Properties

Label 43.8.a.a.1.1
Level $43$
Weight $8$
Character 43.1
Self dual yes
Analytic conductor $13.433$
Analytic rank $1$
Dimension $11$
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: \(1\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Defining polynomial: \(x^{11} - 2 x^{10} - 977 x^{9} + 2592 x^{8} + 344686 x^{7} - 1160956 x^{6} - 53409536 x^{5} + 209758592 x^{4} + 3410917248 x^{3} - 14180732672 x^{2} - 60918607872 x + 238240894976\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 7 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(19.7860\) of defining polynomial
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q-21.7860 q^{2} +54.2998 q^{3} +346.631 q^{4} -409.924 q^{5} -1182.98 q^{6} +476.571 q^{7} -4763.10 q^{8} +761.467 q^{9} +O(q^{10})\) \(q-21.7860 q^{2} +54.2998 q^{3} +346.631 q^{4} -409.924 q^{5} -1182.98 q^{6} +476.571 q^{7} -4763.10 q^{8} +761.467 q^{9} +8930.61 q^{10} +6867.50 q^{11} +18822.0 q^{12} -8276.59 q^{13} -10382.6 q^{14} -22258.8 q^{15} +59400.2 q^{16} -16111.6 q^{17} -16589.3 q^{18} +27534.5 q^{19} -142092. q^{20} +25877.7 q^{21} -149616. q^{22} -69123.4 q^{23} -258635. q^{24} +89912.4 q^{25} +180314. q^{26} -77406.2 q^{27} +165194. q^{28} -111490. q^{29} +484930. q^{30} -169599. q^{31} -684418. q^{32} +372904. q^{33} +351008. q^{34} -195358. q^{35} +263948. q^{36} -60102.6 q^{37} -599867. q^{38} -449417. q^{39} +1.95251e6 q^{40} -813919. q^{41} -563772. q^{42} +79507.0 q^{43} +2.38049e6 q^{44} -312143. q^{45} +1.50592e6 q^{46} -624677. q^{47} +3.22542e6 q^{48} -596423. q^{49} -1.95883e6 q^{50} -874857. q^{51} -2.86892e6 q^{52} +365470. q^{53} +1.68637e6 q^{54} -2.81515e6 q^{55} -2.26995e6 q^{56} +1.49512e6 q^{57} +2.42892e6 q^{58} +1.43500e6 q^{59} -7.71558e6 q^{60} -153457. q^{61} +3.69489e6 q^{62} +362893. q^{63} +7.30752e6 q^{64} +3.39277e6 q^{65} -8.12409e6 q^{66} -1.27944e6 q^{67} -5.58478e6 q^{68} -3.75339e6 q^{69} +4.25607e6 q^{70} +821210. q^{71} -3.62694e6 q^{72} +4.08244e6 q^{73} +1.30940e6 q^{74} +4.88222e6 q^{75} +9.54431e6 q^{76} +3.27285e6 q^{77} +9.79101e6 q^{78} -6.74067e6 q^{79} -2.43496e7 q^{80} -5.86847e6 q^{81} +1.77321e7 q^{82} +281235. q^{83} +8.97001e6 q^{84} +6.60453e6 q^{85} -1.73214e6 q^{86} -6.05387e6 q^{87} -3.27106e7 q^{88} +9.40951e6 q^{89} +6.80036e6 q^{90} -3.94438e6 q^{91} -2.39603e7 q^{92} -9.20921e6 q^{93} +1.36092e7 q^{94} -1.12870e7 q^{95} -3.71638e7 q^{96} -6.78204e6 q^{97} +1.29937e7 q^{98} +5.22937e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} + O(q^{10}) \) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} - 1333q^{10} + 1333q^{11} + 5089q^{12} - 17967q^{13} - 22352q^{14} - 49504q^{15} - 34406q^{16} - 63095q^{17} - 165931q^{18} - 54524q^{19} - 280995q^{20} - 139788q^{21} - 289358q^{22} - 138139q^{23} - 429583q^{24} + 3455q^{25} - 132946q^{26} - 240356q^{27} - 12704q^{28} - 308658q^{29} + 421284q^{30} - 209523q^{31} - 644934q^{32} + 96814q^{33} + 762435q^{34} - 578892q^{35} + 426161q^{36} - 298472q^{37} - 369707q^{38} + 292298q^{39} + 2633173q^{40} - 1346735q^{41} + 1173266q^{42} + 874577q^{43} + 3134292q^{44} + 1893784q^{45} + 3588111q^{46} + 499284q^{47} + 5647533q^{48} + 2544563q^{49} + 3049745q^{50} + 1258424q^{51} + 983088q^{52} - 2210495q^{53} + 6789698q^{54} - 1855072q^{55} - 469976q^{56} - 1238444q^{57} + 4397067q^{58} - 5824216q^{59} - 2889372q^{60} - 4453034q^{61} + 1002789q^{62} - 6240564q^{63} + 4757538q^{64} - 2162872q^{65} - 258940q^{66} - 6859513q^{67} - 9397005q^{68} - 10040030q^{69} + 845078q^{70} - 10726554q^{71} + 1199517q^{72} - 4456898q^{73} + 1046637q^{74} - 3349114q^{75} + 5861267q^{76} - 17019816q^{77} + 1999122q^{78} - 15541320q^{79} - 15680911q^{80} - 12976697q^{81} + 20233655q^{82} - 11146767q^{83} + 12348278q^{84} - 12471976q^{85} - 1908168q^{86} - 18648900q^{87} - 24463544q^{88} - 13531356q^{89} + 20858990q^{90} - 19746448q^{91} - 26023161q^{92} - 21903110q^{93} + 20288857q^{94} - 12291624q^{95} - 13954503q^{96} - 10999901q^{97} + 29909168q^{98} + 29396057q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −21.7860 −1.92563 −0.962815 0.270160i \(-0.912923\pi\)
−0.962815 + 0.270160i \(0.912923\pi\)
\(3\) 54.2998 1.16111 0.580555 0.814221i \(-0.302835\pi\)
0.580555 + 0.814221i \(0.302835\pi\)
\(4\) 346.631 2.70805
\(5\) −409.924 −1.46659 −0.733294 0.679912i \(-0.762018\pi\)
−0.733294 + 0.679912i \(0.762018\pi\)
\(6\) −1182.98 −2.23587
\(7\) 476.571 0.525151 0.262576 0.964911i \(-0.415428\pi\)
0.262576 + 0.964911i \(0.415428\pi\)
\(8\) −4763.10 −3.28908
\(9\) 761.467 0.348179
\(10\) 8930.61 2.82411
\(11\) 6867.50 1.55569 0.777847 0.628453i \(-0.216312\pi\)
0.777847 + 0.628453i \(0.216312\pi\)
\(12\) 18822.0 3.14435
\(13\) −8276.59 −1.04484 −0.522420 0.852688i \(-0.674971\pi\)
−0.522420 + 0.852688i \(0.674971\pi\)
\(14\) −10382.6 −1.01125
\(15\) −22258.8 −1.70287
\(16\) 59400.2 3.62550
\(17\) −16111.6 −0.795367 −0.397684 0.917523i \(-0.630186\pi\)
−0.397684 + 0.917523i \(0.630186\pi\)
\(18\) −16589.3 −0.670463
\(19\) 27534.5 0.920957 0.460479 0.887671i \(-0.347678\pi\)
0.460479 + 0.887671i \(0.347678\pi\)
\(20\) −142092. −3.97160
\(21\) 25877.7 0.609759
\(22\) −149616. −2.99569
\(23\) −69123.4 −1.18462 −0.592308 0.805711i \(-0.701783\pi\)
−0.592308 + 0.805711i \(0.701783\pi\)
\(24\) −258635. −3.81899
\(25\) 89912.4 1.15088
\(26\) 180314. 2.01198
\(27\) −77406.2 −0.756837
\(28\) 165194. 1.42214
\(29\) −111490. −0.848871 −0.424435 0.905458i \(-0.639527\pi\)
−0.424435 + 0.905458i \(0.639527\pi\)
\(30\) 484930. 3.27910
\(31\) −169599. −1.02249 −0.511244 0.859436i \(-0.670815\pi\)
