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Properties

Label 144.8.a.b.1.1
Level $144$
Weight $8$
Character 144.1
Self dual yes
Analytic conductor $44.983$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [144,8,Mod(1,144)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(144, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("144.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Level: \( N \) \(=\) \( 144 = 2^{4} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 144.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,-390] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(44.9834436697\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 144.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-390.000 q^{5} +64.0000 q^{7} -948.000 q^{11} -5098.00 q^{13} -28386.0 q^{17} +8620.00 q^{19} -15288.0 q^{23} +73975.0 q^{25} -36510.0 q^{29} +276808. q^{31} -24960.0 q^{35} +268526. q^{37} +629718. q^{41} -685772. q^{43} +583296. q^{47} -819447. q^{49} +428058. q^{53} +369720. q^{55} +1.30638e6 q^{59} +300662. q^{61} +1.98822e6 q^{65} +507244. q^{67} +5.56063e6 q^{71} +1.36908e6 q^{73} -60672.0 q^{77} +6.91372e6 q^{79} -4.37675e6 q^{83} +1.10705e7 q^{85} +8.52831e6 q^{89} -326272. q^{91} -3.36180e6 q^{95} -8.82681e6 q^{97} +O(q^{100})\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −390.000 −1.39531 −0.697653 − 0.716436i \(-0.745772\pi\)
−0.697653 + 0.716436i \(0.745772\pi\)
\(6\) 0 0
\(7\) 64.0000 0.0705240 0.0352620 − 0.999378i \(-0.488773\pi\)
0.0352620 + 0.999378i \(0.488773\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −948.000 −0.214750 −0.107375 − 0.994219i \(-0.534245\pi\)
−0.107375 + 0.994219i \(0.534245\pi\)
\(12\) 0 0
\(13\) −5098.00 −0.643573 −0.321787 − 0.946812i \(-0.604283\pi\)
−0.321787 + 0.946812i \(0.604283\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −28386.0 −1.40131 −0.700653 − 0.713502i \(-0.747108\pi\)
−0.700653 + 0.713502i \(0.747108\pi\)
\(18\) 0 0
\(19\) 8620.00 0.288317 0.144158 − 0.989555i \(-0.453953\pi\)
0.144158 + 0.989555i \(0.453953\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −15288.0 −0.262001 −0.131001 − 0.991382i \(-0.541819\pi\)
−0.131001 + 0.991382i \(0.541819\pi\)
\(24\) 0 0
\(25\) 73975.0 0.946880
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −36510.0 −0.277983 −0.138992 − 0.990294i \(-0.544386\pi\)
−0.138992 + 0.990294i \(0.544386\pi\)
\(30\) 0 0
\(31\) 276808. 1.66883 0.834416 − 0.551135i \(-0.185805\pi\)
0.834416 + 0.551135i \(0.185805\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −24960.0 −0.0984026
\(36\) 0 0
\(37\) 268526. 0.871526 0.435763 − 0.900061i \(-0.356479\pi\)
0.435763 + 0.900061i \(0.356479\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 629718. 1.42693 0.713465 − 0.700691i \(-0.247125\pi\)
0.713465 + 0.700691i \(0.247125\pi\)
\(42\) 0 0
\(43\) −685772. −1.31535 −0.657673 − 0.753303i \(-0.728459\pi\)
−0.657673 + 0.753303i \(0.728459\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 583296. 0.819495 0.409748 − 0.912199i \(-0.365617\pi\)
0.409748 + 0.912199i \(0.365617\pi\)
\(48\) 0 0
\(49\) −819447. −0.995026
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 428058. 0.394945 0.197473 − 0.980308i \(-0.436727\pi\)
0.197473 + 0.980308i \(0.436727\pi\)
\(54\) 0 0
\(55\) 369720. 0.299643
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.30638e6 0.828109 0.414054 − 0.910252i \(-0.364112\pi\)
0.414054 + 0.910252i \(0.364112\pi\)
\(60\) 0 0
\(61\) 300662. 0.169599 0.0847997 − 0.996398i \(-0.472975\pi\)
0.0847997 + 0.996398i \(0.472975\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.98822e6 0.897982
\(66\) 0 0
\(67\) 507244. 0.206042 0.103021 − 0.994679i \(-0.467149\pi\)
0.103021 + 0.994679i \(0.467149\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.56063e6 1.84383 0.921913 − 0.387397i \(-0.126626\pi\)
0.921913 + 0.387397i \(0.126626\pi\)
\(72\) 0 0
\(73\) 1.36908e6 0.411907 0.205954 − 0.978562i \(-0.433970\pi\)
0.205954 + 0.978562i \(0.433970\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −60672.0 −0.0151451
\(78\) 0 0
\(79\) 6.91372e6 1.57767 0.788836 − 0.614603i \(-0.210684\pi\)
0.788836 + 0.614603i \(0.210684\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.37675e6 −0.840191 −0.420096 − 0.907480i \(-0.638003\pi\)
