Properties

Label 43.8.a.a.1.11
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 1.11
Root \(-19.3827\) of defining polynomial
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q+17.3827 q^{2} -29.0652 q^{3} +174.160 q^{4} -230.747 q^{5} -505.234 q^{6} -1110.68 q^{7} +802.384 q^{8} -1342.21 q^{9} +O(q^{10})\) \(q+17.3827 q^{2} -29.0652 q^{3} +174.160 q^{4} -230.747 q^{5} -505.234 q^{6} -1110.68 q^{7} +802.384 q^{8} -1342.21 q^{9} -4011.02 q^{10} -607.968 q^{11} -5062.00 q^{12} +13041.8 q^{13} -19306.6 q^{14} +6706.73 q^{15} -8344.81 q^{16} -20609.8 q^{17} -23331.3 q^{18} -24170.2 q^{19} -40186.9 q^{20} +32282.1 q^{21} -10568.2 q^{22} +60316.5 q^{23} -23321.5 q^{24} -24880.7 q^{25} +226703. q^{26} +102577. q^{27} -193435. q^{28} -77712.2 q^{29} +116581. q^{30} +34644.3 q^{31} -247761. q^{32} +17670.7 q^{33} -358255. q^{34} +256286. q^{35} -233759. q^{36} +489896. q^{37} -420144. q^{38} -379064. q^{39} -185148. q^{40} -809882. q^{41} +561152. q^{42} +79507.0 q^{43} -105884. q^{44} +309712. q^{45} +1.04847e6 q^{46} +289158. q^{47} +242544. q^{48} +410063. q^{49} -432494. q^{50} +599029. q^{51} +2.27136e6 q^{52} +892274. q^{53} +1.78308e6 q^{54} +140287. q^{55} -891191. q^{56} +702512. q^{57} -1.35085e6 q^{58} -1.51749e6 q^{59} +1.16804e6 q^{60} -160538. q^{61} +602213. q^{62} +1.49077e6 q^{63} -3.23863e6 q^{64} -3.00937e6 q^{65} +307166. q^{66} -2.42938e6 q^{67} -3.58940e6 q^{68} -1.75311e6 q^{69} +4.45495e6 q^{70} +993231. q^{71} -1.07697e6 q^{72} -2.67982e6 q^{73} +8.51574e6 q^{74} +723162. q^{75} -4.20947e6 q^{76} +675257. q^{77} -6.58917e6 q^{78} -4.46840e6 q^{79} +1.92554e6 q^{80} -46018.4 q^{81} -1.40780e7 q^{82} -4.10044e6 q^{83} +5.62225e6 q^{84} +4.75566e6 q^{85} +1.38205e6 q^{86} +2.25872e6 q^{87} -487824. q^{88} -2.96316e6 q^{89} +5.38364e6 q^{90} -1.44853e7 q^{91} +1.05047e7 q^{92} -1.00695e6 q^{93} +5.02636e6 q^{94} +5.57721e6 q^{95} +7.20123e6 q^{96} -9.83610e6 q^{97} +7.12801e6 q^{98} +816022. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} + O(q^{10}) \) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} - 1333q^{10} + 1333q^{11} + 5089q^{12} - 17967q^{13} - 22352q^{14} - 49504q^{15} - 34406q^{16} - 63095q^{17} - 165931q^{18} - 54524q^{19} - 280995q^{20} - 139788q^{21} - 289358q^{22} - 138139q^{23} - 429583q^{24} + 3455q^{25} - 132946q^{26} - 240356q^{27} - 12704q^{28} - 308658q^{29} + 421284q^{30} - 209523q^{31} - 644934q^{32} + 96814q^{33} + 762435q^{34} - 578892q^{35} + 426161q^{36} - 298472q^{37} - 369707q^{38} + 292298q^{39} + 2633173q^{40} - 1346735q^{41} + 1173266q^{42} + 874577q^{43} + 3134292q^{44} + 1893784q^{45} + 3588111q^{46} + 499284q^{47} + 5647533q^{48} + 2544563q^{49} + 3049745q^{50} + 1258424q^{51} + 983088q^{52} - 2210495q^{53} + 6789698q^{54} - 1855072q^{55} - 469976q^{56} - 1238444q^{57} + 4397067q^{58} - 5824216q^{59} - 2889372q^{60} - 4453034q^{61} + 1002789q^{62} - 6240564q^{63} + 4757538q^{64} - 2162872q^{65} - 258940q^{66} - 6859513q^{67} - 9397005q^{68} - 10040030q^{69} + 845078q^{70} - 10726554q^{71} + 1199517q^{72} - 4456898q^{73} + 1046637q^{74} - 3349114q^{75} + 5861267q^{76} - 17019816q^{77} + 1999122q^{78} - 15541320q^{79} - 15680911q^{80} - 12976697q^{81} + 20233655q^{82} - 11146767q^{83} + 12348278q^{84} - 12471976q^{85} - 1908168q^{86} - 18648900q^{87} - 24463544q^{88} - 13531356q^{89} + 20858990q^{90} - 19746448q^{91} - 26023161q^{92} - 21903110q^{93} + 20288857q^{94} - 12291624q^{95} - 13954503q^{96} - 10999901q^{97} + 29909168q^{98} + 29396057q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 17.3827 1.53643 0.768216 0.640191i \(-0.221145\pi\)
0.768216 + 0.640191i \(0.221145\pi\)
\(3\) −29.0652 −0.621512 −0.310756 0.950490i \(-0.600582\pi\)
−0.310756 + 0.950490i \(0.600582\pi\)
\(4\) 174.160 1.36062
\(5\) −230.747 −0.825547 −0.412773 0.910834i \(-0.635440\pi\)
−0.412773 + 0.910834i \(0.635440\pi\)
\(6\) −505.234 −0.954911
\(7\) −1110.68 −1.22390 −0.611949 0.790897i \(-0.709614\pi\)
−0.611949 + 0.790897i \(0.709614\pi\)
\(8\) 802.384 0.554074
\(9\) −1342.21 −0.613723
\(10\) −4011.02 −1.26840
\(11\) −607.968 −0.137723 −0.0688615 0.997626i \(-0.521937\pi\)
−0.0688615 + 0.997626i \(0.521937\pi\)
\(12\) −5062.00 −0.845644
\(13\) 13041.8 1.64640 0.823202 0.567748i \(-0.192185\pi\)
0.823202 + 0.567748i \(0.192185\pi\)
\(14\) −19306.6 −1.88044
\(15\) 6706.73 0.513087
\(16\) −8344.81 −0.509327
\(17\) −20609.8 −1.01743 −0.508713 0.860936i \(-0.669878\pi\)
−0.508713 + 0.860936i \(0.669878\pi\)
\(18\) −23331.3 −0.942944
\(19\) −24170.2 −0.808430 −0.404215 0.914664i \(-0.632455\pi\)
−0.404215 + 0.914664i \(0.632455\pi\)
\(20\) −40186.9 −1.12326
\(21\) 32282.1 0.760667
\(22\) −10568.2 −0.211602
\(23\) 60316.5 1.03369 0.516843 0.856080i \(-0.327107\pi\)
0.516843 + 0.856080i \(0.327107\pi\)
\(24\) −23321.5 −0.344363
\(25\) −24880.7 −0.318472
\(26\) 226703. 2.52959
\(27\) 102577. 1.00295
\(28\) −193435. −1.66526
\(29\) −77712.2 −0.591693 −0.295846 0.955236i \(-0.595602\pi\)
−0.295846 + 0.955236i \(0.595602\pi\)
\(30\) 116581. 0.788324
\(31\) 34644.3 0.208865 0.104433 0.994532i \(-0.466697\pi\)
0.104433 + 0.994532i \(0.466697\pi\)
\(32\) −247761. −1.33662
\(33\) 17670.7 0.0855965
\(34\) −358255. −1.56320
\(35\) 256286. 1.01038
\(36\) −233759. −0.835046
\(37\) 489896. 1.59000 0.795002 0.606607i \(-0.207470\pi\)
0.795002 + 0.606607i \(0.207470\pi\)
\(38\) −420144. −1.24210
\(39\) −379064. −1.02326
\(40\) −185148. −0.457414
\(41\) −809882. −1.83518 −0.917589 0.397529i \(-0.869868\pi\)
−0.917589 + 0.397529i \(0.869868\pi\)
