Properties

Label 177.8.a.a.1.16
Level $177$
Weight $8$
Character 177.1
Self dual yes
Analytic conductor $55.292$
Analytic rank $1$
Dimension $16$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 177 = 3 \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 177.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.2921495107\)
Analytic rank: \(1\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Defining polynomial: \(x^{16} - 6 x^{15} - 1493 x^{14} + 8791 x^{13} + 890490 x^{12} - 5107725 x^{11} - 269092298 x^{10} + 1488374176 x^{9} + 42885295136 x^{8} - 226132003872 x^{7} - 3353576629440 x^{6} + 16796366777600 x^{5} + 99470801612800 x^{4} - 494039551757568 x^{3} - 493048066650624 x^{2} + 3193975642099712 x - 2385018853548032\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{5} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Root \(-19.6562\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+19.6562 q^{2} -27.0000 q^{3} +258.365 q^{4} -436.761 q^{5} -530.717 q^{6} +956.841 q^{7} +2562.48 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+19.6562 q^{2} -27.0000 q^{3} +258.365 q^{4} -436.761 q^{5} -530.717 q^{6} +956.841 q^{7} +2562.48 q^{8} +729.000 q^{9} -8585.06 q^{10} +1972.61 q^{11} -6975.86 q^{12} -5308.65 q^{13} +18807.8 q^{14} +11792.6 q^{15} +17297.8 q^{16} -19184.9 q^{17} +14329.3 q^{18} -21442.5 q^{19} -112844. q^{20} -25834.7 q^{21} +38773.9 q^{22} -34320.3 q^{23} -69186.9 q^{24} +112635. q^{25} -104348. q^{26} -19683.0 q^{27} +247214. q^{28} +40010.4 q^{29} +231796. q^{30} -95465.0 q^{31} +12010.9 q^{32} -53260.4 q^{33} -377103. q^{34} -417911. q^{35} +188348. q^{36} -269178. q^{37} -421478. q^{38} +143334. q^{39} -1.11919e6 q^{40} -360504. q^{41} -507811. q^{42} -929097. q^{43} +509653. q^{44} -318399. q^{45} -674606. q^{46} +54052.8 q^{47} -467040. q^{48} +92001.6 q^{49} +2.21398e6 q^{50} +517994. q^{51} -1.37157e6 q^{52} +680499. q^{53} -386892. q^{54} -861559. q^{55} +2.45188e6 q^{56} +578948. q^{57} +786452. q^{58} +205379. q^{59} +3.04678e6 q^{60} -1.27358e6 q^{61} -1.87648e6 q^{62} +697537. q^{63} -1.97803e6 q^{64} +2.31861e6 q^{65} -1.04690e6 q^{66} +630075. q^{67} -4.95672e6 q^{68} +926648. q^{69} -8.21453e6 q^{70} +2.05155e6 q^{71} +1.86805e6 q^{72} -5.36743e6 q^{73} -5.29101e6 q^{74} -3.04116e6 q^{75} -5.54000e6 q^{76} +1.88747e6 q^{77} +2.81739e6 q^{78} +4.57098e6 q^{79} -7.55500e6 q^{80} +531441. q^{81} -7.08613e6 q^{82} +9.49376e6 q^{83} -6.67478e6 q^{84} +8.37924e6 q^{85} -1.82625e7 q^{86} -1.08028e6 q^{87} +5.05477e6 q^{88} -2.45830e6 q^{89} -6.25851e6 q^{90} -5.07953e6 q^{91} -8.86717e6 q^{92} +2.57756e6 q^{93} +1.06247e6 q^{94} +9.36526e6 q^{95} -324293. q^{96} -1.41914e7 q^{97} +1.80840e6 q^{98} +1.43803e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} + O(q^{10}) \) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} - 3479q^{10} + 898q^{11} - 26298q^{12} - 8172q^{13} - 13315q^{14} + 1836q^{15} + 3138q^{16} - 44985q^{17} - 4374q^{18} - 40137q^{19} + 130657q^{20} + 63261q^{21} + 109394q^{22} - 2833q^{23} - 22113q^{24} + 285746q^{25} - 129420q^{26} - 314928q^{27} + 112890q^{28} + 144375q^{29} + 93933q^{30} - 141759q^{31} - 36224q^{32} - 24246q^{33} - 341332q^{34} - 78859q^{35} + 710046q^{36} - 297971q^{37} + 329075q^{38} + 220644q^{39} - 203048q^{40} + 659077q^{41} + 359505q^{42} - 1431608q^{43} + 254916q^{44} - 49572q^{45} + 873113q^{46} - 1574073q^{47} - 84726q^{48} + 1893545q^{49} + 302533q^{50} + 1214595q^{51} - 4972548q^{52} + 587736q^{53} + 118098q^{54} - 4624036q^{55} - 5798506q^{56} + 1083699q^{57} - 6991380q^{58} + 3286064q^{59} - 3527739q^{60} - 6117131q^{61} - 11570258q^{62} - 1708047q^{63} - 19063011q^{64} - 5335514q^{65} - 2953638q^{66} - 16518710q^{67} - 17284669q^{68} + 76491q^{69} - 39189486q^{70} - 10882582q^{71} + 597051q^{72} - 21097441q^{73} - 16717030q^{74} - 7715142q^{75} - 40864952q^{76} - 3404601q^{77} + 3494340q^{78} - 3784458q^{79} - 27466195q^{80} + 8503056q^{81} - 24990117q^{82} - 1951425q^{83} - 3048030q^{84} - 23238675q^{85} - 35910572q^{86} - 3898125q^{87} - 27843055q^{88} + 10499443q^{89} - 2536191q^{90} + 699217q^{91} - 20062766q^{92} + 3827493q^{93} - 59358988q^{94} - 29236333q^{95} + 978048q^{96} - 25158976q^{97} + 2120460q^{98} + 654642q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 19.6562 1.73738 0.868688 0.495359i \(-0.164964\pi\)
0.868688 + 0.495359i \(0.164964\pi\)
\(3\) −27.0000 −0.577350
\(4\) 258.365 2.01848
\(5\) −436.761 −1.56260 −0.781302 0.624153i \(-0.785444\pi\)
−0.781302 + 0.624153i \(0.785444\pi\)
\(6\) −530.717 −1.00307
\(7\) 956.841 1.05438 0.527189 0.849748i \(-0.323246\pi\)
0.527189 + 0.849748i \(0.323246\pi\)
\(8\) 2562.48 1.76948
\(9\) 729.000 0.333333
\(10\) −8585.06 −2.71483
\(11\) 1972.61 0.446855 0.223427 0.974721i \(-0.428275\pi\)
0.223427 + 0.974721i \(0.428275\pi\)
\(12\) −6975.86 −1.16537
\(13\) −5308.65 −0.670166 −0.335083 0.942189i \(-0.608764\pi\)
−0.335083 + 0.942189i \(0.608764\pi\)
\(14\) 18807.8 1.83185
\(15\) 11792.6 0.902170
\(16\) 17297.8 1.05577
\(17\) −19184.9 −0.947086 −0.473543 0.880771i \(-0.657025\pi\)
−0.473543 + 0.880771i \(0.657025\pi\)
\(18\) 14329.3 0.579125
\(19\) −21442.5 −0.717196 −0.358598 0.933492i \(-0.616745\pi\)
−0.358598 + 0.933492i \(0.616745\pi\)
\(20\) −112844. −3.15408
\(21\) −25834.7 −0.608746
\(22\) 38773.9 0.776355
\(23\) −34320.3 −0.588171 −0.294086 0.955779i \(-0.595015\pi\)
−0.294086 + 0.955779i \(0.595015\pi\)
\(24\) −69186.9 −1.02161
\(25\) 112635. 1.44173
\(26\) −104348. −1.16433
\(27\) −19683.0 −0.192450
\(28\) 247214. 2.12824
\(29\) 40010.4 0.304635 0.152318 0.988332i \(-0.451326\pi\)
