Properties

Label 177.10.a.a.1.9
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 1.9
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-16.6893 q^{2} +81.0000 q^{3} -233.469 q^{4} -1238.34 q^{5} -1351.83 q^{6} +5162.18 q^{7} +12441.3 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q-16.6893 q^{2} +81.0000 q^{3} -233.469 q^{4} -1238.34 q^{5} -1351.83 q^{6} +5162.18 q^{7} +12441.3 q^{8} +6561.00 q^{9} +20667.0 q^{10} +35034.8 q^{11} -18911.0 q^{12} -150094. q^{13} -86152.9 q^{14} -100306. q^{15} -88100.2 q^{16} -77825.8 q^{17} -109498. q^{18} -975192. q^{19} +289114. q^{20} +418136. q^{21} -584704. q^{22} +1.25531e6 q^{23} +1.00775e6 q^{24} -419640. q^{25} +2.50496e6 q^{26} +531441. q^{27} -1.20521e6 q^{28} +4.34772e6 q^{29} +1.67402e6 q^{30} +8.89787e6 q^{31} -4.89963e6 q^{32} +2.83782e6 q^{33} +1.29885e6 q^{34} -6.39253e6 q^{35} -1.53179e6 q^{36} +1.60624e7 q^{37} +1.62752e7 q^{38} -1.21576e7 q^{39} -1.54066e7 q^{40} +4.19074e6 q^{41} -6.97838e6 q^{42} +1.36518e7 q^{43} -8.17953e6 q^{44} -8.12475e6 q^{45} -2.09502e7 q^{46} -4.18175e7 q^{47} -7.13611e6 q^{48} -1.37055e7 q^{49} +7.00347e6 q^{50} -6.30389e6 q^{51} +3.50424e7 q^{52} +1.42148e7 q^{53} -8.86935e6 q^{54} -4.33849e7 q^{55} +6.42243e7 q^{56} -7.89906e7 q^{57} -7.25601e7 q^{58} +1.21174e7 q^{59} +2.34182e7 q^{60} +1.20881e7 q^{61} -1.48499e8 q^{62} +3.38690e7 q^{63} +1.26878e8 q^{64} +1.85868e8 q^{65} -4.73610e7 q^{66} +1.80246e8 q^{67} +1.81699e7 q^{68} +1.01680e8 q^{69} +1.06687e8 q^{70} -4.01126e8 q^{71} +8.16275e7 q^{72} -3.63565e8 q^{73} -2.68070e8 q^{74} -3.39908e7 q^{75} +2.27677e8 q^{76} +1.80856e8 q^{77} +2.02902e8 q^{78} +3.93427e8 q^{79} +1.09098e8 q^{80} +4.30467e7 q^{81} -6.99402e7 q^{82} -3.21042e8 q^{83} -9.76218e7 q^{84} +9.63748e7 q^{85} -2.27838e8 q^{86} +3.52165e8 q^{87} +4.35879e8 q^{88} +9.37868e8 q^{89} +1.35596e8 q^{90} -7.74813e8 q^{91} -2.93077e8 q^{92} +7.20727e8 q^{93} +6.97903e8 q^{94} +1.20762e9 q^{95} -3.96870e8 q^{96} -1.46116e9 q^{97} +2.28735e8 q^{98} +2.29863e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21q - 66q^{2} + 1701q^{3} + 5206q^{4} - 2964q^{5} - 5346q^{6} - 30775q^{7} - 24621q^{8} + 137781q^{9} + O(q^{10}) \) \( 21q - 66q^{2} + 1701q^{3} + 5206q^{4} - 2964q^{5} - 5346q^{6} - 30775q^{7} - 24621q^{8} + 137781q^{9} - 54663q^{10} - 151769q^{11} + 421686q^{12} - 153611q^{13} - 286771q^{14} - 240084q^{15} + 805530q^{16} - 723621q^{17} - 433026q^{18} - 549388q^{19} - 527311q^{20} - 2492775q^{21} + 2973158q^{22} + 169962q^{23} - 1994301q^{24} + 8035779q^{25} - 2337392q^{26} + 11160261q^{27} - 22659054q^{28} - 16845442q^{29} - 4427703q^{30} - 19307976q^{31} - 44923568q^{32} - 12293289q^{33} - 35547496q^{34} - 34882596q^{35} + 34156566q^{36} - 41561129q^{37} - 52335371q^{38} - 12442491q^{39} - 125735038q^{40} - 68169291q^{41} - 23228451q^{42} - 25719587q^{43} - 126277032q^{44} - 19446804q^{45} - 292814271q^{46} - 174095332q^{47} + 65247930q^{48} + 7479350q^{49} - 227877439q^{50} - 58613301q^{51} - 232397708q^{52} - 228390500q^{53} - 35075106q^{54} - 29426208q^{55} + 326778474q^{56} - 44500428q^{57} + 480343762q^{58} + 254464581q^{59} - 42712191q^{60} - 183928964q^{61} - 21753862q^{62} - 201914775q^{63} + 310571245q^{64} + 5308466q^{65} + 240825798q^{66} - 82724114q^{67} - 138336205q^{68} + 13766922q^{69} + 1030274876q^{70} - 404721965q^{71} - 161538381q^{72} + 154162574q^{73} + 36352054q^{74} + 650898099q^{75} + 1068940636q^{76} - 448535481q^{77} - 189328752q^{78} + 272529635q^{79} - 345587859q^{80} + 903981141q^{81} - 38412637q^{82} + 432518643q^{83} - 1835383374q^{84} - 126211490q^{85} - 3699273072q^{86} - 1364480802q^{87} + 170111045q^{88} - 1255621070q^{89} - 358643943q^{90} + 1448885849q^{91} + 1568933320q^{92} - 1563946056q^{93} - 1908445164q^{94} - 2896546490q^{95} - 3638809008q^{96} + 1007235486q^{97} - 9506868248q^{98} - 995756409q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −16.6893 −0.737568 −0.368784 0.929515i \(-0.620226\pi\)
−0.368784 + 0.929515i \(0.620226\pi\)
\(3\) 81.0000 0.577350
\(4\) −233.469 −0.455994
\(5\) −1238.34 −0.886084 −0.443042 0.896501i \(-0.646101\pi\)
−0.443042 + 0.896501i \(0.646101\pi\)
\(6\) −1351.83 −0.425835
\(7\) 5162.18 0.812628 0.406314 0.913734i \(-0.366814\pi\)
0.406314 + 0.913734i \(0.366814\pi\)
\(8\) 12441.3 1.07389
\(9\) 6561.00 0.333333
\(10\) 20667.0 0.653547
\(11\) 35034.8 0.721493 0.360747 0.932664i \(-0.382522\pi\)
0.360747 + 0.932664i \(0.382522\pi\)
\(12\) −18911.0 −0.263268
\(13\) −150094. −1.45754 −0.728768 0.684761i \(-0.759907\pi\)
−0.728768 + 0.684761i \(0.759907\pi\)
\(14\) −86152.9 −0.599368
\(15\) −100306. −0.511581
\(16\) −88100.2 −0.336075
\(17\) −77825.8 −0.225997 −0.112999 0.993595i \(-0.536046\pi\)
−0.112999 + 0.993595i \(0.536046\pi\)
\(18\) −109498. −0.245856
\(19\) −975192. −1.71672 −0.858359 0.513050i \(-0.828516\pi\)
−0.858359 + 0.513050i \(0.828516\pi\)
\(20\) 289114. 0.404049
\(21\) 418136. 0.469171
\(22\) −584704. −0.532150
\(23\) 1.25531e6 0.935356 0.467678 0.883899i \(-0.345091\pi\)
0.467678 + 0.883899i \(0.345091\pi\)
\(24\) 1.00775e6 0.620013
\(25\) −419640. −0.214856
\(26\) 2.50496e6 1.07503
\(27\) 531441. 0.192450
\(28\) −1.20521e6 −0.370553
\(29\) 4.34772e6 1.14149 0.570743 0.821129i \(-0.306655\pi\)
0.570743 + 0.821129i \(0.306655\pi\)
\(30\) 1.67402e6 0.377325
\(31\) 8.89787e6 1.73045 0.865223 0.501387i \(-0.167177\pi\)
0.865223 + 0.501387i \(0.167177\pi\)
\(32\) −4.89963e6 −0.826016
\(33\) 2.83782e6 0.416554
\(34\) 1.29885e6 0.166688
\(35\) −6.39253e6 −0.720056
\(36\) −1.53179e6 −0.151998
\(37\) 1.60624e7 1.40898 0.704488 0.709716i \(-0.251177\pi\)
0.704488 + 0.709716i \(0.251177\pi\)
\(38\) 1.62752e7 1.26620
