Properties

Label 338.10.a.c.1.1
Level $338$
Weight $10$
Character 338.1
Self dual yes
Analytic conductor $174.082$
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] = [338,10,Mod(1,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 338.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: \(174.082112623\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 338.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+16.0000 q^{2} -273.000 q^{3} +256.000 q^{4} -1015.00 q^{5} -4368.00 q^{6} -3955.00 q^{7} +4096.00 q^{8} +54846.0 q^{9} +O(q^{10})\) \(q+16.0000 q^{2} -273.000 q^{3} +256.000 q^{4} -1015.00 q^{5} -4368.00 q^{6} -3955.00 q^{7} +4096.00 q^{8} +54846.0 q^{9} -16240.0 q^{10} +50998.0 q^{11} -69888.0 q^{12} -63280.0 q^{14} +277095. q^{15} +65536.0 q^{16} +509757. q^{17} +877536. q^{18} +626574. q^{19} -259840. q^{20} +1.07972e6 q^{21} +815968. q^{22} +653524. q^{23} -1.11821e6 q^{24} -922900. q^{25} -9.59950e6 q^{27} -1.01248e6 q^{28} -4.94301e6 q^{29} +4.43352e6 q^{30} -4.07170e6 q^{31} +1.04858e6 q^{32} -1.39225e7 q^{33} +8.15611e6 q^{34} +4.01432e6 q^{35} +1.40406e7 q^{36} -2.34888e6 q^{37} +1.00252e7 q^{38} -4.15744e6 q^{40} +1.33510e7 q^{41} +1.72754e7 q^{42} -7.83485e6 q^{43} +1.30555e7 q^{44} -5.56687e7 q^{45} +1.04564e7 q^{46} +3.96377e7 q^{47} -1.78913e7 q^{48} -2.47116e7 q^{49} -1.47664e7 q^{50} -1.39164e8 q^{51} +7.32009e7 q^{53} -1.53592e8 q^{54} -5.17630e7 q^{55} -1.61997e7 q^{56} -1.71055e8 q^{57} -7.90881e7 q^{58} +1.41142e8 q^{59} +7.09363e7 q^{60} -1.32061e8 q^{61} -6.51472e7 q^{62} -2.16916e8 q^{63} +1.67772e7 q^{64} -2.22759e8 q^{66} +1.85673e8 q^{67} +1.30498e8 q^{68} -1.78412e8 q^{69} +6.42292e7 q^{70} -2.24453e8 q^{71} +2.24649e8 q^{72} +1.72524e8 q^{73} -3.75821e7 q^{74} +2.51952e8 q^{75} +1.60403e8 q^{76} -2.01697e8 q^{77} -6.43288e8 q^{79} -6.65190e7 q^{80} +1.54113e9 q^{81} +2.13615e8 q^{82} -7.20077e8 q^{83} +2.76407e8 q^{84} -5.17403e8 q^{85} -1.25358e8 q^{86} +1.34944e9 q^{87} +2.08888e8 q^{88} +7.30281e7 q^{89} -8.90699e8 q^{90} +1.67302e8 q^{92} +1.11157e9 q^{93} +6.34203e8 q^{94} -6.35973e8 q^{95} -2.86261e8 q^{96} +1.58798e7 q^{97} -3.95385e8 q^{98} +2.79704e9 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0000 0.707107
\(3\) −273.000 −1.94588 −0.972942 0.231049i \(-0.925784\pi\)
−0.972942 + 0.231049i \(0.925784\pi\)
\(4\) 256.000 0.500000
\(5\) −1015.00 −0.726275 −0.363137 0.931736i \(-0.618294\pi\)
−0.363137 + 0.931736i \(0.618294\pi\)
\(6\) −4368.00 −1.37595
\(7\) −3955.00 −0.622595 −0.311297 0.950313i \(-0.600763\pi\)
−0.311297 + 0.950313i \(0.600763\pi\)
\(8\) 4096.00 0.353553
\(9\) 54846.0 2.78647
\(10\) −16240.0 −0.513554
\(11\) 50998.0 1.05023 0.525117 0.851030i \(-0.324022\pi\)
0.525117 + 0.851030i \(0.324022\pi\)
\(12\) −69888.0 −0.972942
\(13\) 0 0
\(14\) −63280.0 −0.440241
\(15\) 277095. 1.41325
\(16\) 65536.0 0.250000
\(17\) 509757. 1.48028 0.740139 0.672454i \(-0.234760\pi\)
0.740139 + 0.672454i \(0.234760\pi\)
\(18\) 877536. 1.97033
\(19\) 626574. 1.10301 0.551507 0.834170i \(-0.314053\pi\)
0.551507 + 0.834170i \(0.314053\pi\)
\(20\) −259840. −0.363137
\(21\) 1.07972e6 1.21150
\(22\) 815968. 0.742628
\(23\) 653524. 0.486952 0.243476 0.969907i \(-0.421712\pi\)
0.243476 + 0.969907i \(0.421712\pi\)
\(24\) −1.11821e6 −0.687974
\(25\) −922900. −0.472525
\(26\) 0 0
\(27\) −9.59950e6 −3.47626
\(28\) −1.01248e6 −0.311297
\(29\) −4.94301e6 −1.29778 −0.648889 0.760883i \(-0.724766\pi\)
−0.648889 + 0.760883i \(0.724766\pi\)
\(30\) 4.43352e6 0.999316
\(31\) −4.07170e6 −0.791860 −0.395930 0.918281i \(-0.629578\pi\)
−0.395930 + 0.918281i \(0.629578\pi\)
\(32\) 1.04858e6 0.176777
\(33\) −1.39225e7 −2.04363
\(34\) 8.15611e6 1.04671
\(35\) 4.01432e6 0.452175
\(36\) 1.40406e7 1.39323
\(37\) −2.34888e6 −0.206041 −0.103020 0.994679i \(-0.532851\pi\)
−0.103020 + 0.994679i \(0.532851\pi\)
\(38\) 1.00252e7 0.779949
\(39\) 0 0
\(40\) −4.15744e6 −0.256777
\(41\) 1.33510e7 0.737879 0.368939 0.929453i \(-0.379721\pi\)
0.368939 + 0.929453i \(0.379721\pi\)
\(42\) 1.72754e7 0.856658
\(43\) −7.83485e6 −0.349480 −0.174740 0.984615i \(-0.555909\pi\)
−0.174740 + 0.984615i \(0.555909\pi\)
\(44\) 1.30555e7 0.525117
\(45\) −5.56687e7 −2.02374
\(46\) 1.04564e7 0.344327
\(47\) 3.96377e7 1.18486 0.592431 0.805621i \(-0.298168\pi\)
0.592431 + 0.805621i \(0.298168\pi\)
\(48\) −1.78913e7 −0.486471
\(49\) −2.47116e7 −0.612376
\(50\) −1.47664e7 −0.334125
\(51\) −1.39164e8 −2.88045
\(52\) 0 0
\(53\) 7.32009e7 1.27431 0.637155 0.770736i \(-0.280111\pi\)
0.637155 + 0.770736i \(0.280111\pi\)
\(54\) −1.53592e8 −2.45808
\(55\) −5.17630e7 −0.762759
\(56\) −1.61997e7 −0.220120
\(57\) −1.71055e8 −2.14634
\(58\) −7.90881e7 −0.917667
\(59\) 1.41142e8 1.51643 0.758213 0.652007i \(-0.226073\pi\)
0.758213 + 0.652007i \(0.226073\pi\)
\(60\) 7.09363e7 0.706623
\(61\) −1.32061e8 −1.22121 −0.610606 0.791934i \(-0.709074\pi\)
−0.610606 + 0.791934i \(0.709074\pi\)
\(62\) −6.51472e7 −0.559929
\(63\) −2.16916e8 −1.73484
\(64\) 1.67772e7 0.125000
\(65\) 0 0
\(66\) −2.22759e8 −1.44507
\(67\) 1.85673e8 1.12567 0.562837 0.826568i \(-0.309710\pi\)
0.562837 + 0.826568i \(0.309710\pi\)
\(68\) 1.30498e8 0.740139
\(69\) −1.78412e8 −0.947552
\(70\) 6.42292e7 0.319736
\(71\) −2.24453e8 −1.04824 −0.524122 0.851643i \(-0.675606\pi\)
−0.524122 + 0.851643i \(0.675606\pi\)
