Properties

Label 43.10.a.a.1.9
Level 43
Weight 10
Character 43.1
Self dual yes
Analytic conductor 22.147
Analytic rank 1
Dimension 15
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 43 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 43.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(22.1465409550\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(-0.220103\) of \(x^{15} - 2 x^{14} - 5425 x^{13} + 14888 x^{12} + 11288030 x^{11} - 37600244 x^{10} - 11474166224 x^{9} + 47465836576 x^{8} + 5986976782464 x^{7} - 32493903147264 x^{6} - 1516975415483904 x^{5} + 10892588268404224 x^{4} + 139803541742443008 x^{3} - 1349125586394823680 x^{2} + 2103623681144094720 x + 529838441422848000\)
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.77990 q^{2} -203.939 q^{3} -508.832 q^{4} +1139.29 q^{5} +362.990 q^{6} +4324.96 q^{7} +1816.98 q^{8} +21908.1 q^{9} +O(q^{10})\) \(q-1.77990 q^{2} -203.939 q^{3} -508.832 q^{4} +1139.29 q^{5} +362.990 q^{6} +4324.96 q^{7} +1816.98 q^{8} +21908.1 q^{9} -2027.82 q^{10} +30380.9 q^{11} +103771. q^{12} +34108.8 q^{13} -7697.98 q^{14} -232345. q^{15} +257288. q^{16} -566042. q^{17} -38994.1 q^{18} +246718. q^{19} -579707. q^{20} -882027. q^{21} -54074.8 q^{22} -1.19112e6 q^{23} -370552. q^{24} -655146. q^{25} -60710.1 q^{26} -453781. q^{27} -2.20068e6 q^{28} +7.14721e6 q^{29} +413551. q^{30} -5.75774e6 q^{31} -1.38824e6 q^{32} -6.19584e6 q^{33} +1.00750e6 q^{34} +4.92738e6 q^{35} -1.11475e7 q^{36} -8.31491e6 q^{37} -439132. q^{38} -6.95611e6 q^{39} +2.07006e6 q^{40} -4.50031e6 q^{41} +1.56992e6 q^{42} -3.41880e6 q^{43} -1.54588e7 q^{44} +2.49596e7 q^{45} +2.12007e6 q^{46} -2.85694e7 q^{47} -5.24710e7 q^{48} -2.16483e7 q^{49} +1.16609e6 q^{50} +1.15438e8 q^{51} -1.73556e7 q^{52} +2.07892e7 q^{53} +807684. q^{54} +3.46126e7 q^{55} +7.85834e6 q^{56} -5.03154e7 q^{57} -1.27213e7 q^{58} -3.59484e6 q^{59} +1.18225e8 q^{60} -5.73421e7 q^{61} +1.02482e7 q^{62} +9.47516e7 q^{63} -1.29261e8 q^{64} +3.88598e7 q^{65} +1.10280e7 q^{66} +1.95331e8 q^{67} +2.88020e8 q^{68} +2.42915e8 q^{69} -8.77022e6 q^{70} -4.27590e6 q^{71} +3.98064e7 q^{72} -1.36748e8 q^{73} +1.47997e7 q^{74} +1.33610e8 q^{75} -1.25538e8 q^{76} +1.31396e8 q^{77} +1.23812e7 q^{78} -3.21883e8 q^{79} +2.93125e8 q^{80} -3.38673e8 q^{81} +8.01009e6 q^{82} -5.39331e8 q^{83} +4.48804e8 q^{84} -6.44885e8 q^{85} +6.08511e6 q^{86} -1.45759e9 q^{87} +5.52013e7 q^{88} +2.05645e8 q^{89} -4.44256e7 q^{90} +1.47519e8 q^{91} +6.06079e8 q^{92} +1.17423e9 q^{93} +5.08506e7 q^{94} +2.81083e8 q^{95} +2.83116e8 q^{96} -1.10879e9 q^{97} +3.85318e7 q^{98} +6.65587e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15q - 32q^{2} - 317q^{3} + 3242q^{4} - 4717q^{5} + 687q^{6} - 9680q^{7} - 20394q^{8} + 69516q^{9} + O(q^{10}) \) \( 15q - 32q^{2} - 317q^{3} + 3242q^{4} - 4717q^{5} + 687q^{6} - 9680q^{7} - 20394q^{8} + 69516q^{9} - 36237q^{10} - 104484q^{11} - 266395q^{12} - 116174q^{13} + 416064q^{14} + 415388q^{15} + 996762q^{16} - 884265q^{17} - 588735q^{18} - 689535q^{19} - 3077879q^{20} - 2070198q^{21} - 7276218q^{22} - 2504077q^{23} - 11534895q^{24} + 1315350q^{25} - 13343414q^{26} - 12546986q^{27} - 28059568q^{28} - 18406221q^{29} - 39503820q^{30} - 12033699q^{31} - 18952630q^{32} - 14197716q^{33} - 30383125q^{34} - 27855546q^{35} - 18372959q^{36} - 8722847q^{37} - 63941843q^{38} - 30955510q^{39} - 39665611q^{40} - 18689389q^{41} - 73185310q^{42} - 51282015q^{43} - 68723220q^{44} - 216992888q^{45} - 2067521q^{46} - 104960741q^{47} - 145362479q^{48} + 92663095q^{49} - 42446347q^{50} + 37433407q^{51} + 149226080q^{52} - 215907800q^{53} + 419158122q^{54} + 384379852q^{55} + 430441344q^{56} + 258744488q^{57} + 295963139q^{58} + 185924544q^{59} + 973236172q^{60} + 247538102q^{61} + 139798853q^{62} + 405429926q^{63} + 848556290q^{64} + 94294394q^{65} + 667230492q^{66} + 467904656q^{67} - 88234341q^{68} + 163914994q^{69} + 647526126q^{70} - 8252944q^{71} + 889796745q^{72} - 715627902q^{73} + 725122989q^{74} - 18301762q^{75} + 346300359q^{76} - 1236779964q^{77} + 2058642146q^{78} + 560681783q^{79} - 1157214179q^{80} - 752010645q^{81} + 941346367q^{82} - 1442854698q^{83} + 1895248718q^{84} + 699302088q^{85} + 109401632q^{86} - 2094576907q^{87} - 1464507256q^{88} - 396710008q^{89} + 1411356270q^{90} - 3278076852q^{91} + 155864647q^{92} - 1424759183q^{93} + 4666638949q^{94} - 3854114395q^{95} - 952489551q^{96} - 3063837815q^{97} - 6161086984q^{98} - 6576160348q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.77990 −0.0786611 −0.0393305 0.999226i \(-0.512523\pi\)
−0.0393305 + 0.999226i \(0.512523\pi\)
\(3\) −203.939 −1.45363 −0.726816 0.686832i \(-0.759001\pi\)
−0.726816 + 0.686832i \(0.759001\pi\)
\(4\) −508.832 −0.993812
\(5\) 1139.29 0.815209 0.407604 0.913159i \(-0.366364\pi\)
0.407604 + 0.913159i \(0.366364\pi\)
\(6\) 362.990 0.114344
\(7\) 4324.96 0.680833 0.340417 0.940275i \(-0.389432\pi\)
0.340417 + 0.940275i \(0.389432\pi\)
\(8\) 1816.98 0.156835
\(9\) 21908.1 1.11305
\(10\) −2027.82 −0.0641252
\(11\) 30380.9 0.625652 0.312826 0.949810i \(-0.398724\pi\)
0.312826 + 0.949810i \(0.398724\pi\)
\(12\) 103771. 1.44464
\(13\) 34108.8 0.331224 0.165612 0.986191i \(-0.447040\pi\)
0.165612 + 0.986191i \(0.447040\pi\)
\(14\) −7697.98 −0.0535551
\(15\) −232345. −1.18501
\(16\) 257288. 0.981476
\(17\) −566042. −1.64372 −0.821861 0.569688i \(-0.807064\pi\)
−0.821861 + 0.569688i \(0.807064\pi\)
\(18\) −38994.1 −0.0875534
\(19\) 246718. 0.434319 0.217160 0.976136i \(-0.430321\pi\)
0.217160 + 0.976136i \(0.430321\pi\)
\(20\) −579707. −0.810165
\(21\) −882027. −0.989681
\(22\) −54074.8 −0.0492145
\(23\) −1.19112e6 −0.887523 −0.443761 0.896145i \(-0.646356\pi\)
−0.443761 + 0.896145i \(0.646356\pi\)
\(24\) −370552. −0.227981
\(25\) −655146. −0.335435
