Properties

Label 384.8.d.d.193.2
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 193.2
Root \(1.01122i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.8.d.d.193.7

$q$-expansion

\(f(q)\) \(=\) \(q-27.0000i q^{3} -69.8856i q^{5} -860.192 q^{7} -729.000 q^{9} +O(q^{10})\) \(q-27.0000i q^{3} -69.8856i q^{5} -860.192 q^{7} -729.000 q^{9} -6106.68i q^{11} -5338.61i q^{13} -1886.91 q^{15} +38616.6 q^{17} +13583.6i q^{19} +23225.2i q^{21} +99683.6 q^{23} +73241.0 q^{25} +19683.0i q^{27} -81173.3i q^{29} -153071. q^{31} -164880. q^{33} +60115.1i q^{35} +467160. i q^{37} -144143. q^{39} +384378. q^{41} -450696. i q^{43} +50946.6i q^{45} +503097. q^{47} -83612.9 q^{49} -1.04265e6i q^{51} -1.61153e6i q^{53} -426769. q^{55} +366756. q^{57} +1.58458e6i q^{59} -2.61079e6i q^{61} +627080. q^{63} -373092. q^{65} +1.52096e6i q^{67} -2.69146e6i q^{69} +1.15058e6 q^{71} +3.52252e6 q^{73} -1.97751e6i q^{75} +5.25292e6i q^{77} -7.64873e6 q^{79} +531441. q^{81} -6.34999e6i q^{83} -2.69875e6i q^{85} -2.19168e6 q^{87} -1.70342e6 q^{89} +4.59223e6i q^{91} +4.13291e6i q^{93} +949296. q^{95} -2.70830e6 q^{97} +4.45177e6i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2880 q^{7} - 5832 q^{9} + O(q^{10}) \) \( 8 q + 2880 q^{7} - 5832 q^{9} - 6048 q^{15} + 22896 q^{17} + 207360 q^{23} - 204696 q^{25} - 17856 q^{31} + 7776 q^{33} - 116640 q^{39} + 687056 q^{41} - 1987200 q^{47} + 4815560 q^{49} + 2077056 q^{55} + 878688 q^{57} - 2099520 q^{63} + 12871808 q^{65} - 6336000 q^{71} + 8920752 q^{73} + 1251648 q^{79} + 4251528 q^{81} - 13385952 q^{87} + 4447408 q^{89} - 31607424 q^{95} - 14157584 q^{97} + O(q^{100}) \)

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
\(5\) − 69.8856i − 0.250030i −0.992155 0.125015i \(-0.960102\pi\)
0.992155 0.125015i \(-0.0398980\pi\)
\(6\) 0 0
\(7\) −860.192 −0.947877 −0.473939 0.880558i \(-0.657168\pi\)
−0.473939 + 0.880558i \(0.657168\pi\)
\(8\) 0 0
\(9\) −729.000 −0.333333
\(10\) 0 0
\(11\) − 6106.68i − 1.38335i −0.722211 0.691673i \(-0.756874\pi\)
0.722211 0.691673i \(-0.243126\pi\)
\(12\) 0 0
\(13\) − 5338.61i − 0.673949i −0.941514 0.336974i \(-0.890596\pi\)
0.941514 0.336974i \(-0.109404\pi\)
\(14\) 0 0
\(15\) −1886.91 −0.144355
\(16\) 0 0
\(17\) 38616.6 1.90635 0.953175 0.302419i \(-0.0977941\pi\)
0.953175 + 0.302419i \(0.0977941\pi\)
\(18\) 0 0
\(19\) 13583.6i 0.454335i 0.973856 + 0.227167i \(0.0729465\pi\)
−0.973856 + 0.227167i \(0.927054\pi\)
\(20\) 0 0
\(21\) 23225.2i 0.547257i
\(22\) 0 0
\(23\) 99683.6 1.70835 0.854173 0.519988i \(-0.174064\pi\)
0.854173 + 0.519988i \(0.174064\pi\)
\(24\) 0 0
\(25\) 73241.0 0.937485
\(26\) 0 0
\(27\) 19683.0i 0.192450i
\(28\) 0 0
\(29\) − 81173.3i − 0.618045i −0.951055 0.309023i \(-0.899998\pi\)
0.951055 0.309023i \(-0.100002\pi\)
\(30\) 0 0
\(31\) −153071. −0.922839 −0.461419 0.887182i \(-0.652660\pi\)
−0.461419 + 0.887182i \(0.652660\pi\)
\(32\) 0 0
\(33\) −164880. −0.798675
\(34\) 0 0
\(35\) 60115.1i 0.236998i
\(36\) 0 0
\(37\) 467160.i 1.51621i 0.652131 + 0.758106i \(0.273875\pi\)
−0.652131 + 0.758106i \(0.726125\pi\)
\(38\) 0 0
\(39\) −144143. −0.389104
\(40\) 0 0
\(41\) 384378. 0.870994 0.435497 0.900190i \(-0.356573\pi\)
0.435497 + 0.900190i \(0.356573\pi\)
\(42\) 0 0
\(43\) − 450696.i − 0.864458i −0.901764 0.432229i \(-0.857727\pi\)
0.901764 0.432229i \(-0.142273\pi\)
\(44\) 0 0
\(45\) 50946.6i 0.0833435i
\(46\) 0 0
\(47\) 503097. 0.706821 0.353410 0.935468i \(-0.385022\pi\)
0.353410 + 0.935468i \(0.385022\pi\)
\(48\) 0 0
\(49\) −83612.9 −0.101528
\(50\) 0 0
\(51\) − 1.04265e6i − 1.10063i
\(52\) 0 0
\(53\) − 1.61153e6i − 1.48687i −0.668810 0.743434i \(-0.733196\pi\)
0.668810 0.743434i \(-0.266804\pi\)
\(54\) 0 0
\(55\) −426769. −0.345879
\(56\) 0 0
\(57\) 366756. 0.262310
\(58\) 0 0
\(59\) 1.58458e6i 1.00446i 0.864735 + 0.502228i \(0.167486\pi\)
−0.864735 + 0.502228i \(0.832514\pi\)
\(60\) 0 0
\(61\) − 2.61079e6i − 1.47271i −0.676596 0.736355i \(-0.736545\pi\)
0.676596 0.736355i \(-0.263455\pi\)
\(62\) 0 0
\(63\) 627080. 0.315959
\(64\) 0 0
\(65\) −373092. −0.168508
\(66\) 0 0
\(67\) 1.52096e6i 0.617810i 0.951093 + 0.308905i \(0.0999625\pi\)
−0.951093 + 0.308905i \(0.900038\pi\)
\(68\) 0 0
\(69\) − 2.69146e6i − 0.986315i
\(70\) 0 0
\(71\) 1.15058e6 0.381515 0.190758 0.981637i \(-0.438906\pi\)
0.190758 + 0.981637i \(0.438906\pi\)
\(72\) 0 0
\(73\) 3.52252e6 1.05980 0.529899 0.848061i \(-0.322230\pi\)
