Properties

Label 252.9.c.a.197.8
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 197.8
Root \(-221.696i\) of defining polynomial
Character \(\chi\) \(=\) 252.197
Dual form 252.9.c.a.197.9

$q$-expansion

\(f(q)\) \(=\) \(q-221.696i q^{5} -907.493 q^{7} +O(q^{10})\) \(q-221.696i q^{5} -907.493 q^{7} -19481.4i q^{11} -19141.3 q^{13} +134659. i q^{17} -194587. q^{19} -42602.7i q^{23} +341476. q^{25} -814906. i q^{29} -203649. q^{31} +201187. i q^{35} +1.43832e6 q^{37} +2.77633e6i q^{41} -5.04485e6 q^{43} +5.42366e6i q^{47} +823543. q^{49} -1.34818e7i q^{53} -4.31894e6 q^{55} +1.38302e7i q^{59} +2.26176e7 q^{61} +4.24355e6i q^{65} +3.91134e7 q^{67} -3.02970e7i q^{71} -5.42017e7 q^{73} +1.76792e7i q^{77} -2.10178e7 q^{79} +1.30858e7i q^{83} +2.98533e7 q^{85} +1.78236e7i q^{89} +1.73706e7 q^{91} +4.31392e7i q^{95} +6.89791e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 95480q^{13} - 287560q^{19} - 1754520q^{25} - 3554264q^{31} - 182920q^{37} + 8472416q^{43} + 13176688q^{49} - 18692072q^{55} + 34224568q^{61} + 22683096q^{67} + 2137296q^{73} - 90245624q^{79} - 56204456q^{85} - 25661888q^{91} + 134041152q^{97} + O(q^{100}) \)

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
\(5\) − 221.696i − 0.354713i −0.984147 0.177357i \(-0.943245\pi\)
0.984147 0.177357i \(-0.0567546\pi\)
\(6\) 0 0
\(7\) −907.493 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 19481.4i − 1.33060i −0.746574 0.665302i \(-0.768303\pi\)
0.746574 0.665302i \(-0.231697\pi\)
\(12\) 0 0
\(13\) −19141.3 −0.670191 −0.335095 0.942184i \(-0.608769\pi\)
−0.335095 + 0.942184i \(0.608769\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 134659.i 1.61227i 0.591729 + 0.806137i \(0.298446\pi\)
−0.591729 + 0.806137i \(0.701554\pi\)
\(18\) 0 0
\(19\) −194587. −1.49314 −0.746570 0.665307i \(-0.768301\pi\)
−0.746570 + 0.665307i \(0.768301\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 42602.7i − 0.152239i −0.997099 0.0761194i \(-0.975747\pi\)
0.997099 0.0761194i \(-0.0242530\pi\)
\(24\) 0 0
\(25\) 341476. 0.874178
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 814906.i − 1.15217i −0.817391 0.576084i \(-0.804580\pi\)
0.817391 0.576084i \(-0.195420\pi\)
\(30\) 0 0
\(31\) −203649. −0.220514 −0.110257 0.993903i \(-0.535167\pi\)
−0.110257 + 0.993903i \(0.535167\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 201187.i 0.134069i
\(36\) 0 0
\(37\) 1.43832e6 0.767449 0.383725 0.923448i \(-0.374641\pi\)
0.383725 + 0.923448i \(0.374641\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.77633e6i 0.982507i 0.871017 + 0.491253i \(0.163461\pi\)
−0.871017 + 0.491253i \(0.836539\pi\)
\(42\) 0 0
\(43\) −5.04485e6 −1.47562 −0.737810 0.675009i \(-0.764140\pi\)
−0.737810 + 0.675009i \(0.764140\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.42366e6i 1.11148i 0.831357 + 0.555739i \(0.187565\pi\)
−0.831357 + 0.555739i \(0.812435\pi\)
\(48\) 0 0
\(49\) 823543. 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 1.34818e7i − 1.70861i −0.519772 0.854305i \(-0.673983\pi\)
0.519772 0.854305i \(-0.326017\pi\)
\(54\) 0 0
\(55\) −4.31894e6 −0.471983
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.38302e7i 1.14135i 0.821175 + 0.570676i \(0.193319\pi\)
−0.821175 + 0.570676i \(0.806681\pi\)
\(60\) 0 0
\(61\) 2.26176e7 1.63353 0.816766 0.576969i \(-0.195765\pi\)
0.816766 + 0.576969i \(0.195765\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.24355e6i 0.237726i
\(66\) 0 0
\(67\) 3.91134e7 1.94100 0.970502 0.241091i \(-0.0775054\pi\)
0.970502 + 0.241091i \(0.0775054\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 3.02970e7i − 1.19225i −0.802893 0.596123i \(-0.796707\pi\)
0.802893 0.596123i \(-0.203293\pi\)
\(72\) 0 0
\(73\) −5.42017e7 −1.90863 −0.954315 0.298802i \(-0.903413\pi\)
−0.954315 + 0.298802i \(0.903413\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.76792e7i 0.502921i
\(78\) 0 0
\(79\) −2.10178e7 −0.539609 −0.269804 0.962915i \(-0.586959\pi\)
−0.269804 + 0.962915i \(0.586959\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.30858e7i 0.275732i 0.990451 + 0.137866i \(0.0440244\pi\)