−0.511244 + 0.859436i \(0.670815\pi\)
\(32\) −684418. −3.69230
\(33\) 372904. 1.80633
\(34\) 351008. 1.53158
\(35\) −195358. −0.770180
\(36\) 263948. 0.942886
\(37\) −60102.6 −0.195068 −0.0975342 0.995232i \(-0.531096\pi\)
−0.0975342 + 0.995232i \(0.531096\pi\)
\(38\) −599867. −1.77342
\(39\) −449417. −1.21317
\(40\) 1.95251e6 4.82372
\(41\) −813919. −1.84433 −0.922163 0.386802i \(-0.873580\pi\)
−0.922163 + 0.386802i \(0.873580\pi\)
\(42\) −563772. −1.17417
\(43\) 79507.0 0.152499
\(44\) 2.38049e6 4.21290
\(45\) −312143. −0.510634
\(46\) 1.50592e6 2.28113
\(47\) −624677. −0.877633 −0.438817 0.898577i \(-0.644602\pi\)
−0.438817 + 0.898577i \(0.644602\pi\)
\(48\) 3.22542e6 4.20961
\(49\) −596423. −0.724216
\(50\) −1.95883e6 −2.21617
\(51\) −874857. −0.923510
\(52\) −2.86892e6 −2.82948
\(53\) 365470. 0.337199 0.168600 0.985685i \(-0.446076\pi\)
0.168600 + 0.985685i \(0.446076\pi\)
\(54\) 1.68637e6 1.45739
\(55\) −2.81515e6 −2.28156
\(56\) −2.26995e6 −1.72726
\(57\) 1.49512e6 1.06933
\(58\) 2.42892e6 1.63461
\(59\) 1.43500e6 0.909643 0.454822 0.890583i \(-0.349703\pi\)
0.454822 + 0.890583i \(0.349703\pi\)
\(60\) −7.71558e6 −4.61147
\(61\) −153457. −0.0865630 −0.0432815 0.999063i \(-0.513781\pi\)
−0.0432815 + 0.999063i \(0.513781\pi\)
\(62\) 3.69489e6 1.96893
\(63\) 362893. 0.182846
\(64\) 7.30752e6 3.48450
\(65\) 3.39277e6 1.53235
\(66\) −8.12409e6 −3.47833
\(67\) −1.27944e6 −0.519708 −0.259854 0.965648i \(-0.583674\pi\)
−0.259854 + 0.965648i \(0.583674\pi\)
\(68\) −5.58478e6 −2.15390
\(69\) −3.75339e6 −1.37547
\(70\) 4.25607e6 1.48308
\(71\) 821210. 0.272302 0.136151 0.990688i \(-0.456527\pi\)
0.136151 + 0.990688i \(0.456527\pi\)
\(72\) −3.62694e6 −1.14519
\(73\) 4.08244e6 1.22826 0.614130 0.789205i \(-0.289507\pi\)
0.614130 + 0.789205i \(0.289507\pi\)
\(74\) 1.30940e6 0.375630
\(75\) 4.88222e6 1.33630
\(76\) 9.54431e6 2.49400
\(77\) 3.27285e6 0.816975
\(78\) 9.79101e6 2.33613
\(79\) −6.74067e6 −1.53818 −0.769091 0.639139i \(-0.779291\pi\)
−0.769091 + 0.639139i \(0.779291\pi\)
\(80\) −2.43496e7 −5.31712
\(81\) −5.86847e6 −1.22695
\(82\) 1.77321e7 3.55149
\(83\) 281235. 0.0539878 0.0269939 0.999636i \(-0.491407\pi\)
0.0269939 + 0.999636i \(0.491407\pi\)
\(84\) 8.97001e6 1.65126
\(85\) 6.60453e6 1.16648
\(86\) −1.73214e6 −0.293656
\(87\) −6.05387e6 −0.985633
\(88\) −3.27106e7 −5.11681
\(89\) 9.40951e6 1.41482 0.707411 0.706802i \(-0.249863\pi\)
0.707411 + 0.706802i \(0.249863\pi\)
\(90\) 6.80036e6 0.983293
\(91\) −3.94438e6 −0.548699
\(92\) −2.39603e7 −3.20801
\(93\) −9.20921e6 −1.18722
\(94\) 1.36092e7 1.69000
\(95\) −1.12870e7 −1.35066
\(96\) −3.71638e7 −4.28717
\(97\) −6.78204e6 −0.754500 −0.377250 0.926112i \(-0.623130\pi\)
−0.377250 + 0.926112i \(0.623130\pi\)
\(98\) 1.29937e7 1.39457
\(99\) 5.22937e6 0.541660
\(100\) 3.11664e7 3.11664
\(101\) 1.37928e7 1.33208 0.666038 0.745918i \(-0.267989\pi\)
0.666038 + 0.745918i \(0.267989\pi\)
\(102\) 1.90597e7 1.77834
\(103\) −1.39555e7 −1.25839 −0.629197 0.777246i \(-0.716616\pi\)
−0.629197 + 0.777246i \(0.716616\pi\)
\(104\) 3.94222e7 3.43656
\(105\) −1.06079e7 −0.894264
\(106\) −7.96214e6 −0.649321
\(107\) 4.53042e6 0.357516 0.178758 0.983893i \(-0.442792\pi\)
0.178758 + 0.983893i \(0.442792\pi\)
\(108\) −2.68314e7 −2.04956
\(109\) 1.43535e7 1.06161 0.530806 0.847493i \(-0.321889\pi\)
0.530806 + 0.847493i \(0.321889\pi\)
\(110\) 6.13310e7 4.39345
\(111\) −3.26356e6 −0.226496
\(112\) 2.83084e7 1.90394
\(113\) −1.27510e7 −0.831322 −0.415661 0.909520i \(-0.636450\pi\)
−0.415661 + 0.909520i \(0.636450\pi\)
\(114\) −3.25727e7 −2.05914
\(115\) 2.83353e7 1.73734
\(116\) −3.86458e7 −2.29879
\(117\) −6.30235e6 −0.363791
\(118\) −3.12630e7 −1.75164
\(119\) −7.67832e6 −0.417688
\(120\) 1.06021e8 5.60088
\(121\) 2.76754e7 1.42019
\(122\) 3.34322e6 0.166688
\(123\) −4.41956e7 −2.14147
\(124\) −5.87884e7 −2.76895
\(125\) −4.83193e6 −0.221277
\(126\) −7.90599e6 −0.352095
\(127\) −1.14084e7 −0.494210 −0.247105 0.968989i \(-0.579479\pi\)
−0.247105 + 0.968989i \(0.579479\pi\)
\(128\) −7.15964e7 −3.01756
\(129\) 4.31721e6 0.177068
\(130\) −7.39150e7 −2.95074
\(131\) 3.48381e6 0.135396 0.0676978 0.997706i \(-0.478435\pi\)
0.0676978 + 0.997706i \(0.478435\pi\)
\(132\) 1.29260e8 4.89165
\(133\) 1.31221e7 0.483642
\(134\) 2.78740e7 1.00077
\(135\) 3.17306e7 1.10997
\(136\) 7.67412e7 2.61603
\(137\) 2.52123e7 0.837703 0.418852 0.908055i \(-0.362433\pi\)
0.418852 + 0.908055i \(0.362433\pi\)
\(138\) 8.17714e7 2.64865
\(139\) 2.46536e7 0.778625 0.389313 0.921106i \(-0.372713\pi\)
0.389313 + 0.921106i \(0.372713\pi\)
\(140\) −6.77170e7 −2.08569
\(141\) −3.39198e7 −1.01903
\(142\) −1.78909e7 −0.524352
\(143\) −5.68395e7 −1.62545
\(144\) 4.52313e7 1.26232
\(145\) 4.57023e7 1.24494
\(146\) −8.89402e7 −2.36518
\(147\) −3.23857e7 −0.840896
\(148\) −2.08334e7 −0.528256
\(149\) −4.27780e7 −1.05942 −0.529710 0.848179i \(-0.677699\pi\)
−0.529710 + 0.848179i \(0.677699\pi\)
\(150\) −1.06364e8 −2.57322
\(151\) 6.03840e7 1.42726 0.713629 0.700524i \(-0.247050\pi\)
0.713629 + 0.700524i \(0.247050\pi\)
\(152\) −1.31149e8 −3.02910
\(153\) −1.22685e7 −0.276930
\(154\) −7.13024e7 −1.57319
\(155\) 6.95228e7 1.49957
\(156\) −1.55782e8 −3.28534
\(157\) −5.70927e7 −1.17742 −0.588711 0.808344i \(-0.700364\pi\)
−0.588711 + 0.808344i \(0.700364\pi\)
\(158\) 1.46852e8 2.96197
\(159\) 1.98450e7 0.391526
\(160\) 2.80559e8 5.41508
\(161\) −3.29422e7 −0.622103
\(162\) 1.27851e8 2.36265
\(163\) −2.03054e7 −0.367245 −0.183622 0.982997i \(-0.558782\pi\)
−0.183622 + 0.982997i \(0.558782\pi\)
\(164\) −2.82129e8 −4.99453
