Properties

Label 43.8.a.a.1.7
Level $43$
Weight $8$
Character 43.1
Self dual yes
Analytic conductor $13.433$
Analytic rank $1$
Dimension $11$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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 1.7
Root \(-5.14919\) of defining polynomial
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.14919 q^{2} +34.1076 q^{3} -118.083 q^{4} -39.2891 q^{5} +107.411 q^{6} +434.472 q^{7} -774.961 q^{8} -1023.67 q^{9} +O(q^{10})\) \(q+3.14919 q^{2} +34.1076 q^{3} -118.083 q^{4} -39.2891 q^{5} +107.411 q^{6} +434.472 q^{7} -774.961 q^{8} -1023.67 q^{9} -123.729 q^{10} -6392.85 q^{11} -4027.51 q^{12} +442.964 q^{13} +1368.23 q^{14} -1340.05 q^{15} +12674.1 q^{16} -26318.9 q^{17} -3223.74 q^{18} +7748.98 q^{19} +4639.35 q^{20} +14818.8 q^{21} -20132.3 q^{22} -58609.7 q^{23} -26432.1 q^{24} -76581.4 q^{25} +1394.98 q^{26} -109508. q^{27} -51303.5 q^{28} +27040.0 q^{29} -4220.09 q^{30} +100436. q^{31} +139108. q^{32} -218045. q^{33} -82883.2 q^{34} -17070.0 q^{35} +120878. q^{36} +358755. q^{37} +24403.0 q^{38} +15108.4 q^{39} +30447.5 q^{40} -74304.4 q^{41} +46667.2 q^{42} +79507.0 q^{43} +754884. q^{44} +40219.1 q^{45} -184573. q^{46} +462198. q^{47} +432282. q^{48} -634777. q^{49} -241169. q^{50} -897674. q^{51} -52306.4 q^{52} +1.17404e6 q^{53} -344863. q^{54} +251169. q^{55} -336699. q^{56} +264299. q^{57} +85154.2 q^{58} +1.70392e6 q^{59} +158237. q^{60} +2.64618e6 q^{61} +316293. q^{62} -444757. q^{63} -1.18420e6 q^{64} -17403.6 q^{65} -686664. q^{66} -2.11768e6 q^{67} +3.10780e6 q^{68} -1.99904e6 q^{69} -53756.6 q^{70} -5.06201e6 q^{71} +793306. q^{72} +899577. q^{73} +1.12979e6 q^{74} -2.61201e6 q^{75} -915020. q^{76} -2.77751e6 q^{77} +47579.4 q^{78} -4.05836e6 q^{79} -497952. q^{80} -1.49629e6 q^{81} -233999. q^{82} -6.07269e6 q^{83} -1.74984e6 q^{84} +1.03404e6 q^{85} +250383. q^{86} +922270. q^{87} +4.95421e6 q^{88} -3.93741e6 q^{89} +126658. q^{90} +192455. q^{91} +6.92079e6 q^{92} +3.42563e6 q^{93} +1.45555e6 q^{94} -304450. q^{95} +4.74464e6 q^{96} -1.24981e7 q^{97} -1.99904e6 q^{98} +6.54418e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} + O(q^{10}) \) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} - 1333q^{10} + 1333q^{11} + 5089q^{12} - 17967q^{13} - 22352q^{14} - 49504q^{15} - 34406q^{16} - 63095q^{17} - 165931q^{18} - 54524q^{19} - 280995q^{20} - 139788q^{21} - 289358q^{22} - 138139q^{23} - 429583q^{24} + 3455q^{25} - 132946q^{26} - 240356q^{27} - 12704q^{28} - 308658q^{29} + 421284q^{30} - 209523q^{31} - 644934q^{32} + 96814q^{33} + 762435q^{34} - 578892q^{35} + 426161q^{36} - 298472q^{37} - 369707q^{38} + 292298q^{39} + 2633173q^{40} - 1346735q^{41} + 1173266q^{42} + 874577q^{43} + 3134292q^{44} + 1893784q^{45} + 3588111q^{46} + 499284q^{47} + 5647533q^{48} + 2544563q^{49} + 3049745q^{50} + 1258424q^{51} + 983088q^{52} - 2210495q^{53} + 6789698q^{54} - 1855072q^{55} - 469976q^{56} - 1238444q^{57} + 4397067q^{58} - 5824216q^{59} - 2889372q^{60} - 4453034q^{61} + 1002789q^{62} - 6240564q^{63} + 4757538q^{64} - 2162872q^{65} - 258940q^{66} - 6859513q^{67} - 9397005q^{68} - 10040030q^{69} + 845078q^{70} - 10726554q^{71} + 1199517q^{72} - 4456898q^{73} + 1046637q^{74} - 3349114q^{75} + 5861267q^{76} - 17019816q^{77} + 1999122q^{78} - 15541320q^{79} - 15680911q^{80} - 12976697q^{81} + 20233655q^{82} - 11146767q^{83} + 12348278q^{84} - 12471976q^{85} - 1908168q^{86} - 18648900q^{87} - 24463544q^{88} - 13531356q^{89} + 20858990q^{90} - 19746448q^{91} - 26023161q^{92} - 21903110q^{93} + 20288857q^{94} - 12291624q^{95} - 13954503q^{96} - 10999901q^{97} + 29909168q^{98} + 29396057q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.14919 0.278352 0.139176 0.990268i \(-0.455555\pi\)
0.139176 + 0.990268i \(0.455555\pi\)
\(3\) 34.1076 0.729334 0.364667 0.931138i \(-0.381183\pi\)
0.364667 + 0.931138i \(0.381183\pi\)
\(4\) −118.083 −0.922520
\(5\) −39.2891 −0.140565 −0.0702824 0.997527i \(-0.522390\pi\)
−0.0702824 + 0.997527i \(0.522390\pi\)
\(6\) 107.411 0.203012
\(7\) 434.472 0.478760 0.239380 0.970926i \(-0.423056\pi\)
0.239380 + 0.970926i \(0.423056\pi\)
\(8\) −774.961 −0.535137
\(9\) −1023.67 −0.468071
\(10\) −123.729 −0.0391265
\(11\) −6392.85 −1.44817 −0.724086 0.689710i \(-0.757738\pi\)
−0.724086 + 0.689710i \(0.757738\pi\)
\(12\) −4027.51 −0.672826
\(13\) 442.964 0.0559200 0.0279600 0.999609i \(-0.491099\pi\)
0.0279600 + 0.999609i \(0.491099\pi\)
\(14\) 1368.23 0.133264
\(15\) −1340.05 −0.102519
\(16\) 12674.1 0.773564
\(17\) −26318.9 −1.29926 −0.649630 0.760250i \(-0.725076\pi\)
−0.649630 + 0.760250i \(0.725076\pi\)
\(18\) −3223.74 −0.130289
\(19\) 7748.98 0.259183 0.129592 0.991567i \(-0.458633\pi\)
0.129592 + 0.991567i \(0.458633\pi\)
\(20\) 4639.35 0.129674
\(21\) 14818.8 0.349176
\(22\) −20132.3 −0.403101
\(23\) −58609.7 −1.00444 −0.502218 0.864741i \(-0.667483\pi\)
−0.502218 + 0.864741i \(0.667483\pi\)
\(24\) −26432.1 −0.390294
\(25\) −76581.4 −0.980242
\(26\) 1394.98 0.0155654
\(27\) −109508. −1.07071
\(28\) −51303.5 −0.441666
\(29\) 27040.0 0.205880 0.102940 0.994688i \(-0.467175\pi\)
0.102940 + 0.994688i \(0.467175\pi\)
\(30\) −4220.09 −0.0285363
\(31\) 100436. 0.605514 0.302757 0.953068i \(-0.402093\pi\)
0.302757 + 0.953068i \(0.402093\pi\)
\(32\) 139108. 0.750460
\(33\) −218045. −1.05620
\(34\) −82883.2 −0.361652
\(35\) −17070.0 −0.0672969
\(36\) 120878. 0.431805
\(37\) 358755. 1.16437 0.582186 0.813055i \(-0.302197\pi\)
0.582186 + 0.813055i \(0.302197\pi\)
\(38\) 24403.0 0.0721441
\(39\) 15108.4 0.0407843
\(40\) 30447.5 0.0752214
