Properties

Label 177.10.a.b.1.17
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 1.17
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+29.8017 q^{2} -81.0000 q^{3} +376.142 q^{4} +2386.14 q^{5} -2413.94 q^{6} +4536.93 q^{7} -4048.81 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q+29.8017 q^{2} -81.0000 q^{3} +376.142 q^{4} +2386.14 q^{5} -2413.94 q^{6} +4536.93 q^{7} -4048.81 q^{8} +6561.00 q^{9} +71111.0 q^{10} -38636.2 q^{11} -30467.5 q^{12} -159530. q^{13} +135208. q^{14} -193277. q^{15} -313246. q^{16} -283077. q^{17} +195529. q^{18} -454290. q^{19} +897527. q^{20} -367491. q^{21} -1.15143e6 q^{22} -2.42107e6 q^{23} +327954. q^{24} +3.74054e6 q^{25} -4.75427e6 q^{26} -531441. q^{27} +1.70653e6 q^{28} +6.69743e6 q^{29} -5.75999e6 q^{30} +4.53852e6 q^{31} -7.26227e6 q^{32} +3.12953e6 q^{33} -8.43617e6 q^{34} +1.08258e7 q^{35} +2.46787e6 q^{36} -1.75422e7 q^{37} -1.35386e7 q^{38} +1.29219e7 q^{39} -9.66103e6 q^{40} -1.25230e7 q^{41} -1.09519e7 q^{42} +23258.2 q^{43} -1.45327e7 q^{44} +1.56555e7 q^{45} -7.21520e7 q^{46} -5.96424e6 q^{47} +2.53729e7 q^{48} -1.97699e7 q^{49} +1.11474e8 q^{50} +2.29292e7 q^{51} -6.00059e7 q^{52} +8.21208e7 q^{53} -1.58378e7 q^{54} -9.21914e7 q^{55} -1.83692e7 q^{56} +3.67975e7 q^{57} +1.99595e8 q^{58} -1.21174e7 q^{59} -7.26997e7 q^{60} -6.81954e7 q^{61} +1.35255e8 q^{62} +2.97668e7 q^{63} -5.60462e7 q^{64} -3.80661e8 q^{65} +9.32654e7 q^{66} -6.03338e7 q^{67} -1.06477e8 q^{68} +1.96107e8 q^{69} +3.22626e8 q^{70} -7.46554e7 q^{71} -2.65642e7 q^{72} +1.87509e8 q^{73} -5.22788e8 q^{74} -3.02984e8 q^{75} -1.70877e8 q^{76} -1.75290e8 q^{77} +3.85096e8 q^{78} -1.18977e8 q^{79} -7.47449e8 q^{80} +4.30467e7 q^{81} -3.73207e8 q^{82} -3.24381e8 q^{83} -1.38229e8 q^{84} -6.75461e8 q^{85} +693134. q^{86} -5.42492e8 q^{87} +1.56431e8 q^{88} -2.07673e7 q^{89} +4.66560e8 q^{90} -7.23777e8 q^{91} -9.10665e8 q^{92} -3.67620e8 q^{93} -1.77745e8 q^{94} -1.08400e9 q^{95} +5.88244e8 q^{96} -6.27270e8 q^{97} -5.89175e8 q^{98} -2.53492e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21q + 20q^{2} - 1701q^{3} + 4950q^{4} + 2058q^{5} - 1620q^{6} - 17167q^{7} - 2853q^{8} + 137781q^{9} + O(q^{10}) \) \( 21q + 20q^{2} - 1701q^{3} + 4950q^{4} + 2058q^{5} - 1620q^{6} - 17167q^{7} - 2853q^{8} + 137781q^{9} - 31559q^{10} - 38751q^{11} - 400950q^{12} - 58915q^{13} + 3453q^{14} - 166698q^{15} + 1655714q^{16} - 64233q^{17} + 131220q^{18} - 1937236q^{19} - 1065507q^{20} + 1390527q^{21} - 5386882q^{22} - 1838574q^{23} + 231093q^{24} + 4565755q^{25} - 839702q^{26} - 11160261q^{27} - 4471034q^{28} + 15658544q^{29} + 2556279q^{30} - 14282802q^{31} - 2205286q^{32} + 3138831q^{33} + 19005532q^{34} - 8633300q^{35} + 32476950q^{36} + 7531195q^{37} + 26649773q^{38} + 4772115q^{39} + 17775672q^{40} + 18338245q^{41} - 279693q^{42} - 22480305q^{43} - 80230922q^{44} + 13502538q^{45} - 83894107q^{46} - 110397260q^{47} - 134112834q^{48} + 130653638q^{49} + 65575693q^{50} + 5202873q^{51} + 177908014q^{52} + 145498338q^{53} - 10628820q^{54} + 86448944q^{55} + 354387888q^{56} + 156916116q^{57} + 115508368q^{58} - 254464581q^{59} + 86306067q^{60} + 287595506q^{61} + 819899030q^{62} - 112632687q^{63} + 822446413q^{64} + 77238206q^{65} + 436337442q^{66} - 392860610q^{67} + 167325073q^{68} + 148924494q^{69} - 424902116q^{70} - 248960491q^{71} - 18718533q^{72} - 758406074q^{73} - 923266846q^{74} - 369826155q^{75} - 2312747568q^{76} - 878126795q^{77} + 68015862q^{78} - 1925801029q^{79} - 1898919861q^{80} + 903981141q^{81} - 3249102191q^{82} - 1650336307q^{83} + 362153754q^{84} - 2342480762q^{85} - 3609864952q^{86} - 1268342064q^{87} - 5987792887q^{88} - 574997526q^{89} - 207058599q^{90} - 4481387117q^{91} - 5317166770q^{92} + 1156906962q^{93} - 5360726568q^{94} - 2789231462q^{95} + 178628166q^{96} - 4651540898q^{97} - 5566652976q^{98} - 254245311q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 29.8017 1.31706 0.658531 0.752554i \(-0.271178\pi\)
0.658531 + 0.752554i \(0.271178\pi\)
\(3\) −81.0000 −0.577350
\(4\) 376.142 0.734652
\(5\) 2386.14 1.70738 0.853691 0.520779i \(-0.174359\pi\)
0.853691 + 0.520779i \(0.174359\pi\)
\(6\) −2413.94 −0.760406
\(7\) 4536.93 0.714202 0.357101 0.934066i \(-0.383765\pi\)
0.357101 + 0.934066i \(0.383765\pi\)
\(8\) −4048.81 −0.349480
\(9\) 6561.00 0.333333
\(10\) 71111.0 2.24873
\(11\) −38636.2 −0.795660 −0.397830 0.917459i \(-0.630237\pi\)
−0.397830 + 0.917459i \(0.630237\pi\)
\(12\) −30467.5 −0.424151
\(13\) −159530. −1.54916 −0.774582 0.632473i \(-0.782040\pi\)
−0.774582 + 0.632473i \(0.782040\pi\)
\(14\) 135208. 0.940648
\(15\) −193277. −0.985758
\(16\) −313246. −1.19494
\(17\) −283077. −0.822023 −0.411011 0.911630i \(-0.634824\pi\)
−0.411011 + 0.911630i \(0.634824\pi\)
\(18\) 195529. 0.439021
\(19\) −454290. −0.799727 −0.399863 0.916575i \(-0.630942\pi\)
−0.399863 + 0.916575i \(0.630942\pi\)
\(20\) 897527. 1.25433
\(21\) −367491. −0.412345
\(22\) −1.15143e6 −1.04793
\(23\) −2.42107e6 −1.80398 −0.901990 0.431757i \(-0.857894\pi\)
−0.901990 + 0.431757i \(0.857894\pi\)
\(24\) 327954. 0.201772
\(25\) 3.74054e6 1.91516
\(26\) −4.75427e6 −2.04034
\(27\) −531441. −0.192450
\(28\) 1.70653e6 0.524690
\(29\) 6.69743e6 1.75840 0.879199 0.476454i \(-0.158078\pi\)
0.879199 + 0.476454i \(0.158078\pi\)
\(30\) −5.75999e6 −1.29830
\(31\) 4.53852e6 0.882645 0.441323 0.897349i \(-0.354509\pi\)
0.441323 + 0.897349i \(0.354509\pi\)
\(32\) −7.26227e6 −1.22433
\(33\) 3.12953e6 0.459374
\(34\) −8.43617e6 −1.08266
\(35\) 1.08258e7 1.21942
\(36\) 2.46787e6 0.244884
\(37\) −1.75422e7 −1.53878 −0.769391 0.638779i \(-0.779440\pi\)
−0.769391 + 0.638779i \(0.779440\pi\)
\(38\) −1.35386e7 −1.05329
