Properties

Label 16.8.a.a.1.1
Level $16$
Weight $8$
Character 16.1
Self dual yes
Analytic conductor $4.998$
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] = [16,8,Mod(1,16)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(16, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("16.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 16.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: \(4.99816040775\)
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 \(\chi\) \(=\) 16.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-44.0000 q^{3} +430.000 q^{5} +1224.00 q^{7} -251.000 q^{9} +O(q^{10})\) \(q-44.0000 q^{3} +430.000 q^{5} +1224.00 q^{7} -251.000 q^{9} +3164.00 q^{11} +6118.00 q^{13} -18920.0 q^{15} -16270.0 q^{17} +5476.00 q^{19} -53856.0 q^{21} -1576.00 q^{23} +106775. q^{25} +107272. q^{27} +122838. q^{29} -251360. q^{31} -139216. q^{33} +526320. q^{35} -52338.0 q^{37} -269192. q^{39} -319398. q^{41} -710788. q^{43} -107930. q^{45} -284112. q^{47} +674633. q^{49} +715880. q^{51} +296062. q^{53} +1.36052e6 q^{55} -240944. q^{57} +897548. q^{59} -884810. q^{61} -307224. q^{63} +2.63074e6 q^{65} -4.65969e6 q^{67} +69344.0 q^{69} +2.71079e6 q^{71} -5.67085e6 q^{73} -4.69810e6 q^{75} +3.87274e6 q^{77} +5.12418e6 q^{79} -4.17103e6 q^{81} +1.56356e6 q^{83} -6.99610e6 q^{85} -5.40487e6 q^{87} +1.16057e7 q^{89} +7.48843e6 q^{91} +1.10598e7 q^{93} +2.35468e6 q^{95} +1.09316e7 q^{97} -794164. q^{99} +O(q^{100})\)

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
\(3\) −44.0000 −0.940867 −0.470434 0.882435i \(-0.655902\pi\)
−0.470434 + 0.882435i \(0.655902\pi\)
\(4\) 0 0
\(5\) 430.000 1.53841 0.769207 0.638999i \(-0.220651\pi\)
0.769207 + 0.638999i \(0.220651\pi\)
\(6\) 0 0
\(7\) 1224.00 1.34877 0.674386 0.738379i \(-0.264409\pi\)
0.674386 + 0.738379i \(0.264409\pi\)
\(8\) 0 0
\(9\) −251.000 −0.114769
\(10\) 0 0
\(11\) 3164.00 0.716741 0.358370 0.933580i \(-0.383333\pi\)
0.358370 + 0.933580i \(0.383333\pi\)
\(12\) 0 0
\(13\) 6118.00 0.772339 0.386169 0.922428i \(-0.373798\pi\)
0.386169 + 0.922428i \(0.373798\pi\)
\(14\) 0 0
\(15\) −18920.0 −1.44744
\(16\) 0 0
\(17\) −16270.0 −0.803186 −0.401593 0.915818i \(-0.631543\pi\)
−0.401593 + 0.915818i \(0.631543\pi\)
\(18\) 0 0
\(19\) 5476.00 0.183158 0.0915790 0.995798i \(-0.470809\pi\)
0.0915790 + 0.995798i \(0.470809\pi\)
\(20\) 0 0
\(21\) −53856.0 −1.26901
\(22\) 0 0
\(23\) −1576.00 −0.0270090 −0.0135045 0.999909i \(-0.504299\pi\)
−0.0135045 + 0.999909i \(0.504299\pi\)
\(24\) 0 0
\(25\) 106775. 1.36672
\(26\) 0 0
\(27\) 107272. 1.04885
\(28\) 0 0
\(29\) 122838. 0.935276 0.467638 0.883920i \(-0.345105\pi\)
0.467638 + 0.883920i \(0.345105\pi\)
\(30\) 0 0
\(31\) −251360. −1.51541 −0.757705 0.652597i \(-0.773679\pi\)
−0.757705 + 0.652597i \(0.773679\pi\)
\(32\) 0 0
\(33\) −139216. −0.674358
\(34\) 0 0
\(35\) 526320. 2.07497
\(36\) 0 0
\(37\) −52338.0 −0.169868 −0.0849339 0.996387i \(-0.527068\pi\)
−0.0849339 + 0.996387i \(0.527068\pi\)
\(38\) 0 0
\(39\) −269192. −0.726668
\(40\) 0 0
\(41\) −319398. −0.723750 −0.361875 0.932227i \(-0.617863\pi\)
−0.361875 + 0.932227i \(0.617863\pi\)
\(42\) 0 0
\(43\) −710788. −1.36333 −0.681664 0.731665i \(-0.738743\pi\)
−0.681664 + 0.731665i \(0.738743\pi\)
\(44\) 0 0
\(45\) −107930. −0.176562
\(46\) 0 0
\(47\) −284112. −0.399160 −0.199580 0.979882i \(-0.563958\pi\)
−0.199580 + 0.979882i \(0.563958\pi\)
\(48\) 0 0
\(49\) 674633. 0.819184
\(50\) 0 0
\(51\) 715880. 0.755692
\(52\) 0 0
\(53\) 296062. 0.273160 0.136580 0.990629i \(-0.456389\pi\)
0.136580 + 0.990629i \(0.456389\pi\)
\(54\) 0 0
\(55\) 1.36052e6 1.10264
\(56\) 0 0
\(57\) −240944. −0.172327
\(58\) 0 0
\(59\) 897548. 0.568952 0.284476 0.958683i \(-0.408180\pi\)
0.284476 + 0.958683i \(0.408180\pi\)
\(60\) 0 0
\(61\) −884810. −0.499109 −0.249555 0.968361i \(-0.580284\pi\)
−0.249555 + 0.968361i \(0.580284\pi\)
\(62\) 0 0
\(63\) −307224. −0.154797
\(64\) 0 0
\(65\) 2.63074e6 1.18818
\(66\) 0 0
\(67\) −4.65969e6 −1.89276 −0.946380 0.323057i \(-0.895290\pi\)
−0.946380 + 0.323057i \(0.895290\pi\)
\(68\) 0 0
\(69\) 69344.0 0.0254119
\(70\) 0 0
\(71\) 2.71079e6 0.898860 0.449430 0.893316i \(-0.351627\pi\)
0.449430 + 0.893316i \(0.351627\pi\)
\(72\) 0 0
\(73\) −5.67085e6 −1.70615 −0.853077 0.521784i \(-0.825267\pi\)
−0.853077 + 0.521784i \(0.825267\pi\)
\(74\) 0 0
\(75\) −4.69810e6 −1.28590
\(76\) 0 0
\(77\) 3.87274e6 0.966719
