Properties

Label 177.10.a.b.1.18
Level $177$
Weight $10$
Character 177.1
Self dual yes
Analytic conductor $91.161$
Analytic rank $1$
Dimension $21$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(91.1613430010\)
Analytic rank: \(1\)
Dimension: \(21\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.18
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+33.2747 q^{2} -81.0000 q^{3} +595.206 q^{4} -1222.14 q^{5} -2695.25 q^{6} +7343.10 q^{7} +2768.67 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q+33.2747 q^{2} -81.0000 q^{3} +595.206 q^{4} -1222.14 q^{5} -2695.25 q^{6} +7343.10 q^{7} +2768.67 q^{8} +6561.00 q^{9} -40666.2 q^{10} -29821.5 q^{11} -48211.7 q^{12} -22797.1 q^{13} +244340. q^{14} +98993.0 q^{15} -212619. q^{16} +633639. q^{17} +218315. q^{18} +943799. q^{19} -727423. q^{20} -594791. q^{21} -992302. q^{22} -2.11142e6 q^{23} -224262. q^{24} -459508. q^{25} -758567. q^{26} -531441. q^{27} +4.37066e6 q^{28} -3.29075e6 q^{29} +3.29396e6 q^{30} -5.35294e6 q^{31} -8.49239e6 q^{32} +2.41554e6 q^{33} +2.10842e7 q^{34} -8.97427e6 q^{35} +3.90515e6 q^{36} +1.22347e6 q^{37} +3.14046e7 q^{38} +1.84657e6 q^{39} -3.38369e6 q^{40} +1.04674e7 q^{41} -1.97915e7 q^{42} -2.91320e7 q^{43} -1.77500e7 q^{44} -8.01844e6 q^{45} -7.02569e7 q^{46} -2.38542e7 q^{47} +1.72221e7 q^{48} +1.35675e7 q^{49} -1.52900e7 q^{50} -5.13248e7 q^{51} -1.35690e7 q^{52} +4.57699e7 q^{53} -1.76835e7 q^{54} +3.64459e7 q^{55} +2.03306e7 q^{56} -7.64477e7 q^{57} -1.09499e8 q^{58} -1.21174e7 q^{59} +5.89213e7 q^{60} -1.05999e8 q^{61} -1.78117e8 q^{62} +4.81781e7 q^{63} -1.73721e8 q^{64} +2.78612e7 q^{65} +8.03765e7 q^{66} -3.01366e8 q^{67} +3.77146e8 q^{68} +1.71025e8 q^{69} -2.98616e8 q^{70} +6.25203e7 q^{71} +1.81652e7 q^{72} +1.58794e8 q^{73} +4.07105e7 q^{74} +3.72201e7 q^{75} +5.61755e8 q^{76} -2.18982e8 q^{77} +6.14439e7 q^{78} -4.83457e8 q^{79} +2.59849e8 q^{80} +4.30467e7 q^{81} +3.48300e8 q^{82} +2.41279e7 q^{83} -3.54024e8 q^{84} -7.74393e8 q^{85} -9.69358e8 q^{86} +2.66550e8 q^{87} -8.25660e7 q^{88} +3.08334e7 q^{89} -2.66811e8 q^{90} -1.67402e8 q^{91} -1.25673e9 q^{92} +4.33588e8 q^{93} -7.93741e8 q^{94} -1.15345e9 q^{95} +6.87884e8 q^{96} -1.35366e9 q^{97} +4.51456e8 q^{98} -1.95659e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21q + 20q^{2} - 1701q^{3} + 4950q^{4} + 2058q^{5} - 1620q^{6} - 17167q^{7} - 2853q^{8} + 137781q^{9} + O(q^{10}) \) \( 21q + 20q^{2} - 1701q^{3} + 4950q^{4} + 2058q^{5} - 1620q^{6} - 17167q^{7} - 2853q^{8} + 137781q^{9} - 31559q^{10} - 38751q^{11} - 400950q^{12} - 58915q^{13} + 3453q^{14} - 166698q^{15} + 1655714q^{16} - 64233q^{17} + 131220q^{18} - 1937236q^{19} - 1065507q^{20} + 1390527q^{21} - 5386882q^{22} - 1838574q^{23} + 231093q^{24} + 4565755q^{25} - 839702q^{26} - 11160261q^{27} - 4471034q^{28} + 15658544q^{29} + 2556279q^{30} - 14282802q^{31} - 2205286q^{32} + 3138831q^{33} + 19005532q^{34} - 8633300q^{35} + 32476950q^{36} + 7531195q^{37} + 26649773q^{38} + 4772115q^{39} + 17775672q^{40} + 18338245q^{41} - 279693q^{42} - 22480305q^{43} - 80230922q^{44} + 13502538q^{45} - 83894107q^{46} - 110397260q^{47} - 134112834q^{48} + 130653638q^{49} + 65575693q^{50} + 5202873q^{51} + 177908014q^{52} + 145498338q^{53} - 10628820q^{54} + 86448944q^{55} + 354387888q^{56} + 156916116q^{57} + 115508368q^{58} - 254464581q^{59} + 86306067q^{60} + 287595506q^{61} + 819899030q^{62} - 112632687q^{63} + 822446413q^{64} + 77238206q^{65} + 436337442q^{66} - 392860610q^{67} + 167325073q^{68} + 148924494q^{69} - 424902116q^{70} - 248960491q^{71} - 18718533q^{72} - 758406074q^{73} - 923266846q^{74} - 369826155q^{75} - 2312747568q^{76} - 878126795q^{77} + 68015862q^{78} - 1925801029q^{79} - 1898919861q^{80} + 903981141q^{81} - 3249102191q^{82} - 1650336307q^{83} + 362153754q^{84} - 2342480762q^{85} - 3609864952q^{86} - 1268342064q^{87} - 5987792887q^{88} - 574997526q^{89} - 207058599q^{90} - 4481387117q^{91} - 5317166770q^{92} + 1156906962q^{93} - 5360726568q^{94} - 2789231462q^{95} + 178628166q^{96} - 4651540898q^{97} - 5566652976q^{98} - 254245311q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 33.2747 1.47055 0.735274 0.677770i \(-0.237053\pi\)
0.735274 + 0.677770i \(0.237053\pi\)
\(3\) −81.0000 −0.577350
\(4\) 595.206 1.16251
\(5\) −1222.14 −0.874490 −0.437245 0.899343i \(-0.644046\pi\)
−0.437245 + 0.899343i \(0.644046\pi\)
\(6\) −2695.25 −0.849022
\(7\) 7343.10 1.15595 0.577974 0.816055i \(-0.303843\pi\)
0.577974 + 0.816055i \(0.303843\pi\)
\(8\) 2768.67 0.238983
\(9\) 6561.00 0.333333
\(10\) −40666.2 −1.28598
\(11\) −29821.5 −0.614133 −0.307067 0.951688i \(-0.599347\pi\)
−0.307067 + 0.951688i \(0.599347\pi\)
\(12\) −48211.7 −0.671177
\(13\) −22797.1 −0.221378 −0.110689 0.993855i \(-0.535306\pi\)
−0.110689 + 0.993855i \(0.535306\pi\)
\(14\) 244340. 1.69988
\(15\) 98993.0 0.504887
\(16\) −212619. −0.811077
\(17\) 633639. 1.84002 0.920008 0.391899i \(-0.128182\pi\)
0.920008 + 0.391899i \(0.128182\pi\)
\(18\) 218315. 0.490183
\(19\) 943799. 1.66145 0.830727 0.556680i \(-0.187925\pi\)
0.830727 + 0.556680i \(0.187925\pi\)
\(20\) −727423. −1.01661
\(21\) −594791. −0.667387
\(22\) −992302. −0.903113
\(23\) −2.11142e6 −1.57326 −0.786628 0.617427i \(-0.788175\pi\)
−0.786628 + 0.617427i \(0.788175\pi\)
\(24\) −224262. −0.137977
\(25\) −459508. −0.235268
\(26\) −758567. −0.325547
\(27\) −531441. −0.192450
\(28\) 4.37066e6 1.34380
\(29\) −3.29075e6 −0.863979 −0.431990 0.901879i \(-0.642188\pi\)
−0.431990 + 0.901879i \(0.642188\pi\)
\(30\) 3.29396e6 0.742460
\(31\) −5.35294e6 −1.04103 −0.520517 0.853852i \(-0.674261\pi\)
−0.520517 + 0.853852i \(0.674261\pi\)
\(32\) −8.49239e6 −1.43171
\(33\) 2.41554e6 0.354570
