34\) 4.56116e6 0.585356
\(35\) 1.11311e7 1.25381
\(36\) −2.32661e6 −0.230867
\(37\) 8.60765e6 0.755052 0.377526 0.925999i \(-0.376775\pi\)
0.377526 + 0.925999i \(0.376775\pi\)
\(38\) 8.82061e6 0.686234
\(39\) 7.69243e6 0.532443
\(40\) 2.58623e7 1.59734
\(41\) −1.66287e7 −0.919033 −0.459516 0.888169i \(-0.651977\pi\)
−0.459516 + 0.888169i \(0.651977\pi\)
\(42\) 4.75500e6 0.235792
\(43\) 3.19196e7 1.42380 0.711900 0.702281i \(-0.247835\pi\)
0.711900 + 0.702281i \(0.247835\pi\)
\(44\) −9.70831e6 −0.390487
\(45\) 1.56073e7 0.567375
\(46\) −1.60465e7 −0.528408
\(47\) 4.06112e7 1.21396 0.606982 0.794716i \(-0.292380\pi\)
0.606982 + 0.794716i \(0.292380\pi\)
\(48\) −3.65855e6 −0.0994771
\(49\) −1.84579e7 −0.457403
\(50\) −4.64875e7 −1.05189
\(51\) 2.94493e7 0.609550
\(52\) 3.36769e7 0.638730
\(53\) −6.97055e7 −1.21346 −0.606731 0.794907i \(-0.707519\pi\)
−0.606731 + 0.794907i \(0.707519\pi\)
\(54\) 6.66716e6 0.106701
\(55\) 6.51248e7 0.959654
\(56\) 5.08733e7 0.691264
\(57\) 5.69506e7 0.714597
\(58\) 7.57409e7 0.878830
\(59\) −1.21174e7 −0.130189
\(60\) 6.83275e7 0.680636
\(61\) −6.34803e7 −0.587022 −0.293511 0.955956i \(-0.594824\pi\)
−0.293511 + 0.955956i \(0.594824\pi\)
\(62\) 3.66070e7 0.314631
\(63\) 3.07008e7 0.245537
\(64\) 5.38170e7 0.400968
\(65\) −2.25910e8 −1.56973
\(66\) 2.78202e7 0.180473
\(67\) −1.72165e8 −1.04378 −0.521888 0.853014i \(-0.674772\pi\)
−0.521888 + 0.853014i \(0.674772\pi\)
\(68\) 1.28927e8 0.731229
\(69\) −1.03605e8 −0.550248
\(70\) −1.39644e8 −0.695153
\(71\) −1.91685e8 −0.895212 −0.447606 0.894231i \(-0.647723\pi\)
−0.447606 + 0.894231i \(0.647723\pi\)
\(72\) 7.13313e7 0.312813
\(73\) 1.49098e8 0.614494 0.307247 0.951630i \(-0.400592\pi\)
0.307247 + 0.951630i \(0.400592\pi\)
\(74\) −1.07987e8 −0.418627
\(75\) −3.00148e8 −1.09537
\(76\) 2.49326e8 0.857246
\(77\) 1.28106e8 0.415300
\(78\) −9.65048e7 −0.295205
\(79\) −7.22461e7 −0.208685 −0.104343 0.994541i \(-0.533274\pi\)
−0.104343 + 0.994541i \(0.533274\pi\)
\(80\) 1.07444e8 0.293275
\(81\) 4.30467e7 0.111111
\(82\) 2.08614e8 0.509544
\(83\) −1.57839e8 −0.365059 −0.182530 0.983200i \(-0.558429\pi\)
−0.182530 + 0.983200i \(0.558429\pi\)
\(84\) 1.34406e8 0.294552
\(85\) −8.64861e8 −1.79706
\(86\) −4.00445e8 −0.789405
\(87\) 4.89024e8 0.915152
\(88\) 2.97646e8 0.529089
\(89\) 5.53997e8 0.935949 0.467975 0.883742i \(-0.344984\pi\)
0.467975 + 0.883742i \(0.344984\pi\)
\(90\) −1.95800e8 −0.314573
\(91\) −4.44384e8 −0.679316
\(92\) −4.53574e8 −0.660089
\(93\) 2.36355e8 0.327635
\(94\) −5.09485e8 −0.673064
\(95\) −1.67251e9 −2.10675
\(96\) 4.96783e8 0.596961
\(97\) 6.86877e8 0.787783 0.393891 0.919157i \(-0.371129\pi\)
0.393891 + 0.919157i \(0.371129\pi\)
\(98\) 2.31562e8 0.253600
\(99\) 1.79622e8 0.187932
\(100\) −1.31403e9 −1.31403
\(101\) −1.08954e9 −1.04183 −0.520917 0.853608i \(-0.674410\pi\)
−0.520917 + 0.853608i \(0.674410\pi\)
\(102\) −3.69454e8 −0.337956
\(103\) 4.39108e8 0.384419 0.192209 0.981354i \(-0.438435\pi\)
0.192209 + 0.981354i \(0.438435\pi\)
\(104\) −1.03250e9 −0.865444
\(105\) −9.01615e8 −0.723885
\(106\) 8.74486e8 0.672785
\(107\) −5.80854e8 −0.428391 −0.214195 0.976791i \(-0.568713\pi\)
−0.214195 + 0.976791i \(0.568713\pi\)
\(108\) 1.88455e8 0.133291
\(109\) 2.45513e9 1.66592 0.832960 0.553333i \(-0.186644\pi\)
0.832960 + 0.553333i \(0.186644\pi\)
\(110\) −8.17019e8 −0.532065
\(111\) −6.97220e8 −0.435930
\(112\) 2.11351e8 0.126918
\(113\) −1.05448e9 −0.608396 −0.304198 0.952609i \(-0.598388\pi\)
−0.304198 + 0.952609i \(0.598388\pi\)
\(114\) −7.14470e8 −0.396198
\(115\) 3.04264e9 1.62222
\(116\) 2.14091e9 1.09784
\(117\) −6.23087e8 −0.307406
\(118\) 1.52017e8 0.0721813
\(119\) −1.70126e9 −0.777693
\(120\) −2.09485e9 −0.922224
\(121\) −1.60843e9 −0.682133
\(122\) 7.96387e8 0.325466
\(123\) 1.34692e9 0.530604
\(124\) 1.03474e9 0.393038
\(125\) 4.16861e9 1.52720
\(126\) −3.85155e8 −0.136134
\(127\) −3.43386e9 −1.17129 −0.585646 0.810567i \(-0.699159\pi\)
−0.585646 + 0.810567i \(0.699159\pi\)
\(128\) 2.46500e9 0.811656
\(129\) −2.58549e9 −0.822031
\(130\) 2.83414e9 0.870313
\(131\) −2.83733e9 −0.841760 −0.420880 0.907116i \(-0.638279\pi\)
−0.420880 + 0.907116i \(0.638279\pi\)
\(132\) 7.86373e8 0.225448
\(133\) −3.28998e9 −0.911718
\(134\) 2.15988e9 0.578706
\(135\) −1.26419e9 −0.327574
\(136\) −3.95276e9 −0.990775
\(137\) −2.92430e9 −0.709217 −0.354608 0.935015i \(-0.615386\pi\)
−0.354608 + 0.935015i \(0.615386\pi\)
\(138\) 1.29976e9 0.305077
\(139\) 2.71055e9 0.615873 0.307937 0.951407i \(-0.400362\pi\)
0.307937 + 0.951407i \(0.400362\pi\)
\(140\) −3.94721e9 −0.868388
\(141\) −3.28951e9 −0.700882
\(142\) 2.40477e9 0.496337
\(143\) −2.59997e9 −0.519944
\(144\) 2.96342e8 0.0574331
\(145\) −1.43616e10 −2.69802
\(146\) −1.87049e9 −0.340697
\(147\) 1.49509e9 0.264082
\(148\) −3.05238e9 −0.522951
\(149\) −1.64768e9 −0.273864 −0.136932 0.990580i \(-0.543724\pi\)
−0.136932 + 0.990580i \(0.543724\pi\)
\(150\) 3.76548e9 0.607310
