Properties

Label 43.8.a.a.1.2
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 1.2
Root \(14.1572\) of defining polynomial
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q-16.1572 q^{2} -48.3227 q^{3} +133.057 q^{4} -272.935 q^{5} +780.762 q^{6} +1101.65 q^{7} -81.7025 q^{8} +148.083 q^{9} +O(q^{10})\) \(q-16.1572 q^{2} -48.3227 q^{3} +133.057 q^{4} -272.935 q^{5} +780.762 q^{6} +1101.65 q^{7} -81.7025 q^{8} +148.083 q^{9} +4409.88 q^{10} +685.243 q^{11} -6429.66 q^{12} +13204.0 q^{13} -17799.7 q^{14} +13189.0 q^{15} -15711.2 q^{16} -1155.38 q^{17} -2392.62 q^{18} -12969.6 q^{19} -36315.9 q^{20} -53234.8 q^{21} -11071.6 q^{22} -106988. q^{23} +3948.09 q^{24} -3631.41 q^{25} -213341. q^{26} +98526.0 q^{27} +146582. q^{28} +71411.7 q^{29} -213097. q^{30} +229489. q^{31} +264307. q^{32} -33112.8 q^{33} +18667.8 q^{34} -300680. q^{35} +19703.5 q^{36} -279218. q^{37} +209553. q^{38} -638054. q^{39} +22299.5 q^{40} +345820. q^{41} +860128. q^{42} +79507.0 q^{43} +91176.2 q^{44} -40417.2 q^{45} +1.72863e6 q^{46} -735243. q^{47} +759206. q^{48} +390095. q^{49} +58673.6 q^{50} +55831.2 q^{51} +1.75688e6 q^{52} -714374. q^{53} -1.59191e6 q^{54} -187027. q^{55} -90007.8 q^{56} +626726. q^{57} -1.15382e6 q^{58} -915243. q^{59} +1.75488e6 q^{60} -575.793 q^{61} -3.70792e6 q^{62} +163136. q^{63} -2.25945e6 q^{64} -3.60384e6 q^{65} +535012. q^{66} -3.55836e6 q^{67} -153731. q^{68} +5.16993e6 q^{69} +4.85816e6 q^{70} -1.66804e6 q^{71} -12098.8 q^{72} -1.03269e6 q^{73} +4.51140e6 q^{74} +175480. q^{75} -1.72569e6 q^{76} +754900. q^{77} +1.03092e7 q^{78} +1.58783e6 q^{79} +4.28813e6 q^{80} -5.08490e6 q^{81} -5.58750e6 q^{82} -8.26293e6 q^{83} -7.08325e6 q^{84} +315344. q^{85} -1.28461e6 q^{86} -3.45080e6 q^{87} -55986.1 q^{88} +7.66713e6 q^{89} +653030. q^{90} +1.45462e7 q^{91} -1.42354e7 q^{92} -1.10895e7 q^{93} +1.18795e7 q^{94} +3.53986e6 q^{95} -1.27720e7 q^{96} -1.26607e7 q^{97} -6.30287e6 q^{98} +101473. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} + O(q^{10}) \) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} - 1333q^{10} + 1333q^{11} + 5089q^{12} - 17967q^{13} - 22352q^{14} - 49504q^{15} - 34406q^{16} - 63095q^{17} - 165931q^{18} - 54524q^{19} - 280995q^{20} - 139788q^{21} - 289358q^{22} - 138139q^{23} - 429583q^{24} + 3455q^{25} - 132946q^{26} - 240356q^{27} - 12704q^{28} - 308658q^{29} + 421284q^{30} - 209523q^{31} - 644934q^{32} + 96814q^{33} + 762435q^{34} - 578892q^{35} + 426161q^{36} - 298472q^{37} - 369707q^{38} + 292298q^{39} + 2633173q^{40} - 1346735q^{41} + 1173266q^{42} + 874577q^{43} + 3134292q^{44} + 1893784q^{45} + 3588111q^{46} + 499284q^{47} + 5647533q^{48} + 2544563q^{49} + 3049745q^{50} + 1258424q^{51} + 983088q^{52} - 2210495q^{53} + 6789698q^{54} - 1855072q^{55} - 469976q^{56} - 1238444q^{57} + 4397067q^{58} - 5824216q^{59} - 2889372q^{60} - 4453034q^{61} + 1002789q^{62} - 6240564q^{63} + 4757538q^{64} - 2162872q^{65} - 258940q^{66} - 6859513q^{67} - 9397005q^{68} - 10040030q^{69} + 845078q^{70} - 10726554q^{71} + 1199517q^{72} - 4456898q^{73} + 1046637q^{74} - 3349114q^{75} + 5861267q^{76} - 17019816q^{77} + 1999122q^{78} - 15541320q^{79} - 15680911q^{80} - 12976697q^{81} + 20233655q^{82} - 11146767q^{83} + 12348278q^{84} - 12471976q^{85} - 1908168q^{86} - 18648900q^{87} - 24463544q^{88} - 13531356q^{89} + 20858990q^{90} - 19746448q^{91} - 26023161q^{92} - 21903110q^{93} + 20288857q^{94} - 12291624q^{95} - 13954503q^{96} - 10999901q^{97} + 29909168q^{98} + 29396057q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.1572 −1.42811 −0.714056 0.700088i \(-0.753144\pi\)
−0.714056 + 0.700088i \(0.753144\pi\)
\(3\) −48.3227 −1.03330 −0.516650 0.856196i \(-0.672821\pi\)
−0.516650 + 0.856196i \(0.672821\pi\)
\(4\) 133.057 1.03951
\(5\) −272.935 −0.976482 −0.488241 0.872709i \(-0.662361\pi\)
−0.488241 + 0.872709i \(0.662361\pi\)
\(6\) 780.762 1.47567
\(7\) 1101.65 1.21395 0.606976 0.794720i \(-0.292382\pi\)
0.606976 + 0.794720i \(0.292382\pi\)
\(8\) −81.7025 −0.0564184
\(9\) 148.083 0.0677108
\(10\) 4409.88 1.39453
\(11\) 685.243 0.155228 0.0776140 0.996983i \(-0.475270\pi\)
0.0776140 + 0.996983i \(0.475270\pi\)
\(12\) −6429.66 −1.07412
\(13\) 13204.0 1.66688 0.833440 0.552609i \(-0.186368\pi\)
0.833440 + 0.552609i \(0.186368\pi\)
\(14\) −17799.7 −1.73366
\(15\) 13189.0 1.00900
\(16\) −15711.2 −0.958934
\(17\) −1155.38 −0.0570367 −0.0285184 0.999593i \(-0.509079\pi\)
−0.0285184 + 0.999593i \(0.509079\pi\)
\(18\) −2392.62 −0.0966986
\(19\) −12969.6 −0.433800 −0.216900 0.976194i \(-0.569594\pi\)
−0.216900 + 0.976194i \(0.569594\pi\)
\(20\) −36315.9 −1.01506
\(21\) −53234.8 −1.25438
\(22\) −11071.6 −0.221683
\(23\) −106988. −1.83352 −0.916761 0.399436i \(-0.869206\pi\)
−0.916761 + 0.399436i \(0.869206\pi\)
\(24\) 3948.09 0.0582971
\(25\) −3631.41 −0.0464820
\(26\) −213341. −2.38049
\(27\) 98526.0 0.963335
\(28\) 146582. 1.26191
\(29\) 71411.7 0.543721 0.271861 0.962337i \(-0.412361\pi\)
0.271861 + 0.962337i \(0.412361\pi\)
\(30\) −213097. −1.44097
\(31\) 229489. 1.38356 0.691778 0.722111i \(-0.256828\pi\)
0.691778 + 0.722111i \(0.256828\pi\)
\(32\) 264307. 1.42588
\(33\) −33112.8 −0.160397
\(34\) 18667.8 0.0814549
\(35\) −300680. −1.18540
\(36\) 19703.5 0.0703857
\(37\) −279218. −0.906229 −0.453115 0.891452i \(-0.649687\pi\)
−0.453115 + 0.891452i \(0.649687\pi\)
\(38\) 209553. 0.619515
\(39\) −638054. −1.72239
\(40\) 22299.5 0.0550915
\(41\) 345820. 0.783622 0.391811 0.920046i \(-0.371849\pi\)
0.391811 + 0.920046i \(0.371849\pi\)
\(42\) 860128. 1.79139
