Properties

Label 91.10.a.a.1.11
Level $91$
Weight $10$
Character 91.1
Self dual yes
Analytic conductor $46.868$
Analytic rank $1$
Dimension $12$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 91.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(46.8682610909\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \(x^{12} - 3 x^{11} - 4522 x^{10} + 11094 x^{9} + 7471016 x^{8} - 18339296 x^{7} - 5497728352 x^{6} + 13724467264 x^{5} + 1698856105344 x^{4} - 3404524011264 x^{3} - 154369782114304 x^{2} + 70325953652224 x + 170905444356096\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(31.8951\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

\(f(q)\) \(=\) \(q+29.8951 q^{2} -70.1432 q^{3} +381.717 q^{4} +1113.83 q^{5} -2096.94 q^{6} +2401.00 q^{7} -3894.82 q^{8} -14762.9 q^{9} +O(q^{10})\) \(q+29.8951 q^{2} -70.1432 q^{3} +381.717 q^{4} +1113.83 q^{5} -2096.94 q^{6} +2401.00 q^{7} -3894.82 q^{8} -14762.9 q^{9} +33298.0 q^{10} -33485.4 q^{11} -26774.9 q^{12} +28561.0 q^{13} +71778.1 q^{14} -78127.5 q^{15} -311875. q^{16} -157227. q^{17} -441339. q^{18} +309963. q^{19} +425168. q^{20} -168414. q^{21} -1.00105e6 q^{22} -796674. q^{23} +273195. q^{24} -712510. q^{25} +853834. q^{26} +2.41615e6 q^{27} +916503. q^{28} -1.10860e6 q^{29} -2.33563e6 q^{30} -2.97639e6 q^{31} -7.32939e6 q^{32} +2.34877e6 q^{33} -4.70031e6 q^{34} +2.67430e6 q^{35} -5.63526e6 q^{36} +4.80851e6 q^{37} +9.26637e6 q^{38} -2.00336e6 q^{39} -4.33816e6 q^{40} -2.00231e7 q^{41} -5.03475e6 q^{42} -3.06223e7 q^{43} -1.27819e7 q^{44} -1.64434e7 q^{45} -2.38167e7 q^{46} -6.06687e7 q^{47} +2.18759e7 q^{48} +5.76480e6 q^{49} -2.13006e7 q^{50} +1.10284e7 q^{51} +1.09022e7 q^{52} -4.36135e7 q^{53} +7.22310e7 q^{54} -3.72970e7 q^{55} -9.35146e6 q^{56} -2.17418e7 q^{57} -3.31416e7 q^{58} -3.11877e6 q^{59} -2.98226e7 q^{60} +3.50947e6 q^{61} -8.89795e7 q^{62} -3.54458e7 q^{63} -5.94328e7 q^{64} +3.18121e7 q^{65} +7.02168e7 q^{66} +2.57742e8 q^{67} -6.00162e7 q^{68} +5.58813e7 q^{69} +7.99486e7 q^{70} -1.34306e8 q^{71} +5.74990e7 q^{72} +2.83905e8 q^{73} +1.43751e8 q^{74} +4.99777e7 q^{75} +1.18318e8 q^{76} -8.03984e7 q^{77} -5.98906e7 q^{78} +1.55074e7 q^{79} -3.47376e8 q^{80} +1.21102e8 q^{81} -5.98593e8 q^{82} +4.31445e8 q^{83} -6.42864e7 q^{84} -1.75124e8 q^{85} -9.15456e8 q^{86} +7.77605e7 q^{87} +1.30420e8 q^{88} -5.70691e8 q^{89} -4.91577e8 q^{90} +6.85750e7 q^{91} -3.04104e8 q^{92} +2.08774e8 q^{93} -1.81370e9 q^{94} +3.45246e8 q^{95} +5.14107e8 q^{96} +2.89619e7 q^{97} +1.72339e8 q^{98} +4.94342e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 21 q^{2} - 323 q^{3} + 2945 q^{4} - 5202 q^{5} - 9897 q^{6} + 28812 q^{7} - 25431 q^{8} + 78519 q^{9} + O(q^{10}) \) \( 12 q - 21 q^{2} - 323 q^{3} + 2945 q^{4} - 5202 q^{5} - 9897 q^{6} + 28812 q^{7} - 25431 q^{8} + 78519 q^{9} - 65812 q^{10} - 80061 q^{11} - 184395 q^{12} + 342732 q^{13} - 50421 q^{14} + 160096 q^{15} + 385497 q^{16} - 1493598 q^{17} + 1520858 q^{18} - 109038 q^{19} - 622260 q^{20} - 775523 q^{21} + 4636975 q^{22} - 3367443 q^{23} - 5963895 q^{24} - 51480 q^{25} - 599781 q^{26} - 8158937 q^{27} + 7070945 q^{28} - 13333098 q^{29} + 2915424 q^{30} - 3954765 q^{31} + 4389297 q^{32} - 5790219 q^{33} + 14879968 q^{34} - 12490002 q^{35} + 80697058 q^{36} + 580535 q^{37} - 19134246 q^{38} - 9225203 q^{39} + 12365024 q^{40} - 27018171 q^{41} - 23762697 q^{42} + 31237588 q^{43} - 125053839 q^{44} - 62765470 q^{45} - 114008121 q^{46} - 21983709 q^{47} - 309724207 q^{48} + 69177612 q^{49} - 131331747 q^{50} - 176522692 q^{51} + 84112145 q^{52} - 196548234 q^{53} - 456152547 q^{54} - 309055872 q^{55} - 61059831 q^{56} - 274411494 q^{57} - 521980612 q^{58} - 215907906 q^{59} - 177006648 q^{60} - 218340705 q^{61} - 673289997 q^{62} + 188524119 q^{63} - 386667247 q^{64} - 148574322 q^{65} - 777397365 q^{66} + 14544775 q^{67} - 1246637448 q^{68} - 65252625 q^{69} - 158014612 q^{70} - 552451776 q^{71} + 369379470 q^{72} - 349395159 q^{73} + 73591023 q^{74} + 329300747 q^{75} - 1036299002 q^{76} - 192226461 q^{77} - 282668217 q^{78} + 962249727 q^{79} - 1494536184 q^{80} + 874458108 q^{81} - 1417698067 q^{82} - 2032575912 q^{83} - 442732395 q^{84} - 411671064 q^{85} - 2139249420 q^{86} - 759642172 q^{87} + 558651957 q^{88} - 280821684 q^{89} - 5764700804 q^{90} + 822899532 q^{91} - 4491569571 q^{92} - 1729557923 q^{93} - 1591372165 q^{94} - 1282463328 q^{95} - 2148993055 q^{96} - 2115165937 q^{97} - 121060821 q^{98} - 3595669198 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 29.8951 1.32119 0.660595 0.750743i \(-0.270304\pi\)
0.660595 + 0.750743i \(0.270304\pi\)
\(3\) −70.1432 −0.499965 −0.249983 0.968250i \(-0.580425\pi\)
−0.249983 + 0.968250i \(0.580425\pi\)
\(4\) 381.717 0.745541
\(5\) 1113.83 0.796991 0.398496 0.917170i \(-0.369532\pi\)
0.398496 + 0.917170i \(0.369532\pi\)
\(6\) −2096.94 −0.660549
\(7\) 2401.00 0.377964
\(8\) −3894.82 −0.336188
\(9\) −14762.9 −0.750035
\(10\) 33298.0 1.05298
\(11\) −33485.4 −0.689585 −0.344793 0.938679i \(-0.612051\pi\)
−0.344793 + 0.938679i \(0.612051\pi\)
\(12\) −26774.9 −0.372745
\(13\) 28561.0 0.277350
\(14\) 71778.1 0.499363
\(15\) −78127.5 −0.398468
\(16\) −311875. −1.18971
\(17\) −157227. −0.456569 −0.228285 0.973594i \(-0.573312\pi\)
−0.228285 + 0.973594i \(0.573312\pi\)
\(18\) −441339. −0.990938
\(19\) 309963. 0.545655 0.272828 0.962063i \(-0.412041\pi\)
0.272828 + 0.962063i \(0.412041\pi\)
\(20\) 425168. 0.594190
\(21\) −168414. −0.188969
\(22\) −1.00105e6 −0.911073
\(23\) −796674. −0.593616 −0.296808 0.954937i \(-0.595922\pi\)
−0.296808 + 0.954937i \(0.595922\pi\)
\(24\) 273195. 0.168082
\(25\) −712510. −0.364805
\(26\) 853834. 0.366432
\(27\) 2.41615e6 0.874957
\(28\) 916503. 0.281788
