Properties

Label 177.12.a.d.1.13
Level $177$
Weight $12$
Character 177.1
Self dual yes
Analytic conductor $135.997$
Analytic rank $0$
Dimension $28$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(135.996742959\)
Analytic rank: \(0\)
Dimension: \(28\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-7.28530 q^{2} +243.000 q^{3} -1994.92 q^{4} +6136.55 q^{5} -1770.33 q^{6} -33718.8 q^{7} +29453.9 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-7.28530 q^{2} +243.000 q^{3} -1994.92 q^{4} +6136.55 q^{5} -1770.33 q^{6} -33718.8 q^{7} +29453.9 q^{8} +59049.0 q^{9} -44706.7 q^{10} +984083. q^{11} -484767. q^{12} -2.05640e6 q^{13} +245652. q^{14} +1.49118e6 q^{15} +3.87102e6 q^{16} +6.23478e6 q^{17} -430190. q^{18} +175904. q^{19} -1.22420e7 q^{20} -8.19368e6 q^{21} -7.16934e6 q^{22} +8.59617e6 q^{23} +7.15730e6 q^{24} -1.11708e7 q^{25} +1.49815e7 q^{26} +1.43489e7 q^{27} +6.72665e7 q^{28} -9.86614e7 q^{29} -1.08637e7 q^{30} +3.66929e7 q^{31} -8.85232e7 q^{32} +2.39132e8 q^{33} -4.54223e7 q^{34} -2.06917e8 q^{35} -1.17798e8 q^{36} +5.41747e8 q^{37} -1.28151e6 q^{38} -4.99704e8 q^{39} +1.80746e8 q^{40} -2.76260e8 q^{41} +5.96934e7 q^{42} -4.46177e8 q^{43} -1.96317e9 q^{44} +3.62357e8 q^{45} -6.26257e7 q^{46} +1.06691e8 q^{47} +9.40659e8 q^{48} -8.40367e8 q^{49} +8.13828e7 q^{50} +1.51505e9 q^{51} +4.10235e9 q^{52} -2.52362e8 q^{53} -1.04536e8 q^{54} +6.03888e9 q^{55} -9.93152e8 q^{56} +4.27446e7 q^{57} +7.18778e8 q^{58} +7.14924e8 q^{59} -2.97480e9 q^{60} +1.18084e10 q^{61} -2.67319e8 q^{62} -1.99106e9 q^{63} -7.28294e9 q^{64} -1.26192e10 q^{65} -1.74215e9 q^{66} -3.55842e9 q^{67} -1.24379e10 q^{68} +2.08887e9 q^{69} +1.50746e9 q^{70} -3.31645e9 q^{71} +1.73923e9 q^{72} -8.37143e9 q^{73} -3.94679e9 q^{74} -2.71451e9 q^{75} -3.50915e8 q^{76} -3.31821e10 q^{77} +3.64050e9 q^{78} -2.44696e9 q^{79} +2.37548e10 q^{80} +3.48678e9 q^{81} +2.01264e9 q^{82} +5.29385e10 q^{83} +1.63458e10 q^{84} +3.82601e10 q^{85} +3.25053e9 q^{86} -2.39747e10 q^{87} +2.89851e10 q^{88} +5.01235e10 q^{89} -2.63988e9 q^{90} +6.93393e10 q^{91} -1.71487e10 q^{92} +8.91637e9 q^{93} -7.77278e8 q^{94} +1.07944e9 q^{95} -2.15111e10 q^{96} -9.10022e10 q^{97} +6.12233e9 q^{98} +5.81091e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + O(q^{10}) \) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + 616281q^{10} + 1082362q^{11} + 7099002q^{12} + 503712q^{13} + 1321669q^{14} + 6454566q^{15} + 34870338q^{16} + 13513579q^{17} + 5668704q^{18} + 35971687q^{19} + 96105997q^{20} + 34586919q^{21} - 47598882q^{22} + 61380539q^{23} + 80756433q^{24} + 294744746q^{25} + 62820734q^{26} + 401769396q^{27} + 148068294q^{28} + 322339307q^{29} + 149756283q^{30} + 151247077q^{31} + 466383494q^{32} + 263013966q^{33} + 684479860q^{34} + 960297361q^{35} + 1725057486q^{36} + 863508437q^{37} + 992640509q^{38} + 122402016q^{39} + 3067680252q^{40} + 3081170377q^{41} + 321165567q^{42} + 2554238300q^{43} + 4350123570q^{44} + 1568459538q^{45} - 1987059155q^{46} + 6203398333q^{47} + 8473492134q^{48} + 10327857997q^{49} + 17577682253q^{50} + 3283799697q^{51} + 32137181618q^{52} + 14571770754q^{53} + 1377495072q^{54} + 18251419334q^{55} + 33498842836q^{56} + 8741119941q^{57} + 11860778276q^{58} + 20017880372q^{59} + 23353757271q^{60} + 2761613771q^{61} + 13785829526q^{62} + 8404621317q^{63} + 86547545293q^{64} + 32034985256q^{65} - 11566528326q^{66} + 39381333296q^{67} + 38995496621q^{68} + 14915470977q^{69} + 8551800364q^{70} + 26130020296q^{71} + 19623813219q^{72} + 41382402799q^{73} + 23815315058q^{74} + 71622973278q^{75} + 10611720128q^{76} - 8426124313q^{77} + 15265438362q^{78} + 59825111206q^{79} + 4009687655q^{80} + 97629963228q^{81} - 39592715115q^{82} + 35433122727q^{83} + 35980595442q^{84} - 8950496085q^{85} - 182032360688q^{86} + 78328451601q^{87} - 220003602335q^{88} + 102303043039q^{89} + 36390776769q^{90} - 111146323655q^{91} - 163000203526q^{92} + 36753039711q^{93} - 81314346008q^{94} + 208102168887q^{95} + 113331189042q^{96} - 171891031490q^{97} + 72304707792q^{98} + 63912393738q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −7.28530 −0.160984 −0.0804920 0.996755i \(-0.525649\pi\)
−0.0804920 + 0.996755i \(0.525649\pi\)
\(3\) 243.000 0.577350
\(4\) −1994.92 −0.974084
\(5\) 6136.55 0.878192 0.439096 0.898440i \(-0.355299\pi\)
0.439096 + 0.898440i \(0.355299\pi\)
\(6\) −1770.33 −0.0929441
\(7\) −33718.8 −0.758287 −0.379143 0.925338i \(-0.623781\pi\)
−0.379143 + 0.925338i \(0.623781\pi\)
\(8\) 29453.9 0.317796
\(9\) 59049.0 0.333333
\(10\) −44706.7 −0.141375
\(11\) 984083. 1.84235 0.921174 0.389150i \(-0.127231\pi\)
0.921174 + 0.389150i \(0.127231\pi\)
\(12\) −484767. −0.562388
\(13\) −2.05640e6 −1.53610 −0.768048 0.640393i \(-0.778772\pi\)
−0.768048 + 0.640393i \(0.778772\pi\)
\(14\) 245652. 0.122072
\(15\) 1.49118e6 0.507025
\(16\) 3.87102e6 0.922924
\(17\) 6.23478e6 1.06501 0.532503 0.846428i \(-0.321252\pi\)
0.532503 + 0.846428i \(0.321252\pi\)
\(18\) −430190. −0.0536613
\(19\) 175904. 0.0162979 0.00814893 0.999967i \(-0.497406\pi\)
0.00814893 + 0.999967i \(0.497406\pi\)
\(20\) −1.22420e7 −0.855433
\(21\) −8.19368e6 −0.437797
\(22\) −7.16934e6 −0.296589
\(23\) 8.59617e6 0.278485 0.139242 0.990258i \(-0.455533\pi\)
0.139242 + 0.990258i \(0.455533\pi\)
\(24\) 7.15730e6 0.183480
\(25\) −1.11708e7 −0.228778
\(26\) 1.49815e7 0.247287
\(27\) 1.43489e7 0.192450
\(28\) 6.72665e7 0.738635
\(29\) −9.86614e7 −0.893219 −0.446610 0.894729i \(-0.647369\pi\)
−0.446610 + 0.894729i \(0.647369\pi\)
\(30\) −1.08637e7 −0.0816228
\(31\) 3.66929e7 0.230193 0.115097 0.993354i \(-0.463282\pi\)
0.115097 + 0.993354i \(0.463282\pi\)
\(32\) −8.85232e7 −0.466372
\(33\) 2.39132e8 1.06368
\(34\) −4.54223e7 −0.171449
\(35\) −2.06917e8 −0.665921