\(72\) 2.24649e8 0.985164
\(73\) 1.72524e8 0.711043 0.355521 0.934668i \(-0.384303\pi\)
0.355521 + 0.934668i \(0.384303\pi\)
\(74\) −3.75821e7 −0.145693
\(75\) 2.51952e8 0.919479
\(76\) 1.60403e8 0.551507
\(77\) −2.01697e8 −0.653870
\(78\) 0 0
\(79\) −6.43288e8 −1.85816 −0.929081 0.369876i \(-0.879400\pi\)
−0.929081 + 0.369876i \(0.879400\pi\)
\(80\) −6.65190e7 −0.181569
\(81\) 1.54113e9 3.97792
\(82\) 2.13615e8 0.521759
\(83\) −7.20077e8 −1.66544 −0.832718 0.553698i \(-0.813216\pi\)
−0.832718 + 0.553698i \(0.813216\pi\)
\(84\) 2.76407e8 0.605748
\(85\) −5.17403e8 −1.07509
\(86\) −1.25358e8 −0.247120
\(87\) 1.34944e9 2.52532
\(88\) 2.08888e8 0.371314
\(89\) 7.30281e7 0.123377 0.0616886 0.998095i \(-0.480351\pi\)
0.0616886 + 0.998095i \(0.480351\pi\)
\(90\) −8.90699e8 −1.43100
\(91\) 0 0
\(92\) 1.67302e8 0.243476
\(93\) 1.11157e9 1.54087
\(94\) 6.34203e8 0.837824
\(95\) −6.35973e8 −0.801092
\(96\) −2.86261e8 −0.343987
\(97\) 1.58798e7 0.0182126 0.00910629 0.999959i \(-0.497101\pi\)
0.00910629 + 0.999959i \(0.497101\pi\)
\(98\) −3.95385e8 −0.433015
\(99\) 2.79704e9 2.92644
\(100\) −2.36262e8 −0.236262
\(101\) 1.15275e9 1.10228 0.551138 0.834414i \(-0.314194\pi\)
0.551138 + 0.834414i \(0.314194\pi\)
\(102\) −2.22662e9 −2.03678
\(103\) −1.13892e8 −0.0997069 −0.0498534 0.998757i \(-0.515875\pi\)
−0.0498534 + 0.998757i \(0.515875\pi\)
\(104\) 0 0
\(105\) −1.09591e9 −0.879880
\(106\) 1.17121e9 0.901073
\(107\) 4.16136e8 0.306908 0.153454 0.988156i \(-0.450960\pi\)
0.153454 + 0.988156i \(0.450960\pi\)
\(108\) −2.45747e9 −1.73813
\(109\) 2.97358e8 0.201772 0.100886 0.994898i \(-0.467832\pi\)
0.100886 + 0.994898i \(0.467832\pi\)
\(110\) −8.28208e8 −0.539352
\(111\) 6.41245e8 0.400932
\(112\) −2.59195e8 −0.155649
\(113\) −5.73150e8 −0.330685 −0.165343 0.986236i \(-0.552873\pi\)
−0.165343 + 0.986236i \(0.552873\pi\)
\(114\) −2.73688e9 −1.51769
\(115\) −6.63327e8 −0.353661
\(116\) −1.26541e9 −0.648889
\(117\) 0 0
\(118\) 2.25827e9 1.07227
\(119\) −2.01609e9 −0.921613
\(120\) 1.13498e9 0.499658
\(121\) 2.42848e8 0.102991
\(122\) −2.11298e9 −0.863527
\(123\) −3.64481e9 −1.43583
\(124\) −1.04236e9 −0.395930
\(125\) 2.91917e9 1.06946
\(126\) −3.47065e9 −1.22672
\(127\) −3.35094e9 −1.14301 −0.571506 0.820598i \(-0.693640\pi\)
−0.571506 + 0.820598i \(0.693640\pi\)
\(128\) 2.68435e8 0.0883883
\(129\) 2.13891e9 0.680048
\(130\) 0 0
\(131\) 3.38073e9 1.00297 0.501486 0.865166i \(-0.332787\pi\)
0.501486 + 0.865166i \(0.332787\pi\)
\(132\) −3.56415e9 −1.02182
\(133\) −2.47810e9 −0.686731
\(134\) 2.97077e9 0.795971
\(135\) 9.74349e9 2.52472
\(136\) 2.08796e9 0.523357
\(137\) 6.52392e7 0.0158222 0.00791109 0.999969i \(-0.497482\pi\)
0.00791109 + 0.999969i \(0.497482\pi\)
\(138\) −2.85459e9 −0.670021
\(139\) −3.81657e9 −0.867174 −0.433587 0.901112i \(-0.642752\pi\)
−0.433587 + 0.901112i \(0.642752\pi\)
\(140\) 1.02767e9 0.226087
\(141\) −1.08211e10 −2.30560
\(142\) −3.59124e9 −0.741220
\(143\) 0 0
\(144\) 3.59439e9 0.696616
\(145\) 5.01715e9 0.942543
\(146\) 2.76038e9 0.502783
\(147\) 6.74626e9 1.19161
\(148\) −6.01314e8 −0.103020
\(149\) 5.25362e9 0.873214 0.436607 0.899652i \(-0.356180\pi\)
0.436607 + 0.899652i \(0.356180\pi\)
\(150\) 4.03123e9 0.650170
\(151\) 7.46867e9 1.16909 0.584544 0.811362i \(-0.301273\pi\)
0.584544 + 0.811362i \(0.301273\pi\)
\(152\) 2.56645e9 0.389974
\(153\) 2.79581e10 4.12474
\(154\) −3.22715e9 −0.462356
\(155\) 4.13278e9 0.575108
\(156\) 0 0
\(157\) 2.50101e9 0.328524 0.164262 0.986417i \(-0.447476\pi\)
0.164262 + 0.986417i \(0.447476\pi\)
\(158\) −1.02926e10 −1.31392
\(159\) −1.99839e10 −2.47966
\(160\) −1.06430e9 −0.128388
\(161\) −2.58469e9 −0.303174
\(162\) 2.46581e10 2.81282
\(163\) −6.36713e9 −0.706479 −0.353240 0.935533i \(-0.614920\pi\)
−0.353240 + 0.935533i \(0.614920\pi\)
\(164\) 3.41785e9 0.368939
\(165\) 1.41313e10 1.48424
\(166\) −1.15212e10 −1.17764
\(167\) 1.08910e10 1.08354 0.541770 0.840527i \(-0.317754\pi\)
0.541770 + 0.840527i \(0.317754\pi\)
\(168\) 4.42251e9 0.428329
\(169\) 0 0
\(170\) −8.27845e9 −0.760202
\(171\) 3.43651e10 3.07351
\(172\) −2.00572e9 −0.174740
\(173\) −3.90042e9 −0.331058 −0.165529 0.986205i \(-0.552933\pi\)
−0.165529 + 0.986205i \(0.552933\pi\)
\(174\) 2.15911e10 1.78567
\(175\) 3.65007e9 0.294191
\(176\) 3.34220e9 0.262558
\(177\) −3.85317e10 −2.95079
\(178\) 1.16845e9 0.0872409
\(179\) 7.76362e9 0.565231 0.282615 0.959233i \(-0.408798\pi\)
0.282615 + 0.959233i \(0.408798\pi\)
\(180\) −1.42512e10 −1.01187
\(181\) 1.18648e10 0.821685 0.410842 0.911706i \(-0.365235\pi\)
0.410842 + 0.911706i \(0.365235\pi\)
\(182\) 0 0
\(183\) 3.60527e10 2.37634
\(184\) 2.67683e9 0.172164
\(185\) 2.38412e9 0.149642
\(186\) 1.77852e10 1.08956
\(187\) 2.59966e10 1.55464
\(188\) 1.01472e10 0.592431
\(189\) 3.79660e10 2.16430
\(190\) −1.01756e10 −0.566457
\(191\) −1.90377e10 −1.03505 −0.517527 0.855667i \(-0.673147\pi\)
−0.517527 + 0.855667i \(0.673147\pi\)
\(192\) −4.58018e9 −0.243236
\(193\) 1.51640e10 0.786693 0.393347 0.919390i \(-0.371317\pi\)
0.393347 + 0.919390i \(0.371317\pi\)
\(194\) 2.54076e8 0.0128782
\(195\) 0 0
\(196\) −6.32616e9 −0.306188
\(197\) 2.18786e10 1.03496 0.517478 0.855697i \(-0.326871\pi\)
0.517478 + 0.855697i \(0.326871\pi\)
\(198\) 4.47526e10 2.06931
