Properties

Label 177.12.a.d.1.12
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 1.12
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-12.4112 q^{2} +243.000 q^{3} -1893.96 q^{4} -2629.42 q^{5} -3015.92 q^{6} +23296.0 q^{7} +48924.4 q^{8} +59049.0 q^{9} +O(q^{10})\) \(q-12.4112 q^{2} +243.000 q^{3} -1893.96 q^{4} -2629.42 q^{5} -3015.92 q^{6} +23296.0 q^{7} +48924.4 q^{8} +59049.0 q^{9} +32634.3 q^{10} -219870. q^{11} -460233. q^{12} +1.40491e6 q^{13} -289131. q^{14} -638950. q^{15} +3.27163e6 q^{16} -5.70617e6 q^{17} -732868. q^{18} +820391. q^{19} +4.98003e6 q^{20} +5.66094e6 q^{21} +2.72885e6 q^{22} +2.97055e7 q^{23} +1.18886e7 q^{24} -4.19143e7 q^{25} -1.74366e7 q^{26} +1.43489e7 q^{27} -4.41218e7 q^{28} +9.68327e7 q^{29} +7.93012e6 q^{30} +6.91538e7 q^{31} -1.40802e8 q^{32} -5.34284e7 q^{33} +7.08203e7 q^{34} -6.12552e7 q^{35} -1.11837e8 q^{36} -4.51038e8 q^{37} -1.01820e7 q^{38} +3.41394e8 q^{39} -1.28643e8 q^{40} +4.49616e8 q^{41} -7.02589e7 q^{42} -1.79394e9 q^{43} +4.16426e8 q^{44} -1.55265e8 q^{45} -3.68681e8 q^{46} +5.98279e8 q^{47} +7.95005e8 q^{48} -1.43462e9 q^{49} +5.20205e8 q^{50} -1.38660e9 q^{51} -2.66085e9 q^{52} -3.80930e9 q^{53} -1.78087e8 q^{54} +5.78132e8 q^{55} +1.13974e9 q^{56} +1.99355e8 q^{57} -1.20181e9 q^{58} +7.14924e8 q^{59} +1.21015e9 q^{60} +9.73343e9 q^{61} -8.58280e8 q^{62} +1.37561e9 q^{63} -4.95277e9 q^{64} -3.69412e9 q^{65} +6.63109e8 q^{66} +2.05666e10 q^{67} +1.08073e10 q^{68} +7.21845e9 q^{69} +7.60249e8 q^{70} -1.31509e10 q^{71} +2.88894e9 q^{72} +2.61072e10 q^{73} +5.59791e9 q^{74} -1.01852e10 q^{75} -1.55379e9 q^{76} -5.12210e9 q^{77} -4.23710e9 q^{78} -3.90581e10 q^{79} -8.60249e9 q^{80} +3.48678e9 q^{81} -5.58027e9 q^{82} +2.57619e10 q^{83} -1.07216e10 q^{84} +1.50039e10 q^{85} +2.22649e10 q^{86} +2.35303e10 q^{87} -1.07570e10 q^{88} +4.36938e10 q^{89} +1.92702e9 q^{90} +3.27289e10 q^{91} -5.62612e10 q^{92} +1.68044e10 q^{93} -7.42534e9 q^{94} -2.15716e9 q^{95} -3.42149e10 q^{96} -4.40290e10 q^{97} +1.78053e10 q^{98} -1.29831e10 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + O(q^{10}) \) \( 28q + 96q^{2} + 6804q^{3} + 29214q^{4} + 26562q^{5} + 23328q^{6} + 142333q^{7} + 332331q^{8} + 1653372q^{9} + 616281q^{10} + 1082362q^{11} + 7099002q^{12} + 503712q^{13} + 1321669q^{14} + 6454566q^{15} + 34870338q^{16} + 13513579q^{17} + 5668704q^{18} + 35971687q^{19} + 96105997q^{20} + 34586919q^{21} - 47598882q^{22} + 61380539q^{23} + 80756433q^{24} + 294744746q^{25} + 62820734q^{26} + 401769396q^{27} + 148068294q^{28} + 322339307q^{29} + 149756283q^{30} + 151247077q^{31} + 466383494q^{32} + 263013966q^{33} + 684479860q^{34} + 960297361q^{35} + 1725057486q^{36} + 863508437q^{37} + 992640509q^{38} + 122402016q^{39} + 3067680252q^{40} + 3081170377q^{41} + 321165567q^{42} + 2554238300q^{43} + 4350123570q^{44} + 1568459538q^{45} - 1987059155q^{46} + 6203398333q^{47} + 8473492134q^{48} + 10327857997q^{49} + 17577682253q^{50} + 3283799697q^{51} + 32137181618q^{52} + 14571770754q^{53} + 1377495072q^{54} + 18251419334q^{55} + 33498842836q^{56} + 8741119941q^{57} + 11860778276q^{58} + 20017880372q^{59} + 23353757271q^{60} + 2761613771q^{61} + 13785829526q^{62} + 8404621317q^{63} + 86547545293q^{64} + 32034985256q^{65} - 11566528326q^{66} + 39381333296q^{67} + 38995496621q^{68} + 14915470977q^{69} + 8551800364q^{70} + 26130020296q^{71} + 19623813219q^{72} + 41382402799q^{73} + 23815315058q^{74} + 71622973278q^{75} + 10611720128q^{76} - 8426124313q^{77} + 15265438362q^{78} + 59825111206q^{79} + 4009687655q^{80} + 97629963228q^{81} - 39592715115q^{82} + 35433122727q^{83} + 35980595442q^{84} - 8950496085q^{85} - 182032360688q^{86} + 78328451601q^{87} - 220003602335q^{88} + 102303043039q^{89} + 36390776769q^{90} - 111146323655q^{91} - 163000203526q^{92} + 36753039711q^{93} - 81314346008q^{94} + 208102168887q^{95} + 113331189042q^{96} - 171891031490q^{97} + 72304707792q^{98} + 63912393738q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −12.4112 −0.274251 −0.137125 0.990554i \(-0.543786\pi\)
−0.137125 + 0.990554i \(0.543786\pi\)
\(3\) 243.000 0.577350
\(4\) −1893.96 −0.924786
\(5\) −2629.42 −0.376293 −0.188146 0.982141i \(-0.560248\pi\)
−0.188146 + 0.982141i \(0.560248\pi\)
\(6\) −3015.92 −0.158339
\(7\) 23296.0 0.523893 0.261947 0.965082i \(-0.415636\pi\)
0.261947 + 0.965082i \(0.415636\pi\)
\(8\) 48924.4 0.527874
\(9\) 59049.0 0.333333
\(10\) 32634.3 0.103199
\(11\) −219870. −0.411629 −0.205815 0.978591i \(-0.565984\pi\)
−0.205815 + 0.978591i \(0.565984\pi\)
\(12\) −460233. −0.533926
\(13\) 1.40491e6 1.04945 0.524724 0.851272i \(-0.324168\pi\)
0.524724 + 0.851272i \(0.324168\pi\)
\(14\) −289131. −0.143678
\(15\) −638950. −0.217253
\(16\) 3.27163e6 0.780016
\(17\) −5.70617e6 −0.974711 −0.487355 0.873204i \(-0.662038\pi\)
−0.487355 + 0.873204i \(0.662038\pi\)
\(18\) −732868. −0.0914170
\(19\) 820391. 0.0760109 0.0380055 0.999278i \(-0.487900\pi\)
0.0380055 + 0.999278i \(0.487900\pi\)
\(20\) 4.98003e6 0.347990
\(21\) 5.66094e6 0.302470
\(22\) 2.72885e6 0.112890
\(23\) 2.97055e7 0.962353 0.481176 0.876624i \(-0.340210\pi\)
0.481176 + 0.876624i \(0.340210\pi\)
\(24\) 1.18886e7 0.304768
\(25\) −4.19143e7 −0.858404
\(26\) −1.74366e7 −0.287812
\(27\) 1.43489e7 0.192450
\(28\) −4.41218e7 −0.484489
\(29\) 9.68327e7 0.876664 0.438332 0.898813i \(-0.355569\pi\)
0.438332 + 0.898813i \(0.355569\pi\)
\(30\) 7.93012e6 0.0595817
\(31\) 6.91538e7 0.433837 0.216919 0.976190i \(-0.430399\pi\)
0.216919 + 0.976190i \(0.430399\pi\)
\(32\) −1.40802e8 −0.741795
\(33\) −5.34284e7 −0.237654
\(34\) 7.08203e7 0.267315
\(35\) −6.12552e7 −0.197137
\(36\) −1.11837e8 −0.308262
\(37\) −4.51038e8 −1.06931 −0.534655 0.845071i \(-0.679558\pi\)
