Properties

Label 177.12.a.d.1.2
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 1.2
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-79.8853 q^{2} +243.000 q^{3} +4333.66 q^{4} -5377.20 q^{5} -19412.1 q^{6} -58607.4 q^{7} -182591. q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-79.8853 q^{2} +243.000 q^{3} +4333.66 q^{4} -5377.20 q^{5} -19412.1 q^{6} -58607.4 q^{7} -182591. q^{8} +59049.0 q^{9} +429560. q^{10} -126129. q^{11} +1.05308e6 q^{12} +176292. q^{13} +4.68187e6 q^{14} -1.30666e6 q^{15} +5.71097e6 q^{16} +5.10282e6 q^{17} -4.71715e6 q^{18} -2.83908e6 q^{19} -2.33030e7 q^{20} -1.42416e7 q^{21} +1.00759e7 q^{22} -2.26275e7 q^{23} -4.43695e7 q^{24} -1.99138e7 q^{25} -1.40831e7 q^{26} +1.43489e7 q^{27} -2.53984e8 q^{28} -1.40162e8 q^{29} +1.04383e8 q^{30} -5.94487e7 q^{31} -8.22767e7 q^{32} -3.06495e7 q^{33} -4.07640e8 q^{34} +3.15144e8 q^{35} +2.55898e8 q^{36} +1.51143e8 q^{37} +2.26800e8 q^{38} +4.28390e7 q^{39} +9.81827e8 q^{40} +4.61623e6 q^{41} +1.13769e9 q^{42} -1.18990e9 q^{43} -5.46602e8 q^{44} -3.17519e8 q^{45} +1.80761e9 q^{46} +4.27766e8 q^{47} +1.38776e9 q^{48} +1.45750e9 q^{49} +1.59082e9 q^{50} +1.23999e9 q^{51} +7.63990e8 q^{52} -9.35982e7 q^{53} -1.14627e9 q^{54} +6.78224e8 q^{55} +1.07012e10 q^{56} -6.89896e8 q^{57} +1.11969e10 q^{58} +7.14924e8 q^{59} -5.66262e9 q^{60} -2.35775e8 q^{61} +4.74908e9 q^{62} -3.46071e9 q^{63} -5.12336e9 q^{64} -9.47959e8 q^{65} +2.44844e9 q^{66} -9.33033e9 q^{67} +2.21139e10 q^{68} -5.49849e9 q^{69} -2.51754e10 q^{70} +3.93536e9 q^{71} -1.07818e10 q^{72} -2.05323e10 q^{73} -1.20741e10 q^{74} -4.83905e9 q^{75} -1.23036e10 q^{76} +7.39212e9 q^{77} -3.42220e9 q^{78} -4.02922e10 q^{79} -3.07090e10 q^{80} +3.48678e9 q^{81} -3.68769e8 q^{82} -1.16799e10 q^{83} -6.17182e10 q^{84} -2.74389e10 q^{85} +9.50558e10 q^{86} -3.40594e10 q^{87} +2.30301e10 q^{88} -3.48789e9 q^{89} +2.53651e10 q^{90} -1.03320e10 q^{91} -9.80600e10 q^{92} -1.44460e10 q^{93} -3.41722e10 q^{94} +1.52663e10 q^{95} -1.99932e10 q^{96} -7.99476e10 q^{97} -1.16433e11 q^{98} -7.44782e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + O(q^{10}) \) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + 616281q^{10} + 1082362q^{11} + 7099002q^{12} + 503712q^{13} + 1321669q^{14} + 6454566q^{15} + 34870338q^{16} + 13513579q^{17} + 5668704q^{18} + 35971687q^{19} + 96105997q^{20} + 34586919q^{21} - 47598882q^{22} + 61380539q^{23} + 80756433q^{24} + 294744746q^{25} + 62820734q^{26} + 401769396q^{27} + 148068294q^{28} + 322339307q^{29} + 149756283q^{30} + 151247077q^{31} + 466383494q^{32} + 263013966q^{33} + 684479860q^{34} + 960297361q^{35} + 1725057486q^{36} + 863508437q^{37} + 992640509q^{38} + 122402016q^{39} + 3067680252q^{40} + 3081170377q^{41} + 321165567q^{42} + 2554238300q^{43} + 4350123570q^{44} + 1568459538q^{45} - 1987059155q^{46} + 6203398333q^{47} + 8473492134q^{48} + 10327857997q^{49} + 17577682253q^{50} + 3283799697q^{51} + 32137181618q^{52} + 14571770754q^{53} + 1377495072q^{54} + 18251419334q^{55} + 33498842836q^{56} + 8741119941q^{57} + 11860778276q^{58} + 20017880372q^{59} + 23353757271q^{60} + 2761613771q^{61} + 13785829526q^{62} + 8404621317q^{63} + 86547545293q^{64} + 32034985256q^{65} - 11566528326q^{66} + 39381333296q^{67} + 38995496621q^{68} + 14915470977q^{69} + 8551800364q^{70} + 26130020296q^{71} + 19623813219q^{72} + 41382402799q^{73} + 23815315058q^{74} + 71622973278q^{75} + 10611720128q^{76} - 8426124313q^{77} + 15265438362q^{78} + 59825111206q^{79} + 4009687655q^{80} + 97629963228q^{81} - 39592715115q^{82} + 35433122727q^{83} + 35980595442q^{84} - 8950496085q^{85} - 182032360688q^{86} + 78328451601q^{87} - 220003602335q^{88} + 102303043039q^{89} + 36390776769q^{90} - 111146323655q^{91} - 163000203526q^{92} + 36753039711q^{93} - 81314346008q^{94} + 208102168887q^{95} + 113331189042q^{96} - 171891031490q^{97} + 72304707792q^{98} + 63912393738q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −79.8853 −1.76523 −0.882616 0.470094i \(-0.844220\pi\)
−0.882616 + 0.470094i \(0.844220\pi\)
\(3\) 243.000 0.577350
\(4\) 4333.66 2.11604
\(5\) −5377.20 −0.769523 −0.384761 0.923016i \(-0.625716\pi\)
−0.384761 + 0.923016i \(0.625716\pi\)
\(6\) −19412.1 −1.01916
\(7\) −58607.4 −1.31799 −0.658996 0.752146i \(-0.729019\pi\)
−0.658996 + 0.752146i \(0.729019\pi\)
\(8\) −182591. −1.97008
\(9\) 59049.0 0.333333
\(10\) 429560. 1.35839
\(11\) −126129. −0.236133 −0.118067 0.993006i \(-0.537670\pi\)
−0.118067 + 0.993006i \(0.537670\pi\)
\(12\) 1.05308e6 1.22170
\(13\) 176292. 0.131687 0.0658437 0.997830i \(-0.479026\pi\)
0.0658437 + 0.997830i \(0.479026\pi\)
\(14\) 4.68187e6 2.32656
\(15\) −1.30666e6 −0.444284
\(16\) 5.71097e6 1.36160
\(17\) 5.10282e6 0.871648 0.435824 0.900032i \(-0.356457\pi\)
0.435824 + 0.900032i \(0.356457\pi\)
\(18\) −4.71715e6 −0.588411
\(19\) −2.83908e6 −0.263046 −0.131523 0.991313i \(-0.541987\pi\)
−0.131523 + 0.991313i \(0.541987\pi\)
\(20\) −2.33030e7 −1.62834
\(21\) −1.42416e7 −0.760943
\(22\) 1.00759e7 0.416830
\(23\) −2.26275e7 −0.733051 −0.366526 0.930408i \(-0.619453\pi\)
−0.366526 + 0.930408i \(0.619453\pi\)
\(24\) −4.43695e7 −1.13743
\(25\) −1.99138e7 −0.407835
\(26\) −1.40831e7 −0.232459
\(27\) 1.43489e7 0.192450
\(28\) −2.53984e8 −2.78893
\(29\) −1.40162e8 −1.26894 −0.634471 0.772947i \(-0.718782\pi\)
−0.634471 + 0.772947i \(0.718782\pi\)
\(30\) 1.04383e8 0.784265
\(31\) −5.94487e7 −0.372952 −0.186476 0.982459i \(-0.559707\pi\)
−0.186476 + 0.982459i \(0.559707\pi\)
\(32\) −8.22767e7 −0.433463
\(33\) −3.06495e7 −0.136331
\(34\) −4.07640e8 −1.53866
\(35\) 3.15144e8 1.01423
\(36\) 2.55898e8 0.705348
\(37\) 1.51143e8 0.358327 0.179163 0.983819i \(-0.442661\pi\)