\(41\) −74304.4 −0.168372 −0.0841862 0.996450i \(-0.526829\pi\)
−0.0841862 + 0.996450i \(0.526829\pi\)
\(42\) 46667.2 0.0971939
\(43\) 79507.0 0.152499
\(44\) 754884. 1.33597
\(45\) 40219.1 0.0657944
\(46\) −184573. −0.279587
\(47\) 462198. 0.649359 0.324680 0.945824i \(-0.394743\pi\)
0.324680 + 0.945824i \(0.394743\pi\)
\(48\) 432282. 0.564187
\(49\) −634777. −0.770788
\(50\) −241169. −0.272852
\(51\) −897674. −0.947595
\(52\) −52306.4 −0.0515873
\(53\) 1.17404e6 1.08322 0.541612 0.840629i \(-0.317814\pi\)
0.541612 + 0.840629i \(0.317814\pi\)
\(54\) −344863. −0.298035
\(55\) 251169. 0.203562
\(56\) −336699. −0.256202
\(57\) 264299. 0.189031
\(58\) 85154.2 0.0573071
\(59\) 1.70392e6 1.08011 0.540055 0.841630i \(-0.318404\pi\)
0.540055 + 0.841630i \(0.318404\pi\)
\(60\) 158237. 0.0945756
\(61\) 2.64618e6 1.49267 0.746336 0.665570i \(-0.231811\pi\)
0.746336 + 0.665570i \(0.231811\pi\)
\(62\) 316293. 0.168546
\(63\) −444757. −0.224094
\(64\) −1.18420e6 −0.564672
\(65\) −17403.6 −0.00786038
\(66\) −686664. −0.293995
\(67\) −2.11768e6 −0.860199 −0.430099 0.902782i \(-0.641522\pi\)
−0.430099 + 0.902782i \(0.641522\pi\)
\(68\) 3.10780e6 1.19859
\(69\) −1.99904e6 −0.732570
\(70\) −53756.6 −0.0187322
\(71\) −5.06201e6 −1.67849 −0.839246 0.543752i \(-0.817003\pi\)
−0.839246 + 0.543752i \(0.817003\pi\)
\(72\) 793306. 0.250482
\(73\) 899577. 0.270650 0.135325 0.990801i \(-0.456792\pi\)
0.135325 + 0.990801i \(0.456792\pi\)
\(74\) 1.12979e6 0.324105
\(75\) −2.61201e6 −0.714924
\(76\) −915020. −0.239102
\(77\) −2.77751e6 −0.693327
\(78\) 47579.4 0.0113524
\(79\) −4.05836e6 −0.926095 −0.463047 0.886333i \(-0.653244\pi\)
−0.463047 + 0.886333i \(0.653244\pi\)
\(80\) −497952. −0.108736
\(81\) −1.49629e6 −0.312838
\(82\) −233999. −0.0468668
\(83\) −6.07269e6 −1.16576 −0.582878 0.812559i \(-0.698074\pi\)
−0.582878 + 0.812559i \(0.698074\pi\)
\(84\) −1.74984e6 −0.322122
\(85\) 1.03404e6 0.182630
\(86\) 250383. 0.0424483
\(87\) 922270. 0.150155
\(88\) 4.95421e6 0.774970
\(89\) −3.93741e6 −0.592033 −0.296017 0.955183i \(-0.595658\pi\)
−0.296017 + 0.955183i \(0.595658\pi\)
\(90\) 126658. 0.0183140
\(91\) 192455. 0.0267723
\(92\) 6.92079e6 0.926613
\(93\) 3.42563e6 0.441622
\(94\) 1.45555e6 0.180750
\(95\) −304450. −0.0364320
\(96\) 4.74464e6 0.547336
\(97\) −1.24981e7 −1.39041 −0.695203 0.718813i \(-0.744686\pi\)
−0.695203 + 0.718813i \(0.744686\pi\)
\(98\) −1.99904e6 −0.214550
\(99\) 6.54418e6 0.677848
\(100\) 9.04293e6 0.904293
\(101\) 8.38183e6 0.809495 0.404747 0.914429i \(-0.367359\pi\)
0.404747 + 0.914429i \(0.367359\pi\)
\(102\) −2.82695e6 −0.263765
\(103\) 1.30426e7 1.17607 0.588036 0.808835i \(-0.299902\pi\)
0.588036 + 0.808835i \(0.299902\pi\)
\(104\) −343280. −0.0299248
\(105\) −582216. −0.0490819
\(106\) 3.69728e6 0.301517
\(107\) −8.79724e6 −0.694230 −0.347115 0.937823i \(-0.612839\pi\)
−0.347115 + 0.937823i \(0.612839\pi\)
\(108\) 1.29310e7 0.987756
\(109\) −1.73854e7 −1.28585 −0.642927 0.765927i \(-0.722280\pi\)
−0.642927 + 0.765927i \(0.722280\pi\)
\(110\) 790979. 0.0566618
\(111\) 1.22363e7 0.849217
\(112\) 5.50652e6 0.370352
\(113\) −7.28232e6 −0.474783 −0.237391 0.971414i \(-0.576292\pi\)
−0.237391 + 0.971414i \(0.576292\pi\)
\(114\) 832328. 0.0526172
\(115\) 2.30272e6 0.141188
\(116\) −3.19296e6 −0.189929
\(117\) −453450. −0.0261745
\(118\) 5.36598e6 0.300650
\(119\) −1.14348e7 −0.622035
\(120\) 1.03849e6 0.0548616
\(121\) 2.13813e7 1.09720
\(122\) 8.33331e6 0.415488
\(123\) −2.53434e6 −0.122800
\(124\) −1.18598e7 −0.558599
\(125\) 6.07827e6 0.278352
\(126\) −1.40062e6 −0.0623770
\(127\) 9.14216e6 0.396037 0.198018 0.980198i \(-0.436549\pi\)
0.198018 + 0.980198i \(0.436549\pi\)
\(128\) −2.15351e7 −0.907637
\(129\) 2.71179e6 0.111222
\(130\) −54807.4 −0.00218795
\(131\) −3.19287e7 −1.24089 −0.620443 0.784251i \(-0.713047\pi\)
−0.620443 + 0.784251i \(0.713047\pi\)
\(132\) 2.57473e7 0.974367
\(133\) 3.36671e6 0.124087
\(134\) −6.66899e6 −0.239438
\(135\) 4.30248e6 0.150505
\(136\) 2.03961e7 0.695282
\(137\) 1.07371e7 0.356750 0.178375 0.983963i \(-0.442916\pi\)
0.178375 + 0.983963i \(0.442916\pi\)
\(138\) −6.29535e6 −0.203912
\(139\) −7.67775e6 −0.242483 −0.121242 0.992623i \(-0.538688\pi\)
−0.121242 + 0.992623i \(0.538688\pi\)
\(140\) 2.01567e6 0.0620827
\(141\) 1.57644e7 0.473600
\(142\) −1.59413e7 −0.467211
\(143\) −2.83180e6 −0.0809817
\(144\) −1.29741e7 −0.362083
\(145\) −1.06238e6 −0.0289395
\(146\) 2.83294e6 0.0753360
\(147\) −2.16507e7 −0.562162
\(148\) −4.23627e7 −1.07416
\(149\) −5.77368e7 −1.42988 −0.714942 0.699184i \(-0.753547\pi\)
−0.714942 + 0.699184i \(0.753547\pi\)
\(150\) −8.22571e6 −0.199000
\(151\) −1.59611e7 −0.377263 −0.188632 0.982048i \(-0.560405\pi\)
−0.188632 + 0.982048i \(0.560405\pi\)
\(152\) −6.00516e6 −0.138699
\(153\) 2.69419e7 0.608147
\(154\) −8.74691e6 −0.192989
\(155\) −3.94604e6 −0.0851139
\(156\) −1.78404e6 −0.0376244
\(157\) 5.56041e7 1.14672 0.573361 0.819303i \(-0.305639\pi\)
0.573361 + 0.819303i \(0.305639\pi\)
\(158\) −1.27805e7 −0.257780
\(159\) 4.00437e7 0.790032
\(160\) −5.46543e6 −0.105488
\(161\) −2.54643e7 −0.480884
\(162\) −4.71211e6 −0.0870789
\(163\) −1.30748e7 −0.236472 −0.118236 0.992986i \(-0.537724\pi\)
−0.118236 + 0.992986i \(0.537724\pi\)
\(164\) 8.77405e6 0.155327
\(165\) 8.56677e6 0.148465
\(166\) −1.91241e7 −0.324490
\(167\) 4.97944e7 0.827319 0.413660 0.910432i \(-0.364250\pi\)
0.413660 + 0.910432i \(0.364250\pi\)
\(168\) −1.14840e7 −0.186857
\(169\) −6.25523e7 −0.996873
\(170\) 3.25640e6 0.0508355
\(171\) −7.93242e6 −0.121316
\(172\) −9.38839e6 −0.140683
