Properties

Label 177.10.a.a.1.11
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 1.11
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.19317 q^{2} +81.0000 q^{3} -501.804 q^{4} -2710.01 q^{5} -258.647 q^{6} -1841.65 q^{7} +3237.25 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q-3.19317 q^{2} +81.0000 q^{3} -501.804 q^{4} -2710.01 q^{5} -258.647 q^{6} -1841.65 q^{7} +3237.25 q^{8} +6561.00 q^{9} +8653.54 q^{10} +14149.8 q^{11} -40646.1 q^{12} +130013. q^{13} +5880.69 q^{14} -219511. q^{15} +246586. q^{16} -550189. q^{17} -20950.4 q^{18} -76696.3 q^{19} +1.35989e6 q^{20} -149173. q^{21} -45182.8 q^{22} +1.37026e6 q^{23} +262217. q^{24} +5.39105e6 q^{25} -415155. q^{26} +531441. q^{27} +924144. q^{28} +1.57961e6 q^{29} +700937. q^{30} +2.58891e6 q^{31} -2.44486e6 q^{32} +1.14614e6 q^{33} +1.75685e6 q^{34} +4.99088e6 q^{35} -3.29233e6 q^{36} +346183. q^{37} +244905. q^{38} +1.05311e7 q^{39} -8.77299e6 q^{40} -2.25179e7 q^{41} +476336. q^{42} +3.52017e7 q^{43} -7.10043e6 q^{44} -1.77804e7 q^{45} -4.37546e6 q^{46} +3.28211e7 q^{47} +1.99735e7 q^{48} -3.69620e7 q^{49} -1.72146e7 q^{50} -4.45653e7 q^{51} -6.52412e7 q^{52} -5.90435e7 q^{53} -1.69698e6 q^{54} -3.83462e7 q^{55} -5.96187e6 q^{56} -6.21240e6 q^{57} -5.04398e6 q^{58} +1.21174e7 q^{59} +1.10151e8 q^{60} -1.93757e7 q^{61} -8.26685e6 q^{62} -1.20830e7 q^{63} -1.18445e8 q^{64} -3.52338e8 q^{65} -3.65981e6 q^{66} -2.82027e8 q^{67} +2.76087e8 q^{68} +1.10991e8 q^{69} -1.59368e7 q^{70} +6.18639e7 q^{71} +2.12396e7 q^{72} +3.00367e8 q^{73} -1.10542e6 q^{74} +4.36675e8 q^{75} +3.84865e7 q^{76} -2.60589e7 q^{77} -3.36276e7 q^{78} -2.20861e8 q^{79} -6.68252e8 q^{80} +4.30467e7 q^{81} +7.19035e7 q^{82} -2.37823e8 q^{83} +7.48557e7 q^{84} +1.49102e9 q^{85} -1.12405e8 q^{86} +1.27949e8 q^{87} +4.58065e7 q^{88} -6.63983e8 q^{89} +5.67759e7 q^{90} -2.39439e8 q^{91} -6.87599e8 q^{92} +2.09702e8 q^{93} -1.04803e8 q^{94} +2.07848e8 q^{95} -1.98034e8 q^{96} +1.03674e9 q^{97} +1.18026e8 q^{98} +9.28370e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21q - 66q^{2} + 1701q^{3} + 5206q^{4} - 2964q^{5} - 5346q^{6} - 30775q^{7} - 24621q^{8} + 137781q^{9} + O(q^{10}) \) \( 21q - 66q^{2} + 1701q^{3} + 5206q^{4} - 2964q^{5} - 5346q^{6} - 30775q^{7} - 24621q^{8} + 137781q^{9} - 54663q^{10} - 151769q^{11} + 421686q^{12} - 153611q^{13} - 286771q^{14} - 240084q^{15} + 805530q^{16} - 723621q^{17} - 433026q^{18} - 549388q^{19} - 527311q^{20} - 2492775q^{21} + 2973158q^{22} + 169962q^{23} - 1994301q^{24} + 8035779q^{25} - 2337392q^{26} + 11160261q^{27} - 22659054q^{28} - 16845442q^{29} - 4427703q^{30} - 19307976q^{31} - 44923568q^{32} - 12293289q^{33} - 35547496q^{34} - 34882596q^{35} + 34156566q^{36} - 41561129q^{37} - 52335371q^{38} - 12442491q^{39} - 125735038q^{40} - 68169291q^{41} - 23228451q^{42} - 25719587q^{43} - 126277032q^{44} - 19446804q^{45} - 292814271q^{46} - 174095332q^{47} + 65247930q^{48} + 7479350q^{49} - 227877439q^{50} - 58613301q^{51} - 232397708q^{52} - 228390500q^{53} - 35075106q^{54} - 29426208q^{55} + 326778474q^{56} - 44500428q^{57} + 480343762q^{58} + 254464581q^{59} - 42712191q^{60} - 183928964q^{61} - 21753862q^{62} - 201914775q^{63} + 310571245q^{64} + 5308466q^{65} + 240825798q^{66} - 82724114q^{67} - 138336205q^{68} + 13766922q^{69} + 1030274876q^{70} - 404721965q^{71} - 161538381q^{72} + 154162574q^{73} + 36352054q^{74} + 650898099q^{75} + 1068940636q^{76} - 448535481q^{77} - 189328752q^{78} + 272529635q^{79} - 345587859q^{80} + 903981141q^{81} - 38412637q^{82} + 432518643q^{83} - 1835383374q^{84} - 126211490q^{85} - 3699273072q^{86} - 1364480802q^{87} + 170111045q^{88} - 1255621070q^{89} - 358643943q^{90} + 1448885849q^{91} + 1568933320q^{92} - 1563946056q^{93} - 1908445164q^{94} - 2896546490q^{95} - 3638809008q^{96} + 1007235486q^{97} - 9506868248q^{98} - 995756409q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.19317 −0.141120 −0.0705598 0.997508i \(-0.522479\pi\)
−0.0705598 + 0.997508i \(0.522479\pi\)
\(3\) 81.0000 0.577350
\(4\) −501.804 −0.980085
\(5\) −2710.01 −1.93913 −0.969564 0.244838i \(-0.921265\pi\)
−0.969564 + 0.244838i \(0.921265\pi\)
\(6\) −258.647 −0.0814755
\(7\) −1841.65 −0.289911 −0.144956 0.989438i \(-0.546304\pi\)
−0.144956 + 0.989438i \(0.546304\pi\)
\(8\) 3237.25 0.279429
\(9\) 6561.00 0.333333
\(10\) 8653.54 0.273649
\(11\) 14149.8 0.291396 0.145698 0.989329i \(-0.453457\pi\)
0.145698 + 0.989329i \(0.453457\pi\)
\(12\) −40646.1 −0.565852
\(13\) 130013. 1.26253 0.631267 0.775565i \(-0.282535\pi\)
0.631267 + 0.775565i \(0.282535\pi\)
\(14\) 5880.69 0.0409121
\(15\) −219511. −1.11956
\(16\) 246586. 0.940652
\(17\) −550189. −1.59769 −0.798844 0.601539i \(-0.794555\pi\)
−0.798844 + 0.601539i \(0.794555\pi\)
\(18\) −20950.4 −0.0470399
\(19\) −76696.3 −0.135015 −0.0675077 0.997719i \(-0.521505\pi\)
−0.0675077 + 0.997719i \(0.521505\pi\)
\(20\) 1.35989e6 1.90051
\(21\) −149173. −0.167380
\(22\) −45182.8 −0.0411217
\(23\) 1.37026e6 1.02100 0.510501 0.859877i \(-0.329460\pi\)
0.510501 + 0.859877i \(0.329460\pi\)
\(24\) 262217. 0.161328
\(25\) 5.39105e6 2.76022
\(26\) −415155. −0.178168
\(27\) 531441. 0.192450
\(28\) 924144. 0.284138
\(29\) 1.57961e6 0.414725 0.207362 0.978264i \(-0.433512\pi\)
0.207362 + 0.978264i \(0.433512\pi\)
\(30\) 700937. 0.157991
\(31\) 2.58891e6 0.503489 0.251745 0.967794i \(-0.418996\pi\)
0.251745 + 0.967794i \(0.418996\pi\)
\(32\) −2.44486e6 −0.412173
\(33\) 1.14614e6 0.168238
\(34\) 1.75685e6 0.225465
\(35\) 4.99088e6 0.562175
\(36\) −3.29233e6 −0.326695
\(37\) 346183. 0.0303667 0.0151833 0.999885i \(-0.495167\pi\)
0.0151833 + 0.999885i \(0.495167\pi\)
\(38\) 244905. 0.0190533
\(39\) 1.05311e7 0.728925