0.179163 + 0.983819i \(0.442661\pi\)
\(38\) 2.26800e8 0.464338
\(39\) 4.28390e7 0.0760298
\(40\) 9.81827e8 1.51602
\(41\) 4.61623e6 0.00622266 0.00311133 0.999995i \(-0.499010\pi\)
0.00311133 + 0.999995i \(0.499010\pi\)
\(42\) 1.13769e9 1.34324
\(43\) −1.18990e9 −1.23434 −0.617170 0.786829i \(-0.711721\pi\)
−0.617170 + 0.786829i \(0.711721\pi\)
\(44\) −5.46602e8 −0.499668
\(45\) −3.17519e8 −0.256508
\(46\) 1.80761e9 1.29401
\(47\) 4.27766e8 0.272062 0.136031 0.990705i \(-0.456565\pi\)
0.136031 + 0.990705i \(0.456565\pi\)
\(48\) 1.38776e9 0.786120
\(49\) 1.45750e9 0.737104
\(50\) 1.59082e9 0.719923
\(51\) 1.23999e9 0.503246
\(52\) 7.63990e8 0.278657
\(53\) −9.35982e7 −0.0307433 −0.0153716 0.999882i \(-0.504893\pi\)
−0.0153716 + 0.999882i \(0.504893\pi\)
\(54\) −1.14627e9 −0.339719
\(55\) 6.78224e8 0.181710
\(56\) 1.07012e10 2.59655
\(57\) −6.89896e8 −0.151870
\(58\) 1.11969e10 2.23998
\(59\) 7.14924e8 0.130189
\(60\) −5.66262e9 −0.940125
\(61\) −2.35775e8 −0.0357423 −0.0178712 0.999840i \(-0.505689\pi\)
−0.0178712 + 0.999840i \(0.505689\pi\)
\(62\) 4.74908e9 0.658347
\(63\) −3.46071e9 −0.439331
\(64\) −5.12336e9 −0.596438
\(65\) −9.47959e8 −0.101337
\(66\) 2.44844e9 0.240657
\(67\) −9.33033e9 −0.844278 −0.422139 0.906531i \(-0.638721\pi\)
−0.422139 + 0.906531i \(0.638721\pi\)
\(68\) 2.21139e10 1.84445
\(69\) −5.49849e9 −0.423227
\(70\) −2.51754e10 −1.79034
\(71\) 3.93536e9 0.258859 0.129430 0.991589i \(-0.458685\pi\)
0.129430 + 0.991589i \(0.458685\pi\)
\(72\) −1.07818e10 −0.656693
\(73\) −2.05323e10 −1.15921 −0.579605 0.814898i \(-0.696793\pi\)
−0.579605 + 0.814898i \(0.696793\pi\)
\(74\) −1.20741e10 −0.632530
\(75\) −4.83905e9 −0.235463
\(76\) −1.23036e10 −0.556618
\(77\) 7.39212e9 0.311222
\(78\) −3.42220e9 −0.134210
\(79\) −4.02922e10 −1.47323 −0.736617 0.676310i \(-0.763578\pi\)
−0.736617 + 0.676310i \(0.763578\pi\)
\(80\) −3.07090e10 −1.04778
\(81\) 3.48678e9 0.111111
\(82\) −3.68769e8 −0.0109844
\(83\) −1.16799e10 −0.325470 −0.162735 0.986670i \(-0.552032\pi\)
−0.162735 + 0.986670i \(0.552032\pi\)
\(84\) −6.17182e10 −1.61019
\(85\) −2.74389e10 −0.670753
\(86\) 9.50558e10 2.17890
\(87\) −3.40594e10 −0.732624
\(88\) 2.30301e10 0.465201
\(89\) −3.48789e9 −0.0662091 −0.0331046 0.999452i \(-0.510539\pi\)
−0.0331046 + 0.999452i \(0.510539\pi\)
\(90\) 2.53651e10 0.452795
\(91\) −1.03320e10 −0.173563
\(92\) −9.80600e10 −1.55117
\(93\) −1.44460e10 −0.215324
\(94\) −3.41722e10 −0.480253
\(95\) 1.52663e10 0.202420
\(96\) −1.99932e10 −0.250260
\(97\) −7.99476e10 −0.945281 −0.472640 0.881255i \(-0.656699\pi\)
−0.472640 + 0.881255i \(0.656699\pi\)
\(98\) −1.16433e11 −1.30116
\(99\) −7.44782e9 −0.0787110
\(100\) −8.62996e10 −0.862996
\(101\) −2.66255e10 −0.252076 −0.126038 0.992025i \(-0.540226\pi\)
−0.126038 + 0.992025i \(0.540226\pi\)
\(102\) −9.90566e10 −0.888347
\(103\) −3.83751e10 −0.326170 −0.163085 0.986612i \(-0.552145\pi\)
−0.163085 + 0.986612i \(0.552145\pi\)
\(104\) −3.21893e10 −0.259435
\(105\) 7.65799e10 0.585563
\(106\) 7.47712e9 0.0542690
\(107\) −2.47057e9 −0.0170289 −0.00851445 0.999964i \(-0.502710\pi\)
−0.00851445 + 0.999964i \(0.502710\pi\)
\(108\) 6.21833e10 0.407233
\(109\) 1.54440e11 0.961419 0.480710 0.876880i \(-0.340379\pi\)
0.480710 + 0.876880i \(0.340379\pi\)
\(110\) −5.41801e10 −0.320760
\(111\) 3.67278e10 0.206880
\(112\) −3.34705e11 −1.79458
\(113\) 6.20470e10 0.316803 0.158401 0.987375i \(-0.449366\pi\)
0.158401 + 0.987375i \(0.449366\pi\)
\(114\) 5.51125e10 0.268086
\(115\) 1.21673e11 0.564099
\(116\) −6.07415e11 −2.68514
\(117\) 1.04099e10 0.0438958
\(118\) −5.71119e10 −0.229814
\(119\) −2.99063e11 −1.14883
\(120\) 2.38584e11 0.875275
\(121\) −2.69403e11 −0.944241
\(122\) 1.88349e10 0.0630935
\(123\) 1.12174e9 0.00359266
\(124\) −2.57631e11 −0.789184
\(125\) 3.69639e11 1.08336
\(126\) 2.76460e11 0.775521
\(127\) 1.67974e10 0.0451151 0.0225576 0.999746i \(-0.492819\pi\)
0.0225576 + 0.999746i \(0.492819\pi\)
\(128\) 5.77784e11 1.48631
\(129\) −2.89146e11 −0.712647
\(130\) 7.57280e10 0.178882
\(131\) 2.57616e11 0.583420 0.291710 0.956507i \(-0.405776\pi\)
0.291710 + 0.956507i \(0.405776\pi\)
\(132\) −1.32824e11 −0.288484
\(133\) 1.66391e11 0.346693
\(134\) 7.45356e11 1.49035
\(135\) −7.71570e10 −0.148095
\(136\) −9.31727e11 −1.71722
\(137\) −3.64707e11 −0.645626 −0.322813 0.946463i \(-0.604628\pi\)
−0.322813 + 0.946463i \(0.604628\pi\)
\(138\) 4.39249e11 0.747094
\(139\) −8.98927e10 −0.146941 −0.0734705 0.997297i \(-0.523407\pi\)
−0.0734705 + 0.997297i \(0.523407\pi\)
\(140\) 1.36573e12 2.14615
\(141\) 1.03947e11 0.157075
\(142\) −3.14378e11 −0.456947
\(143\) −2.22356e10 −0.0310958
\(144\) 3.37227e11 0.453867
\(145\) 7.53680e11 0.976479
\(146\) 1.64023e12 2.04627
\(147\) 3.54172e11 0.425567
\(148\) 6.55003e11 0.758235
\(149\) 2.74713e11 0.306446 0.153223 0.988192i \(-0.451035\pi\)
0.153223 + 0.988192i \(0.451035\pi\)
\(150\) 3.86569e11 0.415648
\(151\) 5.01401e11 0.519770 0.259885 0.965640i \(-0.416315\pi\)
0.259885 + 0.965640i \(0.416315\pi\)
\(152\) 5.18389e11 0.518222
\(153\) 3.01317e11 0.290549
\(154\) −5.90521e11 −0.549378
\(155\) 3.19668e11 0.286995
\(156\) 1.85650e11 0.160882
\(157\) −1.56386e12 −1.30843 −0.654215 0.756309i \(-0.727001\pi\)
−0.654215 + 0.756309i \(0.727001\pi\)
\(158\) 3.21875e12 2.60060
\(159\) −2.27444e10 −0.0177496
\(160\) 4.42419e11 0.333560
\(161\) 1.32614e12 0.966156
\(162\) −2.78543e11 −0.196137
\(163\) −5.56452e11 −0.378788 −0.189394 0.981901i \(-0.560652\pi\)
−0.189394 + 0.981901i \(0.560652\pi\)
\(164\) 2.00052e10 0.0131674
\(165\) 1.64808e11 0.104910
\(166\) 9.33055e11 0.574530
\(167\) 1.46011e11 0.0869853 0.0434927 0.999054i \(-0.486151\pi\)
0.0434927 + 0.999054i \(0.486151\pi\)
