Properties

Label 252.9.c.a.197.12
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 197.12
Root \(640.282i\) of defining polynomial
Character \(\chi\) \(=\) 252.197
Dual form 252.9.c.a.197.5

$q$-expansion

\(f(q)\) \(=\) \(q+640.282i q^{5} -907.493 q^{7} +O(q^{10})\) \(q+640.282i q^{5} -907.493 q^{7} +8116.74i q^{11} +47941.1 q^{13} -29809.0i q^{17} -7034.96 q^{19} -351363. i q^{23} -19336.0 q^{25} -387709. i q^{29} +63727.1 q^{31} -581051. i q^{35} +2.45976e6 q^{37} +3.12120e6i q^{41} +2.12121e6 q^{43} -2.22184e6i q^{47} +823543. q^{49} +1.76942e6i q^{53} -5.19700e6 q^{55} -1.12039e7i q^{59} -4.08608e6 q^{61} +3.06958e7i q^{65} -2.48165e7 q^{67} +1.73411e7i q^{71} -3.58546e6 q^{73} -7.36589e6i q^{77} +5.14806e7 q^{79} +4.80626e7i q^{83} +1.90862e7 q^{85} -2.69599e7i q^{89} -4.35062e7 q^{91} -4.50436e6i q^{95} +1.47587e8 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 95480q^{13} - 287560q^{19} - 1754520q^{25} - 3554264q^{31} - 182920q^{37} + 8472416q^{43} + 13176688q^{49} - 18692072q^{55} + 34224568q^{61} + 22683096q^{67} + 2137296q^{73} - 90245624q^{79} - 56204456q^{85} - 25661888q^{91} + 134041152q^{97} + O(q^{100}) \)

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
\(5\) 640.282i 1.02445i 0.858851 + 0.512226i \(0.171179\pi\)
−0.858851 + 0.512226i \(0.828821\pi\)
\(6\) 0 0
\(7\) −907.493 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 8116.74i 0.554385i 0.960814 + 0.277192i \(0.0894039\pi\)
−0.960814 + 0.277192i \(0.910596\pi\)
\(12\) 0 0
\(13\) 47941.1 1.67855 0.839275 0.543707i \(-0.182980\pi\)
0.839275 + 0.543707i \(0.182980\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 29809.0i − 0.356904i −0.983949 0.178452i \(-0.942891\pi\)
0.983949 0.178452i \(-0.0571090\pi\)
\(18\) 0 0
\(19\) −7034.96 −0.0539818 −0.0269909 0.999636i \(-0.508593\pi\)
−0.0269909 + 0.999636i \(0.508593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 351363.i − 1.25558i −0.778382 0.627791i \(-0.783959\pi\)
0.778382 0.627791i \(-0.216041\pi\)
\(24\) 0 0
\(25\) −19336.0 −0.0495001
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 387709.i − 0.548168i −0.961706 0.274084i \(-0.911625\pi\)
0.961706 0.274084i \(-0.0883746\pi\)
\(30\) 0 0
\(31\) 63727.1 0.0690045 0.0345022 0.999405i \(-0.489015\pi\)
0.0345022 + 0.999405i \(0.489015\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 581051.i − 0.387206i
\(36\) 0 0
\(37\) 2.45976e6 1.31246 0.656230 0.754561i \(-0.272150\pi\)
0.656230 + 0.754561i \(0.272150\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.12120e6i 1.10455i 0.833662 + 0.552276i \(0.186240\pi\)
−0.833662 + 0.552276i \(0.813760\pi\)
\(42\) 0 0
\(43\) 2.12121e6 0.620454 0.310227 0.950663i \(-0.399595\pi\)
0.310227 + 0.950663i \(0.399595\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 2.22184e6i − 0.455325i −0.973740 0.227663i \(-0.926892\pi\)
0.973740 0.227663i \(-0.0731083\pi\)
\(48\) 0 0
\(49\) 823543. 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 1.76942e6i 0.224248i 0.993694 + 0.112124i \(0.0357653\pi\)
−0.993694 + 0.112124i \(0.964235\pi\)
\(54\) 0 0
\(55\) −5.19700e6 −0.567940
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 1.12039e7i − 0.924617i −0.886719 0.462308i \(-0.847021\pi\)
0.886719 0.462308i \(-0.152979\pi\)
\(60\) 0 0
\(61\) −4.08608e6 −0.295112 −0.147556 0.989054i \(-0.547141\pi\)
−0.147556 + 0.989054i \(0.547141\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.06958e7i 1.71959i
\(66\) 0 0
\(67\) −2.48165e7 −1.23152 −0.615759 0.787935i \(-0.711150\pi\)
−0.615759 + 0.787935i \(0.711150\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.73411e7i 0.682406i 0.939990 + 0.341203i \(0.110834\pi\)
−0.939990 + 0.341203i \(0.889166\pi\)
\(72\) 0 0
\(73\) −3.58546e6 −0.126257 −0.0631283 0.998005i \(-0.520108\pi\)
−0.0631283 + 0.998005i \(0.520108\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 7.36589e6i − 0.209538i
\(78\) 0 0
\(79\) 5.14806e7 1.32171 0.660853 0.750515i \(-0.270195\pi\)
0.660853 + 0.750515i \(0.270195\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.80626e7i 1.01273i 0.862319 + 0.506366i \(0.169012\pi\)
