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Properties

Label 252.8.a.a.1.1
Level $252$
Weight $8$
Character 252.1
Self dual yes
Analytic conductor $78.721$
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] = [252,8,Mod(1,252)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("252.1"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(252, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 252 = 2^{2} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 252.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,0,0,-100,0,-343] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(78.7210264220\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 252.1

$q$-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)
 
\(f(q)\) \(=\) \(q-100.000 q^{5} -343.000 q^{7} -2774.00 q^{11} -3294.00 q^{13} -5900.00 q^{17} +6644.00 q^{19} -1982.00 q^{23} -68125.0 q^{25} +208106. q^{29} -117792. q^{31} +34300.0 q^{35} -335686. q^{37} +265488. q^{41} -93292.0 q^{43} +657516. q^{47} +117649. q^{49} +608718. q^{53} +277400. q^{55} +536120. q^{59} -1.79709e6 q^{61} +329400. q^{65} +2.12318e6 q^{67} +1.19121e6 q^{71} +1.05643e6 q^{73} +951482. q^{77} +998484. q^{79} -3.89800e6 q^{83} +590000. q^{85} +4.62235e6 q^{89} +1.12984e6 q^{91} -664400. q^{95} +1.52877e7 q^{97} +O(q^{100})\)

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
\(5\) −100.000 −0.357771 −0.178885 − 0.983870i \(-0.557249\pi\)
−0.178885 + 0.983870i \(0.557249\pi\)
\(6\) 0 0
\(7\) −343.000 −0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2774.00 −0.628394 −0.314197 − 0.949358i \(-0.601735\pi\)
−0.314197 + 0.949358i \(0.601735\pi\)
\(12\) 0 0
\(13\) −3294.00 −0.415836 −0.207918 − 0.978146i \(-0.566669\pi\)
−0.207918 + 0.978146i \(0.566669\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −5900.00 −0.291260 −0.145630 − 0.989339i \(-0.546521\pi\)
−0.145630 + 0.989339i \(0.546521\pi\)
\(18\) 0 0
\(19\) 6644.00 0.222225 0.111112 − 0.993808i \(-0.464559\pi\)
0.111112 + 0.993808i \(0.464559\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −1982.00 −0.0339669 −0.0169835 − 0.999856i \(-0.505406\pi\)
−0.0169835 + 0.999856i \(0.505406\pi\)
\(24\) 0 0
\(25\) −68125.0 −0.872000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 208106. 1.58450 0.792249 − 0.610198i \(-0.208910\pi\)
0.792249 + 0.610198i \(0.208910\pi\)
\(30\) 0 0
\(31\) −117792. −0.710150 −0.355075 − 0.934838i \(-0.615545\pi\)
−0.355075 + 0.934838i \(0.615545\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 34300.0 0.135225
\(36\) 0 0
\(37\) −335686. −1.08950 −0.544750 − 0.838599i \(-0.683375\pi\)
−0.544750 + 0.838599i \(0.683375\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 265488. 0.601591 0.300796 − 0.953689i \(-0.402748\pi\)
0.300796 + 0.953689i \(0.402748\pi\)
\(42\) 0 0
\(43\) −93292.0 −0.178939 −0.0894695 − 0.995990i \(-0.528517\pi\)
−0.0894695 + 0.995990i \(0.528517\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 657516. 0.923770 0.461885 − 0.886940i \(-0.347173\pi\)
0.461885 + 0.886940i \(0.347173\pi\)
\(48\) 0 0
\(49\) 117649. 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 608718. 0.561630 0.280815 − 0.959762i \(-0.409395\pi\)
0.280815 + 0.959762i \(0.409395\pi\)
\(54\) 0 0
\(55\) 277400. 0.224821
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 536120. 0.339844 0.169922 − 0.985457i \(-0.445648\pi\)
0.169922 + 0.985457i \(0.445648\pi\)
\(60\) 0 0
\(61\) −1.79709e6 −1.01371 −0.506857 − 0.862030i \(-0.669193\pi\)
−0.506857 + 0.862030i \(0.669193\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 329400. 0.148774
\(66\) 0 0
\(67\) 2.12318e6 0.862431 0.431215 − 0.902249i \(-0.358085\pi\)
0.431215 + 0.902249i \(0.358085\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.19121e6 0.394990 0.197495 − 0.980304i \(-0.436719\pi\)
0.197495 + 0.980304i \(0.436719\pi\)
\(72\) 0 0
\(73\) 1.05643e6 0.317842 0.158921 − 0.987291i \(-0.449199\pi\)
0.158921 + 0.987291i \(0.449199\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 951482. 0.237511
\(78\) 0 0
\(79\) 998484. 0.227849 0.113924 − 0.993489i \(-0.463658\pi\)
