Properties

Label 43.8.a.a.1.9
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 1.9
Root \(-14.3182\) of defining polynomial
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q+12.3182 q^{2} -46.7197 q^{3} +23.7379 q^{4} +330.186 q^{5} -575.503 q^{6} -467.397 q^{7} -1284.32 q^{8} -4.26512 q^{9} +O(q^{10})\) \(q+12.3182 q^{2} -46.7197 q^{3} +23.7379 q^{4} +330.186 q^{5} -575.503 q^{6} -467.397 q^{7} -1284.32 q^{8} -4.26512 q^{9} +4067.29 q^{10} -359.251 q^{11} -1109.03 q^{12} -11439.7 q^{13} -5757.49 q^{14} -15426.2 q^{15} -18859.0 q^{16} -19000.7 q^{17} -52.5385 q^{18} -3671.03 q^{19} +7837.90 q^{20} +21836.7 q^{21} -4425.32 q^{22} -3785.73 q^{23} +60003.2 q^{24} +30897.5 q^{25} -140917. q^{26} +102375. q^{27} -11095.0 q^{28} +115539. q^{29} -190023. q^{30} +15139.6 q^{31} -67915.2 q^{32} +16784.1 q^{33} -234054. q^{34} -154328. q^{35} -101.245 q^{36} -357942. q^{37} -45220.5 q^{38} +534461. q^{39} -424064. q^{40} +540673. q^{41} +268988. q^{42} +79507.0 q^{43} -8527.85 q^{44} -1408.28 q^{45} -46633.3 q^{46} +289961. q^{47} +881086. q^{48} -605083. q^{49} +380601. q^{50} +887708. q^{51} -271555. q^{52} -329115. q^{53} +1.26108e6 q^{54} -118619. q^{55} +600288. q^{56} +171510. q^{57} +1.42323e6 q^{58} +2.44911e6 q^{59} -366185. q^{60} -1.25749e6 q^{61} +186492. q^{62} +1993.50 q^{63} +1.57735e6 q^{64} -3.77723e6 q^{65} +206750. q^{66} -1.70199e6 q^{67} -451036. q^{68} +176868. q^{69} -1.90104e6 q^{70} -3.75585e6 q^{71} +5477.78 q^{72} -5.83440e6 q^{73} -4.40920e6 q^{74} -1.44352e6 q^{75} -87142.4 q^{76} +167913. q^{77} +6.58360e6 q^{78} +3.48456e6 q^{79} -6.22696e6 q^{80} -4.77362e6 q^{81} +6.66012e6 q^{82} +5.17957e6 q^{83} +518356. q^{84} -6.27376e6 q^{85} +979383. q^{86} -5.39795e6 q^{87} +461394. q^{88} +5.98803e6 q^{89} -17347.5 q^{90} +5.34690e6 q^{91} -89865.1 q^{92} -707317. q^{93} +3.57180e6 q^{94} -1.21212e6 q^{95} +3.17298e6 q^{96} -5.18336e6 q^{97} -7.45353e6 q^{98} +1532.25 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} + O(q^{10}) \) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} - 1333q^{10} + 1333q^{11} + 5089q^{12} - 17967q^{13} - 22352q^{14} - 49504q^{15} - 34406q^{16} - 63095q^{17} - 165931q^{18} - 54524q^{19} - 280995q^{20} - 139788q^{21} - 289358q^{22} - 138139q^{23} - 429583q^{24} + 3455q^{25} - 132946q^{26} - 240356q^{27} - 12704q^{28} - 308658q^{29} + 421284q^{30} - 209523q^{31} - 644934q^{32} + 96814q^{33} + 762435q^{34} - 578892q^{35} + 426161q^{36} - 298472q^{37} - 369707q^{38} + 292298q^{39} + 2633173q^{40} - 1346735q^{41} + 1173266q^{42} + 874577q^{43} + 3134292q^{44} + 1893784q^{45} + 3588111q^{46} + 499284q^{47} + 5647533q^{48} + 2544563q^{49} + 3049745q^{50} + 1258424q^{51} + 983088q^{52} - 2210495q^{53} + 6789698q^{54} - 1855072q^{55} - 469976q^{56} - 1238444q^{57} + 4397067q^{58} - 5824216q^{59} - 2889372q^{60} - 4453034q^{61} + 1002789q^{62} - 6240564q^{63} + 4757538q^{64} - 2162872q^{65} - 258940q^{66} - 6859513q^{67} - 9397005q^{68} - 10040030q^{69} + 845078q^{70} - 10726554q^{71} + 1199517q^{72} - 4456898q^{73} + 1046637q^{74} - 3349114q^{75} + 5861267q^{76} - 17019816q^{77} + 1999122q^{78} - 15541320q^{79} - 15680911q^{80} - 12976697q^{81} + 20233655q^{82} - 11146767q^{83} + 12348278q^{84} - 12471976q^{85} - 1908168q^{86} - 18648900q^{87} - 24463544q^{88} - 13531356q^{89} + 20858990q^{90} - 19746448q^{91} - 26023161q^{92} - 21903110q^{93} + 20288857q^{94} - 12291624q^{95} - 13954503q^{96} - 10999901q^{97} + 29909168q^{98} + 29396057q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 12.3182 1.08878 0.544392 0.838831i \(-0.316760\pi\)
0.544392 + 0.838831i \(0.316760\pi\)
\(3\) −46.7197 −0.999024 −0.499512 0.866307i \(-0.666487\pi\)
−0.499512 + 0.866307i \(0.666487\pi\)
\(4\) 23.7379 0.185452
\(5\) 330.186 1.18131 0.590654 0.806925i \(-0.298870\pi\)
0.590654 + 0.806925i \(0.298870\pi\)
\(6\) −575.503 −1.08772
\(7\) −467.397 −0.515042 −0.257521 0.966273i \(-0.582906\pi\)
−0.257521 + 0.966273i \(0.582906\pi\)
\(8\) −1284.32 −0.886867
\(9\) −4.26512 −0.00195021
\(10\) 4067.29 1.28619
\(11\) −359.251 −0.0813811 −0.0406906 0.999172i \(-0.512956\pi\)
−0.0406906 + 0.999172i \(0.512956\pi\)
\(12\) −1109.03 −0.185271
\(13\) −11439.7 −1.44416 −0.722078 0.691812i \(-0.756813\pi\)
−0.722078 + 0.691812i \(0.756813\pi\)
\(14\) −5757.49 −0.560770
\(15\) −15426.2 −1.18016
\(16\) −18859.0 −1.15106
\(17\) −19000.7 −0.937990 −0.468995 0.883201i \(-0.655384\pi\)
−0.468995 + 0.883201i \(0.655384\pi\)
\(18\) −52.5385 −0.00212336
\(19\) −3671.03 −0.122786 −0.0613932 0.998114i \(-0.519554\pi\)
−0.0613932 + 0.998114i \(0.519554\pi\)
\(20\) 7837.90 0.219076
\(21\) 21836.7 0.514540
\(22\) −4425.32 −0.0886065
\(23\) −3785.73 −0.0648787 −0.0324393 0.999474i \(-0.510328\pi\)
−0.0324393 + 0.999474i \(0.510328\pi\)
\(24\) 60003.2 0.886002
\(25\) 30897.5 0.395488
\(26\) −140917. −1.57237
\(27\) 102375. 1.00097
\(28\) −11095.0 −0.0955157
\(29\) 115539. 0.879702 0.439851 0.898071i \(-0.355031\pi\)
0.439851 + 0.898071i \(0.355031\pi\)
\(30\) −190023. −1.28493
\(31\) 15139.6 0.0912741 0.0456371 0.998958i \(-0.485468\pi\)
0.0456371 + 0.998958i \(0.485468\pi\)
\(32\) −67915.2 −0.366389
\(33\) 16784.1 0.0813017
\(34\) −234054. −1.02127
\(35\) −154328. −0.608424
\(36\) −101.245 −0.000361671 0
\(37\) −357942. −1.16173 −0.580867 0.813998i \(-0.697286\pi\)
−0.580867 + 0.813998i \(0.697286\pi\)
\(38\) −45220.5 −0.133688
\(39\) 534461. 1.44275
\(40\) −424064. −1.04766
\(41\) 540673. 1.22516 0.612578 0.790410i \(-0.290133\pi\)
0.612578 + 0.790410i \(0.290133\pi\)
\(42\) 268988. 0.560223