\(40\) −8.77299e6 −0.541848
\(41\) −2.25179e7 −1.24452 −0.622258 0.782812i \(-0.713785\pi\)
−0.622258 + 0.782812i \(0.713785\pi\)
\(42\) 476336. 0.0236206
\(43\) 3.52017e7 1.57020 0.785102 0.619366i \(-0.212610\pi\)
0.785102 + 0.619366i \(0.212610\pi\)
\(44\) −7.10043e6 −0.285593
\(45\) −1.77804e7 −0.646376
\(46\) −4.37546e6 −0.144083
\(47\) 3.28211e7 0.981099 0.490549 0.871413i \(-0.336796\pi\)
0.490549 + 0.871413i \(0.336796\pi\)
\(48\) 1.99735e7 0.543086
\(49\) −3.69620e7 −0.915952
\(50\) −1.72146e7 −0.389521
\(51\) −4.45653e7 −0.922425
\(52\) −6.52412e7 −1.23739
\(53\) −5.90435e7 −1.02785 −0.513926 0.857835i \(-0.671809\pi\)
−0.513926 + 0.857835i \(0.671809\pi\)
\(54\) −1.69698e6 −0.0271585
\(55\) −3.83462e7 −0.565055
\(56\) −5.96187e6 −0.0810095
\(57\) −6.21240e6 −0.0779512
\(58\) −5.04398e6 −0.0585258
\(59\) 1.21174e7 0.130189
\(60\) 1.10151e8 1.09726
\(61\) −1.93757e7 −0.179173 −0.0895865 0.995979i \(-0.528555\pi\)
−0.0895865 + 0.995979i \(0.528555\pi\)
\(62\) −8.26685e6 −0.0710522
\(63\) −1.20830e7 −0.0966370
\(64\) −1.18445e8 −0.882487
\(65\) −3.52338e8 −2.44822
\(66\) −3.65981e6 −0.0237416
\(67\) −2.82027e8 −1.70984 −0.854918 0.518763i \(-0.826393\pi\)
−0.854918 + 0.518763i \(0.826393\pi\)
\(68\) 2.76087e8 1.56587
\(69\) 1.10991e8 0.589475
\(70\) −1.59368e7 −0.0793339
\(71\) 6.18639e7 0.288918 0.144459 0.989511i \(-0.453856\pi\)
0.144459 + 0.989511i \(0.453856\pi\)
\(72\) 2.12396e7 0.0931430
\(73\) 3.00367e8 1.23794 0.618970 0.785414i \(-0.287550\pi\)
0.618970 + 0.785414i \(0.287550\pi\)
\(74\) −1.10542e6 −0.00428534
\(75\) 4.36675e8 1.59361
\(76\) 3.84865e7 0.132327
\(77\) −2.60589e7 −0.0844790
\(78\) −3.36276e7 −0.102866
\(79\) −2.20861e8 −0.637964 −0.318982 0.947761i \(-0.603341\pi\)
−0.318982 + 0.947761i \(0.603341\pi\)
\(80\) −6.68252e8 −1.82405
\(81\) 4.30467e7 0.111111
\(82\) 7.19035e7 0.175626
\(83\) −2.37823e8 −0.550051 −0.275025 0.961437i \(-0.588686\pi\)
−0.275025 + 0.961437i \(0.588686\pi\)
\(84\) 7.48557e7 0.164047
\(85\) 1.49102e9 3.09812
\(86\) −1.12405e8 −0.221587
\(87\) 1.27949e8 0.239441
\(88\) 4.58065e7 0.0814245
\(89\) −6.63983e8 −1.12177 −0.560883 0.827895i \(-0.689538\pi\)
−0.560883 + 0.827895i \(0.689538\pi\)
\(90\) 5.67759e7 0.0912163
\(91\) −2.39439e8 −0.366023
\(92\) −6.87599e8 −1.00067
\(93\) 2.09702e8 0.290690
\(94\) −1.04803e8 −0.138452
\(95\) 2.07848e8 0.261812
\(96\) −1.98034e8 −0.237968
\(97\) 1.03674e9 1.18904 0.594520 0.804081i \(-0.297342\pi\)
0.594520 + 0.804081i \(0.297342\pi\)
\(98\) 1.18026e8 0.129259
\(99\) 9.28370e7 0.0971321
\(100\) −2.70525e9 −2.70525
\(101\) 1.36477e9 1.30501 0.652504 0.757785i \(-0.273719\pi\)
0.652504 + 0.757785i \(0.273719\pi\)
\(102\) 1.42305e8 0.130172
\(103\) 1.23730e9 1.08320 0.541598 0.840638i \(-0.317820\pi\)
0.541598 + 0.840638i \(0.317820\pi\)
\(104\) 4.20886e8 0.352789
\(105\) 4.04262e8 0.324572
\(106\) 1.88536e8 0.145050
\(107\) −7.61706e8 −0.561772 −0.280886 0.959741i \(-0.590628\pi\)
−0.280886 + 0.959741i \(0.590628\pi\)
\(108\) −2.66679e8 −0.188617
\(109\) −6.01385e8 −0.408069 −0.204034 0.978964i \(-0.565405\pi\)
−0.204034 + 0.978964i \(0.565405\pi\)
\(110\) 1.22446e8 0.0797403
\(111\) 2.80408e7 0.0175322
\(112\) −4.54125e8 −0.272706
\(113\) 3.88144e8 0.223944 0.111972 0.993711i \(-0.464283\pi\)
0.111972 + 0.993711i \(0.464283\pi\)
\(114\) 1.98373e7 0.0110004
\(115\) −3.71341e9 −1.97985
\(116\) −7.92656e8 −0.406465
\(117\) 8.53018e8 0.420845
\(118\) −3.86928e7 −0.0183722
\(119\) 1.01325e9 0.463187
\(120\) −7.10612e8 −0.312836
\(121\) −2.15773e9 −0.915088
\(122\) 6.18698e7 0.0252848
\(123\) −1.82395e9 −0.718522
\(124\) −1.29913e9 −0.493462
\(125\) −9.31683e9 −3.41329
\(126\) 3.85832e7 0.0136374
\(127\) −3.50531e9 −1.19567 −0.597833 0.801621i \(-0.703971\pi\)
−0.597833 + 0.801621i \(0.703971\pi\)
\(128\) 1.62999e9 0.536710
\(129\) 2.85134e9 0.906558
\(130\) 1.12508e9 0.345491
\(131\) −1.38898e9 −0.412073 −0.206037 0.978544i \(-0.566057\pi\)
−0.206037 + 0.978544i \(0.566057\pi\)
\(132\) −5.75135e8 −0.164887
\(133\) 1.41247e8 0.0391425
\(134\) 9.00562e8 0.241291
\(135\) −1.44021e9 −0.373185
\(136\) −1.78110e9 −0.446440
\(137\) −3.63297e9 −0.881089 −0.440544 0.897731i \(-0.645215\pi\)
−0.440544 + 0.897731i \(0.645215\pi\)
\(138\) −3.54412e8 −0.0831865
\(139\) −1.45150e9 −0.329801 −0.164900 0.986310i \(-0.552730\pi\)
−0.164900 + 0.986310i \(0.552730\pi\)
\(140\) −2.50444e9 −0.550979
\(141\) 2.65851e9 0.566438
\(142\) −1.97542e8 −0.0407720
\(143\) 1.83967e9 0.367898
\(144\) 1.61785e9 0.313551
\(145\) −4.28077e9 −0.804204
\(146\) −9.59125e8 −0.174698
\(147\) −2.99392e9 −0.528825
\(148\) −1.73716e8 −0.0297619
\(149\) 3.34366e9 0.555756 0.277878 0.960616i \(-0.410369\pi\)
0.277878 + 0.960616i \(0.410369\pi\)
\(150\) −1.39438e9 −0.224890
\(151\) 4.11423e9 0.644009 0.322005 0.946738i \(-0.395643\pi\)
0.322005 + 0.946738i \(0.395643\pi\)
\(152\) −2.48285e8 −0.0377272
\(153\) −3.60979e9 −0.532562
\(154\) 8.32107e7 0.0119216
\(155\) −7.01599e9 −0.976330
\(156\) −5.28454e9 −0.714408
\(157\) −1.30658e10 −1.71627 −0.858137 0.513421i \(-0.828378\pi\)
−0.858137 + 0.513421i \(0.828378\pi\)
\(158\) 7.05246e8 0.0900292
\(159\) −4.78252e9 −0.593431
\(160\) 6.62562e9 0.799257
\(161\) −2.52352e9 −0.295999
\(162\) −1.37456e8 −0.0156800
\(163\) 7.82966e9 0.868758 0.434379 0.900730i \(-0.356968\pi\)
0.434379 + 0.900730i \(0.356968\pi\)
\(164\) 1.12996e10 1.21973
\(165\) −3.10604e9 −0.326234
\(166\) 7.59411e8 0.0776230
\(167\) 1.17540e10 1.16939 0.584697 0.811252i \(-0.301213\pi\)
0.584697 + 0.811252i \(0.301213\pi\)
\(168\) −4.82911e8 −0.0467709
\(169\) 6.29901e9 0.593994
\(170\) −4.76108e9 −0.437206
\(171\) −5.03205e8 −0.0450051
\(172\) −1.76644e10 −1.53893