\(29\) −1.10860e6 −0.291060 −0.145530 0.989354i \(-0.546489\pi\)
−0.145530 + 0.989354i \(0.546489\pi\)
\(30\) −2.33563e6 −0.526452
\(31\) −2.97639e6 −0.578845 −0.289423 0.957201i \(-0.593463\pi\)
−0.289423 + 0.957201i \(0.593463\pi\)
\(32\) −7.32939e6 −1.23564
\(33\) 2.34877e6 0.344769
\(34\) −4.70031e6 −0.603214
\(35\) 2.67430e6 0.301234
\(36\) −5.63526e6 −0.559182
\(37\) 4.80851e6 0.421797 0.210898 0.977508i \(-0.432361\pi\)
0.210898 + 0.977508i \(0.432361\pi\)
\(38\) 9.26637e6 0.720914
\(39\) −2.00336e6 −0.138665
\(40\) −4.33816e6 −0.267939
\(41\) −2.00231e7 −1.10664 −0.553318 0.832970i \(-0.686638\pi\)
−0.553318 + 0.832970i \(0.686638\pi\)
\(42\) −5.03475e6 −0.249664
\(43\) −3.06223e7 −1.36593 −0.682966 0.730450i \(-0.739310\pi\)
−0.682966 + 0.730450i \(0.739310\pi\)
\(44\) −1.27819e7 −0.514114
\(45\) −1.64434e7 −0.597771
\(46\) −2.38167e7 −0.784279
\(47\) −6.06687e7 −1.81353 −0.906764 0.421638i \(-0.861455\pi\)
−0.906764 + 0.421638i \(0.861455\pi\)
\(48\) 2.18759e7 0.594814
\(49\) 5.76480e6 0.142857
\(50\) −2.13006e7 −0.481977
\(51\) 1.10284e7 0.228269
\(52\) 1.09022e7 0.206776
\(53\) −4.36135e7 −0.759241 −0.379620 0.925142i \(-0.623945\pi\)
−0.379620 + 0.925142i \(0.623945\pi\)
\(54\) 7.22310e7 1.15598
\(55\) −3.72970e7 −0.549593
\(56\) −9.35146e6 −0.127067
\(57\) −2.17418e7 −0.272809
\(58\) −3.31416e7 −0.384545
\(59\) −3.11877e6 −0.0335081 −0.0167540 0.999860i \(-0.505333\pi\)
−0.0167540 + 0.999860i \(0.505333\pi\)
\(60\) −2.98226e7 −0.297074
\(61\) 3.50947e6 0.0324532 0.0162266 0.999868i \(-0.494835\pi\)
0.0162266 + 0.999868i \(0.494835\pi\)
\(62\) −8.89795e7 −0.764764
\(63\) −3.54458e7 −0.283486
\(64\) −5.94328e7 −0.442809
\(65\) 3.18121e7 0.221046
\(66\) 7.02168e7 0.455505
\(67\) 2.57742e8 1.56260 0.781302 0.624154i \(-0.214556\pi\)
0.781302 + 0.624154i \(0.214556\pi\)
\(68\) −6.00162e7 −0.340391
\(69\) 5.58813e7 0.296787
\(70\) 7.99486e7 0.397988
\(71\) −1.34306e8 −0.627237 −0.313619 0.949549i \(-0.601541\pi\)
−0.313619 + 0.949549i \(0.601541\pi\)
\(72\) 5.74990e7 0.252153
\(73\) 2.83905e8 1.17009 0.585047 0.810999i \(-0.301076\pi\)
0.585047 + 0.810999i \(0.301076\pi\)
\(74\) 1.43751e8 0.557273
\(75\) 4.99777e7 0.182390
\(76\) 1.18318e8 0.406808
\(77\) −8.03984e7 −0.260639
\(78\) −5.98906e7 −0.183203
\(79\) 1.55074e7 0.0447936 0.0223968 0.999749i \(-0.492870\pi\)
0.0223968 + 0.999749i \(0.492870\pi\)
\(80\) −3.47376e8 −0.948188
\(81\) 1.21102e8 0.312587
\(82\) −5.98593e8 −1.46207
\(83\) 4.31445e8 0.997869 0.498935 0.866640i \(-0.333725\pi\)
0.498935 + 0.866640i \(0.333725\pi\)
\(84\) −6.42864e7 −0.140884
\(85\) −1.75124e8 −0.363881
\(86\) −9.15456e8 −1.80466
\(87\) 7.77605e7 0.145520
\(88\) 1.30420e8 0.231831
\(89\) −5.70691e8 −0.964153 −0.482076 0.876129i \(-0.660117\pi\)
−0.482076 + 0.876129i \(0.660117\pi\)
\(90\) −4.91577e8 −0.789769
\(91\) 6.85750e7 0.104828
\(92\) −3.04104e8 −0.442565
\(93\) 2.08774e8 0.289403
\(94\) −1.81370e9 −2.39601
\(95\) 3.45246e8 0.434882
\(96\) 5.14107e8 0.617779
\(97\) 2.89619e7 0.0332165 0.0166083 0.999862i \(-0.494713\pi\)
0.0166083 + 0.999862i \(0.494713\pi\)
\(98\) 1.72339e8 0.188741
\(99\) 4.94342e8 0.517213
\(100\) −2.71977e8 −0.271977
\(101\) −4.10378e8 −0.392408 −0.196204 0.980563i \(-0.562861\pi\)
−0.196204 + 0.980563i \(0.562861\pi\)
\(102\) 3.29695e8 0.301586
\(103\) −4.17058e8 −0.365114 −0.182557 0.983195i \(-0.558437\pi\)
−0.182557 + 0.983195i \(0.558437\pi\)
\(104\) −1.11240e8 −0.0932418
\(105\) −1.87584e8 −0.150607
\(106\) −1.30383e9 −1.00310
\(107\) −3.24207e7 −0.0239108 −0.0119554 0.999929i \(-0.503806\pi\)
−0.0119554 + 0.999929i \(0.503806\pi\)
\(108\) 9.22285e8 0.652316
\(109\) 2.58850e9 1.75642 0.878209 0.478276i \(-0.158738\pi\)
0.878209 + 0.478276i \(0.158738\pi\)
\(110\) −1.11500e9 −0.726117
\(111\) −3.37285e8 −0.210884
\(112\) −7.48812e8 −0.449668
\(113\) 9.93406e8 0.573157 0.286579 0.958057i \(-0.407482\pi\)
0.286579 + 0.958057i \(0.407482\pi\)
\(114\) −6.49973e8 −0.360432
\(115\) −8.87359e8 −0.473107
\(116\) −4.23170e8 −0.216997
\(117\) −4.21644e8 −0.208022
\(118\) −9.32359e7 −0.0442705
\(119\) −3.77502e8 −0.172567
\(120\) 3.04293e8 0.133960
\(121\) −1.23668e9 −0.524472
\(122\) 1.04916e8 0.0428768
\(123\) 1.40449e9 0.553279
\(124\) −1.13614e9 −0.431553
\(125\) −2.96906e9 −1.08774
\(126\) −1.05966e9 −0.374539
\(127\) −2.79338e8 −0.0952824 −0.0476412 0.998865i \(-0.515170\pi\)
−0.0476412 + 0.998865i \(0.515170\pi\)
\(128\) 1.97590e9 0.650609
\(129\) 2.14794e9 0.682919
\(130\) 9.51025e8 0.292043
\(131\) 3.52969e9 1.04717 0.523583 0.851975i \(-0.324595\pi\)
0.523583 + 0.851975i \(0.324595\pi\)
\(132\) 8.96566e8 0.257039
\(133\) 7.44221e8 0.206238
\(134\) 7.70523e9 2.06449
\(135\) 2.69118e9 0.697333
\(136\) 6.12370e8 0.153493
\(137\) −8.74657e7 −0.0212127 −0.0106063 0.999944i \(-0.503376\pi\)
−0.0106063 + 0.999944i \(0.503376\pi\)
\(138\) 1.67058e9 0.392112
\(139\) 2.64429e9 0.600818 0.300409 0.953811i \(-0.402877\pi\)
0.300409 + 0.953811i \(0.402877\pi\)
\(140\) 1.02083e9 0.224583
\(141\) 4.25550e9 0.906701
\(142\) −4.01508e9 −0.828699
\(143\) −9.56376e8 −0.191257
\(144\) 4.60419e9 0.892323
\(145\) −1.23479e9 −0.231972
\(146\) 8.48738e9 1.54592
\(147\) −4.04362e8 −0.0714236
\(148\) 1.83549e9 0.314467
\(149\) 5.42368e9 0.901480 0.450740 0.892655i \(-0.351160\pi\)
0.450740 + 0.892655i \(0.351160\pi\)
\(150\) 1.49409e9 0.240972
\(151\) 1.01732e9 0.159243 0.0796215 0.996825i \(-0.474629\pi\)
0.0796215 + 0.996825i \(0.474629\pi\)
\(152\) −1.20725e9 −0.183443
\(153\) 2.32113e9 0.342443
\(154\) −2.40352e9 −0.344353
\(155\) −3.31519e9 −0.461334
\(156\) −7.64717e8 −0.103381
\(157\) 9.64063e9 1.26636 0.633180 0.774005i \(-0.281749\pi\)
0.633180 + 0.774005i \(0.281749\pi\)
\(158\) 4.63594e8 0.0591808
\(159\) 3.05919e9 0.379594
\(160\) −8.16369e9 −0.984797
