Properties

Label 43.8.a.a.1.4
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 1.4
Root \(11.4671\) of defining polynomial
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q-13.4671 q^{2} -87.0645 q^{3} +53.3630 q^{4} +131.138 q^{5} +1172.51 q^{6} -712.280 q^{7} +1005.14 q^{8} +5393.23 q^{9} +O(q^{10})\) \(q-13.4671 q^{2} -87.0645 q^{3} +53.3630 q^{4} +131.138 q^{5} +1172.51 q^{6} -712.280 q^{7} +1005.14 q^{8} +5393.23 q^{9} -1766.05 q^{10} +6103.13 q^{11} -4646.02 q^{12} -7927.36 q^{13} +9592.35 q^{14} -11417.5 q^{15} -20366.9 q^{16} -16334.5 q^{17} -72631.2 q^{18} +39583.9 q^{19} +6997.93 q^{20} +62014.3 q^{21} -82191.5 q^{22} +29296.2 q^{23} -87512.4 q^{24} -60927.8 q^{25} +106759. q^{26} -279148. q^{27} -38009.4 q^{28} +71148.2 q^{29} +153760. q^{30} +107693. q^{31} +145624. q^{32} -531366. q^{33} +219979. q^{34} -93407.1 q^{35} +287799. q^{36} +386329. q^{37} -533080. q^{38} +690192. q^{39} +131813. q^{40} -768609. q^{41} -835153. q^{42} +79507.0 q^{43} +325681. q^{44} +707258. q^{45} -394535. q^{46} +929733. q^{47} +1.77323e6 q^{48} -316201. q^{49} +820521. q^{50} +1.42216e6 q^{51} -423028. q^{52} -1.50722e6 q^{53} +3.75932e6 q^{54} +800354. q^{55} -715944. q^{56} -3.44635e6 q^{57} -958161. q^{58} -1.77396e6 q^{59} -609271. q^{60} -1.22781e6 q^{61} -1.45031e6 q^{62} -3.84148e6 q^{63} +645821. q^{64} -1.03958e6 q^{65} +7.15596e6 q^{66} -831926. q^{67} -871658. q^{68} -2.55066e6 q^{69} +1.25792e6 q^{70} -21047.1 q^{71} +5.42097e6 q^{72} -631213. q^{73} -5.20274e6 q^{74} +5.30464e6 q^{75} +2.11231e6 q^{76} -4.34713e6 q^{77} -9.29489e6 q^{78} -3.76509e6 q^{79} -2.67087e6 q^{80} +1.25089e7 q^{81} +1.03509e7 q^{82} +1.71861e6 q^{83} +3.30927e6 q^{84} -2.14208e6 q^{85} -1.07073e6 q^{86} -6.19448e6 q^{87} +6.13453e6 q^{88} -6.08728e6 q^{89} -9.52472e6 q^{90} +5.64650e6 q^{91} +1.56333e6 q^{92} -9.37623e6 q^{93} -1.25208e7 q^{94} +5.19096e6 q^{95} -1.26787e7 q^{96} -7.22067e6 q^{97} +4.25831e6 q^{98} +3.29156e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} + O(q^{10}) \) \( 11q - 24q^{2} - 68q^{3} + 602q^{4} - 752q^{5} - 681q^{6} - 12q^{7} - 3810q^{8} + 2721q^{9} - 1333q^{10} + 1333q^{11} + 5089q^{12} - 17967q^{13} - 22352q^{14} - 49504q^{15} - 34406q^{16} - 63095q^{17} - 165931q^{18} - 54524q^{19} - 280995q^{20} - 139788q^{21} - 289358q^{22} - 138139q^{23} - 429583q^{24} + 3455q^{25} - 132946q^{26} - 240356q^{27} - 12704q^{28} - 308658q^{29} + 421284q^{30} - 209523q^{31} - 644934q^{32} + 96814q^{33} + 762435q^{34} - 578892q^{35} + 426161q^{36} - 298472q^{37} - 369707q^{38} + 292298q^{39} + 2633173q^{40} - 1346735q^{41} + 1173266q^{42} + 874577q^{43} + 3134292q^{44} + 1893784q^{45} + 3588111q^{46} + 499284q^{47} + 5647533q^{48} + 2544563q^{49} + 3049745q^{50} + 1258424q^{51} + 983088q^{52} - 2210495q^{53} + 6789698q^{54} - 1855072q^{55} - 469976q^{56} - 1238444q^{57} + 4397067q^{58} - 5824216q^{59} - 2889372q^{60} - 4453034q^{61} + 1002789q^{62} - 6240564q^{63} + 4757538q^{64} - 2162872q^{65} - 258940q^{66} - 6859513q^{67} - 9397005q^{68} - 10040030q^{69} + 845078q^{70} - 10726554q^{71} + 1199517q^{72} - 4456898q^{73} + 1046637q^{74} - 3349114q^{75} + 5861267q^{76} - 17019816q^{77} + 1999122q^{78} - 15541320q^{79} - 15680911q^{80} - 12976697q^{81} + 20233655q^{82} - 11146767q^{83} + 12348278q^{84} - 12471976q^{85} - 1908168q^{86} - 18648900q^{87} - 24463544q^{88} - 13531356q^{89} + 20858990q^{90} - 19746448q^{91} - 26023161q^{92} - 21903110q^{93} + 20288857q^{94} - 12291624q^{95} - 13954503q^{96} - 10999901q^{97} + 29909168q^{98} + 29396057q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −13.4671 −1.19034 −0.595168 0.803602i \(-0.702914\pi\)
−0.595168 + 0.803602i \(0.702914\pi\)
\(3\) −87.0645 −1.86173 −0.930865 0.365363i \(-0.880945\pi\)
−0.930865 + 0.365363i \(0.880945\pi\)
\(4\) 53.3630 0.416898
\(5\) 131.138 0.469174 0.234587 0.972095i \(-0.424626\pi\)
0.234587 + 0.972095i \(0.424626\pi\)
\(6\) 1172.51 2.21608
\(7\) −712.280 −0.784887 −0.392444 0.919776i \(-0.628370\pi\)
−0.392444 + 0.919776i \(0.628370\pi\)
\(8\) 1005.14 0.694086
\(9\) 5393.23 2.46604
\(10\) −1766.05 −0.558475
\(11\) 6103.13 1.38254 0.691271 0.722596i \(-0.257051\pi\)
0.691271 + 0.722596i \(0.257051\pi\)
\(12\) −4646.02 −0.776152
\(13\) −7927.36 −1.00075 −0.500377 0.865808i \(-0.666805\pi\)
−0.500377 + 0.865808i \(0.666805\pi\)
\(14\) 9592.35 0.934279
\(15\) −11417.5 −0.873476
\(16\) −20366.9 −1.24309
\(17\) −16334.5 −0.806371 −0.403185 0.915118i \(-0.632097\pi\)
−0.403185 + 0.915118i \(0.632097\pi\)
\(18\) −72631.2 −2.93541
\(19\) 39583.9 1.32398 0.661989 0.749514i \(-0.269713\pi\)
0.661989 + 0.749514i \(0.269713\pi\)
\(20\) 6997.93 0.195598
\(21\) 62014.3 1.46125
\(22\) −82191.5 −1.64569
\(23\) 29296.2 0.502069 0.251035 0.967978i \(-0.419229\pi\)
0.251035 + 0.967978i \(0.419229\pi\)
\(24\) −87512.4 −1.29220
\(25\) −60927.8 −0.779875
\(26\) 106759. 1.19123
\(27\) −279148. −2.72937
\(28\) −38009.4 −0.327218
\(29\) 71148.2 0.541715 0.270858 0.962619i \(-0.412693\pi\)
0.270858 + 0.962619i \(0.412693\pi\)
\(30\) 153760. 1.03973
\(31\) 107693. 0.649264 0.324632 0.945840i \(-0.394760\pi\)
0.324632 + 0.945840i \(0.394760\pi\)
\(32\) 145624. 0.785613
\(33\) −531366. −2.57392
\(34\) 219979. 0.959852
\(35\) −93407.1 −0.368249
\(36\) 287799. 1.02809
\(37\) 386329. 1.25387 0.626933 0.779073i \(-0.284310\pi\)
0.626933 + 0.779073i \(0.284310\pi\)
\(38\) −533080. −1.57598
\(39\) 690192. 1.86313
\(40\) 131813. 0.325648
\(41\) −768609. −1.74165 −0.870827 0.491590i \(-0.836416\pi\)