\(36\) −1.17798e8 −0.324695
\(37\) 5.41747e8 1.28436 0.642181 0.766553i \(-0.278030\pi\)
0.642181 + 0.766553i \(0.278030\pi\)
\(38\) −1.28151e6 −0.00262369
\(39\) −4.99704e8 −0.886865
\(40\) 1.80746e8 0.279086
\(41\) −2.76260e8 −0.372397 −0.186199 0.982512i \(-0.559617\pi\)
−0.186199 + 0.982512i \(0.559617\pi\)
\(42\) 5.96934e7 0.0704783
\(43\) −4.46177e8 −0.462839 −0.231420 0.972854i \(-0.574337\pi\)
−0.231420 + 0.972854i \(0.574337\pi\)
\(44\) −1.96317e9 −1.79460
\(45\) 3.62357e8 0.292731
\(46\) −6.26257e7 −0.0448316
\(47\) 1.06691e8 0.0678564 0.0339282 0.999424i \(-0.489198\pi\)
0.0339282 + 0.999424i \(0.489198\pi\)
\(48\) 9.40659e8 0.532850
\(49\) −8.40367e8 −0.425002
\(50\) 8.13828e7 0.0368296
\(51\) 1.51505e9 0.614881
\(52\) 4.10235e9 1.49629
\(53\) −2.52362e8 −0.0828910 −0.0414455 0.999141i \(-0.513196\pi\)
−0.0414455 + 0.999141i \(0.513196\pi\)
\(54\) −1.04536e8 −0.0309814
\(55\) 6.03888e9 1.61794
\(56\) −9.93152e8 −0.240980
\(57\) 4.27446e7 0.00940957
\(58\) 7.18778e8 0.143794
\(59\) 7.14924e8 0.130189
\(60\) −2.97480e9 −0.493885
\(61\) 1.18084e10 1.79009 0.895046 0.445974i \(-0.147143\pi\)
0.895046 + 0.445974i \(0.147143\pi\)
\(62\) −2.67319e8 −0.0370574
\(63\) −1.99106e9 −0.252762
\(64\) −7.28294e9 −0.847846
\(65\) −1.26192e10 −1.34899
\(66\) −1.74215e9 −0.171236
\(67\) −3.55842e9 −0.321992 −0.160996 0.986955i \(-0.551471\pi\)
−0.160996 + 0.986955i \(0.551471\pi\)
\(68\) −1.24379e10 −1.03741
\(69\) 2.08887e9 0.160783
\(70\) 1.50746e9 0.107203
\(71\) −3.31645e9 −0.218149 −0.109074 0.994034i \(-0.534789\pi\)
−0.109074 + 0.994034i \(0.534789\pi\)
\(72\) 1.73923e9 0.105932
\(73\) −8.37143e9 −0.472633 −0.236316 0.971676i \(-0.575940\pi\)
−0.236316 + 0.971676i \(0.575940\pi\)
\(74\) −3.94679e9 −0.206762
\(75\) −2.71451e9 −0.132085
\(76\) −3.50915e8 −0.0158755
\(77\) −3.31821e10 −1.39703
\(78\) 3.64050e9 0.142771
\(79\) −2.44696e9 −0.0894700 −0.0447350 0.998999i \(-0.514244\pi\)
−0.0447350 + 0.998999i \(0.514244\pi\)
\(80\) 2.37548e10 0.810505
\(81\) 3.48678e9 0.111111
\(82\) 2.01264e9 0.0599500
\(83\) 5.29385e10 1.47517 0.737585 0.675255i \(-0.235966\pi\)
0.737585 + 0.675255i \(0.235966\pi\)
\(84\) 1.63458e10 0.426451
\(85\) 3.82601e10 0.935280
\(86\) 3.25053e9 0.0745097
\(87\) −2.39747e10 −0.515700
\(88\) 2.89851e10 0.585491
\(89\) 5.01235e10 0.951473 0.475737 0.879588i \(-0.342182\pi\)
0.475737 + 0.879588i \(0.342182\pi\)
\(90\) −2.63988e9 −0.0471250
\(91\) 6.93393e10 1.16480
\(92\) −1.71487e10 −0.271268
\(93\) 8.91637e9 0.132902
\(94\) −7.77278e8 −0.0109238
\(95\) 1.07944e9 0.0143126
\(96\) −2.15111e10 −0.269260
\(97\) −9.10022e10 −1.07599 −0.537994 0.842949i \(-0.680818\pi\)
−0.537994 + 0.842949i \(0.680818\pi\)
\(98\) 6.12233e9 0.0684184
\(99\) 5.81091e10 0.614116
\(100\) 2.22849e10 0.222849
\(101\) 8.70346e10 0.823995 0.411997 0.911185i \(-0.364831\pi\)
0.411997 + 0.911185i \(0.364831\pi\)
\(102\) −1.10376e10 −0.0989860
\(103\) 7.17170e10 0.609561 0.304781 0.952423i \(-0.401417\pi\)
0.304781 + 0.952423i \(0.401417\pi\)
\(104\) −6.05689e10 −0.488165
\(105\) −5.02809e10 −0.384470
\(106\) 1.83854e9 0.0133441
\(107\) −1.58378e11 −1.09165 −0.545825 0.837899i \(-0.683784\pi\)
−0.545825 + 0.837899i \(0.683784\pi\)
\(108\) −2.86250e10 −0.187463
\(109\) 7.66736e10 0.477310 0.238655 0.971104i \(-0.423294\pi\)
0.238655 + 0.971104i \(0.423294\pi\)
\(110\) −4.39951e10 −0.260462
\(111\) 1.31645e11 0.741526
\(112\) −1.30526e11 −0.699841
\(113\) −6.87594e10 −0.351075 −0.175538 0.984473i \(-0.556166\pi\)
−0.175538 + 0.984473i \(0.556166\pi\)
\(114\) −3.11408e8 −0.00151479
\(115\) 5.27508e10 0.244563
\(116\) 1.96822e11 0.870071
\(117\) −1.21428e11 −0.512032
\(118\) −5.20844e9 −0.0209583
\(119\) −2.10229e11 −0.807580
\(120\) 4.39212e10 0.161130
\(121\) 6.83107e11 2.39425
\(122\) −8.60275e10 −0.288176
\(123\) −6.71312e10 −0.215004
\(124\) −7.31995e10 −0.224228
\(125\) −3.68187e11 −1.07910
\(126\) 1.45055e10 0.0406907
\(127\) 3.88670e11 1.04390 0.521952 0.852975i \(-0.325204\pi\)
0.521952 + 0.852975i \(0.325204\pi\)
\(128\) 2.34354e11 0.602861
\(129\) −1.08421e11 −0.267220
\(130\) 9.19346e10 0.217165
\(131\) −4.12904e11 −0.935097 −0.467549 0.883967i \(-0.654863\pi\)
−0.467549 + 0.883967i \(0.654863\pi\)
\(132\) −4.77050e11 −1.03611
\(133\) −5.93127e9 −0.0123584
\(134\) 2.59242e10 0.0518356
\(135\) 8.80529e10 0.169008
\(136\) 1.83639e11 0.338454
\(137\) 3.63576e11 0.643624 0.321812 0.946804i \(-0.395708\pi\)
0.321812 + 0.946804i \(0.395708\pi\)
\(138\) −1.52180e10 −0.0258835
\(139\) 1.27317e11 0.208115 0.104057 0.994571i \(-0.466817\pi\)
0.104057 + 0.994571i \(0.466817\pi\)
\(140\) 4.12785e11 0.648663
\(141\) 2.59260e10 0.0391769
\(142\) 2.41614e10 0.0351185
\(143\) −2.02366e12 −2.83002
\(144\) 2.28580e11 0.307641
\(145\) −6.05441e11 −0.784418
\(146\) 6.09884e10 0.0760863
\(147\) −2.04209e11 −0.245375
\(148\) −1.08074e12 −1.25108
\(149\) 3.81557e11 0.425633 0.212816 0.977092i \(-0.431736\pi\)
0.212816 + 0.977092i \(0.431736\pi\)
\(150\) 1.97760e10 0.0212636
\(151\) 6.92029e10 0.0717383 0.0358691 0.999356i \(-0.488580\pi\)
0.0358691 + 0.999356i \(0.488580\pi\)
\(152\) 5.18106e9 0.00517939
\(153\) 3.68157e11 0.355002
\(154\) 2.41742e11 0.224899
\(155\) 2.25168e11 0.202154
\(156\) 9.96872e11 0.863881
\(157\) 1.78601e12 1.49429 0.747145 0.664661i \(-0.231424\pi\)
0.747145 + 0.664661i \(0.231424\pi\)
\(158\) 1.78268e10 0.0144032
\(159\) −6.13241e10 −0.0478572
\(160\) −5.43228e11 −0.409564
\(161\) −2.89853e11 −0.211171
\(162\) −2.54023e10 −0.0178871
\(163\) 1.34003e12 0.912182 0.456091 0.889933i \(-0.349249\pi\)
0.456091 + 0.889933i \(0.349249\pi\)
\(164\) 5.51118e11 0.362746
\(165\) 1.46745e12 0.934116
\(166\) −3.85673e11 −0.237479
\(167\) −7.41664e11 −0.441842 −0.220921 0.975292i \(-0.570906\pi\)