−0.420096 + 0.907480i \(0.638003\pi\)
\(84\) 0 0
\(85\) 1.10705e7 1.95525
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 8.52831e6 1.28232 0.641162 − 0.767405i \(-0.278453\pi\)
0.641162 + 0.767405i \(0.278453\pi\)
\(90\) 0 0
\(91\) −326272. −0.0453874
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −3.36180e6 −0.402290
\(96\) 0 0
\(97\) −8.82681e6 −0.981981 −0.490990 − 0.871165i \(-0.663365\pi\)
−0.490990 + 0.871165i \(0.663365\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.19864e7 −1.15762 −0.578808 − 0.815464i \(-0.696482\pi\)
−0.578808 + 0.815464i \(0.696482\pi\)
\(102\) 0 0
\(103\) −7.20939e6 −0.650082 −0.325041 − 0.945700i \(-0.605378\pi\)
−0.325041 + 0.945700i \(0.605378\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.14261e7 0.901683 0.450842 − 0.892604i \(-0.351124\pi\)
0.450842 + 0.892604i \(0.351124\pi\)
\(108\) 0 0
\(109\) 4.02095e6 0.297397 0.148698 − 0.988883i \(-0.452492\pi\)
0.148698 + 0.988883i \(0.452492\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.77063e7 1.15439 0.577197 − 0.816605i \(-0.304147\pi\)
0.577197 + 0.816605i \(0.304147\pi\)
\(114\) 0 0
\(115\) 5.96232e6 0.365572
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.81670e6 −0.0988257
\(120\) 0 0
\(121\) −1.85885e7 −0.953882
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.61850e6 0.0741187
\(126\) 0 0
\(127\) −1.67883e7 −0.727267 −0.363633 − 0.931542i \(-0.618464\pi\)
−0.363633 + 0.931542i \(0.618464\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.68268e7 0.653960 0.326980 − 0.945031i \(-0.393969\pi\)
0.326980 + 0.945031i \(0.393969\pi\)
\(132\) 0 0
\(133\) 551680. 0.0203332
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.80449e7 −0.931820 −0.465910 − 0.884832i \(-0.654273\pi\)
−0.465910 + 0.884832i \(0.654273\pi\)
\(138\) 0 0
\(139\) 1.18273e7 0.373537 0.186769 − 0.982404i \(-0.440199\pi\)
0.186769 + 0.982404i \(0.440199\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 4.83290e6 0.138208
\(144\) 0 0
\(145\) 1.42389e7 0.387872
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −2.07846e7 −0.514743 −0.257371 − 0.966313i \(-0.582856\pi\)
−0.257371 + 0.966313i \(0.582856\pi\)
\(150\) 0 0
\(151\) −76112.0 −0.00179901 −0.000899505 − 1.00000i \(-0.500286\pi\)
−0.000899505 1.00000i \(0.500286\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.07955e8 −2.32853
\(156\) 0 0
\(157\) −3.21825e7 −0.663698 −0.331849 − 0.943332i \(-0.607672\pi\)
−0.331849 + 0.943332i \(0.607672\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −978432. −0.0184774
\(162\) 0 0
\(163\) −5.83435e7 −1.05520 −0.527601 − 0.849492i \(-0.676908\pi\)
−0.527601 + 0.849492i \(0.676908\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −2.58365e7 −0.429266 −0.214633 − 0.976695i \(-0.568855\pi\)
−0.214633 + 0.976695i \(0.568855\pi\)
\(168\) 0 0
\(169\) −3.67589e7 −0.585813
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −6.35201e7 −0.932716 −0.466358 − 0.884596i \(-0.654434\pi\)
−0.466358 + 0.884596i \(0.654434\pi\)
\(174\) 0 0
\(175\) 4.73440e6 0.0667777
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −8.09559e7 −1.05503 −0.527513 − 0.849547i \(-0.676875\pi\)
−0.527513 + 0.849547i \(0.676875\pi\)
\(180\) 0 0
\(181\) 6.45032e7 0.808549 0.404274 − 0.914638i \(-0.367524\pi\)
0.404274 + 0.914638i \(0.367524\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.04725e8 −1.21605
\(186\) 0 0
\(187\) 2.69099e7 0.300931
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 5.68274e7 0.590121 0.295060 − 0.955479i \(-0.404660\pi\)
0.295060 + 0.955479i \(0.404660\pi\)
\(192\) 0 0
\(193\) 1.16377e8 1.16524 0.582621 − 0.812744i \(-0.302027\pi\)
0.582621 + 0.812744i \(0.302027\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.18816e8 1.10724 0.553622 − 0.832768i \(-0.313245\pi\)
0.553622 + 0.832768i \(0.313245\pi\)
\(198\) 0 0
\(199\) 9.50106e7 0.854646 0.427323 − 0.904099i \(-0.359457\pi\)
0.427323 + 0.904099i \(0.359457\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −2.33664e6 −0.0196045
\(204\) 0 0
\(205\) −2.45590e8 −1.99100
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −8.17176e6 −0.0619161
\(210\) 0 0
\(211\) −1.79246e8 −1.31360 −0.656798 − 0.754067i \(-0.728090\pi\)