\(39\) −1.21576e7 −0.841508
\(40\) −1.54066e7 −0.951560
\(41\) 4.19074e6 0.231613 0.115807 0.993272i \(-0.463055\pi\)
0.115807 + 0.993272i \(0.463055\pi\)
\(42\) −6.97838e6 −0.346045
\(43\) 1.36518e7 0.608951 0.304475 0.952520i \(-0.401519\pi\)
0.304475 + 0.952520i \(0.401519\pi\)
\(44\) −8.17953e6 −0.328997
\(45\) −8.12475e6 −0.295361
\(46\) −2.09502e7 −0.689888
\(47\) −4.18175e7 −1.25002 −0.625011 0.780616i \(-0.714906\pi\)
−0.625011 + 0.780616i \(0.714906\pi\)
\(48\) −7.13611e6 −0.194033
\(49\) −1.37055e7 −0.339636
\(50\) 7.00347e6 0.158471
\(51\) −6.30389e6 −0.130480
\(52\) 3.50424e7 0.664627
\(53\) 1.42148e7 0.247457 0.123728 0.992316i \(-0.460515\pi\)
0.123728 + 0.992316i \(0.460515\pi\)
\(54\) −8.86935e6 −0.141945
\(55\) −4.33849e7 −0.639303
\(56\) 6.42243e7 0.872676
\(57\) −7.89906e7 −0.991148
\(58\) −7.25601e7 −0.841923
\(59\) 1.21174e7 0.130189
\(60\) 2.34182e7 0.233278
\(61\) 1.20881e7 0.111783 0.0558914 0.998437i \(-0.482200\pi\)
0.0558914 + 0.998437i \(0.482200\pi\)
\(62\) −1.48499e8 −1.27632
\(63\) 3.38690e7 0.270876
\(64\) 1.26878e8 0.945318
\(65\) 1.85868e8 1.29150
\(66\) −4.73610e7 −0.307237
\(67\) 1.80246e8 1.09277 0.546385 0.837534i \(-0.316004\pi\)
0.546385 + 0.837534i \(0.316004\pi\)
\(68\) 1.81699e7 0.103053
\(69\) 1.01680e8 0.540028
\(70\) 1.06687e8 0.531090
\(71\) −4.01126e8 −1.87335 −0.936674 0.350201i \(-0.886113\pi\)
−0.936674 + 0.350201i \(0.886113\pi\)
\(72\) 8.16275e7 0.357965
\(73\) −3.63565e8 −1.49840 −0.749202 0.662341i \(-0.769563\pi\)
−0.749202 + 0.662341i \(0.769563\pi\)
\(74\) −2.68070e8 −1.03922
\(75\) −3.39908e7 −0.124047
\(76\) 2.27677e8 0.782813
\(77\) 1.80856e8 0.586305
\(78\) 2.02902e8 0.620669
\(79\) 3.93427e8 1.13643 0.568214 0.822881i \(-0.307635\pi\)
0.568214 + 0.822881i \(0.307635\pi\)
\(80\) 1.09098e8 0.297791
\(81\) 4.30467e7 0.111111
\(82\) −6.99402e7 −0.170830
\(83\) −3.21042e8 −0.742523 −0.371261 0.928528i \(-0.621075\pi\)
−0.371261 + 0.928528i \(0.621075\pi\)
\(84\) −9.76218e7 −0.213939
\(85\) 9.63748e7 0.200253
\(86\) −2.27838e8 −0.449142
\(87\) 3.52165e8 0.659037
\(88\) 4.35879e8 0.774807
\(89\) 9.37868e8 1.58448 0.792240 0.610210i \(-0.208915\pi\)
0.792240 + 0.610210i \(0.208915\pi\)
\(90\) 1.35596e8 0.217849
\(91\) −7.74813e8 −1.18443
\(92\) −2.93077e8 −0.426517
\(93\) 7.20727e8 0.999074
\(94\) 6.97903e8 0.921976
\(95\) 1.20762e9 1.52116
\(96\) −3.96870e8 −0.476900
\(97\) −1.46116e9 −1.67581 −0.837903 0.545820i \(-0.816218\pi\)
−0.837903 + 0.545820i \(0.816218\pi\)
\(98\) 2.28735e8 0.250505
\(99\) 2.29863e8 0.240498
\(100\) 9.79729e7 0.0979729
\(101\) −1.32906e9 −1.27086 −0.635430 0.772159i \(-0.719177\pi\)
−0.635430 + 0.772159i \(0.719177\pi\)
\(102\) 1.05207e8 0.0962376
\(103\) −2.90148e8 −0.254011 −0.127005 0.991902i \(-0.540537\pi\)
−0.127005 + 0.991902i \(0.540537\pi\)
\(104\) −1.86737e9 −1.56524
\(105\) −5.17795e8 −0.415725
\(106\) −2.37234e8 −0.182516
\(107\) −2.13497e9 −1.57458 −0.787288 0.616585i \(-0.788516\pi\)
−0.787288 + 0.616585i \(0.788516\pi\)
\(108\) −1.24075e8 −0.0877561
\(109\) −1.57453e8 −0.106839 −0.0534196 0.998572i \(-0.517012\pi\)
−0.0534196 + 0.998572i \(0.517012\pi\)
\(110\) 7.24062e8 0.471529
\(111\) 1.30106e9 0.813473
\(112\) −4.54789e8 −0.273104
\(113\) −2.00171e9 −1.15491 −0.577455 0.816422i \(-0.695954\pi\)
−0.577455 + 0.816422i \(0.695954\pi\)
\(114\) 1.31829e9 0.731038
\(115\) −1.55450e9 −0.828804
\(116\) −1.01506e9 −0.520510
\(117\) −9.84769e8 −0.485845
\(118\) −2.02230e8 −0.0960231
\(119\) −4.01751e8 −0.183652
\(120\) −1.24793e9 −0.549383
\(121\) −1.13051e9 −0.479448
\(122\) −2.01742e8 −0.0824474
\(123\) 3.39450e8 0.133722
\(124\) −2.07738e9 −0.789073
\(125\) 2.93829e9 1.07646
\(126\) −5.65249e8 −0.199789
\(127\) 6.34005e8 0.216260 0.108130 0.994137i \(-0.465514\pi\)
0.108130 + 0.994137i \(0.465514\pi\)
\(128\) 3.91104e8 0.128780
\(129\) 1.10580e9 0.351578
\(130\) −3.10199e9 −0.952567
\(131\) −2.57578e9 −0.764167 −0.382083 0.924128i \(-0.624793\pi\)
−0.382083 + 0.924128i \(0.624793\pi\)
\(132\) −6.62542e8 −0.189946
\(133\) −5.03411e9 −1.39505
\(134\) −3.00817e9 −0.805991
\(135\) −6.58104e8 −0.170527
\(136\) −9.68256e8 −0.242697
\(137\) 2.21465e9 0.537110 0.268555 0.963264i \(-0.413454\pi\)
0.268555 + 0.963264i \(0.413454\pi\)
\(138\) −1.69697e9 −0.398307
\(139\) 6.77480e9 1.53932 0.769662 0.638451i \(-0.220425\pi\)
0.769662 + 0.638451i \(0.220425\pi\)
\(140\) 1.49246e9 0.328341
\(141\) −3.38722e9 −0.721701
\(142\) 6.69450e9 1.38172
\(143\) −5.25852e9 −1.05160
\(144\) −5.78025e8 −0.112025
\(145\) −5.38395e9 −1.01145
\(146\) 6.06763e9 1.10517
\(147\) −1.11015e9 −0.196089
\(148\) −3.75008e9 −0.642485
\(149\) −6.74700e9 −1.12143 −0.560716 0.828008i \(-0.689474\pi\)
−0.560716 + 0.828008i \(0.689474\pi\)
\(150\) 5.67281e8 0.0914930
\(151\) −6.62600e9 −1.03718 −0.518591 0.855022i \(-0.673543\pi\)
−0.518591 + 0.855022i \(0.673543\pi\)
\(152\) −1.21327e10 −1.84357
\(153\) −5.10615e8 −0.0753325
\(154\) −3.01835e9 −0.432440
\(155\) −1.10186e10 −1.53332
\(156\) 2.83843e9 0.383723
\(157\) −5.54638e9 −0.728553 −0.364277 0.931291i \(-0.618684\pi\)
−0.364277 + 0.931291i \(0.618684\pi\)
\(158\) −6.56600e9 −0.838192
\(159\) 1.15140e9 0.142869
\(160\) 6.06740e9 0.731919
\(161\) 6.48015e9 0.760097
\(162\) −7.18417e8 −0.0819520
\(163\) −5.30888e9 −0.589059 −0.294530 0.955642i \(-0.595163\pi\)
−0.294530 + 0.955642i \(0.595163\pi\)
\(164\) −9.78407e8 −0.105614
\(165\) −3.51418e9 −0.369102
\(166\) 5.35794e9 0.547661
\(167\) 3.80378e9 0.378435 0.189217 0.981935i \(-0.439405\pi\)
0.189217 + 0.981935i \(0.439405\pi\)
\(168\) 5.20217e9 0.503840
\(169\) 1.19238e10 1.12441
\(170\) −1.60842e9 −0.147700
\(171\) −6.39824e9 −0.572239
