Properties

Label 252.9.c.a.197.6
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 197.6
Root \(-491.437i\) of defining polynomial
Character \(\chi\) \(=\) 252.197
Dual form 252.9.c.a.197.11

$q$-expansion

\(f(q)\) \(=\) \(q-491.437i q^{5} -907.493 q^{7} +O(q^{10})\) \(q-491.437i q^{5} -907.493 q^{7} +5999.21i q^{11} -47598.9 q^{13} +106621. i q^{17} +137232. q^{19} -80032.4i q^{23} +149114. q^{25} -46364.8i q^{29} +941453. q^{31} +445976. i q^{35} -1.33907e6 q^{37} +4.74791e6i q^{41} +2.35655e6 q^{43} -2.73617e6i q^{47} +823543. q^{49} -1.15514e7i q^{53} +2.94824e6 q^{55} -1.65119e7i q^{59} -1.01747e7 q^{61} +2.33919e7i q^{65} -1.21775e7 q^{67} -5.40380e6i q^{71} +3.94313e7 q^{73} -5.44424e6i q^{77} -7.34681e7 q^{79} +3.14266e7i q^{83} +5.23977e7 q^{85} -8.79931e7i q^{89} +4.31957e7 q^{91} -6.74408e7i q^{95} -1.00964e7 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16q + O(q^{10}) \) \( 16q - 95480q^{13} - 287560q^{19} - 1754520q^{25} - 3554264q^{31} - 182920q^{37} + 8472416q^{43} + 13176688q^{49} - 18692072q^{55} + 34224568q^{61} + 22683096q^{67} + 2137296q^{73} - 90245624q^{79} - 56204456q^{85} - 25661888q^{91} + 134041152q^{97} + O(q^{100}) \)

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
\(5\) − 491.437i − 0.786300i −0.919474 0.393150i \(-0.871385\pi\)
0.919474 0.393150i \(-0.128615\pi\)
\(6\) 0 0
\(7\) −907.493 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5999.21i 0.409754i 0.978788 + 0.204877i \(0.0656795\pi\)
−0.978788 + 0.204877i \(0.934321\pi\)
\(12\) 0 0
\(13\) −47598.9 −1.66657 −0.833285 0.552843i \(-0.813543\pi\)
−0.833285 + 0.552843i \(0.813543\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 106621.i 1.27658i 0.769795 + 0.638291i \(0.220358\pi\)
−0.769795 + 0.638291i \(0.779642\pi\)
\(18\) 0 0
\(19\) 137232. 1.05303 0.526514 0.850166i \(-0.323499\pi\)
0.526514 + 0.850166i \(0.323499\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 80032.4i − 0.285993i −0.989723 0.142996i \(-0.954326\pi\)
0.989723 0.142996i \(-0.0456737\pi\)
\(24\) 0 0
\(25\) 149114. 0.381733
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 46364.8i − 0.0655536i −0.999463 0.0327768i \(-0.989565\pi\)
0.999463 0.0327768i \(-0.0104350\pi\)
\(30\) 0 0
\(31\) 941453. 1.01942 0.509708 0.860347i \(-0.329753\pi\)
0.509708 + 0.860347i \(0.329753\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 445976.i 0.297193i
\(36\) 0 0
\(37\) −1.33907e6 −0.714492 −0.357246 0.934010i \(-0.616284\pi\)
−0.357246 + 0.934010i \(0.616284\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.74791e6i 1.68022i 0.542415 + 0.840111i \(0.317510\pi\)
−0.542415 + 0.840111i \(0.682490\pi\)
\(42\) 0 0
\(43\) 2.35655e6 0.689292 0.344646 0.938733i \(-0.387999\pi\)
0.344646 + 0.938733i \(0.387999\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 2.73617e6i − 0.560727i −0.959894 0.280364i \(-0.909545\pi\)
0.959894 0.280364i \(-0.0904551\pi\)
\(48\) 0 0
\(49\) 823543. 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 1.15514e7i − 1.46397i −0.681322 0.731984i \(-0.738595\pi\)
0.681322 0.731984i \(-0.261405\pi\)
\(54\) 0 0
\(55\) 2.94824e6 0.322189
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 1.65119e7i − 1.36267i −0.731974 0.681333i \(-0.761401\pi\)
0.731974 0.681333i \(-0.238599\pi\)
\(60\) 0 0
\(61\) −1.01747e7 −0.734859 −0.367429 0.930051i \(-0.619762\pi\)
−0.367429 + 0.930051i \(0.619762\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 2.33919e7i 1.31042i
\(66\) 0 0
\(67\) −1.21775e7 −0.604310 −0.302155 0.953259i \(-0.597706\pi\)
−0.302155 + 0.953259i \(0.597706\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 5.40380e6i − 0.212650i −0.994331 0.106325i \(-0.966092\pi\)
0.994331 0.106325i \(-0.0339084\pi\)
\(72\) 0 0
\(73\) 3.94313e7 1.38851 0.694257 0.719727i \(-0.255733\pi\)
0.694257 + 0.719727i \(0.255733\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 5.44424e6i − 0.154873i
\(78\) 0 0
\(79\) −7.34681e7 −1.88621 −0.943106 0.332493i \(-0.892110\pi\)
−0.943106 + 0.332493i \(0.892110\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.14266e7i 0.662194i 0.943597 + 0.331097i \(0.107419\pi\)