\(168\) 2.60038e12 1.49912
\(169\) −1.76108e12 −0.982658
\(170\) 2.19197e12 1.18404
\(171\) −1.67645e11 −0.0876821
\(172\) −5.15664e12 −2.61192
\(173\) −7.04663e11 −0.345723 −0.172861 0.984946i \(-0.555301\pi\)
−0.172861 + 0.984946i \(0.555301\pi\)
\(174\) 2.72084e12 1.29325
\(175\) 1.16710e12 0.537523
\(176\) −7.20321e11 −0.321519
\(177\) 1.73727e11 0.0751646
\(178\) 2.78631e11 0.116874
\(179\) −2.74310e11 −0.111571 −0.0557853 0.998443i \(-0.517766\pi\)
−0.0557853 + 0.998443i \(0.517766\pi\)
\(180\) −1.37602e12 −0.542782
\(181\) −5.00809e12 −1.91620 −0.958098 0.286442i \(-0.907528\pi\)
−0.958098 + 0.286442i \(0.907528\pi\)
\(182\) 8.25376e11 0.306379
\(183\) −5.72932e10 −0.0206358
\(184\) 4.13157e12 1.44417
\(185\) −8.12728e11 −0.275741
\(186\) 1.15403e12 0.380097
\(187\) −6.43616e11 −0.205825
\(188\) 1.85379e12 0.575695
\(189\) −8.40952e11 −0.253648
\(190\) −1.21955e12 −0.357319
\(191\) 1.75218e12 0.498763 0.249381 0.968405i \(-0.419773\pi\)
0.249381 + 0.968405i \(0.419773\pi\)
\(192\) −1.24498e12 −0.344354
\(193\) −1.82880e12 −0.491589 −0.245794 0.969322i \(-0.579049\pi\)
−0.245794 + 0.969322i \(0.579049\pi\)
\(194\) 6.38663e12 1.66864
\(195\) −2.30354e11 −0.0585067
\(196\) 6.31629e12 1.55975
\(197\) 6.33272e12 1.52064 0.760319 0.649550i \(-0.225042\pi\)
0.760319 + 0.649550i \(0.225042\pi\)
\(198\) 5.94971e11 0.138943
\(199\) 5.27508e12 1.19822 0.599110 0.800666i \(-0.295521\pi\)
0.599110 + 0.800666i \(0.295521\pi\)
\(200\) 3.63607e12 0.803466
\(201\) −2.26727e12 −0.487444
\(202\) 2.12699e12 0.444972
\(203\) 8.21453e12 1.67246
\(204\) 5.37368e12 1.06489
\(205\) −2.48224e10 −0.00478848
\(206\) 3.06560e12 0.575766
\(207\) −1.33613e12 −0.244350
\(208\) 1.00680e12 0.179306
\(209\) 3.58091e11 0.0621139
\(210\) −6.11761e12 −1.03366
\(211\) 6.90227e12 1.13616 0.568079 0.822974i \(-0.307687\pi\)
0.568079 + 0.822974i \(0.307687\pi\)
\(212\) −4.05623e11 −0.0650541
\(213\) 9.56293e11 0.149453
\(214\) 1.97362e11 0.0300600
\(215\) 6.39835e12 0.949854
\(216\) −2.61998e12 −0.379142
\(217\) 3.48413e12 0.491548
\(218\) −1.23375e13 −1.69713
\(219\) −4.98935e12 −0.669270
\(220\) 2.93919e12 0.384506
\(221\) 8.99587e11 0.114785
\(222\) −2.93401e12 −0.365191
\(223\) 1.34382e13 1.63179 0.815893 0.578202i \(-0.196246\pi\)
0.815893 + 0.578202i \(0.196246\pi\)
\(224\) 4.82202e12 0.571301
\(225\) −1.17589e12 −0.135945
\(226\) −4.95664e12 −0.559231
\(227\) −6.57119e11 −0.0723605 −0.0361803 0.999345i \(-0.511519\pi\)
−0.0361803 + 0.999345i \(0.511519\pi\)
\(228\) −2.98977e12 −0.321363
\(229\) 2.99837e12 0.314623 0.157312 0.987549i \(-0.449717\pi\)
0.157312 + 0.987549i \(0.449717\pi\)
\(230\) −9.71987e12 −0.995767
\(231\) 1.79628e12 0.179684
\(232\) 2.55923e13 2.49991
\(233\) 1.19755e13 1.14245 0.571224 0.820794i \(-0.306469\pi\)
0.571224 + 0.820794i \(0.306469\pi\)
\(234\) −8.31596e11 −0.0774863
\(235\) −2.30018e12 −0.209358
\(236\) 3.09824e12 0.275486
\(237\) −9.79100e12 −0.850572
\(238\) 2.38907e13 2.02795
\(239\) −6.84363e12 −0.567673 −0.283836 0.958873i \(-0.591607\pi\)
−0.283836 + 0.958873i \(0.591607\pi\)
\(240\) −7.46229e12 −0.604938
\(241\) −1.27423e12 −0.100961 −0.0504807 0.998725i \(-0.516075\pi\)
−0.0504807 + 0.998725i \(0.516075\pi\)
\(242\) 2.15213e13 1.66680
\(243\) 8.47289e11 0.0641500
\(244\) −1.02177e12 −0.0756324
\(245\) −7.83726e12 −0.567219
\(246\) −8.96109e10 −0.00634187
\(247\) −5.00507e11 −0.0346399
\(248\) 1.08548e13 0.734745
\(249\) −2.83822e12 −0.187910
\(250\) −2.95287e13 −1.91238
\(251\) −5.37908e12 −0.340802 −0.170401 0.985375i \(-0.554506\pi\)
−0.170401 + 0.985375i \(0.554506\pi\)
\(252\) −1.49975e13 −0.929644
\(253\) 2.85400e12 0.173098
\(254\) −1.34187e12 −0.0796387
\(255\) −6.66766e12 −0.387260
\(256\) −3.56638e13 −2.02725
\(257\) −9.51524e12 −0.529404 −0.264702 0.964330i \(-0.585274\pi\)
−0.264702 + 0.964330i \(0.585274\pi\)
\(258\) 2.30986e13 1.25799
\(259\) −8.85811e12 −0.472272
\(260\) −4.10813e12 −0.214433
\(261\) −8.27643e12 −0.422980
\(262\) −2.05798e13 −1.02987
\(263\) −1.31906e13 −0.646409 −0.323205 0.946329i \(-0.604760\pi\)
−0.323205 + 0.946329i \(0.604760\pi\)
\(264\) 5.59630e12 0.268584
\(265\) 5.03296e11 0.0236576
\(266\) −1.32922e13 −0.611994
\(267\) −8.47558e11 −0.0382259
\(268\) −4.04345e13 −1.78653
\(269\) 2.67796e12 0.115922 0.0579612 0.998319i \(-0.481540\pi\)
0.0579612 + 0.998319i \(0.481540\pi\)
\(270\) 6.16371e12 0.261422
\(271\) 2.63328e13 1.09437 0.547187 0.837011i \(-0.315699\pi\)
0.547187 + 0.837011i \(0.315699\pi\)
\(272\) 2.91420e13 1.18684
\(273\) −2.51068e12 −0.100207
\(274\) 2.91347e13 1.13968
\(275\) 2.51172e12 0.0963032
\(276\) −2.38286e13 −0.895568
\(277\) −3.57231e12 −0.131617 −0.0658083 0.997832i \(-0.520963\pi\)
−0.0658083 + 0.997832i \(0.520963\pi\)
\(278\) 7.18110e12 0.259385
\(279\) −3.51039e12 −0.124317
\(280\) −5.75423e13 −1.99810
\(281\) 1.76159e12 0.0599820 0.0299910 0.999550i \(-0.490452\pi\)
0.0299910 + 0.999550i \(0.490452\pi\)
\(282\) −8.30385e12 −0.277274
\(283\) 4.50842e13 1.47638 0.738192 0.674591i \(-0.235680\pi\)
0.738192 + 0.674591i \(0.235680\pi\)
\(284\) 1.70545e13 0.547758
\(285\) 3.70971e12 0.116867
\(286\) 1.77630e12 0.0548912
\(287\) −2.70545e11 −0.00820143
\(288\) −4.85836e12 −0.144488
\(289\) −8.23310e12 −0.240229
\(290\) −6.02080e13 −1.72371
\(291\) −1.94273e13 −0.545758
\(292\) −8.89800e13 −2.45294
\(293\) −2.38021e12 −0.0643938 −0.0321969 0.999482i \(-0.510250\pi\)
−0.0321969 + 0.999482i \(0.510250\pi\)
\(294\) −2.82931e13 −0.751225
\(295\) −3.84429e12 −0.100183
\(296\) −2.75973e13 −0.705932
\(297\) −1.80982e12 −0.0454438
\(298\) −2.19455e13 −0.540948
\(299\) −3.98906e12 −0.0965336
\(300\) −2.09708e13 −0.498251
\(301\) 6.97371e13 1.62685
\(302\) −4.00545e13 −0.917516
