Properties

Label 252.9.c.a.197.3
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 197.3
Root \(-899.394i\) of defining polynomial
Character \(\chi\) \(=\) 252.197
Dual form 252.9.c.a.197.14

$q$-expansion

\(f(q)\) \(=\) \(q-899.394i q^{5} +907.493 q^{7} +O(q^{10})\) \(q-899.394i q^{5} +907.493 q^{7} +21934.5i q^{11} +20852.6 q^{13} +54914.5i q^{17} -112287. q^{19} -364864. i q^{23} -418285. q^{25} -493187. i q^{29} -947146. q^{31} -816194. i q^{35} -2.31637e6 q^{37} +336127. i q^{41} -164303. q^{43} -1.21616e6i q^{47} +823543. q^{49} +2.50757e6i q^{53} +1.97278e7 q^{55} +1.27740e7i q^{59} -1.01258e7 q^{61} -1.87547e7i q^{65} +7.53739e6 q^{67} +4.21448e7i q^{71} -3.47745e7 q^{73} +1.99054e7i q^{77} +1.95932e6 q^{79} +2.01258e7i q^{83} +4.93898e7 q^{85} -1.15659e7i q^{89} +1.89236e7 q^{91} +1.00990e8i q^{95} -9.91217e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 95480q^{13} - 287560q^{19} - 1754520q^{25} - 3554264q^{31} - 182920q^{37} + 8472416q^{43} + 13176688q^{49} - 18692072q^{55} + 34224568q^{61} + 22683096q^{67} + 2137296q^{73} - 90245624q^{79} - 56204456q^{85} - 25661888q^{91} + 134041152q^{97} + O(q^{100}) \)

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
\(5\) − 899.394i − 1.43903i −0.694477 0.719515i \(-0.744364\pi\)
0.694477 0.719515i \(-0.255636\pi\)
\(6\) 0 0
\(7\) 907.493 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 21934.5i 1.49816i 0.662481 + 0.749079i \(0.269504\pi\)
−0.662481 + 0.749079i \(0.730496\pi\)
\(12\) 0 0
\(13\) 20852.6 0.730108 0.365054 0.930986i \(-0.381051\pi\)
0.365054 + 0.930986i \(0.381051\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 54914.5i 0.657493i 0.944418 + 0.328747i \(0.106626\pi\)
−0.944418 + 0.328747i \(0.893374\pi\)
\(18\) 0 0
\(19\) −112287. −0.861620 −0.430810 0.902443i \(-0.641772\pi\)
−0.430810 + 0.902443i \(0.641772\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 364864.i − 1.30383i −0.758294 0.651913i \(-0.773967\pi\)
0.758294 0.651913i \(-0.226033\pi\)
\(24\) 0 0
\(25\) −418285. −1.07081
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 493187.i − 0.697299i −0.937253 0.348650i \(-0.886640\pi\)
0.937253 0.348650i \(-0.113360\pi\)
\(30\) 0 0
\(31\) −947146. −1.02558 −0.512791 0.858514i \(-0.671388\pi\)
−0.512791 + 0.858514i \(0.671388\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 816194.i − 0.543902i
\(36\) 0 0
\(37\) −2.31637e6 −1.23595 −0.617976 0.786197i \(-0.712047\pi\)
−0.617976 + 0.786197i \(0.712047\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 336127.i 0.118951i 0.998230 + 0.0594755i \(0.0189428\pi\)
−0.998230 + 0.0594755i \(0.981057\pi\)
\(42\) 0 0
\(43\) −164303. −0.0480587 −0.0240294 0.999711i \(-0.507650\pi\)
−0.0240294 + 0.999711i \(0.507650\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 1.21616e6i − 0.249229i −0.992205 0.124614i \(-0.960231\pi\)
0.992205 0.124614i \(-0.0397694\pi\)
\(48\) 0 0
\(49\) 823543. 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.50757e6i 0.317797i 0.987295 + 0.158898i \(0.0507943\pi\)
−0.987295 + 0.158898i \(0.949206\pi\)
\(54\) 0 0
\(55\) 1.97278e7 2.15589
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.27740e7i 1.05419i 0.849807 + 0.527094i \(0.176718\pi\)
−0.849807 + 0.527094i \(0.823282\pi\)
\(60\) 0 0
\(61\) −1.01258e7 −0.731323 −0.365661 0.930748i \(-0.619157\pi\)
−0.365661 + 0.930748i \(0.619157\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 1.87547e7i − 1.05065i
\(66\) 0 0
\(67\) 7.53739e6 0.374043 0.187022 0.982356i \(-0.440117\pi\)
0.187022 + 0.982356i \(0.440117\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.21448e7i 1.65848i 0.558892 + 0.829240i \(0.311226\pi\)
−0.558892 + 0.829240i \(0.688774\pi\)
\(72\) 0 0
\(73\) −3.47745e7 −1.22453 −0.612266 0.790652i \(-0.709742\pi\)
−0.612266 + 0.790652i \(0.709742\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.99054e7i 0.566250i
\(78\) 0 0
\(79\) 1.95932e6 0.0503033 0.0251517 0.999684i \(-0.491993\pi\)
0.0251517 + 0.999684i \(0.491993\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.01258e7i 0.424073i 0.977262 + 0.212037i \(0.0680096\pi\)