\(39\) 1.29219e7 0.894410
\(40\) −9.66103e6 −0.596696
\(41\) −1.25230e7 −0.692119 −0.346060 0.938213i \(-0.612480\pi\)
−0.346060 + 0.938213i \(0.612480\pi\)
\(42\) −1.09519e7 −0.543083
\(43\) 23258.2 0.00103745 0.000518726 1.00000i \(-0.499835\pi\)
0.000518726 1.00000i \(0.499835\pi\)
\(44\) −1.45327e7 −0.584533
\(45\) 1.56555e7 0.569128
\(46\) −7.21520e7 −2.37595
\(47\) −5.96424e6 −0.178285 −0.0891425 0.996019i \(-0.528413\pi\)
−0.0891425 + 0.996019i \(0.528413\pi\)
\(48\) 2.53729e7 0.689898
\(49\) −1.97699e7 −0.489916
\(50\) 1.11474e8 2.52238
\(51\) 2.29292e7 0.474595
\(52\) −6.00059e7 −1.13810
\(53\) 8.21208e7 1.42959 0.714795 0.699334i \(-0.246520\pi\)
0.714795 + 0.699334i \(0.246520\pi\)
\(54\) −1.58378e7 −0.253469
\(55\) −9.21914e7 −1.35850
\(56\) −1.83692e7 −0.249599
\(57\) 3.67975e7 0.461722
\(58\) 1.99595e8 2.31592
\(59\) −1.21174e7 −0.130189
\(60\) −7.26997e7 −0.724189
\(61\) −6.81954e7 −0.630624 −0.315312 0.948988i \(-0.602109\pi\)
−0.315312 + 0.948988i \(0.602109\pi\)
\(62\) 1.35255e8 1.16250
\(63\) 2.97668e7 0.238067
\(64\) −5.60462e7 −0.417577
\(65\) −3.80661e8 −2.64502
\(66\) 9.32654e7 0.605025
\(67\) −6.03338e7 −0.365784 −0.182892 0.983133i \(-0.558546\pi\)
−0.182892 + 0.983133i \(0.558546\pi\)
\(68\) −1.06477e8 −0.603901
\(69\) 1.96107e8 1.04153
\(70\) 3.22626e8 1.60605
\(71\) −7.46554e7 −0.348657 −0.174328 0.984688i \(-0.555775\pi\)
−0.174328 + 0.984688i \(0.555775\pi\)
\(72\) −2.65642e7 −0.116493
\(73\) 1.87509e8 0.772805 0.386402 0.922330i \(-0.373718\pi\)
0.386402 + 0.922330i \(0.373718\pi\)
\(74\) −5.22788e8 −2.02667
\(75\) −3.02984e8 −1.10572
\(76\) −1.70877e8 −0.587520
\(77\) −1.75290e8 −0.568262
\(78\) 3.85096e8 1.17799
\(79\) −1.18977e8 −0.343670 −0.171835 0.985126i \(-0.554970\pi\)
−0.171835 + 0.985126i \(0.554970\pi\)
\(80\) −7.47449e8 −2.04022
\(81\) 4.30467e7 0.111111
\(82\) −3.73207e8 −0.911564
\(83\) −3.24381e8 −0.750246 −0.375123 0.926975i \(-0.622400\pi\)
−0.375123 + 0.926975i \(0.622400\pi\)
\(84\) −1.38229e8 −0.302930
\(85\) −6.75461e8 −1.40351
\(86\) 693134. 0.00136639
\(87\) −5.42492e8 −1.01521
\(88\) 1.56431e8 0.278067
\(89\) −2.07673e7 −0.0350852 −0.0175426 0.999846i \(-0.505584\pi\)
−0.0175426 + 0.999846i \(0.505584\pi\)
\(90\) 4.66560e8 0.749576
\(91\) −7.23777e8 −1.10642
\(92\) −9.10665e8 −1.32530
\(93\) −3.67620e8 −0.509595
\(94\) −1.77745e8 −0.234812
\(95\) −1.08400e9 −1.36544
\(96\) 5.88244e8 0.706866
\(97\) −6.27270e8 −0.719419 −0.359709 0.933064i \(-0.617124\pi\)
−0.359709 + 0.933064i \(0.617124\pi\)
\(98\) −5.89175e8 −0.645249
\(99\) −2.53492e8 −0.265220
\(100\) 1.40697e9 1.40697
\(101\) 1.09342e9 1.04554 0.522771 0.852473i \(-0.324898\pi\)
0.522771 + 0.852473i \(0.324898\pi\)
\(102\) 6.83330e8 0.625071
\(103\) −1.16308e8 −0.101822 −0.0509111 0.998703i \(-0.516213\pi\)
−0.0509111 + 0.998703i \(0.516213\pi\)
\(104\) 6.45907e8 0.541402
\(105\) −8.76886e8 −0.704030
\(106\) 2.44734e9 1.88286
\(107\) −1.88346e9 −1.38908 −0.694542 0.719452i \(-0.744393\pi\)
−0.694542 + 0.719452i \(0.744393\pi\)
\(108\) −1.99897e8 −0.141384
\(109\) 2.51601e9 1.70723 0.853617 0.520902i \(-0.174404\pi\)
0.853617 + 0.520902i \(0.174404\pi\)
\(110\) −2.74746e9 −1.78922
\(111\) 1.42092e9 0.888416
\(112\) −1.42118e9 −0.853428
\(113\) −6.06458e8 −0.349903 −0.174952 0.984577i \(-0.555977\pi\)
−0.174952 + 0.984577i \(0.555977\pi\)
\(114\) 1.09663e9 0.608117
\(115\) −5.77701e9 −3.08008
\(116\) 2.51918e9 1.29181
\(117\) −1.04668e9 −0.516388
\(118\) −3.61118e8 −0.171467
\(119\) −1.28430e9 −0.587090
\(120\) 7.82543e8 0.344503
\(121\) −8.65191e8 −0.366925
\(122\) −2.03234e9 −0.830571
\(123\) 1.01436e9 0.399595
\(124\) 1.70712e9 0.648437
\(125\) 4.26502e9 1.56252
\(126\) 8.87102e8 0.313549
\(127\) 2.93800e9 1.00216 0.501078 0.865402i \(-0.332937\pi\)
0.501078 + 0.865402i \(0.332937\pi\)
\(128\) 2.04801e9 0.674353
\(129\) −1.88391e6 −0.000598973 0
\(130\) −1.13443e10 −3.48365
\(131\) −2.26879e9 −0.673089 −0.336545 0.941668i \(-0.609258\pi\)
−0.336545 + 0.941668i \(0.609258\pi\)
\(132\) 1.17715e9 0.337480
\(133\) −2.06108e9 −0.571166
\(134\) −1.79805e9 −0.481760
\(135\) −1.26809e9 −0.328586
\(136\) 1.14612e9 0.287281
\(137\) 6.51841e9 1.58088 0.790441 0.612538i \(-0.209852\pi\)
0.790441 + 0.612538i \(0.209852\pi\)
\(138\) 5.84431e9 1.37176
\(139\) 5.94315e9 1.35036 0.675181 0.737652i \(-0.264066\pi\)
0.675181 + 0.737652i \(0.264066\pi\)
\(140\) 4.07202e9 0.895846
\(141\) 4.83104e8 0.102933
\(142\) −2.22486e9 −0.459203
\(143\) 6.16364e9 1.23261
\(144\) −2.05521e9 −0.398313
\(145\) 1.59810e10 3.00226
\(146\) 5.58809e9 1.01783
\(147\) 1.60136e9 0.282853
\(148\) −6.59836e9 −1.13047
\(149\) −1.09007e10 −1.81182 −0.905911 0.423469i \(-0.860812\pi\)
−0.905911 + 0.423469i \(0.860812\pi\)
\(150\) −9.02943e9 −1.45630
\(151\) 7.24743e9 1.13446 0.567228 0.823561i \(-0.308016\pi\)
0.567228 + 0.823561i \(0.308016\pi\)
\(152\) 1.83933e9 0.279489
\(153\) −1.85727e9 −0.274008
\(154\) −5.22394e9 −0.748436
\(155\) 1.08295e10 1.50701
\(156\) 4.86048e9 0.657080
\(157\) −1.21328e10 −1.59372 −0.796861 0.604163i \(-0.793507\pi\)
−0.796861 + 0.604163i \(0.793507\pi\)
\(158\) −3.54573e9 −0.452635
\(159\) −6.65179e9 −0.825375
\(160\) −1.73288e10 −2.09040
\(161\) −1.09842e10 −1.28841
\(162\) 1.28287e9 0.146340
\(163\) 1.76217e9 0.195525 0.0977626 0.995210i \(-0.468831\pi\)
0.0977626 + 0.995210i \(0.468831\pi\)
\(164\) −4.71042e9 −0.508466
\(165\) 7.46750e9 0.784328
\(166\) −9.66711e9 −0.988121
\(167\) −1.15642e10 −1.15051 −0.575254 0.817975i \(-0.695097\pi\)
−0.575254 + 0.817975i \(0.695097\pi\)
\(168\) 1.48790e9 0.144106
\(169\) 1.48453e10 1.39991
\(170\) −2.01299e10 −1.84851
\(171\) −2.98059e9 −0.266576