\(43\) 79507.0 0.152499
\(44\) 91176.2 0.161360
\(45\) −40417.2 −0.0661184
\(46\) 1.72863e6 2.61848
\(47\) −735243. −1.03297 −0.516486 0.856296i \(-0.672760\pi\)
−0.516486 + 0.856296i \(0.672760\pi\)
\(48\) 759206. 0.990867
\(49\) 390095. 0.473680
\(50\) 58673.6 0.0663816
\(51\) 55831.2 0.0589361
\(52\) 1.75688e6 1.73273
\(53\) −714374. −0.659113 −0.329557 0.944136i \(-0.606899\pi\)
−0.329557 + 0.944136i \(0.606899\pi\)
\(54\) −1.59191e6 −1.37575
\(55\) −187027. −0.151577
\(56\) −90007.8 −0.0684892
\(57\) 626726. 0.448245
\(58\) −1.15382e6 −0.776495
\(59\) −915243. −0.580169 −0.290084 0.957001i \(-0.593683\pi\)
−0.290084 + 0.957001i \(0.593683\pi\)
\(60\) 1.75488e6 1.04886
\(61\) −575.793 −0.000324797 0 −0.000162399 1.00000i \(-0.500052\pi\)
−0.000162399 1.00000i \(0.500052\pi\)
\(62\) −3.70792e6 −1.97587
\(63\) 163136. 0.0821976
\(64\) −2.25945e6 −1.07739
\(65\) −3.60384e6 −1.62768
\(66\) 535012. 0.229065
\(67\) −3.55836e6 −1.44540 −0.722700 0.691162i \(-0.757099\pi\)
−0.722700 + 0.691162i \(0.757099\pi\)
\(68\) −153731. −0.0592900
\(69\) 5.16993e6 1.89458
\(70\) 4.85816e6 1.69289
\(71\) −1.66804e6 −0.553099 −0.276549 0.961000i \(-0.589191\pi\)
−0.276549 + 0.961000i \(0.589191\pi\)
\(72\) −12098.8 −0.00382013
\(73\) −1.03269e6 −0.310700 −0.155350 0.987860i \(-0.549650\pi\)
−0.155350 + 0.987860i \(0.549650\pi\)
\(74\) 4.51140e6 1.29420
\(75\) 175480. 0.0480299
\(76\) −1.72569e6 −0.450937
\(77\) 754900. 0.188439
\(78\) 1.03092e7 2.45977
\(79\) 1.58783e6 0.362334 0.181167 0.983452i \(-0.442013\pi\)
0.181167 + 0.983452i \(0.442013\pi\)
\(80\) 4.28813e6 0.936382
\(81\) −5.08490e6 −1.06313
\(82\) −5.58750e6 −1.11910
\(83\) −8.26293e6 −1.58621 −0.793105 0.609085i \(-0.791537\pi\)
−0.793105 + 0.609085i \(0.791537\pi\)
\(84\) −7.08325e6 −1.30393
\(85\) 315344. 0.0556954
\(86\) −1.28461e6 −0.217785
\(87\) −3.45080e6 −0.561827
\(88\) −55986.1 −0.00875771
\(89\) 7.66713e6 1.15284 0.576418 0.817155i \(-0.304450\pi\)
0.576418 + 0.817155i \(0.304450\pi\)
\(90\) 653030. 0.0944245
\(91\) 1.45462e7 2.02351
\(92\) −1.42354e7 −1.90596
\(93\) −1.10895e7 −1.42963
\(94\) 1.18795e7 1.47520
\(95\) 3.53986e6 0.423598
\(96\) −1.27720e7 −1.47337
\(97\) −1.26607e7 −1.40850 −0.704251 0.709951i \(-0.748717\pi\)
−0.704251 + 0.709951i \(0.748717\pi\)
\(98\) −6.30287e6 −0.676468
\(99\) 101473. 0.0105106
\(100\) −483183. −0.0483183
\(101\) −7.42880e6 −0.717454 −0.358727 0.933443i \(-0.616789\pi\)
−0.358727 + 0.933443i \(0.616789\pi\)
\(102\) −902079. −0.0841674
\(103\) 1.57866e7 1.42350 0.711752 0.702431i \(-0.247902\pi\)
0.711752 + 0.702431i \(0.247902\pi\)
\(104\) −1.07880e6 −0.0940427
\(105\) 1.45297e7 1.22488
\(106\) 1.15423e7 0.941288
\(107\) −1.40430e6 −0.110819 −0.0554097 0.998464i \(-0.517646\pi\)
−0.0554097 + 0.998464i \(0.517646\pi\)
\(108\) 1.31095e7 1.00139
\(109\) −6.10839e6 −0.451787 −0.225894 0.974152i \(-0.572530\pi\)
−0.225894 + 0.974152i \(0.572530\pi\)
\(110\) 3.02184e6 0.216470
\(111\) 1.34926e7 0.936407
\(112\) −1.73083e7 −1.16410
\(113\) −8.10527e6 −0.528437 −0.264218 0.964463i \(-0.585114\pi\)
−0.264218 + 0.964463i \(0.585114\pi\)
\(114\) −1.01262e7 −0.640145
\(115\) 2.92007e7 1.79040
\(116\) 9.50180e6 0.565201
\(117\) 1.95530e6 0.112866
\(118\) 1.47878e7 0.828546
\(119\) −1.27283e6 −0.0692398
\(120\) −1.07757e6 −0.0569261
\(121\) −1.90176e7 −0.975904
\(122\) 9303.23 0.000463847 0
\(123\) −1.67109e7 −0.809717
\(124\) 3.05351e7 1.43821
\(125\) 2.23142e7 1.02187
\(126\) −2.63584e6 −0.117387
\(127\) 3.81416e7 1.65229 0.826145 0.563457i \(-0.190529\pi\)
0.826145 + 0.563457i \(0.190529\pi\)
\(128\) 2.67514e6 0.112749
\(129\) −3.84199e6 −0.157577
\(130\) 5.82282e7 2.32451
\(131\) 476451. 0.0185169 0.00925845 0.999957i \(-0.497053\pi\)
0.00925845 + 0.999957i \(0.497053\pi\)
\(132\) −4.40588e6 −0.166734
\(133\) −1.42880e7 −0.526612
\(134\) 5.74933e7 2.06419
\(135\) −2.68912e7 −0.940680
\(136\) 94397.6 0.00321792
\(137\) −5.02778e7 −1.67053 −0.835265 0.549848i \(-0.814686\pi\)
−0.835265 + 0.549848i \(0.814686\pi\)
\(138\) −8.35319e7 −2.70567
\(139\) −5.83450e7 −1.84269 −0.921344 0.388748i \(-0.872908\pi\)
−0.921344 + 0.388748i \(0.872908\pi\)
\(140\) −4.00075e7 −1.23223
\(141\) 3.55289e7 1.06737
\(142\) 2.69510e7 0.789887
\(143\) 9.04796e6 0.258747
\(144\) −2.32656e6 −0.0649301
\(145\) −1.94908e7 −0.530934
\(146\) 1.66855e7 0.443714
\(147\) −1.88505e7 −0.489453
\(148\) −3.71519e7 −0.942030
\(149\) −6.73082e7 −1.66693 −0.833463 0.552575i \(-0.813645\pi\)
−0.833463 + 0.552575i \(0.813645\pi\)
\(150\) −2.83527e6 −0.0685922
\(151\) 5.87220e7 1.38797 0.693987 0.719988i \(-0.255853\pi\)
0.693987 + 0.719988i \(0.255853\pi\)
\(152\) 1.05965e6 0.0244743
\(153\) −171093. −0.00386200
\(154\) −1.21971e7 −0.269113
\(155\) −6.26357e7 −1.35102
\(156\) −8.48974e7 −1.79043
\(157\) −5.66713e6 −0.116873 −0.0584365 0.998291i \(-0.518612\pi\)
−0.0584365 + 0.998291i \(0.518612\pi\)
\(158\) −2.56550e7 −0.517454
\(159\) 3.45205e7 0.681062
\(160\) −7.21387e7 −1.39235
\(161\) −1.17863e8 −2.22581
\(162\) 8.21580e7 1.51826
\(163\) 2.44165e7 0.441598 0.220799 0.975319i \(-0.429134\pi\)
0.220799 + 0.975319i \(0.429134\pi\)
\(164\) 4.60136e7 0.814579
\(165\) 9.03765e6 0.156625
\(166\) 1.33506e8 2.26529
\(167\) 1.01135e7 0.168033 0.0840164 0.996464i \(-0.473225\pi\)
0.0840164 + 0.996464i \(0.473225\pi\)
\(168\) 4.34942e6 0.0707699
\(169\) 1.11598e8 1.77849
\(170\) −5.09510e6 −0.0795392
\(171\) −1.92058e6 −0.0293729
\(172\) 1.05789e7 0.158523
\(173\) −4.24908e7 −0.623927 −0.311963 0.950094i \(-0.600987\pi\)
−0.311963 + 0.950094i \(0.600987\pi\)
