Properties

Label 177.10.a.a.1.20
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 1.20
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+37.7088 q^{2} +81.0000 q^{3} +909.951 q^{4} -1229.57 q^{5} +3054.41 q^{6} -1968.46 q^{7} +15006.2 q^{8} +6561.00 q^{9} +O(q^{10})\) \(q+37.7088 q^{2} +81.0000 q^{3} +909.951 q^{4} -1229.57 q^{5} +3054.41 q^{6} -1968.46 q^{7} +15006.2 q^{8} +6561.00 q^{9} -46365.5 q^{10} -54823.4 q^{11} +73706.0 q^{12} +34302.8 q^{13} -74228.3 q^{14} -99595.0 q^{15} +99972.0 q^{16} -122669. q^{17} +247407. q^{18} +838314. q^{19} -1.11885e6 q^{20} -159446. q^{21} -2.06732e6 q^{22} -2.32811e6 q^{23} +1.21551e6 q^{24} -441287. q^{25} +1.29352e6 q^{26} +531441. q^{27} -1.79121e6 q^{28} -6.13023e6 q^{29} -3.75561e6 q^{30} +3.50327e6 q^{31} -3.91338e6 q^{32} -4.44070e6 q^{33} -4.62569e6 q^{34} +2.42036e6 q^{35} +5.97019e6 q^{36} -9.25273e6 q^{37} +3.16118e7 q^{38} +2.77852e6 q^{39} -1.84512e7 q^{40} -1.75636e7 q^{41} -6.01249e6 q^{42} +2.04392e7 q^{43} -4.98866e7 q^{44} -8.06720e6 q^{45} -8.77902e7 q^{46} +1.13961e7 q^{47} +8.09773e6 q^{48} -3.64788e7 q^{49} -1.66404e7 q^{50} -9.93618e6 q^{51} +3.12138e7 q^{52} -5.64902e7 q^{53} +2.00400e7 q^{54} +6.74091e7 q^{55} -2.95392e7 q^{56} +6.79034e7 q^{57} -2.31163e8 q^{58} +1.21174e7 q^{59} -9.06266e7 q^{60} -1.93157e8 q^{61} +1.32104e8 q^{62} -1.29151e7 q^{63} -1.98754e8 q^{64} -4.21776e7 q^{65} -1.67453e8 q^{66} +2.97914e8 q^{67} -1.11623e8 q^{68} -1.88577e8 q^{69} +9.12688e7 q^{70} -3.55939e8 q^{71} +9.84560e7 q^{72} +4.51678e8 q^{73} -3.48909e8 q^{74} -3.57443e7 q^{75} +7.62824e8 q^{76} +1.07918e8 q^{77} +1.04775e8 q^{78} +3.03934e8 q^{79} -1.22922e8 q^{80} +4.30467e7 q^{81} -6.62302e8 q^{82} +6.28705e8 q^{83} -1.45088e8 q^{84} +1.50830e8 q^{85} +7.70735e8 q^{86} -4.96548e8 q^{87} -8.22693e8 q^{88} -8.08718e8 q^{89} -3.04204e8 q^{90} -6.75237e7 q^{91} -2.11847e9 q^{92} +2.83765e8 q^{93} +4.29732e8 q^{94} -1.03076e9 q^{95} -3.16984e8 q^{96} +5.73218e8 q^{97} -1.37557e9 q^{98} -3.59696e8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 21q - 66q^{2} + 1701q^{3} + 5206q^{4} - 2964q^{5} - 5346q^{6} - 30775q^{7} - 24621q^{8} + 137781q^{9} + O(q^{10}) \) \( 21q - 66q^{2} + 1701q^{3} + 5206q^{4} - 2964q^{5} - 5346q^{6} - 30775q^{7} - 24621q^{8} + 137781q^{9} - 54663q^{10} - 151769q^{11} + 421686q^{12} - 153611q^{13} - 286771q^{14} - 240084q^{15} + 805530q^{16} - 723621q^{17} - 433026q^{18} - 549388q^{19} - 527311q^{20} - 2492775q^{21} + 2973158q^{22} + 169962q^{23} - 1994301q^{24} + 8035779q^{25} - 2337392q^{26} + 11160261q^{27} - 22659054q^{28} - 16845442q^{29} - 4427703q^{30} - 19307976q^{31} - 44923568q^{32} - 12293289q^{33} - 35547496q^{34} - 34882596q^{35} + 34156566q^{36} - 41561129q^{37} - 52335371q^{38} - 12442491q^{39} - 125735038q^{40} - 68169291q^{41} - 23228451q^{42} - 25719587q^{43} - 126277032q^{44} - 19446804q^{45} - 292814271q^{46} - 174095332q^{47} + 65247930q^{48} + 7479350q^{49} - 227877439q^{50} - 58613301q^{51} - 232397708q^{52} - 228390500q^{53} - 35075106q^{54} - 29426208q^{55} + 326778474q^{56} - 44500428q^{57} + 480343762q^{58} + 254464581q^{59} - 42712191q^{60} - 183928964q^{61} - 21753862q^{62} - 201914775q^{63} + 310571245q^{64} + 5308466q^{65} + 240825798q^{66} - 82724114q^{67} - 138336205q^{68} + 13766922q^{69} + 1030274876q^{70} - 404721965q^{71} - 161538381q^{72} + 154162574q^{73} + 36352054q^{74} + 650898099q^{75} + 1068940636q^{76} - 448535481q^{77} - 189328752q^{78} + 272529635q^{79} - 345587859q^{80} + 903981141q^{81} - 38412637q^{82} + 432518643q^{83} - 1835383374q^{84} - 126211490q^{85} - 3699273072q^{86} - 1364480802q^{87} + 170111045q^{88} - 1255621070q^{89} - 358643943q^{90} + 1448885849q^{91} + 1568933320q^{92} - 1563946056q^{93} - 1908445164q^{94} - 2896546490q^{95} - 3638809008q^{96} + 1007235486q^{97} - 9506868248q^{98} - 995756409q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 37.7088 1.66651 0.833254 0.552891i \(-0.186475\pi\)
0.833254 + 0.552891i \(0.186475\pi\)
\(3\) 81.0000 0.577350
\(4\) 909.951 1.77725
\(5\) −1229.57 −0.879807 −0.439904 0.898045i \(-0.644987\pi\)
−0.439904 + 0.898045i \(0.644987\pi\)
\(6\) 3054.41 0.962159
\(7\) −1968.46 −0.309875 −0.154937 0.987924i \(-0.549518\pi\)
−0.154937 + 0.987924i \(0.549518\pi\)
\(8\) 15006.2 1.29529
\(9\) 6561.00 0.333333
\(10\) −46365.5 −1.46621
\(11\) −54823.4 −1.12901 −0.564507 0.825429i \(-0.690934\pi\)
−0.564507 + 0.825429i \(0.690934\pi\)
\(12\) 73706.0 1.02609
\(13\) 34302.8 0.333107 0.166554 0.986032i \(-0.446736\pi\)
0.166554 + 0.986032i \(0.446736\pi\)
\(14\) −74228.3 −0.516409
\(15\) −99595.0 −0.507957
\(16\) 99972.0 0.381363
\(17\) −122669. −0.356217 −0.178108 0.984011i \(-0.556998\pi\)
−0.178108 + 0.984011i \(0.556998\pi\)
\(18\) 247407. 0.555503
\(19\) 838314. 1.47576 0.737879 0.674933i \(-0.235827\pi\)
0.737879 + 0.674933i \(0.235827\pi\)
\(20\) −1.11885e6 −1.56364
\(21\) −159446. −0.178906
\(22\) −2.06732e6 −1.88151
\(23\) −2.32811e6 −1.73472 −0.867358 0.497684i \(-0.834184\pi\)
−0.867358 + 0.497684i \(0.834184\pi\)
\(24\) 1.21551e6 0.747836
\(25\) −441287. −0.225939
\(26\) 1.29352e6 0.555126
\(27\) 531441. 0.192450
\(28\) −1.79121e6 −0.550724
\(29\) −6.13023e6 −1.60948 −0.804740 0.593628i \(-0.797695\pi\)
−0.804740 + 0.593628i \(0.797695\pi\)
\(30\) −3.75561e6 −0.846514
\(31\) 3.50327e6 0.681312 0.340656 0.940188i \(-0.389351\pi\)
0.340656 + 0.940188i \(0.389351\pi\)
\(32\) −3.91338e6 −0.659746
\(33\) −4.44070e6 −0.651836
\(34\) −4.62569e6 −0.593638
\(35\) 2.42036e6 0.272630
\(36\) 5.97019e6 0.592416
\(37\) −9.25273e6 −0.811638 −0.405819 0.913953i \(-0.633014\pi\)
−0.405819 + 0.913953i \(0.633014\pi\)