−0.656798 + 0.754067i \(0.728090\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 2.67451e8 1.83531
\(216\) 0 0
\(217\) 1.77157e7 0.117693
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.44712e8 0.901843
\(222\) 0 0
\(223\) 2.06537e8 1.24718 0.623592 − 0.781750i \(-0.285673\pi\)
0.623592 + 0.781750i \(0.285673\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.33954e7 0.246237 0.123118 − 0.992392i \(-0.460710\pi\)
0.123118 + 0.992392i \(0.460710\pi\)
\(228\) 0 0
\(229\) −3.61931e7 −0.199160 −0.0995799 − 0.995030i \(-0.531750\pi\)
−0.0995799 + 0.995030i \(0.531750\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −9.22347e7 −0.477693 −0.238846 − 0.971057i \(-0.576769\pi\)
−0.238846 + 0.971057i \(0.576769\pi\)
\(234\) 0 0
\(235\) −2.27485e8 −1.14345
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 4.98468e7 0.236181 0.118090 − 0.993003i \(-0.462323\pi\)
0.118090 + 0.993003i \(0.462323\pi\)
\(240\) 0 0
\(241\) 1.99374e8 0.917506 0.458753 − 0.888564i \(-0.348296\pi\)
0.458753 + 0.888564i \(0.348296\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.19584e8 1.38837
\(246\) 0 0
\(247\) −4.39448e7 −0.185553
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3.94678e8 −1.57538 −0.787689 − 0.616073i \(-0.788723\pi\)
−0.787689 + 0.616073i \(0.788723\pi\)
\(252\) 0 0
\(253\) 1.44930e7 0.0562649
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.42885e8 0.525076 0.262538 − 0.964922i \(-0.415441\pi\)
0.262538 + 0.964922i \(0.415441\pi\)
\(258\) 0 0
\(259\) 1.71857e7 0.0614635
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.40241e8 1.49226 0.746131 − 0.665799i \(-0.231909\pi\)
0.746131 + 0.665799i \(0.231909\pi\)
\(264\) 0 0
\(265\) −1.66943e8 −0.551070
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2.75405e8 −0.862657 −0.431329 − 0.902195i \(-0.641955\pi\)
−0.431329 + 0.902195i \(0.641955\pi\)
\(270\) 0 0
\(271\) 4.24670e8 1.29616 0.648080 − 0.761572i \(-0.275572\pi\)
0.648080 + 0.761572i \(0.275572\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −7.01283e7 −0.203343
\(276\) 0 0
\(277\) 5.16158e8 1.45916 0.729581 − 0.683894i \(-0.239715\pi\)
0.729581 + 0.683894i \(0.239715\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 3.11043e8 0.836273 0.418137 − 0.908384i \(-0.362683\pi\)
0.418137 + 0.908384i \(0.362683\pi\)
\(282\) 0 0
\(283\) 5.94308e8 1.55869 0.779344 − 0.626596i \(-0.215552\pi\)
0.779344 + 0.626596i \(0.215552\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.03020e7 0.100633
\(288\) 0 0
\(289\) 3.95426e8 0.963658
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.15515e8 −0.268288 −0.134144 − 0.990962i \(-0.542828\pi\)
−0.134144 + 0.990962i \(0.542828\pi\)
\(294\) 0 0
\(295\) −5.09488e8 −1.15547
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7.79382e7 0.168617
\(300\) 0 0
\(301\) −4.38894e7 −0.0927635
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.17258e8 −0.236643
\(306\) 0 0
\(307\) 2.60600e8 0.514032 0.257016 − 0.966407i \(-0.417261\pi\)
0.257016 + 0.966407i \(0.417261\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5.76795e8 1.08733 0.543663 − 0.839303i \(-0.317037\pi\)
0.543663 + 0.839303i \(0.317037\pi\)
\(312\) 0 0
\(313\) −4.60074e8 −0.848053 −0.424026 − 0.905650i \(-0.639384\pi\)
−0.424026 + 0.905650i \(0.639384\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.25561e7 −0.110297 −0.0551483 − 0.998478i \(-0.517563\pi\)
−0.0551483 + 0.998478i \(0.517563\pi\)
\(318\) 0 0
\(319\) 3.46115e7 0.0596970
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2.44687e8 −0.404020
\(324\) 0 0
\(325\) −3.77125e8 −0.609387
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.73309e7 0.0577941
\(330\) 0 0
\(331\) −6.84236e8 −1.03707 −0.518535 − 0.855057i \(-0.673522\pi\)
−0.518535 + 0.855057i \(0.673522\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.97825e8 −0.287491
\(336\) 0 0
\(337\) −6.26313e8 −0.891429 −0.445714 − 0.895175i \(-0.647050\pi\)
−0.445714 + 0.895175i \(0.647050\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −2.62414e8 −0.358382
\(342\) 0 0
\(343\) −1.05151e8 −0.140697
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.25340e9 −1.61041 −0.805203 − 0.593000i \(-0.797943\pi\)
−0.805203 + 0.593000i \(0.797943\pi\)
\(348\) 0 0
\(349\) 2.65350e8 0.334142 0.167071 − 0.985945i \(-0.446569\pi\)
