Properties

Label 69.8.a.a.1.2
Level $69$
Weight $8$
Character 69.1
Self dual yes
Analytic conductor $21.555$
Analytic rank $1$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 69.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(21.5545667584\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: \(\mathbb{Q}[x]/(x^{5} - \cdots)\)
Defining polynomial: \(x^{5} - 455 x^{3} - 474 x^{2} + 42284 x + 127016\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-9.64907\) of defining polynomial
Character \(\chi\) \(=\) 69.1

$q$-expansion

\(f(q)\) \(=\) \(q-9.64907 q^{2} -27.0000 q^{3} -34.8955 q^{4} -306.939 q^{5} +260.525 q^{6} +733.069 q^{7} +1571.79 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-9.64907 q^{2} -27.0000 q^{3} -34.8955 q^{4} -306.939 q^{5} +260.525 q^{6} +733.069 q^{7} +1571.79 q^{8} +729.000 q^{9} +2961.68 q^{10} +3882.47 q^{11} +942.178 q^{12} -5973.30 q^{13} -7073.43 q^{14} +8287.36 q^{15} -10699.7 q^{16} +14030.7 q^{17} -7034.17 q^{18} +11012.4 q^{19} +10710.8 q^{20} -19792.9 q^{21} -37462.2 q^{22} +12167.0 q^{23} -42438.3 q^{24} +16086.7 q^{25} +57636.8 q^{26} -19683.0 q^{27} -25580.8 q^{28} +116.864 q^{29} -79965.3 q^{30} -65552.7 q^{31} -97947.0 q^{32} -104827. q^{33} -135383. q^{34} -225008. q^{35} -25438.8 q^{36} -382473. q^{37} -106260. q^{38} +161279. q^{39} -482444. q^{40} -370414. q^{41} +190983. q^{42} +368875. q^{43} -135480. q^{44} -223759. q^{45} -117400. q^{46} +830724. q^{47} +288892. q^{48} -286153. q^{49} -155222. q^{50} -378828. q^{51} +208441. q^{52} -545061. q^{53} +189923. q^{54} -1.19168e6 q^{55} +1.15223e6 q^{56} -297336. q^{57} -1127.63 q^{58} +1.87374e6 q^{59} -289191. q^{60} -1.20373e6 q^{61} +632523. q^{62} +534407. q^{63} +2.31466e6 q^{64} +1.83344e6 q^{65} +1.01148e6 q^{66} -4.04596e6 q^{67} -489607. q^{68} -328509. q^{69} +2.17111e6 q^{70} +2.77968e6 q^{71} +1.14583e6 q^{72} -2.58877e6 q^{73} +3.69051e6 q^{74} -434342. q^{75} -384284. q^{76} +2.84611e6 q^{77} -1.55619e6 q^{78} +1.28761e6 q^{79} +3.28415e6 q^{80} +531441. q^{81} +3.57415e6 q^{82} +5.20417e6 q^{83} +690681. q^{84} -4.30657e6 q^{85} -3.55930e6 q^{86} -3155.33 q^{87} +6.10242e6 q^{88} -2.72740e6 q^{89} +2.15906e6 q^{90} -4.37884e6 q^{91} -424573. q^{92} +1.76992e6 q^{93} -8.01571e6 q^{94} -3.38015e6 q^{95} +2.64457e6 q^{96} -7.55774e6 q^{97} +2.76111e6 q^{98} +2.83032e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9} + O(q^{10}) \) \( 5 q - 135 q^{3} + 270 q^{4} - 266 q^{5} - 496 q^{7} + 1422 q^{8} + 3645 q^{9} + 1452 q^{10} - 1148 q^{11} - 7290 q^{12} - 642 q^{13} + 5756 q^{14} + 7182 q^{15} - 22606 q^{16} - 5798 q^{17} - 6036 q^{19} - 27376 q^{20} + 13392 q^{21} - 97896 q^{22} + 60835 q^{23} - 38394 q^{24} - 262477 q^{25} - 355992 q^{26} - 98415 q^{27} - 507124 q^{28} - 169162 q^{29} - 39204 q^{30} - 199640 q^{31} - 284794 q^{32} + 30996 q^{33} - 1027740 q^{34} - 137680 q^{35} + 196830 q^{36} - 202002 q^{37} - 554924 q^{38} + 17334 q^{39} - 340904 q^{40} + 541282 q^{41} - 155412 q^{42} - 909596 q^{43} - 1236032 q^{44} - 193914 q^{45} + 80208 q^{47} + 610362 q^{48} + 850589 q^{49} - 941416 q^{50} + 156546 q^{51} + 146940 q^{52} - 278138 q^{53} - 933560 q^{55} - 539932 q^{56} + 162972 q^{57} - 3522712 q^{58} - 3177380 q^{59} + 739152 q^{60} + 147782 q^{61} + 4606456 q^{62} - 361584 q^{63} - 4142622 q^{64} + 3877332 q^{65} + 2643192 q^{66} - 464916 q^{67} + 7513072 q^{68} - 1642545 q^{69} + 2093200 q^{70} + 1576792 q^{71} + 1036638 q^{72} - 38190 q^{73} + 12164864 q^{74} + 7086879 q^{75} + 6889436 q^{76} + 10332384 q^{77} + 9611784 q^{78} - 3913336 q^{79} + 6334776 q^{80} + 2657205 q^{81} + 6799360 q^{82} + 15774716 q^{83} + 13692348 q^{84} - 8520740 q^{85} + 24874084 q^{86} + 4567374 q^{87} + 53216 q^{88} + 1116482 q^{89} + 1058508 q^{90} - 27369552 q^{91} + 3285090 q^{92} + 5390280 q^{93} - 7153744 q^{94} - 6067832 q^{95} + 7689438 q^{96} - 15738566 q^{97} + 11730488 q^{98} - 836892 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −9.64907 −0.852865 −0.426433 0.904519i \(-0.640230\pi\)
−0.426433 + 0.904519i \(0.640230\pi\)
\(3\) −27.0000 −0.577350
\(4\) −34.8955 −0.272621
\(5\) −306.939 −1.09814 −0.549070 0.835777i \(-0.685018\pi\)
−0.549070 + 0.835777i \(0.685018\pi\)
\(6\) 260.525 0.492402
\(7\) 733.069 0.807796 0.403898 0.914804i \(-0.367655\pi\)
0.403898 + 0.914804i \(0.367655\pi\)
\(8\) 1571.79 1.08537
\(9\) 729.000 0.333333
\(10\) 2961.68 0.936565
\(11\) 3882.47 0.879495 0.439747 0.898122i \(-0.355068\pi\)
0.439747 + 0.898122i \(0.355068\pi\)
\(12\) 942.178 0.157398
\(13\) −5973.30 −0.754071 −0.377036 0.926199i \(-0.623057\pi\)
−0.377036 + 0.926199i \(0.623057\pi\)
\(14\) −7073.43 −0.688941
\(15\) 8287.36 0.634011
\(16\) −10699.7 −0.653057
\(17\) 14030.7 0.692640 0.346320 0.938117i \(-0.387431\pi\)
0.346320 + 0.938117i \(0.387431\pi\)
\(18\) −7034.17 −0.284288
\(19\) 11012.4 0.368337 0.184169 0.982895i \(-0.441041\pi\)
0.184169 + 0.982895i \(0.441041\pi\)
\(20\) 10710.8 0.299376
\(21\) −19792.9 −0.466381
\(22\) −37462.2 −0.750091
\(23\) 12167.0 0.208514
\(24\) −42438.3 −0.626641
\(25\) 16086.7 0.205910
\(26\) 57636.8 0.643121
\(27\) −19683.0 −0.192450
\(28\) −25580.8 −0.220222
\(29\) 116.864 0.000889790 0 0.000444895 1.00000i \(-0.499858\pi\)
0.000444895 1.00000i \(0.499858\pi\)
\(30\) −79965.3 −0.540726
\(31\) −65552.7 −0.395207 −0.197604 0.980282i \(-0.563316\pi\)
−0.197604 + 0.980282i \(0.563316\pi\)
\(32\) −97947.0 −0.528404
\(33\) −104827. −0.507777
\(34\) −135383. −0.590728
\(35\) −225008. −0.887072
\(36\) −25438.8 −0.0908736
\(37\) −382473. −1.24135 −0.620676 0.784067i \(-0.713142\pi\)