\(78\) 0 0
\(79\) 5.12418e6 1.16931 0.584654 0.811282i \(-0.301230\pi\)
0.584654 + 0.811282i \(0.301230\pi\)
\(80\) 0 0
\(81\) −4.17103e6 −0.872059
\(82\) 0 0
\(83\) 1.56356e6 0.300151 0.150076 0.988675i \(-0.452048\pi\)
0.150076 + 0.988675i \(0.452048\pi\)
\(84\) 0 0
\(85\) −6.99610e6 −1.23563
\(86\) 0 0
\(87\) −5.40487e6 −0.879970
\(88\) 0 0
\(89\) 1.16057e7 1.74504 0.872520 0.488579i \(-0.162484\pi\)
0.872520 + 0.488579i \(0.162484\pi\)
\(90\) 0 0
\(91\) 7.48843e6 1.04171
\(92\) 0 0
\(93\) 1.10598e7 1.42580
\(94\) 0 0
\(95\) 2.35468e6 0.281773
\(96\) 0 0
\(97\) 1.09316e7 1.21614 0.608070 0.793884i \(-0.291944\pi\)
0.608070 + 0.793884i \(0.291944\pi\)
\(98\) 0 0
\(99\) −794164. −0.0822597
\(100\) 0 0
\(101\) −1.47908e7 −1.42846 −0.714230 0.699911i \(-0.753223\pi\)
−0.714230 + 0.699911i \(0.753223\pi\)
\(102\) 0 0
\(103\) −8.76105e6 −0.789998 −0.394999 0.918682i \(-0.629255\pi\)
−0.394999 + 0.918682i \(0.629255\pi\)
\(104\) 0 0
\(105\) −2.31581e7 −1.95227
\(106\) 0 0
\(107\) 4.29510e6 0.338946 0.169473 0.985535i \(-0.445794\pi\)
0.169473 + 0.985535i \(0.445794\pi\)
\(108\) 0 0
\(109\) 3.74465e6 0.276961 0.138480 0.990365i \(-0.455778\pi\)
0.138480 + 0.990365i \(0.455778\pi\)
\(110\) 0 0
\(111\) 2.30287e6 0.159823
\(112\) 0 0
\(113\) 3.90760e6 0.254763 0.127381 0.991854i \(-0.459343\pi\)
0.127381 + 0.991854i \(0.459343\pi\)
\(114\) 0 0
\(115\) −677680. −0.0415511
\(116\) 0 0
\(117\) −1.53562e6 −0.0886406
\(118\) 0 0
\(119\) −1.99145e7 −1.08331
\(120\) 0 0
\(121\) −9.47628e6 −0.486283
\(122\) 0 0
\(123\) 1.40535e7 0.680953
\(124\) 0 0
\(125\) 1.23195e7 0.564167
\(126\) 0 0
\(127\) −4.23320e7 −1.83382 −0.916909 0.399097i \(-0.869324\pi\)
−0.916909 + 0.399097i \(0.869324\pi\)
\(128\) 0 0
\(129\) 3.12747e7 1.28271
\(130\) 0 0
\(131\) 2.72098e7 1.05749 0.528745 0.848781i \(-0.322663\pi\)
0.528745 + 0.848781i \(0.322663\pi\)
\(132\) 0 0
\(133\) 6.70262e6 0.247038
\(134\) 0 0
\(135\) 4.61270e7 1.61357
\(136\) 0 0
\(137\) −5.01235e7 −1.66540 −0.832702 0.553721i \(-0.813207\pi\)
−0.832702 + 0.553721i \(0.813207\pi\)
\(138\) 0 0
\(139\) 1.22073e7 0.385539 0.192770 0.981244i \(-0.438253\pi\)
0.192770 + 0.981244i \(0.438253\pi\)
\(140\) 0 0
\(141\) 1.25009e7 0.375557
\(142\) 0 0
\(143\) 1.93574e7 0.553567
\(144\) 0 0
\(145\) 5.28203e7 1.43884
\(146\) 0 0
\(147\) −2.96839e7 −0.770743
\(148\) 0 0
\(149\) −4.66478e7 −1.15526 −0.577629 0.816299i \(-0.696022\pi\)
−0.577629 + 0.816299i \(0.696022\pi\)
\(150\) 0 0
\(151\) 3.04359e7 0.719395 0.359697 0.933069i \(-0.382880\pi\)
0.359697 + 0.933069i \(0.382880\pi\)
\(152\) 0 0
\(153\) 4.08377e6 0.0921810
\(154\) 0 0
\(155\) −1.08085e8 −2.33133
\(156\) 0 0
\(157\) 6.24060e6 0.128700 0.0643498 0.997927i \(-0.479503\pi\)
0.0643498 + 0.997927i \(0.479503\pi\)
\(158\) 0 0
\(159\) −1.30267e7 −0.257007
\(160\) 0 0
\(161\) −1.92902e6 −0.0364290
\(162\) 0 0
\(163\) 2.74060e6 0.0495665 0.0247833 0.999693i \(-0.492110\pi\)
0.0247833 + 0.999693i \(0.492110\pi\)
\(164\) 0 0
\(165\) −5.98629e7 −1.03744
\(166\) 0 0
\(167\) −4.27680e7 −0.710578 −0.355289 0.934757i \(-0.615617\pi\)
−0.355289 + 0.934757i \(0.615617\pi\)
\(168\) 0 0
\(169\) −2.53186e7 −0.403493
\(170\) 0 0
\(171\) −1.37448e6 −0.0210209
\(172\) 0 0
\(173\) 7.21411e7 1.05931 0.529653 0.848214i \(-0.322322\pi\)
0.529653 + 0.848214i \(0.322322\pi\)
\(174\) 0 0
\(175\) 1.30693e8 1.84339
\(176\) 0 0
\(177\) −3.94921e7 −0.535308
\(178\) 0 0
\(179\) 7.21119e7 0.939769 0.469885 0.882728i \(-0.344296\pi\)
0.469885 + 0.882728i \(0.344296\pi\)
\(180\) 0 0
\(181\) 1.44881e7 0.181608 0.0908041 0.995869i \(-0.471056\pi\)
0.0908041 + 0.995869i \(0.471056\pi\)
\(182\) 0 0
\(183\) 3.89316e7 0.469595
\(184\) 0 0
\(185\) −2.25053e7 −0.261327
\(186\) 0 0
\(187\) −5.14783e7 −0.575676
\(188\) 0 0
\(189\) 1.31301e8 1.41466
\(190\) 0 0
\(191\) −1.10503e8 −1.14751 −0.573755 0.819027i \(-0.694514\pi\)
−0.573755 + 0.819027i \(0.694514\pi\)
\(192\) 0 0
\(193\) 3.43354e6 0.0343788 0.0171894 0.999852i \(-0.494528\pi\)
0.0171894 + 0.999852i \(0.494528\pi\)
\(194\) 0 0
\(195\) −1.15753e8 −1.11792
\(196\) 0 0
\(197\) −3.77557e7 −0.351844 −0.175922 0.984404i \(-0.556291\pi\)
−0.175922 + 0.984404i \(0.556291\pi\)
\(198\) 0 0
\(199\) 1.37979e8 1.24116 0.620580 0.784143i \(-0.286897\pi\)
0.620580 + 0.784143i \(0.286897\pi\)
\(200\) 0 0
\(201\) 2.05026e8 1.78084
\(202\) 0 0
\(203\) 1.50354e8 1.26147
\(204\) 0 0
\(205\) −1.37341e8 −1.11343