−0.870827 + 0.491590i \(0.836416\pi\)
\(42\) −835153. −1.73938
\(43\) 79507.0 0.152499
\(44\) 325681. 0.576380
\(45\) 707258. 1.15700
\(46\) −394535. −0.597631
\(47\) 929733. 1.30622 0.653109 0.757264i \(-0.273464\pi\)
0.653109 + 0.757264i \(0.273464\pi\)
\(48\) 1.77323e6 2.31431
\(49\) −316201. −0.383952
\(50\) 820521. 0.928313
\(51\) 1.42216e6 1.50124
\(52\) −423028. −0.417213
\(53\) −1.50722e6 −1.39063 −0.695314 0.718707i \(-0.744734\pi\)
−0.695314 + 0.718707i \(0.744734\pi\)
\(54\) 3.75932e6 3.24886
\(55\) 800354. 0.648653
\(56\) −715944. −0.544780
\(57\) −3.44635e6 −2.46489
\(58\) −958161. −0.644823
\(59\) −1.77396e6 −1.12450 −0.562252 0.826966i \(-0.690065\pi\)
−0.562252 + 0.826966i \(0.690065\pi\)
\(60\) −609271. −0.364151
\(61\) −1.22781e6 −0.692591 −0.346295 0.938125i \(-0.612561\pi\)
−0.346295 + 0.938125i \(0.612561\pi\)
\(62\) −1.45031e6 −0.772842
\(63\) −3.84148e6 −1.93556
\(64\) 645821. 0.307952
\(65\) −1.03958e6 −0.469528
\(66\) 7.15596e6 3.06383
\(67\) −831926. −0.337927 −0.168963 0.985622i \(-0.554042\pi\)
−0.168963 + 0.985622i \(0.554042\pi\)
\(68\) −871658. −0.336175
\(69\) −2.55066e6 −0.934717
\(70\) 1.25792e6 0.438340
\(71\) −21047.1 −0.00697891 −0.00348945 0.999994i \(-0.501111\pi\)
−0.00348945 + 0.999994i \(0.501111\pi\)
\(72\) 5.42097e6 1.71164
\(73\) −631213. −0.189909 −0.0949545 0.995482i \(-0.530271\pi\)
−0.0949545 + 0.995482i \(0.530271\pi\)
\(74\) −5.20274e6 −1.49252
\(75\) 5.30464e6 1.45192
\(76\) 2.11231e6 0.551964
\(77\) −4.34713e6 −1.08514
\(78\) −9.29489e6 −2.21775
\(79\) −3.76509e6 −0.859173 −0.429587 0.903026i \(-0.641341\pi\)
−0.429587 + 0.903026i \(0.641341\pi\)
\(80\) −2.67087e6 −0.583228
\(81\) 1.25089e7 2.61531
\(82\) 1.03509e7 2.07315
\(83\) 1.71861e6 0.329916 0.164958 0.986301i \(-0.447251\pi\)
0.164958 + 0.986301i \(0.447251\pi\)
\(84\) 3.30927e6 0.609192
\(85\) −2.14208e6 −0.378329
\(86\) −1.07073e6 −0.181524
\(87\) −6.19448e6 −1.00853
\(88\) 6.13453e6 0.959604
\(89\) −6.08728e6 −0.915288 −0.457644 0.889135i \(-0.651307\pi\)
−0.457644 + 0.889135i \(0.651307\pi\)
\(90\) −9.52472e6 −1.37722
\(91\) 5.64650e6 0.785479
\(92\) 1.56333e6 0.209312
\(93\) −9.37623e6 −1.20875
\(94\) −1.25208e7 −1.55484
\(95\) 5.19096e6 0.621176
\(96\) −1.26787e7 −1.46260
\(97\) −7.22067e6 −0.803298 −0.401649 0.915794i \(-0.631563\pi\)
−0.401649 + 0.915794i \(0.631563\pi\)
\(98\) 4.25831e6 0.457032
\(99\) 3.29156e7 3.40940
\(100\) −3.25129e6 −0.325129
\(101\) 9.01277e6 0.870429 0.435214 0.900327i \(-0.356673\pi\)
0.435214 + 0.900327i \(0.356673\pi\)
\(102\) −1.91523e7 −1.78699
\(103\) −3.97490e6 −0.358423 −0.179211 0.983811i \(-0.557355\pi\)
−0.179211 + 0.983811i \(0.557355\pi\)
\(104\) −7.96815e6 −0.694609
\(105\) 8.13244e6 0.685580
\(106\) 2.02979e7 1.65531
\(107\) −2.21874e7 −1.75091 −0.875454 0.483301i \(-0.839438\pi\)
−0.875454 + 0.483301i \(0.839438\pi\)
\(108\) −1.48962e7 −1.13787
\(109\) 1.24187e7 0.918506 0.459253 0.888305i \(-0.348117\pi\)
0.459253 + 0.888305i \(0.348117\pi\)
\(110\) −1.07785e7 −0.772115
\(111\) −3.36355e7 −2.33436
\(112\) 1.45069e7 0.975689
\(113\) 2.56219e7 1.67046 0.835231 0.549899i \(-0.185334\pi\)
0.835231 + 0.549899i \(0.185334\pi\)
\(114\) 4.64123e7 2.93404
\(115\) 3.84185e6 0.235558
\(116\) 3.79668e6 0.225840
\(117\) −4.27541e7 −2.46790
\(118\) 2.38901e7 1.33854
\(119\) 1.16347e7 0.632910
\(120\) −1.14762e7 −0.606268
\(121\) 1.77610e7 0.911422
\(122\) 1.65351e7 0.824416
\(123\) 6.69185e7 3.24249
\(124\) 5.74682e6 0.270677
\(125\) −1.82351e7 −0.835072
\(126\) 5.17337e7 2.30397
\(127\) 3.48811e7 1.51104 0.755522 0.655124i \(-0.227383\pi\)
0.755522 + 0.655124i \(0.227383\pi\)
\(128\) −2.73372e7 −1.15218
\(129\) −6.92224e6 −0.283911
\(130\) 1.40001e7 0.558896
\(131\) 7.90037e6 0.307042 0.153521 0.988145i \(-0.450939\pi\)
0.153521 + 0.988145i \(0.450939\pi\)
\(132\) −2.83553e7 −1.07306
\(133\) −2.81948e7 −1.03917
\(134\) 1.12036e7 0.402246
\(135\) −3.66070e7 −1.28055
\(136\) −1.64185e7 −0.559691
\(137\) −2.61766e7 −0.869744 −0.434872 0.900492i \(-0.643206\pi\)
−0.434872 + 0.900492i \(0.643206\pi\)
\(138\) 3.43500e7 1.11263
\(139\) −3.82597e7 −1.20834 −0.604171 0.796855i \(-0.706496\pi\)
−0.604171 + 0.796855i \(0.706496\pi\)
\(140\) −4.98448e6 −0.153522
\(141\) −8.09467e7 −2.43183
\(142\) 283443. 0.00830724
\(143\) −4.83817e7 −1.38358
\(144\) −1.09843e8 −3.06552
\(145\) 9.33025e6 0.254159
\(146\) 8.50061e6 0.226055
\(147\) 2.75299e7 0.714815
\(148\) 2.06157e7 0.522735
\(149\) −7.50017e6 −0.185746 −0.0928729 0.995678i \(-0.529605\pi\)
−0.0928729 + 0.995678i \(0.529605\pi\)
\(150\) −7.14382e7 −1.72827
\(151\) 7.15245e7 1.69058 0.845290 0.534308i \(-0.179428\pi\)
0.845290 + 0.534308i \(0.179428\pi\)
\(152\) 3.97875e7 0.918955
\(153\) −8.80957e7 −1.98854
\(154\) 5.85433e7 1.29168
\(155\) 1.41227e7 0.304618
\(156\) 3.68307e7 0.776737
\(157\) −1.03218e7 −0.212865 −0.106433 0.994320i \(-0.533943\pi\)
−0.106433 + 0.994320i \(0.533943\pi\)
\(158\) 5.07049e7 1.02270
\(159\) 1.31225e8 2.58897
\(160\) 1.90969e7 0.368589
\(161\) −2.08671e7 −0.394068
\(162\) −1.68459e8 −3.11309
\(163\) −9.73117e7 −1.75998 −0.879992 0.474989i \(-0.842452\pi\)
−0.879992 + 0.474989i \(0.842452\pi\)
\(164\) −4.10153e7 −0.726093
\(165\) −6.96824e7 −1.20762
\(166\) −2.31447e7 −0.392711
\(167\) −1.15556e8 −1.91992 −0.959961 0.280133i \(-0.909621\pi\)
−0.959961 + 0.280133i \(0.909621\pi\)
\(168\) 6.23333e7 1.01423
\(169\) 94596.7 0.00150755
\(170\) 2.88476e7 0.450338
\(171\) 2.13485e8 3.26498
\(172\) 4.24273e6 0.0635764
\(173\) 4.06579e6 0.0597012 0.0298506 0.999554i \(-0.490497\pi\)
