Properties

Label 43.8.a.b.1.9
Level 43
Weight 8
Character 43.1
Self dual yes
Analytic conductor 13.433
Analytic rank 0
Dimension 13
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: \(0\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.9
Root \(7.08185\) of \(x^{13} - 3 x^{12} - 1279 x^{11} + 3765 x^{10} + 598742 x^{9} - 1518614 x^{8} - 124677082 x^{7} + 193428526 x^{6} + 11160446785 x^{5} + 1754605765 x^{4} - 349352939351 x^{3} - 481872751923 x^{2} + 2098464001560 x + 1551032970660\)
Character \(\chi\) \(=\) 43.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.08185 q^{2} -1.19690 q^{3} -62.6836 q^{4} +164.718 q^{5} -9.67319 q^{6} +1314.61 q^{7} -1541.08 q^{8} -2185.57 q^{9} +O(q^{10})\) \(q+8.08185 q^{2} -1.19690 q^{3} -62.6836 q^{4} +164.718 q^{5} -9.67319 q^{6} +1314.61 q^{7} -1541.08 q^{8} -2185.57 q^{9} +1331.23 q^{10} +8636.39 q^{11} +75.0263 q^{12} +8425.12 q^{13} +10624.5 q^{14} -197.152 q^{15} -4431.25 q^{16} +6309.90 q^{17} -17663.4 q^{18} +16664.4 q^{19} -10325.1 q^{20} -1573.46 q^{21} +69798.0 q^{22} +44105.5 q^{23} +1844.52 q^{24} -50992.9 q^{25} +68090.6 q^{26} +5233.54 q^{27} -82404.5 q^{28} -23876.9 q^{29} -1593.35 q^{30} +27160.3 q^{31} +161445. q^{32} -10336.9 q^{33} +50995.7 q^{34} +216540. q^{35} +136999. q^{36} -114297. q^{37} +134680. q^{38} -10084.1 q^{39} -253844. q^{40} -792610. q^{41} -12716.5 q^{42} -79507.0 q^{43} -541361. q^{44} -360003. q^{45} +356454. q^{46} +490927. q^{47} +5303.78 q^{48} +904653. q^{49} -412117. q^{50} -7552.34 q^{51} -528117. q^{52} -1.64370e6 q^{53} +42296.7 q^{54} +1.42257e6 q^{55} -2.02591e6 q^{56} -19945.7 q^{57} -192970. q^{58} -625984. q^{59} +12358.2 q^{60} -1.22257e6 q^{61} +219505. q^{62} -2.87317e6 q^{63} +1.87198e6 q^{64} +1.38777e6 q^{65} -83541.5 q^{66} +1.75201e6 q^{67} -395528. q^{68} -52790.0 q^{69} +1.75005e6 q^{70} +1.15151e6 q^{71} +3.36813e6 q^{72} +2.26726e6 q^{73} -923731. q^{74} +61033.5 q^{75} -1.04459e6 q^{76} +1.13535e7 q^{77} -81497.8 q^{78} +4.91491e6 q^{79} -729909. q^{80} +4.77357e6 q^{81} -6.40576e6 q^{82} -9.34319e6 q^{83} +98630.2 q^{84} +1.03936e6 q^{85} -642564. q^{86} +28578.3 q^{87} -1.33093e7 q^{88} +9.44626e6 q^{89} -2.90949e6 q^{90} +1.10757e7 q^{91} -2.76469e6 q^{92} -32508.2 q^{93} +3.96760e6 q^{94} +2.74494e6 q^{95} -193234. q^{96} -6.08401e6 q^{97} +7.31127e6 q^{98} -1.88754e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + O(q^{10}) \) \( 13q + 16q^{2} + 94q^{3} + 922q^{4} + 998q^{5} + 183q^{6} + 1360q^{7} + 3870q^{8} + 10011q^{9} + 4667q^{10} + 1620q^{11} - 19681q^{12} + 13550q^{13} + 44160q^{14} + 31412q^{15} + 114026q^{16} + 110880q^{17} + 159267q^{18} + 105058q^{19} + 167251q^{20} + 129840q^{21} + 201504q^{22} + 160184q^{23} + 161289q^{24} + 270149q^{25} + 272104q^{26} + 252544q^{27} + 208172q^{28} + 285546q^{29} + 107580q^{30} - 99616q^{31} + 200126q^{32} + 531468q^{33} - 80941q^{34} - 187104q^{35} - 608975q^{36} + 176038q^{37} + 652165q^{38} - 794680q^{39} - 895387q^{40} - 410260q^{41} - 3413218q^{42} - 1033591q^{43} - 2177076q^{44} - 1051178q^{45} - 3975765q^{46} - 424556q^{47} - 2360477q^{48} - 1561359q^{49} - 4063801q^{50} - 2375738q^{51} - 4172312q^{52} + 3992458q^{53} - 10438626q^{54} + 406960q^{55} + 1559556q^{56} - 3116152q^{57} - 4052005q^{58} + 2248836q^{59} - 2911436q^{60} + 6210394q^{61} + 885317q^{62} + 11622368q^{63} - 3096318q^{64} + 5600420q^{65} - 2174604q^{66} - 1993648q^{67} + 9327135q^{68} + 13366240q^{69} - 1105098q^{70} + 4978064q^{71} + 11370663q^{72} + 8224814q^{73} - 3613563q^{74} + 27115592q^{75} + 10687121q^{76} + 17261892q^{77} - 15226630q^{78} + 6945708q^{79} + 15822799q^{80} + 35113185q^{81} - 508449q^{82} + 22937328q^{83} - 14010106q^{84} - 575532q^{85} - 1272112q^{86} + 9081380q^{87} + 11202656q^{88} + 9291302q^{89} + 2841402q^{90} + 25581108q^{91} - 14388137q^{92} + 25930480q^{93} - 24645805q^{94} + 30750464q^{95} - 22461255q^{96} + 10001852q^{97} - 32304856q^{98} + 5055452q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.08185 0.714342 0.357171 0.934039i \(-0.383741\pi\)
0.357171 + 0.934039i \(0.383741\pi\)
\(3\) −1.19690 −0.0255938 −0.0127969 0.999918i \(-0.504073\pi\)
−0.0127969 + 0.999918i \(0.504073\pi\)
\(4\) −62.6836 −0.489716
\(5\) 164.718 0.589314 0.294657 0.955603i \(-0.404795\pi\)
0.294657 + 0.955603i \(0.404795\pi\)
\(6\) −9.67319 −0.0182827
\(7\) 1314.61 1.44862 0.724308 0.689476i \(-0.242159\pi\)
0.724308 + 0.689476i \(0.242159\pi\)
\(8\) −1541.08 −1.06417
\(9\) −2185.57 −0.999345
\(10\) 1331.23 0.420972
\(11\) 8636.39 1.95640 0.978201 0.207662i \(-0.0665854\pi\)
0.978201 + 0.207662i \(0.0665854\pi\)
\(12\) 75.0263 0.0125337
\(13\) 8425.12 1.06359 0.531795 0.846873i \(-0.321518\pi\)
0.531795 + 0.846873i \(0.321518\pi\)
\(14\) 10624.5 1.03481
\(15\) −197.152 −0.0150828
\(16\) −4431.25 −0.270462
\(17\) 6309.90 0.311495 0.155748 0.987797i \(-0.450221\pi\)
0.155748 + 0.987797i \(0.450221\pi\)
\(18\) −17663.4 −0.713874
\(19\) 16664.4 0.557382 0.278691 0.960381i \(-0.410099\pi\)
0.278691 + 0.960381i \(0.410099\pi\)
\(20\) −10325.1 −0.288597
\(21\) −1573.46 −0.0370756
\(22\) 69798.0 1.39754
\(23\) 44105.5 0.755867 0.377933 0.925833i \(-0.376635\pi\)
0.377933 + 0.925833i \(0.376635\pi\)
\(24\) 1844.52 0.0272360
\(25\) −50992.9 −0.652709
\(26\) 68090.6 0.759767
\(27\) 5233.54 0.0511708
\(28\) −82404.5 −0.709411
\(29\) −23876.9 −0.181796 −0.0908981 0.995860i \(-0.528974\pi\)
−0.0908981 + 0.995860i \(0.528974\pi\)
\(30\) −1593.35 −0.0107743
\(31\) 27160.3 0.163745 0.0818725 0.996643i \(-0.473910\pi\)
0.0818725 + 0.996643i \(0.473910\pi\)
\(32\) 161445. 0.870964
\(33\) −10336.9 −0.0500717
\(34\) 50995.7 0.222514
