Properties

Label 16.48.a.d.1.3
Level $16$
Weight $48$
Character 16.1
Self dual yes
Analytic conductor $223.852$
Analytic rank $1$
Dimension $4$
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] = [16,48,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, 48, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("16.1");
 
S:= CuspForms(chi, 48);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 16 = 2^{4} \)
Weight: \( k \) \(=\) \( 48 \)
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: \(223.852260248\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 832803191366x^{2} + 3710135215485780x + 13175318942671469337000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{46}\cdot 3^{7}\cdot 5^{3}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 1)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-906006.\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q+2.01642e10 q^{3} -3.11682e16 q^{5} +1.26592e20 q^{7} -2.61822e22 q^{9} +O(q^{10})\) \(q+2.01642e10 q^{3} -3.11682e16 q^{5} +1.26592e20 q^{7} -2.61822e22 q^{9} +1.78518e23 q^{11} -1.12359e26 q^{13} -6.28481e26 q^{15} +4.31126e28 q^{17} +7.19841e29 q^{19} +2.55262e30 q^{21} -8.88061e30 q^{23} +2.60913e32 q^{25} -1.06408e33 q^{27} +2.84937e34 q^{29} -1.11855e35 q^{31} +3.59967e33 q^{33} -3.94564e36 q^{35} -9.87647e36 q^{37} -2.26563e36 q^{39} -3.56038e37 q^{41} +3.37056e38 q^{43} +8.16052e38 q^{45} +1.22952e39 q^{47} +1.07822e40 q^{49} +8.69331e38 q^{51} -1.08443e39 q^{53} -5.56408e39 q^{55} +1.45150e40 q^{57} +1.45567e40 q^{59} +4.79341e41 q^{61} -3.31446e42 q^{63} +3.50203e42 q^{65} -3.35317e42 q^{67} -1.79070e41 q^{69} -9.20990e42 q^{71} -5.22911e43 q^{73} +5.26110e42 q^{75} +2.25989e43 q^{77} +7.68215e43 q^{79} +6.74698e44 q^{81} +1.71031e45 q^{83} -1.34374e45 q^{85} +5.74552e44 q^{87} -9.70124e45 q^{89} -1.42238e46 q^{91} -2.25547e45 q^{93} -2.24361e46 q^{95} +5.35657e46 q^{97} -4.67400e45 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 38461494960 q^{3} - 31\!\cdots\!00 q^{5}+ \cdots - 17\!\cdots\!72 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 38461494960 q^{3} - 31\!\cdots\!00 q^{5}+ \cdots - 53\!\cdots\!16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) 2.01642e10 0.123661 0.0618303 0.998087i \(-0.480306\pi\)
0.0618303 + 0.998087i \(0.480306\pi\)
\(4\) 0 0
\(5\) −3.11682e16 −1.16927 −0.584637 0.811295i \(-0.698763\pi\)
−0.584637 + 0.811295i \(0.698763\pi\)
\(6\) 0 0
\(7\) 1.26592e20 1.74824 0.874122 0.485707i \(-0.161438\pi\)
0.874122 + 0.485707i \(0.161438\pi\)
\(8\) 0 0
\(9\) −2.61822e22 −0.984708
\(10\) 0 0
\(11\) 1.78518e23 0.0601110 0.0300555 0.999548i \(-0.490432\pi\)
0.0300555 + 0.999548i \(0.490432\pi\)
