Properties

Label 531.8.a.d.1.13
Level $531$
Weight $8$
Character 531.1
Self dual yes
Analytic conductor $165.876$
Analytic rank $0$
Dimension $17$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 531 = 3^{2} \cdot 59 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 531.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(165.876448532\)
Analytic rank: \(0\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
Defining polynomial: \(x^{17} - 2 x^{16} - 1639 x^{15} + 1625 x^{14} + 1070274 x^{13} - 274939 x^{12} - 357079564 x^{11} - 89298188 x^{10} + 64650816672 x^{9} + 33122051904 x^{8} - 6210397064704 x^{7} - 2735256748800 x^{6} + 288860762071040 x^{5} - 34502173230080 x^{4} - 5633463408885760 x^{3} + 4719471961341952 x^{2} + 37636623107620864 x - 58321181718347776\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: multiple of \( 2^{10}\cdot 3^{5} \)
Twist minimal: no (minimal twist has level 177)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Root \(-11.3335\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q+13.3335 q^{2} +49.7814 q^{4} -152.093 q^{5} -1328.47 q^{7} -1042.93 q^{8} +O(q^{10})\) \(q+13.3335 q^{2} +49.7814 q^{4} -152.093 q^{5} -1328.47 q^{7} -1042.93 q^{8} -2027.93 q^{10} -5132.02 q^{11} -12251.2 q^{13} -17713.2 q^{14} -20277.8 q^{16} -26601.9 q^{17} +33197.2 q^{19} -7571.39 q^{20} -68427.7 q^{22} +97057.0 q^{23} -54992.8 q^{25} -163351. q^{26} -66133.3 q^{28} -209269. q^{29} -308874. q^{31} -136879. q^{32} -354695. q^{34} +202052. q^{35} +287842. q^{37} +442634. q^{38} +158622. q^{40} -581611. q^{41} +540792. q^{43} -255479. q^{44} +1.29411e6 q^{46} +650609. q^{47} +941302. q^{49} -733244. q^{50} -609883. q^{52} +49948.0 q^{53} +780544. q^{55} +1.38550e6 q^{56} -2.79028e6 q^{58} +205379. q^{59} -20025.0 q^{61} -4.11836e6 q^{62} +770486. q^{64} +1.86332e6 q^{65} -1.38680e6 q^{67} -1.32428e6 q^{68} +2.69405e6 q^{70} -1.24373e6 q^{71} -4.83751e6 q^{73} +3.83794e6 q^{74} +1.65260e6 q^{76} +6.81777e6 q^{77} -2.94089e6 q^{79} +3.08411e6 q^{80} -7.75490e6 q^{82} +8.62802e6 q^{83} +4.04595e6 q^{85} +7.21064e6 q^{86} +5.35232e6 q^{88} +879309. q^{89} +1.62754e7 q^{91} +4.83163e6 q^{92} +8.67488e6 q^{94} -5.04907e6 q^{95} -1.62017e7 q^{97} +1.25508e7 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) \) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 13.3335 1.17852 0.589262 0.807942i \(-0.299419\pi\)
0.589262 + 0.807942i \(0.299419\pi\)
\(3\) 0 0
\(4\) 49.7814 0.388917
\(5\) −152.093 −0.544144 −0.272072 0.962277i \(-0.587709\pi\)
−0.272072 + 0.962277i \(0.587709\pi\)
\(6\) 0 0
\(7\) −1328.47 −1.46390 −0.731948 0.681361i \(-0.761389\pi\)
−0.731948 + 0.681361i \(0.761389\pi\)
\(8\) −1042.93 −0.720176
\(9\) 0 0
\(10\) −2027.93 −0.641286
\(11\) −5132.02 −1.16256 −0.581279 0.813705i \(-0.697447\pi\)
−0.581279 + 0.813705i \(0.697447\pi\)
\(12\) 0 0
\(13\) −12251.2 −1.54660 −0.773299 0.634041i \(-0.781395\pi\)
−0.773299 + 0.634041i \(0.781395\pi\)
\(14\) −17713.2 −1.72524
\(15\) 0 0
\(16\) −20277.8 −1.23766
\(17\) −26601.9 −1.31323 −0.656615 0.754226i \(-0.728012\pi\)
−0.656615 + 0.754226i \(0.728012\pi\)
\(18\) 0 0
\(19\) 33197.2 1.11036 0.555181 0.831730i \(-0.312649\pi\)
0.555181 + 0.831730i \(0.312649\pi\)
\(20\) −7571.39 −0.211627
\(21\) 0 0
\(22\) −68427.7 −1.37010
\(23\) 97057.0 1.66333 0.831667 0.555274i \(-0.187387\pi\)
0.831667 + 0.555274i \(0.187387\pi\)
\(24\) 0 0
\(25\) −54992.8 −0.703907
\(26\) −163351. −1.82270
\(27\) 0 0
\(28\) −66133.3 −0.569334
\(29\) −209269. −1.59335 −0.796675 0.604408i \(-0.793410\pi\)
−0.796675 + 0.604408i \(0.793410\pi\)
\(30\) 0 0
\(31\) −308874. −1.86215 −0.931077 0.364823i \(-0.881129\pi\)
−0.931077 + 0.364823i \(0.881129\pi\)
\(32\) −136879. −0.738436
\(33\) 0 0
\(34\) −354695. −1.54767
\(35\) 202052. 0.796570
\(36\) 0 0
\(37\) 287842. 0.934219 0.467110 0.884199i \(-0.345295\pi\)
0.467110 + 0.884199i \(0.345295\pi\)
\(38\) 442634. 1.30859
\(39\) 0 0
\(40\) 158622. 0.391879
\(41\) −581611. −1.31792 −0.658960 0.752178i \(-0.729004\pi\)
−0.658960 + 0.752178i \(0.729004\pi\)
\(42\) 0 0
\(43\) 540792. 1.03727 0.518634 0.854996i \(-0.326441\pi\)
0.518634 + 0.854996i \(0.326441\pi\)
\(44\) −255479. −0.452138
\(45\) 0 0
\(46\) 1.29411e6 1.96028
\(47\) 650609. 0.914066 0.457033 0.889450i \(-0.348912\pi\)
0.457033 + 0.889450i \(0.348912\pi\)
\(48\) 0 0
\(49\) 941302. 1.14299
\(50\) −733244. −0.829571
\(51\) 0 0
\(52\) −609883. −0.601499
\(53\) 49948.0 0.0460843 0.0230421 0.999734i \(-0.492665\pi\)
0.0230421 + 0.999734i \(0.492665\pi\)
\(54\) 0 0
\(55\) 780544. 0.632599
\(56\) 1.38550e6 1.05426
\(57\) 0 0
\(58\) −2.79028e6 −1.87780
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) −20025.0 −0.0112958 −0.00564791 0.999984i \(-0.501798\pi\)
