Properties

Label 177.8.a.d.1.15
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
Dimension $18$
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: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
Defining polynomial: \(x^{18} - 6 x^{17} - 1798 x^{16} + 11087 x^{15} + 1326765 x^{14} - 8403720 x^{13} - 518334228 x^{12} + 3375594921 x^{11} + 115310342333 x^{10} - 774932111214 x^{9} - 14600047830166 x^{8} + 102185027148481 x^{7} + 988557475638619 x^{6} - 7379206238519716 x^{5} - 30152342836849520 x^{4} + 260578770749067175 x^{3} + 182609347488069978 x^{2} - 3481290425710753600 x + 5164646074739714048\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Root \(15.9283\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+16.9283 q^{2} +27.0000 q^{3} +158.568 q^{4} -421.077 q^{5} +457.065 q^{6} -62.1753 q^{7} +517.469 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+16.9283 q^{2} +27.0000 q^{3} +158.568 q^{4} -421.077 q^{5} +457.065 q^{6} -62.1753 q^{7} +517.469 q^{8} +729.000 q^{9} -7128.13 q^{10} +8422.53 q^{11} +4281.34 q^{12} +11389.2 q^{13} -1052.52 q^{14} -11369.1 q^{15} -11536.9 q^{16} +1307.83 q^{17} +12340.7 q^{18} -7903.89 q^{19} -66769.4 q^{20} -1678.73 q^{21} +142579. q^{22} -2032.53 q^{23} +13971.7 q^{24} +99180.7 q^{25} +192800. q^{26} +19683.0 q^{27} -9859.03 q^{28} +64202.2 q^{29} -192459. q^{30} +320926. q^{31} -261536. q^{32} +227408. q^{33} +22139.4 q^{34} +26180.6 q^{35} +115596. q^{36} -143662. q^{37} -133800. q^{38} +307509. q^{39} -217894. q^{40} +577414. q^{41} -28418.2 q^{42} +817246. q^{43} +1.33555e6 q^{44} -306965. q^{45} -34407.3 q^{46} +264964. q^{47} -311495. q^{48} -819677. q^{49} +1.67896e6 q^{50} +35311.5 q^{51} +1.80597e6 q^{52} +1.60310e6 q^{53} +333200. q^{54} -3.54653e6 q^{55} -32173.8 q^{56} -213405. q^{57} +1.08684e6 q^{58} +205379. q^{59} -1.80277e6 q^{60} -274208. q^{61} +5.43273e6 q^{62} -45325.8 q^{63} -2.95064e6 q^{64} -4.79573e6 q^{65} +3.84964e6 q^{66} -1.66313e6 q^{67} +207381. q^{68} -54878.3 q^{69} +443194. q^{70} -2.97718e6 q^{71} +377235. q^{72} -2.72516e6 q^{73} -2.43196e6 q^{74} +2.67788e6 q^{75} -1.25331e6 q^{76} -523673. q^{77} +5.20561e6 q^{78} +1.33414e6 q^{79} +4.85790e6 q^{80} +531441. q^{81} +9.77465e6 q^{82} +8.00004e6 q^{83} -266194. q^{84} -550698. q^{85} +1.38346e7 q^{86} +1.73346e6 q^{87} +4.35840e6 q^{88} -1.07983e7 q^{89} -5.19640e6 q^{90} -708128. q^{91} -322295. q^{92} +8.66499e6 q^{93} +4.48540e6 q^{94} +3.32814e6 q^{95} -7.06146e6 q^{96} -1.36112e7 q^{97} -1.38758e7 q^{98} +6.14002e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + O(q^{10}) \) \( 18q + 24q^{2} + 486q^{3} + 1358q^{4} + 678q^{5} + 648q^{6} + 3081q^{7} + 4107q^{8} + 13122q^{9} + 3609q^{10} + 15070q^{11} + 36666q^{12} + 13662q^{13} + 20861q^{14} + 18306q^{15} + 60482q^{16} + 71919q^{17} + 17496q^{18} + 56231q^{19} + 143053q^{20} + 83187q^{21} + 274198q^{22} + 150029q^{23} + 110889q^{24} + 399672q^{25} + 182846q^{26} + 354294q^{27} + 434150q^{28} + 591285q^{29} + 97443q^{30} + 426733q^{31} + 1205630q^{32} + 406890q^{33} + 403548q^{34} + 912879q^{35} + 989982q^{36} + 7703q^{37} - 417859q^{38} + 368874q^{39} + 618020q^{40} + 770959q^{41} + 563247q^{42} + 793050q^{43} + 2591274q^{44} + 494262q^{45} - 4068019q^{46} + 1410373q^{47} + 1633014q^{48} + 1637427q^{49} + 1021549q^{50} + 1941813q^{51} - 3749190q^{52} + 1037934q^{53} + 472392q^{54} + 331974q^{55} - 391748q^{56} + 1518237q^{57} + 653724q^{58} + 3696822q^{59} + 3862431q^{60} - 1374623q^{61} + 5251718q^{62} + 2246049q^{63} + 5077197q^{64} + 3257170q^{65} + 7403346q^{66} - 2436904q^{67} + 14119909q^{68} + 4050783q^{69} + 5185580q^{70} + 14289172q^{71} + 2994003q^{72} + 5482515q^{73} + 14934154q^{74} + 10791144q^{75} + 3822912q^{76} + 23157109q^{77} + 4936842q^{78} + 19786414q^{79} + 31978143q^{80} + 9565938q^{81} + 9749509q^{82} + 30227337q^{83} + 11722050q^{84} + 9946981q^{85} + 44295864q^{86} + 15964695q^{87} + 39970897q^{88} + 31061677q^{89} + 2630961q^{90} + 26377785q^{91} + 4719698q^{92} + 11521791q^{93} + 44488296q^{94} + 15534599q^{95} + 32552010q^{96} + 12084118q^{97} + 42274744q^{98} + 10986030q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.9283 1.49627 0.748133 0.663548i \(-0.230950\pi\)
0.748133 + 0.663548i \(0.230950\pi\)
\(3\) 27.0000 0.577350
\(4\) 158.568 1.23881
\(5\) −421.077 −1.50649 −0.753245 0.657740i \(-0.771513\pi\)
−0.753245 + 0.657740i \(0.771513\pi\)
\(6\) 457.065 0.863870
\(7\) −62.1753 −0.0685133 −0.0342567 0.999413i \(-0.510906\pi\)
−0.0342567 + 0.999413i \(0.510906\pi\)
\(8\) 517.469 0.357330
\(9\) 729.000 0.333333
\(10\) −7128.13 −2.25411
\(11\) 8422.53 1.90795 0.953977 0.299878i \(-0.0969461\pi\)
0.953977 + 0.299878i \(0.0969461\pi\)
\(12\) 4281.34 0.715230
\(13\) 11389.2 1.43778 0.718889 0.695125i \(-0.244651\pi\)
0.718889 + 0.695125i \(0.244651\pi\)
\(14\) −1052.52 −0.102514
\(15\) −11369.1 −0.869773
\(16\) −11536.9 −0.704154
\(17\) 1307.83 0.0645626 0.0322813 0.999479i \(-0.489723\pi\)
0.0322813 + 0.999479i \(0.489723\pi\)
\(18\) 12340.7 0.498756
\(19\) −7903.89 −0.264365 −0.132182 0.991225i \(-0.542198\pi\)
−0.132182 + 0.991225i \(0.542198\pi\)
\(20\) −66769.4 −1.86626
\(21\) −1678.73 −0.0395562
\(22\) 142579. 2.85481
\(23\) −2032.53 −0.0348329 −0.0174164 0.999848i \(-0.505544\pi\)
−0.0174164 + 0.999848i \(0.505544\pi\)
\(24\) 13971.7 0.206304
\(25\) 99180.7 1.26951
\(26\) 192800. 2.15130
\(27\) 19683.0 0.192450
\(28\) −9859.03 −0.0848753
\(29\) 64202.2 0.488829 0.244414 0.969671i \(-0.421404\pi\)
0.244414 + 0.969671i \(0.421404\pi\)
\(30\) −192459. −1.30141
\(31\) 320926. 1.93481 0.967405 0.253233i \(-0.0814939\pi\)