−0.620676 + 0.784067i \(0.713142\pi\)
\(38\) −106260. −0.314142
\(39\) 161279. 0.435363
\(40\) −482444. −1.19189
\(41\) −370414. −0.839352 −0.419676 0.907674i \(-0.637856\pi\)
−0.419676 + 0.907674i \(0.637856\pi\)
\(42\) 190983. 0.397760
\(43\) 368875. 0.707522 0.353761 0.935336i \(-0.384903\pi\)
0.353761 + 0.935336i \(0.384903\pi\)
\(44\) −135480. −0.239769
\(45\) −223759. −0.366046
\(46\) −117400. −0.177835
\(47\) 830724. 1.16712 0.583558 0.812071i \(-0.301660\pi\)
0.583558 + 0.812071i \(0.301660\pi\)
\(48\) 288892. 0.377043
\(49\) −286153. −0.347466
\(50\) −155222. −0.175614
\(51\) −378828. −0.399896
\(52\) 208441. 0.205576
\(53\) −545061. −0.502897 −0.251449 0.967871i \(-0.580907\pi\)
−0.251449 + 0.967871i \(0.580907\pi\)
\(54\) 189923. 0.164134
\(55\) −1.19168e6 −0.965808
\(56\) 1.15223e6 0.876760
\(57\) −297336. −0.212660
\(58\) −1127.63 −0.000758871 0
\(59\) 1.87374e6 1.18776 0.593880 0.804554i \(-0.297596\pi\)
0.593880 + 0.804554i \(0.297596\pi\)
\(60\) −289191. −0.172845
\(61\) −1.20373e6 −0.679010 −0.339505 0.940604i \(-0.610260\pi\)
−0.339505 + 0.940604i \(0.610260\pi\)
\(62\) 632523. 0.337058
\(63\) 534407. 0.269265
\(64\) 2.31466e6 1.10371
\(65\) 1.83344e6 0.828076
\(66\) 1.01148e6 0.433065
\(67\) −4.04596e6 −1.64346 −0.821731 0.569876i \(-0.806991\pi\)
−0.821731 + 0.569876i \(0.806991\pi\)
\(68\) −489607. −0.188828
\(69\) −328509. −0.120386
\(70\) 2.17111e6 0.756553
\(71\) 2.77968e6 0.921702 0.460851 0.887478i \(-0.347544\pi\)
0.460851 + 0.887478i \(0.347544\pi\)
\(72\) 1.14583e6 0.361791
\(73\) −2.58877e6 −0.778867 −0.389434 0.921055i \(-0.627329\pi\)
−0.389434 + 0.921055i \(0.627329\pi\)
\(74\) 3.69051e6 1.05871
\(75\) −434342. −0.118882
\(76\) −384284. −0.100416
\(77\) 2.84611e6 0.710452
\(78\) −1.55619e6 −0.371306
\(79\) 1.28761e6 0.293825 0.146913 0.989149i \(-0.453066\pi\)
0.146913 + 0.989149i \(0.453066\pi\)
\(80\) 3.28415e6 0.717148
\(81\) 531441. 0.111111
\(82\) 3.57415e6 0.715854
\(83\) 5.20417e6 0.999029 0.499514 0.866306i \(-0.333512\pi\)
0.499514 + 0.866306i \(0.333512\pi\)
\(84\) 690681. 0.127145
\(85\) −4.30657e6 −0.760615
\(86\) −3.55930e6 −0.603421
\(87\) −3155.33 −0.000513720 0
\(88\) 6.10242e6 0.954581
\(89\) −2.72740e6 −0.410094 −0.205047 0.978752i \(-0.565735\pi\)
−0.205047 + 0.978752i \(0.565735\pi\)
\(90\) 2.15906e6 0.312188
\(91\) −4.37884e6 −0.609136
\(92\) −424573. −0.0568454
\(93\) 1.76992e6 0.228173
\(94\) −8.01571e6 −0.995393
\(95\) −3.38015e6 −0.404486
\(96\) 2.64457e6 0.305074
\(97\) −7.55774e6 −0.840797 −0.420398 0.907340i \(-0.638110\pi\)
−0.420398 + 0.907340i \(0.638110\pi\)
\(98\) 2.76111e6 0.296342
\(99\) 2.83032e6 0.293165
\(100\) −561354. −0.0561354
\(101\) −1.41037e7 −1.36209 −0.681047 0.732240i \(-0.738475\pi\)
−0.681047 + 0.732240i \(0.738475\pi\)
\(102\) 3.65534e6 0.341057
\(103\) −228114. −0.0205694 −0.0102847 0.999947i \(-0.503274\pi\)
−0.0102847 + 0.999947i \(0.503274\pi\)
\(104\) −9.38877e6 −0.818450
\(105\) 6.07520e6 0.512151
\(106\) 5.25933e6 0.428904
\(107\) −1.72470e7 −1.36104 −0.680519 0.732730i \(-0.738246\pi\)
−0.680519 + 0.732730i \(0.738246\pi\)
\(108\) 686847. 0.0524659
\(109\) −3.22193e6 −0.238299 −0.119150 0.992876i \(-0.538017\pi\)
−0.119150 + 0.992876i \(0.538017\pi\)
\(110\) 1.14986e7 0.823704
\(111\) 1.03268e7 0.716695
\(112\) −7.84361e6 −0.527537
\(113\) 1.74509e6 0.113774 0.0568869 0.998381i \(-0.481883\pi\)
0.0568869 + 0.998381i \(0.481883\pi\)
\(114\) 2.86901e6 0.181370
\(115\) −3.73453e6 −0.228978
\(116\) −4078.02 −0.000242575 0
\(117\) −4.35453e6 −0.251357
\(118\) −1.80799e7 −1.01300
\(119\) 1.02854e7 0.559511
\(120\) 1.30260e7 0.688139
\(121\) −4.41363e6 −0.226489
\(122\) 1.16149e7 0.579104
\(123\) 1.00012e7 0.484600
\(124\) 2.28749e6 0.107742
\(125\) 1.90420e7 0.872021
\(126\) −5.15653e6 −0.229647
\(127\) −1.91418e7 −0.829222 −0.414611 0.909999i \(-0.636082\pi\)
−0.414611 + 0.909999i \(0.636082\pi\)
\(128\) −9.79707e6 −0.412916
\(129\) −9.95963e6 −0.408488
\(130\) −1.76910e7 −0.706237
\(131\) −3.17455e7 −1.23376 −0.616882 0.787056i \(-0.711604\pi\)
−0.616882 + 0.787056i \(0.711604\pi\)
\(132\) 3.65797e6 0.138430
\(133\) 8.07287e6 0.297541
\(134\) 3.90397e7 1.40165
\(135\) 6.04149e6 0.211337
\(136\) 2.20533e7 0.751773
\(137\) −3.41056e7 −1.13319 −0.566597 0.823995i \(-0.691740\pi\)
−0.566597 + 0.823995i \(0.691740\pi\)
\(138\) 3.16981e6 0.102673
\(139\) −3.40795e7 −1.07632 −0.538160 0.842843i \(-0.680880\pi\)
−0.538160 + 0.842843i \(0.680880\pi\)
\(140\) 7.85174e6 0.241834
\(141\) −2.24295e7 −0.673835
\(142\) −2.68213e7 −0.786087
\(143\) −2.31911e7 −0.663202
\(144\) −7.80007e6 −0.217686
\(145\) −35870.1 −0.000977113 0
\(146\) 2.49792e7 0.664269
\(147\) 7.72614e6 0.200610
\(148\) 1.33466e7 0.338419
\(149\) −5.32875e7 −1.31970 −0.659848 0.751399i \(-0.729379\pi\)
−0.659848 + 0.751399i \(0.729379\pi\)
\(150\) 4.19100e6 0.101391
\(151\) 6.20237e7 1.46601 0.733007 0.680221i \(-0.238116\pi\)
0.733007 + 0.680221i \(0.238116\pi\)
\(152\) 1.73092e7 0.399784
\(153\) 1.02284e7 0.230880
\(154\) −2.74624e7 −0.605920
\(155\) 2.01207e7 0.433992
\(156\) −5.62791e6 −0.118689
\(157\) 1.23372e7 0.254429 0.127214 0.991875i \(-0.459396\pi\)
0.127214 + 0.991875i \(0.459396\pi\)
\(158\) −1.24242e7 −0.250593
\(159\) 1.47166e7 0.290348
\(160\) 3.00638e7 0.580262
\(161\) 8.91925e6 0.168437
\(162\) −5.12791e6 −0.0947628
\(163\) −9.91990e7 −1.79412 −0.897058 0.441912i \(-0.854300\pi\)
−0.897058 + 0.441912i \(0.854300\pi\)
\(164\) 1.29258e7 0.228825
\(165\) 3.21754e7 0.557609
\(166\) −5.02154e7 −0.852037
\(167\) 4.36229e7 0.724782 0.362391 0.932026i \(-0.381961\pi\)
0.362391 + 0.932026i \(0.381961\pi\)