\(38\) 3.16118e7 2.45936
\(39\) 2.77852e6 0.192320
\(40\) −1.84512e7 −1.13961
\(41\) −1.75636e7 −0.970703 −0.485352 0.874319i \(-0.661308\pi\)
−0.485352 + 0.874319i \(0.661308\pi\)
\(42\) −6.01249e6 −0.298149
\(43\) 2.04392e7 0.911706 0.455853 0.890055i \(-0.349334\pi\)
0.455853 + 0.890055i \(0.349334\pi\)
\(44\) −4.98866e7 −2.00654
\(45\) −8.06720e6 −0.293269
\(46\) −8.77902e7 −2.89092
\(47\) 1.13961e7 0.340655 0.170328 0.985387i \(-0.445517\pi\)
0.170328 + 0.985387i \(0.445517\pi\)
\(48\) 8.09773e6 0.220180
\(49\) −3.64788e7 −0.903978
\(50\) −1.66404e7 −0.376529
\(51\) −9.93618e6 −0.205662
\(52\) 3.12138e7 0.592014
\(53\) −5.64902e7 −0.983403 −0.491701 0.870764i \(-0.663625\pi\)
−0.491701 + 0.870764i \(0.663625\pi\)
\(54\) 2.00400e7 0.320720
\(55\) 6.74091e7 0.993314
\(56\) −2.95392e7 −0.401378
\(57\) 6.79034e7 0.852030
\(58\) −2.31163e8 −2.68221
\(59\) 1.21174e7 0.130189
\(60\) −9.06266e7 −0.902766
\(61\) −1.93157e8 −1.78618 −0.893090 0.449878i \(-0.851468\pi\)
−0.893090 + 0.449878i \(0.851468\pi\)
\(62\) 1.32104e8 1.13541
\(63\) −1.29151e7 −0.103292
\(64\) −1.98754e8 −1.48083
\(65\) −4.21776e7 −0.293070
\(66\) −1.67453e8 −1.08629
\(67\) 2.97914e8 1.80615 0.903075 0.429482i \(-0.141304\pi\)
0.903075 + 0.429482i \(0.141304\pi\)
\(68\) −1.11623e8 −0.633086
\(69\) −1.88577e8 −1.00154
\(70\) 9.12688e7 0.454340
\(71\) −3.55939e8 −1.66231 −0.831157 0.556037i \(-0.812321\pi\)
−0.831157 + 0.556037i \(0.812321\pi\)
\(72\) 9.84560e7 0.431763
\(73\) 4.51678e8 1.86156 0.930778 0.365586i \(-0.119131\pi\)
0.930778 + 0.365586i \(0.119131\pi\)
\(74\) −3.48909e8 −1.35260
\(75\) −3.57443e7 −0.130446
\(76\) 7.62824e8 2.62279
\(77\) 1.07918e8 0.349853
\(78\) 1.04775e8 0.320502
\(79\) 3.03934e8 0.877926 0.438963 0.898505i \(-0.355346\pi\)
0.438963 + 0.898505i \(0.355346\pi\)
\(80\) −1.22922e8 −0.335526
\(81\) 4.30467e7 0.111111
\(82\) −6.62302e8 −1.61768
\(83\) 6.28705e8 1.45411 0.727053 0.686582i \(-0.240890\pi\)
0.727053 + 0.686582i \(0.240890\pi\)
\(84\) −1.45088e8 −0.317961
\(85\) 1.50830e8 0.313402
\(86\) 7.70735e8 1.51937
\(87\) −4.96548e8 −0.929233
\(88\) −8.22693e8 −1.46240
\(89\) −8.08718e8 −1.36629 −0.683144 0.730284i \(-0.739388\pi\)
−0.683144 + 0.730284i \(0.739388\pi\)
\(90\) −3.04204e8 −0.488735
\(91\) −6.75237e7 −0.103222
\(92\) −2.11847e9 −3.08302
\(93\) 2.83765e8 0.393356
\(94\) 4.29732e8 0.567704
\(95\) −1.03076e9 −1.29838
\(96\) −3.16984e8 −0.380905
\(97\) 5.73218e8 0.657427 0.328713 0.944430i \(-0.393385\pi\)
0.328713 + 0.944430i \(0.393385\pi\)
\(98\) −1.37557e9 −1.50649
\(99\) −3.59696e8 −0.376338
\(100\) −4.01550e8 −0.401550
\(101\) −9.04619e8 −0.865007 −0.432503 0.901632i \(-0.642370\pi\)
−0.432503 + 0.901632i \(0.642370\pi\)
\(102\) −3.74681e8 −0.342737
\(103\) 9.29642e8 0.813857 0.406929 0.913460i \(-0.366600\pi\)
0.406929 + 0.913460i \(0.366600\pi\)
\(104\) 5.14756e8 0.431471
\(105\) 1.96049e8 0.157403
\(106\) −2.13017e9 −1.63885
\(107\) −5.27465e8 −0.389015 −0.194508 0.980901i \(-0.562311\pi\)
−0.194508 + 0.980901i \(0.562311\pi\)
\(108\) 4.83585e8 0.342032
\(109\) 6.39618e7 0.0434012 0.0217006 0.999765i \(-0.493092\pi\)
0.0217006 + 0.999765i \(0.493092\pi\)
\(110\) 2.54191e9 1.65537
\(111\) −7.49471e8 −0.468599
\(112\) −1.96791e8 −0.118175
\(113\) 2.05416e9 1.18517 0.592587 0.805506i \(-0.298107\pi\)
0.592587 + 0.805506i \(0.298107\pi\)
\(114\) 2.56055e9 1.41991
\(115\) 2.86257e9 1.52622
\(116\) −5.57821e9 −2.86044
\(117\) 2.25060e8 0.111036
\(118\) 4.56931e8 0.216961
\(119\) 2.41469e8 0.110383
\(120\) −1.49455e9 −0.657952
\(121\) 6.47659e8 0.274671
\(122\) −7.28370e9 −2.97668
\(123\) −1.42265e9 −0.560436
\(124\) 3.18781e9 1.21086
\(125\) 2.94409e9 1.07859
\(126\) −4.87012e8 −0.172136
\(127\) 2.40390e9 0.819974 0.409987 0.912091i \(-0.365533\pi\)
0.409987 + 0.912091i \(0.365533\pi\)
\(128\) −5.49113e9 −1.80808
\(129\) 1.65557e9 0.526374
\(130\) −1.59046e9 −0.488404
\(131\) 2.24064e9 0.664740 0.332370 0.943149i \(-0.392152\pi\)
0.332370 + 0.943149i \(0.392152\pi\)
\(132\) −4.04082e9 −1.15847
\(133\) −1.65019e9 −0.457300
\(134\) 1.12340e10 3.00996
\(135\) −6.53443e8 −0.169319
\(136\) −1.84080e9 −0.461404
\(137\) 2.68246e9 0.650566 0.325283 0.945617i \(-0.394540\pi\)
0.325283 + 0.945617i \(0.394540\pi\)
\(138\) −7.11101e9 −1.66907
\(139\) 2.86442e9 0.650833 0.325417 0.945571i \(-0.394495\pi\)
0.325417 + 0.945571i \(0.394495\pi\)
\(140\) 2.20241e9 0.484531
\(141\) 9.23082e8 0.196677
\(142\) −1.34220e10 −2.77026
\(143\) −1.88060e9 −0.376083
\(144\) 6.55916e8 0.127121
\(145\) 7.53753e9 1.41603
\(146\) 1.70322e10 3.10230
\(147\) −2.95478e9 −0.521912
\(148\) −8.41954e9 −1.44248
\(149\) −4.90150e9 −0.814687 −0.407344 0.913275i \(-0.633545\pi\)
−0.407344 + 0.913275i \(0.633545\pi\)
\(150\) −1.34787e9 −0.217389
\(151\) −1.12831e10 −1.76617 −0.883084 0.469216i \(-0.844537\pi\)
−0.883084 + 0.469216i \(0.844537\pi\)
\(152\) 1.25799e10 1.91154
\(153\) −8.04831e8 −0.118739
\(154\) 4.06945e9 0.583032
\(155\) −4.30751e9 −0.599424
\(156\) 2.52832e9 0.341800
\(157\) −9.88726e9 −1.29876 −0.649378 0.760466i \(-0.724971\pi\)
−0.649378 + 0.760466i \(0.724971\pi\)
\(158\) 1.14610e10 1.46307
\(159\) −4.57570e9 −0.567768
\(160\) 4.81176e9 0.580449
\(161\) 4.58280e9 0.537545
\(162\) 1.62324e9 0.185168
\(163\) 3.26956e9 0.362781 0.181390 0.983411i \(-0.441940\pi\)
0.181390 + 0.983411i \(0.441940\pi\)
\(164\) −1.59820e10 −1.72518
\(165\) 5.46014e9 0.573490
\(166\) 2.37077e10 2.42328
\(167\) 1.42782e10 1.42052 0.710261 0.703938i \(-0.248577\pi\)
0.710261 + 0.703938i \(0.248577\pi\)
\(168\) −2.39268e9 −0.231735
\(169\) −9.42782e9 −0.889040
\(170\) 5.68761e9 0.522287
\(171\) 5.50018e9 0.491920