\(34\) 2.10842e7 2.70583
\(35\) −8.97427e6 −1.01086
\(36\) 3.90515e6 0.387504
\(37\) 1.22347e6 0.107321 0.0536605 0.998559i \(-0.482911\pi\)
0.0536605 + 0.998559i \(0.482911\pi\)
\(38\) 3.14046e7 2.44325
\(39\) 1.84657e6 0.127813
\(40\) −3.38369e6 −0.208988
\(41\) 1.04674e7 0.578511 0.289255 0.957252i \(-0.406592\pi\)
0.289255 + 0.957252i \(0.406592\pi\)
\(42\) −1.97915e7 −0.981425
\(43\) −2.91320e7 −1.29946 −0.649729 0.760166i \(-0.725117\pi\)
−0.649729 + 0.760166i \(0.725117\pi\)
\(44\) −1.77500e7 −0.713938
\(45\) −8.01844e6 −0.291497
\(46\) −7.02569e7 −2.31355
\(47\) −2.38542e7 −0.713057 −0.356528 0.934285i \(-0.616040\pi\)
−0.356528 + 0.934285i \(0.616040\pi\)
\(48\) 1.72221e7 0.468276
\(49\) 1.35675e7 0.336216
\(50\) −1.52900e7 −0.345973
\(51\) −5.13248e7 −1.06233
\(52\) −1.35690e7 −0.257355
\(53\) 4.57699e7 0.796781 0.398391 0.917216i \(-0.369569\pi\)
0.398391 + 0.917216i \(0.369569\pi\)
\(54\) −1.76835e7 −0.283007
\(55\) 3.64459e7 0.537053
\(56\) 2.03306e7 0.276252
\(57\) −7.64477e7 −0.959241
\(58\) −1.09499e8 −1.27052
\(59\) −1.21174e7 −0.130189
\(60\) 5.89213e7 0.586937
\(61\) −1.05999e8 −0.980207 −0.490103 0.871664i \(-0.663041\pi\)
−0.490103 + 0.871664i \(0.663041\pi\)
\(62\) −1.78117e8 −1.53089
\(63\) 4.81781e7 0.385316
\(64\) −1.73721e8 −1.29432
\(65\) 2.78612e7 0.193593
\(66\) 8.03765e7 0.521412
\(67\) −3.01366e8 −1.82708 −0.913540 0.406749i \(-0.866663\pi\)
−0.913540 + 0.406749i \(0.866663\pi\)
\(68\) 3.77146e8 2.13904
\(69\) 1.71025e8 0.908320
\(70\) −2.98616e8 −1.48653
\(71\) 6.25203e7 0.291983 0.145992 0.989286i \(-0.453363\pi\)
0.145992 + 0.989286i \(0.453363\pi\)
\(72\) 1.81652e7 0.0796609
\(73\) 1.58794e8 0.654456 0.327228 0.944945i \(-0.393886\pi\)
0.327228 + 0.944945i \(0.393886\pi\)
\(74\) 4.07105e7 0.157821
\(75\) 3.72201e7 0.135832
\(76\) 5.61755e8 1.93146
\(77\) −2.18982e8 −0.709906
\(78\) 6.14439e7 0.187955
\(79\) −4.83457e8 −1.39648 −0.698242 0.715862i \(-0.746034\pi\)
−0.698242 + 0.715862i \(0.746034\pi\)
\(80\) 2.59849e8 0.709278
\(81\) 4.30467e7 0.111111
\(82\) 3.48300e8 0.850728
\(83\) 2.41279e7 0.0558043 0.0279021 0.999611i \(-0.491117\pi\)
0.0279021 + 0.999611i \(0.491117\pi\)
\(84\) −3.54024e8 −0.775846
\(85\) −7.74393e8 −1.60908
\(86\) −9.69358e8 −1.91091
\(87\) 2.66550e8 0.498819
\(88\) −8.25660e7 −0.146767
\(89\) 3.08334e7 0.0520914 0.0260457 0.999661i \(-0.491708\pi\)
0.0260457 + 0.999661i \(0.491708\pi\)
\(90\) −2.66811e8 −0.428660
\(91\) −1.67402e8 −0.255902
\(92\) −1.25673e9 −1.82893
\(93\) 4.33588e8 0.601041
\(94\) −7.93741e8 −1.04858
\(95\) −1.15345e9 −1.45292
\(96\) 6.87884e8 0.826599
\(97\) −1.35366e9 −1.55251 −0.776257 0.630416i \(-0.782884\pi\)
−0.776257 + 0.630416i \(0.782884\pi\)
\(98\) 4.51456e8 0.494422
\(99\) −1.95659e8 −0.204711
\(100\) −2.73502e8 −0.273502
\(101\) −4.41470e7 −0.0422138 −0.0211069 0.999777i \(-0.506719\pi\)
−0.0211069 + 0.999777i \(0.506719\pi\)
\(102\) −1.70782e9 −1.56221
\(103\) −1.73960e9 −1.52294 −0.761470 0.648200i \(-0.775522\pi\)
−0.761470 + 0.648200i \(0.775522\pi\)
\(104\) −6.31177e7 −0.0529055
\(105\) 7.26916e8 0.583623
\(106\) 1.52298e9 1.17171
\(107\) 1.62889e9 1.20134 0.600670 0.799497i \(-0.294901\pi\)
0.600670 + 0.799497i \(0.294901\pi\)
\(108\) −3.16317e8 −0.223726
\(109\) −1.93862e9 −1.31545 −0.657725 0.753258i \(-0.728481\pi\)
−0.657725 + 0.753258i \(0.728481\pi\)
\(110\) 1.21273e9 0.789762
\(111\) −9.91009e7 −0.0619618
\(112\) −1.56128e9 −0.937563
\(113\) −5.01522e8 −0.289359 −0.144679 0.989479i \(-0.546215\pi\)
−0.144679 + 0.989479i \(0.546215\pi\)
\(114\) −2.54378e9 −1.41061
\(115\) 2.58044e9 1.37580
\(116\) −1.95867e9 −1.00439
\(117\) −1.49572e8 −0.0737927
\(118\) −4.03202e8 −0.191449
\(119\) 4.65288e9 2.12696
\(120\) 2.74079e8 0.120659
\(121\) −1.46863e9 −0.622840
\(122\) −3.52709e9 −1.44144
\(123\) −8.47859e8 −0.334003
\(124\) −3.18610e9 −1.21021
\(125\) 2.94857e9 1.08023
\(126\) 1.60311e9 0.566626
\(127\) 5.57053e8 0.190011 0.0950057 0.995477i \(-0.469713\pi\)
0.0950057 + 0.995477i \(0.469713\pi\)
\(128\) −1.43241e9 −0.471654
\(129\) 2.35969e9 0.750242
\(130\) 9.27073e8 0.284688
\(131\) 2.71370e8 0.0805084 0.0402542 0.999189i \(-0.487183\pi\)
0.0402542 + 0.999189i \(0.487183\pi\)
\(132\) 1.43775e9 0.412192
\(133\) 6.93041e9 1.92055
\(134\) −1.00279e10 −2.68681
\(135\) 6.49493e8 0.168296
\(136\) 1.75434e9 0.439732
\(137\) −9.01699e8 −0.218685 −0.109343 0.994004i \(-0.534875\pi\)
−0.109343 + 0.994004i \(0.534875\pi\)
\(138\) 5.69081e9 1.33573
\(139\) 4.40010e9 0.999761 0.499880 0.866095i \(-0.333377\pi\)
0.499880 + 0.866095i \(0.333377\pi\)
\(140\) −5.34154e9 −1.17514
\(141\) 1.93219e9 0.411683
\(142\) 2.08034e9 0.429376
\(143\) 6.79844e8 0.135956
\(144\) −1.39499e9 −0.270359
\(145\) 4.02174e9 0.755541
\(146\) 5.28381e9 0.962409
\(147\) −1.09897e9 −0.194115
\(148\) 7.28216e8 0.124762
\(149\) 9.19443e9 1.52822 0.764111 0.645084i \(-0.223178\pi\)
0.764111 + 0.645084i \(0.223178\pi\)
\(150\) 1.23849e9 0.199748
\(151\) 3.37031e9 0.527563 0.263781 0.964582i \(-0.415030\pi\)
0.263781 + 0.964582i \(0.415030\pi\)
\(152\) 2.61307e9 0.397059
\(153\) 4.15731e9 0.613339
\(154\) −7.28658e9 −1.04395
\(155\) 6.54202e9 0.910373
\(156\) 1.09909e9 0.148584
\(157\) 4.84811e9 0.636830 0.318415 0.947951i \(-0.396849\pi\)
0.318415 + 0.947951i \(0.396849\pi\)
\(158\) −1.60869e10 −2.05360
\(159\) −3.70737e9 −0.460022
\(160\) 1.03789e10 1.25202
\(161\) −1.55044e10 −1.81860
\(162\) 1.43237e9 0.163394
\(163\) −8.02089e8 −0.0889976 −0.0444988 0.999009i \(-0.514169\pi\)
−0.0444988 + 0.999009i \(0.514169\pi\)
\(164\) 6.23026e9 0.672526
\(165\) −2.95212e9 −0.310068
\(166\) 8.02848e8 0.0820629
\(167\) −1.29522e10 −1.28860 −0.644300 0.764773i \(-0.722851\pi\)