\(35\) 216540. 0.853690
\(36\) 136999. 0.489395
\(37\) −114297. −0.370961 −0.185481 0.982648i \(-0.559384\pi\)
−0.185481 + 0.982648i \(0.559384\pi\)
\(38\) 134680. 0.398161
\(39\) −10084.1 −0.0272213
\(40\) −253844. −0.627128
\(41\) −792610. −1.79604 −0.898020 0.439954i \(-0.854995\pi\)
−0.898020 + 0.439954i \(0.854995\pi\)
\(42\) −12716.5 −0.0264846
\(43\) −79507.0 −0.152499
\(44\) −541361. −0.958081
\(45\) −360003. −0.588928
\(46\) 356454. 0.539947
\(47\) 490927. 0.689722 0.344861 0.938654i \(-0.387926\pi\)
0.344861 + 0.938654i \(0.387926\pi\)
\(48\) 5303.78 0.00692215
\(49\) 904653. 1.09849
\(50\) −412117. −0.466257
\(51\) −7552.34 −0.00797234
\(52\) −528117. −0.520857
\(53\) −1.64370e6 −1.51655 −0.758275 0.651935i \(-0.773958\pi\)
−0.758275 + 0.651935i \(0.773958\pi\)
\(54\) 42296.7 0.0365534
\(55\) 1.42257e6 1.15293
\(56\) −2.02591e6 −1.54157
\(57\) −19945.7 −0.0142655
\(58\) −192970. −0.129865
\(59\) −625984. −0.396809 −0.198404 0.980120i \(-0.563576\pi\)
−0.198404 + 0.980120i \(0.563576\pi\)
\(60\) 12358.2 0.00738628
\(61\) −1.22257e6 −0.689635 −0.344817 0.938670i \(-0.612059\pi\)
−0.344817 + 0.938670i \(0.612059\pi\)
\(62\) 219505. 0.116970
\(63\) −2.87317e6 −1.44767
\(64\) 1.87198e6 0.892628
\(65\) 1.38777e6 0.626789
\(66\) −83541.5 −0.0357683
\(67\) 1.75201e6 0.711665 0.355833 0.934550i \(-0.384197\pi\)
0.355833 + 0.934550i \(0.384197\pi\)
\(68\) −395528. −0.152544
\(69\) −52790.0 −0.0193455
\(70\) 1.75005e6 0.609826
\(71\) 1.15151e6 0.381823 0.190911 0.981607i \(-0.438856\pi\)
0.190911 + 0.981607i \(0.438856\pi\)
\(72\) 3.36813e6 1.06347
\(73\) 2.26726e6 0.682137 0.341068 0.940038i \(-0.389211\pi\)
0.341068 + 0.940038i \(0.389211\pi\)
\(74\) −923731. −0.264993
\(75\) 61033.5 0.0167053
\(76\) −1.04459e6 −0.272959
\(77\) 1.13535e7 2.83407
\(78\) −81497.8 −0.0194453
\(79\) 4.91491e6 1.12156 0.560778 0.827966i \(-0.310502\pi\)
0.560778 + 0.827966i \(0.310502\pi\)
\(80\) −729909. −0.159387
\(81\) 4.77357e6 0.998035
\(82\) −6.40576e6 −1.28299
\(83\) −9.34319e6 −1.79358 −0.896792 0.442452i \(-0.854109\pi\)
−0.896792 + 0.442452i \(0.854109\pi\)
\(84\) 98630.2 0.0181565
\(85\) 1.03936e6 0.183569
\(86\) −642564. −0.108936
\(87\) 28578.3 0.00465286
\(88\) −1.33093e7 −2.08194
\(89\) 9.44626e6 1.42035 0.710174 0.704026i \(-0.248616\pi\)
0.710174 + 0.704026i \(0.248616\pi\)
\(90\) −2.90949e6 −0.420696
\(91\) 1.10757e7 1.54073
\(92\) −2.76469e6 −0.370160
\(93\) −32508.2 −0.00419086
\(94\) 3.96760e6 0.492697
\(95\) 2.74494e6 0.328473
\(96\) −193234. −0.0222913
\(97\) −6.08401e6 −0.676845 −0.338422 0.940994i \(-0.609893\pi\)
−0.338422 + 0.940994i \(0.609893\pi\)
\(98\) 7.31127e6 0.784697
\(99\) −1.88754e7 −1.95512
\(100\) 3.19642e6 0.319642
\(101\) 8.43016e6 0.814163 0.407081 0.913392i \(-0.366547\pi\)
0.407081 + 0.913392i \(0.366547\pi\)
\(102\) −61036.9 −0.00569498
\(103\) −1.11690e7 −1.00712 −0.503561 0.863959i \(-0.667977\pi\)
−0.503561 + 0.863959i \(0.667977\pi\)
\(104\) −1.29838e7 −1.13184
\(105\) −259178. −0.0218492
\(106\) −1.32841e7 −1.08333
\(107\) 9.97447e6 0.787131 0.393565 0.919297i \(-0.371242\pi\)
0.393565 + 0.919297i \(0.371242\pi\)
\(108\) −328057. −0.0250592
\(109\) −1.68276e7 −1.24460 −0.622301 0.782778i \(-0.713802\pi\)
−0.622301 + 0.782778i \(0.713802\pi\)
\(110\) 1.14970e7 0.823589
\(111\) 136802. 0.00949430
\(112\) −5.82536e6 −0.391796
\(113\) −1.00006e7 −0.652006 −0.326003 0.945369i \(-0.605702\pi\)
−0.326003 + 0.945369i \(0.605702\pi\)
\(114\) −161198. −0.0101905
\(115\) 7.26498e6 0.445443
\(116\) 1.49669e6 0.0890285
\(117\) −1.84137e7 −1.06289
\(118\) −5.05911e6 −0.283457
\(119\) 8.29505e6 0.451237
\(120\) 303826. 0.0160506
\(121\) 5.51001e7 2.82751
\(122\) −9.88063e6 −0.492635
\(123\) 948677. 0.0459675
\(124\) −1.70250e6 −0.0801886
\(125\) −2.12681e7 −0.973965
\(126\) −2.32205e7 −1.03413
\(127\) 1.37203e7 0.594362 0.297181 0.954821i \(-0.403954\pi\)
0.297181 + 0.954821i \(0.403954\pi\)
\(128\) −5.53594e6 −0.233322
\(129\) 95162.2 0.00390302
\(130\) 1.12158e7 0.447741
\(131\) −2.22841e7 −0.866054 −0.433027 0.901381i \(-0.642554\pi\)
−0.433027 + 0.901381i \(0.642554\pi\)
\(132\) 647956. 0.0245209
\(133\) 2.19072e7 0.807433
\(134\) 1.41595e7 0.508372
\(135\) 862060. 0.0301557
\(136\) −9.72405e6 −0.331483
\(137\) −1.65210e6 −0.0548928 −0.0274464 0.999623i \(-0.508738\pi\)
−0.0274464 + 0.999623i \(0.508738\pi\)
\(138\) −426641. −0.0138193
\(139\) 167822. 0.00530027 0.00265014 0.999996i \(-0.499156\pi\)
0.00265014 + 0.999996i \(0.499156\pi\)
\(140\) −1.35735e7 −0.418066
\(141\) −587592. −0.0176526
\(142\) 9.30630e6 0.272752
\(143\) 7.27626e7 2.08081
\(144\) 9.68480e6 0.270285
\(145\) −3.93296e6 −0.107135
\(146\) 1.83237e7 0.487279
\(147\) −1.08278e6 −0.0281145
\(148\) 7.16455e6 0.181666
\(149\) −3.24074e7 −0.802587 −0.401294 0.915949i \(-0.631439\pi\)
−0.401294 + 0.915949i \(0.631439\pi\)
\(150\) 493264. 0.0119333
\(151\) −2.03174e7 −0.480229 −0.240114 0.970745i \(-0.577185\pi\)
−0.240114 + 0.970745i \(0.577185\pi\)
\(152\) −2.56812e7 −0.593147
\(153\) −1.37907e7 −0.311291
\(154\) 9.17571e7 2.02450
\(155\) 4.47379e6 0.0964973
\(156\) 632105. 0.0133307
\(157\) −7.75567e7 −1.59945 −0.799725 0.600366i \(-0.795022\pi\)
−0.799725 + 0.600366i \(0.795022\pi\)
\(158\) 3.97216e7 0.801174
\(159\) 1.96735e6 0.0388143
\(160\) 2.65930e7 0.513271
\(161\) 5.79814e7 1.09496
\(162\) 3.85793e7 0.712938
\(163\) −8.55135e7 −1.54660 −0.773300 0.634040i \(-0.781395\pi\)
−0.773300 + 0.634040i \(0.781395\pi\)
\(164\) 4.96837e7 0.879550
\(165\) −1.70268e6 −0.0295080
\(166\) −7.55103e7 −1.28123
\(167\) −5.30786e7 −0.881885 −0.440942 0.897535i \(-0.645356\pi\)