\(161\) −1.91282e9 −0.224366
\(162\) 3.62037e9 0.412986
\(163\) 3.18440e9 0.353332 0.176666 0.984271i \(-0.443469\pi\)
0.176666 + 0.984271i \(0.443469\pi\)
\(164\) −7.64317e9 −0.825042
\(165\) 2.61613e9 0.274778
\(166\) 1.28981e10 1.31837
\(167\) −3.55621e9 −0.353805 −0.176902 0.984228i \(-0.556608\pi\)
−0.176902 + 0.984228i \(0.556608\pi\)
\(168\) 6.55942e8 0.0635292
\(169\) 8.15731e8 0.0769231
\(170\) −5.23534e9 −0.480756
\(171\) −4.57596e9 −0.409260
\(172\) −1.16890e10 −1.01836
\(173\) −1.13899e10 −0.966749 −0.483375 0.875414i \(-0.660589\pi\)
−0.483375 + 0.875414i \(0.660589\pi\)
\(174\) 2.32466e9 0.192259
\(175\) −1.71074e9 −0.137883
\(176\) 1.04433e10 0.820406
\(177\) 2.18760e8 0.0167529
\(178\) −1.70609e10 −1.27383
\(179\) 7.23644e9 0.526849 0.263425 0.964680i \(-0.415148\pi\)
0.263425 + 0.964680i \(0.415148\pi\)
\(180\) −6.27672e9 −0.445663
\(181\) −5.99941e9 −0.415484 −0.207742 0.978184i \(-0.566612\pi\)
−0.207742 + 0.978184i \(0.566612\pi\)
\(182\) 2.05006e9 0.138498
\(183\) −2.46165e8 −0.0162255
\(184\) 3.10290e9 0.199567
\(185\) 5.35586e9 0.336168
\(186\) 6.24131e9 0.382356
\(187\) 5.26480e9 0.314843
\(188\) −2.31583e10 −1.35206
\(189\) 5.80117e9 0.330703
\(190\) 1.03211e10 0.574562
\(191\) −2.95421e10 −1.60617 −0.803084 0.595866i \(-0.796809\pi\)
−0.803084 + 0.595866i \(0.796809\pi\)
\(192\) 4.16881e9 0.221389
\(193\) −1.89971e9 −0.0985552 −0.0492776 0.998785i \(-0.515692\pi\)
−0.0492776 + 0.998785i \(0.515692\pi\)
\(194\) 8.65818e8 0.0438853
\(195\) −2.23140e9 −0.110515
\(196\) 2.20052e9 0.106506
\(197\) −1.73402e10 −0.820271 −0.410135 0.912025i \(-0.634519\pi\)
−0.410135 + 0.912025i \(0.634519\pi\)
\(198\) 1.47784e10 0.683336
\(199\) −1.86147e10 −0.841427 −0.420714 0.907194i \(-0.638220\pi\)
−0.420714 + 0.907194i \(0.638220\pi\)
\(200\) 2.77510e9 0.122643
\(201\) −1.80789e10 −0.781247
\(202\) −1.22683e10 −0.518445
\(203\) −2.66174e9 −0.110010
\(204\) 4.20973e9 0.170184
\(205\) −2.23023e10 −0.881978
\(206\) −1.24680e10 −0.482385
\(207\) 1.17613e10 0.445233
\(208\) −8.90747e9 −0.329966
\(209\) −1.03792e10 −0.376276
\(210\) −5.60785e9 −0.198980
\(211\) 4.92241e10 1.70965 0.854823 0.518919i \(-0.173665\pi\)
0.854823 + 0.518919i \(0.173665\pi\)
\(212\) −1.66480e10 −0.566045
\(213\) 9.42063e9 0.313597
\(214\) −9.69219e8 −0.0315907
\(215\) −3.41080e10 −1.08864
\(216\) −9.41046e9 −0.294150
\(217\) −7.14632e9 −0.218783
\(218\) 7.73833e10 2.32056
\(219\) −1.99140e10 −0.585006
\(220\) −1.42369e10 −0.409745
\(221\) −4.49056e9 −0.126629
\(222\) −1.00832e10 −0.278617
\(223\) −4.21527e10 −1.14144 −0.570721 0.821144i \(-0.693336\pi\)
−0.570721 + 0.821144i \(0.693336\pi\)
\(224\) −1.75979e10 −0.467029
\(225\) 1.05187e10 0.273617
\(226\) 2.96980e10 0.757249
\(227\) 3.70523e10 0.926188 0.463094 0.886309i \(-0.346739\pi\)
0.463094 + 0.886309i \(0.346739\pi\)
\(228\) −8.29921e9 −0.203390
\(229\) −5.49043e10 −1.31931 −0.659654 0.751569i \(-0.729297\pi\)
−0.659654 + 0.751569i \(0.729297\pi\)
\(230\) −2.65277e10 −0.625063
\(231\) 5.63940e9 0.130310
\(232\) 4.31778e9 0.0978509
\(233\) −4.97221e10 −1.10522 −0.552609 0.833441i \(-0.686368\pi\)
−0.552609 + 0.833441i \(0.686368\pi\)
\(234\) −1.26051e10 −0.274837
\(235\) −6.75746e10 −1.44537
\(236\) −1.19049e9 −0.0249816
\(237\) −1.08774e9 −0.0223952
\(238\) −1.12854e10 −0.227994
\(239\) 7.56250e10 1.49925 0.749627 0.661861i \(-0.230233\pi\)
0.749627 + 0.661861i \(0.230233\pi\)
\(240\) 2.43660e10 0.474061
\(241\) 1.96589e10 0.375389 0.187694 0.982227i \(-0.439899\pi\)
0.187694 + 0.982227i \(0.439899\pi\)
\(242\) −3.69706e10 −0.692927
\(243\) −5.60516e10 −1.03124
\(244\) 1.33962e9 0.0241952
\(245\) 6.42100e9 0.113856
\(246\) 4.19873e10 0.730987
\(247\) 8.85285e9 0.151338
\(248\) 1.15925e10 0.194601
\(249\) −3.02629e10 −0.498900
\(250\) −8.87604e10 −1.43711
\(251\) 8.25702e10 1.31308 0.656540 0.754291i \(-0.272019\pi\)
0.656540 + 0.754291i \(0.272019\pi\)
\(252\) −1.35303e10 −0.211351
\(253\) 2.66769e10 0.409349
\(254\) −8.35083e9 −0.125886
\(255\) 1.22837e10 0.181928
\(256\) 8.94993e10 1.30239
\(257\) −3.85626e10 −0.551401 −0.275700 0.961244i \(-0.588910\pi\)
−0.275700 + 0.961244i \(0.588910\pi\)
\(258\) 6.42130e10 0.902265
\(259\) 1.15452e10 0.159424
\(260\) 1.21432e10 0.164799
\(261\) 1.63661e10 0.218305
\(262\) 1.05520e11 1.38350
\(263\) 7.66052e10 0.987318 0.493659 0.869655i \(-0.335659\pi\)
0.493659 + 0.869655i \(0.335659\pi\)
\(264\) −9.14804e9 −0.115907
\(265\) −4.85780e10 −0.605108
\(266\) 2.22486e10 0.272480
\(267\) 4.00301e10 0.482043
\(268\) 9.83846e10 1.16499
\(269\) 1.41446e11 1.64705 0.823525 0.567280i \(-0.192004\pi\)
0.823525 + 0.567280i \(0.192004\pi\)
\(270\) 8.04530e10 0.921308
\(271\) −1.05413e11 −1.18722 −0.593612 0.804752i \(-0.702298\pi\)
−0.593612 + 0.804752i \(0.702298\pi\)
\(272\) 4.90351e10 0.543185
\(273\) −4.81007e9 −0.0524106
\(274\) −2.61480e9 −0.0280260
\(275\) 2.38587e10 0.251564
\(276\) 2.13308e10 0.221267
\(277\) −1.83792e11 −1.87572 −0.937861 0.347012i \(-0.887196\pi\)
−0.937861 + 0.347012i \(0.887196\pi\)
\(278\) 7.90514e10 0.793794
\(279\) 4.39403e10 0.434154
\(280\) −1.04159e10 −0.101271
\(281\) −1.34769e11 −1.28947 −0.644734 0.764407i \(-0.723032\pi\)
−0.644734 + 0.764407i \(0.723032\pi\)
\(282\) 1.27219e11 1.19792
\(283\) 5.13736e10 0.476103 0.238052 0.971252i \(-0.423491\pi\)
0.238052 + 0.971252i \(0.423491\pi\)
\(284\) −5.12668e10 −0.467631
\(285\) −2.42166e10 −0.217426
\(286\) −2.85910e10 −0.252686
\(287\) −4.80755e10 −0.418269
\(288\) 1.08203e11 0.926775
\(289\) −9.38676e10 −0.791545
\(290\) −3.69141e10 −0.306479
\(291\) −2.03148e9 −0.0166071
\(292\) 1.08372e11 0.872353
\(293\) −1.56721e11 −1.24229 −0.621147 0.783694i \(-0.713333\pi\)
−0.621147 + 0.783694i \(0.713333\pi\)
\(294\) −1.20884e10 −0.0943641
\(295\) −3.47378e9 −0.0267056
