Properties

Label 177.12.a.d.1.17
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 1.17
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+23.7768 q^{2} +243.000 q^{3} -1482.66 q^{4} +6921.85 q^{5} +5777.76 q^{6} +24486.4 q^{7} -83947.8 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q+23.7768 q^{2} +243.000 q^{3} -1482.66 q^{4} +6921.85 q^{5} +5777.76 q^{6} +24486.4 q^{7} -83947.8 q^{8} +59049.0 q^{9} +164579. q^{10} -888355. q^{11} -360287. q^{12} -914574. q^{13} +582209. q^{14} +1.68201e6 q^{15} +1.04049e6 q^{16} -3.69383e6 q^{17} +1.40400e6 q^{18} +4.55382e6 q^{19} -1.02628e7 q^{20} +5.95021e6 q^{21} -2.11222e7 q^{22} +7.76606e6 q^{23} -2.03993e7 q^{24} -916085. q^{25} -2.17456e7 q^{26} +1.43489e7 q^{27} -3.63052e7 q^{28} +2.05865e8 q^{29} +3.99928e7 q^{30} +3.06950e8 q^{31} +1.96665e8 q^{32} -2.15870e8 q^{33} -8.78273e7 q^{34} +1.69492e8 q^{35} -8.75499e7 q^{36} +2.90102e8 q^{37} +1.08275e8 q^{38} -2.22241e8 q^{39} -5.81075e8 q^{40} -7.24652e7 q^{41} +1.41477e8 q^{42} -5.44136e8 q^{43} +1.31713e9 q^{44} +4.08728e8 q^{45} +1.84652e8 q^{46} -4.59684e8 q^{47} +2.52838e8 q^{48} -1.37774e9 q^{49} -2.17816e7 q^{50} -8.97600e8 q^{51} +1.35601e9 q^{52} +2.43404e9 q^{53} +3.41171e8 q^{54} -6.14907e9 q^{55} -2.05558e9 q^{56} +1.10658e9 q^{57} +4.89480e9 q^{58} +7.14924e8 q^{59} -2.49386e9 q^{60} +1.80781e9 q^{61} +7.29829e9 q^{62} +1.44590e9 q^{63} +2.54513e9 q^{64} -6.33055e9 q^{65} -5.13270e9 q^{66} +1.65711e9 q^{67} +5.47671e9 q^{68} +1.88715e9 q^{69} +4.02996e9 q^{70} +2.44530e10 q^{71} -4.95704e9 q^{72} -3.35655e10 q^{73} +6.89770e9 q^{74} -2.22609e8 q^{75} -6.75179e9 q^{76} -2.17527e10 q^{77} -5.28419e9 q^{78} +2.48737e10 q^{79} +7.20210e9 q^{80} +3.48678e9 q^{81} -1.72299e9 q^{82} +6.34936e9 q^{83} -8.82216e9 q^{84} -2.55681e10 q^{85} -1.29378e10 q^{86} +5.00251e10 q^{87} +7.45755e10 q^{88} +8.37878e10 q^{89} +9.71825e9 q^{90} -2.23947e10 q^{91} -1.15145e10 q^{92} +7.45889e10 q^{93} -1.09298e10 q^{94} +3.15209e10 q^{95} +4.77895e10 q^{96} +1.27019e11 q^{97} -3.27582e10 q^{98} -5.24565e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + O(q^{10}) \) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + 616281q^{10} + 1082362q^{11} + 7099002q^{12} + 503712q^{13} + 1321669q^{14} + 6454566q^{15} + 34870338q^{16} + 13513579q^{17} + 5668704q^{18} + 35971687q^{19} + 96105997q^{20} + 34586919q^{21} - 47598882q^{22} + 61380539q^{23} + 80756433q^{24} + 294744746q^{25} + 62820734q^{26} + 401769396q^{27} + 148068294q^{28} + 322339307q^{29} + 149756283q^{30} + 151247077q^{31} + 466383494q^{32} + 263013966q^{33} + 684479860q^{34} + 960297361q^{35} + 1725057486q^{36} + 863508437q^{37} + 992640509q^{38} + 122402016q^{39} + 3067680252q^{40} + 3081170377q^{41} + 321165567q^{42} + 2554238300q^{43} + 4350123570q^{44} + 1568459538q^{45} - 1987059155q^{46} + 6203398333q^{47} + 8473492134q^{48} + 10327857997q^{49} + 17577682253q^{50} + 3283799697q^{51} + 32137181618q^{52} + 14571770754q^{53} + 1377495072q^{54} + 18251419334q^{55} + 33498842836q^{56} + 8741119941q^{57} + 11860778276q^{58} + 20017880372q^{59} + 23353757271q^{60} + 2761613771q^{61} + 13785829526q^{62} + 8404621317q^{63} + 86547545293q^{64} + 32034985256q^{65} - 11566528326q^{66} + 39381333296q^{67} + 38995496621q^{68} + 14915470977q^{69} + 8551800364q^{70} + 26130020296q^{71} + 19623813219q^{72} + 41382402799q^{73} + 23815315058q^{74} + 71622973278q^{75} + 10611720128q^{76} - 8426124313q^{77} + 15265438362q^{78} + 59825111206q^{79} + 4009687655q^{80} + 97629963228q^{81} - 39592715115q^{82} + 35433122727q^{83} + 35980595442q^{84} - 8950496085q^{85} - 182032360688q^{86} + 78328451601q^{87} - 220003602335q^{88} + 102303043039q^{89} + 36390776769q^{90} - 111146323655q^{91} - 163000203526q^{92} + 36753039711q^{93} - 81314346008q^{94} + 208102168887q^{95} + 113331189042q^{96} - 171891031490q^{97} + 72304707792q^{98} + 63912393738q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 23.7768 0.525398 0.262699 0.964878i \(-0.415387\pi\)
0.262699 + 0.964878i \(0.415387\pi\)
\(3\) 243.000 0.577350
\(4\) −1482.66 −0.723957
\(5\) 6921.85 0.990575 0.495287 0.868729i \(-0.335063\pi\)
0.495287 + 0.868729i \(0.335063\pi\)
\(6\) 5777.76 0.303338
\(7\) 24486.4 0.550664 0.275332 0.961349i \(-0.411212\pi\)
0.275332 + 0.961349i \(0.411212\pi\)
\(8\) −83947.8 −0.905763
\(9\) 59049.0 0.333333
\(10\) 164579. 0.520446
\(11\) −888355. −1.66313 −0.831567 0.555425i \(-0.812556\pi\)
−0.831567 + 0.555425i \(0.812556\pi\)
\(12\) −360287. −0.417977
\(13\) −914574. −0.683172 −0.341586 0.939850i \(-0.610964\pi\)
−0.341586 + 0.939850i \(0.610964\pi\)
\(14\) 582209. 0.289317
\(15\) 1.68201e6 0.571909
\(16\) 1.04049e6 0.248072
\(17\) −3.69383e6 −0.630968 −0.315484 0.948931i \(-0.602167\pi\)
−0.315484 + 0.948931i \(0.602167\pi\)
\(18\) 1.40400e6 0.175133
\(19\) 4.55382e6 0.421921 0.210961 0.977495i \(-0.432341\pi\)
0.210961 + 0.977495i \(0.432341\pi\)
\(20\) −1.02628e7 −0.717134
\(21\) 5.95021e6 0.317926
\(22\) −2.11222e7 −0.873806
\(23\) 7.76606e6 0.251593 0.125796 0.992056i \(-0.459851\pi\)
0.125796 + 0.992056i \(0.459851\pi\)
\(24\) −2.03993e7 −0.522943
\(25\) −916085. −0.0187614
\(26\) −2.17456e7 −0.358937
\(27\) 1.43489e7 0.192450
\(28\) −3.63052e7 −0.398657
\(29\) 2.05865e8 1.86377 0.931886 0.362750i \(-0.118162\pi\)
0.931886 + 0.362750i \(0.118162\pi\)
\(30\) 3.99928e7 0.300479
\(31\) 3.06950e8 1.92566 0.962828 0.270114i \(-0.0870615\pi\)
0.962828 + 0.270114i \(0.0870615\pi\)
\(32\) 1.96665e8 1.03610
\(33\) −2.15870e8 −0.960210
\(34\) −8.78273e7 −0.331509
\(35\) 1.69492e8 0.545474
\(36\) −8.75499e7 −0.241319
\(37\) 2.90102e8 0.687767 0.343884 0.939012i \(-0.388257\pi\)
0.343884 + 0.939012i \(0.388257\pi\)