0.113924 + 0.993489i \(0.463658\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.89800e6 −0.748288 −0.374144 − 0.927371i \(-0.622063\pi\)
−0.374144 + 0.927371i \(0.622063\pi\)
\(84\) 0 0
\(85\) 590000. 0.104204
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 4.62235e6 0.695021 0.347511 − 0.937676i \(-0.387027\pi\)
0.347511 + 0.937676i \(0.387027\pi\)
\(90\) 0 0
\(91\) 1.12984e6 0.157171
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −664400. −0.0795055
\(96\) 0 0
\(97\) 1.52877e7 1.70075 0.850377 − 0.526174i \(-0.176374\pi\)
0.850377 + 0.526174i \(0.176374\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −2.23869e6 −0.216207 −0.108103 − 0.994140i \(-0.534478\pi\)
−0.108103 + 0.994140i \(0.534478\pi\)
\(102\) 0 0
\(103\) 1.24502e7 1.12266 0.561328 − 0.827593i \(-0.310291\pi\)
0.561328 + 0.827593i \(0.310291\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 7.58860e6 0.598851 0.299425 − 0.954120i \(-0.403205\pi\)
0.299425 + 0.954120i \(0.403205\pi\)
\(108\) 0 0
\(109\) 5.17097e6 0.382454 0.191227 − 0.981546i \(-0.438753\pi\)
0.191227 + 0.981546i \(0.438753\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 9.63868e6 0.628410 0.314205 − 0.949355i \(-0.398262\pi\)
0.314205 + 0.949355i \(0.398262\pi\)
\(114\) 0 0
\(115\) 198200. 0.0121524
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.02370e6 0.110086
\(120\) 0 0
\(121\) −1.17921e7 −0.605121
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.46250e7 0.669747
\(126\) 0 0
\(127\) 8.08309e6 0.350158 0.175079 − 0.984554i \(-0.443982\pi\)
0.175079 + 0.984554i \(0.443982\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 1.97585e7 0.767901 0.383951 − 0.923354i \(-0.374563\pi\)
0.383951 + 0.923354i \(0.374563\pi\)
\(132\) 0 0
\(133\) −2.27889e6 −0.0839930
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.80896e7 1.59783 0.798913 − 0.601447i \(-0.205409\pi\)
0.798913 + 0.601447i \(0.205409\pi\)
\(138\) 0 0
\(139\) 1.37173e7 0.433229 0.216615 − 0.976257i \(-0.430499\pi\)
0.216615 + 0.976257i \(0.430499\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 9.13756e6 0.261309
\(144\) 0 0
\(145\) −2.08106e7 −0.566887
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 7.07104e7 1.75118 0.875591 − 0.483053i \(-0.160472\pi\)
0.875591 + 0.483053i \(0.160472\pi\)
\(150\) 0 0
\(151\) −2.56970e7 −0.607383 −0.303691 − 0.952770i \(-0.598219\pi\)
−0.303691 + 0.952770i \(0.598219\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.17792e7 0.254071
\(156\) 0 0
\(157\) 2.09180e7 0.431390 0.215695 − 0.976461i \(-0.430798\pi\)
0.215695 + 0.976461i \(0.430798\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 679826. 0.0128383
\(162\) 0 0
\(163\) 5.44160e7 0.984169 0.492084 − 0.870547i \(-0.336235\pi\)
0.492084 + 0.870547i \(0.336235\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −6.41888e6 −0.106648 −0.0533238 − 0.998577i \(-0.516982\pi\)
−0.0533238 + 0.998577i \(0.516982\pi\)
\(168\) 0 0
\(169\) −5.18981e7 −0.827081
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 5.46532e7 0.802517 0.401259 − 0.915965i \(-0.368573\pi\)
0.401259 + 0.915965i \(0.368573\pi\)
\(174\) 0 0
\(175\) 2.33669e7 0.329585
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 3.76658e7 0.490865 0.245433 − 0.969414i \(-0.421070\pi\)
0.245433 + 0.969414i \(0.421070\pi\)
\(180\) 0 0
\(181\) 1.76788e7 0.221604 0.110802 − 0.993842i \(-0.464658\pi\)
0.110802 + 0.993842i \(0.464658\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 3.35686e7 0.389791
\(186\) 0 0
\(187\) 1.63666e7 0.183026
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4.09177e7 −0.424907 −0.212454 − 0.977171i \(-0.568145\pi\)
−0.212454 + 0.977171i \(0.568145\pi\)
\(192\) 0 0
\(193\) −1.63827e8 −1.64034 −0.820171 − 0.572118i \(-0.806122\pi\)
−0.820171 + 0.572118i \(0.806122\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.02076e7 −0.747453 −0.373726 − 0.927539i \(-0.621920\pi\)
−0.373726 + 0.927539i \(0.621920\pi\)
\(198\) 0 0
\(199\) 9.83219e7 0.884432 0.442216 − 0.896908i \(-0.354192\pi\)
0.442216 + 0.896908i \(0.354192\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −7.13804e7 −0.598884
\(204\) 0 0
