Properties

Label 177.8.a.a.1.11
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 1.11
Root \(-7.02227\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q+7.02227 q^{2} -27.0000 q^{3} -78.6878 q^{4} -266.773 q^{5} -189.601 q^{6} +665.758 q^{7} -1451.42 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+7.02227 q^{2} -27.0000 q^{3} -78.6878 q^{4} -266.773 q^{5} -189.601 q^{6} +665.758 q^{7} -1451.42 q^{8} +729.000 q^{9} -1873.35 q^{10} +2997.05 q^{11} +2124.57 q^{12} +11110.8 q^{13} +4675.13 q^{14} +7202.88 q^{15} -120.197 q^{16} +17329.7 q^{17} +5119.23 q^{18} -7569.90 q^{19} +20991.8 q^{20} -17975.5 q^{21} +21046.1 q^{22} -84837.6 q^{23} +39188.2 q^{24} -6956.93 q^{25} +78023.0 q^{26} -19683.0 q^{27} -52387.0 q^{28} +59721.5 q^{29} +50580.6 q^{30} -56202.0 q^{31} +184937. q^{32} -80920.4 q^{33} +121694. q^{34} -177607. q^{35} -57363.4 q^{36} -375672. q^{37} -53157.8 q^{38} -299992. q^{39} +387199. q^{40} +560321. q^{41} -126228. q^{42} -599895. q^{43} -235831. q^{44} -194478. q^{45} -595752. q^{46} -508735. q^{47} +3245.32 q^{48} -380309. q^{49} -48853.4 q^{50} -467903. q^{51} -874284. q^{52} +95112.1 q^{53} -138219. q^{54} -799533. q^{55} -966292. q^{56} +204387. q^{57} +419380. q^{58} +205379. q^{59} -566779. q^{60} +1.08522e6 q^{61} -394665. q^{62} +485337. q^{63} +1.31406e6 q^{64} -2.96407e6 q^{65} -568244. q^{66} -731050. q^{67} -1.36364e6 q^{68} +2.29062e6 q^{69} -1.24720e6 q^{70} +4.30967e6 q^{71} -1.05808e6 q^{72} -804495. q^{73} -2.63807e6 q^{74} +187837. q^{75} +595658. q^{76} +1.99531e6 q^{77} -2.10662e6 q^{78} -3.53260e6 q^{79} +32065.4 q^{80} +531441. q^{81} +3.93472e6 q^{82} -8.87854e6 q^{83} +1.41445e6 q^{84} -4.62312e6 q^{85} -4.21262e6 q^{86} -1.61248e6 q^{87} -4.34997e6 q^{88} -5.83811e6 q^{89} -1.36568e6 q^{90} +7.39710e6 q^{91} +6.67568e6 q^{92} +1.51745e6 q^{93} -3.57247e6 q^{94} +2.01945e6 q^{95} -4.99331e6 q^{96} +1.47129e7 q^{97} -2.67063e6 q^{98} +2.18485e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} + O(q^{10}) \) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} - 3479q^{10} + 898q^{11} - 26298q^{12} - 8172q^{13} - 13315q^{14} + 1836q^{15} + 3138q^{16} - 44985q^{17} - 4374q^{18} - 40137q^{19} + 130657q^{20} + 63261q^{21} + 109394q^{22} - 2833q^{23} - 22113q^{24} + 285746q^{25} - 129420q^{26} - 314928q^{27} + 112890q^{28} + 144375q^{29} + 93933q^{30} - 141759q^{31} - 36224q^{32} - 24246q^{33} - 341332q^{34} - 78859q^{35} + 710046q^{36} - 297971q^{37} + 329075q^{38} + 220644q^{39} - 203048q^{40} + 659077q^{41} + 359505q^{42} - 1431608q^{43} + 254916q^{44} - 49572q^{45} + 873113q^{46} - 1574073q^{47} - 84726q^{48} + 1893545q^{49} + 302533q^{50} + 1214595q^{51} - 4972548q^{52} + 587736q^{53} + 118098q^{54} - 4624036q^{55} - 5798506q^{56} + 1083699q^{57} - 6991380q^{58} + 3286064q^{59} - 3527739q^{60} - 6117131q^{61} - 11570258q^{62} - 1708047q^{63} - 19063011q^{64} - 5335514q^{65} - 2953638q^{66} - 16518710q^{67} - 17284669q^{68} + 76491q^{69} - 39189486q^{70} - 10882582q^{71} + 597051q^{72} - 21097441q^{73} - 16717030q^{74} - 7715142q^{75} - 40864952q^{76} - 3404601q^{77} + 3494340q^{78} - 3784458q^{79} - 27466195q^{80} + 8503056q^{81} - 24990117q^{82} - 1951425q^{83} - 3048030q^{84} - 23238675q^{85} - 35910572q^{86} - 3898125q^{87} - 27843055q^{88} + 10499443q^{89} - 2536191q^{90} + 699217q^{91} - 20062766q^{92} + 3827493q^{93} - 59358988q^{94} - 29236333q^{95} + 978048q^{96} - 25158976q^{97} + 2120460q^{98} + 654642q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.02227 0.620686 0.310343 0.950625i \(-0.399556\pi\)
0.310343 + 0.950625i \(0.399556\pi\)
\(3\) −27.0000 −0.577350
\(4\) −78.6878 −0.614748
\(5\) −266.773 −0.954438 −0.477219 0.878784i \(-0.658355\pi\)
−0.477219 + 0.878784i \(0.658355\pi\)
\(6\) −189.601 −0.358354
\(7\) 665.758 0.733623 0.366812 0.930295i \(-0.380449\pi\)
0.366812 + 0.930295i \(0.380449\pi\)
\(8\) −1451.42 −1.00225
\(9\) 729.000 0.333333
\(10\) −1873.35 −0.592407
\(11\) 2997.05 0.678922 0.339461 0.940620i \(-0.389755\pi\)
0.339461 + 0.940620i \(0.389755\pi\)
\(12\) 2124.57 0.354925
\(13\) 11110.8 1.40263 0.701316 0.712851i \(-0.252596\pi\)
0.701316 + 0.712851i \(0.252596\pi\)
\(14\) 4675.13 0.455350
\(15\) 7202.88 0.551045
\(16\) −120.197 −0.00733625
\(17\) 17329.7 0.855502 0.427751 0.903897i \(-0.359306\pi\)
0.427751 + 0.903897i \(0.359306\pi\)
\(18\) 5119.23 0.206895
\(19\) −7569.90 −0.253193 −0.126597 0.991954i \(-0.540405\pi\)
−0.126597 + 0.991954i \(0.540405\pi\)
\(20\) 20991.8 0.586739
\(21\) −17975.5 −0.423558
\(22\) 21046.1 0.421397
\(23\) −84837.6 −1.45392 −0.726961 0.686679i \(-0.759068\pi\)
−0.726961 + 0.686679i \(0.759068\pi\)
\(24\) 39188.2 0.578651
\(25\) −6956.93 −0.0890487
\(26\) 78023.0 0.870594
\(27\) −19683.0 −0.192450
\(28\) −52387.0 −0.450994
\(29\) 59721.5 0.454713 0.227357 0.973812i \(-0.426992\pi\)
0.227357 + 0.973812i \(0.426992\pi\)
\(30\) 50580.6 0.342026
\(31\) −56202.0 −0.338833 −0.169416 0.985545i \(-0.554188\pi\)
−0.169416 + 0.985545i \(0.554188\pi\)
\(32\) 184937. 0.997699
\(33\) −80920.4 −0.391976
\(34\) 121694. 0.530998
\(35\) −177607. −0.700198
\(36\) −57363.4 −0.204916
\(37\) −375672. −1.21928 −0.609639 0.792679i \(-0.708686\pi\)
−0.609639 + 0.792679i \(0.708686\pi\)
\(38\) −53157.8 −0.157154
\(39\) −299992. −0.809810
\(40\) 387199. 0.956587
\(41\) 560321. 1.26968 0.634839 0.772645i \(-0.281066\pi\)
0.634839 + 0.772645i \(0.281066\pi\)
\(42\) −126228. −0.262897
\(43\) −599895. −1.15063 −0.575315 0.817932i \(-0.695120\pi\)
−0.575315 + 0.817932i \(0.695120\pi\)
