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Properties

Label 180.8.a.c.1.1
Level $180$
Weight $8$
Character 180.1
Self dual yes
Analytic conductor $56.229$
Analytic rank $1$
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] = [180,8,Mod(1,180)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(180, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("180.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 180 = 2^{2} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 180.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: \(56.2293045871\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 20)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 180.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+125.000 q^{5} -706.000 q^{7} +3840.00 q^{11} -4054.00 q^{13} -858.000 q^{17} +21044.0 q^{19} -85338.0 q^{23} +15625.0 q^{25} +83106.0 q^{29} -145564. q^{31} -88250.0 q^{35} -498886. q^{37} +689514. q^{41} +867890. q^{43} -235638. q^{47} -325107. q^{49} -1.83544e6 q^{53} +480000. q^{55} -629508. q^{59} -2.66796e6 q^{61} -506750. q^{65} -3.37331e6 q^{67} +2.60005e6 q^{71} -1.62849e6 q^{73} -2.71104e6 q^{77} -4.24353e6 q^{79} -1.25138e6 q^{83} -107250. q^{85} -6.29947e6 q^{89} +2.86212e6 q^{91} +2.63050e6 q^{95} +3.97651e6 q^{97} +O(q^{100})\)

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
\(7\) −706.000 −0.777968 −0.388984 − 0.921245i \(-0.627174\pi\)
−0.388984 + 0.921245i \(0.627174\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3840.00 0.869875 0.434937 − 0.900461i \(-0.356770\pi\)
0.434937 + 0.900461i \(0.356770\pi\)
\(12\) 0 0
\(13\) −4054.00 −0.511778 −0.255889 − 0.966706i \(-0.582368\pi\)
−0.255889 + 0.966706i \(0.582368\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −858.000 −0.0423561 −0.0211781 − 0.999776i \(-0.506742\pi\)
−0.0211781 + 0.999776i \(0.506742\pi\)
\(18\) 0 0
\(19\) 21044.0 0.703867 0.351934 − 0.936025i \(-0.385524\pi\)
0.351934 + 0.936025i \(0.385524\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −85338.0 −1.46250 −0.731249 − 0.682111i \(-0.761062\pi\)
−0.731249 + 0.682111i \(0.761062\pi\)
\(24\) 0 0
\(25\) 15625.0 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 83106.0 0.632761 0.316380 − 0.948632i \(-0.397532\pi\)
0.316380 + 0.948632i \(0.397532\pi\)
\(30\) 0 0
\(31\) −145564. −0.877583 −0.438791 − 0.898589i \(-0.644593\pi\)
−0.438791 + 0.898589i \(0.644593\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −88250.0 −0.347918
\(36\) 0 0
\(37\) −498886. −1.61918 −0.809590 − 0.586995i \(-0.800311\pi\)
−0.809590 + 0.586995i \(0.800311\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 689514. 1.56243 0.781213 − 0.624264i \(-0.214601\pi\)
0.781213 + 0.624264i \(0.214601\pi\)
\(42\) 0 0
\(43\) 867890. 1.66466 0.832329 − 0.554282i \(-0.187007\pi\)
0.832329 + 0.554282i \(0.187007\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −235638. −0.331057 −0.165529 − 0.986205i \(-0.552933\pi\)
−0.165529 + 0.986205i \(0.552933\pi\)
\(48\) 0 0
\(49\) −325107. −0.394766
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.83544e6 −1.69346 −0.846730 − 0.532022i \(-0.821432\pi\)
−0.846730 + 0.532022i \(0.821432\pi\)
\(54\) 0 0
\(55\) 480000. 0.389020
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −629508. −0.399043 −0.199521 − 0.979893i \(-0.563939\pi\)
−0.199521 + 0.979893i \(0.563939\pi\)
\(60\) 0 0
\(61\) −2.66796e6 −1.50496 −0.752479 − 0.658616i \(-0.771142\pi\)
−0.752479 + 0.658616i \(0.771142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −506750. −0.228874
\(66\) 0 0
\(67\) −3.37331e6 −1.37023 −0.685116 − 0.728434i \(-0.740248\pi\)
−0.685116 + 0.728434i \(0.740248\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.60005e6 0.862140 0.431070 − 0.902318i \(-0.358136\pi\)
0.431070 + 0.902318i \(0.358136\pi\)
\(72\) 0 0
\(73\) −1.62849e6 −0.489955 −0.244977 − 0.969529i \(-0.578781\pi\)
−0.244977 + 0.969529i \(0.578781\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −2.71104e6 −0.676735
\(78\) 0 0
\(79\) −4.24353e6 −0.968350 −0.484175 − 0.874971i \(-0.660880\pi\)
−0.484175 + 0.874971i \(0.660880\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1.25138e6 −0.240223 −0.120112 − 0.992760i \(-0.538325\pi\)
