Properties

Label 384.10.d.b.193.6
Level $384$
Weight $10$
Character 384.193
Analytic conductor $197.774$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 384.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(197.773761087\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
Defining polynomial: \( x^{8} + 13062x^{6} + 45211107x^{4} + 45928424926x^{2} + 852972309225 \) Copy content Toggle raw display
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 193.6
Root \(-4.34998i\) of defining polynomial
Character \(\chi\) \(=\) 384.193
Dual form 384.10.d.b.193.3

$q$-expansion

\(f(q)\) \(=\) \(q+81.0000i q^{3} -1428.10i q^{5} -6382.13 q^{7} -6561.00 q^{9} +O(q^{10})\) \(q+81.0000i q^{3} -1428.10i q^{5} -6382.13 q^{7} -6561.00 q^{9} -4073.61i q^{11} -154396. i q^{13} +115676. q^{15} +255714. q^{17} -1.01702e6i q^{19} -516952. i q^{21} +1.56457e6 q^{23} -86331.5 q^{25} -531441. i q^{27} +7.26136e6i q^{29} +611897. q^{31} +329962. q^{33} +9.11429e6i q^{35} -464986. i q^{37} +1.25061e7 q^{39} +2.04743e7 q^{41} -2.95237e7i q^{43} +9.36973e6i q^{45} -1.31748e6 q^{47} +377951. q^{49} +2.07129e7i q^{51} +5.76562e7i q^{53} -5.81750e6 q^{55} +8.23785e7 q^{57} -6.35883e7i q^{59} -5.01862e7i q^{61} +4.18731e7 q^{63} -2.20493e8 q^{65} -2.90029e8i q^{67} +1.26730e8i q^{69} +3.18003e8 q^{71} -3.49122e8 q^{73} -6.99285e6i q^{75} +2.59983e7i q^{77} +4.29993e8 q^{79} +4.30467e7 q^{81} -3.40898e8i q^{83} -3.65185e8i q^{85} -5.88170e8 q^{87} +5.24211e8 q^{89} +9.85376e8i q^{91} +4.95636e7i q^{93} -1.45240e9 q^{95} -9.61784e8 q^{97} +2.67270e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 13632 q^{7} - 52488 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 13632 q^{7} - 52488 q^{9} + 90720 q^{15} + 8304 q^{17} + 4612608 q^{23} - 4754904 q^{25} + 7499328 q^{31} - 6213024 q^{33} + 23211360 q^{39} - 43518896 q^{41} - 49382016 q^{47} - 74106808 q^{49} - 19030656 q^{55} + 38141280 q^{57} - 89439552 q^{63} - 110270336 q^{65} + 741751296 q^{71} - 1507903440 q^{73} - 1008373440 q^{79} + 344373768 q^{81} - 423468000 q^{87} - 1337034448 q^{89} + 543950208 q^{95} - 904817936 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 81.0000i 0.577350i
\(4\) 0 0
\(5\) − 1428.10i − 1.02186i −0.859622 0.510931i \(-0.829301\pi\)
0.859622 0.510931i \(-0.170699\pi\)
\(6\) 0 0
\(7\) −6382.13 −1.00467 −0.502336 0.864672i \(-0.667526\pi\)
−0.502336 + 0.864672i \(0.667526\pi\)
\(8\) 0 0
\(9\) −6561.00 −0.333333
\(10\) 0 0
\(11\) − 4073.61i − 0.0838904i −0.999120 0.0419452i \(-0.986645\pi\)
0.999120 0.0419452i \(-0.0133555\pi\)
\(12\) 0 0
\(13\) − 154396.i − 1.49931i −0.661829 0.749655i \(-0.730219\pi\)
0.661829 0.749655i \(-0.269781\pi\)
\(14\) 0 0
\(15\) 115676. 0.589972
\(16\) 0 0
\(17\) 255714. 0.742566 0.371283 0.928520i \(-0.378918\pi\)
0.371283 + 0.928520i \(0.378918\pi\)
\(18\) 0 0
\(19\) − 1.01702e6i − 1.79035i −0.445716 0.895175i \(-0.647051\pi\)
0.445716 0.895175i \(-0.352949\pi\)
\(20\) 0 0
\(21\) − 516952.i − 0.580048i
\(22\) 0 0
\(23\) 1.56457e6 1.16579 0.582894 0.812548i \(-0.301920\pi\)
0.582894 + 0.812548i \(0.301920\pi\)
\(24\) 0 0
\(25\) −86331.5 −0.0442017
\(26\) 0 0
\(27\) − 531441.i − 0.192450i
\(28\) 0 0
\(29\) 7.26136e6i 1.90646i 0.302253 + 0.953228i \(0.402261\pi\)
−0.302253 + 0.953228i \(0.597739\pi\)
\(30\) 0 0
\(31\) 611897. 0.119001 0.0595005 0.998228i \(-0.481049\pi\)
0.0595005 + 0.998228i \(0.481049\pi\)
\(32\) 0 0
\(33\) 329962. 0.0484342
\(34\) 0 0
\(35\) 9.11429e6i 1.02664i
\(36\) 0 0
\(37\) − 464986.i − 0.0407880i −0.999792 0.0203940i \(-0.993508\pi\)
0.999792 0.0203940i \(-0.00649205\pi\)
\(38\) 0 0
\(39\) 1.25061e7 0.865627
\(40\) 0 0
\(41\) 2.04743e7 1.13157 0.565785 0.824553i \(-0.308573\pi\)
0.565785 + 0.824553i \(0.308573\pi\)
\(42\) 0 0
\(43\) − 2.95237e7i − 1.31693i −0.752611 0.658465i \(-0.771206\pi\)
0.752611 0.658465i \(-0.228794\pi\)
\(44\) 0 0
\(45\) 9.36973e6i 0.340621i
\(46\) 0 0
\(47\) −1.31748e6 −0.0393824 −0.0196912 0.999806i \(-0.506268\pi\)
−0.0196912 + 0.999806i \(0.506268\pi\)
\(48\) 0 0
\(49\) 377951. 0.00936598
\(50\) 0 0
\(51\) 2.07129e7i 0.428721i
\(52\) 0 0
\(53\) 5.76562e7i 1.00370i 0.864954 + 0.501851i \(0.167347\pi\)
−0.864954 + 0.501851i \(0.832653\pi\)
\(54\) 0 0
\(55\) −5.81750e6 −0.0857244
\(56\) 0 0
\(57\) 8.23785e7 1.03366
\(58\) 0 0
\(59\) − 6.35883e7i − 0.683193i −0.939847 0.341596i \(-0.889032\pi\)
0.939847 0.341596i \(-0.110968\pi\)
\(60\) 0 0
\(61\) − 5.01862e7i − 0.464088i −0.972705 0.232044i \(-0.925459\pi\)
0.972705 0.232044i \(-0.0745413\pi\)
\(62\) 0 0