\(174\) 5.57555e7 0.802353
\(175\) −4.00055e6 −0.0564270
\(176\) −1.07660e7 −0.148853
\(177\) 4.42270e7 0.599489
\(178\) −1.23880e8 −1.64638
\(179\) 8.27387e7 1.07826 0.539129 0.842223i \(-0.318753\pi\)
0.539129 + 0.842223i \(0.318753\pi\)
\(180\) −5.37778e6 −0.0687304
\(181\) 1.19734e7 0.150087 0.0750435 0.997180i \(-0.476090\pi\)
0.0750435 + 0.997180i \(0.476090\pi\)
\(182\) −2.35027e8 −2.88980
\(183\) 27823.9 0.000335613 0
\(184\) 8.74116e6 0.103444
\(185\) 7.62085e7 0.884917
\(186\) 1.79176e8 2.04167
\(187\) −791718. −0.00885370
\(188\) −9.78290e7 −1.07378
\(189\) 1.08541e8 1.16944
\(190\) −5.71944e7 −0.604945
\(191\) 1.61956e8 1.68183 0.840914 0.541169i \(-0.182018\pi\)
0.840914 + 0.541169i \(0.182018\pi\)
\(192\) 1.09183e8 1.11327
\(193\) −8.69257e7 −0.870357 −0.435178 0.900344i \(-0.643315\pi\)
−0.435178 + 0.900344i \(0.643315\pi\)
\(194\) 2.04562e8 2.01150
\(195\) 1.74147e8 1.68188
\(196\) 5.19048e7 0.492393
\(197\) −1.25816e8 −1.17247 −0.586237 0.810139i \(-0.699391\pi\)
−0.586237 + 0.810139i \(0.699391\pi\)
\(198\) −1.63953e6 −0.0150103
\(199\) −1.43871e8 −1.29416 −0.647079 0.762423i \(-0.724009\pi\)
−0.647079 + 0.762423i \(0.724009\pi\)
\(200\) 296695. 0.00262244
\(201\) 1.71950e8 1.49353
\(202\) 1.20029e8 1.02460
\(203\) 7.86708e7 0.660051
\(204\) 7.42872e6 0.0612644
\(205\) −9.43864e7 −0.765193
\(206\) −2.55068e8 −2.03292
\(207\) −1.58431e7 −0.124149
\(208\) −2.07451e8 −1.59843
\(209\) −8.88733e6 −0.0673379
\(210\) −2.34759e8 −1.74926
\(211\) −1.83659e8 −1.34593 −0.672966 0.739673i \(-0.734980\pi\)
−0.672966 + 0.739673i \(0.734980\pi\)
\(212\) −9.50523e7 −0.685152
\(213\) 8.06042e7 0.571517
\(214\) 2.26896e7 0.158262
\(215\) −2.17003e7 −0.148912
\(216\) −8.04982e6 −0.0543498
\(217\) 2.52817e8 1.67957
\(218\) 9.86948e7 0.645203
\(219\) 4.99025e7 0.321046
\(220\) −2.48852e7 −0.157566
\(221\) −1.52557e7 −0.0950734
\(222\) −2.18003e8 −1.33730
\(223\) 6.03715e7 0.364557 0.182278 0.983247i \(-0.441653\pi\)
0.182278 + 0.983247i \(0.441653\pi\)
\(224\) 2.91175e8 1.73095
\(225\) −537752. −0.00314733
\(226\) 1.30959e8 0.754667
\(227\) 5.49050e7 0.311545 0.155773 0.987793i \(-0.450213\pi\)
0.155773 + 0.987793i \(0.450213\pi\)
\(228\) 8.33901e7 0.465954
\(229\) −1.05868e8 −0.582560 −0.291280 0.956638i \(-0.594081\pi\)
−0.291280 + 0.956638i \(0.594081\pi\)
\(230\) −4.71803e8 −2.55690
\(231\) −3.64788e7 −0.194715
\(232\) −5.83451e6 −0.0306758
\(233\) 4.21892e7 0.218502 0.109251 0.994014i \(-0.465155\pi\)
0.109251 + 0.994014i \(0.465155\pi\)
\(234\) −3.15922e7 −0.161185
\(235\) 2.00674e8 1.00868
\(236\) −1.21779e8 −0.603089
\(237\) −7.67282e7 −0.374400
\(238\) 2.05654e7 0.0988823
\(239\) 1.65597e8 0.784621 0.392310 0.919833i \(-0.371676\pi\)
0.392310 + 0.919833i \(0.371676\pi\)
\(240\) −2.07214e8 −0.967564
\(241\) −1.17817e8 −0.542185 −0.271093 0.962553i \(-0.587385\pi\)
−0.271093 + 0.962553i \(0.587385\pi\)
\(242\) 3.07272e8 1.39370
\(243\) 3.02398e7 0.135194
\(244\) −76613.1 −0.000337628 0
\(245\) −1.06471e8 −0.462540
\(246\) 2.70003e8 1.15637
\(247\) −1.71251e8 −0.723092
\(248\) −1.87498e7 −0.0780579
\(249\) 3.99287e8 1.63903
\(250\) −3.60536e8 −1.45935
\(251\) −1.68271e8 −0.671662 −0.335831 0.941922i \(-0.609017\pi\)
−0.335831 + 0.941922i \(0.609017\pi\)
\(252\) 2.17064e7 0.0854449
\(253\) −7.33125e7 −0.284614
\(254\) −6.16264e8 −2.35966
\(255\) −1.52383e7 −0.0575501
\(256\) 2.45986e8 0.916371
\(257\) 4.08556e8 1.50136 0.750681 0.660665i \(-0.229726\pi\)
0.750681 + 0.660665i \(0.229726\pi\)
\(258\) 6.20760e7 0.225038
\(259\) −3.07602e8 −1.10012
\(260\) −4.79515e8 −1.69198
\(261\) 1.05749e7 0.0368158
\(262\) −7.69813e6 −0.0264442
\(263\) 2.68865e8 0.911357 0.455679 0.890144i \(-0.349397\pi\)
0.455679 + 0.890144i \(0.349397\pi\)
\(264\) 2.70540e6 0.00904935
\(265\) 1.94978e8 0.643613
\(266\) 2.30855e8 0.752061
\(267\) −3.70496e8 −1.19123
\(268\) −4.73464e8 −1.50250
\(269\) −4.68785e8 −1.46839 −0.734194 0.678939i \(-0.762440\pi\)
−0.734194 + 0.678939i \(0.762440\pi\)
\(270\) 4.34488e8 1.34340
\(271\) 7.19335e7 0.219553 0.109776 0.993956i \(-0.464987\pi\)
0.109776 + 0.993956i \(0.464987\pi\)
\(272\) 1.81524e7 0.0546944
\(273\) −7.02914e8 −2.09090
\(274\) 8.12351e8 2.38570
\(275\) −2.48840e6 −0.00721532
\(276\) 6.87894e8 1.96943
\(277\) −1.86677e8 −0.527729 −0.263864 0.964560i \(-0.584997\pi\)
−0.263864 + 0.964560i \(0.584997\pi\)
\(278\) 9.42695e8 2.63157
\(279\) 3.39836e7 0.0936816
\(280\) 2.45663e7 0.0668785
\(281\) 4.97318e8 1.33709 0.668547 0.743670i \(-0.266917\pi\)
0.668547 + 0.743670i \(0.266917\pi\)
\(282\) −5.74050e8 −1.52433
\(283\) −5.31540e8 −1.39407 −0.697033 0.717039i \(-0.745497\pi\)
−0.697033 + 0.717039i \(0.745497\pi\)
\(284\) −2.21944e8 −0.574949
\(285\) −1.71056e8 −0.437704
\(286\) −1.46190e8 −0.369519
\(287\) 3.80973e8 0.951279
\(288\) 3.91395e7 0.0965477
\(289\) −4.09004e8 −0.996747
\(290\) 3.14917e8 0.758234
\(291\) 6.11800e8 1.45541
\(292\) −1.37407e8 −0.322974
\(293\) −6.12847e8 −1.42336 −0.711680 0.702503i \(-0.752066\pi\)
−0.711680 + 0.702503i \(0.752066\pi\)
\(294\) 3.04572e8 0.698995
\(295\) 2.49802e8 0.566525
\(296\) 2.28128e7 0.0511280
\(297\) 6.75142e7 0.149537
\(298\) 1.08752e9 2.38056
\(299\) −1.41267e9 −3.05626
\(300\) 2.33487e7 0.0499274
\(301\) 8.75891e7 0.185126
\(302\) −9.48785e8 −1.98218
\(303\) 3.58980e8 0.741346
\(304\) 2.03768e8 0.415985
\(305\) 157154. 0.000317159 0
\(306\) 2.76439e6 0.00551537
\(307\) −5.15243e8 −1.01631 −0.508157 0.861264i \(-0.669673\pi\)
−0.508157 + 0.861264i \(0.669673\pi\)
\(308\) 1.00444e8 0.195884
\(309\) −7.62852e8 −1.47091
