Properties

Label 177.8.a.b.1.12
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(1\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(7.01814\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+5.01814 q^{2} +27.0000 q^{3} -102.818 q^{4} -77.5687 q^{5} +135.490 q^{6} -216.829 q^{7} -1158.28 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+5.01814 q^{2} +27.0000 q^{3} -102.818 q^{4} -77.5687 q^{5} +135.490 q^{6} -216.829 q^{7} -1158.28 q^{8} +729.000 q^{9} -389.251 q^{10} +3376.21 q^{11} -2776.09 q^{12} +12448.6 q^{13} -1088.08 q^{14} -2094.36 q^{15} +7348.33 q^{16} -26362.1 q^{17} +3658.23 q^{18} +41401.4 q^{19} +7975.48 q^{20} -5854.37 q^{21} +16942.3 q^{22} -42118.5 q^{23} -31273.5 q^{24} -72108.1 q^{25} +62468.7 q^{26} +19683.0 q^{27} +22293.9 q^{28} -151717. q^{29} -10509.8 q^{30} -239434. q^{31} +185135. q^{32} +91157.8 q^{33} -132289. q^{34} +16819.1 q^{35} -74954.5 q^{36} -411153. q^{37} +207758. q^{38} +336111. q^{39} +89846.2 q^{40} -536752. q^{41} -29378.1 q^{42} +583229. q^{43} -347136. q^{44} -56547.6 q^{45} -211357. q^{46} -587465. q^{47} +198405. q^{48} -776528. q^{49} -361849. q^{50} -711776. q^{51} -1.27994e6 q^{52} -267710. q^{53} +98772.1 q^{54} -261889. q^{55} +251148. q^{56} +1.11784e6 q^{57} -761339. q^{58} -205379. q^{59} +215338. q^{60} -498824. q^{61} -1.20151e6 q^{62} -158068. q^{63} -11553.8 q^{64} -965620. q^{65} +457443. q^{66} +2.87144e6 q^{67} +2.71050e6 q^{68} -1.13720e6 q^{69} +84400.8 q^{70} +1.08176e6 q^{71} -844385. q^{72} -4746.50 q^{73} -2.06322e6 q^{74} -1.94692e6 q^{75} -4.25682e6 q^{76} -732060. q^{77} +1.68666e6 q^{78} -1.41454e6 q^{79} -570000. q^{80} +531441. q^{81} -2.69350e6 q^{82} +3.72243e6 q^{83} +601936. q^{84} +2.04487e6 q^{85} +2.92672e6 q^{86} -4.09637e6 q^{87} -3.91060e6 q^{88} +1.04746e7 q^{89} -283764. q^{90} -2.69921e6 q^{91} +4.33055e6 q^{92} -6.46471e6 q^{93} -2.94798e6 q^{94} -3.21145e6 q^{95} +4.99864e6 q^{96} -4.51283e6 q^{97} -3.89673e6 q^{98} +2.46126e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q - 32q^{2} + 459q^{3} + 1166q^{4} - 1072q^{5} - 864q^{6} - 2407q^{7} - 6645q^{8} + 12393q^{9} - 6391q^{10} - 8888q^{11} + 31482q^{12} - 12702q^{13} - 17555q^{14} - 28944q^{15} + 139226q^{16} - 36167q^{17} - 23328q^{18} - 71037q^{19} - 274883q^{20} - 64989q^{21} - 325182q^{22} - 269995q^{23} - 179415q^{24} + 97329q^{25} - 336906q^{26} + 334611q^{27} - 901362q^{28} - 543825q^{29} - 172557q^{30} - 633109q^{31} - 837062q^{32} - 239976q^{33} - 529288q^{34} - 287621q^{35} + 850014q^{36} - 867607q^{37} - 1727169q^{38} - 342954q^{39} - 815662q^{40} - 1428939q^{41} - 473985q^{42} - 477060q^{43} - 1667926q^{44} - 781488q^{45} + 5305549q^{46} - 1217849q^{47} + 3759102q^{48} + 4350738q^{49} + 4561369q^{50} - 976509q^{51} + 4175994q^{52} - 3487068q^{53} - 629856q^{54} - 960484q^{55} - 5363196q^{56} - 1917999q^{57} - 3082906q^{58} - 3491443q^{59} - 7421841q^{60} + 998917q^{61} - 5742614q^{62} - 1754703q^{63} + 17531621q^{64} - 6075816q^{65} - 8779914q^{66} - 356026q^{67} - 16149231q^{68} - 7289865q^{69} - 548798q^{70} - 12879428q^{71} - 4844205q^{72} - 6176157q^{73} - 5971906q^{74} + 2627883q^{75} - 17624580q^{76} + 239687q^{77} - 9096462q^{78} - 18886490q^{79} - 70463349q^{80} + 9034497q^{81} - 19351611q^{82} - 22824893q^{83} - 24336774q^{84} - 7973079q^{85} - 27502196q^{86} - 14683275q^{87} - 62527651q^{88} - 30609647q^{89} - 4659039q^{90} - 36301521q^{91} - 41388548q^{92} - 17093943q^{93} + 1010176q^{94} - 29303629q^{95} - 22600674q^{96} - 26249806q^{97} - 93110852q^{98} - 6479352q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.01814 0.443545 0.221773 0.975098i \(-0.428816\pi\)
0.221773 + 0.975098i \(0.428816\pi\)
\(3\) 27.0000 0.577350
\(4\) −102.818 −0.803268
\(5\) −77.5687 −0.277518 −0.138759 0.990326i \(-0.544311\pi\)
−0.138759 + 0.990326i \(0.544311\pi\)
\(6\) 135.490 0.256081
\(7\) −216.829 −0.238932 −0.119466 0.992838i \(-0.538118\pi\)
−0.119466 + 0.992838i \(0.538118\pi\)
\(8\) −1158.28 −0.799831
\(9\) 729.000 0.333333
\(10\) −389.251 −0.123092
\(11\) 3376.21 0.764813 0.382407 0.923994i \(-0.375095\pi\)
0.382407 + 0.923994i \(0.375095\pi\)
\(12\) −2776.09 −0.463767
\(13\) 12448.6 1.57151 0.785756 0.618536i \(-0.212274\pi\)
0.785756 + 0.618536i \(0.212274\pi\)
\(14\) −1088.08 −0.105977
\(15\) −2094.36 −0.160225
\(16\) 7348.33 0.448506
\(17\) −26362.1 −1.30139 −0.650696 0.759338i \(-0.725523\pi\)
−0.650696 + 0.759338i \(0.725523\pi\)
\(18\) 3658.23 0.147848
\(19\) 41401.4 1.38477 0.692384 0.721529i \(-0.256560\pi\)
0.692384 + 0.721529i \(0.256560\pi\)
\(20\) 7975.48 0.222921
\(21\) −5854.37 −0.137947
\(22\) 16942.3 0.339229
\(23\) −42118.5 −0.721815 −0.360907 0.932602i \(-0.617533\pi\)
−0.360907 + 0.932602i \(0.617533\pi\)
\(24\) −31273.5 −0.461783
\(25\) −72108.1 −0.922984
\(26\) 62468.7 0.697037
\(27\) 19683.0 0.192450
\(28\) 22293.9 0.191926
\(29\) −151717. −1.15516 −0.577580 0.816334i \(-0.696003\pi\)
−0.577580 + 0.816334i \(0.696003\pi\)
\(30\) −10509.8 −0.0710672
\(31\) −239434. −1.44351 −0.721754 0.692149i \(-0.756664\pi\)
−0.721754 + 0.692149i \(0.756664\pi\)
\(32\) 185135. 0.998764
\(33\) 91157.8 0.441565
\(34\) −132289. −0.577226
\(35\) 16819.1 0.0663079
\(36\) −74954.5 −0.267756
\(37\) −411153. −1.33444 −0.667218 0.744863i \(-0.732515\pi\)
−0.667218 + 0.744863i \(0.732515\pi\)
\(38\) 207758. 0.614208
\(39\) 336111. 0.907313
\(40\) 89846.2 0.221968
\(41\) −536752. −1.21627 −0.608135 0.793834i \(-0.708082\pi\)
−0.608135 + 0.793834i \(0.708082\pi\)
\(42\) −29378.1 −0.0611858
\(43\) 583229. 1.11866 0.559331 0.828944i \(-0.311058\pi\)
