Properties

Label 177.8.a.c.1.16
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

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: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1669 x^{15} + 2385 x^{14} + 1108684 x^{13} - 848131 x^{12} - 377920980 x^{11} + 12724944 x^{10} + 71331230512 x^{9} + 50741131904 x^{8} - 7480805165760 x^{7} - 10751966150272 x^{6} + 413177144536320 x^{5} + 886760582981376 x^{4} - 10454479722123264 x^{3} - 29180140031461376 x^{2} + 79787300207378432 x + 248723246810300416\)
Coefficient ring: \(\Z[a_1, \ldots, a_{31}]\)
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 1.16
Root \(21.5848\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+21.5848 q^{2} -27.0000 q^{3} +337.903 q^{4} -113.334 q^{5} -582.789 q^{6} -647.523 q^{7} +4530.72 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+21.5848 q^{2} -27.0000 q^{3} +337.903 q^{4} -113.334 q^{5} -582.789 q^{6} -647.523 q^{7} +4530.72 q^{8} +729.000 q^{9} -2446.30 q^{10} +797.052 q^{11} -9123.39 q^{12} +863.046 q^{13} -13976.6 q^{14} +3060.03 q^{15} +54543.1 q^{16} +30234.1 q^{17} +15735.3 q^{18} +36679.8 q^{19} -38296.1 q^{20} +17483.1 q^{21} +17204.2 q^{22} +55059.3 q^{23} -122330. q^{24} -65280.3 q^{25} +18628.7 q^{26} -19683.0 q^{27} -218800. q^{28} +201168. q^{29} +66050.1 q^{30} -17365.2 q^{31} +597369. q^{32} -21520.4 q^{33} +652597. q^{34} +73386.6 q^{35} +246332. q^{36} -157061. q^{37} +791727. q^{38} -23302.2 q^{39} -513487. q^{40} +264460. q^{41} +377370. q^{42} +97565.6 q^{43} +269327. q^{44} -82620.8 q^{45} +1.18844e6 q^{46} +236490. q^{47} -1.47266e6 q^{48} -404257. q^{49} -1.40906e6 q^{50} -816321. q^{51} +291626. q^{52} -524092. q^{53} -424854. q^{54} -90333.5 q^{55} -2.93375e6 q^{56} -990355. q^{57} +4.34217e6 q^{58} -205379. q^{59} +1.03399e6 q^{60} +2.50976e6 q^{61} -374823. q^{62} -472044. q^{63} +5.91257e6 q^{64} -97812.8 q^{65} -464514. q^{66} -1.86827e6 q^{67} +1.02162e7 q^{68} -1.48660e6 q^{69} +1.58404e6 q^{70} -3.17472e6 q^{71} +3.30290e6 q^{72} +303181. q^{73} -3.39014e6 q^{74} +1.76257e6 q^{75} +1.23942e7 q^{76} -516110. q^{77} -502974. q^{78} +3.54094e6 q^{79} -6.18161e6 q^{80} +531441. q^{81} +5.70830e6 q^{82} +9.87483e6 q^{83} +5.90761e6 q^{84} -3.42656e6 q^{85} +2.10593e6 q^{86} -5.43154e6 q^{87} +3.61122e6 q^{88} -6.49191e6 q^{89} -1.78335e6 q^{90} -558842. q^{91} +1.86047e7 q^{92} +468859. q^{93} +5.10459e6 q^{94} -4.15709e6 q^{95} -1.61290e7 q^{96} -8435.50 q^{97} -8.72581e6 q^{98} +581051. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + 6521q^{10} - 1764q^{11} - 31482q^{12} + 18192q^{13} - 7827q^{14} + 8586q^{15} + 139226q^{16} - 15507q^{17} + 1458q^{18} + 52083q^{19} + 721q^{20} - 84915q^{21} - 234434q^{22} + 63823q^{23} - 63585q^{24} + 202153q^{25} - 367956q^{26} - 334611q^{27} + 182306q^{28} - 502955q^{29} - 176067q^{30} + 347531q^{31} - 243908q^{32} + 47628q^{33} - 330872q^{34} + 92641q^{35} + 850014q^{36} + 447615q^{37} + 775669q^{38} - 491184q^{39} + 2203270q^{40} + 940335q^{41} + 211329q^{42} + 478562q^{43} - 596924q^{44} - 231822q^{45} - 3078663q^{46} + 703121q^{47} - 3759102q^{48} + 1895082q^{49} - 876967q^{50} + 418689q^{51} + 6278296q^{52} - 1005974q^{53} - 39366q^{54} + 5212846q^{55} + 3425294q^{56} - 1406241q^{57} + 6710166q^{58} - 3491443q^{59} - 19467q^{60} + 11510749q^{61} + 5996234q^{62} + 2292705q^{63} + 29496941q^{64} + 11094180q^{65} + 6329718q^{66} + 14007144q^{67} + 19688159q^{68} - 1723221q^{69} + 30909708q^{70} + 5229074q^{71} + 1716795q^{72} + 5452211q^{73} + 12819662q^{74} - 5458131q^{75} + 41929340q^{76} + 9930777q^{77} + 9934812q^{78} + 15275654q^{79} + 36576105q^{80} + 9034497q^{81} + 32025935q^{82} + 7826609q^{83} - 4922262q^{84} + 11836945q^{85} + 51649136q^{86} + 13579785q^{87} + 30223741q^{88} - 6436185q^{89} + 4753809q^{90} + 11633535q^{91} + 43357972q^{92} - 9383337q^{93} - 4494252q^{94} + 23741055q^{95} + 6585516q^{96} + 26377540q^{97} + 26517816q^{98} - 1285956q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 21.5848 1.90784 0.953922 0.300054i \(-0.0970047\pi\)
0.953922 + 0.300054i \(0.0970047\pi\)
\(3\) −27.0000 −0.577350
\(4\) 337.903 2.63987
\(5\) −113.334 −0.405478 −0.202739 0.979233i \(-0.564984\pi\)
−0.202739 + 0.979233i \(0.564984\pi\)
\(6\) −582.789 −1.10149
\(7\) −647.523 −0.713530 −0.356765 0.934194i \(-0.616120\pi\)
−0.356765 + 0.934194i \(0.616120\pi\)
\(8\) 4530.72 3.12862
\(9\) 729.000 0.333333
\(10\) −2446.30 −0.773588
\(11\) 797.052 0.180556 0.0902781 0.995917i \(-0.471224\pi\)
0.0902781 + 0.995917i \(0.471224\pi\)
\(12\) −9123.39 −1.52413
\(13\) 863.046 0.108951 0.0544756 0.998515i \(-0.482651\pi\)
0.0544756 + 0.998515i \(0.482651\pi\)
\(14\) −13976.6 −1.36130
\(15\) 3060.03 0.234103
\(16\) 54543.1 3.32905
\(17\) 30234.1 1.49254 0.746270 0.665644i \(-0.231843\pi\)
0.746270 + 0.665644i \(0.231843\pi\)
\(18\) 15735.3 0.635948
\(19\) 36679.8 1.22685 0.613423 0.789755i \(-0.289792\pi\)
0.613423 + 0.789755i \(0.289792\pi\)
\(20\) −38296.1 −1.07041
\(21\) 17483.1 0.411957
\(22\) 17204.2 0.344473
\(23\) 55059.3 0.943591 0.471795 0.881708i \(-0.343606\pi\)
0.471795 + 0.881708i \(0.343606\pi\)
\(24\) −122330. −1.80631
\(25\) −65280.3 −0.835588
\(26\) 18628.7 0.207862
\(27\) −19683.0 −0.192450
\(28\) −218800. −1.88363
\(29\) 201168. 1.53167 0.765836 0.643036i \(-0.222325\pi\)
0.765836 + 0.643036i \(0.222325\pi\)
\(30\) 66050.1 0.446631
\(31\) −17365.2 −0.104692 −0.0523459 0.998629i \(-0.516670\pi\)
−0.0523459 + 0.998629i \(0.516670\pi\)
\(32\) 597369. 3.22268
\(33\) −21520.4 −0.104244
\(34\) 652597. 2.84753
\(35\) 73386.6 0.289320