\(168\) −3.11102e7 −0.506198
\(169\) −2.70682e7 −0.431376
\(170\) 4.15543e7 0.648702
\(171\) 8.02806e6 0.122779
\(172\) −1.28721e7 −0.192885
\(173\) 3.86357e7 0.567318 0.283659 0.958925i \(-0.408452\pi\)
0.283659 + 0.958925i \(0.408452\pi\)
\(174\) 30446.0 0.000438134 0
\(175\) 1.17927e7 0.166333
\(176\) −4.15412e7 −0.574360
\(177\) −5.05911e7 −0.685753
\(178\) 2.63169e7 0.349755
\(179\) 5.49433e7 0.716027 0.358014 0.933716i \(-0.383454\pi\)
0.358014 + 0.933716i \(0.383454\pi\)
\(180\) 7.80817e6 0.0997919
\(181\) 7.42706e7 0.930983 0.465492 0.885052i \(-0.345878\pi\)
0.465492 + 0.885052i \(0.345878\pi\)
\(182\) 4.22517e7 0.519511
\(183\) 3.25008e7 0.392027
\(184\) 1.91240e7 0.226316
\(185\) 1.17396e8 1.36318
\(186\) −1.70781e7 −0.194601
\(187\) 5.44736e7 0.609173
\(188\) −2.89885e7 −0.318180
\(189\) −1.44290e7 −0.155460
\(190\) 3.26153e7 0.344972
\(191\) −1.76294e8 −1.83072 −0.915358 0.402642i \(-0.868092\pi\)
−0.915358 + 0.402642i \(0.868092\pi\)
\(192\) −6.24958e7 −0.637230
\(193\) 5.76947e7 0.577677 0.288839 0.957378i \(-0.406731\pi\)
0.288839 + 0.957378i \(0.406731\pi\)
\(194\) 7.29252e7 0.717086
\(195\) −4.95029e7 −0.478090
\(196\) 9.98545e6 0.0947265
\(197\) 3.71726e7 0.346411 0.173205 0.984886i \(-0.444588\pi\)
0.173205 + 0.984886i \(0.444588\pi\)
\(198\) −2.73099e7 −0.250030
\(199\) 9.70573e7 0.873057 0.436528 0.899690i \(-0.356208\pi\)
0.436528 + 0.899690i \(0.356208\pi\)
\(200\) 2.52850e7 0.223490
\(201\) 1.09241e8 0.948853
\(202\) 1.36087e8 1.16168
\(203\) 85669.3 0.000718768 0
\(204\) 1.32194e7 0.109020
\(205\) 1.13695e8 0.921725
\(206\) 2.20109e6 0.0175429
\(207\) 8.86974e6 0.0695048
\(208\) 6.39124e7 0.492452
\(209\) 4.27554e7 0.323951
\(210\) −5.86201e7 −0.436796
\(211\) −1.26885e8 −0.929870 −0.464935 0.885345i \(-0.653922\pi\)
−0.464935 + 0.885345i \(0.653922\pi\)
\(212\) 1.90202e7 0.137100
\(213\) −7.50513e7 −0.532145
\(214\) 1.66418e8 1.16078
\(215\) −1.13222e8 −0.776957
\(216\) −3.09375e7 −0.208880
\(217\) −4.80546e7 −0.319247
\(218\) 3.10886e7 0.203237
\(219\) 6.98968e7 0.449679
\(220\) 4.15843e7 0.263299
\(221\) −8.38094e7 −0.522300
\(222\) −9.96438e7 −0.611245
\(223\) 2.83332e7 0.171091 0.0855457 0.996334i \(-0.472737\pi\)
0.0855457 + 0.996334i \(0.472737\pi\)
\(224\) −7.18019e7 −0.426843
\(225\) 1.17272e7 0.0686368
\(226\) −1.68385e7 −0.0970338
\(227\) −2.73863e8 −1.55397 −0.776987 0.629517i \(-0.783253\pi\)
−0.776987 + 0.629517i \(0.783253\pi\)
\(228\) 1.03757e7 0.0579754
\(229\) −1.97587e8 −1.08726 −0.543632 0.839324i \(-0.682951\pi\)
−0.543632 + 0.839324i \(0.682951\pi\)
\(230\) 3.60347e7 0.195287
\(231\) −7.68451e7 −0.410180
\(232\) 183685. 0.000965755 0
\(233\) 1.14212e8 0.591517 0.295758 0.955263i \(-0.404428\pi\)
0.295758 + 0.955263i \(0.404428\pi\)
\(234\) 4.20172e7 0.214374
\(235\) −2.54982e8 −1.28166
\(236\) −6.53852e7 −0.323808
\(237\) −3.47654e7 −0.169640
\(238\) −9.92450e7 −0.477188
\(239\) −3.98583e8 −1.88854 −0.944270 0.329173i \(-0.893230\pi\)
−0.944270 + 0.329173i \(0.893230\pi\)
\(240\) −8.86722e7 −0.414045
\(241\) 1.02788e8 0.473024 0.236512 0.971629i \(-0.423996\pi\)
0.236512 + 0.971629i \(0.423996\pi\)
\(242\) 4.25874e7 0.193165
\(243\) −1.43489e7 −0.0641500
\(244\) 4.20049e7 0.185112
\(245\) 8.78317e7 0.381566
\(246\) −9.65021e7 −0.413299
\(247\) −6.57805e7 −0.277753
\(248\) −1.03035e8 −0.428947
\(249\) −1.40513e8 −0.576790
\(250\) −1.83737e8 −0.743717
\(251\) −4.18803e8 −1.67168 −0.835838 0.548977i \(-0.815018\pi\)
−0.835838 + 0.548977i \(0.815018\pi\)
\(252\) −1.86484e7 −0.0734073
\(253\) 4.72380e7 0.183387
\(254\) 1.84701e8 0.707214
\(255\) 1.16277e8 0.439141
\(256\) −2.01744e8 −0.751553
\(257\) 3.90111e7 0.143358 0.0716791 0.997428i \(-0.477164\pi\)
0.0716791 + 0.997428i \(0.477164\pi\)
\(258\) 9.61011e7 0.348385
\(259\) −2.80379e8 −1.00276
\(260\) −6.39787e7 −0.225751
\(261\) 85193.8 0.000296597 0
\(262\) 3.06314e8 1.05223
\(263\) 3.22526e8 1.09325 0.546625 0.837377i \(-0.315912\pi\)
0.546625 + 0.837377i \(0.315912\pi\)
\(264\) −1.64765e8 −0.551127
\(265\) 1.67301e8 0.552252
\(266\) −7.78956e7 −0.253763
\(267\) 7.36398e7 0.236768
\(268\) 1.41186e8 0.448042
\(269\) 9.39578e7 0.294306 0.147153 0.989114i \(-0.452989\pi\)
0.147153 + 0.989114i \(0.452989\pi\)
\(270\) −5.82947e7 −0.180242
\(271\) 3.27712e8 1.00023 0.500115 0.865959i \(-0.333291\pi\)
0.500115 + 0.865959i \(0.333291\pi\)
\(272\) −1.50124e8 −0.452333
\(273\) 1.18229e8 0.351685
\(274\) 3.29088e8 0.966462
\(275\) 6.24562e7 0.181097
\(276\) 1.14635e7 0.0328197
\(277\) 4.33983e8 1.22685 0.613427 0.789751i \(-0.289790\pi\)
0.613427 + 0.789751i \(0.289790\pi\)
\(278\) 3.28836e8 0.917956
\(279\) −4.77879e7 −0.131736
\(280\) −3.53665e8 −0.962805
\(281\) −4.53055e8 −1.21809 −0.609044 0.793136i \(-0.708447\pi\)
−0.609044 + 0.793136i \(0.708447\pi\)
\(282\) 2.16424e8 0.574691
\(283\) 5.47208e8 1.43516 0.717580 0.696477i \(-0.245250\pi\)
0.717580 + 0.696477i \(0.245250\pi\)
\(284\) −9.69982e7 −0.251275
\(285\) 9.12640e7 0.233530
\(286\) 2.23773e8 0.565622
\(287\) −2.71539e8 −0.678025
\(288\) −7.14034e7 −0.176135
\(289\) −2.13479e8 −0.520250
\(290\) 346113. 0.000833346 0
\(291\) 2.04059e8 0.485434
\(292\) 9.03363e7 0.212335
\(293\) 7.13317e6 0.0165671 0.00828354 0.999966i \(-0.497363\pi\)
0.00828354 + 0.999966i \(0.497363\pi\)
\(294\) −7.45501e7 −0.171093
\(295\) −5.75126e8 −1.30433
\(296\) −6.01168e8 −1.34733
\(297\) −7.64186e7 −0.169259
\(298\) 5.14175e8 1.12552
\(299\) −7.26771e7 −0.157235
\(300\) 1.51566e7 0.0324098
\(301\) 2.70411e8 0.571533
\(302\) −5.98471e8 −1.25031
\(303\) 3.80799e8 0.786405
\(304\) −1.17830e8 −0.240545
\(305\) 3.69473e8 0.745648