\(172\) 8.74837e6 0.000762165 0
\(173\) −1.84753e10 −1.56814 −0.784069 0.620674i \(-0.786859\pi\)
−0.784069 + 0.620674i \(0.786859\pi\)
\(174\) −1.61672e10 −1.33710
\(175\) 1.69706e10 1.36781
\(176\) 1.21026e10 0.950765
\(177\) 9.81506e8 0.0751646
\(178\) −6.18900e8 −0.0462094
\(179\) −2.58461e9 −0.188173 −0.0940863 0.995564i \(-0.529993\pi\)
−0.0940863 + 0.995564i \(0.529993\pi\)
\(180\) 5.88867e9 0.418111
\(181\) 1.33589e10 0.925162 0.462581 0.886577i \(-0.346923\pi\)
0.462581 + 0.886577i \(0.346923\pi\)
\(182\) −2.15698e10 −1.45722
\(183\) 5.52382e9 0.364091
\(184\) 9.80245e9 0.630455
\(185\) −4.18582e10 −2.62729
\(186\) −1.09557e10 −0.671169
\(187\) 1.09370e10 0.654051
\(188\) −2.24340e9 −0.130977
\(189\) −2.41111e9 −0.137448
\(190\) −3.23050e10 −1.79837
\(191\) 2.64984e10 1.44069 0.720343 0.693618i \(-0.243984\pi\)
0.720343 + 0.693618i \(0.243984\pi\)
\(192\) 4.53974e9 0.241088
\(193\) −1.36817e10 −0.709793 −0.354896 0.934906i \(-0.615484\pi\)
−0.354896 + 0.934906i \(0.615484\pi\)
\(194\) −1.86937e10 −0.947519
\(195\) 3.08335e10 1.52710
\(196\) −7.43627e9 −0.359917
\(197\) −2.46152e9 −0.116441 −0.0582204 0.998304i \(-0.518543\pi\)
−0.0582204 + 0.998304i \(0.518543\pi\)
\(198\) −7.55450e9 −0.349311
\(199\) 1.05143e10 0.475271 0.237635 0.971354i \(-0.423628\pi\)
0.237635 + 0.971354i \(0.423628\pi\)
\(200\) −1.51447e10 −0.669309
\(201\) 4.88704e9 0.211185
\(202\) 3.25858e10 1.37704
\(203\) 3.03858e10 1.25585
\(204\) 8.62463e9 0.348662
\(205\) −2.98816e10 −1.18171
\(206\) −3.46618e9 −0.134106
\(207\) −1.58846e10 −0.601327
\(208\) 4.99721e10 1.85116
\(209\) 1.75520e10 0.636310
\(210\) −2.61327e10 −0.927251
\(211\) −1.77059e10 −0.614961 −0.307481 0.951554i \(-0.599486\pi\)
−0.307481 + 0.951554i \(0.599486\pi\)
\(212\) 3.08891e10 1.05025
\(213\) 6.04708e9 0.201297
\(214\) −5.61302e10 −1.82951
\(215\) 5.54973e7 0.00177133
\(216\) 2.15170e9 0.0672575
\(217\) 2.05909e10 0.630387
\(218\) 7.49813e10 2.24853
\(219\) −1.51882e10 −0.446179
\(220\) −3.46770e10 −0.998021
\(221\) 4.51592e10 1.27345
\(222\) 4.23458e10 1.17010
\(223\) −4.47265e10 −1.21114 −0.605568 0.795794i \(-0.707054\pi\)
−0.605568 + 0.795794i \(0.707054\pi\)
\(224\) −3.29484e10 −0.874417
\(225\) 2.45417e10 0.638385
\(226\) −1.80735e10 −0.460844
\(227\) −7.40431e10 −1.85084 −0.925419 0.378946i \(-0.876287\pi\)
−0.925419 + 0.378946i \(0.876287\pi\)
\(228\) 1.38411e10 0.339205
\(229\) 4.82154e10 1.15858 0.579290 0.815121i \(-0.303330\pi\)
0.579290 + 0.815121i \(0.303330\pi\)
\(230\) −1.72165e11 −4.05666
\(231\) 1.41985e10 0.328086
\(232\) −2.71166e10 −0.614525
\(233\) −1.60111e10 −0.355894 −0.177947 0.984040i \(-0.556946\pi\)
−0.177947 + 0.984040i \(0.556946\pi\)
\(234\) −3.11927e10 −0.680115
\(235\) −1.42315e10 −0.304401
\(236\) −4.55784e9 −0.0956435
\(237\) 9.63716e9 0.198418
\(238\) −3.82743e10 −0.773234
\(239\) −1.89080e10 −0.374848 −0.187424 0.982279i \(-0.560014\pi\)
−0.187424 + 0.982279i \(0.560014\pi\)
\(240\) 6.05433e10 1.17792
\(241\) 2.01485e10 0.384740 0.192370 0.981323i \(-0.438383\pi\)
0.192370 + 0.981323i \(0.438383\pi\)
\(242\) −2.57842e10 −0.483263
\(243\) −3.48678e9 −0.0641500
\(244\) −2.56511e10 −0.463289
\(245\) −4.71736e10 −0.836473
\(246\) 3.02297e10 0.526291
\(247\) 7.24728e10 1.23891
\(248\) −1.83756e10 −0.308467
\(249\) 2.62749e10 0.433155
\(250\) 1.27105e11 2.05794
\(251\) 3.70895e10 0.589819 0.294909 0.955525i \(-0.404711\pi\)
0.294909 + 0.955525i \(0.404711\pi\)
\(252\) 1.11965e10 0.174897
\(253\) 9.35409e10 1.43535
\(254\) 8.75574e10 1.31990
\(255\) 5.47123e10 0.810316
\(256\) 8.97299e10 1.30574
\(257\) 2.31046e10 0.330370 0.165185 0.986263i \(-0.447178\pi\)
0.165185 + 0.986263i \(0.447178\pi\)
\(258\) −5.61438e7 −0.000788884 0
\(259\) −7.95879e10 −1.09900
\(260\) −1.43182e11 −1.94317
\(261\) 4.39419e10 0.586133
\(262\) −6.76137e10 −0.886500
\(263\) −1.11735e11 −1.44009 −0.720043 0.693930i \(-0.755878\pi\)
−0.720043 + 0.693930i \(0.755878\pi\)
\(264\) −1.26709e10 −0.160542
\(265\) 1.95952e11 2.44086
\(266\) −6.14237e10 −0.752261
\(267\) 1.68215e9 0.0202565
\(268\) −2.26941e10 −0.268724
\(269\) 2.85710e10 0.332691 0.166345 0.986068i \(-0.446803\pi\)
0.166345 + 0.986068i \(0.446803\pi\)
\(270\) −3.77913e10 −0.432768
\(271\) 4.59693e10 0.517733 0.258866 0.965913i \(-0.416651\pi\)
0.258866 + 0.965913i \(0.416651\pi\)
\(272\) 8.86726e10 0.982267
\(273\) 5.86259e10 0.638790
\(274\) 1.94260e11 2.08212
\(275\) −1.44520e11 −1.52381
\(276\) 7.37638e10 0.765161
\(277\) −3.17922e10 −0.324460 −0.162230 0.986753i \(-0.551869\pi\)
−0.162230 + 0.986753i \(0.551869\pi\)
\(278\) 1.77116e11 1.77851
\(279\) 2.97772e10 0.294215
\(280\) −4.38314e10 −0.426162
\(281\) 4.11361e9 0.0393590 0.0196795 0.999806i \(-0.493735\pi\)
0.0196795 + 0.999806i \(0.493735\pi\)
\(282\) 1.43973e10 0.135569
\(283\) −6.70679e10 −0.621550 −0.310775 0.950484i \(-0.600588\pi\)
−0.310775 + 0.950484i \(0.600588\pi\)
\(284\) −2.80810e10 −0.256141
\(285\) 8.78039e10 0.788337
\(286\) 1.83687e11 1.62342
\(287\) −5.68160e10 −0.494313
\(288\) −4.76478e10 −0.408109
\(289\) −3.84555e10 −0.324278
\(290\) 4.76261e11 3.95416
\(291\) 5.08089e10 0.415357
\(292\) 7.05300e10 0.567742
\(293\) 2.36249e11 1.87269 0.936344 0.351084i \(-0.114187\pi\)
0.936344 + 0.351084i \(0.114187\pi\)
\(294\) 4.77232e10 0.372535
\(295\) −2.89137e10 −0.222282
\(296\) 7.10251e10 0.537773
\(297\) 2.05329e10 0.153125
\(298\) −3.24859e11 −2.38628
\(299\) 3.86233e11 2.79466
\(300\) −1.13965e11 −0.812316
\(301\) 1.05521e8 0.000740950 0
\(302\) 2.15986e11 1.49415
\(303\) −8.85671e10 −0.603644
\(304\) 1.42304e11 0.955624
\(305\) −1.62724e11 −1.07672
\(306\) −5.53497e10 −0.360885