\(26\) −60710.1 −0.0260544
\(27\) −453781. −0.164327
\(28\) −2.20068e6 −0.676621
\(29\) 7.14721e6 1.87649 0.938243 0.345976i \(-0.112452\pi\)
0.938243 + 0.345976i \(0.112452\pi\)
\(30\) 413551. 0.0932144
\(31\) −5.75774e6 −1.11976 −0.559879 0.828574i \(-0.689152\pi\)
−0.559879 + 0.828574i \(0.689152\pi\)
\(32\) −1.38824e6 −0.234039
\(33\) −6.19584e6 −0.909468
\(34\) 1.00750e6 0.129297
\(35\) 4.92738e6 0.555021
\(36\) −1.11475e7 −1.10616
\(37\) −8.31491e6 −0.729374 −0.364687 0.931130i \(-0.618824\pi\)
−0.364687 + 0.931130i \(0.618824\pi\)
\(38\) −439132. −0.0341640
\(39\) −6.95611e6 −0.481477
\(40\) 2.07006e6 0.127854
\(41\) −4.50031e6 −0.248723 −0.124361 0.992237i \(-0.539688\pi\)
−0.124361 + 0.992237i \(0.539688\pi\)
\(42\) 1.56992e6 0.0778494
\(43\) −3.41880e6 −0.152499
\(44\) −1.54588e7 −0.621781
\(45\) 2.49596e7 0.907365
\(46\) 2.12007e6 0.0698135
\(47\) −2.85694e7 −0.854006 −0.427003 0.904250i \(-0.640431\pi\)
−0.427003 + 0.904250i \(0.640431\pi\)
\(48\) −5.24710e7 −1.42670
\(49\) −2.16483e7 −0.536466
\(50\) 1.16609e6 0.0263856
\(51\) 1.15438e8 2.38937
\(52\) −1.73556e7 −0.329174
\(53\) 2.07892e7 0.361907 0.180953 0.983492i \(-0.442082\pi\)
0.180953 + 0.983492i \(0.442082\pi\)
\(54\) 807684. 0.0129262
\(55\) 3.46126e7 0.510037
\(56\) 7.85834e6 0.106779
\(57\) −5.03154e7 −0.631341
\(58\) −1.27213e7 −0.147606
\(59\) −3.59484e6 −0.0386229 −0.0193115 0.999814i \(-0.506147\pi\)
−0.0193115 + 0.999814i \(0.506147\pi\)
\(60\) 1.18225e8 1.17768
\(61\) −5.73421e7 −0.530261 −0.265130 0.964213i \(-0.585415\pi\)
−0.265130 + 0.964213i \(0.585415\pi\)
\(62\) 1.02482e7 0.0880813
\(63\) 9.47516e7 0.757799
\(64\) −1.29261e8 −0.963066
\(65\) 3.88598e7 0.270016
\(66\) 1.10280e7 0.0715397
\(67\) 1.95331e8 1.18422 0.592111 0.805856i \(-0.298295\pi\)
0.592111 + 0.805856i \(0.298295\pi\)
\(68\) 2.88020e8 1.63355
\(69\) 2.42915e8 1.29013
\(70\) −8.77022e6 −0.0436586
\(71\) −4.27590e6 −0.0199694 −0.00998471 0.999950i \(-0.503178\pi\)
−0.00998471 + 0.999950i \(0.503178\pi\)
\(72\) 3.98064e7 0.174565
\(73\) −1.36748e8 −0.563595 −0.281798 0.959474i \(-0.590931\pi\)
−0.281798 + 0.959474i \(0.590931\pi\)
\(74\) 1.47997e7 0.0573733
\(75\) 1.33610e8 0.487599
\(76\) −1.25538e8 −0.431632
\(77\) 1.31396e8 0.425965
\(78\) 1.23812e7 0.0378735
\(79\) −3.21883e8 −0.929770 −0.464885 0.885371i \(-0.653904\pi\)
−0.464885 + 0.885371i \(0.653904\pi\)
\(80\) 2.93125e8 0.800108
\(81\) −3.38673e8 −0.874175
\(82\) 8.01009e6 0.0195648
\(83\) −5.39331e8 −1.24739 −0.623697 0.781666i \(-0.714370\pi\)
−0.623697 + 0.781666i \(0.714370\pi\)
\(84\) 4.48804e8 0.983557
\(85\) −6.44885e8 −1.33998
\(86\) 6.08511e6 0.0119957
\(87\) −1.45759e9 −2.72772
\(88\) 5.52013e7 0.0981244
\(89\) 2.05645e8 0.347426 0.173713 0.984796i \(-0.444423\pi\)
0.173713 + 0.984796i \(0.444423\pi\)
\(90\) −4.44256e7 −0.0713743
\(91\) 1.47519e8 0.225508
\(92\) 6.06079e8 0.882031
\(93\) 1.17423e9 1.62772
\(94\) 5.08506e7 0.0671770
\(95\) 2.81083e8 0.354061
\(96\) 2.83116e8 0.340207
\(97\) −1.10879e9 −1.27168 −0.635840 0.771821i \(-0.719346\pi\)
−0.635840 + 0.771821i \(0.719346\pi\)
\(98\) 3.85318e7 0.0421990
\(99\) 6.65587e8 0.696380
\(100\) 3.33359e8 0.333359
\(101\) −1.88058e9 −1.79823 −0.899117 0.437709i \(-0.855790\pi\)
−0.899117 + 0.437709i \(0.855790\pi\)
\(102\) −2.05468e8 −0.187950
\(103\) 9.90610e8 0.867232 0.433616 0.901098i \(-0.357237\pi\)
0.433616 + 0.901098i \(0.357237\pi\)
\(104\) 6.19748e7 0.0519476
\(105\) −1.00488e9 −0.806797
\(106\) −3.70027e7 −0.0284680
\(107\) 1.83523e6 0.00135352 0.000676759 1.00000i \(-0.499785\pi\)
0.000676759 1.00000i \(0.499785\pi\)
\(108\) 2.30898e8 0.163311
\(109\) −1.28953e9 −0.875007 −0.437503 0.899217i \(-0.644137\pi\)
−0.437503 + 0.899217i \(0.644137\pi\)
\(110\) −6.16068e7 −0.0401201
\(111\) 1.69573e9 1.06024
\(112\) 1.11276e9 0.668221
\(113\) −1.37637e9 −0.794115 −0.397057 0.917794i \(-0.629969\pi\)
−0.397057 + 0.917794i \(0.629969\pi\)
\(114\) 8.95561e7 0.0496619
\(115\) −1.35703e9 −0.723517
\(116\) −3.63673e9 −1.86488
\(117\) 7.47258e8 0.368667
\(118\) 6.39844e6 0.00303812
\(119\) −2.44811e9 −1.11910
\(120\) −4.22166e8 −0.185852
\(121\) −1.43495e9 −0.608559
\(122\) 1.02063e8 0.0417109
\(123\) 9.17789e8 0.361551
\(124\) 2.92972e9 1.11283
\(125\) −2.97157e9 −1.08866
\(126\) −1.68648e8 −0.0596092
\(127\) −1.41410e9 −0.482352 −0.241176 0.970481i \(-0.577533\pi\)
−0.241176 + 0.970481i \(0.577533\pi\)
\(128\) 9.40848e8 0.309795
\(129\) 6.97227e8 0.221677
\(130\) −6.91664e7 −0.0212398
\(131\) 6.55771e9 1.94550 0.972750 0.231857i \(-0.0744800\pi\)
0.972750 + 0.231857i \(0.0744800\pi\)
\(132\) 3.15264e9 0.903841
\(133\) 1.06704e9 0.295699
\(134\) −3.47668e8 −0.0931522
\(135\) −5.16988e8 −0.133961
\(136\) −1.02848e9 −0.257794
\(137\) 8.93137e8 0.216609 0.108304 0.994118i \(-0.465458\pi\)
0.108304 + 0.994118i \(0.465458\pi\)
\(138\) −4.32364e8 −0.101483
\(139\) 4.97678e9 1.13079 0.565395 0.824820i \(-0.308724\pi\)
0.565395 + 0.824820i \(0.308724\pi\)
\(140\) −2.50721e9 −0.551587
\(141\) 5.82642e9 1.24141
\(142\) 7.61067e6 0.00157082
\(143\) 1.03625e9 0.207231
\(144\) 5.63669e9 1.09243
\(145\) 8.14273e9 1.52973
\(146\) 2.43397e8 0.0443330
\(147\) 4.41494e9 0.779824
\(148\) 4.23089e9 0.724861
\(149\) −3.83286e9 −0.637066 −0.318533 0.947912i \(-0.603190\pi\)
−0.318533 + 0.947912i \(0.603190\pi\)
\(150\) −2.37812e8 −0.0383550
\(151\) 6.11983e9 0.957951 0.478975 0.877828i \(-0.341008\pi\)
0.478975 + 0.877828i \(0.341008\pi\)
\(152\) 4.48280e8 0.0681166
\(153\) −1.24009e10 −1.82954
\(154\) −2.33871e8 −0.0335069
\(155\) −6.55972e9 −0.912836
\(156\) 3.53949e9 0.478498
\(157\) −1.04568e10 −1.37357 −0.686783 0.726863i \(-0.740978\pi\)
−0.686783 + 0.726863i \(0.740978\pi\)
\(158\) 5.72918e8 0.0731367
\(159\) −4.23973e9 −0.526079
\(160\) −1.58160e9 −0.190791
