Properties

Label 384.8.d.d.193.1
Level $384$
Weight $8$
Character 384.193
Analytic conductor $119.956$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 384.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(119.955849786\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \(x^{8} + 449 x^{6} + 50632 x^{4} + 69129 x^{2} + 18225\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{32}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 193.1
Root \(15.2503i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.8.d.d.193.8

$q$-expansion

\(f(q)\) \(=\) \(q-27.0000i q^{3} -522.419i q^{5} +1309.01 q^{7} -729.000 q^{9} +O(q^{10})\) \(q-27.0000i q^{3} -522.419i q^{5} +1309.01 q^{7} -729.000 q^{9} +2794.57i q^{11} +7351.17i q^{13} -14105.3 q^{15} +5644.22 q^{17} -20285.7i q^{19} -35343.4i q^{21} +79560.7 q^{23} -194797. q^{25} +19683.0i q^{27} -140530. i q^{29} +321339. q^{31} +75453.4 q^{33} -683854. i q^{35} -504903. i q^{37} +198482. q^{39} +262029. q^{41} -503139. i q^{43} +380844. i q^{45} -107852. q^{47} +889976. q^{49} -152394. i q^{51} +1.13491e6i q^{53} +1.45994e6 q^{55} -547714. q^{57} +1.80023e6i q^{59} -1.00077e6i q^{61} -954271. q^{63} +3.84039e6 q^{65} -4.70272e6i q^{67} -2.14814e6i q^{69} -4.38848e6 q^{71} -1.51938e6 q^{73} +5.25951e6i q^{75} +3.65813e6i q^{77} +144735. q^{79} +531441. q^{81} +1.45912e6i q^{83} -2.94865e6i q^{85} -3.79430e6 q^{87} +8.58397e6 q^{89} +9.62279e6i q^{91} -8.67616e6i q^{93} -1.05976e7 q^{95} -8.08726e6 q^{97} -2.03724e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2880 q^{7} - 5832 q^{9} + O(q^{10}) \) \( 8 q + 2880 q^{7} - 5832 q^{9} - 6048 q^{15} + 22896 q^{17} + 207360 q^{23} - 204696 q^{25} - 17856 q^{31} + 7776 q^{33} - 116640 q^{39} + 687056 q^{41} - 1987200 q^{47} + 4815560 q^{49} + 2077056 q^{55} + 878688 q^{57} - 2099520 q^{63} + 12871808 q^{65} - 6336000 q^{71} + 8920752 q^{73} + 1251648 q^{79} + 4251528 q^{81} - 13385952 q^{87} + 4447408 q^{89} - 31607424 q^{95} - 14157584 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/384\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(133\) \(257\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 27.0000i − 0.577350i
\(4\) 0 0
\(5\) − 522.419i − 1.86906i −0.355880 0.934532i \(-0.615819\pi\)
0.355880 0.934532i \(-0.384181\pi\)
\(6\) 0 0
\(7\) 1309.01 1.44245 0.721226 0.692700i \(-0.243579\pi\)
0.721226 + 0.692700i \(0.243579\pi\)
\(8\) 0 0
\(9\) −729.000 −0.333333
\(10\) 0 0
\(11\) 2794.57i 0.633054i 0.948584 + 0.316527i \(0.102517\pi\)
−0.948584 + 0.316527i \(0.897483\pi\)
\(12\) 0 0
\(13\) 7351.17i 0.928015i 0.885831 + 0.464007i \(0.153589\pi\)
−0.885831 + 0.464007i \(0.846411\pi\)
\(14\) 0 0
\(15\) −14105.3 −1.07910
\(16\) 0 0
\(17\) 5644.22 0.278633 0.139316 0.990248i \(-0.455509\pi\)
0.139316 + 0.990248i \(0.455509\pi\)
\(18\) 0 0
\(19\) − 20285.7i − 0.678504i −0.940695 0.339252i \(-0.889826\pi\)
0.940695 0.339252i \(-0.110174\pi\)
\(20\) 0 0
\(21\) − 35343.4i − 0.832800i
\(22\) 0 0
\(23\) 79560.7 1.36349 0.681744 0.731591i \(-0.261222\pi\)
0.681744 + 0.731591i \(0.261222\pi\)
\(24\) 0 0
\(25\) −194797. −2.49340
\(26\) 0 0
\(27\) 19683.0i 0.192450i
\(28\) 0 0
\(29\) − 140530.i − 1.06998i −0.844859 0.534989i \(-0.820316\pi\)
0.844859 0.534989i \(-0.179684\pi\)
\(30\) 0 0
\(31\) 321339. 1.93730 0.968652 0.248423i \(-0.0799123\pi\)
0.968652 + 0.248423i \(0.0799123\pi\)
\(32\) 0 0
\(33\) 75453.4 0.365494
\(34\) 0 0
\(35\) − 683854.i − 2.69603i
\(36\) 0 0
\(37\) − 504903.i − 1.63871i −0.573288 0.819354i \(-0.694332\pi\)
0.573288 0.819354i \(-0.305668\pi\)
\(38\) 0 0
\(39\) 198482. 0.535790
\(40\) 0 0
\(41\) 262029. 0.593753 0.296876 0.954916i \(-0.404055\pi\)
0.296876 + 0.954916i \(0.404055\pi\)
\(42\) 0 0
\(43\) − 503139.i − 0.965046i −0.875883 0.482523i \(-0.839720\pi\)
0.875883 0.482523i \(-0.160280\pi\)
\(44\) 0 0
\(45\) 380844.i 0.623021i
\(46\) 0 0
\(47\) −107852. −0.151525 −0.0757625 0.997126i \(-0.524139\pi\)
−0.0757625 + 0.997126i \(0.524139\pi\)
\(48\) 0 0
\(49\) 889976. 1.08067
\(50\) 0 0
\(51\) − 152394.i − 0.160869i
\(52\) 0 0
\(53\) 1.13491e6i 1.04712i 0.851989 + 0.523560i \(0.175397\pi\)
−0.851989 + 0.523560i \(0.824603\pi\)
\(54\) 0 0
\(55\) 1.45994e6 1.18322
\(56\) 0 0
\(57\) −547714. −0.391735
\(58\) 0 0
\(59\) 1.80023e6i 1.14116i 0.821242 + 0.570579i \(0.193281\pi\)
−0.821242 + 0.570579i \(0.806719\pi\)
\(60\) 0 0
\(61\) − 1.00077e6i − 0.564518i −0.959338 0.282259i \(-0.908916\pi\)
0.959338 0.282259i \(-0.0910838\pi\)
