Properties

Label 384.8.d.d.193.3
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 193.3
Root \(-0.596953i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.8.d.d.193.6

$q$-expansion

\(f(q)\) \(=\) \(q-27.0000i q^{3} +135.999i q^{5} -678.568 q^{7} -729.000 q^{9} +O(q^{10})\) \(q-27.0000i q^{3} +135.999i q^{5} -678.568 q^{7} -729.000 q^{9} +5738.26i q^{11} +7354.35i q^{13} +3671.96 q^{15} -15568.9 q^{17} +8423.97i q^{19} +18321.3i q^{21} -34677.9 q^{23} +59629.4 q^{25} +19683.0i q^{27} +133310. i q^{29} -5442.43 q^{31} +154933. q^{33} -92284.3i q^{35} -416604. i q^{37} +198567. q^{39} +432630. q^{41} +410933. i q^{43} -99143.0i q^{45} -859348. q^{47} -363089. q^{49} +420361. i q^{51} -756051. i q^{53} -780395. q^{55} +227447. q^{57} -2.13194e6i q^{59} -296171. i q^{61} +494676. q^{63} -1.00018e6 q^{65} +2.17068e6i q^{67} +936302. i q^{69} +594319. q^{71} +1.08088e6 q^{73} -1.60999e6i q^{75} -3.89379e6i q^{77} +1.51011e6 q^{79} +531441. q^{81} -3.23443e6i q^{83} -2.11735e6i q^{85} +3.59936e6 q^{87} +4.95231e6 q^{89} -4.99042e6i q^{91} +146945. i q^{93} -1.14565e6 q^{95} +1.35947e7 q^{97} -4.18319e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2880 q^{7} - 5832 q^{9} + O(q^{10}) \) \( 8 q + 2880 q^{7} - 5832 q^{9} - 6048 q^{15} + 22896 q^{17} + 207360 q^{23} - 204696 q^{25} - 17856 q^{31} + 7776 q^{33} - 116640 q^{39} + 687056 q^{41} - 1987200 q^{47} + 4815560 q^{49} + 2077056 q^{55} + 878688 q^{57} - 2099520 q^{63} + 12871808 q^{65} - 6336000 q^{71} + 8920752 q^{73} + 1251648 q^{79} + 4251528 q^{81} - 13385952 q^{87} + 4447408 q^{89} - 31607424 q^{95} - 14157584 q^{97} + O(q^{100}) \)

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
\(5\) 135.999i 0.486564i 0.969956 + 0.243282i \(0.0782240\pi\)
−0.969956 + 0.243282i \(0.921776\pi\)
\(6\) 0 0
\(7\) −678.568 −0.747739 −0.373870 0.927481i \(-0.621969\pi\)
−0.373870 + 0.927481i \(0.621969\pi\)
\(8\) 0 0
\(9\) −729.000 −0.333333
\(10\) 0 0
\(11\) 5738.26i 1.29989i 0.759983 + 0.649943i \(0.225207\pi\)
−0.759983 + 0.649943i \(0.774793\pi\)
\(12\) 0 0
\(13\) 7354.35i 0.928416i 0.885726 + 0.464208i \(0.153661\pi\)
−0.885726 + 0.464208i \(0.846339\pi\)
\(14\) 0 0
\(15\) 3671.96 0.280918
\(16\) 0 0
\(17\) −15568.9 −0.768576 −0.384288 0.923213i \(-0.625553\pi\)
−0.384288 + 0.923213i \(0.625553\pi\)
\(18\) 0 0
\(19\) 8423.97i 0.281760i 0.990027 + 0.140880i \(0.0449932\pi\)
−0.990027 + 0.140880i \(0.955007\pi\)
\(20\) 0 0
\(21\) 18321.3i 0.431707i
\(22\) 0 0
\(23\) −34677.9 −0.594299 −0.297149 0.954831i \(-0.596036\pi\)
−0.297149 + 0.954831i \(0.596036\pi\)
\(24\) 0 0
\(25\) 59629.4 0.763256
\(26\) 0 0
\(27\) 19683.0i 0.192450i
\(28\) 0 0
\(29\) 133310.i 1.01501i 0.861650 + 0.507503i \(0.169431\pi\)
−0.861650 + 0.507503i \(0.830569\pi\)
\(30\) 0 0
\(31\) −5442.43 −0.0328115 −0.0164058 0.999865i \(-0.505222\pi\)
−0.0164058 + 0.999865i \(0.505222\pi\)
\(32\) 0 0
\(33\) 154933. 0.750490
\(34\) 0 0
\(35\) − 92284.3i − 0.363823i
\(36\) 0 0
\(37\) − 416604.i − 1.35213i −0.736843 0.676064i \(-0.763684\pi\)
0.736843 0.676064i \(-0.236316\pi\)
\(38\) 0 0
\(39\) 198567. 0.536021
\(40\) 0 0
\(41\) 432630. 0.980331 0.490165 0.871629i \(-0.336936\pi\)
0.490165 + 0.871629i \(0.336936\pi\)
\(42\) 0 0
\(43\) 410933.i 0.788190i 0.919070 + 0.394095i \(0.128942\pi\)
−0.919070 + 0.394095i \(0.871058\pi\)
\(44\) 0 0
\(45\) − 99143.0i − 0.162188i
\(46\) 0 0
\(47\) −859348. −1.20733 −0.603665 0.797238i \(-0.706294\pi\)
−0.603665 + 0.797238i \(0.706294\pi\)
\(48\) 0 0
\(49\) −363089. −0.440886
\(50\) 0 0
\(51\) 420361.i 0.443738i
\(52\) 0 0
\(53\) − 756051.i − 0.697566i −0.937203 0.348783i \(-0.886595\pi\)
0.937203 0.348783i \(-0.113405\pi\)
\(54\) 0 0
\(55\) −780395. −0.632478
\(56\) 0 0
\(57\) 227447. 0.162674
\(58\) 0 0
\(59\) − 2.13194e6i − 1.35143i −0.737164 0.675714i \(-0.763836\pi\)
0.737164 0.675714i \(-0.236164\pi\)
\(60\) 0 0
\(61\) − 296171.i − 0.167066i −0.996505 0.0835331i \(-0.973380\pi\)
0.996505 0.0835331i \(-0.0266204\pi\)
\(62\) 0 0
\(63\) 494676. 0.249246
\(64\) 0 0
\(65\) −1.00018e6 −0.451733
\(66\) 0 0
\(67\) 2.17068e6i 0.881728i 0.897574 + 0.440864i \(0.145328\pi\)
−0.897574 + 0.440864i \(0.854672\pi\)
\(68\) 0 0
\(69\) 936302.i 0.343119i
\(70\) 0 0
\(71\) 594319. 0.197068 0.0985339 0.995134i \(-0.468585\pi\)
0.0985339 + 0.995134i \(0.468585\pi\)
\(72\) 0 0
\(73\) 1.08088e6 0.325198 0.162599 0.986692i \(-0.448012\pi\)