−0.644300 + 0.764773i \(0.722851\pi\)
\(168\) −1.64678e9 −0.159494
\(169\) −1.00848e10 −0.950992
\(170\) −2.57677e10 −2.36622
\(171\) 6.19226e9 0.553818
\(172\) −1.73395e10 −1.51064
\(173\) −1.23942e10 −1.05199 −0.525994 0.850488i \(-0.676307\pi\)
−0.525994 + 0.850488i \(0.676307\pi\)
\(174\) 8.86939e9 0.733537
\(175\) −3.37421e9 −0.271958
\(176\) 6.34062e9 0.498109
\(177\) 9.81506e8 0.0751646
\(178\) 1.02597e9 0.0766030
\(179\) −4.64159e9 −0.337931 −0.168966 0.985622i \(-0.554043\pi\)
−0.168966 + 0.985622i \(0.554043\pi\)
\(180\) −4.77262e9 −0.338868
\(181\) 1.20747e10 0.836227 0.418114 0.908395i \(-0.362691\pi\)
0.418114 + 0.908395i \(0.362691\pi\)
\(182\) −5.57024e9 −0.376316
\(183\) 8.58592e9 0.565923
\(184\) −5.84583e9 −0.375981
\(185\) −1.49524e9 −0.0938511
\(186\) 1.44275e10 0.883860
\(187\) −1.88961e10 −1.13002
\(188\) −1.41982e10 −0.828937
\(189\) −3.90243e9 −0.222462
\(190\) −3.83807e10 −2.13659
\(191\) 2.29459e10 1.24754 0.623771 0.781607i \(-0.285600\pi\)
0.623771 + 0.781607i \(0.285600\pi\)
\(192\) 1.40714e10 0.747278
\(193\) 8.99321e9 0.466559 0.233280 0.972410i \(-0.425054\pi\)
0.233280 + 0.972410i \(0.425054\pi\)
\(194\) −4.50425e10 −2.28305
\(195\) −2.25676e9 −0.111771
\(196\) 8.07549e9 0.390856
\(197\) 2.70257e10 1.27844 0.639219 0.769025i \(-0.279258\pi\)
0.639219 + 0.769025i \(0.279258\pi\)
\(198\) −6.51049e9 −0.301038
\(199\) −4.03112e10 −1.82216 −0.911080 0.412230i \(-0.864750\pi\)
−0.911080 + 0.412230i \(0.864750\pi\)
\(200\) −1.27223e9 −0.0562250
\(201\) 2.44106e10 1.05487
\(202\) −1.46898e9 −0.0620775
\(203\) −2.41643e10 −0.998715
\(204\) −3.05488e10 −1.23498
\(205\) −1.27926e10 −0.505902
\(206\) −5.78848e10 −2.23956
\(207\) −1.38530e10 −0.524419
\(208\) 4.84710e9 0.179555
\(209\) −2.81455e10 −1.02035
\(210\) 2.41879e10 0.858246
\(211\) 2.34009e10 0.812759 0.406379 0.913704i \(-0.366791\pi\)
0.406379 + 0.913704i \(0.366791\pi\)
\(212\) 2.72426e10 0.926268
\(213\) −5.06414e9 −0.168577
\(214\) 5.42010e10 1.76663
\(215\) 3.56032e10 1.13636
\(216\) −1.47139e9 −0.0459922
\(217\) −3.93072e10 −1.20338
\(218\) −6.45072e10 −1.93443
\(219\) −1.28623e10 −0.377850
\(220\) 2.16929e10 0.624331
\(221\) −1.44451e10 −0.407339
\(222\) −3.29755e9 −0.0911178
\(223\) 5.25145e10 1.42202 0.711012 0.703180i \(-0.248237\pi\)
0.711012 + 0.703180i \(0.248237\pi\)
\(224\) −6.23605e10 −1.65498
\(225\) −3.01483e9 −0.0784227
\(226\) −1.66880e10 −0.425516
\(227\) 5.93675e10 1.48400 0.741998 0.670402i \(-0.233878\pi\)
0.741998 + 0.670402i \(0.233878\pi\)
\(228\) −4.55022e10 −1.11513
\(229\) 1.29521e9 0.0311230 0.0155615 0.999879i \(-0.495046\pi\)
0.0155615 + 0.999879i \(0.495046\pi\)
\(230\) 8.58635e10 2.02317
\(231\) 1.77376e10 0.409865
\(232\) −9.11099e9 −0.206476
\(233\) −4.18809e10 −0.930925 −0.465462 0.885068i \(-0.654112\pi\)
−0.465462 + 0.885068i \(0.654112\pi\)
\(234\) −4.97696e9 −0.108516
\(235\) 2.91530e10 0.623560
\(236\) −7.21233e9 −0.151346
\(237\) 3.91600e10 0.806260
\(238\) 1.54823e11 3.12780
\(239\) −2.82303e10 −0.559661 −0.279831 0.960049i \(-0.590278\pi\)
−0.279831 + 0.960049i \(0.590278\pi\)
\(240\) −2.10478e10 −0.409502
\(241\) 9.42595e10 1.79990 0.899950 0.435993i \(-0.143603\pi\)
0.899950 + 0.435993i \(0.143603\pi\)
\(242\) −4.88681e10 −0.915917
\(243\) −3.48678e9 −0.0641500
\(244\) −6.30913e10 −1.13950
\(245\) −1.65814e10 −0.294018
\(246\) −2.82123e10 −0.491168
\(247\) −2.15159e10 −0.367809
\(248\) −1.48205e10 −0.248789
\(249\) −1.95436e9 −0.0322186
\(250\) 9.81127e10 1.58853
\(251\) 6.70119e10 1.06566 0.532832 0.846221i \(-0.321128\pi\)
0.532832 + 0.846221i \(0.321128\pi\)
\(252\) 2.86759e10 0.447935
\(253\) 6.29657e10 0.966189
\(254\) 1.85358e10 0.279421
\(255\) 6.27258e10 0.929000
\(256\) 4.12821e10 0.600733
\(257\) 8.43337e10 1.20587 0.602937 0.797789i \(-0.293997\pi\)
0.602937 + 0.797789i \(0.293997\pi\)
\(258\) 7.85180e10 1.10327
\(259\) 8.98405e9 0.124058
\(260\) 1.65832e10 0.225054
\(261\) −2.15906e10 −0.287993
\(262\) 9.02976e9 0.118391
\(263\) −8.29087e10 −1.06856 −0.534280 0.845307i \(-0.679417\pi\)
−0.534280 + 0.845307i \(0.679417\pi\)
\(264\) 6.68784e9 0.0847361
\(265\) −5.59371e10 −0.696777
\(266\) 2.30607e11 2.82427
\(267\) −2.49750e9 −0.0300750
\(268\) −1.79375e11 −2.12400
\(269\) −2.24386e10 −0.261282 −0.130641 0.991430i \(-0.541704\pi\)
−0.130641 + 0.991430i \(0.541704\pi\)
\(270\) 2.16117e10 0.247487
\(271\) −1.48344e11 −1.67074 −0.835368 0.549691i \(-0.814746\pi\)
−0.835368 + 0.549691i \(0.814746\pi\)
\(272\) −1.34724e11 −1.49240
\(273\) 1.35595e10 0.147745
\(274\) −3.00038e10 −0.321587
\(275\) 1.37032e10 0.144486
\(276\) 1.01795e11 1.05593
\(277\) −1.66355e11 −1.69777 −0.848883 0.528581i \(-0.822724\pi\)
−0.848883 + 0.528581i \(0.822724\pi\)
\(278\) 1.46412e11 1.47020
\(279\) −3.51206e10 −0.347011
\(280\) −2.48468e10 −0.241579
\(281\) 7.65570e10 0.732498 0.366249 0.930517i \(-0.380642\pi\)
0.366249 + 0.930517i \(0.380642\pi\)
\(282\) 6.42930e10 0.605400
\(283\) 1.42902e11 1.32434 0.662170 0.749354i \(-0.269636\pi\)
0.662170 + 0.749354i \(0.269636\pi\)
\(284\) 3.72125e10 0.339434
\(285\) 9.34295e10 0.838846
\(286\) 2.26216e10 0.199929
\(287\) 7.68632e10 0.668729
\(288\) −5.57186e10 −0.477237
\(289\) 2.82910e11 2.38566
\(290\) 1.33822e11 1.11106
\(291\) 1.09646e11 0.896345
\(292\) 9.45150e10 0.760813
\(293\) −1.62226e11 −1.28592 −0.642962 0.765898i \(-0.722295\pi\)
−0.642962 + 0.765898i \(0.722295\pi\)
\(294\) −3.65679e10 −0.285455
\(295\) 1.48091e10 0.113849
\(296\) 3.38738e9 0.0256479
\(297\) 1.58484e10 0.118190
\(298\) 3.05942e11 2.24733
\(299\) 4.81343e10 0.348284
\(300\) 2.21537e10 0.157907
\(301\) −2.13919e11 −1.50211
\(302\) 1.12146e11 0.775807
\(303\) 3.57590e9 0.0243722