\(303\) −6.47000e12 −0.145536
\(304\) −1.62139e13 −0.358164
\(305\) 1.26781e12 0.0275045
\(306\) −2.40708e13 −0.512887
\(307\) 7.90940e13 1.65532 0.827661 0.561228i \(-0.189671\pi\)
0.827661 + 0.561228i \(0.189671\pi\)
\(308\) 3.20349e13 0.658559
\(309\) −9.32514e12 −0.188314
\(310\) −2.55368e13 −0.506613
\(311\) 1.06285e12 0.0207152 0.0103576 0.999946i \(-0.496703\pi\)
0.0103576 + 0.999946i \(0.496703\pi\)
\(312\) −7.82199e12 −0.149785
\(313\) 3.83621e12 0.0721786 0.0360893 0.999349i \(-0.488510\pi\)
0.0360893 + 0.999349i \(0.488510\pi\)
\(314\) 1.24930e14 2.30968
\(315\) 1.86089e13 0.338075
\(316\) −1.74613e14 −3.11743
\(317\) 8.24316e13 1.44633 0.723166 0.690674i \(-0.242686\pi\)
0.723166 + 0.690674i \(0.242686\pi\)
\(318\) 1.81694e12 0.0313322
\(319\) 1.76786e13 0.299639
\(320\) 2.75494e13 0.458973
\(321\) −6.00349e11 −0.00983164
\(322\) −1.05939e14 −1.70549
\(323\) −1.44873e13 −0.229284
\(324\) 1.51105e13 0.235116
\(325\) −3.51065e12 −0.0537067
\(326\) 4.44523e13 0.668649
\(327\) 3.75288e13 0.555076
\(328\) −8.42880e11 −0.0122591
\(329\) −2.50702e13 −0.358576
\(330\) −1.31658e13 −0.185191
\(331\) −2.46273e13 −0.340693 −0.170346 0.985384i \(-0.554489\pi\)
−0.170346 + 0.985384i \(0.554489\pi\)
\(332\) −5.06168e13 −0.688709
\(333\) 8.92486e12 0.119442
\(334\) −1.16642e13 −0.153549
\(335\) 5.01711e13 0.649691
\(336\) −8.13332e13 −1.03610
\(337\) 1.35539e14 1.69863 0.849315 0.527887i \(-0.177016\pi\)
0.849315 + 0.527887i \(0.177016\pi\)
\(338\) 1.40685e14 1.73462
\(339\) 1.50774e13 0.182906
\(340\) −1.18911e14 −1.41934
\(341\) 7.49824e12 0.0880664
\(342\) 1.33923e13 0.154779
\(343\) 3.04659e13 0.346494
\(344\) 2.17265e14 2.43175
\(345\) 2.95665e13 0.325683
\(346\) 5.62922e13 0.610281
\(347\) 1.50378e13 0.160462 0.0802312 0.996776i \(-0.474434\pi\)
0.0802312 + 0.996776i \(0.474434\pi\)
\(348\) −1.47602e14 −1.55026
\(349\) −1.11845e14 −1.15631 −0.578156 0.815926i \(-0.696228\pi\)
−0.578156 + 0.815926i \(0.696228\pi\)
\(350\) −9.32337e13 −0.948853
\(351\) 2.52960e12 0.0253433
\(352\) 1.03775e13 0.102355
\(353\) 9.34930e13 0.907858 0.453929 0.891038i \(-0.350022\pi\)
0.453929 + 0.891038i \(0.350022\pi\)
\(354\) −1.38782e13 −0.132683
\(355\) −2.11613e13 −0.199198
\(356\) −1.51153e13 −0.140101
\(357\) −7.26723e13 −0.663275
\(358\) 2.19133e13 0.196948
\(359\) 2.13254e14 1.88746 0.943731 0.330715i \(-0.107290\pi\)
0.943731 + 0.330715i \(0.107290\pi\)
\(360\) 5.79759e13 0.505340
\(361\) −1.08430e14 −0.930807
\(362\) 4.00073e14 3.38253
\(363\) −6.54649e13 −0.545158
\(364\) −4.47754e13 −0.367267
\(365\) 1.10406e14 0.892039
\(366\) 4.57689e12 0.0364271
\(367\) 1.48764e14 1.16636 0.583181 0.812343i \(-0.301808\pi\)
0.583181 + 0.812343i \(0.301808\pi\)
\(368\) −1.29225e14 −0.998123
\(369\) 2.72584e11 0.00207422
\(370\) 6.49250e13 0.486746
\(371\) 5.48554e12 0.0405194
\(372\) −6.26042e13 −0.455635
\(373\) 2.19989e14 1.57762 0.788808 0.614639i \(-0.210698\pi\)
0.788808 + 0.614639i \(0.210698\pi\)
\(374\) 5.14155e13 0.363329
\(375\) 8.98224e13 0.625479
\(376\) −7.81060e13 −0.535983
\(377\) −2.47095e13 −0.167104
\(378\) 6.71797e13 0.447747
\(379\) −2.90112e13 −0.190568 −0.0952840 0.995450i \(-0.530376\pi\)
−0.0952840 + 0.995450i \(0.530376\pi\)
\(380\) 6.61589e13 0.428330
\(381\) 4.08177e12 0.0260472
\(382\) −1.39973e14 −0.880432
\(383\) 2.44855e14 1.51815 0.759076 0.651002i \(-0.225651\pi\)
0.759076 + 0.651002i \(0.225651\pi\)
\(384\) 1.40401e14 0.858124
\(385\) −3.97489e13 −0.239492
\(386\) 1.46094e14 0.867768
\(387\) −7.02626e13 −0.411447
\(388\) −3.46466e14 −2.00026
\(389\) −2.12542e14 −1.20982 −0.604911 0.796293i \(-0.706791\pi\)
−0.604911 + 0.796293i \(0.706791\pi\)
\(390\) 1.84019e13 0.103278
\(391\) −1.15464e14 −0.638963
\(392\) −2.66125e14 −1.45215
\(393\) 6.26008e13 0.336838
\(394\) −5.05891e14 −2.68428
\(395\) 2.16659e14 1.13369
\(396\) −3.22763e13 −0.166556
\(397\) 1.74540e14 0.888277 0.444138 0.895958i \(-0.353510\pi\)
0.444138 + 0.895958i \(0.353510\pi\)
\(398\) −4.21401e14 −2.11514
\(399\) 4.04330e13 0.200163
\(400\) −1.13727e14 −0.555308
\(401\) 6.36342e13 0.306476 0.153238 0.988189i \(-0.451030\pi\)
0.153238 + 0.988189i \(0.451030\pi\)
\(402\) 1.81121e14 0.860452
\(403\) −1.04803e13 −0.0491131
\(404\) −1.15386e14 −0.533403
\(405\) −1.87492e13 −0.0855025
\(406\) −6.56220e14 −2.95227
\(407\) −1.90636e13 −0.0846128
\(408\) −2.26410e14 −0.991435
\(409\) −3.40083e14 −1.46929 −0.734643 0.678453i \(-0.762650\pi\)
−0.734643 + 0.678453i \(0.762650\pi\)
\(410\) 1.98295e12 0.00845278
\(411\) −8.86238e13 −0.372752
\(412\) −1.66304e14 −0.690191
\(413\) −4.18998e13 −0.171588
\(414\) 1.06737e14 0.431335
\(415\) 6.28054e13 0.250456
\(416\) −1.45047e13 −0.0570816
\(417\) −2.18439e13 −0.0848364
\(418\) −2.86062e13 −0.109646
\(419\) −4.07914e13 −0.154309 −0.0771545 0.997019i \(-0.524583\pi\)
−0.0771545 + 0.997019i \(0.524583\pi\)
\(420\) 3.31871e14 1.23908
\(421\) 3.69687e14 1.36233 0.681165 0.732130i \(-0.261474\pi\)
0.681165 + 0.732130i \(0.261474\pi\)
\(422\) −5.51390e14 −2.00558
\(423\) 2.52591e13 0.0906873
\(424\) 1.70901e13 0.0605666
\(425\) −1.01617e14 −0.355488
\(426\) −7.63938e13 −0.263818
\(427\) 1.38181e13 0.0471081
\(428\) −1.07066e13 −0.0360339
\(429\) −5.40326e12 −0.0179531
\(430\) −5.11134e14 −1.67671
\(431\) −3.53871e14 −1.14609 −0.573047 0.819523i \(-0.694239\pi\)
−0.573047 + 0.819523i \(0.694239\pi\)
\(432\) 8.19461e13 0.262040
\(433\) 4.32180e13 0.136452 0.0682261 0.997670i \(-0.478266\pi\)
0.0682261 + 0.997670i \(0.478266\pi\)
\(434\) −2.78331e14 −0.867697
\(435\) 1.83144e14 0.563771
\(436\) 6.69289e14 2.03441
\(437\) 6.42413e13 0.192826
\(438\) 3.98576e14 1.18142
\(439\) −1.72499e14 −0.504930 −0.252465 0.967606i \(-0.581241\pi\)