\(310\) 1.01202e9 1.92940
\(311\) 6.25714e8 1.17954 0.589772 0.807570i \(-0.299218\pi\)
0.589772 + 0.807570i \(0.299218\pi\)
\(312\) 5.21306e7 0.0971744
\(313\) −6.49995e8 −1.19813 −0.599066 0.800699i \(-0.704461\pi\)
−0.599066 + 0.800699i \(0.704461\pi\)
\(314\) 9.15652e7 0.166908
\(315\) −4.45257e7 −0.0802645
\(316\) 2.11271e8 0.376648
\(317\) 7.98060e8 1.40711 0.703555 0.710641i \(-0.251595\pi\)
0.703555 + 0.710641i \(0.251595\pi\)
\(318\) −5.57756e8 −0.972634
\(319\) 4.89343e7 0.0844008
\(320\) 6.16683e8 1.05205
\(321\) 6.78594e7 0.114510
\(322\) 1.90435e9 3.17870
\(323\) 1.49849e7 0.0247425
\(324\) −6.76580e8 −1.10513
\(325\) −4.79492e7 −0.0774800
\(326\) −3.94504e8 −0.630652
\(327\) 2.95174e8 0.466832
\(328\) −2.82543e7 −0.0442106
\(329\) −8.09983e8 −1.25398
\(330\) −1.46023e8 −0.223678
\(331\) 8.36375e8 1.26766 0.633831 0.773472i \(-0.281482\pi\)
0.633831 + 0.773472i \(0.281482\pi\)
\(332\) −1.09944e9 −1.64887
\(333\) −4.13476e7 −0.0613615
\(334\) −1.63406e8 −0.239970
\(335\) 9.71202e8 1.41141
\(336\) 8.36382e8 1.20287
\(337\) 1.03465e9 1.47262 0.736310 0.676644i \(-0.236566\pi\)
0.736310 + 0.676644i \(0.236566\pi\)
\(338\) −1.80311e9 −2.53989
\(339\) 3.91669e8 0.546034
\(340\) 4.19587e7 0.0578956
\(341\) 1.57256e8 0.214767
\(342\) 3.10313e7 0.0419478
\(343\) −4.77509e8 −0.638928
\(344\) −6.49592e6 −0.00860372
\(345\) −1.41106e9 −1.85002
\(346\) 6.86534e8 0.891037
\(347\) 1.89330e8 0.243258 0.121629 0.992576i \(-0.461188\pi\)
0.121629 + 0.992576i \(0.461188\pi\)
\(348\) −4.59153e8 −0.584023
\(349\) 3.29547e8 0.414981 0.207490 0.978237i \(-0.433470\pi\)
0.207490 + 0.978237i \(0.433470\pi\)
\(350\) 6.46379e7 0.0805841
\(351\) 1.30094e9 1.60577
\(352\) 1.81115e8 0.221337
\(353\) −1.18738e9 −1.43674 −0.718371 0.695661i \(-0.755112\pi\)
−0.718371 + 0.695661i \(0.755112\pi\)
\(354\) −7.14587e8 −0.856138
\(355\) 4.55267e8 0.540091
\(356\) 1.02016e9 1.19838
\(357\) 6.15066e7 0.0715456
\(358\) −1.33683e9 −1.53987
\(359\) −7.45557e7 −0.0850453 −0.0425226 0.999096i \(-0.513539\pi\)
−0.0425226 + 0.999096i \(0.513539\pi\)
\(360\) 3.30218e6 0.00373029
\(361\) −7.25661e8 −0.811818
\(362\) −1.93458e8 −0.214341
\(363\) 9.18982e8 1.00840
\(364\) 1.93548e9 2.10345
\(365\) 2.81858e8 0.303393
\(366\) −449557. −0.000479293 0
\(367\) −1.32724e9 −1.40158 −0.700789 0.713368i \(-0.747169\pi\)
−0.700789 + 0.713368i \(0.747169\pi\)
\(368\) 1.68090e9 1.75823
\(369\) 5.12102e7 0.0530596
\(370\) −1.23132e9 −1.26376
\(371\) −7.86992e8 −0.800132
\(372\) −1.47554e9 −1.48611
\(373\) 1.54991e8 0.154641 0.0773207 0.997006i \(-0.475363\pi\)
0.0773207 + 0.997006i \(0.475363\pi\)
\(374\) 1.27920e7 0.0126441
\(375\) −1.07828e9 −1.05590
\(376\) 6.00712e7 0.0582786
\(377\) 9.42921e8 0.906318
\(378\) −1.75373e9 −1.67010
\(379\) −1.15983e9 −1.09435 −0.547174 0.837019i \(-0.684296\pi\)
−0.547174 + 0.837019i \(0.684296\pi\)
\(380\) 4.71002e8 0.440332
\(381\) −1.84311e9 −1.70731
\(382\) −2.61677e9 −2.40184
\(383\) −8.33044e8 −0.757656 −0.378828 0.925467i \(-0.623673\pi\)
−0.378828 + 0.925467i \(0.623673\pi\)
\(384\) −1.29270e8 −0.116503
\(385\) −2.06039e8 −0.184008
\(386\) 1.40448e9 1.24297
\(387\) 1.17737e7 0.0103258
\(388\) −1.68459e9 −1.46415
\(389\) −3.11695e8 −0.268477 −0.134238 0.990949i \(-0.542859\pi\)
−0.134238 + 0.990949i \(0.542859\pi\)
\(390\) −2.81374e9 −2.40192
\(391\) 1.23612e8 0.104578
\(392\) −3.18718e7 −0.0267242
\(393\) −2.30234e7 −0.0191335
\(394\) 2.03284e9 1.67443
\(395\) −4.33375e8 −0.353813
\(396\) 1.35017e7 0.0109258
\(397\) −2.87056e8 −0.230250 −0.115125 0.993351i \(-0.536727\pi\)
−0.115125 + 0.993351i \(0.536727\pi\)
\(398\) 2.32456e9 1.84820
\(399\) 6.90435e8 0.544148
\(400\) 5.70537e7 0.0445732
\(401\) 8.00363e8 0.619843 0.309921 0.950762i \(-0.399697\pi\)
0.309921 + 0.950762i \(0.399697\pi\)
\(402\) −2.77823e9 −2.13293
\(403\) 3.03018e9 2.30622
\(404\) −9.88452e8 −0.745797
\(405\) 1.38785e9 1.03812
\(406\) −1.27110e9 −0.942627
\(407\) −1.91332e8 −0.140672
\(408\) −4.56155e6 −0.00332508
\(409\) −1.90519e9 −1.37691 −0.688456 0.725278i \(-0.741711\pi\)
−0.688456 + 0.725278i \(0.741711\pi\)
\(410\) 1.52502e9 1.09278
\(411\) 2.42956e9 1.72616
\(412\) 2.10052e9 1.47974
\(413\) −1.00828e9 −0.704297
\(414\) 2.55981e8 0.177299
\(415\) 2.25524e9 1.54891
\(416\) 3.48992e9 2.37678
\(417\) 2.81939e9 1.90405
\(418\) 1.43595e8 0.0961660
\(419\) 1.79896e9 1.19474 0.597369 0.801967i \(-0.296213\pi\)
0.597369 + 0.801967i \(0.296213\pi\)
\(420\) 1.93327e9 1.27327
\(421\) 3.47444e7 0.0226933 0.0113467 0.999936i \(-0.496388\pi\)
0.0113467 + 0.999936i \(0.496388\pi\)
\(422\) 2.96742e9 1.92214
\(423\) −1.08877e8 −0.0699433
\(424\) 5.83661e7 0.0371861
\(425\) 4.19567e6 0.00265118
\(426\) −1.30234e9 −0.816191
\(427\) −634324. −0.000394288 0
\(428\) −1.86851e8 −0.115197
\(429\) −4.37222e8 −0.267363
\(430\) 3.50616e8 0.212663
\(431\) 1.91702e9 1.15334 0.576669 0.816978i \(-0.304352\pi\)
0.576669 + 0.816978i \(0.304352\pi\)
\(432\) −1.54796e9 −0.923775
\(433\) −7.40677e8 −0.438451 −0.219226 0.975674i \(-0.570353\pi\)
−0.219226 + 0.975674i \(0.570353\pi\)
\(434\) −4.08483e9 −2.39861
\(435\) 9.41846e8 0.548615
\(436\) −8.12763e8 −0.469636
\(437\) 1.38759e9 0.795381
\(438\) −8.06287e8 −0.458490
\(439\) 2.96692e9 1.67371 0.836856 0.547423i \(-0.184391\pi\)
0.836856 + 0.547423i \(0.184391\pi\)
\(440\) 1.52806e7 0.00855175
\(441\) 5.77667e7 0.0320732
\(442\) 2.46490e8 0.135776
\(443\) 1.34329e9 0.734102 0.367051 0.930201i \(-0.380367\pi\)
0.367051 + 0.930201i \(0.380367\pi\)
\(444\) 1.79528e9 0.973401
\(445\) −2.09263e9 −1.12572
\(446\) −9.75438e8 −0.520628