\(205\) −2.65488e7 −0.215232
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −1.84305e7 −0.139645
\(210\) 0 0
\(211\) 1.36321e8 0.999021 0.499510 − 0.866308i \(-0.333513\pi\)
0.499510 + 0.866308i \(0.333513\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 9.32920e6 0.0640191
\(216\) 0 0
\(217\) 4.04027e7 0.268411
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.94346e7 0.121116
\(222\) 0 0
\(223\) 1.26358e8 0.763019 0.381510 − 0.924365i \(-0.375404\pi\)
0.381510 + 0.924365i \(0.375404\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.34007e8 −1.32782 −0.663909 − 0.747814i \(-0.731104\pi\)
−0.663909 + 0.747814i \(0.731104\pi\)
\(228\) 0 0
\(229\) 6.83606e7 0.376168 0.188084 − 0.982153i \(-0.439772\pi\)
0.188084 + 0.982153i \(0.439772\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.80385e8 −0.934230 −0.467115 − 0.884197i \(-0.654707\pi\)
−0.467115 + 0.884197i \(0.654707\pi\)
\(234\) 0 0
\(235\) −6.57516e7 −0.330498
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2.61131e8 −1.23727 −0.618637 − 0.785677i \(-0.712315\pi\)
−0.618637 + 0.785677i \(0.712315\pi\)
\(240\) 0 0
\(241\) 1.97756e7 0.0910061 0.0455031 − 0.998964i \(-0.485511\pi\)
0.0455031 + 0.998964i \(0.485511\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.17649e7 −0.0511101
\(246\) 0 0
\(247\) −2.18853e7 −0.0924089
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.53770e8 −0.613779 −0.306890 − 0.951745i \(-0.599288\pi\)
−0.306890 + 0.951745i \(0.599288\pi\)
\(252\) 0 0
\(253\) 5.49807e6 0.0213446
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1.43360e8 0.526821 0.263411 − 0.964684i \(-0.415153\pi\)
0.263411 + 0.964684i \(0.415153\pi\)
\(258\) 0 0
\(259\) 1.15140e8 0.411792
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 5.07301e8 1.71957 0.859786 − 0.510654i \(-0.170597\pi\)
0.859786 + 0.510654i \(0.170597\pi\)
\(264\) 0 0
\(265\) −6.08718e7 −0.200935
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −5.27886e8 −1.65351 −0.826755 − 0.562562i \(-0.809816\pi\)
−0.826755 + 0.562562i \(0.809816\pi\)
\(270\) 0 0
\(271\) −5.05836e8 −1.54389 −0.771947 − 0.635687i \(-0.780717\pi\)
−0.771947 + 0.635687i \(0.780717\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.88979e8 0.547960
\(276\) 0 0
\(277\) −5.88586e8 −1.66391 −0.831956 − 0.554841i \(-0.812779\pi\)
−0.831956 + 0.554841i \(0.812779\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −8.65142e7 −0.232603 −0.116301 − 0.993214i \(-0.537104\pi\)
−0.116301 + 0.993214i \(0.537104\pi\)
\(282\) 0 0
\(283\) −1.57301e8 −0.412552 −0.206276 − 0.978494i \(-0.566134\pi\)
−0.206276 + 0.978494i \(0.566134\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −9.10624e7 −0.227380
\(288\) 0 0
\(289\) −3.75529e8 −0.915168
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −6.80964e8 −1.58157 −0.790783 − 0.612097i \(-0.790326\pi\)
−0.790783 + 0.612097i \(0.790326\pi\)
\(294\) 0 0
\(295\) −5.36120e7 −0.121586
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.52871e6 0.0141247
\(300\) 0 0
\(301\) 3.19992e7 0.0676326
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1.79709e8 0.362677
\(306\) 0 0
\(307\) 2.81734e8 0.555718 0.277859 − 0.960622i \(-0.410375\pi\)
0.277859 + 0.960622i \(0.410375\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 4.19719e8 0.791220 0.395610 − 0.918419i \(-0.370533\pi\)
0.395610 + 0.918419i \(0.370533\pi\)
\(312\) 0 0
\(313\) −2.28684e8 −0.421532 −0.210766 − 0.977537i \(-0.567596\pi\)
−0.210766 + 0.977537i \(0.567596\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.66003e8 −0.469006 −0.234503 − 0.972115i \(-0.575346\pi\)
−0.234503 + 0.972115i \(0.575346\pi\)
\(318\) 0 0
\(319\) −5.77286e8 −0.995689
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −3.91996e7 −0.0647251
\(324\) 0 0
\(325\) 2.24404e8 0.362609
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2.25528e8 −0.349152
\(330\) 0 0
\(331\) −7.16107e8 −1.08538 −0.542688 − 0.839935i \(-0.682593\pi\)
−0.542688 + 0.839935i \(0.682593\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2.12318e8 −0.308553
\(336\) 0 0
\(337\) 5.70266e8 0.811657 0.405829 − 0.913949i \(-0.366983\pi\)