0.0298506 + 0.999554i \(0.490497\pi\)
\(174\) 8.34218e7 1.20049
\(175\) 4.33976e7 0.612114
\(176\) −1.24302e8 −1.71863
\(177\) 1.54449e8 2.09352
\(178\) 8.19780e7 1.08950
\(179\) −8.18724e7 −1.06697 −0.533485 0.845810i \(-0.679118\pi\)
−0.533485 + 0.845810i \(0.679118\pi\)
\(180\) 3.77414e7 0.482352
\(181\) −1.09002e8 −1.36634 −0.683168 0.730261i \(-0.739398\pi\)
−0.683168 + 0.730261i \(0.739398\pi\)
\(182\) −7.60420e7 −0.934983
\(183\) 1.06899e8 1.28942
\(184\) 2.94469e7 0.348479
\(185\) 5.06625e7 0.588282
\(186\) 1.26271e8 1.43882
\(187\) −9.96917e7 −1.11484
\(188\) 4.96133e7 0.544560
\(189\) 1.98832e8 2.14225
\(190\) −6.99072e7 −0.739408
\(191\) −2.66371e7 −0.276612 −0.138306 0.990390i \(-0.544166\pi\)
−0.138306 + 0.990390i \(0.544166\pi\)
\(192\) −5.62281e7 −0.573323
\(193\) 1.74825e8 1.75047 0.875233 0.483701i \(-0.160708\pi\)
0.875233 + 0.483701i \(0.160708\pi\)
\(194\) 9.72416e7 0.956194
\(195\) 9.05106e7 0.874134
\(196\) −1.68734e7 −0.160069
\(197\) −5.70061e7 −0.531238 −0.265619 0.964078i \(-0.585576\pi\)
−0.265619 + 0.964078i \(0.585576\pi\)
\(198\) −4.43277e8 −4.05833
\(199\) −3.66798e7 −0.329945 −0.164972 0.986298i \(-0.552753\pi\)
−0.164972 + 0.986298i \(0.552753\pi\)
\(200\) −6.12412e7 −0.541301
\(201\) 7.24312e7 0.629129
\(202\) −1.21376e8 −1.03610
\(203\) −5.06774e7 −0.425185
\(204\) 7.58905e7 0.625867
\(205\) −1.00794e8 −0.817139
\(206\) 5.35304e7 0.426643
\(207\) 1.58001e8 1.23812
\(208\) 1.61455e8 1.24403
\(209\) 2.41585e8 1.83045
\(210\) −1.09520e8 −0.816070
\(211\) −1.29934e8 −0.952213 −0.476106 0.879388i \(-0.657952\pi\)
−0.476106 + 0.879388i \(0.657952\pi\)
\(212\) −8.04297e7 −0.579750
\(213\) 1.83245e6 0.0129928
\(214\) 2.98800e8 2.08417
\(215\) 1.04264e7 0.0715484
\(216\) −2.80584e8 −1.89442
\(217\) −7.67075e7 −0.509599
\(218\) −1.67243e8 −1.09333
\(219\) 5.49562e7 0.353559
\(220\) 4.27093e7 0.270423
\(221\) 1.29490e8 0.806979
\(222\) 4.52974e8 2.77867
\(223\) −2.02158e8 −1.22074 −0.610370 0.792116i \(-0.708979\pi\)
−0.610370 + 0.792116i \(0.708979\pi\)
\(224\) −1.03725e8 −0.616617
\(225\) −3.28597e8 −1.92320
\(226\) −3.45053e8 −1.98841
\(227\) −3.00569e6 −0.0170551 −0.00852753 0.999964i \(-0.502714\pi\)
−0.00852753 + 0.999964i \(0.502714\pi\)
\(228\) −1.83907e8 −1.02761
\(229\) 2.04655e8 1.12616 0.563078 0.826404i \(-0.309617\pi\)
0.563078 + 0.826404i \(0.309617\pi\)
\(230\) −5.17386e7 −0.280393
\(231\) 3.78481e8 2.02024
\(232\) 7.15142e7 0.375997
\(233\) 5.47940e7 0.283784 0.141892 0.989882i \(-0.454682\pi\)
0.141892 + 0.989882i \(0.454682\pi\)
\(234\) 5.75774e8 2.93762
\(235\) 1.21924e8 0.612844
\(236\) −9.46637e7 −0.468804
\(237\) 3.27806e8 1.59955
\(238\) −1.56686e8 −0.753376
\(239\) −2.12739e8 −1.00799 −0.503994 0.863707i \(-0.668137\pi\)
−0.503994 + 0.863707i \(0.668137\pi\)
\(240\) 2.32538e8 1.08581
\(241\) 1.29896e8 0.597774 0.298887 0.954289i \(-0.403385\pi\)
0.298887 + 0.954289i \(0.403385\pi\)
\(242\) −2.39190e8 −1.08490
\(243\) −4.78586e8 −2.13963
\(244\) −6.55196e7 −0.288740
\(245\) −4.14660e7 −0.180140
\(246\) −9.01199e8 −3.85965
\(247\) −3.13796e8 −1.32497
\(248\) 1.08247e8 0.450645
\(249\) −1.49630e8 −0.614214
\(250\) 2.45575e8 0.994016
\(251\) 4.05186e8 1.61732 0.808661 0.588274i \(-0.200193\pi\)
0.808661 + 0.588274i \(0.200193\pi\)
\(252\) −2.04993e8 −0.806933
\(253\) 1.78798e8 0.694132
\(254\) −4.69747e8 −1.79865
\(255\) 1.86499e8 0.704346
\(256\) 2.85488e8 1.06353
\(257\) −4.32001e8 −1.58752 −0.793758 0.608233i \(-0.791879\pi\)
−0.793758 + 0.608233i \(0.791879\pi\)
\(258\) 9.32225e7 0.337949
\(259\) −2.75174e8 −0.984144
\(260\) −5.54751e7 −0.195745
\(261\) 3.83718e8 1.33589
\(262\) −1.06395e8 −0.365483
\(263\) 9.96029e7 0.337619 0.168809 0.985649i \(-0.446008\pi\)
0.168809 + 0.985649i \(0.446008\pi\)
\(264\) −5.34100e8 −1.78652
\(265\) −1.97654e8 −0.652447
\(266\) 3.79702e8 1.23696
\(267\) 5.29986e8 1.70402
\(268\) −4.43941e7 −0.140881
\(269\) −4.54907e8 −1.42492 −0.712459 0.701713i \(-0.752419\pi\)
−0.712459 + 0.701713i \(0.752419\pi\)
\(270\) 4.92991e8 1.52428
\(271\) −2.69553e8 −0.822718 −0.411359 0.911473i \(-0.634946\pi\)
−0.411359 + 0.911473i \(0.634946\pi\)
\(272\) 3.32683e8 1.00240
\(273\) −4.91610e8 −1.46235
\(274\) 3.52523e8 1.03529
\(275\) −3.71850e8 −1.07821
\(276\) −1.36111e8 −0.389682
\(277\) −9.07532e7 −0.256556 −0.128278 0.991738i \(-0.540945\pi\)
−0.128278 + 0.991738i \(0.540945\pi\)
\(278\) 5.15248e8 1.43833
\(279\) 5.80812e8 1.60111
\(280\) −9.38876e7 −0.255597
\(281\) −2.39385e7 −0.0643614 −0.0321807 0.999482i \(-0.510245\pi\)
−0.0321807 + 0.999482i \(0.510245\pi\)
\(282\) 1.09012e9 2.89469
\(283\) −3.89906e8 −1.02261 −0.511303 0.859401i \(-0.670837\pi\)
−0.511303 + 0.859401i \(0.670837\pi\)
\(284\) −1.12313e6 −0.00290950
\(285\) −4.51948e8 −1.15646
\(286\) 6.51562e8 1.64693
\(287\) 5.47464e8 1.36700
\(288\) 7.85384e8 1.93735
\(289\) −1.43522e8 −0.349766
\(290\) −1.25651e8 −0.302534
\(291\) 6.28664e8 1.49552
\(292\) −3.36834e7 −0.0791728
\(293\) −3.58463e8 −0.832544 −0.416272 0.909240i \(-0.636664\pi\)
−0.416272 + 0.909240i \(0.636664\pi\)
\(294\) −3.70748e8 −0.850869
\(295\) −2.32634e8 −0.527589
\(296\) 3.88317e8 0.870292
\(297\) −1.70368e9 −3.77346
\(298\) 1.01006e8 0.221100
\(299\) −2.32242e8 −0.502448
\(300\) 2.83072e8 0.605302
\(301\) −5.66312e7 −0.119694
\(302\) −9.63228e8 −2.01236
\(303\) −7.84692e8 −1.62050
\(304\) −8.06199e8 −1.64583
\(305\) −1.61013e8 −0.324946
\(306\) 1.18639e9 2.36703
\(307\) 3.60914e8 0.711901 0.355951 0.934505i \(-0.384157\pi\)
0.355951 + 0.934505i \(0.384157\pi\)
