Properties

Label 43.8.a.b.1.8
Level 43
Weight 8
Character 43.1
Self dual yes
Analytic conductor 13.433
Analytic rank 0
Dimension 13
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.4325560958\)
Analytic rank: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(2.37903\) of \(x^{13} - 3 x^{12} - 1279 x^{11} + 3765 x^{10} + 598742 x^{9} - 1518614 x^{8} - 124677082 x^{7} + 193428526 x^{6} + 11160446785 x^{5} + 1754605765 x^{4} - 349352939351 x^{3} - 481872751923 x^{2} + 2098464001560 x + 1551032970660\)
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.37903 q^{2} -66.5036 q^{3} -116.582 q^{4} -241.175 q^{5} -224.718 q^{6} -173.086 q^{7} -826.450 q^{8} +2235.73 q^{9} +O(q^{10})\) \(q+3.37903 q^{2} -66.5036 q^{3} -116.582 q^{4} -241.175 q^{5} -224.718 q^{6} -173.086 q^{7} -826.450 q^{8} +2235.73 q^{9} -814.939 q^{10} +682.590 q^{11} +7753.13 q^{12} +168.040 q^{13} -584.863 q^{14} +16039.0 q^{15} +12129.9 q^{16} +5562.09 q^{17} +7554.59 q^{18} -13825.2 q^{19} +28116.8 q^{20} +11510.8 q^{21} +2306.49 q^{22} -24641.2 q^{23} +54961.9 q^{24} -19959.4 q^{25} +567.813 q^{26} -3240.56 q^{27} +20178.7 q^{28} -33649.4 q^{29} +54196.4 q^{30} +22707.6 q^{31} +146773. q^{32} -45394.7 q^{33} +18794.5 q^{34} +41744.1 q^{35} -260646. q^{36} +181778. q^{37} -46715.6 q^{38} -11175.3 q^{39} +199320. q^{40} +335134. q^{41} +38895.5 q^{42} -79507.0 q^{43} -79577.8 q^{44} -539203. q^{45} -83263.2 q^{46} -743538. q^{47} -806683. q^{48} -793584. q^{49} -67443.4 q^{50} -369899. q^{51} -19590.5 q^{52} -12693.7 q^{53} -10949.9 q^{54} -164624. q^{55} +143047. q^{56} +919423. q^{57} -113702. q^{58} -830705. q^{59} -1.86987e6 q^{60} +2.22282e6 q^{61} +76729.7 q^{62} -386973. q^{63} -1.05668e6 q^{64} -40527.2 q^{65} -153390. q^{66} +3.88681e6 q^{67} -648441. q^{68} +1.63873e6 q^{69} +141055. q^{70} +3.88756e6 q^{71} -1.84772e6 q^{72} +5.22595e6 q^{73} +614232. q^{74} +1.32737e6 q^{75} +1.61177e6 q^{76} -118147. q^{77} -37761.6 q^{78} -2.64353e6 q^{79} -2.92544e6 q^{80} -4.67403e6 q^{81} +1.13243e6 q^{82} +6.76661e6 q^{83} -1.34196e6 q^{84} -1.34144e6 q^{85} -268657. q^{86} +2.23781e6 q^{87} -564127. q^{88} +327679. q^{89} -1.82198e6 q^{90} -29085.4 q^{91} +2.87272e6 q^{92} -1.51014e6 q^{93} -2.51244e6 q^{94} +3.33429e6 q^{95} -9.76093e6 q^{96} +5.05740e6 q^{97} -2.68154e6 q^{98} +1.52608e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + O(q^{10}) \) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + 4667q^{10} + 1620q^{11} - 19681q^{12} + 13550q^{13} + 44160q^{14} + 31412q^{15} + 114026q^{16} + 110880q^{17} + 159267q^{18} + 105058q^{19} + 167251q^{20} + 129840q^{21} + 201504q^{22} + 160184q^{23} + 161289q^{24} + 270149q^{25} + 272104q^{26} + 252544q^{27} + 208172q^{28} + 285546q^{29} + 107580q^{30} - 99616q^{31} + 200126q^{32} + 531468q^{33} - 80941q^{34} - 187104q^{35} - 608975q^{36} + 176038q^{37} + 652165q^{38} - 794680q^{39} - 895387q^{40} - 410260q^{41} - 3413218q^{42} - 1033591q^{43} - 2177076q^{44} - 1051178q^{45} - 3975765q^{46} - 424556q^{47} - 2360477q^{48} - 1561359q^{49} - 4063801q^{50} - 2375738q^{51} - 4172312q^{52} + 3992458q^{53} - 10438626q^{54} + 406960q^{55} + 1559556q^{56} - 3116152q^{57} - 4052005q^{58} + 2248836q^{59} - 2911436q^{60} + 6210394q^{61} + 885317q^{62} + 11622368q^{63} - 3096318q^{64} + 5600420q^{65} - 2174604q^{66} - 1993648q^{67} + 9327135q^{68} + 13366240q^{69} - 1105098q^{70} + 4978064q^{71} + 11370663q^{72} + 8224814q^{73} - 3613563q^{74} + 27115592q^{75} + 10687121q^{76} + 17261892q^{77} - 15226630q^{78} + 6945708q^{79} + 15822799q^{80} + 35113185q^{81} - 508449q^{82} + 22937328q^{83} - 14010106q^{84} - 575532q^{85} - 1272112q^{86} + 9081380q^{87} + 11202656q^{88} + 9291302q^{89} + 2841402q^{90} + 25581108q^{91} - 14388137q^{92} + 25930480q^{93} - 24645805q^{94} + 30750464q^{95} - 22461255q^{96} + 10001852q^{97} - 32304856q^{98} + 5055452q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.37903 0.298667 0.149333 0.988787i \(-0.452287\pi\)
0.149333 + 0.988787i \(0.452287\pi\)
\(3\) −66.5036 −1.42207 −0.711035 0.703157i \(-0.751773\pi\)
−0.711035 + 0.703157i \(0.751773\pi\)
\(4\) −116.582 −0.910798
\(5\) −241.175 −0.862856 −0.431428 0.902147i \(-0.641990\pi\)
−0.431428 + 0.902147i \(0.641990\pi\)
\(6\) −224.718 −0.424725
\(7\) −173.086 −0.190730 −0.0953650 0.995442i \(-0.530402\pi\)
−0.0953650 + 0.995442i \(0.530402\pi\)
\(8\) −826.450 −0.570692
\(9\) 2235.73 1.02228
\(10\) −814.939 −0.257706
\(11\) 682.590 0.154627 0.0773135 0.997007i \(-0.475366\pi\)
0.0773135 + 0.997007i \(0.475366\pi\)
\(12\) 7753.13 1.29522
\(13\) 168.040 0.0212135 0.0106067 0.999944i \(-0.496624\pi\)
0.0106067 + 0.999944i \(0.496624\pi\)
\(14\) −584.863 −0.0569647
\(15\) 16039.0 1.22704
\(16\) 12129.9 0.740351
\(17\) 5562.09 0.274579 0.137289 0.990531i \(-0.456161\pi\)
0.137289 + 0.990531i \(0.456161\pi\)
\(18\) 7554.59 0.305321
\(19\) −13825.2 −0.462416 −0.231208 0.972904i \(-0.574268\pi\)
−0.231208 + 0.972904i \(0.574268\pi\)
\(20\) 28116.8 0.785887
\(21\) 11510.8 0.271231
\(22\) 2306.49 0.0461820
\(23\) −24641.2 −0.422293 −0.211146 0.977454i \(-0.567720\pi\)
−0.211146 + 0.977454i \(0.567720\pi\)
\(24\) 54961.9 0.811564
\(25\) −19959.4 −0.255480
\(26\) 567.813 0.00633576
\(27\) −3240.56 −0.0316845
\(28\) 20178.7 0.173716
\(29\) −33649.4 −0.256203 −0.128102 0.991761i \(-0.540888\pi\)
−0.128102 + 0.991761i \(0.540888\pi\)
\(30\) 54196.4 0.366476
\(31\) 22707.6 0.136901 0.0684503 0.997655i \(-0.478195\pi\)
0.0684503 + 0.997655i \(0.478195\pi\)
\(32\) 146773. 0.791811
\(33\) −45394.7 −0.219890
\(34\) 18794.5 0.0820076
\(35\) 41744.1 0.164572
\(36\) −260646. −0.931091