\(306\) −9.86942e7 −0.196909
\(307\) 9.59422e8 1.89245 0.946227 0.323503i \(-0.104860\pi\)
0.946227 + 0.323503i \(0.104860\pi\)
\(308\) −9.93165e7 −0.193684
\(309\) 6.15907e6 0.0118757
\(310\) −1.94146e8 −0.370137
\(311\) 8.24708e8 1.55467 0.777336 0.629085i \(-0.216570\pi\)
0.777336 + 0.629085i \(0.216570\pi\)
\(312\) 2.53497e8 0.472532
\(313\) −5.01561e7 −0.0924526 −0.0462263 0.998931i \(-0.514720\pi\)
−0.0462263 + 0.998931i \(0.514720\pi\)
\(314\) −1.19042e8 −0.216994
\(315\) −1.64031e8 −0.295691
\(316\) −4.49317e7 −0.0801029
\(317\) 3.58019e8 0.631247 0.315623 0.948885i \(-0.397786\pi\)
0.315623 + 0.948885i \(0.397786\pi\)
\(318\) −1.42002e8 −0.247628
\(319\) 453720. 0.000782565 0
\(320\) −7.10459e8 −1.21203
\(321\) 4.65669e8 0.785796
\(322\) −8.60624e7 −0.143654
\(323\) 1.54512e8 0.255125
\(324\) −1.85449e7 −0.0302912
\(325\) −9.60909e7 −0.155271
\(326\) 9.57178e8 1.53014
\(327\) 8.69920e7 0.137582
\(328\) −5.82213e8 −0.911011
\(329\) 6.08978e8 0.942792
\(330\) −3.10463e8 −0.475566
\(331\) −4.42027e8 −0.669963 −0.334981 0.942225i \(-0.608730\pi\)
−0.334981 + 0.942225i \(0.608730\pi\)
\(332\) −1.81602e8 −0.272356
\(333\) −2.78823e8 −0.413784
\(334\) −4.20921e8 −0.618141
\(335\) 1.24186e9 1.80475
\(336\) 2.11777e8 0.304573
\(337\) 9.16771e8 1.30484 0.652418 0.757859i \(-0.273755\pi\)
0.652418 + 0.757859i \(0.273755\pi\)
\(338\) 2.61183e8 0.367906
\(339\) −4.71174e7 −0.0656874
\(340\) 1.50280e8 0.207359
\(341\) −2.54506e8 −0.347582
\(342\) −7.74633e7 −0.104714
\(343\) −8.13484e8 −1.08848
\(344\) 5.79794e8 0.767926
\(345\) 1.00832e8 0.132200
\(346\) −3.72798e8 −0.483846
\(347\) 7.77286e8 0.998683 0.499341 0.866405i \(-0.333575\pi\)
0.499341 + 0.866405i \(0.333575\pi\)
\(348\) 110107. 0.000140051 0
\(349\) −1.30337e9 −1.64126 −0.820631 0.571458i \(-0.806378\pi\)
−0.820631 + 0.571458i \(0.806378\pi\)
\(350\) −1.13788e8 −0.141860
\(351\) 1.17572e8 0.145121
\(352\) −3.80276e8 −0.464729
\(353\) 8.94399e8 1.08223 0.541115 0.840949i \(-0.318002\pi\)
0.541115 + 0.840949i \(0.318002\pi\)
\(354\) 4.88157e8 0.584855
\(355\) −8.53193e8 −1.01216
\(356\) 9.51739e7 0.111800
\(357\) −2.77707e8 −0.323034
\(358\) −5.30152e8 −0.610675
\(359\) −1.09890e8 −0.125351 −0.0626757 0.998034i \(-0.519963\pi\)
−0.0626757 + 0.998034i \(0.519963\pi\)
\(360\) −3.51702e8 −0.397297
\(361\) −7.72598e8 −0.864328
\(362\) −7.16642e8 −0.794003
\(363\) 1.19168e8 0.130763
\(364\) 1.52802e8 0.166063
\(365\) 7.94595e8 0.855305
\(366\) −3.13603e8 −0.334346
\(367\) 3.67335e8 0.387910 0.193955 0.981010i \(-0.437868\pi\)
0.193955 + 0.981010i \(0.437868\pi\)
\(368\) −1.30183e8 −0.136172
\(369\) −2.70032e8 −0.279784
\(370\) −1.13276e9 −1.16261
\(371\) −3.99567e8 −0.406238
\(372\) −6.17623e7 −0.0622047
\(373\) 1.07675e7 0.0107432 0.00537160 0.999986i \(-0.498290\pi\)
0.00537160 + 0.999986i \(0.498290\pi\)
\(374\) −5.25620e8 −0.519542
\(375\) −5.14133e8 −0.503462
\(376\) 1.30572e9 1.26676
\(377\) −698063. −0.000670965 0
\(378\) 1.39226e8 0.132587
\(379\) −1.58481e9 −1.49534 −0.747670 0.664071i \(-0.768827\pi\)
−0.747670 + 0.664071i \(0.768827\pi\)
\(380\) 1.17952e8 0.110271
\(381\) 5.16830e8 0.478751
\(382\) 1.70107e9 1.56135
\(383\) 1.17134e9 1.06534 0.532669 0.846324i \(-0.321189\pi\)
0.532669 + 0.846324i \(0.321189\pi\)
\(384\) 2.64521e8 0.238397
\(385\) −8.73584e8 −0.780175
\(386\) −5.56700e8 −0.492681
\(387\) 2.68910e8 0.235841
\(388\) 2.63731e8 0.229219
\(389\) −1.30652e8 −0.112536 −0.0562680 0.998416i \(-0.517920\pi\)
−0.0562680 + 0.998416i \(0.517920\pi\)
\(390\) 4.77657e8 0.407746
\(391\) 1.70711e8 0.144425
\(392\) −4.49773e8 −0.377131
\(393\) 8.57127e8 0.712314
\(394\) −3.58681e8 −0.295442
\(395\) −3.95218e8 −0.322661
\(396\) −9.87652e7 −0.0799229
\(397\) −2.35598e9 −1.88975 −0.944874 0.327434i \(-0.893816\pi\)
−0.944874 + 0.327434i \(0.893816\pi\)
\(398\) −9.36513e8 −0.744600
\(399\) −2.17967e8 −0.171785
\(400\) −1.72123e8 −0.134471
\(401\) 5.75742e8 0.445884 0.222942 0.974832i \(-0.428434\pi\)
0.222942 + 0.974832i \(0.428434\pi\)
\(402\) −1.05407e9 −0.809244
\(403\) 3.91566e8 0.298014
\(404\) 4.92154e8 0.371335
\(405\) −1.63120e8 −0.122015
\(406\) −826629. −0.000613012 0
\(407\) −1.48494e9 −1.09176
\(408\) −5.95438e8 −0.434036
\(409\) 2.19710e9 1.58788 0.793942 0.607994i \(-0.208025\pi\)
0.793942 + 0.607994i \(0.208025\pi\)
\(410\) −1.09705e9 −0.786108
\(411\) 9.20852e8 0.654250
\(412\) 7.96014e6 0.00560764
\(413\) 1.37358e9 0.959467
\(414\) −8.55848e7 −0.0592782
\(415\) −1.59736e9 −1.09707
\(416\) 5.85067e8 0.398455
\(417\) 9.20147e8 0.621414
\(418\) −4.12550e8 −0.276286
\(419\) −1.88674e9 −1.25304 −0.626518 0.779407i \(-0.715521\pi\)
−0.626518 + 0.779407i \(0.715521\pi\)
\(420\) −2.11997e8 −0.139623
\(421\) −4.52763e8 −0.295722 −0.147861 0.989008i \(-0.547239\pi\)
−0.147861 + 0.989008i \(0.547239\pi\)
\(422\) 1.22432e9 0.793054
\(423\) 6.05598e8 0.389039
\(424\) −8.56721e8 −0.545832
\(425\) 2.25708e8 0.142622
\(426\) 7.24175e8 0.453848
\(427\) −8.82420e8 −0.548501
\(428\) 6.01842e8 0.371047
\(429\) 6.26160e8 0.382900
\(430\) 1.09249e9 0.662640
\(431\) 4.63280e8 0.278723 0.139361 0.990242i \(-0.455495\pi\)
0.139361 + 0.990242i \(0.455495\pi\)
\(432\) 2.10602e8 0.125681
\(433\) −1.17035e9 −0.692802 −0.346401 0.938087i \(-0.612596\pi\)
−0.346401 + 0.938087i \(0.612596\pi\)
\(434\) 4.63682e8 0.272274
\(435\) 968493. 0.000564137 0
\(436\) 1.12431e8 0.0649654
\(437\) 1.33988e8 0.0768036
\(438\) −6.74439e8 −0.383516
\(439\) 1.64913e9 0.930313 0.465157 0.885228i \(-0.345998\pi\)
0.465157 + 0.885228i \(0.345998\pi\)
\(440\) −1.87307e9 −1.04826
\(441\) −2.08606e8 −0.115822
\(442\) 8.08683e8 0.445451
