Properties

Label 384.8.d.d.193.4
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 193.4
Root \(-14.6646i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.8.d.d.193.5

$q$-expansion

\(f(q)\) \(=\) \(q-27.0000i q^{3} +344.306i q^{5} +1669.75 q^{7} -729.000 q^{9} +O(q^{10})\) \(q-27.0000i q^{3} +344.306i q^{5} +1669.75 q^{7} -729.000 q^{9} -2282.14i q^{11} -11526.9i q^{13} +9296.26 q^{15} -17243.9 q^{17} +14550.2i q^{19} -45083.1i q^{21} -40886.4 q^{23} -40421.6 q^{25} +19683.0i q^{27} -159495. i q^{29} -171754. q^{31} -61617.9 q^{33} +574903. i q^{35} -9029.61i q^{37} -311227. q^{39} -735509. q^{41} -280634. i q^{43} -250999. i q^{45} -529498. q^{47} +1.96451e6 q^{49} +465585. i q^{51} -540438. i q^{53} +785756. q^{55} +392855. q^{57} -727412. i q^{59} +2.21789e6i q^{61} -1.21724e6 q^{63} +3.96878e6 q^{65} -2.59799e6i q^{67} +1.10393e6i q^{69} -524418. q^{71} +1.37636e6 q^{73} +1.09138e6i q^{75} -3.81060e6i q^{77} +6.61970e6 q^{79} +531441. q^{81} +6.38593e6i q^{83} -5.93718e6i q^{85} -4.30636e6 q^{87} -9.60915e6 q^{89} -1.92470e7i q^{91} +4.63736e6i q^{93} -5.00971e6 q^{95} -9.87790e6 q^{97} +1.66368e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2880 q^{7} - 5832 q^{9} + O(q^{10}) \) \( 8 q + 2880 q^{7} - 5832 q^{9} - 6048 q^{15} + 22896 q^{17} + 207360 q^{23} - 204696 q^{25} - 17856 q^{31} + 7776 q^{33} - 116640 q^{39} + 687056 q^{41} - 1987200 q^{47} + 4815560 q^{49} + 2077056 q^{55} + 878688 q^{57} - 2099520 q^{63} + 12871808 q^{65} - 6336000 q^{71} + 8920752 q^{73} + 1251648 q^{79} + 4251528 q^{81} - 13385952 q^{87} + 4447408 q^{89} - 31607424 q^{95} - 14157584 q^{97} + O(q^{100}) \)

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
\(5\) 344.306i 1.23183i 0.787814 + 0.615913i \(0.211213\pi\)
−0.787814 + 0.615913i \(0.788787\pi\)
\(6\) 0 0
\(7\) 1669.75 1.83995 0.919977 0.391972i \(-0.128207\pi\)
0.919977 + 0.391972i \(0.128207\pi\)
\(8\) 0 0
\(9\) −729.000 −0.333333
\(10\) 0 0
\(11\) − 2282.14i − 0.516974i −0.966015 0.258487i \(-0.916776\pi\)
0.966015 0.258487i \(-0.0832239\pi\)
\(12\) 0 0
\(13\) − 11526.9i − 1.45516i −0.686022 0.727581i \(-0.740645\pi\)
0.686022 0.727581i \(-0.259355\pi\)
\(14\) 0 0
\(15\) 9296.26 0.711195
\(16\) 0 0
\(17\) −17243.9 −0.851264 −0.425632 0.904896i \(-0.639948\pi\)
−0.425632 + 0.904896i \(0.639948\pi\)
\(18\) 0 0
\(19\) 14550.2i 0.486666i 0.969943 + 0.243333i \(0.0782407\pi\)
−0.969943 + 0.243333i \(0.921759\pi\)
\(20\) 0 0
\(21\) − 45083.1i − 1.06230i
\(22\) 0 0
\(23\) −40886.4 −0.700698 −0.350349 0.936619i \(-0.613937\pi\)
−0.350349 + 0.936619i \(0.613937\pi\)
\(24\) 0 0
\(25\) −40421.6 −0.517397
\(26\) 0 0
\(27\) 19683.0i 0.192450i
\(28\) 0 0
\(29\) − 159495.i − 1.21438i −0.794558 0.607188i \(-0.792297\pi\)
0.794558 0.607188i \(-0.207703\pi\)
\(30\) 0 0
\(31\) −171754. −1.03548 −0.517739 0.855539i \(-0.673226\pi\)
−0.517739 + 0.855539i \(0.673226\pi\)
\(32\) 0 0
\(33\) −61617.9 −0.298475
\(34\) 0 0
\(35\) 574903.i 2.26651i
\(36\) 0 0
\(37\) − 9029.61i − 0.0293064i −0.999893 0.0146532i \(-0.995336\pi\)
0.999893 0.0146532i \(-0.00466443\pi\)
\(38\) 0 0
\(39\) −311227. −0.840138
\(40\) 0 0
\(41\) −735509. −1.66665 −0.833325 0.552784i \(-0.813565\pi\)
−0.833325 + 0.552784i \(0.813565\pi\)
\(42\) 0 0
\(43\) − 280634.i − 0.538271i −0.963102 0.269135i \(-0.913262\pi\)
0.963102 0.269135i \(-0.0867379\pi\)
\(44\) 0 0
\(45\) − 250999.i − 0.410609i
\(46\) 0 0
\(47\) −529498. −0.743912 −0.371956 0.928250i \(-0.621313\pi\)
−0.371956 + 0.928250i \(0.621313\pi\)
\(48\) 0 0
\(49\) 1.96451e6 2.38543
\(50\) 0 0
\(51\) 465585.i 0.491477i
\(52\) 0 0
\(53\) − 540438.i − 0.498632i −0.968422 0.249316i \(-0.919794\pi\)
0.968422 0.249316i \(-0.0802058\pi\)
\(54\) 0 0
\(55\) 785756. 0.636822
\(56\) 0 0
\(57\) 392855. 0.280977
\(58\) 0 0
\(59\) − 727412.i − 0.461104i −0.973060 0.230552i \(-0.925947\pi\)
0.973060 0.230552i \(-0.0740531\pi\)
\(60\) 0 0
\(61\) 2.21789e6i 1.25108i 0.780193 + 0.625539i \(0.215121\pi\)
−0.780193 + 0.625539i \(0.784879\pi\)
\(62\) 0 0
\(63\) −1.21724e6 −0.613318
\(64\) 0 0
\(65\) 3.96878e6 1.79251
\(66\) 0 0
\(67\) − 2.59799e6i − 1.05530i −0.849462 0.527649i \(-0.823074\pi\)
0.849462 0.527649i \(-0.176926\pi\)
\(68\) 0 0
\(69\) 1.10393e6i 0.404548i
\(70\) 0 0
\(71\) −524418. −0.173889 −0.0869447 0.996213i \(-0.527710\pi\)
−0.0869447 + 0.996213i \(0.527710\pi\)
\(72\) 0 0
\(73\) 1.37636e6 0.414097 0.207049 0.978331i \(-0.433614\pi\)
