Properties

Label 177.8.a.a.1.9
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 1.9
Root \(1.05882\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.05882 q^{2} -27.0000 q^{3} -126.879 q^{4} +151.597 q^{5} +28.5880 q^{6} -1574.54 q^{7} +269.870 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-1.05882 q^{2} -27.0000 q^{3} -126.879 q^{4} +151.597 q^{5} +28.5880 q^{6} -1574.54 q^{7} +269.870 q^{8} +729.000 q^{9} -160.513 q^{10} +1597.23 q^{11} +3425.73 q^{12} +8756.36 q^{13} +1667.15 q^{14} -4093.12 q^{15} +15954.8 q^{16} +19841.7 q^{17} -771.876 q^{18} -4839.26 q^{19} -19234.5 q^{20} +42512.6 q^{21} -1691.17 q^{22} -30741.2 q^{23} -7286.48 q^{24} -55143.3 q^{25} -9271.37 q^{26} -19683.0 q^{27} +199776. q^{28} +14740.2 q^{29} +4333.86 q^{30} +72269.3 q^{31} -51436.5 q^{32} -43125.1 q^{33} -21008.7 q^{34} -238696. q^{35} -92494.7 q^{36} +357556. q^{37} +5123.88 q^{38} -236422. q^{39} +40911.4 q^{40} +562020. q^{41} -45013.0 q^{42} +682037. q^{43} -202654. q^{44} +110514. q^{45} +32549.3 q^{46} -1.24121e6 q^{47} -430778. q^{48} +1.65563e6 q^{49} +58386.6 q^{50} -535726. q^{51} -1.11100e6 q^{52} -1.94069e6 q^{53} +20840.7 q^{54} +242135. q^{55} -424921. q^{56} +130660. q^{57} -15607.1 q^{58} +205379. q^{59} +519331. q^{60} -2.48399e6 q^{61} -76519.8 q^{62} -1.14784e6 q^{63} -1.98775e6 q^{64} +1.32744e6 q^{65} +45661.5 q^{66} -1.19534e6 q^{67} -2.51749e6 q^{68} +830013. q^{69} +252735. q^{70} -5.50733e6 q^{71} +196735. q^{72} -1.03549e6 q^{73} -378585. q^{74} +1.48887e6 q^{75} +614000. q^{76} -2.51490e6 q^{77} +250327. q^{78} -1.05062e6 q^{79} +2.41869e6 q^{80} +531441. q^{81} -595075. q^{82} +8.29947e6 q^{83} -5.39395e6 q^{84} +3.00794e6 q^{85} -722151. q^{86} -397985. q^{87} +431043. q^{88} +1.51623e6 q^{89} -117014. q^{90} -1.37872e7 q^{91} +3.90041e6 q^{92} -1.95127e6 q^{93} +1.31421e6 q^{94} -733618. q^{95} +1.38878e6 q^{96} +1.02696e7 q^{97} -1.75301e6 q^{98} +1.16438e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} + O(q^{10}) \) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} - 3479q^{10} + 898q^{11} - 26298q^{12} - 8172q^{13} - 13315q^{14} + 1836q^{15} + 3138q^{16} - 44985q^{17} - 4374q^{18} - 40137q^{19} + 130657q^{20} + 63261q^{21} + 109394q^{22} - 2833q^{23} - 22113q^{24} + 285746q^{25} - 129420q^{26} - 314928q^{27} + 112890q^{28} + 144375q^{29} + 93933q^{30} - 141759q^{31} - 36224q^{32} - 24246q^{33} - 341332q^{34} - 78859q^{35} + 710046q^{36} - 297971q^{37} + 329075q^{38} + 220644q^{39} - 203048q^{40} + 659077q^{41} + 359505q^{42} - 1431608q^{43} + 254916q^{44} - 49572q^{45} + 873113q^{46} - 1574073q^{47} - 84726q^{48} + 1893545q^{49} + 302533q^{50} + 1214595q^{51} - 4972548q^{52} + 587736q^{53} + 118098q^{54} - 4624036q^{55} - 5798506q^{56} + 1083699q^{57} - 6991380q^{58} + 3286064q^{59} - 3527739q^{60} - 6117131q^{61} - 11570258q^{62} - 1708047q^{63} - 19063011q^{64} - 5335514q^{65} - 2953638q^{66} - 16518710q^{67} - 17284669q^{68} + 76491q^{69} - 39189486q^{70} - 10882582q^{71} + 597051q^{72} - 21097441q^{73} - 16717030q^{74} - 7715142q^{75} - 40864952q^{76} - 3404601q^{77} + 3494340q^{78} - 3784458q^{79} - 27466195q^{80} + 8503056q^{81} - 24990117q^{82} - 1951425q^{83} - 3048030q^{84} - 23238675q^{85} - 35910572q^{86} - 3898125q^{87} - 27843055q^{88} + 10499443q^{89} - 2536191q^{90} + 699217q^{91} - 20062766q^{92} + 3827493q^{93} - 59358988q^{94} - 29236333q^{95} + 978048q^{96} - 25158976q^{97} + 2120460q^{98} + 654642q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.05882 −0.0935869 −0.0467935 0.998905i \(-0.514900\pi\)
−0.0467935 + 0.998905i \(0.514900\pi\)
\(3\) −27.0000 −0.577350
\(4\) −126.879 −0.991241
\(5\) 151.597 0.542370 0.271185 0.962527i \(-0.412584\pi\)
0.271185 + 0.962527i \(0.412584\pi\)
\(6\) 28.5880 0.0540324
\(7\) −1574.54 −1.73504 −0.867522 0.497398i \(-0.834289\pi\)
−0.867522 + 0.497398i \(0.834289\pi\)
\(8\) 269.870 0.186354
\(9\) 729.000 0.333333
\(10\) −160.513 −0.0507587
\(11\) 1597.23 0.361819 0.180910 0.983500i \(-0.442096\pi\)
0.180910 + 0.983500i \(0.442096\pi\)
\(12\) 3425.73 0.572294
\(13\) 8756.36 1.10541 0.552703 0.833378i \(-0.313596\pi\)
0.552703 + 0.833378i \(0.313596\pi\)
\(14\) 1667.15 0.162378
\(15\) −4093.12 −0.313138
\(16\) 15954.8 0.973801
\(17\) 19841.7 0.979507 0.489754 0.871861i \(-0.337087\pi\)
0.489754 + 0.871861i \(0.337087\pi\)
\(18\) −771.876 −0.0311956
\(19\) −4839.26 −0.161861 −0.0809304 0.996720i \(-0.525789\pi\)
−0.0809304 + 0.996720i \(0.525789\pi\)
\(20\) −19234.5 −0.537620
\(21\) 42512.6 1.00173
\(22\) −1691.17 −0.0338616
\(23\) −30741.2 −0.526834 −0.263417 0.964682i \(-0.584849\pi\)
−0.263417 + 0.964682i \(0.584849\pi\)
\(24\) −7286.48 −0.107592
\(25\) −55143.3 −0.705835
\(26\) −9271.37 −0.103452
\(27\) −19683.0 −0.192450
\(28\) 199776. 1.71985
\(29\) 14740.2 0.112230 0.0561152 0.998424i \(-0.482129\pi\)
0.0561152 + 0.998424i \(0.482129\pi\)
\(30\) 4333.86 0.0293056
\(31\) 72269.3 0.435700 0.217850 0.975982i \(-0.430096\pi\)
0.217850 + 0.975982i \(0.430096\pi\)
\(32\) −51436.5 −0.277489
\(33\) −43125.1 −0.208897
\(34\) −21008.7 −0.0916691
\(35\) −238696. −0.941036
\(36\) −92494.7 −0.330414
\(37\) 357556. 1.16048 0.580240 0.814445i \(-0.302959\pi\)
0.580240 + 0.814445i \(0.302959\pi\)
\(38\) 5123.88 0.0151480
\(39\) −236422. −0.638207
\(40\) 40911.4 0.101073
\(41\) 562020. 1.27353 0.636764 0.771059i \(-0.280273\pi\)
0.636764 + 0.771059i \(0.280273\pi\)
\(42\) −45013.0 −0.0937487
\(43\) 682037. 1.30818 0.654092 0.756415i \(-0.273051\pi\)
0.654092 + 0.756415i \(0.273051\pi\)
\(44\) −202654. −0.358650