0.405829 + 0.913949i \(0.366983\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 3.26755e8 0.446254
\(342\) 0 0
\(343\) −4.03536e7 −0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 1.64641e8 0.211536 0.105768 − 0.994391i \(-0.466270\pi\)
0.105768 + 0.994391i \(0.466270\pi\)
\(348\) 0 0
\(349\) 1.31564e9 1.65671 0.828357 − 0.560200i \(-0.189276\pi\)
0.828357 + 0.560200i \(0.189276\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.56118e9 −1.88904 −0.944518 − 0.328459i \(-0.893471\pi\)
−0.944518 + 0.328459i \(0.893471\pi\)
\(354\) 0 0
\(355\) −1.19121e8 −0.141316
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 8.60716e8 0.981814 0.490907 − 0.871212i \(-0.336665\pi\)
0.490907 + 0.871212i \(0.336665\pi\)
\(360\) 0 0
\(361\) −8.49729e8 −0.950616
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.05643e8 −0.113714
\(366\) 0 0
\(367\) 9.12133e8 0.963223 0.481612 − 0.876385i \(-0.340052\pi\)
0.481612 + 0.876385i \(0.340052\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.08790e8 −0.212276
\(372\) 0 0
\(373\) 2.36004e8 0.235472 0.117736 − 0.993045i \(-0.462436\pi\)
0.117736 + 0.993045i \(0.462436\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.85501e8 −0.658891
\(378\) 0 0
\(379\) 8.29313e8 0.782495 0.391247 − 0.920286i \(-0.372044\pi\)
0.391247 + 0.920286i \(0.372044\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.21663e9 1.10653 0.553265 − 0.833005i \(-0.313382\pi\)
0.553265 + 0.833005i \(0.313382\pi\)
\(384\) 0 0
\(385\) −9.51482e7 −0.0849744
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.18213e8 0.101822 0.0509109 − 0.998703i \(-0.483788\pi\)
0.0509109 + 0.998703i \(0.483788\pi\)
\(390\) 0 0
\(391\) 1.16938e7 0.00989321
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −9.98484e7 −0.0815176
\(396\) 0 0
\(397\) 3.62565e7 0.0290817 0.0145408 − 0.999894i \(-0.495371\pi\)
0.0145408 + 0.999894i \(0.495371\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −5.06850e8 −0.392531 −0.196265 − 0.980551i \(-0.562881\pi\)
−0.196265 + 0.980551i \(0.562881\pi\)
\(402\) 0 0
\(403\) 3.88007e8 0.295306
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 9.31193e8 0.684635
\(408\) 0 0
\(409\) 1.68121e9 1.21504 0.607519 − 0.794305i \(-0.292165\pi\)
0.607519 + 0.794305i \(0.292165\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −1.83889e8 −0.128449
\(414\) 0 0
\(415\) 3.89800e8 0.267716
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4.94962e8 −0.328718 −0.164359 − 0.986401i \(-0.552556\pi\)
−0.164359 + 0.986401i \(0.552556\pi\)
\(420\) 0 0
\(421\) −6.57487e7 −0.0429437 −0.0214719 − 0.999769i \(-0.506835\pi\)
−0.0214719 + 0.999769i \(0.506835\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 4.01938e8 0.253979
\(426\) 0 0
\(427\) 6.16402e8 0.383148
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.33418e9 0.802682 0.401341 − 0.915929i \(-0.368544\pi\)
0.401341 + 0.915929i \(0.368544\pi\)
\(432\) 0 0
\(433\) −2.47903e7 −0.0146749 −0.00733745 − 0.999973i \(-0.502336\pi\)
−0.00733745 + 0.999973i \(0.502336\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.31684e7 −0.00754828
\(438\) 0 0
\(439\) −6.89327e8 −0.388866 −0.194433 − 0.980916i \(-0.562287\pi\)
−0.194433 + 0.980916i \(0.562287\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 5.74617e8 0.314026 0.157013 − 0.987597i \(-0.449814\pi\)
0.157013 + 0.987597i \(0.449814\pi\)
\(444\) 0 0
\(445\) −4.62235e8 −0.248658
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −9.56884e7 −0.0498881 −0.0249440 − 0.999689i \(-0.507941\pi\)
−0.0249440 + 0.999689i \(0.507941\pi\)
\(450\) 0 0
\(451\) −7.36464e8 −0.378036
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1.12984e8 −0.0562313
\(456\) 0 0
\(457\) −4.73089e7 −0.0231866 −0.0115933 − 0.999933i \(-0.503690\pi\)
−0.0115933 + 0.999933i \(0.503690\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.13567e9 1.01527 0.507634 − 0.861573i \(-0.330520\pi\)
0.507634 + 0.861573i \(0.330520\pi\)
\(462\) 0 0
\(463\) 2.92675e8 0.137042 0.0685208 − 0.997650i \(-0.478172\pi\)
0.0685208 + 0.997650i \(0.478172\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 3.53834e9 1.60765 0.803824 − 0.594868i \(-0.202796\pi\)