−0.977262 + 0.212037i \(0.931990\pi\)
\(84\) 0 0
\(85\) 4.93898e7 0.946153
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 1.15659e7i − 0.184341i −0.995743 0.0921703i \(-0.970620\pi\)
0.995743 0.0921703i \(-0.0293804\pi\)
\(90\) 0 0
\(91\) 1.89236e7 0.275955
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.00990e8i 1.23990i
\(96\) 0 0
\(97\) −9.91217e7 −1.11965 −0.559825 0.828611i \(-0.689131\pi\)
−0.559825 + 0.828611i \(0.689131\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 1.13987e8i − 1.09539i −0.836679 0.547694i \(-0.815506\pi\)
0.836679 0.547694i \(-0.184494\pi\)
\(102\) 0 0
\(103\) −1.40209e8 −1.24574 −0.622868 0.782327i \(-0.714033\pi\)
−0.622868 + 0.782327i \(0.714033\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 8.47465e7i − 0.646527i −0.946309 0.323263i \(-0.895220\pi\)
0.946309 0.323263i \(-0.104780\pi\)
\(108\) 0 0
\(109\) −6.16339e7 −0.436630 −0.218315 0.975878i \(-0.570056\pi\)
−0.218315 + 0.975878i \(0.570056\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.48925e8i 0.913382i 0.889625 + 0.456691i \(0.150966\pi\)
−0.889625 + 0.456691i \(0.849034\pi\)
\(114\) 0 0
\(115\) −3.28157e8 −1.87625
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 4.98345e7i 0.248509i
\(120\) 0 0
\(121\) −2.66765e8 −1.24448
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.48770e7i 0.101896i
\(126\) 0 0
\(127\) 4.37067e8 1.68009 0.840046 0.542515i \(-0.182528\pi\)
0.840046 + 0.542515i \(0.182528\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.99853e8i 1.35774i 0.734261 + 0.678868i \(0.237529\pi\)
−0.734261 + 0.678868i \(0.762471\pi\)
\(132\) 0 0
\(133\) −1.01900e8 −0.325662
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.51026e8i 1.28032i 0.768240 + 0.640161i \(0.221133\pi\)
−0.768240 + 0.640161i \(0.778867\pi\)
\(138\) 0 0
\(139\) −6.59839e7 −0.176758 −0.0883789 0.996087i \(-0.528169\pi\)
−0.0883789 + 0.996087i \(0.528169\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.57392e8i 1.09382i
\(144\) 0 0
\(145\) −4.43569e8 −1.00344
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.48851e8i 1.51932i 0.650319 + 0.759662i \(0.274635\pi\)
−0.650319 + 0.759662i \(0.725365\pi\)
\(150\) 0 0
\(151\) 8.69152e8 1.67181 0.835907 0.548871i \(-0.184942\pi\)
0.835907 + 0.548871i \(0.184942\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.51858e8i 1.47584i
\(156\) 0 0
\(157\) 3.43252e8 0.564956 0.282478 0.959274i \(-0.408844\pi\)
0.282478 + 0.959274i \(0.408844\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 3.31112e8i − 0.492800i
\(162\) 0 0
\(163\) −1.03494e9 −1.46610 −0.733052 0.680173i \(-0.761905\pi\)
−0.733052 + 0.680173i \(0.761905\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.55929e8i 1.10045i 0.835015 + 0.550227i \(0.185459\pi\)
−0.835015 + 0.550227i \(0.814541\pi\)
\(168\) 0 0
\(169\) −3.80900e8 −0.466943
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 1.21109e9i − 1.35204i −0.736882 0.676021i \(-0.763703\pi\)
0.736882 0.676021i \(-0.236297\pi\)
\(174\) 0 0
\(175\) −3.79590e8 −0.404728
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.65570e8i 0.161276i 0.996743 + 0.0806381i \(0.0256958\pi\)
−0.996743 + 0.0806381i \(0.974304\pi\)
\(180\) 0 0
\(181\) −1.87197e9 −1.74416 −0.872078 0.489367i \(-0.837228\pi\)
−0.872078 + 0.489367i \(0.837228\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 2.08333e9i 1.77857i
\(186\) 0 0
\(187\) −1.20452e9 −0.985029
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 5.54193e8i − 0.416416i −0.978085 0.208208i \(-0.933237\pi\)
0.978085 0.208208i \(-0.0667631\pi\)
\(192\) 0 0
\(193\) −2.65475e9 −1.91335 −0.956677 0.291152i \(-0.905961\pi\)
−0.956677 + 0.291152i \(0.905961\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.36750e9i 1.57190i 0.618292 + 0.785949i \(0.287825\pi\)
−0.618292 + 0.785949i \(0.712175\pi\)
\(198\) 0 0
\(199\) 2.24978e9 1.43459 0.717294 0.696771i \(-0.245380\pi\)
0.717294 + 0.696771i \(0.245380\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 4.47563e8i − 0.263554i
\(204\) 0 0
\(205\) 3.02311e8 0.171174
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 2.46297e9i − 1.29084i
