Properties

Label 252.9.c.a.197.7
Level $252$
Weight $9$
Character 252.197
Analytic conductor $102.659$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 252.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(102.659409735\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Defining polynomial: \(x^{16} + 4002260 x^{14} + 6534459751956 x^{12} + 5613923146579405376 x^{10} + 2733728904154246859079616 x^{8} + 757873148017661341349205888000 x^{6} + 113644318422397913452531577312640000 x^{4} + 8098650340007618970326973663348480000000 x^{2} + 199066230990417435753898292645889849600000000\)
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{37}\cdot 7^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 197.7
Root \(-348.567i\) of defining polynomial
Character \(\chi\) \(=\) 252.197
Dual form 252.9.c.a.197.10

$q$-expansion

\(f(q)\) \(=\) \(q-348.567i q^{5} +907.493 q^{7} +O(q^{10})\) \(q-348.567i q^{5} +907.493 q^{7} +11889.3i q^{11} -4095.93 q^{13} -67449.6i q^{17} -116874. q^{19} +68357.8i q^{23} +269126. q^{25} -774693. i q^{29} +807878. q^{31} -316322. i q^{35} +1.02479e6 q^{37} -850867. i q^{41} -3.69010e6 q^{43} +5.47190e6i q^{47} +823543. q^{49} +541263. i q^{53} +4.14422e6 q^{55} -8.95168e6i q^{59} +650282. q^{61} +1.42770e6i q^{65} -3.39158e7 q^{67} -1.63485e7i q^{71} +1.67469e7 q^{73} +1.07895e7i q^{77} -1.38232e7 q^{79} -1.98832e6i q^{83} -2.35107e7 q^{85} -6.05729e7i q^{89} -3.71702e6 q^{91} +4.07384e7i q^{95} -3.52027e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 95480q^{13} - 287560q^{19} - 1754520q^{25} - 3554264q^{31} - 182920q^{37} + 8472416q^{43} + 13176688q^{49} - 18692072q^{55} + 34224568q^{61} + 22683096q^{67} + 2137296q^{73} - 90245624q^{79} - 56204456q^{85} - 25661888q^{91} + 134041152q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/252\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\) \(127\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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 0
\(3\) 0 0
\(4\) 0 0
\(5\) − 348.567i − 0.557707i −0.960334 0.278853i \(-0.910046\pi\)
0.960334 0.278853i \(-0.0899544\pi\)
\(6\) 0 0
\(7\) 907.493 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 11889.3i 0.812056i 0.913861 + 0.406028i \(0.133087\pi\)
−0.913861 + 0.406028i \(0.866913\pi\)
\(12\) 0 0
\(13\) −4095.93 −0.143410 −0.0717049 0.997426i \(-0.522844\pi\)
−0.0717049 + 0.997426i \(0.522844\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 67449.6i − 0.807576i −0.914852 0.403788i \(-0.867693\pi\)
0.914852 0.403788i \(-0.132307\pi\)
\(18\) 0 0
\(19\) −116874. −0.896816 −0.448408 0.893829i \(-0.648009\pi\)
−0.448408 + 0.893829i \(0.648009\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 68357.8i 0.244274i 0.992513 + 0.122137i \(0.0389747\pi\)
−0.992513 + 0.122137i \(0.961025\pi\)
\(24\) 0 0
\(25\) 269126. 0.688963
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 774693.i − 1.09531i −0.836704 0.547656i \(-0.815520\pi\)
0.836704 0.547656i \(-0.184480\pi\)
\(30\) 0 0
\(31\) 807878. 0.874781 0.437390 0.899272i \(-0.355903\pi\)
0.437390 + 0.899272i \(0.355903\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 316322.i − 0.210793i
\(36\) 0 0
\(37\) 1.02479e6 0.546799 0.273399 0.961901i \(-0.411852\pi\)
0.273399 + 0.961901i \(0.411852\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 850867.i − 0.301111i −0.988602 0.150555i \(-0.951894\pi\)
0.988602 0.150555i \(-0.0481062\pi\)
\(42\) 0 0
\(43\) −3.69010e6 −1.07936 −0.539678 0.841872i \(-0.681454\pi\)
−0.539678 + 0.841872i \(0.681454\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.47190e6i 1.12137i 0.828031 + 0.560683i \(0.189461\pi\)
−0.828031 + 0.560683i \(0.810539\pi\)
\(48\) 0 0
\(49\) 823543. 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 541263.i 0.0685969i 0.999412 + 0.0342984i \(0.0109197\pi\)
−0.999412 + 0.0342984i \(0.989080\pi\)
\(54\) 0 0
\(55\) 4.14422e6 0.452889
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 8.95168e6i − 0.738748i −0.929281 0.369374i \(-0.879572\pi\)
0.929281 0.369374i \(-0.120428\pi\)
\(60\) 0 0
\(61\) 650282. 0.0469659 0.0234829 0.999724i \(-0.492524\pi\)
0.0234829 + 0.999724i \(0.492524\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.42770e6i 0.0799806i
