Properties

Label 400.8.a.b.1.1
Level $400$
Weight $8$
Character 400.1
Self dual yes
Analytic conductor $124.954$
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] = [400,8,Mod(1,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, 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("400.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 400.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: \(124.954010194\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 8)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 400.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-84.0000 q^{3} -456.000 q^{7} +4869.00 q^{9} +O(q^{10})\) \(q-84.0000 q^{3} -456.000 q^{7} +4869.00 q^{9} +2524.00 q^{11} +10778.0 q^{13} +11150.0 q^{17} -4124.00 q^{19} +38304.0 q^{21} +81704.0 q^{23} -225288. q^{27} +99798.0 q^{29} +40480.0 q^{31} -212016. q^{33} +419442. q^{37} -905352. q^{39} +141402. q^{41} -690428. q^{43} -682032. q^{47} -615607. q^{49} -936600. q^{51} -1.81312e6 q^{53} +346416. q^{57} +966028. q^{59} +1.88767e6 q^{61} -2.22026e6 q^{63} +2.96587e6 q^{67} -6.86314e6 q^{69} +2.54823e6 q^{71} +1.68033e6 q^{73} -1.15094e6 q^{77} -4.03806e6 q^{79} +8.27569e6 q^{81} -5.38576e6 q^{83} -8.38303e6 q^{87} -6.47305e6 q^{89} -4.91477e6 q^{91} -3.40032e6 q^{93} +6.06576e6 q^{97} +1.22894e7 q^{99} +O(q^{100})\)

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
\(3\) −84.0000 −1.79620 −0.898100 0.439790i \(-0.855053\pi\)
−0.898100 + 0.439790i \(0.855053\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −456.000 −0.502483 −0.251242 0.967924i \(-0.580839\pi\)
−0.251242 + 0.967924i \(0.580839\pi\)
\(8\) 0 0
\(9\) 4869.00 2.22634
\(10\) 0 0
\(11\) 2524.00 0.571762 0.285881 0.958265i \(-0.407714\pi\)
0.285881 + 0.958265i \(0.407714\pi\)
\(12\) 0 0
\(13\) 10778.0 1.36062 0.680309 0.732925i \(-0.261845\pi\)
0.680309 + 0.732925i \(0.261845\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 11150.0 0.550432 0.275216 0.961382i \(-0.411251\pi\)
0.275216 + 0.961382i \(0.411251\pi\)
\(18\) 0 0
\(19\) −4124.00 −0.137937 −0.0689685 0.997619i \(-0.521971\pi\)
−0.0689685 + 0.997619i \(0.521971\pi\)
\(20\) 0 0
\(21\) 38304.0 0.902561
\(22\) 0 0
\(23\) 81704.0 1.40022 0.700109 0.714036i \(-0.253135\pi\)
0.700109 + 0.714036i \(0.253135\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −225288. −2.20275
\(28\) 0 0
\(29\) 99798.0 0.759852 0.379926 0.925017i \(-0.375949\pi\)
0.379926 + 0.925017i \(0.375949\pi\)
\(30\) 0 0
\(31\) 40480.0 0.244048 0.122024 0.992527i \(-0.461062\pi\)
0.122024 + 0.992527i \(0.461062\pi\)
\(32\) 0 0
\(33\) −212016. −1.02700
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 419442. 1.36134 0.680669 0.732591i \(-0.261689\pi\)
0.680669 + 0.732591i \(0.261689\pi\)
\(38\) 0 0
\(39\) −905352. −2.44394
\(40\) 0 0
\(41\) 141402. 0.320414 0.160207 0.987083i \(-0.448784\pi\)
0.160207 + 0.987083i \(0.448784\pi\)
\(42\) 0 0
\(43\) −690428. −1.32428 −0.662138 0.749382i \(-0.730351\pi\)
−0.662138 + 0.749382i \(0.730351\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −682032. −0.958213 −0.479107 0.877757i \(-0.659039\pi\)
−0.479107 + 0.877757i \(0.659039\pi\)
\(48\) 0 0
\(49\) −615607. −0.747510
\(50\) 0 0
\(51\) −936600. −0.988686
\(52\) 0 0
\(53\) −1.81312e6 −1.67286 −0.836432 0.548071i \(-0.815362\pi\)
−0.836432 + 0.548071i \(0.815362\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 346416. 0.247763
\(58\) 0 0
\(59\) 966028. 0.612361 0.306181 0.951973i \(-0.400949\pi\)
0.306181 + 0.951973i \(0.400949\pi\)
\(60\) 0 0
\(61\) 1.88767e6 1.06481 0.532404 0.846490i \(-0.321289\pi\)
0.532404 + 0.846490i \(0.321289\pi\)
\(62\) 0 0
\(63\) −2.22026e6 −1.11870
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 2.96587e6 1.20473 0.602365 0.798220i \(-0.294225\pi\)
0.602365 + 0.798220i \(0.294225\pi\)
\(68\) 0 0
\(69\) −6.86314e6 −2.51507
\(70\) 0 0
\(71\) 2.54823e6 0.844957 0.422479 0.906373i \(-0.361160\pi\)
0.422479 + 0.906373i \(0.361160\pi\)
\(72\) 0 0
\(73\) 1.68033e6 0.505549 0.252775 0.967525i \(-0.418657\pi\)
0.252775 + 0.967525i \(0.418657\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.15094e6 −0.287301
\(78\) 0 0
\(79\) −4.03806e6 −0.921464 −0.460732 0.887539i \(-0.652413\pi\)
−0.460732 + 0.887539i \(0.652413\pi\)