−0.990451 + 0.137866i \(0.955976\pi\)
\(84\) 0 0
\(85\) 2.98533e7 0.571895
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.78236e7i 0.284077i 0.989861 + 0.142038i \(0.0453657\pi\)
−0.989861 + 0.142038i \(0.954634\pi\)
\(90\) 0 0
\(91\) 1.73706e7 0.253308
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 4.31392e7i 0.529636i
\(96\) 0 0
\(97\) 6.89791e7 0.779167 0.389583 0.920991i \(-0.372619\pi\)
0.389583 + 0.920991i \(0.372619\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.90594e8i 1.83157i 0.401670 + 0.915785i \(0.368430\pi\)
−0.401670 + 0.915785i \(0.631570\pi\)
\(102\) 0 0
\(103\) −2.00729e7 −0.178345 −0.0891727 0.996016i \(-0.528422\pi\)
−0.0891727 + 0.996016i \(0.528422\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 1.00201e7i − 0.0764427i −0.999269 0.0382214i \(-0.987831\pi\)
0.999269 0.0382214i \(-0.0121692\pi\)
\(108\) 0 0
\(109\) 1.41310e8 1.00107 0.500537 0.865715i \(-0.333136\pi\)
0.500537 + 0.865715i \(0.333136\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.29538e8i 1.40780i 0.710299 + 0.703901i \(0.248560\pi\)
−0.710299 + 0.703901i \(0.751440\pi\)
\(114\) 0 0
\(115\) −9.44484e6 −0.0540012
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 1.22202e8i − 0.609382i
\(120\) 0 0
\(121\) −1.65165e8 −0.770508
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 1.62304e8i − 0.664796i
\(126\) 0 0
\(127\) 4.60959e8 1.77193 0.885967 0.463748i \(-0.153496\pi\)
0.885967 + 0.463748i \(0.153496\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.50919e8i 1.19158i 0.803141 + 0.595788i \(0.203160\pi\)
−0.803141 + 0.595788i \(0.796840\pi\)
\(132\) 0 0
\(133\) 1.76587e8 0.564354
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.62021e8i 0.743796i 0.928274 + 0.371898i \(0.121293\pi\)
−0.928274 + 0.371898i \(0.878707\pi\)
\(138\) 0 0
\(139\) 3.38052e8 0.905576 0.452788 0.891618i \(-0.350430\pi\)
0.452788 + 0.891618i \(0.350430\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.72899e8i 0.891759i
\(144\) 0 0
\(145\) −1.80661e8 −0.408689
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.73781e8i 1.56990i 0.619557 + 0.784952i \(0.287312\pi\)
−0.619557 + 0.784952i \(0.712688\pi\)
\(150\) 0 0
\(151\) −3.78380e8 −0.727814 −0.363907 0.931435i \(-0.618557\pi\)
−0.363907 + 0.931435i \(0.618557\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.51482e7i 0.0782192i
\(156\) 0 0
\(157\) 7.38439e8 1.21539 0.607695 0.794170i \(-0.292094\pi\)
0.607695 + 0.794170i \(0.292094\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.86616e7i 0.0575409i
\(162\) 0 0
\(163\) 1.26252e9 1.78850 0.894250 0.447567i \(-0.147709\pi\)
0.894250 + 0.447567i \(0.147709\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.70075e8i 0.861505i 0.902470 + 0.430752i \(0.141752\pi\)
−0.902470 + 0.430752i \(0.858248\pi\)
\(168\) 0 0
\(169\) −4.49341e8 −0.550845
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 1.97871e8i − 0.220902i −0.993882 0.110451i \(-0.964771\pi\)
0.993882 0.110451i \(-0.0352294\pi\)
\(174\) 0 0
\(175\) −3.09887e8 −0.330408
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.26692e9i 1.23406i 0.786938 + 0.617031i \(0.211665\pi\)
−0.786938 + 0.617031i \(0.788335\pi\)
\(180\) 0 0
\(181\) 9.41676e8 0.877379 0.438689 0.898639i \(-0.355443\pi\)
0.438689 + 0.898639i \(0.355443\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 3.18870e8i − 0.272224i
\(186\) 0 0
\(187\) 2.62334e9 2.14530
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.87580e8i 0.291225i 0.989342 + 0.145612i \(0.0465152\pi\)
−0.989342 + 0.145612i \(0.953485\pi\)
\(192\) 0 0
\(193\) 5.95489e8 0.429185 0.214593 0.976704i \(-0.431158\pi\)
0.214593 + 0.976704i \(0.431158\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.92777e9i 1.27994i 0.768400 + 0.639970i \(0.221053\pi\)
−0.768400 + 0.639970i \(0.778947\pi\)
\(198\) 0 0
\(199\) −1.50314e9 −0.958491 −0.479246 0.877681i \(-0.659090\pi\)
−0.479246 + 0.877681i \(0.659090\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 7.39521e8i 0.435478i
\(204\) 0 0
\(205\) 6.15501e8 0.348508
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.79083e9i 1.98678i