0.207049 + 0.978331i \(0.433614\pi\)
\(74\) 0 0
\(75\) 1.09138e6i 0.298719i
\(76\) 0 0
\(77\) − 3.81060e6i − 0.951208i
\(78\) 0 0
\(79\) 6.61970e6 1.51058 0.755290 0.655391i \(-0.227496\pi\)
0.755290 + 0.655391i \(0.227496\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) 6.38593e6i 1.22589i 0.790126 + 0.612944i \(0.210015\pi\)
−0.790126 + 0.612944i \(0.789985\pi\)
\(84\) 0 0
\(85\) − 5.93718e6i − 1.04861i
\(86\) 0 0
\(87\) −4.30636e6 −0.701120
\(88\) 0 0
\(89\) −9.60915e6 −1.44484 −0.722420 0.691454i \(-0.756970\pi\)
−0.722420 + 0.691454i \(0.756970\pi\)
\(90\) 0 0
\(91\) − 1.92470e7i − 2.67743i
\(92\) 0 0
\(93\) 4.63736e6i 0.597834i
\(94\) 0 0
\(95\) −5.00971e6 −0.599488
\(96\) 0 0
\(97\) −9.87790e6 −1.09891 −0.549457 0.835522i \(-0.685165\pi\)
−0.549457 + 0.835522i \(0.685165\pi\)
\(98\) 0 0
\(99\) 1.66368e6i 0.172325i
\(100\) 0 0
\(101\) − 1.59806e7i − 1.54336i −0.636008 0.771682i \(-0.719415\pi\)
0.636008 0.771682i \(-0.280585\pi\)
\(102\) 0 0
\(103\) 1.40912e7 1.27062 0.635312 0.772255i \(-0.280871\pi\)
0.635312 + 0.772255i \(0.280871\pi\)
\(104\) 0 0
\(105\) 1.55224e7 1.30857
\(106\) 0 0
\(107\) − 2.31609e7i − 1.82773i −0.406018 0.913865i \(-0.633083\pi\)
0.406018 0.913865i \(-0.366917\pi\)
\(108\) 0 0
\(109\) − 2.10486e6i − 0.155679i −0.996966 0.0778395i \(-0.975198\pi\)
0.996966 0.0778395i \(-0.0248022\pi\)
\(110\) 0 0
\(111\) −243799. −0.0169201
\(112\) 0 0
\(113\) −2.30926e7 −1.50556 −0.752780 0.658272i \(-0.771288\pi\)
−0.752780 + 0.658272i \(0.771288\pi\)
\(114\) 0 0
\(115\) − 1.40774e7i − 0.863139i
\(116\) 0 0
\(117\) 8.40312e6i 0.485054i
\(118\) 0 0
\(119\) −2.87929e7 −1.56629
\(120\) 0 0
\(121\) 1.42790e7 0.732738
\(122\) 0 0
\(123\) 1.98587e7i 0.962240i
\(124\) 0 0
\(125\) 1.29815e7i 0.594483i
\(126\) 0 0
\(127\) −4.56266e7 −1.97654 −0.988269 0.152724i \(-0.951195\pi\)
−0.988269 + 0.152724i \(0.951195\pi\)
\(128\) 0 0
\(129\) −7.57712e6 −0.310771
\(130\) 0 0
\(131\) 1.78031e7i 0.691904i 0.938252 + 0.345952i \(0.112444\pi\)
−0.938252 + 0.345952i \(0.887556\pi\)
\(132\) 0 0
\(133\) 2.42951e7i 0.895443i
\(134\) 0 0
\(135\) −6.77698e6 −0.237065
\(136\) 0 0
\(137\) 3.05756e6 0.101591 0.0507953 0.998709i \(-0.483824\pi\)
0.0507953 + 0.998709i \(0.483824\pi\)
\(138\) 0 0
\(139\) 5.22305e7i 1.64958i 0.565441 + 0.824789i \(0.308706\pi\)
−0.565441 + 0.824789i \(0.691294\pi\)
\(140\) 0 0
\(141\) 1.42964e7i 0.429498i
\(142\) 0 0
\(143\) −2.63061e7 −0.752280
\(144\) 0 0
\(145\) 5.49150e7 1.49590
\(146\) 0 0
\(147\) − 5.30417e7i − 1.37723i
\(148\) 0 0
\(149\) − 2.81604e6i − 0.0697407i −0.999392 0.0348703i \(-0.988898\pi\)
0.999392 0.0348703i \(-0.0111018\pi\)
\(150\) 0 0
\(151\) −5.76673e7 −1.36305 −0.681523 0.731797i \(-0.738682\pi\)
−0.681523 + 0.731797i \(0.738682\pi\)
\(152\) 0 0
\(153\) 1.25708e7 0.283755
\(154\) 0 0
\(155\) − 5.91360e7i − 1.27553i
\(156\) 0 0
\(157\) 9.17674e7i 1.89252i 0.323412 + 0.946258i \(0.395170\pi\)
−0.323412 + 0.946258i \(0.604830\pi\)
\(158\) 0 0
\(159\) −1.45918e7 −0.287885
\(160\) 0 0
\(161\) −6.82698e7 −1.28925
\(162\) 0 0
\(163\) − 9.20744e7i − 1.66526i −0.553829 0.832630i \(-0.686834\pi\)
0.553829 0.832630i \(-0.313166\pi\)
\(164\) 0 0
\(165\) − 2.12154e7i − 0.367670i
\(166\) 0 0
\(167\) 1.62432e7 0.269875 0.134938 0.990854i \(-0.456917\pi\)
0.134938 + 0.990854i \(0.456917\pi\)
\(168\) 0 0
\(169\) −7.01211e7 −1.11749
\(170\) 0 0
\(171\) − 1.06071e7i − 0.162222i
\(172\) 0 0
\(173\) − 4.74152e7i − 0.696236i −0.937451 0.348118i \(-0.886821\pi\)
0.937451 0.348118i \(-0.113179\pi\)
\(174\) 0 0
\(175\) −6.74939e7 −0.951987
\(176\) 0 0
\(177\) −1.96401e7 −0.266218
\(178\) 0 0
\(179\) − 8.00081e7i − 1.04267i −0.853351 0.521337i \(-0.825433\pi\)
0.853351 0.521337i \(-0.174567\pi\)
\(180\) 0 0
\(181\) − 1.96745e7i − 0.246620i −0.992368 0.123310i \(-0.960649\pi\)
0.992368 0.123310i \(-0.0393509\pi\)
\(182\) 0 0
\(183\) 5.98829e7 0.722311
\(184\) 0 0
\(185\) 3.10895e6 0.0361004
\(186\) 0 0
\(187\) 3.93530e7i 0.440081i
\(188\) 0 0
\(189\) 3.28656e7i 0.354099i
\(190\) 0 0
\(191\) −8.78405e7 −0.912174 −0.456087 0.889935i \(-0.650749\pi\)
−0.456087 + 0.889935i \(0.650749\pi\)
\(192\) 0 0
\(193\) 1.29628e8 1.29792 0.648959 0.760824i \(-0.275205\pi\)
0.648959 + 0.760824i \(0.275205\pi\)
\(194\) 0 0
\(195\) − 1.07157e8i − 1.03490i
\(196\) 0 0
\(197\) − 1.21765e8i − 1.13472i −0.823469 0.567362i \(-0.807964\pi\)