\(308\) −2.31976e8 −0.452393
\(309\) 3.46072e8 0.667287
\(310\) −1.90191e8 −0.362598
\(311\) 5.61989e8 1.05942 0.529708 0.848180i \(-0.322302\pi\)
0.529708 + 0.848180i \(0.322302\pi\)
\(312\) 6.93743e8 1.29318
\(313\) 5.21602e8 0.961466 0.480733 0.876867i \(-0.340371\pi\)
0.480733 + 0.876867i \(0.340371\pi\)
\(314\) 1.39004e8 0.253381
\(315\) −5.03765e8 −0.908116
\(316\) −2.00917e8 −0.358188
\(317\) −1.50341e8 −0.265076 −0.132538 0.991178i \(-0.542313\pi\)
−0.132538 + 0.991178i \(0.542313\pi\)
\(318\) −1.76722e9 −3.08174
\(319\) 4.34227e8 0.748944
\(320\) 8.46919e7 0.144483
\(321\) 1.93174e9 3.25972
\(322\) 2.81019e8 0.469073
\(323\) −6.46583e8 −1.06762
\(324\) 6.67514e8 1.09032
\(325\) 4.82997e8 0.780463
\(326\) 1.31051e9 2.09497
\(327\) −1.08122e9 −1.71001
\(328\) −7.72563e8 −1.20886
\(329\) −6.62230e8 −1.02523
\(330\) 9.38420e8 1.43747
\(331\) 7.31126e8 1.10814 0.554070 0.832470i \(-0.313074\pi\)
0.554070 + 0.832470i \(0.313074\pi\)
\(332\) 9.17100e7 0.137541
\(333\) 2.08356e9 3.09208
\(334\) 1.55620e9 2.28535
\(335\) −1.09097e8 −0.158547
\(336\) −1.26304e9 −1.81647
\(337\) −3.27389e8 −0.465972 −0.232986 0.972480i \(-0.574850\pi\)
−0.232986 + 0.972480i \(0.574850\pi\)
\(338\) −1.27394e6 −0.00179449
\(339\) −2.23076e9 −3.10995
\(340\) −1.14308e8 −0.157725
\(341\) 6.57264e8 0.897635
\(342\) −2.87502e9 −3.88642
\(343\) 8.11816e8 1.08625
\(344\) 7.99160e7 0.105847
\(345\) −3.34489e8 −0.438545
\(346\) −5.47544e7 −0.0710645
\(347\) 7.03971e8 0.904485 0.452243 0.891895i \(-0.350624\pi\)
0.452243 + 0.891895i \(0.350624\pi\)
\(348\) −3.30556e8 −0.420453
\(349\) −3.71304e8 −0.467563 −0.233782 0.972289i \(-0.575110\pi\)
−0.233782 + 0.972289i \(0.575110\pi\)
\(350\) −5.84440e8 −0.728621
\(351\) 2.21291e9 2.73142
\(352\) 8.88763e8 1.08614
\(353\) 1.27959e9 1.54832 0.774160 0.632990i \(-0.218172\pi\)
0.774160 + 0.632990i \(0.218172\pi\)
\(354\) −2.07998e9 −2.49200
\(355\) −2.76008e6 −0.00327432
\(356\) −3.24835e8 −0.381582
\(357\) −1.01297e9 −1.17831
\(358\) 1.10258e9 1.27005
\(359\) −3.54999e8 −0.404946 −0.202473 0.979288i \(-0.564898\pi\)
−0.202473 + 0.979288i \(0.564898\pi\)
\(360\) 7.10897e8 0.803059
\(361\) 6.73010e8 0.752915
\(362\) 1.46794e9 1.62640
\(363\) −1.54636e9 −1.69682
\(364\) 3.01314e8 0.327465
\(365\) −8.27761e7 −0.0891005
\(366\) −1.43962e9 −1.53484
\(367\) 1.73415e9 1.83128 0.915639 0.402001i \(-0.131685\pi\)
0.915639 + 0.402001i \(0.131685\pi\)
\(368\) −5.96671e8 −0.624119
\(369\) −4.14528e9 −4.29498
\(370\) −6.82278e8 −0.700253
\(371\) 1.07356e9 1.09149
\(372\) −5.00344e8 −0.503928
\(373\) −3.01486e8 −0.300805 −0.150403 0.988625i \(-0.548057\pi\)
−0.150403 + 0.988625i \(0.548057\pi\)
\(374\) 1.34256e9 1.32704
\(375\) 1.58763e9 1.55468
\(376\) 9.34516e8 0.906628
\(377\) −5.64018e8 −0.542123
\(378\) −2.67769e9 −2.54999
\(379\) −9.79815e8 −0.924499 −0.462250 0.886750i \(-0.652958\pi\)
−0.462250 + 0.886750i \(0.652958\pi\)
\(380\) 2.77005e8 0.258967
\(381\) −3.03690e9 −2.81315
\(382\) 3.58725e8 0.329261
\(383\) −4.89981e8 −0.445639 −0.222820 0.974860i \(-0.571526\pi\)
−0.222820 + 0.974860i \(0.571526\pi\)
\(384\) 2.38010e9 2.14504
\(385\) −5.70076e8 −0.509120
\(386\) −2.35439e9 −2.08364
\(387\) 4.28799e8 0.376067
\(388\) −3.85317e8 −0.334894
\(389\) −1.81229e9 −1.56101 −0.780504 0.625151i \(-0.785037\pi\)
−0.780504 + 0.625151i \(0.785037\pi\)
\(390\) −1.21892e9 −1.04051
\(391\) −4.78539e8 −0.404854
\(392\) −3.17828e8 −0.266496
\(393\) −6.87841e8 −0.571629
\(394\) 7.67707e8 0.632352
\(395\) −4.93748e8 −0.403102
\(396\) 1.75647e9 1.42137
\(397\) −1.72818e9 −1.38619 −0.693096 0.720845i \(-0.743754\pi\)
−0.693096 + 0.720845i \(0.743754\pi\)
\(398\) 4.93971e8 0.392745
\(399\) 2.45476e9 1.93466
\(400\) 1.24091e9 0.969458
\(401\) −8.93294e8 −0.691814 −0.345907 0.938269i \(-0.612429\pi\)
−0.345907 + 0.938269i \(0.612429\pi\)
\(402\) −9.75439e8 −0.748874
\(403\) −8.53721e8 −0.649753
\(404\) 4.80948e8 0.362880
\(405\) 1.64040e9 1.22703
\(406\) 6.82478e8 0.506113
\(407\) 2.35782e9 1.73352
\(408\) 1.42947e9 1.04199
\(409\) −7.17563e8 −0.518595 −0.259298 0.965797i \(-0.583491\pi\)
−0.259298 + 0.965797i \(0.583491\pi\)
\(410\) 1.35740e9 0.972670
\(411\) 2.27905e9 1.61923
\(412\) −2.12113e8 −0.149426
\(413\) 1.26355e9 0.882610
\(414\) −2.12782e9 −1.47378
\(415\) 2.25375e8 0.154788
\(416\) −1.15442e9 −0.786205
\(417\) 3.33106e9 2.24961
\(418\) −3.25346e9 −2.17885
\(419\) −8.34959e8 −0.554519 −0.277259 0.960795i \(-0.589426\pi\)
−0.277259 + 0.960795i \(0.589426\pi\)
\(420\) 4.33971e8 0.285817
\(421\) 2.75014e6 0.00179626 0.000898128 1.00000i \(-0.499714\pi\)
0.000898128 1.00000i \(0.499714\pi\)
\(422\) 1.74983e9 1.13345
\(423\) 5.01426e9 3.22118
\(424\) −1.51497e9 −0.965215
\(425\) 9.95225e8 0.628869
\(426\) −2.46778e7 −0.0154658
\(427\) 8.74544e8 0.543606
\(428\) −1.18399e9 −0.729951
\(429\) 4.21233e9 2.57586
\(430\) −1.40414e8 −0.0851666
\(431\) −1.21897e9 −0.733370 −0.366685 0.930345i \(-0.619507\pi\)
−0.366685 + 0.930345i \(0.619507\pi\)
\(432\) 5.68537e9 3.39286
\(433\) −1.12096e9 −0.663561 −0.331781 0.943357i \(-0.607649\pi\)
−0.331781 + 0.943357i \(0.607649\pi\)
\(434\) 1.03303e9 0.606594
\(435\) −8.12333e8 −0.473175
\(436\) 6.62697e8 0.382924
\(437\) 1.15966e9 0.664728
\(438\) −7.40101e8 −0.420854
\(439\) −5.95770e8 −0.336088 −0.168044 0.985779i \(-0.553745\pi\)
−0.168044 + 0.985779i \(0.553745\pi\)
\(440\) 8.04471e8 0.450221
\(441\) −1.70534e9 −0.946840
\(442\) −1.74385e9 −0.960575
\(443\) 2.21773e9 1.21198 0.605991 0.795472i \(-0.292777\pi\)
0.605991 + 0.795472i \(0.292777\pi\)