0.167071 + 0.985945i \(0.446569\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 5.69636e8 0.689264 0.344632 − 0.938738i \(-0.388004\pi\)
0.344632 + 0.938738i \(0.388004\pi\)
\(354\) 0 0
\(355\) −2.16865e9 −2.57270
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 9.32541e8 1.06374 0.531872 − 0.846825i \(-0.321489\pi\)
0.531872 + 0.846825i \(0.321489\pi\)
\(360\) 0 0
\(361\) −8.19567e8 −0.916874
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −5.33942e8 −0.574737
\(366\) 0 0
\(367\) 8.52565e8 0.900318 0.450159 − 0.892948i \(-0.351367\pi\)
0.450159 + 0.892948i \(0.351367\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 2.73957e7 0.0278531
\(372\) 0 0
\(373\) 3.81183e8 0.380323 0.190162 − 0.981753i \(-0.439099\pi\)
0.190162 + 0.981753i \(0.439099\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.86128e8 0.178903
\(378\) 0 0
\(379\) 1.48353e9 1.39978 0.699889 − 0.714251i \(-0.253233\pi\)
0.699889 + 0.714251i \(0.253233\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −7.61930e8 −0.692978 −0.346489 − 0.938054i \(-0.612626\pi\)
−0.346489 + 0.938054i \(0.612626\pi\)
\(384\) 0 0
\(385\) 2.36621e7 0.0211320
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.60902e9 −1.38592 −0.692959 − 0.720977i \(-0.743693\pi\)
−0.692959 + 0.720977i \(0.743693\pi\)
\(390\) 0 0
\(391\) 4.33965e8 0.367144
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.69635e9 −2.20134
\(396\) 0 0
\(397\) 1.88016e9 1.50809 0.754046 − 0.656822i \(-0.228100\pi\)
0.754046 + 0.656822i \(0.228100\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −2.68592e8 −0.208012 −0.104006 − 0.994577i \(-0.533166\pi\)
−0.104006 + 0.994577i \(0.533166\pi\)
\(402\) 0 0
\(403\) −1.41117e9 −1.07402
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −2.54563e8 −0.187161
\(408\) 0 0
\(409\) 8.99478e7 0.0650069 0.0325034 − 0.999472i \(-0.489652\pi\)
0.0325034 + 0.999472i \(0.489652\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.36083e7 0.0584015
\(414\) 0 0
\(415\) 1.70693e9 1.17232
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.69054e9 1.12273 0.561367 − 0.827567i \(-0.310276\pi\)
0.561367 + 0.827567i \(0.310276\pi\)
\(420\) 0 0
\(421\) −1.13333e9 −0.740232 −0.370116 − 0.928985i \(-0.620682\pi\)
−0.370116 + 0.928985i \(0.620682\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.09985e9 −1.32687
\(426\) 0 0
\(427\) 1.92424e7 0.0119608
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2.19943e9 1.32324 0.661621 − 0.749839i \(-0.269869\pi\)
0.661621 + 0.749839i \(0.269869\pi\)
\(432\) 0 0
\(433\) −1.51738e8 −0.0898227 −0.0449114 − 0.998991i \(-0.514301\pi\)
−0.0449114 + 0.998991i \(0.514301\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.31783e8 −0.0755393
\(438\) 0 0
\(439\) −9.90763e8 −0.558912 −0.279456 − 0.960158i \(-0.590154\pi\)
−0.279456 + 0.960158i \(0.590154\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −1.77376e9 −0.969351 −0.484675 − 0.874694i \(-0.661062\pi\)
−0.484675 + 0.874694i \(0.661062\pi\)
\(444\) 0 0
\(445\) −3.32604e9 −1.78924
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2.77010e8 0.144422 0.0722110 − 0.997389i \(-0.476994\pi\)
0.0722110 + 0.997389i \(0.476994\pi\)
\(450\) 0 0
\(451\) −5.96973e8 −0.306434
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.27246e8 0.0633293
\(456\) 0 0
\(457\) 2.94758e9 1.44464 0.722320 − 0.691559i \(-0.243076\pi\)
0.722320 + 0.691559i \(0.243076\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.76687e9 1.31533 0.657667 − 0.753309i \(-0.271543\pi\)
0.657667 + 0.753309i \(0.271543\pi\)
\(462\) 0 0
\(463\) −4.63553e8 −0.217053 −0.108527 − 0.994094i \(-0.534613\pi\)
−0.108527 + 0.994094i \(0.534613\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4.17922e8 −0.189883 −0.0949415 − 0.995483i \(-0.530266\pi\)
−0.0949415 + 0.995483i \(0.530266\pi\)
\(468\) 0 0
\(469\) 3.24636e7 0.0145309
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6.50112e8 0.282471
\(474\) 0 0
\(475\) 6.37664e8 0.273001
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −1.50973e9 −0.627660 −0.313830 − 0.949479i \(-0.601612\pi\)
−0.313830 + 0.949479i \(0.601612\pi\)
\(480\) 0 0
\(481\) −1.36895e9 −0.560891
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 3.44246e9 1.37016