\(45\) 110514. 0.180790
\(46\) 32549.3 0.0493047
\(47\) −1.24121e6 −1.74382 −0.871911 0.489665i \(-0.837119\pi\)
−0.871911 + 0.489665i \(0.837119\pi\)
\(48\) −430778. −0.562224
\(49\) 1.65563e6 2.01038
\(50\) 58386.6 0.0660569
\(51\) −535726. −0.565519
\(52\) −1.11100e6 −1.09572
\(53\) −1.94069e6 −1.79057 −0.895286 0.445493i \(-0.853028\pi\)
−0.895286 + 0.445493i \(0.853028\pi\)
\(54\) 20840.7 0.0180108
\(55\) 242135. 0.196240
\(56\) −424921. −0.323333
\(57\) 130660. 0.0934503
\(58\) −15607.1 −0.0105033
\(59\) 205379. 0.130189
\(60\) 519331. 0.310395
\(61\) −2.48399e6 −1.40119 −0.700593 0.713561i \(-0.747081\pi\)
−0.700593 + 0.713561i \(0.747081\pi\)
\(62\) −76519.8 −0.0407758
\(63\) −1.14784e6 −0.578348
\(64\) −1.98775e6 −0.947832
\(65\) 1.32744e6 0.599539
\(66\) 45661.5 0.0195500
\(67\) −1.19534e6 −0.485546 −0.242773 0.970083i \(-0.578057\pi\)
−0.242773 + 0.970083i \(0.578057\pi\)
\(68\) −2.51749e6 −0.970928
\(69\) 830013. 0.304168
\(70\) 252735. 0.0880687
\(71\) −5.50733e6 −1.82615 −0.913076 0.407789i \(-0.866300\pi\)
−0.913076 + 0.407789i \(0.866300\pi\)
\(72\) 196735. 0.0621181
\(73\) −1.03549e6 −0.311542 −0.155771 0.987793i \(-0.549786\pi\)
−0.155771 + 0.987793i \(0.549786\pi\)
\(74\) −378585. −0.108606
\(75\) 1.48887e6 0.407514
\(76\) 614000. 0.160443
\(77\) −2.51490e6 −0.627773
\(78\) 250327. 0.0597278
\(79\) −1.05062e6 −0.239745 −0.119873 0.992789i \(-0.538249\pi\)
−0.119873 + 0.992789i \(0.538249\pi\)
\(80\) 2.41869e6 0.528161
\(81\) 531441. 0.111111
\(82\) −595075. −0.119185
\(83\) 8.29947e6 1.59323 0.796613 0.604490i \(-0.206623\pi\)
0.796613 + 0.604490i \(0.206623\pi\)
\(84\) −5.39395e6 −0.992955
\(85\) 3.00794e6 0.531256
\(86\) −722151. −0.122429
\(87\) −397985. −0.0647962
\(88\) 431043. 0.0674266
\(89\) 1.51623e6 0.227982 0.113991 0.993482i \(-0.463637\pi\)
0.113991 + 0.993482i \(0.463637\pi\)
\(90\) −117014. −0.0169196
\(91\) −1.37872e7 −1.91793
\(92\) 3.90041e6 0.522219
\(93\) −1.95127e6 −0.251552
\(94\) 1.31421e6 0.163199
\(95\) −733618. −0.0877884
\(96\) 1.38878e6 0.160208
\(97\) 1.02696e7 1.14249 0.571247 0.820778i \(-0.306460\pi\)
0.571247 + 0.820778i \(0.306460\pi\)
\(98\) −1.75301e6 −0.188145
\(99\) 1.16438e6 0.120606
\(100\) 6.99653e6 0.699653
\(101\) −1.21422e7 −1.17267 −0.586333 0.810070i \(-0.699429\pi\)
−0.586333 + 0.810070i \(0.699429\pi\)
\(102\) 567235. 0.0529252
\(103\) −2.11994e7 −1.91159 −0.955793 0.294041i \(-0.905000\pi\)
−0.955793 + 0.294041i \(0.905000\pi\)
\(104\) 2.36308e6 0.205997
\(105\) 6.44478e6 0.543308
\(106\) 2.05484e6 0.167574
\(107\) 58058.2 0.00458163 0.00229082 0.999997i \(-0.499271\pi\)
0.00229082 + 0.999997i \(0.499271\pi\)
\(108\) 2.49736e6 0.190765
\(109\) 1.06543e7 0.788011 0.394005 0.919108i \(-0.371089\pi\)
0.394005 + 0.919108i \(0.371089\pi\)
\(110\) −256376. −0.0183655
\(111\) −9.65401e6 −0.670004
\(112\) −2.51214e7 −1.68959
\(113\) −4.86803e6 −0.317379 −0.158690 0.987329i \(-0.550727\pi\)
−0.158690 + 0.987329i \(0.550727\pi\)
\(114\) −138345. −0.00874573
\(115\) −4.66028e6 −0.285739
\(116\) −1.87022e6 −0.111247
\(117\) 6.38339e6 0.368469
\(118\) −217458. −0.0121840
\(119\) −3.12416e7 −1.69949
\(120\) −1.10461e6 −0.0583545
\(121\) −1.69360e7 −0.869087
\(122\) 2.63009e6 0.131133
\(123\) −1.51745e7 −0.735271
\(124\) −9.16945e6 −0.431884
\(125\) −2.02031e7 −0.925194
\(126\) 1.21535e6 0.0541258
\(127\) −3.32597e7 −1.44081 −0.720403 0.693556i \(-0.756043\pi\)
−0.720403 + 0.693556i \(0.756043\pi\)
\(128\) 8.68852e6 0.366194
\(129\) −1.84150e7 −0.755280
\(130\) −1.40551e6 −0.0561090
\(131\) 3.81285e6 0.148184 0.0740918 0.997251i \(-0.476394\pi\)
0.0740918 + 0.997251i \(0.476394\pi\)
\(132\) 5.47166e6 0.207067
\(133\) 7.61961e6 0.280836
\(134\) 1.26565e6 0.0454407
\(135\) −2.98388e6 −0.104379
\(136\) 5.35467e6 0.182535
\(137\) 4.60178e7 1.52899 0.764494 0.644630i \(-0.222989\pi\)
0.764494 + 0.644630i \(0.222989\pi\)
\(138\) −878830. −0.0284661
\(139\) 4.52736e7 1.42986 0.714930 0.699196i \(-0.246458\pi\)
0.714930 + 0.699196i \(0.246458\pi\)
\(140\) 3.02855e7 0.932794
\(141\) 3.35126e7 1.00680
\(142\) 5.83125e6 0.170904
\(143\) 1.39859e7 0.399957
\(144\) 1.16310e7 0.324600
\(145\) 2.23457e6 0.0608704
\(146\) 1.09639e6 0.0291562
\(147\) −4.47021e7 −1.16069
\(148\) −4.53663e7 −1.15032
\(149\) 2.55791e7 0.633481 0.316740 0.948512i \(-0.397412\pi\)
0.316740 + 0.948512i \(0.397412\pi\)
\(150\) −1.57644e6 −0.0381380
\(151\) 2.20103e7 0.520243 0.260121 0.965576i \(-0.416237\pi\)
0.260121 + 0.965576i \(0.416237\pi\)
\(152\) −1.30597e6 −0.0301634
\(153\) 1.44646e7 0.326502
\(154\) 2.66281e6 0.0587513
\(155\) 1.09558e7 0.236311
\(156\) 2.99969e7 0.632617
\(157\) 2.45776e7 0.506862 0.253431 0.967353i \(-0.418441\pi\)
0.253431 + 0.967353i \(0.418441\pi\)
\(158\) 1.11241e6 0.0224370
\(159\) 5.23987e7 1.03379
\(160\) −7.79762e6 −0.150502
\(161\) 4.84033e7 0.914080
\(162\) −562698. −0.0103985
\(163\) −7.12800e7 −1.28917 −0.644586 0.764532i \(-0.722970\pi\)
−0.644586 + 0.764532i \(0.722970\pi\)
\(164\) −7.13085e7 −1.26237
\(165\) −6.53764e6 −0.113299
\(166\) −8.78761e6 −0.149105
\(167\) 1.72745e7 0.287010 0.143505 0.989650i \(-0.454163\pi\)
0.143505 + 0.989650i \(0.454163\pi\)
\(168\) 1.14729e7 0.186676
\(169\) 1.39253e7 0.221923
\(170\) −3.18486e6 −0.0497186
\(171\) −3.52782e6 −0.0539536
\(172\) −8.65361e7 −1.29673
\(173\) −4.27240e7 −0.627351 −0.313676 0.949530i \(-0.601560\pi\)
−0.313676 + 0.949530i \(0.601560\pi\)
\(174\) 421393. 0.00606408
\(175\) 8.68254e7 1.22465
\(176\) 2.54833e7 0.352340
\(177\) −5.54523e6 −0.0751646
\(178\) −1.60541e6 −0.0213361