\(296\) −1.87283e10 −0.141803
\(297\) −8.09056e10 −0.603357
\(298\) 1.62142e11 1.19103
\(299\) −2.27538e10 −0.164639
\(300\) 1.90774e10 0.135979
\(301\) −7.35240e10 −0.516274
\(302\) 3.04128e10 0.210390
\(303\) 2.87852e10 0.196190
\(304\) −9.66697e10 −0.649171
\(305\) 3.90895e9 0.0258649
\(306\) 6.93904e10 0.452432
\(307\) −1.92332e11 −1.23574 −0.617871 0.786279i \(-0.712005\pi\)
−0.617871 + 0.786279i \(0.712005\pi\)
\(308\) −3.06894e10 −0.194317
\(309\) 2.92537e10 0.182544
\(310\) −9.91080e10 −0.609510
\(311\) 1.48197e11 0.898290 0.449145 0.893459i \(-0.351729\pi\)
0.449145 + 0.893459i \(0.351729\pi\)
\(312\) 7.80273e9 0.0466177
\(313\) −8.52035e10 −0.501774 −0.250887 0.968016i \(-0.580722\pi\)
−0.250887 + 0.968016i \(0.580722\pi\)
\(314\) 2.88208e11 1.67310
\(315\) −3.94806e10 −0.225936
\(316\) 5.91942e9 0.0333955
\(317\) −8.99689e10 −0.500410 −0.250205 0.968193i \(-0.580498\pi\)
−0.250205 + 0.968193i \(0.580498\pi\)
\(318\) 9.14548e10 0.501516
\(319\) 3.71218e10 0.200711
\(320\) −6.61980e10 −0.352915
\(321\) 2.27409e9 0.0119546
\(322\) −5.71838e10 −0.296430
\(323\) −4.87345e10 −0.249129
\(324\) 4.62269e10 0.233046
\(325\) −2.03500e10 −0.101179
\(326\) 9.51978e10 0.466818
\(327\) −1.81565e11 −0.878149
\(328\) 7.79865e10 0.372038
\(329\) −1.45666e11 −0.685449
\(330\) 7.82094e10 0.363033
\(331\) 8.80235e10 0.403063 0.201531 0.979482i \(-0.435408\pi\)
0.201531 + 0.979482i \(0.435408\pi\)
\(332\) 1.64690e11 0.743953
\(333\) −7.09878e10 −0.316362
\(334\) −1.06313e11 −0.467443
\(335\) 2.87081e11 1.24538
\(336\) 5.25241e10 0.224818
\(337\) 1.33487e11 0.563772 0.281886 0.959448i \(-0.409040\pi\)
0.281886 + 0.959448i \(0.409040\pi\)
\(338\) 2.43864e10 0.101630
\(339\) −6.96807e10 −0.286559
\(340\) −6.68477e10 −0.271289
\(341\) 9.96656e10 0.399163
\(342\) −1.36799e11 −0.540710
\(343\) 1.38413e10 0.0539949
\(344\) 1.19268e11 0.459210
\(345\) 6.22422e10 0.236537
\(346\) −3.40503e11 −1.27726
\(347\) −1.00900e11 −0.373601 −0.186801 0.982398i \(-0.559812\pi\)
−0.186801 + 0.982398i \(0.559812\pi\)
\(348\) 2.96825e10 0.108491
\(349\) −6.41636e10 −0.231513 −0.115756 0.993278i \(-0.536929\pi\)
−0.115756 + 0.993278i \(0.536929\pi\)
\(350\) −5.11427e10 −0.182170
\(351\) 6.90076e10 0.242669
\(352\) 2.45427e11 0.852082
\(353\) −1.38158e11 −0.473577 −0.236788 0.971561i \(-0.576095\pi\)
−0.236788 + 0.971561i \(0.576095\pi\)
\(354\) 6.53987e9 0.0221337
\(355\) −1.49594e11 −0.499902
\(356\) −2.17842e11 −0.718816
\(357\) 2.64792e10 0.0862775
\(358\) 2.16334e11 0.696068
\(359\) −1.31907e10 −0.0419125 −0.0209563 0.999780i \(-0.506671\pi\)
−0.0209563 + 0.999780i \(0.506671\pi\)
\(360\) 6.40440e10 0.200964
\(361\) −2.26611e11 −0.702260
\(362\) −1.79353e11 −0.548934
\(363\) 8.67445e10 0.262218
\(364\) 2.61762e10 0.0781540
\(365\) 3.16222e11 0.932554
\(366\) −7.35914e9 −0.0214369
\(367\) −5.69665e11 −1.63916 −0.819581 0.572963i \(-0.805794\pi\)
−0.819581 + 0.572963i \(0.805794\pi\)
\(368\) 2.48463e11 0.706231
\(369\) 2.95600e11 0.830015
\(370\) 1.60114e11 0.444142
\(371\) −1.04716e11 −0.286966
\(372\) 7.96925e10 0.215762
\(373\) 4.96716e11 1.32867 0.664337 0.747433i \(-0.268714\pi\)
0.664337 + 0.747433i \(0.268714\pi\)
\(374\) 1.57392e11 0.415968
\(375\) 2.08259e11 0.543831
\(376\) 2.36294e11 0.609687
\(377\) −3.16626e10 −0.0807255
\(378\) 1.73427e11 0.436921
\(379\) −7.50458e11 −1.86832 −0.934158 0.356861i \(-0.883847\pi\)
−0.934158 + 0.356861i \(0.883847\pi\)
\(380\) 1.31786e11 0.324223
\(381\) 1.95936e10 0.0476379
\(382\) −8.83164e11 −2.12205
\(383\) 1.65558e11 0.393149 0.196574 0.980489i \(-0.437018\pi\)
0.196574 + 0.980489i \(0.437018\pi\)
\(384\) −1.38596e11 −0.325282
\(385\) −8.95500e10 −0.207727
\(386\) −5.67920e10 −0.130210
\(387\) 4.52074e11 1.02450
\(388\) 1.10552e10 0.0247643
\(389\) 7.52301e11 1.66578 0.832891 0.553437i \(-0.186684\pi\)
0.832891 + 0.553437i \(0.186684\pi\)
\(390\) −6.67079e10 −0.146011
\(391\) 1.25259e11 0.271027
\(392\) −2.24529e10 −0.0480269
\(393\) −2.47583e11 −0.523546
\(394\) −5.18388e11 −1.08373
\(395\) 1.72725e10 0.0357001
\(396\) 1.88699e11 0.385604
\(397\) −5.82953e10 −0.117781 −0.0588906 0.998264i \(-0.518756\pi\)
−0.0588906 + 0.998264i \(0.518756\pi\)
\(398\) −5.56487e11 −1.11168
\(399\) −5.22020e10 −0.103112
\(400\) 2.22214e11 0.434012
\(401\) −1.84595e11 −0.356508 −0.178254 0.983984i \(-0.557045\pi\)
−0.178254 + 0.983984i \(0.557045\pi\)
\(402\) −5.40469e11 −1.03218
\(403\) −8.50087e10 −0.160543
\(404\) −1.56648e11 −0.292556
\(405\) 1.34887e11 0.249129
\(406\) −7.95730e10 −0.145344
\(407\) −1.61015e11 −0.290865
\(408\) −4.29536e10 −0.0767413
\(409\) −6.99412e11 −1.23588 −0.617942 0.786223i \(-0.712034\pi\)
−0.617942 + 0.786223i \(0.712034\pi\)
\(410\) −6.66731e11 −1.16526
\(411\) 6.13512e9 0.0106056
\(412\) −1.59198e11 −0.272208
\(413\) −7.48817e9 −0.0126649
\(414\) 3.51604e11 0.588237
\(415\) 4.80555e11 0.795293
\(416\) −2.09335e11 −0.342706
\(417\) −1.85479e11 −0.300388
\(418\) −3.10288e11 −0.497132
\(419\) −5.29034e11 −0.838534 −0.419267 0.907863i \(-0.637713\pi\)
−0.419267 + 0.907863i \(0.637713\pi\)
\(420\) −7.16041e10 −0.112284
\(421\) −8.31929e11 −1.29068 −0.645338 0.763897i \(-0.723283\pi\)
−0.645338 + 0.763897i \(0.723283\pi\)
\(422\) 1.47156e12 2.25877
\(423\) 8.95648e11 1.36021
\(424\) 1.69867e11 0.255248
\(425\) 1.12026e11 0.166559
\(426\) 2.81631e11 0.414321
\(427\) 8.42624e9 0.0122662
\(428\) −1.23755e10 −0.0178265
\(429\) 6.70833e10 0.0956217
\(430\) −1.01966e12 −1.43829
\(431\) −6.35897e11 −0.887645 −0.443823 0.896115i \(-0.646378\pi\)
−0.443823 + 0.896115i \(0.646378\pi\)
\(432\) −7.53537e11 −1.04094
\(433\) 9.71952e11 1.32877 0.664384 0.747392i \(-0.268694\pi\)
0.664384 + 0.747392i \(0.268694\pi\)
\(434\) −2.13640e11 −0.289054
\(435\) 8.66119e10 0.115978