\(62\) 0 0
\(63\) −954271. −0.480817
\(64\) 0 0
\(65\) 3.84039e6 1.73452
\(66\) 0 0
\(67\) − 4.70272e6i − 1.91024i −0.296222 0.955119i \(-0.595727\pi\)
0.296222 0.955119i \(-0.404273\pi\)
\(68\) 0 0
\(69\) − 2.14814e6i − 0.787210i
\(70\) 0 0
\(71\) −4.38848e6 −1.45516 −0.727579 0.686024i \(-0.759354\pi\)
−0.727579 + 0.686024i \(0.759354\pi\)
\(72\) 0 0
\(73\) −1.51938e6 −0.457127 −0.228563 0.973529i \(-0.573403\pi\)
−0.228563 + 0.973529i \(0.573403\pi\)
\(74\) 0 0
\(75\) 5.25951e6i 1.43956i
\(76\) 0 0
\(77\) 3.65813e6i 0.913149i
\(78\) 0 0
\(79\) 144735. 0.0330278 0.0165139 0.999864i \(-0.494743\pi\)
0.0165139 + 0.999864i \(0.494743\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 1.45912e6i 0.280102i 0.990144 + 0.140051i \(0.0447267\pi\)
−0.990144 + 0.140051i \(0.955273\pi\)
\(84\) 0 0
\(85\) − 2.94865e6i − 0.520783i
\(86\) 0 0
\(87\) −3.79430e6 −0.617753
\(88\) 0 0
\(89\) 8.58397e6 1.29069 0.645346 0.763890i \(-0.276713\pi\)
0.645346 + 0.763890i \(0.276713\pi\)
\(90\) 0 0
\(91\) 9.62279e6i 1.33862i
\(92\) 0 0
\(93\) − 8.67616e6i − 1.11850i
\(94\) 0 0
\(95\) −1.05976e7 −1.26817
\(96\) 0 0
\(97\) −8.08726e6 −0.899705 −0.449853 0.893103i \(-0.648524\pi\)
−0.449853 + 0.893103i \(0.648524\pi\)
\(98\) 0 0
\(99\) − 2.03724e6i − 0.211018i
\(100\) 0 0
\(101\) − 3.92632e6i − 0.379193i −0.981862 0.189597i \(-0.939282\pi\)
0.981862 0.189597i \(-0.0607180\pi\)
\(102\) 0 0
\(103\) −9.56910e6 −0.862861 −0.431430 0.902146i \(-0.641991\pi\)
−0.431430 + 0.902146i \(0.641991\pi\)
\(104\) 0 0
\(105\) −1.84641e7 −1.55656
\(106\) 0 0
\(107\) 282494.i 0.0222928i 0.999938 + 0.0111464i \(0.00354809\pi\)
−0.999938 + 0.0111464i \(0.996452\pi\)
\(108\) 0 0
\(109\) − 6.12690e6i − 0.453156i −0.973993 0.226578i \(-0.927246\pi\)
0.973993 0.226578i \(-0.0727539\pi\)
\(110\) 0 0
\(111\) −1.36324e7 −0.946108
\(112\) 0 0
\(113\) −2.76431e6 −0.180224 −0.0901118 0.995932i \(-0.528722\pi\)
−0.0901118 + 0.995932i \(0.528722\pi\)
\(114\) 0 0
\(115\) − 4.15640e7i − 2.54844i
\(116\) 0 0
\(117\) − 5.35901e6i − 0.309338i
\(118\) 0 0
\(119\) 7.38836e6 0.401915
\(120\) 0 0
\(121\) 1.16776e7 0.599243
\(122\) 0 0
\(123\) − 7.07478e6i − 0.342803i
\(124\) 0 0
\(125\) 6.09515e7i 2.79126i
\(126\) 0 0
\(127\) 2.26911e7 0.982976 0.491488 0.870884i \(-0.336453\pi\)
0.491488 + 0.870884i \(0.336453\pi\)
\(128\) 0 0
\(129\) −1.35847e7 −0.557170
\(130\) 0 0
\(131\) 2.52080e6i 0.0979690i 0.998800 + 0.0489845i \(0.0155985\pi\)
−0.998800 + 0.0489845i \(0.984401\pi\)
\(132\) 0 0
\(133\) − 2.65543e7i − 0.978710i
\(134\) 0 0
\(135\) 1.02828e7 0.359701
\(136\) 0 0
\(137\) 5.09088e7 1.69149 0.845747 0.533584i \(-0.179155\pi\)
0.845747 + 0.533584i \(0.179155\pi\)
\(138\) 0 0
\(139\) − 5.49354e7i − 1.73500i −0.497434 0.867502i \(-0.665724\pi\)
0.497434 0.867502i \(-0.334276\pi\)
\(140\) 0 0
\(141\) 2.91200e6i 0.0874830i
\(142\) 0 0
\(143\) −2.05434e7 −0.587483
\(144\) 0 0
\(145\) −7.34154e7 −1.99986
\(146\) 0 0
\(147\) − 2.40293e7i − 0.623923i
\(148\) 0 0
\(149\) 7.39556e7i 1.83155i 0.401690 + 0.915776i \(0.368423\pi\)
−0.401690 + 0.915776i \(0.631577\pi\)
\(150\) 0 0
\(151\) −1.70345e7 −0.402634 −0.201317 0.979526i \(-0.564522\pi\)
−0.201317 + 0.979526i \(0.564522\pi\)
\(152\) 0 0
\(153\) −4.11463e6 −0.0928776
\(154\) 0 0
\(155\) − 1.67874e8i − 3.62094i
\(156\) 0 0
\(157\) 4.59236e7i 0.947080i 0.880772 + 0.473540i \(0.157024\pi\)
−0.880772 + 0.473540i \(0.842976\pi\)
\(158\) 0 0
\(159\) 3.06426e7 0.604556
\(160\) 0 0
\(161\) 1.04146e8 1.96676
\(162\) 0 0
\(163\) 6.55609e7i 1.18574i 0.805299 + 0.592868i \(0.202005\pi\)
−0.805299 + 0.592868i \(0.797995\pi\)
\(164\) 0 0
\(165\) − 3.94183e7i − 0.683131i
\(166\) 0 0
\(167\) −2.06517e6 −0.0343121 −0.0171561 0.999853i \(-0.505461\pi\)
−0.0171561 + 0.999853i \(0.505461\pi\)
\(168\) 0 0
\(169\) 8.70875e6 0.138788
\(170\) 0 0
\(171\) 1.47883e7i 0.226168i
\(172\) 0 0
\(173\) − 7.19310e7i − 1.05622i −0.849176 0.528111i \(-0.822901\pi\)
0.849176 0.528111i \(-0.177099\pi\)
\(174\) 0 0
\(175\) −2.54992e8 −3.59661
\(176\) 0 0
\(177\) 4.86062e7 0.658848
\(178\) 0 0
\(179\) 6.97037e7i 0.908386i 0.890903 + 0.454193i \(0.150072\pi\)
−0.890903 + 0.454193i \(0.849928\pi\)
\(180\) 0 0
\(181\) − 5.59928e7i − 0.701870i −0.936400 0.350935i \(-0.885864\pi\)
0.936400 0.350935i \(-0.114136\pi\)
\(182\) 0 0
\(183\) −2.70207e7 −0.325925
\(184\) 0 0
\(185\) −2.63771e8 −3.06285
\(186\) 0 0
\(187\) 1.57732e7i 0.176390i
\(188\) 0 0
\(189\) 2.57653e7i 0.277600i
\(190\) 0 0
