Properties

Label 531.8.a.d.1.8
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 1.8
Root \(2.41303\) of defining polynomial
Character \(\chi\) \(=\) 531.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.413031 q^{2} -127.829 q^{4} -231.152 q^{5} +302.150 q^{7} +105.665 q^{8} +O(q^{10})\) \(q-0.413031 q^{2} -127.829 q^{4} -231.152 q^{5} +302.150 q^{7} +105.665 q^{8} +95.4729 q^{10} -1281.53 q^{11} -14851.9 q^{13} -124.797 q^{14} +16318.5 q^{16} +21166.2 q^{17} -1755.31 q^{19} +29548.0 q^{20} +529.311 q^{22} -91962.2 q^{23} -24693.8 q^{25} +6134.31 q^{26} -38623.7 q^{28} -62655.4 q^{29} -103275. q^{31} -20265.2 q^{32} -8742.32 q^{34} -69842.6 q^{35} -37921.6 q^{37} +724.998 q^{38} -24424.8 q^{40} +45664.2 q^{41} -637229. q^{43} +163817. q^{44} +37983.3 q^{46} -229428. q^{47} -732248. q^{49} +10199.3 q^{50} +1.89852e6 q^{52} +1.07832e6 q^{53} +296227. q^{55} +31926.8 q^{56} +25878.6 q^{58} +205379. q^{59} -2.75987e6 q^{61} +42655.6 q^{62} -2.08040e6 q^{64} +3.43305e6 q^{65} -3.11037e6 q^{67} -2.70567e6 q^{68} +28847.1 q^{70} -4.01814e6 q^{71} -5.86422e6 q^{73} +15662.8 q^{74} +224380. q^{76} -387214. q^{77} +4.62456e6 q^{79} -3.77206e6 q^{80} -18860.7 q^{82} -9.51599e6 q^{83} -4.89262e6 q^{85} +263195. q^{86} -135413. q^{88} +6.95658e6 q^{89} -4.48752e6 q^{91} +1.17555e7 q^{92} +94760.7 q^{94} +405743. q^{95} -9.56241e6 q^{97} +302441. q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} + O(q^{10}) \) \( 17q + 32q^{2} + 1166q^{4} + 1072q^{5} - 2407q^{7} + 6645q^{8} - 6391q^{10} + 8888q^{11} - 12702q^{13} + 17555q^{14} + 139226q^{16} + 36167q^{17} - 71037q^{19} + 274883q^{20} - 325182q^{22} + 269995q^{23} + 97329q^{25} + 336906q^{26} - 901362q^{28} + 543825q^{29} - 633109q^{31} + 837062q^{32} - 529288q^{34} + 287621q^{35} - 867607q^{37} + 1727169q^{38} - 815662q^{40} + 1428939q^{41} - 477060q^{43} + 1667926q^{44} + 5305549q^{46} + 1217849q^{47} + 4350738q^{49} - 4561369q^{50} + 4175994q^{52} + 3487068q^{53} - 960484q^{55} + 5363196q^{56} - 3082906q^{58} + 3491443q^{59} + 998917q^{61} + 5742614q^{62} + 17531621q^{64} + 6075816q^{65} - 356026q^{67} + 16149231q^{68} - 548798q^{70} + 12879428q^{71} - 6176157q^{73} + 5971906q^{74} - 17624580q^{76} - 239687q^{77} - 18886490q^{79} + 70463349q^{80} - 19351611q^{82} + 22824893q^{83} - 7973079q^{85} + 27502196q^{86} - 62527651q^{88} + 30609647q^{89} - 36301521q^{91} + 41388548q^{92} + 1010176q^{94} + 29303629q^{95} - 26249806q^{97} + 93110852q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.413031 −0.0365071 −0.0182536 0.999833i \(-0.505811\pi\)
−0.0182536 + 0.999833i \(0.505811\pi\)
\(3\) 0 0
\(4\) −127.829 −0.998667
\(5\) −231.152 −0.826994 −0.413497 0.910505i \(-0.635693\pi\)
−0.413497 + 0.910505i \(0.635693\pi\)
\(6\) 0 0
\(7\) 302.150 0.332950 0.166475 0.986046i \(-0.446761\pi\)
0.166475 + 0.986046i \(0.446761\pi\)
\(8\) 105.665 0.0729656
\(9\) 0 0
\(10\) 95.4729 0.0301912
\(11\) −1281.53 −0.290304 −0.145152 0.989409i \(-0.546367\pi\)
−0.145152 + 0.989409i \(0.546367\pi\)
\(12\) 0 0
\(13\) −14851.9 −1.87491 −0.937457 0.348100i \(-0.886827\pi\)
−0.937457 + 0.348100i \(0.886827\pi\)
\(14\) −124.797 −0.0121551
\(15\) 0 0
\(16\) 16318.5 0.996003
\(17\) 21166.2 1.04489 0.522447 0.852672i \(-0.325019\pi\)
0.522447 + 0.852672i \(0.325019\pi\)
\(18\) 0 0
\(19\) −1755.31 −0.0587106 −0.0293553 0.999569i \(-0.509345\pi\)
−0.0293553 + 0.999569i \(0.509345\pi\)
\(20\) 29548.0 0.825892
\(21\) 0 0
\(22\) 529.311 0.0105982
\(23\) −91962.2 −1.57602 −0.788011 0.615662i \(-0.788889\pi\)
−0.788011 + 0.615662i \(0.788889\pi\)
\(24\) 0 0
\(25\) −24693.8 −0.316081
\(26\) 6134.31 0.0684478
\(27\) 0 0
\(28\) −38623.7 −0.332507
\(29\) −62655.4 −0.477052 −0.238526 0.971136i \(-0.576664\pi\)
−0.238526 + 0.971136i \(0.576664\pi\)
\(30\) 0 0
\(31\) −103275. −0.622626 −0.311313 0.950307i \(-0.600769\pi\)
−0.311313 + 0.950307i \(0.600769\pi\)
\(32\) −20265.2 −0.109327
\(33\) 0 0
\(34\) −8742.32 −0.0381461
\(35\) −69842.6 −0.275348
\(36\) 0 0
\(37\) −37921.6 −0.123078 −0.0615390 0.998105i \(-0.519601\pi\)
−0.0615390 + 0.998105i \(0.519601\pi\)
\(38\) 724.998 0.00214335
\(39\) 0 0
\(40\) −24424.8 −0.0603421
\(41\) 45664.2 0.103474 0.0517371 0.998661i \(-0.483524\pi\)
0.0517371 + 0.998661i \(0.483524\pi\)
\(42\) 0 0
\(43\) −637229. −1.22224 −0.611119 0.791539i \(-0.709280\pi\)
−0.611119 + 0.791539i \(0.709280\pi\)
\(44\) 163817. 0.289917
\(45\) 0 0
\(46\) 37983.3 0.0575360
\(47\) −229428. −0.322332 −0.161166 0.986927i \(-0.551525\pi\)
−0.161166 + 0.986927i \(0.551525\pi\)
\(48\) 0 0
\(49\) −732248. −0.889144
\(50\) 10199.3 0.0115392
\(51\) 0 0
\(52\) 1.89852e6 1.87242
\(53\) 1.07832e6 0.994906 0.497453 0.867491i \(-0.334269\pi\)
0.497453 + 0.867491i \(0.334269\pi\)
\(54\) 0 0
\(55\) 296227. 0.240080