0.152318 + 0.988332i \(0.451326\pi\)
\(30\) 231796. 1.56741
\(31\) −95465.0 −0.575544 −0.287772 0.957699i \(-0.592915\pi\)
−0.287772 + 0.957699i \(0.592915\pi\)
\(32\) 12010.9 0.0647961
\(33\) −53260.4 −0.257992
\(34\) −377103. −1.64544
\(35\) −417911. −1.64758
\(36\) 188348. 0.672826
\(37\) −269178. −0.873642 −0.436821 0.899549i \(-0.643896\pi\)
−0.436821 + 0.899549i \(0.643896\pi\)
\(38\) −421478. −1.24604
\(39\) 143334. 0.386920
\(40\) −1.11919e6 −2.76499
\(41\) −360504. −0.816896 −0.408448 0.912782i \(-0.633930\pi\)
−0.408448 + 0.912782i \(0.633930\pi\)
\(42\) −507811. −1.05762
\(43\) −929097. −1.78206 −0.891028 0.453949i \(-0.850015\pi\)
−0.891028 + 0.453949i \(0.850015\pi\)
\(44\) 509653. 0.901967
\(45\) −318399. −0.520868
\(46\) −674606. −1.02187
\(47\) 54052.8 0.0759408 0.0379704 0.999279i \(-0.487911\pi\)
0.0379704 + 0.999279i \(0.487911\pi\)
\(48\) −467040. −0.609550
\(49\) 92001.6 0.111714
\(50\) 2.21398e6 2.50483
\(51\) 517994. 0.546800
\(52\) −1.37157e6 −1.35271
\(53\) 680499. 0.627859 0.313929 0.949446i \(-0.398354\pi\)
0.313929 + 0.949446i \(0.398354\pi\)
\(54\) −386892. −0.334358
\(55\) −861559. −0.698258
\(56\) 2.45188e6 1.86570
\(57\) 578948. 0.414073
\(58\) 786452. 0.529266
\(59\) 205379. 0.130189
\(60\) 3.04678e6 1.82101
\(61\) −1.27358e6 −0.718408 −0.359204 0.933259i \(-0.616952\pi\)
−0.359204 + 0.933259i \(0.616952\pi\)
\(62\) −1.87648e6 −0.999936
\(63\) 697537. 0.351460
\(64\) −1.97803e6 −0.943197
\(65\) 2.31861e6 1.04720
\(66\) −1.04690e6 −0.448229
\(67\) 630075. 0.255936 0.127968 0.991778i \(-0.459155\pi\)
0.127968 + 0.991778i \(0.459155\pi\)
\(68\) −4.95672e6 −1.91167
\(69\) 926648. 0.339581
\(70\) −8.21453e6 −2.86246
\(71\) 2.05155e6 0.680265 0.340133 0.940377i \(-0.389528\pi\)
0.340133 + 0.940377i \(0.389528\pi\)
\(72\) 1.86805e6 0.589826
\(73\) −5.36743e6 −1.61487 −0.807433 0.589959i \(-0.799144\pi\)
−0.807433 + 0.589959i \(0.799144\pi\)
\(74\) −5.29101e6 −1.51784
\(75\) −3.04116e6 −0.832385
\(76\) −5.54000e6 −1.44764
\(77\) 1.88747e6 0.471154
\(78\) 2.81739e6 0.672227
\(79\) 4.57098e6 1.04307 0.521536 0.853229i \(-0.325359\pi\)
0.521536 + 0.853229i \(0.325359\pi\)
\(80\) −7.55500e6 −1.64975
\(81\) 531441. 0.111111
\(82\) −7.08613e6 −1.41926
\(83\) 9.49376e6 1.82249 0.911245 0.411865i \(-0.135122\pi\)
0.911245 + 0.411865i \(0.135122\pi\)
\(84\) −6.67478e6 −1.22874
\(85\) 8.37924e6 1.47992
\(86\) −1.82625e7 −3.09610
\(87\) −1.08028e6 −0.175881
\(88\) 5.05477e6 0.790700
\(89\) −2.45830e6 −0.369633 −0.184816 0.982773i \(-0.559169\pi\)
−0.184816 + 0.982773i \(0.559169\pi\)
\(90\) −6.25851e6 −0.904944
\(91\) −5.07953e6 −0.706609
\(92\) −8.86717e6 −1.18721
\(93\) 2.57756e6 0.332290
\(94\) 1.06247e6 0.131938
\(95\) 9.36526e6 1.12069
\(96\) −324293. −0.0374101
\(97\) −1.41914e7 −1.57879 −0.789397 0.613883i \(-0.789607\pi\)
−0.789397 + 0.613883i \(0.789607\pi\)
\(98\) 1.80840e6 0.194090
\(99\) 1.43803e6 0.148952
\(100\) 2.91011e7 2.91011
\(101\) 3.81799e6 0.368731 0.184366 0.982858i \(-0.440977\pi\)
0.184366 + 0.982858i \(0.440977\pi\)
\(102\) 1.01818e7 0.949998
\(103\) 1.77672e7 1.60210 0.801049 0.598599i \(-0.204276\pi\)
0.801049 + 0.598599i \(0.204276\pi\)
\(104\) −1.36033e7 −1.18584
\(105\) 1.12836e7 0.951229
\(106\) 1.33760e7 1.09083
\(107\) 2.35965e7 1.86210 0.931051 0.364888i \(-0.118893\pi\)
0.931051 + 0.364888i \(0.118893\pi\)
\(108\) −5.08540e6 −0.388456
\(109\) 3.99327e6 0.295349 0.147675 0.989036i \(-0.452821\pi\)
0.147675 + 0.989036i \(0.452821\pi\)
\(110\) −1.69350e7 −1.21314
\(111\) 7.26780e6 0.504397
\(112\) 1.65512e7 1.11318
\(113\) −2.69156e6 −0.175481 −0.0877405 0.996143i \(-0.527965\pi\)
−0.0877405 + 0.996143i \(0.527965\pi\)
\(114\) 1.13799e7 0.719401
\(115\) 1.49898e7 0.919079
\(116\) 1.03373e7 0.614900
\(117\) −3.87001e6 −0.223389
\(118\) 4.03696e6 0.226187
\(119\) −1.83569e7 −0.998587
\(120\) 3.02182e7 1.59637
\(121\) −1.55960e7 −0.800321
\(122\) −2.50337e7 −1.24815
\(123\) 9.73362e6 0.471635
\(124\) −2.46648e7 −1.16172
\(125\) −1.50728e7 −0.690255
\(126\) 1.37109e7 0.610618
\(127\) 1.72960e6 0.0749261 0.0374631 0.999298i \(-0.488072\pi\)
0.0374631 + 0.999298i \(0.488072\pi\)
\(128\) −4.04178e7 −1.70348
\(129\) 2.50856e7 1.02887
\(130\) 4.55750e7 1.81939
\(131\) 1.21153e7 0.470854 0.235427 0.971892i \(-0.424351\pi\)
0.235427 + 0.971892i \(0.424351\pi\)
\(132\) −1.37606e7 −0.520751
\(133\) −2.05171e7 −0.756196
\(134\) 1.23849e7 0.444657
\(135\) 8.59677e6 0.300723
\(136\) −4.91610e7 −1.67585
\(137\) −3.48264e7 −1.15714 −0.578571 0.815632i \(-0.696389\pi\)
−0.578571 + 0.815632i \(0.696389\pi\)
\(138\) 1.82144e7 0.589980
\(139\) 3.29388e7 1.04029 0.520147 0.854077i \(-0.325877\pi\)
0.520147 + 0.854077i \(0.325877\pi\)
\(140\) −1.07974e8 −3.32560
\(141\) −1.45942e6 −0.0438445
\(142\) 4.03257e7 1.18188
\(143\) −1.04719e7 −0.299467
\(144\) 1.26101e7 0.351924
\(145\) −1.74750e7 −0.476025
\(146\) −1.05503e8 −2.80563
\(147\) −2.48404e6 −0.0644983
\(148\) −6.95461e7 −1.76343
\(149\) 3.14675e7 0.779311 0.389656 0.920961i \(-0.372594\pi\)
0.389656 + 0.920961i \(0.372594\pi\)
\(150\) −5.97775e7 −1.44617
\(151\) −2.53600e6 −0.0599419 −0.0299710 0.999551i \(-0.509541\pi\)
−0.0299710 + 0.999551i \(0.509541\pi\)
\(152\) −5.49460e7 −1.26906
\(153\) −1.39858e7 −0.315695
\(154\) 3.71005e7 0.818572
\(155\) 4.16954e7 0.899348
\(156\) 3.70324e7 0.780990
\(157\) −5.81479e7 −1.19918 −0.599591 0.800306i \(-0.704670\pi\)
−0.599591 + 0.800306i \(0.704670\pi\)
\(158\) 8.98479e7 1.81221
\(159\) −1.83735e7 −0.362494
\(160\) −5.24588e6 −0.101251
\(161\) −3.28391e7 −0.620155
\(162\) 1.04461e7 0.193042