\(173\) 4.88623e9 0.414731 0.207366 0.978264i \(-0.433511\pi\)
0.207366 + 0.978264i \(0.433511\pi\)
\(174\) −4.08562e8 −0.0337899
\(175\) −9.92840e9 −0.800218
\(176\) 3.48915e9 0.274103
\(177\) 9.81506e8 0.0751646
\(178\) 2.12021e9 0.158303
\(179\) 1.90828e10 1.38933 0.694664 0.719335i \(-0.255553\pi\)
0.694664 + 0.719335i \(0.255553\pi\)
\(180\) 8.92227e9 0.633504
\(181\) −1.44737e10 −1.00237 −0.501183 0.865341i \(-0.667102\pi\)
−0.501183 + 0.865341i \(0.667102\pi\)
\(182\) 7.64569e8 0.0516530
\(183\) −1.56943e9 −0.103446
\(184\) 4.43586e9 0.285297
\(185\) −9.38159e8 −0.0588849
\(186\) −6.69615e8 −0.0410220
\(187\) −7.78508e9 −0.465560
\(188\) −1.64697e10 −0.961561
\(189\) −9.78726e8 −0.0557934
\(190\) −6.63695e8 −0.0369468
\(191\) −8.99432e9 −0.489010 −0.244505 0.969648i \(-0.578626\pi\)
−0.244505 + 0.969648i \(0.578626\pi\)
\(192\) −9.59407e9 −0.509504
\(193\) −2.05992e9 −0.106867 −0.0534333 0.998571i \(-0.517016\pi\)
−0.0534333 + 0.998571i \(0.517016\pi\)
\(194\) −3.31049e9 −0.167797
\(195\) −2.85394e10 −1.41348
\(196\) 1.85476e10 0.897711
\(197\) −3.47766e9 −0.164509 −0.0822543 0.996611i \(-0.526212\pi\)
−0.0822543 + 0.996611i \(0.526212\pi\)
\(198\) −2.96444e8 −0.0137072
\(199\) −1.91700e10 −0.866531 −0.433266 0.901266i \(-0.642639\pi\)
−0.433266 + 0.901266i \(0.642639\pi\)
\(200\) 1.74522e10 0.771285
\(201\) −2.28442e10 −0.987174
\(202\) −4.35794e9 −0.184162
\(203\) −2.90909e9 −0.120233
\(204\) 2.23630e10 0.904055
\(205\) 6.10238e10 2.41328
\(206\) −3.95091e9 −0.152860
\(207\) 8.99024e9 0.340334
\(208\) 3.20596e10 1.18761
\(209\) −1.08524e9 −0.0393430
\(210\) −1.29088e9 −0.0458034
\(211\) 3.62374e10 1.25859 0.629297 0.777165i \(-0.283343\pi\)
0.629297 + 0.777165i \(0.283343\pi\)
\(212\) 2.96282e10 1.00738
\(213\) 5.01098e9 0.166807
\(214\) 2.43226e9 0.0792771
\(215\) −9.53972e10 −3.04483
\(216\) 1.72041e9 0.0537761
\(217\) −4.76786e9 −0.145967
\(218\) 1.92033e9 0.0575865
\(219\) 2.43298e10 0.714725
\(220\) 1.92423e10 0.553802
\(221\) −7.15320e10 −2.01714
\(222\) −8.95391e7 −0.00247414
\(223\) 3.47716e10 0.941570 0.470785 0.882248i \(-0.343971\pi\)
0.470785 + 0.882248i \(0.343971\pi\)
\(224\) 4.50257e9 0.119494
\(225\) 3.53707e10 0.920073
\(226\) −1.23941e9 −0.0316029
\(227\) −5.98792e9 −0.149679 −0.0748393 0.997196i \(-0.523844\pi\)
−0.0748393 + 0.997196i \(0.523844\pi\)
\(228\) 3.11741e9 0.0763988
\(229\) 6.32536e10 1.51994 0.759968 0.649960i \(-0.225214\pi\)
0.759968 + 0.649960i \(0.225214\pi\)
\(230\) 1.18576e10 0.279396
\(231\) −2.11077e9 −0.0487740
\(232\) 5.11360e9 0.115886
\(233\) −4.49276e9 −0.0998646 −0.0499323 0.998753i \(-0.515901\pi\)
−0.0499323 + 0.998753i \(0.515901\pi\)
\(234\) −2.72384e9 −0.0593895
\(235\) −8.89456e10 −1.90248
\(236\) −6.08054e9 −0.127596
\(237\) −1.78897e10 −0.368329
\(238\) −3.23549e9 −0.0653648
\(239\) −4.08140e10 −0.809131 −0.404566 0.914509i \(-0.632577\pi\)
−0.404566 + 0.914509i \(0.632577\pi\)
\(240\) −5.41285e10 −1.05311
\(241\) 1.11274e10 0.212480 0.106240 0.994341i \(-0.466119\pi\)
0.106240 + 0.994341i \(0.466119\pi\)
\(242\) 6.89001e9 0.129137
\(243\) 3.48678e9 0.0641500
\(244\) 9.72278e9 0.175605
\(245\) 1.00167e11 1.77615
\(246\) 5.82419e9 0.101398
\(247\) −9.97155e9 −0.170462
\(248\) 8.38096e9 0.140689
\(249\) −1.92637e10 −0.317572
\(250\) 2.97502e10 0.481682
\(251\) 2.74625e10 0.436725 0.218362 0.975868i \(-0.429929\pi\)
0.218362 + 0.975868i \(0.429929\pi\)
\(252\) 6.06331e9 0.0947125
\(253\) 1.93889e10 0.297516
\(254\) 1.11931e10 0.168732
\(255\) 1.20773e11 1.78870
\(256\) 5.54392e10 0.806746
\(257\) −8.63869e10 −1.23523 −0.617616 0.786480i \(-0.711902\pi\)
−0.617616 + 0.786480i \(0.711902\pi\)
\(258\) −9.10482e9 −0.127933
\(259\) −6.37545e8 −0.00880364
\(260\) 1.76805e11 2.39946
\(261\) 1.03638e10 0.138242
\(262\) 4.43525e9 0.0581516
\(263\) −1.48732e10 −0.191692 −0.0958459 0.995396i \(-0.530556\pi\)
−0.0958459 + 0.995396i \(0.530556\pi\)
\(264\) 3.71033e9 0.0470105
\(265\) 1.60009e11 1.99314
\(266\) −4.51027e8 −0.00552377
\(267\) −5.37827e10 −0.647652
\(268\) 1.41522e11 1.67578
\(269\) −1.12574e11 −1.31085 −0.655427 0.755259i \(-0.727511\pi\)
−0.655427 + 0.755259i \(0.727511\pi\)
\(270\) 4.59885e9 0.0526638
\(271\) −1.46802e11 −1.65337 −0.826683 0.562668i \(-0.809775\pi\)
−0.826683 + 0.562668i \(0.809775\pi\)
\(272\) −1.35669e11 −1.50287
\(273\) −1.93945e10 −0.211323
\(274\) 1.16007e10 0.124339
\(275\) 7.62824e10 0.804317
\(276\) −5.56955e10 −0.577736
\(277\) −1.43398e10 −0.146347 −0.0731736 0.997319i \(-0.523313\pi\)
−0.0731736 + 0.997319i \(0.523313\pi\)
\(278\) 4.63490e9 0.0465413
\(279\) 1.69859e10 0.167830
\(280\) 1.61567e10 0.157088
\(281\) −1.86643e11 −1.78580 −0.892902 0.450251i \(-0.851335\pi\)
−0.892902 + 0.450251i \(0.851335\pi\)
\(282\) −8.48908e9 −0.0799355
\(283\) 1.15819e11 1.07334 0.536672 0.843791i \(-0.319681\pi\)
0.536672 + 0.843791i \(0.319681\pi\)
\(284\) −3.10435e10 −0.283164
\(285\) 1.68357e10 0.151157
\(286\) −5.87438e9 −0.0519176
\(287\) 4.14700e10 0.360799
\(288\) −1.60408e10 −0.137391
\(289\) 1.84120e11 1.55260
\(290\) 1.36692e10 0.113489
\(291\) 8.39759e10 0.686493
\(292\) −1.50725e11 −1.21329
\(293\) −2.26078e11 −1.79206 −0.896032 0.443990i \(-0.853563\pi\)
−0.896032 + 0.443990i \(0.853563\pi\)
\(294\) 9.56010e9 0.0746276
\(295\) −3.28382e10 −0.252453
\(296\) 1.12068e9 0.00848533
\(297\) 7.51979e9 0.0560792
\(298\) −1.06769e10 −0.0784280
\(299\) 1.78152e11 1.28905
\(300\) −2.19125e11 −1.56188
\(301\) −6.48291e10 −0.455220
\(302\) −1.31374e10 −0.0908824
\(303\) 1.10546e11 0.753447
\(304\) −1.89123e10 −0.127003
\(305\) 5.25083e10 0.347439
\(306\) 1.15267e10 0.0751550
\(307\) −2.72211e10 −0.174897 −0.0874487 0.996169i \(-0.527871\pi\)