0.559331 + 0.828944i \(0.311058\pi\)
\(44\) −347136. −0.614350
\(45\) −56547.6 −0.0925061
\(46\) −211357. −0.320158
\(47\) −587465. −0.825353 −0.412677 0.910878i \(-0.635406\pi\)
−0.412677 + 0.910878i \(0.635406\pi\)
\(48\) 198405. 0.258945
\(49\) −776528. −0.942912
\(50\) −361849. −0.409385
\(51\) −711776. −0.751359
\(52\) −1.27994e6 −1.26234
\(53\) −267710. −0.247001 −0.123501 0.992344i \(-0.539412\pi\)
−0.123501 + 0.992344i \(0.539412\pi\)
\(54\) 98772.1 0.0853603
\(55\) −261889. −0.212250
\(56\) 251148. 0.191105
\(57\) 1.11784e6 0.799496
\(58\) −761339. −0.512366
\(59\) −205379. −0.130189
\(60\) 215338. 0.128704
\(61\) −498824. −0.281380 −0.140690 0.990054i \(-0.544932\pi\)
−0.140690 + 0.990054i \(0.544932\pi\)
\(62\) −1.20151e6 −0.640261
\(63\) −158068. −0.0796439
\(64\) −11553.8 −0.00550929
\(65\) −965620. −0.436124
\(66\) 457443. 0.195854
\(67\) 2.87144e6 1.16637 0.583187 0.812338i \(-0.301805\pi\)
0.583187 + 0.812338i \(0.301805\pi\)
\(68\) 2.71050e6 1.04537
\(69\) −1.13720e6 −0.416740
\(70\) 84400.8 0.0294106
\(71\) 1.08176e6 0.358697 0.179349 0.983786i \(-0.442601\pi\)
0.179349 + 0.983786i \(0.442601\pi\)
\(72\) −844385. −0.266610
\(73\) −4746.50 −0.00142805 −0.000714025 1.00000i \(-0.500227\pi\)
−0.000714025 1.00000i \(0.500227\pi\)
\(74\) −2.06322e6 −0.591883
\(75\) −1.94692e6 −0.532885
\(76\) −4.25682e6 −1.11234
\(77\) −732060. −0.182738
\(78\) 1.68666e6 0.402435
\(79\) −1.41454e6 −0.322791 −0.161396 0.986890i \(-0.551600\pi\)
−0.161396 + 0.986890i \(0.551600\pi\)
\(80\) −570000. −0.124469
\(81\) 531441. 0.111111
\(82\) −2.69350e6 −0.539471
\(83\) 3.72243e6 0.714583 0.357292 0.933993i \(-0.383700\pi\)
0.357292 + 0.933993i \(0.383700\pi\)
\(84\) 601936. 0.110809
\(85\) 2.04487e6 0.361160
\(86\) 2.92672e6 0.496178
\(87\) −4.09637e6 −0.666932
\(88\) −3.91060e6 −0.611721
\(89\) 1.04746e7 1.57497 0.787484 0.616336i \(-0.211384\pi\)
0.787484 + 0.616336i \(0.211384\pi\)
\(90\) −283764. −0.0410307
\(91\) −2.69921e6 −0.375484
\(92\) 4.33055e6 0.579810
\(93\) −6.46471e6 −0.833410
\(94\) −2.94798e6 −0.366081
\(95\) −3.21145e6 −0.384299
\(96\) 4.99864e6 0.576637
\(97\) −4.51283e6 −0.502051 −0.251025 0.967980i \(-0.580768\pi\)
−0.251025 + 0.967980i \(0.580768\pi\)
\(98\) −3.89673e6 −0.418224
\(99\) 2.46126e6 0.254938
\(100\) 7.41403e6 0.741403
\(101\) −1.70633e7 −1.64793 −0.823963 0.566644i \(-0.808242\pi\)
−0.823963 + 0.566644i \(0.808242\pi\)
\(102\) −3.57179e6 −0.333262
\(103\) −4.18892e6 −0.377721 −0.188861 0.982004i \(-0.560479\pi\)
−0.188861 + 0.982004i \(0.560479\pi\)
\(104\) −1.44189e7 −1.25694
\(105\) 454116. 0.0382829
\(106\) −1.34341e6 −0.109556
\(107\) −1.20271e7 −0.949116 −0.474558 0.880224i \(-0.657392\pi\)
−0.474558 + 0.880224i \(0.657392\pi\)
\(108\) −2.02377e6 −0.154589
\(109\) −1.96024e7 −1.44983 −0.724913 0.688840i \(-0.758120\pi\)
−0.724913 + 0.688840i \(0.758120\pi\)
\(110\) −1.31419e6 −0.0941424
\(111\) −1.11011e7 −0.770437
\(112\) −1.59333e6 −0.107162
\(113\) −7.54916e6 −0.492180 −0.246090 0.969247i \(-0.579146\pi\)
−0.246090 + 0.969247i \(0.579146\pi\)
\(114\) 5.60947e6 0.354613
\(115\) 3.26708e6 0.200317
\(116\) 1.55993e7 0.927903
\(117\) 9.07501e6 0.523837
\(118\) −1.03062e6 −0.0577447
\(119\) 5.71605e6 0.310944
\(120\) 2.42585e6 0.128153
\(121\) −8.08835e6 −0.415060
\(122\) −2.50317e6 −0.124805
\(123\) −1.44923e7 −0.702214
\(124\) 2.46182e7 1.15952
\(125\) 1.16534e7 0.533663
\(126\) −793208. −0.0353257
\(127\) −1.36062e7 −0.589417 −0.294709 0.955587i \(-0.595223\pi\)
−0.294709 + 0.955587i \(0.595223\pi\)
\(128\) −2.37552e7 −1.00121
\(129\) 1.57472e7 0.645860
\(130\) −4.84562e6 −0.193441
\(131\) −1.29419e7 −0.502978 −0.251489 0.967860i \(-0.580920\pi\)
−0.251489 + 0.967860i \(0.580920\pi\)
\(132\) −9.37268e6 −0.354695
\(133\) −8.97700e6 −0.330865
\(134\) 1.44093e7 0.517340
\(135\) −1.52679e6 −0.0534084
\(136\) 3.05346e7 1.04089
\(137\) −7.65865e6 −0.254466 −0.127233 0.991873i \(-0.540610\pi\)
−0.127233 + 0.991873i \(0.540610\pi\)
\(138\) −5.70663e6 −0.184843
\(139\) 4.87660e6 0.154016 0.0770080 0.997030i \(-0.475463\pi\)
0.0770080 + 0.997030i \(0.475463\pi\)
\(140\) −1.72931e6 −0.0532630
\(141\) −1.58616e7 −0.476518
\(142\) 5.42844e6 0.159098
\(143\) 4.20290e7 1.20191
\(144\) 5.35693e6 0.149502
\(145\) 1.17685e7 0.320578
\(146\) −23818.6 −0.000633405 0
\(147\) −2.09663e7 −0.544390
\(148\) 4.22740e7 1.07191
\(149\) 1.05059e6 0.0260186 0.0130093 0.999915i \(-0.495859\pi\)
0.0130093 + 0.999915i \(0.495859\pi\)
\(150\) −9.76991e6 −0.236359
\(151\) 5.81661e7 1.37484 0.687418 0.726262i \(-0.258744\pi\)
0.687418 + 0.726262i \(0.258744\pi\)
\(152\) −4.79543e7 −1.10758
\(153\) −1.92179e7 −0.433797
\(154\) −3.67358e6 −0.0810526
\(155\) 1.85726e7 0.400600
\(156\) −3.45584e7 −0.728815
\(157\) 4.08109e7 0.841642 0.420821 0.907144i \(-0.361742\pi\)
0.420821 + 0.907144i \(0.361742\pi\)
\(158\) −7.09838e6 −0.143173
\(159\) −7.22818e6 −0.142606
\(160\) −1.43607e7 −0.277175
\(161\) 9.13250e6 0.172464
\(162\) 2.66685e6 0.0492828
\(163\) −7.46918e7 −1.35088 −0.675439 0.737416i \(-0.736046\pi\)
−0.675439 + 0.737416i \(0.736046\pi\)
\(164\) 5.51879e7 0.976990
\(165\) −7.07099e6 −0.122542
\(166\) 1.86797e7 0.316950
\(167\) −5.71426e7 −0.949407 −0.474704 0.880146i \(-0.657445\pi\)
−0.474704 + 0.880146i \(0.657445\pi\)
\(168\) 6.78100e6 0.110334
\(169\) 9.22184e7 1.46965
\(170\) 1.02615e7 0.160191
\(171\) 3.01816e7 0.461589
\(172\) −5.99665e7 −0.898585
\(173\) 6.74021e7 0.989720 0.494860 0.868973i \(-0.335219\pi\)
0.494860 + 0.868973i \(0.335219\pi\)