0.529899 + 0.848061i \(0.322230\pi\)
\(74\) 0 0
\(75\) − 1.97751e6i − 0.541257i
\(76\) 0 0
\(77\) 5.25292e6i 1.31124i
\(78\) 0 0
\(79\) −7.64873e6 −1.74540 −0.872699 0.488259i \(-0.837632\pi\)
−0.872699 + 0.488259i \(0.837632\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) − 6.34999e6i − 1.21899i −0.792790 0.609495i \(-0.791372\pi\)
0.792790 0.609495i \(-0.208628\pi\)
\(84\) 0 0
\(85\) − 2.69875e6i − 0.476646i
\(86\) 0 0
\(87\) −2.19168e6 −0.356829
\(88\) 0 0
\(89\) −1.70342e6 −0.256128 −0.128064 0.991766i \(-0.540876\pi\)
−0.128064 + 0.991766i \(0.540876\pi\)
\(90\) 0 0
\(91\) 4.59223e6i 0.638821i
\(92\) 0 0
\(93\) 4.13291e6i 0.532801i
\(94\) 0 0
\(95\) 949296. 0.113598
\(96\) 0 0
\(97\) −2.70830e6 −0.301297 −0.150649 0.988587i \(-0.548136\pi\)
−0.150649 + 0.988587i \(0.548136\pi\)
\(98\) 0 0
\(99\) 4.45177e6i 0.461115i
\(100\) 0 0
\(101\) 621207.i 0.0599945i 0.999550 + 0.0299973i \(0.00954986\pi\)
−0.999550 + 0.0299973i \(0.990450\pi\)
\(102\) 0 0
\(103\) 772068. 0.0696186 0.0348093 0.999394i \(-0.488918\pi\)
0.0348093 + 0.999394i \(0.488918\pi\)
\(104\) 0 0
\(105\) 1.62311e6 0.136831
\(106\) 0 0
\(107\) − 1.30766e7i − 1.03193i −0.856609 0.515967i \(-0.827433\pi\)
0.856609 0.515967i \(-0.172567\pi\)
\(108\) 0 0
\(109\) 8.05610e6i 0.595843i 0.954590 + 0.297922i \(0.0962934\pi\)
−0.954590 + 0.297922i \(0.903707\pi\)
\(110\) 0 0
\(111\) 1.26133e7 0.875386
\(112\) 0 0
\(113\) 2.54504e7 1.65928 0.829642 0.558296i \(-0.188545\pi\)
0.829642 + 0.558296i \(0.188545\pi\)
\(114\) 0 0
\(115\) − 6.96645e6i − 0.427139i
\(116\) 0 0
\(117\) 3.89185e6i 0.224650i
\(118\) 0 0
\(119\) −3.32177e7 −1.80699
\(120\) 0 0
\(121\) −1.78044e7 −0.913646
\(122\) 0 0
\(123\) − 1.03782e7i − 0.502869i
\(124\) 0 0
\(125\) − 1.05783e7i − 0.484430i
\(126\) 0 0
\(127\) −3.00240e7 −1.30063 −0.650317 0.759663i \(-0.725364\pi\)
−0.650317 + 0.759663i \(0.725364\pi\)
\(128\) 0 0
\(129\) −1.21688e7 −0.499095
\(130\) 0 0
\(131\) − 2.90785e7i − 1.13012i −0.825051 0.565058i \(-0.808854\pi\)
0.825051 0.565058i \(-0.191146\pi\)
\(132\) 0 0
\(133\) − 1.16845e7i − 0.430654i
\(134\) 0 0
\(135\) 1.37556e6 0.0481184
\(136\) 0 0
\(137\) −1.43011e7 −0.475169 −0.237584 0.971367i \(-0.576356\pi\)
−0.237584 + 0.971367i \(0.576356\pi\)
\(138\) 0 0
\(139\) − 2.91992e7i − 0.922188i −0.887351 0.461094i \(-0.847457\pi\)
0.887351 0.461094i \(-0.152543\pi\)
\(140\) 0 0
\(141\) − 1.35836e7i − 0.408083i
\(142\) 0 0
\(143\) −3.26012e7 −0.932304
\(144\) 0 0
\(145\) −5.67285e6 −0.154530
\(146\) 0 0
\(147\) 2.25755e6i 0.0586174i
\(148\) 0 0
\(149\) 5.91575e6i 0.146507i 0.997313 + 0.0732534i \(0.0233382\pi\)
−0.997313 + 0.0732534i \(0.976662\pi\)
\(150\) 0 0
\(151\) 3.16205e7 0.747394 0.373697 0.927551i \(-0.378090\pi\)
0.373697 + 0.927551i \(0.378090\pi\)
\(152\) 0 0
\(153\) −2.81515e7 −0.635450
\(154\) 0 0
\(155\) 1.06974e7i 0.230738i
\(156\) 0 0
\(157\) 5.33595e7i 1.10043i 0.835023 + 0.550216i \(0.185454\pi\)
−0.835023 + 0.550216i \(0.814546\pi\)
\(158\) 0 0
\(159\) −4.35113e7 −0.858443
\(160\) 0 0
\(161\) −8.57470e7 −1.61930
\(162\) 0 0
\(163\) − 3.05332e7i − 0.552225i −0.961125 0.276112i \(-0.910954\pi\)
0.961125 0.276112i \(-0.0890462\pi\)
\(164\) 0 0
\(165\) 1.15228e7i 0.199693i
\(166\) 0 0
\(167\) 1.15038e7 0.191133 0.0955664 0.995423i \(-0.469534\pi\)
0.0955664 + 0.995423i \(0.469534\pi\)
\(168\) 0 0
\(169\) 3.42477e7 0.545793
\(170\) 0 0
\(171\) − 9.90242e6i − 0.151445i
\(172\) 0 0
\(173\) 1.06369e8i 1.56191i 0.624590 + 0.780953i \(0.285266\pi\)
−0.624590 + 0.780953i \(0.714734\pi\)
\(174\) 0 0
\(175\) −6.30013e7 −0.888621
\(176\) 0 0
\(177\) 4.27836e7 0.579923
\(178\) 0 0
\(179\) − 1.03733e8i − 1.35186i −0.736966 0.675930i \(-0.763742\pi\)
0.736966 0.675930i \(-0.236258\pi\)
\(180\) 0 0
\(181\) − 9.35733e7i − 1.17294i −0.809970 0.586472i \(-0.800517\pi\)
0.809970 0.586472i \(-0.199483\pi\)
\(182\) 0 0
\(183\) −7.04913e7 −0.850269
\(184\) 0 0
\(185\) 3.26478e7 0.379099
\(186\) 0 0
\(187\) − 2.35819e8i − 2.63714i
\(188\) 0 0
\(189\) − 1.69312e7i − 0.182419i
\(190\) 0 0
\(191\) −1.06409e8 −1.10500 −0.552498 0.833514i \(-0.686325\pi\)
−0.552498 + 0.833514i \(0.686325\pi\)
\(192\) 0 0
\(193\) 4.87045e7 0.487661 0.243831 0.969818i \(-0.421596\pi\)
0.243831 + 0.969818i \(0.421596\pi\)
\(194\) 0 0
\(195\) 1.00735e7i 0.0972880i
\(196\) 0 0
\(197\) − 2.38283e7i − 0.222056i −0.993817 0.111028i \(-0.964586\pi\)
0.993817 0.111028i \(-0.0354143\pi\)
\(198\) 0 0
