Properties

Label 288.8.c.b.287.2
Level $288$
Weight $8$
Character 288.287
Analytic conductor $89.967$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [288,8,Mod(287,288)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(288, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("288.287");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 288 = 2^{5} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 288.c (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(89.9668873394\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 1360 x^{14} + 710372 x^{12} - 180776904 x^{10} + 23779111124 x^{8} - 1575523372272 x^{6} + \cdots + 18\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{98}\cdot 3^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 287.2
Root \(20.5167 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 288.287
Dual form 288.8.c.b.287.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-445.722i q^{5} +201.952i q^{7} +O(q^{10})\) \(q-445.722i q^{5} +201.952i q^{7} +5367.66 q^{11} +1173.08 q^{13} +7661.23i q^{17} +29821.5i q^{19} +78582.2 q^{23} -120543. q^{25} +250673. i q^{29} -12849.8i q^{31} +90014.2 q^{35} -98047.2 q^{37} +120977. i q^{41} +722010. i q^{43} +803354. q^{47} +782759. q^{49} -962157. i q^{53} -2.39248e6i q^{55} -3.09055e6 q^{59} +775974. q^{61} -522869. i q^{65} -252638. i q^{67} +163095. q^{71} +2.39257e6 q^{73} +1.08401e6i q^{77} -6.38854e6i q^{79} +3.03995e6 q^{83} +3.41478e6 q^{85} +2.42945e6i q^{89} +236906. i q^{91} +1.32921e7 q^{95} -2.12402e6 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 33280 q^{13} - 55248 q^{25} + 869664 q^{37} + 1042032 q^{49} + 5525344 q^{61} + 29037312 q^{73} + 56734176 q^{85} + 82141696 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/288\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(65\) \(127\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) − 445.722i − 1.59466i −0.603541 0.797332i \(-0.706244\pi\)
0.603541 0.797332i \(-0.293756\pi\)
\(6\) 0 0
\(7\) 201.952i 0.222538i 0.993790 + 0.111269i \(0.0354915\pi\)
−0.993790 + 0.111269i \(0.964508\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5367.66 1.21594 0.607968 0.793961i \(-0.291985\pi\)
0.607968 + 0.793961i \(0.291985\pi\)
\(12\) 0 0
\(13\) 1173.08 0.148090 0.0740452 0.997255i \(-0.476409\pi\)
0.0740452 + 0.997255i \(0.476409\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7661.23i 0.378205i 0.981957 + 0.189103i \(0.0605579\pi\)
−0.981957 + 0.189103i \(0.939442\pi\)
\(18\) 0 0
\(19\) 29821.5i 0.997451i 0.866760 + 0.498725i \(0.166198\pi\)
−0.866760 + 0.498725i \(0.833802\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 78582.2 1.34672 0.673359 0.739316i \(-0.264851\pi\)
0.673359 + 0.739316i \(0.264851\pi\)
\(24\) 0 0
\(25\) −120543. −1.54295
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 250673.i 1.90859i 0.298859 + 0.954297i \(0.403394\pi\)
−0.298859 + 0.954297i \(0.596606\pi\)
\(30\) 0 0
\(31\) − 12849.8i − 0.0774696i −0.999250 0.0387348i \(-0.987667\pi\)
0.999250 0.0387348i \(-0.0123328\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 90014.2 0.354873
\(36\) 0 0
\(37\) −98047.2 −0.318221 −0.159111 0.987261i \(-0.550863\pi\)
−0.159111 + 0.987261i \(0.550863\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 120977.i 0.274132i 0.990562 + 0.137066i \(0.0437673\pi\)
−0.990562 + 0.137066i \(0.956233\pi\)
\(42\) 0 0
\(43\) 722010.i 1.38485i 0.721488 + 0.692426i \(0.243458\pi\)
−0.721488 + 0.692426i \(0.756542\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 803354. 1.12866 0.564332 0.825548i \(-0.309134\pi\)
0.564332 + 0.825548i \(0.309134\pi\)
\(48\) 0 0
\(49\) 782759. 0.950477
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 962157.i − 0.887729i −0.896094 0.443864i \(-0.853607\pi\)
0.896094 0.443864i \(-0.146393\pi\)
\(54\) 0 0
\(55\) − 2.39248e6i − 1.93901i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.09055e6 −1.95909 −0.979543 0.201236i \(-0.935504\pi\)
−0.979543 + 0.201236i \(0.935504\pi\)
\(60\) 0 0
\(61\) 775974. 0.437717 0.218858 0.975757i \(-0.429767\pi\)
0.218858 + 0.975757i \(0.429767\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 522869.i − 0.236154i
\(66\) 0 0