−0.252465 + 0.967606i \(0.581241\pi\)
\(440\) −1.23837e14 −0.357982
\(441\) 8.60637e13 0.245701
\(442\) −7.18638e13 −0.202622
\(443\) 1.12197e14 0.312435 0.156218 0.987723i \(-0.450070\pi\)
0.156218 + 0.987723i \(0.450070\pi\)
\(444\) 1.59166e14 0.437767
\(445\) 1.87551e13 0.0509494
\(446\) −1.07351e15 −2.88048
\(447\) 6.67552e13 0.176927
\(448\) 3.00267e14 0.786101
\(449\) 1.93532e14 0.500492 0.250246 0.968182i \(-0.419488\pi\)
0.250246 + 0.968182i \(0.419488\pi\)
\(450\) 9.39363e13 0.239974
\(451\) −5.82243e11 −0.00146938
\(452\) 2.68890e14 0.670369
\(453\) 1.21840e14 0.300090
\(454\) 5.24941e13 0.127733
\(455\) 5.55574e13 0.133561
\(456\) 1.25968e14 0.299196
\(457\) −4.90137e14 −1.15021 −0.575107 0.818078i \(-0.695040\pi\)
−0.575107 + 0.818078i \(0.695040\pi\)
\(458\) −2.39526e14 −0.555383
\(459\) 7.32199e13 0.167749
\(460\) 5.27289e14 1.19366
\(461\) −1.03639e14 −0.231829 −0.115915 0.993259i \(-0.536980\pi\)
−0.115915 + 0.993259i \(0.536980\pi\)
\(462\) −1.43497e14 −0.317184
\(463\) −3.02702e14 −0.661180 −0.330590 0.943775i \(-0.607248\pi\)
−0.330590 + 0.943775i \(0.607248\pi\)
\(464\) −8.00461e14 −1.72779
\(465\) 7.76793e13 0.165697
\(466\) −9.56667e14 −2.01668
\(467\) 4.70737e14 0.980698 0.490349 0.871526i \(-0.336869\pi\)
0.490349 + 0.871526i \(0.336869\pi\)
\(468\) 4.51128e13 0.0928855
\(469\) 5.46826e14 1.11275
\(470\) 1.83751e14 0.369565
\(471\) −3.80018e14 −0.755422
\(472\) −1.30538e14 −0.256482
\(473\) 1.50082e14 0.291469
\(474\) 7.82157e14 1.50146
\(475\) 5.65368e13 0.107279
\(476\) −1.29604e15 −2.43097
\(477\) −5.52688e12 −0.0102478
\(478\) 5.46706e14 1.00207
\(479\) 9.34660e14 1.69359 0.846795 0.531919i \(-0.178529\pi\)
0.846795 + 0.531919i \(0.178529\pi\)
\(480\) 1.07508e14 0.192581
\(481\) 2.66454e13 0.0471871
\(482\) 1.01793e14 0.178220
\(483\) 3.22252e14 0.557810
\(484\) −1.16750e15 −1.99806
\(485\) 4.29894e14 0.727415
\(486\) −6.76859e13 −0.113240
\(487\) −5.95529e13 −0.0985130 −0.0492565 0.998786i \(-0.515685\pi\)
−0.0492565 + 0.998786i \(0.515685\pi\)
\(488\) 4.30502e13 0.0704152
\(489\) −1.35218e14 −0.218693
\(490\) 6.26081e14 1.00127
\(491\) −1.56801e14 −0.247971 −0.123985 0.992284i \(-0.539568\pi\)
−0.123985 + 0.992284i \(0.539568\pi\)
\(492\) 4.86126e12 0.00760222
\(493\) −7.15222e14 −1.10607
\(494\) 3.99831e13 0.0611475
\(495\) 4.00484e13 0.0605699
\(496\) −3.39510e14 −0.507812
\(497\) −2.30641e14 −0.341175
\(498\) 2.26732e14 0.331705
\(499\) −3.26747e14 −0.472779 −0.236389 0.971658i \(-0.575964\pi\)
−0.236389 + 0.971658i \(0.575964\pi\)
\(500\) 1.60189e15 2.29244
\(501\) 3.54808e13 0.0502210
\(502\) 4.29709e14 0.601595
\(503\) 8.02323e14 1.11103 0.555514 0.831507i \(-0.312521\pi\)
0.555514 + 0.831507i \(0.312521\pi\)
\(504\) 6.31892e14 0.865516
\(505\) 1.43171e14 0.193978
\(506\) −2.27993e14 −0.305557
\(507\) −4.27943e14 −0.567338
\(508\) 7.27943e13 0.0954656
\(509\) 5.48322e14 0.711358 0.355679 0.934608i \(-0.384250\pi\)
0.355679 + 0.934608i \(0.384250\pi\)
\(510\) 5.32648e14 0.683603
\(511\) 1.20334e15 1.52783
\(512\) 1.66571e15 2.09226
\(513\) −4.07376e13 −0.0506233
\(514\) 7.60128e14 0.934522
\(515\) 2.06351e14 0.250995
\(516\) −1.25306e15 −1.50799
\(517\) −5.39539e13 −0.0642428
\(518\) 7.07632e14 0.833670
\(519\) −1.71233e14 −0.199603
\(520\) 1.73088e14 0.199641
\(521\) −1.28307e15 −1.46434 −0.732171 0.681121i \(-0.761493\pi\)
−0.732171 + 0.681121i \(0.761493\pi\)
\(522\) 6.61165e14 0.746659
\(523\) 1.03990e15 1.16207 0.581033 0.813880i \(-0.302649\pi\)
0.581033 + 0.813880i \(0.302649\pi\)
\(524\) 1.11642e15 1.23454
\(525\) 2.83604e14 0.310339
\(526\) 1.05373e15 1.14106
\(527\) −3.03356e14 −0.325083
\(528\) −1.75038e14 −0.185629
\(529\) −4.40804e14 −0.462636
\(530\) −4.02060e13 −0.0417612
\(531\) 4.22156e13 0.0433963
\(532\) 7.21081e14 0.733618
\(533\) 8.13805e11 0.000819447 0
\(534\) 6.77074e13 0.0674775
\(535\) 1.32848e13 0.0131041
\(536\) 1.70363e15 1.66329
\(537\) −6.66573e13 −0.0644153
\(538\) −2.13930e14 −0.204630
\(539\) −1.83833e14 −0.174055
\(540\) −3.34372e14 −0.313375
\(541\) 1.19638e14 0.110990 0.0554952 0.998459i \(-0.482326\pi\)
0.0554952 + 0.998459i \(0.482326\pi\)
\(542\) −2.10360e15 −1.93182
\(543\) −1.21697e15 −1.10632
\(544\) −4.19843e14 −0.377827
\(545\) −8.30453e14 −0.739834
\(546\) 2.00566e14 0.176888
\(547\) −8.13071e14 −0.709901 −0.354951 0.934885i \(-0.615502\pi\)
−0.354951 + 0.934885i \(0.615502\pi\)
\(548\) −1.58052e15 −1.36617
\(549\) −1.39223e13 −0.0119141
\(550\) −2.00649e14 −0.169998
\(551\) 3.97931e14 0.333790
\(552\) 1.00397e15 0.833791
\(553\) 2.36142e15 1.94171
\(554\) 2.85375e14 0.232334
\(555\) −1.97493e14 −0.159199
\(556\) −3.89564e14 −0.310934
\(557\) −2.24978e15 −1.77802 −0.889011 0.457887i \(-0.848607\pi\)
−0.889011 + 0.457887i \(0.848607\pi\)
\(558\) 2.80428e14 0.219449
\(559\) −2.09771e14 −0.162547
\(560\) 1.79978e15 1.38097
\(561\) −1.56399e14 −0.118833
\(562\) −1.40725e14 −0.105882
\(563\) −1.70588e15 −1.27102 −0.635508 0.772094i \(-0.719209\pi\)
−0.635508 + 0.772094i \(0.719209\pi\)
\(564\) 4.50471e14 0.332378
\(565\) −3.33639e14 −0.243787
\(566\) −3.60157e15 −2.60616
\(567\) −2.04351e14 −0.146444
\(568\) −7.18560e14 −0.509973
\(569\) −8.49138e14 −0.596844 −0.298422 0.954434i \(-0.596460\pi\)
−0.298422 + 0.954434i \(0.596460\pi\)
\(570\) −2.96351e14 −0.206298
\(571\) −2.05353e15 −1.41580 −0.707901 0.706312i \(-0.750358\pi\)
−0.707901 + 0.706312i \(0.750358\pi\)
\(572\) −9.63616e13 −0.0658000
\(573\) 4.25779e14 0.287961
\(574\) 2.16126e13 0.0144774
\(575\) 4.50600e14 0.298964
\(576\) −3.02529e14 −0.198813
\(577\) 1.13324e15 0.737659 0.368830 0.929497i \(-0.379759\pi\)
0.368830 + 0.929497i \(0.379759\pi\)
\(578\) 6.57704e14 0.424060
\(579\) −4.44399e14 −0.283819