\(447\) 3.25252e9 1.72244
\(448\) −2.48913e9 −1.30790
\(449\) −3.37789e9 −1.76110 −0.880549 0.473954i \(-0.842826\pi\)
−0.880549 + 0.473954i \(0.842826\pi\)
\(450\) 8.68859e6 0.00449475
\(451\) 2.36971e8 0.121640
\(452\) −1.07846e9 −0.549313
\(453\) −2.83760e9 −1.43419
\(454\) −8.87114e8 −0.444922
\(455\) −3.97018e9 −1.97593
\(456\) −5.12051e7 −0.0252893
\(457\) 2.50731e7 0.0122886 0.00614429 0.999981i \(-0.498044\pi\)
0.00614429 + 0.999981i \(0.498044\pi\)
\(458\) 1.71054e9 0.831961
\(459\) −1.13835e8 −0.0549455
\(460\) 3.88535e9 1.86113
\(461\) −7.94770e8 −0.377823 −0.188911 0.981994i \(-0.560496\pi\)
−0.188911 + 0.981994i \(0.560496\pi\)
\(462\) 5.89397e8 0.278074
\(463\) −3.63682e9 −1.70290 −0.851449 0.524438i \(-0.824276\pi\)
−0.851449 + 0.524438i \(0.824276\pi\)
\(464\) −1.12196e9 −0.521392
\(465\) 3.02673e9 1.39601
\(466\) −6.81662e8 −0.312046
\(467\) −2.11944e8 −0.0962967 −0.0481483 0.998840i \(-0.515332\pi\)
−0.0481483 + 0.998840i \(0.515332\pi\)
\(468\) 2.60165e8 0.117325
\(469\) −3.92008e9 −1.75465
\(470\) −3.24233e9 −1.44051
\(471\) 2.73851e8 0.120765
\(472\) 7.47777e7 0.0327322
\(473\) 5.44816e7 0.0236721
\(474\) 1.23972e9 0.534686
\(475\) 4.70979e7 0.0201639
\(476\) −1.69359e8 −0.0719752
\(477\) −1.05787e8 −0.0446291
\(478\) −2.67559e9 −1.12053
\(479\) 2.49114e9 1.03568 0.517838 0.855479i \(-0.326737\pi\)
0.517838 + 0.855479i \(0.326737\pi\)
\(480\) 3.48594e9 1.43872
\(481\) −3.68681e9 −1.51058
\(482\) 1.90360e9 0.774302
\(483\) 5.69547e9 2.29993
\(484\) −2.53042e9 −1.01446
\(485\) 3.45556e9 1.37538
\(486\) −4.88592e8 −0.193072
\(487\) −6.93003e8 −0.271884 −0.135942 0.990717i \(-0.543406\pi\)
−0.135942 + 0.990717i \(0.543406\pi\)
\(488\) 47043.7 1.83245e−5 0
\(489\) −1.17987e9 −0.456304
\(490\) 1.72027e9 0.660559
\(491\) −2.43700e9 −0.929118 −0.464559 0.885542i \(-0.653787\pi\)
−0.464559 + 0.885542i \(0.653787\pi\)
\(492\) −2.22350e9 −0.841705
\(493\) −8.25078e7 −0.0310121
\(494\) 2.76694e9 1.03266
\(495\) −2.76956e7 −0.0102634
\(496\) −3.60555e9 −1.32674
\(497\) −1.83760e9 −0.671435
\(498\) −6.45138e9 −2.34072
\(499\) −3.89667e9 −1.40392 −0.701959 0.712217i \(-0.747691\pi\)
−0.701959 + 0.712217i \(0.747691\pi\)
\(500\) 2.96905e9 1.06224
\(501\) −4.88712e8 −0.173628
\(502\) 2.71879e9 0.959209
\(503\) 4.29376e9 1.50435 0.752176 0.658962i \(-0.229004\pi\)
0.752176 + 0.658962i \(0.229004\pi\)
\(504\) −1.33287e7 −0.00463745
\(505\) 2.02758e9 0.700581
\(506\) 1.18453e9 0.406461
\(507\) −5.39270e9 −1.83772
\(508\) 5.07500e9 1.71756
\(509\) 3.04510e9 1.02350 0.511751 0.859134i \(-0.328997\pi\)
0.511751 + 0.859134i \(0.328997\pi\)
\(510\) 2.46209e8 0.0821880
\(511\) −1.13767e9 −0.377175
\(512\) −4.31688e9 −1.42143
\(513\) −1.27784e9 −0.417894
\(514\) −6.60114e9 −2.14411
\(515\) −4.30872e9 −1.39003
\(516\) −5.11203e8 −0.163802
\(517\) −5.03820e8 −0.160346
\(518\) 4.97000e9 1.57109
\(519\) 2.05327e9 0.644704
\(520\) 2.94443e8 0.0918310
\(521\) 6.13864e9 1.90169 0.950846 0.309665i \(-0.100217\pi\)
0.950846 + 0.309665i \(0.100217\pi\)
\(522\) −1.70861e8 −0.0525770
\(523\) −5.32103e9 −1.62645 −0.813223 0.581953i \(-0.802289\pi\)
−0.813223 + 0.581953i \(0.802289\pi\)
\(524\) 6.33949e7 0.0192484
\(525\) 1.93317e8 0.0583060
\(526\) −4.34411e9 −1.30152
\(527\) −2.65148e8 −0.0789134
\(528\) 5.20241e8 0.153810
\(529\) 8.04153e9 2.36180
\(530\) −3.15030e9 −0.919151
\(531\) −1.35532e8 −0.0392837
\(532\) −1.90111e9 −0.547416
\(533\) 4.56621e9 1.30620
\(534\) 5.98620e9 1.70121
\(535\) 3.83282e8 0.108213
\(536\) 2.90727e8 0.0815471
\(537\) −3.99816e9 −1.11417
\(538\) 7.57428e9 2.09702
\(539\) 2.67310e8 0.0735284
\(540\) −3.57805e9 −0.977842
\(541\) 2.71071e9 0.736026 0.368013 0.929821i \(-0.380038\pi\)
0.368013 + 0.929821i \(0.380038\pi\)
\(542\) −1.16225e9 −0.313546
\(543\) −5.78588e8 −0.155085
\(544\) −3.05376e8 −0.0813277
\(545\) 1.66720e9 0.441163
\(546\) 1.13572e10 2.98604
\(547\) −2.01310e9 −0.525908 −0.262954 0.964808i \(-0.584697\pi\)
−0.262954 + 0.964808i \(0.584697\pi\)
\(548\) −6.68980e9 −1.73652
\(549\) −85265.4 −2.19923e−5 0
\(550\) 4.02057e7 0.0103043
\(551\) −9.26181e8 −0.235866
\(552\) −4.22396e8 −0.106889
\(553\) 1.74924e9 0.439856
\(554\) 3.01618e9 0.753656
\(555\) −3.68260e9 −0.914385
\(556\) −7.76319e9 −1.91548
\(557\) 1.55537e9 0.381364 0.190682 0.981652i \(-0.438930\pi\)
0.190682 + 0.981652i \(0.438930\pi\)
\(558\) −5.49081e8 −0.133788
\(559\) 1.04981e9 0.254197
\(560\) 4.72403e9 1.13672
\(561\) 3.82579e7 0.00914854
\(562\) −8.03529e9 −1.90952
\(563\) −5.00376e9 −1.18173 −0.590863 0.806772i \(-0.701213\pi\)
−0.590863 + 0.806772i \(0.701213\pi\)
\(564\) 4.72736e9 1.10954
\(565\) 2.21221e9 0.516009
\(566\) 8.58822e9 1.99088
\(567\) −5.60179e9 −1.29058
\(568\) 1.36283e8 0.0312049
\(569\) −4.14184e9 −0.942542 −0.471271 0.881988i \(-0.656205\pi\)
−0.471271 + 0.881988i \(0.656205\pi\)
\(570\) 2.76379e9 0.625090
\(571\) 7.78342e9 1.74962 0.874810 0.484465i \(-0.160986\pi\)
0.874810 + 0.484465i \(0.160986\pi\)
\(572\) 1.20389e9 0.268969
\(573\) −7.82617e9 −1.73783
\(574\) −6.15548e9 −1.35853
\(575\) 3.88516e8 0.0852259
\(576\) −3.34587e8 −0.0729508
\(577\) −2.29128e9 −0.496551 −0.248275 0.968690i \(-0.579864\pi\)
−0.248275 + 0.968690i \(0.579864\pi\)
\(578\) 6.60838e9 1.42347
\(579\) 4.20048e9 0.899341
\(580\) −2.59338e9 −0.551909
\(581\) −9.10288e9 −1.92558
\(582\) −9.88501e9 −2.07848
\(583\) −4.89520e8 −0.102313
\(584\) 8.43735e7 0.0175292
\(585\) −5.33669e8 −0.110211
\(586\) 9.90191e9 2.03272
\(587\) 4.43578e9 0.905184 0.452592 0.891718i \(-0.350499\pi\)
0.452592 + 0.891718i \(0.350499\pi\)