\(443\) −7.14982e8 −0.390735 −0.195367 0.980730i \(-0.562590\pi\)
−0.195367 + 0.980730i \(0.562590\pi\)
\(444\) −3.60358e8 −0.195386
\(445\) 8.37146e8 0.450341
\(446\) −2.73389e8 −0.145918
\(447\) 1.43876e9 0.761926
\(448\) 1.69680e9 0.891576
\(449\) −2.68585e9 −1.40030 −0.700149 0.713997i \(-0.746883\pi\)
−0.700149 + 0.713997i \(0.746883\pi\)
\(450\) −1.13157e8 −0.0585379
\(451\) −1.43812e9 −0.738206
\(452\) −6.08956e7 −0.0310171
\(453\) −1.67464e9 −0.846404
\(454\) 2.64253e9 1.32533
\(455\) 1.34404e9 0.668916
\(456\) −4.67349e8 −0.230815
\(457\) −2.96284e9 −1.45212 −0.726059 0.687632i \(-0.758650\pi\)
−0.726059 + 0.687632i \(0.758650\pi\)
\(458\) 1.90653e9 0.927290
\(459\) −2.76166e8 −0.133299
\(460\) 1.30318e8 0.0624241
\(461\) 7.80077e8 0.370838 0.185419 0.982660i \(-0.440636\pi\)
0.185419 + 0.982660i \(0.440636\pi\)
\(462\) 7.41483e8 0.349828
\(463\) 1.50379e9 0.704131 0.352066 0.935975i \(-0.385479\pi\)
0.352066 + 0.935975i \(0.385479\pi\)
\(464\) −1.25041e6 −0.000581083 0
\(465\) −5.43259e8 −0.250566
\(466\) −1.10204e9 −0.504484
\(467\) −2.49049e9 −1.13156 −0.565779 0.824557i \(-0.691424\pi\)
−0.565779 + 0.824557i \(0.691424\pi\)
\(468\) 1.51954e8 0.0685252
\(469\) −2.96597e9 −1.32758
\(470\) 2.46034e9 1.09308
\(471\) −3.33103e8 −0.146895
\(472\) 2.94513e9 1.28916
\(473\) 1.43214e9 0.622261
\(474\) 3.35454e8 0.144680
\(475\) 1.77154e8 0.0758444
\(476\) −3.58915e8 −0.152534
\(477\) −3.97349e8 −0.167632
\(478\) 3.84595e9 1.61067
\(479\) −2.48477e9 −1.03303 −0.516514 0.856279i \(-0.672771\pi\)
−0.516514 + 0.856279i \(0.672771\pi\)
\(480\) −8.11723e8 −0.335014
\(481\) 2.28463e9 0.936069
\(482\) −9.91810e8 −0.403426
\(483\) −2.40820e8 −0.0972472
\(484\) 1.54016e8 0.0617456
\(485\) 2.31977e9 0.923312
\(486\) 1.38454e8 0.0547113
\(487\) −3.44290e9 −1.35075 −0.675373 0.737476i \(-0.736017\pi\)
−0.675373 + 0.737476i \(0.736017\pi\)
\(488\) −1.89202e9 −0.736980
\(489\) 2.67837e9 1.03583
\(490\) −8.47494e8 −0.325425
\(491\) −2.47473e9 −0.943501 −0.471750 0.881732i \(-0.656378\pi\)
−0.471750 + 0.881732i \(0.656378\pi\)
\(492\) −3.48996e8 −0.132112
\(493\) 1.63968e6 0.000616304 0
\(494\) 6.34721e8 0.236885
\(495\) −8.68736e8 −0.321936
\(496\) 7.01393e8 0.258093
\(497\) 2.03770e9 0.744547
\(498\) 1.35582e9 0.491924
\(499\) −1.61485e9 −0.581808 −0.290904 0.956752i \(-0.593956\pi\)
−0.290904 + 0.956752i \(0.593956\pi\)
\(500\) −6.64479e8 −0.237731
\(501\) −1.17782e9 −0.418453
\(502\) 4.04106e9 1.42571
\(503\) 8.25520e8 0.289228 0.144614 0.989488i \(-0.453806\pi\)
0.144614 + 0.989488i \(0.453806\pi\)
\(504\) 8.39975e8 0.292253
\(505\) 4.32897e9 1.49577
\(506\) −4.55802e8 −0.156405
\(507\) 7.30842e8 0.249055
\(508\) 6.67963e8 0.226063
\(509\) −3.14630e9 −1.05752 −0.528759 0.848772i \(-0.677343\pi\)
−0.528759 + 0.848772i \(0.677343\pi\)
\(510\) −1.12197e9 −0.374528
\(511\) −1.89775e9 −0.629165
\(512\) 3.20066e9 1.05389
\(513\) −2.16758e8 −0.0708865
\(514\) −3.76421e8 −0.122265
\(515\) 7.00171e7 0.0225880
\(516\) 3.47546e8 0.111362
\(517\) 3.22526e9 1.02647
\(518\) 2.70540e9 0.855219
\(519\) −1.04316e9 −0.327541
\(520\) 2.88178e9 0.898772
\(521\) 2.98365e9 0.924307 0.462154 0.886800i \(-0.347077\pi\)
0.462154 + 0.886800i \(0.347077\pi\)
\(522\) −822041. −0.000252957 0
\(523\) 3.33480e9 1.01933 0.509663 0.860374i \(-0.329770\pi\)
0.509663 + 0.860374i \(0.329770\pi\)
\(524\) 1.10777e9 0.336350
\(525\) −3.18403e8 −0.0960327
\(526\) −3.11208e9 −0.932395
\(527\) −9.19749e8 −0.273736
\(528\) 1.12161e9 0.331607
\(529\) 1.48036e8 0.0434783
\(530\) −1.61430e9 −0.470996
\(531\) 1.36596e9 0.395920
\(532\) −2.81706e8 −0.0811159
\(533\) 2.21259e9 0.632931
\(534\) −7.10556e8 −0.201931
\(535\) 5.29378e9 1.49461
\(536\) −6.35940e9 −1.78377
\(537\) −1.48347e9 −0.413398
\(538\) −9.06605e8 −0.251004
\(539\) −1.11098e9 −0.305595
\(540\) −2.10820e8 −0.0576149
\(541\) −2.22555e9 −0.604292 −0.302146 0.953262i \(-0.597703\pi\)
−0.302146 + 0.953262i \(0.597703\pi\)
\(542\) −3.16212e9 −0.853062
\(543\) −2.00531e9 −0.537503
\(544\) −1.37426e9 −0.365994
\(545\) 9.88936e8 0.261686
\(546\) −1.14080e9 −0.299940
\(547\) 1.06154e9 0.277318 0.138659 0.990340i \(-0.455721\pi\)
0.138659 + 0.990340i \(0.455721\pi\)
\(548\) 1.19013e9 0.308932
\(549\) −8.77522e8 −0.226337
\(550\) −6.02644e8 −0.154451
\(551\) 1.28696e6 0.000327743 0
\(552\) −5.16347e8 −0.130664
\(553\) 9.43906e8 0.237351
\(554\) −4.18753e9 −1.04634
\(555\) −3.16970e9 −0.787032
\(556\) 1.18922e9 0.293427
\(557\) −5.05682e9 −1.23989 −0.619947 0.784644i \(-0.712846\pi\)
−0.619947 + 0.784644i \(0.712846\pi\)
\(558\) 4.61109e8 0.112353
\(559\) −2.20340e9 −0.533522
\(560\) 2.40751e9 0.579309
\(561\) −1.47079e9 −0.351706
\(562\) 4.37156e9 1.03887
\(563\) −2.75862e9 −0.651498 −0.325749 0.945456i \(-0.605616\pi\)
−0.325749 + 0.945456i \(0.605616\pi\)
\(564\) 7.82690e8 0.183701
\(565\) −5.35636e8 −0.124940
\(566\) −5.28005e9 −1.22400
\(567\) 3.89583e8 0.0897551
\(568\) 4.36907e9 1.00039
\(569\) 5.48258e9 1.24765 0.623824 0.781565i \(-0.285578\pi\)
0.623824 + 0.781565i \(0.285578\pi\)
\(570\) −8.80612e8 −0.199170
\(571\) −3.26474e9 −0.733875 −0.366938 0.930246i \(-0.619594\pi\)
−0.366938 + 0.930246i \(0.619594\pi\)
\(572\) 8.09265e8 0.180803
\(573\) 4.75994e9 1.05696
\(574\) 2.62010e9 0.578264
\(575\) 1.95727e8 0.0429353
\(576\) 1.68739e9 0.367905
\(577\) −1.20637e9 −0.261436 −0.130718 0.991420i \(-0.541728\pi\)
−0.130718 + 0.991420i \(0.541728\pi\)
\(578\) 2.05987e9 0.443703
\(579\) −1.55776e9 −0.333522
\(580\) 1.25170e6 0.000266381 0
\(581\) 3.81501e9 0.807011
\(582\) −1.96898e9 −0.414010
\(583\) −2.11618e9 −0.442296
\(584\) −4.06900e9 −0.845362