\(444\) −1.79489e9 −0.973192
\(445\) −7.98275e8 −0.429430
\(446\) 2.72248e9 1.45309
\(447\) 6.52998e8 0.345808
\(448\) −4.60005e8 −0.241707
\(449\) −7.44092e8 −0.387940 −0.193970 0.981007i \(-0.562136\pi\)
−0.193970 + 0.981007i \(0.562136\pi\)
\(450\) 4.42525e9 2.28926
\(451\) −4.69092e9 −2.40791
\(452\) 1.36726e9 0.696413
\(453\) −6.22725e9 −3.14740
\(454\) 4.04779e7 0.0203012
\(455\) 7.40472e8 0.368527
\(456\) −3.46408e9 −1.71085
\(457\) 7.56617e8 0.370825 0.185413 0.982661i \(-0.440638\pi\)
0.185413 + 0.982661i \(0.440638\pi\)
\(458\) −2.75612e9 −1.34050
\(459\) 4.55975e9 2.20088
\(460\) 2.05013e8 0.0982038
\(461\) −8.02106e7 −0.0381310 −0.0190655 0.999818i \(-0.506069\pi\)
−0.0190655 + 0.999818i \(0.506069\pi\)
\(462\) −5.09705e9 −2.40476
\(463\) 1.38557e9 0.648778 0.324389 0.945924i \(-0.394841\pi\)
0.324389 + 0.945924i \(0.394841\pi\)
\(464\) −1.44907e9 −0.673403
\(465\) −1.22958e9 −0.567116
\(466\) −7.37917e8 −0.337798
\(467\) 4.12145e9 1.87258 0.936292 0.351224i \(-0.114234\pi\)
0.936292 + 0.351224i \(0.114234\pi\)
\(468\) −2.28149e9 −1.02886
\(469\) 5.92564e8 0.265235
\(470\) −1.64196e9 −0.729490
\(471\) 8.98659e8 0.396298
\(472\) −1.78308e9 −0.780504
\(473\) 4.85242e8 0.210836
\(474\) −4.41460e9 −1.90400
\(475\) −2.41176e9 −1.03254
\(476\) 6.20864e8 0.263859
\(477\) −8.12877e9 −3.42934
\(478\) 2.86498e9 1.19984
\(479\) −3.39610e8 −0.141191 −0.0705955 0.997505i \(-0.522490\pi\)
−0.0705955 + 0.997505i \(0.522490\pi\)
\(480\) −1.66266e9 −0.686214
\(481\) −3.06257e9 −1.25481
\(482\) −1.74933e9 −0.711551
\(483\) 1.81678e9 0.733648
\(484\) 9.47782e8 0.379970
\(485\) −9.46906e8 −0.376887
\(486\) 6.44517e9 2.54687
\(487\) −2.35025e9 −0.922066 −0.461033 0.887383i \(-0.652521\pi\)
−0.461033 + 0.887383i \(0.652521\pi\)
\(488\) −1.23413e9 −0.480718
\(489\) 8.47240e9 3.27661
\(490\) 5.58428e8 0.214428
\(491\) −5.12182e8 −0.195272 −0.0976359 0.995222i \(-0.531128\pi\)
−0.0976359 + 0.995222i \(0.531128\pi\)
\(492\) 3.57097e9 1.35179
\(493\) −1.16217e9 −0.436823
\(494\) 4.22592e9 1.57716
\(495\) 4.31649e9 1.59960
\(496\) −2.19337e9 −0.807096
\(497\) 1.49914e7 0.00547766
\(498\) 2.01508e9 0.731121
\(499\) −2.62730e9 −0.946582 −0.473291 0.880906i \(-0.656934\pi\)
−0.473291 + 0.880906i \(0.656934\pi\)
\(500\) −9.73081e8 −0.348140
\(501\) 1.00608e10 3.57438
\(502\) −5.45669e9 −1.92516
\(503\) −1.27864e9 −0.447982 −0.223991 0.974591i \(-0.571909\pi\)
−0.223991 + 0.974591i \(0.571909\pi\)
\(504\) −3.86125e9 −1.34345
\(505\) 1.18192e9 0.408383
\(506\) −2.40790e9 −0.826250
\(507\) −8.23601e6 −0.00280665
\(508\) 1.86136e9 0.629952
\(509\) −3.34993e9 −1.12596 −0.562980 0.826470i \(-0.690345\pi\)
−0.562980 + 0.826470i \(0.690345\pi\)
\(510\) −2.51160e9 −0.838408
\(511\) 4.49600e8 0.149057
\(512\) −3.45536e8 −0.113776
\(513\) −1.10498e10 −3.61362
\(514\) 5.81780e9 1.88968
\(515\) −5.21261e8 −0.168163
\(516\) −3.69391e8 −0.118362
\(517\) 5.67428e9 1.80590
\(518\) 3.70580e9 1.17146
\(519\) −3.53986e8 −0.111148
\(520\) −1.04493e9 −0.325893
\(521\) 3.15645e9 0.977837 0.488918 0.872330i \(-0.337392\pi\)
0.488918 + 0.872330i \(0.337392\pi\)
\(522\) −5.16758e9 −1.59016
\(523\) 5.04134e8 0.154095 0.0770477 0.997027i \(-0.475451\pi\)
0.0770477 + 0.997027i \(0.475451\pi\)
\(524\) 4.21587e8 0.128005
\(525\) −3.77839e9 −1.13959
\(526\) −1.34136e9 −0.401880
\(527\) −1.75911e9 −0.523548
\(528\) 1.08223e10 3.19962
\(529\) −2.54656e9 −0.747926
\(530\) 2.66183e9 0.776630
\(531\) −9.56736e9 −2.77307
\(532\) −1.50456e9 −0.433230
\(533\) 6.09304e9 1.74297
\(534\) −7.13737e9 −2.02836
\(535\) −2.90962e9 −0.821481
\(536\) −8.36206e8 −0.234550
\(537\) 7.12818e9 1.98641
\(538\) 6.12629e9 1.69613
\(539\) −1.92982e9 −0.530830
\(540\) −1.95346e9 −0.533859
\(541\) −5.65936e9 −1.53666 −0.768329 0.640056i \(-0.778911\pi\)
−0.768329 + 0.640056i \(0.778911\pi\)
\(542\) 3.63010e9 0.979311
\(543\) 9.49016e9 2.54375
\(544\) −2.37870e9 −0.633495
\(545\) 1.62856e9 0.430940
\(546\) 6.62056e9 1.74069
\(547\) 1.34725e9 0.351958 0.175979 0.984394i \(-0.443691\pi\)
0.175979 + 0.984394i \(0.443691\pi\)
\(548\) −1.39686e9 −0.362595
\(549\) −6.62186e9 −1.70796
\(550\) 5.00775e9 1.28343
\(551\) 2.81632e9 0.717218
\(552\) −2.56378e9 −0.648775
\(553\) 2.68180e9 0.674354
\(554\) 1.22218e9 0.305388
\(555\) −4.41091e9 −1.09522
\(556\) −2.04165e9 −0.503756
\(557\) −6.05230e9 −1.48398 −0.741989 0.670413i \(-0.766117\pi\)
−0.741989 + 0.670413i \(0.766117\pi\)
\(558\) −7.82186e9 −1.90586
\(559\) −6.30281e8 −0.152613
\(560\) 1.90241e9 0.457768
\(561\) 8.67960e9 2.07553
\(562\) 3.22383e8 0.0766116
\(563\) −2.52625e9 −0.596619 −0.298310 0.954469i \(-0.596423\pi\)
−0.298310 + 0.954469i \(0.596423\pi\)
\(564\) −4.31956e9 −1.01382
\(565\) 3.36001e9 0.783738
\(566\) 5.25091e9 1.21724
\(567\) −8.90985e9 −2.05272
\(568\) −2.11553e7 −0.00484396
\(569\) 1.07885e9 0.245509 0.122755 0.992437i \(-0.460827\pi\)
0.122755 + 0.992437i \(0.460827\pi\)
\(570\) 6.08643e9 1.37658
\(571\) 8.05110e9 1.80979 0.904897 0.425632i \(-0.139948\pi\)
0.904897 + 0.425632i \(0.139948\pi\)
\(572\) −2.58180e9 −0.576814
\(573\) 2.31915e9 0.514976
\(574\) −7.37276e9 −1.62719
\(575\) −1.78495e9 −0.391551
\(576\) 3.48306e9 0.759421
\(577\) 4.42426e9 0.958795 0.479397 0.877598i \(-0.340855\pi\)
0.479397 + 0.877598i \(0.340855\pi\)
\(578\) 1.93283e9 0.416339
\(579\) −1.52211e10 −3.25890
\(580\) 4.97890e8 0.105958
\(581\) −1.22413e9 −0.258947
\(582\) −8.46629e9 −1.78017
\(583\) −9.19875e9 −1.92260
\(584\) −6.34460e8 −0.131813
\(585\) −5.60669e9 −1.15787
\(586\) 4.82745e9 0.991006