\(37\) 181778. 0.589976 0.294988 0.955501i \(-0.404684\pi\)
0.294988 + 0.955501i \(0.404684\pi\)
\(38\) −46715.6 −0.138108
\(39\) −11175.3 −0.0301670
\(40\) 199320. 0.492425
\(41\) 335134. 0.759407 0.379704 0.925108i \(-0.376026\pi\)
0.379704 + 0.925108i \(0.376026\pi\)
\(42\) 38895.5 0.0810078
\(43\) −79507.0 −0.152499
\(44\) −79577.8 −0.140834
\(45\) −539203. −0.882080
\(46\) −83263.2 −0.126125
\(47\) −743538. −1.04463 −0.522313 0.852754i \(-0.674931\pi\)
−0.522313 + 0.852754i \(0.674931\pi\)
\(48\) −806683. −1.05283
\(49\) −793584. −0.963622
\(50\) −67443.4 −0.0763035
\(51\) −369899. −0.390470
\(52\) −19590.5 −0.0193212
\(53\) −12693.7 −0.0117118 −0.00585591 0.999983i \(-0.501864\pi\)
−0.00585591 + 0.999983i \(0.501864\pi\)
\(54\) −10949.9 −0.00946310
\(55\) −164624. −0.133421
\(56\) 143047. 0.108848
\(57\) 919423. 0.657587
\(58\) −113702. −0.0765194
\(59\) −830705. −0.526581 −0.263290 0.964717i \(-0.584808\pi\)
−0.263290 + 0.964717i \(0.584808\pi\)
\(60\) −1.86987e6 −1.11759
\(61\) 2.22282e6 1.25386 0.626931 0.779075i \(-0.284311\pi\)
0.626931 + 0.779075i \(0.284311\pi\)
\(62\) 76729.7 0.0408877
\(63\) −386973. −0.194980
\(64\) −1.05668e6 −0.503864
\(65\) −40527.2 −0.0183042
\(66\) −153390. −0.0656739
\(67\) 3.88681e6 1.57882 0.789409 0.613868i \(-0.210387\pi\)
0.789409 + 0.613868i \(0.210387\pi\)
\(68\) −648441. −0.250086
\(69\) 1.63873e6 0.600530
\(70\) 141055. 0.0491523
\(71\) 3.88756e6 1.28906 0.644530 0.764579i \(-0.277053\pi\)
0.644530 + 0.764579i \(0.277053\pi\)
\(72\) −1.84772e6 −0.583407
\(73\) 5.22595e6 1.57230 0.786150 0.618035i \(-0.212071\pi\)
0.786150 + 0.618035i \(0.212071\pi\)
\(74\) 614232. 0.176206
\(75\) 1.32737e6 0.363311
\(76\) 1.61177e6 0.421167
\(77\) −118147. −0.0294920
\(78\) −37761.6 −0.00900989
\(79\) −2.64353e6 −0.603240 −0.301620 0.953428i \(-0.597527\pi\)
−0.301620 + 0.953428i \(0.597527\pi\)
\(80\) −2.92544e6 −0.638816
\(81\) −4.67403e6 −0.977223
\(82\) 1.13243e6 0.226810
\(83\) 6.76661e6 1.29897 0.649483 0.760376i \(-0.274985\pi\)
0.649483 + 0.760376i \(0.274985\pi\)
\(84\) −1.34196e6 −0.247037
\(85\) −1.34144e6 −0.236922
\(86\) −268657. −0.0455463
\(87\) 2.23781e6 0.364339
\(88\) −564127. −0.0882444
\(89\) 327679. 0.0492701 0.0246350 0.999697i \(-0.492158\pi\)
0.0246350 + 0.999697i \(0.492158\pi\)
\(90\) −1.82198e6 −0.263448
\(91\) −29085.4 −0.00404604
\(92\) 2.87272e6 0.384624
\(93\) −1.51014e6 −0.194682
\(94\) −2.51244e6 −0.311995
\(95\) 3.33429e6 0.398998
\(96\) −9.76093e6 −1.12601
\(97\) 5.05740e6 0.562634 0.281317 0.959615i \(-0.409229\pi\)
0.281317 + 0.959615i \(0.409229\pi\)
\(98\) −2.68154e6 −0.287802
\(99\) 1.52608e6 0.158072
\(100\) 2.32691e6 0.232691
\(101\) 1.31064e7 1.26578 0.632889 0.774242i \(-0.281869\pi\)
0.632889 + 0.774242i \(0.281869\pi\)
\(102\) −1.24990e6 −0.116620
\(103\) −1.35656e7 −1.22324 −0.611618 0.791153i \(-0.709481\pi\)
−0.611618 + 0.791153i \(0.709481\pi\)
\(104\) −138877. −0.0121064
\(105\) −2.77613e6 −0.234033
\(106\) −42892.6 −0.00349793
\(107\) −1.06605e7 −0.841269 −0.420635 0.907230i \(-0.638193\pi\)
−0.420635 + 0.907230i \(0.638193\pi\)
\(108\) 377791. 0.0288581
\(109\) 9.24482e6 0.683763 0.341881 0.939743i \(-0.388936\pi\)
0.341881 + 0.939743i \(0.388936\pi\)
\(110\) −556269. −0.0398484
\(111\) −1.20889e7 −0.838987
\(112\) −2.09952e6 −0.141207
\(113\) −4.05844e6 −0.264597 −0.132299 0.991210i \(-0.542236\pi\)
−0.132299 + 0.991210i \(0.542236\pi\)
\(114\) 3.10676e6 0.196399
\(115\) 5.94285e6 0.364378
\(116\) 3.92292e6 0.233349
\(117\) 375692. 0.0216861
\(118\) −2.80698e6 −0.157272
\(119\) −962721. −0.0523704
\(120\) −1.32555e7 −0.700262
\(121\) −1.90212e7 −0.976090
\(122\) 7.51098e6 0.374487
\(123\) −2.22876e7 −1.07993
\(124\) −2.64730e6 −0.124689
\(125\) 2.36555e7 1.08330
\(126\) −1.30759e6 −0.0582339
\(127\) −2.45396e7 −1.06305 −0.531525 0.847043i \(-0.678381\pi\)
−0.531525 + 0.847043i \(0.678381\pi\)
\(128\) −2.23575e7 −0.942298
\(129\) 5.28750e6 0.216864
\(130\) −136943. −0.00546684
\(131\) −2.67897e7 −1.04116 −0.520581 0.853812i \(-0.674285\pi\)
−0.520581 + 0.853812i \(0.674285\pi\)
\(132\) 5.29221e6 0.200276
\(133\) 2.39294e6 0.0881965
\(134\) 1.31337e7 0.471540
\(135\) 781543. 0.0273391
\(136\) −4.59679e6 −0.156700
\(137\) 4.80765e7 1.59739 0.798695 0.601736i \(-0.205524\pi\)
0.798695 + 0.601736i \(0.205524\pi\)
\(138\) 5.53730e6 0.179358
\(139\) −5.94300e6 −0.187696 −0.0938478 0.995587i \(-0.529917\pi\)
−0.0938478 + 0.995587i \(0.529917\pi\)
\(140\) −4.86662e6 −0.149892
\(141\) 4.94480e7 1.48553
\(142\) 1.31362e7 0.385000
\(143\) 114703. 0.00328017
\(144\) 2.71192e7 0.756847
\(145\) 8.11541e6 0.221066
\(146\) 1.76587e7 0.469594
\(147\) 5.27762e7 1.37034
\(148\) −2.11920e7 −0.537349
\(149\) 4.65127e7 1.15191 0.575956 0.817481i \(-0.304630\pi\)
0.575956 + 0.817481i \(0.304630\pi\)
\(150\) 4.48523e6 0.108509
\(151\) −1.54586e7 −0.365385 −0.182693 0.983170i \(-0.558481\pi\)
−0.182693 + 0.983170i \(0.558481\pi\)
\(152\) 1.14258e7 0.263897
\(153\) 1.24353e7 0.280697
\(154\) −399221. −0.00880828
\(155\) −5.47652e6 −0.118125
\(156\) 1.30284e6 0.0274761
\(157\) 2.72583e6 0.0562147 0.0281074 0.999605i \(-0.491052\pi\)
0.0281074 + 0.999605i \(0.491052\pi\)
\(158\) −8.93258e6 −0.180168
\(159\) 844180. 0.0166550
\(160\) −3.53980e7 −0.683218
\(161\) 4.26504e6 0.0805439
\(162\) −1.57937e7 −0.291864
\(163\) −5.29222e7 −0.957152 −0.478576 0.878046i \(-0.658847\pi\)
−0.478576 + 0.878046i \(0.658847\pi\)
\(164\) −3.90706e7 −0.691667
\(165\) 1.09481e7 0.189734
\(166\) 2.28646e7 0.387958
\(167\) 3.70870e7 0.616189 0.308095 0.951356i \(-0.400309\pi\)
0.308095 + 0.951356i \(0.400309\pi\)