−0.943597 + 0.331097i \(0.892581\pi\)
\(84\) 0 0
\(85\) 5.23977e7 1.00378
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 8.79931e7i − 1.40245i −0.712938 0.701227i \(-0.752636\pi\)
0.712938 0.701227i \(-0.247364\pi\)
\(90\) 0 0
\(91\) 4.31957e7 0.629905
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 6.74408e7i − 0.827996i
\(96\) 0 0
\(97\) −1.00964e7 −0.114046 −0.0570231 0.998373i \(-0.518161\pi\)
−0.0570231 + 0.998373i \(0.518161\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) − 1.06293e8i − 1.02145i −0.859743 0.510727i \(-0.829376\pi\)
0.859743 0.510727i \(-0.170624\pi\)
\(102\) 0 0
\(103\) 1.33130e8 1.18285 0.591423 0.806361i \(-0.298566\pi\)
0.591423 + 0.806361i \(0.298566\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.17472e7i 0.318487i 0.987239 + 0.159244i \(0.0509055\pi\)
−0.987239 + 0.159244i \(0.949094\pi\)
\(108\) 0 0
\(109\) 2.20761e8 1.56393 0.781964 0.623323i \(-0.214218\pi\)
0.781964 + 0.623323i \(0.214218\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.31146e8i 0.804341i 0.915565 + 0.402171i \(0.131744\pi\)
−0.915565 + 0.402171i \(0.868256\pi\)
\(114\) 0 0
\(115\) −3.93309e7 −0.224876
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 9.67582e7i − 0.482503i
\(120\) 0 0
\(121\) 1.78368e8 0.832102
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 2.65248e8i − 1.08646i
\(126\) 0 0
\(127\) −434.944 −1.67193e−6 0 −8.35966e−7 1.00000i \(-0.500000\pi\)
−8.35966e−7 1.00000i \(0.500000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 3.36763e8i − 1.14351i −0.820425 0.571754i \(-0.806263\pi\)
0.820425 0.571754i \(-0.193737\pi\)
\(132\) 0 0
\(133\) −1.24537e8 −0.398007
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 9.24216e6i 0.0262356i 0.999914 + 0.0131178i \(0.00417565\pi\)
−0.999914 + 0.0131178i \(0.995824\pi\)
\(138\) 0 0
\(139\) 4.35507e7 0.116664 0.0583319 0.998297i \(-0.481422\pi\)
0.0583319 + 0.998297i \(0.481422\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 2.85556e8i − 0.682884i
\(144\) 0 0
\(145\) −2.27854e7 −0.0515448
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 8.91966e8i − 1.80969i −0.425746 0.904843i \(-0.639988\pi\)
0.425746 0.904843i \(-0.360012\pi\)
\(150\) 0 0
\(151\) 3.81506e8 0.733827 0.366914 0.930255i \(-0.380414\pi\)
0.366914 + 0.930255i \(0.380414\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 4.62665e8i − 0.801567i
\(156\) 0 0
\(157\) 2.26816e8 0.373314 0.186657 0.982425i \(-0.440235\pi\)
0.186657 + 0.982425i \(0.440235\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 7.26288e7i 0.108095i
\(162\) 0 0
\(163\) 1.59270e8 0.225623 0.112811 0.993616i \(-0.464014\pi\)
0.112811 + 0.993616i \(0.464014\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 1.47027e9i − 1.89031i −0.326625 0.945154i \(-0.605911\pi\)
0.326625 0.945154i \(-0.394089\pi\)
\(168\) 0 0
\(169\) 1.44993e9 1.77746
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.54399e8i 0.172370i 0.996279 + 0.0861849i \(0.0274676\pi\)
−0.996279 + 0.0861849i \(0.972532\pi\)
\(174\) 0 0
\(175\) −1.35320e8 −0.144282
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.46569e8i 0.434987i 0.976062 + 0.217494i \(0.0697882\pi\)
−0.976062 + 0.217494i \(0.930212\pi\)
\(180\) 0 0
\(181\) 3.55769e8 0.331477 0.165739 0.986170i \(-0.446999\pi\)
0.165739 + 0.986170i \(0.446999\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 6.58071e8i 0.561805i
\(186\) 0 0
\(187\) −6.39644e8 −0.523085
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 2.17843e9i − 1.63686i −0.574608 0.818429i \(-0.694845\pi\)
0.574608 0.818429i \(-0.305155\pi\)
\(192\) 0 0
\(193\) −1.79560e8 −0.129414 −0.0647069 0.997904i \(-0.520611\pi\)
−0.0647069 + 0.997904i \(0.520611\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 1.46094e9i − 0.969991i −0.874517 0.484996i \(-0.838821\pi\)
0.874517 0.484996i \(-0.161179\pi\)
\(198\) 0 0
\(199\) 2.06149e9 1.31452 0.657261 0.753663i \(-0.271715\pi\)
0.657261 + 0.753663i \(0.271715\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 4.20757e7i 0.0247769i
\(204\) 0 0
\(205\) 2.33330e9 1.32116