0.803824 + 0.594868i \(0.202796\pi\)
\(468\) 0 0
\(469\) −7.28249e8 −0.325968
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.58792e8 0.112444
\(474\) 0 0
\(475\) −4.52622e8 −0.193780
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4.15698e9 1.72824 0.864120 − 0.503286i \(-0.167876\pi\)
0.864120 + 0.503286i \(0.167876\pi\)
\(480\) 0 0
\(481\) 1.10575e9 0.453053
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.52877e9 −0.608480
\(486\) 0 0
\(487\) 1.24092e8 0.0486847 0.0243423 − 0.999704i \(-0.492251\pi\)
0.0243423 + 0.999704i \(0.492251\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.29218e9 1.25516 0.627578 − 0.778554i \(-0.284046\pi\)
0.627578 + 0.778554i \(0.284046\pi\)
\(492\) 0 0
\(493\) −1.22783e9 −0.461501
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4.08586e8 −0.149292
\(498\) 0 0
\(499\) −3.60439e8 −0.129861 −0.0649306 − 0.997890i \(-0.520683\pi\)
−0.0649306 + 0.997890i \(0.520683\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 4.29760e9 1.50570 0.752849 − 0.658193i \(-0.228679\pi\)
0.752849 + 0.658193i \(0.228679\pi\)
\(504\) 0 0
\(505\) 2.23869e8 0.0773524
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 5.42588e8 0.182372 0.0911860 − 0.995834i \(-0.470934\pi\)
0.0911860 + 0.995834i \(0.470934\pi\)
\(510\) 0 0
\(511\) −3.62355e8 −0.120133
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.24502e9 −0.401654
\(516\) 0 0
\(517\) −1.82395e9 −0.580492
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 2.36755e8 0.0733445 0.0366722 − 0.999327i \(-0.488324\pi\)
0.0366722 + 0.999327i \(0.488324\pi\)
\(522\) 0 0
\(523\) 6.48324e9 1.98169 0.990846 − 0.135000i \(-0.0431036\pi\)
0.990846 + 0.135000i \(0.0431036\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.94973e8 0.206838
\(528\) 0 0
\(529\) −3.40090e9 −0.998846
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −8.74517e8 −0.250163
\(534\) 0 0
\(535\) −7.58860e8 −0.214251
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3.26358e8 −0.0897706
\(540\) 0 0
\(541\) −1.50099e8 −0.0407556 −0.0203778 − 0.999792i \(-0.506487\pi\)
−0.0203778 + 0.999792i \(0.506487\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5.17097e8 −0.136831
\(546\) 0 0
\(547\) 3.32631e9 0.868974 0.434487 − 0.900678i \(-0.356930\pi\)
0.434487 + 0.900678i \(0.356930\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.38266e9 0.352114
\(552\) 0 0
\(553\) −3.42480e8 −0.0861187
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −4.72512e9 −1.15856 −0.579282 − 0.815127i \(-0.696667\pi\)
−0.579282 + 0.815127i \(0.696667\pi\)
\(558\) 0 0
\(559\) 3.07304e8 0.0744092
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −7.11044e9 −1.67926 −0.839628 − 0.543162i \(-0.817227\pi\)
−0.839628 + 0.543162i \(0.817227\pi\)
\(564\) 0 0
\(565\) −9.63868e8 −0.224827
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −6.55315e9 −1.49127 −0.745636 − 0.666353i \(-0.767854\pi\)
−0.745636 + 0.666353i \(0.767854\pi\)
\(570\) 0 0
\(571\) 8.29277e9 1.86412 0.932059 − 0.362306i \(-0.118010\pi\)
0.932059 + 0.362306i \(0.118010\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.35024e8 0.0296192
\(576\) 0 0
\(577\) −5.88000e9 −1.27427 −0.637135 − 0.770752i \(-0.719881\pi\)
−0.637135 + 0.770752i \(0.719881\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.33702e9 0.282826
\(582\) 0 0
\(583\) −1.68858e9 −0.352925
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.43194e9 0.292207 0.146103 − 0.989269i \(-0.453327\pi\)
0.146103 + 0.989269i \(0.453327\pi\)
\(588\) 0 0
\(589\) −7.82610e8 −0.157813
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −3.48138e9 −0.685584 −0.342792 − 0.939411i \(-0.611373\pi\)
−0.342792 + 0.939411i \(0.611373\pi\)
\(594\) 0 0
\(595\) −2.02370e8 −0.0393855
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −7.93597e9 −1.50871 −0.754356 − 0.656466i \(-0.772051\pi\)
−0.754356 + 0.656466i \(0.772051\pi\)
\(600\) 0 0
\(601\) −4.05169e9 −0.761335 −0.380668 − 0.924712i \(-0.624306\pi\)
−0.380668 + 0.924712i \(0.624306\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.17921e9 0.216495