−0.220921 + 0.975292i \(0.570906\pi\)
\(168\) −2.41336e11 −0.139130
\(169\) 2.43660e12 1.35959
\(170\) −2.78736e11 −0.150565
\(171\) 1.03869e10 0.00543262
\(172\) 8.90089e11 0.450844
\(173\) 3.52135e12 1.72765 0.863825 0.503791i \(-0.168062\pi\)
0.863825 + 0.503791i \(0.168062\pi\)
\(174\) 1.74663e11 0.0830195
\(175\) 3.76667e11 0.173480
\(176\) 3.80941e12 1.70035
\(177\) 1.73727e11 0.0751646
\(178\) −3.65165e11 −0.153172
\(179\) 3.10316e12 1.26215 0.631077 0.775720i \(-0.282613\pi\)
0.631077 + 0.775720i \(0.282613\pi\)
\(180\) −7.22876e11 −0.285144
\(181\) 2.94843e12 1.12813 0.564064 0.825731i \(-0.309237\pi\)
0.564064 + 0.825731i \(0.309237\pi\)
\(182\) −5.05158e11 −0.187514
\(183\) 2.86943e12 1.03351
\(184\) 2.53191e11 0.0885014
\(185\) 3.32446e12 1.12792
\(186\) −6.49584e10 −0.0213951
\(187\) 6.13554e12 1.96211
\(188\) −2.12841e11 −0.0660978
\(189\) −4.83828e11 −0.145932
\(190\) −7.86407e9 −0.00230411
\(191\) 1.24738e12 0.355072 0.177536 0.984114i \(-0.443187\pi\)
0.177536 + 0.984114i \(0.443187\pi\)
\(192\) −1.76975e12 −0.489504
\(193\) 3.29097e12 0.884625 0.442313 0.896861i \(-0.354158\pi\)
0.442313 + 0.896861i \(0.354158\pi\)
\(194\) 6.62979e11 0.173217
\(195\) −3.06646e12 −0.778838
\(196\) 1.67647e12 0.413987
\(197\) 6.40666e12 1.53839 0.769197 0.639012i \(-0.220657\pi\)
0.769197 + 0.639012i \(0.220657\pi\)
\(198\) −4.23342e11 −0.0988629
\(199\) −7.08530e12 −1.60941 −0.804704 0.593676i \(-0.797676\pi\)
−0.804704 + 0.593676i \(0.797676\pi\)
\(200\) −3.29024e11 −0.0727048
\(201\) −8.64695e11 −0.185902
\(202\) −6.34074e11 −0.132650
\(203\) 3.32675e12 0.677316
\(204\) −3.02241e12 −0.598946
\(205\) −1.69528e12 −0.327037
\(206\) −5.22480e11 −0.0981296
\(207\) 5.07595e11 0.0928283
\(208\) −7.96036e12 −1.41770
\(209\) 1.73104e11 0.0300263
\(210\) 3.66312e11 0.0618935
\(211\) 7.38091e12 1.21494 0.607472 0.794341i \(-0.292184\pi\)
0.607472 + 0.794341i \(0.292184\pi\)
\(212\) 5.03444e11 0.0807428
\(213\) −8.05898e11 −0.125948
\(214\) 1.15383e12 0.175738
\(215\) −2.73799e12 −0.406462
\(216\) 4.22632e11 0.0611599
\(217\) −1.23724e12 −0.174552
\(218\) −5.58591e11 −0.0768392
\(219\) −2.03426e12 −0.272875
\(220\) −1.20471e13 −1.57601
\(221\) −1.28212e13 −1.63595
\(222\) −9.59071e11 −0.119374
\(223\) 1.50922e13 1.83263 0.916315 0.400459i \(-0.131149\pi\)
0.916315 + 0.400459i \(0.131149\pi\)
\(224\) 2.98490e12 0.353644
\(225\) −6.59626e11 −0.0762594
\(226\) 5.00933e11 0.0565175
\(227\) 1.46411e13 1.61224 0.806121 0.591751i \(-0.201563\pi\)
0.806121 + 0.591751i \(0.201563\pi\)
\(228\) −8.52723e10 −0.00916571
\(229\) 1.33878e13 1.40480 0.702398 0.711784i \(-0.252113\pi\)
0.702398 + 0.711784i \(0.252113\pi\)
\(230\) −3.84306e11 −0.0393708
\(231\) −8.06326e12 −0.806575
\(232\) −2.90597e12 −0.283861
\(233\) 1.61708e13 1.54268 0.771339 0.636425i \(-0.219587\pi\)
0.771339 + 0.636425i \(0.219587\pi\)
\(234\) 8.84640e11 0.0824289
\(235\) 6.54717e11 0.0595910
\(236\) −1.42622e12 −0.126815
\(237\) −5.94611e11 −0.0516555
\(238\) 1.53159e12 0.130007
\(239\) −3.64028e12 −0.301958 −0.150979 0.988537i \(-0.548243\pi\)
−0.150979 + 0.988537i \(0.548243\pi\)
\(240\) 5.77241e12 0.467945
\(241\) 2.03587e12 0.161308 0.0806540 0.996742i \(-0.474299\pi\)
0.0806540 + 0.996742i \(0.474299\pi\)
\(242\) −4.97664e12 −0.385436
\(243\) 8.47289e11 0.0641500
\(244\) −2.35568e13 −1.74370
\(245\) −5.15696e12 −0.373233
\(246\) 4.89071e11 0.0346122
\(247\) −3.61728e11 −0.0250351
\(248\) 1.08075e12 0.0731544
\(249\) 1.28640e13 0.851689
\(250\) 2.68235e12 0.173718
\(251\) −7.68200e12 −0.486708 −0.243354 0.969937i \(-0.578248\pi\)
−0.243354 + 0.969937i \(0.578248\pi\)
\(252\) 3.97202e12 0.246212
\(253\) 8.45934e12 0.513066
\(254\) −2.83158e12 −0.168052
\(255\) 9.29719e12 0.539984
\(256\) 1.32081e13 0.750795
\(257\) −3.17374e13 −1.76579 −0.882895 0.469571i \(-0.844409\pi\)
−0.882895 + 0.469571i \(0.844409\pi\)
\(258\) 7.89879e11 0.0430182
\(259\) −1.82671e13 −0.973914
\(260\) 2.51743e13 1.31403
\(261\) −5.82586e12 −0.297740
\(262\) 3.00813e12 0.150536
\(263\) −1.09978e13 −0.538949 −0.269474 0.963008i \(-0.586850\pi\)
−0.269474 + 0.963008i \(0.586850\pi\)
\(264\) 7.04338e12 0.338033
\(265\) −1.54864e12 −0.0727943
\(266\) 4.32111e10 0.00198951
\(267\) 1.21800e13 0.549333
\(268\) 7.09877e12 0.313648
\(269\) 5.20193e12 0.225178 0.112589 0.993642i \(-0.464086\pi\)
0.112589 + 0.993642i \(0.464086\pi\)
\(270\) −6.41492e11 −0.0272076
\(271\) −3.29168e13 −1.36800 −0.684001 0.729481i \(-0.739761\pi\)
−0.684001 + 0.729481i \(0.739761\pi\)
\(272\) 2.41350e13 0.982919
\(273\) 1.68494e13 0.672498
\(274\) −2.64876e12 −0.103613
\(275\) −1.09930e13 −0.421489
\(276\) −4.16713e12 −0.156617
\(277\) −4.92559e13 −1.81476 −0.907380 0.420310i \(-0.861921\pi\)
−0.907380 + 0.420310i \(0.861921\pi\)
\(278\) −9.27539e11 −0.0335032
\(279\) 2.16668e12 0.0767311
\(280\) −6.09453e12 −0.211627
\(281\) 4.51468e13 1.53724 0.768621 0.639705i \(-0.220943\pi\)
0.768621 + 0.639705i \(0.220943\pi\)
\(282\) −1.88879e11 −0.00630685
\(283\) −4.94712e13 −1.62004 −0.810022 0.586399i \(-0.800545\pi\)
−0.810022 + 0.586399i \(0.800545\pi\)
\(284\) 6.61607e12 0.212495
\(285\) 2.62305e11 0.00826341
\(286\) 1.47430e13 0.455588
\(287\) 9.31516e12 0.282384
\(288\) −5.22721e12 −0.155457
\(289\) 4.60057e12 0.134237
\(290\) 4.41082e12 0.126279
\(291\) −2.21135e13 −0.621222
\(292\) 1.67004e13 0.460384
\(293\) 3.97795e13 1.07619 0.538093 0.842886i \(-0.319145\pi\)
0.538093 + 0.842886i \(0.319145\pi\)
\(294\) 1.48773e12 0.0395014
\(295\) 4.38717e12 0.114331
\(296\) 1.59566e13 0.408165
\(297\) 1.41205e13 0.354560
\(298\) −2.77976e12 −0.0685200
\(299\) −1.76771e13 −0.427779
\(300\) 5.41524e12 0.128662
\(301\) 1.50446e13 0.350965
\(302\) −5.04164e11 −0.0115487