\(210\) 0 0
\(211\) 1.61666e9 0.815624 0.407812 0.913066i \(-0.366292\pi\)
0.407812 + 0.913066i \(0.366292\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.47773e8i 0.0691580i
\(216\) 0 0
\(217\) −8.59528e8 −0.387633
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.14511e9i 0.480041i
\(222\) 0 0
\(223\) −4.03660e9 −1.63229 −0.816143 0.577850i \(-0.803892\pi\)
−0.816143 + 0.577850i \(0.803892\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 8.28153e8i 0.311894i 0.987765 + 0.155947i \(0.0498429\pi\)
−0.987765 + 0.155947i \(0.950157\pi\)
\(228\) 0 0
\(229\) −2.05916e9 −0.748770 −0.374385 0.927273i \(-0.622146\pi\)
−0.374385 + 0.927273i \(0.622146\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 3.06700e9i 1.04062i 0.853979 + 0.520308i \(0.174183\pi\)
−0.853979 + 0.520308i \(0.825817\pi\)
\(234\) 0 0
\(235\) −1.09380e9 −0.358648
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.90569e9i 1.19703i 0.801111 + 0.598516i \(0.204243\pi\)
−0.801111 + 0.598516i \(0.795757\pi\)
\(240\) 0 0
\(241\) 5.96315e9 1.76769 0.883847 0.467775i \(-0.154944\pi\)
0.883847 + 0.467775i \(0.154944\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 7.40690e8i − 0.205576i
\(246\) 0 0
\(247\) −2.34148e9 −0.629075
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 2.52021e9i − 0.634954i −0.948266 0.317477i \(-0.897164\pi\)
0.948266 0.317477i \(-0.102836\pi\)
\(252\) 0 0
\(253\) 8.00312e9 1.95334
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 2.17920e8i − 0.0499533i −0.999688 0.0249766i \(-0.992049\pi\)
0.999688 0.0249766i \(-0.00795114\pi\)
\(258\) 0 0
\(259\) −2.10209e9 −0.467146
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 3.78536e9i − 0.791196i −0.918424 0.395598i \(-0.870537\pi\)
0.918424 0.395598i \(-0.129463\pi\)
\(264\) 0 0
\(265\) 2.25529e9 0.457320
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 7.04980e9i 1.34638i 0.739469 + 0.673190i \(0.235077\pi\)
−0.739469 + 0.673190i \(0.764923\pi\)
\(270\) 0 0
\(271\) −1.60874e8 −0.0298270 −0.0149135 0.999889i \(-0.504747\pi\)
−0.0149135 + 0.999889i \(0.504747\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 9.17488e9i − 1.60424i
\(276\) 0 0
\(277\) 9.44864e9 1.60491 0.802454 0.596714i \(-0.203527\pi\)
0.802454 + 0.596714i \(0.203527\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 5.14862e9i − 0.825783i −0.910780 0.412891i \(-0.864519\pi\)
0.910780 0.412891i \(-0.135481\pi\)
\(282\) 0 0
\(283\) −4.72901e9 −0.737266 −0.368633 0.929575i \(-0.620174\pi\)
−0.368633 + 0.929575i \(0.620174\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 3.05033e8i 0.0449592i
\(288\) 0 0
\(289\) 3.96016e9 0.567703
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 1.31304e9i − 0.178158i −0.996025 0.0890791i \(-0.971608\pi\)
0.996025 0.0890791i \(-0.0283924\pi\)
\(294\) 0 0
\(295\) 1.14888e10 1.51701
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 7.60837e9i − 0.951934i
\(300\) 0 0
\(301\) −1.49104e8 −0.0181645
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 9.10707e9i 1.05240i
\(306\) 0 0
\(307\) −2.29371e9 −0.258217 −0.129108 0.991630i \(-0.541212\pi\)
−0.129108 + 0.991630i \(0.541212\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 1.29318e10i − 1.38235i −0.722687 0.691176i \(-0.757093\pi\)
0.722687 0.691176i \(-0.242907\pi\)
\(312\) 0 0
\(313\) −1.63693e10 −1.70550 −0.852751 0.522317i \(-0.825068\pi\)
−0.852751 + 0.522317i \(0.825068\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 1.30871e10i − 1.29601i −0.761637 0.648004i \(-0.775604\pi\)
0.761637 0.648004i \(-0.224396\pi\)
\(318\) 0 0
\(319\) 1.08178e10 1.04466
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 6.16619e9i − 0.566509i
\(324\) 0 0
\(325\) −8.72233e9 −0.781806
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 1.10365e9i − 0.0941996i
\(330\) 0 0
\(331\) −1.33867e10 −1.11522 −0.557611 0.830102i \(-0.688282\pi\)
−0.557611 + 0.830102i \(0.688282\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 6.77908e9i − 0.538260i
\(336\) 0 0
\(337\) −6.92070e9 −0.536575 −0.268288 0.963339i \(-0.586458\pi\)