\(174\) −2.05562e7 −0.295815
\(175\) 1.56351e7 0.220530
\(176\) 2.48095e7 0.343024
\(177\) −5.54523e6 −0.0751646
\(178\) 5.25629e7 0.698569
\(179\) 5.40298e7 0.704122 0.352061 0.935977i \(-0.385481\pi\)
0.352061 + 0.935977i \(0.385481\pi\)
\(180\) 5.81413e6 0.0743072
\(181\) −2.98190e7 −0.373782 −0.186891 0.982381i \(-0.559841\pi\)
−0.186891 + 0.982381i \(0.559841\pi\)
\(182\) −1.35450e7 −0.166544
\(183\) −1.34683e7 −0.162455
\(184\) 4.87850e7 0.577330
\(185\) 3.18926e7 0.370330
\(186\) −3.24408e7 −0.369655
\(187\) −8.90040e7 −0.995322
\(188\) 6.04022e7 0.662979
\(189\) −4.26784e6 −0.0459824
\(190\) −1.61155e7 −0.170454
\(191\) −1.03129e8 −1.07093 −0.535466 0.844557i \(-0.679864\pi\)
−0.535466 + 0.844557i \(0.679864\pi\)
\(192\) −311953. −0.00318079
\(193\) 1.72283e8 1.72501 0.862506 0.506047i \(-0.168894\pi\)
0.862506 + 0.506047i \(0.168894\pi\)
\(194\) −2.26460e7 −0.222682
\(195\) −2.60717e7 −0.251796
\(196\) 7.98413e7 0.757410
\(197\) −1.27602e8 −1.18912 −0.594562 0.804050i \(-0.702674\pi\)
−0.594562 + 0.804050i \(0.702674\pi\)
\(198\) 1.23510e7 0.113076
\(199\) −6.47395e7 −0.582349 −0.291175 0.956670i \(-0.594046\pi\)
−0.291175 + 0.956670i \(0.594046\pi\)
\(200\) 8.35213e7 0.738231
\(201\) 7.75289e7 0.673406
\(202\) −8.56260e7 −0.730930
\(203\) 3.28967e7 0.276004
\(204\) 7.31835e7 0.603542
\(205\) 4.16351e7 0.337537
\(206\) −2.10206e7 −0.167536
\(207\) −3.07044e7 −0.240605
\(208\) 9.14762e7 0.704833
\(209\) 1.39780e8 1.05909
\(210\) 2.27882e6 0.0169802
\(211\) 1.42567e8 1.04479 0.522395 0.852704i \(-0.325039\pi\)
0.522395 + 0.852704i \(0.325039\pi\)
\(212\) 2.75255e7 0.198408
\(213\) 2.92076e7 0.207094
\(214\) −6.03539e7 −0.420976
\(215\) −4.52403e7 −0.310449
\(216\) −2.27984e7 −0.153928
\(217\) 5.19161e7 0.344900
\(218\) −9.83675e7 −0.643064
\(219\) −128155. −0.000824485 0
\(220\) 2.69269e7 0.170493
\(221\) −3.28170e8 −2.04515
\(222\) −5.57071e7 −0.341724
\(223\) 2.35480e8 1.42196 0.710979 0.703213i \(-0.248252\pi\)
0.710979 + 0.703213i \(0.248252\pi\)
\(224\) −4.01425e7 −0.238636
\(225\) −5.25668e7 −0.307661
\(226\) −3.78828e7 −0.218304
\(227\) 3.35151e8 1.90174 0.950868 0.309598i \(-0.100194\pi\)
0.950868 + 0.309598i \(0.100194\pi\)
\(228\) −1.14934e8 −0.642209
\(229\) −1.68231e8 −0.925727 −0.462864 0.886430i \(-0.653178\pi\)
−0.462864 + 0.886430i \(0.653178\pi\)
\(230\) 1.63947e7 0.0888496
\(231\) −1.97656e7 −0.105504
\(232\) 1.75731e8 0.923933
\(233\) −2.76156e8 −1.43024 −0.715120 0.699002i \(-0.753628\pi\)
−0.715120 + 0.699002i \(0.753628\pi\)
\(234\) 4.55397e7 0.232346
\(235\) 4.55689e7 0.229051
\(236\) 2.11167e7 0.104577
\(237\) −3.81927e7 −0.186364
\(238\) 2.86840e7 0.137918
\(239\) 1.64969e8 0.781647 0.390824 0.920466i \(-0.372190\pi\)
0.390824 + 0.920466i \(0.372190\pi\)
\(240\) −1.53900e7 −0.0718621
\(241\) 2.19804e8 1.01152 0.505761 0.862674i \(-0.331212\pi\)
0.505761 + 0.862674i \(0.331212\pi\)
\(242\) −4.05885e7 −0.184098
\(243\) 1.43489e7 0.0641500
\(244\) 5.12882e7 0.226023
\(245\) 6.02343e7 0.261675
\(246\) −7.27244e7 −0.311464
\(247\) 5.15388e8 2.17618
\(248\) 2.77331e8 1.15456
\(249\) 1.00506e8 0.412565
\(250\) 5.84784e7 0.236704
\(251\) −2.64722e8 −1.05665 −0.528327 0.849041i \(-0.677180\pi\)
−0.528327 + 0.849041i \(0.677180\pi\)
\(252\) 1.62523e7 0.0639753
\(253\) −1.42201e8 −0.552054
\(254\) −6.82777e7 −0.261433
\(255\) 5.52116e7 0.208516
\(256\) −1.17728e8 −0.438572
\(257\) −2.39775e8 −0.881125 −0.440563 0.897722i \(-0.645221\pi\)
−0.440563 + 0.897722i \(0.645221\pi\)
\(258\) 7.90215e7 0.286468
\(259\) 8.91498e7 0.318839
\(260\) 9.92834e7 0.350324
\(261\) −1.10602e8 −0.385054
\(262\) −6.49444e7 −0.223094
\(263\) 1.27661e8 0.432727 0.216364 0.976313i \(-0.430580\pi\)
0.216364 + 0.976313i \(0.430580\pi\)
\(264\) −1.05586e8 −0.353178
\(265\) 2.07659e7 0.0685474
\(266\) −4.50479e7 −0.146754
\(267\) 2.82814e8 0.909308
\(268\) −2.95236e8 −0.936910
\(269\) 1.98210e8 0.620858 0.310429 0.950597i \(-0.399527\pi\)
0.310429 + 0.950597i \(0.399527\pi\)
\(270\) −7.66163e6 −0.0236891
\(271\) 1.17159e8 0.357587 0.178794 0.983887i \(-0.442781\pi\)
0.178794 + 0.983887i \(0.442781\pi\)
\(272\) −1.93717e8 −0.583683
\(273\) −7.28786e7 −0.216786
\(274\) −3.84322e7 −0.112867
\(275\) −2.43452e8 −0.705910
\(276\) 1.16925e8 0.334754
\(277\) 2.80002e8 0.791555 0.395778 0.918346i \(-0.370475\pi\)
0.395778 + 0.918346i \(0.370475\pi\)
\(278\) 2.44715e7 0.0683131
\(279\) −1.74547e8 −0.481169
\(280\) −1.94812e7 −0.0530351
\(281\) −6.16699e7 −0.165806 −0.0829032 0.996558i \(-0.526419\pi\)
−0.0829032 + 0.996558i \(0.526419\pi\)
\(282\) −7.95956e7 −0.211357
\(283\) −6.20410e8 −1.62715 −0.813573 0.581463i \(-0.802481\pi\)
−0.813573 + 0.581463i \(0.802481\pi\)
\(284\) −1.11225e8 −0.288130
\(285\) −8.67092e7 −0.221875
\(286\) 2.10908e8 0.533103
\(287\) 1.16383e8 0.290605
\(288\) 1.34963e8 0.332921
\(289\) 2.84620e8 0.693622
\(290\) 5.90561e7 0.142191
\(291\) −1.21846e8 −0.289859
\(292\) 488027. 0.00114711
\(293\) −7.47133e8 −1.73525 −0.867623 0.497222i \(-0.834353\pi\)
−0.867623 + 0.497222i \(0.834353\pi\)
\(294\) −1.05212e8 −0.241462
\(295\) 1.59310e7 0.0361298
\(296\) 4.76230e8 1.06732
\(297\) 6.64540e7 0.147188
\(298\) 5.27203e6 0.0115404
\(299\) −5.24316e8 −1.13434
\(300\) 2.00179e8 0.428049
\(301\) −1.26461e8 −0.267284
\(302\) 2.91886e8 0.609802
\(303\) −4.60708e8 −0.951430
\(304\) 3.04231e8 0.621077
\(305\) 3.86932e7 0.0780881
\(306\) −9.64384e7 −0.192409
\(307\) 6.94608e8 1.37011 0.685056 0.728491i \(-0.259778\pi\)
0.685056 + 0.728491i \(0.259778\pi\)
\(308\) 7.52691e7 0.146788
\(309\) −1.13101e8 −0.218077