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.23282e8i 0.431483i
\(210\) 0 0
\(211\) 3.03874e9 1.53307 0.766537 0.642200i \(-0.221978\pi\)
0.766537 + 0.642200i \(0.221978\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 1.15810e9i − 0.541990i
\(216\) 0 0
\(217\) −8.54362e8 −0.385303
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 5.07507e9i − 2.12751i
\(222\) 0 0
\(223\) 1.03325e7 0.00417819 0.00208909 0.999998i \(-0.499335\pi\)
0.00208909 + 0.999998i \(0.499335\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.11020e9i 0.794733i 0.917660 + 0.397366i \(0.130076\pi\)
−0.917660 + 0.397366i \(0.869924\pi\)
\(228\) 0 0
\(229\) −2.82730e9 −1.02809 −0.514044 0.857764i \(-0.671853\pi\)
−0.514044 + 0.857764i \(0.671853\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.03733e9i 0.691256i 0.938372 + 0.345628i \(0.112334\pi\)
−0.938372 + 0.345628i \(0.887666\pi\)
\(234\) 0 0
\(235\) −1.34466e9 −0.440900
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 5.94533e9i 1.82215i 0.412240 + 0.911075i \(0.364746\pi\)
−0.412240 + 0.911075i \(0.635254\pi\)
\(240\) 0 0
\(241\) 1.09550e9 0.324746 0.162373 0.986729i \(-0.448085\pi\)
0.162373 + 0.986729i \(0.448085\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 4.04720e8i − 0.112329i
\(246\) 0 0
\(247\) −6.53208e9 −1.75495
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 4.08687e9i − 1.02967i −0.857290 0.514833i \(-0.827854\pi\)
0.857290 0.514833i \(-0.172146\pi\)
\(252\) 0 0
\(253\) 4.80131e8 0.117187
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 5.61577e9i − 1.28729i −0.765323 0.643646i \(-0.777421\pi\)
0.765323 0.643646i \(-0.222579\pi\)
\(258\) 0 0
\(259\) 1.21520e9 0.270053
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 7.85186e9i 1.64116i 0.571535 + 0.820578i \(0.306348\pi\)
−0.571535 + 0.820578i \(0.693652\pi\)
\(264\) 0 0
\(265\) −5.67679e9 −1.15112
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 4.18792e9i − 0.799815i −0.916556 0.399907i \(-0.869042\pi\)
0.916556 0.399907i \(-0.130958\pi\)
\(270\) 0 0
\(271\) 5.90063e9 1.09401 0.547005 0.837129i \(-0.315768\pi\)
0.547005 + 0.837129i \(0.315768\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 8.94569e8i 0.156417i
\(276\) 0 0
\(277\) 4.65342e9 0.790411 0.395206 0.918593i \(-0.370673\pi\)
0.395206 + 0.918593i \(0.370673\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.79692e9i 0.288206i 0.989563 + 0.144103i \(0.0460296\pi\)
−0.989563 + 0.144103i \(0.953970\pi\)
\(282\) 0 0
\(283\) 1.76770e9 0.275590 0.137795 0.990461i \(-0.455998\pi\)
0.137795 + 0.990461i \(0.455998\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 4.30869e9i − 0.635064i
\(288\) 0 0
\(289\) −4.39237e9 −0.629663
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 7.70982e9i 1.04610i 0.852301 + 0.523051i \(0.175206\pi\)
−0.852301 + 0.523051i \(0.824794\pi\)
\(294\) 0 0
\(295\) −8.11457e9 −1.07146
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.80946e9i 0.476627i
\(300\) 0 0
\(301\) −2.13855e9 −0.260528
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 5.00025e9i 0.577819i
\(306\) 0 0
\(307\) −4.19818e9 −0.472615 −0.236308 0.971678i \(-0.575937\pi\)
−0.236308 + 0.971678i \(0.575937\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5.39143e9i 0.576318i 0.957583 + 0.288159i \(0.0930432\pi\)
−0.957583 + 0.288159i \(0.906957\pi\)
\(312\) 0 0
\(313\) −8.50941e9 −0.886588 −0.443294 0.896376i \(-0.646190\pi\)
−0.443294 + 0.896376i \(0.646190\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.92300e10i 1.90433i 0.305577 + 0.952167i \(0.401151\pi\)
−0.305577 + 0.952167i \(0.598849\pi\)
\(318\) 0 0
\(319\) 2.78152e8 0.0268609
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1.46318e10i 1.34428i
\(324\) 0 0
\(325\) −7.09769e9 −0.636185
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 2.48305e9i 0.211935i
\(330\) 0 0
\(331\) −6.32387e8 −0.0526831 −0.0263415 0.999653i \(-0.508386\pi\)
−0.0263415 + 0.999653i \(0.508386\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 5.98449e9i 0.475169i
\(336\) 0 0
\(337\) 1.67516e10 1.29878 0.649391 0.760455i \(-0.275024\pi\)
