Properties

Label 177.8.a.c.1.7
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $0$
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: \(0\)
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 1.7
Root \(-5.66556\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-5.66556 q^{2} -27.0000 q^{3} -95.9014 q^{4} +117.894 q^{5} +152.970 q^{6} +1197.66 q^{7} +1268.53 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-5.66556 q^{2} -27.0000 q^{3} -95.9014 q^{4} +117.894 q^{5} +152.970 q^{6} +1197.66 q^{7} +1268.53 q^{8} +729.000 q^{9} -667.937 q^{10} -6536.63 q^{11} +2589.34 q^{12} +11916.5 q^{13} -6785.43 q^{14} -3183.14 q^{15} +5088.45 q^{16} +29402.4 q^{17} -4130.20 q^{18} +3457.38 q^{19} -11306.2 q^{20} -32336.9 q^{21} +37033.7 q^{22} -37722.6 q^{23} -34250.2 q^{24} -64225.9 q^{25} -67514.0 q^{26} -19683.0 q^{27} -114857. q^{28} -207751. q^{29} +18034.3 q^{30} +111488. q^{31} -191200. q^{32} +176489. q^{33} -166581. q^{34} +141197. q^{35} -69912.1 q^{36} -49565.3 q^{37} -19588.0 q^{38} -321747. q^{39} +149552. q^{40} +112884. q^{41} +183207. q^{42} -38995.6 q^{43} +626872. q^{44} +85944.9 q^{45} +213720. q^{46} +1.22368e6 q^{47} -137388. q^{48} +610852. q^{49} +363876. q^{50} -793865. q^{51} -1.14281e6 q^{52} -475269. q^{53} +111515. q^{54} -770631. q^{55} +1.51927e6 q^{56} -93349.3 q^{57} +1.17702e6 q^{58} -205379. q^{59} +305268. q^{60} +699413. q^{61} -631640. q^{62} +873096. q^{63} +431937. q^{64} +1.40489e6 q^{65} -999909. q^{66} -3.23969e6 q^{67} -2.81973e6 q^{68} +1.01851e6 q^{69} -799963. q^{70} +5.13542e6 q^{71} +924757. q^{72} +5.08296e6 q^{73} +280816. q^{74} +1.73410e6 q^{75} -331568. q^{76} -7.82867e6 q^{77} +1.82288e6 q^{78} +7.01416e6 q^{79} +599899. q^{80} +531441. q^{81} -639551. q^{82} +6.20880e6 q^{83} +3.10115e6 q^{84} +3.46637e6 q^{85} +220932. q^{86} +5.60927e6 q^{87} -8.29189e6 q^{88} +1.44794e6 q^{89} -486926. q^{90} +1.42720e7 q^{91} +3.61765e6 q^{92} -3.01017e6 q^{93} -6.93283e6 q^{94} +407605. q^{95} +5.16241e6 q^{96} -1.19209e7 q^{97} -3.46082e6 q^{98} -4.76520e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + O(q^{10}) \) \( 17q + 2q^{2} - 459q^{3} + 1166q^{4} - 318q^{5} - 54q^{6} + 3145q^{7} + 2355q^{8} + 12393q^{9} + 6521q^{10} - 1764q^{11} - 31482q^{12} + 18192q^{13} - 7827q^{14} + 8586q^{15} + 139226q^{16} - 15507q^{17} + 1458q^{18} + 52083q^{19} + 721q^{20} - 84915q^{21} - 234434q^{22} + 63823q^{23} - 63585q^{24} + 202153q^{25} - 367956q^{26} - 334611q^{27} + 182306q^{28} - 502955q^{29} - 176067q^{30} + 347531q^{31} - 243908q^{32} + 47628q^{33} - 330872q^{34} + 92641q^{35} + 850014q^{36} + 447615q^{37} + 775669q^{38} - 491184q^{39} + 2203270q^{40} + 940335q^{41} + 211329q^{42} + 478562q^{43} - 596924q^{44} - 231822q^{45} - 3078663q^{46} + 703121q^{47} - 3759102q^{48} + 1895082q^{49} - 876967q^{50} + 418689q^{51} + 6278296q^{52} - 1005974q^{53} - 39366q^{54} + 5212846q^{55} + 3425294q^{56} - 1406241q^{57} + 6710166q^{58} - 3491443q^{59} - 19467q^{60} + 11510749q^{61} + 5996234q^{62} + 2292705q^{63} + 29496941q^{64} + 11094180q^{65} + 6329718q^{66} + 14007144q^{67} + 19688159q^{68} - 1723221q^{69} + 30909708q^{70} + 5229074q^{71} + 1716795q^{72} + 5452211q^{73} + 12819662q^{74} - 5458131q^{75} + 41929340q^{76} + 9930777q^{77} + 9934812q^{78} + 15275654q^{79} + 36576105q^{80} + 9034497q^{81} + 32025935q^{82} + 7826609q^{83} - 4922262q^{84} + 11836945q^{85} + 51649136q^{86} + 13579785q^{87} + 30223741q^{88} - 6436185q^{89} + 4753809q^{90} + 11633535q^{91} + 43357972q^{92} - 9383337q^{93} - 4494252q^{94} + 23741055q^{95} + 6585516q^{96} + 26377540q^{97} + 26517816q^{98} - 1285956q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.66556 −0.500770 −0.250385 0.968146i \(-0.580557\pi\)
−0.250385 + 0.968146i \(0.580557\pi\)
\(3\) −27.0000 −0.577350
\(4\) −95.9014 −0.749230
\(5\) 117.894 0.421791 0.210896 0.977509i \(-0.432362\pi\)
0.210896 + 0.977509i \(0.432362\pi\)
\(6\) 152.970 0.289120
\(7\) 1197.66 1.31975 0.659874 0.751376i \(-0.270609\pi\)
0.659874 + 0.751376i \(0.270609\pi\)
\(8\) 1268.53 0.875961
\(9\) 729.000 0.333333
\(10\) −667.937 −0.211220
\(11\) −6536.63 −1.48074 −0.740371 0.672199i \(-0.765350\pi\)
−0.740371 + 0.672199i \(0.765350\pi\)
\(12\) 2589.34 0.432568
\(13\) 11916.5 1.50435 0.752175 0.658964i \(-0.229005\pi\)
0.752175 + 0.658964i \(0.229005\pi\)
\(14\) −6785.43 −0.660890
\(15\) −3183.14 −0.243521
\(16\) 5088.45 0.310574
\(17\) 29402.4 1.45148 0.725741 0.687968i \(-0.241497\pi\)
0.725741 + 0.687968i \(0.241497\pi\)
\(18\) −4130.20 −0.166923
\(19\) 3457.38 0.115640 0.0578202 0.998327i \(-0.481585\pi\)
0.0578202 + 0.998327i \(0.481585\pi\)
\(20\) −11306.2 −0.316018
\(21\) −32336.9 −0.761957
\(22\) 37033.7 0.741511
\(23\) −37722.6 −0.646479 −0.323239 0.946317i \(-0.604772\pi\)
−0.323239 + 0.946317i \(0.604772\pi\)
\(24\) −34250.2 −0.505737
\(25\) −64225.9 −0.822092
\(26\) −67514.0 −0.753333
\(27\) −19683.0 −0.192450
\(28\) −114857. −0.988795
\(29\) −207751. −1.58179 −0.790896 0.611951i \(-0.790385\pi\)
−0.790896 + 0.611951i \(0.790385\pi\)
\(30\) 18034.3 0.121948
\(31\) 111488. 0.672141 0.336071 0.941837i \(-0.390902\pi\)
0.336071 + 0.941837i \(0.390902\pi\)
\(32\) −191200. −1.03149
\(33\) 176489. 0.854907
\(34\) −166581. −0.726858
\(35\) 141197. 0.556658
\(36\) −69912.1 −0.249743
\(37\) −49565.3 −0.160869 −0.0804344 0.996760i \(-0.525631\pi\)
−0.0804344 + 0.996760i \(0.525631\pi\)
\(38\) −19588.0 −0.0579092
\(39\) −321747. −0.868536
\(40\) 149552. 0.369473
\(41\) 112884. 0.255793 0.127896 0.991788i \(-0.459177\pi\)