−0.0874487 + 0.996169i \(0.527871\pi\)
\(308\) 1.30765e10 0.0827966
\(309\) 1.00221e11 0.625383
\(310\) 2.24033e10 0.137779
\(311\) 1.83257e11 1.11080 0.555402 0.831582i \(-0.312564\pi\)
0.555402 + 0.831582i \(0.312564\pi\)
\(312\) 3.40918e10 0.203683
\(313\) −1.76321e11 −1.03837 −0.519187 0.854661i \(-0.673765\pi\)
−0.519187 + 0.854661i \(0.673765\pi\)
\(314\) 4.17213e10 0.242200
\(315\) 3.27452e10 0.187392
\(316\) 1.10829e11 0.625259
\(317\) −1.66747e11 −0.927454 −0.463727 0.885978i \(-0.653488\pi\)
−0.463727 + 0.885978i \(0.653488\pi\)
\(318\) 1.52714e10 0.0837447
\(319\) 2.23512e10 0.120849
\(320\) 3.20989e11 1.71125
\(321\) −6.16982e10 −0.324339
\(322\) 8.05805e9 0.0417713
\(323\) 4.21975e10 0.215712
\(324\) −2.16010e10 −0.108898
\(325\) 7.00909e11 3.48487
\(326\) −2.50015e10 −0.122599
\(327\) −4.87122e10 −0.235599
\(328\) −7.28961e10 −0.347754
\(329\) −6.04448e10 −0.284431
\(330\) 9.91813e9 0.0460381
\(331\) 2.11596e11 0.968906 0.484453 0.874817i \(-0.339019\pi\)
0.484453 + 0.874817i \(0.339019\pi\)
\(332\) 1.19341e11 0.539097
\(333\) 2.27130e9 0.0101222
\(334\) −3.75325e10 −0.165024
\(335\) 7.64298e11 3.31559
\(336\) −3.67841e10 −0.157447
\(337\) 2.26484e11 0.956538 0.478269 0.878214i \(-0.341264\pi\)
0.478269 + 0.878214i \(0.341264\pi\)
\(338\) −2.01138e10 −0.0838242
\(339\) 3.14397e10 0.129294
\(340\) −7.48199e11 −3.03642
\(341\) 3.66327e10 0.146715
\(342\) 1.60682e9 0.00635111
\(343\) 1.42388e11 0.555456
\(344\) 1.13957e11 0.438760
\(345\) −3.00786e11 −1.14307
\(346\) −1.56026e10 −0.0585267
\(347\) −3.84748e11 −1.42460 −0.712300 0.701875i \(-0.752347\pi\)
−0.712300 + 0.701875i \(0.752347\pi\)
\(348\) −6.42051e10 −0.234673
\(349\) −3.46454e11 −1.25006 −0.625031 0.780600i \(-0.714914\pi\)
−0.625031 + 0.780600i \(0.714914\pi\)
\(350\) 3.17031e10 0.112926
\(351\) 6.90945e10 0.242975
\(352\) −3.45944e10 −0.120106
\(353\) 1.25860e11 0.431421 0.215710 0.976457i \(-0.430793\pi\)
0.215710 + 0.976457i \(0.430793\pi\)
\(354\) −3.13412e9 −0.0106072
\(355\) −1.67652e11 −0.560249
\(356\) 3.33189e11 1.09943
\(357\) 8.20735e10 0.267421
\(358\) −6.09348e10 −0.196061
\(359\) −1.40320e11 −0.445855 −0.222927 0.974835i \(-0.571561\pi\)
−0.222927 + 0.974835i \(0.571561\pi\)
\(360\) −5.75596e10 −0.180616
\(361\) −3.16805e11 −0.981771
\(362\) 4.62171e10 0.141454
\(363\) −1.74776e11 −0.528326
\(364\) 1.20151e11 0.358733
\(365\) −8.14000e11 −2.40053
\(366\) 5.01146e9 0.0145982
\(367\) 5.93748e11 1.70846 0.854230 0.519896i \(-0.174029\pi\)
0.854230 + 0.519896i \(0.174029\pi\)
\(368\) 3.37886e11 0.960407
\(369\) −1.47740e11 −0.414839
\(370\) 2.99570e9 0.00830981
\(371\) 1.08737e11 0.297986
\(372\) −1.05229e11 −0.284901
\(373\) 5.66466e10 0.151525 0.0757624 0.997126i \(-0.475861\pi\)
0.0757624 + 0.997126i \(0.475861\pi\)
\(374\) 2.48591e10 0.0656996
\(375\) −7.54663e11 −1.97066
\(376\) 1.06250e11 0.274147
\(377\) 2.05371e11 0.523604
\(378\) 3.12524e9 0.00787354
\(379\) 7.08119e11 1.76291 0.881454 0.472269i \(-0.156565\pi\)
0.881454 + 0.472269i \(0.156565\pi\)
\(380\) −1.04299e11 −0.256598
\(381\) −2.83930e11 −0.690318
\(382\) 2.87204e10 0.0690090
\(383\) −7.14910e11 −1.69768 −0.848842 0.528647i \(-0.822699\pi\)
−0.848842 + 0.528647i \(0.822699\pi\)
\(384\) 1.32029e11 0.309869
\(385\) 7.06201e10 0.163816
\(386\) 6.57767e9 0.0150810
\(387\) 2.30959e11 0.523401
\(388\) −5.20240e11 −1.16536
\(389\) −8.65936e11 −1.91740 −0.958700 0.284421i \(-0.908199\pi\)
−0.958700 + 0.284421i \(0.908199\pi\)
\(390\) 9.11312e10 0.199470
\(391\) −7.53899e11 −1.63124
\(392\) −1.19655e11 −0.255943
\(393\) −1.12507e11 −0.237911
\(394\) 1.11048e10 0.0232154
\(395\) 5.98535e11 1.23709
\(396\) −4.65859e10 −0.0951977
\(397\) −6.09151e11 −1.23074 −0.615372 0.788237i \(-0.710994\pi\)
−0.615372 + 0.788237i \(0.710994\pi\)
\(398\) 6.12133e10 0.122285
\(399\) 1.14410e10 0.0225989
\(400\) 1.32936e12 2.59641
\(401\) −9.64079e10 −0.186193 −0.0930964 0.995657i \(-0.529676\pi\)
−0.0930964 + 0.995657i \(0.529676\pi\)
\(402\) 7.29455e10 0.139310
\(403\) 3.36594e11 0.635672
\(404\) −6.84846e11 −1.27902
\(405\) −1.16657e11 −0.215459
\(406\) 9.28922e9 0.0169673
\(407\) 4.89842e9 0.00884874
\(408\) −1.44269e11 −0.257752
\(409\) 7.08037e10 0.125113 0.0625563 0.998041i \(-0.480075\pi\)
0.0625563 + 0.998041i \(0.480075\pi\)
\(410\) −1.94860e11 −0.340561
\(411\) −2.94271e11 −0.508697
\(412\) −6.20881e11 −1.06162
\(413\) −2.23159e10 −0.0377432
\(414\) −2.87074e10 −0.0480278
\(415\) 6.44504e11 1.06662
\(416\) −3.17865e11 −0.520383
\(417\) −1.17572e11 −0.190411
\(418\) 3.46536e9 0.00555207
\(419\) −7.01996e11 −1.11268 −0.556342 0.830954i \(-0.687795\pi\)
−0.556342 + 0.830954i \(0.687795\pi\)
\(420\) −2.02860e11 −0.318108
\(421\) −2.42037e11 −0.375503 −0.187751 0.982217i \(-0.560120\pi\)
−0.187751 + 0.982217i \(0.560120\pi\)
\(422\) −1.15712e11 −0.177612
\(423\) 2.15339e11 0.327033
\(424\) −1.91138e11 −0.287211
\(425\) −2.96610e12 −4.40996
\(426\) −1.60009e10 −0.0235397
\(427\) 3.56831e10 0.0519442
\(428\) 3.82227e11 0.550585
\(429\) 1.49013e11 0.212406
\(430\) 3.04620e11 0.429685
\(431\) −3.96082e11 −0.552888 −0.276444 0.961030i \(-0.589156\pi\)
−0.276444 + 0.961030i \(0.589156\pi\)
\(432\) 1.31046e11 0.181029
\(433\) 7.07592e11 0.967358 0.483679 0.875245i \(-0.339300\pi\)
0.483679 + 0.875245i \(0.339300\pi\)
\(434\) 1.52246e10 0.0205988
\(435\) −3.46743e11 −0.464307
\(436\) 3.01777e11 0.399942
\(437\) −1.05094e11 −0.137851
\(438\) −7.76891e10 −0.100862
\(439\) −6.89241e11 −0.885688 −0.442844 0.896599i \(-0.646030\pi\)
−0.442844 + 0.896599i \(0.646030\pi\)
\(440\) −1.24136e11 −0.157893
\(441\) −2.42507e11 −0.305317
\(442\) 2.28414e11 0.284657