\(199\) −1.99543e8 −1.79494 −0.897472 0.441072i \(-0.854598\pi\)
−0.897472 + 0.441072i \(0.854598\pi\)
\(200\) 0 0
\(201\) 4.10658e7 0.356693
\(202\) 0 0
\(203\) 6.98246e7i 0.585831i
\(204\) 0 0
\(205\) − 2.68625e7i − 0.217775i
\(206\) 0 0
\(207\) −7.26693e7 −0.569449
\(208\) 0 0
\(209\) 8.29505e7 0.628503
\(210\) 0 0
\(211\) − 3.62815e7i − 0.265886i −0.991124 0.132943i \(-0.957557\pi\)
0.991124 0.132943i \(-0.0424428\pi\)
\(212\) 0 0
\(213\) − 3.10656e7i − 0.220268i
\(214\) 0 0
\(215\) −3.14972e7 −0.216141
\(216\) 0 0
\(217\) 1.31670e8 0.874738
\(218\) 0 0
\(219\) − 9.51079e7i − 0.611874i
\(220\) 0 0
\(221\) − 2.06159e8i − 1.28478i
\(222\) 0 0
\(223\) 1.00575e7 0.0607328 0.0303664 0.999539i \(-0.490333\pi\)
0.0303664 + 0.999539i \(0.490333\pi\)
\(224\) 0 0
\(225\) −5.33927e7 −0.312495
\(226\) 0 0
\(227\) − 3.54925e7i − 0.201394i −0.994917 0.100697i \(-0.967893\pi\)
0.994917 0.100697i \(-0.0321072\pi\)
\(228\) 0 0
\(229\) 1.92621e8i 1.05994i 0.848017 + 0.529969i \(0.177796\pi\)
−0.848017 + 0.529969i \(0.822204\pi\)
\(230\) 0 0
\(231\) 1.41829e8 0.757046
\(232\) 0 0
\(233\) 7.85858e7 0.407004 0.203502 0.979075i \(-0.434768\pi\)
0.203502 + 0.979075i \(0.434768\pi\)
\(234\) 0 0
\(235\) − 3.51593e7i − 0.176727i
\(236\) 0 0
\(237\) 2.06516e8i 1.00771i
\(238\) 0 0
\(239\) −3.94334e8 −1.86841 −0.934205 0.356737i \(-0.883889\pi\)
−0.934205 + 0.356737i \(0.883889\pi\)
\(240\) 0 0
\(241\) −2.64677e8 −1.21802 −0.609012 0.793161i \(-0.708434\pi\)
−0.609012 + 0.793161i \(0.708434\pi\)
\(242\) 0 0
\(243\) − 1.43489e7i − 0.0641500i
\(244\) 0 0
\(245\) 5.84334e6i 0.0253852i
\(246\) 0 0
\(247\) 7.25174e7 0.306198
\(248\) 0 0
\(249\) −1.71450e8 −0.703784
\(250\) 0 0
\(251\) − 3.14631e8i − 1.25586i −0.778268 0.627932i \(-0.783901\pi\)
0.778268 0.627932i \(-0.216099\pi\)
\(252\) 0 0
\(253\) − 6.08736e8i − 2.36324i
\(254\) 0 0
\(255\) −7.28661e7 −0.275192
\(256\) 0 0
\(257\) −8.74334e6 −0.0321301 −0.0160650 0.999871i \(-0.505114\pi\)
−0.0160650 + 0.999871i \(0.505114\pi\)
\(258\) 0 0
\(259\) − 4.01848e8i − 1.43718i
\(260\) 0 0
\(261\) 5.91754e7i 0.206015i
\(262\) 0 0
\(263\) 1.27808e8 0.433224 0.216612 0.976258i \(-0.430499\pi\)
0.216612 + 0.976258i \(0.430499\pi\)
\(264\) 0 0
\(265\) −1.12623e8 −0.371762
\(266\) 0 0
\(267\) 4.59923e7i 0.147875i
\(268\) 0 0
\(269\) − 7.00994e7i − 0.219574i −0.993955 0.109787i \(-0.964983\pi\)
0.993955 0.109787i \(-0.0350169\pi\)
\(270\) 0 0
\(271\) −4.25699e8 −1.29930 −0.649651 0.760232i \(-0.725085\pi\)
−0.649651 + 0.760232i \(0.725085\pi\)
\(272\) 0 0
\(273\) 1.23990e8 0.368823
\(274\) 0 0
\(275\) − 4.47259e8i − 1.29687i
\(276\) 0 0
\(277\) 2.13084e7i 0.0602382i 0.999546 + 0.0301191i \(0.00958866\pi\)
−0.999546 + 0.0301191i \(0.990411\pi\)
\(278\) 0 0
\(279\) 1.11588e8 0.307613
\(280\) 0 0
\(281\) −4.21595e8 −1.13350 −0.566752 0.823889i \(-0.691800\pi\)
−0.566752 + 0.823889i \(0.691800\pi\)
\(282\) 0 0
\(283\) 6.62621e8i 1.73785i 0.494944 + 0.868925i \(0.335189\pi\)
−0.494944 + 0.868925i \(0.664811\pi\)
\(284\) 0 0
\(285\) − 2.56310e7i − 0.0655856i
\(286\) 0 0
\(287\) −3.30639e8 −0.825596
\(288\) 0 0
\(289\) 1.08090e9 2.63417
\(290\) 0 0
\(291\) 7.31240e7i 0.173954i
\(292\) 0 0
\(293\) 1.55155e8i 0.360354i 0.983634 + 0.180177i \(0.0576670\pi\)
−0.983634 + 0.180177i \(0.942333\pi\)
\(294\) 0 0
\(295\) 1.10739e8 0.251145
\(296\) 0 0
\(297\) 1.20198e8 0.266225
\(298\) 0 0
\(299\) − 5.32172e8i − 1.15134i
\(300\) 0 0
\(301\) 3.87685e8i 0.819400i
\(302\) 0 0
\(303\) 1.67726e7 0.0346379
\(304\) 0 0
\(305\) −1.82457e8 −0.368222
\(306\) 0 0
\(307\) 6.43544e8i 1.26939i 0.772764 + 0.634693i \(0.218873\pi\)
−0.772764 + 0.634693i \(0.781127\pi\)
\(308\) 0 0
\(309\) − 2.08458e7i − 0.0401943i
\(310\) 0 0
\(311\) 5.58443e8 1.05273 0.526366 0.850258i \(-0.323554\pi\)
0.526366 + 0.850258i \(0.323554\pi\)
\(312\) 0 0
\(313\) 4.78709e8 0.882403 0.441201 0.897408i \(-0.354552\pi\)
0.441201 + 0.897408i \(0.354552\pi\)
\(314\) 0 0
\(315\) − 4.38239e7i − 0.0789994i
\(316\) 0 0
\(317\) 5.26905e8i 0.929020i 0.885568 + 0.464510i \(0.153770\pi\)
−0.885568 + 0.464510i \(0.846230\pi\)
\(318\) 0 0
\(319\) −4.95700e8 −0.854971
\(320\) 0 0
\(321\) −3.53068e8 −0.595787
\(322\) 0 0
\(323\) 5.24551e8i 0.866122i
\(324\) 0 0
\(325\) − 3.91005e8i − 0.631817i
\(326\) 0 0
\(327\) 2.17515e8 0.344010
\(328\) 0 0
\(329\) −4.32760e8 −0.669979
\(330\) 0 0
\(331\) − 5.87362e8i − 0.890241i −0.895471 0.445121i \(-0.853161\pi\)
