Properties

Label 177.12.a.d.1.3
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 1.3
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-76.9358 q^{2} +243.000 q^{3} +3871.11 q^{4} +1565.71 q^{5} -18695.4 q^{6} +51126.9 q^{7} -140263. q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-76.9358 q^{2} +243.000 q^{3} +3871.11 q^{4} +1565.71 q^{5} -18695.4 q^{6} +51126.9 q^{7} -140263. q^{8} +59049.0 q^{9} -120459. q^{10} +913812. q^{11} +940680. q^{12} +32876.1 q^{13} -3.93349e6 q^{14} +380468. q^{15} +2.86317e6 q^{16} +6.52534e6 q^{17} -4.54298e6 q^{18} +2.05323e7 q^{19} +6.06104e6 q^{20} +1.24238e7 q^{21} -7.03049e7 q^{22} -4.51667e6 q^{23} -3.40838e7 q^{24} -4.63767e7 q^{25} -2.52935e6 q^{26} +1.43489e7 q^{27} +1.97918e8 q^{28} +1.26164e7 q^{29} -2.92716e7 q^{30} +2.56107e8 q^{31} +6.69776e7 q^{32} +2.22056e8 q^{33} -5.02032e8 q^{34} +8.00500e7 q^{35} +2.28585e8 q^{36} -2.39779e7 q^{37} -1.57967e9 q^{38} +7.98888e6 q^{39} -2.19611e8 q^{40} +6.93270e8 q^{41} -9.55838e8 q^{42} +8.57433e8 q^{43} +3.53747e9 q^{44} +9.24537e7 q^{45} +3.47494e8 q^{46} +2.59899e9 q^{47} +6.95750e8 q^{48} +6.36634e8 q^{49} +3.56803e9 q^{50} +1.58566e9 q^{51} +1.27267e8 q^{52} +3.92590e9 q^{53} -1.10394e9 q^{54} +1.43077e9 q^{55} -7.17119e9 q^{56} +4.98934e9 q^{57} -9.70650e8 q^{58} +7.14924e8 q^{59} +1.47283e9 q^{60} +1.27105e9 q^{61} -1.97038e10 q^{62} +3.01899e9 q^{63} -1.10167e10 q^{64} +5.14744e7 q^{65} -1.70841e10 q^{66} +9.16382e9 q^{67} +2.52603e10 q^{68} -1.09755e9 q^{69} -6.15870e9 q^{70} -1.96637e10 q^{71} -8.28236e9 q^{72} -3.06456e10 q^{73} +1.84475e9 q^{74} -1.12695e10 q^{75} +7.94828e10 q^{76} +4.67204e10 q^{77} -6.14631e8 q^{78} -3.72081e10 q^{79} +4.48289e9 q^{80} +3.48678e9 q^{81} -5.33373e10 q^{82} -4.45375e10 q^{83} +4.80941e10 q^{84} +1.02168e10 q^{85} -6.59672e10 q^{86} +3.06578e9 q^{87} -1.28174e11 q^{88} +2.20586e10 q^{89} -7.11299e9 q^{90} +1.68085e9 q^{91} -1.74845e10 q^{92} +6.22340e10 q^{93} -1.99955e11 q^{94} +3.21476e10 q^{95} +1.62756e10 q^{96} -8.48502e10 q^{97} -4.89799e10 q^{98} +5.39597e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + O(q^{10}) \) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + 616281q^{10} + 1082362q^{11} + 7099002q^{12} + 503712q^{13} + 1321669q^{14} + 6454566q^{15} + 34870338q^{16} + 13513579q^{17} + 5668704q^{18} + 35971687q^{19} + 96105997q^{20} + 34586919q^{21} - 47598882q^{22} + 61380539q^{23} + 80756433q^{24} + 294744746q^{25} + 62820734q^{26} + 401769396q^{27} + 148068294q^{28} + 322339307q^{29} + 149756283q^{30} + 151247077q^{31} + 466383494q^{32} + 263013966q^{33} + 684479860q^{34} + 960297361q^{35} + 1725057486q^{36} + 863508437q^{37} + 992640509q^{38} + 122402016q^{39} + 3067680252q^{40} + 3081170377q^{41} + 321165567q^{42} + 2554238300q^{43} + 4350123570q^{44} + 1568459538q^{45} - 1987059155q^{46} + 6203398333q^{47} + 8473492134q^{48} + 10327857997q^{49} + 17577682253q^{50} + 3283799697q^{51} + 32137181618q^{52} + 14571770754q^{53} + 1377495072q^{54} + 18251419334q^{55} + 33498842836q^{56} + 8741119941q^{57} + 11860778276q^{58} + 20017880372q^{59} + 23353757271q^{60} + 2761613771q^{61} + 13785829526q^{62} + 8404621317q^{63} + 86547545293q^{64} + 32034985256q^{65} - 11566528326q^{66} + 39381333296q^{67} + 38995496621q^{68} + 14915470977q^{69} + 8551800364q^{70} + 26130020296q^{71} + 19623813219q^{72} + 41382402799q^{73} + 23815315058q^{74} + 71622973278q^{75} + 10611720128q^{76} - 8426124313q^{77} + 15265438362q^{78} + 59825111206q^{79} + 4009687655q^{80} + 97629963228q^{81} - 39592715115q^{82} + 35433122727q^{83} + 35980595442q^{84} - 8950496085q^{85} - 182032360688q^{86} + 78328451601q^{87} - 220003602335q^{88} + 102303043039q^{89} + 36390776769q^{90} - 111146323655q^{91} - 163000203526q^{92} + 36753039711q^{93} - 81314346008q^{94} + 208102168887q^{95} + 113331189042q^{96} - 171891031490q^{97} + 72304707792q^{98} + 63912393738q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −76.9358 −1.70006 −0.850028 0.526737i \(-0.823415\pi\)
−0.850028 + 0.526737i \(0.823415\pi\)
\(3\) 243.000 0.577350
\(4\) 3871.11 1.89019
\(5\) 1565.71 0.224066 0.112033 0.993704i \(-0.464264\pi\)
0.112033 + 0.993704i \(0.464264\pi\)
\(6\) −18695.4 −0.981528
\(7\) 51126.9 1.14977 0.574884 0.818235i \(-0.305047\pi\)
0.574884 + 0.818235i \(0.305047\pi\)
\(8\) −140263. −1.51338
\(9\) 59049.0 0.333333
\(10\) −120459. −0.380925
\(11\) 913812. 1.71079 0.855396 0.517974i \(-0.173314\pi\)
0.855396 + 0.517974i \(0.173314\pi\)
\(12\) 940680. 1.09130
\(13\) 32876.1 0.0245579 0.0122790 0.999925i \(-0.496091\pi\)
0.0122790 + 0.999925i \(0.496091\pi\)
\(14\) −3.93349e6 −1.95467
\(15\) 380468. 0.129365
\(16\) 2.86317e6 0.682633
\(17\) 6.52534e6 1.11464 0.557319 0.830298i \(-0.311830\pi\)
0.557319 + 0.830298i \(0.311830\pi\)
\(18\) −4.54298e6 −0.566685
\(19\) 2.05323e7 1.90236 0.951179 0.308638i \(-0.0998733\pi\)
0.951179 + 0.308638i \(0.0998733\pi\)
\(20\) 6.06104e6 0.423528
\(21\) 1.24238e7 0.663819
\(22\) −7.03049e7 −2.90844
\(23\) −4.51667e6 −0.146324 −0.0731620 0.997320i \(-0.523309\pi\)
−0.0731620 + 0.997320i \(0.523309\pi\)
\(24\) −3.40838e7 −0.873748
\(25\) −4.63767e7 −0.949794
\(26\) −2.52935e6 −0.0417498
\(27\) 1.43489e7 0.192450
\(28\) 1.97918e8 2.17328
\(29\) 1.26164e7 0.114221 0.0571104 0.998368i \(-0.481811\pi\)
0.0571104 + 0.998368i \(0.481811\pi\)
\(30\) −2.92716e7 −0.219927
\(31\) 2.56107e8 1.60669 0.803344 0.595515i \(-0.203052\pi\)
0.803344 + 0.595515i \(0.203052\pi\)
\(32\) 6.69776e7 0.352862
\(33\) 2.22056e8 0.987726
\(34\) −5.02032e8 −1.89495
\(35\) 8.00500e7 0.257624
\(36\) 2.28585e8 0.630064
\(37\) −2.39779e7 −0.0568461 −0.0284231 0.999596i \(-0.509049\pi\)