−0.534655 + 0.845071i \(0.679558\pi\)
\(38\) −1.01820e7 −0.0208461
\(39\) 3.41394e8 0.605900
\(40\) −1.28643e8 −0.198635
\(41\) 4.49616e8 0.606081 0.303041 0.952978i \(-0.401998\pi\)
0.303041 + 0.952978i \(0.401998\pi\)
\(42\) −7.02589e7 −0.0829526
\(43\) −1.79394e9 −1.86094 −0.930469 0.366371i \(-0.880600\pi\)
−0.930469 + 0.366371i \(0.880600\pi\)
\(44\) 4.16426e8 0.380669
\(45\) −1.55265e8 −0.125431
\(46\) −3.68681e8 −0.263926
\(47\) 5.98279e8 0.380509 0.190255 0.981735i \(-0.439069\pi\)
0.190255 + 0.981735i \(0.439069\pi\)
\(48\) 7.95005e8 0.450343
\(49\) −1.43462e9 −0.725536
\(50\) 5.20205e8 0.235418
\(51\) −1.38660e9 −0.562750
\(52\) −2.66085e9 −0.970516
\(53\) −3.80930e9 −1.25120 −0.625602 0.780143i \(-0.715146\pi\)
−0.625602 + 0.780143i \(0.715146\pi\)
\(54\) −1.78087e8 −0.0527796
\(55\) 5.78132e8 0.154893
\(56\) 1.13974e9 0.276550
\(57\) 1.99355e8 0.0438849
\(58\) −1.20181e9 −0.240426
\(59\) 7.14924e8 0.130189
\(60\) 1.21015e9 0.200912
\(61\) 9.73343e9 1.47554 0.737772 0.675050i \(-0.235878\pi\)
0.737772 + 0.675050i \(0.235878\pi\)
\(62\) −8.58280e8 −0.118980
\(63\) 1.37561e9 0.174631
\(64\) −4.95277e9 −0.576579
\(65\) −3.69412e9 −0.394900
\(66\) 6.63109e8 0.0651769
\(67\) 2.05666e10 1.86102 0.930511 0.366265i \(-0.119364\pi\)
0.930511 + 0.366265i \(0.119364\pi\)
\(68\) 1.08073e10 0.901399
\(69\) 7.21845e9 0.555615
\(70\) 7.60249e8 0.0540650
\(71\) −1.31509e10 −0.865038 −0.432519 0.901625i \(-0.642375\pi\)
−0.432519 + 0.901625i \(0.642375\pi\)
\(72\) 2.88894e9 0.175958
\(73\) 2.61072e10 1.47396 0.736978 0.675917i \(-0.236252\pi\)
0.736978 + 0.675917i \(0.236252\pi\)
\(74\) 5.59791e9 0.293259
\(75\) −1.01852e10 −0.495600
\(76\) −1.55379e9 −0.0702939
\(77\) −5.12210e9 −0.215650
\(78\) −4.23710e9 −0.166169
\(79\) −3.90581e10 −1.42811 −0.714056 0.700089i \(-0.753144\pi\)
−0.714056 + 0.700089i \(0.753144\pi\)
\(80\) −8.60249e9 −0.293514
\(81\) 3.48678e9 0.111111
\(82\) −5.58027e9 −0.166218
\(83\) 2.57619e10 0.717875 0.358938 0.933362i \(-0.383139\pi\)
0.358938 + 0.933362i \(0.383139\pi\)
\(84\) −1.07216e10 −0.279720
\(85\) 1.50039e10 0.366776
\(86\) 2.22649e10 0.510364
\(87\) 2.35303e10 0.506142
\(88\) −1.07570e10 −0.217289
\(89\) 4.36938e10 0.829420 0.414710 0.909954i \(-0.363883\pi\)
0.414710 + 0.909954i \(0.363883\pi\)
\(90\) 1.92702e9 0.0343995
\(91\) 3.27289e10 0.549799
\(92\) −5.62612e10 −0.889971
\(93\) 1.68044e10 0.250476
\(94\) −7.42534e9 −0.104355
\(95\) −2.15716e9 −0.0286024
\(96\) −3.42149e10 −0.428275
\(97\) −4.40290e10 −0.520589 −0.260294 0.965529i \(-0.583820\pi\)
−0.260294 + 0.965529i \(0.583820\pi\)
\(98\) 1.78053e10 0.198979
\(99\) −1.29831e10 −0.137210
\(100\) 7.93840e10 0.793840
\(101\) −9.64384e10 −0.913025 −0.456512 0.889717i \(-0.650902\pi\)
−0.456512 + 0.889717i \(0.650902\pi\)
\(102\) 1.72093e10 0.154335
\(103\) −5.45856e10 −0.463953 −0.231976 0.972721i \(-0.574519\pi\)
−0.231976 + 0.972721i \(0.574519\pi\)
\(104\) 6.87346e10 0.553977
\(105\) −1.48850e10 −0.113817
\(106\) 4.72779e10 0.343144
\(107\) 6.01740e10 0.414761 0.207381 0.978260i \(-0.433506\pi\)
0.207381 + 0.978260i \(0.433506\pi\)
\(108\) −2.71763e10 −0.177975
\(109\) 1.50982e10 0.0939894 0.0469947 0.998895i \(-0.485036\pi\)
0.0469947 + 0.998895i \(0.485036\pi\)
\(110\) −7.17529e9 −0.0424796
\(111\) −1.09602e11 −0.617366
\(112\) 7.62159e10 0.408645
\(113\) 1.65861e11 0.846863 0.423431 0.905928i \(-0.360826\pi\)
0.423431 + 0.905928i \(0.360826\pi\)
\(114\) −2.47423e9 −0.0120355
\(115\) −7.81085e10 −0.362126
\(116\) −1.83397e11 −0.810727
\(117\) 8.29588e10 0.349816
\(118\) −8.87305e9 −0.0357044
\(119\) −1.32931e11 −0.510644
\(120\) −3.12603e10 −0.114682
\(121\) −2.36969e11 −0.830561
\(122\) −1.20803e11 −0.404669
\(123\) 1.09257e11 0.349921
\(124\) −1.30975e11 −0.401207
\(125\) 2.38600e11 0.699304
\(126\) −1.70729e10 −0.0478927
\(127\) 4.89021e11 1.31343 0.656715 0.754139i \(-0.271945\pi\)
0.656715 + 0.754139i \(0.271945\pi\)
\(128\) 3.49832e11 0.899922
\(129\) −4.35928e11 −1.07441
\(130\) 4.58483e10 0.108302
\(131\) −5.90891e11 −1.33818 −0.669091 0.743181i \(-0.733316\pi\)
−0.669091 + 0.743181i \(0.733316\pi\)
\(132\) 1.01191e11 0.219779
\(133\) 1.91119e10 0.0398216
\(134\) −2.55256e11 −0.510387
\(135\) −3.77294e10 −0.0724176
\(136\) −2.79171e11 −0.514525
\(137\) 5.20144e11 0.920789 0.460394 0.887714i \(-0.347708\pi\)
0.460394 + 0.887714i \(0.347708\pi\)
\(138\) −8.95894e10 −0.152378
\(139\) 1.06455e10 0.0174014 0.00870072 0.999962i \(-0.497230\pi\)
0.00870072 + 0.999962i \(0.497230\pi\)
\(140\) 1.16015e11 0.182310
\(141\) 1.45382e11 0.219687
\(142\) 1.63218e11 0.237237
\(143\) −3.08898e11 −0.431984
\(144\) 1.93186e11 0.260005
\(145\) −2.54614e11 −0.329882
\(146\) −3.24021e11 −0.404234
\(147\) −3.48613e11 −0.418888
\(148\) 8.54249e11 0.988883
\(149\) 1.50398e12 1.67771 0.838856 0.544353i \(-0.183225\pi\)
0.838856 + 0.544353i \(0.183225\pi\)
\(150\) 1.26410e11 0.135919
\(151\) 1.56843e12 1.62590 0.812949 0.582335i \(-0.197861\pi\)
0.812949 + 0.582335i \(0.197861\pi\)
\(152\) 4.01371e10 0.0401242
\(153\) −3.36944e11 −0.324904
\(154\) 6.35713e10 0.0591421
\(155\) −1.81835e11 −0.163250
\(156\) −6.46588e11 −0.560328
\(157\) 1.64554e12 1.37676 0.688382 0.725348i \(-0.258321\pi\)
0.688382 + 0.725348i \(0.258321\pi\)
\(158\) 4.84757e11 0.391661
\(159\) −9.25660e11 −0.722382
\(160\) 3.70228e11 0.279132
\(161\) 6.92021e11 0.504170
\(162\) −4.32751e10 −0.0304723
\(163\) −1.67635e11 −0.114113 −0.0570563 0.998371i \(-0.518171\pi\)
−0.0570563 + 0.998371i \(0.518171\pi\)
\(164\) −8.51557e11 −0.560496
\(165\) 1.40486e11 0.0894275
\(166\) −3.19736e11 −0.196878
\(167\) 2.31740e12 1.38058 0.690289 0.723534i \(-0.257483\pi\)
0.690289 + 0.723534i \(0.257483\pi\)
\(168\) 2.76958e11 0.159666