0.895471 0.445121i \(-0.146839\pi\)
\(332\) 0 0
\(333\) − 3.40560e8i − 0.505404i
\(334\) 0 0
\(335\) 1.06293e8 0.154471
\(336\) 0 0
\(337\) −1.12323e9 −1.59869 −0.799344 0.600874i \(-0.794820\pi\)
−0.799344 + 0.600874i \(0.794820\pi\)
\(338\) 0 0
\(339\) − 6.87161e8i − 0.957988i
\(340\) 0 0
\(341\) 9.34753e8i 1.27661i
\(342\) 0 0
\(343\) 7.80328e8 1.04411
\(344\) 0 0
\(345\) −1.88094e8 −0.246609
\(346\) 0 0
\(347\) 1.10945e9i 1.42546i 0.701440 + 0.712728i \(0.252541\pi\)
−0.701440 + 0.712728i \(0.747459\pi\)
\(348\) 0 0
\(349\) 4.43892e8i 0.558970i 0.960150 + 0.279485i \(0.0901637\pi\)
−0.960150 + 0.279485i \(0.909836\pi\)
\(350\) 0 0
\(351\) 1.05080e8 0.129701
\(352\) 0 0
\(353\) 1.12493e8 0.136118 0.0680589 0.997681i \(-0.478319\pi\)
0.0680589 + 0.997681i \(0.478319\pi\)
\(354\) 0 0
\(355\) − 8.04089e7i − 0.0953905i
\(356\) 0 0
\(357\) 8.96877e8i 1.04326i
\(358\) 0 0
\(359\) 4.86420e8 0.554856 0.277428 0.960746i \(-0.410518\pi\)
0.277428 + 0.960746i \(0.410518\pi\)
\(360\) 0 0
\(361\) 7.09358e8 0.793580
\(362\) 0 0
\(363\) 4.80718e8i 0.527494i
\(364\) 0 0
\(365\) − 2.46173e8i − 0.264982i
\(366\) 0 0
\(367\) −1.24608e9 −1.31587 −0.657935 0.753075i \(-0.728570\pi\)
−0.657935 + 0.753075i \(0.728570\pi\)
\(368\) 0 0
\(369\) −2.80212e8 −0.290331
\(370\) 0 0
\(371\) 1.38622e9i 1.40937i
\(372\) 0 0
\(373\) − 1.04875e9i − 1.04638i −0.852215 0.523192i \(-0.824741\pi\)
0.852215 0.523192i \(-0.175259\pi\)
\(374\) 0 0
\(375\) −2.85614e8 −0.279686
\(376\) 0 0
\(377\) −4.33353e8 −0.416531
\(378\) 0 0
\(379\) 1.55519e8i 0.146739i 0.997305 + 0.0733697i \(0.0233753\pi\)
−0.997305 + 0.0733697i \(0.976625\pi\)
\(380\) 0 0
\(381\) 8.10647e8i 0.750922i
\(382\) 0 0
\(383\) 1.81354e8 0.164942 0.0824711 0.996593i \(-0.473719\pi\)
0.0824711 + 0.996593i \(0.473719\pi\)
\(384\) 0 0
\(385\) 3.67104e8 0.327851
\(386\) 0 0
\(387\) 3.28557e8i 0.288153i
\(388\) 0 0
\(389\) − 2.90627e8i − 0.250330i −0.992136 0.125165i \(-0.960054\pi\)
0.992136 0.125165i \(-0.0399460\pi\)
\(390\) 0 0
\(391\) 3.84944e9 3.25671
\(392\) 0 0
\(393\) −7.85120e8 −0.652472
\(394\) 0 0
\(395\) 5.34536e8i 0.436403i
\(396\) 0 0
\(397\) − 2.05323e9i − 1.64691i −0.567381 0.823455i \(-0.692043\pi\)
0.567381 0.823455i \(-0.307957\pi\)
\(398\) 0 0
\(399\) −3.15481e8 −0.248638
\(400\) 0 0
\(401\) −4.73752e8 −0.366898 −0.183449 0.983029i \(-0.558726\pi\)
−0.183449 + 0.983029i \(0.558726\pi\)
\(402\) 0 0
\(403\) 8.17185e8i 0.621946i
\(404\) 0 0
\(405\) − 3.71401e7i − 0.0277812i
\(406\) 0 0
\(407\) 2.85280e9 2.09745
\(408\) 0 0
\(409\) 1.63944e7 0.0118485 0.00592424 0.999982i \(-0.498114\pi\)
0.00592424 + 0.999982i \(0.498114\pi\)
\(410\) 0 0
\(411\) 3.86130e8i 0.274339i
\(412\) 0 0
\(413\) − 1.36304e9i − 0.952102i
\(414\) 0 0
\(415\) −4.43773e8 −0.304784
\(416\) 0 0
\(417\) −7.88379e8 −0.532425
\(418\) 0 0
\(419\) 1.55726e9i 1.03422i 0.855920 + 0.517109i \(0.172992\pi\)
−0.855920 + 0.517109i \(0.827008\pi\)
\(420\) 0 0
\(421\) − 1.48339e9i − 0.968877i −0.874825 0.484439i \(-0.839024\pi\)
0.874825 0.484439i \(-0.160976\pi\)
\(422\) 0 0
\(423\) −3.66758e8 −0.235607
\(424\) 0 0
\(425\) 2.82832e9 1.78717
\(426\) 0 0
\(427\) 2.24578e9i 1.39595i
\(428\) 0 0
\(429\) 8.80233e8i 0.538266i
\(430\) 0 0
\(431\) −8.81971e8 −0.530620 −0.265310 0.964163i \(-0.585474\pi\)
−0.265310 + 0.964163i \(0.585474\pi\)
\(432\) 0 0
\(433\) 2.54428e9 1.50611 0.753057 0.657955i \(-0.228578\pi\)
0.753057 + 0.657955i \(0.228578\pi\)
\(434\) 0 0
\(435\) 1.53167e8i 0.0892180i
\(436\) 0 0
\(437\) 1.35406e9i 0.776162i
\(438\) 0 0
\(439\) −6.23338e7 −0.0351640 −0.0175820 0.999845i \(-0.505597\pi\)
−0.0175820 + 0.999845i \(0.505597\pi\)
\(440\) 0 0
\(441\) 6.09538e7 0.0338428
\(442\) 0 0
\(443\) − 2.60350e9i − 1.42280i −0.702786 0.711401i \(-0.748061\pi\)
0.702786 0.711401i \(-0.251939\pi\)
\(444\) 0 0
\(445\) 1.19045e8i 0.0640397i
\(446\) 0 0
\(447\) 1.59725e8 0.0845857
\(448\) 0 0
\(449\) −1.34950e9 −0.703577 −0.351789 0.936079i \(-0.614426\pi\)
−0.351789 + 0.936079i \(0.614426\pi\)
\(450\) 0 0
\(451\) − 2.34728e9i − 1.20489i
\(452\) 0 0
\(453\) − 8.53753e8i − 0.431508i
\(454\) 0 0
\(455\) 3.20931e8 0.159725
\(456\) 0 0
\(457\) 1.69773e9 0.832072 0.416036 0.909348i \(-0.363419\pi\)
0.416036 + 0.909348i \(0.363419\pi\)
\(458\) 0 0
\(459\) 7.60090e8i 0.366877i
\(460\) 0 0
\(461\) 4.08982e9i 1.94425i 0.234473 + 0.972123i \(0.424663\pi\)
−0.234473 + 0.972123i \(0.575337\pi\)