\(436\) 9.88073e11 1.30948
\(437\) −2.46939e11 −0.323910
\(438\) −5.95332e11 −0.772904
\(439\) 9.59753e11 1.23330 0.616651 0.787237i \(-0.288489\pi\)
0.616651 + 0.787237i \(0.288489\pi\)
\(440\) 1.45265e11 0.184767
\(441\) −8.51054e10 −0.107148
\(442\) −1.34246e11 −0.167302
\(443\) 4.97592e11 0.613842 0.306921 0.951735i \(-0.400701\pi\)
0.306921 + 0.951735i \(0.400701\pi\)
\(444\) −1.28747e11 −0.157223
\(445\) −6.35652e11 −0.768421
\(446\) −1.26016e12 −1.50806
\(447\) −3.80434e11 −0.450709
\(448\) −1.42698e11 −0.167366
\(449\) −3.20380e11 −0.372011 −0.186006 0.982549i \(-0.559554\pi\)
−0.186006 + 0.982549i \(0.559554\pi\)
\(450\) 3.14459e11 0.361499
\(451\) 6.70482e11 0.763120
\(452\) 3.79200e11 0.427312
\(453\) −7.13579e10 −0.0796160
\(454\) 1.10768e12 1.22367
\(455\) 7.63808e10 0.0835474
\(456\) 8.46803e10 0.0917151
\(457\) −1.61657e11 −0.173370 −0.0866848 0.996236i \(-0.527627\pi\)
−0.0866848 + 0.996236i \(0.527627\pi\)
\(458\) −1.64137e12 −1.74306
\(459\) −3.79883e11 −0.399478
\(460\) −3.38720e11 −0.352721
\(461\) 8.13199e11 0.838576 0.419288 0.907853i \(-0.362280\pi\)
0.419288 + 0.907853i \(0.362280\pi\)
\(462\) 1.68590e11 0.172165
\(463\) 2.02415e11 0.204705 0.102353 0.994748i \(-0.467363\pi\)
0.102353 + 0.994748i \(0.467363\pi\)
\(464\) 3.45744e11 0.346277
\(465\) 2.32538e11 0.230651
\(466\) −1.48645e12 −1.46020
\(467\) −3.73598e11 −0.363478 −0.181739 0.983347i \(-0.558173\pi\)
−0.181739 + 0.983347i \(0.558173\pi\)
\(468\) −1.60949e11 −0.155089
\(469\) 6.18839e11 0.590608
\(470\) −2.02015e12 −1.90960
\(471\) −6.76225e11 −0.633136
\(472\) 1.21470e10 0.0112650
\(473\) 1.02540e12 0.941927
\(474\) −3.25180e10 −0.0295883
\(475\) −2.20852e11 −0.199058
\(476\) −1.44099e11 −0.128656
\(477\) 6.43863e11 0.569457
\(478\) 2.26082e12 1.98080
\(479\) −1.12289e12 −0.974600 −0.487300 0.873235i \(-0.662018\pi\)
−0.487300 + 0.873235i \(0.662018\pi\)
\(480\) 5.72627e11 0.492364
\(481\) 1.37336e11 0.116985
\(482\) 5.87703e11 0.495960
\(483\) 1.34171e11 0.112175
\(484\) −4.72061e11 −0.391015
\(485\) 3.22586e10 0.0264733
\(486\) −1.67567e12 −1.36246
\(487\) 3.09213e11 0.249102 0.124551 0.992213i \(-0.460251\pi\)
0.124551 + 0.992213i \(0.460251\pi\)
\(488\) −1.36688e10 −0.0109104
\(489\) −2.23364e11 −0.176654
\(490\) 1.91957e11 0.150425
\(491\) −7.16983e11 −0.556727 −0.278363 0.960476i \(-0.589792\pi\)
−0.278363 + 0.960476i \(0.589792\pi\)
\(492\) 5.36116e11 0.412493
\(493\) 1.74301e11 0.132889
\(494\) 2.64657e11 0.199946
\(495\) 5.50613e11 0.412214
\(496\) 9.28263e11 0.688658
\(497\) −3.22468e11 −0.237073
\(498\) −9.04713e11 −0.659141
\(499\) 1.84974e12 1.33554 0.667770 0.744367i \(-0.267249\pi\)
0.667770 + 0.744367i \(0.267249\pi\)
\(500\) −1.13334e12 −0.810953
\(501\) 2.49444e11 0.176890
\(502\) 2.46844e12 1.73483
\(503\) 2.12992e12 1.48357 0.741784 0.670639i \(-0.233980\pi\)
0.741784 + 0.670639i \(0.233980\pi\)
\(504\) 1.38055e11 0.0953048
\(505\) −4.57091e11 −0.312746
\(506\) 7.97510e11 0.540827
\(507\) −5.72180e10 −0.0384589
\(508\) −1.06628e11 −0.0710370
\(509\) −9.55257e11 −0.630798 −0.315399 0.948959i \(-0.602138\pi\)
−0.315399 + 0.948959i \(0.602138\pi\)
\(510\) 3.67224e11 0.240361
\(511\) 6.81657e11 0.442254
\(512\) 1.66393e12 1.07009
\(513\) 7.48916e11 0.477425
\(514\) −1.15283e12 −0.728505
\(515\) −4.64531e11 −0.290993
\(516\) 8.19907e11 0.509144
\(517\) 2.03151e12 1.25058
\(518\) 3.45146e11 0.210630
\(519\) 7.98927e11 0.483341
\(520\) −1.23902e11 −0.0743129
\(521\) 1.31386e12 0.781230 0.390615 0.920554i \(-0.372262\pi\)
0.390615 + 0.920554i \(0.372262\pi\)
\(522\) 4.89267e11 0.288422
\(523\) −3.94633e11 −0.230640 −0.115320 0.993328i \(-0.536789\pi\)
−0.115320 + 0.993328i \(0.536789\pi\)
\(524\) 1.34734e12 0.780705
\(525\) 1.19997e11 0.0689369
\(526\) 2.29012e12 1.30443
\(527\) 4.67969e11 0.264283
\(528\) −7.32524e11 −0.410175
\(529\) −1.16646e12 −0.647620
\(530\) −1.45224e12 −0.799462
\(531\) 4.60422e10 0.0251322
\(532\) 2.84082e11 0.153759
\(533\) −5.71881e11 −0.306925
\(534\) 1.19670e12 0.636870
\(535\) −3.61111e10 −0.0190567
\(536\) −1.00386e12 −0.525329
\(537\) −5.07587e11 −0.263406
\(538\) 4.22856e12 2.17607
\(539\) −1.93037e11 −0.0985122
\(540\) 1.02727e12 0.519890
\(541\) 2.62837e12 1.31916 0.659582 0.751632i \(-0.270733\pi\)
0.659582 + 0.751632i \(0.270733\pi\)
\(542\) −3.15133e12 −1.56855
\(543\) 4.20818e11 0.207728
\(544\) 1.15238e12 0.564156
\(545\) 2.88314e12 1.39985
\(546\) −1.43797e11 −0.0692443
\(547\) 1.85373e12 0.885325 0.442663 0.896688i \(-0.354034\pi\)
0.442663 + 0.896688i \(0.354034\pi\)
\(548\) −3.33872e10 −0.0158149
\(549\) −5.18101e10 −0.0243410
\(550\) 7.13257e11 0.332364
\(551\) −3.43624e11 −0.158818
\(552\) −2.17648e11 −0.0997764
\(553\) 3.72332e10 0.0169304
\(554\) −5.49449e12 −2.47818
\(555\) −3.75677e11 −0.168072
\(556\) 1.00937e12 0.447934
\(557\) −4.50145e12 −1.98155 −0.990774 0.135527i \(-0.956727\pi\)
−0.990774 + 0.135527i \(0.956727\pi\)
\(558\) 1.31360e12 0.573600
\(559\) −8.74602e11 −0.378841
\(560\) −8.34049e11 −0.358381
\(561\) −3.69290e11 −0.157411
\(562\) −4.02893e12 −1.70363
\(563\) 4.31343e11 0.180940 0.0904700 0.995899i \(-0.471163\pi\)
0.0904700 + 0.995899i \(0.471163\pi\)
\(564\) 1.62440e12 0.675983
\(565\) 1.10648e12 0.456801
\(566\) 1.53582e12 0.629023
\(567\) 2.90767e11 0.118147
\(568\) 5.23096e11 0.210870
\(569\) −6.31363e11 −0.252507 −0.126254 0.991998i \(-0.540295\pi\)
−0.126254 + 0.991998i \(0.540295\pi\)
\(570\) −7.23958e11 −0.287261
\(571\) 4.45553e12 1.75403 0.877015 0.480463i \(-0.159531\pi\)
0.877015 + 0.480463i \(0.159531\pi\)
\(572\) −3.65065e11 −0.142590
\(573\) 2.07218e12 0.803028
\(574\) −1.43722e12 −0.552612
\(575\) 5.67639e11 0.216554
\(576\) 8.77403e11 0.332122
\(577\) −7.52025e11 −0.282450 −0.141225 0.989978i \(-0.545104\pi\)