\(210\) 0 0
\(211\) −2.59815e9 −1.31079 −0.655397 0.755285i \(-0.727499\pi\)
−0.655397 + 0.755285i \(0.727499\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.11842e9i 0.523422i
\(216\) 0 0
\(217\) 1.84810e8 0.0833464
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 2.57755e9i − 1.08053i
\(222\) 0 0
\(223\) −1.32831e9 −0.537130 −0.268565 0.963262i \(-0.586549\pi\)
−0.268565 + 0.963262i \(0.586549\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 1.87981e9i − 0.707964i −0.935252 0.353982i \(-0.884827\pi\)
0.935252 0.353982i \(-0.115173\pi\)
\(228\) 0 0
\(229\) −5.36880e7 −0.0195225 −0.00976125 0.999952i \(-0.503107\pi\)
−0.00976125 + 0.999952i \(0.503107\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 1.03976e9i − 0.352784i −0.984320 0.176392i \(-0.943557\pi\)
0.984320 0.176392i \(-0.0564426\pi\)
\(234\) 0 0
\(235\) 1.20240e9 0.394256
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.06002e9i 0.631364i 0.948865 + 0.315682i \(0.102233\pi\)
−0.948865 + 0.315682i \(0.897767\pi\)
\(240\) 0 0
\(241\) 5.91153e9 1.75239 0.876197 0.481954i \(-0.160073\pi\)
0.876197 + 0.481954i \(0.160073\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 1.82576e8i − 0.0506733i
\(246\) 0 0
\(247\) 3.72466e9 1.00069
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 6.27398e9i − 1.58070i −0.612658 0.790348i \(-0.709900\pi\)
0.612658 0.790348i \(-0.290100\pi\)
\(252\) 0 0
\(253\) −8.29959e8 −0.202570
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 4.16469e9i − 0.954663i −0.878723 0.477331i \(-0.841604\pi\)
0.878723 0.477331i \(-0.158396\pi\)
\(258\) 0 0
\(259\) −1.30527e9 −0.290069
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 5.30124e9i 1.10804i 0.832504 + 0.554019i \(0.186906\pi\)
−0.832504 + 0.554019i \(0.813094\pi\)
\(264\) 0 0
\(265\) −2.98885e9 −0.606067
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 6.25848e9i 1.19525i 0.801775 + 0.597626i \(0.203889\pi\)
−0.801775 + 0.597626i \(0.796111\pi\)
\(270\) 0 0
\(271\) −1.62468e9 −0.301226 −0.150613 0.988593i \(-0.548125\pi\)
−0.150613 + 0.988593i \(0.548125\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 6.65242e9i − 1.16319i
\(276\) 0 0
\(277\) −2.99114e9 −0.508063 −0.254032 0.967196i \(-0.581757\pi\)
−0.254032 + 0.967196i \(0.581757\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.36189e9i 0.378822i 0.981898 + 0.189411i \(0.0606579\pi\)
−0.981898 + 0.189411i \(0.939342\pi\)
\(282\) 0 0
\(283\) −6.63475e9 −1.03438 −0.517188 0.855872i \(-0.673021\pi\)
−0.517188 + 0.855872i \(0.673021\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 2.51950e9i − 0.371353i
\(288\) 0 0
\(289\) −1.11572e10 −1.59943
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 6.55688e9i − 0.889665i −0.895614 0.444833i \(-0.853263\pi\)
0.895614 0.444833i \(-0.146737\pi\)
\(294\) 0 0
\(295\) 3.06609e9 0.404853
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.15471e8i 0.102029i
\(300\) 0 0
\(301\) 4.57816e9 0.557732
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 5.01423e9i − 0.579436i
\(306\) 0 0
\(307\) 1.12109e10 1.26208 0.631042 0.775749i \(-0.282628\pi\)
0.631042 + 0.775749i \(0.282628\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 7.89689e8i − 0.0844140i −0.999109 0.0422070i \(-0.986561\pi\)
0.999109 0.0422070i \(-0.0134389\pi\)
\(312\) 0 0
\(313\) −1.75039e9 −0.182371 −0.0911857 0.995834i \(-0.529066\pi\)
−0.0911857 + 0.995834i \(0.529066\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.03386e8i 0.0102383i 0.999987 + 0.00511913i \(0.00162948\pi\)
−0.999987 + 0.00511913i \(0.998371\pi\)
\(318\) 0 0
\(319\) −1.58755e10 −1.53308
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 2.62029e10i − 2.40735i
\(324\) 0 0
\(325\) −6.53630e9 −0.585866
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 4.92193e9i − 0.420099i
\(330\) 0 0
\(331\) −1.71555e10 −1.42919 −0.714596 0.699538i \(-0.753389\pi\)
−0.714596 + 0.699538i \(0.753389\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 8.67128e9i − 0.688500i
\(336\) 0 0
\(337\) 3.46235e9 0.268443 0.134221 0.990951i \(-0.457147\pi\)