\(179\) 5.80252e7 0.756191 0.378095 0.925767i \(-0.376579\pi\)
0.378095 + 0.925767i \(0.376579\pi\)
\(180\) −1.40219e7 −0.179207
\(181\) −4.33616e7 −0.543539 −0.271769 0.962362i \(-0.587609\pi\)
−0.271769 + 0.962362i \(0.587609\pi\)
\(182\) 1.45981e7 0.179493
\(183\) 6.70678e7 0.808976
\(184\) −8.29612e6 −0.0981777
\(185\) 5.42044e7 0.629410
\(186\) 2.06603e6 0.0235419
\(187\) 3.16917e7 0.354405
\(188\) 1.57483e8 1.72855
\(189\) 3.09917e7 0.333910
\(190\) 776766. 0.00821585
\(191\) −6.93621e7 −0.720287 −0.360143 0.932897i \(-0.617272\pi\)
−0.360143 + 0.932897i \(0.617272\pi\)
\(192\) 5.36692e7 0.547231
\(193\) 3.17140e7 0.317541 0.158771 0.987315i \(-0.449247\pi\)
0.158771 + 0.987315i \(0.449247\pi\)
\(194\) −1.08736e7 −0.106922
\(195\) −3.58408e7 −0.346144
\(196\) −2.10065e8 −1.99277
\(197\) −1.33959e8 −1.24836 −0.624178 0.781282i \(-0.714566\pi\)
−0.624178 + 0.781282i \(0.714566\pi\)
\(198\) −1.23286e6 −0.0112872
\(199\) 7.57957e7 0.681803 0.340902 0.940099i \(-0.389268\pi\)
0.340902 + 0.940099i \(0.389268\pi\)
\(200\) −1.48815e7 −0.131535
\(201\) 3.22742e7 0.280330
\(202\) 1.28564e7 0.109746
\(203\) −2.32090e7 −0.194725
\(204\) 6.79723e7 0.560566
\(205\) 8.52006e7 0.690723
\(206\) 2.24463e7 0.178899
\(207\) −2.24103e7 −0.175611
\(208\) 1.39706e8 1.07645
\(209\) −7.72939e6 −0.0585644
\(210\) −6.82384e6 −0.0508465
\(211\) −2.11972e8 −1.55343 −0.776713 0.629854i \(-0.783115\pi\)
−0.776713 + 0.629854i \(0.783115\pi\)
\(212\) 2.46233e8 1.77489
\(213\) 1.48698e8 1.05433
\(214\) −61472.9 −0.000428781 0
\(215\) 1.03395e8 0.709519
\(216\) −5.31184e6 −0.0358639
\(217\) −1.13791e8 −0.755959
\(218\) −1.12809e7 −0.0737475
\(219\) 2.79583e7 0.179869
\(220\) −3.07218e7 −0.194521
\(221\) 1.73741e8 1.08275
\(222\) 1.02218e7 0.0627036
\(223\) −1.80114e8 −1.08763 −0.543814 0.839206i \(-0.683020\pi\)
−0.543814 + 0.839206i \(0.683020\pi\)
\(224\) 8.09888e7 0.481456
\(225\) −4.01995e7 −0.235278
\(226\) 5.15434e6 0.0297025
\(227\) 7.73940e7 0.439154 0.219577 0.975595i \(-0.429532\pi\)
0.219577 + 0.975595i \(0.429532\pi\)
\(228\) −1.65780e7 −0.0926318
\(229\) 1.39410e8 0.767134 0.383567 0.923513i \(-0.374696\pi\)
0.383567 + 0.923513i \(0.374696\pi\)
\(230\) 4.93437e6 0.0267414
\(231\) 6.79022e7 0.362445
\(232\) 3.97793e6 0.0209146
\(233\) 1.24112e8 0.642787 0.321393 0.946946i \(-0.395849\pi\)
0.321393 + 0.946946i \(0.395849\pi\)
\(234\) −6.75883e6 −0.0344839
\(235\) −1.88163e8 −0.945796
\(236\) −2.60583e7 −0.129049
\(237\) 2.83667e7 0.138417
\(238\) 3.30790e7 0.159050
\(239\) −1.55356e8 −0.736096 −0.368048 0.929807i \(-0.619974\pi\)
−0.368048 + 0.929807i \(0.619974\pi\)
\(240\) −6.53047e7 −0.304934
\(241\) 3.44462e8 1.58519 0.792597 0.609746i \(-0.208729\pi\)
0.792597 + 0.609746i \(0.208729\pi\)
\(242\) 1.79321e7 0.0813352
\(243\) −1.43489e7 −0.0641500
\(244\) 3.15166e8 1.38891
\(245\) 2.50989e8 1.09037
\(246\) 1.60670e7 0.0688118
\(247\) −4.23743e7 −0.178922
\(248\) 1.95033e7 0.0811945
\(249\) −2.24086e8 −0.919849
\(250\) 2.13913e7 0.0865860
\(251\) −4.67431e8 −1.86578 −0.932888 0.360166i \(-0.882720\pi\)
−0.932888 + 0.360166i \(0.882720\pi\)
\(252\) 1.45637e8 0.573283
\(253\) −4.91007e7 −0.190619
\(254\) 3.52159e7 0.134841
\(255\) −8.12145e7 −0.306721
\(256\) 2.45232e8 0.913561
\(257\) −4.09247e8 −1.50390 −0.751952 0.659218i \(-0.770887\pi\)
−0.751952 + 0.659218i \(0.770887\pi\)
\(258\) 1.94981e7 0.0706843
\(259\) −5.62986e8 −2.01349
\(260\) −1.68424e8 −0.594288
\(261\) 1.07456e7 0.0374101
\(262\) −4.03710e6 −0.0138680
\(263\) 1.04382e8 0.353818 0.176909 0.984227i \(-0.443390\pi\)
0.176909 + 0.984227i \(0.443390\pi\)
\(264\) −1.16382e7 −0.0389287
\(265\) −2.94204e8 −0.971152
\(266\) −8.06776e6 −0.0262825
\(267\) −4.09382e7 −0.131625
\(268\) 1.51664e8 0.481293
\(269\) −1.95159e8 −0.611302 −0.305651 0.952144i \(-0.598874\pi\)
−0.305651 + 0.952144i \(0.598874\pi\)
\(270\) 3.15938e6 0.00976853
\(271\) 3.63467e8 1.10936 0.554681 0.832063i \(-0.312840\pi\)
0.554681 + 0.832063i \(0.312840\pi\)
\(272\) 3.16570e8 0.953845
\(273\) 3.72256e8 1.10732
\(274\) −4.87244e7 −0.143093
\(275\) −8.80763e7 −0.255385
\(276\) −1.05311e8 −0.301504
\(277\) −3.76920e8 −1.06554 −0.532771 0.846260i \(-0.678849\pi\)
−0.532771 + 0.846260i \(0.678849\pi\)
\(278\) −4.79364e7 −0.133816
\(279\) 5.26843e7 0.145233
\(280\) −6.44167e7 −0.175366
\(281\) 3.38642e8 0.910475 0.455238 0.890370i \(-0.349554\pi\)
0.455238 + 0.890370i \(0.349554\pi\)
\(282\) −3.54837e7 −0.0942229
\(283\) −2.79439e8 −0.732882 −0.366441 0.930441i \(-0.619424\pi\)
−0.366441 + 0.930441i \(0.619424\pi\)
\(284\) 6.98764e8 1.81016
\(285\) 1.98077e7 0.0506847
\(286\) −1.48085e7 −0.0374308
\(287\) −8.84923e8 −2.20963
\(288\) −3.74972e7 −0.0924964
\(289\) −1.66455e7 −0.0405653
\(290\) −2.36600e6 −0.00569667
\(291\) −2.77280e8 −0.659619
\(292\) 1.31382e8 0.308813
\(293\) 6.25978e8 1.45386 0.726930 0.686712i \(-0.240946\pi\)
0.726930 + 0.686712i \(0.240946\pi\)
\(294\) 4.73313e7 0.108626
\(295\) 3.11349e7 0.0706106
\(296\) 9.64935e7 0.216260
\(297\) −3.14382e7 −0.0696322
\(298\) −2.70836e7 −0.0592855
\(299\) −2.69181e8 −0.582365
\(300\) −1.88906e8 −0.403945
\(301\) −1.07390e9 −2.26976
\(302\) −2.33048e7 −0.0486879
\(303\) 3.27841e8 0.677039
\(304\) −7.72092e7 −0.157620
\(305\) −3.76566e8 −0.759962
\(306\) −1.53153e7 −0.0305564
\(307\) −2.80506e8 −0.553297 −0.276649 0.960971i \(-0.589224\pi\)
−0.276649 + 0.960971i \(0.589224\pi\)
\(308\) 3.19087e8 0.622275
\(309\) 5.72385e8 1.10365
\(310\) −1.16002e7 −0.0221156
\(311\) 3.02914e8 0.571029 0.285515 0.958374i \(-0.407835\pi\)