\(169\) 1.81623e11 0.101343
\(170\) −1.86217e11 −0.100589
\(171\) 4.84433e10 0.0253370
\(172\) 3.39766e12 1.72097
\(173\) −7.66240e10 −0.0375934 −0.0187967 0.999823i \(-0.505984\pi\)
−0.0187967 + 0.999823i \(0.505984\pi\)
\(174\) −2.92039e11 −0.138810
\(175\) −9.76436e11 −0.449712
\(176\) −7.19332e11 −0.321078
\(177\) 1.73727e11 0.0751646
\(178\) −5.42291e11 −0.227469
\(179\) 4.03824e12 1.64248 0.821240 0.570582i \(-0.193282\pi\)
0.821240 + 0.570582i \(0.193282\pi\)
\(180\) 2.94066e11 0.115997
\(181\) 4.73737e12 1.81261 0.906306 0.422622i \(-0.138890\pi\)
0.906306 + 0.422622i \(0.138890\pi\)
\(182\) −4.06205e11 −0.150783
\(183\) 2.36522e12 0.851905
\(184\) 1.45333e12 0.508001
\(185\) 1.18597e12 0.402373
\(186\) −2.08562e11 −0.0686932
\(187\) 1.25462e12 0.401219
\(188\) −1.13312e12 −0.351890
\(189\) 3.34273e11 0.100823
\(190\) 2.67729e10 0.00784422
\(191\) 2.61020e12 0.743002 0.371501 0.928433i \(-0.378843\pi\)
0.371501 + 0.928433i \(0.378843\pi\)
\(192\) −1.20352e12 −0.332888
\(193\) −2.73486e12 −0.735139 −0.367569 0.929996i \(-0.619810\pi\)
−0.367569 + 0.929996i \(0.619810\pi\)
\(194\) 5.46452e11 0.142772
\(195\) −8.97670e11 −0.227996
\(196\) 2.71712e12 0.670966
\(197\) −2.98094e12 −0.715796 −0.357898 0.933761i \(-0.616506\pi\)
−0.357898 + 0.933761i \(0.616506\pi\)
\(198\) 1.61136e11 0.0376299
\(199\) 2.64798e12 0.601481 0.300741 0.953706i \(-0.402766\pi\)
0.300741 + 0.953706i \(0.402766\pi\)
\(200\) −2.05063e12 −0.453129
\(201\) 4.99769e12 1.07446
\(202\) 1.19691e12 0.250398
\(203\) 2.25582e12 0.459278
\(204\) 2.62617e12 0.520423
\(205\) −1.18223e12 −0.228064
\(206\) 6.77472e11 0.127239
\(207\) 1.75408e12 0.320784
\(208\) 4.59635e12 0.818587
\(209\) −1.80379e11 −0.0312883
\(210\) 1.84740e11 0.0312145
\(211\) 5.05129e12 0.831474 0.415737 0.909485i \(-0.363524\pi\)
0.415737 + 0.909485i \(0.363524\pi\)
\(212\) 7.21467e12 1.15710
\(213\) −3.19567e12 −0.499430
\(214\) −7.46830e11 −0.113749
\(215\) 4.71703e12 0.700257
\(216\) 7.02012e11 0.101589
\(217\) 1.61101e12 0.227284
\(218\) −1.87386e11 −0.0257767
\(219\) 6.34405e12 0.850989
\(220\) −1.09496e12 −0.143243
\(221\) −8.01668e12 −1.02291
\(222\) 1.36029e12 0.169313
\(223\) −1.15280e13 −1.39983 −0.699917 0.714224i \(-0.746780\pi\)
−0.699917 + 0.714224i \(0.746780\pi\)
\(224\) −3.28013e12 −0.388621
\(225\) −2.47499e12 −0.286135
\(226\) −2.05853e12 −0.232253
\(227\) −1.82623e12 −0.201101 −0.100550 0.994932i \(-0.532060\pi\)
−0.100550 + 0.994932i \(0.532060\pi\)
\(228\) −3.77571e11 −0.0405842
\(229\) −1.06757e12 −0.112022 −0.0560108 0.998430i \(-0.517838\pi\)
−0.0560108 + 0.998430i \(0.517838\pi\)
\(230\) 9.69418e11 0.0993135
\(231\) −1.24467e12 −0.124505
\(232\) 4.73748e12 0.462768
\(233\) 1.52920e13 1.45884 0.729420 0.684066i \(-0.239790\pi\)
0.729420 + 0.684066i \(0.239790\pi\)
\(234\) −1.02962e12 −0.0959374
\(235\) −1.57313e12 −0.143183
\(236\) −1.35404e12 −0.120397
\(237\) −9.49112e12 −0.824521
\(238\) 1.64983e12 0.140045
\(239\) −1.01553e13 −0.842372 −0.421186 0.906974i \(-0.638386\pi\)
−0.421186 + 0.906974i \(0.638386\pi\)
\(240\) −2.09041e12 −0.169461
\(241\) −1.22766e13 −0.972714 −0.486357 0.873760i \(-0.661674\pi\)
−0.486357 + 0.873760i \(0.661674\pi\)
\(242\) 2.94106e12 0.227782
\(243\) 8.47289e11 0.0641500
\(244\) −1.84348e13 −1.36456
\(245\) 3.77223e12 0.273014
\(246\) −1.35601e12 −0.0959662
\(247\) 1.15258e12 0.0797696
\(248\) 3.38331e12 0.229011
\(249\) 6.26015e12 0.414465
\(250\) −2.96131e12 −0.191785
\(251\) 1.41141e13 0.894225 0.447112 0.894478i \(-0.352452\pi\)
0.447112 + 0.894478i \(0.352452\pi\)
\(252\) −2.60535e12 −0.161496
\(253\) −6.53136e12 −0.396133
\(254\) −6.06933e12 −0.360209
\(255\) 3.64596e12 0.211759
\(256\) 5.80145e12 0.329774
\(257\) 1.81778e13 1.01137 0.505683 0.862719i \(-0.331240\pi\)
0.505683 + 0.862719i \(0.331240\pi\)
\(258\) 5.41038e12 0.294659
\(259\) −1.05074e13 −0.560204
\(260\) 6.99652e12 0.365198
\(261\) 5.71787e12 0.292221
\(262\) 7.33365e12 0.366998
\(263\) 1.38536e13 0.678902 0.339451 0.940624i \(-0.389759\pi\)
0.339451 + 0.940624i \(0.389759\pi\)
\(264\) −2.61395e12 −0.125452
\(265\) 1.00163e13 0.470819
\(266\) −2.37201e11 −0.0109211
\(267\) 1.06176e13 0.478866
\(268\) −3.89524e13 −1.72105
\(269\) −1.54482e13 −0.668715 −0.334357 0.942446i \(-0.608519\pi\)
−0.334357 + 0.942446i \(0.608519\pi\)
\(270\) 4.68266e11 0.0198606
\(271\) 3.07070e13 1.27616 0.638082 0.769969i \(-0.279728\pi\)
0.638082 + 0.769969i \(0.279728\pi\)
\(272\) −1.86685e13 −0.760290
\(273\) 7.95313e12 0.317427
\(274\) −6.45559e12 −0.252527
\(275\) 9.21569e12 0.353344
\(276\) −1.36715e13 −0.513825
\(277\) 1.11150e13 0.409514 0.204757 0.978813i \(-0.434360\pi\)
0.204757 + 0.978813i \(0.434360\pi\)
\(278\) −1.32123e11 −0.00477236
\(279\) 4.08346e12 0.144612
\(280\) −2.99687e12 −0.104064
\(281\) 3.48370e13 1.18620 0.593098 0.805130i \(-0.297905\pi\)
0.593098 + 0.805130i \(0.297905\pi\)
\(282\) −1.80436e12 −0.0602494
\(283\) 1.58891e13 0.520324 0.260162 0.965565i \(-0.416224\pi\)
0.260162 + 0.965565i \(0.416224\pi\)
\(284\) 2.49073e13 0.799975
\(285\) −5.24189e11 −0.0165136
\(286\) 3.83379e12 0.118472
\(287\) 1.04743e13 0.317522
\(288\) −8.31421e12 −0.247265
\(289\) −1.71150e12 −0.0499389
\(290\) 3.16006e12 0.0904704
\(291\) −1.06991e13 −0.300562
\(292\) −4.94461e13 −1.36309
\(293\) −6.51516e13 −1.76260 −0.881299 0.472560i \(-0.843330\pi\)
−0.881299 + 0.472560i \(0.843330\pi\)
\(294\) 4.32670e12 0.114880
\(295\) −1.87984e12 −0.0489891
\(296\) −2.20668e13 −0.564461
\(297\) −3.15489e12 −0.0792181
\(298\) −1.86662e13 −0.460114
\(299\) 4.17337e13 1.00994
\(300\) 1.92903e13 0.458324
\(301\) −4.17917e13 −0.974933
\(302\) −1.94661e13 −0.445904
\(303\) −2.34345e13 −0.527135
\(304\) 2.68401e12 0.0592898