−0.00564791 + 0.999984i \(0.501798\pi\)
\(62\) −4.11836e6 −2.19459
\(63\) 0 0
\(64\) 770486. 0.367396
\(65\) 1.86332e6 0.841573
\(66\) 0 0
\(67\) −1.38680e6 −0.563314 −0.281657 0.959515i \(-0.590884\pi\)
−0.281657 + 0.959515i \(0.590884\pi\)
\(68\) −1.32428e6 −0.510738
\(69\) 0 0
\(70\) 2.69405e6 0.938776
\(71\) −1.24373e6 −0.412403 −0.206201 0.978510i \(-0.566110\pi\)
−0.206201 + 0.978510i \(0.566110\pi\)
\(72\) 0 0
\(73\) −4.83751e6 −1.45543 −0.727716 0.685879i \(-0.759418\pi\)
−0.727716 + 0.685879i \(0.759418\pi\)
\(74\) 3.83794e6 1.10100
\(75\) 0 0
\(76\) 1.65260e6 0.431839
\(77\) 6.81777e6 1.70186
\(78\) 0 0
\(79\) −2.94089e6 −0.671095 −0.335547 0.942023i \(-0.608921\pi\)
−0.335547 + 0.942023i \(0.608921\pi\)
\(80\) 3.08411e6 0.673466
\(81\) 0 0
\(82\) −7.75490e6 −1.55320
\(83\) 8.62802e6 1.65630 0.828148 0.560510i \(-0.189395\pi\)
0.828148 + 0.560510i \(0.189395\pi\)
\(84\) 0 0
\(85\) 4.04595e6 0.714586
\(86\) 7.21064e6 1.22244
\(87\) 0 0
\(88\) 5.35232e6 0.837245
\(89\) 879309. 0.132214 0.0661069 0.997813i \(-0.478942\pi\)
0.0661069 + 0.997813i \(0.478942\pi\)
\(90\) 0 0
\(91\) 1.62754e7 2.26406
\(92\) 4.83163e6 0.646899
\(93\) 0 0
\(94\) 8.67488e6 1.07725
\(95\) −5.04907e6 −0.604197
\(96\) 0 0
\(97\) −1.62017e7 −1.80244 −0.901220 0.433362i \(-0.857327\pi\)
−0.901220 + 0.433362i \(0.857327\pi\)
\(98\) 1.25508e7 1.34704
\(99\) 0 0
\(100\) −2.73761e6 −0.273761
\(101\) −617795. −0.0596650 −0.0298325 0.999555i \(-0.509497\pi\)
−0.0298325 + 0.999555i \(0.509497\pi\)
\(102\) 0 0
\(103\) −6.98654e6 −0.629988 −0.314994 0.949094i \(-0.602002\pi\)
−0.314994 + 0.949094i \(0.602002\pi\)
\(104\) 1.27771e7 1.11382
\(105\) 0 0
\(106\) 665980. 0.0543114
\(107\) 7.47019e6 0.589506 0.294753 0.955573i \(-0.404763\pi\)
0.294753 + 0.955573i \(0.404763\pi\)
\(108\) 0 0
\(109\) −2.25015e7 −1.66425 −0.832125 0.554588i \(-0.812876\pi\)
−0.832125 + 0.554588i \(0.812876\pi\)
\(110\) 1.04074e7 0.745532
\(111\) 0 0
\(112\) 2.69386e7 1.81181
\(113\) −7.33737e6 −0.478372 −0.239186 0.970974i \(-0.576881\pi\)
−0.239186 + 0.970974i \(0.576881\pi\)
\(114\) 0 0
\(115\) −1.47617e7 −0.905094
\(116\) −1.04177e7 −0.619681
\(117\) 0 0
\(118\) 2.73841e6 0.153431
\(119\) 3.53399e7 1.92243
\(120\) 0 0
\(121\) 6.85051e6 0.351539
\(122\) −267002. −0.0133124
\(123\) 0 0
\(124\) −1.53762e7 −0.724223
\(125\) 2.02463e7 0.927171
\(126\) 0 0
\(127\) −1.12559e7 −0.487605 −0.243803 0.969825i \(-0.578395\pi\)
−0.243803 + 0.969825i \(0.578395\pi\)
\(128\) 2.77938e7 1.17142
\(129\) 0 0
\(130\) 2.48446e7 0.991813
\(131\) −4.30156e6 −0.167177 −0.0835885 0.996500i \(-0.526638\pi\)
−0.0835885 + 0.996500i \(0.526638\pi\)
\(132\) 0 0
\(133\) −4.41017e7 −1.62545
\(134\) −1.84908e7 −0.663879
\(135\) 0 0
\(136\) 2.77438e7 0.945756
\(137\) 1.85481e7 0.616281 0.308140 0.951341i \(-0.400293\pi\)
0.308140 + 0.951341i \(0.400293\pi\)
\(138\) 0 0
\(139\) −4.12956e7 −1.30422 −0.652112 0.758123i \(-0.726117\pi\)
−0.652112 + 0.758123i \(0.726117\pi\)
\(140\) 1.00584e7 0.309800
\(141\) 0 0
\(142\) −1.65832e7 −0.486026
\(143\) 6.28736e7 1.79801
\(144\) 0 0
\(145\) 3.18283e7 0.867012
\(146\) −6.45008e7 −1.71526
\(147\) 0 0
\(148\) 1.43292e7 0.363334
\(149\) −2.77964e7 −0.688394 −0.344197 0.938897i \(-0.611849\pi\)
−0.344197 + 0.938897i \(0.611849\pi\)
\(150\) 0 0
\(151\) 5.05159e7 1.19401 0.597006 0.802236i \(-0.296357\pi\)
0.597006 + 0.802236i \(0.296357\pi\)
\(152\) −3.46223e7 −0.799655
\(153\) 0 0
\(154\) 9.09045e7 2.00568
\(155\) 4.69775e7 1.01328
\(156\) 0 0
\(157\) −3.27653e7 −0.675717 −0.337858 0.941197i \(-0.609703\pi\)
−0.337858 + 0.941197i \(0.609703\pi\)
\(158\) −3.92122e7 −0.790901
\(159\) 0 0
\(160\) 2.08184e7 0.401816
\(161\) −1.28938e8 −2.43495
\(162\) 0 0
\(163\) 3.27745e7 0.592761 0.296381 0.955070i \(-0.404220\pi\)
0.296381 + 0.955070i \(0.404220\pi\)
\(164\) −2.89534e7 −0.512562
\(165\) 0 0
\(166\) 1.15041e8 1.95198
\(167\) 4.91242e7 0.816183 0.408092 0.912941i \(-0.366194\pi\)
0.408092 + 0.912941i \(0.366194\pi\)
\(168\) 0 0
\(169\) 8.73439e7 1.39197
\(170\) 5.39466e7 0.842157
\(171\) 0 0
\(172\) 2.69214e7 0.403411
\(173\) −2.08181e7 −0.305688 −0.152844 0.988250i \(-0.548843\pi\)
−0.152844 + 0.988250i \(0.548843\pi\)
\(174\) 0 0
\(175\) 7.30565e7 1.03045
\(176\) 1.04066e8 1.43885
\(177\) 0 0
\(178\) 1.17242e7 0.155817
\(179\) −8.64093e7 −1.12609 −0.563047 0.826425i \(-0.690371\pi\)
−0.563047 + 0.826425i \(0.690371\pi\)
\(180\) 0 0
\(181\) −6.54723e7 −0.820696 −0.410348 0.911929i \(-0.634593\pi\)
−0.410348 + 0.911929i \(0.634593\pi\)
\(182\) 2.17008e8 2.66825
\(183\) 0 0
\(184\) −1.01223e8 −1.19789
\(185\) −4.37788e7 −0.508350
\(186\) 0 0
\(187\) 1.36521e8 1.52671
\(188\) 3.23882e7 0.355496