\(172\) 1.85986e10 1.62033
\(173\) 1.09311e10 0.927808 0.463904 0.885886i \(-0.346448\pi\)
0.463904 + 0.885886i \(0.346448\pi\)
\(174\) −1.87242e10 −1.54857
\(175\) 8.68658e8 0.0700128
\(176\) −5.48080e9 −0.430564
\(177\) 9.81506e8 0.0751646
\(178\) −3.04958e10 −2.27693
\(179\) −1.02017e9 −0.0742738 −0.0371369 0.999310i \(-0.511824\pi\)
−0.0371369 + 0.999310i \(0.511824\pi\)
\(180\) −7.34075e9 −0.521212
\(181\) 9.70649e9 0.672216 0.336108 0.941823i \(-0.390889\pi\)
0.336108 + 0.941823i \(0.390889\pi\)
\(182\) −2.54624e9 −0.172019
\(183\) −1.56457e10 −1.03125
\(184\) −3.49362e10 −2.24696
\(185\) 1.13769e10 0.714085
\(186\) 1.07004e10 0.655531
\(187\) 6.72513e9 0.402174
\(188\) 1.03699e10 0.605429
\(189\) −1.04612e9 −0.0596354
\(190\) −3.88688e10 −2.16377
\(191\) −3.09671e10 −1.68365 −0.841823 0.539753i \(-0.818518\pi\)
−0.841823 + 0.539753i \(0.818518\pi\)
\(192\) −1.60991e10 −0.854960
\(193\) −3.31776e10 −1.72122 −0.860611 0.509263i \(-0.829918\pi\)
−0.860611 + 0.509263i \(0.829918\pi\)
\(194\) 2.16154e10 1.09561
\(195\) −3.41638e9 −0.169204
\(196\) −3.31939e10 −1.60659
\(197\) −1.47501e10 −0.697745 −0.348872 0.937170i \(-0.613435\pi\)
−0.348872 + 0.937170i \(0.613435\pi\)
\(198\) −1.35637e10 −0.627170
\(199\) −4.72288e8 −0.0213485 −0.0106743 0.999943i \(-0.503398\pi\)
−0.0106743 + 0.999943i \(0.503398\pi\)
\(200\) −6.62206e9 −0.292657
\(201\) 2.41310e10 1.04278
\(202\) −3.41121e10 −1.44154
\(203\) 1.20671e10 0.498737
\(204\) −9.04144e9 −0.365512
\(205\) 2.15957e10 0.854032
\(206\) 3.50557e10 1.35630
\(207\) −1.52747e10 −0.578239
\(208\) 3.42931e9 0.127035
\(209\) −4.59592e10 −1.66615
\(210\) 7.39277e9 0.262313
\(211\) −4.37403e10 −1.51918 −0.759592 0.650400i \(-0.774601\pi\)
−0.759592 + 0.650400i \(0.774601\pi\)
\(212\) −5.14033e10 −1.74775
\(213\) −2.88311e10 −0.959738
\(214\) −1.98900e10 −0.648297
\(215\) −2.51313e10 −0.802126
\(216\) 7.97493e9 0.249279
\(217\) −6.89606e9 −0.211121
\(218\) 2.41192e9 0.0723284
\(219\) 3.65859e10 1.07477
\(220\) 6.13390e10 1.76537
\(221\) −4.20788e9 −0.118658
\(222\) −2.82616e10 −0.780925
\(223\) −5.12163e9 −0.138687 −0.0693436 0.997593i \(-0.522090\pi\)
−0.0693436 + 0.997593i \(0.522090\pi\)
\(224\) 7.70334e9 0.204439
\(225\) −2.89529e9 −0.0753130
\(226\) 7.74600e10 1.97510
\(227\) 2.08665e10 0.521594 0.260797 0.965394i \(-0.416015\pi\)
0.260797 + 0.965394i \(0.416015\pi\)
\(228\) 6.17888e10 1.51427
\(229\) 6.41955e7 0.00154257 0.000771285 1.00000i \(-0.499754\pi\)
0.000771285 1.00000i \(0.499754\pi\)
\(230\) 1.07944e11 2.54345
\(231\) 8.74135e9 0.201988
\(232\) −9.19917e10 −2.08474
\(233\) 3.05598e10 0.679279 0.339640 0.940556i \(-0.389695\pi\)
0.339640 + 0.940556i \(0.389695\pi\)
\(234\) 8.48675e9 0.185042
\(235\) −1.40123e10 −0.299711
\(236\) 1.10262e10 0.231378
\(237\) 2.46187e10 0.506871
\(238\) 9.10551e9 0.183953
\(239\) 7.82466e10 1.55123 0.775613 0.631209i \(-0.217441\pi\)
0.775613 + 0.631209i \(0.217441\pi\)
\(240\) −9.95671e9 −0.193716
\(241\) 2.17115e10 0.414584 0.207292 0.978279i \(-0.433535\pi\)
0.207292 + 0.978279i \(0.433535\pi\)
\(242\) 2.44224e10 0.457741
\(243\) 3.48678e9 0.0641500
\(244\) −1.75763e11 −3.17449
\(245\) 4.48531e10 0.795326
\(246\) −5.36465e10 −0.933971
\(247\) 2.87565e10 0.491586
\(248\) 5.25710e10 0.882497
\(249\) 5.09251e10 0.839528
\(250\) 1.11018e11 1.79748
\(251\) −1.21004e11 −1.92428 −0.962141 0.272551i \(-0.912133\pi\)
−0.962141 + 0.272551i \(0.912133\pi\)
\(252\) −1.17521e10 −0.183575
\(253\) 1.27635e11 1.95852
\(254\) 9.06482e10 1.36649
\(255\) 1.22172e10 0.180943
\(256\) −1.05302e11 −1.53234
\(257\) 7.96267e10 1.13857 0.569285 0.822140i \(-0.307220\pi\)
0.569285 + 0.822140i \(0.307220\pi\)
\(258\) 6.24296e10 0.877206
\(259\) 1.82137e10 0.251506
\(260\) −3.83795e10 −0.520858
\(261\) −4.02204e10 −0.536493
\(262\) 8.44919e10 1.10779
\(263\) 7.06283e10 0.910286 0.455143 0.890418i \(-0.349588\pi\)
0.455143 + 0.890418i \(0.349588\pi\)
\(264\) −6.66382e10 −0.844317
\(265\) 6.94585e10 0.865205
\(266\) −6.22266e10 −0.762094
\(267\) −6.55061e10 −0.788826
\(268\) 2.71087e11 3.20998
\(269\) −4.68471e10 −0.545503 −0.272752 0.962084i \(-0.587934\pi\)
−0.272752 + 0.962084i \(0.587934\pi\)
\(270\) −2.46405e10 −0.282171
\(271\) −1.82719e10 −0.205789 −0.102894 0.994692i \(-0.532810\pi\)
−0.102894 + 0.994692i \(0.532810\pi\)
\(272\) −1.22635e10 −0.135848
\(273\) −5.46942e9 −0.0595950
\(274\) 1.01152e11 1.08417
\(275\) 2.41929e10 0.255088
\(276\) −1.71596e11 −1.77998
\(277\) −1.08957e10 −0.111198 −0.0555988 0.998453i \(-0.517707\pi\)
−0.0555988 + 0.998453i \(0.517707\pi\)
\(278\) 1.08014e11 1.08462
\(279\) 2.29850e10 0.227104
\(280\) 3.63205e10 0.353135
\(281\) 1.25313e11 1.19899 0.599497 0.800377i \(-0.295367\pi\)
0.599497 + 0.800377i \(0.295367\pi\)
\(282\) 3.48083e10 0.327764
\(283\) 4.91058e10 0.455087 0.227543 0.973768i \(-0.426931\pi\)
0.227543 + 0.973768i \(0.426931\pi\)
\(284\) −3.23887e11 −2.95435
\(285\) −8.34919e10 −0.749622
\(286\) −7.09149e10 −0.626744
\(287\) 3.45733e10 0.300796
\(288\) −2.56757e10 −0.219915
\(289\) −1.03540e11 −0.873110
\(290\) 2.84231e11 2.35983
\(291\) 4.64307e10 0.379565
\(292\) 4.11005e11 3.30845
\(293\) 1.93751e11 1.53581 0.767907 0.640561i \(-0.221298\pi\)
0.767907 + 0.640561i \(0.221298\pi\)
\(294\) −1.11421e11 −0.869770
\(295\) −1.48991e10 −0.114541
\(296\) −1.38849e11 −1.05131
\(297\) −2.91354e10 −0.217279
\(298\) −1.84830e11 −1.35768
\(299\) −7.98607e10 −0.577847
\(300\) −3.25255e10 −0.231835
\(301\) −4.02337e10 −0.282515
\(302\) −4.25471e11 −2.94333
\(303\) −7.32741e10 −0.499412
\(304\) 8.38079e10 0.562799
\(305\) 2.37499e11 1.57149
\(306\) −3.03492e10 −0.197879
\(307\) 1.59101e11 1.02224 0.511118 0.859511i \(-0.329232\pi\)