\(307\) 2.97016e11 1.90835 0.954174 0.299252i \(-0.0967371\pi\)
0.954174 + 0.299252i \(0.0967371\pi\)
\(308\) −6.59338e10 −0.417475
\(309\) 9.42095e9 0.0587870
\(310\) 3.22739e11 1.98483
\(311\) 5.87677e10 0.356219 0.178109 0.984011i \(-0.443002\pi\)
0.178109 + 0.984011i \(0.443002\pi\)
\(312\) −5.23185e10 −0.312579
\(313\) −6.36206e10 −0.374670 −0.187335 0.982296i \(-0.559985\pi\)
−0.187335 + 0.982296i \(0.559985\pi\)
\(314\) −3.61578e11 −2.09903
\(315\) 7.10278e10 0.406472
\(316\) −4.47523e10 −0.252478
\(317\) 4.71059e10 0.262004 0.131002 0.991382i \(-0.458181\pi\)
0.131002 + 0.991382i \(0.458181\pi\)
\(318\) −1.98235e11 −1.08707
\(319\) −2.58763e11 −1.39909
\(320\) −1.33734e11 −0.712963
\(321\) 1.52560e11 0.801988
\(322\) −3.27349e11 −1.69691
\(323\) 1.28599e11 0.657394
\(324\) 1.61917e10 0.0816280
\(325\) −5.96728e11 −2.96689
\(326\) 5.25156e10 0.257519
\(327\) −2.03797e11 −0.985671
\(328\) 5.07032e10 0.241882
\(329\) −2.70594e10 −0.127332
\(330\) 2.22544e11 1.03301
\(331\) 2.06454e11 0.945362 0.472681 0.881234i \(-0.343286\pi\)
0.472681 + 0.881234i \(0.343286\pi\)
\(332\) −1.22013e11 −0.551170
\(333\) −1.15095e11 −0.512927
\(334\) −3.44631e11 −1.51529
\(335\) −1.43965e11 −0.624533
\(336\) 1.15115e11 0.492727
\(337\) −1.31797e11 −0.556636 −0.278318 0.960489i \(-0.589777\pi\)
−0.278318 + 0.960489i \(0.589777\pi\)
\(338\) 4.42416e11 1.84377
\(339\) 4.91231e10 0.202017
\(340\) −2.54069e11 −1.03109
\(341\) −1.75351e11 −0.702285
\(342\) −8.88268e10 −0.351096
\(343\) −2.72776e11 −1.06410
\(344\) −9.41680e7 −0.000362569 0
\(345\) 4.67938e11 1.77829
\(346\) −5.50596e11 −2.06533
\(347\) −9.11544e10 −0.337517 −0.168758 0.985657i \(-0.553976\pi\)
−0.168758 + 0.985657i \(0.553976\pi\)
\(348\) −2.04054e11 −0.745827
\(349\) 1.77044e10 0.0638804 0.0319402 0.999490i \(-0.489831\pi\)
0.0319402 + 0.999490i \(0.489831\pi\)
\(350\) 5.05752e11 1.80149
\(351\) 8.47808e10 0.298137
\(352\) 2.80587e11 0.974148
\(353\) 5.36706e11 1.83972 0.919858 0.392253i \(-0.128304\pi\)
0.919858 + 0.392253i \(0.128304\pi\)
\(354\) 2.92506e10 0.0989964
\(355\) −1.78138e11 −0.595291
\(356\) −7.81143e9 −0.0257754
\(357\) 1.04028e11 0.338957
\(358\) −7.70258e10 −0.247835
\(359\) −2.70489e11 −0.859457 −0.429728 0.902958i \(-0.641391\pi\)
−0.429728 + 0.902958i \(0.641391\pi\)
\(360\) −6.33860e10 −0.198899
\(361\) −1.16309e11 −0.360437
\(362\) 3.98119e11 1.21850
\(363\) 7.00805e10 0.211844
\(364\) −2.72243e11 −0.812830
\(365\) 4.47423e11 1.31947
\(366\) 1.64619e11 0.479530
\(367\) −5.34031e11 −1.53663 −0.768315 0.640072i \(-0.778905\pi\)
−0.768315 + 0.640072i \(0.778905\pi\)
\(368\) 7.58390e11 2.15565
\(369\) −8.21634e10 −0.230706
\(370\) −1.24745e12 −3.46030
\(371\) 3.72576e11 1.02102
\(372\) −1.38277e11 −0.374375
\(373\) −2.50709e11 −0.670625 −0.335313 0.942107i \(-0.608842\pi\)
−0.335313 + 0.942107i \(0.608842\pi\)
\(374\) 3.25942e11 0.861425
\(375\) −3.45467e11 −0.902122
\(376\) 2.41481e10 0.0623071
\(377\) −1.06844e12 −2.72405
\(378\) −7.18552e10 −0.181028
\(379\) −9.62945e10 −0.239731 −0.119866 0.992790i \(-0.538246\pi\)
−0.119866 + 0.992790i \(0.538246\pi\)
\(380\) −4.07737e11 −1.00312
\(381\) −2.37978e11 −0.578595
\(382\) 7.89697e11 1.89747
\(383\) −2.04722e11 −0.486150 −0.243075 0.970007i \(-0.578156\pi\)
−0.243075 + 0.970007i \(0.578156\pi\)
\(384\) −1.65889e11 −0.389338
\(385\) −4.18266e11 −0.970240
\(386\) −4.07737e11 −0.934841
\(387\) 1.52597e8 0.000345817 0
\(388\) −2.35942e11 −0.528522
\(389\) −2.39385e11 −0.530059 −0.265030 0.964240i \(-0.585382\pi\)
−0.265030 + 0.964240i \(0.585382\pi\)
\(390\) 9.18892e11 2.01129
\(391\) 6.85348e11 1.48291
\(392\) 8.00444e10 0.171216
\(393\) 1.83772e11 0.388608
\(394\) −7.33574e10 −0.153360
\(395\) −2.83896e11 −0.586777
\(396\) −9.53490e10 −0.194844
\(397\) 7.87958e11 1.59201 0.796005 0.605290i \(-0.206943\pi\)
0.796005 + 0.605290i \(0.206943\pi\)
\(398\) 3.13344e11 0.625961
\(399\) 1.66948e11 0.329763
\(400\) −1.17171e12 −2.28849
\(401\) 2.39918e11 0.463355 0.231678 0.972793i \(-0.425579\pi\)
0.231678 + 0.972793i \(0.425579\pi\)
\(402\) 1.45642e11 0.278144
\(403\) −7.24030e11 −1.36736
\(404\) 4.11281e11 0.768109
\(405\) 1.02715e11 0.189709
\(406\) 9.05548e11 1.65403
\(407\) 6.77765e11 1.22435
\(408\) −9.28360e10 −0.165862
\(409\) 6.11043e11 1.07973 0.539867 0.841750i \(-0.318474\pi\)
0.539867 + 0.841750i \(0.318474\pi\)
\(410\) −8.90523e11 −1.55639
\(411\) −5.27992e11 −0.912722
\(412\) −4.37483e10 −0.0748038
\(413\) −5.49756e10 −0.0929812
\(414\) −4.73389e11 −0.791984
\(415\) −7.74018e11 −1.28096
\(416\) 1.15855e12 1.89668
\(417\) −4.81395e11 −0.779632
\(418\) 5.23080e11 0.838060
\(419\) −6.22265e11 −0.986308 −0.493154 0.869942i \(-0.664156\pi\)
−0.493154 + 0.869942i \(0.664156\pi\)
\(420\) −3.29833e11 −0.517217
\(421\) −7.96680e11 −1.23599 −0.617994 0.786183i \(-0.712054\pi\)
−0.617994 + 0.786183i \(0.712054\pi\)
\(422\) −5.27667e11 −0.809942
\(423\) −3.91314e10 −0.0594284
\(424\) −3.32492e11 −0.499614
\(425\) −1.05886e12 −1.57430
\(426\) 1.80213e11 0.265121
\(427\) −3.09398e11 −0.450393
\(428\) −7.08446e11 −1.02049
\(429\) −4.99255e11 −0.711646
\(430\) 1.65391e9 0.00233295
\(431\) 1.12004e12 1.56346 0.781729 0.623619i \(-0.214338\pi\)
0.781729 + 0.623619i \(0.214338\pi\)
\(432\) 1.66472e11 0.229966
\(433\) 9.65636e11 1.32013 0.660067 0.751207i \(-0.270528\pi\)
0.660067 + 0.751207i \(0.270528\pi\)
\(434\) 6.13645e11 0.830259
\(435\) −1.29446e12 −1.73336
\(436\) 9.46375e11 1.25422
\(437\) 1.09987e12 1.44269
\(438\) −4.52636e11 −0.587645
\(439\) −6.43221e11 −0.826552 −0.413276 0.910606i \(-0.635615\pi\)
−0.413276 + 0.910606i \(0.635615\pi\)
\(440\) 3.73266e11 0.474767
\(441\) −1.29710e11 −0.163305
