Properties

Label 360.8.a.b.1.1
Level $360$
Weight $8$
Character 360.1
Self dual yes
Analytic conductor $112.459$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [360,8,Mod(1,360)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(360, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("360.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 360.a (trivial)

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: yes
Analytic conductor: \(112.458609174\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 360.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-125.000 q^{5} +776.000 q^{7} +O(q^{10})\) \(q-125.000 q^{5} +776.000 q^{7} +124.000 q^{11} -13082.0 q^{13} +15950.0 q^{17} -20516.0 q^{19} +29224.0 q^{23} +15625.0 q^{25} +221482. q^{29} -109760. q^{31} -97000.0 q^{35} +73422.0 q^{37} -12762.0 q^{41} +290548. q^{43} -1.26915e6 q^{47} -221367. q^{49} +395778. q^{53} -15500.0 q^{55} -421492. q^{59} -2.12225e6 q^{61} +1.63525e6 q^{65} -3.13287e6 q^{67} +5.37655e6 q^{71} +4.98547e6 q^{73} +96224.0 q^{77} +3.86750e6 q^{79} +6.19020e6 q^{83} -1.99375e6 q^{85} -1.12439e6 q^{89} -1.01516e7 q^{91} +2.56450e6 q^{95} +9.96810e6 q^{97} +O(q^{100})\)

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
\(5\) −125.000 −0.447214
\(6\) 0 0
\(7\) 776.000 0.855103 0.427552 0.903991i \(-0.359376\pi\)
0.427552 + 0.903991i \(0.359376\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 124.000 0.0280897 0.0140449 0.999901i \(-0.495529\pi\)
0.0140449 + 0.999901i \(0.495529\pi\)
\(12\) 0 0
\(13\) −13082.0 −1.65148 −0.825738 0.564053i \(-0.809241\pi\)
−0.825738 + 0.564053i \(0.809241\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 15950.0 0.787389 0.393695 0.919241i \(-0.371197\pi\)
0.393695 + 0.919241i \(0.371197\pi\)
\(18\) 0 0
\(19\) −20516.0 −0.686207 −0.343103 0.939298i \(-0.611478\pi\)
−0.343103 + 0.939298i \(0.611478\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 29224.0 0.500832 0.250416 0.968138i \(-0.419433\pi\)
0.250416 + 0.968138i \(0.419433\pi\)
\(24\) 0 0
\(25\) 15625.0 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 221482. 1.68634 0.843171 0.537646i \(-0.180686\pi\)
0.843171 + 0.537646i \(0.180686\pi\)
\(30\) 0 0
\(31\) −109760. −0.661726 −0.330863 0.943679i \(-0.607340\pi\)
−0.330863 + 0.943679i \(0.607340\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −97000.0 −0.382414
\(36\) 0 0
\(37\) 73422.0 0.238298 0.119149 0.992876i \(-0.461983\pi\)
0.119149 + 0.992876i \(0.461983\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −12762.0 −0.0289185 −0.0144592 0.999895i \(-0.504603\pi\)
−0.0144592 + 0.999895i \(0.504603\pi\)
\(42\) 0 0
\(43\) 290548. 0.557286 0.278643 0.960395i \(-0.410115\pi\)
0.278643 + 0.960395i \(0.410115\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.26915e6 −1.78308 −0.891541 0.452941i \(-0.850375\pi\)
−0.891541 + 0.452941i \(0.850375\pi\)
\(48\) 0 0
\(49\) −221367. −0.268798
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 395778. 0.365162 0.182581 0.983191i \(-0.441555\pi\)
0.182581 + 0.983191i \(0.441555\pi\)
\(54\) 0 0
\(55\) −15500.0 −0.0125621
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −421492. −0.267182 −0.133591 0.991037i \(-0.542651\pi\)
−0.133591 + 0.991037i \(0.542651\pi\)
\(60\) 0 0
\(61\) −2.12225e6 −1.19713 −0.598566 0.801073i \(-0.704263\pi\)
−0.598566 + 0.801073i \(0.704263\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.63525e6 0.738563
\(66\) 0 0
\(67\) −3.13287e6 −1.27257 −0.636283 0.771456i \(-0.719529\pi\)
−0.636283 + 0.771456i \(0.719529\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.37655e6 1.78279 0.891394 0.453230i \(-0.149728\pi\)
0.891394 + 0.453230i \(0.149728\pi\)
\(72\) 0 0
\(73\) 4.98547e6 1.49995 0.749973 0.661468i \(-0.230066\pi\)
0.749973 + 0.661468i \(0.230066\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 96224.0 0.0240196
\(78\) 0 0
\(79\) 3.86750e6 0.882543 0.441272 0.897374i \(-0.354528\pi\)
0.441272 + 0.897374i \(0.354528\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.19020e6 1.18831 0.594157 0.804349i \(-0.297486\pi\)