\(191\) −9.89901e7 −1.02796 −0.513979 0.857803i \(-0.671829\pi\)
−0.513979 + 0.857803i \(0.671829\pi\)
\(192\) 0 0
\(193\) −6.30846e7 −0.631645 −0.315822 0.948818i \(-0.602280\pi\)
−0.315822 + 0.948818i \(0.602280\pi\)
\(194\) 0 0
\(195\) − 1.03691e8i − 1.00142i
\(196\) 0 0
\(197\) − 5.83443e7i − 0.543709i −0.962338 0.271855i \(-0.912363\pi\)
0.962338 0.271855i \(-0.0876370\pi\)
\(198\) 0 0
\(199\) −1.34085e8 −1.20613 −0.603067 0.797691i \(-0.706055\pi\)
−0.603067 + 0.797691i \(0.706055\pi\)
\(200\) 0 0
\(201\) −1.26974e8 −1.10288
\(202\) 0 0
\(203\) − 1.83955e8i − 1.54339i
\(204\) 0 0
\(205\) − 1.36889e8i − 1.10976i
\(206\) 0 0
\(207\) −5.79997e7 −0.454496
\(208\) 0 0
\(209\) 5.66898e7 0.429530
\(210\) 0 0
\(211\) − 6.48026e7i − 0.474902i −0.971400 0.237451i \(-0.923688\pi\)
0.971400 0.237451i \(-0.0763118\pi\)
\(212\) 0 0
\(213\) 1.18489e8i 0.840135i
\(214\) 0 0
\(215\) −2.62849e8 −1.80373
\(216\) 0 0
\(217\) 4.20637e8 2.79447
\(218\) 0 0
\(219\) 4.10233e7i 0.263922i
\(220\) 0 0
\(221\) 4.14916e7i 0.258576i
\(222\) 0 0
\(223\) −8.00971e6 −0.0483671 −0.0241835 0.999708i \(-0.507699\pi\)
−0.0241835 + 0.999708i \(0.507699\pi\)
\(224\) 0 0
\(225\) 1.42007e8 0.831133
\(226\) 0 0
\(227\) − 6.80971e7i − 0.386401i −0.981159 0.193200i \(-0.938113\pi\)
0.981159 0.193200i \(-0.0618867\pi\)
\(228\) 0 0
\(229\) 1.20257e8i 0.661740i 0.943676 + 0.330870i \(0.107342\pi\)
−0.943676 + 0.330870i \(0.892658\pi\)
\(230\) 0 0
\(231\) 9.87696e7 0.527207
\(232\) 0 0
\(233\) −6.83662e7 −0.354075 −0.177038 0.984204i \(-0.556651\pi\)
−0.177038 + 0.984204i \(0.556651\pi\)
\(234\) 0 0
\(235\) 5.63438e7i 0.283210i
\(236\) 0 0
\(237\) − 3.90785e6i − 0.0190686i
\(238\) 0 0
\(239\) −3.05674e8 −1.44832 −0.724162 0.689630i \(-0.757773\pi\)
−0.724162 + 0.689630i \(0.757773\pi\)
\(240\) 0 0
\(241\) −2.58964e7 −0.119174 −0.0595869 0.998223i \(-0.518978\pi\)
−0.0595869 + 0.998223i \(0.518978\pi\)
\(242\) 0 0
\(243\) − 1.43489e7i − 0.0641500i
\(244\) 0 0
\(245\) − 4.64940e8i − 2.01983i
\(246\) 0 0
\(247\) 1.49124e8 0.629662
\(248\) 0 0
\(249\) 3.93961e7 0.161717
\(250\) 0 0
\(251\) 4.32650e7i 0.172694i 0.996265 + 0.0863472i \(0.0275194\pi\)
−0.996265 + 0.0863472i \(0.972481\pi\)
\(252\) 0 0
\(253\) 2.22338e8i 0.863161i
\(254\) 0 0
\(255\) −7.96135e7 −0.300674
\(256\) 0 0
\(257\) −1.33105e8 −0.489133 −0.244567 0.969632i \(-0.578646\pi\)
−0.244567 + 0.969632i \(0.578646\pi\)
\(258\) 0 0
\(259\) − 6.60925e8i − 2.36376i
\(260\) 0 0
\(261\) 1.02446e8i 0.356660i
\(262\) 0 0
\(263\) 5.73164e8 1.94283 0.971413 0.237394i \(-0.0762934\pi\)
0.971413 + 0.237394i \(0.0762934\pi\)
\(264\) 0 0
\(265\) 5.92900e8 1.95714
\(266\) 0 0
\(267\) − 2.31767e8i − 0.745182i
\(268\) 0 0
\(269\) − 4.24355e8i − 1.32922i −0.747191 0.664609i \(-0.768598\pi\)
0.747191 0.664609i \(-0.231402\pi\)
\(270\) 0 0
\(271\) −5.45956e8 −1.66635 −0.833173 0.553013i \(-0.813478\pi\)
−0.833173 + 0.553013i \(0.813478\pi\)
\(272\) 0 0
\(273\) 2.59815e8 0.772851
\(274\) 0 0
\(275\) − 5.44373e8i − 1.57845i
\(276\) 0 0
\(277\) 4.35552e8i 1.23129i 0.788023 + 0.615645i \(0.211105\pi\)
−0.788023 + 0.615645i \(0.788895\pi\)
\(278\) 0 0
\(279\) −2.34256e8 −0.645768
\(280\) 0 0
\(281\) −4.75803e8 −1.27925 −0.639625 0.768687i \(-0.720910\pi\)
−0.639625 + 0.768687i \(0.720910\pi\)
\(282\) 0 0
\(283\) − 4.81622e8i − 1.26315i −0.775316 0.631573i \(-0.782409\pi\)
0.775316 0.631573i \(-0.217591\pi\)
\(284\) 0 0
\(285\) 2.86136e8i 0.732177i
\(286\) 0 0
\(287\) 3.42999e8 0.856459
\(288\) 0 0
\(289\) −3.78481e8 −0.922364
\(290\) 0 0
\(291\) 2.18356e8i 0.519445i
\(292\) 0 0
\(293\) 5.88523e8i 1.36687i 0.730012 + 0.683435i \(0.239515\pi\)
−0.730012 + 0.683435i \(0.760485\pi\)
\(294\) 0 0
\(295\) 9.40475e8 2.13290
\(296\) 0 0
\(297\) −5.50055e7 −0.121831
\(298\) 0 0
\(299\) 5.84864e8i 1.26534i
\(300\) 0 0
\(301\) − 6.58616e8i − 1.39203i
\(302\) 0 0
\(303\) −1.06011e8 −0.218927
\(304\) 0 0
\(305\) −5.22819e8 −1.05512
\(306\) 0 0
\(307\) − 4.44023e8i − 0.875834i −0.899016 0.437917i \(-0.855716\pi\)
0.899016 0.437917i \(-0.144284\pi\)
\(308\) 0 0
\(309\) 2.58366e8i 0.498173i
\(310\) 0 0
\(311\) −1.57481e8 −0.296871 −0.148435 0.988922i \(-0.547424\pi\)
−0.148435 + 0.988922i \(0.547424\pi\)
\(312\) 0 0
\(313\) 1.55060e8 0.285821 0.142911 0.989736i \(-0.454354\pi\)
0.142911 + 0.989736i \(0.454354\pi\)
\(314\) 0 0
\(315\) 4.98530e8i 0.898678i
\(316\) 0 0
\(317\) − 2.90492e8i − 0.512185i −0.966652 0.256092i \(-0.917565\pi\)
0.966652 0.256092i \(-0.0824352\pi\)