\(151\) −8.11578e9 −1.27038 −0.635191 0.772355i \(-0.719078\pi\)
−0.635191 + 0.772355i \(0.719078\pi\)
\(152\) −7.64405e9 −1.16152
\(153\) −2.38539e9 −0.351924
\(154\) −1.60715e9 −0.230257
\(155\) −6.94121e9 −0.965924
\(156\) −2.72783e9 −0.368771
\(157\) 1.60158e9 0.210378 0.105189 0.994452i \(-0.466455\pi\)
0.105189 + 0.994452i \(0.466455\pi\)
\(158\) 9.06358e8 0.115703
\(159\) 5.64615e9 0.700592
\(160\) −1.45894e10 −1.75994
\(161\) 5.98513e9 0.702033
\(162\) −5.40040e8 −0.0616039
\(163\) −6.76642e9 −0.750783 −0.375392 0.926866i \(-0.622492\pi\)
−0.375392 + 0.926866i \(0.622492\pi\)
\(164\) 5.89674e9 0.636524
\(165\) −5.27511e9 −0.554056
\(166\) 1.98016e9 0.202401
\(167\) −4.09332e9 −0.407241 −0.203621 0.979050i \(-0.565271\pi\)
−0.203621 + 0.979050i \(0.565271\pi\)
\(168\) −4.12074e9 −0.399101
\(169\) −1.58553e9 −0.149515
\(170\) 1.08501e10 0.996350
\(171\) −4.61300e9 −0.412573
\(172\) −1.13191e10 −0.986127
\(173\) 1.52173e10 1.29160 0.645802 0.763505i \(-0.276523\pi\)
0.645802 + 0.763505i \(0.276523\pi\)
\(174\) −6.13501e9 −0.507392
\(175\) 1.73392e10 1.39752
\(176\) 1.23656e9 0.0971419
\(177\) 9.81506e8 0.0751646
\(178\) −6.95013e9 −0.518923
\(179\) 2.22183e10 1.61761 0.808804 0.588079i \(-0.200116\pi\)
0.808804 + 0.588079i \(0.200116\pi\)
\(180\) −5.53452e9 −0.392965
\(181\) −2.47904e10 −1.71684 −0.858420 0.512947i \(-0.828554\pi\)
−0.858420 + 0.512947i \(0.828554\pi\)
\(182\) 5.57499e9 0.376637
\(183\) 5.14190e9 0.338917
\(184\) 1.39061e10 0.894385
\(185\) 2.04758e10 1.28519
\(186\) −2.96517e9 −0.181652
\(187\) −9.95360e9 −0.595241
\(188\) −1.44012e10 −0.840794
\(189\) −2.48677e9 −0.141761
\(190\) 2.09824e10 1.16806
\(191\) −1.94115e10 −1.05538 −0.527691 0.849436i \(-0.676942\pi\)
−0.527691 + 0.849436i \(0.676942\pi\)
\(192\) −4.35917e9 −0.231499
\(193\) −1.35292e10 −0.701884 −0.350942 0.936397i \(-0.614139\pi\)
−0.350942 + 0.936397i \(0.614139\pi\)
\(194\) −8.61717e9 −0.436774
\(195\) 1.82987e10 0.906284
\(196\) 6.54539e9 0.316798
\(197\) 2.17864e10 1.03059 0.515297 0.857012i \(-0.327682\pi\)
0.515297 + 0.857012i \(0.327682\pi\)
\(198\) −2.25344e9 −0.104196
\(199\) 1.59326e10 0.720190 0.360095 0.932916i \(-0.382744\pi\)
0.360095 + 0.932916i \(0.382744\pi\)
\(200\) 4.02866e10 1.78043
\(201\) 1.39453e10 0.602625
\(202\) 1.36688e10 0.577629
\(203\) −2.82504e10 −1.16760
\(204\) −1.04431e10 −0.422175
\(205\) −3.95562e10 −1.56431
\(206\) −5.50880e9 −0.213135
\(207\) 8.39198e9 0.317686
\(208\) −4.28946e9 −0.158898
\(209\) −1.92488e10 −0.697822
\(210\) 1.13112e10 0.401347
\(211\) −2.42054e10 −0.840699 −0.420350 0.907362i \(-0.638093\pi\)
−0.420350 + 0.907362i \(0.638093\pi\)
\(212\) 2.47184e10 0.840446
\(213\) 1.55265e10 0.516851
\(214\) 7.28707e9 0.237515
\(215\) 7.59301e10 2.42349
\(216\) −5.77784e9 −0.180602
\(217\) −1.36540e10 −0.418013
\(218\) −3.08006e10 −0.923645
\(219\) −1.20769e10 −0.354778
\(220\) −2.30941e10 −0.664658
\(221\) 3.45278e10 0.973651
\(222\) 8.74692e9 0.241695
\(223\) −1.04006e10 −0.281636 −0.140818 0.990036i \(-0.544973\pi\)
−0.140818 + 0.990036i \(0.544973\pi\)
\(224\) −2.86986e10 −0.761632
\(225\) 2.43120e10 0.632410
\(226\) 1.32289e10 0.337316
\(227\) 1.85297e10 0.463182 0.231591 0.972813i \(-0.425607\pi\)
0.231591 + 0.972813i \(0.425607\pi\)
\(228\) −2.01954e10 −0.494931
\(229\) −4.59022e10 −1.10300 −0.551498 0.834176i \(-0.685944\pi\)
−0.551498 + 0.834176i \(0.685944\pi\)
\(230\) −3.81712e10 −0.899417
\(231\) −1.03766e10 −0.239773
\(232\) −6.56380e10 −1.48751
\(233\) 7.97085e10 1.77175 0.885876 0.463922i \(-0.153558\pi\)
0.885876 + 0.463922i \(0.153558\pi\)
\(234\) 7.81689e9 0.170437
\(235\) 9.66057e10 2.06632
\(236\) 4.29696e9 0.0901691
\(237\) 5.85193e9 0.120485
\(238\) 2.13430e10 0.431180
\(239\) −2.10848e10 −0.418002 −0.209001 0.977915i \(-0.567021\pi\)
−0.209001 + 0.977915i \(0.567021\pi\)
\(240\) −8.70293e9 −0.169323
\(241\) 8.05924e10 1.53892 0.769462 0.638693i \(-0.220524\pi\)
0.769462 + 0.638693i \(0.220524\pi\)
\(242\) 2.01785e10 0.378198
\(243\) −3.48678e9 −0.0641500
\(244\) 2.25109e10 0.406573
\(245\) −4.39075e10 −0.778558
\(246\) −1.68977e10 −0.294185
\(247\) 6.67716e10 1.14145
\(248\) −3.17241e10 −0.532545
\(249\) 1.27850e10 0.210767
\(250\) −5.22970e10 −0.846733
\(251\) −7.67172e10 −1.22000 −0.610002 0.792400i \(-0.708831\pi\)
−0.610002 + 0.792400i \(0.708831\pi\)
\(252\) −1.08869e10 −0.170060
\(253\) 3.50174e10 0.537331
\(254\) 4.30792e10 0.649405
\(255\) 7.00538e10 1.03753
\(256\) −5.84788e10 −0.850978
\(257\) −1.42887e10 −0.204312 −0.102156 0.994768i \(-0.532574\pi\)
−0.102156 + 0.994768i \(0.532574\pi\)
\(258\) 3.24360e10 0.455763
\(259\) 4.02777e10 0.556180
\(260\) 8.01104e10 1.08720
\(261\) −3.96109e10 −0.528364
\(262\) 3.55955e10 0.466701
\(263\) −1.53883e11 −1.98331 −0.991655 0.128923i \(-0.958848\pi\)
−0.991655 + 0.128923i \(0.958848\pi\)
\(264\) −2.41093e10 −0.305469
\(265\) −1.65815e11 −2.06546
\(266\) 4.12742e10 0.505488
\(267\) −4.48738e10 −0.540371
\(268\) 6.10517e10 0.722922