\(606\) 0 0
\(607\) 5.19159e9 0.942194 0.471097 − 0.882081i \(-0.343858\pi\)
0.471097 + 0.882081i \(0.343858\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −2.16586e9 −0.384137
\(612\) 0 0
\(613\) 3.69807e9 0.648430 0.324215 − 0.945983i \(-0.394900\pi\)
0.324215 + 0.945983i \(0.394900\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.19879e9 1.06245 0.531226 − 0.847230i \(-0.321732\pi\)
0.531226 + 0.847230i \(0.321732\pi\)
\(618\) 0 0
\(619\) −2.85137e9 −0.483211 −0.241605 − 0.970375i \(-0.577674\pi\)
−0.241605 + 0.970375i \(0.577674\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −1.58547e9 −0.262693
\(624\) 0 0
\(625\) 3.85977e9 0.632384
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.98055e9 0.317328
\(630\) 0 0
\(631\) −7.56414e9 −1.19855 −0.599276 − 0.800543i \(-0.704545\pi\)
−0.599276 + 0.800543i \(0.704545\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8.08309e8 −0.125276
\(636\) 0 0
\(637\) −3.87536e8 −0.0594051
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1.12019e10 −1.67991 −0.839957 − 0.542653i \(-0.817420\pi\)
−0.839957 + 0.542653i \(0.817420\pi\)
\(642\) 0 0
\(643\) 7.87742e9 1.16855 0.584273 − 0.811557i \(-0.301380\pi\)
0.584273 + 0.811557i \(0.301380\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.24945e9 −0.181365 −0.0906825 − 0.995880i \(-0.528905\pi\)
−0.0906825 + 0.995880i \(0.528905\pi\)
\(648\) 0 0
\(649\) −1.48720e9 −0.213556
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −7.52696e9 −1.05785 −0.528924 − 0.848669i \(-0.677404\pi\)
−0.528924 + 0.848669i \(0.677404\pi\)
\(654\) 0 0
\(655\) −1.97585e9 −0.274733
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −2.29417e9 −0.312268 −0.156134 − 0.987736i \(-0.549903\pi\)
−0.156134 + 0.987736i \(0.549903\pi\)
\(660\) 0 0
\(661\) −5.08384e9 −0.684679 −0.342339 − 0.939576i \(-0.611219\pi\)
−0.342339 + 0.939576i \(0.611219\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.27889e8 0.0300502
\(666\) 0 0
\(667\) −4.12466e8 −0.0538205
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 4.98513e9 0.637012
\(672\) 0 0
\(673\) −5.62649e9 −0.711516 −0.355758 − 0.934578i \(-0.615777\pi\)
−0.355758 + 0.934578i \(0.615777\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.02745e10 1.27263 0.636314 − 0.771430i \(-0.280458\pi\)
0.636314 + 0.771430i \(0.280458\pi\)
\(678\) 0 0
\(679\) −5.24368e9 −0.642824
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.55246e10 −1.86443 −0.932217 − 0.361901i \(-0.882128\pi\)
−0.932217 + 0.361901i \(0.882128\pi\)
\(684\) 0 0
\(685\) −4.80896e9 −0.571655
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −2.00512e9 −0.233546
\(690\) 0 0
\(691\) 8.23449e9 0.949432 0.474716 − 0.880139i \(-0.342551\pi\)
0.474716 + 0.880139i \(0.342551\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.37173e9 −0.154997
\(696\) 0 0
\(697\) −1.56638e9 −0.175219
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 4.88820e9 0.535964 0.267982 − 0.963424i \(-0.413643\pi\)
0.267982 + 0.963424i \(0.413643\pi\)
\(702\) 0 0
\(703\) −2.23030e9 −0.242114
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 7.67870e8 0.0817184
\(708\) 0 0
\(709\) −7.95106e9 −0.837844 −0.418922 − 0.908022i \(-0.637592\pi\)
−0.418922 + 0.908022i \(0.637592\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.33464e8 0.0241216
\(714\) 0 0
\(715\) −9.13756e8 −0.0934887
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1.30035e10 1.30469 0.652346 − 0.757921i \(-0.273785\pi\)
0.652346 + 0.757921i \(0.273785\pi\)
\(720\) 0 0
\(721\) −4.27043e9 −0.424324
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1.41772e10 −1.38168
\(726\) 0 0
\(727\) −1.73805e10 −1.67761 −0.838805 − 0.544431i \(-0.816745\pi\)
−0.838805 + 0.544431i \(0.816745\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 5.50423e8 0.0521177
\(732\) 0 0
\(733\) 9.47114e9 0.888256 0.444128 − 0.895963i \(-0.353513\pi\)
0.444128 + 0.895963i \(0.353513\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −5.88969e9 −0.541946
\(738\) 0 0
\(739\) 5.52914e9 0.503967 0.251983 − 0.967732i \(-0.418917\pi\)
