Properties

Label 177.8.a.a.1.1
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 1.1
Root \(19.7363\) of defining polynomial
Character \(\chi\) \(=\) 177.1

$q$-expansion

\(f(q)\) \(=\) \(q-19.7363 q^{2} -27.0000 q^{3} +261.522 q^{4} -90.5952 q^{5} +532.880 q^{6} -807.818 q^{7} -2635.23 q^{8} +729.000 q^{9} +O(q^{10})\) \(q-19.7363 q^{2} -27.0000 q^{3} +261.522 q^{4} -90.5952 q^{5} +532.880 q^{6} -807.818 q^{7} -2635.23 q^{8} +729.000 q^{9} +1788.01 q^{10} +4971.97 q^{11} -7061.09 q^{12} -2921.19 q^{13} +15943.4 q^{14} +2446.07 q^{15} +18534.9 q^{16} -29331.0 q^{17} -14387.8 q^{18} +8907.94 q^{19} -23692.6 q^{20} +21811.1 q^{21} -98128.3 q^{22} -21320.0 q^{23} +71151.2 q^{24} -69917.5 q^{25} +57653.6 q^{26} -19683.0 q^{27} -211262. q^{28} +144361. q^{29} -48276.4 q^{30} +124102. q^{31} -28500.9 q^{32} -134243. q^{33} +578885. q^{34} +73184.5 q^{35} +190649. q^{36} +249110. q^{37} -175810. q^{38} +78872.3 q^{39} +238739. q^{40} +769960. q^{41} -430470. q^{42} -214862. q^{43} +1.30028e6 q^{44} -66043.9 q^{45} +420778. q^{46} +826532. q^{47} -500442. q^{48} -170973. q^{49} +1.37991e6 q^{50} +791936. q^{51} -763956. q^{52} +791662. q^{53} +388470. q^{54} -450437. q^{55} +2.12879e6 q^{56} -240514. q^{57} -2.84915e6 q^{58} +205379. q^{59} +639701. q^{60} -2.23090e6 q^{61} -2.44931e6 q^{62} -588900. q^{63} -1.80996e6 q^{64} +264646. q^{65} +2.64947e6 q^{66} -60136.4 q^{67} -7.67069e6 q^{68} +575640. q^{69} -1.44439e6 q^{70} +3.73452e6 q^{71} -1.92108e6 q^{72} +476492. q^{73} -4.91652e6 q^{74} +1.88777e6 q^{75} +2.32962e6 q^{76} -4.01645e6 q^{77} -1.55665e6 q^{78} +1.82860e6 q^{79} -1.67917e6 q^{80} +531441. q^{81} -1.51962e7 q^{82} +4.01281e6 q^{83} +5.70408e6 q^{84} +2.65724e6 q^{85} +4.24058e6 q^{86} -3.89775e6 q^{87} -1.31023e7 q^{88} +5.38781e6 q^{89} +1.30346e6 q^{90} +2.35979e6 q^{91} -5.57565e6 q^{92} -3.35075e6 q^{93} -1.63127e7 q^{94} -807017. q^{95} +769525. q^{96} -1.52550e7 q^{97} +3.37437e6 q^{98} +3.62457e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} + O(q^{10}) \) \( 16q - 6q^{2} - 432q^{3} + 974q^{4} - 68q^{5} + 162q^{6} - 2343q^{7} + 819q^{8} + 11664q^{9} - 3479q^{10} + 898q^{11} - 26298q^{12} - 8172q^{13} - 13315q^{14} + 1836q^{15} + 3138q^{16} - 44985q^{17} - 4374q^{18} - 40137q^{19} + 130657q^{20} + 63261q^{21} + 109394q^{22} - 2833q^{23} - 22113q^{24} + 285746q^{25} - 129420q^{26} - 314928q^{27} + 112890q^{28} + 144375q^{29} + 93933q^{30} - 141759q^{31} - 36224q^{32} - 24246q^{33} - 341332q^{34} - 78859q^{35} + 710046q^{36} - 297971q^{37} + 329075q^{38} + 220644q^{39} - 203048q^{40} + 659077q^{41} + 359505q^{42} - 1431608q^{43} + 254916q^{44} - 49572q^{45} + 873113q^{46} - 1574073q^{47} - 84726q^{48} + 1893545q^{49} + 302533q^{50} + 1214595q^{51} - 4972548q^{52} + 587736q^{53} + 118098q^{54} - 4624036q^{55} - 5798506q^{56} + 1083699q^{57} - 6991380q^{58} + 3286064q^{59} - 3527739q^{60} - 6117131q^{61} - 11570258q^{62} - 1708047q^{63} - 19063011q^{64} - 5335514q^{65} - 2953638q^{66} - 16518710q^{67} - 17284669q^{68} + 76491q^{69} - 39189486q^{70} - 10882582q^{71} + 597051q^{72} - 21097441q^{73} - 16717030q^{74} - 7715142q^{75} - 40864952q^{76} - 3404601q^{77} + 3494340q^{78} - 3784458q^{79} - 27466195q^{80} + 8503056q^{81} - 24990117q^{82} - 1951425q^{83} - 3048030q^{84} - 23238675q^{85} - 35910572q^{86} - 3898125q^{87} - 27843055q^{88} + 10499443q^{89} - 2536191q^{90} + 699217q^{91} - 20062766q^{92} + 3827493q^{93} - 59358988q^{94} - 29236333q^{95} + 978048q^{96} - 25158976q^{97} + 2120460q^{98} + 654642q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −19.7363 −1.74446 −0.872230 0.489096i \(-0.837327\pi\)
−0.872230 + 0.489096i \(0.837327\pi\)
\(3\) −27.0000 −0.577350
\(4\) 261.522 2.04314
\(5\) −90.5952 −0.324123 −0.162062 0.986781i \(-0.551814\pi\)
−0.162062 + 0.986781i \(0.551814\pi\)
\(6\) 532.880 1.00716
\(7\) −807.818 −0.890165 −0.445083 0.895490i \(-0.646826\pi\)
−0.445083 + 0.895490i \(0.646826\pi\)
\(8\) −2635.23 −1.81971
\(9\) 729.000 0.333333
\(10\) 1788.01 0.565420
\(11\) 4971.97 1.12630 0.563150 0.826355i \(-0.309589\pi\)
0.563150 + 0.826355i \(0.309589\pi\)
\(12\) −7061.09 −1.17961
\(13\) −2921.19 −0.368773 −0.184386 0.982854i \(-0.559030\pi\)
−0.184386 + 0.982854i \(0.559030\pi\)
\(14\) 15943.4 1.55286
\(15\) 2446.07 0.187133
\(16\) 18534.9 1.13128
\(17\) −29331.0 −1.44795 −0.723977 0.689824i \(-0.757688\pi\)
−0.723977 + 0.689824i \(0.757688\pi\)
\(18\) −14387.8 −0.581487
\(19\) 8907.94 0.297948 0.148974 0.988841i \(-0.452403\pi\)
0.148974 + 0.988841i \(0.452403\pi\)
\(20\) −23692.6 −0.662229
\(21\) 21811.1 0.513937
\(22\) −98128.3 −1.96479
\(23\) −21320.0 −0.365376 −0.182688 0.983171i \(-0.558480\pi\)
−0.182688 + 0.983171i \(0.558480\pi\)
\(24\) 71151.2 1.05061
\(25\) −69917.5 −0.894944
\(26\) 57653.6 0.643309
\(27\) −19683.0 −0.192450
\(28\) −211262. −1.81873
\(29\) 144361. 1.09915 0.549575 0.835444i \(-0.314790\pi\)
0.549575 + 0.835444i \(0.314790\pi\)
\(30\) −48276.4 −0.326445
\(31\) 124102. 0.748190 0.374095 0.927390i \(-0.377953\pi\)
0.374095 + 0.927390i \(0.377953\pi\)
\(32\) −28500.9 −0.153757
\(33\) −134243. −0.650270
\(34\) 578885. 2.52590
\(35\) 73184.5 0.288523
\(36\) 190649. 0.681047
\(37\) 249110. 0.808510 0.404255 0.914646i \(-0.367531\pi\)
0.404255 + 0.914646i \(0.367531\pi\)
\(38\) −175810. −0.519757
\(39\) 78872.3 0.212911
\(40\) 238739. 0.589812
\(41\) 769960. 1.74472 0.872358 0.488867i \(-0.162590\pi\)
0.872358 + 0.488867i \(0.162590\pi\)
\(42\) −430470. −0.896542
\(43\) −214862. −0.412116 −0.206058 0.978540i \(-0.566064\pi\)