0.594157 + 0.804349i \(0.297486\pi\)
\(84\) 0 0
\(85\) −1.99375e6 −0.352131
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.12439e6 −0.169065 −0.0845325 0.996421i \(-0.526940\pi\)
−0.0845325 + 0.996421i \(0.526940\pi\)
\(90\) 0 0
\(91\) −1.01516e7 −1.41218
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.56450e6 0.306881
\(96\) 0 0
\(97\) 9.96810e6 1.10895 0.554474 0.832201i \(-0.312919\pi\)
0.554474 + 0.832201i \(0.312919\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −425166. −0.0410614 −0.0205307 0.999789i \(-0.506536\pi\)
−0.0205307 + 0.999789i \(0.506536\pi\)
\(102\) 0 0
\(103\) 7.06209e6 0.636800 0.318400 0.947957i \(-0.396855\pi\)
0.318400 + 0.947957i \(0.396855\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.82186e7 −1.43771 −0.718855 0.695160i \(-0.755333\pi\)
−0.718855 + 0.695160i \(0.755333\pi\)
\(108\) 0 0
\(109\) 2.27869e7 1.68536 0.842679 0.538416i \(-0.180977\pi\)
0.842679 + 0.538416i \(0.180977\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 700398. 0.0456636 0.0228318 0.999739i \(-0.492732\pi\)
0.0228318 + 0.999739i \(0.492732\pi\)
\(114\) 0 0
\(115\) −3.65300e6 −0.223979
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 1.23772e7 0.673299
\(120\) 0 0
\(121\) −1.94718e7 −0.999211
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1.95312e6 −0.0894427
\(126\) 0 0
\(127\) 3.69868e7 1.60226 0.801132 0.598488i \(-0.204231\pi\)
0.801132 + 0.598488i \(0.204231\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.86042e7 1.88897 0.944484 0.328559i \(-0.106563\pi\)
0.944484 + 0.328559i \(0.106563\pi\)
\(132\) 0 0
\(133\) −1.59204e7 −0.586778
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.45152e7 0.814542 0.407271 0.913307i \(-0.366480\pi\)
0.407271 + 0.913307i \(0.366480\pi\)
\(138\) 0 0
\(139\) −2.14944e7 −0.678848 −0.339424 0.940633i \(-0.610232\pi\)
−0.339424 + 0.940633i \(0.610232\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.62217e6 −0.0463895
\(144\) 0 0
\(145\) −2.76852e7 −0.754155
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.56239e7 −0.386935 −0.193467 0.981107i \(-0.561973\pi\)
−0.193467 + 0.981107i \(0.561973\pi\)
\(150\) 0 0
\(151\) 7.83038e7 1.85082 0.925408 0.378972i \(-0.123722\pi\)
0.925408 + 0.378972i \(0.123722\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 1.37200e7 0.295933
\(156\) 0 0
\(157\) −6.52688e7 −1.34604 −0.673019 0.739625i \(-0.735003\pi\)
−0.673019 + 0.739625i \(0.735003\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.26778e7 0.428263
\(162\) 0 0
\(163\) −1.07845e7 −0.195049 −0.0975246 0.995233i \(-0.531092\pi\)
−0.0975246 + 0.995233i \(0.531092\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3.90065e7 −0.648081 −0.324041 0.946043i \(-0.605041\pi\)
−0.324041 + 0.946043i \(0.605041\pi\)
\(168\) 0 0
\(169\) 1.08390e8 1.72737
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.39258e7 0.498159 0.249080 0.968483i \(-0.419872\pi\)
0.249080 + 0.968483i \(0.419872\pi\)
\(174\) 0 0
\(175\) 1.21250e7 0.171021
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.44958e8 1.88911 0.944553 0.328358i \(-0.106495\pi\)
0.944553 + 0.328358i \(0.106495\pi\)
\(180\) 0 0
\(181\) 1.39247e8 1.74547 0.872733 0.488199i \(-0.162346\pi\)
0.872733 + 0.488199i \(0.162346\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −9.17775e6 −0.106570
\(186\) 0 0
\(187\) 1.97780e6 0.0221175
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2.42946e7 0.252285 0.126143 0.992012i \(-0.459740\pi\)
0.126143 + 0.992012i \(0.459740\pi\)
\(192\) 0 0
\(193\) 1.04825e8 1.04958 0.524790 0.851232i \(-0.324144\pi\)
0.524790 + 0.851232i \(0.324144\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.05566e8 1.91567 0.957833 0.287324i \(-0.0927657\pi\)
0.957833 + 0.287324i \(0.0927657\pi\)
\(198\) 0 0
\(199\) 4.56329e7 0.410480 0.205240 0.978712i \(-0.434202\pi\)
0.205240 + 0.978712i \(0.434202\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 1.71870e8 1.44200
\(204\) 0 0
\(205\) 1.59525e6 0.0129327
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −2.54398e6 −0.0192754
\(210\) 0 0