\(310\) 9.31998e7 0.177684
\(311\) 1.27745e8 0.240815 0.120407 0.992725i \(-0.461580\pi\)
0.120407 + 0.992725i \(0.461580\pi\)
\(312\) −3.89311e8 −0.725697
\(313\) −9.74255e8 −1.79584 −0.897920 0.440159i \(-0.854922\pi\)
−0.897920 + 0.440159i \(0.854922\pi\)
\(314\) 2.04795e8 0.373306
\(315\) 1.22611e7 0.0221026
\(316\) 1.45441e8 0.259288
\(317\) −8.99019e8 −1.58512 −0.792559 0.609795i \(-0.791252\pi\)
−0.792559 + 0.609795i \(0.791252\pi\)
\(318\) −3.62720e7 −0.0632524
\(319\) −5.12230e8 −0.883482
\(320\) 896215. 0.00152893
\(321\) −3.24733e8 −0.547972
\(322\) 4.58282e7 0.0764957
\(323\) −1.09143e9 −1.80213
\(324\) −5.46418e7 −0.0892519
\(325\) −8.97643e8 −1.45048
\(326\) −3.74814e8 −0.599176
\(327\) −5.29264e8 −0.837058
\(328\) 6.21708e8 0.972810
\(329\) 1.27379e8 0.197203
\(330\) −3.54833e7 −0.0543531
\(331\) −1.16605e9 −1.76733 −0.883666 0.468119i \(-0.844932\pi\)
−0.883666 + 0.468119i \(0.844932\pi\)
\(332\) −3.82733e8 −0.574002
\(333\) −2.99731e8 −0.444812
\(334\) −2.86750e8 −0.421105
\(335\) −2.22734e8 −0.323690
\(336\) −4.30198e7 −0.0618702
\(337\) 1.23311e9 1.75509 0.877544 0.479496i \(-0.159181\pi\)
0.877544 + 0.479496i \(0.159181\pi\)
\(338\) 4.62765e8 0.651857
\(339\) −2.03827e8 −0.284160
\(340\) −2.10250e8 −0.290108
\(341\) −8.08379e8 −1.10401
\(342\) 1.51456e8 0.204736
\(343\) 3.46941e8 0.464223
\(344\) −6.75541e8 −0.894741
\(345\) 8.82112e7 0.115653
\(346\) 3.38233e8 0.438986
\(347\) −5.89436e8 −0.757327 −0.378664 0.925534i \(-0.623616\pi\)
−0.378664 + 0.925534i \(0.623616\pi\)
\(348\) 4.21182e8 0.535725
\(349\) 1.15878e9 1.45919 0.729596 0.683878i \(-0.239708\pi\)
0.729596 + 0.683878i \(0.239708\pi\)
\(350\) 7.84592e7 0.0978150
\(351\) 2.45025e8 0.302438
\(352\) 6.25054e8 0.763868
\(353\) −5.26080e8 −0.636561 −0.318281 0.947997i \(-0.603105\pi\)
−0.318281 + 0.947997i \(0.603105\pi\)
\(354\) −2.78268e7 −0.0333389
\(355\) −8.39110e7 −0.0995450
\(356\) −1.07698e9 −1.26512
\(357\) 1.54333e8 0.179523
\(358\) 2.71129e8 0.312310
\(359\) 1.09995e9 1.25470 0.627351 0.778737i \(-0.284139\pi\)
0.627351 + 0.778737i \(0.284139\pi\)
\(360\) 6.54979e7 0.0739893
\(361\) 8.20202e8 0.917583
\(362\) −1.49636e8 −0.165789
\(363\) −2.18385e8 −0.239635
\(364\) 2.77528e8 0.301614
\(365\) 368180. 0.000396310 0
\(366\) −6.75856e7 −0.0720561
\(367\) 1.21755e9 1.28575 0.642874 0.765972i \(-0.277742\pi\)
0.642874 + 0.765972i \(0.277742\pi\)
\(368\) −3.09501e8 −0.323738
\(369\) −3.91292e8 −0.405423
\(370\) 1.60042e8 0.164258
\(371\) 5.80473e7 0.0590164
\(372\) 6.64690e8 0.669451
\(373\) 5.53595e8 0.552346 0.276173 0.961108i \(-0.410934\pi\)
0.276173 + 0.961108i \(0.410934\pi\)
\(374\) −4.46635e8 −0.441471
\(375\) 3.14642e8 0.308111
\(376\) 6.80449e8 0.660143
\(377\) −1.88866e9 −1.81535
\(378\) −2.14166e7 −0.0203953
\(379\) −2.47132e8 −0.233180 −0.116590 0.993180i \(-0.537196\pi\)
−0.116590 + 0.993180i \(0.537196\pi\)
\(380\) 3.30196e8 0.308695
\(381\) −3.67367e8 −0.340300
\(382\) −5.17514e8 −0.475007
\(383\) −599846. −0.000545562 0 −0.000272781 1.00000i \(-0.500087\pi\)
−0.000272781 1.00000i \(0.500087\pi\)
\(384\) −6.41391e8 −0.578047
\(385\) 5.67850e7 0.0507132
\(386\) 8.64541e8 0.765121
\(387\) 4.25174e8 0.372888
\(388\) 4.64001e8 0.403281
\(389\) −8.92101e8 −0.768406 −0.384203 0.923249i \(-0.625524\pi\)
−0.384203 + 0.923249i \(0.625524\pi\)
\(390\) −1.30832e8 −0.111683
\(391\) 1.11033e9 0.939364
\(392\) 8.99436e8 0.754170
\(393\) −3.49432e8 −0.290395
\(394\) −6.40327e8 −0.527430
\(395\) 1.09724e8 0.0895805
\(396\) −2.53062e8 −0.204783
\(397\) −9.58791e8 −0.769055 −0.384527 0.923114i \(-0.625636\pi\)
−0.384527 + 0.923114i \(0.625636\pi\)
\(398\) −3.24872e8 −0.258298
\(399\) −2.42379e8 −0.191025
\(400\) −5.29874e8 −0.413964
\(401\) 1.82830e9 1.41593 0.707966 0.706247i \(-0.249613\pi\)
0.707966 + 0.706247i \(0.249613\pi\)
\(402\) 3.89051e8 0.298686
\(403\) −2.98061e9 −2.26849
\(404\) 1.75442e9 1.32373
\(405\) −4.12232e7 −0.0308354
\(406\) 1.65080e8 0.122420
\(407\) −1.38814e9 −1.02059
\(408\) 8.24435e8 0.600960
\(409\) −7.15047e8 −0.516776 −0.258388 0.966041i \(-0.583191\pi\)
−0.258388 + 0.966041i \(0.583191\pi\)
\(410\) 2.08931e8 0.149713
\(411\) −2.06784e8 −0.146916
\(412\) 4.30697e8 0.303411
\(413\) 4.45320e7 0.0311062
\(414\) −1.54079e8 −0.106719
\(415\) −2.88744e8 −0.198310
\(416\) 2.30466e9 1.56957
\(417\) 1.31668e8 0.0889212
\(418\) 7.01435e8 0.469754
\(419\) 2.22875e9 1.48017 0.740086 0.672513i \(-0.234785\pi\)
0.740086 + 0.672513i \(0.234785\pi\)
\(420\) −4.66914e7 −0.0307514
\(421\) 1.49109e9 0.973906 0.486953 0.873428i \(-0.338108\pi\)
0.486953 + 0.873428i \(0.338108\pi\)
\(422\) 7.15419e8 0.463412
\(423\) −4.28262e8 −0.275118
\(424\) 3.10083e8 0.197559
\(425\) 1.90092e9 1.20116
\(426\) 1.46568e8 0.0918555
\(427\) 1.08159e8 0.0672306
\(428\) 1.23661e9 0.762394
\(429\) 1.13478e9 0.693925
\(430\) −2.27022e8 −0.137698
\(431\) 2.66273e9 1.60198 0.800988 0.598680i \(-0.204308\pi\)
0.800988 + 0.598680i \(0.204308\pi\)
\(432\) 1.44637e8 0.0863151
\(433\) 7.09435e7 0.0419957 0.0209979 0.999780i \(-0.493316\pi\)
0.0209979 + 0.999780i \(0.493316\pi\)
\(434\) 2.60522e8 0.152979
\(435\) 3.17750e8 0.185086
\(436\) 2.01548e9 1.16460
\(437\) −1.74376e9 −0.999546
\(438\) −643103. −0.000365697 0
\(439\) 6.26733e7 0.0353555 0.0176777 0.999844i \(-0.494373\pi\)
0.0176777 + 0.999844i \(0.494373\pi\)
\(440\) 3.03340e8 0.169764
\(441\) −5.66089e8 −0.314304
\(442\) −1.64680e9 −0.907119
\(443\) −6.41984e8 −0.350842 −0.175421 0.984494i \(-0.556129\pi\)
−0.175421 + 0.984494i \(0.556129\pi\)
\(444\) 1.14140e9 0.618867