\(442\) 1.34582e12 1.67721
\(443\) 5.95044e11 0.734061 0.367030 0.930209i \(-0.380374\pi\)
0.367030 + 0.930209i \(0.380374\pi\)
\(444\) 5.34467e11 0.652676
\(445\) −4.95536e10 −0.0599039
\(446\) −1.33293e12 −1.59514
\(447\) 8.82955e11 1.04606
\(448\) −2.54278e11 −0.298234
\(449\) −7.83154e10 −0.0909366 −0.0454683 0.998966i \(-0.514478\pi\)
−0.0454683 + 0.998966i \(0.514478\pi\)
\(450\) 7.31384e11 0.840793
\(451\) 4.83841e11 0.550691
\(452\) −2.28114e11 −0.257057
\(453\) −5.87042e11 −0.654979
\(454\) −2.20661e12 −2.43767
\(455\) −1.72703e12 −1.88908
\(456\) −1.48986e11 −0.161363
\(457\) −1.38358e11 −0.148382 −0.0741909 0.997244i \(-0.523637\pi\)
−0.0741909 + 0.997244i \(0.523637\pi\)
\(458\) 1.43690e12 1.52592
\(459\) 1.50439e11 0.158198
\(460\) −2.17297e12 −2.26279
\(461\) 1.35849e12 1.40088 0.700442 0.713710i \(-0.252986\pi\)
0.700442 + 0.713710i \(0.252986\pi\)
\(462\) 4.23139e11 0.432110
\(463\) −1.12512e12 −1.13785 −0.568924 0.822390i \(-0.692640\pi\)
−0.568924 + 0.822390i \(0.692640\pi\)
\(464\) −2.09794e12 −2.10118
\(465\) −8.77192e11 −0.870074
\(466\) −4.77159e11 −0.468734
\(467\) −2.51270e11 −0.244464 −0.122232 0.992502i \(-0.539005\pi\)
−0.122232 + 0.992502i \(0.539005\pi\)
\(468\) −3.93699e11 −0.379365
\(469\) −2.73730e11 −0.261243
\(470\) −4.24124e11 −0.400915
\(471\) 9.82756e11 0.920135
\(472\) 4.90609e10 0.0454984
\(473\) −8.98608e8 −0.000825458 0
\(474\) 2.87204e11 0.261329
\(475\) −1.69929e12 −1.53160
\(476\) −4.83079e11 −0.431307
\(477\) 5.38795e11 0.476530
\(478\) −5.63491e11 −0.493698
\(479\) −9.84792e11 −0.854742 −0.427371 0.904076i \(-0.640560\pi\)
−0.427371 + 0.904076i \(0.640560\pi\)
\(480\) 1.40363e12 1.20689
\(481\) 2.79851e12 2.38382
\(482\) 6.00461e11 0.506726
\(483\) 8.89722e11 0.743862
\(484\) −3.25434e11 −0.269562
\(485\) −1.49675e12 −1.22832
\(486\) −1.03912e11 −0.0844896
\(487\) 6.52730e11 0.525839 0.262920 0.964818i \(-0.415315\pi\)
0.262920 + 0.964818i \(0.415315\pi\)
\(488\) 2.76110e11 0.220391
\(489\) −1.42736e11 −0.112887
\(490\) −1.40586e12 −1.10169
\(491\) −1.94676e12 −1.51163 −0.755816 0.654784i \(-0.772760\pi\)
−0.755816 + 0.654784i \(0.772760\pi\)
\(492\) 3.81544e11 0.293563
\(493\) −1.89589e12 −1.44544
\(494\) 2.15981e12 1.63172
\(495\) −6.04868e11 −0.452832
\(496\) −1.42167e12 −1.05471
\(497\) −3.38706e11 −0.249012
\(498\) 7.83036e11 0.570492
\(499\) 1.39758e12 1.00908 0.504540 0.863388i \(-0.331662\pi\)
0.504540 + 0.863388i \(0.331662\pi\)
\(500\) 1.60425e12 1.14791
\(501\) 9.36696e11 0.664246
\(502\) 1.10533e12 0.776828
\(503\) 1.86980e12 1.30238 0.651191 0.758914i \(-0.274270\pi\)
0.651191 + 0.758914i \(0.274270\pi\)
\(504\) −1.20520e11 −0.0831998
\(505\) 2.60906e12 1.78514
\(506\) 2.78768e12 1.89045
\(507\) −1.20247e12 −0.808238
\(508\) 1.10510e12 0.736235
\(509\) 2.58542e12 1.70726 0.853632 0.520876i \(-0.174395\pi\)
0.853632 + 0.520876i \(0.174395\pi\)
\(510\) 1.63052e12 1.06724
\(511\) 8.50716e11 0.551939
\(512\) 1.62552e12 1.04539
\(513\) 2.41428e11 0.153907
\(514\) 6.88558e11 0.435117
\(515\) −2.77527e11 −0.173849
\(516\) −7.08618e8 −0.000440036 0
\(517\) 2.30436e11 0.141854
\(518\) −2.37185e12 −1.44745
\(519\) 1.49650e12 0.905365
\(520\) 1.54122e12 0.924381
\(521\) −6.28243e11 −0.373558 −0.186779 0.982402i \(-0.559805\pi\)
−0.186779 + 0.982402i \(0.559805\pi\)
\(522\) 1.30954e12 0.771973
\(523\) 9.72209e10 0.0568201 0.0284101 0.999596i \(-0.490956\pi\)
0.0284101 + 0.999596i \(0.490956\pi\)
\(524\) −8.53385e11 −0.494486
\(525\) −1.37462e12 −0.789704
\(526\) −3.32989e12 −1.89668
\(527\) −1.28475e12 −0.725555
\(528\) −9.80314e11 −0.548924
\(529\) 4.06042e12 2.25434
\(530\) 5.83970e12 3.21476
\(531\) −7.95020e10 −0.0433963
\(532\) −7.75258e11 −0.419608
\(533\) 1.99779e12 1.07221
\(534\) 5.01309e10 0.0266790
\(535\) −4.49419e12 −2.37170
\(536\) 2.44280e11 0.127834
\(537\) 2.09353e11 0.108642
\(538\) 8.51465e11 0.438174
\(539\) 7.63832e11 0.389806
\(540\) −4.76982e11 −0.241396
\(541\) −2.95963e12 −1.48542 −0.742710 0.669613i \(-0.766460\pi\)
−0.742710 + 0.669613i \(0.766460\pi\)
\(542\) 1.36996e12 0.681886
\(543\) −1.08207e12 −0.534142
\(544\) 2.05578e12 1.00643
\(545\) 6.00355e12 2.91490
\(546\) 1.74715e12 0.841325
\(547\) −7.61194e10 −0.0363540 −0.0181770 0.999835i \(-0.505786\pi\)
−0.0181770 + 0.999835i \(0.505786\pi\)
\(548\) 2.45185e12 1.16140
\(549\) −4.47430e11 −0.210208
\(550\) −4.30695e12 −2.00696
\(551\) −3.04257e12 −1.40624
\(552\) −7.93998e11 −0.363994
\(553\) −5.39792e11 −0.245450
\(554\) −9.47461e11 −0.427334
\(555\) 3.39051e12 1.51687
\(556\) 2.23547e12 0.992046
\(557\) −2.53535e11 −0.111607 −0.0558034 0.998442i \(-0.517772\pi\)
−0.0558034 + 0.998442i \(0.517772\pi\)
\(558\) 8.87411e11 0.387499
\(559\) −3.71038e9 −0.00160718
\(560\) −3.39112e12 −1.45713
\(561\) −8.85898e11 −0.377616
\(562\) 1.22592e11 0.0518383
\(563\) 3.97944e12 1.66930 0.834650 0.550781i \(-0.185670\pi\)
0.834650 + 0.550781i \(0.185670\pi\)
\(564\) 1.81715e11 0.0756199
\(565\) −1.44709e12 −0.597419
\(566\) −1.99874e12 −0.818619
\(567\) 1.95300e11 0.0793558
\(568\) 3.02265e11 0.121849
\(569\) −4.78419e12 −1.91339 −0.956695 0.291093i \(-0.905981\pi\)
−0.956695 + 0.291093i \(0.905981\pi\)
\(570\) 2.61671e12 1.03829
\(571\) −3.60842e12 −1.42054 −0.710272 0.703927i \(-0.751428\pi\)
−0.710272 + 0.703927i \(0.751428\pi\)
\(572\) 2.31840e12 0.905537
\(573\) −2.14637e12 −0.831780
\(574\) −1.69321e12 −0.651040
\(575\) −9.05610e12 −3.45490
\(576\) −3.67719e11 −0.139192
\(577\) −2.72207e12 −1.02237 −0.511185 0.859471i \(-0.670793\pi\)
−0.511185 + 0.859471i \(0.670793\pi\)
\(578\) −1.14604e12 −0.427094
\(579\) 1.10822e12 0.409799
\(580\) 6.01112e12 2.20561
\(581\) −1.47169e12 −0.535827