0.134221 + 0.990951i \(0.457147\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.96737e9i 0.293417i
\(342\) 0 0
\(343\) −7.47359e8 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 9.02435e9i − 0.622441i −0.950338 0.311220i \(-0.899262\pi\)
0.950338 0.311220i \(-0.100738\pi\)
\(348\) 0 0
\(349\) −1.16999e10 −0.788646 −0.394323 0.918972i \(-0.629021\pi\)
−0.394323 + 0.918972i \(0.629021\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 2.18232e10i − 1.40546i −0.711456 0.702731i \(-0.751964\pi\)
0.711456 0.702731i \(-0.248036\pi\)
\(354\) 0 0
\(355\) −6.71672e9 −0.422906
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4.38983e9i 0.264283i 0.991231 + 0.132142i \(0.0421854\pi\)
−0.991231 + 0.132142i \(0.957815\pi\)
\(360\) 0 0
\(361\) 2.08807e10 1.22947
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.20163e10i 0.677017i
\(366\) 0 0
\(367\) −2.80399e10 −1.54565 −0.772826 0.634618i \(-0.781158\pi\)
−0.772826 + 0.634618i \(0.781158\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.22346e10i 0.645794i
\(372\) 0 0
\(373\) 2.38135e10 1.23023 0.615117 0.788436i \(-0.289109\pi\)
0.615117 + 0.788436i \(0.289109\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.55984e10i 0.772172i
\(378\) 0 0
\(379\) −1.46006e10 −0.707644 −0.353822 0.935313i \(-0.615118\pi\)
−0.353822 + 0.935313i \(0.615118\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 1.86050e10i − 0.864639i −0.901721 0.432319i \(-0.857695\pi\)
0.901721 0.432319i \(-0.142305\pi\)
\(384\) 0 0
\(385\) 3.91941e9 0.178393
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 2.88545e10i − 1.26013i −0.776542 0.630065i \(-0.783028\pi\)
0.776542 0.630065i \(-0.216972\pi\)
\(390\) 0 0
\(391\) 5.73682e9 0.245451
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 4.65956e9i 0.191406i
\(396\) 0 0
\(397\) −1.40951e10 −0.567421 −0.283711 0.958910i \(-0.591566\pi\)
−0.283711 + 0.958910i \(0.591566\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 1.40495e10i − 0.543354i −0.962388 0.271677i \(-0.912422\pi\)
0.962388 0.271677i \(-0.0875783\pi\)
\(402\) 0 0
\(403\) 3.89811e9 0.147786
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 2.80205e10i − 1.02117i
\(408\) 0 0
\(409\) −1.92375e9 −0.0687474 −0.0343737 0.999409i \(-0.510944\pi\)
−0.0343737 + 0.999409i \(0.510944\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 1.25508e10i − 0.431391i
\(414\) 0 0
\(415\) 2.90107e9 0.0978060
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8.86258e9i 0.287544i 0.989611 + 0.143772i \(0.0459232\pi\)
−0.989611 + 0.143772i \(0.954077\pi\)
\(420\) 0 0
\(421\) −7.06857e9 −0.225011 −0.112505 0.993651i \(-0.535888\pi\)
−0.112505 + 0.993651i \(0.535888\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.59827e10i 1.40942i
\(426\) 0 0
\(427\) −2.05253e10 −0.617417
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 2.15168e10i − 0.623546i −0.950157 0.311773i \(-0.899077\pi\)
0.950157 0.311773i \(-0.100923\pi\)
\(432\) 0 0
\(433\) −1.27339e10 −0.362250 −0.181125 0.983460i \(-0.557974\pi\)
−0.181125 + 0.983460i \(0.557974\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8.28995e9i 0.227314i
\(438\) 0 0
\(439\) 6.14824e10 1.65536 0.827681 0.561199i \(-0.189660\pi\)
0.827681 + 0.561199i \(0.189660\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.95753e10i 0.508269i 0.967169 + 0.254135i \(0.0817906\pi\)
−0.967169 + 0.254135i \(0.918209\pi\)
\(444\) 0 0
\(445\) 3.95142e9 0.100766
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 5.91515e10i 1.45539i 0.685899 + 0.727697i \(0.259409\pi\)
−0.685899 + 0.727697i \(0.740591\pi\)
\(450\) 0 0
\(451\) 5.40867e10 1.30733
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 3.85099e9i − 0.0898518i
\(456\) 0 0
\(457\) −1.05670e10 −0.242262 −0.121131 0.992637i \(-0.538652\pi\)
−0.121131 + 0.992637i \(0.538652\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.94330e10i 0.430265i 0.976585 + 0.215132i \(0.0690183\pi\)
−0.976585 + 0.215132i \(0.930982\pi\)
\(462\) 0 0
\(463\) 5.45181e10 1.18636 0.593180 0.805070i \(-0.297872\pi\)