\(42\) 561152. 1.16871
\(43\) 79507.0 0.152499
\(44\) −105884. −0.187389
\(45\) 309712. 0.506657
\(46\) 1.04847e6 1.58819
\(47\) 289158. 0.406249 0.203125 0.979153i \(-0.434890\pi\)
0.203125 + 0.979153i \(0.434890\pi\)
\(48\) 242544. 0.316553
\(49\) 410063. 0.497925
\(50\) −432494. −0.489311
\(51\) 599029. 0.632342
\(52\) 2.27136e6 2.24014
\(53\) 892274. 0.823252 0.411626 0.911353i \(-0.364961\pi\)
0.411626 + 0.911353i \(0.364961\pi\)
\(54\) 1.78308e6 1.54096
\(55\) 140287. 0.113697
\(56\) −891191. −0.678129
\(57\) 702512. 0.502449
\(58\) −1.35085e6 −0.909096
\(59\) −1.51749e6 −0.961930 −0.480965 0.876740i \(-0.659714\pi\)
−0.480965 + 0.876740i \(0.659714\pi\)
\(60\) 1.16804e6 0.698119
\(61\) −160538. −0.0905570 −0.0452785 0.998974i \(-0.514418\pi\)
−0.0452785 + 0.998974i \(0.514418\pi\)
\(62\) 602213. 0.320907
\(63\) 1.49077e6 0.751134
\(64\) −3.23863e6 −1.54430
\(65\) −3.00937e6 −1.35918
\(66\) 307166. 0.131513
\(67\) −2.42938e6 −0.986811 −0.493406 0.869799i \(-0.664248\pi\)
−0.493406 + 0.869799i \(0.664248\pi\)
\(68\) −3.58940e6 −1.38433
\(69\) −1.75311e6 −0.642449
\(70\) 4.45495e6 1.55239
\(71\) 993231. 0.329341 0.164671 0.986349i \(-0.447344\pi\)
0.164671 + 0.986349i \(0.447344\pi\)
\(72\) −1.07697e6 −0.340048
\(73\) −2.67982e6 −0.806262 −0.403131 0.915142i \(-0.632078\pi\)
−0.403131 + 0.915142i \(0.632078\pi\)
\(74\) 8.51574e6 2.44293
\(75\) 723162. 0.197934
\(76\) −4.20947e6 −1.09997
\(77\) 675257. 0.168559
\(78\) −6.58917e6 −1.57217
\(79\) −4.46840e6 −1.01966 −0.509832 0.860274i \(-0.670292\pi\)
−0.509832 + 0.860274i \(0.670292\pi\)
\(80\) 1.92554e6 0.420473
\(81\) −46018.4 −0.00962130
\(82\) −1.40780e7 −2.81963
\(83\) −4.10044e6 −0.787150 −0.393575 0.919292i \(-0.628762\pi\)
−0.393575 + 0.919292i \(0.628762\pi\)
\(84\) 5.62225e6 1.03498
\(85\) 4.75566e6 0.839932
\(86\) 1.38205e6 0.234304
\(87\) 2.25872e6 0.367744
\(88\) −487824. −0.0763087
\(89\) −2.96316e6 −0.445543 −0.222772 0.974871i \(-0.571510\pi\)
−0.222772 + 0.974871i \(0.571510\pi\)
\(90\) 5.38364e6 0.778444
\(91\) −1.44853e7 −2.01503
\(92\) 1.05047e7 1.40646
\(93\) −1.00695e6 −0.129812
\(94\) 5.02636e6 0.624175
\(95\) 5.57721e6 0.667397
\(96\) 7.20123e6 0.830725
\(97\) −9.83610e6 −1.09426 −0.547132 0.837046i \(-0.684280\pi\)
−0.547132 + 0.837046i \(0.684280\pi\)
\(98\) 7.12801e6 0.765028
\(99\) 816022. 0.0845238
\(100\) −4.33321e6 −0.433321
\(101\) −1.11991e7 −1.08158 −0.540788 0.841159i \(-0.681874\pi\)
−0.540788 + 0.841159i \(0.681874\pi\)
\(102\) 1.04128e7 0.971550
\(103\) 2.09902e6 0.189272 0.0946359 0.995512i \(-0.469831\pi\)
0.0946359 + 0.995512i \(0.469831\pi\)
\(104\) 1.04646e7 0.912229
\(105\) −7.44901e6 −0.627966
\(106\) 1.55102e7 1.26487
\(107\) 1.39300e7 1.09928 0.549639 0.835402i \(-0.314765\pi\)
0.549639 + 0.835402i \(0.314765\pi\)
\(108\) 1.78649e7 1.36463
\(109\) 2.48819e7 1.84031 0.920154 0.391557i \(-0.128063\pi\)
0.920154 + 0.391557i \(0.128063\pi\)
\(110\) 2.43857e6 0.174687
\(111\) −1.42390e7 −0.988206
\(112\) 9.26840e6 0.623364
\(113\) −1.72023e7 −1.12153 −0.560766 0.827974i \(-0.689493\pi\)
−0.560766 + 0.827974i \(0.689493\pi\)
\(114\) 1.22116e7 0.771978
\(115\) −1.39179e7 −0.853357
\(116\) −1.35343e7 −0.805071
\(117\) −1.75049e7 −1.01044
\(118\) −2.63781e7 −1.47794
\(119\) 2.28908e7 1.24522
\(120\) 5.38137e6 0.284288
\(121\) −1.91175e7 −0.981032
\(122\) −2.79058e6 −0.139135
\(123\) 2.35394e7 1.14059
\(124\) 6.03365e6 0.284187
\(125\) 2.37683e7 1.08846
\(126\) 2.59136e7 1.15407
\(127\) −3.68734e7 −1.59735 −0.798675 0.601763i \(-0.794465\pi\)
−0.798675 + 0.601763i \(0.794465\pi\)
\(128\) −2.45829e7 −1.03609
\(129\) −2.31089e6 −0.0947797
\(130\) −5.23110e7 −2.08829
\(131\) 2.41863e7 0.939981 0.469991 0.882671i \(-0.344257\pi\)
0.469991 + 0.882671i \(0.344257\pi\)
\(132\) 3.07753e6 0.116465
\(133\) 2.68453e7 0.989435
\(134\) −4.22293e7 −1.51617
\(135\) −2.36695e7 −0.827981
\(136\) −1.65370e7 −0.563729
\(137\) −4.37084e6 −0.145226 −0.0726128 0.997360i \(-0.523134\pi\)
−0.0726128 + 0.997360i \(0.523134\pi\)
\(138\) −3.04739e7 −0.987079
\(139\) 2.60325e7 0.822174 0.411087 0.911596i \(-0.365149\pi\)
0.411087 + 0.911596i \(0.365149\pi\)
\(140\) 4.46347e7 1.37475
\(141\) −8.40445e6 −0.252489
\(142\) 1.72651e7 0.506010
\(143\) −7.92901e6 −0.226748
\(144\) 1.12005e7 0.312586
\(145\) 1.79319e7 0.488470
\(146\) −4.65827e7 −1.23877
\(147\) −1.19186e7 −0.309466
\(148\) 8.53202e7 2.16340
\(149\) 2.65502e7 0.657530 0.328765 0.944412i \(-0.393368\pi\)
0.328765 + 0.944412i \(0.393368\pi\)
\(150\) 1.25705e7 0.304113
\(151\) 5.28071e7 1.24817 0.624084 0.781357i \(-0.285472\pi\)
0.624084 + 0.781357i \(0.285472\pi\)
\(152\) −1.93938e7 −0.447930
\(153\) 2.76627e7 0.624417
\(154\) 1.17378e7 0.258979
\(155\) −7.99408e6 −0.172428
\(156\) −6.60177e7 −1.39227
\(157\) −6.72632e7 −1.38717 −0.693584 0.720376i \(-0.743969\pi\)
−0.693584 + 0.720376i \(0.743969\pi\)
\(158\) −7.76730e7 −1.56664
\(159\) −2.59342e7 −0.511661
\(160\) 5.71702e7 1.10344
\(161\) −6.69922e7 −1.26513
\(162\) −799926. −0.0147825
\(163\) 5.59234e7 1.01143 0.505717 0.862700i \(-0.331228\pi\)
0.505717 + 0.862700i \(0.331228\pi\)
\(164\) −1.41049e8 −2.49699
\(165\) −4.07748e6 −0.0706639
\(166\) −7.12770e7 −1.20940
\(167\) 1.13304e7 0.188252 0.0941258 0.995560i \(-0.469994\pi\)
0.0941258 + 0.995560i \(0.469994\pi\)
\(168\) 2.59027e7 0.421466
\(169\) 1.07341e8 1.71065
\(170\) 8.26664e7 1.29050
\(171\) 3.24415e7 0.496152
\(172\) 1.38469e7 0.207493
\(173\) −3.43489e7 −0.504373 −0.252186 0.967679i \(-0.581150\pi\)
−0.252186 + 0.967679i \(0.581150\pi\)