−0.206058 + 0.978540i \(0.566064\pi\)
\(44\) 1.30028e6 2.30119
\(45\) −66043.9 −0.108041
\(46\) 420778. 0.637383
\(47\) 826532. 1.16123 0.580614 0.814179i \(-0.302813\pi\)
0.580614 + 0.814179i \(0.302813\pi\)
\(48\) −500442. −0.653145
\(49\) −170973. −0.207606
\(50\) 1.37991e6 1.56119
\(51\) 791936. 0.835977
\(52\) −763956. −0.753454
\(53\) 791662. 0.730423 0.365211 0.930925i \(-0.380997\pi\)
0.365211 + 0.930925i \(0.380997\pi\)
\(54\) 388470. 0.335721
\(55\) −450437. −0.365060
\(56\) 2.12879e6 1.61985
\(57\) −240514. −0.172020
\(58\) −2.84915e6 −1.91742
\(59\) 205379. 0.130189
\(60\) 639701. 0.382338
\(61\) −2.23090e6 −1.25842 −0.629210 0.777236i \(-0.716621\pi\)
−0.629210 + 0.777236i \(0.716621\pi\)
\(62\) −2.44931e6 −1.30519
\(63\) −588900. −0.296722
\(64\) −1.80996e6 −0.863057
\(65\) 264646. 0.119528
\(66\) 2.64947e6 1.13437
\(67\) −60136.4 −0.0244273 −0.0122137 0.999925i \(-0.503888\pi\)
−0.0122137 + 0.999925i \(0.503888\pi\)
\(68\) −7.67069e6 −2.95837
\(69\) 575640. 0.210950
\(70\) −1.44439e6 −0.503317
\(71\) 3.73452e6 1.23831 0.619157 0.785268i \(-0.287475\pi\)
0.619157 + 0.785268i \(0.287475\pi\)
\(72\) −1.92108e6 −0.606572
\(73\) 476492. 0.143359 0.0716796 0.997428i \(-0.477164\pi\)
0.0716796 + 0.997428i \(0.477164\pi\)
\(74\) −4.91652e6 −1.41041
\(75\) 1.88777e6 0.516696
\(76\) 2.32962e6 0.608748
\(77\) −4.01645e6 −1.00259
\(78\) −1.55665e6 −0.371415
\(79\) 1.82860e6 0.417277 0.208638 0.977993i \(-0.433097\pi\)
0.208638 + 0.977993i \(0.433097\pi\)
\(80\) −1.67917e6 −0.366674
\(81\) 531441. 0.111111
\(82\) −1.51962e7 −3.04359
\(83\) 4.01281e6 0.770327 0.385164 0.922848i \(-0.374145\pi\)
0.385164 + 0.922848i \(0.374145\pi\)
\(84\) 5.70408e6 1.05005
\(85\) 2.65724e6 0.469316
\(86\) 4.24058e6 0.718920
\(87\) −3.89775e6 −0.634595
\(88\) −1.31023e7 −2.04955
\(89\) 5.38781e6 0.810116 0.405058 0.914291i \(-0.367251\pi\)
0.405058 + 0.914291i \(0.367251\pi\)
\(90\) 1.30346e6 0.188473
\(91\) 2.35979e6 0.328269
\(92\) −5.57565e6 −0.746514
\(93\) −3.35075e6 −0.431968
\(94\) −1.63127e7 −2.02571
\(95\) −807017. −0.0965717
\(96\) 769525. 0.0887715
\(97\) −1.52550e7 −1.69711 −0.848556 0.529106i \(-0.822527\pi\)
−0.848556 + 0.529106i \(0.822527\pi\)
\(98\) 3.37437e6 0.362160
\(99\) 3.62457e6 0.375433
\(100\) −1.82850e7 −1.82850
\(101\) −1.51903e7 −1.46703 −0.733517 0.679671i \(-0.762123\pi\)
−0.733517 + 0.679671i \(0.762123\pi\)
\(102\) −1.56299e7 −1.45833
\(103\) −7.35870e6 −0.663546 −0.331773 0.943359i \(-0.607647\pi\)
−0.331773 + 0.943359i \(0.607647\pi\)
\(104\) 7.69802e6 0.671061
\(105\) −1.97598e6 −0.166579
\(106\) −1.56245e7 −1.27419
\(107\) −3.56527e6 −0.281352 −0.140676 0.990056i \(-0.544928\pi\)
−0.140676 + 0.990056i \(0.544928\pi\)
\(108\) −5.14753e6 −0.393202
\(109\) 1.51208e7 1.11836 0.559179 0.829047i \(-0.311116\pi\)
0.559179 + 0.829047i \(0.311116\pi\)
\(110\) 8.88996e6 0.636833
\(111\) −6.72598e6 −0.466794
\(112\) −1.49728e7 −1.00703
\(113\) −2.38727e7 −1.55642 −0.778212 0.628002i \(-0.783873\pi\)
−0.778212 + 0.628002i \(0.783873\pi\)
\(114\) 4.74687e6 0.300082
\(115\) 1.93149e6 0.118427
\(116\) 3.77536e7 2.24572
\(117\) −2.12955e6 −0.122924
\(118\) −4.05342e6 −0.227109
\(119\) 2.36941e7 1.28892
\(120\) −6.44595e6 −0.340528
\(121\) 5.23332e6 0.268552
\(122\) 4.40297e7 2.19526
\(123\) −2.07889e7 −1.00731
\(124\) 3.24553e7 1.52866
\(125\) 1.34119e7 0.614195
\(126\) 1.16227e7 0.517619
\(127\) 6.26438e6 0.271372 0.135686 0.990752i \(-0.456676\pi\)
0.135686 + 0.990752i \(0.456676\pi\)
\(128\) 3.93701e7 1.65933
\(129\) 5.80127e6 0.237935
\(130\) −5.22314e6 −0.208511
\(131\) −2.94087e7 −1.14295 −0.571474 0.820620i \(-0.693628\pi\)
−0.571474 + 0.820620i \(0.693628\pi\)
\(132\) −3.51075e7 −1.32859
\(133\) −7.19600e6 −0.265222
\(134\) 1.18687e6 0.0426124
\(135\) 1.78319e6 0.0623776
\(136\) 7.72938e7 2.63486
\(137\) −4.62523e6 −0.153678 −0.0768390 0.997044i \(-0.524483\pi\)
−0.0768390 + 0.997044i \(0.524483\pi\)
\(138\) −1.13610e7 −0.367993
\(139\) −1.29866e7 −0.410152 −0.205076 0.978746i \(-0.565744\pi\)
−0.205076 + 0.978746i \(0.565744\pi\)
\(140\) 1.91393e7 0.589493
\(141\) −2.23164e7 −0.670435
\(142\) −7.37056e7 −2.16019
\(143\) −1.45241e7 −0.415349
\(144\) 1.35119e7 0.377093
\(145\) −1.30784e7 −0.356260
\(146\) −9.40419e6 −0.250084
\(147\) 4.61626e6 0.119861
\(148\) 6.51478e7 1.65190
\(149\) −6.03191e7 −1.49384 −0.746918 0.664917i \(-0.768467\pi\)
−0.746918 + 0.664917i \(0.768467\pi\)
\(150\) −3.72577e7 −0.901356
\(151\) −9.08405e6 −0.214714 −0.107357 0.994221i \(-0.534239\pi\)
−0.107357 + 0.994221i \(0.534239\pi\)
\(152\) −2.34745e7 −0.542180
\(153\) −2.13823e7 −0.482652
\(154\) 7.92699e7 1.74898
\(155\) −1.12430e7 −0.242506
\(156\) 2.06268e7 0.435007
\(157\) 5.23075e7 1.07874 0.539368 0.842070i \(-0.318663\pi\)
0.539368 + 0.842070i \(0.318663\pi\)
\(158\) −3.60898e7 −0.727923
\(159\) −2.13749e7 −0.421710
\(160\) 2.58205e6 0.0498361
\(161\) 1.72227e7 0.325245
\(162\) −1.04887e7 −0.193829
\(163\) 1.85380e7 0.335280 0.167640 0.985848i \(-0.446385\pi\)
0.167640 + 0.985848i \(0.446385\pi\)
\(164\) 2.01361e8 3.56470
\(165\) 1.21618e7 0.210768
\(166\) −7.91980e7 −1.34380
\(167\) 1.04169e7 0.173074 0.0865369 0.996249i \(-0.472420\pi\)
0.0865369 + 0.996249i \(0.472420\pi\)
\(168\) −5.74772e7 −0.935219
\(169\) −5.42151e7 −0.864007
\(170\) −5.24442e7 −0.818703
\(171\) 6.49389e6 0.0993158
\(172\) −5.61911e7 −0.842011
\(173\) 7.37218e7 1.08252 0.541259 0.840856i \(-0.317948\pi\)
0.541259 + 0.840856i \(0.317948\pi\)
\(174\) 7.69272e7 1.10703