\(303\) 2.11494e13 0.475734
\(304\) 6.80928e11 0.0150417
\(305\) 7.24626e13 1.57204
\(306\) −2.68214e12 −0.0571496
\(307\) 9.32676e12 0.195195 0.0975977 0.995226i \(-0.468884\pi\)
0.0975977 + 0.995226i \(0.468884\pi\)
\(308\) 6.61958e13 1.36082
\(309\) 1.74272e13 0.351930
\(310\) −1.64042e12 −0.0325435
\(311\) −4.41430e13 −0.860359 −0.430180 0.902743i \(-0.641550\pi\)
−0.430180 + 0.902743i \(0.641550\pi\)
\(312\) −1.47182e13 −0.281842
\(313\) 9.01074e13 1.69538 0.847689 0.530494i \(-0.177993\pi\)
0.847689 + 0.530494i \(0.177993\pi\)
\(314\) −1.30116e13 −0.240557
\(315\) −1.22183e13 −0.221974
\(316\) 4.88150e12 0.0871513
\(317\) −4.48767e13 −0.787400 −0.393700 0.919239i \(-0.628805\pi\)
−0.393700 + 0.919239i \(0.628805\pi\)
\(318\) 4.46765e11 0.00770424
\(319\) −9.70909e13 −1.64562
\(320\) −4.46922e13 −0.744572
\(321\) −3.84858e13 −0.630265
\(322\) 2.11166e12 0.0339952
\(323\) 1.09672e12 0.0173573
\(324\) −6.95587e12 −0.108232
\(325\) 2.29716e13 0.351425
\(326\) −9.76250e12 −0.146847
\(327\) 1.86317e13 0.275575
\(328\) −8.13694e12 −0.118346
\(329\) −3.59751e12 −0.0514546
\(330\) −1.06908e13 −0.150378
\(331\) −4.88286e13 −0.675493 −0.337746 0.941237i \(-0.609665\pi\)
−0.337746 + 0.941237i \(0.609665\pi\)
\(332\) −1.05608e14 −1.43694
\(333\) 3.19896e13 0.428120
\(334\) 5.40325e12 0.0711295
\(335\) −2.18364e13 −0.282771
\(336\) −3.17179e13 −0.404053
\(337\) −7.41414e13 −0.929173 −0.464586 0.885528i \(-0.653797\pi\)
−0.464586 + 0.885528i \(0.653797\pi\)
\(338\) −1.77514e13 −0.218872
\(339\) −1.67085e13 −0.202694
\(340\) −7.63259e13 −0.911041
\(341\) 3.61088e13 0.424096
\(342\) −7.56720e10 −0.000874564 0
\(343\) 9.50093e13 1.08056
\(344\) −1.31417e13 −0.147088
\(345\) 1.28185e13 0.141199
\(346\) −2.56541e13 −0.278124
\(347\) −2.14365e13 −0.228740 −0.114370 0.993438i \(-0.536485\pi\)
−0.114370 + 0.993438i \(0.536485\pi\)
\(348\) 4.78277e13 0.502336
\(349\) −8.64032e13 −0.893285 −0.446642 0.894713i \(-0.647380\pi\)
−0.446642 + 0.894713i \(0.647380\pi\)
\(350\) −2.74413e12 −0.0279274
\(351\) −2.95070e13 −0.295622
\(352\) −8.71142e13 −0.859220
\(353\) −1.26783e14 −1.23112 −0.615558 0.788091i \(-0.711070\pi\)
−0.615558 + 0.788091i \(0.711070\pi\)
\(354\) −1.26565e12 −0.0121003
\(355\) −2.03516e13 −0.191577
\(356\) −9.99926e13 −0.926815
\(357\) −5.10858e13 −0.466256
\(358\) −2.26075e13 −0.203187
\(359\) −7.94965e13 −0.703604 −0.351802 0.936074i \(-0.614431\pi\)
−0.351802 + 0.936074i \(0.614431\pi\)
\(360\) 1.06729e13 0.0930286
\(361\) −1.16459e14 −0.999734
\(362\) −2.14802e13 −0.181610
\(363\) 1.65995e14 1.38232
\(364\) −1.38327e14 −1.13461
\(365\) −5.13717e13 −0.415062
\(366\) −2.09047e13 −0.166379
\(367\) 1.31070e14 1.02764 0.513820 0.857898i \(-0.328230\pi\)
0.513820 + 0.857898i \(0.328230\pi\)
\(368\) 3.32760e13 0.257020
\(369\) −1.63129e13 −0.124132
\(370\) −2.42197e13 −0.181576
\(371\) 8.50937e12 0.0628552
\(372\) −1.77875e13 −0.129458
\(373\) 1.07548e14 0.771264 0.385632 0.922653i \(-0.373983\pi\)
0.385632 + 0.922653i \(0.373983\pi\)
\(374\) −4.46992e13 −0.315869
\(375\) −8.94694e13 −0.623021
\(376\) 3.14248e12 0.0215645
\(377\) 2.02887e14 1.37207
\(378\) 3.52484e12 0.0234928
\(379\) 1.38488e14 0.909695 0.454848 0.890569i \(-0.349694\pi\)
0.454848 + 0.890569i \(0.349694\pi\)
\(380\) −2.15341e12 −0.0139417
\(381\) 9.44469e13 0.602699
\(382\) −9.08757e12 −0.0571610
\(383\) −1.26680e14 −0.785445 −0.392723 0.919657i \(-0.628467\pi\)
−0.392723 + 0.919657i \(0.628467\pi\)
\(384\) 5.69480e13 0.348062
\(385\) −2.03624e14 −1.22686
\(386\) −2.39757e13 −0.142410
\(387\) −2.63463e13 −0.154280
\(388\) 1.81543e14 1.04810
\(389\) −9.22822e13 −0.525285 −0.262643 0.964893i \(-0.584594\pi\)
−0.262643 + 0.964893i \(0.584594\pi\)
\(390\) 2.23401e13 0.125380
\(391\) 5.35952e13 0.296588
\(392\) −2.47521e13 −0.135064
\(393\) −1.00336e14 −0.539879
\(394\) −4.66745e13 −0.247657
\(395\) −1.50159e13 −0.0785719
\(396\) −1.15923e14 −0.598201
\(397\) −1.42869e14 −0.727094 −0.363547 0.931576i \(-0.618434\pi\)
−0.363547 + 0.931576i \(0.618434\pi\)
\(398\) 5.16185e13 0.259089
\(399\) −1.44130e12 −0.00713515
\(400\) −4.32425e13 −0.211145
\(401\) −1.62373e14 −0.782021 −0.391011 0.920386i \(-0.627874\pi\)
−0.391011 + 0.920386i \(0.627874\pi\)
\(402\) 6.29957e12 0.0299273
\(403\) −7.54551e13 −0.353599
\(404\) −1.73627e14 −0.802640
\(405\) 2.13968e13 0.0975769
\(406\) −2.42364e13 −0.109037
\(407\) 5.33124e14 2.36624
\(408\) 4.46242e13 0.195407
\(409\) −3.19251e14 −1.37928 −0.689642 0.724150i \(-0.742232\pi\)
−0.689642 + 0.724150i \(0.742232\pi\)
\(410\) 1.23507e13 0.0526476
\(411\) 8.83490e13 0.371596
\(412\) −1.43070e14 −0.593764
\(413\) −2.41064e13 −0.0987205
\(414\) −3.69798e12 −0.0149439
\(415\) 3.24860e14 1.29548
\(416\) 1.82039e14 0.716392
\(417\) 3.09379e13 0.120155
\(418\) −1.26111e12 −0.00483376
\(419\) −1.20423e14 −0.455548 −0.227774 0.973714i \(-0.573145\pi\)
−0.227774 + 0.973714i \(0.573145\pi\)
\(420\) 1.00307e14 0.374506
\(421\) 3.74027e14 1.37833 0.689163 0.724607i \(-0.257978\pi\)
0.689163 + 0.724607i \(0.257978\pi\)
\(422\) −5.37721e13 −0.195586
\(423\) 6.30001e12 0.0226188
\(424\) −7.43307e12 −0.0263424
\(425\) −6.96476e13 −0.243650
\(426\) 5.87121e12 0.0202756
\(427\) −3.98164e14 −1.35740
\(428\) 3.15952e14 1.06336
\(429\) −4.91750e14 −1.63391
\(430\) 1.99471e13 0.0654339
\(431\) 2.55097e14 0.826191 0.413096 0.910688i \(-0.364447\pi\)
0.413096 + 0.910688i \(0.364447\pi\)
\(432\) 5.55450e13 0.177617
\(433\) −5.01589e13 −0.158367 −0.0791835 0.996860i \(-0.525231\pi\)
−0.0791835 + 0.996860i \(0.525231\pi\)
\(434\) 9.01368e12 0.0281001
\(435\) −1.47122e14 −0.452884
\(436\) −1.52958e14 −0.464940
\(437\) 1.51210e12 0.00453871
\(438\) 1.48202e13 0.0439284
\(439\) 7.04081e13 0.206095 0.103048 0.994676i \(-0.467141\pi\)