\(44\) −235831. −0.417366
\(45\) −194478. −0.318146
\(46\) −595752. −0.902429
\(47\) −508735. −0.714741 −0.357371 0.933963i \(-0.616327\pi\)
−0.357371 + 0.933963i \(0.616327\pi\)
\(48\) 3245.32 0.00423558
\(49\) −380309. −0.461797
\(50\) −48853.4 −0.0552713
\(51\) −467903. −0.493924
\(52\) −874284. −0.862265
\(53\) 95112.1 0.0877547 0.0438773 0.999037i \(-0.486029\pi\)
0.0438773 + 0.999037i \(0.486029\pi\)
\(54\) −138219. −0.119451
\(55\) −799533. −0.647988
\(56\) −966292. −0.735276
\(57\) 204387. 0.146181
\(58\) 419380. 0.282234
\(59\) 205379. 0.130189
\(60\) −566779. −0.338754
\(61\) 1.08522e6 0.612160 0.306080 0.952006i \(-0.400982\pi\)
0.306080 + 0.952006i \(0.400982\pi\)
\(62\) −394665. −0.210309
\(63\) 485337. 0.244541
\(64\) 1.31406e6 0.626594
\(65\) −2.96407e6 −1.33872
\(66\) −568244. −0.243294
\(67\) −731050. −0.296951 −0.148476 0.988916i \(-0.547437\pi\)
−0.148476 + 0.988916i \(0.547437\pi\)
\(68\) −1.36364e6 −0.525918
\(69\) 2.29062e6 0.839422
\(70\) −1.24720e6 −0.434603
\(71\) 4.30967e6 1.42902 0.714512 0.699623i \(-0.246649\pi\)
0.714512 + 0.699623i \(0.246649\pi\)
\(72\) −1.05808e6 −0.334084
\(73\) −804495. −0.242043 −0.121022 0.992650i \(-0.538617\pi\)
−0.121022 + 0.992650i \(0.538617\pi\)
\(74\) −2.63807e6 −0.756790
\(75\) 187837. 0.0514123
\(76\) 595658. 0.155650
\(77\) 1.99531e6 0.498073
\(78\) −2.10662e6 −0.502638
\(79\) −3.53260e6 −0.806121 −0.403060 0.915173i \(-0.632054\pi\)
−0.403060 + 0.915173i \(0.632054\pi\)
\(80\) 32065.4 0.00700199
\(81\) 531441. 0.111111
\(82\) 3.93472e6 0.788072
\(83\) −8.87854e6 −1.70439 −0.852193 0.523227i \(-0.824728\pi\)
−0.852193 + 0.523227i \(0.824728\pi\)
\(84\) 1.41445e6 0.260381
\(85\) −4.62312e6 −0.816523
\(86\) −4.21262e6 −0.714180
\(87\) −1.61248e6 −0.262529
\(88\) −4.34997e6 −0.680451
\(89\) −5.83811e6 −0.877823 −0.438912 0.898530i \(-0.644636\pi\)
−0.438912 + 0.898530i \(0.644636\pi\)
\(90\) −1.36568e6 −0.197469
\(91\) 7.39710e6 1.02900
\(92\) 6.67568e6 0.893796
\(93\) 1.51745e6 0.195625
\(94\) −3.57247e6 −0.443630
\(95\) 2.01945e6 0.241657
\(96\) −4.99331e6 −0.576022
\(97\) 1.47129e7 1.63681 0.818405 0.574642i \(-0.194859\pi\)
0.818405 + 0.574642i \(0.194859\pi\)
\(98\) −2.67063e6 −0.286631
\(99\) 2.18485e6 0.226307
\(100\) 547426. 0.0547426
\(101\) −4.85985e6 −0.469352 −0.234676 0.972074i \(-0.575403\pi\)
−0.234676 + 0.972074i \(0.575403\pi\)
\(102\) −3.28574e6 −0.306572
\(103\) −4.63498e6 −0.417944 −0.208972 0.977922i \(-0.567012\pi\)
−0.208972 + 0.977922i \(0.567012\pi\)
\(104\) −1.61264e7 −1.40579
\(105\) 4.79538e6 0.404259
\(106\) 667902. 0.0544681
\(107\) −1.55941e7 −1.23060 −0.615301 0.788292i \(-0.710965\pi\)
−0.615301 + 0.788292i \(0.710965\pi\)
\(108\) 1.54881e6 0.118308
\(109\) −554937. −0.0410442 −0.0205221 0.999789i \(-0.506533\pi\)
−0.0205221 + 0.999789i \(0.506533\pi\)
\(110\) −5.61454e6 −0.402198
\(111\) 1.01431e7 0.703951
\(112\) −80022.2 −0.00538204
\(113\) −812959. −0.0530022 −0.0265011 0.999649i \(-0.508437\pi\)
−0.0265011 + 0.999649i \(0.508437\pi\)
\(114\) 1.43526e6 0.0907327
\(115\) 2.26324e7 1.38768
\(116\) −4.69935e6 −0.279534
\(117\) 8.09977e6 0.467544
\(118\) 1.44223e6 0.0808065
\(119\) 1.15374e7 0.627616
\(120\) −1.04544e7 −0.552286
\(121\) −1.05049e7 −0.539066
\(122\) 7.62074e6 0.379960
\(123\) −1.51287e7 −0.733049
\(124\) 4.42241e6 0.208297
\(125\) 2.26976e7 1.03943
\(126\) 3.40817e6 0.151783
\(127\) −3.99070e7 −1.72877 −0.864383 0.502834i \(-0.832291\pi\)
−0.864383 + 0.502834i \(0.832291\pi\)
\(128\) −1.44443e7 −0.608780
\(129\) 1.61972e7 0.664316
\(130\) −2.08145e7 −0.830928
\(131\) 4.33560e7 1.68500 0.842500 0.538696i \(-0.181083\pi\)
0.842500 + 0.538696i \(0.181083\pi\)
\(132\) 6.36744e6 0.240966
\(133\) −5.03972e6 −0.185749
\(134\) −5.13363e6 −0.184314
\(135\) 5.25090e6 0.183682
\(136\) −2.51527e7 −0.857429
\(137\) −3.31553e7 −1.10162 −0.550809 0.834632i \(-0.685681\pi\)
−0.550809 + 0.834632i \(0.685681\pi\)
\(138\) 1.60853e7 0.521018
\(139\) −4.77383e7 −1.50770 −0.753850 0.657046i \(-0.771806\pi\)
−0.753850 + 0.657046i \(0.771806\pi\)
\(140\) 1.39755e7 0.430445
\(141\) 1.37358e7 0.412656
\(142\) 3.02636e7 0.886976
\(143\) 3.32996e7 0.952277
\(144\) −87623.7 −0.00244542
\(145\) −1.59321e7 −0.433996
\(146\) −5.64938e6 −0.150233
\(147\) 1.02684e7 0.266618
\(148\) 2.95608e7 0.749549
\(149\) −7.91243e7 −1.95956 −0.979778 0.200086i \(-0.935878\pi\)
−0.979778 + 0.200086i \(0.935878\pi\)
\(150\) 1.31904e6 0.0319109
\(151\) −5.25091e7 −1.24112 −0.620562 0.784157i \(-0.713096\pi\)
−0.620562 + 0.784157i \(0.713096\pi\)
\(152\) 1.09871e7 0.253764
\(153\) 1.26334e7 0.285167
\(154\) 1.40116e7 0.309147
\(155\) 1.49932e7 0.323395
\(156\) 2.36057e7 0.497829
\(157\) −5.32097e7 −1.09734 −0.548671 0.836038i \(-0.684866\pi\)
−0.548671 + 0.836038i \(0.684866\pi\)
\(158\) −2.48069e7 −0.500348
\(159\) −2.56803e6 −0.0506652
\(160\) −4.93364e7 −0.952241
\(161\) −5.64813e7 −1.06663
\(162\) 3.73192e6 0.0689652
\(163\) 6.98231e7 1.26282 0.631412 0.775448i \(-0.282476\pi\)
0.631412 + 0.775448i \(0.282476\pi\)
\(164\) −4.40904e7 −0.780532
\(165\) 2.15874e7 0.374116
\(166\) −6.23475e7 −1.05789
\(167\) 8.36408e7 1.38967 0.694833 0.719171i \(-0.255478\pi\)
0.694833 + 0.719171i \(0.255478\pi\)
\(168\) 2.60899e7 0.424512
\(169\) 6.07013e7 0.967375
\(170\) −3.24647e7 −0.506805
\(171\) −5.51845e6 −0.0843978
\(172\) 4.72044e7 0.707347
\(173\) −2.42624e7 −0.356265 −0.178132 0.984007i \(-0.557006\pi\)
−0.178132 + 0.984007i \(0.557006\pi\)
\(174\) −1.13233e7 −0.162948
\(175\) −4.63163e6 −0.0653282
\(176\) −360237. −0.00498074