\(168\) −9.51314e6 −0.154789
\(169\) −6.27203e7 −0.999550
\(170\) −4.53277e6 −0.0707607
\(171\) −3.09093e7 −0.472719
\(172\) 9.26910e6 0.138895
\(173\) −3.93055e7 −0.577154 −0.288577 0.957457i \(-0.593182\pi\)
−0.288577 + 0.957457i \(0.593182\pi\)
\(174\) 7.56162e6 0.108816
\(175\) 3.45469e6 0.0487277
\(176\) 8.27975e6 0.114478
\(177\) 5.52449e7 0.748834
\(178\) 1.10724e6 0.0147153
\(179\) −4.76706e7 −0.621248 −0.310624 0.950533i \(-0.600538\pi\)
−0.310624 + 0.950533i \(0.600538\pi\)
\(180\) 6.28614e7 0.803397
\(181\) 2.04178e7 0.255938 0.127969 0.991778i \(-0.459154\pi\)
0.127969 + 0.991778i \(0.459154\pi\)
\(182\) −98280.5 −0.00120842
\(183\) −1.47826e8 −1.78308
\(184\) 2.03647e7 0.240999
\(185\) −4.38403e7 −0.509064
\(186\) −5.10280e6 −0.0581451
\(187\) 3.79663e6 0.0424573
\(188\) 8.66833e7 0.951444
\(189\) 560895. 0.00604318
\(190\) 1.12667e7 0.119167
\(191\) 7.99910e7 0.830662 0.415331 0.909670i \(-0.363666\pi\)
0.415331 + 0.909670i \(0.363666\pi\)
\(192\) 7.02729e7 0.716529
\(193\) −3.82721e7 −0.383205 −0.191603 0.981473i \(-0.561368\pi\)
−0.191603 + 0.981473i \(0.561368\pi\)
\(194\) 1.70891e7 0.168040
\(195\) 2.69520e6 0.0260298
\(196\) 9.25178e7 0.877665
\(197\) 1.23397e8 1.14993 0.574967 0.818177i \(-0.305015\pi\)
0.574967 + 0.818177i \(0.305015\pi\)
\(198\) 5.15669e6 0.0472109
\(199\) −7.95502e7 −0.715576 −0.357788 0.933803i \(-0.616469\pi\)
−0.357788 + 0.933803i \(0.616469\pi\)
\(200\) 1.64954e7 0.145801
\(201\) −2.58487e8 −2.24519
\(202\) 4.42868e7 0.378046
\(203\) 5.82424e6 0.0488656
\(204\) 4.31236e7 0.355639
\(205\) −8.08260e7 −0.655259
\(206\) −4.58387e7 −0.365340
\(207\) −5.50909e7 −0.431702
\(208\) 2.03831e6 0.0157054
\(209\) −9.43691e6 −0.0715019
\(210\) −9.38064e6 −0.0698980
\(211\) −2.30074e8 −1.68608 −0.843041 0.537849i \(-0.819237\pi\)
−0.843041 + 0.537849i \(0.819237\pi\)
\(212\) 1.47986e6 0.0106671
\(213\) −2.58537e8 −1.83313
\(214\) −3.60222e7 −0.251259
\(215\) 1.91751e7 0.131584
\(216\) 2.67816e6 0.0180821
\(217\) −3.93037e6 −0.0261110
\(218\) 3.12385e7 0.204217
\(219\) −3.47545e8 −2.23592
\(220\) 1.91922e7 0.121519
\(221\) 934655. 0.00582477
\(222\) −4.08486e7 −0.250578
\(223\) 1.94009e8 1.17153 0.585767 0.810479i \(-0.300793\pi\)
0.585767 + 0.810479i \(0.300793\pi\)
\(224\) −2.54044e7 −0.151022
\(225\) −4.46238e7 −0.261172
\(226\) −1.37136e7 −0.0790264
\(227\) 2.48770e8 1.41159 0.705793 0.708418i \(-0.250591\pi\)
0.705793 + 0.708418i \(0.250591\pi\)
\(228\) −1.07188e8 −0.598929
\(229\) 2.61570e8 1.43934 0.719670 0.694316i \(-0.244293\pi\)
0.719670 + 0.694316i \(0.244293\pi\)
\(230\) 2.00811e7 0.108828
\(231\) 7.85718e6 0.0419397
\(232\) 2.78096e7 0.146213
\(233\) 1.70926e8 0.885244 0.442622 0.896708i \(-0.354048\pi\)
0.442622 + 0.896708i \(0.354048\pi\)
\(234\) 1.26948e6 0.00647692
\(235\) 1.79323e8 0.901362
\(236\) 9.68454e7 0.479609
\(237\) 1.75804e8 0.857848
\(238\) −3.25306e6 −0.0156413
\(239\) 8.66501e7 0.410560 0.205280 0.978703i \(-0.434190\pi\)
0.205280 + 0.978703i \(0.434190\pi\)
\(240\) 1.94552e8 0.908441
\(241\) −1.19315e8 −0.549079 −0.274540 0.961576i \(-0.588525\pi\)
−0.274540 + 0.961576i \(0.588525\pi\)
\(242\) −6.42733e7 −0.291526
\(243\) 3.17927e8 1.42136
\(244\) −2.59141e8 −1.14202
\(245\) 1.91393e8 0.831467
\(246\) −7.53105e7 −0.322539
\(247\) −2.32318e6 −0.00980944
\(248\) −1.87667e7 −0.0781281
\(249\) −4.50004e8 −1.84722
\(250\) 7.99328e7 0.323545
\(251\) 4.55167e8 1.81682 0.908412 0.418077i \(-0.137296\pi\)
0.908412 + 0.418077i \(0.137296\pi\)
\(252\) 4.51142e7 0.177587
\(253\) −1.68198e7 −0.0652979
\(254\) −8.29199e7 −0.317498
\(255\) 8.92106e7 0.336919
\(256\) 5.97082e7 0.222431
\(257\) 5.12052e8 1.88169 0.940845 0.338837i \(-0.110033\pi\)
0.940845 + 0.338837i \(0.110033\pi\)
\(258\) 1.78666e7 0.0647699
\(259\) −3.14632e7 −0.112526
\(260\) 4.72475e6 0.0166714
\(261\) −7.52309e7 −0.261912
\(262\) −9.05232e7 −0.310961
\(263\) 9.38421e7 0.318092 0.159046 0.987271i \(-0.449158\pi\)
0.159046 + 0.987271i \(0.449158\pi\)
\(264\) 3.75164e7 0.125490
\(265\) 3.06142e6 0.0101056
\(266\) 8.08582e6 0.0263414
\(267\) −2.17918e7 −0.0700655
\(268\) −4.53133e8 −1.43798
\(269\) −5.02612e6 −0.0157434 −0.00787172 0.999969i \(-0.502506\pi\)
−0.00787172 + 0.999969i \(0.502506\pi\)
\(270\) 2.64086e6 0.00816529
\(271\) −4.90350e8 −1.49663 −0.748314 0.663345i \(-0.769136\pi\)
−0.748314 + 0.663345i \(0.769136\pi\)
\(272\) 6.74677e7 0.203285
\(273\) 1.93428e6 0.00575375
\(274\) 1.62452e8 0.477088
\(275\) −1.36241e7 −0.0395041
\(276\) −1.91046e8 −0.546961
\(277\) 5.00692e7 0.141544 0.0707721 0.997493i \(-0.477454\pi\)
0.0707721 + 0.997493i \(0.477454\pi\)
\(278\) −2.00816e7 −0.0560584
\(279\) 5.07680e7 0.139951
\(280\) −3.44994e7 −0.0939202
\(281\) 5.08518e8 1.36721 0.683604 0.729853i \(-0.260412\pi\)
0.683604 + 0.729853i \(0.260412\pi\)
\(282\) 1.67086e8 0.443679
\(283\) 9.23331e7 0.242161 0.121081 0.992643i \(-0.461364\pi\)
0.121081 + 0.992643i \(0.461364\pi\)
\(284\) −4.53220e8 −1.17407
\(285\) −2.21742e8 −0.567403
\(286\) 387583. 0.000979679 0
\(287\) −5.80070e7 −0.144842
\(288\) 3.28144e8 0.809452
\(289\) −3.79402e8 −0.924606
\(290\) 2.74222e7 0.0660252
\(291\) −3.36335e8 −0.800105
\(292\) −6.09253e8 −1.43205
\(293\) −2.46782e8 −0.573162 −0.286581 0.958056i \(-0.592519\pi\)
−0.286581 + 0.958056i \(0.592519\pi\)
\(294\) 1.78332e8 0.409274
\(295\) 2.00346e8 0.454363
\(296\) −1.50230e8 −0.336695
\(297\) −2.21197e6 −0.00489927
\(298\) 1.57168e8 0.344038
\(299\) −4.14071e6 −0.00895830
\(300\) −1.54748e8 −0.330903
\(301\) 1.37616e7 0.0290860
\(302\) −5.22351e7 −0.109128
\(303\) −8.71621e8 −1.80002
\(304\) −1.67698e8 −0.342350