0.162599 + 0.986692i \(0.448012\pi\)
\(74\) 0 0
\(75\) − 1.60999e6i − 0.440666i
\(76\) 0 0
\(77\) − 3.89379e6i − 0.971976i
\(78\) 0 0
\(79\) 1.51011e6 0.344599 0.172300 0.985045i \(-0.444880\pi\)
0.172300 + 0.985045i \(0.444880\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) − 3.23443e6i − 0.620903i −0.950589 0.310452i \(-0.899520\pi\)
0.950589 0.310452i \(-0.100480\pi\)
\(84\) 0 0
\(85\) − 2.11735e6i − 0.373961i
\(86\) 0 0
\(87\) 3.59936e6 0.586014
\(88\) 0 0
\(89\) 4.95231e6 0.744634 0.372317 0.928106i \(-0.378563\pi\)
0.372317 + 0.928106i \(0.378563\pi\)
\(90\) 0 0
\(91\) − 4.99042e6i − 0.694213i
\(92\) 0 0
\(93\) 146945.i 0.0189437i
\(94\) 0 0
\(95\) −1.14565e6 −0.137094
\(96\) 0 0
\(97\) 1.35947e7 1.51240 0.756201 0.654339i \(-0.227053\pi\)
0.756201 + 0.654339i \(0.227053\pi\)
\(98\) 0 0
\(99\) − 4.18319e6i − 0.433296i
\(100\) 0 0
\(101\) − 4.08245e6i − 0.394272i −0.980376 0.197136i \(-0.936836\pi\)
0.980376 0.197136i \(-0.0631641\pi\)
\(102\) 0 0
\(103\) −2.12174e7 −1.91321 −0.956603 0.291396i \(-0.905880\pi\)
−0.956603 + 0.291396i \(0.905880\pi\)
\(104\) 0 0
\(105\) −2.49168e6 −0.210053
\(106\) 0 0
\(107\) 3.69166e6i 0.291325i 0.989334 + 0.145663i \(0.0465314\pi\)
−0.989334 + 0.145663i \(0.953469\pi\)
\(108\) 0 0
\(109\) 1.98319e7i 1.46680i 0.679795 + 0.733402i \(0.262069\pi\)
−0.679795 + 0.733402i \(0.737931\pi\)
\(110\) 0 0
\(111\) −1.12483e7 −0.780651
\(112\) 0 0
\(113\) −1.75650e7 −1.14518 −0.572591 0.819841i \(-0.694062\pi\)
−0.572591 + 0.819841i \(0.694062\pi\)
\(114\) 0 0
\(115\) − 4.71614e6i − 0.289164i
\(116\) 0 0
\(117\) − 5.36132e6i − 0.309472i
\(118\) 0 0
\(119\) 1.05646e7 0.574695
\(120\) 0 0
\(121\) −1.34404e7 −0.689705
\(122\) 0 0
\(123\) − 1.16810e7i − 0.565994i
\(124\) 0 0
\(125\) 1.87344e7i 0.857936i
\(126\) 0 0
\(127\) −1.41817e7 −0.614349 −0.307174 0.951653i \(-0.599383\pi\)
−0.307174 + 0.951653i \(0.599383\pi\)
\(128\) 0 0
\(129\) 1.10952e7 0.455062
\(130\) 0 0
\(131\) − 1.82585e7i − 0.709602i −0.934942 0.354801i \(-0.884549\pi\)
0.934942 0.354801i \(-0.115451\pi\)
\(132\) 0 0
\(133\) − 5.71623e6i − 0.210683i
\(134\) 0 0
\(135\) −2.67686e6 −0.0936392
\(136\) 0 0
\(137\) −3.65417e7 −1.21413 −0.607066 0.794651i \(-0.707654\pi\)
−0.607066 + 0.794651i \(0.707654\pi\)
\(138\) 0 0
\(139\) − 2.79526e7i − 0.882816i −0.897307 0.441408i \(-0.854479\pi\)
0.897307 0.441408i \(-0.145521\pi\)
\(140\) 0 0
\(141\) 2.32024e7i 0.697053i
\(142\) 0 0
\(143\) −4.22011e7 −1.20684
\(144\) 0 0
\(145\) −1.81299e7 −0.493865
\(146\) 0 0
\(147\) 9.80340e6i 0.254546i
\(148\) 0 0
\(149\) − 3.80729e7i − 0.942898i −0.881893 0.471449i \(-0.843731\pi\)
0.881893 0.471449i \(-0.156269\pi\)
\(150\) 0 0
\(151\) −1.94179e7 −0.458967 −0.229484 0.973313i \(-0.573704\pi\)
−0.229484 + 0.973313i \(0.573704\pi\)
\(152\) 0 0
\(153\) 1.13497e7 0.256192
\(154\) 0 0
\(155\) − 740163.i − 0.0159649i
\(156\) 0 0
\(157\) − 3.65812e7i − 0.754413i −0.926129 0.377206i \(-0.876885\pi\)
0.926129 0.377206i \(-0.123115\pi\)
\(158\) 0 0
\(159\) −2.04134e7 −0.402740
\(160\) 0 0
\(161\) 2.35313e7 0.444380
\(162\) 0 0
\(163\) − 4.06341e7i − 0.734910i −0.930041 0.367455i \(-0.880229\pi\)
0.930041 0.367455i \(-0.119771\pi\)
\(164\) 0 0
\(165\) 2.10707e7i 0.365161i
\(166\) 0 0
\(167\) −8.24852e7 −1.37047 −0.685233 0.728324i \(-0.740300\pi\)
−0.685233 + 0.728324i \(0.740300\pi\)
\(168\) 0 0
\(169\) 8.66207e6 0.138044
\(170\) 0 0
\(171\) − 6.14107e6i − 0.0939200i
\(172\) 0 0
\(173\) 2.91738e7i 0.428383i 0.976792 + 0.214191i \(0.0687117\pi\)
−0.976792 + 0.214191i \(0.931288\pi\)
\(174\) 0 0
\(175\) −4.04626e7 −0.570716
\(176\) 0 0
\(177\) −5.75624e7 −0.780247
\(178\) 0 0
\(179\) − 1.41818e8i − 1.84818i −0.382175 0.924090i \(-0.624825\pi\)
0.382175 0.924090i \(-0.375175\pi\)
\(180\) 0 0
\(181\) 2.83773e7i 0.355711i 0.984057 + 0.177855i \(0.0569159\pi\)
−0.984057 + 0.177855i \(0.943084\pi\)
\(182\) 0 0
\(183\) −7.99663e6 −0.0964557
\(184\) 0 0
\(185\) 5.66576e7 0.657896
\(186\) 0 0
\(187\) − 8.93384e7i − 0.999062i
\(188\) 0 0
\(189\) − 1.33562e7i − 0.143902i
\(190\) 0 0
\(191\) −1.53282e8 −1.59174 −0.795872 0.605465i \(-0.792987\pi\)
−0.795872 + 0.605465i \(0.792987\pi\)
\(192\) 0 0
\(193\) 2.43290e7 0.243598 0.121799 0.992555i \(-0.461134\pi\)
0.121799 + 0.992555i \(0.461134\pi\)
\(194\) 0 0
\(195\) 2.70049e7i 0.260808i
\(196\) 0 0
\(197\) 7.99556e7i 0.745104i 0.928011 + 0.372552i \(0.121517\pi\)
−0.928011 + 0.372552i \(0.878483\pi\)