\(443\) 1.07328e12 1.32403 0.662015 0.749491i \(-0.269702\pi\)
0.662015 + 0.749491i \(0.269702\pi\)
\(444\) −1.40710e10 −0.0171831
\(445\) 1.79940e12 2.17525
\(446\) −1.11032e11 −0.132874
\(447\) 2.70836e11 0.320866
\(448\) 2.18134e11 0.255843
\(449\) −6.19979e11 −0.719894 −0.359947 0.932973i \(-0.617205\pi\)
−0.359947 + 0.932973i \(0.617205\pi\)
\(450\) −1.12945e11 −0.129840
\(451\) −3.18624e11 −0.362647
\(452\) −1.94772e11 −0.219485
\(453\) 3.33253e11 0.371819
\(454\) 1.91205e10 0.0211226
\(455\) 6.48882e11 0.709765
\(456\) −2.01111e10 −0.0217818
\(457\) 4.70420e10 0.0504502 0.0252251 0.999682i \(-0.491970\pi\)
0.0252251 + 0.999682i \(0.491970\pi\)
\(458\) −2.01980e11 −0.214493
\(459\) −2.92393e11 −0.307475
\(460\) 1.86340e12 1.94042
\(461\) 6.86441e11 0.707863 0.353931 0.935271i \(-0.384845\pi\)
0.353931 + 0.935271i \(0.384845\pi\)
\(462\) 6.74007e9 0.00688296
\(463\) 9.62356e11 0.973243 0.486621 0.873613i \(-0.338229\pi\)
0.486621 + 0.873613i \(0.338229\pi\)
\(464\) 3.89511e11 0.390112
\(465\) −5.68296e11 −0.563684
\(466\) 1.43462e10 0.0140929
\(467\) 1.04065e11 0.101246 0.0506230 0.998718i \(-0.483879\pi\)
0.0506230 + 0.998718i \(0.483879\pi\)
\(468\) −4.28048e11 −0.412464
\(469\) 5.19394e11 0.495700
\(470\) 2.84019e11 0.268477
\(471\) −1.05833e12 −0.990891
\(472\) 3.92269e10 0.0363785
\(473\) 4.98098e11 0.457551
\(474\) 5.71249e10 0.0519784
\(475\) −4.13474e11 −0.372672
\(476\) −5.08454e11 −0.453963
\(477\) −3.87384e11 −0.342617
\(478\) 1.30326e11 0.114184
\(479\) −1.00881e12 −0.875585 −0.437793 0.899076i \(-0.644240\pi\)
−0.437793 + 0.899076i \(0.644240\pi\)
\(480\) 5.36675e11 0.461451
\(481\) 4.50084e10 0.0383390
\(482\) −3.55318e10 −0.0299851
\(483\) −2.04405e11 −0.170895
\(484\) 1.08276e12 0.896865
\(485\) −2.80958e12 −2.30570
\(486\) −1.11339e10 −0.00905283
\(487\) 1.55674e12 1.25411 0.627053 0.778976i \(-0.284261\pi\)
0.627053 + 0.778976i \(0.284261\pi\)
\(488\) −6.27239e10 −0.0500661
\(489\) 6.34203e11 0.501578
\(490\) −3.19852e11 −0.250649
\(491\) 2.06409e12 1.60273 0.801367 0.598173i \(-0.204106\pi\)
0.801367 + 0.598173i \(0.204106\pi\)
\(492\) 9.15265e11 0.704212
\(493\) −8.69086e11 −0.662600
\(494\) 3.18409e10 0.0240555
\(495\) −2.51589e11 −0.188352
\(496\) 6.38391e11 0.473608
\(497\) −1.13931e11 −0.0837606
\(498\) 6.15123e10 0.0448156
\(499\) −1.98601e12 −1.43393 −0.716966 0.697108i \(-0.754470\pi\)
−0.716966 + 0.697108i \(0.754470\pi\)
\(500\) 4.67522e12 3.34531
\(501\) 9.52073e11 0.675150
\(502\) −8.76924e10 −0.0616304
\(503\) −1.09056e12 −0.759617 −0.379808 0.925065i \(-0.624010\pi\)
−0.379808 + 0.925065i \(0.624010\pi\)
\(504\) −3.91158e10 −0.0270032
\(505\) −3.69854e12 −2.53058
\(506\) −6.19120e10 −0.0419853
\(507\) 5.10219e11 0.342942
\(508\) 1.75898e12 1.17185
\(509\) −2.18830e12 −1.44503 −0.722513 0.691357i \(-0.757013\pi\)
−0.722513 + 0.691357i \(0.757013\pi\)
\(510\) −3.85648e11 −0.252421
\(511\) −5.53170e11 −0.358893
\(512\) −1.01158e12 −0.650557
\(513\) −4.07596e10 −0.0259837
\(514\) 2.75848e11 0.174316
\(515\) −3.35310e12 −2.10046
\(516\) −1.43081e12 −0.888504
\(517\) 4.64413e11 0.285888
\(518\) 2.03579e9 0.00124237
\(519\) 3.95785e11 0.239445
\(520\) −1.14061e12 −0.684102
\(521\) 1.77724e12 1.05676 0.528379 0.849008i \(-0.322800\pi\)
0.528379 + 0.849008i \(0.322800\pi\)
\(522\) −3.30935e10 −0.0195086
\(523\) −1.31100e12 −0.766207 −0.383104 0.923705i \(-0.625145\pi\)
−0.383104 + 0.923705i \(0.625145\pi\)
\(524\) 6.96994e11 0.403867
\(525\) −8.04201e11 −0.462006
\(526\) 4.74927e10 0.0270515
\(527\) −1.42439e12 −0.804418
\(528\) 2.82621e11 0.158253
\(529\) 7.64464e10 0.0424431
\(530\) −5.10935e11 −0.281271
\(531\) 7.95020e10 0.0433963
\(532\) −7.08785e10 −0.0383629
\(533\) −2.92763e12 −1.57124
\(534\) 1.71737e11 0.0913964
\(535\) 2.06423e12 1.08935
\(536\) −9.12993e11 −0.477778
\(537\) 1.54571e12 0.802128
\(538\) 3.59469e11 0.184987
\(539\) −5.23005e11 −0.266905
\(540\) 7.22704e11 0.365753
\(541\) 2.91894e12 1.46500 0.732499 0.680768i \(-0.238354\pi\)
0.732499 + 0.680768i \(0.238354\pi\)
\(542\) 4.68763e11 0.233322
\(543\) −1.17237e12 −0.578716
\(544\) 1.34514e12 0.658524
\(545\) 1.62976e12 0.791298
\(546\) 6.19301e10 0.0298219
\(547\) −6.69857e10 −0.0319918 −0.0159959 0.999872i \(-0.505092\pi\)
−0.0159959 + 0.999872i \(0.505092\pi\)
\(548\) 1.82304e12 0.863542
\(549\) −1.27124e11 −0.0597243
\(550\) −2.43583e11 −0.113505
\(551\) −1.21150e11 −0.0559942
\(552\) 3.59305e11 0.164716
\(553\) 4.06747e11 0.184953
\(554\) 4.57895e10 0.0206525
\(555\) −7.59909e10 −0.0339972
\(556\) 7.28370e11 0.323233
\(557\) −6.11851e11 −0.269338 −0.134669 0.990891i \(-0.542997\pi\)
−0.134669 + 0.990891i \(0.542997\pi\)
\(558\) −5.42388e10 −0.0236841
\(559\) 4.57670e12 1.98244
\(560\) 1.23068e12 0.528811
\(561\) −6.30591e11 −0.268791
\(562\) 5.95984e11 0.252012
\(563\) −3.59339e12 −1.50736 −0.753680 0.657241i \(-0.771723\pi\)
−0.753680 + 0.657241i \(0.771723\pi\)
\(564\) −1.33405e12 −0.555157
\(565\) −1.05188e12 −0.434257
\(566\) −3.69829e11 −0.151470
\(567\) −7.92768e10 −0.0322123
\(568\) 2.00269e11 0.0807321
\(569\) 2.36321e12 0.945141 0.472571 0.881293i \(-0.343326\pi\)
0.472571 + 0.881293i \(0.343326\pi\)
\(570\) −5.37593e10 −0.0213313
\(571\) 3.97794e11 0.156601 0.0783007 0.996930i \(-0.475051\pi\)
0.0783007 + 0.996930i \(0.475051\pi\)
\(572\) −9.23152e11 −0.360571
\(573\) −7.28540e11 −0.282330
\(574\) −1.32421e11 −0.0509158
\(575\) 7.38711e12 2.81819
\(576\) −7.77120e11 −0.294162
\(577\) −3.37457e12 −1.26744 −0.633719 0.773563i \(-0.718473\pi\)
−0.633719 + 0.773563i \(0.718473\pi\)
\(578\) −5.87927e11 −0.219103
\(579\) −1.66853e11 −0.0616995
\(580\) 2.14811e12 0.788188
\(581\) 4.37986e11 0.159466
\(582\) −2.68149e11 −0.0968776