\(175\) 5.64806e7 0.796648
\(176\) 9.21549e7 1.27416
\(177\) −5.54523e6 −0.0751646
\(178\) −1.06335e8 −1.41321
\(179\) −9.01187e7 −1.17444 −0.587218 0.809429i \(-0.699777\pi\)
−0.587218 + 0.809429i \(0.699777\pi\)
\(180\) −1.72719e7 −0.220743
\(181\) 5.95539e7 0.746509 0.373255 0.927729i \(-0.378242\pi\)
0.373255 + 0.927729i \(0.378242\pi\)
\(182\) −4.65736e7 −0.572651
\(183\) 6.02343e7 0.726549
\(184\) 5.61831e7 0.664880
\(185\) −2.25682e7 −0.262057
\(186\) 6.61314e7 0.753551
\(187\) −1.45833e8 −1.63083
\(188\) 2.16156e8 2.37255
\(189\) 1.59003e7 0.171312
\(190\) 1.59275e7 0.168465
\(191\) 8.79003e7 0.912795 0.456398 0.889776i \(-0.349139\pi\)
0.456398 + 0.889776i \(0.349139\pi\)
\(192\) 4.88690e7 0.498286
\(193\) 9.19947e7 0.921112 0.460556 0.887631i \(-0.347650\pi\)
0.460556 + 0.887631i \(0.347650\pi\)
\(194\) 3.01077e8 2.96054
\(195\) −7.14545e6 −0.0690094
\(196\) −4.47131e7 −0.424168
\(197\) −9.51132e7 −0.886357 −0.443179 0.896433i \(-0.646149\pi\)
−0.443179 + 0.896433i \(0.646149\pi\)
\(198\) −7.15356e7 −0.654928
\(199\) −9.84642e7 −0.885712 −0.442856 0.896593i \(-0.646035\pi\)
−0.442856 + 0.896593i \(0.646035\pi\)
\(200\) 1.84249e8 1.62854
\(201\) 1.62368e6 0.0141031
\(202\) 2.99800e8 2.55918
\(203\) −1.16618e8 −0.978425
\(204\) 2.07109e8 1.70802
\(205\) −6.97547e7 −0.565503
\(206\) 1.45234e8 1.15753
\(207\) −1.55423e7 −0.121792
\(208\) −5.41440e7 −0.417185
\(209\) 4.42900e7 0.335578
\(210\) 3.89986e7 0.290590
\(211\) −1.85405e8 −1.35873 −0.679364 0.733802i \(-0.737744\pi\)
−0.679364 + 0.733802i \(0.737744\pi\)
\(212\) 2.07037e8 1.49236
\(213\) −1.00832e8 −0.714940
\(214\) 7.03654e7 0.490807
\(215\) 1.94655e7 0.133576
\(216\) 5.18692e7 0.350204
\(217\) −1.00252e8 −0.666013
\(218\) −2.98428e8 −1.95093
\(219\) −1.28653e7 −0.0827685
\(220\) −1.17799e8 −0.745869
\(221\) 8.56814e7 0.533966
\(222\) 1.32746e8 0.814303
\(223\) −5.33953e7 −0.322431 −0.161215 0.986919i \(-0.551541\pi\)
−0.161215 + 0.986919i \(0.551541\pi\)
\(224\) 2.30236e7 0.136869
\(225\) −5.09699e7 −0.298315
\(226\) 4.71160e8 2.71512
\(227\) −7.79437e7 −0.442273 −0.221137 0.975243i \(-0.570977\pi\)
−0.221137 + 0.975243i \(0.570977\pi\)
\(228\) −6.28998e7 −0.351461
\(229\) −2.50546e7 −0.137868 −0.0689340 0.997621i \(-0.521960\pi\)
−0.0689340 + 0.997621i \(0.521960\pi\)
\(230\) −3.81205e7 −0.206591
\(231\) 1.08444e8 0.578847
\(232\) −3.80424e8 −2.00014
\(233\) 2.23037e8 1.15513 0.577567 0.816344i \(-0.304002\pi\)
0.577567 + 0.816344i \(0.304002\pi\)
\(234\) 4.20295e7 0.214436
\(235\) −7.48799e7 −0.376381
\(236\) 5.37111e7 0.265994
\(237\) −4.93722e7 −0.240915
\(238\) −4.67634e8 −2.24847
\(239\) −1.46524e8 −0.694249 −0.347124 0.937819i \(-0.612842\pi\)
−0.347124 + 0.937819i \(0.612842\pi\)
\(240\) 4.53376e7 0.211699
\(241\) −5.35245e7 −0.246316 −0.123158 0.992387i \(-0.539302\pi\)
−0.123158 + 0.992387i \(0.539302\pi\)
\(242\) −1.03286e8 −0.468478
\(243\) −1.43489e7 −0.0641500
\(244\) −5.83429e8 −2.57113
\(245\) 1.54893e7 0.0672900
\(246\) 4.10297e8 1.75722
\(247\) −2.60218e7 −0.109875
\(248\) −3.27037e8 −1.36149
\(249\) −1.08346e8 −0.444749
\(250\) −2.64702e8 −1.07144
\(251\) −3.32032e8 −1.32532 −0.662661 0.748919i \(-0.730573\pi\)
−0.662661 + 0.748919i \(0.730573\pi\)
\(252\) −1.54010e8 −0.606244
\(253\) −1.06002e8 −0.411523
\(254\) −1.23636e8 −0.473398
\(255\) −7.17456e7 −0.270960
\(256\) −5.45345e8 −2.03157
\(257\) 3.09916e8 1.13888 0.569441 0.822032i \(-0.307160\pi\)
0.569441 + 0.822032i \(0.307160\pi\)
\(258\) −1.14496e8 −0.415069
\(259\) −2.01236e8 −0.719708
\(260\) 6.92108e7 0.244212
\(261\) 1.05239e8 0.366383
\(262\) 5.80419e8 1.99383
\(263\) −8.42828e7 −0.285689 −0.142845 0.989745i \(-0.545625\pi\)
−0.142845 + 0.989745i \(0.545625\pi\)
\(264\) 3.53762e8 1.18331
\(265\) −7.17208e7 −0.236747
\(266\) 1.42022e8 0.462670
\(267\) −1.45471e8 −0.467721
\(268\) −1.57270e7 −0.0499084
\(269\) −1.91871e8 −0.601001 −0.300501 0.953782i \(-0.597154\pi\)
−0.300501 + 0.953782i \(0.597154\pi\)
\(270\) −3.51935e7 −0.108815
\(271\) −7.09645e6 −0.0216595 −0.0108298 0.999941i \(-0.503447\pi\)
−0.0108298 + 0.999941i \(0.503447\pi\)
\(272\) −5.43646e8 −1.63804
\(273\) −6.37145e7 −0.189526
\(274\) 9.12851e7 0.268085
\(275\) −3.47628e8 −1.00798
\(276\) 1.50542e8 0.431000
\(277\) −2.59631e8 −0.733968 −0.366984 0.930227i \(-0.619610\pi\)
−0.366984 + 0.930227i \(0.619610\pi\)
\(278\) 2.56308e8 0.715493
\(279\) 9.04702e7 0.249397
\(280\) −1.92858e8 −0.525030
\(281\) −4.92572e7 −0.132433 −0.0662167 0.997805i \(-0.521093\pi\)
−0.0662167 + 0.997805i \(0.521093\pi\)
\(282\) 4.40443e8 1.16955
\(283\) −1.62830e8 −0.427053 −0.213526 0.976937i \(-0.568495\pi\)
−0.213526 + 0.976937i \(0.568495\pi\)
\(284\) 9.76658e8 2.53005
\(285\) 2.17895e7 0.0557557
\(286\) 2.86652e8 0.724559
\(287\) −6.21988e8 −1.55309
\(288\) −2.07772e7 −0.0512522
\(289\) 4.49966e8 1.09657
\(290\) 2.58120e8 0.621482
\(291\) 4.11884e8 0.979828
\(292\) 1.24613e8 0.292903
\(293\) −5.14011e8 −1.19381 −0.596905 0.802312i \(-0.703603\pi\)
−0.596905 + 0.802312i \(0.703603\pi\)
\(294\) −9.11079e7 −0.209093
\(295\) −1.86064e7 −0.0421973
\(296\) −6.56462e8 −1.47126
\(297\) −9.78633e7 −0.216757
\(298\) 1.19048e9 2.60594
\(299\) 6.22799e7 0.134741
\(300\) 4.93694e8 1.05568
\(301\) 1.73569e8 0.366852
\(302\) 1.79286e8 0.374560
\(303\) 4.10137e8 0.846993
\(304\) 1.65108e8 0.337062
\(305\) 2.02109e8 0.407883
\(306\) 4.22007e8 0.841966
\(307\) 1.27895e8 0.252272 0.126136 0.992013i \(-0.459742\pi\)
0.126136 + 0.992013i \(0.459742\pi\)
\(308\) −1.05039e9 −2.04844
\(309\) 1.98685e8 0.383098