\(445\) −8.12500e8 −0.437082
\(446\) 1.18167e9 0.630703
\(447\) 2.83661e7 0.0150218
\(448\) 2.50520e6 0.00131634
\(449\) −2.77507e9 −1.44681 −0.723406 0.690423i \(-0.757425\pi\)
−0.723406 + 0.690423i \(0.757425\pi\)
\(450\) −2.63788e8 −0.136462
\(451\) −1.81219e9 −0.930219
\(452\) 7.76191e8 0.395352
\(453\) 1.57048e9 0.793762
\(454\) 1.68184e9 0.843506
\(455\) 2.09374e8 0.104204
\(456\) −1.29477e9 −0.639462
\(457\) 3.79393e9 1.85944 0.929721 0.368266i \(-0.120048\pi\)
0.929721 + 0.368266i \(0.120048\pi\)
\(458\) −8.44209e8 −0.410602
\(459\) −5.18885e8 −0.250453
\(460\) −3.35916e8 −0.160908
\(461\) −3.74187e9 −1.77884 −0.889418 0.457095i \(-0.848890\pi\)
−0.889418 + 0.457095i \(0.848890\pi\)
\(462\) −9.91867e7 −0.0467958
\(463\) 1.68652e9 0.789691 0.394846 0.918748i \(-0.370798\pi\)
0.394846 + 0.918748i \(0.370798\pi\)
\(464\) −1.11487e9 −0.518097
\(465\) 5.01459e8 0.231287
\(466\) −1.38579e9 −0.634376
\(467\) −1.90310e8 −0.0864675 −0.0432338 0.999065i \(-0.513766\pi\)
−0.0432338 + 0.999065i \(0.513766\pi\)
\(468\) −9.33077e8 −0.420782
\(469\) −6.22610e8 −0.278684
\(470\) 2.28671e8 0.101594
\(471\) 1.10189e9 0.485922
\(472\) 2.37886e8 0.104129
\(473\) 1.96910e9 0.855568
\(474\) −1.91656e8 −0.0826607
\(475\) −2.98537e9 −1.27812
\(476\) −5.87714e8 −0.249771
\(477\) −1.95161e8 −0.0823338
\(478\) 8.27840e8 0.346696
\(479\) 1.04486e9 0.434395 0.217197 0.976128i \(-0.430308\pi\)
0.217197 + 0.976128i \(0.430308\pi\)
\(480\) −3.87738e8 −0.160027
\(481\) −5.11827e9 −2.09708
\(482\) 1.10301e9 0.448656
\(483\) 2.46578e8 0.0995723
\(484\) 8.31630e8 0.333404
\(485\) 3.50054e8 0.139328
\(486\) 7.20049e7 0.0284534
\(487\) −4.52457e9 −1.77511 −0.887557 0.460697i \(-0.847599\pi\)
−0.887557 + 0.460697i \(0.847599\pi\)
\(488\) 5.77778e8 0.225056
\(489\) −2.01668e9 −0.779930
\(490\) 3.02264e8 0.116065
\(491\) 1.28164e9 0.488630 0.244315 0.969696i \(-0.421437\pi\)
0.244315 + 0.969696i \(0.421437\pi\)
\(492\) 1.49007e9 0.564065
\(493\) 3.99958e9 1.50332
\(494\) 2.58629e9 0.965235
\(495\) −1.90917e8 −0.0707499
\(496\) −1.75944e9 −0.647423
\(497\) −2.34557e8 −0.0857041
\(498\) 5.04351e8 0.182991
\(499\) −3.26234e9 −1.17538 −0.587689 0.809087i \(-0.699962\pi\)
−0.587689 + 0.809087i \(0.699962\pi\)
\(500\) −1.19818e9 −0.428674
\(501\) −1.54285e9 −0.548140
\(502\) −1.32841e9 −0.468674
\(503\) 5.11806e8 0.179315 0.0896577 0.995973i \(-0.471423\pi\)
0.0896577 + 0.995973i \(0.471423\pi\)
\(504\) 1.83087e8 0.0637016
\(505\) 1.32358e9 0.457330
\(506\) −7.13586e8 −0.244861
\(507\) 2.48990e9 0.848504
\(508\) 1.39896e9 0.473460
\(509\) 5.38317e9 1.80936 0.904682 0.426088i \(-0.140108\pi\)
0.904682 + 0.426088i \(0.140108\pi\)
\(510\) 2.77059e8 0.0924863
\(511\) 1.02918e6 0.000341206 0
\(512\) 2.44989e9 0.806681
\(513\) 8.14903e8 0.266499
\(514\) −1.20322e9 −0.390819
\(515\) 3.24929e8 0.104825
\(516\) −1.61910e9 −0.518799
\(517\) −1.98341e9 −0.631241
\(518\) 4.47366e8 0.141419
\(519\) 1.81986e9 0.571415
\(520\) 1.11846e9 0.348825
\(521\) 1.92620e9 0.596720 0.298360 0.954453i \(-0.403560\pi\)
0.298360 + 0.954453i \(0.403560\pi\)
\(522\) −5.55016e8 −0.170789
\(523\) 3.47164e9 1.06115 0.530577 0.847637i \(-0.321975\pi\)
0.530577 + 0.847637i \(0.321975\pi\)
\(524\) 1.33067e9 0.404026
\(525\) 4.22148e8 0.127323
\(526\) 6.40623e8 0.191934
\(527\) 6.31197e9 1.87857
\(528\) 6.69857e8 0.198045
\(529\) −1.63086e9 −0.478984
\(530\) 1.04206e8 0.0304039
\(531\) −1.49721e8 −0.0433963
\(532\) 9.23000e8 0.265773
\(533\) −6.68179e9 −1.91138
\(534\) 1.41920e9 0.403319
\(535\) 9.32930e8 0.263397
\(536\) −3.32593e9 −0.932902
\(537\) 1.45880e9 0.406525
\(538\) 9.94645e8 0.275379
\(539\) −2.62173e9 −0.721152
\(540\) 1.56981e8 0.0429013
\(541\) −1.13960e9 −0.309430 −0.154715 0.987959i \(-0.549446\pi\)
−0.154715 + 0.987959i \(0.549446\pi\)
\(542\) 5.87919e8 0.158606
\(543\) −8.05114e8 −0.215803
\(544\) −4.88053e9 −1.29978
\(545\) 1.52053e9 0.402353
\(546\) −3.65715e8 −0.0961543
\(547\) −3.40030e9 −0.888303 −0.444152 0.895952i \(-0.646495\pi\)
−0.444152 + 0.895952i \(0.646495\pi\)
\(548\) 7.87449e8 0.204405
\(549\) −3.63643e8 −0.0937933
\(550\) −1.22168e9 −0.313103
\(551\) −6.28131e9 −1.59963
\(552\) 1.31719e9 0.333321
\(553\) 3.06714e8 0.0771250
\(554\) 1.40509e9 0.351091
\(555\) 8.61101e8 0.213810
\(556\) −5.01404e8 −0.123716
\(557\) 1.94268e9 0.476331 0.238165 0.971225i \(-0.423454\pi\)
0.238165 + 0.971225i \(0.423454\pi\)
\(558\) −8.75903e8 −0.213420
\(559\) 7.26036e9 1.75799
\(560\) 1.23592e8 0.0297395
\(561\) −2.40311e9 −0.574650
\(562\) −3.09468e8 −0.0735427
\(563\) 6.16858e9 1.45682 0.728410 0.685142i \(-0.240260\pi\)
0.728410 + 0.685142i \(0.240260\pi\)
\(564\) 1.63086e9 0.382771
\(565\) 5.85579e8 0.136589
\(566\) −3.11331e9 −0.721713
\(567\) −1.15232e8 −0.0265480
\(568\) −1.25298e9 −0.286897
\(569\) −5.43476e9 −1.23677 −0.618383 0.785877i \(-0.712212\pi\)
−0.618383 + 0.785877i \(0.712212\pi\)
\(570\) −4.35119e8 −0.0984116
\(571\) −5.20959e9 −1.17105 −0.585527 0.810653i \(-0.699112\pi\)
−0.585527 + 0.810653i \(0.699112\pi\)
\(572\) −4.32135e9 −0.965458
\(573\) −2.78447e9 −0.618303
\(574\) 5.84027e8 0.128897
\(575\) 3.03709e9 0.666223
\(576\) −8.42274e6 −0.00183643
\(577\) 2.67350e9 0.579381 0.289691 0.957120i \(-0.406448\pi\)
0.289691 + 0.957120i \(0.406448\pi\)
\(578\) 1.42826e9 0.307653
\(579\) 4.65164e9 0.995936
\(580\) −1.21002e9 −0.257510
\(581\) −8.07129e8 −0.170737
\(582\) −6.11443e8 −0.128566
\(583\) −9.03847e8 −0.188910
\(584\) 5.49777e6 0.00114220
\(585\) −7.03937e8 −0.145375
\(586\) −3.74922e9 −0.769660
\(587\) 4.23836e9 0.864898 0.432449 0.901658i \(-0.357650\pi\)