\(172\) −3.18727e9 −0.277678
\(173\) −8.19884e8 −0.0695897 −0.0347949 0.999394i \(-0.511078\pi\)
−0.0347949 + 0.999394i \(0.511078\pi\)
\(174\) −5.87737e9 −0.486084
\(175\) −2.16625e9 −0.174598
\(176\) −3.08657e9 −0.242476
\(177\) 9.81506e8 0.0751646
\(178\) −1.56523e10 −1.16866
\(179\) 1.23784e10 0.901209 0.450604 0.892724i \(-0.351209\pi\)
0.450604 + 0.892724i \(0.351209\pi\)
\(180\) 1.89688e9 0.134683
\(181\) −5.53512e9 −0.383331 −0.191665 0.981460i \(-0.561389\pi\)
−0.191665 + 0.981460i \(0.561389\pi\)
\(182\) 1.29311e10 0.873600
\(183\) 9.79139e8 0.0645378
\(184\) 1.56178e10 1.00447
\(185\) −1.98908e10 −1.24847
\(186\) −1.20284e10 −0.736885
\(187\) −2.72661e9 −0.163056
\(188\) 9.76309e9 0.570003
\(189\) 2.74339e9 0.156390
\(190\) −2.01543e10 −1.12196
\(191\) 5.39092e9 0.293098 0.146549 0.989203i \(-0.453183\pi\)
0.146549 + 0.989203i \(0.453183\pi\)
\(192\) 1.02772e10 0.545780
\(193\) 1.87568e10 0.973086 0.486543 0.873657i \(-0.338258\pi\)
0.486543 + 0.873657i \(0.338258\pi\)
\(194\) 2.43856e10 1.23602
\(195\) 1.50553e10 0.745647
\(196\) 3.19982e9 0.154872
\(197\) −4.09941e10 −1.93920 −0.969601 0.244693i \(-0.921313\pi\)
−0.969601 + 0.244693i \(0.921313\pi\)
\(198\) −3.83624e9 −0.177383
\(199\) 3.27876e10 1.48208 0.741039 0.671462i \(-0.234333\pi\)
0.741039 + 0.671462i \(0.234333\pi\)
\(200\) −5.22087e9 −0.230732
\(201\) 1.45999e10 0.630911
\(202\) 2.21810e10 0.937344
\(203\) 2.24437e10 0.927603
\(204\) 1.47176e9 0.0594979
\(205\) −5.18955e9 −0.205229
\(206\) 4.84235e9 0.187350
\(207\) 8.23611e9 0.311785
\(208\) 1.32233e10 0.489842
\(209\) −3.41656e10 −1.23860
\(210\) 8.64161e9 0.306625
\(211\) −3.75829e10 −1.30533 −0.652664 0.757647i \(-0.726349\pi\)
−0.652664 + 0.757647i \(0.726349\pi\)
\(212\) −3.31872e9 −0.112839
\(213\) −3.24912e10 −1.08158
\(214\) 3.56310e10 1.16136
\(215\) −1.69056e10 −0.539581
\(216\) 6.61183e9 0.206671
\(217\) 4.59324e10 1.40621
\(218\) 2.62776e9 0.0788011
\(219\) −2.94488e10 −0.865104
\(220\) 1.01290e10 0.291518
\(221\) 1.16812e10 0.329399
\(222\) −2.17137e10 −0.599991
\(223\) 4.06649e10 1.10115 0.550577 0.834784i \(-0.314408\pi\)
0.550577 + 0.834784i \(0.314408\pi\)
\(224\) −2.52928e10 −0.671243
\(225\) −2.75326e9 −0.0716185
\(226\) 3.34070e10 0.851824
\(227\) −2.68733e10 −0.671746 −0.335873 0.941907i \(-0.609031\pi\)
−0.335873 + 0.941907i \(0.609031\pi\)
\(228\) 1.84418e10 0.451957
\(229\) −1.51366e10 −0.363720 −0.181860 0.983324i \(-0.558212\pi\)
−0.181860 + 0.983324i \(0.558212\pi\)
\(230\) 2.59435e10 0.611299
\(231\) 1.46493e10 0.338504
\(232\) 5.40913e10 1.22583
\(233\) −7.80036e10 −1.73386 −0.866928 0.498433i \(-0.833909\pi\)
−0.866928 + 0.498433i \(0.833909\pi\)
\(234\) 1.64350e10 0.358344
\(235\) 5.17843e10 1.10762
\(236\) −2.82903e9 −0.0593654
\(237\) 3.18676e10 0.656117
\(238\) 6.70492e9 0.135456
\(239\) −7.06123e10 −1.39988 −0.699938 0.714203i \(-0.746789\pi\)
−0.699938 + 0.714203i \(0.746789\pi\)
\(240\) 8.83693e9 0.171930
\(241\) −1.02931e11 −1.96548 −0.982742 0.184981i \(-0.940778\pi\)
−0.982742 + 0.184981i \(0.940778\pi\)
\(242\) 1.88674e10 0.353625
\(243\) 3.48678e9 0.0641500
\(244\) −2.82220e9 −0.0509723
\(245\) 1.69721e10 0.300946
\(246\) −5.66516e9 −0.0986289
\(247\) 1.46371e11 2.50218
\(248\) 1.10701e11 1.85832
\(249\) −2.60044e10 −0.428696
\(250\) −4.90378e10 −0.793965
\(251\) 6.64645e9 0.105696 0.0528479 0.998603i \(-0.483170\pi\)
0.0528479 + 0.998603i \(0.483170\pi\)
\(252\) −7.90737e9 −0.123518
\(253\) 4.39796e10 0.674853
\(254\) −1.05811e10 −0.159506
\(255\) 7.80636e9 0.115616
\(256\) −7.14890e10 −1.04030
\(257\) −5.93058e10 −0.848005 −0.424002 0.905661i \(-0.639375\pi\)
−0.424002 + 0.905661i \(0.639375\pi\)
\(258\) −1.84549e10 −0.259312
\(259\) 8.29172e10 1.14497
\(260\) −4.33943e10 −0.588915
\(261\) 2.85254e10 0.380495
\(262\) 4.29879e10 0.563625
\(263\) −5.21758e10 −0.672462 −0.336231 0.941779i \(-0.609152\pi\)
−0.336231 + 0.941779i \(0.609152\pi\)
\(264\) 3.53062e10 0.447335
\(265\) −1.76028e10 −0.219268
\(266\) 8.40156e10 1.02895
\(267\) 7.59673e10 0.914799
\(268\) −4.20818e10 −0.498296
\(269\) 7.23099e9 0.0842000 0.0421000 0.999113i \(-0.486595\pi\)
0.0421000 + 0.999113i \(0.486595\pi\)
\(270\) 1.09833e10 0.125775
\(271\) 3.44520e8 0.00388018 0.00194009 0.999998i \(-0.499382\pi\)
0.00194009 + 0.999998i \(0.499382\pi\)
\(272\) 6.85647e9 0.0759522
\(273\) −6.27599e10 −0.683833
\(274\) −3.69609e10 −0.396155
\(275\) −1.47020e10 −0.155017
\(276\) −2.37392e10 −0.246250
\(277\) −5.13370e10 −0.523928 −0.261964 0.965078i \(-0.584370\pi\)
−0.261964 + 0.965078i \(0.584370\pi\)
\(278\) −1.13066e11 −1.13536
\(279\) 5.83789e10 0.576816
\(280\) −7.95315e10 −0.773264
\(281\) −1.88645e11 −1.80496 −0.902479 0.430735i \(-0.858254\pi\)
−0.902479 + 0.430735i \(0.858254\pi\)
\(282\) 5.65301e10 0.532303
\(283\) 3.40056e10 0.315145 0.157573 0.987507i \(-0.449633\pi\)
0.157573 + 0.987507i \(0.449633\pi\)
\(284\) 9.36505e10 0.854236
\(285\) 9.78171e10 0.878240
\(286\) 8.77607e10 0.775627
\(287\) 2.16333e10 0.188215
\(288\) −3.21465e10 −0.275339
\(289\) −1.12531e11 −0.948925
\(290\) 8.98541e10 0.746014
\(291\) −1.18354e11 −0.967527
\(292\) 8.48811e10 0.683264
\(293\) 1.12493e11 0.891704 0.445852 0.895107i \(-0.352901\pi\)
0.445852 + 0.895107i \(0.352901\pi\)
\(294\) 1.85275e10 0.144629
\(295\) −1.50054e10 −0.115358
\(296\) 1.99838e11 1.51309
\(297\) 1.86189e10 0.138851
\(298\) 1.12602e11 0.827132
\(299\) −1.88415e11 −1.36331
\(300\) 7.93580e9 0.0565647
\(301\) 7.04731e10 0.494850
\(302\) 1.10583e11 0.764992
\(303\) −1.07654e11 −0.733731
\(304\) 8.59146e10 0.576947
\(305\) −1.49692e10 −0.0990489
\(306\) 8.52178e9 0.0555628
\(307\) −2.81159e11 −1.80646 −0.903231 0.429154i \(-0.858812\pi\)