\(305\) −2.55933e13 −0.555236
\(306\) 4.18187e12 0.0891051
\(307\) −6.94381e12 −0.145324 −0.0726619 0.997357i \(-0.523149\pi\)
−0.0726619 + 0.997357i \(0.523149\pi\)
\(308\) 9.70107e12 0.199430
\(309\) −1.32643e13 −0.267863
\(310\) 2.25678e12 0.0447714
\(311\) 9.11794e13 1.77711 0.888555 0.458770i \(-0.151710\pi\)
0.888555 + 0.458770i \(0.151710\pi\)
\(312\) 1.67025e13 0.319839
\(313\) 4.52842e13 0.852026 0.426013 0.904717i \(-0.359918\pi\)
0.426013 + 0.904717i \(0.359918\pi\)
\(314\) −2.04230e13 −0.377579
\(315\) −3.61706e12 −0.0657124
\(316\) 7.39746e13 1.32070
\(317\) 5.55014e13 0.973818 0.486909 0.873453i \(-0.338124\pi\)
0.486909 + 0.873453i \(0.338124\pi\)
\(318\) 1.14885e13 0.198114
\(319\) −2.12906e13 −0.360860
\(320\) 1.30229e13 0.216962
\(321\) 1.46223e13 0.239462
\(322\) −8.58880e12 −0.138269
\(323\) −4.68129e12 −0.0740887
\(324\) −6.60384e12 −0.102754
\(325\) −5.88859e13 −0.900851
\(326\) 2.08055e12 0.0312955
\(327\) 3.66886e12 0.0542648
\(328\) 2.19972e13 0.319935
\(329\) 1.39375e13 0.199346
\(330\) −1.74360e12 −0.0245256
\(331\) −1.22787e14 −1.69862 −0.849311 0.527892i \(-0.822982\pi\)
−0.849311 + 0.527892i \(0.822982\pi\)
\(332\) −4.87921e13 −0.663881
\(333\) −2.66333e13 −0.356437
\(334\) −2.87617e13 −0.378625
\(335\) −5.40784e13 −0.700289
\(336\) 1.85205e13 0.235932
\(337\) 1.15972e14 1.45341 0.726706 0.686948i \(-0.241050\pi\)
0.726706 + 0.686948i \(0.241050\pi\)
\(338\) −2.25416e12 −0.0277934
\(339\) 4.03042e13 0.488936
\(340\) −2.84169e13 −0.339190
\(341\) −1.52048e13 −0.178580
\(342\) −6.01238e11 −0.00694869
\(343\) −7.94849e13 −0.903997
\(344\) −8.77675e13 −0.982341
\(345\) −1.89804e13 −0.209074
\(346\) 9.50994e11 0.0103100
\(347\) 8.80751e13 0.939812 0.469906 0.882716i \(-0.344288\pi\)
0.469906 + 0.882716i \(0.344288\pi\)
\(348\) −4.45656e13 −0.468073
\(349\) −1.17002e14 −1.20963 −0.604816 0.796365i \(-0.706753\pi\)
−0.604816 + 0.796365i \(0.706753\pi\)
\(350\) 1.21187e13 0.123334
\(351\) 2.01590e13 0.201967
\(352\) 3.09581e13 0.305344
\(353\) −7.06831e13 −0.686364 −0.343182 0.939269i \(-0.611505\pi\)
−0.343182 + 0.939269i \(0.611505\pi\)
\(354\) −2.15615e12 −0.0206140
\(355\) 3.45793e13 0.325507
\(356\) −8.27544e13 −0.767037
\(357\) −3.23023e13 −0.294821
\(358\) −5.01193e13 −0.450452
\(359\) 3.49060e12 0.0308945 0.0154472 0.999881i \(-0.495083\pi\)
0.0154472 + 0.999881i \(0.495083\pi\)
\(360\) −7.59624e12 −0.0662117
\(361\) −1.15817e14 −0.994222
\(362\) −5.87963e13 −0.497110
\(363\) −5.75834e13 −0.479525
\(364\) −6.19874e13 −0.508447
\(365\) −6.86469e13 −0.554639
\(366\) −2.93552e13 −0.233636
\(367\) 2.40276e13 0.188385 0.0941927 0.995554i \(-0.469973\pi\)
0.0941927 + 0.995554i \(0.469973\pi\)
\(368\) 9.71854e13 0.750651
\(369\) 2.65494e13 0.202027
\(370\) −1.47193e13 −0.110351
\(371\) −8.87416e13 −0.655497
\(372\) −3.18269e13 −0.231637
\(373\) 4.62694e13 0.331814 0.165907 0.986141i \(-0.446945\pi\)
0.165907 + 0.986141i \(0.446945\pi\)
\(374\) −1.55713e13 −0.110035
\(375\) 5.79799e13 0.403743
\(376\) 2.92704e13 0.200861
\(377\) 1.36042e14 0.920014
\(378\) −4.14872e12 −0.0276509
\(379\) −1.43879e14 −0.945110 −0.472555 0.881301i \(-0.656668\pi\)
−0.472555 + 0.881301i \(0.656668\pi\)
\(380\) 4.08557e12 0.0264511
\(381\) 1.18832e14 0.758309
\(382\) −3.23956e13 −0.203769
\(383\) −1.48728e14 −0.922149 −0.461074 0.887362i \(-0.652536\pi\)
−0.461074 + 0.887362i \(0.652536\pi\)
\(384\) 8.50092e13 0.519570
\(385\) 1.34682e13 0.0811474
\(386\) 3.39428e13 0.201612
\(387\) −1.05930e14 −0.620313
\(388\) 8.33893e13 0.481433
\(389\) −1.76673e14 −1.00565 −0.502827 0.864387i \(-0.667707\pi\)
−0.502827 + 0.864387i \(0.667707\pi\)
\(390\) 1.11411e13 0.0625280
\(391\) −1.69505e14 −0.938016
\(392\) −7.01880e13 −0.382992
\(393\) −1.43586e14 −0.772600
\(394\) 3.69970e13 0.196308
\(395\) 1.02700e14 0.537388
\(396\) 2.45895e13 0.126890
\(397\) −6.00013e13 −0.305361 −0.152680 0.988276i \(-0.548790\pi\)
−0.152680 + 0.988276i \(0.548790\pi\)
\(398\) −3.28645e13 −0.164957
\(399\) 4.64418e12 0.0229910
\(400\) −1.37128e14 −0.669569
\(401\) 2.97986e14 1.43517 0.717583 0.696473i \(-0.245248\pi\)
0.717583 + 0.696473i \(0.245248\pi\)
\(402\) −6.20272e13 −0.294672
\(403\) 9.71551e13 0.455290
\(404\) 1.82651e14 0.844353
\(405\) −9.16824e12 −0.0418103
\(406\) −2.79974e13 −0.125957
\(407\) 9.91697e13 0.440159
\(408\) −6.78386e13 −0.297061
\(409\) −1.83676e14 −0.793549 −0.396774 0.917916i \(-0.629870\pi\)
−0.396774 + 0.917916i \(0.629870\pi\)
\(410\) 1.46729e13 0.0625467
\(411\) 1.26395e14 0.531618
\(412\) 1.03383e14 0.429057
\(413\) 1.66549e13 0.0682051
\(414\) −2.17702e13 −0.0879754
\(415\) −6.77391e13 −0.270131
\(416\) −1.97815e14 −0.778476
\(417\) 2.58686e12 0.0100467
\(418\) 2.23872e12 0.00858085
\(419\) −3.37184e14 −1.27553 −0.637763 0.770233i \(-0.720140\pi\)
−0.637763 + 0.770233i \(0.720140\pi\)
\(420\) 2.81917e13 0.105257
\(421\) 4.41343e14 1.62639 0.813195 0.581991i \(-0.197726\pi\)
0.813195 + 0.581991i \(0.197726\pi\)
\(422\) −6.26924e13 −0.228032
\(423\) 3.53278e13 0.126836
\(424\) −1.86368e14 −0.660478
\(425\) 2.39170e14 0.836695
\(426\) 3.96620e13 0.136969
\(427\) 2.26750e14 0.773027
\(428\) −1.13967e14 −0.383565
\(429\) −7.50623e13 −0.249406
\(430\) −5.85440e13 −0.192046
\(431\) −4.62208e14 −1.49697 −0.748483 0.663154i \(-0.769218\pi\)
−0.748483 + 0.663154i \(0.769218\pi\)
\(432\) 4.69443e13 0.150114
\(433\) −5.20774e13 −0.164424 −0.0822121 0.996615i \(-0.526198\pi\)
−0.0822121 + 0.996615i \(0.526198\pi\)
\(434\) −1.99945e13 −0.0623329
\(435\) −6.18713e13 −0.190457
\(436\) −2.85954e13 −0.0869202
\(437\) 2.43702e13 0.0731494
\(438\) −7.87371e13 −0.233384
\(439\) −2.17454e14 −0.636520 −0.318260 0.948004i \(-0.603098\pi\)
−0.318260 + 0.948004i \(0.603098\pi\)