\(269\) 1.59562e10 0.185800 0.0928998 0.995675i \(-0.470386\pi\)
0.0928998 + 0.995675i \(0.470386\pi\)
\(270\) 1.58598e10 0.181619
\(271\) −4.94957e10 −0.557450 −0.278725 0.960371i \(-0.589912\pi\)
−0.278725 + 0.960371i \(0.589912\pi\)
\(272\) −1.64215e10 −0.181909
\(273\) 3.59951e10 0.392203
\(274\) 3.66866e10 0.393214
\(275\) 1.01447e11 1.06965
\(276\) 3.67395e10 0.381103
\(277\) −4.78501e10 −0.488342 −0.244171 0.969732i \(-0.578516\pi\)
−0.244171 + 0.969732i \(0.578516\pi\)
\(278\) −3.40051e10 −0.341462
\(279\) −1.91447e10 −0.189160
\(280\) 1.21017e11 1.17662
\(281\) 1.87649e11 1.79543 0.897716 0.440575i \(-0.145225\pi\)
0.897716 + 0.440575i \(0.145225\pi\)
\(282\) 4.12683e10 0.388594
\(283\) 9.35083e10 0.866585 0.433292 0.901253i \(-0.357352\pi\)
0.433292 + 0.901253i \(0.357352\pi\)
\(284\) 6.79739e10 0.620026
\(285\) 1.35474e11 1.21633
\(286\) 3.26178e10 0.288275
\(287\) −7.78105e10 −0.676970
\(288\) −4.02394e10 −0.344655
\(289\) 1.35964e10 0.114653
\(290\) 1.80172e11 1.49588
\(291\) −5.56371e10 −0.454826
\(292\) −5.28718e10 −0.425600
\(293\) −6.02379e10 −0.477491 −0.238746 0.971082i \(-0.576736\pi\)
−0.238746 + 0.971082i \(0.576736\pi\)
\(294\) −1.87565e10 −0.146416
\(295\) −2.88247e10 −0.221598
\(296\) 9.35826e10 0.708570
\(297\) −1.45494e10 −0.108503
\(298\) 2.06709e10 0.151840
\(299\) −1.21471e11 −0.878926
\(300\) 1.06436e11 0.758653
\(301\) 1.49361e11 1.04879
\(302\) 1.01816e11 0.704344
\(303\) 8.82530e10 0.601503
\(304\) −3.17568e10 −0.213258
\(305\) −1.51006e11 −0.999185
\(306\) 2.99258e10 0.195119
\(307\) −2.53708e10 −0.163009 −0.0815044 0.996673i \(-0.525972\pi\)
−0.0815044 + 0.996673i \(0.525972\pi\)
\(308\) −4.54280e10 −0.287637
\(309\) −3.55678e10 −0.221944
\(310\) 8.70805e10 0.535542
\(311\) −1.97030e11 −1.19429 −0.597147 0.802132i \(-0.703699\pi\)
−0.597147 + 0.802132i \(0.703699\pi\)
\(312\) 8.36323e10 0.499664
\(313\) 1.03704e10 0.0610728 0.0305364 0.999534i \(-0.490278\pi\)
0.0305364 + 0.999534i \(0.490278\pi\)
\(314\) −2.00925e10 −0.116641
\(315\) 7.30309e10 0.417935
\(316\) 2.56193e10 0.144536
\(317\) 2.39312e11 1.33106 0.665530 0.746371i \(-0.268206\pi\)
0.665530 + 0.746371i \(0.268206\pi\)
\(318\) −7.08334e10 −0.388433
\(319\) −1.65286e11 −0.893670
\(320\) 1.28019e11 0.682497
\(321\) 4.70492e10 0.247332
\(322\) −7.50861e10 −0.389232
\(323\) 2.55625e11 1.30675
\(324\) −1.52649e10 −0.0769558
\(325\) −3.51908e11 −1.74966
\(326\) 8.48876e10 0.416260
\(327\) −1.98865e11 −0.961820
\(328\) −1.80788e11 −0.862455
\(329\) 1.90032e11 0.894220
\(330\) 6.61785e10 0.307188
\(331\) 1.79055e11 0.819901 0.409951 0.912108i \(-0.365546\pi\)
0.409951 + 0.912108i \(0.365546\pi\)
\(332\) 5.59717e10 0.252841
\(333\) 5.64748e10 0.251684
\(334\) 5.13525e10 0.225789
\(335\) −4.09544e11 −1.77664
\(336\) −1.71194e10 −0.0732760
\(337\) −3.60414e11 −1.52218 −0.761091 0.648645i \(-0.775336\pi\)
−0.761091 + 0.648645i \(0.775336\pi\)
\(338\) 1.98911e10 0.0828962
\(339\) 8.54131e10 0.351258
\(340\) 3.06690e11 1.24464
\(341\) −7.98857e10 −0.319944
\(342\) 5.78720e10 0.228745
\(343\) −2.75196e11 −1.07354
\(344\) 3.47030e11 1.33615
\(345\) −2.46454e11 −0.936591
\(346\) −1.90907e11 −0.716111
\(347\) −6.49091e10 −0.240338 −0.120169 0.992753i \(-0.538344\pi\)
−0.120169 + 0.992753i \(0.538344\pi\)
\(348\) −1.73414e11 −0.633836
\(349\) −3.82648e11 −1.38066 −0.690328 0.723497i \(-0.742534\pi\)
−0.690328 + 0.723497i \(0.742534\pi\)
\(350\) −2.17528e11 −0.774835
\(351\) 5.04700e10 0.177481
\(352\) −1.67908e11 −0.582947
\(353\) −3.78040e11 −1.29584 −0.647919 0.761709i \(-0.724361\pi\)
−0.647919 + 0.761709i \(0.724361\pi\)
\(354\) −1.23134e10 −0.0416739
\(355\) −4.55979e11 −1.52376
\(356\) −1.96454e11 −0.648240
\(357\) 1.37802e11 0.449001
\(358\) −2.78739e11 −0.896858
\(359\) 1.86227e11 0.591723 0.295861 0.955231i \(-0.404393\pi\)
0.295861 + 0.955231i \(0.404393\pi\)
\(360\) 1.69682e11 0.532446
\(361\) 1.71653e11 0.531948
\(362\) 3.11006e11 0.951876
\(363\) 1.30283e11 0.393829
\(364\) 1.57584e11 0.470496
\(365\) 3.54672e11 1.04595
\(366\) −6.45074e10 −0.187908
\(367\) 3.89606e11 1.12106 0.560529 0.828135i \(-0.310598\pi\)
0.560529 + 0.828135i \(0.310598\pi\)
\(368\) 5.77721e10 0.164211
\(369\) −1.09101e11 −0.306344
\(370\) −2.56878e11 −0.712556
\(371\) −3.26172e11 −0.893850
\(372\) −8.38142e10 −0.226921
\(373\) −2.61650e11 −0.699893 −0.349946 0.936770i \(-0.613800\pi\)
−0.349946 + 0.936770i \(0.613800\pi\)
\(374\) 1.24872e11 0.330022
\(375\) −3.37657e11 −0.881729
\(376\) 4.41526e11 1.13923
\(377\) 5.73355e11 1.46180
\(378\) 3.11975e10 0.0785972
\(379\) −5.94167e11 −1.47922 −0.739608 0.673037i \(-0.764989\pi\)
−0.739608 + 0.673037i \(0.764989\pi\)
\(380\) 5.93094e11 1.45914
\(381\) 2.78142e11 0.676246
\(382\) 2.43526e11 0.585141
\(383\) −7.50953e11 −1.78327 −0.891637 0.452750i \(-0.850443\pi\)
−0.891637 + 0.452750i \(0.850443\pi\)
\(384\) −1.99665e11 −0.468610
\(385\) 3.04738e11 0.706892
\(386\) 1.69730e11 0.389149
\(387\) 2.09424e11 0.474600
\(388\) −2.43575e11 −0.545620