\(80\) 0 0
\(81\) 8.27569e6 1.73024
\(82\) 0 0
\(83\) −5.38576e6 −1.03389 −0.516945 0.856019i \(-0.672931\pi\)
−0.516945 + 0.856019i \(0.672931\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −8.38303e6 −1.36485
\(88\) 0 0
\(89\) −6.47305e6 −0.973293 −0.486647 0.873599i \(-0.661780\pi\)
−0.486647 + 0.873599i \(0.661780\pi\)
\(90\) 0 0
\(91\) −4.91477e6 −0.683688
\(92\) 0 0
\(93\) −3.40032e6 −0.438359
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 6.06576e6 0.674814 0.337407 0.941359i \(-0.390450\pi\)
0.337407 + 0.941359i \(0.390450\pi\)
\(98\) 0 0
\(99\) 1.22894e7 1.27293
\(100\) 0 0
\(101\) 9.70069e6 0.936866 0.468433 0.883499i \(-0.344819\pi\)
0.468433 + 0.883499i \(0.344819\pi\)
\(102\) 0 0
\(103\) 4.10159e6 0.369847 0.184924 0.982753i \(-0.440796\pi\)
0.184924 + 0.982753i \(0.440796\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 72900.0 0.00575287 0.00287643 0.999996i \(-0.499084\pi\)
0.00287643 + 0.999996i \(0.499084\pi\)
\(108\) 0 0
\(109\) 9.55841e6 0.706957 0.353478 0.935443i \(-0.384999\pi\)
0.353478 + 0.935443i \(0.384999\pi\)
\(110\) 0 0
\(111\) −3.52331e7 −2.44524
\(112\) 0 0
\(113\) −9.33890e6 −0.608865 −0.304433 0.952534i \(-0.598467\pi\)
−0.304433 + 0.952534i \(0.598467\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 5.24781e7 3.02920
\(118\) 0 0
\(119\) −5.08440e6 −0.276583
\(120\) 0 0
\(121\) −1.31166e7 −0.673089
\(122\) 0 0
\(123\) −1.18778e7 −0.575529
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −3.59794e7 −1.55862 −0.779311 0.626637i \(-0.784431\pi\)
−0.779311 + 0.626637i \(0.784431\pi\)
\(128\) 0 0
\(129\) 5.79960e7 2.37867
\(130\) 0 0
\(131\) 676052. 0.0262743 0.0131371 0.999914i \(-0.495818\pi\)
0.0131371 + 0.999914i \(0.495818\pi\)
\(132\) 0 0
\(133\) 1.88054e6 0.0693111
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 2.95841e7 0.982962 0.491481 0.870888i \(-0.336456\pi\)
0.491481 + 0.870888i \(0.336456\pi\)
\(138\) 0 0
\(139\) 3.19084e7 1.00775 0.503876 0.863776i \(-0.331907\pi\)
0.503876 + 0.863776i \(0.331907\pi\)
\(140\) 0 0
\(141\) 5.72907e7 1.72114
\(142\) 0 0
\(143\) 2.72037e7 0.777949
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 5.17110e7 1.34268
\(148\) 0 0
\(149\) −1.16603e7 −0.288773 −0.144386 0.989521i \(-0.546121\pi\)
−0.144386 + 0.989521i \(0.546121\pi\)
\(150\) 0 0
\(151\) 1.76295e7 0.416698 0.208349 0.978055i \(-0.433191\pi\)
0.208349 + 0.978055i \(0.433191\pi\)
\(152\) 0 0
\(153\) 5.42894e7 1.22545
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −6.34658e6 −0.130885 −0.0654427 0.997856i \(-0.520846\pi\)
−0.0654427 + 0.997856i \(0.520846\pi\)
\(158\) 0 0
\(159\) 1.52302e8 3.00480
\(160\) 0 0
\(161\) −3.72570e7 −0.703587
\(162\) 0 0
\(163\) 8.04234e7 1.45454 0.727271 0.686351i \(-0.240789\pi\)
0.727271 + 0.686351i \(0.240789\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.14767e8 1.90682 0.953411 0.301673i \(-0.0975451\pi\)
0.953411 + 0.301673i \(0.0975451\pi\)
\(168\) 0 0
\(169\) 5.34168e7 0.851283
\(170\) 0 0
\(171\) −2.00798e7 −0.307095
\(172\) 0 0
\(173\) 6.33755e7 0.930594 0.465297 0.885155i \(-0.345947\pi\)
0.465297 + 0.885155i \(0.345947\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −8.11464e7 −1.09992
\(178\) 0 0
\(179\) 1.13228e7 0.147559 0.0737796 0.997275i \(-0.476494\pi\)
0.0737796 + 0.997275i \(0.476494\pi\)
\(180\) 0 0
\(181\) −5.22650e6 −0.0655143 −0.0327571 0.999463i \(-0.510429\pi\)
−0.0327571 + 0.999463i \(0.510429\pi\)
\(182\) 0 0
\(183\) −1.58564e8 −1.91261
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2.81426e7 0.314716
\(188\) 0 0
\(189\) 1.02731e8 1.10684
\(190\) 0 0
\(191\) 8.50301e7 0.882990 0.441495 0.897264i \(-0.354448\pi\)
0.441495 + 0.897264i \(0.354448\pi\)
\(192\) 0 0
\(193\) −1.15092e8 −1.15237 −0.576186 0.817319i \(-0.695460\pi\)
−0.576186 + 0.817319i \(0.695460\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.38522e8 1.29088 0.645441 0.763810i \(-0.276674\pi\)
0.645441 + 0.763810i \(0.276674\pi\)
\(198\) 0 0
\(199\) 2.19614e7 0.197548 0.0987742 0.995110i \(-0.468508\pi\)
0.0987742 + 0.995110i \(0.468508\pi\)
\(200\) 0 0
\(201\) −2.49133e8 −2.16394
\(202\) 0 0
\(203\) −4.55079e7 −0.381813
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 3.97817e8 3.11736
\(208\) 0 0
\(209\) −1.04090e7 −0.0788671