0.593180 + 0.805070i \(0.297872\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 6.48698e10i − 1.36388i −0.731410 0.681938i \(-0.761137\pi\)
0.731410 0.681938i \(-0.238863\pi\)
\(468\) 0 0
\(469\) −3.54951e10 −0.733631
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9.82807e10i 1.96347i
\(474\) 0 0
\(475\) −6.64469e10 −1.30527
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 3.58840e10i − 0.681647i −0.940127 0.340823i \(-0.889294\pi\)
0.940127 0.340823i \(-0.110706\pi\)
\(480\) 0 0
\(481\) −2.75314e10 −0.514337
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 1.52924e10i − 0.276381i
\(486\) 0 0
\(487\) −8.43804e10 −1.50012 −0.750060 0.661370i \(-0.769975\pi\)
−0.750060 + 0.661370i \(0.769975\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.79053e10i 1.34042i 0.742172 + 0.670210i \(0.233796\pi\)
−0.742172 + 0.670210i \(0.766204\pi\)
\(492\) 0 0
\(493\) 1.09734e11 1.85761
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.74943e10i 0.450627i
\(498\) 0 0
\(499\) −2.86613e10 −0.462268 −0.231134 0.972922i \(-0.574244\pi\)
−0.231134 + 0.972922i \(0.574244\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 2.24664e10i 0.350963i 0.984483 + 0.175481i \(0.0561482\pi\)
−0.984483 + 0.175481i \(0.943852\pi\)
\(504\) 0 0
\(505\) 4.22538e10 0.649682
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1.20669e11i 1.79773i 0.438227 + 0.898864i \(0.355607\pi\)
−0.438227 + 0.898864i \(0.644393\pi\)
\(510\) 0 0
\(511\) 4.91877e10 0.721394
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 4.45008e9i 0.0632615i
\(516\) 0 0
\(517\) 1.05660e11 1.47894
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4.64705e10i 0.630705i 0.948975 + 0.315353i \(0.102123\pi\)
−0.948975 + 0.315353i \(0.897877\pi\)
\(522\) 0 0
\(523\) 2.55571e10 0.341590 0.170795 0.985307i \(-0.445366\pi\)
0.170795 + 0.985307i \(0.445366\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 2.74231e10i − 0.355529i
\(528\) 0 0
\(529\) 7.64960e10 0.976823
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 5.31426e10i − 0.658467i
\(534\) 0 0
\(535\) −2.22141e9 −0.0271152
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 1.60438e10i − 0.190086i
\(540\) 0 0
\(541\) 1.38276e11 1.61421 0.807104 0.590410i \(-0.201034\pi\)
0.807104 + 0.590410i \(0.201034\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 3.13278e10i − 0.355094i
\(546\) 0 0
\(547\) 6.33492e9 0.0707606 0.0353803 0.999374i \(-0.488736\pi\)
0.0353803 + 0.999374i \(0.488736\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.58571e11i 1.72035i
\(552\) 0 0
\(553\) 1.90735e10 0.203953
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 8.12825e10i 0.844455i 0.906490 + 0.422227i \(0.138752\pi\)
−0.906490 + 0.422227i \(0.861248\pi\)
\(558\) 0 0
\(559\) 9.65651e10 0.988947
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 6.53719e10i − 0.650665i −0.945600 0.325332i \(-0.894524\pi\)
0.945600 0.325332i \(-0.105476\pi\)
\(564\) 0 0
\(565\) 5.08877e10 0.499366
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 5.59552e10i 0.533815i 0.963722 + 0.266908i \(0.0860019\pi\)
−0.963722 + 0.266908i \(0.913998\pi\)
\(570\) 0 0
\(571\) −1.07046e11 −1.00699 −0.503496 0.863997i \(-0.667953\pi\)
−0.503496 + 0.863997i \(0.667953\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 1.45478e10i − 0.133084i
\(576\) 0 0
\(577\) −9.68411e10 −0.873688 −0.436844 0.899537i \(-0.643904\pi\)
−0.436844 + 0.899537i \(0.643904\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 1.18753e10i − 0.104217i
\(582\) 0 0
\(583\) −2.62643e11 −2.27348
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.19009e11i 1.00236i 0.865342 + 0.501182i \(0.167101\pi\)
−0.865342 + 0.501182i \(0.832899\pi\)
\(588\) 0 0
\(589\) 3.96276e10 0.329258
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.43498e11i 1.16045i 0.814456 + 0.580226i \(0.197036\pi\)
−0.814456 + 0.580226i \(0.802964\pi\)
\(594\) 0 0
\(595\) −2.70916e10 −0.216156
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 1.87645e11i − 1.45757i −0.684741 0.728787i \(-0.740085\pi\)
0.684741 0.728787i \(-0.259915\pi\)
\(600\) 0 0
\(601\) −3.22645e10 −0.247302 −0.123651 0.992326i \(-0.539460\pi\)