\(66\) 0 0
\(67\) −3.39158e7 −1.68307 −0.841536 0.540201i \(-0.818348\pi\)
−0.841536 + 0.540201i \(0.818348\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 1.63485e7i − 0.643346i −0.946851 0.321673i \(-0.895755\pi\)
0.946851 0.321673i \(-0.104245\pi\)
\(72\) 0 0
\(73\) 1.67469e7 0.589716 0.294858 0.955541i \(-0.404728\pi\)
0.294858 + 0.955541i \(0.404728\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.07895e7i 0.306928i
\(78\) 0 0
\(79\) −1.38232e7 −0.354895 −0.177448 0.984130i \(-0.556784\pi\)
−0.177448 + 0.984130i \(0.556784\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 1.98832e6i − 0.0418961i −0.999781 0.0209480i \(-0.993332\pi\)
0.999781 0.0209480i \(-0.00666845\pi\)
\(84\) 0 0
\(85\) −2.35107e7 −0.450391
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 6.05729e7i − 0.965424i −0.875779 0.482712i \(-0.839652\pi\)
0.875779 0.482712i \(-0.160348\pi\)
\(90\) 0 0
\(91\) −3.71702e6 −0.0542038
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.07384e7i 0.500160i
\(96\) 0 0
\(97\) −3.52027e7 −0.397639 −0.198819 0.980036i \(-0.563711\pi\)
−0.198819 + 0.980036i \(0.563711\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 1.19699e8i − 1.15029i −0.818053 0.575143i \(-0.804946\pi\)
0.818053 0.575143i \(-0.195054\pi\)
\(102\) 0 0
\(103\) 1.19816e7 0.106455 0.0532273 0.998582i \(-0.483049\pi\)
0.0532273 + 0.998582i \(0.483049\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 1.98152e8i − 1.51169i −0.654749 0.755846i \(-0.727226\pi\)
0.654749 0.755846i \(-0.272774\pi\)
\(108\) 0 0
\(109\) −2.18524e7 −0.154808 −0.0774038 0.997000i \(-0.524663\pi\)
−0.0774038 + 0.997000i \(0.524663\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.09321e8i 0.670485i 0.942132 + 0.335243i \(0.108818\pi\)
−0.942132 + 0.335243i \(0.891182\pi\)
\(114\) 0 0
\(115\) 2.38272e7 0.136233
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 6.12100e7i − 0.305235i
\(120\) 0 0
\(121\) 7.30030e7 0.340564
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 2.29967e8i − 0.941946i
\(126\) 0 0
\(127\) −3.13988e8 −1.20698 −0.603488 0.797372i \(-0.706223\pi\)
−0.603488 + 0.797372i \(0.706223\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 1.95386e8i − 0.663451i −0.943376 0.331726i \(-0.892369\pi\)
0.943376 0.331726i \(-0.107631\pi\)
\(132\) 0 0
\(133\) −1.06062e8 −0.338964
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 4.32944e8i − 1.22899i −0.788919 0.614497i \(-0.789359\pi\)
0.788919 0.614497i \(-0.210641\pi\)
\(138\) 0 0
\(139\) −5.25420e8 −1.40750 −0.703748 0.710450i \(-0.748492\pi\)
−0.703748 + 0.710450i \(0.748492\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 4.86978e7i − 0.116457i
\(144\) 0 0
\(145\) −2.70032e8 −0.610863
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 2.49656e8i − 0.506521i −0.967398 0.253261i \(-0.918497\pi\)
0.967398 0.253261i \(-0.0815030\pi\)
\(150\) 0 0
\(151\) −1.90108e8 −0.365672 −0.182836 0.983143i \(-0.558528\pi\)
−0.182836 + 0.983143i \(0.558528\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 2.81599e8i − 0.487871i
\(156\) 0 0
\(157\) −4.70501e8 −0.774393 −0.387197 0.921997i \(-0.626557\pi\)
−0.387197 + 0.921997i \(0.626557\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.20342e7i 0.0923267i
\(162\) 0 0
\(163\) −4.63201e8 −0.656174 −0.328087 0.944648i \(-0.606404\pi\)
−0.328087 + 0.944648i \(0.606404\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 4.36026e8i − 0.560592i −0.959914 0.280296i \(-0.909567\pi\)
0.959914 0.280296i \(-0.0904325\pi\)
\(168\) 0 0
\(169\) −7.98954e8 −0.979434
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 1.28581e9i − 1.43546i −0.696322 0.717729i \(-0.745181\pi\)
0.696322 0.717729i \(-0.254819\pi\)
\(174\) 0 0
\(175\) 2.44230e8 0.260404
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 5.77990e8i − 0.562999i −0.959561 0.281500i \(-0.909168\pi\)
0.959561 0.281500i \(-0.0908318\pi\)
\(180\) 0 0
\(181\) 1.36134e9 1.26839 0.634195 0.773173i \(-0.281332\pi\)
0.634195 + 0.773173i \(0.281332\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 3.57207e8i − 0.304953i
\(186\) 0 0
\(187\) 8.01930e8 0.655798
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 3.97105e8i − 0.298381i −0.988808 0.149191i \(-0.952333\pi\)