\(67\) − 252638.i − 0.102621i −0.998683 0.0513106i \(-0.983660\pi\)
0.998683 0.0513106i \(-0.0163398\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 163095. 0.0540799 0.0270400 0.999634i \(-0.491392\pi\)
0.0270400 + 0.999634i \(0.491392\pi\)
\(72\) 0 0
\(73\) 2.39257e6 0.719839 0.359919 0.932983i \(-0.382804\pi\)
0.359919 + 0.932983i \(0.382804\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 1.08401e6i 0.270592i
\(78\) 0 0
\(79\) − 6.38854e6i − 1.45783i −0.684605 0.728915i \(-0.740025\pi\)
0.684605 0.728915i \(-0.259975\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.03995e6 0.583571 0.291786 0.956484i \(-0.405751\pi\)
0.291786 + 0.956484i \(0.405751\pi\)
\(84\) 0 0
\(85\) 3.41478e6 0.603110
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.42945e6i 0.365294i 0.983179 + 0.182647i \(0.0584665\pi\)
−0.983179 + 0.182647i \(0.941533\pi\)
\(90\) 0 0
\(91\) 236906.i 0.0329557i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.32921e7 1.59060
\(96\) 0 0
\(97\) −2.12402e6 −0.236297 −0.118148 0.992996i \(-0.537696\pi\)
−0.118148 + 0.992996i \(0.537696\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 1.19855e7i 1.15753i 0.815495 + 0.578764i \(0.196465\pi\)
−0.815495 + 0.578764i \(0.803535\pi\)
\(102\) 0 0
\(103\) 1.34950e7i 1.21686i 0.793606 + 0.608432i \(0.208201\pi\)
−0.793606 + 0.608432i \(0.791799\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.51239e7 1.19350 0.596749 0.802428i \(-0.296459\pi\)
0.596749 + 0.802428i \(0.296459\pi\)
\(108\) 0 0
\(109\) 1.49832e7 1.10818 0.554091 0.832456i \(-0.313066\pi\)
0.554091 + 0.832456i \(0.313066\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.58082e7i 1.68261i 0.540560 + 0.841306i \(0.318213\pi\)
−0.540560 + 0.841306i \(0.681787\pi\)
\(114\) 0 0
\(115\) − 3.50258e7i − 2.14756i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.54720e6 −0.0841650
\(120\) 0 0
\(121\) 9.32463e6 0.478501
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.89067e7i 0.865825i
\(126\) 0 0
\(127\) − 1.30174e7i − 0.563913i −0.959427 0.281956i \(-0.909017\pi\)
0.959427 0.281956i \(-0.0909833\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.59125e7 0.618426 0.309213 0.950993i \(-0.399934\pi\)
0.309213 + 0.950993i \(0.399934\pi\)
\(132\) 0 0
\(133\) −6.02249e6 −0.221971
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 1.22241e7i − 0.406156i −0.979163 0.203078i \(-0.934905\pi\)
0.979163 0.203078i \(-0.0650946\pi\)
\(138\) 0 0
\(139\) − 5.36512e7i − 1.69445i −0.531238 0.847223i \(-0.678273\pi\)
0.531238 0.847223i \(-0.321727\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 6.29671e6 0.180068
\(144\) 0 0
\(145\) 1.11730e8 3.04357
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 4.26997e7i − 1.05748i −0.848783 0.528741i \(-0.822664\pi\)
0.848783 0.528741i \(-0.177336\pi\)
\(150\) 0 0
\(151\) − 9.72945e6i − 0.229969i −0.993367 0.114984i \(-0.963318\pi\)
0.993367 0.114984i \(-0.0366818\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −5.72745e6 −0.123538
\(156\) 0 0
\(157\) 8.74323e7 1.80311 0.901557 0.432661i \(-0.142425\pi\)
0.901557 + 0.432661i \(0.142425\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 1.58698e7i 0.299696i
\(162\) 0 0
\(163\) − 1.07309e8i − 1.94080i −0.241498 0.970401i \(-0.577639\pi\)
0.241498 0.970401i \(-0.422361\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.78896e7 0.961817 0.480909 0.876771i \(-0.340307\pi\)
0.480909 + 0.876771i \(0.340307\pi\)
\(168\) 0 0
\(169\) −6.13724e7 −0.978069
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.07783e8i 1.58267i 0.611384 + 0.791334i \(0.290613\pi\)
−0.611384 + 0.791334i \(0.709387\pi\)
\(174\) 0 0
\(175\) − 2.43439e7i − 0.343365i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −4.65555e7 −0.606716 −0.303358 0.952877i \(-0.598108\pi\)
−0.303358 + 0.952877i \(0.598108\pi\)
\(180\) 0 0
\(181\) 4.00341e7 0.501828 0.250914 0.968009i \(-0.419269\pi\)
0.250914 + 0.968009i \(0.419269\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.37018e7i 0.507456i
\(186\) 0 0
\(187\) 4.11229e7i 0.459873i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.76797e8 1.83593 0.917967 0.396657i \(-0.129830\pi\)
0.917967 + 0.396657i \(0.129830\pi\)
\(192\) 0 0
\(193\) 3.32665e7 0.333087 0.166543 0.986034i \(-0.446739\pi\)