\(486\) 0 0
\(487\) −9.29460e8 −0.364653 −0.182326 − 0.983238i \(-0.558363\pi\)
−0.182326 + 0.983238i \(0.558363\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 5.12803e9 1.95508 0.977541 − 0.210743i \(-0.0675885\pi\)
0.977541 + 0.210743i \(0.0675885\pi\)
\(492\) 0 0
\(493\) 1.03637e9 0.389540
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.55880e8 0.130034
\(498\) 0 0
\(499\) 4.10649e8 0.147951 0.0739757 − 0.997260i \(-0.476431\pi\)
0.0739757 + 0.997260i \(0.476431\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 5.02041e9 1.75894 0.879470 − 0.475954i \(-0.157897\pi\)
0.879470 + 0.475954i \(0.157897\pi\)
\(504\) 0 0
\(505\) 4.67470e9 1.61523
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 3.24926e9 1.09212 0.546062 − 0.837745i \(-0.316126\pi\)
0.546062 + 0.837745i \(0.316126\pi\)
\(510\) 0 0
\(511\) 8.76212e7 0.0290493
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.81166e9 0.907064
\(516\) 0 0
\(517\) −5.52965e8 −0.175987
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 2.10950e9 0.653503 0.326752 − 0.945110i \(-0.394046\pi\)
0.326752 + 0.945110i \(0.394046\pi\)
\(522\) 0 0
\(523\) 5.28911e9 1.61669 0.808345 − 0.588709i \(-0.200364\pi\)
0.808345 + 0.588709i \(0.200364\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7.85747e9 −2.33854
\(528\) 0 0
\(529\) −3.17110e9 −0.931355
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −3.21030e9 −0.918334
\(534\) 0 0
\(535\) −4.45617e9 −1.25812
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 7.76836e8 0.213682
\(540\) 0 0
\(541\) 3.04614e9 0.827101 0.413551 − 0.910481i \(-0.364288\pi\)
0.413551 + 0.910481i \(0.364288\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1.56817e9 −0.414959
\(546\) 0 0
\(547\) 4.85537e9 1.26843 0.634215 − 0.773157i \(-0.281323\pi\)
0.634215 + 0.773157i \(0.281323\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3.14716e8 −0.0801472
\(552\) 0 0
\(553\) 4.42478e8 0.111264
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −1.27762e9 −0.313263 −0.156631 − 0.987657i \(-0.550063\pi\)
−0.156631 + 0.987657i \(0.550063\pi\)
\(558\) 0 0
\(559\) 3.49607e9 0.846522
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4.71265e9 1.11297 0.556487 − 0.830856i \(-0.312149\pi\)
0.556487 + 0.830856i \(0.312149\pi\)
\(564\) 0 0
\(565\) −6.90546e9 −1.61073
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −4.57800e9 −1.04180 −0.520898 − 0.853619i \(-0.674403\pi\)
−0.520898 + 0.853619i \(0.674403\pi\)
\(570\) 0 0
\(571\) −4.95119e9 −1.11297 −0.556485 − 0.830858i \(-0.687850\pi\)
−0.556485 + 0.830858i \(0.687850\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.13093e9 −0.248084
\(576\) 0 0
\(577\) 8.51847e9 1.84606 0.923031 − 0.384725i \(-0.125704\pi\)
0.923031 + 0.384725i \(0.125704\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −2.80112e8 −0.0592536
\(582\) 0 0
\(583\) −4.05799e8 −0.0848147
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −5.62247e8 −0.114735 −0.0573673 − 0.998353i \(-0.518271\pi\)
−0.0573673 + 0.998353i \(0.518271\pi\)
\(588\) 0 0
\(589\) 2.38608e9 0.481152
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −3.62110e9 −0.713099 −0.356549 − 0.934277i \(-0.616047\pi\)
−0.356549 + 0.934277i \(0.616047\pi\)
\(594\) 0 0
\(595\) 7.08515e8 0.137892
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −7.48104e9 −1.42222 −0.711112 − 0.703079i \(-0.751808\pi\)
−0.711112 + 0.703079i \(0.751808\pi\)
\(600\) 0 0
\(601\) −5.81270e9 −1.09224 −0.546119 − 0.837707i \(-0.683895\pi\)
−0.546119 + 0.837707i \(0.683895\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 7.24950e9 1.33096
\(606\) 0 0
\(607\) −3.84051e9 −0.696993 −0.348497 − 0.937310i \(-0.613308\pi\)
−0.348497 + 0.937310i \(0.613308\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.97364e9 −0.527405
\(612\) 0 0
\(613\) 1.70484e9 0.298932 0.149466 − 0.988767i \(-0.452245\pi\)
0.149466 + 0.988767i \(0.452245\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.80809e9 0.481297 0.240649 − 0.970612i \(-0.422640\pi\)
0.240649 + 0.970612i \(0.422640\pi\)
\(618\) 0 0
\(619\) 2.54365e9 0.431063 0.215532 − 0.976497i \(-0.430852\pi\)
0.215532 + 0.976497i \(0.430852\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 5.45812e8 0.0904346