\(189\) 0 0
\(190\) −6.73215e7 −0.712060
\(191\) 8.45503e7 0.878007 0.439004 0.898485i \(-0.355331\pi\)
0.439004 + 0.898485i \(0.355331\pi\)
\(192\) 0 0
\(193\) −2.04209e7 −0.204467 −0.102234 0.994760i \(-0.532599\pi\)
−0.102234 + 0.994760i \(0.532599\pi\)
\(194\) −2.16026e8 −2.12422
\(195\) 0 0
\(196\) 4.68593e7 0.444529
\(197\) −3.23191e7 −0.301181 −0.150591 0.988596i \(-0.548118\pi\)
−0.150591 + 0.988596i \(0.548118\pi\)
\(198\) 0 0
\(199\) −7.22071e7 −0.649522 −0.324761 0.945796i \(-0.605284\pi\)
−0.324761 + 0.945796i \(0.605284\pi\)
\(200\) 5.73534e7 0.506937
\(201\) 0 0
\(202\) −8.23736e6 −0.0703166
\(203\) 2.78008e8 2.33250
\(204\) 0 0
\(205\) 8.84589e7 0.717139
\(206\) −9.31549e7 −0.742455
\(207\) 0 0
\(208\) 2.48428e8 1.91416
\(209\) −1.70369e8 −1.29086
\(210\) 0 0
\(211\) −1.90681e8 −1.39739 −0.698696 0.715419i \(-0.746236\pi\)
−0.698696 + 0.715419i \(0.746236\pi\)
\(212\) 2.48648e6 0.0179230
\(213\) 0 0
\(214\) 9.96036e7 0.694747
\(215\) −8.22507e7 −0.564423
\(216\) 0 0
\(217\) 4.10331e8 2.72600
\(218\) −3.00023e8 −1.96136
\(219\) 0 0
\(220\) 3.88566e7 0.246028
\(221\) 3.25905e8 2.03104
\(222\) 0 0
\(223\) 2.03359e8 1.22800 0.613998 0.789307i \(-0.289560\pi\)
0.613998 + 0.789307i \(0.289560\pi\)
\(224\) 1.81841e8 1.08099
\(225\) 0 0
\(226\) −9.78326e7 −0.563773
\(227\) −3.52165e7 −0.199828 −0.0999139 0.994996i \(-0.531857\pi\)
−0.0999139 + 0.994996i \(0.531857\pi\)
\(228\) 0 0
\(229\) −3.50737e8 −1.93000 −0.965000 0.262249i \(-0.915536\pi\)
−0.965000 + 0.262249i \(0.915536\pi\)
\(230\) −1.96824e8 −1.06667
\(231\) 0 0
\(232\) 2.18252e8 1.14749
\(233\) 2.71870e8 1.40804 0.704020 0.710180i \(-0.251387\pi\)
0.704020 + 0.710180i \(0.251387\pi\)
\(234\) 0 0
\(235\) −9.89530e7 −0.497384
\(236\) 1.02240e7 0.0506327
\(237\) 0 0
\(238\) 4.71203e8 2.26563
\(239\) 7.12076e7 0.337391 0.168696 0.985668i \(-0.446045\pi\)
0.168696 + 0.985668i \(0.446045\pi\)
\(240\) 0 0
\(241\) −2.36356e8 −1.08770 −0.543848 0.839184i \(-0.683033\pi\)
−0.543848 + 0.839184i \(0.683033\pi\)
\(242\) 9.13410e7 0.414297
\(243\) 0 0
\(244\) −996870. −0.00439313
\(245\) −1.43165e8 −0.621952
\(246\) 0 0
\(247\) −4.06707e8 −1.71728
\(248\) 3.22133e8 1.34108
\(249\) 0 0
\(250\) 2.69953e8 1.09269
\(251\) −3.04691e7 −0.121619 −0.0608094 0.998149i \(-0.519368\pi\)
−0.0608094 + 0.998149i \(0.519368\pi\)
\(252\) 0 0
\(253\) −4.98099e8 −1.93372
\(254\) −1.50081e8 −0.574654
\(255\) 0 0
\(256\) 2.71966e8 1.01315
\(257\) −2.69817e8 −0.991524 −0.495762 0.868458i \(-0.665111\pi\)
−0.495762 + 0.868458i \(0.665111\pi\)
\(258\) 0 0
\(259\) −3.82391e8 −1.36760
\(260\) 9.27588e7 0.327302
\(261\) 0 0
\(262\) −5.73548e7 −0.197022
\(263\) 3.30750e8 1.12113 0.560564 0.828111i \(-0.310585\pi\)
0.560564 + 0.828111i \(0.310585\pi\)
\(264\) 0 0
\(265\) −7.59674e6 −0.0250765
\(266\) −5.88029e8 −1.91564
\(267\) 0 0
\(268\) −6.90366e7 −0.219082
\(269\) −3.87843e8 −1.21485 −0.607426 0.794377i \(-0.707798\pi\)
−0.607426 + 0.794377i \(0.707798\pi\)
\(270\) 0 0
\(271\) 3.62357e8 1.10597 0.552986 0.833191i \(-0.313488\pi\)
0.552986 + 0.833191i \(0.313488\pi\)
\(272\) 5.39428e8 1.62533
\(273\) 0 0
\(274\) 2.47311e8 0.726301
\(275\) 2.82224e8 0.818333
\(276\) 0 0
\(277\) 3.92785e8 1.11039 0.555195 0.831721i \(-0.312644\pi\)
0.555195 + 0.831721i \(0.312644\pi\)
\(278\) −5.50614e8 −1.53706
\(279\) 0 0
\(280\) −2.10725e8 −0.573670
\(281\) −2.62473e7 −0.0705686 −0.0352843 0.999377i \(-0.511234\pi\)
−0.0352843 + 0.999377i \(0.511234\pi\)
\(282\) 0 0
\(283\) 3.02721e8 0.793945 0.396972 0.917831i \(-0.370061\pi\)
0.396972 + 0.917831i \(0.370061\pi\)
\(284\) −6.19146e7 −0.160390
\(285\) 0 0
\(286\) 8.38323e8 2.11900
\(287\) 7.72656e8 1.92930
\(288\) 0 0
\(289\) 2.97321e8 0.724574
\(290\) 4.24381e8 1.02179
\(291\) 0 0
\(292\) −2.40818e8 −0.566042
\(293\) −4.44159e8 −1.03158 −0.515789 0.856716i \(-0.672501\pi\)
−0.515789 + 0.856716i \(0.672501\pi\)
\(294\) 0 0
\(295\) −3.12367e7 −0.0708415
\(296\) −3.00198e8 −0.672802
\(297\) 0 0
\(298\) −3.70623e8 −0.811288
\(299\) −1.18907e9 −2.57251
\(300\) 0 0
\(301\) −7.18429e8 −1.51845
\(302\) 6.73552e8 1.40717
\(303\) 0 0
\(304\) −6.73168e8 −1.37425
\(305\) 3.04566e6 0.00614655
\(306\) 0 0
\(307\) −7.17875e8 −1.41600 −0.708002 0.706211i \(-0.750403\pi\)
−0.708002 + 0.706211i \(0.750403\pi\)
\(308\) 3.39398e8 0.661883
\(309\) 0 0
\(310\) 6.26374e8 1.19417
\(311\) 9.91134e8 1.86841 0.934203 0.356743i \(-0.116113\pi\)
0.934203 + 0.356743i \(0.116113\pi\)
\(312\) 0 0
\(313\) −1.77188e7 −0.0326610 −0.0163305 0.999867i \(-0.505198\pi\)
−0.0163305 + 0.999867i \(0.505198\pi\)
\(314\) −4.36875e8 −0.796348
\(315\) 0 0
\(316\) −1.46401e8 −0.261000
\(317\) −6.67310e8 −1.17658 −0.588289 0.808651i \(-0.700198\pi\)