−0.123651 + 0.992326i \(0.539460\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.66165e10i 0.273310i
\(606\) 0 0
\(607\) 4.80704e10 0.354097 0.177049 0.984202i \(-0.443345\pi\)
0.177049 + 0.984202i \(0.443345\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 1.03816e11i − 0.744902i
\(612\) 0 0
\(613\) 1.76119e11 1.24728 0.623640 0.781712i \(-0.285653\pi\)
0.623640 + 0.781712i \(0.285653\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 8.55999e10i 0.590653i 0.955396 + 0.295327i \(0.0954285\pi\)
−0.955396 + 0.295327i \(0.904572\pi\)
\(618\) 0 0
\(619\) −7.64495e10 −0.520729 −0.260365 0.965510i \(-0.583843\pi\)
−0.260365 + 0.965510i \(0.583843\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 1.61748e10i − 0.107371i
\(624\) 0 0
\(625\) 9.74070e10 0.638367
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.93683e11i 1.23734i
\(630\) 0 0
\(631\) −1.98446e10 −0.125177 −0.0625885 0.998039i \(-0.519936\pi\)
−0.0625885 + 0.998039i \(0.519936\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 1.02193e11i − 0.628529i
\(636\) 0 0
\(637\) −1.57637e10 −0.0957415
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 2.72891e11i − 1.61643i −0.588888 0.808214i \(-0.700434\pi\)
0.588888 0.808214i \(-0.299566\pi\)
\(642\) 0 0
\(643\) −4.55006e10 −0.266179 −0.133089 0.991104i \(-0.542490\pi\)
−0.133089 + 0.991104i \(0.542490\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 9.73260e10i 0.555408i 0.960667 + 0.277704i \(0.0895734\pi\)
−0.960667 + 0.277704i \(0.910427\pi\)
\(648\) 0 0
\(649\) 2.69431e11 1.51869
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.60986e11i 0.885392i 0.896672 + 0.442696i \(0.145978\pi\)
−0.896672 + 0.442696i \(0.854022\pi\)
\(654\) 0 0
\(655\) 7.77973e10 0.422668
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 2.02994e11i − 1.07632i −0.842843 0.538160i \(-0.819120\pi\)
0.842843 0.538160i \(-0.180880\pi\)
\(660\) 0 0
\(661\) −2.97091e11 −1.55627 −0.778133 0.628099i \(-0.783833\pi\)
−0.778133 + 0.628099i \(0.783833\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 3.91485e10i − 0.200184i
\(666\) 0 0
\(667\) −3.47172e10 −0.175405
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 4.40623e11i − 2.17359i
\(672\) 0 0
\(673\) 2.16935e10 0.105747 0.0528737 0.998601i \(-0.483162\pi\)
0.0528737 + 0.998601i \(0.483162\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.42130e10i 0.0676596i 0.999428 + 0.0338298i \(0.0107704\pi\)
−0.999428 + 0.0338298i \(0.989230\pi\)
\(678\) 0 0
\(679\) −6.25980e10 −0.294497
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 7.61548e9i − 0.0349957i −0.999847 0.0174978i \(-0.994430\pi\)
0.999847 0.0174978i \(-0.00557002\pi\)
\(684\) 0 0
\(685\) 5.80890e10 0.263834
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 2.58059e11i 1.14509i
\(690\) 0 0
\(691\) −1.09041e11 −0.478273 −0.239136 0.970986i \(-0.576864\pi\)
−0.239136 + 0.970986i \(0.576864\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 7.49448e10i − 0.321220i
\(696\) 0 0
\(697\) −3.73857e11 −1.58407
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 3.22792e11i 1.33675i 0.743823 + 0.668376i \(0.233010\pi\)
−0.743823 + 0.668376i \(0.766990\pi\)
\(702\) 0 0
\(703\) −2.79880e11 −1.14591
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1.72962e11i − 0.692268i
\(708\) 0 0
\(709\) −1.28670e11 −0.509205 −0.254602 0.967046i \(-0.581945\pi\)
−0.254602 + 0.967046i \(0.581945\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8.67600e9i 0.0335708i
\(714\) 0 0
\(715\) 8.26702e10 0.316319
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 1.19347e11i − 0.446577i −0.974752 0.223288i \(-0.928321\pi\)
0.974752 0.223288i \(-0.0716792\pi\)
\(720\) 0 0
\(721\) 1.82160e10 0.0674082
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 2.78271e11i − 1.00720i
\(726\) 0 0
\(727\) −1.26145e11 −0.451578 −0.225789 0.974176i \(-0.572496\pi\)
−0.225789 + 0.974176i \(0.572496\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 6.79333e11i − 2.37910i
\(732\) 0 0
\(733\) −2.14006e10 −0.0741327 −0.0370664 0.999313i \(-0.511801\pi\)