\(304\) −2.00670e11 −1.34757
\(305\) 1.29545e11 0.857180
\(306\) 1.38333e11 0.901944
\(307\) −2.18138e11 −1.40155 −0.700774 0.713384i \(-0.747162\pi\)
−0.700774 + 0.713384i \(0.747162\pi\)
\(308\) −1.30340e11 −0.825275
\(309\) 1.40908e11 0.879270
\(310\) 2.17684e11 1.33875
\(311\) −1.16507e11 −0.706202 −0.353101 0.935585i \(-0.614873\pi\)
−0.353101 + 0.935585i \(0.614873\pi\)
\(312\) 5.11253e9 0.0305450
\(313\) 1.14756e11 0.675814 0.337907 0.941179i \(-0.390281\pi\)
0.337907 + 0.941179i \(0.390281\pi\)
\(314\) 1.61319e11 0.936490
\(315\) −5.88802e10 −0.336955
\(316\) −2.87757e11 −1.62343
\(317\) −3.51023e11 −1.95240 −0.976200 0.216872i \(-0.930415\pi\)
−0.976200 + 0.216872i \(0.930415\pi\)
\(318\) −1.23362e11 −0.676484
\(319\) 9.81350e10 0.530598
\(320\) 2.12311e11 1.13187
\(321\) −1.31940e11 −0.693594
\(322\) −5.15904e11 −2.67434
\(323\) 5.98028e11 3.05710
\(324\) 2.56217e10 0.129168
\(325\) 1.04755e10 0.0520832
\(326\) −2.66893e10 −0.130875
\(327\) 1.57029e11 0.759476
\(328\) 2.89808e10 0.138254
\(329\) −1.75164e11 −0.824256
\(330\) −9.82310e10 −0.455970
\(331\) 2.31178e11 1.05857 0.529285 0.848444i \(-0.322460\pi\)
0.529285 + 0.848444i \(0.322460\pi\)
\(332\) 1.43611e10 0.0648732
\(333\) 8.02717e9 0.0357737
\(334\) −4.30979e11 −1.89495
\(335\) 3.68310e11 1.59776
\(336\) 1.26464e11 0.541302
\(337\) −1.88549e11 −0.796324 −0.398162 0.917315i \(-0.630352\pi\)
−0.398162 + 0.917315i \(0.630352\pi\)
\(338\) −3.35569e11 −1.39848
\(339\) 4.06232e10 0.167061
\(340\) −4.60924e11 −1.87057
\(341\) 1.59633e11 0.639333
\(342\) 2.06046e11 0.814416
\(343\) −1.96693e11 −0.767300
\(344\) −8.06569e10 −0.310548
\(345\) −2.09016e11 −0.794316
\(346\) −4.12413e11 −1.54700
\(347\) 6.04999e10 0.224012 0.112006 0.993708i \(-0.464272\pi\)
0.112006 + 0.993708i \(0.464272\pi\)
\(348\) 1.58653e11 0.579883
\(349\) −1.82709e11 −0.659245 −0.329622 0.944113i \(-0.606921\pi\)
−0.329622 + 0.944113i \(0.606921\pi\)
\(350\) −1.12276e11 −0.399927
\(351\) 1.21153e10 0.0426042
\(352\) 2.53256e11 0.879261
\(353\) 4.28924e10 0.147026 0.0735130 0.997294i \(-0.476579\pi\)
0.0735130 + 0.997294i \(0.476579\pi\)
\(354\) 3.26593e10 0.110533
\(355\) −7.64083e10 −0.255336
\(356\) 1.83522e10 0.0605569
\(357\) −3.76883e11 −1.22800
\(358\) −1.54448e11 −0.496944
\(359\) −7.82389e10 −0.248598 −0.124299 0.992245i \(-0.539668\pi\)
−0.124299 + 0.992245i \(0.539668\pi\)
\(360\) −2.22004e10 −0.0696626
\(361\) 5.68068e11 1.76043
\(362\) 4.01784e11 1.22971
\(363\) 1.18959e11 0.359597
\(364\) −9.96385e10 −0.297489
\(365\) −1.94067e11 −0.572315
\(366\) 2.85694e11 0.832217
\(367\) −8.56690e10 −0.246505 −0.123253 0.992375i \(-0.539333\pi\)
−0.123253 + 0.992375i \(0.539333\pi\)
\(368\) 4.48928e11 1.27603
\(369\) 6.86766e10 0.192837
\(370\) −4.97538e10 −0.138013
\(371\) 3.36093e11 0.921038
\(372\) 2.58074e11 0.698718
\(373\) 2.10755e11 0.563752 0.281876 0.959451i \(-0.409043\pi\)
0.281876 + 0.959451i \(0.409043\pi\)
\(374\) −6.28761e11 −1.66174
\(375\) −2.38834e11 −0.623671
\(376\) −6.60444e10 −0.170408
\(377\) 7.50195e10 0.191266
\(378\) −1.29852e11 −0.327142
\(379\) −5.11215e11 −1.27270 −0.636351 0.771399i \(-0.719557\pi\)
−0.636351 + 0.771399i \(0.719557\pi\)
\(380\) −6.86541e11 −1.68904
\(381\) −4.51213e10 −0.109703
\(382\) 7.63519e11 1.83457
\(383\) 8.17990e11 1.94247 0.971233 0.238132i \(-0.0765350\pi\)
0.971233 + 0.238132i \(0.0765350\pi\)
\(384\) 1.16025e11 0.272309
\(385\) 2.67626e11 0.620806
\(386\) 2.99246e11 0.686098
\(387\) −1.91135e11 −0.433152
\(388\) −8.05705e11 −1.80482
\(389\) 1.37600e11 0.304681 0.152340 0.988328i \(-0.451319\pi\)
0.152340 + 0.988328i \(0.451319\pi\)
\(390\) −7.50929e10 −0.164364
\(391\) −1.33788e12 −2.89482
\(392\) 3.75641e10 0.0803499
\(393\) −2.19810e10 −0.0464815
\(394\) 8.99273e11 1.88000
\(395\) 5.90850e11 1.22121
\(396\) −1.16457e11 −0.237979
\(397\) 2.84094e11 0.573991 0.286995 0.957932i \(-0.407344\pi\)
0.286995 + 0.957932i \(0.407344\pi\)
\(398\) −1.34134e12 −2.67957
\(399\) −5.61363e11 −1.10883
\(400\) 9.77001e10 0.190821
\(401\) −6.62746e11 −1.27996 −0.639982 0.768390i \(-0.721058\pi\)
−0.639982 + 0.768390i \(0.721058\pi\)
\(402\) 8.12257e11 1.55123
\(403\) 1.22032e11 0.230462
\(404\) −2.62766e10 −0.0490741
\(405\) −5.26090e10 −0.0971655
\(406\) −8.04059e11 −1.46866
\(407\) −3.64856e10 −0.0659094
\(408\) −1.42101e11 −0.253879
\(409\) −4.50389e11 −0.795853 −0.397926 0.917417i \(-0.630270\pi\)
−0.397926 + 0.917417i \(0.630270\pi\)
\(410\) −4.25670e11 −0.743953
\(411\) 7.30376e10 0.126258
\(412\) −1.03542e12 −1.77044
\(413\) −8.89790e10 −0.150492
\(414\) −4.60955e11 −0.771183
\(415\) −2.94875e10 −0.0488002
\(416\) 1.93602e11 0.316949
\(417\) −3.56408e11 −0.577212
\(418\) −9.36534e11 −1.50048
\(419\) 9.29357e11 1.47306 0.736528 0.676407i \(-0.236464\pi\)
0.736528 + 0.676407i \(0.236464\pi\)
\(420\) 4.32665e11 0.678469
\(421\) −6.37533e11 −0.989084 −0.494542 0.869154i \(-0.664664\pi\)
−0.494542 + 0.869154i \(0.664664\pi\)
\(422\) 7.78658e11 1.19520
\(423\) −1.56507e11 −0.237686
\(424\) 1.26722e11 0.190417
\(425\) −2.91162e11 −0.432897
\(426\) −1.68508e11 −0.247900
\(427\) −7.78362e11 −1.13307
\(428\) 9.69528e11 1.39657
\(429\) −5.50674e10 −0.0784940
\(430\) 1.18469e12 1.67107
\(431\) −2.65041e11 −0.369969 −0.184984 0.982741i \(-0.559223\pi\)
−0.184984 + 0.982741i \(0.559223\pi\)
\(432\) 1.12994e11 0.156092
\(433\) −2.08274e11 −0.284733 −0.142367 0.989814i \(-0.545471\pi\)
−0.142367 + 0.989814i \(0.545471\pi\)
\(434\) −1.30793e12 −1.76963
\(435\) −3.25761e11 −0.436212
\(436\) −1.15388e12 −1.52923
\(437\) −1.99276e12 −2.61389
\(438\) −4.27989e11 −0.555647
\(439\) 2.97100e11 0.381779 0.190890 0.981611i \(-0.438863\pi\)
0.190890 + 0.981611i \(0.438863\pi\)