0.166543 + 0.986034i \(0.446739\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.11714e7i 0.756434i 0.925717 + 0.378217i \(0.123463\pi\)
−0.925717 + 0.378217i \(0.876537\pi\)
\(198\) 0 0
\(199\) − 1.43040e8i − 1.28668i −0.765580 0.643341i \(-0.777548\pi\)
0.765580 0.643341i \(-0.222452\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −5.06237e7 −0.424735
\(204\) 0 0
\(205\) 5.39223e7 0.437149
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 1.60072e8i 1.21284i
\(210\) 0 0
\(211\) 1.51433e8i 1.10977i 0.831928 + 0.554884i \(0.187237\pi\)
−0.831928 + 0.554884i \(0.812763\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3.21816e8 2.20837
\(216\) 0 0
\(217\) 2.59504e6 0.0172399
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 8.98726e6i 0.0560086i
\(222\) 0 0
\(223\) 2.36793e8i 1.42989i 0.699181 + 0.714945i \(0.253548\pi\)
−0.699181 + 0.714945i \(0.746452\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1.75301e8 −0.994705 −0.497353 0.867548i \(-0.665694\pi\)
−0.497353 + 0.867548i \(0.665694\pi\)
\(228\) 0 0
\(229\) −3.08750e8 −1.69896 −0.849480 0.527621i \(-0.823084\pi\)
−0.849480 + 0.527621i \(0.823084\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 1.91707e8i − 0.992869i −0.868074 0.496435i \(-0.834642\pi\)
0.868074 0.496435i \(-0.165358\pi\)
\(234\) 0 0
\(235\) − 3.58072e8i − 1.79984i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.13941e8 0.539868 0.269934 0.962879i \(-0.412998\pi\)
0.269934 + 0.962879i \(0.412998\pi\)
\(240\) 0 0
\(241\) 1.68043e8 0.773323 0.386662 0.922222i \(-0.373628\pi\)
0.386662 + 0.922222i \(0.373628\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 3.48893e8i − 1.51569i
\(246\) 0 0
\(247\) 3.49831e7i 0.147713i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −3.42282e8 −1.36624 −0.683118 0.730308i \(-0.739377\pi\)
−0.683118 + 0.730308i \(0.739377\pi\)
\(252\) 0 0
\(253\) 4.21803e8 1.63752
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.19015e8i 0.437355i 0.975797 + 0.218678i \(0.0701743\pi\)
−0.975797 + 0.218678i \(0.929826\pi\)
\(258\) 0 0
\(259\) − 1.98008e7i − 0.0708163i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3.17137e8 −1.07498 −0.537492 0.843269i \(-0.680628\pi\)
−0.537492 + 0.843269i \(0.680628\pi\)
\(264\) 0 0
\(265\) −4.28854e8 −1.41563
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 9.03521e7i 0.283012i 0.989937 + 0.141506i \(0.0451945\pi\)
−0.989937 + 0.141506i \(0.954805\pi\)
\(270\) 0 0
\(271\) 1.29807e8i 0.396191i 0.980183 + 0.198095i \(0.0634756\pi\)
−0.980183 + 0.198095i \(0.936524\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −6.47035e8 −1.87613
\(276\) 0 0
\(277\) −5.01224e8 −1.41694 −0.708472 0.705739i \(-0.750615\pi\)
−0.708472 + 0.705739i \(0.750615\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 5.39332e8i − 1.45005i −0.688720 0.725027i \(-0.741827\pi\)
0.688720 0.725027i \(-0.258173\pi\)
\(282\) 0 0
\(283\) − 2.61973e8i − 0.687076i −0.939139 0.343538i \(-0.888375\pi\)
0.939139 0.343538i \(-0.111625\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.44316e7 −0.0610049
\(288\) 0 0
\(289\) 3.51644e8 0.856961
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 4.42675e8i − 1.02813i −0.857751 0.514065i \(-0.828139\pi\)
0.857751 0.514065i \(-0.171861\pi\)
\(294\) 0 0
\(295\) 1.37753e9i 3.12408i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 9.21834e7 0.199436
\(300\) 0 0
\(301\) −1.45811e8 −0.308182
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 3.45869e8i − 0.698011i
\(306\) 0 0
\(307\) − 6.00549e8i − 1.18458i −0.805725 0.592290i \(-0.798224\pi\)
0.805725 0.592290i \(-0.201776\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 7.60256e8 1.43317 0.716586 0.697499i \(-0.245704\pi\)
0.716586 + 0.697499i \(0.245704\pi\)
\(312\) 0 0
\(313\) −9.79664e8 −1.80581 −0.902905 0.429840i \(-0.858570\pi\)
−0.902905 + 0.429840i \(0.858570\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 5.73197e8i 1.01064i 0.862932 + 0.505320i \(0.168626\pi\)
−0.862932 + 0.505320i \(0.831374\pi\)
\(318\) 0 0
\(319\) 1.34553e9i 2.32073i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −2.28469e8 −0.377241
\(324\) 0 0
\(325\) −1.41407e8 −0.228496