\(63\) 4.18731e7 0.334891
\(64\) 0 0
\(65\) −2.20493e8 −1.53209
\(66\) 0 0
\(67\) − 2.90029e8i − 1.75835i −0.476499 0.879175i \(-0.658094\pi\)
0.476499 0.879175i \(-0.341906\pi\)
\(68\) 0 0
\(69\) 1.26730e8i 0.673068i
\(70\) 0 0
\(71\) 3.18003e8 1.48514 0.742572 0.669766i \(-0.233606\pi\)
0.742572 + 0.669766i \(0.233606\pi\)
\(72\) 0 0
\(73\) −3.49122e8 −1.43888 −0.719440 0.694554i \(-0.755602\pi\)
−0.719440 + 0.694554i \(0.755602\pi\)
\(74\) 0 0
\(75\) − 6.99285e6i − 0.0255199i
\(76\) 0 0
\(77\) 2.59983e7i 0.0842824i
\(78\) 0 0
\(79\) 4.29993e8 1.24205 0.621025 0.783791i \(-0.286716\pi\)
0.621025 + 0.783791i \(0.286716\pi\)
\(80\) 0 0
\(81\) 4.30467e7 0.111111
\(82\) 0 0
\(83\) − 3.40898e8i − 0.788447i −0.919015 0.394223i \(-0.871014\pi\)
0.919015 0.394223i \(-0.128986\pi\)
\(84\) 0 0
\(85\) − 3.65185e8i − 0.758800i
\(86\) 0 0
\(87\) −5.88170e8 −1.10069
\(88\) 0 0
\(89\) 5.24211e8 0.885628 0.442814 0.896614i \(-0.353980\pi\)
0.442814 + 0.896614i \(0.353980\pi\)
\(90\) 0 0
\(91\) 9.85376e8i 1.50632i
\(92\) 0 0
\(93\) 4.95636e7i 0.0687053i
\(94\) 0 0
\(95\) −1.45240e9 −1.82949
\(96\) 0 0
\(97\) −9.61784e8 −1.10307 −0.551537 0.834150i \(-0.685958\pi\)
−0.551537 + 0.834150i \(0.685958\pi\)
\(98\) 0 0
\(99\) 2.67270e7i 0.0279635i
\(100\) 0 0
\(101\) 2.61811e8i 0.250346i 0.992135 + 0.125173i \(0.0399486\pi\)
−0.992135 + 0.125173i \(0.960051\pi\)
\(102\) 0 0
\(103\) 4.87298e8 0.426606 0.213303 0.976986i \(-0.431578\pi\)
0.213303 + 0.976986i \(0.431578\pi\)
\(104\) 0 0
\(105\) −7.38257e8 −0.592729
\(106\) 0 0
\(107\) 1.63558e9i 1.20627i 0.797640 + 0.603134i \(0.206082\pi\)
−0.797640 + 0.603134i \(0.793918\pi\)
\(108\) 0 0
\(109\) 1.81852e9i 1.23396i 0.786981 + 0.616978i \(0.211643\pi\)
−0.786981 + 0.616978i \(0.788357\pi\)
\(110\) 0 0
\(111\) 3.76639e7 0.0235489
\(112\) 0 0
\(113\) −2.89653e9 −1.67119 −0.835594 0.549348i \(-0.814876\pi\)
−0.835594 + 0.549348i \(0.814876\pi\)
\(114\) 0 0
\(115\) − 2.23435e9i − 1.19127i
\(116\) 0 0
\(117\) 1.01299e9i 0.499770i
\(118\) 0 0
\(119\) −1.63200e9 −0.746036
\(120\) 0 0
\(121\) 2.34135e9 0.992962
\(122\) 0 0
\(123\) 1.65842e9i 0.653312i
\(124\) 0 0
\(125\) − 2.66596e9i − 0.976694i
\(126\) 0 0
\(127\) 4.08260e8 0.139258 0.0696290 0.997573i \(-0.477818\pi\)
0.0696290 + 0.997573i \(0.477818\pi\)
\(128\) 0 0
\(129\) 2.39142e9 0.760330
\(130\) 0 0
\(131\) 1.67425e9i 0.496706i 0.968670 + 0.248353i \(0.0798893\pi\)
−0.968670 + 0.248353i \(0.920111\pi\)
\(132\) 0 0
\(133\) 6.49074e9i 1.79871i
\(134\) 0 0
\(135\) −7.58948e8 −0.196657
\(136\) 0 0
\(137\) −1.57362e9 −0.381642 −0.190821 0.981625i \(-0.561115\pi\)
−0.190821 + 0.981625i \(0.561115\pi\)
\(138\) 0 0
\(139\) − 5.36040e9i − 1.21795i −0.793188 0.608977i \(-0.791580\pi\)
0.793188 0.608977i \(-0.208420\pi\)
\(140\) 0 0
\(141\) − 1.06715e8i − 0.0227374i
\(142\) 0 0
\(143\) −6.28950e8 −0.125778
\(144\) 0 0
\(145\) 1.03699e10 1.94813
\(146\) 0 0
\(147\) 3.06140e7i 0.00540745i
\(148\) 0 0
\(149\) − 1.83174e9i − 0.304457i −0.988345 0.152228i \(-0.951355\pi\)
0.988345 0.152228i \(-0.0486449\pi\)
\(150\) 0 0
\(151\) −4.73373e9 −0.740981 −0.370490 0.928836i \(-0.620810\pi\)
−0.370490 + 0.928836i \(0.620810\pi\)
\(152\) 0 0
\(153\) −1.67774e9 −0.247522
\(154\) 0 0
\(155\) − 8.73847e8i − 0.121603i
\(156\) 0 0
\(157\) − 8.73612e8i − 0.114755i −0.998353 0.0573773i \(-0.981726\pi\)
0.998353 0.0573773i \(-0.0182738\pi\)
\(158\) 0 0
\(159\) −4.67015e9 −0.579487
\(160\) 0 0
\(161\) −9.98528e9 −1.17123
\(162\) 0 0
\(163\) − 8.32642e9i − 0.923877i −0.886912 0.461939i \(-0.847154\pi\)
0.886912 0.461939i \(-0.152846\pi\)
\(164\) 0 0
\(165\) − 4.71218e8i − 0.0494930i
\(166\) 0 0
\(167\) −1.58090e9 −0.157283 −0.0786414 0.996903i \(-0.525058\pi\)
−0.0786414 + 0.996903i \(0.525058\pi\)
\(168\) 0 0
\(169\) −1.32337e10 −1.24793
\(170\) 0 0
\(171\) 6.67266e9i 0.596783i
\(172\) 0 0
\(173\) − 1.19275e10i − 1.01238i −0.862423 0.506189i \(-0.831054\pi\)
0.862423 0.506189i \(-0.168946\pi\)
\(174\) 0 0
\(175\) 5.50979e8 0.0444082
\(176\) 0 0
\(177\) 5.15065e9 0.394441
\(178\) 0 0
\(179\) − 1.27369e10i − 0.927309i −0.886016 0.463654i \(-0.846538\pi\)
0.886016 0.463654i \(-0.153462\pi\)
\(180\) 0 0
\(181\) − 8.46716e9i − 0.586387i −0.956053 0.293193i \(-0.905282\pi\)
0.956053 0.293193i \(-0.0947180\pi\)
\(182\) 0 0
\(183\) 4.06508e9 0.267941
\(184\) 0 0
\(185\) −6.64044e8 −0.0416797
\(186\) 0 0
\(187\) − 1.04168e9i − 0.0622942i
\(188\) 0 0
\(189\) 3.39172e9i 0.193349i
\(190\) 0 0
\(191\) 3.90757e9 0.212450 0.106225 0.994342i \(-0.466124\pi\)