\(440\) 1.00907e11 0.128346
\(441\) 8.90166e10 0.112072
\(442\) −4.80658e11 −0.599012
\(443\) 4.93616e11 0.608937 0.304469 0.952522i \(-0.401521\pi\)
0.304469 + 0.952522i \(0.401521\pi\)
\(444\) −5.89855e10 −0.0720314
\(445\) −3.76826e10 −0.0455534
\(446\) 1.74740e12 2.09116
\(447\) −7.44749e11 −0.882320
\(448\) −1.27565e12 −1.49617
\(449\) −2.14905e11 −0.249539 −0.124769 0.992186i \(-0.539819\pi\)
−0.124769 + 0.992186i \(0.539819\pi\)
\(450\) −1.00318e11 −0.115324
\(451\) −3.12154e11 −0.355283
\(452\) −2.98509e11 −0.336383
\(453\) −2.72995e11 −0.304589
\(454\) 1.97544e12 2.18229
\(455\) 2.04587e11 0.223783
\(456\) −2.11659e11 −0.229242
\(457\) 6.45951e11 0.692750 0.346375 0.938096i \(-0.387413\pi\)
0.346375 + 0.938096i \(0.387413\pi\)
\(458\) 4.30978e10 0.0457679
\(459\) −3.36742e11 −0.354111
\(460\) 1.53590e12 1.59938
\(461\) −4.25233e11 −0.438503 −0.219252 0.975668i \(-0.570362\pi\)
−0.219252 + 0.975668i \(0.570362\pi\)
\(462\) 5.90213e11 0.602726
\(463\) 1.46536e12 1.48194 0.740969 0.671539i \(-0.234367\pi\)
0.740969 + 0.671539i \(0.234367\pi\)
\(464\) 6.99675e11 0.700754
\(465\) −5.29904e11 −0.525604
\(466\) −1.39358e12 −1.36897
\(467\) −6.74336e10 −0.0656070 −0.0328035 0.999462i \(-0.510444\pi\)
−0.0328035 + 0.999462i \(0.510444\pi\)
\(468\) −8.90261e10 −0.0857850
\(469\) −2.21296e12 −2.11201
\(470\) 9.70059e11 0.916976
\(471\) −3.92697e11 −0.367674
\(472\) −3.35490e10 −0.0311129
\(473\) 8.68760e11 0.798040
\(474\) 1.30304e12 1.18564
\(475\) −4.33683e11 −0.390887
\(476\) 2.76942e12 2.47262
\(477\) 3.00297e11 0.265594
\(478\) −9.39355e11 −0.823009
\(479\) −6.79998e11 −0.590199 −0.295099 0.955467i \(-0.595353\pi\)
−0.295099 + 0.955467i \(0.595353\pi\)
\(480\) −8.40688e11 −0.722852
\(481\) −2.78915e10 −0.0237585
\(482\) 3.13646e12 2.64684
\(483\) 1.25585e12 1.04997
\(484\) −8.74135e11 −0.724060
\(485\) 1.65435e12 1.35766
\(486\) −1.16022e11 −0.0943357
\(487\) −1.21750e12 −0.980821 −0.490410 0.871492i \(-0.663153\pi\)
−0.490410 + 0.871492i \(0.663153\pi\)
\(488\) −2.93476e11 −0.234252
\(489\) 6.49692e10 0.0513828
\(490\) −5.51741e11 −0.432367
\(491\) −9.98749e11 −0.775514 −0.387757 0.921762i \(-0.626750\pi\)
−0.387757 + 0.921762i \(0.626750\pi\)
\(492\) −5.04651e11 −0.388283
\(493\) −2.08514e12 −1.58974
\(494\) −7.15935e11 −0.540882
\(495\) 2.39122e11 0.179018
\(496\) 1.13814e12 0.844358
\(497\) 4.59093e11 0.337518
\(498\) −6.50306e10 −0.0473790
\(499\) −1.86731e12 −1.34823 −0.674116 0.738626i \(-0.735475\pi\)
−0.674116 + 0.738626i \(0.735475\pi\)
\(500\) 1.75501e12 1.25578
\(501\) 1.04912e12 0.743973
\(502\) 2.22980e12 1.56711
\(503\) 9.42114e11 0.656217 0.328108 0.944640i \(-0.393589\pi\)
0.328108 + 0.944640i \(0.393589\pi\)
\(504\) 1.33389e11 0.0920839
\(505\) 5.39536e10 0.0369155
\(506\) 2.09517e12 1.42083
\(507\) 8.16868e11 0.549055
\(508\) 3.31562e11 0.220891
\(509\) −1.88541e12 −1.24502 −0.622508 0.782613i \(-0.713886\pi\)
−0.622508 + 0.782613i \(0.713886\pi\)
\(510\) 2.08718e12 1.36614
\(511\) 1.16604e12 0.756517
\(512\) 2.10704e12 1.35506
\(513\) −5.01573e11 −0.319747
\(514\) 2.80618e12 1.77330
\(515\) 2.12603e12 1.33180
\(516\) 1.40450e12 0.872166
\(517\) 7.11367e11 0.437912
\(518\) 2.98942e11 0.182433
\(519\) 1.00393e12 0.607366
\(520\) 7.71384e10 0.0462653
\(521\) −7.90673e11 −0.470140 −0.235070 0.971978i \(-0.575532\pi\)
−0.235070 + 0.971978i \(0.575532\pi\)
\(522\) −7.18420e11 −0.423508
\(523\) 6.50634e11 0.380259 0.190129 0.981759i \(-0.439109\pi\)
0.190129 + 0.981759i \(0.439109\pi\)
\(524\) 1.61521e11 0.0935920
\(525\) 2.73311e11 0.157015
\(526\) −2.75876e12 −1.57137
\(527\) −3.39183e12 −1.91552
\(528\) −5.13590e11 −0.287584
\(529\) 2.65694e12 1.47513
\(530\) −1.86129e12 −1.02464
\(531\) −7.95020e10 −0.0433963
\(532\) 4.12503e12 2.23267
\(533\) −2.38626e11 −0.128070
\(534\) −8.31037e10 −0.0442267
\(535\) −1.99073e12 −1.05056
\(536\) −8.34383e11 −0.436641
\(537\) 3.75969e11 0.195105
\(538\) −7.46636e11 −0.384228
\(539\) −4.04605e11 −0.206482
\(540\) 3.86583e11 0.195646
\(541\) 1.93440e12 0.970862 0.485431 0.874275i \(-0.338663\pi\)
0.485431 + 0.874275i \(0.338663\pi\)
\(542\) −4.93610e12 −2.45690
\(543\) −9.78054e11 −0.482796
\(544\) −5.38111e12 −2.63437
\(545\) 2.36926e12 1.15035
\(546\) 4.51189e11 0.217266
\(547\) −4.16068e11 −0.198711 −0.0993554 0.995052i \(-0.531678\pi\)
−0.0993554 + 0.995052i \(0.531678\pi\)
\(548\) −5.36697e11 −0.254224
\(549\) −6.95460e11 −0.326736
\(550\) 4.55971e11 0.212474
\(551\) −3.10580e12 −1.43546
\(552\) 4.73512e11 0.217073
\(553\) −3.55007e12 −1.61426
\(554\) −5.53542e12 −2.49665
\(555\) 1.21115e11 0.0541849
\(556\) 2.61897e12 1.16223
\(557\) −1.43212e9 −0.000630422 0 −0.000315211 1.00000i \(-0.500100\pi\)
−0.000315211 1.00000i \(0.500100\pi\)
\(558\) −1.16863e12 −0.510297
\(559\) 6.64125e11 0.287671
\(560\) 1.90810e12 0.819889
\(561\) 1.53058e12 0.652415
\(562\) 2.54741e12 1.07717
\(563\) −7.08013e11 −0.296998 −0.148499 0.988913i \(-0.547444\pi\)
−0.148499 + 0.988913i \(0.547444\pi\)
\(564\) 1.15005e12 0.478587
\(565\) 6.12928e11 0.253041
\(566\) 4.75502e12 1.94751
\(567\) 3.16096e11 0.128439
\(568\) 1.73098e11 0.0697790
\(569\) −3.86045e12 −1.54395 −0.771975 0.635653i \(-0.780731\pi\)
−0.771975 + 0.635653i \(0.780731\pi\)
\(570\) 3.10884e12 1.23356
\(571\) 8.54976e11 0.336582 0.168291 0.985737i \(-0.446175\pi\)
0.168291 + 0.985737i \(0.446175\pi\)
\(572\) 4.04648e11 0.158050
\(573\) −1.85862e12 −0.720269
\(574\) 2.55760e12 0.983398
\(575\) 9.70214e11 0.370137
\(576\) −1.13978e12 −0.431441
\(577\) −8.16514e11 −0.306671 −0.153335 0.988174i \(-0.549002\pi\)
−0.153335 + 0.988174i \(0.549002\pi\)
\(578\) 9.41376e12 3.50823
\(579\) −7.28450e11 −0.269368