\(165\) −1.52862e8 −2.64915
\(166\) −6.12699e6 −0.103961
\(167\) 3.18949e7 0.529925 0.264962 0.964259i \(-0.414640\pi\)
0.264962 + 0.964259i \(0.414640\pi\)
\(168\) −1.23258e8 −2.00555
\(169\) 5.75340e6 0.0916898
\(170\) −1.43886e8 −2.24620
\(171\) 2.09666e7 0.320658
\(172\) 2.75596e7 0.412974
\(173\) −2.81474e7 −0.413311 −0.206656 0.978414i \(-0.566258\pi\)
−0.206656 + 0.978414i \(0.566258\pi\)
\(174\) 1.31890e8 1.89797
\(175\) 4.28496e7 0.604385
\(176\) 4.07931e8 5.64017
\(177\) 7.79204e7 1.05620
\(178\) −2.04996e8 −2.72443
\(179\) −3.87229e6 −0.0504641 −0.0252321 0.999682i \(-0.508032\pi\)
−0.0252321 + 0.999682i \(0.508032\pi\)
\(180\) −1.08198e8 −1.38283
\(181\) −5.34400e7 −0.669871 −0.334935 0.942241i \(-0.608715\pi\)
−0.334935 + 0.942241i \(0.608715\pi\)
\(182\) 8.59324e7 1.05659
\(183\) −8.33268e6 −0.100509
\(184\) 3.29242e8 3.89630
\(185\) 2.46375e7 0.286085
\(186\) 2.00632e8 2.28615
\(187\) −1.10647e8 −1.23735
\(188\) −2.16532e8 −2.37668
\(189\) −3.68895e7 −0.397454
\(190\) 2.45900e8 2.60088
\(191\) −1.82945e7 −0.189978 −0.0949891 0.995478i \(-0.530282\pi\)
−0.0949891 + 0.995478i \(0.530282\pi\)
\(192\) 3.96797e8 4.04589
\(193\) 8.24942e7 0.825986 0.412993 0.910734i \(-0.364483\pi\)
0.412993 + 0.910734i \(0.364483\pi\)
\(194\) 1.47754e8 1.45289
\(195\) 1.84227e8 1.77923
\(196\) −2.06739e8 −1.96122
\(197\) 7.08196e7 0.659966 0.329983 0.943987i \(-0.392957\pi\)
0.329983 + 0.943987i \(0.392957\pi\)
\(198\) −1.13927e8 −1.04304
\(199\) 1.61028e8 1.44849 0.724247 0.689540i \(-0.242187\pi\)
0.724247 + 0.689540i \(0.242187\pi\)
\(200\) −4.28262e8 −3.78533
\(201\) −6.94736e7 −0.603439
\(202\) −3.00491e8 −2.56509
\(203\) −5.31327e7 −0.445785
\(204\) −3.03252e8 −2.50091
\(205\) 3.33645e8 2.70486
\(206\) 3.04036e8 2.42320
\(207\) −5.26352e7 −0.412458
\(208\) −4.91631e8 −3.78807
\(209\) 1.89093e8 1.43273
\(210\) 2.31103e8 1.72202
\(211\) −2.05501e8 −1.50600 −0.753002 0.658018i \(-0.771395\pi\)
−0.753002 + 0.658018i \(0.771395\pi\)
\(212\) 1.26683e8 0.913153
\(213\) 4.45915e7 0.316172
\(214\) −9.86999e7 −0.688444
\(215\) −3.25918e7 −0.223652
\(216\) 3.68693e8 2.48930
\(217\) −8.08261e7 −0.536961
\(218\) −3.12706e8 −2.04427
\(219\) 2.21676e8 1.42615
\(220\) −9.75818e8 −6.17859
\(221\) 1.33349e8 0.831031
\(222\) 7.10999e7 0.436148
\(223\) −8.77217e7 −0.529712 −0.264856 0.964288i \(-0.585324\pi\)
−0.264856 + 0.964288i \(0.585324\pi\)
\(224\) −3.26174e8 −1.93901
\(225\) 6.84653e7 0.400711
\(226\) 2.77793e8 1.60082
\(227\) −1.96811e8 −1.11676 −0.558378 0.829586i \(-0.688576\pi\)
−0.558378 + 0.829586i \(0.688576\pi\)
\(228\) 5.18254e8 2.89581
\(229\) −1.91603e8 −1.05434 −0.527168 0.849761i \(-0.676746\pi\)
−0.527168 + 0.849761i \(0.676746\pi\)
\(230\) −6.17314e8 −3.34548
\(231\) 1.77715e8 0.948598
\(232\) 5.31036e8 2.79201
\(233\) −7.48220e7 −0.387511 −0.193755 0.981050i \(-0.562067\pi\)
−0.193755 + 0.981050i \(0.562067\pi\)
\(234\) 1.37303e8 0.700527
\(235\) 2.56070e8 1.28713
\(236\) 4.97417e8 2.46336
\(237\) −3.66017e8 −1.78600
\(238\) 1.67280e8 0.804313
\(239\) 3.09828e8 1.46801 0.734004 0.679145i \(-0.237649\pi\)
0.734004 + 0.679145i \(0.237649\pi\)
\(240\) −1.32218e9 −6.17376
\(241\) −2.29329e8 −1.05536 −0.527678 0.849445i \(-0.676937\pi\)
−0.527678 + 0.849445i \(0.676937\pi\)
\(242\) −6.02937e8 −2.73475
\(243\) −1.49369e8 −0.667788
\(244\) −5.31929e7 −0.234417
\(245\) 2.44488e8 1.06213
\(246\) 9.62847e8 4.12367
\(247\) −2.27892e8 −0.962253
\(248\) 8.07818e8 3.36305
\(249\) 1.52710e7 0.0626859
\(250\) 1.05269e8 0.426097
\(251\) 7.43216e7 0.296659 0.148329 0.988938i \(-0.452610\pi\)
0.148329 + 0.988938i \(0.452610\pi\)
\(252\) 1.25790e8 0.495158
\(253\) −4.74705e8 −1.84290
\(254\) 2.48544e8 0.951666
\(255\) 3.58625e8 1.35441
\(256\) 6.24437e8 2.32621
\(257\) 8.21908e7 0.302035 0.151017 0.988531i \(-0.451745\pi\)
0.151017 + 0.988531i \(0.451745\pi\)
\(258\) −9.40549e7 −0.340967
\(259\) −2.86431e7 −0.102440
\(260\) 1.17604e9 4.14968
\(261\) −8.48957e7 −0.295559
\(262\) −7.58983e7 −0.260722
\(263\) 4.73912e8 1.60640 0.803198 0.595713i \(-0.203130\pi\)
0.803198 + 0.595713i \(0.203130\pi\)
\(264\) −1.77618e9 −5.94118
\(265\) −1.49815e8 −0.494532
\(266\) −2.85879e8 −0.931315
\(267\) 5.10934e8 1.64277
\(268\) −4.43495e8 −1.40740
\(269\) −1.61866e8 −0.507016 −0.253508 0.967333i \(-0.581584\pi\)
−0.253508 + 0.967333i \(0.581584\pi\)
\(270\) −6.91284e8 −2.13739
\(271\) 9.56080e7 0.291811 0.145905 0.989299i \(-0.453390\pi\)
0.145905 + 0.989299i \(0.453390\pi\)
\(272\) −9.57033e8 −2.88361
\(273\) −2.14179e8 −0.637100
\(274\) −5.49275e8 −1.61311
\(275\) 6.17473e8 1.79042
\(276\) −1.30104e9 −3.72485
\(277\) 5.77704e8 1.63315 0.816575 0.577240i \(-0.195870\pi\)
0.816575 + 0.577240i \(0.195870\pi\)
\(278\) −5.37104e8 −1.49934
\(279\) −1.29144e8 −0.356008
\(280\) 9.30507e8 2.53318
\(281\) 1.72244e8 0.463096 0.231548 0.972823i \(-0.425621\pi\)
0.231548 + 0.972823i \(0.425621\pi\)
\(282\) 7.38978e8 1.96227
\(283\) −3.33472e8 −0.874595 −0.437298 0.899317i \(-0.644064\pi\)
−0.437298 + 0.899317i \(0.644064\pi\)
\(284\) 2.84657e8 0.737407
\(285\) −6.12884e8 −1.56827
\(286\) 1.23831e9 3.13002
\(287\) −3.87890e8 −0.968549
\(288\) −5.21162e8 −1.28558
\(289\) −1.50755e8 −0.367391
\(290\) −9.95670e8 −2.39730
\(291\) −3.68263e8 −0.876058
\(292\) 1.41510e9 3.32619
\(293\) −2.96954e8 −0.689688 −0.344844 0.938660i \(-0.612068\pi\)
−0.344844 + 0.938660i \(0.612068\pi\)
\(294\) 7.05555e8 1.61925
\(295\) −5.88242e8 −1.33407
\(296\) 2.86274e8 0.641596
\(297\) −5.31587e8 −1.17741
\(298\) 9.31962e8 2.04005
\(299\) 5.72106e8 1.23773
\(300\) 1.69233e9 3.61877