0.106225 + 0.994342i \(0.466124\pi\)
\(192\) 0 0
\(193\) 1.82705e10 0.947858 0.473929 0.880563i \(-0.342835\pi\)
0.473929 + 0.880563i \(0.342835\pi\)
\(194\) 0 0
\(195\) − 1.78599e10i − 0.884552i
\(196\) 0 0
\(197\) − 1.05198e10i − 0.497633i −0.968551 0.248816i \(-0.919958\pi\)
0.968551 0.248816i \(-0.0800416\pi\)
\(198\) 0 0
\(199\) 3.02525e10 1.36748 0.683741 0.729725i \(-0.260352\pi\)
0.683741 + 0.729725i \(0.260352\pi\)
\(200\) 0 0
\(201\) 2.34924e10 1.01518
\(202\) 0 0
\(203\) − 4.63429e10i − 1.91536i
\(204\) 0 0
\(205\) − 2.92392e10i − 1.15631i
\(206\) 0 0
\(207\) −1.02651e10 −0.388596
\(208\) 0 0
\(209\) −4.14294e9 −0.150193
\(210\) 0 0
\(211\) − 1.53444e10i − 0.532940i −0.963843 0.266470i \(-0.914143\pi\)
0.963843 0.266470i \(-0.0858573\pi\)
\(212\) 0 0
\(213\) 2.57582e10i 0.857448i
\(214\) 0 0
\(215\) −4.21626e10 −1.34572
\(216\) 0 0
\(217\) −3.90520e9 −0.119557
\(218\) 0 0
\(219\) − 2.82789e10i − 0.830738i
\(220\) 0 0
\(221\) − 3.94814e10i − 1.11334i
\(222\) 0 0
\(223\) −6.26805e10 −1.69731 −0.848654 0.528948i \(-0.822586\pi\)
−0.848654 + 0.528948i \(0.822586\pi\)
\(224\) 0 0
\(225\) 5.66421e8 0.0147339
\(226\) 0 0
\(227\) 4.85806e10i 1.21436i 0.794565 + 0.607179i \(0.207699\pi\)
−0.794565 + 0.607179i \(0.792301\pi\)
\(228\) 0 0
\(229\) 6.64074e10i 1.59572i 0.602843 + 0.797860i \(0.294035\pi\)
−0.602843 + 0.797860i \(0.705965\pi\)
\(230\) 0 0
\(231\) −2.10586e9 −0.0486605
\(232\) 0 0
\(233\) 1.53115e10 0.340342 0.170171 0.985415i \(-0.445568\pi\)
0.170171 + 0.985415i \(0.445568\pi\)
\(234\) 0 0
\(235\) 1.88148e9i 0.0402434i
\(236\) 0 0
\(237\) 3.48294e10i 0.717098i
\(238\) 0 0
\(239\) −5.49921e10 −1.09021 −0.545105 0.838368i \(-0.683510\pi\)
−0.545105 + 0.838368i \(0.683510\pi\)
\(240\) 0 0
\(241\) −1.88950e10 −0.360802 −0.180401 0.983593i \(-0.557740\pi\)
−0.180401 + 0.983593i \(0.557740\pi\)
\(242\) 0 0
\(243\) 3.48678e9i 0.0641500i
\(244\) 0 0
\(245\) − 5.39750e8i − 0.00957074i
\(246\) 0 0
\(247\) −1.57024e11 −2.68429
\(248\) 0 0
\(249\) 2.76127e10 0.455210
\(250\) 0 0
\(251\) − 1.10372e11i − 1.75520i −0.479395 0.877599i \(-0.659144\pi\)
0.479395 0.877599i \(-0.340856\pi\)
\(252\) 0 0
\(253\) − 6.37344e9i − 0.0977984i
\(254\) 0 0
\(255\) 2.95800e10 0.438093
\(256\) 0 0
\(257\) 2.31328e10 0.330772 0.165386 0.986229i \(-0.447113\pi\)
0.165386 + 0.986229i \(0.447113\pi\)
\(258\) 0 0
\(259\) 2.96760e9i 0.0409785i
\(260\) 0 0
\(261\) − 4.76418e10i − 0.635485i
\(262\) 0 0
\(263\) 1.88727e10 0.243239 0.121619 0.992577i \(-0.461191\pi\)
0.121619 + 0.992577i \(0.461191\pi\)
\(264\) 0 0
\(265\) 8.23385e10 1.02564
\(266\) 0 0
\(267\) 4.24611e10i 0.511317i
\(268\) 0 0
\(269\) 6.86708e10i 0.799626i 0.916597 + 0.399813i \(0.130925\pi\)
−0.916597 + 0.399813i \(0.869075\pi\)
\(270\) 0 0
\(271\) −1.15508e11 −1.30092 −0.650460 0.759540i \(-0.725424\pi\)
−0.650460 + 0.759540i \(0.725424\pi\)
\(272\) 0 0
\(273\) −7.98155e10 −0.869672
\(274\) 0 0
\(275\) 3.51681e8i 0.00370810i
\(276\) 0 0
\(277\) − 4.52642e10i − 0.461950i −0.972960 0.230975i \(-0.925808\pi\)
0.972960 0.230975i \(-0.0741916\pi\)
\(278\) 0 0
\(279\) −4.01465e9 −0.0396670
\(280\) 0 0
\(281\) −9.00026e10 −0.861146 −0.430573 0.902556i \(-0.641688\pi\)
−0.430573 + 0.902556i \(0.641688\pi\)
\(282\) 0 0
\(283\) − 9.87406e10i − 0.915076i −0.889190 0.457538i \(-0.848731\pi\)
0.889190 0.457538i \(-0.151269\pi\)
\(284\) 0 0
\(285\) − 1.17644e11i − 1.05626i
\(286\) 0 0
\(287\) −1.30670e11 −1.13686
\(288\) 0 0
\(289\) −5.31980e10 −0.448595
\(290\) 0 0
\(291\) − 7.79045e10i − 0.636860i
\(292\) 0 0
\(293\) 8.54059e10i 0.676992i 0.940968 + 0.338496i \(0.109918\pi\)
−0.940968 + 0.338496i \(0.890082\pi\)
\(294\) 0 0
\(295\) −9.08101e10 −0.698128
\(296\) 0 0
\(297\) −2.16488e9 −0.0161447
\(298\) 0 0
\(299\) − 2.41563e11i − 1.74788i
\(300\) 0 0
\(301\) 1.88424e11i 1.32308i
\(302\) 0 0
\(303\) −2.12067e10 −0.144537
\(304\) 0 0
\(305\) −7.16707e10 −0.474234
\(306\) 0 0
\(307\) − 1.83582e11i − 1.17953i −0.807576 0.589763i \(-0.799221\pi\)
0.807576 0.589763i \(-0.200779\pi\)
\(308\) 0 0
\(309\) 3.94711e10i 0.246301i
\(310\) 0 0
\(311\) 1.13932e11 0.690598 0.345299 0.938493i \(-0.387778\pi\)
0.345299 + 0.938493i \(0.387778\pi\)
\(312\) 0 0
\(313\) −2.41679e11 −1.42328 −0.711640 0.702545i \(-0.752047\pi\)
−0.711640 + 0.702545i \(0.752047\pi\)
\(314\) 0 0
\(315\) − 5.97988e10i − 0.342212i
\(316\) 0 0
\(317\) − 1.06227e11i − 0.590840i −0.955367 0.295420i \(-0.904540\pi\)
0.955367 0.295420i \(-0.0954596\pi\)
\(318\) 0 0
\(319\) 2.95799e10 0.159933
\(320\) 0 0
\(321\) −1.32482e11 −0.696440