−0.440942 + 0.897535i \(0.645356\pi\)
\(168\) 2.42482e6 0.0394546
\(169\) 8.23412e6 0.131224
\(170\) 8.39993e6 0.131131
\(171\) −3.64213e7 −0.557017
\(172\) 4.98379e6 0.0746810
\(173\) 7.43872e7 1.09229 0.546144 0.837691i \(-0.316095\pi\)
0.546144 + 0.837691i \(0.316095\pi\)
\(174\) 230966. 0.00332373
\(175\) −6.70357e7 −0.945525
\(176\) −3.82700e7 −0.529133
\(177\) 749243. 0.0101558
\(178\) 7.63433e7 1.01461
\(179\) 1.45355e8 1.89428 0.947139 0.320824i \(-0.103960\pi\)
0.947139 + 0.320824i \(0.103960\pi\)
\(180\) 2.25663e7 0.288408
\(181\) −4.70208e7 −0.589406 −0.294703 0.955589i \(-0.595221\pi\)
−0.294703 + 0.955589i \(0.595221\pi\)
\(182\) 8.95125e7 1.10061
\(183\) 1.46330e6 0.0176504
\(184\) −6.79699e7 −0.804368
\(185\) −1.88268e7 −0.218613
\(186\) −262727. −0.00299370
\(187\) 5.44948e7 0.609410
\(188\) −3.07731e7 −0.337768
\(189\) 6.88006e6 0.0741269
\(190\) 2.21842e7 0.234642
\(191\) −6.43802e7 −0.668552 −0.334276 0.942475i \(-0.608492\pi\)
−0.334276 + 0.942475i \(0.608492\pi\)
\(192\) −2.24057e6 −0.0228457
\(193\) −1.29604e8 −1.29768 −0.648842 0.760923i \(-0.724746\pi\)
−0.648842 + 0.760923i \(0.724746\pi\)
\(194\) −4.91701e7 −0.483499
\(195\) −1.66103e6 −0.0160419
\(196\) −5.67070e7 −0.537948
\(197\) 8.33653e7 0.776879 0.388440 0.921474i \(-0.373014\pi\)
0.388440 + 0.921474i \(0.373014\pi\)
\(198\) −1.52548e8 −1.39662
\(199\) −1.89130e8 −1.70127 −0.850636 0.525755i \(-0.823783\pi\)
−0.850636 + 0.525755i \(0.823783\pi\)
\(200\) 7.85840e7 0.694591
\(201\) −2.09699e6 −0.0182142
\(202\) 6.81313e7 0.581590
\(203\) −3.13888e7 −0.263353
\(204\) 473408. 0.00390418
\(205\) −1.30557e8 −1.05843
\(206\) −9.02659e7 −0.719430
\(207\) −9.63955e7 −0.755371
\(208\) −3.73338e7 −0.287661
\(209\) 1.43921e8 1.09046
\(210\) −2.09463e6 −0.0156078
\(211\) 1.41316e8 1.03563 0.517814 0.855493i \(-0.326746\pi\)
0.517814 + 0.855493i \(0.326746\pi\)
\(212\) 1.03033e8 0.742679
\(213\) −1.37824e6 −0.00977229
\(214\) 8.06122e7 0.562280
\(215\) −1.30963e7 −0.0898696
\(216\) −8.06529e6 −0.0544542
\(217\) 3.57051e7 0.237204
\(218\) −1.35998e8 −0.889070
\(219\) −2.71369e6 −0.0174585
\(220\) −8.91720e7 −0.564611
\(221\) 5.31617e7 0.331303
\(222\) 1.10562e6 0.00678218
\(223\) −1.10823e8 −0.669211 −0.334605 0.942358i \(-0.608603\pi\)
−0.334605 + 0.942358i \(0.608603\pi\)
\(224\) 2.12237e8 1.26169
\(225\) 1.11448e8 0.652281
\(226\) −8.08234e7 −0.465755
\(227\) −1.43855e8 −0.816269 −0.408134 0.912922i \(-0.633820\pi\)
−0.408134 + 0.912922i \(0.633820\pi\)
\(228\) 1.25027e6 0.00698606
\(229\) −3.52277e8 −1.93847 −0.969236 0.246132i \(-0.920840\pi\)
−0.969236 + 0.246132i \(0.920840\pi\)
\(230\) 5.87145e7 0.318198
\(231\) −1.35890e7 −0.0725347
\(232\) 3.67961e7 0.193461
\(233\) 3.78934e6 0.0196254 0.00981269 0.999952i \(-0.496876\pi\)
0.00981269 + 0.999952i \(0.496876\pi\)
\(234\) −1.48817e8 −0.759269
\(235\) 8.08647e7 0.406463
\(236\) 3.92390e7 0.194324
\(237\) −5.88267e6 −0.0287048
\(238\) 6.70394e7 0.322337
\(239\) −9.27614e6 −0.0439516 −0.0219758 0.999759i \(-0.506996\pi\)
−0.0219758 + 0.999759i \(0.506996\pi\)
\(240\) 873630. 0.00407932
\(241\) 3.57012e8 1.64294 0.821472 0.570249i \(-0.193153\pi\)
0.821472 + 0.570249i \(0.193153\pi\)
\(242\) 4.45311e8 2.01981
\(243\) −1.71593e7 −0.0767143
\(244\) 7.66351e7 0.337725
\(245\) 1.49013e8 0.647355
\(246\) 7.66707e6 0.0328365
\(247\) 1.40400e8 0.592826
\(248\) −4.18561e7 −0.174252
\(249\) 1.11829e7 0.0459046
\(250\) −1.71885e8 −0.695744
\(251\) 4.81876e8 1.92343 0.961716 0.274047i \(-0.0883625\pi\)
0.961716 + 0.274047i \(0.0883625\pi\)
\(252\) 1.80101e8 0.708946
\(253\) 3.80912e8 1.47878
\(254\) 1.10886e8 0.424577
\(255\) −1.24401e6 −0.00469822
\(256\) −2.84354e8 −1.05930
\(257\) −1.70067e6 −0.00624962 −0.00312481 0.999995i \(-0.500995\pi\)
−0.00312481 + 0.999995i \(0.500995\pi\)
\(258\) 769087. 0.00278809
\(259\) −1.50256e8 −0.537380
\(260\) −8.69906e7 −0.306948
\(261\) 5.21846e7 0.181677
\(262\) −1.80097e8 −0.618658
\(263\) −3.62647e8 −1.22925 −0.614623 0.788821i \(-0.710692\pi\)
−0.614623 + 0.788821i \(0.710692\pi\)
\(264\) 1.59300e7 0.0532846
\(265\) −2.70747e8 −0.893724
\(266\) 1.77051e8 0.576783
\(267\) −1.13063e7 −0.0363521
\(268\) −1.09823e8 −0.348514
\(269\) 3.91885e8 1.22751 0.613756 0.789496i \(-0.289658\pi\)
0.613756 + 0.789496i \(0.289658\pi\)
\(270\) 6.96704e6 0.0215415
\(271\) 5.27272e6 0.0160932 0.00804660 0.999968i \(-0.497439\pi\)
0.00804660 + 0.999968i \(0.497439\pi\)
\(272\) −2.79608e7 −0.0842477
\(273\) −1.32566e7 −0.0394332
\(274\) −1.33521e7 −0.0392122
\(275\) −4.40394e8 −1.27696
\(276\) 3.30907e6 0.00947380
\(277\) 5.31633e8 1.50291 0.751455 0.659784i \(-0.229352\pi\)
0.751455 + 0.659784i \(0.229352\pi\)
\(278\) 1.35632e6 0.00378620
\(279\) −5.93606e7 −0.163638
\(280\) −3.33705e8 −0.908468
\(281\) 5.83130e8 1.56781 0.783905 0.620881i \(-0.213225\pi\)
0.783905 + 0.620881i \(0.213225\pi\)
\(282\) −4.74883e6 −0.0126100
\(283\) −2.16632e8 −0.568159 −0.284079 0.958801i \(-0.591688\pi\)
−0.284079 + 0.958801i \(0.591688\pi\)
\(284\) −7.21806e7 −0.186985
\(285\) −3.28543e6 −0.00840688
\(286\) 5.88057e8 1.48641
\(287\) −1.04197e9 −2.60177
\(288\) −3.52849e8 −0.870393
\(289\) −3.70524e8 −0.902971
\(290\) −3.17856e7 −0.0765311
\(291\) 7.28198e6 0.0173230
\(292\) −1.42120e8 −0.334053
\(293\) −9.62588e7 −0.223565 −0.111782 0.993733i \(-0.535656\pi\)
−0.111782 + 0.993733i \(0.535656\pi\)
\(294\) −8.75088e6 −0.0200834
\(295\) −1.03111e8 −0.233845
\(296\) 1.76140e8 0.394764
\(297\) 4.51989e7 0.100111
\(298\) −2.61912e8 −0.573321
\(299\) 3.71594e8 0.803932
\(300\) −3.82580e6 −0.00818085
\(301\) −1.04521e8 −0.220912
\(302\) −1.64202e8 −0.343047