0.988808 0.149191i \(-0.0476668\pi\)
\(192\) 0 0
\(193\) −1.44780e9 −1.04347 −0.521734 0.853108i \(-0.674715\pi\)
−0.521734 + 0.853108i \(0.674715\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 1.27633e9i − 0.847420i −0.905798 0.423710i \(-0.860728\pi\)
0.905798 0.423710i \(-0.139272\pi\)
\(198\) 0 0
\(199\) −2.33138e9 −1.48663 −0.743313 0.668944i \(-0.766746\pi\)
−0.743313 + 0.668944i \(0.766746\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 7.03029e8i − 0.413989i
\(204\) 0 0
\(205\) −2.96584e8 −0.167931
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 1.38955e9i − 0.728265i
\(210\) 0 0
\(211\) −5.69510e8 −0.287324 −0.143662 0.989627i \(-0.545888\pi\)
−0.143662 + 0.989627i \(0.545888\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.28625e9i 0.601964i
\(216\) 0 0
\(217\) 7.33144e8 0.330636
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.76269e8i 0.115814i
\(222\) 0 0
\(223\) −1.10060e8 −0.0445049 −0.0222525 0.999752i \(-0.507084\pi\)
−0.0222525 + 0.999752i \(0.507084\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 1.62031e9i − 0.610232i −0.952315 0.305116i \(-0.901305\pi\)
0.952315 0.305116i \(-0.0986953\pi\)
\(228\) 0 0
\(229\) −1.86805e9 −0.679278 −0.339639 0.940556i \(-0.610305\pi\)
−0.339639 + 0.940556i \(0.610305\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 1.23197e9i − 0.417999i −0.977916 0.209000i \(-0.932979\pi\)
0.977916 0.209000i \(-0.0670207\pi\)
\(234\) 0 0
\(235\) 1.90732e9 0.625393
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.64262e9i 0.503437i 0.967800 + 0.251719i \(0.0809958\pi\)
−0.967800 + 0.251719i \(0.919004\pi\)
\(240\) 0 0
\(241\) −1.10426e9 −0.327342 −0.163671 0.986515i \(-0.552334\pi\)
−0.163671 + 0.986515i \(0.552334\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 2.87060e8i − 0.0796724i
\(246\) 0 0
\(247\) 4.78707e8 0.128612
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 1.57251e9i − 0.396187i −0.980183 0.198093i \(-0.936525\pi\)
0.980183 0.198093i \(-0.0634749\pi\)
\(252\) 0 0
\(253\) −8.12727e8 −0.198364
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 1.88202e9i − 0.431412i −0.976458 0.215706i \(-0.930795\pi\)
0.976458 0.215706i \(-0.0692053\pi\)
\(258\) 0 0
\(259\) 9.29989e8 0.206671
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 1.28482e9i 0.268547i 0.990944 + 0.134274i \(0.0428701\pi\)
−0.990944 + 0.134274i \(0.957130\pi\)
\(264\) 0 0
\(265\) 1.88666e8 0.0382570
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 9.82675e9i 1.87673i 0.345652 + 0.938363i \(0.387658\pi\)
−0.345652 + 0.938363i \(0.612342\pi\)
\(270\) 0 0
\(271\) 2.37009e9 0.439428 0.219714 0.975564i \(-0.429488\pi\)
0.219714 + 0.975564i \(0.429488\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.19973e9i 0.559477i
\(276\) 0 0
\(277\) −3.04785e9 −0.517695 −0.258847 0.965918i \(-0.583343\pi\)
−0.258847 + 0.965918i \(0.583343\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7.73650e9i 1.24085i 0.784266 + 0.620425i \(0.213040\pi\)
−0.784266 + 0.620425i \(0.786960\pi\)
\(282\) 0 0
\(283\) 5.94321e9 0.926564 0.463282 0.886211i \(-0.346672\pi\)
0.463282 + 0.886211i \(0.346672\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 7.72155e8i − 0.113809i
\(288\) 0 0
\(289\) 2.42631e9 0.347820
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 3.90739e9i 0.530172i 0.964225 + 0.265086i \(0.0854003\pi\)
−0.964225 + 0.265086i \(0.914600\pi\)
\(294\) 0 0
\(295\) −3.12026e9 −0.412005
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 2.79988e8i − 0.0350312i
\(300\) 0 0
\(301\) −3.34874e9 −0.407958
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 2.26667e8i − 0.0261932i
\(306\) 0 0
\(307\) 6.72279e9 0.756826 0.378413 0.925637i \(-0.376470\pi\)
0.378413 + 0.925637i \(0.376470\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 7.74568e8i − 0.0827977i −0.999143 0.0413988i \(-0.986819\pi\)
0.999143 0.0413988i \(-0.0131814\pi\)
\(312\) 0 0
\(313\) 1.68797e10 1.75868 0.879339 0.476197i \(-0.157985\pi\)
0.879339 + 0.476197i \(0.157985\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 6.08895e9i − 0.602983i −0.953469 0.301492i \(-0.902515\pi\)
0.953469 0.301492i \(-0.0974845\pi\)