\(301\) 3.78907e7 0.0800848
\(302\) −1.31553e9 −2.74837
\(303\) 7.48949e8 1.54669
\(304\) 1.63555e9 3.33893
\(305\) 6.29056e7 0.126952
\(306\) 2.67281e8 0.533265
\(307\) 3.16973e8 0.625228 0.312614 0.949880i \(-0.398795\pi\)
0.312614 + 0.949880i \(0.398795\pi\)
\(308\) 1.13447e9 2.21241
\(309\) −7.57783e8 −1.46113
\(310\) −1.51462e9 −2.88761
\(311\) −8.80151e8 −1.65919 −0.829594 0.558367i \(-0.811428\pi\)
−0.829594 + 0.558367i \(0.811428\pi\)
\(312\) 2.14062e9 3.99023
\(313\) −8.29846e8 −1.52965 −0.764825 0.644238i \(-0.777175\pi\)
−0.764825 + 0.644238i \(0.777175\pi\)
\(314\) 1.24382e9 2.26728
\(315\) −1.48758e8 −0.268160
\(316\) −2.33652e9 −4.16548
\(317\) −2.04074e8 −0.359817 −0.179908 0.983683i \(-0.557580\pi\)
−0.179908 + 0.983683i \(0.557580\pi\)
\(318\) −4.32343e8 −0.753934
\(319\) −7.65655e8 −1.32058
\(320\) −2.99553e9 −5.11032
\(321\) 2.46001e8 0.415116
\(322\) 7.17680e8 1.19794
\(323\) −4.43625e8 −0.732499
\(324\) −2.03419e9 −3.32265
\(325\) −7.44168e8 −1.20248
\(326\) 4.42375e8 0.707178
\(327\) 7.79393e8 1.23265
\(328\) 3.87677e9 6.06614
\(329\) −2.97703e8 −0.460890
\(330\) 3.33026e9 5.10128
\(331\) −4.43600e8 −0.672347 −0.336174 0.941800i \(-0.609133\pi\)
−0.336174 + 0.941800i \(0.609133\pi\)
\(332\) 9.74847e7 0.146202
\(333\) −4.57661e7 −0.0679187
\(334\) −6.94864e8 −1.02044
\(335\) 5.24475e8 0.762198
\(336\) 1.53714e9 2.21068
\(337\) 9.56240e8 1.36101 0.680506 0.732742i \(-0.261760\pi\)
0.680506 + 0.732742i \(0.261760\pi\)
\(338\) −1.25344e8 −0.176561
\(339\) −6.92376e8 −0.965257
\(340\) 2.28933e9 3.15888
\(341\) −1.16472e9 −1.59068
\(342\) −4.56779e8 −0.617468
\(343\) −6.76714e8 −0.905474
\(344\) −3.78700e8 −0.501580
\(345\) 1.53860e9 2.01725
\(346\) 6.13220e8 0.795885
\(347\) 7.21806e8 0.927401 0.463700 0.885992i \(-0.346521\pi\)
0.463700 + 0.885992i \(0.346521\pi\)
\(348\) −2.09846e9 −2.66915
\(349\) 4.87486e8 0.613865 0.306933 0.951731i \(-0.400697\pi\)
0.306933 + 0.951731i \(0.400697\pi\)
\(350\) −9.33523e8 −1.16382
\(351\) 6.40659e8 0.790773
\(352\) −4.70024e9 −5.74409
\(353\) 2.31713e8 0.280375 0.140187 0.990125i \(-0.455229\pi\)
0.140187 + 0.990125i \(0.455229\pi\)
\(354\) −1.69758e9 −2.03385
\(355\) −3.36633e8 −0.399354
\(356\) 3.26163e9 3.83142
\(357\) −4.16931e8 −0.484982
\(358\) 8.43619e7 0.0971753
\(359\) −1.05857e8 −0.120750 −0.0603751 0.998176i \(-0.519230\pi\)
−0.0603751 + 0.998176i \(0.519230\pi\)
\(360\) 1.48677e9 1.67952
\(361\) −1.35724e8 −0.151838
\(362\) 1.16424e9 1.28992
\(363\) 1.50277e9 1.64899
\(364\) −1.36724e9 −1.48591
\(365\) −1.67349e9 −1.80135
\(366\) 1.81536e8 0.193544
\(367\) 2.03126e8 0.214504 0.107252 0.994232i \(-0.465795\pi\)
0.107252 + 0.994232i \(0.465795\pi\)
\(368\) −4.10595e9 −4.29483
\(369\) −6.19772e8 −0.642155
\(370\) −5.36752e8 −0.550894
\(371\) 1.74172e8 0.177080
\(372\) −3.19220e9 −3.21506
\(373\) −7.70434e8 −0.768696 −0.384348 0.923188i \(-0.625574\pi\)
−0.384348 + 0.923188i \(0.625574\pi\)
\(374\) 2.41055e9 2.38268
\(375\) −2.62373e8 −0.256927
\(376\) 2.97540e9 2.88661
\(377\) 9.22754e8 0.886934
\(378\) 8.03676e8 0.765349
\(379\) 9.79682e8 0.924375 0.462187 0.886782i \(-0.347065\pi\)
0.462187 + 0.886782i \(0.347065\pi\)
\(380\) −3.91244e9 −3.65767
\(381\) −6.19473e8 −0.573833
\(382\) 3.98564e8 0.365828
\(383\) −1.47413e9 −1.34072 −0.670362 0.742034i \(-0.733861\pi\)
−0.670362 + 0.742034i \(0.733861\pi\)
\(384\) −3.88767e9 −3.50372
\(385\) −1.34162e9 −1.19816
\(386\) −1.79722e9 −1.59054
\(387\) 6.05419e7 0.0530967
\(388\) −2.35086e9 −2.04323
\(389\) 1.14843e9 0.989194 0.494597 0.869122i \(-0.335316\pi\)
0.494597 + 0.869122i \(0.335316\pi\)
\(390\) −4.01357e9 −3.42613
\(391\) 1.11369e9 0.942205
\(392\) 2.84082e9 2.38201
\(393\) 1.89170e8 0.157209
\(394\) −1.54288e9 −1.27085
\(395\) 2.76316e9 2.25588
\(396\) 1.81266e9 1.46684
\(397\) 1.53079e9 1.22786 0.613928 0.789362i \(-0.289588\pi\)
0.613928 + 0.789362i \(0.289588\pi\)
\(398\) −3.50817e9 −2.78927
\(399\) 7.12529e8 0.561562
\(400\) 5.34082e9 4.17251
\(401\) −5.05503e8 −0.391488 −0.195744 0.980655i \(-0.562712\pi\)
−0.195744 + 0.980655i \(0.562712\pi\)
\(402\) 1.51355e9 1.16200
\(403\) 1.40370e9 1.06834
\(404\) 4.78103e9 3.60734
\(405\) 2.40562e9 1.79943
\(406\) 1.15755e9 0.858418
\(407\) −4.12754e8 −0.303467
\(408\) 4.16703e9 3.03750
\(409\) 1.60782e9 1.16200 0.580998 0.813905i \(-0.302662\pi\)
0.580998 + 0.813905i \(0.302662\pi\)
\(410\) −7.26879e9 −5.20857
\(411\) 1.36902e9 0.972666
\(412\) −4.83742e9 −3.40780
\(413\) 6.83881e8 0.477700
\(414\) 1.14671e9 0.794242
\(415\) −1.15285e8 −0.0791779
\(416\) 5.66465e9 3.85786
\(417\) 1.33868e9 0.904070
\(418\) −4.11959e9 −2.75891
\(419\) 2.86738e9 1.90430 0.952152 0.305625i \(-0.0988654\pi\)
0.952152 + 0.305625i \(0.0988654\pi\)
\(420\) −3.67702e9 −2.42172
\(421\) 1.98494e9 1.29646 0.648231 0.761444i \(-0.275509\pi\)
0.648231 + 0.761444i \(0.275509\pi\)
\(422\) 4.47706e9 2.90001
\(423\) −4.75671e8 −0.305573
\(424\) −1.74077e9 −1.10908
\(425\) −1.44863e9 −0.915371
\(426\) −9.71472e8 −0.608831
\(427\) −7.31331e7 −0.0454586
\(428\) 1.57038e9 0.968172
\(429\) −3.08637e9 −1.88733
\(430\) 7.10046e8 0.430672
\(431\) 7.78770e7 0.0468531 0.0234266 0.999726i \(-0.492542\pi\)
0.0234266 + 0.999726i \(0.492542\pi\)
\(432\) −4.59794e9 −2.74391
\(433\) −1.06362e9 −0.629619 −0.314810 0.949155i \(-0.601941\pi\)
−0.314810 + 0.949155i \(0.601941\pi\)
\(434\) 1.76088e9 1.03399
\(435\) 2.48162e9 1.44552
\(436\) 4.97537e9 2.87490
\(437\) −1.90328e9 −1.09098
\(438\) −4.82944e9 −2.74623
\(439\) −1.17090e9 −0.660531 −0.330266 0.943888i \(-0.607138\pi\)
−0.330266 + 0.943888i \(0.607138\pi\)