\(585\) 1.33658e9 0.276025
\(586\) −6.88285e7 −0.0141295
\(587\) −4.66286e9 −0.951523 −0.475761 0.879574i \(-0.657827\pi\)
−0.475761 + 0.879574i \(0.657827\pi\)
\(588\) −2.69607e8 −0.0546904
\(589\) −7.21894e8 −0.145569
\(590\) 5.54943e9 1.11241
\(591\) −1.00366e9 −0.200000
\(592\) 4.09235e9 0.810674
\(593\) 4.02461e9 0.792560 0.396280 0.918130i \(-0.370301\pi\)
0.396280 + 0.918130i \(0.370301\pi\)
\(594\) 7.37368e8 0.144355
\(595\) −3.15701e9 −0.614421
\(596\) 1.85949e9 0.359776
\(597\) −2.62055e9 −0.504060
\(598\) 7.01267e8 0.134100
\(599\) 4.66630e9 0.887113 0.443557 0.896246i \(-0.353717\pi\)
0.443557 + 0.896246i \(0.353717\pi\)
\(600\) −6.82694e8 −0.129032
\(601\) −7.89419e9 −1.48336 −0.741681 0.670753i \(-0.765971\pi\)
−0.741681 + 0.670753i \(0.765971\pi\)
\(602\) −2.60921e9 −0.487440
\(603\) −2.94950e9 −0.547821
\(604\) −2.16434e9 −0.399666
\(605\) 1.35472e9 0.248717
\(606\) −3.67435e9 −0.670698
\(607\) −7.35070e9 −1.33404 −0.667019 0.745041i \(-0.732430\pi\)
−0.667019 + 0.745041i \(0.732430\pi\)
\(608\) −1.07863e9 −0.194631
\(609\) −2.31307e6 −0.000414981 0
\(610\) −3.56507e9 −0.635937
\(611\) −4.96216e9 −0.880089
\(612\) −3.56923e8 −0.0629427
\(613\) 3.32089e9 0.582294 0.291147 0.956678i \(-0.405963\pi\)
0.291147 + 0.956678i \(0.405963\pi\)
\(614\) −9.25753e9 −1.61401
\(615\) −3.06976e9 −0.532158
\(616\) 4.47349e9 0.771106
\(617\) −6.89774e8 −0.118225 −0.0591125 0.998251i \(-0.518827\pi\)
−0.0591125 + 0.998251i \(0.518827\pi\)
\(618\) −5.94293e7 −0.0101284
\(619\) −1.04648e10 −1.77342 −0.886711 0.462325i \(-0.847016\pi\)
−0.886711 + 0.462325i \(0.847016\pi\)
\(620\) −7.02121e8 −0.118315
\(621\) −2.39483e8 −0.0401286
\(622\) −7.95767e9 −1.32593
\(623\) −1.99937e9 −0.331272
\(624\) −1.72564e9 −0.284317
\(625\) −7.10151e9 −1.16351
\(626\) 4.83960e8 0.0788496
\(627\) −1.15440e9 −0.187033
\(628\) −4.30511e8 −0.0693626
\(629\) −5.36636e9 −0.859810
\(630\) 1.58274e9 0.252184
\(631\) 3.22289e9 0.510673 0.255336 0.966852i \(-0.417814\pi\)
0.255336 + 0.966852i \(0.417814\pi\)
\(632\) 2.02385e9 0.318910
\(633\) 3.42590e9 0.536861
\(634\) −3.45455e9 −0.538368
\(635\) 5.87538e9 0.910601
\(636\) −5.13544e8 −0.0791549
\(637\) 1.70928e9 0.262014
\(638\) −4.37798e6 −0.000667423 0
\(639\) 2.02639e9 0.307234
\(640\) 3.00711e9 0.453439
\(641\) 5.23897e9 0.785675 0.392837 0.919608i \(-0.371494\pi\)
0.392837 + 0.919608i \(0.371494\pi\)
\(642\) −4.49327e9 −0.670178
\(643\) 6.78028e9 1.00579 0.502897 0.864346i \(-0.332268\pi\)
0.502897 + 0.864346i \(0.332268\pi\)
\(644\) −3.11241e8 −0.0459194
\(645\) 3.05700e9 0.448577
\(646\) −1.49090e9 −0.217587
\(647\) −8.41553e8 −0.122156 −0.0610782 0.998133i \(-0.519454\pi\)
−0.0610782 + 0.998133i \(0.519454\pi\)
\(648\) 8.35313e8 0.120597
\(649\) 7.27475e9 1.04463
\(650\) 9.27188e8 0.132425
\(651\) 1.29747e9 0.184317
\(652\) 3.46160e9 0.489114
\(653\) 7.33932e9 1.03148 0.515739 0.856746i \(-0.327518\pi\)
0.515739 + 0.856746i \(0.327518\pi\)
\(654\) −8.39392e8 −0.117339
\(655\) 9.74393e9 1.35485
\(656\) 3.96332e9 0.548145
\(657\) −1.88721e9 −0.259622
\(658\) −5.87607e9 −0.804074
\(659\) 1.14657e10 1.56064 0.780319 0.625382i \(-0.215057\pi\)
0.780319 + 0.625382i \(0.215057\pi\)
\(660\) −1.12278e9 −0.152016
\(661\) 1.40403e10 1.89091 0.945453 0.325758i \(-0.105619\pi\)
0.945453 + 0.325758i \(0.105619\pi\)
\(662\) 4.26515e9 0.571388
\(663\) 2.26285e9 0.301550
\(664\) 8.17986e9 1.08432
\(665\) −2.47788e9 −0.326742
\(666\) 2.69038e9 0.352902
\(667\) 1.42188e6 0.000185534 0
\(668\) −1.52224e9 −0.197591
\(669\) −7.64995e8 −0.0987796
\(670\) −1.19828e10 −1.53921
\(671\) −4.67346e9 −0.597186
\(672\) 1.93865e9 0.246438
\(673\) 1.07497e10 1.35939 0.679693 0.733496i \(-0.262113\pi\)
0.679693 + 0.733496i \(0.262113\pi\)
\(674\) −8.84599e9 −1.11285
\(675\) −3.16635e8 −0.0396275
\(676\) 9.44558e8 0.117602
\(677\) 1.29380e10 1.60254 0.801269 0.598304i \(-0.204159\pi\)
0.801269 + 0.598304i \(0.204159\pi\)
\(678\) 4.54639e8 0.0560225
\(679\) −5.54034e9 −0.679192
\(680\) −6.76902e9 −0.825552
\(681\) 7.39431e9 0.897187
\(682\) 2.45575e9 0.296441
\(683\) −1.12095e10 −1.34621 −0.673104 0.739548i \(-0.735039\pi\)
−0.673104 + 0.739548i \(0.735039\pi\)
\(684\) −2.80143e8 −0.0334721
\(685\) 1.04684e10 1.24440
\(686\) 7.84936e9 0.928325
\(687\) 5.33486e9 0.627732
\(688\) −3.94685e9 −0.462052
\(689\) 3.25581e9 0.379221
\(690\) −9.72938e8 −0.112749
\(691\) 2.40363e9 0.277137 0.138568 0.990353i \(-0.455750\pi\)
0.138568 + 0.990353i \(0.455750\pi\)
\(692\) −1.34821e9 −0.154663
\(693\) 2.07482e9 0.236817
\(694\) −7.50008e9 −0.851742
\(695\) 1.04603e10 1.18195
\(696\) −4.95951e6 −0.000557579 0
\(697\) −5.19716e9 −0.581368
\(698\) 1.25763e10 1.39978
\(699\) −3.08373e9 −0.341512
\(700\) −4.11511e8 −0.0453460
\(701\) 1.28427e9 0.140813 0.0704064 0.997518i \(-0.477570\pi\)
0.0704064 + 0.997518i \(0.477570\pi\)
\(702\) −1.13446e9 −0.123769
\(703\) −4.21196e9 −0.457236
\(704\) 8.98658e9 0.970711
\(705\) 6.88451e9 0.739965
\(706\) −8.63011e9 −0.922997
\(707\) −1.03389e10 −1.10029
\(708\) 1.76540e9 0.186951
\(709\) −1.19734e10 −1.26170 −0.630851 0.775904i \(-0.717294\pi\)
−0.630851 + 0.775904i \(0.717294\pi\)
\(710\) 8.23252e9 0.863234
\(711\) 9.38667e8 0.0979418
\(712\) −4.28690e9 −0.445106
\(713\) −7.97580e8 −0.0824064
\(714\) 2.67961e9 0.275504
\(715\) 7.11827e9 0.728288
\(716\) −1.91727e9 −0.195204
\(717\) 1.07617e10 1.09035
\(718\) 1.06034e9 0.106908
\(719\) 3.94160e8 0.0395477 0.0197738 0.999804i \(-0.493705\pi\)
0.0197738 + 0.999804i \(0.493705\pi\)
\(720\) 2.39415e9 0.239049
\(721\) −1.67223e8 −0.0166159
\(722\) 7.45485e9 0.737155