\(163\) −4.39268e7 −0.794461 −0.397230 0.917719i \(-0.630029\pi\)
−0.397230 + 0.917719i \(0.630029\pi\)
\(164\) −9.31417e7 −1.64889
\(165\) 2.32621e7 0.403139
\(166\) 1.86611e8 3.16635
\(167\) −9.02250e7 −1.49906 −0.749530 0.661970i \(-0.769720\pi\)
−0.749530 + 0.661970i \(0.769720\pi\)
\(168\) −6.62009e7 −1.07716
\(169\) −3.45668e7 −0.550878
\(170\) 1.64704e8 2.57118
\(171\) −1.56316e7 −0.239065
\(172\) −2.40046e8 −3.59704
\(173\) 2.25125e7 0.330569 0.165285 0.986246i \(-0.447146\pi\)
0.165285 + 0.986246i \(0.447146\pi\)
\(174\) −2.12342e7 −0.305572
\(175\) 1.07774e8 1.52013
\(176\) 3.41217e7 0.471777
\(177\) −5.54523e6 −0.0751646
\(178\) −4.83208e7 −0.642191
\(179\) 1.26881e8 1.65353 0.826766 0.562546i \(-0.190178\pi\)
0.826766 + 0.562546i \(0.190178\pi\)
\(180\) −8.22632e7 −1.05136
\(181\) 9.29702e7 1.16538 0.582691 0.812694i \(-0.302000\pi\)
0.582691 + 0.812694i \(0.302000\pi\)
\(182\) −9.98442e7 −1.22765
\(183\) 3.43866e7 0.414773
\(184\) −8.79450e7 −1.04076
\(185\) 1.17566e8 1.36516
\(186\) 5.06649e7 0.577314
\(187\) −3.78444e7 −0.423210
\(188\) 1.39653e7 0.153285
\(189\) −1.88335e7 −0.202915
\(190\) 1.84085e8 1.94707
\(191\) −1.55664e8 −1.61649 −0.808243 0.588850i \(-0.799581\pi\)
−0.808243 + 0.588850i \(0.799581\pi\)
\(192\) 5.34067e7 0.544555
\(193\) 1.85473e7 0.185708 0.0928539 0.995680i \(-0.470401\pi\)
0.0928539 + 0.995680i \(0.470401\pi\)
\(194\) −2.78949e8 −2.74296
\(195\) −6.26025e7 −0.604604
\(196\) 2.37700e7 0.225493
\(197\) 3.60337e7 0.335798 0.167899 0.985804i \(-0.446302\pi\)
0.167899 + 0.985804i \(0.446302\pi\)
\(198\) 2.82662e7 0.258785
\(199\) 3.41912e7 0.307559 0.153780 0.988105i \(-0.450855\pi\)
0.153780 + 0.988105i \(0.450855\pi\)
\(200\) 2.88626e8 2.55112
\(201\) −1.70120e7 −0.147765
\(202\) 7.50471e7 0.640625
\(203\) 3.82836e7 0.321201
\(204\) 1.33831e8 1.10370
\(205\) 1.57454e8 1.27649
\(206\) 3.49235e8 2.78345
\(207\) −2.50195e7 −0.196057
\(208\) −9.18278e7 −0.707543
\(209\) −4.22977e7 −0.320483
\(210\) 2.21792e8 1.65264
\(211\) 1.66203e8 1.21801 0.609005 0.793166i \(-0.291569\pi\)
0.609005 + 0.793166i \(0.291569\pi\)
\(212\) 1.75817e8 1.26732
\(213\) −5.53919e7 −0.392751
\(214\) 4.63816e8 3.23517
\(215\) 4.05793e8 2.78465
\(216\) −5.04373e7 −0.340536
\(217\) −9.13449e7 −0.606841
\(218\) 7.84923e7 0.513133
\(219\) 1.44921e8 0.932343
\(220\) −2.22597e8 −1.40942
\(221\) 1.01846e8 0.634705
\(222\) 1.42857e8 0.876328
\(223\) −2.57838e8 −1.55697 −0.778485 0.627663i \(-0.784012\pi\)
−0.778485 + 0.627663i \(0.784012\pi\)
\(224\) 1.14925e7 0.0683197
\(225\) 8.21112e7 0.480578
\(226\) −5.29058e7 −0.304877
\(227\) −2.27009e8 −1.28811 −0.644055 0.764979i \(-0.722749\pi\)
−0.644055 + 0.764979i \(0.722749\pi\)
\(228\) 1.49580e8 0.835798
\(229\) −7.40017e7 −0.407209 −0.203605 0.979053i \(-0.565266\pi\)
−0.203605 + 0.979053i \(0.565266\pi\)
\(230\) 2.94642e8 1.59679
\(231\) −5.09618e7 −0.272021
\(232\) 1.02526e8 0.539046
\(233\) 6.77414e7 0.350839 0.175420 0.984494i \(-0.443872\pi\)
0.175420 + 0.984494i \(0.443872\pi\)
\(234\) −7.60695e7 −0.388110
\(235\) −2.36082e7 −0.118665
\(236\) 5.30628e7 0.262783
\(237\) −1.23416e8 −0.602218
\(238\) −3.60827e8 −1.73492
\(239\) 1.28770e8 0.610128 0.305064 0.952332i \(-0.401322\pi\)
0.305064 + 0.952332i \(0.401322\pi\)
\(240\) 2.03985e8 0.952486
\(241\) −2.97076e8 −1.36712 −0.683561 0.729893i \(-0.739570\pi\)
−0.683561 + 0.729893i \(0.739570\pi\)
\(242\) −3.06557e8 −1.39046
\(243\) −1.43489e7 −0.0641500
\(244\) −3.29048e8 −1.45009
\(245\) −4.01827e7 −0.174565
\(246\) 1.91326e8 0.819408
\(247\) 1.13831e8 0.480640
\(248\) −2.44627e8 −1.01841
\(249\) −2.56332e8 −1.05221
\(250\) −2.96274e8 −1.19923
\(251\) −6.81145e7 −0.271883 −0.135941 0.990717i \(-0.543406\pi\)
−0.135941 + 0.990717i \(0.543406\pi\)
\(252\) 1.80219e8 0.709413
\(253\) −6.77005e7 −0.262827
\(254\) 3.39974e7 0.130175
\(255\) −2.26240e8 −0.854433
\(256\) −5.41272e8 −2.01640
\(257\) −1.12934e8 −0.415010 −0.207505 0.978234i \(-0.566534\pi\)
−0.207505 + 0.978234i \(0.566534\pi\)
\(258\) 4.93087e8 1.78754
\(259\) −2.57560e8 −0.921149
\(260\) 5.99049e8 2.11376
\(261\) 2.91676e7 0.101545
\(262\) 2.38141e8 0.818050
\(263\) 5.13397e8 1.74024 0.870119 0.492842i \(-0.164042\pi\)
0.870119 + 0.492842i \(0.164042\pi\)
\(264\) −1.36479e8 −0.456511
\(265\) −2.97216e8 −0.981095
\(266\) −4.03287e8 −1.31380
\(267\) 6.63742e7 0.213408
\(268\) 1.62789e8 0.516600
\(269\) −7.41447e7 −0.232245 −0.116123 0.993235i \(-0.537047\pi\)
−0.116123 + 0.993235i \(0.537047\pi\)
\(270\) 1.68980e8 0.522470
\(271\) 1.72102e8 0.525285 0.262642 0.964893i \(-0.415406\pi\)
0.262642 + 0.964893i \(0.415406\pi\)
\(272\) −3.31857e8 −0.999907
\(273\) 1.37147e8 0.407961
\(274\) −6.84554e8 −2.01039
\(275\) 2.22186e8 0.644246
\(276\) 2.39414e8 0.685436
\(277\) 4.25097e8 1.20174 0.600868 0.799348i \(-0.294822\pi\)
0.600868 + 0.799348i \(0.294822\pi\)
\(278\) 6.47451e8 1.80738
\(279\) −6.95940e7 −0.191848
\(280\) −1.07089e9 −2.91535
\(281\) 3.70445e7 0.0995983 0.0497991 0.998759i \(-0.484142\pi\)
0.0497991 + 0.998759i \(0.484142\pi\)
\(282\) −2.86867e7 −0.0761743
\(283\) −2.60297e7 −0.0682679 −0.0341339 0.999417i \(-0.510867\pi\)
−0.0341339 + 0.999417i \(0.510867\pi\)
\(284\) 5.30049e8 1.37310
\(285\) −2.52862e8 −0.647033
\(286\) −2.05837e8 −0.520287
\(287\) −3.44945e8 −0.861318
\(288\) 8.75591e6 0.0215987
\(289\) −4.22765e7 −0.103028
\(290\) −3.43492e8 −0.827034
\(291\) 3.83169e8 0.911517
\(292\) −1.38676e9 −3.25957
\(293\) 6.51952e8 1.51418 0.757092 0.653308i \(-0.226619\pi\)
0.757092 + 0.653308i \(0.226619\pi\)
\(294\) −4.88268e7 −0.112058
\(295\) −8.97016e7 −0.203434
\(296\) −6.89762e8 −1.54589