\(43\) 79507.0 0.152499
\(44\) −8527.85 −0.0150923
\(45\) −1408.28 −0.00230380
\(46\) −46633.3 −0.0706389
\(47\) 289961. 0.407378 0.203689 0.979036i \(-0.434707\pi\)
0.203689 + 0.979036i \(0.434707\pi\)
\(48\) 881086. 1.14994
\(49\) −605083. −0.734731
\(50\) 380601. 0.430601
\(51\) 887708. 0.937075
\(52\) −271555. −0.267822
\(53\) −329115. −0.303656 −0.151828 0.988407i \(-0.548516\pi\)
−0.151828 + 0.988407i \(0.548516\pi\)
\(54\) 1.26108e6 1.08984
\(55\) −118619. −0.0961361
\(56\) 600288. 0.456774
\(57\) 171510. 0.122667
\(58\) 1.42323e6 0.957806
\(59\) 2.44911e6 1.55248 0.776240 0.630437i \(-0.217124\pi\)
0.776240 + 0.630437i \(0.217124\pi\)
\(60\) −366185. −0.218862
\(61\) −1.25749e6 −0.709333 −0.354666 0.934993i \(-0.615406\pi\)
−0.354666 + 0.934993i \(0.615406\pi\)
\(62\) 186492. 0.0993779
\(63\) 1993.50 0.00100444
\(64\) 1.57735e6 0.752141
\(65\) −3.77723e6 −1.70599
\(66\) 206750. 0.0885201
\(67\) −1.70199e6 −0.691347 −0.345673 0.938355i \(-0.612350\pi\)
−0.345673 + 0.938355i \(0.612350\pi\)
\(68\) −451036. −0.173952
\(69\) 176868. 0.0648154
\(70\) −1.90104e6 −0.662442
\(71\) −3.75585e6 −1.24539 −0.622693 0.782466i \(-0.713961\pi\)
−0.622693 + 0.782466i \(0.713961\pi\)
\(72\) 5477.78 0.00172958
\(73\) −5.83440e6 −1.75536 −0.877680 0.479247i \(-0.840910\pi\)
−0.877680 + 0.479247i \(0.840910\pi\)
\(74\) −4.40920e6 −1.26488
\(75\) −1.44352e6 −0.395102
\(76\) −87142.4 −0.0227710
\(77\) 167913. 0.0419147
\(78\) 6.58360e6 1.57084
\(79\) 3.48456e6 0.795158 0.397579 0.917568i \(-0.369851\pi\)
0.397579 + 0.917568i \(0.369851\pi\)
\(80\) −6.22696e6 −1.35976
\(81\) −4.77362e6 −0.998046
\(82\) 6.66012e6 1.33393
\(83\) 5.17957e6 0.994307 0.497154 0.867663i \(-0.334379\pi\)
0.497154 + 0.867663i \(0.334379\pi\)
\(84\) 518356. 0.0954225
\(85\) −6.27376e6 −1.10806
\(86\) 979383. 0.166038
\(87\) −5.39795e6 −0.878844
\(88\) 461394. 0.0721743
\(89\) 5.98803e6 0.900365 0.450183 0.892936i \(-0.351359\pi\)
0.450183 + 0.892936i \(0.351359\pi\)
\(90\) −17347.5 −0.00250834
\(91\) 5.34690e6 0.743801
\(92\) −89865.1 −0.0120319
\(93\) −707317. −0.0911851
\(94\) 3.57180e6 0.443547
\(95\) −1.21212e6 −0.145049
\(96\) 3.17298e6 0.366031
\(97\) −5.18336e6 −0.576647 −0.288324 0.957533i \(-0.593098\pi\)
−0.288324 + 0.957533i \(0.593098\pi\)
\(98\) −7.45353e6 −0.799964
\(99\) 1532.25 0.000158711 0
\(100\) 733440. 0.0733440
\(101\) −930045. −0.0898213 −0.0449106 0.998991i \(-0.514300\pi\)
−0.0449106 + 0.998991i \(0.514300\pi\)
\(102\) 1.09350e7 1.02027
\(103\) −1.70172e7 −1.53447 −0.767233 0.641368i \(-0.778367\pi\)
−0.767233 + 0.641368i \(0.778367\pi\)
\(104\) 1.46923e7 1.28077
\(105\) 7.21016e6 0.607830
\(106\) −4.05410e6 −0.330616
\(107\) 1.27256e7 1.00423 0.502116 0.864800i \(-0.332555\pi\)
0.502116 + 0.864800i \(0.332555\pi\)
\(108\) 2.43017e6 0.185632
\(109\) −1.67891e7 −1.24175 −0.620874 0.783910i \(-0.713222\pi\)
−0.620874 + 0.783910i \(0.713222\pi\)
\(110\) −1.46118e6 −0.104672
\(111\) 1.67230e7 1.16060
\(112\) 8.81463e6 0.592845
\(113\) 1.36793e7 0.891842 0.445921 0.895072i \(-0.352876\pi\)
0.445921 + 0.895072i \(0.352876\pi\)
\(114\) 2.11269e6 0.133558
\(115\) −1.24999e6 −0.0766417
\(116\) 2.74265e6 0.163143
\(117\) 48791.8 0.00281641
\(118\) 3.01686e7 1.69032
\(119\) 8.88087e6 0.483105
\(120\) 1.98122e7 1.04664
\(121\) −1.93581e7 −0.993377
\(122\) −1.54900e7 −0.772311
\(123\) −2.52601e7 −1.22396
\(124\) 359381. 0.0169270
\(125\) −1.55938e7 −0.714115
\(126\) 24556.4 0.00109362
\(127\) 1.29034e6 0.0558972 0.0279486 0.999609i \(-0.491103\pi\)
0.0279486 + 0.999609i \(0.491103\pi\)
\(128\) 2.81233e7 1.18531
\(129\) −3.71455e6 −0.152350
\(130\) −4.65287e7 −1.85746
\(131\) −3.96207e7 −1.53983 −0.769915 0.638146i \(-0.779701\pi\)
−0.769915 + 0.638146i \(0.779701\pi\)
\(132\) 398419. 0.0150776
\(133\) 1.71583e6 0.0632402
\(134\) −2.09655e7 −0.752728
\(135\) 3.38029e7 1.18246
\(136\) 2.44030e7 0.831873
\(137\) −1.20603e7 −0.400715 −0.200358 0.979723i \(-0.564210\pi\)
−0.200358 + 0.979723i \(0.564210\pi\)
\(138\) 2.17870e6 0.0705700
\(139\) 3.45946e7 1.09259 0.546294 0.837594i \(-0.316038\pi\)
0.546294 + 0.837594i \(0.316038\pi\)
\(140\) −3.66341e6 −0.112833
\(141\) −1.35469e7 −0.406981
\(142\) −4.62653e7 −1.35596
\(143\) 4.10973e6 0.117527
\(144\) 80435.7 0.00224481
\(145\) 3.81493e7 1.03920
\(146\) −7.18693e7 −1.91121
\(147\) 2.82693e7 0.734014
\(148\) −8.49678e6 −0.215446
\(149\) 3.46074e7 0.857072 0.428536 0.903525i \(-0.359030\pi\)
0.428536 + 0.903525i \(0.359030\pi\)
\(150\) −1.77816e7 −0.430181
\(151\) −2.74859e7 −0.649666 −0.324833 0.945771i \(-0.605308\pi\)
−0.324833 + 0.945771i \(0.605308\pi\)
\(152\) 4.71478e6 0.108895
\(153\) 81040.2 0.00182928
\(154\) 2.06838e6 0.0456361
\(155\) 4.99887e6 0.107823
\(156\) 1.26870e7 0.267560
\(157\) −3.91834e7 −0.808078 −0.404039 0.914742i \(-0.632394\pi\)
−0.404039 + 0.914742i \(0.632394\pi\)
\(158\) 4.29235e7 0.865756
\(159\) 1.53762e7 0.303360
\(160\) −2.24246e7 −0.432818
\(161\) 1.76944e6 0.0334153
\(162\) −5.88024e7 −1.08666
\(163\) 4.81506e7 0.870854 0.435427 0.900224i \(-0.356597\pi\)
0.435427 + 0.900224i \(0.356597\pi\)
\(164\) 1.28344e7 0.227208
\(165\) 5.54187e6 0.0960423
\(166\) 6.38030e7 1.08259
\(167\) −5.80547e7 −0.964561 −0.482280 0.876017i \(-0.660191\pi\)
−0.482280 + 0.876017i \(0.660191\pi\)
\(168\) −2.80453e7 −0.456329
\(169\) 6.81189e7 1.08559
\(170\) −7.72813e7 −1.20643
\(171\) 15657.4 0.000239460 0
\(172\) 1.88733e6 0.0282812
\(173\) −3.04236e7 −0.446735 −0.223367 0.974734i \(-0.571705\pi\)
−0.223367 + 0.974734i \(0.571705\pi\)
\(174\) −6.64930e7 −0.956872
\(175\) −1.44414e7 −0.203693