\(389\) 6.24639e11 1.38311 0.691554 0.722325i \(-0.256926\pi\)
0.691554 + 0.722325i \(0.256926\pi\)
\(390\) −2.29565e11 −0.502476
\(391\) −4.65033e11 −1.00621
\(392\) −2.00674e11 −0.429244
\(393\) 2.29823e11 0.485991
\(394\) −2.73320e11 −0.571398
\(395\) −1.71858e11 −0.355209
\(396\) −6.36962e10 −0.130162
\(397\) 5.02778e11 1.01583 0.507913 0.861408i \(-0.330417\pi\)
0.507913 + 0.861408i \(0.330417\pi\)
\(398\) −1.99881e11 −0.399298
\(399\) 2.66488e11 0.526381
\(400\) 1.67369e11 0.326892
\(401\) −8.79008e11 −1.69763 −0.848815 0.528690i \(-0.822683\pi\)
−0.848815 + 0.528690i \(0.822683\pi\)
\(402\) −1.74950e11 −0.334116
\(403\) 2.77113e11 0.523341
\(404\) 3.86365e11 0.721576
\(405\) 1.02399e11 0.189125
\(406\) 3.54414e11 0.647356
\(407\) 2.35654e11 0.425696
\(408\) 3.20173e11 0.572024
\(409\) 3.67932e11 0.650150 0.325075 0.945688i \(-0.394611\pi\)
0.325075 + 0.945688i \(0.394611\pi\)
\(410\) 4.96250e11 0.867307
\(411\) 2.36868e11 0.409466
\(412\) −1.55713e11 −0.266249
\(413\) −5.67006e10 −0.0958987
\(414\) −1.05281e11 −0.176136
\(415\) −3.75466e11 −0.621376
\(416\) 5.82452e11 0.953542
\(417\) −2.19555e11 −0.355575
\(418\) 2.41484e11 0.386897
\(419\) −1.62115e11 −0.256957 −0.128478 0.991712i \(-0.541009\pi\)
−0.128478 + 0.991712i \(0.541009\pi\)
\(420\) 3.19724e11 0.501364
\(421\) −6.72976e11 −1.04407 −0.522035 0.852924i \(-0.674827\pi\)
−0.522035 + 0.852924i \(0.674827\pi\)
\(422\) 3.03667e11 0.466113
\(423\) 2.66450e11 0.404655
\(424\) −7.57840e11 −1.13876
\(425\) −1.34723e12 −2.00304
\(426\) −1.94787e11 −0.286560
\(427\) −2.97042e11 −0.432407
\(428\) 2.05978e11 0.296704
\(429\) 2.10598e11 0.300190
\(430\) −9.52575e11 −1.34367
\(431\) −5.75800e11 −0.803755 −0.401877 0.915693i \(-0.631642\pi\)
−0.401877 + 0.915693i \(0.631642\pi\)
\(432\) −2.40037e10 −0.0331590
\(433\) 1.04209e12 1.42465 0.712326 0.701849i \(-0.247642\pi\)
0.712326 + 0.701849i \(0.247642\pi\)
\(434\) 1.71295e11 0.231761
\(435\) 1.16329e12 1.55770
\(436\) −8.70617e11 −1.15382
\(437\) −8.99306e11 −1.17962
\(438\) 1.51510e11 0.196702
\(439\) −7.00416e11 −0.900048 −0.450024 0.893017i \(-0.648584\pi\)
−0.450024 + 0.893017i \(0.648584\pi\)
\(440\) 7.08039e11 0.900575
\(441\) −1.21102e11 −0.152468
\(442\) −4.33166e11 −0.539826
\(443\) −2.07557e11 −0.256048 −0.128024 0.991771i \(-0.540863\pi\)
−0.128024 + 0.991771i \(0.540863\pi\)
\(444\) 2.47243e11 0.301926
\(445\) 1.31784e12 1.59310
\(446\) 1.30480e11 0.156149
\(447\) 1.33462e11 0.158115
\(448\) 2.51825e11 0.295357
\(449\) −7.72881e11 −0.897437 −0.448718 0.893673i \(-0.648119\pi\)
−0.448718 + 0.893673i \(0.648119\pi\)
\(450\) −3.05004e11 −0.350630
\(451\) −4.55248e11 −0.518148
\(452\) 3.73932e11 0.421376
\(453\) 6.57378e11 0.733455
\(454\) −2.32463e11 −0.256805
\(455\) −1.05710e12 −1.15628
\(456\) 6.19168e11 0.670605
\(457\) 1.64560e11 0.176482 0.0882412 0.996099i \(-0.471875\pi\)
0.0882412 + 0.996099i \(0.471875\pi\)
\(458\) 5.75863e11 0.611540
\(459\) 1.93217e11 0.203183
\(460\) −1.07896e12 −1.12355
\(461\) −1.67596e12 −1.72827 −0.864133 0.503264i \(-0.832132\pi\)
−0.864133 + 0.503264i \(0.832132\pi\)
\(462\) 1.30179e11 0.132939
\(463\) 1.94865e11 0.197070 0.0985348 0.995134i \(-0.468584\pi\)
0.0985348 + 0.995134i \(0.468584\pi\)
\(464\) −2.72690e11 −0.273110
\(465\) 5.62238e11 0.557676
\(466\) −9.99977e11 −0.982321
\(467\) 1.16272e12 1.13122 0.565611 0.824672i \(-0.308641\pi\)
0.565611 + 0.824672i \(0.308641\pi\)
\(468\) 2.20954e11 0.212910
\(469\) −8.05608e11 −0.768858
\(470\) −1.21196e12 −1.14564
\(471\) −1.29728e11 −0.121462
\(472\) −1.31740e11 −0.122174
\(473\) 8.73871e11 0.802735
\(474\) −7.34150e10 −0.0668009
\(475\) −2.60533e12 −2.34824
\(476\) 6.03286e11 0.538632
\(477\) −4.57338e11 −0.404487
\(478\) 2.64517e11 0.231755
\(479\) −2.02118e12 −1.75427 −0.877133 0.480247i \(-0.840547\pi\)
−0.877133 + 0.480247i \(0.840547\pi\)
\(480\) 1.18174e12 1.01610
\(481\) −8.17454e11 −0.696323
\(482\) −1.01107e12 −0.853233
\(483\) −4.84796e11 −0.405319
\(484\) 5.70370e11 0.472446
\(485\) 1.63394e12 1.34090
\(486\) 4.37432e10 0.0355670
\(487\) −1.82880e12 −1.47328 −0.736642 0.676283i \(-0.763590\pi\)
−0.736642 + 0.676283i \(0.763590\pi\)
\(488\) −6.90159e11 −0.550884
\(489\) 5.48080e11 0.433465
\(490\) 5.50838e11 0.431660
\(491\) −3.89531e11 −0.302465 −0.151232 0.988498i \(-0.548324\pi\)
−0.151232 + 0.988498i \(0.548324\pi\)
\(492\) −4.77636e11 −0.367497
\(493\) 2.19500e12 1.67349
\(494\) −8.37678e11 −0.632857
\(495\) 4.27284e11 0.319885
\(496\) −1.31796e11 −0.0977766
\(497\) −8.96950e11 −0.659424
\(498\) −1.60393e11 −0.116857
\(499\) −7.68705e11 −0.555019 −0.277509 0.960723i \(-0.589509\pi\)
−0.277509 + 0.960723i \(0.589509\pi\)
\(500\) −1.47824e12 −1.05774
\(501\) 3.31559e11 0.235121
\(502\) 9.62450e11 0.676412
\(503\) −1.59173e12 −1.10870 −0.554349 0.832285i \(-0.687033\pi\)
−0.554349 + 0.832285i \(0.687033\pi\)
\(504\) 3.33780e11 0.230421
\(505\) −2.59180e12 −1.77333
\(506\) −4.39309e11 −0.297915
\(507\) 1.28428e11 0.0863224