−0.903231 + 0.429154i \(0.858812\pi\)
\(308\) −4.22242e10 −0.267352
\(309\) −2.35020e10 −0.146653
\(310\) 1.83892e11 1.13093
\(311\) 2.44428e11 1.48159 0.740796 0.671730i \(-0.234449\pi\)
0.740796 + 0.671730i \(0.234449\pi\)
\(312\) −1.51257e11 −0.903691
\(313\) 2.00787e11 1.18246 0.591230 0.806503i \(-0.298643\pi\)
0.591230 + 0.806503i \(0.298643\pi\)
\(314\) 9.25650e10 0.537357
\(315\) −4.19414e10 −0.240019
\(316\) −9.18529e10 −0.518204
\(317\) −3.84791e10 −0.214022 −0.107011 0.994258i \(-0.534128\pi\)
−0.107011 + 0.994258i \(0.534128\pi\)
\(318\) −1.92160e10 −0.105376
\(319\) 1.52321e11 0.823574
\(320\) −1.57119e11 −0.837631
\(321\) −1.72932e11 −0.909082
\(322\) −1.08149e11 −0.560623
\(323\) 7.58951e10 0.387974
\(324\) −1.00501e10 −0.0506660
\(325\) 6.29855e10 0.313160
\(326\) 8.86013e10 0.434471
\(327\) −1.27537e10 −0.0616836
\(328\) 5.21383e10 0.248728
\(329\) −2.15869e11 −1.01580
\(330\) 5.86490e10 0.272238
\(331\) 2.96271e11 1.35663 0.678317 0.734769i \(-0.262710\pi\)
0.678317 + 0.734769i \(0.262710\pi\)
\(332\) 7.49532e10 0.338586
\(333\) 1.05386e11 0.469659
\(334\) −6.34822e10 −0.279121
\(335\) −2.23206e11 −0.968285
\(336\) −3.68379e10 −0.157677
\(337\) 3.27569e11 1.38347 0.691733 0.722153i \(-0.256847\pi\)
0.691733 + 0.722153i \(0.256847\pi\)
\(338\) −1.98999e11 −0.829328
\(339\) −1.62139e11 −0.666788
\(340\) −2.25005e10 −0.0913140
\(341\) 3.11735e11 1.24851
\(342\) 1.06782e11 0.422065
\(343\) −2.79063e11 −1.08863
\(344\) 1.69847e11 0.653949
\(345\) −1.25915e11 −0.478510
\(346\) 1.36833e10 0.0513271
\(347\) 2.08345e9 0.00771438 0.00385719 0.999993i \(-0.498772\pi\)
0.00385719 + 0.999993i \(0.498772\pi\)
\(348\) −8.22196e10 −0.300517
\(349\) 1.21788e10 0.0439432 0.0219716 0.999759i \(-0.493006\pi\)
0.0219716 + 0.999759i \(0.493006\pi\)
\(350\) 3.61532e10 0.128778
\(351\) −7.97663e10 −0.280503
\(352\) −1.71657e11 −0.595965
\(353\) −3.66036e11 −1.25469 −0.627346 0.778741i \(-0.715859\pi\)
−0.627346 + 0.778741i \(0.715859\pi\)
\(354\) −1.63806e10 −0.0554390
\(355\) 4.96731e11 1.65994
\(356\) −2.18963e11 −0.722513
\(357\) −3.25418e10 −0.106031
\(358\) −2.06586e11 −0.664702
\(359\) 4.38842e11 1.39439 0.697193 0.716883i \(-0.254432\pi\)
0.697193 + 0.716883i \(0.254432\pi\)
\(360\) −1.01083e11 −0.317187
\(361\) 6.28312e11 1.94712
\(362\) 9.23771e10 0.282732
\(363\) −9.15715e10 −0.276809
\(364\) 1.80895e11 0.540095
\(365\) 4.50217e11 1.32771
\(366\) −1.63411e10 −0.0476010
\(367\) −5.63859e11 −1.62246 −0.811229 0.584729i \(-0.801201\pi\)
−0.811229 + 0.584729i \(0.801201\pi\)
\(368\) −1.10593e11 −0.314350
\(369\) 2.74954e10 0.0772043
\(370\) 3.31962e11 0.920832
\(371\) 7.33793e10 0.201090
\(372\) −1.68267e11 −0.455572
\(373\) 3.13939e11 0.839760 0.419880 0.907580i \(-0.362072\pi\)
0.419880 + 0.907580i \(0.362072\pi\)
\(374\) 4.55051e10 0.120265
\(375\) 2.38001e11 0.621497
\(376\) −5.20265e11 −1.34239
\(377\) −6.52567e11 −1.66375
\(378\) −4.57852e10 −0.115348
\(379\) 2.46108e11 0.612701 0.306351 0.951919i \(-0.400892\pi\)
0.306351 + 0.951919i \(0.400892\pi\)
\(380\) −2.81942e11 −0.693638
\(381\) 5.13544e10 0.124858
\(382\) −8.99704e10 −0.216179
\(383\) 4.38027e11 1.04017 0.520087 0.854113i \(-0.325899\pi\)
0.520087 + 0.854113i \(0.325899\pi\)
\(384\) 3.16795e10 0.0743511
\(385\) −2.23961e11 −0.519516
\(386\) −3.13037e11 −0.717716
\(387\) 8.95695e10 0.202984
\(388\) 3.41134e11 0.764157
\(389\) −6.89652e11 −1.52706 −0.763531 0.645771i \(-0.776536\pi\)
−0.763531 + 0.645771i \(0.776536\pi\)
\(390\) −2.51261e11 −0.549965
\(391\) −9.76958e10 −0.211388
\(392\) −1.70515e11 −0.364733
\(393\) −2.08638e11 −0.441192
\(394\) 6.84160e11 1.43029
\(395\) −4.87196e11 −1.00697
\(396\) −5.36659e10 −0.109666
\(397\) 1.59295e11 0.321844 0.160922 0.986967i \(-0.448553\pi\)
0.160922 + 0.986967i \(0.448553\pi\)
\(398\) −5.47201e11 −1.09313
\(399\) −4.07763e11 −0.805434
\(400\) 3.69703e10 0.0722077
\(401\) 1.49763e11 0.289238 0.144619 0.989487i \(-0.453804\pi\)
0.144619 + 0.989487i \(0.453804\pi\)
\(402\) −2.43662e11 −0.465339
\(403\) −1.33552e12 −2.52219
\(404\) 3.10293e11 0.579504
\(405\) −5.33065e10 −0.0984538
\(406\) −3.74568e11 −0.684170
\(407\) 5.62744e11 1.01657
\(408\) −7.84287e10 −0.140121
\(409\) −7.42138e11 −1.31138 −0.655692 0.755029i \(-0.727623\pi\)
−0.655692 + 0.755029i \(0.727623\pi\)
\(410\) 8.66098e10 0.151370
\(411\) 1.79387e11 0.310100
\(412\) 6.77406e10 0.115827
\(413\) 6.25520e10 0.105795
\(414\) −1.37455e11 −0.229963
\(415\) 3.97559e11 0.657937
\(416\) 7.35406e11 1.20395
\(417\) 5.48759e11 0.888729
\(418\) 5.70199e11 0.913551
\(419\) −2.21707e11 −0.351411 −0.175706 0.984443i \(-0.556221\pi\)
−0.175706 + 0.984443i \(0.556221\pi\)
\(420\) 1.20889e11 0.189568
\(421\) −2.98540e11 −0.463163 −0.231581 0.972816i \(-0.574390\pi\)
−0.231581 + 0.972816i \(0.574390\pi\)
\(422\) 6.27231e11 0.962768
\(423\) −2.74365e11 −0.416674
\(424\) 1.76851e11 0.265742
\(425\) 3.26588e10 0.0485568
\(426\) 5.42254e11 0.797737
\(427\) 6.24011e10 0.0908378
\(428\) 4.98448e11 0.717998
\(429\) −4.25940e11 −0.607142
\(430\) 2.82141e11 0.397978
\(431\) −9.42123e11 −1.31510 −0.657552 0.753409i \(-0.728408\pi\)
−0.657552 + 0.753409i \(0.728408\pi\)
\(432\) −4.68200e10 −0.0646778
\(433\) 9.76275e10 0.133468 0.0667339 0.997771i \(-0.478742\pi\)
0.0667339 + 0.997771i \(0.478742\pi\)
\(434\) −7.66577e11 −1.03717
\(435\) −4.36100e11 −0.583962
\(436\) 3.67603e10 0.0487180
\(437\) −1.22417e12 −1.60574
\(438\) 4.91478e11 0.638073
\(439\) −5.14664e11 −0.661353 −0.330677 0.943744i \(-0.607277\pi\)
−0.330677 + 0.943744i \(0.607277\pi\)
\(440\) −5.39766e11 −0.686544
\(441\) −8.99220e10 −0.113212
\(442\) −1.94951e11 −0.242954