0.285515 + 0.958374i \(0.407835\pi\)
\(312\) −6.38030e7 −0.118932
\(313\) −7.65931e8 −1.41184 −0.705918 0.708294i \(-0.749465\pi\)
−0.705918 + 0.708294i \(0.749465\pi\)
\(314\) −2.60231e7 −0.0474357
\(315\) −1.74009e8 −0.313679
\(316\) 1.33301e8 0.237645
\(317\) −1.06128e9 −1.87120 −0.935601 0.353058i \(-0.885142\pi\)
−0.935601 + 0.353058i \(0.885142\pi\)
\(318\) −5.54806e7 −0.0967489
\(319\) 2.35434e7 0.0406071
\(320\) −3.01337e8 −0.514076
\(321\) −1.56757e6 −0.00264521
\(322\) −5.12501e7 −0.0855459
\(323\) −9.60192e7 −0.158544
\(324\) −6.74287e7 −0.110138
\(325\) −4.82855e8 −0.780234
\(326\) 7.54724e7 0.120650
\(327\) −2.87666e8 −0.454958
\(328\) 1.51672e8 0.237327
\(329\) 1.95433e9 3.02561
\(330\) 6.92215e6 0.0106033
\(331\) −3.08627e8 −0.467774 −0.233887 0.972264i \(-0.575145\pi\)
−0.233887 + 0.972264i \(0.575145\pi\)
\(332\) −1.05303e9 −1.57927
\(333\) 2.60658e8 0.386827
\(334\) −1.82905e7 −0.0268604
\(335\) −1.81210e8 −0.263346
\(336\) 6.78278e8 0.975484
\(337\) −7.55380e8 −1.07513 −0.537565 0.843222i \(-0.680656\pi\)
−0.537565 + 0.843222i \(0.680656\pi\)
\(338\) −1.47443e7 −0.0207691
\(339\) 1.31437e8 0.183239
\(340\) −3.81645e8 −0.526603
\(341\) 1.15430e8 0.157645
\(342\) 3.73531e6 0.00504935
\(343\) −1.31016e9 −1.75306
\(344\) 1.84061e8 0.243785
\(345\) 1.25827e8 0.164971
\(346\) 4.52368e7 0.0587119
\(347\) −2.59655e8 −0.333614 −0.166807 0.985990i \(-0.553346\pi\)
−0.166807 + 0.985990i \(0.553346\pi\)
\(348\) 5.04959e7 0.0642287
\(349\) −4.22817e8 −0.532431 −0.266215 0.963914i \(-0.585773\pi\)
−0.266215 + 0.963914i \(0.585773\pi\)
\(350\) −9.19321e7 −0.114612
\(351\) −1.72351e8 −0.212736
\(352\) −8.21556e7 −0.100401
\(353\) 1.55553e9 1.88221 0.941105 0.338116i \(-0.109790\pi\)
0.941105 + 0.338116i \(0.109790\pi\)
\(354\) 5.87138e6 0.00703442
\(355\) −8.34895e8 −0.990450
\(356\) −1.92378e8 −0.225985
\(357\) 8.43522e8 0.981201
\(358\) −6.14380e7 −0.0707696
\(359\) −7.61688e8 −0.868853 −0.434427 0.900707i \(-0.643049\pi\)
−0.434427 + 0.900707i \(0.643049\pi\)
\(360\) 2.98244e7 0.0336910
\(361\) −8.70453e8 −0.973801
\(362\) 4.59119e7 0.0508681
\(363\) 4.57273e8 0.501767
\(364\) 1.74931e9 1.90113
\(365\) −1.56977e8 −0.168971
\(366\) −7.10124e7 −0.0757095
\(367\) 1.37332e8 0.145024 0.0725122 0.997368i \(-0.476898\pi\)
0.0725122 + 0.997368i \(0.476898\pi\)
\(368\) −4.90469e8 −0.513031
\(369\) 4.09713e8 0.424509
\(370\) −5.73924e7 −0.0589045
\(371\) 3.05570e9 3.10672
\(372\) 2.47575e8 0.249348
\(373\) −2.94107e8 −0.293443 −0.146722 0.989178i \(-0.546872\pi\)
−0.146722 + 0.989178i \(0.546872\pi\)
\(374\) −3.35556e7 −0.0331677
\(375\) 5.45483e8 0.534161
\(376\) −3.34964e8 −0.324968
\(377\) 1.29070e8 0.124060
\(378\) −3.28145e7 −0.0312496
\(379\) 6.95112e8 0.655870 0.327935 0.944700i \(-0.393647\pi\)
0.327935 + 0.944700i \(0.393647\pi\)
\(380\) 9.30806e7 0.0870195
\(381\) 8.98012e8 0.831850
\(382\) 7.34417e7 0.0674094
\(383\) −5.22431e8 −0.475153 −0.237576 0.971369i \(-0.576353\pi\)
−0.237576 + 0.971369i \(0.576353\pi\)
\(384\) −2.34590e8 −0.211422
\(385\) −3.81251e8 −0.340485
\(386\) −3.35792e7 −0.0297177
\(387\) 4.97205e8 0.436061
\(388\) −1.30300e9 −1.13249
\(389\) −1.26347e9 −1.08829 −0.544143 0.838993i \(-0.683145\pi\)
−0.544143 + 0.838993i \(0.683145\pi\)
\(390\) 3.79488e7 0.0323946
\(391\) −6.09958e8 −0.516038
\(392\) 4.46806e8 0.374643
\(393\) −1.02947e8 −0.0855538
\(394\) 1.41837e8 0.116830
\(395\) −1.59271e8 −0.130031
\(396\) −1.47735e8 −0.119550
\(397\) 7.32040e8 0.587175 0.293588 0.955932i \(-0.405151\pi\)
0.293588 + 0.955932i \(0.405151\pi\)
\(398\) −8.02537e7 −0.0638079
\(399\) −2.05730e8 −0.162141
\(400\) −8.79799e8 −0.687343
\(401\) −1.29683e9 −1.00434 −0.502168 0.864770i \(-0.667464\pi\)
−0.502168 + 0.864770i \(0.667464\pi\)
\(402\) −3.41724e7 −0.0262352
\(403\) 6.32816e8 0.481626
\(404\) 1.54060e9 1.16240
\(405\) 8.05649e7 0.0602633
\(406\) 2.45741e7 0.0182237
\(407\) 5.71097e8 0.419884
\(408\) −1.44576e8 −0.105387
\(409\) 2.56134e9 1.85113 0.925563 0.378594i \(-0.123592\pi\)
0.925563 + 0.378594i \(0.123592\pi\)
\(410\) −9.02116e7 −0.0646426
\(411\) −1.24248e9 −0.882762
\(412\) 2.68976e9 1.89484
\(413\) −3.23378e8 −0.225884
\(414\) 2.37284e7 0.0164349
\(415\) 1.25818e9 0.864118
\(416\) −4.50396e8 −0.306738
\(417\) −1.22239e9 −0.825530
\(418\) 8.18400e6 0.00548086
\(419\) −1.65931e9 −1.10199 −0.550995 0.834509i \(-0.685752\pi\)
−0.550995 + 0.834509i \(0.685752\pi\)
\(420\) −8.17707e8 −0.538549
\(421\) −2.44008e9 −1.59374 −0.796870 0.604151i \(-0.793512\pi\)
−0.796870 + 0.604151i \(0.793512\pi\)
\(422\) 2.24440e8 0.145380
\(423\) −9.04840e8 −0.581274
\(424\) −5.23734e8 −0.333680
\(425\) −1.09414e9 −0.691370
\(426\) −1.57444e8 −0.0986714
\(427\) 3.91115e9 2.43112
\(428\) −7.36636e6 −0.00454150
\(429\) −3.77619e8 −0.230916
\(430\) −1.09476e8 −0.0664017
\(431\) 5.66963e8 0.341102 0.170551 0.985349i \(-0.445445\pi\)
0.170551 + 0.985349i \(0.445445\pi\)
\(432\) −3.14038e8 −0.187408
\(433\) −1.17068e9 −0.692996 −0.346498 0.938051i \(-0.612629\pi\)
−0.346498 + 0.938051i \(0.612629\pi\)
\(434\) 1.20484e8 0.0707479
\(435\) −6.03334e7 −0.0351435
\(436\) −1.35181e9 −0.781109
\(437\) 1.48765e8 0.0852737
\(438\) −2.96026e7 −0.0168334
\(439\) −1.09400e9 −0.617151 −0.308575 0.951200i \(-0.599852\pi\)
−0.308575 + 0.951200i \(0.599852\pi\)
\(440\) 6.53448e7 0.0365701
\(441\) 1.20696e9 0.670127
\(442\) −1.83960e8 −0.101332
\(443\) 1.26000e8 0.0688584 0.0344292 0.999407i \(-0.489039\pi\)
0.0344292 + 0.999407i \(0.489039\pi\)
\(444\) 1.22489e9 0.664135
\(445\) 2.29856e8 0.123650
\(446\) 1.90707e8 0.101788