\(580\) 3.26619e15 2.06627
\(581\) 6.84530e14 0.428967
\(582\) 1.55195e15 0.963390
\(583\) 1.18055e13 0.00725950
\(584\) 3.74901e15 2.28373
\(585\) −5.59760e13 −0.0337788
\(586\) 1.90144e14 0.113670
\(587\) 5.14998e13 0.0304998 0.0152499 0.999884i \(-0.495146\pi\)
0.0152499 + 0.999884i \(0.495146\pi\)
\(588\) 1.53486e15 0.900520
\(589\) 1.68780e14 0.0981038
\(590\) 3.07103e14 0.176847
\(591\) 1.53885e15 0.877941
\(592\) 8.63174e14 0.487898
\(593\) −5.81234e14 −0.325500 −0.162750 0.986667i \(-0.552036\pi\)
−0.162750 + 0.986667i \(0.552036\pi\)
\(594\) 1.44578e14 0.0802189
\(595\) 1.60812e15 0.884048
\(596\) 1.19051e15 0.648454
\(597\) 1.28184e15 0.691793
\(598\) 3.18667e14 0.170404
\(599\) 1.73245e15 0.917935 0.458968 0.888453i \(-0.348219\pi\)
0.458968 + 0.888453i \(0.348219\pi\)
\(600\) 8.83565e14 0.463881
\(601\) −6.41826e14 −0.333893 −0.166947 0.985966i \(-0.553391\pi\)
−0.166947 + 0.985966i \(0.553391\pi\)
\(602\) −5.57097e15 −2.87177
\(603\) −5.50947e14 −0.281426
\(604\) 2.17290e15 1.09986
\(605\) 1.44864e15 0.726615
\(606\) 5.16858e14 0.256905
\(607\) 1.01087e15 0.497916 0.248958 0.968514i \(-0.419912\pi\)
0.248958 + 0.968514i \(0.419912\pi\)
\(608\) 2.33590e14 0.114021
\(609\) 1.99613e15 0.965592
\(610\) −1.01279e14 −0.0485519
\(611\) 7.54118e13 0.0358272
\(612\) 1.30580e15 0.614816
\(613\) 2.84554e15 1.32780 0.663899 0.747823i \(-0.268901\pi\)
0.663899 + 0.747823i \(0.268901\pi\)
\(614\) −6.31845e15 −2.92203
\(615\) −6.03185e12 −0.00276463
\(616\) −1.34973e15 −0.613131
\(617\) 2.05036e15 0.923128 0.461564 0.887107i \(-0.347288\pi\)
0.461564 + 0.887107i \(0.347288\pi\)
\(618\) 7.44942e14 0.332419
\(619\) −1.26840e14 −0.0560991 −0.0280496 0.999607i \(-0.508930\pi\)
−0.0280496 + 0.999607i \(0.508930\pi\)
\(620\) 1.38533e15 0.607295
\(621\) −3.24680e14 −0.141076
\(622\) −8.49060e13 −0.0365671
\(623\) 2.04416e14 0.0872631
\(624\) 2.44652e14 0.103522
\(625\) −1.01527e15 −0.425836
\(626\) −3.06457e14 −0.127412
\(627\) 8.70162e13 0.0358615
\(628\) −6.77724e15 −2.76870
\(629\) 7.71257e14 0.312335
\(630\) −1.48658e15 −0.596781
\(631\) 3.72069e15 1.48068 0.740342 0.672230i \(-0.234663\pi\)
0.740342 + 0.672230i \(0.234663\pi\)
\(632\) 7.35697e15 2.90239
\(633\) 1.67725e15 0.655961
\(634\) −6.58508e15 −2.55311
\(635\) −9.03232e13 −0.0347171
\(636\) −9.85663e13 −0.0375590
\(637\) 2.56945e14 0.0970674
\(638\) −1.41226e15 −0.528932
\(639\) 2.32379e14 0.0862865
\(640\) −3.10686e15 −1.14375
\(641\) 2.79910e15 1.02164 0.510821 0.859687i \(-0.329341\pi\)
0.510821 + 0.859687i \(0.329341\pi\)
\(642\) 4.79591e13 0.0173551
\(643\) −1.50932e15 −0.541527 −0.270763 0.962646i \(-0.587276\pi\)
−0.270763 + 0.962646i \(0.587276\pi\)
\(644\) 5.74704e15 2.04443
\(645\) 1.55480e15 0.548398
\(646\) 1.15732e15 0.404739
\(647\) −3.98677e15 −1.38244 −0.691222 0.722643i \(-0.742927\pi\)
−0.691222 + 0.722643i \(0.742927\pi\)
\(648\) −6.36654e14 −0.218898
\(649\) −9.01730e13 −0.0307419
\(650\) 2.80449e14 0.0948048
\(651\) 8.46645e14 0.283796
\(652\) −2.41147e15 −0.801532
\(653\) 2.85864e13 0.00942186 0.00471093 0.999989i \(-0.498500\pi\)
0.00471093 + 0.999989i \(0.498500\pi\)
\(654\) −2.99800e15 −0.979838
\(655\) −1.38526e15 −0.448955
\(656\) 2.63631e13 0.00847278
\(657\) −1.21241e15 −0.386403
\(658\) 2.00274e15 0.632969
\(659\) −6.88757e14 −0.215872 −0.107936 0.994158i \(-0.534424\pi\)
−0.107936 + 0.994158i \(0.534424\pi\)
\(660\) 7.14224e14 0.221995
\(661\) −2.13260e15 −0.657357 −0.328679 0.944442i \(-0.606603\pi\)
−0.328679 + 0.944442i \(0.606603\pi\)
\(662\) 1.96736e15 0.601402
\(663\) 2.18600e14 0.0662713
\(664\) 2.13264e15 0.641201
\(665\) −8.94717e14 −0.266788
\(666\) −7.12965e14 −0.210843
\(667\) 3.17152e15 0.930199
\(668\) 6.32763e14 0.184065
\(669\) 3.26548e15 0.942112
\(670\) −4.00793e15 −1.14686
\(671\) 2.97381e13 0.00843995
\(672\) 1.17175e15 0.329841
\(673\) 2.04667e15 0.571432 0.285716 0.958314i \(-0.407769\pi\)
0.285716 + 0.958314i \(0.407769\pi\)
\(674\) −1.08275e16 −2.99847
\(675\) −2.85741e14 −0.0784878
\(676\) −7.63193e15 −2.07935
\(677\) −4.20708e15 −1.13695 −0.568477 0.822699i \(-0.692467\pi\)
−0.568477 + 0.822699i \(0.692467\pi\)
\(678\) −1.20446e15 −0.322872
\(679\) 4.68552e15 1.24587
\(680\) 5.01009e15 1.32144
\(681\) −1.59680e14 −0.0417774
\(682\) −5.98999e14 −0.155458
\(683\) −1.59000e15 −0.409340 −0.204670 0.978831i \(-0.565612\pi\)
−0.204670 + 0.978831i \(0.565612\pi\)
\(684\) −7.26515e14 −0.185539
\(685\) 1.96110e15 0.496824
\(686\) −2.43378e15 −0.611643
\(687\) 7.28605e14 0.181648
\(688\) −6.79550e15 −1.68068
\(689\) −1.65006e13 −0.00404850
\(690\) −2.36193e15 −0.574906
\(691\) −6.78840e15 −1.63922 −0.819612 0.572920i \(-0.805811\pi\)
−0.819612 + 0.572920i \(0.805811\pi\)
\(692\) −3.05377e15 −0.731565
\(693\) 4.36497e14 0.103741
\(694\) −1.20130e15 −0.283254
\(695\) 4.83371e14 0.113074
\(696\) 6.21892e15 1.44333
\(697\) 2.35558e13 0.00542398
\(698\) 8.93474e15 2.04116
\(699\) 2.91005e15 0.659592
\(700\) 5.05779e15 1.13742
\(701\) 5.58259e15 1.24562 0.622812 0.782372i \(-0.285990\pi\)
0.622812 + 0.782372i \(0.285990\pi\)
\(702\) −2.02078e14 −0.0447367
\(703\) −4.29107e14 −0.0942565
\(704\) 6.46207e14 0.140839
\(705\) −5.58945e14 −0.120873
\(706\) −7.46871e15 −1.60258
\(707\) 1.56045e15 0.332234
\(708\) 7.52872e14 0.159052
\(709\) 1.57903e15 0.331007 0.165504 0.986209i \(-0.447075\pi\)
0.165504 + 0.986209i \(0.447075\pi\)
\(710\) 1.69047e15 0.351631
\(711\) −2.37921e15 −0.491078
\(712\) 6.36856e14 0.130437
\(713\) 1.34518e15 0.273393
\(714\) 5.80545e15 1.17083
\(715\) 1.19566e14 0.0239289
\(716\) −1.18877e15 −0.236088
\(717\) −1.66300e15 −0.327746
\(718\) −1.70359e16 −3.33181
\(719\) −6.92912e15 −1.34484 −0.672418 0.740172i \(-0.734744\pi\)