\(443\) 9.70725e11 1.19751 0.598756 0.800932i \(-0.295662\pi\)
0.598756 + 0.800932i \(0.295662\pi\)
\(444\) −3.03757e11 −0.370939
\(445\) −1.16140e12 −1.40398
\(446\) −6.78667e11 −0.812175
\(447\) −5.46507e11 −0.647459
\(448\) 6.54969e11 0.768192
\(449\) −9.41306e11 −1.09301 −0.546503 0.837457i \(-0.684041\pi\)
−0.546503 + 0.837457i \(0.684041\pi\)
\(450\) 4.59498e10 0.0528235
\(451\) 1.46821e11 0.167107
\(452\) 4.67337e11 0.526632
\(453\) −5.36706e11 −0.598818
\(454\) 4.48496e11 0.495458
\(455\) 9.59482e11 1.04951
\(456\) −9.82747e11 −1.06439
\(457\) −9.91302e11 −1.06312 −0.531561 0.847020i \(-0.678394\pi\)
−0.531561 + 0.847020i \(0.678394\pi\)
\(458\) 2.52618e11 0.268268
\(459\) −4.13598e10 −0.0434932
\(460\) 3.62929e11 0.377930
\(461\) −8.44394e10 −0.0870745 −0.0435372 0.999052i \(-0.513863\pi\)
−0.0435372 + 0.999052i \(0.513863\pi\)
\(462\) −2.44486e11 −0.249669
\(463\) 1.87338e12 1.89457 0.947285 0.320391i \(-0.103814\pi\)
0.947285 + 0.320391i \(0.103814\pi\)
\(464\) −3.83035e11 −0.383625
\(465\) −8.92505e11 −0.885263
\(466\) 1.30182e12 1.27884
\(467\) −4.33922e11 −0.422169 −0.211084 0.977468i \(-0.567699\pi\)
−0.211084 + 0.977468i \(0.567699\pi\)
\(468\) 2.29913e11 0.221542
\(469\) 9.30461e11 0.888015
\(470\) −8.64241e11 −0.816948
\(471\) −4.49257e11 −0.420630
\(472\) 1.50756e11 0.139809
\(473\) 4.78288e11 0.439354
\(474\) −5.31846e11 −0.483931
\(475\) 4.09229e11 0.368846
\(476\) 9.37963e10 0.0837441
\(477\) 9.32633e10 0.0824856
\(478\) 1.17847e12 1.03250
\(479\) −5.72342e11 −0.496759 −0.248380 0.968663i \(-0.579898\pi\)
−0.248380 + 0.968663i \(0.579898\pi\)
\(480\) 4.91460e11 0.422574
\(481\) −2.41088e12 −2.05363
\(482\) 1.71784e12 1.44968
\(483\) 5.24892e11 0.438842
\(484\) 2.63940e11 0.218625
\(485\) 1.80941e12 1.48490
\(486\) −5.81918e10 −0.0473150
\(487\) −2.71725e8 −0.000218902 0 −0.000109451 1.00000i \(-0.500035\pi\)
−0.000109451 1.00000i \(0.500035\pi\)
\(488\) 1.50392e11 0.120043
\(489\) −4.30020e11 −0.340094
\(490\) −2.83252e11 −0.221968
\(491\) −1.17741e10 −0.00914240 −0.00457120 0.999990i \(-0.501455\pi\)
−0.00457120 + 0.999990i \(0.501455\pi\)
\(492\) −7.92509e10 −0.0609764
\(493\) −3.38365e11 −0.257973
\(494\) −2.44282e12 −1.84552
\(495\) −2.84649e11 −0.213101
\(496\) −7.83903e11 −0.581561
\(497\) −2.07069e12 −1.52234
\(498\) 4.33993e11 0.316192
\(499\) 6.69043e11 0.483060 0.241530 0.970393i \(-0.422351\pi\)
0.241530 + 0.970393i \(0.422351\pi\)
\(500\) −6.85999e11 −0.490861
\(501\) 3.08106e11 0.218489
\(502\) −1.10924e11 −0.0779578
\(503\) 1.37546e12 0.958058 0.479029 0.877799i \(-0.340989\pi\)
0.479029 + 0.877799i \(0.340989\pi\)
\(504\) 4.21376e11 0.290892
\(505\) 1.64582e12 1.12609
\(506\) −7.33987e11 −0.497750
\(507\) 9.65827e11 0.649178
\(508\) −1.48020e11 −0.0986131
\(509\) −1.63049e11 −0.107668 −0.0538341 0.998550i \(-0.517144\pi\)
−0.0538341 + 0.998550i \(0.517144\pi\)
\(510\) −1.30282e11 −0.0852746
\(511\) −1.87679e12 −1.21765
\(512\) 9.92852e11 0.638513
\(513\) −5.18257e11 −0.330383
\(514\) 9.89770e11 0.625461
\(515\) 3.59302e11 0.225075
\(516\) −2.58169e11 −0.160317
\(517\) −1.46507e12 −0.901883
\(518\) −1.38383e12 −0.844496
\(519\) −6.64106e10 −0.0401777
\(520\) 2.31244e12 1.38693
\(521\) −2.29938e12 −1.36723 −0.683614 0.729843i \(-0.739593\pi\)
−0.683614 + 0.729843i \(0.739593\pi\)
\(522\) −4.76067e11 −0.280641
\(523\) 1.95322e12 1.14155 0.570773 0.821108i \(-0.306644\pi\)
0.570773 + 0.821108i \(0.306644\pi\)
\(524\) 6.01365e11 0.348455
\(525\) −1.75467e11 −0.100804
\(526\) 8.70775e11 0.495987
\(527\) −6.92484e11 −0.391076
\(528\) −2.50012e11 −0.139994
\(529\) −2.25340e11 −0.125109
\(530\) 2.93777e11 0.161725
\(531\) 7.95020e10 0.0433963
\(532\) 1.17531e12 0.636136
\(533\) −6.29006e11 −0.337584
\(534\) −1.26784e12 −0.674726
\(535\) 2.64381e12 1.39521
\(536\) 2.24250e12 1.17352
\(537\) 1.00265e12 0.520313
\(538\) −1.20680e11 −0.0621032
\(539\) −4.80170e11 −0.245045
\(540\) 1.53647e11 0.0777592
\(541\) −1.65773e12 −0.832005 −0.416002 0.909363i \(-0.636569\pi\)
−0.416002 + 0.909363i \(0.636569\pi\)
\(542\) −5.74977e9 −0.00286190
\(543\) −4.48345e11 −0.221316
\(544\) 3.81318e11 0.186677
\(545\) 1.94980e11 0.0946684
\(546\) 1.04742e12 0.504373
\(547\) 1.49021e12 0.711711 0.355856 0.934541i \(-0.384189\pi\)
0.355856 + 0.934541i \(0.384189\pi\)
\(548\) −5.17053e11 −0.244919
\(549\) 7.93102e10 0.0372609
\(550\) 2.45365e11 0.114335
\(551\) −4.23986e12 −1.95961
\(552\) 1.26504e12 0.579933
\(553\) 2.03094e12 0.923493
\(554\) 8.56777e11 0.386433
\(555\) −1.61115e12 −0.720805
\(556\) −1.58171e12 −0.701923
\(557\) −1.32765e12 −0.584433 −0.292216 0.956352i \(-0.594393\pi\)
−0.292216 + 0.956352i \(0.594393\pi\)
\(558\) −9.74300e11 −0.425440
\(559\) −2.04906e12 −0.887567
\(560\) 5.63183e11 0.241993
\(561\) −2.20855e11 −0.0941402
\(562\) 3.14834e12 1.33128
\(563\) 1.28033e12 0.537072 0.268536 0.963270i \(-0.413460\pi\)
0.268536 + 0.963270i \(0.413460\pi\)
\(564\) 7.90810e11 0.329091
\(565\) 2.47880e12 1.02335
\(566\) −5.67527e11 −0.232441
\(567\) 2.22215e11 0.0902920
\(568\) −4.99054e12 −2.01178
\(569\) 2.33110e12 0.932302 0.466151 0.884705i \(-0.345640\pi\)
0.466151 + 0.884705i \(0.345640\pi\)
\(570\) −1.63249e12 −0.647761
\(571\) 2.22629e12 0.876433 0.438217 0.898869i \(-0.355610\pi\)
0.438217 + 0.898869i \(0.355610\pi\)
\(572\) 1.22770e12 0.479524
\(573\) 4.36664e11 0.169220
\(574\) −3.61044e11 −0.138821
\(575\) −5.26780e11 −0.200966
\(576\) 8.32449e11 0.315106
\(577\) 3.83505e12 1.44039 0.720194 0.693772i \(-0.244053\pi\)
0.720194 + 0.693772i \(0.244053\pi\)
\(578\) 1.87806e12 0.699896
\(579\) 1.51930e12 0.561811
\(580\) 1.25699e12 0.461216
\(581\) −1.65727e12 −0.603395
\(582\) 1.97523e12 0.713616