−0.862319 + 0.506366i \(0.830988\pi\)
\(84\) 0 0
\(85\) 1.90862e7 0.365631
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 2.69599e7i − 0.429693i −0.976648 0.214847i \(-0.931075\pi\)
0.976648 0.214847i \(-0.0689252\pi\)
\(90\) 0 0
\(91\) −4.35062e7 −0.634433
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 4.50436e6i − 0.0553017i
\(96\) 0 0
\(97\) 1.47587e8 1.66709 0.833547 0.552449i \(-0.186307\pi\)
0.833547 + 0.552449i \(0.186307\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 1.04528e8i − 1.00450i −0.864723 0.502249i \(-0.832506\pi\)
0.864723 0.502249i \(-0.167494\pi\)
\(102\) 0 0
\(103\) −4.86644e7 −0.432377 −0.216188 0.976352i \(-0.569363\pi\)
−0.216188 + 0.976352i \(0.569363\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 2.18081e8i 1.66373i 0.554979 + 0.831865i \(0.312726\pi\)
−0.554979 + 0.831865i \(0.687274\pi\)
\(108\) 0 0
\(109\) 1.04183e8 0.738061 0.369031 0.929417i \(-0.379690\pi\)
0.369031 + 0.929417i \(0.379690\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.22264e8i 0.749867i 0.927052 + 0.374934i \(0.122334\pi\)
−0.927052 + 0.374934i \(0.877666\pi\)
\(114\) 0 0
\(115\) 2.24972e8 1.28628
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.70515e7i 0.134897i
\(120\) 0 0
\(121\) 1.48477e8 0.692658
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 2.37730e8i 0.973741i
\(126\) 0 0
\(127\) 1.59142e8 0.611746 0.305873 0.952072i \(-0.401052\pi\)
0.305873 + 0.952072i \(0.401052\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.75418e8i 0.935206i 0.883939 + 0.467603i \(0.154882\pi\)
−0.883939 + 0.467603i \(0.845118\pi\)
\(132\) 0 0
\(133\) 6.38417e6 0.0204032
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.92582e8i 0.546682i 0.961917 + 0.273341i \(0.0881287\pi\)
−0.961917 + 0.273341i \(0.911871\pi\)
\(138\) 0 0
\(139\) −5.26604e8 −1.41067 −0.705334 0.708875i \(-0.749203\pi\)
−0.705334 + 0.708875i \(0.749203\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.89126e8i 0.930563i
\(144\) 0 0
\(145\) 2.48243e8 0.561571
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 2.53599e7i − 0.0514519i −0.999669 0.0257260i \(-0.991810\pi\)
0.999669 0.0257260i \(-0.00818973\pi\)
\(150\) 0 0
\(151\) 2.06642e8 0.397475 0.198738 0.980053i \(-0.436316\pi\)
0.198738 + 0.980053i \(0.436316\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 4.08033e7i 0.0706917i
\(156\) 0 0
\(157\) 2.42458e8 0.399060 0.199530 0.979892i \(-0.436059\pi\)
0.199530 + 0.979892i \(0.436059\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.18860e8i 0.474566i
\(162\) 0 0
\(163\) −2.62352e8 −0.371650 −0.185825 0.982583i \(-0.559496\pi\)
−0.185825 + 0.982583i \(0.559496\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.62142e8i 0.979873i 0.871758 + 0.489937i \(0.162980\pi\)
−0.871758 + 0.489937i \(0.837020\pi\)
\(168\) 0 0
\(169\) 1.48262e9 1.81753
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.60297e9i 1.78954i 0.446529 + 0.894769i \(0.352660\pi\)
−0.446529 + 0.894769i \(0.647340\pi\)
\(174\) 0 0
\(175\) 1.75472e7 0.0187093
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.28649e9i 1.25312i 0.779373 + 0.626560i \(0.215538\pi\)
−0.779373 + 0.626560i \(0.784462\pi\)
\(180\) 0 0
\(181\) 2.94339e8 0.274242 0.137121 0.990554i \(-0.456215\pi\)
0.137121 + 0.990554i \(0.456215\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 1.57494e9i 1.34455i
\(186\) 0 0
\(187\) 2.41952e8 0.197862
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 6.83713e8i − 0.513736i −0.966446 0.256868i \(-0.917309\pi\)
0.966446 0.256868i \(-0.0826906\pi\)
\(192\) 0 0
\(193\) 1.74277e9 1.25606 0.628032 0.778187i \(-0.283861\pi\)
0.628032 + 0.778187i \(0.283861\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 2.35657e9i − 1.56464i −0.622876 0.782321i \(-0.714036\pi\)
0.622876 0.782321i \(-0.285964\pi\)
\(198\) 0 0
\(199\) 2.71407e9 1.73065 0.865324 0.501212i \(-0.167112\pi\)
0.865324 + 0.501212i \(0.167112\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 3.51843e8i 0.207188i
\(204\) 0 0
\(205\) −1.99845e9 −1.13156
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 5.71009e7i − 0.0299267i