\(36\) 246332. 0.879957
\(37\) −157061. −0.509757 −0.254879 0.966973i \(-0.582036\pi\)
−0.254879 + 0.966973i \(0.582036\pi\)
\(38\) 791727. 2.34063
\(39\) −23302.2 −0.0629030
\(40\) −513487. −1.26858
\(41\) 264460. 0.599261 0.299630 0.954055i \(-0.403137\pi\)
0.299630 + 0.954055i \(0.403137\pi\)
\(42\) 377370. 0.785949
\(43\) 97565.6 0.187136 0.0935680 0.995613i \(-0.470173\pi\)
0.0935680 + 0.995613i \(0.470173\pi\)
\(44\) 269327. 0.476645
\(45\) −82620.8 −0.135159
\(46\) 1.18844e6 1.80022
\(47\) 236490. 0.332254 0.166127 0.986104i \(-0.446874\pi\)
0.166127 + 0.986104i \(0.446874\pi\)
\(48\) −1.47266e6 −1.92203
\(49\) −404257. −0.490876
\(50\) −1.40906e6 −1.59417
\(51\) −816321. −0.861718
\(52\) 291626. 0.287617
\(53\) −524092. −0.483551 −0.241775 0.970332i \(-0.577730\pi\)
−0.241775 + 0.970332i \(0.577730\pi\)
\(54\) −424854. −0.367165
\(55\) −90333.5 −0.0732115
\(56\) −2.93375e6 −2.23236
\(57\) −990355. −0.708319
\(58\) 4.34217e6 2.92219
\(59\) −205379. −0.130189
\(60\) 1.03399e6 0.618001
\(61\) 2.50976e6 1.41572 0.707860 0.706353i \(-0.249661\pi\)
0.707860 + 0.706353i \(0.249661\pi\)
\(62\) −374823. −0.199736
\(63\) −472044. −0.237843
\(64\) 5.91257e6 2.81933
\(65\) −97812.8 −0.0441773
\(66\) −464514. −0.198882
\(67\) −1.86827e6 −0.758888 −0.379444 0.925215i \(-0.623885\pi\)
−0.379444 + 0.925215i \(0.623885\pi\)
\(68\) 1.02162e7 3.94011
\(69\) −1.48660e6 −0.544782
\(70\) 1.58404e6 0.551978
\(71\) −3.17472e6 −1.05269 −0.526345 0.850271i \(-0.676438\pi\)
−0.526345 + 0.850271i \(0.676438\pi\)
\(72\) 3.30290e6 1.04287
\(73\) 303181. 0.0912162 0.0456081 0.998959i \(-0.485477\pi\)
0.0456081 + 0.998959i \(0.485477\pi\)
\(74\) −3.39014e6 −0.972538
\(75\) 1.76257e6 0.482427
\(76\) 1.23942e7 3.23871
\(77\) −516110. −0.128832
\(78\) −502974. −0.120009
\(79\) 3.54094e6 0.808023 0.404011 0.914754i \(-0.367616\pi\)
0.404011 + 0.914754i \(0.367616\pi\)
\(80\) −6.18161e6 −1.34985
\(81\) 531441. 0.111111
\(82\) 5.70830e6 1.14330
\(83\) 9.87483e6 1.89564 0.947821 0.318804i \(-0.103281\pi\)
0.947821 + 0.318804i \(0.103281\pi\)
\(84\) 5.90761e6 1.08751
\(85\) −3.42656e6 −0.605191
\(86\) 2.10593e6 0.357026
\(87\) −5.43154e6 −0.884311
\(88\) 3.61122e6 0.564891
\(89\) −6.49191e6 −0.976130 −0.488065 0.872807i \(-0.662297\pi\)
−0.488065 + 0.872807i \(0.662297\pi\)
\(90\) −1.78335e6 −0.257863
\(91\) −558842. −0.0777399
\(92\) 1.86047e7 2.49096
\(93\) 468859. 0.0604439
\(94\) 5.10459e6 0.633889
\(95\) −4.15709e6 −0.497458
\(96\) −1.61290e7 −1.86062
\(97\) −8435.50 −0.000938447 0 −0.000469223 1.00000i \(-0.500149\pi\)
−0.000469223 1.00000i \(0.500149\pi\)
\(98\) −8.72581e6 −0.936514
\(99\) 581051. 0.0601854
\(100\) −2.20584e7 −2.20584
\(101\) 3.74750e6 0.361923 0.180962 0.983490i \(-0.442079\pi\)
0.180962 + 0.983490i \(0.442079\pi\)
\(102\) −1.76201e7 −1.64402
\(103\) −7.16189e6 −0.645799 −0.322900 0.946433i \(-0.604658\pi\)
−0.322900 + 0.946433i \(0.604658\pi\)
\(104\) 3.91022e6 0.340867
\(105\) −1.98144e6 −0.167039
\(106\) −1.13124e7 −0.922539
\(107\) −1.89118e7 −1.49242 −0.746208 0.665712i \(-0.768128\pi\)
−0.746208 + 0.665712i \(0.768128\pi\)
\(108\) −6.65095e6 −0.508043
\(109\) −1.60040e7 −1.18368 −0.591842 0.806054i \(-0.701599\pi\)
−0.591842 + 0.806054i \(0.701599\pi\)
\(110\) −1.94983e6 −0.139676
\(111\) 4.24066e6 0.294309
\(112\) −3.53179e7 −2.37537
\(113\) −4.77754e6 −0.311480 −0.155740 0.987798i \(-0.549776\pi\)
−0.155740 + 0.987798i \(0.549776\pi\)
\(114\) −2.13766e7 −1.35136
\(115\) −6.24012e6 −0.382605
\(116\) 6.79754e7 4.04342
\(117\) 629160. 0.0363171
\(118\) −4.43306e6 −0.248380
\(119\) −1.95773e7 −1.06497
\(120\) 1.38641e7 0.732418
\(121\) −1.88519e7 −0.967399
\(122\) 5.41726e7 2.70097
\(123\) −7.14041e6 −0.345983
\(124\) −5.86775e6 −0.276373
\(125\) 1.62528e7 0.744290
\(126\) −1.01890e7 −0.453768
\(127\) 2.70787e7 1.17304 0.586522 0.809933i \(-0.300497\pi\)
0.586522 + 0.809933i \(0.300497\pi\)
\(128\) 5.11584e7 2.15617
\(129\) −2.63427e6 −0.108043
\(130\) −2.11127e6 −0.0842834
\(131\) 6.72224e6 0.261255 0.130627 0.991432i \(-0.458301\pi\)
0.130627 + 0.991432i \(0.458301\pi\)
\(132\) −7.27182e6 −0.275191
\(133\) −2.37510e7 −0.875390
\(134\) −4.03262e7 −1.44784
\(135\) 2.23076e6 0.0780342
\(136\) 1.36982e8 4.66959
\(137\) 4.39105e7 1.45897 0.729485 0.683997i \(-0.239760\pi\)
0.729485 + 0.683997i \(0.239760\pi\)
\(138\) −3.20880e7 −1.03936
\(139\) 8.55233e6 0.270105 0.135052 0.990838i \(-0.456880\pi\)
0.135052 + 0.990838i \(0.456880\pi\)
\(140\) 2.47976e7 0.763768
\(141\) −6.38523e6 −0.191827
\(142\) −6.85256e7 −2.00837
\(143\) 687893. 0.0196718
\(144\) 3.97619e7 1.10968
\(145\) −2.27993e7 −0.621059
\(146\) 6.54410e6 0.174026
\(147\) 1.09149e7 0.283407
\(148\) −5.30716e7 −1.34569
\(149\) −4.40487e7 −1.09089 −0.545445 0.838146i \(-0.683639\pi\)
−0.545445 + 0.838146i \(0.683639\pi\)
\(150\) 3.80447e7 0.920395
\(151\) −5.79273e7 −1.36919 −0.684596 0.728923i \(-0.740021\pi\)
−0.684596 + 0.728923i \(0.740021\pi\)
\(152\) 1.66186e8 3.83833
\(153\) 2.20407e7 0.497513
\(154\) −1.11401e7 −0.245792
\(155\) 1.96807e6 0.0424502
\(156\) −7.87391e6 −0.166056
\(157\) −6.48864e7 −1.33815 −0.669075 0.743195i \(-0.733309\pi\)
−0.669075 + 0.743195i \(0.733309\pi\)
\(158\) 7.64304e7 1.54158
\(159\) 1.41505e7 0.279178
\(160\) −6.77025e7 −1.30673
\(161\) −3.56522e7 −0.673280
\(162\) 1.14710e7 0.211983
\(163\) 6.81238e7 1.23209 0.616045 0.787711i \(-0.288734\pi\)
0.616045 + 0.787711i \(0.288734\pi\)
\(164\) 8.93618e7 1.58197
\(165\) 2.43900e6 0.0422687
\(166\) 2.13146e8 3.61659
\(167\) 3.52225e7 0.585210 0.292605 0.956233i \(-0.405478\pi\)