\(318\) 0 0
\(319\) 9.21058e9 0.889455
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 7.88310e9i 0.724247i
\(324\) 0 0
\(325\) −1.10232e9 −0.0988041
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 4.96571e9i 0.423836i
\(330\) 0 0
\(331\) 6.68142e9 0.556618 0.278309 0.960492i \(-0.410226\pi\)
0.278309 + 0.960492i \(0.410226\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.18219e10i 0.938661i
\(336\) 0 0
\(337\) 8.18497e9 0.634596 0.317298 0.948326i \(-0.397224\pi\)
0.317298 + 0.948326i \(0.397224\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 9.60512e9i 0.710371i
\(342\) 0 0
\(343\) 7.47359e8 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.41926e10i 0.978916i 0.872027 + 0.489458i \(0.162805\pi\)
−0.872027 + 0.489458i \(0.837195\pi\)
\(348\) 0 0
\(349\) −4.87708e9 −0.328744 −0.164372 0.986398i \(-0.552560\pi\)
−0.164372 + 0.986398i \(0.552560\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 1.09772e10i − 0.706957i −0.935443 0.353479i \(-0.884999\pi\)
0.935443 0.353479i \(-0.115001\pi\)
\(354\) 0 0
\(355\) −5.69854e9 −0.358798
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.83944e10i 1.10741i 0.832713 + 0.553704i \(0.186786\pi\)
−0.832713 + 0.553704i \(0.813214\pi\)
\(360\) 0 0
\(361\) −3.32406e9 −0.195722
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 5.83741e9i − 0.328889i
\(366\) 0 0
\(367\) 1.73627e10 0.957093 0.478547 0.878062i \(-0.341164\pi\)
0.478547 + 0.878062i \(0.341164\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 4.91192e8i 0.0259272i
\(372\) 0 0
\(373\) 8.25097e9 0.426255 0.213128 0.977024i \(-0.431635\pi\)
0.213128 + 0.977024i \(0.431635\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.17309e9i 0.157078i
\(378\) 0 0
\(379\) −1.09004e10 −0.528308 −0.264154 0.964481i \(-0.585093\pi\)
−0.264154 + 0.964481i \(0.585093\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.93716e10i 1.36500i 0.730886 + 0.682500i \(0.239107\pi\)
−0.730886 + 0.682500i \(0.760893\pi\)
\(384\) 0 0
\(385\) 3.76085e9 0.171176
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 8.08142e9i 0.352931i 0.984307 + 0.176465i \(0.0564663\pi\)
−0.984307 + 0.176465i \(0.943534\pi\)
\(390\) 0 0
\(391\) 4.61070e9 0.197270
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4.81831e9i 0.197928i
\(396\) 0 0
\(397\) −4.23249e9 −0.170386 −0.0851930 0.996364i \(-0.527151\pi\)
−0.0851930 + 0.996364i \(0.527151\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.74199e9i 0.144719i 0.997379 + 0.0723595i \(0.0230529\pi\)
−0.997379 + 0.0723595i \(0.976947\pi\)
\(402\) 0 0
\(403\) −3.30901e9 −0.125452
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.21840e10i 0.444032i
\(408\) 0 0
\(409\) −9.46732e9 −0.338325 −0.169162 0.985588i \(-0.554106\pi\)
−0.169162 + 0.985588i \(0.554106\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 8.12358e9i − 0.279221i
\(414\) 0 0
\(415\) −6.93061e8 −0.0233657
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.06975e10i 0.671525i 0.941947 + 0.335762i \(0.108994\pi\)
−0.941947 + 0.335762i \(0.891006\pi\)
\(420\) 0 0
\(421\) 2.83855e10 0.903582 0.451791 0.892124i \(-0.350785\pi\)
0.451791 + 0.892124i \(0.350785\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 1.81525e10i − 0.556390i
\(426\) 0 0
\(427\) 5.90126e8 0.0177514
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 3.33726e10i − 0.967123i −0.875310 0.483561i \(-0.839343\pi\)
0.875310 0.483561i \(-0.160657\pi\)
\(432\) 0 0
\(433\) 1.35508e10 0.385490 0.192745 0.981249i \(-0.438261\pi\)
0.192745 + 0.981249i \(0.438261\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 7.98924e9i − 0.219068i
\(438\) 0 0
\(439\) −1.08521e10 −0.292183 −0.146091 0.989271i \(-0.546669\pi\)
−0.146091 + 0.989271i \(0.546669\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 2.77764e10i − 0.721209i −0.932719 0.360605i \(-0.882570\pi\)
0.932719 0.360605i \(-0.117430\pi\)
\(444\) 0 0
\(445\) −2.11137e10 −0.538424
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 4.44594e10i − 1.09390i −0.837165 0.546950i \(-0.815789\pi\)
0.837165 0.546950i \(-0.184211\pi\)
\(450\) 0 0
\(451\) 1.01162e10 0.244519
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.29563e9i 0.0302298i
\(456\) 0 0