−0.0284231 + 0.999596i \(0.509049\pi\)
\(38\) −1.57967e9 −3.23412
\(39\) 7.98888e6 0.0141785
\(40\) −2.19611e8 −0.339097
\(41\) 6.93270e8 0.934526 0.467263 0.884118i \(-0.345240\pi\)
0.467263 + 0.884118i \(0.345240\pi\)
\(42\) −9.55838e8 −1.12853
\(43\) 8.57433e8 0.889454 0.444727 0.895666i \(-0.353301\pi\)
0.444727 + 0.895666i \(0.353301\pi\)
\(44\) 3.53747e9 3.23373
\(45\) 9.24537e7 0.0746888
\(46\) 3.47494e8 0.248759
\(47\) 2.59899e9 1.65298 0.826488 0.562954i \(-0.190335\pi\)
0.826488 + 0.562954i \(0.190335\pi\)
\(48\) 6.95750e8 0.394118
\(49\) 6.36634e8 0.321967
\(50\) 3.56803e9 1.61470
\(51\) 1.58566e9 0.643537
\(52\) 1.27267e8 0.0464192
\(53\) 3.92590e9 1.28950 0.644751 0.764393i \(-0.276961\pi\)
0.644751 + 0.764393i \(0.276961\pi\)
\(54\) −1.10394e9 −0.327176
\(55\) 1.43077e9 0.383331
\(56\) −7.17119e9 −1.74003
\(57\) 4.98934e9 1.09833
\(58\) −9.70650e8 −0.194182
\(59\) 7.14924e8 0.130189
\(60\) 1.47283e9 0.244524
\(61\) 1.27105e9 0.192685 0.0963423 0.995348i \(-0.469286\pi\)
0.0963423 + 0.995348i \(0.469286\pi\)
\(62\) −1.97038e10 −2.73146
\(63\) 3.01899e9 0.383256
\(64\) −1.10167e10 −1.28252
\(65\) 5.14744e7 0.00550260
\(66\) −1.70841e10 −1.67919
\(67\) 9.16382e9 0.829211 0.414606 0.910001i \(-0.363920\pi\)
0.414606 + 0.910001i \(0.363920\pi\)
\(68\) 2.52603e10 2.10688
\(69\) −1.09755e9 −0.0844802
\(70\) −6.15870e9 −0.437976
\(71\) −1.96637e10 −1.29343 −0.646717 0.762730i \(-0.723859\pi\)
−0.646717 + 0.762730i \(0.723859\pi\)
\(72\) −8.28236e9 −0.504459
\(73\) −3.06456e10 −1.73018 −0.865092 0.501613i \(-0.832740\pi\)
−0.865092 + 0.501613i \(0.832740\pi\)
\(74\) 1.84475e9 0.0966416
\(75\) −1.12695e10 −0.548364
\(76\) 7.94828e10 3.59582
\(77\) 4.67204e10 1.96701
\(78\) −6.14631e8 −0.0241043
\(79\) −3.72081e10 −1.36047 −0.680234 0.732995i \(-0.738122\pi\)
−0.680234 + 0.732995i \(0.738122\pi\)
\(80\) 4.48289e9 0.152955
\(81\) 3.48678e9 0.111111
\(82\) −5.33373e10 −1.58875
\(83\) −4.45375e10 −1.24107 −0.620535 0.784179i \(-0.713085\pi\)
−0.620535 + 0.784179i \(0.713085\pi\)
\(84\) 4.80941e10 1.25475
\(85\) 1.02168e10 0.249753
\(86\) −6.59672e10 −1.51212
\(87\) 3.06578e9 0.0659454
\(88\) −1.28174e11 −2.58907
\(89\) 2.20586e10 0.418728 0.209364 0.977838i \(-0.432861\pi\)
0.209364 + 0.977838i \(0.432861\pi\)
\(90\) −7.11299e9 −0.126975
\(91\) 1.68085e9 0.0282359
\(92\) −1.74845e10 −0.276580
\(93\) 6.22340e10 0.927622
\(94\) −1.99955e11 −2.81015
\(95\) 3.21476e10 0.426254
\(96\) 1.62756e10 0.203725
\(97\) −8.48502e10 −1.00325 −0.501624 0.865086i \(-0.667264\pi\)
−0.501624 + 0.865086i \(0.667264\pi\)
\(98\) −4.89799e10 −0.547362
\(99\) 5.39597e10 0.570264
\(100\) −1.79529e11 −1.79529
\(101\) 4.47202e8 0.00423386 0.00211693 0.999998i \(-0.499326\pi\)
0.00211693 + 0.999998i \(0.499326\pi\)
\(102\) −1.21994e11 −1.09405
\(103\) −1.78642e11 −1.51838 −0.759189 0.650870i \(-0.774404\pi\)
−0.759189 + 0.650870i \(0.774404\pi\)
\(104\) −4.61128e9 −0.0371653
\(105\) 1.94521e10 0.148739
\(106\) −3.02042e11 −2.19222
\(107\) 2.67706e11 1.84522 0.922608 0.385740i \(-0.126054\pi\)
0.922608 + 0.385740i \(0.126054\pi\)
\(108\) 5.55462e10 0.363768
\(109\) −1.91231e11 −1.19046 −0.595228 0.803557i \(-0.702938\pi\)
−0.595228 + 0.803557i \(0.702938\pi\)
\(110\) −1.10077e11 −0.651684
\(111\) −5.82662e9 −0.0328201
\(112\) 1.46385e11 0.784869
\(113\) −1.46819e11 −0.749639 −0.374819 0.927098i \(-0.622295\pi\)
−0.374819 + 0.927098i \(0.622295\pi\)
\(114\) −3.83859e11 −1.86722
\(115\) −7.07180e9 −0.0327863
\(116\) 4.88394e10 0.215899
\(117\) 1.94130e9 0.00818597
\(118\) −5.50033e10 −0.221328
\(119\) 3.33620e11 1.28158
\(120\) −5.33654e10 −0.195777
\(121\) 5.49741e11 1.92681
\(122\) −9.77888e10 −0.327575
\(123\) 1.68465e11 0.539549
\(124\) 9.91418e11 3.03695
\(125\) −1.49063e11 −0.436883
\(126\) −2.32269e11 −0.651557
\(127\) −4.33041e11 −1.16308 −0.581538 0.813519i \(-0.697549\pi\)
−0.581538 + 0.813519i \(0.697549\pi\)
\(128\) 7.10411e11 1.82749
\(129\) 2.08356e11 0.513527
\(130\) −3.96022e9 −0.00935473
\(131\) −2.63061e11 −0.595751 −0.297875 0.954605i \(-0.596278\pi\)
−0.297875 + 0.954605i \(0.596278\pi\)
\(132\) 8.59605e11 1.86699
\(133\) 1.04975e12 2.18727
\(134\) −7.05026e11 −1.40971
\(135\) 2.24662e10 0.0431216
\(136\) −9.15260e11 −1.68687
\(137\) −1.61673e11 −0.286202 −0.143101 0.989708i \(-0.545707\pi\)
−0.143101 + 0.989708i \(0.545707\pi\)
\(138\) 8.44410e10 0.143621
\(139\) −9.90647e11 −1.61934 −0.809669 0.586887i \(-0.800353\pi\)
−0.809669 + 0.586887i \(0.800353\pi\)
\(140\) 3.09882e11 0.486959
\(141\) 6.31555e11 0.954346
\(142\) 1.51284e12 2.19891
\(143\) 3.00426e10 0.0420135
\(144\) 1.69067e11 0.227544
\(145\) 1.97536e10 0.0255930
\(146\) 2.35774e12 2.94141
\(147\) 1.54702e11 0.185888
\(148\) −9.28210e10 −0.107450
\(149\) −1.05631e12 −1.17834 −0.589168 0.808011i \(-0.700544\pi\)
−0.589168 + 0.808011i \(0.700544\pi\)
\(150\) 8.67030e11 0.932250
\(151\) −8.33063e11 −0.863585 −0.431792 0.901973i \(-0.642119\pi\)
−0.431792 + 0.901973i \(0.642119\pi\)
\(152\) −2.87991e12 −2.87898
\(153\) 3.85315e11 0.371546
\(154\) −3.59447e12 −3.34404
\(155\) 4.00989e11 0.360005
\(156\) 3.09259e10 0.0268001
\(157\) 1.19086e12 0.996350 0.498175 0.867076i \(-0.334004\pi\)
0.498175 + 0.867076i \(0.334004\pi\)
\(158\) 2.86263e12 2.31287
\(159\) 9.53993e11 0.744494
\(160\) 1.04868e11 0.0790644
\(161\) −2.30924e11 −0.168239
\(162\) −2.68258e11 −0.188895
\(163\) 1.90143e12 1.29434 0.647170 0.762346i \(-0.275953\pi\)
0.647170 + 0.762346i \(0.275953\pi\)
\(164\) 2.68373e12 1.76643
\(165\) 3.47676e11 0.221316
\(166\) 3.42653e12 2.10989
\(167\) 1.16025e12 0.691213 0.345607 0.938379i \(-0.387673\pi\)
0.345607 + 0.938379i \(0.387673\pi\)
\(168\) −1.74260e12 −1.00461