0.967405 + 0.253233i \(0.0814939\pi\)
\(32\) −261536. −1.41093
\(33\) 227408. 1.10156
\(34\) 22139.4 0.0966029
\(35\) 26180.6 0.103215
\(36\) 115596. 0.412938
\(37\) −143662. −0.466269 −0.233135 0.972444i \(-0.574898\pi\)
−0.233135 + 0.972444i \(0.574898\pi\)
\(38\) −133800. −0.395560
\(39\) 307509. 0.830102
\(40\) −217894. −0.538314
\(41\) 577414. 1.30841 0.654205 0.756318i \(-0.273004\pi\)
0.654205 + 0.756318i \(0.273004\pi\)
\(42\) −28418.2 −0.0591866
\(43\) 817246. 1.56752 0.783760 0.621064i \(-0.213299\pi\)
0.783760 + 0.621064i \(0.213299\pi\)
\(44\) 1.33555e6 2.36360
\(45\) −306965. −0.502163
\(46\) −34407.3 −0.0521193
\(47\) 264964. 0.372259 0.186129 0.982525i \(-0.440406\pi\)
0.186129 + 0.982525i \(0.440406\pi\)
\(48\) −311495. −0.406543
\(49\) −819677. −0.995306
\(50\) 1.67896e6 1.89953
\(51\) 35311.5 0.0372752
\(52\) 1.80597e6 1.78114
\(53\) 1.60310e6 1.47909 0.739545 0.673107i \(-0.235041\pi\)
0.739545 + 0.673107i \(0.235041\pi\)
\(54\) 333200. 0.287957
\(55\) −3.54653e6 −2.87432
\(56\) −32173.8 −0.0244818
\(57\) −213405. −0.152631
\(58\) 1.08684e6 0.731418
\(59\) 205379. 0.130189
\(60\) −1.80277e6 −1.07749
\(61\) −274208. −0.154677 −0.0773386 0.997005i \(-0.524642\pi\)
−0.0773386 + 0.997005i \(0.524642\pi\)
\(62\) 5.43273e6 2.89499
\(63\) −45325.8 −0.0228378
\(64\) −2.95064e6 −1.40698
\(65\) −4.79573e6 −2.16600
\(66\) 3.84964e6 1.64823
\(67\) −1.66313e6 −0.675562 −0.337781 0.941225i \(-0.609676\pi\)
−0.337781 + 0.941225i \(0.609676\pi\)
\(68\) 207381. 0.0799811
\(69\) −54878.3 −0.0201108
\(70\) 443194. 0.154437
\(71\) −2.97718e6 −0.987190 −0.493595 0.869692i \(-0.664317\pi\)
−0.493595 + 0.869692i \(0.664317\pi\)
\(72\) 377235. 0.119110
\(73\) −2.72516e6 −0.819904 −0.409952 0.912107i \(-0.634454\pi\)
−0.409952 + 0.912107i \(0.634454\pi\)
\(74\) −2.43196e6 −0.697663
\(75\) 2.67788e6 0.732954
\(76\) −1.25331e6 −0.327499
\(77\) −523673. −0.130720
\(78\) 5.20561e6 1.24205
\(79\) 1.33414e6 0.304442 0.152221 0.988346i \(-0.451357\pi\)
0.152221 + 0.988346i \(0.451357\pi\)
\(80\) 4.85790e6 1.06080
\(81\) 531441. 0.111111
\(82\) 9.77465e6 1.95773
\(83\) 8.00004e6 1.53574 0.767872 0.640603i \(-0.221316\pi\)
0.767872 + 0.640603i \(0.221316\pi\)
\(84\) −266194. −0.0490028
\(85\) −550698. −0.0972629
\(86\) 1.38346e7 2.34543
\(87\) 1.73346e6 0.282225
\(88\) 4.35840e6 0.681769
\(89\) −1.07983e7 −1.62365 −0.811825 0.583901i \(-0.801526\pi\)
−0.811825 + 0.583901i \(0.801526\pi\)
\(90\) −5.19640e6 −0.751370
\(91\) −708128. −0.0985069
\(92\) −322295. −0.0431515
\(93\) 8.66499e6 1.11706
\(94\) 4.48540e6 0.556998
\(95\) 3.32814e6 0.398263
\(96\) −7.06146e6 −0.814602
\(97\) −1.36112e7 −1.51425 −0.757124 0.653272i \(-0.773396\pi\)
−0.757124 + 0.653272i \(0.773396\pi\)
\(98\) −1.38758e7 −1.48924
\(99\) 6.14002e6 0.635985
\(100\) 1.57269e7 1.57269
\(101\) −8.15059e6 −0.787162 −0.393581 0.919290i \(-0.628764\pi\)
−0.393581 + 0.919290i \(0.628764\pi\)
\(102\) 597764. 0.0557737
\(103\) −5.09641e6 −0.459551 −0.229776 0.973244i \(-0.573799\pi\)
−0.229776 + 0.973244i \(0.573799\pi\)
\(104\) 5.89356e6 0.513761
\(105\) 706876. 0.0595910
\(106\) 2.71378e7 2.21311
\(107\) −2.19007e7 −1.72828 −0.864140 0.503251i \(-0.832137\pi\)
−0.864140 + 0.503251i \(0.832137\pi\)
\(108\) 3.12110e6 0.238410
\(109\) 19486.8 0.00144128 0.000720639 1.00000i \(-0.499771\pi\)
0.000720639 1.00000i \(0.499771\pi\)
\(110\) −6.00368e7 −4.30074
\(111\) −3.87888e6 −0.269201
\(112\) 717307. 0.0482439
\(113\) 7.14160e6 0.465608 0.232804 0.972524i \(-0.425210\pi\)
0.232804 + 0.972524i \(0.425210\pi\)
\(114\) −3.61259e6 −0.228377
\(115\) 855851. 0.0524754
\(116\) 1.01804e7 0.605568
\(117\) 8.30273e6 0.479259
\(118\) 3.47672e6 0.194797
\(119\) −81314.9 −0.00442340
\(120\) −5.88314e6 −0.310796
\(121\) 5.14518e7 2.64029
\(122\) −4.64189e6 −0.231438
\(123\) 1.55902e7 0.755410
\(124\) 5.08886e7 2.39687
\(125\) −8.86607e6 −0.406019
\(126\) −767290. −0.0341714
\(127\) −3.64410e7 −1.57862 −0.789308 0.613997i \(-0.789561\pi\)
−0.789308 + 0.613997i \(0.789561\pi\)
\(128\) −1.64729e7 −0.694280
\(129\) 2.20656e7 0.905008
\(130\) −8.11837e7 −3.24091
\(131\) 1.30081e7 0.505549 0.252774 0.967525i \(-0.418657\pi\)
0.252774 + 0.967525i \(0.418657\pi\)
\(132\) 3.60597e7 1.36463
\(133\) 491427. 0.0181125
\(134\) −2.81541e7 −1.01082
\(135\) −8.28806e6 −0.289924
\(136\) 676763. 0.0230701
\(137\) −1.31043e7 −0.435404 −0.217702 0.976015i \(-0.569856\pi\)
−0.217702 + 0.976015i \(0.569856\pi\)
\(138\) −928998. −0.0300911
\(139\) 8.55786e6 0.270280 0.135140 0.990827i \(-0.456852\pi\)
0.135140 + 0.990827i \(0.456852\pi\)
\(140\) 4.15141e6 0.127864
\(141\) 7.15403e6 0.214924
\(142\) −5.03986e7 −1.47710
\(143\) 9.59260e7 2.74322
\(144\) −8.41036e6 −0.234718
\(145\) −2.70340e7 −0.736416
\(146\) −4.61325e7 −1.22679
\(147\) −2.21313e7 −0.574640
\(148\) −2.27803e7 −0.577621
\(149\) 8.02680e7 1.98788 0.993941 0.109916i \(-0.0350582\pi\)
0.993941 + 0.109916i \(0.0350582\pi\)
\(150\) 4.53320e7 1.09669
\(151\) 606581. 0.0143374 0.00716869 0.999974i \(-0.497718\pi\)
0.00716869 + 0.999974i \(0.497718\pi\)
\(152\) −4.09002e6 −0.0944653
\(153\) 953410. 0.0215209
\(154\) −8.86492e6 −0.195592
\(155\) −1.35134e8 −2.91477
\(156\) 4.87611e7 1.02834
\(157\) 6.17847e7 1.27418 0.637092 0.770788i \(-0.280137\pi\)
0.637092 + 0.770788i \(0.280137\pi\)
\(158\) 2.25847e7 0.455527
\(159\) 4.32837e7 0.853953
\(160\) 1.10127e8 2.12555
\(161\) 126373. 0.00238652
\(162\) 8.99641e6 0.166252
\(163\) −1.21598e6 −0.0219922 −0.0109961 0.999940i \(-0.503500\pi\)
−0.0109961 + 0.999940i \(0.503500\pi\)
\(164\) 9.15595e7 1.62088
\(165\) −9.57564e7 −1.65949