\(56\) 31926.8 0.0242939
\(57\) 0 0
\(58\) 25878.6 0.0174158
\(59\) 205379. 0.130189
\(60\) 0 0
\(61\) −2.75987e6 −1.55680 −0.778402 0.627767i \(-0.783969\pi\)
−0.778402 + 0.627767i \(0.783969\pi\)
\(62\) 42655.6 0.0227303
\(63\) 0 0
\(64\) −2.08040e6 −0.992012
\(65\) 3.43305e6 1.55054
\(66\) 0 0
\(67\) −3.11037e6 −1.26343 −0.631715 0.775201i \(-0.717649\pi\)
−0.631715 + 0.775201i \(0.717649\pi\)
\(68\) −2.70567e6 −1.04350
\(69\) 0 0
\(70\) 28847.1 0.0100522
\(71\) −4.01814e6 −1.33236 −0.666179 0.745792i \(-0.732071\pi\)
−0.666179 + 0.745792i \(0.732071\pi\)
\(72\) 0 0
\(73\) −5.86422e6 −1.76433 −0.882165 0.470940i \(-0.843915\pi\)
−0.882165 + 0.470940i \(0.843915\pi\)
\(74\) 15662.8 0.00449322
\(75\) 0 0
\(76\) 224380. 0.0586323
\(77\) −387214. −0.0966570
\(78\) 0 0
\(79\) 4.62456e6 1.05530 0.527649 0.849462i \(-0.323074\pi\)
0.527649 + 0.849462i \(0.323074\pi\)
\(80\) −3.77206e6 −0.823689
\(81\) 0 0
\(82\) −18860.7 −0.00377755
\(83\) −9.51599e6 −1.82676 −0.913378 0.407113i \(-0.866535\pi\)
−0.913378 + 0.407113i \(0.866535\pi\)
\(84\) 0 0
\(85\) −4.89262e6 −0.864122
\(86\) 263195. 0.0446204
\(87\) 0 0
\(88\) −135413. −0.0211822
\(89\) 6.95658e6 1.04600 0.522999 0.852333i \(-0.324813\pi\)
0.522999 + 0.852333i \(0.324813\pi\)
\(90\) 0 0
\(91\) −4.48752e6 −0.624254
\(92\) 1.17555e7 1.57392
\(93\) 0 0
\(94\) 94760.7 0.0117674
\(95\) 405743. 0.0485533
\(96\) 0 0
\(97\) −9.56241e6 −1.06382 −0.531908 0.846802i \(-0.678525\pi\)
−0.531908 + 0.846802i \(0.678525\pi\)
\(98\) 302441. 0.0324601
\(99\) 0 0
\(100\) 3.15660e6 0.315660
\(101\) 9.74084e6 0.940744 0.470372 0.882468i \(-0.344120\pi\)
0.470372 + 0.882468i \(0.344120\pi\)
\(102\) 0 0
\(103\) −1.17131e7 −1.05619 −0.528093 0.849186i \(-0.677093\pi\)
−0.528093 + 0.849186i \(0.677093\pi\)
\(104\) −1.56934e6 −0.136804
\(105\) 0 0
\(106\) −445380. −0.0363212
\(107\) 2.32996e7 1.83867 0.919337 0.393471i \(-0.128726\pi\)
0.919337 + 0.393471i \(0.128726\pi\)
\(108\) 0 0
\(109\) 1.84368e7 1.36362 0.681808 0.731531i \(-0.261194\pi\)
0.681808 + 0.731531i \(0.261194\pi\)
\(110\) −122351. −0.00876463
\(111\) 0 0
\(112\) 4.93064e6 0.331620
\(113\) 1.69409e7 1.10449 0.552247 0.833681i \(-0.313771\pi\)
0.552247 + 0.833681i \(0.313771\pi\)
\(114\) 0 0
\(115\) 2.12572e7 1.30336
\(116\) 8.00921e6 0.476416
\(117\) 0 0
\(118\) −84827.9 −0.00475282
\(119\) 6.39538e6 0.347898
\(120\) 0 0
\(121\) −1.78449e7 −0.915723
\(122\) 1.13991e6 0.0568344
\(123\) 0 0
\(124\) 1.32015e7 0.621796
\(125\) 2.37668e7 1.08839
\(126\) 0 0
\(127\) −1.02426e7 −0.443708 −0.221854 0.975080i \(-0.571211\pi\)
−0.221854 + 0.975080i \(0.571211\pi\)
\(128\) 3.45322e6 0.145542
\(129\) 0 0
\(130\) −1.41796e6 −0.0566059
\(131\) −2.19669e7 −0.853728 −0.426864 0.904316i \(-0.640382\pi\)
−0.426864 + 0.904316i \(0.640382\pi\)
\(132\) 0 0
\(133\) −530367. −0.0195477
\(134\) 1.28468e6 0.0461242
\(135\) 0 0
\(136\) 2.23654e6 0.0762414
\(137\) −1.76506e6 −0.0586459 −0.0293229 0.999570i \(-0.509335\pi\)
−0.0293229 + 0.999570i \(0.509335\pi\)
\(138\) 0 0
\(139\) −2.43007e7 −0.767481 −0.383740 0.923441i \(-0.625364\pi\)
−0.383740 + 0.923441i \(0.625364\pi\)
\(140\) 8.92793e6 0.274981
\(141\) 0 0
\(142\) 1.65962e6 0.0486406
\(143\) 1.90332e7 0.544296
\(144\) 0 0
\(145\) 1.44829e7 0.394519
\(146\) 2.42210e6 0.0644107
\(147\) 0 0
\(148\) 4.84749e6 0.122914
\(149\) −7.68149e7 −1.90236 −0.951181 0.308633i \(-0.900129\pi\)
−0.951181 + 0.308633i \(0.900129\pi\)
\(150\) 0 0
\(151\) 4.28340e7 1.01244 0.506220 0.862404i \(-0.331042\pi\)
0.506220 + 0.862404i \(0.331042\pi\)
\(152\) −185476. −0.00428385
\(153\) 0 0
\(154\) 159931. 0.00352867
\(155\) 2.38721e7 0.514908
\(156\) 0 0
\(157\) 5.51222e7 1.13678 0.568392 0.822758i \(-0.307566\pi\)
0.568392 + 0.822758i \(0.307566\pi\)
\(158\) −1.91009e6 −0.0385259
\(159\) 0 0
\(160\) 4.68435e6 0.0904126
\(161\) −2.77864e7 −0.524737
\(162\) 0 0
\(163\) 3.76340e7 0.680650 0.340325 0.940308i \(-0.389463\pi\)
0.340325 + 0.940308i \(0.389463\pi\)
\(164\) −5.83723e6 −0.103336
\(165\) 0 0
\(166\) 3.93040e6 0.0666896
\(167\) −9.25680e7 −1.53799 −0.768995 0.639255i \(-0.779243\pi\)
−0.768995 + 0.639255i \(0.779243\pi\)
\(168\) 0 0
\(169\) 1.57832e8 2.51531
\(170\) 2.02080e6 0.0315466
\(171\) 0 0
\(172\) 8.14566e7 1.22061
\(173\) 1.09599e8 1.60933 0.804665 0.593729i \(-0.202345\pi\)
0.804665 + 0.593729i \(0.202345\pi\)
\(174\) 0 0
\(175\) −7.46124e6 −0.105239
\(176\) −2.09126e7 −0.289144
\(177\) 0 0
\(178\) −2.87328e6 −0.0381864
\(179\) −4.35682e7 −0.567786 −0.283893 0.958856i \(-0.591626\pi\)
−0.283893 + 0.958856i \(0.591626\pi\)
\(180\) 0 0
\(181\) −1.29762e8 −1.62656 −0.813282 0.581870i \(-0.802321\pi\)
−0.813282 + 0.581870i \(0.802321\pi\)
\(182\) 1.85348e6 0.0227897
\(183\) 0 0
\(184\) −9.71724e6 −0.114995