\(176\) 6.77510e6 0.0936745
\(177\) −1.14422e8 −1.55097
\(178\) 7.37617e7 0.980304
\(179\) 6.08790e6 0.0793382 0.0396691 0.999213i \(-0.487370\pi\)
0.0396691 + 0.999213i \(0.487370\pi\)
\(180\) −33429.6 −0.000427245 0
\(181\) 1.51213e8 1.89546 0.947728 0.319079i \(-0.103373\pi\)
0.947728 + 0.319079i \(0.103373\pi\)
\(182\) 6.58641e7 0.809840
\(183\) 5.87496e7 0.708641
\(184\) 4.86209e6 0.0575388
\(185\) −1.18187e8 −1.37237
\(186\) −8.71287e6 −0.0992809
\(187\) 6.82602e6 0.0763347
\(188\) 6.88307e6 0.0755491
\(189\) −4.78500e7 −0.515543
\(190\) −1.49311e7 −0.157927
\(191\) 1.15042e8 1.19465 0.597323 0.802000i \(-0.296231\pi\)
0.597323 + 0.802000i \(0.296231\pi\)
\(192\) −7.36936e7 −0.751407
\(193\) 5.16665e7 0.517319 0.258659 0.965969i \(-0.416719\pi\)
0.258659 + 0.965969i \(0.416719\pi\)
\(194\) −6.38496e7 −0.627845
\(195\) 1.76471e8 1.70433
\(196\) −1.43634e7 −0.136257
\(197\) −7.84505e7 −0.731078 −0.365539 0.930796i \(-0.619115\pi\)
−0.365539 + 0.930796i \(0.619115\pi\)
\(198\) 18874.5 0.000172802 0
\(199\) −7.08100e7 −0.636955 −0.318478 0.947930i \(-0.603172\pi\)
−0.318478 + 0.947930i \(0.603172\pi\)
\(200\) −3.96823e7 −0.350745
\(201\) 7.95167e7 0.690672
\(202\) −1.14565e7 −0.0977960
\(203\) −5.40026e7 −0.453084
\(204\) 2.10723e7 0.173783
\(205\) 1.78522e8 1.44729
\(206\) −2.09621e8 −1.67070
\(207\) 16146.6 0.000126527 0
\(208\) 2.15741e8 1.66231
\(209\) 1.31882e6 0.00999250
\(210\) 8.88161e7 0.661796
\(211\) −1.44130e8 −1.05625 −0.528124 0.849167i \(-0.677104\pi\)
−0.528124 + 0.849167i \(0.677104\pi\)
\(212\) −7.81248e6 −0.0563136
\(213\) 1.75472e8 1.24417
\(214\) 1.56756e8 1.09339
\(215\) 2.62521e7 0.180148
\(216\) −1.31483e8 −0.887730
\(217\) −7.07619e6 −0.0470101
\(218\) −2.06811e8 −1.35200
\(219\) 2.72582e8 1.75365
\(220\) −2.81577e6 −0.0178286
\(221\) 2.17363e8 1.35460
\(222\) 2.05997e8 1.26364
\(223\) 1.64343e8 0.992396 0.496198 0.868209i \(-0.334729\pi\)
0.496198 + 0.868209i \(0.334729\pi\)
\(224\) 3.17434e7 0.188706
\(225\) −131781. −0.000771285 0
\(226\) 1.68504e8 0.971024
\(227\) −3.36542e8 −1.90963 −0.954814 0.297204i \(-0.903946\pi\)
−0.954814 + 0.297204i \(0.903946\pi\)
\(228\) 4.07127e6 0.0227488
\(229\) −5.78696e7 −0.318439 −0.159220 0.987243i \(-0.550898\pi\)
−0.159220 + 0.987243i \(0.550898\pi\)
\(230\) −1.53976e7 −0.0834463
\(231\) −7.84485e6 −0.0418738
\(232\) −1.48389e8 −0.780179
\(233\) −2.71239e8 −1.40477 −0.702386 0.711797i \(-0.747882\pi\)
−0.702386 + 0.711797i \(0.747882\pi\)
\(234\) 601027. 0.00306647
\(235\) 9.57411e7 0.481239
\(236\) 5.81366e7 0.287911
\(237\) −1.62798e8 −0.794382
\(238\) 1.09396e8 0.525997
\(239\) −1.16728e8 −0.553075 −0.276537 0.961003i \(-0.589187\pi\)
−0.276537 + 0.961003i \(0.589187\pi\)
\(240\) 2.90922e8 1.35843
\(241\) −4.09158e8 −1.88292 −0.941460 0.337126i \(-0.890545\pi\)
−0.941460 + 0.337126i \(0.890545\pi\)
\(242\) −2.38457e8 −1.08157
\(243\) −872436. −0.00390042
\(244\) −2.98501e7 −0.131547
\(245\) −1.99790e8 −0.867944
\(246\) −3.11159e8 −1.33263
\(247\) 4.19956e7 0.177323
\(248\) −1.94441e7 −0.0809480
\(249\) −2.41988e8 −0.993337
\(250\) −1.92088e8 −0.777517
\(251\) −3.43101e8 −1.36950 −0.684752 0.728776i \(-0.740090\pi\)
−0.684752 + 0.728776i \(0.740090\pi\)
\(252\) 47321.5 0.000186276 0
\(253\) 1.36003e6 0.00527990
\(254\) 1.58946e7 0.0608601
\(255\) 2.93108e8 1.10697
\(256\) 1.44527e8 0.538405
\(257\) 7.47286e7 0.274613 0.137307 0.990529i \(-0.456155\pi\)
0.137307 + 0.990529i \(0.456155\pi\)
\(258\) −4.57565e7 −0.165876
\(259\) 1.67301e8 0.598343
\(260\) −8.96634e7 −0.316380
\(261\) −492787. −0.00171561
\(262\) −4.88056e8 −1.67654
\(263\) 2.57577e8 0.873097 0.436549 0.899681i \(-0.356201\pi\)
0.436549 + 0.899681i \(0.356201\pi\)
\(264\) −2.15562e7 −0.0721038
\(265\) −1.08669e8 −0.358711
\(266\) 2.11359e7 0.0688550
\(267\) −2.79759e8 −0.899487
\(268\) −4.04017e7 −0.128212
\(269\) −3.45824e8 −1.08323 −0.541617 0.840625i \(-0.682188\pi\)
−0.541617 + 0.840625i \(0.682188\pi\)
\(270\) 4.16390e8 1.28744
\(271\) −2.24146e8 −0.684129 −0.342065 0.939676i \(-0.611126\pi\)
−0.342065 + 0.939676i \(0.611126\pi\)
\(272\) 3.58333e8 1.07968
\(273\) −2.49806e8 −0.743076
\(274\) −1.48561e8 −0.436293
\(275\) −1.11000e7 −0.0321852
\(276\) 4.19847e6 0.0120201
\(277\) 2.45845e8 0.694995 0.347497 0.937681i \(-0.387032\pi\)
0.347497 + 0.937681i \(0.387032\pi\)
\(278\) 4.26143e8 1.18959
\(279\) −64572.0 −0.000178004 0
\(280\) 1.98206e8 0.539591
\(281\) 4.79601e8 1.28946 0.644731 0.764410i \(-0.276970\pi\)
0.644731 + 0.764410i \(0.276970\pi\)
\(282\) −1.66874e8 −0.443114
\(283\) 5.71921e8 1.49997 0.749987 0.661452i \(-0.230060\pi\)
0.749987 + 0.661452i \(0.230060\pi\)
\(284\) −8.91559e7 −0.230959
\(285\) 5.66300e7 0.144907
\(286\) 5.06245e7 0.127962
\(287\) −2.52709e8 −0.631007
\(288\) 289666. 0.000714536 0
\(289\) −4.93122e7 −0.120174
\(290\) 4.69930e8 1.13146
\(291\) 2.42165e8 0.576085
\(292\) −1.38496e8 −0.325535
\(293\) 4.33629e8 1.00712 0.503560 0.863960i \(-0.332023\pi\)
0.503560 + 0.863960i \(0.332023\pi\)
\(294\) 3.48227e8 0.799184
\(295\) 8.08661e8 1.83396
\(296\) 4.59713e8 1.03030
\(297\) −3.67785e7 −0.0814603
\(298\) 4.26301e8 0.933167
\(299\) 4.33077e7 0.0936949
\(300\) −3.42661e7 −0.0732725
\(301\) −3.71614e7 −0.0785432
\(302\) −3.38576e8 −0.707346
\(303\) 4.34515e7 0.0897336
\(304\) 6.92318e7 0.141335
\(305\) −4.15205e8 −0.837940
\(306\) 998269. 0.00199169
\(307\) −5.42296e8 −1.06968 −0.534838 0.844955i \(-0.679627\pi\)
−0.534838 + 0.844955i \(0.679627\pi\)
\(308\) 3.98589e6 0.00777317
\(309\) 7.95038e8 1.53297
\(310\) 6.15770e7 0.117396