0.292605 + 0.956233i \(0.405478\pi\)
\(168\) 7.92112e7 1.28885
\(169\) −6.20037e7 −0.988130
\(170\) −7.39617e7 −1.15461
\(171\) 2.67396e7 0.408948
\(172\) 3.29678e7 0.494015
\(173\) −5.35823e7 −0.786792 −0.393396 0.919369i \(-0.628700\pi\)
−0.393396 + 0.919369i \(0.628700\pi\)
\(174\) −1.17239e8 −1.68713
\(175\) 4.22705e7 0.596217
\(176\) 4.34737e7 0.601080
\(177\) 5.54523e6 0.0751646
\(178\) −1.40127e8 −1.86230
\(179\) −1.25733e8 −1.63857 −0.819283 0.573390i \(-0.805628\pi\)
−0.819283 + 0.573390i \(0.805628\pi\)
\(180\) −2.79179e7 −0.356803
\(181\) 6.35900e7 0.797102 0.398551 0.917146i \(-0.369513\pi\)
0.398551 + 0.917146i \(0.369513\pi\)
\(182\) −1.20625e7 −0.148316
\(183\) −6.77634e7 −0.817366
\(184\) 2.49459e8 2.95213
\(185\) 1.78005e7 0.206695
\(186\) 1.01202e7 0.115317
\(187\) 2.40982e7 0.269487
\(188\) 7.99108e7 0.877108
\(189\) 1.27452e7 0.137319
\(190\) −8.97299e7 −0.949073
\(191\) 2.21042e7 0.229540 0.114770 0.993392i \(-0.463387\pi\)
0.114770 + 0.993392i \(0.463387\pi\)
\(192\) −1.59639e8 −1.62774
\(193\) −2.73971e6 −0.0274318 −0.0137159 0.999906i \(-0.504366\pi\)
−0.0137159 + 0.999906i \(0.504366\pi\)
\(194\) −182078. −0.00179041
\(195\) 2.64095e6 0.0255058
\(196\) −1.36600e8 −1.29585
\(197\) 1.45954e7 0.136014 0.0680072 0.997685i \(-0.478336\pi\)
0.0680072 + 0.997685i \(0.478336\pi\)
\(198\) 1.25419e7 0.114824
\(199\) −1.26330e8 −1.13637 −0.568184 0.822901i \(-0.692354\pi\)
−0.568184 + 0.822901i \(0.692354\pi\)
\(200\) −2.95767e8 −2.61424
\(201\) 5.04433e7 0.438144
\(202\) 8.08890e7 0.690493
\(203\) −1.30261e8 −1.09289
\(204\) −2.75838e8 −2.27482
\(205\) −2.99724e7 −0.242987
\(206\) −1.54588e8 −1.23208
\(207\) 4.01383e7 0.314530
\(208\) 4.70732e7 0.362704
\(209\) 2.92357e7 0.221514
\(210\) −4.27690e7 −0.318685
\(211\) 2.01964e8 1.48008 0.740041 0.672562i \(-0.234806\pi\)
0.740041 + 0.672562i \(0.234806\pi\)
\(212\) −1.77092e8 −1.27651
\(213\) 8.57173e7 0.607771
\(214\) −4.08208e8 −2.84730
\(215\) −1.10575e7 −0.0758795
\(216\) −8.91782e7 −0.602103
\(217\) 1.12443e7 0.0747007
\(218\) −3.45443e8 −2.25829
\(219\) −8.18588e6 −0.0526637
\(220\) −3.05240e7 −0.193269
\(221\) 2.60934e7 0.162614
\(222\) 9.15337e7 0.561495
\(223\) −2.44131e8 −1.47420 −0.737100 0.675784i \(-0.763805\pi\)
−0.737100 + 0.675784i \(0.763805\pi\)
\(224\) −3.86810e8 −2.29948
\(225\) −4.75893e7 −0.278529
\(226\) −1.03122e8 −0.594255
\(227\) 6.69826e7 0.380077 0.190038 0.981777i \(-0.439139\pi\)
0.190038 + 0.981777i \(0.439139\pi\)
\(228\) −3.34644e8 −1.86987
\(229\) −5.42627e6 −0.0298591 −0.0149296 0.999889i \(-0.504752\pi\)
−0.0149296 + 0.999889i \(0.504752\pi\)
\(230\) −1.34692e8 −0.729951
\(231\) 1.39350e7 0.0743813
\(232\) 9.11436e8 4.79202
\(233\) −1.14071e8 −0.590784 −0.295392 0.955376i \(-0.595450\pi\)
−0.295392 + 0.955376i \(0.595450\pi\)
\(234\) 1.35803e7 0.0692873
\(235\) −2.68025e7 −0.134722
\(236\) −6.93983e7 −0.343682
\(237\) −9.56053e7 −0.466512
\(238\) −4.22571e8 −2.03180
\(239\) 2.35111e8 1.11399 0.556993 0.830517i \(-0.311955\pi\)
0.556993 + 0.830517i \(0.311955\pi\)
\(240\) 1.66903e8 0.779338
\(241\) −2.14221e8 −0.985830 −0.492915 0.870078i \(-0.664069\pi\)
−0.492915 + 0.870078i \(0.664069\pi\)
\(242\) −4.06914e8 −1.84565
\(243\) −1.43489e7 −0.0641500
\(244\) 8.48055e8 3.73732
\(245\) 4.58163e7 0.199039
\(246\) −1.54124e8 −0.660082
\(247\) 3.16564e7 0.133666
\(248\) −7.86767e7 −0.327541
\(249\) −2.66620e8 −1.09445
\(250\) 3.50813e8 1.41999
\(251\) −1.53834e8 −0.614036 −0.307018 0.951704i \(-0.599331\pi\)
−0.307018 + 0.951704i \(0.599331\pi\)
\(252\) −1.59505e8 −0.627875
\(253\) 4.38852e7 0.170371
\(254\) 5.84488e8 2.23799
\(255\) 9.25173e7 0.349407
\(256\) 3.47435e8 1.29430
\(257\) 4.74131e8 1.74234 0.871169 0.490984i \(-0.163363\pi\)
0.871169 + 0.490984i \(0.163363\pi\)
\(258\) −5.68602e7 −0.206129
\(259\) 1.01701e8 0.363727
\(260\) −3.30513e7 −0.116622
\(261\) 1.46651e8 0.510557
\(262\) 1.45098e8 0.498434
\(263\) −1.50757e8 −0.511014 −0.255507 0.966807i \(-0.582242\pi\)
−0.255507 + 0.966807i \(0.582242\pi\)
\(264\) −9.75030e7 −0.326140
\(265\) 5.93977e7 0.196069
\(266\) −5.12661e8 −1.67011
\(267\) 1.75282e8 0.563569
\(268\) −6.31295e8 −2.00337
\(269\) 5.50062e8 1.72297 0.861486 0.507781i \(-0.169534\pi\)
0.861486 + 0.507781i \(0.169534\pi\)
\(270\) 4.81505e7 0.148877
\(271\) 4.96505e8 1.51541 0.757707 0.652595i \(-0.226319\pi\)
0.757707 + 0.652595i \(0.226319\pi\)
\(272\) 1.64906e9 4.96873
\(273\) 1.50887e7 0.0448832
\(274\) 9.47799e8 2.78349
\(275\) −5.20318e7 −0.150871
\(276\) −5.02328e8 −1.43815
\(277\) 1.08273e8 0.306083 0.153041 0.988220i \(-0.451093\pi\)
0.153041 + 0.988220i \(0.451093\pi\)
\(278\) 1.84600e8 0.515318
\(279\) −1.26592e7 −0.0348973
\(280\) 3.32495e8 0.905173
\(281\) −6.35664e8 −1.70905 −0.854527 0.519407i \(-0.826153\pi\)
−0.854527 + 0.519407i \(0.826153\pi\)
\(282\) −1.37824e8 −0.365976
\(283\) 1.82266e8 0.478027 0.239014 0.971016i \(-0.423176\pi\)
0.239014 + 0.971016i \(0.423176\pi\)
\(284\) −1.07275e9 −2.77897
\(285\) 1.12241e8 0.287208
\(286\) 1.48480e7 0.0375308
\(287\) −1.71244e8 −0.427590
\(288\) 4.35482e8 1.07423
\(289\) 5.03762e8 1.22767
\(290\) −4.92117e8 −1.18488
\(291\) 227758. 0.000541812 0
\(292\) 1.02446e8 0.240799
\(293\) −2.94539e8 −0.684080 −0.342040 0.939685i \(-0.611118\pi\)
−0.342040 + 0.939685i \(0.611118\pi\)
\(294\) 2.35597e8 0.540697
\(295\) 2.32765e7 0.0527887
\(296\) −7.11602e8 −1.59484
\(297\) −1.56884e7 −0.0347481
\(298\) −9.50782e8 −2.08125
\(299\) 4.75187e7 0.102805
\(300\) 5.95578e8 1.27354
\(301\) −6.31760e7 −0.133527
\(302\) −1.25035e9 −2.61220
\(303\) −1.01182e8 −0.208957