0.103048 + 0.994676i \(0.467141\pi\)
\(440\) 1.77869e14 0.514174
\(441\) −4.96228e13 −0.141667
\(442\) 9.34061e13 0.263362
\(443\) 1.17079e14 0.326030 0.163015 0.986624i \(-0.447878\pi\)
0.163015 + 0.986624i \(0.447878\pi\)
\(444\) −2.62621e14 −0.722309
\(445\) 3.07586e14 0.835576
\(446\) −1.09951e14 −0.295024
\(447\) 9.27184e13 0.245739
\(448\) 2.45572e14 0.642910
\(449\) −5.74915e13 −0.148679 −0.0743393 0.997233i \(-0.523685\pi\)
−0.0743393 + 0.997233i \(0.523685\pi\)
\(450\) 4.80557e12 0.0122765
\(451\) −2.71863e14 −0.686086
\(452\) 1.37170e14 0.341977
\(453\) 1.68163e13 0.0414181
\(454\) −1.06665e14 −0.259545
\(455\) 4.25504e14 1.02292
\(456\) 1.25900e12 0.00299032
\(457\) 2.90433e14 0.681565 0.340783 0.940142i \(-0.389308\pi\)
0.340783 + 0.940142i \(0.389308\pi\)
\(458\) −9.75340e13 −0.226150
\(459\) 8.94623e13 0.204960
\(460\) −1.05234e14 −0.238225
\(461\) −4.64431e14 −1.03888 −0.519441 0.854506i \(-0.673860\pi\)
−0.519441 + 0.854506i \(0.673860\pi\)
\(462\) 5.87433e13 0.129846
\(463\) 1.51143e14 0.330135 0.165068 0.986282i \(-0.447216\pi\)
0.165068 + 0.986282i \(0.447216\pi\)
\(464\) −3.81921e14 −0.824374
\(465\) 5.47158e13 0.116714
\(466\) −1.17810e14 −0.248346
\(467\) 5.53556e14 1.15324 0.576619 0.817013i \(-0.304372\pi\)
0.576619 + 0.817013i \(0.304372\pi\)
\(468\) 2.42240e14 0.498762
\(469\) 1.19986e14 0.244162
\(470\) −4.76981e12 −0.00959319
\(471\) 4.33999e14 0.862729
\(472\) 2.10573e13 0.0413735
\(473\) −4.39075e14 −0.852711
\(474\) 4.33192e12 0.00831571
\(475\) −1.96499e12 −0.00372860
\(476\) 4.19392e14 0.786650
\(477\) −1.49018e13 −0.0276303
\(478\) 2.65205e13 0.0486104
\(479\) 9.79817e14 1.77541 0.887707 0.460409i \(-0.152297\pi\)
0.887707 + 0.460409i \(0.152297\pi\)
\(480\) −1.32004e14 −0.236462
\(481\) −1.11405e15 −1.97290
\(482\) −1.48319e13 −0.0259680
\(483\) −7.04342e13 −0.121920
\(484\) −1.36275e15 −2.33220
\(485\) −5.58440e14 −0.944925
\(486\) −6.17275e12 −0.0103271
\(487\) 1.35144e14 0.223557 0.111778 0.993733i \(-0.464345\pi\)
0.111778 + 0.993733i \(0.464345\pi\)
\(488\) 3.47802e14 0.568884
\(489\) 3.25626e14 0.526648
\(490\) 3.75700e13 0.0600845
\(491\) 4.49871e14 0.711443 0.355721 0.934592i \(-0.384235\pi\)
0.355721 + 0.934592i \(0.384235\pi\)
\(492\) 1.33922e14 0.209432
\(493\) −6.15132e14 −0.951284
\(494\) 2.63530e12 0.00403024
\(495\) 3.56590e14 0.539312
\(496\) 1.42039e14 0.212451
\(497\) 1.11827e14 0.165419
\(498\) −9.37185e13 −0.137108
\(499\) −7.83753e14 −1.13403 −0.567017 0.823706i \(-0.691903\pi\)
−0.567017 + 0.823706i \(0.691903\pi\)
\(500\) 7.34505e14 1.05114
\(501\) −1.80224e14 −0.255098
\(502\) 5.59657e13 0.0783522
\(503\) 2.94819e14 0.408255 0.204127 0.978944i \(-0.434564\pi\)
0.204127 + 0.978944i \(0.434564\pi\)
\(504\) −5.86446e13 −0.0803268
\(505\) 5.34093e14 0.723626
\(506\) −6.16288e13 −0.0825954
\(507\) 5.92094e14 0.784959
\(508\) −7.75368e14 −1.01685
\(509\) −1.18940e15 −1.54306 −0.771528 0.636195i \(-0.780507\pi\)
−0.771528 + 0.636195i \(0.780507\pi\)
\(510\) −6.77329e13 −0.0869288
\(511\) 2.82275e14 0.358391
\(512\) −5.76182e14 −0.723727
\(513\) 2.52403e12 0.00313652
\(514\) 2.31216e14 0.284264
\(515\) 4.40095e14 0.535312
\(516\) 2.16292e14 0.260295
\(517\) 1.04993e14 0.125015
\(518\) 1.33081e14 0.156785
\(519\) 8.55689e14 0.997460
\(520\) −3.71685e14 −0.428703
\(521\) 1.62699e15 1.85685 0.928427 0.371515i \(-0.121161\pi\)
0.928427 + 0.371515i \(0.121161\pi\)
\(522\) 4.24431e13 0.0479313
\(523\) 5.12180e14 0.572352 0.286176 0.958177i \(-0.407616\pi\)
0.286176 + 0.958177i \(0.407616\pi\)
\(524\) 8.23712e14 0.910864
\(525\) 9.15301e13 0.100158
\(526\) 8.01220e13 0.0867621
\(527\) 2.28772e14 0.245157
\(528\) 9.25686e14 0.981696
\(529\) −8.78916e14 −0.922446
\(530\) 1.12823e13 0.0117187
\(531\) 4.22156e13 0.0433963
\(532\) 1.18324e13 0.0120382
\(533\) 5.68100e14 0.572038
\(534\) −8.87351e13 −0.0884338
\(535\) −9.71894e14 −0.958679
\(536\) −1.04809e14 −0.102328
\(537\) 7.54068e14 0.728705
\(538\) −3.78976e13 −0.0362501
\(539\) −8.26990e14 −0.783001
\(540\) −1.75659e14 −0.164628
\(541\) 4.66542e14 0.432819 0.216409 0.976303i \(-0.430565\pi\)
0.216409 + 0.976303i \(0.430565\pi\)
\(542\) 2.39809e14 0.220226
\(543\) 7.16468e14 0.651325
\(544\) −5.51923e14 −0.496689
\(545\) 4.70512e14 0.419170
\(546\) −1.22753e14 −0.108261
\(547\) 3.44855e14 0.301097 0.150548 0.988603i \(-0.451896\pi\)
0.150548 + 0.988603i \(0.451896\pi\)
\(548\) −7.25307e14 −0.626944
\(549\) 6.97272e14 0.596697
\(550\) 8.00874e13 0.0678530
\(551\) −1.73549e13 −0.0145576
\(552\) 6.15254e13 0.0510963
\(553\) 8.25086e13 0.0678439
\(554\) 3.58844e14 0.292147
\(555\) 8.07844e14 0.651203
\(556\) −2.53987e14 −0.202721
\(557\) −4.50069e14 −0.355693 −0.177846 0.984058i \(-0.556913\pi\)
−0.177846 + 0.984058i \(0.556913\pi\)
\(558\) −1.57849e13 −0.0123525
\(559\) 9.17516e14 0.710965
\(560\) −8.00983e14 −0.614595
\(561\) 1.49094e15 1.13283
\(562\) −3.28908e14 −0.247471
\(563\) 1.65669e15 1.23437 0.617186 0.786817i \(-0.288273\pi\)
0.617186 + 0.786817i \(0.288273\pi\)
\(564\) −5.17204e13 −0.0381616
\(565\) −4.21946e14 −0.308312
\(566\) 3.60413e14 0.260801
\(567\) −1.17570e14 −0.0842541
\(568\) −9.76825e13 −0.0693268
\(569\) −1.00863e15 −0.708946 −0.354473 0.935066i \(-0.615340\pi\)
−0.354473 + 0.935066i \(0.615340\pi\)
\(570\) −1.91097e12 −0.00133028
\(571\) 1.20071e15 0.827828 0.413914 0.910316i \(-0.364161\pi\)
0.413914 + 0.910316i \(0.364161\pi\)
\(572\) 4.03705e15 2.75668
\(573\) 3.03114e14 0.205001
\(574\) −6.78638e13 −0.0454593
\(575\) −9.60262e13 −0.0637113
\(576\) −4.30050e14 −0.282615
\(577\) 9.68634e14 0.630511 0.315256 0.949007i \(-0.397910\pi\)
0.315256 + 0.949007i \(0.397910\pi\)
\(578\) −3.35165e13 −0.0216100
\(579\) 7.99707e14 0.510738