\(303\) −1.00901e7 −0.0208375
\(304\) −7.38444e7 −0.150751
\(305\) −2.01380e8 −0.406411
\(306\) −1.11455e8 −0.222368
\(307\) 7.18187e8 1.41662 0.708310 0.705902i \(-0.249458\pi\)
0.708310 + 0.705902i \(0.249458\pi\)
\(308\) −7.11677e8 −1.38789
\(309\) 1.33682e7 0.0257761
\(310\) 3.61565e7 0.0689320
\(311\) −1.67567e8 −0.315883 −0.157942 0.987448i \(-0.550486\pi\)
−0.157942 + 0.987448i \(0.550486\pi\)
\(312\) 1.55403e7 0.0289680
\(313\) 6.19823e8 1.14252 0.571258 0.820770i \(-0.306455\pi\)
0.571258 + 0.820770i \(0.306455\pi\)
\(314\) −6.26802e8 −1.14255
\(315\) −4.73263e8 −0.853131
\(316\) −3.08084e8 −0.549244
\(317\) 2.63030e8 0.463765 0.231883 0.972744i \(-0.425511\pi\)
0.231883 + 0.972744i \(0.425511\pi\)
\(318\) 1.58998e7 0.0277266
\(319\) −2.06210e8 −0.355666
\(320\) 3.08349e8 0.526038
\(321\) −1.19385e7 −0.0201457
\(322\) 4.68598e8 0.782176
\(323\) 1.05151e8 0.173622
\(324\) −2.99225e8 −0.488754
\(325\) −4.29621e8 −0.694215
\(326\) −6.91108e8 −1.10480
\(327\) 2.01410e7 0.0318541
\(328\) 1.22147e9 1.91129
\(329\) 6.45377e8 0.999143
\(330\) −1.37608e7 −0.0210788
\(331\) −9.80269e8 −1.48575 −0.742877 0.669427i \(-0.766540\pi\)
−0.742877 + 0.669427i \(0.766540\pi\)
\(332\) 5.85665e8 0.878347
\(333\) 2.49804e8 0.370718
\(334\) −4.28973e8 −0.629967
\(335\) 2.88589e8 0.419394
\(336\) 6.97240e6 0.0100275
\(337\) −6.28527e8 −0.894580 −0.447290 0.894389i \(-0.647611\pi\)
−0.447290 + 0.894389i \(0.647611\pi\)
\(338\) 6.65470e7 0.0937389
\(339\) 1.19697e7 0.0166873
\(340\) −6.51507e7 −0.0898965
\(341\) 2.34567e8 0.320351
\(342\) −2.94351e8 −0.397901
\(343\) 1.06628e8 0.142673
\(344\) 1.22526e8 0.162284
\(345\) −8.69548e6 −0.0114006
\(346\) 6.01187e8 0.780267
\(347\) 1.41484e9 1.81783 0.908915 0.416982i \(-0.136912\pi\)
0.908915 + 0.416982i \(0.136912\pi\)
\(348\) −1.79139e6 −0.00227858
\(349\) 1.08261e9 1.36327 0.681637 0.731691i \(-0.261269\pi\)
0.681637 + 0.731691i \(0.261269\pi\)
\(350\) −5.41773e8 −0.675428
\(351\) 4.40932e7 0.0544248
\(352\) 1.39430e9 1.70395
\(353\) −1.04347e9 −1.26261 −0.631306 0.775533i \(-0.717481\pi\)
−0.631306 + 0.775533i \(0.717481\pi\)
\(354\) 6.05527e6 0.00725474
\(355\) 1.89674e8 0.225014
\(356\) −5.92126e8 −0.695567
\(357\) −9.92838e6 −0.0115489
\(358\) 1.17474e9 1.35316
\(359\) 7.36087e8 0.839650 0.419825 0.907605i \(-0.362091\pi\)
0.419825 + 0.907605i \(0.362091\pi\)
\(360\) 5.54792e8 0.626717
\(361\) −6.16168e8 −0.689325
\(362\) −3.80015e8 −0.421037
\(363\) −6.59495e7 −0.0723666
\(364\) −6.94267e8 −0.754522
\(365\) 3.73459e8 0.401993
\(366\) 1.18262e7 0.0126084
\(367\) 5.85636e8 0.618439 0.309219 0.950991i \(-0.399932\pi\)
0.309219 + 0.950991i \(0.399932\pi\)
\(368\) −1.95443e8 −0.204433
\(369\) 1.73230e9 1.79486
\(370\) −1.52155e8 −0.156164
\(371\) −2.16082e9 −2.19690
\(372\) 2.03773e6 0.00205233
\(373\) −1.55931e9 −1.55579 −0.777896 0.628393i \(-0.783713\pi\)
−0.777896 + 0.628393i \(0.783713\pi\)
\(374\) 4.40419e8 0.435327
\(375\) 2.54558e7 0.0249274
\(376\) −7.56556e8 −0.733979
\(377\) −2.01166e8 −0.193357
\(378\) 5.56036e7 0.0529519
\(379\) −2.07398e7 −0.0195689 −0.00978446 0.999952i \(-0.503115\pi\)
−0.00978446 + 0.999952i \(0.503115\pi\)
\(380\) −1.72063e8 −0.160859
\(381\) −1.64219e7 −0.0152120
\(382\) −5.20311e8 −0.477575
\(383\) 1.16713e9 1.06151 0.530755 0.847525i \(-0.321908\pi\)
0.530755 + 0.847525i \(0.321908\pi\)
\(384\) 6.62598e6 0.00597161
\(385\) 1.87013e9 1.67016
\(386\) −1.04744e9 −0.926990
\(387\) 1.73768e8 0.152399
\(388\) 3.81368e8 0.331462
\(389\) −1.12210e9 −0.966512 −0.483256 0.875479i \(-0.660546\pi\)
−0.483256 + 0.875479i \(0.660546\pi\)
\(390\) −1.34242e7 −0.0114594
\(391\) 2.78301e8 0.235449
\(392\) −1.39414e9 −1.16897
\(393\) 2.66719e7 0.0221656
\(394\) 6.73746e8 0.554957
\(395\) 8.09575e8 0.660948
\(396\) 1.18318e9 0.957453
\(397\) 6.25123e8 0.501417 0.250708 0.968063i \(-0.419337\pi\)
0.250708 + 0.968063i \(0.419337\pi\)
\(398\) −1.52852e9 −1.21529
\(399\) −2.62208e7 −0.0206653
\(400\) 2.25962e8 0.176533
\(401\) 6.80350e8 0.526899 0.263449 0.964673i \(-0.415140\pi\)
0.263449 + 0.964673i \(0.415140\pi\)
\(402\) −1.69476e7 −0.0130112
\(403\) 2.28828e8 0.174158
\(404\) −5.28433e8 −0.398708
\(405\) 7.86295e8 0.588156
\(406\) −2.53680e8 −0.188124
\(407\) −9.87113e8 −0.725749
\(408\) 1.16387e7 0.00848390
\(409\) 1.45254e9 1.04977 0.524887 0.851172i \(-0.324108\pi\)
0.524887 + 0.851172i \(0.324108\pi\)
\(410\) −1.05515e9 −0.756082
\(411\) 1.97741e6 0.00140491
\(412\) 7.00111e8 0.493204
\(413\) −8.22925e8 −0.574824
\(414\) −7.79054e8 −0.539593
\(415\) −1.53899e9 −1.05698
\(416\) 1.36019e9 0.926349
\(417\) −200867. −0.000135654 0
\(418\) 1.16315e9 0.778963
\(419\) −2.07021e9 −1.37488 −0.687441 0.726240i \(-0.741266\pi\)
−0.687441 + 0.726240i \(0.741266\pi\)
\(420\) 1.62462e7 0.0106999
\(421\) −1.86742e9 −1.21971 −0.609853 0.792514i \(-0.708772\pi\)
−0.609853 + 0.792514i \(0.708772\pi\)
\(422\) 1.14210e9 0.739792
\(423\) −1.07295e9 −0.689271
\(424\) 2.53307e9 1.61386
\(425\) −3.21760e8 −0.203316
\(426\) −1.11387e7 −0.00698076
\(427\) −1.60720e9 −0.999016
\(428\) −6.25236e8 −0.385470
\(429\) −8.70898e7 −0.0532558
\(430\) −1.05842e8 −0.0641976
\(431\) 1.03410e9 0.622144 0.311072 0.950386i \(-0.399312\pi\)
0.311072 + 0.950386i \(0.399312\pi\)
\(432\) −2.31911e7 −0.0138398
\(433\) 1.88745e9 1.11730 0.558648 0.829405i \(-0.311320\pi\)
0.558648 + 0.829405i \(0.311320\pi\)
\(434\) 2.88564e8 0.169444
\(435\) 4.70738e6 0.00274199
\(436\) 1.05482e9 0.609501
\(437\) 7.34993e8 0.421307
\(438\) −2.19317e7 −0.0124713
\(439\) −2.19197e9 −1.23654 −0.618270 0.785965i \(-0.712166\pi\)
−0.618270 + 0.785965i \(0.712166\pi\)