\(440\) 1.34088e10 7.50424
\(441\) −4.54156e8 −0.252157
\(442\) −2.90515e9 −1.60026
\(443\) −1.61819e8 −0.0884335 −0.0442167 0.999022i \(-0.514079\pi\)
−0.0442167 + 0.999022i \(0.514079\pi\)
\(444\) −1.13125e9 −0.613364
\(445\) −3.85718e9 −2.07496
\(446\) 1.91111e9 1.02003
\(447\) −2.32283e9 −1.23010
\(448\) 3.48255e9 1.82989
\(449\) 1.17696e9 0.613618 0.306809 0.951771i \(-0.400739\pi\)
0.306809 + 0.951771i \(0.400739\pi\)
\(450\) −1.49159e9 −0.771622
\(451\) −5.58959e9 −2.86921
\(452\) −4.41989e9 −2.25127
\(453\) 3.27884e9 1.65721
\(454\) 4.28773e9 2.15046
\(455\) 1.61689e9 0.804714
\(456\) −7.12139e9 −3.51712
\(457\) 7.63423e8 0.374161 0.187081 0.982345i \(-0.440097\pi\)
0.187081 + 0.982345i \(0.440097\pi\)
\(458\) 4.17428e9 2.03026
\(459\) 1.24714e9 0.601963
\(460\) 9.82190e9 4.70482
\(461\) −3.36981e9 −1.60196 −0.800980 0.598691i \(-0.795688\pi\)
−0.800980 + 0.598691i \(0.795688\pi\)
\(462\) −3.87170e9 −1.82665
\(463\) −2.92488e9 −1.36954 −0.684769 0.728761i \(-0.740097\pi\)
−0.684769 + 0.728761i \(0.740097\pi\)
\(464\) −6.62251e9 −3.07758
\(465\) 3.77507e9 1.74116
\(466\) 1.63007e9 0.746202
\(467\) −2.12116e9 −0.963748 −0.481874 0.876240i \(-0.660044\pi\)
−0.481874 + 0.876240i \(0.660044\pi\)
\(468\) −2.18459e9 −0.985165
\(469\) −6.09746e8 −0.272925
\(470\) −5.57875e9 −2.47853
\(471\) −3.10012e9 −1.36712
\(472\) −6.83507e9 −2.99189
\(473\) 5.46014e8 0.237241
\(474\) 7.97405e9 3.43918
\(475\) 2.47569e9 1.05991
\(476\) −2.66154e9 −1.13112
\(477\) 2.78293e8 0.117406
\(478\) −6.74992e9 −2.82684
\(479\) 3.77482e8 0.156936 0.0784678 0.996917i \(-0.474997\pi\)
0.0784678 + 0.996917i \(0.474997\pi\)
\(480\) 1.52343e10 6.28750
\(481\) 4.97444e8 0.203815
\(482\) 4.99616e9 2.03223
\(483\) −1.78875e9 −0.722330
\(484\) 9.59315e9 3.84594
\(485\) 2.78012e9 1.10654
\(486\) 3.25416e9 1.28591
\(487\) −5.49674e8 −0.215652 −0.107826 0.994170i \(-0.534389\pi\)
−0.107826 + 0.994170i \(0.534389\pi\)
\(488\) 7.30931e8 0.284713
\(489\) −1.10258e9 −0.426412
\(490\) −5.32642e9 −2.04526
\(491\) −3.90286e9 −1.48798 −0.743991 0.668190i \(-0.767069\pi\)
−0.743991 + 0.668190i \(0.767069\pi\)
\(492\) −1.53196e10 −5.79921
\(493\) 1.79628e9 0.675164
\(494\) 4.96485e9 1.85294
\(495\) −2.14364e9 −0.794391
\(496\) −1.00742e10 −3.70703
\(497\) 3.91365e8 0.142999
\(498\) −3.32694e8 −0.120710
\(499\) 5.05727e9 1.82207 0.911033 0.412334i \(-0.135286\pi\)
0.911033 + 0.412334i \(0.135286\pi\)
\(500\) −1.67490e9 −0.599229
\(501\) 1.73189e9 0.615301
\(502\) −1.61917e9 −0.571255
\(503\) 2.90093e9 1.01636 0.508182 0.861249i \(-0.330318\pi\)
0.508182 + 0.861249i \(0.330318\pi\)
\(504\) −1.72849e9 −0.601396
\(505\) −5.65402e9 −1.95361
\(506\) 1.03419e10 3.54875
\(507\) 3.12408e8 0.106462
\(508\) −3.95450e9 −1.33835
\(509\) −1.02191e9 −0.343479 −0.171739 0.985142i \(-0.554939\pi\)
−0.171739 + 0.985142i \(0.554939\pi\)
\(510\) −7.81300e9 −2.60809
\(511\) 1.94557e9 0.645022
\(512\) −4.43967e9 −1.46186
\(513\) −2.13134e9 −0.697014
\(514\) −1.79061e9 −0.581608
\(515\) 5.72071e9 1.84554
\(516\) 1.49648e9 0.479509
\(517\) −4.28997e9 −1.36533
\(518\) 6.24020e8 0.197262
\(519\) −1.52840e9 −0.479900
\(520\) −1.61601e10 −5.04002
\(521\) −5.15409e8 −0.159669 −0.0798344 0.996808i \(-0.525439\pi\)
−0.0798344 + 0.996808i \(0.525439\pi\)
\(522\) 1.84954e9 0.569137
\(523\) 4.99074e8 0.152549 0.0762745 0.997087i \(-0.475697\pi\)
0.0762745 + 0.997087i \(0.475697\pi\)
\(524\) 1.20760e9 0.366659
\(525\) 2.32672e9 0.701758
\(526\) −1.03247e10 −3.09332
\(527\) 2.73252e9 0.813253
\(528\) 2.21506e10 6.54887
\(529\) 1.37322e9 0.403316
\(530\) 3.26387e9 0.952286
\(531\) 1.09271e9 0.316718
\(532\) 4.54854e9 1.30973
\(533\) 6.73647e9 1.92702
\(534\) −1.11312e10 −3.16336
\(535\) −1.85713e9 −0.524328
\(536\) 6.09412e9 1.70936
\(537\) −2.10265e8 −0.0585944
\(538\) 3.52641e9 0.976325
\(539\) −4.09594e9 −1.12666
\(540\) 1.09988e10 3.00585
\(541\) 3.98160e9 1.08110 0.540551 0.841311i \(-0.318216\pi\)
0.540551 + 0.841311i \(0.318216\pi\)
\(542\) −2.08292e9 −0.561920
\(543\) −2.90178e9 −0.777794
\(544\) 1.10271e10 2.93673
\(545\) −5.88385e9 −1.55695
\(546\) 4.66611e9 1.22682
\(547\) 4.26163e9 1.11332 0.556660 0.830740i \(-0.312083\pi\)
0.556660 + 0.830740i \(0.312083\pi\)
\(548\) 8.73935e9 2.26854
\(549\) −1.16852e8 −0.0301394
\(550\) −1.34523e10 −3.44768
\(551\) −3.06981e9 −0.781774
\(552\) 1.78777e10 4.52404
\(553\) −3.21240e9 −0.807778
\(554\) −1.25859e10 −3.14484
\(555\) 1.33781e9 0.332176
\(556\) 8.54570e9 2.10856
\(557\) −6.93171e9 −1.69960 −0.849800 0.527104i \(-0.823278\pi\)
−0.849800 + 0.527104i \(0.823278\pi\)
\(558\) 2.81354e9 0.685541
\(559\) −6.58047e8 −0.159337
\(560\) −1.16043e10 −2.79229
\(561\) −6.00808e9 −1.43670
\(562\) −3.75250e9 −0.891752
\(563\) 8.38570e8 0.198043 0.0990216 0.995085i \(-0.468429\pi\)
0.0990216 + 0.995085i \(0.468429\pi\)
\(564\) −1.17577e10 −2.75959
\(565\) 5.22693e9 1.21921
\(566\) 7.26503e9 1.68415
\(567\) −2.79674e9 −0.644334
\(568\) −3.91150e9 −0.895622
\(569\) 7.43736e9 1.69249 0.846244 0.532795i \(-0.178858\pi\)
0.846244 + 0.532795i \(0.178858\pi\)
\(570\) 1.33523e10 3.01991
\(571\) −2.43113e9 −0.546490 −0.273245 0.961944i \(-0.588097\pi\)
−0.273245 + 0.961944i \(0.588097\pi\)
\(572\) −1.97023e10 −4.40181
\(573\) −9.93387e8 −0.220586
\(574\) 8.45058e9 1.86507
\(575\) −6.21505e9 −1.36335
\(576\) 5.56443e9 1.21323
\(577\) −2.78106e8 −0.0602692 −0.0301346 0.999546i \(-0.509594\pi\)
−0.0301346 + 0.999546i \(0.509594\pi\)
\(578\) 3.28434e9 0.707459
\(579\) 4.47942e9 0.959061
\(580\) 1.58418e10 3.37137
\(581\) 1.34028e8 0.0283518
\(582\) 8.02299e9 1.68696