\(440\) 2.82847e13 0.0817641
\(441\) −8.47130e13 −0.241845
\(442\) 9.94964e13 0.280534
\(443\) −2.78979e14 −0.776875 −0.388438 0.921475i \(-0.626985\pi\)
−0.388438 + 0.921475i \(0.626985\pi\)
\(444\) 2.07582e14 0.570932
\(445\) −1.14890e14 −0.312105
\(446\) 1.43076e14 0.383906
\(447\) 3.65467e14 0.968628
\(448\) −1.15380e14 −0.302066
\(449\) 6.68167e14 1.72795 0.863973 0.503539i \(-0.167969\pi\)
0.863973 + 0.503539i \(0.167969\pi\)
\(450\) 3.07176e13 0.0784727
\(451\) −9.88571e13 −0.249481
\(452\) −3.14135e14 −0.783167
\(453\) 3.81130e14 0.938712
\(454\) 2.26657e13 0.0551521
\(455\) −8.60583e13 −0.206885
\(456\) 9.75333e12 0.0231657
\(457\) 7.36922e13 0.172935 0.0864674 0.996255i \(-0.472442\pi\)
0.0864674 + 0.996255i \(0.472442\pi\)
\(458\) 1.32498e13 0.0307220
\(459\) −8.18773e13 −0.187583
\(460\) 1.47935e14 0.334890
\(461\) 5.66378e13 0.126693 0.0633463 0.997992i \(-0.479823\pi\)
0.0633463 + 0.997992i \(0.479823\pi\)
\(462\) 1.54478e13 0.0341457
\(463\) −6.02905e14 −1.31690 −0.658451 0.752624i \(-0.728788\pi\)
−0.658451 + 0.752624i \(0.728788\pi\)
\(464\) 3.16800e14 0.683812
\(465\) −4.41858e13 −0.0942523
\(466\) −1.89792e14 −0.400088
\(467\) 9.34292e14 1.94644 0.973218 0.229886i \(-0.0738354\pi\)
0.973218 + 0.229886i \(0.0738354\pi\)
\(468\) −1.57121e14 −0.323505
\(469\) 4.79121e14 0.974976
\(470\) 1.95244e13 0.0392680
\(471\) 3.99865e14 0.794875
\(472\) 3.49772e13 0.0687234
\(473\) 3.94434e14 0.766016
\(474\) 1.17796e14 0.226126
\(475\) −3.43861e13 −0.0652481
\(476\) 2.51767e14 0.472237
\(477\) −2.24935e14 −0.417068
\(478\) 1.26039e14 0.231021
\(479\) 2.84742e14 0.515948 0.257974 0.966152i \(-0.416945\pi\)
0.257974 + 0.966152i \(0.416945\pi\)
\(480\) 8.99654e13 0.161157
\(481\) −6.33669e14 −1.12219
\(482\) 1.52367e14 0.266768
\(483\) 1.68161e14 0.291083
\(484\) 4.48810e14 0.768092
\(485\) 1.15771e14 0.195894
\(486\) −1.05158e13 −0.0175932
\(487\) −1.18679e15 −1.96320 −0.981602 0.190941i \(-0.938846\pi\)
−0.981602 + 0.190941i \(0.938846\pi\)
\(488\) 4.76202e14 0.778902
\(489\) −4.07354e13 −0.0658829
\(490\) −4.68178e13 −0.0748743
\(491\) −3.24362e14 −0.512959 −0.256479 0.966550i \(-0.582563\pi\)
−0.256479 + 0.966550i \(0.582563\pi\)
\(492\) −2.06928e14 −0.323602
\(493\) −5.52544e14 −0.854493
\(494\) −1.43049e13 −0.0218769
\(495\) 3.41381e13 0.0516310
\(496\) 2.26245e14 0.338400
\(497\) −3.06364e14 −0.453187
\(498\) −7.76958e13 −0.113668
\(499\) 1.00957e15 1.46078 0.730388 0.683033i \(-0.239339\pi\)
0.730388 + 0.683033i \(0.239339\pi\)
\(500\) −4.51900e14 −0.646707
\(501\) 5.63129e14 0.797077
\(502\) −1.75172e14 −0.245242
\(503\) −1.07256e15 −1.48524 −0.742619 0.669714i \(-0.766416\pi\)
−0.742619 + 0.669714i \(0.766416\pi\)
\(504\) 6.73008e13 0.0921833
\(505\) 2.53578e14 0.343564
\(506\) 8.10618e13 0.108640
\(507\) 4.41344e13 0.0585105
\(508\) −9.26187e14 −1.21464
\(509\) 1.40656e15 1.82478 0.912391 0.409319i \(-0.134234\pi\)
0.912391 + 0.409319i \(0.134234\pi\)
\(510\) −4.52507e13 −0.0580750
\(511\) 6.08194e14 0.772196
\(512\) −7.88459e14 −0.990363
\(513\) 1.17717e13 0.0146283
\(514\) −2.25608e14 −0.277368
\(515\) 1.43529e14 0.174582
\(516\) 8.25631e14 0.993602
\(517\) −1.31544e14 −0.156629
\(518\) 1.30409e14 0.153636
\(519\) −1.86196e13 −0.0217045
\(520\) −1.80732e14 −0.208458
\(521\) −9.68497e14 −1.10533 −0.552663 0.833405i \(-0.686388\pi\)
−0.552663 + 0.833405i \(0.686388\pi\)
\(522\) −7.09655e13 −0.0801419
\(523\) 5.77418e14 0.645254 0.322627 0.946526i \(-0.395434\pi\)
0.322627 + 0.946526i \(0.395434\pi\)
\(524\) 1.11913e15 1.23753
\(525\) −2.37274e14 −0.259641
\(526\) −1.71940e14 −0.186189
\(527\) −3.94603e14 −0.422866
\(528\) −1.74798e14 −0.185374
\(529\) −7.03905e13 −0.0738767
\(530\) −1.24314e14 −0.129122
\(531\) 4.22156e13 0.0433963
\(532\) −3.61972e13 −0.0368265
\(533\) 6.31672e14 0.636051
\(534\) −1.31777e14 −0.131329
\(535\) −1.58223e14 −0.156072
\(536\) 1.00621e15 0.982385
\(537\) 9.81292e14 0.948287
\(538\) 1.91731e14 0.183396
\(539\) 3.15430e14 0.298652
\(540\) 7.14580e13 0.0669708
\(541\) −3.42211e14 −0.317474 −0.158737 0.987321i \(-0.550742\pi\)
−0.158737 + 0.987321i \(0.550742\pi\)
\(542\) −3.81110e14 −0.349989
\(543\) 1.15118e15 1.04651
\(544\) 8.03440e14 0.723035
\(545\) −3.96995e13 −0.0353675
\(546\) −9.87077e13 −0.0870546
\(547\) 1.11343e15 0.972148 0.486074 0.873918i \(-0.338429\pi\)
0.486074 + 0.873918i \(0.338429\pi\)
\(548\) −9.85132e14 −0.851533
\(549\) 5.74750e14 0.491848
\(550\) −1.14378e14 −0.0969049
\(551\) 7.94407e13 0.0666360
\(552\) 3.53158e14 0.293295
\(553\) −9.09899e14 −0.748178
\(554\) −1.37950e14 −0.112310
\(555\) 2.88191e14 0.232310
\(556\) −2.01622e13 −0.0160926
\(557\) 1.77163e15 1.40013 0.700066 0.714078i \(-0.253154\pi\)
0.700066 + 0.714078i \(0.253154\pi\)
\(558\) −5.06806e13 −0.0396601
\(559\) −2.52033e15 −1.95296
\(560\) −2.00404e14 −0.153770
\(561\) 3.04872e14 0.231644
\(562\) −4.32369e14 −0.325315
\(563\) 1.13752e15 0.847542 0.423771 0.905769i \(-0.360706\pi\)
0.423771 + 0.905769i \(0.360706\pi\)
\(564\) −2.75348e14 −0.203164
\(565\) −4.36119e14 −0.318668
\(566\) −1.97202e14 −0.142699
\(567\) 8.12283e13 0.0582104
\(568\) −6.43400e14 −0.456631
\(569\) 2.22030e15 1.56061 0.780303 0.625402i \(-0.215065\pi\)
0.780303 + 0.625402i \(0.215065\pi\)
\(570\) 6.50580e12 0.00452886
\(571\) 1.21610e15 0.838437 0.419218 0.907885i \(-0.362304\pi\)
0.419218 + 0.907885i \(0.362304\pi\)
\(572\) 5.85042e14 0.399493
\(573\) 6.34278e14 0.428972
\(574\) −1.29998e14 −0.0870807
\(575\) −1.24509e15 −0.826088
\(576\) −2.92456e14 −0.192193
\(577\) −2.57058e15 −1.67326 −0.836631 0.547767i \(-0.815478\pi\)
−0.836631 + 0.547767i \(0.815478\pi\)
\(578\) 2.12417e13 0.0136958
\(579\) −6.64570e14 −0.424432
\(580\) 4.82230e14 0.305070