\(199\) 2.40126e8 0.0108543 0.00542713 0.999985i \(-0.498272\pi\)
0.00542713 + 0.999985i \(0.498272\pi\)
\(200\) −3.78020e9 −0.167063
\(201\) −5.06888e10 −2.19043
\(202\) 1.84441e10 0.779427
\(203\) 1.95496e10 0.807989
\(204\) −3.56259e10 −1.44022
\(205\) −1.35512e10 −0.535903
\(206\) −1.82227e9 −0.0705034
\(207\) 3.58432e10 1.35688
\(208\) 0 0
\(209\) 3.19540e10 1.15842
\(210\) −1.75346e10 −0.622169
\(211\) −3.76473e10 −1.30756 −0.653781 0.756684i \(-0.726818\pi\)
−0.653781 + 0.756684i \(0.726818\pi\)
\(212\) 1.87394e10 0.637155
\(213\) 6.12756e10 2.03976
\(214\) 6.65818e9 0.217017
\(215\) 7.95237e9 0.253819
\(216\) −3.93195e10 −1.22904
\(217\) 1.61036e10 0.493007
\(218\) 4.75772e9 0.142674
\(219\) −4.70990e10 −1.38361
\(220\) −1.32513e10 −0.381379
\(221\) 0 0
\(222\) 1.02599e10 0.283502
\(223\) −4.95888e10 −1.34280 −0.671401 0.741095i \(-0.734307\pi\)
−0.671401 + 0.741095i \(0.734307\pi\)
\(224\) −4.14712e9 −0.110060
\(225\) −5.06174e10 −1.31667
\(226\) −9.17039e9 −0.233830
\(227\) −4.95601e10 −1.23884 −0.619421 0.785059i \(-0.712633\pi\)
−0.619421 + 0.785059i \(0.712633\pi\)
\(228\) −4.37900e10 −1.07317
\(229\) 4.36828e10 1.04966 0.524832 0.851206i \(-0.324128\pi\)
0.524832 + 0.851206i \(0.324128\pi\)
\(230\) −1.06132e10 −0.250076
\(231\) 5.50633e10 1.27236
\(232\) −2.02466e10 −0.458834
\(233\) 4.54109e10 1.00939 0.504695 0.863298i \(-0.331605\pi\)
0.504695 + 0.863298i \(0.331605\pi\)
\(234\) 0 0
\(235\) −4.02322e10 −0.860536
\(236\) 3.61323e10 0.758213
\(237\) 1.75618e11 3.61577
\(238\) −3.22574e10 −0.651678
\(239\) −1.52935e10 −0.303192 −0.151596 0.988443i \(-0.548441\pi\)
−0.151596 + 0.988443i \(0.548441\pi\)
\(240\) 1.81597e10 0.353312
\(241\) −3.11366e10 −0.594558 −0.297279 0.954791i \(-0.596079\pi\)
−0.297279 + 0.954791i \(0.596079\pi\)
\(242\) 3.88557e9 0.0728259
\(243\) −2.31781e11 −4.26433
\(244\) −3.38077e10 −0.610606
\(245\) 2.50823e10 0.444753
\(246\) −5.83170e10 −1.01528
\(247\) 0 0
\(248\) −1.66777e10 −0.279965
\(249\) 1.96581e11 3.24074
\(250\) 4.67066e10 0.756221
\(251\) 1.14721e11 1.82436 0.912182 0.409786i \(-0.134397\pi\)
0.912182 + 0.409786i \(0.134397\pi\)
\(252\) −5.55305e10 −0.867419
\(253\) 3.33284e10 0.511414
\(254\) −5.36151e10 −0.808231
\(255\) 1.41251e11 2.09200
\(256\) 4.29497e9 0.0625000
\(257\) −4.34944e10 −0.621920 −0.310960 0.950423i \(-0.600651\pi\)
−0.310960 + 0.950423i \(0.600651\pi\)
\(258\) 3.42226e10 0.480866
\(259\) 9.28983e9 0.128280
\(260\) 0 0
\(261\) −2.71104e11 −3.61621
\(262\) 5.40916e10 0.709209
\(263\) 6.65824e10 0.858141 0.429071 0.903271i \(-0.358841\pi\)
0.429071 + 0.903271i \(0.358841\pi\)
\(264\) −5.70264e10 −0.722534
\(265\) −7.42989e10 −0.925499
\(266\) −3.96496e10 −0.485592
\(267\) −1.99367e10 −0.240078
\(268\) 4.75323e10 0.562837
\(269\) 4.33594e10 0.504891 0.252446 0.967611i \(-0.418765\pi\)
0.252446 + 0.967611i \(0.418765\pi\)
\(270\) 1.55896e11 1.78524
\(271\) −6.11570e10 −0.688786 −0.344393 0.938826i \(-0.611915\pi\)
−0.344393 + 0.938826i \(0.611915\pi\)
\(272\) 3.34074e10 0.370069
\(273\) 0 0
\(274\) 1.04383e9 0.0111880
\(275\) −4.70661e10 −0.496262
\(276\) −4.56735e10 −0.473776
\(277\) −5.62976e10 −0.574555 −0.287277 0.957847i \(-0.592750\pi\)
−0.287277 + 0.957847i \(0.592750\pi\)
\(278\) −6.10651e10 −0.613185
\(279\) −2.23316e11 −2.20649
\(280\) 1.64427e10 0.159868
\(281\) −1.06179e11 −1.01592 −0.507961 0.861380i \(-0.669601\pi\)
−0.507961 + 0.861380i \(0.669601\pi\)
\(282\) −1.73137e11 −1.63031
\(283\) 4.87202e10 0.451513 0.225756 0.974184i \(-0.427515\pi\)
0.225756 + 0.974184i \(0.427515\pi\)
\(284\) −5.74599e10 −0.524122
\(285\) 1.73621e11 1.55883
\(286\) 0 0
\(287\) −5.28030e10 −0.459399
\(288\) 5.75102e10 0.492582
\(289\) 1.41264e11 1.19122
\(290\) 8.02744e10 0.666479
\(291\) −4.33518e9 −0.0354396
\(292\) 4.41661e10 0.355521
\(293\) 1.95097e11 1.54648 0.773242 0.634111i \(-0.218634\pi\)
0.773242 + 0.634111i \(0.218634\pi\)
\(294\) 1.07940e11 0.842598
\(295\) −1.43259e11 −1.10134
\(296\) −9.62102e9 −0.0728465
\(297\) −4.89555e11 −3.65088
\(298\) 8.40580e10 0.617456
\(299\) 0 0
\(300\) 6.44996e10 0.459739
\(301\) 3.09868e10 0.217584
\(302\) 1.19499e11 0.826670
\(303\) −3.14702e11 −2.14490
\(304\) 4.10632e10 0.275754
\(305\) 1.34042e11 0.886936
\(306\) 4.47330e11 2.91663
\(307\) −2.19416e11 −1.40976 −0.704880 0.709327i \(-0.748999\pi\)
−0.704880 + 0.709327i \(0.748999\pi\)
\(308\) −5.16345e10 −0.326935
\(309\) 3.10925e10 0.194018
\(310\) 6.61244e10 0.406663
\(311\) −5.37193e10 −0.325618 −0.162809 0.986658i \(-0.552055\pi\)
−0.162809 + 0.986658i \(0.552055\pi\)
\(312\) 0 0
\(313\) 2.75468e11 1.62226 0.811132 0.584862i \(-0.198852\pi\)
0.811132 + 0.584862i \(0.198852\pi\)
\(314\) 4.00161e10 0.232301
\(315\) 2.20170e11 1.25997
\(316\) −1.64682e11 −0.929081
\(317\) 1.42713e11 0.793776 0.396888 0.917867i \(-0.370090\pi\)
0.396888 + 0.917867i \(0.370090\pi\)
\(318\) −3.19742e11 −1.75338
\(319\) −2.52083e11 −1.36297
\(320\) −1.70289e10 −0.0907844
\(321\) −1.13605e11 −0.597208
\(322\) −4.13550e10 −0.214376
\(323\) 3.19400e11 1.63277
\(324\) 3.94529e11 1.98896
\(325\) 0 0
\(326\) −1.01874e11 −0.499556
\(327\) −8.11787e10 −0.392624
\(328\) 5.46855e10 0.260880
\(329\) −1.56767e11 −0.737689
\(330\) 2.26101e11 1.04952
\(331\) 1.75508e11 0.803656 0.401828 0.915715i \(-0.368375\pi\)
0.401828 + 0.915715i \(0.368375\pi\)