\(210\) 0 0
\(211\) 2.44112e9 1.23157 0.615786 0.787913i \(-0.288839\pi\)
0.615786 + 0.787913i \(0.288839\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.35817e9i 0.635625i
\(216\) 0 0
\(217\) −5.78319e7 −0.0260812
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 1.42908e9i − 0.599082i
\(222\) 0 0
\(223\) −3.86582e8 −0.156323 −0.0781614 0.996941i \(-0.524905\pi\)
−0.0781614 + 0.996941i \(0.524905\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 1.81490e9i − 0.683516i −0.939788 0.341758i \(-0.888978\pi\)
0.939788 0.341758i \(-0.111022\pi\)
\(228\) 0 0
\(229\) −2.65732e9 −0.966276 −0.483138 0.875544i \(-0.660503\pi\)
−0.483138 + 0.875544i \(0.660503\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.24408e9i 1.77929i 0.456657 + 0.889643i \(0.349047\pi\)
−0.456657 + 0.889643i \(0.650953\pi\)
\(234\) 0 0
\(235\) 1.42261e9 0.466458
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.39464e9i 0.733919i 0.930237 + 0.366959i \(0.119601\pi\)
−0.930237 + 0.366959i \(0.880399\pi\)
\(240\) 0 0
\(241\) −2.18983e9 −0.649145 −0.324572 0.945861i \(-0.605220\pi\)
−0.324572 + 0.945861i \(0.605220\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 5.27300e8i 0.146350i
\(246\) 0 0
\(247\) −3.37263e8 −0.0906111
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.22517e8i 0.0308676i 0.999881 + 0.0154338i \(0.00491292\pi\)
−0.999881 + 0.0154338i \(0.995087\pi\)
\(252\) 0 0
\(253\) 2.85193e9 0.696075
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.54143e9i 0.811795i 0.913919 + 0.405897i \(0.133041\pi\)
−0.913919 + 0.405897i \(0.866959\pi\)
\(258\) 0 0
\(259\) −2.23221e9 −0.496063
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 3.24479e9i − 0.678209i −0.940749 0.339105i \(-0.889876\pi\)
0.940749 0.339105i \(-0.110124\pi\)
\(264\) 0 0
\(265\) −1.13293e9 −0.229731
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 4.16580e9i − 0.795590i −0.917474 0.397795i \(-0.869776\pi\)
0.917474 0.397795i \(-0.130224\pi\)
\(270\) 0 0
\(271\) −7.78226e9 −1.44287 −0.721437 0.692480i \(-0.756518\pi\)
−0.721437 + 0.692480i \(0.756518\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 1.56945e8i − 0.0274421i
\(276\) 0 0
\(277\) −6.42407e9 −1.09117 −0.545583 0.838057i \(-0.683692\pi\)
−0.545583 + 0.838057i \(0.683692\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 1.04039e10i − 1.66867i −0.551261 0.834333i \(-0.685853\pi\)
0.551261 0.834333i \(-0.314147\pi\)
\(282\) 0 0
\(283\) −7.82909e8 −0.122058 −0.0610289 0.998136i \(-0.519438\pi\)
−0.0610289 + 0.998136i \(0.519438\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 2.83246e9i − 0.417481i
\(288\) 0 0
\(289\) 6.08718e9 0.872619
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.18853e10i 1.61264i 0.591477 + 0.806322i \(0.298545\pi\)
−0.591477 + 0.806322i \(0.701455\pi\)
\(294\) 0 0
\(295\) 7.17366e9 0.947225
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 1.68447e10i − 2.10756i
\(300\) 0 0
\(301\) −1.92498e9 −0.234510
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 2.61624e9i − 0.302328i
\(306\) 0 0
\(307\) −1.09190e10 −1.22922 −0.614611 0.788831i \(-0.710687\pi\)
−0.614611 + 0.788831i \(0.710687\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 1.01076e10i − 1.08046i −0.841519 0.540228i \(-0.818338\pi\)
0.841519 0.540228i \(-0.181662\pi\)
\(312\) 0 0
\(313\) 7.14089e9 0.744003 0.372002 0.928232i \(-0.378672\pi\)
0.372002 + 0.928232i \(0.378672\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 2.17626e9i − 0.215514i −0.994177 0.107757i \(-0.965633\pi\)
0.994177 0.107757i \(-0.0343668\pi\)
\(318\) 0 0
\(319\) 3.14693e9 0.303896
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.09705e8i 0.0192663i
\(324\) 0 0
\(325\) −9.26987e8 −0.0830884
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.01631e9i 0.172097i
\(330\) 0 0
\(331\) −8.58963e9 −0.715587 −0.357794 0.933801i \(-0.616471\pi\)
−0.357794 + 0.933801i \(0.616471\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 1.58895e10i − 1.26163i
\(336\) 0 0
\(337\) −1.44960e10 −1.12390 −0.561952 0.827170i \(-0.689949\pi\)