\(206\) 0 0
\(207\) 395576. 0.00309980
\(208\) 0 0
\(209\) 1.73261e7 0.131277
\(210\) 0 0
\(211\) −1.05441e8 −0.772720 −0.386360 0.922348i \(-0.626268\pi\)
−0.386360 + 0.922348i \(0.626268\pi\)
\(212\) 0 0
\(213\) −1.19275e8 −0.845708
\(214\) 0 0
\(215\) −3.05639e8 −2.09736
\(216\) 0 0
\(217\) −3.07665e8 −2.04394
\(218\) 0 0
\(219\) 2.49518e8 1.60527
\(220\) 0 0
\(221\) −9.95399e7 −0.620332
\(222\) 0 0
\(223\) −1.32420e8 −0.799625 −0.399812 0.916597i \(-0.630925\pi\)
−0.399812 + 0.916597i \(0.630925\pi\)
\(224\) 0 0
\(225\) −2.68005e7 −0.156857
\(226\) 0 0
\(227\) 2.23501e8 1.26820 0.634102 0.773249i \(-0.281370\pi\)
0.634102 + 0.773249i \(0.281370\pi\)
\(228\) 0 0
\(229\) 3.58523e8 1.97284 0.986422 0.164228i \(-0.0525132\pi\)
0.986422 + 0.164228i \(0.0525132\pi\)
\(230\) 0 0
\(231\) −1.70400e8 −0.909554
\(232\) 0 0
\(233\) 2.05337e8 1.06346 0.531731 0.846913i \(-0.321542\pi\)
0.531731 + 0.846913i \(0.321542\pi\)
\(234\) 0 0
\(235\) −1.22168e8 −0.614074
\(236\) 0 0
\(237\) −2.25464e8 −1.10016
\(238\) 0 0
\(239\) 1.27881e8 0.605917 0.302959 0.953004i \(-0.402026\pi\)
0.302959 + 0.953004i \(0.402026\pi\)
\(240\) 0 0
\(241\) 2.13448e8 0.982275 0.491138 0.871082i \(-0.336581\pi\)
0.491138 + 0.871082i \(0.336581\pi\)
\(242\) 0 0
\(243\) −5.10785e7 −0.228358
\(244\) 0 0
\(245\) 2.90092e8 1.26024
\(246\) 0 0
\(247\) 3.35022e7 0.141460
\(248\) 0 0
\(249\) −6.87965e7 −0.282402
\(250\) 0 0
\(251\) 2.81046e8 1.12181 0.560906 0.827880i \(-0.310453\pi\)
0.560906 + 0.827880i \(0.310453\pi\)
\(252\) 0 0
\(253\) −4.98646e6 −0.0193585
\(254\) 0 0
\(255\) 3.07828e8 1.16257
\(256\) 0 0
\(257\) −4.11663e8 −1.51278 −0.756389 0.654122i \(-0.773038\pi\)
−0.756389 + 0.654122i \(0.773038\pi\)
\(258\) 0 0
\(259\) −6.40617e7 −0.229113
\(260\) 0 0
\(261\) −3.08323e7 −0.107341
\(262\) 0 0
\(263\) 1.01818e8 0.345127 0.172563 0.984998i \(-0.444795\pi\)
0.172563 + 0.984998i \(0.444795\pi\)
\(264\) 0 0
\(265\) 1.27307e8 0.420233
\(266\) 0 0
\(267\) −5.10650e8 −1.64185
\(268\) 0 0
\(269\) −5.30511e6 −0.0166173 −0.00830867 0.999965i \(-0.502645\pi\)
−0.00830867 + 0.999965i \(0.502645\pi\)
\(270\) 0 0
\(271\) −2.51186e8 −0.766660 −0.383330 0.923611i \(-0.625223\pi\)
−0.383330 + 0.923611i \(0.625223\pi\)
\(272\) 0 0
\(273\) −3.29491e8 −0.980109
\(274\) 0 0
\(275\) 3.37836e8 0.979584
\(276\) 0 0
\(277\) −1.19651e8 −0.338251 −0.169125 0.985595i \(-0.554094\pi\)
−0.169125 + 0.985595i \(0.554094\pi\)
\(278\) 0 0
\(279\) 6.30914e7 0.173922
\(280\) 0 0
\(281\) 6.21696e8 1.67150 0.835749 0.549111i \(-0.185034\pi\)
0.835749 + 0.549111i \(0.185034\pi\)
\(282\) 0 0
\(283\) −3.41726e8 −0.896243 −0.448122 0.893973i \(-0.647907\pi\)
−0.448122 + 0.893973i \(0.647907\pi\)
\(284\) 0 0
\(285\) −1.03606e8 −0.265111
\(286\) 0 0
\(287\) −3.90943e8 −0.976173
\(288\) 0 0
\(289\) −1.45626e8 −0.354892
\(290\) 0 0
\(291\) −4.80991e8 −1.14423
\(292\) 0 0
\(293\) 3.71317e6 0.00862398 0.00431199 0.999991i \(-0.498627\pi\)
0.00431199 + 0.999991i \(0.498627\pi\)
\(294\) 0 0
\(295\) 3.85946e8 0.875284
\(296\) 0 0
\(297\) 3.39409e8 0.751753
\(298\) 0 0
\(299\) −9.64197e6 −0.0208601
\(300\) 0 0
\(301\) −8.70005e8 −1.83882
\(302\) 0 0
\(303\) 6.50797e8 1.34399
\(304\) 0 0
\(305\) −3.80468e8 −0.767837
\(306\) 0 0
\(307\) 3.23276e7 0.0637659 0.0318830 0.999492i \(-0.489850\pi\)
0.0318830 + 0.999492i \(0.489850\pi\)
\(308\) 0 0
\(309\) 3.85486e8 0.743283
\(310\) 0 0
\(311\) 8.26997e8 1.55899 0.779494 0.626410i \(-0.215476\pi\)
0.779494 + 0.626410i \(0.215476\pi\)
\(312\) 0 0
\(313\) 3.53027e8 0.650732 0.325366 0.945588i \(-0.394513\pi\)
0.325366 + 0.945588i \(0.394513\pi\)
\(314\) 0 0
\(315\) −1.32106e8 −0.238142
\(316\) 0 0
\(317\) 9.30187e8 1.64007 0.820036 0.572312i \(-0.193953\pi\)
0.820036 + 0.572312i \(0.193953\pi\)
\(318\) 0 0
\(319\) 3.88659e8 0.670350
\(320\) 0 0
\(321\) −1.88984e8 −0.318903
\(322\) 0 0
\(323\) −8.90945e7 −0.147110
\(324\) 0 0
\(325\) 6.53249e8 1.05557
\(326\) 0 0
\(327\) −1.64764e8 −0.260583
\(328\) 0 0
\(329\) −3.47753e8 −0.538376
\(330\) 0 0
\(331\) 3.21754e8 0.487670 0.243835 0.969817i \(-0.421594\pi\)
0.243835 + 0.969817i \(0.421594\pi\)
\(332\) 0 0
\(333\) 1.31368e7 0.0194956
\(334\) 0 0
\(335\) −2.00367e9 −2.91185
\(336\) 0 0
\(337\) −2.47705e8 −0.352558 −0.176279 0.984340i \(-0.556406\pi\)
−0.176279 + 0.984340i \(0.556406\pi\)
\(338\) 0 0
\(339\) −1.71934e8 −0.239698
\(340\) 0 0
\(341\) −7.95303e8 −1.08616