\(318\) 0 0
\(319\) 3.92720e8 0.677354
\(320\) 0 0
\(321\) 7.62733e6 0.0128708
\(322\) 0 0
\(323\) − 1.14497e8i − 0.189054i
\(324\) 0 0
\(325\) − 1.43198e9i − 2.31391i
\(326\) 0 0
\(327\) −1.65426e8 −0.261630
\(328\) 0 0
\(329\) −1.41179e8 −0.218568
\(330\) 0 0
\(331\) 4.85485e8i 0.735831i 0.929859 + 0.367915i \(0.119928\pi\)
−0.929859 + 0.367915i \(0.880072\pi\)
\(332\) 0 0
\(333\) 3.68074e8i 0.546236i
\(334\) 0 0
\(335\) −2.45679e9 −3.57036
\(336\) 0 0
\(337\) −2.59275e8 −0.369026 −0.184513 0.982830i \(-0.559071\pi\)
−0.184513 + 0.982830i \(0.559071\pi\)
\(338\) 0 0
\(339\) 7.46363e7i 0.104052i
\(340\) 0 0
\(341\) 8.98005e8i 1.22642i
\(342\) 0 0
\(343\) 8.69613e7 0.116358
\(344\) 0 0
\(345\) −1.12223e9 −1.47134
\(346\) 0 0
\(347\) 8.57834e8i 1.10217i 0.834448 + 0.551087i \(0.185787\pi\)
−0.834448 + 0.551087i \(0.814213\pi\)
\(348\) 0 0
\(349\) 1.37991e7i 0.0173765i 0.999962 + 0.00868827i \(0.00276560\pi\)
−0.999962 + 0.00868827i \(0.997234\pi\)
\(350\) 0 0
\(351\) −1.44693e8 −0.178597
\(352\) 0 0
\(353\) −1.74329e8 −0.210940 −0.105470 0.994422i \(-0.533635\pi\)
−0.105470 + 0.994422i \(0.533635\pi\)
\(354\) 0 0
\(355\) 2.29263e9i 2.71978i
\(356\) 0 0
\(357\) − 1.99486e8i − 0.232045i
\(358\) 0 0
\(359\) 1.45142e9 1.65563 0.827813 0.561004i \(-0.189585\pi\)
0.827813 + 0.561004i \(0.189585\pi\)
\(360\) 0 0
\(361\) 4.82362e8 0.539632
\(362\) 0 0
\(363\) − 3.15294e8i − 0.345973i
\(364\) 0 0
\(365\) 7.93753e8i 0.854399i
\(366\) 0 0
\(367\) −9.49474e8 −1.00266 −0.501328 0.865257i \(-0.667155\pi\)
−0.501328 + 0.865257i \(0.667155\pi\)
\(368\) 0 0
\(369\) −1.91019e8 −0.197918
\(370\) 0 0
\(371\) 1.48562e9i 1.51042i
\(372\) 0 0
\(373\) 6.16143e8i 0.614753i 0.951588 + 0.307377i \(0.0994511\pi\)
−0.951588 + 0.307377i \(0.900549\pi\)
\(374\) 0 0
\(375\) 1.64569e9 1.61153
\(376\) 0 0
\(377\) 1.03306e9 0.992956
\(378\) 0 0
\(379\) − 2.44694e8i − 0.230880i −0.993314 0.115440i \(-0.963172\pi\)
0.993314 0.115440i \(-0.0368278\pi\)
\(380\) 0 0
\(381\) − 6.12660e8i − 0.567521i
\(382\) 0 0
\(383\) 1.25530e9 1.14170 0.570848 0.821056i \(-0.306615\pi\)
0.570848 + 0.821056i \(0.306615\pi\)
\(384\) 0 0
\(385\) 1.91108e9 1.70673
\(386\) 0 0
\(387\) 3.66788e8i 0.321682i
\(388\) 0 0
\(389\) 9.17850e8i 0.790584i 0.918556 + 0.395292i \(0.129357\pi\)
−0.918556 + 0.395292i \(0.870643\pi\)
\(390\) 0 0
\(391\) 4.49058e8 0.379912
\(392\) 0 0
\(393\) 6.80616e7 0.0565625
\(394\) 0 0
\(395\) − 7.56124e7i − 0.0617310i
\(396\) 0 0
\(397\) 2.91845e8i 0.234091i 0.993127 + 0.117046i \(0.0373424\pi\)
−0.993127 + 0.117046i \(0.962658\pi\)
\(398\) 0 0
\(399\) −7.16966e8 −0.565058
\(400\) 0 0
\(401\) 2.30913e9 1.78831 0.894157 0.447754i \(-0.147776\pi\)
0.894157 + 0.447754i \(0.147776\pi\)
\(402\) 0 0
\(403\) 2.36222e9i 1.79785i
\(404\) 0 0
\(405\) − 2.77635e8i − 0.207674i
\(406\) 0 0
\(407\) 1.41099e9 1.03739
\(408\) 0 0
\(409\) −1.85520e9 −1.34078 −0.670392 0.742007i \(-0.733874\pi\)
−0.670392 + 0.742007i \(0.733874\pi\)
\(410\) 0 0
\(411\) − 1.37454e9i − 0.976585i
\(412\) 0 0
\(413\) 2.35653e9i 1.64607i
\(414\) 0 0
\(415\) 7.62270e8 0.523528
\(416\) 0 0
\(417\) −1.48326e9 −1.00170
\(418\) 0 0
\(419\) − 2.05244e9i − 1.36308i −0.731782 0.681539i \(-0.761311\pi\)
0.731782 0.681539i \(-0.238689\pi\)
\(420\) 0 0
\(421\) − 1.56223e9i − 1.02037i −0.860065 0.510184i \(-0.829577\pi\)
0.860065 0.510184i \(-0.170423\pi\)
\(422\) 0 0
\(423\) 7.86239e7 0.0505083
\(424\) 0 0
\(425\) −1.09947e9 −0.694743
\(426\) 0 0
\(427\) − 1.31002e9i − 0.814290i
\(428\) 0 0
\(429\) 5.54671e8i 0.339184i
\(430\) 0 0
\(431\) 1.14071e9 0.686283 0.343141 0.939284i \(-0.388509\pi\)
0.343141 + 0.939284i \(0.388509\pi\)
\(432\) 0 0
\(433\) 1.21428e9 0.718802 0.359401 0.933183i \(-0.382981\pi\)
0.359401 + 0.933183i \(0.382981\pi\)
\(434\) 0 0
\(435\) 1.98222e9i 1.15462i
\(436\) 0 0
\(437\) − 1.61395e9i − 0.925132i
\(438\) 0 0
\(439\) 2.88103e9 1.62526 0.812630 0.582781i \(-0.198035\pi\)
0.812630 + 0.582781i \(0.198035\pi\)
\(440\) 0 0
\(441\) −6.48792e8 −0.360222
\(442\) 0 0
\(443\) − 2.98450e8i − 0.163102i −0.996669 0.0815508i \(-0.974013\pi\)
0.996669 0.0815508i \(-0.0259873\pi\)
\(444\) 0 0
\(445\) − 4.48443e9i − 2.41239i
\(446\) 0 0
\(447\) 1.99680e9 1.05745
\(448\) 0 0
\(449\) −6.76245e7 −0.0352567 −0.0176284 0.999845i \(-0.505612\pi\)
−0.0176284 + 0.999845i \(0.505612\pi\)
\(450\) 0 0
\(451\) 7.32258e8i 0.375877i
\(452\) 0 0
\(453\) 4.59932e8i 0.232461i
\(454\) 0 0
\(455\) 5.02713e9 2.50196
\(456\) 0 0