−0.672418 + 0.740172i \(0.734744\pi\)
\(720\) −1.81334e15 −0.349261
\(721\) 2.24906e15 0.429890
\(722\) 8.66195e15 1.64309
\(723\) −3.09639e14 −0.0582901
\(724\) −2.17033e16 −4.05475
\(725\) 2.79116e15 0.517518
\(726\) 5.22969e15 0.962330
\(727\) 2.98687e15 0.545478 0.272739 0.962088i \(-0.412071\pi\)
0.272739 + 0.962088i \(0.412071\pi\)
\(728\) 1.88653e15 0.341933
\(729\) 2.05891e14 0.0370370
\(730\) −8.81985e15 −1.57466
\(731\) −6.07186e15 −1.07591
\(732\) −2.48289e14 −0.0436664
\(733\) −5.90090e15 −1.03002 −0.515011 0.857184i \(-0.672212\pi\)
−0.515011 + 0.857184i \(0.672212\pi\)
\(734\) −1.18840e16 −2.05890
\(735\) −1.90445e15 −0.327484
\(736\) 1.86172e15 0.317750
\(737\) 1.17683e15 0.199362
\(738\) −2.17754e13 −0.00366148
\(739\) −7.63713e15 −1.27464 −0.637318 0.770601i \(-0.719956\pi\)
−0.637318 + 0.770601i \(0.719956\pi\)
\(740\) −3.52209e15 −0.583479
\(741\) −1.21623e14 −0.0199994
\(742\) −4.38214e14 −0.0715261
\(743\) −3.21264e15 −0.520504 −0.260252 0.965541i \(-0.583806\pi\)
−0.260252 + 0.965541i \(0.583806\pi\)
\(744\) 2.63771e15 0.424205
\(745\) −1.47719e15 −0.235817
\(746\) −1.75739e16 −2.78486
\(747\) −6.89688e14 −0.108490
\(748\) −2.78921e15 −0.435535
\(749\) 1.44794e14 0.0224440
\(750\) −7.17549e15 −1.10412
\(751\) −1.15415e16 −1.76296 −0.881480 0.472221i \(-0.843452\pi\)
−0.881480 + 0.472221i \(0.843452\pi\)
\(752\) 2.44296e15 0.370440
\(753\) −1.30712e15 −0.196762
\(754\) 1.97392e15 0.294977
\(755\) −2.69613e15 −0.399975
\(756\) −3.64440e15 −0.536730
\(757\) 2.06420e14 0.0301804 0.0150902 0.999886i \(-0.495196\pi\)
0.0150902 + 0.999886i \(0.495196\pi\)
\(758\) 2.31757e15 0.336397
\(759\) 6.93522e14 0.0999379
\(760\) −2.78748e15 −0.398784
\(761\) −1.46112e15 −0.207525 −0.103763 0.994602i \(-0.533088\pi\)
−0.103763 + 0.994602i \(0.533088\pi\)
\(762\) −3.26074e14 −0.0459794
\(763\) −9.05130e15 −1.26714
\(764\) 7.59333e15 1.05540
\(765\) −1.62024e15 −0.223584
\(766\) −1.95603e16 −2.67989
\(767\) 1.26036e14 0.0171442
\(768\) −8.66630e15 −1.17043
\(769\) 6.54282e14 0.0877345 0.0438672 0.999037i \(-0.486032\pi\)
0.0438672 + 0.999037i \(0.486032\pi\)
\(770\) 3.17535e15 0.422759
\(771\) −2.31220e15 −0.305652
\(772\) −7.92541e15 −1.04022
\(773\) −2.28734e15 −0.298087 −0.149044 0.988831i \(-0.547619\pi\)
−0.149044 + 0.988831i \(0.547619\pi\)
\(774\) 5.61295e15 0.726299
\(775\) 1.18385e15 0.152103
\(776\) 1.45977e16 1.86228
\(777\) −2.15252e15 −0.272666
\(778\) 1.69790e16 2.13562
\(779\) −1.31058e13 −0.00163685
\(780\) −9.98276e14 −0.123803
\(781\) −4.96365e14 −0.0611253
\(782\) 9.22390e15 1.12792
\(783\) −2.01117e15 −0.244208
\(784\) 8.32371e15 1.00364
\(785\) 8.40920e15 1.00687
\(786\) −5.00088e15 −0.594597
\(787\) −4.31298e15 −0.509233 −0.254617 0.967042i \(-0.581949\pi\)
−0.254617 + 0.967042i \(0.581949\pi\)
\(788\) 2.74439e16 3.21774
\(789\) −3.20531e15 −0.373205
\(790\) −1.73079e16 −2.00122
\(791\) −3.63641e15 −0.417544
\(792\) 1.35990e15 0.155067
\(793\) −4.15652e13 −0.00470682
\(794\) −1.39432e16 −1.56801
\(795\) 1.22301e14 0.0136587
\(796\) 2.28604e16 2.53549
\(797\) 5.27892e15 0.581466 0.290733 0.956804i \(-0.406101\pi\)
0.290733 + 0.956804i \(0.406101\pi\)
\(798\) −3.23000e15 −0.353335
\(799\) 2.18281e15 0.237142
\(800\) 1.63844e15 0.176781
\(801\) −2.05956e14 −0.0220697
\(802\) −5.08344e15 −0.541002
\(803\) 2.58973e15 0.273728
\(804\) −9.82557e15 −1.03145
\(805\) −7.13093e15 −0.743479
\(806\) 8.37225e14 0.0866961
\(807\) 6.50745e14 0.0669278
\(808\) 4.86157e15 0.496608
\(809\) 3.78268e15 0.383781 0.191890 0.981416i \(-0.438538\pi\)
0.191890 + 0.981416i \(0.438538\pi\)
\(810\) 1.49778e15 0.150932
\(811\) −7.41965e15 −0.742624 −0.371312 0.928508i \(-0.621092\pi\)
−0.371312 + 0.928508i \(0.621092\pi\)
\(812\) 3.55990e16 3.53899
\(813\) 6.39886e15 0.631837
\(814\) 1.52290e15 0.149361
\(815\) 2.99216e15 0.291486
\(816\) 7.08152e15 0.685221
\(817\) 3.37823e15 0.324689
\(818\) 2.71676e16 2.59363
\(819\) −6.10095e14 −0.0578544
\(820\) −1.07572e14 −0.0101326
\(821\) −4.41256e15 −0.412861 −0.206430 0.978461i \(-0.566185\pi\)
−0.206430 + 0.978461i \(0.566185\pi\)
\(822\) 7.07974e15 0.657994
\(823\) 4.36342e15 0.402836 0.201418 0.979505i \(-0.435445\pi\)
0.201418 + 0.979505i \(0.435445\pi\)
\(824\) 7.00692e15 0.642581
\(825\) 6.10347e14 0.0556007
\(826\) 3.34718e15 0.302893
\(827\) 1.61404e16 1.45089 0.725445 0.688280i \(-0.241634\pi\)
0.725445 + 0.688280i \(0.241634\pi\)
\(828\) −5.79035e15 −0.517056
\(829\) 7.10217e15 0.630001 0.315000 0.949092i \(-0.397995\pi\)
0.315000 + 0.949092i \(0.397995\pi\)
\(830\) −5.01722e15 −0.442114
\(831\) −8.68071e14 −0.0759888
\(832\) −9.03208e14 −0.0785434
\(833\) 7.43734e15 0.642496
\(834\) 1.74501e15 0.149756
\(835\) −7.85133e14 −0.0669372
\(836\) 1.55185e15 0.131436
\(837\) −8.53024e14 −0.0717747
\(838\) 3.25863e15 0.272391
\(839\) −5.77411e15 −0.479506 −0.239753 0.970834i \(-0.577067\pi\)
−0.239753 + 0.970834i \(0.577067\pi\)
\(840\) −1.39828e16 −1.15361
\(841\) 7.44490e15 0.610212
\(842\) −2.95325e16 −2.40483
\(843\) 4.28067e14 0.0346306
\(844\) 2.99121e16 2.40416
\(845\) 9.46969e15 0.756178
\(846\) −2.01783e15 −0.160084
\(847\) 1.57890e16 1.24450
\(848\) −5.34536e14 −0.0418600
\(849\) 1.09555e16 0.852391
\(850\) 8.11767e15 0.627520
\(851\) −3.42000e15 −0.262672
\(852\) 4.14425e15 0.316248
\(853\) 3.94735e15 0.299286 0.149643 0.988740i \(-0.452188\pi\)
0.149643 + 0.988740i \(0.452188\pi\)
\(854\) −1.10387e15 −0.0831568
\(855\) 9.01459e14 0.0674734
\(856\) 4.51103e14 0.0335483
\(857\) −1.61729e16 −1.19507 −0.597536 0.801842i \(-0.703854\pi\)
−0.597536 + 0.801842i \(0.703854\pi\)
\(858\) 4.31641e14 0.0316915
\(859\) 1.41118e16 1.02949 0.514743 0.857345i \(-0.327888\pi\)