\(508\) 1.21769e12 0.811239
\(509\) 1.48575e12 0.981105 0.490552 0.871412i \(-0.336795\pi\)
0.490552 + 0.871412i \(0.336795\pi\)
\(510\) −8.78855e11 −0.575243
\(511\) 6.97671e11 0.452644
\(512\) −5.28439e11 −0.339844
\(513\) 3.73653e11 0.238199
\(514\) 1.79258e11 0.113277
\(515\) 1.04455e12 0.654329
\(516\) 9.16845e11 0.569341
\(517\) 1.11182e12 0.684429
\(518\) −5.05301e11 −0.308366
\(519\) −1.23260e12 −0.745708
\(520\) −2.45610e12 −1.47309
\(521\) 2.56479e11 0.152505 0.0762523 0.997089i \(-0.475705\pi\)
0.0762523 + 0.997089i \(0.475705\pi\)
\(522\) 4.96936e11 0.292943
\(523\) −4.08961e10 −0.0239015 −0.0119507 0.999929i \(-0.503804\pi\)
−0.0119507 + 0.999929i \(0.503804\pi\)
\(524\) 1.00615e12 0.583005
\(525\) −1.40448e12 −0.806860
\(526\) 1.93053e12 1.09962
\(527\) 1.06089e12 0.599130
\(528\) −1.00161e11 −0.0560849
\(529\) −1.65134e11 −0.0916822
\(530\) 2.08022e12 1.14516
\(531\) −7.95020e10 −0.0433963
\(532\) 1.16667e12 0.631458
\(533\) 1.57920e12 0.847548
\(534\) 5.62960e11 0.299600
\(535\) −1.38173e12 −0.729175
\(536\) −1.87178e12 −0.979519
\(537\) −1.79969e12 −0.933926
\(538\) −2.00178e11 −0.103014
\(539\) −5.05326e11 −0.257883
\(540\) 4.48296e11 0.226879
\(541\) 1.22129e12 0.612957 0.306478 0.951878i \(-0.400849\pi\)
0.306478 + 0.951878i \(0.400849\pi\)
\(542\) 6.20945e11 0.309070
\(543\) 2.00802e12 0.991218
\(544\) 2.22983e12 1.09163
\(545\) 5.84023e12 2.83561
\(546\) −4.51574e11 −0.217451
\(547\) 2.08681e12 0.996645 0.498323 0.866992i \(-0.333949\pi\)
0.498323 + 0.866992i \(0.333949\pi\)
\(548\) 1.03699e12 0.491205
\(549\) −4.16494e11 −0.195674
\(550\) −1.27270e12 −0.593053
\(551\) 4.24481e12 1.96190
\(552\) −1.12639e12 −0.516373
\(553\) −3.38060e11 −0.153720
\(554\) 6.00300e11 0.270754
\(555\) −1.65854e12 −0.742007
\(556\) −9.61195e11 −0.426555
\(557\) 3.43626e12 1.51265 0.756323 0.654199i \(-0.226994\pi\)
0.756323 + 0.654199i \(0.226994\pi\)
\(558\) 2.40179e11 0.104877
\(559\) −3.03135e12 −1.31305
\(560\) 5.02759e11 0.216030
\(561\) 8.06241e11 0.343662
\(562\) −2.35414e12 −0.995450
\(563\) −5.32352e11 −0.223311 −0.111656 0.993747i \(-0.535615\pi\)
−0.111656 + 0.993747i \(0.535615\pi\)
\(564\) 1.16650e12 0.485432
\(565\) −2.50840e12 −1.03557
\(566\) −1.17310e12 −0.480465
\(567\) 2.01428e11 0.0818457
\(568\) −2.08401e12 −0.840101
\(569\) 4.30475e11 0.172164 0.0860820 0.996288i \(-0.472565\pi\)
0.0860820 + 0.996288i \(0.472565\pi\)
\(570\) −1.69957e12 −0.674378
\(571\) 4.07772e12 1.60530 0.802649 0.596452i \(-0.203423\pi\)
0.802649 + 0.596452i \(0.203423\pi\)
\(572\) 9.21981e11 0.360114
\(573\) 1.57234e12 0.609325
\(574\) 9.76166e11 0.375336
\(575\) 4.73963e12 1.80817
\(576\) 3.53093e11 0.133656
\(577\) 2.87140e12 1.07845 0.539227 0.842160i \(-0.318716\pi\)
0.539227 + 0.842160i \(0.318716\pi\)
\(578\) −1.70573e11 −0.0635675
\(579\) 1.09587e12 0.405233
\(580\) 5.09278e12 1.86866
\(581\) −7.38574e11 −0.268907
\(582\) 6.97991e11 0.252172
\(583\) −1.90835e12 −0.684146
\(584\) 1.62099e12 0.576665
\(585\) −1.48219e12 −0.523243
\(586\) 7.55710e11 0.264738
\(587\) 1.97794e12 0.687609 0.343804 0.939041i \(-0.388284\pi\)
0.343804 + 0.939041i \(0.388284\pi\)
\(588\) −5.30176e11 −0.182904
\(589\) 2.05160e12 0.702382
\(590\) 3.61618e11 0.122862
\(591\) −1.76470e12 −0.595014
\(592\) 3.88784e11 0.130095
\(593\) −1.12402e12 −0.373275 −0.186637 0.982429i \(-0.559759\pi\)
−0.186637 + 0.982429i \(0.559759\pi\)
\(594\) 1.82528e11 0.0601578
\(595\) −4.04694e12 −1.32373
\(596\) 5.84287e11 0.189679
\(597\) −1.29054e12 −0.415802
\(598\) 1.52391e12 0.487307
\(599\) 4.59815e12 1.45936 0.729680 0.683789i \(-0.239669\pi\)
0.729680 + 0.683789i \(0.239669\pi\)
\(600\) −3.26321e12 −1.02793
\(601\) −3.40227e12 −1.06374 −0.531868 0.846827i \(-0.678510\pi\)
−0.531868 + 0.846827i \(0.678510\pi\)
\(602\) −1.87380e12 −0.581485
\(603\) −1.12957e12 −0.347925
\(604\) 2.87795e12 0.879868
\(605\) −3.82613e12 −1.16108
\(606\) −1.10717e12 −0.333494
\(607\) −4.76313e12 −1.42411 −0.712055 0.702124i \(-0.752235\pi\)
−0.712055 + 0.702124i \(0.752235\pi\)
\(608\) 4.31216e12 1.27976
\(609\) 2.28828e12 0.674112
\(610\) 1.89444e12 0.553983
\(611\) −3.85678e12 −1.11954
\(612\) 8.45890e11 0.243743
\(613\) −1.30004e12 −0.371863 −0.185932 0.982563i \(-0.559530\pi\)
−0.185932 + 0.982563i \(0.559530\pi\)
\(614\) 3.18287e11 0.0903777
\(615\) 3.20405e12 0.903154
\(616\) 1.39277e12 0.389733
\(617\) 6.20428e12 1.72349 0.861744 0.507343i \(-0.169372\pi\)
0.861744 + 0.507343i \(0.169372\pi\)
\(618\) 4.46213e11 0.123054
\(619\) 4.36576e12 1.19523 0.597615 0.801783i \(-0.296115\pi\)
0.597615 + 0.801783i \(0.296115\pi\)
\(620\) 2.46144e12 0.669001
\(621\) −6.79750e11 −0.183416
\(622\) 2.47183e12 0.662158
\(623\) 2.59231e12 0.689431
\(624\) 3.47446e11 0.0917395
\(625\) 2.67889e12 0.702255
\(626\) −1.30102e11 −0.0338609
\(627\) 1.55915e12 0.402888
\(628\) −5.67939e11 −0.145708
\(629\) −3.12950e12 −0.797163
\(630\) −9.16203e11 −0.231718
\(631\) 4.38403e12 1.10088 0.550441 0.834874i \(-0.314459\pi\)