\(173\) 9.03870e7 1.32722 0.663612 0.748077i \(-0.269022\pi\)
0.663612 + 0.748077i \(0.269022\pi\)
\(174\) 2.90441e6 0.0417960
\(175\) −3.32724e7 −0.469301
\(176\) −8.10234e7 −1.12025
\(177\) 5.81167e7 0.787761
\(178\) −1.23997e7 −0.164794
\(179\) −1.32357e8 −1.72489 −0.862447 0.506147i \(-0.831069\pi\)
−0.862447 + 0.506147i \(0.831069\pi\)
\(180\) −4.74918e6 −0.0606966
\(181\) −1.38366e7 −0.173441 −0.0867207 0.996233i \(-0.527639\pi\)
−0.0867207 + 0.996233i \(0.527639\pi\)
\(182\) 606079. 0.00745211
\(183\) 9.02547e7 1.08866
\(184\) 4.54203e7 0.537511
\(185\) −1.40951e7 −0.163670
\(186\) 1.07880e7 0.122926
\(187\) 1.68253e8 1.88155
\(188\) −5.45775e7 −0.599047
\(189\) −4.75782e7 −0.512616
\(190\) −958772. −0.0101409
\(191\) 3.54814e7 0.368455 0.184227 0.982884i \(-0.441022\pi\)
0.184227 + 0.982884i \(0.441022\pi\)
\(192\) −4.03903e7 −0.411835
\(193\) 1.17213e8 1.17361 0.586807 0.809727i \(-0.300385\pi\)
0.586807 + 0.809727i \(0.300385\pi\)
\(194\) −3.93588e7 −0.387022
\(195\) −593596. −0.00573284
\(196\) 7.49562e7 0.711068
\(197\) −3.73724e7 −0.348272 −0.174136 0.984722i \(-0.555713\pi\)
−0.174136 + 0.984722i \(0.555713\pi\)
\(198\) 2.06089e7 0.188680
\(199\) 1.42971e8 1.28606 0.643031 0.765840i \(-0.277677\pi\)
0.643031 + 0.765840i \(0.277677\pi\)
\(200\) 5.93476e7 0.524564
\(201\) −7.22290e7 −0.627373
\(202\) 2.63960e7 0.225324
\(203\) 1.17481e7 0.0985672
\(204\) 1.06000e8 0.874176
\(205\) 2.91935e6 0.0236672
\(206\) 4.10736e7 0.327362
\(207\) 5.99972e7 0.470148
\(208\) 5.61416e6 0.0432577
\(209\) −4.95380e7 −0.375342
\(210\) −1.83351e6 −0.0136620
\(211\) 1.59039e8 1.16551 0.582754 0.812648i \(-0.301975\pi\)
0.582754 + 0.812648i \(0.301975\pi\)
\(212\) −1.38634e8 −0.999295
\(213\) −1.72653e8 −1.22418
\(214\) −2.77042e7 −0.193240
\(215\) −3.12375e6 −0.0214359
\(216\) 8.48647e7 0.572979
\(217\) 4.36367e7 0.289896
\(218\) −5.47499e7 −0.357920
\(219\) 3.06824e7 0.197395
\(220\) −2.96587e7 −0.187790
\(221\) −1.16583e7 −0.0726546
\(222\) 3.85344e7 0.236381
\(223\) −4.10300e7 −0.247762 −0.123881 0.992297i \(-0.539534\pi\)
−0.123881 + 0.992297i \(0.539534\pi\)
\(224\) 6.04385e7 0.359291
\(225\) 7.83942e7 0.458823
\(226\) −2.29334e7 −0.132157
\(227\) 4.91817e7 0.279070 0.139535 0.990217i \(-0.455439\pi\)
0.139535 + 0.990217i \(0.455439\pi\)
\(228\) −3.12091e7 −0.174385
\(229\) −1.27086e8 −0.699315 −0.349657 0.936878i \(-0.613702\pi\)
−0.349657 + 0.936878i \(0.613702\pi\)
\(230\) 7.25171e6 0.0393000
\(231\) −9.47342e7 −0.505667
\(232\) −2.09550e7 −0.110174
\(233\) −7.98934e7 −0.413776 −0.206888 0.978365i \(-0.566334\pi\)
−0.206888 + 0.978365i \(0.566334\pi\)
\(234\) −1.42800e6 −0.00728573
\(235\) −1.81593e7 −0.0912771
\(236\) −2.01204e8 −0.996423
\(237\) −1.38421e8 −0.675433
\(238\) −3.60104e7 −0.173144
\(239\) −1.85347e8 −0.878199 −0.439100 0.898438i \(-0.644702\pi\)
−0.439100 + 0.898438i \(0.644702\pi\)
\(240\) −1.69840e7 −0.0793048
\(241\) 1.85587e7 0.0854061 0.0427031 0.999088i \(-0.486403\pi\)
0.0427031 + 0.999088i \(0.486403\pi\)
\(242\) 6.73338e7 0.305407
\(243\) 1.88460e8 0.842552
\(244\) −3.12467e8 −1.37702
\(245\) 2.49398e7 0.108346
\(246\) −7.98113e6 −0.0341815
\(247\) 3.43252e6 0.0144935
\(248\) −7.78341e7 −0.324033
\(249\) −2.07125e8 −0.850226
\(250\) 1.91416e7 0.0774799
\(251\) 1.86162e8 0.743075 0.371537 0.928418i \(-0.378831\pi\)
0.371537 + 0.928418i \(0.378831\pi\)
\(252\) 5.25180e7 0.206731
\(253\) 3.74683e8 1.45460
\(254\) 2.87904e7 0.110238
\(255\) 3.52688e7 0.133199
\(256\) 8.37598e7 0.312029
\(257\) −4.61094e8 −1.69443 −0.847214 0.531251i \(-0.821722\pi\)
−0.847214 + 0.531251i \(0.821722\pi\)
\(258\) 8.53995e6 0.0309590
\(259\) 1.55869e8 0.557456
\(260\) 2.05507e6 0.00725136
\(261\) −2.76801e7 −0.0963666
\(262\) −1.00550e8 −0.345403
\(263\) −4.24659e8 −1.43945 −0.719723 0.694261i \(-0.755731\pi\)
−0.719723 + 0.694261i \(0.755731\pi\)
\(264\) 1.68976e8 0.565212
\(265\) −4.61270e7 −0.152263
\(266\) 1.06024e7 0.0345398
\(267\) −1.34296e8 −0.431790
\(268\) 2.50061e8 0.793551
\(269\) −2.01202e7 −0.0630230 −0.0315115 0.999503i \(-0.510032\pi\)
−0.0315115 + 0.999503i \(0.510032\pi\)
\(270\) 1.35493e7 0.0418933
\(271\) 3.31931e8 1.01311 0.506554 0.862208i \(-0.330919\pi\)
0.506554 + 0.862208i \(0.330919\pi\)
\(272\) −3.33567e8 −1.00506
\(273\) 6.56419e6 0.0195259
\(274\) 3.38131e7 0.0993020
\(275\) 4.89573e8 1.41956
\(276\) 2.36051e8 0.675811
\(277\) −6.27194e8 −1.77306 −0.886528 0.462674i \(-0.846890\pi\)
−0.886528 + 0.462674i \(0.846890\pi\)
\(278\) −2.41787e7 −0.0674957
\(279\) −1.02814e8 −0.283424
\(280\) 1.32286e7 0.0360130
\(281\) −5.46595e8 −1.46958 −0.734791 0.678294i \(-0.762720\pi\)
−0.734791 + 0.678294i \(0.762720\pi\)
\(282\) 4.96453e7 0.131827
\(283\) 4.74316e8 1.24398 0.621992 0.783023i \(-0.286324\pi\)
0.621992 + 0.783023i \(0.286324\pi\)
\(284\) 5.97736e8 1.54844
\(285\) −1.03841e7 −0.0265711
\(286\) −8.91789e6 −0.0225414
\(287\) −3.22831e7 −0.0806100
\(288\) −1.42401e8 −0.351269
\(289\) 2.82345e8 0.688078
\(290\) −3.34563e6 −0.00805536
\(291\) −4.26279e8 −1.01407
\(292\) −1.06224e8 −0.249680
\(293\) −2.61274e8 −0.606820 −0.303410 0.952860i \(-0.598125\pi\)
−0.303410 + 0.952860i \(0.598125\pi\)
\(294\) −6.81823e7 −0.156479
\(295\) −6.69455e7 −0.151825
\(296\) −2.78021e8 −0.623099
\(297\) 7.00070e8 1.55058
\(298\) −1.81824e8 −0.398011
\(299\) −2.59620e7 −0.0561680
\(300\) 3.08432e8 0.659532
\(301\) 3.45435e7 0.0730103
\(302\) −5.02647e7 −0.105012
\(303\) 2.85884e8 0.590392
\(304\) 9.82111e7 0.200495
\(305\) −1.03966e8 −0.209817
\(306\) 8.48452e7 0.169279
\(307\) 6.32337e8 1.24728 0.623640 0.781711i \(-0.285653\pi\)