−0.268288 + 0.963339i \(0.586458\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 2.07752e10i − 1.53648i
\(342\) 0 0
\(343\) 7.47359e8 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 5.48176e9i − 0.378096i −0.981968 0.189048i \(-0.939460\pi\)
0.981968 0.189048i \(-0.0605402\pi\)
\(348\) 0 0
\(349\) 1.62728e10 1.09689 0.548443 0.836188i \(-0.315221\pi\)
0.548443 + 0.836188i \(0.315221\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 2.57447e10i − 1.65802i −0.559237 0.829008i \(-0.688906\pi\)
0.559237 0.829008i \(-0.311094\pi\)
\(354\) 0 0
\(355\) 3.79048e10 2.38660
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 5.68498e9i 0.342256i 0.985249 + 0.171128i \(0.0547412\pi\)
−0.985249 + 0.171128i \(0.945259\pi\)
\(360\) 0 0
\(361\) −4.37515e9 −0.257611
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 3.12760e10i 1.76214i
\(366\) 0 0
\(367\) 1.12442e10 0.619820 0.309910 0.950766i \(-0.399701\pi\)
0.309910 + 0.950766i \(0.399701\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.27560e9i 0.120116i
\(372\) 0 0
\(373\) 1.82711e10 0.943910 0.471955 0.881623i \(-0.343549\pi\)
0.471955 + 0.881623i \(0.343549\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 1.02842e10i − 0.509104i
\(378\) 0 0
\(379\) 2.30958e10 1.11938 0.559689 0.828703i \(-0.310921\pi\)
0.559689 + 0.828703i \(0.310921\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.12755e10i 0.988745i 0.869250 + 0.494372i \(0.164602\pi\)
−0.869250 + 0.494372i \(0.835398\pi\)
\(384\) 0 0
\(385\) 1.79028e10 0.814852
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 3.68614e9i − 0.160981i −0.996755 0.0804903i \(-0.974351\pi\)
0.996755 0.0804903i \(-0.0256486\pi\)
\(390\) 0 0
\(391\) 2.00363e10 0.857257
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 1.76220e9i − 0.0723880i
\(396\) 0 0
\(397\) 2.57692e9 0.103738 0.0518692 0.998654i \(-0.483482\pi\)
0.0518692 + 0.998654i \(0.483482\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.27654e10i 0.493693i 0.969055 + 0.246847i \(0.0793944\pi\)
−0.969055 + 0.246847i \(0.920606\pi\)
\(402\) 0 0
\(403\) −1.97505e10 −0.748785
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 5.08085e10i − 1.85165i
\(408\) 0 0
\(409\) −1.66538e10 −0.595142 −0.297571 0.954700i \(-0.596177\pi\)
−0.297571 + 0.954700i \(0.596177\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.15923e10i 0.398446i
\(414\) 0 0
\(415\) 1.81010e10 0.610254
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.06318e10i 0.669391i 0.942326 + 0.334696i \(0.108633\pi\)
−0.942326 + 0.334696i \(0.891367\pi\)
\(420\) 0 0
\(421\) −5.94606e10 −1.89278 −0.946391 0.323023i \(-0.895301\pi\)
−0.946391 + 0.323023i \(0.895301\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 2.29699e10i − 0.704050i
\(426\) 0 0
\(427\) −9.18907e9 −0.276414
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 3.73736e10i − 1.08307i −0.840679 0.541534i \(-0.817844\pi\)
0.840679 0.541534i \(-0.182156\pi\)
\(432\) 0 0
\(433\) −3.44971e10 −0.981365 −0.490683 0.871338i \(-0.663253\pi\)
−0.490683 + 0.871338i \(0.663253\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.09696e10i 1.12340i
\(438\) 0 0
\(439\) 5.40944e10 1.45644 0.728222 0.685341i \(-0.240347\pi\)
0.728222 + 0.685341i \(0.240347\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.18604e10i 0.827249i 0.910448 + 0.413625i \(0.135737\pi\)
−0.910448 + 0.413625i \(0.864263\pi\)
\(444\) 0 0
\(445\) −1.04023e10 −0.265272
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 5.50863e10i − 1.35537i −0.735352 0.677686i \(-0.762983\pi\)
0.735352 0.677686i \(-0.237017\pi\)
\(450\) 0 0
\(451\) −7.37279e9 −0.178207
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 1.70198e10i − 0.397107i
\(456\) 0 0
\(457\) −4.41618e10 −1.01247 −0.506234 0.862396i \(-0.668963\pi\)
−0.506234 + 0.862396i \(0.668963\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 7.85244e10i 1.73861i 0.494280 + 0.869303i \(0.335432\pi\)
−0.494280 + 0.869303i \(0.664568\pi\)
\(462\) 0 0
\(463\) −7.61686e10 −1.65750 −0.828748 0.559623i \(-0.810946\pi\)