\(174\) 3.92628e7 0.565014
\(175\) 2.76344e7 0.389778
\(176\) 5.07338e6 0.0701460
\(177\) 4.41062e7 0.597851
\(178\) −5.15078e7 −0.684547
\(179\) 1.06394e8 1.38654 0.693268 0.720680i \(-0.256170\pi\)
0.693268 + 0.720680i \(0.256170\pi\)
\(180\) 5.39394e7 0.689369
\(181\) −1.03201e8 −1.29363 −0.646815 0.762647i \(-0.723899\pi\)
−0.646815 + 0.762647i \(0.723899\pi\)
\(182\) −2.51794e8 −3.09596
\(183\) 4.66606e6 0.0562823
\(184\) 4.83970e7 0.572739
\(185\) −1.13042e8 −1.31262
\(186\) −1.75035e7 −0.199448
\(187\) 1.25301e7 0.140123
\(188\) 5.03597e7 0.552752
\(189\) −1.13930e8 −1.22751
\(190\) 9.69471e7 1.02541
\(191\) 6.61411e7 0.686839 0.343419 0.939182i \(-0.388415\pi\)
0.343419 + 0.939182i \(0.388415\pi\)
\(192\) 9.41315e7 0.959800
\(193\) 5.13623e7 0.514273 0.257136 0.966375i \(-0.417221\pi\)
0.257136 + 0.966375i \(0.417221\pi\)
\(194\) −1.70978e8 −1.68126
\(195\) 8.74679e7 0.844749
\(196\) 7.14164e7 0.677488
\(197\) −9.19611e7 −0.856983 −0.428492 0.903546i \(-0.640955\pi\)
−0.428492 + 0.903546i \(0.640955\pi\)
\(198\) 1.41847e7 0.129865
\(199\) 6.18358e7 0.556230 0.278115 0.960548i \(-0.410290\pi\)
0.278115 + 0.960548i \(0.410290\pi\)
\(200\) −1.99639e7 −0.176457
\(201\) 7.06106e7 0.613315
\(202\) −1.94670e8 −1.66177
\(203\) 8.63132e7 0.724171
\(204\) 1.04327e8 0.860379
\(205\) 1.86878e8 1.51503
\(206\) 3.64867e7 0.290803
\(207\) −8.09576e7 −0.634397
\(208\) −1.08832e8 −0.838558
\(209\) 1.46947e7 0.111339
\(210\) −1.29484e8 −0.964827
\(211\) 1.45231e8 1.06431 0.532157 0.846646i \(-0.321382\pi\)
0.532157 + 0.846646i \(0.321382\pi\)
\(212\) 1.55398e8 1.12014
\(213\) −2.88685e7 −0.204689
\(214\) 2.42142e8 1.68897
\(215\) −1.83460e7 −0.125895
\(216\) 8.23065e7 0.555707
\(217\) −3.84787e7 −0.255630
\(218\) 4.32515e8 2.82751
\(219\) 7.78897e7 0.501101
\(220\) 2.44324e7 0.154699
\(221\) −2.68789e8 −1.67509
\(222\) −2.47512e8 −1.51831
\(223\) 1.18865e8 0.717775 0.358888 0.933381i \(-0.383156\pi\)
0.358888 + 0.933381i \(0.383156\pi\)
\(224\) 2.75183e8 1.63589
\(225\) 3.33951e7 0.195454
\(226\) −2.99023e8 −1.72316
\(227\) −2.33026e8 −1.32225 −0.661125 0.750276i \(-0.729921\pi\)
−0.661125 + 0.750276i \(0.729921\pi\)
\(228\) 1.22349e8 0.683644
\(229\) 2.52042e8 1.38691 0.693457 0.720498i \(-0.256087\pi\)
0.693457 + 0.720498i \(0.256087\pi\)
\(230\) −2.41931e8 −1.31112
\(231\) −1.96265e7 −0.104761
\(232\) −6.23551e7 −0.327841
\(233\) 1.61379e8 0.835799 0.417899 0.908493i \(-0.362766\pi\)
0.417899 + 0.908493i \(0.362766\pi\)
\(234\) −3.04283e8 −1.55247
\(235\) −6.67224e7 −0.335378
\(236\) −2.64286e8 −1.30882
\(237\) 1.29875e8 0.633733
\(238\) 3.97906e8 1.91320
\(239\) −3.28589e8 −1.55690 −0.778449 0.627707i \(-0.783993\pi\)
−0.778449 + 0.627707i \(0.783993\pi\)
\(240\) −5.59664e7 −0.261329
\(241\) −5.95088e7 −0.273855 −0.136928 0.990581i \(-0.543723\pi\)
−0.136928 + 0.990581i \(0.543723\pi\)
\(242\) −3.32315e8 −1.50729
\(243\) −2.22999e8 −0.996968
\(244\) −2.79592e7 −0.123214
\(245\) −9.46209e7 −0.411060
\(246\) 4.09180e8 1.75243
\(247\) −3.15223e8 −1.33100
\(248\) 2.77981e7 0.115727
\(249\) 1.19180e8 0.489223
\(250\) 4.13158e8 1.67235
\(251\) −1.34270e8 −0.535944 −0.267972 0.963427i \(-0.586354\pi\)
−0.267972 + 0.963427i \(0.586354\pi\)
\(252\) 2.59631e8 1.02201
\(253\) −3.66705e7 −0.142362
\(254\) −6.40960e8 −2.45422
\(255\) −1.38224e8 −0.522028
\(256\) −1.27732e7 −0.0475837
\(257\) −2.54950e8 −0.936892 −0.468446 0.883492i \(-0.655186\pi\)
−0.468446 + 0.883492i \(0.655186\pi\)
\(258\) −4.01696e7 −0.145623
\(259\) −5.44117e8 −1.94600
\(260\) −5.24111e8 −1.84934
\(261\) 1.04306e8 0.363135
\(262\) 4.20424e8 1.44422
\(263\) 1.82117e8 0.617314 0.308657 0.951173i \(-0.400121\pi\)
0.308657 + 0.951173i \(0.400121\pi\)
\(264\) 1.41787e7 0.0474268
\(265\) −2.05890e8 −0.679633
\(266\) 4.66645e8 1.52020
\(267\) 8.61249e7 0.276911
\(268\) −4.23101e8 −1.34268
\(269\) −4.86909e8 −1.52516 −0.762579 0.646895i \(-0.776067\pi\)
−0.762579 + 0.646895i \(0.776067\pi\)
\(270\) −4.11440e8 −1.27214
\(271\) −5.57858e7 −0.170267 −0.0851336 0.996370i \(-0.527132\pi\)
−0.0851336 + 0.996370i \(0.527132\pi\)
\(272\) 1.71985e8 0.518202
\(273\) 4.21018e8 1.25237
\(274\) −7.59772e7 −0.223129
\(275\) 1.51266e7 0.0438610
\(276\) −3.05322e8 −0.874131
\(277\) 6.88409e8 1.94611 0.973055 0.230572i \(-0.0740596\pi\)
0.973055 + 0.230572i \(0.0740596\pi\)
\(278\) 4.52516e8 1.26322
\(279\) −4.65000e7 −0.128185
\(280\) 2.05640e8 0.559828
\(281\) 2.36816e8 0.636707 0.318353 0.947972i \(-0.396870\pi\)
0.318353 + 0.947972i \(0.396870\pi\)
\(282\) −1.46092e8 −0.387932
\(283\) 3.02454e8 0.793245 0.396623 0.917982i \(-0.370182\pi\)
0.396623 + 0.917982i \(0.370182\pi\)
\(284\) 1.72981e8 0.448109
\(285\) −1.62103e8 −0.414795
\(286\) −1.37828e8 −0.348383
\(287\) 8.99518e8 2.24607
\(288\) 3.32548e8 0.820314
\(289\) 1.44250e7 0.0351539
\(290\) 3.11705e8 0.750501
\(291\) 2.85889e8 0.680098
\(292\) −4.66718e8 −1.09702
\(293\) −6.95562e8 −1.61547 −0.807735 0.589546i \(-0.799307\pi\)
−0.807735 + 0.589546i \(0.799307\pi\)
\(294\) −2.07177e8 −0.475474
\(295\) 3.50157e8 0.794118
\(296\) 3.93085e8 0.880979
\(297\) −6.23638e7 −0.138129
\(298\) 4.61515e8 1.01025
\(299\) 7.86637e8 1.70187
\(300\) 1.25946e8 0.269314
\(301\) −8.83067e7 −0.186643
\(302\) 9.17932e8 1.91773
\(303\) 3.25503e8 0.672212
\(304\) 2.01696e8 0.411755
\(305\) 3.70436e7 0.0747591
\(306\) 4.80854e8 0.959374
\(307\) 6.66587e8 1.31484 0.657420 0.753525i \(-0.271648\pi\)
0.657420 + 0.753525i \(0.271648\pi\)
\(308\) 1.17603e8 0.229345
\(309\) −6.10085e7 −0.117635