\(210\) 0 0
\(211\) 6.10208e7 0.447187 0.223594 0.974682i \(-0.428221\pi\)
0.223594 + 0.974682i \(0.428221\pi\)
\(212\) 0 0
\(213\) −2.14051e8 −1.51771
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −1.84589e7 −0.122630
\(218\) 0 0
\(219\) −1.41147e8 −0.908068
\(220\) 0 0
\(221\) 1.20175e8 0.748928
\(222\) 0 0
\(223\) −4.22448e7 −0.255098 −0.127549 0.991832i \(-0.540711\pi\)
−0.127549 + 0.991832i \(0.540711\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.39102e8 −1.35673 −0.678364 0.734726i \(-0.737311\pi\)
−0.678364 + 0.734726i \(0.737311\pi\)
\(228\) 0 0
\(229\) −4.67889e7 −0.257465 −0.128733 0.991679i \(-0.541091\pi\)
−0.128733 + 0.991679i \(0.541091\pi\)
\(230\) 0 0
\(231\) 9.66793e7 0.516050
\(232\) 0 0
\(233\) −3.45225e8 −1.78795 −0.893977 0.448113i \(-0.852096\pi\)
−0.893977 + 0.448113i \(0.852096\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 3.39197e8 1.65513
\(238\) 0 0
\(239\) −2.34413e8 −1.11068 −0.555340 0.831624i \(-0.687412\pi\)
−0.555340 + 0.831624i \(0.687412\pi\)
\(240\) 0 0
\(241\) −1.09557e8 −0.504175 −0.252087 0.967705i \(-0.581117\pi\)
−0.252087 + 0.967705i \(0.581117\pi\)
\(242\) 0 0
\(243\) −2.02453e8 −0.905112
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −4.44485e7 −0.187680
\(248\) 0 0
\(249\) 4.52404e8 1.85707
\(250\) 0 0
\(251\) 3.94031e8 1.57280 0.786398 0.617720i \(-0.211943\pi\)
0.786398 + 0.617720i \(0.211943\pi\)
\(252\) 0 0
\(253\) 2.06221e8 0.800591
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3.19064e8 −1.17250 −0.586248 0.810131i \(-0.699396\pi\)
−0.586248 + 0.810131i \(0.699396\pi\)
\(258\) 0 0
\(259\) −1.91266e8 −0.684050
\(260\) 0 0
\(261\) 4.85916e8 1.69169
\(262\) 0 0
\(263\) 2.19359e8 0.743549 0.371774 0.928323i \(-0.378750\pi\)
0.371774 + 0.928323i \(0.378750\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 5.43736e8 1.74823
\(268\) 0 0
\(269\) −1.48033e8 −0.463687 −0.231844 0.972753i \(-0.574476\pi\)
−0.231844 + 0.972753i \(0.574476\pi\)
\(270\) 0 0
\(271\) 3.69934e8 1.12910 0.564549 0.825399i \(-0.309050\pi\)
0.564549 + 0.825399i \(0.309050\pi\)
\(272\) 0 0
\(273\) 4.12841e8 1.22804
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 3.95860e8 1.11908 0.559541 0.828803i \(-0.310977\pi\)
0.559541 + 0.828803i \(0.310977\pi\)
\(278\) 0 0
\(279\) 1.97097e8 0.543332
\(280\) 0 0
\(281\) −5.97760e8 −1.60714 −0.803572 0.595208i \(-0.797070\pi\)
−0.803572 + 0.595208i \(0.797070\pi\)
\(282\) 0 0
\(283\) 8.05797e7 0.211336 0.105668 0.994401i \(-0.466302\pi\)
0.105668 + 0.994401i \(0.466302\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.44793e7 −0.161003
\(288\) 0 0
\(289\) −2.86016e8 −0.697025
\(290\) 0 0
\(291\) −5.09524e8 −1.21210
\(292\) 0 0
\(293\) −7.54530e8 −1.75243 −0.876213 0.481924i \(-0.839938\pi\)
−0.876213 + 0.481924i \(0.839938\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −5.68627e8 −1.25945
\(298\) 0 0
\(299\) 8.80606e8 1.90516
\(300\) 0 0
\(301\) 3.14835e8 0.665427
\(302\) 0 0
\(303\) −8.14858e8 −1.68280
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 8.20472e8 1.61838 0.809188 0.587549i \(-0.199907\pi\)
0.809188 + 0.587549i \(0.199907\pi\)
\(308\) 0 0
\(309\) −3.44534e8 −0.664320
\(310\) 0 0
\(311\) −6.53503e8 −1.23193 −0.615965 0.787773i \(-0.711234\pi\)
−0.615965 + 0.787773i \(0.711234\pi\)
\(312\) 0 0
\(313\) −6.63587e8 −1.22319 −0.611594 0.791172i \(-0.709471\pi\)
−0.611594 + 0.791172i \(0.709471\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.54718e8 0.625426 0.312713 0.949848i \(-0.398762\pi\)
0.312713 + 0.949848i \(0.398762\pi\)
\(318\) 0 0
\(319\) 2.51890e8 0.434454
\(320\) 0 0
\(321\) −6.12360e6 −0.0103333
\(322\) 0 0
\(323\) −4.59826e7 −0.0759250
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −8.02906e8 −1.26984
\(328\) 0 0
\(329\) 3.11007e8 0.481486
\(330\) 0 0
\(331\) 3.05543e8 0.463100 0.231550 0.972823i \(-0.425620\pi\)
0.231550 + 0.972823i \(0.425620\pi\)
\(332\) 0 0
\(333\) 2.04226e9 3.03080
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −3.54965e7 −0.0505220 −0.0252610 0.999681i \(-0.508042\pi\)
−0.0252610 + 0.999681i \(0.508042\pi\)
\(338\) 0 0
\(339\) 7.84467e8 1.09364
\(340\) 0 0
\(341\) 1.02172e8 0.139537
\(342\) 0 0
\(343\) 6.56252e8 0.878095