0.127896 + 0.991788i \(0.459177\pi\)
\(42\) 183207. 0.381565
\(43\) −38995.6 −0.0747956 −0.0373978 0.999300i \(-0.511907\pi\)
−0.0373978 + 0.999300i \(0.511907\pi\)
\(44\) 626872. 1.10942
\(45\) 85944.9 0.140597
\(46\) 213720. 0.323737
\(47\) 1.22368e6 1.71919 0.859597 0.510972i \(-0.170715\pi\)
0.859597 + 0.510972i \(0.170715\pi\)
\(48\) −137388. −0.179310
\(49\) 610852. 0.741737
\(50\) 363876. 0.411679
\(51\) −793865. −0.838013
\(52\) −1.14281e6 −1.12710
\(53\) −475269. −0.438504 −0.219252 0.975668i \(-0.570362\pi\)
−0.219252 + 0.975668i \(0.570362\pi\)
\(54\) 111515. 0.0963732
\(55\) −770631. −0.624564
\(56\) 1.51927e6 1.15605
\(57\) −93349.3 −0.0667650
\(58\) 1.17702e6 0.792114
\(59\) −205379. −0.130189
\(60\) 305268. 0.182453
\(61\) 699413. 0.394529 0.197265 0.980350i \(-0.436794\pi\)
0.197265 + 0.980350i \(0.436794\pi\)
\(62\) −631640. −0.336588
\(63\) 873096. 0.439916
\(64\) 431937. 0.205964
\(65\) 1.40489e6 0.634521
\(66\) −999909. −0.428111
\(67\) −3.23969e6 −1.31596 −0.657979 0.753036i \(-0.728589\pi\)
−0.657979 + 0.753036i \(0.728589\pi\)
\(68\) −2.81973e6 −1.08749
\(69\) 1.01851e6 0.373245
\(70\) −799963. −0.278758
\(71\) 5.13542e6 1.70283 0.851417 0.524490i \(-0.175744\pi\)
0.851417 + 0.524490i \(0.175744\pi\)
\(72\) 924757. 0.291987
\(73\) 5.08296e6 1.52928 0.764639 0.644459i \(-0.222917\pi\)
0.764639 + 0.644459i \(0.222917\pi\)
\(74\) 280816. 0.0805583
\(75\) 1.73410e6 0.474635
\(76\) −331568. −0.0866412
\(77\) −7.82867e6 −1.95421
\(78\) 1.82288e6 0.434937
\(79\) 7.01416e6 1.60059 0.800296 0.599605i \(-0.204675\pi\)
0.800296 + 0.599605i \(0.204675\pi\)
\(80\) 599899. 0.130998
\(81\) 531441. 0.111111
\(82\) −639551. −0.128093
\(83\) 6.20880e6 1.19188 0.595942 0.803027i \(-0.296779\pi\)
0.595942 + 0.803027i \(0.296779\pi\)
\(84\) 3.10115e6 0.570881
\(85\) 3.46637e6 0.612222
\(86\) 220932. 0.0374554
\(87\) 5.60927e6 0.913248
\(88\) −8.29189e6 −1.29707
\(89\) 1.44794e6 0.217714 0.108857 0.994057i \(-0.465281\pi\)
0.108857 + 0.994057i \(0.465281\pi\)
\(90\) −486926. −0.0704068
\(91\) 1.42720e7 1.98536
\(92\) 3.61765e6 0.484361
\(93\) −3.01017e6 −0.388061
\(94\) −6.93283e6 −0.860921
\(95\) 407605. 0.0487761
\(96\) 5.16241e6 0.595530
\(97\) −1.19209e7 −1.32620 −0.663098 0.748532i \(-0.730759\pi\)
−0.663098 + 0.748532i \(0.730759\pi\)
\(98\) −3.46082e6 −0.371439
\(99\) −4.76520e6 −0.493581
\(100\) 6.15936e6 0.615936
\(101\) −1.33614e7 −1.29041 −0.645205 0.764009i \(-0.723228\pi\)
−0.645205 + 0.764009i \(0.723228\pi\)
\(102\) 4.49769e6 0.419652
\(103\) 2.58316e6 0.232928 0.116464 0.993195i \(-0.462844\pi\)
0.116464 + 0.993195i \(0.462844\pi\)
\(104\) 1.51165e7 1.31775
\(105\) −3.81233e6 −0.321387
\(106\) 2.69266e6 0.219590
\(107\) −4.52462e6 −0.357058 −0.178529 0.983935i \(-0.557134\pi\)
−0.178529 + 0.983935i \(0.557134\pi\)
\(108\) 1.88763e6 0.144189
\(109\) −312613. −0.0231214 −0.0115607 0.999933i \(-0.503680\pi\)
−0.0115607 + 0.999933i \(0.503680\pi\)
\(110\) 4.36606e6 0.312763
\(111\) 1.33826e6 0.0928777
\(112\) 6.09425e6 0.409880
\(113\) 4.97336e6 0.324246 0.162123 0.986771i \(-0.448166\pi\)
0.162123 + 0.986771i \(0.448166\pi\)
\(114\) 528876. 0.0334339
\(115\) −4.44728e6 −0.272679
\(116\) 1.99236e7 1.18513
\(117\) 8.68716e6 0.501450
\(118\) 1.16359e6 0.0651947
\(119\) 3.52141e7 1.91559
\(120\) −4.03791e6 −0.213315
\(121\) 2.32403e7 1.19260
\(122\) −3.96257e6 −0.197568
\(123\) −3.04786e6 −0.147682
\(124\) −1.06918e7 −0.503588
\(125\) −1.67824e7 −0.768543
\(126\) −4.94658e6 −0.220297
\(127\) 4.34322e7 1.88148 0.940738 0.339134i \(-0.110134\pi\)
0.940738 + 0.339134i \(0.110134\pi\)
\(128\) 2.20265e7 0.928347
\(129\) 1.05288e6 0.0431833
\(130\) −7.95951e6 −0.317749
\(131\) −3.43447e7 −1.33478 −0.667391 0.744707i \(-0.732589\pi\)
−0.667391 + 0.744707i \(0.732589\pi\)
\(132\) −1.69255e7 −0.640521
\(133\) 4.14077e6 0.152616
\(134\) 1.83547e7 0.658992
\(135\) −2.32051e6 −0.0811738
\(136\) 3.72978e7 1.27144
\(137\) −5.15704e7 −1.71348 −0.856738 0.515752i \(-0.827513\pi\)
−0.856738 + 0.515752i \(0.827513\pi\)
\(138\) −5.77044e6 −0.186910
\(139\) 6.25955e7 1.97693 0.988466 0.151444i \(-0.0483924\pi\)
0.988466 + 0.151444i \(0.0483924\pi\)
\(140\) −1.35410e7 −0.417065
\(141\) −3.30393e7 −0.992577
\(142\) −2.90951e7 −0.852728
\(143\) −7.78940e7 −2.22755
\(144\) 3.70948e6 0.103525
\(145\) −2.44926e7 −0.667186
\(146\) −2.87978e7 −0.765816
\(147\) −1.64930e7 −0.428242
\(148\) 4.75338e6 0.120528
\(149\) −4.84710e6 −0.120041 −0.0600206 0.998197i \(-0.519117\pi\)
−0.0600206 + 0.998197i \(0.519117\pi\)
\(150\) −9.82466e6 −0.237683
\(151\) 7.66941e7 1.81277 0.906385 0.422452i \(-0.138831\pi\)
0.906385 + 0.422452i \(0.138831\pi\)
\(152\) 4.38578e6 0.101297
\(153\) 2.14344e7 0.483827
\(154\) 4.43538e7 0.978608
\(155\) 1.31437e7 0.283503
\(156\) 3.08560e7 0.650733
\(157\) −4.97181e6 −0.102534 −0.0512668 0.998685i \(-0.516326\pi\)
−0.0512668 + 0.998685i \(0.516326\pi\)
\(158\) −3.97392e7 −0.801529
\(159\) 1.28323e7 0.253170
\(160\) −2.25414e7 −0.435073
\(161\) −4.51790e7 −0.853190
\(162\) −3.01091e6 −0.0556411
\(163\) 1.80568e7 0.326577 0.163288 0.986578i \(-0.447790\pi\)
0.163288 + 0.986578i \(0.447790\pi\)
\(164\) −1.08257e7 −0.191648
\(165\) 2.08070e7 0.360592
\(166\) −3.51763e7 −0.596860
\(167\) −3.89288e7 −0.646790 −0.323395 0.946264i \(-0.604824\pi\)
−0.323395 + 0.946264i \(0.604824\pi\)
\(168\) −4.10202e7 −0.667445
\(169\) 7.92556e7 1.26307
\(170\) −1.96390e7 −0.306582
\(171\) 2.52043e6 0.0385468
\(172\) 3.73973e6 0.0560391
\(173\) 9.04299e7 1.32786 0.663928 0.747797i \(-0.268888\pi\)
0.663928 + 0.747797i \(0.268888\pi\)