\(457\) 2.61798e9 1.28310 0.641548 0.767083i \(-0.278293\pi\)
0.641548 + 0.767083i \(0.278293\pi\)
\(458\) 0 0
\(459\) 1.11095e8i 0.0536229i
\(460\) 0 0
\(461\) 2.33589e9i 1.11045i 0.831700 + 0.555225i \(0.187368\pi\)
−0.831700 + 0.555225i \(0.812632\pi\)
\(462\) 0 0
\(463\) −3.00500e9 −1.40706 −0.703528 0.710667i \(-0.748393\pi\)
−0.703528 + 0.710667i \(0.748393\pi\)
\(464\) 0 0
\(465\) −4.53259e9 −2.09055
\(466\) 0 0
\(467\) 1.57079e9i 0.713690i 0.934164 + 0.356845i \(0.116148\pi\)
−0.934164 + 0.356845i \(0.883852\pi\)
\(468\) 0 0
\(469\) − 6.15593e9i − 2.75543i
\(470\) 0 0
\(471\) 1.23994e9 0.546797
\(472\) 0 0
\(473\) 1.40606e9 0.610926
\(474\) 0 0
\(475\) 3.95159e9i 1.69178i
\(476\) 0 0
\(477\) − 8.27351e8i − 0.349040i
\(478\) 0 0
\(479\) −1.47580e9 −0.613556 −0.306778 0.951781i \(-0.599251\pi\)
−0.306778 + 0.951781i \(0.599251\pi\)
\(480\) 0 0
\(481\) 3.71163e9 1.52075
\(482\) 0 0
\(483\) − 2.81194e9i − 1.13551i
\(484\) 0 0
\(485\) 4.22494e9i 1.68161i
\(486\) 0 0
\(487\) 3.56484e9 1.39859 0.699293 0.714836i \(-0.253498\pi\)
0.699293 + 0.714836i \(0.253498\pi\)
\(488\) 0 0
\(489\) 1.77014e9 0.684585
\(490\) 0 0
\(491\) 1.80845e9i 0.689480i 0.938698 + 0.344740i \(0.112033\pi\)
−0.938698 + 0.344740i \(0.887967\pi\)
\(492\) 0 0
\(493\) − 7.93180e8i − 0.298131i
\(494\) 0 0
\(495\) −1.06429e9 −0.394406
\(496\) 0 0
\(497\) −5.74458e9 −2.09899
\(498\) 0 0
\(499\) − 3.73339e9i − 1.34509i −0.740055 0.672546i \(-0.765201\pi\)
0.740055 0.672546i \(-0.234799\pi\)
\(500\) 0 0
\(501\) 5.57595e7i 0.0198101i
\(502\) 0 0
\(503\) −3.16793e8 −0.110991 −0.0554955 0.998459i \(-0.517674\pi\)
−0.0554955 + 0.998459i \(0.517674\pi\)
\(504\) 0 0
\(505\) −2.05118e9 −0.708736
\(506\) 0 0
\(507\) − 2.35136e8i − 0.0801294i
\(508\) 0 0
\(509\) − 4.48810e9i − 1.50852i −0.656578 0.754258i \(-0.727997\pi\)
0.656578 0.754258i \(-0.272003\pi\)
\(510\) 0 0
\(511\) −1.98889e9 −0.659383
\(512\) 0 0
\(513\) 3.99284e8 0.130578
\(514\) 0 0
\(515\) 4.99908e9i 1.61274i
\(516\) 0 0
\(517\) − 3.01399e8i − 0.0959235i
\(518\) 0 0
\(519\) −1.94214e9 −0.609810
\(520\) 0 0
\(521\) 4.33409e9 1.34266 0.671329 0.741159i \(-0.265724\pi\)
0.671329 + 0.741159i \(0.265724\pi\)
\(522\) 0 0
\(523\) 5.74767e9i 1.75686i 0.477875 + 0.878428i \(0.341407\pi\)
−0.477875 + 0.878428i \(0.658593\pi\)
\(524\) 0 0
\(525\) 6.88478e9i 2.07650i
\(526\) 0 0
\(527\) 1.81371e9 0.539796
\(528\) 0 0
\(529\) 2.92508e9 0.859097
\(530\) 0 0
\(531\) − 1.31237e9i − 0.380386i
\(532\) 0 0
\(533\) 1.92622e9i 0.551011i
\(534\) 0 0
\(535\) 1.47580e8 0.0416667
\(536\) 0 0
\(537\) 1.88200e9 0.524457
\(538\) 0 0
\(539\) 2.48710e9i 0.684120i
\(540\) 0 0
\(541\) 4.47248e9i 1.21439i 0.794553 + 0.607195i \(0.207705\pi\)
−0.794553 + 0.607195i \(0.792295\pi\)
\(542\) 0 0
\(543\) −1.51180e9 −0.405225
\(544\) 0 0
\(545\) −3.20081e9 −0.846978
\(546\) 0 0
\(547\) − 8.14796e8i − 0.212860i −0.994320 0.106430i \(-0.966058\pi\)
0.994320 0.106430i \(-0.0339420\pi\)
\(548\) 0 0
\(549\) 7.29558e8i 0.188173i
\(550\) 0 0
\(551\) −2.85075e9 −0.725985
\(552\) 0 0
\(553\) 1.89460e8 0.0476410
\(554\) 0 0
\(555\) 7.12181e9i 1.76834i
\(556\) 0 0
\(557\) 5.85518e8i 0.143565i 0.997420 + 0.0717823i \(0.0228687\pi\)
−0.997420 + 0.0717823i \(0.977131\pi\)
\(558\) 0 0
\(559\) 3.69866e9 0.895578
\(560\) 0 0
\(561\) 4.25875e8 0.101839
\(562\) 0 0
\(563\) − 4.19201e9i − 0.990018i −0.868888 0.495009i \(-0.835165\pi\)
0.868888 0.495009i \(-0.164835\pi\)
\(564\) 0 0
\(565\) 1.44413e9i 0.336849i
\(566\) 0 0
\(567\) 6.95664e8 0.160272
\(568\) 0 0
\(569\) −7.74762e9 −1.76309 −0.881547 0.472096i \(-0.843497\pi\)
−0.881547 + 0.472096i \(0.843497\pi\)
\(570\) 0 0
\(571\) − 4.48502e9i − 1.00818i −0.863651 0.504090i \(-0.831828\pi\)
0.863651 0.504090i \(-0.168172\pi\)
\(572\) 0 0
\(573\) 2.67273e9i 0.593492i
\(574\) 0 0
\(575\) −1.54982e10 −3.39972
\(576\) 0 0
\(577\) 6.51505e9 1.41190 0.705948 0.708264i \(-0.250521\pi\)
0.705948 + 0.708264i \(0.250521\pi\)
\(578\) 0 0
\(579\) 1.70328e9i 0.364680i
\(580\) 0 0
\(581\) 1.91000e9i 0.404034i
\(582\) 0 0
\(583\) −3.17159e9 −0.662884
\(584\) 0 0
\(585\) −2.79965e9 −0.578173
\(586\) 0 0
\(587\) − 1.46081e9i − 0.298099i −0.988830 0.149050i \(-0.952379\pi\)
0.988830 0.149050i \(-0.0476214\pi\)
\(588\) 0 0
\(589\) − 6.51859e9i − 1.31447i
\(590\) 0 0
\(591\) −1.57530e9 −0.313911
\(592\) 0 0
\(593\) −6.59477e9 −1.29870 −0.649349 0.760490i \(-0.724959\pi\)
−0.649349 + 0.760490i \(0.724959\pi\)
\(594\) 0 0
\(595\) − 3.85982e9i − 0.751204i