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.90594e8 −0.244882 −0.122441 0.992476i \(-0.539072\pi\)
−0.122441 + 0.992476i \(0.539072\pi\)
\(348\) 0 0
\(349\) 8.60864e8 1.08404 0.542020 0.840366i \(-0.317660\pi\)
0.542020 + 0.840366i \(0.317660\pi\)
\(350\) 0 0
\(351\) −2.42815e9 −2.99710
\(352\) 0 0
\(353\) 1.04544e9 1.26500 0.632498 0.774562i \(-0.282030\pi\)
0.632498 + 0.774562i \(0.282030\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 4.27090e8 0.496798
\(358\) 0 0
\(359\) −7.63303e8 −0.870696 −0.435348 0.900262i \(-0.643375\pi\)
−0.435348 + 0.900262i \(0.643375\pi\)
\(360\) 0 0
\(361\) −8.76864e8 −0.980973
\(362\) 0 0
\(363\) 1.10179e9 1.20900
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.38692e9 −1.46460 −0.732302 0.680980i \(-0.761554\pi\)
−0.732302 + 0.680980i \(0.761554\pi\)
\(368\) 0 0
\(369\) 6.88486e8 0.713351
\(370\) 0 0
\(371\) 8.26782e8 0.840586
\(372\) 0 0
\(373\) −4.77105e8 −0.476029 −0.238015 0.971262i \(-0.576497\pi\)
−0.238015 + 0.971262i \(0.576497\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1.07562e9 1.03387
\(378\) 0 0
\(379\) 3.92468e8 0.370311 0.185156 0.982709i \(-0.440721\pi\)
0.185156 + 0.982709i \(0.440721\pi\)
\(380\) 0 0
\(381\) 3.02227e9 2.79960
\(382\) 0 0
\(383\) 2.10409e9 1.91368 0.956839 0.290617i \(-0.0938605\pi\)
0.956839 + 0.290617i \(0.0938605\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.36169e9 −2.94829
\(388\) 0 0
\(389\) −1.26019e9 −1.08546 −0.542730 0.839907i \(-0.682609\pi\)
−0.542730 + 0.839907i \(0.682609\pi\)
\(390\) 0 0
\(391\) 9.11000e8 0.770725
\(392\) 0 0
\(393\) −5.67884e7 −0.0471939
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 9.81298e8 0.787107 0.393554 0.919302i \(-0.371246\pi\)
0.393554 + 0.919302i \(0.371246\pi\)
\(398\) 0 0
\(399\) −1.57966e8 −0.124497
\(400\) 0 0
\(401\) 9.09981e8 0.704737 0.352369 0.935861i \(-0.385376\pi\)
0.352369 + 0.935861i \(0.385376\pi\)
\(402\) 0 0
\(403\) 4.36293e8 0.332056
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.05867e9 0.778361
\(408\) 0 0
\(409\) −3.55609e7 −0.0257004 −0.0128502 0.999917i \(-0.504090\pi\)
−0.0128502 + 0.999917i \(0.504090\pi\)
\(410\) 0 0
\(411\) −2.48507e9 −1.76560
\(412\) 0 0
\(413\) −4.40509e8 −0.307701
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −2.68031e9 −1.81013
\(418\) 0 0
\(419\) −2.65360e9 −1.76233 −0.881163 0.472813i \(-0.843239\pi\)
−0.881163 + 0.472813i \(0.843239\pi\)
\(420\) 0 0
\(421\) −1.12113e9 −0.732264 −0.366132 0.930563i \(-0.619318\pi\)
−0.366132 + 0.930563i \(0.619318\pi\)
\(422\) 0 0
\(423\) −3.32081e9 −2.13331
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −8.60778e8 −0.535049
\(428\) 0 0
\(429\) −2.28511e9 −1.39735
\(430\) 0 0
\(431\) 1.06344e9 0.639799 0.319900 0.947451i \(-0.396351\pi\)
0.319900 + 0.947451i \(0.396351\pi\)
\(432\) 0 0
\(433\) 7.05962e8 0.417901 0.208951 0.977926i \(-0.432995\pi\)
0.208951 + 0.977926i \(0.432995\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −3.36947e8 −0.193142
\(438\) 0 0
\(439\) −1.48506e9 −0.837760 −0.418880 0.908042i \(-0.637577\pi\)
−0.418880 + 0.908042i \(0.637577\pi\)
\(440\) 0 0
\(441\) −2.99739e9 −1.66421
\(442\) 0 0
\(443\) −7.22153e8 −0.394654 −0.197327 0.980338i \(-0.563226\pi\)
−0.197327 + 0.980338i \(0.563226\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 9.79462e8 0.518694
\(448\) 0 0
\(449\) −1.22968e9 −0.641109 −0.320554 0.947230i \(-0.603869\pi\)
−0.320554 + 0.947230i \(0.603869\pi\)
\(450\) 0 0
\(451\) 3.56899e8 0.183201
\(452\) 0 0
\(453\) −1.48088e9 −0.748473
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.85551e7 0.0434017 0.0217009 0.999765i \(-0.493092\pi\)
0.0217009 + 0.999765i \(0.493092\pi\)
\(458\) 0 0
\(459\) −2.51196e9 −1.21246
\(460\) 0 0
\(461\) 2.10937e8 0.100277 0.0501384 0.998742i \(-0.484034\pi\)
0.0501384 + 0.998742i \(0.484034\pi\)
\(462\) 0 0
\(463\) −3.29775e9 −1.54413 −0.772066 0.635543i \(-0.780776\pi\)
−0.772066 + 0.635543i \(0.780776\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.82873e7 0.0401134 0.0200567 0.999799i \(-0.493615\pi\)
0.0200567 + 0.999799i \(0.493615\pi\)
\(468\) 0 0
\(469\) −1.35244e9 −0.605357
\(470\) 0 0
\(471\) 5.33113e8 0.235096
\(472\) 0 0
\(473\) −1.74264e9 −0.757171
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −8.82807e9 −3.72436