\(310\) −1.38959e8 −0.264924
\(311\) −8.68038e8 −1.63635 −0.818177 0.574967i \(-0.805015\pi\)
−0.818177 + 0.574967i \(0.805015\pi\)
\(312\) −3.04155e8 −0.566962
\(313\) 6.01957e8 1.10958 0.554792 0.831989i \(-0.312798\pi\)
0.554792 + 0.831989i \(0.312798\pi\)
\(314\) −1.16922e9 −2.13129
\(315\) −3.43990e8 −0.620096
\(316\) −7.78215e8 −1.38738
\(317\) −8.81210e8 −1.55372 −0.776859 0.629675i \(-0.783188\pi\)
−0.776859 + 0.629675i \(0.783188\pi\)
\(318\) −4.50807e8 −0.786132
\(319\) 4.72466e7 0.0814897
\(320\) 7.47305e8 1.27489
\(321\) −4.04879e8 −0.683215
\(322\) −1.16451e9 −1.94378
\(323\) 4.98143e8 0.822517
\(324\) −8.01455e6 −0.0130910
\(325\) −3.24489e8 −0.524334
\(326\) 9.72103e8 1.55400
\(327\) −7.23198e8 −1.14377
\(328\) −6.49837e8 −1.01682
\(329\) −3.21161e8 −0.497208
\(330\) −7.08777e7 −0.108570
\(331\) −8.76684e8 −1.32876 −0.664378 0.747397i \(-0.731303\pi\)
−0.664378 + 0.747397i \(0.731303\pi\)
\(332\) −7.14133e8 −1.07102
\(333\) −6.57545e8 −0.975822
\(334\) 1.96954e8 0.289236
\(335\) 5.60574e8 0.814659
\(336\) −2.69388e8 −0.387428
\(337\) −3.84104e8 −0.546694 −0.273347 0.961916i \(-0.588131\pi\)
−0.273347 + 0.961916i \(0.588131\pi\)
\(338\) 1.86587e9 2.62829
\(339\) 4.99989e8 0.697046
\(340\) 8.28244e8 1.14283
\(341\) −2.10626e7 −0.0287655
\(342\) 5.63922e8 0.762304
\(343\) 4.59244e8 0.614488
\(344\) 6.37952e7 0.0844954
\(345\) 4.04526e8 0.530371
\(346\) −5.97078e8 −0.774934
\(347\) −9.79801e8 −1.25888 −0.629441 0.777049i \(-0.716716\pi\)
−0.629441 + 0.777049i \(0.716716\pi\)
\(348\) 3.93379e8 0.500361
\(349\) −1.02828e9 −1.29485 −0.647427 0.762128i \(-0.724155\pi\)
−0.647427 + 0.762128i \(0.724155\pi\)
\(350\) 4.80362e8 0.598867
\(351\) 1.33780e9 1.65126
\(352\) 1.50631e8 0.184083
\(353\) −1.21223e9 −1.46680 −0.733402 0.679795i \(-0.762069\pi\)
−0.733402 + 0.679795i \(0.762069\pi\)
\(354\) 7.66686e8 0.918557
\(355\) −2.29185e8 −0.271887
\(356\) −5.16063e8 −0.606217
\(357\) −6.65328e8 −0.773922
\(358\) 1.84942e9 2.13032
\(359\) 1.13679e9 1.29673 0.648365 0.761330i \(-0.275453\pi\)
0.648365 + 0.761330i \(0.275453\pi\)
\(360\) 2.48508e8 0.280725
\(361\) −3.09674e8 −0.346441
\(362\) −1.79392e9 −1.98757
\(363\) 5.55656e8 0.609723
\(364\) −2.52275e9 −2.74170
\(365\) 6.18362e8 0.665607
\(366\) 8.11090e7 0.0864739
\(367\) 1.53409e9 1.62002 0.810009 0.586418i \(-0.199462\pi\)
0.810009 + 0.586418i \(0.199462\pi\)
\(368\) −5.03330e8 −0.526484
\(369\) 1.08703e9 1.12629
\(370\) −1.96498e9 −2.01676
\(371\) −9.91029e8 −1.00758
\(372\) −1.75369e8 −0.176626
\(373\) −8.11683e8 −0.809852 −0.404926 0.914350i \(-0.632703\pi\)
−0.404926 + 0.914350i \(0.632703\pi\)
\(374\) 2.17808e8 0.215289
\(375\) −6.90831e8 −0.676491
\(376\) 2.32016e8 0.225092
\(377\) −1.01351e9 −0.974166
\(378\) −1.98042e9 −1.88598
\(379\) 2.21198e8 0.208710 0.104355 0.994540i \(-0.466722\pi\)
0.104355 + 0.994540i \(0.466722\pi\)
\(380\) 9.71325e8 0.908076
\(381\) 1.07173e9 0.992772
\(382\) 1.14971e9 1.05528
\(383\) 1.31039e9 1.19180 0.595901 0.803058i \(-0.296795\pi\)
0.595901 + 0.803058i \(0.296795\pi\)
\(384\) 7.14507e8 0.643943
\(385\) −1.55814e8 −0.139153
\(386\) 8.92817e8 0.790145
\(387\) −1.06715e8 −0.0935919
\(388\) −1.71305e9 −1.48888
\(389\) 2.22960e8 0.192045 0.0960226 0.995379i \(-0.469388\pi\)
0.0960226 + 0.995379i \(0.469388\pi\)
\(390\) 1.52043e9 1.29790
\(391\) −1.24311e9 −1.05170
\(392\) 3.29028e8 0.275887
\(393\) −7.02980e8 −0.584210
\(394\) −1.59854e9 −1.31670
\(395\) 1.03107e9 0.841780
\(396\) 1.42118e8 0.115005
\(397\) −6.55657e8 −0.525909 −0.262954 0.964808i \(-0.584697\pi\)
−0.262954 + 0.964808i \(0.584697\pi\)
\(398\) 1.07488e9 0.854610
\(399\) −7.80265e8 −0.614946
\(400\) 2.07624e8 0.162207
\(401\) −7.45083e8 −0.577031 −0.288516 0.957475i \(-0.593162\pi\)
−0.288516 + 0.957475i \(0.593162\pi\)
\(402\) 1.22741e9 0.942317
\(403\) 4.51825e8 0.343877
\(404\) −1.95043e9 −1.47162
\(405\) 1.06186e7 0.00794283
\(406\) 1.50036e9 1.11264
\(407\) −2.97841e8 −0.218980
\(408\) 4.80651e8 0.350364
\(409\) −4.87681e8 −0.352456 −0.176228 0.984349i \(-0.556390\pi\)
−0.176228 + 0.984349i \(0.556390\pi\)
\(410\) 3.24846e9 2.32773
\(411\) 1.27040e8 0.0902594
\(412\) 3.65565e8 0.257528
\(413\) 1.68544e9 1.17730
\(414\) −1.40726e9 −0.974708
\(415\) 9.46167e8 0.649829
\(416\) −3.23125e9 −2.20062
\(417\) −7.56641e8 −0.510991
\(418\) 2.55434e8 0.171065
\(419\) 1.39810e9 0.928518 0.464259 0.885700i \(-0.346321\pi\)
0.464259 + 0.885700i \(0.346321\pi\)
\(420\) −1.29732e9 −0.854426
\(421\) −2.54250e9 −1.66063 −0.830316 0.557293i \(-0.811840\pi\)
−0.830316 + 0.557293i \(0.811840\pi\)
\(422\) 2.52451e9 1.63525
\(423\) −3.88111e8 −0.249325
\(424\) 7.15947e8 0.456142
\(425\) 5.12785e8 0.324022
\(426\) −5.01814e8 −0.314491
\(427\) 1.78306e8 0.110833
\(428\) 2.42605e9 1.49570
\(429\) 2.30459e8 0.140926
\(430\) −3.18904e8 −0.193429
\(431\) 6.45199e8 0.388171 0.194086 0.980985i \(-0.437826\pi\)
0.194086 + 0.980985i \(0.437826\pi\)
\(432\) −8.55989e8 −0.510829
\(433\) −1.41525e9 −0.837770 −0.418885 0.908039i \(-0.637579\pi\)
−0.418885 + 0.908039i \(0.637579\pi\)
\(434\) −6.68865e8 −0.392757
\(435\) −5.21195e8 −0.303590
\(436\) 4.33342e9 2.50397
\(437\) −1.45786e9 −0.835663
\(438\) 1.35394e9 0.769908
\(439\) 3.20780e9 1.80959 0.904797 0.425844i \(-0.140023\pi\)
0.904797 + 0.425844i \(0.140023\pi\)
\(440\) 1.12564e8 0.0629964
\(441\) −5.50391e8 −0.305588
\(442\) −4.67230e9 −2.57367
\(443\) −3.24734e9 −1.77466 −0.887328 0.461138i \(-0.847441\pi\)
−0.887328 + 0.461138i \(0.847441\pi\)
\(444\) −2.47985e9 −1.34458
\(445\) 6.83741e8 0.367817