\(38\) 1.08275e8 0.221676
\(39\) −2.22241e8 −0.394430
\(40\) −5.81075e8 −0.897226
\(41\) −7.24652e7 −0.0976828 −0.0488414 0.998807i \(-0.515553\pi\)
−0.0488414 + 0.998807i \(0.515553\pi\)
\(42\) 1.41477e8 0.167037
\(43\) −5.44136e8 −0.564457 −0.282229 0.959347i \(-0.591074\pi\)
−0.282229 + 0.959347i \(0.591074\pi\)
\(44\) 1.31713e9 1.20404
\(45\) 4.08728e8 0.330192
\(46\) 1.84652e8 0.132186
\(47\) −4.59684e8 −0.292362 −0.146181 0.989258i \(-0.546698\pi\)
−0.146181 + 0.989258i \(0.546698\pi\)
\(48\) 2.52838e8 0.143224
\(49\) −1.37774e9 −0.696769
\(50\) −2.17816e7 −0.00985721
\(51\) −8.97600e8 −0.364290
\(52\) 1.35601e9 0.494588
\(53\) 2.43404e9 0.799485 0.399743 0.916627i \(-0.369100\pi\)
0.399743 + 0.916627i \(0.369100\pi\)
\(54\) 3.41171e8 0.101113
\(55\) −6.14907e9 −1.64746
\(56\) −2.05558e9 −0.498771
\(57\) 1.10658e9 0.243596
\(58\) 4.89480e9 0.979222
\(59\) 7.14924e8 0.130189
\(60\) −2.49386e9 −0.414037
\(61\) 1.80781e9 0.274055 0.137028 0.990567i \(-0.456245\pi\)
0.137028 + 0.990567i \(0.456245\pi\)
\(62\) 7.29829e9 1.01174
\(63\) 1.44590e9 0.183555
\(64\) 2.54513e9 0.296293
\(65\) −6.33055e9 −0.676734
\(66\) −5.13270e9 −0.504492
\(67\) 1.65711e9 0.149948 0.0749738 0.997186i \(-0.476113\pi\)
0.0749738 + 0.997186i \(0.476113\pi\)
\(68\) 5.47671e9 0.456794
\(69\) 1.88715e9 0.145257
\(70\) 4.02996e9 0.286591
\(71\) 2.44530e10 1.60846 0.804231 0.594317i \(-0.202578\pi\)
0.804231 + 0.594317i \(0.202578\pi\)
\(72\) −4.95704e9 −0.301921
\(73\) −3.35655e10 −1.89503 −0.947517 0.319705i \(-0.896416\pi\)
−0.947517 + 0.319705i \(0.896416\pi\)
\(74\) 6.89770e9 0.361351
\(75\) −2.22609e8 −0.0108319
\(76\) −6.75179e9 −0.305453
\(77\) −2.17527e10 −0.915827
\(78\) −5.28419e9 −0.207233
\(79\) 2.48737e10 0.909477 0.454739 0.890625i \(-0.349733\pi\)
0.454739 + 0.890625i \(0.349733\pi\)
\(80\) 7.20210e9 0.245733
\(81\) 3.48678e9 0.111111
\(82\) −1.72299e9 −0.0513223
\(83\) 6.34936e9 0.176930 0.0884648 0.996079i \(-0.471804\pi\)
0.0884648 + 0.996079i \(0.471804\pi\)
\(84\) −8.82216e9 −0.230165
\(85\) −2.55681e10 −0.625021
\(86\) −1.29378e10 −0.296565
\(87\) 5.00251e10 1.07605
\(88\) 7.45755e10 1.50640
\(89\) 8.37878e10 1.59051 0.795254 0.606276i \(-0.207337\pi\)
0.795254 + 0.606276i \(0.207337\pi\)
\(90\) 9.71825e9 0.173482
\(91\) −2.23947e10 −0.376198
\(92\) −1.15145e10 −0.182142
\(93\) 7.45889e10 1.11178
\(94\) −1.09298e10 −0.153606
\(95\) 3.15209e10 0.417944
\(96\) 4.77895e10 0.598192
\(97\) 1.27019e11 1.50184 0.750921 0.660392i \(-0.229610\pi\)
0.750921 + 0.660392i \(0.229610\pi\)
\(98\) −3.27582e10 −0.366081
\(99\) −5.24565e10 −0.554378
\(100\) 1.35825e9 0.0135825
\(101\) 3.15002e10 0.298227 0.149113 0.988820i \(-0.452358\pi\)
0.149113 + 0.988820i \(0.452358\pi\)
\(102\) −2.13420e10 −0.191397
\(103\) 1.13132e11 0.961568 0.480784 0.876839i \(-0.340352\pi\)
0.480784 + 0.876839i \(0.340352\pi\)
\(104\) 7.67765e10 0.618792
\(105\) 4.11864e10 0.314929
\(106\) 5.78736e10 0.420048
\(107\) −2.48504e11 −1.71286 −0.856431 0.516262i \(-0.827323\pi\)
−0.856431 + 0.516262i \(0.827323\pi\)
\(108\) −2.12746e10 −0.139326
\(109\) 2.02030e11 1.25768 0.628840 0.777535i \(-0.283530\pi\)
0.628840 + 0.777535i \(0.283530\pi\)
\(110\) −1.46205e11 −0.865570
\(111\) 7.04948e10 0.397083
\(112\) 2.54778e10 0.136604
\(113\) 1.51738e11 0.774755 0.387377 0.921921i \(-0.373381\pi\)
0.387377 + 0.921921i \(0.373381\pi\)
\(114\) 2.63109e10 0.127985
\(115\) 5.37555e10 0.249221
\(116\) −3.05228e11 −1.34929
\(117\) −5.40047e10 −0.227724
\(118\) 1.69986e10 0.0684009
\(119\) −9.04487e10 −0.347451
\(120\) −1.41201e11 −0.518014
\(121\) 5.03864e11 1.76601
\(122\) 4.29839e10 0.143988
\(123\) −1.76090e10 −0.0563972
\(124\) −4.55104e11 −1.39409
\(125\) −3.44322e11 −1.00916
\(126\) 3.43788e10 0.0964391
\(127\) −2.49462e11 −0.670014 −0.335007 0.942216i \(-0.608739\pi\)
−0.335007 + 0.942216i \(0.608739\pi\)
\(128\) −3.42254e11 −0.880428
\(129\) −1.32225e11 −0.325890
\(130\) −1.50520e11 −0.355554
\(131\) −3.18121e11 −0.720444 −0.360222 0.932867i \(-0.617299\pi\)
−0.360222 + 0.932867i \(0.617299\pi\)
\(132\) 3.20063e11 0.695151
\(133\) 1.11507e11 0.232337
\(134\) 3.94007e10 0.0787821
\(135\) 9.93210e10 0.190636
\(136\) 3.10089e11 0.571508
\(137\) 6.28463e11 1.11254 0.556271 0.831001i \(-0.312232\pi\)
0.556271 + 0.831001i \(0.312232\pi\)
\(138\) 4.48704e10 0.0763177
\(139\) 1.09931e12 1.79696 0.898480 0.439015i \(-0.144673\pi\)
0.898480 + 0.439015i \(0.144673\pi\)
\(140\) −2.51299e11 −0.394900
\(141\) −1.11703e11 −0.168795
\(142\) 5.81413e11 0.845082
\(143\) 8.12467e11 1.13621
\(144\) 6.14397e10 0.0826905
\(145\) 1.42497e12 1.84621
\(146\) −7.98079e11 −0.995646
\(147\) −3.34791e11 −0.402280
\(148\) −4.30124e11 −0.497914
\(149\) −6.56389e11 −0.732212 −0.366106 0.930573i \(-0.619309\pi\)
−0.366106 + 0.930573i \(0.619309\pi\)
\(150\) −5.29292e9 −0.00569106
\(151\) −1.51015e12 −1.56548 −0.782741 0.622348i \(-0.786179\pi\)
−0.782741 + 0.622348i \(0.786179\pi\)
\(152\) −3.82284e11 −0.382161
\(153\) −2.18117e11 −0.210323
\(154\) −5.17208e11 −0.481173
\(155\) 2.12466e12 1.90751
\(156\) 3.29510e11 0.285550
\(157\) −1.32963e12 −1.11246 −0.556229 0.831029i \(-0.687752\pi\)
−0.556229 + 0.831029i \(0.687752\pi\)
\(158\) 5.91417e11 0.477837
\(159\) 5.91472e11 0.461583
\(160\) 1.36128e12 1.02633
\(161\) 1.90163e11 0.138543
\(162\) 8.29045e10 0.0583775
\(163\) 2.23220e12 1.51950 0.759752 0.650213i \(-0.225320\pi\)
0.759752 + 0.650213i \(0.225320\pi\)
\(164\) 1.07442e11 0.0707182
\(165\) −1.49422e12 −0.951160
\(166\) 1.50967e11 0.0929584
\(167\) 6.78703e11 0.404333 0.202167 0.979351i \(-0.435202\pi\)
0.202167 + 0.979351i \(0.435202\pi\)
\(168\) −4.99507e11 −0.287966
\(169\) −9.55715e11 −0.533275