\(177\) −5.54523e6 −0.0751646
\(178\) −4.09967e7 −0.544853
\(179\) −1.12557e7 −0.146686 −0.0733430 0.997307i \(-0.523367\pi\)
−0.0733430 + 0.997307i \(0.523367\pi\)
\(180\) 1.53030e7 0.195580
\(181\) 1.11658e7 0.139963 0.0699816 0.997548i \(-0.477706\pi\)
0.0699816 + 0.997548i \(0.477706\pi\)
\(182\) 5.19444e7 0.638688
\(183\) −2.93011e7 −0.353431
\(184\) 1.23135e8 1.45720
\(185\) 1.00219e8 1.16373
\(186\) 1.06560e7 0.121422
\(187\) 5.19381e7 0.580819
\(188\) 4.00312e7 0.439386
\(189\) −1.31041e7 −0.141186
\(190\) 1.41811e7 0.149993
\(191\) −3.59428e7 −0.373246 −0.186623 0.982432i \(-0.559754\pi\)
−0.186623 + 0.982432i \(0.559754\pi\)
\(192\) −3.54797e7 −0.361764
\(193\) 2.78406e7 0.278758 0.139379 0.990239i \(-0.455489\pi\)
0.139379 + 0.990239i \(0.455489\pi\)
\(194\) 1.03318e8 1.01595
\(195\) 8.00298e7 0.772913
\(196\) 2.99257e7 0.283889
\(197\) 8.97822e7 0.836679 0.418339 0.908291i \(-0.362612\pi\)
0.418339 + 0.908291i \(0.362612\pi\)
\(198\) 1.53426e7 0.140466
\(199\) 1.97256e7 0.177437 0.0887183 0.996057i \(-0.471723\pi\)
0.0887183 + 0.996057i \(0.471723\pi\)
\(200\) 1.00974e7 0.0892493
\(201\) 1.97383e7 0.171445
\(202\) −3.41272e7 −0.291320
\(203\) 3.97601e7 0.333588
\(204\) 3.68183e7 0.303639
\(205\) −1.49479e8 −1.21183
\(206\) −3.25481e7 −0.259412
\(207\) −6.18466e7 −0.484641
\(208\) −1.33549e6 −0.0102901
\(209\) −2.26874e7 −0.171898
\(210\) 3.36744e7 0.250918
\(211\) 1.59619e8 1.16976 0.584881 0.811119i \(-0.301141\pi\)
0.584881 + 0.811119i \(0.301141\pi\)
\(212\) −7.48416e6 −0.0539470
\(213\) −1.16361e8 −0.825048
\(214\) −1.09506e8 −0.763818
\(215\) 1.60036e8 1.09820
\(216\) 2.85682e7 0.192884
\(217\) −3.74169e7 −0.248576
\(218\) −3.89692e6 −0.0254756
\(219\) 2.17214e7 0.139744
\(220\) 6.29135e7 0.398350
\(221\) 1.92547e8 1.19995
\(222\) 7.12279e7 0.436933
\(223\) −1.22127e8 −0.737471 −0.368736 0.929534i \(-0.620209\pi\)
−0.368736 + 0.929534i \(0.620209\pi\)
\(224\) 1.23123e8 0.731935
\(225\) −5.07160e6 −0.0296829
\(226\) −5.70881e6 −0.0328977
\(227\) −1.05761e7 −0.0600118 −0.0300059 0.999550i \(-0.509553\pi\)
−0.0300059 + 0.999550i \(0.509553\pi\)
\(228\) −1.60828e7 −0.0898647
\(229\) 5.44892e7 0.299838 0.149919 0.988698i \(-0.452099\pi\)
0.149919 + 0.988698i \(0.452099\pi\)
\(230\) 1.58931e8 0.861313
\(231\) −5.38734e7 −0.287562
\(232\) −8.66808e7 −0.455738
\(233\) 1.03393e8 0.535484 0.267742 0.963491i \(-0.413722\pi\)
0.267742 + 0.963491i \(0.413722\pi\)
\(234\) 5.68788e7 0.290198
\(235\) 1.35717e8 0.682176
\(236\) −1.61608e7 −0.0800334
\(237\) 9.53803e7 0.465414
\(238\) 8.10188e7 0.389553
\(239\) −9.19771e7 −0.435800 −0.217900 0.975971i \(-0.569921\pi\)
−0.217900 + 0.975971i \(0.569921\pi\)
\(240\) −865766. −0.00404260
\(241\) −3.22890e8 −1.48592 −0.742958 0.669338i \(-0.766578\pi\)
−0.742958 + 0.669338i \(0.766578\pi\)
\(242\) −7.37679e7 −0.334591
\(243\) −1.43489e7 −0.0641500
\(244\) −8.53939e7 −0.376325
\(245\) 1.01456e8 0.440756
\(246\) −1.06238e8 −0.454993
\(247\) −8.41076e7 −0.355137
\(248\) 8.15725e7 0.339596
\(249\) 2.39721e8 0.984028
\(250\) 1.59389e8 0.645160
\(251\) 2.78313e8 1.11090 0.555451 0.831549i \(-0.312546\pi\)
0.555451 + 0.831549i \(0.312546\pi\)
\(252\) −3.81901e7 −0.150331
\(253\) −2.54263e8 −0.987099
\(254\) −2.80238e8 −1.07302
\(255\) 1.24824e8 0.471420
\(256\) −2.69632e8 −1.00446
\(257\) −1.56960e8 −0.576795 −0.288398 0.957511i \(-0.593123\pi\)
−0.288398 + 0.957511i \(0.593123\pi\)
\(258\) 1.13741e8 0.412332
\(259\) −2.50107e8 −0.894491
\(260\) 2.33236e8 0.822978
\(261\) 4.35370e7 0.151571
\(262\) 3.04458e8 1.04586
\(263\) 2.42199e8 0.820971 0.410486 0.911867i \(-0.365359\pi\)
0.410486 + 0.911867i \(0.365359\pi\)
\(264\) 1.17449e8 0.392858
\(265\) −2.53734e7 −0.0837563
\(266\) −3.53902e7 −0.115292
\(267\) 1.57629e8 0.506811
\(268\) 5.75247e7 0.182550
\(269\) 5.70264e7 0.178625 0.0893126 0.996004i \(-0.471533\pi\)
0.0893126 + 0.996004i \(0.471533\pi\)
\(270\) 3.68732e7 0.114009
\(271\) 2.57235e8 0.785124 0.392562 0.919726i \(-0.371589\pi\)
0.392562 + 0.919726i \(0.371589\pi\)
\(272\) −2.08298e6 −0.00627617
\(273\) −1.99722e8 −0.594095
\(274\) −2.32825e8 −0.683759
\(275\) −2.08503e7 −0.0604571
\(276\) −1.80243e8 −0.516033
\(277\) −6.22571e8 −1.75999 −0.879994 0.474984i \(-0.842454\pi\)
−0.879994 + 0.474984i \(0.842454\pi\)
\(278\) −3.35231e8 −0.935809
\(279\) −4.09712e7 −0.112944
\(280\) 2.57781e8 0.701775
\(281\) 3.00467e8 0.807838 0.403919 0.914795i \(-0.367648\pi\)
0.403919 + 0.914795i \(0.367648\pi\)
\(282\) 9.64567e7 0.256130
\(283\) −4.37742e8 −1.14806 −0.574032 0.818833i \(-0.694621\pi\)
−0.574032 + 0.818833i \(0.694621\pi\)
\(284\) −3.39118e8 −0.878490
\(285\) −5.45251e7 −0.139521
\(286\) 2.33839e8 0.591065
\(287\) 3.73038e8 0.931465
\(288\) 1.34819e8 0.332566
\(289\) −1.10019e8 −0.268117
\(290\) −1.11880e8 −0.269375
\(291\) −3.97249e8 −0.945013
\(292\) 6.33039e7 0.148796
\(293\) −3.94385e8 −0.915976 −0.457988 0.888958i \(-0.651430\pi\)
−0.457988 + 0.888958i \(0.651430\pi\)
\(294\) 7.21071e7 0.165486
\(295\) −5.47897e7 −0.124257
\(296\) 5.45257e8 1.22203
\(297\) −5.89909e7 −0.130659
\(298\) −5.55632e8 −1.21627
\(299\) −9.42614e8 −2.03932
\(300\) −1.47805e7 −0.0316056
\(301\) −3.99385e8 −0.844129
\(302\) −3.68733e8 −0.770349
\(303\) 1.31216e8 0.270980
\(304\) 909879. 0.00185749
\(305\) −2.89509e8 −0.584269
\(306\) 8.87150e7 0.176999
\(307\) −2.75368e8 −0.543162 −0.271581 0.962416i \(-0.587546\pi\)
−0.271581 + 0.962416i \(0.587546\pi\)
\(308\) −1.57006e8 −0.306189
\(309\) 1.25145e8 0.241300
\(310\) 1.05286e8 0.200727