\(581\) 6.00151e14 0.376090
\(582\) 1.32788e14 0.0824294
\(583\) 8.37551e14 0.515032
\(584\) 1.27728e15 0.778064
\(585\) −2.18134e14 −0.131633
\(586\) 8.08608e14 0.483394
\(587\) 8.26236e14 0.489322 0.244661 0.969609i \(-0.421323\pi\)
0.244661 + 0.969609i \(0.421323\pi\)
\(588\) 6.60260e14 0.387382
\(589\) 5.67332e13 0.0329764
\(590\) 2.33310e13 0.0134353
\(591\) −7.24369e14 −0.413265
\(592\) −1.47563e15 −0.834079
\(593\) 2.20813e15 1.23659 0.618293 0.785948i \(-0.287824\pi\)
0.618293 + 0.785948i \(0.287824\pi\)
\(594\) 3.91560e13 0.0217256
\(595\) 3.49533e14 0.192152
\(596\) −2.84848e15 −1.55153
\(597\) 6.43458e14 0.347265
\(598\) −5.17965e14 −0.276977
\(599\) 2.14568e15 1.13689 0.568443 0.822722i \(-0.307546\pi\)
0.568443 + 0.822722i \(0.307546\pi\)
\(600\) −4.98303e14 −0.261614
\(601\) 1.03034e15 0.536006 0.268003 0.963418i \(-0.413636\pi\)
0.268003 + 0.963418i \(0.413636\pi\)
\(602\) 5.18685e14 0.267376
\(603\) 1.21444e15 0.620340
\(604\) −2.97056e15 −1.50361
\(605\) 6.23092e14 0.312534
\(606\) 2.90850e14 0.144567
\(607\) 6.29663e14 0.310149 0.155074 0.987903i \(-0.450438\pi\)
0.155074 + 0.987903i \(0.450438\pi\)
\(608\) −1.15513e14 −0.0563845
\(609\) 5.48164e14 0.265164
\(610\) 3.17643e14 0.152274
\(611\) 8.40530e14 0.399325
\(612\) 6.38159e14 0.300466
\(613\) 4.61880e14 0.215524 0.107762 0.994177i \(-0.465631\pi\)
0.107762 + 0.994177i \(0.465631\pi\)
\(614\) 8.61808e13 0.0398552
\(615\) −2.87282e14 −0.131673
\(616\) −2.50596e14 −0.113836
\(617\) 2.25606e15 1.01574 0.507870 0.861434i \(-0.330433\pi\)
0.507870 + 0.861434i \(0.330433\pi\)
\(618\) 1.64626e14 0.0734617
\(619\) 3.97848e13 0.0175962 0.00879810 0.999961i \(-0.497199\pi\)
0.00879810 + 0.999961i \(0.497199\pi\)
\(620\) 3.44388e14 0.150971
\(621\) 4.26242e14 0.185205
\(622\) −1.13164e15 −0.487374
\(623\) 1.01789e15 0.434528
\(624\) 1.11691e15 0.472612
\(625\) 1.41921e15 0.595261
\(626\) −5.62030e14 −0.233669
\(627\) −4.38322e13 −0.0180643
\(628\) −3.11658e15 −1.27321
\(629\) 2.57370e15 1.04227
\(630\) 4.48919e13 0.0180217
\(631\) −3.85979e15 −1.53604 −0.768018 0.640428i \(-0.778757\pi\)
−0.768018 + 0.640428i \(0.778757\pi\)
\(632\) −1.91089e15 −0.753864
\(633\) 1.22746e15 0.480052
\(634\) −6.88837e14 −0.267070
\(635\) −1.28584e15 −0.494234
\(636\) 1.75316e15 0.668050
\(637\) −2.01552e15 −0.761413
\(638\) 2.64241e14 0.0989663
\(639\) −7.76548e14 −0.288346
\(640\) −9.19857e14 −0.338634
\(641\) 3.34095e15 1.21941 0.609707 0.792627i \(-0.291287\pi\)
0.609707 + 0.792627i \(0.291287\pi\)
\(642\) −1.81480e14 −0.0656728
\(643\) −2.23408e15 −0.801563 −0.400781 0.916174i \(-0.631261\pi\)
−0.400781 + 0.916174i \(0.631261\pi\)
\(644\) −1.31066e15 −0.466250
\(645\) 1.14624e15 0.404294
\(646\) 5.81004e13 0.0203189
\(647\) −5.72708e15 −1.98591 −0.992956 0.118486i \(-0.962196\pi\)
−0.992956 + 0.118486i \(0.962196\pi\)
\(648\) 1.70589e14 0.0586527
\(649\) −1.57190e14 −0.0535896
\(650\) 7.30844e14 0.247059
\(651\) 3.91475e14 0.131223
\(652\) 3.17495e14 0.105530
\(653\) −5.26368e15 −1.73487 −0.867435 0.497551i \(-0.834233\pi\)
−0.867435 + 0.497551i \(0.834233\pi\)
\(654\) −4.55349e13 −0.0148822
\(655\) 1.55370e15 0.503548
\(656\) 1.47098e15 0.472753
\(657\) 1.54160e15 0.491319
\(658\) −1.72981e14 −0.0546709
\(659\) −3.25334e13 −0.0101967 −0.00509835 0.999987i \(-0.501623\pi\)
−0.00509835 + 0.999987i \(0.501623\pi\)
\(660\) −2.66075e14 −0.0827014
\(661\) 3.05435e15 0.941479 0.470739 0.882272i \(-0.343987\pi\)
0.470739 + 0.882272i \(0.343987\pi\)
\(662\) 1.52393e15 0.465849
\(663\) −1.94805e15 −0.590577
\(664\) 1.26039e15 0.378948
\(665\) −5.02532e13 −0.0149846
\(666\) 3.30551e14 0.0977530
\(667\) 2.87647e15 0.843660
\(668\) −4.38908e15 −1.27674
\(669\) −2.80130e15 −0.808194
\(670\) 6.71176e14 0.192055
\(671\) −2.14009e15 −0.607377
\(672\) −7.97071e14 −0.224371
\(673\) 7.74289e14 0.216182 0.108091 0.994141i \(-0.465526\pi\)
0.108091 + 0.994141i \(0.465526\pi\)
\(674\) −1.43935e15 −0.398600
\(675\) −6.01424e14 −0.165200
\(676\) −3.43987e14 −0.0937208
\(677\) −7.12871e14 −0.192652 −0.0963260 0.995350i \(-0.530709\pi\)
−0.0963260 + 0.995350i \(0.530709\pi\)
\(678\) −5.00223e14 −0.134091
\(679\) −1.02570e15 −0.272733
\(680\) 7.34059e14 0.193612
\(681\) −4.43774e14 −0.116106
\(682\) 1.88710e14 0.0489757
\(683\) 3.21628e15 0.828018 0.414009 0.910273i \(-0.364128\pi\)
0.414009 + 0.910273i \(0.364128\pi\)
\(684\) −9.17498e13 −0.0234313
\(685\) −1.36768e15 −0.346486
\(686\) 9.86501e14 0.247922
\(687\) −2.59420e14 −0.0646757
\(688\) −5.86911e15 −1.45156
\(689\) −5.35174e15 −1.31307
\(690\) 2.35569e14 0.0573387
\(691\) −5.82369e15 −1.40627 −0.703135 0.711056i \(-0.748217\pi\)
−0.703135 + 0.711056i \(0.748217\pi\)
\(692\) 1.45123e14 0.0347658
\(693\) −3.02455e14 −0.0718833
\(694\) −1.09312e15 −0.257744
\(695\) −2.79916e13 −0.00654804
\(696\) 1.15121e15 0.267179
\(697\) −2.56559e15 −0.590754
\(698\) 1.45213e15 0.331743
\(699\) 3.71597e15 0.842262
\(700\) 1.84933e15 0.415888
\(701\) −5.89396e15 −1.31510 −0.657549 0.753412i \(-0.728407\pi\)
−0.657549 + 0.753412i \(0.728407\pi\)
\(702\) −2.50197e14 −0.0553895
\(703\) −3.70027e14 −0.0812792
\(704\) 1.08897e15 0.237337
\(705\) −3.82270e14 −0.0826667
\(706\) 8.77260e14 0.188236
\(707\) −2.24663e15 −0.478328
\(708\) −3.29032e14 −0.0695112
\(709\) −4.66229e14 −0.0977339 −0.0488669 0.998805i \(-0.515561\pi\)
−0.0488669 + 0.998805i \(0.515561\pi\)
\(710\) −4.29170e14 −0.0892706
\(711\) −2.30634e15 −0.476037
\(712\) 2.13769e15 0.437830
\(713\) 2.05425e15 0.417504
\(714\) 4.00909e14 0.0808548
\(715\) 8.12225e14 0.162552
\(716\) −7.64827e15 −1.51894
\(717\) −2.46774e15 −0.486344
\(718\) −4.33225e13 −0.00847284
\(719\) 7.59546e15 1.47416 0.737081 0.675805i \(-0.236204\pi\)