−0.0370664 + 0.999313i \(0.511801\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 7.61984e11i − 2.58271i
\(738\) 0 0
\(739\) 9.32712e9 0.0312730 0.0156365 0.999878i \(-0.495023\pi\)
0.0156365 + 0.999878i \(0.495023\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 5.47858e11i 1.79768i 0.438275 + 0.898841i \(0.355590\pi\)
−0.438275 + 0.898841i \(0.644410\pi\)
\(744\) 0 0
\(745\) 1.71544e11 0.556866
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 9.09315e9i 0.0288926i
\(750\) 0 0
\(751\) −3.25787e10 −0.102418 −0.0512088 0.998688i \(-0.516307\pi\)
−0.0512088 + 0.998688i \(0.516307\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 8.38853e10i 0.258165i
\(756\) 0 0
\(757\) 1.83680e11 0.559342 0.279671 0.960096i \(-0.409775\pi\)
0.279671 + 0.960096i \(0.409775\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.15436e11i 0.344194i 0.985080 + 0.172097i \(0.0550543\pi\)
−0.985080 + 0.172097i \(0.944946\pi\)
\(762\) 0 0
\(763\) −1.28238e11 −0.378371
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 2.64728e11i − 0.764924i
\(768\) 0 0
\(769\) 1.80169e11 0.515198 0.257599 0.966252i \(-0.417069\pi\)
0.257599 + 0.966252i \(0.417069\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 6.65686e11i − 1.86445i −0.361874 0.932227i \(-0.617863\pi\)
0.361874 0.932227i \(-0.382137\pi\)
\(774\) 0 0
\(775\) −6.95413e10 −0.192768
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 5.40239e11i − 1.46702i
\(780\) 0 0
\(781\) −5.90227e11 −1.58641
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 1.63709e11i − 0.431115i
\(786\) 0 0
\(787\) 4.36920e11 1.13895 0.569473 0.822010i \(-0.307147\pi\)
0.569473 + 0.822010i \(0.307147\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 2.08304e11i − 0.532099i
\(792\) 0 0
\(793\) −4.32931e11 −1.09478
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.59644e11i 0.891333i 0.895199 + 0.445667i \(0.147033\pi\)
−0.895199 + 0.445667i \(0.852967\pi\)
\(798\) 0 0
\(799\) −7.30343e11 −1.79201
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.05592e12i 2.53963i
\(804\) 0 0
\(805\) 8.57112e9 0.0204105
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 4.25571e11i 0.993522i 0.867887 + 0.496761i \(0.165477\pi\)
−0.867887 + 0.496761i \(0.834523\pi\)
\(810\) 0 0
\(811\) −1.09416e11 −0.252929 −0.126465 0.991971i \(-0.540363\pi\)
−0.126465 + 0.991971i \(0.540363\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 2.79896e11i − 0.634405i
\(816\) 0 0
\(817\) 9.81665e11 2.20331
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 6.79455e11i 1.49551i 0.663977 + 0.747753i \(0.268867\pi\)
−0.663977 + 0.747753i \(0.731133\pi\)
\(822\) 0 0
\(823\) −7.80059e11 −1.70031 −0.850155 0.526533i \(-0.823492\pi\)
−0.850155 + 0.526533i \(0.823492\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 6.01868e11i − 1.28671i −0.765569 0.643353i \(-0.777543\pi\)
0.765569 0.643353i \(-0.222457\pi\)
\(828\) 0 0
\(829\) 1.26918e11 0.268722 0.134361 0.990932i \(-0.457102\pi\)
0.134361 + 0.990932i \(0.457102\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.10897e11i 0.230325i
\(834\) 0 0
\(835\) 1.48553e11 0.305587
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 8.99511e11i − 1.81534i −0.419682 0.907671i \(-0.637858\pi\)
0.419682 0.907671i \(-0.362142\pi\)
\(840\) 0 0
\(841\) −1.63826e11 −0.327490
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 9.96170e10i 0.195392i
\(846\) 0 0
\(847\) 1.49886e11 0.291225
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 6.12764e10i − 0.116836i
\(852\) 0 0
\(853\) 3.68583e11 0.696209 0.348105 0.937456i \(-0.386825\pi\)
0.348105 + 0.937456i \(0.386825\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.55843e11i 0.474297i 0.971473 + 0.237148i \(0.0762127\pi\)
−0.971473 + 0.237148i \(0.923787\pi\)
\(858\) 0 0
\(859\) −1.52059e11 −0.279279 −0.139640 0.990202i \(-0.544594\pi\)
−0.139640 + 0.990202i \(0.544594\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 8.82719e11i − 1.59140i −0.605691 0.795700i \(-0.707103\pi\)
0.605691 0.795700i \(-0.292897\pi\)
\(864\) 0 0
\(865\) −4.38673e10 −0.0783567
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 4.09456e11i 0.718006i