\(297\) −3.88269e7 −0.0859973
\(298\) 6.18532e8 1.35396
\(299\) 1.82194e8 0.394172
\(300\) −7.85729e8 −1.68015
\(301\) −8.88998e8 −1.87896
\(302\) −4.98481e7 −0.104142
\(303\) −1.03086e8 −0.212887
\(304\) −3.70908e8 −0.757196
\(305\) 5.56250e8 1.12259
\(306\) −2.74908e8 −0.548482
\(307\) 8.07706e8 1.59320 0.796598 0.604510i \(-0.206631\pi\)
0.796598 + 0.604510i \(0.206631\pi\)
\(308\) 4.87657e8 0.951014
\(309\) −4.79715e8 −0.924972
\(310\) 8.19573e8 1.56251
\(311\) −1.60231e8 −0.302055 −0.151027 0.988530i \(-0.548258\pi\)
−0.151027 + 0.988530i \(0.548258\pi\)
\(312\) 3.67289e8 0.684647
\(313\) 2.15368e8 0.396987 0.198494 0.980102i \(-0.436395\pi\)
0.198494 + 0.980102i \(0.436395\pi\)
\(314\) −1.14296e9 −2.08343
\(315\) −3.04657e8 −0.549192
\(316\) 1.18098e9 2.10542
\(317\) −8.64021e8 −1.52341 −0.761705 0.647924i \(-0.775638\pi\)
−0.761705 + 0.647924i \(0.775638\pi\)
\(318\) −3.61152e8 −0.629789
\(319\) 7.89250e7 0.136128
\(320\) 8.63926e8 1.47384
\(321\) −6.37104e8 −1.07509
\(322\) −6.45490e8 −1.07744
\(323\) 4.11373e8 0.679246
\(324\) 1.37306e8 0.224275
\(325\) −5.97942e8 −0.966201
\(326\) −8.63432e8 −1.38028
\(327\) −1.07818e8 −0.170520
\(328\) −9.23784e8 −1.44548
\(329\) 5.17199e7 0.0800704
\(330\) 4.57244e8 0.700405
\(331\) 9.12406e8 1.38290 0.691449 0.722425i \(-0.256973\pi\)
0.691449 + 0.722425i \(0.256973\pi\)
\(332\) 2.45286e9 3.67865
\(333\) −1.96231e8 −0.291214
\(334\) −1.77348e9 −2.60443
\(335\) −2.75193e8 −0.399926
\(336\) −4.46883e8 −0.642697
\(337\) 5.45796e8 0.776829 0.388414 0.921485i \(-0.373023\pi\)
0.388414 + 0.921485i \(0.373023\pi\)
\(338\) −6.79450e8 −0.957082
\(339\) 7.26722e7 0.101314
\(340\) 2.16490e9 2.98719
\(341\) −1.88315e8 −0.257185
\(342\) −3.07257e8 −0.415347
\(343\) −6.99969e8 −0.936589
\(344\) −2.38079e9 −3.15331
\(345\) −4.04724e8 −0.530631
\(346\) 4.42510e8 0.574323
\(347\) −5.84036e8 −0.750389 −0.375194 0.926946i \(-0.622424\pi\)
−0.375194 + 0.926946i \(0.622424\pi\)
\(348\) −2.79107e8 −0.355012
\(349\) −7.64386e8 −0.962551 −0.481276 0.876569i \(-0.659826\pi\)
−0.481276 + 0.876569i \(0.659826\pi\)
\(350\) 2.11843e9 2.64104
\(351\) 1.04490e8 0.128973
\(352\) 2.36927e7 0.0289545
\(353\) −1.06373e9 −1.28712 −0.643559 0.765396i \(-0.722543\pi\)
−0.643559 + 0.765396i \(0.722543\pi\)
\(354\) −1.08998e8 −0.130589
\(355\) −8.96039e8 −1.06299
\(356\) −6.35140e8 −0.746095
\(357\) 4.95637e8 0.576535
\(358\) 2.49400e9 2.87281
\(359\) −1.09179e9 −1.24540 −0.622700 0.782461i \(-0.713964\pi\)
−0.622700 + 0.782461i \(0.713964\pi\)
\(360\) −8.15890e8 −0.921665
\(361\) −4.34091e8 −0.485630
\(362\) 1.82744e9 2.02471
\(363\) 4.21092e8 0.462065
\(364\) −1.31237e9 −1.42627
\(365\) 2.34429e9 2.52340
\(366\) 6.75909e8 0.720617
\(367\) −1.92123e8 −0.202884 −0.101442 0.994841i \(-0.532346\pi\)
−0.101442 + 0.994841i \(0.532346\pi\)
\(368\) −5.93665e8 −0.620975
\(369\) −2.62808e8 −0.272299
\(370\) 2.31091e9 2.37179
\(371\) 6.51129e8 0.662001
\(372\) 6.65950e8 0.670721
\(373\) −1.24365e9 −1.24085 −0.620423 0.784267i \(-0.713039\pi\)
−0.620423 + 0.784267i \(0.713039\pi\)
\(374\) −7.43876e8 −0.735275
\(375\) 4.06966e8 0.398519
\(376\) 1.38509e8 0.134376
\(377\) −2.12401e8 −0.204156
\(378\) −3.70194e8 −0.352540
\(379\) −2.03376e9 −1.91894 −0.959472 0.281805i \(-0.909067\pi\)
−0.959472 + 0.281805i \(0.909067\pi\)
\(380\) 2.41966e9 2.26210
\(381\) −4.66993e7 −0.0432586
\(382\) −3.05976e9 −2.80844
\(383\) −8.58688e8 −0.780979 −0.390490 0.920607i \(-0.627694\pi\)
−0.390490 + 0.920607i \(0.627694\pi\)
\(384\) 1.09128e9 0.983507
\(385\) −8.24375e8 −0.736228
\(386\) 3.64569e8 0.322645
\(387\) −6.77311e8 −0.594019
\(388\) −3.66657e9 −3.18676
\(389\) −1.79662e9 −1.54751 −0.773755 0.633485i \(-0.781624\pi\)
−0.773755 + 0.633485i \(0.781624\pi\)
\(390\) −1.23053e9 −1.05042
\(391\) 6.58433e8 0.557049
\(392\) 2.35752e8 0.197676
\(393\) −3.27114e8 −0.271848
\(394\) 7.08285e8 0.583407
\(395\) −1.99643e9 −1.62991
\(396\) 3.71537e8 0.300656
\(397\) −1.32366e9 −1.06172 −0.530860 0.847460i \(-0.678131\pi\)
−0.530860 + 0.847460i \(0.678131\pi\)
\(398\) 6.72069e8 0.534347
\(399\) 5.53961e8 0.436590
\(400\) 1.94834e9 1.52214
\(401\) −1.25074e9 −0.968635 −0.484317 0.874892i \(-0.660932\pi\)
−0.484317 + 0.874892i \(0.660932\pi\)
\(402\) −3.34392e8 −0.256723
\(403\) 5.06790e8 0.385710
\(404\) 9.86435e8 0.744276
\(405\) −2.32113e8 −0.173623
\(406\) 7.52510e8 0.558047
\(407\) −5.30983e8 −0.390391
\(408\) 1.32735e9 0.967551
\(409\) 1.84026e9 1.32999 0.664993 0.746850i \(-0.268435\pi\)
0.664993 + 0.746850i \(0.268435\pi\)
\(410\) 3.09495e9 2.21774
\(411\) 9.40313e8 0.668076
\(412\) 4.59043e9 3.23380
\(413\) 1.96515e8 0.137268
\(414\) −4.91788e8 −0.340625
\(415\) −4.14651e9 −2.84783
\(416\) −6.37614e7 −0.0434242
\(417\) −8.89348e8 −0.600614
\(418\) −8.31411e8 −0.556799
\(419\) 1.97514e9 1.31174 0.655871 0.754873i \(-0.272301\pi\)
0.655871 + 0.754873i \(0.272301\pi\)
\(420\) 2.91529e9 1.92003
\(421\) 1.04914e9 0.685247 0.342624 0.939473i \(-0.388684\pi\)
0.342624 + 0.939473i \(0.388684\pi\)
\(422\) 3.26692e9 2.11614
\(423\) 3.94045e7 0.0253136
\(424\) 1.74376e9 1.11098
\(425\) −2.16091e9 −1.36545
\(426\) −1.08879e9 −0.682357
\(427\) −1.21861e9 −0.757474
\(428\) 6.09650e9 3.75861
\(429\) 2.82741e8 0.172897
\(430\) 7.97635e9 4.83798
\(431\) 1.15205e9 0.693109 0.346554 0.938030i \(-0.387352\pi\)
0.346554 + 0.938030i \(0.387352\pi\)
\(432\) −3.40472e8 −0.203183
\(433\) 2.37623e9 1.40663 0.703317 0.710877i \(-0.251702\pi\)
0.703317 + 0.710877i \(0.251702\pi\)
\(434\) −1.79549e9 −1.05431
\(435\) 4.71825e8 0.274833
\(436\) 1.03172e9 0.596155
\(437\) 7.35913e8 0.421834
\(438\) 2.84858e9 1.61983