0.623640 + 0.781711i \(0.285653\pi\)
\(308\) 3.27976e8 0.639608
\(309\) 4.44851e8 0.857749
\(310\) −1.24268e7 −0.0236916
\(311\) 9.11382e8 1.71806 0.859031 0.511923i \(-0.171067\pi\)
0.859031 + 0.511923i \(0.171067\pi\)
\(312\) −1.17085e7 −0.0218252
\(313\) −4.76864e8 −0.879001 −0.439501 0.898242i \(-0.644845\pi\)
−0.439501 + 0.898242i \(0.644845\pi\)
\(314\) 1.75108e8 0.319192
\(315\) 1.74741e7 0.0314997
\(316\) 4.79221e8 0.854341
\(317\) 2.70951e8 0.477731 0.238865 0.971053i \(-0.423225\pi\)
0.238865 + 0.971053i \(0.423225\pi\)
\(318\) 1.26105e8 0.219907
\(319\) −1.72863e8 −0.298150
\(320\) 4.65262e7 0.0793730
\(321\) −3.00053e8 −0.506326
\(322\) −8.01919e7 −0.133855
\(323\) −2.03945e8 −0.336747
\(324\) 1.76686e8 0.288599
\(325\) −3.39228e7 −0.0548151
\(326\) −4.11751e7 −0.0658223
\(327\) −5.92974e8 −0.937818
\(328\) 5.75830e7 0.0901023
\(329\) 2.00812e8 0.310888
\(330\) 2.69784e7 0.0413254
\(331\) −4.79247e8 −0.726376 −0.363188 0.931716i \(-0.618312\pi\)
−0.363188 + 0.931716i \(0.618312\pi\)
\(332\) 7.17079e8 1.07543
\(333\) −3.67248e8 −0.545009
\(334\) 1.56812e8 0.230286
\(335\) 8.32017e7 0.120914
\(336\) 1.87814e8 0.270110
\(337\) −4.14532e8 −0.590002 −0.295001 0.955497i \(-0.595320\pi\)
−0.295001 + 0.955497i \(0.595320\pi\)
\(338\) −1.96989e8 −0.277481
\(339\) −2.48382e8 −0.346275
\(340\) −1.22103e8 −0.168480
\(341\) −6.42073e8 −0.876888
\(342\) −2.49807e7 −0.0337686
\(343\) −6.33599e8 −0.847784
\(344\) −6.16148e7 −0.0816076
\(345\) 7.85403e7 0.102974
\(346\) 2.84646e8 0.369435
\(347\) −3.46330e7 −0.0444976 −0.0222488 0.999752i \(-0.507083\pi\)
−0.0222488 + 0.999752i \(0.507083\pi\)
\(348\) −1.08904e8 −0.138521
\(349\) −1.29687e9 −1.63308 −0.816541 0.577287i \(-0.804111\pi\)
−0.816541 + 0.577287i \(0.804111\pi\)
\(350\) −1.04781e8 −0.130631
\(351\) −4.85082e7 −0.0598743
\(352\) −8.89297e8 −1.08679
\(353\) 2.44931e8 0.296369 0.148184 0.988960i \(-0.452657\pi\)
0.148184 + 0.988960i \(0.452657\pi\)
\(354\) 1.83021e8 0.219275
\(355\) 1.98882e8 0.235937
\(356\) 4.64940e8 0.546163
\(357\) −3.90014e8 −0.453671
\(358\) −4.16819e8 −0.480128
\(359\) 1.89464e8 0.216121 0.108060 0.994144i \(-0.465536\pi\)
0.108060 + 0.994144i \(0.465536\pi\)
\(360\) −3.11683e7 −0.0352090
\(361\) −8.33825e8 −0.932824
\(362\) −4.35740e7 −0.0482777
\(363\) 7.29265e8 0.800225
\(364\) −2.27256e7 −0.0246980
\(365\) −3.53435e7 −0.0380439
\(366\) 2.84229e8 0.303030
\(367\) −4.82269e8 −0.509282 −0.254641 0.967036i \(-0.581957\pi\)
−0.254641 + 0.967036i \(0.581957\pi\)
\(368\) −7.42824e8 −0.776996
\(369\) 7.60633e7 0.0788103
\(370\) −4.43883e7 −0.0455578
\(371\) 5.10088e8 0.518604
\(372\) −4.04508e8 −0.407405
\(373\) −1.14493e9 −1.14235 −0.571176 0.820828i \(-0.693512\pi\)
−0.571176 + 0.820828i \(0.693512\pi\)
\(374\) 5.29860e8 0.523733
\(375\) 2.07315e8 0.203012
\(376\) −3.58185e8 −0.347496
\(377\) 1.19778e7 0.0115128
\(378\) −1.49833e8 −0.142688
\(379\) 1.22401e9 1.15491 0.577454 0.816423i \(-0.304046\pi\)
0.577454 + 0.816423i \(0.304046\pi\)
\(380\) 3.59503e7 0.0336093
\(381\) 3.11817e8 0.288843
\(382\) 1.11738e8 0.102560
\(383\) 1.02057e9 0.928210 0.464105 0.885780i \(-0.346376\pi\)
0.464105 + 0.885780i \(0.346376\pi\)
\(384\) −7.34511e8 −0.661971
\(385\) 1.09126e8 0.0974574
\(386\) 3.69126e8 0.326678
\(387\) −8.13891e7 −0.0713802
\(388\) 1.47580e9 1.28268
\(389\) −5.65923e7 −0.0487454 −0.0243727 0.999703i \(-0.507759\pi\)
−0.0243727 + 0.999703i \(0.507759\pi\)
\(390\) −1.86935e6 −0.00159575
\(391\) 1.54254e9 1.30502
\(392\) 4.91928e8 0.412477
\(393\) −1.08901e9 −0.905021
\(394\) −1.17693e8 −0.0969422
\(395\) 1.59449e8 0.130176
\(396\) −7.72754e8 −0.625328
\(397\) −9.60632e8 −0.770532 −0.385266 0.922806i \(-0.625890\pi\)
−0.385266 + 0.922806i \(0.625890\pi\)
\(398\) 4.50243e8 0.357978
\(399\) 1.14830e8 0.0905007
\(400\) −9.70598e8 −0.758279
\(401\) 1.62607e9 1.25931 0.629656 0.776874i \(-0.283196\pi\)
0.629656 + 0.776874i \(0.283196\pi\)
\(402\) −2.27463e8 −0.174630
\(403\) 4.44896e7 0.0338603
\(404\) −9.89748e8 −0.746775
\(405\) 5.87879e7 0.0439740
\(406\) 3.69971e7 0.0274364
\(407\) −2.29347e9 −1.68621
\(408\) 6.95662e8 0.507093
\(409\) −1.13080e9 −0.817247 −0.408623 0.912703i \(-0.633991\pi\)
−0.408623 + 0.912703i \(0.633991\pi\)
\(410\) 9.19359e6 0.00658782
\(411\) 3.66216e8 0.260190
\(412\) −1.54010e9 −1.08495
\(413\) 7.40306e8 0.517114
\(414\) 1.88943e8 0.130867
\(415\) 2.38590e8 0.163864
\(416\) 6.16199e7 0.0419657
\(417\) −2.61869e8 −0.176851
\(418\) −1.56005e8 −0.104477
\(419\) −5.90861e7 −0.0392406 −0.0196203 0.999808i \(-0.506246\pi\)
−0.0196203 + 0.999808i \(0.506246\pi\)
\(420\) 6.87496e7 0.0452791
\(421\) 2.62456e9 1.71423 0.857116 0.515124i \(-0.172254\pi\)
0.857116 + 0.515124i \(0.172254\pi\)
\(422\) 5.00845e8 0.324421
\(423\) −4.73139e8 −0.303947
\(424\) −9.09837e8 −0.579673
\(425\) 2.01554e9 1.27359
\(426\) −5.43718e8 −0.340753
\(427\) 1.14969e9 0.714632
\(428\) 1.03880e9 0.640441
\(429\) −9.65859e7 −0.0590627
\(430\) −9.83730e6 −0.00596673
\(431\) −5.24926e8 −0.315811 −0.157906 0.987454i \(-0.550474\pi\)
−0.157906 + 0.987454i \(0.550474\pi\)
\(432\) −1.38792e9 −0.828266
\(433\) 2.25218e9 1.33320 0.666600 0.745415i \(-0.267749\pi\)
0.666600 + 0.745415i \(0.267749\pi\)
\(434\) 1.37420e8 0.0806931
\(435\) −3.62351e7 −0.0211066
\(436\) 2.05291e9 1.18623
\(437\) −4.54166e8 −0.260333
\(438\) 9.66248e7 0.0549451
\(439\) 3.75255e8 0.211690 0.105845 0.994383i \(-0.466245\pi\)
0.105845 + 0.994383i \(0.466245\pi\)
\(440\) −1.94646e8 −0.108933
\(441\) 6.49804e8 0.360784
\(442\) −3.67143e7 −0.0202235
\(443\) 5.78363e7 0.0316073 0.0158036 0.999875i \(-0.494969\pi\)