\(580\) 2.39377e12 0.878326
\(581\) 1.77173e11 0.0645068
\(582\) 3.64844e12 1.31812
\(583\) −1.36493e12 −0.489330
\(584\) 4.39647e11 0.156404
\(585\) 1.82797e11 0.0645309
\(586\) −5.39801e12 −1.89101
\(587\) −6.88105e11 −0.239212 −0.119606 0.992821i \(-0.538163\pi\)
−0.119606 + 0.992821i \(0.538163\pi\)
\(588\) −6.54115e11 −0.225661
\(589\) −5.05210e12 −1.72963
\(590\) 4.92767e11 0.167420
\(591\) −2.18908e12 −0.738106
\(592\) −2.60132e11 −0.0870456
\(593\) 1.01206e12 0.336093 0.168046 0.985779i \(-0.446254\pi\)
0.168046 + 0.985779i \(0.446254\pi\)
\(594\) 5.27350e11 0.173804
\(595\) −5.68645e12 −1.86001
\(596\) 5.47258e12 1.77658
\(597\) 3.26520e12 1.05202
\(598\) 1.60165e12 0.512169
\(599\) −2.62579e11 −0.0833373 −0.0416687 0.999131i \(-0.513267\pi\)
−0.0416687 + 0.999131i \(0.513267\pi\)
\(600\) 1.03050e11 0.0324615
\(601\) −4.47857e12 −1.40024 −0.700122 0.714023i \(-0.746871\pi\)
−0.700122 + 0.714023i \(0.746871\pi\)
\(602\) −7.11810e12 −2.20892
\(603\) −1.97726e12 −0.609027
\(604\) 2.00603e12 0.613298
\(605\) 1.79486e12 0.544667
\(606\) 1.18987e11 0.0358404
\(607\) −4.13002e12 −1.23482 −0.617408 0.786643i \(-0.711817\pi\)
−0.617408 + 0.786643i \(0.711817\pi\)
\(608\) −8.01511e12 −2.37872
\(609\) 1.95731e12 0.576608
\(610\) 4.31058e12 1.26053
\(611\) 5.43806e11 0.157855
\(612\) 2.47445e12 0.713014
\(613\) 6.67693e12 1.90987 0.954937 0.296809i \(-0.0959225\pi\)
0.954937 + 0.296809i \(0.0959225\pi\)
\(614\) −7.25847e12 −2.06104
\(615\) 1.03620e12 0.292082
\(616\) −6.06290e11 −0.169655
\(617\) 3.41841e12 0.949601 0.474801 0.880093i \(-0.342520\pi\)
0.474801 + 0.880093i \(0.342520\pi\)
\(618\) 4.68867e12 1.29301
\(619\) 2.23977e12 0.613190 0.306595 0.951840i \(-0.400810\pi\)
0.306595 + 0.951840i \(0.400810\pi\)
\(620\) 3.89385e12 1.05832
\(621\) 1.12210e12 0.302773
\(622\) −3.87673e12 −1.03850
\(623\) 2.26413e11 0.0602150
\(624\) −3.92615e11 −0.103666
\(625\) −2.70607e12 −0.709381
\(626\) 3.81848e12 0.993817
\(627\) 2.27979e12 0.589102
\(628\) 2.88562e12 0.740323
\(629\) 7.75237e11 0.197472
\(630\) −1.95922e12 −0.495508
\(631\) 1.76983e11 0.0444425 0.0222212 0.999753i \(-0.492926\pi\)
0.0222212 + 0.999753i \(0.492926\pi\)
\(632\) −1.33853e12 −0.333735
\(633\) −1.89547e12 −0.469246
\(634\) −1.16802e13 −2.87110
\(635\) −6.80795e11 −0.166163
\(636\) −2.20665e12 −0.534781
\(637\) −3.09301e11 −0.0744309
\(638\) 3.26541e12 0.780270
\(639\) 4.10195e11 0.0973278
\(640\) 1.75060e12 0.412456
\(641\) −5.42293e12 −1.26874 −0.634371 0.773029i \(-0.718741\pi\)
−0.634371 + 0.773029i \(0.718741\pi\)
\(642\) −4.39028e12 −1.01996
\(643\) 7.75462e12 1.78900 0.894502 0.447064i \(-0.147531\pi\)
0.894502 + 0.447064i \(0.147531\pi\)
\(644\) −9.22830e12 −2.11415
\(645\) −2.88386e12 −0.656079
\(646\) 1.98992e13 4.49562
\(647\) −5.81831e12 −1.30535 −0.652676 0.757637i \(-0.726354\pi\)
−0.652676 + 0.757637i \(0.726354\pi\)
\(648\) 1.19182e11 0.0265536
\(649\) 3.61358e11 0.0799533
\(650\) 3.48568e11 0.0765909
\(651\) 3.18388e12 0.694772
\(652\) −4.77408e11 −0.103461
\(653\) 1.39312e12 0.299833 0.149917 0.988699i \(-0.452099\pi\)
0.149917 + 0.988699i \(0.452099\pi\)
\(654\) 5.22508e12 1.11685
\(655\) −3.31651e11 −0.0704037
\(656\) −2.22557e12 −0.469217
\(657\) 1.04185e12 0.218152
\(658\) −5.82852e12 −1.21211
\(659\) 1.30966e12 0.270504 0.135252 0.990811i \(-0.456816\pi\)
0.135252 + 0.990811i \(0.456816\pi\)
\(660\) −1.75712e12 −0.360458
\(661\) 8.50546e12 1.73297 0.866486 0.499202i \(-0.166373\pi\)
0.866486 + 0.499202i \(0.166373\pi\)
\(662\) 7.69237e12 1.55668
\(663\) 1.17006e12 0.235178
\(664\) 6.68021e10 0.0133363
\(665\) −8.46991e12 −1.67950
\(666\) 2.67102e11 0.0526069
\(667\) 6.94815e12 1.35926
\(668\) −7.70921e12 −1.49801
\(669\) −4.25367e12 −0.821006
\(670\) 1.22554e13 2.34959
\(671\) 3.16105e12 0.601977
\(672\) 5.05120e12 0.955505
\(673\) 7.37062e12 1.38496 0.692478 0.721439i \(-0.256519\pi\)
0.692478 + 0.721439i \(0.256519\pi\)
\(674\) −6.27392e12 −1.17103
\(675\) 2.44201e11 0.0452774
\(676\) −6.00253e12 −1.10554
\(677\) −1.27182e12 −0.232689 −0.116344 0.993209i \(-0.537118\pi\)
−0.116344 + 0.993209i \(0.537118\pi\)
\(678\) 1.35173e12 0.245672
\(679\) −9.94004e12 −1.79463
\(680\) −2.14404e12 −0.384541
\(681\) −4.80877e12 −0.856785
\(682\) 5.31173e12 0.940170
\(683\) 1.72717e12 0.303698 0.151849 0.988404i \(-0.451477\pi\)
0.151849 + 0.988404i \(0.451477\pi\)
\(684\) 3.68568e12 0.643820
\(685\) 1.10200e12 0.191238
\(686\) −6.54490e12 −1.12835
\(687\) −1.04912e11 −0.0179689
\(688\) 6.19401e12 1.05396
\(689\) −1.04342e12 −0.176390
\(690\) −6.95494e12 −1.16808
\(691\) −9.94197e12 −1.65890 −0.829452 0.558578i \(-0.811347\pi\)
−0.829452 + 0.558578i \(0.811347\pi\)
\(692\) −7.37711e12 −1.22295
\(693\) −1.43674e12 −0.236635
\(694\) 2.01312e12 0.329421
\(695\) −5.37752e12 −0.874280
\(696\) 7.37990e11 0.119209
\(697\) 6.63255e12 1.06447
\(698\) −6.07960e12 −0.969451
\(699\) 3.39236e12 0.537470
\(700\) −2.00835e12 −0.316154
\(701\) 1.02802e13 1.60795 0.803973 0.594665i \(-0.202715\pi\)
0.803973 + 0.594665i \(0.202715\pi\)
\(702\) 4.03134e11 0.0626516
\(703\) 1.15471e12 0.178309
\(704\) 5.18062e12 0.794887
\(705\) −2.36140e12 −0.360013
\(706\) 1.42723e12 0.216209
\(707\) −3.24176e11 −0.0487970
\(708\) 5.84199e11 0.0873798
\(709\) 3.73020e12 0.554401 0.277200 0.960812i \(-0.410593\pi\)
0.277200 + 0.960812i \(0.410593\pi\)
\(710\) −2.54246e12 −0.375485
\(711\) −3.17196e12 −0.465495
\(712\) 8.53675e10 0.0124489
\(713\) 1.13023e13 1.63781
\(714\) −1.25407e13 −1.80584
\(715\) −8.30862e11 −0.118892
\(716\) −2.76270e12 −0.392849
\(717\) 2.28666e12 0.323120
\(718\) −2.60338e12 −0.365575
\(719\) −6.22941e12 −0.869295 −0.434647 0.900601i \(-0.643127\pi\)