\(211\) −1.19090e7 −0.0872743 −0.0436372 0.999047i \(-0.513895\pi\)
−0.0436372 + 0.999047i \(0.513895\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.63185e7 −0.249226
\(216\) 0 0
\(217\) −8.51738e7 −0.565844
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.08658e8 −1.30035
\(222\) 0 0
\(223\) −1.68668e8 −1.01851 −0.509255 0.860616i \(-0.670079\pi\)
−0.509255 + 0.860616i \(0.670079\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1.05791e8 0.600286 0.300143 0.953894i \(-0.402966\pi\)
0.300143 + 0.953894i \(0.402966\pi\)
\(228\) 0 0
\(229\) −3.48457e8 −1.91746 −0.958728 0.284324i \(-0.908231\pi\)
−0.958728 + 0.284324i \(0.908231\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.21828e8 1.14887 0.574434 0.818551i \(-0.305222\pi\)
0.574434 + 0.818551i \(0.305222\pi\)
\(234\) 0 0
\(235\) 1.58644e8 0.797418
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 2.89900e8 1.37359 0.686794 0.726852i \(-0.259018\pi\)
0.686794 + 0.726852i \(0.259018\pi\)
\(240\) 0 0
\(241\) −8.63833e7 −0.397530 −0.198765 0.980047i \(-0.563693\pi\)
−0.198765 + 0.980047i \(0.563693\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.76709e7 0.120210
\(246\) 0 0
\(247\) 2.68390e8 1.13325
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −2.84765e8 −1.13666 −0.568328 0.822802i \(-0.692410\pi\)
−0.568328 + 0.822802i \(0.692410\pi\)
\(252\) 0 0
\(253\) 3.62378e6 0.0140682
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 8.45535e7 0.310717 0.155359 0.987858i \(-0.450347\pi\)
0.155359 + 0.987858i \(0.450347\pi\)
\(258\) 0 0
\(259\) 5.69755e7 0.203769
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.57270e7 0.222792 0.111396 0.993776i \(-0.464468\pi\)
0.111396 + 0.993776i \(0.464468\pi\)
\(264\) 0 0
\(265\) −4.94722e7 −0.163306
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −2.81943e8 −0.883136 −0.441568 0.897228i \(-0.645578\pi\)
−0.441568 + 0.897228i \(0.645578\pi\)
\(270\) 0 0
\(271\) 3.09940e8 0.945987 0.472993 0.881066i \(-0.343173\pi\)
0.472993 + 0.881066i \(0.343173\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 1.93750e6 0.00561794
\(276\) 0 0
\(277\) −3.25764e8 −0.920924 −0.460462 0.887680i \(-0.652316\pi\)
−0.460462 + 0.887680i \(0.652316\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 2.96310e8 0.796662 0.398331 0.917242i \(-0.369590\pi\)
0.398331 + 0.917242i \(0.369590\pi\)
\(282\) 0 0
\(283\) 5.00092e8 1.31159 0.655794 0.754940i \(-0.272334\pi\)
0.655794 + 0.754940i \(0.272334\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −9.90331e6 −0.0247283
\(288\) 0 0
\(289\) −1.55936e8 −0.380018
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.07288e8 −0.249180 −0.124590 0.992208i \(-0.539762\pi\)
−0.124590 + 0.992208i \(0.539762\pi\)
\(294\) 0 0
\(295\) 5.26865e7 0.119487
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −3.82308e8 −0.827113
\(300\) 0 0
\(301\) 2.25465e8 0.476537
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 2.65281e8 0.535374
\(306\) 0 0
\(307\) 8.06445e8 1.59071 0.795354 0.606145i \(-0.207285\pi\)
0.795354 + 0.606145i \(0.207285\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3.60964e7 −0.0680460 −0.0340230 0.999421i \(-0.510832\pi\)
−0.0340230 + 0.999421i \(0.510832\pi\)
\(312\) 0 0
\(313\) −5.31349e8 −0.979433 −0.489717 0.871882i \(-0.662900\pi\)
−0.489717 + 0.871882i \(0.662900\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −1.18418e7 −0.0208791 −0.0104396 0.999946i \(-0.503323\pi\)
−0.0104396 + 0.999946i \(0.503323\pi\)
\(318\) 0 0
\(319\) 2.74638e7 0.0473688
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −3.27230e8 −0.540312
\(324\) 0 0
\(325\) −2.04406e8 −0.330295
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −9.84862e8 −1.52472
\(330\) 0 0
\(331\) −3.74592e8 −0.567755 −0.283877 0.958861i \(-0.591621\pi\)
−0.283877 + 0.958861i \(0.591621\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 3.91608e8 0.569109
\(336\) 0 0
\(337\) 1.66536e8 0.237030 0.118515 0.992952i \(-0.462187\pi\)
0.118515 + 0.992952i \(0.462187\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.36102e7 −0.0185877
\(342\) 0 0
\(343\) −8.10850e8 −1.08495