\(311\) −8.11532e8 −1.52983 −0.764917 0.644129i \(-0.777220\pi\)
−0.764917 + 0.644129i \(0.777220\pi\)
\(312\) −6.86420e8 −1.27952
\(313\) −8.40388e8 −1.54908 −0.774541 0.632524i \(-0.782019\pi\)
−0.774541 + 0.632524i \(0.782019\pi\)
\(314\) −4.82669e8 −0.879823
\(315\) 658226. 0.00118656
\(316\) 8.27160e7 0.147464
\(317\) 3.45739e8 0.609595 0.304797 0.952417i \(-0.401411\pi\)
0.304797 + 0.952417i \(0.401411\pi\)
\(318\) 1.89406e8 0.330293
\(319\) −4.15075e7 −0.0715911
\(320\) 5.20820e8 0.888510
\(321\) −5.94535e8 −1.00325
\(322\) 2.17963e7 0.0363820
\(323\) 6.97521e7 0.115172
\(324\) −1.13316e8 −0.185090
\(325\) −3.53459e8 −0.571146
\(326\) 5.93129e8 0.948173
\(327\) 7.84380e8 1.24054
\(328\) −6.94398e8 −1.08655
\(329\) −1.35527e8 −0.209817
\(330\) 6.82659e7 0.104569
\(331\) 5.89707e8 0.893796 0.446898 0.894585i \(-0.352529\pi\)
0.446898 + 0.894585i \(0.352529\pi\)
\(332\) 1.22952e8 0.184396
\(333\) 1.52667e6 0.00226563
\(334\) −7.15129e8 −1.05020
\(335\) −5.61974e8 −0.816693
\(336\) −4.11817e8 −0.592266
\(337\) 4.92765e8 0.701351 0.350676 0.936497i \(-0.385952\pi\)
0.350676 + 0.936497i \(0.385952\pi\)
\(338\) 8.39101e8 1.18197
\(339\) −6.39091e8 −0.890972
\(340\) −1.48926e8 −0.205491
\(341\) −5.43891e6 −0.00742799
\(342\) 192871. 0.000260720 0
\(343\) 6.67736e8 0.893460
\(344\) −1.02113e8 −0.135246
\(345\) 5.83993e7 0.0765669
\(346\) −3.74764e8 −0.486398
\(347\) 9.83108e8 1.26313 0.631565 0.775323i \(-0.282413\pi\)
0.631565 + 0.775323i \(0.282413\pi\)
\(348\) −1.28136e8 −0.162983
\(349\) 1.25957e9 1.58611 0.793053 0.609153i \(-0.208491\pi\)
0.793053 + 0.609153i \(0.208491\pi\)
\(350\) −1.77892e8 −0.221778
\(351\) −1.17115e9 −1.44556
\(352\) 2.43986e7 0.0298171
\(353\) −8.97531e7 −0.108602 −0.0543010 0.998525i \(-0.517293\pi\)
−0.0543010 + 0.998525i \(0.517293\pi\)
\(354\) −1.40947e9 −1.68867
\(355\) −1.24013e9 −1.47118
\(356\) 1.42143e8 0.166975
\(357\) −4.14912e8 −0.482634
\(358\) 7.49920e7 0.0863822
\(359\) 1.81494e8 0.207029 0.103515 0.994628i \(-0.466991\pi\)
0.103515 + 0.994628i \(0.466991\pi\)
\(360\) 1.80868e6 0.00204317
\(361\) −8.80395e8 −0.984923
\(362\) 1.86267e9 2.06374
\(363\) 9.04406e8 0.992408
\(364\) 1.26924e8 0.137940
\(365\) −1.92643e9 −2.07362
\(366\) 7.23689e8 0.771557
\(367\) 1.07414e9 1.13430 0.567150 0.823614i \(-0.308046\pi\)
0.567150 + 0.823614i \(0.308046\pi\)
\(368\) 7.13949e7 0.0746792
\(369\) −2.30603e6 −0.00238932
\(370\) −1.45585e9 −1.49421
\(371\) 1.53827e8 0.156396
\(372\) −1.67902e7 −0.0169105
\(373\) −1.39148e9 −1.38834 −0.694169 0.719812i \(-0.744228\pi\)
−0.694169 + 0.719812i \(0.744228\pi\)
\(374\) 8.40842e7 0.0831120
\(375\) 7.28540e8 0.713418
\(376\) −3.72404e8 −0.361290
\(377\) −1.32173e9 −1.27043
\(378\) −5.89425e8 −0.561316
\(379\) −1.85979e8 −0.175480 −0.0877400 0.996143i \(-0.527964\pi\)
−0.0877400 + 0.996143i \(0.527964\pi\)
\(380\) −2.87732e7 −0.0268996
\(381\) −6.02843e7 −0.0558427
\(382\) 1.41711e9 1.30071
\(383\) 8.12564e8 0.739029 0.369515 0.929225i \(-0.379524\pi\)
0.369515 + 0.929225i \(0.379524\pi\)
\(384\) −1.31391e9 −1.18415
\(385\) 5.54424e7 0.0495142
\(386\) 6.36437e8 0.563249
\(387\) −339107. −0.000297405 0
\(388\) −1.23042e8 −0.106940
\(389\) −3.04526e8 −0.262301 −0.131151 0.991362i \(-0.541867\pi\)
−0.131151 + 0.991362i \(0.541867\pi\)
\(390\) 2.17381e9 1.85565
\(391\) 7.19315e7 0.0608556
\(392\) 7.77121e8 0.651609
\(393\) 1.85107e9 1.53833
\(394\) −9.66368e8 −0.795986
\(395\) 1.15055e9 0.939326
\(396\) 36372.3 2.94332e−5 0
\(397\) −2.15202e9 −1.72616 −0.863078 0.505071i \(-0.831466\pi\)
−0.863078 + 0.505071i \(0.831466\pi\)
\(398\) −8.72251e8 −0.693507
\(399\) −8.01631e7 −0.0631785
\(400\) −5.82694e8 −0.455230
\(401\) 1.32277e9 1.02442 0.512211 0.858860i \(-0.328827\pi\)
0.512211 + 0.858860i \(0.328827\pi\)
\(402\) 9.79502e8 0.751994
\(403\) −1.73193e8 −0.131814
\(404\) −2.20773e7 −0.0166575
\(405\) −1.57618e9 −1.17900
\(406\) −6.65215e8 −0.493311
\(407\) 1.28591e8 0.0945433
\(408\) −1.14010e9 −0.831061
\(409\) 5.69273e8 0.411423 0.205712 0.978613i \(-0.434049\pi\)
0.205712 + 0.978613i \(0.434049\pi\)
\(410\) 2.19907e9 1.57578
\(411\) 5.63454e8 0.400325
\(412\) −4.03951e8 −0.284570
\(413\) −1.14471e9 −0.799594
\(414\) 198897. 0.000137761 0
\(415\) 1.71022e9 1.17458
\(416\) 7.76931e8 0.529122
\(417\) −1.61625e9 −1.09152
\(418\) 1.62455e7 0.0108797
\(419\) −2.17734e9 −1.44603 −0.723016 0.690831i \(-0.757245\pi\)
−0.723016 + 0.690831i \(0.757245\pi\)
\(420\) 1.71154e8 0.112723
\(421\) −2.30881e8 −0.150800 −0.0754001 0.997153i \(-0.524023\pi\)
−0.0754001 + 0.997153i \(0.524023\pi\)
\(422\) −1.77542e9 −1.15003
\(423\) −1.23672e6 −0.000794474 0
\(424\) 4.22689e8 0.269302
\(425\) −5.87074e8 −0.370964
\(426\) 2.16150e9 1.35463
\(427\) 5.87747e8 0.365336
\(428\) 3.02078e8 0.186237
\(429\) −1.92006e8 −0.117412
\(430\) 3.23378e8 0.196142
\(431\) 2.33302e9 1.40361 0.701807 0.712367i \(-0.252377\pi\)
0.701807 + 0.712367i \(0.252377\pi\)
\(432\) −1.93069e9 −1.15218
\(433\) −2.85667e8 −0.169104 −0.0845518 0.996419i \(-0.526946\pi\)
−0.0845518 + 0.996419i \(0.526946\pi\)
\(434\) −8.71659e7 −0.0511838
\(435\) −1.78233e9 −1.03818
\(436\) −3.98536e8 −0.230285
\(437\) 1.38975e7 0.00796622
\(438\) 3.35771e9 1.90934
\(439\) 2.53362e9 1.42927 0.714636 0.699496i \(-0.246592\pi\)
0.714636 + 0.699496i \(0.246592\pi\)
\(440\) 1.52346e8 0.0852600
\(441\) 2.58075e6 0.00143288
\(442\) 2.67752e9 1.47487
\(443\) −6.54414e8 −0.357635 −0.178817 0.983882i \(-0.557227\pi\)
−0.178817 + 0.983882i \(0.557227\pi\)
\(444\) 3.96968e8 0.215236
\(445\) 1.97716e9 1.06361
\(446\) 2.02441e9 1.08051