\(723\) −2.77528e9 −0.273101
\(724\) −2.59171e9 −0.253805
\(725\) 1.87996e6 0.000183217 0
\(726\) −1.14986e9 −0.111524
\(727\) −1.16145e10 −1.12106 −0.560530 0.828134i \(-0.689402\pi\)
−0.560530 + 0.828134i \(0.689402\pi\)
\(728\) −6.88261e9 −0.661140
\(729\) 3.87420e8 0.0370370
\(730\) −7.66710e9 −0.729460
\(731\) 5.17557e9 0.490057
\(732\) −1.13413e9 −0.106875
\(733\) 7.63834e9 0.716366 0.358183 0.933651i \(-0.383396\pi\)
0.358183 + 0.933651i \(0.383396\pi\)
\(734\) −3.54444e9 −0.330835
\(735\) −2.37146e9 −0.220297
\(736\) −1.19172e9 −0.110180
\(737\) −1.57083e10 −1.44542
\(738\) 2.60556e9 0.238618
\(739\) 7.94049e9 0.723755 0.361878 0.932226i \(-0.382136\pi\)
0.361878 + 0.932226i \(0.382136\pi\)
\(740\) −4.09659e9 −0.371631
\(741\) 1.77607e9 0.160361
\(742\) 3.85545e9 0.346467
\(743\) 5.34188e9 0.477786 0.238893 0.971046i \(-0.423216\pi\)
0.238893 + 0.971046i \(0.423216\pi\)
\(744\) 2.78195e9 0.247653
\(745\) 1.63560e10 1.44921
\(746\) −1.03896e8 −0.00916250
\(747\) 3.79384e9 0.333010
\(748\) −1.90088e9 −0.166073
\(749\) −1.26432e10 −1.09944
\(750\) 4.96091e9 0.429385
\(751\) −1.70844e10 −1.47184 −0.735921 0.677068i \(-0.763250\pi\)
−0.735921 + 0.677068i \(0.763250\pi\)
\(752\) −8.88849e9 −0.762194
\(753\) 1.13077e10 0.965142
\(754\) 6.73566e6 0.000572243 0
\(755\) −1.90375e10 −1.60989
\(756\) 5.03506e8 0.0423817
\(757\) −1.18119e10 −0.989655 −0.494828 0.868991i \(-0.664769\pi\)
−0.494828 + 0.868991i \(0.664769\pi\)
\(758\) 1.52919e10 1.27532
\(759\) −1.27542e9 −0.105879
\(760\) −5.31288e9 −0.439018
\(761\) −1.10997e10 −0.912987 −0.456494 0.889727i \(-0.650895\pi\)
−0.456494 + 0.889727i \(0.650895\pi\)
\(762\) −4.98692e9 −0.408310
\(763\) −2.36189e9 −0.192497
\(764\) 6.15186e9 0.499091
\(765\) −3.13949e9 −0.253538
\(766\) −1.13023e10 −0.908589
\(767\) −1.11924e10 −0.895655
\(768\) 5.44708e9 0.433910
\(769\) −2.76235e9 −0.219047 −0.109523 0.993984i \(-0.534932\pi\)
−0.109523 + 0.993984i \(0.534932\pi\)
\(770\) 8.42928e9 0.665385
\(771\) −1.05330e9 −0.0827679
\(772\) −2.01328e9 −0.157487
\(773\) 1.66549e9 0.129692 0.0648460 0.997895i \(-0.479344\pi\)
0.0648460 + 0.997895i \(0.479344\pi\)
\(774\) −2.59473e9 −0.201140
\(775\) −1.05453e9 −0.0813772
\(776\) −1.18792e10 −0.912579
\(777\) 7.57024e9 0.578943
\(778\) 1.26067e9 0.0959780
\(779\) −4.07916e9 −0.309164
\(780\) 1.72743e9 0.130337
\(781\) 1.07920e10 0.810632
\(782\) −1.64720e9 −0.123175
\(783\) −2.30023e6 −0.000171240 0
\(784\) 3.06175e9 0.226915
\(785\) −3.78676e9 −0.279398
\(786\) −8.27048e9 −0.607508
\(787\) 3.50520e9 0.256331 0.128165 0.991753i \(-0.459091\pi\)
0.128165 + 0.991753i \(0.459091\pi\)
\(788\) −1.29716e9 −0.0944388
\(789\) −8.70820e9 −0.631188
\(790\) 3.81348e9 0.275186
\(791\) 1.27927e9 0.0919060
\(792\) 4.44866e9 0.318194
\(793\) 7.19027e9 0.512022
\(794\) 2.27330e10 1.61170
\(795\) −4.51712e9 −0.318843
\(796\) −3.38686e9 −0.238013
\(797\) −2.46740e10 −1.72638 −0.863189 0.504881i \(-0.831536\pi\)
−0.863189 + 0.504881i \(0.831536\pi\)
\(798\) 2.10318e9 0.146510
\(799\) 1.16556e10 0.808391
\(800\) −1.57565e9 −0.108804
\(801\) −1.98827e9 −0.136698
\(802\) −5.55537e9 −0.380279
\(803\) −1.00508e10 −0.685009
\(804\) −3.81201e9 −0.258677
\(805\) −2.73767e9 −0.184967
\(806\) −3.77825e9 −0.254166
\(807\) −2.53686e9 −0.169918
\(808\) −2.21680e10 −1.47838
\(809\) 2.43326e9 0.161573 0.0807865 0.996731i \(-0.474257\pi\)
0.0807865 + 0.996731i \(0.474257\pi\)
\(810\) 1.57396e9 0.104063
\(811\) −4.26801e9 −0.280965 −0.140482 0.990083i \(-0.544865\pi\)
−0.140482 + 0.990083i \(0.544865\pi\)
\(812\) −2.98947e6 −0.000195951 0
\(813\) −8.84823e9 −0.577484
\(814\) 1.43283e10 0.931127
\(815\) 3.04481e10 1.97019
\(816\) 4.05334e9 0.261155
\(817\) 4.06221e9 0.260606
\(818\) −2.12000e10 −1.35425
\(819\) −3.19217e9 −0.203045
\(820\) −3.96743e9 −0.251282
\(821\) 1.69378e10 1.06821 0.534103 0.845419i \(-0.320649\pi\)
0.534103 + 0.845419i \(0.320649\pi\)
\(822\) −8.88537e9 −0.557987
\(823\) −1.49405e10 −0.934258 −0.467129 0.884189i \(-0.654712\pi\)
−0.467129 + 0.884189i \(0.654712\pi\)
\(824\) −3.58547e8 −0.0223255
\(825\) −1.68632e9 −0.104556
\(826\) −1.32538e10 −0.818296
\(827\) 2.62579e10 1.61432 0.807162 0.590330i \(-0.201002\pi\)
0.807162 + 0.590330i \(0.201002\pi\)
\(828\) −3.09514e8 −0.0189485
\(829\) 1.26847e10 0.773282 0.386641 0.922230i \(-0.373635\pi\)
0.386641 + 0.922230i \(0.373635\pi\)
\(830\) 1.54131e10 0.935656
\(831\) −1.17175e10 −0.708325
\(832\) −1.38261e10 −0.832280
\(833\) −4.01492e9 −0.240669
\(834\) −8.87856e9 −0.529982
\(835\) −1.33896e10 −0.795911
\(836\) −1.49197e9 −0.0883157
\(837\) 1.29027e9 0.0760576
\(838\) 1.82053e10 1.06867
\(839\) 1.31687e10 0.769796 0.384898 0.922959i \(-0.374237\pi\)
0.384898 + 0.922959i \(0.374237\pi\)
\(840\) 9.54894e9 0.555876
\(841\) −1.72499e10 −0.999999
\(842\) 4.36875e9 0.252211
\(843\) 1.22325e10 0.703264
\(844\) 4.42772e9 0.253502
\(845\) 8.30830e9 0.473711
\(846\) −5.84346e9 −0.331798
\(847\) −3.23549e9 −0.182957
\(848\) 5.83198e9 0.328421
\(849\) −1.47746e10 −0.828589
\(850\) −2.17787e9 −0.121637
\(851\) −4.65355e9 −0.258840
\(852\) 2.61895e9 0.145074
\(853\) −1.52335e8 −0.00840386 −0.00420193 0.999991i \(-0.501338\pi\)
−0.00420193 + 0.999991i \(0.501338\pi\)
\(854\) 8.51453e9 0.467798
\(855\) −2.46413e9 −0.134829
\(856\) −2.71087e10 −1.47724
\(857\) −8.43291e9 −0.457661 −0.228831 0.973466i \(-0.573490\pi\)
−0.228831 + 0.973466i \(0.573490\pi\)
\(858\) −6.04187e9 −0.326562
\(859\) 3.11811e10 1.67847 0.839237 0.543766i \(-0.183002\pi\)
0.839237 + 0.543766i \(0.183002\pi\)
\(860\) 3.95094e9 0.211815
\(861\) 7.33155e9 0.391458
\(862\) −4.47022e9 −0.237713