\(332\) −1.84340e11 −0.832718
\(333\) −1.28827e11 −0.574126
\(334\) 1.74257e11 0.766179
\(335\) −1.88458e11 −0.817548
\(336\) 7.07602e10 0.302874
\(337\) −4.30700e11 −1.81903 −0.909517 0.415667i \(-0.863548\pi\)
−0.909517 + 0.415667i \(0.863548\pi\)
\(338\) 0 0
\(339\) 1.56470e11 0.643475
\(340\) −1.32455e11 −0.537544
\(341\) −2.07649e11 −0.831638
\(342\) 5.49841e11 2.17330
\(343\) 2.57333e11 1.00386
\(344\) −3.20915e10 −0.123560
\(345\) 1.81088e11 0.688184
\(346\) −6.24068e10 −0.234093
\(347\) 4.56474e10 0.169018 0.0845091 0.996423i \(-0.473068\pi\)
0.0845091 + 0.996423i \(0.473068\pi\)
\(348\) 3.45457e11 1.26266
\(349\) −7.42442e10 −0.267885 −0.133942 0.990989i \(-0.542764\pi\)
−0.133942 + 0.990989i \(0.542764\pi\)
\(350\) 5.84011e10 0.208025
\(351\) 0 0
\(352\) 5.34753e10 0.185657
\(353\) 5.28812e11 1.81266 0.906328 0.422575i \(-0.138874\pi\)
0.906328 + 0.422575i \(0.138874\pi\)
\(354\) −6.16507e11 −2.08652
\(355\) 2.27819e11 0.761313
\(356\) 1.86952e10 0.0616886
\(357\) 5.50392e11 1.79335
\(358\) 1.24218e11 0.399679
\(359\) −1.29531e11 −0.411575 −0.205788 0.978597i \(-0.565976\pi\)
−0.205788 + 0.978597i \(0.565976\pi\)
\(360\) −2.28019e11 −0.715500
\(361\) 6.99073e10 0.216641
\(362\) 1.89836e11 0.581019
\(363\) −6.62976e10 −0.200409
\(364\) 0 0
\(365\) −1.75112e11 −0.516413
\(366\) 5.76844e11 1.68032
\(367\) 4.26366e11 1.22683 0.613416 0.789760i \(-0.289795\pi\)
0.613416 + 0.789760i \(0.289795\pi\)
\(368\) 4.28293e10 0.121738
\(369\) 7.32247e11 2.05607
\(370\) 3.81459e10 0.105813
\(371\) −2.89510e11 −0.793379
\(372\) 2.84563e11 0.770433
\(373\) 7.26602e10 0.194360 0.0971799 0.995267i \(-0.469018\pi\)
0.0971799 + 0.995267i \(0.469018\pi\)
\(374\) 4.15945e11 1.09929
\(375\) −7.96932e11 −2.08104
\(376\) 1.62356e11 0.418912
\(377\) 0 0
\(378\) 6.07456e11 1.53039
\(379\) −3.39342e10 −0.0844814 −0.0422407 0.999107i \(-0.513450\pi\)
−0.0422407 + 0.999107i \(0.513450\pi\)
\(380\) −1.62809e11 −0.400546
\(381\) 9.14808e11 2.22417
\(382\) −3.04602e11 −0.731894
\(383\) 2.41078e11 0.572484 0.286242 0.958157i \(-0.407594\pi\)
0.286242 + 0.958157i \(0.407594\pi\)
\(384\) −7.32829e10 −0.171993
\(385\) 2.04723e11 0.474889
\(386\) 2.42624e11 0.556276
\(387\) −4.29710e11 −0.973814
\(388\) 4.06522e9 0.00910629
\(389\) 3.01321e11 0.667199 0.333600 0.942715i \(-0.391737\pi\)
0.333600 + 0.942715i \(0.391737\pi\)
\(390\) 0 0
\(391\) 3.33138e11 0.720824
\(392\) −1.01219e11 −0.216508
\(393\) −9.22938e11 −1.95167
\(394\) 3.50058e11 0.731824
\(395\) 6.52937e11 1.34954
\(396\) 7.16041e11 1.46322
\(397\) −8.41446e11 −1.70008 −0.850039 0.526719i \(-0.823422\pi\)
−0.850039 + 0.526719i \(0.823422\pi\)
\(398\) 3.84202e9 0.00767512
\(399\) 6.76521e11 1.33630
\(400\) −6.04832e10 −0.118131
\(401\) 2.99661e11 0.578736 0.289368 0.957218i \(-0.406555\pi\)
0.289368 + 0.957218i \(0.406555\pi\)
\(402\) −8.11020e11 −1.54887
\(403\) 0 0
\(404\) 2.95105e11 0.551138
\(405\) −1.56425e12 −2.88907
\(406\) 3.12793e11 0.571335
\(407\) −1.19788e11 −0.216391
\(408\) −5.70014e11 −1.01839
\(409\) 1.00237e11 0.177122 0.0885609 0.996071i \(-0.471773\pi\)
0.0885609 + 0.996071i \(0.471773\pi\)
\(410\) −2.16820e11 −0.378941
\(411\) −1.78103e10 −0.0307881
\(412\) −2.91563e10 −0.0498534
\(413\) −5.58215e11 −0.944118
\(414\) 5.73491e11 0.959456
\(415\) 7.30878e11 1.20956
\(416\) 0 0
\(417\) 1.04192e12 1.68742
\(418\) 5.11264e11 0.819129
\(419\) 8.52438e10 0.135114 0.0675569 0.997715i \(-0.478480\pi\)
0.0675569 + 0.997715i \(0.478480\pi\)
\(420\) −2.80553e11 −0.439940
\(421\) 4.68176e11 0.726339 0.363170 0.931723i \(-0.381695\pi\)
0.363170 + 0.931723i \(0.381695\pi\)
\(422\) −6.02356e11 −0.924586
\(423\) 2.17397e12 3.30158
\(424\) 2.99831e11 0.450537
\(425\) −4.70455e11 −0.699468
\(426\) 9.80409e11 1.44233
\(427\) 5.22302e11 0.760320
\(428\) 1.06531e11 0.153454
\(429\) 0 0
\(430\) 1.27238e11 0.179477
\(431\) −5.60478e11 −0.782367 −0.391184 0.920313i \(-0.627934\pi\)
−0.391184 + 0.920313i \(0.627934\pi\)
\(432\) −6.29113e11 −0.869064
\(433\) 7.10331e11 0.971103 0.485551 0.874208i \(-0.338619\pi\)
0.485551 + 0.874208i \(0.338619\pi\)
\(434\) 2.57657e11 0.348609
\(435\) −1.36968e12 −1.83408
\(436\) 7.61236e10 0.100886
\(437\) 4.09481e11 0.537115
\(438\) −7.53583e11 −0.978358
\(439\) −5.86774e10 −0.0754016 −0.0377008 0.999289i \(-0.512003\pi\)
−0.0377008 + 0.999289i \(0.512003\pi\)
\(440\) −2.12021e11 −0.269676
\(441\) −1.35533e12 −1.70636
\(442\) 0 0
\(443\) −8.95096e11 −1.10421 −0.552106 0.833774i \(-0.686176\pi\)
−0.552106 + 0.833774i \(0.686176\pi\)
\(444\) 1.64159e11 0.200466
\(445\) −7.41235e10 −0.0896058
\(446\) −7.93421e11 −0.949504
\(447\) −1.43424e12 −1.69917
\(448\) −6.63539e10 −0.0778243
\(449\) −7.15234e11 −0.830500 −0.415250 0.909707i \(-0.636306\pi\)
−0.415250 + 0.909707i \(0.636306\pi\)
\(450\) −8.09878e11 −0.931029
\(451\) 6.80872e11 0.774946
\(452\) −1.46726e11 −0.165343
\(453\) −2.03895e12 −2.27491
\(454\) −7.92962e11 −0.875994
\(455\) 0 0
\(456\) −7.00640e11 −0.758845
\(457\) −3.54038e11 −0.379688 −0.189844 0.981814i \(-0.560798\pi\)
−0.189844 + 0.981814i \(0.560798\pi\)
\(458\) 6.98924e11 0.742225
\(459\) −4.89341e12 −5.14582
\(460\) −1.69812e11 −0.176831
\(461\) −8.80144e11 −0.907611 −0.453805 0.891101i \(-0.649934\pi\)
−0.453805 + 0.891101i \(0.649934\pi\)
\(462\) 8.81013e11 0.899691