0.649391 + 0.760455i \(0.275024\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.64797e9i 0.417710i
\(342\) 0 0
\(343\) −7.47359e8 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.77019e9i 0.535937i 0.963428 + 0.267969i \(0.0863524\pi\)
−0.963428 + 0.267969i \(0.913648\pi\)
\(348\) 0 0
\(349\) −4.05365e9 −0.273240 −0.136620 0.990624i \(-0.543624\pi\)
−0.136620 + 0.990624i \(0.543624\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 1.37233e9i − 0.0883812i −0.999023 0.0441906i \(-0.985929\pi\)
0.999023 0.0441906i \(-0.0140709\pi\)
\(354\) 0 0
\(355\) −2.65563e9 −0.167207
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 8.16787e9i − 0.491735i −0.969303 0.245867i \(-0.920927\pi\)
0.969303 0.245867i \(-0.0790727\pi\)
\(360\) 0 0
\(361\) 1.84899e9 0.108869
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 1.93780e10i − 1.09179i
\(366\) 0 0
\(367\) −2.92285e10 −1.61117 −0.805587 0.592477i \(-0.798150\pi\)
−0.805587 + 0.592477i \(0.798150\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.04828e10i 0.553328i
\(372\) 0 0
\(373\) −8.42210e9 −0.435096 −0.217548 0.976050i \(-0.569806\pi\)
−0.217548 + 0.976050i \(0.569806\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.20692e9i 0.109250i
\(378\) 0 0
\(379\) −2.32101e10 −1.12491 −0.562457 0.826826i \(-0.690144\pi\)
−0.562457 + 0.826826i \(0.690144\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.47468e10i 0.685336i 0.939457 + 0.342668i \(0.111331\pi\)
−0.939457 + 0.342668i \(0.888669\pi\)
\(384\) 0 0
\(385\) −2.67550e9 −0.121776
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 6.70446e9i − 0.292796i −0.989226 0.146398i \(-0.953232\pi\)
0.989226 0.146398i \(-0.0467680\pi\)
\(390\) 0 0
\(391\) 8.53317e9 0.365093
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3.61050e10i 1.48313i
\(396\) 0 0
\(397\) −1.04252e10 −0.419682 −0.209841 0.977735i \(-0.567295\pi\)
−0.209841 + 0.977735i \(0.567295\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 2.66893e10i − 1.03219i −0.856532 0.516094i \(-0.827385\pi\)
0.856532 0.516094i \(-0.172615\pi\)
\(402\) 0 0
\(403\) −4.48122e10 −1.69893
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 8.03339e9i − 0.292766i
\(408\) 0 0
\(409\) 3.20725e10 1.14614 0.573072 0.819505i \(-0.305752\pi\)
0.573072 + 0.819505i \(0.305752\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 1.49844e10i 0.515039i
\(414\) 0 0
\(415\) 1.54442e10 0.520683
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 6.50024e9i − 0.210898i −0.994425 0.105449i \(-0.966372\pi\)
0.994425 0.105449i \(-0.0336280\pi\)
\(420\) 0 0
\(421\) 1.35499e10 0.431327 0.215664 0.976468i \(-0.430809\pi\)
0.215664 + 0.976468i \(0.430809\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.58988e10i 0.487314i
\(426\) 0 0
\(427\) 9.23350e9 0.277751
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.10549e9i 0.176934i 0.996079 + 0.0884671i \(0.0281968\pi\)
−0.996079 + 0.0884671i \(0.971803\pi\)
\(432\) 0 0
\(433\) 4.28877e10 1.22006 0.610030 0.792378i \(-0.291157\pi\)
0.610030 + 0.792378i \(0.291157\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 1.09830e10i − 0.301158i
\(438\) 0 0
\(439\) 2.18244e10 0.587604 0.293802 0.955866i \(-0.405079\pi\)
0.293802 + 0.955866i \(0.405079\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.20937e10i 0.314011i 0.987598 + 0.157005i \(0.0501840\pi\)
−0.987598 + 0.157005i \(0.949816\pi\)
\(444\) 0 0
\(445\) −4.32431e10 −1.10275
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 3.64601e10i − 0.897083i −0.893762 0.448542i \(-0.851944\pi\)
0.893762 0.448542i \(-0.148056\pi\)
\(450\) 0 0
\(451\) −2.84837e10 −0.688478
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 2.12280e10i − 0.495294i
\(456\) 0 0
\(457\) 6.95062e10 1.59353 0.796763 0.604293i \(-0.206544\pi\)
0.796763 + 0.604293i \(0.206544\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 2.44898e9i − 0.0542227i −0.999632 0.0271114i \(-0.991369\pi\)
0.999632 0.0271114i \(-0.00863087\pi\)
\(462\) 0 0
\(463\) −4.19882e10 −0.913698 −0.456849 0.889544i \(-0.651022\pi\)