−0.561952 + 0.827170i \(0.689949\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.17257e8i 0.0382550i
\(342\) 0 0
\(343\) −7.47359e8 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 1.48761e10i − 1.02605i −0.858373 0.513027i \(-0.828524\pi\)
0.858373 0.513027i \(-0.171476\pi\)
\(348\) 0 0
\(349\) 2.99091e9 0.201605 0.100803 0.994906i \(-0.467859\pi\)
0.100803 + 0.994906i \(0.467859\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 1.22457e10i − 0.788653i −0.918970 0.394327i \(-0.870978\pi\)
0.918970 0.394327i \(-0.129022\pi\)
\(354\) 0 0
\(355\) −1.11032e10 −0.699092
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 1.69094e10i − 1.01800i −0.860765 0.509002i \(-0.830015\pi\)
0.860765 0.509002i \(-0.169985\pi\)
\(360\) 0 0
\(361\) −1.69341e10 −0.997086
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 2.29571e9i − 0.129344i
\(366\) 0 0
\(367\) 1.25574e9 0.0692204 0.0346102 0.999401i \(-0.488981\pi\)
0.0346102 + 0.999401i \(0.488981\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 1.60574e9i − 0.0847576i
\(372\) 0 0
\(373\) −7.27627e9 −0.375901 −0.187950 0.982179i \(-0.560184\pi\)
−0.187950 + 0.982179i \(0.560184\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 1.85872e10i − 0.920127i
\(378\) 0 0
\(379\) −1.54709e10 −0.749824 −0.374912 0.927060i \(-0.622327\pi\)
−0.374912 + 0.927060i \(0.622327\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.53417e10i 1.17772i 0.808237 + 0.588858i \(0.200422\pi\)
−0.808237 + 0.588858i \(0.799578\pi\)
\(384\) 0 0
\(385\) 4.71624e9 0.214661
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.80420e10i 0.787929i 0.919126 + 0.393964i \(0.128897\pi\)
−0.919126 + 0.393964i \(0.871103\pi\)
\(390\) 0 0
\(391\) −1.04738e10 −0.448123
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3.29621e10i 1.35402i
\(396\) 0 0
\(397\) 2.15193e10 0.866296 0.433148 0.901323i \(-0.357403\pi\)
0.433148 + 0.901323i \(0.357403\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 2.66386e10i − 1.03023i −0.857121 0.515115i \(-0.827749\pi\)
0.857121 0.515115i \(-0.172251\pi\)
\(402\) 0 0
\(403\) 3.05515e9 0.115828
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.99652e10i 0.727607i
\(408\) 0 0
\(409\) 3.17363e8 0.0113413 0.00567066 0.999984i \(-0.498195\pi\)
0.00567066 + 0.999984i \(0.498195\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.01675e10i 0.349472i
\(414\) 0 0
\(415\) −3.07736e10 −1.03749
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 2.22236e10i − 0.721039i −0.932752 0.360519i \(-0.882599\pi\)
0.932752 0.360519i \(-0.117401\pi\)
\(420\) 0 0
\(421\) −2.61883e10 −0.833642 −0.416821 0.908989i \(-0.636856\pi\)
−0.416821 + 0.908989i \(0.636856\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 5.76386e8i 0.0176668i
\(426\) 0 0
\(427\) 3.70808e9 0.111542
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 9.43747e8i 0.0273493i 0.999906 + 0.0136747i \(0.00435292\pi\)
−0.999906 + 0.0136747i \(0.995647\pi\)
\(432\) 0 0
\(433\) 5.68198e10 1.61640 0.808198 0.588910i \(-0.200443\pi\)
0.808198 + 0.588910i \(0.200443\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.47183e9i 0.0677785i
\(438\) 0 0
\(439\) 7.47760e9 0.201328 0.100664 0.994920i \(-0.467903\pi\)
0.100664 + 0.994920i \(0.467903\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 3.70177e10i − 0.961157i −0.876952 0.480579i \(-0.840427\pi\)
0.876952 0.480579i \(-0.159573\pi\)
\(444\) 0 0
\(445\) 1.72620e10 0.440200
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 1.00631e9i − 0.0247597i −0.999923 0.0123798i \(-0.996059\pi\)
0.999923 0.0123798i \(-0.00394072\pi\)
\(450\) 0 0
\(451\) −2.53340e10 −0.612346
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 2.78562e10i − 0.649945i
\(456\) 0 0
\(457\) 2.29431e10 0.526003 0.263001 0.964795i \(-0.415288\pi\)
0.263001 + 0.964795i \(0.415288\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.78011e10i 0.836953i 0.908228 + 0.418477i \(0.137436\pi\)
−0.908228 + 0.418477i \(0.862564\pi\)
\(462\) 0 0
\(463\) −1.70770e9 −0.0371610 −0.0185805 0.999827i \(-0.505915\pi\)