\(587\) −3.26125e9 −0.665503 −0.332752 0.943015i \(-0.607977\pi\)
−0.332752 + 0.943015i \(0.607977\pi\)
\(588\) 1.46908e9 0.298005
\(589\) 4.26290e9 0.859611
\(590\) 3.13290e9 0.628008
\(591\) 4.96320e9 0.989022
\(592\) −7.86831e9 −1.55867
\(593\) −1.10562e9 −0.217727 −0.108864 0.994057i \(-0.534721\pi\)
−0.108864 + 0.994057i \(0.534721\pi\)
\(594\) 2.29436e10 4.49169
\(595\) 1.52576e9 0.296945
\(596\) −4.00231e8 −0.0774371
\(597\) 3.19351e9 0.614268
\(598\) 3.12762e9 0.598081
\(599\) 4.87422e9 0.926640 0.463320 0.886191i \(-0.346658\pi\)
0.463320 + 0.886191i \(0.346658\pi\)
\(600\) 5.33193e9 1.00776
\(601\) 4.41328e9 0.829280 0.414640 0.909986i \(-0.363908\pi\)
0.414640 + 0.909986i \(0.363908\pi\)
\(602\) 7.62659e8 0.142476
\(603\) −4.48676e9 −0.833341
\(604\) 3.81676e9 0.704800
\(605\) 2.32915e9 0.427616
\(606\) 1.05675e10 1.92894
\(607\) 9.64653e9 1.75070 0.875348 0.483494i \(-0.160632\pi\)
0.875348 + 0.483494i \(0.160632\pi\)
\(608\) 5.76436e9 1.04013
\(609\) 4.41220e9 0.791580
\(610\) 2.16838e9 0.386795
\(611\) −7.37033e9 −1.30720
\(612\) −4.70105e9 −0.829020
\(613\) −1.75264e9 −0.307313 −0.153656 0.988124i \(-0.549105\pi\)
−0.153656 + 0.988124i \(0.549105\pi\)
\(614\) −4.86047e9 −0.847401
\(615\) 8.77558e9 1.52129
\(616\) −4.36950e9 −0.753181
\(617\) 3.17834e9 0.544757 0.272378 0.962190i \(-0.412190\pi\)
0.272378 + 0.962190i \(0.412190\pi\)
\(618\) −4.66060e9 −0.794295
\(619\) 4.18253e9 0.708797 0.354398 0.935095i \(-0.384686\pi\)
0.354398 + 0.935095i \(0.384686\pi\)
\(620\) 7.53628e8 0.126995
\(621\) −8.17798e9 −1.37033
\(622\) −7.56836e9 −1.26106
\(623\) 4.33584e9 0.718398
\(624\) −1.40570e10 −2.31605
\(625\) 2.36866e9 0.388081
\(626\) −7.02447e9 −1.14447
\(627\) −2.10335e10 −3.40781
\(628\) −5.50800e8 −0.0887433
\(629\) −6.31050e9 −1.01108
\(630\) 6.78426e9 1.08096
\(631\) 3.38318e9 0.536071 0.268035 0.963409i \(-0.413626\pi\)
0.268035 + 0.963409i \(0.413626\pi\)
\(632\) −3.78446e9 −0.596341
\(633\) 1.13126e10 1.77276
\(634\) 2.02466e9 0.315529
\(635\) 4.57424e9 0.708943
\(636\) 7.00257e9 1.07934
\(637\) 2.50664e9 0.384241
\(638\) −5.84778e9 −0.891494
\(639\) −1.13512e8 −0.0172103
\(640\) −3.58496e9 −0.540573
\(641\) 2.99478e9 0.449120 0.224560 0.974460i \(-0.427906\pi\)
0.224560 + 0.974460i \(0.427906\pi\)
\(642\) −2.60149e10 −3.88016
\(643\) 1.17542e9 0.174363 0.0871817 0.996192i \(-0.472214\pi\)
0.0871817 + 0.996192i \(0.472214\pi\)
\(644\) −1.11353e9 −0.164286
\(645\) −9.07770e8 −0.133204
\(646\) 8.70760e9 1.27082
\(647\) −5.64077e9 −0.818792 −0.409396 0.912357i \(-0.634261\pi\)
−0.409396 + 0.912357i \(0.634261\pi\)
\(648\) 1.25733e10 1.81525
\(649\) −1.08267e10 −1.55467
\(650\) −6.50457e9 −0.929013
\(651\) 6.67850e9 0.948736
\(652\) −5.19285e9 −0.733734
\(653\) −7.03306e9 −0.988435 −0.494217 0.869338i \(-0.664545\pi\)
−0.494217 + 0.869338i \(0.664545\pi\)
\(654\) 1.45610e10 2.03549
\(655\) 1.03604e9 0.144056
\(656\) 1.56541e10 2.16504
\(657\) −3.40427e9 −0.468323
\(658\) 8.91832e9 1.22037
\(659\) −5.30583e9 −0.722194 −0.361097 0.932528i \(-0.617598\pi\)
−0.361097 + 0.932528i \(0.617598\pi\)
\(660\) −3.71846e9 −0.503454
\(661\) −4.29135e9 −0.577949 −0.288974 0.957337i \(-0.593314\pi\)
−0.288974 + 0.957337i \(0.593314\pi\)
\(662\) −9.84615e9 −1.31906
\(663\) −1.12739e10 −1.50238
\(664\) 1.72745e9 0.228990
\(665\) −3.69741e9 −0.487553
\(666\) −2.80595e10 −3.68062
\(667\) 2.08437e9 0.271978
\(668\) −6.16640e9 −0.800413
\(669\) 1.76008e10 2.27269
\(670\) 1.46922e9 0.188724
\(671\) −7.49349e9 −0.957536
\(672\) 9.03077e9 1.14797
\(673\) 3.84315e9 0.485998 0.242999 0.970027i \(-0.421869\pi\)
0.242999 + 0.970027i \(0.421869\pi\)
\(674\) 4.40898e9 0.554662
\(675\) 1.70079e10 2.12857
\(676\) 5.04796e6 0.000628496 0
\(677\) −1.03024e10 −1.27608 −0.638041 0.770002i \(-0.720255\pi\)
−0.638041 + 0.770002i \(0.720255\pi\)
\(678\) 3.00419e10 3.70188
\(679\) 5.14314e9 0.630498
\(680\) −2.15310e9 −0.262593
\(681\) 2.61689e8 0.0317519
\(682\) −8.85145e9 −1.06849
\(683\) 2.78807e8 0.0334835 0.0167417 0.999860i \(-0.494671\pi\)
0.0167417 + 0.999860i \(0.494671\pi\)
\(684\) 1.13922e10 1.36116
\(685\) −3.43276e9 −0.408062
\(686\) −1.09328e10 −1.29300
\(687\) −1.78182e10 −2.09660
\(688\) −1.61931e9 −0.189570
\(689\) 1.19483e10 1.39167
\(690\) 4.50460e9 0.522016
\(691\) 1.57009e9 0.181030 0.0905151 0.995895i \(-0.471149\pi\)
0.0905151 + 0.995895i \(0.471149\pi\)
\(692\) 2.16963e8 0.0248893
\(693\) −2.34451e10 −2.67600
\(694\) −9.48045e9 −1.07664
\(695\) −5.01731e9 −0.566923
\(696\) −6.22635e9 −0.700005
\(697\) 1.25548e10 1.40442
\(698\) 5.00039e9 0.556557
\(699\) −4.77061e9 −0.528328
\(700\) 2.31583e9 0.255189
\(701\) 1.55305e10 1.70284 0.851418 0.524488i \(-0.175743\pi\)
0.851418 + 0.524488i \(0.175743\pi\)
\(702\) −2.98015e10 −3.25131
\(703\) 1.52924e10 1.66009
\(704\) 3.94153e9 0.425756
\(705\) −1.06152e10 −1.14095
\(706\) −1.72324e10 −1.84302
\(707\) −6.41961e9 −0.683189
\(708\) 8.24185e9 0.872787
\(709\) 1.58882e10 1.67422 0.837112 0.547032i \(-0.184242\pi\)
0.837112 + 0.547032i \(0.184242\pi\)
\(710\) 3.71702e7 0.00389754
\(711\) −2.03060e10 −2.11875
\(712\) −6.11859e9 −0.635289
\(713\) 3.15499e9 0.325975
\(714\) 1.36418e10 1.40258
\(715\) −6.34470e9 −0.649142
\(716\) −4.36896e9 −0.444818
\(717\) 1.85220e10 1.87660
\(718\) 4.78082e9 0.482021
\(719\) 6.03906e9 0.605924 0.302962 0.953003i \(-0.402024\pi\)
0.302962 + 0.953003i \(0.402024\pi\)
\(720\) −1.44046e10 −1.43826
\(721\) 2.83124e9 0.281322
\(722\) −9.06349e9 −0.896222
\(723\) −1.13093e10 −1.11289
\(724\) −5.81665e9 −0.569624