\(463\) −8.27556e11 −0.836918 −0.418459 0.908236i \(-0.637430\pi\)
−0.418459 + 0.908236i \(0.637430\pi\)
\(464\) −3.23945e11 −0.324444
\(465\) −1.12825e12 −1.11909
\(466\) 7.26575e11 0.713746
\(467\) 5.77865e11 0.562213 0.281106 0.959677i \(-0.409299\pi\)
0.281106 + 0.959677i \(0.409299\pi\)
\(468\) 0 0
\(469\) −7.34337e11 −0.700838
\(470\) −6.43716e11 −0.608491
\(471\) −6.82775e11 −0.639269
\(472\) 5.78116e11 0.536137
\(473\) −3.99562e11 −0.367036
\(474\) 2.80988e12 2.55673
\(475\) −5.78265e11 −0.521202
\(476\) −5.16119e11 −0.460806
\(477\) 4.01478e12 3.55082
\(478\) −2.44696e11 −0.214389
\(479\) 7.35347e11 0.638238 0.319119 0.947715i \(-0.396613\pi\)
0.319119 + 0.947715i \(0.396613\pi\)
\(480\) 2.90555e11 0.249829
\(481\) 0 0
\(482\) −4.98185e11 −0.420416
\(483\) 7.05620e11 0.589941
\(484\) 6.21692e10 0.0514957
\(485\) −1.61180e10 −0.0132273
\(486\) −3.70850e12 −3.01533
\(487\) 1.19742e12 0.964644 0.482322 0.875994i \(-0.339794\pi\)
0.482322 + 0.875994i \(0.339794\pi\)
\(488\) −5.40923e11 −0.431764
\(489\) 1.73823e12 1.37473
\(490\) 4.01316e11 0.314488
\(491\) 2.88203e11 0.223786 0.111893 0.993720i \(-0.464309\pi\)
0.111893 + 0.993720i \(0.464309\pi\)
\(492\) −9.33072e11 −0.717914
\(493\) −2.51973e12 −1.92107
\(494\) 0 0
\(495\) −2.83899e12 −2.12540
\(496\) −2.66843e11 −0.197965
\(497\) 8.87710e11 0.652631
\(498\) 3.14530e12 2.29155
\(499\) −5.86545e11 −0.423495 −0.211748 0.977324i \(-0.567916\pi\)
−0.211748 + 0.977324i \(0.567916\pi\)
\(500\) 7.47306e11 0.534729
\(501\) −2.97325e12 −2.10844
\(502\) 1.83554e12 1.29002
\(503\) 1.12388e12 0.782825 0.391413 0.920215i \(-0.371986\pi\)
0.391413 + 0.920215i \(0.371986\pi\)
\(504\) −8.88488e11 −0.613358
\(505\) −1.17005e12 −0.800556
\(506\) 5.33255e11 0.361624
\(507\) 0 0
\(508\) −8.57842e11 −0.571506
\(509\) 2.43786e12 1.60983 0.804914 0.593391i \(-0.202211\pi\)
0.804914 + 0.593391i \(0.202211\pi\)
\(510\) 2.26002e12 1.47927
\(511\) −6.82331e11 −0.442691
\(512\) 6.87195e10 0.0441942
\(513\) −6.01480e12 −3.83436
\(514\) −6.95911e11 −0.439764
\(515\) 1.15600e11 0.0724146
\(516\) 5.47562e11 0.340024
\(517\) 2.02144e12 1.24438
\(518\) 1.48637e11 0.0907077
\(519\) 1.06482e12 0.644201
\(520\) 0 0
\(521\) −8.90970e11 −0.529777 −0.264889 0.964279i \(-0.585335\pi\)
−0.264889 + 0.964279i \(0.585335\pi\)
\(522\) −4.33767e12 −2.55705
\(523\) 6.25163e11 0.365372 0.182686 0.983171i \(-0.441521\pi\)
0.182686 + 0.983171i \(0.441521\pi\)
\(524\) 8.65466e11 0.501486
\(525\) −9.96469e11 −0.572462
\(526\) 1.06532e12 0.606797
\(527\) −2.07558e12 −1.17217
\(528\) −9.12422e11 −0.510908
\(529\) −1.37406e12 −0.762878
\(530\) −1.18878e12 −0.654427
\(531\) 7.74105e12 4.22547
\(532\) −6.34394e11 −0.343365
\(533\) 0 0
\(534\) −3.18987e11 −0.169761
\(535\) −4.22378e11 −0.222900
\(536\) 7.60517e11 0.397986
\(537\) −2.11947e12 −1.09987
\(538\) 6.93750e11 0.357012
\(539\) −1.26024e12 −0.643138
\(540\) 2.49433e12 1.26236
\(541\) 3.66933e12 1.84161 0.920807 0.390019i \(-0.127531\pi\)
0.920807 + 0.390019i \(0.127531\pi\)
\(542\) −9.78512e11 −0.487045
\(543\) −3.23908e12 −1.59890
\(544\) 5.34519e11 0.261679
\(545\) −3.01818e11 −0.146542
\(546\) 0 0
\(547\) −5.84419e11 −0.279114 −0.139557 0.990214i \(-0.544568\pi\)
−0.139557 + 0.990214i \(0.544568\pi\)
\(548\) 1.67012e10 0.00791109
\(549\) −7.24303e12 −3.40287
\(550\) −7.53057e11 −0.350910
\(551\) −3.09716e12 −1.43147
\(552\) −7.30776e11 −0.335010
\(553\) 2.54420e12 1.15688
\(554\) −9.00762e11 −0.406271
\(555\) −6.50864e11 −0.291187
\(556\) −9.77041e11 −0.433587
\(557\) −3.78849e11 −0.166770 −0.0833850 0.996517i \(-0.526573\pi\)
−0.0833850 + 0.996517i \(0.526573\pi\)
\(558\) −3.57306e12 −1.56022
\(559\) 0 0
\(560\) 2.63083e11 0.113044
\(561\) −7.09707e12 −3.02514
\(562\) −1.69886e12 −0.718365
\(563\) 3.41564e12 1.43280 0.716398 0.697692i \(-0.245790\pi\)
0.716398 + 0.697692i \(0.245790\pi\)
\(564\) −2.77020e12 −1.15280
\(565\) 5.81747e11 0.240168
\(566\) 7.79523e11 0.319268
\(567\) −6.09517e12 −2.47663
\(568\) −9.19358e11 −0.370610
\(569\) −1.45515e12 −0.581973 −0.290987 0.956727i \(-0.593984\pi\)
−0.290987 + 0.956727i \(0.593984\pi\)
\(570\) 2.77793e12 1.10226
\(571\) 2.05220e12 0.807898 0.403949 0.914782i \(-0.367637\pi\)
0.403949 + 0.914782i \(0.367637\pi\)
\(572\) 0 0
\(573\) 5.19728e12 2.01410
\(574\) −8.44849e11 −0.324844
\(575\) −6.03137e11 −0.230097
\(576\) 9.20163e11 0.348308
\(577\) 4.21231e12 1.58208 0.791042 0.611762i \(-0.209539\pi\)
0.791042 + 0.611762i \(0.209539\pi\)
\(578\) 2.26023e12 0.842320
\(579\) −4.13977e12 −1.53081
\(580\) 1.28439e12 0.471271
\(581\) 2.84791e12 1.03689
\(582\) −6.93629e10 −0.0250596
\(583\) 3.73310e12 1.33832
\(584\) 7.06657e11 0.251392
\(585\) 0 0
\(586\) 3.12155e12 1.09353
\(587\) 2.68770e12 0.934350 0.467175 0.884165i \(-0.345272\pi\)
0.467175 + 0.884165i \(0.345272\pi\)
\(588\) 1.72704e12 0.595806
\(589\) −2.55122e12 −0.873432
\(590\) −2.29214e12 −0.778766
\(591\) −5.97286e12 −2.01390
\(592\) −1.53936e11 −0.0515102
\(593\) −6.65599e11 −0.221038 −0.110519 0.993874i \(-0.535251\pi\)
−0.110519 + 0.993874i \(0.535251\pi\)
\(594\) −7.83288e12 −2.58156
\(595\) 2.04633e12 0.669344
\(596\) 1.34493e12 0.436607
\(597\) −6.55544e10 −0.0211211
\(598\) 0 0
\(599\) −3.38921e12 −1.07567 −0.537833 0.843051i \(-0.680757\pi\)
−0.537833 + 0.843051i \(0.680757\pi\)