0.0158036 + 0.999875i \(0.494969\pi\)
\(444\) −1.44489e9 −0.783420
\(445\) 1.54697e8 0.0832190
\(446\) −1.29211e8 −0.0689650
\(447\) −1.96926e9 −1.04286
\(448\) −5.14503e8 −0.270343
\(449\) 7.38749e8 0.385154 0.192577 0.981282i \(-0.438315\pi\)
0.192577 + 0.981282i \(0.438315\pi\)
\(450\) 2.46878e8 0.127714
\(451\) 4.75016e8 0.243832
\(452\) 8.59915e8 0.437997
\(453\) −5.44396e8 −0.275151
\(454\) 1.54883e8 0.0776797
\(455\) −7.56139e6 −0.00376324
\(456\) −2.04822e8 −0.101158
\(457\) −1.98291e9 −0.971845 −0.485922 0.874002i \(-0.661516\pi\)
−0.485922 + 0.874002i \(0.661516\pi\)
\(458\) −4.00217e8 −0.194656
\(459\) 2.88214e9 1.39114
\(460\) −2.71911e8 −0.130249
\(461\) 1.72573e9 0.820389 0.410195 0.911998i \(-0.365461\pi\)
0.410195 + 0.911998i \(0.365461\pi\)
\(462\) −2.98336e8 −0.140753
\(463\) 3.52996e9 1.65286 0.826429 0.563040i \(-0.190368\pi\)
0.826429 + 0.563040i \(0.190368\pi\)
\(464\) 3.42707e8 0.159261
\(465\) −1.34590e8 −0.0620765
\(466\) −2.51600e8 −0.115175
\(467\) 3.16059e9 1.43602 0.718009 0.696034i \(-0.245054\pi\)
0.718009 + 0.696034i \(0.245054\pi\)
\(468\) 5.35446e7 0.0241465
\(469\) −9.20073e8 −0.411829
\(470\) −5.71871e7 −0.0254071
\(471\) 1.89652e9 0.836344
\(472\) −1.32047e9 −0.578007
\(473\) −5.08276e8 −0.220844
\(474\) −4.35914e8 −0.188008
\(475\) −5.93428e8 −0.254062
\(476\) 1.35025e9 0.573840
\(477\) −1.20183e9 −0.507026
\(478\) −5.83693e8 −0.244448
\(479\) −1.20495e9 −0.500950 −0.250475 0.968123i \(-0.580587\pi\)
−0.250475 + 0.968123i \(0.580587\pi\)
\(480\) −1.86413e8 −0.0769362
\(481\) 1.58916e8 0.0651117
\(482\) 5.84450e7 0.0237730
\(483\) −8.68525e8 −0.350725
\(484\) −2.52476e9 −1.01219
\(485\) 4.91037e8 0.195442
\(486\) 5.93496e8 0.234526
\(487\) −1.76020e9 −0.690576 −0.345288 0.938497i \(-0.612219\pi\)
−0.345288 + 0.938497i \(0.612219\pi\)
\(488\) −2.05068e9 −0.798784
\(489\) −4.45951e8 −0.172467
\(490\) 7.85402e7 0.0301582
\(491\) −3.34075e9 −1.27368 −0.636839 0.770997i \(-0.719758\pi\)
−0.636839 + 0.770997i \(0.719758\pi\)
\(492\) 2.99262e8 0.113285
\(493\) −7.11664e8 −0.267492
\(494\) 1.08097e7 0.00403430
\(495\) −2.57115e8 −0.0952815
\(496\) 1.27293e9 0.468404
\(497\) −2.19930e9 −0.803595
\(498\) −6.52276e8 −0.236662
\(499\) −3.65908e9 −1.31832 −0.659159 0.752004i \(-0.729087\pi\)
−0.659159 + 0.752004i \(0.729087\pi\)
\(500\) −7.17738e8 −0.256786
\(501\) 1.69837e9 0.603392
\(502\) 5.86259e8 0.206836
\(503\) −9.88140e7 −0.0346203 −0.0173101 0.999850i \(-0.505510\pi\)
−0.0173101 + 0.999850i \(0.505510\pi\)
\(504\) 3.44669e8 0.119921
\(505\) −3.29314e8 −0.113786
\(506\) 1.17995e9 0.404889
\(507\) −2.13351e9 −0.727054
\(508\) −1.07953e9 −0.365352
\(509\) −3.29148e9 −1.10632 −0.553158 0.833076i \(-0.686578\pi\)
−0.553158 + 0.833076i \(0.686578\pi\)
\(510\) 1.11068e8 0.0370761
\(511\) 3.90841e8 0.129577
\(512\) 3.02027e9 0.994491
\(513\) −8.48578e8 −0.277511
\(514\) −1.45207e9 −0.471647
\(515\) −5.12431e8 −0.165314
\(516\) −3.20215e8 −0.102605
\(517\) −2.95476e9 −0.940384
\(518\) 4.90861e8 0.155169
\(519\) 3.08288e9 0.967990
\(520\) 1.34871e7 0.00420638
\(521\) 1.48897e9 0.461267 0.230634 0.973041i \(-0.425920\pi\)
0.230634 + 0.973041i \(0.425920\pi\)
\(522\) −8.71700e7 −0.0268238
\(523\) 5.83188e9 1.78259 0.891297 0.453420i \(-0.149796\pi\)
0.891297 + 0.453420i \(0.149796\pi\)
\(524\) 3.77023e9 1.14474
\(525\) −1.13484e9 −0.342277
\(526\) −1.33733e9 −0.400673
\(527\) −2.64337e9 −0.786720
\(528\) −2.76351e9 −0.817039
\(529\) 3.02763e7 0.00889216
\(530\) −1.45263e8 −0.0423827
\(531\) −1.74426e9 −0.505568
\(532\) −3.97550e8 −0.114472
\(533\) −3.29142e7 −0.00941538
\(534\) −4.22923e8 −0.120190
\(535\) 3.45635e8 0.0975843
\(536\) 1.64112e9 0.460324
\(537\) −4.51439e9 −1.25802
\(538\) −6.33623e7 −0.0175426
\(539\) 4.05803e9 1.11623
\(540\) −5.08048e8 −0.138844
\(541\) 3.86680e9 1.04993 0.524966 0.851123i \(-0.324078\pi\)
0.524966 + 0.851123i \(0.324078\pi\)
\(542\) 1.04532e9 0.282001
\(543\) −4.71931e8 −0.126497
\(544\) −3.66117e9 −0.975043
\(545\) 6.83056e8 0.180746
\(546\) 2.06719e7 0.00543508
\(547\) 3.69733e9 0.965901 0.482950 0.875648i \(-0.339565\pi\)
0.482950 + 0.875648i \(0.339565\pi\)
\(548\) −1.26786e9 −0.329109
\(549\) −2.70882e9 −0.698677
\(550\) 1.54176e9 0.395136
\(551\) 2.09533e8 0.0533607
\(552\) 1.54918e9 0.392025
\(553\) −1.76324e9 −0.443378
\(554\) −1.97515e9 −0.493534
\(555\) −4.80751e8 −0.119370
\(556\) 9.06608e8 0.223696
\(557\) 2.06821e9 0.507110 0.253555 0.967321i \(-0.418400\pi\)
0.253555 + 0.967321i \(0.418400\pi\)
\(558\) −3.23780e8 −0.0788915
\(559\) 3.52188e7 0.00852771
\(560\) −2.16346e8 −0.0520584
\(561\) 5.73869e9 1.37228
\(562\) −1.72133e9 −0.409061
\(563\) 3.92675e9 0.927372 0.463686 0.886000i \(-0.346527\pi\)
0.463686 + 0.886000i \(0.346527\pi\)
\(564\) −1.86151e9 −0.436906
\(565\) 2.86115e8 0.0667377
\(566\) 1.49371e9 0.346265
\(567\) −6.50097e8 −0.149774
\(568\) 3.92286e9 0.898223
\(569\) −3.55278e9 −0.808491 −0.404245 0.914651i \(-0.632466\pi\)
−0.404245 + 0.914651i \(0.632466\pi\)
\(570\) −3.27014e7 −0.00739612
\(571\) 1.16477e8 0.0261828 0.0130914 0.999914i \(-0.495833\pi\)
0.0130914 + 0.999914i \(0.495833\pi\)
\(572\) 3.34386e8 0.0747072
\(573\) 1.21019e9 0.268727
\(574\) −1.01666e8 −0.0224380
\(575\) 4.48841e9 0.984590
\(576\) 1.21224e9 0.264307
\(577\) 6.38024e9 1.38268 0.691340 0.722529i \(-0.257021\pi\)
0.691340 + 0.722529i \(0.257021\pi\)
\(578\) 8.89159e8 0.191528
\(579\) 3.99786e9 0.855958
\(580\) 1.25448e8 0.0266973
\(581\) −2.63841e9 −0.558118
\(582\) −1.34243e9 −0.282269
\(583\) −7.50547e9 −1.56869
\(584\) −6.97137e8 −0.144835