\(457\) −8.26792e10 −1.89553 −0.947766 0.318966i \(-0.896665\pi\)
−0.947766 + 0.318966i \(0.896665\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 5.06508e10i − 1.12146i −0.828000 0.560728i \(-0.810521\pi\)
0.828000 0.560728i \(-0.189479\pi\)
\(462\) 0 0
\(463\) 7.76023e10 1.68869 0.844346 0.535798i \(-0.179989\pi\)
0.844346 + 0.535798i \(0.179989\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.58099e10i 0.542648i 0.962488 + 0.271324i \(0.0874615\pi\)
−0.962488 + 0.271324i \(0.912539\pi\)
\(468\) 0 0
\(469\) −3.07783e10 −0.636142
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 4.38728e10i − 0.876497i
\(474\) 0 0
\(475\) −3.14538e10 −0.617873
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.61244e10i 0.686213i 0.939296 + 0.343107i \(0.111479\pi\)
−0.939296 + 0.343107i \(0.888521\pi\)
\(480\) 0 0
\(481\) −4.19746e9 −0.0784163
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1.22705e10i 0.221766i
\(486\) 0 0
\(487\) 9.41764e9 0.167427 0.0837137 0.996490i \(-0.473322\pi\)
0.0837137 + 0.996490i \(0.473322\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 2.29411e10i − 0.394720i −0.980331 0.197360i \(-0.936763\pi\)
0.980331 0.197360i \(-0.0632368\pi\)
\(492\) 0 0
\(493\) −5.22528e10 −0.884548
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 1.48361e10i − 0.243162i
\(498\) 0 0
\(499\) 1.95129e10 0.314717 0.157359 0.987542i \(-0.449702\pi\)
0.157359 + 0.987542i \(0.449702\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 1.96172e10i − 0.306454i −0.988191 0.153227i \(-0.951033\pi\)
0.988191 0.153227i \(-0.0489666\pi\)
\(504\) 0 0
\(505\) −4.17232e10 −0.641522
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 5.09459e10i − 0.758993i −0.925193 0.379496i \(-0.876097\pi\)
0.925193 0.379496i \(-0.123903\pi\)
\(510\) 0 0
\(511\) 1.51977e10 0.222892
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 4.17638e9i − 0.0593705i
\(516\) 0 0
\(517\) −6.50572e10 −0.910612
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 1.02686e11i − 1.39368i −0.717229 0.696838i \(-0.754590\pi\)
0.717229 0.696838i \(-0.245410\pi\)
\(522\) 0 0
\(523\) −3.84556e10 −0.513987 −0.256994 0.966413i \(-0.582732\pi\)
−0.256994 + 0.966413i \(0.582732\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 5.44911e10i − 0.706452i
\(528\) 0 0
\(529\) 7.36382e10 0.940330
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 3.48509e9i 0.0431822i
\(534\) 0 0
\(535\) −6.90692e10 −0.843081
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 9.79136e9i 0.116008i
\(540\) 0 0
\(541\) −6.67196e10 −0.778869 −0.389434 0.921054i \(-0.627329\pi\)
−0.389434 + 0.921054i \(0.627329\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 7.61701e9i 0.0863373i
\(546\) 0 0
\(547\) −1.55267e10 −0.173432 −0.0867158 0.996233i \(-0.527637\pi\)
−0.0867158 + 0.996233i \(0.527637\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 9.05414e10i 0.982293i
\(552\) 0 0
\(553\) −1.25445e10 −0.134138
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 1.18540e11i − 1.23152i −0.787933 0.615762i \(-0.788848\pi\)
0.787933 0.615762i \(-0.211152\pi\)
\(558\) 0 0
\(559\) 1.51144e10 0.154790
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 7.71879e10i − 0.768273i −0.923276 0.384136i \(-0.874499\pi\)
0.923276 0.384136i \(-0.125501\pi\)
\(564\) 0 0
\(565\) 3.81056e10 0.373934
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 1.17585e11i − 1.12177i −0.827895 0.560883i \(-0.810462\pi\)
0.827895 0.560883i \(-0.189538\pi\)
\(570\) 0 0
\(571\) −1.04156e11 −0.979807 −0.489904 0.871777i \(-0.662968\pi\)
−0.489904 + 0.871777i \(0.662968\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.83969e10i 0.168295i
\(576\) 0 0
\(577\) −3.22534e10 −0.290986 −0.145493 0.989359i \(-0.546477\pi\)
−0.145493 + 0.989359i \(0.546477\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 1.80438e9i − 0.0158352i
\(582\) 0 0
\(583\) −6.43524e9 −0.0557045
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.16532e11i 0.981509i 0.871298 + 0.490754i \(0.163279\pi\)
−0.871298 + 0.490754i \(0.836721\pi\)
\(588\) 0 0
\(589\) −9.44199e10 −0.784517
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5.84175e10i 0.472415i 0.971703 + 0.236208i \(0.0759046\pi\)