−0.456849 + 0.889544i \(0.651022\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 5.55739e10i − 1.16843i −0.811599 0.584216i \(-0.801402\pi\)
0.811599 0.584216i \(-0.198598\pi\)
\(468\) 0 0
\(469\) 1.10510e10 0.228408
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.41375e10i 0.282440i
\(474\) 0 0
\(475\) 2.04632e10 0.401976
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 3.59246e10i − 0.682417i −0.939988 0.341208i \(-0.889164\pi\)
0.939988 0.341208i \(-0.110836\pi\)
\(480\) 0 0
\(481\) 6.37385e10 1.19075
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.96176e9i 0.0896745i
\(486\) 0 0
\(487\) 9.03441e10 1.60614 0.803071 0.595884i \(-0.203198\pi\)
0.803071 + 0.595884i \(0.203198\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4.45844e10i 0.767108i 0.923518 + 0.383554i \(0.125300\pi\)
−0.923518 + 0.383554i \(0.874700\pi\)
\(492\) 0 0
\(493\) 4.94348e9 0.0836846
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.90391e9i 0.0803742i
\(498\) 0 0
\(499\) −4.06435e10 −0.655525 −0.327762 0.944760i \(-0.606295\pi\)
−0.327762 + 0.944760i \(0.606295\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 1.09003e10i − 0.170282i −0.996369 0.0851409i \(-0.972866\pi\)
0.996369 0.0851409i \(-0.0271340\pi\)
\(504\) 0 0
\(505\) −5.22363e10 −0.803169
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 7.87110e10i − 1.17264i −0.810080 0.586319i \(-0.800576\pi\)
0.810080 0.586319i \(-0.199424\pi\)
\(510\) 0 0
\(511\) −3.57837e10 −0.524809
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 6.54253e10i − 0.930072i
\(516\) 0 0
\(517\) 1.64149e10 0.229760
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 4.77689e10i − 0.648327i −0.946001 0.324164i \(-0.894917\pi\)
0.946001 0.324164i \(-0.105083\pi\)
\(522\) 0 0
\(523\) 1.39395e11 1.86312 0.931562 0.363582i \(-0.118446\pi\)
0.931562 + 0.363582i \(0.118446\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.00379e11i 1.30137i
\(528\) 0 0
\(529\) 7.19058e10 0.918208
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 2.25995e11i − 2.80021i
\(534\) 0 0
\(535\) 2.05161e10 0.250426
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.94061e9i 0.0585363i
\(540\) 0 0
\(541\) −8.97193e10 −1.04736 −0.523681 0.851914i \(-0.675442\pi\)
−0.523681 + 0.851914i \(0.675442\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 1.08490e11i − 1.22972i
\(546\) 0 0
\(547\) 3.81538e9 0.0426176 0.0213088 0.999773i \(-0.493217\pi\)
0.0213088 + 0.999773i \(0.493217\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 6.36272e9i − 0.0690298i
\(552\) 0 0
\(553\) 6.66718e10 0.712921
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 7.53991e10i − 0.783332i −0.920108 0.391666i \(-0.871899\pi\)
0.920108 0.391666i \(-0.128101\pi\)
\(558\) 0 0
\(559\) −1.12169e11 −1.14875
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 5.47694e10i − 0.545136i −0.962137 0.272568i \(-0.912127\pi\)
0.962137 0.272568i \(-0.0878729\pi\)
\(564\) 0 0
\(565\) 6.44499e10 0.632453
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 2.13904e10i − 0.204066i −0.994781 0.102033i \(-0.967465\pi\)
0.994781 0.102033i \(-0.0325347\pi\)
\(570\) 0 0
\(571\) −8.98254e10 −0.844996 −0.422498 0.906364i \(-0.638847\pi\)
−0.422498 + 0.906364i \(0.638847\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 1.19340e10i − 0.109173i
\(576\) 0 0
\(577\) 6.66116e10 0.600962 0.300481 0.953788i \(-0.402853\pi\)
0.300481 + 0.953788i \(0.402853\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 2.85194e10i − 0.250286i
\(582\) 0 0
\(583\) 6.92993e10 0.599867
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.87389e11i 1.57831i 0.614195 + 0.789154i \(0.289481\pi\)
−0.614195 + 0.789154i \(0.710519\pi\)
\(588\) 0 0
\(589\) 1.29197e11 1.07348
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 1.25162e11i − 1.01217i −0.862483 0.506085i \(-0.831092\pi\)
0.862483 0.506085i \(-0.168908\pi\)
\(594\) 0 0
\(595\) −4.75506e10 −0.379392
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.83190e11i 1.42296i 0.702704 + 0.711482i \(0.251976\pi\)
−0.702704 + 0.711482i \(0.748024\pi\)
\(600\) 0 0
\(601\) 8.92487e10 0.684076 0.342038 0.939686i \(-0.388883\pi\)