\(583\) 4.98012e11 0.178538
\(584\) −4.52323e12 −1.60913
\(585\) 1.21948e12 0.430499
\(586\) −1.87742e12 −0.657692
\(587\) −3.47195e11 −0.120699 −0.0603493 0.998177i \(-0.519221\pi\)
−0.0603493 + 0.998177i \(0.519221\pi\)
\(588\) 2.59185e11 0.0894154
\(589\) −8.67713e12 −2.97069
\(590\) 2.50429e11 0.0850845
\(591\) −3.32052e12 −1.11960
\(592\) −1.41510e12 −0.473522
\(593\) 2.82977e12 0.939733 0.469867 0.882737i \(-0.344302\pi\)
0.469867 + 0.882737i \(0.344302\pi\)
\(594\) −3.10736e11 −0.102412
\(595\) 4.97504e11 0.162731
\(596\) 1.57522e12 0.511366
\(597\) 2.65580e12 0.855678
\(598\) 3.14451e12 1.00554
\(599\) 3.70847e12 1.17699 0.588497 0.808499i \(-0.299720\pi\)
0.588497 + 0.808499i \(0.299720\pi\)
\(600\) −4.22891e11 −0.133213
\(601\) −4.83713e12 −1.51235 −0.756175 0.654370i \(-0.772934\pi\)
−0.756175 + 0.654370i \(0.772934\pi\)
\(602\) −1.17614e12 −0.364986
\(603\) 1.18259e12 0.364256
\(604\) 1.54696e12 0.472949
\(605\) 1.39996e12 0.424831
\(606\) 1.79666e12 0.541176
\(607\) 3.15555e12 0.943466 0.471733 0.881741i \(-0.343629\pi\)
0.471733 + 0.881741i \(0.343629\pi\)
\(608\) 4.77808e12 1.41804
\(609\) 1.81794e12 0.535552
\(610\) 2.49825e11 0.0730553
\(611\) 6.27657e12 1.82195
\(612\) 1.19213e11 0.0343512
\(613\) 3.36352e12 0.962104 0.481052 0.876692i \(-0.340255\pi\)
0.481052 + 0.876692i \(0.340255\pi\)
\(614\) 4.69233e12 1.33239
\(615\) −4.20354e11 −0.118489
\(616\) 2.25008e12 0.629630
\(617\) −3.33692e11 −0.0926964 −0.0463482 0.998925i \(-0.514758\pi\)
−0.0463482 + 0.998925i \(0.514758\pi\)
\(618\) 3.92231e11 0.108167
\(619\) 1.06153e12 0.290619 0.145310 0.989386i \(-0.453582\pi\)
0.145310 + 0.989386i \(0.453582\pi\)
\(620\) 2.57250e12 0.699185
\(621\) 6.67125e11 0.180009
\(622\) −4.07931e12 −1.09277
\(623\) 4.84144e12 1.28759
\(624\) 1.07109e12 0.282810
\(625\) −2.81899e12 −0.738982
\(626\) −3.35098e12 −0.872144
\(627\) −2.76742e12 −0.715106
\(628\) 1.29491e12 0.332216
\(629\) −1.25007e12 −0.318425
\(630\) 6.99970e11 0.177030
\(631\) 6.83930e12 1.71743 0.858716 0.512451i \(-0.171262\pi\)
0.858716 + 0.512451i \(0.171262\pi\)
\(632\) 4.89475e12 1.22040
\(633\) −3.04422e12 −0.753632
\(634\) 6.42187e11 0.157855
\(635\) −7.85113e11 −0.191624
\(636\) −2.68816e11 −0.0651475
\(637\) 2.05712e12 0.495031
\(638\) −2.54213e12 −0.607441
\(639\) −2.63179e12 −0.624450
\(640\) −4.84320e11 −0.114110
\(641\) 4.26714e12 0.998335 0.499167 0.866506i \(-0.333639\pi\)
0.499167 + 0.866506i \(0.333639\pi\)
\(642\) 2.88611e12 0.670510
\(643\) −2.12525e12 −0.490299 −0.245149 0.969485i \(-0.578837\pi\)
−0.245149 + 0.969485i \(0.578837\pi\)
\(644\) −1.51291e12 −0.346599
\(645\) −1.36935e12 −0.311527
\(646\) −1.26663e12 −0.286157
\(647\) −6.71843e12 −1.50729 −0.753647 0.657279i \(-0.771707\pi\)
−0.753647 + 0.657279i \(0.771707\pi\)
\(648\) 5.35558e11 0.119322
\(649\) 4.24529e11 0.0939304
\(650\) −1.05118e12 −0.230976
\(651\) 3.72052e12 0.811875
\(652\) 1.23946e12 0.268608
\(653\) −7.51268e12 −1.61691 −0.808454 0.588559i \(-0.799695\pi\)
−0.808454 + 0.588559i \(0.799695\pi\)
\(654\) 2.12849e11 0.0454958
\(655\) 3.18969e12 0.677116
\(656\) −3.69205e11 −0.0778394
\(657\) −2.38535e12 −0.499468
\(658\) 3.60270e12 0.749224
\(659\) −2.68952e12 −0.555507 −0.277754 0.960652i \(-0.589590\pi\)
−0.277754 + 0.960652i \(0.589590\pi\)
\(660\) 8.20452e11 0.168308
\(661\) 3.74262e11 0.0762552 0.0381276 0.999273i \(-0.487861\pi\)
0.0381276 + 0.999273i \(0.487861\pi\)
\(662\) −4.94454e12 −1.00061
\(663\) 9.46178e11 0.190179
\(664\) −3.99418e12 −0.797391
\(665\) 6.23394e12 1.23613
\(666\) −1.75881e12 −0.346405
\(667\) 5.45775e12 1.06770
\(668\) −8.88064e11 −0.172564
\(669\) 3.29386e12 0.635751
\(670\) 3.72513e12 0.714176
\(671\) 4.23505e11 0.0806505
\(672\) −2.04871e12 −0.387543
\(673\) −9.54535e10 −0.0179359 −0.00896797 0.999960i \(-0.502855\pi\)
−0.00896797 + 0.999960i \(0.502855\pi\)
\(674\) −5.46689e12 −1.02040
\(675\) −2.23014e11 −0.0413490
\(676\) −2.78384e12 −0.512724
\(677\) 7.02943e12 1.28609 0.643044 0.765829i \(-0.277671\pi\)
0.643044 + 0.765829i \(0.277671\pi\)
\(678\) 2.70597e12 0.491801
\(679\) −7.54274e12 −1.36181
\(680\) 1.19903e12 0.215050
\(681\) −2.17674e12 −0.387833
\(682\) −5.20262e12 −0.920857
\(683\) 1.65634e12 0.291243 0.145622 0.989340i \(-0.453482\pi\)
0.145622 + 0.989340i \(0.453482\pi\)
\(684\) 1.49379e12 0.260938
\(685\) −2.74249e12 −0.475924
\(686\) 4.65735e12 0.802935
\(687\) −1.22606e12 −0.209994
\(688\) −1.20273e12 −0.204653
\(689\) −2.13356e12 −0.360677
\(690\) 2.10143e12 0.352934
\(691\) −6.66811e12 −1.11263 −0.556316 0.830971i \(-0.687786\pi\)
−0.556316 + 0.830971i \(0.687786\pi\)
\(692\) 1.91418e11 0.0317325
\(693\) 1.18659e12 0.195435
\(694\) −3.47713e10 −0.00568988
\(695\) −8.38951e12 −1.36397
\(696\) 4.38140e12 0.707736
\(697\) −3.26147e11 −0.0523439
\(698\) −2.03256e11 −0.0324111
\(699\) −6.31829e12 −1.00104
\(700\) 5.05753e11 0.0796155
\(701\) −6.53867e12 −1.02272 −0.511362 0.859366i \(-0.670859\pi\)
−0.511362 + 0.859366i \(0.670859\pi\)
\(702\) 1.33124e12 0.206890
\(703\) −1.56640e13 −2.41882
\(704\) 4.44516e12 0.682040
\(705\) 4.19453e12 0.639488
\(706\) 6.10886e12 0.925420
\(707\) −6.86083e12 −1.03274
\(708\) −2.29151e11 −0.0342746
\(709\) −2.15440e10 −0.00320198 −0.00160099 0.999999i \(-0.500510\pi\)
−0.00160099 + 0.999999i \(0.500510\pi\)
\(710\) −8.29006e12 −1.22432
\(711\) 2.58127e12 0.378809
\(712\) 1.16683e13 1.70156
\(713\) 1.11696e13 1.61858
\(714\) 5.43098e11 0.0782053
\(715\) 6.51183e12 0.931807
\(716\) −2.88997e12 −0.410946
\(717\) −5.71960e12 −0.808219
\(718\) −7.32395e12 −1.02845
\(719\) −2.79294e12 −0.389746 −0.194873 0.980828i \(-0.562429\pi\)
−0.194873 + 0.980828i \(0.562429\pi\)
\(720\) 7.15792e11 0.0992637