\(310\) 2.21896e8 0.423042
\(311\) 6.02895e8 1.13653 0.568264 0.822846i \(-0.307615\pi\)
0.568264 + 0.822846i \(0.307615\pi\)
\(312\) −2.07846e8 −0.387437
\(313\) −8.09308e8 −1.49179 −0.745897 0.666061i \(-0.767979\pi\)
−0.745897 + 0.666061i \(0.767979\pi\)
\(314\) −1.03236e9 −1.88181
\(315\) 5.33515e7 0.0961744
\(316\) 4.78219e8 0.852555
\(317\) 9.39100e7 0.165579 0.0827893 0.996567i \(-0.473617\pi\)
0.0827893 + 0.996567i \(0.473617\pi\)
\(318\) 4.21861e8 0.735655
\(319\) 7.17759e8 1.23797
\(320\) 1.63974e8 0.279737
\(321\) 9.62624e7 0.162439
\(322\) −3.39912e8 −0.567376
\(323\) −2.61278e8 −0.431415
\(324\) 1.38983e8 0.227016
\(325\) 2.04243e8 0.330031
\(326\) −3.65873e8 −0.584882
\(327\) −4.08260e8 −0.645684
\(328\) −2.02902e9 −3.17489
\(329\) −6.67688e8 −1.03368
\(330\) −2.40029e8 −0.367675
\(331\) 9.98609e8 1.51355 0.756776 0.653674i \(-0.226773\pi\)
0.756776 + 0.653674i \(0.226773\pi\)
\(332\) 1.04944e9 1.57389
\(333\) 1.81601e8 0.269503
\(334\) −2.05591e8 −0.301920
\(335\) 5.44807e6 0.00791746
\(336\) 4.04266e8 0.581406
\(337\) 1.07195e9 1.52570 0.762851 0.646575i \(-0.223799\pi\)
0.762851 + 0.646575i \(0.223799\pi\)
\(338\) 1.07001e9 1.50722
\(339\) 6.44564e8 0.898602
\(340\) 6.94927e8 0.958878
\(341\) 6.17030e8 0.842687
\(342\) −1.28165e8 −0.173252
\(343\) 8.03388e8 1.07497
\(344\) 5.66210e8 0.749934
\(345\) −5.21502e7 −0.0683737
\(346\) −1.45500e9 −1.88841
\(347\) −1.06764e9 −1.37174 −0.685869 0.727725i \(-0.740578\pi\)
−0.685869 + 0.727725i \(0.740578\pi\)
\(348\) −1.01935e9 −1.29657
\(349\) −1.09465e9 −1.37843 −0.689216 0.724556i \(-0.742045\pi\)
−0.689216 + 0.724556i \(0.742045\pi\)
\(350\) −1.11472e9 −1.38972
\(351\) 5.74979e7 0.0709703
\(352\) −1.41706e8 −0.173176
\(353\) −2.65164e8 −0.320851 −0.160426 0.987048i \(-0.551287\pi\)
−0.160426 + 0.987048i \(0.551287\pi\)
\(354\) 1.09442e8 0.131122
\(355\) −3.38330e8 −0.401366
\(356\) 1.40903e9 1.65518
\(357\) −6.39740e8 −0.744158
\(358\) 1.77861e9 2.04876
\(359\) −7.74082e8 −0.882992 −0.441496 0.897263i \(-0.645552\pi\)
−0.441496 + 0.897263i \(0.645552\pi\)
\(360\) 1.74041e8 0.196604
\(361\) −8.14520e8 −0.911227
\(362\) −1.17537e9 −1.30226
\(363\) −1.41300e8 −0.155049
\(364\) 6.17138e8 0.670699
\(365\) −4.31679e7 −0.0464660
\(366\) −1.18880e9 −1.26744
\(367\) 6.18045e8 0.652663 0.326331 0.945255i \(-0.394187\pi\)
0.326331 + 0.945255i \(0.394187\pi\)
\(368\) −3.95164e8 −0.413342
\(369\) 5.61301e8 0.581572
\(370\) 4.45413e8 0.457148
\(371\) −6.39519e8 −0.650197
\(372\) −8.76294e8 −0.882571
\(373\) −2.27726e7 −0.0227212 −0.0113606 0.999935i \(-0.503616\pi\)
−0.0113606 + 0.999935i \(0.503616\pi\)
\(374\) 2.87820e9 2.84492
\(375\) −3.62122e8 −0.354606
\(376\) −2.17810e9 −2.11310
\(377\) −4.21707e8 −0.405337
\(378\) −3.13813e8 −0.298847
\(379\) −9.75855e8 −0.920763 −0.460382 0.887721i \(-0.652287\pi\)
−0.460382 + 0.887721i \(0.652287\pi\)
\(380\) −2.11053e8 −0.197310
\(381\) −1.69138e8 −0.156677
\(382\) −1.73483e9 −1.59233
\(383\) −8.32929e8 −0.757552 −0.378776 0.925488i \(-0.623655\pi\)
−0.378776 + 0.925488i \(0.623655\pi\)
\(384\) −1.06299e9 −0.958012
\(385\) 3.63871e8 0.324964
\(386\) −1.81564e9 −1.60684
\(387\) −1.56634e8 −0.137372
\(388\) −3.98951e9 −3.46744
\(389\) −1.77468e8 −0.152861 −0.0764304 0.997075i \(-0.524352\pi\)
−0.0764304 + 0.997075i \(0.524352\pi\)
\(390\) 1.41025e8 0.120384
\(391\) 6.25336e8 0.529048
\(392\) 4.50552e8 0.377784
\(393\) 7.94035e8 0.659881
\(394\) 1.87718e9 1.54621
\(395\) −1.65663e8 −0.135249
\(396\) 9.47903e8 0.767063
\(397\) −1.50894e9 −1.21033 −0.605166 0.796099i \(-0.706893\pi\)
−0.605166 + 0.796099i \(0.706893\pi\)
\(398\) 1.94332e9 1.54509
\(399\) 1.94292e8 0.153126
\(400\) −1.29591e9 −1.01243
\(401\) 1.77996e9 1.37850 0.689248 0.724525i \(-0.257941\pi\)
0.689248 + 0.724525i \(0.257941\pi\)
\(402\) −3.20455e7 −0.0246023
\(403\) −3.62525e8 −0.275912
\(404\) −3.97258e9 −2.99736
\(405\) −4.81460e7 −0.0360137
\(406\) 2.30160e9 1.70682
\(407\) 1.23857e9 0.910625
\(408\) −2.08693e9 −1.52124
\(409\) 1.01746e9 0.735335 0.367667 0.929957i \(-0.380157\pi\)
0.367667 + 0.929957i \(0.380157\pi\)
\(410\) 1.37670e9 0.986497
\(411\) 1.24881e8 0.0887260
\(412\) −1.92446e9 −1.35572
\(413\) −1.65909e8 −0.115890
\(414\) 3.06747e8 0.212461
\(415\) −3.63541e8 −0.249681
\(416\) 8.32568e7 0.0567013
\(417\) 3.50639e8 0.236801
\(418\) −8.74122e8 −0.585403
\(419\) −1.28968e9 −0.856514 −0.428257 0.903657i \(-0.640872\pi\)
−0.428257 + 0.903657i \(0.640872\pi\)
\(420\) −5.16762e8 −0.340344
\(421\) 5.35636e8 0.349851 0.174925 0.984582i \(-0.444032\pi\)
0.174925 + 0.984582i \(0.444032\pi\)
\(422\) 3.65921e9 2.37025
\(423\) 6.02542e8 0.387076
\(424\) −2.08621e9 −1.32916
\(425\) 2.05075e9 1.29584
\(426\) 1.99005e9 1.24718
\(427\) 1.80216e9 1.12020
\(428\) −9.32397e8 −0.574841
\(429\) 3.92151e8 0.239802
\(430\) −3.84176e8 −0.233019
\(431\) −2.75089e9 −1.65502 −0.827509 0.561452i \(-0.810243\pi\)
−0.827509 + 0.561452i \(0.810243\pi\)
\(432\) −3.64822e8 −0.217715
\(433\) 8.00172e8 0.473670 0.236835 0.971550i \(-0.423890\pi\)
0.236835 + 0.971550i \(0.423890\pi\)
\(434\) 1.97860e9 1.16183
\(435\) 3.53117e8 0.205687
\(436\) 3.95441e9 2.28496
\(437\) −1.89917e8 −0.108863
\(438\) 2.53913e8 0.144386
\(439\) 7.72445e8 0.435754 0.217877 0.975976i \(-0.430087\pi\)
0.217877 + 0.975976i \(0.430087\pi\)
\(440\) 1.18700e9 0.664305
\(441\) −1.24639e8 −0.0692020
\(442\) −1.69104e9 −0.931483
\(443\) 2.85561e9 1.56058 0.780289 0.625419i \(-0.215072\pi\)
0.780289 + 0.625419i \(0.215072\pi\)
\(444\) −1.75899e9 −0.953724