−0.120112 + 0.992760i \(0.538325\pi\)
\(84\) 0 0
\(85\) −107250. −0.0189422
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −6.29947e6 −0.947194 −0.473597 − 0.880742i \(-0.657045\pi\)
−0.473597 + 0.880742i \(0.657045\pi\)
\(90\) 0 0
\(91\) 2.86212e6 0.398147
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.63050e6 0.314779
\(96\) 0 0
\(97\) 3.97651e6 0.442386 0.221193 − 0.975230i \(-0.429005\pi\)
0.221193 + 0.975230i \(0.429005\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −1.25053e7 −1.20773 −0.603865 − 0.797086i \(-0.706374\pi\)
−0.603865 + 0.797086i \(0.706374\pi\)
\(102\) 0 0
\(103\) −2.17226e7 −1.95876 −0.979382 − 0.202015i \(-0.935251\pi\)
−0.979382 + 0.202015i \(0.935251\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.04592e6 −0.240367 −0.120184 − 0.992752i \(-0.538348\pi\)
−0.120184 + 0.992752i \(0.538348\pi\)
\(108\) 0 0
\(109\) −1.37261e7 −1.01521 −0.507603 − 0.861591i \(-0.669468\pi\)
−0.507603 + 0.861591i \(0.669468\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 2.03152e7 1.32449 0.662243 − 0.749289i \(-0.269605\pi\)
0.662243 + 0.749289i \(0.269605\pi\)
\(114\) 0 0
\(115\) −1.06672e7 −0.654049
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 605748. 0.0329517
\(120\) 0 0
\(121\) −4.74157e6 −0.243318
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.95312e6 0.0894427
\(126\) 0 0
\(127\) −1.44019e7 −0.623888 −0.311944 − 0.950100i \(-0.600980\pi\)
−0.311944 + 0.950100i \(0.600980\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.51283e7 −1.36524 −0.682618 − 0.730775i \(-0.739159\pi\)
−0.682618 + 0.730775i \(0.739159\pi\)
\(132\) 0 0
\(133\) −1.48571e7 −0.547586
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.66669e7 −0.886036 −0.443018 − 0.896513i \(-0.646092\pi\)
−0.443018 + 0.896513i \(0.646092\pi\)
\(138\) 0 0
\(139\) 5.37105e7 1.69632 0.848159 − 0.529742i \(-0.177711\pi\)
0.848159 + 0.529742i \(0.177711\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.55674e7 −0.445183
\(144\) 0 0
\(145\) 1.03882e7 0.282979
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 3.57914e7 0.886393 0.443197 − 0.896424i \(-0.353844\pi\)
0.443197 + 0.896424i \(0.353844\pi\)
\(150\) 0 0
\(151\) −8.13922e6 −0.192382 −0.0961908 − 0.995363i \(-0.530666\pi\)
−0.0961908 + 0.995363i \(0.530666\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.81955e7 −0.392467
\(156\) 0 0
\(157\) 5.10735e6 0.105329 0.0526643 − 0.998612i \(-0.483229\pi\)
0.0526643 + 0.998612i \(0.483229\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 6.02486e7 1.13778
\(162\) 0 0
\(163\) 8.85622e7 1.60174 0.800870 − 0.598839i \(-0.204371\pi\)
0.800870 + 0.598839i \(0.204371\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.12170e8 1.86367 0.931834 − 0.362885i \(-0.118208\pi\)
0.931834 + 0.362885i \(0.118208\pi\)
\(168\) 0 0
\(169\) −4.63136e7 −0.738083
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.03754e7 −0.152350 −0.0761749 − 0.997094i \(-0.524271\pi\)
−0.0761749 + 0.997094i \(0.524271\pi\)
\(174\) 0 0
\(175\) −1.10312e7 −0.155594
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1.91537e7 −0.249613 −0.124806 − 0.992181i \(-0.539831\pi\)
−0.124806 + 0.992181i \(0.539831\pi\)
\(180\) 0 0
\(181\) −2.46040e7 −0.308411 −0.154206 − 0.988039i \(-0.549282\pi\)
−0.154206 + 0.988039i \(0.549282\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −6.23608e7 −0.724120
\(186\) 0 0
\(187\) −3.29472e6 −0.0368445
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.15013e8 1.19434 0.597171 − 0.802114i \(-0.296291\pi\)
0.597171 + 0.802114i \(0.296291\pi\)
\(192\) 0 0
\(193\) 1.31761e8 1.31928 0.659640 − 0.751582i \(-0.270709\pi\)
0.659640 + 0.751582i \(0.270709\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.46842e8 −1.36841 −0.684206 − 0.729288i \(-0.739851\pi\)
−0.684206 + 0.729288i \(0.739851\pi\)
\(198\) 0 0
\(199\) 1.27107e8 1.14336 0.571679 − 0.820477i \(-0.306292\pi\)
0.571679 + 0.820477i \(0.306292\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −5.86728e7 −0.492267
\(204\) 0 0
\(205\) 8.61892e7 0.698738