\(462\) 0 0
\(463\) 3.44026e9 1.61086 0.805430 0.592691i \(-0.201935\pi\)
0.805430 + 0.592691i \(0.201935\pi\)
\(464\) 0 0
\(465\) 2.88831e8 0.133217
\(466\) 0 0
\(467\) 2.41766e8i 0.109847i 0.998491 + 0.0549233i \(0.0174914\pi\)
−0.998491 + 0.0549233i \(0.982509\pi\)
\(468\) 0 0
\(469\) − 1.30831e9i − 0.585608i
\(470\) 0 0
\(471\) 1.44071e9 0.635334
\(472\) 0 0
\(473\) −2.75226e9 −1.19584
\(474\) 0 0
\(475\) 9.94874e8i 0.425932i
\(476\) 0 0
\(477\) 1.17480e9i 0.495622i
\(478\) 0 0
\(479\) −1.03213e9 −0.429103 −0.214551 0.976713i \(-0.568829\pi\)
−0.214551 + 0.976713i \(0.568829\pi\)
\(480\) 0 0
\(481\) 2.49399e9 1.02185
\(482\) 0 0
\(483\) 2.31517e9i 0.934905i
\(484\) 0 0
\(485\) 1.89271e8i 0.0753335i
\(486\) 0 0
\(487\) 4.67124e9 1.83265 0.916327 0.400431i \(-0.131139\pi\)
0.916327 + 0.400431i \(0.131139\pi\)
\(488\) 0 0
\(489\) −8.24397e8 −0.318827
\(490\) 0 0
\(491\) 4.59929e9i 1.75350i 0.480947 + 0.876750i \(0.340293\pi\)
−0.480947 + 0.876750i \(0.659707\pi\)
\(492\) 0 0
\(493\) − 3.13464e9i − 1.17821i
\(494\) 0 0
\(495\) 3.11115e8 0.115293
\(496\) 0 0
\(497\) −9.89718e8 −0.361630
\(498\) 0 0
\(499\) 4.60308e9i 1.65843i 0.558932 + 0.829213i \(0.311211\pi\)
−0.558932 + 0.829213i \(0.688789\pi\)
\(500\) 0 0
\(501\) − 3.10604e8i − 0.110351i
\(502\) 0 0
\(503\) 2.74041e9 0.960125 0.480063 0.877234i \(-0.340614\pi\)
0.480063 + 0.877234i \(0.340614\pi\)
\(504\) 0 0
\(505\) 4.34135e7 0.0150005
\(506\) 0 0
\(507\) − 9.24688e8i − 0.315114i
\(508\) 0 0
\(509\) 3.06630e9i 1.03063i 0.857001 + 0.515315i \(0.172325\pi\)
−0.857001 + 0.515315i \(0.827675\pi\)
\(510\) 0 0
\(511\) −3.03004e9 −1.00456
\(512\) 0 0
\(513\) −2.67365e8 −0.0874368
\(514\) 0 0
\(515\) − 5.39564e7i − 0.0174068i
\(516\) 0 0
\(517\) − 3.07225e9i − 0.977777i
\(518\) 0 0
\(519\) 2.87197e9 0.901767
\(520\) 0 0
\(521\) −2.75950e9 −0.854865 −0.427433 0.904047i \(-0.640582\pi\)
−0.427433 + 0.904047i \(0.640582\pi\)
\(522\) 0 0
\(523\) − 2.03158e9i − 0.620979i −0.950577 0.310490i \(-0.899507\pi\)
0.950577 0.310490i \(-0.100493\pi\)
\(524\) 0 0
\(525\) 1.70104e9i 0.513045i
\(526\) 0 0
\(527\) −5.91107e9 −1.75925
\(528\) 0 0
\(529\) 6.53198e9 1.91845
\(530\) 0 0
\(531\) − 1.15516e9i − 0.334819i
\(532\) 0 0
\(533\) − 2.05205e9i − 0.587005i
\(534\) 0 0
\(535\) −9.13867e8 −0.258015
\(536\) 0 0
\(537\) −2.80079e9 −0.780497
\(538\) 0 0
\(539\) 5.10597e8i 0.140449i
\(540\) 0 0
\(541\) − 8.68196e8i − 0.235737i −0.993029 0.117868i \(-0.962394\pi\)
0.993029 0.117868i \(-0.0376061\pi\)
\(542\) 0 0
\(543\) −2.52648e9 −0.677199
\(544\) 0 0
\(545\) 5.63005e8 0.148979
\(546\) 0 0
\(547\) − 2.66370e8i − 0.0695873i −0.999395 0.0347937i \(-0.988923\pi\)
0.999395 0.0347937i \(-0.0110774\pi\)
\(548\) 0 0
\(549\) 1.90326e9i 0.490903i
\(550\) 0 0
\(551\) 1.10262e9 0.280800
\(552\) 0 0
\(553\) 6.57937e9 1.65442
\(554\) 0 0
\(555\) − 8.81491e8i − 0.218873i
\(556\) 0 0
\(557\) − 6.92953e9i − 1.69907i −0.527535 0.849533i \(-0.676884\pi\)
0.527535 0.849533i \(-0.323116\pi\)
\(558\) 0 0
\(559\) −2.40609e9 −0.582600
\(560\) 0 0
\(561\) −6.36712e9 −1.52255
\(562\) 0 0
\(563\) − 6.05174e9i − 1.42923i −0.699520 0.714613i \(-0.746603\pi\)
0.699520 0.714613i \(-0.253397\pi\)
\(564\) 0 0
\(565\) − 1.77862e9i − 0.414871i
\(566\) 0 0
\(567\) −4.57141e8 −0.105320
\(568\) 0 0
\(569\) −2.35823e9 −0.536654 −0.268327 0.963328i \(-0.586471\pi\)
−0.268327 + 0.963328i \(0.586471\pi\)
\(570\) 0 0
\(571\) 1.32039e9i 0.296808i 0.988927 + 0.148404i \(0.0474136\pi\)
−0.988927 + 0.148404i \(0.952586\pi\)
\(572\) 0 0
\(573\) 2.87304e9i 0.637970i
\(574\) 0 0
\(575\) 7.30092e9 1.60155
\(576\) 0 0
\(577\) −6.48171e9 −1.40467 −0.702334 0.711847i \(-0.747859\pi\)
−0.702334 + 0.711847i \(0.747859\pi\)
\(578\) 0 0
\(579\) − 1.31502e9i − 0.281551i
\(580\) 0 0
\(581\) 5.46221e9i 1.15545i
\(582\) 0 0
\(583\) −9.84109e9 −2.05685
\(584\) 0 0
\(585\) 2.71984e8 0.0561692
\(586\) 0 0
\(587\) 1.51129e8i 0.0308399i 0.999881 + 0.0154200i \(0.00490852\pi\)
−0.999881 + 0.0154200i \(0.995091\pi\)
\(588\) 0 0
\(589\) − 2.07924e9i − 0.419278i
\(590\) 0 0
\(591\) −6.43365e8 −0.128204
\(592\) 0 0
\(593\) −4.81216e9 −0.947652 −0.473826 0.880619i \(-0.657127\pi\)
−0.473826 + 0.880619i \(0.657127\pi\)
\(594\) 0 0
\(595\) 2.32144e9i 0.451802i
\(596\) 0 0
\(597\) 5.38766e9i 1.03631i
\(598\) 0 0
\(599\) 3.95101e9 0.751129 0.375565 0.926796i \(-0.377449\pi\)
0.375565 + 0.926796i \(0.377449\pi\)
\(600\) 0 0