\(198\) 0 0
\(199\) 1.00913e8 0.907744 0.453872 0.891067i \(-0.350042\pi\)
0.453872 + 0.891067i \(0.350042\pi\)
\(200\) 0 0
\(201\) 5.86084e7 0.509066
\(202\) 0 0
\(203\) − 9.04597e7i − 0.758960i
\(204\) 0 0
\(205\) 5.88370e7i 0.476993i
\(206\) 0 0
\(207\) 2.52802e7 0.198100
\(208\) 0 0
\(209\) −4.83389e7 −0.366256
\(210\) 0 0
\(211\) − 2.05256e8i − 1.50421i −0.659046 0.752103i \(-0.729040\pi\)
0.659046 0.752103i \(-0.270960\pi\)
\(212\) 0 0
\(213\) − 1.60466e7i − 0.113777i
\(214\) 0 0
\(215\) −5.58863e7 −0.383505
\(216\) 0 0
\(217\) 3.69305e6 0.0245345
\(218\) 0 0
\(219\) − 2.91837e7i − 0.187753i
\(220\) 0 0
\(221\) − 1.14499e8i − 0.713558i
\(222\) 0 0
\(223\) −3.00391e8 −1.81393 −0.906965 0.421207i \(-0.861607\pi\)
−0.906965 + 0.421207i \(0.861607\pi\)
\(224\) 0 0
\(225\) −4.34698e7 −0.254419
\(226\) 0 0
\(227\) − 1.75962e8i − 0.998457i −0.866470 0.499229i \(-0.833617\pi\)
0.866470 0.499229i \(-0.166383\pi\)
\(228\) 0 0
\(229\) − 1.94371e8i − 1.06956i −0.844990 0.534782i \(-0.820394\pi\)
0.844990 0.534782i \(-0.179606\pi\)
\(230\) 0 0
\(231\) −1.05132e8 −0.561171
\(232\) 0 0
\(233\) 2.25030e7 0.116545 0.0582725 0.998301i \(-0.481441\pi\)
0.0582725 + 0.998301i \(0.481441\pi\)
\(234\) 0 0
\(235\) − 1.16870e8i − 0.587443i
\(236\) 0 0
\(237\) − 4.07730e7i − 0.198955i
\(238\) 0 0
\(239\) −7.68892e7 −0.364312 −0.182156 0.983270i \(-0.558308\pi\)
−0.182156 + 0.983270i \(0.558308\pi\)
\(240\) 0 0
\(241\) 2.93504e8 1.35068 0.675342 0.737505i \(-0.263996\pi\)
0.675342 + 0.737505i \(0.263996\pi\)
\(242\) 0 0
\(243\) − 1.43489e7i − 0.0641500i
\(244\) 0 0
\(245\) − 4.93796e7i − 0.214519i
\(246\) 0 0
\(247\) −6.19528e7 −0.261590
\(248\) 0 0
\(249\) −8.73295e7 −0.358479
\(250\) 0 0
\(251\) 1.21218e6i 0.00483850i 0.999997 + 0.00241925i \(0.000770072\pi\)
−0.999997 + 0.00241925i \(0.999230\pi\)
\(252\) 0 0
\(253\) − 1.98990e8i − 0.772521i
\(254\) 0 0
\(255\) −5.71685e7 −0.215907
\(256\) 0 0
\(257\) 6.15812e7 0.226299 0.113149 0.993578i \(-0.463906\pi\)
0.113149 + 0.993578i \(0.463906\pi\)
\(258\) 0 0
\(259\) 2.82694e8i 1.01104i
\(260\) 0 0
\(261\) − 9.71828e7i − 0.338335i
\(262\) 0 0
\(263\) 1.13314e8 0.384094 0.192047 0.981386i \(-0.438487\pi\)
0.192047 + 0.981386i \(0.438487\pi\)
\(264\) 0 0
\(265\) 1.02822e8 0.339410
\(266\) 0 0
\(267\) − 1.33712e8i − 0.429914i
\(268\) 0 0
\(269\) 1.25653e8i 0.393587i 0.980445 + 0.196793i \(0.0630528\pi\)
−0.980445 + 0.196793i \(0.936947\pi\)
\(270\) 0 0
\(271\) −4.75707e8 −1.45193 −0.725967 0.687729i \(-0.758608\pi\)
−0.725967 + 0.687729i \(0.758608\pi\)
\(272\) 0 0
\(273\) −1.34741e8 −0.400804
\(274\) 0 0
\(275\) 3.42168e8i 0.992146i
\(276\) 0 0
\(277\) − 8.33181e7i − 0.235538i −0.993041 0.117769i \(-0.962426\pi\)
0.993041 0.117769i \(-0.0375742\pi\)
\(278\) 0 0
\(279\) 3.96753e6 0.0109372
\(280\) 0 0
\(281\) −1.28781e8 −0.346241 −0.173121 0.984901i \(-0.555385\pi\)
−0.173121 + 0.984901i \(0.555385\pi\)
\(282\) 0 0
\(283\) − 3.36835e8i − 0.883414i −0.897159 0.441707i \(-0.854373\pi\)
0.897159 0.441707i \(-0.145627\pi\)
\(284\) 0 0
\(285\) 3.09325e7i 0.0791513i
\(286\) 0 0
\(287\) −2.93568e8 −0.733032
\(288\) 0 0
\(289\) −1.67948e8 −0.409290
\(290\) 0 0
\(291\) − 3.67056e8i − 0.873186i
\(292\) 0 0
\(293\) 6.88149e8i 1.59825i 0.601163 + 0.799127i \(0.294704\pi\)
−0.601163 + 0.799127i \(0.705296\pi\)
\(294\) 0 0
\(295\) 2.89941e8 0.657556
\(296\) 0 0
\(297\) −1.12946e8 −0.250163
\(298\) 0 0
\(299\) − 2.55033e8i − 0.551756i
\(300\) 0 0
\(301\) − 2.78846e8i − 0.589361i
\(302\) 0 0
\(303\) −1.10226e8 −0.227633
\(304\) 0 0
\(305\) 4.02789e7 0.0812883
\(306\) 0 0
\(307\) − 1.82538e8i − 0.360054i −0.983662 0.180027i \(-0.942381\pi\)
0.983662 0.180027i \(-0.0576186\pi\)
\(308\) 0 0
\(309\) 5.72869e8i 1.10459i
\(310\) 0 0
\(311\) 6.63147e8 1.25011 0.625055 0.780580i \(-0.285076\pi\)
0.625055 + 0.780580i \(0.285076\pi\)
\(312\) 0 0
\(313\) −6.30887e8 −1.16291 −0.581455 0.813578i \(-0.697516\pi\)
−0.581455 + 0.813578i \(0.697516\pi\)
\(314\) 0 0
\(315\) 6.72753e7i 0.121274i
\(316\) 0 0
\(317\) − 5.83229e7i − 0.102833i −0.998677 0.0514164i \(-0.983626\pi\)
0.998677 0.0514164i \(-0.0163736\pi\)
\(318\) 0 0
\(319\) −7.64965e8 −1.31939
\(320\) 0 0
\(321\) 9.96747e7 0.168197
\(322\) 0 0
\(323\) − 1.31152e8i − 0.216554i
\(324\) 0 0
\(325\) 4.38535e8i 0.708619i
\(326\) 0 0
\(327\) 5.35462e8 0.846860
\(328\) 0 0
\(329\) 5.83125e8 0.902768
\(330\) 0 0