\(166\) 1.35427e8 2.29788
\(167\) 1.21038e7 0.201102 0.100551 0.994932i \(-0.467940\pi\)
0.100551 + 0.994932i \(0.467940\pi\)
\(168\) −868692. −0.0141346
\(169\) 6.69656e7 1.06721
\(170\) −9.32240e6 −0.145531
\(171\) −5.76194e6 −0.0881215
\(172\) 1.29589e8 1.94187
\(173\) −1.01126e8 −1.48492 −0.742460 0.669890i \(-0.766341\pi\)
−0.742460 + 0.669890i \(0.766341\pi\)
\(174\) 2.93445e7 0.422284
\(175\) −6.16659e6 −0.0869785
\(176\) −9.71695e7 −1.34349
\(177\) 5.54523e6 0.0751646
\(178\) −1.82798e8 −2.42941
\(179\) −7.67192e7 −0.999813 −0.499906 0.866079i \(-0.666632\pi\)
−0.499906 + 0.866079i \(0.666632\pi\)
\(180\) −4.86749e7 −0.622087
\(181\) 4.41123e7 0.552949 0.276474 0.961021i \(-0.410834\pi\)
0.276474 + 0.961021i \(0.410834\pi\)
\(182\) −1.19874e7 −0.147393
\(183\) −7.40363e6 −0.0893030
\(184\) −1.05177e6 −0.0124468
\(185\) 6.04929e7 0.702430
\(186\) 1.46684e8 1.67142
\(187\) 1.10153e7 0.123183
\(188\) 4.20149e7 0.461159
\(189\) −1.22380e6 −0.0131854
\(190\) 5.63399e7 0.595907
\(191\) 3.96720e7 0.411972 0.205986 0.978555i \(-0.433960\pi\)
0.205986 + 0.978555i \(0.433960\pi\)
\(192\) −7.96674e7 −0.812318
\(193\) −6.49743e7 −0.650565 −0.325283 0.945617i \(-0.605459\pi\)
−0.325283 + 0.945617i \(0.605459\pi\)
\(194\) −2.30416e8 −2.26572
\(195\) −1.29485e8 −1.25054
\(196\) −1.29975e8 −1.23300
\(197\) 1.60147e8 1.49241 0.746204 0.665717i \(-0.231874\pi\)
0.746204 + 0.665717i \(0.231874\pi\)
\(198\) 1.03940e8 0.951603
\(199\) 3.62822e7 0.326368 0.163184 0.986596i \(-0.447824\pi\)
0.163184 + 0.986596i \(0.447824\pi\)
\(200\) 5.13229e7 0.453635
\(201\) −4.49046e7 −0.390036
\(202\) −1.37976e8 −1.17780
\(203\) −3.99179e6 −0.0334913
\(204\) 5.59928e6 0.0461771
\(205\) −2.43136e8 −1.97111
\(206\) −8.62737e7 −0.687611
\(207\) −1.48171e6 −0.0116110
\(208\) −1.31396e8 −1.01242
\(209\) −6.65707e7 −0.504396
\(210\) 1.19662e7 0.0891640
\(211\) −2.00331e8 −1.46811 −0.734056 0.679089i \(-0.762375\pi\)
−0.734056 + 0.679089i \(0.762375\pi\)
\(212\) 2.54200e8 1.83232
\(213\) −8.03838e7 −0.569954
\(214\) −3.70742e8 −2.58597
\(215\) −3.44123e8 −2.36145
\(216\) 1.01853e7 0.0687682
\(217\) −1.99537e7 −0.132560
\(218\) 329879. 0.00215654
\(219\) −7.35795e7 −0.473372
\(220\) −5.62367e8 −3.56074
\(221\) 1.48952e7 0.0928267
\(222\) −6.56630e7 −0.402796
\(223\) 1.92870e8 1.16465 0.582327 0.812955i \(-0.302142\pi\)
0.582327 + 0.812955i \(0.302142\pi\)
\(224\) 1.62611e7 0.0966676
\(225\) 7.23027e7 0.423171
\(226\) 1.20895e8 0.696674
\(227\) −3.48084e7 −0.197512 −0.0987560 0.995112i \(-0.531486\pi\)
−0.0987560 + 0.995112i \(0.531486\pi\)
\(228\) −3.38393e7 −0.189081
\(229\) −2.38784e8 −1.31396 −0.656978 0.753910i \(-0.728165\pi\)
−0.656978 + 0.753910i \(0.728165\pi\)
\(230\) 1.44881e7 0.0785172
\(231\) −1.41392e7 −0.0754714
\(232\) 3.32226e7 0.174673
\(233\) 3.17570e8 1.64473 0.822364 0.568962i \(-0.192655\pi\)
0.822364 + 0.568962i \(0.192655\pi\)
\(234\) 1.40551e8 0.717100
\(235\) −1.11570e8 −0.560804
\(236\) 3.25666e7 0.161280
\(237\) 3.60217e7 0.175770
\(238\) −1.37653e6 −0.00661858
\(239\) −1.63040e8 −0.772505 −0.386252 0.922393i \(-0.626231\pi\)
−0.386252 + 0.922393i \(0.626231\pi\)
\(240\) 1.31163e8 0.612453
\(241\) 9.18889e7 0.422867 0.211433 0.977392i \(-0.432187\pi\)
0.211433 + 0.977392i \(0.432187\pi\)
\(242\) 8.70993e8 3.95058
\(243\) 1.43489e7 0.0641500
\(244\) −4.34808e7 −0.191616
\(245\) 3.45147e8 1.49942
\(246\) 2.63915e8 1.13030
\(247\) −9.00191e7 −0.380098
\(248\) 1.66069e8 0.691366
\(249\) 2.16001e8 0.886662
\(250\) −1.50088e8 −0.607513
\(251\) −1.98730e7 −0.0793242 −0.0396621 0.999213i \(-0.512628\pi\)
−0.0396621 + 0.999213i \(0.512628\pi\)
\(252\) −7.18723e6 −0.0282918
\(253\) −1.71190e7 −0.0664596
\(254\) −6.16884e8 −2.36203
\(255\) −1.48688e7 −0.0561548
\(256\) 9.88239e7 0.368148
\(257\) 5.09658e8 1.87289 0.936447 0.350810i \(-0.114094\pi\)
0.936447 + 0.350810i \(0.114094\pi\)
\(258\) 3.73534e8 1.35413
\(259\) 8.93225e6 0.0319456
\(260\) −7.60451e8 −2.68327
\(261\) 4.68034e7 0.162943
\(262\) 2.20205e8 0.756436
\(263\) −1.92628e8 −0.652941 −0.326471 0.945207i \(-0.605859\pi\)
−0.326471 + 0.945207i \(0.605859\pi\)
\(264\) 1.17677e8 0.393620
\(265\) −6.75028e8 −2.22823
\(266\) 8.31903e6 0.0271011
\(267\) −2.91555e8 −0.937415
\(268\) −2.63720e8 −0.836896
\(269\) 4.06864e8 1.27443 0.637215 0.770686i \(-0.280086\pi\)
0.637215 + 0.770686i \(0.280086\pi\)
\(270\) −1.40303e8 −0.433804
\(271\) −1.77838e8 −0.542789 −0.271395 0.962468i \(-0.587485\pi\)
−0.271395 + 0.962468i \(0.587485\pi\)
\(272\) −1.50883e7 −0.0454620
\(273\) −1.91195e7 −0.0568730
\(274\) −2.21834e8 −0.651481
\(275\) 8.35352e8 2.42217
\(276\) −8.70195e6 −0.0249135
\(277\) −5.24499e8 −1.48274 −0.741371 0.671095i \(-0.765824\pi\)
−0.741371 + 0.671095i \(0.765824\pi\)
\(278\) 1.44870e8 0.404411
\(279\) 2.33955e8 0.644937
\(280\) 1.35476e7 0.0368817
\(281\) −9.91059e7 −0.266457 −0.133229 0.991085i \(-0.542534\pi\)
−0.133229 + 0.991085i \(0.542534\pi\)
\(282\) 1.21106e8 0.321583
\(283\) 3.36868e8 0.883502 0.441751 0.897138i \(-0.354357\pi\)
0.441751 + 0.897138i \(0.354357\pi\)
\(284\) −4.72086e8 −1.22294
\(285\) 8.98599e7 0.229937
\(286\) 1.62387e9 4.10458
\(287\) −3.59009e7 −0.0896434
\(288\) −1.90659e8 −0.470310
\(289\) −4.08628e8 −0.995832
\(290\) −4.57641e8 −1.10187
\(291\) −3.67504e8 −0.874251
\(292\) −4.32125e8 −1.01571
\(293\) 1.54762e8 0.359441 0.179720 0.983718i \(-0.442481\pi\)
0.179720 + 0.983718i \(0.442481\pi\)
\(294\) −3.74646e8 −0.859815
\(295\) −8.64803e7 −0.196128
\(296\) −7.43407e7 −0.166612
\(297\) 1.65781e8 0.367186
\(298\) 1.35880e9 2.97440
\(299\) −2.31489e7 −0.0500819
\(300\) 4.24627e8 0.907994