\(624\) 0 0
\(625\) −6.41051e9 −1.05030
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −7.62238e9 −1.22127
\(630\) 0 0
\(631\) 1.51146e8 0.0239494 0.0119747 − 0.999928i \(-0.496188\pi\)
0.0119747 + 0.999928i \(0.496188\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 6.54744e9 1.01476
\(636\) 0 0
\(637\) 4.17754e9 0.640373
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.23625e10 1.85397 0.926987 − 0.375094i \(-0.122390\pi\)
0.926987 + 0.375094i \(0.122390\pi\)
\(642\) 0 0
\(643\) −2.86744e9 −0.425359 −0.212680 − 0.977122i \(-0.568219\pi\)
−0.212680 + 0.977122i \(0.568219\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −4.10640e9 −0.596068 −0.298034 − 0.954555i \(-0.596331\pi\)
−0.298034 + 0.954555i \(0.596331\pi\)
\(648\) 0 0
\(649\) −1.23845e9 −0.177837
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −6.91100e9 −0.971280 −0.485640 − 0.874159i \(-0.661413\pi\)
−0.485640 + 0.874159i \(0.661413\pi\)
\(654\) 0 0
\(655\) −6.56244e9 −0.912475
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 3.42444e9 0.466112 0.233056 − 0.972463i \(-0.425127\pi\)
0.233056 + 0.972463i \(0.425127\pi\)
\(660\) 0 0
\(661\) −6.76437e9 −0.911008 −0.455504 − 0.890234i \(-0.650541\pi\)
−0.455504 + 0.890234i \(0.650541\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.15155e8 −0.0283711
\(666\) 0 0
\(667\) 5.58165e8 0.0728320
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −2.85028e8 −0.0364215
\(672\) 0 0
\(673\) −1.74959e9 −0.221250 −0.110625 − 0.993862i \(-0.535285\pi\)
−0.110625 + 0.993862i \(0.535285\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −8.30011e9 −1.02807 −0.514036 − 0.857769i \(-0.671850\pi\)
−0.514036 + 0.857769i \(0.671850\pi\)
\(678\) 0 0
\(679\) −5.64916e8 −0.0692532
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.21232e10 −1.45594 −0.727969 − 0.685610i \(-0.759536\pi\)
−0.727969 + 0.685610i \(0.759536\pi\)
\(684\) 0 0
\(685\) 1.09375e10 1.30017
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.18224e9 −0.254176
\(690\) 0 0
\(691\) −8.21846e9 −0.947583 −0.473791 − 0.880637i \(-0.657115\pi\)
−0.473791 + 0.880637i \(0.657115\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4.61265e9 −0.521199
\(696\) 0 0
\(697\) −1.78752e10 −1.99957
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −4.72231e9 −0.517775 −0.258888 − 0.965907i \(-0.583356\pi\)
−0.258888 + 0.965907i \(0.583356\pi\)
\(702\) 0 0
\(703\) 2.31469e9 0.251275
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −7.67131e8 −0.0816397
\(708\) 0 0
\(709\) 2.78975e9 0.293970 0.146985 − 0.989139i \(-0.453043\pi\)
0.146985 + 0.989139i \(0.453043\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −4.23184e9 −0.437236
\(714\) 0 0
\(715\) −1.88483e9 −0.192842
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.51985e9 0.152493 0.0762463 − 0.997089i \(-0.475706\pi\)
0.0762463 + 0.997089i \(0.475706\pi\)
\(720\) 0 0
\(721\) −4.61401e8 −0.0458464
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −2.70083e9 −0.263217
\(726\) 0 0
\(727\) 8.11761e9 0.783534 0.391767 − 0.920065i \(-0.371864\pi\)
0.391767 + 0.920065i \(0.371864\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.94663e10 1.84320
\(732\) 0 0
\(733\) −1.03241e10 −0.968249 −0.484124 − 0.874999i \(-0.660862\pi\)
−0.484124 + 0.874999i \(0.660862\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −4.80867e8 −0.0442475
\(738\) 0 0
\(739\) 1.35365e10 1.23382 0.616908 − 0.787035i \(-0.288385\pi\)
0.616908 + 0.787035i \(0.288385\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.71936e10 −1.53782 −0.768910 − 0.639356i \(-0.779201\pi\)
−0.768910 + 0.639356i \(0.779201\pi\)
\(744\) 0 0
\(745\) 8.10601e9 0.718224
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 7.31269e8 0.0635903
\(750\) 0 0
\(751\) −1.12478e10 −0.969013 −0.484506 − 0.874788i \(-0.661001\pi\)
−0.484506 + 0.874788i \(0.661001\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2.96837e7 0.00251017
\(756\) 0 0
\(757\) 1.63068e10 1.36626 0.683131 − 0.730296i \(-0.260618\pi\)
0.683131 + 0.730296i \(0.260618\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −6.14069e9 −0.505093 −0.252546 − 0.967585i \(-0.581268\pi\)
−0.252546 + 0.967585i \(0.581268\pi\)
\(762\) 0 0
\(763\) 2.57341e8 0.0209736