\(342\) 0 0
\(343\) −1.82266e8 −0.243880
\(344\) 0 0
\(345\) 2.98179e7 0.0390940
\(346\) 0 0
\(347\) 2.23076e8 0.286615 0.143308 0.989678i \(-0.454226\pi\)
0.143308 + 0.989678i \(0.454226\pi\)
\(348\) 0 0
\(349\) −5.57087e8 −0.701510 −0.350755 0.936467i \(-0.614075\pi\)
−0.350755 + 0.936467i \(0.614075\pi\)
\(350\) 0 0
\(351\) 6.56290e8 0.810067
\(352\) 0 0
\(353\) 5.64509e8 0.683061 0.341530 0.939871i \(-0.389055\pi\)
0.341530 + 0.939871i \(0.389055\pi\)
\(354\) 0 0
\(355\) 1.16564e9 1.38282
\(356\) 0 0
\(357\) 8.76237e8 1.01926
\(358\) 0 0
\(359\) −1.66005e9 −1.89361 −0.946803 0.321813i \(-0.895708\pi\)
−0.946803 + 0.321813i \(0.895708\pi\)
\(360\) 0 0
\(361\) −8.63885e8 −0.966453
\(362\) 0 0
\(363\) 4.16956e8 0.457527
\(364\) 0 0
\(365\) −2.43847e9 −2.62477
\(366\) 0 0
\(367\) 7.00412e8 0.739643 0.369822 0.929103i \(-0.379419\pi\)
0.369822 + 0.929103i \(0.379419\pi\)
\(368\) 0 0
\(369\) 8.01689e7 0.0830642
\(370\) 0 0
\(371\) 3.62380e8 0.368430
\(372\) 0 0
\(373\) 4.33445e8 0.432467 0.216233 0.976342i \(-0.430623\pi\)
0.216233 + 0.976342i \(0.430623\pi\)
\(374\) 0 0
\(375\) −5.42058e8 −0.530807
\(376\) 0 0
\(377\) 7.51523e8 0.722350
\(378\) 0 0
\(379\) −1.38421e9 −1.30606 −0.653032 0.757330i \(-0.726504\pi\)
−0.653032 + 0.757330i \(0.726504\pi\)
\(380\) 0 0
\(381\) 1.86261e9 1.72538
\(382\) 0 0
\(383\) 4.70821e8 0.428213 0.214107 0.976810i \(-0.431316\pi\)
0.214107 + 0.976810i \(0.431316\pi\)
\(384\) 0 0
\(385\) 1.66528e9 1.48722
\(386\) 0 0
\(387\) 1.78408e8 0.156468
\(388\) 0 0
\(389\) 1.52817e7 0.0131628 0.00658140 0.999978i \(-0.497905\pi\)
0.00658140 + 0.999978i \(0.497905\pi\)
\(390\) 0 0
\(391\) 2.56415e7 0.0216933
\(392\) 0 0
\(393\) −1.19723e9 −0.994957
\(394\) 0 0
\(395\) 2.20340e9 1.79888
\(396\) 0 0
\(397\) 1.02601e9 0.822969 0.411485 0.911417i \(-0.365010\pi\)
0.411485 + 0.911417i \(0.365010\pi\)
\(398\) 0 0
\(399\) −2.94915e8 −0.232430
\(400\) 0 0
\(401\) −2.18349e8 −0.169101 −0.0845504 0.996419i \(-0.526945\pi\)
−0.0845504 + 0.996419i \(0.526945\pi\)
\(402\) 0 0
\(403\) −1.53782e9 −1.17041
\(404\) 0 0
\(405\) −1.79354e9 −1.34159
\(406\) 0 0
\(407\) −1.65597e8 −0.121751
\(408\) 0 0
\(409\) 7.50779e8 0.542601 0.271300 0.962495i \(-0.412546\pi\)
0.271300 + 0.962495i \(0.412546\pi\)
\(410\) 0 0
\(411\) 2.20544e9 1.56692
\(412\) 0 0
\(413\) 1.09860e9 0.767386
\(414\) 0 0
\(415\) 6.72329e8 0.461757
\(416\) 0 0
\(417\) −5.37122e8 −0.362741
\(418\) 0 0
\(419\) −2.38283e8 −0.158250 −0.0791251 0.996865i \(-0.525213\pi\)
−0.0791251 + 0.996865i \(0.525213\pi\)
\(420\) 0 0
\(421\) −1.23640e9 −0.807552 −0.403776 0.914858i \(-0.632303\pi\)
−0.403776 + 0.914858i \(0.632303\pi\)
\(422\) 0 0
\(423\) 7.13121e7 0.0458112
\(424\) 0 0
\(425\) −1.73723e9 −1.09773
\(426\) 0 0
\(427\) −1.08301e9 −0.673184
\(428\) 0 0
\(429\) −8.51723e8 −0.520833
\(430\) 0 0
\(431\) −2.24030e9 −1.34783 −0.673915 0.738809i \(-0.735389\pi\)
−0.673915 + 0.738809i \(0.735389\pi\)
\(432\) 0 0
\(433\) −1.19040e9 −0.704670 −0.352335 0.935874i \(-0.614612\pi\)
−0.352335 + 0.935874i \(0.614612\pi\)
\(434\) 0 0
\(435\) −2.32409e9 −1.35376
\(436\) 0 0
\(437\) −8.63018e6 −0.00494692
\(438\) 0 0
\(439\) 1.38822e9 0.783127 0.391564 0.920151i \(-0.371934\pi\)
0.391564 + 0.920151i \(0.371934\pi\)
\(440\) 0 0
\(441\) −1.69333e8 −0.0940170
\(442\) 0 0
\(443\) −2.85788e9 −1.56182 −0.780911 0.624642i \(-0.785245\pi\)
−0.780911 + 0.624642i \(0.785245\pi\)
\(444\) 0 0
\(445\) 4.99044e9 2.68460
\(446\) 0 0
\(447\) 2.05250e9 1.08694
\(448\) 0 0
\(449\) 2.96866e7 0.0154774 0.00773870 0.999970i \(-0.497537\pi\)
0.00773870 + 0.999970i \(0.497537\pi\)
\(450\) 0 0
\(451\) −1.01058e9 −0.518741
\(452\) 0 0
\(453\) −1.33918e9 −0.676855
\(454\) 0 0
\(455\) 3.22003e9 1.60258
\(456\) 0 0
\(457\) 1.83015e9 0.896974 0.448487 0.893789i \(-0.351963\pi\)
0.448487 + 0.893789i \(0.351963\pi\)
\(458\) 0 0
\(459\) −1.74532e9 −0.842422
\(460\) 0 0
\(461\) −8.13408e8 −0.386683 −0.193342 0.981132i \(-0.561933\pi\)
−0.193342 + 0.981132i \(0.561933\pi\)
\(462\) 0 0
\(463\) 1.01303e9 0.474337 0.237168 0.971469i \(-0.423781\pi\)
0.237168 + 0.971469i \(0.423781\pi\)
\(464\) 0 0
\(465\) 4.75573e9 2.19347
\(466\) 0 0
\(467\) 3.80251e9 1.72767 0.863837 0.503771i \(-0.168055\pi\)
0.863837 + 0.503771i \(0.168055\pi\)
\(468\) 0 0
\(469\) −5.70346e9 −2.55290
\(470\) 0 0
\(471\) −2.74586e8 −0.121089
\(472\) 0 0
\(473\) −2.24893e9 −0.977153