\(305\) −5.36090e8 −1.08190
\(306\) 4.20193e7 0.0838348
\(307\) 3.84464e8 0.758352 0.379176 0.925325i \(-0.376207\pi\)
0.379176 + 0.925325i \(0.376207\pi\)
\(308\) 1.37738e7 0.0268613
\(309\) 9.02164e8 1.73953
\(310\) −1.85053e7 −0.0352802
\(311\) −5.62608e8 −1.06058 −0.530291 0.847815i \(-0.677917\pi\)
−0.530291 + 0.847815i \(0.677917\pi\)
\(312\) 9.23581e6 0.0172161
\(313\) −2.64846e8 −0.488188 −0.244094 0.969752i \(-0.578491\pi\)
−0.244094 + 0.969752i \(0.578491\pi\)
\(314\) 9.21066e6 0.0167895
\(315\) 9.33285e7 0.168239
\(316\) 3.08189e8 0.549429
\(317\) 9.81092e7 0.172983 0.0864913 0.996253i \(-0.472435\pi\)
0.0864913 + 0.996253i \(0.472435\pi\)
\(318\) 2.85251e6 0.00497430
\(319\) −2.29687e7 −0.0396159
\(320\) 2.54845e8 0.434762
\(321\) 7.08963e8 1.19634
\(322\) 1.44117e7 0.0240558
\(323\) −7.68968e7 −0.126970
\(324\) 5.44908e8 0.890053
\(325\) −3.35398e6 −0.00541962
\(326\) −1.78826e8 −0.285870
\(327\) −6.14813e8 −0.972358
\(328\) −2.76971e8 −0.433388
\(329\) 1.28696e8 0.199241
\(330\) 3.69939e7 0.0566671
\(331\) 6.54036e8 0.991296 0.495648 0.868523i \(-0.334931\pi\)
0.495648 + 0.868523i \(0.334931\pi\)
\(332\) −7.88866e8 −1.18310
\(333\) 4.06405e8 0.603121
\(334\) 1.25318e8 0.184035
\(335\) −9.37404e8 −1.36229
\(336\) 1.39626e8 0.200806
\(337\) 6.62977e8 0.943613 0.471806 0.881702i \(-0.343602\pi\)
0.471806 + 0.881702i \(0.343602\pi\)
\(338\) −2.11934e8 −0.298532
\(339\) 2.69901e8 0.376275
\(340\) 1.56388e8 0.215788
\(341\) 1.55000e7 0.0211685
\(342\) −1.04443e8 −0.141185
\(343\) 2.79902e8 0.374522
\(344\) 6.57086e7 0.0870297
\(345\) −3.95221e8 −0.518170
\(346\) −1.32814e8 −0.172377
\(347\) −5.98992e8 −0.769605 −0.384802 0.922999i \(-0.625730\pi\)
−0.384802 + 0.922999i \(0.625730\pi\)
\(348\) −2.60888e8 −0.331839
\(349\) 1.09709e9 1.38151 0.690753 0.723091i \(-0.257279\pi\)
0.690753 + 0.723091i \(0.257279\pi\)
\(350\) 1.16735e7 0.0145534
\(351\) −544544. −0.000672137 0
\(352\) 1.00186e8 0.122435
\(353\) 1.20482e9 1.45784 0.728921 0.684598i \(-0.240022\pi\)
0.728921 + 0.684598i \(0.240022\pi\)
\(354\) 1.86674e8 0.223652
\(355\) −9.37585e8 −1.11227
\(356\) −3.82015e7 −0.0448751
\(357\) 6.40244e7 0.0744743
\(358\) −1.61080e8 −0.185546
\(359\) −5.38564e8 −0.614338 −0.307169 0.951655i \(-0.599382\pi\)
−0.307169 + 0.951655i \(0.599382\pi\)
\(360\) 4.45624e8 0.503396
\(361\) −7.02737e8 −0.786172
\(362\) 6.89924e7 0.0764401
\(363\) 1.26498e9 1.38807
\(364\) 3.39084e6 0.00368513
\(365\) −1.26037e9 −1.35667
\(366\) −4.99507e8 −0.532547
\(367\) 1.10179e9 1.16350 0.581750 0.813368i \(-0.302368\pi\)
0.581750 + 0.813368i \(0.302368\pi\)
\(368\) −2.98895e8 −0.312645
\(369\) 7.49268e8 0.776327
\(370\) −1.48138e8 −0.152041
\(371\) 2.19711e6 0.00223379
\(372\) 1.76055e8 0.177316
\(373\) 1.71818e9 1.71431 0.857154 0.515060i \(-0.172231\pi\)
0.857154 + 0.515060i \(0.172231\pi\)
\(374\) 1.28289e7 0.0126806
\(375\) −1.57318e9 −1.54052
\(376\) 6.14498e8 0.596160
\(377\) −5.65446e6 −0.00543496
\(378\) 1.89528e6 0.00180490
\(379\) 1.07347e8 0.101287 0.0506433 0.998717i \(-0.483873\pi\)
0.0506433 + 0.998717i \(0.483873\pi\)
\(380\) −3.88719e8 −0.363407
\(381\) 1.63197e9 1.51173
\(382\) 2.70292e8 0.248091
\(383\) −6.46792e8 −0.588259 −0.294130 0.955766i \(-0.595030\pi\)
−0.294130 + 0.955766i \(0.595030\pi\)
\(384\) 1.48685e9 1.34001
\(385\) 2.84941e7 0.0254473
\(386\) −1.29322e8 −0.114451
\(387\) −1.77756e8 −0.155896
\(388\) −5.89603e8 −0.512446
\(389\) −5.05273e7 −0.0435214 −0.0217607 0.999763i \(-0.506927\pi\)
−0.0217607 + 0.999763i \(0.506927\pi\)
\(390\) 9.10717e6 0.00777423
\(391\) −1.37056e8 −0.115953
\(392\) 6.55858e8 0.549932
\(393\) 1.78161e9 1.48060
\(394\) 4.16962e8 0.343447
\(395\) 6.37555e8 0.520509
\(396\) −1.77914e8 −0.143972
\(397\) −1.12623e9 −0.903359 −0.451679 0.892180i \(-0.649175\pi\)
−0.451679 + 0.892180i \(0.649175\pi\)
\(398\) −2.68803e8 −0.213719
\(399\) −1.59139e8 −0.125422
\(400\) −2.42106e8 −0.189145
\(401\) −90676.9 −7.02250e−5 0 −3.51125e−5 1.00000i \(-0.500011\pi\)
−3.51125e−5 1.00000i \(0.500011\pi\)
\(402\) −8.73435e8 −0.670563
\(403\) 3.81579e6 0.00290413
\(404\) −1.52797e9 −1.15287
\(405\) 1.12726e9 0.843202
\(406\) 1.96803e7 0.0145945
\(407\) 1.24080e8 0.0912262
\(408\) 3.05703e8 0.222838
\(409\) −3.28251e8 −0.237233 −0.118616 0.992940i \(-0.537846\pi\)
−0.118616 + 0.992940i \(0.537846\pi\)
\(410\) −2.73114e8 −0.195704
\(411\) −3.19726e9 −2.27160
\(412\) 1.58151e9 1.11412
\(413\) 1.43783e8 0.100435
\(414\) −1.86154e8 −0.128935
\(415\) −1.63194e9 −1.12082
\(416\) 2.46638e7 0.0167970
\(417\) 3.95231e8 0.266916
\(418\) −3.18876e7 −0.0213553
\(419\) 2.00896e9 1.33421 0.667103 0.744965i \(-0.267534\pi\)
0.667103 + 0.744965i \(0.267534\pi\)
\(420\) 3.23648e8 0.213157
\(421\) −1.17015e9 −0.764283 −0.382141 0.924104i \(-0.624813\pi\)
−0.382141 + 0.924104i \(0.624813\pi\)
\(422\) −7.77427e8 −0.503577
\(423\) −1.66235e9 −1.06790
\(424\) 1.04908e7 0.00668384
\(425\) −1.11016e8 −0.0701495
\(426\) −8.73604e8 −0.547496
\(427\) −3.84739e8 −0.239149
\(428\) 1.24283e9 0.766227
\(429\) −7.62813e6 −0.00466463
\(430\) 6.47934e7 0.0392999
\(431\) −2.84974e9 −1.71449 −0.857245 0.514909i \(-0.827826\pi\)
−0.857245 + 0.514909i \(0.827826\pi\)
\(432\) −3.93077e7 −0.0234576
\(433\) 1.92069e9 1.13697 0.568486 0.822693i \(-0.307529\pi\)
0.568486 + 0.822693i \(0.307529\pi\)
\(434\) −1.32808e7 −0.00779850
\(435\) −5.39704e8 −0.314372
\(436\) −1.07778e9 −0.622770
\(437\) 3.40668e8 0.195275
\(438\) −1.17436e9 −0.667795
\(439\) 7.86877e8 0.443896 0.221948 0.975059i \(-0.428759\pi\)
0.221948 + 0.975059i \(0.428759\pi\)
\(440\) 1.36053e8 0.0761422
\(441\) −1.77424e9 −0.985092