\(331\) 1.07610e9i 1.63100i 0.578755 + 0.815502i \(0.303539\pi\)
−0.578755 + 0.815502i \(0.696461\pi\)
\(332\) 0 0
\(333\) 3.03705e8i 0.450709i
\(334\) 0 0
\(335\) −2.95210e8 −0.429017
\(336\) 0 0
\(337\) 8.16112e8 1.16157 0.580785 0.814057i \(-0.302746\pi\)
0.580785 + 0.814057i \(0.302746\pi\)
\(338\) 0 0
\(339\) 4.74256e8i 0.661171i
\(340\) 0 0
\(341\) − 3.12300e7i − 0.0426513i
\(342\) 0 0
\(343\) 8.05210e8 1.07741
\(344\) 0 0
\(345\) −1.27336e8 −0.166949
\(346\) 0 0
\(347\) − 1.57951e8i − 0.202941i −0.994839 0.101471i \(-0.967645\pi\)
0.994839 0.101471i \(-0.0323548\pi\)
\(348\) 0 0
\(349\) − 6.59184e8i − 0.830076i −0.909804 0.415038i \(-0.863768\pi\)
0.909804 0.415038i \(-0.136232\pi\)
\(350\) 0 0
\(351\) −1.44756e8 −0.178674
\(352\) 0 0
\(353\) −1.19233e9 −1.44273 −0.721365 0.692555i \(-0.756485\pi\)
−0.721365 + 0.692555i \(0.756485\pi\)
\(354\) 0 0
\(355\) 8.08266e7i 0.0958860i
\(356\) 0 0
\(357\) − 2.85243e8i − 0.331800i
\(358\) 0 0
\(359\) 8.82599e8 1.00678 0.503388 0.864061i \(-0.332087\pi\)
0.503388 + 0.864061i \(0.332087\pi\)
\(360\) 0 0
\(361\) 8.22908e8 0.920611
\(362\) 0 0
\(363\) 3.62891e8i 0.398201i
\(364\) 0 0
\(365\) 1.46998e8i 0.158229i
\(366\) 0 0
\(367\) 1.48390e9 1.56702 0.783508 0.621381i \(-0.213428\pi\)
0.783508 + 0.621381i \(0.213428\pi\)
\(368\) 0 0
\(369\) −3.15387e8 −0.326777
\(370\) 0 0
\(371\) 5.13032e8i 0.521597i
\(372\) 0 0
\(373\) − 6.64518e8i − 0.663019i −0.943452 0.331509i \(-0.892442\pi\)
0.943452 0.331509i \(-0.107558\pi\)
\(374\) 0 0
\(375\) 5.05829e8 0.495330
\(376\) 0 0
\(377\) −9.80406e8 −0.942348
\(378\) 0 0
\(379\) 6.05654e8i 0.571462i 0.958310 + 0.285731i \(0.0922364\pi\)
−0.958310 + 0.285731i \(0.907764\pi\)
\(380\) 0 0
\(381\) 3.82906e8i 0.354694i
\(382\) 0 0
\(383\) −1.16691e9 −1.06131 −0.530656 0.847587i \(-0.678055\pi\)
−0.530656 + 0.847587i \(0.678055\pi\)
\(384\) 0 0
\(385\) 5.29551e8 0.472928
\(386\) 0 0
\(387\) − 2.99570e8i − 0.262730i
\(388\) 0 0
\(389\) − 5.72847e7i − 0.0493418i −0.999696 0.0246709i \(-0.992146\pi\)
0.999696 0.0246709i \(-0.00785379\pi\)
\(390\) 0 0
\(391\) 5.39897e8 0.456764
\(392\) 0 0
\(393\) −4.92979e8 −0.409689
\(394\) 0 0
\(395\) 2.05373e8i 0.167670i
\(396\) 0 0
\(397\) − 5.03578e7i − 0.0403924i −0.999796 0.0201962i \(-0.993571\pi\)
0.999796 0.0201962i \(-0.00642909\pi\)
\(398\) 0 0
\(399\) −1.54338e8 −0.121638
\(400\) 0 0
\(401\) −1.40624e9 −1.08907 −0.544533 0.838740i \(-0.683293\pi\)
−0.544533 + 0.838740i \(0.683293\pi\)
\(402\) 0 0
\(403\) − 4.00255e7i − 0.0304627i
\(404\) 0 0
\(405\) 7.22753e7i 0.0540626i
\(406\) 0 0
\(407\) 2.39058e9 1.75761
\(408\) 0 0
\(409\) 1.66336e9 1.20214 0.601068 0.799198i \(-0.294742\pi\)
0.601068 + 0.799198i \(0.294742\pi\)
\(410\) 0 0
\(411\) 9.86625e8i 0.700980i
\(412\) 0 0
\(413\) 1.44666e9i 1.01051i
\(414\) 0 0
\(415\) 4.39878e8 0.302109
\(416\) 0 0
\(417\) −7.54720e8 −0.509694
\(418\) 0 0
\(419\) 1.28045e9i 0.850380i 0.905104 + 0.425190i \(0.139793\pi\)
−0.905104 + 0.425190i \(0.860207\pi\)
\(420\) 0 0
\(421\) − 2.29065e9i − 1.49614i −0.663621 0.748069i \(-0.730981\pi\)
0.663621 0.748069i \(-0.269019\pi\)
\(422\) 0 0
\(423\) 6.26464e8 0.402444
\(424\) 0 0
\(425\) −9.28364e8 −0.586620
\(426\) 0 0
\(427\) 2.00972e8i 0.124922i
\(428\) 0 0
\(429\) 1.13943e9i 0.696767i
\(430\) 0 0
\(431\) −1.04952e9 −0.631422 −0.315711 0.948855i \(-0.602243\pi\)
−0.315711 + 0.948855i \(0.602243\pi\)
\(432\) 0 0
\(433\) 2.09479e7 0.0124003 0.00620015 0.999981i \(-0.498026\pi\)
0.00620015 + 0.999981i \(0.498026\pi\)
\(434\) 0 0
\(435\) 4.89509e8i 0.285133i
\(436\) 0 0
\(437\) − 2.92125e8i − 0.167450i
\(438\) 0 0
\(439\) 3.25105e9 1.83400 0.916998 0.398893i \(-0.130605\pi\)
0.916998 + 0.398893i \(0.130605\pi\)
\(440\) 0 0
\(441\) 2.64692e8 0.146962
\(442\) 0 0
\(443\) 2.03069e9i 1.10976i 0.831930 + 0.554881i \(0.187236\pi\)
−0.831930 + 0.554881i \(0.812764\pi\)
\(444\) 0 0
\(445\) 6.73507e8i 0.362312i
\(446\) 0 0
\(447\) −1.02797e9 −0.544382
\(448\) 0 0
\(449\) 3.32188e9 1.73189 0.865947 0.500135i \(-0.166716\pi\)
0.865947 + 0.500135i \(0.166716\pi\)
\(450\) 0 0
\(451\) 2.48254e9i 1.27432i
\(452\) 0 0
\(453\) 5.24282e8i 0.264985i
\(454\) 0 0
\(455\) 6.78691e8 0.337779
\(456\) 0 0
\(457\) 3.82371e9 1.87404 0.937019 0.349279i \(-0.113573\pi\)
0.937019 + 0.349279i \(0.113573\pi\)
\(458\) 0 0
\(459\) − 3.06443e8i − 0.147913i
\(460\) 0 0
\(461\) − 3.70370e9i − 1.76069i −0.474337 0.880343i \(-0.657312\pi\)
0.474337 0.880343i \(-0.342688\pi\)