\(596\) 0 0
\(597\) 3.62030e9i 0.696362i
\(598\) 0 0
\(599\) −1.13100e9 −0.215015 −0.107508 0.994204i \(-0.534287\pi\)
−0.107508 + 0.994204i \(0.534287\pi\)
\(600\) 0 0
\(601\) 7.83427e9 1.47210 0.736051 0.676926i \(-0.236688\pi\)
0.736051 + 0.676926i \(0.236688\pi\)
\(602\) 0 0
\(603\) 3.42829e9i 0.636746i
\(604\) 0 0
\(605\) − 6.10058e9i − 1.12002i
\(606\) 0 0
\(607\) −1.94589e9 −0.353149 −0.176575 0.984287i \(-0.556502\pi\)
−0.176575 + 0.984287i \(0.556502\pi\)
\(608\) 0 0
\(609\) −4.96680e9 −0.891078
\(610\) 0 0
\(611\) − 7.92836e8i − 0.140617i
\(612\) 0 0
\(613\) − 7.86592e9i − 1.37923i −0.724174 0.689617i \(-0.757779\pi\)
0.724174 0.689617i \(-0.242221\pi\)
\(614\) 0 0
\(615\) −3.69600e9 −0.640721
\(616\) 0 0
\(617\) −7.05302e9 −1.20886 −0.604432 0.796657i \(-0.706600\pi\)
−0.604432 + 0.796657i \(0.706600\pi\)
\(618\) 0 0
\(619\) − 5.09288e9i − 0.863070i −0.902096 0.431535i \(-0.857972\pi\)
0.902096 0.431535i \(-0.142028\pi\)
\(620\) 0 0
\(621\) 1.56599e9i 0.262403i
\(622\) 0 0
\(623\) 1.12365e10 1.86176
\(624\) 0 0
\(625\) 1.66237e10 2.72363
\(626\) 0 0
\(627\) − 1.53063e9i − 0.247989i
\(628\) 0 0
\(629\) − 2.84978e9i − 0.456598i
\(630\) 0 0
\(631\) 6.95936e9 1.10272 0.551362 0.834266i \(-0.314108\pi\)
0.551362 + 0.834266i \(0.314108\pi\)
\(632\) 0 0
\(633\) −1.74967e9 −0.274185
\(634\) 0 0
\(635\) − 1.18543e10i − 1.83724i
\(636\) 0 0
\(637\) 6.54237e9i 1.00288i
\(638\) 0 0
\(639\) 3.19920e9 0.485052
\(640\) 0 0
\(641\) −4.26493e9 −0.639601 −0.319801 0.947485i \(-0.603616\pi\)
−0.319801 + 0.947485i \(0.603616\pi\)
\(642\) 0 0
\(643\) − 7.80345e9i − 1.15757i −0.815479 0.578787i \(-0.803526\pi\)
0.815479 0.578787i \(-0.196474\pi\)
\(644\) 0 0
\(645\) 7.09693e9i 1.04139i
\(646\) 0 0
\(647\) 4.27680e9 0.620803 0.310401 0.950606i \(-0.399537\pi\)
0.310401 + 0.950606i \(0.399537\pi\)
\(648\) 0 0
\(649\) −5.03087e9 −0.722415
\(650\) 0 0
\(651\) − 1.13572e10i − 1.61339i
\(652\) 0 0
\(653\) 4.79185e9i 0.673452i 0.941603 + 0.336726i \(0.109320\pi\)
−0.941603 + 0.336726i \(0.890680\pi\)
\(654\) 0 0
\(655\) 1.31691e9 0.183110
\(656\) 0 0
\(657\) 1.10763e9 0.152376
\(658\) 0 0
\(659\) 4.04254e9i 0.550244i 0.961409 + 0.275122i \(0.0887183\pi\)
−0.961409 + 0.275122i \(0.911282\pi\)
\(660\) 0 0
\(661\) 4.02211e8i 0.0541688i 0.999633 + 0.0270844i \(0.00862228\pi\)
−0.999633 + 0.0270844i \(0.991378\pi\)
\(662\) 0 0
\(663\) 1.12027e9 0.149289
\(664\) 0 0
\(665\) −1.38725e10 −1.82927
\(666\) 0 0
\(667\) − 1.11806e10i − 1.45890i
\(668\) 0 0
\(669\) 2.16262e8i 0.0279247i
\(670\) 0 0
\(671\) 2.79671e9 0.357370
\(672\) 0 0
\(673\) 4.17076e9 0.527427 0.263713 0.964601i \(-0.415053\pi\)
0.263713 + 0.964601i \(0.415053\pi\)
\(674\) 0 0
\(675\) − 3.83418e9i − 0.479855i
\(676\) 0 0
\(677\) − 7.81235e8i − 0.0967657i −0.998829 0.0483828i \(-0.984593\pi\)
0.998829 0.0483828i \(-0.0154068\pi\)
\(678\) 0 0
\(679\) −1.05863e10 −1.29778
\(680\) 0 0
\(681\) −1.83862e9 −0.223089
\(682\) 0 0
\(683\) − 1.16917e10i − 1.40412i −0.712117 0.702061i \(-0.752263\pi\)
0.712117 0.702061i \(-0.247737\pi\)
\(684\) 0 0
\(685\) − 2.65957e10i − 3.16151i
\(686\) 0 0
\(687\) 3.24695e9 0.382056
\(688\) 0 0
\(689\) −8.34294e9 −0.971744
\(690\) 0 0
\(691\) 2.33629e9i 0.269373i 0.990888 + 0.134687i \(0.0430028\pi\)
−0.990888 + 0.134687i \(0.956997\pi\)
\(692\) 0 0
\(693\) − 2.66678e9i − 0.304383i
\(694\) 0 0
\(695\) −2.86993e10 −3.24283
\(696\) 0 0
\(697\) 1.47895e9 0.165439
\(698\) 0 0
\(699\) 1.84589e9i 0.204426i
\(700\) 0 0
\(701\) − 1.38336e10i − 1.51678i −0.651801 0.758390i \(-0.725986\pi\)
0.651801 0.758390i \(-0.274014\pi\)
\(702\) 0 0
\(703\) −1.02423e10 −1.11187
\(704\) 0 0
\(705\) 1.52128e9 0.163511
\(706\) 0 0
\(707\) − 5.13961e9i − 0.546968i
\(708\) 0 0
\(709\) − 4.96747e8i − 0.0523448i −0.999657 0.0261724i \(-0.991668\pi\)
0.999657 0.0261724i \(-0.00833188\pi\)
\(710\) 0 0
\(711\) −1.05512e8 −0.0110093
\(712\) 0 0
\(713\) 2.55660e10 2.64149
\(714\) 0 0
\(715\) 1.07322e10i 1.09804i
\(716\) 0 0
\(717\) 8.25320e9i 0.836191i
\(718\) 0 0
\(719\) −2.84708e9 −0.285660 −0.142830 0.989747i \(-0.545620\pi\)
−0.142830 + 0.989747i \(0.545620\pi\)
\(720\) 0 0
\(721\) −1.25261e10 −1.24463
\(722\) 0 0
\(723\) 6.99204e8i 0.0688050i
\(724\) 0 0
\(725\) 2.73747e10i 2.66788i
\(726\) 0 0
\(727\) −9.04361e8 −0.0872914 −0.0436457 0.999047i \(-0.513897\pi\)
−0.0436457 + 0.999047i \(0.513897\pi\)
\(728\) 0 0
\(729\) −3.87420e8 −0.0370370
\(730\) 0 0
\(731\) − 2.83982e9i − 0.268894i