\(442\) 3.15823e6 0.00173966
\(443\) −1.97285e9 −1.07816 −0.539078 0.842256i \(-0.681227\pi\)
−0.539078 + 0.842256i \(0.681227\pi\)
\(444\) 1.40935e9 0.764148
\(445\) −7.90281e7 −0.0425130
\(446\) 6.55562e8 0.349898
\(447\) −3.09326e9 −1.63810
\(448\) 1.82896e8 0.0961019
\(449\) 5.76219e7 0.0300417 0.0150209 0.999887i \(-0.495219\pi\)
0.0150209 + 0.999887i \(0.495219\pi\)
\(450\) −1.50785e8 −0.0780036
\(451\) 2.28759e8 0.117425
\(452\) 4.73142e8 0.240995
\(453\) 1.02805e9 0.519603
\(454\) 8.40601e8 0.421594
\(455\) 7.01469e6 0.00349115
\(456\) −7.59857e8 −0.375280
\(457\) 4.53691e8 0.222358 0.111179 0.993800i \(-0.464537\pi\)
0.111179 + 0.993800i \(0.464537\pi\)
\(458\) 8.83852e8 0.429883
\(459\) −1.80243e7 −0.00869988
\(460\) −6.92830e8 −0.331875
\(461\) 2.09843e9 0.997563 0.498781 0.866728i \(-0.333781\pi\)
0.498781 + 0.866728i \(0.333781\pi\)
\(462\) 2.65497e7 0.0125260
\(463\) −2.12353e8 −0.0994318 −0.0497159 0.998763i \(-0.515832\pi\)
−0.0497159 + 0.998763i \(0.515832\pi\)
\(464\) −4.08165e8 −0.189680
\(465\) 3.64208e8 0.167983
\(466\) 5.77565e8 0.264393
\(467\) 1.04433e9 0.474491 0.237246 0.971450i \(-0.423755\pi\)
0.237246 + 0.971450i \(0.423755\pi\)
\(468\) −4.37990e7 −0.0197517
\(469\) −6.72753e8 −0.301128
\(470\) 6.05939e8 0.269207
\(471\) −1.81277e8 −0.0799412
\(472\) 6.86537e8 0.300515
\(473\) −5.42707e7 −0.0235804
\(474\) 5.94048e8 0.256211
\(475\) 2.75942e8 0.118138
\(476\) 1.12236e8 0.0476989
\(477\) −2.83798e7 −0.0119728
\(478\) 2.92793e8 0.122621
\(479\) 1.22092e9 0.507589 0.253794 0.967258i \(-0.418321\pi\)
0.253794 + 0.967258i \(0.418321\pi\)
\(480\) 2.35410e9 0.971583
\(481\) 3.05460e7 0.0125154
\(482\) −4.03168e8 −0.163992
\(483\) −2.83641e8 −0.114539
\(484\) 2.21754e9 0.889021
\(485\) −1.21972e9 −0.485472
\(486\) 1.07428e9 0.424514
\(487\) −8.85642e8 −0.347462 −0.173731 0.984793i \(-0.555582\pi\)
−0.173731 + 0.984793i \(0.555582\pi\)
\(488\) −1.83705e9 −0.715569
\(489\) 3.51951e9 1.36114
\(490\) 6.46723e8 0.248332
\(491\) −1.70646e9 −0.650597 −0.325298 0.945611i \(-0.605465\pi\)
−0.325298 + 0.945611i \(0.605465\pi\)
\(492\) 2.59834e9 0.983598
\(493\) −1.87161e8 −0.0703480
\(494\) −7.85010e6 −0.00292975
\(495\) −3.68054e8 −0.136393
\(496\) 2.75441e8 0.101355
\(497\) −6.72883e8 −0.245862
\(498\) −1.52058e9 −0.551703
\(499\) −1.84137e9 −0.663422 −0.331711 0.943381i \(-0.607626\pi\)
−0.331711 + 0.943381i \(0.607626\pi\)
\(500\) −2.75781e9 −0.986666
\(501\) −2.46642e9 −0.876264
\(502\) 1.53802e9 0.542625
\(503\) −1.71136e9 −0.599587 −0.299794 0.954004i \(-0.596918\pi\)
−0.299794 + 0.954004i \(0.596918\pi\)
\(504\) 3.19814e8 0.111273
\(505\) −3.16093e9 −1.09218
\(506\) −5.68346e7 −0.0195023
\(507\) 4.17112e9 1.42143
\(508\) 2.86087e9 0.968224
\(509\) 2.79743e8 0.0940257 0.0470128 0.998894i \(-0.485030\pi\)
0.0470128 + 0.998894i \(0.485030\pi\)
\(510\) 3.01445e8 0.100627
\(511\) −9.04540e8 −0.299885
\(512\) 3.06352e9 1.00873
\(513\) 4.48012e7 0.0146514
\(514\) 1.73024e9 0.561999
\(515\) 3.27170e9 1.05548
\(516\) −6.16428e8 −0.197519
\(517\) −5.07532e8 −0.161527
\(518\) −1.06315e8 −0.0336078
\(519\) 2.61395e9 0.820753
\(520\) 3.34937e7 0.0104460
\(521\) −3.76777e9 −1.16722 −0.583609 0.812035i \(-0.698360\pi\)
−0.583609 + 0.812035i \(0.698360\pi\)
\(522\) −2.54208e8 −0.0782243
\(523\) 8.71991e8 0.266536 0.133268 0.991080i \(-0.457453\pi\)
0.133268 + 0.991080i \(0.457453\pi\)
\(524\) 3.12320e9 0.948288
\(525\) −2.29749e8 −0.0692942
\(526\) 3.17095e8 0.0950035
\(527\) 1.26302e8 0.0375900
\(528\) −5.50633e8 −0.162796
\(529\) −2.79764e9 −0.821669
\(530\) 1.03446e7 0.00301821
\(531\) −1.85723e9 −0.538313
\(532\) −2.78974e8 −0.0803292
\(533\) 5.63159e7 0.0161097
\(534\) −7.36352e7 −0.0209262
\(535\) 2.57106e9 0.725894
\(536\) −3.21226e9 −0.901018
\(537\) 3.17026e9 0.883457
\(538\) −1.69834e7 −0.00470205
\(539\) −5.41692e8 −0.149002
\(540\) −9.11139e7 −0.0249004
\(541\) −1.47997e9 −0.401848 −0.200924 0.979607i \(-0.564394\pi\)
−0.200924 + 0.979607i \(0.564394\pi\)
\(542\) −1.65691e9 −0.446993
\(543\) −1.35786e9 −0.363961
\(544\) 8.16365e8 0.217414
\(545\) −2.22962e9 −0.589989
\(546\) 6.53601e6 0.00171846
\(547\) 6.46090e9 1.68786 0.843932 0.536450i \(-0.180235\pi\)
0.843932 + 0.536450i \(0.180235\pi\)
\(548\) −5.60486e9 −1.45490
\(549\) 4.96962e9 1.28180
\(550\) −4.60362e7 −0.0117986
\(551\) 4.65209e8 0.118472
\(552\) −1.35433e9 −0.342718
\(553\) 4.57559e8 0.115056
\(554\) 1.69185e8 0.0422745
\(555\) 2.91554e9 0.723925
\(556\) 6.92848e8 0.170953
\(557\) 3.65569e9 0.896347 0.448174 0.893946i \(-0.352075\pi\)
0.448174 + 0.893946i \(0.352075\pi\)
\(558\) 1.71547e8 0.0417987
\(559\) −1.33604e7 −0.00323502
\(560\) 5.06352e8 0.121841
\(561\) −2.52489e8 −0.0603772
\(562\) 1.71830e9 0.408340
\(563\) 7.39782e9 1.74713 0.873564 0.486710i \(-0.161803\pi\)
0.873564 + 0.486710i \(0.161803\pi\)
\(564\) −5.76475e9 −1.35302
\(565\) 9.78797e8 0.228309
\(566\) 3.11996e8 0.0723256
\(567\) 8.09009e8 0.186386
\(568\) −3.21288e9 −0.735657
\(569\) −1.65447e9 −0.376502 −0.188251 0.982121i \(-0.560282\pi\)
−0.188251 + 0.982121i \(0.560282\pi\)
\(570\) −7.49274e8 −0.169464
\(571\) −1.49420e9 −0.335878 −0.167939 0.985797i \(-0.553711\pi\)
−0.167939 + 0.985797i \(0.553711\pi\)
\(572\) −1.33723e7 −0.00298758
\(573\) −5.31969e9 −1.18126
\(574\) −1.96007e8 −0.0432594
\(575\) 4.91823e8 0.107888
\(576\) −2.36245e9 −0.515090
\(577\) −5.15169e9 −1.11644 −0.558219 0.829694i \(-0.688515\pi\)
−0.558219 + 0.829694i \(0.688515\pi\)
\(578\) −1.28201e9 −0.276149
\(579\) 2.54523e9 0.544944
\(580\) −9.46113e8 −0.201347
\(581\) −1.17121e9 −0.247752
\(582\) −1.13649e9 −0.238965
\(583\) −8.66462e6 −0.00181096