\(439\) −6.70393e8 −0.378185 −0.189092 0.981959i \(-0.560555\pi\)
−0.189092 + 0.981959i \(0.560555\pi\)
\(440\) −2.20773e9 −1.23555
\(441\) 6.70691e7 0.0372381
\(442\) 2.00191e9 1.10272
\(443\) −2.24468e9 −1.22671 −0.613354 0.789808i \(-0.710180\pi\)
−0.613354 + 0.789808i \(0.710180\pi\)
\(444\) 1.87775e9 1.01811
\(445\) 1.07369e9 0.577590
\(446\) −5.06811e9 −2.70504
\(447\) −8.49624e8 −0.449936
\(448\) −1.89266e9 −0.994487
\(449\) −5.76894e8 −0.300769 −0.150385 0.988628i \(-0.548051\pi\)
−0.150385 + 0.988628i \(0.548051\pi\)
\(450\) 1.61399e9 0.834945
\(451\) −7.11134e8 −0.365034
\(452\) −6.95406e8 −0.354204
\(453\) 6.84721e7 0.0346075
\(454\) −4.46213e9 −2.23793
\(455\) 2.21854e9 1.10415
\(456\) 1.48354e9 0.732694
\(457\) 9.02311e8 0.442231 0.221116 0.975248i \(-0.429030\pi\)
0.221116 + 0.975248i \(0.429030\pi\)
\(458\) −1.45459e9 −0.707476
\(459\) 3.77617e8 0.182267
\(460\) 3.87284e9 1.85514
\(461\) 1.56907e9 0.745913 0.372956 0.927849i \(-0.378344\pi\)
0.372956 + 0.927849i \(0.378344\pi\)
\(462\) −1.00171e9 −0.472603
\(463\) 6.01283e8 0.281543 0.140772 0.990042i \(-0.455042\pi\)
0.140772 + 0.990042i \(0.455042\pi\)
\(464\) 6.92092e8 0.321626
\(465\) −1.12578e9 −0.519239
\(466\) 1.33154e9 0.609540
\(467\) −2.18611e9 −0.993260 −0.496630 0.867962i \(-0.665429\pi\)
−0.496630 + 0.867962i \(0.665429\pi\)
\(468\) −9.99874e8 −0.450905
\(469\) 6.02882e8 0.269853
\(470\) −4.64046e8 −0.206167
\(471\) 1.56999e9 0.692348
\(472\) 5.26279e8 0.230366
\(473\) −1.83274e9 −0.796320
\(474\) −2.42589e9 −1.04628
\(475\) −2.41519e9 −1.03401
\(476\) −4.74279e9 −2.01563
\(477\) 4.96084e8 0.209286
\(478\) 2.53112e9 1.06002
\(479\) 1.17364e9 0.487936 0.243968 0.969783i \(-0.421551\pi\)
0.243968 + 0.969783i \(0.421551\pi\)
\(480\) 1.41639e8 0.0584571
\(481\) 1.42897e9 0.585485
\(482\) −5.83937e9 −2.37521
\(483\) 8.86655e8 0.358047
\(484\) −4.02946e9 −1.61543
\(485\) 6.19827e9 2.46703
\(486\) −2.82045e8 −0.111453
\(487\) 1.37532e9 0.539574 0.269787 0.962920i \(-0.413047\pi\)
0.269787 + 0.962920i \(0.413047\pi\)
\(488\) −3.26352e9 −1.27121
\(489\) 1.18602e9 0.458682
\(490\) −7.89838e8 −0.303286
\(491\) 4.93551e9 1.88168 0.940842 0.338845i \(-0.110036\pi\)
0.940842 + 0.338845i \(0.110036\pi\)
\(492\) 2.51483e9 0.951985
\(493\) −7.67598e8 −0.288516
\(494\) 2.23748e9 0.835053
\(495\) −6.28077e8 −0.232753
\(496\) −1.65133e9 −0.607643
\(497\) 1.96301e9 0.717257
\(498\) −5.03850e9 −1.82809
\(499\) −4.54580e9 −1.63779 −0.818895 0.573943i \(-0.805413\pi\)
−0.818895 + 0.573943i \(0.805413\pi\)
\(500\) −3.89429e9 −1.39326
\(501\) 2.43607e9 0.865483
\(502\) −1.33887e9 −0.472362
\(503\) 1.13170e9 0.396500 0.198250 0.980151i \(-0.436474\pi\)
0.198250 + 0.980151i \(0.436474\pi\)
\(504\) 1.78742e9 0.621900
\(505\) −1.66755e9 −0.576181
\(506\) −1.33073e9 −0.456630
\(507\) 9.33302e8 0.318049
\(508\) 4.46869e8 0.151237
\(509\) 3.02720e9 1.01749 0.508743 0.860918i \(-0.330110\pi\)
0.508743 + 0.860918i \(0.330110\pi\)
\(510\) −4.44700e9 −1.48447
\(511\) −5.13578e9 −1.70268
\(512\) −5.46586e9 −1.79976
\(513\) 4.22053e8 0.138024
\(514\) −2.21985e9 −0.721028
\(515\) −7.76003e9 −2.50345
\(516\) 6.48124e9 2.07675
\(517\) 1.06625e8 0.0339345
\(518\) −5.06265e9 −1.60038
\(519\) −6.07838e8 −0.190854
\(520\) 5.94139e9 1.85301
\(521\) 9.37018e8 0.290279 0.145140 0.989411i \(-0.453637\pi\)
0.145140 + 0.989411i \(0.453637\pi\)
\(522\) 5.73324e8 0.176422
\(523\) 2.52142e9 0.770708 0.385354 0.922769i \(-0.374079\pi\)
0.385354 + 0.922769i \(0.374079\pi\)
\(524\) 3.13018e9 0.950407
\(525\) −2.90990e9 −0.877649
\(526\) 1.00914e10 3.02345
\(527\) 1.83149e9 0.545089
\(528\) −9.21287e8 −0.272381
\(529\) −2.22694e9 −0.654055
\(530\) −5.84212e9 −1.70453
\(531\) 1.49721e8 0.0433963
\(532\) −5.30089e9 −1.52637
\(533\) 1.91379e9 0.547456
\(534\) 1.30466e9 0.370769
\(535\) −1.03060e10 −2.90973
\(536\) 1.61455e9 0.452872
\(537\) −3.42580e9 −0.954667
\(538\) −1.45740e9 −0.403497
\(539\) 1.81483e8 0.0499201
\(540\) 2.22111e9 0.607003
\(541\) 5.53185e8 0.150204 0.0751018 0.997176i \(-0.476072\pi\)
0.0751018 + 0.997176i \(0.476072\pi\)
\(542\) 3.38288e9 0.912617
\(543\) −2.51019e9 −0.672834
\(544\) −2.30428e8 −0.0613675
\(545\) −1.74410e9 −0.461514
\(546\) 2.69579e9 0.708781
\(547\) 4.50977e8 0.117814 0.0589072 0.998263i \(-0.481238\pi\)
0.0589072 + 0.998263i \(0.481238\pi\)
\(548\) −8.99792e9 −2.33566
\(549\) −9.28439e8 −0.239469
\(550\) 4.36732e9 1.11930
\(551\) −8.57924e8 −0.218483
\(552\) 2.37452e9 0.600881
\(553\) 4.37370e9 1.09979
\(554\) 8.35578e9 2.08787
\(555\) −3.17429e9 −0.788173
\(556\) 8.51024e9 2.09981
\(557\) −3.86605e9 −0.947926 −0.473963 0.880545i \(-0.657177\pi\)
−0.473963 + 0.880545i \(0.657177\pi\)
\(558\) −1.36795e9 −0.333312
\(559\) 4.93225e9 1.19427
\(560\) −7.22893e9 −1.73947
\(561\) 1.02180e9 0.244340
\(562\) 7.28153e8 0.173040
\(563\) 2.10838e9 0.497930 0.248965 0.968512i \(-0.419910\pi\)
0.248965 + 0.968512i \(0.419910\pi\)
\(564\) −3.77064e8 −0.0884990
\(565\) 1.17557e9 0.274207
\(566\) −5.11644e8 −0.118607
\(567\) 5.08504e8 0.117153
\(568\) 5.25706e9 1.20371
\(569\) −5.98974e9 −1.36306 −0.681530 0.731790i \(-0.738685\pi\)
−0.681530 + 0.731790i \(0.738685\pi\)
\(570\) −4.97030e9 −1.12414
\(571\) 5.77991e9 1.29925 0.649627 0.760253i \(-0.274925\pi\)
0.649627 + 0.760253i \(0.274925\pi\)
\(572\) −2.70557e9 −0.604467
\(573\) 4.20293e9 0.933278
\(574\) −6.78030e9 −1.49643
\(575\) −3.86568e9 −0.847986
\(576\) −1.44198e9 −0.314399
\(577\) 8.69442e8 0.188419 0.0942096 0.995552i \(-0.469968\pi\)
0.0942096 + 0.995552i \(0.469968\pi\)
\(578\) −8.30994e8 −0.178999
\(579\) −5.00777e8 −0.107219