0.251983 + 0.967732i \(0.418917\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.53701e10 1.37472 0.687362 − 0.726315i \(-0.258769\pi\)
0.687362 + 0.726315i \(0.258769\pi\)
\(744\) 0 0
\(745\) −7.07104e9 −0.626522
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −2.60289e9 −0.226344
\(750\) 0 0
\(751\) −8.51950e9 −0.733964 −0.366982 − 0.930228i \(-0.619609\pi\)
−0.366982 + 0.930228i \(0.619609\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2.56970e9 0.217304
\(756\) 0 0
\(757\) −4.72648e8 −0.0396006 −0.0198003 − 0.999804i \(-0.506303\pi\)
−0.0198003 + 0.999804i \(0.506303\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.43465e10 1.18005 0.590025 − 0.807385i \(-0.299118\pi\)
0.590025 + 0.807385i \(0.299118\pi\)
\(762\) 0 0
\(763\) −1.77364e9 −0.144554
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −1.76598e9 −0.141319
\(768\) 0 0
\(769\) 1.39480e10 1.10603 0.553017 − 0.833170i \(-0.313476\pi\)
0.553017 + 0.833170i \(0.313476\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 4.32337e9 0.336662 0.168331 − 0.985731i \(-0.446162\pi\)
0.168331 + 0.985731i \(0.446162\pi\)
\(774\) 0 0
\(775\) 8.02458e9 0.619250
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.76390e9 0.133688
\(780\) 0 0
\(781\) −3.30443e9 −0.248209
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −2.09180e9 −0.154339
\(786\) 0 0
\(787\) 6.71201e9 0.490842 0.245421 − 0.969417i \(-0.421074\pi\)
0.245421 + 0.969417i \(0.421074\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −3.30607e9 −0.237517
\(792\) 0 0
\(793\) 5.91961e9 0.421539
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −5.61333e9 −0.392750 −0.196375 − 0.980529i \(-0.562917\pi\)
−0.196375 + 0.980529i \(0.562917\pi\)
\(798\) 0 0
\(799\) −3.87934e9 −0.269057
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −2.93054e9 −0.199730
\(804\) 0 0
\(805\) −6.79826e7 −0.00459317
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −4.15844e9 −0.276128 −0.138064 − 0.990423i \(-0.544088\pi\)
−0.138064 + 0.990423i \(0.544088\pi\)
\(810\) 0 0
\(811\) −1.11994e10 −0.737259 −0.368629 − 0.929576i \(-0.620173\pi\)
−0.368629 + 0.929576i \(0.620173\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −5.44160e9 −0.352107
\(816\) 0 0
\(817\) −6.19832e8 −0.0397646
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −2.32872e10 −1.46864 −0.734322 − 0.678801i \(-0.762500\pi\)
−0.734322 + 0.678801i \(0.762500\pi\)
\(822\) 0 0
\(823\) 9.31943e9 0.582761 0.291380 − 0.956607i \(-0.405886\pi\)
0.291380 + 0.956607i \(0.405886\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.16608e10 0.716902 0.358451 − 0.933548i \(-0.383305\pi\)
0.358451 + 0.933548i \(0.383305\pi\)
\(828\) 0 0
\(829\) −2.31007e8 −0.0140826 −0.00704132 − 0.999975i \(-0.502241\pi\)
−0.00704132 + 0.999975i \(0.502241\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −6.94129e8 −0.0416086
\(834\) 0 0
\(835\) 6.41888e8 0.0381554
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 2.77032e10 1.61943 0.809716 − 0.586822i \(-0.199621\pi\)
0.809716 + 0.586822i \(0.199621\pi\)
\(840\) 0 0
\(841\) 2.60582e10 1.51063
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 5.18981e9 0.295905
\(846\) 0 0
\(847\) 4.04469e9 0.228714
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 6.65330e8 0.0370070
\(852\) 0 0
\(853\) 2.87010e10 1.58335 0.791673 − 0.610944i \(-0.209210\pi\)
0.791673 + 0.610944i \(0.209210\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.44528e9 −0.0784366 −0.0392183 − 0.999231i \(-0.512487\pi\)
−0.0392183 + 0.999231i \(0.512487\pi\)
\(858\) 0 0
\(859\) 2.94906e10 1.58748 0.793738 − 0.608260i \(-0.208132\pi\)
0.793738 + 0.608260i \(0.208132\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −8.87258e9 −0.469907 −0.234954 − 0.972007i \(-0.575494\pi\)
−0.234954 + 0.972007i \(0.575494\pi\)
\(864\) 0 0
\(865\) −5.46532e9 −0.287117
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −2.76979e9 −0.143179
\(870\) 0 0
\(871\) −6.99374e9 −0.358630
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −5.01638e9 −0.253141
\(876\) 0 0
\(877\) 1.57308e10 0.787504 0.393752 − 0.919217i \(-0.371177\pi\)