0.342038 + 0.939686i \(0.388883\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 8.76568e10i − 0.654281i
\(606\) 0 0
\(607\) −1.61679e11 −1.19097 −0.595484 0.803367i \(-0.703040\pi\)
−0.595484 + 0.803367i \(0.703040\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.30239e11i 0.934492i
\(612\) 0 0
\(613\) 3.43758e10 0.243450 0.121725 0.992564i \(-0.461157\pi\)
0.121725 + 0.992564i \(0.461157\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1.50411e11i 1.03786i 0.854817 + 0.518930i \(0.173669\pi\)
−0.854817 + 0.518930i \(0.826331\pi\)
\(618\) 0 0
\(619\) −1.96128e11 −1.33591 −0.667953 0.744203i \(-0.732829\pi\)
−0.667953 + 0.744203i \(0.732829\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 7.98531e10i 0.530078i
\(624\) 0 0
\(625\) −7.21049e10 −0.472547
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 1.42774e11i − 0.912109i
\(630\) 0 0
\(631\) 2.34299e11 1.47792 0.738962 0.673747i \(-0.235316\pi\)
0.738962 + 0.673747i \(0.235316\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 213748.i 0 1.31464e-6i
\(636\) 0 0
\(637\) −3.91998e10 −0.238082
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6.73910e9i 0.0399181i 0.999801 + 0.0199591i \(0.00635359\pi\)
−0.999801 + 0.0199591i \(0.993646\pi\)
\(642\) 0 0
\(643\) 1.90111e11 1.11215 0.556074 0.831133i \(-0.312307\pi\)
0.556074 + 0.831133i \(0.312307\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.82979e11i 1.04420i 0.852885 + 0.522099i \(0.174851\pi\)
−0.852885 + 0.522099i \(0.825149\pi\)
\(648\) 0 0
\(649\) 9.90584e10 0.558358
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 2.11840e11i − 1.16508i −0.812802 0.582540i \(-0.802059\pi\)
0.812802 0.582540i \(-0.197941\pi\)
\(654\) 0 0
\(655\) −1.65498e11 −0.899140
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 2.25407e11i − 1.19516i −0.801810 0.597579i \(-0.796129\pi\)
0.801810 0.597579i \(-0.203871\pi\)
\(660\) 0 0
\(661\) 1.85024e11 0.969222 0.484611 0.874730i \(-0.338961\pi\)
0.484611 + 0.874730i \(0.338961\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 6.12020e10i 0.312953i
\(666\) 0 0
\(667\) −3.71069e9 −0.0187478
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 6.10404e10i − 0.301112i
\(672\) 0 0
\(673\) 2.14867e10 0.104739 0.0523696 0.998628i \(-0.483323\pi\)
0.0523696 + 0.998628i \(0.483323\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 2.67075e11i − 1.27139i −0.771940 0.635695i \(-0.780714\pi\)
0.771940 0.635695i \(-0.219286\pi\)
\(678\) 0 0
\(679\) 9.16244e9 0.0431054
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 2.89664e11i 1.33110i 0.746351 + 0.665552i \(0.231804\pi\)
−0.746351 + 0.665552i \(0.768196\pi\)
\(684\) 0 0
\(685\) 4.54194e9 0.0206291
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 5.49835e11i 2.43981i
\(690\) 0 0
\(691\) −1.34359e11 −0.589326 −0.294663 0.955601i \(-0.595207\pi\)
−0.294663 + 0.955601i \(0.595207\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 2.14024e10i − 0.0917327i
\(696\) 0 0
\(697\) −5.06229e11 −2.14494
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 4.57810e11i 1.89589i 0.318434 + 0.947945i \(0.396843\pi\)
−0.318434 + 0.947945i \(0.603157\pi\)
\(702\) 0 0
\(703\) −1.83763e11 −0.752381
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 9.64601e10i 0.386074i
\(708\) 0 0
\(709\) −9.69596e10 −0.383712 −0.191856 0.981423i \(-0.561451\pi\)
−0.191856 + 0.981423i \(0.561451\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 7.53468e10i − 0.291546i
\(714\) 0 0
\(715\) −1.40333e11 −0.536952
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.45199e11i 0.543310i 0.962395 + 0.271655i \(0.0875709\pi\)
−0.962395 + 0.271655i \(0.912429\pi\)
\(720\) 0 0
\(721\) −1.20815e11 −0.447074
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 6.91367e9i − 0.0250240i
\(726\) 0 0
\(727\) −4.37599e11 −1.56653 −0.783265 0.621688i \(-0.786447\pi\)
−0.783265 + 0.621688i \(0.786447\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 2.51259e11i 0.879938i
\(732\) 0 0
\(733\) 4.99110e10 0.172894 0.0864471 0.996256i \(-0.472449\pi\)