−0.0185805 + 0.999827i \(0.505915\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 4.86432e10i − 1.02271i −0.859368 0.511357i \(-0.829143\pi\)
0.859368 0.511357i \(-0.170857\pi\)
\(468\) 0 0
\(469\) 2.25208e10 0.465470
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.72173e10i 0.343970i
\(474\) 0 0
\(475\) 1.36028e8 0.00267210
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.55978e10i 0.676210i 0.941108 + 0.338105i \(0.109786\pi\)
−0.941108 + 0.338105i \(0.890214\pi\)
\(480\) 0 0
\(481\) 1.17924e11 2.20303
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.44970e10i 1.70786i
\(486\) 0 0
\(487\) −2.70179e10 −0.480325 −0.240162 0.970733i \(-0.577201\pi\)
−0.240162 + 0.970733i \(0.577201\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 6.67674e10i 1.14878i 0.818580 + 0.574392i \(0.194762\pi\)
−0.818580 + 0.574392i \(0.805238\pi\)
\(492\) 0 0
\(493\) −1.15572e10 −0.195643
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 1.57369e10i − 0.257925i
\(498\) 0 0
\(499\) 9.30718e10 1.50112 0.750561 0.660802i \(-0.229784\pi\)
0.750561 + 0.660802i \(0.229784\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 1.79146e10i 0.279856i 0.990162 + 0.139928i \(0.0446871\pi\)
−0.990162 + 0.139928i \(0.955313\pi\)
\(504\) 0 0
\(505\) 6.69277e10 1.02906
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1.00413e11i 1.49596i 0.663724 + 0.747978i \(0.268975\pi\)
−0.663724 + 0.747978i \(0.731025\pi\)
\(510\) 0 0
\(511\) 3.25378e9 0.0477205
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 3.11589e10i − 0.442949i
\(516\) 0 0
\(517\) 1.80341e10 0.252425
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3.66810e10i 0.497841i 0.968524 + 0.248921i \(0.0800758\pi\)
−0.968524 + 0.248921i \(0.919924\pi\)
\(522\) 0 0
\(523\) −2.96773e9 −0.0396659 −0.0198329 0.999803i \(-0.506313\pi\)
−0.0198329 + 0.999803i \(0.506313\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 1.89964e9i − 0.0246280i
\(528\) 0 0
\(529\) −4.51453e10 −0.576487
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.49634e11i 1.85405i
\(534\) 0 0
\(535\) −1.39633e11 −1.70441
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 6.68449e9i 0.0791978i
\(540\) 0 0
\(541\) 4.71697e10 0.550648 0.275324 0.961351i \(-0.411215\pi\)
0.275324 + 0.961351i \(0.411215\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 6.67067e10i 0.756107i
\(546\) 0 0
\(547\) −7.66653e10 −0.856347 −0.428173 0.903697i \(-0.640843\pi\)
−0.428173 + 0.903697i \(0.640843\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 2.72751e9i 0.0295911i
\(552\) 0 0
\(553\) −4.67182e10 −0.499558
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.34358e10i 0.243478i 0.992562 + 0.121739i \(0.0388471\pi\)
−0.992562 + 0.121739i \(0.961153\pi\)
\(558\) 0 0
\(559\) 1.01693e11 1.04146
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 1.18235e11i − 1.17682i −0.808561 0.588412i \(-0.799753\pi\)
0.808561 0.588412i \(-0.200247\pi\)
\(564\) 0 0
\(565\) −7.82833e10 −0.768202
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.28655e11i 1.22738i 0.789549 + 0.613688i \(0.210315\pi\)
−0.789549 + 0.613688i \(0.789685\pi\)
\(570\) 0 0
\(571\) 1.13973e11 1.07216 0.536078 0.844168i \(-0.319905\pi\)
0.536078 + 0.844168i \(0.319905\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6.79395e9i 0.0621514i
\(576\) 0 0
\(577\) 1.31966e11 1.19058 0.595291 0.803510i \(-0.297037\pi\)
0.595291 + 0.803510i \(0.297037\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 4.36164e10i − 0.382777i
\(582\) 0 0
\(583\) −1.43619e10 −0.124319
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.91244e10i 0.582209i 0.956691 + 0.291105i \(0.0940228\pi\)
−0.956691 + 0.291105i \(0.905977\pi\)
\(588\) 0 0
\(589\) −4.48317e8 −0.00372498
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5.96912e10i 0.482716i 0.970436 + 0.241358i \(0.0775928\pi\)
−0.970436 + 0.241358i \(0.922407\pi\)
\(594\) 0 0
\(595\) −1.73206e10 −0.138196
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 1.24032e11i − 0.963443i −0.876324 0.481721i \(-0.840012\pi\)
0.876324 0.481721i \(-0.159988\pi\)
\(600\) 0 0
\(601\) 1.51053e11 1.15779 0.578896 0.815401i \(-0.303484\pi\)