\(311\) −2.81733e8 −0.531100 −0.265550 0.964097i \(-0.585554\pi\)
−0.265550 + 0.964097i \(0.585554\pi\)
\(312\) 4.35413e8 0.811634
\(313\) −8.03742e8 −1.48153 −0.740767 0.671762i \(-0.765538\pi\)
−0.740767 + 0.671762i \(0.765538\pi\)
\(314\) −3.73653e8 −0.681105
\(315\) −1.29475e8 −0.233399
\(316\) 2.77973e8 0.495561
\(317\) 2.22203e8 0.391780 0.195890 0.980626i \(-0.437240\pi\)
0.195890 + 0.980626i \(0.437240\pi\)
\(318\) −1.80334e7 −0.0314472
\(319\) 1.78988e8 0.308715
\(320\) −3.50557e8 −0.598045
\(321\) 4.21041e8 0.710488
\(322\) −3.96627e8 −0.662043
\(323\) −1.31184e8 −0.216607
\(324\) −4.18179e7 −0.0683054
\(325\) −7.72971e7 −0.124903
\(326\) 4.90317e8 0.783818
\(327\) 1.49833e7 0.0236969
\(328\) −8.13259e8 −1.27254
\(329\) −3.38694e8 −0.524351
\(330\) 1.51592e8 0.232209
\(331\) −2.85912e8 −0.433346 −0.216673 0.976244i \(-0.569521\pi\)
−0.216673 + 0.976244i \(0.569521\pi\)
\(332\) 6.98633e8 1.04777
\(333\) −2.73865e8 −0.406426
\(334\) 5.87348e8 0.862547
\(335\) 1.95025e8 0.283421
\(336\) 2.16060e6 0.00310732
\(337\) −4.83392e8 −0.688010 −0.344005 0.938968i \(-0.611784\pi\)
−0.344005 + 0.938968i \(0.611784\pi\)
\(338\) 4.26261e8 0.600436
\(339\) 2.19499e7 0.0306008
\(340\) 3.63783e8 0.501956
\(341\) −1.68440e8 −0.230041
\(342\) −3.87521e7 −0.0523846
\(343\) −8.01474e8 −1.07241
\(344\) 8.70697e8 1.15322
\(345\) −6.11075e8 −0.801176
\(346\) −1.70377e8 −0.221129
\(347\) 6.40094e8 0.822414 0.411207 0.911542i \(-0.365107\pi\)
0.411207 + 0.911542i \(0.365107\pi\)
\(348\) 1.26883e8 0.161389
\(349\) 4.24832e7 0.0534968 0.0267484 0.999642i \(-0.491485\pi\)
0.0267484 + 0.999642i \(0.491485\pi\)
\(350\) −3.25246e7 −0.0405483
\(351\) −2.18694e8 −0.269937
\(352\) 5.54266e8 0.677359
\(353\) 1.12511e9 1.36139 0.680694 0.732568i \(-0.261678\pi\)
0.680694 + 0.732568i \(0.261678\pi\)
\(354\) −3.89401e7 −0.0466537
\(355\) −1.14971e9 −1.36391
\(356\) 4.59388e8 0.539640
\(357\) −3.11510e8 −0.362354
\(358\) −7.90408e7 −0.0910460
\(359\) 1.35745e7 0.0154843 0.00774215 0.999970i \(-0.497536\pi\)
0.00774215 + 0.999970i \(0.497536\pi\)
\(360\) 2.82268e8 0.318862
\(361\) −8.36568e8 −0.935893
\(362\) 7.84090e7 0.0868733
\(363\) 2.83631e8 0.311230
\(364\) −5.82061e8 −0.632578
\(365\) 2.14618e8 0.231015
\(366\) −2.05760e8 −0.219370
\(367\) 8.41482e8 0.888615 0.444308 0.895874i \(-0.353450\pi\)
0.444308 + 0.895874i \(0.353450\pi\)
\(368\) 1.01972e7 0.0106663
\(369\) 4.08474e8 0.423226
\(370\) 7.03767e8 0.722309
\(371\) 6.33216e7 0.0643789
\(372\) −1.19405e8 −0.120260
\(373\) −1.40652e9 −1.40334 −0.701672 0.712500i \(-0.747563\pi\)
−0.701672 + 0.712500i \(0.747563\pi\)
\(374\) 3.64723e8 0.360506
\(375\) −6.12835e8 −0.600115
\(376\) 7.38386e8 0.716351
\(377\) 6.63554e8 0.637795
\(378\) −9.20206e7 −0.0876322
\(379\) −7.42481e8 −0.700564 −0.350282 0.936644i \(-0.613914\pi\)
−0.350282 + 0.936644i \(0.613914\pi\)
\(380\) −1.58906e8 −0.148558
\(381\) 1.07749e9 0.998104
\(382\) −2.52400e8 −0.231669
\(383\) −7.94805e8 −0.722878 −0.361439 0.932396i \(-0.617714\pi\)
−0.361439 + 0.932396i \(0.617714\pi\)
\(384\) 3.89995e8 0.351479
\(385\) −5.32296e8 −0.475379
\(386\) 1.95504e8 0.173021
\(387\) −4.37323e8 −0.383543
\(388\) −1.15773e9 −1.00623
\(389\) −8.55397e8 −0.736790 −0.368395 0.929669i \(-0.620093\pi\)
−0.368395 + 0.929669i \(0.620093\pi\)
\(390\) 5.61990e8 0.479736
\(391\) −1.47021e9 −1.24383
\(392\) 5.51987e8 0.462837
\(393\) −1.17061e9 −0.972835
\(394\) 6.30475e8 0.519315
\(395\) 9.42405e8 0.769392
\(396\) −1.71921e8 −0.139122
\(397\) 3.23507e8 0.259487 0.129744 0.991548i \(-0.458585\pi\)
0.129744 + 0.991548i \(0.458585\pi\)
\(398\) 1.38518e8 0.110133
\(399\) 1.36072e8 0.107242
\(400\) 836203. 0.000653284 0
\(401\) −2.15538e8 −0.166924 −0.0834621 0.996511i \(-0.526598\pi\)
−0.0834621 + 0.996511i \(0.526598\pi\)
\(402\) 1.38608e8 0.106414
\(403\) −6.24449e8 −0.475258
\(404\) 3.82411e8 0.288533
\(405\) −1.41774e8 −0.106049
\(406\) 2.79206e8 0.207054
\(407\) −1.12591e9 −0.827795
\(408\) 6.79122e8 0.495037
\(409\) 2.07548e9 1.49998 0.749992 0.661447i \(-0.230057\pi\)
0.749992 + 0.661447i \(0.230057\pi\)
\(410\) −1.04968e9 −0.752165
\(411\) 8.95193e8 0.636019
\(412\) 3.64716e8 0.256930
\(413\) 1.36733e8 0.0955096
\(414\) −4.34303e8 −0.300810
\(415\) 2.36856e9 1.62673
\(416\) 2.05480e9 1.39940
\(417\) 1.28893e9 0.870471
\(418\) −1.59317e8 −0.106695
\(419\) −2.46186e9 −1.63499 −0.817495 0.575936i \(-0.804638\pi\)
−0.817495 + 0.575936i \(0.804638\pi\)
\(420\) −3.77337e8 −0.248518
\(421\) 2.15291e9 1.40617 0.703086 0.711105i \(-0.251805\pi\)
0.703086 + 0.711105i \(0.251805\pi\)
\(422\) 1.12089e9 0.726055
\(423\) −3.70868e8 −0.238247
\(424\) −1.38047e8 −0.0879523
\(425\) −1.20562e8 −0.0761814
\(426\) −8.17118e8 −0.512096
\(427\) 7.22497e8 0.449095
\(428\) 1.22707e9 0.756510
\(429\) −8.99090e8 −0.549797
\(430\) 1.12382e9 0.681640
\(431\) −2.75746e9 −1.65897 −0.829486 0.558528i \(-0.811366\pi\)
−0.829486 + 0.558528i \(0.811366\pi\)
\(432\) 2.36584e6 0.00141186
\(433\) 1.22443e9 0.724814 0.362407 0.932020i \(-0.381955\pi\)
0.362407 + 0.932020i \(0.381955\pi\)
\(434\) −2.62751e8 −0.154288
\(435\) 4.30167e8 0.250567
\(436\) 4.36668e7 0.0252318
\(437\) 6.42212e8 0.368123
\(438\) 1.52533e8 0.0867371
\(439\) −1.80512e9 −1.01831 −0.509155 0.860675i \(-0.670042\pi\)
−0.509155 + 0.860675i \(0.670042\pi\)
\(440\) 1.16046e9 0.649448
\(441\) −2.77246e8 −0.153932
\(442\) 1.35212e9 0.744795
\(443\) 2.60973e9 1.42621 0.713103 0.701059i \(-0.247289\pi\)
0.713103 + 0.701059i \(0.247289\pi\)
\(444\) −7.98142e8 −0.432753
\(445\) 1.55745e9 0.837828