\(169\) −1.79108e12 −0.999397
\(170\) −7.86037e11 −0.424594
\(171\) 1.21241e12 0.634120
\(172\) 3.31922e12 1.68124
\(173\) 5.72770e11 0.281013 0.140507 0.990080i \(-0.455127\pi\)
0.140507 + 0.990080i \(0.455127\pi\)
\(174\) −2.35868e11 −0.112111
\(175\) −2.37110e12 −1.09204
\(176\) 2.61640e12 1.16784
\(177\) 1.73727e11 0.0751646
\(178\) −1.69709e12 −0.711862
\(179\) −2.31921e12 −0.943297 −0.471648 0.881787i \(-0.656341\pi\)
−0.471648 + 0.881787i \(0.656341\pi\)
\(180\) 3.57898e11 0.141176
\(181\) 3.81190e12 1.45851 0.729254 0.684243i \(-0.239867\pi\)
0.729254 + 0.684243i \(0.239867\pi\)
\(182\) −1.29318e11 −0.0480026
\(183\) 3.08864e11 0.111246
\(184\) 6.33520e11 0.221443
\(185\) −3.75424e10 −0.0127373
\(186\) −4.78802e12 −1.57701
\(187\) 5.96293e12 1.90691
\(188\) 1.00610e13 3.12444
\(189\) 7.33615e11 0.221273
\(190\) −2.47330e12 −0.724657
\(191\) −2.53491e11 −0.0721571 −0.0360785 0.999349i \(-0.511487\pi\)
−0.0360785 + 0.999349i \(0.511487\pi\)
\(192\) −2.67707e12 −0.740462
\(193\) −7.22905e12 −1.94319 −0.971596 0.236645i \(-0.923952\pi\)
−0.971596 + 0.236645i \(0.923952\pi\)
\(194\) 6.52801e12 1.70558
\(195\) 1.25083e10 0.00317693
\(196\) 2.46448e12 0.608580
\(197\) −1.65192e12 −0.396666 −0.198333 0.980135i \(-0.563553\pi\)
−0.198333 + 0.980135i \(0.563553\pi\)
\(198\) −4.15143e12 −0.969481
\(199\) −5.73623e12 −1.30297 −0.651486 0.758661i \(-0.725854\pi\)
−0.651486 + 0.758661i \(0.725854\pi\)
\(200\) 6.50491e12 1.43740
\(201\) 2.22681e12 0.478745
\(202\) −3.44059e10 −0.00719780
\(203\) 6.45036e11 0.131327
\(204\) 6.13826e12 1.21641
\(205\) 1.08546e12 0.209396
\(206\) 1.37440e13 2.58133
\(207\) −2.66705e11 −0.0487747
\(208\) 9.41297e10 0.0167640
\(209\) 1.87627e13 3.25454
\(210\) −1.49657e12 −0.252865
\(211\) 5.59098e12 0.920310 0.460155 0.887839i \(-0.347794\pi\)
0.460155 + 0.887839i \(0.347794\pi\)
\(212\) 1.51976e13 2.43740
\(213\) −4.77828e12 −0.746765
\(214\) −2.05962e13 −3.13697
\(215\) 1.34249e12 0.199297
\(216\) −2.01261e12 −0.291249
\(217\) 1.30939e13 1.84732
\(218\) 1.47125e13 2.02384
\(219\) −7.44688e12 −0.998922
\(220\) 5.53866e12 0.724569
\(221\) 2.14527e11 0.0273732
\(222\) 4.48275e11 0.0557961
\(223\) −9.40765e12 −1.14236 −0.571182 0.820823i \(-0.693515\pi\)
−0.571182 + 0.820823i \(0.693515\pi\)
\(224\) 3.42436e12 0.405709
\(225\) −2.73850e12 −0.316598
\(226\) 1.12957e13 1.27443
\(227\) −4.54229e12 −0.500188 −0.250094 0.968222i \(-0.580461\pi\)
−0.250094 + 0.968222i \(0.580461\pi\)
\(228\) 1.93143e13 2.07605
\(229\) −7.76171e11 −0.0814446 −0.0407223 0.999171i \(-0.512966\pi\)
−0.0407223 + 0.999171i \(0.512966\pi\)
\(230\) 5.44075e11 0.0557385
\(231\) 1.13531e13 1.13566
\(232\) −1.76960e12 −0.172859
\(233\) −1.74016e13 −1.66009 −0.830046 0.557695i \(-0.811686\pi\)
−0.830046 + 0.557695i \(0.811686\pi\)
\(234\) −1.49355e11 −0.0139166
\(235\) 4.06927e12 0.370376
\(236\) 2.76755e12 0.246082
\(237\) −9.04157e12 −0.785467
\(238\) −2.56673e13 −2.17875
\(239\) −7.77017e12 −0.644528 −0.322264 0.946650i \(-0.604444\pi\)
−0.322264 + 0.946650i \(0.604444\pi\)
\(240\) 1.08934e12 0.0883086
\(241\) 8.55732e12 0.678023 0.339011 0.940782i \(-0.389908\pi\)
0.339011 + 0.940782i \(0.389908\pi\)
\(242\) −4.22948e13 −3.27569
\(243\) 8.47289e11 0.0641500
\(244\) 4.92036e12 0.364211
\(245\) 9.96785e11 0.0721420
\(246\) −1.29610e13 −0.917263
\(247\) 6.75021e11 0.0467180
\(248\) −3.59222e13 −2.43152
\(249\) −1.08226e13 −0.716532
\(250\) 1.14683e13 0.742726
\(251\) −3.85428e12 −0.244195 −0.122098 0.992518i \(-0.538962\pi\)
−0.122098 + 0.992518i \(0.538962\pi\)
\(252\) 1.16869e13 0.724427
\(253\) −4.12739e12 −0.250330
\(254\) 3.33163e13 1.97730
\(255\) 2.48268e12 0.144195
\(256\) −3.20938e13 −1.82432
\(257\) 1.67310e13 0.930870 0.465435 0.885082i \(-0.345898\pi\)
0.465435 + 0.885082i \(0.345898\pi\)
\(258\) −1.60300e13 −0.873024
\(259\) −1.22591e12 −0.0653599
\(260\) 1.99263e11 0.0104010
\(261\) 7.44984e11 0.0380736
\(262\) 2.02388e13 1.01281
\(263\) 3.61000e13 1.76909 0.884547 0.466451i \(-0.154468\pi\)
0.884547 + 0.466451i \(0.154468\pi\)
\(264\) −3.11462e13 −1.49480
\(265\) 6.14682e12 0.288934
\(266\) −8.07635e13 −3.71849
\(267\) 5.36023e12 0.241753
\(268\) 3.54742e13 1.56737
\(269\) 4.59887e12 0.199074 0.0995368 0.995034i \(-0.468264\pi\)
0.0995368 + 0.995034i \(0.468264\pi\)
\(270\) −1.72846e12 −0.0733091
\(271\) −5.83394e12 −0.242455 −0.121228 0.992625i \(-0.538683\pi\)
−0.121228 + 0.992625i \(0.538683\pi\)
\(272\) 1.86831e13 0.760888
\(273\) 4.08447e11 0.0163020
\(274\) 1.24384e13 0.486560
\(275\) −4.23796e13 −1.62490
\(276\) −4.24875e12 −0.159684
\(277\) 3.31954e13 1.22304 0.611518 0.791230i \(-0.290559\pi\)
0.611518 + 0.791230i \(0.290559\pi\)
\(278\) 7.62162e13 2.75297
\(279\) 1.51229e13 0.535563
\(280\) −1.12280e13 −0.389882
\(281\) −6.70673e12 −0.228363 −0.114182 0.993460i \(-0.536425\pi\)
−0.114182 + 0.993460i \(0.536425\pi\)
\(282\) −4.85892e13 −1.62244
\(283\) −4.61820e13 −1.51233 −0.756166 0.654379i \(-0.772930\pi\)
−0.756166 + 0.654379i \(0.772930\pi\)
\(284\) −7.61204e13 −2.44484
\(285\) 7.81187e12 0.246098
\(286\) −2.31135e12 −0.0714253
\(287\) 3.54448e13 1.07449
\(288\) 3.95496e12 0.117621
\(289\) 8.30812e12 0.242418
\(290\) −1.51976e12 −0.0435096
\(291\) −2.06186e13 −0.579225
\(292\) −1.18633e14 −3.27038
\(293\) −2.57384e13 −0.696321 −0.348160 0.937435i \(-0.613194\pi\)
−0.348160 + 0.937435i \(0.613194\pi\)
\(294\) −1.19021e13 −0.316020
\(295\) 1.11936e12 0.0291709
\(296\) 3.36320e12 0.0860295
\(297\) 1.31122e13 0.329242
\(298\) 8.12684e13 2.00324
\(299\) −1.48490e11 −0.00359341
\(300\) −4.36256e13 −1.03651
\(301\) 4.38379e13 1.02267
\(302\) 6.40924e13 1.46814
\(303\) 1.08670e11 0.00244442
\(304\) 5.87874e13 1.29861