0.511118 + 0.859511i \(0.329232\pi\)
\(308\) 9.82000e10 0.621775
\(309\) 7.53010e10 0.469881
\(310\) −1.62431e11 −0.998944
\(311\) 1.03591e11 0.627913 0.313957 0.949437i \(-0.398345\pi\)
0.313957 + 0.949437i \(0.398345\pi\)
\(312\) 4.16952e10 0.249110
\(313\) −4.45093e10 −0.262120 −0.131060 0.991374i \(-0.541838\pi\)
−0.131060 + 0.991374i \(0.541838\pi\)
\(314\) −3.72836e11 −2.16439
\(315\) 1.58800e10 0.0908767
\(316\) 2.76565e11 1.56029
\(317\) −9.15058e10 −0.508958 −0.254479 0.967078i \(-0.581904\pi\)
−0.254479 + 0.967078i \(0.581904\pi\)
\(318\) −1.72544e11 −0.946190
\(319\) 3.36080e11 1.81712
\(320\) 2.44382e11 1.30285
\(321\) −4.27247e10 −0.224598
\(322\) 1.72812e11 0.895822
\(323\) −1.02835e11 −0.525690
\(324\) 3.91704e10 0.197472
\(325\) −1.51374e10 −0.0752620
\(326\) 1.23291e11 0.604577
\(327\) 5.18091e9 0.0250577
\(328\) −2.63564e11 −1.25734
\(329\) −2.24328e10 −0.105560
\(330\) 2.05895e11 0.955726
\(331\) 1.49265e11 0.683492 0.341746 0.939792i \(-0.388982\pi\)
0.341746 + 0.939792i \(0.388982\pi\)
\(332\) 5.72091e11 2.58431
\(333\) −6.07072e10 −0.270546
\(334\) 5.38412e11 2.36731
\(335\) −3.66305e11 −1.58906
\(336\) −1.59401e10 −0.0682282
\(337\) −1.51793e11 −0.641085 −0.320543 0.947234i \(-0.603865\pi\)
−0.320543 + 0.947234i \(0.603865\pi\)
\(338\) −3.55511e11 −1.48159
\(339\) 1.66387e11 0.684261
\(340\) 1.37248e11 0.556993
\(341\) −1.92061e11 −0.769211
\(342\) 2.07405e11 0.819788
\(343\) 1.51242e11 0.589994
\(344\) 3.06715e11 1.18092
\(345\) 2.31868e11 0.881161
\(346\) 4.12200e11 1.54620
\(347\) −2.24014e11 −0.829453 −0.414726 0.909946i \(-0.636123\pi\)
−0.414726 + 0.909946i \(0.636123\pi\)
\(348\) −4.51835e11 −1.65148
\(349\) −5.50922e11 −1.98781 −0.993906 0.110228i \(-0.964842\pi\)
−0.993906 + 0.110228i \(0.964842\pi\)
\(350\) 3.27560e10 0.116677
\(351\) 1.82299e10 0.0641065
\(352\) 2.14545e11 0.744862
\(353\) 4.45986e11 1.52875 0.764373 0.644774i \(-0.223049\pi\)
0.764373 + 0.644774i \(0.223049\pi\)
\(354\) 3.70114e10 0.125262
\(355\) 4.37652e11 1.46252
\(356\) −7.35894e11 −2.42823
\(357\) 1.95590e10 0.0637294
\(358\) −3.84695e10 −0.123778
\(359\) 2.89771e11 0.920723 0.460362 0.887731i \(-0.347720\pi\)
0.460362 + 0.887731i \(0.347720\pi\)
\(360\) −1.21058e11 −0.379869
\(361\) 3.80082e11 1.17786
\(362\) 3.66020e11 1.12025
\(363\) 5.24604e10 0.158581
\(364\) −6.14433e10 −0.183450
\(365\) −5.55369e11 −1.63781
\(366\) −5.89979e11 −1.71859
\(367\) −6.50943e11 −1.87303 −0.936517 0.350623i \(-0.885970\pi\)
−0.936517 + 0.350623i \(0.885970\pi\)
\(368\) −2.32746e11 −0.661556
\(369\) −1.15235e11 −0.323568
\(370\) 4.29008e11 1.19003
\(371\) 1.11199e11 0.304732
\(372\) 2.58212e11 0.699091
\(373\) −1.12418e11 −0.300708 −0.150354 0.988632i \(-0.548041\pi\)
−0.150354 + 0.988632i \(0.548041\pi\)
\(374\) 2.53596e11 0.670225
\(375\) 2.38472e11 0.622724
\(376\) 1.71012e11 0.441247
\(377\) −2.10284e11 −0.536129
\(378\) −3.94480e10 −0.0993829
\(379\) 3.08628e11 0.768349 0.384174 0.923261i \(-0.374486\pi\)
0.384174 + 0.923261i \(0.374486\pi\)
\(380\) −9.37945e11 −2.30755
\(381\) 1.94716e11 0.473412
\(382\) −1.16773e12 −2.80581
\(383\) −2.45330e10 −0.0582582 −0.0291291 0.999576i \(-0.509273\pi\)
−0.0291291 + 0.999576i \(0.509273\pi\)
\(384\) −4.44781e11 −1.04389
\(385\) −1.32692e11 −0.307803
\(386\) −1.25109e12 −2.86843
\(387\) 1.34101e11 0.303902
\(388\) 5.21601e11 1.16841
\(389\) −6.10513e11 −1.35183 −0.675914 0.736980i \(-0.736251\pi\)
−0.675914 + 0.736980i \(0.736251\pi\)
\(390\) −1.28828e11 −0.281980
\(391\) 2.85587e11 0.617935
\(392\) −5.47409e11 −1.17091
\(393\) 1.81492e11 0.383788
\(394\) −5.56208e11 −1.16280
\(395\) −3.73708e11 −0.772406
\(396\) −3.27306e11 −0.668846
\(397\) −4.47627e11 −0.904396 −0.452198 0.891917i \(-0.649360\pi\)
−0.452198 + 0.891917i \(0.649360\pi\)
\(398\) −1.78094e10 −0.0355775
\(399\) −1.33665e11 −0.264022
\(400\) −4.41164e10 −0.0861648
\(401\) −5.80109e11 −1.12037 −0.560183 0.828369i \(-0.689269\pi\)
−0.560183 + 0.828369i \(0.689269\pi\)
\(402\) 9.09951e11 1.73780
\(403\) 1.20172e11 0.226950
\(404\) −8.23159e11 −1.53733
\(405\) −5.29289e10 −0.0977564
\(406\) 4.55036e11 0.831149
\(407\) 5.07267e11 0.916350
\(408\) −1.49105e11 −0.266392
\(409\) 3.49105e11 0.616880 0.308440 0.951244i \(-0.400193\pi\)
0.308440 + 0.951244i \(0.400193\pi\)
\(410\) 8.14346e11 1.42325
\(411\) 2.17280e11 0.375604
\(412\) 8.45929e11 1.44643
\(413\) −2.38526e10 −0.0403422
\(414\) −5.75992e11 −0.963639
\(415\) −7.73036e11 −1.27933
\(416\) −1.34240e11 −0.219766
\(417\) 2.32018e11 0.375759
\(418\) −1.73307e12 −2.77665
\(419\) −1.19183e11 −0.188908 −0.0944541 0.995529i \(-0.530111\pi\)
−0.0944541 + 0.995529i \(0.530111\pi\)
\(420\) 1.78395e11 0.279744
\(421\) −2.91158e11 −0.451709 −0.225855 0.974161i \(-0.572517\pi\)
−0.225855 + 0.974161i \(0.572517\pi\)
\(422\) −1.64939e12 −2.53173
\(423\) 7.47697e10 0.113552
\(424\) −8.47705e11 −1.27379
\(425\) 5.41322e10 0.0804833
\(426\) −1.08718e12 −1.59941
\(427\) 3.80222e11 0.553492
\(428\) −4.79967e11 −0.691376
\(429\) −1.52328e11 −0.217131
\(430\) −9.47672e11 −1.33675
\(431\) −3.30127e11 −0.460822 −0.230411 0.973093i \(-0.574007\pi\)
−0.230411 + 0.973093i \(0.574007\pi\)
\(432\) 5.31292e10 0.0733933
\(433\) −6.15872e11 −0.841966 −0.420983 0.907069i \(-0.638315\pi\)
−0.420983 + 0.907069i \(0.638315\pi\)
\(434\) −2.60042e11 −0.351836
\(435\) 6.10540e11 0.817546
\(436\) 5.82021e10 0.0771347
\(437\) −1.95169e12 −2.56002
\(438\) 1.37961e12 1.79111
\(439\) 1.71665e11 0.220593 0.110297 0.993899i \(-0.464820\pi\)
0.110297 + 0.993899i \(0.464820\pi\)
\(440\) 1.01156e12 1.28663
\(441\) −2.39337e11 −0.301326
\(442\) −1.58674e11 −0.197745