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.08090e7 0.612276
\(210\) 0 0
\(211\) 8.90501e7 0.652598 0.326299 − 0.945267i \(-0.394198\pi\)
0.326299 + 0.945267i \(0.394198\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 1.08486e8 0.744458
\(216\) 0 0
\(217\) 1.02768e8 0.682731
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 3.47833e6 0.0216769
\(222\) 0 0
\(223\) −4.78436e7 −0.288906 −0.144453 − 0.989512i \(-0.546142\pi\)
−0.144453 + 0.989512i \(0.546142\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.51060e7 0.426171 0.213086 − 0.977034i \(-0.431649\pi\)
0.213086 + 0.977034i \(0.431649\pi\)
\(228\) 0 0
\(229\) −5.53497e7 −0.304573 −0.152286 − 0.988336i \(-0.548664\pi\)
−0.152286 + 0.988336i \(0.548664\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 2.88910e8 1.49629 0.748147 − 0.663533i \(-0.230944\pi\)
0.748147 + 0.663533i \(0.230944\pi\)
\(234\) 0 0
\(235\) −2.94548e7 −0.148053
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 9.79382e7 0.464044 0.232022 − 0.972710i \(-0.425466\pi\)
0.232022 + 0.972710i \(0.425466\pi\)
\(240\) 0 0
\(241\) 8.64399e7 0.397790 0.198895 − 0.980021i \(-0.436265\pi\)
0.198895 + 0.980021i \(0.436265\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −4.06384e7 −0.176545
\(246\) 0 0
\(247\) −8.53124e7 −0.360224
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.35898e8 −0.542443 −0.271221 − 0.962517i \(-0.587428\pi\)
−0.271221 + 0.962517i \(0.587428\pi\)
\(252\) 0 0
\(253\) −3.27698e8 −1.27219
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −4.52157e8 −1.66159 −0.830793 − 0.556581i \(-0.812113\pi\)
−0.830793 + 0.556581i \(0.812113\pi\)
\(258\) 0 0
\(259\) 3.52214e8 1.25967
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6.05973e7 0.205404 0.102702 − 0.994712i \(-0.467251\pi\)
0.102702 + 0.994712i \(0.467251\pi\)
\(264\) 0 0
\(265\) −2.29430e8 −0.757339
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3.74181e7 0.117206 0.0586029 − 0.998281i \(-0.481335\pi\)
0.0586029 + 0.998281i \(0.481335\pi\)
\(270\) 0 0
\(271\) 5.16074e8 1.57514 0.787571 − 0.616224i \(-0.211339\pi\)
0.787571 + 0.616224i \(0.211339\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 6.00000e7 0.173975
\(276\) 0 0
\(277\) −7.86991e7 −0.222480 −0.111240 − 0.993794i \(-0.535482\pi\)
−0.111240 + 0.993794i \(0.535482\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −5.36047e7 −0.144122 −0.0720611 − 0.997400i \(-0.522958\pi\)
−0.0720611 + 0.997400i \(0.522958\pi\)
\(282\) 0 0
\(283\) 1.17665e8 0.308599 0.154299 − 0.988024i \(-0.450688\pi\)
0.154299 + 0.988024i \(0.450688\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −4.86797e8 −1.21552
\(288\) 0 0
\(289\) −4.09603e8 −0.998206
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 2.79303e8 0.648693 0.324347 − 0.945938i \(-0.394856\pi\)
0.324347 + 0.945938i \(0.394856\pi\)
\(294\) 0 0
\(295\) −7.86885e7 −0.178457
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 3.45960e8 0.748475
\(300\) 0 0
\(301\) −6.12730e8 −1.29505
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −3.33495e8 −0.673038
\(306\) 0 0
\(307\) −6.72857e8 −1.32721 −0.663603 − 0.748085i \(-0.730973\pi\)
−0.663603 + 0.748085i \(0.730973\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3.93153e7 −0.0741139 −0.0370570 − 0.999313i \(-0.511798\pi\)
−0.0370570 + 0.999313i \(0.511798\pi\)
\(312\) 0 0
\(313\) −1.92264e8 −0.354399 −0.177199 − 0.984175i \(-0.556704\pi\)
−0.177199 + 0.984175i \(0.556704\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7.31614e7 −0.128995 −0.0644977 − 0.997918i \(-0.520545\pi\)
−0.0644977 + 0.997918i \(0.520545\pi\)
\(318\) 0 0
\(319\) 3.19127e8 0.550423
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −1.80558e7 −0.0298131
\(324\) 0 0
\(325\) −6.33438e7 −0.102356
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 1.66360e8 0.257552
\(330\) 0 0
\(331\) −9.16104e8 −1.38850 −0.694252 − 0.719732i \(-0.744265\pi\)
−0.694252 + 0.719732i \(0.744265\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −4.21663e8 −0.612786
\(336\) 0 0
\(337\) −7.56503e8 −1.07673 −0.538364 − 0.842712i \(-0.680958\pi\)