\(583\) 2.50987e9 0.524579
\(584\) −1.94451e10 −4.03985
\(585\) 2.58348e9 0.533531
\(586\) 6.46945e9 1.32809
\(587\) −4.32874e9 −0.883340 −0.441670 0.897178i \(-0.645614\pi\)
−0.441670 + 0.897178i \(0.645614\pi\)
\(588\) −1.12259e10 −2.27719
\(589\) −4.66983e9 −0.941668
\(590\) 1.28155e10 2.56893
\(591\) 3.84549e9 0.766294
\(592\) −3.57011e9 −0.707221
\(593\) −1.16343e9 −0.229112 −0.114556 0.993417i \(-0.536544\pi\)
−0.114556 + 0.993417i \(0.536544\pi\)
\(594\) 1.15812e10 2.26725
\(595\) 3.14753e9 0.612576
\(596\) −1.48282e10 −2.86897
\(597\) 8.74381e9 1.68186
\(598\) −1.24639e10 −2.38342
\(599\) −6.78491e9 −1.28988 −0.644941 0.764232i \(-0.723118\pi\)
−0.644941 + 0.764232i \(0.723118\pi\)
\(600\) −2.32545e10 −4.39519
\(601\) 2.61686e9 0.491723 0.245862 0.969305i \(-0.420929\pi\)
0.245862 + 0.969305i \(0.420929\pi\)
\(602\) −8.25488e8 −0.154214
\(603\) −9.74254e8 −0.180951
\(604\) 2.09310e10 3.86509
\(605\) −1.13448e10 −2.08283
\(606\) −1.63166e10 −2.97835
\(607\) 1.41096e9 0.256067 0.128034 0.991770i \(-0.459133\pi\)
0.128034 + 0.991770i \(0.459133\pi\)
\(608\) −1.88451e10 −3.40045
\(609\) −2.88509e9 −0.517606
\(610\) −1.37046e9 −0.244463
\(611\) 5.17020e9 0.916986
\(612\) −4.25263e9 −0.749941
\(613\) 2.52910e9 0.443459 0.221730 0.975108i \(-0.428830\pi\)
0.221730 + 0.975108i \(0.428830\pi\)
\(614\) −6.90558e9 −1.20396
\(615\) 1.81168e10 3.14065
\(616\) −1.55889e10 −2.68710
\(617\) −3.77565e9 −0.647133 −0.323567 0.946205i \(-0.604882\pi\)
−0.323567 + 0.946205i \(0.604882\pi\)
\(618\) 1.65091e10 2.81360
\(619\) −9.20238e9 −1.55949 −0.779746 0.626096i \(-0.784652\pi\)
−0.779746 + 0.626096i \(0.784652\pi\)
\(620\) 2.40987e10 4.06091
\(621\) 5.35058e9 0.896562
\(622\) 1.91750e10 3.19498
\(623\) 4.48430e9 0.742996
\(624\) −2.66955e10 −4.39837
\(625\) −5.04368e9 −0.826357
\(626\) 1.80790e10 2.94554
\(627\) 1.02677e10 1.66356
\(628\) −1.97901e10 −3.18852
\(629\) 9.68349e8 0.155151
\(630\) 3.24085e9 0.516377
\(631\) −4.06379e9 −0.643915 −0.321958 0.946754i \(-0.604341\pi\)
−0.321958 + 0.946754i \(0.604341\pi\)
\(632\) 3.21065e10 5.05921
\(633\) −1.11587e10 −1.74864
\(634\) 4.44597e9 0.692874
\(635\) 4.67657e9 0.724802
\(636\) 6.87887e9 1.06027
\(637\) 4.93635e9 0.756690
\(638\) 1.66806e10 2.54296
\(639\) 6.25324e8 0.0948096
\(640\) 2.93490e10 4.42552
\(641\) 4.61167e9 0.691599 0.345800 0.938308i \(-0.387608\pi\)
0.345800 + 0.938308i \(0.387608\pi\)
\(642\) −5.35938e9 −0.799359
\(643\) −8.37065e9 −1.24171 −0.620856 0.783925i \(-0.713215\pi\)
−0.620856 + 0.783925i \(0.713215\pi\)
\(644\) −1.14188e10 −1.68469
\(645\) −1.76973e9 −0.259685
\(646\) 9.66483e9 1.41052
\(647\) 1.08886e10 1.58054 0.790272 0.612756i \(-0.209939\pi\)
0.790272 + 0.612756i \(0.209939\pi\)
\(648\) 2.79521e10 4.03554
\(649\) 9.85489e9 1.41513
\(650\) 1.62125e10 2.31554
\(651\) −4.38884e9 −0.623471
\(652\) −7.03849e9 −0.994518
\(653\) 4.93139e9 0.693064 0.346532 0.938038i \(-0.387359\pi\)
0.346532 + 0.938038i \(0.387359\pi\)
\(654\) −1.69799e10 −2.37363
\(655\) −1.42809e9 −0.198569
\(656\) −4.83470e10 −6.68661
\(657\) 3.10865e9 0.427654
\(658\) 6.48576e9 0.887504
\(659\) −9.90535e9 −1.34825 −0.674126 0.738616i \(-0.735480\pi\)
−0.674126 + 0.738616i \(0.735480\pi\)
\(660\) −5.29867e10 −7.17403
\(661\) −6.13396e9 −0.826107 −0.413053 0.910707i \(-0.635538\pi\)
−0.413053 + 0.910707i \(0.635538\pi\)
\(662\) 9.66428e9 1.29469
\(663\) 7.24083e9 0.964920
\(664\) −1.33955e9 −0.177570
\(665\) −5.37907e9 −0.709303
\(666\) 9.97061e8 0.130786
\(667\) 7.70655e9 1.00559
\(668\) 1.10558e10 1.43506
\(669\) −4.76327e9 −0.615055
\(670\) −1.14262e10 −1.46771
\(671\) −1.05387e9 −0.134666
\(672\) −1.77112e10 −2.25141
\(673\) −5.75452e9 −0.727707 −0.363853 0.931456i \(-0.618539\pi\)
−0.363853 + 0.931456i \(0.618539\pi\)
\(674\) −2.08327e10 −2.62081
\(675\) −6.95977e9 −0.871027
\(676\) 1.99431e9 0.248301
\(677\) −3.89999e9 −0.483062 −0.241531 0.970393i \(-0.577650\pi\)
−0.241531 + 0.970393i \(0.577650\pi\)
\(678\) 1.50841e10 1.85873
\(679\) −3.23212e9 −0.396226
\(680\) −3.14580e10 −3.83663
\(681\) −1.06868e10 −1.29668
\(682\) 2.53747e10 3.06306
\(683\) −1.93635e8 −0.0232547 −0.0116273 0.999932i \(-0.503701\pi\)
−0.0116273 + 0.999932i \(0.503701\pi\)
\(684\) 7.26767e9 0.868358
\(685\) −1.03351e10 −1.22856
\(686\) 1.47429e10 1.74361
\(687\) −1.04040e10 −1.22420
\(688\) 4.72273e9 0.552884
\(689\) −3.02485e9 −0.352319
\(690\) −3.35200e10 −3.88448
\(691\) 1.72931e8 0.0199388 0.00996940 0.999950i \(-0.496827\pi\)
0.00996940 + 0.999950i \(0.496827\pi\)
\(692\) −9.75676e9 −1.11927
\(693\) 2.49217e9 0.284453
\(694\) −1.57253e10 −1.78583
\(695\) −1.01061e10 −1.14192
\(696\) 2.88352e10 3.24183
\(697\) 1.31135e10 1.46692
\(698\) −1.06204e10 −1.18208
\(699\) −4.06282e9 −0.449943
\(700\) 1.48530e10 1.63671
\(701\) −1.41045e10 −1.54649 −0.773243 0.634110i \(-0.781367\pi\)
−0.773243 + 0.634110i \(0.781367\pi\)
\(702\) −1.39574e10 −1.52274
\(703\) −1.65489e9 −0.179650
\(704\) 5.01844e10 5.42082
\(705\) 1.39045e10 1.49450
\(706\) −5.04811e9 −0.539898
\(707\) 6.57327e9 0.699541
\(708\) 2.70096e10 2.86024
\(709\) 7.26226e9 0.765262 0.382631 0.923901i \(-0.375018\pi\)
0.382631 + 0.923901i \(0.375018\pi\)
\(710\) 7.33391e9 0.769008
\(711\) −5.13279e9 −0.535562
\(712\) −4.48184e10 −4.65347
\(713\) 1.17233e10 1.21126
\(714\) 9.08327e9 0.933896
\(715\) 2.32998e10 2.38387
\(716\) −1.34226e9 −0.136660
\(717\) 1.68236e10 1.70452
\(718\) 2.30620e9 0.232520
\(719\) −5.40870e9 −0.542677 −0.271339 0.962484i \(-0.587466\pi\)
−0.271339 + 0.962484i \(0.587466\pi\)
\(720\) −1.85414e10 −1.85131