\(583\) −8.35455e11 −0.299512
\(584\) 9.72364e11 0.345916
\(585\) −2.31169e12 −0.816072
\(586\) 7.21905e11 0.252895
\(587\) 3.37780e12 1.17425 0.587127 0.809495i \(-0.300259\pi\)
0.587127 + 0.809495i \(0.300259\pi\)
\(588\) 1.50236e12 0.518293
\(589\) −1.98560e11 −0.0679788
\(590\) 1.04858e11 0.0356261
\(591\) −2.81690e11 −0.0949791
\(592\) 8.53639e10 0.0285645
\(593\) 3.82059e12 1.26877 0.634386 0.773016i \(-0.281253\pi\)
0.634386 + 0.773016i \(0.281253\pi\)
\(594\) −2.40120e10 −0.00791388
\(595\) −2.74593e12 −0.898179
\(596\) −1.67786e12 −0.544688
\(597\) −1.55277e12 −0.500292
\(598\) −5.68869e11 −0.181910
\(599\) 5.37586e12 1.70619 0.853095 0.521756i \(-0.174723\pi\)
0.853095 + 0.521756i \(0.174723\pi\)
\(600\) 1.41363e12 0.445301
\(601\) −5.38752e12 −1.68443 −0.842216 0.539140i \(-0.818749\pi\)
−0.842216 + 0.539140i \(0.818749\pi\)
\(602\) 2.07011e11 0.0642404
\(603\) −1.85038e12 −0.569945
\(604\) −2.06453e12 −0.631184
\(605\) 5.84748e12 1.77447
\(606\) −3.52993e11 −0.106326
\(607\) 3.90671e12 1.16805 0.584026 0.811735i \(-0.301477\pi\)
0.584026 + 0.811735i \(0.301477\pi\)
\(608\) 1.87512e11 0.0556498
\(609\) −2.35636e11 −0.0694167
\(610\) −1.67668e11 −0.0490305
\(611\) 4.26719e12 1.23867
\(612\) 1.81141e12 0.521957
\(613\) −6.06879e12 −1.73592 −0.867961 0.496633i \(-0.834569\pi\)
−0.867961 + 0.496633i \(0.834569\pi\)
\(614\) 8.69217e10 0.0246815
\(615\) 4.94293e12 1.39331
\(616\) −8.43593e10 −0.0236059
\(617\) −3.25090e12 −0.903068 −0.451534 0.892254i \(-0.649123\pi\)
−0.451534 + 0.892254i \(0.649123\pi\)
\(618\) −3.20023e11 −0.0882539
\(619\) −1.92638e12 −0.527392 −0.263696 0.964606i \(-0.584942\pi\)
−0.263696 + 0.964606i \(0.584942\pi\)
\(620\) 3.52065e12 0.956887
\(621\) 7.28210e11 0.196492
\(622\) −5.85170e11 −0.156756
\(623\) 1.22282e12 0.325212
\(624\) 2.59682e12 0.685665
\(625\) 1.47193e13 3.85858
\(626\) 5.63022e11 0.146535
\(627\) −8.79044e10 −0.0227147
\(628\) 6.55645e12 1.68210
\(629\) −1.90466e11 −0.0485165
\(630\) −1.04561e11 −0.0264446
\(631\) −3.10554e12 −0.779839 −0.389919 0.920849i \(-0.627497\pi\)
−0.389919 + 0.920849i \(0.627497\pi\)
\(632\) −7.14981e11 −0.178266
\(633\) 2.93523e12 0.726650
\(634\) 5.32453e11 0.130882
\(635\) 9.49944e12 2.31855
\(636\) 2.39989e12 0.581613
\(637\) −4.80555e12 −1.15642
\(638\) −7.13714e10 −0.0170542
\(639\) 4.05889e11 0.0963061
\(640\) −4.41729e12 −1.04075
\(641\) −5.08337e12 −1.18930 −0.594649 0.803986i \(-0.702709\pi\)
−0.594649 + 0.803986i \(0.702709\pi\)
\(642\) 1.97013e11 0.0457706
\(643\) 2.93307e12 0.676664 0.338332 0.941027i \(-0.390137\pi\)
0.338332 + 0.941027i \(0.390137\pi\)
\(644\) 1.26631e12 0.290105
\(645\) −7.72717e12 −1.75793
\(646\) −1.34744e11 −0.0304412
\(647\) 1.70042e12 0.381493 0.190746 0.981639i \(-0.438909\pi\)
0.190746 + 0.981639i \(0.438909\pi\)
\(648\) 1.39353e11 0.0310477
\(649\) 1.71458e11 0.0379366
\(650\) −2.23812e12 −0.491784
\(651\) −3.86197e11 −0.0842741
\(652\) −3.92895e12 −0.851457
\(653\) −2.02165e12 −0.435107 −0.217554 0.976048i \(-0.569808\pi\)
−0.217554 + 0.976048i \(0.569808\pi\)
\(654\) 1.55546e11 0.0332476
\(655\) 3.76415e12 0.799063
\(656\) −5.55261e12 −1.17066
\(657\) 1.97071e12 0.412647
\(658\) 1.93011e11 0.0401389
\(659\) −5.03004e11 −0.103893 −0.0519466 0.998650i \(-0.516543\pi\)
−0.0519466 + 0.998650i \(0.516543\pi\)
\(660\) 1.55862e12 0.319738
\(661\) −8.54822e12 −1.74168 −0.870841 0.491565i \(-0.836425\pi\)
−0.870841 + 0.491565i \(0.836425\pi\)
\(662\) −6.75663e11 −0.136732
\(663\) −5.79409e12 −1.16459
\(664\) −7.69893e11 −0.153700
\(665\) −3.82782e11 −0.0759022
\(666\) −7.25266e9 −0.00142845
\(667\) 2.16447e12 0.423434
\(668\) −5.89819e12 −1.14611
\(669\) 2.81650e12 0.543616
\(670\) −2.44053e12 −0.467895
\(671\) −2.74162e11 −0.0522103
\(672\) 3.64708e11 0.0689897
\(673\) −2.43690e12 −0.457900 −0.228950 0.973438i \(-0.573529\pi\)
−0.228950 + 0.973438i \(0.573529\pi\)
\(674\) −7.23201e11 −0.134986
\(675\) 2.86503e12 0.531204
\(676\) −3.16086e12 −0.582164
\(677\) −9.41571e12 −1.72268 −0.861339 0.508030i \(-0.830374\pi\)
−0.861339 + 0.508030i \(0.830374\pi\)
\(678\) −1.00392e11 −0.0182460
\(679\) −1.90931e12 −0.344716
\(680\) 4.82680e12 0.865704
\(681\) −4.85021e11 −0.0864170
\(682\) −1.16974e11 −0.0207043
\(683\) −7.38882e12 −1.29922 −0.649609 0.760269i \(-0.725067\pi\)
−0.649609 + 0.760269i \(0.725067\pi\)
\(684\) 2.52510e11 0.0441089
\(685\) 9.84541e12 1.70854
\(686\) −4.54669e11 −0.0783857
\(687\) 5.12354e12 0.877536
\(688\) 8.68027e12 1.47702
\(689\) −7.67645e12 −1.29770
\(690\) 9.60462e11 0.161309
\(691\) −4.29537e12 −0.716719 −0.358360 0.933584i \(-0.616664\pi\)
−0.358360 + 0.933584i \(0.616664\pi\)
\(692\) −2.45193e12 −0.406472
\(693\) −1.70973e11 −0.0281597
\(694\) 1.22857e12 0.201039
\(695\) 3.93360e12 0.639526
\(696\) 4.14202e11 0.0669068
\(697\) 1.23891e13 1.98835
\(698\) 1.10629e12 0.176408
\(699\) −3.63914e11 −0.0576569
\(700\) 4.98211e12 0.784281
\(701\) 1.56584e12 0.244916 0.122458 0.992474i \(-0.460922\pi\)
0.122458 + 0.992474i \(0.460922\pi\)
\(702\) −2.20631e11 −0.0342885
\(703\) −2.65509e10 −0.00409997
\(704\) −1.67598e12 −0.257153
\(705\) −7.20460e12 −1.09840
\(706\) −4.01892e11 −0.0608819
\(707\) −2.51342e12 −0.378336
\(708\) −4.92523e11 −0.0736677
\(709\) −4.53682e12 −0.674285 −0.337143 0.941454i \(-0.609460\pi\)
−0.337143 + 0.941454i \(0.609460\pi\)
\(710\) 5.35342e11 0.0790622
\(711\) −1.44907e12 −0.212655
\(712\) −2.14948e12 −0.313454
\(713\) 3.54747e12 0.514063
\(714\) −2.62075e11 −0.0377384
\(715\) −4.98552e12 −0.713401
\(716\) −9.57584e12 −1.36166
\(717\) −3.30594e12 −0.467152
\(718\) 4.48065e11 0.0629189
\(719\) −9.46521e11 −0.132084 −0.0660420 0.997817i \(-0.521037\pi\)