\(174\) −3.17797e7 −0.457327
\(175\) −7.69210e7 −1.08496
\(176\) −3.32613e7 −0.459881
\(177\) 5.54523e6 0.0751646
\(178\) −8.20342e6 −0.109025
\(179\) 1.14493e7 0.149208 0.0746042 0.997213i \(-0.476231\pi\)
0.0746042 + 0.997213i \(0.476231\pi\)
\(180\) −8.24223e6 −0.105339
\(181\) −9.25112e7 −1.15963 −0.579815 0.814748i \(-0.696875\pi\)
−0.579815 + 0.814748i \(0.696875\pi\)
\(182\) −8.08589e7 −0.994210
\(183\) −1.88841e7 −0.227781
\(184\) −4.78522e7 −0.566291
\(185\) −5.84347e6 −0.0678531
\(186\) 1.70543e7 0.194329
\(187\) −1.92193e8 −2.14927
\(188\) −1.17353e8 −1.28807
\(189\) −2.35736e7 −0.253986
\(190\) −2.30931e6 −0.0244256
\(191\) 2.78323e7 0.289023 0.144511 0.989503i \(-0.453839\pi\)
0.144511 + 0.989503i \(0.453839\pi\)
\(192\) −1.16623e7 −0.118913
\(193\) −3.52675e7 −0.353122 −0.176561 0.984290i \(-0.556497\pi\)
−0.176561 + 0.984290i \(0.556497\pi\)
\(194\) 6.75386e7 0.664119
\(195\) −3.79321e7 −0.366341
\(196\) −5.85816e7 −0.555731
\(197\) 1.76100e8 1.64107 0.820536 0.571595i \(-0.193675\pi\)
0.820536 + 0.571595i \(0.193675\pi\)
\(198\) 2.69976e7 0.247170
\(199\) 1.94813e8 1.75239 0.876196 0.481954i \(-0.160073\pi\)
0.876196 + 0.481954i \(0.160073\pi\)
\(200\) −8.14724e7 −0.720121
\(201\) 8.74717e7 0.759769
\(202\) 7.57000e7 0.646199
\(203\) −2.48815e8 −2.08757
\(204\) 7.61327e7 0.627864
\(205\) 1.33084e7 0.107891
\(206\) −1.46351e7 −0.116643
\(207\) −2.74998e7 −0.215493
\(208\) 6.06368e7 0.467212
\(209\) −2.25996e7 −0.171234
\(210\) 2.15990e7 0.160941
\(211\) 9.43223e7 0.691235 0.345617 0.938376i \(-0.387670\pi\)
0.345617 + 0.938376i \(0.387670\pi\)
\(212\) 4.55789e7 0.328540
\(213\) −1.38656e8 −0.983131
\(214\) 2.56345e7 0.178804
\(215\) −4.59736e6 −0.0315481
\(216\) −2.49684e7 −0.168579
\(217\) 1.33525e8 0.887058
\(218\) 1.77113e6 0.0115785
\(219\) −1.37240e8 −0.882929
\(220\) 7.39045e7 0.467942
\(221\) 3.50375e8 2.18354
\(222\) −7.58202e6 −0.0465103
\(223\) −3.04202e8 −1.83694 −0.918470 0.395492i \(-0.870574\pi\)
−0.918470 + 0.395492i \(0.870574\pi\)
\(224\) −2.28994e8 −1.36130
\(225\) −4.68207e7 −0.274031
\(226\) −2.81769e7 −0.162373
\(227\) 1.54534e8 0.876866 0.438433 0.898764i \(-0.355534\pi\)
0.438433 + 0.898764i \(0.355534\pi\)
\(228\) 8.95232e6 0.0500223
\(229\) 1.18030e8 0.649485 0.324742 0.945803i \(-0.394722\pi\)
0.324742 + 0.945803i \(0.394722\pi\)
\(230\) 2.51963e7 0.136550
\(231\) 2.11374e8 1.12826
\(232\) −2.63537e8 −1.38559
\(233\) −6.82032e7 −0.353231 −0.176615 0.984280i \(-0.556515\pi\)
−0.176615 + 0.984280i \(0.556515\pi\)
\(234\) −4.92177e7 −0.251111
\(235\) 1.44265e8 0.725141
\(236\) 1.96961e7 0.0975414
\(237\) −1.89382e8 −0.924103
\(238\) −1.99508e8 −0.959270
\(239\) −8.01024e7 −0.379536 −0.189768 0.981829i \(-0.560774\pi\)
−0.189768 + 0.981829i \(0.560774\pi\)
\(240\) −1.61973e7 −0.0756315
\(241\) 9.44904e7 0.434839 0.217419 0.976078i \(-0.430236\pi\)
0.217419 + 0.976078i \(0.430236\pi\)
\(242\) −1.31670e8 −0.597216
\(243\) −1.43489e7 −0.0641500
\(244\) −6.70746e7 −0.295593
\(245\) 7.20159e7 0.312858
\(246\) 1.72679e7 0.0739547
\(247\) 4.12000e7 0.173964
\(248\) 1.41425e8 0.588770
\(249\) −1.67638e8 −0.688135
\(250\) 9.50815e7 0.384863
\(251\) −2.37800e8 −0.949191 −0.474596 0.880204i \(-0.657406\pi\)
−0.474596 + 0.880204i \(0.657406\pi\)
\(252\) −8.37311e7 −0.329598
\(253\) 2.46579e8 0.957268
\(254\) −2.46068e8 −0.942187
\(255\) −9.35921e7 −0.353467
\(256\) −1.80080e8 −0.670852
\(257\) 673684. 0.00247566 0.00123783 0.999999i \(-0.499606\pi\)
0.00123783 + 0.999999i \(0.499606\pi\)
\(258\) −5.96517e6 −0.0216249
\(259\) −5.93625e7 −0.212307
\(260\) −1.34731e8 −0.475402
\(261\) −1.51450e8 −0.527264
\(262\) 1.94582e8 0.668419
\(263\) −3.23561e7 −0.109676 −0.0548380 0.998495i \(-0.517464\pi\)
−0.0548380 + 0.998495i \(0.517464\pi\)
\(264\) 2.23881e8 0.748865
\(265\) −5.60314e7 −0.184957
\(266\) −2.34598e7 −0.0764256
\(267\) −3.90945e7 −0.125697
\(268\) 3.10691e8 0.985955
\(269\) 2.61753e8 0.819896 0.409948 0.912109i \(-0.365547\pi\)
0.409948 + 0.912109i \(0.365547\pi\)
\(270\) 1.31470e7 0.0406494
\(271\) 6.21861e8 1.89802 0.949011 0.315244i \(-0.102086\pi\)
0.949011 + 0.315244i \(0.102086\pi\)
\(272\) 1.49613e8 0.450793
\(273\) −3.85344e8 −1.14625
\(274\) 2.92175e8 0.858057
\(275\) 4.19821e8 1.21731
\(276\) −9.76766e7 −0.279646
\(277\) 2.81765e8 0.796541 0.398270 0.917268i \(-0.369611\pi\)
0.398270 + 0.917268i \(0.369611\pi\)
\(278\) −3.54639e8 −0.989988
\(279\) 8.12745e7 0.224047
\(280\) 1.79113e8 0.487611
\(281\) 2.34019e8 0.629186 0.314593 0.949227i \(-0.398132\pi\)
0.314593 + 0.949227i \(0.398132\pi\)
\(282\) 1.87186e8 0.497053
\(283\) −3.59081e8 −0.941759 −0.470879 0.882198i \(-0.656063\pi\)
−0.470879 + 0.882198i \(0.656063\pi\)
\(284\) −4.92494e8 −1.27581
\(285\) −1.10053e7 −0.0281609
\(286\) 4.41314e8 1.11549
\(287\) 1.35197e8 0.337582
\(288\) −1.39385e8 −0.343829
\(289\) 4.54163e8 1.10680
\(290\) 1.38764e8 0.334107
\(291\) 3.21864e8 0.765680
\(292\) −4.87463e8 −1.14578
\(293\) 2.48310e8 0.576711 0.288355 0.957523i \(-0.406892\pi\)
0.288355 + 0.957523i \(0.406892\pi\)
\(294\) 9.34422e7 0.214451
\(295\) −2.42130e7 −0.0549125
\(296\) −6.28750e7 −0.140915
\(297\) 1.28660e8 0.284969
\(298\) 2.74616e7 0.0601130
\(299\) −4.49523e8 −0.972530
\(300\) −1.66303e8 −0.355611
\(301\) −4.67036e7 −0.0987114
\(302\) −4.34516e8 −0.907781
\(303\) 3.60758e8 0.745019
\(304\) 1.75927e7 0.0359150
\(305\) 8.24567e7 0.166409
\(306\) −1.21438e8 −0.242286
\(307\) 1.92429e8 0.379566 0.189783 0.981826i \(-0.439222\pi\)
0.189783 + 0.981826i \(0.439222\pi\)
\(308\) 7.50780e8 1.46415