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.62239e8i 0.251170i
\(330\) 0 0
\(331\) 6.30547e8i 0.955695i 0.878443 + 0.477847i \(0.158583\pi\)
−0.878443 + 0.477847i \(0.841417\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −1.12606e8 −0.163646
\(336\) 0 0
\(337\) −8.22055e8 −1.17003 −0.585014 0.811023i \(-0.698911\pi\)
−0.585014 + 0.811023i \(0.698911\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 6.89735e7i − 0.0941981i
\(342\) 0 0
\(343\) 3.24395e8i 0.434055i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.29721e9 −1.66670 −0.833351 0.552744i \(-0.813581\pi\)
−0.833351 + 0.552744i \(0.813581\pi\)
\(348\) 0 0
\(349\) −1.22063e9 −1.53707 −0.768536 0.639807i \(-0.779014\pi\)
−0.768536 + 0.639807i \(0.779014\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3.98374e8i 0.482036i 0.970521 + 0.241018i \(0.0774813\pi\)
−0.970521 + 0.241018i \(0.922519\pi\)
\(354\) 0 0
\(355\) − 7.26950e7i − 0.0862393i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3.74326e8 0.426992 0.213496 0.976944i \(-0.431515\pi\)
0.213496 + 0.976944i \(0.431515\pi\)
\(360\) 0 0
\(361\) 4.55145e6 0.00509184
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 1.06642e9i − 1.14790i
\(366\) 0 0
\(367\) − 3.89260e8i − 0.411063i −0.978650 0.205532i \(-0.934108\pi\)
0.978650 0.205532i \(-0.0658924\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.94309e8 0.197553
\(372\) 0 0
\(373\) −2.20587e8 −0.220089 −0.110044 0.993927i \(-0.535099\pi\)
−0.110044 + 0.993927i \(0.535099\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.94060e8i 0.282645i
\(378\) 0 0
\(379\) 3.58994e8i 0.338727i 0.985554 + 0.169364i \(0.0541712\pi\)
−0.985554 + 0.169364i \(0.945829\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.11450e9 1.01364 0.506822 0.862051i \(-0.330820\pi\)
0.506822 + 0.862051i \(0.330820\pi\)
\(384\) 0 0
\(385\) 4.83166e8 0.431503
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 7.82474e8i − 0.673979i −0.941508 0.336989i \(-0.890591\pi\)
0.941508 0.336989i \(-0.109409\pi\)
\(390\) 0 0
\(391\) 6.02036e8i 0.509336i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2.84751e9 −2.32475
\(396\) 0 0
\(397\) 1.16495e8 0.0934415 0.0467207 0.998908i \(-0.485123\pi\)
0.0467207 + 0.998908i \(0.485123\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 1.72842e9i − 1.33858i −0.743000 0.669291i \(-0.766598\pi\)
0.743000 0.669291i \(-0.233402\pi\)
\(402\) 0 0
\(403\) − 1.50739e7i − 0.0114725i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −5.26284e8 −0.386937
\(408\) 0 0
\(409\) 1.63857e9 1.18422 0.592111 0.805857i \(-0.298295\pi\)
0.592111 + 0.805857i \(0.298295\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 6.24141e8i − 0.435971i
\(414\) 0 0
\(415\) − 1.35497e9i − 0.930600i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.04990e9 −1.36139 −0.680697 0.732565i \(-0.738323\pi\)
−0.680697 + 0.732565i \(0.738323\pi\)
\(420\) 0 0
\(421\) 1.85898e9 1.21420 0.607098 0.794627i \(-0.292334\pi\)
0.607098 + 0.794627i \(0.292334\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 9.23509e8i − 0.583552i
\(426\) 0 0
\(427\) 1.56709e8i 0.0974085i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 2.25651e9 1.35758 0.678792 0.734330i \(-0.262504\pi\)
0.678792 + 0.734330i \(0.262504\pi\)
\(432\) 0 0
\(433\) −1.19933e9 −0.709954 −0.354977 0.934875i \(-0.615511\pi\)
−0.354977 + 0.934875i \(0.615511\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.34344e9i 1.34329i
\(438\) 0 0
\(439\) − 2.26073e9i − 1.27533i −0.770312 0.637667i \(-0.779900\pi\)
0.770312 0.637667i \(-0.220100\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.14557e9 1.17255 0.586274 0.810113i \(-0.300594\pi\)
0.586274 + 0.810113i \(0.300594\pi\)
\(444\) 0 0
\(445\) 1.08286e9 0.582521
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 8.44899e8i 0.440496i 0.975444 + 0.220248i \(0.0706867\pi\)
−0.975444 + 0.220248i \(0.929313\pi\)
\(450\) 0 0
\(451\) 6.49365e8i 0.333328i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.05594e8 0.0525533
\(456\) 0 0
\(457\) −8.46931e8 −0.415089 −0.207544 0.978226i \(-0.566547\pi\)
−0.207544 + 0.978226i \(0.566547\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.29935e9i 0.617694i 0.951112 + 0.308847i \(0.0999432\pi\)