\(446\) −8.57609e8 −0.457738
\(447\) 2.13636e9 1.13135
\(448\) 8.74848e8 0.459684
\(449\) 2.04391e9 1.06561 0.532805 0.846238i \(-0.321138\pi\)
0.532805 + 0.846238i \(0.321138\pi\)
\(450\) −3.56141e7 −0.0184238
\(451\) 1.67931e9 0.862012
\(452\) 6.39699e7 0.0325830
\(453\) 1.41775e9 0.716563
\(454\) −7.42685e7 −0.0372485
\(455\) −1.97335e9 −0.982119
\(456\) −2.96651e8 −0.146511
\(457\) −1.13009e9 −0.553866 −0.276933 0.960889i \(-0.589318\pi\)
−0.276933 + 0.960889i \(0.589318\pi\)
\(458\) 3.82638e8 0.186105
\(459\) −3.41101e8 −0.164641
\(460\) −1.78089e9 −0.853072
\(461\) 1.04379e9 0.496202 0.248101 0.968734i \(-0.420194\pi\)
0.248101 + 0.968734i \(0.420194\pi\)
\(462\) −3.78313e8 −0.178486
\(463\) 3.87197e8 0.181300 0.0906501 0.995883i \(-0.471106\pi\)
0.0906501 + 0.995883i \(0.471106\pi\)
\(464\) −7.17835e6 −0.00333589
\(465\) −4.04816e8 −0.186712
\(466\) 7.26055e8 0.332368
\(467\) 3.01337e8 0.136913 0.0684563 0.997654i \(-0.478193\pi\)
0.0684563 + 0.997654i \(0.478193\pi\)
\(468\) −6.37353e8 −0.287422
\(469\) −4.86702e8 −0.217850
\(470\) 9.53040e8 0.423417
\(471\) 1.43666e9 0.633550
\(472\) −2.98090e8 −0.130482
\(473\) −1.79791e9 −0.781187
\(474\) 6.69786e8 0.288876
\(475\) 5.26633e7 0.0225465
\(476\) −9.07853e8 −0.385826
\(477\) 6.93367e7 0.0292516
\(478\) −6.45888e8 −0.270495
\(479\) −4.36417e9 −1.81438 −0.907189 0.420723i \(-0.861776\pi\)
−0.907189 + 0.420723i \(0.861776\pi\)
\(480\) 1.33208e9 0.549777
\(481\) −4.17402e9 −1.71020
\(482\) −2.26742e9 −0.922288
\(483\) 1.52500e9 0.615820
\(484\) 8.26604e8 0.331390
\(485\) −3.92502e9 −1.56223
\(486\) −1.00762e8 −0.0398171
\(487\) 3.93974e9 1.54567 0.772835 0.634608i \(-0.218838\pi\)
0.772835 + 0.634608i \(0.218838\pi\)
\(488\) −1.57511e9 −0.613539
\(489\) −1.88522e9 −0.729092
\(490\) 7.12454e8 0.273571
\(491\) −1.95275e9 −0.744495 −0.372248 0.928133i \(-0.621413\pi\)
−0.372248 + 0.928133i \(0.621413\pi\)
\(492\) 1.19044e9 0.450640
\(493\) 1.03496e9 0.389008
\(494\) −5.90626e8 −0.220429
\(495\) −5.82860e8 −0.215996
\(496\) 6.75531e6 0.00248576
\(497\) 2.86920e9 1.04837
\(498\) 1.68338e9 0.610773
\(499\) 1.78306e9 0.642414 0.321207 0.947009i \(-0.395912\pi\)
0.321207 + 0.947009i \(0.395912\pi\)
\(500\) −1.78602e9 −0.638987
\(501\) −2.25830e9 −0.802325
\(502\) 1.95439e9 0.689522
\(503\) 1.58061e8 0.0553781 0.0276890 0.999617i \(-0.491185\pi\)
0.0276890 + 0.999617i \(0.491185\pi\)
\(504\) −7.04427e8 −0.245092
\(505\) 1.29648e9 0.447967
\(506\) −1.78550e9 −0.612679
\(507\) −1.63894e9 −0.558514
\(508\) 3.14019e9 1.06276
\(509\) 3.07905e9 1.03492 0.517458 0.855709i \(-0.326878\pi\)
0.517458 + 0.855709i \(0.326878\pi\)
\(510\) 8.76548e8 0.292604
\(511\) −5.35599e8 −0.177569
\(512\) −4.45593e7 −0.0146721
\(513\) 1.48998e8 0.0487271
\(514\) −1.10221e9 −0.358009
\(515\) 1.23649e9 0.398901
\(516\) −1.27452e9 −0.408387
\(517\) −1.52470e9 −0.485253
\(518\) −1.75632e9 −0.555199
\(519\) 6.55086e8 0.205690
\(520\) 4.30209e9 1.34174
\(521\) 1.45110e9 0.449538 0.224769 0.974412i \(-0.427837\pi\)
0.224769 + 0.974412i \(0.427837\pi\)
\(522\) 3.05728e8 0.0940782
\(523\) −6.11321e8 −0.186859 −0.0934293 0.995626i \(-0.529783\pi\)
−0.0934293 + 0.995626i \(0.529783\pi\)
\(524\) −3.41159e9 −1.03585
\(525\) 1.25054e8 0.0377173
\(526\) 1.70079e9 0.509566
\(527\) −9.73966e8 −0.289872
\(528\) 9.72639e6 0.00287563
\(529\) 3.79259e9 1.11389
\(530\) −1.78179e8 −0.0519864
\(531\) 1.49721e8 0.0433963
\(532\) 3.96564e8 0.114189
\(533\) 6.22561e9 1.78089
\(534\) 1.10691e9 0.314571
\(535\) 4.16010e9 1.17453
\(536\) 1.06106e9 0.297620
\(537\) 3.03905e8 0.0846892
\(538\) 4.00454e8 0.110870
\(539\) −1.13981e9 −0.313524
\(540\) −4.13182e8 −0.112918
\(541\) 4.85916e9 1.31938 0.659691 0.751537i \(-0.270687\pi\)
0.659691 + 0.751537i \(0.270687\pi\)
\(542\) 1.80637e9 0.487316
\(543\) −3.01476e8 −0.0808078
\(544\) 3.20492e9 0.853533
\(545\) 1.48043e8 0.0391741
\(546\) −1.40250e9 −0.368747
\(547\) −4.87899e9 −1.27460 −0.637300 0.770616i \(-0.719949\pi\)
−0.637300 + 0.770616i \(0.719949\pi\)
\(548\) 2.60892e9 0.677217
\(549\) 7.91129e8 0.204053
\(550\) −1.46416e8 −0.0375249
\(551\) −4.52086e8 −0.115130
\(552\) −3.32464e9 −0.841313
\(553\) −2.35186e9 −0.591389
\(554\) −4.37186e9 −1.09240
\(555\) −2.70592e9 −0.671877
\(556\) 3.75642e9 0.926856
\(557\) 3.74696e9 0.918725 0.459362 0.888249i \(-0.348078\pi\)
0.459362 + 0.888249i \(0.348078\pi\)
\(558\) −2.87711e8 −0.0701030
\(559\) −6.66531e9 −1.61391
\(560\) 2.13478e7 0.00513682
\(561\) −1.40233e9 −0.335336
\(562\) 2.10996e9 0.501414
\(563\) 1.61119e9 0.380512 0.190256 0.981735i \(-0.439068\pi\)
0.190256 + 0.981735i \(0.439068\pi\)
\(564\) −1.08084e9 −0.253680
\(565\) 2.16876e8 0.0505873
\(566\) −3.07394e9 −0.712588
\(567\) 3.53811e8 0.0815137
\(568\) −6.25513e9 −1.43224
\(569\) 1.15056e9 0.261828 0.130914 0.991394i \(-0.458209\pi\)
0.130914 + 0.991394i \(0.458209\pi\)
\(570\) −3.82890e8 −0.0865987
\(571\) −4.25054e9 −0.955472 −0.477736 0.878503i \(-0.658543\pi\)
−0.477736 + 0.878503i \(0.658543\pi\)
\(572\) −2.62027e9 −0.585410
\(573\) 9.70456e8 0.215494
\(574\) 2.61957e9 0.578148
\(575\) 5.90209e8 0.129470
\(576\) 9.57953e8 0.208865
\(577\) 2.31080e9 0.500780 0.250390 0.968145i \(-0.419441\pi\)
0.250390 + 0.968145i \(0.419441\pi\)
\(578\) −7.72580e8 −0.166416
\(579\) −7.51695e8 −0.160941
\(580\) 1.25366e9 0.266798
\(581\) −5.91096e9 −1.25038
\(582\) −2.78959e9 −0.586557
\(583\) 2.85056e8 0.0595785
\(584\) 1.16766e9 0.242589
\(585\) −2.16080e9 −0.446241
\(586\) −2.76948e9 −0.568534
\(587\) −4.19562e9 −0.856175 −0.428087 0.903737i \(-0.640812\pi\)