\(600\) 1.03199e12 0.325085
\(601\) 3.16187e11 0.0988572 0.0494286 0.998778i \(-0.484260\pi\)
0.0494286 + 0.998778i \(0.484260\pi\)
\(602\) 4.95789e11 0.153855
\(603\) 1.01834e13 3.13665
\(604\) 1.91198e12 0.584544
\(605\) −2.46491e11 −0.0748001
\(606\) −5.03523e12 −1.51667
\(607\) 5.15352e12 1.54083 0.770416 0.637542i \(-0.220049\pi\)
0.770416 + 0.637542i \(0.220049\pi\)
\(608\) 6.57010e11 0.194987
\(609\) −5.33704e12 −1.57225
\(610\) 2.14467e12 0.627158
\(611\) 0 0
\(612\) 7.15728e12 2.06237
\(613\) 5.31719e12 1.52093 0.760466 0.649378i \(-0.224970\pi\)
0.760466 + 0.649378i \(0.224970\pi\)
\(614\) −3.51065e12 −0.996850
\(615\) 3.69948e12 1.04281
\(616\) −8.26151e11 −0.231178
\(617\) 3.65267e12 1.01468 0.507338 0.861747i \(-0.330630\pi\)
0.507338 + 0.861747i \(0.330630\pi\)
\(618\) 4.97480e11 0.137191
\(619\) −3.48472e11 −0.0954024 −0.0477012 0.998862i \(-0.515190\pi\)
−0.0477012 + 0.998862i \(0.515190\pi\)
\(620\) 1.05799e12 0.287554
\(621\) −6.27350e12 −1.69277
\(622\) −8.59509e11 −0.230247
\(623\) −2.88826e11 −0.0768140
\(624\) 0 0
\(625\) −1.16041e12 −0.304196
\(626\) 4.40749e12 1.14711
\(627\) −8.72345e12 −2.25416
\(628\) 6.40258e11 0.164262
\(629\) −1.19736e12 −0.304998
\(630\) 3.52271e12 0.890933
\(631\) −1.68936e12 −0.424220 −0.212110 0.977246i \(-0.568033\pi\)
−0.212110 + 0.977246i \(0.568033\pi\)
\(632\) −2.63491e12 −0.656960
\(633\) 1.02777e13 2.54436
\(634\) 2.28341e12 0.561285
\(635\) 3.40121e12 0.830140
\(636\) −5.11587e12 −1.23983
\(637\) 0 0
\(638\) −4.03333e12 −0.963765
\(639\) −1.23103e13 −2.92089
\(640\) −2.72462e11 −0.0641942
\(641\) 5.11812e12 1.19743 0.598714 0.800963i \(-0.295679\pi\)
0.598714 + 0.800963i \(0.295679\pi\)
\(642\) −1.81768e12 −0.422290
\(643\) 1.43341e12 0.330690 0.165345 0.986236i \(-0.447126\pi\)
0.165345 + 0.986236i \(0.447126\pi\)
\(644\) −6.61680e11 −0.151587
\(645\) −2.17100e12 −0.493902
\(646\) 5.11041e12 1.15454
\(647\) 2.24349e12 0.503333 0.251667 0.967814i \(-0.419021\pi\)
0.251667 + 0.967814i \(0.419021\pi\)
\(648\) 6.31247e12 1.40641
\(649\) 7.19794e12 1.59260
\(650\) 0 0
\(651\) −4.39628e12 −0.959335
\(652\) −1.62998e12 −0.353240
\(653\) 4.70352e12 1.01231 0.506155 0.862443i \(-0.331066\pi\)
0.506155 + 0.862443i \(0.331066\pi\)
\(654\) −1.29886e12 −0.277627
\(655\) −3.43144e12 −0.728434
\(656\) 8.74969e11 0.184470
\(657\) 9.46223e12 1.98130
\(658\) −2.50827e12 −0.521625
\(659\) 8.29484e11 0.171326 0.0856631 0.996324i \(-0.472699\pi\)
0.0856631 + 0.996324i \(0.472699\pi\)
\(660\) 3.61761e12 0.742120
\(661\) 7.92098e11 0.161388 0.0806942 0.996739i \(-0.474286\pi\)
0.0806942 + 0.996739i \(0.474286\pi\)
\(662\) 2.80812e12 0.568271
\(663\) 0 0
\(664\) −2.94944e12 −0.588820
\(665\) 2.51527e12 0.498755
\(666\) −2.06123e12 −0.405968
\(667\) −3.23037e12 −0.631955
\(668\) 2.78811e12 0.541770
\(669\) 1.35377e13 2.61294
\(670\) −3.01533e12 −0.578094
\(671\) −6.73486e12 −1.28256
\(672\) 1.13216e12 0.214164
\(673\) 5.56234e12 1.04518 0.522589 0.852585i \(-0.324966\pi\)
0.522589 + 0.852585i \(0.324966\pi\)
\(674\) −6.89121e12 −1.28625
\(675\) 8.85938e12 1.64262
\(676\) 0 0
\(677\) 2.20470e11 0.0403368 0.0201684 0.999797i \(-0.493580\pi\)
0.0201684 + 0.999797i \(0.493580\pi\)
\(678\) 2.50352e12 0.455006
\(679\) −6.28045e10 −0.0113391
\(680\) −2.11928e12 −0.380101
\(681\) 1.35299e13 2.41064
\(682\) −3.32238e12 −0.588057
\(683\) 3.69879e12 0.650379 0.325190 0.945649i \(-0.394572\pi\)
0.325190 + 0.945649i \(0.394572\pi\)
\(684\) 8.79746e12 1.53676
\(685\) −6.62178e10 −0.0114913
\(686\) 4.11733e12 0.709834
\(687\) −1.19254e13 −2.04252
\(688\) −5.13465e11 −0.0873700
\(689\) 0 0
\(690\) 2.89741e12 0.486619
\(691\) 5.40354e12 0.901628 0.450814 0.892618i \(-0.351134\pi\)
0.450814 + 0.892618i \(0.351134\pi\)
\(692\) −9.98508e11 −0.165529
\(693\) −1.10623e13 −1.82199
\(694\) 7.30358e11 0.119514
\(695\) 3.87382e12 0.629807
\(696\) 5.52731e12 0.892837
\(697\) 6.80575e12 1.09227
\(698\) −1.18791e12 −0.189423
\(699\) −1.23972e13 −1.96416
\(700\) 9.34418e11 0.147096
\(701\) −9.91676e12 −1.55110 −0.775548 0.631288i \(-0.782527\pi\)
−0.775548 + 0.631288i \(0.782527\pi\)
\(702\) 0 0
\(703\) −1.47175e12 −0.227266
\(704\) 8.55604e11 0.131279
\(705\) 1.09834e13 1.67450
\(706\) 8.46100e12 1.28174
\(707\) −4.55914e12 −0.686271
\(708\) −9.86411e12 −1.47539
\(709\) −9.83990e12 −1.46245 −0.731227 0.682134i \(-0.761052\pi\)
−0.731227 + 0.682134i \(0.761052\pi\)
\(710\) 3.64511e12 0.538329
\(711\) −3.52818e13 −5.17771
\(712\) 2.99123e11 0.0436204
\(713\) −2.66095e12 −0.385598
\(714\) 8.80628e12 1.26809
\(715\) 0 0
\(716\) 1.98749e12 0.282615
\(717\) 4.17513e12 0.589976
\(718\) −2.07250e12 −0.291028
\(719\) −8.29065e12 −1.15693 −0.578467 0.815706i \(-0.696349\pi\)
−0.578467 + 0.815706i \(0.696349\pi\)
\(720\) −3.64830e12 −0.505935
\(721\) 4.50442e11 0.0620770
\(722\) 1.11852e12 0.153188
\(723\) 8.50028e12 1.15694
\(724\) 3.03738e12 0.410842
\(725\) 4.56190e12 0.613232
\(726\) −1.06076e12 −0.141711
\(727\) −7.62623e12 −1.01252 −0.506262 0.862380i \(-0.668973\pi\)
−0.506262 + 0.862380i \(0.668973\pi\)
\(728\) 0 0
\(729\) 3.29423e13 4.31996
\(730\) −2.80178e12 −0.365159
\(731\) −3.99387e12 −0.517327
\(732\) 9.22950e12 1.18817
\(733\) −1.20191e12 −0.153781 −0.0768905 0.997040i \(-0.524499\pi\)
−0.0768905 + 0.997040i \(0.524499\pi\)