\(588\) −2.50818e9 −0.508790
\(589\) −2.97638e9 −0.600186
\(590\) −4.03611e9 −0.809061
\(591\) 6.07976e9 1.21152
\(592\) 4.38685e9 0.869014
\(593\) 6.57596e9 1.29499 0.647496 0.762068i \(-0.275816\pi\)
0.647496 + 0.762068i \(0.275816\pi\)
\(594\) −1.09084e9 −0.213555
\(595\) 3.47400e8 0.0676115
\(596\) −8.95581e9 −1.73278
\(597\) 6.95223e9 1.33725
\(598\) 2.28248e10 4.36469
\(599\) −9.99449e9 −1.90006 −0.950029 0.312162i \(-0.898947\pi\)
−0.950029 + 0.312162i \(0.898947\pi\)
\(600\) −1.43371e7 −0.00270977
\(601\) −8.94449e8 −0.168072 −0.0840359 0.996463i \(-0.526781\pi\)
−0.0840359 + 0.996463i \(0.526781\pi\)
\(602\) −1.41520e9 −0.264381
\(603\) −5.26934e8 −0.0978691
\(604\) 7.81335e9 1.44281
\(605\) 5.19057e9 0.952953
\(606\) −5.80012e9 −1.05872
\(607\) −6.24872e9 −1.13405 −0.567023 0.823702i \(-0.691905\pi\)
−0.567023 + 0.823702i \(0.691905\pi\)
\(608\) −3.42796e9 −0.618548
\(609\) −3.80159e9 −0.682032
\(610\) −2.53918e6 −0.000452938 0
\(611\) −9.70817e9 −1.72184
\(612\) −2.27651e7 −0.00401457
\(613\) 4.93146e9 0.864696 0.432348 0.901707i \(-0.357685\pi\)
0.432348 + 0.901707i \(0.357685\pi\)
\(614\) 8.32491e9 1.45141
\(615\) 4.56100e9 0.790674
\(616\) −6.16772e7 −0.0106314
\(617\) −7.71513e9 −1.32235 −0.661173 0.750234i \(-0.729941\pi\)
−0.661173 + 0.750234i \(0.729941\pi\)
\(618\) 1.23256e10 2.10062
\(619\) 1.04934e10 1.77827 0.889137 0.457641i \(-0.151305\pi\)
0.889137 + 0.457641i \(0.151305\pi\)
\(620\) −8.33410e9 −1.40439
\(621\) −1.05411e10 −1.76630
\(622\) −1.01098e10 −1.68452
\(623\) 8.44651e9 1.39949
\(624\) 1.00246e10 1.65166
\(625\) −5.80662e9 −0.951357
\(626\) 1.05021e10 1.71107
\(627\) 4.29460e8 0.0695803
\(628\) −7.54049e8 −0.121490
\(629\) 3.22604e8 0.0516883
\(630\) 7.19412e8 0.114627
\(631\) 5.59798e9 0.887010 0.443505 0.896272i \(-0.353735\pi\)
0.443505 + 0.896272i \(0.353735\pi\)
\(632\) −1.29730e8 −0.0204423
\(633\) 8.87489e9 1.39075
\(634\) −1.28945e10 −2.00951
\(635\) −1.04102e10 −1.61343
\(636\) 4.59318e9 0.707968
\(637\) 5.15083e9 0.789567
\(638\) −7.90644e8 −0.120534
\(639\) −2.47009e8 −0.0374507
\(640\) −7.30139e8 −0.110097
\(641\) 1.08513e10 1.62734 0.813670 0.581327i \(-0.197466\pi\)
0.813670 + 0.581327i \(0.197466\pi\)
\(642\) −1.09642e9 −0.163533
\(643\) −4.25867e9 −0.631736 −0.315868 0.948803i \(-0.602296\pi\)
−0.315868 + 0.948803i \(0.602296\pi\)
\(644\) −1.56825e10 −2.31374
\(645\) 1.04861e9 0.153871
\(646\) −2.42114e8 −0.0353351
\(647\) 3.03007e9 0.439833 0.219917 0.975519i \(-0.429421\pi\)
0.219917 + 0.975519i \(0.429421\pi\)
\(648\) 4.15449e8 0.0599798
\(649\) −6.27164e8 −0.0900585
\(650\) 7.74727e8 0.110650
\(651\) −1.22168e10 −1.73550
\(652\) 3.24878e9 0.459044
\(653\) −3.93756e9 −0.553389 −0.276695 0.960958i \(-0.589239\pi\)
−0.276695 + 0.960958i \(0.589239\pi\)
\(654\) −4.76920e9 −0.666689
\(655\) −1.30040e8 −0.0180814
\(656\) −5.43323e9 −0.751441
\(657\) −1.52925e8 −0.0210377
\(658\) 1.30871e10 1.79082
\(659\) 9.30694e7 0.0126680 0.00633400 0.999980i \(-0.497984\pi\)
0.00633400 + 0.999980i \(0.497984\pi\)
\(660\) 1.20252e9 0.162813
\(661\) 1.17439e10 1.58164 0.790818 0.612051i \(-0.209655\pi\)
0.790818 + 0.612051i \(0.209655\pi\)
\(662\) −1.35135e10 −1.81036
\(663\) 7.37196e8 0.0982394
\(664\) 6.75102e8 0.0894914
\(665\) 3.89970e9 0.514227
\(666\) 6.68064e8 0.0876311
\(667\) −7.64016e9 −0.996925
\(668\) 1.34567e9 0.174671
\(669\) −2.91731e9 −0.376697
\(670\) −1.56919e10 −2.01565
\(671\) −394558. −5.04176e−5 0
\(672\) −1.40704e10 −1.78860
\(673\) 1.30594e9 0.165146 0.0825732 0.996585i \(-0.473686\pi\)
0.0825732 + 0.996585i \(0.473686\pi\)
\(674\) −1.67172e10 −2.10307
\(675\) −3.57788e8 −0.0447778
\(676\) 1.48488e10 1.84875
\(677\) 1.71912e9 0.212935 0.106467 0.994316i \(-0.466046\pi\)
0.106467 + 0.994316i \(0.466046\pi\)
\(678\) −6.32829e9 −0.779798
\(679\) −1.39477e10 −1.70985
\(680\) −2.57644e7 −0.00314224
\(681\) −2.65316e9 −0.321920
\(682\) −2.54082e9 −0.306711
\(683\) −4.52474e9 −0.543402 −0.271701 0.962382i \(-0.587586\pi\)
−0.271701 + 0.962382i \(0.587586\pi\)
\(684\) −2.55546e8 −0.0305333
\(685\) 1.37226e10 1.63124
\(686\) 7.71523e9 0.912461
\(687\) 5.11583e9 0.601960
\(688\) −1.24915e9 −0.146236
\(689\) −9.43261e9 −1.09866
\(690\) 2.27988e10 2.64204
\(691\) 1.43457e10 1.65405 0.827025 0.562165i \(-0.190031\pi\)
0.827025 + 0.562165i \(0.190031\pi\)
\(692\) −5.65368e9 −0.648575
\(693\) 1.11788e8 0.0127594
\(694\) −3.05905e9 −0.347399
\(695\) 1.59244e10 1.79935
\(696\) 2.81939e8 0.0316974
\(697\) −3.99554e8 −0.0446952
\(698\) −5.32457e9 −0.592639
\(699\) −2.03870e9 −0.225779
\(700\) −5.32300e8 −0.0586561
\(701\) 1.24050e7 0.00136014 0.000680069 1.00000i \(-0.499784\pi\)
0.000680069 1.00000i \(0.499784\pi\)
\(702\) −2.10196e10 −2.29321
\(703\) 3.62135e9 0.393122
\(704\) −1.54827e9 −0.167241
\(705\) −9.69709e9 −1.04227
\(706\) 1.91848e10 2.05183
\(707\) −8.18396e9 −0.870954
\(708\) 5.88470e9 0.623172
\(709\) −1.13921e8 −0.0120044 −0.00600220 0.999982i \(-0.501911\pi\)
−0.00600220 + 0.999982i \(0.501911\pi\)
\(710\) −7.35586e9 −0.771311
\(711\) 2.35131e8 0.0245339
\(712\) −6.26424e8 −0.0650412
\(713\) −2.45525e10 −2.53678
\(714\) −9.93777e8 −0.102175
\(715\) −2.46951e9 −0.252662
\(716\) 1.10089e10 1.12086
\(717\) −8.00210e9 −0.810750
\(718\) 1.20461e9 0.121454
\(719\) −6.47748e8 −0.0649912 −0.0324956 0.999472i \(-0.510345\pi\)
−0.0324956 + 0.999472i \(0.510345\pi\)
\(720\) 6.35001e8 0.0634031
\(721\) 1.73914e10 1.72807
\(722\) 1.17247e10 1.15937
\(723\) 5.69323e9 0.560241
\(724\) 1.59314e9 0.156016
\(725\) −2.59325e8 −0.0252733