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3.31438e8 −0.425842 −0.212921 0.977069i \(-0.568298\pi\)
−0.212921 + 0.977069i \(0.568298\pi\)
\(348\) 0 0
\(349\) −4.29412e6 −0.00540736 −0.00270368 0.999996i \(-0.500861\pi\)
−0.00270368 + 0.999996i \(0.500861\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.90007e7 0.0592912 0.0296456 0.999560i \(-0.490562\pi\)
0.0296456 + 0.999560i \(0.490562\pi\)
\(354\) 0 0
\(355\) −6.72069e8 −0.797287
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.14432e9 1.30532 0.652658 0.757653i \(-0.273654\pi\)
0.652658 + 0.757653i \(0.273654\pi\)
\(360\) 0 0
\(361\) −4.72965e8 −0.529120
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −6.23183e8 −0.670796
\(366\) 0 0
\(367\) 5.58047e8 0.589304 0.294652 0.955605i \(-0.404796\pi\)
0.294652 + 0.955605i \(0.404796\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 3.07124e8 0.312252
\(372\) 0 0
\(373\) −8.89487e8 −0.887480 −0.443740 0.896156i \(-0.646349\pi\)
−0.443740 + 0.896156i \(0.646349\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −2.89743e9 −2.78495
\(378\) 0 0
\(379\) 2.40167e8 0.226608 0.113304 0.993560i \(-0.463857\pi\)
0.113304 + 0.993560i \(0.463857\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4.54872e7 0.0413708 0.0206854 0.999786i \(-0.493415\pi\)
0.0206854 + 0.999786i \(0.493415\pi\)
\(384\) 0 0
\(385\) −1.20280e7 −0.0107419
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −7.34802e8 −0.632917 −0.316458 0.948606i \(-0.602494\pi\)
−0.316458 + 0.948606i \(0.602494\pi\)
\(390\) 0 0
\(391\) 4.66123e8 0.394350
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4.83438e8 −0.394685
\(396\) 0 0
\(397\) 8.23610e8 0.660625 0.330312 0.943872i \(-0.392846\pi\)
0.330312 + 0.943872i \(0.392846\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −3.84193e8 −0.297539 −0.148770 0.988872i \(-0.547531\pi\)
−0.148770 + 0.988872i \(0.547531\pi\)
\(402\) 0 0
\(403\) 1.43588e9 1.09282
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 9.10433e6 0.00669372
\(408\) 0 0
\(409\) −1.97006e9 −1.42379 −0.711897 0.702284i \(-0.752164\pi\)
−0.711897 + 0.702284i \(0.752164\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −3.27078e8 −0.228468
\(414\) 0 0
\(415\) −7.73774e8 −0.531430
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −2.66755e9 −1.77159 −0.885794 0.464078i \(-0.846386\pi\)
−0.885794 + 0.464078i \(0.846386\pi\)
\(420\) 0 0
\(421\) −7.63075e8 −0.498402 −0.249201 0.968452i \(-0.580168\pi\)
−0.249201 + 0.968452i \(0.580168\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.49219e8 0.157478
\(426\) 0 0
\(427\) −1.64687e9 −1.02367
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 6.70874e8 0.403618 0.201809 0.979425i \(-0.435318\pi\)
0.201809 + 0.979425i \(0.435318\pi\)
\(432\) 0 0
\(433\) −1.38362e9 −0.819048 −0.409524 0.912299i \(-0.634305\pi\)
−0.409524 + 0.912299i \(0.634305\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −5.99560e8 −0.343675
\(438\) 0 0
\(439\) 1.86303e9 1.05098 0.525489 0.850801i \(-0.323883\pi\)
0.525489 + 0.850801i \(0.323883\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 3.57996e8 0.195644 0.0978218 0.995204i \(-0.468812\pi\)
0.0978218 + 0.995204i \(0.468812\pi\)
\(444\) 0 0
\(445\) 1.40549e8 0.0756081
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 2.31697e9 1.20797 0.603987 0.796994i \(-0.293578\pi\)
0.603987 + 0.796994i \(0.293578\pi\)
\(450\) 0 0
\(451\) −1.58249e6 −0.000812311 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1.26895e9 0.631547
\(456\) 0 0
\(457\) −2.21599e9 −1.08608 −0.543039 0.839708i \(-0.682726\pi\)
−0.543039 + 0.839708i \(0.682726\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −2.52028e9 −1.19810 −0.599052 0.800710i \(-0.704456\pi\)
−0.599052 + 0.800710i \(0.704456\pi\)
\(462\) 0 0
\(463\) 1.53493e9 0.718711 0.359355 0.933201i \(-0.382997\pi\)
0.359355 + 0.933201i \(0.382997\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.53046e9 0.695364 0.347682 0.937613i \(-0.386969\pi\)
0.347682 + 0.937613i \(0.386969\pi\)
\(468\) 0 0
\(469\) −2.43111e9 −1.08818