\(721\) −6.65080e9 −0.660846
\(722\) 2.95688e9 0.292384
\(723\) −1.24525e10 −1.22538
\(724\) −1.85239e10 −1.81405
\(725\) −1.00243e10 −0.976947
\(726\) −3.27393e10 −3.17535
\(727\) −1.56561e10 −1.51116 −0.755582 0.655054i \(-0.772646\pi\)
−0.755582 + 0.655054i \(0.772646\pi\)
\(728\) 1.87875e10 1.80471
\(729\) 4.72362e9 0.451574
\(730\) 3.64587e10 3.46874
\(731\) −1.28099e9 −0.121292
\(732\) −2.88836e9 −0.272184
\(733\) 1.25157e10 1.17379 0.586895 0.809663i \(-0.300350\pi\)
0.586895 + 0.809663i \(0.300350\pi\)
\(734\) −4.42531e9 −0.413055
\(735\) 1.32756e10 1.23325
\(736\) 4.73093e10 4.37396
\(737\) −8.78659e9 −0.808507
\(738\) 1.35024e10 1.23655
\(739\) −1.12760e10 −1.02778 −0.513888 0.857857i \(-0.671795\pi\)
−0.513888 + 0.857857i \(0.671795\pi\)
\(740\) 8.54011e9 0.774733
\(741\) −1.23745e10 −1.11728
\(742\) −3.79452e9 −0.340992
\(743\) 6.56019e9 0.586753 0.293377 0.955997i \(-0.405221\pi\)
0.293377 + 0.955997i \(0.405221\pi\)
\(744\) 4.38644e10 3.90487
\(745\) 1.75357e10 1.55373
\(746\) 1.67847e10 1.48022
\(747\) 2.14151e8 0.0187974
\(748\) −3.83535e10 −3.35081
\(749\) 2.15907e9 0.187750
\(750\) 5.71606e9 0.494746
\(751\) 1.28366e10 1.10589 0.552943 0.833219i \(-0.313505\pi\)
0.552943 + 0.833219i \(0.313505\pi\)
\(752\) −3.71060e10 −3.18186
\(753\) 4.03565e9 0.344454
\(754\) −2.01031e10 −1.70791
\(755\) −2.47528e10 −2.09320
\(756\) −1.27870e10 −1.07633
\(757\) −3.54032e9 −0.296624 −0.148312 0.988941i \(-0.547384\pi\)
−0.148312 + 0.988941i \(0.547384\pi\)
\(758\) −2.13434e10 −1.78000
\(759\) −2.57764e10 −2.13981
\(760\) 5.37613e10 4.44244
\(761\) −1.14256e10 −0.939795 −0.469897 0.882721i \(-0.655709\pi\)
−0.469897 + 0.882721i \(0.655709\pi\)
\(762\) 1.34959e10 1.10499
\(763\) 6.84047e9 0.557506
\(764\) −6.34144e9 −0.514471
\(765\) 5.02913e9 0.406142
\(766\) 3.21154e10 2.58174
\(767\) −1.18769e10 −0.950431
\(768\) 3.39068e10 2.70099
\(769\) 1.70491e10 1.35195 0.675974 0.736926i \(-0.263723\pi\)
0.675974 + 0.736926i \(0.263723\pi\)
\(770\) 2.92285e10 2.30722
\(771\) 4.46294e9 0.350696
\(772\) 2.85950e10 2.23681
\(773\) −1.14110e10 −0.888576 −0.444288 0.895884i \(-0.646543\pi\)
−0.444288 + 0.895884i \(0.646543\pi\)
\(774\) −1.31897e9 −0.102245
\(775\) −1.52491e10 −1.17676
\(776\) 3.23035e10 2.48161
\(777\) −1.55532e9 −0.118945
\(778\) −2.50198e10 −1.90482
\(779\) −2.24108e10 −1.69854
\(780\) 6.38586e10 4.81824
\(781\) 5.63966e9 0.423618
\(782\) −2.42629e10 −1.81434
\(783\) 8.62999e9 0.642457
\(784\) −3.54277e10 −2.62565
\(785\) 2.34037e10 1.72679
\(786\) −4.12126e9 −0.302727
\(787\) −1.06817e10 −0.781142 −0.390571 0.920573i \(-0.627722\pi\)
−0.390571 + 0.920573i \(0.627722\pi\)
\(788\) 2.45483e10 1.78722
\(789\) 2.57333e10 1.86520
\(790\) −6.01982e10 −4.34399
\(791\) −6.07675e9 −0.436570
\(792\) −2.49080e10 −1.78156
\(793\) 1.27010e9 0.0904444
\(794\) −3.33497e10 −2.36440
\(795\) −8.13492e9 −0.574206
\(796\) 5.58174e10 3.92260
\(797\) 1.80601e10 1.26362 0.631810 0.775123i \(-0.282312\pi\)
0.631810 + 0.775123i \(0.282312\pi\)
\(798\) −1.55232e10 −1.08136
\(799\) 1.00646e10 0.698041
\(800\) −6.15377e10 −4.24939
\(801\) 7.16503e9 0.492611
\(802\) 1.10129e10 0.753862
\(803\) 2.80362e10 1.91080
\(804\) −2.40817e10 −1.63415
\(805\) 1.35038e10 0.912368
\(806\) −3.05811e10 −2.05722
\(807\) −8.78926e9 −0.588701
\(808\) −6.56967e10 −4.38131
\(809\) 1.87987e10 1.24827 0.624134 0.781317i \(-0.285452\pi\)
0.624134 + 0.781317i \(0.285452\pi\)
\(810\) −5.24090e10 −3.46504
\(811\) −1.93491e9 −0.127376 −0.0636881 0.997970i \(-0.520286\pi\)
−0.0636881 + 0.997970i \(0.520286\pi\)
\(812\) −1.84174e10 −1.20721
\(813\) 5.19149e9 0.338825
\(814\) 8.99228e9 0.584365
\(815\) 8.32368e9 0.538596
\(816\) −5.19667e10 −3.34819
\(817\) 2.18918e9 0.140445
\(818\) −3.50279e10 −2.23758
\(819\) −3.00351e9 −0.191045
\(820\) 1.15652e11 7.32492
\(821\) 1.74599e10 1.10113 0.550567 0.834791i \(-0.314411\pi\)
0.550567 + 0.834791i \(0.314411\pi\)
\(822\) −2.98255e10 −1.87300
\(823\) −1.78070e10 −1.11350 −0.556752 0.830679i \(-0.687953\pi\)
−0.556752 + 0.830679i \(0.687953\pi\)
\(824\) 6.64716e10 4.13896
\(825\) 3.35287e10 2.07887
\(826\) −1.48990e10 −0.919874
\(827\) −2.44106e10 −1.50075 −0.750377 0.661010i \(-0.770128\pi\)
−0.750377 + 0.661010i \(0.770128\pi\)
\(828\) −1.82450e10 −1.11696
\(829\) 1.80753e10 1.10191 0.550953 0.834536i \(-0.314264\pi\)
0.550953 + 0.834536i \(0.314264\pi\)
\(830\) 2.51160e9 0.152467
\(831\) 3.13692e10 1.89627
\(832\) −6.04814e10 −3.64074
\(833\) 9.60934e9 0.576018
\(834\) −2.91646e10 −1.74091
\(835\) −1.30745e10 −0.777181
\(836\) 6.55455e10 3.87990
\(837\) 1.31280e10 0.773857
\(838\) −6.24688e10 −3.66699
\(839\) −3.99758e9 −0.233684 −0.116842 0.993150i \(-0.537277\pi\)
−0.116842 + 0.993150i \(0.537277\pi\)
\(840\) 5.05264e10 2.94131
\(841\) −4.81993e9 −0.279418
\(842\) −4.32439e10 −2.49651
\(843\) 9.35279e9 0.537706
\(844\) −7.12331e10 −4.07834
\(845\) −2.35846e9 −0.134471
\(846\) 1.03630e10 0.588421
\(847\) 1.31893e10 0.745812
\(848\) 2.17090e10 1.22252
\(849\) −1.81075e10 −1.01550
\(850\) 3.15600e10 1.76267
\(851\) 4.15450e9 0.231081
\(852\) 1.54568e10 0.856212
\(853\) 1.49532e10 0.824923 0.412461 0.910975i \(-0.364669\pi\)
0.412461 + 0.910975i \(0.364669\pi\)
\(854\) 1.59328e9 0.0875366
\(855\) −8.59470e9 −0.470272
\(856\) −2.15788e10 −1.17590
\(857\) −2.77267e10 −1.50475 −0.752377 0.658732i \(-0.771093\pi\)
−0.752377 + 0.658732i \(0.771093\pi\)
\(858\) 6.72398e10 3.63430
\(859\) −1.83528e10 −0.987930 −0.493965 0.869482i \(-0.664453\pi\)
−0.493965 + 0.869482i \(0.664453\pi\)
\(860\) −1.12973e10 −0.605663
\(861\) −2.10623e10 −1.12459