\(304\) 2.00063e9 4.08422
\(305\) −2.84442e8 −0.574042
\(306\) 4.75743e8 0.949178
\(307\) −6.80308e8 −1.34190 −0.670952 0.741501i \(-0.734114\pi\)
−0.670952 + 0.741501i \(0.734114\pi\)
\(308\) −1.74395e8 −0.340100
\(309\) 1.93371e8 0.372852
\(310\) 4.24804e7 0.0809884
\(311\) −1.83957e8 −0.346780 −0.173390 0.984853i \(-0.555472\pi\)
−0.173390 + 0.984853i \(0.555472\pi\)
\(312\) −1.05576e8 −0.196800
\(313\) −4.45917e8 −0.821956 −0.410978 0.911645i \(-0.634813\pi\)
−0.410978 + 0.911645i \(0.634813\pi\)
\(314\) −1.40056e9 −2.55298
\(315\) 5.34989e7 0.0964401
\(316\) 1.19650e9 2.13308
\(317\) −5.41308e8 −0.954414 −0.477207 0.878791i \(-0.658351\pi\)
−0.477207 + 0.878791i \(0.658351\pi\)
\(318\) 3.05435e8 0.532628
\(319\) 1.60341e8 0.276553
\(320\) −6.70098e8 −1.14318
\(321\) 5.10619e8 0.861647
\(322\) −7.69545e8 −1.28451
\(323\) 1.10898e9 1.83111
\(324\) 1.79576e8 0.293319
\(325\) −5.63399e7 −0.0910383
\(326\) 1.47044e9 2.35063
\(327\) 4.32108e8 0.683401
\(328\) 1.19819e9 1.87486
\(329\) −1.53133e8 −0.237073
\(330\) 5.26454e7 0.0806420
\(331\) 3.83779e7 0.0581679 0.0290839 0.999577i \(-0.490741\pi\)
0.0290839 + 0.999577i \(0.490741\pi\)
\(332\) 3.33674e9 5.00425
\(333\) −1.14498e8 −0.169919
\(334\) 7.60270e8 1.11649
\(335\) 2.11739e8 0.307712
\(336\) 9.53583e8 1.37142
\(337\) 8.56217e8 1.21865 0.609325 0.792920i \(-0.291440\pi\)
0.609325 + 0.792920i \(0.291440\pi\)
\(338\) −1.33834e9 −1.88520
\(339\) 1.28994e8 0.179833
\(340\) −1.15785e9 −1.59763
\(341\) −1.38409e7 −0.0189028
\(342\) 5.77169e8 0.780210
\(343\) 7.95029e8 1.06378
\(344\) 4.42043e8 0.585477
\(345\) 1.68483e8 0.220897
\(346\) −1.15656e9 −1.50108
\(347\) −2.36529e8 −0.303901 −0.151950 0.988388i \(-0.548555\pi\)
−0.151950 + 0.988388i \(0.548555\pi\)
\(348\) −1.83533e9 −2.33447
\(349\) −2.69181e8 −0.338966 −0.169483 0.985533i \(-0.554210\pi\)
−0.169483 + 0.985533i \(0.554210\pi\)
\(350\) 9.12400e8 1.13749
\(351\) −1.69873e7 −0.0209677
\(352\) 4.76134e8 0.581875
\(353\) −4.00304e8 −0.484371 −0.242186 0.970230i \(-0.577864\pi\)
−0.242186 + 0.970230i \(0.577864\pi\)
\(354\) 1.19693e8 0.143402
\(355\) 3.59805e8 0.426842
\(356\) −2.19364e9 −2.57686
\(357\) 5.28586e8 0.614861
\(358\) −2.71392e9 −3.12613
\(359\) −6.09817e8 −0.695615 −0.347808 0.937566i \(-0.613074\pi\)
−0.347808 + 0.937566i \(0.613074\pi\)
\(360\) −3.74332e8 −0.422862
\(361\) 4.51538e8 0.505149
\(362\) 1.37258e9 1.52075
\(363\) 5.09001e8 0.558528
\(364\) −1.88835e8 −0.205223
\(365\) −3.43608e7 −0.0369861
\(366\) −1.46266e9 −1.55941
\(367\) −3.65526e8 −0.386000 −0.193000 0.981199i \(-0.561822\pi\)
−0.193000 + 0.981199i \(0.561822\pi\)
\(368\) 3.00311e9 3.14126
\(369\) 1.92791e8 0.199754
\(370\) 3.84219e8 0.394342
\(371\) 3.39361e8 0.345028
\(372\) 1.58429e8 0.159564
\(373\) −1.64569e9 −1.64198 −0.820990 0.570942i \(-0.806578\pi\)
−0.820990 + 0.570942i \(0.806578\pi\)
\(374\) 5.20154e8 0.514140
\(375\) −4.38825e8 −0.429716
\(376\) 1.07147e9 1.03950
\(377\) 1.73617e8 0.166878
\(378\) 2.75102e8 0.261983
\(379\) 1.32916e9 1.25412 0.627062 0.778969i \(-0.284257\pi\)
0.627062 + 0.778969i \(0.284257\pi\)
\(380\) −1.40469e9 −1.31323
\(381\) −7.31124e8 −0.677257
\(382\) 4.77115e8 0.437927
\(383\) 1.43631e9 1.30633 0.653163 0.757217i \(-0.273442\pi\)
0.653163 + 0.757217i \(0.273442\pi\)
\(384\) −1.38128e9 −1.24486
\(385\) 5.84930e7 0.0522386
\(386\) −5.91360e7 −0.0523355
\(387\) 7.11254e7 0.0623787
\(388\) −2.85038e6 −0.00247738
\(389\) 1.62608e9 1.40062 0.700309 0.713840i \(-0.253046\pi\)
0.700309 + 0.713840i \(0.253046\pi\)
\(390\) 5.70043e7 0.0486610
\(391\) 1.66467e9 1.40835
\(392\) −1.83158e9 −1.53576
\(393\) −1.81500e8 −0.150836
\(394\) 3.15039e8 0.259494
\(395\) −4.01310e8 −0.327635
\(396\) 1.96339e8 0.158882
\(397\) 1.66052e9 1.33192 0.665958 0.745989i \(-0.268023\pi\)
0.665958 + 0.745989i \(0.268023\pi\)
\(398\) −2.72680e9 −2.16802
\(399\) 6.41278e8 0.505407
\(400\) −3.56059e9 −2.78171
\(401\) 1.65806e9 1.28409 0.642045 0.766667i \(-0.278086\pi\)
0.642045 + 0.766667i \(0.278086\pi\)
\(402\) 1.08881e9 0.835911
\(403\) −1.49869e7 −0.0114063
\(404\) 1.26629e9 0.955431
\(405\) −6.02306e7 −0.0450531
\(406\) −2.81165e9 −2.08507
\(407\) −1.25186e8 −0.0920398
\(408\) −3.69852e9 −2.69599
\(409\) 2.60200e9 1.88051 0.940256 0.340468i \(-0.110585\pi\)
0.940256 + 0.340468i \(0.110585\pi\)
\(410\) −6.46947e8 −0.463581
\(411\) −1.18558e9 −0.842337
\(412\) −2.42003e9 −1.70483
\(413\) 1.32988e8 0.0928936
\(414\) 8.66376e8 0.600075
\(415\) −1.11916e9 −0.768640
\(416\) 5.15557e8 0.351115
\(417\) −2.30913e8 −0.155945
\(418\) 6.31047e8 0.422615
\(419\) −1.66480e9 −1.10564 −0.552819 0.833302i \(-0.686448\pi\)
−0.552819 + 0.833302i \(0.686448\pi\)
\(420\) −6.69535e8 −0.440962
\(421\) 2.00138e9 1.30720 0.653601 0.756839i \(-0.273257\pi\)
0.653601 + 0.756839i \(0.273257\pi\)
\(422\) 4.35936e9 2.82377
\(423\) 1.72401e8 0.110751
\(424\) −2.37451e9 −1.51284
\(425\) −1.97369e9 −1.24715
\(426\) 1.85019e9 1.15953
\(427\) −1.62512e9 −1.01016
\(428\) −6.39037e9 −3.93979
\(429\) −1.85731e7 −0.0113575
\(430\) −2.38675e8 −0.144766
\(431\) 4.72430e8 0.284228 0.142114 0.989850i \(-0.454610\pi\)
0.142114 + 0.989850i \(0.454610\pi\)
\(432\) −1.07357e9 −0.640675
\(433\) 6.77203e8 0.400877 0.200439 0.979706i \(-0.435763\pi\)
0.200439 + 0.979706i \(0.435763\pi\)
\(434\) 2.42707e8 0.142517
\(435\) 6.15580e8 0.358568
\(436\) −5.40781e9 −3.12477
\(437\) 2.01957e9 1.15764
\(438\) −1.76691e8 −0.100474
\(439\) 1.96136e9 1.10645 0.553224 0.833033i \(-0.313397\pi\)
0.553224 + 0.833033i \(0.313397\pi\)
\(440\) −4.09276e8 −0.229051
\(441\) −2.94703e8 −0.163625