−0.0660420 + 0.997817i \(0.521037\pi\)
\(720\) −4.38440e12 −0.608015
\(721\) −2.27866e12 −0.314030
\(722\) 1.01161e12 0.138547
\(723\) 9.01323e11 0.122676
\(724\) 7.26296e12 0.982404
\(725\) 8.51577e12 1.14473
\(726\) 5.58090e11 0.0745572
\(727\) −1.17432e13 −1.55913 −0.779567 0.626319i \(-0.784561\pi\)
−0.779567 + 0.626319i \(0.784561\pi\)
\(728\) −7.75123e11 −0.102277
\(729\) 2.82430e11 0.0370370
\(730\) 2.59924e12 0.338761
\(731\) −1.93676e13 −2.50870
\(732\) 7.87545e11 0.101385
\(733\) −1.11342e13 −1.42460 −0.712299 0.701876i \(-0.752346\pi\)
−0.712299 + 0.701876i \(0.752346\pi\)
\(734\) −1.89594e12 −0.241097
\(735\) 8.11356e12 1.02546
\(736\) −3.35009e12 −0.420829
\(737\) −3.99063e12 −0.498240
\(738\) 4.71759e11 0.0585419
\(739\) −1.07908e13 −1.33092 −0.665462 0.746432i \(-0.731765\pi\)
−0.665462 + 0.746432i \(0.731765\pi\)
\(740\) 4.70772e11 0.0577122
\(741\) −8.07696e11 −0.0984161
\(742\) −3.47216e11 −0.0420516
\(743\) 2.76883e12 0.333309 0.166655 0.986015i \(-0.446704\pi\)
0.166655 + 0.986015i \(0.446704\pi\)
\(744\) 6.78858e11 0.0812271
\(745\) −9.06137e12 −1.07768
\(746\) −1.80882e11 −0.0213831
\(747\) −1.56036e12 −0.183350
\(748\) 3.90658e12 0.456288
\(749\) 1.40279e12 0.162864
\(750\) 2.40977e12 0.278099
\(751\) 1.09208e13 1.25278 0.626390 0.779510i \(-0.284532\pi\)
0.626390 + 0.779510i \(0.284532\pi\)
\(752\) 8.09324e12 0.922873
\(753\) 2.22446e12 0.252143
\(754\) −6.55785e11 −0.0738908
\(755\) −1.11496e13 −1.24882
\(756\) 4.91128e11 0.0546823
\(757\) 4.41484e12 0.488634 0.244317 0.969695i \(-0.421436\pi\)
0.244317 + 0.969695i \(0.421436\pi\)
\(758\) −2.26115e12 −0.248781
\(759\) 1.57050e12 0.171771
\(760\) 6.72856e11 0.0731579
\(761\) 5.46578e12 0.590774 0.295387 0.955378i \(-0.404551\pi\)
0.295387 + 0.955378i \(0.404551\pi\)
\(762\) 9.06638e11 0.0974174
\(763\) 1.10754e12 0.118304
\(764\) 4.51338e12 0.479272
\(765\) 9.78258e12 1.03271
\(766\) 2.28283e12 0.239576
\(767\) 1.57542e12 0.164368
\(768\) 4.49057e12 0.465775
\(769\) −6.66656e12 −0.687437 −0.343719 0.939073i \(-0.611687\pi\)
−0.343719 + 0.939073i \(0.611687\pi\)
\(770\) −2.25502e11 −0.0231176
\(771\) −6.99734e12 −0.713162
\(772\) 1.03367e12 0.104738
\(773\) −1.37755e13 −1.38771 −0.693855 0.720114i \(-0.744089\pi\)
−0.693855 + 0.720114i \(0.744089\pi\)
\(774\) −7.37491e11 −0.0738622
\(775\) 1.39570e13 1.38974
\(776\) 3.35618e12 0.332252
\(777\) −5.16412e10 −0.00508278
\(778\) 2.76508e12 0.270583
\(779\) 1.72704e12 0.168029
\(780\) 1.43212e13 1.38533
\(781\) 8.75364e11 0.0841897
\(782\) 2.40733e12 0.230200
\(783\) 8.39471e11 0.0798138
\(784\) −9.11431e12 −0.861592
\(785\) 3.54084e13 3.32808
\(786\) 3.59255e11 0.0335739
\(787\) −1.16028e13 −1.07815 −0.539074 0.842259i \(-0.681226\pi\)
−0.539074 + 0.842259i \(0.681226\pi\)
\(788\) 1.74510e12 0.161232
\(789\) −1.20473e12 −0.110673
\(790\) −1.91123e12 −0.174578
\(791\) −7.14824e11 −0.0649239
\(792\) 3.00536e11 0.0271415
\(793\) −2.51910e12 −0.226212
\(794\) 1.94512e12 0.173682
\(795\) 1.29607e13 1.15074
\(796\) 9.61960e12 0.849275
\(797\) 4.99260e12 0.438293 0.219147 0.975692i \(-0.429673\pi\)
0.219147 + 0.975692i \(0.429673\pi\)
\(798\) −3.65332e10 −0.00318915
\(799\) −1.80578e13 −1.56749
\(800\) −1.31804e13 −1.13769
\(801\) −4.35639e12 −0.373922
\(802\) 3.07847e11 0.0262755
\(803\) 4.25014e12 0.360731
\(804\) 1.14633e13 0.967515
\(805\) 6.83878e12 0.573981
\(806\) −1.07480e12 −0.0897059
\(807\) −9.11852e12 −0.756822
\(808\) 4.41810e12 0.364657
\(809\) −2.00895e13 −1.64892 −0.824462 0.565918i \(-0.808522\pi\)
−0.824462 + 0.565918i \(0.808522\pi\)
\(810\) 3.72507e11 0.0304054
\(811\) −2.34770e13 −1.90567 −0.952837 0.303482i \(-0.901851\pi\)
−0.952837 + 0.303482i \(0.901851\pi\)
\(812\) 1.45979e12 0.117839
\(813\) −1.18909e13 −0.954571
\(814\) −1.56415e10 −0.00124873
\(815\) −2.12185e13 −1.68463
\(816\) −1.09892e13 −0.867681
\(817\) −2.69984e12 −0.212002
\(818\) −2.26088e11 −0.0176558
\(819\) −1.57096e12 −0.122008
\(820\) −3.06220e13 −2.36522
\(821\) −1.67424e13 −1.28609 −0.643047 0.765827i \(-0.722330\pi\)
−0.643047 + 0.765827i \(0.722330\pi\)
\(822\) 9.39658e11 0.0717871
\(823\) 1.40349e13 1.06638 0.533189 0.845996i \(-0.320993\pi\)
0.533189 + 0.845996i \(0.320993\pi\)
\(824\) 4.00544e12 0.302676
\(825\) 6.17887e12 0.464373
\(826\) 7.12585e10 0.00532631
\(827\) 1.64500e13 1.22290 0.611450 0.791283i \(-0.290587\pi\)
0.611450 + 0.791283i \(0.290587\pi\)
\(828\) −4.51134e12 −0.333556
\(829\) 7.32077e12 0.538346 0.269173 0.963092i \(-0.413250\pi\)
0.269173 + 0.963092i \(0.413250\pi\)
\(830\) −2.05801e12 −0.150521
\(831\) −1.16152e12 −0.0844936
\(832\) −1.53995e13 −1.11417
\(833\) 2.03361e13 1.46340
\(834\) 3.75427e11 0.0268707
\(835\) −3.18535e13 −2.26760
\(836\) 5.44577e11 0.0385595
\(837\) 1.37586e12 0.0968965
\(838\) 2.24159e12 0.157021
\(839\) −3.76125e12 −0.262062 −0.131031 0.991378i \(-0.541829\pi\)
−0.131031 + 0.991378i \(0.541829\pi\)
\(840\) 1.30870e12 0.0906947
\(841\) −1.20120e13 −0.828004
\(842\) 7.72867e11 0.0529908
\(843\) −1.51181e13 −1.03103
\(844\) −1.81841e13 −1.23353
\(845\) −1.70704e13 −1.15183
\(846\) −6.87615e11 −0.0461508
\(847\) 3.97377e12 0.265294
\(848\) −1.45593e13 −0.966851
\(849\) 9.38130e12 0.619696
\(850\) 9.47126e12 0.622333
\(851\) 4.74358e11 0.0310044
\(852\) −2.51453e12 −0.163485
\(853\) 1.07001e13 0.692016 0.346008 0.938232i \(-0.387537\pi\)
0.346008 + 0.938232i \(0.387537\pi\)
\(854\) −1.13942e11 −0.00733035
\(855\) 1.36369e12 0.0872707
\(856\) −2.46583e12 −0.156975
\(857\) −1.89336e13 −1.19900 −0.599501 0.800374i \(-0.704634\pi\)
−0.599501 + 0.800374i \(0.704634\pi\)
\(858\) −4.75824e11 −0.0299746
\(859\) 3.57770e12 0.224199 0.112100 0.993697i \(-0.464242\pi\)
0.112100 + 0.993697i \(0.464242\pi\)