−0.971703 + 0.236208i \(0.924095\pi\)
\(594\) 0 0
\(595\) −2.13358e10 −0.170232
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 4.11787e10i − 0.319864i −0.987128 0.159932i \(-0.948873\pi\)
0.987128 0.159932i \(-0.0511275\pi\)
\(600\) 0 0
\(601\) −7.25375e10 −0.555987 −0.277994 0.960583i \(-0.589669\pi\)
−0.277994 + 0.960583i \(0.589669\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 2.54464e10i − 0.189935i
\(606\) 0 0
\(607\) 8.61757e10 0.634790 0.317395 0.948293i \(-0.397192\pi\)
0.317395 + 0.948293i \(0.397192\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 2.24125e10i − 0.160815i
\(612\) 0 0
\(613\) −1.89534e11 −1.34229 −0.671143 0.741328i \(-0.734196\pi\)
−0.671143 + 0.741328i \(0.734196\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 1.36770e11i − 0.943739i −0.881669 0.471869i \(-0.843579\pi\)
0.881669 0.471869i \(-0.156421\pi\)
\(618\) 0 0
\(619\) 2.05400e11 1.39907 0.699534 0.714599i \(-0.253391\pi\)
0.699534 + 0.714599i \(0.253391\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 5.49695e10i − 0.364896i
\(624\) 0 0
\(625\) 2.49685e10 0.163633
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 6.91216e10i − 0.441582i
\(630\) 0 0
\(631\) 1.44468e11 0.911288 0.455644 0.890162i \(-0.349409\pi\)
0.455644 + 0.890162i \(0.349409\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.09446e11i 0.673138i
\(636\) 0 0
\(637\) −3.37317e9 −0.0204871
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 1.48167e10i − 0.0877648i −0.999037 0.0438824i \(-0.986027\pi\)
0.999037 0.0438824i \(-0.0139727\pi\)
\(642\) 0 0
\(643\) 3.20057e11 1.87234 0.936168 0.351553i \(-0.114346\pi\)
0.936168 + 0.351553i \(0.114346\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 2.36291e10i − 0.134843i −0.997725 0.0674216i \(-0.978523\pi\)
0.997725 0.0674216i \(-0.0214773\pi\)
\(648\) 0 0
\(649\) 1.06429e11 0.599905
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 1.18818e10i − 0.0653473i −0.999466 0.0326737i \(-0.989598\pi\)
0.999466 0.0326737i \(-0.0104022\pi\)
\(654\) 0 0
\(655\) −6.81052e10 −0.370011
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 3.02462e11i 1.60372i 0.597510 + 0.801862i \(0.296157\pi\)
−0.597510 + 0.801862i \(0.703843\pi\)
\(660\) 0 0
\(661\) −1.94947e8 −0.00102120 −0.000510600 1.00000i \(-0.500163\pi\)
−0.000510600 1.00000i \(0.500163\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 3.69698e10i 0.189043i
\(666\) 0 0
\(667\) 5.29563e10 0.267556
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 7.73141e9i 0.0381389i
\(672\) 0 0
\(673\) −5.04705e10 −0.246024 −0.123012 0.992405i \(-0.539255\pi\)
−0.123012 + 0.992405i \(0.539255\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 9.18944e10i − 0.437456i −0.975786 0.218728i \(-0.929809\pi\)
0.975786 0.218728i \(-0.0701907\pi\)
\(678\) 0 0
\(679\) −3.19462e10 −0.150293
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 2.76743e11i 1.27173i 0.771802 + 0.635863i \(0.219356\pi\)
−0.771802 + 0.635863i \(0.780644\pi\)
\(684\) 0 0
\(685\) −1.50910e11 −0.685418
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 2.21697e9i − 0.00983747i
\(690\) 0 0
\(691\) 3.33822e11 1.46421 0.732104 0.681193i \(-0.238538\pi\)
0.732104 + 0.681193i \(0.238538\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1.83144e11i 0.784970i
\(696\) 0 0
\(697\) −5.73906e10 −0.243170
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 4.07724e11i 1.68847i 0.535971 + 0.844236i \(0.319945\pi\)
−0.535971 + 0.844236i \(0.680055\pi\)
\(702\) 0 0
\(703\) −1.19771e11 −0.490378
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1.08626e11i − 0.434767i
\(708\) 0 0
\(709\) −4.90558e10 −0.194136 −0.0970678 0.995278i \(-0.530946\pi\)
−0.0970678 + 0.995278i \(0.530946\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.52248e10i 0.213686i
\(714\) 0 0
\(715\) −1.69744e10 −0.0649488
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 4.69821e11i 1.75799i 0.476830 + 0.878996i \(0.341786\pi\)
−0.476830 + 0.878996i \(0.658214\pi\)
\(720\) 0 0
\(721\) 1.08732e10 0.0402361
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 2.08490e11i − 0.754630i
\(726\) 0 0
\(727\) 1.92122e11 0.687764 0.343882 0.939013i \(-0.388258\pi\)