\(446\) 2.06621e9 1.10281
\(447\) −7.71687e8 −0.408663
\(448\) 3.59707e9 1.89006
\(449\) −9.00041e8 −0.469246 −0.234623 0.972086i \(-0.575385\pi\)
−0.234623 + 0.972086i \(0.575385\pi\)
\(450\) 5.80499e8 0.300302
\(451\) 4.92383e8 0.252746
\(452\) −2.99595e9 −1.52598
\(453\) −1.53485e9 −0.775751
\(454\) −4.05063e9 −2.03155
\(455\) 3.34244e9 1.66350
\(456\) 5.63685e8 0.278394
\(457\) 3.56466e9 1.74708 0.873538 0.486756i \(-0.161820\pi\)
0.873538 + 0.486756i \(0.161820\pi\)
\(458\) 4.38119e9 2.13090
\(459\) −2.11410e9 −1.02042
\(460\) −2.42394e9 −1.16110
\(461\) 2.12032e9 1.00797 0.503986 0.863712i \(-0.331866\pi\)
0.503986 + 0.863712i \(0.331866\pi\)
\(462\) −3.41163e8 −0.160959
\(463\) −1.32103e9 −0.618555 −0.309278 0.950972i \(-0.600087\pi\)
−0.309278 + 0.950972i \(0.600087\pi\)
\(464\) 6.48494e8 0.301365
\(465\) 2.32350e8 0.107166
\(466\) 2.80521e9 1.28415
\(467\) 2.75292e7 0.0125079 0.00625395 0.999980i \(-0.498009\pi\)
0.00625395 + 0.999980i \(0.498009\pi\)
\(468\) −3.04865e9 −1.37482
\(469\) 2.69826e9 1.20776
\(470\) −1.15982e9 −0.515285
\(471\) 1.95502e9 0.862141
\(472\) −1.21761e9 −0.532980
\(473\) −4.83377e7 −0.0210026
\(474\) 2.25758e9 0.973688
\(475\) 6.01370e8 0.257463
\(476\) 3.98667e9 1.69428
\(477\) −1.19762e9 −0.505249
\(478\) −5.71178e9 −2.39207
\(479\) 3.09426e9 1.28642 0.643209 0.765691i \(-0.277603\pi\)
0.643209 + 0.765691i \(0.277603\pi\)
\(480\) −1.66167e9 −0.685803
\(481\) 6.38914e9 2.61779
\(482\) −1.03443e9 −0.420760
\(483\) 1.94715e9 0.786291
\(484\) −3.32951e9 −1.33482
\(485\) 2.26965e9 0.903366
\(486\) −3.87634e9 −1.53177
\(487\) 2.45364e9 0.962629 0.481315 0.876548i \(-0.340159\pi\)
0.481315 + 0.876548i \(0.340159\pi\)
\(488\) −1.28813e8 −0.0501753
\(489\) −1.62543e9 −0.628618
\(490\) −1.64477e9 −0.631566
\(491\) −3.55312e9 −1.35464 −0.677321 0.735688i \(-0.736859\pi\)
−0.677321 + 0.735688i \(0.736859\pi\)
\(492\) 4.09962e9 1.55191
\(493\) 1.60163e9 0.602003
\(494\) −5.47945e9 −2.04500
\(495\) −1.88295e8 −0.0697783
\(496\) −2.89100e8 −0.106381
\(497\) −1.10316e9 −0.403080
\(498\) 2.07168e9 0.751658
\(499\) 1.61280e9 0.581069 0.290534 0.956865i \(-0.406167\pi\)
0.290534 + 0.956865i \(0.406167\pi\)
\(500\) 4.13948e9 1.48099
\(501\) −3.29322e8 −0.117001
\(502\) −2.33397e9 −0.823442
\(503\) −3.82508e9 −1.34015 −0.670074 0.742294i \(-0.733738\pi\)
−0.670074 + 0.742294i \(0.733738\pi\)
\(504\) 1.19617e9 0.416184
\(505\) 2.58415e9 0.892891
\(506\) −6.37435e8 −0.218730
\(507\) −3.11988e9 −1.06319
\(508\) −6.42186e9 −2.17339
\(509\) 9.34320e8 0.314039 0.157019 0.987596i \(-0.449811\pi\)
0.157019 + 0.987596i \(0.449811\pi\)
\(510\) −2.40272e9 −0.802060
\(511\) 2.97642e9 0.986782
\(512\) 2.92457e9 0.962981
\(513\) −2.47931e9 −0.810813
\(514\) −4.43174e9 −1.43947
\(515\) −4.84343e8 −0.156253
\(516\) −4.02464e8 −0.128959
\(517\) −1.75799e8 −0.0559499
\(518\) −9.45825e9 −2.98990
\(519\) 9.98359e8 0.313474
\(520\) −2.41467e9 −0.753088
\(521\) 1.99243e9 0.617237 0.308619 0.951186i \(-0.400133\pi\)
0.308619 + 0.951186i \(0.400133\pi\)
\(522\) 1.81313e9 0.557933
\(523\) −2.41120e9 −0.737017 −0.368509 0.929624i \(-0.620132\pi\)
−0.368509 + 0.929624i \(0.620132\pi\)
\(524\) 4.21228e9 1.27896
\(525\) −8.03200e8 −0.242251
\(526\) 3.16570e9 0.948460
\(527\) −7.14012e8 −0.212505
\(528\) −1.47459e8 −0.0435966
\(529\) 2.33258e8 0.0685081
\(530\) −3.57893e9 −1.04421
\(531\) 2.03679e9 0.590358
\(532\) 4.67537e9 1.34625
\(533\) −1.05623e10 −3.02145
\(534\) 1.49709e9 0.425454
\(535\) −3.21431e9 −0.907506
\(536\) −1.94930e9 −0.546766
\(537\) −3.09236e9 −0.861749
\(538\) −8.46382e9 −2.34330
\(539\) −2.49305e8 −0.0685757
\(540\) −4.12227e9 −1.12657
\(541\) 4.57719e9 1.24282 0.621410 0.783486i \(-0.286560\pi\)
0.621410 + 0.783486i \(0.286560\pi\)
\(542\) −9.69710e8 −0.261604
\(543\) 2.99957e9 0.804006
\(544\) 5.10630e9 1.35991
\(545\) −5.74143e9 −1.51926
\(546\) 7.31844e9 1.92417
\(547\) 1.95785e9 0.511475 0.255737 0.966746i \(-0.417682\pi\)
0.255737 + 0.966746i \(0.417682\pi\)
\(548\) −7.61225e8 −0.197597
\(549\) 2.15475e8 0.0555769
\(550\) 2.62943e8 0.0673894
\(551\) 1.87832e9 0.478342
\(552\) −1.40667e9 −0.355964
\(553\) 4.96295e9 1.24796
\(554\) 1.19664e10 2.99007
\(555\) 3.28560e9 0.815810
\(556\) 4.53381e9 1.11867
\(557\) −5.46100e9 −1.33899 −0.669497 0.742814i \(-0.733490\pi\)
−0.669497 + 0.742814i \(0.733490\pi\)
\(558\) −8.08298e8 −0.196948
\(559\) 1.03692e9 0.251074
\(560\) −2.13866e9 −0.514616
\(561\) −3.64190e8 −0.0870880
\(562\) 4.11652e9 0.978256
\(563\) 1.67608e9 0.395836 0.197918 0.980219i \(-0.436582\pi\)
0.197918 + 0.980219i \(0.436582\pi\)
\(564\) −1.46372e9 −0.343542
\(565\) 3.96938e9 0.925877
\(566\) 5.25749e9 1.21877
\(567\) 5.11116e7 0.0117755
\(568\) 7.96953e8 0.182479
\(569\) −6.70164e8 −0.152506 −0.0762532 0.997088i \(-0.524296\pi\)
−0.0762532 + 0.997088i \(0.524296\pi\)
\(570\) −2.81779e9 −0.637304
\(571\) −4.11196e9 −0.924320 −0.462160 0.886797i \(-0.652925\pi\)
−0.462160 + 0.886797i \(0.652925\pi\)
\(572\) −1.38092e9 −0.308518
\(573\) −1.92241e9 −0.426878
\(574\) 1.56361e10 3.45094
\(575\) −1.50071e9 −0.329201
\(576\) 4.34693e9 0.947772
\(577\) 3.72343e9 0.806915 0.403458 0.914998i \(-0.367808\pi\)
0.403458 + 0.914998i \(0.367808\pi\)
\(578\) 2.50746e8 0.0540116
\(579\) −1.49286e9 −0.319627
\(580\) 3.12301e9 0.664624
\(581\) 4.55427e9 0.963391
\(582\) 4.96953e9 1.04492
\(583\) −5.42474e8 −0.113381
\(584\) −2.15025e9 −0.446729
\(585\) 4.03921e9 0.834162
\(586\) −1.20908e10 −2.48206
\(587\) −5.57358e9 −1.13737 −0.568684 0.822556i \(-0.692547\pi\)