−0.588289 + 0.808651i \(0.700198\pi\)
\(318\) 0 0
\(319\) 1.07397e9 1.85236
\(320\) −1.17185e8 −0.199917
\(321\) 0 0
\(322\) −1.71919e9 −2.86964
\(323\) −8.83109e8 −1.45816
\(324\) 0 0
\(325\) 6.73728e8 1.08866
\(326\) 4.36998e8 0.698583
\(327\) 0 0
\(328\) 6.06577e8 0.949134
\(329\) −8.64318e8 −1.33810
\(330\) 0 0
\(331\) 1.21309e9 1.83863 0.919313 0.393526i \(-0.128745\pi\)
0.919313 + 0.393526i \(0.128745\pi\)
\(332\) 4.29515e8 0.644161
\(333\) 0 0
\(334\) 6.54996e8 0.961891
\(335\) 2.10922e8 0.306524
\(336\) 0 0
\(337\) 4.03850e8 0.574799 0.287399 0.957811i \(-0.407209\pi\)
0.287399 + 0.957811i \(0.407209\pi\)
\(338\) 1.16460e9 1.64047
\(339\) 0 0
\(340\) 2.01413e8 0.277915
\(341\) 1.58515e9 2.16486
\(342\) 0 0
\(343\) −1.56440e8 −0.209324
\(344\) −5.64006e8 −0.747015
\(345\) 0 0
\(346\) −2.77577e8 −0.360261
\(347\) 5.73452e8 0.736791 0.368395 0.929669i \(-0.379907\pi\)
0.368395 + 0.929669i \(0.379907\pi\)
\(348\) 0 0
\(349\) −3.21244e8 −0.404525 −0.202263 0.979331i \(-0.564829\pi\)
−0.202263 + 0.979331i \(0.564829\pi\)
\(350\) 9.74096e8 1.21441
\(351\) 0 0
\(352\) 7.02468e8 0.858474
\(353\) −3.98442e8 −0.482118 −0.241059 0.970510i \(-0.577495\pi\)
−0.241059 + 0.970510i \(0.577495\pi\)
\(354\) 0 0
\(355\) 1.89162e8 0.224407
\(356\) 4.37732e7 0.0514202
\(357\) 0 0
\(358\) −1.15214e9 −1.32713
\(359\) −1.40704e9 −1.60500 −0.802501 0.596651i \(-0.796498\pi\)
−0.802501 + 0.596651i \(0.796498\pi\)
\(360\) 0 0
\(361\) 2.08186e8 0.232903
\(362\) −8.72973e8 −0.967210
\(363\) 0 0
\(364\) 8.10214e8 0.880531
\(365\) 7.35751e8 0.791965
\(366\) 0 0
\(367\) −9.65705e8 −1.01980 −0.509898 0.860235i \(-0.670317\pi\)
−0.509898 + 0.860235i \(0.670317\pi\)
\(368\) −1.96811e9 −2.05864
\(369\) 0 0
\(370\) −5.83723e8 −0.599102
\(371\) −6.63547e7 −0.0674626
\(372\) 0 0
\(373\) −6.50998e8 −0.649529 −0.324765 0.945795i \(-0.605285\pi\)
−0.324765 + 0.945795i \(0.605285\pi\)
\(374\) 1.82030e9 1.79926
\(375\) 0 0
\(376\) −6.78537e8 −0.658288
\(377\) 2.56380e9 2.46427
\(378\) 0 0
\(379\) −3.97991e7 −0.0375522 −0.0187761 0.999824i \(-0.505977\pi\)
−0.0187761 + 0.999824i \(0.505977\pi\)
\(380\) −2.51349e8 −0.234982
\(381\) 0 0
\(382\) 1.12735e9 1.03475
\(383\) −6.27745e7 −0.0570936 −0.0285468 0.999592i \(-0.509088\pi\)
−0.0285468 + 0.999592i \(0.509088\pi\)
\(384\) 0 0
\(385\) −1.03693e9 −0.926058
\(386\) −2.72281e8 −0.240970
\(387\) 0 0
\(388\) −8.06545e8 −0.701000
\(389\) 4.14596e8 0.357109 0.178555 0.983930i \(-0.442858\pi\)
0.178555 + 0.983930i \(0.442858\pi\)
\(390\) 0 0
\(391\) −2.58190e9 −2.18434
\(392\) −9.81708e8 −0.823154
\(393\) 0 0
\(394\) −4.30926e8 −0.354949
\(395\) 4.47288e8 0.365172
\(396\) 0 0
\(397\) −5.40817e8 −0.433794 −0.216897 0.976194i \(-0.569594\pi\)
−0.216897 + 0.976194i \(0.569594\pi\)
\(398\) −9.62771e8 −0.765477
\(399\) 0 0
\(400\) 1.11513e9 0.871198
\(401\) −1.82362e9 −1.41231 −0.706153 0.708059i \(-0.749571\pi\)
−0.706153 + 0.708059i \(0.749571\pi\)
\(402\) 0 0
\(403\) 3.78408e9 2.88000
\(404\) −3.07547e7 −0.0232047
\(405\) 0 0
\(406\) 3.70681e9 2.74890
\(407\) −1.47721e9 −1.08608
\(408\) 0 0
\(409\) 1.97200e9 1.42520 0.712599 0.701571i \(-0.247518\pi\)
0.712599 + 0.701571i \(0.247518\pi\)
\(410\) 1.17946e9 0.845165
\(411\) 0 0
\(412\) −3.47800e8 −0.245013
\(413\) −2.72841e8 −0.190583
\(414\) 0 0
\(415\) −1.31226e9 −0.901263
\(416\) 1.67694e9 1.14206
\(417\) 0 0
\(418\) −2.27161e9 −1.52131
\(419\) −4.90370e8 −0.325668 −0.162834 0.986653i \(-0.552064\pi\)
−0.162834 + 0.986653i \(0.552064\pi\)
\(420\) 0 0
\(421\) −1.72536e8 −0.112692 −0.0563459 0.998411i \(-0.517945\pi\)
−0.0563459 + 0.998411i \(0.517945\pi\)
\(422\) −2.54244e9 −1.64686
\(423\) 0 0
\(424\) −5.20921e7 −0.0331888
\(425\) 1.46291e9 0.924392
\(426\) 0 0
\(427\) 2.66027e7 0.0165359
\(428\) 3.71876e8 0.229269
\(429\) 0 0
\(430\) −1.09669e9 −0.665186
\(431\) −1.99685e9 −1.20136 −0.600681 0.799489i \(-0.705104\pi\)
−0.600681 + 0.799489i \(0.705104\pi\)
\(432\) 0 0
\(433\) −6.53171e8 −0.386651 −0.193326 0.981135i \(-0.561927\pi\)
−0.193326 + 0.981135i \(0.561927\pi\)
\(434\) 5.47114e9 3.21265
\(435\) 0 0
\(436\) −1.12015e9 −0.647255
\(437\) 3.22203e9 1.84690
\(438\) 0 0
\(439\) −2.18962e9 −1.23521 −0.617607 0.786487i \(-0.711898\pi\)
−0.617607 + 0.786487i \(0.711898\pi\)
\(440\) −8.14050e8 −0.455582
\(441\) 0 0
\(442\) 4.34545e9 2.39363
\(443\) −2.98274e9 −1.63006 −0.815028 0.579422i \(-0.803278\pi\)
−0.815028 + 0.579422i \(0.803278\pi\)
\(444\) 0 0
\(445\) −1.33737e8 −0.0719433
\(446\) 2.71149e9 1.44722
\(447\) 0 0
\(448\) −1.02357e9 −0.537830
\(449\) 1.54617e9 0.806111 0.403055 0.915176i \(-0.367948\pi\)
0.403055 + 0.915176i \(0.367948\pi\)
\(450\) 0 0
\(451\) 2.98484e9 1.53216
\(452\) −3.65264e8 −0.186047
\(453\) 0 0