0.393752 + 0.919217i \(0.371177\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 9.12522e9 0.449602 0.224801 − 0.974405i \(-0.427827\pi\)
0.224801 + 0.974405i \(0.427827\pi\)
\(882\) 0 0
\(883\) 2.24538e10 1.09756 0.548778 − 0.835968i \(-0.315093\pi\)
0.548778 + 0.835968i \(0.315093\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 3.43569e10 1.65303 0.826516 − 0.562913i \(-0.190320\pi\)
0.826516 + 0.562913i \(0.190320\pi\)
\(888\) 0 0
\(889\) −2.77250e9 −0.132347
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 4.36854e9 0.205284
\(894\) 0 0
\(895\) −3.76658e9 −0.175617
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.45132e10 −1.12523
\(900\) 0 0
\(901\) −3.59144e9 −0.163580
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.76788e9 −0.0792835
\(906\) 0 0
\(907\) −4.01062e10 −1.78478 −0.892392 − 0.451261i \(-0.850974\pi\)
−0.892392 + 0.451261i \(0.850974\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 3.70230e10 1.62240 0.811200 − 0.584769i \(-0.198815\pi\)
0.811200 + 0.584769i \(0.198815\pi\)
\(912\) 0 0
\(913\) 1.08131e10 0.470220
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −6.77718e9 −0.290239
\(918\) 0 0
\(919\) −3.37645e10 −1.43501 −0.717507 − 0.696551i \(-0.754717\pi\)
−0.717507 + 0.696551i \(0.754717\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −3.92386e9 −0.164251
\(924\) 0 0
\(925\) 2.28686e10 0.950044
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 4.77100e10 1.95233 0.976167 − 0.217019i \(-0.0696332\pi\)
0.976167 + 0.217019i \(0.0696332\pi\)
\(930\) 0 0
\(931\) 7.81660e8 0.0317464
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.63666e9 −0.0654814
\(936\) 0 0
\(937\) 3.86353e8 0.0153425 0.00767125 − 0.999971i \(-0.497558\pi\)
0.00767125 + 0.999971i \(0.497558\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 2.79446e10 1.09329 0.546643 − 0.837366i \(-0.315905\pi\)
0.546643 + 0.837366i \(0.315905\pi\)
\(942\) 0 0
\(943\) −5.26197e8 −0.0204342
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 5.50921e9 0.210797 0.105399 − 0.994430i \(-0.466388\pi\)
0.105399 + 0.994430i \(0.466388\pi\)
\(948\) 0 0
\(949\) −3.47988e9 −0.132170
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 7.40406e9 0.277105 0.138553 − 0.990355i \(-0.455755\pi\)
0.138553 + 0.990355i \(0.455755\pi\)
\(954\) 0 0
\(955\) 4.09177e9 0.152019
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1.64947e10 −0.603921
\(960\) 0 0
\(961\) −1.36377e10 −0.495687
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.63827e10 0.586867
\(966\) 0 0
\(967\) 1.97731e10 0.703204 0.351602 − 0.936149i \(-0.385637\pi\)
0.351602 + 0.936149i \(0.385637\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 2.81294e10 0.986037 0.493018 − 0.870019i \(-0.335894\pi\)
0.493018 + 0.870019i \(0.335894\pi\)
\(972\) 0 0
\(973\) −4.70504e9 −0.163745
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −9.63337e8 −0.0330482 −0.0165241 − 0.999863i \(-0.505260\pi\)
−0.0165241 + 0.999863i \(0.505260\pi\)
\(978\) 0 0
\(979\) −1.28224e10 −0.436747
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −4.20434e10 −1.41176 −0.705879 − 0.708332i \(-0.749448\pi\)
−0.705879 + 0.708332i \(0.749448\pi\)
\(984\) 0 0
\(985\) 8.02076e9 0.267417
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 1.84905e8 0.00607800
\(990\) 0 0
\(991\) −2.87955e10 −0.939869 −0.469934 − 0.882701i \(-0.655722\pi\)
−0.469934 + 0.882701i \(0.655722\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −9.83219e9 −0.316424
\(996\) 0 0
\(997\) −3.54828e10 −1.13393 −0.566963 − 0.823743i \(-0.691882\pi\)
−0.566963 + 0.823743i \(0.691882\pi\)
\(998\) 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 252.8.a.a.1.1 1
3.2 odd 2 84.8.a.a.1.1 ✓ 1
12.11 even 2 336.8.a.j.1.1 1
21.2 odd 6 588.8.i.f.361.1 2
21.5 even 6 588.8.i.c.361.1 2
21.11 odd 6 588.8.i.f.373.1 2
21.17 even 6 588.8.i.c.373.1 2
21.20 even 2 588.8.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.8.a.a.1.1 ✓ 1 3.2 odd 2
252.8.a.a.1.1 1 1.1 even 1 trivial
336.8.a.j.1.1 1 12.11 even 2
588.8.a.c.1.1 1 21.20 even 2
588.8.i.c.361.1 2 21.5 even 6
588.8.i.c.373.1 2 21.17 even 6
588.8.i.f.361.1 2 21.2 odd 6
588.8.i.f.373.1 2 21.11 odd 6