0.0864471 + 0.996256i \(0.472449\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 7.30556e10i − 0.247619i
\(738\) 0 0
\(739\) 4.34218e11 1.45590 0.727948 0.685632i \(-0.240474\pi\)
0.727948 + 0.685632i \(0.240474\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.23440e10i 0.106130i 0.998591 + 0.0530650i \(0.0168991\pi\)
−0.998591 + 0.0530650i \(0.983101\pi\)
\(744\) 0 0
\(745\) −4.38345e11 −1.42295
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 3.78853e10i − 0.120377i
\(750\) 0 0
\(751\) −5.35441e11 −1.68326 −0.841631 0.540053i \(-0.818404\pi\)
−0.841631 + 0.540053i \(0.818404\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 1.87486e11i − 0.577008i
\(756\) 0 0
\(757\) 1.72315e11 0.524736 0.262368 0.964968i \(-0.415497\pi\)
0.262368 + 0.964968i \(0.415497\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 6.61349e11i − 1.97193i −0.166946 0.985966i \(-0.553391\pi\)
0.166946 0.985966i \(-0.446609\pi\)
\(762\) 0 0
\(763\) −2.00339e11 −0.591109
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7.85949e11i 2.27098i
\(768\) 0 0
\(769\) 2.88022e11 0.823607 0.411803 0.911273i \(-0.364899\pi\)
0.411803 + 0.911273i \(0.364899\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 2.55828e10i 0.0716524i 0.999358 + 0.0358262i \(0.0114063\pi\)
−0.999358 + 0.0358262i \(0.988594\pi\)
\(774\) 0 0
\(775\) 1.40384e11 0.389145
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 6.51563e11i 1.76932i
\(780\) 0 0
\(781\) 3.24185e10 0.0871343
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 1.11466e11i − 0.293537i
\(786\) 0 0
\(787\) 7.77507e10 0.202677 0.101339 0.994852i \(-0.467687\pi\)
0.101339 + 0.994852i \(0.467687\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 1.19014e11i − 0.304012i
\(792\) 0 0
\(793\) 4.84307e11 1.22469
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.89643e11i 0.717845i 0.933367 + 0.358922i \(0.116856\pi\)
−0.933367 + 0.358922i \(0.883144\pi\)
\(798\) 0 0
\(799\) 2.91734e11 0.715815
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 2.36557e11i 0.568949i
\(804\) 0 0
\(805\) 3.56925e10 0.0849951
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.37144e11i 0.553628i 0.960924 + 0.276814i \(0.0892786\pi\)
−0.960924 + 0.276814i \(0.910721\pi\)
\(810\) 0 0
\(811\) −7.30982e11 −1.68975 −0.844877 0.534961i \(-0.820326\pi\)
−0.844877 + 0.534961i \(0.820326\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 7.82711e10i − 0.177407i
\(816\) 0 0
\(817\) 3.23394e11 0.725844
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 9.00241e11i 1.98146i 0.135835 + 0.990731i \(0.456628\pi\)
−0.135835 + 0.990731i \(0.543372\pi\)
\(822\) 0 0
\(823\) −7.19062e11 −1.56735 −0.783676 0.621169i \(-0.786658\pi\)
−0.783676 + 0.621169i \(0.786658\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 3.80038e11i − 0.812466i −0.913770 0.406233i \(-0.866842\pi\)
0.913770 0.406233i \(-0.133158\pi\)
\(828\) 0 0
\(829\) 2.57067e11 0.544288 0.272144 0.962257i \(-0.412267\pi\)
0.272144 + 0.962257i \(0.412267\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 8.78073e10i 0.182369i
\(834\) 0 0
\(835\) −7.22548e11 −1.48635
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 1.40281e11i − 0.283108i −0.989931 0.141554i \(-0.954790\pi\)
0.989931 0.141554i \(-0.0452098\pi\)
\(840\) 0 0
\(841\) 4.98097e11 0.995703
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 7.12548e11i − 1.39761i
\(846\) 0 0
\(847\) −1.61868e11 −0.314505
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.07169e11i 0.204339i
\(852\) 0 0
\(853\) 9.23072e10 0.174357 0.0871786 0.996193i \(-0.472215\pi\)
0.0871786 + 0.996193i \(0.472215\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 9.32383e9i 0.0172851i 0.999963 + 0.00864254i \(0.00275104\pi\)
−0.999963 + 0.00864254i \(0.997249\pi\)
\(858\) 0 0
\(859\) 4.42594e11 0.812892 0.406446 0.913675i \(-0.366768\pi\)
0.406446 + 0.913675i \(0.366768\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 7.42643e11i 1.33887i 0.742873 + 0.669433i \(0.233463\pi\)
−0.742873 + 0.669433i \(0.766537\pi\)
\(864\) 0 0
\(865\) 7.58776e10 0.135534
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 4.40751e11i − 0.772883i