\(862\) −1.69663e9 −0.0902218
\(863\) −5.98092e9 −0.316760 −0.158380 0.987378i \(-0.550627\pi\)
−0.158380 + 0.987378i \(0.550627\pi\)
\(864\) 5.29782e10 2.79447
\(865\) 1.15383e10 0.606157
\(866\) 2.31720e10 1.21241
\(867\) −8.18594e9 −0.426581
\(868\) −2.80168e10 −1.45412
\(869\) −4.62915e10 −2.39294
\(870\) −5.40647e10 −2.78353
\(871\) 1.05894e10 0.543012
\(872\) −6.83672e10 −3.49173
\(873\) −5.16429e9 −0.262701
\(874\) 4.14649e10 2.10083
\(875\) −2.30276e9 −0.116204
\(876\) 7.68397e10 3.86208
\(877\) 2.39573e10 1.19933 0.599666 0.800251i \(-0.295300\pi\)
0.599666 + 0.800251i \(0.295300\pi\)
\(878\) 2.55092e10 1.27194
\(879\) −1.61246e10 −0.800805
\(880\) −1.67221e11 −8.27181
\(881\) 1.57373e9 0.0775380 0.0387690 0.999248i \(-0.487656\pi\)
0.0387690 + 0.999248i \(0.487656\pi\)
\(882\) 9.89426e9 0.485561
\(883\) 1.75377e10 0.857254 0.428627 0.903482i \(-0.358998\pi\)
0.428627 + 0.903482i \(0.358998\pi\)
\(884\) 4.62229e10 2.25048
\(885\) −3.19414e10 −1.54901
\(886\) 3.52539e9 0.170290
\(887\) 4.02646e10 1.93727 0.968637 0.248479i \(-0.0799307\pi\)
0.968637 + 0.248479i \(0.0799307\pi\)
\(888\) 1.55446e10 0.744964
\(889\) −5.43691e9 −0.259535
\(890\) 8.40326e10 3.99561
\(891\) −4.03017e10 −1.90876
\(892\) −3.04070e10 −1.43449
\(893\) −1.72002e10 −0.808263
\(894\) 5.06053e10 2.36873
\(895\) 1.58734e9 0.0740100
\(896\) −3.41207e10 −1.58467
\(897\) 3.10652e10 1.43715
\(898\) −2.56412e10 −1.18160
\(899\) 1.89086e10 0.867960
\(900\) 2.37322e10 1.08515
\(901\) −5.88831e9 −0.268197
\(902\) 1.21775e11 5.52503
\(903\) 2.05746e9 0.0929873
\(904\) 6.07342e10 2.73429
\(905\) 2.19063e10 0.982424
\(906\) −7.14329e10 −3.19117
\(907\) 2.96929e10 1.32138 0.660688 0.750660i \(-0.270264\pi\)
0.660688 + 0.750660i \(0.270264\pi\)
\(908\) −6.82207e10 −3.02424
\(909\) 1.05028e10 0.463801
\(910\) −3.52257e10 −1.54958
\(911\) 6.10006e9 0.267313 0.133656 0.991028i \(-0.457328\pi\)
0.133656 + 0.991028i \(0.457328\pi\)
\(912\) 8.88103e10 3.87687
\(913\) 1.93138e9 0.0839886
\(914\) −1.66320e10 −0.720496
\(915\) 3.41576e9 0.147406
\(916\) −6.64156e10 −2.85520
\(917\) 1.66028e9 0.0711031
\(918\) −2.71702e10 −1.15916
\(919\) −3.43035e10 −1.45792 −0.728960 0.684556i \(-0.759996\pi\)
−0.728960 + 0.684556i \(0.759996\pi\)
\(920\) −1.34964e11 −5.71426
\(921\) 1.72116e10 0.725959
\(922\) 7.34147e10 3.08478
\(923\) −6.79682e9 −0.284511
\(924\) 6.16015e10 2.56885
\(925\) −5.40397e9 −0.224500
\(926\) 6.37214e10 2.63722
\(927\) −1.06267e10 −0.438146
\(928\) 7.63056e10 3.13428
\(929\) −1.04991e10 −0.429632 −0.214816 0.976655i \(-0.568915\pi\)
−0.214816 + 0.976655i \(0.568915\pi\)
\(930\) −8.22438e10 −3.35284
\(931\) −1.64222e10 −0.666972
\(932\) −2.59356e10 −1.04940
\(933\) −4.77920e10 −1.92650
\(934\) 4.62116e10 1.85582
\(935\) 4.53566e10 1.81468
\(936\) 3.00187e10 1.19654
\(937\) 3.00943e10 1.19507 0.597537 0.801841i \(-0.296146\pi\)
0.597537 + 0.801841i \(0.296146\pi\)
\(938\) 1.32839e10 0.525553
\(939\) −4.50604e10 −1.77609
\(940\) 8.87617e10 3.48561
\(941\) −1.94953e10 −0.762720 −0.381360 0.924427i \(-0.624544\pi\)
−0.381360 + 0.924427i \(0.624544\pi\)
\(942\) 6.75394e10 2.63256
\(943\) 5.62608e10 2.18482
\(944\) 8.52396e10 3.29791
\(945\) 1.51219e10 0.582901
\(946\) −1.18955e10 −0.456839
\(947\) 2.51681e10 0.962998 0.481499 0.876447i \(-0.340092\pi\)
0.481499 + 0.876447i \(0.340092\pi\)
\(948\) −1.26873e11 −4.83659
\(949\) −3.37887e10 −1.28333
\(950\) −5.39355e10 −2.04100
\(951\) −1.10812e10 −0.417787
\(952\) 3.65726e10 1.37381
\(953\) 1.34093e10 0.501859 0.250929 0.968005i \(-0.419264\pi\)
0.250929 + 0.968005i \(0.419264\pi\)
\(954\) −6.06291e9 −0.226080
\(955\) 7.49935e9 0.278620
\(956\) 1.07396e11 3.97544
\(957\) −4.15749e10 −1.53334
\(958\) −8.22383e9 −0.302200
\(959\) 1.20154e10 0.439921
\(960\) −1.62656e11 −5.93365
\(961\) 1.25131e9 0.0454815
\(962\) −1.08373e10 −0.392473
\(963\) 3.44976e9 0.124479
\(964\) −7.94924e10 −2.85796
\(965\) −3.38163e10 −1.21138
\(966\) 3.89698e10 1.39094
\(967\) −4.92333e10 −1.75092 −0.875460 0.483291i \(-0.839441\pi\)
−0.875460 + 0.483291i \(0.839441\pi\)
\(968\) −1.31821e11 −4.67111
\(969\) −2.40887e10 −0.850513
\(970\) −6.05677e10 −2.13079
\(971\) 1.64288e10 0.575889 0.287945 0.957647i \(-0.407028\pi\)
0.287945 + 0.957647i \(0.407028\pi\)
\(972\) −5.17760e10 −1.80841
\(973\) 1.17492e10 0.408896
\(974\) 1.19752e10 0.415266
\(975\) −4.04082e10 −1.39622
\(976\) −9.11538e9 −0.313834
\(977\) 1.72848e10 0.592972 0.296486 0.955037i \(-0.404185\pi\)
0.296486 + 0.955037i \(0.404185\pi\)
\(978\) 2.40208e10 0.821112
\(979\) 6.46198e10 2.20103
\(980\) 8.47471e10 2.87630
\(981\) 1.09297e10 0.369630
\(982\) 8.50278e10 2.86530
\(983\) −8.75111e9 −0.293850 −0.146925 0.989148i \(-0.546938\pi\)
−0.146925 + 0.989148i \(0.546938\pi\)
\(984\) 2.10508e11 7.04346
\(985\) −2.90306e10 −0.967898
\(986\) −3.91338e10 −1.30012
\(987\) −1.61652e10 −0.535144
\(988\) −7.89943e10 −2.60583
\(989\) −5.49580e9 −0.180652
\(990\) 4.67015e10 1.52970
\(991\) −2.33937e10 −0.763555 −0.381778 0.924254i \(-0.624688\pi\)
−0.381778 + 0.924254i \(0.624688\pi\)
\(992\) 1.16077e11 3.77533
\(993\) −2.40874e10 −0.780670
\(994\) −8.52628e9 −0.275364
\(995\) −6.60094e10 −2.12434
\(996\) 5.29340e9 0.169757
\(997\) −8.07234e9 −0.257968 −0.128984 0.991647i \(-0.541172\pi\)
−0.128984 + 0.991647i \(0.541172\pi\)
\(998\) −1.10178e11 −3.50862
\(999\) 4.65231e9 0.147635
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.a.1.1 11
3.2 odd 2 387.8.a.b.1.11 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.a.1.1 11 1.1 even 1 trivial
387.8.a.b.1.11 11 3.2 odd 2