−0.568684 + 0.822556i \(0.692547\pi\)
\(588\) −2.07574e9 −0.421067
\(589\) −8.37359e8 −0.168853
\(590\) 6.08668e9 1.22011
\(591\) 2.67287e9 0.532625
\(592\) −4.08809e9 −0.809832
\(593\) −7.08618e9 −1.39547 −0.697735 0.716356i \(-0.745809\pi\)
−0.697735 + 0.716356i \(0.745809\pi\)
\(594\) −1.08405e9 −0.212226
\(595\) −5.28200e9 −1.02799
\(596\) 4.62397e9 0.894650
\(597\) −1.79727e9 −0.345704
\(598\) 1.36739e10 2.61480
\(599\) −9.43000e9 −1.79274 −0.896371 0.443304i \(-0.853806\pi\)
−0.896371 + 0.443304i \(0.853806\pi\)
\(600\) 5.80254e8 0.109670
\(601\) −8.09030e9 −1.52021 −0.760106 0.649799i \(-0.774853\pi\)
−0.760106 + 0.649799i \(0.774853\pi\)
\(602\) −1.53501e9 −0.286764
\(603\) 3.26075e9 0.605629
\(604\) 9.19688e9 1.69829
\(605\) 4.41132e9 0.809888
\(606\) 5.65814e9 1.03281
\(607\) −5.71746e9 −1.03763 −0.518815 0.854886i \(-0.673627\pi\)
−0.518815 + 0.854886i \(0.673627\pi\)
\(608\) 5.98843e9 1.08056
\(609\) −2.50871e9 −0.450081
\(610\) 6.43920e8 0.114862
\(611\) 3.77115e9 0.668851
\(612\) 4.81773e9 0.849597
\(613\) −2.69282e9 −0.472166 −0.236083 0.971733i \(-0.575864\pi\)
−0.236083 + 0.971733i \(0.575864\pi\)
\(614\) 1.15871e10 2.02016
\(615\) −5.43166e9 −0.941607
\(616\) 5.41816e8 0.0933940
\(617\) 3.24038e9 0.555390 0.277695 0.960669i \(-0.410429\pi\)
0.277695 + 0.960669i \(0.410429\pi\)
\(618\) −1.06049e9 −0.180738
\(619\) −2.96275e9 −0.502086 −0.251043 0.967976i \(-0.580773\pi\)
−0.251043 + 0.967976i \(0.580773\pi\)
\(620\) −1.39225e9 −0.234610
\(621\) 6.18711e9 1.03673
\(622\) −1.50889e10 −2.51415
\(623\) 3.29112e9 0.545299
\(624\) 3.16322e9 0.521174
\(625\) −3.54067e9 −0.580103
\(626\) 1.04637e10 1.70480
\(627\) −4.27105e8 −0.0691988
\(628\) −1.17146e10 −1.88741
\(629\) −1.00967e10 −1.61771
\(630\) −5.97949e9 −0.952736
\(631\) −2.21319e9 −0.350685 −0.175342 0.984508i \(-0.556103\pi\)
−0.175342 + 0.984508i \(0.556103\pi\)
\(632\) −3.58537e9 −0.564969
\(633\) −4.22116e9 −0.661483
\(634\) −1.53179e10 −2.38718
\(635\) 8.50843e9 1.31869
\(636\) −4.51669e9 −0.696178
\(637\) 5.34796e9 0.819786
\(638\) 8.21275e8 0.125203
\(639\) −1.33313e9 −0.202124
\(640\) 5.67243e9 0.855341
\(641\) −3.46476e9 −0.519600 −0.259800 0.965662i \(-0.583657\pi\)
−0.259800 + 0.965662i \(0.583657\pi\)
\(642\) −7.03790e9 −1.04971
\(643\) 6.65219e9 0.986794 0.493397 0.869804i \(-0.335755\pi\)
0.493397 + 0.869804i \(0.335755\pi\)
\(644\) −1.16674e10 −1.72136
\(645\) 5.33232e8 0.0782451
\(646\) 8.65908e9 1.26374
\(647\) −5.79744e9 −0.841533 −0.420766 0.907169i \(-0.638239\pi\)
−0.420766 + 0.907169i \(0.638239\pi\)
\(648\) −3.69244e7 −0.00533091
\(649\) 9.22585e8 0.132480
\(650\) −5.64051e9 −0.805604
\(651\) 1.11839e9 0.158877
\(652\) 9.73962e9 1.37618
\(653\) 5.81937e9 0.817861 0.408931 0.912566i \(-0.365902\pi\)
0.408931 + 0.912566i \(0.365902\pi\)
\(654\) −1.25712e10 −1.75733
\(655\) −5.58092e9 −0.775999
\(656\) 6.75832e9 0.934706
\(657\) 3.59689e9 0.494821
\(658\) −5.58267e9 −0.763926
\(659\) −6.30400e9 −0.858059 −0.429029 0.903290i \(-0.641144\pi\)
−0.429029 + 0.903290i \(0.641144\pi\)
\(660\) −7.10133e8 −0.0961470
\(661\) −5.51668e9 −0.742973 −0.371486 0.928438i \(-0.621152\pi\)
−0.371486 + 0.928438i \(0.621152\pi\)
\(662\) −1.52392e10 −2.04154
\(663\) 7.81242e9 1.04109
\(664\) −3.29013e9 −0.436139
\(665\) −6.19448e9 −0.816825
\(666\) −1.14299e10 −1.49928
\(667\) −4.68733e9 −0.611625
\(668\) 1.97331e9 0.256140
\(669\) −3.45485e9 −0.446106
\(670\) 9.74431e9 1.25167
\(671\) 9.76017e7 0.0124718
\(672\) −7.99825e9 −1.01672
\(673\) 5.21617e9 0.659628 0.329814 0.944046i \(-0.393014\pi\)
0.329814 + 0.944046i \(0.393014\pi\)
\(674\) −6.67678e9 −0.839958
\(675\) −2.55219e9 −0.319411
\(676\) 1.86944e10 2.32755
\(677\) 3.56727e9 0.441851 0.220925 0.975291i \(-0.429092\pi\)
0.220925 + 0.975291i \(0.429092\pi\)
\(678\) 8.69118e9 1.07096
\(679\) 1.09247e10 1.33927
\(680\) 3.81586e9 0.465384
\(681\) 6.77295e9 0.821794
\(682\) −3.66127e8 −0.0441963
\(683\) −6.40635e9 −0.769375 −0.384688 0.923047i \(-0.625691\pi\)
−0.384688 + 0.923047i \(0.625691\pi\)
\(684\) 5.65001e9 0.675076
\(685\) 1.00856e9 0.119890
\(686\) 7.98291e9 0.944120
\(687\) −7.32567e9 −0.861983
\(688\) −6.63471e8 −0.0776716
\(689\) 1.16369e10 1.35541
\(690\) 7.03178e9 0.814880
\(691\) −1.21339e10 −1.39903 −0.699513 0.714620i \(-0.746600\pi\)
−0.699513 + 0.714620i \(0.746600\pi\)
\(692\) −5.98220e9 −0.686261
\(693\) −9.06338e8 −0.103448
\(694\) −1.70316e10 −1.93419
\(695\) −6.00693e9 −0.678744
\(696\) 1.81236e9 0.203757
\(697\) 1.66915e10 1.86716
\(698\) −1.78743e10 −1.98946
\(699\) −4.69052e9 −0.519459
\(700\) 4.81280e9 0.530341
\(701\) 1.62846e10 1.78552 0.892759 0.450535i \(-0.148767\pi\)
0.892759 + 0.450535i \(0.148767\pi\)
\(702\) 2.32546e10 2.53705
\(703\) −1.18409e10 −1.28541
\(704\) 1.96898e9 0.212686
\(705\) 1.93930e9 0.208441
\(706\) −2.10718e10 −2.25364
\(707\) 1.24386e10 1.32374
\(708\) 7.68152e9 0.813450
\(709\) −3.23891e9 −0.341301 −0.170650 0.985332i \(-0.554587\pi\)
−0.170650 + 0.985332i \(0.554587\pi\)
\(710\) −3.98387e9 −0.417735
\(711\) 5.99753e9 0.625791
\(712\) −2.37759e9 −0.246864
\(713\) 2.08962e9 0.215901
\(714\) −1.15652e10 −1.18908
\(715\) 1.82960e9 0.187191
\(716\) 1.85295e10 1.88655
\(717\) 9.55052e9 0.967631
\(718\) 1.97605e10 1.99234
\(719\) 8.54046e9 0.856900 0.428450 0.903565i \(-0.359060\pi\)
0.428450 + 0.903565i \(0.359060\pi\)
\(720\) −2.58449e9 −0.258054
\(721\) −2.33133e9 −0.231649
\(722\) −5.38298e9 −0.532283
\(723\) 1.72964e9 0.170204
\(724\) −1.79735e10 −1.76014
\(725\) 1.93353e9 0.188438