0.343882 + 0.939013i \(0.388258\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.48896e11i 0.871662i
\(732\) 0 0
\(733\) 7.47278e10 0.258861 0.129430 0.991589i \(-0.458685\pi\)
0.129430 + 0.991589i \(0.458685\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 4.03236e11i − 1.36675i
\(738\) 0 0
\(739\) 1.49064e11 0.499799 0.249900 0.968272i \(-0.419602\pi\)
0.249900 + 0.968272i \(0.419602\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 2.90655e11i 0.953723i 0.878978 + 0.476862i \(0.158226\pi\)
−0.878978 + 0.476862i \(0.841774\pi\)
\(744\) 0 0
\(745\) −8.70219e10 −0.282490
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 1.79822e11i − 0.571366i
\(750\) 0 0
\(751\) −3.85693e10 −0.121250 −0.0606250 0.998161i \(-0.519309\pi\)
−0.0606250 + 0.998161i \(0.519309\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 6.62652e10i 0.203938i
\(756\) 0 0
\(757\) 1.75336e11 0.533934 0.266967 0.963706i \(-0.413979\pi\)
0.266967 + 0.963706i \(0.413979\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 5.42227e11i 1.61675i 0.588668 + 0.808375i \(0.299653\pi\)
−0.588668 + 0.808375i \(0.700347\pi\)
\(762\) 0 0
\(763\) −1.98309e10 −0.0585118
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.66654e10i 0.105944i
\(768\) 0 0
\(769\) 2.67008e11 0.763517 0.381759 0.924262i \(-0.375319\pi\)
0.381759 + 0.924262i \(0.375319\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 1.93424e11i − 0.541743i −0.962616 0.270871i \(-0.912688\pi\)
0.962616 0.270871i \(-0.0873118\pi\)
\(774\) 0 0
\(775\) 2.17421e11 0.602692
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 9.94441e10i 0.270041i
\(780\) 0 0
\(781\) 1.94372e11 0.522433
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.64001e11i 0.431884i
\(786\) 0 0
\(787\) −8.52841e10 −0.222315 −0.111158 0.993803i \(-0.535456\pi\)
−0.111158 + 0.993803i \(0.535456\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 9.92079e10i 0.253420i
\(792\) 0 0
\(793\) −2.66351e9 −0.00673536
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 6.61423e11i 1.63925i 0.572897 + 0.819627i \(0.305819\pi\)
−0.572897 + 0.819627i \(0.694181\pi\)
\(798\) 0 0
\(799\) 3.69078e11 0.905588
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.99109e11i 0.478883i
\(804\) 0 0
\(805\) 2.16231e10 0.0514913
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 8.13264e11i − 1.89862i −0.314344 0.949309i \(-0.601785\pi\)
0.314344 0.949309i \(-0.398215\pi\)
\(810\) 0 0
\(811\) 4.22085e10 0.0975702 0.0487851 0.998809i \(-0.484465\pi\)
0.0487851 + 0.998809i \(0.484465\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.61456e11i 0.365953i
\(816\) 0 0
\(817\) 4.31277e11 0.967983
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 5.27883e11i − 1.16189i −0.813943 0.580945i \(-0.802683\pi\)
0.813943 0.580945i \(-0.197317\pi\)
\(822\) 0 0
\(823\) −8.78060e10 −0.191393 −0.0956963 0.995411i \(-0.530508\pi\)
−0.0956963 + 0.995411i \(0.530508\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.95035e11i 0.416955i 0.978027 + 0.208478i \(0.0668509\pi\)
−0.978027 + 0.208478i \(0.933149\pi\)
\(828\) 0 0
\(829\) −7.56585e11 −1.60192 −0.800958 0.598720i \(-0.795676\pi\)
−0.800958 + 0.598720i \(0.795676\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 5.55476e10i − 0.115368i
\(834\) 0 0
\(835\) −1.51984e11 −0.312646
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 8.44106e10i 0.170353i 0.996366 + 0.0851763i \(0.0271454\pi\)
−0.996366 + 0.0851763i \(0.972855\pi\)
\(840\) 0 0
\(841\) −9.99034e10 −0.199708
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.78489e11i 0.546237i
\(846\) 0 0
\(847\) 6.62497e10 0.128721
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 7.00523e10i 0.133569i
\(852\) 0 0
\(853\) 4.56707e11 0.862664 0.431332 0.902193i \(-0.358044\pi\)
0.431332 + 0.902193i \(0.358044\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.28440e11i 0.608881i 0.952531 + 0.304440i \(0.0984694\pi\)
−0.952531 + 0.304440i \(0.901531\pi\)
\(858\) 0 0
\(859\) 1.00764e12 1.85069 0.925344 0.379129i \(-0.123776\pi\)
0.925344 + 0.379129i \(0.123776\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 4.82398e11i 0.869685i 0.900507 + 0.434842i \(0.143196\pi\)
−0.900507 + 0.434842i \(0.856804\pi\)
\(864\) 0 0