0.578896 + 0.815401i \(0.303484\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 9.50674e10i 0.709594i
\(606\) 0 0
\(607\) 1.77767e11 1.30947 0.654735 0.755859i \(-0.272780\pi\)
0.654735 + 0.755859i \(0.272780\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 1.06518e11i − 0.764286i
\(612\) 0 0
\(613\) −9.29216e10 −0.658074 −0.329037 0.944317i \(-0.606724\pi\)
−0.329037 + 0.944317i \(0.606724\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.33802e10i 0.161327i 0.996741 + 0.0806637i \(0.0257040\pi\)
−0.996741 + 0.0806637i \(0.974296\pi\)
\(618\) 0 0
\(619\) 1.29256e11 0.880418 0.440209 0.897895i \(-0.354904\pi\)
0.440209 + 0.897895i \(0.354904\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 2.44659e10i 0.162409i
\(624\) 0 0
\(625\) −1.59767e11 −1.04705
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 7.33230e10i − 0.468422i
\(630\) 0 0
\(631\) 4.31032e10 0.271889 0.135945 0.990716i \(-0.456593\pi\)
0.135945 + 0.990716i \(0.456593\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 1.01896e11i 0.626703i
\(636\) 0 0
\(637\) 3.94815e10 0.239793
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.47970e11i 0.876478i 0.898858 + 0.438239i \(0.144398\pi\)
−0.898858 + 0.438239i \(0.855602\pi\)
\(642\) 0 0
\(643\) −2.78103e11 −1.62690 −0.813451 0.581634i \(-0.802414\pi\)
−0.813451 + 0.581634i \(0.802414\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 2.85426e11i − 1.62883i −0.580282 0.814416i \(-0.697058\pi\)
0.580282 0.814416i \(-0.302942\pi\)
\(648\) 0 0
\(649\) 9.09393e10 0.512593
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 2.60195e11i − 1.43102i −0.698600 0.715512i \(-0.746193\pi\)
0.698600 0.715512i \(-0.253807\pi\)
\(654\) 0 0
\(655\) −1.76345e11 −0.958073
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 2.08589e11i − 1.10599i −0.833185 0.552994i \(-0.813485\pi\)
0.833185 0.552994i \(-0.186515\pi\)
\(660\) 0 0
\(661\) −2.70845e10 −0.141878 −0.0709390 0.997481i \(-0.522600\pi\)
−0.0709390 + 0.997481i \(0.522600\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.08767e9i 0.0209021i
\(666\) 0 0
\(667\) −1.36227e11 −0.688270
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 3.31656e10i − 0.163606i
\(672\) 0 0
\(673\) 4.08197e11 1.98980 0.994900 0.100862i \(-0.0321601\pi\)
0.994900 + 0.100862i \(0.0321601\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2.84230e11i 1.35305i 0.736418 + 0.676527i \(0.236516\pi\)
−0.736418 + 0.676527i \(0.763484\pi\)
\(678\) 0 0
\(679\) −1.33934e11 −0.630102
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 3.10241e11i − 1.42566i −0.701335 0.712831i \(-0.747412\pi\)
0.701335 0.712831i \(-0.252588\pi\)
\(684\) 0 0
\(685\) −1.23307e11 −0.560049
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 8.48280e10i 0.376411i
\(690\) 0 0
\(691\) 2.06970e10 0.0907810 0.0453905 0.998969i \(-0.485547\pi\)
0.0453905 + 0.998969i \(0.485547\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 3.37175e11i − 1.44516i
\(696\) 0 0
\(697\) 9.30398e10 0.394219
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1.29778e11i 0.537440i 0.963218 + 0.268720i \(0.0866007\pi\)
−0.963218 + 0.268720i \(0.913399\pi\)
\(702\) 0 0
\(703\) −1.73043e10 −0.0708489
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.48588e10i 0.379664i
\(708\) 0 0
\(709\) −2.37263e11 −0.938956 −0.469478 0.882944i \(-0.655558\pi\)
−0.469478 + 0.882944i \(0.655558\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 2.23914e10i − 0.0866408i
\(714\) 0 0
\(715\) −2.49150e11 −0.953316
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 3.40426e11i 1.27382i 0.770940 + 0.636908i \(0.219787\pi\)
−0.770940 + 0.636908i \(0.780213\pi\)
\(720\) 0 0
\(721\) 4.41626e10 0.163423
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 7.49672e9i 0.0271343i
\(726\) 0 0
\(727\) −3.88892e11 −1.39217 −0.696084 0.717960i \(-0.745076\pi\)
−0.696084 + 0.717960i \(0.745076\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 6.32311e10i − 0.221443i
\(732\) 0 0
\(733\) −1.53399e11 −0.531383 −0.265691 0.964058i \(-0.585600\pi\)