\(454\) −4.69558e8 −0.235502
\(455\) −2.47538e9 −1.23197
\(456\) 0 0
\(457\) −7.00396e8 −0.343271 −0.171635 0.985161i \(-0.554905\pi\)
−0.171635 + 0.985161i \(0.554905\pi\)
\(458\) −4.67654e9 −2.27455
\(459\) 0 0
\(460\) −7.34857e8 −0.352006
\(461\) 2.39080e9 1.13655 0.568277 0.822838i \(-0.307610\pi\)
0.568277 + 0.822838i \(0.307610\pi\)
\(462\) 0 0
\(463\) 2.71887e9 1.27308 0.636539 0.771244i \(-0.280365\pi\)
0.636539 + 0.771244i \(0.280365\pi\)
\(464\) 4.24352e9 1.97203
\(465\) 0 0
\(466\) 3.62496e9 1.65941
\(467\) −3.63245e9 −1.65041 −0.825203 0.564836i \(-0.808939\pi\)
−0.825203 + 0.564836i \(0.808939\pi\)
\(468\) 0 0
\(469\) 1.84232e9 0.824633
\(470\) −1.31939e9 −0.586178
\(471\) 0 0
\(472\) −2.14195e8 −0.0937589
\(473\) −2.77536e9 −1.20588
\(474\) 0 0
\(475\) −1.82561e9 −0.781592
\(476\) 1.75927e9 0.747666
\(477\) 0 0
\(478\) 9.49444e8 0.397623
\(479\) 7.60336e8 0.316105 0.158053 0.987431i \(-0.449478\pi\)
0.158053 + 0.987431i \(0.449478\pi\)
\(480\) 0 0
\(481\) −3.52642e9 −1.44486
\(482\) −3.15145e9 −1.28188
\(483\) 0 0
\(484\) 3.41028e8 0.136720
\(485\) 2.46417e9 0.980787
\(486\) 0 0
\(487\) −3.11666e9 −1.22275 −0.611375 0.791341i \(-0.709383\pi\)
−0.611375 + 0.791341i \(0.709383\pi\)
\(488\) 2.08846e7 0.00813497
\(489\) 0 0
\(490\) −1.90889e9 −0.732984
\(491\) 2.02161e9 0.770749 0.385374 0.922760i \(-0.374072\pi\)
0.385374 + 0.922760i \(0.374072\pi\)
\(492\) 0 0
\(493\) 5.56694e9 2.09244
\(494\) −5.42281e9 −2.02386
\(495\) 0 0
\(496\) 6.26330e9 2.30471
\(497\) 1.65226e9 0.603715
\(498\) 0 0
\(499\) −3.90803e9 −1.40801 −0.704006 0.710194i \(-0.748607\pi\)
−0.704006 + 0.710194i \(0.748607\pi\)
\(500\) 1.00789e9 0.360593
\(501\) 0 0
\(502\) −4.06258e8 −0.143331
\(503\) 5.07994e9 1.77980 0.889899 0.456158i \(-0.150775\pi\)
0.889899 + 0.456158i \(0.150775\pi\)
\(504\) 0 0
\(505\) 9.39623e7 0.0324664
\(506\) −6.64139e9 −2.27894
\(507\) 0 0
\(508\) −5.60336e8 −0.189638
\(509\) 2.67138e9 0.897892 0.448946 0.893559i \(-0.351800\pi\)
0.448946 + 0.893559i \(0.351800\pi\)
\(510\) 0 0
\(511\) 6.42651e9 2.13060
\(512\) 6.86375e7 0.0226004
\(513\) 0 0
\(514\) −3.59760e9 −1.16853
\(515\) 1.06260e9 0.342804
\(516\) 0 0
\(517\) −3.33894e9 −1.06265
\(518\) −5.09860e9 −1.61175
\(519\) 0 0
\(520\) −1.94331e9 −0.606080
\(521\) −1.52466e9 −0.472326 −0.236163 0.971713i \(-0.575890\pi\)
−0.236163 + 0.971713i \(0.575890\pi\)
\(522\) 0 0
\(523\) 9.55727e8 0.292131 0.146066 0.989275i \(-0.453339\pi\)
0.146066 + 0.989275i \(0.453339\pi\)
\(524\) −2.14138e8 −0.0650180
\(525\) 0 0
\(526\) 4.41005e9 1.32127
\(527\) 8.21663e9 2.44544
\(528\) 0 0
\(529\) 6.01524e9 1.76668
\(530\) −1.01291e8 −0.0295532
\(531\) 0 0
\(532\) −2.19544e9 −0.632167
\(533\) 7.12545e9 2.03829
\(534\) 0 0
\(535\) −1.13616e9 −0.320776
\(536\) 1.44632e9 0.405685
\(537\) 0 0
\(538\) −5.17129e9 −1.43173
\(539\) −4.83079e9 −1.32879
\(540\) 0 0
\(541\) 1.12167e9 0.304561 0.152280 0.988337i \(-0.451338\pi\)
0.152280 + 0.988337i \(0.451338\pi\)
\(542\) 4.83147e9 1.30341
\(543\) 0 0
\(544\) 3.64125e9 0.969737
\(545\) 3.42232e9 0.905591
\(546\) 0 0
\(547\) 5.80841e8 0.151740 0.0758702 0.997118i \(-0.475827\pi\)
0.0758702 + 0.997118i \(0.475827\pi\)
\(548\) 9.23352e8 0.239682
\(549\) 0 0
\(550\) 3.76303e9 0.964424
\(551\) −6.94715e9 −1.76920
\(552\) 0 0
\(553\) 3.90690e9 0.982413
\(554\) 5.23718e9 1.30862
\(555\) 0 0
\(556\) −2.05575e9 −0.507235
\(557\) −3.50741e9 −0.859989 −0.429994 0.902832i \(-0.641484\pi\)
−0.429994 + 0.902832i \(0.641484\pi\)
\(558\) 0 0
\(559\) −6.62537e9 −1.60424
\(560\) −4.09717e9 −0.985883
\(561\) 0 0
\(562\) −3.49967e8 −0.0831668
\(563\) −1.57440e9 −0.371822 −0.185911 0.982567i \(-0.559524\pi\)
−0.185911 + 0.982567i \(0.559524\pi\)
\(564\) 0 0
\(565\) 1.11596e9 0.260303
\(566\) 4.03632e9 0.935682
\(567\) 0 0
\(568\) 1.29712e9 0.297003
\(569\) 2.71007e7 0.00616719 0.00308360 0.999995i \(-0.499018\pi\)
0.00308360 + 0.999995i \(0.499018\pi\)
\(570\) 0 0
\(571\) −8.12943e9 −1.82740 −0.913700 0.406390i \(-0.866787\pi\)
−0.913700 + 0.406390i \(0.866787\pi\)
\(572\) 3.12993e9 0.699276
\(573\) 0 0
\(574\) 1.03022e10 2.27372
\(575\) −5.33743e9 −1.17083
\(576\) 0 0
\(577\) −8.89512e8 −0.192769 −0.0963844 0.995344i \(-0.530728\pi\)
−0.0963844 + 0.995344i \(0.530728\pi\)
\(578\) 3.96431e9 0.853927
\(579\) 0 0
\(580\) 1.58446e9 0.337196
\(581\) −1.14621e10 −2.42464
\(582\) 0 0
\(583\) −2.56334e8 −0.0535756
\(584\) 5.04516e9 1.04817
\(585\) 0 0
\(586\) −5.92218e9 −1.21574
\(587\) −8.09762e8 −0.165243 −0.0826217 0.996581i \(-0.526329\pi\)
−0.0826217 + 0.996581i \(0.526329\pi\)
\(588\) 0 0
\(589\) −1.02538e10 −2.06766
\(590\) −4.16493e8 −0.0834884
\(591\) 0 0
\(592\) −5.83682e9 −1.15625
\(593\) 6.48925e9 1.27792 0.638959 0.769241i \(-0.279365\pi\)