\(580\) −4.51493e9 −0.960845
\(581\) 9.08402e9 1.92159
\(582\) 7.53163e9 1.58365
\(583\) 1.34236e9 0.280562
\(584\) −1.37539e10 −2.85747
\(585\) 1.69027e9 0.349068
\(586\) 1.28149e10 2.63071
\(587\) −1.19117e9 −0.243074 −0.121537 0.992587i \(-0.538782\pi\)
−0.121537 + 0.992587i \(0.538782\pi\)
\(588\) −6.41790e8 −0.130188
\(589\) 2.04701e9 0.412778
\(590\) −1.76319e9 −0.353441
\(591\) −9.72911e8 −0.193873
\(592\) −4.65618e9 −0.922367
\(593\) 5.37577e9 1.05864 0.529321 0.848421i \(-0.322447\pi\)
0.529321 + 0.848421i \(0.322447\pi\)
\(594\) −7.63187e8 −0.149410
\(595\) 8.01760e9 1.56040
\(596\) 8.13012e9 1.57302
\(597\) −9.23164e8 −0.177570
\(598\) 3.58125e9 0.684826
\(599\) −7.00918e9 −1.33252 −0.666260 0.745720i \(-0.732106\pi\)
−0.666260 + 0.745720i \(0.732106\pi\)
\(600\) −7.79290e9 −1.47289
\(601\) 5.18595e9 0.974469 0.487235 0.873271i \(-0.338006\pi\)
0.487235 + 0.873271i \(0.338006\pi\)
\(602\) −1.74743e10 −3.26446
\(603\) 4.59325e8 0.0853119
\(604\) −6.55215e8 −0.120991
\(605\) 6.81172e9 1.25058
\(606\) −2.02627e9 −0.369865
\(607\) 5.60023e9 1.01636 0.508178 0.861252i \(-0.330319\pi\)
0.508178 + 0.861252i \(0.330319\pi\)
\(608\) −2.57543e8 −0.0464715
\(609\) −1.03366e9 −0.185446
\(610\) 1.09337e10 1.95036
\(611\) −2.86947e8 −0.0508929
\(612\) −3.61345e9 −0.637224
\(613\) 3.23965e9 0.568051 0.284025 0.958817i \(-0.408330\pi\)
0.284025 + 0.958817i \(0.408330\pi\)
\(614\) 1.58764e10 2.76798
\(615\) −4.25127e9 −0.736980
\(616\) 4.83661e9 0.833697
\(617\) 1.21036e9 0.207451 0.103726 0.994606i \(-0.466924\pi\)
0.103726 + 0.994606i \(0.466924\pi\)
\(618\) −9.42936e9 −1.60702
\(619\) −9.96098e9 −1.68805 −0.844024 0.536305i \(-0.819820\pi\)
−0.844024 + 0.536305i \(0.819820\pi\)
\(620\) 1.07726e10 1.81531
\(621\) 6.75527e8 0.113194
\(622\) −3.14953e9 −0.524783
\(623\) −2.35221e9 −0.389733
\(624\) 2.47935e9 0.408500
\(625\) −2.21642e9 −0.363138
\(626\) 4.23332e9 0.689717
\(627\) 1.14204e9 0.185031
\(628\) −1.50234e10 −2.42052
\(629\) 5.16416e9 0.827414
\(630\) −5.98839e9 −0.954154
\(631\) 8.86210e9 1.40422 0.702108 0.712071i \(-0.252243\pi\)
0.702108 + 0.712071i \(0.252243\pi\)
\(632\) 1.17130e10 1.84569
\(633\) −4.48749e9 −0.703219
\(634\) −1.69833e10 −2.64674
\(635\) −7.55424e8 −0.117080
\(636\) −4.74706e9 −0.731687
\(637\) −4.88404e8 −0.0748671
\(638\) 1.55136e9 0.236505
\(639\) 1.49558e9 0.226755
\(640\) 1.76529e10 2.66187
\(641\) 1.08624e10 1.62901 0.814507 0.580154i \(-0.197008\pi\)
0.814507 + 0.580154i \(0.197008\pi\)
\(642\) −1.25230e10 −1.86783
\(643\) −5.64420e9 −0.837267 −0.418634 0.908155i \(-0.637491\pi\)
−0.418634 + 0.908155i \(0.637491\pi\)
\(644\) −8.48447e9 −1.25177
\(645\) −1.09564e10 −1.60772
\(646\) 8.08603e9 1.18011
\(647\) 1.50140e7 0.00217937 0.00108968 0.999999i \(-0.499653\pi\)
0.00108968 + 0.999999i \(0.499653\pi\)
\(648\) 1.36181e9 0.196609
\(649\) 4.05132e8 0.0581756
\(650\) −1.17533e10 −1.67865
\(651\) 2.46631e9 0.350360
\(652\) −1.13491e10 −1.60360
\(653\) −3.25704e9 −0.457748 −0.228874 0.973456i \(-0.573504\pi\)
−0.228874 + 0.973456i \(0.573504\pi\)
\(654\) −2.11929e9 −0.296257
\(655\) −5.29151e9 −0.735758
\(656\) −6.23592e9 −0.862457
\(657\) −3.91286e9 −0.538289
\(658\) 1.01661e9 0.139112
\(659\) 9.09574e9 1.23805 0.619026 0.785370i \(-0.287527\pi\)
0.619026 + 0.785370i \(0.287527\pi\)
\(660\) 6.01011e9 0.813727
\(661\) −8.48609e9 −1.14289 −0.571443 0.820642i \(-0.693616\pi\)
−0.571443 + 0.820642i \(0.693616\pi\)
\(662\) 1.79344e10 2.40261
\(663\) −2.74985e9 −0.366447
\(664\) 2.43276e10 3.22486
\(665\) 8.96106e9 1.18164
\(666\) −3.85714e9 −0.505948
\(667\) −1.37317e9 −0.179178
\(668\) −2.33110e10 −3.02582
\(669\) 6.96164e9 0.898918
\(670\) −5.40923e9 −0.694822
\(671\) −2.51227e9 −0.321024
\(672\) −3.10297e8 −0.0394444
\(673\) −1.43232e10 −1.81129 −0.905646 0.424035i \(-0.860613\pi\)
−0.905646 + 0.424035i \(0.860613\pi\)
\(674\) 1.07283e10 1.34964
\(675\) −2.21700e9 −0.277462
\(676\) −8.93084e9 −1.11193
\(677\) −2.38817e9 −0.295805 −0.147902 0.989002i \(-0.547252\pi\)
−0.147902 + 0.989002i \(0.547252\pi\)
\(678\) 1.42846e9 0.176021
\(679\) −1.35789e10 −1.66465
\(680\) 2.14716e10 2.61869
\(681\) 6.12925e9 0.743690
\(682\) −3.70156e9 −0.446827
\(683\) 1.52714e10 1.83403 0.917013 0.398857i \(-0.130593\pi\)
0.917013 + 0.398857i \(0.130593\pi\)
\(684\) −4.03866e9 −0.482548
\(685\) 1.52108e10 1.80816
\(686\) −1.37587e10 −1.62721
\(687\) 1.99805e9 0.235102
\(688\) −1.60713e10 −1.88145
\(689\) −3.61253e9 −0.420770
\(690\) −7.95533e9 −0.921905
\(691\) −1.01080e10 −1.16544 −0.582721 0.812672i \(-0.698012\pi\)
−0.582721 + 0.812672i \(0.698012\pi\)
\(692\) 5.81645e9 0.667247
\(693\) 1.37597e9 0.157051
\(694\) −1.14799e10 −1.30371
\(695\) −1.43864e10 −1.62557
\(696\) −2.76820e9 −0.311218
\(697\) 6.91626e9 0.773671
\(698\) −1.50249e10 −1.67231
\(699\) −1.82902e9 −0.202557
\(700\) 2.78451e10 3.06835
\(701\) 1.18838e10 1.30299 0.651494 0.758654i \(-0.274143\pi\)
0.651494 + 0.758654i \(0.274143\pi\)
\(702\) 2.05388e9 0.224076
\(703\) 5.77185e9 0.626572
\(704\) −3.90188e9 −0.421472
\(705\) 6.37420e8 0.0685116
\(706\) −2.09088e10 −2.23621
\(707\) 3.65321e9 0.388782
\(708\) −1.43269e9 −0.151718
\(709\) −1.60939e10 −1.69589 −0.847946 0.530082i \(-0.822161\pi\)
−0.847946 + 0.530082i \(0.822161\pi\)
\(710\) −1.76127e10 −1.84681
\(711\) 3.33224e9 0.347691
\(712\) −6.29935e9 −0.654057
\(713\) 3.27639e9 0.338518
\(714\) 9.74233e9 1.00166
\(715\) 4.57372e9 0.467949
\(716\) 3.27817e10 3.33762
\(717\) −3.47678e9 −0.352257
\(718\) −2.14604e10 −2.16373
\(719\) 6.07025e9 0.609053 0.304527 0.952504i \(-0.401502\pi\)
0.304527 + 0.952504i \(0.401502\pi\)