\(309\) −6.97454e7 −0.134481
\(310\) −7.44667e7 −0.141970
\(311\) −4.80733e8 −0.906238 −0.453119 0.891450i \(-0.649689\pi\)
−0.453119 + 0.891450i \(0.649689\pi\)
\(312\) −4.08145e8 −0.760804
\(313\) 7.85937e8 1.44871 0.724357 0.689425i \(-0.242137\pi\)
0.724357 + 0.689425i \(0.242137\pi\)
\(314\) 2.81681e7 0.0513457
\(315\) 1.02933e8 0.185553
\(316\) −6.72668e8 −1.19921
\(317\) −2.09039e8 −0.368570 −0.184285 0.982873i \(-0.558997\pi\)
−0.184285 + 0.982873i \(0.558997\pi\)
\(318\) −7.27019e7 −0.126780
\(319\) 1.35799e9 2.34222
\(320\) 5.09229e7 0.0868736
\(321\) 1.22165e8 0.206148
\(322\) 2.55964e8 0.427252
\(323\) 1.01655e8 0.167850
\(324\) −5.09659e7 −0.0832477
\(325\) −7.65351e8 −1.23671
\(326\) −1.02302e8 −0.163540
\(327\) 8.44054e6 0.0133491
\(328\) 1.43196e8 0.224065
\(329\) 1.46555e9 2.26890
\(330\) −1.17884e8 −0.180574
\(331\) 7.95790e8 1.20615 0.603074 0.797685i \(-0.293942\pi\)
0.603074 + 0.797685i \(0.293942\pi\)
\(332\) −5.95432e8 −0.892995
\(333\) −3.61331e7 −0.0536230
\(334\) 2.20554e8 0.323893
\(335\) −3.81941e8 −0.555060
\(336\) −1.64545e8 −0.236644
\(337\) 2.75834e8 0.392593 0.196297 0.980545i \(-0.437108\pi\)
0.196297 + 0.980545i \(0.437108\pi\)
\(338\) −4.49027e8 −0.632506
\(339\) −1.34281e8 −0.187204
\(340\) −3.32430e8 −0.458695
\(341\) −7.28753e8 −0.995268
\(342\) −1.42797e7 −0.0193031
\(343\) −2.54732e8 −0.340843
\(344\) −4.94670e7 −0.0655181
\(345\) 1.20077e8 0.157431
\(346\) −5.12337e8 −0.664950
\(347\) 4.81631e8 0.618816 0.309408 0.950929i \(-0.399869\pi\)
0.309408 + 0.950929i \(0.399869\pi\)
\(348\) −5.37936e8 −0.684232
\(349\) −3.21673e8 −0.405066 −0.202533 0.979275i \(-0.564917\pi\)
−0.202533 + 0.979275i \(0.564917\pi\)
\(350\) 4.35801e8 0.543313
\(351\) −2.34553e8 −0.289512
\(352\) 1.24981e9 1.52737
\(353\) 1.17516e9 1.42195 0.710977 0.703216i \(-0.248253\pi\)
0.710977 + 0.703216i \(0.248253\pi\)
\(354\) −3.14169e7 −0.0376402
\(355\) 6.05437e8 0.718240
\(356\) −1.38860e8 −0.163118
\(357\) −9.50782e8 −1.10597
\(358\) −6.48668e7 −0.0747191
\(359\) 3.28816e8 0.375079 0.187539 0.982257i \(-0.439949\pi\)
0.187539 + 0.982257i \(0.439949\pi\)
\(360\) 1.09023e8 0.123158
\(361\) −8.81918e8 −0.986627
\(362\) 5.24128e8 0.580707
\(363\) −6.27489e8 −0.688546
\(364\) −1.36870e9 −1.48749
\(365\) 5.99251e8 0.645036
\(366\) 1.06989e8 0.114066
\(367\) −6.95555e8 −0.734515 −0.367257 0.930119i \(-0.619703\pi\)
−0.367257 + 0.930119i \(0.619703\pi\)
\(368\) −1.91950e8 −0.200780
\(369\) 8.22923e7 0.0852643
\(370\) 3.31065e7 0.0339788
\(371\) −5.69211e8 −0.578715
\(372\) 2.88679e8 0.290747
\(373\) 1.04776e9 1.04540 0.522699 0.852517i \(-0.324925\pi\)
0.522699 + 0.852517i \(0.324925\pi\)
\(374\) 1.08888e9 1.07629
\(375\) 4.53124e8 0.443718
\(376\) 1.55227e9 1.50595
\(377\) −2.47567e9 −2.37957
\(378\) 1.33558e8 0.127188
\(379\) 9.05666e8 0.854537 0.427268 0.904125i \(-0.359476\pi\)
0.427268 + 0.904125i \(0.359476\pi\)
\(380\) −3.90899e7 −0.0365445
\(381\) −1.17267e9 −1.08627
\(382\) −1.57686e8 −0.144734
\(383\) 1.13004e9 1.02777 0.513887 0.857858i \(-0.328205\pi\)
0.513887 + 0.857858i \(0.328205\pi\)
\(384\) −5.94715e8 −0.535982
\(385\) −9.22955e8 −0.824267
\(386\) 1.99810e8 0.176833
\(387\) −2.84278e7 −0.0249319
\(388\) 1.14323e9 0.993626
\(389\) −9.69975e8 −0.835482 −0.417741 0.908566i \(-0.637178\pi\)
−0.417741 + 0.908566i \(0.637178\pi\)
\(390\) 2.14907e8 0.183453
\(391\) −1.10914e9 −0.938352
\(392\) 7.74883e8 0.649733
\(393\) 9.27307e8 0.770637
\(394\) −9.97706e8 −0.821799
\(395\) 8.26929e8 0.675116
\(396\) 4.56989e8 0.369805
\(397\) −1.37378e9 −1.10192 −0.550961 0.834531i \(-0.685739\pi\)
−0.550961 + 0.834531i \(0.685739\pi\)
\(398\) −1.10372e9 −0.877546
\(399\) −1.11801e8 −0.0881130
\(400\) −3.26811e8 −0.255321
\(401\) 5.72339e8 0.443249 0.221625 0.975132i \(-0.428864\pi\)
0.221625 + 0.975132i \(0.428864\pi\)
\(402\) −4.95577e8 −0.380469
\(403\) 1.32855e9 1.01114
\(404\) 1.28138e9 0.966813
\(405\) 6.26538e7 0.0468657
\(406\) 1.40968e9 1.04539
\(407\) 3.23990e8 0.238205
\(408\) −1.00704e9 −0.734067
\(409\) 1.07412e8 0.0776286 0.0388143 0.999246i \(-0.487642\pi\)
0.0388143 + 0.999246i \(0.487642\pi\)
\(410\) −7.53994e7 −0.0540287
\(411\) 1.39240e9 0.989276
\(412\) −2.47729e8 −0.174516
\(413\) −2.45975e8 −0.171817
\(414\) 1.55802e8 0.107912
\(415\) 7.31982e8 0.502727
\(416\) −2.27845e9 −1.55172
\(417\) −1.69008e9 −1.14138
\(418\) 1.28040e8 0.0857486
\(419\) 4.96522e8 0.329753 0.164877 0.986314i \(-0.447277\pi\)
0.164877 + 0.986314i \(0.447277\pi\)
\(420\) 3.65608e8 0.240793
\(421\) 2.13271e9 1.39298 0.696490 0.717566i \(-0.254744\pi\)
0.696490 + 0.717566i \(0.254744\pi\)
\(422\) −5.34389e8 −0.346150
\(423\) 8.92062e8 0.573065
\(424\) −6.02891e8 −0.384113
\(425\) −1.88840e9 −1.19325
\(426\) 7.85567e8 0.492323
\(427\) 8.37660e8 0.520679
\(428\) 4.33918e8 0.267519
\(429\) 2.10314e9 1.28608
\(430\) 2.60466e7 0.0157984
\(431\) −2.75546e8 −0.165777 −0.0828885 0.996559i \(-0.526415\pi\)
−0.0828885 + 0.996559i \(0.526415\pi\)
\(432\) −1.00156e8 −0.0597701
\(433\) 1.42964e9 0.846289 0.423144 0.906062i \(-0.360926\pi\)
0.423144 + 0.906062i \(0.360926\pi\)
\(434\) −7.56492e8 −0.444212
\(435\) 6.61300e8 0.385200
\(436\) 2.99800e7 0.0173232
\(437\) −1.30421e8 −0.0747591
\(438\) 7.77541e8 0.442144
\(439\) −2.29794e9 −1.29632 −0.648161 0.761503i \(-0.724462\pi\)
−0.648161 + 0.761503i \(0.724462\pi\)
\(440\) −9.77566e8 −0.547094
\(441\) 4.45311e8 0.247246
\(442\) −1.98507e9 −1.09345
\(443\) −5.72726e8 −0.312992 −0.156496 0.987679i \(-0.550020\pi\)
−0.156496 + 0.987679i \(0.550020\pi\)