−0.951112 + 0.308847i \(0.900057\pi\)
\(462\) 0 0
\(463\) − 5.18884e8i − 0.242961i −0.992594 0.121481i \(-0.961236\pi\)
0.992594 0.121481i \(-0.0387642\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.69058e9 0.768118 0.384059 0.923309i \(-0.374526\pi\)
0.384059 + 0.923309i \(0.374526\pi\)
\(468\) 0 0
\(469\) 5.10206e7 0.0228371
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 3.87551e9i 1.68389i
\(474\) 0 0
\(475\) − 3.59477e9i − 1.53902i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −1.84920e9 −0.768792 −0.384396 0.923168i \(-0.625590\pi\)
−0.384396 + 0.923168i \(0.625590\pi\)
\(480\) 0 0
\(481\) −1.15018e8 −0.0471255
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 9.46724e8i 0.376814i
\(486\) 0 0
\(487\) 3.02324e9i 1.18610i 0.805166 + 0.593050i \(0.202076\pi\)
−0.805166 + 0.593050i \(0.797924\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.78758e9 1.06278 0.531388 0.847129i \(-0.321671\pi\)
0.531388 + 0.847129i \(0.321671\pi\)
\(492\) 0 0
\(493\) −1.92046e9 −0.721840
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.29373e7i 0.0120348i
\(498\) 0 0
\(499\) − 1.09189e9i − 0.393392i −0.980465 0.196696i \(-0.936979\pi\)
0.980465 0.196696i \(-0.0630212\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 3.91716e9 1.37241 0.686204 0.727410i \(-0.259276\pi\)
0.686204 + 0.727410i \(0.259276\pi\)
\(504\) 0 0
\(505\) 5.34220e9 1.84587
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1.25898e9i 0.423163i 0.977360 + 0.211581i \(0.0678613\pi\)
−0.977360 + 0.211581i \(0.932139\pi\)
\(510\) 0 0
\(511\) 4.83184e8i 0.160191i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 6.01501e9 1.94049
\(516\) 0 0
\(517\) 4.31213e9 1.37238
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 4.11117e9i 1.27360i 0.771029 + 0.636800i \(0.219742\pi\)
−0.771029 + 0.636800i \(0.780258\pi\)
\(522\) 0 0
\(523\) 2.91561e6i 0 0.000891197i 1.00000 0.000445599i \(0.000141838\pi\)
−1.00000 0.000445599i \(0.999858\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9.84455e7 0.0292994
\(528\) 0 0
\(529\) 2.77034e9 0.813650
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.41916e8i 0.0405964i
\(534\) 0 0
\(535\) − 6.74107e9i − 1.90323i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.20158e9 1.15572
\(540\) 0 0
\(541\) −4.88925e9 −1.32755 −0.663777 0.747931i \(-0.731048\pi\)
−0.663777 + 0.747931i \(0.731048\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 6.67833e9i − 1.76718i
\(546\) 0 0
\(547\) − 3.00310e9i − 0.784539i −0.919850 0.392270i \(-0.871690\pi\)
0.919850 0.392270i \(-0.128310\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −7.47542e9 −1.90373
\(552\) 0 0
\(553\) 1.29018e9 0.324422
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.50407e8i 0.0368786i 0.999830 + 0.0184393i \(0.00586974\pi\)
−0.999830 + 0.0184393i \(0.994130\pi\)
\(558\) 0 0
\(559\) 8.46978e8i 0.205083i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −4.06608e9 −0.960276 −0.480138 0.877193i \(-0.659413\pi\)
−0.480138 + 0.877193i \(0.659413\pi\)
\(564\) 0 0
\(565\) 1.15033e10 2.68320
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 7.52046e9i 1.71140i 0.517472 + 0.855700i \(0.326873\pi\)
−0.517472 + 0.855700i \(0.673127\pi\)
\(570\) 0 0
\(571\) − 3.43788e9i − 0.772795i −0.922332 0.386398i \(-0.873719\pi\)
0.922332 0.386398i \(-0.126281\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −9.47254e9 −2.07792
\(576\) 0 0
\(577\) −3.81016e9 −0.825711 −0.412855 0.910797i \(-0.635469\pi\)
−0.412855 + 0.910797i \(0.635469\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 6.13923e8i 0.129867i
\(582\) 0 0
\(583\) − 5.16453e9i − 1.07942i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7.98289e9 1.62902 0.814511 0.580149i \(-0.197006\pi\)
0.814511 + 0.580149i \(0.197006\pi\)
\(588\) 0 0
\(589\) 3.83201e8 0.0772721
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 1.09226e9i − 0.215096i −0.994200 0.107548i \(-0.965700\pi\)
0.994200 0.107548i \(-0.0343000\pi\)
\(594\) 0 0
\(595\) 6.89620e8i 0.134215i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.96369e9 0.373318 0.186659 0.982425i \(-0.440234\pi\)
0.186659 + 0.982425i \(0.440234\pi\)
\(600\) 0 0