−0.828748 + 0.559623i \(0.810946\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.86924e10i 0.813502i 0.913539 + 0.406751i \(0.133338\pi\)
−0.913539 + 0.406751i \(0.866662\pi\)
\(468\) 0 0
\(469\) 6.84013e9 0.141375
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 3.60391e9i − 0.0719996i
\(474\) 0 0
\(475\) 4.69680e10 0.922630
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.21577e9i 0.0610862i 0.999533 + 0.0305431i \(0.00972368\pi\)
−0.999533 + 0.0305431i \(0.990276\pi\)
\(480\) 0 0
\(481\) −4.83024e10 −0.902378
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 8.91495e10i 1.61121i
\(486\) 0 0
\(487\) −6.36460e10 −1.13150 −0.565751 0.824576i \(-0.691414\pi\)
−0.565751 + 0.824576i \(0.691414\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 7.53919e10i − 1.29717i −0.761140 0.648587i \(-0.775360\pi\)
0.761140 0.648587i \(-0.224640\pi\)
\(492\) 0 0
\(493\) 2.70831e10 0.458470
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.82461e10i 0.626847i
\(498\) 0 0
\(499\) 6.00879e10 0.969136 0.484568 0.874754i \(-0.338977\pi\)
0.484568 + 0.874754i \(0.338977\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 7.02391e10i − 1.09725i −0.836068 0.548626i \(-0.815151\pi\)
0.836068 0.548626i \(-0.184849\pi\)
\(504\) 0 0
\(505\) −1.02519e11 −1.57630
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 7.49035e10i 1.11591i 0.829870 + 0.557957i \(0.188415\pi\)
−0.829870 + 0.557957i \(0.811585\pi\)
\(510\) 0 0
\(511\) −3.15576e10 −0.462829
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.26103e11i 1.79265i
\(516\) 0 0
\(517\) 2.66758e10 0.373384
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 8.53205e10i 1.15798i 0.815333 + 0.578992i \(0.196554\pi\)
−0.815333 + 0.578992i \(0.803446\pi\)
\(522\) 0 0
\(523\) 6.90127e10 0.922407 0.461203 0.887294i \(-0.347418\pi\)
0.461203 + 0.887294i \(0.347418\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 5.20120e10i − 0.674313i
\(528\) 0 0
\(529\) −5.48148e10 −0.699963
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 7.00912e9i 0.0868470i
\(534\) 0 0
\(535\) −7.62205e10 −0.930372
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1.80640e10i 0.214023i
\(540\) 0 0
\(541\) 2.81530e10 0.328652 0.164326 0.986406i \(-0.447455\pi\)
0.164326 + 0.986406i \(0.447455\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5.54332e10i 0.628324i
\(546\) 0 0
\(547\) −1.27027e10 −0.141888 −0.0709440 0.997480i \(-0.522601\pi\)
−0.0709440 + 0.997480i \(0.522601\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 5.53785e10i 0.600807i
\(552\) 0 0
\(553\) 1.77807e9 0.0190129
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.93534e10i 1.03220i 0.856530 + 0.516098i \(0.172616\pi\)
−0.856530 + 0.516098i \(0.827384\pi\)
\(558\) 0 0
\(559\) −3.42615e9 −0.0350881
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 1.28318e11i − 1.27719i −0.769544 0.638594i \(-0.779516\pi\)
0.769544 0.638594i \(-0.220484\pi\)
\(564\) 0 0
\(565\) 1.33942e11 1.31438
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 9.69562e10i 0.924968i 0.886628 + 0.462484i \(0.153042\pi\)
−0.886628 + 0.462484i \(0.846958\pi\)
\(570\) 0 0
\(571\) 4.85788e10 0.456985 0.228493 0.973546i \(-0.426620\pi\)
0.228493 + 0.973546i \(0.426620\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.52617e11i 1.39615i
\(576\) 0 0
\(577\) −1.00425e11 −0.906024 −0.453012 0.891504i \(-0.649650\pi\)
−0.453012 + 0.891504i \(0.649650\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.82640e10i 0.160285i
\(582\) 0 0
\(583\) −5.50024e10 −0.476110
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 5.94122e10i 0.500407i 0.968193 + 0.250203i \(0.0804975\pi\)
−0.968193 + 0.250203i \(0.919502\pi\)
\(588\) 0 0
\(589\) 1.06352e11 0.883662
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 1.44264e11i − 1.16665i −0.812240 0.583324i \(-0.801752\pi\)
0.812240 0.583324i \(-0.198248\pi\)
\(594\) 0 0
\(595\) 4.48209e10 0.357612
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 4.20875e10i − 0.326923i −0.986550 0.163462i \(-0.947734\pi\)
0.986550 0.163462i \(-0.0522660\pi\)
\(600\) 0 0
\(601\) −8.91047e10 −0.682972 −0.341486 0.939887i \(-0.610930\pi\)