0.432449 + 0.901658i \(0.357650\pi\)
\(588\) 2.15571e9 0.437291
\(589\) −9.91288e9 −1.99892
\(590\) 7.99440e7 0.0160252
\(591\) −3.44526e9 −0.686541
\(592\) −3.02129e9 −0.598503
\(593\) −5.59276e9 −1.10137 −0.550687 0.834712i \(-0.685634\pi\)
−0.550687 + 0.834712i \(0.685634\pi\)
\(594\) 3.33476e8 0.0652847
\(595\) −4.43387e8 −0.0862926
\(596\) −1.08020e8 −0.0208999
\(597\) −1.74797e9 −0.336220
\(598\) −2.63109e9 −0.503132
\(599\) −2.76072e9 −0.524842 −0.262421 0.964953i \(-0.584521\pi\)
−0.262421 + 0.964953i \(0.584521\pi\)
\(600\) 2.25507e9 0.426218
\(601\) −3.38464e9 −0.635992 −0.317996 0.948092i \(-0.603010\pi\)
−0.317996 + 0.948092i \(0.603010\pi\)
\(602\) −6.34598e8 −0.118552
\(603\) 2.09328e9 0.388791
\(604\) −5.98054e9 −1.10436
\(605\) 6.27403e8 0.115187
\(606\) −2.31190e9 −0.422003
\(607\) 3.57949e9 0.649622 0.324811 0.945779i \(-0.394699\pi\)
0.324811 + 0.945779i \(0.394699\pi\)
\(608\) 7.66483e9 1.38306
\(609\) 8.88210e8 0.159351
\(610\) 1.94168e8 0.0346356
\(611\) −7.31310e9 −1.29705
\(612\) 1.97596e9 0.348455
\(613\) 9.92114e9 1.73960 0.869801 0.493403i \(-0.164247\pi\)
0.869801 + 0.493403i \(0.164247\pi\)
\(614\) 3.48564e9 0.607706
\(615\) 1.12415e9 0.194877
\(616\) 8.47929e8 0.146160
\(617\) −2.67237e9 −0.458034 −0.229017 0.973422i \(-0.573551\pi\)
−0.229017 + 0.973422i \(0.573551\pi\)
\(618\) −5.67556e8 −0.0967272
\(619\) 5.77104e9 0.977996 0.488998 0.872285i \(-0.337363\pi\)
0.488998 + 0.872285i \(0.337363\pi\)
\(620\) −1.90960e9 −0.321789
\(621\) −8.29019e8 −0.138913
\(622\) 6.41044e8 0.106812
\(623\) −2.27119e9 −0.376309
\(624\) 2.46986e9 0.406936
\(625\) 4.72951e9 0.774882
\(626\) −4.88895e9 −0.796536
\(627\) 3.77406e9 0.611466
\(628\) −4.19610e9 −0.676064
\(629\) 1.08388e10 1.73662
\(630\) 6.15282e7 0.00980352
\(631\) −1.33017e9 −0.210768 −0.105384 0.994432i \(-0.533607\pi\)
−0.105384 + 0.994432i \(0.533607\pi\)
\(632\) 1.63844e9 0.258178
\(633\) 3.84930e9 0.603210
\(634\) −4.51141e9 −0.703072
\(635\) 1.05541e9 0.163574
\(636\) 7.43188e8 0.114551
\(637\) −9.66667e9 −1.48180
\(638\) −2.57044e9 −0.391864
\(639\) 7.88605e8 0.119566
\(640\) 1.84266e9 0.277853
\(641\) 8.97295e8 0.134565 0.0672825 0.997734i \(-0.478567\pi\)
0.0672825 + 0.997734i \(0.478567\pi\)
\(642\) −1.62956e9 −0.243051
\(643\) 7.92605e9 1.17576 0.587879 0.808949i \(-0.299963\pi\)
0.587879 + 0.808949i \(0.299963\pi\)
\(644\) −9.38988e8 −0.138535
\(645\) −1.22149e9 −0.179238
\(646\) −5.47693e9 −0.799325
\(647\) −9.72853e9 −1.41215 −0.706077 0.708135i \(-0.749537\pi\)
−0.706077 + 0.708135i \(0.749537\pi\)
\(648\) −6.15557e8 −0.0888701
\(649\) −6.93403e8 −0.0995702
\(650\) −4.50450e9 −0.643354
\(651\) 1.40173e9 0.199128
\(652\) 7.67968e9 1.08512
\(653\) −4.58349e9 −0.644170 −0.322085 0.946711i \(-0.604384\pi\)
−0.322085 + 0.946711i \(0.604384\pi\)
\(654\) −2.65592e9 −0.371273
\(655\) 1.00389e9 0.139586
\(656\) −3.94423e9 −0.545505
\(657\) −3.46020e6 −0.000476017 0
\(658\) 6.39208e8 0.0874684
\(659\) −1.15331e9 −0.156982 −0.0784908 0.996915i \(-0.525010\pi\)
−0.0784908 + 0.996915i \(0.525010\pi\)
\(660\) 7.27027e8 0.0984344
\(661\) −1.03408e8 −0.0139267 −0.00696334 0.999976i \(-0.502217\pi\)
−0.00696334 + 0.999976i \(0.502217\pi\)
\(662\) −5.85139e9 −0.783892
\(663\) −8.86059e9 −1.18077
\(664\) −4.31161e9 −0.571546
\(665\) 6.96335e8 0.0918211
\(666\) −1.50409e9 −0.197294
\(667\) 6.39011e9 0.833812
\(668\) 5.87530e9 0.762628
\(669\) 6.35796e9 0.820968
\(670\) −1.11771e9 −0.143571
\(671\) −1.68414e9 −0.215203
\(672\) −1.08385e9 −0.137777
\(673\) −8.65901e9 −1.09500 −0.547502 0.836804i \(-0.684421\pi\)
−0.547502 + 0.836804i \(0.684421\pi\)
\(674\) 6.18795e9 0.778461
\(675\) −1.41930e9 −0.177628
\(676\) −9.48174e9 −1.18052
\(677\) 1.41759e10 1.75586 0.877930 0.478789i \(-0.158924\pi\)
0.877930 + 0.478789i \(0.158924\pi\)
\(678\) −1.02283e9 −0.126038
\(679\) 9.78511e8 0.119956
\(680\) −2.36853e9 −0.288867
\(681\) 9.04908e9 1.09797
\(682\) −4.05656e9 −0.489681
\(683\) 2.10450e9 0.252742 0.126371 0.991983i \(-0.459667\pi\)
0.126371 + 0.991983i \(0.459667\pi\)
\(684\) −3.10322e9 −0.370780
\(685\) 5.94072e8 0.0706191
\(686\) 1.74100e9 0.205904
\(687\) −4.54225e9 −0.534469
\(688\) 4.28575e9 0.501727
\(689\) −3.33261e9 −0.388166
\(690\) 4.42656e8 0.0512973
\(691\) 8.19673e9 0.945078 0.472539 0.881310i \(-0.343338\pi\)
0.472539 + 0.881310i \(0.343338\pi\)
\(692\) −6.93017e9 −0.795010
\(693\) −5.33672e8 −0.0609127
\(694\) −2.95787e9 −0.335909
\(695\) −3.78272e8 −0.0427423
\(696\) 4.74474e9 0.533433
\(697\) 1.41499e10 1.58284
\(698\) 5.81493e9 0.647218
\(699\) −7.45621e9 −0.825749
\(700\) −1.60757e9 −0.177145
\(701\) −1.58794e9 −0.174109 −0.0870547 0.996204i \(-0.527745\pi\)
−0.0870547 + 0.996204i \(0.527745\pi\)
\(702\) 1.22957e9 0.134145
\(703\) −1.70223e10 −1.84788
\(704\) −3.90082e7 −0.00421358
\(705\) 1.23036e9 0.132242
\(706\) −2.63994e9 −0.282344
\(707\) 3.69981e9 0.393741
\(708\) 5.70151e8 0.0603773
\(709\) −1.39994e10 −1.47519 −0.737596 0.675242i \(-0.764039\pi\)
−0.737596 + 0.675242i \(0.764039\pi\)
\(710\) −4.21077e8 −0.0441527
\(711\) −1.03120e9 −0.107597
\(712\) −1.21325e10 −1.25971
\(713\) 1.00846e10 1.04195
\(714\) 7.74467e8 0.0796268
\(715\) −3.26014e9 −0.333553
\(716\) −5.55525e9 −0.565598
\(717\) 4.45417e9 0.451284
\(718\) 5.51968e9 0.556517
\(719\) −3.63177e8 −0.0364391 −0.0182195 0.999834i \(-0.505800\pi\)
−0.0182195 + 0.999834i \(0.505800\pi\)
\(720\) −4.15530e8 −0.0414896
\(721\) 9.08277e8 0.0902495
\(722\) 4.11589e9 0.406990
\(723\) 5.93470e9 0.584002
\(724\) 3.06594e9 0.300247