\(732\) 0 0
\(733\) 1.59471e10i 1.49561i 0.663921 + 0.747803i \(0.268891\pi\)
−0.663921 + 0.747803i \(0.731109\pi\)
\(734\) 0 0
\(735\) −1.25534e10 −1.16615
\(736\) 0 0
\(737\) 1.31421e10 1.20928
\(738\) 0 0
\(739\) 1.23619e9i 0.112675i 0.998412 + 0.0563376i \(0.0179423\pi\)
−0.998412 + 0.0563376i \(0.982058\pi\)
\(740\) 0 0
\(741\) − 4.02634e9i − 0.363536i
\(742\) 0 0
\(743\) 1.28465e10 1.14901 0.574505 0.818501i \(-0.305194\pi\)
0.574505 + 0.818501i \(0.305194\pi\)
\(744\) 0 0
\(745\) 3.86358e10 3.42329
\(746\) 0 0
\(747\) − 1.06370e9i − 0.0933673i
\(748\) 0 0
\(749\) 3.69788e8i 0.0321563i
\(750\) 0 0
\(751\) 6.66117e9 0.573866 0.286933 0.957951i \(-0.407364\pi\)
0.286933 + 0.957951i \(0.407364\pi\)
\(752\) 0 0
\(753\) 1.16815e9 0.0997052
\(754\) 0 0
\(755\) 8.89916e9i 0.752549i
\(756\) 0 0
\(757\) − 1.22593e10i − 1.02714i −0.858049 0.513569i \(-0.828323\pi\)
0.858049 0.513569i \(-0.171677\pi\)
\(758\) 0 0
\(759\) 6.00312e9 0.498346
\(760\) 0 0
\(761\) −4.82752e8 −0.0397080 −0.0198540 0.999803i \(-0.506320\pi\)
−0.0198540 + 0.999803i \(0.506320\pi\)
\(762\) 0 0
\(763\) − 8.02020e9i − 0.653656i
\(764\) 0 0
\(765\) 2.14956e9i 0.173594i
\(766\) 0 0
\(767\) −1.32338e10 −1.05901
\(768\) 0 0
\(769\) −1.09177e10 −0.865745 −0.432872 0.901455i \(-0.642500\pi\)
−0.432872 + 0.901455i \(0.642500\pi\)
\(770\) 0 0
\(771\) 3.59383e9i 0.282401i
\(772\) 0 0
\(773\) 1.93076e10i 1.50349i 0.659453 + 0.751746i \(0.270788\pi\)
−0.659453 + 0.751746i \(0.729212\pi\)
\(774\) 0 0
\(775\) −6.25958e10 −4.83047
\(776\) 0 0
\(777\) −1.78450e10 −1.36472
\(778\) 0 0
\(779\) − 5.31544e9i − 0.402864i
\(780\) 0 0
\(781\) − 1.22639e10i − 0.921193i
\(782\) 0 0
\(783\) 2.76605e9 0.205918
\(784\) 0 0
\(785\) 2.39913e10 1.77015
\(786\) 0 0
\(787\) 2.09108e10i 1.52918i 0.644517 + 0.764590i \(0.277058\pi\)
−0.644517 + 0.764590i \(0.722942\pi\)
\(788\) 0 0
\(789\) − 1.54754e10i − 1.12169i
\(790\) 0 0
\(791\) −3.61852e9 −0.259964
\(792\) 0 0
\(793\) 7.35680e9 0.523881
\(794\) 0 0
\(795\) − 1.60083e10i − 1.12995i
\(796\) 0 0
\(797\) 9.05983e9i 0.633893i 0.948443 + 0.316946i \(0.102658\pi\)
−0.948443 + 0.316946i \(0.897342\pi\)
\(798\) 0 0
\(799\) −6.08738e8 −0.0422199
\(800\) 0 0
\(801\) −6.25771e9 −0.430231
\(802\) 0 0
\(803\) − 4.24601e9i − 0.289386i
\(804\) 0 0
\(805\) − 5.44079e10i − 3.67601i
\(806\) 0 0
\(807\) −1.14576e10 −0.767424
\(808\) 0 0
\(809\) −2.28791e10 −1.51921 −0.759606 0.650384i \(-0.774608\pi\)
−0.759606 + 0.650384i \(0.774608\pi\)
\(810\) 0 0
\(811\) 5.84918e9i 0.385054i 0.981292 + 0.192527i \(0.0616683\pi\)
−0.981292 + 0.192527i \(0.938332\pi\)
\(812\) 0 0
\(813\) 1.47408e10i 0.962065i
\(814\) 0 0
\(815\) 3.42503e10 2.21622
\(816\) 0 0
\(817\) −1.02065e10 −0.654788
\(818\) 0 0
\(819\) − 7.01502e9i − 0.446206i
\(820\) 0 0
\(821\) 1.60574e10i 1.01268i 0.862333 + 0.506342i \(0.169003\pi\)
−0.862333 + 0.506342i \(0.830997\pi\)
\(822\) 0 0
\(823\) 1.90253e8 0.0118969 0.00594844 0.999982i \(-0.498107\pi\)
0.00594844 + 0.999982i \(0.498107\pi\)
\(824\) 0 0
\(825\) −1.46981e10 −0.911321
\(826\) 0 0
\(827\) 1.27195e10i 0.781989i 0.920393 + 0.390994i \(0.127869\pi\)
−0.920393 + 0.390994i \(0.872131\pi\)
\(828\) 0 0
\(829\) − 5.37231e9i − 0.327506i −0.986501 0.163753i \(-0.947640\pi\)
0.986501 0.163753i \(-0.0523601\pi\)
\(830\) 0 0
\(831\) 1.17599e10 0.710886
\(832\) 0 0
\(833\) 5.02322e9 0.301109
\(834\) 0 0
\(835\) 1.07888e9i 0.0641316i
\(836\) 0 0
\(837\) 6.32492e9i 0.372834i
\(838\) 0 0
\(839\) −1.73272e10 −1.01289 −0.506445 0.862272i \(-0.669041\pi\)
−0.506445 + 0.862272i \(0.669041\pi\)
\(840\) 0 0
\(841\) −2.49873e9 −0.144855
\(842\) 0 0
\(843\) 1.28467e10i 0.738575i
\(844\) 0 0
\(845\) − 4.54962e9i − 0.259404i
\(846\) 0 0
\(847\) 1.52861e10 0.864379
\(848\) 0 0
\(849\) −1.30038e10 −0.729278
\(850\) 0 0
\(851\) − 4.01704e10i − 2.23436i
\(852\) 0 0
\(853\) 2.20248e10i 1.21504i 0.794305 + 0.607520i \(0.207835\pi\)
−0.794305 + 0.607520i \(0.792165\pi\)
\(854\) 0 0
\(855\) 7.72568e9 0.422723
\(856\) 0 0
\(857\) 2.96218e9 0.160760 0.0803800 0.996764i \(-0.474387\pi\)
0.0803800 + 0.996764i \(0.474387\pi\)
\(858\) 0 0
\(859\) − 3.47029e9i − 0.186806i −0.995628 0.0934028i \(-0.970226\pi\)
0.995628 0.0934028i \(-0.0297744\pi\)
\(860\) 0 0
\(861\) − 9.26099e9i − 0.494477i
\(862\) 0 0
\(863\) 9.13684e9 0.483903 0.241951 0.970288i \(-0.422213\pi\)
0.241951 + 0.970288i \(0.422213\pi\)
\(864\) 0 0
\(865\) −3.75781e10 −1.97414
\(866\) 0 0
\(867\) 1.02190e10i 0.532527i
\(868\) 0 0