\(301\) −5.08125e7 −0.107396
\(302\) 1.02684e7 0.0214525
\(303\) −2.20066e8 −0.454468
\(304\) 9.11860e7 0.186153
\(305\) 1.15463e8 0.233020
\(306\) 1.61396e7 0.0322010
\(307\) 1.34130e8 0.264570 0.132285 0.991212i \(-0.457769\pi\)
0.132285 + 0.991212i \(0.457769\pi\)
\(308\) −8.30380e7 −0.161938
\(309\) −1.37603e8 −0.265322
\(310\) −2.28760e9 −4.36128
\(311\) −1.66463e8 −0.313802 −0.156901 0.987614i \(-0.550150\pi\)
−0.156901 + 0.987614i \(0.550150\pi\)
\(312\) 1.59126e8 0.296620
\(313\) 4.01885e8 0.740793 0.370396 0.928874i \(-0.379222\pi\)
0.370396 + 0.928874i \(0.379222\pi\)
\(314\) 1.04591e9 1.90652
\(315\) 1.90857e7 0.0344049
\(316\) 2.11551e8 0.377147
\(317\) 1.50022e8 0.264513 0.132257 0.991216i \(-0.457778\pi\)
0.132257 + 0.991216i \(0.457778\pi\)
\(318\) 7.32720e8 1.27774
\(319\) 5.40745e8 0.932663
\(320\) 1.24245e9 2.11960
\(321\) −5.91318e8 −0.997823
\(322\) 2.13929e6 0.00357086
\(323\) −1.03370e7 −0.0170681
\(324\) 8.42697e7 0.137646
\(325\) 1.12959e9 1.82528
\(326\) −2.05845e7 −0.0329062
\(327\) 526144. 0.000832122 0
\(328\) 2.98794e8 0.467534
\(329\) −1.64742e7 −0.0255047
\(330\) −1.62099e9 −2.48304
\(331\) 2.12361e8 0.321867 0.160934 0.986965i \(-0.448550\pi\)
0.160934 + 0.986965i \(0.448550\pi\)
\(332\) 1.26855e9 1.90250
\(333\) −1.04730e8 −0.155423
\(334\) 2.04898e8 0.300902
\(335\) 7.00307e8 1.01773
\(336\) 1.93673e7 0.0278536
\(337\) 3.59426e8 0.511570 0.255785 0.966734i \(-0.417666\pi\)
0.255785 + 0.966734i \(0.417666\pi\)
\(338\) 1.13362e9 1.59682
\(339\) 1.92823e8 0.268819
\(340\) −8.73232e7 −0.120491
\(341\) 2.70301e9 3.69153
\(342\) −9.75399e7 −0.131853
\(343\) 1.02168e8 0.136705
\(344\) 4.22899e8 0.560122
\(345\) 2.31080e7 0.0302967
\(346\) −1.71190e9 −2.22184
\(347\) −1.80343e8 −0.231711 −0.115855 0.993266i \(-0.536961\pi\)
−0.115855 + 0.993266i \(0.536961\pi\)
\(348\) 2.74871e8 0.349625
\(349\) 1.08403e9 1.36506 0.682531 0.730856i \(-0.260879\pi\)
0.682531 + 0.730856i \(0.260879\pi\)
\(350\) −1.04390e8 −0.130143
\(351\) 2.24174e8 0.276701
\(352\) −2.20279e9 −2.69199
\(353\) −9.01373e8 −1.09067 −0.545335 0.838218i \(-0.683597\pi\)
−0.545335 + 0.838218i \(0.683597\pi\)
\(354\) 9.38715e7 0.112466
\(355\) 1.25362e9 1.48719
\(356\) −1.71227e9 −2.01140
\(357\) −2.19550e6 −0.00255385
\(358\) −1.29873e9 −1.49599
\(359\) −4.12758e8 −0.470830 −0.235415 0.971895i \(-0.575645\pi\)
−0.235415 + 0.971895i \(0.575645\pi\)
\(360\) −1.58845e8 −0.179438
\(361\) −8.31400e8 −0.930111
\(362\) 7.46748e8 0.827359
\(363\) 1.38920e9 1.52437
\(364\) −1.12287e8 −0.122032
\(365\) 1.14750e9 1.23518
\(366\) −1.25331e8 −0.133621
\(367\) −5.59200e8 −0.590521 −0.295261 0.955417i \(-0.595407\pi\)
−0.295261 + 0.955417i \(0.595407\pi\)
\(368\) 2.34490e7 0.0245277
\(369\) 4.20935e8 0.436136
\(370\) 1.02404e9 1.05102
\(371\) −9.96732e7 −0.101337
\(372\) 1.37399e9 1.38383
\(373\) 7.41533e8 0.739860 0.369930 0.929060i \(-0.379382\pi\)
0.369930 + 0.929060i \(0.379382\pi\)
\(374\) 1.86470e8 0.184314
\(375\) −2.39384e8 −0.234415
\(376\) 1.37111e8 0.133019
\(377\) 7.31212e8 0.702827
\(378\) −2.07168e7 −0.0197289
\(379\) −1.84508e9 −1.74091 −0.870457 0.492244i \(-0.836177\pi\)
−0.870457 + 0.492244i \(0.836177\pi\)
\(380\) 5.27738e8 0.493373
\(381\) −9.83906e8 −0.911415
\(382\) 6.71581e8 0.616420
\(383\) 5.18339e8 0.471431 0.235716 0.971822i \(-0.424257\pi\)
0.235716 + 0.971822i \(0.424257\pi\)
\(384\) −4.44768e8 −0.400843
\(385\) 2.20507e8 0.196929
\(386\) −1.09991e9 −0.973419
\(387\) 5.95772e8 0.522507
\(388\) −2.15831e9 −1.87587
\(389\) 1.36832e9 1.17860 0.589298 0.807916i \(-0.299404\pi\)
0.589298 + 0.807916i \(0.299404\pi\)
\(390\) −2.19196e9 −1.87114
\(391\) −2.65821e6 −0.00224890
\(392\) −4.24157e8 −0.355652
\(393\) 3.51218e8 0.291879
\(394\) 2.71102e9 2.23304
\(395\) −5.61773e8 −0.458639
\(396\) 9.73613e8 0.787867
\(397\) 3.13292e8 0.251294 0.125647 0.992075i \(-0.459899\pi\)
0.125647 + 0.992075i \(0.459899\pi\)
\(398\) 6.14196e8 0.488334
\(399\) 1.32685e7 0.0104573
\(400\) −1.14423e9 −0.893932
\(401\) 7.32537e8 0.567315 0.283658 0.958926i \(-0.408452\pi\)
0.283658 + 0.958926i \(0.408452\pi\)
\(402\) −7.60160e8 −0.583598
\(403\) 3.65509e9 2.78183
\(404\) −1.29242e9 −0.975148
\(405\) −2.23778e8 −0.167388
\(406\) −6.75743e7 −0.0501119
\(407\) −1.21000e9 −0.889620
\(408\) 1.82726e7 0.0133196
\(409\) −4.52805e8 −0.327250 −0.163625 0.986523i \(-0.552319\pi\)
−0.163625 + 0.986523i \(0.552319\pi\)
\(410\) −4.11588e9 −2.94930
\(411\) −3.53817e8 −0.251381
\(412\) −8.08129e8 −0.569299
\(413\) −1.27695e7 −0.00891967
\(414\) −2.50829e7 −0.0173731
\(415\) −3.36863e9 −2.31358
\(416\) −2.97868e9 −2.02861
\(417\) 2.31062e8 0.156046
\(418\) −1.12693e9 −0.754710
\(419\) 2.74019e8 0.181983 0.0909916 0.995852i \(-0.470996\pi\)
0.0909916 + 0.995852i \(0.470996\pi\)
\(420\) 1.12088e8 0.0738222
\(421\) −2.84288e9 −1.85683 −0.928414 0.371546i \(-0.878828\pi\)
−0.928414 + 0.371546i \(0.878828\pi\)
\(422\) −3.39127e9 −2.19669
\(423\) 1.93159e8 0.124086
\(424\) 8.29553e8 0.528523
\(425\) 1.29712e8 0.0819631
\(426\) −1.36076e9 −0.852803
\(427\) 1.70490e7 0.0105975
\(428\) −3.47275e9 −2.14102
\(429\) 2.59000e9 1.58380
\(430\) −5.82543e9 −3.53336
\(431\) 1.06285e9 0.639440 0.319720 0.947512i \(-0.396411\pi\)
0.319720 + 0.947512i \(0.396411\pi\)
\(432\) −2.27080e8 −0.135514
\(433\) −1.53055e9 −0.906022 −0.453011 0.891505i \(-0.649650\pi\)
−0.453011 + 0.891505i \(0.649650\pi\)
\(434\) −3.37782e8 −0.198346
\(435\) −7.29919e8 −0.425170
\(436\) 3.08999e6 0.00178548
\(437\) 1.60649e7 0.00920858
\(438\) −1.24558e9 −0.708290
\(439\) 5.70324e8 0.321733 0.160867 0.986976i \(-0.448571\pi\)