−0.341486 + 0.939887i \(0.610930\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.39927e11i 1.79084i
\(606\) 0 0
\(607\) −2.58228e10 −0.190217 −0.0951083 0.995467i \(-0.530320\pi\)
−0.0951083 + 0.995467i \(0.530320\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 2.53600e10i − 0.181964i
\(612\) 0 0
\(613\) −2.19898e11 −1.55733 −0.778663 0.627442i \(-0.784102\pi\)
−0.778663 + 0.627442i \(0.784102\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.21752e10i 0.429019i 0.976722 + 0.214510i \(0.0688153\pi\)
−0.976722 + 0.214510i \(0.931185\pi\)
\(618\) 0 0
\(619\) −2.05984e11 −1.40304 −0.701521 0.712649i \(-0.747495\pi\)
−0.701521 + 0.712649i \(0.747495\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 1.04960e10i − 0.0696742i
\(624\) 0 0
\(625\) −1.41018e11 −0.924177
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 1.27202e11i − 0.812630i
\(630\) 0 0
\(631\) 1.41366e11 0.891715 0.445858 0.895104i \(-0.352899\pi\)
0.445858 + 0.895104i \(0.352899\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 3.93095e11i − 2.41770i
\(636\) 0 0
\(637\) 1.71730e10 0.104301
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.65042e10i 0.0977605i 0.998805 + 0.0488802i \(0.0155653\pi\)
−0.998805 + 0.0488802i \(0.984435\pi\)
\(642\) 0 0
\(643\) −5.08388e10 −0.297407 −0.148704 0.988882i \(-0.547510\pi\)
−0.148704 + 0.988882i \(0.547510\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 2.01779e11i − 1.15149i −0.817630 0.575745i \(-0.804712\pi\)
0.817630 0.575745i \(-0.195288\pi\)
\(648\) 0 0
\(649\) −2.80191e11 −1.57934
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 4.45752e10i − 0.245155i −0.992459 0.122577i \(-0.960884\pi\)
0.992459 0.122577i \(-0.0391160\pi\)
\(654\) 0 0
\(655\) 3.59625e11 1.95382
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 2.28817e11i − 1.21324i −0.794993 0.606619i \(-0.792525\pi\)
0.794993 0.606619i \(-0.207475\pi\)
\(660\) 0 0
\(661\) 6.44329e10 0.337522 0.168761 0.985657i \(-0.446023\pi\)
0.168761 + 0.985657i \(0.446023\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 9.16481e10i 0.468637i
\(666\) 0 0
\(667\) −1.79946e11 −0.909157
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 2.22104e11i − 1.09564i
\(672\) 0 0
\(673\) 3.53716e11 1.72423 0.862114 0.506715i \(-0.169140\pi\)
0.862114 + 0.506715i \(0.169140\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 6.09182e10i − 0.289996i −0.989432 0.144998i \(-0.953682\pi\)
0.989432 0.144998i \(-0.0463176\pi\)
\(678\) 0 0
\(679\) −8.99523e10 −0.423188
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 6.41129e10i 0.294620i 0.989090 + 0.147310i \(0.0470615\pi\)
−0.989090 + 0.147310i \(0.952938\pi\)
\(684\) 0 0
\(685\) 4.05650e11 1.84242
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 5.22894e10i 0.232026i
\(690\) 0 0
\(691\) 1.88610e11 0.827280 0.413640 0.910441i \(-0.364257\pi\)
0.413640 + 0.910441i \(0.364257\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.93455e10i 0.254360i
\(696\) 0 0
\(697\) −1.84582e10 −0.0782094
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.73439e11i 0.718248i 0.933290 + 0.359124i \(0.116924\pi\)
−0.933290 + 0.359124i \(0.883076\pi\)
\(702\) 0 0
\(703\) 2.60099e11 1.06492
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1.03442e11i − 0.414018i
\(708\) 0 0
\(709\) −3.87397e11 −1.53310 −0.766551 0.642184i \(-0.778029\pi\)
−0.766551 + 0.642184i \(0.778029\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 3.45580e11i 1.33718i
\(714\) 0 0
\(715\) 4.11376e11 1.57404
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 3.88408e11i − 1.45336i −0.686976 0.726680i \(-0.741062\pi\)
0.686976 0.726680i \(-0.258938\pi\)
\(720\) 0 0
\(721\) −1.27238e11 −0.470844
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.06292e11i 0.746674i
\(726\) 0 0
\(727\) −5.15997e10 −0.184718 −0.0923591 0.995726i \(-0.529441\pi\)
−0.0923591 + 0.995726i \(0.529441\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 9.02263e9i − 0.0315983i
\(732\) 0 0
\(733\) 1.68072e11 0.582209 0.291105 0.956691i \(-0.405977\pi\)