\(721\) −1.49780e12 −0.206416
\(722\) −1.04861e13 −1.43613
\(723\) −8.33741e12 −1.13477
\(724\) 1.29228e12 0.174797
\(725\) −1.82448e12 −0.245254
\(726\) 1.52826e12 0.204166
\(727\) −1.32795e13 −1.76310 −0.881548 0.472094i \(-0.843498\pi\)
−0.881548 + 0.472094i \(0.843498\pi\)
\(728\) −9.63970e12 −1.27196
\(729\) 2.82430e11 0.0370370
\(730\) −7.51378e12 −0.979277
\(731\) −1.06246e12 −0.137621
\(732\) −2.28599e11 −0.0294289
\(733\) −5.45238e12 −0.697619 −0.348809 0.937194i \(-0.613414\pi\)
−0.348809 + 0.937194i \(0.613414\pi\)
\(734\) 9.41038e12 1.19667
\(735\) 1.37474e12 0.173751
\(736\) −6.15057e12 −0.772619
\(737\) 6.31487e12 0.788426
\(738\) −4.58878e11 −0.0569434
\(739\) 6.21424e12 0.766457 0.383229 0.923653i \(-0.374812\pi\)
0.383229 + 0.923653i \(0.374812\pi\)
\(740\) 4.64387e12 0.569295
\(741\) 1.18560e13 1.44463
\(742\) −1.22465e12 −0.148318
\(743\) −7.16003e12 −0.861917 −0.430958 0.902372i \(-0.641824\pi\)
−0.430958 + 0.902372i \(0.641824\pi\)
\(744\) 8.96679e12 1.07290
\(745\) 8.35508e12 0.993682
\(746\) −5.23940e12 −0.619379
\(747\) −2.10635e12 −0.247508
\(748\) 6.36579e11 0.0743524
\(749\) −1.10211e13 −1.27955
\(750\) −3.97206e12 −0.458396
\(751\) 6.95263e11 0.0797571 0.0398786 0.999205i \(-0.487303\pi\)
0.0398786 + 0.999205i \(0.487303\pi\)
\(752\) 3.68413e12 0.420102
\(753\) 5.38362e11 0.0610235
\(754\) 1.08909e13 1.22713
\(755\) 8.20524e12 0.919030
\(756\) −6.40497e11 −0.0713130
\(757\) −5.88840e12 −0.651727 −0.325864 0.945417i \(-0.605655\pi\)
−0.325864 + 0.945417i \(0.605655\pi\)
\(758\) −4.10735e12 −0.451909
\(759\) 3.56235e12 0.389627
\(760\) 1.50244e13 1.63356
\(761\) −6.31218e12 −0.682257 −0.341129 0.940017i \(-0.610809\pi\)
−0.341129 + 0.940017i \(0.610809\pi\)
\(762\) −8.57066e11 −0.0920909
\(763\) −8.12798e11 −0.0868205
\(764\) −1.25861e12 −0.133651
\(765\) 6.32315e11 0.0667509
\(766\) −7.31034e12 −0.767199
\(767\) −1.81875e12 −0.189755
\(768\) −5.79061e12 −0.600618
\(769\) −1.64038e13 −1.69151 −0.845757 0.533569i \(-0.820851\pi\)
−0.845757 + 0.533569i \(0.820851\pi\)
\(770\) 3.73774e12 0.383178
\(771\) −4.80377e12 −0.489596
\(772\) −4.37913e12 −0.443721
\(773\) 6.75582e12 0.680566 0.340283 0.940323i \(-0.389477\pi\)
0.340283 + 0.940323i \(0.389477\pi\)
\(774\) −1.49485e12 −0.149714
\(775\) −3.73390e12 −0.371796
\(776\) −1.81787e13 −1.79964
\(777\) 6.71629e12 0.661051
\(778\) 1.15098e13 1.12631
\(779\) −4.08677e12 −0.397614
\(780\) −3.51494e12 −0.340011
\(781\) −1.40534e13 −1.35161
\(782\) 1.63047e12 0.155913
\(783\) 2.31056e12 0.219679
\(784\) 1.20746e12 0.114143
\(785\) 6.86830e12 0.645559
\(786\) 3.48202e12 0.325409
\(787\) 1.56632e13 1.45544 0.727719 0.685875i \(-0.240580\pi\)
0.727719 + 0.685875i \(0.240580\pi\)
\(788\) 9.57084e12 0.884264
\(789\) −4.22624e12 −0.388246
\(790\) 8.13093e12 0.742709
\(791\) −1.03332e13 −0.938512
\(792\) 2.85980e12 0.258269
\(793\) −1.81436e12 −0.162927
\(794\) −2.65852e12 −0.237382
\(795\) −1.42582e12 −0.126594
\(796\) −7.65489e12 −0.675819
\(797\) 1.43951e13 1.26372 0.631860 0.775083i \(-0.282292\pi\)
0.631860 + 0.775083i \(0.282292\pi\)
\(798\) 6.80526e12 0.594062
\(799\) 3.25448e12 0.282502
\(800\) 2.05608e12 0.177474
\(801\) 6.15335e12 0.528160
\(802\) −2.49943e12 −0.213333
\(803\) −1.27374e13 −1.08109
\(804\) −3.40863e12 −0.287692
\(805\) −8.02463e12 −0.673509
\(806\) 2.22888e13 1.86028
\(807\) 5.85710e11 0.0486129
\(808\) −1.65352e13 −1.36477
\(809\) −1.15711e13 −0.949742 −0.474871 0.880055i \(-0.657505\pi\)
−0.474871 + 0.880055i \(0.657505\pi\)
\(810\) 8.89645e11 0.0726163
\(811\) 7.84788e12 0.637028 0.318514 0.947918i \(-0.396816\pi\)
0.318514 + 0.947918i \(0.396816\pi\)
\(812\) −5.23990e12 −0.422981
\(813\) 2.79061e10 0.00224022
\(814\) −9.39177e12 −0.749787
\(815\) 6.57420e12 0.521956
\(816\) 5.55374e11 0.0438510
\(817\) −1.33131e13 −1.04540
\(818\) 1.23857e13 0.967234
\(819\) −5.08355e12 −0.394811
\(820\) 1.21160e12 0.0935830
\(821\) −2.29251e12 −0.176103 −0.0880515 0.996116i \(-0.528064\pi\)
−0.0880515 + 0.996116i \(0.528064\pi\)
\(822\) −2.99383e12 −0.228720
\(823\) −8.03820e12 −0.610745 −0.305372 0.952233i \(-0.598781\pi\)
−0.305372 + 0.952233i \(0.598781\pi\)
\(824\) −3.60982e12 −0.272781
\(825\) −1.19086e12 −0.0894990
\(826\) −1.04395e12 −0.0780311
\(827\) 1.50364e13 1.11782 0.558908 0.829230i \(-0.311221\pi\)
0.558908 + 0.829230i \(0.311221\pi\)
\(828\) −1.92288e12 −0.142172
\(829\) −5.18678e12 −0.381419 −0.190710 0.981647i \(-0.561079\pi\)
−0.190710 + 0.981647i \(0.561079\pi\)
\(830\) −6.63495e12 −0.485273
\(831\) −4.15830e12 −0.302490
\(832\) −1.90437e13 −1.37783
\(833\) 1.06664e12 0.0767569
\(834\) −9.15838e12 −0.655498
\(835\) −4.71037e12 −0.335325
\(836\) 7.97661e12 0.564794
\(837\) 4.72869e12 0.333025
\(838\) 3.70012e12 0.259189
\(839\) −1.41530e13 −0.986099 −0.493050 0.870001i \(-0.664118\pi\)
−0.493050 + 0.870001i \(0.664118\pi\)
\(840\) −6.44205e12 −0.446444
\(841\) 4.39550e12 0.302989
\(842\) 4.98242e12 0.341614
\(843\) −1.52802e13 −1.04209
\(844\) 8.77445e12 0.595222
\(845\) −1.47657e13 −0.996320
\(846\) 4.57894e12 0.307325
\(847\) −5.83591e12 −0.389613
\(848\) −1.25233e12 −0.0831642
\(849\) 2.75445e12 0.181949
\(850\) −5.45051e11 −0.0358139
\(851\) 2.01634e13 1.31790
\(852\) 7.58569e12 0.493193
\(853\) −2.92961e13 −1.89469 −0.947346 0.320210i \(-0.896246\pi\)
−0.947346 + 0.320210i \(0.896246\pi\)
\(854\) −1.04143e12 −0.0669990
\(855\) 7.92319e12 0.507052
\(856\) −2.65618e13 −1.69093
\(857\) −1.94036e13 −1.22877 −0.614384 0.789007i \(-0.710595\pi\)
−0.614384 + 0.789007i \(0.710595\pi\)
\(858\) 7.10862e12 0.447809
\(859\) 2.45006e12 0.153535 0.0767676 0.997049i \(-0.475540\pi\)
0.0767676 + 0.997049i \(0.475540\pi\)