−0.265691 + 0.964058i \(0.585600\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 2.01429e11i − 0.682734i
\(738\) 0 0
\(739\) −3.54937e11 −1.19007 −0.595036 0.803699i \(-0.702862\pi\)
−0.595036 + 0.803699i \(0.702862\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 6.04592e11i − 1.98384i −0.126858 0.991921i \(-0.540489\pi\)
0.126858 0.991921i \(-0.459511\pi\)
\(744\) 0 0
\(745\) 1.62375e10 0.0527100
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 1.97907e11i − 0.628831i
\(750\) 0 0
\(751\) −2.76854e11 −0.870345 −0.435172 0.900347i \(-0.643313\pi\)
−0.435172 + 0.900347i \(0.643313\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1.32309e11i 0.407194i
\(756\) 0 0
\(757\) 5.28734e11 1.61010 0.805052 0.593204i \(-0.202137\pi\)
0.805052 + 0.593204i \(0.202137\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 5.83174e10i 0.173884i 0.996213 + 0.0869419i \(0.0277095\pi\)
−0.996213 + 0.0869419i \(0.972291\pi\)
\(762\) 0 0
\(763\) −9.45456e10 −0.278961
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 5.37128e11i − 1.55202i
\(768\) 0 0
\(769\) −1.31873e11 −0.377095 −0.188547 0.982064i \(-0.560378\pi\)
−0.188547 + 0.982064i \(0.560378\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 4.91940e11i − 1.37783i −0.724844 0.688913i \(-0.758088\pi\)
0.724844 0.688913i \(-0.241912\pi\)
\(774\) 0 0
\(775\) −1.23222e9 −0.00341573
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 2.19575e10i − 0.0596256i
\(780\) 0 0
\(781\) −1.40753e11 −0.378316
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.55241e11i 0.408817i
\(786\) 0 0
\(787\) −6.56249e11 −1.71068 −0.855342 0.518065i \(-0.826653\pi\)
−0.855342 + 0.518065i \(0.826653\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 1.10954e11i − 0.283423i
\(792\) 0 0
\(793\) −1.95891e11 −0.495361
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.56799e11i 0.388607i 0.980941 + 0.194303i \(0.0622446\pi\)
−0.980941 + 0.194303i \(0.937755\pi\)
\(798\) 0 0
\(799\) −6.62309e10 −0.162508
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 2.91023e10i − 0.0699947i
\(804\) 0 0
\(805\) −2.04160e11 −0.486169
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 4.63793e11i − 1.08275i −0.840780 0.541377i \(-0.817903\pi\)
0.840780 0.541377i \(-0.182097\pi\)
\(810\) 0 0
\(811\) 7.29900e11 1.68725 0.843626 0.536931i \(-0.180417\pi\)
0.843626 + 0.536931i \(0.180417\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 1.67979e11i − 0.380737i
\(816\) 0 0
\(817\) −1.49226e10 −0.0334932
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 1.81580e10i − 0.0399664i −0.999800 0.0199832i \(-0.993639\pi\)
0.999800 0.0199832i \(-0.00636127\pi\)
\(822\) 0 0
\(823\) 9.12249e11 1.98845 0.994224 0.107329i \(-0.0342297\pi\)
0.994224 + 0.107329i \(0.0342297\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.47368e11i 0.742623i 0.928508 + 0.371311i \(0.121092\pi\)
−0.928508 + 0.371311i \(0.878908\pi\)
\(828\) 0 0
\(829\) 6.72798e11 1.42451 0.712257 0.701919i \(-0.247673\pi\)
0.712257 + 0.701919i \(0.247673\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 2.45490e10i − 0.0509863i
\(834\) 0 0
\(835\) −4.87986e11 −1.00383
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.22680e11i 0.247587i 0.992308 + 0.123793i \(0.0395060\pi\)
−0.992308 + 0.123793i \(0.960494\pi\)
\(840\) 0 0
\(841\) 3.49928e11 0.699512
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 9.49293e11i 1.86197i
\(846\) 0 0
\(847\) −1.34742e11 −0.261800
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 8.64270e11i − 1.64790i
\(852\) 0 0
\(853\) 5.53404e11 1.04531 0.522656 0.852544i \(-0.324941\pi\)
0.522656 + 0.852544i \(0.324941\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 6.47734e11i − 1.20081i −0.799697 0.600403i \(-0.795007\pi\)
0.799697 0.600403i \(-0.204993\pi\)
\(858\) 0 0
\(859\) 7.66128e11 1.40711 0.703556 0.710640i \(-0.251594\pi\)
0.703556 + 0.710640i \(0.251594\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 4.95413e11i 0.893150i 0.894746 + 0.446575i \(0.147356\pi\)
−0.894746 + 0.446575i \(0.852644\pi\)
\(864\) 0 0
\(865\) −1.02635e12 −1.83329
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 4.17855e11i 0.732733i