0.823469 0.567362i \(-0.192036\pi\)
\(198\) 0 0
\(199\) −1.43807e8 −1.29359 −0.646793 0.762665i \(-0.723890\pi\)
−0.646793 + 0.762665i \(0.723890\pi\)
\(200\) 0 0
\(201\) −7.01457e7 −0.609277
\(202\) 0 0
\(203\) − 2.66315e8i − 2.23440i
\(204\) 0 0
\(205\) − 2.53240e8i − 2.05302i
\(206\) 0 0
\(207\) 2.98062e7 0.233566
\(208\) 0 0
\(209\) 3.32056e7 0.251593
\(210\) 0 0
\(211\) − 1.17120e8i − 0.858303i −0.903233 0.429152i \(-0.858813\pi\)
0.903233 0.429152i \(-0.141187\pi\)
\(212\) 0 0
\(213\) 1.41593e7i 0.100395i
\(214\) 0 0
\(215\) 9.66240e7 0.663056
\(216\) 0 0
\(217\) −2.86785e8 −1.90523
\(218\) 0 0
\(219\) − 3.71617e7i − 0.239079i
\(220\) 0 0
\(221\) 1.98769e8i 1.23873i
\(222\) 0 0
\(223\) −3.15918e8 −1.90769 −0.953844 0.300301i \(-0.902913\pi\)
−0.953844 + 0.300301i \(0.902913\pi\)
\(224\) 0 0
\(225\) 2.94674e7 0.172466
\(226\) 0 0
\(227\) 6.46937e7i 0.367089i 0.983011 + 0.183545i \(0.0587572\pi\)
−0.983011 + 0.183545i \(0.941243\pi\)
\(228\) 0 0
\(229\) − 2.06223e8i − 1.13478i −0.823448 0.567392i \(-0.807952\pi\)
0.823448 0.567392i \(-0.192048\pi\)
\(230\) 0 0
\(231\) −1.02886e8 −0.549180
\(232\) 0 0
\(233\) 3.20225e8 1.65848 0.829238 0.558895i \(-0.188775\pi\)
0.829238 + 0.558895i \(0.188775\pi\)
\(234\) 0 0
\(235\) − 1.82309e8i − 0.916371i
\(236\) 0 0
\(237\) − 1.78732e8i − 0.872134i
\(238\) 0 0
\(239\) −1.24636e8 −0.590540 −0.295270 0.955414i \(-0.595410\pi\)
−0.295270 + 0.955414i \(0.595410\pi\)
\(240\) 0 0
\(241\) 2.47831e8 1.14050 0.570251 0.821470i \(-0.306846\pi\)
0.570251 + 0.821470i \(0.306846\pi\)
\(242\) 0 0
\(243\) − 1.43489e7i − 0.0641500i
\(244\) 0 0
\(245\) 6.76391e8i 2.93844i
\(246\) 0 0
\(247\) 1.67719e8 0.708177
\(248\) 0 0
\(249\) 1.72420e8 0.707767
\(250\) 0 0
\(251\) − 1.73374e8i − 0.692031i −0.938229 0.346016i \(-0.887534\pi\)
0.938229 0.346016i \(-0.112466\pi\)
\(252\) 0 0
\(253\) 9.33086e7i 0.362243i
\(254\) 0 0
\(255\) −1.60304e8 −0.605415
\(256\) 0 0
\(257\) 9.68797e7 0.356014 0.178007 0.984029i \(-0.443035\pi\)
0.178007 + 0.984029i \(0.443035\pi\)
\(258\) 0 0
\(259\) − 1.50771e7i − 0.0539225i
\(260\) 0 0
\(261\) 1.16272e8i 0.404792i
\(262\) 0 0
\(263\) −3.23068e8 −1.09509 −0.547544 0.836777i \(-0.684437\pi\)
−0.547544 + 0.836777i \(0.684437\pi\)
\(264\) 0 0
\(265\) 1.86076e8 0.614229
\(266\) 0 0
\(267\) 2.59447e8i 0.834179i
\(268\) 0 0
\(269\) − 1.84244e8i − 0.577112i −0.957463 0.288556i \(-0.906825\pi\)
0.957463 0.288556i \(-0.0931752\pi\)
\(270\) 0 0
\(271\) 1.62257e8 0.495235 0.247618 0.968858i \(-0.420352\pi\)
0.247618 + 0.968858i \(0.420352\pi\)
\(272\) 0 0
\(273\) −5.19669e8 −1.54582
\(274\) 0 0
\(275\) 9.22480e7i 0.267481i
\(276\) 0 0
\(277\) − 9.60497e7i − 0.271529i −0.990741 0.135765i \(-0.956651\pi\)
0.990741 0.135765i \(-0.0433491\pi\)
\(278\) 0 0
\(279\) 1.25209e8 0.345159
\(280\) 0 0
\(281\) 3.61577e8 0.972139 0.486070 0.873920i \(-0.338430\pi\)
0.486070 + 0.873920i \(0.338430\pi\)
\(282\) 0 0
\(283\) − 3.43417e8i − 0.900677i −0.892858 0.450338i \(-0.851303\pi\)
0.892858 0.450338i \(-0.148697\pi\)
\(284\) 0 0
\(285\) 1.35262e8i 0.346114i
\(286\) 0 0
\(287\) −1.22811e9 −3.06656
\(288\) 0 0
\(289\) −1.12987e8 −0.275350
\(290\) 0 0
\(291\) 2.66703e8i 0.634458i
\(292\) 0 0
\(293\) − 2.82677e8i − 0.656528i −0.944586 0.328264i \(-0.893536\pi\)
0.944586 0.328264i \(-0.106464\pi\)
\(294\) 0 0
\(295\) 2.50452e8 0.568000
\(296\) 0 0
\(297\) 4.49194e7 0.0994917
\(298\) 0 0
\(299\) 4.71293e8i 1.01963i
\(300\) 0 0
\(301\) − 4.68587e8i − 0.990394i
\(302\) 0 0
\(303\) −4.31477e8 −0.891062
\(304\) 0 0
\(305\) −7.63631e8 −1.54111
\(306\) 0 0
\(307\) 1.57819e8i 0.311296i 0.987813 + 0.155648i \(0.0497466\pi\)
−0.987813 + 0.155648i \(0.950253\pi\)
\(308\) 0 0
\(309\) − 3.80462e8i − 0.733595i
\(310\) 0 0
\(311\) 5.79231e8 1.09192 0.545960 0.837811i \(-0.316165\pi\)
0.545960 + 0.837811i \(0.316165\pi\)
\(312\) 0 0
\(313\) −8.81032e7 −0.162400 −0.0812001 0.996698i \(-0.525875\pi\)
−0.0812001 + 0.996698i \(0.525875\pi\)
\(314\) 0 0
\(315\) − 4.19105e8i − 0.755502i
\(316\) 0 0
\(317\) 1.19161e8i 0.210100i 0.994467 + 0.105050i \(0.0335002\pi\)
−0.994467 + 0.105050i \(0.966500\pi\)
\(318\) 0 0
\(319\) −3.63990e8 −0.627801
\(320\) 0 0
\(321\) −6.25344e8 −1.05524
\(322\) 0 0
\(323\) − 2.50902e8i − 0.414281i
\(324\) 0 0
\(325\) 4.65937e8i 0.752896i
\(326\) 0 0
\(327\) −5.68312e7 −0.0898813
\(328\) 0 0
\(329\) −8.84126e8 −1.36876
\(330\) 0 0