\(601\) −4.30889e9 −0.809664 −0.404832 0.914391i \(-0.632670\pi\)
−0.404832 + 0.914391i \(0.632670\pi\)
\(602\) 0 0
\(603\) − 1.10878e9i − 0.205937i
\(604\) 0 0
\(605\) 1.24427e9i 0.228439i
\(606\) 0 0
\(607\) 1.02322e9 0.185698 0.0928489 0.995680i \(-0.470403\pi\)
0.0928489 + 0.995680i \(0.470403\pi\)
\(608\) 0 0
\(609\) 1.88527e9 0.338230
\(610\) 0 0
\(611\) − 2.68584e9i − 0.476361i
\(612\) 0 0
\(613\) − 2.24790e9i − 0.394153i −0.980388 0.197076i \(-0.936855\pi\)
0.980388 0.197076i \(-0.0631447\pi\)
\(614\) 0 0
\(615\) −7.25288e8 −0.125733
\(616\) 0 0
\(617\) 1.76194e9 0.301990 0.150995 0.988535i \(-0.451752\pi\)
0.150995 + 0.988535i \(0.451752\pi\)
\(618\) 0 0
\(619\) − 4.40993e9i − 0.747333i −0.927563 0.373667i \(-0.878100\pi\)
0.927563 0.373667i \(-0.121900\pi\)
\(620\) 0 0
\(621\) 1.96207e9i 0.328772i
\(622\) 0 0
\(623\) 1.46527e9 0.242778
\(624\) 0 0
\(625\) 4.98268e9 0.816362
\(626\) 0 0
\(627\) − 2.23966e9i − 0.362866i
\(628\) 0 0
\(629\) 1.80401e10i 2.89043i
\(630\) 0 0
\(631\) −2.94248e9 −0.466241 −0.233120 0.972448i \(-0.574894\pi\)
−0.233120 + 0.972448i \(0.574894\pi\)
\(632\) 0 0
\(633\) −9.79600e8 −0.153510
\(634\) 0 0
\(635\) 2.09825e9i 0.325198i
\(636\) 0 0
\(637\) 4.46377e8i 0.0684248i
\(638\) 0 0
\(639\) −8.38772e8 −0.127172
\(640\) 0 0
\(641\) −1.15137e9 −0.172669 −0.0863343 0.996266i \(-0.527515\pi\)
−0.0863343 + 0.996266i \(0.527515\pi\)
\(642\) 0 0
\(643\) − 2.10275e9i − 0.311924i −0.987763 0.155962i \(-0.950152\pi\)
0.987763 0.155962i \(-0.0498478\pi\)
\(644\) 0 0
\(645\) 8.50424e8i 0.124789i
\(646\) 0 0
\(647\) −1.11353e10 −1.61636 −0.808181 0.588934i \(-0.799548\pi\)
−0.808181 + 0.588934i \(0.799548\pi\)
\(648\) 0 0
\(649\) 9.67651e9 1.38951
\(650\) 0 0
\(651\) − 3.55509e9i − 0.505030i
\(652\) 0 0
\(653\) − 4.60816e9i − 0.647636i −0.946119 0.323818i \(-0.895033\pi\)
0.946119 0.323818i \(-0.104967\pi\)
\(654\) 0 0
\(655\) −2.03217e9 −0.282563
\(656\) 0 0
\(657\) −2.56791e9 −0.353266
\(658\) 0 0
\(659\) − 1.25838e9i − 0.171282i −0.996326 0.0856411i \(-0.972706\pi\)
0.996326 0.0856411i \(-0.0272938\pi\)
\(660\) 0 0
\(661\) − 1.45629e10i − 1.96129i −0.195791 0.980646i \(-0.562727\pi\)
0.195791 0.980646i \(-0.437273\pi\)
\(662\) 0 0
\(663\) −5.56629e9 −0.741769
\(664\) 0 0
\(665\) −8.16577e8 −0.107677
\(666\) 0 0
\(667\) − 8.09164e9i − 1.05584i
\(668\) 0 0
\(669\) − 2.71553e8i − 0.0350641i
\(670\) 0 0
\(671\) −1.59432e10 −2.03727
\(672\) 0 0
\(673\) 1.22925e10 1.55449 0.777243 0.629200i \(-0.216617\pi\)
0.777243 + 0.629200i \(0.216617\pi\)
\(674\) 0 0
\(675\) 1.44160e9i 0.180419i
\(676\) 0 0
\(677\) 3.50903e9i 0.434636i 0.976101 + 0.217318i \(0.0697309\pi\)
−0.976101 + 0.217318i \(0.930269\pi\)
\(678\) 0 0
\(679\) 2.32965e9 0.285593
\(680\) 0 0
\(681\) −9.58296e8 −0.116275
\(682\) 0 0
\(683\) − 6.38859e9i − 0.767243i −0.923490 0.383621i \(-0.874677\pi\)
0.923490 0.383621i \(-0.125323\pi\)
\(684\) 0 0
\(685\) 9.99442e8i 0.118807i
\(686\) 0 0
\(687\) 5.20078e9 0.611955
\(688\) 0 0
\(689\) −8.60332e9 −1.00207
\(690\) 0 0
\(691\) − 1.38774e10i − 1.60005i −0.599966 0.800026i \(-0.704819\pi\)
0.599966 0.800026i \(-0.295181\pi\)
\(692\) 0 0
\(693\) − 3.82938e9i − 0.437081i
\(694\) 0 0
\(695\) −2.04061e9 −0.230575
\(696\) 0 0
\(697\) 1.48434e10 1.66042
\(698\) 0 0
\(699\) − 2.12182e9i − 0.234984i
\(700\) 0 0
\(701\) − 5.25165e9i − 0.575815i −0.957658 0.287907i \(-0.907040\pi\)
0.957658 0.287907i \(-0.0929595\pi\)
\(702\) 0 0
\(703\) −6.34570e9 −0.688868
\(704\) 0 0
\(705\) −9.49300e8 −0.102033
\(706\) 0 0
\(707\) − 5.34358e8i − 0.0568675i
\(708\) 0 0
\(709\) − 1.13161e10i − 1.19243i −0.802824 0.596215i \(-0.796670\pi\)
0.802824 0.596215i \(-0.203330\pi\)
\(710\) 0 0
\(711\) 5.57592e9 0.581799
\(712\) 0 0
\(713\) −1.52586e10 −1.57653
\(714\) 0 0
\(715\) 2.27836e9i 0.233104i
\(716\) 0 0
\(717\) 1.06470e10i 1.07873i
\(718\) 0 0
\(719\) 7.00326e9 0.702666 0.351333 0.936251i \(-0.385729\pi\)
0.351333 + 0.936251i \(0.385729\pi\)
\(720\) 0 0
\(721\) −6.64126e8 −0.0659899
\(722\) 0 0
\(723\) 7.14627e9i 0.703227i
\(724\) 0 0
\(725\) − 5.94522e9i − 0.579408i
\(726\) 0 0
\(727\) 2.14254e9 0.206804 0.103402 0.994640i \(-0.467027\pi\)
0.103402 + 0.994640i \(0.467027\pi\)
\(728\) 0 0
\(729\) −3.87420e8 −0.0370370
\(730\) 0 0
\(731\) − 1.74043e10i − 1.64796i
\(732\) 0 0
\(733\) − 2.00725e10i − 1.88251i −0.337691 0.941257i \(-0.609646\pi\)
0.337691 0.941257i \(-0.390354\pi\)
\(734\) 0 0