\(478\) 0 0
\(479\) 4.51507e9 1.87711 0.938557 0.345125i \(-0.112164\pi\)
0.938557 + 0.345125i \(0.112164\pi\)
\(480\) 0 0
\(481\) 4.52075e9 1.85226
\(482\) 0 0
\(483\) 3.12959e9 1.26378
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 3.31338e9 1.29993 0.649964 0.759965i \(-0.274784\pi\)
0.649964 + 0.759965i \(0.274784\pi\)
\(488\) 0 0
\(489\) −6.75557e9 −2.61265
\(490\) 0 0
\(491\) 4.01694e9 1.53147 0.765737 0.643154i \(-0.222374\pi\)
0.765737 + 0.643154i \(0.222374\pi\)
\(492\) 0 0
\(493\) 1.11275e9 0.418247
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −1.16199e9 −0.424577
\(498\) 0 0
\(499\) −2.70976e9 −0.976290 −0.488145 0.872763i \(-0.662326\pi\)
−0.488145 + 0.872763i \(0.662326\pi\)
\(500\) 0 0
\(501\) −9.64045e9 −3.42504
\(502\) 0 0
\(503\) 3.04579e8 0.106712 0.0533558 0.998576i \(-0.483008\pi\)
0.0533558 + 0.998576i \(0.483008\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −4.48701e9 −1.52908
\(508\) 0 0
\(509\) −1.88202e8 −0.0632575 −0.0316287 0.999500i \(-0.510069\pi\)
−0.0316287 + 0.999500i \(0.510069\pi\)
\(510\) 0 0
\(511\) −7.66229e8 −0.254030
\(512\) 0 0
\(513\) 9.29088e8 0.303841
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −1.72145e9 −0.547870
\(518\) 0 0
\(519\) −5.32355e9 −1.67153
\(520\) 0 0
\(521\) 4.14963e9 1.28552 0.642758 0.766069i \(-0.277790\pi\)
0.642758 + 0.766069i \(0.277790\pi\)
\(522\) 0 0
\(523\) 2.51360e9 0.768318 0.384159 0.923267i \(-0.374491\pi\)
0.384159 + 0.923267i \(0.374491\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.51352e8 0.134332
\(528\) 0 0
\(529\) 3.27072e9 0.960613
\(530\) 0 0
\(531\) 4.70359e9 1.36332
\(532\) 0 0
\(533\) 1.52403e9 0.435962
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −9.51112e8 −0.265046
\(538\) 0 0
\(539\) −1.55379e9 −0.427398
\(540\) 0 0
\(541\) −1.32416e9 −0.359543 −0.179772 0.983708i \(-0.557536\pi\)
−0.179772 + 0.983708i \(0.557536\pi\)
\(542\) 0 0
\(543\) 4.39026e8 0.117677
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 5.58047e8 0.145786 0.0728929 0.997340i \(-0.476777\pi\)
0.0728929 + 0.997340i \(0.476777\pi\)
\(548\) 0 0
\(549\) 9.19107e9 2.37062
\(550\) 0 0
\(551\) −4.11567e8 −0.104812
\(552\) 0 0
\(553\) 1.84136e9 0.463020
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3.30331e9 0.809946 0.404973 0.914329i \(-0.367281\pi\)
0.404973 + 0.914329i \(0.367281\pi\)
\(558\) 0 0
\(559\) −7.44143e9 −1.80184
\(560\) 0 0
\(561\) −2.36398e9 −0.565293
\(562\) 0 0
\(563\) −1.22011e8 −0.0288152 −0.0144076 0.999896i \(-0.504586\pi\)
−0.0144076 + 0.999896i \(0.504586\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −3.77371e9 −0.869417
\(568\) 0 0
\(569\) 5.00925e8 0.113993 0.0569967 0.998374i \(-0.481848\pi\)
0.0569967 + 0.998374i \(0.481848\pi\)
\(570\) 0 0
\(571\) 6.98702e9 1.57060 0.785300 0.619116i \(-0.212509\pi\)
0.785300 + 0.619116i \(0.212509\pi\)
\(572\) 0 0
\(573\) −7.14253e9 −1.58603
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 8.16573e9 1.76962 0.884809 0.465954i \(-0.154289\pi\)
0.884809 + 0.465954i \(0.154289\pi\)
\(578\) 0 0
\(579\) 9.66769e9 2.06989
\(580\) 0 0
\(581\) 2.45591e9 0.519512
\(582\) 0 0
\(583\) −4.57631e9 −0.956479
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.53182e9 1.74104 0.870519 0.492135i \(-0.163783\pi\)
0.870519 + 0.492135i \(0.163783\pi\)
\(588\) 0 0
\(589\) −1.66940e8 −0.0336632
\(590\) 0 0
\(591\) −1.16358e10 −2.31868
\(592\) 0 0
\(593\) 1.71175e9 0.337092 0.168546 0.985694i \(-0.446093\pi\)
0.168546 + 0.985694i \(0.446093\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −1.84475e9 −0.354836
\(598\) 0 0
\(599\) 4.77362e9 0.907516 0.453758 0.891125i \(-0.350083\pi\)
0.453758 + 0.891125i \(0.350083\pi\)
\(600\) 0 0
\(601\) 7.89998e8 0.148445 0.0742224 0.997242i \(-0.476353\pi\)
0.0742224 + 0.997242i \(0.476353\pi\)
\(602\) 0 0
\(603\) 1.44408e10 2.68214
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −1.82652e9 −0.331485 −0.165743 0.986169i \(-0.553002\pi\)
−0.165743 + 0.986169i \(0.553002\pi\)
\(608\) 0 0
\(609\) 3.82266e9 0.685813
\(610\) 0 0
\(611\) −7.35094e9 −1.30376
\(612\) 0 0
\(613\) 6.90339e9 1.21046 0.605231 0.796050i \(-0.293081\pi\)
0.605231 + 0.796050i \(0.293081\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 5.69235e9 0.975649 0.487825 0.872942i \(-0.337791\pi\)