\(331\) − 9.04996e8i − 1.37167i −0.727758 0.685834i \(-0.759438\pi\)
0.727758 0.685834i \(-0.240562\pi\)
\(332\) 0 0
\(333\) 6.58258e6i 0.00976881i
\(334\) 0 0
\(335\) 8.94503e8 1.29994
\(336\) 0 0
\(337\) 3.49042e8 0.496790 0.248395 0.968659i \(-0.420097\pi\)
0.248395 + 0.968659i \(0.420097\pi\)
\(338\) 0 0
\(339\) 6.23500e8i 0.869236i
\(340\) 0 0
\(341\) 3.91967e8i 0.535315i
\(342\) 0 0
\(343\) 1.90512e9 2.54913
\(344\) 0 0
\(345\) −3.80090e8 −0.498334
\(346\) 0 0
\(347\) 6.41085e8i 0.823688i 0.911254 + 0.411844i \(0.135115\pi\)
−0.911254 + 0.411844i \(0.864885\pi\)
\(348\) 0 0
\(349\) − 4.06717e8i − 0.512158i −0.966656 0.256079i \(-0.917569\pi\)
0.966656 0.256079i \(-0.0824307\pi\)
\(350\) 0 0
\(351\) 2.26884e8 0.280046
\(352\) 0 0
\(353\) 9.01273e8 1.09055 0.545274 0.838258i \(-0.316426\pi\)
0.545274 + 0.838258i \(0.316426\pi\)
\(354\) 0 0
\(355\) − 1.80560e8i − 0.214202i
\(356\) 0 0
\(357\) 7.77409e8i 0.904296i
\(358\) 0 0
\(359\) 1.69288e9 1.93106 0.965532 0.260285i \(-0.0838164\pi\)
0.965532 + 0.260285i \(0.0838164\pi\)
\(360\) 0 0
\(361\) 6.82164e8 0.763156
\(362\) 0 0
\(363\) − 3.85533e8i − 0.423046i
\(364\) 0 0
\(365\) 4.73889e8i 0.510096i
\(366\) 0 0
\(367\) −4.61417e8 −0.487262 −0.243631 0.969868i \(-0.578339\pi\)
−0.243631 + 0.969868i \(0.578339\pi\)
\(368\) 0 0
\(369\) 5.36186e8 0.555550
\(370\) 0 0
\(371\) − 9.02394e8i − 0.917461i
\(372\) 0 0
\(373\) − 6.51827e8i − 0.650356i −0.945653 0.325178i \(-0.894576\pi\)
0.945653 0.325178i \(-0.105424\pi\)
\(374\) 0 0
\(375\) 3.50500e8 0.343225
\(376\) 0 0
\(377\) −1.83848e9 −1.76711
\(378\) 0 0
\(379\) 1.31212e9i 1.23804i 0.785374 + 0.619021i \(0.212471\pi\)
−0.785374 + 0.619021i \(0.787529\pi\)
\(380\) 0 0
\(381\) 1.23192e9i 1.14115i
\(382\) 0 0
\(383\) −2.86348e8 −0.260434 −0.130217 0.991486i \(-0.541567\pi\)
−0.130217 + 0.991486i \(0.541567\pi\)
\(384\) 0 0
\(385\) 1.31201e9 1.17172
\(386\) 0 0
\(387\) 2.04582e8i 0.179424i
\(388\) 0 0
\(389\) 5.07035e8i 0.436731i 0.975867 + 0.218366i \(0.0700725\pi\)
−0.975867 + 0.218366i \(0.929927\pi\)
\(390\) 0 0
\(391\) 7.05040e8 0.596479
\(392\) 0 0
\(393\) 4.80684e8 0.399471
\(394\) 0 0
\(395\) 2.27920e9i 1.86077i
\(396\) 0 0
\(397\) 7.17590e8i 0.575585i 0.957693 + 0.287793i \(0.0929213\pi\)
−0.957693 + 0.287793i \(0.907079\pi\)
\(398\) 0 0
\(399\) 6.55968e8 0.516984
\(400\) 0 0
\(401\) −2.14391e8 −0.166036 −0.0830178 0.996548i \(-0.526456\pi\)
−0.0830178 + 0.996548i \(0.526456\pi\)
\(402\) 0 0
\(403\) 1.97979e9i 1.50679i
\(404\) 0 0
\(405\) 1.82978e8i 0.136870i
\(406\) 0 0
\(407\) −2.06069e7 −0.0151507
\(408\) 0 0
\(409\) 6.86290e8 0.495994 0.247997 0.968761i \(-0.420228\pi\)
0.247997 + 0.968761i \(0.420228\pi\)
\(410\) 0 0
\(411\) − 8.25542e7i − 0.0586534i
\(412\) 0 0
\(413\) − 1.21459e9i − 0.848410i
\(414\) 0 0
\(415\) −2.19871e9 −1.51008
\(416\) 0 0
\(417\) 1.41022e9 0.952384
\(418\) 0 0
\(419\) − 1.08929e9i − 0.723428i −0.932289 0.361714i \(-0.882192\pi\)
0.932289 0.361714i \(-0.117808\pi\)
\(420\) 0 0
\(421\) − 1.47613e9i − 0.964136i −0.876134 0.482068i \(-0.839886\pi\)
0.876134 0.482068i \(-0.160114\pi\)
\(422\) 0 0
\(423\) 3.86004e8 0.247971
\(424\) 0 0
\(425\) 6.97027e8 0.440441
\(426\) 0 0
\(427\) 3.70330e9i 2.30193i
\(428\) 0 0
\(429\) 7.10264e8i 0.434329i
\(430\) 0 0
\(431\) 1.47143e8 0.0885256 0.0442628 0.999020i \(-0.485906\pi\)
0.0442628 + 0.999020i \(0.485906\pi\)
\(432\) 0 0
\(433\) 1.31693e9 0.779572 0.389786 0.920905i \(-0.372549\pi\)
0.389786 + 0.920905i \(0.372549\pi\)
\(434\) 0 0
\(435\) − 1.48270e9i − 0.863659i
\(436\) 0 0
\(437\) − 5.94904e8i − 0.341006i
\(438\) 0 0
\(439\) 6.94899e8 0.392009 0.196004 0.980603i \(-0.437203\pi\)
0.196004 + 0.980603i \(0.437203\pi\)
\(440\) 0 0
\(441\) −1.43212e9 −0.795144
\(442\) 0 0
\(443\) 2.53160e9i 1.38351i 0.722133 + 0.691754i \(0.243162\pi\)
−0.722133 + 0.691754i \(0.756838\pi\)
\(444\) 0 0
\(445\) − 3.30849e9i − 1.77979i
\(446\) 0 0
\(447\) −7.60330e7 −0.0402648
\(448\) 0 0
\(449\) −3.62399e9 −1.88940 −0.944701 0.327934i \(-0.893648\pi\)
−0.944701 + 0.327934i \(0.893648\pi\)
\(450\) 0 0
\(451\) 1.67854e9i 0.861614i
\(452\) 0 0
\(453\) 1.55702e9i 0.786954i
\(454\) 0 0
\(455\) 6.62686e9 3.29813
\(456\) 0 0
\(457\) 2.06640e9 1.01277 0.506383 0.862309i \(-0.330982\pi\)
0.506383 + 0.862309i \(0.330982\pi\)
\(458\) 0 0
\(459\) − 3.39412e8i − 0.163826i
\(460\) 0 0
\(461\) 1.53131e9i 0.727966i 0.931406 + 0.363983i \(0.118583\pi\)