\(584\) −4.31899e9 −0.897300
\(585\) −9.06077e7 −0.0187120
\(586\) −8.33885e8 −0.171184
\(587\) −7.65712e8 −0.156254 −0.0781272 0.996943i \(-0.524894\pi\)
−0.0781272 + 0.996943i \(0.524894\pi\)
\(588\) −6.15276e9 −1.24810
\(589\) −3.13936e8 −0.0633050
\(590\) 6.76974e8 0.135703
\(591\) −8.20634e9 −1.63528
\(592\) 2.20495e9 0.436790
\(593\) 1.51945e9 0.299222 0.149611 0.988745i \(-0.452198\pi\)
0.149611 + 0.988745i \(0.452198\pi\)
\(594\) −7.47431e6 −0.00146325
\(595\) 2.32185e8 0.0451881
\(596\) −5.42255e9 −1.04916
\(597\) 5.29038e9 1.01760
\(598\) −1.39916e7 −0.00267555
\(599\) −8.16357e9 −1.55198 −0.775991 0.630745i \(-0.782750\pi\)
−0.775991 + 0.630745i \(0.782750\pi\)
\(600\) −1.09701e9 −0.207338
\(601\) −8.29079e9 −1.55788 −0.778942 0.627096i \(-0.784243\pi\)
−0.778942 + 0.627096i \(0.784243\pi\)
\(602\) 4.65007e7 0.00868704
\(603\) 8.68985e9 1.61399
\(604\) 1.80220e9 0.332792
\(605\) 4.58746e9 0.842225
\(606\) −2.94523e9 −0.537608
\(607\) −5.48624e9 −0.995668 −0.497834 0.867272i \(-0.665871\pi\)
−0.497834 + 0.867272i \(0.665871\pi\)
\(608\) −2.02916e9 −0.366146
\(609\) −3.87333e8 −0.0694903
\(610\) −1.81146e9 −0.323128
\(611\) −1.24944e8 −0.0221601
\(612\) −1.44974e9 −0.255658
\(613\) −5.20382e8 −0.0912453 −0.0456227 0.998959i \(-0.514527\pi\)
−0.0456227 + 0.998959i \(0.514527\pi\)
\(614\) 1.29911e9 0.226495
\(615\) 5.37522e9 0.931823
\(616\) 9.76424e7 0.0168309
\(617\) 1.65572e9 0.283784 0.141892 0.989882i \(-0.454681\pi\)
0.141892 + 0.989882i \(0.454681\pi\)
\(618\) 3.04844e9 0.519539
\(619\) −4.63680e8 −0.0785781 −0.0392890 0.999228i \(-0.512509\pi\)
−0.0392890 + 0.999228i \(0.512509\pi\)
\(620\) 6.38464e8 0.107588
\(621\) 7.98511e7 0.0133801
\(622\) −1.90107e9 −0.316761
\(623\) −5.67166e7 −0.00939728
\(624\) −1.35555e8 −0.0223342
\(625\) −4.14581e9 −0.679250
\(626\) −8.94921e8 −0.145806
\(627\) 6.27588e8 0.101681
\(628\) −3.17783e8 −0.0512003
\(629\) 1.01106e9 0.161995
\(630\) 3.15360e8 0.0502475
\(631\) −5.00414e8 −0.0792915 −0.0396458 0.999214i \(-0.512623\pi\)
−0.0396458 + 0.999214i \(0.512623\pi\)
\(632\) 2.18475e9 0.344264
\(633\) 1.53007e10 2.39773
\(634\) 3.31514e8 0.0516642
\(635\) 5.91834e9 0.917258
\(636\) −9.84163e7 −0.0151694
\(637\) −1.33354e8 −0.0204418
\(638\) −7.76121e7 −0.0118320
\(639\) 8.69153e9 1.31778
\(640\) 5.39208e9 0.813067
\(641\) 6.72118e9 1.00796 0.503979 0.863716i \(-0.331869\pi\)
0.503979 + 0.863716i \(0.331869\pi\)
\(642\) 2.39561e9 0.357308
\(643\) −2.00229e9 −0.297021 −0.148511 0.988911i \(-0.547448\pi\)
−0.148511 + 0.988911i \(0.547448\pi\)
\(644\) −4.97228e8 −0.0733592
\(645\) −1.27522e9 −0.187122
\(646\) −2.59837e8 −0.0379216
\(647\) 1.06244e10 1.54219 0.771095 0.636720i \(-0.219709\pi\)
0.771095 + 0.636720i \(0.219709\pi\)
\(648\) 3.86285e9 0.557693
\(649\) −5.67031e8 −0.0814236
\(650\) −1.13332e7 −0.00161866
\(651\) 2.61384e8 0.0371317
\(652\) 6.16978e9 0.871772
\(653\) −2.49361e9 −0.350456 −0.175228 0.984528i \(-0.556066\pi\)
−0.175228 + 0.984528i \(0.556066\pi\)
\(654\) −2.07747e9 −0.290411
\(655\) 6.46102e9 0.898372
\(656\) 4.06514e9 0.562228
\(657\) 1.16838e10 1.60733
\(658\) 4.34868e8 0.0595068
\(659\) −7.14359e9 −0.972339 −0.486170 0.873864i \(-0.661606\pi\)
−0.486170 + 0.873864i \(0.661606\pi\)
\(660\) −1.27635e9 −0.172809
\(661\) −1.06348e10 −1.43227 −0.716134 0.697963i \(-0.754090\pi\)
−0.716134 + 0.697963i \(0.754090\pi\)
\(662\) 2.21001e9 0.296067
\(663\) −6.21579e7 −0.00828322
\(664\) −5.59227e9 −0.741310
\(665\) −5.77119e8 −0.0761009
\(666\) 1.37326e9 0.180132
\(667\) 8.29161e8 0.108193
\(668\) −4.32368e9 −0.561224
\(669\) −1.29023e10 −1.66600
\(670\) −3.16752e9 −0.406871
\(671\) 1.51727e9 0.193881
\(672\) 1.68948e9 0.214764
\(673\) 6.11664e9 0.773500 0.386750 0.922185i \(-0.373598\pi\)
0.386750 + 0.922185i \(0.373598\pi\)
\(674\) 2.24022e9 0.281826
\(675\) 6.46795e7 0.00809475
\(676\) 7.31207e9 0.910388
\(677\) 9.09572e9 1.12662 0.563309 0.826246i \(-0.309528\pi\)
0.563309 + 0.826246i \(0.309528\pi\)
\(678\) 9.12004e8 0.112381
\(679\) −8.75366e8 −0.107311
\(680\) 1.10863e9 0.135209
\(681\) −1.65441e10 −2.00737
\(682\) 5.23749e7 0.00632234
\(683\) 7.39960e9 0.888660 0.444330 0.895863i \(-0.353442\pi\)
0.444330 + 0.895863i \(0.353442\pi\)
\(684\) 3.60347e9 0.430551
\(685\) −1.15949e10 −1.37832
\(686\) 9.45798e8 0.111857
\(687\) −1.73953e10 −2.04684
\(688\) −9.64413e8 −0.112903
\(689\) −2.13306e6 −0.000248448 0
\(690\) −1.33546e9 −0.154760
\(691\) 7.73691e9 0.892061 0.446030 0.895018i \(-0.352837\pi\)
0.446030 + 0.895018i \(0.352837\pi\)
\(692\) 4.58232e9 0.525671
\(693\) −2.64144e8 −0.0301491
\(694\) −2.02401e9 −0.229855
\(695\) 1.43331e9 0.161954
\(696\) −1.84944e9 −0.207925
\(697\) 1.86405e9 0.208517
\(698\) 3.70710e9 0.412610
\(699\) −1.13672e10 −1.25888
\(700\) −4.02755e8 −0.0443811
\(701\) 9.43506e9 1.03450 0.517251 0.855833i \(-0.326955\pi\)
0.517251 + 0.855833i \(0.326955\pi\)
\(702\) −1.84003e6 −0.000200745 0
\(703\) −2.51311e9 −0.272814
\(704\) −7.21278e8 −0.0779109
\(705\) −1.19256e10 −1.28180
\(706\) 4.07112e9 0.435409
\(707\) −2.26853e9 −0.241422
\(708\) −6.44057e9 −0.682037
\(709\) −1.29011e10 −1.35945 −0.679726 0.733466i \(-0.737901\pi\)
−0.679726 + 0.733466i \(0.737901\pi\)
\(710\) −3.16813e9 −0.332199
\(711\) −5.91022e9 −0.616680
\(712\) −2.70810e8 −0.0281180
\(713\) −5.59542e8 −0.0578122
\(714\) 2.16340e8 0.0222430
\(715\) −2.76634e7 −0.00283032
\(716\) 5.55754e9 0.565831
\(717\) −5.76254e9 −0.583844
\(718\) −1.81983e9 −0.183482
\(719\) 1.56677e10 1.57201 0.786004 0.618222i \(-0.212147\pi\)
0.786004 + 0.618222i \(0.212147\pi\)
\(720\) −6.54048e9 −0.653049
\(721\) 2.34802e9 0.233308