\(170\) −6.07928e11 −0.328385
\(171\) 2.68899e11 0.140640
\(172\) 8.06772e11 0.408643
\(173\) 3.16596e12 1.55329 0.776645 0.629939i \(-0.216920\pi\)
0.776645 + 0.629939i \(0.216920\pi\)
\(174\) 1.18944e12 0.565354
\(175\) −2.24317e10 −0.0103312
\(176\) −9.24323e11 −0.412576
\(177\) 1.73727e11 0.0751646
\(178\) 1.99220e12 0.835649
\(179\) −1.83525e12 −0.746454 −0.373227 0.927740i \(-0.621749\pi\)
−0.373227 + 0.927740i \(0.621749\pi\)
\(180\) −6.06007e11 −0.239045
\(181\) 4.33122e11 0.165721 0.0828606 0.996561i \(-0.473594\pi\)
0.0828606 + 0.996561i \(0.473594\pi\)
\(182\) −5.32473e11 −0.197654
\(183\) 4.39297e11 0.158226
\(184\) −6.51944e11 −0.227883
\(185\) 2.00804e12 0.681285
\(186\) 1.77348e12 0.584126
\(187\) 3.28143e12 1.04938
\(188\) 6.81557e11 0.211658
\(189\) 3.51354e11 0.105975
\(190\) 7.49465e11 0.219587
\(191\) −6.20522e12 −1.76634 −0.883169 0.469054i \(-0.844595\pi\)
−0.883169 + 0.469054i \(0.844595\pi\)
\(192\) 6.18468e11 0.171065
\(193\) 2.56884e12 0.690513 0.345256 0.938508i \(-0.387792\pi\)
0.345256 + 0.938508i \(0.387792\pi\)
\(194\) 3.02010e12 0.789064
\(195\) −1.53832e12 −0.390712
\(196\) 2.04273e12 0.504431
\(197\) 6.44753e12 1.54821 0.774103 0.633060i \(-0.218201\pi\)
0.774103 + 0.633060i \(0.218201\pi\)
\(198\) −1.24725e12 −0.291269
\(199\) 2.26246e12 0.513912 0.256956 0.966423i \(-0.417280\pi\)
0.256956 + 0.966423i \(0.417280\pi\)
\(200\) 7.69034e10 0.0169934
\(201\) 4.02677e11 0.0865723
\(202\) 7.48974e11 0.156688
\(203\) 5.04089e12 1.02631
\(204\) 1.33084e12 0.263730
\(205\) −5.01593e11 −0.0967622
\(206\) 2.68991e12 0.505206
\(207\) 4.58578e11 0.0838642
\(208\) −9.51603e11 −0.169476
\(209\) −4.04541e12 −0.701711
\(210\) 9.79281e11 0.165463
\(211\) 6.14974e12 1.01229 0.506143 0.862450i \(-0.331071\pi\)
0.506143 + 0.862450i \(0.331071\pi\)
\(212\) −3.60886e12 −0.578793
\(213\) 5.94207e12 0.928646
\(214\) −5.90862e12 −0.899933
\(215\) −3.76643e12 −0.559137
\(216\) −1.20456e12 −0.174314
\(217\) 7.51612e12 1.06039
\(218\) 4.80362e12 0.660782
\(219\) −8.15641e12 −1.09410
\(220\) 9.11700e12 1.19269
\(221\) 3.37828e12 0.431060
\(222\) 1.67614e12 0.208626
\(223\) −1.23110e13 −1.49491 −0.747457 0.664310i \(-0.768726\pi\)
−0.747457 + 0.664310i \(0.768726\pi\)
\(224\) 4.81562e12 0.570542
\(225\) −5.40939e10 −0.00625381
\(226\) 3.60785e12 0.407054
\(227\) 1.40310e13 1.54507 0.772534 0.634974i \(-0.218989\pi\)
0.772534 + 0.634974i \(0.218989\pi\)
\(228\) −1.64069e12 −0.176353
\(229\) −1.64351e12 −0.172455 −0.0862277 0.996275i \(-0.527481\pi\)
−0.0862277 + 0.996275i \(0.527481\pi\)
\(230\) 1.27813e12 0.130940
\(231\) −5.28590e12 −0.528753
\(232\) −1.72819e13 −1.68814
\(233\) −1.93499e13 −1.84596 −0.922978 0.384853i \(-0.874252\pi\)
−0.922978 + 0.384853i \(0.874252\pi\)
\(234\) −1.28406e12 −0.119646
\(235\) −3.18187e12 −0.289607
\(236\) −1.05999e12 −0.0942512
\(237\) 6.04432e12 0.525087
\(238\) −2.15058e12 −0.182550
\(239\) 1.91104e13 1.58519 0.792594 0.609750i \(-0.208730\pi\)
0.792594 + 0.609750i \(0.208730\pi\)
\(240\) 1.75011e12 0.141874
\(241\) −1.44086e12 −0.114164 −0.0570820 0.998369i \(-0.518180\pi\)
−0.0570820 + 0.998369i \(0.518180\pi\)
\(242\) 1.19803e13 0.927858
\(243\) 8.47289e11 0.0641500
\(244\) −2.68037e12 −0.198404
\(245\) −9.53652e12 −0.690202
\(246\) −4.18686e11 −0.0296310
\(247\) −4.16481e12 −0.288245
\(248\) −2.57678e13 −1.74419
\(249\) 1.54289e12 0.102150
\(250\) −8.18687e12 −0.530210
\(251\) 6.54826e12 0.414878 0.207439 0.978248i \(-0.433487\pi\)
0.207439 + 0.978248i \(0.433487\pi\)
\(252\) −2.14378e12 −0.132886
\(253\) −6.89902e12 −0.418432
\(254\) −5.93140e12 −0.352024
\(255\) −6.21305e12 −0.360856
\(256\) −1.33501e13 −0.758867
\(257\) −1.75348e12 −0.0975594 −0.0487797 0.998810i \(-0.515533\pi\)
−0.0487797 + 0.998810i \(0.515533\pi\)
\(258\) −3.14389e12 −0.171222
\(259\) 7.10357e12 0.378728
\(260\) 9.38608e12 0.489926
\(261\) 1.21561e13 0.621258
\(262\) −7.56389e12 −0.378520
\(263\) 1.49939e13 0.734780 0.367390 0.930067i \(-0.380251\pi\)
0.367390 + 0.930067i \(0.380251\pi\)
\(264\) 1.81219e13 0.869723
\(265\) 1.68481e13 0.791950
\(266\) 2.65128e12 0.122069
\(267\) 2.03604e13 0.918280
\(268\) −2.45694e12 −0.108556
\(269\) 2.22020e13 0.961069 0.480534 0.876976i \(-0.340443\pi\)
0.480534 + 0.876976i \(0.340443\pi\)
\(270\) 2.36153e12 0.100160
\(271\) −8.63351e12 −0.358803 −0.179402 0.983776i \(-0.557416\pi\)
−0.179402 + 0.983776i \(0.557416\pi\)
\(272\) −3.84338e12 −0.156525
\(273\) −5.44190e12 −0.217198
\(274\) 1.49428e13 0.584527
\(275\) 8.13809e11 0.0312028
\(276\) −2.79802e12 −0.105160
\(277\) 5.13042e13 1.89023 0.945114 0.326740i \(-0.105950\pi\)
0.945114 + 0.326740i \(0.105950\pi\)
\(278\) 2.61380e13 0.944118
\(279\) 1.81251e13 0.641885
\(280\) −1.42284e13 −0.494070
\(281\) 2.26330e13 0.770651 0.385325 0.922781i \(-0.374089\pi\)
0.385325 + 0.922781i \(0.374089\pi\)
\(282\) −2.65594e12 −0.0886847
\(283\) 2.57270e13 0.842486 0.421243 0.906948i \(-0.361594\pi\)
0.421243 + 0.906948i \(0.361594\pi\)
\(284\) −3.62556e13 −1.16446
\(285\) 7.65957e12 0.241300
\(286\) 1.93178e13 0.596960
\(287\) −1.77441e12 −0.0537904
\(288\) 1.16128e13 0.345366
\(289\) −2.06275e13 −0.601879
\(290\) 3.38811e13 0.969993
\(291\) 3.08656e13 0.867089
\(292\) 4.97663e13 1.37192
\(293\) −5.35154e13 −1.44779 −0.723896 0.689909i \(-0.757651\pi\)
−0.723896 + 0.689909i \(0.757651\pi\)
\(294\) −7.96025e12 −0.211357
\(295\) 4.94860e12 0.128962
\(296\) −2.43535e13 −0.622954
\(297\) −1.27469e13 −0.320070
\(298\) −1.56068e13 −0.384702
\(299\) −7.10264e12 −0.171881
\(300\) 3.30054e11 0.00784185
\(301\) −1.33240e13 −0.310826
\(302\) −3.59066e13 −0.822500
\(303\) 7.65456e12 0.172181
\(304\) 4.73819e12 0.104667
\(305\) 1.25134e13 0.271472