\(445\) −4.88110e8 −0.262577
\(446\) 1.05383e9 0.562467
\(447\) 1.62861e9 0.862466
\(448\) 1.46212e9 0.768263
\(449\) 1.99990e9 1.04267 0.521334 0.853352i \(-0.325434\pi\)
0.521334 + 0.853352i \(0.325434\pi\)
\(450\) 1.00596e9 0.520398
\(451\) 3.82822e9 1.96507
\(452\) −6.24324e9 −3.17999
\(453\) 2.45269e8 0.123965
\(454\) 1.53832e9 0.771528
\(455\) −2.13786e8 −0.106399
\(456\) 6.33811e8 0.313027
\(457\) −1.08512e9 −0.531829 −0.265915 0.963997i \(-0.585674\pi\)
−0.265915 + 0.963997i \(0.585674\pi\)
\(458\) 4.94485e8 0.240505
\(459\) 5.77321e8 0.278659
\(460\) 5.05127e8 0.241962
\(461\) −3.43732e9 −1.63405 −0.817027 0.576600i \(-0.804379\pi\)
−0.817027 + 0.576600i \(0.804379\pi\)
\(462\) −2.14029e9 −1.00978
\(463\) −2.83083e9 −1.32550 −0.662750 0.748840i \(-0.730611\pi\)
−0.662750 + 0.748840i \(0.730611\pi\)
\(464\) 2.67572e9 1.24345
\(465\) 3.03562e8 0.140011
\(466\) −4.40194e9 −2.01508
\(467\) −1.27289e9 −0.578337 −0.289169 0.957278i \(-0.593379\pi\)
−0.289169 + 0.957278i \(0.593379\pi\)
\(468\) −5.56924e8 −0.251151
\(469\) 4.85793e7 0.0217443
\(470\) 1.47785e9 0.656581
\(471\) −1.41230e9 −0.622809
\(472\) −5.41221e8 −0.236907
\(473\) −1.06829e9 −0.464167
\(474\) 9.74426e8 0.420266
\(475\) −6.22821e8 −0.266646
\(476\) 6.19652e9 2.63344
\(477\) 5.77121e8 0.243474
\(478\) 2.89184e9 1.21109
\(479\) −1.83875e9 −0.764448 −0.382224 0.924070i \(-0.624842\pi\)
−0.382224 + 0.924070i \(0.624842\pi\)
\(480\) −6.97153e7 −0.0287729
\(481\) −7.27699e8 −0.298156
\(482\) 1.05638e9 0.429689
\(483\) −4.65013e8 −0.187780
\(484\) 1.36863e9 0.548689
\(485\) 1.38203e9 0.550073
\(486\) 2.83194e8 0.111907
\(487\) −1.80837e9 −0.709473 −0.354736 0.934966i \(-0.615429\pi\)
−0.354736 + 0.934966i \(0.615429\pi\)
\(488\) 5.87893e9 2.28996
\(489\) −5.00527e8 −0.193574
\(490\) −3.05701e8 −0.117385
\(491\) 3.14661e9 1.19966 0.599829 0.800128i \(-0.295235\pi\)
0.599829 + 0.800128i \(0.295235\pi\)
\(492\) −5.43676e9 −2.05808
\(493\) −4.23425e9 −1.59152
\(494\) 5.13575e8 0.191672
\(495\) −3.28368e8 −0.121687
\(496\) 2.30021e9 0.846412
\(497\) −3.01681e9 −1.10230
\(498\) 2.13835e9 0.775846
\(499\) 2.26779e9 0.817056 0.408528 0.912746i \(-0.366042\pi\)
0.408528 + 0.912746i \(0.366042\pi\)
\(500\) 3.50752e9 1.25489
\(501\) −2.81257e8 −0.0999242
\(502\) 6.55308e9 2.31197
\(503\) 1.65622e9 0.580270 0.290135 0.956986i \(-0.406300\pi\)
0.290135 + 0.956986i \(0.406300\pi\)
\(504\) 1.55188e9 0.539949
\(505\) 1.37616e9 0.475500
\(506\) 2.09210e9 0.717885
\(507\) 1.46381e9 0.498834
\(508\) 1.63827e9 0.554451
\(509\) 5.27031e9 1.77143 0.885715 0.464229i \(-0.153668\pi\)
0.885715 + 0.464229i \(0.153668\pi\)
\(510\) 1.41599e9 0.472678
\(511\) −3.84919e8 −0.127613
\(512\) 5.72373e9 1.88466
\(513\) −1.75335e8 −0.0573400
\(514\) −6.11660e9 −1.98673
\(515\) 6.66663e8 0.215071
\(516\) 1.51716e9 0.486135
\(517\) 4.10949e9 1.30789
\(518\) 3.97165e9 1.25550
\(519\) −1.99049e9 −0.624992
\(520\) −6.97403e8 −0.217507
\(521\) −1.06833e9 −0.330957 −0.165478 0.986213i \(-0.552917\pi\)
−0.165478 + 0.986213i \(0.552917\pi\)
\(522\) −2.07703e9 −0.639141
\(523\) −4.88919e9 −1.49445 −0.747225 0.664572i \(-0.768614\pi\)
−0.747225 + 0.664572i \(0.768614\pi\)
\(524\) −7.69102e9 −2.33520
\(525\) −1.52498e9 −0.459945
\(526\) 1.66343e9 0.498374
\(527\) −3.64002e9 −1.08335
\(528\) −2.48818e9 −0.735637
\(529\) −2.95028e9 −0.866500
\(530\) 1.41550e9 0.412995
\(531\) 1.49721e8 0.0433963
\(532\) −1.88191e9 −0.541887
\(533\) −2.24920e9 −0.643404
\(534\) 2.87106e9 0.815920
\(535\) 3.22997e8 0.0911927
\(536\) 1.58473e8 0.0444507
\(537\) 2.43320e9 0.678061
\(538\) 3.78682e9 1.04842
\(539\) −8.50070e8 −0.233827
\(540\) 4.66342e8 0.127446
\(541\) 2.76671e9 0.751231 0.375615 0.926776i \(-0.377431\pi\)
0.375615 + 0.926776i \(0.377431\pi\)
\(542\) 1.40058e8 0.0377842
\(543\) −1.60796e9 −0.430997
\(544\) 8.35960e8 0.222633
\(545\) −1.36987e9 −0.362486
\(546\) 1.25749e9 0.330620
\(547\) −5.33645e9 −1.39411 −0.697054 0.717018i \(-0.745506\pi\)
−0.697054 + 0.717018i \(0.745506\pi\)
\(548\) −1.20960e9 −0.313986
\(549\) −1.62633e9 −0.419473
\(550\) 6.86089e9 1.75837
\(551\) 1.28596e9 0.327489
\(552\) −1.51694e9 −0.383869
\(553\) −1.47718e9 −0.371445
\(554\) 5.12416e9 1.28038
\(555\) 6.09341e8 0.151299
\(556\) −3.39629e9 −0.837998
\(557\) −4.61945e8 −0.113265 −0.0566326 0.998395i \(-0.518036\pi\)
−0.0566326 + 0.998395i \(0.518036\pi\)
\(558\) −1.78555e9 −0.435063
\(559\) 6.27653e8 0.151977
\(560\) 1.35647e9 0.326400
\(561\) 3.93748e9 0.941561
\(562\) 9.72156e8 0.231025
\(563\) 5.71818e9 1.35045 0.675224 0.737612i \(-0.264047\pi\)
0.675224 + 0.737612i \(0.264047\pi\)
\(564\) −5.83622e9 −1.36979
\(565\) 2.16276e9 0.504473
\(566\) 3.21366e9 0.744976
\(567\) −4.29308e8 −0.0989072
\(568\) −9.84131e9 −2.25338
\(569\) −5.60950e9 −1.27653 −0.638265 0.769816i \(-0.720348\pi\)
−0.638265 + 0.769816i \(0.720348\pi\)
\(570\) −4.30043e8 −0.0972636
\(571\) 3.21411e9 0.722494 0.361247 0.932470i \(-0.382351\pi\)
0.361247 + 0.932470i \(0.382351\pi\)
\(572\) −3.79837e9 −0.848615
\(573\) −2.37331e9 −0.527003
\(574\) 1.22757e10 2.70930
\(575\) 1.49064e9 0.326991
\(576\) −1.31946e9 −0.287686
\(577\) 2.42295e9 0.525085 0.262543 0.964920i \(-0.415439\pi\)
0.262543 + 0.964920i \(0.415439\pi\)
\(578\) −8.88068e9 −1.91293
\(579\) −2.48386e9 −0.531804
\(580\) −3.42029e9 −0.727889
\(581\) −3.24162e9 −0.685718
\(582\) −8.12907e9 −1.70927
\(583\) 3.93612e9 0.822675
\(584\) −1.25567e9 −0.260873
\(585\) 1.92927e8 0.0398426
\(586\) 1.01447e10 2.08255
\(587\) −8.83936e9 −1.80380 −0.901898 0.431949i \(-0.857826\pi\)