\(305\) 1.99009e12 0.0431741
\(306\) −2.96445e13 −0.631649
\(307\) −4.18095e13 −0.875012 −0.437506 0.899216i \(-0.644138\pi\)
−0.437506 + 0.899216i \(0.644138\pi\)
\(308\) 1.80860e14 3.71803
\(309\) −4.34101e13 −0.876636
\(310\) −3.08504e13 −0.612028
\(311\) 2.49606e13 0.486488 0.243244 0.969965i \(-0.421788\pi\)
0.243244 + 0.969965i \(0.421788\pi\)
\(312\) −1.12054e12 −0.0214574
\(313\) 3.25889e13 0.613163 0.306582 0.951844i \(-0.400815\pi\)
0.306582 + 0.951844i \(0.400815\pi\)
\(314\) −9.16196e13 −1.69385
\(315\) 4.72687e12 0.0858748
\(316\) −1.44037e14 −2.57155
\(317\) 2.60297e13 0.456712 0.228356 0.973578i \(-0.426665\pi\)
0.228356 + 0.973578i \(0.426665\pi\)
\(318\) −7.33962e13 −1.26568
\(319\) 1.15290e13 0.195408
\(320\) −1.72490e13 −0.287369
\(321\) 6.50525e13 1.06534
\(322\) 1.77663e13 0.286015
\(323\) 1.33980e14 2.12044
\(324\) 1.34977e13 0.210021
\(325\) −1.52468e12 −0.0233250
\(326\) −1.46288e14 −2.20045
\(327\) −4.64692e13 −0.687311
\(328\) −9.72398e13 −1.41429
\(329\) 1.32878e14 1.90054
\(330\) −2.67487e13 −0.376250
\(331\) −1.38408e14 −1.91472 −0.957362 0.288890i \(-0.906714\pi\)
−0.957362 + 0.288890i \(0.906714\pi\)
\(332\) −1.72410e14 −2.34586
\(333\) −1.41587e12 −0.0189487
\(334\) −8.92650e13 −1.17510
\(335\) 1.43479e13 0.185798
\(336\) 3.55716e13 0.453145
\(337\) −1.29114e14 −1.61811 −0.809056 0.587731i \(-0.800021\pi\)
−0.809056 + 0.587731i \(0.800021\pi\)
\(338\) 1.37798e14 1.69903
\(339\) −3.56771e13 −0.432804
\(340\) 3.95503e13 0.472081
\(341\) 2.34034e14 2.74871
\(342\) −9.32778e13 −1.07804
\(343\) −6.85455e13 −0.779581
\(344\) −1.20266e14 −1.34608
\(345\) −1.71845e12 −0.0189292
\(346\) −4.40665e13 −0.477738
\(347\) 5.49638e13 0.586495 0.293248 0.956036i \(-0.405264\pi\)
0.293248 + 0.956036i \(0.405264\pi\)
\(348\) 1.18680e13 0.124650
\(349\) 1.07812e14 1.11462 0.557308 0.830306i \(-0.311834\pi\)
0.557308 + 0.830306i \(0.311834\pi\)
\(350\) 1.82422e14 1.85654
\(351\) 4.71736e11 0.00472617
\(352\) 6.12050e13 0.603673
\(353\) −3.68605e12 −0.0357932 −0.0178966 0.999840i \(-0.505697\pi\)
−0.0178966 + 0.999840i \(0.505697\pi\)
\(354\) −1.33658e13 −0.127784
\(355\) −3.07877e13 −0.289815
\(356\) 8.53912e13 0.791477
\(357\) 8.10697e13 0.739918
\(358\) 1.78430e14 1.60366
\(359\) 1.24750e14 1.10414 0.552068 0.833799i \(-0.313839\pi\)
0.552068 + 0.833799i \(0.313839\pi\)
\(360\) −1.29678e13 −0.113032
\(361\) 3.05084e14 2.61897
\(362\) −2.93271e14 −2.47955
\(363\) 1.33587e14 1.11244
\(364\) 6.50677e12 0.0533713
\(365\) −4.79821e13 −0.387676
\(366\) −2.37627e13 −0.189125
\(367\) −1.29296e14 −1.01373 −0.506864 0.862026i \(-0.669196\pi\)
−0.506864 + 0.862026i \(0.669196\pi\)
\(368\) −1.29320e13 −0.0998855
\(369\) 4.09369e13 0.311509
\(370\) 2.88835e12 0.0216541
\(371\) 2.00719e14 1.48263
\(372\) 2.40915e14 1.75338
\(373\) 9.93102e13 0.712189 0.356094 0.934450i \(-0.384108\pi\)
0.356094 + 0.934450i \(0.384108\pi\)
\(374\) −4.58763e14 −3.24186
\(375\) −3.62224e13 −0.252235
\(376\) −3.64541e14 −2.50157
\(377\) 4.14776e11 0.00280502
\(378\) −5.64413e13 −0.376177
\(379\) 8.97364e13 0.589458 0.294729 0.955581i \(-0.404771\pi\)
0.294729 + 0.955581i \(0.404771\pi\)
\(380\) 1.24447e14 0.805703
\(381\) −1.05229e14 −0.671503
\(382\) 1.95025e13 0.122671
\(383\) 2.51798e14 1.56120 0.780600 0.625031i \(-0.214914\pi\)
0.780600 + 0.625031i \(0.214914\pi\)
\(384\) 1.72630e14 1.05510
\(385\) 7.31506e13 0.440742
\(386\) 5.56172e14 3.30354
\(387\) 5.06305e13 0.296485
\(388\) −3.28465e14 −1.89633
\(389\) 7.40151e13 0.421306 0.210653 0.977561i \(-0.432441\pi\)
0.210653 + 0.977561i \(0.432441\pi\)
\(390\) −9.62334e11 −0.00540096
\(391\) −2.94728e13 −0.163098
\(392\) −8.92960e13 −0.487257
\(393\) −6.39239e13 −0.343957
\(394\) 1.27092e14 0.674354
\(395\) −5.82571e13 −0.304835
\(396\) 2.08884e14 1.07791
\(397\) −2.83014e14 −1.44033 −0.720163 0.693805i \(-0.755933\pi\)
−0.720163 + 0.693805i \(0.755933\pi\)
\(398\) 4.41322e14 2.21513
\(399\) 2.55090e14 1.26282
\(400\) −1.32784e14 −0.648361
\(401\) −1.30178e14 −0.626964 −0.313482 0.949594i \(-0.601496\pi\)
−0.313482 + 0.949594i \(0.601496\pi\)
\(402\) −1.71321e14 −0.813894
\(403\) 8.41978e12 0.0394569
\(404\) 1.73117e12 0.00800281
\(405\) 5.45930e12 0.0248963
\(406\) −4.96263e13 −0.223264
\(407\) −2.19113e13 −0.0972519
\(408\) −2.22408e14 −0.973913
\(409\) 1.36549e14 0.589944 0.294972 0.955506i \(-0.404690\pi\)
0.294972 + 0.955506i \(0.404690\pi\)
\(410\) −8.35107e13 −0.355984
\(411\) −3.92864e13 −0.165239
\(412\) −6.91545e14 −2.87003
\(413\) 3.65519e13 0.149687
\(414\) 2.05192e13 0.0829197
\(415\) −6.97329e13 −0.278082
\(416\) 2.20196e12 0.00866555
\(417\) −2.40727e14 −0.934925
\(418\) −1.44352e15 −5.53290
\(419\) −2.52548e14 −0.955359 −0.477680 0.878534i \(-0.658522\pi\)
−0.477680 + 0.878534i \(0.658522\pi\)
\(420\) 7.53014e13 0.281146
\(421\) −8.40559e13 −0.309754 −0.154877 0.987934i \(-0.549498\pi\)
−0.154877 + 0.987934i \(0.549498\pi\)
\(422\) −4.30146e14 −1.56458
\(423\) 1.53468e14 0.550992
\(424\) −5.50656e14 −1.95150
\(425\) −3.02623e14 −1.05868
\(426\) 3.67621e14 1.26954
\(427\) 6.49846e13 0.221543
\(428\) 1.03632e15 3.48781
\(429\) 7.30034e12 0.0242565
\(430\) −1.03286e14 −0.338816
\(431\) −4.02522e14 −1.30366 −0.651830 0.758365i \(-0.725998\pi\)
−0.651830 + 0.758365i \(0.725998\pi\)
\(432\) 4.10833e13 0.131373
\(433\) 6.12432e14 1.93364 0.966818 0.255467i \(-0.0822291\pi\)
0.966818 + 0.255467i \(0.0822291\pi\)
\(434\) −1.00739e15 −3.14055
\(435\) 4.80012e12 0.0147761
\(436\) −7.40279e14 −2.25019
\(437\) −9.27376e13 −0.278361
\(438\) 5.72931e14 1.69822
\(439\) −2.46527e14 −0.721622 −0.360811 0.932639i \(-0.617500\pi\)
−0.360811 + 0.932639i \(0.617500\pi\)