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 3.60280e7 0.0156540
\(474\) 0 0
\(475\) −3.20562e8 −0.137241
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −3.06870e9 −1.27579 −0.637897 0.770122i \(-0.720196\pi\)
−0.637897 + 0.770122i \(0.720196\pi\)
\(480\) 0 0
\(481\) −9.60507e8 −0.393543
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1.24601e9 −0.495937
\(486\) 0 0
\(487\) −3.08947e9 −1.21208 −0.606041 0.795433i \(-0.707243\pi\)
−0.606041 + 0.795433i \(0.707243\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 1.55851e9 0.594189 0.297095 0.954848i \(-0.403982\pi\)
0.297095 + 0.954848i \(0.403982\pi\)
\(492\) 0 0
\(493\) 3.53264e9 1.32781
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 4.17220e9 1.52447
\(498\) 0 0
\(499\) 2.26641e8 0.0816559 0.0408279 0.999166i \(-0.487000\pi\)
0.0408279 + 0.999166i \(0.487000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9.93458e8 0.348066 0.174033 0.984740i \(-0.444320\pi\)
0.174033 + 0.984740i \(0.444320\pi\)
\(504\) 0 0
\(505\) 5.31458e7 0.0183632
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2.41773e9 0.812635 0.406317 0.913732i \(-0.366813\pi\)
0.406317 + 0.913732i \(0.366813\pi\)
\(510\) 0 0
\(511\) 3.86872e9 1.28261
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −8.82761e8 −0.284785
\(516\) 0 0
\(517\) −1.57375e8 −0.0500862
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −6.43295e9 −1.99287 −0.996433 0.0843875i \(-0.973107\pi\)
−0.996433 + 0.0843875i \(0.973107\pi\)
\(522\) 0 0
\(523\) 2.45552e9 0.750564 0.375282 0.926911i \(-0.377546\pi\)
0.375282 + 0.926911i \(0.377546\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.75067e9 −0.521036
\(528\) 0 0
\(529\) −2.55078e9 −0.749167
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.66952e8 0.0477582
\(534\) 0 0
\(535\) 2.27732e9 0.642963
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −2.74495e7 −0.00755047
\(540\) 0 0
\(541\) 3.81566e9 1.03605 0.518023 0.855367i \(-0.326668\pi\)
0.518023 + 0.855367i \(0.326668\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −2.84836e9 −0.753715
\(546\) 0 0
\(547\) 1.11439e9 0.291128 0.145564 0.989349i \(-0.453500\pi\)
0.145564 + 0.989349i \(0.453500\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −4.54392e9 −1.15718
\(552\) 0 0
\(553\) 3.00118e9 0.754666
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −3.64478e9 −0.893671 −0.446835 0.894616i \(-0.647449\pi\)
−0.446835 + 0.894616i \(0.647449\pi\)
\(558\) 0 0
\(559\) −3.80095e9 −0.920345
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 1.31668e9 0.310957 0.155478 0.987839i \(-0.450308\pi\)
0.155478 + 0.987839i \(0.450308\pi\)
\(564\) 0 0
\(565\) −8.75498e7 −0.0204214
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1.15078e9 −0.261878 −0.130939 0.991390i \(-0.541799\pi\)
−0.130939 + 0.991390i \(0.541799\pi\)
\(570\) 0 0
\(571\) −8.33459e8 −0.187352 −0.0936759 0.995603i \(-0.529862\pi\)
−0.0936759 + 0.995603i \(0.529862\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.56625e8 0.100166
\(576\) 0 0
\(577\) −1.44998e9 −0.314229 −0.157115 0.987580i \(-0.550219\pi\)
−0.157115 + 0.987580i \(0.550219\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 4.80359e9 1.01613
\(582\) 0 0
\(583\) 4.90765e7 0.0102573
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.30473e9 1.28657 0.643284 0.765627i \(-0.277571\pi\)
0.643284 + 0.765627i \(0.277571\pi\)
\(588\) 0 0
\(589\) 2.25184e9 0.454081
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 5.95526e9 1.17276 0.586381 0.810036i \(-0.300552\pi\)
0.586381 + 0.810036i \(0.300552\pi\)
\(594\) 0 0
\(595\) −1.54715e9 −0.301109
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 3.72932e9 0.708983 0.354491 0.935059i \(-0.384654\pi\)
0.354491 + 0.935059i \(0.384654\pi\)
\(600\) 0 0
\(601\) 5.66326e9 1.06416 0.532078 0.846695i \(-0.321411\pi\)
0.532078 + 0.846695i \(0.321411\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 2.43397e9 0.446861
\(606\) 0 0
\(607\) 5.42376e9 0.984329 0.492164 0.870502i \(-0.336206\pi\)
0.492164 + 0.870502i \(0.336206\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.66030e10 2.94472