\(580\) 1.20781e15 0.764090
\(581\) −1.78502e15 −1.11860
\(582\) 1.61104e14 0.100007
\(583\) −2.48346e14 −0.152714
\(584\) −2.46571e14 −0.150201
\(585\) −7.45150e14 −0.449662
\(586\) −2.89806e14 −0.173249
\(587\) −2.05902e15 −1.21942 −0.609708 0.792626i \(-0.708713\pi\)
−0.609708 + 0.792626i \(0.708713\pi\)
\(588\) 4.07382e14 0.239016
\(589\) 6.45442e12 0.00375165
\(590\) −3.19619e13 −0.0184054
\(591\) 1.55682e15 0.888192
\(592\) 2.09712e15 1.18537
\(593\) −2.19888e15 −1.23140 −0.615702 0.787979i \(-0.711127\pi\)
−0.615702 + 0.787979i \(0.711127\pi\)
\(594\) −1.02872e14 −0.0570785
\(595\) −1.29008e15 −0.709210
\(596\) −7.61177e14 −0.414602
\(597\) −1.72173e15 −0.929192
\(598\) 1.28783e14 0.0688656
\(599\) −1.59947e15 −0.847481 −0.423740 0.905784i \(-0.639283\pi\)
−0.423740 + 0.905784i \(0.639283\pi\)
\(600\) −7.99529e13 −0.0419762
\(601\) −2.27264e15 −1.18228 −0.591142 0.806568i \(-0.701323\pi\)
−0.591142 + 0.806568i \(0.701323\pi\)
\(602\) −1.09604e14 −0.0564997
\(603\) −2.10121e14 −0.107331
\(604\) −1.38055e14 −0.0698791
\(605\) 4.19192e15 2.10261
\(606\) −1.54080e14 −0.0765855
\(607\) 6.48922e14 0.319635 0.159818 0.987147i \(-0.448909\pi\)
0.159818 + 0.987147i \(0.448909\pi\)
\(608\) −1.55716e13 −0.00760086
\(609\) 8.08399e14 0.391049
\(610\) −5.27912e14 −0.253074
\(611\) −2.19399e14 −0.104234
\(612\) −7.34446e14 −0.345802
\(613\) −2.15085e15 −1.00364 −0.501819 0.864973i \(-0.667336\pi\)
−0.501819 + 0.864973i \(0.667336\pi\)
\(614\) −6.79483e13 −0.0314233
\(615\) −4.11954e14 −0.188815
\(616\) −9.77344e14 −0.443970
\(617\) 1.76381e15 0.794114 0.397057 0.917794i \(-0.370031\pi\)
0.397057 + 0.917794i \(0.370031\pi\)
\(618\) −1.26963e14 −0.0566552
\(619\) −1.42359e15 −0.629630 −0.314815 0.949153i \(-0.601943\pi\)
−0.314815 + 0.949153i \(0.601943\pi\)
\(620\) −4.49193e14 −0.196915
\(621\) 1.23346e14 0.0535944
\(622\) 3.21595e14 0.138504
\(623\) −1.69011e15 −0.721489
\(624\) −1.93437e15 −0.818509
\(625\) −1.71395e15 −0.718882
\(626\) −6.56460e14 −0.272929
\(627\) 4.20642e13 0.0173357
\(628\) −3.56295e15 −1.45556
\(629\) 3.37767e15 1.36785
\(630\) 8.90138e13 0.0357342
\(631\) 1.68693e12 0.000671327 0 0.000335664 1.00000i \(-0.499893\pi\)
0.000335664 1.00000i \(0.499893\pi\)
\(632\) −7.20725e13 −0.0284332
\(633\) 1.79356e15 0.701448
\(634\) 3.26941e14 0.126759
\(635\) 2.38510e15 0.916749
\(636\) 1.22337e14 0.0466169
\(637\) 1.72813e15 0.652843
\(638\) 7.07337e14 0.264919
\(639\) −1.95833e14 −0.0727162
\(640\) 1.43813e15 0.529428
\(641\) 1.34445e15 0.490709 0.245355 0.969433i \(-0.421096\pi\)
0.245355 + 0.969433i \(0.421096\pi\)
\(642\) 2.80381e14 0.101463
\(643\) −1.23473e15 −0.443008 −0.221504 0.975159i \(-0.571097\pi\)
−0.221504 + 0.975159i \(0.571097\pi\)
\(644\) 5.78234e14 0.205699
\(645\) −6.65331e14 −0.234671
\(646\) −7.98995e12 −0.00279425
\(647\) 9.70146e14 0.336406 0.168203 0.985752i \(-0.446204\pi\)
0.168203 + 0.985752i \(0.446204\pi\)
\(648\) 1.02700e14 0.0353107
\(649\) 7.03545e14 0.239853
\(650\) −1.67355e14 −0.0565739
\(651\) −3.00650e14 −0.100778
\(652\) −2.67325e15 −0.888542
\(653\) −4.52761e15 −1.49227 −0.746133 0.665796i \(-0.768092\pi\)
−0.746133 + 0.665796i \(0.768092\pi\)
\(654\) −1.35738e14 −0.0443631
\(655\) −2.53381e15 −0.821195
\(656\) −1.06941e15 −0.343695
\(657\) −4.94324e14 −0.157544
\(658\) 2.62089e13 0.00828336
\(659\) 3.99949e15 1.25353 0.626766 0.779208i \(-0.284378\pi\)
0.626766 + 0.779208i \(0.284378\pi\)
\(660\) −2.92745e15 −0.909908
\(661\) 1.04030e15 0.320666 0.160333 0.987063i \(-0.448743\pi\)
0.160333 + 0.987063i \(0.448743\pi\)
\(662\) 3.55731e14 0.108743
\(663\) −3.11554e15 −0.944516
\(664\) 1.55925e15 0.468803
\(665\) −3.63976e13 −0.0108531
\(666\) −2.33054e14 −0.0689205
\(667\) −8.48110e14 −0.248748
\(668\) 1.47956e15 0.430391
\(669\) 3.66739e15 1.05807
\(670\) 1.59085e14 0.0455216
\(671\) 1.16204e16 3.29797
\(672\) 7.25331e14 0.204176
\(673\) −3.14094e15 −0.876955 −0.438477 0.898742i \(-0.644482\pi\)
−0.438477 + 0.898742i \(0.644482\pi\)
\(674\) 5.40143e14 0.149582
\(675\) −1.60289e14 −0.0440284
\(676\) −4.86084e15 −1.32435
\(677\) 1.53188e15 0.413988 0.206994 0.978342i \(-0.433632\pi\)
0.206994 + 0.978342i \(0.433632\pi\)
\(678\) 1.21727e14 0.0326304
\(679\) 3.06849e15 0.815907
\(680\) 1.12691e15 0.297228
\(681\) 3.55778e15 0.930829
\(682\) −2.63064e14 −0.0682727
\(683\) −3.66393e15 −0.943264 −0.471632 0.881796i \(-0.656335\pi\)
−0.471632 + 0.881796i \(0.656335\pi\)
\(684\) −2.07212e13 −0.00529183
\(685\) 2.23110e15 0.565225
\(686\) −6.92172e14 −0.173953
\(687\) 3.25323e15 0.811060
\(688\) −1.72716e15 −0.427166
\(689\) 5.18957e14 0.127329
\(690\) −9.33863e13 −0.0227307
\(691\) −6.93874e15 −1.67553 −0.837763 0.546034i \(-0.816137\pi\)
−0.837763 + 0.546034i \(0.816137\pi\)
\(692\) −7.02483e15 −1.68288
\(693\) −1.95937e15 −0.465676
\(694\) 1.56171e14 0.0368235
\(695\) 7.81285e14 0.182765
\(696\) −7.06150e14 −0.163888
\(697\) −1.72242e15 −0.396605
\(698\) 6.29474e14 0.143805
\(699\) 3.92952e15 0.890665
\(700\) −7.51422e14 −0.168984
\(701\) 3.04592e15 0.679625 0.339813 0.940493i \(-0.389636\pi\)
0.339813 + 0.940493i \(0.389636\pi\)
\(702\) 2.14968e14 0.0475904
\(703\) 9.52954e13 0.0209323
\(704\) −7.16701e15 −1.56203
\(705\) 1.59096e14 0.0344049
\(706\) 9.23651e14 0.198190
\(707\) −2.93471e15 −0.624824
\(708\) −3.46571e14 −0.0732166
\(709\) 3.77463e15 0.791262 0.395631 0.918410i \(-0.370526\pi\)
0.395631 + 0.918410i \(0.370526\pi\)
\(710\) 1.48267e14 0.0308408
\(711\) −1.44490e14 −0.0298233
\(712\) 1.47633e15 0.302374
\(713\) 3.15418e14 0.0641053
\(714\) 3.72175e14 0.0750598
\(715\) −1.24183e16 −2.48530
\(716\) −6.19057e15 −1.22944
\(717\) −8.84588e14 −0.174335
\(718\) 5.79156e14 0.113269
\(719\) −1.02169e16 −1.98294 −0.991472 0.130321i \(-0.958399\pi\)