\(722\) −2.37457e9 −0.234803
\(723\) 7.93487e9 0.780828
\(724\) −2.38035e9 −0.233108
\(725\) 6.71622e8 0.0654549
\(726\) 4.27441e9 0.414570
\(727\) 1.29468e10 1.24966 0.624830 0.780761i \(-0.285168\pi\)
0.624830 + 0.780761i \(0.285168\pi\)
\(728\) 2.40377e7 0.00230904
\(729\) −1.09212e10 −1.04405
\(730\) −4.25884e9 −0.405192
\(731\) −4.42225e8 −0.0418729
\(732\) 1.72338e10 1.62403
\(733\) 1.96470e10 1.84261 0.921305 0.388842i \(-0.127125\pi\)
0.921305 + 0.388842i \(0.127125\pi\)
\(734\) 3.72297e9 0.347499
\(735\) −1.27283e10 −1.18240
\(736\) −3.61666e9 −0.334376
\(737\) 2.65310e9 0.244128
\(738\) 2.53180e9 0.231863
\(739\) 5.46414e9 0.498042 0.249021 0.968498i \(-0.419891\pi\)
0.249021 + 0.968498i \(0.419891\pi\)
\(740\) 5.11100e9 0.463655
\(741\) 1.54500e8 0.0139497
\(742\) 7.42410e6 0.000667160 0
\(743\) 1.10547e10 0.988750 0.494375 0.869249i \(-0.335397\pi\)
0.494375 + 0.869249i \(0.335397\pi\)
\(744\) 1.24805e9 0.111104
\(745\) −1.12177e10 −0.993934
\(746\) 5.80580e9 0.512007
\(747\) 1.51283e10 1.32791
\(748\) −4.42619e8 −0.0386700
\(749\) 1.84519e9 0.160455
\(750\) −5.31582e9 −0.460104
\(751\) −1.34448e10 −1.15828 −0.579141 0.815227i \(-0.696612\pi\)
−0.579141 + 0.815227i \(0.696612\pi\)
\(752\) −9.01906e9 −0.773390
\(753\) −3.02702e10 −2.58365
\(754\) −1.91066e7 −0.00162324
\(755\) 3.72824e9 0.315275
\(756\) −6.53903e7 −0.00550411
\(757\) 4.89813e9 0.410388 0.205194 0.978721i \(-0.434217\pi\)
0.205194 + 0.978721i \(0.434217\pi\)
\(758\) 3.62728e8 0.0302509
\(759\) 1.11858e9 0.0928581
\(760\) −2.75562e9 −0.227705
\(761\) 1.08629e10 0.893512 0.446756 0.894656i \(-0.352579\pi\)
0.446756 + 0.894656i \(0.352579\pi\)
\(762\) 5.51447e9 0.451504
\(763\) −1.60015e9 −0.130414
\(764\) −9.32552e9 −0.756565
\(765\) −2.99910e9 −0.242201
\(766\) −2.18553e9 −0.175694
\(767\) −1.39592e8 −0.0111706
\(768\) −3.97081e9 −0.316312
\(769\) 1.74439e9 0.138325 0.0691627 0.997605i \(-0.477967\pi\)
0.0691627 + 0.997605i \(0.477967\pi\)
\(770\) 9.62824e7 0.00760028
\(771\) −3.40533e10 −2.67589
\(772\) 4.46184e9 0.349022
\(773\) −2.29156e10 −1.78444 −0.892221 0.451598i \(-0.850854\pi\)
−0.892221 + 0.451598i \(0.850854\pi\)
\(774\) −6.00643e8 −0.0465611
\(775\) −4.53230e8 −0.0349754
\(776\) −4.17969e9 −0.321091
\(777\) 2.09241e9 0.160020
\(778\) −1.70733e8 −0.0129984
\(779\) −4.63328e9 −0.351162
\(780\) −3.14213e8 −0.0237079
\(781\) 2.65361e9 0.199324
\(782\) −4.63118e8 −0.0346312
\(783\) 1.09043e8 0.00811766
\(784\) −9.62611e9 −0.713419
\(785\) −6.57403e8 −0.0485052
\(786\) 6.02012e9 0.442207
\(787\) 1.46509e10 1.07140 0.535701 0.844407i \(-0.320047\pi\)
0.535701 + 0.844407i \(0.320047\pi\)
\(788\) −1.43859e10 −1.04736
\(789\) −6.24084e9 −0.452349
\(790\) 2.15432e9 0.155459
\(791\) 7.02460e8 0.0504666
\(792\) −1.26123e9 −0.0902105
\(793\) 3.73523e8 0.0265988
\(794\) −3.80556e9 −0.269803
\(795\) −2.03595e8 −0.0143709
\(796\) 9.27414e9 0.651745
\(797\) −1.25977e10 −0.881426 −0.440713 0.897648i \(-0.645274\pi\)
−0.440713 + 0.897648i \(0.645274\pi\)
\(798\) −5.37736e8 −0.0374593
\(799\) −4.13563e9 −0.286832
\(800\) −2.92950e9 −0.202292
\(801\) 7.32600e8 0.0503678
\(802\) −306400. −2.09739e−5 0
\(803\) 3.56718e9 0.243120
\(804\) 3.01350e10 2.04491
\(805\) −1.02862e9 −0.0694978
\(806\) 1.28937e7 0.000867369 0
\(807\) 3.34255e8 0.0223883
\(808\) −1.08318e10 −0.722370
\(809\) 2.20916e10 1.46692 0.733462 0.679731i \(-0.237903\pi\)
0.733462 + 0.679731i \(0.237903\pi\)
\(810\) 3.80905e9 0.251837
\(811\) −7.33733e9 −0.483020 −0.241510 0.970398i \(-0.577643\pi\)
−0.241510 + 0.970398i \(0.577643\pi\)
\(812\) −6.79003e8 −0.0445067
\(813\) 3.26100e10 2.12831
\(814\) 4.19269e8 0.0272463
\(815\) 1.27635e10 0.825884
\(816\) −4.48685e9 −0.289085
\(817\) 1.09920e9 0.0705177
\(818\) −1.10917e9 −0.0708535
\(819\) −6.50271e7 −0.00413619
\(820\) 9.42287e9 0.596808
\(821\) −1.04735e10 −0.660528 −0.330264 0.943889i \(-0.607138\pi\)
−0.330264 + 0.943889i \(0.607138\pi\)
\(822\) −1.08036e10 −0.678451
\(823\) 2.13220e10 1.33330 0.666650 0.745371i \(-0.267727\pi\)
0.666650 + 0.745371i \(0.267727\pi\)
\(824\) 1.12113e10 0.698091
\(825\) 9.06050e8 0.0561776
\(826\) 4.85849e8 0.0299965
\(827\) 4.07306e9 0.250410 0.125205 0.992131i \(-0.460041\pi\)
0.125205 + 0.992131i \(0.460041\pi\)
\(828\) 6.42262e9 0.393193
\(829\) −4.97505e9 −0.303289 −0.151644 0.988435i \(-0.548457\pi\)
−0.151644 + 0.988435i \(0.548457\pi\)
\(830\) −5.51438e9 −0.334752
\(831\) −3.32978e9 −0.201286
\(832\) −1.77565e8 −0.0106887
\(833\) −4.41399e9 −0.264590
\(834\) 1.33550e9 0.0797190
\(835\) −8.94448e9 −0.531682
\(836\) 1.10018e9 0.0651238
\(837\) −7.35852e7 −0.00433762
\(838\) 6.78835e9 0.398483
\(839\) 5.31627e9 0.310771 0.155386 0.987854i \(-0.450338\pi\)
0.155386 + 0.987854i \(0.450338\pi\)
\(840\) 2.29434e9 0.133561
\(841\) −1.61176e10 −0.934360
\(842\) −3.95397e9 −0.228266
\(843\) −3.38183e10 −1.94426
\(844\) 2.68225e10 1.53568
\(845\) 1.51266e10 0.862467
\(846\) −5.61713e9 −0.318947
\(847\) 3.29231e9 0.186170
\(848\) −1.53974e8 −0.00867086
\(849\) −6.14048e9 −0.344370
\(850\) −3.75126e8 −0.0209513
\(851\) −4.47921e9 −0.249143
\(852\) 3.01408e10 1.66961
\(853\) 1.45978e10 0.805315 0.402658 0.915351i \(-0.368087\pi\)
0.402658 + 0.915351i \(0.368087\pi\)
\(854\) −1.30005e9 −0.0714259
\(855\) 7.45456e9 0.407888
\(856\) 8.81039e9 0.480106
\(857\) −1.57329e10 −0.853839 −0.426919 0.904290i \(-0.640401\pi\)
−0.426919 + 0.904290i \(0.640401\pi\)
\(858\) −2.57757e7 −0.00139317
\(859\) 1.26238e10 0.679540 0.339770 0.940509i \(-0.389651\pi\)
0.339770 + 0.940509i \(0.389651\pi\)
\(860\) −2.23548e9 −0.119847
\(861\) 3.85767e9 0.205975