\(12\) 0 0
\(13\) −1.12359e26 −0.746346 −0.373173 0.927762i \(-0.621730\pi\)
−0.373173 + 0.927762i \(0.621730\pi\)
\(14\) 0 0
\(15\) −6.28481e26 −0.144593
\(16\) 0 0
\(17\) 4.31126e28 0.523667 0.261833 0.965113i \(-0.415673\pi\)
0.261833 + 0.965113i \(0.415673\pi\)
\(18\) 0 0
\(19\) 7.19841e29 0.640511 0.320256 0.947331i \(-0.396231\pi\)
0.320256 + 0.947331i \(0.396231\pi\)
\(20\) 0 0
\(21\) 2.55262e30 0.216189
\(22\) 0 0
\(23\) −8.88061e30 −0.0886826 −0.0443413 0.999016i \(-0.514119\pi\)
−0.0443413 + 0.999016i \(0.514119\pi\)
\(24\) 0 0
\(25\) 2.60913e32 0.367202
\(26\) 0 0
\(27\) −1.06408e33 −0.245430
\(28\) 0 0
\(29\) 2.84937e34 1.22573 0.612866 0.790187i \(-0.290017\pi\)
0.612866 + 0.790187i \(0.290017\pi\)
\(30\) 0 0
\(31\) −1.11855e35 −1.00382 −0.501911 0.864919i \(-0.667370\pi\)
−0.501911 + 0.864919i \(0.667370\pi\)
\(32\) 0 0
\(33\) 3.59967e33 0.00743336
\(34\) 0 0
\(35\) −3.94564e36 −2.04418
\(36\) 0 0
\(37\) −9.87647e36 −1.38631 −0.693156 0.720787i \(-0.743780\pi\)
−0.693156 + 0.720787i \(0.743780\pi\)
\(38\) 0 0
\(39\) −2.26563e36 −0.0922935
\(40\) 0 0
\(41\) −3.56038e37 −0.447791 −0.223896 0.974613i \(-0.571877\pi\)
−0.223896 + 0.974613i \(0.571877\pi\)
\(42\) 0 0
\(43\) 3.37056e38 1.38418 0.692091 0.721811i \(-0.256690\pi\)
0.692091 + 0.721811i \(0.256690\pi\)
\(44\) 0 0
\(45\) 8.16052e38 1.15139
\(46\) 0 0
\(47\) 1.22952e39 0.624361 0.312181 0.950023i \(-0.398941\pi\)
0.312181 + 0.950023i \(0.398941\pi\)
\(48\) 0 0
\(49\) 1.07822e40 2.05635
\(50\) 0 0
\(51\) 8.69331e38 0.0647569
\(52\) 0 0
\(53\) −1.08443e39 −0.0327129 −0.0163565 0.999866i \(-0.505207\pi\)
−0.0163565 + 0.999866i \(0.505207\pi\)
\(54\) 0 0
\(55\) −5.56408e39 −0.0702863
\(56\) 0 0
\(57\) 1.45150e40 0.0792060
\(58\) 0 0
\(59\) 1.45567e40 0.0353216 0.0176608 0.999844i \(-0.494378\pi\)
0.0176608 + 0.999844i \(0.494378\pi\)
\(60\) 0 0
\(61\) 4.79341e41 0.531366 0.265683 0.964060i \(-0.414403\pi\)
0.265683 + 0.964060i \(0.414403\pi\)
\(62\) 0 0
\(63\) −3.31446e42 −1.72151
\(64\) 0 0
\(65\) 3.50203e42 0.872683
\(66\) 0 0
\(67\) −3.35317e42 −0.409919 −0.204959 0.978770i \(-0.565706\pi\)
−0.204959 + 0.978770i \(0.565706\pi\)
\(68\) 0 0
\(69\) −1.79070e41 −0.0109665
\(70\) 0 0
\(71\) −9.20990e42 −0.288193 −0.144097 0.989564i \(-0.546028\pi\)
−0.144097 + 0.989564i \(0.546028\pi\)
\(72\) 0 0
\(73\) −5.22911e43 −0.851805 −0.425903 0.904769i \(-0.640043\pi\)
−0.425903 + 0.904769i \(0.640043\pi\)
\(74\) 0 0
\(75\) 5.26110e42 0.0454085
\(76\) 0 0
\(77\) 2.25989e43 0.105089
\(78\) 0 0
\(79\) 7.68215e43 0.195545 0.0977724 0.995209i \(-0.468828\pi\)
0.0977724 + 0.995209i \(0.468828\pi\)
\(80\) 0 0
\(81\) 6.74698e44 0.954358