\(725\) 1.09401e10 1.06619
\(726\) −1.09589e9 −0.106289
\(727\) −7.18128e9 −0.693157 −0.346578 0.938021i \(-0.612657\pi\)
−0.346578 + 0.938021i \(0.612657\pi\)
\(728\) 3.12643e9 0.300324
\(729\) 3.87420e8 0.0370370
\(730\) 1.84758e6 0.000175782 0
\(731\) −1.53751e10 −1.45582
\(732\) 1.38478e9 0.130495
\(733\) 1.12100e10 1.05133 0.525667 0.850691i \(-0.323816\pi\)
0.525667 + 0.850691i \(0.323816\pi\)
\(734\) 6.10985e9 0.570288
\(735\) 1.62633e9 0.151078
\(736\) −7.79760e9 −0.720922
\(737\) 9.69459e9 0.892059
\(738\) −1.96356e9 −0.179824
\(739\) −2.61736e9 −0.238566 −0.119283 0.992860i \(-0.538060\pi\)
−0.119283 + 0.992860i \(0.538060\pi\)
\(740\) −3.27914e9 −0.297474
\(741\) 1.39155e10 1.25642
\(742\) 2.91289e8 0.0261765
\(743\) −2.16513e10 −1.93653 −0.968264 0.249930i \(-0.919593\pi\)
−0.968264 + 0.249930i \(0.919593\pi\)
\(744\) 7.48794e9 0.666587
\(745\) −8.14933e7 −0.00722063
\(746\) 2.77802e9 0.244991
\(747\) 2.71365e9 0.238194
\(748\) 9.15123e9 0.799510
\(749\) 2.60783e9 0.226774
\(750\) 1.57892e9 0.136661
\(751\) −1.23735e10 −1.06599 −0.532995 0.846118i \(-0.678934\pi\)
−0.532995 + 0.846118i \(0.678934\pi\)
\(752\) −4.31689e9 −0.370176
\(753\) −7.14750e9 −0.610059
\(754\) −9.47759e9 −0.805190
\(755\) −4.51187e9 −0.381542
\(756\) 4.38812e8 0.0369362
\(757\) −1.68097e9 −0.140840 −0.0704198 0.997517i \(-0.522434\pi\)
−0.0704198 + 0.997517i \(0.522434\pi\)
\(758\) −1.24014e9 −0.103426
\(759\) −3.83943e9 −0.318728
\(760\) 3.71976e9 0.307374
\(761\) 1.31161e10 1.07884 0.539421 0.842036i \(-0.318643\pi\)
0.539421 + 0.842036i \(0.318643\pi\)
\(762\) −1.84350e9 −0.150939
\(763\) 4.25036e9 0.346409
\(764\) 1.06035e10 0.860246
\(765\) 1.49071e9 0.120387
\(766\) −3.01011e6 −0.000241981 0
\(767\) −2.55668e9 −0.204593
\(768\) −3.17866e9 −0.253209
\(769\) 1.42811e10 1.13245 0.566224 0.824251i \(-0.308404\pi\)
0.566224 + 0.824251i \(0.308404\pi\)
\(770\) 2.84955e8 0.0224936
\(771\) −6.47392e9 −0.508718
\(772\) −1.77138e10 −1.38565
\(773\) −8.34783e9 −0.650048 −0.325024 0.945706i \(-0.605372\pi\)
−0.325024 + 0.945706i \(0.605372\pi\)
\(774\) 2.13358e9 0.165393
\(775\) 1.72651e10 1.33233
\(776\) 5.22711e9 0.401556
\(777\) 2.40704e9 0.184082
\(778\) −4.47669e9 −0.340823
\(779\) −2.22223e10 −1.68425
\(780\) 2.68065e9 0.202260
\(781\) 3.65226e9 0.274336
\(782\) 5.57180e9 0.416651
\(783\) −2.98625e9 −0.222311
\(784\) −5.70618e9 −0.422902
\(785\) −3.16565e9 −0.233571
\(786\) −1.75350e9 −0.128803
\(787\) 1.47026e10 1.07518 0.537591 0.843206i \(-0.319334\pi\)
0.537591 + 0.843206i \(0.319334\pi\)
\(788\) 1.31199e10 0.955184
\(789\) 3.44685e9 0.249835
\(790\) 5.50613e8 0.0397330
\(791\) 1.63687e9 0.117597
\(792\) −2.85083e9 −0.203907
\(793\) −6.20965e9 −0.442192
\(794\) −4.81135e9 −0.341111
\(795\) 5.60681e8 0.0395759
\(796\) 6.65640e9 0.467782
\(797\) 3.94070e9 0.275721 0.137860 0.990452i \(-0.455977\pi\)
0.137860 + 0.990452i \(0.455977\pi\)
\(798\) −1.21629e9 −0.0847282
\(799\) 1.54868e10 1.07411
\(800\) −1.33497e10 −0.921843
\(801\) 7.63597e9 0.524989
\(802\) 9.17467e9 0.628030
\(803\) −1.60252e7 −0.00109219
\(804\) −7.97138e9 −0.540925
\(805\) −7.08397e8 −0.0478620
\(806\) −1.49571e10 −1.00618
\(807\) 5.35167e9 0.358452
\(808\) 1.97640e10 1.31806
\(809\) 2.12121e10 1.40852 0.704260 0.709942i \(-0.251279\pi\)
0.704260 + 0.709942i \(0.251279\pi\)
\(810\) −2.06864e8 −0.0136769
\(811\) 4.13209e9 0.272018 0.136009 0.990708i \(-0.456572\pi\)
0.136009 + 0.990708i \(0.456572\pi\)
\(812\) −3.38238e9 −0.221705
\(813\) 3.16329e9 0.206453
\(814\) −6.96589e9 −0.452680
\(815\) 5.79375e9 0.374893
\(816\) −5.23036e9 −0.336989
\(817\) 2.41465e10 1.54909
\(818\) −3.58821e9 −0.229214
\(819\) −1.96772e9 −0.125161
\(820\) −4.28085e9 −0.271133
\(821\) 4.99283e9 0.314880 0.157440 0.987529i \(-0.449676\pi\)
0.157440 + 0.987529i \(0.449676\pi\)
\(822\) −1.03767e9 −0.0651640
\(823\) 8.56606e9 0.535650 0.267825 0.963468i \(-0.413695\pi\)
0.267825 + 0.963468i \(0.413695\pi\)
\(824\) 4.85193e9 0.302113
\(825\) −6.57321e9 −0.407557
\(826\) 2.23468e8 0.0137970
\(827\) 2.19884e10 1.35184 0.675920 0.736975i \(-0.263747\pi\)
0.675920 + 0.736975i \(0.263747\pi\)
\(828\) 3.15697e9 0.193270
\(829\) −1.74819e9 −0.106573 −0.0532867 0.998579i \(-0.516970\pi\)
−0.0532867 + 0.998579i \(0.516970\pi\)
\(830\) −1.44896e9 −0.0879595
\(831\) 7.56004e9 0.457005
\(832\) −1.43829e8 −0.00865792
\(833\) 2.04709e10 1.22710
\(834\) 6.60730e8 0.0394406
\(835\) 4.43248e9 0.263478
\(836\) −1.43719e10 −0.850732
\(837\) −4.71277e9 −0.277803
\(838\) 1.11842e10 0.656523
\(839\) 7.57544e8 0.0442834 0.0221417 0.999755i \(-0.492952\pi\)
0.0221417 + 0.999755i \(0.492952\pi\)
\(840\) −5.25993e8 −0.0306198
\(841\) 5.76829e9 0.334396
\(842\) 7.48251e9 0.431972
\(843\) −1.66509e9 −0.0957284
\(844\) −1.46584e10 −0.839246
\(845\) −7.15327e9 −0.407855
\(846\) −2.14908e9 −0.122027
\(847\) 1.75379e9 0.0991710
\(848\) −1.96722e9 −0.110782
\(849\) −1.67511e10 −0.939433
\(850\) 9.53908e9 0.532771
\(851\) 1.73172e10 0.963215
\(852\) −3.00307e9 −0.166352
\(853\) −1.72683e9 −0.0952636 −0.0476318 0.998865i \(-0.515167\pi\)
−0.0476318 + 0.998865i \(0.515167\pi\)
\(854\) 5.42759e8 0.0298198
\(855\) −2.34115e9 −0.128100
\(856\) 1.39308e10 0.759132
\(857\) −3.43055e10 −1.86179 −0.930895 0.365286i \(-0.880971\pi\)
−0.930895 + 0.365286i \(0.880971\pi\)
\(858\) 5.69451e9 0.307787
\(859\) −1.16013e10 −0.624499 −0.312249 0.950000i \(-0.601082\pi\)
−0.312249 + 0.950000i \(0.601082\pi\)
\(860\) 4.65153e9 0.249374
\(861\) 3.14234e9 0.167781
\(862\) 1.33619e10 0.710549
\(863\) 1.06656e10 0.564870 0.282435 0.959286i \(-0.408858\pi\)