\(474\) 0 0
\(475\) 5.84700e8 0.250326
\(476\) 0 0
\(477\) −7.43116e7 −0.0313503
\(478\) 0 0
\(479\) 8.67385e8 0.360610 0.180305 0.983611i \(-0.442292\pi\)
0.180305 + 0.983611i \(0.442292\pi\)
\(480\) 0 0
\(481\) −3.20204e8 −0.131195
\(482\) 0 0
\(483\) 8.48771e7 0.0342748
\(484\) 0 0
\(485\) 4.70060e9 1.87093
\(486\) 0 0
\(487\) −1.30043e9 −0.510195 −0.255098 0.966915i \(-0.582108\pi\)
−0.255098 + 0.966915i \(0.582108\pi\)
\(488\) 0 0
\(489\) −1.20586e8 −0.0466355
\(490\) 0 0
\(491\) −1.88652e9 −0.719244 −0.359622 0.933098i \(-0.617094\pi\)
−0.359622 + 0.933098i \(0.617094\pi\)
\(492\) 0 0
\(493\) −1.99857e9 −0.751201
\(494\) 0 0
\(495\) −3.41491e8 −0.126550
\(496\) 0 0
\(497\) 3.31801e9 1.21236
\(498\) 0 0
\(499\) 3.81787e9 1.37553 0.687763 0.725935i \(-0.258593\pi\)
0.687763 + 0.725935i \(0.258593\pi\)
\(500\) 0 0
\(501\) 1.88179e9 0.668559
\(502\) 0 0
\(503\) 4.47974e8 0.156951 0.0784757 0.996916i \(-0.474995\pi\)
0.0784757 + 0.996916i \(0.474995\pi\)
\(504\) 0 0
\(505\) −6.36006e9 −2.19756
\(506\) 0 0
\(507\) 1.11402e9 0.379633
\(508\) 0 0
\(509\) 4.50519e9 1.51426 0.757131 0.653263i \(-0.226600\pi\)
0.757131 + 0.653263i \(0.226600\pi\)
\(510\) 0 0
\(511\) −6.94113e9 −2.30121
\(512\) 0 0
\(513\) 5.87421e8 0.192105
\(514\) 0 0
\(515\) −3.76725e9 −1.21534
\(516\) 0 0
\(517\) −8.98930e8 −0.286094
\(518\) 0 0
\(519\) −3.17421e9 −0.996666
\(520\) 0 0
\(521\) 4.27301e8 0.132374 0.0661868 0.997807i \(-0.478917\pi\)
0.0661868 + 0.997807i \(0.478917\pi\)
\(522\) 0 0
\(523\) −8.14705e8 −0.249026 −0.124513 0.992218i \(-0.539737\pi\)
−0.124513 + 0.992218i \(0.539737\pi\)
\(524\) 0 0
\(525\) −5.75047e9 −1.73439
\(526\) 0 0
\(527\) 4.08963e9 1.21716
\(528\) 0 0
\(529\) −3.40234e9 −0.999271
\(530\) 0 0
\(531\) −2.25285e8 −0.0652981
\(532\) 0 0
\(533\) −1.95408e9 −0.558980
\(534\) 0 0
\(535\) 1.84689e9 0.521439
\(536\) 0 0
\(537\) −3.17292e9 −0.884198
\(538\) 0 0
\(539\) 2.13454e9 0.587142
\(540\) 0 0
\(541\) 5.17489e9 1.40511 0.702556 0.711629i \(-0.252042\pi\)
0.702556 + 0.711629i \(0.252042\pi\)
\(542\) 0 0
\(543\) −6.37475e8 −0.170869
\(544\) 0 0
\(545\) 1.61020e9 0.426080
\(546\) 0 0
\(547\) 6.45830e9 1.68718 0.843592 0.536984i \(-0.180436\pi\)
0.843592 + 0.536984i \(0.180436\pi\)
\(548\) 0 0
\(549\) 2.22087e8 0.0572823
\(550\) 0 0
\(551\) 6.72661e8 0.171303
\(552\) 0 0
\(553\) 6.27199e9 1.57713
\(554\) 0 0
\(555\) 9.90235e8 0.245874
\(556\) 0 0
\(557\) −4.83562e9 −1.18566 −0.592829 0.805328i \(-0.701989\pi\)
−0.592829 + 0.805328i \(0.701989\pi\)
\(558\) 0 0
\(559\) −4.34860e9 −1.05295
\(560\) 0 0
\(561\) 2.26504e9 0.541635
\(562\) 0 0
\(563\) −1.16508e9 −0.275154 −0.137577 0.990491i \(-0.543931\pi\)
−0.137577 + 0.990491i \(0.543931\pi\)
\(564\) 0 0
\(565\) 1.68027e9 0.391931
\(566\) 0 0
\(567\) −5.10534e9 −1.17621
\(568\) 0 0
\(569\) 2.79201e9 0.635367 0.317683 0.948197i \(-0.397095\pi\)
0.317683 + 0.948197i \(0.397095\pi\)
\(570\) 0 0
\(571\) 7.14069e9 1.60514 0.802572 0.596555i \(-0.203464\pi\)
0.802572 + 0.596555i \(0.203464\pi\)
\(572\) 0 0
\(573\) 4.86212e9 1.07965
\(574\) 0 0
\(575\) −1.68277e8 −0.0369138
\(576\) 0 0
\(577\) 4.44897e7 0.00964148 0.00482074 0.999988i \(-0.498466\pi\)
0.00482074 + 0.999988i \(0.498466\pi\)
\(578\) 0 0
\(579\) −1.51076e8 −0.0323459
\(580\) 0 0
\(581\) 1.91379e9 0.404835
\(582\) 0 0
\(583\) 9.36740e8 0.195785
\(584\) 0 0
\(585\) −6.60316e8 −0.136366
\(586\) 0 0
\(587\) 4.21422e8 0.0859972 0.0429986 0.999075i \(-0.486309\pi\)
0.0429986 + 0.999075i \(0.486309\pi\)
\(588\) 0 0
\(589\) −1.37645e9 −0.277559
\(590\) 0 0
\(591\) 1.66125e9 0.331039
\(592\) 0 0
\(593\) −4.60547e9 −0.906948 −0.453474 0.891270i \(-0.649815\pi\)
−0.453474 + 0.891270i \(0.649815\pi\)
\(594\) 0 0
\(595\) −8.56323e9 −1.66659
\(596\) 0 0
\(597\) −6.07109e9 −1.16777
\(598\) 0 0
\(599\) 5.96438e8 0.113389 0.0566946 0.998392i \(-0.481944\pi\)
0.0566946 + 0.998392i \(0.481944\pi\)
\(600\) 0 0
\(601\) 2.18925e9 0.411372 0.205686 0.978618i \(-0.434058\pi\)
0.205686 + 0.978618i \(0.434058\pi\)
\(602\) 0 0
\(603\) 1.16958e9 0.217230
\(604\) 0 0
\(605\) −4.07480e9 −0.748105
\(606\) 0 0
\(607\) −9.88184e8 −0.179340 −0.0896701 0.995972i \(-0.528581\pi\)
−0.0896701 + 0.995972i \(0.528581\pi\)
\(608\) 0 0
\(609\) −6.61556e9 −1.18688
\(610\) 0 0
\(611\) −1.73820e9 −0.308287
\(612\) 0 0
\(613\) 3.18993e8 0.0559332 0.0279666 0.999609i \(-0.491097\pi\)
0.0279666 + 0.999609i \(0.491097\pi\)