0.160867 + 0.986976i \(0.448571\pi\)
\(440\) −1.83522e9 −1.02708
\(441\) −5.97545e8 −0.331769
\(442\) 2.52151e8 0.138893
\(443\) −1.53721e9 −0.840078 −0.420039 0.907506i \(-0.637984\pi\)
−0.420039 + 0.907506i \(0.637984\pi\)
\(444\) −6.15067e8 −0.333490
\(445\) 4.54693e9 2.44601
\(446\) 3.26496e9 1.74263
\(447\) 2.16724e9 1.14770
\(448\) 1.83457e8 0.0963966
\(449\) −3.53047e9 −1.84065 −0.920323 0.391160i \(-0.872074\pi\)
−0.920323 + 0.391160i \(0.872074\pi\)
\(450\) 1.22396e9 0.633177
\(451\) 4.86328e9 2.49639
\(452\) 1.13243e9 0.576802
\(453\) 1.63777e7 0.00827769
\(454\) −5.89248e8 −0.295531
\(455\) 2.98176e8 0.148400
\(456\) −1.10430e8 −0.0545396
\(457\) 2.13769e8 0.104770 0.0523850 0.998627i \(-0.483318\pi\)
0.0523850 + 0.998627i \(0.483318\pi\)
\(458\) −4.04221e9 −1.96603
\(459\) 2.57421e7 0.0124251
\(460\) 1.35711e8 0.0650073
\(461\) −2.72011e9 −1.29310 −0.646551 0.762871i \(-0.723789\pi\)
−0.646551 + 0.762871i \(0.723789\pi\)
\(462\) −2.39353e8 −0.112925
\(463\) −2.70333e9 −1.26580 −0.632901 0.774233i \(-0.718136\pi\)
−0.632901 + 0.774233i \(0.718136\pi\)
\(464\) −7.40691e8 −0.344210
\(465\) −3.64863e9 −1.68285
\(466\) 5.37593e9 2.46095
\(467\) 2.43270e9 1.10530 0.552650 0.833414i \(-0.313617\pi\)
0.552650 + 0.833414i \(0.313617\pi\)
\(468\) 1.31655e9 0.593713
\(469\) 1.03406e8 0.0462850
\(470\) −1.88870e9 −0.839112
\(471\) 1.66819e9 0.735650
\(472\) 1.06277e8 0.0465204
\(473\) 6.88327e9 2.99076
\(474\) 6.09786e8 0.262999
\(475\) −7.83913e8 −0.335614
\(476\) −1.28940e7 −0.00547977
\(477\) 1.16866e9 0.493030
\(478\) −2.75999e9 −1.15587
\(479\) 3.67141e8 0.152636 0.0763182 0.997084i \(-0.475684\pi\)
0.0763182 + 0.997084i \(0.475684\pi\)
\(480\) 2.97342e9 1.22719
\(481\) −1.63620e9 −0.670392
\(482\) 1.55552e9 0.632721
\(483\) 3.41208e6 0.00137786
\(484\) 8.15862e9 3.27083
\(485\) 5.73138e9 2.28120
\(486\) 2.42903e8 0.0959856
\(487\) −3.04793e9 −1.19579 −0.597894 0.801575i \(-0.703996\pi\)
−0.597894 + 0.801575i \(0.703996\pi\)
\(488\) −1.41894e8 −0.0552708
\(489\) −3.28314e7 −0.0126972
\(490\) 5.84276e9 2.24353
\(491\) −4.09474e9 −1.56114 −0.780568 0.625071i \(-0.785070\pi\)
−0.780568 + 0.625071i \(0.785070\pi\)
\(492\) 2.47211e9 0.935813
\(493\) 8.39657e7 0.0315600
\(494\) −1.52387e9 −0.568727
\(495\) −2.58542e9 −0.958105
\(496\) −3.70247e9 −1.36240
\(497\) 1.85107e8 0.0676356
\(498\) 3.65654e9 1.32668
\(499\) −3.30636e9 −1.19124 −0.595618 0.803268i \(-0.703093\pi\)
−0.595618 + 0.803268i \(0.703093\pi\)
\(500\) −1.40588e9 −0.502982
\(501\) 3.26804e8 0.116106
\(502\) −3.36417e8 −0.118690
\(503\) −4.42202e9 −1.54929 −0.774645 0.632397i \(-0.782071\pi\)
−0.774645 + 0.632397i \(0.782071\pi\)
\(504\) −2.34547e7 −0.00816062
\(505\) 3.43202e9 1.18585
\(506\) −2.89797e8 −0.0994412
\(507\) 1.80807e9 0.616152
\(508\) −5.77838e9 −1.95561
\(509\) 3.77691e9 1.26947 0.634737 0.772728i \(-0.281108\pi\)
0.634737 + 0.772728i \(0.281108\pi\)
\(510\) −2.51705e8 −0.0840225
\(511\) 1.69438e8 0.0561743
\(512\) 3.78145e9 1.24513
\(513\) −1.55572e8 −0.0508770
\(514\) 8.62766e9 2.80235
\(515\) 2.14598e9 0.692310
\(516\) 3.49891e9 1.12114
\(517\) 2.23167e9 0.710252
\(518\) 1.51208e8 0.0477992
\(519\) −2.73041e9 −0.857319
\(520\) −2.48164e9 −0.773976
\(521\) −4.54529e9 −1.40809 −0.704044 0.710157i \(-0.748624\pi\)
−0.704044 + 0.710157i \(0.748624\pi\)
\(522\) 7.92303e8 0.243806
\(523\) −4.00555e9 −1.22435 −0.612176 0.790721i \(-0.709706\pi\)
−0.612176 + 0.790721i \(0.709706\pi\)
\(524\) 2.06267e9 0.626281
\(525\) −1.66498e8 −0.0502171
\(526\) −3.26087e9 −0.976974
\(527\) 4.19717e8 0.124916
\(528\) −2.62358e9 −0.775666
\(529\) −3.40069e9 −0.998787
\(530\) −1.14271e10 −3.33403
\(531\) 1.49721e8 0.0433963
\(532\) 7.79247e7 0.0224380
\(533\) 6.57629e9 1.88120
\(534\) −4.93554e9 −1.40262
\(535\) 9.22186e9 2.60364
\(536\) −8.60620e8 −0.241399
\(537\) −2.07142e9 −0.577242
\(538\) 6.88752e9 1.90689
\(539\) −6.90376e9 −1.89900
\(540\) −1.31422e9 −0.359162
\(541\) −1.98627e9 −0.539320 −0.269660 0.962956i \(-0.586911\pi\)
−0.269660 + 0.962956i \(0.586911\pi\)
\(542\) −3.01049e9 −0.812157
\(543\) 1.19103e9 0.319245
\(544\) −3.42045e8 −0.0910934
\(545\) −8.20544e6 −0.00217127
\(546\) −3.23660e8 −0.0850972
\(547\) −1.31698e9 −0.344052 −0.172026 0.985092i \(-0.555031\pi\)
−0.172026 + 0.985092i \(0.555031\pi\)
\(548\) −2.07793e9 −0.539385
\(549\) −1.99898e8 −0.0515591
\(550\) 1.41411e10 3.62422
\(551\) −5.07447e8 −0.129229
\(552\) −2.83978e7 −0.00718618
\(553\) −8.29503e7 −0.0208584
\(554\) −8.87889e9 −2.21858
\(555\) 1.63331e9 0.405548
\(556\) 1.35700e9 0.334826
\(557\) 3.87423e9 0.949931 0.474965 0.880004i \(-0.342461\pi\)
0.474965 + 0.880004i \(0.342461\pi\)
\(558\) 3.96046e9 0.964998
\(559\) 9.30778e9 2.25375
\(560\) −3.02042e8 −0.0726789
\(561\) 2.97412e8 0.0711195
\(562\) −1.67770e9 −0.398691
\(563\) −5.14028e9 −1.21397 −0.606984 0.794714i \(-0.707621\pi\)
−0.606984 + 0.794714i \(0.707621\pi\)
\(564\) 1.13440e9 0.266250
\(565\) −3.00716e9 −0.701435
\(566\) 5.70262e9 1.32196
\(567\) −3.30425e7 −0.00761259
\(568\) −1.54060e9 −0.352752
\(569\) 5.94344e9 1.35252 0.676262 0.736661i \(-0.263599\pi\)
0.676262 + 0.736661i \(0.263599\pi\)
\(570\) 1.52118e9 0.344047
\(571\) −6.34028e9 −1.42522 −0.712611 0.701560i \(-0.752487\pi\)
−0.712611 + 0.701560i \(0.752487\pi\)
\(572\) 1.52108e10 3.39833
\(573\) 1.07114e9 0.237852
\(574\) −6.07742e8 −0.134131
\(575\) −2.01588e8 −0.0442208
\(576\) −2.15102e9 −0.468992
\(577\) 2.74120e9 0.594052 0.297026 0.954869i \(-0.404005\pi\)
0.297026 + 0.954869i \(0.404005\pi\)
\(578\) −6.91739e9 −1.49003
\(579\) −1.75431e9 −0.375604
\(580\) −4.28674e9 −0.912282
\(581\) −4.97405e8 −0.105219