\(726\) −1.48482e10 −1.44011
\(727\) −5.68079e9 −0.548325 −0.274163 0.961683i \(-0.588401\pi\)
−0.274163 + 0.961683i \(0.588401\pi\)
\(728\) −1.18846e9 −0.114163
\(729\) 9.65941e9 0.923430
\(730\) −4.55405e9 −0.433279
\(731\) −9.18610e7 −0.00869802
\(732\) 3.70215e6 0.000348872 0
\(733\) −1.04141e10 −0.976697 −0.488348 0.872649i \(-0.662400\pi\)
−0.488348 + 0.872649i \(0.662400\pi\)
\(734\) 2.14445e10 2.00161
\(735\) 5.14495e9 0.477943
\(736\) −2.82776e10 −2.61439
\(737\) −2.43834e9 −0.224367
\(738\) −8.27416e8 −0.0757751
\(739\) 7.74337e9 0.705788 0.352894 0.935663i \(-0.385198\pi\)
0.352894 + 0.935663i \(0.385198\pi\)
\(740\) 1.01401e10 0.919876
\(741\) 8.27531e9 0.747172
\(742\) 1.27156e10 1.14268
\(743\) −5.39810e9 −0.482815 −0.241407 0.970424i \(-0.577609\pi\)
−0.241407 + 0.970424i \(0.577609\pi\)
\(744\) 9.06043e8 0.0806573
\(745\) 1.83708e10 1.62772
\(746\) −2.50423e9 −0.220845
\(747\) −1.22360e9 −0.107403
\(748\) −1.05343e8 −0.00920347
\(749\) −1.54705e9 −0.134529
\(750\) 1.74221e10 1.50794
\(751\) −1.23730e10 −1.06595 −0.532973 0.846132i \(-0.678925\pi\)
−0.532973 + 0.846132i \(0.678925\pi\)
\(752\) 1.15515e10 0.990552
\(753\) 8.13130e9 0.694029
\(754\) −1.52350e10 −1.29432
\(755\) −1.60273e10 −1.35533
\(756\) 1.44422e10 1.21564
\(757\) 2.74811e9 0.230249 0.115125 0.993351i \(-0.463273\pi\)
0.115125 + 0.993351i \(0.463273\pi\)
\(758\) 1.87396e10 1.56285
\(759\) 3.54266e9 0.294092
\(760\) −2.89215e8 −0.0238987
\(761\) −6.98646e9 −0.574660 −0.287330 0.957832i \(-0.592768\pi\)
−0.287330 + 0.957832i \(0.592768\pi\)
\(762\) 2.97795e10 2.43824
\(763\) −6.72933e9 −0.548448
\(764\) 2.15494e10 1.74827
\(765\) 4.66973e7 0.00377117
\(766\) 1.34597e10 1.08202
\(767\) −1.20849e10 −0.967072
\(768\) −1.18867e10 −0.946887
\(769\) 1.64020e10 1.30064 0.650318 0.759662i \(-0.274636\pi\)
0.650318 + 0.759662i \(0.274636\pi\)
\(770\) 3.32902e9 0.262784
\(771\) −1.97425e10 −1.55136
\(772\) −1.15660e10 −0.904741
\(773\) −6.42318e9 −0.500175 −0.250087 0.968223i \(-0.580459\pi\)
−0.250087 + 0.968223i \(0.580459\pi\)
\(774\) −1.90230e8 −0.0147464
\(775\) −8.33370e8 −0.0643105
\(776\) 1.03441e9 0.0794654
\(777\) 1.48641e10 1.13675
\(778\) 5.03613e9 0.383415
\(779\) −4.48515e9 −0.339935
\(780\) 2.31715e10 1.74833
\(781\) −1.14301e9 −0.0858564
\(782\) −1.99722e9 −0.149349
\(783\) 7.03590e9 0.523786
\(784\) −6.12886e9 −0.454227
\(785\) 1.54676e9 0.114124
\(786\) 3.71994e8 0.0273248
\(787\) 8.36518e9 0.611735 0.305868 0.952074i \(-0.401053\pi\)
0.305868 + 0.952074i \(0.401053\pi\)
\(788\) −1.67406e10 −1.21879
\(789\) −1.29923e10 −0.941706
\(790\) 7.00214e9 0.505285
\(791\) −8.92919e9 −0.641497
\(792\) −8.29061e6 −0.000592991 0
\(793\) −7.60279e6 −0.000541398 0
\(794\) 4.63804e9 0.328823
\(795\) −9.42185e9 −0.665046
\(796\) −1.91430e10 −1.34528
\(797\) 1.20395e10 0.842376 0.421188 0.906973i \(-0.361613\pi\)
0.421188 + 0.906973i \(0.361613\pi\)
\(798\) −1.11555e10 −0.777105
\(799\) 8.49487e8 0.0589173
\(800\) −9.59808e8 −0.0662780
\(801\) 1.13537e9 0.0780595
\(802\) −1.29317e10 −0.885206
\(803\) −7.07645e8 −0.0482293
\(804\) 2.28790e10 1.55254
\(805\) 3.21690e10 2.17346
\(806\) −4.89594e10 −3.29354
\(807\) 2.26530e10 1.51729
\(808\) 6.06952e8 0.0404776
\(809\) −2.68334e10 −1.78178 −0.890892 0.454215i \(-0.849920\pi\)
−0.890892 + 0.454215i \(0.849920\pi\)
\(810\) −2.24238e10 −1.48256
\(811\) 7.68247e9 0.505741 0.252870 0.967500i \(-0.418625\pi\)
0.252870 + 0.967500i \(0.418625\pi\)
\(812\) 1.04677e10 0.686127
\(813\) −3.47602e9 −0.226864
\(814\) 3.09141e9 0.200896
\(815\) −6.66412e9 −0.431213
\(816\) −8.77173e8 −0.0565158
\(817\) −1.03117e9 −0.0661538
\(818\) 3.07826e10 1.96639
\(819\) 2.15406e9 0.137014
\(820\) −1.25587e10 −0.795422
\(821\) 1.61885e10 1.02096 0.510478 0.859891i \(-0.329469\pi\)
0.510478 + 0.859891i \(0.329469\pi\)
\(822\) −3.92550e10 −2.46515
\(823\) 2.11454e10 1.32226 0.661131 0.750271i \(-0.270077\pi\)
0.661131 + 0.750271i \(0.270077\pi\)
\(824\) −1.28981e9 −0.0803118
\(825\) 1.20246e8 0.00745559
\(826\) 1.62910e10 1.00582
\(827\) 2.52181e10 1.55040 0.775198 0.631718i \(-0.217650\pi\)
0.775198 + 0.631718i \(0.217650\pi\)
\(828\) −2.10803e9 −0.129054
\(829\) 1.48168e10 0.903260 0.451630 0.892205i \(-0.350843\pi\)
0.451630 + 0.892205i \(0.350843\pi\)
\(830\) −3.64385e10 −2.21201
\(831\) 9.02072e9 0.545303
\(832\) −2.98338e10 −1.79588
\(833\) −4.50709e8 −0.0270171
\(834\) −4.55536e10 −2.71920
\(835\) −2.76033e9 −0.164081
\(836\) −1.18252e9 −0.0699981
\(837\) 2.26106e10 1.33283
\(838\) −2.90662e10 −1.70622
\(839\) −1.45298e10 −0.849361 −0.424680 0.905343i \(-0.639614\pi\)
−0.424680 + 0.905343i \(0.639614\pi\)
\(840\) −1.18711e9 −0.0691056
\(841\) −1.21503e10 −0.704367
\(842\) −5.61375e8 −0.0324086
\(843\) −2.40317e10 −1.38162
\(844\) −2.44370e10 −1.39910
\(845\) −3.04589e10 −1.73667
\(846\) 1.75916e9 0.0998869
\(847\) −2.09508e10 −1.18470
\(848\) 1.12237e10 0.632046
\(849\) 2.56854e10 1.44049
\(850\) −6.77904e7 −0.00378619
\(851\) 2.98729e10 1.66159
\(852\) 1.07249e10 0.594095
\(853\) −1.93358e10 −1.06669 −0.533347 0.845896i \(-0.679066\pi\)
−0.533347 + 0.845896i \(0.679066\pi\)
\(854\) 1.02489e7 0.000563088 0
\(855\) 5.24195e8 0.0286821
\(856\) 1.14735e8 0.00625224
\(857\) 3.62840e9 0.196916 0.0984582 0.995141i \(-0.468609\pi\)
0.0984582 + 0.995141i \(0.468609\pi\)
\(858\) 7.06431e9 0.381825
\(859\) 5.66924e9 0.305175 0.152588 0.988290i \(-0.451239\pi\)
0.152588 + 0.988290i \(0.451239\pi\)
\(860\) −2.88736e9 −0.154795
\(861\) −1.84097e10 −0.982957
\(862\) −3.09738e10 −1.64710