−0.931406 + 0.363983i \(0.881417\pi\)
\(462\) 0 0
\(463\) −2.09022e8 −0.0978721 −0.0489360 0.998802i \(-0.515583\pi\)
−0.0489360 + 0.998802i \(0.515583\pi\)
\(464\) 0 0
\(465\) −1.59667e9 −0.736428
\(466\) 0 0
\(467\) 1.40416e9i 0.637979i 0.947758 + 0.318989i \(0.103343\pi\)
−0.947758 + 0.318989i \(0.896657\pi\)
\(468\) 0 0
\(469\) − 4.33798e9i − 1.94170i
\(470\) 0 0
\(471\) 2.47772e9 1.09264
\(472\) 0 0
\(473\) −6.40447e8 −0.278272
\(474\) 0 0
\(475\) − 5.88142e8i − 0.251799i
\(476\) 0 0
\(477\) 3.93979e8i 0.166211i
\(478\) 0 0
\(479\) −3.95376e9 −1.64375 −0.821876 0.569666i \(-0.807073\pi\)
−0.821876 + 0.569666i \(0.807073\pi\)
\(480\) 0 0
\(481\) −1.04083e8 −0.0426456
\(482\) 0 0
\(483\) 1.84329e9i 0.744351i
\(484\) 0 0
\(485\) − 3.40102e9i − 1.35367i
\(486\) 0 0
\(487\) −7.40238e7 −0.0290416 −0.0145208 0.999895i \(-0.504622\pi\)
−0.0145208 + 0.999895i \(0.504622\pi\)
\(488\) 0 0
\(489\) −2.48601e9 −0.961439
\(490\) 0 0
\(491\) − 4.08459e9i − 1.55727i −0.627479 0.778634i \(-0.715913\pi\)
0.627479 0.778634i \(-0.284087\pi\)
\(492\) 0 0
\(493\) 2.75031e9i 1.03375i
\(494\) 0 0
\(495\) −5.72816e8 −0.212274
\(496\) 0 0
\(497\) −8.75644e8 −0.319949
\(498\) 0 0
\(499\) 3.35811e9i 1.20988i 0.796270 + 0.604941i \(0.206803\pi\)
−0.796270 + 0.604941i \(0.793197\pi\)
\(500\) 0 0
\(501\) − 4.38565e8i − 0.155812i
\(502\) 0 0
\(503\) −2.31504e9 −0.811093 −0.405546 0.914074i \(-0.632919\pi\)
−0.405546 + 0.914074i \(0.632919\pi\)
\(504\) 0 0
\(505\) 5.50222e9 1.90116
\(506\) 0 0
\(507\) 1.89327e9i 0.645186i
\(508\) 0 0
\(509\) − 2.61529e9i − 0.879039i −0.898233 0.439520i \(-0.855149\pi\)
0.898233 0.439520i \(-0.144851\pi\)
\(510\) 0 0
\(511\) 2.29817e9 0.761920
\(512\) 0 0
\(513\) −2.86391e8 −0.0936589
\(514\) 0 0
\(515\) 4.85168e9i 1.56519i
\(516\) 0 0
\(517\) 1.20839e9i 0.384583i
\(518\) 0 0
\(519\) −1.28021e9 −0.401972
\(520\) 0 0
\(521\) 9.75395e8 0.302168 0.151084 0.988521i \(-0.451724\pi\)
0.151084 + 0.988521i \(0.451724\pi\)
\(522\) 0 0
\(523\) − 2.45790e9i − 0.751291i −0.926763 0.375646i \(-0.877421\pi\)
0.926763 0.375646i \(-0.122579\pi\)
\(524\) 0 0
\(525\) 1.82233e9i 0.549630i
\(526\) 0 0
\(527\) 2.96171e9 0.881465
\(528\) 0 0
\(529\) −1.73313e9 −0.509022
\(530\) 0 0
\(531\) 5.30284e8i 0.153701i
\(532\) 0 0
\(533\) 8.47814e9i 2.42524i
\(534\) 0 0
\(535\) 7.97444e9 2.25145
\(536\) 0 0
\(537\) −2.16022e9 −0.601988
\(538\) 0 0
\(539\) − 4.48329e9i − 1.23321i
\(540\) 0 0
\(541\) − 1.30734e9i − 0.354977i −0.984123 0.177488i \(-0.943203\pi\)
0.984123 0.177488i \(-0.0567972\pi\)
\(542\) 0 0
\(543\) −5.31211e8 −0.142386
\(544\) 0 0
\(545\) 7.24715e8 0.191770
\(546\) 0 0
\(547\) − 5.91915e9i − 1.54634i −0.634201 0.773168i \(-0.718671\pi\)
0.634201 0.773168i \(-0.281329\pi\)
\(548\) 0 0
\(549\) − 1.61684e9i − 0.417026i
\(550\) 0 0
\(551\) 2.32068e9 0.590995
\(552\) 0 0
\(553\) 1.10532e10 2.77940
\(554\) 0 0
\(555\) − 8.39416e7i − 0.0208426i
\(556\) 0 0
\(557\) 4.93049e9i 1.20892i 0.796636 + 0.604459i \(0.206611\pi\)
−0.796636 + 0.604459i \(0.793389\pi\)
\(558\) 0 0
\(559\) −3.23484e9 −0.783271
\(560\) 0 0
\(561\) 1.06253e9 0.254081
\(562\) 0 0
\(563\) 6.84747e8i 0.161715i 0.996726 + 0.0808576i \(0.0257659\pi\)
−0.996726 + 0.0808576i \(0.974234\pi\)
\(564\) 0 0
\(565\) − 7.95092e9i − 1.85459i
\(566\) 0 0
\(567\) 8.87371e8 0.204439
\(568\) 0 0
\(569\) −5.26374e9 −1.19785 −0.598924 0.800806i \(-0.704405\pi\)
−0.598924 + 0.800806i \(0.704405\pi\)
\(570\) 0 0
\(571\) − 3.59028e9i − 0.807053i −0.914968 0.403527i \(-0.867784\pi\)
0.914968 0.403527i \(-0.132216\pi\)
\(572\) 0 0
\(573\) 2.37169e9i 0.526644i
\(574\) 0 0
\(575\) 1.65269e9 0.362539
\(576\) 0 0
\(577\) 3.91183e9 0.847744 0.423872 0.905722i \(-0.360671\pi\)
0.423872 + 0.905722i \(0.360671\pi\)
\(578\) 0 0
\(579\) − 3.49995e9i − 0.749353i
\(580\) 0 0
\(581\) 1.06629e10i 2.25558i
\(582\) 0 0
\(583\) −1.23336e9 −0.257780
\(584\) 0 0
\(585\) −2.89324e9 −0.597502
\(586\) 0 0
\(587\) 3.88957e9i 0.793721i 0.917879 + 0.396860i \(0.129900\pi\)
−0.917879 + 0.396860i \(0.870100\pi\)
\(588\) 0 0
\(589\) − 2.49905e9i − 0.503932i
\(590\) 0 0
\(591\) −3.28765e9 −0.655133
\(592\) 0 0
\(593\) 4.76138e9 0.937652 0.468826 0.883291i \(-0.344677\pi\)
0.468826 + 0.883291i \(0.344677\pi\)
\(594\) 0 0
\(595\) − 9.91357e9i − 1.92939i
\(596\) 0 0
\(597\) 3.88280e9i 0.746852i
\(598\) 0 0
\(599\) −5.22390e9 −0.993118 −0.496559 0.868003i \(-0.665403\pi\)