−0.141225 + 0.989978i \(0.545104\pi\)
\(578\) −2.80618e12 −1.04578
\(579\) 1.33252e11 0.0492742
\(580\) −4.71339e11 −0.172945
\(581\) 1.03590e12 0.377159
\(582\) −6.07313e10 −0.0219411
\(583\) 1.46041e12 0.523561
\(584\) −1.10576e12 −0.393372
\(585\) −4.69639e11 −0.165792
\(586\) −4.68520e12 −1.64130
\(587\) −3.19638e12 −1.11119 −0.555593 0.831454i \(-0.687509\pi\)
−0.555593 + 0.831454i \(0.687509\pi\)
\(588\) −1.54352e11 −0.0532493
\(589\) −9.22571e11 −0.315850
\(590\) −1.03849e11 −0.0352832
\(591\) 1.21630e12 0.410107
\(592\) −1.49966e12 −0.501816
\(593\) −4.40773e12 −1.46376 −0.731878 0.681436i \(-0.761356\pi\)
−0.731878 + 0.681436i \(0.761356\pi\)
\(594\) −2.41868e12 −0.797149
\(595\) −4.20472e11 −0.137534
\(596\) 2.07031e12 0.672090
\(597\) 1.30569e12 0.420684
\(598\) −6.80228e11 −0.217520
\(599\) −5.68564e12 −1.80451 −0.902254 0.431205i \(-0.858089\pi\)
−0.902254 + 0.431205i \(0.858089\pi\)
\(600\) −1.94654e11 −0.0613174
\(601\) −1.17360e12 −0.366933 −0.183466 0.983026i \(-0.558732\pi\)
−0.183466 + 0.983026i \(0.558732\pi\)
\(602\) −2.19801e12 −0.682096
\(603\) −3.80503e12 −1.17201
\(604\) 3.88328e11 0.118722
\(605\) −1.37745e12 −0.417999
\(606\) 8.60537e11 0.259205
\(607\) −4.69756e11 −0.140450 −0.0702252 0.997531i \(-0.522372\pi\)
−0.0702252 + 0.997531i \(0.522372\pi\)
\(608\) −2.27184e12 −0.674235
\(609\) 1.86703e11 0.0550013
\(610\) 1.16858e11 0.0341724
\(611\) −1.73276e12 −0.502982
\(612\) 8.86015e11 0.255305
\(613\) −3.95664e12 −1.13176 −0.565880 0.824488i \(-0.691463\pi\)
−0.565880 + 0.824488i \(0.691463\pi\)
\(614\) −5.74977e12 −1.63265
\(615\) 1.56436e12 0.440959
\(616\) 3.13137e11 0.0876237
\(617\) −3.22974e12 −0.897189 −0.448594 0.893735i \(-0.648075\pi\)
−0.448594 + 0.893735i \(0.648075\pi\)
\(618\) 8.74544e11 0.241176
\(619\) 2.67612e11 0.0732651 0.0366326 0.999329i \(-0.488337\pi\)
0.0366326 + 0.999329i \(0.488337\pi\)
\(620\) −1.26547e12 −0.343944
\(621\) −1.92488e12 −0.519388
\(622\) 4.43035e12 1.18681
\(623\) −1.37023e12 −0.364416
\(624\) 6.24798e11 0.164972
\(625\) −1.91541e12 −0.502112
\(626\) −2.54717e12 −0.662938
\(627\) 7.28032e11 0.188125
\(628\) 3.67999e12 0.944123
\(629\) −7.56027e11 −0.192579
\(630\) −1.18028e12 −0.298504
\(631\) 3.92920e12 0.986670 0.493335 0.869839i \(-0.335778\pi\)
0.493335 + 0.869839i \(0.335778\pi\)
\(632\) −6.03984e10 −0.0150591
\(633\) −3.45273e12 −0.854764
\(634\) −2.68963e12 −0.661136
\(635\) −3.11134e11 −0.0759393
\(636\) 1.16774e12 0.283003
\(637\) 1.64648e11 0.0396214
\(638\) 1.10976e12 0.265177
\(639\) 1.98275e12 0.470450
\(640\) 2.20081e12 0.518529
\(641\) −8.01874e11 −0.187605 −0.0938026 0.995591i \(-0.529902\pi\)
−0.0938026 + 0.995591i \(0.529902\pi\)
\(642\) 6.79841e10 0.0157943
\(643\) −2.60093e12 −0.600039 −0.300019 0.953933i \(-0.596993\pi\)
−0.300019 + 0.953933i \(0.596993\pi\)
\(644\) −7.30154e11 −0.167274
\(645\) 2.39244e12 0.544280
\(646\) −1.45692e12 −0.329147
\(647\) 4.79153e12 1.07499 0.537496 0.843266i \(-0.319370\pi\)
0.537496 + 0.843266i \(0.319370\pi\)
\(648\) −4.71672e11 −0.105088
\(649\) 1.04433e11 0.0231067
\(650\) −6.08365e11 −0.133676
\(651\) 5.01265e11 0.109384
\(652\) 1.21554e12 0.263423
\(653\) 5.97413e12 1.28578 0.642888 0.765960i \(-0.277736\pi\)
0.642888 + 0.765960i \(0.277736\pi\)
\(654\) −5.42791e12 −1.16020
\(655\) 3.93147e12 0.834581
\(656\) 6.24472e12 1.31657
\(657\) −4.19128e12 −0.877611
\(658\) −4.35469e12 −0.905608
\(659\) 7.35551e11 0.151925 0.0759624 0.997111i \(-0.475797\pi\)
0.0759624 + 0.997111i \(0.475797\pi\)
\(660\) 9.98621e11 0.204858
\(661\) 2.43174e12 0.495463 0.247731 0.968829i \(-0.420315\pi\)
0.247731 + 0.968829i \(0.420315\pi\)
\(662\) 2.63147e12 0.532522
\(663\) 3.14982e11 0.0633103
\(664\) −1.68040e12 −0.335472
\(665\) 8.28934e11 0.164370
\(666\) −2.12219e12 −0.417974
\(667\) 8.83190e11 0.172778
\(668\) −1.35747e12 −0.263776
\(669\) 2.95673e12 0.570681
\(670\) 8.58230e12 1.64538
\(671\) −1.17516e11 −0.0223792
\(672\) 1.23437e12 0.233498
\(673\) −2.68993e12 −0.505445 −0.252722 0.967539i \(-0.581326\pi\)
−0.252722 + 0.967539i \(0.581326\pi\)
\(674\) 3.99060e12 0.744849
\(675\) −1.72153e12 −0.319189
\(676\) 3.11378e11 0.0573493
\(677\) −7.99672e12 −1.46306 −0.731531 0.681808i \(-0.761194\pi\)
−0.731531 + 0.681808i \(0.761194\pi\)
\(678\) −2.08311e12 −0.378598
\(679\) 6.95375e10 0.0125547
\(680\) 6.82076e11 0.122333
\(681\) −2.59897e12 −0.463062
\(682\) 2.97951e12 0.527370
\(683\) 3.73678e12 0.657060 0.328530 0.944494i \(-0.393447\pi\)
0.328530 + 0.944494i \(0.393447\pi\)
\(684\) −1.74672e12 −0.305120
\(685\) −9.74218e10 −0.0169063
\(686\) 4.13787e11 0.0713375
\(687\) 3.85116e12 0.659609
\(688\) 9.55032e12 1.62506
\(689\) −1.24564e12 −0.210575
\(690\) 1.86074e12 0.312510
\(691\) −2.29823e11 −0.0383479 −0.0191740 0.999816i \(-0.506104\pi\)
−0.0191740 + 0.999816i \(0.506104\pi\)
\(692\) −4.34773e12 −0.720752
\(693\) 1.18692e12 0.195488
\(694\) −3.01641e12 −0.493598
\(695\) 2.94529e12 0.478846
\(696\) −3.02863e11 −0.0489221
\(697\) 3.14817e12 0.505255
\(698\) −1.91818e12 −0.305872
\(699\) 3.48767e12 0.552570
\(700\) −6.53018e11 −0.102798
\(701\) 9.79164e12 1.53153 0.765763 0.643123i \(-0.222362\pi\)
0.765763 + 0.643123i \(0.222362\pi\)
\(702\) 2.06299e12 0.320612
\(703\) 1.49046e12 0.230156
\(704\) 1.99013e12 0.305355
\(705\) 4.73990e12 0.722633
\(706\) −4.13025e12 −0.625685
\(707\) −9.85318e11 −0.148316
\(708\) 8.35046e10 0.0124900
\(709\) 5.44585e12 0.809389 0.404695 0.914452i \(-0.367378\pi\)
0.404695 + 0.914452i \(0.367378\pi\)
\(710\) −4.47211e12 −0.660466
\(711\) −2.28934e11 −0.0335967
\(712\) 2.22274e12 0.324137
\(713\) 2.37122e12 0.343612
\(714\) 7.91597e11 0.113989
\(715\) −1.06524e12 −0.152430
\(716\) 2.76227e12 0.392788
\(717\) −5.30458e12 −0.749575