\(863\) 1.07163e10 0.567553 0.283776 0.958891i \(-0.408413\pi\)
0.283776 + 0.958891i \(0.408413\pi\)
\(864\) 1.92789e9 0.101691
\(865\) −1.18588e10 −0.622995
\(866\) 1.12928e10 0.590866
\(867\) 5.76393e9 0.300367
\(868\) 1.67689e9 0.0870332
\(869\) 4.99910e9 0.258418
\(870\) −9.34506e6 −0.000481133 0
\(871\) 2.41677e10 1.23929
\(872\) −5.06419e9 −0.258644
\(873\) −5.50959e9 −0.280266
\(874\) −1.29286e9 −0.0655031
\(875\) 1.39591e10 0.704415
\(876\) −2.43908e9 −0.122592
\(877\) 1.75959e9 0.0880874 0.0440437 0.999030i \(-0.485976\pi\)
0.0440437 + 0.999030i \(0.485976\pi\)
\(878\) −1.59126e10 −0.793432
\(879\) −1.92596e8 −0.00956501
\(880\) 1.27506e10 0.630728
\(881\) −2.66596e10 −1.31353 −0.656763 0.754097i \(-0.728075\pi\)
−0.656763 + 0.754097i \(0.728075\pi\)
\(882\) 2.01285e9 0.0987806
\(883\) 2.55224e10 1.24755 0.623776 0.781603i \(-0.285598\pi\)
0.623776 + 0.781603i \(0.285598\pi\)
\(884\) 2.92457e9 0.142390
\(885\) 1.55284e10 0.753053
\(886\) 6.89891e9 0.333244
\(887\) −1.78494e10 −0.858800 −0.429400 0.903114i \(-0.641275\pi\)
−0.429400 + 0.903114i \(0.641275\pi\)
\(888\) 1.62315e10 0.777883
\(889\) −1.40323e10 −0.669842
\(890\) −8.07768e9 −0.384080
\(891\) 2.06330e9 0.0977216
\(892\) −9.88699e8 −0.0466431
\(893\) 9.14829e9 0.429892
\(894\) −1.38827e10 −0.649821
\(895\) −1.68643e10 −0.786298
\(896\) −7.18193e9 −0.333551
\(897\) 1.96228e9 0.0907795
\(898\) 2.59160e10 1.19427
\(899\) −7.66074e6 −0.000351651 0
\(900\) −4.09227e8 −0.0187118
\(901\) −7.64757e9 −0.348327
\(902\) 1.38765e10 0.629590
\(903\) −7.30109e9 −0.329975
\(904\) 2.74291e9 0.123487
\(905\) −2.27966e10 −1.02235
\(906\) 1.61587e10 0.721868
\(907\) −8.37198e9 −0.372566 −0.186283 0.982496i \(-0.559644\pi\)
−0.186283 + 0.982496i \(0.559644\pi\)
\(908\) 9.55659e9 0.423645
\(909\) −1.02816e10 −0.454031
\(910\) −1.29687e10 −0.570495
\(911\) −2.02213e10 −0.886126 −0.443063 0.896490i \(-0.646108\pi\)
−0.443063 + 0.896490i \(0.646108\pi\)
\(912\) 3.18140e9 0.138879
\(913\) 2.02050e10 0.878641
\(914\) 2.85887e10 1.23846
\(915\) −9.97578e9 −0.430500
\(916\) 6.89490e9 0.296411
\(917\) −2.32716e10 −0.996629
\(918\) 2.66474e9 0.113686
\(919\) 1.96145e10 0.833629 0.416815 0.908992i \(-0.363146\pi\)
0.416815 + 0.908992i \(0.363146\pi\)
\(920\) −5.86990e9 −0.248527
\(921\) −2.59044e10 −1.09261
\(922\) −7.52702e9 −0.316275
\(923\) −1.66039e10 −0.695029
\(924\) 2.68154e9 0.111824
\(925\) −6.15275e9 −0.255607
\(926\) −1.45102e10 −0.600529
\(927\) −1.66295e8 −0.00685646
\(928\) −1.14465e7 −0.000470169 0
\(929\) 1.66505e10 0.681355 0.340677 0.940180i \(-0.389344\pi\)
0.340677 + 0.940180i \(0.389344\pi\)
\(930\) 5.24194e9 0.213699
\(931\) −3.15124e9 −0.127985
\(932\) −3.98549e9 −0.161260
\(933\) −2.22671e10 −0.897591
\(934\) 2.40310e10 0.965066
\(935\) −1.67201e10 −0.668957
\(936\) −6.84441e9 −0.272817
\(937\) 4.92026e10 1.95389 0.976944 0.213496i \(-0.0684849\pi\)
0.976944 + 0.213496i \(0.0684849\pi\)
\(938\) 2.86188e10 1.13225
\(939\) 1.35422e9 0.0533775
\(940\) 8.89771e9 0.349406
\(941\) −9.05047e9 −0.354085 −0.177042 0.984203i \(-0.556653\pi\)
−0.177042 + 0.984203i \(0.556653\pi\)
\(942\) 3.21414e9 0.125281
\(943\) −4.50683e9 −0.175017
\(944\) −2.00485e10 −0.775675
\(945\) 4.42882e9 0.170717
\(946\) −1.38189e10 −0.530705
\(947\) −4.80454e9 −0.183834 −0.0919171 0.995767i \(-0.529299\pi\)
−0.0919171 + 0.995767i \(0.529299\pi\)
\(948\) 1.21316e9 0.0462474
\(949\) 1.54635e10 0.587321
\(950\) −1.70937e9 −0.0646851
\(951\) −9.66653e9 −0.364451
\(952\) 1.61666e10 0.607279
\(953\) 3.86498e10 1.44651 0.723256 0.690580i \(-0.242645\pi\)
0.723256 + 0.690580i \(0.242645\pi\)
\(954\) 3.83405e9 0.142968
\(955\) 5.41116e10 2.01038
\(956\) 1.39087e10 0.514855
\(957\) −1.22504e7 −0.000451814 0
\(958\) 2.39757e10 0.881034
\(959\) −2.50018e10 −0.915389
\(960\) 1.91824e10 0.699767
\(961\) −2.32155e10 −0.843811
\(962\) −2.20445e10 −0.798341
\(963\) −1.25731e10 −0.453680
\(964\) −3.58684e9 −0.128956
\(965\) −1.77088e10 −0.634370
\(966\) 2.32369e9 0.0829387
\(967\) 2.31025e10 0.821612 0.410806 0.911723i \(-0.365247\pi\)
0.410806 + 0.911723i \(0.365247\pi\)
\(968\) −6.93730e9 −0.245825
\(969\) −4.17182e9 −0.147296
\(970\) −2.23836e10 −0.787461
\(971\) 3.14383e10 1.10203 0.551013 0.834497i \(-0.314242\pi\)
0.551013 + 0.834497i \(0.314242\pi\)
\(972\) 5.00712e8 0.0174886
\(973\) −2.49826e10 −0.869447
\(974\) 3.32208e10 1.15200
\(975\) 2.59446e9 0.0896458
\(976\) 1.28796e10 0.443432
\(977\) 3.59756e10 1.23418 0.617089 0.786894i \(-0.288312\pi\)
0.617089 + 0.786894i \(0.288312\pi\)
\(978\) −2.58438e10 −0.883427
\(979\) −1.05890e10 −0.360676
\(980\) −3.06493e9 −0.104023
\(981\) −2.34878e9 −0.0794331
\(982\) 2.38788e10 0.804679
\(983\) 3.43584e10 1.15371 0.576853 0.816848i \(-0.304281\pi\)
0.576853 + 0.816848i \(0.304281\pi\)
\(984\) 1.57198e10 0.525972
\(985\) −1.14097e10 −0.380407
\(986\) −1.58214e7 −0.000525624 0
\(987\) −1.64424e10 −0.544321
\(988\) 2.29544e9 0.0757211
\(989\) 4.48810e9 0.147528
\(990\) 8.38249e9 0.274568
\(991\) −5.34838e9 −0.174568 −0.0872839 0.996183i \(-0.527819\pi\)
−0.0872839 + 0.996183i \(0.527819\pi\)
\(992\) 6.42069e9 0.208829
\(993\) 1.19347e10 0.386803
\(994\) −1.96619e10 −0.634998
\(995\) −2.97907e10 −0.958738
\(996\) 4.90325e9 0.157245
\(997\) −5.37617e10 −1.71807 −0.859034 0.511919i \(-0.828935\pi\)
−0.859034 + 0.511919i \(0.828935\pi\)
\(998\) 1.55818e10 0.496204
\(999\) 7.52823e9 0.238898
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 69.8.a.a.1.2 5
3.2 odd 2 207.8.a.a.1.4 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
69.8.a.a.1.2 5 1.1 even 1 trivial
207.8.a.a.1.4 5 3.2 odd 2