0.638959 + 0.769241i \(0.279365\pi\)
\(594\) 0 0
\(595\) −5.37495e9 −1.04608
\(596\) −1.38374e9 −0.267728
\(597\) 0 0
\(598\) −1.58544e10 −3.03176
\(599\) −6.51857e8 −0.123925 −0.0619624 0.998078i \(-0.519736\pi\)
−0.0619624 + 0.998078i \(0.519736\pi\)
\(600\) 0 0
\(601\) −2.97476e9 −0.558974 −0.279487 0.960150i \(-0.590164\pi\)
−0.279487 + 0.960150i \(0.590164\pi\)
\(602\) −9.57915e9 −1.78953
\(603\) 0 0
\(604\) 2.51475e9 0.464372
\(605\) −1.04191e9 −0.191288
\(606\) 0 0
\(607\) −4.76572e9 −0.864905 −0.432452 0.901657i \(-0.642352\pi\)
−0.432452 + 0.901657i \(0.642352\pi\)
\(608\) −4.54402e9 −0.819931
\(609\) 0 0
\(610\) 4.06091e7 0.00724385
\(611\) −7.97076e9 −1.41369
\(612\) 0 0
\(613\) 4.10844e9 0.720385 0.360193 0.932878i \(-0.382711\pi\)
0.360193 + 0.932878i \(0.382711\pi\)
\(614\) −9.57176e9 −1.66879
\(615\) 0 0
\(616\) −7.11042e9 −1.22564
\(617\) −8.36433e9 −1.43362 −0.716809 0.697270i \(-0.754398\pi\)
−0.716809 + 0.697270i \(0.754398\pi\)
\(618\) 0 0
\(619\) −1.37677e8 −0.0233315 −0.0116658 0.999932i \(-0.503713\pi\)
−0.0116658 + 0.999932i \(0.503713\pi\)
\(620\) 2.33861e9 0.394082
\(621\) 0 0
\(622\) 1.32153e10 2.20196
\(623\) −1.16814e9 −0.193547
\(624\) 0 0
\(625\) 1.21700e9 0.199393
\(626\) −2.36253e8 −0.0384917
\(627\) 0 0
\(628\) −1.63110e9 −0.262798
\(629\) −7.65715e9 −1.22685
\(630\) 0 0
\(631\) −1.86229e9 −0.295083 −0.147542 0.989056i \(-0.547136\pi\)
−0.147542 + 0.989056i \(0.547136\pi\)
\(632\) 3.06713e9 0.483306
\(633\) 0 0
\(634\) −8.89756e9 −1.38662
\(635\) 1.71195e9 0.265328
\(636\) 0 0
\(637\) −1.15321e10 −1.76775
\(638\) 1.43198e10 2.18305
\(639\) 0 0
\(640\) −4.22724e9 −0.637422
\(641\) −1.04508e10 −1.56727 −0.783637 0.621219i \(-0.786638\pi\)
−0.783637 + 0.621219i \(0.786638\pi\)
\(642\) 0 0
\(643\) −1.17114e10 −1.73728 −0.868642 0.495440i \(-0.835007\pi\)
−0.868642 + 0.495440i \(0.835007\pi\)
\(644\) −6.41870e9 −0.946993
\(645\) 0 0
\(646\) −1.17749e10 −1.71848
\(647\) 2.82698e7 0.00410353 0.00205176 0.999998i \(-0.499347\pi\)
0.00205176 + 0.999998i \(0.499347\pi\)
\(648\) 0 0
\(649\) −1.05401e9 −0.151352
\(650\) 8.98314e9 1.28301
\(651\) 0 0
\(652\) 1.63156e9 0.230535
\(653\) −1.03288e10 −1.45163 −0.725813 0.687892i \(-0.758536\pi\)
−0.725813 + 0.687892i \(0.758536\pi\)
\(654\) 0 0
\(655\) 6.54237e8 0.0909684
\(656\) 1.17938e10 1.63114
\(657\) 0 0
\(658\) −1.15244e10 −1.57698
\(659\) 8.85377e9 1.20512 0.602558 0.798075i \(-0.294148\pi\)
0.602558 + 0.798075i \(0.294148\pi\)
\(660\) 0 0
\(661\) 3.94438e8 0.0531219 0.0265610 0.999647i \(-0.491544\pi\)
0.0265610 + 0.999647i \(0.491544\pi\)
\(662\) 1.61746e10 2.16686
\(663\) 0 0
\(664\) −8.99838e9 −1.19282
\(665\) 6.70756e9 0.884481
\(666\) 0 0
\(667\) −2.03110e10 −2.65027
\(668\) 2.44547e9 0.317427
\(669\) 0 0
\(670\) 2.81232e9 0.361246
\(671\) 1.02769e8 0.0131320
\(672\) 0 0
\(673\) −3.32239e8 −0.0420144 −0.0210072 0.999779i \(-0.506687\pi\)
−0.0210072 + 0.999779i \(0.506687\pi\)
\(674\) 5.38472e9 0.677413
\(675\) 0 0
\(676\) 4.34810e9 0.541360
\(677\) 7.35957e9 0.911574 0.455787 0.890089i \(-0.349358\pi\)
0.455787 + 0.890089i \(0.349358\pi\)
\(678\) 0 0
\(679\) 2.15236e10 2.63858
\(680\) −4.21963e9 −0.514628
\(681\) 0 0
\(682\) 2.11355e10 2.55134
\(683\) 1.53063e10 1.83822 0.919111 0.393999i \(-0.128909\pi\)
0.919111 + 0.393999i \(0.128909\pi\)
\(684\) 0 0
\(685\) −2.82104e9 −0.335345
\(686\) −2.08589e9 −0.246693
\(687\) 0 0
\(688\) −1.09661e10 −1.28379
\(689\) −6.11924e8 −0.0712739
\(690\) 0 0
\(691\) −1.49200e10 −1.72026 −0.860132 0.510072i \(-0.829619\pi\)
−0.860132 + 0.510072i \(0.829619\pi\)
\(692\) −1.03635e9 −0.118887
\(693\) 0 0
\(694\) 7.64611e9 0.868325
\(695\) 6.28077e9 0.709686
\(696\) 0 0
\(697\) 1.54719e10 1.73073
\(698\) −4.28329e9 −0.476742
\(699\) 0 0
\(700\) 3.63685e9 0.400758
\(701\) 6.52424e9 0.715347 0.357673 0.933847i \(-0.383570\pi\)
0.357673 + 0.933847i \(0.383570\pi\)
\(702\) 0 0
\(703\) 9.55558e9 1.03732
\(704\) −3.95415e9 −0.427119
\(705\) 0 0
\(706\) −5.31262e9 −0.568188
\(707\) 8.20726e8 0.0873434
\(708\) 0 0
\(709\) 4.55527e9 0.480012 0.240006 0.970771i \(-0.422851\pi\)
0.240006 + 0.970771i \(0.422851\pi\)
\(710\) 2.52219e9 0.264468
\(711\) 0 0
\(712\) −9.17054e8 −0.0952171
\(713\) −2.99784e10 −3.09738
\(714\) 0 0
\(715\) −9.56262e9 −0.978376
\(716\) −4.30157e9 −0.437957
\(717\) 0 0
\(718\) −1.87607e10 −1.89153
\(719\) −6.09158e9 −0.611194 −0.305597 0.952161i \(-0.598856\pi\)
−0.305597 + 0.952161i \(0.598856\pi\)
\(720\) 0 0
\(721\) 9.28145e9 0.922236
\(722\) 2.77584e9 0.274482
\(723\) 0 0
\(724\) −3.25930e9 −0.319183
\(725\) 1.15083e10 1.12157
\(726\) 0 0
\(727\) 1.68599e10 1.62737 0.813684 0.581308i \(-0.197459\pi\)
0.813684 + 0.581308i \(0.197459\pi\)