\(612\) 0 0
\(613\) −2.55053e9 −0.447217 −0.223608 0.974679i \(-0.571784\pi\)
−0.223608 + 0.974679i \(0.571784\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 6.54132e9 1.12116 0.560580 0.828101i \(-0.310578\pi\)
0.560580 + 0.828101i \(0.310578\pi\)
\(618\) 0 0
\(619\) −1.17083e10 −1.98416 −0.992079 0.125612i \(-0.959911\pi\)
−0.992079 + 0.125612i \(0.959911\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −8.72530e8 −0.144568
\(624\) 0 0
\(625\) 2.44141e8 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.17108e9 0.187633
\(630\) 0 0
\(631\) 1.85340e9 0.293675 0.146838 0.989161i \(-0.453091\pi\)
0.146838 + 0.989161i \(0.453091\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −4.62335e9 −0.716554
\(636\) 0 0
\(637\) 2.89592e9 0.443914
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 9.84313e9 1.47615 0.738074 0.674720i \(-0.235735\pi\)
0.738074 + 0.674720i \(0.235735\pi\)
\(642\) 0 0
\(643\) 6.73935e9 0.999723 0.499861 0.866105i \(-0.333384\pi\)
0.499861 + 0.866105i \(0.333384\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −8.26180e9 −1.19925 −0.599625 0.800281i \(-0.704684\pi\)
−0.599625 + 0.800281i \(0.704684\pi\)
\(648\) 0 0
\(649\) −5.22650e7 −0.00750507
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −2.55833e9 −0.359551 −0.179776 0.983708i \(-0.557537\pi\)
−0.179776 + 0.983708i \(0.557537\pi\)
\(654\) 0 0
\(655\) −6.07553e9 −0.844772
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −8.90633e9 −1.21227 −0.606136 0.795361i \(-0.707281\pi\)
−0.606136 + 0.795361i \(0.707281\pi\)
\(660\) 0 0
\(661\) −5.57591e9 −0.750950 −0.375475 0.926833i \(-0.622520\pi\)
−0.375475 + 0.926833i \(0.622520\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1.99005e9 0.262415
\(666\) 0 0
\(667\) 6.47259e9 0.844574
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −2.63159e8 −0.0336271
\(672\) 0 0
\(673\) 3.92812e9 0.496744 0.248372 0.968665i \(-0.420104\pi\)
0.248372 + 0.968665i \(0.420104\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.25393e10 1.55315 0.776576 0.630023i \(-0.216955\pi\)
0.776576 + 0.630023i \(0.216955\pi\)
\(678\) 0 0
\(679\) 7.73524e9 0.948265
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −2.72047e9 −0.326717 −0.163358 0.986567i \(-0.552233\pi\)
−0.163358 + 0.986567i \(0.552233\pi\)
\(684\) 0 0
\(685\) −3.06440e9 −0.364274
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −5.17757e9 −0.603057
\(690\) 0 0
\(691\) −6.85887e9 −0.790823 −0.395412 0.918504i \(-0.629398\pi\)
−0.395412 + 0.918504i \(0.629398\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.68680e9 0.303590
\(696\) 0 0
\(697\) −2.03554e8 −0.0227701
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −1.76168e9 −0.193158 −0.0965791 0.995325i \(-0.530790\pi\)
−0.0965791 + 0.995325i \(0.530790\pi\)
\(702\) 0 0
\(703\) −1.50633e9 −0.163522
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −3.29929e8 −0.0351117
\(708\) 0 0
\(709\) −2.49745e9 −0.263169 −0.131585 0.991305i \(-0.542006\pi\)
−0.131585 + 0.991305i \(0.542006\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −3.20763e9 −0.331414
\(714\) 0 0
\(715\) 2.02771e8 0.0207460
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 8.33572e9 0.836358 0.418179 0.908365i \(-0.362669\pi\)
0.418179 + 0.908365i \(0.362669\pi\)
\(720\) 0 0
\(721\) 5.48018e9 0.544530
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.46066e9 0.337268
\(726\) 0 0
\(727\) 1.99428e10 1.92494 0.962468 0.271396i \(-0.0874852\pi\)
0.962468 + 0.271396i \(0.0874852\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 4.63424e9 0.438801
\(732\) 0 0
\(733\) 1.63020e10 1.52889 0.764446 0.644687i \(-0.223012\pi\)
0.764446 + 0.644687i \(0.223012\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3.88476e8 −0.0357460
\(738\) 0 0
\(739\) −1.63178e10 −1.48732 −0.743662 0.668556i \(-0.766913\pi\)
−0.743662 + 0.668556i \(0.766913\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.49757e10 1.33945 0.669725 0.742609i \(-0.266412\pi\)
0.669725 + 0.742609i \(0.266412\pi\)
\(744\) 0 0
\(745\) 1.95299e9 0.173042