\(447\) −6.90636e8 −0.365740
\(448\) 3.12979e9 1.64453
\(449\) −2.96820e8 −0.154750 −0.0773750 0.997002i \(-0.524654\pi\)
−0.0773750 + 0.997002i \(0.524654\pi\)
\(450\) 4.25638e7 0.0220190
\(451\) 8.97673e8 0.460787
\(452\) 6.17650e8 0.314599
\(453\) −5.94277e8 −0.300362
\(454\) −8.19459e7 −0.0410990
\(455\) −2.09011e9 −1.04023
\(456\) 3.52612e7 0.0174149
\(457\) 2.03128e8 0.0995552 0.0497776 0.998760i \(-0.484149\pi\)
0.0497776 + 0.998760i \(0.484149\pi\)
\(458\) −1.47610e8 −0.0717937
\(459\) −3.90544e8 −0.188506
\(460\) 5.91291e8 0.283236
\(461\) −1.66300e9 −0.790567 −0.395283 0.918559i \(-0.629354\pi\)
−0.395283 + 0.918559i \(0.629354\pi\)
\(462\) −7.18959e7 −0.0339201
\(463\) −8.36357e8 −0.391614 −0.195807 0.980642i \(-0.562733\pi\)
−0.195807 + 0.980642i \(0.562733\pi\)
\(464\) 2.35176e8 0.109290
\(465\) −2.95807e8 −0.136434
\(466\) −1.31411e8 −0.0601564
\(467\) −3.67614e9 −1.67026 −0.835129 0.550054i \(-0.814607\pi\)
−0.835129 + 0.550054i \(0.814607\pi\)
\(468\) −8.09917e8 −0.365241
\(469\) 1.88211e9 0.842444
\(470\) 1.99230e8 0.0885142
\(471\) −6.63594e8 −0.292637
\(472\) 5.54256e7 0.0242612
\(473\) 1.08937e9 0.473326
\(474\) −3.00351e7 −0.0129540
\(475\) 2.66853e8 0.114247
\(476\) 3.96390e9 1.68460
\(477\) −1.41477e9 −0.596857
\(478\) 1.64493e8 0.0688890
\(479\) 4.47354e9 1.85985 0.929924 0.367752i \(-0.119873\pi\)
0.929924 + 0.367752i \(0.119873\pi\)
\(480\) 2.10536e8 0.0868923
\(481\) 3.13089e9 1.28280
\(482\) −3.64722e8 −0.148353
\(483\) −1.30689e9 −0.527744
\(484\) 2.14883e9 0.861475
\(485\) 1.55685e9 0.619654
\(486\) 1.51928e7 0.00600360
\(487\) 5.90180e8 0.231544 0.115772 0.993276i \(-0.463066\pi\)
0.115772 + 0.993276i \(0.463066\pi\)
\(488\) −6.70354e8 −0.261117
\(489\) 1.92456e9 0.744304
\(490\) −2.65751e8 −0.102044
\(491\) 1.22115e9 0.465568 0.232784 0.972528i \(-0.425216\pi\)
0.232784 + 0.972528i \(0.425216\pi\)
\(492\) 1.92533e9 0.728831
\(493\) 2.92471e8 0.109930
\(494\) 4.48666e7 0.0167447
\(495\) 1.76516e8 0.0654133
\(496\) 1.15304e9 0.424285
\(497\) 8.67151e9 3.16846
\(498\) 2.37265e8 0.0860859
\(499\) 1.48542e9 0.535176 0.267588 0.963533i \(-0.413773\pi\)
0.267588 + 0.963533i \(0.413773\pi\)
\(500\) 2.56335e9 0.917090
\(501\) −4.66411e8 −0.165705
\(502\) 4.94923e8 0.174612
\(503\) −2.28973e9 −0.802224 −0.401112 0.916029i \(-0.631376\pi\)
−0.401112 + 0.916029i \(0.631376\pi\)
\(504\) −3.09767e8 −0.107778
\(505\) −1.84073e9 −0.636019
\(506\) 5.19885e7 0.0178394
\(507\) −3.75984e8 −0.128127
\(508\) 4.21996e9 1.42819
\(509\) −4.18910e9 −1.40802 −0.704009 0.710191i \(-0.748608\pi\)
−0.704009 + 0.710191i \(0.748608\pi\)
\(510\) 8.59911e7 0.0287050
\(511\) 1.63042e9 0.540539
\(512\) −1.37179e9 −0.451691
\(513\) 9.52512e7 0.0311501
\(514\) 4.33317e8 0.140746
\(515\) −3.21377e9 −1.03679
\(516\) 2.33648e9 0.748665
\(517\) −1.98249e9 −0.630948
\(518\) 5.96098e8 0.188436
\(519\) 1.15355e9 0.362201
\(520\) 3.58235e8 0.111727
\(521\) 1.43063e9 0.443196 0.221598 0.975138i \(-0.428873\pi\)
0.221598 + 0.975138i \(0.428873\pi\)
\(522\) −1.13776e7 −0.00350110
\(523\) 1.34628e9 0.411509 0.205755 0.978604i \(-0.434035\pi\)
0.205755 + 0.978604i \(0.434035\pi\)
\(524\) −4.83770e8 −0.146886
\(525\) −2.34429e9 −0.707055
\(526\) −1.10521e8 −0.0331127
\(527\) 1.43395e9 0.426772
\(528\) −6.88050e8 −0.203424
\(529\) −2.45980e9 −0.722446
\(530\) 3.11507e8 0.0908871
\(531\) 1.49721e8 0.0433963
\(532\) −9.66768e8 −0.278376
\(533\) 4.92125e9 1.40776
\(534\) 4.33460e7 0.0123184
\(535\) 8.80145e6 0.00248494
\(536\) −3.22586e8 −0.0904835
\(537\) −1.56668e9 −0.436587
\(538\) 2.06637e8 0.0572099
\(539\) 2.64442e9 0.727395
\(540\) 3.78592e8 0.103465
\(541\) −3.77248e9 −1.02432 −0.512162 0.858889i \(-0.671155\pi\)
−0.512162 + 0.858889i \(0.671155\pi\)
\(542\) −3.84845e8 −0.103822
\(543\) 1.17076e9 0.313812
\(544\) −1.02059e9 −0.271803
\(545\) 1.61516e9 0.427393
\(546\) −3.94150e8 −0.103630
\(547\) 1.81833e9 0.475026 0.237513 0.971384i \(-0.423668\pi\)
0.237513 + 0.971384i \(0.423668\pi\)
\(548\) −5.83869e9 −1.51560
\(549\) −1.81083e9 −0.467062
\(550\) 9.32566e7 0.0239007
\(551\) −7.13317e7 −0.0181657
\(552\) 2.23995e8 0.0566829
\(553\) 1.65424e9 0.415969
\(554\) 3.99089e8 0.0997208
\(555\) −1.46352e9 −0.363390
\(556\) −5.74427e9 −1.41734
\(557\) −5.99186e7 −0.0146916 −0.00734579 0.999973i \(-0.502338\pi\)
−0.00734579 + 0.999973i \(0.502338\pi\)
\(558\) −5.57829e7 −0.0135919
\(559\) 5.97216e9 1.44607
\(560\) −3.80833e9 −0.916382
\(561\) −8.55675e8 −0.204616
\(562\) −3.58559e8 −0.0852086
\(563\) −5.52477e9 −1.30477 −0.652386 0.757887i \(-0.726232\pi\)
−0.652386 + 0.757887i \(0.726232\pi\)
\(564\) −4.25204e9 −0.997978
\(565\) −7.37978e8 −0.172137
\(566\) 2.95874e8 0.0685882
\(567\) −8.36775e8 −0.192783
\(568\) −1.48626e9 −0.340311
\(569\) −5.19237e9 −1.18161 −0.590803 0.806816i \(-0.701189\pi\)
−0.590803 + 0.806816i \(0.701189\pi\)
\(570\) −2.09727e7 −0.00474342
\(571\) 1.98094e9 0.445293 0.222647 0.974899i \(-0.428530\pi\)
0.222647 + 0.974899i \(0.428530\pi\)
\(572\) −1.77451e9 −0.396454
\(573\) 1.87278e9 0.415858
\(574\) 9.36970e8 0.206792
\(575\) 1.69517e9 0.371858
\(576\) −1.44907e9 −0.315944
\(577\) −3.25873e9 −0.706209 −0.353104 0.935584i \(-0.614874\pi\)
−0.353104 + 0.935584i \(0.614874\pi\)
\(578\) 1.76245e7 0.00379638
\(579\) −8.56278e8 −0.183333
\(580\) −2.83520e8 −0.0603372
\(581\) −1.30679e10 −2.76432
\(582\) 2.93588e8 0.0617317
\(583\) −3.09973e9 −0.647863
\(584\) −2.79448e8 −0.0580571
\(585\) 9.67703e8 0.199846
\(586\) −6.62795e8 −0.136062
\(587\) −7.32754e8 −0.149529 −0.0747643 0.997201i \(-0.523820\pi\)