\(306\) −5.18612e12 −0.110503
\(307\) 7.26163e13 1.51975 0.759877 0.650067i \(-0.225259\pi\)
0.759877 + 0.650067i \(0.225259\pi\)
\(308\) 3.22519e13 0.663020
\(309\) 2.74910e13 0.555162
\(310\) 5.05177e13 1.00220
\(311\) −5.96904e13 −1.16338 −0.581691 0.813410i \(-0.697609\pi\)
−0.581691 + 0.813410i \(0.697609\pi\)
\(312\) 1.86567e13 0.357260
\(313\) −6.28890e13 −1.18326 −0.591630 0.806209i \(-0.701516\pi\)
−0.591630 + 0.806209i \(0.701516\pi\)
\(314\) −3.16144e13 −0.584483
\(315\) 1.00083e13 0.181825
\(316\) −3.68794e13 −0.658423
\(317\) −9.60991e13 −1.68614 −0.843070 0.537804i \(-0.819254\pi\)
−0.843070 + 0.537804i \(0.819254\pi\)
\(318\) 1.40633e13 0.242515
\(319\) −1.82881e14 −3.09970
\(320\) 1.76170e13 0.293500
\(321\) −6.03864e13 −0.988921
\(322\) 4.52147e12 0.0727901
\(323\) −1.68210e13 −0.266219
\(324\) −5.16973e12 −0.0804397
\(325\) 8.37828e11 0.0128173
\(326\) 5.30746e13 0.798344
\(327\) 4.90933e13 0.726122
\(328\) 6.08330e12 0.0884775
\(329\) −1.12560e13 −0.160993
\(330\) −3.55278e13 −0.499737
\(331\) −4.93954e13 −0.683333 −0.341667 0.939821i \(-0.610991\pi\)
−0.341667 + 0.939821i \(0.610991\pi\)
\(332\) −9.41397e12 −0.128089
\(333\) 1.71302e13 0.229256
\(334\) 1.61374e13 0.212436
\(335\) 1.14703e13 0.148534
\(336\) 6.19111e12 0.0788683
\(337\) 6.72011e13 0.842193 0.421097 0.907016i \(-0.361645\pi\)
0.421097 + 0.907016i \(0.361645\pi\)
\(338\) −2.27238e13 −0.280182
\(339\) 3.68724e13 0.447305
\(340\) 3.79090e13 0.452489
\(341\) −2.72681e14 −3.20262
\(342\) 6.39354e12 0.0738921
\(343\) −8.21537e13 −0.934349
\(344\) 4.56791e13 0.511265
\(345\) 1.30626e13 0.143888
\(346\) 7.52764e13 0.816094
\(347\) −1.31316e14 −1.40121 −0.700607 0.713547i \(-0.747087\pi\)
−0.700607 + 0.713547i \(0.747087\pi\)
\(348\) −7.41705e13 −0.779014
\(349\) −5.29347e13 −0.547269 −0.273635 0.961834i \(-0.588226\pi\)
−0.273635 + 0.961834i \(0.588226\pi\)
\(350\) −5.33353e11 −0.00542801
\(351\) −1.31231e13 −0.131477
\(352\) −1.74708e14 −1.72317
\(353\) 6.86471e13 0.666594 0.333297 0.942822i \(-0.391839\pi\)
0.333297 + 0.942822i \(0.391839\pi\)
\(354\) 4.13066e12 0.0394913
\(355\) 1.69260e14 1.59330
\(356\) −1.24229e14 −1.15146
\(357\) −2.19790e13 −0.200601
\(358\) −4.36363e13 −0.392185
\(359\) 1.03179e14 0.913215 0.456607 0.889668i \(-0.349064\pi\)
0.456607 + 0.889668i \(0.349064\pi\)
\(360\) −3.43119e13 −0.299075
\(361\) −9.57530e13 −0.821983
\(362\) 1.02982e13 0.0870695
\(363\) 1.22439e14 1.01961
\(364\) 3.32038e13 0.272352
\(365\) −2.32335e14 −1.87717
\(366\) 1.04451e13 0.0831315
\(367\) 5.44575e12 0.0426967 0.0213484 0.999772i \(-0.493204\pi\)
0.0213484 + 0.999772i \(0.493204\pi\)
\(368\) 8.08049e12 0.0624129
\(369\) −4.27900e12 −0.0325609
\(370\) 4.77448e13 0.357945
\(371\) 5.96010e13 0.440247
\(372\) −1.10590e14 −0.804880
\(373\) −7.26263e13 −0.520829 −0.260414 0.965497i \(-0.583859\pi\)
−0.260414 + 0.965497i \(0.583859\pi\)
\(374\) 7.80219e13 0.551344
\(375\) −8.36703e13 −0.582638
\(376\) 3.85895e13 0.264811
\(377\) −1.88279e14 −1.27328
\(378\) 8.35406e12 0.0556792
\(379\) −2.08008e14 −1.36636 −0.683179 0.730251i \(-0.739403\pi\)
−0.683179 + 0.730251i \(0.739403\pi\)
\(380\) −4.67349e13 −0.302574
\(381\) −6.06193e13 −0.386833
\(382\) −1.47540e14 −0.928030
\(383\) 2.06489e14 1.28027 0.640137 0.768261i \(-0.278878\pi\)
0.640137 + 0.768261i \(0.278878\pi\)
\(384\) −8.31677e13 −0.508315
\(385\) −1.50569e14 −0.907195
\(386\) 6.10787e13 0.362794
\(387\) −3.21307e13 −0.188152
\(388\) −1.88327e14 −1.08727
\(389\) 2.16026e14 1.22966 0.614828 0.788661i \(-0.289225\pi\)
0.614828 + 0.788661i \(0.289225\pi\)
\(390\) −3.65764e13 −0.205279
\(391\) −2.86865e13 −0.158747
\(392\) 1.15658e14 0.631108
\(393\) −7.73034e13 −0.415949
\(394\) 1.53301e14 0.813424
\(395\) 1.72172e14 0.900905
\(396\) 7.77754e13 0.401346
\(397\) −6.66369e13 −0.339131 −0.169565 0.985519i \(-0.554236\pi\)
−0.169565 + 0.985519i \(0.554236\pi\)
\(398\) 5.37940e13 0.270008
\(399\) 2.70962e13 0.134140
\(400\) −9.53175e11 −0.00465418
\(401\) 9.42589e13 0.453971 0.226986 0.973898i \(-0.427113\pi\)
0.226986 + 0.973898i \(0.427113\pi\)
\(402\) 9.57437e12 0.0454849
\(403\) −2.80729e14 −1.31556
\(404\) −4.67043e13 −0.215903
\(405\) 2.41350e13 0.110064
\(406\) 1.19856e14 0.539222
\(407\) −2.57714e14 −1.14385
\(408\) 7.53516e13 0.329960
\(409\) 1.68108e14 0.726288 0.363144 0.931733i \(-0.381703\pi\)
0.363144 + 0.931733i \(0.381703\pi\)
\(410\) −1.19263e13 −0.0508386
\(411\) 1.52716e14 0.642326
\(412\) −1.67737e14 −0.696135
\(413\) 1.75060e13 0.0716903
\(414\) 1.09035e13 0.0440620
\(415\) 4.39493e13 0.175262
\(416\) −1.79864e14 −0.707835
\(417\) 2.67132e14 1.03747
\(418\) −9.61869e13 −0.368677
\(419\) −5.52023e13 −0.208824 −0.104412 0.994534i \(-0.533296\pi\)
−0.104412 + 0.994534i \(0.533296\pi\)
\(420\) −6.10657e13 −0.227995
\(421\) −2.98720e14 −1.10081 −0.550406 0.834897i \(-0.685527\pi\)
−0.550406 + 0.834897i \(0.685527\pi\)
\(422\) 1.46221e14 0.531853
\(423\) −2.71439e13 −0.0974541
\(424\) −2.04332e14 −0.724144
\(425\) 3.38386e12 0.0118379
\(426\) 1.41283e14 0.487908
\(427\) 4.42668e13 0.150912
\(428\) 3.68448e14 1.24004
\(429\) 1.97429e14 0.655989
\(430\) −8.95536e13 −0.293769
\(431\) −4.61512e14 −1.49471 −0.747357 0.664423i \(-0.768677\pi\)
−0.747357 + 0.664423i \(0.768677\pi\)
\(432\) 1.49299e13 0.0477414
\(433\) −6.14093e13 −0.193888 −0.0969440 0.995290i \(-0.530907\pi\)
−0.0969440 + 0.995290i \(0.530907\pi\)
\(434\) 1.78709e14 0.557126
\(435\) 3.46267e14 1.06591
\(436\) −2.99543e14 −0.910506
\(437\) 3.53653e13 0.106152
\(438\) −1.93933e14 −0.574837
\(439\) 2.71791e14 0.795573 0.397786 0.917478i \(-0.369779\pi\)
0.397786 + 0.917478i \(0.369779\pi\)
\(440\) 5.16201e14 1.49221
\(441\) −8.13542e13 −0.232256