−0.538364 + 0.842712i \(0.680958\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −5.58966e8 −0.763387
\(342\) 0 0
\(343\) 8.10947e8 1.08508
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −3.87175e8 −0.497456 −0.248728 − 0.968573i \(-0.580012\pi\)
−0.248728 + 0.968573i \(0.580012\pi\)
\(348\) 0 0
\(349\) −1.99761e7 −0.0251548 −0.0125774 − 0.999921i \(-0.504004\pi\)
−0.0125774 + 0.999921i \(0.504004\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.36499e9 −1.65165 −0.825827 − 0.563923i \(-0.809291\pi\)
−0.825827 + 0.563923i \(0.809291\pi\)
\(354\) 0 0
\(355\) 3.25006e8 0.385561
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1.09937e9 1.25404 0.627022 − 0.779001i \(-0.284274\pi\)
0.627022 + 0.779001i \(0.284274\pi\)
\(360\) 0 0
\(361\) −4.51022e8 −0.504571
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2.03562e8 −0.219115
\(366\) 0 0
\(367\) 3.64118e8 0.384513 0.192256 − 0.981345i \(-0.438419\pi\)
0.192256 + 0.981345i \(0.438419\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 1.29582e9 1.31746
\(372\) 0 0
\(373\) 9.57949e7 0.0955788 0.0477894 − 0.998857i \(-0.484782\pi\)
0.0477894 + 0.998857i \(0.484782\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3.36912e8 −0.323833
\(378\) 0 0
\(379\) −6.84090e8 −0.645470 −0.322735 − 0.946489i \(-0.604602\pi\)
−0.322735 + 0.946489i \(0.604602\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 8.50349e8 0.773395 0.386698 − 0.922207i \(-0.373616\pi\)
0.386698 + 0.922207i \(0.373616\pi\)
\(384\) 0 0
\(385\) −3.38880e8 −0.302645
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.28862e8 0.110994 0.0554972 − 0.998459i \(-0.482326\pi\)
0.0554972 + 0.998459i \(0.482326\pi\)
\(390\) 0 0
\(391\) 7.32200e7 0.0619457
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −5.30441e8 −0.433059
\(396\) 0 0
\(397\) −1.47383e9 −1.18217 −0.591084 − 0.806610i \(-0.701300\pi\)
−0.591084 + 0.806610i \(0.701300\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.00938e9 0.781717 0.390858 − 0.920451i \(-0.372178\pi\)
0.390858 + 0.920451i \(0.372178\pi\)
\(402\) 0 0
\(403\) 5.90116e8 0.449128
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −1.91572e9 −1.40848
\(408\) 0 0
\(409\) 8.79404e8 0.635561 0.317780 − 0.948164i \(-0.397063\pi\)
0.317780 + 0.948164i \(0.397063\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 4.44433e8 0.310442
\(414\) 0 0
\(415\) −1.56422e8 −0.107431
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 2.38560e9 1.58434 0.792170 − 0.610300i \(-0.208951\pi\)
0.792170 + 0.610300i \(0.208951\pi\)
\(420\) 0 0
\(421\) 1.41780e9 0.926034 0.463017 − 0.886349i \(-0.346767\pi\)
0.463017 + 0.886349i \(0.346767\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.34063e7 −0.00847122
\(426\) 0 0
\(427\) 1.88358e9 1.17081
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −2.47434e9 −1.48864 −0.744319 − 0.667825i \(-0.767226\pi\)
−0.744319 + 0.667825i \(0.767226\pi\)
\(432\) 0 0
\(433\) −1.42266e8 −0.0842160 −0.0421080 − 0.999113i \(-0.513407\pi\)
−0.0421080 + 0.999113i \(0.513407\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.79585e9 −1.02940
\(438\) 0 0
\(439\) 2.80718e9 1.58359 0.791797 − 0.610785i \(-0.209146\pi\)
0.791797 + 0.610785i \(0.209146\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 2.29630e9 1.25492 0.627460 − 0.778649i \(-0.284095\pi\)
0.627460 + 0.778649i \(0.284095\pi\)
\(444\) 0 0
\(445\) −7.87433e8 −0.423598
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −1.63361e8 −0.0851700 −0.0425850 − 0.999093i \(-0.513559\pi\)
−0.0425850 + 0.999093i \(0.513559\pi\)
\(450\) 0 0
\(451\) 2.64773e9 1.35912
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 3.57766e8 0.178057
\(456\) 0 0
\(457\) −3.28894e9 −1.61194 −0.805970 − 0.591956i \(-0.798356\pi\)
−0.805970 + 0.591956i \(0.798356\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 9.55698e8 0.454326 0.227163 − 0.973857i \(-0.427055\pi\)
0.227163 + 0.973857i \(0.427055\pi\)
\(462\) 0 0
\(463\) −1.26696e9 −0.593237 −0.296618 − 0.954996i \(-0.595859\pi\)
−0.296618 + 0.954996i \(0.595859\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −3.93953e9 −1.78993 −0.894963 − 0.446140i \(-0.852798\pi\)