\(442\) 5.63221e8 0.310242
\(443\) 7.15483e7 0.0391008 0.0195504 0.999809i \(-0.493777\pi\)
0.0195504 + 0.999809i \(0.493777\pi\)
\(444\) 1.43293e9 0.776936
\(445\) 7.35757e8 0.395799
\(446\) −5.26952e9 −2.81254
\(447\) 1.18932e9 0.629826
\(448\) −3.82853e9 −2.01168
\(449\) −2.36841e9 −1.23479 −0.617397 0.786652i \(-0.711813\pi\)
−0.617397 + 0.786652i \(0.711813\pi\)
\(450\) −1.02721e9 −0.531391
\(451\) 2.10788e8 0.108200
\(452\) −1.61435e9 −0.822266
\(453\) 1.56404e9 0.790503
\(454\) 1.44580e9 0.725127
\(455\) 6.33360e7 0.0315218
\(456\) −4.48703e9 −2.21606
\(457\) 9.23556e8 0.452644 0.226322 0.974053i \(-0.427330\pi\)
0.226322 + 0.974053i \(0.427330\pi\)
\(458\) −1.17125e8 −0.0569666
\(459\) −5.95098e8 −0.287239
\(460\) −2.10856e9 −1.01003
\(461\) −2.95040e8 −0.140258 −0.0701289 0.997538i \(-0.522341\pi\)
−0.0701289 + 0.997538i \(0.522341\pi\)
\(462\) 3.00783e8 0.141908
\(463\) 2.57536e9 1.20588 0.602940 0.797787i \(-0.293996\pi\)
0.602940 + 0.797787i \(0.293996\pi\)
\(464\) 1.09723e10 5.09901
\(465\) −5.31379e7 −0.0245086
\(466\) −2.46219e9 −1.12712
\(467\) −9.62315e8 −0.437228 −0.218614 0.975811i \(-0.570154\pi\)
−0.218614 + 0.975811i \(0.570154\pi\)
\(468\) 2.12595e8 0.0958724
\(469\) 1.20975e9 0.541489
\(470\) −5.78526e8 −0.257028
\(471\) 1.75193e9 0.772581
\(472\) −9.30515e8 −0.407311
\(473\) 7.77649e7 0.0337886
\(474\) −2.06362e9 −0.890032
\(475\) −2.39447e9 −1.02514
\(476\) −6.61523e9 −2.81139
\(477\) −3.82063e8 −0.161184
\(478\) 5.07482e9 2.12531
\(479\) −1.04097e9 −0.432777 −0.216389 0.976307i \(-0.569428\pi\)
−0.216389 + 0.976307i \(0.569428\pi\)
\(480\) 1.82797e9 0.754439
\(481\) −1.35551e8 −0.0555387
\(482\) −4.62391e9 −1.88081
\(483\) 9.62609e8 0.388718
\(484\) −6.37011e9 −2.55381
\(485\) 956032. 0.000380519 0
\(486\) −3.09718e8 −0.122388
\(487\) 1.54301e9 0.605365 0.302683 0.953091i \(-0.402118\pi\)
0.302683 + 0.953091i \(0.402118\pi\)
\(488\) 1.13710e10 4.42924
\(489\) −1.83934e9 −0.711347
\(490\) 9.88934e8 0.379736
\(491\) 8.27706e8 0.315566 0.157783 0.987474i \(-0.449565\pi\)
0.157783 + 0.987474i \(0.449565\pi\)
\(492\) −2.41277e9 −0.913351
\(493\) 6.08213e9 2.28608
\(494\) 6.83296e8 0.255014
\(495\) −6.58531e7 −0.0244038
\(496\) −9.47149e8 −0.348524
\(497\) 2.05570e9 0.751126
\(498\) −5.75494e9 −2.08804
\(499\) −5.08912e9 −1.83354 −0.916771 0.399413i \(-0.869214\pi\)
−0.916771 + 0.399413i \(0.869214\pi\)
\(500\) 5.49186e9 1.96483
\(501\) −9.51006e8 −0.337871
\(502\) −3.32047e9 −1.17149
\(503\) 1.24283e9 0.435434 0.217717 0.976012i \(-0.430139\pi\)
0.217717 + 0.976012i \(0.430139\pi\)
\(504\) −2.13870e9 −0.744120
\(505\) −4.24721e8 −0.146752
\(506\) 9.47252e8 0.325042
\(507\) 1.67410e9 0.570497
\(508\) 9.14997e9 3.09668
\(509\) −3.51527e9 −1.18154 −0.590768 0.806842i \(-0.701175\pi\)
−0.590768 + 0.806842i \(0.701175\pi\)
\(510\) 1.99697e9 0.666615
\(511\) −1.96317e8 −0.0650854
\(512\) 9.51036e8 0.313150
\(513\) −7.21969e8 −0.236106
\(514\) 1.02340e10 3.32411
\(515\) 8.11689e8 0.261857
\(516\) −8.90130e8 −0.285220
\(517\) 1.88495e8 0.0599905
\(518\) 2.19519e9 0.693934
\(519\) 1.44672e9 0.454255
\(520\) −4.43163e8 −0.138214
\(521\) −4.15959e9 −1.28860 −0.644301 0.764772i \(-0.722852\pi\)
−0.644301 + 0.764772i \(0.722852\pi\)
\(522\) 3.16544e9 0.974064
\(523\) −9.27018e8 −0.283356 −0.141678 0.989913i \(-0.545250\pi\)
−0.141678 + 0.989913i \(0.545250\pi\)
\(524\) 2.27147e9 0.689679
\(525\) −1.14130e9 −0.344226
\(526\) −3.25406e9 −0.974936
\(527\) −5.25020e8 −0.156257
\(528\) −1.17379e9 −0.347034
\(529\) −3.73295e8 −0.109637
\(530\) 1.28209e9 0.374069
\(531\) −1.49721e8 −0.0433963
\(532\) −8.02555e9 −2.31092
\(533\) 2.28241e8 0.0652902
\(534\) 3.78342e9 1.07520
\(535\) 2.14336e9 0.605142
\(536\) −8.46461e9 −2.37427
\(537\) 3.39479e9 0.946026
\(538\) 1.18730e10 3.28716
\(539\) −3.22214e8 −0.0886306
\(540\) 7.53782e8 0.206000
\(541\) −6.33596e9 −1.72037 −0.860186 0.509981i \(-0.829652\pi\)
−0.860186 + 0.509981i \(0.829652\pi\)
\(542\) 1.07170e10 2.89118
\(543\) −1.71693e9 −0.460207
\(544\) 1.80609e10 4.80998
\(545\) 1.81381e9 0.479958
\(546\) 3.25687e8 0.0856301
\(547\) 2.00352e9 0.523406 0.261703 0.965148i \(-0.415716\pi\)
0.261703 + 0.965148i \(0.415716\pi\)
\(548\) 1.48375e10 3.85149
\(549\) 1.82961e9 0.471906
\(550\) −1.12310e9 −0.287838
\(551\) 7.37881e9 1.87912
\(552\) −6.73538e9 −1.70442
\(553\) −2.29284e9 −0.576548
\(554\) 2.33704e9 0.583959
\(555\) −4.80613e8 −0.119336
\(556\) 2.88986e9 0.713042
\(557\) −4.93285e9 −1.20950 −0.604748 0.796417i \(-0.706726\pi\)
−0.604748 + 0.796417i \(0.706726\pi\)
\(558\) −2.73246e8 −0.0665786
\(559\) 8.42036e7 0.0203887
\(560\) 4.00273e9 0.963160
\(561\) −6.50650e8 −0.155589
\(562\) −1.37207e10 −3.26061
\(563\) 4.64054e9 1.09595 0.547973 0.836496i \(-0.315400\pi\)
0.547973 + 0.836496i \(0.315400\pi\)
\(564\) −2.15759e9 −0.506398
\(565\) 5.41460e8 0.126298
\(566\) 3.93417e9 0.912001
\(567\) −3.44120e8 −0.0792811
\(568\) −1.43838e10 −3.29347
\(569\) −4.82571e9 −1.09817 −0.549083 0.835768i \(-0.685023\pi\)
−0.549083 + 0.835768i \(0.685023\pi\)
\(570\) 2.42271e9 0.547947
\(571\) −6.84565e9 −1.53882 −0.769410 0.638755i \(-0.779450\pi\)
−0.769410 + 0.638755i \(0.779450\pi\)
\(572\) 2.32441e8 0.0519310
\(573\) −5.96814e8 −0.132525
\(574\) −3.69626e9 −0.815775
\(575\) −3.59429e9 −0.788453
\(576\) 4.31026e9 0.939778
\(577\) −3.78107e9 −0.819407 −0.409704 0.912219i \(-0.634368\pi\)
−0.409704 + 0.912219i \(0.634368\pi\)
\(578\) 1.08736e10 2.34221
\(579\) 7.39721e7 0.0158377
\(580\) −7.70395e9 −1.63951
\(581\) −6.39418e9 −1.35260
\(582\) 4.91612e6 0.00103369
\(583\) −4.17729e8 −0.0873080