\(442\) 8.03246e13 0.226478
\(443\) 2.08496e14 0.580600 0.290300 0.956936i \(-0.406245\pi\)
0.290300 + 0.956936i \(0.406245\pi\)
\(444\) −1.04520e14 −0.287471
\(445\) 5.79967e14 1.57552
\(446\) −2.92716e14 −0.785425
\(447\) −1.59503e14 −0.422743
\(448\) 6.23213e13 0.163158
\(449\) −1.30410e14 −0.337254 −0.168627 0.985680i \(-0.553933\pi\)
−0.168627 + 0.985680i \(0.553933\pi\)
\(450\) −1.28618e12 −0.00328574
\(451\) 6.43748e13 0.162460
\(452\) −2.24977e14 −0.560889
\(453\) −3.66967e14 −0.903831
\(454\) 3.33613e14 0.811775
\(455\) −1.55013e14 −0.372653
\(456\) −9.28949e13 −0.220641
\(457\) 6.29919e14 1.47824 0.739121 0.673573i \(-0.235241\pi\)
0.739121 + 0.673573i \(0.235241\pi\)
\(458\) −3.90773e13 −0.0906077
\(459\) −5.30024e13 −0.121430
\(460\) −7.97014e13 −0.180426
\(461\) −4.96432e14 −1.11046 −0.555232 0.831696i \(-0.687371\pi\)
−0.555232 + 0.831696i \(0.687371\pi\)
\(462\) −1.25682e14 −0.277806
\(463\) 4.11958e14 0.899824 0.449912 0.893073i \(-0.351455\pi\)
0.449912 + 0.893073i \(0.351455\pi\)
\(464\) 2.14200e14 0.462349
\(465\) 5.16294e14 1.10130
\(466\) −4.60079e14 −0.969861
\(467\) −2.15183e13 −0.0448297 −0.0224149 0.999749i \(-0.507135\pi\)
−0.0224149 + 0.999749i \(0.507135\pi\)
\(468\) 8.00708e13 0.164863
\(469\) 4.05767e13 0.0825707
\(470\) −7.56545e13 −0.152159
\(471\) −3.23101e14 −0.642278
\(472\) −6.00164e13 −0.117920
\(473\) 4.83386e14 0.938768
\(474\) 1.43714e14 0.275879
\(475\) −4.17169e12 −0.00791584
\(476\) 1.34105e14 0.251540
\(477\) 1.43728e14 0.266495
\(478\) 4.54383e14 0.832854
\(479\) 2.18237e14 0.395443 0.197721 0.980258i \(-0.436646\pi\)
0.197721 + 0.980258i \(0.436646\pi\)
\(480\) 3.30792e14 0.592554
\(481\) −2.65320e14 −0.469864
\(482\) −3.42591e13 −0.0599815
\(483\) 4.62097e13 0.0799878
\(484\) −7.47061e14 −1.27852
\(485\) 8.79207e14 1.48769
\(486\) 2.01458e13 0.0337043
\(487\) −6.93472e14 −1.14715 −0.573574 0.819154i \(-0.694443\pi\)
−0.573574 + 0.819154i \(0.694443\pi\)
\(488\) −1.51762e14 −0.248229
\(489\) 5.42425e14 0.877286
\(490\) −2.26748e14 −0.362631
\(491\) 5.82804e14 0.921668 0.460834 0.887486i \(-0.347550\pi\)
0.460834 + 0.887486i \(0.347550\pi\)
\(492\) 2.61083e13 0.0408292
\(493\) −7.60429e14 −1.17598
\(494\) −9.90257e13 −0.151443
\(495\) −3.63096e14 −0.549153
\(496\) 3.19378e14 0.477701
\(497\) 5.98766e14 0.885722
\(498\) 3.66851e13 0.0536695
\(499\) −6.41955e14 −0.928863 −0.464431 0.885609i \(-0.653741\pi\)
−0.464431 + 0.885609i \(0.653741\pi\)
\(500\) 5.10514e14 0.730588
\(501\) 1.64925e14 0.233442
\(502\) 1.55697e14 0.217976
\(503\) 1.02609e15 1.42090 0.710448 0.703750i \(-0.248492\pi\)
0.710448 + 0.703750i \(0.248492\pi\)
\(504\) −1.21380e14 −0.166257
\(505\) 2.18040e14 0.295416
\(506\) −1.64037e14 −0.219843
\(507\) −2.32239e14 −0.307887
\(508\) 3.69868e14 0.485062
\(509\) 4.61233e14 0.598374 0.299187 0.954194i \(-0.403285\pi\)
0.299187 + 0.954194i \(0.403285\pi\)
\(510\) −1.47726e14 −0.189593
\(511\) −8.21899e14 −1.04353
\(512\) 3.83513e14 0.481721
\(513\) 6.53424e13 0.0811988
\(514\) −4.16921e13 −0.0512575
\(515\) 7.83082e14 0.952506
\(516\) 1.96045e14 0.235930
\(517\) 4.08363e14 0.486237
\(518\) 1.68900e14 0.198983
\(519\) 7.69329e14 0.896792
\(520\) 5.31436e14 0.612960
\(521\) −4.43782e13 −0.0506479 −0.0253240 0.999679i \(-0.508062\pi\)
−0.0253240 + 0.999679i \(0.508062\pi\)
\(522\) 2.89033e14 0.326407
\(523\) 3.21587e14 0.359367 0.179684 0.983724i \(-0.442493\pi\)
0.179684 + 0.983724i \(0.442493\pi\)
\(524\) 4.71667e14 0.521571
\(525\) −5.45090e12 −0.00596474
\(526\) 3.56506e14 0.386052
\(527\) −1.13382e15 −1.21503
\(528\) −2.24610e14 −0.238201
\(529\) −8.92498e14 −0.936701
\(530\) 4.00593e14 0.416089
\(531\) 4.22156e13 0.0433963
\(532\) −1.65327e14 −0.168202
\(533\) 6.62748e13 0.0667342
\(534\) 4.84106e14 0.482462
\(535\) −1.72011e15 −1.69672
\(536\) −1.39111e14 −0.135817
\(537\) −4.45965e14 −0.430965
\(538\) 5.27892e14 0.504943
\(539\) 1.22392e15 1.15882
\(540\) −1.47260e14 −0.138012
\(541\) −8.14845e14 −0.755945 −0.377972 0.925817i \(-0.623379\pi\)
−0.377972 + 0.925817i \(0.623379\pi\)
\(542\) −2.05277e14 −0.188514
\(543\) 1.05249e14 0.0956792
\(544\) −7.26445e14 −0.653746
\(545\) 1.39842e15 1.24583
\(546\) −1.29391e14 −0.114115
\(547\) 8.04498e14 0.702416 0.351208 0.936297i \(-0.385771\pi\)
0.351208 + 0.936297i \(0.385771\pi\)
\(548\) −9.31799e14 −0.805433
\(549\) 1.06749e14 0.0913518
\(550\) 1.93498e13 0.0163939
\(551\) 9.37471e14 0.786365
\(552\) −1.58422e14 −0.131568
\(553\) 6.09069e14 0.500816
\(554\) 1.21985e15 0.993122
\(555\) 4.87955e14 0.393340
\(556\) −1.62991e15 −1.30092
\(557\) −1.28858e15 −1.01837 −0.509186 0.860656i \(-0.670053\pi\)
−0.509186 + 0.860656i \(0.670053\pi\)
\(558\) 4.30957e14 0.337245
\(559\) 4.97653e14 0.385622
\(560\) 1.76354e14 0.135316
\(561\) 7.97388e14 0.605862
\(562\) 5.38140e14 0.404898
\(563\) −1.13486e15 −0.845564 −0.422782 0.906231i \(-0.638946\pi\)
−0.422782 + 0.906231i \(0.638946\pi\)
\(564\) 1.65618e14 0.122201
\(565\) 1.05031e15 0.767453
\(566\) 6.11704e14 0.442640
\(567\) 8.53789e13 0.0611849
\(568\) −2.05277e15 −1.45689
\(569\) −3.63483e14 −0.255486 −0.127743 0.991807i \(-0.540773\pi\)
−0.127743 + 0.991807i \(0.540773\pi\)
\(570\) 1.82120e14 0.126779
\(571\) 5.35440e14 0.369158 0.184579 0.982818i \(-0.440908\pi\)
0.184579 + 0.982818i \(0.440908\pi\)
\(572\) −1.20462e15 −0.822565
\(573\) −1.50787e15 −1.01980
\(574\) −4.21899e13 −0.0282613
\(575\) −7.11438e12 −0.00472024
\(576\) 1.50288e14 0.0987642
\(577\) −1.71645e15 −1.11728 −0.558642 0.829409i \(-0.688678\pi\)
−0.558642 + 0.829409i \(0.688678\pi\)
\(578\) −4.90456e14 −0.316226
\(579\) 6.24228e14 0.398668
\(580\) −2.11275e15 −1.33657
\(581\) 1.55473e14 0.0974287