\(860\) 4.78707e13 2.98419
\(861\) 3.35907e12 0.208307
\(862\) 1.26476e12 0.0780234
\(863\) 2.26658e13 1.39099 0.695493 0.718533i \(-0.255186\pi\)
0.695493 + 0.718533i \(0.255186\pi\)
\(864\) −1.29930e12 −0.0793228
\(865\) −1.32418e13 −0.804217
\(866\) −2.25946e12 −0.136513
\(867\) 1.49137e13 0.896397
\(868\) 2.39253e12 0.143060
\(869\) −3.12514e12 −0.185900
\(870\) 1.10721e12 0.0655229
\(871\) −3.66673e13 −2.15873
\(872\) −1.94683e12 −0.114026
\(873\) 6.80205e12 0.396347
\(874\) 3.35582e11 0.0194535
\(875\) 1.71583e13 0.989550
\(876\) −1.22088e13 −0.700492
\(877\) 1.66771e12 0.0951969 0.0475984 0.998867i \(-0.484843\pi\)
0.0475984 + 0.998867i \(0.484843\pi\)
\(878\) 2.20087e12 0.124988
\(879\) −1.83123e13 −1.03465
\(880\) −9.45565e12 −0.531520
\(881\) 2.99101e13 1.67273 0.836365 0.548172i \(-0.184676\pi\)
0.836365 + 0.548172i \(0.184676\pi\)
\(882\) 7.74368e11 0.0430862
\(883\) 1.23317e13 0.682654 0.341327 0.939945i \(-0.389124\pi\)
0.341327 + 0.939945i \(0.389124\pi\)
\(884\) 3.58950e13 1.97696
\(885\) −2.65990e12 −0.145754
\(886\) −3.42718e12 −0.186847
\(887\) 3.44036e12 0.186616 0.0933078 0.995637i \(-0.470256\pi\)
0.0933078 + 0.995637i \(0.470256\pi\)
\(888\) 9.07750e10 0.00489901
\(889\) 6.45554e12 0.346637
\(890\) −5.74581e12 −0.306970
\(891\) 6.09103e11 0.0323774
\(892\) −1.74485e13 −0.922819
\(893\) −2.51726e12 −0.132463
\(894\) −8.64828e11 −0.0452804
\(895\) −5.17148e13 −2.69408
\(896\) −3.00186e12 −0.155598
\(897\) 1.44303e13 0.744233
\(898\) 1.97970e12 0.101591
\(899\) 4.08948e12 0.208809
\(900\) −1.77491e13 −0.901750
\(901\) 3.24851e13 1.64219
\(902\) 1.01742e12 0.0511766
\(903\) −5.25116e12 −0.262821
\(904\) 1.25652e12 0.0625765
\(905\) 3.92240e13 1.94372
\(906\) −1.06413e12 −0.0524710
\(907\) 2.75178e13 1.35015 0.675074 0.737750i \(-0.264112\pi\)
0.675074 + 0.737750i \(0.264112\pi\)
\(908\) 3.00476e12 0.146698
\(909\) 8.95425e12 0.435003
\(910\) −2.07199e12 −0.100162
\(911\) −4.63688e12 −0.223046 −0.111523 0.993762i \(-0.535573\pi\)
−0.111523 + 0.993762i \(0.535573\pi\)
\(912\) −1.53189e12 −0.0733250
\(913\) −3.36516e12 −0.160283
\(914\) −1.50213e11 −0.00711952
\(915\) 4.25317e12 0.200594
\(916\) −3.17409e13 −1.48967
\(917\) 2.55800e12 0.119465
\(918\) 9.33661e11 0.0433908
\(919\) 9.65278e12 0.446409 0.223204 0.974772i \(-0.428348\pi\)
0.223204 + 0.974772i \(0.428348\pi\)
\(920\) −1.20212e13 −0.553228
\(921\) −2.20491e12 −0.100977
\(922\) −2.19192e12 −0.0998933
\(923\) 8.04314e12 0.364769
\(924\) 1.05919e12 0.0478026
\(925\) 1.86629e12 0.0838187
\(926\) −3.07297e12 −0.137344
\(927\) 8.11791e12 0.361065
\(928\) −3.86194e12 −0.170938
\(929\) −1.76404e12 −0.0777029 −0.0388515 0.999245i \(-0.512370\pi\)
−0.0388515 + 0.999245i \(0.512370\pi\)
\(930\) 1.81467e12 0.0795469
\(931\) 2.83485e12 0.123668
\(932\) 2.25448e12 0.0978758
\(933\) 1.48438e13 0.641323
\(934\) −3.32297e11 −0.0142878
\(935\) 2.10977e13 0.902780
\(936\) 2.76143e12 0.117596
\(937\) 4.03584e13 1.71043 0.855217 0.518270i \(-0.173424\pi\)
0.855217 + 0.518270i \(0.173424\pi\)
\(938\) −1.65851e12 −0.0699530
\(939\) −1.42820e13 −0.599505
\(940\) 4.46333e13 1.86459
\(941\) −4.23441e13 −1.76051 −0.880257 0.474497i \(-0.842630\pi\)
−0.880257 + 0.474497i \(0.842630\pi\)
\(942\) 3.37942e12 0.139834
\(943\) −3.08553e13 −1.27065
\(944\) 2.98798e12 0.122463
\(945\) 2.65236e12 0.108191
\(946\) −1.59051e12 −0.0645695
\(947\) 7.00700e12 0.283111 0.141556 0.989930i \(-0.454790\pi\)
0.141556 + 0.989930i \(0.454790\pi\)
\(948\) 8.97712e12 0.360994
\(949\) 3.90518e13 1.56294
\(950\) 1.32029e12 0.0525913
\(951\) −1.35065e13 −0.535466
\(952\) 3.28015e12 0.129428
\(953\) 4.88462e13 1.91828 0.959142 0.282925i \(-0.0913046\pi\)
0.959142 + 0.282925i \(0.0913046\pi\)
\(954\) 1.23698e12 0.0483500
\(955\) 2.43747e13 0.948254
\(956\) 2.04806e13 0.793018
\(957\) 1.81045e12 0.0697723
\(958\) 3.22130e12 0.123562
\(959\) 6.69065e12 0.255437
\(960\) 2.60001e13 0.987993
\(961\) −1.97371e13 −0.746499
\(962\) −1.43720e11 −0.00541038
\(963\) −4.99755e12 −0.187257
\(964\) −5.58379e12 −0.208249
\(965\) 5.58241e12 0.207228
\(966\) 6.52702e11 0.0241167
\(967\) 1.66396e13 0.611962 0.305981 0.952038i \(-0.401016\pi\)
0.305981 + 0.952038i \(0.401016\pi\)
\(968\) −6.98511e12 −0.255702
\(969\) 3.41800e12 0.124542
\(970\) 8.97147e12 0.325380
\(971\) 5.12205e13 1.84909 0.924543 0.381078i \(-0.124447\pi\)
0.924543 + 0.381078i \(0.124447\pi\)
\(972\) −1.74968e12 −0.0628725
\(973\) 2.67315e12 0.0956129
\(974\) −4.97093e12 −0.176979
\(975\) 5.67736e13 2.01199
\(976\) −4.77778e12 −0.168539
\(977\) 1.79228e13 0.629333 0.314667 0.949202i \(-0.398107\pi\)
0.314667 + 0.949202i \(0.398107\pi\)
\(978\) −2.02512e12 −0.0707824
\(979\) −9.39525e12 −0.326878
\(980\) −5.02644e13 −1.74078
\(981\) −3.94569e12 −0.136023
\(982\) −6.59099e12 −0.226177
\(983\) −2.58483e13 −0.882961 −0.441480 0.897271i \(-0.645547\pi\)
−0.441480 + 0.897271i \(0.645547\pi\)
\(984\) −5.90458e12 −0.200776
\(985\) 9.42450e12 0.319003
\(986\) 2.77514e12 0.0935059
\(987\) −4.89603e12 −0.164217
\(988\) 5.00376e12 0.167067
\(989\) 4.82354e13 1.60318
\(990\) 8.03369e11 0.0265801
\(991\) −1.55358e13 −0.511686 −0.255843 0.966718i \(-0.582353\pi\)
−0.255843 + 0.966718i \(0.582353\pi\)
\(992\) −6.32955e12 −0.207525
\(993\) 1.71393e13 0.559398
\(994\) 3.63803e11 0.0118203
\(995\) 5.19511e13 1.68032
\(996\) 9.66658e12 0.311248
\(997\) 5.75049e13 1.84322 0.921609 0.388120i \(-0.126875\pi\)
0.921609 + 0.388120i \(0.126875\pi\)
\(998\) 6.34167e12 0.202356
\(999\) 1.83976e11 0.00584407
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.a.1.11 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.10.a.a.1.11 21 1.1 even 1 trivial