−0.991472 + 0.130321i \(0.958399\pi\)
\(720\) 1.40269e15 0.270168
\(721\) −2.41821e15 −0.462222
\(722\) 8.48441e14 0.160941
\(723\) 4.94716e14 0.0931312
\(724\) −5.88189e15 −1.09889
\(725\) 1.10213e15 0.204349
\(726\) −1.20932e15 −0.222531
\(727\) 9.05981e15 1.65455 0.827275 0.561797i \(-0.189890\pi\)
0.827275 + 0.561797i \(0.189890\pi\)
\(728\) 2.04231e15 0.370169
\(729\) 2.05891e14 0.0370370
\(730\) 3.74258e14 0.0668184
\(731\) −2.78181e15 −0.492927
\(732\) −5.72430e15 −1.00673
\(733\) 5.86732e15 1.02416 0.512080 0.858938i \(-0.328875\pi\)
0.512080 + 0.858938i \(0.328875\pi\)
\(734\) −9.54887e14 −0.165434
\(735\) −1.25314e15 −0.215486
\(736\) −7.60960e14 −0.129878
\(737\) −3.50178e15 −0.593222
\(738\) 1.18844e14 0.0199833
\(739\) −1.14387e16 −1.90912 −0.954560 0.298018i \(-0.903674\pi\)
−0.954560 + 0.298018i \(0.903674\pi\)
\(740\) −6.63205e15 −1.09869
\(741\) −8.78999e13 −0.0144540
\(742\) −6.19933e13 −0.0101187
\(743\) 1.91244e15 0.309848 0.154924 0.987926i \(-0.450487\pi\)
0.154924 + 0.987926i \(0.450487\pi\)
\(744\) 2.62622e14 0.0422357
\(745\) 2.34145e15 0.373787
\(746\) −7.83518e14 −0.124161
\(747\) 3.12596e15 0.491723
\(748\) −1.22399e16 −1.91126
\(749\) 5.34032e15 0.827784
\(750\) 6.51812e14 0.100296
\(751\) 8.64165e15 1.32001 0.660005 0.751261i \(-0.270554\pi\)
0.660005 + 0.751261i \(0.270554\pi\)
\(752\) 4.13005e14 0.0626263
\(753\) −1.86673e15 −0.281001
\(754\) −1.47809e15 −0.220881
\(755\) 4.24667e14 0.0630000
\(756\) 9.65201e14 0.142150
\(757\) 7.45030e14 0.108930 0.0544649 0.998516i \(-0.482655\pi\)
0.0544649 + 0.998516i \(0.482655\pi\)
\(758\) −1.00893e15 −0.146446
\(759\) 2.05562e15 0.296219
\(760\) 3.17938e13 0.00454850
\(761\) 1.21503e16 1.72573 0.862865 0.505434i \(-0.168668\pi\)
0.862865 + 0.505434i \(0.168668\pi\)
\(762\) −6.88074e14 −0.0970248
\(763\) −2.58535e15 −0.361938
\(764\) −2.48844e15 −0.345870
\(765\) 2.25922e15 0.311760
\(766\) 9.22904e14 0.126444
\(767\) −1.47017e15 −0.199983
\(768\) 3.20957e15 0.433472
\(769\) 1.05690e16 1.41723 0.708614 0.705596i \(-0.249321\pi\)
0.708614 + 0.705596i \(0.249321\pi\)
\(770\) 1.48346e15 0.197505
\(771\) −7.71218e15 −1.01948
\(772\) −6.56524e15 −0.861699
\(773\) −1.01248e15 −0.131947 −0.0659736 0.997821i \(-0.521015\pi\)
−0.0659736 + 0.997821i \(0.521015\pi\)
\(774\) 1.91941e14 0.0248366
\(775\) −4.09889e14 −0.0526632
\(776\) −2.68037e15 −0.341945
\(777\) −4.43890e15 −0.562289
\(778\) 6.72304e14 0.0845625
\(779\) −4.85952e13 −0.00606928
\(780\) 6.11736e15 0.758654
\(781\) −3.26366e15 −0.401906
\(782\) −3.90457e14 −0.0477459
\(783\) −1.41568e15 −0.171900
\(784\) −3.25308e15 −0.392244
\(785\) 1.09599e16 1.31227
\(786\) 7.30976e14 0.0869118
\(787\) −3.03453e14 −0.0358286 −0.0179143 0.999840i \(-0.505703\pi\)
−0.0179143 + 0.999840i \(0.505703\pi\)
\(788\) −1.27808e16 −1.49852
\(789\) −2.67246e15 −0.311162
\(790\) 1.09395e14 0.0126488
\(791\) 2.31849e15 0.266216
\(792\) 1.71154e15 0.195164
\(793\) −2.42827e16 −2.74975
\(794\) 1.04084e15 0.117050
\(795\) −3.76319e14 −0.0420278
\(796\) 1.41346e16 1.56770
\(797\) 5.23026e14 0.0576106 0.0288053 0.999585i \(-0.490830\pi\)
0.0288053 + 0.999585i \(0.490830\pi\)
\(798\) 1.05003e13 0.00114864
\(799\) 6.65196e14 0.0722674
\(800\) 9.88877e14 0.106696
\(801\) 2.95974e15 0.317158
\(802\) 1.18293e15 0.125893
\(803\) −8.23817e15 −0.870754
\(804\) 1.72500e15 0.181085
\(805\) −1.77870e15 −0.185449
\(806\) 5.49713e14 0.0569237
\(807\) 1.26407e15 0.130007
\(808\) 2.56351e15 0.261862
\(809\) 1.63033e16 1.65409 0.827046 0.562135i \(-0.190020\pi\)
0.827046 + 0.562135i \(0.190020\pi\)
\(810\) −1.55882e14 −0.0157083
\(811\) 7.45401e15 0.746063 0.373031 0.927819i \(-0.378318\pi\)
0.373031 + 0.927819i \(0.378318\pi\)
\(812\) −6.63661e15 −0.659763
\(813\) −7.99878e15 −0.789816
\(814\) −3.88397e15 −0.380927
\(815\) 8.22314e15 0.801071
\(816\) 5.86480e15 0.567489
\(817\) −7.84842e13 −0.00754329
\(818\) 2.32584e15 0.222043
\(819\) 4.09441e15 0.388267
\(820\) 3.38196e15 0.318561
\(821\) 9.56085e15 0.894559 0.447280 0.894394i \(-0.352393\pi\)
0.447280 + 0.894394i \(0.352393\pi\)
\(822\) −6.43649e14 −0.0598211
\(823\) 1.47807e15 0.136457 0.0682285 0.997670i \(-0.478265\pi\)
0.0682285 + 0.997670i \(0.478265\pi\)
\(824\) 2.11235e15 0.193716
\(825\) −2.67130e15 −0.243347
\(826\) 1.75623e14 0.0158924
\(827\) 6.56307e14 0.0589966 0.0294983 0.999565i \(-0.490609\pi\)
0.0294983 + 0.999565i \(0.490609\pi\)
\(828\) −1.01261e15 −0.0904226
\(829\) −6.73203e15 −0.597167 −0.298584 0.954383i \(-0.596514\pi\)
−0.298584 + 0.954383i \(0.596514\pi\)
\(830\) −2.36670e15 −0.208552
\(831\) −1.19692e16 −1.04775
\(832\) 1.49766e16 1.30237
\(833\) −5.23950e15 −0.452629
\(834\) −2.25392e14 −0.0193431
\(835\) −4.55126e15 −0.388022
\(836\) −3.45329e14 −0.0292482
\(837\) 5.26503e14 0.0443007
\(838\) 8.77321e14 0.0733359
\(839\) −8.01856e15 −0.665895 −0.332947 0.942945i \(-0.608043\pi\)
−0.332947 + 0.942945i \(0.608043\pi\)
\(840\) −1.48097e15 −0.122183
\(841\) −2.46644e15 −0.202159
\(842\) −2.72490e15 −0.221888
\(843\) 1.09707e16 0.887527
\(844\) −1.47244e16 −1.18346
\(845\) 1.49523e16 1.19398
\(846\) −4.58975e13 −0.00364126
\(847\) −2.30336e16 −1.81553
\(848\) −9.76901e14 −0.0765021
\(849\) −1.20215e16 −0.935333
\(850\) 5.07404e14 0.0392238
\(851\) 4.65695e15 0.357675
\(852\) 1.60770e15 0.122684
\(853\) 2.59125e16 1.96467 0.982333 0.187139i \(-0.0599215\pi\)
0.982333 + 0.187139i \(0.0599215\pi\)
\(854\) 2.90075e15 0.218520
\(855\) 6.37401e13 0.00477088
\(856\) −4.66485e15 −0.346922
\(857\) −2.30391e16 −1.70244 −0.851218 0.524813i \(-0.824135\pi\)
−0.851218 + 0.524813i \(0.824135\pi\)
\(858\) 3.58255e15 0.263034
\(859\) −9.62049e15 −0.701834 −0.350917 0.936406i \(-0.614130\pi\)
−0.350917 + 0.936406i \(0.614130\pi\)