\(161\) −5.15154e9 −0.604255
\(162\) 6.02803e8 0.0687635
\(163\) 8.35163e9 0.926674 0.463337 0.886182i \(-0.346652\pi\)
0.463337 + 0.886182i \(0.346652\pi\)
\(164\) 2.28990e9 0.247184
\(165\) −7.05885e9 −0.741407
\(166\) 9.59953e8 0.0981213
\(167\) −8.28252e9 −0.824021 −0.412011 0.911179i \(-0.635173\pi\)
−0.412011 + 0.911179i \(0.635173\pi\)
\(168\) −1.60262e9 −0.155217
\(169\) −9.44109e9 −0.890291
\(170\) 1.14783e9 0.105404
\(171\) 5.40511e9 0.483417
\(172\) 1.73960e9 0.151555
\(173\) 6.37085e9 0.540742 0.270371 0.962756i \(-0.412854\pi\)
0.270371 + 0.962756i \(0.412854\pi\)
\(174\) 2.59437e9 0.214565
\(175\) −2.83348e9 −0.228375
\(176\) 7.81663e9 0.614063
\(177\) 7.33127e8 0.0561435
\(178\) −3.66026e8 −0.0273289
\(179\) −1.73831e10 −1.26558 −0.632789 0.774325i \(-0.718090\pi\)
−0.632789 + 0.774325i \(0.718090\pi\)
\(180\) −1.27003e10 −0.901750
\(181\) −1.20834e10 −0.836826 −0.418413 0.908257i \(-0.637414\pi\)
−0.418413 + 0.908257i \(0.637414\pi\)
\(182\) −2.62569e8 −0.0177387
\(183\) 1.16943e10 0.770804
\(184\) −2.16423e9 −0.139195
\(185\) −9.47309e9 −0.594592
\(186\) −2.09000e9 −0.128038
\(187\) −1.71969e10 −1.02840
\(188\) 1.45370e10 0.848722
\(189\) −1.96259e9 −0.111879
\(190\) −5.00298e8 −0.0278508
\(191\) 2.09279e10 1.13782 0.568912 0.822398i \(-0.307365\pi\)
0.568912 + 0.822398i \(0.307365\pi\)
\(192\) 2.63612e10 1.39994
\(193\) 2.13946e10 1.10993 0.554965 0.831874i \(-0.312732\pi\)
0.554965 + 0.831874i \(0.312732\pi\)
\(194\) 1.97354e9 0.100032
\(195\) −7.92502e9 −0.392504
\(196\) 1.10154e10 0.533147
\(197\) −2.74114e10 −1.29668 −0.648340 0.761351i \(-0.724536\pi\)
−0.648340 + 0.761351i \(0.724536\pi\)
\(198\) −1.18468e9 −0.0547780
\(199\) 4.11872e9 0.186176 0.0930880 0.995658i \(-0.470326\pi\)
0.0930880 + 0.995658i \(0.470326\pi\)
\(200\) −1.19038e9 −0.0526080
\(201\) −3.98355e10 −1.72142
\(202\) 3.34724e9 0.141451
\(203\) 3.09114e10 1.27757
\(204\) −5.87386e10 −2.37458
\(205\) −5.12716e9 −0.202761
\(206\) −1.76318e9 −0.0682174
\(207\) −2.60951e10 −0.987854
\(208\) 8.77578e9 0.325088
\(209\) 7.49550e9 0.271733
\(210\) 1.78859e9 0.0634635
\(211\) 3.83930e10 1.33346 0.666732 0.745298i \(-0.267693\pi\)
0.666732 + 0.745298i \(0.267693\pi\)
\(212\) −1.05782e10 −0.359667
\(213\) 8.72023e8 0.0290282
\(214\) −3.26653e6 −0.000106469 0
\(215\) −3.89500e9 −0.124318
\(216\) −8.24510e8 −0.0257723
\(217\) −2.49020e10 −0.762368
\(218\) 2.29523e9 0.0688290
\(219\) 2.78882e10 0.819260
\(220\) −1.76120e10 −0.506881
\(221\) −1.93070e10 −0.544440
\(222\) −3.01823e9 −0.0833997
\(223\) 3.85620e10 1.04421 0.522105 0.852881i \(-0.325147\pi\)
0.522105 + 0.852881i \(0.325147\pi\)
\(224\) −6.00407e9 −0.159342
\(225\) −1.43530e10 −0.373354
\(226\) 2.44980e9 0.0624659
\(227\) −1.12890e10 −0.282188 −0.141094 0.989996i \(-0.545062\pi\)
−0.141094 + 0.989996i \(0.545062\pi\)
\(228\) 2.56021e10 0.627434
\(229\) 3.48087e10 0.836428 0.418214 0.908349i \(-0.362656\pi\)
0.418214 + 0.908349i \(0.362656\pi\)
\(230\) 2.41537e9 0.0569126
\(231\) −2.67968e10 −0.619196
\(232\) 1.29863e10 0.294299
\(233\) 2.66211e10 0.591730 0.295865 0.955230i \(-0.404392\pi\)
0.295865 + 0.955230i \(0.404392\pi\)
\(234\) −1.33004e9 −0.0289997
\(235\) −3.25488e10 −0.696193
\(236\) 1.82917e9 0.0383839
\(237\) 6.56444e10 1.35154
\(238\) 4.35738e9 0.0880297
\(239\) 5.93716e10 1.17703 0.588516 0.808485i \(-0.299712\pi\)
0.588516 + 0.808485i \(0.299712\pi\)
\(240\) −5.97797e10 −1.16306
\(241\) 5.18516e10 0.990114 0.495057 0.868860i \(-0.335147\pi\)
0.495057 + 0.868860i \(0.335147\pi\)
\(242\) 2.55406e9 0.0478699
\(243\) 7.80004e10 1.43506
\(244\) 2.91775e10 0.526980
\(245\) −2.46637e10 −0.437332
\(246\) −1.63357e9 −0.0284400
\(247\) 8.41524e9 0.143857
\(248\) −1.04617e10 −0.175618
\(249\) 1.09991e11 1.81325
\(250\) 5.28909e9 0.0856350
\(251\) −7.95240e10 −1.26464 −0.632320 0.774708i \(-0.717897\pi\)
−0.632320 + 0.774708i \(0.717897\pi\)
\(252\) −4.82126e10 −0.753110
\(253\) −3.61872e10 −0.555281
\(254\) 2.51695e9 0.0379423
\(255\) 1.31517e11 1.94783
\(256\) 6.45068e10 0.938697
\(257\) −1.13156e10 −0.161800 −0.0808999 0.996722i \(-0.525779\pi\)
−0.0808999 + 0.996722i \(0.525779\pi\)
\(258\) −1.24099e9 −0.0174373
\(259\) −3.59617e10 −0.496582
\(260\) −1.97731e10 −0.268346
\(261\) 1.56582e11 2.08862
\(262\) −1.16720e10 −0.153035
\(263\) −1.29655e11 −1.67105 −0.835525 0.549452i \(-0.814837\pi\)
−0.835525 + 0.549452i \(0.814837\pi\)
\(264\) −1.12577e10 −0.142637
\(265\) 2.36849e10 0.295030
\(266\) −1.89923e9 −0.0232600
\(267\) −4.19389e10 −0.505030
\(268\) −9.93904e10 −1.17690
\(269\) 2.31009e10 0.268995 0.134497 0.990914i \(-0.457058\pi\)
0.134497 + 0.990914i \(0.457058\pi\)
\(270\) 9.20185e8 0.0105375
\(271\) 8.96402e10 1.00958 0.504790 0.863242i \(-0.331570\pi\)
0.504790 + 0.863242i \(0.331570\pi\)
\(272\) −1.45636e11 −1.61327
\(273\) −3.00849e10 −0.327806
\(274\) −1.58969e9 −0.0170387
\(275\) −1.99039e10 −0.209866
\(276\) −1.23603e11 −1.28215
\(277\) −9.97287e10 −1.01780 −0.508899 0.860826i \(-0.669947\pi\)
−0.508899 + 0.860826i \(0.669947\pi\)
\(278\) −8.85815e9 −0.0889491
\(279\) −1.26141e11 −1.24634
\(280\) 8.95292e9 0.0870470
\(281\) 1.31117e10 0.125453 0.0627265 0.998031i \(-0.480020\pi\)
0.0627265 + 0.998031i \(0.480020\pi\)
\(282\) −1.03704e10 −0.0976507
\(283\) −9.75983e10 −0.904489 −0.452245 0.891894i \(-0.649377\pi\)
−0.452245 + 0.891894i \(0.649377\pi\)
\(284\) 2.17572e9 0.0198459
\(285\) −5.73237e10 −0.514674
\(286\) −1.84443e9 −0.0163010
\(287\) −1.94637e10 −0.169339
\(288\) −3.04136e10 −0.260497
\(289\) 2.01816e11 1.70182
\(290\) −1.44932e10 −0.120330
\(291\) 2.26126e11 1.84855
\(292\) 6.95816e10 0.560108
\(293\) −9.70939e10 −0.769640 −0.384820 0.922992i \(-0.625736\pi\)
−0.384820 + 0.922992i \(0.625736\pi\)
\(294\) −7.85814e9 −0.0613418