\(720\) −5.50759e9 −0.549918
\(721\) 1.70004e10 1.68922
\(722\) −8.53256e9 −0.843721
\(723\) 8.02104e9 0.789308
\(724\) 2.40202e10 2.35230
\(725\) 4.50659e9 0.439203
\(726\) 8.27705e9 0.802781
\(727\) −1.05256e10 −1.01596 −0.507979 0.861369i \(-0.669607\pi\)
−0.507979 + 0.861369i \(0.669607\pi\)
\(728\) −1.30162e10 −1.25033
\(729\) 3.87420e8 0.0370370
\(730\) 4.60797e10 4.38409
\(731\) 1.78247e10 1.68776
\(732\) 8.88430e9 0.837210
\(733\) −8.12696e9 −0.762192 −0.381096 0.924536i \(-0.624453\pi\)
−0.381096 + 0.924536i \(0.624453\pi\)
\(734\) −3.77639e9 −0.352485
\(735\) 1.08493e9 0.100785
\(736\) −4.12216e8 −0.0381112
\(737\) 1.24289e9 0.114366
\(738\) −5.16579e9 −0.473086
\(739\) −5.50602e9 −0.501860 −0.250930 0.968005i \(-0.580736\pi\)
−0.250930 + 0.968005i \(0.580736\pi\)
\(740\) 3.03751e10 2.75554
\(741\) −3.07343e9 −0.277498
\(742\) 1.27987e10 1.15014
\(743\) −4.28129e9 −0.382925 −0.191462 0.981500i \(-0.561323\pi\)
−0.191462 + 0.981500i \(0.561323\pi\)
\(744\) 6.60493e9 0.587981
\(745\) −1.37438e10 −1.21776
\(746\) −2.44455e10 −2.15582
\(747\) 6.92095e9 0.607497
\(748\) −9.77767e9 −0.854240
\(749\) 2.25781e10 1.96336
\(750\) 7.99940e9 0.692378
\(751\) 7.38153e9 0.635926 0.317963 0.948103i \(-0.397001\pi\)
0.317963 + 0.948103i \(0.397001\pi\)
\(752\) 9.34992e8 0.0801762
\(753\) 1.83909e9 0.156971
\(754\) −4.17500e9 −0.354696
\(755\) 1.10763e9 0.0936655
\(756\) −4.86592e9 −0.409580
\(757\) −2.14892e10 −1.80046 −0.900232 0.435410i \(-0.856604\pi\)
−0.900232 + 0.435410i \(0.856604\pi\)
\(758\) −3.99759e10 −3.33393
\(759\) 1.82791e9 0.151743
\(760\) 2.39983e10 1.98304
\(761\) 9.48902e9 0.780504 0.390252 0.920708i \(-0.372388\pi\)
0.390252 + 0.920708i \(0.372388\pi\)
\(762\) −9.17929e8 −0.0751565
\(763\) 3.82092e9 0.311410
\(764\) −4.02182e10 −3.26284
\(765\) 6.10847e9 0.493307
\(766\) −1.68785e10 −1.35685
\(767\) −1.09029e9 −0.0872482
\(768\) 1.46144e10 1.16417
\(769\) −2.02433e9 −0.160524 −0.0802620 0.996774i \(-0.525576\pi\)
−0.0802620 + 0.996774i \(0.525576\pi\)
\(770\) −1.62041e10 −1.27911
\(771\) 3.04921e9 0.239606
\(772\) 4.79198e9 0.374847
\(773\) −5.39051e9 −0.419761 −0.209880 0.977727i \(-0.567307\pi\)
−0.209880 + 0.977727i \(0.567307\pi\)
\(774\) −1.33133e10 −1.03203
\(775\) −1.07527e10 −0.829781
\(776\) −3.63652e10 −2.79364
\(777\) 6.95413e9 0.531826
\(778\) −3.53147e10 −2.68861
\(779\) 7.73012e9 0.585875
\(780\) −1.61743e10 −1.22038
\(781\) 4.04691e9 0.303980
\(782\) 1.29423e10 0.967803
\(783\) −7.87526e8 −0.0586271
\(784\) 1.59142e9 0.117945
\(785\) 2.53968e10 1.87385
\(786\) −6.42981e9 −0.472301
\(787\) −3.44573e8 −0.0251982 −0.0125991 0.999921i \(-0.504011\pi\)
−0.0125991 + 0.999921i \(0.504011\pi\)
\(788\) 9.30986e9 0.677800
\(789\) −1.38617e10 −1.00473
\(790\) −3.92421e10 −2.83177
\(791\) −2.57540e9 −0.185023
\(792\) 3.68492e9 0.263567
\(793\) 6.76098e9 0.481453
\(794\) −2.60181e10 −1.84461
\(795\) 8.02482e9 0.566436
\(796\) 8.83382e9 0.620802
\(797\) 7.94031e9 0.555563 0.277781 0.960644i \(-0.410401\pi\)
0.277781 + 0.960644i \(0.410401\pi\)
\(798\) 1.08888e10 0.758522
\(799\) −1.03700e9 −0.0719225
\(800\) 1.35285e9 0.0934188
\(801\) −1.79210e9 −0.123211
\(802\) −2.45847e10 −1.68288
\(803\) −1.05878e10 −0.721611
\(804\) −4.39532e9 −0.298259
\(805\) 1.43428e10 0.969057
\(806\) 9.96156e9 0.670123
\(807\) 2.00191e9 0.134087
\(808\) 9.78352e9 0.652462
\(809\) 1.83573e10 1.21896 0.609478 0.792803i \(-0.291379\pi\)
0.609478 + 0.792803i \(0.291379\pi\)
\(810\) −4.56245e9 −0.301648
\(811\) −2.86473e10 −1.88587 −0.942934 0.332979i \(-0.891946\pi\)
−0.942934 + 0.332979i \(0.891946\pi\)
\(812\) 9.89115e9 0.648337
\(813\) −4.64677e9 −0.303273
\(814\) −1.04371e10 −0.678256
\(815\) 1.91855e10 1.24143
\(816\) 8.96014e9 0.577297
\(817\) 1.99222e10 1.27808
\(818\) 3.61724e10 2.31069
\(819\) −3.70298e9 −0.235536
\(820\) 4.06807e10 2.57656
\(821\) −1.87449e10 −1.18218 −0.591088 0.806607i \(-0.701302\pi\)
−0.591088 + 0.806607i \(0.701302\pi\)
\(822\) 1.84829e10 1.16070
\(823\) 2.20970e10 1.38176 0.690881 0.722969i \(-0.257223\pi\)
0.690881 + 0.722969i \(0.257223\pi\)
\(824\) 4.55281e10 2.83488
\(825\) −5.99901e9 −0.371956
\(826\) 3.86273e9 0.238487
\(827\) 7.00279e9 0.430529 0.215264 0.976556i \(-0.430939\pi\)
0.215264 + 0.976556i \(0.430939\pi\)
\(828\) −6.46417e9 −0.395737
\(829\) −1.55766e10 −0.949581 −0.474791 0.880099i \(-0.657476\pi\)
−0.474791 + 0.880099i \(0.657476\pi\)
\(830\) −8.15045e10 −4.94776
\(831\) −1.14776e10 −0.693823
\(832\) 1.05007e10 0.632099
\(833\) −1.76505e9 −0.105803
\(834\) −1.74812e10 −1.04349
\(835\) 3.94068e10 2.34244
\(836\) −1.09282e10 −0.646887
\(837\) 1.87904e9 0.110763
\(838\) 3.88237e10 2.27899
\(839\) −2.48824e10 −1.45454 −0.727268 0.686353i \(-0.759210\pi\)
−0.727268 + 0.686353i \(0.759210\pi\)
\(840\) 2.89140e10 1.68318
\(841\) −1.56490e10 −0.907197
\(842\) 2.06221e10 1.19053
\(843\) −1.00020e9 −0.0575031
\(844\) 4.29411e10 2.45853
\(845\) 1.50974e10 0.860804
\(846\) 7.74541e8 0.0439793
\(847\) −1.49229e10 −0.843841
\(848\) 1.17711e10 0.662876
\(849\) 7.02802e8 0.0394145
\(850\) −4.24751e10 −2.37229
\(851\) 9.23827e9 0.513851
\(852\) −1.43113e10 −0.792760
\(853\) −1.67988e10 −0.926736 −0.463368 0.886166i \(-0.653359\pi\)
−0.463368 + 0.886166i \(0.653359\pi\)
\(854\) −2.39533e10 −1.31602
\(855\) 6.82727e9 0.373565
\(856\) 6.04654e10 3.29495
\(857\) −2.93017e10 −1.59023 −0.795115 0.606458i \(-0.792590\pi\)
−0.795115 + 0.606458i \(0.792590\pi\)
\(858\) 5.55761e9 0.300388
\(859\) 2.65222e10 1.42769 0.713844 0.700305i \(-0.246952\pi\)
0.713844 + 0.700305i \(0.246952\pi\)
\(860\) 1.04843e11 5.62075
\(861\) 9.31352e9 0.497282