\(860\) 3.94693e12 0.246046
\(861\) 1.75230e12 0.108666
\(862\) 1.57233e13 0.969978
\(863\) 2.43811e13 1.49625 0.748127 0.663556i \(-0.230953\pi\)
0.748127 + 0.663556i \(0.230953\pi\)
\(864\) −2.60386e12 −0.158967
\(865\) 1.01530e12 0.0616623
\(866\) −1.62933e12 −0.0984415
\(867\) −9.11501e12 −0.547862
\(868\) −1.07238e13 −0.641223
\(869\) 1.37836e13 0.819925
\(870\) 7.27818e12 0.430711
\(871\) −2.70539e13 −1.59275
\(872\) −1.95892e12 −0.114734
\(873\) −9.58664e12 −0.558602
\(874\) 2.04305e13 1.18434
\(875\) 1.51680e13 0.874764
\(876\) 6.87537e12 0.394482
\(877\) 2.54386e13 1.45209 0.726046 0.687646i \(-0.241356\pi\)
0.726046 + 0.687646i \(0.241356\pi\)
\(878\) 8.58936e12 0.487793
\(879\) 9.11192e12 0.514826
\(880\) 3.82222e12 0.214854
\(881\) 2.62242e13 1.46660 0.733299 0.679906i \(-0.237979\pi\)
0.733299 + 0.679906i \(0.237979\pi\)
\(882\) 1.50073e12 0.0835015
\(883\) −3.51862e13 −1.94782 −0.973910 0.226933i \(-0.927130\pi\)
−0.973910 + 0.226933i \(0.927130\pi\)
\(884\) −2.72720e12 −0.150204
\(885\) −1.21544e12 −0.0666021
\(886\) −1.62007e13 −0.883245
\(887\) −7.20872e12 −0.391023 −0.195511 0.980701i \(-0.562637\pi\)
−0.195511 + 0.980701i \(0.562637\pi\)
\(888\) 1.61869e13 0.873584
\(889\) 3.27284e12 0.175739
\(890\) 1.93829e13 1.03553
\(891\) 1.50813e12 0.0801659
\(892\) −9.49399e12 −0.502120
\(893\) 4.07801e13 2.14594
\(894\) 9.12080e12 0.477545
\(895\) −1.53286e13 −0.798547
\(896\) 2.01895e12 0.104650
\(897\) −1.52616e13 −0.787110
\(898\) 1.57097e13 0.806165
\(899\) 3.86854e13 1.97528
\(900\) 6.42800e11 0.0326576
\(901\) −1.10628e12 −0.0559246
\(902\) −2.45034e12 −0.123253
\(903\) 5.70832e12 0.285702
\(904\) −2.49039e13 −1.24025
\(905\) 6.85436e12 0.339663
\(906\) 8.95722e12 0.441668
\(907\) 1.82394e13 0.894905 0.447453 0.894308i \(-0.352331\pi\)
0.447453 + 0.894308i \(0.352331\pi\)
\(908\) 6.27409e12 0.306312
\(909\) −8.71994e12 −0.423620
\(910\) −1.60130e13 −0.774083
\(911\) 3.59657e13 1.73004 0.865020 0.501737i \(-0.167306\pi\)
0.865020 + 0.501737i \(0.167306\pi\)
\(912\) 6.95908e12 0.333100
\(913\) −1.12476e13 −0.535725
\(914\) 1.65441e13 0.784125
\(915\) −1.21251e12 −0.0571859
\(916\) 3.53392e12 0.165854
\(917\) −1.32966e13 −0.620983
\(918\) 6.90264e11 0.0320792
\(919\) 3.31634e13 1.53369 0.766847 0.641830i \(-0.221824\pi\)
0.766847 + 0.641830i \(0.221824\pi\)
\(920\) −1.93401e13 −0.890048
\(921\) −2.27739e13 −1.04296
\(922\) 1.40923e12 0.0642233
\(923\) 6.02068e13 2.73047
\(924\) −3.42016e12 −0.154356
\(925\) −6.74044e12 −0.302726
\(926\) −3.12653e13 −1.39737
\(927\) −1.90366e12 −0.0846703
\(928\) −2.13022e13 −0.942885
\(929\) 4.54694e11 0.0200285 0.0100142 0.999950i \(-0.496812\pi\)
0.0100142 + 0.999950i \(0.496812\pi\)
\(930\) 1.48952e13 0.652941
\(931\) 1.33655e13 0.583059
\(932\) 1.82114e13 0.790628
\(933\) 1.97986e13 0.855398
\(934\) 7.24184e12 0.311378
\(935\) 3.37647e12 0.144481
\(936\) −1.22518e13 −0.521746
\(937\) −2.05486e13 −0.870871 −0.435436 0.900220i \(-0.643406\pi\)
−0.435436 + 0.900220i \(0.643406\pi\)
\(938\) −1.55287e13 −0.654971
\(939\) 1.62637e13 0.682693
\(940\) −1.20900e13 −0.505070
\(941\) 1.05780e13 0.439796 0.219898 0.975523i \(-0.429428\pi\)
0.219898 + 0.975523i \(0.429428\pi\)
\(942\) 7.49776e12 0.310243
\(943\) 5.26069e12 0.216641
\(944\) −1.06754e12 −0.0437533
\(945\) −3.39725e12 −0.138575
\(946\) −7.98227e12 −0.324053
\(947\) −5.86586e12 −0.237005 −0.118502 0.992954i \(-0.537809\pi\)
−0.118502 + 0.992954i \(0.537809\pi\)
\(948\) −7.44009e12 −0.299185
\(949\) 5.45690e13 2.18398
\(950\) −6.82973e12 −0.272049
\(951\) −3.11680e12 −0.123565
\(952\) −4.99831e12 −0.197223
\(953\) −1.52754e12 −0.0599894 −0.0299947 0.999550i \(-0.509549\pi\)
−0.0299947 + 0.999550i \(0.509549\pi\)
\(954\) −1.55650e12 −0.0608387
\(955\) −6.67579e12 −0.259709
\(956\) 1.64858e13 0.638335
\(957\) 1.23380e13 0.475491
\(958\) 9.55196e12 0.366393
\(959\) 1.14324e13 0.436470
\(960\) −1.27266e13 −0.483606
\(961\) 5.27324e13 1.99445
\(962\) 4.02358e13 1.51469
\(963\) −1.40075e13 −0.524859
\(964\) 2.40312e13 0.896249
\(965\) −2.32273e13 −0.862235
\(966\) −8.76006e12 −0.323676
\(967\) −1.28750e13 −0.473510 −0.236755 0.971569i \(-0.576084\pi\)
−0.236755 + 0.971569i \(0.576084\pi\)
\(968\) −1.40651e13 −0.514876
\(969\) 6.14750e12 0.223997
\(970\) −3.01976e13 −1.09522
\(971\) −1.83107e13 −0.661025 −0.330513 0.943802i \(-0.607222\pi\)
−0.330513 + 0.943802i \(0.607222\pi\)
\(972\) −8.14056e11 −0.0292520
\(973\) 3.49727e13 1.25090
\(974\) 4.53489e9 0.000161455 0
\(975\) 5.10183e12 0.180803
\(976\) −1.06497e12 −0.0375675
\(977\) −4.64360e13 −1.63053 −0.815265 0.579088i \(-0.803409\pi\)
−0.815265 + 0.579088i \(0.803409\pi\)
\(978\) 7.17670e12 0.250842
\(979\) 3.28580e13 1.14319
\(980\) −3.96246e12 −0.137230
\(981\) −1.03305e12 −0.0356130
\(982\) 1.96501e11 0.00674314
\(983\) −1.65522e13 −0.565413 −0.282706 0.959206i \(-0.591232\pi\)
−0.282706 + 0.959206i \(0.591232\pi\)
\(984\) 4.22320e12 0.143603
\(985\) 5.07646e13 1.71829
\(986\) 5.64705e12 0.190272
\(987\) −1.74854e13 −0.586474
\(988\) −3.41730e13 −1.14098
\(989\) 1.71373e13 0.569586
\(990\) 4.75057e12 0.157176
\(991\) −5.38376e13 −1.77319 −0.886593 0.462551i \(-0.846934\pi\)
−0.886593 + 0.462551i \(0.846934\pi\)
\(992\) −4.35962e13 −1.42938
\(993\) 2.39979e13 0.783253
\(994\) 3.45582e13 1.12283
\(995\) −4.06022e13 −1.31325
\(996\) 6.07121e12 0.195483
\(997\) −1.14792e11 −0.00367946 −0.00183973 0.999998i \(-0.500586\pi\)
−0.00183973 + 0.999998i \(0.500586\pi\)
\(998\) −1.11658e13 −0.356290
\(999\) 8.53624e12 0.271158
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.1.9 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.10.a.a.1.9 21 1.1 even 1 trivial