\(443\) 1.14879e12 1.41717 0.708587 0.705623i \(-0.249333\pi\)
0.708587 + 0.705623i \(0.249333\pi\)
\(444\) −6.81982e11 −0.832817
\(445\) 9.94374e11 1.20207
\(446\) −1.93130e11 −0.231123
\(447\) −3.97022e11 −0.470360
\(448\) 3.91240e11 0.458873
\(449\) 4.41175e11 0.512274 0.256137 0.966641i \(-0.417550\pi\)
0.256137 + 0.966641i \(0.417550\pi\)
\(450\) −1.09178e11 −0.125510
\(451\) 9.62897e11 1.09594
\(452\) 1.86919e12 2.10635
\(453\) −9.13930e11 −1.01970
\(454\) 7.86850e11 0.869241
\(455\) 8.30250e10 0.0908150
\(456\) 1.01897e12 1.10363
\(457\) −1.18772e12 −1.27377 −0.636885 0.770959i \(-0.719777\pi\)
−0.636885 + 0.770959i \(0.719777\pi\)
\(458\) 2.42073e9 0.00257071
\(459\) −6.51913e10 −0.0685540
\(460\) 2.60480e12 2.71246
\(461\) 3.70002e11 0.381549 0.190774 0.981634i \(-0.438900\pi\)
0.190774 + 0.981634i \(0.438900\pi\)
\(462\) 3.29625e11 0.336614
\(463\) 1.60791e12 1.62610 0.813049 0.582196i \(-0.197806\pi\)
0.813049 + 0.582196i \(0.197806\pi\)
\(464\) −6.12851e11 −0.613796
\(465\) −3.48908e11 −0.346077
\(466\) 1.15237e12 1.13202
\(467\) −1.16153e12 −1.13007 −0.565035 0.825067i \(-0.691137\pi\)
−0.565035 + 0.825067i \(0.691137\pi\)
\(468\) 2.04794e11 0.197338
\(469\) −5.86432e11 −0.559680
\(470\) −5.28385e11 −0.499471
\(471\) −8.00868e11 −0.749837
\(472\) 1.81836e11 0.168632
\(473\) −1.12054e12 −1.02933
\(474\) 9.28340e11 0.844704
\(475\) −3.69937e11 −0.333432
\(476\) 2.19725e11 0.196177
\(477\) −3.70632e11 −0.327801
\(478\) 2.95058e12 2.58513
\(479\) −1.05602e12 −0.916560 −0.458280 0.888808i \(-0.651534\pi\)
−0.458280 + 0.888808i \(0.651534\pi\)
\(480\) 3.89753e11 0.335123
\(481\) −3.17394e11 −0.270362
\(482\) 8.18713e11 0.690907
\(483\) 3.71207e11 0.310352
\(484\) 5.89338e11 0.488158
\(485\) −7.04811e11 −0.578409
\(486\) 1.31482e11 0.106907
\(487\) −3.50079e11 −0.282023 −0.141012 0.990008i \(-0.545036\pi\)
−0.141012 + 0.990008i \(0.545036\pi\)
\(488\) −2.89855e12 −2.31362
\(489\) 2.64834e11 0.209452
\(490\) 1.69136e12 1.32542
\(491\) −1.76362e12 −1.36942 −0.684711 0.728814i \(-0.740072\pi\)
−0.684711 + 0.728814i \(0.740072\pi\)
\(492\) −1.29454e12 −0.996034
\(493\) 7.51988e11 0.573324
\(494\) 1.08437e12 0.819232
\(495\) 4.42271e11 0.331105
\(496\) 3.50229e11 0.259827
\(497\) 7.00653e11 0.515109
\(498\) 1.92032e12 1.39908
\(499\) −1.33296e12 −0.962418 −0.481209 0.876606i \(-0.659802\pi\)
−0.481209 + 0.876606i \(0.659802\pi\)
\(500\) 2.67898e12 1.91692
\(501\) 1.15653e12 0.820139
\(502\) −4.56292e12 −3.20683
\(503\) −1.31118e12 −0.913285 −0.456643 0.889650i \(-0.650948\pi\)
−0.456643 + 0.889650i \(0.650948\pi\)
\(504\) −1.93807e11 −0.133793
\(505\) 1.11229e12 0.761039
\(506\) 4.81296e12 3.26388
\(507\) −7.63653e11 −0.513287
\(508\) 2.18743e12 1.45730
\(509\) 1.55199e12 1.02485 0.512424 0.858732i \(-0.328748\pi\)
0.512424 + 0.858732i \(0.328748\pi\)
\(510\) 4.60696e11 0.301543
\(511\) −8.89111e11 −0.576849
\(512\) −1.15933e12 −0.745578
\(513\) 4.45514e11 0.284010
\(514\) 3.00263e12 1.89744
\(515\) −1.14306e12 −0.716038
\(516\) 1.50649e12 0.935497
\(517\) −6.24772e11 −0.384604
\(518\) 6.86815e11 0.419137
\(519\) 8.85422e11 0.535670
\(520\) −6.32927e11 −0.379611
\(521\) −1.16071e12 −0.690167 −0.345083 0.938572i \(-0.612149\pi\)
−0.345083 + 0.938572i \(0.612149\pi\)
\(522\) −1.51666e12 −0.894070
\(523\) −1.29386e12 −0.756188 −0.378094 0.925767i \(-0.623420\pi\)
−0.378094 + 0.925767i \(0.623420\pi\)
\(524\) 2.03888e12 1.18141
\(525\) 7.03613e10 0.0404219
\(526\) 2.66331e12 1.51700
\(527\) −4.29743e11 −0.242695
\(528\) −4.43945e11 −0.248586
\(529\) 3.61895e12 2.00924
\(530\) 2.61920e12 1.44187
\(531\) 7.95020e10 0.0433963
\(532\) −1.50159e12 −0.812736
\(533\) −6.02481e11 −0.323348
\(534\) −2.47016e12 −1.31459
\(535\) 6.48554e11 0.342258
\(536\) 4.47057e12 2.33949
\(537\) −8.26341e10 −0.0428820
\(538\) −1.76655e12 −0.909085
\(539\) 1.99989e12 1.02060
\(540\) −5.94601e11 −0.300922
\(541\) −1.05100e12 −0.527492 −0.263746 0.964592i \(-0.584958\pi\)
−0.263746 + 0.964592i \(0.584958\pi\)
\(542\) −6.89011e11 −0.342949
\(543\) 7.86226e11 0.388104
\(544\) 4.80050e11 0.235013
\(545\) −7.86454e10 −0.0381847
\(546\) −2.06245e11 −0.0993155
\(547\) −9.08834e11 −0.434052 −0.217026 0.976166i \(-0.569636\pi\)
−0.217026 + 0.976166i \(0.569636\pi\)
\(548\) 2.44091e12 1.15622
\(549\) −1.26730e12 −0.595393
\(550\) 9.12284e11 0.425107
\(551\) −5.13905e12 −2.37520
\(552\) −2.82983e12 −1.29728
\(553\) −5.98284e11 −0.272047
\(554\) −4.10863e11 −0.185312
\(555\) 9.21526e11 0.412277
\(556\) 2.60648e12 1.15669
\(557\) −2.33074e12 −1.02599 −0.512997 0.858390i \(-0.671465\pi\)
−0.512997 + 0.858390i \(0.671465\pi\)
\(558\) 8.66735e11 0.378471
\(559\) 7.01120e11 0.303696
\(560\) 2.41968e11 0.103971
\(561\) 5.44736e11 0.232195
\(562\) 4.72539e12 1.99813
\(563\) −3.08395e11 −0.129366 −0.0646829 0.997906i \(-0.520604\pi\)
−0.0646829 + 0.997906i \(0.520604\pi\)
\(564\) 8.39960e11 0.349544
\(565\) −2.52574e12 −1.04273
\(566\) 1.85172e12 0.758405
\(567\) −8.47359e10 −0.0344305
\(568\) −5.34131e12 −2.15318
\(569\) −2.54066e12 −1.01611 −0.508056 0.861324i \(-0.669636\pi\)
−0.508056 + 0.861324i \(0.669636\pi\)
\(570\) −3.14838e12 −1.24925
\(571\) −3.83142e12 −1.50833 −0.754167 0.656683i \(-0.771959\pi\)
−0.754167 + 0.656683i \(0.771959\pi\)
\(572\) −1.71125e12 −0.668392
\(573\) −2.50834e12 −0.972054
\(574\) 1.30372e12 0.501280
\(575\) 1.02737e12 0.391940
\(576\) −1.30403e12 −0.493612
\(577\) −3.89311e12 −1.46219 −0.731097 0.682273i \(-0.760991\pi\)
−0.731097 + 0.682273i \(0.760991\pi\)
\(578\) −3.90437e12 −1.45504
\(579\) −2.68738e12 −0.993748
\(580\) 6.85878e12 2.51664
\(581\) −1.23758e12 −0.450590
\(582\) 1.75084e12 0.632549