\(582\) 7.33885e14 0.455566
\(583\) −2.16229e15 −1.32965
\(584\) 2.81775e15 1.71645
\(585\) −3.73812e14 −0.225578
\(586\) −1.27242e15 −0.760667
\(587\) 1.33142e15 0.788508 0.394254 0.919002i \(-0.371003\pi\)
0.394254 + 0.919002i \(0.371003\pi\)
\(588\) 4.96383e14 0.291234
\(589\) 1.39780e15 0.812475
\(590\) 1.17662e14 0.0677563
\(591\) 1.56675e15 0.893857
\(592\) 3.01848e14 0.170615
\(593\) 7.24920e14 0.405965 0.202983 0.979182i \(-0.434936\pi\)
0.202983 + 0.979182i \(0.434936\pi\)
\(594\) −3.03081e14 −0.168164
\(595\) −6.26072e14 −0.344177
\(596\) 9.73205e14 0.530090
\(597\) 5.49777e14 0.296707
\(598\) −1.68878e14 −0.0903059
\(599\) −1.70755e15 −0.904743 −0.452371 0.891830i \(-0.649422\pi\)
−0.452371 + 0.891830i \(0.649422\pi\)
\(600\) 1.86875e13 0.00981115
\(601\) −1.03547e15 −0.538675 −0.269337 0.963046i \(-0.586805\pi\)
−0.269337 + 0.963046i \(0.586805\pi\)
\(602\) −3.16801e14 −0.163307
\(603\) 9.78506e13 0.0499825
\(604\) 2.23905e15 1.13334
\(605\) 3.48767e15 1.74937
\(606\) 1.82001e14 0.0904636
\(607\) 3.90639e15 1.92414 0.962072 0.272797i \(-0.0879488\pi\)
0.962072 + 0.272797i \(0.0879488\pi\)
\(608\) 8.95576e14 0.437152
\(609\) 1.22494e15 0.592542
\(610\) 2.97528e14 0.142631
\(611\) 4.20415e14 0.199734
\(612\) 3.23394e14 0.152265
\(613\) −1.81046e15 −0.844805 −0.422402 0.906408i \(-0.638813\pi\)
−0.422402 + 0.906408i \(0.638813\pi\)
\(614\) 1.72658e15 0.798475
\(615\) −1.21887e14 −0.0558657
\(616\) 1.82609e15 0.829522
\(617\) −1.18367e15 −0.532921 −0.266460 0.963846i \(-0.585854\pi\)
−0.266460 + 0.963846i \(0.585854\pi\)
\(618\) 6.53649e14 0.291681
\(619\) −3.52172e15 −1.55760 −0.778800 0.627273i \(-0.784171\pi\)
−0.778800 + 0.627273i \(0.784171\pi\)
\(620\) −3.15017e15 −1.38095
\(621\) 1.11435e14 0.0484190
\(622\) −1.41925e15 −0.611238
\(623\) 2.05167e15 0.875835
\(624\) −2.31239e14 −0.0978468
\(625\) −2.33862e15 −0.980887
\(626\) −1.49530e15 −0.621682
\(627\) −9.83035e14 −0.405133
\(628\) 1.97140e15 0.805372
\(629\) −1.07159e15 −0.433959
\(630\) 2.37965e14 0.0955302
\(631\) −1.23880e15 −0.492992 −0.246496 0.969144i \(-0.579279\pi\)
−0.246496 + 0.969144i \(0.579279\pi\)
\(632\) −2.08810e15 −0.823771
\(633\) 1.49439e15 0.584444
\(634\) −2.28493e15 −0.885894
\(635\) −1.72674e15 −0.663699
\(636\) −8.76954e14 −0.334166
\(637\) 1.26005e15 0.476014
\(638\) −4.34832e15 −1.62858
\(639\) 1.44392e15 0.536154
\(640\) −2.36903e15 −0.872130
\(641\) 2.54832e15 0.930112 0.465056 0.885281i \(-0.346034\pi\)
0.465056 + 0.885281i \(0.346034\pi\)
\(642\) −1.43579e15 −0.519577
\(643\) 1.45911e14 0.0523513 0.0261757 0.999657i \(-0.491667\pi\)
0.0261757 + 0.999657i \(0.491667\pi\)
\(644\) −2.81948e14 −0.100299
\(645\) −9.15243e14 −0.322818
\(646\) −3.99950e14 −0.139871
\(647\) 1.99009e14 0.0690080 0.0345040 0.999405i \(-0.489015\pi\)
0.0345040 + 0.999405i \(0.489015\pi\)
\(648\) −2.92708e14 −0.100640
\(649\) −6.35107e14 −0.216521
\(650\) 1.99209e13 0.00673418
\(651\) 1.82642e15 0.612216
\(652\) −3.30961e15 −1.10006
\(653\) 4.24825e14 0.140019 0.0700096 0.997546i \(-0.477697\pi\)
0.0700096 + 0.997546i \(0.477697\pi\)
\(654\) 1.16728e15 0.381503
\(655\) −2.20199e15 −0.713654
\(656\) −7.53991e13 −0.0242323
\(657\) −1.98201e15 −0.631678
\(658\) −2.67632e14 −0.0845855
\(659\) −4.66835e15 −1.46317 −0.731583 0.681752i \(-0.761218\pi\)
−0.731583 + 0.681752i \(0.761218\pi\)
\(660\) 2.21543e15 0.688599
\(661\) 2.49961e14 0.0770484 0.0385242 0.999258i \(-0.487734\pi\)
0.0385242 + 0.999258i \(0.487734\pi\)
\(662\) −1.17446e15 −0.359022
\(663\) 8.20922e14 0.248873
\(664\) −5.33015e14 −0.160256
\(665\) 7.71834e14 0.230147
\(666\) 4.07302e14 0.120450
\(667\) 1.59876e15 0.468911
\(668\) −1.00629e15 −0.292720
\(669\) −2.99157e15 −0.863089
\(670\) 2.72726e14 0.0780396
\(671\) −1.60598e15 −0.455791
\(672\) 1.17019e15 0.329403
\(673\) 4.01308e15 1.12046 0.560228 0.828338i \(-0.310713\pi\)
0.560228 + 0.828338i \(0.310713\pi\)
\(674\) 1.59783e15 0.442486
\(675\) −1.31448e13 −0.00361064
\(676\) 1.41700e15 0.386069
\(677\) 3.65865e15 0.988743 0.494371 0.869251i \(-0.335398\pi\)
0.494371 + 0.869251i \(0.335398\pi\)
\(678\) 8.76708e14 0.235013
\(679\) 3.11024e15 0.827010
\(680\) 2.14639e15 0.566121
\(681\) 3.40954e15 0.892045
\(682\) −6.48348e15 −1.68265
\(683\) 6.60913e15 1.70149 0.850747 0.525575i \(-0.176150\pi\)
0.850747 + 0.525575i \(0.176150\pi\)
\(684\) −3.98686e14 −0.101818
\(685\) 4.35013e15 1.10206
\(686\) −1.95335e15 −0.490905
\(687\) −3.99373e14 −0.0995672
\(688\) −5.66167e14 −0.140026
\(689\) −2.22611e15 −0.546186
\(690\) 3.10586e14 0.0755984
\(691\) −4.05146e15 −0.978322 −0.489161 0.872193i \(-0.662697\pi\)
−0.489161 + 0.872193i \(0.662697\pi\)
\(692\) −4.69406e15 −1.12451
\(693\) −1.28447e15 −0.305276
\(694\) −3.12226e15 −0.736195
\(695\) 7.60925e15 1.78002
\(696\) −4.19950e15 −0.974646
\(697\) 2.67674e14 0.0616348
\(698\) −1.25862e15 −0.287534
\(699\) −4.70203e15 −1.06576
\(700\) 3.32586e13 0.00747938
\(701\) 3.60994e15 0.805473 0.402736 0.915316i \(-0.368059\pi\)
0.402736 + 0.915316i \(0.368059\pi\)
\(702\) −3.12026e14 −0.0690775
\(703\) 1.32107e15 0.290183
\(704\) −2.26098e15 −0.492774
\(705\) −7.73193e14 −0.167205
\(706\) 1.63221e15 0.350227
\(707\) 7.71329e14 0.164223
\(708\) −2.57578e14 −0.0544160
\(709\) −4.47467e15 −0.938009 −0.469004 0.883196i \(-0.655387\pi\)
−0.469004 + 0.883196i \(0.655387\pi\)
\(710\) 4.02446e15 0.837117
\(711\) 1.46877e15 0.303159
\(712\) −7.03381e15 −1.44062
\(713\) 2.38380e15 0.484481
\(714\) −5.22591e14 −0.105395
\(715\) 5.62377e15 1.12550
\(716\) 2.72106e15 0.540401
\(717\) 4.64382e15 0.915208
\(718\) 2.45327e15 0.479801
\(719\) 5.79191e15 1.12412 0.562061 0.827096i \(-0.310009\pi\)
0.562061 + 0.827096i \(0.310009\pi\)