\(440\) −2.00683e14 −0.580124
\(441\) 3.75926e13 0.107322
\(442\) −1.65048e13 −0.0465359
\(443\) −2.24582e14 −0.625396 −0.312698 0.949853i \(-0.601233\pi\)
−0.312698 + 0.949853i \(0.601233\pi\)
\(444\) −2.25555e13 −0.0620363
\(445\) 3.45373e13 0.0938229
\(446\) 7.23785e14 1.94208
\(447\) −2.56684e14 −0.680312
\(448\) −5.63252e14 −1.47460
\(449\) −2.31424e14 −0.598486 −0.299243 0.954177i \(-0.596734\pi\)
−0.299243 + 0.954177i \(0.596734\pi\)
\(450\) 2.10688e14 0.538235
\(451\) 6.33519e14 1.59878
\(452\) −5.68354e14 −1.41696
\(453\) −2.02434e14 −0.498591
\(454\) 3.49465e14 0.850347
\(455\) 2.63173e12 0.00632671
\(456\) −6.99818e14 −1.66218
\(457\) −1.17751e14 −0.276329 −0.138165 0.990409i \(-0.544120\pi\)
−0.138165 + 0.990409i \(0.544120\pi\)
\(458\) 5.97153e13 0.138460
\(459\) 9.36314e13 0.214512
\(460\) −2.73757e13 −0.0619723
\(461\) 8.83754e13 0.197686 0.0988431 0.995103i \(-0.468486\pi\)
0.0988431 + 0.995103i \(0.468486\pi\)
\(462\) −8.73456e14 −1.93068
\(463\) −5.39769e13 −0.117900 −0.0589498 0.998261i \(-0.518775\pi\)
−0.0589498 + 0.998261i \(0.518775\pi\)
\(464\) 3.61228e13 0.0779709
\(465\) 9.74404e13 0.207849
\(466\) 1.33881e15 2.82225
\(467\) −2.11045e14 −0.439675 −0.219838 0.975536i \(-0.570553\pi\)
−0.219838 + 0.975536i \(0.570553\pi\)
\(468\) 7.51499e12 0.0154731
\(469\) 4.68518e14 0.953401
\(470\) −3.13072e14 −0.629661
\(471\) 2.89379e14 0.575243
\(472\) −1.00277e14 −0.197025
\(473\) 7.83533e14 1.52167
\(474\) 6.95620e14 1.33534
\(475\) −9.52219e14 −1.80685
\(476\) 1.29148e15 2.42242
\(477\) 2.31820e14 0.429834
\(478\) 5.97804e14 1.09573
\(479\) 1.49975e14 0.271752 0.135876 0.990726i \(-0.456615\pi\)
0.135876 + 0.990726i \(0.456615\pi\)
\(480\) 2.54828e13 0.0456479
\(481\) −7.88297e11 −0.00139602
\(482\) −6.58364e14 −1.15268
\(483\) −5.61144e13 −0.0971327
\(484\) 2.12811e15 3.64204
\(485\) −1.32851e14 −0.224794
\(486\) −6.51868e13 −0.109059
\(487\) 4.89329e13 0.0809452 0.0404726 0.999181i \(-0.487114\pi\)
0.0404726 + 0.999181i \(0.487114\pi\)
\(488\) −1.78280e14 −0.291604
\(489\) 4.62047e14 0.747287
\(490\) −7.66884e13 −0.122645
\(491\) 9.34061e14 1.47716 0.738579 0.674166i \(-0.235497\pi\)
0.738579 + 0.674166i \(0.235497\pi\)
\(492\) 6.52145e14 1.01985
\(493\) 8.23260e13 0.127315
\(494\) −5.19332e13 −0.0794232
\(495\) 8.44853e13 0.127777
\(496\) 7.33277e14 1.09678
\(497\) −1.00534e15 −1.48715
\(498\) 8.32646e14 1.21815
\(499\) 9.58838e14 1.38737 0.693685 0.720278i \(-0.255986\pi\)
0.693685 + 0.720278i \(0.255986\pi\)
\(500\) −5.77040e14 −0.825793
\(501\) 2.81942e14 0.399072
\(502\) 2.96532e14 0.415146
\(503\) −1.14612e15 −1.58710 −0.793552 0.608502i \(-0.791771\pi\)
−0.793552 + 0.608502i \(0.791771\pi\)
\(504\) −4.23452e14 −0.580011
\(505\) 7.00189e11 0.000948665 0
\(506\) 3.17544e14 0.425575
\(507\) −4.35232e14 −0.577002
\(508\) −1.67635e15 −2.19844
\(509\) 1.03284e15 1.33994 0.669971 0.742388i \(-0.266307\pi\)
0.669971 + 0.742388i \(0.266307\pi\)
\(510\) −1.91007e14 −0.245139
\(511\) −1.56681e15 −1.98931
\(512\) 1.01424e15 1.27395
\(513\) 2.94616e14 0.366109
\(514\) −1.28721e15 −1.58253
\(515\) −2.79702e14 −0.340217
\(516\) 8.06570e14 0.970664
\(517\) 2.37499e15 2.82790
\(518\) 9.43166e13 0.111115
\(519\) 1.39183e14 0.162243
\(520\) −7.21993e12 −0.00832750
\(521\) −8.31000e14 −0.948404 −0.474202 0.880416i \(-0.657263\pi\)
−0.474202 + 0.880416i \(0.657263\pi\)
\(522\) −5.73159e13 −0.0647273
\(523\) 1.40683e15 1.57211 0.786053 0.618159i \(-0.212121\pi\)
0.786053 + 0.618159i \(0.212121\pi\)
\(524\) −1.01834e15 −1.12608
\(525\) −5.76176e14 −0.630492
\(526\) −2.77738e15 −3.00756
\(527\) 1.67118e15 1.79088
\(528\) 6.35785e14 0.674254
\(529\) −9.32409e14 −0.978589
\(530\) −4.72910e14 −0.491204
\(531\) 4.22156e13 0.0433963
\(532\) 4.06371e15 4.13436
\(533\) 2.27920e13 0.0229500
\(534\) −4.12394e14 −0.410994
\(535\) 4.19150e14 0.413451
\(536\) −1.28534e15 −1.25491
\(537\) −5.63568e14 −0.544613
\(538\) −3.53818e14 −0.338437
\(539\) 5.81764e14 0.550819
\(540\) 8.69693e13 0.0815080
\(541\) −1.07327e15 −0.995693 −0.497846 0.867265i \(-0.665876\pi\)
−0.497846 + 0.867265i \(0.665876\pi\)
\(542\) 4.48839e14 0.412187
\(543\) 9.26291e14 0.842070
\(544\) 4.37051e14 0.393313
\(545\) −2.99413e14 −0.266741
\(546\) −3.14242e13 −0.0277143
\(547\) 5.35752e14 0.467771 0.233885 0.972264i \(-0.424856\pi\)
0.233885 + 0.972264i \(0.424856\pi\)
\(548\) −6.25853e14 −0.540977
\(549\) 7.50539e13 0.0642282
\(550\) 3.26051e15 2.76242
\(551\) 2.59043e14 0.217289
\(552\) 1.53945e14 0.127850
\(553\) −1.90233e15 −1.56422
\(554\) −2.55392e15 −2.07923
\(555\) −9.12280e12 −0.00735388
\(556\) −3.83491e15 −3.06086
\(557\) 1.83578e14 0.145083 0.0725414 0.997365i \(-0.476889\pi\)
0.0725414 + 0.997365i \(0.476889\pi\)
\(558\) −1.16349e15 −0.910487
\(559\) 2.81890e13 0.0218431
\(560\) 2.29197e14 0.175863
\(561\) 1.44899e15 1.10096
\(562\) 5.15988e14 0.388230
\(563\) −1.07635e15 −0.801968 −0.400984 0.916085i \(-0.631332\pi\)
−0.400984 + 0.916085i \(0.631332\pi\)
\(564\) 2.44482e15 1.80390
\(565\) −2.29877e14 −0.167969
\(566\) 3.55305e15 2.57105
\(567\) 1.78269e14 0.127752
\(568\) 2.75808e15 1.95745
\(569\) 7.24643e14 0.509339 0.254669 0.967028i \(-0.418033\pi\)
0.254669 + 0.967028i \(0.418033\pi\)
\(570\) −6.01012e14 −0.418381
\(571\) 4.18065e13 0.0288234 0.0144117 0.999896i \(-0.495412\pi\)
0.0144117 + 0.999896i \(0.495412\pi\)
\(572\) 1.16298e14 0.0794135
\(573\) −6.15983e13 −0.0416599
\(574\) −2.72697e15 −1.82669
\(575\) 2.09468e14 0.138978
\(576\) −6.50528e14 −0.427506
\(577\) −6.03500e14 −0.392835 −0.196417 0.980520i \(-0.562931\pi\)
−0.196417 + 0.980520i \(0.562931\pi\)
\(578\) −6.39192e14 −0.412124
\(579\) −1.75666e15 −1.12190
\(580\) 7.64683e13 0.0483757