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −6.65993e9 −0.532949
\(768\) 0 0
\(769\) 2.45069e10 1.94333 0.971664 − 0.236368i \(-0.0759569\pi\)
0.971664 + 0.236368i \(0.0759569\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.01722e10 0.792110 0.396055 − 0.918227i \(-0.370379\pi\)
0.396055 + 0.918227i \(0.370379\pi\)
\(774\) 0 0
\(775\) 2.04769e10 1.58018
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5.42817e9 0.411408
\(780\) 0 0
\(781\) −5.27148e9 −0.395962
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.25512e10 0.926062
\(786\) 0 0
\(787\) 9.79135e9 0.716030 0.358015 − 0.933716i \(-0.383454\pi\)
0.358015 + 0.933716i \(0.383454\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.13320e9 0.0814124
\(792\) 0 0
\(793\) −1.53277e9 −0.109150
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 9.75782e9 0.682729 0.341365 − 0.939931i \(-0.389111\pi\)
0.341365 + 0.939931i \(0.389111\pi\)
\(798\) 0 0
\(799\) −1.65574e10 −1.14836
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1.29789e9 −0.0884572
\(804\) 0 0
\(805\) 3.81588e8 0.0257816
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.78706e9 0.185066 0.0925330 − 0.995710i \(-0.470504\pi\)
0.0925330 + 0.995710i \(0.470504\pi\)
\(810\) 0 0
\(811\) 7.99983e9 0.526633 0.263316 − 0.964710i \(-0.415184\pi\)
0.263316 + 0.964710i \(0.415184\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2.27540e10 1.47233
\(816\) 0 0
\(817\) −5.91135e9 −0.379236
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.02402e10 0.645813 0.322906 − 0.946431i \(-0.395340\pi\)
0.322906 + 0.946431i \(0.395340\pi\)
\(822\) 0 0
\(823\) −2.78682e10 −1.74265 −0.871324 − 0.490707i \(-0.836738\pi\)
−0.871324 + 0.490707i \(0.836738\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.35125e10 1.44554 0.722769 − 0.691090i \(-0.242869\pi\)
0.722769 + 0.691090i \(0.242869\pi\)
\(828\) 0 0
\(829\) −1.28598e10 −0.783960 −0.391980 − 0.919974i \(-0.628210\pi\)
−0.391980 + 0.919974i \(0.628210\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.32608e10 1.39434
\(834\) 0 0
\(835\) 1.00762e10 0.598957
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −7.99832e9 −0.467554 −0.233777 − 0.972290i \(-0.575109\pi\)
−0.233777 + 0.972290i \(0.575109\pi\)
\(840\) 0 0
\(841\) −1.59169e10 −0.922725
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.43360e10 0.817389
\(846\) 0 0
\(847\) −1.18966e9 −0.0672716
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −4.10523e9 −0.228341
\(852\) 0 0
\(853\) 4.20827e9 0.232157 0.116079 − 0.993240i \(-0.462968\pi\)
0.116079 + 0.993240i \(0.462968\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −3.19307e10 −1.73291 −0.866453 − 0.499259i \(-0.833606\pi\)
−0.866453 + 0.499259i \(0.833606\pi\)
\(858\) 0 0
\(859\) −2.18002e10 −1.17350 −0.586752 − 0.809767i \(-0.699594\pi\)
−0.586752 + 0.809767i \(0.699594\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.04728e10 0.554657 0.277329 − 0.960775i \(-0.410551\pi\)
0.277329 + 0.960775i \(0.410551\pi\)
\(864\) 0 0
\(865\) 2.47728e10 1.30142
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −6.55421e9 −0.338806
\(870\) 0 0
\(871\) −2.58593e9 −0.132603
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 1.03584e8 0.00522714
\(876\) 0 0
\(877\) −1.77787e10 −0.890024 −0.445012 − 0.895525i \(-0.646801\pi\)
−0.445012 + 0.895525i \(0.646801\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 7.64253e9 0.376549 0.188274 − 0.982116i \(-0.439711\pi\)
0.188274 + 0.982116i \(0.439711\pi\)
\(882\) 0 0
\(883\) 2.76375e10 1.35094 0.675472 − 0.737386i \(-0.263940\pi\)
0.675472 + 0.737386i \(0.263940\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 3.23087e10 1.55449 0.777243 − 0.629200i \(-0.216617\pi\)
0.777243 + 0.629200i \(0.216617\pi\)
\(888\) 0 0
\(889\) −1.07445e9 −0.0512897
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5.02801e9 0.236274
\(894\) 0 0
\(895\) 3.15728e10 1.47208
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1.01063e10 −0.463908
\(900\) 0 0
\(901\) −1.21509e10 −0.553439
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.51562e10 −1.12817
\(906\) 0 0
\(907\) −2.27142e10 −1.01082 −0.505409 − 0.862880i \(-0.668658\pi\)
−0.505409 + 0.862880i \(0.668658\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 7.50925e9 0.329065 0.164533 − 0.986372i \(-0.447388\pi\)