\(601\) 7.03584e9 1.32207 0.661036 0.750354i \(-0.270117\pi\)
0.661036 + 0.750354i \(0.270117\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 4.15619e9i − 0.763048i
\(606\) 0 0
\(607\) − 3.97206e9i − 0.720868i −0.932785 0.360434i \(-0.882629\pi\)
0.932785 0.360434i \(-0.117371\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.42401e8 0.167144
\(612\) 0 0
\(613\) −1.11811e9 −0.196054 −0.0980268 0.995184i \(-0.531253\pi\)
−0.0980268 + 0.995184i \(0.531253\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 7.93863e9i − 1.36065i −0.732909 0.680327i \(-0.761838\pi\)
0.732909 0.680327i \(-0.238162\pi\)
\(618\) 0 0
\(619\) 5.53963e9i 0.938779i 0.882991 + 0.469389i \(0.155526\pi\)
−0.882991 + 0.469389i \(0.844474\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −4.90631e8 −0.0812917
\(624\) 0 0
\(625\) −9.90310e8 −0.162252
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 7.51163e8i − 0.120353i
\(630\) 0 0
\(631\) − 2.90603e9i − 0.460466i −0.973136 0.230233i \(-0.926051\pi\)
0.973136 0.230233i \(-0.0739489\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5.80215e9 −0.899251
\(636\) 0 0
\(637\) 9.18241e8 0.140757
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.87322e9i 0.430889i 0.976516 + 0.215444i \(0.0691200\pi\)
−0.976516 + 0.215444i \(0.930880\pi\)
\(642\) 0 0
\(643\) 1.04113e10i 1.54443i 0.635364 + 0.772213i \(0.280850\pi\)
−0.635364 + 0.772213i \(0.719150\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 4.77707e9 0.693420 0.346710 0.937972i \(-0.387299\pi\)
0.346710 + 0.937972i \(0.387299\pi\)
\(648\) 0 0
\(649\) −1.65890e10 −2.38212
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 5.13626e9i 0.721856i 0.932594 + 0.360928i \(0.117540\pi\)
−0.932594 + 0.360928i \(0.882460\pi\)
\(654\) 0 0
\(655\) − 7.09253e9i − 0.986181i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.18123e10 1.60781 0.803906 0.594757i \(-0.202752\pi\)
0.803906 + 0.594757i \(0.202752\pi\)
\(660\) 0 0
\(661\) −6.45411e9 −0.869223 −0.434612 0.900618i \(-0.643114\pi\)
−0.434612 + 0.900618i \(0.643114\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.68436e9i 0.353968i
\(666\) 0 0
\(667\) 1.96984e10i 2.57034i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.16517e9 0.532235
\(672\) 0 0
\(673\) 7.48253e9 0.946228 0.473114 0.881001i \(-0.343130\pi\)
0.473114 + 0.881001i \(0.343130\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 3.62586e9i 0.449108i 0.974462 + 0.224554i \(0.0720925\pi\)
−0.974462 + 0.224554i \(0.927908\pi\)
\(678\) 0 0
\(679\) − 4.28950e8i − 0.0525850i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 4.06326e9 0.487980 0.243990 0.969778i \(-0.421544\pi\)
0.243990 + 0.969778i \(0.421544\pi\)
\(684\) 0 0
\(685\) −5.44853e9 −0.647683
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 1.12869e9i − 0.131464i
\(690\) 0 0
\(691\) 1.03231e10i 1.19025i 0.803635 + 0.595123i \(0.202897\pi\)
−0.803635 + 0.595123i \(0.797103\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −2.39135e10 −2.70207
\(696\) 0 0
\(697\) −9.26835e8 −0.103678
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 8.70619e9i 0.954586i 0.878744 + 0.477293i \(0.158382\pi\)
−0.878744 + 0.477293i \(0.841618\pi\)
\(702\) 0 0
\(703\) − 2.92391e9i − 0.317410i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.42049e9 −0.257594
\(708\) 0 0
\(709\) −1.36353e10 −1.43682 −0.718410 0.695620i \(-0.755130\pi\)
−0.718410 + 0.695620i \(0.755130\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 1.00977e9i − 0.104330i
\(714\) 0 0
\(715\) − 2.80658e9i − 0.287149i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.82007e9 0.182615 0.0913075 0.995823i \(-0.470895\pi\)
0.0913075 + 0.995823i \(0.470895\pi\)
\(720\) 0 0
\(721\) −2.72533e9 −0.270799
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 3.02168e10i − 2.94487i
\(726\) 0 0
\(727\) 1.04415e10i 1.00784i 0.863750 + 0.503920i \(0.168109\pi\)
−0.863750 + 0.503920i \(0.831891\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −5.53149e9 −0.523758
\(732\) 0 0
\(733\) −7.80483e9 −0.731981 −0.365990 0.930619i \(-0.619270\pi\)
−0.365990 + 0.930619i \(0.619270\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 1.35608e9i − 0.124781i
\(738\) 0 0
\(739\) 3.97888e9i 0.362664i 0.983422 + 0.181332i \(0.0580409\pi\)