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1.41376e10 −1.22939
\(750\) 0 0
\(751\) −8.72538e9 −0.751700 −0.375850 0.926681i \(-0.622649\pi\)
−0.375850 + 0.926681i \(0.622649\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −9.78797e9 −0.827710
\(756\) 0 0
\(757\) 4.69261e9 0.393168 0.196584 0.980487i \(-0.437015\pi\)
0.196584 + 0.980487i \(0.437015\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 2.02825e10 1.66831 0.834153 0.551533i \(-0.185957\pi\)
0.834153 + 0.551533i \(0.185957\pi\)
\(762\) 0 0
\(763\) 1.76826e10 1.44116
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 5.51396e9 0.441245
\(768\) 0 0
\(769\) −1.96632e10 −1.55924 −0.779618 0.626255i \(-0.784587\pi\)
−0.779618 + 0.626255i \(0.784587\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.23325e10 0.960337 0.480169 0.877176i \(-0.340575\pi\)
0.480169 + 0.877176i \(0.340575\pi\)
\(774\) 0 0
\(775\) −1.71500e9 −0.132345
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.61825e8 0.0198441
\(780\) 0 0
\(781\) 6.66692e8 0.0500780
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 8.15861e9 0.601966
\(786\) 0 0
\(787\) −8.92456e9 −0.652642 −0.326321 0.945259i \(-0.605809\pi\)
−0.326321 + 0.945259i \(0.605809\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.43509e8 0.0390471
\(792\) 0 0
\(793\) 2.77633e10 1.97704
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.07737e9 −0.0753806 −0.0376903 0.999289i \(-0.512000\pi\)
−0.0376903 + 0.999289i \(0.512000\pi\)
\(798\) 0 0
\(799\) −2.02430e10 −1.40398
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 6.18198e8 0.0421331
\(804\) 0 0
\(805\) −2.83473e9 −0.191525
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −2.40339e7 −0.00159589 −0.000797947 1.00000i \(-0.500254\pi\)
−0.000797947 1.00000i \(0.500254\pi\)
\(810\) 0 0
\(811\) −2.45618e10 −1.61691 −0.808456 0.588556i \(-0.799697\pi\)
−0.808456 + 0.588556i \(0.799697\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.34806e9 0.0872286
\(816\) 0 0
\(817\) −5.96088e9 −0.382414
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.96936e10 −1.24201 −0.621005 0.783807i \(-0.713275\pi\)
−0.621005 + 0.783807i \(0.713275\pi\)
\(822\) 0 0
\(823\) −2.62460e10 −1.64121 −0.820604 0.571497i \(-0.806363\pi\)
−0.820604 + 0.571497i \(0.806363\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.66841e10 −1.64052 −0.820262 0.571988i \(-0.806172\pi\)
−0.820262 + 0.571988i \(0.806172\pi\)
\(828\) 0 0
\(829\) 6.66586e9 0.406364 0.203182 0.979141i \(-0.434872\pi\)
0.203182 + 0.979141i \(0.434872\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −3.53080e9 −0.211649
\(834\) 0 0
\(835\) 4.87581e9 0.289831
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1.93131e10 −1.12898 −0.564490 0.825440i \(-0.690927\pi\)
−0.564490 + 0.825440i \(0.690927\pi\)
\(840\) 0 0
\(841\) 3.18044e10 1.84375
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.35488e10 −0.772505
\(846\) 0 0
\(847\) −1.51101e10 −0.854429
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 2.14568e9 0.119347
\(852\) 0 0
\(853\) 2.38350e10 1.31490 0.657451 0.753497i \(-0.271635\pi\)
0.657451 + 0.753497i \(0.271635\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2.28013e10 −1.23745 −0.618723 0.785609i \(-0.712350\pi\)
−0.618723 + 0.785609i \(0.712350\pi\)
\(858\) 0 0
\(859\) −2.27333e10 −1.22373 −0.611866 0.790961i \(-0.709581\pi\)
−0.611866 + 0.790961i \(0.709581\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1.04541e9 0.0553668 0.0276834 0.999617i \(-0.491187\pi\)
0.0276834 + 0.999617i \(0.491187\pi\)
\(864\) 0 0
\(865\) −4.24072e9 −0.222784
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 4.79570e8 0.0247904
\(870\) 0 0
\(871\) 4.09842e10 2.10161
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.51562e9 −0.0764828
\(876\) 0 0
\(877\) −2.09543e10 −1.04900 −0.524498 0.851412i \(-0.675747\pi\)
−0.524498 + 0.851412i \(0.675747\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −8.44270e9 −0.415974 −0.207987 0.978132i \(-0.566691\pi\)
−0.207987 + 0.978132i \(0.566691\pi\)
\(882\) 0 0