\(440\) −2.19229e9 −1.22691
\(441\) −1.97718e9 −1.09777
\(442\) 4.29645e8 0.236664
\(443\) −1.23285e9 −0.673745 −0.336873 0.941550i \(-0.609369\pi\)
−0.336873 + 0.941550i \(0.609369\pi\)
\(444\) −8.57527e6 −0.00464951
\(445\) 1.55597e9 0.837031
\(446\) −8.95655e8 −0.478045
\(447\) 3.87885e7 0.0205412
\(448\) 2.46092e9 1.29308
\(449\) 2.06378e9 1.07597 0.537985 0.842954i \(-0.319186\pi\)
0.537985 + 0.842954i \(0.319186\pi\)
\(450\) 9.00709e8 0.465952
\(451\) −6.84529e9 −3.51378
\(452\) 6.26874e8 0.319298
\(453\) 2.43179e7 0.0122909
\(454\) −1.16261e9 −0.583095
\(455\) 1.82438e9 0.907976
\(456\) 3.07379e7 0.0151809
\(457\) −3.67119e9 −1.79928 −0.899642 0.436628i \(-0.856173\pi\)
−0.899642 + 0.436628i \(0.856173\pi\)
\(458\) −2.84705e9 −1.38473
\(459\) 3.30231e7 0.0159395
\(460\) −4.55395e8 −0.218140
\(461\) 1.50027e9 0.713206 0.356603 0.934256i \(-0.383935\pi\)
0.356603 + 0.934256i \(0.383935\pi\)
\(462\) −1.09824e8 −0.0518146
\(463\) 1.00012e9 0.468293 0.234147 0.972201i \(-0.424770\pi\)
0.234147 + 0.972201i \(0.424770\pi\)
\(464\) 1.05805e8 0.0491690
\(465\) −5.35470e6 −0.00246973
\(466\) 3.06249e7 0.0140192
\(467\) −1.13114e9 −0.513933 −0.256967 0.966420i \(-0.582723\pi\)
−0.256967 + 0.966420i \(0.582723\pi\)
\(468\) 1.15424e9 0.520516
\(469\) 2.30321e9 1.03093
\(470\) 6.53536e8 0.290354
\(471\) 9.28279e7 0.0409360
\(472\) 9.64690e8 0.422271
\(473\) −6.86654e8 −0.298348
\(474\) −4.75429e7 −0.0205051
\(475\) −8.49768e8 −0.363808
\(476\) −5.19964e8 −0.220978
\(477\) 3.59242e9 1.51556
\(478\) −7.49684e7 −0.0313964
\(479\) 4.27056e8 0.177546 0.0887730 0.996052i \(-0.471705\pi\)
0.0887730 + 0.996052i \(0.471705\pi\)
\(480\) −3.18292e7 −0.0131366
\(481\) −9.62965e8 −0.394551
\(482\) 2.88532e9 1.17362
\(483\) −6.93982e7 −0.0280242
\(484\) −3.45387e9 −1.38467
\(485\) −1.00215e9 −0.398874
\(486\) −1.38679e8 −0.0548002
\(487\) 1.86591e9 0.732049 0.366025 0.930605i \(-0.380719\pi\)
0.366025 + 0.930605i \(0.380719\pi\)
\(488\) 1.88407e9 0.733886
\(489\) 1.02351e8 0.0395834
\(490\) 1.20430e9 0.462433
\(491\) −2.67497e9 −1.01985 −0.509923 0.860220i \(-0.670326\pi\)
−0.509923 + 0.860220i \(0.670326\pi\)
\(492\) −5.94666e7 −0.0225110
\(493\) −1.50661e8 −0.0566287
\(494\) 1.13469e9 0.423481
\(495\) −3.10913e9 −1.15218
\(496\) −1.20354e8 −0.0442868
\(497\) 1.51378e9 0.553115
\(498\) 9.03785e7 0.0327916
\(499\) −2.01727e9 −0.726794 −0.363397 0.931634i \(-0.618383\pi\)
−0.363397 + 0.931634i \(0.618383\pi\)
\(500\) 1.33316e9 0.476966
\(501\) 6.35299e7 0.0225708
\(502\) 3.89445e9 1.37399
\(503\) 2.78344e9 0.975201 0.487601 0.873067i \(-0.337872\pi\)
0.487601 + 0.873067i \(0.337872\pi\)
\(504\) 4.42777e9 1.54056
\(505\) 1.38860e9 0.479798
\(506\) 3.07848e9 1.05635
\(507\) −9.85545e6 −0.00335852
\(508\) −8.60039e8 −0.291068
\(509\) −6.34132e8 −0.213141 −0.106571 0.994305i \(-0.533987\pi\)
−0.106571 + 0.994305i \(0.533987\pi\)
\(510\) −1.00539e7 −0.00335613
\(511\) 2.98056e9 0.988154
\(512\) −1.58950e9 −0.523380
\(513\) 8.72140e7 0.0285217
\(514\) −1.37446e7 −0.00446437
\(515\) −1.83973e9 −0.593512
\(516\) −5.96511e6 −0.00191137
\(517\) 4.23984e9 1.34937
\(518\) −1.21434e9 −0.383873
\(519\) −8.90343e7 −0.0279558
\(520\) −2.13866e9 −0.667007
\(521\) −2.55557e9 −0.791692 −0.395846 0.918317i \(-0.629549\pi\)
−0.395846 + 0.918317i \(0.629549\pi\)
\(522\) 4.21748e8 0.129780
\(523\) 1.86691e9 0.570647 0.285323 0.958431i \(-0.407899\pi\)
0.285323 + 0.958431i \(0.407899\pi\)
\(524\) 1.39685e9 0.424120
\(525\) 8.02352e7 0.0241996
\(526\) −2.93086e9 −0.878101
\(527\) 1.71379e8 0.0510058
\(528\) 4.58055e7 0.0135425
\(529\) −1.45953e9 −0.428666
\(530\) −2.18814e9 −0.638425
\(531\) 1.36813e9 0.396549
\(532\) −1.37322e9 −0.395413
\(533\) −6.67783e9 −1.91025
\(534\) −9.13755e7 −0.0259678
\(535\) 1.64298e9 0.463867
\(536\) −2.69999e9 −0.757330
\(537\) −1.73976e8 −0.0484817
\(538\) 3.16716e9 0.876863
\(539\) 7.81294e9 2.14909
\(540\) −5.40371e7 −0.0147677
\(541\) −2.51318e8 −0.0682391 −0.0341195 0.999418i \(-0.510863\pi\)
−0.0341195 + 0.999418i \(0.510863\pi\)
\(542\) 4.26134e7 0.0114960
\(543\) 5.62793e7 0.0150851
\(544\) 1.01870e9 0.271301
\(545\) −2.77182e9 −0.733461
\(546\) −1.07138e8 −0.0281688
\(547\) 1.95845e9 0.511631 0.255815 0.966726i \(-0.417656\pi\)
0.255815 + 0.966726i \(0.417656\pi\)
\(548\) 1.03560e8 0.0268819
\(549\) 2.67201e9 0.689183
\(550\) −3.55920e9 −0.912186
\(551\) −3.97895e8 −0.101330
\(552\) 8.13534e7 0.0205868
\(553\) 6.46118e9 1.62470
\(554\) 4.29658e9 1.07359
\(555\) 2.25338e7 0.00559513
\(556\) −1.05197e7 −0.00259563
\(557\) −9.85537e7 −0.0241646 −0.0120823 0.999927i \(-0.503846\pi\)
−0.0120823 + 0.999927i \(0.503846\pi\)
\(558\) −4.79744e8 −0.116893
\(559\) −6.69856e8 −0.162196
\(560\) −9.59544e8 −0.230891
\(561\) −6.52250e7 −0.0155971
\(562\) 4.71277e9 1.11995
\(563\) −8.32439e8 −0.196595 −0.0982976 0.995157i \(-0.531340\pi\)
−0.0982976 + 0.995157i \(0.531340\pi\)
\(564\) 3.68324e7 0.00864477
\(565\) −1.64728e9 −0.384236
\(566\) −1.75079e9 −0.405859
\(567\) 6.27538e9 1.44577
\(568\) −1.77456e9 −0.406323
\(569\) −4.82012e8 −0.109689 −0.0548447 0.998495i \(-0.517466\pi\)
−0.0548447 + 0.998495i \(0.517466\pi\)
\(570\) −2.65523e7 −0.00600538
\(571\) −3.91868e9 −0.880874 −0.440437 0.897784i \(-0.645176\pi\)
−0.440437 + 0.897784i \(0.645176\pi\)
\(572\) −4.56103e9 −1.01901
\(573\) 7.70568e7 0.0171108
\(574\) −8.42107e9 −1.85856
\(575\) −2.24907e9 −0.493361
\(576\) −4.09133e9 −0.892043
\(577\) −6.04095e9 −1.30915 −0.654576 0.755996i \(-0.727153\pi\)
−0.654576 + 0.755996i \(0.727153\pi\)
\(578\) −2.99452e9 −0.645030
\(579\) 1.55124e8 0.0332127
\(580\) 2.46532e8 0.0524658
\(581\) −1.22826e10 −2.59822