−0.983422 + 0.181332i \(0.941959\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.20404e10 −1.07691 −0.538455 0.842654i \(-0.680992\pi\)
−0.538455 + 0.842654i \(0.680992\pi\)
\(744\) 0 0
\(745\) −1.90322e10 −1.68633
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 3.05430e9i 0.265598i
\(750\) 0 0
\(751\) 4.23550e9i 0.364893i 0.983216 + 0.182446i \(0.0584016\pi\)
−0.983216 + 0.182446i \(0.941598\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −4.33663e9 −0.366723
\(756\) 0 0
\(757\) 8.69080e9 0.728155 0.364078 0.931369i \(-0.381384\pi\)
0.364078 + 0.931369i \(0.381384\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 7.08162e9i 0.582487i 0.956649 + 0.291243i \(0.0940690\pi\)
−0.956649 + 0.291243i \(0.905931\pi\)
\(762\) 0 0
\(763\) 3.02587e9i 0.246612i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −3.62547e9 −0.290122
\(768\) 0 0
\(769\) 1.91172e10 1.51594 0.757969 0.652290i \(-0.226192\pi\)
0.757969 + 0.652290i \(0.226192\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 1.56030e10i − 1.21501i −0.794316 0.607505i \(-0.792170\pi\)
0.794316 0.607505i \(-0.207830\pi\)
\(774\) 0 0
\(775\) 1.54896e9i 0.119532i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3.60772e9 −0.273434
\(780\) 0 0
\(781\) 8.75438e8 0.0657578
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 3.89705e10i − 2.87536i
\(786\) 0 0
\(787\) − 2.59880e9i − 0.190047i −0.995475 0.0950234i \(-0.969707\pi\)
0.995475 0.0950234i \(-0.0302926\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −5.21201e9 −0.374445
\(792\) 0 0
\(793\) 9.10282e8 0.0648216
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.40901e8i 0.0168552i 0.999964 + 0.00842760i \(0.00268262\pi\)
−0.999964 + 0.00842760i \(0.997317\pi\)
\(798\) 0 0
\(799\) 6.15468e9i 0.426866i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1.28425e10 0.875278
\(804\) 0 0
\(805\) 7.07352e9 0.477914
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 1.61647e10i − 1.07337i −0.843783 0.536684i \(-0.819677\pi\)
0.843783 0.536684i \(-0.180323\pi\)
\(810\) 0 0
\(811\) − 3.42220e9i − 0.225285i −0.993636 0.112642i \(-0.964069\pi\)
0.993636 0.112642i \(-0.0359315\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −4.78302e10 −3.09493
\(816\) 0 0
\(817\) −2.15314e10 −1.38132
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 3.94367e9i 0.248714i 0.992238 + 0.124357i \(0.0396868\pi\)
−0.992238 + 0.124357i \(0.960313\pi\)
\(822\) 0 0
\(823\) 2.27646e10i 1.42351i 0.702427 + 0.711755i \(0.252100\pi\)
−0.702427 + 0.711755i \(0.747900\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 5.20469e9 0.319982 0.159991 0.987118i \(-0.448853\pi\)
0.159991 + 0.987118i \(0.448853\pi\)
\(828\) 0 0
\(829\) 6.42188e9 0.391491 0.195745 0.980655i \(-0.437287\pi\)
0.195745 + 0.980655i \(0.437287\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 5.99690e9i 0.359475i
\(834\) 0 0
\(835\) − 2.58026e10i − 1.53377i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.63779e10 0.957396 0.478698 0.877980i \(-0.341109\pi\)
0.478698 + 0.877980i \(0.341109\pi\)
\(840\) 0 0
\(841\) −4.55868e10 −2.64273
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 2.73550e10i 1.55969i
\(846\) 0 0
\(847\) 1.88312e9i 0.106485i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −7.70477e9 −0.428555
\(852\) 0 0
\(853\) −2.95535e10 −1.63037 −0.815187 0.579198i \(-0.803366\pi\)
−0.815187 + 0.579198i \(0.803366\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 3.36480e10i 1.82611i 0.407840 + 0.913054i \(0.366282\pi\)
−0.407840 + 0.913054i \(0.633718\pi\)
\(858\) 0 0
\(859\) 1.03722e10i 0.558336i 0.960242 + 0.279168i \(0.0900586\pi\)
−0.960242 + 0.279168i \(0.909941\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 2.13251e10 1.12942 0.564708 0.825291i \(-0.308989\pi\)
0.564708 + 0.825291i \(0.308989\pi\)
\(864\) 0 0
\(865\) 4.80414e10 2.52382
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 3.42915e10i − 1.77263i
\(870\) 0 0
\(871\) − 2.96365e8i − 0.0151972i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.81823e9 −0.192679
\(876\) 0 0
\(877\) −7.22632e9 −0.361759 −0.180879 0.983505i \(-0.557894\pi\)