−0.0747643 + 0.997201i \(0.523820\pi\)
\(588\) 5.67176e9 1.15053
\(589\) −3.49730e8 −0.0705227
\(590\) −3.29661e7 −0.00660823
\(591\) 3.61688e9 0.720739
\(592\) 5.70472e9 1.13008
\(593\) 7.42096e9 1.46140 0.730699 0.682700i \(-0.239194\pi\)
0.730699 + 0.682700i \(0.239194\pi\)
\(594\) 3.32872e7 0.00651666
\(595\) −4.73613e9 −0.921752
\(596\) −3.24545e9 −0.627933
\(597\) −2.04648e9 −0.393639
\(598\) 2.85013e8 0.0545018
\(599\) −1.05376e9 −0.200331 −0.100165 0.994971i \(-0.531937\pi\)
−0.100165 + 0.994971i \(0.531937\pi\)
\(600\) 4.01801e8 0.0759419
\(601\) −7.07267e9 −1.32899 −0.664496 0.747292i \(-0.731354\pi\)
−0.664496 + 0.747292i \(0.731354\pi\)
\(602\) 1.13706e9 0.212420
\(603\) −8.71404e8 −0.161849
\(604\) −2.79264e9 −0.515686
\(605\) −2.56745e9 −0.471367
\(606\) −3.47123e8 −0.0633620
\(607\) 4.55310e9 0.826317 0.413159 0.910659i \(-0.364426\pi\)
0.413159 + 0.910659i \(0.364426\pi\)
\(608\) 2.48914e8 0.0449146
\(609\) 6.26644e8 0.112424
\(610\) 3.98714e8 0.0711225
\(611\) −1.08685e10 −1.92763
\(612\) −1.83525e9 −0.323643
\(613\) 6.47628e9 1.13557 0.567785 0.823177i \(-0.307800\pi\)
0.567785 + 0.823177i \(0.307800\pi\)
\(614\) 2.97004e8 0.0517814
\(615\) −2.30042e9 −0.398789
\(616\) −6.78694e8 −0.116988
\(617\) 4.44035e9 0.761061 0.380531 0.924768i \(-0.375741\pi\)
0.380531 + 0.924768i \(0.375741\pi\)
\(618\) −6.06049e8 −0.103288
\(619\) 2.96260e9 0.502060 0.251030 0.967979i \(-0.419231\pi\)
0.251030 + 0.967979i \(0.419231\pi\)
\(620\) −1.39006e9 −0.234241
\(621\) 6.05079e8 0.101389
\(622\) −3.20730e8 −0.0534409
\(623\) −2.38737e9 −0.395559
\(624\) −3.77205e9 −0.621486
\(625\) 1.24534e9 0.204037
\(626\) 8.10979e8 0.132129
\(627\) 2.08694e8 0.0338121
\(628\) −3.11837e9 −0.502423
\(629\) 7.09452e9 1.13670
\(630\) 1.84244e8 0.0293562
\(631\) 1.60390e9 0.254142 0.127071 0.991894i \(-0.459442\pi\)
0.127071 + 0.991894i \(0.459442\pi\)
\(632\) −2.83530e8 −0.0446775
\(633\) 5.72325e9 0.896871
\(634\) 1.12369e9 0.175120
\(635\) −5.04207e9 −0.781450
\(636\) −6.64830e9 −1.02473
\(637\) 1.44973e10 2.22229
\(638\) −2.49281e7 −0.00380030
\(639\) −4.01484e9 −0.608717
\(640\) 1.31715e9 0.198613
\(641\) −6.32307e8 −0.0948254 −0.0474127 0.998875i \(-0.515098\pi\)
−0.0474127 + 0.998875i \(0.515098\pi\)
\(642\) 1.65977e6 0.000247557 0
\(643\) −4.72453e9 −0.700842 −0.350421 0.936592i \(-0.613961\pi\)
−0.350421 + 0.936592i \(0.613961\pi\)
\(644\) −6.14136e9 −0.906074
\(645\) −2.79166e9 −0.409641
\(646\) 1.01667e8 0.0148376
\(647\) −6.52515e9 −0.947165 −0.473582 0.880750i \(-0.657039\pi\)
−0.473582 + 0.880750i \(0.657039\pi\)
\(648\) 1.43420e8 0.0207060
\(649\) 3.28037e8 0.0471049
\(650\) 5.11254e8 0.0730197
\(651\) 3.07235e9 0.436453
\(652\) 9.04393e9 1.27788
\(653\) −7.88114e9 −1.10762 −0.553812 0.832641i \(-0.686828\pi\)
−0.553812 + 0.832641i \(0.686828\pi\)
\(654\) 3.04585e8 0.0425781
\(655\) 5.78017e8 0.0803703
\(656\) 8.96689e9 1.24016
\(657\) −7.54873e8 −0.103847
\(658\) −2.06928e9 −0.283157
\(659\) 7.14701e9 0.972805 0.486402 0.873735i \(-0.338309\pi\)
0.486402 + 0.873735i \(0.338309\pi\)
\(660\) 8.29488e8 0.112307
\(661\) −5.05842e9 −0.681255 −0.340627 0.940198i \(-0.610639\pi\)
−0.340627 + 0.940198i \(0.610639\pi\)
\(662\) 3.26779e8 0.0437776
\(663\) −4.69101e9 −0.625128
\(664\) 2.23978e9 0.296904
\(665\) 1.15511e9 0.152317
\(666\) −2.75989e8 −0.0362019
\(667\) −4.53132e8 −0.0591267
\(668\) −2.19177e9 −0.284496
\(669\) 4.86308e9 0.627942
\(670\) 1.91868e8 0.0246457
\(671\) −3.96750e9 −0.506977
\(672\) −2.18670e9 −0.277969
\(673\) −1.46795e10 −1.85634 −0.928169 0.372158i \(-0.878618\pi\)
−0.928169 + 0.372158i \(0.878618\pi\)
\(674\) 7.99808e8 0.100618
\(675\) 1.08539e9 0.135838
\(676\) −1.76683e9 −0.219979
\(677\) 8.90911e9 1.10350 0.551752 0.834008i \(-0.313959\pi\)
0.551752 + 0.834008i \(0.313959\pi\)
\(678\) −1.39167e8 −0.0171488
\(679\) −1.61700e10 −1.98228
\(680\) 8.11753e8 0.0990017
\(681\) −2.08964e9 −0.253546
\(682\) −1.22219e8 −0.0147535
\(683\) −1.04640e10 −1.25668 −0.628338 0.777940i \(-0.716265\pi\)
−0.628338 + 0.777940i \(0.716265\pi\)
\(684\) 4.47606e8 0.0534810
\(685\) 6.97617e9 0.829278
\(686\) 1.38722e9 0.164063
\(687\) −3.76408e9 −0.442905
\(688\) 1.08817e10 1.27391
\(689\) −1.69934e10 −1.97931
\(690\) −1.33228e8 −0.0154392
\(691\) −9.56029e9 −1.10230 −0.551148 0.834408i \(-0.685810\pi\)
−0.551148 + 0.834408i \(0.685810\pi\)
\(692\) 5.42078e9 0.621857
\(693\) −1.83336e9 −0.209258
\(694\) 2.74927e8 0.0312219
\(695\) 6.86335e9 0.775513
\(696\) −1.07404e8 −0.0120750
\(697\) 1.11514e10 1.24743
\(698\) 4.47685e8 0.0498285
\(699\) −3.35101e9 −0.371113
\(700\) −1.10163e10 −1.21393
\(701\) 4.24136e9 0.465041 0.232521 0.972591i \(-0.425303\pi\)
0.232521 + 0.972591i \(0.425303\pi\)
\(702\) 1.82488e8 0.0199093
\(703\) −1.73031e9 −0.187836
\(704\) −3.17488e9 −0.342944
\(705\) 5.08041e9 0.546056
\(706\) −1.64702e9 −0.176150
\(707\) 1.91185e10 2.03463
\(708\) 7.03573e8 0.0745063
\(709\) 1.04133e10 1.09730 0.548652 0.836051i \(-0.315141\pi\)
0.548652 + 0.836051i \(0.315141\pi\)
\(710\) 8.84000e8 0.0926932
\(711\) −7.65900e8 −0.0799151
\(712\) 4.09184e8 0.0424853
\(713\) −2.22164e9 −0.229542
\(714\) −8.93134e8 −0.0918275
\(715\) 2.12022e9 0.216925
\(716\) −7.36218e9 −0.749567
\(717\) 4.19460e9 0.424985
\(718\) 8.06487e8 0.0813133
\(719\) −6.99544e9 −0.701882 −0.350941 0.936398i \(-0.614138\pi\)
−0.350941 + 0.936398i \(0.614138\pi\)
\(720\) 1.76323e9 0.176054
\(721\) 3.33794e10 3.31669
\(722\) 9.21649e8 0.0911351
\(723\) −9.30049e9 −0.915212
\(724\) 5.50168e9 0.538778