−0.496559 + 0.868003i \(0.665403\pi\)
\(600\) 0 0
\(601\) −8.36008e9 −1.57091 −0.785453 0.618922i \(-0.787570\pi\)
−0.785453 + 0.618922i \(0.787570\pi\)
\(602\) 0 0
\(603\) 1.89393e9i 0.351766i
\(604\) 0 0
\(605\) 4.91634e9i 0.902606i
\(606\) 0 0
\(607\) 5.78582e9 1.05004 0.525018 0.851091i \(-0.324058\pi\)
0.525018 + 0.851091i \(0.324058\pi\)
\(608\) 0 0
\(609\) −7.19052e9 −1.29003
\(610\) 0 0
\(611\) 6.10347e9i 1.08251i
\(612\) 0 0
\(613\) − 1.73833e9i − 0.304803i −0.988319 0.152402i \(-0.951299\pi\)
0.988319 0.152402i \(-0.0487007\pi\)
\(614\) 0 0
\(615\) −6.83748e9 −1.18531
\(616\) 0 0
\(617\) −4.44016e9 −0.761028 −0.380514 0.924775i \(-0.624253\pi\)
−0.380514 + 0.924775i \(0.624253\pi\)
\(618\) 0 0
\(619\) 1.64970e9i 0.279568i 0.990182 + 0.139784i \(0.0446408\pi\)
−0.990182 + 0.139784i \(0.955359\pi\)
\(620\) 0 0
\(621\) − 8.04766e8i − 0.134849i
\(622\) 0 0
\(623\) −1.60448e10 −2.65844
\(624\) 0 0
\(625\) −7.62755e9 −1.24970
\(626\) 0 0
\(627\) − 8.96551e8i − 0.145258i
\(628\) 0 0
\(629\) 1.55706e8i 0.0249475i
\(630\) 0 0
\(631\) −3.28962e9 −0.521246 −0.260623 0.965441i \(-0.583928\pi\)
−0.260623 + 0.965441i \(0.583928\pi\)
\(632\) 0 0
\(633\) −3.16223e9 −0.495541
\(634\) 0 0
\(635\) − 1.57095e10i − 2.43475i
\(636\) 0 0
\(637\) − 2.26447e10i − 3.47119i
\(638\) 0 0
\(639\) 3.82300e8 0.0579631
\(640\) 0 0
\(641\) 7.66402e9 1.14935 0.574677 0.818381i \(-0.305128\pi\)
0.574677 + 0.818381i \(0.305128\pi\)
\(642\) 0 0
\(643\) − 1.16781e7i − 0.00173234i −1.00000 0.000866170i \(-0.999724\pi\)
1.00000 0.000866170i \(-0.000275710\pi\)
\(644\) 0 0
\(645\) − 2.60885e9i − 0.382816i
\(646\) 0 0
\(647\) −1.10751e9 −0.160762 −0.0803811 0.996764i \(-0.525614\pi\)
−0.0803811 + 0.996764i \(0.525614\pi\)
\(648\) 0 0
\(649\) −1.66006e9 −0.238379
\(650\) 0 0
\(651\) 7.74321e9i 1.09999i
\(652\) 0 0
\(653\) − 5.58523e9i − 0.784955i −0.919762 0.392477i \(-0.871618\pi\)
0.919762 0.392477i \(-0.128382\pi\)
\(654\) 0 0
\(655\) −6.12971e9 −0.852306
\(656\) 0 0
\(657\) −1.00337e9 −0.138032
\(658\) 0 0
\(659\) 7.10525e9i 0.967120i 0.875311 + 0.483560i \(0.160657\pi\)
−0.875311 + 0.483560i \(0.839343\pi\)
\(660\) 0 0
\(661\) − 3.08453e9i − 0.415416i −0.978191 0.207708i \(-0.933400\pi\)
0.978191 0.207708i \(-0.0666005\pi\)
\(662\) 0 0
\(663\) 5.36676e9 0.715179
\(664\) 0 0
\(665\) −8.36495e9 −1.10303
\(666\) 0 0
\(667\) 6.52116e9i 0.850911i
\(668\) 0 0
\(669\) 8.52979e9i 1.10140i
\(670\) 0 0
\(671\) 5.06153e9 0.646775
\(672\) 0 0
\(673\) 1.17552e10 1.48655 0.743273 0.668988i \(-0.233272\pi\)
0.743273 + 0.668988i \(0.233272\pi\)
\(674\) 0 0
\(675\) − 7.95619e8i − 0.0995731i
\(676\) 0 0
\(677\) 4.23386e9i 0.524416i 0.965011 + 0.262208i \(0.0844507\pi\)
−0.965011 + 0.262208i \(0.915549\pi\)
\(678\) 0 0
\(679\) −1.64936e10 −2.02195
\(680\) 0 0
\(681\) 1.74673e9 0.211939
\(682\) 0 0
\(683\) 3.23179e9i 0.388124i 0.980989 + 0.194062i \(0.0621663\pi\)
−0.980989 + 0.194062i \(0.937834\pi\)
\(684\) 0 0
\(685\) 1.05274e9i 0.125142i
\(686\) 0 0
\(687\) −5.56803e9 −0.655168
\(688\) 0 0
\(689\) −6.22958e9 −0.725590
\(690\) 0 0
\(691\) − 2.19695e9i − 0.253307i −0.991947 0.126653i \(-0.959576\pi\)
0.991947 0.126653i \(-0.0404236\pi\)
\(692\) 0 0
\(693\) 2.77793e9i 0.317069i
\(694\) 0 0
\(695\) −1.79833e10 −2.03199
\(696\) 0 0
\(697\) 1.26830e10 1.41876
\(698\) 0 0
\(699\) − 8.64607e9i − 0.957522i
\(700\) 0 0
\(701\) 5.54482e9i 0.607959i 0.952679 + 0.303980i \(0.0983155\pi\)
−0.952679 + 0.303980i \(0.901684\pi\)
\(702\) 0 0
\(703\) 1.31382e8 0.0142624
\(704\) 0 0
\(705\) −4.92235e9 −0.529067
\(706\) 0 0
\(707\) − 2.66836e10i − 2.83972i
\(708\) 0 0
\(709\) − 1.08253e10i − 1.14071i −0.821397 0.570357i \(-0.806805\pi\)
0.821397 0.570357i \(-0.193195\pi\)
\(710\) 0 0
\(711\) −4.82576e9 −0.503527
\(712\) 0 0
\(713\) 7.02240e9 0.725558
\(714\) 0 0
\(715\) − 9.05734e9i − 0.926679i
\(716\) 0 0
\(717\) 3.36516e9i 0.340948i
\(718\) 0 0
\(719\) 1.08562e10 1.08925 0.544623 0.838681i \(-0.316673\pi\)
0.544623 + 0.838681i \(0.316673\pi\)
\(720\) 0 0
\(721\) 2.35287e10 2.33789
\(722\) 0 0
\(723\) − 6.69144e9i − 0.658469i
\(724\) 0 0
\(725\) 6.44704e9i 0.628315i
\(726\) 0 0
\(727\) −1.04714e10 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(728\) 0 0
\(729\) −3.87420e8 −0.0370370
\(730\) 0 0
\(731\) 4.83922e9i 0.458210i
\(732\) 0 0
\(733\) 1.93042e10i 1.81045i 0.424931 + 0.905226i \(0.360298\pi\)
−0.424931 + 0.905226i \(0.639702\pi\)