−0.901898 + 0.431949i \(0.857826\pi\)
\(588\) 1.20725e9 0.244894
\(589\) 1.10549e9 0.222921
\(590\) 3.67221e8 0.0736114
\(591\) 2.56806e9 0.511739
\(592\) 4.61723e9 0.914651
\(593\) −2.57682e9 −0.507450 −0.253725 0.967276i \(-0.581656\pi\)
−0.253725 + 0.967276i \(0.581656\pi\)
\(594\) 1.93146e9 0.378123
\(595\) −2.14657e9 −0.417769
\(596\) −1.57748e10 −3.05211
\(597\) 2.65853e9 0.511366
\(598\) −1.22917e9 −0.235050
\(599\) −6.40768e9 −1.21817 −0.609084 0.793106i \(-0.708463\pi\)
−0.609084 + 0.793106i \(0.708463\pi\)
\(600\) −4.97471e9 −0.940240
\(601\) 2.70821e9 0.508887 0.254443 0.967088i \(-0.418108\pi\)
0.254443 + 0.967088i \(0.418108\pi\)
\(602\) −3.42562e9 −0.639958
\(603\) −4.38394e7 −0.00814244
\(604\) −2.37568e9 −0.438691
\(605\) −4.74114e8 −0.0870439
\(606\) −8.09459e9 −1.47754
\(607\) −8.53113e9 −1.54827 −0.774134 0.633022i \(-0.781814\pi\)
−0.774134 + 0.633022i \(0.781814\pi\)
\(608\) −2.53885e8 −0.0458114
\(609\) 3.14867e9 0.564894
\(610\) −3.98888e9 −0.711535
\(611\) −2.41446e9 −0.428229
\(612\) −5.59193e9 −0.986125
\(613\) −2.93110e9 −0.513947 −0.256973 0.966418i \(-0.582725\pi\)
−0.256973 + 0.966418i \(0.582725\pi\)
\(614\) −2.52418e9 −0.440079
\(615\) 1.88338e9 0.326493
\(616\) 1.05843e10 1.82443
\(617\) 7.69933e9 1.31964 0.659819 0.751425i \(-0.270633\pi\)
0.659819 + 0.751425i \(0.270633\pi\)
\(618\) −3.92131e9 −0.668300
\(619\) 1.43713e9 0.243545 0.121772 0.992558i \(-0.461142\pi\)
0.121772 + 0.992558i \(0.461142\pi\)
\(620\) −2.94030e9 −0.495473
\(621\) 4.19642e8 0.0703166
\(622\) −1.18989e10 −1.98263
\(623\) −4.35237e9 −0.721137
\(624\) 1.46189e9 0.240862
\(625\) 4.24725e9 0.695869
\(626\) 1.59728e10 2.60237
\(627\) −1.19583e9 −0.193746
\(628\) 1.36796e10 2.20401
\(629\) −7.30664e9 −1.17069
\(630\) −1.05296e9 −0.167772
\(631\) −7.31824e9 −1.15959 −0.579794 0.814763i \(-0.696867\pi\)
−0.579794 + 0.814763i \(0.696867\pi\)
\(632\) −4.81878e9 −0.759325
\(633\) 5.00593e9 0.784462
\(634\) −1.85344e9 −0.288845
\(635\) −5.67523e8 −0.0879580
\(636\) −5.59000e9 −0.861612
\(637\) 4.99444e8 0.0765595
\(638\) −1.41659e10 −2.15959
\(639\) 2.72246e9 0.412771
\(640\) −3.56674e9 −0.537826
\(641\) 1.21948e10 1.82883 0.914414 0.404779i \(-0.132652\pi\)
0.914414 + 0.404779i \(0.132652\pi\)
\(642\) −1.89986e9 −0.283368
\(643\) 1.07659e10 1.59702 0.798510 0.601982i \(-0.205622\pi\)
0.798510 + 0.601982i \(0.205622\pi\)
\(644\) 4.50411e9 0.664521
\(645\) −5.25567e8 −0.0771204
\(646\) 5.15667e9 0.752585
\(647\) −1.11488e10 −1.61831 −0.809156 0.587594i \(-0.800075\pi\)
−0.809156 + 0.587594i \(0.800075\pi\)
\(648\) −1.40047e9 −0.202191
\(649\) 1.02114e9 0.146632
\(650\) −4.03100e9 −0.575726
\(651\) 2.70680e9 0.384523
\(652\) 4.84810e9 0.685023
\(653\) 1.17825e10 1.65593 0.827965 0.560780i \(-0.189498\pi\)
0.827965 + 0.560780i \(0.189498\pi\)
\(654\) 8.05755e9 1.12637
\(655\) 2.66429e9 0.370456
\(656\) 1.42711e10 1.97376
\(657\) 3.47363e8 0.0477864
\(658\) 1.31777e10 1.80322
\(659\) −1.18824e10 −1.61736 −0.808680 0.588249i \(-0.799818\pi\)
−0.808680 + 0.588249i \(0.799818\pi\)
\(660\) 3.18057e9 0.430627
\(661\) −3.51347e9 −0.473185 −0.236592 0.971609i \(-0.576031\pi\)
−0.236592 + 0.971609i \(0.576031\pi\)
\(662\) −1.97089e10 −2.64033
\(663\) −2.31340e9 −0.308286
\(664\) −1.05747e10 −1.40178
\(665\) 6.51923e8 0.0859648
\(666\) −3.58414e9 −0.470138
\(667\) −3.07778e9 −0.401603
\(668\) 2.72425e9 0.353614
\(669\) 1.44167e9 0.186155
\(670\) −1.07525e8 −0.0138117
\(671\) −1.10920e10 −1.41736
\(672\) −6.21637e8 −0.0790213
\(673\) −7.05934e8 −0.0892712 −0.0446356 0.999003i \(-0.514213\pi\)
−0.0446356 + 0.999003i \(0.514213\pi\)
\(674\) −2.11563e10 −2.66152
\(675\) 1.37619e9 0.172232
\(676\) −1.41784e10 −1.76529
\(677\) −1.39303e10 −1.72544 −0.862720 0.505681i \(-0.831241\pi\)
−0.862720 + 0.505681i \(0.831241\pi\)
\(678\) −1.27213e10 −1.56757
\(679\) 1.23232e10 1.51071
\(680\) −7.00245e9 −0.854021
\(681\) 2.10448e9 0.255346
\(682\) −1.21779e10 −1.47003
\(683\) −2.00962e9 −0.241347 −0.120673 0.992692i \(-0.538505\pi\)
−0.120673 + 0.992692i \(0.538505\pi\)
\(684\) 1.69829e9 0.202916
\(685\) 4.19024e8 0.0498106
\(686\) −1.58559e10 −1.87524
\(687\) 6.76474e8 0.0795981
\(688\) −3.98244e9 −0.466219
\(689\) −2.31260e9 −0.269360
\(690\) 1.02925e9 0.119275
\(691\) −1.48060e9 −0.170712 −0.0853558 0.996351i \(-0.527203\pi\)
−0.0853558 + 0.996351i \(0.527203\pi\)
\(692\) 1.92799e10 2.21173
\(693\) −2.92799e9 −0.334198
\(694\) 2.10712e10 2.39294
\(695\) 1.17653e9 0.132940
\(696\) 1.02715e10 1.15478
\(697\) −2.25837e10 −2.52627
\(698\) 2.16043e10 2.40462
\(699\) −6.02201e9 −0.666917
\(700\) 1.47709e10 1.62766
\(701\) −9.38403e9 −1.02891 −0.514454 0.857518i \(-0.672005\pi\)
−0.514454 + 0.857518i \(0.672005\pi\)
\(702\) −1.13480e9 −0.123805
\(703\) 2.21906e9 0.240894
\(704\) −8.99908e9 −0.972061
\(705\) 2.02176e9 0.217304
\(706\) 5.23337e9 0.559712
\(707\) 1.22710e10 1.30590
\(708\) −1.45020e9 −0.153572
\(709\) 5.90748e9 0.622501 0.311251 0.950328i \(-0.399252\pi\)
0.311251 + 0.950328i \(0.399252\pi\)
\(710\) 6.67738e9 0.700167
\(711\) 1.33305e9 0.139092
\(712\) −1.41981e10 −1.47418
\(713\) −2.64585e9 −0.273371
\(714\) 1.26261e10 1.29815
\(715\) 1.31581e9 0.134624
\(716\) −2.35680e10 −2.39954
\(717\) 3.95614e9 0.400825
\(718\) 1.52775e10 1.54034
\(719\) −1.54549e10 −1.55065 −0.775327 0.631560i \(-0.782415\pi\)
−0.775327 + 0.631560i \(0.782415\pi\)
\(720\) −1.22412e9 −0.122225
\(721\) 5.94450e9 0.590665
\(722\) 1.60756e10 1.58960
\(723\) 1.44516e9 0.142211
\(724\) 1.55746e10 1.52522