\(725\) −4.33490e9 −0.422470
\(726\) 2.08249e10 2.01979
\(727\) 1.96027e10 1.89211 0.946054 0.324008i \(-0.105030\pi\)
0.946054 + 0.324008i \(0.105030\pi\)
\(728\) 5.67555e9 0.545190
\(729\) 1.43108e10 1.36810
\(730\) 1.11475e9 0.106059
\(731\) −1.29871e9 −0.122970
\(732\) 5.70443e9 0.537556
\(733\) −4.74673e9 −0.445174 −0.222587 0.974913i \(-0.571450\pi\)
−0.222587 + 0.974913i \(0.571450\pi\)
\(734\) −2.33539e10 −2.17984
\(735\) 3.61022e9 0.335373
\(736\) 4.26623e9 0.394432
\(737\) −5.07735e9 −0.467198
\(738\) 5.58249e10 5.11247
\(739\) −9.87972e9 −0.900512 −0.450256 0.892900i \(-0.648667\pi\)
−0.450256 + 0.892900i \(0.648667\pi\)
\(740\) 2.70350e9 0.245254
\(741\) 2.73205e10 2.46674
\(742\) −1.44578e10 −1.29923
\(743\) −1.28591e10 −1.15014 −0.575069 0.818105i \(-0.695025\pi\)
−0.575069 + 0.818105i \(0.695025\pi\)
\(744\) −9.42447e9 −0.838980
\(745\) −9.83559e8 −0.0871472
\(746\) 4.06014e9 0.358059
\(747\) 9.26884e9 0.813585
\(748\) −5.31985e9 −0.464776
\(749\) 1.58036e10 1.37427
\(750\) −2.13808e10 −1.85059
\(751\) 7.42726e9 0.639866 0.319933 0.947440i \(-0.396340\pi\)
0.319933 + 0.947440i \(0.396340\pi\)
\(752\) −1.89357e10 −1.62375
\(753\) −3.52773e10 −3.01102
\(754\) 7.59569e9 0.645309
\(755\) 9.37960e9 0.793177
\(756\) 1.06103e10 0.893099
\(757\) 7.19774e9 0.603060 0.301530 0.953457i \(-0.402503\pi\)
0.301530 + 0.953457i \(0.402503\pi\)
\(758\) 1.31953e10 1.10046
\(759\) −1.55670e10 −1.29229
\(760\) 5.21766e9 0.431150
\(761\) −1.01426e10 −0.834265 −0.417132 0.908846i \(-0.636965\pi\)
−0.417132 + 0.908846i \(0.636965\pi\)
\(762\) 4.08983e10 3.34860
\(763\) −8.84556e9 −0.720924
\(764\) −1.42144e9 −0.115319
\(765\) −1.15527e10 −0.932973
\(766\) 6.59862e9 0.530460
\(767\) 1.40628e10 1.12535
\(768\) −2.48559e10 −1.98000
\(769\) −3.34842e9 −0.265521 −0.132760 0.991148i \(-0.542384\pi\)
−0.132760 + 0.991148i \(0.542384\pi\)
\(770\) 7.67727e9 0.606023
\(771\) 3.76119e10 2.95553
\(772\) 9.32920e9 0.729767
\(773\) −1.64501e10 −1.28097 −0.640485 0.767970i \(-0.721267\pi\)
−0.640485 + 0.767970i \(0.721267\pi\)
\(774\) −5.77468e9 −0.447646
\(775\) −6.56149e9 −0.506345
\(776\) −7.25782e9 −0.557558
\(777\) 2.39579e10 1.83221
\(778\) 2.44064e10 1.85812
\(779\) −3.04245e10 −2.30591
\(780\) 4.82992e9 0.364425
\(781\) −1.28453e8 −0.00964863
\(782\) 6.44453e9 0.481912
\(783\) −1.98609e10 −1.47854
\(784\) 6.44002e9 0.477288
\(785\) −1.35358e9 −0.0998710
\(786\) 9.26324e9 0.680431
\(787\) −1.67903e9 −0.122786 −0.0613929 0.998114i \(-0.519554\pi\)
−0.0613929 + 0.998114i \(0.519554\pi\)
\(788\) −3.04202e9 −0.221472
\(789\) −8.67187e9 −0.628555
\(790\) 6.64935e9 0.479827
\(791\) −1.82500e10 −1.31112
\(792\) 3.30849e10 2.36642
\(793\) 9.73330e9 0.693113
\(794\) 2.32736e10 1.65003
\(795\) 1.72086e10 1.21468
\(796\) −1.95734e9 −0.137553
\(797\) −1.88216e10 −1.31690 −0.658449 0.752625i \(-0.728787\pi\)
−0.658449 + 0.752625i \(0.728787\pi\)
\(798\) −3.30586e10 −2.30289
\(799\) −1.51867e10 −1.05330
\(800\) −8.87255e9 −0.612680
\(801\) −3.28301e10 −2.25714
\(802\) 1.20301e10 0.823491
\(803\) −3.85237e9 −0.262557
\(804\) 3.86515e9 0.262283
\(805\) −2.73647e9 −0.184887
\(806\) 1.14972e10 0.773424
\(807\) 3.96063e10 2.65281
\(808\) 9.05913e9 0.604153
\(809\) 1.07428e10 0.713342 0.356671 0.934230i \(-0.383912\pi\)
0.356671 + 0.934230i \(0.383912\pi\)
\(810\) −2.20914e10 −1.46058
\(811\) 2.81229e10 1.85135 0.925673 0.378326i \(-0.123500\pi\)
0.925673 + 0.378326i \(0.123500\pi\)
\(812\) −2.70430e9 −0.177259
\(813\) 2.34685e10 1.53168
\(814\) −3.17530e10 −2.06347
\(815\) −1.27613e10 −0.825739
\(816\) −2.89648e10 −1.86619
\(817\) 3.14719e9 0.201905
\(818\) 9.66350e9 0.617302
\(819\) 3.04528e10 1.93702
\(820\) −5.37867e9 −0.340664
\(821\) 1.62322e10 1.02371 0.511853 0.859073i \(-0.328959\pi\)
0.511853 + 0.859073i \(0.328959\pi\)
\(822\) −3.06923e10 −1.92743
\(823\) −6.36841e9 −0.398228 −0.199114 0.979976i \(-0.563806\pi\)
−0.199114 + 0.979976i \(0.563806\pi\)
\(824\) −3.99535e9 −0.248776
\(825\) 3.23749e10 2.00734
\(826\) −1.70164e10 −1.05060
\(827\) −1.77050e10 −1.08850 −0.544248 0.838924i \(-0.683185\pi\)
−0.544248 + 0.838924i \(0.683185\pi\)
\(828\) 8.43140e9 0.516171
\(829\) −3.43518e9 −0.209415 −0.104708 0.994503i \(-0.533391\pi\)
−0.104708 + 0.994503i \(0.533391\pi\)
\(830\) −3.03515e9 −0.184250
\(831\) 7.90138e9 0.477639
\(832\) −5.11966e9 −0.308184
\(833\) 5.16499e9 0.309608
\(834\) −4.48598e10 −2.67779
\(835\) −1.51538e10 −0.900778
\(836\) 1.28917e10 0.763113
\(837\) −3.00623e10 −1.77208
\(838\) 1.12445e10 0.660063
\(839\) −2.12784e10 −1.24386 −0.621930 0.783073i \(-0.713651\pi\)
−0.621930 + 0.783073i \(0.713651\pi\)
\(840\) 8.17428e9 0.475852
\(841\) −1.21878e10 −0.706545
\(842\) −3.70365e7 −0.00213815
\(843\) 2.08420e9 0.119823
\(844\) −6.93367e9 −0.396976
\(845\) 1.24052e7 0.000707305 0
\(846\) −6.75276e10 −3.83429
\(847\) −1.26508e10 −0.715363
\(848\) 3.06973e10 1.72868
\(849\) 3.39470e10 1.90381
\(850\) −1.34028e10 −0.748565
\(851\) 1.13180e10 0.629528
\(852\) 9.77852e7 0.00541669
\(853\) −8.50432e9 −0.469157 −0.234578 0.972097i \(-0.575371\pi\)
−0.234578 + 0.972097i \(0.575371\pi\)
\(854\) −1.17776e10 −0.647073
\(855\) 2.79960e10 1.53184
\(856\) −2.23016e10 −1.21528
\(857\) −8.11675e8 −0.0440503 −0.0220252 0.999757i \(-0.507011\pi\)
−0.0220252 + 0.999757i \(0.507011\pi\)
\(858\) −5.67279e10 −3.06614
\(859\) −1.84624e10 −0.993832 −0.496916 0.867799i \(-0.665534\pi\)
−0.496916 + 0.867799i \(0.665534\pi\)
\(860\) 5.56384e8 0.0298284
\(861\) −4.76647e10 −2.54499
\(862\) 1.64160e10 0.872957