\(582\) −6.22122e9 −1.30811
\(583\) 1.35021e10 2.82204
\(584\) −1.41019e9 −0.292976
\(585\) −3.49609e9 −0.722000
\(586\) 2.61986e9 0.537819
\(587\) −7.75794e8 −0.158312 −0.0791558 0.996862i \(-0.525222\pi\)
−0.0791558 + 0.996862i \(0.525222\pi\)
\(588\) −3.50932e9 −0.711872
\(589\) −2.53656e9 −0.511495
\(590\) −1.46397e9 −0.293460
\(591\) 4.32398e9 0.861642
\(592\) 1.65741e9 0.328325
\(593\) 1.63585e9 0.322145 0.161072 0.986943i \(-0.448505\pi\)
0.161072 + 0.986943i \(0.448505\pi\)
\(594\) 2.80639e9 0.549408
\(595\) 3.42398e7 0.00666381
\(596\) 1.27280e10 2.46262
\(597\) 9.79619e8 0.188429
\(598\) −3.91872e8 −0.0749360
\(599\) −4.17190e9 −0.793121 −0.396561 0.918009i \(-0.629796\pi\)
−0.396561 + 0.918009i \(0.629796\pi\)
\(600\) 1.38572e9 0.261906
\(601\) −4.54954e9 −0.854883 −0.427442 0.904043i \(-0.640585\pi\)
−0.427442 + 0.904043i \(0.640585\pi\)
\(602\) −8.60171e8 −0.160693
\(603\) −1.21242e9 −0.225187
\(604\) 9.61845e7 0.0177613
\(605\) −2.16652e10 −3.97757
\(606\) −3.72535e9 −0.680006
\(607\) −5.26426e9 −0.955381 −0.477691 0.878528i \(-0.658526\pi\)
−0.477691 + 0.878528i \(0.658526\pi\)
\(608\) 2.06715e9 0.373000
\(609\) −1.07778e8 −0.0193362
\(610\) 1.95459e9 0.348660
\(611\) 3.01773e9 0.535225
\(612\) 1.51181e8 0.0266604
\(613\) −1.30761e9 −0.229281 −0.114641 0.993407i \(-0.536572\pi\)
−0.114641 + 0.993407i \(0.536572\pi\)
\(614\) 2.27059e9 0.395867
\(615\) −6.56466e9 −1.13802
\(616\) −2.70985e8 −0.0467103
\(617\) −5.09050e9 −0.872493 −0.436247 0.899827i \(-0.643692\pi\)
−0.436247 + 0.899827i \(0.643692\pi\)
\(618\) −2.32939e9 −0.396993
\(619\) 8.73085e9 1.47958 0.739792 0.672836i \(-0.234924\pi\)
0.739792 + 0.672836i \(0.234924\pi\)
\(620\) −2.14280e10 −3.61086
\(621\) −4.00063e7 −0.00670359
\(622\) −2.81794e9 −0.469532
\(623\) 6.71391e8 0.111242
\(624\) −3.54768e9 −0.584519
\(625\) −4.01519e9 −0.657850
\(626\) 6.80324e9 1.10842
\(627\) −1.79741e9 −0.291213
\(628\) 9.79709e9 1.57848
\(629\) −1.87886e8 −0.0301035
\(630\) 3.23088e8 0.0514789
\(631\) −7.67163e9 −1.21558 −0.607792 0.794096i \(-0.707945\pi\)
−0.607792 + 0.794096i \(0.707945\pi\)
\(632\) 6.90373e8 0.108786
\(633\) −5.40893e9 −0.847615
\(634\) 2.53962e9 0.395782
\(635\) 1.53444e10 2.37817
\(636\) 6.86341e9 1.05789
\(637\) −9.33548e9 −1.43103
\(638\) 9.15390e9 1.39551
\(639\) −2.17036e9 −0.329063
\(640\) 6.93635e9 1.04593
\(641\) −3.46528e9 −0.519679 −0.259840 0.965652i \(-0.583670\pi\)
−0.259840 + 0.965652i \(0.583670\pi\)
\(642\) −1.00100e10 −1.49301
\(643\) 5.49488e9 0.815117 0.407559 0.913179i \(-0.366380\pi\)
0.407559 + 0.913179i \(0.366380\pi\)
\(644\) 2.00388e7 0.00295645
\(645\) −9.29133e9 −1.36339
\(646\) −1.74988e8 −0.0255384
\(647\) 8.98807e9 1.30467 0.652336 0.757930i \(-0.273789\pi\)
0.652336 + 0.757930i \(0.273789\pi\)
\(648\) 2.75004e8 0.0397033
\(649\) 1.72981e9 0.248395
\(650\) 1.91221e10 2.73110
\(651\) −5.38749e8 −0.0765337
\(652\) −1.92815e8 −0.0272443
\(653\) −2.09537e9 −0.294486 −0.147243 0.989100i \(-0.547040\pi\)
−0.147243 + 0.989100i \(0.547040\pi\)
\(654\) 8.90673e6 0.00124508
\(655\) −5.47739e9 −0.761604
\(656\) −6.66154e9 −0.921321
\(657\) −1.98665e9 −0.273301
\(658\) −2.78881e8 −0.0381618
\(659\) 1.34881e10 1.83591 0.917953 0.396690i \(-0.129841\pi\)
0.917953 + 0.396690i \(0.129841\pi\)
\(660\) −1.51839e10 −2.05580
\(661\) −2.15284e9 −0.289939 −0.144969 0.989436i \(-0.546308\pi\)
−0.144969 + 0.989436i \(0.546308\pi\)
\(662\) 3.59492e9 0.481599
\(663\) 4.02170e8 0.0535935
\(664\) 4.13977e9 0.548767
\(665\) −2.06928e8 −0.0272863
\(666\) −1.77290e9 −0.232554
\(667\) −1.30493e8 −0.0170273
\(668\) 1.91928e9 0.249127
\(669\) 5.20748e9 0.672413
\(670\) 1.18550e10 1.52279
\(671\) −2.30953e9 −0.295117
\(672\) 4.39049e8 0.0558110
\(673\) −1.17042e10 −1.48010 −0.740048 0.672554i \(-0.765197\pi\)
−0.740048 + 0.672554i \(0.765197\pi\)
\(674\) 6.08448e9 0.765445
\(675\) 1.95217e9 0.244318
\(676\) 1.06186e10 1.32207
\(677\) −6.54348e9 −0.810491 −0.405245 0.914208i \(-0.632814\pi\)
−0.405245 + 0.914208i \(0.632814\pi\)
\(678\) 3.26417e9 0.402225
\(679\) 8.46284e8 0.103746
\(680\) −2.84969e8 −0.0347549
\(681\) −9.39827e8 −0.114034
\(682\) 4.57574e10 5.52352
\(683\) 1.02317e10 1.22878 0.614390 0.789003i \(-0.289402\pi\)
0.614390 + 0.789003i \(0.289402\pi\)
\(684\) −9.13660e8 −0.109166
\(685\) 5.51793e9 0.655932
\(686\) 1.72953e9 0.204547
\(687\) −6.44716e9 −0.758612
\(688\) −9.42844e9 −1.10377
\(689\) 1.82580e10 2.12660
\(690\) 3.91179e8 0.0453319
\(691\) −1.10128e10 −1.26977 −0.634887 0.772605i \(-0.718953\pi\)
−0.634887 + 0.772605i \(0.718953\pi\)
\(692\) −1.60354e10 −1.83954
\(693\) −3.81758e8 −0.0435734
\(694\) −3.05291e9 −0.346701
\(695\) −3.60352e9 −0.407174
\(696\) 8.97011e8 0.100848
\(697\) 7.55161e8 0.0844743
\(698\) 1.83508e10 2.04250
\(699\) 8.57439e9 0.949584
\(700\) −9.77826e8 −0.107750
\(701\) 5.07374e9 0.556308 0.278154 0.960536i \(-0.410277\pi\)
0.278154 + 0.960536i \(0.410277\pi\)
\(702\) 3.79489e9 0.414018
\(703\) 1.13549e9 0.123265
\(704\) −2.48519e10 −2.68445
\(705\) −3.01240e9 −0.323780
\(706\) −1.52587e10 −1.63193
\(707\) 5.06766e8 0.0539311
\(708\) 8.79298e8 0.0931150
\(709\) −9.55775e8 −0.100715 −0.0503575 0.998731i \(-0.516036\pi\)
−0.0503575 + 0.998731i \(0.516036\pi\)
\(710\) 2.12217e10 2.22524
\(711\) 9.72585e8 0.101481
\(712\) −5.58781e9 −0.580178
\(713\) −6.52291e8 −0.0673950
\(714\) −3.71662e7 −0.00382124
\(715\) −4.03922e10 −4.13263
\(716\) −1.21652e10 −1.23858
\(717\) −4.40208e9 −0.446006
\(718\) −6.98730e9 −0.704488
\(719\) 9.39305e9 0.942444 0.471222 0.882015i \(-0.343813\pi\)
0.471222 + 0.882015i \(0.343813\pi\)