\(462\) 0 0
\(463\) −2.76589e9 −1.29510 −0.647548 0.762025i \(-0.724206\pi\)
−0.647548 + 0.762025i \(0.724206\pi\)
\(464\) 0 0
\(465\) −1.99844e7 −0.00921734
\(466\) 0 0
\(467\) 2.83943e9i 1.29009i 0.764143 + 0.645047i \(0.223162\pi\)
−0.764143 + 0.645047i \(0.776838\pi\)
\(468\) 0 0
\(469\) − 1.47296e9i − 0.659302i
\(470\) 0 0
\(471\) −9.87692e8 −0.435560
\(472\) 0 0
\(473\) −2.35804e9 −1.02456
\(474\) 0 0
\(475\) 5.02316e8i 0.215055i
\(476\) 0 0
\(477\) 5.51161e8i 0.232522i
\(478\) 0 0
\(479\) 1.74825e9 0.726823 0.363412 0.931629i \(-0.381612\pi\)
0.363412 + 0.931629i \(0.381612\pi\)
\(480\) 0 0
\(481\) 3.06385e9 1.25534
\(482\) 0 0
\(483\) − 6.35344e8i − 0.256563i
\(484\) 0 0
\(485\) 1.84886e9i 0.735880i
\(486\) 0 0
\(487\) 3.08994e9 1.21227 0.606134 0.795363i \(-0.292720\pi\)
0.606134 + 0.795363i \(0.292720\pi\)
\(488\) 0 0
\(489\) −1.09712e9 −0.424300
\(490\) 0 0
\(491\) 2.56083e9i 0.976326i 0.872752 + 0.488163i \(0.162333\pi\)
−0.872752 + 0.488163i \(0.837667\pi\)
\(492\) 0 0
\(493\) − 2.07549e9i − 0.780110i
\(494\) 0 0
\(495\) 5.68908e8 0.210826
\(496\) 0 0
\(497\) −4.03286e8 −0.147355
\(498\) 0 0
\(499\) − 3.17346e9i − 1.14336i −0.820478 0.571678i \(-0.806293\pi\)
0.820478 0.571678i \(-0.193707\pi\)
\(500\) 0 0
\(501\) 2.22710e9i 0.791239i
\(502\) 0 0
\(503\) 5.17025e9 1.81144 0.905719 0.423878i \(-0.139331\pi\)
0.905719 + 0.423878i \(0.139331\pi\)
\(504\) 0 0
\(505\) 5.55208e8 0.191839
\(506\) 0 0
\(507\) − 2.33876e8i − 0.0796999i
\(508\) 0 0
\(509\) 3.39559e9i 1.14131i 0.821190 + 0.570655i \(0.193310\pi\)
−0.821190 + 0.570655i \(0.806690\pi\)
\(510\) 0 0
\(511\) −7.33450e8 −0.243163
\(512\) 0 0
\(513\) −1.65809e8 −0.0542247
\(514\) 0 0
\(515\) − 2.88554e9i − 0.930896i
\(516\) 0 0
\(517\) − 4.93116e9i − 1.56939i
\(518\) 0 0
\(519\) 7.87694e8 0.247327
\(520\) 0 0
\(521\) 2.69324e8 0.0834341 0.0417170 0.999129i \(-0.486717\pi\)
0.0417170 + 0.999129i \(0.486717\pi\)
\(522\) 0 0
\(523\) 8.32211e8i 0.254377i 0.991879 + 0.127188i \(0.0405953\pi\)
−0.991879 + 0.127188i \(0.959405\pi\)
\(524\) 0 0
\(525\) 1.09249e9i 0.329503i
\(526\) 0 0
\(527\) 8.47326e7 0.0252182
\(528\) 0 0
\(529\) −2.20227e9 −0.646809
\(530\) 0 0
\(531\) 1.55418e9i 0.450476i
\(532\) 0 0
\(533\) 3.18171e9i 0.910155i
\(534\) 0 0
\(535\) −5.02061e8 −0.141748
\(536\) 0 0
\(537\) −3.82907e9 −1.06705
\(538\) 0 0
\(539\) − 2.08350e9i − 0.573102i
\(540\) 0 0
\(541\) − 4.71332e9i − 1.27978i −0.768465 0.639891i \(-0.778979\pi\)
0.768465 0.639891i \(-0.221021\pi\)
\(542\) 0 0
\(543\) 7.66188e8 0.205370
\(544\) 0 0
\(545\) −2.69712e9 −0.713694
\(546\) 0 0
\(547\) 5.08805e9i 1.32922i 0.747192 + 0.664608i \(0.231401\pi\)
−0.747192 + 0.664608i \(0.768599\pi\)
\(548\) 0 0
\(549\) 2.15909e8i 0.0556887i
\(550\) 0 0
\(551\) −1.12300e9 −0.285988
\(552\) 0 0
\(553\) −1.02471e9 −0.257670
\(554\) 0 0
\(555\) − 1.52976e9i − 0.379837i
\(556\) 0 0
\(557\) 4.34605e9i 1.06562i 0.846236 + 0.532809i \(0.178863\pi\)
−0.846236 + 0.532809i \(0.821137\pi\)
\(558\) 0 0
\(559\) −3.02214e9 −0.731768
\(560\) 0 0
\(561\) −2.41214e9 −0.576809
\(562\) 0 0
\(563\) 4.99896e8i 0.118059i 0.998256 + 0.0590296i \(0.0188006\pi\)
−0.998256 + 0.0590296i \(0.981199\pi\)
\(564\) 0 0
\(565\) − 2.38882e9i − 0.557204i
\(566\) 0 0
\(567\) −3.60619e8 −0.0830821
\(568\) 0 0
\(569\) −3.36621e9 −0.766034 −0.383017 0.923741i \(-0.625115\pi\)
−0.383017 + 0.923741i \(0.625115\pi\)
\(570\) 0 0
\(571\) − 4.57042e9i − 1.02738i −0.857977 0.513688i \(-0.828279\pi\)
0.857977 0.513688i \(-0.171721\pi\)
\(572\) 0 0
\(573\) 4.13860e9i 0.918993i
\(574\) 0 0
\(575\) −2.06782e9 −0.453602
\(576\) 0 0
\(577\) 7.07908e9 1.53413 0.767064 0.641570i \(-0.221717\pi\)
0.767064 + 0.641570i \(0.221717\pi\)
\(578\) 0 0
\(579\) − 6.56882e8i − 0.140641i
\(580\) 0 0
\(581\) 2.19478e9i 0.464274i
\(582\) 0 0
\(583\) 4.33841e9 0.906757
\(584\) 0 0
\(585\) 7.29133e8 0.150578
\(586\) 0 0
\(587\) − 3.62621e9i − 0.739980i −0.929036 0.369990i \(-0.879361\pi\)
0.929036 0.369990i \(-0.120639\pi\)
\(588\) 0 0
\(589\) − 4.58468e7i − 0.00924497i
\(590\) 0 0
\(591\) 2.15880e9 0.430186
\(592\) 0 0
\(593\) −4.79267e9 −0.943814 −0.471907 0.881648i \(-0.656434\pi\)
−0.471907 + 0.881648i \(0.656434\pi\)
\(594\) 0 0
\(595\) 1.43677e9i 0.279626i
\(596\) 0 0
\(597\) − 2.72466e9i − 0.524086i
\(598\) 0 0
\(599\) −8.85908e9 −1.68421 −0.842103 0.539317i \(-0.818682\pi\)
−0.842103 + 0.539317i \(0.818682\pi\)