0.291105 + 0.956691i \(0.405977\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.65329e11i 0.560376i
\(738\) 0 0
\(739\) 2.48752e11 0.834043 0.417021 0.908897i \(-0.363074\pi\)
0.417021 + 0.908897i \(0.363074\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 1.00329e11i − 0.329209i −0.986360 0.164605i \(-0.947365\pi\)
0.986360 0.164605i \(-0.0526348\pi\)
\(744\) 0 0
\(745\) 6.73512e11 2.18635
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 7.69068e10i − 0.244364i
\(750\) 0 0
\(751\) 3.60192e11 1.13233 0.566166 0.824291i \(-0.308426\pi\)
0.566166 + 0.824291i \(0.308426\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 7.81710e11i − 2.40579i
\(756\) 0 0
\(757\) 3.05727e11 0.931000 0.465500 0.885048i \(-0.345874\pi\)
0.465500 + 0.885048i \(0.345874\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 4.98951e11i − 1.48771i −0.668339 0.743857i \(-0.732995\pi\)
0.668339 0.743857i \(-0.267005\pi\)
\(762\) 0 0
\(763\) −5.59323e10 −0.165031
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.66371e11i 0.769670i
\(768\) 0 0
\(769\) −5.27348e11 −1.50797 −0.753983 0.656894i \(-0.771870\pi\)
−0.753983 + 0.656894i \(0.771870\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 7.10274e10i − 0.198933i −0.995041 0.0994667i \(-0.968286\pi\)
0.995041 0.0994667i \(-0.0317137\pi\)
\(774\) 0 0
\(775\) 3.96177e11 1.09820
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 3.77427e10i − 0.102491i
\(780\) 0 0
\(781\) −9.24426e11 −2.48467
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 3.08719e11i − 0.812989i
\(786\) 0 0
\(787\) −4.06243e10 −0.105898 −0.0529489 0.998597i \(-0.516862\pi\)
−0.0529489 + 0.998597i \(0.516862\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.35148e11i 0.345226i
\(792\) 0 0
\(793\) −2.11149e11 −0.533944
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 1.67239e10i − 0.0414482i −0.999785 0.0207241i \(-0.993403\pi\)
0.999785 0.0207241i \(-0.00659715\pi\)
\(798\) 0 0
\(799\) 6.67846e10 0.163866
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 7.62763e11i − 1.83454i
\(804\) 0 0
\(805\) −2.97800e11 −0.709154
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 5.87647e11i − 1.37190i −0.727648 0.685950i \(-0.759387\pi\)
0.727648 0.685950i \(-0.240613\pi\)
\(810\) 0 0
\(811\) −4.74855e11 −1.09768 −0.548842 0.835926i \(-0.684931\pi\)
−0.548842 + 0.835926i \(0.684931\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 9.30819e11i 2.10977i
\(816\) 0 0
\(817\) 1.84492e10 0.0414084
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 5.95199e11i − 1.31005i −0.755605 0.655027i \(-0.772657\pi\)
0.755605 0.655027i \(-0.227343\pi\)
\(822\) 0 0
\(823\) 6.88252e11 1.50020 0.750098 0.661326i \(-0.230006\pi\)
0.750098 + 0.661326i \(0.230006\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 1.12237e11i − 0.239946i −0.992777 0.119973i \(-0.961719\pi\)
0.992777 0.119973i \(-0.0382808\pi\)
\(828\) 0 0
\(829\) −4.65905e11 −0.986460 −0.493230 0.869899i \(-0.664184\pi\)
−0.493230 + 0.869899i \(0.664184\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 4.52244e10i 0.0939276i
\(834\) 0 0
\(835\) 7.69818e11 1.58359
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 8.26729e11i − 1.66846i −0.551418 0.834229i \(-0.685913\pi\)
0.551418 0.834229i \(-0.314087\pi\)
\(840\) 0 0
\(841\) 2.57013e11 0.513774
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 3.42579e11i 0.671945i
\(846\) 0 0
\(847\) −2.42087e11 −0.470368
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 8.45161e11i 1.61147i
\(852\) 0 0
\(853\) 4.95037e11 0.935064 0.467532 0.883976i \(-0.345143\pi\)
0.467532 + 0.883976i \(0.345143\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.44761e11i 0.639138i 0.947563 + 0.319569i \(0.103538\pi\)
−0.947563 + 0.319569i \(0.896462\pi\)
\(858\) 0 0
\(859\) 8.07526e11 1.48315 0.741573 0.670872i \(-0.234080\pi\)
0.741573 + 0.670872i \(0.234080\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 7.70866e11i 1.38975i 0.719132 + 0.694873i \(0.244539\pi\)
−0.719132 + 0.694873i \(0.755461\pi\)
\(864\) 0 0
\(865\) −1.08924e12 −1.94563
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 4.29767e10i 0.0753623i