\(582\) 5.88519e7 0.0123746
\(583\) −1.41956e10 −2.96698
\(584\) −3.49402e9 −0.725907
\(585\) −3.03307e9 −0.626378
\(586\) −7.77949e8 −0.159702
\(587\) 9.41828e9 1.92193 0.960967 0.276665i \(-0.0892291\pi\)
0.960967 + 0.276665i \(0.0892291\pi\)
\(588\) 6.78727e7 0.0137681
\(589\) 4.52611e8 0.0912686
\(590\) −8.33329e8 −0.167045
\(591\) −9.97802e7 −0.0198833
\(592\) 5.06479e8 0.100331
\(593\) 2.77653e9 0.546778 0.273389 0.961904i \(-0.411855\pi\)
0.273389 + 0.961904i \(0.411855\pi\)
\(594\) 3.65291e8 0.0715132
\(595\) 1.36635e9 0.265920
\(596\) 2.03141e9 0.393040
\(597\) 2.26370e8 0.0435420
\(598\) 3.00317e9 0.574282
\(599\) 2.48861e9 0.473110 0.236555 0.971618i \(-0.423982\pi\)
0.236555 + 0.971618i \(0.423982\pi\)
\(600\) −9.40574e7 −0.0177772
\(601\) 9.96150e8 0.187182 0.0935910 0.995611i \(-0.470165\pi\)
0.0935910 + 0.995611i \(0.470165\pi\)
\(602\) −8.44720e8 −0.157807
\(603\) −3.82914e9 −0.711199
\(604\) 1.27357e9 0.235176
\(605\) 9.07599e9 1.66629
\(606\) −8.15466e7 −0.0148851
\(607\) 4.07704e9 0.739920 0.369960 0.929048i \(-0.379371\pi\)
0.369960 + 0.929048i \(0.379371\pi\)
\(608\) 2.69039e9 0.485460
\(609\) 3.75693e7 0.00674020
\(610\) −1.62752e9 −0.290317
\(611\) 4.13612e9 0.733582
\(612\) 8.64453e8 0.152444
\(613\) 7.37622e9 1.29337 0.646684 0.762758i \(-0.276155\pi\)
0.646684 + 0.762758i \(0.276155\pi\)
\(614\) 5.80428e9 1.01195
\(615\) 1.56265e8 0.0270893
\(616\) −1.74966e10 −3.01593
\(617\) −3.57832e9 −0.613312 −0.306656 0.951820i \(-0.599210\pi\)
−0.306656 + 0.951820i \(0.599210\pi\)
\(618\) 1.08040e8 0.0184129
\(619\) −2.12273e9 −0.359731 −0.179865 0.983691i \(-0.557566\pi\)
−0.179865 + 0.983691i \(0.557566\pi\)
\(620\) −2.80434e8 −0.0472563
\(621\) 2.30828e8 0.0386783
\(622\) −1.35425e9 −0.225649
\(623\) 1.24181e10 2.05754
\(624\) 4.46850e7 0.00736233
\(625\) 4.80577e8 0.0787377
\(626\) 5.00932e9 0.816147
\(627\) −1.72259e8 −0.0279091
\(628\) 4.86154e9 0.783277
\(629\) −7.21203e8 −0.115553
\(630\) −3.82484e9 −0.609427
\(631\) −1.03343e10 −1.63749 −0.818744 0.574159i \(-0.805329\pi\)
−0.818744 + 0.574159i \(0.805329\pi\)
\(632\) −7.57425e9 −1.19352
\(633\) −1.69142e8 −0.0265056
\(634\) 2.12577e9 0.331287
\(635\) 2.25999e9 0.350266
\(636\) −1.23321e8 −0.0190080
\(637\) 7.62181e9 1.16834
\(638\) −1.66656e9 −0.254067
\(639\) −2.51669e9 −0.381573
\(640\) −9.11871e8 −0.137500
\(641\) 4.06584e9 0.609744 0.304872 0.952393i \(-0.401386\pi\)
0.304872 + 0.952393i \(0.401386\pi\)
\(642\) −9.64850e7 −0.0143909
\(643\) 1.27525e10 1.89172 0.945860 0.324574i \(-0.105221\pi\)
0.945860 + 0.324574i \(0.105221\pi\)
\(644\) −3.63449e9 −0.536220
\(645\) 1.56750e7 0.00230010
\(646\) 8.49815e8 0.124025
\(647\) 6.06984e9 0.881074 0.440537 0.897734i \(-0.354788\pi\)
0.440537 + 0.897734i \(0.354788\pi\)
\(648\) −7.35644e9 −1.06208
\(649\) −5.40625e9 −0.776318
\(650\) −3.47213e9 −0.495907
\(651\) −4.27356e7 −0.00607094
\(652\) 5.36030e9 0.757395
\(653\) 8.96474e9 1.25992 0.629958 0.776629i \(-0.283072\pi\)
0.629958 + 0.776629i \(0.283072\pi\)
\(654\) 1.62777e8 0.0227547
\(655\) −3.67059e9 −0.510378
\(656\) 3.51226e9 0.485761
\(657\) −4.95525e9 −0.681690
\(658\) 5.21584e9 0.713730
\(659\) 7.66829e9 1.04376 0.521879 0.853020i \(-0.325231\pi\)
0.521879 + 0.853020i \(0.325231\pi\)
\(660\) 1.06730e8 0.0144505
\(661\) 9.45702e9 1.27365 0.636824 0.771010i \(-0.280248\pi\)
0.636824 + 0.771010i \(0.280248\pi\)
\(662\) −7.92239e9 −1.06134
\(663\) −6.36294e7 −0.00847931
\(664\) 1.43986e10 1.90867
\(665\) 3.60852e9 0.475832
\(666\) 2.01888e9 0.264819
\(667\) −1.05310e9 −0.137414
\(668\) 3.32716e9 0.431873
\(669\) 1.32644e8 0.0171276
\(670\) 2.33233e9 0.299591
\(671\) −1.05586e10 −1.34920
\(672\) −2.54027e8 −0.0322915
\(673\) 7.72168e9 0.976470 0.488235 0.872712i \(-0.337641\pi\)
0.488235 + 0.872712i \(0.337641\pi\)
\(674\) −5.07966e9 −0.639036
\(675\) −2.66873e8 −0.0333996
\(676\) −5.16145e8 −0.0642626
\(677\) −3.61144e9 −0.447321 −0.223661 0.974667i \(-0.571801\pi\)
−0.223661 + 0.974667i \(0.571801\pi\)
\(678\) 9.67377e7 0.0119204
\(679\) −7.99810e9 −0.980489
\(680\) −1.60173e9 −0.195347
\(681\) 1.72180e8 0.0208914
\(682\) 1.89573e9 0.228840
\(683\) 1.07988e10 1.29689 0.648444 0.761262i \(-0.275420\pi\)
0.648444 + 0.761262i \(0.275420\pi\)
\(684\) 2.28302e9 0.272780
\(685\) −2.72132e8 −0.0323491
\(686\) 8.61751e8 0.101917
\(687\) 4.21641e8 0.0496129
\(688\) 3.52316e8 0.0412451
\(689\) −1.38484e10 −1.61299
\(690\) −7.02756e7 −0.00814390
\(691\) 3.51258e9 0.404998 0.202499 0.979282i \(-0.435094\pi\)
0.202499 + 0.979282i \(0.435094\pi\)
\(692\) −4.66286e9 −0.534911
\(693\) −2.48138e10 −2.83222
\(694\) 1.14345e10 1.29855
\(695\) 2.76434e7 0.00312353
\(696\) −4.40414e7 −0.00495141
\(697\) −5.00129e9 −0.559458
\(698\) 8.74949e9 0.973843
\(699\) −4.53547e6 −0.000502288 0
\(700\) 4.20204e9 0.463039
\(701\) −2.46256e9 −0.270006 −0.135003 0.990845i \(-0.543104\pi\)
−0.135003 + 0.990845i \(0.543104\pi\)
\(702\) 3.56355e8 0.0388779
\(703\) −1.90469e9 −0.206767
\(704\) 1.61671e10 1.74634
\(705\) −9.67871e7 −0.0104029
\(706\) −8.43320e9 −0.901937
\(707\) 1.10824e10 1.17941
\(708\) −4.69653e7 −0.00497348
\(709\) −1.34920e10 −1.42172 −0.710859 0.703335i \(-0.751693\pi\)
−0.710859 + 0.703335i \(0.751693\pi\)
\(710\) 1.53292e9 0.160737
\(711\) −1.07419e10 −1.12082
\(712\) −1.45574e10 −1.51149
\(713\) 1.19792e9 0.123769
\(714\) −8.02397e7 −0.00824984
\(715\) 1.19853e10 1.22625
\(716\) −9.11137e9 −0.927658
\(717\) 1.11026e7 0.00112489
\(718\) 5.94894e9 0.599797
\(719\) 7.00395e9 0.702736 0.351368 0.936237i \(-0.385717\pi\)
0.351368 + 0.936237i \(0.385717\pi\)
\(720\) 1.59526e9 0.159283