\(870\) 0 0
\(871\) −7.48682e11 −1.30084
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.47289e11i 0.251269i
\(876\) 0 0
\(877\) −6.67630e11 −1.12859 −0.564297 0.825572i \(-0.690853\pi\)
−0.564297 + 0.825572i \(0.690853\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 9.80883e11i 1.62822i 0.580709 + 0.814111i \(0.302775\pi\)
−0.580709 + 0.814111i \(0.697225\pi\)
\(882\) 0 0
\(883\) 5.60537e11 0.922065 0.461033 0.887383i \(-0.347479\pi\)
0.461033 + 0.887383i \(0.347479\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 5.43060e11i 0.877311i 0.898655 + 0.438655i \(0.144545\pi\)
−0.898655 + 0.438655i \(0.855455\pi\)
\(888\) 0 0
\(889\) −4.18317e11 −0.669728
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 1.05538e12i − 1.65959i
\(894\) 0 0
\(895\) 2.80871e11 0.437739
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 1.65955e11i 0.254069i
\(900\) 0 0
\(901\) 1.81544e12 2.75475
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 2.08766e11i − 0.311218i
\(906\) 0 0
\(907\) −1.56480e11 −0.231222 −0.115611 0.993295i \(-0.536883\pi\)
−0.115611 + 0.993295i \(0.536883\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.05213e12i 1.52755i 0.645483 + 0.763775i \(0.276656\pi\)
−0.645483 + 0.763775i \(0.723344\pi\)
\(912\) 0 0
\(913\) 2.54929e11 0.366891
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 3.18457e11i − 0.450374i
\(918\) 0 0
\(919\) 1.04578e12 1.46615 0.733077 0.680146i \(-0.238084\pi\)
0.733077 + 0.680146i \(0.238084\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 5.79924e11i 0.799033i
\(924\) 0 0
\(925\) 4.91153e11 0.670888
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.30474e12i 1.75171i 0.482578 + 0.875853i \(0.339700\pi\)
−0.482578 + 0.875853i \(0.660300\pi\)
\(930\) 0 0
\(931\) −1.60251e11 −0.213306
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 5.81583e11i − 0.760966i
\(936\) 0 0
\(937\) −7.72155e11 −1.00172 −0.500860 0.865528i \(-0.666983\pi\)
−0.500860 + 0.865528i \(0.666983\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 1.85580e11i − 0.236686i −0.992973 0.118343i \(-0.962242\pi\)
0.992973 0.118343i \(-0.0377582\pi\)
\(942\) 0 0
\(943\) 1.18279e11 0.149576
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 4.65591e11i − 0.578902i −0.957193 0.289451i \(-0.906527\pi\)
0.957193 0.289451i \(-0.0934727\pi\)
\(948\) 0 0
\(949\) 1.03749e12 1.27915
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 3.01476e11i 0.365495i 0.983160 + 0.182748i \(0.0584991\pi\)
−0.983160 + 0.182748i \(0.941501\pi\)
\(954\) 0 0
\(955\) 8.59249e10 0.103301
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 2.37782e11i − 0.281128i
\(960\) 0 0
\(961\) −8.11418e11 −0.951374
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 1.32017e11i − 0.152238i
\(966\) 0 0
\(967\) 8.13827e11 0.930735 0.465368 0.885118i \(-0.345922\pi\)
0.465368 + 0.885118i \(0.345922\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.34616e12i 1.51433i 0.653225 + 0.757164i \(0.273416\pi\)
−0.653225 + 0.757164i \(0.726584\pi\)
\(972\) 0 0
\(973\) −3.06780e11 −0.342275
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 8.92037e11i − 0.979050i −0.871989 0.489525i \(-0.837170\pi\)
0.871989 0.489525i \(-0.162830\pi\)
\(978\) 0 0
\(979\) 3.47229e11 0.377994
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 3.53170e11i 0.378242i 0.981954 + 0.189121i \(0.0605638\pi\)
−0.981954 + 0.189121i \(0.939436\pi\)
\(984\) 0 0
\(985\) 4.27377e11 0.454011
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.14924e11i 0.224647i
\(990\) 0 0
\(991\) 7.61424e11 0.789463 0.394732 0.918796i \(-0.370838\pi\)
0.394732 + 0.918796i \(0.370838\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 3.33241e11i 0.339990i
\(996\) 0 0
\(997\) −1.57737e12 −1.59644 −0.798221 0.602365i \(-0.794225\pi\)
−0.798221 + 0.602365i \(0.794225\pi\)
\(998\) 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.197.8 16
3.2 odd 2 inner 252.9.c.a.197.9 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.9.c.a.197.8 16 1.1 even 1 trivial
252.9.c.a.197.9 yes 16 3.2 odd 2 inner