\(870\) 0 0
\(871\) 1.57174e11 0.273092
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.25757e10i 0.0385131i
\(876\) 0 0
\(877\) −5.63983e11 −0.953384 −0.476692 0.879071i \(-0.658164\pi\)
−0.476692 + 0.879071i \(0.658164\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 7.69781e11i − 1.27780i −0.769289 0.638901i \(-0.779389\pi\)
0.769289 0.638901i \(-0.220611\pi\)
\(882\) 0 0
\(883\) −8.40219e11 −1.38213 −0.691066 0.722792i \(-0.742859\pi\)
−0.691066 + 0.722792i \(0.742859\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 1.09144e12i − 1.76322i −0.471983 0.881608i \(-0.656461\pi\)
0.471983 0.881608i \(-0.343539\pi\)
\(888\) 0 0
\(889\) 3.96635e11 0.635015
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.36559e11i 0.214740i
\(894\) 0 0
\(895\) 1.48913e11 0.232081
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 4.67120e11i 0.715137i
\(900\) 0 0
\(901\) −1.37702e11 −0.208949
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.68364e12i 2.50989i
\(906\) 0 0
\(907\) 1.73121e10 0.0255812 0.0127906 0.999918i \(-0.495929\pi\)
0.0127906 + 0.999918i \(0.495929\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 1.09440e12i − 1.58892i −0.607319 0.794458i \(-0.707755\pi\)
0.607319 0.794458i \(-0.292245\pi\)
\(912\) 0 0
\(913\) −4.41450e11 −0.635329
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3.62864e11i 0.513176i
\(918\) 0 0
\(919\) −1.09637e12 −1.53707 −0.768537 0.639805i \(-0.779015\pi\)
−0.768537 + 0.639805i \(0.779015\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 8.78828e11i 1.21087i
\(924\) 0 0
\(925\) 9.68903e11 1.32347
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 6.77872e11i − 0.910090i −0.890468 0.455045i \(-0.849623\pi\)
0.890468 0.455045i \(-0.150377\pi\)
\(930\) 0 0
\(931\) −9.24733e10 −0.123089
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.08334e12i 1.41749i
\(936\) 0 0
\(937\) 1.32341e12 1.71687 0.858434 0.512924i \(-0.171438\pi\)
0.858434 + 0.512924i \(0.171438\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 5.32278e11i 0.678859i 0.940631 + 0.339430i \(0.110234\pi\)
−0.940631 + 0.339430i \(0.889766\pi\)
\(942\) 0 0
\(943\) 1.22641e11 0.155091
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 3.55370e11i − 0.441856i −0.975290 0.220928i \(-0.929091\pi\)
0.975290 0.220928i \(-0.0709086\pi\)
\(948\) 0 0
\(949\) −7.25139e11 −0.894039
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 8.27352e11i 1.00304i 0.865146 + 0.501520i \(0.167226\pi\)
−0.865146 + 0.501520i \(0.832774\pi\)
\(954\) 0 0
\(955\) −4.98438e11 −0.599236
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 4.09303e11i 0.483917i
\(960\) 0 0
\(961\) 4.41948e10 0.0518177
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.38767e12i 2.75337i
\(966\) 0 0
\(967\) −5.09343e11 −0.582511 −0.291256 0.956645i \(-0.594073\pi\)
−0.291256 + 0.956645i \(0.594073\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.25774e12i 1.41486i 0.706784 + 0.707429i \(0.250145\pi\)
−0.706784 + 0.707429i \(0.749855\pi\)
\(972\) 0 0
\(973\) −5.98799e10 −0.0668082
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.03465e12i 1.13557i 0.823177 + 0.567785i \(0.192199\pi\)
−0.823177 + 0.567785i \(0.807801\pi\)
\(978\) 0 0
\(979\) 2.53694e11 0.276171
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 9.31373e11i − 0.997492i −0.866748 0.498746i \(-0.833794\pi\)
0.866748 0.498746i \(-0.166206\pi\)
\(984\) 0 0
\(985\) 2.12931e12 2.26201
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 5.99484e10i 0.0626603i
\(990\) 0 0
\(991\) −7.90752e11 −0.819871 −0.409936 0.912114i \(-0.634449\pi\)
−0.409936 + 0.912114i \(0.634449\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 2.02344e12i − 2.06442i
\(996\) 0 0
\(997\) −1.61160e12 −1.63109 −0.815543 0.578696i \(-0.803562\pi\)
−0.815543 + 0.578696i \(0.803562\pi\)
\(998\) 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.197.3 16
3.2 odd 2 inner 252.9.c.a.197.14 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.9.c.a.197.3 16 1.1 even 1 trivial
252.9.c.a.197.14 yes 16 3.2 odd 2 inner