0.487825 + 0.872942i \(0.337791\pi\)
\(618\) 0 0
\(619\) 4.28594e9 0.726321 0.363161 0.931727i \(-0.381698\pi\)
0.363161 + 0.931727i \(0.381698\pi\)
\(620\) 0 0
\(621\) −1.84069e10 −3.08433
\(622\) 0 0
\(623\) 2.95171e9 0.489064
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 8.74354e8 0.141661
\(628\) 0 0
\(629\) 4.67678e9 0.749324
\(630\) 0 0
\(631\) 5.61602e8 0.0889869 0.0444935 0.999010i \(-0.485833\pi\)
0.0444935 + 0.999010i \(0.485833\pi\)
\(632\) 0 0
\(633\) −5.12575e9 −0.803238
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −6.63501e9 −1.01708
\(638\) 0 0
\(639\) 1.24073e10 1.88116
\(640\) 0 0
\(641\) 5.17445e9 0.775998 0.387999 0.921660i \(-0.373166\pi\)
0.387999 + 0.921660i \(0.373166\pi\)
\(642\) 0 0
\(643\) −1.04374e10 −1.54830 −0.774148 0.633004i \(-0.781822\pi\)
−0.774148 + 0.633004i \(0.781822\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −9.71623e8 −0.141037 −0.0705185 0.997510i \(-0.522465\pi\)
−0.0705185 + 0.997510i \(0.522465\pi\)
\(648\) 0 0
\(649\) 2.43825e9 0.350125
\(650\) 0 0
\(651\) 1.55055e9 0.220268
\(652\) 0 0
\(653\) 7.25223e9 1.01924 0.509619 0.860400i \(-0.329786\pi\)
0.509619 + 0.860400i \(0.329786\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 8.18151e9 1.12552
\(658\) 0 0
\(659\) −3.81924e9 −0.519851 −0.259925 0.965629i \(-0.583698\pi\)
−0.259925 + 0.965629i \(0.583698\pi\)
\(660\) 0 0
\(661\) 1.07881e10 1.45292 0.726459 0.687210i \(-0.241165\pi\)
0.726459 + 0.687210i \(0.241165\pi\)
\(662\) 0 0
\(663\) −1.00947e10 −1.34523
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 8.15390e9 1.06396
\(668\) 0 0
\(669\) 3.54857e9 0.458207
\(670\) 0 0
\(671\) 4.76448e9 0.608817
\(672\) 0 0
\(673\) 6.34833e9 0.802798 0.401399 0.915903i \(-0.368524\pi\)
0.401399 + 0.915903i \(0.368524\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −8.82566e9 −1.09317 −0.546584 0.837404i \(-0.684072\pi\)
−0.546584 + 0.837404i \(0.684072\pi\)
\(678\) 0 0
\(679\) −2.76599e9 −0.339083
\(680\) 0 0
\(681\) 2.00846e10 2.43696
\(682\) 0 0
\(683\) 4.92331e9 0.591268 0.295634 0.955301i \(-0.404469\pi\)
0.295634 + 0.955301i \(0.404469\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 3.93027e9 0.462459
\(688\) 0 0
\(689\) −1.95418e10 −2.27613
\(690\) 0 0
\(691\) 5.68449e9 0.655418 0.327709 0.944779i \(-0.393723\pi\)
0.327709 + 0.944779i \(0.393723\pi\)
\(692\) 0 0
\(693\) −5.60395e9 −0.639628
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.57663e9 0.176366
\(698\) 0 0
\(699\) 2.89989e10 3.21152
\(700\) 0 0
\(701\) −1.70567e9 −0.187017 −0.0935085 0.995618i \(-0.529808\pi\)
−0.0935085 + 0.995618i \(0.529808\pi\)
\(702\) 0 0
\(703\) −1.72978e9 −0.187779
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −4.42351e9 −0.470760
\(708\) 0 0
\(709\) 4.52189e9 0.476495 0.238248 0.971204i \(-0.423427\pi\)
0.238248 + 0.971204i \(0.423427\pi\)
\(710\) 0 0
\(711\) −1.96613e10 −2.05149
\(712\) 0 0
\(713\) 3.30738e9 0.341720
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1.96907e10 1.99500
\(718\) 0 0
\(719\) 3.09206e9 0.310239 0.155120 0.987896i \(-0.450424\pi\)
0.155120 + 0.987896i \(0.450424\pi\)
\(720\) 0 0
\(721\) −1.87033e9 −0.185842
\(722\) 0 0
\(723\) 9.20280e9 0.905599
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.44622e10 1.39593 0.697965 0.716132i \(-0.254089\pi\)
0.697965 + 0.716132i \(0.254089\pi\)
\(728\) 0 0
\(729\) −1.09288e9 −0.104478
\(730\) 0 0
\(731\) −7.69827e9 −0.728924
\(732\) 0 0
\(733\) −3.15415e9 −0.295814 −0.147907 0.989001i \(-0.547254\pi\)
−0.147907 + 0.989001i \(0.547254\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 7.48585e9 0.688819
\(738\) 0 0
\(739\) −1.54236e10 −1.40582 −0.702912 0.711277i \(-0.748117\pi\)
−0.702912 + 0.711277i \(0.748117\pi\)
\(740\) 0 0
\(741\) 3.73367e9 0.337111
\(742\) 0 0
\(743\) −1.59520e10 −1.42677 −0.713385 0.700772i \(-0.752839\pi\)
−0.713385 + 0.700772i \(0.752839\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −2.62233e10 −2.30179
\(748\) 0 0
\(749\) −3.32424e7 −0.00289072
\(750\) 0 0
\(751\) −6.13964e9 −0.528936 −0.264468 0.964395i \(-0.585196\pi\)
−0.264468 + 0.964395i \(0.585196\pi\)
\(752\) 0 0
\(753\) −3.30986e10 −2.82506
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.42818e10 1.19660 0.598299 0.801273i \(-0.295843\pi\)