\(185\) 8.76564e6 0.101785
\(186\) 0 0
\(187\) −2.71251e7 −0.303337
\(188\) 2.93276e7 0.321902
\(189\) 0 0
\(190\) −167585. −0.00177254
\(191\) −3.07331e7 −0.319146 −0.159573 0.987186i \(-0.551012\pi\)
−0.159573 + 0.987186i \(0.551012\pi\)
\(192\) 0 0
\(193\) 1.24448e8 1.24606 0.623029 0.782199i \(-0.285902\pi\)
0.623029 + 0.782199i \(0.285902\pi\)
\(194\) 3.94957e6 0.0388369
\(195\) 0 0
\(196\) 9.36029e7 0.887959
\(197\) 5.43655e7 0.506630 0.253315 0.967384i \(-0.418479\pi\)
0.253315 + 0.967384i \(0.418479\pi\)
\(198\) 0 0
\(199\) 1.40875e8 1.26721 0.633606 0.773656i \(-0.281574\pi\)
0.633606 + 0.773656i \(0.281574\pi\)
\(200\) −2.60929e6 −0.0230630
\(201\) 0 0
\(202\) −4.02327e6 −0.0343439
\(203\) −1.89313e7 −0.158835
\(204\) 0 0
\(205\) −1.05554e7 −0.0855726
\(206\) 4.83786e6 0.0385584
\(207\) 0 0
\(208\) −2.42362e8 −1.86742
\(209\) 2.24948e6 0.0170439
\(210\) 0 0
\(211\) 2.07381e8 1.51978 0.759891 0.650051i \(-0.225252\pi\)
0.759891 + 0.650051i \(0.225252\pi\)
\(212\) −1.37841e8 −0.993580
\(213\) 0 0
\(214\) −9.62345e6 −0.0671247
\(215\) 1.47297e8 1.01078
\(216\) 0 0
\(217\) −3.12044e7 −0.207304
\(218\) −7.61496e6 −0.0497817
\(219\) 0 0
\(220\) −3.78666e7 −0.239760
\(221\) −3.14360e8 −1.95909
\(222\) 0 0
\(223\) −2.57344e8 −1.55398 −0.776992 0.629511i \(-0.783255\pi\)
−0.776992 + 0.629511i \(0.783255\pi\)
\(224\) −6.12314e6 −0.0364004
\(225\) 0 0
\(226\) −6.99714e6 −0.0403219
\(227\) 1.28065e8 0.726676 0.363338 0.931657i \(-0.381637\pi\)
0.363338 + 0.931657i \(0.381637\pi\)
\(228\) 0 0
\(229\) 9.10898e7 0.501240 0.250620 0.968086i \(-0.419366\pi\)
0.250620 + 0.968086i \(0.419366\pi\)
\(230\) −8.77990e6 −0.0475819
\(231\) 0 0
\(232\) −6.62052e6 −0.0348084
\(233\) 6.28725e7 0.325623 0.162812 0.986657i \(-0.447944\pi\)
0.162812 + 0.986657i \(0.447944\pi\)
\(234\) 0 0
\(235\) 5.30326e7 0.266566
\(236\) −2.62535e7 −0.130015
\(237\) 0 0
\(238\) −2.64149e6 −0.0127008
\(239\) 2.11572e8 1.00246 0.501229 0.865315i \(-0.332881\pi\)
0.501229 + 0.865315i \(0.332881\pi\)
\(240\) 0 0
\(241\) 1.41219e7 0.0649882 0.0324941 0.999472i \(-0.489655\pi\)
0.0324941 + 0.999472i \(0.489655\pi\)
\(242\) 7.37048e6 0.0334304
\(243\) 0 0
\(244\) 3.52792e8 1.55473
\(245\) 1.69261e8 0.735317
\(246\) 0 0
\(247\) 2.60698e7 0.110077
\(248\) −1.09126e7 −0.0454303
\(249\) 0 0
\(250\) −9.81641e6 −0.0397340
\(251\) −8.14251e7 −0.325013 −0.162506 0.986707i \(-0.551958\pi\)
−0.162506 + 0.986707i \(0.551958\pi\)
\(252\) 0 0
\(253\) 1.17852e8 0.457526
\(254\) 4.23051e6 0.0161985
\(255\) 0 0
\(256\) 2.64865e8 0.986699
\(257\) −2.21346e8 −0.813402 −0.406701 0.913561i \(-0.633321\pi\)
−0.406701 + 0.913561i \(0.633321\pi\)
\(258\) 0 0
\(259\) −1.14580e7 −0.0409789
\(260\) −4.38845e8 −1.54848
\(261\) 0 0
\(262\) 9.07301e6 0.0311672
\(263\) −4.49267e8 −1.52286 −0.761429 0.648249i \(-0.775502\pi\)
−0.761429 + 0.648249i \(0.775502\pi\)
\(264\) 0 0
\(265\) −2.49256e8 −0.822781
\(266\) 219058. 0.000713631 0
\(267\) 0 0
\(268\) 3.97597e8 1.26175
\(269\) 4.00214e8 1.25360 0.626801 0.779180i \(-0.284364\pi\)
0.626801 + 0.779180i \(0.284364\pi\)
\(270\) 0 0
\(271\) 2.23102e8 0.680944 0.340472 0.940255i \(-0.389413\pi\)
0.340472 + 0.940255i \(0.389413\pi\)
\(272\) 3.45402e8 1.04072
\(273\) 0 0
\(274\) 729025. 0.00214099
\(275\) 3.16458e7 0.0917597
\(276\) 0 0
\(277\) 2.84338e8 0.803815 0.401907 0.915680i \(-0.368347\pi\)
0.401907 + 0.915680i \(0.368347\pi\)
\(278\) 1.00370e7 0.0280185
\(279\) 0 0
\(280\) −7.37995e6 −0.0200909
\(281\) 3.25571e8 0.875332 0.437666 0.899138i \(-0.355805\pi\)
0.437666 + 0.899138i \(0.355805\pi\)
\(282\) 0 0
\(283\) −4.66681e7 −0.122396 −0.0611980 0.998126i \(-0.519492\pi\)
−0.0611980 + 0.998126i \(0.519492\pi\)
\(284\) 5.13636e8 1.33058
\(285\) 0 0
\(286\) −7.86129e6 −0.0198707
\(287\) 1.37974e7 0.0344518
\(288\) 0 0
\(289\) 3.76710e7 0.0918048
\(290\) −5.98190e6 −0.0144028
\(291\) 0 0
\(292\) 7.49619e8 1.76198
\(293\) 7.70273e7 0.178899 0.0894495 0.995991i \(-0.471489\pi\)
0.0894495 + 0.995991i \(0.471489\pi\)
\(294\) 0 0
\(295\) −4.74737e7 −0.107665
\(296\) −4.00700e6 −0.00898046
\(297\) 0 0
\(298\) 3.17269e7 0.0694498
\(299\) 1.36582e9 2.95491
\(300\) 0 0
\(301\) −1.92539e8 −0.406945
\(302\) −1.76918e7 −0.0369613
\(303\) 0 0
\(304\) −2.86441e7 −0.0584759
\(305\) 6.37948e8 1.28747
\(306\) 0 0
\(307\) −1.67909e8 −0.331200 −0.165600 0.986193i \(-0.552956\pi\)
−0.165600 + 0.986193i \(0.552956\pi\)
\(308\) 4.94973e7 0.0965281
\(309\) 0 0
\(310\) −9.85992e6 −0.0187978
\(311\) 2.22728e8 0.419868 0.209934 0.977716i \(-0.432675\pi\)
0.209934 + 0.977716i \(0.432675\pi\)
\(312\) 0 0
\(313\) −2.23952e8 −0.412809 −0.206404 0.978467i \(-0.566176\pi\)
−0.206404 + 0.978467i \(0.566176\pi\)
\(314\) −2.27672e7 −0.0415007
\(315\) 0 0
\(316\) −5.91154e8 −1.05389