−0.894963 + 0.446140i \(0.852798\pi\)
\(468\) 0 0
\(469\) 2.38155e9 1.06600
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 3.33270e9 1.44804
\(474\) 0 0
\(475\) 3.28812e8 0.140773
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 3.23581e9 1.34527 0.672634 − 0.739975i \(-0.265163\pi\)
0.672634 + 0.739975i \(0.265163\pi\)
\(480\) 0 0
\(481\) 2.02248e9 0.828662
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.97064e8 0.197841
\(486\) 0 0
\(487\) 1.47239e9 0.577658 0.288829 − 0.957381i \(-0.406734\pi\)
0.288829 + 0.957381i \(0.406734\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 3.21107e9 1.22423 0.612116 − 0.790768i \(-0.290318\pi\)
0.612116 + 0.790768i \(0.290318\pi\)
\(492\) 0 0
\(493\) −7.13049e7 −0.0268013
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.83564e9 −0.670717
\(498\) 0 0
\(499\) −4.19098e8 −0.150995 −0.0754976 − 0.997146i \(-0.524055\pi\)
−0.0754976 + 0.997146i \(0.524055\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 3.70706e9 1.29880 0.649399 − 0.760448i \(-0.275021\pi\)
0.649399 + 0.760448i \(0.275021\pi\)
\(504\) 0 0
\(505\) −1.56317e9 −0.540114
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 2.28893e9 0.769343 0.384672 − 0.923053i \(-0.374315\pi\)
0.384672 + 0.923053i \(0.374315\pi\)
\(510\) 0 0
\(511\) 1.14972e9 0.381169
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.71533e9 −0.875986
\(516\) 0 0
\(517\) −9.04850e8 −0.287978
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.90038e9 −0.588719 −0.294359 − 0.955695i \(-0.595106\pi\)
−0.294359 + 0.955695i \(0.595106\pi\)
\(522\) 0 0
\(523\) 6.07040e9 1.85550 0.927751 − 0.373200i \(-0.121739\pi\)
0.927751 + 0.373200i \(0.121739\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.24894e8 0.0371710
\(528\) 0 0
\(529\) 3.87775e9 1.13890
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −2.79529e9 −0.799616
\(534\) 0 0
\(535\) −3.80740e8 −0.107495
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −1.24841e9 −0.343397
\(540\) 0 0
\(541\) −4.10261e9 −1.11396 −0.556981 − 0.830526i \(-0.688040\pi\)
−0.556981 + 0.830526i \(0.688040\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1.71576e9 −0.454014
\(546\) 0 0
\(547\) −2.53877e9 −0.663234 −0.331617 − 0.943414i \(-0.607594\pi\)
−0.331617 + 0.943414i \(0.607594\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.74888e9 0.445379
\(552\) 0 0
\(553\) 2.99593e9 0.753345
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −3.19358e9 −0.783041 −0.391520 − 0.920169i \(-0.628051\pi\)
−0.391520 + 0.920169i \(0.628051\pi\)
\(558\) 0 0
\(559\) −3.51843e9 −0.851936
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −3.52214e9 −0.831815 −0.415908 − 0.909407i \(-0.636536\pi\)
−0.415908 + 0.909407i \(0.636536\pi\)
\(564\) 0 0
\(565\) 2.53940e9 0.592328
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 5.35081e9 1.21766 0.608831 − 0.793300i \(-0.291639\pi\)
0.608831 + 0.793300i \(0.291639\pi\)
\(570\) 0 0
\(571\) −2.59117e9 −0.582464 −0.291232 − 0.956652i \(-0.594065\pi\)
−0.291232 + 0.956652i \(0.594065\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −1.33341e9 −0.292499
\(576\) 0 0
\(577\) −3.50238e8 −0.0759011 −0.0379505 − 0.999280i \(-0.512083\pi\)
−0.0379505 + 0.999280i \(0.512083\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 8.83473e8 0.186886
\(582\) 0 0
\(583\) −7.04810e9 −1.47310
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.42501e9 0.494857 0.247429 − 0.968906i \(-0.420414\pi\)
0.247429 + 0.968906i \(0.420414\pi\)
\(588\) 0 0
\(589\) −3.06325e9 −0.617702
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −7.57894e9 −1.49251 −0.746254 − 0.665661i \(-0.768150\pi\)
−0.746254 + 0.665661i \(0.768150\pi\)
\(594\) 0 0
\(595\) 7.57185e7 0.0147364
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 3.06431e9 0.582557 0.291279 − 0.956638i \(-0.405919\pi\)
0.291279 + 0.956638i \(0.405919\pi\)
\(600\) 0 0
\(601\) −8.51705e9 −1.60040 −0.800200 − 0.599733i \(-0.795273\pi\)
−0.800200 + 0.599733i \(0.795273\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −5.92696e8 −0.108815
\(606\) 0 0
\(607\) −5.51105e9 −1.00017 −0.500085 − 0.865976i \(-0.666698\pi\)