\(614\) 0 0
\(615\) 6.04301e9 1.04759
\(616\) 0 0
\(617\) 3.83798e9 0.657816 0.328908 0.944362i \(-0.393319\pi\)
0.328908 + 0.944362i \(0.393319\pi\)
\(618\) 0 0
\(619\) −2.96834e9 −0.503032 −0.251516 0.967853i \(-0.580929\pi\)
−0.251516 + 0.967853i \(0.580929\pi\)
\(620\) 0 0
\(621\) −1.69061e8 −0.0283284
\(622\) 0 0
\(623\) 1.42053e10 2.35366
\(624\) 0 0
\(625\) −3.04441e9 −0.498796
\(626\) 0 0
\(627\) −7.62347e8 −0.123514
\(628\) 0 0
\(629\) 8.51539e8 0.136436
\(630\) 0 0
\(631\) −8.10788e9 −1.28471 −0.642354 0.766408i \(-0.722042\pi\)
−0.642354 + 0.766408i \(0.722042\pi\)
\(632\) 0 0
\(633\) 4.63941e9 0.727026
\(634\) 0 0
\(635\) −1.82028e10 −2.82117
\(636\) 0 0
\(637\) 4.12740e9 0.632687
\(638\) 0 0
\(639\) −6.80409e8 −0.103161
\(640\) 0 0
\(641\) −1.10357e10 −1.65499 −0.827495 0.561473i \(-0.810235\pi\)
−0.827495 + 0.561473i \(0.810235\pi\)
\(642\) 0 0
\(643\) −6.76532e9 −1.00358 −0.501788 0.864991i \(-0.667324\pi\)
−0.501788 + 0.864991i \(0.667324\pi\)
\(644\) 0 0
\(645\) 1.34481e10 1.97334
\(646\) 0 0
\(647\) −6.76242e9 −0.981606 −0.490803 0.871271i \(-0.663297\pi\)
−0.490803 + 0.871271i \(0.663297\pi\)
\(648\) 0 0
\(649\) 2.83984e9 0.407791
\(650\) 0 0
\(651\) 1.35372e10 1.92308
\(652\) 0 0
\(653\) −9.17211e9 −1.28906 −0.644530 0.764579i \(-0.722947\pi\)
−0.644530 + 0.764579i \(0.722947\pi\)
\(654\) 0 0
\(655\) 1.17002e10 1.62686
\(656\) 0 0
\(657\) 1.42338e9 0.195814
\(658\) 0 0
\(659\) 5.62299e9 0.765364 0.382682 0.923880i \(-0.375000\pi\)
0.382682 + 0.923880i \(0.375000\pi\)
\(660\) 0 0
\(661\) 3.39966e9 0.457857 0.228928 0.973443i \(-0.426478\pi\)
0.228928 + 0.973443i \(0.426478\pi\)
\(662\) 0 0
\(663\) 4.37975e9 0.583650
\(664\) 0 0
\(665\) 2.88213e9 0.380047
\(666\) 0 0
\(667\) −1.93593e8 −0.0252609
\(668\) 0 0
\(669\) 5.82647e9 0.752340
\(670\) 0 0
\(671\) −2.79954e9 −0.357732
\(672\) 0 0
\(673\) 1.14073e10 1.44254 0.721271 0.692653i \(-0.243558\pi\)
0.721271 + 0.692653i \(0.243558\pi\)
\(674\) 0 0
\(675\) 1.14540e10 1.43348
\(676\) 0 0
\(677\) 3.10086e9 0.384080 0.192040 0.981387i \(-0.438490\pi\)
0.192040 + 0.981387i \(0.438490\pi\)
\(678\) 0 0
\(679\) 1.33803e10 1.64029
\(680\) 0 0
\(681\) −9.83405e9 −1.19321
\(682\) 0 0
\(683\) −6.43067e9 −0.772296 −0.386148 0.922437i \(-0.626195\pi\)
−0.386148 + 0.922437i \(0.626195\pi\)
\(684\) 0 0
\(685\) −2.15531e10 −2.56208
\(686\) 0 0
\(687\) −1.57750e10 −1.85618
\(688\) 0 0
\(689\) 1.81131e9 0.210972
\(690\) 0 0
\(691\) 4.53885e9 0.523326 0.261663 0.965159i \(-0.415729\pi\)
0.261663 + 0.965159i \(0.415729\pi\)
\(692\) 0 0
\(693\) −9.72057e8 −0.110949
\(694\) 0 0
\(695\) 5.24915e9 0.593119
\(696\) 0 0
\(697\) 5.19661e9 0.581306
\(698\) 0 0
\(699\) −9.03484e9 −1.00058
\(700\) 0 0
\(701\) −7.84396e9 −0.860047 −0.430024 0.902818i \(-0.641495\pi\)
−0.430024 + 0.902818i \(0.641495\pi\)
\(702\) 0 0
\(703\) −2.86603e8 −0.0311126
\(704\) 0 0
\(705\) 5.37540e9 0.577762
\(706\) 0 0
\(707\) −1.81040e10 −1.92666
\(708\) 0 0
\(709\) 4.05387e9 0.427177 0.213588 0.976924i \(-0.431485\pi\)
0.213588 + 0.976924i \(0.431485\pi\)
\(710\) 0 0
\(711\) −1.28617e9 −0.134201
\(712\) 0 0
\(713\) 3.96143e8 0.0409297
\(714\) 0 0
\(715\) 8.32366e9 0.851615
\(716\) 0 0
\(717\) −5.62676e9 −0.570087
\(718\) 0 0
\(719\) −7.84938e9 −0.787561 −0.393780 0.919205i \(-0.628833\pi\)
−0.393780 + 0.919205i \(0.628833\pi\)
\(720\) 0 0
\(721\) −1.07235e10 −1.06553
\(722\) 0 0
\(723\) −9.39173e9 −0.924190
\(724\) 0 0
\(725\) 1.31160e10 1.27826
\(726\) 0 0
\(727\) −6.44318e9 −0.621913 −0.310957 0.950424i \(-0.600649\pi\)
−0.310957 + 0.950424i \(0.600649\pi\)
\(728\) 0 0
\(729\) 1.13695e10 1.08691
\(730\) 0 0
\(731\) 1.15645e10 1.09501
\(732\) 0 0
\(733\) 9.48714e9 0.889757 0.444879 0.895591i \(-0.353247\pi\)
0.444879 + 0.895591i \(0.353247\pi\)
\(734\) 0 0
\(735\) −1.27641e10 −1.18572
\(736\) 0 0
\(737\) −1.47433e10 −1.35662
\(738\) 0 0
\(739\) 2.07907e10 1.89502 0.947508 0.319733i \(-0.103593\pi\)
0.947508 + 0.319733i \(0.103593\pi\)
\(740\) 0 0
\(741\) −1.47410e9 −0.133095
\(742\) 0 0
\(743\) 5.87068e9 0.525083 0.262541 0.964921i \(-0.415439\pi\)
0.262541 + 0.964921i \(0.415439\pi\)
\(744\) 0 0
\(745\) −2.00585e10 −1.77727
\(746\) 0 0
\(747\) −3.92453e8 −0.0344481
\(748\) 0 0
\(749\) 5.25720e9 0.457160
\(750\) 0 0
\(751\) −7.08413e9 −0.610305 −0.305152 0.952304i \(-0.598707\pi\)
−0.305152 + 0.952304i \(0.598707\pi\)
\(752\) 0 0
\(753\) −1.23660e10 −1.05548