\(582\) 1.51419e12 0.547050
\(583\) −3.17284e12 −1.13747
\(584\) −7.59189e11 −0.270080
\(585\) −2.49752e12 −0.881672
\(586\) 7.04062e12 2.46645
\(587\) −1.09398e12 −0.380311 −0.190155 0.981754i \(-0.560899\pi\)
−0.190155 + 0.981754i \(0.560899\pi\)
\(588\) 6.02338e11 0.207798
\(589\) −2.06180e12 −0.705875
\(590\) −8.61678e11 −0.292760
\(591\) 1.99383e11 0.0672271
\(592\) 5.49503e12 1.83875
\(593\) −3.44381e12 −1.14365 −0.571825 0.820376i \(-0.693764\pi\)
−0.571825 + 0.820376i \(0.693764\pi\)
\(594\) 6.11914e11 0.201675
\(595\) −3.06452e12 −1.00239
\(596\) −4.10020e12 −1.33106
\(597\) −8.51657e11 −0.274398
\(598\) 1.15104e13 3.68074
\(599\) 5.37863e12 1.70707 0.853535 0.521036i \(-0.174454\pi\)
0.853535 + 0.521036i \(0.174454\pi\)
\(600\) 1.22672e12 0.386426
\(601\) −3.41951e12 −1.06912 −0.534562 0.845129i \(-0.679524\pi\)
−0.534562 + 0.845129i \(0.679524\pi\)
\(602\) 3.14470e9 0.000975877 0
\(603\) −3.95850e11 −0.121928
\(604\) 2.72606e12 0.833431
\(605\) −2.06447e12 −0.626482
\(606\) −2.63945e12 −0.795036
\(607\) −4.54496e12 −1.35888 −0.679440 0.733731i \(-0.737777\pi\)
−0.679440 + 0.733731i \(0.737777\pi\)
\(608\) 3.29917e12 0.979127
\(609\) −2.46125e12 −0.725066
\(610\) −4.84944e12 −1.41810
\(611\) 9.51476e11 0.276193
\(612\) −6.98595e11 −0.201300
\(613\) 4.16258e12 1.19067 0.595334 0.803478i \(-0.297020\pi\)
0.595334 + 0.803478i \(0.297020\pi\)
\(614\) 8.85159e12 2.51341
\(615\) 2.42041e12 0.682262
\(616\) 7.09715e11 0.198596
\(617\) −3.97173e12 −1.10331 −0.551653 0.834074i \(-0.686003\pi\)
−0.551653 + 0.834074i \(0.686003\pi\)
\(618\) 2.80760e11 0.0774261
\(619\) −5.61410e12 −1.53700 −0.768498 0.639853i \(-0.778995\pi\)
−0.768498 + 0.639853i \(0.778995\pi\)
\(620\) 4.07344e12 1.10713
\(621\) 1.28665e12 0.347176
\(622\) 1.75138e12 0.469162
\(623\) −9.42196e10 −0.0250579
\(624\) −4.04774e12 −1.06877
\(625\) 2.87119e12 0.752666
\(626\) −1.89600e12 −0.493463
\(627\) −1.42171e12 −0.367374
\(628\) −4.56365e12 −1.17083
\(629\) 4.96579e12 1.26491
\(630\) 2.11675e12 0.535349
\(631\) −1.35635e12 −0.340596 −0.170298 0.985393i \(-0.554473\pi\)
−0.170298 + 0.985393i \(0.554473\pi\)
\(632\) 4.81717e11 0.120106
\(633\) 1.43418e12 0.355048
\(634\) 1.40384e12 0.345076
\(635\) 7.01048e12 1.71106
\(636\) −2.50201e12 −0.606363
\(637\) 3.15389e12 0.758959
\(638\) −7.71159e12 −1.84268
\(639\) −4.89814e11 −0.116219
\(640\) 4.88684e12 1.15138
\(641\) 7.66190e12 1.79257 0.896283 0.443482i \(-0.146257\pi\)
0.896283 + 0.443482i \(0.146257\pi\)
\(642\) 4.54654e12 1.05627
\(643\) −2.08161e10 −0.00480231 −0.00240115 0.999997i \(-0.500764\pi\)
−0.00240115 + 0.999997i \(0.500764\pi\)
\(644\) −4.13162e12 −0.946530
\(645\) −4.49528e9 −0.00102268
\(646\) 3.83246e12 0.865828
\(647\) 3.70962e12 0.832262 0.416131 0.909305i \(-0.363386\pi\)
0.416131 + 0.909305i \(0.363386\pi\)
\(648\) −1.74288e11 −0.0388311
\(649\) 4.68169e11 0.103586
\(650\) −1.77835e13 −3.90758
\(651\) −1.66787e12 −0.363954
\(652\) 6.62825e11 0.143643
\(653\) 1.71707e12 0.369555 0.184777 0.982780i \(-0.440844\pi\)
0.184777 + 0.982780i \(0.440844\pi\)
\(654\) −6.07349e12 −1.29819
\(655\) −5.41364e12 −1.14922
\(656\) 3.92278e12 0.827040
\(657\) 1.23025e12 0.257602
\(658\) −8.06415e11 −0.167704
\(659\) −8.75824e12 −1.80897 −0.904487 0.426502i \(-0.859746\pi\)
−0.904487 + 0.426502i \(0.859746\pi\)
\(660\) 2.80884e12 0.576208
\(661\) −1.83195e12 −0.373256 −0.186628 0.982431i \(-0.559756\pi\)
−0.186628 + 0.982431i \(0.559756\pi\)
\(662\) 6.15269e12 1.24510
\(663\) −3.65790e12 −0.735226
\(664\) 1.31336e12 0.262196
\(665\) −4.91803e12 −0.975200
\(666\) −3.43001e12 −0.675557
\(667\) −1.62149e13 −3.17212
\(668\) −4.34976e12 −0.845223
\(669\) 3.62285e12 0.699250
\(670\) −4.29040e12 −0.822548
\(671\) 2.63481e12 0.501762
\(672\) 2.66882e12 0.504845
\(673\) 8.48178e12 1.59375 0.796873 0.604147i \(-0.206486\pi\)
0.796873 + 0.604147i \(0.206486\pi\)
\(674\) −3.92778e12 −0.733124
\(675\) −1.98788e12 −0.368572
\(676\) 5.58395e12 1.02845
\(677\) −3.79185e12 −0.693748 −0.346874 0.937912i \(-0.612757\pi\)
−0.346874 + 0.937912i \(0.612757\pi\)
\(678\) 1.46395e12 0.266068
\(679\) −2.84588e12 −0.513810
\(680\) 2.73481e12 0.490498
\(681\) 5.99749e12 1.06858
\(682\) −5.22576e12 −0.924953
\(683\) 6.53274e12 1.14869 0.574344 0.818614i \(-0.305257\pi\)
0.574344 + 0.818614i \(0.305257\pi\)
\(684\) −1.12113e12 −0.195840
\(685\) 1.55538e13 2.69917
\(686\) −8.12919e12 −1.40149
\(687\) −3.90545e12 −0.668907
\(688\) −7.28553e9 −0.00123969
\(689\) −1.31007e13 −2.21467
\(690\) 1.39453e13 2.34211
\(691\) 1.88280e12 0.314162 0.157081 0.987586i \(-0.449792\pi\)
0.157081 + 0.987586i \(0.449792\pi\)
\(692\) −6.94933e12 −1.15204
\(693\) −1.15008e12 −0.189421
\(694\) −2.71656e12 −0.444530
\(695\) 1.41812e13 2.30558
\(696\) 2.19645e12 0.354796
\(697\) 3.54497e12 0.568938
\(698\) 5.27622e11 0.0841344
\(699\) 1.29690e12 0.205475
\(700\) 6.38334e12 1.00486
\(701\) 3.95934e12 0.619287 0.309643 0.950853i \(-0.399790\pi\)
0.309643 + 0.950853i \(0.399790\pi\)
\(702\) 2.52661e12 0.392665
\(703\) 7.96925e12 1.23060
\(704\) 2.16541e12 0.332249
\(705\) 1.15275e12 0.175746
\(706\) 1.59948e13 2.42302
\(707\) 4.96078e12 0.746728
\(708\) 3.69185e11 0.0552198
\(709\) −4.42959e12 −0.658347 −0.329174 0.944269i \(-0.606770\pi\)
−0.329174 + 0.944269i \(0.606770\pi\)
\(710\) −5.30882e12 −0.784035
\(711\) −7.80610e11 −0.114557
\(712\) 8.40827e10 0.0122616
\(713\) −1.09881e13 −1.59227
\(714\) 3.10022e12 0.446427
\(715\) 1.47073e13 2.10453
\(716\) −9.72180e11 −0.138241
\(717\) 1.53155e12 0.216419
\(718\) −8.06103e12 −1.13196
\(719\) −1.03766e13 −1.44803 −0.724014 0.689786i \(-0.757705\pi\)
−0.724014 + 0.689786i \(0.757705\pi\)