\(447\) −1.61685e9 −0.856236
\(448\) −7.37251e8 −0.387385
\(449\) 2.92747e9 1.52627 0.763133 0.646241i \(-0.223660\pi\)
0.763133 + 0.646241i \(0.223660\pi\)
\(450\) −1.62331e6 −0.000839764 0
\(451\) −1.94237e8 −0.0997046
\(452\) 3.24716e8 0.165394
\(453\) 1.28413e9 0.649032
\(454\) −4.14559e9 −2.07917
\(455\) 1.76547e9 0.878658
\(456\) −2.20273e8 −0.108789
\(457\) −2.45680e9 −1.20410 −0.602050 0.798459i \(-0.705649\pi\)
−0.602050 + 0.798459i \(0.705649\pi\)
\(458\) −7.12849e8 −0.346712
\(459\) −1.94520e9 −0.938903
\(460\) −2.96721e7 −0.0142134
\(461\) 1.25968e9 0.598832 0.299416 0.954123i \(-0.403208\pi\)
0.299416 + 0.954123i \(0.403208\pi\)
\(462\) −9.66344e7 −0.0455916
\(463\) 1.60779e9 0.752829 0.376414 0.926451i \(-0.377157\pi\)
0.376414 + 0.926451i \(0.377157\pi\)
\(464\) −2.17895e9 −1.01259
\(465\) −2.33546e8 −0.107718
\(466\) −3.34117e9 −1.52949
\(467\) 1.74456e9 0.792643 0.396321 0.918112i \(-0.370287\pi\)
0.396321 + 0.918112i \(0.370287\pi\)
\(468\) 1.15821e6 0.000522309 0
\(469\) 7.95507e8 0.356073
\(470\) 1.17936e9 0.523966
\(471\) 1.83064e9 0.807290
\(472\) −3.14544e9 −1.37684
\(473\) −2.85630e7 −0.0124105
\(474\) −2.00537e9 −0.864911
\(475\) −1.13426e8 −0.0485605
\(476\) 2.10813e8 0.0895928
\(477\) 1.40371e6 0.000592194 0
\(478\) −1.43788e9 −0.602179
\(479\) 2.90237e9 1.20664 0.603322 0.797498i \(-0.293844\pi\)
0.603322 + 0.797498i \(0.293844\pi\)
\(480\) 1.04767e9 0.432396
\(481\) 4.09476e9 1.67773
\(482\) −5.04009e9 −2.05009
\(483\) −8.26677e7 −0.0333827
\(484\) −4.59520e8 −0.184224
\(485\) −1.71147e9 −0.681198
\(486\) −1.07468e7 −0.00424672
\(487\) 9.92687e8 0.389458 0.194729 0.980857i \(-0.437617\pi\)
0.194729 + 0.980857i \(0.437617\pi\)
\(488\) 1.61502e9 0.629084
\(489\) −2.24959e9 −0.870005
\(490\) −2.46105e9 −0.945004
\(491\) 4.20047e9 1.60145 0.800723 0.599034i \(-0.204449\pi\)
0.800723 + 0.599034i \(0.204449\pi\)
\(492\) −5.99621e8 −0.226986
\(493\) −2.19532e9 −0.825152
\(494\) 5.17310e8 0.193066
\(495\) 505926. 0.000187486 0
\(496\) −2.85517e8 −0.105062
\(497\) 1.75547e9 0.641427
\(498\) −2.98086e9 −1.08153
\(499\) 3.12871e9 1.12723 0.563616 0.826037i \(-0.309410\pi\)
0.563616 + 0.826037i \(0.309410\pi\)
\(500\) −3.70165e8 −0.132434
\(501\) 2.71230e9 0.963620
\(502\) −4.22638e9 −1.49110
\(503\) 3.86209e8 0.135311 0.0676556 0.997709i \(-0.478448\pi\)
0.0676556 + 0.997709i \(0.478448\pi\)
\(504\) −2.56030e6 −0.000890808 0
\(505\) −3.07087e8 −0.106107
\(506\) 1.67531e7 0.00574867
\(507\) −3.18250e9 −1.08453
\(508\) 3.06299e7 0.0103663
\(509\) −1.34663e9 −0.452621 −0.226311 0.974055i \(-0.572666\pi\)
−0.226311 + 0.974055i \(0.572666\pi\)
\(510\) 3.61056e9 1.20526
\(511\) 2.72698e9 0.904085
\(512\) −1.81947e9 −0.599102
\(513\) −3.75823e8 −0.122906
\(514\) 9.20522e8 0.298994
\(515\) −5.61883e9 −1.81268
\(516\) −8.81754e7 −0.0282536
\(517\) −1.04169e8 −0.0331529
\(518\) 2.06085e9 0.651466
\(519\) 1.42138e9 0.446299
\(520\) 4.85118e9 1.51299
\(521\) −4.59610e9 −1.42383 −0.711914 0.702267i \(-0.752171\pi\)
−0.711914 + 0.702267i \(0.752171\pi\)
\(522\) −6.07025e6 −0.00186793
\(523\) −3.76968e9 −1.15226 −0.576128 0.817359i \(-0.695437\pi\)
−0.576128 + 0.817359i \(0.695437\pi\)
\(524\) −9.40511e8 −0.285565
\(525\) 6.74698e8 0.203494
\(526\) 3.17289e9 0.950615
\(527\) −2.87662e8 −0.0856142
\(528\) −3.16531e8 −0.0935831
\(529\) −3.39049e9 −0.995791
\(530\) −1.33860e9 −0.390559
\(531\) −1.04457e7 −0.00302767
\(532\) 4.07301e7 0.0117280
\(533\) −6.18515e9 −1.76932
\(534\) −3.44613e9 −0.979348
\(535\) 4.20180e9 1.18631
\(536\) 2.18591e9 0.613133
\(537\) −2.84425e8 −0.0792608
\(538\) −4.25993e9 −1.17941
\(539\) 2.17377e8 0.0597933
\(540\) 8.02408e8 0.219289
\(541\) 6.00863e9 1.63149 0.815746 0.578410i \(-0.196327\pi\)
0.815746 + 0.578410i \(0.196327\pi\)
\(542\) −2.76107e9 −0.744869
\(543\) −7.06463e9 −1.89361
\(544\) 1.29044e9 0.343669
\(545\) −5.54350e9 −1.46689
\(546\) −3.07715e9 −0.809050
\(547\) 2.57778e9 0.673427 0.336714 0.941607i \(-0.390685\pi\)
0.336714 + 0.941607i \(0.390685\pi\)
\(548\) −2.86286e8 −0.0743135
\(549\) 5.36334e6 0.00138335
\(550\) −1.36731e8 −0.0350428
\(551\) −4.24147e8 −0.108015
\(552\) −2.27156e8 −0.0574826
\(553\) −1.62867e9 −0.409540
\(554\) 3.02836e9 0.756700
\(555\) 5.52168e9 1.37103
\(556\) 8.21202e8 0.202623
\(557\) −1.73000e9 −0.424181 −0.212091 0.977250i \(-0.568027\pi\)
−0.212091 + 0.977250i \(0.568027\pi\)
\(558\) −795411. −0.000193808 0
\(559\) −9.09538e8 −0.220232
\(560\) 2.91046e9 0.700332
\(561\) −3.18910e8 −0.0762602
\(562\) 5.90782e9 1.40395
\(563\) −4.68663e9 −1.10683 −0.553416 0.832905i \(-0.686676\pi\)
−0.553416 + 0.832905i \(0.686676\pi\)
\(564\) −3.21575e8 −0.0754754
\(565\) 4.51669e9 1.05354
\(566\) 7.04504e9 1.63315
\(567\) 2.23118e9 0.514036
\(568\) 4.82372e9 1.10449
\(569\) 8.45042e9 1.92303 0.961513 0.274759i \(-0.0885981\pi\)
0.961513 + 0.274759i \(0.0885981\pi\)
\(570\) 6.97579e8 0.157773
\(571\) 3.74129e8 0.0840998 0.0420499 0.999116i \(-0.486611\pi\)
0.0420499 + 0.999116i \(0.486611\pi\)
\(572\) 9.75563e7 0.0217956
\(573\) −5.37473e9 −1.19348
\(574\) −3.11292e9 −0.687031
\(575\) −1.16969e8 −0.0256587
\(576\) −6.72760e6 −0.00146684
\(577\) 5.59761e7 0.0121307 0.00606537 0.999982i \(-0.498069\pi\)
0.00606537 + 0.999982i \(0.498069\pi\)
\(578\) −6.07437e8 −0.130844
\(579\) −2.41384e9 −0.516814
\(580\) 9.05583e8 0.192722
\(581\) −2.42092e9 −0.512111
\(582\) 2.98304e9 0.627232
\(583\) 1.18235e8 0.0247119
\(584\) 7.49324e9 1.55677
\(585\) 1.61103e7 0.00332705
\(586\) 5.34153e9 1.09654
\(587\) −4.41966e9 −0.901894 −0.450947 0.892551i \(-0.648914\pi\)