\(322\) 0 0
\(323\) − 2.60066e11i − 1.32945i
\(324\) 0 0
\(325\) 1.33293e10i 0.0662721i
\(326\) 0 0
\(327\) −1.47300e11 −0.712424
\(328\) 0 0
\(329\) 8.40830e9 0.0395664
\(330\) 0 0
\(331\) 6.82287e10i 0.312422i 0.987724 + 0.156211i \(0.0499279\pi\)
−0.987724 + 0.156211i \(0.950072\pi\)
\(332\) 0 0
\(333\) 3.05077e9i 0.0135960i
\(334\) 0 0
\(335\) −4.14190e11 −1.79679
\(336\) 0 0
\(337\) 3.87135e11 1.63504 0.817518 0.575903i \(-0.195349\pi\)
0.817518 + 0.575903i \(0.195349\pi\)
\(338\) 0 0
\(339\) − 2.34619e11i − 0.964861i
\(340\) 0 0
\(341\) − 2.49263e9i − 0.00998304i
\(342\) 0 0
\(343\) 2.55130e11 0.995262
\(344\) 0 0
\(345\) 1.80983e11 0.687782
\(346\) 0 0
\(347\) 2.81800e11i 1.04342i 0.853124 + 0.521709i \(0.174705\pi\)
−0.853124 + 0.521709i \(0.825295\pi\)
\(348\) 0 0
\(349\) − 1.80174e11i − 0.650096i −0.945697 0.325048i \(-0.894619\pi\)
0.945697 0.325048i \(-0.105381\pi\)
\(350\) 0 0
\(351\) −8.20525e10 −0.288542
\(352\) 0 0
\(353\) 1.18821e11 0.407294 0.203647 0.979044i \(-0.434721\pi\)
0.203647 + 0.979044i \(0.434721\pi\)
\(354\) 0 0
\(355\) − 4.54139e11i − 1.51761i
\(356\) 0 0
\(357\) − 1.32192e11i − 0.430724i
\(358\) 0 0
\(359\) −4.95475e11 −1.57433 −0.787167 0.616740i \(-0.788453\pi\)
−0.787167 + 0.616740i \(0.788453\pi\)
\(360\) 0 0
\(361\) −7.11639e11 −2.20535
\(362\) 0 0
\(363\) 1.89650e11i 0.573287i
\(364\) 0 0
\(365\) 4.98580e11i 1.47034i
\(366\) 0 0
\(367\) −5.16234e11 −1.48542 −0.742709 0.669614i \(-0.766460\pi\)
−0.742709 + 0.669614i \(0.766460\pi\)
\(368\) 0 0
\(369\) −1.34332e11 −0.377190
\(370\) 0 0
\(371\) − 3.67969e11i − 1.00839i
\(372\) 0 0
\(373\) − 2.26879e11i − 0.606881i −0.952850 0.303441i \(-0.901865\pi\)
0.952850 0.303441i \(-0.0981354\pi\)
\(374\) 0 0
\(375\) 2.15943e11 0.563894
\(376\) 0 0
\(377\) 1.12113e12 2.85837
\(378\) 0 0
\(379\) − 5.96753e11i − 1.48566i −0.669482 0.742828i \(-0.733484\pi\)
0.669482 0.742828i \(-0.266516\pi\)
\(380\) 0 0
\(381\) 3.30691e10i 0.0804007i
\(382\) 0 0
\(383\) 3.71694e11 0.882654 0.441327 0.897346i \(-0.354508\pi\)
0.441327 + 0.897346i \(0.354508\pi\)
\(384\) 0 0
\(385\) 3.71281e10 0.0861249
\(386\) 0 0
\(387\) 1.93705e11i 0.438977i
\(388\) 0 0
\(389\) 2.71241e11i 0.600596i 0.953845 + 0.300298i \(0.0970862\pi\)
−0.953845 + 0.300298i \(0.902914\pi\)
\(390\) 0 0
\(391\) 4.00083e11 0.865674
\(392\) 0 0
\(393\) −1.35614e11 −0.286773
\(394\) 0 0
\(395\) − 6.14071e11i − 1.26920i
\(396\) 0 0
\(397\) 3.00982e11i 0.608112i 0.952654 + 0.304056i \(0.0983410\pi\)
−0.952654 + 0.304056i \(0.901659\pi\)
\(398\) 0 0
\(399\) −5.25750e11 −1.03849
\(400\) 0 0
\(401\) 3.10723e11 0.600101 0.300050 0.953923i \(-0.402997\pi\)
0.300050 + 0.953923i \(0.402997\pi\)
\(402\) 0 0
\(403\) − 9.44746e10i − 0.178419i
\(404\) 0 0
\(405\) − 6.14748e10i − 0.113540i
\(406\) 0 0
\(407\) −1.89417e9 −0.00342172
\(408\) 0 0
\(409\) −4.46642e11 −0.789231 −0.394616 0.918846i \(-0.629122\pi\)
−0.394616 + 0.918846i \(0.629122\pi\)
\(410\) 0 0
\(411\) − 1.27463e11i − 0.220341i
\(412\) 0 0
\(413\) 4.05829e11i 0.686384i
\(414\) 0 0
\(415\) −4.86834e11 −0.805684
\(416\) 0 0
\(417\) 4.34193e11 0.703186
\(418\) 0 0
\(419\) 8.23618e11i 1.30546i 0.757592 + 0.652728i \(0.226376\pi\)
−0.757592 + 0.652728i \(0.773624\pi\)
\(420\) 0 0
\(421\) − 6.04524e11i − 0.937872i −0.883232 0.468936i \(-0.844637\pi\)
0.883232 0.468936i \(-0.155363\pi\)
\(422\) 0 0
\(423\) 8.64395e9 0.0131275
\(424\) 0 0
\(425\) −2.20762e10 −0.0328227
\(426\) 0 0
\(427\) 3.20295e11i 0.466256i
\(428\) 0 0
\(429\) − 5.09450e10i − 0.0726179i
\(430\) 0 0
\(431\) −1.20184e12 −1.67765 −0.838823 0.544404i \(-0.816756\pi\)
−0.838823 + 0.544404i \(0.816756\pi\)
\(432\) 0 0
\(433\) 1.53930e11 0.210440 0.105220 0.994449i \(-0.466445\pi\)
0.105220 + 0.994449i \(0.466445\pi\)
\(434\) 0 0
\(435\) 8.39963e11i 1.12476i
\(436\) 0 0
\(437\) − 1.59120e12i − 2.08717i
\(438\) 0 0
\(439\) −4.56899e11 −0.587124 −0.293562 0.955940i \(-0.594841\pi\)
−0.293562 + 0.955940i \(0.594841\pi\)
\(440\) 0 0
\(441\) −2.47974e9 −0.00312199
\(442\) 0 0
\(443\) 1.43507e12i 1.77034i 0.465271 + 0.885168i \(0.345957\pi\)
−0.465271 + 0.885168i \(0.654043\pi\)
\(444\) 0 0
\(445\) − 7.48623e11i − 0.904989i
\(446\) 0 0
\(447\) 1.48371e11 0.175778
\(448\) 0 0
\(449\) −9.54602e11 −1.10844 −0.554222 0.832369i \(-0.686984\pi\)
−0.554222 + 0.832369i \(0.686984\pi\)
\(450\) 0 0
\(451\) − 8.34043e10i − 0.0949279i
\(452\) 0 0
\(453\) − 3.83432e11i − 0.427805i
\(454\) 0 0
\(455\) 1.40721e12 1.53925
\(456\) 0 0
\(457\) 4.57857e11 0.491029 0.245515 0.969393i \(-0.421043\pi\)