−0.428087 + 0.903737i \(0.640812\pi\)
\(588\) −8.07994e8 −0.163903
\(589\) 4.25443e8 0.0857902
\(590\) −3.84748e8 −0.0771248
\(591\) −2.42412e9 −0.483057
\(592\) 4.51547e7 0.00894493
\(593\) −7.48887e9 −1.47477 −0.737386 0.675472i \(-0.763940\pi\)
−0.737386 + 0.675472i \(0.763940\pi\)
\(594\) −4.14250e8 −0.0810980
\(595\) −3.07788e9 −0.599020
\(596\) 6.22611e9 1.20463
\(597\) −5.32590e8 −0.102443
\(598\) −6.61928e9 −1.26578
\(599\) 2.12227e9 0.403467 0.201733 0.979440i \(-0.435343\pi\)
0.201733 + 0.979440i \(0.435343\pi\)
\(600\) −2.72630e8 −0.0515281
\(601\) −6.18903e8 −0.116295 −0.0581477 0.998308i \(-0.518519\pi\)
−0.0581477 + 0.998308i \(0.518519\pi\)
\(602\) −2.80459e9 −0.523939
\(603\) −5.32935e8 −0.0989838
\(604\) 4.13183e9 0.762979
\(605\) 2.80242e9 0.514504
\(606\) 9.21434e8 0.168194
\(607\) −4.99735e8 −0.0906942 −0.0453471 0.998971i \(-0.514439\pi\)
−0.0453471 + 0.998971i \(0.514439\pi\)
\(608\) −1.39996e9 −0.252611
\(609\) −1.07352e9 −0.192597
\(610\) −2.03301e9 −0.362648
\(611\) −5.65245e9 −1.00252
\(612\) −9.94093e8 −0.175306
\(613\) 7.99751e9 1.40231 0.701154 0.713010i \(-0.252669\pi\)
0.701154 + 0.713010i \(0.252669\pi\)
\(614\) −1.93371e9 −0.337133
\(615\) 4.03593e9 0.699649
\(616\) −2.89603e9 −0.499195
\(617\) 1.58063e9 0.270914 0.135457 0.990783i \(-0.456750\pi\)
0.135457 + 0.990783i \(0.456750\pi\)
\(618\) 8.78798e8 0.149772
\(619\) 6.43846e9 1.09110 0.545550 0.838078i \(-0.316321\pi\)
0.545550 + 0.838078i \(0.316321\pi\)
\(620\) −1.17978e9 −0.198806
\(621\) 1.66986e9 0.279807
\(622\) −1.97840e9 −0.329646
\(623\) −3.88677e9 −0.643992
\(624\) 3.60581e7 0.00594096
\(625\) −5.51161e9 −0.903022
\(626\) −5.64409e9 −0.919568
\(627\) 6.12559e8 0.0992456
\(628\) 4.18695e9 0.674589
\(629\) −6.51030e9 −1.04310
\(630\) −9.09209e8 −0.144868
\(631\) −1.07401e10 −1.70179 −0.850896 0.525334i \(-0.823940\pi\)
−0.850896 + 0.525334i \(0.823940\pi\)
\(632\) 5.12728e9 0.807936
\(633\) −4.30973e9 −0.675362
\(634\) 1.56037e9 0.243173
\(635\) 1.06461e10 1.65000
\(636\) 2.02072e8 0.0311463
\(637\) −4.22554e9 −0.647731
\(638\) 1.25690e9 0.191615
\(639\) 3.14175e9 0.476342
\(640\) 3.85335e9 0.581043
\(641\) 4.76542e9 0.714658 0.357329 0.933978i \(-0.383687\pi\)
0.357329 + 0.933978i \(0.383687\pi\)
\(642\) 2.95666e9 0.440990
\(643\) −5.32986e9 −0.790637 −0.395319 0.918544i \(-0.629366\pi\)
−0.395319 + 0.918544i \(0.629366\pi\)
\(644\) 4.44439e9 0.655709
\(645\) −4.32097e9 −0.634048
\(646\) −9.21211e8 −0.134445
\(647\) 3.64298e9 0.528801 0.264400 0.964413i \(-0.414826\pi\)
0.264400 + 0.964413i \(0.414826\pi\)
\(648\) −7.71342e8 −0.111361
\(649\) 6.15531e8 0.0883881
\(650\) −5.42801e8 −0.0775253
\(651\) 1.01026e9 0.143515
\(652\) −5.49423e9 −0.776319
\(653\) 3.76641e9 0.529337 0.264668 0.964340i \(-0.414738\pi\)
0.264668 + 0.964340i \(0.414738\pi\)
\(654\) 1.05217e8 0.0147083
\(655\) −1.15662e10 −1.60823
\(656\) −6.73490e7 −0.00931467
\(657\) −5.86477e8 −0.0806811
\(658\) −2.37840e9 −0.325458
\(659\) −4.55405e9 −0.619867 −0.309934 0.950758i \(-0.600307\pi\)
−0.309934 + 0.950758i \(0.600307\pi\)
\(660\) −1.69866e9 −0.229987
\(661\) −1.46628e9 −0.197475 −0.0987377 0.995113i \(-0.531480\pi\)
−0.0987377 + 0.995113i \(0.531480\pi\)
\(662\) −2.00775e9 −0.268972
\(663\) −5.19878e9 −0.692794
\(664\) 1.28865e10 1.70823
\(665\) 1.34446e9 0.177285
\(666\) −1.92315e9 −0.252263
\(667\) −5.06663e9 −0.661118
\(668\) −6.58151e9 −0.854295
\(669\) 3.29743e9 0.425779
\(670\) 1.36952e9 0.175916
\(671\) 3.25247e9 0.415609
\(672\) −3.32433e9 −0.422583
\(673\) 1.29610e10 1.63903 0.819514 0.573059i \(-0.194244\pi\)
0.819514 + 0.573059i \(0.194244\pi\)
\(674\) −3.39451e9 −0.427039
\(675\) 1.36933e8 0.0171374
\(676\) −4.77645e9 −0.594692
\(677\) 2.05035e9 0.253962 0.126981 0.991905i \(-0.459471\pi\)
0.126981 + 0.991905i \(0.459471\pi\)
\(678\) 1.54138e8 0.0189935
\(679\) 9.79525e9 1.20080
\(680\) 6.71007e9 0.818362
\(681\) 2.85556e8 0.0346478
\(682\) −1.18283e9 −0.142783
\(683\) 1.07567e9 0.129184 0.0645918 0.997912i \(-0.479425\pi\)
0.0645918 + 0.997912i \(0.479425\pi\)
\(684\) 4.34235e8 0.0518834
\(685\) 8.84495e9 1.05143
\(686\) −5.62817e9 −0.665629
\(687\) −1.47121e9 −0.173111
\(688\) 7.21056e7 0.00844130
\(689\) 1.05677e9 0.123087
\(690\) −4.29113e9 −0.497279
\(691\) 4.59832e9 0.530183 0.265092 0.964223i \(-0.414598\pi\)
0.265092 + 0.964223i \(0.414598\pi\)
\(692\) 1.90916e9 0.219013
\(693\) 1.45458e9 0.166024
\(694\) 4.49491e9 0.510461
\(695\) 1.27353e10 1.43901
\(696\) 2.34038e9 0.263120
\(697\) 9.71022e9 1.08621
\(698\) 2.98328e8 0.0332048
\(699\) −2.79162e9 −0.309162
\(700\) 3.64453e8 0.0401604
\(701\) 9.38304e9 1.02880 0.514400 0.857551i \(-0.328015\pi\)
0.514400 + 0.857551i \(0.328015\pi\)
\(702\) −1.53573e9 −0.167546
\(703\) 2.84380e9 0.308713
\(704\) 3.93832e9 0.425409
\(705\) −3.66436e9 −0.393854
\(706\) 7.90079e9 0.844995
\(707\) −3.23549e9 −0.344327
\(708\) 4.36342e8 0.0462073
\(709\) −1.12461e10 −1.18506 −0.592528 0.805550i \(-0.701870\pi\)
−0.592528 + 0.805550i \(0.701870\pi\)
\(710\) −8.07354e9 −0.846564
\(711\) −2.57527e9 −0.268707
\(712\) 8.47353e9 0.879800
\(713\) 4.76804e9 0.492636
\(714\) −2.18751e9 −0.224908
\(715\) −8.88345e9 −0.908889
\(716\) 8.85689e8 0.0901750
\(717\) 2.48338e9 0.251609
\(718\) 9.53234e7 0.00961090
\(719\) −6.98063e9 −0.700396 −0.350198 0.936676i \(-0.613886\pi\)
−0.350198 + 0.936676i \(0.613886\pi\)
\(720\) 2.33757e7 0.00233400
\(721\) −3.08578e9 −0.306613
\(722\) −5.87461e9 −0.580896
\(723\) 8.71802e9 0.857894
\(724\) −8.78610e8 −0.0860421