0.598299 + 0.801273i \(0.295843\pi\)
\(758\) 0 0
\(759\) −1.73226e10 −1.43802
\(760\) 0 0
\(761\) 1.47536e10 1.21353 0.606767 0.794880i \(-0.292466\pi\)
0.606767 + 0.794880i \(0.292466\pi\)
\(762\) 0 0
\(763\) −4.35863e9 −0.355234
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.04118e10 0.833190
\(768\) 0 0
\(769\) 1.97592e10 1.56685 0.783424 0.621487i \(-0.213471\pi\)
0.783424 + 0.621487i \(0.213471\pi\)
\(770\) 0 0
\(771\) 2.68014e10 2.10604
\(772\) 0 0
\(773\) −1.01370e10 −0.789374 −0.394687 0.918816i \(-0.629147\pi\)
−0.394687 + 0.918816i \(0.629147\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1.60663e10 1.22869
\(778\) 0 0
\(779\) −5.83142e8 −0.0441970
\(780\) 0 0
\(781\) 6.43174e9 0.483114
\(782\) 0 0
\(783\) −2.24833e10 −1.67376
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −1.27882e10 −0.935188 −0.467594 0.883943i \(-0.654879\pi\)
−0.467594 + 0.883943i \(0.654879\pi\)
\(788\) 0 0
\(789\) −1.84261e10 −1.33556
\(790\) 0 0
\(791\) 4.25854e9 0.305945
\(792\) 0 0
\(793\) 2.03453e10 1.44880
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 7.38617e9 0.516791 0.258396 0.966039i \(-0.416806\pi\)
0.258396 + 0.966039i \(0.416806\pi\)
\(798\) 0 0
\(799\) −7.60466e9 −0.527431
\(800\) 0 0
\(801\) −3.15173e10 −2.16688
\(802\) 0 0
\(803\) 4.24114e9 0.289054
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1.24348e10 0.832875
\(808\) 0 0
\(809\) 1.53742e10 1.02087 0.510437 0.859915i \(-0.329484\pi\)
0.510437 + 0.859915i \(0.329484\pi\)
\(810\) 0 0
\(811\) −9.77882e9 −0.643744 −0.321872 0.946783i \(-0.604312\pi\)
−0.321872 + 0.946783i \(0.604312\pi\)
\(812\) 0 0
\(813\) −3.10745e10 −2.02809
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 2.84733e9 0.182667
\(818\) 0 0
\(819\) −2.39300e10 −1.52212
\(820\) 0 0
\(821\) 1.83470e10 1.15708 0.578540 0.815654i \(-0.303623\pi\)
0.578540 + 0.815654i \(0.303623\pi\)
\(822\) 0 0
\(823\) −3.16960e10 −1.98201 −0.991004 0.133829i \(-0.957273\pi\)
−0.991004 + 0.133829i \(0.957273\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −6.12845e9 −0.376774 −0.188387 0.982095i \(-0.560326\pi\)
−0.188387 + 0.982095i \(0.560326\pi\)
\(828\) 0 0
\(829\) −1.24652e10 −0.759904 −0.379952 0.925006i \(-0.624060\pi\)
−0.379952 + 0.925006i \(0.624060\pi\)
\(830\) 0 0
\(831\) −3.32522e10 −2.01010
\(832\) 0 0
\(833\) −6.86402e9 −0.411454
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −9.11966e9 −0.537575
\(838\) 0 0
\(839\) 1.82237e10 1.06530 0.532648 0.846337i \(-0.321197\pi\)
0.532648 + 0.846337i \(0.321197\pi\)
\(840\) 0 0
\(841\) −7.29024e9 −0.422625
\(842\) 0 0
\(843\) 5.02118e10 2.88675
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 5.98117e9 0.338216
\(848\) 0 0
\(849\) −6.76870e9 −0.379602
\(850\) 0 0
\(851\) 3.42701e10 1.90617
\(852\) 0 0
\(853\) 2.48619e10 1.37155 0.685777 0.727812i \(-0.259463\pi\)
0.685777 + 0.727812i \(0.259463\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −2.96761e10 −1.61055 −0.805275 0.592902i \(-0.797982\pi\)
−0.805275 + 0.592902i \(0.797982\pi\)
\(858\) 0 0
\(859\) −1.14772e10 −0.617819 −0.308910 0.951091i \(-0.599964\pi\)
−0.308910 + 0.951091i \(0.599964\pi\)
\(860\) 0 0
\(861\) 5.41626e9 0.289194
\(862\) 0 0
\(863\) 2.13485e10 1.13066 0.565328 0.824866i \(-0.308750\pi\)
0.565328 + 0.824866i \(0.308750\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 2.40254e10 1.25200
\(868\) 0 0
\(869\) −1.01921e10 −0.526858
\(870\) 0 0
\(871\) 3.19661e10 1.63918
\(872\) 0 0
\(873\) 2.95342e10 1.50236
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 7.92753e9 0.396862 0.198431 0.980115i \(-0.436415\pi\)
0.198431 + 0.980115i \(0.436415\pi\)
\(878\) 0 0
\(879\) 6.33805e10 3.14771
\(880\) 0 0
\(881\) 7.32045e9 0.360680 0.180340 0.983604i \(-0.442280\pi\)
0.180340 + 0.983604i \(0.442280\pi\)
\(882\) 0 0
\(883\) −3.54988e9 −0.173521 −0.0867604 0.996229i \(-0.527651\pi\)
−0.0867604 + 0.996229i \(0.527651\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 5.80634e9 0.279364 0.139682 0.990196i \(-0.455392\pi\)
0.139682 + 0.990196i \(0.455392\pi\)
\(888\) 0 0
\(889\) 1.64066e10 0.783182
\(890\) 0 0
\(891\) 2.08878e10 0.989285
\(892\) 0 0
\(893\) 2.81270e9 0.132173
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −7.39709e10 −3.42206