\(720\) 4.25277e14 0.0819111
\(721\) 2.77020e15 0.529501
\(722\) −2.27670e15 −0.431868
\(723\) −3.50130e14 −0.0659126
\(724\) −6.42175e14 −0.119975
\(725\) −1.88590e14 −0.0349670
\(726\) 2.91120e15 0.535699
\(727\) −1.72964e15 −0.315875 −0.157938 0.987449i \(-0.550484\pi\)
−0.157938 + 0.987449i \(0.550484\pi\)
\(728\) 1.87998e15 0.340747
\(729\) 2.05891e14 0.0370370
\(730\) −5.52418e15 −0.986262
\(731\) 2.00995e15 0.356155
\(732\) −6.51331e14 −0.114549
\(733\) −1.53166e15 −0.267356 −0.133678 0.991025i \(-0.542679\pi\)
−0.133678 + 0.991025i \(0.542679\pi\)
\(734\) 1.29482e14 0.0224328
\(735\) −2.31737e15 −0.398488
\(736\) 1.52731e15 0.260675
\(737\) −1.47210e15 −0.249383
\(738\) −1.01741e14 −0.0171074
\(739\) 4.50150e15 0.751299 0.375650 0.926762i \(-0.377420\pi\)
0.375650 + 0.926762i \(0.377420\pi\)
\(740\) −2.97726e15 −0.493221
\(741\) −1.01205e15 −0.166418
\(742\) 1.41712e15 0.231305
\(743\) 9.19848e15 1.49031 0.745157 0.666889i \(-0.232375\pi\)
0.745157 + 0.666889i \(0.232375\pi\)
\(744\) −6.26158e15 −1.00701
\(745\) −4.54343e15 −0.725311
\(746\) −1.72682e15 −0.273642
\(747\) 3.74923e14 0.0589765
\(748\) −4.86526e15 −0.759709
\(749\) −6.08497e15 −0.943211
\(750\) −1.98941e15 −0.306117
\(751\) −4.33394e15 −0.662008 −0.331004 0.943629i \(-0.607387\pi\)
−0.331004 + 0.943629i \(0.607387\pi\)
\(752\) −4.78296e14 −0.0725268
\(753\) 1.59123e15 0.239530
\(754\) −4.47666e15 −0.668977
\(755\) −1.04531e16 −1.55073
\(756\) −5.20940e14 −0.0767216
\(757\) 5.41348e15 0.791496 0.395748 0.918359i \(-0.370485\pi\)
0.395748 + 0.918359i \(0.370485\pi\)
\(758\) −4.94577e15 −0.717882
\(759\) −1.67646e15 −0.241582
\(760\) −2.64611e15 −0.378559
\(761\) −4.17764e15 −0.593356 −0.296678 0.954978i \(-0.595879\pi\)
−0.296678 + 0.954978i \(0.595879\pi\)
\(762\) −1.44133e15 −0.203241
\(763\) 4.94699e15 0.692559
\(764\) 9.20027e15 1.27875
\(765\) −1.50977e15 −0.208340
\(766\) 4.90963e15 0.672653
\(767\) −6.53851e14 −0.0889415
\(768\) −3.24408e15 −0.438132
\(769\) −2.00420e15 −0.268748 −0.134374 0.990931i \(-0.542902\pi\)
−0.134374 + 0.990931i \(0.542902\pi\)
\(770\) −3.58004e15 −0.476638
\(771\) −4.26096e14 −0.0563259
\(772\) −3.80873e15 −0.499902
\(773\) −8.64697e15 −1.12688 −0.563439 0.826158i \(-0.690522\pi\)
−0.563439 + 0.826158i \(0.690522\pi\)
\(774\) −7.63965e14 −0.0988549
\(775\) −2.81193e14 −0.0361281
\(776\) −1.06630e16 −1.36031
\(777\) 1.72617e15 0.218659
\(778\) 5.13641e15 0.646058
\(779\) −3.29994e14 −0.0412144
\(780\) 2.28082e15 0.282859
\(781\) −2.17229e16 −2.67509
\(782\) −6.82072e14 −0.0834052
\(783\) 2.95393e15 0.358683
\(784\) −1.43352e15 −0.172849
\(785\) −9.20352e15 −1.10197
\(786\) −1.83803e15 −0.218538
\(787\) −2.22935e15 −0.263219 −0.131610 0.991302i \(-0.542015\pi\)
−0.131610 + 0.991302i \(0.542015\pi\)
\(788\) −9.55952e15 −1.12084
\(789\) 3.64351e15 0.424226
\(790\) 4.09370e15 0.473334
\(791\) 3.71553e15 0.426629
\(792\) 4.40361e15 0.502135
\(793\) −1.65337e15 −0.187227
\(794\) −1.58441e15 −0.178179
\(795\) 4.09408e15 0.457233
\(796\) −3.35447e15 −0.372050
\(797\) 2.86409e15 0.315476 0.157738 0.987481i \(-0.449580\pi\)
0.157738 + 0.987481i \(0.449580\pi\)
\(798\) 6.44260e14 0.0704766
\(799\) 1.69799e15 0.184471
\(800\) −1.80162e14 −0.0194387
\(801\) 4.94759e15 0.530169
\(802\) 2.24117e15 0.238515
\(803\) 2.98181e16 3.15169
\(804\) −5.97035e14 −0.0626746
\(805\) 1.31628e15 0.137237
\(806\) −6.67483e15 −0.691190
\(807\) 5.39509e15 0.554873
\(808\) −2.64438e15 −0.270123
\(809\) −1.24502e16 −1.26316 −0.631581 0.775310i \(-0.717594\pi\)
−0.631581 + 0.775310i \(0.717594\pi\)
\(810\) 5.73853e14 0.0578273
\(811\) −1.62012e16 −1.62156 −0.810778 0.585354i \(-0.800956\pi\)
−0.810778 + 0.585354i \(0.800956\pi\)
\(812\) −7.47396e15 −0.743006
\(813\) −2.09794e15 −0.207155
\(814\) −6.12761e15 −0.600975
\(815\) 1.54510e16 1.50518
\(816\) −9.33941e14 −0.0903699
\(817\) −2.47790e15 −0.238156
\(818\) 3.99706e15 0.381590
\(819\) −1.32238e15 −0.125399
\(820\) 7.43695e14 0.0700517
\(821\) −7.47532e15 −0.699427 −0.349713 0.936857i \(-0.613721\pi\)
−0.349713 + 0.936857i \(0.613721\pi\)
\(822\) 3.63111e15 0.337477
\(823\) −5.76646e13 −0.00532366 −0.00266183 0.999996i \(-0.500847\pi\)
−0.00266183 + 0.999996i \(0.500847\pi\)
\(824\) −9.49718e15 −0.870953
\(825\) 1.97756e14 0.0180149
\(826\) 4.16235e14 0.0376659
\(827\) −4.43940e13 −0.00399065 −0.00199532 0.999998i \(-0.500635\pi\)
−0.00199532 + 0.999998i \(0.500635\pi\)
\(828\) −6.79918e14 −0.0607141
\(829\) 1.05447e16 0.935372 0.467686 0.883895i \(-0.345088\pi\)
0.467686 + 0.883895i \(0.345088\pi\)
\(830\) 1.04497e15 0.0920822
\(831\) 1.24669e16 1.09132
\(832\) −2.32771e15 −0.202419
\(833\) 5.08914e15 0.439639
\(834\) 6.35154e15 0.545087
\(835\) 4.69788e15 0.400522
\(836\) 5.99799e15 0.508009
\(837\) 4.40440e15 0.370593
\(838\) −1.31253e15 −0.109716
\(839\) −6.14078e15 −0.509956 −0.254978 0.966947i \(-0.582068\pi\)
−0.254978 + 0.966947i \(0.582068\pi\)
\(840\) −3.45751e15 −0.285251
\(841\) 3.01798e16 2.47365
\(842\) −7.10261e15 −0.578364
\(843\) 5.49982e15 0.444935
\(844\) −9.11800e15 −0.732852
\(845\) −6.61532e15 −0.528249
\(846\) −6.45394e14 −0.0512022
\(847\) 1.23378e16 0.972478
\(848\) 2.53259e15 0.198329
\(849\) 6.25165e15 0.486410
\(850\) 8.04573e13 0.00621959
\(851\) 2.25295e15 0.173037
\(852\) −8.81010e15 −0.672300
\(853\) 1.02253e16 0.775277 0.387638 0.921812i \(-0.373291\pi\)
0.387638 + 0.921812i \(0.373291\pi\)
\(854\) 1.05252e15 0.0792890
\(855\) 1.86128e15 0.139315
\(856\) 2.08614e16 1.55145
\(857\) −1.43797e15 −0.106257 −0.0531283 0.998588i \(-0.516919\pi\)
−0.0531283 + 0.998588i \(0.516919\pi\)
\(858\) 4.69424e15 0.344655
\(859\) −5.05945e15 −0.369097 −0.184549 0.982823i \(-0.559082\pi\)