\(295\) −4.09556e9 −0.0314857
\(296\) −1.51080e10 −0.114392
\(297\) −1.37863e10 −0.102812
\(298\) 6.82209e9 0.0501123
\(299\) −4.06276e10 −0.293969
\(300\) −6.79849e10 −0.484582
\(301\) −1.47862e10 −0.103826
\(302\) −1.08927e10 −0.0753534
\(303\) 3.83524e11 2.61397
\(304\) 6.34775e10 0.426274
\(305\) −6.53293e10 −0.432273
\(306\) 2.20723e10 0.143913
\(307\) −2.39809e11 −1.54079 −0.770393 0.637569i \(-0.779940\pi\)
−0.770393 + 0.637569i \(0.779940\pi\)
\(308\) −6.68585e10 −0.423329
\(309\) −2.02024e11 −1.26064
\(310\) 1.16756e10 0.0718047
\(311\) −5.16196e10 −0.312891 −0.156445 0.987687i \(-0.550004\pi\)
−0.156445 + 0.987687i \(0.550004\pi\)
\(312\) −1.26391e10 −0.0755127
\(313\) 9.12143e10 0.537172 0.268586 0.963256i \(-0.413444\pi\)
0.268586 + 0.963256i \(0.413444\pi\)
\(314\) 1.86120e10 0.108046
\(315\) 1.07949e11 0.617764
\(316\) 1.63784e11 0.924017
\(317\) −3.41738e11 −1.90076 −0.950378 0.311099i \(-0.899303\pi\)
−0.950378 + 0.311099i \(0.899303\pi\)
\(318\) 7.54628e9 0.0413819
\(319\) 2.17138e11 1.17403
\(320\) −1.47265e11 −0.785100
\(321\) −3.74276e8 −0.00196752
\(322\) 9.16920e9 0.0475313
\(323\) −1.39653e11 −0.713901
\(324\) 1.72328e11 0.868766
\(325\) −2.23462e10 −0.111104
\(326\) −1.48650e10 −0.0728932
\(327\) 2.62985e11 1.27194
\(328\) −8.17696e9 −0.0390085
\(329\) −1.23562e11 −0.581436
\(330\) 1.25640e10 0.0583198
\(331\) −3.21682e11 −1.47299 −0.736497 0.676441i \(-0.763521\pi\)
−0.736497 + 0.676441i \(0.763521\pi\)
\(332\) 2.74429e11 1.23968
\(333\) −1.82164e11 −0.811826
\(334\) 1.47420e10 0.0648184
\(335\) 2.22538e11 0.965389
\(336\) −2.26935e11 −0.971348
\(337\) −4.05004e11 −1.71051 −0.855253 0.518211i \(-0.826598\pi\)
−0.855253 + 0.518211i \(0.826598\pi\)
\(338\) 1.68042e10 0.0700312
\(339\) 2.80696e11 1.15435
\(340\) 3.28138e11 1.33169
\(341\) −1.74925e11 −0.700579
\(342\) −9.62054e9 −0.0380261
\(343\) −2.68156e11 −1.04608
\(344\) −6.21188e9 −0.0239172
\(345\) 2.76751e11 1.05173
\(346\) −1.13395e10 −0.0425353
\(347\) −8.23031e10 −0.304743 −0.152371 0.988323i \(-0.548691\pi\)
−0.152371 + 0.988323i \(0.548691\pi\)
\(348\) 7.41670e11 2.71084
\(349\) 3.73070e11 1.34610 0.673048 0.739599i \(-0.264985\pi\)
0.673048 + 0.739599i \(0.264985\pi\)
\(350\) 5.04330e9 0.0179642
\(351\) −1.54779e10 −0.0544291
\(352\) −4.21759e10 −0.146427
\(353\) 1.33171e11 0.456483 0.228242 0.973605i \(-0.426702\pi\)
0.228242 + 0.973605i \(0.426702\pi\)
\(354\) −1.30489e9 −0.00441631
\(355\) −4.87149e9 −0.0162792
\(356\) −1.04639e11 −0.345276
\(357\) 4.99265e11 1.62676
\(358\) 3.09401e10 0.0995517
\(359\) −2.99339e11 −0.951125 −0.475563 0.879682i \(-0.657755\pi\)
−0.475563 + 0.879682i \(0.657755\pi\)
\(360\) 4.53510e10 0.142307
\(361\) −2.61818e11 −0.811367
\(362\) 2.15072e10 0.0658256
\(363\) 2.92642e11 0.884621
\(364\) −7.50624e10 −0.224113
\(365\) −1.55795e11 −0.459448
\(366\) −2.08146e10 −0.0606323
\(367\) −4.71753e11 −1.35743 −0.678716 0.734401i \(-0.737463\pi\)
−0.678716 + 0.734401i \(0.737463\pi\)
\(368\) −3.06460e11 −0.871082
\(369\) −9.85933e10 −0.276840
\(370\) 1.68611e10 0.0467712
\(371\) 8.99125e10 0.246398
\(372\) −5.97484e11 −1.61764
\(373\) 5.78431e11 1.54725 0.773627 0.633641i \(-0.218441\pi\)
0.773627 + 0.633641i \(0.218441\pi\)
\(374\) 3.06086e10 0.0808950
\(375\) 6.06020e11 1.58251
\(376\) −5.19099e10 −0.133938
\(377\) 2.43783e11 0.621536
\(378\) 3.49320e9 0.00880056
\(379\) 2.82375e11 0.702990 0.351495 0.936190i \(-0.385673\pi\)
0.351495 + 0.936190i \(0.385673\pi\)
\(380\) −1.43024e11 −0.351870
\(381\) 2.88390e11 0.701162
\(382\) −3.72495e10 −0.0895024
\(383\) 1.02581e11 0.243596 0.121798 0.992555i \(-0.461134\pi\)
0.121798 + 0.992555i \(0.461134\pi\)
\(384\) −1.91875e11 −0.450328
\(385\) 1.49698e11 0.347250
\(386\) −3.80801e10 −0.0873083
\(387\) −7.48994e10 −0.169738
\(388\) 5.64189e11 1.26381
\(389\) 3.65492e11 0.809290 0.404645 0.914474i \(-0.367395\pi\)
0.404645 + 0.914474i \(0.367395\pi\)
\(390\) 1.41057e10 0.0308748
\(391\) 6.74223e11 1.45884
\(392\) −3.93345e10 −0.0841369
\(393\) −1.33737e12 −2.82804
\(394\) 4.87894e10 0.101998
\(395\) −3.66717e11 −0.757957
\(396\) −3.38672e11 −0.692071
\(397\) 7.94172e11 1.60457 0.802283 0.596944i \(-0.203619\pi\)
0.802283 + 0.596944i \(0.203619\pi\)
\(398\) −7.33090e9 −0.0146448
\(399\) −2.17612e11 −0.429838
\(400\) −1.68561e11 −0.329221
\(401\) −6.31476e11 −1.21957 −0.609786 0.792566i \(-0.708744\pi\)
−0.609786 + 0.792566i \(0.708744\pi\)
\(402\) 7.09031e10 0.135409
\(403\) −1.96389e11 −0.370890
\(404\) 9.56900e11 1.78711
\(405\) −3.85847e11 −0.712635
\(406\) −5.50191e10 −0.100495
\(407\) −2.52614e11 −0.456334
\(408\) 2.09748e11 0.374737
\(409\) −1.73599e11 −0.306755 −0.153377 0.988168i \(-0.549015\pi\)
−0.153377 + 0.988168i \(0.549015\pi\)
\(410\) 9.12581e9 0.0159494
\(411\) −1.82145e11 −0.314869
\(412\) −5.04054e11 −0.861866
\(413\) −1.55475e10 −0.0262958
\(414\) 4.64466e10 0.0777056
\(415\) −6.14453e11 −1.01689
\(416\) −4.73511e10 −0.0775193
\(417\) −1.01496e12 −1.64375
\(418\) −1.33412e10 −0.0213748
\(419\) −7.33980e11 −1.16338 −0.581689 0.813411i \(-0.697608\pi\)
−0.581689 + 0.813411i \(0.697608\pi\)
\(420\) 5.11317e11 0.801805
\(421\) 9.33067e11 1.44758 0.723791 0.690019i \(-0.242398\pi\)
0.723791 + 0.690019i \(0.242398\pi\)
\(422\) −6.83356e10 −0.104892
\(423\) −6.25901e11 −0.950548
\(424\) 3.77735e10 0.0567598
\(425\) 3.70840e11 0.551362
\(426\) −1.55211e9 −0.00228339
\(427\) −2.48002e11 −0.361019
\(428\) −9.33825e8 −0.00134514
\(429\) −2.11333e11 −0.301237
\(430\) 6.93270e9 0.00977900
\(431\) 6.98145e11 0.974536 0.487268 0.873252i \(-0.337993\pi\)
0.487268 + 0.873252i \(0.337993\pi\)
\(432\) −1.16752e11 −0.161283
\(433\) 5.56024e11 0.760147 0.380074 0.924956i \(-0.375899\pi\)
0.380074 + 0.924956i \(0.375899\pi\)
\(434\) 4.43229e10 0.0599687