\(317\) −7.57737e8 −1.33601 −0.668007 0.744155i \(-0.732853\pi\)
−0.668007 + 0.744155i \(0.732853\pi\)
\(318\) 0 0
\(319\) 8.02947e7 0.138490
\(320\) 4.80888e8 0.820388
\(321\) 0 0
\(322\) 1.14766e7 0.0191566
\(323\) −3.71533e7 −0.0613464
\(324\) 0 0
\(325\) 3.66751e8 0.592625
\(326\) −1.55440e7 −0.0248486
\(327\) 0 0
\(328\) 4.82513e6 0.00755006
\(329\) −6.93215e7 −0.107320
\(330\) 0 0
\(331\) 2.98249e8 0.452044 0.226022 0.974122i \(-0.427428\pi\)
0.226022 + 0.974122i \(0.427428\pi\)
\(332\) 1.21642e9 1.82432
\(333\) 0 0
\(334\) 3.82335e7 0.0561476
\(335\) 7.18969e8 1.04485
\(336\) 0 0
\(337\) −1.29895e9 −1.84879 −0.924393 0.381442i \(-0.875428\pi\)
−0.924393 + 0.381442i \(0.875428\pi\)
\(338\) −6.51894e7 −0.0918266
\(339\) 0 0
\(340\) 6.25420e8 0.862970
\(341\) 1.32349e8 0.180751
\(342\) 0 0
\(343\) −4.70083e8 −0.628991
\(344\) −6.73331e7 −0.0891814
\(345\) 0 0
\(346\) −4.52678e7 −0.0587521
\(347\) −1.13535e9 −1.45874 −0.729369 0.684120i \(-0.760186\pi\)
−0.729369 + 0.684120i \(0.760186\pi\)
\(348\) 0 0
\(349\) 2.24671e8 0.282916 0.141458 0.989944i \(-0.454821\pi\)
0.141458 + 0.989944i \(0.454821\pi\)
\(350\) 3.08172e6 0.00384199
\(351\) 0 0
\(352\) 2.59705e7 0.0317381
\(353\) −1.00698e9 −1.21846 −0.609228 0.792995i \(-0.708520\pi\)
−0.609228 + 0.792995i \(0.708520\pi\)
\(354\) 0 0
\(355\) 9.28800e8 1.10185
\(356\) −8.89256e8 −1.04460
\(357\) 0 0
\(358\) 1.79950e7 0.0207282
\(359\) 5.34366e7 0.0609548 0.0304774 0.999535i \(-0.490297\pi\)
0.0304774 + 0.999535i \(0.490297\pi\)
\(360\) 0 0
\(361\) −8.90791e8 −0.996553
\(362\) 5.35956e7 0.0593812
\(363\) 0 0
\(364\) 5.73637e8 0.623422
\(365\) 1.35552e9 1.45909
\(366\) 0 0
\(367\) 1.32145e8 0.139547 0.0697733 0.997563i \(-0.477772\pi\)
0.0697733 + 0.997563i \(0.477772\pi\)
\(368\) −1.50069e9 −1.56972
\(369\) 0 0
\(370\) −3.62048e6 −0.00371587
\(371\) 3.25814e8 0.331254
\(372\) 0 0
\(373\) 1.16722e8 0.116458 0.0582292 0.998303i \(-0.481455\pi\)
0.0582292 + 0.998303i \(0.481455\pi\)
\(374\) 1.12035e7 0.0110740
\(375\) 0 0
\(376\) −2.42426e7 −0.0235191
\(377\) 9.30555e8 0.894432
\(378\) 0 0
\(379\) 1.54193e9 1.45488 0.727440 0.686172i \(-0.240710\pi\)
0.727440 + 0.686172i \(0.240710\pi\)
\(380\) −5.18659e7 −0.0484886
\(381\) 0 0
\(382\) 1.26937e7 0.0116511
\(383\) 1.11000e9 1.00955 0.504774 0.863252i \(-0.331576\pi\)
0.504774 + 0.863252i \(0.331576\pi\)
\(384\) 0 0
\(385\) 8.95052e7 0.0799347
\(386\) −5.14010e7 −0.0454900
\(387\) 0 0
\(388\) 1.22236e9 1.06240
\(389\) −3.52125e8 −0.303301 −0.151650 0.988434i \(-0.548459\pi\)
−0.151650 + 0.988434i \(0.548459\pi\)
\(390\) 0 0
\(391\) −1.94649e9 −1.64678
\(392\) −7.73734e7 −0.0648769
\(393\) 0 0
\(394\) −2.24546e7 −0.0184956
\(395\) −1.06897e9 −0.872725
\(396\) 0 0
\(397\) 1.23251e9 0.988611 0.494305 0.869288i \(-0.335422\pi\)
0.494305 + 0.869288i \(0.335422\pi\)
\(398\) −5.81859e7 −0.0462623
\(399\) 0 0
\(400\) −4.02967e8 −0.314818
\(401\) 1.95405e9 1.51332 0.756661 0.653808i \(-0.226829\pi\)
0.756661 + 0.653808i \(0.226829\pi\)
\(402\) 0 0
\(403\) 1.53383e9 1.16737
\(404\) −1.24517e9 −0.939491
\(405\) 0 0
\(406\) 7.81924e6 0.00579860
\(407\) 4.85975e7 0.0357301
\(408\) 0 0
\(409\) −8.06706e8 −0.583020 −0.291510 0.956568i \(-0.594158\pi\)
−0.291510 + 0.956568i \(0.594158\pi\)
\(410\) 4.35969e6 0.00312401
\(411\) 0 0
\(412\) 1.49728e9 1.05478
\(413\) 6.20553e7 0.0433465
\(414\) 0 0
\(415\) 2.19964e9 1.51072
\(416\) 3.00978e8 0.204979
\(417\) 0 0
\(418\) −929105. −0.000622225 0
\(419\) 1.37468e9 0.912960 0.456480 0.889734i \(-0.349110\pi\)
0.456480 + 0.889734i \(0.349110\pi\)
\(420\) 0 0
\(421\) −8.51303e8 −0.556028 −0.278014 0.960577i \(-0.589676\pi\)
−0.278014 + 0.960577i \(0.589676\pi\)
\(422\) −8.56549e7 −0.0554829
\(423\) 0 0
\(424\) 1.13941e8 0.0725939
\(425\) −5.22675e8 −0.330271
\(426\) 0 0
\(427\) −8.33894e8 −0.518338
\(428\) −2.97837e9 −1.83622
\(429\) 0 0
\(430\) −6.08381e7 −0.0369008
\(431\) −3.76775e8 −0.226679 −0.113339 0.993556i \(-0.536155\pi\)
−0.113339 + 0.993556i \(0.536155\pi\)
\(432\) 0 0
\(433\) 1.14074e9 0.675270 0.337635 0.941277i \(-0.390373\pi\)
0.337635 + 0.941277i \(0.390373\pi\)
\(434\) 1.28884e7 0.00756806
\(435\) 0 0
\(436\) −2.35676e9 −1.36180
\(437\) 1.61422e8 0.0925291
\(438\) 0 0
\(439\) 1.15020e9 0.648854 0.324427 0.945911i \(-0.394829\pi\)
0.324427 + 0.945911i \(0.394829\pi\)
\(440\) 3.13010e7 0.0175176
\(441\) 0 0
\(442\) 1.29840e8 0.0715207
\(443\) −4.99658e8 −0.273061 −0.136531 0.990636i \(-0.543595\pi\)
−0.136531 + 0.990636i \(0.543595\pi\)
\(444\) 0 0
\(445\) −1.60803e9 −0.865034
\(446\) 1.06291e8 0.0567315
\(447\) 0 0
\(448\) −6.28593e8 −0.330291
\(449\) −1.70526e9 −0.889054 −0.444527 0.895766i \(-0.646628\pi\)
−0.444527 + 0.895766i \(0.646628\pi\)
\(450\) 0 0
\(451\) −5.85199e7 −0.0300390