\(583\) 3.09698e12 1.11027
\(584\) 6.77799e12 2.41125
\(585\) −2.76727e11 −0.0976901
\(586\) 7.30609e12 2.55945
\(587\) −1.18968e12 −0.413578 −0.206789 0.978386i \(-0.566301\pi\)
−0.206789 + 0.978386i \(0.566301\pi\)
\(588\) −2.68870e12 −0.927567
\(589\) 2.93684e12 1.00545
\(590\) −5.61827e11 −0.190884
\(591\) −1.19476e12 −0.402843
\(592\) −9.25014e11 −0.309528
\(593\) 3.51766e12 1.16817 0.584087 0.811691i \(-0.301453\pi\)
0.584087 + 0.811691i \(0.301453\pi\)
\(594\) −1.09866e12 −0.362097
\(595\) −2.96903e11 −0.0971154
\(596\) −4.46013e12 −1.44790
\(597\) −3.82553e10 −0.0123256
\(598\) −3.01145e12 −0.962986
\(599\) 4.65537e12 1.47752 0.738760 0.673968i \(-0.235412\pi\)
0.738760 + 0.673968i \(0.235412\pi\)
\(600\) −5.36387e11 −0.168965
\(601\) −3.04398e12 −0.951713 −0.475857 0.879523i \(-0.657862\pi\)
−0.475857 + 0.879523i \(0.657862\pi\)
\(602\) −1.51716e12 −0.470813
\(603\) 1.95461e12 0.602050
\(604\) −1.02671e13 −3.13892
\(605\) −7.96341e11 −0.241657
\(606\) −2.76308e12 −0.832274
\(607\) 1.65582e12 0.495066 0.247533 0.968879i \(-0.420380\pi\)
0.247533 + 0.968879i \(0.420380\pi\)
\(608\) −3.28064e12 −0.973626
\(609\) 9.77437e11 0.287946
\(610\) 8.95580e12 2.61891
\(611\) 3.90917e11 0.113475
\(612\) −7.32357e11 −0.211029
\(613\) 9.71802e11 0.277975 0.138988 0.990294i \(-0.455615\pi\)
0.138988 + 0.990294i \(0.455615\pi\)
\(614\) 5.99951e12 1.70356
\(615\) 1.74925e12 0.493076
\(616\) 1.61944e12 0.453161
\(617\) −6.68130e12 −1.85600 −0.927999 0.372583i \(-0.878472\pi\)
−0.927999 + 0.372583i \(0.878472\pi\)
\(618\) 2.83951e12 0.783060
\(619\) −4.90937e12 −1.34406 −0.672028 0.740525i \(-0.734577\pi\)
−0.672028 + 0.740525i \(0.734577\pi\)
\(620\) −3.91963e12 −1.06532
\(621\) −1.23725e12 −0.333846
\(622\) 3.90628e12 1.04642
\(623\) 1.59193e12 0.423378
\(624\) 2.77775e11 0.0733435
\(625\) −2.75807e12 −0.723012
\(626\) −1.67839e12 −0.436826
\(627\) −3.72270e12 −0.961953
\(628\) −8.99692e12 −2.30821
\(629\) 1.13502e12 0.289119
\(630\) 5.98814e11 0.151447
\(631\) 5.31433e12 1.33449 0.667246 0.744837i \(-0.267473\pi\)
0.667246 + 0.744837i \(0.267473\pi\)
\(632\) 4.56091e12 1.13717
\(633\) −3.54296e12 −0.877101
\(634\) −3.45057e12 −0.848183
\(635\) −2.95576e12 −0.721419
\(636\) −4.16367e12 −1.00906
\(637\) −1.25132e12 −0.301122
\(638\) 1.26732e13 3.02825
\(639\) −2.33532e12 −0.554105
\(640\) 6.75172e12 1.59076
\(641\) −1.72255e11 −0.0403005 −0.0201503 0.999797i \(-0.506414\pi\)
−0.0201503 + 0.999797i \(0.506414\pi\)
\(642\) −1.61109e12 −0.374294
\(643\) −7.19928e12 −1.66089 −0.830443 0.557104i \(-0.811913\pi\)
−0.830443 + 0.557104i \(0.811913\pi\)
\(644\) 4.17012e12 0.955350
\(645\) −2.03564e12 −0.463108
\(646\) −3.87778e12 −0.876067
\(647\) −3.07938e11 −0.0690866 −0.0345433 0.999403i \(-0.510998\pi\)
−0.0345433 + 0.999403i \(0.510998\pi\)
\(648\) 6.45970e11 0.143921
\(649\) −6.64315e11 −0.146985
\(650\) −5.70812e11 −0.125425
\(651\) −5.58581e11 −0.121891
\(652\) 2.97514e12 0.644752
\(653\) −1.58359e12 −0.340827 −0.170413 0.985373i \(-0.554510\pi\)
−0.170413 + 0.985373i \(0.554510\pi\)
\(654\) 1.95366e11 0.0417588
\(655\) −2.75502e12 −0.584843
\(656\) −1.75587e12 −0.370190
\(657\) 2.96346e12 0.620518
\(658\) −8.45912e11 −0.175917
\(659\) 2.99583e12 0.618776 0.309388 0.950936i \(-0.399876\pi\)
0.309388 + 0.950936i \(0.399876\pi\)
\(660\) 4.96846e12 1.01923
\(661\) 8.52932e10 0.0173783 0.00868916 0.999962i \(-0.497234\pi\)
0.00868916 + 0.999962i \(0.497234\pi\)
\(662\) 5.62861e12 1.13904
\(663\) −3.40839e11 −0.0685075
\(664\) 9.43451e12 1.88349
\(665\) 2.02902e12 0.402336
\(666\) −2.28919e12 −0.450867
\(667\) 1.42718e13 2.79199
\(668\) 1.29924e13 2.52462
\(669\) −4.14852e11 −0.0800711
\(670\) −1.38129e13 −2.64819
\(671\) 1.05895e13 2.01662
\(672\) 6.23970e11 0.118033
\(673\) 5.60383e12 1.05297 0.526487 0.850183i \(-0.323509\pi\)
0.526487 + 0.850183i \(0.323509\pi\)
\(674\) −5.72391e12 −1.06837
\(675\) −2.34518e11 −0.0434820
\(676\) −8.57885e12 −1.58004
\(677\) 5.42123e10 0.00991856 0.00495928 0.999988i \(-0.498421\pi\)
0.00495928 + 0.999988i \(0.498421\pi\)
\(678\) 6.27426e12 1.14033
\(679\) −1.12836e12 −0.203720
\(680\) 2.26339e12 0.405947
\(681\) 1.69019e12 0.301143
\(682\) −7.24240e12 −1.28190
\(683\) 3.94513e12 0.693694 0.346847 0.937922i \(-0.387252\pi\)
0.346847 + 0.937922i \(0.387252\pi\)
\(684\) 5.00489e12 0.874263
\(685\) −3.29827e12 −0.572373
\(686\) 5.70314e12 0.983230
\(687\) 5.19984e9 0.000890604 0
\(688\) 2.04334e12 0.347691
\(689\) −1.93777e12 −0.327579
\(690\) 8.74347e12 1.46846
\(691\) −6.44422e12 −1.07527 −0.537637 0.843176i \(-0.680683\pi\)
−0.537637 + 0.843176i \(0.680683\pi\)
\(692\) 9.94680e12 1.64894
\(693\) 7.08049e11 0.116618
\(694\) −8.44728e12 −1.38229
\(695\) −3.52200e12 −0.572608
\(696\) −7.45132e12 −1.20363
\(697\) 2.15451e12 0.345781
\(698\) −2.07746e13 −3.31271
\(699\) 2.47534e12 0.392182
\(700\) 7.90436e11 0.124430
\(701\) −3.63823e12 −0.569062 −0.284531 0.958667i \(-0.591838\pi\)
−0.284531 + 0.958667i \(0.591838\pi\)
\(702\) 6.87427e11 0.106834
\(703\) −7.75669e12 −1.19778
\(704\) 1.08964e13 1.67188
\(705\) −1.13499e12 −0.173038
\(706\) 1.68176e13 2.54767
\(707\) 1.78071e12 0.268044
\(708\) 8.93123e11 0.133586
\(709\) −3.78886e12 −0.563119 −0.281560 0.959544i \(-0.590852\pi\)
−0.281560 + 0.959544i \(0.590852\pi\)
\(710\) 1.65033e13 2.43730
\(711\) 1.99411e12 0.292642
\(712\) −1.21358e13 −1.76974
\(713\) −8.15601e12 −1.18188
\(714\) 7.37546e11 0.106206
\(715\) 2.31232e12 0.330880
\(716\) −9.28309e11 −0.132003
\(717\) 6.33798e12 0.895601
\(718\) 1.09269e13 1.53439
\(719\) 4.05107e11 0.0565314 0.0282657 0.999600i \(-0.491002\pi\)
0.0282657 + 0.999600i \(0.491002\pi\)