−0.500085 + 0.865976i \(0.666698\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.55276e8 0.169428
\(612\) 0 0
\(613\) 1.31574e9 0.230706 0.115353 − 0.993325i \(-0.463200\pi\)
0.115353 + 0.993325i \(0.463200\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 9.70804e8 0.166392 0.0831962 − 0.996533i \(-0.473487\pi\)
0.0831962 + 0.996533i \(0.473487\pi\)
\(618\) 0 0
\(619\) 6.96530e9 1.18038 0.590191 − 0.807264i \(-0.299052\pi\)
0.590191 + 0.807264i \(0.299052\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 4.44742e9 0.736886
\(624\) 0 0
\(625\) 2.44141e8 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4.28044e8 0.0685822
\(630\) 0 0
\(631\) 1.57127e9 0.248971 0.124485 − 0.992221i \(-0.460272\pi\)
0.124485 + 0.992221i \(0.460272\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1.80024e9 −0.279011
\(636\) 0 0
\(637\) 1.31798e9 0.202033
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −4.19744e9 −0.629479 −0.314740 − 0.949178i \(-0.601917\pi\)
−0.314740 + 0.949178i \(0.601917\pi\)
\(642\) 0 0
\(643\) 5.02531e9 0.745460 0.372730 − 0.927940i \(-0.378422\pi\)
0.372730 + 0.927940i \(0.378422\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.28047e10 −1.85867 −0.929337 − 0.369232i \(-0.879621\pi\)
−0.929337 + 0.369232i \(0.879621\pi\)
\(648\) 0 0
\(649\) −2.41731e9 −0.347117
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.11817e10 1.57148 0.785742 − 0.618554i \(-0.212281\pi\)
0.785742 + 0.618554i \(0.212281\pi\)
\(654\) 0 0
\(655\) −4.39104e9 −0.610552
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −4.11341e9 −0.559891 −0.279945 − 0.960016i \(-0.590316\pi\)
−0.279945 + 0.960016i \(0.590316\pi\)
\(660\) 0 0
\(661\) 4.55818e9 0.613884 0.306942 − 0.951728i \(-0.400694\pi\)
0.306942 + 0.951728i \(0.400694\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.85713e9 −0.244888
\(666\) 0 0
\(667\) −7.09210e9 −0.925410
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −1.02450e10 −1.30913
\(672\) 0 0
\(673\) −7.02401e9 −0.888244 −0.444122 − 0.895966i \(-0.646484\pi\)
−0.444122 + 0.895966i \(0.646484\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −6.27731e9 −0.777523 −0.388762 − 0.921338i \(-0.627097\pi\)
−0.388762 + 0.921338i \(0.627097\pi\)
\(678\) 0 0
\(679\) −2.80742e9 −0.344162
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −6.99148e9 −0.839647 −0.419823 − 0.907606i \(-0.637908\pi\)
−0.419823 + 0.907606i \(0.637908\pi\)
\(684\) 0 0
\(685\) −3.33337e9 −0.396247
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 7.44088e9 0.866677
\(690\) 0 0
\(691\) −4.17342e9 −0.481193 −0.240596 − 0.970625i \(-0.577343\pi\)
−0.240596 + 0.970625i \(0.577343\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.71381e9 0.758616
\(696\) 0 0
\(697\) −5.91603e8 −0.0661783
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 6.73860e9 0.738851 0.369425 − 0.929260i \(-0.379555\pi\)
0.369425 + 0.929260i \(0.379555\pi\)
\(702\) 0 0
\(703\) −1.04986e10 −1.13969
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.82876e9 0.939576
\(708\) 0 0
\(709\) −7.10555e9 −0.748749 −0.374374 − 0.927278i \(-0.622142\pi\)
−0.374374 + 0.927278i \(0.622142\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1.24221e10 1.28346
\(714\) 0 0
\(715\) −1.94592e9 −0.199092
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −1.77811e10 −1.78405 −0.892024 − 0.451988i \(-0.850715\pi\)
−0.892024 + 0.451988i \(0.850715\pi\)
\(720\) 0 0
\(721\) 1.53362e10 1.52386
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.29853e9 0.126552
\(726\) 0 0
\(727\) −3.27207e9 −0.315829 −0.157915 − 0.987453i \(-0.550477\pi\)
−0.157915 + 0.987453i \(0.550477\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −7.44650e8 −0.0705084
\(732\) 0 0
\(733\) −7.15509e9 −0.671044 −0.335522 − 0.942032i \(-0.608913\pi\)
−0.335522 + 0.942032i \(0.608913\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.29535e10 −1.19193
\(738\) 0 0
\(739\) 1.70559e9 0.155460 0.0777299 − 0.996974i \(-0.475233\pi\)
0.0777299 + 0.996974i \(0.475233\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 9.59074e9 0.857810 0.428905 − 0.903350i \(-0.358899\pi\)