−0.434647 + 0.900601i \(0.643127\pi\)
\(720\) 1.70487e12 0.236426
\(721\) −1.27741e13 −1.76044
\(722\) 1.89023e13 2.58879
\(723\) −7.63502e12 −1.03917
\(724\) 7.18697e12 0.972125
\(725\) 1.51212e12 0.203267
\(726\) 3.95831e12 0.528805
\(727\) −1.29128e13 −1.71442 −0.857209 0.514969i \(-0.827803\pi\)
−0.857209 + 0.514969i \(0.827803\pi\)
\(728\) −4.63480e11 −0.0611561
\(729\) 2.82430e11 0.0370370
\(730\) −6.45754e12 −0.841616
\(731\) −1.84592e13 −2.39102
\(732\) 5.11040e12 0.657892
\(733\) −1.19680e13 −1.53128 −0.765640 0.643269i \(-0.777578\pi\)
−0.765640 + 0.643269i \(0.777578\pi\)
\(734\) −2.85061e12 −0.362498
\(735\) 1.34309e12 0.169751
\(736\) 1.79310e13 2.25245
\(737\) 8.98719e12 1.12207
\(738\) 2.28519e12 0.283576
\(739\) 1.09774e13 1.35394 0.676968 0.736012i \(-0.263294\pi\)
0.676968 + 0.736012i \(0.263294\pi\)
\(740\) −8.89979e11 −0.109103
\(741\) 1.74279e12 0.212355
\(742\) 1.11834e13 1.35443
\(743\) 5.36386e12 0.645696 0.322848 0.946451i \(-0.395360\pi\)
0.322848 + 0.946451i \(0.395360\pi\)
\(744\) 1.20046e12 0.143638
\(745\) −1.12368e13 −1.33641
\(746\) 7.01281e12 0.829024
\(747\) 1.58303e11 0.0186014
\(748\) −1.12471e13 −1.31366
\(749\) 1.19611e13 1.38869
\(750\) −7.94713e12 −0.917138
\(751\) 7.01860e12 0.805138 0.402569 0.915390i \(-0.368117\pi\)
0.402569 + 0.915390i \(0.368117\pi\)
\(752\) 5.07185e12 0.578344
\(753\) −5.42796e12 −0.615261
\(754\) 2.49625e12 0.281266
\(755\) −4.11898e12 −0.461348
\(756\) −2.32275e12 −0.258615
\(757\) 1.79609e13 1.98791 0.993954 0.109801i \(-0.0350212\pi\)
0.993954 + 0.109801i \(0.0350212\pi\)
\(758\) −1.70105e13 −1.87157
\(759\) −5.10022e12 −0.557829
\(760\) −3.19353e12 −0.347224
\(761\) 1.18244e13 1.27805 0.639027 0.769185i \(-0.279337\pi\)
0.639027 + 0.769185i \(0.279337\pi\)
\(762\) −1.50140e12 −0.161324
\(763\) −1.42355e13 −1.52059
\(764\) 1.36576e13 1.45028
\(765\) −5.08079e12 −0.536358
\(766\) 2.72184e13 2.85649
\(767\) 2.76241e11 0.0288210
\(768\) −3.34385e12 −0.346834
\(769\) −6.84476e12 −0.705813 −0.352907 0.935659i \(-0.614807\pi\)
−0.352907 + 0.935659i \(0.614807\pi\)
\(770\) 8.90519e12 0.912925
\(771\) −6.83103e12 −0.696212
\(772\) 5.35281e12 0.542381
\(773\) −2.32880e12 −0.234598 −0.117299 0.993097i \(-0.537424\pi\)
−0.117299 + 0.993097i \(0.537424\pi\)
\(774\) −6.35996e12 −0.636972
\(775\) 2.45972e12 0.244922
\(776\) −3.74783e12 −0.371024
\(777\) −7.27708e11 −0.0716246
\(778\) 4.57860e12 0.448048
\(779\) 9.87912e12 0.961169
\(780\) −1.34324e12 −0.129935
\(781\) −1.86445e12 −0.179317
\(782\) −4.45175e13 −4.25697
\(783\) 1.74884e12 0.166273
\(784\) −2.88472e12 −0.272697
\(785\) −5.92505e12 −0.556901
\(786\) −7.31410e11 −0.0683533
\(787\) −8.23251e12 −0.764973 −0.382486 0.923961i \(-0.624932\pi\)
−0.382486 + 0.923961i \(0.624932\pi\)
\(788\) 1.60859e13 1.48620
\(789\) 6.71560e12 0.616934
\(790\) 1.96604e13 1.79585
\(791\) −3.68272e12 −0.334484
\(792\) −5.41715e11 −0.0489224
\(793\) 2.41647e12 0.216996
\(794\) 9.45315e12 0.844081
\(795\) 4.53091e12 0.402284
\(796\) −2.39935e13 −2.11828
\(797\) −6.43290e12 −0.564735 −0.282367 0.959306i \(-0.591120\pi\)
−0.282367 + 0.959306i \(0.591120\pi\)
\(798\) −1.86792e13 −1.63059
\(799\) −1.51149e13 −1.31204
\(800\) 3.90232e12 0.336836
\(801\) 2.02298e11 0.0173638
\(802\) −2.20527e13 −1.88225
\(803\) −4.73547e12 −0.401923
\(804\) 1.45294e13 1.22629
\(805\) 1.89485e13 1.59035
\(806\) 4.06056e12 0.338906
\(807\) 1.81752e12 0.150851
\(808\) −1.22228e11 −0.0100884
\(809\) 9.69199e12 0.795508 0.397754 0.917492i \(-0.369790\pi\)
0.397754 + 0.917492i \(0.369790\pi\)
\(810\) −1.75055e12 −0.142887
\(811\) 2.19755e13 1.78379 0.891895 0.452242i \(-0.149376\pi\)
0.891895 + 0.452242i \(0.149376\pi\)
\(812\) −1.43827e13 −1.16102
\(813\) 1.20159e13 0.964600
\(814\) −1.21405e12 −0.0969229
\(815\) 9.80262e11 0.0778275
\(816\) 1.09126e13 0.861635
\(817\) −2.74947e13 −2.15899
\(818\) −1.49866e13 −1.17034
\(819\) −1.09832e12 −0.0853005
\(820\) −7.61423e12 −0.588117
\(821\) 1.92611e13 1.47958 0.739789 0.672839i \(-0.234925\pi\)
0.739789 + 0.672839i \(0.234925\pi\)
\(822\) 2.43031e12 0.185668
\(823\) −3.16275e12 −0.240307 −0.120153 0.992755i \(-0.538339\pi\)
−0.120153 + 0.992755i \(0.538339\pi\)
\(824\) −4.81639e12 −0.363956
\(825\) −1.10996e12 −0.0834190
\(826\) −2.96075e12 −0.221305
\(827\) −1.34577e13 −1.00045 −0.500226 0.865895i \(-0.666750\pi\)
−0.500226 + 0.865895i \(0.666750\pi\)
\(828\) −8.24541e12 −0.609643
\(829\) 8.05129e12 0.592066 0.296033 0.955178i \(-0.404336\pi\)
0.296033 + 0.955178i \(0.404336\pi\)
\(830\) −9.81189e11 −0.0717631
\(831\) 1.34748e13 0.980205
\(832\) 3.96034e12 0.286535
\(833\) 8.59692e12 0.618644
\(834\) −1.18594e13 −0.848818
\(835\) 1.58293e13 1.12687
\(836\) −1.67524e13 −1.18617
\(837\) 2.84477e12 0.200347
\(838\) 3.09241e13 2.16620
\(839\) 1.60923e12 0.112122 0.0560608 0.998427i \(-0.482146\pi\)
0.0560608 + 0.998427i \(0.482146\pi\)
\(840\) 2.01259e12 0.139476
\(841\) −3.67814e12 −0.253540
\(842\) −2.12137e13 −1.45450
\(843\) −6.20112e12 −0.422908
\(844\) 1.39284e13 0.944842
\(845\) 1.23250e13 0.831632
\(846\) −5.20773e12 −0.349528
\(847\) −1.07843e13 −0.719971
\(848\) −9.73156e12 −0.646251
\(849\) −1.15751e13 −0.764608
\(850\) −9.68834e12 −0.636596
\(851\) −2.58325e12 −0.168843
\(852\) −3.01421e12 −0.195973
\(853\) 4.39684e11 0.0284361 0.0142180 0.999899i \(-0.495474\pi\)
0.0142180 + 0.999899i \(0.495474\pi\)
\(854\) −2.58998e13 −1.66623
\(855\) −7.56779e12 −0.484308
\(856\) 4.50987e12 0.287099
\(857\) −1.45210e13 −0.919567 −0.459783 0.888031i \(-0.652073\pi\)
−0.459783 + 0.888031i \(0.652073\pi\)
\(858\) −1.83235e12 −0.115429
\(859\) −3.00853e13 −1.88532 −0.942659 0.333757i \(-0.891683\pi\)
−0.942659 + 0.333757i \(0.891683\pi\)