\(725\) −4.15478e8 −0.0404917
\(726\) 1.99173e9 0.193176
\(727\) 7.76937e9 0.749921 0.374960 0.927041i \(-0.377656\pi\)
0.374960 + 0.927041i \(0.377656\pi\)
\(728\) −1.07363e10 −1.03132
\(729\) 3.87420e8 0.0370370
\(730\) 1.50710e9 0.143388
\(731\) −1.03960e10 −0.984366
\(732\) 2.30564e9 0.217271
\(733\) −1.54005e10 −1.44434 −0.722172 0.691713i \(-0.756856\pi\)
−0.722172 + 0.691713i \(0.756856\pi\)
\(734\) 5.90911e9 0.551551
\(735\) −2.73932e9 −0.254471
\(736\) −1.56896e10 −1.45058
\(737\) −2.19099e9 −0.201607
\(738\) 2.86841e9 0.262691
\(739\) 8.56257e9 0.780456 0.390228 0.920718i \(-0.372396\pi\)
0.390228 + 0.920718i \(0.372396\pi\)
\(740\) −7.88604e9 −0.715398
\(741\) 2.27090e9 0.205038
\(742\) 4.44661e8 0.0399591
\(743\) −3.55238e9 −0.317730 −0.158865 0.987300i \(-0.550783\pi\)
−0.158865 + 0.987300i \(0.550783\pi\)
\(744\) −2.20246e9 −0.196066
\(745\) 2.11083e10 1.87027
\(746\) −9.87694e9 −0.871037
\(747\) −6.47246e9 −0.568129
\(748\) −4.08689e9 −0.357057
\(749\) −1.03819e10 −0.902798
\(750\) −4.30349e9 −0.372483
\(751\) 1.09272e10 0.941391 0.470696 0.882296i \(-0.344003\pi\)
0.470696 + 0.882296i \(0.344003\pi\)
\(752\) 6.11484e7 0.00524352
\(753\) −7.51445e9 −0.641379
\(754\) 4.65965e9 0.395871
\(755\) 1.40080e10 1.18458
\(756\) 1.03113e9 0.0867938
\(757\) 6.79624e9 0.569421 0.284710 0.958614i \(-0.408103\pi\)
0.284710 + 0.958614i \(0.408103\pi\)
\(758\) −5.21390e9 −0.434831
\(759\) 6.86509e9 0.569902
\(760\) −2.93106e9 −0.242202
\(761\) 1.52139e10 1.25139 0.625696 0.780067i \(-0.284815\pi\)
0.625696 + 0.780067i \(0.284815\pi\)
\(762\) 7.56642e9 0.619509
\(763\) −3.69454e8 −0.0301109
\(764\) 2.82826e9 0.229452
\(765\) −3.37025e9 −0.272174
\(766\) −5.58133e9 −0.448680
\(767\) 2.28192e9 0.182607
\(768\) 7.28005e9 0.579923
\(769\) −1.81575e10 −1.43984 −0.719921 0.694056i \(-0.755822\pi\)
−0.719921 + 0.694056i \(0.755822\pi\)
\(770\) −3.73792e9 −0.295062
\(771\) 4.23791e9 0.333013
\(772\) −2.19071e9 −0.171366
\(773\) 1.26937e10 0.988459 0.494230 0.869331i \(-0.335450\pi\)
0.494230 + 0.869331i \(0.335450\pi\)
\(774\) −3.07100e9 −0.238060
\(775\) 3.90993e8 0.0301726
\(776\) −2.13546e10 −1.64050
\(777\) 6.75288e9 0.516435
\(778\) −6.00682e9 −0.457316
\(779\) −4.24157e9 −0.321474
\(780\) −6.29737e9 −0.475147
\(781\) 1.29163e10 0.970196
\(782\) −1.03242e10 −0.772030
\(783\) −1.17550e9 −0.0875096
\(784\) 4.57121e7 0.00338786
\(785\) 1.41949e10 1.04734
\(786\) −8.22035e9 −0.603826
\(787\) −2.24532e10 −1.64198 −0.820988 0.570946i \(-0.806577\pi\)
−0.820988 + 0.570946i \(0.806577\pi\)
\(788\) −7.06477e9 −0.514347
\(789\) −6.53938e9 −0.473988
\(790\) 6.61782e9 0.477551
\(791\) −5.41234e8 −0.0388837
\(792\) −3.17113e9 −0.226817
\(793\) 1.20577e10 0.858635
\(794\) 2.27175e9 0.161060
\(795\) 6.85081e8 0.0483567
\(796\) −1.55216e9 −0.109079
\(797\) 4.23290e9 0.296165 0.148083 0.988975i \(-0.452690\pi\)
0.148083 + 0.988975i \(0.452690\pi\)
\(798\) 9.55536e8 0.0665636
\(799\) −8.81624e9 −0.611462
\(800\) −1.28660e9 −0.0888438
\(801\) −4.25598e9 −0.292608
\(802\) −1.51357e9 −0.103608
\(803\) −2.41111e9 −0.164328
\(804\) −1.55317e9 −0.105395
\(805\) 1.50677e10 1.01803
\(806\) −4.38504e9 −0.294986
\(807\) −1.53971e9 −0.103129
\(808\) 7.05367e9 0.470409
\(809\) −5.59705e9 −0.371654 −0.185827 0.982582i \(-0.559496\pi\)
−0.185827 + 0.982582i \(0.559496\pi\)
\(810\) −9.95577e8 −0.0658230
\(811\) 2.20508e10 1.45162 0.725808 0.687897i \(-0.241466\pi\)
0.725808 + 0.687897i \(0.241466\pi\)
\(812\) −3.12863e9 −0.205073
\(813\) −6.94535e9 −0.453291
\(814\) −7.90643e9 −0.513801
\(815\) −1.86270e10 −1.20529
\(816\) 5.62406e7 0.00362355
\(817\) 4.54114e9 0.291332
\(818\) 1.45746e10 0.931020
\(819\) 5.39249e9 0.343001
\(820\) 1.17622e10 0.744969
\(821\) 1.92550e10 1.21435 0.607173 0.794569i \(-0.292303\pi\)
0.607173 + 0.794569i \(0.292303\pi\)
\(822\) 6.28628e9 0.394768
\(823\) −2.63648e10 −1.64864 −0.824318 0.566127i \(-0.808441\pi\)
−0.824318 + 0.566127i \(0.808441\pi\)
\(824\) 6.72729e9 0.418885
\(825\) 5.62957e8 0.0349049
\(826\) 9.60173e8 0.0592815
\(827\) 2.32586e10 1.42993 0.714965 0.699160i \(-0.246443\pi\)
0.714965 + 0.699160i \(0.246443\pi\)
\(828\) 4.86657e9 0.297932
\(829\) 1.16870e10 0.712464 0.356232 0.934398i \(-0.384061\pi\)
0.356232 + 0.934398i \(0.384061\pi\)
\(830\) 1.66326e10 1.00969
\(831\) 1.68094e10 1.01613
\(832\) 1.46003e10 0.878881
\(833\) −6.59067e9 −0.395068
\(834\) 9.05124e9 0.540290
\(835\) −2.23132e10 −1.32635
\(836\) 1.78522e9 0.105674
\(837\) 1.10622e9 0.0652084
\(838\) −1.72879e10 −1.01482
\(839\) 3.29655e10 1.92705 0.963524 0.267623i \(-0.0862383\pi\)
0.963524 + 0.267623i \(0.0862383\pi\)
\(840\) −6.96009e9 −0.405170
\(841\) −1.36832e10 −0.793236
\(842\) 1.51183e10 0.872791
\(843\) −8.11260e9 −0.466405
\(844\) −1.25601e10 −0.719109
\(845\) −1.61935e10 −0.923299
\(846\) −2.60433e9 −0.147877
\(847\) −6.99369e9 −0.395471
\(848\) −1.14322e7 −0.000643790 0
\(849\) 1.18190e10 0.662835
\(850\) −8.46617e8 −0.0472847
\(851\) 3.18711e10 1.77274
\(852\) 9.15619e9 0.507197
\(853\) −6.75557e9 −0.372684 −0.186342 0.982485i \(-0.559663\pi\)
−0.186342 + 0.982485i \(0.559663\pi\)
\(854\) 5.07356e9 0.278747
\(855\) 1.47218e9 0.0805524
\(856\) 2.26336e10 1.23337
\(857\) 3.29149e10 1.78632 0.893162 0.449735i \(-0.148482\pi\)
0.893162 + 0.449735i \(0.148482\pi\)
\(858\) −6.31365e9 −0.341252
\(859\) 1.99567e10 1.07427 0.537135 0.843496i \(-0.319506\pi\)
0.537135 + 0.843496i \(0.319506\pi\)
\(860\) −1.25929e10 −0.675119
\(861\) −1.00720e10 −0.537782
\(862\) −1.93636e10 −1.02970