0.514743 + 0.857345i \(0.327888\pi\)
\(860\) 2.77283e16 2.00993
\(861\) −6.57425e13 −0.00473510
\(862\) 2.82691e16 2.02312
\(863\) 9.26132e14 0.0658587 0.0329294 0.999458i \(-0.489516\pi\)
0.0329294 + 0.999458i \(0.489516\pi\)
\(864\) −1.18058e15 −0.0834200
\(865\) 3.78912e15 0.266041
\(866\) −3.45248e15 −0.240870
\(867\) −2.00064e15 −0.138696
\(868\) 1.50991e16 1.04014
\(869\) 5.08203e15 0.347879
\(870\) −1.46305e16 −0.995186
\(871\) −1.64486e15 −0.111181
\(872\) −2.81992e16 −1.89407
\(873\) −4.72082e15 −0.315094
\(874\) −5.13194e15 −0.340383
\(875\) −2.16636e16 −1.42786
\(876\) −2.16222e16 −1.41621
\(877\) 2.38244e16 1.55069 0.775345 0.631538i \(-0.217576\pi\)
0.775345 + 0.631538i \(0.217576\pi\)
\(878\) 1.37801e16 0.891318
\(879\) −5.78392e14 −0.0371778
\(880\) 3.87331e15 0.247416
\(881\) 1.15892e16 0.735676 0.367838 0.929890i \(-0.380098\pi\)
0.367838 + 0.929890i \(0.380098\pi\)
\(882\) −6.87522e15 −0.433720
\(883\) 1.53134e16 0.960039 0.480019 0.877258i \(-0.340630\pi\)
0.480019 + 0.877258i \(0.340630\pi\)
\(884\) 3.89851e15 0.242891
\(885\) −9.34163e14 −0.0578409
\(886\) −8.96287e15 −0.551520
\(887\) −2.03630e16 −1.24527 −0.622633 0.782514i \(-0.713937\pi\)
−0.622633 + 0.782514i \(0.713937\pi\)
\(888\) −6.70615e15 −0.407570
\(889\) −9.84453e14 −0.0594614
\(890\) −1.49826e15 −0.0899376
\(891\) −4.39786e14 −0.0262370
\(892\) 5.82365e16 3.45293
\(893\) −1.21446e15 −0.0715649
\(894\) −5.33275e15 −0.312317
\(895\) 1.47502e15 0.0858561
\(896\) −3.38624e16 −1.95895
\(897\) −9.69341e14 −0.0557337
\(898\) −1.54603e16 −0.883485
\(899\) 8.33246e15 0.473255
\(900\) −5.09591e15 −0.287665
\(901\) −4.77615e14 −0.0267973
\(902\) 4.65126e13 0.00259379
\(903\) 1.69461e16 0.939264
\(904\) −1.13292e16 −0.624126
\(905\) 2.69295e16 1.47456
\(906\) −9.73325e15 −0.529728
\(907\) 2.90182e16 1.56975 0.784876 0.619653i \(-0.212727\pi\)
0.784876 + 0.619653i \(0.212727\pi\)
\(908\) −2.84773e15 −0.153118
\(909\) −1.57221e15 −0.0840252
\(910\) −4.43822e15 −0.235766
\(911\) −3.54949e16 −1.87420 −0.937098 0.349066i \(-0.886499\pi\)
−0.937098 + 0.349066i \(0.886499\pi\)
\(912\) −3.93997e15 −0.206786
\(913\) 1.47318e15 0.0768542
\(914\) 3.91547e16 2.03039
\(915\) 3.08077e14 0.0158798
\(916\) 1.29939e16 0.665757
\(917\) −1.50982e16 −0.768943
\(918\) −5.84919e15 −0.296116
\(919\) 2.20469e16 1.10946 0.554730 0.832030i \(-0.312822\pi\)
0.554730 + 0.832030i \(0.312822\pi\)
\(920\) −2.22163e16 −1.11132
\(921\) 1.92198e16 0.955701
\(922\) 8.27923e15 0.409232
\(923\) 6.93774e14 0.0340885
\(924\) 7.78448e15 0.380219
\(925\) −3.00984e15 −0.146138
\(926\) 2.41814e16 1.16714
\(927\) −2.26601e15 −0.108723
\(928\) 1.15321e16 0.550039
\(929\) −2.57160e16 −1.21932 −0.609660 0.792663i \(-0.708694\pi\)
−0.609660 + 0.792663i \(0.708694\pi\)
\(930\) −6.20544e15 −0.292493
\(931\) −4.13794e15 −0.193893
\(932\) 5.18978e16 2.41747
\(933\) 2.58272e14 0.0119599
\(934\) −3.76049e16 −1.73116
\(935\) 3.46086e15 0.158387
\(936\) −1.90074e15 −0.0864782
\(937\) 2.62394e16 1.18682 0.593412 0.804899i \(-0.297780\pi\)
0.593412 + 0.804899i \(0.297780\pi\)
\(938\) −4.36833e16 −1.96427
\(939\) 9.32199e14 0.0416723
\(940\) −9.96822e15 −0.443011
\(941\) −2.10008e16 −0.927881 −0.463941 0.885866i \(-0.653565\pi\)
−0.463941 + 0.885866i \(0.653565\pi\)
\(942\) 3.03579e16 1.33350
\(943\) −1.04454e14 −0.00456153
\(944\) 4.08291e15 0.177265
\(945\) 4.52197e15 0.195188
\(946\) −1.19893e16 −0.514510
\(947\) −1.77925e16 −0.759122 −0.379561 0.925167i \(-0.623925\pi\)
−0.379561 + 0.925167i \(0.623925\pi\)
\(948\) −4.24309e16 −1.79985
\(949\) −3.61968e15 −0.152653
\(950\) −4.51646e15 −0.189373
\(951\) 2.00309e16 0.835040
\(952\) 5.46061e16 2.26328
\(953\) −1.08891e15 −0.0448725 −0.0224363 0.999748i \(-0.507142\pi\)
−0.0224363 + 0.999748i \(0.507142\pi\)
\(954\) 4.41516e14 0.0180897
\(955\) −9.42181e15 −0.383810
\(956\) −2.96580e16 −1.20122
\(957\) 4.29589e15 0.172997
\(958\) −7.46656e16 −2.98958
\(959\) 2.13745e16 0.850930
\(960\) 6.69450e15 0.264988
\(961\) −2.18743e16 −0.860907
\(962\) −2.12857e15 −0.0832963
\(963\) −1.45885e14 −0.00567630
\(964\) −5.52210e15 −0.213639
\(965\) 9.83385e15 0.378289
\(966\) −2.57432e16 −0.984665
\(967\) 1.66647e16 0.633801 0.316901 0.948459i \(-0.397358\pi\)
0.316901 + 0.948459i \(0.397358\pi\)
\(968\) 4.91905e16 1.86023
\(969\) −3.52041e15 −0.132377
\(970\) −3.43422e16 −1.28406
\(971\) 1.14441e16 0.425478 0.212739 0.977109i \(-0.431762\pi\)
0.212739 + 0.977109i \(0.431762\pi\)
\(972\) 3.67186e15 0.135744
\(973\) 5.26837e15 0.193667
\(974\) 4.75740e15 0.173898
\(975\) −8.53087e14 −0.0310076
\(976\) −1.34650e15 −0.0486668
\(977\) 3.65010e16 1.31185 0.655925 0.754826i \(-0.272279\pi\)
0.655925 + 0.754826i \(0.272279\pi\)
\(978\) 1.08019e16 0.386044
\(979\) 4.39926e14 0.0156342
\(980\) −3.39640e16 −1.20026
\(981\) 9.11950e15 0.320473
\(982\) 1.25261e16 0.437726
\(983\) −3.58113e16 −1.24444 −0.622222 0.782841i \(-0.713770\pi\)
−0.622222 + 0.782841i \(0.713770\pi\)
\(984\) −2.04820e14 −0.00707781
\(985\) −3.40523e16 −1.17017
\(986\) 5.71357e16 1.95247
\(987\) −6.09207e15 −0.207024
\(988\) −2.16903e15 −0.0732996
\(989\) 2.69246e16 0.904835
\(990\) −3.19928e15 −0.106920
\(991\) 4.00599e16 1.33139 0.665694 0.746224i \(-0.268135\pi\)
0.665694 + 0.746224i \(0.268135\pi\)
\(992\) 4.89124e15 0.161661
\(993\) −5.98443e15 −0.196699
\(994\) 1.84248e16 0.602253
\(995\) −2.83652e16 −0.922058
\(996\) −1.22999e16 −0.397626
\(997\) 1.73367e16 0.557370 0.278685 0.960383i \(-0.410102\pi\)
0.278685 + 0.960383i \(0.410102\pi\)
\(998\) 2.61022e16 0.834564
\(999\) 2.16874e15 0.0689600
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.1.2 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.d.1.2 28 1.1 even 1 trivial