\(734\) 6.82185e12 0.867501
\(735\) −6.84746e12 −0.865439
\(736\) 6.85270e11 0.0860818
\(737\) 9.46896e12 1.18222
\(738\) 1.17159e13 1.45386
\(739\) −8.41989e12 −1.03850 −0.519250 0.854622i \(-0.673789\pi\)
−0.519250 + 0.854622i \(0.673789\pi\)
\(740\) 6.10334e11 0.0748212
\(741\) 0 0
\(742\) −4.63215e12 −0.561003
\(743\) −7.66300e12 −0.922464 −0.461232 0.887280i \(-0.652592\pi\)
−0.461232 + 0.887280i \(0.652592\pi\)
\(744\) 4.55301e12 0.544779
\(745\) −5.33243e12 −0.634193
\(746\) 1.16256e12 0.137433
\(747\) −3.94934e13 −4.64068
\(748\) 6.65513e12 0.777319
\(749\) −1.64582e12 −0.191079
\(750\) −1.27509e13 −1.47152
\(751\) −5.77487e12 −0.662464 −0.331232 0.943549i \(-0.607464\pi\)
−0.331232 + 0.943549i \(0.607464\pi\)
\(752\) 2.59770e12 0.296216
\(753\) −3.13188e13 −3.55000
\(754\) 0 0
\(755\) −7.58070e12 −0.849079
\(756\) 9.71930e12 1.08215
\(757\) 3.74312e12 0.414288 0.207144 0.978310i \(-0.433583\pi\)
0.207144 + 0.978310i \(0.433583\pi\)
\(758\) −5.42947e11 −0.0597374
\(759\) −9.09866e12 −0.995152
\(760\) −2.60494e12 −0.283229
\(761\) 1.00523e13 1.08651 0.543255 0.839568i \(-0.317192\pi\)
0.543255 + 0.839568i \(0.317192\pi\)
\(762\) 1.46369e13 1.57272
\(763\) −1.17605e12 −0.125622
\(764\) −4.87364e12 −0.517527
\(765\) −2.83775e13 −2.99570
\(766\) 3.85725e12 0.404807
\(767\) 0 0
\(768\) −1.17253e12 −0.121618
\(769\) −2.83829e12 −0.292677 −0.146338 0.989235i \(-0.546749\pi\)
−0.146338 + 0.989235i \(0.546749\pi\)
\(770\) 3.27556e12 0.335797
\(771\) 1.18740e13 1.21019
\(772\) 3.88198e12 0.393347
\(773\) 1.09262e13 1.10068 0.550340 0.834941i \(-0.314498\pi\)
0.550340 + 0.834941i \(0.314498\pi\)
\(774\) −6.87536e12 −0.688591
\(775\) 3.75777e12 0.374173
\(776\) 6.50436e10 0.00643912
\(777\) −2.53612e12 −0.249618
\(778\) 4.82113e12 0.471781
\(779\) 8.36536e12 0.813891
\(780\) 0 0
\(781\) −1.14466e13 −1.10090
\(782\) 5.33021e12 0.509700
\(783\) 4.74504e13 4.51140
\(784\) −1.61950e12 −0.153094
\(785\) −2.53852e12 −0.238598
\(786\) −1.47670e13 −1.38004
\(787\) −1.15918e13 −1.07712 −0.538561 0.842587i \(-0.681032\pi\)
−0.538561 + 0.842587i \(0.681032\pi\)
\(788\) 5.60092e12 0.517478
\(789\) −1.81770e13 −1.66984
\(790\) 1.04470e13 0.954266
\(791\) 2.26681e12 0.205883
\(792\) 1.14567e13 1.03465
\(793\) 0 0
\(794\) −1.34631e13 −1.20214
\(795\) 2.02836e13 1.80091
\(796\) 6.14722e10 0.00542713
\(797\) 1.28149e12 0.112500 0.0562499 0.998417i \(-0.482086\pi\)
0.0562499 + 0.998417i \(0.482086\pi\)
\(798\) 1.08243e13 0.944906
\(799\) 2.02056e13 1.75392
\(800\) −9.67731e11 −0.0835314
\(801\) 4.00530e12 0.343786
\(802\) 4.79458e12 0.409228
\(803\) 8.79836e12 0.746761
\(804\) −1.29763e13 −1.09522
\(805\) 2.62346e12 0.220187
\(806\) 0 0
\(807\) −1.18371e13 −0.982460
\(808\) 4.72168e12 0.389714
\(809\) 1.27199e13 1.04404 0.522019 0.852934i \(-0.325179\pi\)
0.522019 + 0.852934i \(0.325179\pi\)
\(810\) −2.50279e13 −2.04288
\(811\) −1.43302e13 −1.16321 −0.581606 0.813470i \(-0.697576\pi\)
−0.581606 + 0.813470i \(0.697576\pi\)
\(812\) 5.00469e12 0.403995
\(813\) 1.66959e13 1.34030
\(814\) −1.91661e12 −0.153012
\(815\) 6.46263e12 0.513098
\(816\) −9.12023e12 −0.720112
\(817\) −4.90911e12 −0.385482
\(818\) 1.60379e12 0.125244
\(819\) 0 0
\(820\) −3.46911e12 −0.267951
\(821\) 1.40225e12 0.107716 0.0538581 0.998549i \(-0.482848\pi\)
0.0538581 + 0.998549i \(0.482848\pi\)
\(822\) −2.84965e11 −0.0217705
\(823\) 1.77090e13 1.34554 0.672768 0.739854i \(-0.265105\pi\)
0.672768 + 0.739854i \(0.265105\pi\)
\(824\) −4.66501e11 −0.0352517
\(825\) 1.28490e13 0.965668
\(826\) −8.93144e12 −0.667592
\(827\) 8.46614e12 0.629377 0.314688 0.949195i \(-0.398100\pi\)
0.314688 + 0.949195i \(0.398100\pi\)
\(828\) 9.17585e12 0.678438
\(829\) 2.10818e13 1.55029 0.775143 0.631786i \(-0.217678\pi\)
0.775143 + 0.631786i \(0.217678\pi\)
\(830\) 1.16941e13 0.855291
\(831\) 1.53693e13 1.11802
\(832\) 0 0
\(833\) −1.25969e13 −0.906486
\(834\) 1.66708e13 1.19319
\(835\) −1.10544e13 −0.786948
\(836\) 8.18023e12 0.579212
\(837\) 3.90863e13 2.75271
\(838\) 1.36390e12 0.0955399
\(839\) −1.29136e13 −0.899746 −0.449873 0.893093i \(-0.648531\pi\)
−0.449873 + 0.893093i \(0.648531\pi\)
\(840\) −4.48885e12 −0.311084
\(841\) 9.92616e12 0.684226
\(842\) 7.49081e12 0.513599
\(843\) 2.89869e13 1.97687
\(844\) −9.63770e12 −0.653781
\(845\) 0 0
\(846\) 3.47835e13 2.33457
\(847\) −9.60465e11 −0.0641219
\(848\) 4.79730e12 0.318578
\(849\) −1.33006e13 −0.878592
\(850\) −7.52728e12 −0.494598
\(851\) −1.53505e12 −0.100332
\(852\) 1.56865e13 1.01988
\(853\) 9.78896e12 0.633091 0.316545 0.948577i \(-0.397477\pi\)
0.316545 + 0.948577i \(0.397477\pi\)
\(854\) 8.35684e12 0.537628
\(855\) −3.48806e13 −2.23221
\(856\) 1.70449e12 0.108508
\(857\) 1.16792e11 0.00739603 0.00369802 0.999993i \(-0.498823\pi\)
0.00369802 + 0.999993i \(0.498823\pi\)
\(858\) 0 0
\(859\) −1.69224e13 −1.06045 −0.530227 0.847856i \(-0.677893\pi\)
−0.530227 + 0.847856i \(0.677893\pi\)
\(860\) 2.03581e12 0.126909
\(861\) 1.44152e13 0.893938
\(862\) −8.96764e12 −0.553217
\(863\) 2.35604e13 1.44589 0.722945 0.690906i \(-0.242788\pi\)
0.722945 + 0.690906i \(0.242788\pi\)
\(864\) −1.00658e13 −0.614521
\(865\) 3.95893e12 0.240439
\(866\) 1.13653e13 0.686673
\(867\) −3.85652e13 −2.31798
\(868\) 4.12251e12 0.246504
\(869\) −3.28064e13 −1.95151
\(870\) −2.19149e13 −1.29689