\(725\) −8.12824e8 −0.0792161
\(726\) −4.84168e8 −0.0469589
\(727\) 1.85412e10 1.78965 0.894824 0.446418i \(-0.147301\pi\)
0.894824 + 0.446418i \(0.147301\pi\)
\(728\) −3.72076e9 −0.357414
\(729\) 3.87420e8 0.0370370
\(730\) 1.66210e8 0.0158135
\(731\) 1.35328e10 1.28137
\(732\) −8.50949e9 −0.801890
\(733\) −7.68407e9 −0.720655 −0.360327 0.932826i \(-0.617335\pi\)
−0.360327 + 0.932826i \(0.617335\pi\)
\(734\) −1.45409e8 −0.0135724
\(735\) −6.77671e9 −0.629526
\(736\) 1.58122e9 0.146191
\(737\) −1.90923e9 −0.175680
\(738\) −4.33810e8 −0.0397285
\(739\) −1.62766e9 −0.148357 −0.0741785 0.997245i \(-0.523633\pi\)
−0.0741785 + 0.997245i \(0.523633\pi\)
\(740\) −6.87740e9 −0.623897
\(741\) 1.14411e9 0.103301
\(742\) −3.23542e9 −0.290748
\(743\) −1.23451e9 −0.110417 −0.0552084 0.998475i \(-0.517582\pi\)
−0.0552084 + 0.998475i \(0.517582\pi\)
\(744\) −5.26589e8 −0.0468777
\(745\) 3.87772e9 0.343581
\(746\) 3.11405e8 0.0274625
\(747\) 6.05032e9 0.531075
\(748\) −4.02101e9 −0.351301
\(749\) −9.14149e7 −0.00794934
\(750\) −5.77566e8 −0.0499905
\(751\) 1.87234e10 1.61304 0.806520 0.591206i \(-0.201348\pi\)
0.806520 + 0.591206i \(0.201348\pi\)
\(752\) −1.98032e10 −1.69814
\(753\) 1.26206e10 1.07721
\(754\) −1.36662e8 −0.0116104
\(755\) 3.33669e9 0.282164
\(756\) −3.93219e9 −0.330985
\(757\) −3.90822e9 −0.327449 −0.163724 0.986506i \(-0.552351\pi\)
−0.163724 + 0.986506i \(0.552351\pi\)
\(758\) −7.35995e8 −0.0613809
\(759\) 1.32572e9 0.110054
\(760\) −1.97981e8 −0.0163597
\(761\) 8.54296e9 0.702687 0.351344 0.936247i \(-0.385725\pi\)
0.351344 + 0.936247i \(0.385725\pi\)
\(762\) −9.50829e8 −0.0778502
\(763\) −1.67756e10 −1.36723
\(764\) 8.80059e9 0.713978
\(765\) 2.19279e9 0.177085
\(766\) 5.53158e8 0.0444681
\(767\) 1.79837e9 0.143912
\(768\) −6.62127e9 −0.527445
\(769\) −8.98742e9 −0.712677 −0.356338 0.934357i \(-0.615975\pi\)
−0.356338 + 0.934357i \(0.615975\pi\)
\(770\) 4.03674e8 0.0318650
\(771\) 1.10497e10 0.868279
\(772\) −4.02384e9 −0.314760
\(773\) 7.14394e9 0.556301 0.278150 0.960538i \(-0.410279\pi\)
0.278150 + 0.960538i \(0.410279\pi\)
\(774\) −5.26448e8 −0.0408096
\(775\) −3.98517e9 −0.307532
\(776\) 2.77146e9 0.212908
\(777\) 1.52006e10 1.16249
\(778\) 1.33779e9 0.101849
\(779\) −2.71976e9 −0.206134
\(780\) 4.54745e9 0.343112
\(781\) −8.79645e9 −0.660737
\(782\) 6.45833e8 0.0482944
\(783\) −2.90131e8 −0.0215987
\(784\) 2.64153e10 1.95771
\(785\) 3.72588e9 0.274907
\(786\) 1.09002e8 0.00800672
\(787\) 2.12694e9 0.155541 0.0777705 0.996971i \(-0.475220\pi\)
0.0777705 + 0.996971i \(0.475220\pi\)
\(788\) 1.69965e10 1.23742
\(789\) −2.81831e9 −0.204277
\(790\) 1.68638e8 0.0121692
\(791\) 7.66490e9 0.550667
\(792\) 3.14230e8 0.0224755
\(793\) −2.17507e10 −1.54888
\(794\) −7.75095e8 −0.0549519
\(795\) 7.94349e9 0.560695
\(796\) −9.61688e9 −0.675832
\(797\) −7.00833e9 −0.490355 −0.245177 0.969478i \(-0.578846\pi\)
−0.245177 + 0.969478i \(0.578846\pi\)
\(798\) 2.17830e8 0.0151742
\(799\) −2.46277e10 −1.70809
\(800\) 2.83638e9 0.195862
\(801\) 1.10533e9 0.0759939
\(802\) 1.37311e9 0.0939926
\(803\) −1.65391e9 −0.112722
\(804\) −4.09492e9 −0.277875
\(805\) 7.33780e9 0.495770
\(806\) −6.70035e8 −0.0450739
\(807\) 5.26930e9 0.352935
\(808\) −3.27682e9 −0.218531
\(809\) 2.24886e10 1.49329 0.746643 0.665225i \(-0.231665\pi\)
0.746643 + 0.665225i \(0.231665\pi\)
\(810\) −8.53033e7 −0.00563986
\(811\) −7.32750e9 −0.482373 −0.241186 0.970479i \(-0.577536\pi\)
−0.241186 + 0.970479i \(0.577536\pi\)
\(812\) 2.94474e9 0.193019
\(813\) −9.81362e9 −0.640490
\(814\) −6.04686e8 −0.0392957
\(815\) −1.08058e10 −0.699209
\(816\) −8.54738e9 −0.550703
\(817\) −3.30056e9 −0.211743
\(818\) −2.71199e9 −0.173241
\(819\) −1.00509e10 −0.639310
\(820\) −1.08102e10 −0.684673
\(821\) −1.86005e10 −1.17307 −0.586534 0.809924i \(-0.699508\pi\)
−0.586534 + 0.809924i \(0.699508\pi\)
\(822\) 1.31556e9 0.0826150
\(823\) −3.44677e9 −0.215532 −0.107766 0.994176i \(-0.534370\pi\)
−0.107766 + 0.994176i \(0.534370\pi\)
\(824\) −5.72108e9 −0.356232
\(825\) 2.37806e9 0.147446
\(826\) 3.42397e8 0.0211398
\(827\) 5.34418e9 0.328558 0.164279 0.986414i \(-0.447470\pi\)
0.164279 + 0.986414i \(0.447470\pi\)
\(828\) 2.84340e9 0.174073
\(829\) 1.24427e10 0.758532 0.379266 0.925288i \(-0.376176\pi\)
0.379266 + 0.925288i \(0.376176\pi\)
\(830\) −1.33218e9 −0.0808701
\(831\) 1.01769e10 0.615191
\(832\) −1.74054e10 −1.04774
\(833\) 3.28506e10 1.96918
\(834\) 1.29428e9 0.0772588
\(835\) 2.61876e9 0.155666
\(836\) 9.80697e8 0.0580514
\(837\) −1.42248e9 −0.0838505
\(838\) 1.75690e9 0.103132
\(839\) 2.08037e10 1.21611 0.608056 0.793894i \(-0.291950\pi\)
0.608056 + 0.793894i \(0.291950\pi\)
\(840\) 1.73925e9 0.101248
\(841\) −1.70326e10 −0.987404
\(842\) 2.58360e9 0.149153
\(843\) −9.14332e9 −0.525663
\(844\) 2.68948e10 1.53982
\(845\) 2.11104e9 0.120364
\(846\) 9.58059e8 0.0543996
\(847\) 2.66665e10 1.50790
\(848\) −3.09633e10 −1.74366
\(849\) 7.54485e9 0.423130
\(850\) 1.15849e9 0.0647032
\(851\) −1.09917e10 −0.611380
\(852\) −1.88666e10 −1.04509
\(853\) 3.30134e10 1.82125 0.910623 0.413239i \(-0.135603\pi\)
0.910623 + 0.413239i \(0.135603\pi\)
\(854\) −4.14118e9 −0.227521
\(855\) −5.34807e8 −0.0292628
\(856\) 1.56681e7 0.000853806 0
\(857\) 7.37991e9 0.400515 0.200257 0.979743i \(-0.435822\pi\)
0.200257 + 0.979743i \(0.435822\pi\)
\(858\) 3.99829e8 0.0216107
\(859\) 8.66866e9 0.466633 0.233317 0.972401i \(-0.425042\pi\)
0.233317 + 0.972401i \(0.425042\pi\)
\(860\) −1.31186e10 −0.703305
\(861\) 2.38929e10 1.27573
\(862\) −6.00309e8 −0.0319227