\(444\) −1.28341e8 −0.0695867
\(445\) 1.70704e8 0.0918300
\(446\) 1.72347e9 0.919884
\(447\) 1.30872e8 0.0693058
\(448\) 5.17315e8 0.271820
\(449\) −2.51611e9 −1.31180 −0.655899 0.754848i \(-0.727710\pi\)
−0.655899 + 0.754848i \(0.727710\pi\)
\(450\) 2.65266e8 0.137226
\(451\) −7.37880e8 −0.378763
\(452\) −4.76952e8 −0.242935
\(453\) −2.07074e9 −1.04660
\(454\) −8.75521e8 −0.439108
\(455\) 1.68259e9 0.837409
\(456\) −1.18416e8 −0.0584836
\(457\) −9.31683e8 −0.456627 −0.228314 0.973588i \(-0.573321\pi\)
−0.228314 + 0.973588i \(0.573321\pi\)
\(458\) −6.68708e8 −0.325242
\(459\) −5.78727e8 −0.279338
\(460\) 4.26500e8 0.204299
\(461\) −2.12369e8 −0.100957 −0.0504786 0.998725i \(-0.516075\pi\)
−0.0504786 + 0.998725i \(0.516075\pi\)
\(462\) −1.19755e9 −0.565000
\(463\) 3.14522e9 1.47271 0.736355 0.676596i \(-0.236545\pi\)
0.736355 + 0.676596i \(0.236545\pi\)
\(464\) −1.05713e9 −0.491264
\(465\) −3.54881e8 −0.163681
\(466\) 3.86409e8 0.176887
\(467\) 3.87688e9 1.76146 0.880732 0.473615i \(-0.157051\pi\)
0.880732 + 0.473615i \(0.157051\pi\)
\(468\) −8.33111e8 −0.375701
\(469\) −3.88006e9 −1.73673
\(470\) −8.17341e8 −0.363129
\(471\) 1.34239e8 0.0591978
\(472\) −2.60529e8 −0.114040
\(473\) 2.54900e8 0.110753
\(474\) 1.07296e9 0.462763
\(475\) −2.22054e8 −0.0950671
\(476\) −3.37709e9 −1.43522
\(477\) −3.46471e8 −0.146168
\(478\) 4.53825e8 0.190060
\(479\) −3.88318e9 −1.61441 −0.807205 0.590272i \(-0.799021\pi\)
−0.807205 + 0.590272i \(0.799021\pi\)
\(480\) 6.08619e8 0.251189
\(481\) −5.90648e8 −0.242003
\(482\) −5.35342e8 −0.217754
\(483\) 1.21983e9 0.492589
\(484\) −2.22878e9 −0.893528
\(485\) −1.40541e9 −0.559378
\(486\) 8.12947e7 0.0321244
\(487\) −1.79849e9 −0.705597 −0.352799 0.935699i \(-0.614770\pi\)
−0.352799 + 0.935699i \(0.614770\pi\)
\(488\) 8.87224e8 0.345592
\(489\) −4.87535e8 −0.188549
\(490\) −4.08011e8 −0.156670
\(491\) −1.24531e9 −0.474780 −0.237390 0.971414i \(-0.576292\pi\)
−0.237390 + 0.971414i \(0.576292\pi\)
\(492\) 2.92294e8 0.110648
\(493\) −6.10837e9 −2.29594
\(494\) −2.33421e8 −0.0871157
\(495\) −5.61790e8 −0.208188
\(496\) 5.67299e8 0.208750
\(497\) 6.15051e9 2.24731
\(498\) 9.49761e8 0.344597
\(499\) 3.02782e9 1.09088 0.545442 0.838148i \(-0.316362\pi\)
0.545442 + 0.838148i \(0.316362\pi\)
\(500\) 1.60945e9 0.575815
\(501\) 1.05108e9 0.373424
\(502\) 1.34727e9 0.475326
\(503\) −3.67205e9 −1.28653 −0.643267 0.765642i \(-0.722421\pi\)
−0.643267 + 0.765642i \(0.722421\pi\)
\(504\) 1.10755e9 0.385350
\(505\) −1.57523e9 −0.544284
\(506\) −1.39701e9 −0.479371
\(507\) −2.13990e9 −0.729232
\(508\) −4.16521e9 −1.40966
\(509\) −7.51140e8 −0.252469 −0.126235 0.992000i \(-0.540289\pi\)
−0.126235 + 0.992000i \(0.540289\pi\)
\(510\) 5.30252e8 0.177005
\(511\) 6.08767e9 2.01826
\(512\) −1.79913e9 −0.592405
\(513\) −6.80516e7 −0.0222550
\(514\) −3.81680e6 −0.00123973
\(515\) 3.04540e8 0.0982469
\(516\) −1.00973e8 −0.0323542
\(517\) −7.99873e9 −2.54568
\(518\) 3.36322e8 0.106317
\(519\) −2.44161e9 −0.766638
\(520\) 1.78214e9 0.555816
\(521\) 2.30813e9 0.715036 0.357518 0.933906i \(-0.383623\pi\)
0.357518 + 0.933906i \(0.383623\pi\)
\(522\) 8.58051e8 0.264038
\(523\) −9.93626e8 −0.303716 −0.151858 0.988402i \(-0.548526\pi\)
−0.151858 + 0.988402i \(0.548526\pi\)
\(524\) 3.29371e9 1.00006
\(525\) 2.07687e9 0.626399
\(526\) 1.83316e8 0.0549224
\(527\) 3.27800e9 0.975601
\(528\) 8.98055e8 0.265512
\(529\) −1.98183e9 −0.582065
\(530\) 3.17450e8 0.0926210
\(531\) −1.49721e8 −0.0433963
\(532\) −3.97106e8 −0.114345
\(533\) 1.34519e9 0.384802
\(534\) 2.21492e8 0.0629455
\(535\) −5.33427e8 −0.150604
\(536\) −4.10964e9 −1.15273
\(537\) −3.09131e8 −0.0861456
\(538\) −1.48298e9 −0.410579
\(539\) −3.99291e9 −1.09832
\(540\) 2.22540e8 0.0608178
\(541\) −7.57542e8 −0.205691 −0.102846 0.994697i \(-0.532795\pi\)
−0.102846 + 0.994697i \(0.532795\pi\)
\(542\) −3.52320e9 −0.950472
\(543\) 2.49780e9 0.669512
\(544\) −5.62175e9 −1.49719
\(545\) −3.68552e7 −0.00975240
\(546\) 2.18319e9 0.574007
\(547\) −7.20248e9 −1.88160 −0.940798 0.338969i \(-0.889922\pi\)
−0.940798 + 0.338969i \(0.889922\pi\)
\(548\) 4.94567e9 1.28379
\(549\) 5.09872e8 0.131510
\(550\) −2.37852e9 −0.609590
\(551\) −7.18273e8 −0.182919
\(552\) 1.29201e9 0.326948
\(553\) 8.40060e9 2.11238
\(554\) −1.59636e9 −0.398884
\(555\) 1.57774e8 0.0391750
\(556\) −6.00300e9 −1.48118
\(557\) −2.72425e9 −0.667966 −0.333983 0.942579i \(-0.608393\pi\)
−0.333983 + 0.942579i \(0.608393\pi\)
\(558\) −4.60466e8 −0.112196
\(559\) −4.64693e8 −0.112519
\(560\) 7.18477e8 0.172884
\(561\) 5.18920e9 1.24088
\(562\) −1.32585e9 −0.315077
\(563\) 4.08861e9 0.965599 0.482799 0.875731i \(-0.339620\pi\)
0.482799 + 0.875731i \(0.339620\pi\)
\(564\) 3.16852e9 0.743668
\(565\) 5.86330e8 0.136764
\(566\) 2.03440e9 0.471604
\(567\) 6.36487e8 0.146639
\(568\) 6.51443e9 1.49162
\(569\) −6.62224e9 −1.50699 −0.753497 0.657451i \(-0.771635\pi\)
−0.753497 + 0.657451i \(0.771635\pi\)
\(570\) 6.23515e7 0.0141021
\(571\) 1.73743e9 0.390555 0.195277 0.980748i \(-0.437439\pi\)
0.195277 + 0.980748i \(0.437439\pi\)
\(572\) 7.47014e9 1.66895
\(573\) −7.51472e8 −0.166867
\(574\) −7.65966e8 −0.169051
\(575\) 2.42277e9 0.531465
\(576\) 3.14882e8 0.0686545
\(577\) 3.85948e9 0.836399 0.418200 0.908355i \(-0.362661\pi\)
0.418200 + 0.908355i \(0.362661\pi\)
\(578\) −2.57309e9 −0.554252
\(579\) 9.52223e8 0.203875
\(580\) 2.34887e9 0.499875
\(581\) 7.43604e9 1.57299
\(582\) −1.82354e9 −0.383430
\(583\) 3.10665e9 0.649311
\(584\) 6.44787e9 1.33959
\(585\) 1.02417e9 0.211507
\(586\) −1.40682e9 −0.288799