\(728\) −1.69741e10 −1.63052
\(729\) 0 0
\(730\) 9.81011e9 0.933349
\(731\) −1.43861e10 −1.36217
\(732\) 0 0
\(733\) −1.44578e9 −0.135593 −0.0677966 0.997699i \(-0.521597\pi\)
−0.0677966 + 0.997699i \(0.521597\pi\)
\(734\) −1.28762e10 −1.20185
\(735\) 0 0
\(736\) −1.32851e10 −1.22827
\(737\) 7.11707e9 0.654885
\(738\) 0 0
\(739\) 8.56684e8 0.0780845 0.0390423 0.999238i \(-0.487569\pi\)
0.0390423 + 0.999238i \(0.487569\pi\)
\(740\) −2.17937e9 −0.197706
\(741\) 0 0
\(742\) −8.84738e8 −0.0795062
\(743\) −1.09080e10 −0.975630 −0.487815 0.872947i \(-0.662206\pi\)
−0.487815 + 0.872947i \(0.662206\pi\)
\(744\) 0 0
\(745\) 4.22764e9 0.374585
\(746\) −8.68006e9 −0.765485
\(747\) 0 0
\(748\) 6.79622e9 0.593762
\(749\) −9.92396e9 −0.862976
\(750\) 0 0
\(751\) −5.27485e9 −0.454433 −0.227217 0.973844i \(-0.572963\pi\)
−0.227217 + 0.973844i \(0.572963\pi\)
\(752\) −1.31929e10 −1.13130
\(753\) 0 0
\(754\) 3.41843e10 2.90420
\(755\) −7.68311e9 −0.649715
\(756\) 0 0
\(757\) −2.07054e10 −1.73479 −0.867395 0.497620i \(-0.834208\pi\)
−0.867395 + 0.497620i \(0.834208\pi\)
\(758\) −5.30659e8 −0.0442562
\(759\) 0 0
\(760\) 5.26580e9 0.435128
\(761\) −1.35780e10 −1.11683 −0.558417 0.829561i \(-0.688591\pi\)
−0.558417 + 0.829561i \(0.688591\pi\)
\(762\) 0 0
\(763\) 2.98927e10 2.43629
\(764\) 4.20903e9 0.341472
\(765\) 0 0
\(766\) −8.37001e8 −0.0672861
\(767\) −2.51614e9 −0.201350
\(768\) 0 0
\(769\) 1.62978e10 1.29237 0.646184 0.763182i \(-0.276364\pi\)
0.646184 + 0.763182i \(0.276364\pi\)
\(770\) −1.38259e10 −1.09138
\(771\) 0 0
\(772\) −1.01658e9 −0.0795208
\(773\) 2.64697e9 0.206120 0.103060 0.994675i \(-0.467137\pi\)
0.103060 + 0.994675i \(0.467137\pi\)
\(774\) 0 0
\(775\) 1.69858e10 1.31078
\(776\) 1.68972e10 1.29807
\(777\) 0 0
\(778\) 5.52800e9 0.420862
\(779\) −1.93079e10 −1.46337
\(780\) 0 0
\(781\) 6.38285e9 0.479442
\(782\) −3.44257e10 −2.57430
\(783\) 0 0
\(784\) −1.90876e10 −1.41463
\(785\) 4.98336e9 0.367687
\(786\) 0 0
\(787\) −7.25919e9 −0.530856 −0.265428 0.964131i \(-0.585513\pi\)
−0.265428 + 0.964131i \(0.585513\pi\)
\(788\) −1.60889e9 −0.117134
\(789\) 0 0
\(790\) 5.96390e9 0.430364
\(791\) 9.74751e9 0.700287
\(792\) 0 0
\(793\) 2.45330e8 0.0174701
\(794\) −7.21097e9 −0.511237
\(795\) 0 0
\(796\) −3.59457e9 −0.252610
\(797\) 3.67130e9 0.256871 0.128436 0.991718i \(-0.459004\pi\)
0.128436 + 0.991718i \(0.459004\pi\)
\(798\) 0 0
\(799\) −1.73074e10 −1.20038
\(800\) 7.52737e9 0.519791
\(801\) 0 0
\(802\) −2.43152e10 −1.66444
\(803\) 2.48262e10 1.69202
\(804\) 0 0
\(805\) 1.96105e10 1.32496
\(806\) 5.04550e10 3.39415
\(807\) 0 0
\(808\) 6.44315e8 0.0429693
\(809\) 5.97080e9 0.396472 0.198236 0.980154i \(-0.436479\pi\)
0.198236 + 0.980154i \(0.436479\pi\)
\(810\) 0 0
\(811\) 2.51680e10 1.65682 0.828409 0.560123i \(-0.189246\pi\)
0.828409 + 0.560123i \(0.189246\pi\)
\(812\) 1.38396e10 0.907148
\(813\) 0 0
\(814\) −1.96964e10 −1.27997
\(815\) −4.98477e9 −0.322548
\(816\) 0 0
\(817\) 1.79528e10 1.15174
\(818\) 2.62936e10 1.67963
\(819\) 0 0
\(820\) 4.40361e9 0.278907
\(821\) −1.74411e10 −1.09995 −0.549975 0.835181i \(-0.685363\pi\)
−0.549975 + 0.835181i \(0.685363\pi\)
\(822\) 0 0
\(823\) 1.10753e10 0.692557 0.346278 0.938132i \(-0.387445\pi\)
0.346278 + 0.938132i \(0.387445\pi\)
\(824\) 7.28644e9 0.453702
\(825\) 0 0
\(826\) −3.63791e9 −0.224606
\(827\) 7.16696e9 0.440622 0.220311 0.975430i \(-0.429293\pi\)
0.220311 + 0.975430i \(0.429293\pi\)
\(828\) 0 0
\(829\) 1.09285e9 0.0666224 0.0333112 0.999445i \(-0.489395\pi\)
0.0333112 + 0.999445i \(0.489395\pi\)
\(830\) −1.74970e10 −1.06216
\(831\) 0 0
\(832\) −9.43940e9 −0.568215
\(833\) −2.50404e10 −1.50101
\(834\) 0 0
\(835\) −7.47144e9 −0.444121
\(836\) −8.48121e9 −0.502037
\(837\) 0 0
\(838\) −6.53834e9 −0.383807
\(839\) 2.38628e10 1.39494 0.697469 0.716615i \(-0.254309\pi\)
0.697469 + 0.716615i \(0.254309\pi\)
\(840\) 0 0
\(841\) 2.65435e10 1.53877
\(842\) −2.30050e9 −0.132810
\(843\) 0 0
\(844\) −9.49235e9 −0.543469
\(845\) −1.32844e10 −0.757431
\(846\) 0 0
\(847\) −9.10073e9 −0.514617
\(848\) −1.01284e9 −0.0570367
\(849\) 0 0
\(850\) 1.95057e10 1.08942
\(851\) 2.79371e10 1.55392
\(852\) 0 0
\(853\) 1.71276e10 0.944878 0.472439 0.881363i \(-0.343374\pi\)
0.472439 + 0.881363i \(0.343374\pi\)
\(854\) 3.54706e8 0.0194879
\(855\) 0 0
\(856\) −7.79086e9 −0.424548
\(857\) −3.37860e8 −0.0183360 −0.00916798 0.999958i \(-0.502918\pi\)
−0.00916798 + 0.999958i \(0.502918\pi\)
\(858\) 0 0
\(859\) 3.39086e9 0.182530 0.0912649 0.995827i \(-0.470909\pi\)
0.0912649 + 0.995827i \(0.470909\pi\)
\(860\) −4.09455e9 −0.219514
\(861\) 0 0
\(862\) −2.66249e10 −1.41583
\(863\) −9.19359e9 −0.486909 −0.243454 0.969912i \(-0.578281\pi\)