\(720\) −8.06493e11 −0.111842
\(721\) −1.82997e12 −0.252194
\(722\) 1.43324e13 1.96292
\(723\) 1.75863e12 0.239360
\(724\) 8.83243e12 1.19469
\(725\) 2.70519e12 0.363644
\(726\) 1.97822e12 0.264277
\(727\) −6.44546e12 −0.855755 −0.427877 0.903837i \(-0.640738\pi\)
−0.427877 + 0.903837i \(0.640738\pi\)
\(728\) −1.01328e12 −0.133702
\(729\) 2.82430e11 0.0370370
\(730\) −2.09423e13 −2.72942
\(731\) −2.50725e12 −0.324765
\(732\) −1.42368e13 −1.83279
\(733\) −6.47343e12 −0.828260 −0.414130 0.910218i \(-0.635914\pi\)
−0.414130 + 0.910218i \(0.635914\pi\)
\(734\) −2.45462e13 −3.12142
\(735\) 3.63310e12 0.459182
\(736\) 9.11078e12 1.14447
\(737\) −1.63327e13 −2.03917
\(738\) −4.34536e12 −0.539228
\(739\) 8.07204e12 0.995597 0.497799 0.867293i \(-0.334142\pi\)
0.497799 + 0.867293i \(0.334142\pi\)
\(740\) 1.03524e13 1.26911
\(741\) 2.32927e12 0.283817
\(742\) 4.19317e12 0.507838
\(743\) −2.00695e12 −0.241594 −0.120797 0.992677i \(-0.538545\pi\)
−0.120797 + 0.992677i \(0.538545\pi\)
\(744\) 4.25825e12 0.509510
\(745\) 6.02673e12 0.716768
\(746\) −4.23913e12 −0.501132
\(747\) 4.12494e12 0.484702
\(748\) 6.11954e12 0.714762
\(749\) 1.03830e12 0.120546
\(750\) 8.99247e12 1.03777
\(751\) −1.30296e13 −1.49469 −0.747345 0.664436i \(-0.768672\pi\)
−0.747345 + 0.664436i \(0.768672\pi\)
\(752\) 1.13929e12 0.129913
\(753\) −9.80135e12 −1.11099
\(754\) −7.92954e12 −0.893464
\(755\) 1.38733e13 1.55389
\(756\) −9.51920e11 −0.105987
\(757\) −1.01558e13 −1.12405 −0.562024 0.827121i \(-0.689977\pi\)
−0.562024 + 0.827121i \(0.689977\pi\)
\(758\) 1.16380e13 1.28046
\(759\) 1.03384e13 1.13075
\(760\) −1.54679e13 −1.68178
\(761\) 4.34966e11 0.0470137 0.0235069 0.999724i \(-0.492517\pi\)
0.0235069 + 0.999724i \(0.492517\pi\)
\(762\) 7.34250e12 0.788945
\(763\) −1.25906e11 −0.0134489
\(764\) −2.81786e13 −2.99226
\(765\) 9.89594e11 0.104467
\(766\) −9.25111e11 −0.0970877
\(767\) 4.15659e11 0.0433669
\(768\) −8.52942e12 −0.884696
\(769\) 7.77469e12 0.801705 0.400852 0.916143i \(-0.368714\pi\)
0.400852 + 0.916143i \(0.368714\pi\)
\(770\) −5.00367e12 −0.512956
\(771\) 6.44976e12 0.657354
\(772\) −3.01900e13 −3.05904
\(773\) 1.37897e13 1.38914 0.694571 0.719424i \(-0.255594\pi\)
0.694571 + 0.719424i \(0.255594\pi\)
\(774\) 5.05680e12 0.506455
\(775\) −1.54595e12 −0.153935
\(776\) 8.60185e12 0.851558
\(777\) 1.47531e12 0.145207
\(778\) −2.30217e13 −2.25283
\(779\) −1.47238e13 −1.43252
\(780\) −3.10874e12 −0.300718
\(781\) 1.95138e13 1.87678
\(782\) 1.07691e13 1.02979
\(783\) −3.25785e12 −0.309744
\(784\) −3.64685e12 −0.344743
\(785\) 1.21571e13 1.14266
\(786\) 6.84384e12 0.639586
\(787\) 1.68157e13 1.56253 0.781265 0.624200i \(-0.214575\pi\)
0.781265 + 0.624200i \(0.214575\pi\)
\(788\) −1.34219e13 −1.24007
\(789\) 5.72089e12 0.525554
\(790\) −1.40921e13 −1.28722
\(791\) −4.04355e12 −0.367256
\(792\) −5.39769e12 −0.487467
\(793\) −6.62581e12 −0.594990
\(794\) −1.68795e13 −1.50718
\(795\) 5.62614e12 0.499526
\(796\) −4.29759e11 −0.0379416
\(797\) 4.29486e12 0.377040 0.188520 0.982069i \(-0.439631\pi\)
0.188520 + 0.982069i \(0.439631\pi\)
\(798\) −5.04036e12 −0.439995
\(799\) −1.39794e12 −0.121347
\(800\) 1.72692e12 0.149062
\(801\) −5.30600e12 −0.455429
\(802\) −2.18752e13 −1.86710
\(803\) −2.47625e13 −2.10172
\(804\) 2.19580e13 1.85328
\(805\) −5.63487e12 −0.472936
\(806\) 4.53154e12 0.378214
\(807\) −3.79461e12 −0.314946
\(808\) −1.35749e13 −1.12043
\(809\) −1.42375e13 −1.16860 −0.584300 0.811538i \(-0.698631\pi\)
−0.584300 + 0.811538i \(0.698631\pi\)
\(810\) −1.99588e12 −0.162912
\(811\) −6.09093e12 −0.494413 −0.247206 0.968963i \(-0.579513\pi\)
−0.247206 + 0.968963i \(0.579513\pi\)
\(812\) 1.09805e13 0.886379
\(813\) −1.48002e12 −0.118812
\(814\) 1.91284e13 1.52710
\(815\) −4.02014e12 −0.319177
\(816\) −9.93340e11 −0.0784318
\(817\) 1.71344e13 1.34546
\(818\) 1.31643e13 1.02804
\(819\) −4.43023e11 −0.0344072
\(820\) 1.96510e13 1.51783
\(821\) 1.97006e13 1.51334 0.756668 0.653799i \(-0.226826\pi\)
0.756668 + 0.653799i \(0.226826\pi\)
\(822\) 8.19334e12 0.625948
\(823\) 1.67540e13 1.27297 0.636486 0.771289i \(-0.280387\pi\)
0.636486 + 0.771289i \(0.280387\pi\)
\(824\) 1.39504e13 1.05418
\(825\) 1.95962e12 0.147275
\(826\) −8.99451e11 −0.0672307
\(827\) −7.37166e12 −0.548013 −0.274006 0.961728i \(-0.588349\pi\)
−0.274006 + 0.961728i \(0.588349\pi\)
\(828\) −1.38993e13 −1.02767
\(829\) 1.37833e13 1.01358 0.506790 0.862070i \(-0.330832\pi\)
0.506790 + 0.862070i \(0.330832\pi\)
\(830\) −2.91502e13 −2.13202
\(831\) −8.82551e11 −0.0642000
\(832\) −6.81782e12 −0.493277
\(833\) 4.47481e12 0.322012
\(834\) 8.74911e12 0.626205
\(835\) −1.75560e13 −1.24979
\(836\) −4.18206e13 −2.96116
\(837\) 1.86178e12 0.131119
\(838\) −4.49424e12 −0.314817
\(839\) −1.73944e13 −1.21194 −0.605969 0.795488i \(-0.707214\pi\)
−0.605969 + 0.795488i \(0.707214\pi\)
\(840\) 2.94196e12 0.203883
\(841\) 2.30725e13 1.59042
\(842\) −1.09792e13 −0.752777
\(843\) 1.01503e13 0.692240
\(844\) −3.98015e13 −2.69997
\(845\) 1.15921e13 0.782183
\(846\) 2.81947e12 0.189235
\(847\) −1.27489e12 −0.0851135
\(848\) −5.64743e12 −0.375033
\(849\) 3.97757e12 0.262744
\(850\) 2.04126e12 0.134126
\(851\) 2.15414e13 1.40796
\(852\) −2.62349e13 −1.70569
\(853\) 2.90733e13 1.88029 0.940143 0.340781i \(-0.110691\pi\)
0.940143 + 0.340781i \(0.110691\pi\)
\(854\) 1.43377e13 0.922399
\(855\) −6.76284e12 −0.432794
\(856\) −7.91527e12 −0.503887
\(857\) −1.80812e13 −1.14502 −0.572509 0.819898i \(-0.694030\pi\)
−0.572509 + 0.819898i \(0.694030\pi\)
\(858\) −5.74411e12 −0.361851
\(859\) 2.42753e13 1.52123 0.760615 0.649203i \(-0.224897\pi\)
0.760615 + 0.649203i \(0.224897\pi\)