\(452\) −2.16555e9 −1.10302
\(453\) 0 0
\(454\) −5.28949e7 −0.0265289
\(455\) 1.03730e9 0.516254
\(456\) 0 0
\(457\) −6.10097e8 −0.299015 −0.149507 0.988761i \(-0.547769\pi\)
−0.149507 + 0.988761i \(0.547769\pi\)
\(458\) −3.76229e7 −0.0182988
\(459\) 0 0
\(460\) −2.71730e9 −1.30162
\(461\) −2.09113e9 −0.994093 −0.497046 0.867724i \(-0.665582\pi\)
−0.497046 + 0.867724i \(0.665582\pi\)
\(462\) 0 0
\(463\) −3.87258e8 −0.181329 −0.0906643 0.995882i \(-0.528899\pi\)
−0.0906643 + 0.995882i \(0.528899\pi\)
\(464\) −1.02244e9 −0.475146
\(465\) 0 0
\(466\) −2.59683e7 −0.0118876
\(467\) −2.46901e8 −0.112180 −0.0560898 0.998426i \(-0.517863\pi\)
−0.0560898 + 0.998426i \(0.517863\pi\)
\(468\) 0 0
\(469\) −9.39800e8 −0.420659
\(470\) −2.19041e7 −0.00973157
\(471\) 0 0
\(472\) 2.17015e7 0.00949931
\(473\) 8.16627e8 0.354821
\(474\) 0 0
\(475\) 4.33453e7 0.0185573
\(476\) −8.17518e8 −0.347434
\(477\) 0 0
\(478\) −8.73860e7 −0.0365969
\(479\) 3.29359e9 1.36929 0.684645 0.728877i \(-0.259957\pi\)
0.684645 + 0.728877i \(0.259957\pi\)
\(480\) 0 0
\(481\) 5.63209e8 0.230761
\(482\) −5.83280e6 −0.00237253
\(483\) 0 0
\(484\) 2.28110e9 0.914503
\(485\) 2.21037e9 0.879769
\(486\) 0 0
\(487\) 1.64421e9 0.645070 0.322535 0.946558i \(-0.395465\pi\)
0.322535 + 0.946558i \(0.395465\pi\)
\(488\) −2.91623e8 −0.113593
\(489\) 0 0
\(490\) −6.99099e7 −0.0268443
\(491\) −2.85157e9 −1.08717 −0.543587 0.839353i \(-0.682934\pi\)
−0.543587 + 0.839353i \(0.682934\pi\)
\(492\) 0 0
\(493\) −1.32618e9 −0.498469
\(494\) −1.07676e7 −0.00401861
\(495\) 0 0
\(496\) −1.68529e9 −0.620138
\(497\) −1.21408e9 −0.443609
\(498\) 0 0
\(499\) 3.73412e9 1.34535 0.672676 0.739937i \(-0.265145\pi\)
0.672676 + 0.739937i \(0.265145\pi\)
\(500\) −3.03809e9 −1.08694
\(501\) 0 0
\(502\) 3.36311e7 0.0118653
\(503\) 4.65904e9 1.63233 0.816166 0.577817i \(-0.196095\pi\)
0.816166 + 0.577817i \(0.196095\pi\)
\(504\) 0 0
\(505\) −2.25161e9 −0.777990
\(506\) −4.86766e7 −0.0167030
\(507\) 0 0
\(508\) 1.30931e9 0.443117
\(509\) −4.17061e9 −1.40180 −0.700902 0.713258i \(-0.747219\pi\)
−0.700902 + 0.713258i \(0.747219\pi\)
\(510\) 0 0
\(511\) −1.77187e9 −0.587435
\(512\) −5.51410e8 −0.181564
\(513\) 0 0
\(514\) 9.14226e7 0.0296950
\(515\) 2.70750e9 0.873460
\(516\) 0 0
\(517\) 2.94018e8 0.0935743
\(518\) 4.73251e6 0.00149602
\(519\) 0 0
\(520\) 3.62755e8 0.113136
\(521\) 1.08528e9 0.336209 0.168104 0.985769i \(-0.446235\pi\)
0.168104 + 0.985769i \(0.446235\pi\)
\(522\) 0 0
\(523\) −4.07601e9 −1.24589 −0.622945 0.782266i \(-0.714064\pi\)
−0.622945 + 0.782266i \(0.714064\pi\)
\(524\) 2.80802e9 0.852590
\(525\) 0 0
\(526\) 1.85561e8 0.0555952
\(527\) −2.18593e9 −0.650579
\(528\) 0 0
\(529\) 5.05223e9 1.48384
\(530\) 1.02950e8 0.0300374
\(531\) 0 0
\(532\) 6.77965e7 0.0195217
\(533\) −6.78202e8 −0.194005
\(534\) 0 0
\(535\) −5.38574e9 −1.52057
\(536\) −3.28659e8 −0.0921869
\(537\) 0 0
\(538\) −1.65301e8 −0.0457654
\(539\) 9.38396e8 0.258122
\(540\) 0 0
\(541\) −4.61079e9 −1.25194 −0.625972 0.779846i \(-0.715297\pi\)
−0.625972 + 0.779846i \(0.715297\pi\)
\(542\) −9.21481e7 −0.0248593
\(543\) 0 0
\(544\) −4.28939e8 −0.114235
\(545\) −4.26169e9 −1.12770
\(546\) 0 0
\(547\) −3.70849e9 −0.968817 −0.484409 0.874842i \(-0.660965\pi\)
−0.484409 + 0.874842i \(0.660965\pi\)
\(548\) 2.25627e8 0.0585677
\(549\) 0 0
\(550\) −1.30707e7 −0.00334988
\(551\) 1.09980e8 0.0280080
\(552\) 0 0
\(553\) 1.39731e9 0.351362
\(554\) −1.17441e8 −0.0293450
\(555\) 0 0
\(556\) 3.10635e9 0.766458
\(557\) 4.17975e9 1.02484 0.512421 0.858734i \(-0.328749\pi\)
0.512421 + 0.858734i \(0.328749\pi\)
\(558\) 0 0
\(559\) 9.46409e9 2.29159
\(560\) −1.13973e9 −0.274248
\(561\) 0 0
\(562\) −1.34471e8 −0.0319559
\(563\) −5.00396e9 −1.18178 −0.590888 0.806754i \(-0.701222\pi\)
−0.590888 + 0.806754i \(0.701222\pi\)
\(564\) 0 0
\(565\) −3.91593e9 −0.913410
\(566\) 1.92754e7 0.00446833
\(567\) 0 0
\(568\) −4.24579e8 −0.0972163
\(569\) −5.07854e8 −0.115570 −0.0577851 0.998329i \(-0.518404\pi\)
−0.0577851 + 0.998329i \(0.518404\pi\)
\(570\) 0 0
\(571\) −5.27683e9 −1.18617 −0.593085 0.805140i \(-0.702090\pi\)
−0.593085 + 0.805140i \(0.702090\pi\)
\(572\) −2.43300e9 −0.543571
\(573\) 0 0
\(574\) −5.69877e6 −0.00125774
\(575\) 2.27090e9 0.498150
\(576\) 0 0
\(577\) −1.54218e9 −0.334209 −0.167105 0.985939i \(-0.553442\pi\)
−0.167105 + 0.985939i \(0.553442\pi\)
\(578\) −1.55593e7 −0.00335153
\(579\) 0 0
\(580\) −1.85134e9 −0.393993
\(581\) −2.87526e9 −0.608219
\(582\) 0 0
\(583\) −1.38190e9 −0.288826
\(584\) −6.19645e8 −0.128735
\(585\) 0 0
\(586\) −3.18147e7 −0.00653109
\(587\) −9.40246e8 −0.191870 −0.0959352 0.995388i \(-0.530584\pi\)
−0.0959352 + 0.995388i \(0.530584\pi\)
\(588\) 0 0
\(589\) 1.81279e8 0.0365547
\(590\) 1.96081e7 0.00393056
\(591\) 0 0