−0.243454 + 0.969912i \(0.578281\pi\)
\(864\) 0 0
\(865\) 3.16628e9 0.166339
\(866\) −8.70904e9 −0.455678
\(867\) 0 0
\(868\) 2.04269e10 1.06019
\(869\) 1.50927e10 0.780186
\(870\) 0 0
\(871\) 1.69899e10 0.871221
\(872\) 2.34674e10 1.19855
\(873\) 0 0
\(874\) 4.29608e10 2.17662
\(875\) −2.68966e10 −1.35728
\(876\) 0 0
\(877\) −1.78531e10 −0.893747 −0.446874 0.894597i \(-0.647463\pi\)
−0.446874 + 0.894597i \(0.647463\pi\)
\(878\) −2.91952e10 −1.45573
\(879\) 0 0
\(880\) −1.58277e10 −0.782942
\(881\) 1.00212e10 0.493746 0.246873 0.969048i \(-0.420597\pi\)
0.246873 + 0.969048i \(0.420597\pi\)
\(882\) 0 0
\(883\) 4.80376e8 0.0234811 0.0117406 0.999931i \(-0.496263\pi\)
0.0117406 + 0.999931i \(0.496263\pi\)
\(884\) 1.62240e10 0.789906
\(885\) 0 0
\(886\) −3.97703e10 −1.92106
\(887\) 2.86366e9 0.137781 0.0688903 0.997624i \(-0.478054\pi\)
0.0688903 + 0.997624i \(0.478054\pi\)
\(888\) 0 0
\(889\) 1.49532e10 0.713804
\(890\) −1.78317e9 −0.0847869
\(891\) 0 0
\(892\) 1.01235e10 0.477589
\(893\) 2.15984e10 1.01494
\(894\) 0 0
\(895\) 1.31422e10 0.612758
\(896\) −3.69234e10 −1.71484
\(897\) 0 0
\(898\) 2.06158e10 0.950020
\(899\) 6.46377e10 2.96706
\(900\) 0 0
\(901\) −1.32871e9 −0.0605193
\(902\) 3.97983e10 1.80568
\(903\) 0 0
\(904\) 7.65233e9 0.344512
\(905\) 9.95787e9 0.446577
\(906\) 0 0
\(907\) −5.67695e9 −0.252633 −0.126316 0.991990i \(-0.540315\pi\)
−0.126316 + 0.991990i \(0.540315\pi\)
\(908\) −1.75313e9 −0.0777164
\(909\) 0 0
\(910\) −3.30054e10 −1.45191
\(911\) −2.67566e10 −1.17251 −0.586256 0.810126i \(-0.699399\pi\)
−0.586256 + 0.810126i \(0.699399\pi\)
\(912\) 0 0
\(913\) −4.42792e10 −1.92554
\(914\) −9.33871e9 −0.404553
\(915\) 0 0
\(916\) −1.74602e10 −0.750610
\(917\) 5.71452e9 0.244730
\(918\) 0 0
\(919\) 2.00200e10 0.850862 0.425431 0.904991i \(-0.360122\pi\)
0.425431 + 0.904991i \(0.360122\pi\)
\(920\) 1.53953e10 0.651826
\(921\) 0 0
\(922\) 3.18776e10 1.33945
\(923\) 1.52372e10 0.637822
\(924\) 0 0
\(925\) −1.58293e10 −0.657604
\(926\) 3.62520e10 1.50035
\(927\) 0 0
\(928\) 2.86446e10 1.17659
\(929\) −2.42743e10 −0.993326 −0.496663 0.867943i \(-0.665441\pi\)
−0.496663 + 0.867943i \(0.665441\pi\)
\(930\) 0 0
\(931\) 3.12486e10 1.26913
\(932\) 1.35340e10 0.547611
\(933\) 0 0
\(934\) −4.84332e10 −1.94504
\(935\) −2.07639e10 −0.830748
\(936\) 0 0
\(937\) 2.31846e10 0.920683 0.460342 0.887742i \(-0.347727\pi\)
0.460342 + 0.887742i \(0.347727\pi\)
\(938\) 2.45645e10 0.971849
\(939\) 0 0
\(940\) −4.92602e9 −0.193441
\(941\) 4.82845e10 1.88905 0.944527 0.328434i \(-0.106521\pi\)
0.944527 + 0.328434i \(0.106521\pi\)
\(942\) 0 0
\(943\) −5.64495e10 −2.19214
\(944\) −4.16464e9 −0.161130
\(945\) 0 0
\(946\) −3.70052e10 −1.42116
\(947\) 9.48483e9 0.362915 0.181457 0.983399i \(-0.441919\pi\)
0.181457 + 0.983399i \(0.441919\pi\)
\(948\) 0 0
\(949\) 5.92654e10 2.25097
\(950\) −2.43417e10 −0.921124
\(951\) 0 0
\(952\) −3.68569e10 −1.38449
\(953\) −3.13620e10 −1.17376 −0.586880 0.809674i \(-0.699644\pi\)
−0.586880 + 0.809674i \(0.699644\pi\)
\(954\) 0 0
\(955\) −1.28595e10 −0.477762
\(956\) 3.54481e9 0.131217
\(957\) 0 0
\(958\) 1.01379e10 0.372537
\(959\) −2.46407e10 −0.902171
\(960\) 0 0
\(961\) 6.78906e10 2.46762
\(962\) −4.70194e10 −1.70280
\(963\) 0 0
\(964\) −1.17661e10 −0.423024
\(965\) 3.10587e9 0.111260
\(966\) 0 0
\(967\) −4.56505e10 −1.62350 −0.811752 0.584003i \(-0.801486\pi\)
−0.811752 + 0.584003i \(0.801486\pi\)
\(968\) −7.14457e9 −0.253170
\(969\) 0 0
\(970\) 3.28559e10 1.15588
\(971\) 4.75876e10 1.66812 0.834059 0.551675i \(-0.186011\pi\)
0.834059 + 0.551675i \(0.186011\pi\)
\(972\) 0 0
\(973\) 5.48602e10 1.90925
\(974\) −4.15558e10 −1.44104
\(975\) 0 0
\(976\) 4.06063e8 0.0139804
\(977\) 3.64883e10 1.25176 0.625882 0.779918i \(-0.284739\pi\)
0.625882 + 0.779918i \(0.284739\pi\)
\(978\) 0 0
\(979\) −4.51264e9 −0.153706
\(980\) −7.12697e9 −0.241888
\(981\) 0 0
\(982\) 2.69551e10 0.908346
\(983\) 2.58715e10 0.868729 0.434365 0.900737i \(-0.356973\pi\)
0.434365 + 0.900737i \(0.356973\pi\)
\(984\) 0 0
\(985\) 4.91551e9 0.163886
\(986\) 7.42266e10 2.46598
\(987\) 0 0
\(988\) −2.02464e10 −0.667881
\(989\) 5.24877e10 1.72532
\(990\) 0 0
\(991\) 5.08417e10 1.65944 0.829722 0.558177i \(-0.188499\pi\)
0.829722 + 0.558177i \(0.188499\pi\)
\(992\) 4.22785e10 1.37508
\(993\) 0 0
\(994\) 2.20304e10 0.711492
\(995\) 1.09822e10 0.353434
\(996\) 0 0
\(997\) −7.59411e9 −0.242686 −0.121343 0.992611i \(-0.538720\pi\)
−0.121343 + 0.992611i \(0.538720\pi\)
\(998\) −5.21076e10 −1.65937
\(999\) 0 0
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 531.8.a.d.1.13 17
3.2 odd 2 177.8.a.b.1.5 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.5 17 3.2 odd 2
531.8.a.d.1.13 17 1.1 even 1 trivial