\(581\) −2.27707e15 −1.42694
\(582\) 1.58631e15 0.984716
\(583\) 3.58753e15 2.20607
\(584\) 4.29843e15 2.61842
\(585\) 3.03951e12 0.00183420
\(586\) 1.98020e15 1.18378
\(587\) 5.57063e14 0.329910 0.164955 0.986301i \(-0.447252\pi\)
0.164955 + 0.986301i \(0.447252\pi\)
\(588\) 5.98869e14 0.351364
\(589\) 5.25846e15 3.05650
\(590\) −8.61192e13 −0.0495922
\(591\) −4.01416e14 −0.229015
\(592\) −6.86527e13 −0.0388050
\(593\) 1.34398e15 0.752649 0.376324 0.926488i \(-0.377188\pi\)
0.376324 + 0.926488i \(0.377188\pi\)
\(594\) −1.00880e15 −0.559730
\(595\) 5.22353e14 0.287158
\(596\) −4.08911e15 −2.22728
\(597\) −1.39390e15 −0.752271
\(598\) 1.14242e13 0.00610900
\(599\) −1.30206e14 −0.0689895 −0.0344947 0.999405i \(-0.510982\pi\)
−0.0344947 + 0.999405i \(0.510982\pi\)
\(600\) 1.58069e15 0.829881
\(601\) −8.49953e14 −0.442166 −0.221083 0.975255i \(-0.570959\pi\)
−0.221083 + 0.975255i \(0.570959\pi\)
\(602\) −3.37270e15 −1.73859
\(603\) 5.41115e14 0.276404
\(604\) −3.22488e15 −1.63234
\(605\) 8.60736e14 0.431733
\(606\) −8.36062e12 −0.00415565
\(607\) −1.32016e15 −0.650264 −0.325132 0.945669i \(-0.605409\pi\)
−0.325132 + 0.945669i \(0.605409\pi\)
\(608\) 1.37520e15 0.671270
\(609\) 1.56744e14 0.0758220
\(610\) −1.53109e14 −0.0733984
\(611\) 8.54446e13 0.0405936
\(612\) 1.49160e15 0.702293
\(613\) 3.37466e15 1.57470 0.787349 0.616508i \(-0.211453\pi\)
0.787349 + 0.616508i \(0.211453\pi\)
\(614\) 3.21665e15 1.48757
\(615\) 2.63767e14 0.120895
\(616\) −6.55312e15 −2.97683
\(617\) −3.99533e15 −1.79880 −0.899402 0.437123i \(-0.855998\pi\)
−0.899402 + 0.437123i \(0.855998\pi\)
\(618\) 3.33979e15 1.49033
\(619\) −3.73341e15 −1.65123 −0.825615 0.564234i \(-0.809172\pi\)
−0.825615 + 0.564234i \(0.809172\pi\)
\(620\) 1.55227e15 0.680478
\(621\) −6.48093e13 −0.0281601
\(622\) −1.92036e15 −0.827058
\(623\) 1.12779e15 0.481441
\(624\) 2.28735e13 0.00967872
\(625\) 2.03110e15 0.851904
\(626\) −2.50725e15 −1.04241
\(627\) 4.55933e15 1.87901
\(628\) 4.60995e15 1.88329
\(629\) −1.56464e14 −0.0633628
\(630\) −3.63665e14 −0.145992
\(631\) 2.36236e14 0.0940122 0.0470061 0.998895i \(-0.485032\pi\)
0.0470061 + 0.998895i \(0.485032\pi\)
\(632\) 5.21890e15 2.05890
\(633\) 1.35861e15 0.531341
\(634\) −2.00261e15 −0.776436
\(635\) −6.78017e14 −0.260606
\(636\) 3.69301e15 1.40724
\(637\) 2.09300e13 0.00790684
\(638\) −8.86992e14 −0.332205
\(639\) −1.16112e15 −0.431145
\(640\) 1.11230e15 0.409479
\(641\) −2.64828e15 −0.966597 −0.483298 0.875456i \(-0.660561\pi\)
−0.483298 + 0.875456i \(0.660561\pi\)
\(642\) −5.00487e15 −1.81113
\(643\) 3.97895e15 1.42760 0.713802 0.700347i \(-0.246971\pi\)
0.713802 + 0.700347i \(0.246971\pi\)
\(644\) −8.93931e14 −0.318003
\(645\) 3.26225e14 0.115064
\(646\) −1.03079e16 −3.60487
\(647\) −4.68166e15 −1.62340 −0.811701 0.584073i \(-0.801458\pi\)
−0.811701 + 0.584073i \(0.801458\pi\)
\(648\) −4.89065e14 −0.168153
\(649\) 6.53307e14 0.222726
\(650\) 1.17303e14 0.0396537
\(651\) 3.18183e15 1.06655
\(652\) 7.36064e15 2.44655
\(653\) −2.30151e15 −0.758561 −0.379281 0.925282i \(-0.623828\pi\)
−0.379281 + 0.925282i \(0.623828\pi\)
\(654\) 3.57515e15 1.16847
\(655\) −4.11878e14 −0.133488
\(656\) 1.98495e15 0.637938
\(657\) −1.80959e15 −0.576728
\(658\) −1.02231e16 −3.23103
\(659\) −2.06137e15 −0.646078 −0.323039 0.946386i \(-0.604705\pi\)
−0.323039 + 0.946386i \(0.604705\pi\)
\(660\) 1.34589e15 0.418330
\(661\) −3.47857e15 −1.07224 −0.536121 0.844141i \(-0.680111\pi\)
−0.536121 + 0.844141i \(0.680111\pi\)
\(662\) 1.06485e16 3.25514
\(663\) 5.21302e13 0.0158039
\(664\) 6.24695e15 1.87821
\(665\) 1.64361e15 0.490094
\(666\) 1.08931e14 0.0322139
\(667\) −5.69840e13 −0.0167132
\(668\) 4.49147e15 1.30653
\(669\) −2.28606e15 −0.659544
\(670\) −1.10387e15 −0.315868
\(671\) 1.16150e15 0.329643
\(672\) 8.32119e14 0.234236
\(673\) −8.65934e14 −0.241770 −0.120885 0.992667i \(-0.538573\pi\)
−0.120885 + 0.992667i \(0.538573\pi\)
\(674\) 9.93348e15 2.75088
\(675\) −6.65455e14 −0.182788
\(676\) −6.93347e15 −1.88905
\(677\) 5.97805e15 1.61556 0.807778 0.589487i \(-0.200670\pi\)
0.807778 + 0.589487i \(0.200670\pi\)
\(678\) 2.74485e15 0.735791
\(679\) −4.33813e15 −1.15350
\(680\) −1.43303e15 −0.377970
\(681\) −1.10378e15 −0.288784
\(682\) −1.80056e16 −4.67296
\(683\) −5.49833e15 −1.41552 −0.707762 0.706451i \(-0.750295\pi\)
−0.707762 + 0.706451i \(0.750295\pi\)
\(684\) 4.69338e15 1.19861
\(685\) −2.53133e14 −0.0641283
\(686\) 5.27360e15 1.32533
\(687\) −1.88610e14 −0.0470221
\(688\) 2.45498e15 0.607171
\(689\) 1.29068e14 0.0316674
\(690\) 1.32210e14 0.0321806
\(691\) 1.75970e15 0.424922 0.212461 0.977170i \(-0.431852\pi\)
0.212461 + 0.977170i \(0.431852\pi\)
\(692\) 2.21726e15 0.531169
\(693\) 2.75879e15 0.655672
\(694\) −4.22868e15 −0.997075
\(695\) −1.55107e15 −0.362839
\(696\) −4.30014e14 −0.0998002
\(697\) 4.52382e15 1.04166
\(698\) −8.29456e15 −1.89491
\(699\) −4.22860e15 −0.958455
\(700\) −9.17878e15 −2.06417
\(701\) −2.42930e13 −0.00542040 −0.00271020 0.999996i \(-0.500863\pi\)
−0.00271020 + 0.999996i \(0.500863\pi\)
\(702\) −3.62933e13 −0.00803476
\(703\) −4.92320e14 −0.108142
\(704\) −1.00672e16 −2.19412
\(705\) 9.88832e14 0.213837
\(706\) 2.83589e14 0.0608504
\(707\) 2.28641e13 0.00486796
\(708\) 6.72515e14 0.142076
\(709\) 9.48390e14 0.198807 0.0994037 0.995047i \(-0.468306\pi\)
0.0994037 + 0.995047i \(0.468306\pi\)
\(710\) 2.36867e15 0.492702
\(711\) −2.19710e15 −0.453489
\(712\) −3.09399e15 −0.633694
\(713\) −1.15675e15 −0.235097
\(714\) −6.23716e15 −1.25790
\(715\) 4.70380e13 0.00941380
\(716\) −8.97792e15 −1.78301
\(717\) −1.88815e15 −0.372118
\(718\) −9.59777e15 −1.87709
\(719\) 4.17730e14 0.0810749 0.0405375 0.999178i \(-0.487093\pi\)