\(720\) −4.90401e12 −0.680073
\(721\) −5.27681e11 −0.0727216
\(722\) −3.46620e12 −0.474718
\(723\) −1.63203e12 −0.222129
\(724\) 5.02485e12 0.679672
\(725\) 2.50520e13 3.36761
\(726\) 2.08852e12 0.279012
\(727\) −1.05595e13 −1.40197 −0.700985 0.713176i \(-0.747256\pi\)
−0.700985 + 0.713176i \(0.747256\pi\)
\(728\) 2.93044e12 0.386670
\(729\) 2.82430e11 0.0370370
\(730\) 1.33340e13 1.73783
\(731\) −6.58385e9 −0.000852809 0
\(732\) 2.07774e12 0.267480
\(733\) 7.65386e12 0.979292 0.489646 0.871921i \(-0.337126\pi\)
0.489646 + 0.871921i \(0.337126\pi\)
\(734\) −1.59150e13 −2.02384
\(735\) 3.82107e12 0.482938
\(736\) 1.75825e13 2.20866
\(737\) 2.33107e12 0.291039
\(738\) −2.44861e12 −0.303855
\(739\) −1.13401e13 −1.39867 −0.699335 0.714794i \(-0.746520\pi\)
−0.699335 + 0.714794i \(0.746520\pi\)
\(740\) −1.57446e13 −1.93014
\(741\) −5.87030e12 −0.715284
\(742\) 1.11034e13 1.34474
\(743\) −1.41428e13 −1.70249 −0.851245 0.524768i \(-0.824152\pi\)
−0.851245 + 0.524768i \(0.824152\pi\)
\(744\) 1.48842e12 0.178093
\(745\) −2.60105e13 −3.09347
\(746\) −7.47155e12 −0.883255
\(747\) −2.12826e12 −0.250082
\(748\) 4.11387e12 0.480499
\(749\) −8.54511e12 −0.992086
\(750\) −1.02955e13 −1.18815
\(751\) 2.00665e12 0.230193 0.115097 0.993354i \(-0.463282\pi\)
0.115097 + 0.993354i \(0.463282\pi\)
\(752\) 1.86828e12 0.213040
\(753\) −3.00425e12 −0.340532
\(754\) −3.18414e13 −3.58774
\(755\) 1.72934e13 1.93695
\(756\) −9.06919e11 −0.100977
\(757\) −2.68492e12 −0.297166 −0.148583 0.988900i \(-0.547471\pi\)
−0.148583 + 0.988900i \(0.547471\pi\)
\(758\) −2.86974e12 −0.315741
\(759\) −7.57681e12 −0.828702
\(760\) 4.38890e12 0.477194
\(761\) 1.32884e13 1.43629 0.718145 0.695894i \(-0.244992\pi\)
0.718145 + 0.695894i \(0.244992\pi\)
\(762\) −7.09215e12 −0.762045
\(763\) 1.14150e13 1.21931
\(764\) 9.96715e12 1.05840
\(765\) −4.43170e12 −0.467836
\(766\) −6.10107e12 −0.640290
\(767\) 1.93308e12 0.201684
\(768\) −7.26812e12 −0.753870
\(769\) 9.49050e12 0.978635 0.489317 0.872106i \(-0.337246\pi\)
0.489317 + 0.872106i \(0.337246\pi\)
\(770\) −1.24650e13 −1.27787
\(771\) −1.87148e12 −0.190739
\(772\) −5.14625e12 −0.521450
\(773\) −1.65964e13 −1.67188 −0.835940 0.548821i \(-0.815077\pi\)
−0.835940 + 0.548821i \(0.815077\pi\)
\(774\) 4.54765e9 0.000455463 0
\(775\) 1.69765e13 1.69040
\(776\) 2.53970e12 0.251423
\(777\) 6.44662e12 0.634508
\(778\) −7.13409e12 −0.698121
\(779\) 5.68906e12 0.553506
\(780\) 1.15978e13 1.12189
\(781\) 2.88440e12 0.277412
\(782\) 2.04245e13 1.95309
\(783\) −3.55929e12 −0.338404
\(784\) 6.19283e12 0.585419
\(785\) −2.89505e13 −2.72109
\(786\) 5.47671e12 0.511821
\(787\) −1.57061e13 −1.45943 −0.729715 0.683752i \(-0.760347\pi\)
−0.729715 + 0.683752i \(0.760347\pi\)
\(788\) −9.25879e11 −0.0855434
\(789\) 9.05053e12 0.831434
\(790\) −8.46060e12 −0.772822
\(791\) −2.75146e12 −0.249902
\(792\) 1.02634e12 0.0926891
\(793\) 1.08792e13 0.976940
\(794\) 2.34825e13 2.09678
\(795\) −1.58721e13 −1.40923
\(796\) 3.95486e12 0.349158
\(797\) −9.19907e12 −0.807572 −0.403786 0.914853i \(-0.632306\pi\)
−0.403786 + 0.914853i \(0.632306\pi\)
\(798\) 4.97532e12 0.434318
\(799\) 1.68834e12 0.146554
\(800\) −2.71648e13 −2.34478
\(801\) −1.36254e11 −0.0116951
\(802\) 7.14998e12 0.610267
\(803\) −7.24465e12 −0.614890
\(804\) 1.83822e12 0.155148
\(805\) −2.62099e13 −2.19980
\(806\) −2.15773e13 −1.80090
\(807\) −2.31425e12 −0.192079
\(808\) −4.42705e12 −0.365396
\(809\) −4.51270e12 −0.370397 −0.185199 0.982701i \(-0.559293\pi\)
−0.185199 + 0.982701i \(0.559293\pi\)
\(810\) 3.06110e12 0.249859
\(811\) 1.23066e13 0.998954 0.499477 0.866327i \(-0.333526\pi\)
0.499477 + 0.866327i \(0.333526\pi\)
\(812\) 1.14294e13 0.922614
\(813\) −3.72351e12 −0.298913
\(814\) 2.01986e13 1.61254
\(815\) 4.20478e12 0.333836
\(816\) −7.18248e12 −0.567112
\(817\) −1.05660e10 −0.000829677 0
\(818\) 1.82101e13 1.42208
\(819\) −4.74870e12 −0.368805
\(820\) −1.12397e13 −0.868147
\(821\) 1.38659e13 1.06513 0.532566 0.846389i \(-0.321228\pi\)
0.532566 + 0.846389i \(0.321228\pi\)
\(822\) −1.57350e13 −1.20211
\(823\) 1.32295e13 1.00518 0.502591 0.864525i \(-0.332380\pi\)
0.502591 + 0.864525i \(0.332380\pi\)
\(824\) 4.70909e11 0.0355848
\(825\) 1.17061e13 0.879774
\(826\) −1.63837e12 −0.122462
\(827\) 5.96201e12 0.443218 0.221609 0.975136i \(-0.428869\pi\)
0.221609 + 0.975136i \(0.428869\pi\)
\(828\) −5.97487e12 −0.441766
\(829\) 1.03751e13 0.762954 0.381477 0.924378i \(-0.375415\pi\)
0.381477 + 0.924378i \(0.375415\pi\)
\(830\) −2.30671e13 −1.68710
\(831\) 2.57517e12 0.187327
\(832\) 8.94105e12 0.646895
\(833\) 5.59639e12 0.402722
\(834\) −1.43464e13 −1.02682
\(835\) −2.75937e13 −1.96436
\(836\) 6.60205e12 0.467466
\(837\) −2.41195e12 −0.169865
\(838\) −1.85446e13 −1.29903
\(839\) 9.79768e12 0.682644 0.341322 0.939946i \(-0.389125\pi\)
0.341322 + 0.939946i \(0.389125\pi\)
\(840\) 3.55035e12 0.246045
\(841\) 3.03485e13 2.09197
\(842\) −2.37424e13 −1.62787
\(843\) −3.33202e11 −0.0227239
\(844\) −6.65994e12 −0.451782
\(845\) 3.54230e13 2.39018
\(846\) −1.16618e12 −0.0782708
\(847\) −3.92531e12 −0.262059
\(848\) −2.57240e13 −1.70827
\(849\) 5.43250e12 0.358852
\(850\) −3.15558e13 −2.07345
\(851\) 4.24709e13 2.77593
\(852\) 2.27456e12 0.147883
\(853\) −2.47801e12 −0.160263 −0.0801313 0.996784i \(-0.525534\pi\)
−0.0801313 + 0.996784i \(0.525534\pi\)
\(854\) −9.22058e12 −0.593195
\(855\) −7.11211e12 −0.455146
\(856\) 7.62575e12 0.485457
\(857\) −1.95528e12 −0.123821 −0.0619105 0.998082i \(-0.519719\pi\)
−0.0619105 + 0.998082i \(0.519719\pi\)
\(858\) −1.48786e13 −0.937282
\(859\) −2.05939e13 −1.29054 −0.645268 0.763956i \(-0.723254\pi\)