0.737081 + 0.675805i \(0.236204\pi\)
\(720\) −5.07969e14 −0.0978382
\(721\) −1.27163e15 −0.243062
\(722\) 1.43743e15 0.272666
\(723\) −2.98322e15 −0.561596
\(724\) −8.97240e15 −1.67628
\(725\) −4.05867e15 −0.752531
\(726\) 7.14678e14 0.131510
\(727\) 6.98199e15 1.27509 0.637543 0.770414i \(-0.279951\pi\)
0.637543 + 0.770414i \(0.279951\pi\)
\(728\) 1.60124e15 0.290225
\(729\) 2.05891e14 0.0370370
\(730\) 8.51989e14 0.152110
\(731\) 1.02365e16 1.81388
\(732\) −4.47965e15 −0.787831
\(733\) −1.14815e15 −0.200413 −0.100207 0.994967i \(-0.531950\pi\)
−0.100207 + 0.994967i \(0.531950\pi\)
\(734\) −2.98211e14 −0.0516649
\(735\) 9.16652e14 0.157625
\(736\) −4.18260e15 −0.713868
\(737\) −4.52198e15 −0.766051
\(738\) −3.29509e14 −0.0554061
\(739\) 8.01991e15 1.33852 0.669261 0.743028i \(-0.266611\pi\)
0.669261 + 0.743028i \(0.266611\pi\)
\(740\) −2.24618e15 −0.372109
\(741\) 2.80077e14 0.0460550
\(742\) 1.10139e15 0.179771
\(743\) −4.20001e15 −0.680474 −0.340237 0.940340i \(-0.610507\pi\)
−0.340237 + 0.940340i \(0.610507\pi\)
\(744\) 8.22144e14 0.132220
\(745\) −3.95460e15 −0.631311
\(746\) −5.74257e14 −0.0910003
\(747\) 1.52122e15 0.239292
\(748\) −2.37620e15 −0.371042
\(749\) 1.40182e15 0.217291
\(750\) −7.19598e14 −0.110727
\(751\) 6.91910e15 1.05689 0.528445 0.848967i \(-0.322775\pi\)
0.528445 + 0.848967i \(0.322775\pi\)
\(752\) 1.95734e15 0.296804
\(753\) 3.42972e15 0.516281
\(754\) −1.68844e15 −0.252315
\(755\) −4.12408e15 −0.611813
\(756\) −6.33100e14 −0.0932400
\(757\) −1.00977e16 −1.47636 −0.738182 0.674601i \(-0.764316\pi\)
−0.738182 + 0.674601i \(0.764316\pi\)
\(758\) 1.78571e15 0.259197
\(759\) −1.58712e15 −0.228707
\(760\) −1.05538e14 −0.0150985
\(761\) 3.19854e14 0.0454293 0.0227146 0.999742i \(-0.492769\pi\)
0.0227146 + 0.999742i \(0.492769\pi\)
\(762\) −1.47485e15 −0.207967
\(763\) 3.51728e14 0.0492404
\(764\) −4.94362e15 −0.687118
\(765\) 8.85968e14 0.122259
\(766\) 1.84590e15 0.252900
\(767\) 1.00441e15 0.136627
\(768\) 1.40975e15 0.190395
\(769\) −2.19159e14 −0.0293876 −0.0146938 0.999892i \(-0.504677\pi\)
−0.0146938 + 0.999892i \(0.504677\pi\)
\(770\) −1.67156e14 −0.0222548
\(771\) 4.41720e15 0.583913
\(772\) 5.17971e15 0.679846
\(773\) 1.29588e16 1.68879 0.844396 0.535720i \(-0.179960\pi\)
0.844396 + 0.535720i \(0.179960\pi\)
\(774\) 1.31472e15 0.170121
\(775\) −2.89853e15 −0.372407
\(776\) −2.15409e15 −0.274805
\(777\) −2.55330e15 −0.323434
\(778\) 2.19272e15 0.275801
\(779\) 3.68861e14 0.0460688
\(780\) 1.70015e15 0.210847
\(781\) 2.89149e15 0.356075
\(782\) 2.10376e15 0.257252
\(783\) 1.38944e15 0.168714
\(784\) −4.69354e15 −0.565930
\(785\) −4.32681e15 −0.518066
\(786\) 1.78208e15 0.211886
\(787\) −1.52401e16 −1.79939 −0.899696 0.436517i \(-0.856212\pi\)
−0.899696 + 0.436517i \(0.856212\pi\)
\(788\) 5.64579e15 0.661959
\(789\) 3.36643e15 0.391964
\(790\) −1.27463e15 −0.147379
\(791\) 3.86390e15 0.443666
\(792\) −6.35191e14 −0.0724295
\(793\) 1.36746e16 1.54851
\(794\) 7.44687e14 0.0837454
\(795\) 2.43395e15 0.271827
\(796\) −5.01517e15 −0.556242
\(797\) −1.44420e16 −1.59077 −0.795384 0.606106i \(-0.792731\pi\)
−0.795384 + 0.606106i \(0.792731\pi\)
\(798\) −5.76398e13 −0.00630531
\(799\) −3.41388e15 −0.370887
\(800\) 5.90161e15 0.636759
\(801\) 2.58007e15 0.276473
\(802\) −3.69836e15 −0.393595
\(803\) −5.74019e15 −0.606723
\(804\) −9.46543e15 −0.993647
\(805\) −1.81962e15 −0.189716
\(806\) −1.20581e15 −0.124864
\(807\) −3.75392e15 −0.386083
\(808\) −4.71819e15 −0.481962
\(809\) −2.52611e15 −0.256293 −0.128146 0.991755i \(-0.540903\pi\)
−0.128146 + 0.991755i \(0.540903\pi\)
\(810\) 1.13789e14 0.0114665
\(811\) −1.71297e16 −1.71450 −0.857248 0.514905i \(-0.827827\pi\)
−0.857248 + 0.514905i \(0.827827\pi\)
\(812\) −4.27243e15 −0.424734
\(813\) 7.46180e15 0.736793
\(814\) −1.23081e15 −0.120714
\(815\) 4.40784e14 0.0429397
\(816\) −4.53644e15 −0.438954
\(817\) −1.47173e15 −0.141452
\(818\) 2.27963e15 0.217631
\(819\) 1.93261e15 0.183266
\(820\) 2.23910e15 0.210910
\(821\) 1.56247e16 1.46192 0.730960 0.682420i \(-0.239073\pi\)
0.730960 + 0.682420i \(0.239073\pi\)
\(822\) −1.56871e15 −0.145797
\(823\) −1.41870e15 −0.130976 −0.0654881 0.997853i \(-0.520860\pi\)
−0.0654881 + 0.997853i \(0.520860\pi\)
\(824\) −2.67057e15 −0.244909
\(825\) 2.23941e15 0.204003
\(826\) −2.06707e14 −0.0187053
\(827\) 1.58941e16 1.42874 0.714372 0.699766i \(-0.246713\pi\)
0.714372 + 0.699766i \(0.246713\pi\)
\(828\) −3.32217e15 −0.296657
\(829\) 9.82076e15 0.871154 0.435577 0.900151i \(-0.356544\pi\)
0.435577 + 0.900151i \(0.356544\pi\)
\(830\) 8.40722e14 0.0740837
\(831\) 2.70094e15 0.236433
\(832\) −6.95822e15 −0.605090
\(833\) 8.18620e15 0.707188
\(834\) −3.21060e13 −0.00275532
\(835\) −6.09344e15 −0.519502
\(836\) 3.41632e14 0.0289350
\(837\) 9.92281e14 0.0834920
\(838\) 4.18485e15 0.349814
\(839\) −1.39823e16 −1.16115 −0.580576 0.814206i \(-0.697173\pi\)
−0.580576 + 0.814206i \(0.697173\pi\)
\(840\) −7.28240e14 −0.0600812
\(841\) −2.82394e15 −0.231461
\(842\) −5.47759e15 −0.446039
\(843\) 8.46540e15 0.684851
\(844\) −9.56695e15 −0.768936
\(845\) −4.77564e14 −0.0381347
\(846\) −4.38459e14 −0.0347850
\(847\) −5.52044e15 −0.435126
\(848\) −1.24626e16 −0.975959
\(849\) 3.86105e15 0.300409
\(850\) −2.96838e15 −0.229464
\(851\) −1.33983e16 −1.02905
\(852\) 6.05248e15 0.461866
\(853\) 2.98160e15 0.226063 0.113032 0.993591i \(-0.463944\pi\)
0.113032 + 0.993591i \(0.463944\pi\)
\(854\) −2.81424e15 −0.212003
\(855\) −1.27378e14 −0.00953412
\(856\) 2.94398e15 0.218942
\(857\) 1.96566e16 1.45249 0.726247 0.687434i \(-0.241263\pi\)
0.726247 + 0.687434i \(0.241263\pi\)
\(858\) 9.31612e14 0.0683998
\(859\) 2.47331e15 0.180433 0.0902164 0.995922i \(-0.471244\pi\)