\(584\) 1.37363e9 0.285381
\(585\) −7.13055e7 −0.0147258
\(586\) −6.35757e9 −1.30512
\(587\) 4.65957e9 0.950852 0.475426 0.879756i \(-0.342294\pi\)
0.475426 + 0.879756i \(0.342294\pi\)
\(588\) 3.68820e9 0.748158
\(589\) −6.36951e8 −0.128441
\(590\) 5.02419e8 0.100713
\(591\) −3.94076e8 −0.0785279
\(592\) −8.56661e9 −1.69701
\(593\) −7.46187e9 −1.46945 −0.734727 0.678363i \(-0.762690\pi\)
−0.734727 + 0.678363i \(0.762690\pi\)
\(594\) −3.38630e8 −0.0662939
\(595\) 2.21878e9 0.431822
\(596\) −1.48842e10 −2.87981
\(597\) 3.41090e9 0.656083
\(598\) 1.02568e9 0.196137
\(599\) −4.56957e9 −0.868723 −0.434362 0.900739i \(-0.643026\pi\)
−0.434362 + 0.900739i \(0.643026\pi\)
\(600\) 7.98571e9 1.50933
\(601\) −3.26243e9 −0.613028 −0.306514 0.951866i \(-0.599163\pi\)
−0.306514 + 0.951866i \(0.599163\pi\)
\(602\) −1.36364e9 −0.254749
\(603\) −1.36197e9 −0.252963
\(604\) −1.95738e10 −3.61449
\(605\) 2.13657e9 0.392259
\(606\) −2.18400e9 −0.398657
\(607\) 7.83444e9 1.42183 0.710915 0.703278i \(-0.248281\pi\)
0.710915 + 0.703278i \(0.248281\pi\)
\(608\) 2.19114e10 3.95373
\(609\) 3.51704e9 0.630982
\(610\) −6.13962e9 −1.09518
\(611\) 2.04102e8 0.0361995
\(612\) 7.44761e9 1.31337
\(613\) 1.83847e9 0.322363 0.161181 0.986925i \(-0.448470\pi\)
0.161181 + 0.986925i \(0.448470\pi\)
\(614\) −1.46843e10 −2.56014
\(615\) 8.09254e8 0.140288
\(616\) −2.33835e9 −0.403067
\(617\) −7.28601e9 −1.24880 −0.624399 0.781106i \(-0.714656\pi\)
−0.624399 + 0.781106i \(0.714656\pi\)
\(618\) 4.17388e9 0.711344
\(619\) 2.62525e8 0.0444891 0.0222446 0.999753i \(-0.492919\pi\)
0.0222446 + 0.999753i \(0.492919\pi\)
\(620\) 6.65018e8 0.112063
\(621\) −1.08373e9 −0.181594
\(622\) −3.97067e9 −0.661603
\(623\) 4.20366e9 0.696497
\(624\) −1.27098e9 −0.209407
\(625\) 3.25803e9 0.533795
\(626\) −9.62502e9 −1.56816
\(627\) −7.89365e8 −0.127891
\(628\) −2.19253e10 −3.53254
\(629\) −4.74861e9 −0.760833
\(630\) 1.15476e9 0.183993
\(631\) 1.62131e9 0.256899 0.128450 0.991716i \(-0.459000\pi\)
0.128450 + 0.991716i \(0.459000\pi\)
\(632\) 1.60430e10 2.52799
\(633\) −5.45303e9 −0.854526
\(634\) −1.16840e10 −1.82087
\(635\) −3.06895e9 −0.475643
\(636\) 4.78150e9 0.736994
\(637\) −3.48892e8 −0.0534815
\(638\) 3.46094e9 0.527620
\(639\) −2.31437e9 −0.350897
\(640\) −5.79801e9 −0.874277
\(641\) 1.50371e9 0.225508 0.112754 0.993623i \(-0.464033\pi\)
0.112754 + 0.993623i \(0.464033\pi\)
\(642\) 1.10216e10 1.64389
\(643\) 7.26580e9 1.07782 0.538908 0.842365i \(-0.318837\pi\)
0.538908 + 0.842365i \(0.318837\pi\)
\(644\) −1.20470e10 −1.77737
\(645\) 2.98554e8 0.0438090
\(646\) 2.39371e10 3.49348
\(647\) −1.31988e10 −1.91588 −0.957942 0.286962i \(-0.907355\pi\)
−0.957942 + 0.286962i \(0.907355\pi\)
\(648\) 2.40781e9 0.347624
\(649\) −1.63698e8 −0.0235064
\(650\) −1.21609e9 −0.173687
\(651\) −3.03597e8 −0.0431285
\(652\) 2.30193e10 3.25256
\(653\) −1.08050e10 −1.51855 −0.759276 0.650768i \(-0.774447\pi\)
−0.759276 + 0.650768i \(0.774447\pi\)
\(654\) 9.32697e9 1.30382
\(655\) −7.61861e8 −0.105933
\(656\) 1.44244e10 1.99497
\(657\) 2.21019e8 0.0304054
\(658\) −3.30534e9 −0.452299
\(659\) −1.38422e10 −1.88411 −0.942053 0.335464i \(-0.891107\pi\)
−0.942053 + 0.335464i \(0.891107\pi\)
\(660\) 8.24148e8 0.111584
\(661\) 1.08222e10 1.45751 0.728756 0.684773i \(-0.240099\pi\)
0.728756 + 0.684773i \(0.240099\pi\)
\(662\) 8.28379e8 0.110975
\(663\) −7.04522e8 −0.0938852
\(664\) 4.47401e10 5.93074
\(665\) 2.69181e9 0.354951
\(666\) −2.47141e9 −0.324179
\(667\) 1.10762e10 1.44527
\(668\) 1.19018e10 1.54488
\(669\) 6.59154e9 0.851130
\(670\) 4.57035e9 0.587067
\(671\) 2.00041e9 0.255617
\(672\) 1.04439e10 1.32761
\(673\) −7.77779e9 −0.983566 −0.491783 0.870718i \(-0.663655\pi\)
−0.491783 + 0.870718i \(0.663655\pi\)
\(674\) 1.84813e10 2.32500
\(675\) 1.28491e9 0.160809
\(676\) −2.09513e10 −2.60853
\(677\) −9.88602e9 −1.22451 −0.612253 0.790662i \(-0.709737\pi\)
−0.612253 + 0.790662i \(0.709737\pi\)
\(678\) 2.78430e9 0.343093
\(679\) 5.46218e6 0.000669610 0
\(680\) −1.55248e10 −1.89341
\(681\) −1.80853e9 −0.219437
\(682\) −2.98754e8 −0.0360635
\(683\) 1.44552e10 1.73601 0.868003 0.496559i \(-0.165403\pi\)
0.868003 + 0.496559i \(0.165403\pi\)
\(684\) 9.03540e9 1.07957
\(685\) −4.97657e9 −0.591580
\(686\) 1.71605e10 2.02953
\(687\) 1.46509e8 0.0172392
\(688\) 5.32153e9 0.622984
\(689\) −4.52315e8 −0.0526834
\(690\) 3.63668e9 0.421437
\(691\) −9.78421e9 −1.12811 −0.564056 0.825736i \(-0.690760\pi\)
−0.564056 + 0.825736i \(0.690760\pi\)
\(692\) −1.81056e10 −2.07703
\(693\) −3.76244e8 −0.0429441
\(694\) −5.10544e9 −0.579795
\(695\) −9.69273e8 −0.109521
\(696\) −2.46088e10 −2.76667
\(697\) 7.99570e9 0.894420
\(698\) −5.81022e9 −0.646694
\(699\) 3.07991e9 0.341089
\(700\) 1.42833e10 1.57393
\(701\) −1.39642e7 −0.00153110 −0.000765548 1.00000i \(-0.500244\pi\)
−0.000765548 1.00000i \(0.500244\pi\)
\(702\) −3.66668e8 −0.0400031
\(703\) −5.76099e9 −0.625393
\(704\) 4.71263e9 0.509048
\(705\) 7.23667e8 0.0777815
\(706\) −8.64047e9 −0.924105
\(707\) −2.42659e9 −0.258243
\(708\) 1.87375e9 0.198425
\(709\) 1.10083e10 1.16000 0.580002 0.814615i \(-0.303052\pi\)
0.580002 + 0.814615i \(0.303052\pi\)
\(710\) 7.76631e9 0.814349
\(711\) 2.58134e9 0.269341
\(712\) −2.94130e10 −3.05394
\(713\) −9.56114e8 −0.0987862
\(714\) 1.14094e10 1.17306
\(715\) −7.79619e7 −0.00797648
\(716\) −4.24856e10 −4.32560
\(717\) −6.34799e9 −0.643161
\(718\) −1.31628e10 −1.32713
\(719\) −1.00251e10 −1.00586 −0.502932 0.864326i \(-0.667746\pi\)
−0.502932 + 0.864326i \(0.667746\pi\)
\(720\) −4.50639e9 −0.449951
\(721\) 4.63749e9 0.460797