0.550441 + 0.834874i \(0.314459\pi\)
\(632\) −7.85461e11 −0.195838
\(633\) 1.96063e12 0.485378
\(634\) −3.00227e12 −0.737986
\(635\) −8.16843e12 −1.99369
\(636\) −2.00219e12 −0.485232
\(637\) 1.75291e12 0.421825
\(638\) 2.07358e12 0.495482
\(639\) −1.25765e12 −0.298404
\(640\) 5.86372e12 1.38154
\(641\) −2.65746e12 −0.621734 −0.310867 0.950453i \(-0.600619\pi\)
−0.310867 + 0.950453i \(0.600619\pi\)
\(642\) −5.90252e11 −0.137129
\(643\) −4.60843e12 −1.06317 −0.531586 0.847004i \(-0.678404\pi\)
−0.531586 + 0.847004i \(0.678404\pi\)
\(644\) −2.12240e12 −0.486229
\(645\) −6.15033e12 −1.39920
\(646\) −3.20692e12 −0.724507
\(647\) −1.13960e12 −0.255672 −0.127836 0.991795i \(-0.540803\pi\)
−0.127836 + 0.991795i \(0.540803\pi\)
\(648\) 4.68005e11 0.104271
\(649\) −3.31740e11 −0.0734001
\(650\) 4.41483e12 0.970072
\(651\) 1.10597e12 0.241340
\(652\) 2.39945e12 0.519994
\(653\) −1.92790e12 −0.414929 −0.207465 0.978243i \(-0.566521\pi\)
−0.207465 + 0.978243i \(0.566521\pi\)
\(654\) 2.49485e12 0.533267
\(655\) −6.74941e12 −1.43278
\(656\) −7.51073e11 −0.158349
\(657\) 9.78229e11 0.204831
\(658\) −2.38403e12 −0.495787
\(659\) −6.04821e12 −1.24923 −0.624615 0.780933i \(-0.714744\pi\)
−0.624615 + 0.780933i \(0.714744\pi\)
\(660\) 1.87062e12 0.383740
\(661\) 8.21417e12 1.67362 0.836810 0.547493i \(-0.184418\pi\)
0.836810 + 0.547493i \(0.184418\pi\)
\(662\) −2.24633e12 −0.454582
\(663\) −2.79675e12 −0.562138
\(664\) −1.71603e12 −0.342585
\(665\) −7.82617e12 −1.55186
\(666\) −7.08501e11 −0.139542
\(667\) −7.72217e12 −1.51068
\(668\) 1.45154e12 0.282056
\(669\) 8.42451e11 0.162602
\(670\) 5.13791e12 0.985030
\(671\) −1.73792e12 −0.330961
\(672\) 2.32459e12 0.439728
\(673\) 3.76497e12 0.707447 0.353724 0.935350i \(-0.384915\pi\)
0.353724 + 0.935350i \(0.384915\pi\)
\(674\) 4.52155e12 0.843951
\(675\) −1.96927e12 −0.365122
\(676\) 5.62248e11 0.103554
\(677\) 6.21117e12 1.13638 0.568191 0.822897i \(-0.307644\pi\)
0.568191 + 0.822897i \(0.307644\pi\)
\(678\) −1.07154e12 −0.194749
\(679\) 3.21410e12 0.580290
\(680\) −9.40279e12 −1.68642
\(681\) −1.50091e12 −0.267419
\(682\) 1.00220e12 0.177388
\(683\) 8.19423e12 1.44084 0.720419 0.693539i \(-0.243950\pi\)
0.720419 + 0.693539i \(0.243950\pi\)
\(684\) 1.63583e12 0.285749
\(685\) −6.95630e12 −1.20718
\(686\) 3.45245e12 0.595208
\(687\) 3.71808e12 0.636816
\(688\) 1.44172e12 0.245320
\(689\) 6.61981e12 1.11908
\(690\) 3.09187e12 0.519279
\(691\) −3.72136e12 −0.620942 −0.310471 0.950583i \(-0.600487\pi\)
−0.310471 + 0.950583i \(0.600487\pi\)
\(692\) −5.39623e12 −0.894568
\(693\) 8.40504e11 0.138433
\(694\) 8.14312e11 0.133252
\(695\) 6.44784e12 1.04829
\(696\) 5.31668e12 0.858813
\(697\) 6.04572e12 0.970288
\(698\) 4.80049e12 0.765483
\(699\) −6.45639e12 −1.02292
\(700\) −6.14871e12 −0.967927
\(701\) 3.58404e12 0.560585 0.280293 0.959915i \(-0.409568\pi\)
0.280293 + 0.959915i \(0.409568\pi\)
\(702\) −6.33168e11 −0.0984016
\(703\) −6.05199e12 −0.934542
\(704\) 1.47336e12 0.226064
\(705\) −7.82506e12 −1.19299
\(706\) 4.74267e12 0.718458
\(707\) −5.09828e12 −0.767426
\(708\) −3.48054e11 −0.0520592
\(709\) −1.86082e12 −0.276565 −0.138282 0.990393i \(-0.544158\pi\)
−0.138282 + 0.990393i \(0.544158\pi\)
\(710\) 5.72046e12 0.844828
\(711\) −4.74006e11 −0.0695618
\(712\) 6.02307e12 0.878330
\(713\) −3.73227e12 −0.540842
\(714\) −1.72878e12 −0.248942
\(715\) −6.18479e12 −0.885009
\(716\) −7.87890e12 −1.12036
\(717\) 1.70787e12 0.241334
\(718\) −2.33630e12 −0.328072
\(719\) 8.32527e12 1.16176 0.580882 0.813988i \(-0.302708\pi\)
0.580882 + 0.813988i \(0.302708\pi\)
\(720\) 7.04937e11 0.0977584
\(721\) 2.05471e12 0.283167
\(722\) −2.15346e12 −0.294930
\(723\) −6.52798e12 −0.888498
\(724\) 8.79098e12 1.18909
\(725\) −2.23715e13 −3.00728
\(726\) −1.63446e12 −0.218353
\(727\) 1.05496e13 1.40066 0.700329 0.713821i \(-0.253037\pi\)
0.700329 + 0.713821i \(0.253037\pi\)
\(728\) −4.83135e12 −0.637496
\(729\) 2.82430e11 0.0370370
\(730\) −4.44952e12 −0.579909
\(731\) −1.16051e13 −1.50321
\(732\) −1.82338e12 −0.234735
\(733\) −3.93351e12 −0.503283 −0.251641 0.967821i \(-0.580970\pi\)
−0.251641 + 0.967821i \(0.580970\pi\)
\(734\) −4.88777e12 −0.621554
\(735\) 3.55650e12 0.449501
\(736\) −7.84468e12 −0.985429
\(737\) −4.71340e12 −0.588478
\(738\) 1.36872e12 0.169848
\(739\) −6.24664e12 −0.770454 −0.385227 0.922822i \(-0.625877\pi\)
−0.385227 + 0.922822i \(0.625877\pi\)
\(740\) −7.26098e12 −0.890128
\(741\) −5.40850e12 −0.659014
\(742\) 4.09197e12 0.495581
\(743\) −8.32453e12 −1.00210 −0.501049 0.865419i \(-0.667052\pi\)
−0.501049 + 0.865419i \(0.667052\pi\)
\(744\) 2.56965e12 0.307465
\(745\) −3.91949e12 −0.466151
\(746\) 3.28251e12 0.388045
\(747\) −1.03558e12 −0.121686
\(748\) 3.52967e12 0.412265
\(749\) −2.71798e12 −0.315558
\(750\) 4.23605e12 0.488861
\(751\) 9.51081e12 1.09103 0.545516 0.838100i \(-0.316334\pi\)
0.545516 + 0.838100i \(0.316334\pi\)
\(752\) 1.83430e12 0.209165
\(753\) 6.21409e12 0.704369