\(734\) 0 0
\(735\) 1.82626e10 1.69651
\(736\) 0 0
\(737\) −5.92898e9 −0.545562
\(738\) 0 0
\(739\) − 2.64009e9i − 0.240637i −0.992735 0.120319i \(-0.961608\pi\)
0.992735 0.120319i \(-0.0383916\pi\)
\(740\) 0 0
\(741\) − 4.52840e9i − 0.408866i
\(742\) 0 0
\(743\) 2.00716e10 1.79523 0.897617 0.440777i \(-0.145297\pi\)
0.897617 + 0.440777i \(0.145297\pi\)
\(744\) 0 0
\(745\) 9.69578e8 0.0859084
\(746\) 0 0
\(747\) − 4.65534e9i − 0.408629i
\(748\) 0 0
\(749\) − 3.86728e10i − 3.36294i
\(750\) 0 0
\(751\) −1.78345e10 −1.53646 −0.768229 0.640175i \(-0.778862\pi\)
−0.768229 + 0.640175i \(0.778862\pi\)
\(752\) 0 0
\(753\) −4.68110e9 −0.399545
\(754\) 0 0
\(755\) − 1.98552e10i − 1.67904i
\(756\) 0 0
\(757\) 2.59023e9i 0.217021i 0.994095 + 0.108511i \(0.0346082\pi\)
−0.994095 + 0.108511i \(0.965392\pi\)
\(758\) 0 0
\(759\) 2.51933e9 0.209141
\(760\) 0 0
\(761\) 1.47456e10 1.21287 0.606436 0.795132i \(-0.292599\pi\)
0.606436 + 0.795132i \(0.292599\pi\)
\(762\) 0 0
\(763\) − 3.51458e9i − 0.286442i
\(764\) 0 0
\(765\) 4.32820e9i 0.349536i
\(766\) 0 0
\(767\) −8.38482e9 −0.670980
\(768\) 0 0
\(769\) 2.16839e9 0.171947 0.0859737 0.996297i \(-0.472600\pi\)
0.0859737 + 0.996297i \(0.472600\pi\)
\(770\) 0 0
\(771\) − 2.61575e9i − 0.205545i
\(772\) 0 0
\(773\) 2.19037e10i 1.70565i 0.522198 + 0.852824i \(0.325112\pi\)
−0.522198 + 0.852824i \(0.674888\pi\)
\(774\) 0 0
\(775\) 6.94258e9 0.535753
\(776\) 0 0
\(777\) −4.07083e8 −0.0311322
\(778\) 0 0
\(779\) − 1.07018e10i − 0.811101i
\(780\) 0 0
\(781\) 1.19680e9i 0.0898963i
\(782\) 0 0
\(783\) 3.13933e9 0.233707
\(784\) 0 0
\(785\) −3.15961e10 −2.33125
\(786\) 0 0
\(787\) 9.86968e9i 0.721758i 0.932613 + 0.360879i \(0.117523\pi\)
−0.932613 + 0.360879i \(0.882477\pi\)
\(788\) 0 0
\(789\) 8.72284e9i 0.632249i
\(790\) 0 0
\(791\) −3.85588e10 −2.77016
\(792\) 0 0
\(793\) 2.55654e10 1.82052
\(794\) 0 0
\(795\) − 5.02405e9i − 0.354625i
\(796\) 0 0
\(797\) 1.69316e10i 1.18466i 0.805695 + 0.592331i \(0.201792\pi\)
−0.805695 + 0.592331i \(0.798208\pi\)
\(798\) 0 0
\(799\) 9.13060e9 0.633265
\(800\) 0 0
\(801\) 7.00507e9 0.481614
\(802\) 0 0
\(803\) − 3.14105e9i − 0.214077i
\(804\) 0 0
\(805\) − 2.35057e10i − 1.58814i
\(806\) 0 0
\(807\) −4.97459e9 −0.333196
\(808\) 0 0
\(809\) −5.74007e8 −0.0381151 −0.0190576 0.999818i \(-0.506067\pi\)
−0.0190576 + 0.999818i \(0.506067\pi\)
\(810\) 0 0
\(811\) 2.00117e10i 1.31738i 0.752415 + 0.658689i \(0.228889\pi\)
−0.752415 + 0.658689i \(0.771111\pi\)
\(812\) 0 0
\(813\) − 4.38095e9i − 0.285924i
\(814\) 0 0
\(815\) 3.17018e10 2.05131
\(816\) 0 0
\(817\) 4.08328e9 0.261958
\(818\) 0 0
\(819\) 1.40311e10i 0.892477i
\(820\) 0 0
\(821\) − 8.15447e9i − 0.514274i −0.966375 0.257137i \(-0.917221\pi\)
0.966375 0.257137i \(-0.0827791\pi\)
\(822\) 0 0
\(823\) −9.80708e9 −0.613254 −0.306627 0.951830i \(-0.599200\pi\)
−0.306627 + 0.951830i \(0.599200\pi\)
\(824\) 0 0
\(825\) 2.49070e9 0.154430
\(826\) 0 0
\(827\) 1.05065e10i 0.645937i 0.946410 + 0.322969i \(0.104681\pi\)
−0.946410 + 0.322969i \(0.895319\pi\)
\(828\) 0 0
\(829\) − 1.45134e10i − 0.884766i −0.896826 0.442383i \(-0.854133\pi\)
0.896826 0.442383i \(-0.145867\pi\)
\(830\) 0 0
\(831\) −2.59334e9 −0.156767
\(832\) 0 0
\(833\) −3.38757e10 −2.03063
\(834\) 0 0
\(835\) 5.59262e9i 0.332439i
\(836\) 0 0
\(837\) − 3.38063e9i − 0.199278i
\(838\) 0 0
\(839\) −1.98293e9 −0.115915 −0.0579576 0.998319i \(-0.518459\pi\)
−0.0579576 + 0.998319i \(0.518459\pi\)
\(840\) 0 0
\(841\) −8.18867e9 −0.474709
\(842\) 0 0
\(843\) − 9.76257e9i − 0.561265i
\(844\) 0 0
\(845\) − 2.41431e10i − 1.37656i
\(846\) 0 0
\(847\) 2.38423e10 1.34820
\(848\) 0 0
\(849\) −9.27225e9 −0.520006
\(850\) 0 0
\(851\) 3.69188e8i 0.0205350i
\(852\) 0 0
\(853\) − 7.37240e9i − 0.406712i −0.979105 0.203356i \(-0.934815\pi\)
0.979105 0.203356i \(-0.0651849\pi\)
\(854\) 0 0
\(855\) 3.65208e9 0.199829
\(856\) 0 0
\(857\) −2.41520e10 −1.31075 −0.655374 0.755304i \(-0.727489\pi\)
−0.655374 + 0.755304i \(0.727489\pi\)
\(858\) 0 0
\(859\) − 3.64471e10i − 1.96195i −0.194145 0.980973i \(-0.562193\pi\)
0.194145 0.980973i \(-0.437807\pi\)
\(860\) 0 0
\(861\) 3.31590e10i 1.77048i
\(862\) 0 0
\(863\) 4.59737e9 0.243485 0.121742 0.992562i \(-0.461152\pi\)
0.121742 + 0.992562i \(0.461152\pi\)
\(864\) 0 0
\(865\) 1.63253e10 0.857642
\(866\) 0 0
\(867\) 3.05064e9i 0.158973i
\(868\) 0 0
\(869\) − 1.51071e10i − 0.780930i
\(870\) 0 0