\(592\) −6.18824e8 −0.122586
\(593\) −3.28316e9 −0.646547 −0.323274 0.946306i \(-0.604783\pi\)
−0.323274 + 0.946306i \(0.604783\pi\)
\(594\) 0 0
\(595\) −1.47830e9 −0.287710
\(596\) 9.81920e9 1.89983
\(597\) 0 0
\(598\) −5.64125e8 −0.107875
\(599\) −7.40433e8 −0.140764 −0.0703820 0.997520i \(-0.522422\pi\)
−0.0703820 + 0.997520i \(0.522422\pi\)
\(600\) 0 0
\(601\) −2.01096e9 −0.377870 −0.188935 0.981990i \(-0.560504\pi\)
−0.188935 + 0.981990i \(0.560504\pi\)
\(602\) 7.95245e7 0.0148564
\(603\) 0 0
\(604\) −5.47545e9 −1.01109
\(605\) 4.12487e9 0.757298
\(606\) 0 0
\(607\) −4.88190e9 −0.885990 −0.442995 0.896524i \(-0.646084\pi\)
−0.442995 + 0.896524i \(0.646084\pi\)
\(608\) 3.55718e7 0.00641864
\(609\) 0 0
\(610\) −2.63493e8 −0.0470017
\(611\) 3.40744e9 0.604344
\(612\) 0 0
\(613\) −6.53926e9 −1.14661 −0.573306 0.819341i \(-0.694339\pi\)
−0.573306 + 0.819341i \(0.694339\pi\)
\(614\) 6.93517e7 0.0120912
\(615\) 0 0
\(616\) −4.09151e7 −0.00705264
\(617\) −1.08583e9 −0.186108 −0.0930541 0.995661i \(-0.529663\pi\)
−0.0930541 + 0.995661i \(0.529663\pi\)
\(618\) 0 0
\(619\) −1.46677e9 −0.248568 −0.124284 0.992247i \(-0.539663\pi\)
−0.124284 + 0.992247i \(0.539663\pi\)
\(620\) −3.05156e9 −0.514222
\(621\) 0 0
\(622\) −9.19935e7 −0.0153282
\(623\) 2.10193e9 0.348266
\(624\) 0 0
\(625\) −3.56453e9 −0.584012
\(626\) 9.24990e7 0.0150705
\(627\) 0 0
\(628\) −7.04624e9 −1.13527
\(629\) −8.02657e8 −0.128603
\(630\) 0 0
\(631\) 5.56162e9 0.881248 0.440624 0.897692i \(-0.354757\pi\)
0.440624 + 0.897692i \(0.354757\pi\)
\(632\) 4.88656e8 0.0770005
\(633\) 0 0
\(634\) 3.12969e8 0.0487741
\(635\) 2.36760e9 0.366944
\(636\) 0 0
\(637\) 1.08753e10 1.66707
\(638\) −3.31642e7 −0.00505588
\(639\) 0 0
\(640\) −7.98218e8 −0.120363
\(641\) 8.98365e8 0.134726 0.0673628 0.997729i \(-0.478542\pi\)
0.0673628 + 0.997729i \(0.478542\pi\)
\(642\) 0 0
\(643\) 6.66060e9 0.988042 0.494021 0.869450i \(-0.335527\pi\)
0.494021 + 0.869450i \(0.335527\pi\)
\(644\) 3.55192e9 0.524038
\(645\) 0 0
\(646\) 1.53455e7 0.00223958
\(647\) −2.89308e9 −0.419948 −0.209974 0.977707i \(-0.567338\pi\)
−0.209974 + 0.977707i \(0.567338\pi\)
\(648\) 0 0
\(649\) −2.63199e8 −0.0377944
\(650\) −1.51480e8 −0.0216350
\(651\) 0 0
\(652\) −4.81074e9 −0.679743
\(653\) −5.97319e9 −0.839480 −0.419740 0.907644i \(-0.637879\pi\)
−0.419740 + 0.907644i \(0.637879\pi\)
\(654\) 0 0
\(655\) 5.07769e9 0.706028
\(656\) 7.45172e8 0.103061
\(657\) 0 0
\(658\) 2.86320e7 0.00391796
\(659\) −8.14255e9 −1.10831 −0.554155 0.832414i \(-0.686959\pi\)
−0.554155 + 0.832414i \(0.686959\pi\)
\(660\) 0 0
\(661\) 3.91035e9 0.526635 0.263318 0.964709i \(-0.415183\pi\)
0.263318 + 0.964709i \(0.415183\pi\)
\(662\) −1.23186e8 −0.0165028
\(663\) 0 0
\(664\) −1.00551e9 −0.133290
\(665\) 1.22595e8 0.0161658
\(666\) 0 0
\(667\) 5.76193e9 0.751844
\(668\) 1.18329e10 1.53594
\(669\) 0 0
\(670\) −2.96956e8 −0.0381444
\(671\) 3.53685e9 0.451947
\(672\) 0 0
\(673\) −8.56370e9 −1.08295 −0.541475 0.840717i \(-0.682134\pi\)
−0.541475 + 0.840717i \(0.682134\pi\)
\(674\) 5.36505e8 0.0674939
\(675\) 0 0
\(676\) −2.01755e10 −2.51195
\(677\) 1.32503e10 1.64122 0.820608 0.571492i \(-0.193635\pi\)
0.820608 + 0.571492i \(0.193635\pi\)
\(678\) 0 0
\(679\) −2.88928e9 −0.354198
\(680\) −5.16981e8 −0.0630512
\(681\) 0 0
\(682\) −5.46643e7 −0.00659870
\(683\) −1.52557e10 −1.83214 −0.916070 0.401018i \(-0.868656\pi\)
−0.916070 + 0.401018i \(0.868656\pi\)
\(684\) 0 0
\(685\) 4.07997e8 0.0484998
\(686\) 1.94159e8 0.0229627
\(687\) 0 0
\(688\) −1.03986e10 −1.21735
\(689\) −1.60151e10 −1.86536
\(690\) 0 0
\(691\) 2.37047e9 0.273314 0.136657 0.990618i \(-0.456364\pi\)
0.136657 + 0.990618i \(0.456364\pi\)
\(692\) −1.40100e10 −1.60719
\(693\) 0 0
\(694\) 4.68936e8 0.0532544
\(695\) 5.61716e9 0.634702
\(696\) 0 0
\(697\) 9.66540e8 0.108120
\(698\) −9.27960e7 −0.0103285
\(699\) 0 0
\(700\) 9.53766e8 0.105099
\(701\) 8.71012e9 0.955017 0.477508 0.878627i \(-0.341540\pi\)
0.477508 + 0.878627i \(0.341540\pi\)
\(702\) 0 0
\(703\) 6.65641e7 0.00722598
\(704\) 2.66609e9 0.287985
\(705\) 0 0
\(706\) 4.15914e8 0.0444823
\(707\) 2.94320e9 0.313221
\(708\) 0 0
\(709\) −8.22562e9 −0.866775 −0.433388 0.901208i \(-0.642682\pi\)
−0.433388 + 0.901208i \(0.642682\pi\)
\(710\) −3.83623e8 −0.0402254
\(711\) 0 0
\(712\) 7.35071e8 0.0763219
\(713\) 9.49736e9 0.981272
\(714\) 0 0
\(715\) −4.39955e9 −0.450130
\(716\) 5.56930e9 0.567029
\(717\) 0 0
\(718\) −2.20710e7 −0.00222529
\(719\) 5.78538e9 0.580471 0.290236 0.956955i \(-0.406266\pi\)
0.290236 + 0.956955i \(0.406266\pi\)
\(720\) 0 0
\(721\) −3.53911e9 −0.351658
\(722\) 3.67924e8 0.0363813
\(723\) 0 0
\(724\) 1.65873e10 1.62440
\(725\) 1.54720e9 0.150787
\(726\) 0 0
\(727\) −1.00474e10 −0.969798 −0.484899 0.874570i \(-0.661144\pi\)