\(754\) −7.19298e12 −0.810472
\(755\) −1.93058e13 −2.16235
\(756\) 8.81837e11 0.0981839
\(757\) 4.64714e12 0.514345 0.257172 0.966366i \(-0.417209\pi\)
0.257172 + 0.966366i \(0.417209\pi\)
\(758\) 7.45408e12 0.820129
\(759\) −2.83641e12 −0.310228
\(760\) −1.81836e13 −1.97706
\(761\) 8.45553e12 0.913924 0.456962 0.889486i \(-0.348938\pi\)
0.456962 + 0.889486i \(0.348938\pi\)
\(762\) −3.48941e12 −0.374934
\(763\) 1.14882e13 1.22714
\(764\) 6.88357e12 0.730960
\(765\) −5.67436e12 −0.599018
\(766\) 9.42103e12 0.988710
\(767\) 1.15076e12 0.120063
\(768\) 4.73678e12 0.491312
\(769\) 1.48631e13 1.53264 0.766321 0.642457i \(-0.222085\pi\)
0.766321 + 0.642457i \(0.222085\pi\)
\(770\) −3.82307e12 −0.391926
\(771\) 1.15738e12 0.117959
\(772\) 4.79763e12 0.486126
\(773\) −1.81427e13 −1.82765 −0.913827 0.406103i \(-0.866887\pi\)
−0.913827 + 0.406103i \(0.866887\pi\)
\(774\) −2.62732e12 −0.263135
\(775\) −1.08126e13 −1.07664
\(776\) 7.46775e12 0.739285
\(777\) −3.26249e12 −0.321111
\(778\) −7.83637e12 −0.766843
\(779\) 1.16915e13 1.13750
\(780\) −6.48894e12 −0.627694
\(781\) −5.24782e12 −0.504718
\(782\) 5.83404e12 0.557878
\(783\) 3.20849e12 0.305051
\(784\) −8.33691e11 −0.0788103
\(785\) 3.80982e12 0.358089
\(786\) −2.88323e12 −0.269450
\(787\) 1.05260e13 0.978087 0.489044 0.872259i \(-0.337346\pi\)
0.489044 + 0.872259i \(0.337346\pi\)
\(788\) −7.72573e12 −0.713792
\(789\) 1.24645e13 1.14506
\(790\) 2.15604e12 0.196940
\(791\) −4.93423e12 −0.448152
\(792\) 1.95286e12 0.176363
\(793\) 6.02861e12 0.541362
\(794\) −6.30757e12 −0.563209
\(795\) 1.34310e13 1.19250
\(796\) −5.64988e12 −0.498805
\(797\) 5.90203e12 0.518131 0.259065 0.965860i \(-0.416586\pi\)
0.259065 + 0.965860i \(0.416586\pi\)
\(798\) −3.34321e12 −0.291844
\(799\) −1.47651e13 −1.28167
\(800\) −2.27264e13 −1.96167
\(801\) 3.63477e12 0.311983
\(802\) 1.10275e13 0.941225
\(803\) 4.08188e12 0.346450
\(804\) −4.94519e12 −0.417379
\(805\) 1.42374e13 1.19495
\(806\) −3.47651e12 −0.290158
\(807\) −1.29245e12 −0.107271
\(808\) −1.18455e13 −0.977695
\(809\) 1.36585e13 1.12107 0.560536 0.828130i \(-0.310595\pi\)
0.560536 + 0.828130i \(0.310595\pi\)
\(810\) −1.28464e12 −0.104858
\(811\) −1.76828e13 −1.43535 −0.717675 0.696379i \(-0.754793\pi\)
−0.717675 + 0.696379i \(0.754793\pi\)
\(812\) 1.00179e13 0.808679
\(813\) 4.00915e12 0.321844
\(814\) −2.95638e12 −0.236021
\(815\) −1.60959e13 −1.27793
\(816\) 1.33014e12 0.105025
\(817\) −2.24425e13 −1.76226
\(818\) −4.61587e12 −0.360466
\(819\) −2.91560e12 −0.226439
\(820\) 1.40271e13 1.08344
\(821\) 2.37038e12 0.182085 0.0910423 0.995847i \(-0.470980\pi\)
0.0910423 + 0.995847i \(0.470980\pi\)
\(822\) −2.97161e12 −0.227022
\(823\) −4.92072e12 −0.373878 −0.186939 0.982372i \(-0.559857\pi\)
−0.186939 + 0.982372i \(0.559857\pi\)
\(824\) 4.77400e12 0.360753
\(825\) −8.21723e12 −0.617565
\(826\) 7.11334e11 0.0531696
\(827\) 1.33372e13 0.991496 0.495748 0.868466i \(-0.334894\pi\)
0.495748 + 0.868466i \(0.334894\pi\)
\(828\) −2.97590e12 −0.220030
\(829\) −1.21354e13 −0.892395 −0.446198 0.894934i \(-0.647222\pi\)
−0.446198 + 0.894934i \(0.647222\pi\)
\(830\) 4.71039e12 0.344513
\(831\) 3.87586e12 0.281944
\(832\) −5.11090e12 −0.369779
\(833\) 6.71076e12 0.482913
\(834\) 2.75441e12 0.197143
\(835\) −9.73716e12 −0.693175
\(836\) 6.82585e12 0.483313
\(837\) 1.55072e12 0.109212
\(838\) 2.03380e12 0.142466
\(839\) 1.56293e12 0.108896 0.0544479 0.998517i \(-0.482660\pi\)
0.0544479 + 0.998517i \(0.482660\pi\)
\(840\) −9.80238e12 −0.679321
\(841\) 2.19422e13 1.51251
\(842\) 8.44277e12 0.578869
\(843\) −1.51996e13 −1.03659
\(844\) 8.58352e12 0.582270
\(845\) −3.77164e12 −0.254493
\(846\) −3.34273e12 −0.224355
\(847\) −7.52632e12 −0.502467
\(848\) −3.14841e12 −0.209079
\(849\) −7.57417e12 −0.500323
\(850\) 1.69015e13 1.11056
\(851\) 1.10098e13 0.719608
\(852\) −5.50589e12 −0.357972
\(853\) −1.21011e13 −0.782624 −0.391312 0.920258i \(-0.627979\pi\)
−0.391312 + 0.920258i \(0.627979\pi\)
\(854\) 3.72652e12 0.239742
\(855\) −1.09734e13 −0.702251
\(856\) −6.31506e12 −0.402018
\(857\) −4.10860e12 −0.260184 −0.130092 0.991502i \(-0.541527\pi\)
−0.130092 + 0.991502i \(0.541527\pi\)
\(858\) −2.64204e12 −0.166436
\(859\) 2.13660e13 1.33892 0.669460 0.742848i \(-0.266526\pi\)
0.669460 + 0.742848i \(0.266526\pi\)
\(860\) −2.69257e13 −1.67851
\(861\) 6.30265e12 0.390849
\(862\) 7.22365e12 0.445630
\(863\) −2.75466e13 −1.69052 −0.845259 0.534356i \(-0.820554\pi\)
−0.845259 + 0.534356i \(0.820554\pi\)
\(864\) 3.25939e12 0.198987
\(865\) 3.61988e13 2.19847
\(866\) −1.30734e13 −0.789876
\(867\) −1.10131e12 −0.0661948
\(868\) 4.84186e12 0.289517
\(869\) −1.97790e12 −0.117656
\(870\) −1.45939e13 −0.863646
\(871\) 1.63502e13 0.962589
\(872\) 2.66922e13 1.56336
\(873\) 4.50660e12 0.262594
\(874\) 1.12822e13 0.654021
\(875\) 1.95061e13 1.12495
\(876\) 4.28262e12 0.245720
\(877\) −2.30078e13 −1.31334 −0.656670 0.754178i \(-0.728035\pi\)
−0.656670 + 0.754178i \(0.728035\pi\)