\(725\) −1.00934e10 −0.983678
\(726\) 2.78873e9 0.270476
\(727\) −1.56689e8 −0.0151240 −0.00756201 0.999971i \(-0.502407\pi\)
−0.00756201 + 0.999971i \(0.502407\pi\)
\(728\) −6.21860e9 −0.597355
\(729\) 3.87420e8 0.0370370
\(730\) 8.51975e8 0.0810581
\(731\) 6.30211e9 0.596726
\(732\) 1.57526e10 1.48444
\(733\) 7.57137e9 0.710086 0.355043 0.934850i \(-0.384466\pi\)
0.355043 + 0.934850i \(0.384466\pi\)
\(734\) −1.21979e10 −1.13854
\(735\) −4.18211e8 −0.0388499
\(736\) 6.07640e8 0.0561790
\(737\) −2.98996e8 −0.0275125
\(738\) −1.10780e10 −1.01453
\(739\) −7.72084e9 −0.703734 −0.351867 0.936050i \(-0.614453\pi\)
−0.351867 + 0.936050i \(0.614453\pi\)
\(740\) −5.90208e9 −0.535419
\(741\) 7.02589e8 0.0634363
\(742\) 1.26217e10 1.13424
\(743\) 5.80399e9 0.519118 0.259559 0.965727i \(-0.416423\pi\)
0.259559 + 0.965727i \(0.416423\pi\)
\(744\) 8.82999e9 0.786058
\(745\) 5.46462e9 0.484187
\(746\) 4.49446e8 0.0396362
\(747\) 2.92534e9 0.256776
\(748\) −3.81384e10 −3.33202
\(749\) 2.88009e9 0.250450
\(750\) 7.14696e9 0.618596
\(751\) 1.00286e10 0.863975 0.431987 0.901880i \(-0.357813\pi\)
0.431987 + 0.901880i \(0.357813\pi\)
\(752\) 1.53197e10 1.31367
\(753\) 8.96486e9 0.765176
\(754\) 8.32294e9 0.707093
\(755\) 8.22971e8 0.0695938
\(756\) 4.15827e9 0.350015
\(757\) −1.32049e10 −1.10637 −0.553184 0.833059i \(-0.686587\pi\)
−0.553184 + 0.833059i \(0.686587\pi\)
\(758\) 1.92598e10 1.60623
\(759\) 2.86207e9 0.237593
\(760\) 2.12667e9 0.175733
\(761\) 2.09392e10 1.72232 0.861161 0.508332i \(-0.169738\pi\)
0.861161 + 0.508332i \(0.169738\pi\)
\(762\) 3.33817e9 0.273316
\(763\) −1.22148e10 −0.995523
\(764\) 2.29878e10 1.86497
\(765\) 1.93713e9 0.156439
\(766\) 1.64390e10 1.32152
\(767\) −5.99952e8 −0.0480101
\(768\) 1.47243e10 1.17293
\(769\) 1.74920e10 1.38706 0.693532 0.720426i \(-0.256054\pi\)
0.693532 + 0.720426i \(0.256054\pi\)
\(770\) −7.18147e9 −0.566886
\(771\) −8.36774e9 −0.657533
\(772\) 2.40586e10 1.88196
\(773\) 1.87442e10 1.45962 0.729808 0.683652i \(-0.239610\pi\)
0.729808 + 0.683652i \(0.239610\pi\)
\(774\) 3.09138e9 0.239640
\(775\) −8.67689e9 −0.669589
\(776\) 4.02003e10 3.08826
\(777\) 5.43337e9 0.415523
\(778\) 3.50256e9 0.266659
\(779\) 6.85876e9 0.519834
\(780\) −1.86869e9 −0.140996
\(781\) 1.85679e10 1.39471
\(782\) −1.23418e10 −0.922902
\(783\) −2.84146e9 −0.211532
\(784\) −3.16896e9 −0.234861
\(785\) −4.73881e9 −0.349644
\(786\) −1.56713e10 −1.15114
\(787\) −1.93622e10 −1.41594 −0.707968 0.706245i \(-0.750388\pi\)
−0.707968 + 0.706245i \(0.750388\pi\)
\(788\) −2.48742e10 −1.81095
\(789\) 2.27564e9 0.164943
\(790\) 3.26957e9 0.235937
\(791\) 1.92848e10 1.38547
\(792\) −9.55156e9 −0.683182
\(793\) 6.51689e9 0.464071
\(794\) 2.97809e10 2.11138
\(795\) 1.93646e9 0.136686
\(796\) −2.57505e10 −1.80963
\(797\) 2.19937e10 1.53884 0.769420 0.638744i \(-0.220546\pi\)
0.769420 + 0.638744i \(0.220546\pi\)
\(798\) −3.83461e9 −0.267123
\(799\) −2.42430e10 −1.68141
\(800\) 1.99271e9 0.137604
\(801\) 3.92771e9 0.270039
\(802\) −3.51299e10 −2.40473
\(803\) 2.36910e9 0.161465
\(804\) 4.24628e8 0.0288146
\(805\) −1.56029e9 −0.105419
\(806\) 7.15491e9 0.481318
\(807\) 5.18051e9 0.346988
\(808\) 4.00298e10 2.66958
\(809\) 9.05373e9 0.601185 0.300592 0.953753i \(-0.402816\pi\)
0.300592 + 0.953753i \(0.402816\pi\)
\(810\) 9.50224e8 0.0628244
\(811\) −2.27877e9 −0.150012 −0.0750062 0.997183i \(-0.523898\pi\)
−0.0750062 + 0.997183i \(0.523898\pi\)
\(812\) −3.04980e10 −1.99906
\(813\) 1.91604e8 0.0125051
\(814\) −2.44448e10 −1.58855
\(815\) −1.67946e9 −0.108672
\(816\) 1.46784e10 0.945724
\(817\) −1.91398e9 −0.122789
\(818\) −2.00809e10 −1.28276
\(819\) 1.72029e9 0.109423
\(820\) −1.82424e10 −1.15540
\(821\) 1.34643e10 0.849148 0.424574 0.905393i \(-0.360424\pi\)
0.424574 + 0.905393i \(0.360424\pi\)
\(822\) −2.46470e9 −0.154779
\(823\) 1.64535e10 1.02887 0.514434 0.857530i \(-0.328002\pi\)
0.514434 + 0.857530i \(0.328002\pi\)
\(824\) 1.93919e10 1.20746
\(825\) 9.38595e9 0.581955
\(826\) 3.27443e9 0.202165
\(827\) 1.52598e10 0.938165 0.469082 0.883154i \(-0.344585\pi\)
0.469082 + 0.883154i \(0.344585\pi\)
\(828\) −4.06465e9 −0.248838
\(829\) −1.06615e10 −0.649946 −0.324973 0.945723i \(-0.605355\pi\)
−0.324973 + 0.945723i \(0.605355\pi\)
\(830\) 7.17496e9 0.435558
\(831\) 7.01004e9 0.423757
\(832\) 5.28725e9 0.318272
\(833\) 5.01479e9 0.300604
\(834\) −6.92032e9 −0.413090
\(835\) −9.43722e8 −0.0560972
\(836\) 1.15828e10 0.685633
\(837\) −2.44270e9 −0.143989
\(838\) 2.54536e10 1.49415
\(839\) −1.09099e10 −0.637757 −0.318878 0.947796i \(-0.603306\pi\)
−0.318878 + 0.947796i \(0.603306\pi\)
\(840\) 5.20716e9 0.303126
\(841\) 3.59025e9 0.208132
\(842\) −1.05715e10 −0.610300
\(843\) 1.32994e9 0.0764605
\(844\) −4.84874e10 −2.77607
\(845\) 4.91163e9 0.280045
\(846\) −1.18920e10 −0.675238
\(847\) −4.22757e9 −0.239056
\(848\) 1.46734e10 0.826312
\(849\) 4.39641e9 0.246559
\(850\) −4.04742e10 −2.26054
\(851\) −5.31103e9 −0.295410
\(852\) −2.63698e10 −1.46072
\(853\) −3.09781e10 −1.70897 −0.854483 0.519480i \(-0.826126\pi\)
−0.854483 + 0.519480i \(0.826126\pi\)
\(854\) −3.55680e10 −1.95415
\(855\) −5.88315e8 −0.0321906
\(856\) 9.39531e9 0.511980
\(857\) 4.23395e9 0.229781 0.114890 0.993378i \(-0.463348\pi\)
0.114890 + 0.993378i \(0.463348\pi\)
\(858\) −7.73960e9 −0.418324
\(859\) 1.85088e10 0.996329 0.498165 0.867082i \(-0.334007\pi\)
0.498165 + 0.867082i \(0.334007\pi\)
\(860\) 5.09064e9 0.272915
\(861\) 1.67937e10 0.896674
\(862\) 5.42924e10 2.88711
\(863\) −1.05929e10 −0.561020 −0.280510 0.959851i \(-0.590504\pi\)