\(862\) −9.62936e9 −0.512061
\(863\) 5.25485e9 0.278306 0.139153 0.990271i \(-0.455562\pi\)
0.139153 + 0.990271i \(0.455562\pi\)
\(864\) −4.75626e8 −0.0250881
\(865\) 9.47952e9 0.498000
\(866\) 6.49008e9 0.339576
\(867\) 2.52316e10 1.31485
\(868\) 4.58211e8 0.0237819
\(869\) −1.80445e9 −0.0932771
\(870\) −1.82368e9 −0.0938924
\(871\) 6.53141e8 0.0334922
\(872\) −7.64038e9 −0.390218
\(873\) 1.13070e10 0.575170
\(874\) 1.15113e9 0.0583221
\(875\) −4.09445e9 −0.206617
\(876\) 4.05175e10 2.03647
\(877\) −2.81828e10 −1.41086 −0.705432 0.708778i \(-0.749247\pi\)
−0.705432 + 0.708778i \(0.749247\pi\)
\(878\) 2.65888e9 0.132577
\(879\) 1.64119e10 0.815076
\(880\) −1.99687e9 −0.0987782
\(881\) 7.05789e9 0.347744 0.173872 0.984768i \(-0.444372\pi\)
0.173872 + 0.984768i \(0.444372\pi\)
\(882\) −5.99520e9 −0.294214
\(883\) −3.54924e10 −1.73489 −0.867447 0.497530i \(-0.834240\pi\)
−0.867447 + 0.497530i \(0.834240\pi\)
\(884\) −1.08964e8 −0.00530519
\(885\) −1.33237e10 −0.646136
\(886\) −6.66633e9 −0.322010
\(887\) −2.92725e10 −1.40840 −0.704202 0.709999i \(-0.748695\pi\)
−0.704202 + 0.709999i \(0.748695\pi\)
\(888\) 9.99085e9 0.478803
\(889\) 4.24745e9 0.202755
\(890\) −2.67038e8 −0.0126972
\(891\) −3.19044e9 −0.151105
\(892\) −2.26180e10 −1.06703
\(893\) 1.02795e10 0.483051
\(894\) −1.04522e10 −0.489246
\(895\) 1.14970e10 0.536047
\(896\) 3.86977e9 0.179724
\(897\) 2.75372e8 0.0127393
\(898\) 1.94706e8 0.00897248
\(899\) −7.64098e8 −0.0350744
\(900\) 5.20233e9 0.237875
\(901\) −7.06038e7 −0.00321582
\(902\) 7.72983e8 0.0350709
\(903\) −9.15193e8 −0.0413624
\(904\) 3.35410e9 0.151003
\(905\) −4.92428e9 −0.220837
\(906\) 3.47382e9 0.155188
\(907\) −3.76454e10 −1.67528 −0.837639 0.546224i \(-0.816065\pi\)
−0.837639 + 0.546224i \(0.816065\pi\)
\(908\) −2.90021e10 −1.28567
\(909\) 2.93023e10 1.29398
\(910\) 2.37028e7 0.00104269
\(911\) 6.61567e9 0.289908 0.144954 0.989438i \(-0.453697\pi\)
0.144954 + 0.989438i \(0.453697\pi\)
\(912\) 1.11525e10 0.486845
\(913\) 4.61882e9 0.200855
\(914\) 1.53303e9 0.0664110
\(915\) 3.56519e10 1.53854
\(916\) −3.04944e10 −1.31095
\(917\) 4.63692e9 0.198581
\(918\) −6.09046e7 −0.00259837
\(919\) 2.38719e10 1.01457 0.507285 0.861778i \(-0.330649\pi\)
0.507285 + 0.861778i \(0.330649\pi\)
\(920\) −4.91147e9 −0.207948
\(921\) −2.55682e10 −1.07843
\(922\) 7.09064e9 0.297939
\(923\) 6.53267e8 0.0273454
\(924\) −9.16007e8 −0.0381986
\(925\) −3.62817e9 −0.150727
\(926\) −7.17548e8 −0.0296970
\(927\) −3.03291e10 −1.25049
\(928\) −4.93883e9 −0.202864
\(929\) −2.31224e10 −0.946190 −0.473095 0.881012i \(-0.656863\pi\)
−0.473095 + 0.881012i \(0.656863\pi\)
\(930\) 1.23067e9 0.0501708
\(931\) 1.09714e10 0.445594
\(932\) −1.99270e10 −0.806279
\(933\) 3.74155e10 1.50822
\(934\) 3.52882e9 0.141715
\(935\) −9.15653e8 −0.0366345
\(936\) −3.10491e8 −0.0123761
\(937\) −3.07105e10 −1.21954 −0.609772 0.792577i \(-0.708739\pi\)
−0.609772 + 0.792577i \(0.708739\pi\)
\(938\) −2.27325e9 −0.0899369
\(939\) 1.76132e10 0.694238
\(940\) −2.09059e10 −0.820958
\(941\) 4.74404e8 0.0185603 0.00928015 0.999957i \(-0.497046\pi\)
0.00928015 + 0.999957i \(0.497046\pi\)
\(942\) −6.12542e8 −0.0238758
\(943\) −8.25809e9 −0.320692
\(944\) −1.00764e10 −0.389855
\(945\) −1.35274e8 −0.00521439
\(946\) −1.83382e8 −0.00704268
\(947\) −3.03487e10 −1.16122 −0.580611 0.814181i \(-0.697186\pi\)
−0.580611 + 0.814181i \(0.697186\pi\)
\(948\) −2.04957e10 −0.781327
\(949\) 8.78171e8 0.0333539
\(950\) 9.32416e8 0.0352839
\(951\) −6.52461e9 −0.245993
\(952\) 7.95641e8 0.0298874
\(953\) 1.34857e10 0.504717 0.252359 0.967634i \(-0.418794\pi\)
0.252359 + 0.967634i \(0.418794\pi\)
\(954\) −9.58961e7 −0.00357587
\(955\) −1.92919e10 −0.716741
\(956\) −1.01019e10 −0.373937
\(957\) 1.52750e9 0.0563366
\(958\) 4.12552e9 0.151600
\(959\) −8.32137e9 −0.304670
\(960\) −1.69481e10 −0.618261
\(961\) −2.69970e10 −0.981258
\(962\) 1.03216e8 0.00373795
\(963\) −2.38340e10 −0.860013
\(964\) 1.39100e10 0.500100
\(965\) 9.23028e9 0.330651
\(966\) −9.58430e8 −0.0342090
\(967\) 5.31442e10 1.89001 0.945004 0.327060i \(-0.106058\pi\)
0.945004 + 0.327060i \(0.106058\pi\)
\(968\) 1.57201e10 0.557047
\(969\) 5.11391e9 0.180559
\(970\) −4.12147e9 −0.144995
\(971\) 1.88877e10 0.662083 0.331042 0.943616i \(-0.392600\pi\)
0.331042 + 0.943616i \(0.392600\pi\)
\(972\) −3.70646e10 −1.29457
\(973\) 1.02865e9 0.0357992
\(974\) −2.99261e9 −0.103775
\(975\) 2.23052e8 0.00770707
\(976\) 2.69626e10 0.928299
\(977\) −4.88964e9 −0.167744 −0.0838718 0.996477i \(-0.526729\pi\)
−0.0838718 + 0.996477i \(0.526729\pi\)
\(978\) 1.18925e10 0.406526
\(979\) 2.23670e8 0.00761848
\(980\) −2.23130e10 −0.757298
\(981\) 2.06689e10 0.698998
\(982\) −5.76619e9 −0.194312
\(983\) 4.22030e10 1.41712 0.708560 0.705651i \(-0.249345\pi\)
0.708560 + 0.705651i \(0.249345\pi\)
\(984\) 1.84196e10 0.616307
\(985\) −2.97603e10 −0.992227
\(986\) −6.32423e8 −0.0210106
\(987\) −8.55875e9 −0.283335
\(988\) 2.70842e8 0.00893442
\(989\) 1.95915e9 0.0643991
\(990\) −1.24367e9 −0.0407362
\(991\) 4.76499e10 1.55527 0.777633 0.628719i \(-0.216420\pi\)
0.777633 + 0.628719i \(0.216420\pi\)
\(992\) 3.33286e9 0.108399
\(993\) −4.34957e10 −1.40969
\(994\) −2.27369e9 −0.0734310
\(995\) 1.91856e10 0.617439
\(996\) 5.24624e10 1.68244
\(997\) −6.81917e8 −0.0217921 −0.0108960 0.999941i \(-0.503468\pi\)
−0.0108960 + 0.999941i \(0.503468\pi\)
\(998\) −6.22205e9 −0.198142
\(999\) −5.89061e8 −0.0186931
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 43.8.a.b.1.8 13
3.2 odd 2 387.8.a.d.1.6 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.8 13 1.1 even 1 trivial
387.8.a.d.1.6 13 3.2 odd 2