−0.450947 + 0.892551i \(0.648914\pi\)
\(588\) 6.71053e8 0.136124
\(589\) −5.55778e7 −0.0112072
\(590\) 9.96124e9 1.99678
\(591\) 3.66519e9 0.730365
\(592\) 6.75042e9 1.33723
\(593\) 4.30470e9 0.847718 0.423859 0.905728i \(-0.360675\pi\)
0.423859 + 0.905728i \(0.360675\pi\)
\(594\) −4.53044e8 −0.0886927
\(595\) 2.93234e9 0.570695
\(596\) 8.21506e8 0.158946
\(597\) 3.30823e9 0.636334
\(598\) 5.33472e8 0.102014
\(599\) −2.79184e9 −0.530758 −0.265379 0.964144i \(-0.585497\pi\)
−0.265379 + 0.964144i \(0.585497\pi\)
\(600\) 1.85395e9 0.350403
\(601\) 9.53719e9 1.79209 0.896045 0.443963i \(-0.146428\pi\)
0.896045 + 0.443963i \(0.146428\pi\)
\(602\) −4.57761e8 −0.0855167
\(603\) 7.25920e6 0.00134827
\(604\) −6.52455e8 −0.120482
\(605\) −6.39177e9 −1.17348
\(606\) 5.35244e8 0.0977006
\(607\) −1.03426e10 −1.87703 −0.938515 0.345240i \(-0.887798\pi\)
−0.938515 + 0.345240i \(0.887798\pi\)
\(608\) 2.49319e8 0.0449876
\(609\) 2.52299e9 0.452642
\(610\) −5.11457e9 −0.912336
\(611\) −3.31708e9 −0.588317
\(612\) 1.92372e6 0.000339244 0
\(613\) −2.76574e7 −0.00484954 −0.00242477 0.999997i \(-0.500772\pi\)
−0.00242477 + 0.999997i \(0.500772\pi\)
\(614\) −6.68011e9 −1.16465
\(615\) −8.34052e9 −1.44587
\(616\) −2.15654e8 −0.0371728
\(617\) 5.52656e9 0.947234 0.473617 0.880731i \(-0.342948\pi\)
0.473617 + 0.880731i \(0.342948\pi\)
\(618\) 9.79344e9 1.66907
\(619\) −1.07166e10 −1.81609 −0.908046 0.418871i \(-0.862426\pi\)
−0.908046 + 0.418871i \(0.862426\pi\)
\(620\) 1.18662e8 0.0199960
\(621\) −3.87565e8 −0.0649418
\(622\) −9.99661e9 −1.66566
\(623\) −2.79879e9 −0.463726
\(624\) −1.00794e10 −1.66069
\(625\) −7.56273e9 −1.23908
\(626\) −1.03521e10 −1.68662
\(627\) −6.16150e7 −0.00998275
\(628\) −9.30130e8 −0.149860
\(629\) 6.80115e9 1.08970
\(630\) 8.10816e6 0.00129190
\(631\) −6.12871e7 −0.00971105 −0.00485552 0.999988i \(-0.501546\pi\)
−0.00485552 + 0.999988i \(0.501546\pi\)
\(632\) −4.47530e9 −0.705199
\(633\) 6.73372e9 1.05522
\(634\) 4.25888e9 0.663717
\(635\) 4.26051e8 0.0660318
\(636\) 3.64997e8 0.0562587
\(637\) 6.92198e9 1.06107
\(638\) −5.11297e8 −0.0779473
\(639\) 1.60191e7 0.00242877
\(640\) 9.28591e9 1.40021
\(641\) 9.04843e9 1.35697 0.678485 0.734614i \(-0.262637\pi\)
0.678485 + 0.734614i \(0.262637\pi\)
\(642\) −7.32360e9 −1.09233
\(643\) −6.45538e9 −0.957599 −0.478799 0.877924i \(-0.658928\pi\)
−0.478799 + 0.877924i \(0.658928\pi\)
\(644\) 4.20027e7 0.00619693
\(645\) −1.22649e9 −0.179972
\(646\) 8.59220e8 0.125398
\(647\) 2.44098e8 0.0354322 0.0177161 0.999843i \(-0.494360\pi\)
0.0177161 + 0.999843i \(0.494360\pi\)
\(648\) 6.13086e9 0.885134
\(649\) −8.79845e8 −0.126343
\(650\) −4.35397e9 −0.621855
\(651\) 3.30598e8 0.0469642
\(652\) 1.14299e9 0.161502
\(653\) 3.48013e9 0.489102 0.244551 0.969636i \(-0.421359\pi\)
0.244551 + 0.969636i \(0.421359\pi\)
\(654\) 9.66215e9 1.35068
\(655\) −1.30822e10 −1.81901
\(656\) −1.01965e10 −1.41023
\(657\) 2.48844e7 0.00342333
\(658\) −1.66945e9 −0.228446
\(659\) −9.27568e9 −1.26254 −0.631272 0.775561i \(-0.717467\pi\)
−0.631272 + 0.775561i \(0.717467\pi\)
\(660\) 1.31552e8 0.0178113
\(661\) −3.16224e9 −0.425882 −0.212941 0.977065i \(-0.568304\pi\)
−0.212941 + 0.977065i \(0.568304\pi\)
\(662\) 7.26412e9 0.973151
\(663\) −1.01551e10 −1.35328
\(664\) −6.65223e9 −0.881819
\(665\) 5.66542e8 0.0747062
\(666\) 1.88058e7 0.00246678
\(667\) −4.37399e8 −0.0570739
\(668\) −1.37809e9 −0.178880
\(669\) −7.67808e9 −0.991428
\(670\) −6.92250e9 −0.889203
\(671\) 4.51754e8 0.0577263
\(672\) −1.48304e9 −0.188522
\(673\) −2.80981e9 −0.355323 −0.177662 0.984092i \(-0.556853\pi\)
−0.177662 + 0.984092i \(0.556853\pi\)
\(674\) 6.06998e9 0.763620
\(675\) 3.16314e9 0.395872
\(676\) 1.61700e9 0.201324
\(677\) −1.54447e10 −1.91302 −0.956511 0.291696i \(-0.905780\pi\)
−0.956511 + 0.291696i \(0.905780\pi\)
\(678\) −7.87245e9 −0.970076
\(679\) 2.42269e9 0.296998
\(680\) 8.05752e9 0.982698
\(681\) 1.57232e10 1.90776
\(682\) −6.69975e7 −0.00808748
\(683\) −8.31895e9 −0.999070 −0.499535 0.866294i \(-0.666496\pi\)
−0.499535 + 0.866294i \(0.666496\pi\)
\(684\) 371673. 4.44083e−5 0
\(685\) −3.98214e9 −0.473368
\(686\) 8.22530e9 0.972786
\(687\) 2.70365e9 0.318128
\(688\) −1.49942e9 −0.175535
\(689\) 3.76498e9 0.438526
\(690\) 7.19374e8 0.0833649
\(691\) 9.72394e9 1.12116 0.560582 0.828099i \(-0.310577\pi\)
0.560582 + 0.828099i \(0.310577\pi\)
\(692\) −7.22192e8 −0.0828479
\(693\) −716168. −8.17427e−5 0
\(694\) 1.21101e10 1.37528
\(695\) 1.14226e10 1.29068
\(696\) 6.93270e9 0.779418
\(697\) −1.02732e10 −1.14918
\(698\) 1.55156e10 1.72693
\(699\) 1.26722e10 1.40340
\(700\) −3.42808e8 −0.0377753
\(701\) −2.48982e8 −0.0272995 −0.0136498 0.999907i \(-0.504345\pi\)
−0.0136498 + 0.999907i \(0.504345\pi\)
\(702\) −1.44264e10 −1.57390
\(703\) 1.31402e9 0.142645
\(704\) −5.66666e8 −0.0612101
\(705\) −4.47300e9 −0.480769
\(706\) −1.10560e9 −0.118244
\(707\) 4.34701e8 0.0462618
\(708\) −2.71613e9 −0.287630
\(709\) 7.54233e9 0.794774 0.397387 0.917651i \(-0.369917\pi\)
0.397387 + 0.917651i \(0.369917\pi\)
\(710\) −1.52761e10 −1.60180
\(711\) −1.48621e7 −0.00155073
\(712\) −7.69055e9 −0.798505
\(713\) −5.73143e7 −0.00592174
\(714\) −5.11097e9 −0.525484
\(715\) 1.35697e9 0.138836
\(716\) 1.44514e8 0.0147134
\(717\) 5.45352e9 0.552535
\(718\) 2.23568e9 0.225410
\(719\) 6.70726e9 0.672967 0.336484 0.941689i \(-0.390762\pi\)
0.336484 + 0.941689i \(0.390762\pi\)
\(720\) 2.65587e7 0.00265181
\(721\) 7.95378e9 0.790315
\(722\) −1.08449e10 −1.07237
\(723\) 1.91158e10 1.88108
\(724\) 3.58947e9 0.351516
\(725\) 3.56986e9 0.347911
\(726\) 1.11406e10 1.08052