\(754\) 0 0
\(755\) 1.30874e10 1.10673
\(756\) 0 0
\(757\) −1.12028e10 −0.938622 −0.469311 0.883033i \(-0.655498\pi\)
−0.469311 + 0.883033i \(0.655498\pi\)
\(758\) 0 0
\(759\) 2.19404e8 0.0182137
\(760\) 0 0
\(761\) 2.08007e10 1.71093 0.855465 0.517861i \(-0.173272\pi\)
0.855465 + 0.517861i \(0.173272\pi\)
\(762\) 0 0
\(763\) 4.58345e9 0.373557
\(764\) 0 0
\(765\) 1.75602e9 0.141813
\(766\) 0 0
\(767\) 5.49120e9 0.439424
\(768\) 0 0
\(769\) 1.94604e10 1.54315 0.771576 0.636137i \(-0.219469\pi\)
0.771576 + 0.636137i \(0.219469\pi\)
\(770\) 0 0
\(771\) 1.81132e10 1.42332
\(772\) 0 0
\(773\) 1.81342e7 0.00141211 0.000706057 1.00000i \(-0.499775\pi\)
0.000706057 1.00000i \(0.499775\pi\)
\(774\) 0 0
\(775\) −2.68390e10 −2.07114
\(776\) 0 0
\(777\) 2.81872e9 0.215565
\(778\) 0 0
\(779\) −1.74902e9 −0.132561
\(780\) 0 0
\(781\) 8.57695e9 0.644249
\(782\) 0 0
\(783\) 1.31771e10 0.980964
\(784\) 0 0
\(785\) 2.68346e9 0.197993
\(786\) 0 0
\(787\) −3.48980e9 −0.255205 −0.127603 0.991825i \(-0.540728\pi\)
−0.127603 + 0.991825i \(0.540728\pi\)
\(788\) 0 0
\(789\) −4.47998e9 −0.324718
\(790\) 0 0
\(791\) 4.78290e9 0.343617
\(792\) 0 0
\(793\) −5.41327e9 −0.385481
\(794\) 0 0
\(795\) −5.60149e9 −0.395384
\(796\) 0 0
\(797\) 9.60101e9 0.671758 0.335879 0.941905i \(-0.390967\pi\)
0.335879 + 0.941905i \(0.390967\pi\)
\(798\) 0 0
\(799\) 4.62250e9 0.320600
\(800\) 0 0
\(801\) −2.91302e9 −0.200277
\(802\) 0 0
\(803\) −1.79426e10 −1.22287
\(804\) 0 0
\(805\) −8.29480e8 −0.0560429
\(806\) 0 0
\(807\) 2.33425e8 0.0156347
\(808\) 0 0
\(809\) 1.28054e10 0.850305 0.425152 0.905122i \(-0.360221\pi\)
0.425152 + 0.905122i \(0.360221\pi\)
\(810\) 0 0
\(811\) −1.41860e10 −0.933871 −0.466936 0.884291i \(-0.654642\pi\)
−0.466936 + 0.884291i \(0.654642\pi\)
\(812\) 0 0
\(813\) 1.10522e10 0.721325
\(814\) 0 0
\(815\) 1.17846e9 0.0762539
\(816\) 0 0
\(817\) −3.89228e9 −0.249704
\(818\) 0 0
\(819\) −1.87960e9 −0.119556
\(820\) 0 0
\(821\) −1.88610e10 −1.18950 −0.594749 0.803912i \(-0.702749\pi\)
−0.594749 + 0.803912i \(0.702749\pi\)
\(822\) 0 0
\(823\) 7.98216e9 0.499138 0.249569 0.968357i \(-0.419711\pi\)
0.249569 + 0.968357i \(0.419711\pi\)
\(824\) 0 0
\(825\) −1.48648e10 −0.921658
\(826\) 0 0
\(827\) 1.69379e10 1.04134 0.520668 0.853759i \(-0.325683\pi\)
0.520668 + 0.853759i \(0.325683\pi\)
\(828\) 0 0
\(829\) −1.93636e10 −1.18044 −0.590221 0.807242i \(-0.700960\pi\)
−0.590221 + 0.807242i \(0.700960\pi\)
\(830\) 0 0
\(831\) 5.26466e9 0.318249
\(832\) 0 0
\(833\) −1.09763e10 −0.657957
\(834\) 0 0
\(835\) −1.83903e10 −1.09316
\(836\) 0 0
\(837\) −2.69639e10 −1.58944
\(838\) 0 0
\(839\) 9.16289e9 0.535631 0.267815 0.963470i \(-0.413698\pi\)
0.267815 + 0.963470i \(0.413698\pi\)
\(840\) 0 0
\(841\) −2.16070e9 −0.125259
\(842\) 0 0
\(843\) −2.73546e10 −1.57266
\(844\) 0 0
\(845\) −1.08870e10 −0.620740
\(846\) 0 0
\(847\) −1.15990e10 −0.655884
\(848\) 0 0
\(849\) 1.50360e10 0.843246
\(850\) 0 0
\(851\) 8.24847e7 0.00458796
\(852\) 0 0
\(853\) 2.16335e10 1.19345 0.596727 0.802445i \(-0.296468\pi\)
0.596727 + 0.802445i \(0.296468\pi\)
\(854\) 0 0
\(855\) −5.91025e8 −0.0323388
\(856\) 0 0
\(857\) −1.34574e10 −0.730345 −0.365172 0.930940i \(-0.618990\pi\)
−0.365172 + 0.930940i \(0.618990\pi\)
\(858\) 0 0
\(859\) 2.10347e10 1.13230 0.566148 0.824304i \(-0.308433\pi\)
0.566148 + 0.824304i \(0.308433\pi\)
\(860\) 0 0
\(861\) 1.72015e10 0.918449
\(862\) 0 0
\(863\) 3.15666e10 1.67182 0.835912 0.548863i \(-0.184939\pi\)
0.835912 + 0.548863i \(0.184939\pi\)
\(864\) 0 0
\(865\) 3.10207e10 1.62965
\(866\) 0 0
\(867\) 6.40753e9 0.333906
\(868\) 0 0
\(869\) 1.62129e10 0.838091
\(870\) 0 0
\(871\) −2.85080e10 −1.46185
\(872\) 0 0
\(873\) −2.74384e9 −0.139575
\(874\) 0 0
\(875\) 1.50791e10 0.760933
\(876\) 0 0
\(877\) 3.09323e10 1.54851 0.774254 0.632875i \(-0.218125\pi\)
0.774254 + 0.632875i \(0.218125\pi\)
\(878\) 0 0
\(879\) −1.63379e8 −0.00811402
\(880\) 0 0
\(881\) 2.59947e10 1.28077 0.640383 0.768056i \(-0.278776\pi\)
0.640383 + 0.768056i \(0.278776\pi\)
\(882\) 0 0
\(883\) −2.77295e10 −1.35544 −0.677718 0.735322i \(-0.737031\pi\)
−0.677718 + 0.735322i \(0.737031\pi\)
\(884\) 0 0
\(885\) −1.69816e10 −0.823526
\(886\) 0 0
\(887\) 6.18068e9 0.297374 0.148687 0.988884i \(-0.452495\pi\)
0.148687 + 0.988884i \(0.452495\pi\)
\(888\) 0 0
\(889\) −5.18144e10 −2.47340
\(890\) 0 0
\(891\) −1.31971e10 −0.625040
\(892\) 0 0