\(863\) −9.44417e8 −0.0500180 −0.0250090 0.999687i \(-0.507961\pi\)
−0.0250090 + 0.999687i \(0.507961\pi\)
\(864\) 1.01242e9 0.0534028
\(865\) −6.47684e9 −0.340257
\(866\) 1.23953e9 0.0648554
\(867\) 4.49429e8 0.0234204
\(868\) 1.44377e10 0.749338
\(869\) −1.67807e9 −0.0867445
\(870\) 6.38819e7 0.00328898
\(871\) −1.04668e10 −0.536725
\(872\) 2.87527e9 0.146849
\(873\) 7.48656e9 0.380831
\(874\) −1.57514e8 −0.00798050
\(875\) 3.18106e10 1.60525
\(876\) −3.54731e9 −0.178293
\(877\) −1.94376e10 −0.973071 −0.486535 0.873661i \(-0.661740\pi\)
−0.486535 + 0.873661i \(0.661740\pi\)
\(878\) 1.15834e9 0.0577573
\(879\) −1.69014e10 −0.839386
\(880\) 3.86320e9 0.191099
\(881\) 3.61733e10 1.78227 0.891134 0.453741i \(-0.149911\pi\)
0.891134 + 0.453741i \(0.149911\pi\)
\(882\) −1.27795e9 −0.0627151
\(883\) −2.10672e10 −1.02978 −0.514890 0.857256i \(-0.672167\pi\)
−0.514890 + 0.857256i \(0.672167\pi\)
\(884\) −2.20441e10 −1.07327
\(885\) −8.40641e8 −0.0407670
\(886\) −1.33410e8 −0.00644424
\(887\) 3.08019e10 1.48199 0.740993 0.671512i \(-0.234355\pi\)
0.740993 + 0.671512i \(0.234355\pi\)
\(888\) −2.60532e9 −0.124858
\(889\) 5.23688e10 2.49986
\(890\) −2.43375e8 −0.0115721
\(891\) 8.48831e8 0.0402022
\(892\) 2.28527e10 1.07810
\(893\) 6.00653e9 0.282256
\(894\) 7.31256e8 0.0342285
\(895\) 8.79645e9 0.410135
\(896\) −1.36804e10 −0.635363
\(897\) 7.26789e9 0.336229
\(898\) 3.14277e8 0.0144826
\(899\) 1.06526e9 0.0488988
\(900\) 5.10047e9 0.233218
\(901\) −3.85067e10 −1.75388
\(902\) −9.50469e8 −0.0431236
\(903\) 2.89952e10 1.31044
\(904\) −1.31373e9 −0.0591449
\(905\) −6.57349e9 −0.294799
\(906\) 6.29230e8 0.0281100
\(907\) 1.57628e10 0.701466 0.350733 0.936475i \(-0.385932\pi\)
0.350733 + 0.936475i \(0.385932\pi\)
\(908\) −9.81966e9 −0.435307
\(909\) −8.85170e9 −0.390889
\(910\) 2.21304e9 0.0973517
\(911\) 8.64623e9 0.378890 0.189445 0.981891i \(-0.439331\pi\)
0.189445 + 0.981891i \(0.439331\pi\)
\(912\) 2.08465e9 0.0910020
\(913\) 1.32561e10 0.576460
\(914\) −2.15075e8 −0.00931706
\(915\) 1.01673e10 0.438764
\(916\) −1.76883e10 −0.760415
\(917\) −6.00348e9 −0.257105
\(918\) 4.13514e8 0.0176417
\(919\) −1.72145e10 −0.731628 −0.365814 0.930688i \(-0.619209\pi\)
−0.365814 + 0.930688i \(0.619209\pi\)
\(920\) −1.25767e9 −0.0532486
\(921\) 7.57367e9 0.319446
\(922\) 1.76081e9 0.0739867
\(923\) −4.82242e10 −2.01864
\(924\) −8.61536e9 −0.359270
\(925\) −1.97168e10 −0.819107
\(926\) 8.85547e8 0.0366499
\(927\) −1.54544e10 −0.637195
\(928\) −7.58184e8 −0.0311427
\(929\) 2.13454e10 0.873471 0.436735 0.899590i \(-0.356135\pi\)
0.436735 + 0.899590i \(0.356135\pi\)
\(930\) 3.13205e8 0.0127684
\(931\) −8.01205e9 −0.325402
\(932\) −1.57472e10 −0.637157
\(933\) −8.17868e9 −0.329684
\(934\) 3.89236e9 0.156314
\(935\) 4.80436e9 0.192219
\(936\) 1.72268e9 0.0686657
\(937\) 2.94917e9 0.117115 0.0585573 0.998284i \(-0.481350\pi\)
0.0585573 + 0.998284i \(0.481350\pi\)
\(938\) −1.99281e9 −0.0788417
\(939\) 2.06801e10 0.815124
\(940\) 2.38740e10 0.937513
\(941\) 8.66911e9 0.339165 0.169582 0.985516i \(-0.445758\pi\)
0.169582 + 0.985516i \(0.445758\pi\)
\(942\) 7.02623e8 0.0273870
\(943\) −1.72772e10 −0.670937
\(944\) 3.27677e9 0.126778
\(945\) 4.69825e9 0.181103
\(946\) −1.15344e9 −0.0442971
\(947\) 1.86313e10 0.712884 0.356442 0.934317i \(-0.383990\pi\)
0.356442 + 0.934317i \(0.383990\pi\)
\(948\) −3.59913e9 −0.137205
\(949\) −9.06713e9 −0.344380
\(950\) −2.82548e8 −0.0106920
\(951\) 2.86544e10 1.08034
\(952\) −8.43115e9 −0.316707
\(953\) 3.78151e10 1.41527 0.707637 0.706576i \(-0.249761\pi\)
0.707637 + 0.706576i \(0.249761\pi\)
\(954\) 1.49798e9 0.0558580
\(955\) −1.05151e10 −0.390662
\(956\) 1.97114e10 0.729649
\(957\) −6.35672e8 −0.0234445
\(958\) −4.73666e9 −0.174057
\(959\) −7.24570e10 −2.65286
\(960\) 8.13609e9 0.296802
\(961\) −2.22898e10 −0.810165
\(962\) −3.31503e9 −0.120054
\(963\) 4.23244e7 0.00152721
\(964\) −4.37050e10 −1.57131
\(965\) 4.80775e9 0.172225
\(966\) 1.38375e9 0.0493900
\(967\) −3.55682e10 −1.26494 −0.632470 0.774585i \(-0.717959\pi\)
−0.632470 + 0.774585i \(0.717959\pi\)
\(968\) −4.57052e9 −0.161958
\(969\) 2.59252e9 0.0915353
\(970\) −1.64841e9 −0.0579915
\(971\) 2.79517e10 0.979806 0.489903 0.871777i \(-0.337032\pi\)
0.489903 + 0.871777i \(0.337032\pi\)
\(972\) 1.82057e9 0.0635882
\(973\) −7.12852e10 −2.48087
\(974\) −6.24891e8 −0.0216695
\(975\) 1.30371e10 0.450468
\(976\) −3.96315e10 −1.36448
\(977\) −6.61358e9 −0.226885 −0.113443 0.993545i \(-0.536188\pi\)
−0.113443 + 0.993545i \(0.536188\pi\)
\(978\) −2.03775e9 −0.0696571
\(979\) 2.42176e9 0.0824882
\(980\) −3.18453e10 −1.08082
\(981\) 7.76698e9 0.262670
\(982\) −1.29297e9 −0.0435711
\(983\) 1.77476e10 0.595939 0.297970 0.954575i \(-0.403691\pi\)
0.297970 + 0.954575i \(0.403691\pi\)
\(984\) −4.09515e9 −0.137021
\(985\) −2.03077e10 −0.677071
\(986\) −3.09672e8 −0.0102881
\(987\) −5.27670e10 −1.74684
\(988\) 5.37641e9 0.177355
\(989\) −2.09667e10 −0.689195
\(990\) −1.86898e8 −0.00612183
\(991\) −4.22116e9 −0.137776 −0.0688880 0.997624i \(-0.521945\pi\)
−0.0688880 + 0.997624i \(0.521945\pi\)
\(992\) −3.71728e9 −0.120902
\(993\) 8.33294e9 0.270070
\(994\) −9.18153e9 −0.296526
\(995\) 1.14904e10 0.369790
\(996\) 2.84318e10 0.911793
\(997\) −2.02379e10 −0.646742 −0.323371 0.946272i \(-0.604816\pi\)
−0.323371 + 0.946272i \(0.604816\pi\)
\(998\) −1.57278e9 −0.0500854
\(999\) −7.03777e9 −0.223335
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.1.9 16
3.2 odd 2 531.8.a.b.1.8 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.9 16 1.1 even 1 trivial
531.8.a.b.1.8 16 3.2 odd 2