\(726\) 9.65883e9 0.936799
\(727\) −9.39101e9 −0.906445 −0.453223 0.891397i \(-0.649726\pi\)
−0.453223 + 0.891397i \(0.649726\pi\)
\(728\) −1.16228e10 −1.11648
\(729\) 6.58217e9 0.629249
\(730\) 1.07488e10 1.02266
\(731\) −1.63862e9 −0.155156
\(732\) 8.12640e8 0.0765790
\(733\) 6.63206e9 0.621991 0.310996 0.950411i \(-0.399338\pi\)
0.310996 + 0.950411i \(0.399338\pi\)
\(734\) 2.66667e10 2.48905
\(735\) 2.75018e9 0.255479
\(736\) −1.49441e10 −1.38165
\(737\) 1.47699e9 0.135907
\(738\) 1.88956e10 1.73047
\(739\) 9.73593e9 0.887405 0.443703 0.896174i \(-0.353665\pi\)
0.443703 + 0.896174i \(0.353665\pi\)
\(740\) −1.96874e10 −1.78598
\(741\) 9.16204e9 0.827234
\(742\) −1.72268e10 −1.54807
\(743\) −1.04679e9 −0.0936265 −0.0468132 0.998904i \(-0.514907\pi\)
−0.0468132 + 0.998904i \(0.514907\pi\)
\(744\) −8.07957e8 −0.0719255
\(745\) −6.12638e9 −0.542822
\(746\) −1.41093e10 −1.24428
\(747\) 5.50367e9 0.483092
\(748\) 2.18224e9 0.190654
\(749\) −1.54717e10 −1.34540
\(750\) −1.20085e10 −1.03938
\(751\) 1.53753e10 1.32460 0.662300 0.749239i \(-0.269580\pi\)
0.662300 + 0.749239i \(0.269580\pi\)
\(752\) −2.41297e9 −0.206914
\(753\) 3.90258e9 0.333096
\(754\) −1.76176e10 −1.49674
\(755\) −1.21851e10 −1.03042
\(756\) −1.98421e10 −1.67017
\(757\) −1.40954e9 −0.118098 −0.0590489 0.998255i \(-0.518807\pi\)
−0.0590489 + 0.998255i \(0.518807\pi\)
\(758\) 3.84502e9 0.320669
\(759\) 1.06584e9 0.0884800
\(760\) 4.47506e9 0.369787
\(761\) −1.37074e10 −1.12748 −0.563740 0.825952i \(-0.690638\pi\)
−0.563740 + 0.825952i \(0.690638\pi\)
\(762\) 1.86297e10 1.52533
\(763\) −2.76358e10 −2.25235
\(764\) 1.15191e10 0.934529
\(765\) −6.38310e9 −0.515486
\(766\) 2.27781e10 1.83112
\(767\) −1.97908e10 −1.58373
\(768\) 3.71255e8 0.0295738
\(769\) −3.22094e9 −0.255412 −0.127706 0.991812i \(-0.540761\pi\)
−0.127706 + 0.991812i \(0.540761\pi\)
\(770\) −2.70847e9 −0.213799
\(771\) 7.41019e9 0.582290
\(772\) 8.94524e9 0.699732
\(773\) 1.47558e10 1.14904 0.574518 0.818492i \(-0.305190\pi\)
0.574518 + 0.818492i \(0.305190\pi\)
\(774\) −1.85500e9 −0.143798
\(775\) −8.61973e8 −0.0665178
\(776\) −7.89234e9 −0.606303
\(777\) 1.58149e10 1.20946
\(778\) 3.87566e9 0.295064
\(779\) 1.95750e10 1.48361
\(780\) 1.52334e10 1.14939
\(781\) −6.03853e8 −0.0453579
\(782\) −2.16087e10 −1.61586
\(783\) −7.97152e9 −0.593437
\(784\) −3.42190e9 −0.253607
\(785\) 1.55208e10 1.14517
\(786\) −1.22197e10 −0.897599
\(787\) −5.55719e9 −0.406390 −0.203195 0.979138i \(-0.565133\pi\)
−0.203195 + 0.979138i \(0.565133\pi\)
\(788\) −1.60159e10 −1.16603
\(789\) −5.29328e9 −0.383668
\(790\) 1.79228e10 1.29334
\(791\) 1.91062e10 1.37264
\(792\) 6.54764e8 0.0468324
\(793\) −2.09370e9 −0.149094
\(794\) −1.13971e10 −0.808023
\(795\) 5.98424e9 0.422400
\(796\) 1.07693e10 0.756820
\(797\) −1.17611e10 −0.822891 −0.411446 0.911434i \(-0.634976\pi\)
−0.411446 + 0.911434i \(0.634976\pi\)
\(798\) −1.35631e10 −0.944822
\(799\) −5.95949e9 −0.413328
\(800\) 6.16446e9 0.425677
\(801\) 3.97719e9 0.273440
\(802\) −1.29516e10 −0.886569
\(803\) 1.62925e9 0.111041
\(804\) 1.22975e10 0.834491
\(805\) 1.54583e10 1.04442
\(806\) 7.85396e9 0.528343
\(807\) 1.41521e10 0.947904
\(808\) −8.98595e9 −0.599273
\(809\) 1.25815e10 0.835433 0.417717 0.908577i \(-0.362830\pi\)
0.417717 + 0.908577i \(0.362830\pi\)
\(810\) 1.84581e8 0.0122036
\(811\) 2.43090e10 1.60027 0.800136 0.599818i \(-0.204760\pi\)
0.800136 + 0.599818i \(0.204760\pi\)
\(812\) 1.50323e10 0.985325
\(813\) 1.62143e9 0.105823
\(814\) −5.17730e9 −0.336448
\(815\) −1.29042e10 −0.834986
\(816\) −4.99878e9 −0.322069
\(817\) −1.92170e9 −0.123284
\(818\) −8.47724e9 −0.541524
\(819\) 1.94423e10 1.23667
\(820\) 3.25467e10 2.06138
\(821\) −9.75294e9 −0.615084 −0.307542 0.951534i \(-0.599506\pi\)
−0.307542 + 0.951534i \(0.599506\pi\)
\(822\) 2.20830e9 0.138677
\(823\) −1.00621e10 −0.629202 −0.314601 0.949224i \(-0.601871\pi\)
−0.314601 + 0.949224i \(0.601871\pi\)
\(824\) 1.68422e9 0.104871
\(825\) −4.39660e8 −0.0272601
\(826\) 2.92976e10 1.80885
\(827\) −1.80699e10 −1.11093 −0.555464 0.831541i \(-0.687459\pi\)
−0.555464 + 0.831541i \(0.687459\pi\)
\(828\) −1.40996e10 −0.863176
\(829\) −5.71510e9 −0.348404 −0.174202 0.984710i \(-0.555735\pi\)
−0.174202 + 0.984710i \(0.555735\pi\)
\(830\) 1.64470e10 0.998419
\(831\) −2.00088e10 −1.20953
\(832\) −4.22376e10 −2.54254
\(833\) −8.45131e9 −0.506601
\(834\) −1.31525e10 −0.785103
\(835\) −2.61447e9 −0.155410
\(836\) 2.55923e9 0.151491
\(837\) 3.55372e9 0.209481
\(838\) 2.43029e10 1.42660
\(839\) −1.09882e10 −0.642332 −0.321166 0.947023i \(-0.604075\pi\)
−0.321166 + 0.947023i \(0.604075\pi\)
\(840\) −5.97697e9 −0.347940
\(841\) −1.12107e10 −0.649900
\(842\) −4.41956e10 −2.55145
\(843\) −6.88312e9 −0.395721
\(844\) 2.52933e10 1.44813
\(845\) −2.47686e10 −1.41222
\(846\) −6.74644e9 −0.383070
\(847\) 2.12334e10 1.20068
\(848\) −7.44586e9 −0.419305
\(849\) −8.79091e9 −0.493011
\(850\) 8.91362e9 0.497838
\(851\) 2.95488e10 1.64357
\(852\) −5.02773e9 −0.278505
\(853\) 2.69850e10 1.48868 0.744338 0.667803i \(-0.232765\pi\)
0.744338 + 0.667803i \(0.232765\pi\)
\(854\) 3.09944e9 0.170287
\(855\) −7.48579e9 −0.409597
\(856\) 1.11772e10 0.609082
\(857\) 4.73891e9 0.257185 0.128593 0.991698i \(-0.458954\pi\)
0.128593 + 0.991698i \(0.458954\pi\)
\(858\) 4.00600e9 0.216524
\(859\) −1.73970e10 −0.936479 −0.468239 0.883602i \(-0.655111\pi\)
−0.468239 + 0.883602i \(0.655111\pi\)
\(860\) −3.19514e9 −0.171295
\(861\) −2.61447e10 −1.39596
\(862\) 1.12153e10 0.596399
\(863\) 9.17138e9 0.485732 0.242866 0.970060i \(-0.421912\pi\)