\(722\) 9.74636e9 0.963745
\(723\) 5.78396e9 0.569169
\(724\) 2.14873e10 2.10425
\(725\) −1.31323e10 −1.27985
\(726\) 1.09867e10 1.06559
\(727\) 1.36958e9 0.132195 0.0660977 0.997813i \(-0.478945\pi\)
0.0660977 + 0.997813i \(0.478945\pi\)
\(728\) −2.53196e9 −0.243219
\(729\) 3.87420e8 0.0370370
\(730\) −7.41672e8 −0.0705637
\(731\) 2.94981e9 0.279308
\(732\) −2.28975e10 −2.15774
\(733\) −2.47923e9 −0.232516 −0.116258 0.993219i \(-0.537090\pi\)
−0.116258 + 0.993219i \(0.537090\pi\)
\(734\) −7.88981e9 −0.736428
\(735\) −1.23704e9 −0.114915
\(736\) 3.28907e10 3.04089
\(737\) −1.48911e9 −0.137022
\(738\) 4.16135e9 0.381099
\(739\) −4.61946e9 −0.421052 −0.210526 0.977588i \(-0.567518\pi\)
−0.210526 + 0.977588i \(0.567518\pi\)
\(740\) 6.01484e9 0.545649
\(741\) −8.54722e8 −0.0771723
\(742\) 7.32505e9 0.658259
\(743\) −1.55272e10 −1.38878 −0.694389 0.719600i \(-0.744325\pi\)
−0.694389 + 0.719600i \(0.744325\pi\)
\(744\) 2.12427e9 0.189106
\(745\) 4.99224e9 0.442332
\(746\) −3.55220e10 −3.13264
\(747\) 7.19875e9 0.631880
\(748\) 8.14285e9 0.711411
\(749\) 1.22458e10 1.06488
\(750\) −9.47194e9 −0.819831
\(751\) −5.50391e9 −0.474167 −0.237084 0.971489i \(-0.576192\pi\)
−0.237084 + 0.971489i \(0.576192\pi\)
\(752\) 1.28989e10 1.10609
\(753\) 4.15352e9 0.354514
\(754\) 3.74749e9 0.318376
\(755\) 6.56516e9 0.555177
\(756\) 4.30664e9 0.362504
\(757\) −1.57056e10 −1.31589 −0.657946 0.753065i \(-0.728574\pi\)
−0.657946 + 0.753065i \(0.728574\pi\)
\(758\) 2.86897e10 2.39268
\(759\) −1.18490e9 −0.0983638
\(760\) −1.88346e10 −1.55636
\(761\) −1.80765e10 −1.48685 −0.743427 0.668817i \(-0.766801\pi\)
−0.743427 + 0.668817i \(0.766801\pi\)
\(762\) −1.57812e10 −1.29210
\(763\) 1.03630e10 0.844594
\(764\) 7.46910e9 0.605956
\(765\) −2.49797e9 −0.201730
\(766\) 3.10024e10 2.49227
\(767\) −1.77251e8 −0.0141842
\(768\) −9.38075e9 −0.747262
\(769\) −1.35958e10 −1.07811 −0.539053 0.842272i \(-0.681218\pi\)
−0.539053 + 0.842272i \(0.681218\pi\)
\(770\) 1.26256e9 0.0996630
\(771\) −1.28015e10 −1.00594
\(772\) −9.25757e8 −0.0724163
\(773\) 1.25983e9 0.0981032 0.0490516 0.998796i \(-0.484380\pi\)
0.0490516 + 0.998796i \(0.484380\pi\)
\(774\) 1.53523e9 0.119009
\(775\) 1.13360e9 0.0874792
\(776\) −3.82189e7 −0.00293604
\(777\) −2.74592e9 −0.209998
\(778\) 3.50987e10 2.67216
\(779\) 9.70033e9 0.735200
\(780\) 8.92385e8 0.0673319
\(781\) −2.53041e9 −0.190070
\(782\) 3.59316e10 2.68691
\(783\) −3.95959e9 −0.294770
\(784\) −2.20494e10 −1.63415
\(785\) 7.35386e9 0.542590
\(786\) −3.91765e9 −0.287771
\(787\) 1.93512e10 1.41513 0.707566 0.706647i \(-0.249793\pi\)
0.707566 + 0.706647i \(0.249793\pi\)
\(788\) 4.93184e9 0.359060
\(789\) 4.07045e9 0.295034
\(790\) −8.66220e9 −0.625077
\(791\) 3.09357e9 0.222250
\(792\) 2.63258e9 0.188297
\(793\) 2.16603e9 0.154244
\(794\) 3.58420e10 2.54109
\(795\) −1.60374e9 −0.113200
\(796\) −4.26872e10 −2.99987
\(797\) 1.07925e10 0.755122 0.377561 0.925985i \(-0.376763\pi\)
0.377561 + 0.925985i \(0.376763\pi\)
\(798\) 1.38419e10 0.964238
\(799\) 7.15006e9 0.495902
\(800\) −3.89964e10 −2.69284
\(801\) −4.73260e9 −0.325377
\(802\) 3.57890e10 2.44985
\(803\) 2.41651e8 0.0164696
\(804\) 1.70450e10 1.15664
\(805\) 4.04062e9 0.273000
\(806\) −3.23490e8 −0.0217615
\(807\) −1.48517e10 −0.994759
\(808\) 1.69789e10 1.13232
\(809\) −4.14105e9 −0.274973 −0.137487 0.990504i \(-0.543902\pi\)
−0.137487 + 0.990504i \(0.543902\pi\)
\(810\) −1.30006e9 −0.0859542
\(811\) 3.15148e9 0.207463 0.103732 0.994605i \(-0.466922\pi\)
0.103732 + 0.994605i \(0.466922\pi\)
\(812\) −4.40156e10 −2.88510
\(813\) −1.34056e10 −0.874925
\(814\) −2.70212e9 −0.175598
\(815\) −7.72077e9 −0.499585
\(816\) −4.45247e10 −2.86870
\(817\) 3.57869e9 0.229587
\(818\) 5.61637e10 3.58773
\(819\) −4.07396e8 −0.0259133
\(820\) −1.01278e10 −0.641454
\(821\) −3.16518e10 −1.99617 −0.998085 0.0618543i \(-0.980299\pi\)
−0.998085 + 0.0618543i \(0.980299\pi\)
\(822\) −2.55906e10 −1.60705
\(823\) −2.99731e9 −0.187427 −0.0937135 0.995599i \(-0.529874\pi\)
−0.0937135 + 0.995599i \(0.529874\pi\)
\(824\) −3.24485e10 −2.02046
\(825\) 1.40486e9 0.0871052
\(826\) 2.87051e9 0.177227
\(827\) −6.29398e9 −0.386951 −0.193475 0.981105i \(-0.561976\pi\)
−0.193475 + 0.981105i \(0.561976\pi\)
\(828\) 1.35629e10 0.830319
\(829\) 1.29264e10 0.788018 0.394009 0.919107i \(-0.371088\pi\)
0.394009 + 0.919107i \(0.371088\pi\)
\(830\) −2.41568e10 −1.46645
\(831\) −2.92336e9 −0.176717
\(832\) 5.10282e9 0.307170
\(833\) −1.22223e10 −0.732651
\(834\) −4.98421e9 −0.297519
\(835\) −3.99192e9 −0.237290
\(836\) 9.87886e9 0.584769
\(837\) 3.41798e8 0.0201480
\(838\) −3.59343e10 −2.10938
\(839\) 2.72788e9 0.159462 0.0797312 0.996816i \(-0.474594\pi\)
0.0797312 + 0.996816i \(0.474594\pi\)
\(840\) −8.97735e9 −0.522602
\(841\) 2.32187e10 1.34602
\(842\) 4.31994e10 2.49394
\(843\) 1.71629e10 0.986723
\(844\) 6.82444e10 3.90722
\(845\) 7.02715e9 0.400664
\(846\) 3.72125e9 0.211296
\(847\) 1.22070e10 0.690268
\(848\) −2.85856e10 −1.60976
\(849\) −4.92118e9 −0.275989
\(850\) −4.26017e10 −2.37936
\(851\) −8.64770e9 −0.481002
\(852\) 2.89642e10 1.60444
\(853\) −1.22649e10 −0.676618 −0.338309 0.941035i \(-0.609855\pi\)
−0.338309 + 0.941035i \(0.609855\pi\)
\(854\) −3.50780e10 −1.92722
\(855\) −3.03052e9 −0.165819
\(856\) −8.56842e10 −4.66920
\(857\) −3.61684e10 −1.96289 −0.981447 0.191732i \(-0.938589\pi\)
−0.981447 + 0.191732i \(0.938589\pi\)
\(858\) −4.00897e8 −0.0216684
\(859\) −9.13867e9 −0.491934 −0.245967 0.969278i \(-0.579106\pi\)
−0.245967 + 0.969278i \(0.579106\pi\)
\(860\) −3.73638e9 −0.200312
\(861\) 4.62358e9 0.246869