\(893\) −1.55580e9 −0.0731093
\(894\) 0 0
\(895\) 3.10081e10 1.44575
\(896\) 0 0
\(897\) 4.24247e8 0.0196266
\(898\) 0 0
\(899\) −3.08766e10 −1.41733
\(900\) 0 0
\(901\) −4.81693e9 −0.219398
\(902\) 0 0
\(903\) 3.82802e10 1.73008
\(904\) 0 0
\(905\) 6.22987e9 0.279389
\(906\) 0 0
\(907\) 9.47637e9 0.421713 0.210856 0.977517i \(-0.432375\pi\)
0.210856 + 0.977517i \(0.432375\pi\)
\(908\) 0 0
\(909\) 3.71250e9 0.163943
\(910\) 0 0
\(911\) −2.63833e10 −1.15615 −0.578075 0.815983i \(-0.696196\pi\)
−0.578075 + 0.815983i \(0.696196\pi\)
\(912\) 0 0
\(913\) 4.94709e9 0.215131
\(914\) 0 0
\(915\) 1.67406e10 0.722433
\(916\) 0 0
\(917\) 3.33048e10 1.42631
\(918\) 0 0
\(919\) 2.34537e10 0.996797 0.498398 0.866948i \(-0.333922\pi\)
0.498398 + 0.866948i \(0.333922\pi\)
\(920\) 0 0
\(921\) −1.42241e9 −0.0599952
\(922\) 0 0
\(923\) 1.65846e10 0.694224
\(924\) 0 0
\(925\) −5.58839e9 −0.232162
\(926\) 0 0
\(927\) 2.19902e9 0.0906673
\(928\) 0 0
\(929\) −4.04287e10 −1.65438 −0.827189 0.561924i \(-0.810061\pi\)
−0.827189 + 0.561924i \(0.810061\pi\)
\(930\) 0 0
\(931\) 3.69429e9 0.150040
\(932\) 0 0
\(933\) −3.63879e10 −1.46680
\(934\) 0 0
\(935\) −2.21357e10 −0.885629
\(936\) 0 0
\(937\) −1.09610e10 −0.435273 −0.217636 0.976030i \(-0.569835\pi\)
−0.217636 + 0.976030i \(0.569835\pi\)
\(938\) 0 0
\(939\) −1.55332e10 −0.612252
\(940\) 0 0
\(941\) −1.87938e9 −0.0735276 −0.0367638 0.999324i \(-0.511705\pi\)
−0.0367638 + 0.999324i \(0.511705\pi\)
\(942\) 0 0
\(943\) 5.03371e8 0.0195478
\(944\) 0 0
\(945\) 5.64594e10 2.17633
\(946\) 0 0
\(947\) 1.97695e10 0.756435 0.378218 0.925717i \(-0.376537\pi\)
0.378218 + 0.925717i \(0.376537\pi\)
\(948\) 0 0
\(949\) −3.46943e10 −1.31773
\(950\) 0 0
\(951\) −4.09282e10 −1.54309
\(952\) 0 0
\(953\) 2.81049e9 0.105186 0.0525929 0.998616i \(-0.483251\pi\)
0.0525929 + 0.998616i \(0.483251\pi\)
\(954\) 0 0
\(955\) −4.75162e10 −1.76535
\(956\) 0 0
\(957\) −1.71010e10 −0.630711
\(958\) 0 0
\(959\) −6.13512e10 −2.24625
\(960\) 0 0
\(961\) 3.56692e10 1.29647
\(962\) 0 0
\(963\) −1.07807e9 −0.0389005
\(964\) 0 0
\(965\) 1.47642e9 0.0528889
\(966\) 0 0
\(967\) 2.06586e10 0.734697 0.367348 0.930083i \(-0.380266\pi\)
0.367348 + 0.930083i \(0.380266\pi\)
\(968\) 0 0
\(969\) 3.92016e9 0.138411
\(970\) 0 0
\(971\) 4.53900e10 1.59108 0.795542 0.605899i \(-0.207186\pi\)
0.795542 + 0.605899i \(0.207186\pi\)
\(972\) 0 0
\(973\) 1.49418e10 0.520004
\(974\) 0 0
\(975\) −2.87430e10 −0.993152
\(976\) 0 0
\(977\) −2.14529e10 −0.735960 −0.367980 0.929834i \(-0.619950\pi\)
−0.367980 + 0.929834i \(0.619950\pi\)
\(978\) 0 0
\(979\) 3.67204e10 1.25074
\(980\) 0 0
\(981\) −9.39906e8 −0.0317865
\(982\) 0 0
\(983\) −3.89213e10 −1.30692 −0.653462 0.756960i \(-0.726684\pi\)
−0.653462 + 0.756960i \(0.726684\pi\)
\(984\) 0 0
\(985\) −1.62349e10 −0.541282
\(986\) 0 0
\(987\) 1.53011e10 0.506540
\(988\) 0 0
\(989\) 1.12020e9 0.0368222
\(990\) 0 0
\(991\) 3.27057e10 1.06749 0.533747 0.845644i \(-0.320784\pi\)
0.533747 + 0.845644i \(0.320784\pi\)
\(992\) 0 0
\(993\) −1.41572e10 −0.458833
\(994\) 0 0
\(995\) 5.93311e10 1.90942
\(996\) 0 0
\(997\) −9.21065e9 −0.294345 −0.147173 0.989111i \(-0.547017\pi\)
−0.147173 + 0.989111i \(0.547017\pi\)
\(998\) 0 0
\(999\) −5.61440e9 −0.178166
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 16.8.a.a.1.1 1
3.2 odd 2 144.8.a.a.1.1 1
4.3 odd 2 8.8.a.b.1.1 1
5.2 odd 4 400.8.c.f.49.2 2
5.3 odd 4 400.8.c.f.49.1 2
5.4 even 2 400.8.a.p.1.1 1
8.3 odd 2 64.8.a.b.1.1 1
8.5 even 2 64.8.a.f.1.1 1
12.11 even 2 72.8.a.a.1.1 1
16.3 odd 4 256.8.b.g.129.2 2
16.5 even 4 256.8.b.a.129.2 2
16.11 odd 4 256.8.b.g.129.1 2
16.13 even 4 256.8.b.a.129.1 2
20.3 even 4 200.8.c.c.49.2 2
20.7 even 4 200.8.c.c.49.1 2
20.19 odd 2 200.8.a.b.1.1 1
24.5 odd 2 576.8.a.z.1.1 1
24.11 even 2 576.8.a.y.1.1 1
28.27 even 2 392.8.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8.8.a.b.1.1 1 4.3 odd 2
16.8.a.a.1.1 1 1.1 even 1 trivial
64.8.a.b.1.1 1 8.3 odd 2
64.8.a.f.1.1 1 8.5 even 2
72.8.a.a.1.1 1 12.11 even 2
144.8.a.a.1.1 1 3.2 odd 2
200.8.a.b.1.1 1 20.19 odd 2
200.8.c.c.49.1 2 20.7 even 4
200.8.c.c.49.2 2 20.3 even 4
256.8.b.a.129.1 2 16.13 even 4
256.8.b.a.129.2 2 16.5 even 4
256.8.b.g.129.1 2 16.11 odd 4
256.8.b.g.129.2 2 16.3 odd 4
392.8.a.b.1.1 1 28.27 even 2
400.8.a.p.1.1 1 5.4 even 2
400.8.c.f.49.1 2 5.3 odd 4
400.8.c.f.49.2 2 5.2 odd 4
576.8.a.y.1.1 1 24.11 even 2
576.8.a.z.1.1 1 24.5 odd 2