0.242866 + 0.970060i \(0.421912\pi\)
\(864\) −2.54147e10 −1.34056
\(865\) 7.92592e9 0.416383
\(866\) −2.46009e10 −1.28718
\(867\) −4.19266e8 −0.0218486
\(868\) −6.70144e9 −0.347816
\(869\) 2.71664e9 0.140431
\(870\) −9.05979e9 −0.466445
\(871\) −3.16836e10 −1.62469
\(872\) 1.99648e10 1.01967
\(873\) 1.32021e10 0.671575
\(874\) −2.53416e10 −1.28394
\(875\) −2.63989e10 −1.33216
\(876\) 1.35653e10 0.681811
\(877\) −4.84972e9 −0.242783 −0.121392 0.992605i \(-0.538736\pi\)
−0.121392 + 0.992605i \(0.538736\pi\)
\(878\) 5.57603e10 2.78032
\(879\) 2.02167e10 1.00403
\(880\) −1.17067e9 −0.0579088
\(881\) −5.10633e9 −0.251590 −0.125795 0.992056i \(-0.540148\pi\)
−0.125795 + 0.992056i \(0.540148\pi\)
\(882\) −9.56730e9 −0.469515
\(883\) 2.88460e10 1.41001 0.705006 0.709201i \(-0.250944\pi\)
0.705006 + 0.709201i \(0.250944\pi\)
\(884\) −4.68123e10 −2.27917
\(885\) −1.01774e10 −0.493554
\(886\) −5.64476e10 −2.72664
\(887\) −7.47479e9 −0.359638 −0.179819 0.983700i \(-0.557551\pi\)
−0.179819 + 0.983700i \(0.557551\pi\)
\(888\) −1.14251e10 −0.547539
\(889\) 4.09544e10 1.95499
\(890\) 1.18853e10 0.565126
\(891\) 2.79777e7 0.00132507
\(892\) 2.07016e10 0.976622
\(893\) −6.98900e9 −0.328424
\(894\) −1.34140e10 −0.627882
\(895\) −2.45501e10 −1.14465
\(896\) 2.73037e10 1.26807
\(897\) −2.28638e10 −1.05773
\(898\) −1.56452e10 −0.720964
\(899\) −2.69229e9 −0.123584
\(900\) 5.81609e9 0.265939
\(901\) −1.83896e10 −0.837597
\(902\) 8.55896e9 0.388328
\(903\) 2.56665e9 0.116001
\(904\) −1.38029e10 −0.621412
\(905\) 2.38134e10 1.06795
\(906\) −2.66799e10 −1.19189
\(907\) 3.37329e9 0.150116 0.0750581 0.997179i \(-0.476086\pi\)
0.0750581 + 0.997179i \(0.476086\pi\)
\(908\) −4.05837e10 −1.79908
\(909\) 1.50315e10 0.663788
\(910\) 5.81007e10 2.55586
\(911\) −1.87342e10 −0.820958 −0.410479 0.911870i \(-0.634638\pi\)
−0.410479 + 0.911870i \(0.634638\pi\)
\(912\) −5.86233e9 −0.255911
\(913\) 2.49294e9 0.108409
\(914\) 6.19636e10 2.68426
\(915\) −1.07668e9 −0.0464637
\(916\) 4.38956e10 1.88707
\(917\) −2.68632e10 −1.15044
\(918\) −3.67488e10 −1.56781
\(919\) 2.69873e10 1.14698 0.573489 0.819214i \(-0.305590\pi\)
0.573489 + 0.819214i \(0.305590\pi\)
\(920\) −1.11675e10 −0.472823
\(921\) −1.93745e10 −0.817188
\(922\) 3.68570e10 1.54868
\(923\) 1.29535e10 0.542229
\(924\) −3.41815e9 −0.142541
\(925\) −1.21889e10 −0.506372
\(926\) −2.29631e10 −0.950368
\(927\) −2.81733e9 −0.116160
\(928\) 1.92541e10 0.790868
\(929\) 1.85412e10 0.758721 0.379360 0.925249i \(-0.376144\pi\)
0.379360 + 0.925249i \(0.376144\pi\)
\(930\) 4.03888e9 0.164653
\(931\) −9.91129e9 −0.402537
\(932\) 2.81058e10 1.13721
\(933\) 2.52297e10 1.01701
\(934\) 4.78533e8 0.0192176
\(935\) −2.89129e9 −0.115678
\(936\) −1.40457e10 −0.559856
\(937\) 4.42644e10 1.75779 0.878893 0.477020i \(-0.158283\pi\)
0.878893 + 0.477020i \(0.158283\pi\)
\(938\) 4.69032e10 1.85563
\(939\) −1.74960e10 −0.689620
\(940\) −1.16204e10 −0.456323
\(941\) 3.74450e10 1.46498 0.732488 0.680780i \(-0.238359\pi\)
0.732488 + 0.680780i \(0.238359\pi\)
\(942\) 3.39837e10 1.32462
\(943\) −4.88493e10 −1.89700
\(944\) 1.26632e10 0.489937
\(945\) 2.62891e10 1.01336
\(946\) −8.40242e8 −0.0322690
\(947\) −3.20475e10 −1.22622 −0.613111 0.789996i \(-0.710082\pi\)
−0.613111 + 0.789996i \(0.710082\pi\)
\(948\) 2.26190e10 0.862272
\(949\) −3.49498e10 −1.32743
\(950\) 1.04535e10 0.395574
\(951\) 2.56126e10 0.965654
\(952\) 1.83673e10 0.689946
\(953\) 3.72778e10 1.39517 0.697583 0.716504i \(-0.254259\pi\)
0.697583 + 0.716504i \(0.254259\pi\)
\(954\) −2.08179e10 −0.776280
\(955\) −1.52619e10 −0.567018
\(956\) −5.72270e10 −2.11835
\(957\) −1.37323e9 −0.0506468
\(958\) 5.37867e10 1.97649
\(959\) 4.85460e9 0.177741
\(960\) −2.17206e10 −0.792360
\(961\) −2.63124e10 −0.956375
\(962\) 1.11061e11 4.02205
\(963\) −1.86970e10 −0.674653
\(964\) −1.03640e10 −0.372614
\(965\) −1.18517e10 −0.424556
\(966\) 3.38467e10 1.20808
\(967\) 9.26852e8 0.0329623 0.0164812 0.999864i \(-0.494754\pi\)
0.0164812 + 0.999864i \(0.494754\pi\)
\(968\) −1.53396e10 −0.543564
\(969\) −1.44786e10 −0.511204
\(970\) 3.94528e10 1.38796
\(971\) −1.22212e10 −0.428398 −0.214199 0.976790i \(-0.568714\pi\)
−0.214199 + 0.976790i \(0.568714\pi\)
\(972\) −3.88375e10 −1.35650
\(973\) −2.89137e10 −1.00626
\(974\) 4.26510e10 1.47901
\(975\) 9.43135e9 0.325880
\(976\) 1.33966e9 0.0461231
\(977\) −1.62117e10 −0.556157 −0.278078 0.960558i \(-0.589697\pi\)
−0.278078 + 0.960558i \(0.589697\pi\)
\(978\) −2.82544e10 −0.965829
\(979\) 1.80151e9 0.0613616
\(980\) −1.64792e10 −0.559298
\(981\) −3.33968e10 −1.12944
\(982\) −6.17630e10 −2.08132
\(983\) −7.23772e9 −0.243033 −0.121516 0.992589i \(-0.538776\pi\)
−0.121516 + 0.992589i \(0.538776\pi\)
\(984\) 1.88877e10 0.631969
\(985\) 2.12198e10 0.707480
\(986\) 2.78408e10 0.924937
\(987\) 9.33463e9 0.309020
\(988\) −5.48992e10 −1.81099
\(989\) 4.79559e9 0.157636
\(990\) −3.27308e9 −0.107210
\(991\) 4.83836e10 1.57921 0.789606 0.613614i \(-0.210285\pi\)
0.789606 + 0.613614i \(0.210285\pi\)
\(992\) −8.58351e9 −0.279173
\(993\) 2.54810e10 0.825838
\(994\) −1.91759e10 −0.619305
\(995\) −1.42685e10 −0.459194
\(996\) 2.07564e10 0.665649
\(997\) −4.51500e10 −1.44286 −0.721431 0.692486i \(-0.756515\pi\)
−0.721431 + 0.692486i \(0.756515\pi\)
\(998\) 2.80348e10 0.892772
\(999\) 5.02523e10 1.59469
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.1.11 11
3.2 odd 2 387.8.a.b.1.1 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.a.1.11 11 1.1 even 1 trivial
387.8.a.b.1.1 11 3.2 odd 2