\(860\) 5.46208e15 0.395928
\(861\) 2.26358e15 0.163034
\(862\) −1.85846e15 −0.133004
\(863\) −1.14684e16 −0.815536 −0.407768 0.913085i \(-0.633693\pi\)
−0.407768 + 0.913085i \(0.633693\pi\)
\(864\) −1.27021e15 −0.0897533
\(865\) 2.16090e16 1.51721
\(866\) 3.65423e14 0.0254945
\(867\) 1.11794e15 0.0775019
\(868\) 2.46820e15 0.170029
\(869\) −2.40801e15 −0.164835
\(870\) 1.07183e15 0.0729071
\(871\) 7.31751e15 0.494611
\(872\) 2.25834e15 0.151687
\(873\) −5.37359e15 −0.358663
\(874\) −1.10161e13 −0.000730659 0
\(875\) 1.24148e16 0.818270
\(876\) 4.05819e15 0.265803
\(877\) −4.36497e15 −0.284108 −0.142054 0.989859i \(-0.545371\pi\)
−0.142054 + 0.989859i \(0.545371\pi\)
\(878\) −5.12944e14 −0.0331780
\(879\) 9.66641e15 0.621336
\(880\) 2.33766e16 1.49323
\(881\) −7.32263e15 −0.464836 −0.232418 0.972616i \(-0.574664\pi\)
−0.232418 + 0.972616i \(0.574664\pi\)
\(882\) 3.61517e14 0.0228061
\(883\) 9.83832e15 0.616790 0.308395 0.951258i \(-0.400208\pi\)
0.308395 + 0.951258i \(0.400208\pi\)
\(884\) 2.55773e16 1.59355
\(885\) 1.06608e15 0.0660090
\(886\) −8.52954e14 −0.0524856
\(887\) 1.98191e16 1.21200 0.606001 0.795464i \(-0.292773\pi\)
0.606001 + 0.795464i \(0.292773\pi\)
\(888\) 3.87745e15 0.235654
\(889\) −1.31055e16 −0.791579
\(890\) −2.24086e15 −0.134514
\(891\) 3.43128e15 0.204705
\(892\) −3.01077e16 −1.78514
\(893\) 1.87674e13 0.00110591
\(894\) −6.75481e14 −0.0395601
\(895\) 1.90427e16 1.10841
\(896\) −7.90214e15 −0.457142
\(897\) −4.29554e15 −0.246979
\(898\) 4.18843e14 0.0239349
\(899\) −3.62017e15 −0.205613
\(900\) 1.31590e15 0.0742831
\(901\) −1.57342e15 −0.0882794
\(902\) 1.98060e15 0.110449
\(903\) 3.65583e15 0.202630
\(904\) −2.02523e15 −0.111570
\(905\) 1.80932e16 0.990713
\(906\) −1.22512e14 −0.00666765
\(907\) 2.93743e16 1.58901 0.794505 0.607257i \(-0.207730\pi\)
0.794505 + 0.607257i \(0.207730\pi\)
\(908\) −2.92078e16 −1.57046
\(909\) 5.13931e15 0.274665
\(910\) −3.09993e15 −0.164674
\(911\) 4.72649e15 0.249567 0.124784 0.992184i \(-0.460176\pi\)
0.124784 + 0.992184i \(0.460176\pi\)
\(912\) 1.65465e14 0.00868432
\(913\) 5.20958e16 2.71778
\(914\) −2.11589e15 −0.109721
\(915\) 1.76084e16 0.907621
\(916\) −2.67076e16 −1.36839
\(917\) 1.39226e16 0.709072
\(918\) −6.51760e14 −0.0329953
\(919\) 1.45271e16 0.731046 0.365523 0.930802i \(-0.380890\pi\)
0.365523 + 0.930802i \(0.380890\pi\)
\(920\) 1.55372e15 0.0777212
\(921\) 2.26640e15 0.112696
\(922\) 3.38352e15 0.167243
\(923\) 6.81994e15 0.335097
\(924\) 1.60856e16 0.785671
\(925\) −6.05176e15 −0.293834
\(926\) −1.10112e15 −0.0531465
\(927\) 4.23482e15 0.203187
\(928\) 8.73382e15 0.416572
\(929\) −2.01875e15 −0.0957185 −0.0478592 0.998854i \(-0.515240\pi\)
−0.0478592 + 0.998854i \(0.515240\pi\)
\(930\) −3.98621e14 −0.0187890
\(931\) −1.47824e14 −0.00692661
\(932\) −3.22596e16 −1.50270
\(933\) −1.07268e16 −0.496729
\(934\) −4.03282e15 −0.185653
\(935\) 3.76511e16 1.72311
\(936\) −3.57653e15 −0.162722
\(937\) −4.90720e15 −0.221955 −0.110978 0.993823i \(-0.535398\pi\)
−0.110978 + 0.993823i \(0.535398\pi\)
\(938\) −8.74132e14 −0.0393062
\(939\) 2.18961e16 0.978827
\(940\) −1.30611e15 −0.0580466
\(941\) −2.30046e15 −0.101642 −0.0508208 0.998708i \(-0.516184\pi\)
−0.0508208 + 0.998708i \(0.516184\pi\)
\(942\) −3.16182e15 −0.138885
\(943\) −2.37478e15 −0.103707
\(944\) 2.76749e15 0.120154
\(945\) −2.96904e15 −0.128157
\(946\) 3.19879e15 0.137273
\(947\) −1.29652e16 −0.553167 −0.276583 0.960990i \(-0.589202\pi\)
−0.276583 + 0.960990i \(0.589202\pi\)
\(948\) 1.18620e15 0.0503168
\(949\) 1.72150e16 0.726009
\(950\) 1.43155e13 0.000600244 0
\(951\) −1.09050e16 −0.454606
\(952\) −6.19208e15 −0.256645
\(953\) −3.89460e16 −1.60491 −0.802457 0.596710i \(-0.796474\pi\)
−0.802457 + 0.596710i \(0.796474\pi\)
\(954\) 1.08564e14 0.00444804
\(955\) 7.65464e15 0.311822
\(956\) 7.26208e15 0.294132
\(957\) −2.35931e16 −0.950100
\(958\) −7.13826e15 −0.285813
\(959\) −1.22594e16 −0.488051
\(960\) −1.08602e16 −0.429879
\(961\) −2.40621e16 −0.947011
\(962\) 8.11617e15 0.317606
\(963\) −9.35205e15 −0.363883
\(964\) −4.06140e15 −0.157128
\(965\) 2.01952e16 0.776871
\(966\) 5.13135e14 0.0196271
\(967\) −6.35337e15 −0.241634 −0.120817 0.992675i \(-0.538551\pi\)
−0.120817 + 0.992675i \(0.538551\pi\)
\(968\) 2.01202e16 0.760882
\(969\) 2.66503e14 0.0100212
\(970\) 4.06841e15 0.152118
\(971\) −4.09349e16 −1.52190 −0.760952 0.648808i \(-0.775268\pi\)
−0.760952 + 0.648808i \(0.775268\pi\)
\(972\) −1.69028e15 −0.0624875
\(973\) −4.29296e15 −0.157811
\(974\) −9.84566e14 −0.0359891
\(975\) 5.58210e15 0.202896
\(976\) 4.57104e16 1.65212
\(977\) 7.17746e14 0.0257959 0.0128980 0.999917i \(-0.495894\pi\)
0.0128980 + 0.999917i \(0.495894\pi\)
\(978\) −2.37229e15 −0.0847820
\(979\) 4.93257e16 1.75295
\(980\) 1.02877e16 0.363560
\(981\) 4.52750e15 0.159103
\(982\) −3.27745e15 −0.114531
\(983\) 2.84140e16 0.987387 0.493694 0.869636i \(-0.335646\pi\)
0.493694 + 0.869636i \(0.335646\pi\)
\(984\) −1.97728e15 −0.0683273
\(985\) 3.93148e16 1.35101
\(986\) 4.48142e15 0.153141
\(987\) −8.74194e14 −0.0297073
\(988\) 7.21620e14 0.0243863
\(989\) −3.83541e15 −0.128894
\(990\) −2.59786e15 −0.0868206
\(991\) −6.22968e15 −0.207043 −0.103522 0.994627i \(-0.533011\pi\)
−0.103522 + 0.994627i \(0.533011\pi\)
\(992\) −3.24817e15 −0.107356
\(993\) −1.18654e16 −0.389996
\(994\) −8.14693e14 −0.0266298
\(995\) −4.34793e16 −1.41337
\(996\) −2.56628e16 −0.829617
\(997\) 5.19909e16 1.67149 0.835745 0.549118i \(-0.185036\pi\)
0.835745 + 0.549118i \(0.185036\pi\)
\(998\) 5.70988e15 0.182561
\(999\) 7.77348e15 0.247175
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 177.12.a.d.1.13 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.d.1.13 28 1.1 even 1 trivial