0.282435 + 0.959286i \(0.408858\pi\)
\(864\) 3.64401e9 0.192212
\(865\) −5.22830e9 −0.274665
\(866\) 3.56005e8 0.0186270
\(867\) 7.68474e9 0.400463
\(868\) −5.33792e9 −0.277047
\(869\) −4.77580e9 −0.246875
\(870\) 1.59452e9 0.0820940
\(871\) 3.57453e10 1.83297
\(872\) 2.27050e10 1.15962
\(873\) −3.28985e9 −0.167350
\(874\) −8.75046e9 −0.443344
\(875\) −2.52679e9 −0.127509
\(876\) 1.31767e7 0.000662282 0
\(877\) −5.25940e9 −0.263292 −0.131646 0.991297i \(-0.542026\pi\)
−0.131646 + 0.991297i \(0.542026\pi\)
\(878\) 3.14503e8 0.0156817
\(879\) −2.01726e10 −1.00184
\(880\) −1.92444e9 −0.0951954
\(881\) −2.79482e10 −1.37701 −0.688507 0.725230i \(-0.741734\pi\)
−0.688507 + 0.725230i \(0.741734\pi\)
\(882\) −2.84072e9 −0.139408
\(883\) 3.72423e9 0.182043 0.0910214 0.995849i \(-0.470987\pi\)
0.0910214 + 0.995849i \(0.470987\pi\)
\(884\) 3.37419e10 1.64281
\(885\) 4.30137e8 0.0208596
\(886\) −3.22157e9 −0.155614
\(887\) 6.26596e9 0.301477 0.150739 0.988574i \(-0.451835\pi\)
0.150739 + 0.988574i \(0.451835\pi\)
\(888\) 1.28582e10 0.616219
\(889\) 2.95021e9 0.140830
\(890\) −4.07724e9 −0.193866
\(891\) 1.79426e9 0.0849793
\(892\) −2.42116e10 −1.14221
\(893\) −2.43219e10 −1.14292
\(894\) 1.42345e8 0.00666286
\(895\) −4.19102e9 −0.195407
\(896\) 5.15081e9 0.239220
\(897\) −1.41565e10 −0.654912
\(898\) −1.39257e10 −0.641727
\(899\) 3.63263e10 1.66748
\(900\) 5.40483e9 0.247134
\(901\) 7.05740e9 0.321446
\(902\) −9.09382e9 −0.412594
\(903\) −3.41444e9 −0.154316
\(904\) 8.74403e9 0.393661
\(905\) 2.31302e9 0.103731
\(906\) 7.88092e9 0.352069
\(907\) 8.21053e9 0.365381 0.182690 0.983170i \(-0.441519\pi\)
0.182690 + 0.983170i \(0.441519\pi\)
\(908\) −3.44596e10 −1.52760
\(909\) −1.24391e10 −0.549309
\(910\) 1.05067e9 0.0462191
\(911\) −1.57505e10 −0.690207 −0.345103 0.938565i \(-0.612156\pi\)
−0.345103 + 0.938565i \(0.612156\pi\)
\(912\) 8.21423e9 0.358579
\(913\) 1.25677e10 0.546523
\(914\) 1.90385e10 0.824746
\(915\) 1.04472e9 0.0450842
\(916\) 1.72973e10 0.743607
\(917\) 2.80618e9 0.120177
\(918\) −2.60384e9 −0.111087
\(919\) −6.94140e9 −0.295014 −0.147507 0.989061i \(-0.547125\pi\)
−0.147507 + 0.989061i \(0.547125\pi\)
\(920\) −3.78419e9 −0.160220
\(921\) 1.87544e10 0.791034
\(922\) −1.87773e10 −0.788994
\(923\) 1.34664e10 0.563697
\(924\) 2.03227e9 0.0847478
\(925\) 2.96475e10 1.23166
\(926\) 8.46318e9 0.350264
\(927\) −3.05372e9 −0.125907
\(928\) −2.80881e10 −1.15373
\(929\) −1.64084e9 −0.0671448 −0.0335724 0.999436i \(-0.510688\pi\)
−0.0335724 + 0.999436i \(0.510688\pi\)
\(930\) 2.51639e9 0.102586
\(931\) −3.21493e10 −1.30571
\(932\) 2.83939e10 1.14887
\(933\) 3.44912e9 0.139035
\(934\) −9.55004e8 −0.0383523
\(935\) 6.90393e9 0.276220
\(936\) −1.05114e10 −0.418981
\(937\) 3.84565e10 1.52715 0.763575 0.645719i \(-0.223442\pi\)
0.763575 + 0.645719i \(0.223442\pi\)
\(938\) −3.12435e9 −0.123609
\(939\) −2.63049e10 −1.03683
\(940\) −4.68532e9 −0.183989
\(941\) 3.13483e10 1.22645 0.613226 0.789907i \(-0.289871\pi\)
0.613226 + 0.789907i \(0.289871\pi\)
\(942\) 5.52946e9 0.215529
\(943\) 2.26072e10 0.877921
\(944\) −1.50919e9 −0.0583905
\(945\) 3.31051e8 0.0127610
\(946\) 9.88124e9 0.379483
\(947\) −4.56816e10 −1.74790 −0.873950 0.486016i \(-0.838450\pi\)
−0.873950 + 0.486016i \(0.838450\pi\)
\(948\) 3.92691e9 0.149700
\(949\) −5.90871e7 −0.00224420
\(950\) −1.49810e10 −0.566903
\(951\) −2.42735e10 −0.915168
\(952\) −6.62078e9 −0.248702
\(953\) −2.57557e10 −0.963937 −0.481968 0.876189i \(-0.660078\pi\)
−0.481968 + 0.876189i \(0.660078\pi\)
\(954\) −9.79345e8 −0.0365188
\(955\) 7.99955e9 0.297204
\(956\) −1.69619e10 −0.627872
\(957\) −1.38302e10 −0.510079
\(958\) 5.24327e9 0.192674
\(959\) 1.66061e9 0.0608000
\(960\) 2.41978e7 0.000882728 0
\(961\) 2.98159e10 1.08372
\(962\) −2.56842e10 −0.930151
\(963\) −8.76779e9 −0.316372
\(964\) −2.25998e10 −0.812522
\(965\) −1.33638e10 −0.478723
\(966\) 1.23736e9 0.0441648
\(967\) 2.61784e10 0.931002 0.465501 0.885047i \(-0.345874\pi\)
0.465501 + 0.885047i \(0.345874\pi\)
\(968\) 9.36857e9 0.331978
\(969\) −2.94685e10 −1.04046
\(970\) 1.75662e9 0.0617984
\(971\) −7.63541e9 −0.267648 −0.133824 0.991005i \(-0.542726\pi\)
−0.133824 + 0.991005i \(0.542726\pi\)
\(972\) −1.47533e9 −0.0515296
\(973\) −1.05739e9 −0.0367993
\(974\) −2.27050e10 −0.787344
\(975\) −2.42364e10 −0.837435
\(976\) −3.66552e9 −0.126201
\(977\) −1.18991e10 −0.408211 −0.204105 0.978949i \(-0.565429\pi\)
−0.204105 + 0.978949i \(0.565429\pi\)
\(978\) −1.01200e10 −0.345934
\(979\) 3.53644e10 1.20456
\(980\) −6.19319e9 −0.210195
\(981\) −1.42901e10 −0.483275
\(982\) 6.43144e9 0.216729
\(983\) −3.88296e10 −1.30384 −0.651921 0.758287i \(-0.726037\pi\)
−0.651921 + 0.758287i \(0.726037\pi\)
\(984\) 1.67861e10 0.561652
\(985\) 9.89796e9 0.330004
\(986\) 2.00705e10 0.666789
\(987\) 3.43924e9 0.113855
\(988\) −5.29913e10 −1.74806
\(989\) −2.45647e10 −0.807467
\(990\) −9.58048e8 −0.0313808
\(991\) −4.18861e10 −1.36714 −0.683569 0.729886i \(-0.739573\pi\)
−0.683569 + 0.729886i \(0.739573\pi\)
\(992\) −4.43275e10 −1.44172
\(993\) −3.14833e10 −1.02037
\(994\) −1.17704e9 −0.0380136
\(995\) 5.02176e9 0.161613
\(996\) −1.03338e10 −0.331400
\(997\) −2.75354e10 −0.879950 −0.439975 0.898010i \(-0.645013\pi\)
−0.439975 + 0.898010i \(0.645013\pi\)
\(998\) −1.63709e10 −0.521333
\(999\) −8.09273e9 −0.256812
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 177.8.a.b.1.12 17
3.2 odd 2 531.8.a.d.1.6 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.12 17 1.1 even 1 trivial
531.8.a.d.1.6 17 3.2 odd 2