0.0405375 + 0.999178i \(0.487093\pi\)
\(720\) 2.64710e14 0.0509850
\(721\) −9.13344e15 −1.74578
\(722\) −2.34719e16 −4.45240
\(723\) 2.07943e15 0.391457
\(724\) 1.47563e16 2.75686
\(725\) −5.85105e14 −0.108486
\(726\) −1.02776e16 −1.89122
\(727\) −1.11557e15 −0.203731 −0.101866 0.994798i \(-0.532481\pi\)
−0.101866 + 0.994798i \(0.532481\pi\)
\(728\) −2.35761e14 −0.0427315
\(729\) 2.05891e14 0.0370370
\(730\) 3.69154e15 0.659071
\(731\) 5.59504e15 0.991419
\(732\) 1.19565e15 0.210277
\(733\) −1.69075e15 −0.295126 −0.147563 0.989053i \(-0.547143\pi\)
−0.147563 + 0.989053i \(0.547143\pi\)
\(734\) 9.94749e15 1.72340
\(735\) 2.42219e14 0.0416512
\(736\) −3.02516e14 −0.0516321
\(737\) 8.37401e15 1.41861
\(738\) −3.14951e15 −0.529582
\(739\) −2.83296e15 −0.472820 −0.236410 0.971653i \(-0.575971\pi\)
−0.236410 + 0.971653i \(0.575971\pi\)
\(740\) −1.45331e14 −0.0240759
\(741\) 1.64030e14 0.0269726
\(742\) −1.54425e16 −2.52055
\(743\) 4.85393e15 0.786421 0.393210 0.919448i \(-0.371364\pi\)
0.393210 + 0.919448i \(0.371364\pi\)
\(744\) −8.72909e15 −1.40384
\(745\) −1.65388e15 −0.264025
\(746\) −7.64051e15 −1.21076
\(747\) −2.62990e15 −0.413690
\(748\) 2.30832e16 3.60443
\(749\) 1.36870e16 2.12157
\(750\) 2.78679e15 0.428813
\(751\) 3.71833e15 0.567974 0.283987 0.958828i \(-0.408343\pi\)
0.283987 + 0.958828i \(0.408343\pi\)
\(752\) 7.44135e15 1.12838
\(753\) −9.36589e14 −0.140986
\(754\) −3.19111e13 −0.00476870
\(755\) −1.30434e15 −0.193500
\(756\) 2.83991e15 0.418248
\(757\) 6.65316e15 0.972749 0.486374 0.873750i \(-0.338319\pi\)
0.486374 + 0.873750i \(0.338319\pi\)
\(758\) −6.90394e15 −1.00211
\(759\) −1.00296e15 −0.144528
\(760\) −4.50911e15 −0.645083
\(761\) −5.97347e15 −0.848421 −0.424210 0.905564i \(-0.639448\pi\)
−0.424210 + 0.905564i \(0.639448\pi\)
\(762\) 8.09587e15 1.14159
\(763\) −9.77707e15 −1.36875
\(764\) −9.81292e14 −0.136391
\(765\) 6.03291e14 0.0832509
\(766\) −1.93722e16 −2.65413
\(767\) 2.35039e13 0.00319717
\(768\) −7.79878e15 −1.05327
\(769\) −3.70785e15 −0.497195 −0.248598 0.968607i \(-0.579970\pi\)
−0.248598 + 0.968607i \(0.579970\pi\)
\(770\) −5.62790e15 −0.749286
\(771\) 4.06563e15 0.537438
\(772\) −2.79845e16 −3.67301
\(773\) 3.54842e15 0.462432 0.231216 0.972902i \(-0.425730\pi\)
0.231216 + 0.972902i \(0.425730\pi\)
\(774\) −3.89530e15 −0.504041
\(775\) −1.18774e16 −1.52602
\(776\) 1.19013e16 1.51829
\(777\) −2.97897e14 −0.0377355
\(778\) −5.69441e15 −0.716244
\(779\) 1.42344e16 1.77780
\(780\) 4.84210e13 0.00600500
\(781\) −1.79689e16 −2.21280
\(782\) 2.26751e15 0.277276
\(783\) 1.81031e14 0.0219818
\(784\) 1.82279e15 0.219785
\(785\) 1.86454e15 0.223249
\(786\) 4.91803e15 0.584746
\(787\) 8.76529e15 1.03492 0.517458 0.855709i \(-0.326878\pi\)
0.517458 + 0.855709i \(0.326878\pi\)
\(788\) −6.39477e15 −0.749774
\(789\) 8.77231e15 1.02139
\(790\) 4.48206e15 0.518237
\(791\) −7.50642e15 −0.861911
\(792\) −7.56853e15 −0.863024
\(793\) 4.17870e13 0.00473193
\(794\) 2.17739e16 2.44863
\(795\) 1.49368e15 0.166816
\(796\) −2.22056e16 −2.46287
\(797\) 7.06335e15 0.778019 0.389009 0.921234i \(-0.372817\pi\)
0.389009 + 0.921234i \(0.372817\pi\)
\(798\) −1.96255e16 −2.14687
\(799\) 1.69593e16 1.84247
\(800\) −3.10620e15 −0.335146
\(801\) 1.30254e15 0.139576
\(802\) 1.00153e16 1.06587
\(803\) −2.80043e16 −2.95999
\(804\) 8.62023e15 0.904921
\(805\) −3.61559e14 −0.0376966
\(806\) −6.47783e14 −0.0670790
\(807\) 1.11753e15 0.114935
\(808\) −6.27257e13 −0.00640742
\(809\) −9.88415e15 −1.00282 −0.501409 0.865210i \(-0.667185\pi\)
−0.501409 + 0.865210i \(0.667185\pi\)
\(810\) −4.20015e14 −0.0423250
\(811\) −1.34709e16 −1.34828 −0.674141 0.738603i \(-0.735486\pi\)
−0.674141 + 0.738603i \(0.735486\pi\)
\(812\) 2.49701e15 0.248234
\(813\) −1.41765e15 −0.139981
\(814\) 1.68576e15 0.165334
\(815\) 2.97709e15 0.290018
\(816\) 4.54000e15 0.439299
\(817\) 1.76051e16 1.69206
\(818\) −1.05055e16 −1.00294
\(819\) 9.92526e13 0.00941197
\(820\) 4.20194e15 0.395798
\(821\) 2.04293e16 1.91147 0.955733 0.294235i \(-0.0950649\pi\)
0.955733 + 0.294235i \(0.0950649\pi\)
\(822\) 3.02253e15 0.280916
\(823\) 1.35924e16 1.25486 0.627431 0.778672i \(-0.284106\pi\)
0.627431 + 0.778672i \(0.284106\pi\)
\(824\) 2.50568e16 2.29788
\(825\) −1.02982e16 −0.938137
\(826\) −2.81215e15 −0.254476
\(827\) 9.45269e14 0.0849718 0.0424859 0.999097i \(-0.486472\pi\)
0.0424859 + 0.999097i \(0.486472\pi\)
\(828\) −1.03245e15 −0.0921935
\(829\) 1.79154e16 1.58919 0.794595 0.607140i \(-0.207683\pi\)
0.794595 + 0.607140i \(0.207683\pi\)
\(830\) 5.36495e15 0.472755
\(831\) 8.06649e15 0.706121
\(832\) −3.62187e14 −0.0314960
\(833\) 4.15425e15 0.358877
\(834\) 1.85205e16 1.58943
\(835\) 1.81662e15 0.154878
\(836\) 7.26324e16 6.15171
\(837\) 3.67485e15 0.309207
\(838\) 1.94300e16 1.62416
\(839\) 1.21669e15 0.101039 0.0505197 0.998723i \(-0.483912\pi\)
0.0505197 + 0.998723i \(0.483912\pi\)
\(840\) −2.72841e15 −0.225099
\(841\) −1.20413e16 −0.986954
\(842\) 6.46691e15 0.526599
\(843\) −1.62974e15 −0.131846
\(844\) 2.16433e16 1.73956
\(845\) −2.80431e15 −0.223931
\(846\) −1.18072e16 −0.936718
\(847\) 2.81066e16 2.21539
\(848\) 1.12405e16 0.880256
\(849\) −1.12222e16 −0.873146
\(850\) 2.32826e16 1.79981
\(851\) 1.08300e14 0.00831795
\(852\) −1.84973e16 −1.41153
\(853\) 5.65881e15 0.429048 0.214524 0.976719i \(-0.431180\pi\)
0.214524 + 0.976719i \(0.431180\pi\)
\(854\) −4.99964e15 −0.376635
\(855\) 1.89828e15 0.142085
\(856\) −3.75491e16 −2.79250
\(857\) −2.30030e16 −1.69977 −0.849885 0.526969i \(-0.823329\pi\)
−0.849885 + 0.526969i \(0.823329\pi\)
\(858\) −5.61657e14 −0.0412374
\(859\) −1.57086e16 −1.14597 −0.572986 0.819565i \(-0.694215\pi\)