\(718\) −3.94338e11 −0.0553744
\(719\) −7.91110e12 −1.10397 −0.551985 0.833854i \(-0.686129\pi\)
−0.551985 + 0.833854i \(0.686129\pi\)
\(720\) 5.12828e12 0.711174
\(721\) −1.00136e12 −0.138000
\(722\) −6.77455e12 −0.927819
\(723\) −1.37893e12 −0.187681
\(724\) −2.29008e12 −0.309761
\(725\) 7.89886e11 0.106180
\(726\) 2.59324e12 0.346439
\(727\) −7.47380e12 −0.992286 −0.496143 0.868241i \(-0.665251\pi\)
−0.496143 + 0.868241i \(0.665251\pi\)
\(728\) −2.67087e11 −0.0352421
\(729\) 1.54797e12 0.202997
\(730\) 9.45349e12 1.23208
\(731\) 4.81464e12 0.623642
\(732\) −9.39656e10 −0.0120968
\(733\) −2.41459e12 −0.308941 −0.154470 0.987997i \(-0.549367\pi\)
−0.154470 + 0.987997i \(0.549367\pi\)
\(734\) −1.70302e13 −2.16564
\(735\) −4.50390e11 −0.0569240
\(736\) 5.83914e12 0.733498
\(737\) −8.63059e12 −1.07755
\(738\) 8.83699e12 1.09661
\(739\) −7.85537e12 −0.968872 −0.484436 0.874827i \(-0.660975\pi\)
−0.484436 + 0.874827i \(0.660975\pi\)
\(740\) 2.04442e12 0.250627
\(741\) −6.20967e11 −0.0756635
\(742\) −3.13050e12 −0.379136
\(743\) 9.59756e12 1.15534 0.577672 0.816269i \(-0.303961\pi\)
0.577672 + 0.816269i \(0.303961\pi\)
\(744\) −8.13136e11 −0.0972937
\(745\) 6.04105e12 0.718471
\(746\) 1.48494e13 1.75543
\(747\) −6.36939e12 −0.748437
\(748\) 2.00966e12 0.234729
\(749\) −7.78420e10 −0.00903745
\(750\) 6.22594e12 0.718504
\(751\) −6.26106e12 −0.718237 −0.359119 0.933292i \(-0.616923\pi\)
−0.359119 + 0.933292i \(0.616923\pi\)
\(752\) 1.89211e13 2.15757
\(753\) −5.79174e12 −0.656495
\(754\) −9.46557e11 −0.106654
\(755\) 1.13312e12 0.126915
\(756\) 2.21441e12 0.246552
\(757\) 1.57603e12 0.174435 0.0872175 0.996189i \(-0.472203\pi\)
0.0872175 + 0.996189i \(0.472203\pi\)
\(758\) −2.24350e13 −2.46840
\(759\) −1.87121e12 −0.204660
\(760\) −1.34467e12 −0.146202
\(761\) −2.47789e12 −0.267825 −0.133912 0.990993i \(-0.542754\pi\)
−0.133912 + 0.990993i \(0.542754\pi\)
\(762\) 5.85754e11 0.0629387
\(763\) 6.21498e12 0.663864
\(764\) −1.12767e13 −1.19746
\(765\) 2.58534e12 0.272924
\(766\) 4.94938e12 0.519424
\(767\) −8.90752e10 −0.00929346
\(768\) −6.27777e12 −0.651148
\(769\) 8.56237e12 0.882929 0.441464 0.897279i \(-0.354459\pi\)
0.441464 + 0.897279i \(0.354459\pi\)
\(770\) −2.67711e12 −0.274446
\(771\) 2.70490e12 0.275681
\(772\) −7.25152e11 −0.0734770
\(773\) −1.18439e13 −1.19313 −0.596563 0.802567i \(-0.703467\pi\)
−0.596563 + 0.802567i \(0.703467\pi\)
\(774\) 1.35148e13 1.35355
\(775\) 2.12071e12 0.211166
\(776\) −1.12801e11 −0.0111670
\(777\) −8.09820e11 −0.0797066
\(778\) 2.24901e13 2.20081
\(779\) −6.20642e12 −0.603841
\(780\) −8.51764e11 −0.0823936
\(781\) 4.49728e12 0.432534
\(782\) 3.74462e12 0.358078
\(783\) −2.67853e12 −0.254665
\(784\) −1.79790e12 −0.169959
\(785\) 1.07380e13 1.00928
\(786\) −7.40153e12 −0.691704
\(787\) 2.00273e13 1.86096 0.930479 0.366345i \(-0.119391\pi\)
0.930479 + 0.366345i \(0.119391\pi\)
\(788\) −6.61907e12 −0.611546
\(789\) −5.37333e12 −0.493625
\(790\) 5.16364e11 0.0471666
\(791\) 2.38517e12 0.216633
\(792\) −1.92537e12 −0.173881
\(793\) 1.00234e11 0.00900089
\(794\) −1.74274e12 −0.155611
\(795\) 3.40741e12 0.302533
\(796\) −7.10554e12 −0.627319
\(797\) −1.83555e13 −1.61140 −0.805702 0.592321i \(-0.798212\pi\)
−0.805702 + 0.592321i \(0.798212\pi\)
\(798\) −1.56058e12 −0.136230
\(799\) 9.53875e12 0.828001
\(800\) 5.22227e12 0.450769
\(801\) 8.42507e12 0.723148
\(802\) −5.51848e12 −0.471015
\(803\) −9.50668e12 −0.806880
\(804\) −6.90101e12 −0.582452
\(805\) −2.13055e12 −0.178818
\(806\) −2.54134e12 −0.212107
\(807\) −9.92151e12 −0.823468
\(808\) 1.59835e12 0.131923
\(809\) 3.82815e11 0.0314211 0.0157105 0.999877i \(-0.494999\pi\)
0.0157105 + 0.999877i \(0.494999\pi\)
\(810\) 4.03247e12 0.329146
\(811\) 1.28299e13 1.04143 0.520713 0.853732i \(-0.325666\pi\)
0.520713 + 0.853732i \(0.325666\pi\)
\(812\) −1.01603e12 −0.0820172
\(813\) 7.39400e12 0.593570
\(814\) −4.81356e12 −0.384288
\(815\) 3.54687e12 0.281602
\(816\) −3.43948e12 −0.271573
\(817\) −9.49176e12 −0.745328
\(818\) −2.09090e13 −1.63284
\(819\) −1.01237e12 −0.0786250
\(820\) −8.51318e12 −0.657551
\(821\) 1.24894e13 0.959394 0.479697 0.877434i \(-0.340747\pi\)
0.479697 + 0.877434i \(0.340747\pi\)
\(822\) 1.83410e11 0.0140120
\(823\) −1.15679e13 −0.878931 −0.439465 0.898260i \(-0.644832\pi\)
−0.439465 + 0.898260i \(0.644832\pi\)
\(824\) 1.62436e12 0.122747
\(825\) −1.67352e12 −0.125773
\(826\) −2.23859e11 −0.0167327
\(827\) 7.47785e12 0.555907 0.277953 0.960595i \(-0.410344\pi\)
0.277953 + 0.960595i \(0.410344\pi\)
\(828\) 4.48947e12 0.331939
\(829\) −6.46169e12 −0.475172 −0.237586 0.971367i \(-0.576356\pi\)
−0.237586 + 0.971367i \(0.576356\pi\)
\(830\) 1.43663e13 1.05073
\(831\) 1.28918e13 0.937796
\(832\) −1.69746e12 −0.122813
\(833\) −9.06381e11 −0.0652242
\(834\) −5.54492e12 −0.396870
\(835\) −3.96101e12 −0.281979
\(836\) −3.96193e12 −0.280529
\(837\) −7.19140e12 −0.506465
\(838\) −1.58155e13 −1.10786
\(839\) 4.80572e12 0.334834 0.167417 0.985886i \(-0.446457\pi\)
0.167417 + 0.985886i \(0.446457\pi\)
\(840\) 7.30607e11 0.0506322
\(841\) −1.32782e13 −0.915284
\(842\) −2.48706e13 −1.70523
\(843\) 9.45311e12 0.644690
\(844\) 1.87897e13 1.27461
\(845\) 9.08584e11 0.0613070
\(846\) 2.67755e13 1.79709
\(847\) −2.96926e12 −0.198232
\(848\) 1.36020e13 0.903276
\(849\) −3.60351e12 −0.238035
\(850\) 3.34902e12 0.220056
\(851\) −3.83082e12 −0.250385
\(852\) 3.59601e12 0.233799
\(853\) 8.41348e12 0.544133 0.272066 0.962278i \(-0.412293\pi\)
0.272066 + 0.962278i \(0.412293\pi\)
\(854\) 2.51903e11 0.0162059
\(855\) −5.09684e12 −0.326177
\(856\) 1.26273e11 0.00803854
\(857\) 3.95026e12 0.250157 0.125078 0.992147i \(-0.460082\pi\)
0.125078 + 0.992147i \(0.460082\pi\)
\(858\) 2.00546e12 0.126334
\(859\) 7.00057e12 0.438696 0.219348 0.975647i \(-0.429607\pi\)