\(600\) 0 0
\(601\) 6.62334e9 1.24456 0.622280 0.782794i \(-0.286206\pi\)
0.622280 + 0.782794i \(0.286206\pi\)
\(602\) 0 0
\(603\) − 1.58243e9i − 0.293909i
\(604\) 0 0
\(605\) − 1.82788e9i − 0.335585i
\(606\) 0 0
\(607\) −5.63072e9 −1.02189 −0.510944 0.859614i \(-0.670704\pi\)
−0.510944 + 0.859614i \(0.670704\pi\)
\(608\) 0 0
\(609\) −2.44241e9 −0.438186
\(610\) 0 0
\(611\) − 6.31994e9i − 1.12091i
\(612\) 0 0
\(613\) − 9.64204e8i − 0.169066i −0.996421 0.0845332i \(-0.973060\pi\)
0.996421 0.0845332i \(-0.0269399\pi\)
\(614\) 0 0
\(615\) 1.58860e9 0.275392
\(616\) 0 0
\(617\) 1.64837e9 0.282525 0.141263 0.989972i \(-0.454884\pi\)
0.141263 + 0.989972i \(0.454884\pi\)
\(618\) 0 0
\(619\) 3.74163e9i 0.634080i 0.948412 + 0.317040i \(0.102689\pi\)
−0.948412 + 0.317040i \(0.897311\pi\)
\(620\) 0 0
\(621\) − 6.82564e8i − 0.114373i
\(622\) 0 0
\(623\) −3.36048e9 −0.556792
\(624\) 0 0
\(625\) 2.11069e9 0.345815
\(626\) 0 0
\(627\) 1.30515e9i 0.211458i
\(628\) 0 0
\(629\) 6.48608e9i 1.03921i
\(630\) 0 0
\(631\) −6.99751e9 −1.10877 −0.554384 0.832261i \(-0.687046\pi\)
−0.554384 + 0.832261i \(0.687046\pi\)
\(632\) 0 0
\(633\) −5.54191e9 −0.868453
\(634\) 0 0
\(635\) − 1.92869e9i − 0.298920i
\(636\) 0 0
\(637\) − 2.67028e9i − 0.409326i
\(638\) 0 0
\(639\) −4.33259e8 −0.0656893
\(640\) 0 0
\(641\) −2.96243e9 −0.444268 −0.222134 0.975016i \(-0.571302\pi\)
−0.222134 + 0.975016i \(0.571302\pi\)
\(642\) 0 0
\(643\) − 5.11629e9i − 0.758956i −0.925201 0.379478i \(-0.876104\pi\)
0.925201 0.379478i \(-0.123896\pi\)
\(644\) 0 0
\(645\) 1.50893e9i 0.221417i
\(646\) 0 0
\(647\) −2.13323e9 −0.309651 −0.154826 0.987942i \(-0.549482\pi\)
−0.154826 + 0.987942i \(0.549482\pi\)
\(648\) 0 0
\(649\) 1.22336e10 1.75670
\(650\) 0 0
\(651\) − 9.97125e7i − 0.0141650i
\(652\) 0 0
\(653\) − 2.79110e9i − 0.392265i −0.980577 0.196132i \(-0.937162\pi\)
0.980577 0.196132i \(-0.0628383\pi\)
\(654\) 0 0
\(655\) 2.48313e9 0.345267
\(656\) 0 0
\(657\) −7.87961e8 −0.108399
\(658\) 0 0
\(659\) − 8.87521e9i − 1.20804i −0.796971 0.604018i \(-0.793566\pi\)
0.796971 0.604018i \(-0.206434\pi\)
\(660\) 0 0
\(661\) − 3.28619e9i − 0.442576i −0.975209 0.221288i \(-0.928974\pi\)
0.975209 0.221288i \(-0.0710260\pi\)
\(662\) 0 0
\(663\) −3.09148e9 −0.411973
\(664\) 0 0
\(665\) 7.77400e8 0.102511
\(666\) 0 0
\(667\) − 4.62290e9i − 0.603217i
\(668\) 0 0
\(669\) 8.11057e9i 1.04727i
\(670\) 0 0
\(671\) 1.69951e9 0.217167
\(672\) 0 0
\(673\) −2.69108e9 −0.340310 −0.170155 0.985417i \(-0.554427\pi\)
−0.170155 + 0.985417i \(0.554427\pi\)
\(674\) 0 0
\(675\) 1.17368e9i 0.146889i
\(676\) 0 0
\(677\) 1.40413e10i 1.73919i 0.493767 + 0.869594i \(0.335619\pi\)
−0.493767 + 0.869594i \(0.664381\pi\)
\(678\) 0 0
\(679\) −9.22490e9 −1.13088
\(680\) 0 0
\(681\) −4.75098e9 −0.576459
\(682\) 0 0
\(683\) 6.15198e9i 0.738827i 0.929265 + 0.369413i \(0.120441\pi\)
−0.929265 + 0.369413i \(0.879559\pi\)
\(684\) 0 0
\(685\) − 4.96962e9i − 0.590753i
\(686\) 0 0
\(687\) −5.24802e9 −0.617514
\(688\) 0 0
\(689\) 5.56026e9 0.647631
\(690\) 0 0
\(691\) − 4.44942e9i − 0.513015i −0.966542 0.256508i \(-0.917428\pi\)
0.966542 0.256508i \(-0.0825719\pi\)
\(692\) 0 0
\(693\) 2.83858e9i 0.323992i
\(694\) 0 0
\(695\) 3.80151e9 0.429546
\(696\) 0 0
\(697\) −6.73557e9 −0.753459
\(698\) 0 0
\(699\) − 6.07580e8i − 0.0672873i
\(700\) 0 0
\(701\) − 1.06110e10i − 1.16344i −0.813391 0.581718i \(-0.802381\pi\)
0.813391 0.581718i \(-0.197619\pi\)
\(702\) 0 0
\(703\) 3.50946e9 0.380975
\(704\) 0 0
\(705\) −3.15549e9 −0.339161
\(706\) 0 0
\(707\) 2.77022e9i 0.294813i
\(708\) 0 0
\(709\) − 5.58133e9i − 0.588133i −0.955785 0.294067i \(-0.904991\pi\)
0.955785 0.294067i \(-0.0950088\pi\)
\(710\) 0 0
\(711\) −1.10087e9 −0.114866
\(712\) 0 0
\(713\) 1.88732e8 0.0194999
\(714\) 0 0
\(715\) − 5.73930e9i − 0.587202i
\(716\) 0 0
\(717\) 2.07601e9i 0.210335i
\(718\) 0 0
\(719\) −1.18370e10 −1.18765 −0.593826 0.804593i \(-0.702383\pi\)
−0.593826 + 0.804593i \(0.702383\pi\)
\(720\) 0 0
\(721\) 1.43974e10 1.43058
\(722\) 0 0
\(723\) − 7.92459e9i − 0.779818i
\(724\) 0 0
\(725\) 7.94917e9i 0.774710i
\(726\) 0 0
\(727\) −1.87993e10 −1.81456 −0.907278 0.420532i \(-0.861843\pi\)
−0.907278 + 0.420532i \(0.861843\pi\)
\(728\) 0 0
\(729\) −3.87420e8 −0.0370370
\(730\) 0 0
\(731\) − 6.39777e9i − 0.605784i
\(732\) 0 0
\(733\) − 1.66065e10i − 1.55745i −0.627362 0.778727i \(-0.715866\pi\)
0.627362 0.778727i \(-0.284134\pi\)