\(865\) −4.48189e11 −0.800565
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 1.64348e11i − 0.288195i
\(870\) 0 0
\(871\) 1.38917e11 0.241369
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 2.08694e11i − 0.356022i
\(876\) 0 0
\(877\) −1.47431e10 −0.0249225 −0.0124613 0.999922i \(-0.503967\pi\)
−0.0124613 + 0.999922i \(0.503967\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 6.80542e11i − 1.12967i −0.825204 0.564835i \(-0.808940\pi\)
0.825204 0.564835i \(-0.191060\pi\)
\(882\) 0 0
\(883\) −1.59710e11 −0.262717 −0.131358 0.991335i \(-0.541934\pi\)
−0.131358 + 0.991335i \(0.541934\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 6.26548e11i 1.01218i 0.862479 + 0.506092i \(0.168911\pi\)
−0.862479 + 0.506092i \(0.831089\pi\)
\(888\) 0 0
\(889\) −2.84942e11 −0.456194
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 6.39523e11i − 1.00566i
\(894\) 0 0
\(895\) −2.01468e11 −0.313989
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 6.25858e11i − 0.958158i
\(900\) 0 0
\(901\) 3.65079e10 0.0553972
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 4.74519e11i − 0.707390i
\(906\) 0 0
\(907\) −9.67535e11 −1.42968 −0.714838 0.699290i \(-0.753500\pi\)
−0.714838 + 0.699290i \(0.753500\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 5.30084e11i − 0.769611i −0.922998 0.384805i \(-0.874269\pi\)
0.922998 0.384805i \(-0.125731\pi\)
\(912\) 0 0
\(913\) 2.36397e10 0.0340220
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 1.77312e11i − 0.250761i
\(918\) 0 0
\(919\) −1.29433e12 −1.81461 −0.907304 0.420476i \(-0.861863\pi\)
−0.907304 + 0.420476i \(0.861863\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 6.69623e10i 0.0922621i
\(924\) 0 0
\(925\) 2.75798e11 0.376724
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.17539e12i 1.57804i 0.614369 + 0.789019i \(0.289411\pi\)
−0.614369 + 0.789019i \(0.710589\pi\)
\(930\) 0 0
\(931\) −9.62507e10 −0.128117
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 2.79526e11i − 0.365743i
\(936\) 0 0
\(937\) −7.22945e11 −0.937880 −0.468940 0.883230i \(-0.655364\pi\)
−0.468940 + 0.883230i \(0.655364\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 1.19339e12i − 1.52203i −0.648733 0.761016i \(-0.724701\pi\)
0.648733 0.761016i \(-0.275299\pi\)
\(942\) 0 0
\(943\) 5.81633e10 0.0735534
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 1.56063e12i − 1.94044i −0.242226 0.970220i \(-0.577877\pi\)
0.242226 0.970220i \(-0.422123\pi\)
\(948\) 0 0
\(949\) −6.85941e10 −0.0845711
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 8.68527e11i 1.05296i 0.850188 + 0.526480i \(0.176488\pi\)
−0.850188 + 0.526480i \(0.823512\pi\)
\(954\) 0 0
\(955\) −1.38418e11 −0.166409
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 3.92894e11i − 0.464516i
\(960\) 0 0
\(961\) −2.00224e11 −0.234759
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 5.04655e11i 0.581949i
\(966\) 0 0
\(967\) 1.38012e12 1.57838 0.789189 0.614151i \(-0.210501\pi\)
0.789189 + 0.614151i \(0.210501\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 8.75507e11i 0.984879i 0.870347 + 0.492439i \(0.163895\pi\)
−0.870347 + 0.492439i \(0.836105\pi\)
\(972\) 0 0
\(973\) −4.76815e11 −0.531984
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 7.13074e11i − 0.782630i −0.920257 0.391315i \(-0.872020\pi\)
0.920257 0.391315i \(-0.127980\pi\)
\(978\) 0 0
\(979\) 7.20170e11 0.783979
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1.62031e12i − 1.73534i −0.497140 0.867670i \(-0.665617\pi\)
0.497140 0.867670i \(-0.334383\pi\)
\(984\) 0 0
\(985\) −4.44887e11 −0.472612
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 2.52247e11i − 0.263658i
\(990\) 0 0
\(991\) −1.45491e12 −1.50849 −0.754243 0.656595i \(-0.771996\pi\)
−0.754243 + 0.656595i \(0.771996\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 8.12643e11i 0.829101i
\(996\) 0 0
\(997\) −2.93410e11 −0.296958 −0.148479 0.988916i \(-0.547438\pi\)
−0.148479 + 0.988916i \(0.547438\pi\)
\(998\) 0 0
\(999\) 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 252.9.c.a.197.7 16
3.2 odd 2 inner 252.9.c.a.197.10 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.9.c.a.197.7 16 1.1 even 1 trivial
252.9.c.a.197.10 yes 16 3.2 odd 2 inner