\(435\) −1.66062e12 −2.22366
\(436\) 6.56153e11 0.869593
\(437\) −2.93870e11 −0.385468
\(438\) −4.96381e10 −0.0644439
\(439\) −1.04816e12 −1.34691 −0.673456 0.739228i \(-0.735191\pi\)
−0.673456 + 0.739228i \(0.735191\pi\)
\(440\) 6.28902e10 0.0799919
\(441\) −4.74274e11 −0.597111
\(442\) 3.43645e10 0.0428262
\(443\) 5.51262e11 0.680051 0.340026 0.940416i \(-0.389564\pi\)
0.340026 + 0.940416i \(0.389564\pi\)
\(444\) −8.62844e11 −1.05368
\(445\) 2.34289e11 0.283225
\(446\) −6.86364e10 −0.0821387
\(447\) 7.81668e11 0.926059
\(448\) −5.59046e11 −0.655687
\(449\) −1.10157e12 −1.27910 −0.639550 0.768749i \(-0.720879\pi\)
−0.639550 + 0.768749i \(0.720879\pi\)
\(450\) 2.55468e10 0.0293684
\(451\) −1.36723e11 −0.155614
\(452\) 7.00343e11 0.789201
\(453\) −1.24807e12 −1.39251
\(454\) 2.00932e10 0.0221972
\(455\) 1.68067e11 0.183836
\(456\) −9.14218e10 −0.0990165
\(457\) −7.85228e11 −0.842118 −0.421059 0.907033i \(-0.638341\pi\)
−0.421059 + 0.907033i \(0.638341\pi\)
\(458\) −6.19559e10 −0.0657943
\(459\) 2.56859e11 0.270109
\(460\) 6.90499e11 0.719040
\(461\) 1.57045e12 1.61946 0.809728 0.586806i \(-0.199615\pi\)
0.809728 + 0.586806i \(0.199615\pi\)
\(462\) 4.76955e10 0.0487066
\(463\) 1.25678e11 0.127100 0.0635501 0.997979i \(-0.479758\pi\)
0.0635501 + 0.997979i \(0.479758\pi\)
\(464\) 1.83889e12 1.84173
\(465\) 1.33778e12 1.32693
\(466\) −4.73827e10 −0.0465461
\(467\) −7.17571e11 −0.698134 −0.349067 0.937098i \(-0.613501\pi\)
−0.349067 + 0.937098i \(0.613501\pi\)
\(468\) −3.80229e11 −0.366386
\(469\) 8.44796e11 0.806258
\(470\) 5.79335e10 0.0547633
\(471\) 2.13254e12 1.99666
\(472\) −6.53173e9 −0.00605744
\(473\) −1.03866e11 −0.0954111
\(474\) −1.16840e11 −0.106314
\(475\) −1.61636e11 −0.145686
\(476\) 1.24568e12 1.11218
\(477\) 4.55452e11 0.402819
\(478\) −1.05675e11 −0.0925866
\(479\) 1.77215e12 1.53812 0.769062 0.639174i \(-0.220724\pi\)
0.769062 + 0.639174i \(0.220724\pi\)
\(480\) 3.22550e11 0.277340
\(481\) −2.83612e11 −0.241586
\(482\) −9.22904e10 −0.0778834
\(483\) 1.05060e12 0.878365
\(484\) 7.30149e11 0.604794
\(485\) −1.26324e12 −1.03668
\(486\) −1.38833e11 −0.112883
\(487\) 2.21711e12 1.78610 0.893051 0.449956i \(-0.148560\pi\)
0.893051 + 0.449956i \(0.148560\pi\)
\(488\) −1.04189e11 −0.0831637
\(489\) −1.70322e12 −1.34704
\(490\) 4.38989e10 0.0344010
\(491\) −1.36110e12 −1.05688 −0.528438 0.848972i \(-0.677222\pi\)
−0.528438 + 0.848972i \(0.677222\pi\)
\(492\) −4.67000e11 −0.359314
\(493\) −4.04562e12 −3.08442
\(494\) −1.49783e10 −0.0113159
\(495\) 7.58296e11 0.567695
\(496\) −1.48140e12 −1.09902
\(497\) −1.84931e10 −0.0135958
\(498\) −1.95772e11 −0.142632
\(499\) 2.40786e12 1.73852 0.869258 0.494359i \(-0.164597\pi\)
0.869258 + 0.494359i \(0.164597\pi\)
\(500\) 1.51203e12 1.08192
\(501\) 1.68913e12 1.19782
\(502\) 1.41545e11 0.0994778
\(503\) −2.31596e12 −1.61315 −0.806576 0.591130i \(-0.798682\pi\)
−0.806576 + 0.591130i \(0.798682\pi\)
\(504\) 1.72161e11 0.118850
\(505\) −2.14253e12 −1.46594
\(506\) 6.44095e10 0.0436790
\(507\) 1.92541e12 1.29416
\(508\) 7.19540e11 0.479367
\(509\) 8.58304e10 0.0566776 0.0283388 0.999598i \(-0.490978\pi\)
0.0283388 + 0.999598i \(0.490978\pi\)
\(510\) −2.34087e11 −0.153219
\(511\) −5.91428e11 −0.383714
\(512\) −5.96529e11 −0.383634
\(513\) −1.11956e11 −0.0713705
\(514\) 2.01406e10 0.0127273
\(515\) 1.12859e12 0.706975
\(516\) −3.54771e11 −0.220305
\(517\) −8.67964e11 −0.534311
\(518\) 6.40080e10 0.0390616
\(519\) −1.29926e12 −0.786040
\(520\) 7.06072e10 0.0423481
\(521\) −1.42834e12 −0.849301 −0.424650 0.905357i \(-0.639603\pi\)
−0.424650 + 0.905357i \(0.639603\pi\)
\(522\) −2.78699e11 −0.164293
\(523\) 2.06838e12 1.20885 0.604426 0.796661i \(-0.293403\pi\)
0.604426 + 0.796661i \(0.293403\pi\)
\(524\) −3.33677e12 −1.93346
\(525\) 5.77857e11 0.331973
\(526\) 2.30773e11 0.131447
\(527\) 3.25912e12 1.84057
\(528\) −1.59412e12 −0.892621
\(529\) −3.82390e11 −0.212303
\(530\) −4.21567e10 −0.0232073
\(531\) −7.87560e10 −0.0429891
\(532\) −5.42946e11 −0.293869
\(533\) −1.53500e11 −0.0823828
\(534\) 7.46470e10 0.0397262
\(535\) 2.09086e9 0.00110340
\(536\) 3.54911e11 0.185728
\(537\) 3.54509e12 1.83968
\(538\) −4.11172e10 −0.0211594
\(539\) −6.57695e11 −0.335641
\(540\) 2.63060e11 0.133132
\(541\) 1.23040e11 0.0617529 0.0308764 0.999523i \(-0.490170\pi\)
0.0308764 + 0.999523i \(0.490170\pi\)
\(542\) −1.59550e11 −0.0794147
\(543\) 2.46427e12 1.21644
\(544\) 7.85801e11 0.384696
\(545\) −1.46914e12 −0.713313
\(546\) 5.35480e10 0.0257855
\(547\) 1.80654e12 0.862787 0.431393 0.902164i \(-0.358022\pi\)
0.431393 + 0.902164i \(0.358022\pi\)
\(548\) −4.54457e11 −0.215268
\(549\) −1.25626e12 −0.590205
\(550\) 3.54269e10 0.0165082
\(551\) 1.76334e12 0.814994
\(552\) 4.41371e11 0.202338
\(553\) −1.39213e12 −0.633019
\(554\) 1.77507e11 0.0800610
\(555\) 1.93193e12 0.864318
\(556\) −2.53234e12 −1.12379
\(557\) −6.38748e11 −0.281178 −0.140589 0.990068i \(-0.544900\pi\)
−0.140589 + 0.990068i \(0.544900\pi\)
\(558\) 2.24518e11 0.0980386
\(559\) −1.16611e11 −0.0505111
\(560\) 1.26775e12 0.544740
\(561\) 3.50711e12 1.49491
\(562\) −2.33375e10 −0.00986827
\(563\) 1.90057e12 0.797252 0.398626 0.917114i \(-0.369487\pi\)
0.398626 + 0.917114i \(0.369487\pi\)
\(564\) −2.96467e12 −1.23373
\(565\) −1.56809e12 −0.647369
\(566\) 1.73715e11 0.0711481
\(567\) −1.46475e12 −0.595167
\(568\) −7.76921e9 −0.00313191
\(569\) 1.34358e12 0.537353 0.268677 0.963230i \(-0.413414\pi\)
0.268677 + 0.963230i \(0.413414\pi\)
\(570\) 1.02030e11 0.0404848
\(571\) 1.95896e12 0.771193 0.385597 0.922667i \(-0.373996\pi\)
0.385597 + 0.922667i \(0.373996\pi\)
\(572\) −5.27279e11 −0.205949
\(573\) −4.26801e12 −1.65398
\(574\) 3.46433e10 0.0133204
\(575\) 7.80356e11 0.297706
\(576\) −2.83185e12 −1.07194
\(577\) −2.94220e12 −1.10505 −0.552523 0.833497i \(-0.686335\pi\)