\(735\) 1.57770e8 0.0146561
\(736\) 0 0
\(737\) 9.28799e9 0.854645
\(738\) 0 0
\(739\) 7.84304e8i 0.0714873i 0.999361 + 0.0357436i \(0.0113800\pi\)
−0.999361 + 0.0357436i \(0.988620\pi\)
\(740\) 0 0
\(741\) − 1.95797e9i − 0.176784i
\(742\) 0 0
\(743\) −2.16696e10 −1.93816 −0.969079 0.246749i \(-0.920638\pi\)
−0.969079 + 0.246749i \(0.920638\pi\)
\(744\) 0 0
\(745\) 4.13426e8 0.0366312
\(746\) 0 0
\(747\) 4.62914e9i 0.406330i
\(748\) 0 0
\(749\) 1.12484e10i 0.978146i
\(750\) 0 0
\(751\) −1.78319e9 −0.153623 −0.0768117 0.997046i \(-0.524474\pi\)
−0.0768117 + 0.997046i \(0.524474\pi\)
\(752\) 0 0
\(753\) −8.49502e9 −0.725074
\(754\) 0 0
\(755\) − 2.20982e9i − 0.186871i
\(756\) 0 0
\(757\) 9.55236e9i 0.800341i 0.916441 + 0.400171i \(0.131049\pi\)
−0.916441 + 0.400171i \(0.868951\pi\)
\(758\) 0 0
\(759\) −1.64359e10 −1.36441
\(760\) 0 0
\(761\) 2.69468e9 0.221647 0.110823 0.993840i \(-0.464651\pi\)
0.110823 + 0.993840i \(0.464651\pi\)
\(762\) 0 0
\(763\) − 6.92979e9i − 0.564786i
\(764\) 0 0
\(765\) 1.96739e9i 0.158882i
\(766\) 0 0
\(767\) 8.45944e9 0.676952
\(768\) 0 0
\(769\) 6.87358e9 0.545056 0.272528 0.962148i \(-0.412140\pi\)
0.272528 + 0.962148i \(0.412140\pi\)
\(770\) 0 0
\(771\) 2.36070e8i 0.0185503i
\(772\) 0 0
\(773\) 1.50956e10i 1.17550i 0.809043 + 0.587749i \(0.199986\pi\)
−0.809043 + 0.587749i \(0.800014\pi\)
\(774\) 0 0
\(775\) −1.12110e10 −0.865147
\(776\) 0 0
\(777\) −1.08499e10 −0.829758
\(778\) 0 0
\(779\) 5.22123e9i 0.395723i
\(780\) 0 0
\(781\) − 7.02622e9i − 0.527768i
\(782\) 0 0
\(783\) 1.59773e9 0.118943
\(784\) 0 0
\(785\) 3.72906e9 0.275141
\(786\) 0 0
\(787\) − 6.42984e9i − 0.470207i −0.971970 0.235103i \(-0.924457\pi\)
0.971970 0.235103i \(-0.0755429\pi\)
\(788\) 0 0
\(789\) − 3.45081e9i − 0.250122i
\(790\) 0 0
\(791\) −2.18922e10 −1.57280
\(792\) 0 0
\(793\) −1.39380e10 −0.992531
\(794\) 0 0
\(795\) 3.04081e9i 0.214637i
\(796\) 0 0
\(797\) − 2.25537e10i − 1.57803i −0.614377 0.789013i \(-0.710593\pi\)
0.614377 0.789013i \(-0.289407\pi\)
\(798\) 0 0
\(799\) 1.94279e10 1.34745
\(800\) 0 0
\(801\) 1.24179e9 0.0853759
\(802\) 0 0
\(803\) − 2.15109e10i − 1.46607i
\(804\) 0 0
\(805\) 5.99248e9i 0.404875i
\(806\) 0 0
\(807\) −1.89268e9 −0.126771
\(808\) 0 0
\(809\) 1.28393e10 0.852556 0.426278 0.904592i \(-0.359824\pi\)
0.426278 + 0.904592i \(0.359824\pi\)
\(810\) 0 0
\(811\) 1.20292e10i 0.791891i 0.918274 + 0.395946i \(0.129583\pi\)
−0.918274 + 0.395946i \(0.870417\pi\)
\(812\) 0 0
\(813\) 1.14939e10i 0.750153i
\(814\) 0 0
\(815\) −2.13383e9 −0.138073
\(816\) 0 0
\(817\) 6.12206e9 0.392754
\(818\) 0 0
\(819\) − 3.34774e9i − 0.212940i
\(820\) 0 0
\(821\) − 1.28268e10i − 0.808944i −0.914550 0.404472i \(-0.867455\pi\)
0.914550 0.404472i \(-0.132545\pi\)
\(822\) 0 0
\(823\) −1.15343e10 −0.721257 −0.360628 0.932710i \(-0.617438\pi\)
−0.360628 + 0.932710i \(0.617438\pi\)
\(824\) 0 0
\(825\) −1.20760e10 −0.748746
\(826\) 0 0
\(827\) 1.84870e8i 0.0113657i 0.999984 + 0.00568287i \(0.00180892\pi\)
−0.999984 + 0.00568287i \(0.998191\pi\)
\(828\) 0 0
\(829\) 7.96965e9i 0.485845i 0.970046 + 0.242923i \(0.0781062\pi\)
−0.970046 + 0.242923i \(0.921894\pi\)
\(830\) 0 0
\(831\) 5.75328e8 0.0347786
\(832\) 0 0
\(833\) −3.22884e9 −0.193548
\(834\) 0 0
\(835\) − 8.03953e8i − 0.0477890i
\(836\) 0 0
\(837\) − 3.01289e9i − 0.177600i
\(838\) 0 0
\(839\) 4.77023e9 0.278851 0.139426 0.990233i \(-0.455474\pi\)
0.139426 + 0.990233i \(0.455474\pi\)
\(840\) 0 0
\(841\) 1.06608e10 0.618020
\(842\) 0 0
\(843\) 1.13831e10i 0.654429i
\(844\) 0 0
\(845\) − 2.39342e9i − 0.136465i
\(846\) 0 0
\(847\) 1.53152e10 0.866025
\(848\) 0 0
\(849\) 1.78908e10 1.00335
\(850\) 0 0
\(851\) 4.65682e10i 2.59022i
\(852\) 0 0
\(853\) − 5.90759e9i − 0.325904i −0.986634 0.162952i \(-0.947899\pi\)
0.986634 0.162952i \(-0.0521015\pi\)
\(854\) 0 0
\(855\) −6.92037e8 −0.0378659
\(856\) 0 0
\(857\) 2.61048e9 0.141673 0.0708367 0.997488i \(-0.477433\pi\)
0.0708367 + 0.997488i \(0.477433\pi\)
\(858\) 0 0
\(859\) − 2.45743e10i − 1.32283i −0.750019 0.661416i \(-0.769956\pi\)
0.750019 0.661416i \(-0.230044\pi\)
\(860\) 0 0
\(861\) 8.92725e9i 0.476658i
\(862\) 0 0
\(863\) 3.17420e10 1.68111 0.840556 0.541725i \(-0.182229\pi\)
0.840556 + 0.541725i \(0.182229\pi\)
\(864\) 0 0
\(865\) 7.43368e9 0.390524
\(866\) 0 0
\(867\) − 2.91844e10i − 1.52084i
\(868\) 0 0
\(869\) 4.67083e10i 2.41449i
\(870\) 0 0
\(871\) 8.11980e9 0.416372
\(872\) 0 0