\(860\) 2.11913e13 1.32103
\(861\) −6.22592e12 −0.386091
\(862\) −8.81916e12 −0.544057
\(863\) −2.40297e13 −1.47469 −0.737344 0.675518i \(-0.763920\pi\)
−0.737344 + 0.675518i \(0.763920\pi\)
\(864\) 4.51321e12 0.275533
\(865\) 1.51474e13 0.919953
\(866\) −6.93024e12 −0.418714
\(867\) −2.29157e13 −1.37736
\(868\) −2.33959e13 −1.39895
\(869\) 1.44174e13 0.857627
\(870\) −1.08396e13 −0.641470
\(871\) 6.87027e12 0.404476
\(872\) −5.36741e12 −0.314370
\(873\) −8.88134e12 −0.517505
\(874\) −6.63084e13 −3.84385
\(875\) 2.16516e13 1.24869
\(876\) −7.65572e12 −0.439256
\(877\) 8.54455e12 0.487743 0.243872 0.969808i \(-0.421582\pi\)
0.243872 + 0.969808i \(0.421582\pi\)
\(878\) 9.88592e12 0.561425
\(879\) 1.31403e13 0.742428
\(880\) −7.74910e12 −0.435591
\(881\) 5.82575e10 0.00325807 0.00162904 0.999999i \(-0.499481\pi\)
0.00162904 + 0.999999i \(0.499481\pi\)
\(882\) 2.96200e12 0.164807
\(883\) 1.99889e13 1.10654 0.553268 0.833003i \(-0.313380\pi\)
0.553268 + 0.833003i \(0.313380\pi\)
\(884\) −8.59784e12 −0.473537
\(885\) −1.19953e12 −0.0657307
\(886\) 1.64249e13 0.895472
\(887\) 3.41855e13 1.85432 0.927162 0.374660i \(-0.122241\pi\)
0.927162 + 0.374660i \(0.122241\pi\)
\(888\) −2.74378e11 −0.0148078
\(889\) 4.09050e12 0.219643
\(890\) −1.25388e12 −0.0669885
\(891\) −1.28372e12 −0.0682370
\(892\) 3.12569e13 1.65312
\(893\) −2.25135e13 −1.18471
\(894\) −2.47813e13 −1.29749
\(895\) 5.67265e12 0.295517
\(896\) −1.05184e13 −0.545207
\(897\) −3.89888e12 −0.201082
\(898\) −7.15090e12 −0.366959
\(899\) 1.76152e13 0.899431
\(900\) −1.79445e12 −0.0911674
\(901\) 2.90016e13 1.46609
\(902\) −1.03868e13 −0.522460
\(903\) 1.73274e13 0.867241
\(904\) −1.38855e12 −0.0691517
\(905\) −1.47570e13 −0.731272
\(906\) −9.08385e12 −0.447912
\(907\) −2.53806e13 −1.24529 −0.622643 0.782506i \(-0.713941\pi\)
−0.622643 + 0.782506i \(0.713941\pi\)
\(908\) 3.53359e13 1.72516
\(909\) −2.89648e11 −0.0140713
\(910\) 6.80759e12 0.329084
\(911\) 2.89755e13 1.39379 0.696897 0.717172i \(-0.254564\pi\)
0.696897 + 0.717172i \(0.254564\pi\)
\(912\) 1.62542e13 0.778018
\(913\) −7.19529e11 −0.0342712
\(914\) 2.14938e13 1.01872
\(915\) −1.04932e13 −0.494893
\(916\) 7.70919e11 0.0361809
\(917\) 1.99270e12 0.0930635
\(918\) −1.12050e13 −0.520738
\(919\) 1.24959e13 0.577895 0.288948 0.957345i \(-0.406695\pi\)
0.288948 + 0.957345i \(0.406695\pi\)
\(920\) 7.14440e12 0.328791
\(921\) 1.76691e13 0.809184
\(922\) −1.41495e13 −0.644840
\(923\) −1.42528e12 −0.0646387
\(924\) 1.05575e13 0.476473
\(925\) −5.62193e11 −0.0252492
\(926\) 4.87594e13 2.17926
\(927\) −1.14135e13 −0.507647
\(928\) 2.79463e13 1.23697
\(929\) 3.35765e13 1.47899 0.739494 0.673164i \(-0.235065\pi\)
0.739494 + 0.673164i \(0.235065\pi\)
\(930\) −1.76324e13 −0.772926
\(931\) 1.28050e13 0.558608
\(932\) −2.49278e13 −1.08221
\(933\) 9.43704e12 0.407726
\(934\) −2.24383e12 −0.0964783
\(935\) 2.30936e13 0.988186
\(936\) −4.14115e11 −0.0176352
\(937\) −2.67690e13 −1.13450 −0.567249 0.823547i \(-0.691992\pi\)
−0.567249 + 0.823547i \(0.691992\pi\)
\(938\) −7.36356e13 −3.10581
\(939\) −9.29527e12 −0.390181
\(940\) 1.73521e13 0.724897
\(941\) −1.27689e13 −0.530886 −0.265443 0.964127i \(-0.585518\pi\)
−0.265443 + 0.964127i \(0.585518\pi\)
\(942\) −1.30669e13 −0.540683
\(943\) −2.21011e13 −0.910146
\(944\) 2.57638e12 0.105593
\(945\) 4.76930e12 0.194541
\(946\) 2.89077e13 1.17356
\(947\) 1.65573e13 0.668983 0.334492 0.942399i \(-0.391435\pi\)
0.334492 + 0.942399i \(0.391435\pi\)
\(948\) 2.33083e13 0.937288
\(949\) −3.62004e12 −0.144882
\(950\) −1.44307e13 −0.574818
\(951\) 2.84329e13 1.12722
\(952\) 1.28823e13 0.508307
\(953\) 2.27052e13 0.891676 0.445838 0.895114i \(-0.352906\pi\)
0.445838 + 0.895114i \(0.352906\pi\)
\(954\) 9.99228e12 0.390568
\(955\) −2.80431e13 −1.09096
\(956\) −1.68029e13 −0.650613
\(957\) −7.94893e12 −0.306341
\(958\) −2.26268e13 −0.867915
\(959\) −6.62127e12 −0.252789
\(960\) −1.71972e13 −0.653486
\(961\) 2.21433e12 0.0837503
\(962\) −9.28082e11 −0.0349381
\(963\) 1.06872e13 0.400447
\(964\) 5.61039e13 2.09241
\(965\) −1.09909e13 −0.408001
\(966\) 4.17882e13 1.54403
\(967\) −4.78565e13 −1.76004 −0.880019 0.474939i \(-0.842470\pi\)
−0.880019 + 0.474939i \(0.842470\pi\)
\(968\) −4.06614e12 −0.148848
\(969\) −4.84402e13 −1.76502
\(970\) 5.50481e13 1.99650
\(971\) 5.29607e13 1.91191 0.955955 0.293514i \(-0.0948248\pi\)
0.955955 + 0.293514i \(0.0948248\pi\)
\(972\) −2.07536e12 −0.0745752
\(973\) 3.23104e13 1.15567
\(974\) −4.05121e13 −1.44234
\(975\) −8.48512e11 −0.0300703
\(976\) 2.25374e13 0.795023
\(977\) 1.40423e13 0.493074 0.246537 0.969133i \(-0.420707\pi\)
0.246537 + 0.969133i \(0.420707\pi\)
\(978\) 2.16183e12 0.0755609
\(979\) −9.19498e11 −0.0319911
\(980\) −9.86935e12 −0.341799
\(981\) −1.27193e13 −0.438484
\(982\) −3.32331e13 −1.14043
\(983\) −7.60200e12 −0.259679 −0.129840 0.991535i \(-0.541446\pi\)
−0.129840 + 0.991535i \(0.541446\pi\)
\(984\) −2.34744e12 −0.0798210
\(985\) −3.30291e13 −1.11798
\(986\) −6.93826e13 −2.33778
\(987\) 1.41883e13 0.475885
\(988\) −1.28064e13 −0.427583
\(989\) 6.15098e13 2.04438
\(990\) 7.95671e12 0.263254
\(991\) −2.87997e13 −0.948542 −0.474271 0.880379i \(-0.657288\pi\)
−0.474271 + 0.880379i \(0.657288\pi\)
\(992\) 4.54593e13 1.49046
\(993\) −1.87254e13 −0.611166
\(994\) 1.52762e13 0.496336
\(995\) 4.92657e13 1.59346
\(996\) −1.16325e12 −0.0374545
\(997\) −4.03019e13 −1.29181 −0.645904 0.763419i \(-0.723519\pi\)
−0.645904 + 0.763419i \(0.723519\pi\)
\(998\) −6.21343e13 −1.98264
\(999\) −6.50201e11 −0.0206539
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 177.10.a.b.1.18 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.10.a.b.1.18 21 1.1 even 1 trivial