−0.484899 + 0.874570i \(0.661144\pi\)
\(728\) −4.74176e8 −0.0455491
\(729\) 0 0
\(730\) −5.59874e8 −0.0532672
\(731\) −1.34877e10 −1.27711
\(732\) 0 0
\(733\) 7.89455e9 0.740395 0.370198 0.928953i \(-0.379290\pi\)
0.370198 + 0.928953i \(0.379290\pi\)
\(734\) −5.45800e7 −0.00509445
\(735\) 0 0
\(736\) 1.86364e9 0.172301
\(737\) 3.98603e9 0.366779
\(738\) 0 0
\(739\) 8.45416e9 0.770575 0.385288 0.922797i \(-0.374102\pi\)
0.385288 + 0.922797i \(0.374102\pi\)
\(740\) −1.12051e9 −0.101649
\(741\) 0 0
\(742\) −1.34571e8 −0.0120931
\(743\) −5.05511e9 −0.452137 −0.226068 0.974111i \(-0.572587\pi\)
−0.226068 + 0.974111i \(0.572587\pi\)
\(744\) 0 0
\(745\) 1.77559e10 1.57324
\(746\) −4.82097e7 −0.00425156
\(747\) 0 0
\(748\) 3.46739e9 0.302933
\(749\) 7.03997e9 0.612187
\(750\) 0 0
\(751\) −1.42778e10 −1.23005 −0.615023 0.788509i \(-0.710853\pi\)
−0.615023 + 0.788509i \(0.710853\pi\)
\(752\) −3.74392e9 −0.321043
\(753\) 0 0
\(754\) −3.84348e8 −0.0326532
\(755\) −9.90116e9 −0.837282
\(756\) 0 0
\(757\) 1.01019e10 0.846388 0.423194 0.906039i \(-0.360909\pi\)
0.423194 + 0.906039i \(0.360909\pi\)
\(758\) −6.36864e8 −0.0531135
\(759\) 0 0
\(760\) 4.28731e7 0.00354272
\(761\) 3.03041e9 0.249261 0.124631 0.992203i \(-0.460225\pi\)
0.124631 + 0.992203i \(0.460225\pi\)
\(762\) 0 0
\(763\) 5.57067e9 0.454016
\(764\) 3.92859e9 0.318720
\(765\) 0 0
\(766\) −4.58464e8 −0.0368557
\(767\) −3.05028e9 −0.244093
\(768\) 0 0
\(769\) −1.80433e10 −1.43078 −0.715392 0.698723i \(-0.753752\pi\)
−0.715392 + 0.698723i \(0.753752\pi\)
\(770\) −3.69684e7 −0.00291819
\(771\) 0 0
\(772\) −1.59081e10 −1.24440
\(773\) 1.03732e10 0.807766 0.403883 0.914811i \(-0.367660\pi\)
0.403883 + 0.914811i \(0.367660\pi\)
\(774\) 0 0
\(775\) 2.55024e9 0.196800
\(776\) −1.01042e9 −0.0776220
\(777\) 0 0
\(778\) 1.45439e8 0.0110726
\(779\) −8.01549e7 −0.00607503
\(780\) 0 0
\(781\) 5.14936e9 0.386789
\(782\) 8.03963e8 0.0601191
\(783\) 0 0
\(784\) −1.19492e10 −0.885590
\(785\) −1.27416e10 −0.940113
\(786\) 0 0
\(787\) −1.10014e10 −0.804518 −0.402259 0.915526i \(-0.631775\pi\)
−0.402259 + 0.915526i \(0.631775\pi\)
\(788\) −6.94950e9 −0.505955
\(789\) 0 0
\(790\) 4.41520e8 0.0318607
\(791\) 5.11871e9 0.367742
\(792\) 0 0
\(793\) 4.09894e10 2.91887
\(794\) −5.09067e8 −0.0360913
\(795\) 0 0
\(796\) −1.80080e10 −1.26552
\(797\) −1.83467e10 −1.28367 −0.641834 0.766843i \(-0.721826\pi\)
−0.641834 + 0.766843i \(0.721826\pi\)
\(798\) 0 0
\(799\) −4.85612e9 −0.336803
\(800\) 5.00426e8 0.0345561
\(801\) 0 0
\(802\) −8.07085e8 −0.0552470
\(803\) 7.51515e9 0.512193
\(804\) 0 0
\(805\) 6.42288e9 0.433954
\(806\) −6.33518e8 −0.0426174
\(807\) 0 0
\(808\) 1.02927e9 0.0686420
\(809\) −1.83197e10 −1.21646 −0.608231 0.793760i \(-0.708121\pi\)
−0.608231 + 0.793760i \(0.708121\pi\)
\(810\) 0 0
\(811\) −8.15379e9 −0.536768 −0.268384 0.963312i \(-0.586490\pi\)
−0.268384 + 0.963312i \(0.586490\pi\)
\(812\) 2.41998e9 0.158623
\(813\) 0 0
\(814\) −2.00723e7 −0.00130440
\(815\) −8.69918e9 −0.562894
\(816\) 0 0
\(817\) 1.11853e9 0.0717583
\(818\) 3.33195e8 0.0212844
\(819\) 0 0
\(820\) 1.34929e9 0.0854586
\(821\) 1.71557e10 1.08195 0.540977 0.841038i \(-0.318055\pi\)
0.540977 + 0.841038i \(0.318055\pi\)
\(822\) 0 0
\(823\) 3.13687e10 1.96154 0.980771 0.195160i \(-0.0625227\pi\)
0.980771 + 0.195160i \(0.0625227\pi\)
\(824\) −1.23767e9 −0.0770653
\(825\) 0 0
\(826\) −2.56308e7 −0.00158246
\(827\) 3.08831e10 1.89868 0.949338 0.314256i \(-0.101755\pi\)
0.949338 + 0.314256i \(0.101755\pi\)
\(828\) 0 0
\(829\) 2.08621e10 1.27180 0.635899 0.771773i \(-0.280630\pi\)
0.635899 + 0.771773i \(0.280630\pi\)
\(830\) −9.08519e8 −0.0551519
\(831\) 0 0
\(832\) 3.08980e10 1.85994
\(833\) −1.54989e10 −0.929062
\(834\) 0 0
\(835\) 2.13973e10 1.27191
\(836\) −2.87550e8 −0.0170212
\(837\) 0 0
\(838\) −5.67784e8 −0.0333295
\(839\) −3.33343e10 −1.94861 −0.974303 0.225241i \(-0.927683\pi\)
−0.974303 + 0.225241i \(0.927683\pi\)
\(840\) 0 0
\(841\) −1.33242e10 −0.772421
\(842\) 3.51615e8 0.0202990
\(843\) 0 0
\(844\) −2.65094e10 −1.51776
\(845\) −3.64831e10 −2.08014
\(846\) 0 0
\(847\) −5.39183e9 −0.304891
\(848\) 1.75966e10 0.990930
\(849\) 0 0
\(850\) 2.15881e8 0.0120573
\(851\) 3.48735e9 0.193973
\(852\) 0 0
\(853\) −5.30610e9 −0.292721 −0.146360 0.989231i \(-0.546756\pi\)
−0.146360 + 0.989231i \(0.546756\pi\)
\(854\) 3.44424e8 0.0189231
\(855\) 0 0
\(856\) 2.46196e9 0.134160
\(857\) 3.22916e9 0.175250 0.0876248 0.996154i \(-0.472072\pi\)
0.0876248 + 0.996154i \(0.472072\pi\)
\(858\) 0 0
\(859\) −1.09723e10 −0.590637 −0.295318 0.955399i \(-0.595426\pi\)
−0.295318 + 0.955399i \(0.595426\pi\)
\(860\) −1.88288e10 −1.00944
\(861\) 0 0
\(862\) 1.55620e8 0.00827540
\(863\) −1.83908e10 −0.974009 −0.487005 0.873399i \(-0.661910\pi\)