\(878\) 8.78701e12 0.499018
\(879\) 4.87927e12 0.275680
\(880\) 2.94151e12 0.165348
\(881\) −8.63759e12 −0.483060 −0.241530 0.970393i \(-0.577649\pi\)
−0.241530 + 0.970393i \(0.577649\pi\)
\(882\) 1.51928e12 0.0845335
\(883\) 2.79887e13 1.54938 0.774692 0.632338i \(-0.217905\pi\)
0.774692 + 0.632338i \(0.217905\pi\)
\(884\) −1.22440e13 −0.674352
\(885\) 2.33480e12 0.127940
\(886\) 2.60390e12 0.141962
\(887\) −2.43474e13 −1.32068 −0.660338 0.750968i \(-0.729587\pi\)
−0.660338 + 0.750968i \(0.729587\pi\)
\(888\) −7.58019e12 −0.409093
\(889\) −1.60680e13 −0.862787
\(890\) −1.65329e13 −0.883272
\(891\) 1.17850e12 0.0626441
\(892\) 3.68819e12 0.195061
\(893\) −2.85535e13 −1.50255
\(894\) −1.67434e12 −0.0876647
\(895\) 5.28528e13 2.75337
\(896\) 1.15344e13 0.597875
\(897\) 9.83915e12 0.507448
\(898\) 9.69612e12 0.497570
\(899\) 1.76167e13 0.899509
\(900\) −8.62132e12 −0.438009
\(901\) 2.53430e13 1.28114
\(902\) 5.71129e12 0.287279
\(903\) −1.20982e13 −0.605518
\(904\) −1.14644e13 −0.570942
\(905\) −5.89712e13 −2.92228
\(906\) −8.24709e12 −0.406653
\(907\) 2.29403e13 1.12556 0.562778 0.826608i \(-0.309733\pi\)
0.562778 + 0.826608i \(0.309733\pi\)
\(908\) −6.57086e12 −0.320801
\(909\) −7.14849e12 −0.347278
\(910\) 1.32617e13 0.641083
\(911\) 3.68575e13 1.77293 0.886467 0.462792i \(-0.153152\pi\)
0.886467 + 0.462792i \(0.153152\pi\)
\(912\) 2.57230e12 0.123125
\(913\) −4.32120e12 −0.205819
\(914\) −2.06448e12 −0.0978480
\(915\) 1.22315e13 0.576880
\(916\) 1.62775e13 0.763938
\(917\) −1.32767e13 −0.620050
\(918\) −2.42399e12 −0.112652
\(919\) 1.18969e12 0.0550192 0.0275096 0.999622i \(-0.491242\pi\)
0.0275096 + 0.999622i \(0.491242\pi\)
\(920\) 3.30797e13 1.52235
\(921\) 2.05503e12 0.0941131
\(922\) 2.10257e13 0.958210
\(923\) 1.82040e13 0.825581
\(924\) 3.67967e12 0.166067
\(925\) 3.18959e13 1.43251
\(926\) −2.44467e12 −0.109262
\(927\) 2.88099e12 0.128140
\(928\) 3.70277e13 1.63893
\(929\) 3.55367e13 1.56533 0.782667 0.622441i \(-0.213859\pi\)
0.782667 + 0.622441i \(0.213859\pi\)
\(930\) −7.05352e12 −0.309195
\(931\) 1.29776e13 0.566137
\(932\) −2.82656e13 −1.22712
\(933\) 1.59595e13 0.689526
\(934\) −1.45868e13 −0.627189
\(935\) −2.36775e13 −1.01317
\(936\) −6.77421e12 −0.288481
\(937\) −3.70448e13 −1.57000 −0.784999 0.619498i \(-0.787336\pi\)
−0.784999 + 0.619498i \(0.787336\pi\)
\(938\) 1.01067e13 0.426282
\(939\) −8.40005e11 −0.0352604
\(940\) −3.42575e13 −1.43114
\(941\) −4.72046e13 −1.96260 −0.981298 0.192496i \(-0.938342\pi\)
−0.981298 + 0.192496i \(0.938342\pi\)
\(942\) 1.62749e12 0.0673426
\(943\) −2.12693e13 −0.875891
\(944\) −5.47308e11 −0.0224315
\(945\) −5.91550e12 −0.241295
\(946\) −1.09631e13 −0.445064
\(947\) 3.75098e13 1.51555 0.757774 0.652517i \(-0.226287\pi\)
0.757774 + 0.652517i \(0.226287\pi\)
\(948\) −2.07517e12 −0.0834479
\(949\) −1.41595e13 −0.566697
\(950\) 3.26850e13 1.30195
\(951\) −1.93843e13 −0.768488
\(952\) −1.84961e13 −0.729817
\(953\) −1.68178e13 −0.660467 −0.330233 0.943899i \(-0.607127\pi\)
−0.330233 + 0.943899i \(0.607127\pi\)
\(954\) 5.73750e12 0.224262
\(955\) −4.61760e13 −1.79639
\(956\) 7.47692e12 0.289509
\(957\) 1.33881e13 0.515960
\(958\) 2.53566e13 0.972626
\(959\) −1.36836e13 −0.522417
\(960\) −1.03696e13 −0.394040
\(961\) −1.79251e13 −0.677965
\(962\) 1.02553e13 0.386066
\(963\) −3.81099e12 −0.142797
\(964\) −2.85790e13 −1.06586
\(965\) −3.21832e13 −1.19469
\(966\) 6.08197e12 0.224723
\(967\) 1.43412e13 0.527430 0.263715 0.964601i \(-0.415052\pi\)
0.263715 + 0.964601i \(0.415052\pi\)
\(968\) −1.74869e13 −0.640139
\(969\) −2.07056e13 −0.754451
\(970\) −2.04985e13 −0.743444
\(971\) −1.95428e13 −0.705504 −0.352752 0.935717i \(-0.614754\pi\)
−0.352752 + 0.935717i \(0.614754\pi\)
\(972\) 1.23646e12 0.0444304
\(973\) 1.26835e13 0.453659
\(974\) 2.29431e13 0.816840
\(975\) 2.85045e13 1.01017
\(976\) −2.86723e12 −0.101144
\(977\) 1.37309e13 0.482140 0.241070 0.970508i \(-0.422502\pi\)
0.241070 + 0.970508i \(0.422502\pi\)
\(978\) −6.87590e12 −0.240328
\(979\) 1.51669e13 0.527686
\(980\) 1.55701e13 0.539231
\(981\) 1.61081e13 0.555307
\(982\) 4.88683e12 0.167697
\(983\) −3.86404e12 −0.131993 −0.0659965 0.997820i \(-0.521023\pi\)
−0.0659965 + 0.997820i \(0.521023\pi\)
\(984\) 1.46438e13 0.497938
\(985\) 5.18254e13 1.75420
\(986\) −2.75372e13 −0.927843
\(987\) −1.53926e13 −0.516278
\(988\) −2.36780e13 −0.790568
\(989\) 4.08274e13 1.35696
\(990\) −5.36046e12 −0.177355
\(991\) −3.55386e13 −1.17049 −0.585246 0.810855i \(-0.699002\pi\)
−0.585246 + 0.810855i \(0.699002\pi\)
\(992\) 1.78962e13 0.586756
\(993\) −1.45035e13 −0.473370
\(994\) 1.12526e13 0.365608
\(995\) 3.79003e13 1.22585
\(996\) −4.53370e12 −0.145978
\(997\) −4.68278e12 −0.150098 −0.0750491 0.997180i \(-0.523911\pi\)
−0.0750491 + 0.997180i \(0.523911\pi\)
\(998\) 9.64374e12 0.307722
\(999\) −4.57446e12 −0.145310
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.1.8 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.10.a.b.1.8 21 1.1 even 1 trivial