\(585\) 1.78156e7 0.00367922
\(586\) −8.22803e8 −0.168910
\(587\) −3.91131e9 −0.798157 −0.399079 0.916917i \(-0.630670\pi\)
−0.399079 + 0.916917i \(0.630670\pi\)
\(588\) 2.55657e9 0.518606
\(589\) 7.78278e8 0.156939
\(590\) −2.10824e8 −0.0422609
\(591\) −1.27468e9 −0.254007
\(592\) 4.54689e9 0.900716
\(593\) −9.01946e9 −1.77619 −0.888094 0.459661i \(-0.847971\pi\)
−0.888094 + 0.459661i \(0.847971\pi\)
\(594\) 2.20465e9 0.431606
\(595\) 4.49263e8 0.0874362
\(596\) 6.81771e9 1.31910
\(597\) 4.87639e9 0.937969
\(598\) −8.17593e7 −0.0156345
\(599\) 3.06365e9 0.582432 0.291216 0.956657i \(-0.405940\pi\)
0.291216 + 0.956657i \(0.405940\pi\)
\(600\) 2.02420e9 0.382582
\(601\) 4.13537e9 0.777059 0.388529 0.921436i \(-0.372983\pi\)
0.388529 + 0.921436i \(0.372983\pi\)
\(602\) 1.08784e8 0.0203225
\(603\) 2.16781e9 0.402635
\(604\) 1.88473e9 0.348033
\(605\) −8.40052e8 −0.154228
\(606\) 9.00303e8 0.164337
\(607\) −3.50688e9 −0.636445 −0.318222 0.948016i \(-0.603086\pi\)
−0.318222 + 0.948016i \(0.603086\pi\)
\(608\) 1.07795e9 0.194507
\(609\) 4.00700e8 0.0718885
\(610\) −3.27408e8 −0.0584030
\(611\) 2.04737e8 0.0363122
\(612\) −3.18137e9 −0.561028
\(613\) −2.51080e9 −0.440251 −0.220125 0.975472i \(-0.570647\pi\)
−0.220125 + 0.975472i \(0.570647\pi\)
\(614\) 1.99135e9 0.347183
\(615\) 9.95719e7 0.0172613
\(616\) 2.15246e9 0.371025
\(617\) −8.91772e9 −1.52847 −0.764233 0.644941i \(-0.776882\pi\)
−0.764233 + 0.644941i \(0.776882\pi\)
\(618\) 1.40092e9 0.238756
\(619\) −1.01712e10 −1.72368 −0.861838 0.507183i \(-0.830687\pi\)
−0.861838 + 0.507183i \(0.830687\pi\)
\(620\) 4.65959e8 0.0785193
\(621\) 6.41825e9 1.07546
\(622\) 2.87012e9 0.478226
\(623\) −1.71069e9 −0.283442
\(624\) 1.91485e8 0.0315493
\(625\) 5.74411e9 0.941115
\(626\) −1.50174e9 −0.244672
\(627\) −1.68962e9 −0.273750
\(628\) −6.56588e9 −1.05787
\(629\) −9.44203e9 −1.51282
\(630\) 5.50292e7 0.00876801
\(631\) −8.57149e9 −1.35817 −0.679084 0.734061i \(-0.737623\pi\)
−0.679084 + 0.734061i \(0.737623\pi\)
\(632\) 3.14507e9 0.495588
\(633\) 5.42444e9 0.850045
\(634\) 8.53276e8 0.132977
\(635\) −3.59187e8 −0.0556688
\(636\) −4.72847e9 −0.728820
\(637\) −2.81184e8 −0.0431025
\(638\) −5.44378e8 −0.0829905
\(639\) 5.18184e9 0.785654
\(640\) 8.46094e8 0.127582
\(641\) 3.31058e9 0.496479 0.248240 0.968699i \(-0.420148\pi\)
0.248240 + 0.968699i \(0.420148\pi\)
\(642\) −9.44924e8 −0.140937
\(643\) −3.42873e9 −0.508622 −0.254311 0.967122i \(-0.581849\pi\)
−0.254311 + 0.967122i \(0.581849\pi\)
\(644\) 3.00689e9 0.443626
\(645\) −1.06544e8 −0.0156340
\(646\) −6.42260e8 −0.0937340
\(647\) 2.83989e8 0.0412228 0.0206114 0.999788i \(-0.493439\pi\)
0.0206114 + 0.999788i \(0.493439\pi\)
\(648\) 1.15957e9 0.167411
\(649\) −1.08929e10 −1.56418
\(650\) −1.06829e8 −0.0152579
\(651\) 1.48834e9 0.211431
\(652\) 1.54391e9 0.218150
\(653\) 3.48386e8 0.0489626 0.0244813 0.999700i \(-0.492207\pi\)
0.0244813 + 0.999700i \(0.492207\pi\)
\(654\) −1.86739e9 −0.261043
\(655\) 1.25445e9 0.174425
\(656\) −9.41739e8 −0.130247
\(657\) −9.20872e8 −0.126684
\(658\) 6.32395e8 0.0865361
\(659\) −9.43143e9 −1.28375 −0.641873 0.766811i \(-0.721842\pi\)
−0.641873 + 0.766811i \(0.721842\pi\)
\(660\) −1.01159e9 −0.136962
\(661\) 1.05208e10 1.41692 0.708458 0.705753i \(-0.249391\pi\)
0.708458 + 0.705753i \(0.249391\pi\)
\(662\) −1.50924e9 −0.202188
\(663\) −3.97637e8 −0.0529895
\(664\) 4.70610e9 0.623840
\(665\) −1.32275e8 −0.0174422
\(666\) −1.15653e9 −0.151704
\(667\) −1.58481e9 −0.206793
\(668\) −5.87986e9 −0.763219
\(669\) −1.39943e9 −0.180701
\(670\) 2.62018e8 0.0336565
\(671\) −1.69166e10 −2.16164
\(672\) 2.06141e9 0.262043
\(673\) −2.80231e8 −0.0354376 −0.0177188 0.999843i \(-0.505640\pi\)
−0.0177188 + 0.999843i \(0.505640\pi\)
\(674\) −1.30544e9 −0.164228
\(675\) 8.38630e9 1.04956
\(676\) 7.38634e9 0.919636
\(677\) −3.28856e9 −0.407329 −0.203664 0.979041i \(-0.565285\pi\)
−0.203664 + 0.979041i \(0.565285\pi\)
\(678\) −7.82203e8 −0.0963864
\(679\) −5.43006e9 −0.665672
\(680\) −8.01344e8 −0.0977322
\(681\) 1.67747e9 0.203535
\(682\) −2.02201e9 −0.244083
\(683\) −2.54212e9 −0.305298 −0.152649 0.988280i \(-0.548780\pi\)
−0.152649 + 0.988280i \(0.548780\pi\)
\(684\) 9.36680e8 0.111917
\(685\) −4.21849e8 −0.0501465
\(686\) −1.99532e9 −0.235982
\(687\) −4.33459e9 −0.510034
\(688\) 1.00768e9 0.117967
\(689\) 5.20058e8 0.0605738
\(690\) 2.47338e8 0.0286629
\(691\) 1.50328e10 1.73328 0.866638 0.498937i \(-0.166276\pi\)
0.866638 + 0.498937i \(0.166276\pi\)
\(692\) −1.06731e10 −1.22439
\(693\) 2.84326e9 0.324527
\(694\) −1.09066e8 −0.0123860
\(695\) 3.01651e8 0.0340846
\(696\) −7.14724e8 −0.0803537
\(697\) 1.95561e9 0.218760
\(698\) −4.08410e9 −0.454571
\(699\) −2.72497e9 −0.301781
\(700\) 3.92890e9 0.432940
\(701\) −3.96542e9 −0.434787 −0.217393 0.976084i \(-0.569755\pi\)
−0.217393 + 0.976084i \(0.569755\pi\)
\(702\) −1.52762e8 −0.0166661
\(703\) 2.77999e9 0.301786
\(704\) 7.57043e9 0.817742
\(705\) −6.19370e8 −0.0665715
\(706\) 7.71335e8 0.0824947
\(707\) 3.64167e9 0.387554
\(708\) −6.86257e9 −0.726725
\(709\) 8.82480e9 0.929914 0.464957 0.885333i \(-0.346070\pi\)
0.464957 + 0.885333i \(0.346070\pi\)
\(710\) 6.26317e8 0.0656734
\(711\) 4.15443e9 0.433479
\(712\) 3.05134e9 0.316819
\(713\) −5.88654e9 −0.608200
\(714\) −1.22823e9 −0.126280
\(715\) 1.11259e8 0.0113832
\(716\) 1.56291e10 1.59125
\(717\) −6.32174e9 −0.640501
\(718\) 5.96659e8 0.0601576
\(719\) −1.63954e10 −1.64501 −0.822507 0.568755i \(-0.807425\pi\)
−0.822507 + 0.568755i \(0.807425\pi\)
\(720\) 5.09740e8 0.0508961
\(721\) 5.66664e9 0.563056