−0.487005 + 0.873399i \(0.661910\pi\)
\(864\) 0 0
\(865\) −2.53340e10 −1.33091
\(866\) −4.71159e8 −0.0246522
\(867\) 0 0
\(868\) 3.98884e9 0.207027
\(869\) −5.92650e9 −0.306358
\(870\) 0 0
\(871\) 4.61951e10 2.36882
\(872\) 1.94813e9 0.0994970
\(873\) 0 0
\(874\) −6.66724e7 −0.00337797
\(875\) 7.18113e9 0.362380
\(876\) 0 0
\(877\) 2.93823e10 1.47091 0.735456 0.677572i \(-0.236968\pi\)
0.735456 + 0.677572i \(0.236968\pi\)
\(878\) −4.75068e8 −0.0236878
\(879\) 0 0
\(880\) 4.83399e9 0.239120
\(881\) −7.96639e9 −0.392506 −0.196253 0.980553i \(-0.562877\pi\)
−0.196253 + 0.980553i \(0.562877\pi\)
\(882\) 0 0
\(883\) −2.11047e10 −1.03161 −0.515806 0.856706i \(-0.672507\pi\)
−0.515806 + 0.856706i \(0.672507\pi\)
\(884\) 4.01844e10 1.95648
\(885\) 0 0
\(886\) 2.06374e8 0.00996868
\(887\) 3.91929e10 1.88571 0.942854 0.333206i \(-0.108130\pi\)
0.942854 + 0.333206i \(0.108130\pi\)
\(888\) 0 0
\(889\) −3.09480e9 −0.147733
\(890\) 6.64165e8 0.0315799
\(891\) 0 0
\(892\) 3.28961e10 1.55191
\(893\) 4.02717e8 0.0189243
\(894\) 0 0
\(895\) 1.00709e10 0.469555
\(896\) 1.04339e9 0.0484584
\(897\) 0 0
\(898\) 7.04325e8 0.0324568
\(899\) 6.47071e9 0.297025
\(900\) 0 0
\(901\) 2.28240e10 1.03957
\(902\) 2.41706e7 0.00109664
\(903\) 0 0
\(904\) 1.79007e9 0.0805901
\(905\) 2.99946e10 1.34516
\(906\) 0 0
\(907\) 7.45412e9 0.331719 0.165860 0.986149i \(-0.446960\pi\)
0.165860 + 0.986149i \(0.446960\pi\)
\(908\) −1.63705e10 −0.725707
\(909\) 0 0
\(910\) −4.28436e8 −0.0188470
\(911\) −3.46618e9 −0.151893 −0.0759463 0.997112i \(-0.524198\pi\)
−0.0759463 + 0.997112i \(0.524198\pi\)
\(912\) 0 0
\(913\) 1.21950e10 0.530315
\(914\) 2.51989e8 0.0109162
\(915\) 0 0
\(916\) −1.16440e10 −0.500572
\(917\) −6.63730e9 −0.284249
\(918\) 0 0
\(919\) −1.94364e9 −0.0826060 −0.0413030 0.999147i \(-0.513151\pi\)
−0.0413030 + 0.999147i \(0.513151\pi\)
\(920\) 2.24616e9 0.0951005
\(921\) 0 0
\(922\) 8.63700e8 0.0362915
\(923\) 5.96772e10 2.49806
\(924\) 0 0
\(925\) 9.36428e8 0.0389026
\(926\) 1.59949e8 0.00661979
\(927\) 0 0
\(928\) 1.26973e9 0.0521546
\(929\) −3.65536e10 −1.49581 −0.747904 0.663807i \(-0.768939\pi\)
−0.747904 + 0.663807i \(0.768939\pi\)
\(930\) 0 0
\(931\) 1.28532e9 0.0522022
\(932\) −8.03696e9 −0.325189
\(933\) 0 0
\(934\) 1.01978e8 0.00409536
\(935\) 6.27002e9 0.250858
\(936\) 0 0
\(937\) −2.70613e10 −1.07463 −0.537317 0.843380i \(-0.680562\pi\)
−0.537317 + 0.843380i \(0.680562\pi\)
\(938\) 3.88167e8 0.0153571
\(939\) 0 0
\(940\) −6.77912e9 −0.266211
\(941\) 3.52735e10 1.38002 0.690010 0.723800i \(-0.257606\pi\)
0.690010 + 0.723800i \(0.257606\pi\)
\(942\) 0 0
\(943\) −4.19938e9 −0.163078
\(944\) 3.35148e9 0.129669
\(945\) 0 0
\(946\) −3.37292e8 −0.0129535
\(947\) 1.60900e10 0.615648 0.307824 0.951443i \(-0.400399\pi\)
0.307824 + 0.951443i \(0.400399\pi\)
\(948\) 0 0
\(949\) 8.70950e10 3.30797
\(950\) −1.79030e7 −0.000677474 0
\(951\) 0 0
\(952\) 6.75771e8 0.0253846
\(953\) −2.34617e10 −0.878081 −0.439040 0.898467i \(-0.644681\pi\)
−0.439040 + 0.898467i \(0.644681\pi\)
\(954\) 0 0
\(955\) 7.10400e9 0.263932
\(956\) −2.70452e10 −1.00112
\(957\) 0 0
\(958\) −1.36035e9 −0.0499888
\(959\) −5.33313e8 −0.0195262
\(960\) 0 0
\(961\) −1.68470e10 −0.612337
\(962\) −2.32623e8 −0.00842441
\(963\) 0 0
\(964\) −1.80520e9 −0.0649016
\(965\) −2.87664e10 −1.03048
\(966\) 0 0
\(967\) 2.75570e10 0.980029 0.490014 0.871714i \(-0.336992\pi\)
0.490014 + 0.871714i \(0.336992\pi\)
\(968\) −1.88559e9 −0.0668163
\(969\) 0 0
\(970\) −9.12951e8 −0.0321179
\(971\) 3.36601e10 1.17991 0.589954 0.807437i \(-0.299146\pi\)
0.589954 + 0.807437i \(0.299146\pi\)
\(972\) 0 0
\(973\) −7.34247e9 −0.255533
\(974\) −6.79111e8 −0.0235496
\(975\) 0 0
\(976\) −4.50370e10 −1.55058
\(977\) 3.08878e9 0.105963 0.0529817 0.998595i \(-0.483128\pi\)
0.0529817 + 0.998595i \(0.483128\pi\)
\(978\) 0 0
\(979\) −8.91505e9 −0.303658
\(980\) −2.16365e10 −0.734337
\(981\) 0 0
\(982\) 1.17779e9 0.0396896
\(983\) 4.00750e10 1.34566 0.672832 0.739796i \(-0.265078\pi\)
0.672832 + 0.739796i \(0.265078\pi\)
\(984\) 0 0
\(985\) −1.25667e10 −0.418980
\(986\) 5.47754e8 0.0181977
\(987\) 0 0
\(988\) −3.33248e9 −0.109931
\(989\) 5.86010e10 1.92627
\(990\) 0 0
\(991\) −3.18674e9 −0.104013 −0.0520066 0.998647i \(-0.516562\pi\)
−0.0520066 + 0.998647i \(0.516562\pi\)
\(992\) 2.09288e9 0.0680697
\(993\) 0 0
\(994\) 5.01453e8 0.0161949
\(995\) −3.25636e10 −1.04798
\(996\) 0 0
\(997\) −4.16475e8 −0.0133093 −0.00665465 0.999978i \(-0.502118\pi\)
−0.00665465 + 0.999978i \(0.502118\pi\)
\(998\) −1.54231e9 −0.0491150
\(999\) 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.1.8 17
3.2 odd 2 177.8.a.b.1.10 17
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
177.8.a.b.1.10 17 3.2 odd 2
531.8.a.d.1.8 17 1.1 even 1 trivial