\(863\) 1.08133e10 0.572689 0.286344 0.958127i \(-0.407560\pi\)
0.286344 + 0.958127i \(0.407560\pi\)
\(864\) −3.64012e9 −0.192007
\(865\) 6.47257e9 0.340033
\(866\) 8.59828e9 0.449882
\(867\) 2.97050e9 0.154797
\(868\) 2.94425e9 0.152811
\(869\) −1.05874e10 −0.547293
\(870\) 3.02075e9 0.155524
\(871\) −8.12255e9 −0.416513
\(872\) 8.05445e8 0.0411366
\(873\) 1.07257e10 0.545603
\(874\) 4.50978e9 0.228489
\(875\) 1.51111e10 0.762549
\(876\) −1.70921e9 −0.0859073
\(877\) 3.04464e10 1.52418 0.762091 0.647470i \(-0.224173\pi\)
0.762091 + 0.647470i \(0.224173\pi\)
\(878\) −1.26760e10 −0.632051
\(879\) 1.06484e10 0.528839
\(880\) 9.61016e7 0.00475380
\(881\) −1.83036e10 −0.901821 −0.450910 0.892569i \(-0.648901\pi\)
−0.450910 + 0.892569i \(0.648901\pi\)
\(882\) −1.94689e9 −0.0955437
\(883\) −2.58285e10 −1.26251 −0.631257 0.775574i \(-0.717461\pi\)
−0.631257 + 0.775574i \(0.717461\pi\)
\(884\) −1.51511e10 −0.737669
\(885\) 1.47932e9 0.0717399
\(886\) 1.83262e10 0.885227
\(887\) −1.56417e10 −0.752578 −0.376289 0.926502i \(-0.622800\pi\)
−0.376289 + 0.926502i \(0.622800\pi\)
\(888\) −1.47219e10 −0.705536
\(889\) −2.65684e10 −1.26826
\(890\) 1.09368e10 0.520028
\(891\) 1.59276e9 0.0754357
\(892\) 9.60991e9 0.453359
\(893\) 3.85107e9 0.180968
\(894\) 1.50021e10 0.702214
\(895\) 3.00273e9 0.140003
\(896\) −9.61638e9 −0.446615
\(897\) 2.54506e10 1.17740
\(898\) 1.43528e10 0.661410
\(899\) −3.35647e9 −0.154072
\(900\) 3.99073e8 0.0182475
\(901\) 1.64827e9 0.0750743
\(902\) 1.17926e10 0.535039
\(903\) 1.07834e10 0.487358
\(904\) 1.17994e9 0.0531216
\(905\) −2.97873e9 −0.133586
\(906\) 9.95579e9 0.444761
\(907\) 3.32082e10 1.47781 0.738906 0.673808i \(-0.235343\pi\)
0.738906 + 0.673808i \(0.235343\pi\)
\(908\) 8.32213e8 0.0368922
\(909\) −3.54283e9 −0.156451
\(910\) −1.38574e10 −0.609588
\(911\) −6.12880e9 −0.268572 −0.134286 0.990943i \(-0.542874\pi\)
−0.134286 + 0.990943i \(0.542874\pi\)
\(912\) −2.45667e7 −0.00107242
\(913\) −2.66094e10 −1.15715
\(914\) −7.93576e9 −0.343777
\(915\) 7.81675e9 0.337328
\(916\) −4.28764e9 −0.184325
\(917\) 2.88646e10 1.23616
\(918\) −2.39530e9 −0.102191
\(919\) 1.60841e10 0.683585 0.341793 0.939775i \(-0.388966\pi\)
0.341793 + 0.939775i \(0.388966\pi\)
\(920\) −3.28491e10 −1.39080
\(921\) 7.43494e9 0.313595
\(922\) 7.32974e9 0.307986
\(923\) 4.78839e10 2.00439
\(924\) 4.23918e9 0.176779
\(925\) 2.61353e9 0.108575
\(926\) 2.71900e9 0.112531
\(927\) −3.37890e9 −0.139315
\(928\) 1.10447e10 0.453667
\(929\) −1.02473e10 −0.419327 −0.209664 0.977774i \(-0.567237\pi\)
−0.209664 + 0.977774i \(0.567237\pi\)
\(930\) −2.84273e9 −0.115890
\(931\) 2.87890e9 0.116924
\(932\) −8.13579e9 −0.329188
\(933\) 7.60679e9 0.306631
\(934\) 2.11607e9 0.0849798
\(935\) −1.38557e10 −0.554355
\(936\) −1.17561e10 −0.468597
\(937\) −1.95793e10 −0.777516 −0.388758 0.921340i \(-0.627096\pi\)
−0.388758 + 0.921340i \(0.627096\pi\)
\(938\) −3.41775e9 −0.135217
\(939\) 2.17010e10 0.855364
\(940\) −1.06793e10 −0.419367
\(941\) 7.05944e8 0.0276189 0.0138095 0.999905i \(-0.495604\pi\)
0.0138095 + 0.999905i \(0.495604\pi\)
\(942\) 1.00886e10 0.393236
\(943\) −4.75363e10 −1.84601
\(944\) −2.46860e7 −0.000955098 0
\(945\) 3.49583e9 0.134753
\(946\) −1.26254e10 −0.484872
\(947\) 1.93347e10 0.739798 0.369899 0.929072i \(-0.379392\pi\)
0.369899 + 0.929072i \(0.379392\pi\)
\(948\) −7.50526e9 −0.286112
\(949\) −8.93858e9 −0.339498
\(950\) 3.69815e8 0.0139943
\(951\) −5.99948e9 −0.226194
\(952\) −1.67456e10 −0.629030
\(953\) −2.96357e10 −1.10915 −0.554574 0.832134i \(-0.687119\pi\)
−0.554574 + 0.832134i \(0.687119\pi\)
\(954\) 4.86901e8 0.0181560
\(955\) 9.58859e9 0.356240
\(956\) 7.23747e9 0.267907
\(957\) −4.83269e9 −0.178237
\(958\) −3.06464e10 −1.12616
\(959\) −2.20734e10 −0.808172
\(960\) 9.46505e9 0.345282
\(961\) −2.43540e10 −0.885192
\(962\) −2.93111e10 −1.06150
\(963\) −1.13681e10 −0.410201
\(964\) 2.54075e10 0.913465
\(965\) −7.42712e9 −0.266057
\(966\) 1.07089e10 0.382231
\(967\) −2.65671e10 −0.944824 −0.472412 0.881378i \(-0.656617\pi\)
−0.472412 + 0.881378i \(0.656617\pi\)
\(968\) 1.52469e10 0.540280
\(969\) 3.54198e9 0.125058
\(970\) −2.75625e10 −0.969657
\(971\) −1.88208e10 −0.659736 −0.329868 0.944027i \(-0.607004\pi\)
−0.329868 + 0.944027i \(0.607004\pi\)
\(972\) 1.12908e9 0.0394361
\(973\) −3.17821e10 −1.10608
\(974\) 2.76659e10 0.959376
\(975\) 2.08702e9 0.0721125
\(976\) −1.30441e8 −0.00449096
\(977\) 1.00691e10 0.345430 0.172715 0.984972i \(-0.444746\pi\)
0.172715 + 0.984972i \(0.444746\pi\)
\(978\) −1.32386e10 −0.452537
\(979\) −1.74971e10 −0.595973
\(980\) −7.98338e9 −0.270954
\(981\) −4.04549e8 −0.0136814
\(982\) −1.37127e10 −0.462098
\(983\) −4.43932e10 −1.49066 −0.745332 0.666694i \(-0.767709\pi\)
−0.745332 + 0.666694i \(0.767709\pi\)
\(984\) 2.19580e10 0.734700
\(985\) −2.39515e10 −0.798558
\(986\) 7.26775e9 0.241452
\(987\) 9.14474e9 0.302734
\(988\) 6.61824e9 0.218320
\(989\) 5.08936e10 1.67292
\(990\) −4.09300e9 −0.134066
\(991\) 1.29610e10 0.423040 0.211520 0.977374i \(-0.432159\pi\)
0.211520 + 0.977374i \(0.432159\pi\)
\(992\) −1.03938e10 −0.338053
\(993\) 7.71963e9 0.250193
\(994\) 2.01483e10 0.650707
\(995\) −5.26225e9 −0.169352
\(996\) −1.88631e10 −0.604930
\(997\) −5.75699e10 −1.83977 −0.919883 0.392194i \(-0.871716\pi\)
−0.919883 + 0.392194i \(0.871716\pi\)
\(998\) 1.25211e10 0.398737
\(999\) 7.39436e9 0.234650
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.1.11 16
3.2 odd 2 531.8.a.b.1.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.11 16 1.1 even 1 trivial
531.8.a.b.1.6 16 3.2 odd 2