0.245515 + 0.969393i \(0.421043\pi\)
\(458\) 0 0
\(459\) − 1.35897e11i − 0.142907i
\(460\) 0 0
\(461\) − 1.58756e12i − 1.63710i −0.574437 0.818549i \(-0.694779\pi\)
0.574437 0.818549i \(-0.305221\pi\)
\(462\) 0 0
\(463\) −7.26819e11 −0.735041 −0.367521 0.930015i \(-0.619793\pi\)
−0.367521 + 0.930015i \(0.619793\pi\)
\(464\) 0 0
\(465\) 7.07816e10 0.0702073
\(466\) 0 0
\(467\) − 8.73721e11i − 0.850054i −0.905181 0.425027i \(-0.860265\pi\)
0.905181 0.425027i \(-0.139735\pi\)
\(468\) 0 0
\(469\) 1.85100e12i 1.76657i
\(470\) 0 0
\(471\) 7.07626e10 0.0662536
\(472\) 0 0
\(473\) −1.20268e11 −0.110478
\(474\) 0 0
\(475\) 8.78007e10i 0.0791365i
\(476\) 0 0
\(477\) − 3.78282e11i − 0.334567i
\(478\) 0 0
\(479\) −6.89800e11 −0.598706 −0.299353 0.954142i \(-0.596771\pi\)
−0.299353 + 0.954142i \(0.596771\pi\)
\(480\) 0 0
\(481\) −7.17921e10 −0.0611538
\(482\) 0 0
\(483\) − 8.08807e11i − 0.676212i
\(484\) 0 0
\(485\) 1.37352e12i 1.12719i
\(486\) 0 0
\(487\) 2.05185e12 1.65297 0.826485 0.562959i \(-0.190337\pi\)
0.826485 + 0.562959i \(0.190337\pi\)
\(488\) 0 0
\(489\) 6.74440e11 0.533401
\(490\) 0 0
\(491\) − 5.51164e11i − 0.427971i −0.976837 0.213985i \(-0.931356\pi\)
0.976837 0.213985i \(-0.0686445\pi\)
\(492\) 0 0
\(493\) 1.85683e12i 1.41567i
\(494\) 0 0
\(495\) 3.81686e10 0.0285748
\(496\) 0 0
\(497\) −2.02954e12 −1.49208
\(498\) 0 0
\(499\) 1.99167e12i 1.43802i 0.695000 + 0.719009i \(0.255404\pi\)
−0.695000 + 0.719009i \(0.744596\pi\)
\(500\) 0 0
\(501\) − 1.28053e11i − 0.0908073i
\(502\) 0 0
\(503\) 2.42693e11 0.169044 0.0845221 0.996422i \(-0.473064\pi\)
0.0845221 + 0.996422i \(0.473064\pi\)
\(504\) 0 0
\(505\) 3.73891e11 0.255819
\(506\) 0 0
\(507\) − 1.07193e12i − 0.720494i
\(508\) 0 0
\(509\) 7.46973e11i 0.493259i 0.969110 + 0.246629i \(0.0793230\pi\)
−0.969110 + 0.246629i \(0.920677\pi\)
\(510\) 0 0
\(511\) 2.22814e12 1.44560
\(512\) 0 0
\(513\) −5.40485e11 −0.344553
\(514\) 0 0
\(515\) − 6.95908e11i − 0.435933i
\(516\) 0 0
\(517\) 5.36688e9i 0.00330381i
\(518\) 0 0
\(519\) 9.66128e11 0.584496
\(520\) 0 0
\(521\) 1.38178e12 0.821619 0.410809 0.911721i \(-0.365246\pi\)
0.410809 + 0.911721i \(0.365246\pi\)
\(522\) 0 0
\(523\) − 7.83681e10i − 0.0458017i −0.999738 0.0229008i \(-0.992710\pi\)
0.999738 0.0229008i \(-0.00729020\pi\)
\(524\) 0 0
\(525\) 4.46293e10i 0.0256391i
\(526\) 0 0
\(527\) 1.56471e11 0.0883661
\(528\) 0 0
\(529\) 6.46722e11 0.359060
\(530\) 0 0
\(531\) 4.17203e11i 0.227731i
\(532\) 0 0
\(533\) − 3.16115e12i − 1.69658i
\(534\) 0 0
\(535\) 2.33576e12 1.23264
\(536\) 0 0
\(537\) 1.03169e12 0.535382
\(538\) 0 0
\(539\) − 1.53963e9i 0 0.000785716i
\(540\) 0 0
\(541\) 1.88658e12i 0.946865i 0.880830 + 0.473433i \(0.156985\pi\)
−0.880830 + 0.473433i \(0.843015\pi\)
\(542\) 0 0
\(543\) 6.85840e11 0.338551
\(544\) 0 0
\(545\) 2.59702e12 1.26093
\(546\) 0 0
\(547\) 3.29535e12i 1.57383i 0.617061 + 0.786915i \(0.288323\pi\)
−0.617061 + 0.786915i \(0.711677\pi\)
\(548\) 0 0
\(549\) 3.29272e11i 0.154696i
\(550\) 0 0
\(551\) 7.38493e12 3.41322
\(552\) 0 0
\(553\) −2.74427e12 −1.24785
\(554\) 0 0
\(555\) − 5.37876e10i − 0.0240638i
\(556\) 0 0
\(557\) − 1.26338e12i − 0.556143i −0.960560 0.278072i \(-0.910305\pi\)
0.960560 0.278072i \(-0.0896952\pi\)
\(558\) 0 0
\(559\) −4.55835e12 −1.97449
\(560\) 0 0
\(561\) 8.43762e10 0.0359656
\(562\) 0 0
\(563\) 2.18128e12i 0.915004i 0.889209 + 0.457502i \(0.151256\pi\)
−0.889209 + 0.457502i \(0.848744\pi\)
\(564\) 0 0
\(565\) 4.13652e12i 1.70772i
\(566\) 0 0
\(567\) −2.74730e11 −0.111630
\(568\) 0 0
\(569\) −3.77066e12 −1.50804 −0.754020 0.656852i \(-0.771888\pi\)
−0.754020 + 0.656852i \(0.771888\pi\)
\(570\) 0 0
\(571\) − 4.82244e10i − 0.0189847i −0.999955 0.00949236i \(-0.996978\pi\)
0.999955 0.00949236i \(-0.00302156\pi\)
\(572\) 0 0
\(573\) 3.16513e11i 0.122658i
\(574\) 0 0
\(575\) −1.35072e11 −0.0515298
\(576\) 0 0
\(577\) −3.41311e12 −1.28192 −0.640958 0.767576i \(-0.721463\pi\)
−0.640958 + 0.767576i \(0.721463\pi\)
\(578\) 0 0
\(579\) 1.47991e12i 0.547246i
\(580\) 0 0
\(581\) 2.17565e12i 0.792131i
\(582\) 0 0
\(583\) 2.34869e11 0.0842009
\(584\) 0 0
\(585\) 1.44665e12 0.510696
\(586\) 0 0
\(587\) 5.13080e12i 1.78366i 0.452366 + 0.891832i \(0.350580\pi\)
−0.452366 + 0.891832i \(0.649420\pi\)
\(588\) 0 0
\(589\) − 6.22310e11i − 0.213053i
\(590\) 0 0
\(591\) 8.52103e11 0.287308
\(592\) 0 0
\(593\) 3.28338e12 1.09037 0.545186 0.838315i \(-0.316459\pi\)
0.545186 + 0.838315i \(0.316459\pi\)
\(594\) 0 0
\(595\) 2.33066e12i 0.762345i
\(596\) 0 0