\(722\) −2.62587e9 −0.259653
\(723\) 6.32994e8 0.0622896
\(724\) 1.63386e9 0.160003
\(725\) −2.07076e9 −0.201812
\(726\) 2.29660e9 0.222744
\(727\) −4.74182e9 −0.457693 −0.228847 0.973462i \(-0.573495\pi\)
−0.228847 + 0.973462i \(0.573495\pi\)
\(728\) −1.49145e8 −0.0143268
\(729\) 9.70030e9 0.927339
\(730\) −1.11304e8 −0.0105896
\(731\) −2.09254e9 −0.198135
\(732\) −1.06575e10 −1.00431
\(733\) −1.56780e10 −1.47037 −0.735184 0.677868i \(-0.762904\pi\)
−0.735184 + 0.677868i \(0.762904\pi\)
\(734\) −1.51876e9 −0.141760
\(735\) 8.50637e8 0.0790202
\(736\) −8.15309e9 −0.753789
\(737\) 1.35380e10 1.24572
\(738\) 2.39538e8 0.0219370
\(739\) 2.09854e10 1.91277 0.956383 0.292116i \(-0.0943594\pi\)
0.956383 + 0.292116i \(0.0943594\pi\)
\(740\) 1.66439e9 0.150989
\(741\) 1.17075e8 0.0105706
\(742\) 1.60636e9 0.144354
\(743\) 1.57915e10 1.41242 0.706209 0.708004i \(-0.250404\pi\)
0.706209 + 0.708004i \(0.250404\pi\)
\(744\) −2.65473e9 −0.236328
\(745\) 2.26842e9 0.200991
\(746\) −3.60562e9 −0.317976
\(747\) 6.21645e9 0.545657
\(748\) −1.98677e10 −1.73577
\(749\) −3.82215e9 −0.332370
\(750\) 6.52875e8 0.0565087
\(751\) 1.46937e10 1.26588 0.632939 0.774201i \(-0.281848\pi\)
0.632939 + 0.774201i \(0.281848\pi\)
\(752\) 5.85792e9 0.502321
\(753\) 6.34953e9 0.541950
\(754\) 3.77203e7 0.00320461
\(755\) 6.27098e8 0.0530299
\(756\) 5.61816e9 0.472899
\(757\) 8.22699e9 0.689295 0.344648 0.938732i \(-0.387998\pi\)
0.344648 + 0.938732i \(0.387998\pi\)
\(758\) 3.85464e9 0.321471
\(759\) 1.27795e10 1.06089
\(760\) 2.35937e8 0.0194961
\(761\) 2.23002e10 1.83427 0.917133 0.398581i \(-0.130497\pi\)
0.917133 + 0.398581i \(0.130497\pi\)
\(762\) 9.81971e8 0.0804000
\(763\) −7.55346e9 −0.615616
\(764\) −4.18974e9 −0.339907
\(765\) −1.05852e9 −0.0854840
\(766\) 3.21397e9 0.258369
\(767\) 7.54777e8 0.0603997
\(768\) 2.85684e9 0.227574
\(769\) 1.62540e10 1.28890 0.644448 0.764648i \(-0.277087\pi\)
0.644448 + 0.764648i \(0.277087\pi\)
\(770\) 3.43658e8 0.0271274
\(771\) −1.57268e10 −1.23581
\(772\) −1.38408e10 −1.08268
\(773\) −1.29767e10 −1.01050 −0.505248 0.862974i \(-0.668599\pi\)
−0.505248 + 0.862974i \(0.668599\pi\)
\(774\) −2.56310e8 −0.0198688
\(775\) −7.69154e9 −0.593550
\(776\) 9.68552e9 0.744058
\(777\) 5.31631e9 0.406571
\(778\) −1.78220e8 −0.0135684
\(779\) −5.75783e8 −0.0436393
\(780\) 7.00934e7 0.00528866
\(781\) 3.23607e10 2.43074
\(782\) 4.85776e9 0.363256
\(783\) −2.96111e9 −0.220439
\(784\) −8.04521e9 −0.596254
\(785\) −2.18463e9 −0.161189
\(786\) −3.42951e9 −0.251914
\(787\) −1.18344e10 −0.865436 −0.432718 0.901529i \(-0.642445\pi\)
−0.432718 + 0.901529i \(0.642445\pi\)
\(788\) 4.41302e9 0.321288
\(789\) −1.44841e10 −1.04984
\(790\) 5.02136e8 0.0362348
\(791\) −3.16396e9 −0.227307
\(792\) −5.07148e9 −0.362741
\(793\) 1.17216e9 0.0834701
\(794\) −3.02522e9 −0.214479
\(795\) −1.57328e9 −0.111051
\(796\) −1.68824e10 −1.18642
\(797\) −5.74935e8 −0.0402267 −0.0201134 0.999798i \(-0.506403\pi\)
−0.0201134 + 0.999798i \(0.506403\pi\)
\(798\) 3.61623e8 0.0251910
\(799\) −1.21645e10 −0.843687
\(800\) −1.06531e10 −0.735632
\(801\) 4.03062e9 0.277114
\(802\) 5.12080e9 0.350532
\(803\) −5.75086e9 −0.391948
\(804\) 8.52899e9 0.578764
\(805\) 1.00047e9 0.0675954
\(806\) 1.40106e8 0.00942508
\(807\) −6.86251e8 −0.0459648
\(808\) −6.49559e9 −0.433191
\(809\) 1.85697e10 1.23306 0.616532 0.787330i \(-0.288537\pi\)
0.616532 + 0.787330i \(0.288537\pi\)
\(810\) 1.85134e8 0.0122402
\(811\) −1.35102e10 −0.889384 −0.444692 0.895683i \(-0.646687\pi\)
−0.444692 + 0.895683i \(0.646687\pi\)
\(812\) −1.38725e9 −0.0909303
\(813\) 1.13214e10 0.738895
\(814\) −7.22256e9 −0.469360
\(815\) 5.13697e8 0.0332396
\(816\) −1.13772e10 −0.733026
\(817\) 6.16098e8 0.0395251
\(818\) −3.56110e9 −0.227482
\(819\) −1.97011e8 −0.0125313
\(820\) −3.44724e8 −0.0218335
\(821\) −2.18113e10 −1.37557 −0.687783 0.725917i \(-0.741416\pi\)
−0.687783 + 0.725917i \(0.741416\pi\)
\(822\) 1.15328e9 0.0724244
\(823\) 2.25550e10 1.41040 0.705202 0.709006i \(-0.250856\pi\)
0.705202 + 0.709006i \(0.250856\pi\)
\(824\) −1.01075e10 −0.629359
\(825\) 1.66982e10 1.03533
\(826\) 2.33137e9 0.143940
\(827\) −2.31862e10 −1.42548 −0.712739 0.701430i \(-0.752545\pi\)
−0.712739 + 0.701430i \(0.752545\pi\)
\(828\) −7.08462e9 −0.433721
\(829\) 2.89637e10 1.76568 0.882842 0.469671i \(-0.155627\pi\)
0.882842 + 0.469671i \(0.155627\pi\)
\(830\) 7.51366e8 0.0456119
\(831\) −2.13921e10 −1.29315
\(832\) −5.24559e8 −0.0315764
\(833\) 1.67066e10 1.00145
\(834\) −8.24677e8 −0.0492269
\(835\) −1.95638e9 −0.116292
\(836\) 5.84958e9 0.346260
\(837\) −1.09986e10 −0.648333
\(838\) −1.86073e8 −0.0109227
\(839\) −9.53603e9 −0.557444 −0.278722 0.960372i \(-0.589911\pi\)
−0.278722 + 0.960372i \(0.589911\pi\)
\(840\) 4.51195e8 0.0262655
\(841\) −1.65187e10 −0.957613
\(842\) 8.26525e9 0.477160
\(843\) −1.86430e10 −1.07182
\(844\) −1.87798e10 −1.07521
\(845\) 2.45762e9 0.140125
\(846\) −1.49000e9 −0.0846041
\(847\) 9.28957e9 0.525296
\(848\) 1.48799e10 0.837942
\(849\) 1.61778e10 0.907281
\(850\) 6.34731e9 0.354506
\(851\) −2.10265e10 −1.16954
\(852\) 2.03873e10 1.12933
\(853\) 2.23351e10 1.23216 0.616078 0.787685i \(-0.288721\pi\)
0.616078 + 0.787685i \(0.288721\pi\)
\(854\) 3.62059e9 0.198919
\(855\) 3.11657e8 0.0170528
\(856\) 6.81752e9 0.371508
\(857\) −2.98724e10 −1.62120 −0.810602 0.585598i \(-0.800860\pi\)
−0.810602 + 0.585598i \(0.800860\pi\)
\(858\) −3.04168e8 −0.0164402
\(859\) −1.94294e10 −1.04588 −0.522941 0.852369i \(-0.675165\pi\)
−0.522941 + 0.852369i \(0.675165\pi\)
\(860\) 3.68861e8 0.0197751
\(861\) −1.10110e9 −0.0587917