−0.280510 + 0.959851i \(0.590504\pi\)
\(864\) 5.60984e8 0.0295905
\(865\) −6.67884e9 −0.350869
\(866\) −1.57924e10 −0.826298
\(867\) −1.21491e10 −0.633107
\(868\) −2.62180e10 −1.36076
\(869\) 9.09175e9 0.469979
\(870\) −6.96923e9 −0.358813
\(871\) 1.75670e8 0.00900812
\(872\) −3.98467e10 −2.03509
\(873\) −1.11209e10 −0.565704
\(874\) 3.74827e9 0.189907
\(875\) −1.08344e10 −0.546735
\(876\) −3.36455e9 −0.169108
\(877\) −9.90824e9 −0.496018 −0.248009 0.968758i \(-0.579776\pi\)
−0.248009 + 0.968758i \(0.579776\pi\)
\(878\) −1.52452e10 −0.760156
\(879\) 1.38783e10 0.689247
\(880\) −8.34879e9 −0.412985
\(881\) 1.77972e9 0.0876873 0.0438437 0.999038i \(-0.486040\pi\)
0.0438437 + 0.999038i \(0.486040\pi\)
\(882\) 2.45991e9 0.120720
\(883\) 3.07682e10 1.50397 0.751985 0.659180i \(-0.229096\pi\)
0.751985 + 0.659180i \(0.229096\pi\)
\(884\) 2.24076e10 1.09097
\(885\) 5.02372e8 0.0243626
\(886\) −5.63591e10 −2.72236
\(887\) −6.77910e9 −0.326167 −0.163083 0.986612i \(-0.552144\pi\)
−0.163083 + 0.986612i \(0.552144\pi\)
\(888\) 1.77245e10 0.849431
\(889\) −5.06048e9 −0.241566
\(890\) 9.63348e9 0.458056
\(891\) 2.64231e9 0.125144
\(892\) −1.39640e10 −0.658771
\(893\) 7.36270e9 0.345985
\(894\) −3.21428e10 −1.50454
\(895\) 8.16432e9 0.380662
\(896\) −3.18039e10 −1.47707
\(897\) −1.68156e9 −0.0777925
\(898\) −3.94707e10 −1.81889
\(899\) 1.79155e10 0.822374
\(900\) −1.33297e10 −0.609499
\(901\) −2.32202e10 −1.05762
\(902\) −7.55549e10 −3.42799
\(903\) −4.68637e9 −0.211802
\(904\) 6.29101e10 2.83225
\(905\) −5.39530e9 −0.241961
\(906\) −4.84071e9 −0.216252
\(907\) 3.35163e10 1.49153 0.745763 0.666211i \(-0.232085\pi\)
0.745763 + 0.666211i \(0.232085\pi\)
\(908\) −2.03840e10 −0.903626
\(909\) −1.10737e10 −0.489011
\(910\) 4.21935e9 0.185610
\(911\) 2.19745e10 0.962953 0.481476 0.876459i \(-0.340101\pi\)
0.481476 + 0.876459i \(0.340101\pi\)
\(912\) −4.45791e9 −0.194603
\(913\) 1.99516e10 0.867619
\(914\) 2.14163e10 0.927755
\(915\) −5.45694e9 −0.235491
\(916\) −6.55233e9 −0.281683
\(917\) 2.37569e10 1.01741
\(918\) −1.13942e10 −0.486109
\(919\) 1.02526e10 0.435741 0.217871 0.975978i \(-0.430089\pi\)
0.217871 + 0.975978i \(0.430089\pi\)
\(920\) −5.08992e9 −0.215503
\(921\) −3.45317e9 −0.145650
\(922\) 6.78399e10 2.85054
\(923\) −1.09093e10 −0.456656
\(924\) 2.83605e10 1.18267
\(925\) −1.74172e10 −0.723571
\(926\) 5.58701e10 2.31228
\(927\) −5.36449e9 −0.221182
\(928\) −4.11443e9 −0.169002
\(929\) −3.38955e10 −1.38703 −0.693517 0.720440i \(-0.743940\pi\)
−0.693517 + 0.720440i \(0.743940\pi\)
\(930\) −5.99119e9 −0.244243
\(931\) −1.52301e9 −0.0618557
\(932\) 5.83292e10 2.36010
\(933\) −1.62782e10 −0.656175
\(934\) 2.51221e10 1.00889
\(935\) 1.32117e10 0.528591
\(936\) 5.61185e9 0.223687
\(937\) −5.25645e9 −0.208739 −0.104370 0.994539i \(-0.533283\pi\)
−0.104370 + 0.994539i \(0.533283\pi\)
\(938\) −9.58776e8 −0.0379321
\(939\) 2.18513e10 0.861288
\(940\) −1.95827e10 −0.768999
\(941\) 2.78632e10 1.09010 0.545052 0.838402i \(-0.316510\pi\)
0.545052 + 0.838402i \(0.316510\pi\)
\(942\) 2.78736e10 1.08646
\(943\) −1.64156e10 −0.637477
\(944\) 3.80668e9 0.147280
\(945\) −1.44049e9 −0.0555263
\(946\) 2.10840e10 0.809720
\(947\) 2.23930e10 0.856817 0.428409 0.903585i \(-0.359074\pi\)
0.428409 + 0.903585i \(0.359074\pi\)
\(948\) −1.29119e10 −0.492223
\(949\) −1.39193e9 −0.0528670
\(950\) 1.22922e10 0.465154
\(951\) −2.53557e9 −0.0955969
\(952\) −6.24393e10 −2.34546
\(953\) −4.95205e9 −0.185336 −0.0926680 0.995697i \(-0.529540\pi\)
−0.0926680 + 0.995697i \(0.529540\pi\)
\(954\) −1.13902e10 −0.424731
\(955\) −7.96334e9 −0.295858
\(956\) −3.83191e10 −1.41845
\(957\) −1.93795e10 −0.714744
\(958\) 3.62901e10 1.33355
\(959\) 3.73635e9 0.136799
\(960\) −4.42729e9 −0.161506
\(961\) −1.21114e10 −0.440211
\(962\) 1.43621e10 0.520122
\(963\) −2.59909e9 −0.0937840
\(964\) −1.39978e10 −0.503258
\(965\) −8.33428e9 −0.298554
\(966\) 9.17763e9 0.327575
\(967\) 2.63566e9 0.0937338 0.0468669 0.998901i \(-0.485076\pi\)
0.0468669 + 0.998901i \(0.485076\pi\)
\(968\) −1.37910e10 −0.488688
\(969\) 7.05452e9 0.249077
\(970\) −2.72761e10 −0.959581
\(971\) 2.95178e10 1.03471 0.517353 0.855772i \(-0.326918\pi\)
0.517353 + 0.855772i \(0.326918\pi\)
\(972\) −3.75255e9 −0.131067
\(973\) 1.04908e10 0.365103
\(974\) 3.56905e10 1.23765
\(975\) −5.51455e9 −0.190543
\(976\) −4.13494e10 −1.42362
\(977\) 3.98464e9 0.136697 0.0683483 0.997662i \(-0.478227\pi\)
0.0683483 + 0.997662i \(0.478227\pi\)
\(978\) 9.87856e9 0.337682
\(979\) 2.67880e10 0.912434
\(980\) 4.05079e9 0.137483
\(981\) 1.10230e10 0.372786
\(982\) −6.21024e10 −2.09275
\(983\) −2.75939e10 −0.926566 −0.463283 0.886210i \(-0.653329\pi\)
−0.463283 + 0.886210i \(0.653329\pi\)
\(984\) 5.47836e10 1.83302
\(985\) 8.61680e9 0.287289
\(986\) 8.35684e10 2.77634
\(987\) 1.80276e10 0.596798
\(988\) −6.80528e9 −0.224490
\(989\) 4.58086e9 0.150577
\(990\) 6.48078e9 0.212278
\(991\) 7.84374e8 0.0256015 0.0128008 0.999918i \(-0.495925\pi\)
0.0128008 + 0.999918i \(0.495925\pi\)
\(992\) −3.53702e9 −0.115039
\(993\) −2.69624e10 −0.873850
\(994\) 5.95408e10 1.92292
\(995\) 8.92038e9 0.287080
\(996\) −2.83348e10 −0.908683
\(997\) 8.21987e9 0.262683 0.131342 0.991337i \(-0.458072\pi\)
0.131342 + 0.991337i \(0.458072\pi\)
\(998\) −4.47579e10 −1.42532
\(999\) −4.90324e9 −0.155598
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.1.1 16
3.2 odd 2 531.8.a.b.1.16 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.a.1.1 16 1.1 even 1 trivial
531.8.a.b.1.16 16 3.2 odd 2