\(863\) 1.83495e10 0.971823 0.485911 0.874008i \(-0.338488\pi\)
0.485911 + 0.874008i \(0.338488\pi\)
\(864\) −4.06507e10 −2.14423
\(865\) 5.33180e8 0.0280103
\(866\) 1.50960e10 0.789860
\(867\) 1.24957e10 0.651170
\(868\) −4.09334e9 −0.212451
\(869\) −2.29789e10 −1.18784
\(870\) 1.09398e10 0.563237
\(871\) 6.59498e9 0.338182
\(872\) 1.24825e10 0.637523
\(873\) −3.89427e10 −1.98096
\(874\) −1.56172e10 −0.791250
\(875\) 1.29885e10 0.655437
\(876\) 2.93263e9 0.147398
\(877\) 3.41189e10 1.70803 0.854016 0.520247i \(-0.174160\pi\)
0.854016 + 0.520247i \(0.174160\pi\)
\(878\) 8.02330e9 0.400057
\(879\) 3.12094e10 1.54997
\(880\) −1.63007e10 −0.806337
\(881\) 1.72277e10 0.848811 0.424406 0.905472i \(-0.360483\pi\)
0.424406 + 0.905472i \(0.360483\pi\)
\(882\) 2.29660e10 1.12706
\(883\) 2.58479e10 1.26347 0.631733 0.775186i \(-0.282344\pi\)
0.631733 + 0.775186i \(0.282344\pi\)
\(884\) 6.90995e9 0.336428
\(885\) 2.02541e10 0.982228
\(886\) −2.98664e10 −1.44266
\(887\) −3.51605e10 −1.69170 −0.845848 0.533424i \(-0.820905\pi\)
−0.845848 + 0.533424i \(0.820905\pi\)
\(888\) −3.38086e10 −1.62025
\(889\) −2.48451e10 −1.18600
\(890\) 1.07505e10 0.511166
\(891\) 7.63436e10 3.61577
\(892\) −1.07877e10 −0.508925
\(893\) 3.68024e10 1.72940
\(894\) −8.79400e9 −0.411628
\(895\) −1.07366e10 −0.500595
\(896\) 1.94718e10 0.904330
\(897\) 2.02200e10 0.935422
\(898\) 1.00208e10 0.461778
\(899\) 7.66216e9 0.351716
\(900\) −1.75349e10 −0.801780
\(901\) 2.46197e10 1.12136
\(902\) 6.31731e10 2.86622
\(903\) 4.93057e9 0.222838
\(904\) 2.57537e10 1.15945
\(905\) −1.42943e10 −0.641050
\(906\) 8.38630e10 3.74647
\(907\) 1.84400e8 0.00820607 0.00410303 0.999992i \(-0.498694\pi\)
0.00410303 + 0.999992i \(0.498694\pi\)
\(908\) −1.60392e8 −0.00711023
\(909\) 4.86079e10 2.14651
\(910\) −9.97202e9 −0.438670
\(911\) −1.19294e10 −0.522761 −0.261380 0.965236i \(-0.584178\pi\)
−0.261380 + 0.965236i \(0.584178\pi\)
\(912\) 7.01913e10 3.06409
\(913\) 1.04889e10 0.456123
\(914\) −1.01894e10 −0.441407
\(915\) 1.40185e10 0.604962
\(916\) 1.09210e10 0.469493
\(917\) −5.62727e9 −0.240993
\(918\) −6.14067e10 −2.61979
\(919\) 3.79246e10 1.61182 0.805910 0.592038i \(-0.201677\pi\)
0.805910 + 0.592038i \(0.201677\pi\)
\(920\) 3.86161e9 0.163498
\(921\) −3.14228e10 −1.32537
\(922\) 1.08020e9 0.0453887
\(923\) 1.66848e8 0.00698417
\(924\) 2.01969e10 0.842234
\(925\) −2.35382e10 −0.977860
\(926\) −1.86597e10 −0.772263
\(927\) −2.14375e10 −0.883885
\(928\) 1.03609e10 0.425578
\(929\) −3.05711e10 −1.25099 −0.625497 0.780226i \(-0.715104\pi\)
−0.625497 + 0.780226i \(0.715104\pi\)
\(930\) 1.65589e10 0.675059
\(931\) −1.25165e10 −0.508344
\(932\) 2.92397e9 0.118309
\(933\) −4.89293e10 −1.97234
\(934\) −5.55040e10 −2.22900
\(935\) −1.30734e10 −0.523055
\(936\) −4.29740e10 −1.71293
\(937\) −1.29971e10 −0.516129 −0.258065 0.966128i \(-0.583085\pi\)
−0.258065 + 0.966128i \(0.583085\pi\)
\(938\) −7.98012e9 −0.315718
\(939\) −4.54130e10 −1.78999
\(940\) 6.50621e9 0.255494
\(941\) −3.94818e10 −1.54466 −0.772330 0.635221i \(-0.780909\pi\)
−0.772330 + 0.635221i \(0.780909\pi\)
\(942\) −1.21023e10 −0.471728
\(943\) −2.25173e10 −0.874431
\(944\) 3.61299e10 1.39787
\(945\) 2.60744e10 1.00509
\(946\) −6.53480e9 −0.250965
\(947\) 1.20683e10 0.461766 0.230883 0.972982i \(-0.425839\pi\)
0.230883 + 0.972982i \(0.425839\pi\)
\(948\) 1.74927e10 0.666849
\(949\) 5.00385e9 0.190052
\(950\) 3.24794e10 1.22907
\(951\) 1.30894e10 0.493499
\(952\) 1.16946e10 0.439294
\(953\) 8.06727e9 0.301927 0.150963 0.988539i \(-0.451762\pi\)
0.150963 + 0.988539i \(0.451762\pi\)
\(954\) 1.09471e11 4.08206
\(955\) −3.49314e9 −0.129779
\(956\) −1.13524e10 −0.420228
\(957\) −3.78057e10 −1.39433
\(958\) 4.57357e9 0.168065
\(959\) 1.86451e10 0.682651
\(960\) −7.37366e9 −0.268988
\(961\) −1.59148e10 −0.578456
\(962\) 4.12440e10 1.49365
\(963\) −1.19662e11 −4.31781
\(964\) 6.93165e9 0.249211
\(965\) 2.29263e10 0.821274
\(966\) −2.44668e10 −0.873287
\(967\) −2.31765e10 −0.824244 −0.412122 0.911129i \(-0.635212\pi\)
−0.412122 + 0.911129i \(0.635212\pi\)
\(968\) 1.78524e10 0.632606
\(969\) 5.62944e10 1.98761
\(970\) 1.27521e10 0.448622
\(971\) 6.13552e9 0.215072 0.107536 0.994201i \(-0.465704\pi\)
0.107536 + 0.994201i \(0.465704\pi\)
\(972\) −2.55388e10 −0.892007
\(973\) 2.72516e10 0.948412
\(974\) 3.16510e10 1.09757
\(975\) −4.20519e10 −1.45301
\(976\) 2.50066e10 0.860956
\(977\) −2.15287e10 −0.738563 −0.369282 0.929317i \(-0.620396\pi\)
−0.369282 + 0.929317i \(0.620396\pi\)
\(978\) −1.14099e11 −3.90027
\(979\) −3.71514e10 −1.26542
\(980\) −2.21275e9 −0.0751003
\(981\) 6.69766e10 2.26507
\(982\) 6.89761e9 0.232439
\(983\) 5.33254e8 0.0179059 0.00895296 0.999960i \(-0.497150\pi\)
0.00895296 + 0.999960i \(0.497150\pi\)
\(984\) 6.72628e10 2.25057
\(985\) −7.47568e9 −0.249243
\(986\) 1.56511e10 0.519966
\(987\) 5.76567e10 1.90871
\(988\) −1.67451e10 −0.552380
\(989\) 2.32925e9 0.0765648
\(990\) −5.81306e10 −1.90406
\(991\) −2.96780e10 −0.968671 −0.484336 0.874882i \(-0.660939\pi\)
−0.484336 + 0.874882i \(0.660939\pi\)
\(992\) 1.56827e10 0.510070
\(993\) −6.36551e10 −2.06306
\(994\) −2.01891e8 −0.00652025
\(995\) −4.81012e9 −0.154802
\(996\) −7.98469e9 −0.256065
\(997\) 3.33968e10 1.06726 0.533632 0.845717i \(-0.320827\pi\)
0.533632 + 0.845717i \(0.320827\pi\)
\(998\) 3.53822e10 1.12675
\(999\) −1.07843e11 −3.42226
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.1.4 11
3.2 odd 2 387.8.a.b.1.8 11
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.a.1.4 11 1.1 even 1 trivial
387.8.a.b.1.8 11 3.2 odd 2