−0.552523 + 0.833497i \(0.686335\pi\)
\(578\) −3.59211e11 −0.133867
\(579\) −4.36319e12 −1.61343
\(580\) −4.14328e12 −1.52026
\(581\) −2.33258e12 −0.849267
\(582\) −4.02481e11 −0.145409
\(583\) 6.31594e11 0.226428
\(584\) −2.48467e11 −0.0883917
\(585\) 8.51343e11 0.300541
\(586\) 1.72817e11 0.0605407
\(587\) −5.99434e11 −0.208387 −0.104193 0.994557i \(-0.533226\pi\)
−0.104193 + 0.994557i \(0.533226\pi\)
\(588\) −2.24646e12 −0.774999
\(589\) −1.42054e12 −0.486333
\(590\) 7.28967e9 0.00247670
\(591\) 5.59024e12 1.88490
\(592\) −2.13933e12 −0.715862
\(593\) 5.53945e11 0.183959 0.0919794 0.995761i \(-0.470681\pi\)
0.0919794 + 0.995761i \(0.470681\pi\)
\(594\) 2.45381e10 0.00808728
\(595\) −2.78910e12 −0.912301
\(596\) 1.95028e12 0.633124
\(597\) −8.39968e11 −0.270631
\(598\) 7.23129e10 0.0231239
\(599\) −1.23783e12 −0.392864 −0.196432 0.980517i \(-0.562935\pi\)
−0.196432 + 0.980517i \(0.562935\pi\)
\(600\) 2.42766e11 0.0764727
\(601\) −4.88544e12 −1.52745 −0.763727 0.645539i \(-0.776633\pi\)
−0.763727 + 0.645539i \(0.776633\pi\)
\(602\) 2.63179e10 0.00816707
\(603\) 4.27932e12 1.31809
\(604\) −3.11397e12 −0.952024
\(605\) −1.63482e12 −0.496103
\(606\) −6.82633e11 −0.205618
\(607\) 1.73678e12 0.519272 0.259636 0.965707i \(-0.416397\pi\)
0.259636 + 0.965707i \(0.416397\pi\)
\(608\) −3.42503e11 −0.101648
\(609\) −6.30403e12 −1.85712
\(610\) 1.16279e11 0.0340031
\(611\) −9.74468e11 −0.282867
\(612\) 6.30997e12 1.81822
\(613\) 5.18063e12 1.48187 0.740936 0.671575i \(-0.234382\pi\)
0.740936 + 0.671575i \(0.234382\pi\)
\(614\) 4.26835e11 0.121200
\(615\) 1.04563e12 0.294740
\(616\) 2.38743e11 0.0668064
\(617\) 4.99397e12 1.38728 0.693638 0.720324i \(-0.256007\pi\)
0.693638 + 0.720324i \(0.256007\pi\)
\(618\) 3.59582e11 0.0991630
\(619\) 1.19615e11 0.0327475 0.0163738 0.999866i \(-0.494788\pi\)
0.0163738 + 0.999866i \(0.494788\pi\)
\(620\) 3.33780e12 0.907188
\(621\) 5.40507e11 0.145844
\(622\) 9.18775e10 0.0246123
\(623\) 8.89404e11 0.236539
\(624\) −1.78972e12 −0.472558
\(625\) −2.10590e12 −0.552049
\(626\) −1.62352e11 −0.0422545
\(627\) −1.52862e12 −0.395000
\(628\) 5.32074e12 1.36507
\(629\) 4.70659e12 1.19889
\(630\) −1.92139e11 −0.0485940
\(631\) 6.27030e12 1.57455 0.787275 0.616602i \(-0.211491\pi\)
0.787275 + 0.616602i \(0.211491\pi\)
\(632\) −5.84853e11 −0.145821
\(633\) −7.82983e12 −1.93837
\(634\) 6.08258e11 0.149515
\(635\) −1.61107e12 −0.393217
\(636\) 2.15731e12 0.522824
\(637\) −7.38399e11 −0.177690
\(638\) −3.86484e11 −0.0923503
\(639\) −9.36769e10 −0.0222269
\(640\) 1.07190e12 0.252548
\(641\) 8.09157e12 1.89309 0.946546 0.322570i \(-0.104547\pi\)
0.946546 + 0.322570i \(0.104547\pi\)
\(642\) 6.66172e8 0.000154767 0
\(643\) 8.41338e11 0.194098 0.0970490 0.995280i \(-0.469060\pi\)
0.0970490 + 0.995280i \(0.469060\pi\)
\(644\) 2.62127e12 0.600516
\(645\) 7.94343e11 0.180713
\(646\) 2.48567e11 0.0561562
\(647\) 2.04314e12 0.458384 0.229192 0.973381i \(-0.426392\pi\)
0.229192 + 0.973381i \(0.426392\pi\)
\(648\) −6.15361e11 −0.137102
\(649\) −1.09214e11 −0.0241645
\(650\) 3.97740e10 0.00873955
\(651\) 5.07848e12 1.10820
\(652\) −4.24958e12 −0.920941
\(653\) 9.39158e11 0.202129 0.101065 0.994880i \(-0.467775\pi\)
0.101065 + 0.994880i \(0.467775\pi\)
\(654\) −4.68086e11 −0.100052
\(655\) 7.47112e12 1.58599
\(656\) −1.15788e12 −0.244115
\(657\) −2.99588e12 −0.627307
\(658\) 2.19927e11 0.0457364
\(659\) −4.13790e12 −0.854664 −0.427332 0.904095i \(-0.640546\pi\)
−0.427332 + 0.904095i \(0.640546\pi\)
\(660\) 3.59177e12 0.736819
\(661\) 1.12840e12 0.229909 0.114954 0.993371i \(-0.463328\pi\)
0.114954 + 0.993371i \(0.463328\pi\)
\(662\) 5.72561e11 0.115867
\(663\) 3.93745e12 0.791415
\(664\) −9.79950e11 −0.195636
\(665\) 1.21567e12 0.241056
\(666\) 3.24233e11 0.0638591
\(667\) −8.51317e12 −1.66542
\(668\) 4.21441e12 0.818923
\(669\) −7.86430e12 −1.51790
\(670\) −3.96094e11 −0.0759385
\(671\) −1.74210e12 −0.331759
\(672\) 1.22446e12 0.231624
\(673\) −3.96514e12 −0.745059 −0.372529 0.928020i \(-0.621509\pi\)
−0.372529 + 0.928020i \(0.621509\pi\)
\(674\) 7.20865e11 0.134550
\(675\) 2.97293e11 0.0551211
\(676\) 4.80393e12 0.884782
\(677\) 5.57988e11 0.102088 0.0510442 0.998696i \(-0.483745\pi\)
0.0510442 + 0.998696i \(0.483745\pi\)
\(678\) −4.99610e11 −0.0908024
\(679\) −4.79548e12 −0.865802
\(680\) −1.17174e12 −0.210156
\(681\) 2.30227e12 0.410198
\(682\) 3.11348e11 0.0551083
\(683\) 1.00796e13 1.77236 0.886178 0.463344i \(-0.153351\pi\)
0.886178 + 0.463344i \(0.153351\pi\)
\(684\) −2.75029e12 −0.480426
\(685\) 1.01754e12 0.176581
\(686\) 4.77290e11 0.0822855
\(687\) −7.09885e12 −1.21586
\(688\) −8.79616e11 −0.149674
\(689\) 7.09095e11 0.119872
\(690\) −4.92588e11 −0.0827299
\(691\) −1.07053e13 −1.78626 −0.893132 0.449794i \(-0.851497\pi\)
−0.893132 + 0.449794i \(0.851497\pi\)
\(692\) −3.24169e12 −0.537396
\(693\) 2.87863e12 0.474119
\(694\) 1.46491e11 0.0239714
\(695\) 5.66999e12 0.921830
\(696\) −2.64841e12 −0.427803
\(697\) 2.54737e12 0.408831
\(698\) −6.64026e11 −0.105885
\(699\) −5.42907e12 −0.860158
\(700\) 1.44176e12 0.226962
\(701\) 9.07245e12 1.41904 0.709518 0.704687i \(-0.248913\pi\)
0.709518 + 0.704687i \(0.248913\pi\)
\(702\) 2.75491e10 0.00428145
\(703\) −2.05144e12 −0.316781
\(704\) −3.92705e12 −0.602544
\(705\) 6.63797e12 1.01201
\(706\) −2.37031e11 −0.0359074
\(707\) −8.13344e12 −1.22430
\(708\) −3.73039e11 −0.0557961
\(709\) 1.23204e12 0.183112 0.0915559 0.995800i \(-0.470816\pi\)
0.0915559 + 0.995800i \(0.470816\pi\)
\(710\) 8.67075e9 0.00128054
\(711\) −7.05183e12 −1.03488
\(712\) 3.73651e11 0.0544887
\(713\) 6.85814e12 0.993811
\(714\) −8.88639e11 −0.127963
\(715\) 1.18059e12 0.168936
\(716\) 8.84508e12 1.25775
\(717\) −1.21082e13 −1.71097
\(718\) 5.32792e11 0.0748165