0.164533 + 0.986372i \(0.447388\pi\)
\(912\) 0 0
\(913\) 4.14916e9 0.180431
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.07691e9 0.0461199
\(918\) 0 0
\(919\) 2.49374e10 1.05986 0.529928 − 0.848043i \(-0.322219\pi\)
0.529928 + 0.848043i \(0.322219\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −2.83481e10 −1.18664
\(924\) 0 0
\(925\) 1.98642e10 0.825230
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 8.66205e9 0.354459 0.177229 − 0.984170i \(-0.443287\pi\)
0.177229 + 0.984170i \(0.443287\pi\)
\(930\) 0 0
\(931\) −7.06363e9 −0.286883
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.04949e10 −0.419891
\(936\) 0 0
\(937\) 2.82655e10 1.12245 0.561226 − 0.827663i \(-0.310330\pi\)
0.561226 + 0.827663i \(0.310330\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 4.67082e10 1.82738 0.913691 − 0.406410i \(-0.133220\pi\)
0.913691 + 0.406410i \(0.133220\pi\)
\(942\) 0 0
\(943\) −9.62713e9 −0.373857
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4.67392e10 −1.78837 −0.894184 − 0.447701i \(-0.852243\pi\)
−0.894184 + 0.447701i \(0.852243\pi\)
\(948\) 0 0
\(949\) −6.97958e9 −0.265093
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −3.82420e10 −1.43125 −0.715625 − 0.698484i \(-0.753858\pi\)
−0.715625 + 0.698484i \(0.753858\pi\)
\(954\) 0 0
\(955\) −2.21627e10 −0.823399
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1.79487e9 −0.0657157
\(960\) 0 0
\(961\) 4.91101e10 1.78500
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −4.53870e10 −1.62587
\(966\) 0 0
\(967\) 4.90012e10 1.74267 0.871333 − 0.490692i \(-0.163256\pi\)
0.871333 + 0.490692i \(0.163256\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 2.72929e10 0.956713 0.478357 − 0.878166i \(-0.341233\pi\)
0.478357 + 0.878166i \(0.341233\pi\)
\(972\) 0 0
\(973\) 7.56947e8 0.0263433
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −3.94482e9 −0.135331 −0.0676653 − 0.997708i \(-0.521555\pi\)
−0.0676653 + 0.997708i \(0.521555\pi\)
\(978\) 0 0
\(979\) −8.08484e9 −0.275380
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 4.74320e8 0.0159270 0.00796351 − 0.999968i \(-0.497465\pi\)
0.00796351 + 0.999968i \(0.497465\pi\)
\(984\) 0 0
\(985\) −4.63383e10 −1.54494
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.04841e10 0.344622
\(990\) 0 0
\(991\) −1.22197e10 −0.398843 −0.199421 − 0.979914i \(-0.563906\pi\)
−0.199421 + 0.979914i \(0.563906\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −3.70541e10 −1.19249
\(996\) 0 0
\(997\) −3.60690e10 −1.15266 −0.576330 − 0.817217i \(-0.695516\pi\)
−0.576330 + 0.817217i \(0.695516\pi\)
\(998\) 0 0
\(999\) 0 0
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 144.8.a.b.1.1 1
3.2 odd 2 48.8.a.g.1.1 1
4.3 odd 2 9.8.a.a.1.1 1
8.3 odd 2 576.8.a.w.1.1 1
8.5 even 2 576.8.a.x.1.1 1
12.11 even 2 3.8.a.a.1.1 ✓ 1
20.3 even 4 225.8.b.f.199.2 2
20.7 even 4 225.8.b.f.199.1 2
20.19 odd 2 225.8.a.i.1.1 1
24.5 odd 2 192.8.a.a.1.1 1
24.11 even 2 192.8.a.i.1.1 1
28.27 even 2 441.8.a.a.1.1 1
36.7 odd 6 81.8.c.c.28.1 2
36.11 even 6 81.8.c.a.28.1 2
36.23 even 6 81.8.c.a.55.1 2
36.31 odd 6 81.8.c.c.55.1 2
60.23 odd 4 75.8.b.c.49.1 2
60.47 odd 4 75.8.b.c.49.2 2
60.59 even 2 75.8.a.a.1.1 1
84.11 even 6 147.8.e.b.79.1 2
84.23 even 6 147.8.e.b.67.1 2
84.47 odd 6 147.8.e.a.67.1 2
84.59 odd 6 147.8.e.a.79.1 2
84.83 odd 2 147.8.a.b.1.1 1
132.131 odd 2 363.8.a.b.1.1 1
156.155 even 2 507.8.a.a.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.8.a.a.1.1 ✓ 1 12.11 even 2
9.8.a.a.1.1 1 4.3 odd 2
48.8.a.g.1.1 1 3.2 odd 2
75.8.a.a.1.1 1 60.59 even 2
75.8.b.c.49.1 2 60.23 odd 4
75.8.b.c.49.2 2 60.47 odd 4
81.8.c.a.28.1 2 36.11 even 6
81.8.c.a.55.1 2 36.23 even 6
81.8.c.c.28.1 2 36.7 odd 6
81.8.c.c.55.1 2 36.31 odd 6
144.8.a.b.1.1 1 1.1 even 1 trivial
147.8.a.b.1.1 1 84.83 odd 2
147.8.e.a.67.1 2 84.47 odd 6
147.8.e.a.79.1 2 84.59 odd 6
147.8.e.b.67.1 2 84.23 even 6
147.8.e.b.79.1 2 84.11 even 6
192.8.a.a.1.1 1 24.5 odd 2
192.8.a.i.1.1 1 24.11 even 2
225.8.a.i.1.1 1 20.19 odd 2
225.8.b.f.199.1 2 20.7 even 4
225.8.b.f.199.2 2 20.3 even 4
363.8.a.b.1.1 1 132.131 odd 2
441.8.a.a.1.1 1 28.27 even 2
507.8.a.a.1.1 1 156.155 even 2
576.8.a.w.1.1 1 8.3 odd 2
576.8.a.x.1.1 1 8.5 even 2