\(898\) 0 0
\(899\) 4.03982e9 0.185440
\(900\) 0 0
\(901\) −2.02163e10 −0.920798
\(902\) 0 0
\(903\) −2.64462e10 −1.19524
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.78240e10 0.793196 0.396598 0.917992i \(-0.370191\pi\)
0.396598 + 0.917992i \(0.370191\pi\)
\(908\) 0 0
\(909\) 4.72326e10 2.08578
\(910\) 0 0
\(911\) −1.87703e10 −0.822538 −0.411269 0.911514i \(-0.634914\pi\)
−0.411269 + 0.911514i \(0.634914\pi\)
\(912\) 0 0
\(913\) −1.35937e10 −0.591138
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −3.08280e8 −0.0132024
\(918\) 0 0
\(919\) −3.75844e10 −1.59736 −0.798681 0.601754i \(-0.794469\pi\)
−0.798681 + 0.601754i \(0.794469\pi\)
\(920\) 0 0
\(921\) −6.89197e10 −2.90693
\(922\) 0 0
\(923\) 2.74648e10 1.14966
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1.99707e10 0.823404
\(928\) 0 0
\(929\) −1.92372e10 −0.787205 −0.393602 0.919281i \(-0.628771\pi\)
−0.393602 + 0.919281i \(0.628771\pi\)
\(930\) 0 0
\(931\) 2.53876e9 0.103109
\(932\) 0 0
\(933\) 5.48943e10 2.21279
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.04732e9 −0.0415900 −0.0207950 0.999784i \(-0.506620\pi\)
−0.0207950 + 0.999784i \(0.506620\pi\)
\(938\) 0 0
\(939\) 5.57413e10 2.19709
\(940\) 0 0
\(941\) −7.97861e9 −0.312150 −0.156075 0.987745i \(-0.549884\pi\)
−0.156075 + 0.987745i \(0.549884\pi\)
\(942\) 0 0
\(943\) 1.15531e10 0.448650
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4.26943e9 −0.163360 −0.0816799 0.996659i \(-0.526029\pi\)
−0.0816799 + 0.996659i \(0.526029\pi\)
\(948\) 0 0
\(949\) 1.81106e10 0.687860
\(950\) 0 0
\(951\) −2.97963e10 −1.12339
\(952\) 0 0
\(953\) 1.06048e10 0.396897 0.198449 0.980111i \(-0.436410\pi\)
0.198449 + 0.980111i \(0.436410\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −2.11588e10 −0.780367
\(958\) 0 0
\(959\) −1.34904e10 −0.493922
\(960\) 0 0
\(961\) −2.58740e10 −0.940441
\(962\) 0 0
\(963\) 3.54950e8 0.0128078
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.65090e10 −0.587123 −0.293562 0.955940i \(-0.594841\pi\)
−0.293562 + 0.955940i \(0.594841\pi\)
\(968\) 0 0
\(969\) 3.86254e9 0.136377
\(970\) 0 0
\(971\) 2.46094e10 0.862649 0.431324 0.902197i \(-0.358046\pi\)
0.431324 + 0.902197i \(0.358046\pi\)
\(972\) 0 0
\(973\) −1.45503e10 −0.506379
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2.53886e10 0.870979 0.435489 0.900194i \(-0.356575\pi\)
0.435489 + 0.900194i \(0.356575\pi\)
\(978\) 0 0
\(979\) −1.63380e10 −0.556492
\(980\) 0 0
\(981\) 4.65399e10 1.57392
\(982\) 0 0
\(983\) 1.87585e10 0.629884 0.314942 0.949111i \(-0.398015\pi\)
0.314942 + 0.949111i \(0.398015\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −2.61246e10 −0.864846
\(988\) 0 0
\(989\) −5.64107e10 −1.85428
\(990\) 0 0
\(991\) 3.59792e9 0.117434 0.0587170 0.998275i \(-0.481299\pi\)
0.0587170 + 0.998275i \(0.481299\pi\)
\(992\) 0 0
\(993\) −2.56656e10 −0.831821
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.34287e10 0.429143 0.214571 0.976708i \(-0.431165\pi\)
0.214571 + 0.976708i \(0.431165\pi\)
\(998\) 0 0
\(999\) −9.44952e10 −2.99868
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 400.8.a.b.1.1 1
4.3 odd 2 200.8.a.i.1.1 1
5.2 odd 4 400.8.c.b.49.2 2
5.3 odd 4 400.8.c.b.49.1 2
5.4 even 2 16.8.a.c.1.1 1
15.14 odd 2 144.8.a.g.1.1 1
20.3 even 4 200.8.c.a.49.2 2
20.7 even 4 200.8.c.a.49.1 2
20.19 odd 2 8.8.a.a.1.1 1
40.19 odd 2 64.8.a.g.1.1 1
40.29 even 2 64.8.a.a.1.1 1
60.59 even 2 72.8.a.d.1.1 1
80.19 odd 4 256.8.b.e.129.1 2
80.29 even 4 256.8.b.c.129.2 2
80.59 odd 4 256.8.b.e.129.2 2
80.69 even 4 256.8.b.c.129.1 2
120.29 odd 2 576.8.a.k.1.1 1
120.59 even 2 576.8.a.j.1.1 1
140.139 even 2 392.8.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.8.a.a.1.1 1 20.19 odd 2
16.8.a.c.1.1 1 5.4 even 2
64.8.a.a.1.1 1 40.29 even 2
64.8.a.g.1.1 1 40.19 odd 2
72.8.a.d.1.1 1 60.59 even 2
144.8.a.g.1.1 1 15.14 odd 2
200.8.a.i.1.1 1 4.3 odd 2
200.8.c.a.49.1 2 20.7 even 4
200.8.c.a.49.2 2 20.3 even 4
256.8.b.c.129.1 2 80.69 even 4
256.8.b.c.129.2 2 80.29 even 4
256.8.b.e.129.1 2 80.19 odd 4
256.8.b.e.129.2 2 80.59 odd 4
392.8.a.d.1.1 1 140.139 even 2
400.8.a.b.1.1 1 1.1 even 1 trivial
400.8.c.b.49.1 2 5.3 odd 4
400.8.c.b.49.2 2 5.2 odd 4
576.8.a.j.1.1 1 120.59 even 2
576.8.a.k.1.1 1 120.29 odd 2