\(727\) 1.58709e10 1.53190 0.765951 0.642899i \(-0.222269\pi\)
0.765951 + 0.642899i \(0.222269\pi\)
\(728\) −6.86713e9 −0.659653
\(729\) 1.04807e10 1.00194
\(730\) −2.37302e10 −2.25773
\(731\) −1.51069e9 −0.143042
\(732\) 1.39459e9 0.131419
\(733\) −2.20320e9 −0.206629 −0.103314 0.994649i \(-0.532945\pi\)
−0.103314 + 0.994649i \(0.532945\pi\)
\(734\) 1.32314e10 1.23501
\(735\) 9.33412e9 0.867097
\(736\) 2.57108e8 0.0237708
\(737\) 6.11443e8 0.0562626
\(738\) −2.84062e7 −0.00260145
\(739\) −1.48017e10 −1.34913 −0.674567 0.738214i \(-0.735670\pi\)
−0.674567 + 0.738214i \(0.735670\pi\)
\(740\) −2.80551e9 −0.254508
\(741\) −1.96202e9 −0.177150
\(742\) 1.89487e9 0.170281
\(743\) −2.07320e10 −1.85430 −0.927152 0.374686i \(-0.877750\pi\)
−0.927152 + 0.374686i \(0.877750\pi\)
\(744\) 9.08422e8 0.0808691
\(745\) 1.14269e10 1.01247
\(746\) −1.71405e10 −1.51160
\(747\) −2.20915e7 −0.00193911
\(748\) 1.62035e8 0.0141564
\(749\) −5.94790e9 −0.517222
\(750\) 8.97430e9 0.776759
\(751\) −2.94907e9 −0.254065 −0.127033 0.991899i \(-0.540545\pi\)
−0.127033 + 0.991899i \(0.540545\pi\)
\(752\) −5.46837e9 −0.468917
\(753\) 1.60296e10 1.36817
\(754\) −1.62814e10 −1.38322
\(755\) −9.07543e9 −0.767455
\(756\) −1.13586e9 −0.0956086
\(757\) −9.04947e9 −0.758207 −0.379103 0.925354i \(-0.623768\pi\)
−0.379103 + 0.925354i \(0.623768\pi\)
\(758\) −2.29093e9 −0.191060
\(759\) −6.35401e7 −0.00527475
\(760\) 1.55675e9 0.128639
\(761\) 2.07707e10 1.70846 0.854230 0.519896i \(-0.174029\pi\)
0.854230 + 0.519896i \(0.174029\pi\)
\(762\) −7.42593e8 −0.0608007
\(763\) 7.84716e9 0.639553
\(764\) 2.73085e9 0.221550
\(765\) 2.67583e7 0.00216094
\(766\) 1.00093e10 0.804644
\(767\) −2.80172e10 −2.24202
\(768\) −6.75226e9 −0.537879
\(769\) −1.18493e9 −0.0939617 −0.0469808 0.998896i \(-0.514960\pi\)
−0.0469808 + 0.998896i \(0.514960\pi\)
\(770\) 6.82951e8 0.0539103
\(771\) −3.49130e9 −0.274345
\(772\) 1.22645e9 0.0959378
\(773\) −1.66232e9 −0.129446 −0.0647228 0.997903i \(-0.520616\pi\)
−0.0647228 + 0.997903i \(0.520616\pi\)
\(774\) −4.17718e6 −0.000323810 0
\(775\) 4.67775e8 0.0360978
\(776\) 6.65710e9 0.511410
\(777\) −7.81627e9 −0.597759
\(778\) −3.75121e9 −0.285590
\(779\) −1.98483e9 −0.150433
\(780\) 4.18905e9 0.316071
\(781\) 1.34929e9 0.101351
\(782\) 8.86066e8 0.0662586
\(783\) 1.18283e10 0.880558
\(784\) 1.14112e10 0.845719
\(785\) −1.29378e10 −0.954589
\(786\) 2.28018e10 1.67491
\(787\) 2.65010e10 1.93799 0.968993 0.247088i \(-0.0794736\pi\)
0.968993 + 0.247088i \(0.0794736\pi\)
\(788\) −1.86225e9 −0.135580
\(789\) −1.20340e10 −0.872246
\(790\) 1.41727e10 1.02272
\(791\) −6.39365e9 −0.459336
\(792\) −1.96790e6 −0.000140755 0
\(793\) 1.43853e10 1.02439
\(794\) −2.65090e10 −1.87941
\(795\) 5.07698e9 0.358361
\(796\) −1.68088e9 −0.118125
\(797\) −2.48128e7 −0.00173609 −0.000868045 1.00000i \(-0.500276\pi\)
−0.000868045 1.00000i \(0.500276\pi\)
\(798\) −9.87465e8 −0.0687878
\(799\) −5.50947e9 −0.382117
\(800\) −2.09841e9 −0.144902
\(801\) −2.55396e7 −0.00175590
\(802\) 1.62941e10 1.11537
\(803\) 2.09601e9 0.142853
\(804\) 1.88756e9 0.128087
\(805\) 5.84243e8 0.0394737
\(806\) −2.13342e9 −0.143517
\(807\) 1.61568e10 1.08218
\(808\) 1.19448e9 0.0796596
\(809\) −2.21936e8 −0.0147369 −0.00736847 0.999973i \(-0.502345\pi\)
−0.00736847 + 0.999973i \(0.502345\pi\)
\(810\) −1.94157e10 −1.28368
\(811\) 1.47954e10 0.973987 0.486994 0.873405i \(-0.338093\pi\)
0.486994 + 0.873405i \(0.338093\pi\)
\(812\) −1.28191e9 −0.0840253
\(813\) 1.04720e10 0.683462
\(814\) 1.58401e9 0.102937
\(815\) 1.58986e10 1.02875
\(816\) −1.67412e10 −1.07863
\(817\) −2.91873e8 −0.0187248
\(818\) 7.01241e9 0.447951
\(819\) −2.28051e7 −0.00145057
\(820\) 4.23774e9 0.268402
\(821\) −2.55026e10 −1.60836 −0.804181 0.594384i \(-0.797396\pi\)
−0.804181 + 0.594384i \(0.797396\pi\)
\(822\) 6.94074e9 0.435867
\(823\) 1.87610e9 0.117316 0.0586578 0.998278i \(-0.481318\pi\)
0.0586578 + 0.998278i \(0.481318\pi\)
\(824\) 2.18555e10 1.36087
\(825\) 5.18587e8 0.0321538
\(826\) −1.41007e10 −0.870585
\(827\) −2.34241e10 −1.44010 −0.720052 0.693920i \(-0.755882\pi\)
−0.720052 + 0.693920i \(0.755882\pi\)
\(828\) 383285. 2.34647e−5 0
\(829\) 2.88836e10 1.76080 0.880402 0.474228i \(-0.157273\pi\)
0.880402 + 0.474228i \(0.157273\pi\)
\(830\) 2.10668e10 1.27887
\(831\) −1.14858e10 −0.694317
\(832\) −1.80445e10 −1.08621
\(833\) 1.14970e10 0.689171
\(834\) −1.99093e10 −1.18843
\(835\) −1.91688e10 −1.13944
\(836\) 3.13060e7 0.00185313
\(837\) 1.54992e9 0.0913629
\(838\) −2.68209e10 −1.57442
\(839\) 8.35473e9 0.488388 0.244194 0.969726i \(-0.421477\pi\)
0.244194 + 0.969726i \(0.421477\pi\)
\(840\) −9.26016e9 −0.539065
\(841\) −3.90062e9 −0.226124
\(842\) −2.84404e9 −0.164189
\(843\) −2.24069e10 −1.28820
\(844\) −3.42134e9 −0.195883
\(845\) 2.24919e10 1.28241
\(846\) −1.52341e7 −0.000865012 0
\(847\) 9.04793e9 0.511631
\(848\) 6.20676e9 0.349526
\(849\) −2.67200e10 −1.49851
\(850\) −7.23169e9 −0.403899
\(851\) 1.35507e9 0.0753718
\(852\) 4.16534e9 0.230734
\(853\) −2.54143e10 −1.40203 −0.701014 0.713147i \(-0.747269\pi\)
−0.701014 + 0.713147i \(0.747269\pi\)
\(854\) 7.23998e9 0.397773
\(855\) 5.16984e6 0.000282876 0
\(856\) −1.63437e10 −0.890620
\(857\) −2.93738e10 −1.59414 −0.797071 0.603885i \(-0.793619\pi\)
−0.797071 + 0.603885i \(0.793619\pi\)
\(858\) −2.36516e9 −0.127837
\(859\) −7.87923e9 −0.424138 −0.212069 0.977255i \(-0.568020\pi\)
−0.212069 + 0.977255i \(0.568020\pi\)
\(860\) 6.23168e8 0.0334088
\(861\) 1.18065e10 0.630392
\(862\) 2.87386e10 1.52823
\(863\) −1.41785e10 −0.750916 −0.375458 0.926839i \(-0.622515\pi\)