\(862\) 1.01973e10 0.542262
\(863\) 2.82611e10 1.49676 0.748378 0.663272i \(-0.230833\pi\)
0.748378 + 0.663272i \(0.230833\pi\)
\(864\) −1.17580e10 −0.620206
\(865\) 6.07272e9 0.319027
\(866\) 1.46173e10 0.764811
\(867\) −1.36016e10 −0.708798
\(868\) 3.79950e9 0.197200
\(869\) 2.82231e9 0.145893
\(870\) 1.32872e10 0.684093
\(871\) −1.61240e9 −0.0826818
\(872\) −7.25097e10 −3.70330
\(873\) −6.14948e6 −0.000312816 0
\(874\) 4.35919e10 2.20860
\(875\) −1.05240e10 −0.531073
\(876\) −2.76604e9 −0.139025
\(877\) −8.41638e9 −0.421334 −0.210667 0.977558i \(-0.567564\pi\)
−0.210667 + 0.977558i \(0.567564\pi\)
\(878\) 4.23355e10 2.11093
\(879\) 7.95257e9 0.394954
\(880\) −4.92707e9 −0.243724
\(881\) 2.07127e10 1.02052 0.510259 0.860021i \(-0.329550\pi\)
0.510259 + 0.860021i \(0.329550\pi\)
\(882\) −6.36111e9 −0.312171
\(883\) −1.60168e10 −0.782910 −0.391455 0.920197i \(-0.628028\pi\)
−0.391455 + 0.920197i \(0.628028\pi\)
\(884\) 8.81705e9 0.429280
\(885\) −6.28466e8 −0.0304776
\(886\) 1.54435e9 0.0745983
\(887\) 2.24851e10 1.08184 0.540920 0.841074i \(-0.318076\pi\)
0.540920 + 0.841074i \(0.318076\pi\)
\(888\) 1.92132e10 0.920779
\(889\) −1.75341e10 −0.837002
\(890\) 1.58812e10 0.755122
\(891\) 4.23586e8 0.0200618
\(892\) −8.24928e10 −3.89170
\(893\) 8.67441e9 0.407624
\(894\) 2.56711e10 1.20161
\(895\) 1.42499e10 0.664402
\(896\) −3.31263e10 −1.53849
\(897\) −1.28301e9 −0.0593547
\(898\) −5.11216e10 −2.35580
\(899\) −3.49331e9 −0.160354
\(900\) −1.60806e10 −0.735281
\(901\) −1.58454e10 −0.721718
\(902\) 4.54982e9 0.206429
\(903\) 1.70575e9 0.0770919
\(904\) −2.16457e10 −0.974501
\(905\) −7.20694e9 −0.323207
\(906\) 3.37594e10 1.50816
\(907\) 1.63521e10 0.727693 0.363847 0.931459i \(-0.381463\pi\)
0.363847 + 0.931459i \(0.381463\pi\)
\(908\) 2.26336e10 1.00335
\(909\) 2.73193e9 0.120641
\(910\) 1.36710e9 0.0601387
\(911\) 5.02950e9 0.220399 0.110200 0.993909i \(-0.464851\pi\)
0.110200 + 0.993909i \(0.464851\pi\)
\(912\) −5.40170e10 −2.35803
\(913\) 7.87075e9 0.342270
\(914\) 1.99348e10 0.863574
\(915\) 7.67993e9 0.331424
\(916\) −1.83356e9 −0.0788243
\(917\) −4.35280e9 −0.186413
\(918\) −1.28451e10 −0.548008
\(919\) −2.98229e10 −1.26749 −0.633746 0.773541i \(-0.718483\pi\)
−0.633746 + 0.773541i \(0.718483\pi\)
\(920\) −2.82723e10 −1.19702
\(921\) 1.83683e10 0.774749
\(922\) −6.36837e9 −0.267590
\(923\) −2.73993e9 −0.114692
\(924\) 4.70867e9 0.196357
\(925\) 1.02530e10 0.425947
\(926\) 5.55885e10 2.30063
\(927\) −5.22102e9 −0.215266
\(928\) 1.20172e11 4.93610
\(929\) 1.92466e10 0.787589 0.393794 0.919199i \(-0.371162\pi\)
0.393794 + 0.919199i \(0.371162\pi\)
\(930\) −1.14697e9 −0.0467587
\(931\) −1.48281e10 −0.602228
\(932\) −3.85449e10 −1.55959
\(933\) 4.96683e9 0.200214
\(934\) −2.07714e10 −0.834164
\(935\) −2.73115e9 −0.109271
\(936\) 2.85055e9 0.113622
\(937\) 2.73206e10 1.08493 0.542465 0.840078i \(-0.317491\pi\)
0.542465 + 0.840078i \(0.317491\pi\)
\(938\) 2.61121e10 1.03308
\(939\) 1.20397e10 0.474556
\(940\) −9.05664e9 −0.355648
\(941\) 4.38771e10 1.71662 0.858309 0.513132i \(-0.171515\pi\)
0.858309 + 0.513132i \(0.171515\pi\)
\(942\) 3.78151e10 1.47396
\(943\) 1.45610e10 0.565457
\(944\) −1.12020e10 −0.433405
\(945\) −1.44447e9 −0.0556797
\(946\) 1.67854e9 0.0644633
\(947\) −7.76685e9 −0.297180 −0.148590 0.988899i \(-0.547474\pi\)
−0.148590 + 0.988899i \(0.547474\pi\)
\(948\) −3.23054e10 −1.23153
\(949\) 2.61659e8 0.00993811
\(950\) −5.16842e10 −1.95580
\(951\) 1.46153e10 0.551031
\(952\) −8.86992e10 −3.33189
\(953\) −8.96167e9 −0.335401 −0.167700 0.985838i \(-0.553634\pi\)
−0.167700 + 0.985838i \(0.553634\pi\)
\(954\) −8.24675e9 −0.307513
\(955\) −2.50517e9 −0.0930734
\(956\) 7.94448e10 2.94078
\(957\) −4.32922e9 −0.159668
\(958\) −2.24692e10 −0.825672
\(959\) −2.84331e10 −1.04102
\(960\) 1.80926e10 0.660013
\(961\) −2.72111e10 −0.989040
\(962\) −2.92585e9 −0.105959
\(963\) −1.37867e10 −0.497472
\(964\) −7.23859e10 −2.60246
\(965\) 3.10503e8 0.0111230
\(966\) 2.07777e10 0.741614
\(967\) 3.13646e10 1.11544 0.557721 0.830029i \(-0.311676\pi\)
0.557721 + 0.830029i \(0.311676\pi\)
\(968\) −8.54126e10 −3.02662
\(969\) −2.99425e10 −1.05719
\(970\) 2.06358e7 0.000725971 0
\(971\) −3.71209e10 −1.30122 −0.650611 0.759411i \(-0.725487\pi\)
−0.650611 + 0.759411i \(0.725487\pi\)
\(972\) −4.84854e9 −0.169348
\(973\) −5.53783e9 −0.192728
\(974\) 3.33056e10 1.15494
\(975\) 1.52118e9 0.0525610
\(976\) 1.36890e11 4.71299
\(977\) 3.64354e10 1.24995 0.624975 0.780644i \(-0.285109\pi\)
0.624975 + 0.780644i \(0.285109\pi\)
\(978\) −3.97018e10 −1.35714
\(979\) −5.17439e9 −0.176246
\(980\) 1.54815e10 0.525437
\(981\) −1.16669e10 −0.394562
\(982\) 1.78659e10 0.602052
\(983\) −4.88356e9 −0.163983 −0.0819915 0.996633i \(-0.526128\pi\)
−0.0819915 + 0.996633i \(0.526128\pi\)
\(984\) −3.23512e10 −1.08245
\(985\) −1.65416e9 −0.0551508
\(986\) 1.31282e11 4.36149
\(987\) 4.13458e9 0.136874
\(988\) 1.06968e10 0.352862
\(989\) 5.37190e9 0.176580
\(990\) −1.42143e9 −0.0465587
\(991\) 2.89018e10 0.943336 0.471668 0.881776i \(-0.343652\pi\)
0.471668 + 0.881776i \(0.343652\pi\)
\(992\) −1.03734e10 −0.337389
\(993\) −1.03620e9 −0.0335832
\(994\) 4.43719e10 1.43303
\(995\) 1.43175e10 0.460772
\(996\) −9.00919e10 −2.88920
\(997\) −8.79812e9 −0.281162 −0.140581 0.990069i \(-0.544897\pi\)
−0.140581 + 0.990069i \(0.544897\pi\)
\(998\) −1.09848e11 −3.49811
\(999\) 3.09144e9 0.0981028
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.c.1.16 17
3.2 odd 2 531.8.a.c.1.2 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.16 17 1.1 even 1 trivial
531.8.a.c.1.2 17 3.2 odd 2