\(720\) 3.54141e9 0.353600
\(721\) 3.16871e8 0.0314854
\(722\) −1.40742e10 −1.39169
\(723\) 2.48100e9 0.244142
\(724\) 6.99481e9 0.685001
\(725\) 6.36762e9 0.620574
\(726\) 2.35168e10 2.28087
\(727\) −1.28780e9 −0.124302 −0.0621508 0.998067i \(-0.519796\pi\)
−0.0621508 + 0.998067i \(0.519796\pi\)
\(728\) −3.66434e8 −0.0351995
\(729\) 3.87420e8 0.0370370
\(730\) 1.94253e10 1.84815
\(731\) 1.06882e9 0.101203
\(732\) −1.17398e9 −0.110630
\(733\) 6.05304e8 0.0567687 0.0283844 0.999597i \(-0.490964\pi\)
0.0283844 + 0.999597i \(0.490964\pi\)
\(734\) −9.46631e9 −0.883578
\(735\) 9.31897e9 0.865690
\(736\) 5.31579e8 0.0491468
\(737\) −1.40078e10 −1.28894
\(738\) 7.12572e9 0.652576
\(739\) 6.67329e9 0.608253 0.304126 0.952632i \(-0.401635\pi\)
0.304126 + 0.952632i \(0.401635\pi\)
\(740\) 9.59224e9 0.870180
\(741\) −2.43051e9 −0.219449
\(742\) −1.68730e9 −0.151628
\(743\) −6.01656e9 −0.538130 −0.269065 0.963122i \(-0.586715\pi\)
−0.269065 + 0.963122i \(0.586715\pi\)
\(744\) 4.48386e9 0.399160
\(745\) −3.37990e10 −2.99472
\(746\) 1.25529e10 1.10703
\(747\) 5.83203e9 0.511915
\(748\) 1.74667e9 0.152600
\(749\) 1.36168e9 0.118410
\(750\) −4.05237e9 −0.350748
\(751\) 6.84419e9 0.589634 0.294817 0.955554i \(-0.404741\pi\)
0.294817 + 0.955554i \(0.404741\pi\)
\(752\) −3.05685e9 −0.262127
\(753\) −5.36571e8 −0.0457979
\(754\) 1.23782e10 1.05162
\(755\) −2.55417e8 −0.0215991
\(756\) −1.94055e8 −0.0163343
\(757\) 4.04396e8 0.0338822 0.0169411 0.999856i \(-0.494607\pi\)
0.0169411 + 0.999856i \(0.494607\pi\)
\(758\) −3.12341e10 −2.60487
\(759\) −4.62214e8 −0.0383704
\(760\) 1.72221e9 0.142311
\(761\) 3.58785e9 0.295112 0.147556 0.989054i \(-0.452859\pi\)
0.147556 + 0.989054i \(0.452859\pi\)
\(762\) −1.66559e10 −1.36372
\(763\) −1.21160e6 −9.87467e−5 0
\(764\) 6.29072e9 0.510356
\(765\) −4.01459e8 −0.0324210
\(766\) 8.77461e9 0.705387
\(767\) 2.33910e9 0.187183
\(768\) 2.66824e9 0.212550
\(769\) −1.00703e10 −0.798550 −0.399275 0.916831i \(-0.630738\pi\)
−0.399275 + 0.916831i \(0.630738\pi\)
\(770\) 3.73281e9 0.294658
\(771\) 1.37608e10 1.08132
\(772\) −1.03029e10 −0.805929
\(773\) 1.18136e10 0.919927 0.459964 0.887938i \(-0.347862\pi\)
0.459964 + 0.887938i \(0.347862\pi\)
\(774\) 1.00854e10 0.781809
\(775\) 3.18296e10 2.45627
\(776\) −7.04340e9 −0.541086
\(777\) 2.41171e8 0.0184438
\(778\) 2.31634e10 1.76349
\(779\) −4.56381e9 −0.345897
\(780\) −2.05322e10 −1.54919
\(781\) −2.50754e10 −1.88351
\(782\) −4.49990e7 −0.00336496
\(783\) 1.26369e9 0.0940751
\(784\) 9.45649e9 0.700848
\(785\) −2.60161e10 −1.91955
\(786\) 5.94553e9 0.436728
\(787\) −1.44585e10 −1.05734 −0.528668 0.848829i \(-0.677308\pi\)
−0.528668 + 0.848829i \(0.677308\pi\)
\(788\) 2.53943e10 1.84882
\(789\) −5.20095e9 −0.376976
\(790\) −9.50988e9 −0.686247
\(791\) −4.44031e8 −0.0319004
\(792\) 3.17727e9 0.227256
\(793\) −3.12302e9 −0.222392
\(794\) 5.30351e9 0.376003
\(795\) −1.82257e10 −1.28647
\(796\) 5.75320e9 0.404309
\(797\) 2.14577e10 1.50134 0.750669 0.660678i \(-0.229731\pi\)
0.750669 + 0.660678i \(0.229731\pi\)
\(798\) 2.24614e8 0.0156468
\(799\) 3.46529e8 0.0240340
\(800\) −2.59393e10 −1.79120
\(801\) −7.87200e9 −0.541217
\(802\) 1.24006e10 0.848855
\(803\) −2.29528e10 −1.56434
\(804\) −7.12044e9 −0.483182
\(805\) −5.32128e7 −0.00359526
\(806\) 6.18745e10 4.16236
\(807\) 1.09853e10 0.735792
\(808\) −4.21768e9 −0.281277
\(809\) 1.76950e9 0.117498 0.0587491 0.998273i \(-0.481289\pi\)
0.0587491 + 0.998273i \(0.481289\pi\)
\(810\) −3.78818e9 −0.250457
\(811\) −1.15307e10 −0.759074 −0.379537 0.925177i \(-0.623917\pi\)
−0.379537 + 0.925177i \(0.623917\pi\)
\(812\) −6.32971e8 −0.0414895
\(813\) −4.80161e9 −0.313379
\(814\) −2.04833e10 −1.33111
\(815\) 5.12020e8 0.0331311
\(816\) −4.07383e8 −0.0262475
\(817\) −6.45942e9 −0.414397
\(818\) −7.66523e9 −0.489653
\(819\) −5.16225e8 −0.0328356
\(820\) −3.85536e10 −2.44183
\(821\) 1.88474e10 1.18864 0.594321 0.804228i \(-0.297421\pi\)
0.594321 + 0.804228i \(0.297421\pi\)
\(822\) −5.98953e9 −0.376133
\(823\) 2.70001e10 1.68836 0.844180 0.536059i \(-0.180088\pi\)
0.844180 + 0.536059i \(0.180088\pi\)
\(824\) −2.63723e9 −0.164211
\(825\) 2.25545e10 1.39844
\(826\) −2.16166e8 −0.0133462
\(827\) 2.57017e10 1.58013 0.790064 0.613024i \(-0.210047\pi\)
0.790064 + 0.613024i \(0.210047\pi\)
\(828\) −2.34953e8 −0.0143838
\(829\) 1.20559e10 0.734952 0.367476 0.930033i \(-0.380222\pi\)
0.367476 + 0.930033i \(0.380222\pi\)
\(830\) −5.70253e10 −3.46174
\(831\) −1.41615e10 −0.856061
\(832\) −3.36055e10 −2.02292
\(833\) −1.07200e9 −0.0642595
\(834\) 3.91150e9 0.233487
\(835\) −5.09665e9 −0.302958
\(836\) −1.05560e10 −0.624853
\(837\) 6.31678e9 0.372354
\(838\) 4.63868e9 0.272295
\(839\) 1.49873e10 0.876106 0.438053 0.898949i \(-0.355668\pi\)
0.438053 + 0.898949i \(0.355668\pi\)
\(840\) 3.65786e8 0.0212936
\(841\) −1.31280e10 −0.761047
\(842\) −4.81253e10 −2.77831
\(843\) −2.67586e9 −0.153839
\(844\) −3.17661e10 −1.81872
\(845\) −2.81977e10 −1.60774
\(846\) 3.26986e9 0.185666
\(847\) −3.19903e9 −0.180895
\(848\) −1.84947e10 −1.04151
\(849\) 9.09545e9 0.510090
\(850\) 2.19580e9 0.122639
\(851\) 2.91998e8 0.0162415
\(852\) −1.27463e10 −0.706067
\(853\) 7.24550e9 0.399712 0.199856 0.979825i \(-0.435953\pi\)
0.199856 + 0.979825i \(0.435953\pi\)
\(854\) 2.88611e8 0.0158566
\(855\) 2.42622e9 0.132754
\(856\) −1.13329e10 −0.617566
\(857\) 3.25083e10 1.76426 0.882128 0.471010i \(-0.156110\pi\)
0.882128 + 0.471010i \(0.156110\pi\)
\(858\) 4.38444e10 2.36978
\(859\) −1.44247e10 −0.776480 −0.388240 0.921558i \(-0.626917\pi\)
−0.388240 + 0.921558i \(0.626917\pi\)
\(860\) −5.45670e10 −2.92540
\(861\) −9.69324e8 −0.0517557