\(870\) 0 0
\(871\) 5.79637e11 1.00713
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 2.40711e11i 0.410642i
\(876\) 0 0
\(877\) −7.41714e11 −1.25383 −0.626914 0.779088i \(-0.715682\pi\)
−0.626914 + 0.779088i \(0.715682\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 3.16039e11i − 0.524610i −0.964985 0.262305i \(-0.915517\pi\)
0.964985 0.262305i \(-0.0844827\pi\)
\(882\) 0 0
\(883\) −4.17818e11 −0.687296 −0.343648 0.939098i \(-0.611663\pi\)
−0.343648 + 0.939098i \(0.611663\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 1.10372e12i − 1.78306i −0.452963 0.891529i \(-0.649633\pi\)
0.452963 0.891529i \(-0.350367\pi\)
\(888\) 0 0
\(889\) 394709. 6.31931e−7 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 3.75489e11i − 0.590462i
\(894\) 0 0
\(895\) 2.19461e11 0.342030
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 4.36503e10i − 0.0668265i
\(900\) 0 0
\(901\) 1.23163e12 1.86888
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 1.74838e11i − 0.260640i
\(906\) 0 0
\(907\) 6.80731e11 1.00588 0.502940 0.864321i \(-0.332252\pi\)
0.502940 + 0.864321i \(0.332252\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 2.03180e10i − 0.0294990i −0.999891 0.0147495i \(-0.995305\pi\)
0.999891 0.0147495i \(-0.00469508\pi\)
\(912\) 0 0
\(913\) −1.88535e11 −0.271337
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3.05610e11i 0.432205i
\(918\) 0 0
\(919\) −3.05292e11 −0.428010 −0.214005 0.976833i \(-0.568651\pi\)
−0.214005 + 0.976833i \(0.568651\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2.57215e11i 0.354397i
\(924\) 0 0
\(925\) −1.99675e11 −0.272745
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 8.77155e11i − 1.17764i −0.808263 0.588821i \(-0.799592\pi\)
0.808263 0.588821i \(-0.200408\pi\)
\(930\) 0 0
\(931\) 1.13016e11 0.150433
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 3.14345e11i 0.411301i
\(936\) 0 0
\(937\) 9.38579e11 1.21762 0.608811 0.793315i \(-0.291647\pi\)
0.608811 + 0.793315i \(0.291647\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 1.85800e11i − 0.236967i −0.992956 0.118484i \(-0.962197\pi\)
0.992956 0.118484i \(-0.0378033\pi\)
\(942\) 0 0
\(943\) 3.79986e11 0.480531
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 4.46197e11i − 0.554787i −0.960756 0.277394i \(-0.910529\pi\)
0.960756 0.277394i \(-0.0894706\pi\)
\(948\) 0 0
\(949\) −1.87689e12 −2.31406
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 8.78364e11i − 1.06488i −0.846466 0.532442i \(-0.821274\pi\)
0.846466 0.532442i \(-0.178726\pi\)
\(954\) 0 0
\(955\) −1.07056e12 −1.28706
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 8.38720e9i − 0.00991613i
\(960\) 0 0
\(961\) 3.34427e10 0.0392109
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 8.82425e10i 0.101758i
\(966\) 0 0
\(967\) 1.05174e12 1.20282 0.601412 0.798939i \(-0.294605\pi\)
0.601412 + 0.798939i \(0.294605\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 1.39307e12i − 1.56710i −0.621327 0.783552i \(-0.713406\pi\)
0.621327 0.783552i \(-0.286594\pi\)
\(972\) 0 0
\(973\) −3.95220e10 −0.0440948
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 1.65500e11i − 0.181643i −0.995867 0.0908216i \(-0.971051\pi\)
0.995867 0.0908216i \(-0.0289493\pi\)
\(978\) 0 0
\(979\) 5.27889e11 0.574661
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 7.64045e11i 0.818286i 0.912470 + 0.409143i \(0.134172\pi\)
−0.912470 + 0.409143i \(0.865828\pi\)
\(984\) 0 0
\(985\) −7.17961e11 −0.762704
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 1.88601e11i − 0.197132i
\(990\) 0 0
\(991\) 3.64848e10 0.0378283 0.0189142 0.999821i \(-0.493979\pi\)
0.0189142 + 0.999821i \(0.493979\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 1.01309e12i − 1.03361i
\(996\) 0 0
\(997\) −5.76241e11 −0.583208 −0.291604 0.956539i \(-0.594189\pi\)
−0.291604 + 0.956539i \(0.594189\pi\)
\(998\) 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.197.6 16
3.2 odd 2 inner 252.9.c.a.197.11 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
252.9.c.a.197.6 16 1.1 even 1 trivial
252.9.c.a.197.11 yes 16 3.2 odd 2 inner