−0.180879 + 0.983505i \(0.557894\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 2.44432e10i 1.20432i 0.798376 + 0.602159i \(0.205693\pi\)
−0.798376 + 0.602159i \(0.794307\pi\)
\(882\) 0 0
\(883\) − 4.58740e9i − 0.224235i −0.993695 0.112118i \(-0.964237\pi\)
0.993695 0.112118i \(-0.0357633\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.97698e10 0.951196 0.475598 0.879663i \(-0.342232\pi\)
0.475598 + 0.879663i \(0.342232\pi\)
\(888\) 0 0
\(889\) 2.62889e9 0.125492
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2.39572e10i 1.12579i
\(894\) 0 0
\(895\) 2.07508e10i 0.967508i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3.22110e9 0.147858
\(900\) 0 0
\(901\) 7.37131e9 0.335744
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 1.78441e10i − 0.800247i
\(906\) 0 0
\(907\) 1.22458e10i 0.544957i 0.962162 + 0.272479i \(0.0878434\pi\)
−0.962162 + 0.272479i \(0.912157\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 3.56276e10 1.56125 0.780625 0.625000i \(-0.214901\pi\)
0.780625 + 0.625000i \(0.214901\pi\)
\(912\) 0 0
\(913\) 1.63175e10 0.709585
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3.21354e9i 0.137623i
\(918\) 0 0
\(919\) − 1.02923e10i − 0.437430i −0.975789 0.218715i \(-0.929814\pi\)
0.975789 0.218715i \(-0.0701865\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.91324e8 0.00800872
\(924\) 0 0
\(925\) 1.18189e10 0.491000
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.80492e10i 0.738590i 0.929312 + 0.369295i \(0.120401\pi\)
−0.929312 + 0.369295i \(0.879599\pi\)
\(930\) 0 0
\(931\) 2.33430e10i 0.948054i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 1.83294e10 0.733343
\(936\) 0 0
\(937\) −2.03829e10 −0.809425 −0.404712 0.914444i \(-0.632628\pi\)
−0.404712 + 0.914444i \(0.632628\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 4.71378e8i − 0.0184419i −0.999957 0.00922094i \(-0.997065\pi\)
0.999957 0.00922094i \(-0.00293516\pi\)
\(942\) 0 0
\(943\) 9.50666e9i 0.369179i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.91127e10 −0.731302 −0.365651 0.930752i \(-0.619154\pi\)
−0.365651 + 0.930752i \(0.619154\pi\)
\(948\) 0 0
\(949\) 2.80669e9 0.106601
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 2.26355e10i 0.847159i 0.905859 + 0.423580i \(0.139227\pi\)
−0.905859 + 0.423580i \(0.860773\pi\)
\(954\) 0 0
\(955\) − 7.88021e10i − 2.92770i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2.46867e9 0.0903852
\(960\) 0 0
\(961\) 2.73475e10 0.993998
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 1.48276e10i − 0.531161i
\(966\) 0 0
\(967\) − 5.25242e10i − 1.86796i −0.357328 0.933979i \(-0.616312\pi\)
0.357328 0.933979i \(-0.383688\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 4.09805e9 0.143651 0.0718257 0.997417i \(-0.477117\pi\)
0.0718257 + 0.997417i \(0.477117\pi\)
\(972\) 0 0
\(973\) 1.08349e10 0.377078
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 5.11726e10i − 1.75552i −0.479097 0.877762i \(-0.659036\pi\)
0.479097 0.877762i \(-0.340964\pi\)
\(978\) 0 0
\(979\) 1.30405e10i 0.444174i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 3.41382e10 1.14631 0.573157 0.819446i \(-0.305719\pi\)
0.573157 + 0.819446i \(0.305719\pi\)
\(984\) 0 0
\(985\) 3.61799e10 1.20626
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 5.67371e10i 1.86501i
\(990\) 0 0
\(991\) − 5.50572e10i − 1.79703i −0.438938 0.898517i \(-0.644645\pi\)
0.438938 0.898517i \(-0.355355\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −6.37560e10 −2.05182
\(996\) 0 0
\(997\) 4.48885e10 1.43450 0.717252 0.696814i \(-0.245400\pi\)
0.717252 + 0.696814i \(0.245400\pi\)
\(998\) 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 288.8.c.b.287.2 yes 16
3.2 odd 2 inner 288.8.c.b.287.16 yes 16
4.3 odd 2 inner 288.8.c.b.287.1 16
8.3 odd 2 576.8.c.g.575.15 16
8.5 even 2 576.8.c.g.575.16 16
12.11 even 2 inner 288.8.c.b.287.15 yes 16
24.5 odd 2 576.8.c.g.575.2 16
24.11 even 2 576.8.c.g.575.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.8.c.b.287.1 16 4.3 odd 2 inner
288.8.c.b.287.2 yes 16 1.1 even 1 trivial
288.8.c.b.287.15 yes 16 12.11 even 2 inner
288.8.c.b.287.16 yes 16 3.2 odd 2 inner
576.8.c.g.575.1 16 24.11 even 2
576.8.c.g.575.2 16 24.5 odd 2
576.8.c.g.575.15 16 8.3 odd 2
576.8.c.g.575.16 16 8.5 even 2