0.428905 + 0.903350i \(0.358899\pi\)
\(744\) 0 0
\(745\) 4.47392e9 0.396407
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2.15042e9 0.186998
\(750\) 0 0
\(751\) 6.33249e9 0.545550 0.272775 − 0.962078i \(-0.412059\pi\)
0.272775 + 0.962078i \(0.412059\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1.01740e9 −0.0860357
\(756\) 0 0
\(757\) 1.95934e10 1.64163 0.820813 − 0.571197i \(-0.193521\pi\)
0.820813 + 0.571197i \(0.193521\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.16891e10 0.961471 0.480736 − 0.876866i \(-0.340370\pi\)
0.480736 + 0.876866i \(0.340370\pi\)
\(762\) 0 0
\(763\) 9.69063e9 0.789798
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.55203e9 0.204221
\(768\) 0 0
\(769\) 2.10168e10 1.66657 0.833287 − 0.552841i \(-0.186456\pi\)
0.833287 + 0.552841i \(0.186456\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −2.12393e10 −1.65391 −0.826956 − 0.562267i \(-0.809929\pi\)
−0.826956 + 0.562267i \(0.809929\pi\)
\(774\) 0 0
\(775\) −2.27444e9 −0.175517
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.45101e10 1.09974
\(780\) 0 0
\(781\) 9.98420e9 0.749954
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 6.38418e8 0.0471044
\(786\) 0 0
\(787\) −8.33601e9 −0.609602 −0.304801 − 0.952416i \(-0.598590\pi\)
−0.304801 + 0.952416i \(0.598590\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −1.43426e10 −1.03041
\(792\) 0 0
\(793\) 1.08159e10 0.770205
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.60775e10 1.12490 0.562451 − 0.826830i \(-0.309858\pi\)
0.562451 + 0.826830i \(0.309858\pi\)
\(798\) 0 0
\(799\) 2.02177e8 0.0140223
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −6.25342e9 −0.426200
\(804\) 0 0
\(805\) 7.53108e9 0.508829
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.86150e9 0.123607 0.0618034 − 0.998088i \(-0.480315\pi\)
0.0618034 + 0.998088i \(0.480315\pi\)
\(810\) 0 0
\(811\) −2.58014e10 −1.69852 −0.849258 − 0.527979i \(-0.822950\pi\)
−0.849258 + 0.527979i \(0.822950\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.10703e10 0.716320
\(816\) 0 0
\(817\) 1.82639e10 1.17170
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 6.63649e9 0.418540 0.209270 − 0.977858i \(-0.432891\pi\)
0.209270 + 0.977858i \(0.432891\pi\)
\(822\) 0 0
\(823\) −1.48106e8 −0.00926130 −0.00463065 − 0.999989i \(-0.501474\pi\)
−0.00463065 + 0.999989i \(0.501474\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.01466e10 1.23861 0.619303 − 0.785152i \(-0.287415\pi\)
0.619303 + 0.785152i \(0.287415\pi\)
\(828\) 0 0
\(829\) −1.98564e10 −1.21049 −0.605243 − 0.796041i \(-0.706924\pi\)
−0.605243 + 0.796041i \(0.706924\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.78942e8 0.0167208
\(834\) 0 0
\(835\) 1.40212e10 0.833458
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1.71078e10 −1.00006 −0.500031 − 0.866007i \(-0.666678\pi\)
−0.500031 + 0.866007i \(0.666678\pi\)
\(840\) 0 0
\(841\) −1.03433e10 −0.599614
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −5.78920e9 −0.330081
\(846\) 0 0
\(847\) 3.34755e9 0.189293
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 4.25739e10 2.36805
\(852\) 0 0
\(853\) 2.00983e10 1.10876 0.554379 − 0.832264i \(-0.312956\pi\)
0.554379 + 0.832264i \(0.312956\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.23744e10 1.21428 0.607140 − 0.794595i \(-0.292317\pi\)
0.607140 + 0.794595i \(0.292317\pi\)
\(858\) 0 0
\(859\) −1.88899e10 −1.01684 −0.508421 − 0.861109i \(-0.669771\pi\)
−0.508421 + 0.861109i \(0.669771\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1.66361e10 −0.881075 −0.440538 − 0.897734i \(-0.645212\pi\)
−0.440538 + 0.897734i \(0.645212\pi\)
\(864\) 0 0
\(865\) −1.29692e9 −0.0681329
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1.62951e10 −0.842343
\(870\) 0 0
\(871\) 1.36754e10 0.701255
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.37891e9 −0.0695835
\(876\) 0 0
\(877\) 1.42290e10 0.712323 0.356161 − 0.934424i \(-0.384085\pi\)
0.356161 + 0.934424i \(0.384085\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −2.41869e10 −1.19169 −0.595847 − 0.803098i \(-0.703184\pi\)
−0.595847 + 0.803098i \(0.703184\pi\)