\(863\) −7.37525e9 −0.390606 −0.195303 0.980743i \(-0.562569\pi\)
−0.195303 + 0.980743i \(0.562569\pi\)
\(864\) 2.60411e10 1.37360
\(865\) 1.15972e10 0.609253
\(866\) 1.19673e10 0.626158
\(867\) 1.97642e10 1.02994
\(868\) 3.36391e10 1.74592
\(869\) 1.08805e9 0.0562444
\(870\) −1.52176e10 −0.783483
\(871\) −4.69847e10 −2.40931
\(872\) 4.99071e8 0.0254891
\(873\) −1.87484e9 −0.0953707
\(874\) −2.24196e10 −1.13589
\(875\) 2.45825e10 1.24050
\(876\) 6.63986e9 0.333729
\(877\) −1.97900e10 −0.990711 −0.495355 0.868690i \(-0.664962\pi\)
−0.495355 + 0.868690i \(0.664962\pi\)
\(878\) −4.79373e10 −2.39025
\(879\) 2.96144e10 1.47076
\(880\) 2.93841e9 0.145353
\(881\) 5.73994e9 0.282808 0.141404 0.989952i \(-0.454838\pi\)
0.141404 + 0.989952i \(0.454838\pi\)
\(882\) −9.33350e8 −0.0458041
\(883\) −3.59778e9 −0.175862 −0.0879311 0.996127i \(-0.528026\pi\)
−0.0879311 + 0.996127i \(0.528026\pi\)
\(884\) −2.02987e9 −0.0988293
\(885\) −1.20711e10 −0.585391
\(886\) −2.17039e10 −1.04838
\(887\) −2.01902e9 −0.0971422 −0.0485711 0.998820i \(-0.515467\pi\)
−0.0485711 + 0.998820i \(0.515467\pi\)
\(888\) −1.10238e9 −0.0528306
\(889\) 4.20188e10 2.00580
\(890\) 3.38111e10 1.60766
\(891\) −3.48439e9 −0.165027
\(892\) 8.03284e9 0.378959
\(893\) 9.53581e9 0.448103
\(894\) −5.25517e10 −2.45983
\(895\) −2.25823e10 −1.05290
\(896\) 2.94707e9 0.136871
\(897\) 6.82639e10 3.15804
\(898\) 5.45775e10 2.51505
\(899\) 1.63882e10 0.752268
\(900\) −7.15514e7 −0.00327167
\(901\) 8.25375e8 0.0375937
\(902\) −3.82879e9 −0.173716
\(903\) −4.23254e9 −0.191291
\(904\) 6.62221e8 0.0298135
\(905\) −3.26797e9 −0.146557
\(906\) 4.58479e10 2.04819
\(907\) 9.54457e9 0.424748 0.212374 0.977188i \(-0.431881\pi\)
0.212374 + 0.977188i \(0.431881\pi\)
\(908\) 7.30548e9 0.323853
\(909\) −1.10008e9 −0.0485793
\(910\) 6.41472e10 2.82184
\(911\) 1.73423e10 0.759964 0.379982 0.924994i \(-0.375930\pi\)
0.379982 + 0.924994i \(0.375930\pi\)
\(912\) −9.84660e9 −0.429838
\(913\) −5.66212e9 −0.246224
\(914\) −4.05113e8 −0.0175495
\(915\) −7.59411e6 −0.000327720 0
\(916\) −1.40864e10 −0.605574
\(917\) 5.24883e8 0.0224786
\(918\) 1.83926e9 0.0784683
\(919\) 1.56927e8 0.00666949 0.00333475 0.999994i \(-0.498939\pi\)
0.00333475 + 0.999994i \(0.498939\pi\)
\(920\) −2.38577e9 −0.101012
\(921\) 2.48979e10 1.05016
\(922\) 1.28413e10 0.539574
\(923\) −2.20249e10 −0.921949
\(924\) −4.85375e9 −0.202407
\(925\) 1.01396e9 0.0421234
\(926\) 5.87610e10 2.43193
\(927\) 2.33774e9 0.0963866
\(928\) 1.88746e10 0.775283
\(929\) −9.87406e9 −0.404055 −0.202028 0.979380i \(-0.564753\pi\)
−0.202028 + 0.979380i \(0.564753\pi\)
\(930\) −4.89036e10 −1.99366
\(931\) −5.05938e9 −0.205482
\(932\) 5.61356e9 0.227134
\(933\) −3.02362e10 −1.21882
\(934\) 3.42443e9 0.137522
\(935\) 2.16088e8 0.00864548
\(936\) −1.59753e8 −0.00636770
\(937\) −1.57119e10 −0.623935 −0.311968 0.950093i \(-0.600988\pi\)
−0.311968 + 0.950093i \(0.600988\pi\)
\(938\) 6.33377e10 2.50583
\(939\) 3.14095e10 1.23803
\(940\) 2.67010e10 1.04853
\(941\) 9.28175e9 0.363133 0.181567 0.983379i \(-0.441883\pi\)
0.181567 + 0.983379i \(0.441883\pi\)
\(942\) −4.42468e9 −0.172466
\(943\) −3.69984e10 −1.43679
\(944\) 1.43795e10 0.556344
\(945\) −2.96248e10 −1.14194
\(946\) −8.80273e8 −0.0338064
\(947\) −3.17945e10 −1.21654 −0.608270 0.793730i \(-0.708136\pi\)
−0.608270 + 0.793730i \(0.708136\pi\)
\(948\) −1.02092e10 −0.389191
\(949\) −1.36357e10 −0.517899
\(950\) −7.60973e8 −0.0287963
\(951\) −3.85644e10 −1.45397
\(952\) 1.03993e8 0.00390640
\(953\) −1.99879e10 −0.748070 −0.374035 0.927415i \(-0.622026\pi\)
−0.374035 + 0.927415i \(0.622026\pi\)
\(954\) 1.70923e9 0.0637353
\(955\) −4.42036e10 −1.64228
\(956\) 2.20338e10 0.815618
\(957\) −2.36464e9 −0.0872114
\(958\) −4.02500e10 −1.47906
\(959\) −5.53887e10 −2.02794
\(960\) −2.97998e10 −1.08709
\(961\) 2.51527e10 0.914225
\(962\) 5.95687e10 2.15727
\(963\) −2.07953e8 −0.00750366
\(964\) −1.56763e10 −0.563605
\(965\) 2.37251e10 0.849888
\(966\) −9.20231e10 −3.28456
\(967\) 5.44278e10 1.93566 0.967828 0.251611i \(-0.0809604\pi\)
0.967828 + 0.251611i \(0.0809604\pi\)
\(968\) 1.55379e9 0.0550589
\(969\) −7.24108e8 −0.0255664
\(970\) −5.58323e10 −1.96419
\(971\) 2.25661e10 0.791024 0.395512 0.918461i \(-0.370567\pi\)
0.395512 + 0.918461i \(0.370567\pi\)
\(972\) 4.02360e9 0.140535
\(973\) −6.42759e10 −2.23693
\(974\) 1.11970e10 0.388281
\(975\) 2.31704e9 0.0800602
\(976\) 9.04638e6 0.000311459 0
\(977\) −4.26254e9 −0.146230 −0.0731152 0.997324i \(-0.523294\pi\)
−0.0731152 + 0.997324i \(0.523294\pi\)
\(978\) 1.90635e10 0.651653
\(979\) 5.25385e9 0.178953
\(980\) −1.41666e10 −0.480813
\(981\) −9.04552e8 −0.0305909
\(982\) 3.93753e10 1.32688
\(983\) −1.23804e10 −0.415717 −0.207858 0.978159i \(-0.566649\pi\)
−0.207858 + 0.978159i \(0.566649\pi\)
\(984\) 1.36533e9 0.0456829
\(985\) 3.43396e10 1.14490
\(986\) 1.33310e9 0.0442887
\(987\) 3.91405e10 1.29574
\(988\) −2.27861e10 −0.751658
\(989\) −8.50627e9 −0.279610
\(990\) 4.47484e8 0.0146573
\(991\) 3.54785e10 1.15800 0.578998 0.815329i \(-0.303444\pi\)
0.578998 + 0.815329i \(0.303444\pi\)
\(992\) 6.06557e10 1.97279
\(993\) −4.04159e10 −1.30988
\(994\) 2.96906e10 0.958885
\(995\) 3.92674e10 1.26372
\(996\) 5.31278e10 1.70378
\(997\) −1.77312e10 −0.566636 −0.283318 0.959026i \(-0.591435\pi\)
−0.283318 + 0.959026i \(0.591435\pi\)
\(998\) 6.29595e10 2.00495
\(999\) −2.75103e10 −0.873003
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.1.2 11
3.2 odd 2 387.8.a.b.1.10 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.a.1.2 11 1.1 even 1 trivial
387.8.a.b.1.10 11 3.2 odd 2