\(883\) 1.96174e10 0.958911 0.479455 0.877566i \(-0.340834\pi\)
0.479455 + 0.877566i \(0.340834\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.06979e10 0.514713 0.257356 0.966317i \(-0.417149\pi\)
0.257356 + 0.966317i \(0.417149\pi\)
\(888\) 0 0
\(889\) 2.87018e10 1.37010
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 2.60379e10 1.22356
\(894\) 0 0
\(895\) −1.81197e10 −0.844834
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −2.43099e10 −1.11590
\(900\) 0 0
\(901\) 6.31266e9 0.287525
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −1.74059e10 −0.780596
\(906\) 0 0
\(907\) −3.17099e10 −1.41114 −0.705569 0.708641i \(-0.749308\pi\)
−0.705569 + 0.708641i \(0.749308\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −3.00203e10 −1.31553 −0.657766 0.753223i \(-0.728498\pi\)
−0.657766 + 0.753223i \(0.728498\pi\)
\(912\) 0 0
\(913\) 7.67584e8 0.0333794
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3.77169e10 1.61526
\(918\) 0 0
\(919\) −1.17761e10 −0.500494 −0.250247 0.968182i \(-0.580512\pi\)
−0.250247 + 0.968182i \(0.580512\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −7.03361e10 −2.94423
\(924\) 0 0
\(925\) 1.14722e9 0.0476596
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.88395e10 −0.770928 −0.385464 0.922723i \(-0.625959\pi\)
−0.385464 + 0.922723i \(0.625959\pi\)
\(930\) 0 0
\(931\) 4.54157e9 0.184451
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −2.47225e8 −0.00989126
\(936\) 0 0
\(937\) −2.45667e9 −0.0975568 −0.0487784 0.998810i \(-0.515533\pi\)
−0.0487784 + 0.998810i \(0.515533\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −3.10769e10 −1.21583 −0.607916 0.794001i \(-0.707994\pi\)
−0.607916 + 0.794001i \(0.707994\pi\)
\(942\) 0 0
\(943\) −3.72957e8 −0.0144833
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.82161e10 0.696998 0.348499 0.937309i \(-0.386691\pi\)
0.348499 + 0.937309i \(0.386691\pi\)
\(948\) 0 0
\(949\) −6.52199e10 −2.47713
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 3.90305e10 1.46076 0.730380 0.683041i \(-0.239343\pi\)
0.730380 + 0.683041i \(0.239343\pi\)
\(954\) 0 0
\(955\) −3.03682e9 −0.112825
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.90238e10 0.696518
\(960\) 0 0
\(961\) −1.54654e10 −0.562119
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.31032e10 −0.469386
\(966\) 0 0
\(967\) 1.22959e10 0.437289 0.218645 0.975805i \(-0.429836\pi\)
0.218645 + 0.975805i \(0.429836\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −4.44806e8 −0.0155921 −0.00779603 0.999970i \(-0.502482\pi\)
−0.00779603 + 0.999970i \(0.502482\pi\)
\(972\) 0 0
\(973\) −1.66796e10 −0.580486
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.57833e10 −0.541459 −0.270730 0.962655i \(-0.587265\pi\)
−0.270730 + 0.962655i \(0.587265\pi\)
\(978\) 0 0
\(979\) −1.39425e8 −0.00474898
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 2.45516e10 0.824409 0.412204 0.911091i \(-0.364759\pi\)
0.412204 + 0.911091i \(0.364759\pi\)
\(984\) 0 0
\(985\) −2.56958e10 −0.856712
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 8.49097e9 0.279107
\(990\) 0 0
\(991\) 4.56393e10 1.48964 0.744819 0.667266i \(-0.232536\pi\)
0.744819 + 0.667266i \(0.232536\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −5.70411e9 −0.183572
\(996\) 0 0
\(997\) 4.18036e10 1.33592 0.667960 0.744197i \(-0.267168\pi\)
0.667960 + 0.744197i \(0.267168\pi\)
\(998\) 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 360.8.a.b.1.1 1
3.2 odd 2 40.8.a.a.1.1 1
12.11 even 2 80.8.a.c.1.1 1
15.2 even 4 200.8.c.d.49.2 2
15.8 even 4 200.8.c.d.49.1 2
15.14 odd 2 200.8.a.g.1.1 1
24.5 odd 2 320.8.a.g.1.1 1
24.11 even 2 320.8.a.b.1.1 1
60.23 odd 4 400.8.c.h.49.2 2
60.47 odd 4 400.8.c.h.49.1 2
60.59 even 2 400.8.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.8.a.a.1.1 1 3.2 odd 2
80.8.a.c.1.1 1 12.11 even 2
200.8.a.g.1.1 1 15.14 odd 2
200.8.c.d.49.1 2 15.8 even 4
200.8.c.d.49.2 2 15.2 even 4
320.8.a.b.1.1 1 24.11 even 2
320.8.a.g.1.1 1 24.5 odd 2
360.8.a.b.1.1 1 1.1 even 1 trivial
400.8.a.g.1.1 1 60.59 even 2
400.8.c.h.49.1 2 60.47 odd 4
400.8.c.h.49.2 2 60.23 odd 4