\(860\) −2.28683e13 −1.42558
\(861\) 2.80044e12 0.173665
\(862\) −1.24487e13 −0.767963
\(863\) −1.03192e13 −0.633280 −0.316640 0.948546i \(-0.602555\pi\)
−0.316640 + 0.948546i \(0.602555\pi\)
\(864\) −2.07973e12 −0.126968
\(865\) −1.34406e13 −0.816292
\(866\) −2.32238e13 −1.40314
\(867\) −8.38676e12 −0.504090
\(868\) −6.27508e12 −0.375215
\(869\) −1.66627e13 −0.991190
\(870\) 2.30227e13 1.36245
\(871\) 1.02193e13 0.601642
\(872\) 9.59827e11 0.0562171
\(873\) 3.76089e12 0.219142
\(874\) −7.35957e13 −4.26630
\(875\) −5.79534e12 −0.334228
\(876\) 3.32914e13 1.91013
\(877\) 2.00090e13 1.14216 0.571081 0.820894i \(-0.306524\pi\)
0.571081 + 0.820894i \(0.306524\pi\)
\(878\) 6.47329e12 0.367620
\(879\) 1.56938e13 0.886703
\(880\) 6.73902e12 0.378813
\(881\) 7.75286e12 0.433581 0.216791 0.976218i \(-0.430441\pi\)
0.216791 + 0.976218i \(0.430441\pi\)
\(882\) −9.02511e12 −0.502162
\(883\) −1.02439e13 −0.567078 −0.283539 0.958961i \(-0.591508\pi\)
−0.283539 + 0.958961i \(0.591508\pi\)
\(884\) −3.82897e12 −0.210885
\(885\) −1.20683e12 −0.0661304
\(886\) 4.33194e13 2.36173
\(887\) −1.04930e13 −0.569170 −0.284585 0.958651i \(-0.591856\pi\)
−0.284585 + 0.958651i \(0.591856\pi\)
\(888\) −1.12468e13 −0.606972
\(889\) −4.73199e12 −0.254089
\(890\) 3.74966e13 2.00326
\(891\) −2.35997e12 −0.125446
\(892\) −4.66043e12 −0.246482
\(893\) 9.55349e12 0.502725
\(894\) −1.49712e13 −0.783858
\(895\) 1.25437e12 0.0653467
\(896\) 1.08091e13 0.560277
\(897\) −6.46871e12 −0.333620
\(898\) 1.66361e13 0.853708
\(899\) −2.14759e13 −1.09656
\(900\) −2.63457e12 −0.133850
\(901\) 6.92959e12 0.350305
\(902\) 3.63097e13 1.82639
\(903\) −3.25893e12 −0.163110
\(904\) 3.08253e13 1.53514
\(905\) −1.19348e13 −0.591420
\(906\) −3.44632e13 −1.69933
\(907\) 3.40016e13 1.66827 0.834135 0.551560i \(-0.185967\pi\)
0.834135 + 0.551560i \(0.185967\pi\)
\(908\) 1.89875e13 0.927003
\(909\) −5.93520e12 −0.288336
\(910\) 3.13077e12 0.151344
\(911\) 1.05487e13 0.507419 0.253710 0.967280i \(-0.418349\pi\)
0.253710 + 0.967280i \(0.418349\pi\)
\(912\) 6.78844e12 0.324932
\(913\) −3.44678e13 −1.64170
\(914\) −4.47874e13 −2.12275
\(915\) 1.92374e13 0.907303
\(916\) 5.84148e10 0.00274153
\(917\) −4.41062e12 −0.205986
\(918\) −2.45828e12 −0.114246
\(919\) 1.63341e13 0.755397 0.377698 0.925929i \(-0.376716\pi\)
0.377698 + 0.925929i \(0.376716\pi\)
\(920\) 4.29564e13 1.97689
\(921\) 1.28872e13 0.590188
\(922\) 1.39523e13 0.635854
\(923\) −1.22097e13 −0.553729
\(924\) 7.95420e12 0.358982
\(925\) 4.08311e12 0.183381
\(926\) 6.06322e13 2.70990
\(927\) 6.09938e12 0.271286
\(928\) 2.39899e13 1.06185
\(929\) −2.40994e13 −1.06154 −0.530770 0.847516i \(-0.678097\pi\)
−0.530770 + 0.847516i \(0.678097\pi\)
\(930\) −1.31569e13 −0.576741
\(931\) −3.05806e13 −1.33405
\(932\) 2.78079e13 1.20725
\(933\) 8.39086e12 0.362526
\(934\) −4.37999e13 −1.88327
\(935\) −8.26901e12 −0.353835
\(936\) 3.37731e12 0.143824
\(937\) −1.28907e13 −0.546322 −0.273161 0.961968i \(-0.588069\pi\)
−0.273161 + 0.961968i \(0.588069\pi\)
\(938\) −2.21136e13 −0.932712
\(939\) −3.60525e12 −0.151335
\(940\) −1.27505e13 −0.532661
\(941\) 5.56232e12 0.231261 0.115630 0.993292i \(-0.463111\pi\)
0.115630 + 0.993292i \(0.463111\pi\)
\(942\) −3.01997e13 −1.24961
\(943\) 4.08900e13 1.68390
\(944\) 1.21140e12 0.0496492
\(945\) 1.28628e12 0.0524677
\(946\) −4.22544e13 −1.71538
\(947\) 2.78874e13 1.12677 0.563383 0.826196i \(-0.309500\pi\)
0.563383 + 0.826196i \(0.309500\pi\)
\(948\) 2.24018e13 0.900835
\(949\) 1.54938e13 0.620098
\(950\) −1.39499e13 −0.555666
\(951\) −7.41197e12 −0.293847
\(952\) 3.62355e12 0.142977
\(953\) −1.74771e13 −0.686359 −0.343179 0.939270i \(-0.611504\pi\)
−0.343179 + 0.939270i \(0.611504\pi\)
\(954\) −1.39761e13 −0.546283
\(955\) 3.80762e13 1.48128
\(956\) 7.12006e13 2.75691
\(957\) 2.72225e13 1.04912
\(958\) −3.98211e13 −1.52745
\(959\) −5.28033e12 −0.201594
\(960\) 1.97949e13 0.752200
\(961\) −1.41667e13 −0.535813
\(962\) −1.19686e13 −0.450561
\(963\) −3.46070e12 −0.129672
\(964\) 1.97564e13 0.736819
\(965\) 4.07941e13 1.51434
\(966\) 1.39978e13 0.517203
\(967\) −1.21293e13 −0.446086 −0.223043 0.974809i \(-0.571599\pi\)
−0.223043 + 0.974809i \(0.571599\pi\)
\(968\) 9.71893e12 0.355778
\(969\) −8.32964e12 −0.303507
\(970\) −2.65776e13 −0.963923
\(971\) −2.65153e13 −0.957216 −0.478608 0.878029i \(-0.658858\pi\)
−0.478608 + 0.878029i \(0.658858\pi\)
\(972\) 3.17280e12 0.114011
\(973\) −5.63850e12 −0.201677
\(974\) −1.32010e13 −0.469994
\(975\) −1.22613e12 −0.0434525
\(976\) −1.93102e13 −0.681183
\(977\) −4.50088e13 −1.58042 −0.790209 0.612837i \(-0.790028\pi\)
−0.790209 + 0.612837i \(0.790028\pi\)
\(978\) 9.98656e12 0.349053
\(979\) 4.43367e13 1.54256
\(980\) 4.08141e13 1.41349
\(981\) 4.19654e11 0.0144671
\(982\) −6.65038e13 −2.28215
\(983\) −1.71512e13 −0.585875 −0.292937 0.956132i \(-0.594633\pi\)
−0.292937 + 0.956132i \(0.594633\pi\)
\(984\) −2.13487e13 −0.725927
\(985\) 1.81362e13 0.613881
\(986\) 2.83565e13 0.955448
\(987\) −1.81705e12 −0.0609453
\(988\) 2.61670e13 0.873670
\(989\) −4.75846e13 −1.58155
\(990\) 1.66775e13 0.551789
\(991\) 7.04586e12 0.232061 0.116031 0.993246i \(-0.462983\pi\)
0.116031 + 0.993246i \(0.462983\pi\)
\(992\) −1.37096e13 −0.449493
\(993\) 1.20905e13 0.394614
\(994\) 2.64208e13 0.858434
\(995\) 5.80710e11 0.0187826
\(996\) 4.63394e13 1.49205
\(997\) −1.13340e13 −0.363292 −0.181646 0.983364i \(-0.558142\pi\)
−0.181646 + 0.983364i \(0.558142\pi\)
\(998\) −5.02642e13 −1.60388
\(999\) −4.91728e12 −0.156200
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.1.20 21
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.10.a.a.1.20 21 1.1 even 1 trivial