0.0902164 + 0.995922i \(0.471244\pi\)
\(860\) −8.93389e15 −0.647588
\(861\) 2.54525e15 0.183321
\(862\) 5.73654e15 0.410544
\(863\) 2.05179e16 1.45906 0.729530 0.683949i \(-0.239739\pi\)
0.729530 + 0.683949i \(0.239739\pi\)
\(864\) −2.02035e15 −0.142758
\(865\) 2.01477e14 0.0141461
\(866\) 6.46341e14 0.0450935
\(867\) −4.15895e14 −0.0288322
\(868\) −3.05119e15 −0.210189
\(869\) 8.58771e15 0.587853
\(870\) 7.67895e14 0.0522331
\(871\) 2.88943e16 1.95305
\(872\) 7.38670e14 0.0496146
\(873\) −2.59987e15 −0.173530
\(874\) −3.02462e14 −0.0200613
\(875\) 5.55844e15 0.366361
\(876\) −1.20154e16 −0.786983
\(877\) −1.47558e16 −0.960428 −0.480214 0.877151i \(-0.659441\pi\)
−0.480214 + 0.877151i \(0.659441\pi\)
\(878\) 2.69886e15 0.174566
\(879\) −1.58318e16 −1.01764
\(880\) 1.89143e15 0.120819
\(881\) 1.32702e16 0.842385 0.421192 0.906971i \(-0.361612\pi\)
0.421192 + 0.906971i \(0.361612\pi\)
\(882\) 1.05139e15 0.0663263
\(883\) 1.33127e16 0.834608 0.417304 0.908767i \(-0.362975\pi\)
0.417304 + 0.908767i \(0.362975\pi\)
\(884\) 1.51833e16 0.945973
\(885\) −4.56801e14 −0.0282839
\(886\) 3.46246e15 0.213059
\(887\) 2.10196e16 1.28542 0.642709 0.766111i \(-0.277811\pi\)
0.642709 + 0.766111i \(0.277811\pi\)
\(888\) −5.36222e15 −0.325892
\(889\) 1.13922e16 0.688097
\(890\) 1.42591e15 0.0855950
\(891\) −7.66639e14 −0.0457366
\(892\) 2.18336e16 1.29455
\(893\) 4.90823e14 0.0289229
\(894\) −4.53588e15 −0.265647
\(895\) −1.06182e16 −0.618053
\(896\) 8.14970e15 0.471463
\(897\) 1.01413e16 0.583089
\(898\) −8.29274e15 −0.473890
\(899\) 6.69635e15 0.380329
\(900\) 4.68755e15 0.264613
\(901\) 2.17365e16 1.21956
\(902\) 1.22693e15 0.0684203
\(903\) −1.01554e16 −0.562878
\(904\) 8.11465e15 0.447037
\(905\) −1.24566e16 −0.682073
\(906\) −4.73027e15 −0.257443
\(907\) 1.91783e16 1.03745 0.518727 0.854940i \(-0.326406\pi\)
0.518727 + 0.854940i \(0.326406\pi\)
\(908\) 3.45882e15 0.185975
\(909\) −5.69459e15 −0.304342
\(910\) 1.06808e15 0.0567385
\(911\) −1.43002e16 −0.755077 −0.377538 0.925994i \(-0.623229\pi\)
−0.377538 + 0.925994i \(0.623229\pi\)
\(912\) 6.52215e14 0.0342310
\(913\) −5.66428e15 −0.295498
\(914\) −9.14607e14 −0.0474275
\(915\) −6.21918e15 −0.320566
\(916\) 2.02194e15 0.103596
\(917\) −1.37654e16 −0.701065
\(918\) 1.01619e15 0.0514448
\(919\) −1.10785e16 −0.557501 −0.278750 0.960364i \(-0.589920\pi\)
−0.278750 + 0.960364i \(0.589920\pi\)
\(920\) −3.82141e15 −0.191157
\(921\) −1.68735e15 −0.0839027
\(922\) −7.02942e14 −0.0347456
\(923\) −1.84759e16 −0.907813
\(924\) 2.35736e15 0.115141
\(925\) 1.89049e16 0.917899
\(926\) 7.48276e15 0.361161
\(927\) −3.22323e15 −0.154651
\(928\) −1.36342e16 −0.650304
\(929\) 3.91778e16 1.85761 0.928803 0.370575i \(-0.120839\pi\)
0.928803 + 0.370575i \(0.120839\pi\)
\(930\) 5.48398e14 0.0258488
\(931\) −1.17695e15 −0.0551487
\(932\) −2.89626e16 −1.34912
\(933\) 2.21566e16 1.02602
\(934\) −1.15957e16 −0.533811
\(935\) −3.29892e15 −0.150976
\(936\) 4.05871e15 0.184659
\(937\) −5.36663e15 −0.242736 −0.121368 0.992608i \(-0.538728\pi\)
−0.121368 + 0.992608i \(0.538728\pi\)
\(938\) −5.94645e15 −0.267388
\(939\) 1.10041e16 0.491917
\(940\) 2.97945e15 0.132414
\(941\) 3.20147e16 1.41451 0.707255 0.706959i \(-0.249933\pi\)
0.707255 + 0.706959i \(0.249933\pi\)
\(942\) −4.96280e15 −0.217995
\(943\) 1.33561e16 0.583264
\(944\) 2.33897e15 0.101549
\(945\) −8.78945e14 −0.0379391
\(946\) −4.89539e15 −0.210081
\(947\) 3.95764e16 1.68854 0.844271 0.535917i \(-0.180034\pi\)
0.844271 + 0.535917i \(0.180034\pi\)
\(948\) 1.79758e16 0.762506
\(949\) 3.66784e16 1.54684
\(950\) 4.26772e14 0.0178943
\(951\) 1.34868e16 0.562234
\(952\) −6.50358e15 −0.269556
\(953\) −3.58692e15 −0.147812 −0.0739061 0.997265i \(-0.523547\pi\)
−0.0739061 + 0.997265i \(0.523547\pi\)
\(954\) 2.79171e15 0.114381
\(955\) −6.86332e15 −0.279586
\(956\) 1.92337e16 0.779014
\(957\) −5.17362e15 −0.208343
\(958\) −3.53398e15 −0.141499
\(959\) 1.21173e16 0.482395
\(960\) 3.16458e15 0.125263
\(961\) −2.06262e16 −0.811785
\(962\) 7.86458e15 0.307760
\(963\) 3.55321e15 0.138254
\(964\) 2.32515e16 0.899552
\(965\) 7.19110e15 0.276627
\(966\) −2.08708e15 −0.0798297
\(967\) 6.50498e15 0.247400 0.123700 0.992320i \(-0.460524\pi\)
0.123700 + 0.992320i \(0.460524\pi\)
\(968\) −1.15936e16 −0.438432
\(969\) −1.13755e15 −0.0427751
\(970\) −1.43685e15 −0.0537240
\(971\) −9.73203e14 −0.0361824 −0.0180912 0.999836i \(-0.505759\pi\)
−0.0180912 + 0.999836i \(0.505759\pi\)
\(972\) −1.60473e15 −0.0593251
\(973\) 2.47998e14 0.00911650
\(974\) 1.47295e16 0.538410
\(975\) −1.43093e16 −0.520107
\(976\) 3.18442e16 1.15095
\(977\) −1.99885e16 −0.718389 −0.359195 0.933263i \(-0.616949\pi\)
−0.359195 + 0.933263i \(0.616949\pi\)
\(978\) 5.05574e14 0.0180685
\(979\) −9.60695e15 −0.341414
\(980\) −7.14446e15 −0.252479
\(981\) 8.91533e14 0.0313298
\(982\) 4.02572e15 0.140679
\(983\) −4.23850e16 −1.47288 −0.736440 0.676503i \(-0.763495\pi\)
−0.736440 + 0.676503i \(0.763495\pi\)
\(984\) 5.34532e15 0.184714
\(985\) 7.83817e15 0.269349
\(986\) 6.85772e15 0.234346
\(987\) 3.38682e15 0.115093
\(988\) −2.18294e15 −0.0737699
\(989\) −5.32900e16 −1.79088
\(990\) −4.23694e14 −0.0141599
\(991\) 5.09195e16 1.69231 0.846154 0.532938i \(-0.178912\pi\)
0.846154 + 0.532938i \(0.178912\pi\)
\(992\) −9.73699e15 −0.321818
\(993\) −2.98371e16 −0.980700
\(994\) 3.80234e15 0.124287
\(995\) −6.96265e15 −0.226333
\(996\) −1.18565e16 −0.383292
\(997\) −3.39819e15 −0.109251 −0.0546253 0.998507i \(-0.517396\pi\)
−0.0546253 + 0.998507i \(0.517396\pi\)
\(998\) −1.25300e16 −0.400619
\(999\) −6.47190e15 −0.205789
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.1.12 28
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.12.a.d.1.12 28 1.1 even 1 trivial