\(734\) 0 0
\(735\) −1.33325e9 −0.123853
\(736\) 0 0
\(737\) −1.24559e10 −1.14615
\(738\) 0 0
\(739\) − 1.52701e10i − 1.39183i −0.718123 0.695916i \(-0.754999\pi\)
0.718123 0.695916i \(-0.245001\pi\)
\(740\) 0 0
\(741\) 1.67273e9i 0.151029i
\(742\) 0 0
\(743\) −1.96451e10 −1.75709 −0.878544 0.477662i \(-0.841484\pi\)
−0.878544 + 0.477662i \(0.841484\pi\)
\(744\) 0 0
\(745\) 5.17787e9 0.458780
\(746\) 0 0
\(747\) 2.35790e9i 0.206968i
\(748\) 0 0
\(749\) − 2.50504e9i − 0.217835i
\(750\) 0 0
\(751\) 8.24022e9 0.709903 0.354951 0.934885i \(-0.384497\pi\)
0.354951 + 0.934885i \(0.384497\pi\)
\(752\) 0 0
\(753\) 3.27290e7 0.00279351
\(754\) 0 0
\(755\) − 2.64080e9i − 0.223317i
\(756\) 0 0
\(757\) − 2.20969e10i − 1.85138i −0.378286 0.925689i \(-0.623486\pi\)
0.378286 0.925689i \(-0.376514\pi\)
\(758\) 0 0
\(759\) −5.37274e9 −0.446015
\(760\) 0 0
\(761\) −7.33501e9 −0.603329 −0.301665 0.953414i \(-0.597542\pi\)
−0.301665 + 0.953414i \(0.597542\pi\)
\(762\) 0 0
\(763\) − 1.34573e10i − 1.09679i
\(764\) 0 0
\(765\) 1.54355e9i 0.124654i
\(766\) 0 0
\(767\) 1.56790e10 1.25469
\(768\) 0 0
\(769\) 1.77014e10 1.40367 0.701834 0.712341i \(-0.252365\pi\)
0.701834 + 0.712341i \(0.252365\pi\)
\(770\) 0 0
\(771\) − 1.66269e9i − 0.130654i
\(772\) 0 0
\(773\) − 1.32070e10i − 1.02844i −0.857660 0.514218i \(-0.828082\pi\)
0.857660 0.514218i \(-0.171918\pi\)
\(774\) 0 0
\(775\) −3.24528e8 −0.0250436
\(776\) 0 0
\(777\) 7.63274e9 0.583723
\(778\) 0 0
\(779\) 3.64446e9i 0.276218i
\(780\) 0 0
\(781\) 3.41036e9i 0.256166i
\(782\) 0 0
\(783\) −2.62394e9 −0.195338
\(784\) 0 0
\(785\) 4.97499e9 0.367070
\(786\) 0 0
\(787\) 3.85138e9i 0.281647i 0.990035 + 0.140823i \(0.0449750\pi\)
−0.990035 + 0.140823i \(0.955025\pi\)
\(788\) 0 0
\(789\) − 3.05948e9i − 0.221757i
\(790\) 0 0
\(791\) 1.19191e10 0.856297
\(792\) 0 0
\(793\) 2.17815e9 0.155107
\(794\) 0 0
\(795\) − 2.77619e9i − 0.195959i
\(796\) 0 0
\(797\) − 5.23208e9i − 0.366075i −0.983106 0.183038i \(-0.941407\pi\)
0.983106 0.183038i \(-0.0585930\pi\)
\(798\) 0 0
\(799\) 1.33791e10 0.927926
\(800\) 0 0
\(801\) −3.61023e9 −0.248211
\(802\) 0 0
\(803\) 6.20236e9i 0.422720i
\(804\) 0 0
\(805\) 3.20022e9i 0.216219i
\(806\) 0 0
\(807\) 3.39264e9 0.227237
\(808\) 0 0
\(809\) −1.03141e10 −0.684877 −0.342439 0.939540i \(-0.611253\pi\)
−0.342439 + 0.939540i \(0.611253\pi\)
\(810\) 0 0
\(811\) − 3.64238e9i − 0.239780i −0.992787 0.119890i \(-0.961746\pi\)
0.992787 0.119890i \(-0.0382542\pi\)
\(812\) 0 0
\(813\) 1.28441e10i 0.838275i
\(814\) 0 0
\(815\) 5.52618e9 0.357580
\(816\) 0 0
\(817\) −3.46168e9 −0.222080
\(818\) 0 0
\(819\) 3.63802e9i 0.231404i
\(820\) 0 0
\(821\) 2.43614e10i 1.53639i 0.640215 + 0.768196i \(0.278845\pi\)
−0.640215 + 0.768196i \(0.721155\pi\)
\(822\) 0 0
\(823\) −5.41025e9 −0.338312 −0.169156 0.985589i \(-0.554104\pi\)
−0.169156 + 0.985589i \(0.554104\pi\)
\(824\) 0 0
\(825\) 9.23855e9 0.572816
\(826\) 0 0
\(827\) − 1.73669e10i − 1.06771i −0.845576 0.533856i \(-0.820743\pi\)
0.845576 0.533856i \(-0.179257\pi\)
\(828\) 0 0
\(829\) 2.35643e10i 1.43653i 0.695771 + 0.718264i \(0.255063\pi\)
−0.695771 + 0.718264i \(0.744937\pi\)
\(830\) 0 0
\(831\) −2.24959e9 −0.135988
\(832\) 0 0
\(833\) 5.65290e9 0.338855
\(834\) 0 0
\(835\) − 1.12179e10i − 0.666819i
\(836\) 0 0
\(837\) − 1.07123e8i − 0.00631458i
\(838\) 0 0
\(839\) −2.43054e10 −1.42081 −0.710406 0.703792i \(-0.751489\pi\)
−0.710406 + 0.703792i \(0.751489\pi\)
\(840\) 0 0
\(841\) −5.21603e8 −0.0302381
\(842\) 0 0
\(843\) 3.47708e9i 0.199902i
\(844\) 0 0
\(845\) 1.17803e9i 0.0671673i
\(846\) 0 0
\(847\) 9.12022e9 0.515719
\(848\) 0 0
\(849\) −9.09454e9 −0.510040
\(850\) 0 0
\(851\) 1.44469e10i 0.803568i
\(852\) 0 0
\(853\) − 2.67203e10i − 1.47407i −0.675853 0.737037i \(-0.736224\pi\)
0.675853 0.737037i \(-0.263776\pi\)
\(854\) 0 0
\(855\) 8.35178e8 0.0456980
\(856\) 0 0
\(857\) −6.59602e9 −0.357972 −0.178986 0.983852i \(-0.557282\pi\)
−0.178986 + 0.983852i \(0.557282\pi\)
\(858\) 0 0
\(859\) 2.98762e10i 1.60823i 0.594471 + 0.804117i \(0.297361\pi\)
−0.594471 + 0.804117i \(0.702639\pi\)
\(860\) 0 0
\(861\) 7.92635e9i 0.423216i
\(862\) 0 0
\(863\) 3.81550e9 0.202075 0.101038 0.994883i \(-0.467784\pi\)
0.101038 + 0.994883i \(0.467784\pi\)
\(864\) 0 0
\(865\) −3.96760e9 −0.208436
\(866\) 0 0
\(867\) 4.53459e9i 0.236304i
\(868\) 0 0
\(869\) 8.66541e9i 0.447940i
\(870\) 0 0
\(871\) −1.59640e10 −0.818610
\(872\) 0 0