\(871\) −2.99468e10 −1.53563
\(872\) 0 0
\(873\) 7.20099e9 0.366305
\(874\) 0 0
\(875\) 2.16758e10i 1.09382i
\(876\) 0 0
\(877\) 1.57473e10i 0.788327i 0.919040 + 0.394164i \(0.128966\pi\)
−0.919040 + 0.394164i \(0.871034\pi\)
\(878\) 0 0
\(879\) −7.63227e9 −0.379047
\(880\) 0 0
\(881\) 2.30534e10 1.13584 0.567922 0.823082i \(-0.307748\pi\)
0.567922 + 0.823082i \(0.307748\pi\)
\(882\) 0 0
\(883\) 5.63512e9i 0.275449i 0.990471 + 0.137724i \(0.0439788\pi\)
−0.990471 + 0.137724i \(0.956021\pi\)
\(884\) 0 0
\(885\) − 6.76222e9i − 0.327935i
\(886\) 0 0
\(887\) 1.11246e10 0.535245 0.267623 0.963524i \(-0.413762\pi\)
0.267623 + 0.963524i \(0.413762\pi\)
\(888\) 0 0
\(889\) −7.61848e10 −3.63674
\(890\) 0 0
\(891\) − 1.21282e9i − 0.0574415i
\(892\) 0 0
\(893\) − 7.70429e9i − 0.362036i
\(894\) 0 0
\(895\) 2.75473e10 1.28439
\(896\) 0 0
\(897\) 1.27249e10 0.588683
\(898\) 0 0
\(899\) 2.73939e10i 1.25746i
\(900\) 0 0
\(901\) 9.31926e9i 0.424468i
\(902\) 0 0
\(903\) −1.26519e10 −0.571804
\(904\) 0 0
\(905\) 6.77404e9 0.303793
\(906\) 0 0
\(907\) 7.73698e9i 0.344307i 0.985070 + 0.172154i \(0.0550725\pi\)
−0.985070 + 0.172154i \(0.944927\pi\)
\(908\) 0 0
\(909\) 1.16499e10i 0.514455i
\(910\) 0 0
\(911\) 1.03842e10 0.455051 0.227526 0.973772i \(-0.426936\pi\)
0.227526 + 0.973772i \(0.426936\pi\)
\(912\) 0 0
\(913\) 1.45736e10 0.633752
\(914\) 0 0
\(915\) 2.06180e10i 0.889761i
\(916\) 0 0
\(917\) 2.97266e10i 1.27307i
\(918\) 0 0
\(919\) −8.08020e9 −0.343414 −0.171707 0.985148i \(-0.554928\pi\)
−0.171707 + 0.985148i \(0.554928\pi\)
\(920\) 0 0
\(921\) 4.26110e9 0.179727
\(922\) 0 0
\(923\) 6.04491e9i 0.253037i
\(924\) 0 0
\(925\) 3.64992e8i 0.0151631i
\(926\) 0 0
\(927\) −1.02725e10 −0.423541
\(928\) 0 0
\(929\) 1.28931e10 0.527598 0.263799 0.964578i \(-0.415024\pi\)
0.263799 + 0.964578i \(0.415024\pi\)
\(930\) 0 0
\(931\) 2.85839e10i 1.16091i
\(932\) 0 0
\(933\) − 1.56392e10i − 0.630420i
\(934\) 0 0
\(935\) −1.35495e10 −0.542104
\(936\) 0 0
\(937\) −7.17106e9 −0.284770 −0.142385 0.989811i \(-0.545477\pi\)
−0.142385 + 0.989811i \(0.545477\pi\)
\(938\) 0 0
\(939\) 2.37879e9i 0.0937618i
\(940\) 0 0
\(941\) 2.40976e9i 0.0942781i 0.998888 + 0.0471391i \(0.0150104\pi\)
−0.998888 + 0.0471391i \(0.984990\pi\)
\(942\) 0 0
\(943\) 3.00723e10 1.16782
\(944\) 0 0
\(945\) −1.13158e10 −0.436189
\(946\) 0 0
\(947\) 4.20590e10i 1.60929i 0.593758 + 0.804644i \(0.297644\pi\)
−0.593758 + 0.804644i \(0.702356\pi\)
\(948\) 0 0
\(949\) − 1.58652e10i − 0.602578i
\(950\) 0 0
\(951\) 3.21734e9 0.121301
\(952\) 0 0
\(953\) 2.79027e10 1.04429 0.522145 0.852857i \(-0.325132\pi\)
0.522145 + 0.852857i \(0.325132\pi\)
\(954\) 0 0
\(955\) − 3.02440e10i − 1.12364i
\(956\) 0 0
\(957\) 9.82772e9i 0.362461i
\(958\) 0 0
\(959\) 5.10535e9 0.186922
\(960\) 0 0
\(961\) 1.98684e9 0.0722155
\(962\) 0 0
\(963\) 1.68843e10i 0.609243i
\(964\) 0 0
\(965\) 4.46316e10i 1.59881i
\(966\) 0 0
\(967\) 2.50078e10 0.889371 0.444686 0.895687i \(-0.353315\pi\)
0.444686 + 0.895687i \(0.353315\pi\)
\(968\) 0 0
\(969\) −6.77435e9 −0.239185
\(970\) 0 0
\(971\) 3.65828e10i 1.28236i 0.767391 + 0.641179i \(0.221554\pi\)
−0.767391 + 0.641179i \(0.778446\pi\)
\(972\) 0 0
\(973\) 8.72117e10i 3.03515i
\(974\) 0 0
\(975\) 1.25803e10 0.434685
\(976\) 0 0
\(977\) 2.86412e10 0.982563 0.491281 0.871001i \(-0.336529\pi\)
0.491281 + 0.871001i \(0.336529\pi\)
\(978\) 0 0
\(979\) 2.19295e10i 0.746945i
\(980\) 0 0
\(981\) 1.53444e9i 0.0518930i
\(982\) 0 0
\(983\) −3.81805e10 −1.28205 −0.641024 0.767521i \(-0.721490\pi\)
−0.641024 + 0.767521i \(0.721490\pi\)
\(984\) 0 0
\(985\) 4.19244e10 1.39778
\(986\) 0 0
\(987\) 2.38714e10i 0.790256i
\(988\) 0 0
\(989\) 1.14741e10i 0.377165i
\(990\) 0 0
\(991\) 2.06103e10 0.672709 0.336354 0.941736i \(-0.390806\pi\)
0.336354 + 0.941736i \(0.390806\pi\)
\(992\) 0 0
\(993\) −2.44349e10 −0.791932
\(994\) 0 0
\(995\) − 4.95137e10i − 1.59347i
\(996\) 0 0
\(997\) − 2.00020e10i − 0.639204i −0.947552 0.319602i \(-0.896451\pi\)
0.947552 0.319602i \(-0.103549\pi\)
\(998\) 0 0
\(999\) 1.77730e8 0.00564002
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.193.4 yes 8
4.3 odd 2 384.8.d.c.193.8 yes 8
8.3 odd 2 384.8.d.c.193.1 8
8.5 even 2 inner 384.8.d.d.193.5 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.8.d.c.193.1 8 8.3 odd 2
384.8.d.c.193.8 yes 8 4.3 odd 2
384.8.d.d.193.4 yes 8 1.1 even 1 trivial
384.8.d.d.193.5 yes 8 8.5 even 2 inner