\(862\) 1.79922e10 0.956773
\(863\) −1.11462e10 −0.590324 −0.295162 0.955447i \(-0.595374\pi\)
−0.295162 + 0.955447i \(0.595374\pi\)
\(864\) −5.14781e9 −0.271534
\(865\) 4.25820e10 2.23702
\(866\) −2.59096e10 −1.35565
\(867\) −1.10330e10 −0.574944
\(868\) −3.16402e9 −0.164218
\(869\) 1.12368e10 0.580862
\(870\) −1.23563e10 −0.636167
\(871\) −1.89418e10 −0.971309
\(872\) 1.00838e7 0.000515012 0
\(873\) −9.92260e9 −0.504749
\(874\) 2.71952e8 0.0137785
\(875\) 5.51251e8 0.0278177
\(876\) −1.16674e10 −0.586419
\(877\) −2.20664e10 −1.10467 −0.552334 0.833623i \(-0.686263\pi\)
−0.552334 + 0.833623i \(0.686263\pi\)
\(878\) 9.65463e9 0.481399
\(879\) 4.17857e9 0.207523
\(880\) 4.09158e10 2.02396
\(881\) −5.78724e9 −0.285138 −0.142569 0.989785i \(-0.545536\pi\)
−0.142569 + 0.989785i \(0.545536\pi\)
\(882\) −1.01154e10 −0.496414
\(883\) −3.69412e10 −1.80571 −0.902855 0.429944i \(-0.858533\pi\)
−0.902855 + 0.429944i \(0.858533\pi\)
\(884\) 2.36190e9 0.114995
\(885\) −2.33497e9 −0.113235
\(886\) −2.60224e10 −1.25698
\(887\) 8.57880e9 0.412756 0.206378 0.978472i \(-0.433832\pi\)
0.206378 + 0.978472i \(0.433832\pi\)
\(888\) −2.00720e9 −0.0961934
\(889\) 2.26573e9 0.108156
\(890\) 7.69720e10 3.65989
\(891\) 4.47608e9 0.211995
\(892\) 3.05830e10 1.44279
\(893\) −2.09425e9 −0.0984120
\(894\) 3.66877e10 1.71727
\(895\) 3.23047e10 1.50621
\(896\) 1.02421e9 0.0475674
\(897\) −6.25020e8 −0.0289148
\(898\) −5.97649e10 −2.75410
\(899\) 2.06041e10 0.945791
\(900\) 1.14649e10 0.524230
\(901\) 2.09658e9 0.0954939
\(902\) 8.23273e10 3.73526
\(903\) −1.37194e9 −0.0620051
\(904\) 3.69555e9 0.166376
\(905\) −1.85747e10 −0.833012
\(906\) 2.77247e8 0.0123856
\(907\) 4.99672e9 0.222362 0.111181 0.993800i \(-0.464537\pi\)
0.111181 + 0.993800i \(0.464537\pi\)
\(908\) −5.51951e9 −0.244681
\(909\) −5.94178e9 −0.262387
\(910\) 5.04762e9 0.222046
\(911\) −2.67519e10 −1.17230 −0.586151 0.810202i \(-0.699358\pi\)
−0.586151 + 0.810202i \(0.699358\pi\)
\(912\) 2.46202e9 0.107476
\(913\) 6.73806e10 2.93013
\(914\) 3.61875e9 0.156764
\(915\) 3.11750e9 0.134534
\(916\) −3.78635e10 −1.62775
\(917\) −8.08780e8 −0.0346368
\(918\) 4.35770e8 0.0185912
\(919\) −1.20296e10 −0.511268 −0.255634 0.966774i \(-0.582284\pi\)
−0.255634 + 0.966774i \(0.582284\pi\)
\(920\) 4.42876e8 0.0187510
\(921\) 3.62150e9 0.152749
\(922\) −4.60469e10 −1.93483
\(923\) −3.39077e10 −1.41936
\(924\) −2.24203e9 −0.0934950
\(925\) −1.42485e10 −0.591935
\(926\) −4.57629e10 −1.89398
\(927\) −3.71528e9 −0.153184
\(928\) −1.67911e10 −0.689704
\(929\) 9.92866e9 0.406290 0.203145 0.979149i \(-0.434884\pi\)
0.203145 + 0.979149i \(0.434884\pi\)
\(930\) −6.17652e10 −2.51799
\(931\) 6.47864e9 0.263124
\(932\) 5.03565e10 2.03751
\(933\) −4.49450e9 −0.181174
\(934\) 4.11816e10 1.65382
\(935\) −4.63827e9 −0.185573
\(936\) 4.29641e9 0.171254
\(937\) −4.20534e10 −1.66998 −0.834992 0.550262i \(-0.814528\pi\)
−0.834992 + 0.550262i \(0.814528\pi\)
\(938\) 1.75049e9 0.0692547
\(939\) 1.08509e10 0.427697
\(940\) −1.76915e10 −0.694732
\(941\) −3.81848e9 −0.149392 −0.0746960 0.997206i \(-0.523799\pi\)
−0.0746960 + 0.997206i \(0.523799\pi\)
\(942\) 2.82396e10 1.10073
\(943\) −1.17361e9 −0.0455757
\(944\) −2.36943e9 −0.0916730
\(945\) 5.15313e8 0.0198637
\(946\) 1.16522e11 4.47497
\(947\) −1.89974e10 −0.726892 −0.363446 0.931615i \(-0.618400\pi\)
−0.363446 + 0.931615i \(0.618400\pi\)
\(948\) 5.71189e9 0.217746
\(949\) −3.10375e10 −1.17884
\(950\) −1.32703e10 −0.502169
\(951\) 4.05059e9 0.152717
\(952\) −4.20779e7 −0.00158061
\(953\) −3.31060e10 −1.23903 −0.619514 0.784986i \(-0.712670\pi\)
−0.619514 + 0.784986i \(0.712670\pi\)
\(954\) 1.97834e10 0.737704
\(955\) −1.67050e10 −0.620631
\(956\) −2.58529e10 −0.956990
\(957\) 1.46001e10 0.538473
\(958\) 6.21507e9 0.228385
\(959\) 8.14766e8 0.0298310
\(960\) 3.35461e10 1.22375
\(961\) 7.54807e10 2.74349
\(962\) −2.76981e10 −1.00308
\(963\) −1.59656e10 −0.576093
\(964\) 1.45707e10 0.523853
\(965\) 2.73592e10 0.980070
\(966\) 5.77607e7 0.00206164
\(967\) 4.23227e9 0.150515 0.0752576 0.997164i \(-0.476022\pi\)
0.0752576 + 0.997164i \(0.476022\pi\)
\(968\) 2.66247e10 0.943455
\(969\) −2.79098e8 −0.00985425
\(970\) 9.70227e10 3.41328
\(971\) 2.31036e10 0.809866 0.404933 0.914346i \(-0.367295\pi\)
0.404933 + 0.914346i \(0.367295\pi\)
\(972\) 2.27528e9 0.0794700
\(973\) −5.32088e8 −0.0185178
\(974\) −5.15964e10 −1.78922
\(975\) 3.04989e10 1.05382
\(976\) 3.16350e9 0.108917
\(977\) −5.89335e9 −0.202177 −0.101088 0.994877i \(-0.532232\pi\)
−0.101088 + 0.994877i \(0.532232\pi\)
\(978\) −5.55781e8 −0.0189984
\(979\) −9.09494e10 −3.09785
\(980\) 5.47294e10 1.85750
\(981\) 1.42059e7 0.000480426 0
\(982\) −6.93171e10 −2.33588
\(983\) 4.17317e10 1.40129 0.700646 0.713509i \(-0.252895\pi\)
0.700646 + 0.713509i \(0.252895\pi\)
\(984\) 8.06743e9 0.269931
\(985\) −6.74343e10 −2.24830
\(986\) 1.42140e9 0.0472222
\(987\) −4.44804e8 −0.0147251
\(988\) −1.42742e10 −0.470870
\(989\) −1.66108e9 −0.0546012
\(990\) −4.37669e10 −1.43358
\(991\) 2.97978e10 0.972583 0.486292 0.873797i \(-0.338349\pi\)
0.486292 + 0.873797i \(0.338349\pi\)
\(992\) −8.39335e10 −2.72989
\(993\) 5.73375e9 0.185830
\(994\) 3.13355e9 0.101201
\(995\) −1.52776e10 −0.491670
\(996\) 3.42509e10 1.09841
\(997\) 1.92441e10 0.614986 0.307493 0.951550i \(-0.400510\pi\)
0.307493 + 0.951550i \(0.400510\pi\)
\(998\) −5.59711e10 −1.78241
\(999\) −2.82770e9 −0.0897335
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.d.1.15 18
3.2 odd 2 531.8.a.e.1.4 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.d.1.15 18 1.1 even 1 trivial
531.8.a.e.1.4 18 3.2 odd 2