\(587\) −2.98276e9 −0.608674 −0.304337 0.952564i \(-0.598435\pi\)
−0.304337 + 0.952564i \(0.598435\pi\)
\(588\) 1.58170e9 0.320852
\(589\) 3.85455e8 0.0777267
\(590\) 1.37180e8 0.0274985
\(591\) −4.75470e9 −0.947473
\(592\) −2.52211e8 −0.0499618
\(593\) 2.61405e9 0.514781 0.257390 0.966308i \(-0.417137\pi\)
0.257390 + 0.966308i \(0.417137\pi\)
\(594\) −7.28934e8 −0.142704
\(595\) 4.15155e9 0.807980
\(596\) 4.64844e8 0.0899384
\(597\) −5.25994e9 −1.01174
\(598\) 2.54680e9 0.487014
\(599\) −2.42519e8 −0.0461054 −0.0230527 0.999734i \(-0.507339\pi\)
−0.0230527 + 0.999734i \(0.507339\pi\)
\(600\) 2.19975e9 0.415762
\(601\) 6.57764e9 1.23597 0.617987 0.786188i \(-0.287948\pi\)
0.617987 + 0.786188i \(0.287948\pi\)
\(602\) 2.64602e8 0.0494317
\(603\) −2.36174e9 −0.438653
\(604\) −7.35507e9 −1.35818
\(605\) 2.73990e9 0.503027
\(606\) −2.04390e9 −0.373083
\(607\) 3.79055e9 0.687927 0.343963 0.938983i \(-0.388230\pi\)
0.343963 + 0.938983i \(0.388230\pi\)
\(608\) −6.61053e8 −0.119282
\(609\) 6.71801e9 1.20526
\(610\) −4.67164e8 −0.0833326
\(611\) 1.45820e10 2.58627
\(612\) −2.05558e9 −0.362498
\(613\) −5.87144e9 −1.02952 −0.514758 0.857336i \(-0.672118\pi\)
−0.514758 + 0.857336i \(0.672118\pi\)
\(614\) −1.09022e9 −0.190075
\(615\) −3.59326e8 −0.0622910
\(616\) −9.93089e9 −1.71181
\(617\) −4.89410e9 −0.838832 −0.419416 0.907794i \(-0.637765\pi\)
−0.419416 + 0.907794i \(0.637765\pi\)
\(618\) 3.95147e8 0.0673440
\(619\) 9.45629e9 1.60252 0.801260 0.598316i \(-0.204163\pi\)
0.801260 + 0.598316i \(0.204163\pi\)
\(620\) −1.26050e9 −0.212409
\(621\) 7.42494e8 0.124415
\(622\) 2.72362e9 0.453817
\(623\) 1.73415e9 0.287328
\(624\) −1.63719e9 −0.269745
\(625\) 3.03911e9 0.497928
\(626\) −4.45278e9 −0.725473
\(627\) 6.10189e8 0.0988617
\(628\) 4.76804e8 0.0768212
\(629\) −1.45734e9 −0.233498
\(630\) −5.83173e8 −0.0929193
\(631\) 2.37404e9 0.376171 0.188086 0.982153i \(-0.439772\pi\)
0.188086 + 0.982153i \(0.439772\pi\)
\(632\) 8.89766e9 1.40206
\(633\) −2.54670e9 −0.399085
\(634\) 1.18432e9 0.184569
\(635\) 5.12041e9 0.793590
\(636\) −1.23063e9 −0.189683
\(637\) 7.27925e9 1.11583
\(638\) −7.69377e9 −1.17292
\(639\) 3.74372e9 0.567611
\(640\) 2.59680e9 0.391569
\(641\) 1.19471e10 1.79168 0.895841 0.444375i \(-0.146574\pi\)
0.895841 + 0.444375i \(0.146574\pi\)
\(642\) −6.92133e8 −0.103233
\(643\) 3.95645e9 0.586904 0.293452 0.955974i \(-0.405196\pi\)
0.293452 + 0.955974i \(0.405196\pi\)
\(644\) 4.33272e9 0.639235
\(645\) 1.24129e8 0.0182143
\(646\) −5.75935e8 −0.0840542
\(647\) 8.69167e9 1.26165 0.630824 0.775926i \(-0.282717\pi\)
0.630824 + 0.775926i \(0.282717\pi\)
\(648\) 6.74148e8 0.0973290
\(649\) 1.34249e9 0.192776
\(650\) 4.33615e9 0.619309
\(651\) −3.60516e9 −0.512143
\(652\) −1.73168e9 −0.244681
\(653\) 2.95452e9 0.415231 0.207616 0.978210i \(-0.433430\pi\)
0.207616 + 0.978210i \(0.433430\pi\)
\(654\) −4.78204e7 −0.00668485
\(655\) −4.04904e9 −0.562999
\(656\) 5.74404e8 0.0794427
\(657\) 3.70548e9 0.509759
\(658\) −8.30319e9 −1.13620
\(659\) −6.81330e9 −0.927382 −0.463691 0.885997i \(-0.653475\pi\)
−0.463691 + 0.885997i \(0.653475\pi\)
\(660\) −1.99542e9 −0.270166
\(661\) −9.65769e9 −1.30067 −0.650336 0.759646i \(-0.725372\pi\)
−0.650336 + 0.759646i \(0.725372\pi\)
\(662\) −4.50860e9 −0.604003
\(663\) −9.46013e9 −1.26066
\(664\) 7.87603e9 1.04404
\(665\) 4.88173e8 0.0643722
\(666\) 2.04715e8 0.0268528
\(667\) 7.83690e9 1.02259
\(668\) 3.73332e9 0.484594
\(669\) 8.21345e9 1.06056
\(670\) 2.16391e9 0.277957
\(671\) −4.57180e9 −0.584196
\(672\) 6.18283e9 0.785950
\(673\) −5.74382e9 −0.726354 −0.363177 0.931720i \(-0.618308\pi\)
−0.363177 + 0.931720i \(0.618308\pi\)
\(674\) −1.56275e9 −0.196599
\(675\) 1.26416e9 0.158212
\(676\) −7.60072e9 −0.946327
\(677\) −6.39314e9 −0.791870 −0.395935 0.918279i \(-0.629579\pi\)
−0.395935 + 0.918279i \(0.629579\pi\)
\(678\) 7.60776e8 0.0937460
\(679\) −1.42772e10 −1.75025
\(680\) 4.39719e9 0.536283
\(681\) −4.17241e9 −0.506259
\(682\) 4.12880e9 0.498400
\(683\) −1.10711e10 −1.32959 −0.664794 0.747027i \(-0.731481\pi\)
−0.664794 + 0.747027i \(0.731481\pi\)
\(684\) −2.41713e8 −0.0288804
\(685\) −6.07985e9 −0.722729
\(686\) 1.44320e9 0.170684
\(687\) −3.18681e9 −0.374980
\(688\) −1.98427e8 −0.0232296
\(689\) −5.66356e9 −0.659663
\(690\) −6.80301e8 −0.0788369
\(691\) 1.01541e10 1.17077 0.585383 0.810757i \(-0.300944\pi\)
0.585383 + 0.810757i \(0.300944\pi\)
\(692\) −8.67236e9 −0.994869
\(693\) −5.70710e9 −0.651402
\(694\) −2.72871e9 −0.309884
\(695\) 7.37965e9 0.833852
\(696\) 7.11551e9 0.799970
\(697\) 3.31906e9 0.371279
\(698\) 1.82246e9 0.202845
\(699\) 1.84149e9 0.203938
\(700\) 7.37683e9 0.812880
\(701\) 6.42013e9 0.703933 0.351966 0.936013i \(-0.385513\pi\)
0.351966 + 0.936013i \(0.385513\pi\)
\(702\) 1.32888e9 0.144979
\(703\) −1.71366e8 −0.0186029
\(704\) −2.82341e9 −0.304979
\(705\) −3.89515e9 −0.418660
\(706\) −6.65794e9 −0.712071
\(707\) −1.60025e10 −1.70302
\(708\) −5.31796e8 −0.0563155
\(709\) −1.49237e10 −1.57259 −0.786293 0.617854i \(-0.788002\pi\)
−0.786293 + 0.617854i \(0.788002\pi\)
\(710\) −3.43014e9 −0.359673
\(711\) 5.11332e9 0.533531
\(712\) 1.83676e9 0.190709
\(713\) −4.20560e9 −0.434525
\(714\) 5.38672e9 0.553835
\(715\) −9.18326e9 −0.939562
\(716\) −1.09800e9 −0.111791
\(717\) 2.16276e9 0.219125
\(718\) −1.86293e9 −0.187828
\(719\) −3.46295e9 −0.347452 −0.173726 0.984794i \(-0.555581\pi\)
−0.173726 + 0.984794i \(0.555581\pi\)
\(720\) 4.37326e8 0.0436659
\(721\) 3.09376e9 0.307406
\(722\) 4.99656e9 0.494073
\(723\) −2.55124e9 −0.251054
\(724\) 8.87195e9 0.868828