\(719\) −7.74691e12 −1.08106 −0.540528 0.841326i \(-0.681776\pi\)
−0.540528 + 0.841326i \(0.681776\pi\)
\(720\) 6.42181e12 0.890556
\(721\) 4.28435e12 0.590440
\(722\) 4.66009e11 0.0638230
\(723\) −1.05746e13 −1.43926
\(724\) 6.14841e12 0.831648
\(725\) −4.68246e12 −0.629439
\(726\) −5.20873e11 −0.0695852
\(727\) 3.08284e12 0.409304 0.204652 0.978835i \(-0.434394\pi\)
0.204652 + 0.978835i \(0.434394\pi\)
\(728\) 2.68039e11 0.0353676
\(729\) −9.24122e12 −1.21187
\(730\) 2.77299e11 0.0361406
\(731\) 1.93519e12 0.250665
\(732\) −5.95043e12 −0.766035
\(733\) −8.15400e12 −1.04328 −0.521642 0.853164i \(-0.674680\pi\)
−0.521642 + 0.853164i \(0.674680\pi\)
\(734\) 8.39672e11 0.106777
\(735\) 5.02989e12 0.635720
\(736\) 1.65355e12 0.207715
\(737\) 5.93431e12 0.740912
\(738\) 1.75486e11 0.0217765
\(739\) 1.08477e13 1.33794 0.668969 0.743291i \(-0.266736\pi\)
0.668969 + 0.743291i \(0.266736\pi\)
\(740\) 4.82021e12 0.590913
\(741\) −1.71620e12 −0.209115
\(742\) −1.60035e11 −0.0193819
\(743\) −6.60935e12 −0.795626 −0.397813 0.917467i \(-0.630231\pi\)
−0.397813 + 0.917467i \(0.630231\pi\)
\(744\) 2.13354e12 0.255283
\(745\) −4.36673e12 −0.519342
\(746\) −1.02955e12 −0.121709
\(747\) −1.18157e13 −1.38841
\(748\) 8.75031e12 1.02204
\(749\) 7.93731e9 0.000921521 0
\(750\) −1.07865e12 −0.124482
\(751\) 1.03430e13 1.18650 0.593248 0.805020i \(-0.297845\pi\)
0.593248 + 0.805020i \(0.297845\pi\)
\(752\) −7.35057e12 −0.838186
\(753\) 1.62180e13 1.83832
\(754\) −4.33908e11 −0.0488907
\(755\) 6.97226e12 0.780930
\(756\) 9.98626e11 0.111187
\(757\) 7.48200e12 0.828107 0.414054 0.910253i \(-0.364113\pi\)
0.414054 + 0.910253i \(0.364113\pi\)
\(758\) −5.02598e11 −0.0552979
\(759\) 7.37998e12 0.807174
\(760\) 5.10720e11 0.0555293
\(761\) −8.49259e12 −0.917929 −0.458965 0.888455i \(-0.651780\pi\)
−0.458965 + 0.888455i \(0.651780\pi\)
\(762\) −5.13305e11 −0.0551541
\(763\) −5.57715e12 −0.595734
\(764\) −1.06488e13 −1.13078
\(765\) −1.41282e13 −1.49146
\(766\) −1.82583e11 −0.0191615
\(767\) −1.22616e11 −0.0127928
\(768\) −1.31554e13 −1.36452
\(769\) −1.35879e13 −1.40115 −0.700573 0.713581i \(-0.747072\pi\)
−0.700573 + 0.713581i \(0.747072\pi\)
\(770\) −2.66447e11 −0.0273151
\(771\) 2.30769e12 0.235197
\(772\) −1.08862e13 −1.10306
\(773\) −1.48951e13 −1.50050 −0.750251 0.661153i \(-0.770067\pi\)
−0.750251 + 0.661153i \(0.770067\pi\)
\(774\) 1.33313e11 0.0133518
\(775\) 3.77216e12 0.375606
\(776\) −2.01465e12 −0.199444
\(777\) 7.33398e12 0.721847
\(778\) −6.50537e11 −0.0636596
\(779\) −1.11031e12 −0.108025
\(780\) 4.03250e12 0.390076
\(781\) −1.29906e11 −0.0124939
\(782\) −1.20005e12 −0.114754
\(783\) −3.24327e12 −0.308358
\(784\) −5.56986e12 −0.526528
\(785\) −1.19133e13 −1.11974
\(786\) 2.38038e12 0.222457
\(787\) −1.79246e13 −1.66558 −0.832788 0.553592i \(-0.813257\pi\)
−0.832788 + 0.553592i \(0.813257\pi\)
\(788\) 1.39478e13 1.28866
\(789\) 2.64418e13 2.42909
\(790\) 6.52719e11 0.0596217
\(791\) −5.95276e12 −0.540660
\(792\) 1.20935e12 0.109217
\(793\) −1.95587e12 −0.175635
\(794\) −1.41354e12 −0.126217
\(795\) −4.83028e12 −0.428864
\(796\) −2.09574e12 −0.185024
\(797\) 1.38729e13 1.21788 0.608940 0.793216i \(-0.291595\pi\)
0.608940 + 0.793216i \(0.291595\pi\)
\(798\) 3.87326e11 0.0338115
\(799\) 1.61715e13 1.40375
\(800\) 9.09498e11 0.0785049
\(801\) 4.50528e12 0.386701
\(802\) 1.12396e12 0.0959327
\(803\) −4.15452e12 −0.352615
\(804\) 2.02696e13 1.71077
\(805\) −5.86909e12 −0.492594
\(806\) 3.49553e11 0.0291746
\(807\) −4.71117e12 −0.391019
\(808\) −3.41697e12 −0.282027
\(809\) 9.79702e12 0.804129 0.402064 0.915611i \(-0.368293\pi\)
0.402064 + 0.915611i \(0.368293\pi\)
\(810\) 6.86767e11 0.0560566
\(811\) 1.11554e13 0.905508 0.452754 0.891635i \(-0.350442\pi\)
0.452754 + 0.891635i \(0.350442\pi\)
\(812\) −1.57287e13 −1.26967
\(813\) −1.82811e13 −1.46756
\(814\) 4.49627e11 0.0358957
\(815\) 9.51492e12 0.755433
\(816\) 2.97008e13 2.34511
\(817\) −8.43479e11 −0.0662331
\(818\) 3.08988e11 0.0241297
\(819\) 3.23186e12 0.251001
\(820\) 2.60886e12 0.201506
\(821\) −1.94239e13 −1.49208 −0.746039 0.665902i \(-0.768047\pi\)
−0.746039 + 0.665902i \(0.768047\pi\)
\(822\) 3.24200e11 0.0247679
\(823\) −1.46781e13 −1.11524 −0.557622 0.830095i \(-0.688286\pi\)
−0.557622 + 0.830095i \(0.688286\pi\)
\(824\) 1.79991e12 0.136013
\(825\) 4.05918e12 0.305067
\(826\) 2.76730e10 0.00206845
\(827\) −1.77714e13 −1.32114 −0.660568 0.750766i \(-0.729684\pi\)
−0.660568 + 0.750766i \(0.729684\pi\)
\(828\) 1.32780e13 0.981742
\(829\) 1.08538e13 0.798151 0.399075 0.916918i \(-0.369331\pi\)
0.399075 + 0.916918i \(0.369331\pi\)
\(830\) 1.09366e12 0.0799894
\(831\) 2.03386e13 1.47950
\(832\) −4.40892e12 −0.318990
\(833\) 1.22539e13 0.881802
\(834\) 1.80652e12 0.129299
\(835\) −9.43618e12 −0.671749
\(836\) −3.81395e12 −0.270052
\(837\) 2.61275e12 0.184007
\(838\) 1.30641e12 0.0915126
\(839\) 9.83099e12 0.684965 0.342482 0.939524i \(-0.388732\pi\)
0.342482 + 0.939524i \(0.388732\pi\)
\(840\) −1.82585e12 −0.126534
\(841\) 3.65754e13 2.52120
\(842\) −1.66076e12 −0.113868
\(843\) −2.67399e12 −0.182363
\(844\) −1.95356e13 −1.32521
\(845\) −1.07561e13 −0.725773
\(846\) 1.11404e12 0.0747711
\(847\) −6.20610e12 −0.414327
\(848\) 5.34881e12 0.355203
\(849\) 1.99041e13 1.31479
\(850\) −6.60057e11 −0.0433707
\(851\) 9.90405e12 0.647336
\(852\) −4.43713e11 −0.0288486
\(853\) 2.22566e13 1.43942 0.719712 0.694273i \(-0.244274\pi\)
0.719712 + 0.694273i \(0.244274\pi\)
\(854\) 4.41419e11 0.0283982
\(855\) 6.15799e12 0.394086
\(856\) 3.33457e9 0.000212280 0
\(857\) 6.93260e12 0.439018 0.219509 0.975610i \(-0.429554\pi\)
0.219509 + 0.975610i \(0.429554\pi\)
\(858\) 3.76150e11 0.0236956
\(859\) 3.01241e13 1.88775 0.943875 0.330303i \(-0.107151\pi\)
0.943875 + 0.330303i \(0.107151\pi\)