−0.645268 + 0.763956i \(0.723254\pi\)
\(860\) 2.08748e10 0.00130131
\(861\) 4.60209e12 0.285392
\(862\) 3.33791e13 2.05917
\(863\) 2.06141e13 1.26507 0.632537 0.774530i \(-0.282014\pi\)
0.632537 + 0.774530i \(0.282014\pi\)
\(864\) 3.85947e12 0.235622
\(865\) −4.40847e13 −2.67741
\(866\) 2.87776e13 1.73870
\(867\) 3.11489e12 0.187222
\(868\) 7.74511e12 0.463115
\(869\) 4.59683e12 0.273445
\(870\) −3.85772e13 −2.28294
\(871\) 9.62506e12 0.566659
\(872\) −1.01868e13 −0.596644
\(873\) −4.11552e12 −0.239806
\(874\) 3.27779e13 1.90011
\(875\) 1.93501e13 1.11596
\(876\) −5.71293e12 −0.327786
\(877\) 2.61353e13 1.49186 0.745931 0.666023i \(-0.232005\pi\)
0.745931 + 0.666023i \(0.232005\pi\)
\(878\) −1.91691e13 −1.08862
\(879\) −1.91362e13 −1.08120
\(880\) 2.88786e13 1.62332
\(881\) −1.51843e13 −0.849188 −0.424594 0.905384i \(-0.639583\pi\)
−0.424594 + 0.905384i \(0.639583\pi\)
\(882\) −3.86558e12 −0.215083
\(883\) −2.16816e13 −1.20024 −0.600119 0.799911i \(-0.704880\pi\)
−0.600119 + 0.799911i \(0.704880\pi\)
\(884\) 1.69863e13 0.935541
\(885\) 2.34201e12 0.128335
\(886\) 1.77333e13 0.966804
\(887\) −2.05352e13 −1.11389 −0.556945 0.830550i \(-0.688026\pi\)
−0.556945 + 0.830550i \(0.688026\pi\)
\(888\) −5.75304e12 −0.310484
\(889\) 1.33295e13 0.715741
\(890\) −1.47678e12 −0.0788971
\(891\) −1.66316e12 −0.0884067
\(892\) −1.68235e13 −0.889763
\(893\) 2.70949e12 0.142579
\(894\) 2.63136e13 1.37772
\(895\) −6.16724e12 −0.321283
\(896\) 9.29169e12 0.481625
\(897\) −3.12849e13 −1.61350
\(898\) −2.33393e12 −0.119769
\(899\) 3.03964e13 1.55204
\(900\) 9.23115e12 0.468991
\(901\) −2.32465e13 −1.17516
\(902\) 1.44193e13 0.725295
\(903\) −8.54718e9 −0.000427788 0
\(904\) 2.45543e12 0.122284
\(905\) 3.18762e13 1.57961
\(906\) −1.74949e13 −0.862648
\(907\) −7.71988e12 −0.378772 −0.189386 0.981903i \(-0.560650\pi\)
−0.189386 + 0.981903i \(0.560650\pi\)
\(908\) −2.78507e13 −1.35972
\(909\) 7.17394e12 0.348514
\(910\) −5.14685e13 −2.48803
\(911\) −4.01977e13 −1.93361 −0.966805 0.255516i \(-0.917755\pi\)
−0.966805 + 0.255516i \(0.917755\pi\)
\(912\) −1.15267e13 −0.551730
\(913\) 1.25329e13 0.596941
\(914\) −4.12330e12 −0.195428
\(915\) 1.31806e13 0.621643
\(916\) 1.81358e13 0.851153
\(917\) −1.02933e13 −0.480722
\(918\) 4.48333e12 0.208357
\(919\) −1.82906e13 −0.845879 −0.422939 0.906158i \(-0.639002\pi\)
−0.422939 + 0.906158i \(0.639002\pi\)
\(920\) 2.33900e13 1.07643
\(921\) −2.40583e13 −1.10179
\(922\) 4.04853e13 1.84505
\(923\) 1.19098e13 0.540127
\(924\) 5.34064e12 0.241029
\(925\) −6.56174e13 −2.94701
\(926\) −3.35305e13 −1.49862
\(927\) −7.63097e11 −0.0339407
\(928\) −4.86386e13 −2.15286
\(929\) −7.48341e12 −0.329631 −0.164816 0.986324i \(-0.552703\pi\)
−0.164816 + 0.986324i \(0.552703\pi\)
\(930\) −2.61418e13 −1.14594
\(931\) 8.98124e12 0.391798
\(932\) −6.02245e12 −0.261458
\(933\) −4.76018e12 −0.205663
\(934\) −7.48827e12 −0.321974
\(935\) 2.60972e13 1.11671
\(936\) 4.23780e12 0.180467
\(937\) 2.83293e12 0.120063 0.0600313 0.998196i \(-0.480880\pi\)
0.0600313 + 0.998196i \(0.480880\pi\)
\(938\) −8.15763e12 −0.344074
\(939\) 5.15327e12 0.216316
\(940\) −5.35307e12 −0.223629
\(941\) 2.67700e13 1.11300 0.556500 0.830848i \(-0.312144\pi\)
0.556500 + 0.830848i \(0.312144\pi\)
\(942\) 2.92878e13 1.21187
\(943\) 3.03190e13 1.24857
\(944\) 3.79571e12 0.155568
\(945\) −5.75325e12 −0.234677
\(946\) −2.67801e10 −0.00108718
\(947\) 3.96311e13 1.60126 0.800628 0.599161i \(-0.204499\pi\)
0.800628 + 0.599161i \(0.204499\pi\)
\(948\) 3.62494e12 0.145768
\(949\) −2.99134e13 −1.19720
\(950\) −5.06417e13 −2.01721
\(951\) −3.81558e12 −0.151268
\(952\) 5.19989e12 0.205176
\(953\) 3.16563e13 1.24320 0.621601 0.783334i \(-0.286483\pi\)
0.621601 + 0.783334i \(0.286483\pi\)
\(954\) 1.60570e13 0.627620
\(955\) 6.32289e13 2.45980
\(956\) −7.11209e12 −0.275383
\(957\) 2.09598e13 0.807763
\(958\) −2.93485e13 −1.12575
\(959\) 2.95736e13 1.12907
\(960\) 1.08325e13 0.411630
\(961\) −5.84150e12 −0.220937
\(962\) 8.34004e13 3.13964
\(963\) −1.23574e13 −0.463028
\(964\) 7.57870e12 0.282650
\(965\) −3.26464e13 −1.21189
\(966\) 2.65152e13 0.979712
\(967\) 2.31023e13 0.849642 0.424821 0.905277i \(-0.360337\pi\)
0.424821 + 0.905277i \(0.360337\pi\)
\(968\) 3.50299e12 0.128233
\(969\) −1.04165e13 −0.379546
\(970\) −4.46058e13 −1.61778
\(971\) −2.14504e13 −0.774369 −0.387185 0.922002i \(-0.626552\pi\)
−0.387185 + 0.922002i \(0.626552\pi\)
\(972\) −1.31152e12 −0.0471279
\(973\) 2.69637e13 0.964431
\(974\) 1.94525e13 0.692563
\(975\) 4.83350e13 1.71293
\(976\) 2.13619e13 0.753557
\(977\) −4.35915e13 −1.53065 −0.765325 0.643644i \(-0.777422\pi\)
−0.765325 + 0.643644i \(0.777422\pi\)
\(978\) −4.25376e12 −0.148679
\(979\) 8.02368e11 0.0279159
\(980\) −1.77440e13 −0.614516
\(981\) 1.65075e13 0.569078
\(982\) −5.80168e13 −1.99091
\(983\) 2.79176e13 0.953647 0.476824 0.878999i \(-0.341788\pi\)
0.476824 + 0.878999i \(0.341788\pi\)
\(984\) −4.10696e12 −0.139651
\(985\) −5.87352e12 −0.198809
\(986\) −5.65007e13 −1.90374
\(987\) 2.19181e12 0.0735149
\(988\) 2.72601e13 0.910166
\(989\) −5.63097e10 −0.00187154
\(990\) −1.80261e13 −0.596408
\(991\) 1.63625e13 0.538913 0.269456 0.963013i \(-0.413156\pi\)
0.269456 + 0.963013i \(0.413156\pi\)
\(992\) −3.29599e13 −1.08065
\(993\) −1.67228e13 −0.545805
\(994\) −1.00940e13 −0.327964
\(995\) 2.50886e13 0.811469
\(996\) 9.88307e12 0.318218
\(997\) 1.67058e13 0.535476 0.267738 0.963492i \(-0.413724\pi\)
0.267738 + 0.963492i \(0.413724\pi\)
\(998\) 4.16504e13 1.32902
\(999\) 9.32266e12 0.296139
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.1.17 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.10.a.b.1.17 21 1.1 even 1 trivial