\(721\) −1.46828e10 −1.45893
\(722\) −4.97978e9 −0.492414
\(723\) −4.27308e8 −0.0420492
\(724\) 2.94743e9 0.288642
\(725\) 1.21755e9 0.118660
\(726\) −5.32994e8 −0.0516945
\(727\) 8.00714e9 0.772871 0.386436 0.922316i \(-0.373706\pi\)
0.386436 + 0.922316i \(0.373706\pi\)
\(728\) −1.70686e10 −1.63960
\(729\) −1.04193e10 −0.996072
\(730\) 3.01824e9 0.287160
\(731\) −5.01681e8 −0.0475026
\(732\) −9.17248e7 −0.00864367
\(733\) 9.27809e9 0.870151 0.435075 0.900394i \(-0.356722\pi\)
0.435075 + 0.900394i \(0.356722\pi\)
\(734\) 4.73303e9 0.441777
\(735\) −1.78354e8 −0.0165683
\(736\) 7.12062e9 0.658332
\(737\) 1.51311e10 1.39230
\(738\) 1.40002e10 1.28215
\(739\) −1.50075e10 −1.36789 −0.683947 0.729532i \(-0.739738\pi\)
−0.683947 + 0.729532i \(0.739738\pi\)
\(740\) 1.18013e9 0.107058
\(741\) −1.68045e8 −0.0151727
\(742\) −1.74634e10 −1.56934
\(743\) 2.52291e9 0.225653 0.112827 0.993615i \(-0.464010\pi\)
0.112827 + 0.993615i \(0.464010\pi\)
\(744\) 5.00977e7 0.00445977
\(745\) −5.33809e9 −0.472976
\(746\) −1.26021e10 −1.11137
\(747\) 2.04202e10 1.79241
\(748\) −3.41593e9 −0.298438
\(749\) 1.31125e10 1.14025
\(750\) 2.05730e8 0.0178067
\(751\) −9.71269e9 −0.836758 −0.418379 0.908273i \(-0.637402\pi\)
−0.418379 + 0.908273i \(0.637402\pi\)
\(752\) −2.17542e9 −0.186544
\(753\) −5.76758e8 −0.0492279
\(754\) −1.62579e9 −0.138123
\(755\) −3.34664e9 −0.283006
\(756\) −4.31267e8 −0.0363011
\(757\) −1.39407e10 −1.16802 −0.584008 0.811748i \(-0.698516\pi\)
−0.584008 + 0.811748i \(0.698516\pi\)
\(758\) −1.67616e8 −0.0139789
\(759\) −4.55915e8 −0.0378475
\(760\) −4.23016e9 −0.349550
\(761\) 9.33520e9 0.767852 0.383926 0.923364i \(-0.374572\pi\)
0.383926 + 0.923364i \(0.374572\pi\)
\(762\) −1.32719e8 −0.0108665
\(763\) −2.21217e10 −1.80295
\(764\) 4.03558e9 0.327401
\(765\) −2.27158e9 −0.183448
\(766\) 9.43258e9 0.758281
\(767\) −5.27399e9 −0.422042
\(768\) 3.40344e8 0.0271115
\(769\) −1.02873e10 −0.815754 −0.407877 0.913037i \(-0.633731\pi\)
−0.407877 + 0.913037i \(0.633731\pi\)
\(770\) 1.51141e10 1.19307
\(771\) 2.03554e6 0.000159952 0
\(772\) 8.12408e9 0.635497
\(773\) −6.15947e9 −0.479640 −0.239820 0.970817i \(-0.577088\pi\)
−0.239820 + 0.970817i \(0.577088\pi\)
\(774\) 1.40437e9 0.108865
\(775\) −1.38498e9 −0.106878
\(776\) 9.37594e9 0.720275
\(777\) 1.79842e8 0.0137536
\(778\) −9.06863e9 −0.690420
\(779\) −1.32084e10 −1.00108
\(780\) 1.04119e8 0.00785597
\(781\) 9.94486e9 0.746999
\(782\) 2.24919e9 0.168191
\(783\) −1.24961e8 −0.00930266
\(784\) −4.00875e9 −0.297100
\(785\) −1.27750e10 −0.942579
\(786\) 2.15558e8 0.0158338
\(787\) 6.80084e9 0.497337 0.248669 0.968589i \(-0.420007\pi\)
0.248669 + 0.968589i \(0.420007\pi\)
\(788\) −5.22564e9 −0.380450
\(789\) 4.34053e8 0.0314611
\(790\) 6.54287e9 0.472143
\(791\) −1.31469e10 −0.944506
\(792\) 2.90885e10 2.08057
\(793\) −1.03003e10 −0.733489
\(794\) 5.05215e9 0.358183
\(795\) 3.24058e8 0.0228738
\(796\) 1.18553e10 0.833140
\(797\) −1.73727e10 −1.21552 −0.607762 0.794119i \(-0.707933\pi\)
−0.607762 + 0.794119i \(0.707933\pi\)
\(798\) −2.11913e8 −0.0147621
\(799\) 3.09770e9 0.214845
\(800\) −8.23255e9 −0.568486
\(801\) −2.06454e10 −1.41942
\(802\) 5.49849e9 0.376386
\(803\) 1.95810e10 1.33453
\(804\) 1.31447e8 0.00891979
\(805\) 9.55061e9 0.645276
\(806\) 1.84936e9 0.124408
\(807\) −4.69048e8 −0.0314167
\(808\) −1.29915e10 −0.866404
\(809\) 9.63683e9 0.639903 0.319952 0.947434i \(-0.396333\pi\)
0.319952 + 0.947434i \(0.396333\pi\)
\(810\) 6.35472e9 0.420145
\(811\) 9.66741e9 0.636410 0.318205 0.948022i \(-0.396920\pi\)
0.318205 + 0.948022i \(0.396920\pi\)
\(812\) 1.96756e9 0.128968
\(813\) −6.31094e6 −0.000411886 0
\(814\) −7.97770e9 −0.518433
\(815\) −1.40856e10 −0.911433
\(816\) 3.34663e7 0.00215622
\(817\) −1.32494e9 −0.0850000
\(818\) 1.17392e10 0.749897
\(819\) −2.42068e10 −1.53972
\(820\) 8.18381e9 0.518331
\(821\) 4.33448e9 0.273361 0.136680 0.990615i \(-0.456357\pi\)
0.136680 + 0.990615i \(0.456357\pi\)
\(822\) 1.59811e7 0.00100359
\(823\) 7.05830e9 0.441368 0.220684 0.975345i \(-0.429171\pi\)
0.220684 + 0.975345i \(0.429171\pi\)
\(824\) 1.72122e10 1.07175
\(825\) 5.27109e8 0.0326823
\(826\) −6.65076e9 −0.410621
\(827\) 1.97938e10 1.21691 0.608457 0.793587i \(-0.291789\pi\)
0.608457 + 0.793587i \(0.291789\pi\)
\(828\) 6.04242e9 0.369917
\(829\) 1.55516e10 0.948055 0.474027 0.880510i \(-0.342800\pi\)
0.474027 + 0.880510i \(0.342800\pi\)
\(830\) −1.24379e10 −0.755048
\(831\) −6.36313e8 −0.0384652
\(832\) 1.57716e10 0.949390
\(833\) 5.70827e9 0.342174
\(834\) −1.62338e6 −9.69033e−5 0
\(835\) −8.74302e9 −0.519707
\(836\) −9.02147e9 −0.534017
\(837\) 1.42144e8 0.00837897
\(838\) −1.67311e10 −0.982135
\(839\) −4.02189e9 −0.235106 −0.117553 0.993067i \(-0.537505\pi\)
−0.117553 + 0.993067i \(0.537505\pi\)
\(840\) 3.99413e8 0.0232511
\(841\) −1.66798e10 −0.966950
\(842\) −1.50922e10 −0.871287
\(843\) −6.97950e8 −0.0401262
\(844\) −8.85822e9 −0.507163
\(845\) 1.35631e9 0.0773323
\(846\) −8.67146e9 −0.492375
\(847\) 7.24351e10 4.09597
\(848\) 7.28365e9 0.410170
\(849\) 2.59287e8 0.0145413
\(850\) −2.60042e9 −0.145237
\(851\) −5.04112e9 −0.280397
\(852\) 8.63932e7 0.00478565
\(853\) −7.65536e9 −0.422322 −0.211161 0.977451i \(-0.567725\pi\)
−0.211161 + 0.977451i \(0.567725\pi\)
\(854\) −1.29892e10 −0.713639
\(855\) −5.99925e9 −0.328258
\(856\) −1.53714e10 −0.837638
\(857\) −1.60019e10 −0.868437 −0.434218 0.900808i \(-0.642975\pi\)
−0.434218 + 0.900808i \(0.642975\pi\)
\(858\) −7.03847e8 −0.0380428
\(859\) −9.96999e9 −0.536684 −0.268342 0.963324i \(-0.586476\pi\)
−0.268342 + 0.963324i \(0.586476\pi\)
\(860\) 8.20921e8 0.0440106
\(861\) 1.24714e9 0.0665892