\(862\) 2.26449e10 1.20419
\(863\) 2.67744e10 1.41802 0.709009 0.705199i \(-0.249142\pi\)
0.709009 + 0.705199i \(0.249142\pi\)
\(864\) −2.36410e8 −0.0124700
\(865\) −9.83259e9 −0.516549
\(866\) 4.67076e10 2.44385
\(867\) 1.14147e9 0.0594834
\(868\) −2.36003e10 −1.22489
\(869\) 9.01675e9 0.466102
\(870\) 9.27428e9 0.477488
\(871\) −3.34485e9 −0.171519
\(872\) 1.02327e10 0.522614
\(873\) −1.03456e10 −0.526265
\(874\) 1.44652e10 0.732885
\(875\) −1.44223e10 −0.727790
\(876\) 3.74424e10 1.88191
\(877\) 2.17802e10 1.09034 0.545171 0.838325i \(-0.316465\pi\)
0.545171 + 0.838325i \(0.316465\pi\)
\(878\) −1.31774e10 −0.657049
\(879\) −1.76027e10 −0.874215
\(880\) −1.49031e10 −0.737201
\(881\) −2.03738e10 −1.00382 −0.501910 0.864920i \(-0.667369\pi\)
−0.501910 + 0.864920i \(0.667369\pi\)
\(882\) 1.31832e9 0.0646966
\(883\) 2.67217e10 1.30618 0.653088 0.757282i \(-0.273473\pi\)
0.653088 + 0.757282i \(0.273473\pi\)
\(884\) 2.63135e10 1.28114
\(885\) 2.42194e9 0.117453
\(886\) −4.41218e10 −2.13125
\(887\) 6.91217e9 0.332569 0.166285 0.986078i \(-0.446823\pi\)
0.166285 + 0.986078i \(0.446823\pi\)
\(888\) 1.86236e10 0.892520
\(889\) 1.65495e9 0.0790005
\(890\) 2.11047e10 1.00349
\(891\) 1.04833e9 0.0496506
\(892\) −6.66164e10 −3.14271
\(893\) −1.15903e9 −0.0544645
\(894\) −1.67004e10 −0.781708
\(895\) −5.54169e10 −2.58382
\(896\) −3.86734e10 −1.79612
\(897\) −4.91925e9 −0.227575
\(898\) −1.13395e10 −0.522550
\(899\) −3.81960e9 −0.175331
\(900\) 2.12147e10 0.970035
\(901\) −1.30553e10 −0.594636
\(902\) −1.39782e10 −0.634202
\(903\) 2.40029e10 1.08482
\(904\) −6.89707e9 −0.310510
\(905\) −4.06058e10 −1.82103
\(906\) 1.34590e9 0.0601262
\(907\) −1.77215e10 −0.788632 −0.394316 0.918975i \(-0.629018\pi\)
−0.394316 + 0.918975i \(0.629018\pi\)
\(908\) −5.86512e10 −2.60002
\(909\) 2.78332e9 0.122910
\(910\) 4.36081e10 1.91832
\(911\) −2.97591e10 −1.30408 −0.652042 0.758183i \(-0.726087\pi\)
−0.652042 + 0.758183i \(0.726087\pi\)
\(912\) 1.00145e10 0.437167
\(913\) 1.87275e10 0.814389
\(914\) 1.77360e10 0.768323
\(915\) −1.50187e10 −0.648127
\(916\) −1.91195e10 −0.821942
\(917\) 1.15925e10 0.496458
\(918\) 7.42251e9 0.316666
\(919\) −7.49318e8 −0.0318465 −0.0159233 0.999873i \(-0.505069\pi\)
−0.0159233 + 0.999873i \(0.505069\pi\)
\(920\) 3.84110e10 1.62629
\(921\) −2.18081e10 −0.919832
\(922\) 3.08418e10 1.29593
\(923\) −1.08910e10 −0.455891
\(924\) −1.31667e10 −0.549068
\(925\) −3.03190e10 −1.25956
\(926\) 1.18189e10 0.489147
\(927\) 1.29523e10 0.534033
\(928\) 4.80560e8 0.0197392
\(929\) 3.70776e10 1.51725 0.758623 0.651529i \(-0.225872\pi\)
0.758623 + 0.651529i \(0.225872\pi\)
\(930\) −2.21285e10 −0.902113
\(931\) −1.97274e9 −0.0801211
\(932\) 1.75020e10 0.708161
\(933\) 4.32624e9 0.174391
\(934\) −4.29706e10 −1.72567
\(935\) 1.65290e10 0.661310
\(936\) −9.91680e9 −0.395281
\(937\) 2.52127e10 1.00122 0.500611 0.865672i \(-0.333109\pi\)
0.500611 + 0.865672i \(0.333109\pi\)
\(938\) 1.18504e10 0.468836
\(939\) −5.81495e9 −0.229201
\(940\) −6.09952e9 −0.239524
\(941\) 2.49972e10 0.977975 0.488988 0.872291i \(-0.337366\pi\)
0.488988 + 0.872291i \(0.337366\pi\)
\(942\) 3.08601e10 1.20287
\(943\) 1.23726e10 0.480475
\(944\) 3.55260e9 0.137450
\(945\) 8.22574e9 0.317076
\(946\) −3.60247e10 −1.38351
\(947\) −1.83651e10 −0.702696 −0.351348 0.936245i \(-0.614277\pi\)
−0.351348 + 0.936245i \(0.614277\pi\)
\(948\) −3.18865e10 −1.21556
\(949\) 2.84938e10 1.08223
\(950\) −4.74733e10 −1.79646
\(951\) 2.33286e10 0.879541
\(952\) −4.70393e10 −1.76698
\(953\) −1.85113e10 −0.692805 −0.346403 0.938086i \(-0.612597\pi\)
−0.346403 + 0.938086i \(0.612597\pi\)
\(954\) 9.75111e9 0.363609
\(955\) 6.79881e10 2.52593
\(956\) 3.32696e10 1.23153
\(957\) −2.13097e9 −0.0785935
\(958\) 2.30694e10 0.847728
\(959\) −3.33233e10 −1.22007
\(960\) −2.33260e10 −0.850924
\(961\) −1.83990e10 −0.668749
\(962\) 2.80881e10 1.01721
\(963\) 1.72018e10 0.620701
\(964\) −7.67540e10 −2.75950
\(965\) −8.10075e9 −0.290188
\(966\) 1.74282e10 0.622062
\(967\) −4.59158e10 −1.63294 −0.816469 0.577389i \(-0.804072\pi\)
−0.816469 + 0.577389i \(0.804072\pi\)
\(968\) −3.99644e10 −1.41615
\(969\) −1.11071e10 −0.392163
\(970\) 1.21834e11 4.28616
\(971\) 9.02722e9 0.316437 0.158218 0.987404i \(-0.449425\pi\)
0.158218 + 0.987404i \(0.449425\pi\)
\(972\) −3.70726e9 −0.129485
\(973\) 3.15172e10 1.09686
\(974\) 2.70335e10 0.937444
\(975\) 1.61444e10 0.557836
\(976\) −2.20301e10 −0.758476
\(977\) 2.33504e10 0.801058 0.400529 0.916284i \(-0.368826\pi\)
0.400529 + 0.916284i \(0.368826\pi\)
\(978\) 2.33127e10 0.796904
\(979\) −4.84927e9 −0.165172
\(980\) −1.03818e10 −0.352356
\(981\) 2.91109e9 0.0984497
\(982\) 9.70132e10 3.26919
\(983\) −4.78814e10 −1.60779 −0.803895 0.594771i \(-0.797243\pi\)
−0.803895 + 0.594771i \(0.797243\pi\)
\(984\) 2.49422e10 0.834548
\(985\) −1.57381e10 −0.524719
\(986\) −1.50880e10 −0.501261
\(987\) −1.39644e9 −0.0462287
\(988\) 2.94099e10 0.970162
\(989\) 3.18869e10 1.04815
\(990\) −1.23456e10 −0.404379
\(991\) −1.35826e10 −0.443329 −0.221664 0.975123i \(-0.571149\pi\)
−0.221664 + 0.975123i \(0.571149\pi\)
\(992\) −1.14662e9 −0.0372930
\(993\) −2.46350e10 −0.798417
\(994\) 3.85852e10 1.24615
\(995\) −1.49334e10 −0.480594
\(996\) −6.62271e10 −2.12387
\(997\) 4.46503e10 1.42689 0.713446 0.700710i \(-0.247133\pi\)
0.713446 + 0.700710i \(0.247133\pi\)
\(998\) −8.93530e10 −2.84546
\(999\) 5.29823e9 0.168132
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 177.8.a.a.1.16 16
3.2 odd 2 531.8.a.b.1.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.16 16 1.1 even 1 trivial
531.8.a.b.1.1 16 3.2 odd 2