\(862\) −1.65309e9 −0.0879067
\(863\) 2.20900e10 1.16992 0.584962 0.811060i \(-0.301109\pi\)
0.584962 + 0.811060i \(0.301109\pi\)
\(864\) −1.52335e10 −0.803529
\(865\) −3.55122e9 −0.186561
\(866\) 7.09255e9 0.371099
\(867\) 9.63011e9 0.501839
\(868\) −5.15273e9 −0.267435
\(869\) 2.59445e10 1.34114
\(870\) −1.14111e8 −0.00587505
\(871\) −9.38057e8 −0.0481023
\(872\) 1.34730e10 0.688108
\(873\) 1.27939e10 0.650810
\(874\) −1.43025e9 −0.0724642
\(875\) 2.64083e9 0.133264
\(876\) −3.62306e9 −0.182100
\(877\) −3.02430e10 −1.51400 −0.757001 0.653413i \(-0.773336\pi\)
−0.757001 + 0.653413i \(0.773336\pi\)
\(878\) 1.18175e9 0.0589243
\(879\) −8.91144e9 −0.442575
\(880\) 3.18333e9 0.157468
\(881\) −2.28024e10 −1.12348 −0.561739 0.827314i \(-0.689868\pi\)
−0.561739 + 0.827314i \(0.689868\pi\)
\(882\) 2.04636e9 0.100425
\(883\) −3.29419e10 −1.61023 −0.805113 0.593122i \(-0.797895\pi\)
−0.805113 + 0.593122i \(0.797895\pi\)
\(884\) 1.37664e9 0.0670253
\(885\) −2.28335e9 −0.110731
\(886\) 1.82138e8 0.00879795
\(887\) 2.72521e9 0.131119 0.0655597 0.997849i \(-0.479117\pi\)
0.0655597 + 0.997849i \(0.479117\pi\)
\(888\) −9.48263e9 −0.454447
\(889\) 3.97201e9 0.189607
\(890\) 4.87171e8 0.0231642
\(891\) 9.56557e9 0.453042
\(892\) 4.84493e9 0.228565
\(893\) 3.58156e9 0.168303
\(894\) −6.20158e9 −0.290283
\(895\) 5.20020e9 0.242459
\(896\) −9.35640e9 −0.434541
\(897\) −8.85502e8 −0.0409653
\(898\) 2.32646e9 0.107208
\(899\) 2.71580e9 0.124663
\(900\) −9.25699e9 −0.423274
\(901\) −3.08995e10 −1.40739
\(902\) 1.49592e9 0.0678711
\(903\) 1.17820e9 0.0532489
\(904\) 5.64351e9 0.254074
\(905\) 5.43625e8 0.0243798
\(906\) −1.71441e9 −0.0765888
\(907\) 6.55130e9 0.291542 0.145771 0.989318i \(-0.453434\pi\)
0.145771 + 0.989318i \(0.453434\pi\)
\(908\) −5.80751e9 −0.257448
\(909\) −8.58025e9 −0.378901
\(910\) −2.38123e7 −0.00104750
\(911\) −2.40723e10 −1.05488 −0.527440 0.849592i \(-0.676848\pi\)
−0.527440 + 0.849592i \(0.676848\pi\)
\(912\) 3.34974e9 0.146228
\(913\) 3.88218e10 1.68821
\(914\) −6.24457e9 −0.270515
\(915\) −3.54602e9 −0.153027
\(916\) 1.50066e10 0.645132
\(917\) −1.38721e10 −0.594087
\(918\) 9.07640e9 0.387226
\(919\) 4.57146e10 1.94290 0.971450 0.237244i \(-0.0762441\pi\)
0.971450 + 0.237244i \(0.0762441\pi\)
\(920\) −1.78452e9 −0.0755551
\(921\) 2.15675e10 0.909685
\(922\) 5.43466e9 0.228357
\(923\) −2.24229e9 −0.0938612
\(924\) 1.11865e10 0.466488
\(925\) −2.74739e10 −1.14137
\(926\) 1.11165e10 0.460076
\(927\) −1.33513e10 −0.550485
\(928\) 3.76149e9 0.154505
\(929\) −3.25677e10 −1.33270 −0.666349 0.745640i \(-0.732144\pi\)
−0.666349 + 0.745640i \(0.732144\pi\)
\(930\) −4.23849e8 −0.0172791
\(931\) −4.91888e9 −0.199775
\(932\) 9.43402e9 0.381717
\(933\) 3.10850e10 1.25304
\(934\) 9.95332e9 0.399718
\(935\) −6.61048e9 −0.264480
\(936\) 3.51406e8 0.0140070
\(937\) −3.92369e10 −1.55814 −0.779069 0.626938i \(-0.784308\pi\)
−0.779069 + 0.626938i \(0.784308\pi\)
\(938\) −2.89748e9 −0.114633
\(939\) −1.62647e10 −0.641086
\(940\) 2.14430e9 0.0842049
\(941\) 3.07063e10 1.20133 0.600667 0.799499i \(-0.294902\pi\)
0.600667 + 0.799499i \(0.294902\pi\)
\(942\) 5.97251e9 0.232798
\(943\) 4.35496e9 0.169119
\(944\) 2.15956e10 0.835534
\(945\) 1.86930e9 0.0720558
\(946\) −1.60066e9 −0.0614723
\(947\) 4.74031e10 1.81377 0.906883 0.421382i \(-0.138455\pi\)
0.906883 + 0.421382i \(0.138455\pi\)
\(948\) 1.63451e10 0.623100
\(949\) 3.98480e8 0.0151348
\(950\) −1.86882e9 −0.0707187
\(951\) 9.24148e9 0.348425
\(952\) 8.86153e9 0.332874
\(953\) 4.93561e10 1.84721 0.923604 0.383347i \(-0.125229\pi\)
0.923604 + 0.383347i \(0.125229\pi\)
\(954\) −3.78480e9 −0.141132
\(955\) −1.39403e9 −0.0517918
\(956\) 2.18863e10 0.810157
\(957\) −5.89593e9 −0.217451
\(958\) −3.79462e9 −0.139440
\(959\) 4.66495e9 0.170798
\(960\) 1.58690e9 0.0578895
\(961\) −1.74252e10 −0.633353
\(962\) 5.00456e8 0.0181239
\(963\) 9.00550e9 0.324949
\(964\) −2.19147e9 −0.0787889
\(965\) −4.60519e9 −0.164969
\(966\) −2.73515e9 −0.0976251
\(967\) −6.84804e9 −0.243542 −0.121771 0.992558i \(-0.538857\pi\)
−0.121771 + 0.992558i \(0.538857\pi\)
\(968\) −1.65697e10 −0.587152
\(969\) −6.95606e9 −0.245601
\(970\) 1.54637e9 0.0544017
\(971\) 4.01610e8 0.0140779 0.00703894 0.999975i \(-0.497759\pi\)
0.00703894 + 0.999975i \(0.497759\pi\)
\(972\) −2.22538e10 −0.777271
\(973\) −3.33576e9 −0.116091
\(974\) −5.54322e9 −0.192223
\(975\) −1.15703e9 −0.0399785
\(976\) 3.35378e10 1.15468
\(977\) 3.71330e10 1.27388 0.636940 0.770913i \(-0.280200\pi\)
0.636940 + 0.770913i \(0.280200\pi\)
\(978\) −1.40438e9 −0.0480065
\(979\) 2.51713e10 0.857365
\(980\) −2.94496e9 −0.0999511
\(981\) 1.77969e10 0.601872
\(982\) −1.05207e10 −0.354530
\(983\) 1.12625e10 0.378179 0.189089 0.981960i \(-0.439446\pi\)
0.189089 + 0.981960i \(0.439446\pi\)
\(984\) 1.96402e9 0.0657147
\(985\) 1.46832e9 0.0489548
\(986\) −2.24116e9 −0.0744568
\(987\) 6.84920e9 0.226741
\(988\) −4.05321e8 −0.0133706
\(989\) −4.65988e9 −0.153175
\(990\) −8.09703e8 −0.0265218
\(991\) 4.74895e10 1.55003 0.775015 0.631943i \(-0.217743\pi\)
0.775015 + 0.631943i \(0.217743\pi\)
\(992\) 1.39715e10 0.454414
\(993\) −1.63460e10 −0.529771
\(994\) −6.92602e9 −0.223682
\(995\) −5.61719e9 −0.180775
\(996\) 2.44578e10 0.784351
\(997\) 1.12122e10 0.358310 0.179155 0.983821i \(-0.442664\pi\)
0.179155 + 0.983821i \(0.442664\pi\)
\(998\) −1.15231e10 −0.366956
\(999\) −3.92866e10 −1.24671
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.1.7 11
3.2 odd 2 387.8.a.b.1.5 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.a.1.7 11 1.1 even 1 trivial
387.8.a.b.1.5 11 3.2 odd 2