\(871\) 0 0
\(872\) 1.21798e12 0.0713370
\(873\) 8.70942e11 0.0507487
\(874\) 6.55170e12 0.379798
\(875\) −1.15453e13 −0.665839
\(876\) −1.20573e13 −0.691804
\(877\) −1.08137e13 −0.617269 −0.308635 0.951181i \(-0.599872\pi\)
−0.308635 + 0.951181i \(0.599872\pi\)
\(878\) −9.38838e11 −0.0533170
\(879\) −5.32614e13 −3.00928
\(880\) −3.39234e12 −0.190690
\(881\) 3.00463e13 1.68035 0.840175 0.542315i \(-0.182452\pi\)
0.840175 + 0.542315i \(0.182452\pi\)
\(882\) −2.16853e13 −1.20658
\(883\) 2.20296e13 1.21950 0.609752 0.792592i \(-0.291269\pi\)
0.609752 + 0.792592i \(0.291269\pi\)
\(884\) 0 0
\(885\) 3.91096e13 2.14308
\(886\) −1.43215e13 −0.780796
\(887\) −1.90825e11 −0.0103509 −0.00517545 0.999987i \(-0.501647\pi\)
−0.00517545 + 0.999987i \(0.501647\pi\)
\(888\) 2.62654e12 0.141751
\(889\) 1.32530e13 0.711633
\(890\) −1.18598e12 −0.0633609
\(891\) 7.85945e13 4.17775
\(892\) −1.26947e13 −0.671401
\(893\) 2.48359e13 1.30692
\(894\) −2.29478e13 −1.20150
\(895\) −7.88008e12 −0.410513
\(896\) −1.06166e12 −0.0550301
\(897\) 0 0
\(898\) −1.14437e13 −0.587252
\(899\) 2.01264e13 1.02766
\(900\) −1.29580e13 −0.658337
\(901\) 3.73147e13 1.88633
\(902\) 1.08940e13 0.547969
\(903\) −8.45940e12 −0.423394
\(904\) −2.34762e12 −0.116915
\(905\) −1.20427e13 −0.596769
\(906\) −3.26232e13 −1.60860
\(907\) 2.20950e13 1.08408 0.542040 0.840352i \(-0.317652\pi\)
0.542040 + 0.840352i \(0.317652\pi\)
\(908\) −1.26874e13 −0.619421
\(909\) 6.32239e13 3.07145
\(910\) 0 0
\(911\) −2.54984e12 −0.122653 −0.0613267 0.998118i \(-0.519533\pi\)
−0.0613267 + 0.998118i \(0.519533\pi\)
\(912\) −1.12102e13 −0.536585
\(913\) −3.67225e13 −1.74910
\(914\) −5.66461e12 −0.268480
\(915\) −3.65935e13 −1.72587
\(916\) 1.11828e13 0.524832
\(917\) −1.33708e13 −0.624445
\(918\) −7.82946e13 −3.63864
\(919\) −3.29350e13 −1.52313 −0.761567 0.648086i \(-0.775570\pi\)
−0.761567 + 0.648086i \(0.775570\pi\)
\(920\) −2.71699e12 −0.125038
\(921\) 5.99005e13 2.74323
\(922\) −1.40823e13 −0.641778
\(923\) 0 0
\(924\) 1.40962e13 0.636178
\(925\) 2.16778e12 0.0973595
\(926\) −1.32409e13 −0.591790
\(927\) −6.24651e12 −0.277830
\(928\) −5.18312e12 −0.229417
\(929\) −4.79946e11 −0.0211408 −0.0105704 0.999944i \(-0.503365\pi\)
−0.0105704 + 0.999944i \(0.503365\pi\)
\(930\) −1.80520e13 −0.791318
\(931\) −1.54836e13 −0.675460
\(932\) 1.16252e13 0.504695
\(933\) 1.46654e13 0.633615
\(934\) 9.24585e12 0.397544
\(935\) −2.63865e13 −1.12909
\(936\) 0 0
\(937\) 2.33835e13 0.991019 0.495509 0.868603i \(-0.334981\pi\)
0.495509 + 0.868603i \(0.334981\pi\)
\(938\) −1.17494e13 −0.495567
\(939\) −7.52028e13 −3.15674
\(940\) −1.02995e13 −0.430268
\(941\) −1.63712e13 −0.680655 −0.340327 0.940307i \(-0.610538\pi\)
−0.340327 + 0.940307i \(0.610538\pi\)
\(942\) −1.09244e13 −0.452031
\(943\) 8.72517e12 0.359312
\(944\) 9.24986e12 0.379106
\(945\) −3.85355e13 −1.57187
\(946\) −6.39298e12 −0.259534
\(947\) 3.62894e13 1.46624 0.733120 0.680099i \(-0.238063\pi\)
0.733120 + 0.680099i \(0.238063\pi\)
\(948\) 4.49581e13 1.80788
\(949\) 0 0
\(950\) −9.25224e12 −0.368545
\(951\) −3.89608e13 −1.54460
\(952\) −8.25790e12 −0.325839
\(953\) 2.54579e13 0.999778 0.499889 0.866089i \(-0.333374\pi\)
0.499889 + 0.866089i \(0.333374\pi\)
\(954\) 6.42364e13 2.51081
\(955\) 1.93232e13 0.751734
\(956\) −3.91514e12 −0.151596
\(957\) 6.88188e13 2.65218
\(958\) 1.17656e13 0.451302
\(959\) −2.58021e11 −0.00985080
\(960\) 4.64888e12 0.176656
\(961\) −9.86088e12 −0.372958
\(962\) 0 0
\(963\) 2.28234e13 0.855189
\(964\) −7.97096e12 −0.297279
\(965\) −1.53914e13 −0.571356
\(966\) 1.12899e13 0.417151
\(967\) 7.82018e11 0.0287606 0.0143803 0.999897i \(-0.495422\pi\)
0.0143803 + 0.999897i \(0.495422\pi\)
\(968\) 9.94707e11 0.0364130
\(969\) −8.71963e13 −3.17718
\(970\) −2.57888e11 −0.00935314
\(971\) 4.27952e13 1.54493 0.772464 0.635059i \(-0.219024\pi\)
0.772464 + 0.635059i \(0.219024\pi\)
\(972\) −5.93360e13 −2.13216
\(973\) 1.50945e13 0.539898
\(974\) 1.91588e13 0.682106
\(975\) 0 0
\(976\) −8.65477e12 −0.305303
\(977\) 4.23080e13 1.48558 0.742792 0.669522i \(-0.233501\pi\)
0.742792 + 0.669522i \(0.233501\pi\)
\(978\) 2.78116e13 0.972078
\(979\) 3.72429e12 0.129575
\(980\) 6.42106e12 0.222377
\(981\) 1.63089e13 0.562230
\(982\) 4.61126e12 0.158240
\(983\) −1.87916e13 −0.641910 −0.320955 0.947094i \(-0.604004\pi\)
−0.320955 + 0.947094i \(0.604004\pi\)
\(984\) −1.49292e13 −0.507642
\(985\) −2.22068e13 −0.751662
\(986\) −4.03157e13 −1.35840
\(987\) 4.27974e13 1.43546
\(988\) 0 0
\(989\) −5.12026e12 −0.170180
\(990\) −4.54239e13 −1.50289
\(991\) 2.91914e13 0.961441 0.480721 0.876874i \(-0.340375\pi\)
0.480721 + 0.876874i \(0.340375\pi\)
\(992\) −4.26949e12 −0.139982
\(993\) −4.79136e13 −1.56382
\(994\) 1.42034e13 0.461480
\(995\) −2.43728e11 −0.00788318
\(996\) 5.03248e13 1.62037
\(997\) −1.19441e13 −0.382847 −0.191423 0.981508i \(-0.561310\pi\)
−0.191423 + 0.981508i \(0.561310\pi\)
\(998\) −9.38472e12 −0.299456
\(999\) 2.25481e13 0.716251
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 338.10.a.c.1.1 1
13.12 even 2 26.10.a.a.1.1 1
39.38 odd 2 234.10.a.b.1.1 1
52.51 odd 2 208.10.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.10.a.a.1.1 1 13.12 even 2
208.10.a.c.1.1 1 52.51 odd 2
234.10.a.b.1.1 1 39.38 odd 2
338.10.a.c.1.1 1 1.1 even 1 trivial