\(873\) 1.97435e9 0.100432
\(874\) 0 0
\(875\) 9.09938e9i 0.459181i
\(876\) 0 0
\(877\) 1.25309e10i 0.627313i 0.949537 + 0.313657i \(0.101554\pi\)
−0.949537 + 0.313657i \(0.898446\pi\)
\(878\) 0 0
\(879\) 4.18918e9 0.208050
\(880\) 0 0
\(881\) −3.40090e9 −0.167563 −0.0837815 0.996484i \(-0.526700\pi\)
−0.0837815 + 0.996484i \(0.526700\pi\)
\(882\) 0 0
\(883\) 1.37453e10i 0.671881i 0.941883 + 0.335941i \(0.109054\pi\)
−0.941883 + 0.335941i \(0.890946\pi\)
\(884\) 0 0
\(885\) − 2.98996e9i − 0.144999i
\(886\) 0 0
\(887\) 1.68738e10 0.811858 0.405929 0.913905i \(-0.366948\pi\)
0.405929 + 0.913905i \(0.366948\pi\)
\(888\) 0 0
\(889\) 2.58264e10 1.23284
\(890\) 0 0
\(891\) − 3.24534e9i − 0.153705i
\(892\) 0 0
\(893\) 6.83385e9i 0.321133i
\(894\) 0 0
\(895\) −7.24945e9 −0.338006
\(896\) 0 0
\(897\) −1.43686e10 −0.664725
\(898\) 0 0
\(899\) 1.24253e10i 0.570356i
\(900\) 0 0
\(901\) − 6.22317e10i − 2.83449i
\(902\) 0 0
\(903\) 1.04675e10 0.473081
\(904\) 0 0
\(905\) −6.53943e9 −0.293271
\(906\) 0 0
\(907\) 3.30493e10i 1.47074i 0.677664 + 0.735372i \(0.262992\pi\)
−0.677664 + 0.735372i \(0.737008\pi\)
\(908\) 0 0
\(909\) − 4.52860e8i − 0.0199982i
\(910\) 0 0
\(911\) 2.26308e10 0.991712 0.495856 0.868405i \(-0.334854\pi\)
0.495856 + 0.868405i \(0.334854\pi\)
\(912\) 0 0
\(913\) −3.87774e10 −1.68628
\(914\) 0 0
\(915\) 4.92633e9i 0.212593i
\(916\) 0 0
\(917\) 2.50131e10i 1.07121i
\(918\) 0 0
\(919\) −3.68178e10 −1.56478 −0.782391 0.622788i \(-0.786000\pi\)
−0.782391 + 0.622788i \(0.786000\pi\)
\(920\) 0 0
\(921\) 1.73757e10 0.732880
\(922\) 0 0
\(923\) − 6.14250e9i − 0.257122i
\(924\) 0 0
\(925\) 3.42153e10i 1.42143i
\(926\) 0 0
\(927\) −5.62837e8 −0.0232062
\(928\) 0 0
\(929\) −4.59562e10 −1.88057 −0.940285 0.340389i \(-0.889441\pi\)
−0.940285 + 0.340389i \(0.889441\pi\)
\(930\) 0 0
\(931\) − 1.13576e9i − 0.0461278i
\(932\) 0 0
\(933\) − 1.50780e10i − 0.607795i
\(934\) 0 0
\(935\) −1.64804e10 −0.659366
\(936\) 0 0
\(937\) −3.13165e10 −1.24361 −0.621806 0.783171i \(-0.713601\pi\)
−0.621806 + 0.783171i \(0.713601\pi\)
\(938\) 0 0
\(939\) − 1.29252e10i − 0.509455i
\(940\) 0 0
\(941\) − 1.02564e9i − 0.0401264i −0.999799 0.0200632i \(-0.993613\pi\)
0.999799 0.0200632i \(-0.00638674\pi\)
\(942\) 0 0
\(943\) 3.83162e10 1.48796
\(944\) 0 0
\(945\) −1.18324e9 −0.0456103
\(946\) 0 0
\(947\) − 8.12232e9i − 0.310781i −0.987853 0.155391i \(-0.950336\pi\)
0.987853 0.155391i \(-0.0496637\pi\)
\(948\) 0 0
\(949\) − 1.88054e10i − 0.714249i
\(950\) 0 0
\(951\) 1.42264e10 0.536370
\(952\) 0 0
\(953\) −3.41119e10 −1.27668 −0.638338 0.769757i \(-0.720378\pi\)
−0.638338 + 0.769757i \(0.720378\pi\)
\(954\) 0 0
\(955\) 7.43645e9i 0.276283i
\(956\) 0 0
\(957\) 1.33839e10i 0.493617i
\(958\) 0 0
\(959\) 1.23017e10 0.450402
\(960\) 0 0
\(961\) −4.08200e9 −0.148368
\(962\) 0 0
\(963\) 9.53284e9i 0.343978i
\(964\) 0 0
\(965\) − 3.40374e9i − 0.121930i
\(966\) 0 0
\(967\) −2.99763e10 −1.06607 −0.533035 0.846093i \(-0.678949\pi\)
−0.533035 + 0.846093i \(0.678949\pi\)
\(968\) 0 0
\(969\) 1.41629e10 0.500056
\(970\) 0 0
\(971\) 3.45450e9i 0.121093i 0.998165 + 0.0605463i \(0.0192843\pi\)
−0.998165 + 0.0605463i \(0.980716\pi\)
\(972\) 0 0
\(973\) 2.51169e10i 0.874121i
\(974\) 0 0
\(975\) −1.05571e10 −0.364779
\(976\) 0 0
\(977\) 3.88403e10 1.33245 0.666226 0.745750i \(-0.267909\pi\)
0.666226 + 0.745750i \(0.267909\pi\)
\(978\) 0 0
\(979\) 1.04022e10i 0.354313i
\(980\) 0 0
\(981\) − 5.87289e9i − 0.198614i
\(982\) 0 0
\(983\) 3.64409e10 1.22363 0.611817 0.790999i \(-0.290439\pi\)
0.611817 + 0.790999i \(0.290439\pi\)
\(984\) 0 0
\(985\) −1.66526e9 −0.0555207
\(986\) 0 0
\(987\) 1.16845e10i 0.386813i
\(988\) 0 0
\(989\) − 4.49270e10i − 1.47679i
\(990\) 0 0
\(991\) 4.07339e10 1.32953 0.664764 0.747053i \(-0.268532\pi\)
0.664764 + 0.747053i \(0.268532\pi\)
\(992\) 0 0
\(993\) −1.58588e10 −0.513981
\(994\) 0 0
\(995\) 1.39452e10i 0.448790i
\(996\) 0 0
\(997\) 2.26657e10i 0.724330i 0.932114 + 0.362165i \(0.117962\pi\)
−0.932114 + 0.362165i \(0.882038\pi\)
\(998\) 0 0
\(999\) −9.19512e9 −0.291795
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.193.2 yes 8
4.3 odd 2 384.8.d.c.193.6 yes 8
8.3 odd 2 384.8.d.c.193.3 8
8.5 even 2 inner 384.8.d.d.193.7 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.8.d.c.193.3 8 8.3 odd 2
384.8.d.c.193.6 yes 8 4.3 odd 2
384.8.d.d.193.2 yes 8 1.1 even 1 trivial
384.8.d.d.193.7 yes 8 8.5 even 2 inner