−0.375458 + 0.926839i \(0.622515\pi\)
\(864\) −6.95284e9 −0.366745
\(865\) −1.00454e10 −0.527731
\(866\) −3.51890e9 −0.184117
\(867\) 2.30385e9 0.120057
\(868\) −1.67974e8 −0.00871811
\(869\) −1.25183e9 −0.0647108
\(870\) −2.19550e10 −1.13036
\(871\) 1.94703e10 0.998412
\(872\) 2.15625e10 1.10127
\(873\) 2.21076e7 0.00112459
\(874\) 1.71192e8 0.00867350
\(875\) 7.28852e9 0.367800
\(876\) 6.47051e9 0.325218
\(877\) −9.31500e9 −0.466320 −0.233160 0.972438i \(-0.574907\pi\)
−0.233160 + 0.972438i \(0.574907\pi\)
\(878\) 3.12096e10 1.55617
\(879\) −2.02590e10 −1.00614
\(880\) 2.23704e9 0.110658
\(881\) 1.57869e10 0.777826 0.388913 0.921275i \(-0.372851\pi\)
0.388913 + 0.921275i \(0.372851\pi\)
\(882\) 3.17902e7 0.00156010
\(883\) −9.82559e9 −0.480282 −0.240141 0.970738i \(-0.577194\pi\)
−0.240141 + 0.970738i \(0.577194\pi\)
\(884\) 5.15973e9 0.251214
\(885\) −3.77804e10 −1.83217
\(886\) −8.06120e9 −0.389387
\(887\) −5.02494e9 −0.241768 −0.120884 0.992667i \(-0.538573\pi\)
−0.120884 + 0.992667i \(0.538573\pi\)
\(888\) −2.14777e10 −1.02930
\(889\) −6.03100e8 −0.0287895
\(890\) 2.43550e10 1.15804
\(891\) 1.71493e9 0.0812221
\(892\) 3.90116e9 0.184042
\(893\) −1.06446e9 −0.0500205
\(894\) −1.99167e10 −0.932256
\(895\) 2.01014e9 0.0937228
\(896\) −1.31448e10 −0.610484
\(897\) −2.02332e9 −0.0936035
\(898\) 3.60611e10 1.66178
\(899\) 1.74921e9 0.0802940
\(900\) −3.12821e6 −0.000143036 0
\(901\) 6.25341e9 0.284826
\(902\) −2.39265e9 −0.108557
\(903\) 1.73617e9 0.0784666
\(904\) −1.75686e10 −0.790945
\(905\) 4.99283e10 2.23912
\(906\) 1.58182e10 0.706656
\(907\) −3.57868e9 −0.159257 −0.0796283 0.996825i \(-0.525373\pi\)
−0.0796283 + 0.996825i \(0.525373\pi\)
\(908\) −7.98879e9 −0.354144
\(909\) 3.96675e6 0.000175171 0
\(910\) 2.17474e10 0.956670
\(911\) −2.81588e10 −1.23396 −0.616978 0.786980i \(-0.711643\pi\)
−0.616978 + 0.786980i \(0.711643\pi\)
\(912\) −3.23449e9 −0.141197
\(913\) −1.86077e9 −0.0809178
\(914\) −3.02633e10 −1.31100
\(915\) 1.93983e10 0.837123
\(916\) −1.37370e9 −0.0590552
\(917\) 1.85186e10 0.793078
\(918\) −2.39614e10 −1.02226
\(919\) −4.74290e9 −0.201576 −0.100788 0.994908i \(-0.532136\pi\)
−0.100788 + 0.994908i \(0.532136\pi\)
\(920\) 1.60539e9 0.0679710
\(921\) 2.53359e10 1.06863
\(922\) 1.55169e10 0.652000
\(923\) 4.29659e10 1.79853
\(924\) −1.86220e8 −0.00776559
\(925\) −1.10595e10 −0.459452
\(926\) 1.98051e10 0.819668
\(927\) 7.25803e7 0.00299254
\(928\) −7.84685e9 −0.322313
\(929\) −2.37371e10 −0.971344 −0.485672 0.874141i \(-0.661425\pi\)
−0.485672 + 0.874141i \(0.661425\pi\)
\(930\) −2.87686e9 −0.117281
\(931\) 2.22128e9 0.0902150
\(932\) −6.43862e9 −0.260518
\(933\) 3.79146e10 1.52834
\(934\) 2.14899e10 0.863018
\(935\) 2.25385e9 0.0901748
\(936\) −6.26643e7 −0.00249778
\(937\) −1.59383e10 −0.632926 −0.316463 0.948605i \(-0.602495\pi\)
−0.316463 + 0.948605i \(0.602495\pi\)
\(938\) 9.79921e9 0.387687
\(939\) 3.92627e10 1.54757
\(940\) 2.27269e9 0.0892468
\(941\) −4.38313e10 −1.71483 −0.857415 0.514626i \(-0.827931\pi\)
−0.857415 + 0.514626i \(0.827931\pi\)
\(942\) 2.25502e10 0.878965
\(943\) −2.04684e9 −0.0794865
\(944\) −4.61877e10 −1.78700
\(945\) −1.57994e10 −0.609015
\(946\) −3.51844e8 −0.0135124
\(947\) −3.68371e10 −1.40948 −0.704742 0.709464i \(-0.748937\pi\)
−0.704742 + 0.709464i \(0.748937\pi\)
\(948\) −3.86447e9 −0.147320
\(949\) 6.67440e10 2.53501
\(950\) −1.39720e9 −0.0528720
\(951\) −1.61528e10 −0.609000
\(952\) −1.14059e10 −0.428450
\(953\) 3.58699e10 1.34247 0.671236 0.741244i \(-0.265764\pi\)
0.671236 + 0.741244i \(0.265764\pi\)
\(954\) 1.72912e7 0.000644772 0
\(955\) 3.79852e10 1.41125
\(956\) −2.77088e9 −0.102569
\(957\) 1.93922e9 0.0715213
\(958\) 3.57520e10 1.31377
\(959\) 5.63695e9 0.206385
\(960\) −2.43326e10 −0.887643
\(961\) −2.72834e10 −0.991669
\(962\) 5.04401e10 1.82668
\(963\) −5.42760e7 −0.00195847
\(964\) −9.71254e9 −0.349191
\(965\) 1.70595e10 0.611112
\(966\) −1.01832e9 −0.0363465
\(967\) −6.33260e9 −0.225211 −0.112605 0.993640i \(-0.535920\pi\)
−0.112605 + 0.993640i \(0.535920\pi\)
\(968\) 2.48620e10 0.880994
\(969\) −3.25880e9 −0.115060
\(970\) −2.10822e10 −0.741678
\(971\) −3.71924e10 −1.30373 −0.651864 0.758336i \(-0.726013\pi\)
−0.651864 + 0.758336i \(0.726013\pi\)
\(972\) −2.07098e7 −0.000723341 0
\(973\) −1.61694e10 −0.562729
\(974\) 1.22281e10 0.424036
\(975\) 1.65135e10 0.570589
\(976\) 2.37149e10 0.816484
\(977\) 3.55570e10 1.21982 0.609908 0.792472i \(-0.291206\pi\)
0.609908 + 0.792472i \(0.291206\pi\)
\(978\) −2.77108e10 −0.947248
\(979\) −2.15121e9 −0.0732727
\(980\) −4.74258e9 −0.160962
\(981\) 7.16073e7 0.00242167
\(982\) 5.17422e10 1.74363
\(983\) −2.71260e10 −0.910854 −0.455427 0.890273i \(-0.650513\pi\)
−0.455427 + 0.890273i \(0.650513\pi\)
\(984\) 3.24421e10 1.08549
\(985\) −2.59032e10 −0.863628
\(986\) −2.70424e10 −0.898413
\(987\) 6.33180e9 0.209612
\(988\) 9.96886e8 0.0328849
\(989\) −3.00992e8 −0.00989390
\(990\) 6.23209e6 0.000204132 0
\(991\) −1.89518e10 −0.618575 −0.309287 0.950969i \(-0.600091\pi\)
−0.309287 + 0.950969i \(0.600091\pi\)
\(992\) −1.02821e9 −0.0334418
\(993\) −2.75510e10 −0.892924
\(994\) 2.16243e10 0.698376
\(995\) −2.33804e10 −0.752440
\(996\) −5.74429e9 −0.184216
\(997\) 8.54283e9 0.273004 0.136502 0.990640i \(-0.456414\pi\)
0.136502 + 0.990640i \(0.456414\pi\)
\(998\) 3.85401e10 1.22731
\(999\) −3.66445e10 −1.16286
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.1.9 11
3.2 odd 2 387.8.a.b.1.3 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.a.1.9 11 1.1 even 1 trivial
387.8.a.b.1.3 11 3.2 odd 2