\(597\) 2.45045e12i 0.789516i
\(598\) 0 0
\(599\) 2.83692e12 0.900382 0.450191 0.892932i \(-0.351356\pi\)
0.450191 + 0.892932i \(0.351356\pi\)
\(600\) 0 0
\(601\) −2.83184e11 −0.0885389 −0.0442694 0.999020i \(-0.514096\pi\)
−0.0442694 + 0.999020i \(0.514096\pi\)
\(602\) 0 0
\(603\) 1.90288e12i 0.586117i
\(604\) 0 0
\(605\) − 3.34368e12i − 1.01467i
\(606\) 0 0
\(607\) −2.38906e12 −0.714294 −0.357147 0.934048i \(-0.616251\pi\)
−0.357147 + 0.934048i \(0.616251\pi\)
\(608\) 0 0
\(609\) 3.75378e12 1.10584
\(610\) 0 0
\(611\) 2.03413e11i 0.0590464i
\(612\) 0 0
\(613\) − 6.27251e11i − 0.179419i −0.995968 0.0897097i \(-0.971406\pi\)
0.995968 0.0897097i \(-0.0285939\pi\)
\(614\) 0 0
\(615\) 2.36838e12 0.667595
\(616\) 0 0
\(617\) −6.57062e12 −1.82525 −0.912626 0.408794i \(-0.865949\pi\)
−0.912626 + 0.408794i \(0.865949\pi\)
\(618\) 0 0
\(619\) 1.27976e12i 0.350366i 0.984536 + 0.175183i \(0.0560517\pi\)
−0.984536 + 0.175183i \(0.943948\pi\)
\(620\) 0 0
\(621\) − 8.31476e11i − 0.224356i
\(622\) 0 0
\(623\) −3.34558e12 −0.889765
\(624\) 0 0
\(625\) −3.97586e12 −1.04225
\(626\) 0 0
\(627\) − 3.35578e11i − 0.0867141i
\(628\) 0 0
\(629\) − 1.18904e11i − 0.0302878i
\(630\) 0 0
\(631\) 6.41367e12 1.61055 0.805276 0.592900i \(-0.202017\pi\)
0.805276 + 0.592900i \(0.202017\pi\)
\(632\) 0 0
\(633\) 1.24289e12 0.307693
\(634\) 0 0
\(635\) − 5.83035e11i − 0.142302i
\(636\) 0 0
\(637\) − 5.83542e10i − 0.0140425i
\(638\) 0 0
\(639\) −2.08642e12 −0.495048
\(640\) 0 0
\(641\) 2.19384e11 0.0513266 0.0256633 0.999671i \(-0.491830\pi\)
0.0256633 + 0.999671i \(0.491830\pi\)
\(642\) 0 0
\(643\) − 1.43315e12i − 0.330629i −0.986241 0.165315i \(-0.947136\pi\)
0.986241 0.165315i \(-0.0528639\pi\)
\(644\) 0 0
\(645\) − 3.41517e12i − 0.776952i
\(646\) 0 0
\(647\) 2.24505e12 0.503682 0.251841 0.967769i \(-0.418964\pi\)
0.251841 + 0.967769i \(0.418964\pi\)
\(648\) 0 0
\(649\) −2.59034e11 −0.0573133
\(650\) 0 0
\(651\) − 3.16322e11i − 0.0690262i
\(652\) 0 0
\(653\) − 4.25852e12i − 0.916535i −0.888814 0.458267i \(-0.848470\pi\)
0.888814 0.458267i \(-0.151530\pi\)
\(654\) 0 0
\(655\) 2.39099e12 0.507565
\(656\) 0 0
\(657\) 2.29059e12 0.479627
\(658\) 0 0
\(659\) 2.98594e12i 0.616733i 0.951268 + 0.308367i \(0.0997823\pi\)
−0.951268 + 0.308367i \(0.900218\pi\)
\(660\) 0 0
\(661\) 3.60046e11i 0.0733587i 0.999327 + 0.0366794i \(0.0116780\pi\)
−0.999327 + 0.0366794i \(0.988322\pi\)
\(662\) 0 0
\(663\) 3.19799e12 0.642786
\(664\) 0 0
\(665\) 9.26940e12 1.83804
\(666\) 0 0
\(667\) 1.13609e13i 2.22252i
\(668\) 0 0
\(669\) − 5.07712e12i − 0.979941i
\(670\) 0 0
\(671\) −2.04439e11 −0.0389325
\(672\) 0 0
\(673\) −9.00012e12 −1.69114 −0.845572 0.533862i \(-0.820740\pi\)
−0.845572 + 0.533862i \(0.820740\pi\)
\(674\) 0 0
\(675\) 4.58801e10i 0.00850662i
\(676\) 0 0
\(677\) − 3.09265e12i − 0.565825i −0.959146 0.282913i \(-0.908699\pi\)
0.959146 0.282913i \(-0.0913006\pi\)
\(678\) 0 0
\(679\) 6.13823e12 1.10823
\(680\) 0 0
\(681\) −3.93503e12 −0.701109
\(682\) 0 0
\(683\) 9.11567e12i 1.60286i 0.598090 + 0.801429i \(0.295927\pi\)
−0.598090 + 0.801429i \(0.704073\pi\)
\(684\) 0 0
\(685\) 2.24727e12i 0.389985i
\(686\) 0 0
\(687\) −5.37900e12 −0.921289
\(688\) 0 0
\(689\) 8.90190e12 1.50486
\(690\) 0 0
\(691\) − 8.66620e12i − 1.44603i −0.690832 0.723016i \(-0.742755\pi\)
0.690832 0.723016i \(-0.257245\pi\)
\(692\) 0 0
\(693\) − 1.70575e11i − 0.0280941i
\(694\) 0 0
\(695\) −7.65517e12 −1.24458
\(696\) 0 0
\(697\) 5.23557e12 0.840266
\(698\) 0 0
\(699\) 1.24023e12i 0.196497i
\(700\) 0 0
\(701\) 1.21255e13i 1.89657i 0.317417 + 0.948286i \(0.397184\pi\)
−0.317417 + 0.948286i \(0.602816\pi\)
\(702\) 0 0
\(703\) −4.72899e11 −0.0730247
\(704\) 0 0
\(705\) −1.52400e11 −0.0232345
\(706\) 0 0
\(707\) − 1.67091e12i − 0.251516i
\(708\) 0 0
\(709\) 6.84961e12i 1.01802i 0.860759 + 0.509012i \(0.169989\pi\)
−0.860759 + 0.509012i \(0.830011\pi\)
\(710\) 0 0
\(711\) −2.82118e12 −0.414017
\(712\) 0 0
\(713\) 9.57354e11 0.138730
\(714\) 0 0
\(715\) 8.98201e11i 0.128528i
\(716\) 0 0
\(717\) − 4.45436e12i − 0.629433i
\(718\) 0 0
\(719\) 3.22078e12 0.449450 0.224725 0.974422i \(-0.427852\pi\)
0.224725 + 0.974422i \(0.427852\pi\)
\(720\) 0 0
\(721\) −3.11000e12 −0.428599
\(722\) 0 0
\(723\) − 1.53049e12i − 0.208309i
\(724\) 0 0
\(725\) − 6.26884e11i − 0.0842686i
\(726\) 0 0
\(727\) 1.26034e13 1.67334 0.836669 0.547708i \(-0.184500\pi\)
0.836669 + 0.547708i \(0.184500\pi\)
\(728\) 0 0
\(729\) −2.82430e11 −0.0370370
\(730\) 0 0
\(731\) − 7.54964e12i − 0.977908i
\(732\) 0 0