\(725\) 1.33430e10 1.30038
\(726\) 3.55508e9 0.344803
\(727\) −1.19482e10 −1.15327 −0.576637 0.817000i \(-0.695635\pi\)
−0.576637 + 0.817000i \(0.695635\pi\)
\(728\) 1.81044e10 1.73910
\(729\) 3.87420e8 0.0370370
\(730\) −3.39510e9 −0.323015
\(731\) −1.14656e9 −0.108564
\(732\) 1.81102e9 0.170661
\(733\) −4.90087e9 −0.459631 −0.229815 0.973234i \(-0.573812\pi\)
−0.229815 + 0.973234i \(0.573812\pi\)
\(734\) 3.94071e9 0.367823
\(735\) −1.94443e9 −0.180629
\(736\) 7.21258e9 0.666835
\(737\) 2.11767e10 1.94859
\(738\) −4.66233e8 −0.0426978
\(739\) −1.16073e10 −1.05797 −0.528986 0.848631i \(-0.677428\pi\)
−0.528986 + 0.848631i \(0.677428\pi\)
\(740\) 5.60397e8 0.0508375
\(741\) −1.11240e9 −0.100438
\(742\) 3.22490e9 0.289803
\(743\) −1.83926e10 −1.64506 −0.822531 0.568720i \(-0.807439\pi\)
−0.822531 + 0.568720i \(0.807439\pi\)
\(744\) −3.81848e9 −0.339926
\(745\) −5.71445e8 −0.0506323
\(746\) −5.93617e9 −0.523504
\(747\) 4.52621e9 0.397295
\(748\) 1.84315e10 1.61030
\(749\) −5.41897e9 −0.471227
\(750\) −2.56720e9 −0.222201
\(751\) −1.54442e10 −1.33054 −0.665268 0.746604i \(-0.731683\pi\)
−0.665268 + 0.746604i \(0.731683\pi\)
\(752\) 6.22663e9 0.533938
\(753\) 6.42060e9 0.548016
\(754\) 1.40261e10 1.19162
\(755\) 9.04180e9 0.764611
\(756\) 2.26074e9 0.190294
\(757\) −1.95008e10 −1.63387 −0.816933 0.576732i \(-0.804328\pi\)
−0.816933 + 0.576732i \(0.804328\pi\)
\(758\) −5.13111e9 −0.427926
\(759\) −6.65762e9 −0.552679
\(760\) 5.17059e8 0.0427260
\(761\) −9.37201e9 −0.770880 −0.385440 0.922733i \(-0.625950\pi\)
−0.385440 + 0.922733i \(0.625950\pi\)
\(762\) 6.64383e9 0.543972
\(763\) −3.74405e8 −0.0305144
\(764\) −2.66916e9 −0.216544
\(765\) 2.52699e9 0.204074
\(766\) −6.40231e9 −0.514679
\(767\) −2.44741e9 −0.195850
\(768\) 4.86217e9 0.387317
\(769\) −1.02340e10 −0.811528 −0.405764 0.913978i \(-0.632994\pi\)
−0.405764 + 0.913978i \(0.632994\pi\)
\(770\) 5.22906e9 0.412768
\(771\) −1.81895e7 −0.00142932
\(772\) 3.38220e9 0.264569
\(773\) 2.44487e10 1.90383 0.951913 0.306367i \(-0.0991135\pi\)
0.951913 + 0.306367i \(0.0991135\pi\)
\(774\) 1.61060e8 0.0124851
\(775\) −7.16040e9 −0.552562
\(776\) −1.51220e10 −1.16170
\(777\) 1.60279e9 0.122575
\(778\) 5.49546e9 0.418384
\(779\) 3.90283e8 0.0295800
\(780\) 3.63774e9 0.274474
\(781\) −3.35684e10 −2.52146
\(782\) 6.28388e9 0.469899
\(783\) 4.08915e9 0.304416
\(784\) 3.10829e9 0.230364
\(785\) −5.86148e8 −0.0432478
\(786\) −5.25372e9 −0.385912
\(787\) −3.61895e8 −0.0264649 −0.0132325 0.999912i \(-0.504212\pi\)
−0.0132325 + 0.999912i \(0.504212\pi\)
\(788\) −1.68882e10 −1.22954
\(789\) 8.73615e8 0.0633214
\(790\) −4.68502e9 −0.338078
\(791\) 5.95640e9 0.427924
\(792\) −6.04479e9 −0.432358
\(793\) 8.33458e9 0.593510
\(794\) 7.78325e9 0.551810
\(795\) 1.51285e9 0.106785
\(796\) −1.86828e10 −1.31294
\(797\) −5.22352e9 −0.365476 −0.182738 0.983162i \(-0.558496\pi\)
−0.182738 + 0.983162i \(0.558496\pi\)
\(798\) 6.33415e8 0.0441244
\(799\) 3.59791e10 2.49538
\(800\) 1.22800e10 0.847978
\(801\) 1.05555e9 0.0725714
\(802\) −3.24262e9 −0.221966
\(803\) −3.32254e10 −2.26447
\(804\) −8.38866e9 −0.569241
\(805\) −5.32634e9 −0.359868
\(806\) −7.52697e9 −0.506346
\(807\) −7.06733e9 −0.473367
\(808\) −1.69493e10 −1.13035
\(809\) −2.37703e10 −1.57839 −0.789194 0.614144i \(-0.789502\pi\)
−0.789194 + 0.614144i \(0.789502\pi\)
\(810\) −3.54969e8 −0.0234689
\(811\) 1.74347e10 1.14774 0.573868 0.818948i \(-0.305442\pi\)
0.573868 + 0.818948i \(0.305442\pi\)
\(812\) 2.38617e10 1.56407
\(813\) −1.67903e10 −1.09582
\(814\) −1.83559e9 −0.119286
\(815\) 2.12880e9 0.137747
\(816\) −4.03954e9 −0.260266
\(817\) −1.34823e8 −0.00864940
\(818\) −6.08550e8 −0.0388741
\(819\) 1.04043e10 0.661788
\(820\) −1.27629e9 −0.0808353
\(821\) −2.87309e9 −0.181196 −0.0905978 0.995888i \(-0.528878\pi\)
−0.0905978 + 0.995888i \(0.528878\pi\)
\(822\) −7.88873e9 −0.495400
\(823\) 6.48563e9 0.405558 0.202779 0.979225i \(-0.435003\pi\)
0.202779 + 0.979225i \(0.435003\pi\)
\(824\) 3.27681e9 0.204036
\(825\) −1.13352e10 −0.702812
\(826\) 1.39359e9 0.0860406
\(827\) −1.45803e10 −0.896388 −0.448194 0.893936i \(-0.647933\pi\)
−0.448194 + 0.893936i \(0.647933\pi\)
\(828\) 2.63727e9 0.161454
\(829\) 2.97207e9 0.181183 0.0905916 0.995888i \(-0.471124\pi\)
0.0905916 + 0.995888i \(0.471124\pi\)
\(830\) −4.14709e9 −0.251750
\(831\) −7.60766e9 −0.459883
\(832\) 5.14720e9 0.309841
\(833\) 1.79605e10 1.07662
\(834\) 9.57526e9 0.571570
\(835\) −4.58948e9 −0.272810
\(836\) 2.16733e9 0.128293
\(837\) −2.19441e9 −0.129354
\(838\) −2.81307e9 −0.165130
\(839\) −1.23558e10 −0.722276 −0.361138 0.932512i \(-0.617612\pi\)
−0.361138 + 0.932512i \(0.617612\pi\)
\(840\) −4.83605e9 −0.281523
\(841\) 2.59104e10 1.50206
\(842\) −1.20830e10 −0.697562
\(843\) −6.31851e9 −0.363261
\(844\) −9.04564e9 −0.517894
\(845\) 9.34377e9 0.532750
\(846\) −5.05403e9 −0.286974
\(847\) 2.78341e10 1.57393
\(848\) −2.41838e9 −0.136188
\(849\) 9.69518e9 0.543725
\(850\) 1.06988e10 0.597545
\(851\) 1.86973e9 0.103998
\(852\) 1.32973e10 0.736591
\(853\) −3.00088e9 −0.165549 −0.0827746 0.996568i \(-0.526378\pi\)
−0.0827746 + 0.996568i \(0.526378\pi\)
\(854\) −4.74582e9 −0.260741
\(855\) 2.97144e8 0.0162587
\(856\) −5.73961e9 −0.312769
\(857\) 6.68872e9 0.363003 0.181502 0.983391i \(-0.441904\pi\)
0.181502 + 0.983391i \(0.441904\pi\)
\(858\) −1.19155e10 −0.644029
\(859\) 9.56843e9 0.515068 0.257534 0.966269i \(-0.417090\pi\)
0.257534 + 0.966269i \(0.417090\pi\)
\(860\) 4.40893e8 0.0236368
\(861\) −3.65031e9 −0.194903
\(862\) 1.56113e9 0.0830161