\(862\) 8.35742e9 0.444423
\(863\) 2.21580e10 1.17353 0.586763 0.809759i \(-0.300402\pi\)
0.586763 + 0.809759i \(0.300402\pi\)
\(864\) 8.44930e8 0.0445679
\(865\) 1.22529e10 0.643701
\(866\) 1.52541e10 0.798131
\(867\) 4.43481e8 0.0231104
\(868\) −2.23813e9 −0.116162
\(869\) 4.24471e10 2.19421
\(870\) 3.80443e7 0.00195872
\(871\) 1.47609e10 0.756920
\(872\) 2.59327e10 1.32446
\(873\) 1.32970e10 0.676402
\(874\) 5.94011e9 0.300957
\(875\) −2.79592e10 −1.41090
\(876\) 1.70104e8 0.00854969
\(877\) −8.03090e9 −0.402037 −0.201018 0.979587i \(-0.564425\pi\)
−0.201018 + 0.979587i \(0.564425\pi\)
\(878\) −1.77152e10 −0.883313
\(879\) 1.15212e8 0.00572187
\(880\) −6.30378e9 −0.311825
\(881\) 1.58502e10 0.780944 0.390472 0.920615i \(-0.372312\pi\)
0.390472 + 0.920615i \(0.372312\pi\)
\(882\) −1.59793e10 −0.784183
\(883\) −1.53480e10 −0.750222 −0.375111 0.926980i \(-0.622395\pi\)
−0.375111 + 0.926980i \(0.622395\pi\)
\(884\) −3.33237e9 −0.162245
\(885\) 1.23414e8 0.00598498
\(886\) −9.96368e9 −0.481284
\(887\) 6.55736e9 0.315498 0.157749 0.987479i \(-0.449576\pi\)
0.157749 + 0.987479i \(0.449576\pi\)
\(888\) −2.10823e8 −0.0101035
\(889\) 1.80368e10 0.861002
\(890\) 1.25751e10 0.597926
\(891\) 4.12264e10 1.95256
\(892\) 6.94679e9 0.327723
\(893\) 8.18102e9 0.384439
\(894\) 3.13483e8 0.0146735
\(895\) 2.39426e10 1.11632
\(896\) −7.27760e9 −0.337995
\(897\) −4.44762e8 −0.0205757
\(898\) 1.66791e10 0.768611
\(899\) −6.48503e8 −0.0297682
\(900\) −6.98599e9 −0.319433
\(901\) −1.03716e10 −0.472398
\(902\) −5.53226e10 −2.51004
\(903\) 1.25101e8 0.00565397
\(904\) 1.54117e10 0.693842
\(905\) −7.74518e9 −0.347345
\(906\) 1.96534e8 0.00877988
\(907\) −3.93411e10 −1.75074 −0.875368 0.483457i \(-0.839381\pi\)
−0.875368 + 0.483457i \(0.839381\pi\)
\(908\) 9.01733e9 0.399740
\(909\) −1.84247e10 −0.813629
\(910\) 1.47443e10 0.648605
\(911\) 3.45238e10 1.51288 0.756440 0.654064i \(-0.226937\pi\)
0.756440 + 0.654064i \(0.226937\pi\)
\(912\) 8.83845e7 0.00385829
\(913\) −8.06914e10 −3.50897
\(914\) −2.96700e10 −1.28530
\(915\) 2.41032e8 0.0104016
\(916\) 2.20820e10 0.949301
\(917\) −2.92948e10 −1.25458
\(918\) 2.66888e8 0.0113862
\(919\) −9.91457e9 −0.421376 −0.210688 0.977553i \(-0.567570\pi\)
−0.210688 + 0.977553i \(0.567570\pi\)
\(920\) −1.11959e10 −0.474025
\(921\) −8.59600e8 −0.0362567
\(922\) 1.21249e10 0.509473
\(923\) 9.70157e9 0.406103
\(924\) 8.51809e8 0.0355214
\(925\) 5.82833e9 0.242130
\(926\) 8.08281e9 0.334521
\(927\) 2.44105e10 1.00646
\(928\) −3.85481e9 −0.158338
\(929\) −4.62667e10 −1.89327 −0.946636 0.322304i \(-0.895543\pi\)
−0.946636 + 0.322304i \(0.895543\pi\)
\(930\) −4.32759e7 −0.00176423
\(931\) 1.50755e10 0.612278
\(932\) −2.37530e8 −0.00961086
\(933\) 2.00561e8 0.00808465
\(934\) −9.14169e9 −0.367124
\(935\) 8.97629e9 0.359134
\(936\) 2.83769e10 1.13110
\(937\) −2.62108e10 −1.04086 −0.520430 0.853904i \(-0.674228\pi\)
−0.520430 + 0.853904i \(0.674228\pi\)
\(938\) 1.86142e10 0.736436
\(939\) −7.41868e8 −0.0292413
\(940\) −5.06889e9 −0.199052
\(941\) 2.90508e10 1.13657 0.568283 0.822833i \(-0.307608\pi\)
0.568283 + 0.822833i \(0.307608\pi\)
\(942\) 7.50222e8 0.0292423
\(943\) −3.49584e10 −1.35757
\(944\) 2.77390e9 0.107322
\(945\) 1.13327e9 0.0436840
\(946\) −5.54943e9 −0.213123
\(947\) 6.74455e9 0.258064 0.129032 0.991640i \(-0.458813\pi\)
0.129032 + 0.991640i \(0.458813\pi\)
\(948\) 3.68747e8 0.0140572
\(949\) 1.91019e10 0.725514
\(950\) −6.86770e9 −0.259883
\(951\) −3.14822e8 −0.0118695
\(952\) −1.27833e10 −0.480191
\(953\) 1.83022e10 0.684981 0.342490 0.939521i \(-0.388730\pi\)
0.342490 + 0.939521i \(0.388730\pi\)
\(954\) 2.90334e10 1.08263
\(955\) −1.06046e10 −0.393987
\(956\) 5.81462e8 0.0215238
\(957\) 2.46814e8 0.00910285
\(958\) 3.45141e9 0.126828
\(959\) −2.17187e9 −0.0795186
\(960\) −3.69064e8 −0.0134633
\(961\) −2.67749e10 −0.973188
\(962\) −7.78254e9 −0.281844
\(963\) −2.17999e10 −0.786615
\(964\) −2.23788e10 −0.804576
\(965\) −2.13482e10 −0.764744
\(966\) −5.60866e8 −0.0200188
\(967\) −2.88943e9 −0.102759 −0.0513795 0.998679i \(-0.516362\pi\)
−0.0513795 + 0.998679i \(0.516362\pi\)
\(968\) −8.49135e10 −3.00894
\(969\) −1.25856e8 −0.00444364
\(970\) −8.09922e9 −0.284933
\(971\) −5.32082e10 −1.86514 −0.932569 0.360991i \(-0.882438\pi\)
−0.932569 + 0.360991i \(0.882438\pi\)
\(972\) 1.07560e9 0.0375682
\(973\) 2.20621e8 0.00767806
\(974\) 1.50800e10 0.522933
\(975\) 5.14215e8 0.0177676
\(976\) 5.41751e9 0.186520
\(977\) −1.19053e10 −0.408423 −0.204211 0.978927i \(-0.565463\pi\)
−0.204211 + 0.978927i \(0.565463\pi\)
\(978\) 8.27189e8 0.0282760
\(979\) 8.15816e10 2.77877
\(980\) −9.34067e9 −0.317020
\(981\) 3.67779e10 1.24379
\(982\) −2.16187e10 −0.728518
\(983\) −4.93240e10 −1.65623 −0.828116 0.560557i \(-0.810587\pi\)
−0.828116 + 0.560557i \(0.810587\pi\)
\(984\) −1.46199e9 −0.0489170
\(985\) 1.37318e10 0.457826
\(986\) −1.21762e9 −0.0404522
\(987\) −7.72453e8 −0.0255719
\(988\) −8.80078e9 −0.290317
\(989\) −3.50669e9 −0.115269
\(990\) −2.51275e10 −0.823050
\(991\) 3.40025e10 1.10982 0.554911 0.831910i \(-0.312752\pi\)
0.554911 + 0.831910i \(0.312752\pi\)
\(992\) 4.38489e9 0.142616
\(993\) 1.17329e9 0.0380261
\(994\) 1.22341e10 0.395113
\(995\) −3.11531e10 −1.00258
\(996\) −7.00984e8 −0.0224802
\(997\) −4.30970e10 −1.37725 −0.688627 0.725116i \(-0.741786\pi\)
−0.688627 + 0.725116i \(0.741786\pi\)
\(998\) −1.63033e10 −0.519179
\(999\) −5.98177e8 −0.0189824
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.b.1.9 13
3.2 odd 2 387.8.a.d.1.5 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
43.8.a.b.1.9 13 1.1 even 1 trivial
387.8.a.d.1.5 13 3.2 odd 2