\(733\) 7.29094e12i 0.932858i 0.884559 + 0.466429i \(0.154460\pi\)
−0.884559 + 0.466429i \(0.845540\pi\)
\(734\) 0 0
\(735\) 4.37198e10 0.00552567
\(736\) 0 0
\(737\) −1.18147e12 −0.147509
\(738\) 0 0
\(739\) − 3.39916e10i − 0.00419248i −0.999998 0.00209624i \(-0.999333\pi\)
0.999998 0.00209624i \(-0.000667255\pi\)
\(740\) 0 0
\(741\) − 1.27189e13i − 1.54978i
\(742\) 0 0
\(743\) 9.08965e12 1.09420 0.547101 0.837067i \(-0.315731\pi\)
0.547101 + 0.837067i \(0.315731\pi\)
\(744\) 0 0
\(745\) −2.61590e12 −0.311113
\(746\) 0 0
\(747\) 2.23663e12i 0.262816i
\(748\) 0 0
\(749\) − 1.04385e13i − 1.21190i
\(750\) 0 0
\(751\) −7.25185e12 −0.831896 −0.415948 0.909388i \(-0.636550\pi\)
−0.415948 + 0.909388i \(0.636550\pi\)
\(752\) 0 0
\(753\) 8.94012e12 1.01336
\(754\) 0 0
\(755\) 6.76021e12i 0.757180i
\(756\) 0 0
\(757\) 1.12343e13i 1.24341i 0.783253 + 0.621703i \(0.213559\pi\)
−0.783253 + 0.621703i \(0.786441\pi\)
\(758\) 0 0
\(759\) 5.16249e11 0.0564639
\(760\) 0 0
\(761\) 1.63034e13 1.76217 0.881083 0.472962i \(-0.156815\pi\)
0.881083 + 0.472962i \(0.156815\pi\)
\(762\) 0 0
\(763\) − 1.16060e13i − 1.23972i
\(764\) 0 0
\(765\) 2.39598e12i 0.252933i
\(766\) 0 0
\(767\) −9.81779e12 −1.02432
\(768\) 0 0
\(769\) 4.19893e12 0.432982 0.216491 0.976285i \(-0.430539\pi\)
0.216491 + 0.976285i \(0.430539\pi\)
\(770\) 0 0
\(771\) 1.87375e12i 0.190971i
\(772\) 0 0
\(773\) − 1.11644e13i − 1.12467i −0.826908 0.562337i \(-0.809902\pi\)
0.826908 0.562337i \(-0.190098\pi\)
\(774\) 0 0
\(775\) −5.28260e10 −0.00526005
\(776\) 0 0
\(777\) −2.40376e11 −0.0236590
\(778\) 0 0
\(779\) − 2.08227e13i − 2.02591i
\(780\) 0 0
\(781\) − 1.29542e12i − 0.124589i
\(782\) 0 0
\(783\) 3.85898e12 0.366898
\(784\) 0 0
\(785\) −1.24760e12 −0.117263
\(786\) 0 0
\(787\) − 3.40070e12i − 0.315996i −0.987439 0.157998i \(-0.949496\pi\)
0.987439 0.157998i \(-0.0505040\pi\)
\(788\) 0 0
\(789\) 1.52869e12i 0.140434i
\(790\) 0 0
\(791\) 1.84860e13 1.67900
\(792\) 0 0
\(793\) −7.74856e12 −0.695812
\(794\) 0 0
\(795\) 6.66942e12i 0.592156i
\(796\) 0 0
\(797\) 8.35775e11i 0.0733714i 0.999327 + 0.0366857i \(0.0116800\pi\)
−0.999327 + 0.0366857i \(0.988320\pi\)
\(798\) 0 0
\(799\) −3.36898e11 −0.0292440
\(800\) 0 0
\(801\) −3.43935e12 −0.295209
\(802\) 0 0
\(803\) 1.42219e12i 0.120708i
\(804\) 0 0
\(805\) 1.42599e13i 1.19684i
\(806\) 0 0
\(807\) −5.56234e12 −0.461664
\(808\) 0 0
\(809\) 1.21985e13 1.00124 0.500622 0.865666i \(-0.333105\pi\)
0.500622 + 0.865666i \(0.333105\pi\)
\(810\) 0 0
\(811\) − 1.61845e13i − 1.31373i −0.754009 0.656864i \(-0.771883\pi\)
0.754009 0.656864i \(-0.228117\pi\)
\(812\) 0 0
\(813\) − 9.35616e12i − 0.751087i
\(814\) 0 0
\(815\) −1.18909e13 −0.944075
\(816\) 0 0
\(817\) −3.00261e13 −2.35776
\(818\) 0 0
\(819\) − 6.46506e12i − 0.502105i
\(820\) 0 0
\(821\) − 3.85044e12i − 0.295778i −0.989004 0.147889i \(-0.952752\pi\)
0.989004 0.147889i \(-0.0472479\pi\)
\(822\) 0 0
\(823\) −8.54699e12 −0.649403 −0.324701 0.945817i \(-0.605264\pi\)
−0.324701 + 0.945817i \(0.605264\pi\)
\(824\) 0 0
\(825\) −2.84861e10 −0.00214087
\(826\) 0 0
\(827\) − 4.46202e12i − 0.331709i −0.986150 0.165854i \(-0.946962\pi\)
0.986150 0.165854i \(-0.0530382\pi\)
\(828\) 0 0
\(829\) 5.99259e12i 0.440675i 0.975424 + 0.220338i \(0.0707159\pi\)
−0.975424 + 0.220338i \(0.929284\pi\)
\(830\) 0 0
\(831\) 3.66640e12 0.266707
\(832\) 0 0
\(833\) 9.66476e10 0.00695486
\(834\) 0 0
\(835\) 2.25768e12i 0.160721i
\(836\) 0 0
\(837\) − 3.25187e11i − 0.0229018i
\(838\) 0 0
\(839\) −9.87709e12 −0.688177 −0.344088 0.938937i \(-0.611812\pi\)
−0.344088 + 0.938937i \(0.611812\pi\)
\(840\) 0 0
\(841\) −3.82201e13 −2.63457
\(842\) 0 0
\(843\) − 7.29021e12i − 0.497183i
\(844\) 0 0
\(845\) 1.88990e13i 1.27521i
\(846\) 0 0
\(847\) −1.49428e13 −0.997602
\(848\) 0 0
\(849\) 7.99799e12 0.528319
\(850\) 0 0
\(851\) − 7.27502e11i − 0.0475501i
\(852\) 0 0
\(853\) − 2.87375e13i − 1.85857i −0.369365 0.929284i \(-0.620425\pi\)
0.369365 0.929284i \(-0.379575\pi\)
\(854\) 0 0
\(855\) 9.52919e12 0.609830
\(856\) 0 0
\(857\) −4.90239e12 −0.310452 −0.155226 0.987879i \(-0.549611\pi\)
−0.155226 + 0.987879i \(0.549611\pi\)
\(858\) 0 0
\(859\) − 4.08811e12i − 0.256185i −0.991762 0.128092i \(-0.959115\pi\)
0.991762 0.128092i \(-0.0408854\pi\)
\(860\) 0 0
\(861\) − 1.05842e13i − 0.656365i
\(862\) 0 0
\(863\) −2.90112e13 −1.78040 −0.890200 0.455570i \(-0.849435\pi\)
−0.890200 + 0.455570i \(0.849435\pi\)
\(864\) 0 0
\(865\) −1.70336e13 −1.03451
\(866\) 0 0
\(867\) − 4.30904e12i − 0.258997i
\(868\) 0 0
\(869\) − 1.75162e12i − 0.104196i