\(863\) 3.10651e10 1.64526 0.822632 0.568575i \(-0.192505\pi\)
0.822632 + 0.568575i \(0.192505\pi\)
\(864\) 3.76340e9 0.198510
\(865\) 1.06612e10 0.560078
\(866\) −8.09971e9 −0.423796
\(867\) −1.22624e10 −0.639011
\(868\) −1.28052e10 −0.664610
\(869\) −4.58490e10 −2.37006
\(870\) −3.74664e9 −0.192897
\(871\) −3.86060e10 −1.97966
\(872\) −3.96558e8 −0.0202534
\(873\) −8.69034e9 −0.442066
\(874\) 7.38911e8 0.0374371
\(875\) −2.00996e10 −1.01428
\(876\) 1.31615e10 0.661516
\(877\) −3.23289e10 −1.61843 −0.809213 0.587515i \(-0.800106\pi\)
−0.809213 + 0.587515i \(0.800106\pi\)
\(878\) 1.30191e10 0.649159
\(879\) −6.70438e9 −0.332964
\(880\) −3.92132e9 −0.193974
\(881\) 1.05719e10 0.520881 0.260440 0.965490i \(-0.416132\pi\)
0.260440 + 0.965490i \(0.416132\pi\)
\(882\) −2.52294e9 −0.123813
\(883\) 1.62742e10 0.795493 0.397746 0.917495i \(-0.369792\pi\)
0.397746 + 0.917495i \(0.369792\pi\)
\(884\) −3.36015e10 −1.63597
\(885\) 6.53751e8 0.0317038
\(886\) 3.24481e9 0.156737
\(887\) −5.54686e9 −0.266879 −0.133440 0.991057i \(-0.542602\pi\)
−0.133440 + 0.991057i \(0.542602\pi\)
\(888\) 1.69763e9 0.0813573
\(889\) 5.20171e10 2.48308
\(890\) −9.67136e8 −0.0459857
\(891\) −3.47383e9 −0.164527
\(892\) 2.91734e10 1.37629
\(893\) 4.23072e9 0.198808
\(894\) −7.41462e8 −0.0347062
\(895\) 1.34981e9 0.0629348
\(896\) 2.63803e10 1.22519
\(897\) 1.21371e10 0.561490
\(898\) 1.42552e10 0.656909
\(899\) −2.31616e10 −1.06319
\(900\) 4.49017e9 0.205312
\(901\) −1.39740e10 −0.636481
\(902\) 4.18051e9 0.189673
\(903\) 1.26100e9 0.0569911
\(904\) 6.30884e9 0.284027
\(905\) −1.09065e10 −0.489121
\(906\) 1.17319e10 0.524107
\(907\) 2.33401e10 1.03867 0.519335 0.854571i \(-0.326180\pi\)
0.519335 + 0.854571i \(0.326180\pi\)
\(908\) −1.48200e10 −0.656974
\(909\) −9.74047e9 −0.430137
\(910\) −9.53280e9 −0.419349
\(911\) 1.77936e10 0.779740 0.389870 0.920870i \(-0.372520\pi\)
0.389870 + 0.920870i \(0.372520\pi\)
\(912\) −4.75003e8 −0.0207355
\(913\) −4.05846e10 −1.76487
\(914\) 5.27851e9 0.228665
\(915\) −2.22633e9 −0.0960762
\(916\) −1.13193e10 −0.486613
\(917\) −4.11334e10 −1.76158
\(918\) 3.27882e9 0.139884
\(919\) 2.41659e10 1.02706 0.513532 0.858070i \(-0.328337\pi\)
0.513532 + 0.858070i \(0.328337\pi\)
\(920\) −5.64150e9 −0.238856
\(921\) −5.19559e9 −0.219142
\(922\) 1.20319e9 0.0505563
\(923\) 6.11965e10 2.56166
\(924\) −2.02711e10 −0.845327
\(925\) 3.18338e9 0.132249
\(926\) −1.78194e10 −0.737489
\(927\) 1.88313e9 0.0776426
\(928\) 3.97220e10 1.63160
\(929\) −2.78712e10 −1.14052 −0.570258 0.821466i \(-0.693157\pi\)
−0.570258 + 0.821466i \(0.693157\pi\)
\(930\) 2.01060e9 0.0819664
\(931\) 2.11195e9 0.0857747
\(932\) 6.54078e9 0.264651
\(933\) 1.29798e10 0.523217
\(934\) −2.19647e10 −0.882088
\(935\) −2.26584e10 −0.906543
\(936\) 1.10199e10 0.439251
\(937\) 1.83113e10 0.727162 0.363581 0.931563i \(-0.381554\pi\)
0.363581 + 0.931563i \(0.381554\pi\)
\(938\) 2.19827e10 0.869704
\(939\) −2.12203e10 −0.836416
\(940\) −1.38352e10 −0.543297
\(941\) 3.20558e9 0.125413 0.0627066 0.998032i \(-0.480027\pi\)
0.0627066 + 0.998032i \(0.480027\pi\)
\(942\) −7.60540e8 −0.0296445
\(943\) −4.25828e9 −0.165365
\(944\) −1.04506e9 −0.0404333
\(945\) −2.77919e9 −0.107129
\(946\) −1.44415e9 −0.0554618
\(947\) 1.13660e10 0.434895 0.217448 0.976072i \(-0.430227\pi\)
0.217448 + 0.976072i \(0.430227\pi\)
\(948\) 1.81620e10 0.692365
\(949\) 6.05713e10 2.30057
\(950\) 1.25806e9 0.0476067
\(951\) 5.64405e9 0.212794
\(952\) 4.46701e10 1.67798
\(953\) −2.26241e10 −0.846732 −0.423366 0.905959i \(-0.639152\pi\)
−0.423366 + 0.905959i \(0.639152\pi\)
\(954\) 1.96295e9 0.0731965
\(955\) 3.28127e9 0.121907
\(956\) 7.68193e9 0.284359
\(957\) −3.66657e10 −1.35228
\(958\) 2.20004e10 0.808447
\(959\) −6.17639e10 −2.26136
\(960\) −1.37492e9 −0.0501565
\(961\) −1.50831e10 −0.548226
\(962\) 3.34635e9 0.121188
\(963\) −3.29845e9 −0.119019
\(964\) −9.06176e9 −0.325794
\(965\) −4.15784e9 −0.148944
\(966\) −6.91104e9 −0.246674
\(967\) 4.77814e10 1.69928 0.849642 0.527359i \(-0.176818\pi\)
0.849642 + 0.527359i \(0.176818\pi\)
\(968\) 2.94810e10 1.04467
\(969\) −2.74469e9 −0.0969082
\(970\) 7.96242e9 0.280120
\(971\) 2.36126e9 0.0827707 0.0413854 0.999143i \(-0.486823\pi\)
0.0413854 + 0.999143i \(0.486823\pi\)
\(972\) 1.37608e9 0.0480631
\(973\) 7.49683e10 2.60905
\(974\) 1.01895e10 0.353342
\(975\) 2.06645e10 0.714017
\(976\) 3.55893e9 0.122531
\(977\) −2.09178e10 −0.717604 −0.358802 0.933414i \(-0.616815\pi\)
−0.358802 + 0.933414i \(0.616815\pi\)
\(978\) 2.76216e9 0.0944197
\(979\) −9.46467e9 −0.322379
\(980\) −6.90643e9 −0.234403
\(981\) −2.27895e8 −0.00770713
\(982\) 7.05538e9 0.237755
\(983\) −5.00358e9 −0.168013 −0.0840066 0.996465i \(-0.526772\pi\)
−0.0840066 + 0.996465i \(0.526772\pi\)
\(984\) −3.86630e9 −0.129364
\(985\) 2.07612e10 0.692190
\(986\) 3.46073e10 1.14974
\(987\) −3.95700e10 −1.30995
\(988\) −3.95114e9 −0.130339
\(989\) 1.47102e9 0.0483538
\(990\) 3.18286e9 0.104254
\(991\) 2.88517e10 0.941702 0.470851 0.882213i \(-0.343947\pi\)
0.470851 + 0.882213i \(0.343947\pi\)
\(992\) −2.13165e10 −0.693306
\(993\) −2.14863e10 −0.696370
\(994\) −3.48461e10 −1.12539
\(995\) 2.29673e10 0.739144
\(996\) 1.60767e10 0.515571
\(997\) −4.64145e10 −1.48327 −0.741636 0.670802i \(-0.765950\pi\)
−0.741636 + 0.670802i \(0.765950\pi\)
\(998\) −1.71543e10 −0.546282
\(999\) 9.75595e8 0.0309592
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.1.7 17
3.2 odd 2 531.8.a.c.1.11 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.c.1.7 17 1.1 even 1 trivial
531.8.a.c.1.11 17 3.2 odd 2