\(882\) 0 0
\(883\) 3.23241e10 1.58003 0.790013 − 0.613090i \(-0.210074\pi\)
0.790013 + 0.613090i \(0.210074\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 9.10944e9 0.438287 0.219144 − 0.975693i \(-0.429674\pi\)
0.219144 + 0.975693i \(0.429674\pi\)
\(888\) 0 0
\(889\) 1.01677e10 0.485365
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −4.95877e9 −0.233020
\(894\) 0 0
\(895\) −2.39421e9 −0.111630
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1.20972e10 −0.555300
\(900\) 0 0
\(901\) 1.57481e9 0.0717284
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3.07550e9 −0.137926
\(906\) 0 0
\(907\) 6.03785e9 0.268693 0.134347 − 0.990934i \(-0.457106\pi\)
0.134347 + 0.990934i \(0.457106\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −3.61401e10 −1.58371 −0.791853 − 0.610711i \(-0.790884\pi\)
−0.791853 + 0.610711i \(0.790884\pi\)
\(912\) 0 0
\(913\) −4.80529e9 −0.208964
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.48006e10 1.06211
\(918\) 0 0
\(919\) 1.43783e10 0.611088 0.305544 − 0.952178i \(-0.401162\pi\)
0.305544 + 0.952178i \(0.401162\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1.05406e10 −0.441225
\(924\) 0 0
\(925\) −7.79509e9 −0.323836
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 9.95926e9 0.407542 0.203771 − 0.979019i \(-0.434680\pi\)
0.203771 + 0.979019i \(0.434680\pi\)
\(930\) 0 0
\(931\) −6.84155e9 −0.277863
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −4.11840e8 −0.0164774
\(936\) 0 0
\(937\) −8.59930e9 −0.341487 −0.170744 − 0.985315i \(-0.554617\pi\)
−0.170744 + 0.985315i \(0.554617\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 3.62216e10 1.41711 0.708555 − 0.705656i \(-0.249347\pi\)
0.708555 + 0.705656i \(0.249347\pi\)
\(942\) 0 0
\(943\) −5.88417e10 −2.28504
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.08322e9 0.117972 0.0589861 − 0.998259i \(-0.481213\pi\)
0.0589861 + 0.998259i \(0.481213\pi\)
\(948\) 0 0
\(949\) 6.60191e9 0.250748
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −5.78801e9 −0.216623 −0.108311 − 0.994117i \(-0.534544\pi\)
−0.108311 + 0.994117i \(0.534544\pi\)
\(954\) 0 0
\(955\) 1.43766e10 0.534126
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1.88269e10 0.689307
\(960\) 0 0
\(961\) −6.32374e9 −0.229849
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.64701e10 0.590000
\(966\) 0 0
\(967\) 1.82018e10 0.647323 0.323662 − 0.946173i \(-0.395086\pi\)
0.323662 + 0.946173i \(0.395086\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −4.86220e10 −1.70438 −0.852188 − 0.523235i \(-0.824725\pi\)
−0.852188 + 0.523235i \(0.824725\pi\)
\(972\) 0 0
\(973\) −3.79196e10 −1.31968
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.78750e10 1.29934 0.649668 − 0.760218i \(-0.274908\pi\)
0.649668 + 0.760218i \(0.274908\pi\)
\(978\) 0 0
\(979\) −2.41899e10 −0.823940
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 4.92346e10 1.65323 0.826614 − 0.562769i \(-0.190264\pi\)
0.826614 + 0.562769i \(0.190264\pi\)
\(984\) 0 0
\(985\) −1.83552e10 −0.611973
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −7.40640e10 −2.43456
\(990\) 0 0
\(991\) −1.33475e10 −0.435654 −0.217827 − 0.975987i \(-0.569897\pi\)
−0.217827 + 0.975987i \(0.569897\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.58883e10 0.511325
\(996\) 0 0
\(997\) −1.26641e10 −0.404709 −0.202354 − 0.979312i \(-0.564859\pi\)
−0.202354 + 0.979312i \(0.564859\pi\)
\(998\) 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 180.8.a.c.1.1 1
3.2 odd 2 20.8.a.a.1.1 ✓ 1
12.11 even 2 80.8.a.b.1.1 1
15.2 even 4 100.8.c.a.49.2 2
15.8 even 4 100.8.c.a.49.1 2
15.14 odd 2 100.8.a.a.1.1 1
24.5 odd 2 320.8.a.e.1.1 1
24.11 even 2 320.8.a.d.1.1 1
60.23 odd 4 400.8.c.l.49.2 2
60.47 odd 4 400.8.c.l.49.1 2
60.59 even 2 400.8.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.8.a.a.1.1 ✓ 1 3.2 odd 2
80.8.a.b.1.1 1 12.11 even 2
100.8.a.a.1.1 1 15.14 odd 2
100.8.c.a.49.1 2 15.8 even 4
100.8.c.a.49.2 2 15.2 even 4
180.8.a.c.1.1 1 1.1 even 1 trivial
320.8.a.d.1.1 1 24.11 even 2
320.8.a.e.1.1 1 24.5 odd 2
400.8.a.j.1.1 1 60.59 even 2
400.8.c.l.49.1 2 60.47 odd 4
400.8.c.l.49.2 2 60.23 odd 4