−0.184549 + 0.982823i \(0.559082\pi\)
\(860\) 5.58435e15 0.404792
\(861\) −4.31183e14 −0.0310559
\(862\) −1.09733e16 −0.785319
\(863\) −1.23610e16 −0.879012 −0.439506 0.898240i \(-0.644846\pi\)
−0.439506 + 0.898240i \(0.644846\pi\)
\(864\) 2.82192e15 0.199397
\(865\) 2.19143e16 1.53865
\(866\) −1.46012e15 −0.101868
\(867\) −5.01249e15 −0.347495
\(868\) −1.11439e16 −0.767677
\(869\) −2.20967e16 −1.51258
\(870\) 8.23310e15 0.560025
\(871\) −1.51555e15 −0.102440
\(872\) −1.69600e16 −1.13916
\(873\) 7.50035e15 0.500614
\(874\) 8.40872e14 0.0557721
\(875\) −8.43122e15 −0.555708
\(876\) 1.20932e16 0.792081
\(877\) 6.38167e15 0.415372 0.207686 0.978196i \(-0.433407\pi\)
0.207686 + 0.978196i \(0.433407\pi\)
\(878\) 6.46231e15 0.417992
\(879\) −1.30042e16 −0.835884
\(880\) −6.39802e15 −0.408687
\(881\) −9.60382e15 −0.609644 −0.304822 0.952409i \(-0.598597\pi\)
−0.304822 + 0.952409i \(0.598597\pi\)
\(882\) −1.93434e15 −0.122027
\(883\) 2.03319e16 1.27466 0.637329 0.770592i \(-0.280039\pi\)
0.637329 + 0.770592i \(0.280039\pi\)
\(884\) −5.00885e15 −0.312069
\(885\) 1.20251e15 0.0744562
\(886\) 4.95737e15 0.305046
\(887\) 1.16294e16 0.711178 0.355589 0.934642i \(-0.384280\pi\)
0.355589 + 0.934642i \(0.384280\pi\)
\(888\) −5.91789e15 −0.359663
\(889\) −6.10844e15 −0.368952
\(890\) 1.37897e16 0.827773
\(891\) −3.09750e15 −0.184793
\(892\) 1.82531e16 1.08225
\(893\) −2.09332e15 −0.123354
\(894\) −3.79246e15 −0.222108
\(895\) −1.27033e16 −0.739418
\(896\) −8.38058e15 −0.484820
\(897\) −1.72594e15 −0.0992356
\(898\) −3.10074e15 −0.177192
\(899\) 6.31903e16 3.58899
\(900\) 8.02032e13 0.00452749
\(901\) −8.99092e15 −0.504450
\(902\) 1.53063e15 0.0853559
\(903\) −3.23772e15 −0.179456
\(904\) −1.27381e16 −0.701744
\(905\) 2.99801e15 0.164159
\(906\) −8.72530e15 −0.474871
\(907\) −2.64117e16 −1.42875 −0.714375 0.699763i \(-0.753289\pi\)
−0.714375 + 0.699763i \(0.753289\pi\)
\(908\) −2.08033e16 −1.11856
\(909\) 1.86006e15 0.0994088
\(910\) −3.68570e15 −0.195791
\(911\) 4.28301e15 0.226151 0.113075 0.993586i \(-0.463930\pi\)
0.113075 + 0.993586i \(0.463930\pi\)
\(912\) 1.15138e15 0.0604293
\(913\) −5.64049e15 −0.294257
\(914\) 1.49774e16 0.776665
\(915\) 3.04075e15 0.156735
\(916\) 2.43677e15 0.124850
\(917\) −7.78965e15 −0.396722
\(918\) −1.26023e15 −0.0637990
\(919\) 8.05938e15 0.405571 0.202785 0.979223i \(-0.435001\pi\)
0.202785 + 0.979223i \(0.435001\pi\)
\(920\) −4.51266e15 −0.225735
\(921\) 1.76458e16 0.877430
\(922\) −1.18035e16 −0.583435
\(923\) −2.23641e16 −1.09886
\(924\) 7.83721e15 0.382795
\(925\) −2.65758e14 −0.0129035
\(926\) 9.79504e15 0.472765
\(927\) 6.68032e15 0.320523
\(928\) 4.04863e16 1.93105
\(929\) 2.56238e15 0.121495 0.0607474 0.998153i \(-0.480652\pi\)
0.0607474 + 0.998153i \(0.480652\pi\)
\(930\) 1.22758e16 0.578620
\(931\) −6.27399e15 −0.293982
\(932\) 2.86894e16 1.33639
\(933\) −1.45048e16 −0.671679
\(934\) −5.11637e14 −0.0235534
\(935\) 2.27136e16 1.03949
\(936\) 4.53358e15 0.206264
\(937\) 1.21684e16 0.550385 0.275193 0.961389i \(-0.411258\pi\)
0.275193 + 0.961389i \(0.411258\pi\)
\(938\) 9.64783e14 0.0433825
\(939\) −1.52820e16 −0.683156
\(940\) 4.71764e15 0.209663
\(941\) 2.93994e16 1.29896 0.649479 0.760379i \(-0.274987\pi\)
0.649479 + 0.760379i \(0.274987\pi\)
\(942\) −7.68229e15 −0.337451
\(943\) −5.62769e14 −0.0245763
\(944\) 7.43870e14 0.0322962
\(945\) 2.43202e15 0.104976
\(946\) 1.14934e16 0.493226
\(947\) 1.40267e16 0.598455 0.299228 0.954182i \(-0.403271\pi\)
0.299228 + 0.954182i \(0.403271\pi\)
\(948\) −8.96169e15 −0.380141
\(949\) 3.06981e16 1.29464
\(950\) −9.91894e13 −0.00415897
\(951\) −2.33521e16 −0.973493
\(952\) 7.59297e15 0.314709
\(953\) 3.03827e16 1.25203 0.626015 0.779811i \(-0.284685\pi\)
0.626015 + 0.779811i \(0.284685\pi\)
\(954\) 3.41738e15 0.140016
\(955\) −4.29516e16 −1.74969
\(956\) −2.83343e16 −1.14761
\(957\) −4.44401e16 −1.78961
\(958\) 5.18898e15 0.207765
\(959\) 1.53888e16 0.612637
\(960\) 4.28094e15 0.169452
\(961\) 6.88100e16 2.70815
\(962\) −6.30845e15 −0.246865
\(963\) −1.46739e16 −0.570954
\(964\) 2.13632e15 0.0826499
\(965\) 1.77811e16 0.684005
\(966\) 1.09872e15 0.0420254
\(967\) −3.25574e16 −1.23824 −0.619120 0.785296i \(-0.712511\pi\)
−0.619120 + 0.785296i \(0.712511\pi\)
\(968\) −4.22983e16 −1.59959
\(969\) −4.08751e15 −0.153702
\(970\) 2.09047e16 0.781627
\(971\) 4.36365e16 1.62235 0.811174 0.584805i \(-0.198829\pi\)
0.811174 + 0.584805i \(0.198829\pi\)
\(972\) −1.25624e15 −0.0464419
\(973\) 2.69182e16 0.989520
\(974\) −1.64885e16 −0.602709
\(975\) 2.03592e14 0.00740007
\(976\) 1.88100e15 0.0679853
\(977\) −1.98101e16 −0.711979 −0.355989 0.934490i \(-0.615856\pi\)
−0.355989 + 0.934490i \(0.615856\pi\)
\(978\) 1.28971e16 0.460924
\(979\) −7.44334e16 −2.64523
\(980\) 1.41395e16 0.499677
\(981\) 1.19297e16 0.419227
\(982\) 1.38572e16 0.484242
\(983\) −4.32181e16 −1.50183 −0.750916 0.660398i \(-0.770388\pi\)
−0.750916 + 0.660398i \(0.770388\pi\)
\(984\) 1.47824e15 0.0510825
\(985\) 4.46288e16 1.53361
\(986\) −1.80805e16 −0.617858
\(987\) −2.73522e15 −0.0929495
\(988\) 6.17501e15 0.208677
\(989\) −4.22580e15 −0.142013
\(990\) −8.63326e15 −0.288523
\(991\) 2.96218e16 0.984478 0.492239 0.870460i \(-0.336179\pi\)
0.492239 + 0.870460i \(0.336179\pi\)
\(992\) 6.03663e16 1.99517
\(993\) −1.20031e16 −0.394523
\(994\) 1.42367e16 0.465356
\(995\) 1.56604e16 0.509068
\(996\) −2.28759e15 −0.0739525
\(997\) −4.68798e16 −1.50717 −0.753585 0.657350i \(-0.771677\pi\)
−0.753585 + 0.657350i \(0.771677\pi\)
\(998\) −1.52636e16 −0.488022
\(999\) 4.16265e15 0.132361
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.1.17 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.d.1.17 28 1.1 even 1 trivial