\(870\) 0 0
\(871\) −1.18973e12 −2.06716
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 2.15738e11i − 0.368039i
\(876\) 0 0
\(877\) 8.33732e11 1.40938 0.704690 0.709516i \(-0.251086\pi\)
0.704690 + 0.709516i \(0.251086\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 2.38563e11i − 0.396005i −0.980202 0.198002i \(-0.936555\pi\)
0.980202 0.198002i \(-0.0634453\pi\)
\(882\) 0 0
\(883\) 6.91704e11 1.13783 0.568915 0.822396i \(-0.307363\pi\)
0.568915 + 0.822396i \(0.307363\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.58558e11i 0.256150i 0.991764 + 0.128075i \(0.0408798\pi\)
−0.991764 + 0.128075i \(0.959120\pi\)
\(888\) 0 0
\(889\) −1.44421e11 −0.231218
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.56306e10i 0.0245793i
\(894\) 0 0
\(895\) −8.23714e11 −1.28376
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 2.47075e10i − 0.0378260i
\(900\) 0 0
\(901\) 5.27447e10 0.0800349
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.88460e11i 0.280947i
\(906\) 0 0
\(907\) −1.20962e12 −1.78739 −0.893693 0.448678i \(-0.851895\pi\)
−0.893693 + 0.448678i \(0.851895\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 4.19638e11i 0.609258i 0.952471 + 0.304629i \(0.0985325\pi\)
−0.952471 + 0.304629i \(0.901467\pi\)
\(912\) 0 0
\(913\) −3.90112e11 −0.561443
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 2.49940e11i − 0.353475i
\(918\) 0 0
\(919\) 2.65442e11 0.372140 0.186070 0.982536i \(-0.440425\pi\)
0.186070 + 0.982536i \(0.440425\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 8.31351e11i 1.14545i
\(924\) 0 0
\(925\) −4.75618e10 −0.0649668
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.02149e12i 1.37143i 0.727872 + 0.685713i \(0.240510\pi\)
−0.727872 + 0.685713i \(0.759490\pi\)
\(930\) 0 0
\(931\) −5.79359e9 −0.00771168
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.54918e11i 0.202700i
\(936\) 0 0
\(937\) 9.49784e11 1.23216 0.616079 0.787684i \(-0.288720\pi\)
0.616079 + 0.787684i \(0.288720\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 5.52549e11i 0.704713i 0.935866 + 0.352356i \(0.114620\pi\)
−0.935866 + 0.352356i \(0.885380\pi\)
\(942\) 0 0
\(943\) 1.09667e12 1.38686
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 1.31274e12i − 1.63223i −0.577893 0.816113i \(-0.696125\pi\)
0.577893 0.816113i \(-0.303875\pi\)
\(948\) 0 0
\(949\) −1.71891e11 −0.211928
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 9.34394e11i 1.13281i 0.824126 + 0.566407i \(0.191667\pi\)
−0.824126 + 0.566407i \(0.808333\pi\)
\(954\) 0 0
\(955\) 4.37769e11 0.526298
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 1.74767e11i − 0.206626i
\(960\) 0 0
\(961\) −8.48830e11 −0.995238
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.11587e12i 1.28678i
\(966\) 0 0
\(967\) −4.78441e11 −0.547170 −0.273585 0.961848i \(-0.588209\pi\)
−0.273585 + 0.961848i \(0.588209\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 8.53487e11i 0.960108i 0.877239 + 0.480054i \(0.159383\pi\)
−0.877239 + 0.480054i \(0.840617\pi\)
\(972\) 0 0
\(973\) 4.77889e11 0.533182
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.25702e12i 1.37964i 0.723983 + 0.689818i \(0.242309\pi\)
−0.723983 + 0.689818i \(0.757691\pi\)
\(978\) 0 0
\(979\) 2.18827e11 0.238215
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 1.32959e10i − 0.0142398i −0.999975 0.00711989i \(-0.997734\pi\)
0.999975 0.00711989i \(-0.00226635\pi\)
\(984\) 0 0
\(985\) 1.50887e12 1.60290
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 7.45315e11i − 0.779031i
\(990\) 0 0
\(991\) 1.32471e12 1.37349 0.686746 0.726898i \(-0.259038\pi\)
0.686746 + 0.726898i \(0.259038\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.73777e12i 1.77296i
\(996\) 0 0
\(997\) 9.09754e11 0.920753 0.460377 0.887724i \(-0.347714\pi\)
0.460377 + 0.887724i \(0.347714\pi\)
\(998\) 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.197.12 yes 16
3.2 odd 2 inner 252.9.c.a.197.5 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.9.c.a.197.5 16 3.2 odd 2 inner
252.9.c.a.197.12 yes 16 1.1 even 1 trivial