−0.572986 + 0.819565i \(0.694215\pi\)
\(860\) 5.19694e15 0.376709
\(861\) 8.61308e15 0.620356
\(862\) 3.09683e16 2.21629
\(863\) −2.37841e16 −1.69132 −0.845662 0.533718i \(-0.820794\pi\)
−0.845662 + 0.533718i \(0.820794\pi\)
\(864\) 9.61055e14 0.0679083
\(865\) 8.96792e14 0.0629656
\(866\) −4.71180e16 −3.28729
\(867\) 2.01887e15 0.139960
\(868\) 5.06882e16 3.49179
\(869\) −3.40012e16 −2.32748
\(870\) −3.69301e14 −0.0251203
\(871\) 3.01270e14 0.0203637
\(872\) 2.68226e16 1.80161
\(873\) −5.01032e15 −0.334416
\(874\) 7.13484e15 0.473229
\(875\) −7.62114e15 −0.502314
\(876\) −2.88277e16 −1.88815
\(877\) 9.34304e15 0.608121 0.304061 0.952653i \(-0.401657\pi\)
0.304061 + 0.952653i \(0.401657\pi\)
\(878\) 1.89668e16 1.22680
\(879\) −6.25443e15 −0.402021
\(880\) 4.09652e15 0.261674
\(881\) 2.21334e16 1.40502 0.702509 0.711675i \(-0.252063\pi\)
0.702509 + 0.711675i \(0.252063\pi\)
\(882\) −2.89222e15 −0.182454
\(883\) 8.12953e15 0.509661 0.254831 0.966986i \(-0.417980\pi\)
0.254831 + 0.966986i \(0.417980\pi\)
\(884\) 8.30460e14 0.0517406
\(885\) 2.72006e14 0.0168419
\(886\) 1.72784e16 1.06321
\(887\) 4.28272e15 0.261903 0.130951 0.991389i \(-0.458197\pi\)
0.130951 + 0.991389i \(0.458197\pi\)
\(888\) 8.17256e14 0.0496692
\(889\) −2.21400e16 −1.33727
\(890\) −2.65716e15 −0.159504
\(891\) 3.18627e15 0.190088
\(892\) −3.64181e16 −2.15929
\(893\) 5.33632e16 3.14455
\(894\) 1.97482e16 1.15657
\(895\) −3.63121e15 −0.211361
\(896\) 3.63211e16 2.10119
\(897\) −3.60832e13 −0.00207466
\(898\) 1.78048e16 1.01746
\(899\) 3.23114e15 0.183517
\(900\) −1.06010e16 −0.598431
\(901\) 2.56178e16 1.43733
\(902\) −4.87403e16 −2.71801
\(903\) 1.06526e16 0.590437
\(904\) 2.05933e16 1.13449
\(905\) 5.96833e15 0.326803
\(906\) 1.55744e16 0.847632
\(907\) 2.31261e16 1.25101 0.625507 0.780219i \(-0.284892\pi\)
0.625507 + 0.780219i \(0.284892\pi\)
\(908\) −1.75837e16 −0.945451
\(909\) 2.64069e13 0.00141129
\(910\) −2.02474e14 −0.0107558
\(911\) 2.70095e16 1.42615 0.713075 0.701088i \(-0.247302\pi\)
0.713075 + 0.701088i \(0.247302\pi\)
\(912\) 1.42853e16 0.749754
\(913\) −4.06989e16 −2.12321
\(914\) 9.05929e15 0.469775
\(915\) 4.83592e14 0.0249266
\(916\) −3.00465e15 −0.153946
\(917\) −1.34495e16 −0.684976
\(918\) −7.20361e15 −0.364683
\(919\) −1.76645e16 −0.888926 −0.444463 0.895797i \(-0.646605\pi\)
−0.444463 + 0.895797i \(0.646605\pi\)
\(920\) 9.91909e14 0.0496180
\(921\) −1.01597e16 −0.505188
\(922\) −6.79923e15 −0.336078
\(923\) −6.46465e14 −0.0317640
\(924\) 4.39490e16 2.14661
\(925\) 1.11201e15 0.0539921
\(926\) 4.15275e15 0.200436
\(927\) −1.05487e16 −0.506126
\(928\) 8.45014e14 0.0403042
\(929\) −5.81360e15 −0.275651 −0.137825 0.990457i \(-0.544011\pi\)
−0.137825 + 0.990457i \(0.544011\pi\)
\(930\) −7.49665e15 −0.353355
\(931\) 1.30716e16 0.612497
\(932\) −6.73637e16 −3.13789
\(933\) 6.06542e15 0.280874
\(934\) 1.62369e16 0.747472
\(935\) 9.33623e15 0.427275
\(936\) −2.72292e14 −0.0123884
\(937\) 3.33891e16 1.51021 0.755104 0.655606i \(-0.227587\pi\)
0.755104 + 0.655606i \(0.227587\pi\)
\(938\) −3.60458e16 −1.62084
\(939\) 7.91910e15 0.354010
\(940\) 1.57526e16 0.700082
\(941\) 1.60096e15 0.0707357 0.0353678 0.999374i \(-0.488740\pi\)
0.0353678 + 0.999374i \(0.488740\pi\)
\(942\) −2.22636e16 −0.977946
\(943\) −3.13127e15 −0.136744
\(944\) 2.04695e15 0.0888712
\(945\) 1.14863e15 0.0495798
\(946\) −6.02817e16 −2.58693
\(947\) −2.59697e16 −1.10801 −0.554003 0.832515i \(-0.686900\pi\)
−0.554003 + 0.832515i \(0.686900\pi\)
\(948\) −3.50009e16 −1.48468
\(949\) −1.00751e15 −0.0424897
\(950\) 7.32597e16 3.07175
\(951\) 6.32521e15 0.263683
\(952\) −4.67944e16 −1.93951
\(953\) 1.89690e16 0.781687 0.390843 0.920457i \(-0.372183\pi\)
0.390843 + 0.920457i \(0.372183\pi\)
\(954\) −1.78353e16 −0.730741
\(955\) −3.96893e14 −0.0161680
\(956\) −3.00792e16 −1.21828
\(957\) 2.80155e15 0.112819
\(958\) −1.15384e16 −0.461994
\(959\) −8.26582e15 −0.329066
\(960\) −4.19151e15 −0.165913
\(961\) 4.01822e16 1.58145
\(962\) 6.06483e13 0.00237332
\(963\) 1.58078e16 0.615072
\(964\) 3.31264e16 1.28159
\(965\) −1.13186e16 −0.435404
\(966\) 4.31721e15 0.165131
\(967\) 1.04059e16 0.395761 0.197880 0.980226i \(-0.436594\pi\)
0.197880 + 0.980226i \(0.436594\pi\)
\(968\) −7.71081e16 −2.91599
\(969\) 3.25572e16 1.22424
\(970\) 1.02210e16 0.382162
\(971\) −2.97243e16 −1.10511 −0.552556 0.833476i \(-0.686347\pi\)
−0.552556 + 0.833476i \(0.686347\pi\)
\(972\) 3.27995e15 0.121256
\(973\) −5.06487e16 −1.86186
\(974\) −3.76469e15 −0.137611
\(975\) −3.70498e14 −0.0134667
\(976\) 3.63922e15 0.131533
\(977\) −4.20362e15 −0.151079 −0.0755394 0.997143i \(-0.524068\pi\)
−0.0755394 + 0.997143i \(0.524068\pi\)
\(978\) −3.55479e16 −1.27043
\(979\) 2.01574e16 0.716357
\(980\) 3.85867e15 0.136362
\(981\) −1.12920e16 −0.396819
\(982\) −7.18627e16 −2.51125
\(983\) −5.44435e15 −0.189192 −0.0945958 0.995516i \(-0.530156\pi\)
−0.0945958 + 0.995516i \(0.530156\pi\)
\(984\) −2.36293e16 −0.816540
\(985\) −2.58643e15 −0.0888794
\(986\) −6.33382e15 −0.216442
\(987\) 3.22895e16 1.09728
\(988\) 2.61308e15 0.0883059
\(989\) −3.87274e15 −0.130148
\(990\) −6.49994e15 −0.217228
\(991\) 3.98861e16 1.32561 0.662805 0.748792i \(-0.269366\pi\)
0.662805 + 0.748792i \(0.269366\pi\)
\(992\) 1.71534e16 0.566939
\(993\) −3.36331e16 −1.10547
\(994\) 7.73469e16 2.52824
\(995\) −8.98128e15 −0.291952
\(996\) −4.18956e16 −1.35438
\(997\) −5.81768e16 −1.87036 −0.935182 0.354168i \(-0.884764\pi\)
−0.935182 + 0.354168i \(0.884764\pi\)
\(998\) −7.37690e16 −2.35861
\(999\) −3.44056e14 −0.0109400
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.1.3 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.d.1.3 28 1.1 even 1 trivial