\(860\) 1.98190e12 0.123549
\(861\) 3.96940e12 0.246156
\(862\) −1.24263e12 −0.0766581
\(863\) −2.10193e13 −1.28994 −0.644971 0.764207i \(-0.723131\pi\)
−0.644971 + 0.764207i \(0.723131\pi\)
\(864\) 6.29956e11 0.0384590
\(865\) 7.25824e12 0.440818
\(866\) −9.89664e11 −0.0597940
\(867\) −4.11581e13 −2.47383
\(868\) 1.26709e13 0.757651
\(869\) −9.77908e12 −0.581713
\(870\) 2.95573e12 0.174916
\(871\) 6.66249e12 0.392243
\(872\) −2.34304e12 −0.137232
\(873\) −2.42915e13 −1.41544
\(874\) 5.23058e11 0.0303214
\(875\) −1.28519e13 −0.741195
\(876\) −1.41904e13 −0.814191
\(877\) 1.39399e13 0.795720 0.397860 0.917446i \(-0.369753\pi\)
0.397860 + 0.917446i \(0.369753\pi\)
\(878\) 1.86562e12 0.105949
\(879\) 1.98012e13 1.11877
\(880\) 8.90540e12 0.500589
\(881\) 3.43345e12 0.192017 0.0960083 0.995381i \(-0.469392\pi\)
0.0960083 + 0.995381i \(0.469392\pi\)
\(882\) 8.44158e11 0.0469694
\(883\) 1.64420e13 0.910189 0.455095 0.890443i \(-0.349605\pi\)
0.455095 + 0.890443i \(0.349605\pi\)
\(884\) 9.82402e12 0.541071
\(885\) 8.35244e11 0.0457687
\(886\) −9.81190e11 −0.0534935
\(887\) −2.76514e13 −1.49990 −0.749949 0.661496i \(-0.769922\pi\)
−0.749949 + 0.661496i \(0.769922\pi\)
\(888\) 3.08111e12 0.166283
\(889\) −6.11593e12 −0.328401
\(890\) −4.17010e11 −0.0222788
\(891\) −1.02892e13 −0.546929
\(892\) −1.96216e13 −1.03775
\(893\) −7.04858e12 −0.370911
\(894\) −1.39129e12 −0.0728448
\(895\) −1.98044e13 −1.03171
\(896\) 4.06913e12 0.210919
\(897\) 8.28555e12 0.427322
\(898\) 1.96069e12 0.100615
\(899\) −4.11517e13 −2.10121
\(900\) 7.30326e12 0.371044
\(901\) −1.17676e13 −0.594874
\(902\) 2.43354e11 0.0122408
\(903\) 3.01548e12 0.150925
\(904\) −2.50084e12 −0.124545
\(905\) −1.37665e13 −0.682188
\(906\) 2.22144e12 0.109536
\(907\) −1.33233e13 −0.653699 −0.326850 0.945076i \(-0.605987\pi\)
−0.326850 + 0.945076i \(0.605987\pi\)
\(908\) 5.74420e12 0.280442
\(909\) −4.12000e13 −2.00152
\(910\) −2.99142e11 −0.0144607
\(911\) −2.44420e11 −0.0117572 −0.00587860 0.999983i \(-0.501871\pi\)
−0.00587860 + 0.999983i \(0.501871\pi\)
\(912\) −1.29455e13 −0.619645
\(913\) −1.63853e13 −0.780435
\(914\) 1.39763e12 0.0662419
\(915\) 1.33232e13 0.628366
\(916\) −1.77118e13 −0.831252
\(917\) 2.83618e13 1.32456
\(918\) −4.57183e11 −0.0212470
\(919\) 1.75324e13 0.810813 0.405407 0.914136i \(-0.367130\pi\)
0.405407 + 0.914136i \(0.367130\pi\)
\(920\) −2.46569e12 −0.113473
\(921\) 4.89063e13 2.23974
\(922\) −2.79523e12 −0.127388
\(923\) −1.45846e11 −0.00661434
\(924\) 1.36350e13 0.615365
\(925\) 5.44748e12 0.244657
\(926\) −2.23695e11 −0.00999784
\(927\) 2.17024e13 0.965269
\(928\) −9.92202e12 −0.439172
\(929\) −1.10334e13 −0.486004 −0.243002 0.970026i \(-0.578132\pi\)
−0.243002 + 0.970026i \(0.578132\pi\)
\(930\) −2.38112e12 −0.104378
\(931\) −5.34103e12 −0.232998
\(932\) −1.35457e13 −0.588069
\(933\) 1.05272e13 0.454828
\(934\) 1.27720e12 0.0549159
\(935\) −1.95922e13 −0.838360
\(936\) 1.35775e12 0.0578200
\(937\) −2.57249e13 −1.09025 −0.545124 0.838356i \(-0.683517\pi\)
−0.545124 + 0.838356i \(0.683517\pi\)
\(938\) −1.50365e12 −0.0634211
\(939\) −1.86022e13 −0.780851
\(940\) 1.65619e13 0.691886
\(941\) −8.89212e12 −0.369702 −0.184851 0.982767i \(-0.559180\pi\)
−0.184851 + 0.982767i \(0.559180\pi\)
\(942\) −3.79571e12 −0.157059
\(943\) 5.36041e12 0.220747
\(944\) −9.24908e11 −0.0379075
\(945\) −2.23595e12 −0.0912051
\(946\) 1.84871e11 0.00750514
\(947\) −5.49551e12 −0.222041 −0.111020 0.993818i \(-0.535412\pi\)
−0.111020 + 0.993818i \(0.535412\pi\)
\(948\) −3.34020e13 −1.34318
\(949\) −4.66430e12 −0.186676
\(950\) 2.87696e11 0.0114598
\(951\) 6.96936e13 2.76300
\(952\) −4.44815e12 −0.175515
\(953\) 1.44013e13 0.565565 0.282783 0.959184i \(-0.408742\pi\)
0.282783 + 0.959184i \(0.408742\pi\)
\(954\) −8.10657e11 −0.0316862
\(955\) 2.38429e13 0.927564
\(956\) −3.02102e13 −1.16975
\(957\) −4.42830e13 −1.70661
\(958\) −3.15425e12 −0.120990
\(959\) 3.86278e12 0.147474
\(960\) 3.00331e13 1.14125
\(961\) 6.71190e12 0.253858
\(962\) 5.04799e11 0.0190034
\(963\) 4.02064e10 0.00150653
\(964\) −2.63837e13 −0.983988
\(965\) 2.43746e13 0.904825
\(966\) −1.86996e12 −0.0690931
\(967\) 1.72578e13 0.634698 0.317349 0.948309i \(-0.397207\pi\)
0.317349 + 0.948309i \(0.397207\pi\)
\(968\) −2.60727e12 −0.0954436
\(969\) 2.84806e13 1.03775
\(970\) 2.24843e12 0.0815467
\(971\) 1.22438e13 0.442007 0.221003 0.975273i \(-0.429067\pi\)
0.221003 + 0.975273i \(0.429067\pi\)
\(972\) −3.96891e13 −1.42618
\(973\) 2.15244e13 0.769879
\(974\) −3.94622e12 −0.140497
\(975\) 4.55727e12 0.161504
\(976\) −1.47534e13 −0.520438
\(977\) −5.34352e13 −1.87630 −0.938149 0.346231i \(-0.887461\pi\)
−0.938149 + 0.346231i \(0.887461\pi\)
\(978\) 3.03156e12 0.105960
\(979\) 6.24766e12 0.217368
\(980\) 1.25497e13 0.434626
\(981\) −2.82511e13 −0.973923
\(982\) 2.42262e12 0.0831350
\(983\) −5.98058e12 −0.204293 −0.102146 0.994769i \(-0.532571\pi\)
−0.102146 + 0.994769i \(0.532571\pi\)
\(984\) 1.66760e12 0.0567040
\(985\) −3.12295e13 −1.05706
\(986\) 7.20079e12 0.242624
\(987\) 2.51990e13 0.845194
\(988\) −4.28194e12 −0.142967
\(989\) 4.07220e12 0.135346
\(990\) −1.34969e12 −0.0446555
\(991\) −1.53735e13 −0.506340 −0.253170 0.967422i \(-0.581473\pi\)
−0.253170 + 0.967422i \(0.581473\pi\)
\(992\) 7.99310e12 0.262067
\(993\) 6.56035e13 2.14119
\(994\) 3.29158e10 0.00106946
\(995\) 4.69241e12 0.151772
\(996\) −5.59667e13 −1.80203
\(997\) −1.18916e12 −0.0381165 −0.0190583 0.999818i \(-0.506067\pi\)
−0.0190583 + 0.999818i \(0.506067\pi\)
\(998\) −4.28574e12 −0.136754
\(999\) 3.77315e12 0.119856
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 43.10.a.a.1.9 15
3.2 odd 2 387.10.a.c.1.7 15
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.10.a.a.1.9 15 1.1 even 1 trivial
387.10.a.c.1.7 15 3.2 odd 2