\(873\) −9.91051e9 −0.504134
\(874\) 0 0
\(875\) − 1.27126e10i − 0.641512i
\(876\) 0 0
\(877\) 3.17545e10i 1.58967i 0.606827 + 0.794834i \(0.292442\pi\)
−0.606827 + 0.794834i \(0.707558\pi\)
\(878\) 0 0
\(879\) 1.85800e10 0.922752
\(880\) 0 0
\(881\) −1.61889e10 −0.797629 −0.398814 0.917032i \(-0.630578\pi\)
−0.398814 + 0.917032i \(0.630578\pi\)
\(882\) 0 0
\(883\) 3.12476e10i 1.52740i 0.645569 + 0.763702i \(0.276620\pi\)
−0.645569 + 0.763702i \(0.723380\pi\)
\(884\) 0 0
\(885\) − 7.82840e9i − 0.379640i
\(886\) 0 0
\(887\) −3.19461e10 −1.53704 −0.768520 0.639825i \(-0.779007\pi\)
−0.768520 + 0.639825i \(0.779007\pi\)
\(888\) 0 0
\(889\) 9.62323e9 0.459372
\(890\) 0 0
\(891\) 3.04954e9i 0.144432i
\(892\) 0 0
\(893\) − 7.23912e9i − 0.340177i
\(894\) 0 0
\(895\) 1.92870e10 0.899257
\(896\) 0 0
\(897\) −6.88589e9 −0.318557
\(898\) 0 0
\(899\) − 7.25528e8i − 0.0333039i
\(900\) 0 0
\(901\) 1.17709e10i 0.536133i
\(902\) 0 0
\(903\) −7.52883e9 −0.340268
\(904\) 0 0
\(905\) −3.85928e9 −0.173076
\(906\) 0 0
\(907\) 2.44254e10i 1.08697i 0.839419 + 0.543484i \(0.182895\pi\)
−0.839419 + 0.543484i \(0.817105\pi\)
\(908\) 0 0
\(909\) 2.97611e9i 0.131424i
\(910\) 0 0
\(911\) 3.71437e10 1.62769 0.813843 0.581084i \(-0.197371\pi\)
0.813843 + 0.581084i \(0.197371\pi\)
\(912\) 0 0
\(913\) 1.85600e10 0.807104
\(914\) 0 0
\(915\) − 1.08753e9i − 0.0469319i
\(916\) 0 0
\(917\) 1.23896e10i 0.530597i
\(918\) 0 0
\(919\) 4.47162e9 0.190047 0.0950234 0.995475i \(-0.469707\pi\)
0.0950234 + 0.995475i \(0.469707\pi\)
\(920\) 0 0
\(921\) −4.92851e9 −0.207877
\(922\) 0 0
\(923\) 4.37083e9i 0.182961i
\(924\) 0 0
\(925\) − 2.48418e10i − 1.03202i
\(926\) 0 0
\(927\) 1.54675e10 0.637735
\(928\) 0 0
\(929\) 4.64840e9 0.190217 0.0951083 0.995467i \(-0.469680\pi\)
0.0951083 + 0.995467i \(0.469680\pi\)
\(930\) 0 0
\(931\) − 3.05865e9i − 0.124224i
\(932\) 0 0
\(933\) − 1.79050e10i − 0.721752i
\(934\) 0 0
\(935\) 1.21499e10 0.486107
\(936\) 0 0
\(937\) −1.46668e9 −0.0582434 −0.0291217 0.999576i \(-0.509271\pi\)
−0.0291217 + 0.999576i \(0.509271\pi\)
\(938\) 0 0
\(939\) 1.70339e10i 0.671407i
\(940\) 0 0
\(941\) − 4.42582e10i − 1.73153i −0.500449 0.865766i \(-0.666832\pi\)
0.500449 0.865766i \(-0.333168\pi\)
\(942\) 0 0
\(943\) −1.50027e10 −0.582609
\(944\) 0 0
\(945\) 1.81643e9 0.0700177
\(946\) 0 0
\(947\) 6.90715e9i 0.264286i 0.991231 + 0.132143i \(0.0421858\pi\)
−0.991231 + 0.132143i \(0.957814\pi\)
\(948\) 0 0
\(949\) 7.94916e9i 0.301919i
\(950\) 0 0
\(951\) −1.57472e9 −0.0593706
\(952\) 0 0
\(953\) 2.23061e10 0.834829 0.417415 0.908716i \(-0.362936\pi\)
0.417415 + 0.908716i \(0.362936\pi\)
\(954\) 0 0
\(955\) − 2.08461e10i − 0.774484i
\(956\) 0 0
\(957\) 2.06541e10i 0.761752i
\(958\) 0 0
\(959\) 2.47960e10 0.907854
\(960\) 0 0
\(961\) −2.74830e10 −0.998923
\(962\) 0 0
\(963\) − 2.69122e9i − 0.0971084i
\(964\) 0 0
\(965\) 3.30871e9i 0.118526i
\(966\) 0 0
\(967\) 2.51312e10 0.893759 0.446880 0.894594i \(-0.352535\pi\)
0.446880 + 0.894594i \(0.352535\pi\)
\(968\) 0 0
\(969\) −3.54110e9 −0.125028
\(970\) 0 0
\(971\) 3.22732e10i 1.13129i 0.824648 + 0.565646i \(0.191373\pi\)
−0.824648 + 0.565646i \(0.808627\pi\)
\(972\) 0 0
\(973\) 1.89677e10i 0.660116i
\(974\) 0 0
\(975\) 1.18404e10 0.409121
\(976\) 0 0
\(977\) −4.02679e10 −1.38143 −0.690714 0.723128i \(-0.742704\pi\)
−0.690714 + 0.723128i \(0.742704\pi\)
\(978\) 0 0
\(979\) 2.84176e10i 0.967939i
\(980\) 0 0
\(981\) − 1.44575e10i − 0.488935i
\(982\) 0 0
\(983\) 2.21448e10 0.743590 0.371795 0.928315i \(-0.378742\pi\)
0.371795 + 0.928315i \(0.378742\pi\)
\(984\) 0 0
\(985\) −1.08739e10 −0.362541
\(986\) 0 0
\(987\) − 1.57444e10i − 0.521214i
\(988\) 0 0
\(989\) − 1.42503e10i − 0.468421i
\(990\) 0 0
\(991\) −3.63975e9 −0.118799 −0.0593997 0.998234i \(-0.518919\pi\)
−0.0593997 + 0.998234i \(0.518919\pi\)
\(992\) 0 0
\(993\) 2.90547e10 0.941660
\(994\) 0 0
\(995\) 1.37241e10i 0.441675i
\(996\) 0 0
\(997\) − 1.58095e10i − 0.505226i −0.967567 0.252613i \(-0.918710\pi\)
0.967567 0.252613i \(-0.0812899\pi\)
\(998\) 0 0
\(999\) 8.20002e9 0.260217
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.193.3 yes 8
4.3 odd 2 384.8.d.c.193.7 yes 8
8.3 odd 2 384.8.d.c.193.2 8
8.5 even 2 inner 384.8.d.d.193.6 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.8.d.c.193.2 8 8.3 odd 2
384.8.d.c.193.7 yes 8 4.3 odd 2
384.8.d.d.193.3 yes 8 1.1 even 1 trivial
384.8.d.d.193.6 yes 8 8.5 even 2 inner