\(870\) 0 0
\(871\) −4.47794e13 −2.63631
\(872\) 0 0
\(873\) 6.31026e12 0.367691
\(874\) 0 0
\(875\) 1.70145e13i 0.981257i
\(876\) 0 0
\(877\) 3.38408e13i 1.93171i 0.259082 + 0.965855i \(0.416580\pi\)
−0.259082 + 0.965855i \(0.583420\pi\)
\(878\) 0 0
\(879\) −6.91788e12 −0.390862
\(880\) 0 0
\(881\) 2.28323e13 1.27690 0.638452 0.769662i \(-0.279575\pi\)
0.638452 + 0.769662i \(0.279575\pi\)
\(882\) 0 0
\(883\) 2.70857e12i 0.149940i 0.997186 + 0.0749699i \(0.0238861\pi\)
−0.997186 + 0.0749699i \(0.976114\pi\)
\(884\) 0 0
\(885\) − 7.35562e12i − 0.403065i
\(886\) 0 0
\(887\) −1.71218e13 −0.928735 −0.464368 0.885643i \(-0.653718\pi\)
−0.464368 + 0.885643i \(0.653718\pi\)
\(888\) 0 0
\(889\) −2.60557e12 −0.139909
\(890\) 0 0
\(891\) − 1.75356e11i − 0.00932116i
\(892\) 0 0
\(893\) 1.33990e12i 0.0705082i
\(894\) 0 0
\(895\) −1.81895e13 −0.947582
\(896\) 0 0
\(897\) 1.95666e13 1.00914
\(898\) 0 0
\(899\) 4.44320e12i 0.226870i
\(900\) 0 0
\(901\) 1.47435e13i 0.745315i
\(902\) 0 0
\(903\) −1.52623e13 −0.763882
\(904\) 0 0
\(905\) −1.20919e13 −0.599206
\(906\) 0 0
\(907\) 2.45440e13i 1.20424i 0.798406 + 0.602119i \(0.205677\pi\)
−0.798406 + 0.602119i \(0.794323\pi\)
\(908\) 0 0
\(909\) − 1.71774e12i − 0.0834488i
\(910\) 0 0
\(911\) 1.04402e13 0.502198 0.251099 0.967961i \(-0.419208\pi\)
0.251099 + 0.967961i \(0.419208\pi\)
\(912\) 0 0
\(913\) −1.38868e12 −0.0661432
\(914\) 0 0
\(915\) − 5.80533e12i − 0.273799i
\(916\) 0 0
\(917\) − 1.06853e13i − 0.499027i
\(918\) 0 0
\(919\) 1.14054e13 0.527464 0.263732 0.964596i \(-0.415047\pi\)
0.263732 + 0.964596i \(0.415047\pi\)
\(920\) 0 0
\(921\) 1.48702e13 0.681000
\(922\) 0 0
\(923\) − 4.90985e13i − 2.22669i
\(924\) 0 0
\(925\) 4.01429e10i 0.00180290i
\(926\) 0 0
\(927\) −3.19716e12 −0.142202
\(928\) 0 0
\(929\) 1.77202e13 0.780544 0.390272 0.920700i \(-0.372381\pi\)
0.390272 + 0.920700i \(0.372381\pi\)
\(930\) 0 0
\(931\) − 3.84383e11i − 0.0167684i
\(932\) 0 0
\(933\) 9.22852e12i 0.398717i
\(934\) 0 0
\(935\) −1.48762e12 −0.0636561
\(936\) 0 0
\(937\) 1.13938e13 0.482880 0.241440 0.970416i \(-0.422380\pi\)
0.241440 + 0.970416i \(0.422380\pi\)
\(938\) 0 0
\(939\) − 1.95760e13i − 0.821731i
\(940\) 0 0
\(941\) − 3.57056e12i − 0.148451i −0.997241 0.0742255i \(-0.976352\pi\)
0.997241 0.0742255i \(-0.0236484\pi\)
\(942\) 0 0
\(943\) 3.20334e13 1.31917
\(944\) 0 0
\(945\) 4.84371e12 0.197576
\(946\) 0 0
\(947\) − 3.48838e13i − 1.40945i −0.709482 0.704724i \(-0.751071\pi\)
0.709482 0.704724i \(-0.248929\pi\)
\(948\) 0 0
\(949\) 5.39032e13i 2.15733i
\(950\) 0 0
\(951\) 8.60442e12 0.341122
\(952\) 0 0
\(953\) −2.48299e13 −0.975117 −0.487558 0.873090i \(-0.662112\pi\)
−0.487558 + 0.873090i \(0.662112\pi\)
\(954\) 0 0
\(955\) − 5.58038e12i − 0.217094i
\(956\) 0 0
\(957\) 2.39597e12i 0.0923376i
\(958\) 0 0
\(959\) 1.00430e13 0.383425
\(960\) 0 0
\(961\) −2.60652e13 −0.985839
\(962\) 0 0
\(963\) − 1.07310e13i − 0.402090i
\(964\) 0 0
\(965\) − 2.60921e13i − 0.968580i
\(966\) 0 0
\(967\) 1.43858e13 0.529071 0.264535 0.964376i \(-0.414781\pi\)
0.264535 + 0.964376i \(0.414781\pi\)
\(968\) 0 0
\(969\) 2.10654e13 0.767560
\(970\) 0 0
\(971\) − 9.63124e12i − 0.347693i −0.984773 0.173846i \(-0.944380\pi\)
0.984773 0.173846i \(-0.0556196\pi\)
\(972\) 0 0
\(973\) 3.42108e13i 1.22364i
\(974\) 0 0
\(975\) −1.07967e12 −0.0382622
\(976\) 0 0
\(977\) −9.87031e12 −0.346581 −0.173291 0.984871i \(-0.555440\pi\)
−0.173291 + 0.984871i \(0.555440\pi\)
\(978\) 0 0
\(979\) − 2.13543e12i − 0.0742957i
\(980\) 0 0
\(981\) − 1.19313e13i − 0.411318i
\(982\) 0 0
\(983\) −5.82371e13 −1.98934 −0.994669 0.103116i \(-0.967119\pi\)
−0.994669 + 0.103116i \(0.967119\pi\)
\(984\) 0 0
\(985\) −1.50233e13 −0.508512
\(986\) 0 0
\(987\) 6.81072e11i 0.0228437i
\(988\) 0 0
\(989\) − 4.61918e13i − 1.53526i
\(990\) 0 0
\(991\) −6.04603e13 −1.99131 −0.995655 0.0931229i \(-0.970315\pi\)
−0.995655 + 0.0931229i \(0.970315\pi\)
\(992\) 0 0
\(993\) −5.52652e12 −0.180377
\(994\) 0 0
\(995\) − 4.32034e13i − 1.39738i
\(996\) 0 0
\(997\) 4.77299e13i 1.52990i 0.644091 + 0.764949i \(0.277236\pi\)
−0.644091 + 0.764949i \(0.722764\pi\)
\(998\) 0 0
\(999\) −2.47113e11 −0.00784965
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 384.10.d.b.193.6 yes 8
4.3 odd 2 384.10.d.a.193.2 8
8.3 odd 2 384.10.d.a.193.7 yes 8
8.5 even 2 inner 384.10.d.b.193.3 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.10.d.a.193.2 8 4.3 odd 2
384.10.d.a.193.7 yes 8 8.3 odd 2
384.10.d.b.193.3 yes 8 8.5 even 2 inner
384.10.d.b.193.6 yes 8 1.1 even 1 trivial