\(869\) 4.04472e8i 0.0209083i
\(870\) 0 0
\(871\) 3.45705e10 1.77273
\(872\) 0 0
\(873\) 5.89561e9 0.299902
\(874\) 0 0
\(875\) 7.97864e10i 4.02625i
\(876\) 0 0
\(877\) − 4.62260e9i − 0.231413i −0.993283 0.115706i \(-0.963087\pi\)
0.993283 0.115706i \(-0.0369132\pi\)
\(878\) 0 0
\(879\) 1.58901e10 0.789163
\(880\) 0 0
\(881\) 8.53164e9 0.420356 0.210178 0.977663i \(-0.432596\pi\)
0.210178 + 0.977663i \(0.432596\pi\)
\(882\) 0 0
\(883\) − 2.09924e10i − 1.02612i −0.858352 0.513061i \(-0.828511\pi\)
0.858352 0.513061i \(-0.171489\pi\)
\(884\) 0 0
\(885\) − 2.53928e10i − 1.23143i
\(886\) 0 0
\(887\) −1.79154e10 −0.861973 −0.430986 0.902358i \(-0.641834\pi\)
−0.430986 + 0.902358i \(0.641834\pi\)
\(888\) 0 0
\(889\) 2.97030e10 1.41790
\(890\) 0 0
\(891\) 1.48515e9i 0.0703393i
\(892\) 0 0
\(893\) 2.18785e9i 0.102810i
\(894\) 0 0
\(895\) 3.64146e10 1.69783
\(896\) 0 0
\(897\) 1.57913e10 0.730542
\(898\) 0 0
\(899\) − 4.51577e10i − 2.07287i
\(900\) 0 0
\(901\) 6.40569e9i 0.291762i
\(902\) 0 0
\(903\) −1.77826e10 −0.803691
\(904\) 0 0
\(905\) −2.92517e10 −1.31184
\(906\) 0 0
\(907\) − 2.75245e10i − 1.22488i −0.790516 0.612441i \(-0.790188\pi\)
0.790516 0.612441i \(-0.209812\pi\)
\(908\) 0 0
\(909\) 2.86229e9i 0.126398i
\(910\) 0 0
\(911\) −1.40208e10 −0.614409 −0.307204 0.951644i \(-0.599393\pi\)
−0.307204 + 0.951644i \(0.599393\pi\)
\(912\) 0 0
\(913\) −4.07760e9 −0.177320
\(914\) 0 0
\(915\) 1.41161e10i 0.609174i
\(916\) 0 0
\(917\) 3.29976e9i 0.141316i
\(918\) 0 0
\(919\) 3.39157e10 1.44144 0.720721 0.693226i \(-0.243811\pi\)
0.720721 + 0.693226i \(0.243811\pi\)
\(920\) 0 0
\(921\) −1.19886e10 −0.505663
\(922\) 0 0
\(923\) − 3.22605e10i − 1.35041i
\(924\) 0 0
\(925\) 9.83534e10i 4.08595i
\(926\) 0 0
\(927\) 6.97587e9 0.287620
\(928\) 0 0
\(929\) −9.41796e9 −0.385391 −0.192696 0.981259i \(-0.561723\pi\)
−0.192696 + 0.981259i \(0.561723\pi\)
\(930\) 0 0
\(931\) − 1.80538e10i − 0.733237i
\(932\) 0 0
\(933\) 4.25199e9i 0.171398i
\(934\) 0 0
\(935\) 8.24020e9 0.329683
\(936\) 0 0
\(937\) −2.54198e10 −1.00945 −0.504723 0.863281i \(-0.668405\pi\)
−0.504723 + 0.863281i \(0.668405\pi\)
\(938\) 0 0
\(939\) − 4.18662e9i − 0.165019i
\(940\) 0 0
\(941\) − 8.70006e9i − 0.340376i −0.985412 0.170188i \(-0.945563\pi\)
0.985412 0.170188i \(-0.0544375\pi\)
\(942\) 0 0
\(943\) 2.08472e10 0.809574
\(944\) 0 0
\(945\) 1.34603e10 0.518852
\(946\) 0 0
\(947\) 2.27301e10i 0.869716i 0.900499 + 0.434858i \(0.143201\pi\)
−0.900499 + 0.434858i \(0.856799\pi\)
\(948\) 0 0
\(949\) − 1.11692e10i − 0.424220i
\(950\) 0 0
\(951\) −7.84328e9 −0.295710
\(952\) 0 0
\(953\) 3.29285e10 1.23238 0.616192 0.787596i \(-0.288674\pi\)
0.616192 + 0.787596i \(0.288674\pi\)
\(954\) 0 0
\(955\) 5.17143e10i 1.92132i
\(956\) 0 0
\(957\) − 1.06034e10i − 0.391071i
\(958\) 0 0
\(959\) 6.66403e10 2.43990
\(960\) 0 0
\(961\) 7.57462e10 2.75314
\(962\) 0 0
\(963\) − 2.05938e8i − 0.00743095i
\(964\) 0 0
\(965\) 3.29566e10i 1.18058i
\(966\) 0 0
\(967\) 1.34709e10 0.479074 0.239537 0.970887i \(-0.423004\pi\)
0.239537 + 0.970887i \(0.423004\pi\)
\(968\) 0 0
\(969\) −3.09142e9 −0.109150
\(970\) 0 0
\(971\) 3.31485e10i 1.16198i 0.813912 + 0.580988i \(0.197334\pi\)
−0.813912 + 0.580988i \(0.802666\pi\)
\(972\) 0 0
\(973\) − 7.19112e10i − 2.50266i
\(974\) 0 0
\(975\) −3.86636e10 −1.33594
\(976\) 0 0
\(977\) 2.46700e10 0.846326 0.423163 0.906053i \(-0.360920\pi\)
0.423163 + 0.906053i \(0.360920\pi\)
\(978\) 0 0
\(979\) 2.39885e10i 0.817078i
\(980\) 0 0
\(981\) 4.46651e9i 0.151052i
\(982\) 0 0
\(983\) 7.13043e9 0.239430 0.119715 0.992808i \(-0.461802\pi\)
0.119715 + 0.992808i \(0.461802\pi\)
\(984\) 0 0
\(985\) −3.04802e10 −1.01623
\(986\) 0 0
\(987\) 3.81184e9i 0.126190i
\(988\) 0 0
\(989\) − 4.00301e10i − 1.31583i
\(990\) 0 0
\(991\) 3.96102e10 1.29285 0.646426 0.762976i \(-0.276263\pi\)
0.646426 + 0.762976i \(0.276263\pi\)
\(992\) 0 0
\(993\) 1.31081e10 0.424832
\(994\) 0 0
\(995\) 7.00487e10i 2.25434i
\(996\) 0 0
\(997\) 4.99492e9i 0.159623i 0.996810 + 0.0798116i \(0.0254319\pi\)
−0.996810 + 0.0798116i \(0.974568\pi\)
\(998\) 0 0
\(999\) 9.93800e9 0.315369
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 384.8.d.d.193.1 yes 8
4.3 odd 2 384.8.d.c.193.5 yes 8
8.3 odd 2 384.8.d.c.193.4 8
8.5 even 2 inner 384.8.d.d.193.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.8.d.c.193.4 8 8.3 odd 2
384.8.d.c.193.5 yes 8 4.3 odd 2
384.8.d.d.193.1 yes 8 1.1 even 1 trivial
384.8.d.d.193.8 yes 8 8.5 even 2 inner