0.219348 + 0.975647i \(0.429607\pi\)
\(860\) −1.30196e13 −0.811623
\(861\) 3.37217e12 0.209120
\(862\) −1.90102e13 −1.17275
\(863\) −2.02805e13 −1.24460 −0.622299 0.782780i \(-0.713801\pi\)
−0.622299 + 0.782780i \(0.713801\pi\)
\(864\) −1.77089e13 −1.08113
\(865\) −1.26864e13 −0.770491
\(866\) 2.90566e13 1.75555
\(867\) 6.58417e12 0.395745
\(868\) −2.72787e12 −0.163112
\(869\) −5.19270e11 −0.0308890
\(870\) 2.58927e12 0.153229
\(871\) 7.36137e12 0.433388
\(872\) −1.00817e13 −0.590487
\(873\) −4.27562e11 −0.0249135
\(874\) −7.38228e12 −0.427946
\(875\) −7.12872e12 −0.411126
\(876\) −7.60153e12 −0.436146
\(877\) −5.14430e12 −0.293649 −0.146824 0.989163i \(-0.546905\pi\)
−0.146824 + 0.989163i \(0.546905\pi\)
\(878\) 2.86919e13 1.62942
\(879\) 1.09929e13 0.621104
\(880\) 1.16320e13 0.653857
\(881\) −1.26062e13 −0.705004 −0.352502 0.935811i \(-0.614669\pi\)
−0.352502 + 0.935811i \(0.614669\pi\)
\(882\) −2.54423e12 −0.141563
\(883\) 1.72689e13 0.955963 0.477981 0.878370i \(-0.341369\pi\)
0.477981 + 0.878370i \(0.341369\pi\)
\(884\) −1.71412e12 −0.0944075
\(885\) 2.43662e11 0.0133519
\(886\) 1.48756e13 0.811001
\(887\) −2.25561e12 −0.122351 −0.0611755 0.998127i \(-0.519485\pi\)
−0.0611755 + 0.998127i \(0.519485\pi\)
\(888\) 1.31366e12 0.0708966
\(889\) −6.70690e11 −0.0360134
\(890\) −1.90029e13 −1.01523
\(891\) −4.05516e12 −0.215555
\(892\) −1.60904e13 −0.850991
\(893\) −1.88050e13 −0.989561
\(894\) −1.13731e13 −0.595471
\(895\) 8.06016e12 0.419894
\(896\) 4.74413e12 0.245907
\(897\) 1.59603e12 0.0823140
\(898\) −9.57778e12 −0.491498
\(899\) 3.29962e12 0.168479
\(900\) 4.01518e12 0.203992
\(901\) 6.85721e12 0.346646
\(902\) 2.00441e13 1.00823
\(903\) 5.15721e12 0.258119
\(904\) −3.86914e12 −0.192689
\(905\) −6.68231e12 −0.331137
\(906\) −2.13325e12 −0.105188
\(907\) −1.98957e13 −0.976172 −0.488086 0.872796i \(-0.662305\pi\)
−0.488086 + 0.872796i \(0.662305\pi\)
\(908\) 1.41435e13 0.690511
\(909\) 6.05838e12 0.294320
\(910\) 2.28341e12 0.110382
\(911\) 1.23828e13 0.595643 0.297822 0.954622i \(-0.403740\pi\)
0.297822 + 0.954622i \(0.403740\pi\)
\(912\) 6.78072e12 0.324563
\(913\) −1.44471e13 −0.688116
\(914\) −4.83276e12 −0.229054
\(915\) −2.74186e11 −0.0129316
\(916\) −2.09579e13 −0.983599
\(917\) 8.47477e12 0.395791
\(918\) −1.13566e13 −0.527786
\(919\) −3.21756e13 −1.48801 −0.744007 0.668172i \(-0.767077\pi\)
−0.744007 + 0.668172i \(0.767077\pi\)
\(920\) 3.45610e12 0.159053
\(921\) 1.34908e13 0.617828
\(922\) 2.43107e13 1.10792
\(923\) −3.83590e12 −0.173964
\(924\) 2.15266e12 0.0971517
\(925\) −3.42612e12 −0.153874
\(926\) 6.05123e12 0.270454
\(927\) 6.15699e12 0.273848
\(928\) 8.12534e12 0.359646
\(929\) −4.04390e13 −1.78127 −0.890634 0.454720i \(-0.849739\pi\)
−0.890634 + 0.454720i \(0.849739\pi\)
\(930\) 6.95175e12 0.304734
\(931\) 1.78687e12 0.0779508
\(932\) −1.89798e13 −0.823985
\(933\) −1.03950e13 −0.449114
\(934\) −1.11687e13 −0.480223
\(935\) 5.86409e12 0.250927
\(936\) 1.64223e12 0.0699346
\(937\) 2.69594e13 1.14257 0.571284 0.820752i \(-0.306445\pi\)
0.571284 + 0.820752i \(0.306445\pi\)
\(938\) 1.85002e13 0.780306
\(939\) 5.97645e12 0.250869
\(940\) −2.57944e13 −1.07758
\(941\) 1.75116e13 0.728068 0.364034 0.931386i \(-0.381399\pi\)
0.364034 + 0.931386i \(0.381399\pi\)
\(942\) −2.02158e13 −0.836492
\(943\) 1.59519e13 0.656916
\(944\) 9.72667e11 0.0398649
\(945\) 6.46151e12 0.263567
\(946\) 3.06544e13 1.24446
\(947\) −2.17887e13 −0.880353 −0.440177 0.897911i \(-0.645084\pi\)
−0.440177 + 0.897911i \(0.645084\pi\)
\(948\) −4.15207e11 −0.0166966
\(949\) 8.10862e12 0.324526
\(950\) −6.60238e12 −0.262993
\(951\) 6.31071e12 0.250188
\(952\) 1.47030e12 0.0580150
\(953\) −3.37546e13 −1.32561 −0.662803 0.748794i \(-0.730633\pi\)
−0.662803 + 0.748794i \(0.730633\pi\)
\(954\) 1.92483e13 0.752360
\(955\) −3.29048e13 −1.28010
\(956\) 2.88674e13 1.11776
\(957\) −2.60384e12 −0.100348
\(958\) −3.35688e13 −1.28763
\(959\) −2.10005e11 −0.00801764
\(960\) 4.64334e12 0.176445
\(961\) −1.75807e13 −0.664938
\(962\) 4.10567e12 0.154560
\(963\) 4.78624e11 0.0179340
\(964\) 7.50412e12 0.279868
\(965\) −2.11595e12 −0.0785476
\(966\) 4.01105e12 0.148205
\(967\) −1.66009e13 −0.610539 −0.305270 0.952266i \(-0.598747\pi\)
−0.305270 + 0.952266i \(0.598747\pi\)
\(968\) 4.81664e12 0.176321
\(969\) 3.41839e12 0.124556
\(970\) 9.64374e11 0.0349762
\(971\) −6.48313e12 −0.234045 −0.117022 0.993129i \(-0.537335\pi\)
−0.117022 + 0.993129i \(0.537335\pi\)
\(972\) −2.13958e13 −0.768831
\(973\) 6.34895e12 0.227088
\(974\) 9.24396e12 0.329111
\(975\) 1.42741e12 0.0505859
\(976\) −1.09452e12 −0.0386099
\(977\) −1.49152e13 −0.523725 −0.261862 0.965105i \(-0.584337\pi\)
−0.261862 + 0.965105i \(0.584337\pi\)
\(978\) −6.67748e12 −0.233393
\(979\) 1.91098e13 0.664866
\(980\) 2.45101e12 0.0848842
\(981\) −3.82138e13 −1.31738
\(982\) −2.14343e13 −0.735541
\(983\) −1.85671e12 −0.0634239 −0.0317119 0.999497i \(-0.510096\pi\)
−0.0317119 + 0.999497i \(0.510096\pi\)
\(984\) −5.47022e12 −0.186006
\(985\) −1.93141e13 −0.653748
\(986\) 5.21075e12 0.175571
\(987\) 1.02174e13 0.342701
\(988\) 3.37928e12 0.112828
\(989\) 2.43960e13 0.810839
\(990\) 1.64606e13 0.544613
\(991\) −5.77488e12 −0.190200 −0.0951002 0.995468i \(-0.530317\pi\)
−0.0951002 + 0.995468i \(0.530317\pi\)
\(992\) 2.18151e13 0.715246
\(993\) −6.17425e12 −0.201517
\(994\) −9.64021e12 −0.313219
\(995\) −2.07336e13 −0.670610
\(996\) −1.15519e13 −0.371951
\(997\) −2.38121e13 −0.763256 −0.381628 0.924316i \(-0.624636\pi\)
−0.381628 + 0.924316i \(0.624636\pi\)
\(998\) 5.52980e13 1.76450
\(999\) 1.16181e13 0.369054
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 91.10.a.a.1.11 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.10.a.a.1.11 12 1.1 even 1 trivial