\(82\) 0 0
\(83\) 1.71031e45 1.36377 0.681884 0.731460i \(-0.261161\pi\)
0.681884 + 0.731460i \(0.261161\pi\)
\(84\) 0 0
\(85\) −1.34374e45 −0.612310
\(86\) 0 0
\(87\) 5.74552e44 0.151575
\(88\) 0 0
\(89\) −9.70124e45 −1.50024 −0.750122 0.661299i \(-0.770005\pi\)
−0.750122 + 0.661299i \(0.770005\pi\)
\(90\) 0 0
\(91\) −1.42238e46 −1.30479
\(92\) 0 0
\(93\) −2.25547e45 −0.124133
\(94\) 0 0
\(95\) −2.24361e46 −0.748933
\(96\) 0 0
\(97\) 5.35657e46 1.09585 0.547925 0.836528i \(-0.315418\pi\)
0.547925 + 0.836528i \(0.315418\pi\)
\(98\) 0 0
\(99\) −4.67400e45 −0.0591918
\(100\) 0 0
\(101\) 1.46056e47 1.15602 0.578011 0.816029i \(-0.303829\pi\)
0.578011 + 0.816029i \(0.303829\pi\)
\(102\) 0 0
\(103\) 3.94670e47 1.97042 0.985212 0.171342i \(-0.0548104\pi\)
0.985212 + 0.171342i \(0.0548104\pi\)
\(104\) 0 0
\(105\) −7.95606e46 −0.252784
\(106\) 0 0
\(107\) −5.84600e47 −1.19218 −0.596088 0.802919i \(-0.703279\pi\)
−0.596088 + 0.802919i \(0.703279\pi\)
\(108\) 0 0
\(109\) −7.05835e47 −0.931494 −0.465747 0.884918i \(-0.654214\pi\)
−0.465747 + 0.884918i \(0.654214\pi\)
\(110\) 0 0
\(111\) −1.99151e47 −0.171432
\(112\) 0 0
\(113\) −6.63621e47 −0.375471 −0.187736 0.982220i \(-0.560115\pi\)
−0.187736 + 0.982220i \(0.560115\pi\)
\(114\) 0 0
\(115\) 2.76792e47 0.103694
\(116\) 0 0
\(117\) 2.94181e48 0.734933
\(118\) 0 0
\(119\) 5.45771e48 0.915497
\(120\) 0 0
\(121\) −8.78788e48 −0.996387
\(122\) 0 0
\(123\) −7.17921e47 −0.0553741
\(124\) 0 0
\(125\) 1.40141e49 0.739914
\(126\) 0 0
\(127\) −1.84983e49 −0.672580 −0.336290 0.941758i \(-0.609172\pi\)
−0.336290 + 0.941758i \(0.609172\pi\)
\(128\) 0 0
\(129\) 6.79645e48 0.171169
\(130\) 0 0
\(131\) −3.62008e49 −0.635100 −0.317550 0.948242i \(-0.602860\pi\)
−0.317550 + 0.948242i \(0.602860\pi\)
\(132\) 0 0
\(133\) 9.11260e49 1.11977
\(134\) 0 0
\(135\) 3.31656e49 0.286975
\(136\) 0 0
\(137\) −2.09650e50 −1.28399 −0.641994 0.766709i \(-0.721893\pi\)
−0.641994 + 0.766709i \(0.721893\pi\)
\(138\) 0 0
\(139\) 1.18819e50 0.517653 0.258826 0.965924i \(-0.416664\pi\)
0.258826 + 0.965924i \(0.416664\pi\)
\(140\) 0 0
\(141\) 2.47923e49 0.0772089
\(142\) 0 0
\(143\) −2.00582e49 −0.0448636
\(144\) 0 0
\(145\) −8.88096e50 −1.43322
\(146\) 0 0
\(147\) 2.17414e50 0.254290
\(148\) 0 0
\(149\) 1.81480e51 1.54508 0.772540 0.634966i \(-0.218986\pi\)
0.772540 + 0.634966i \(0.218986\pi\)
\(150\) 0 0
\(151\) −1.57042e51 −0.977366 −0.488683 0.872461i \(-0.662523\pi\)
−0.488683 + 0.872461i \(0.662523\pi\)
\(152\) 0 0
\(153\) −1.12878e51 −0.515659
\(154\) 0 0
\(155\) 3.48633e51 1.17374
\(156\) 0 0
\(157\) 3.70380e51 0.922584 0.461292 0.887248i \(-0.347386\pi\)
0.461292 + 0.887248i \(0.347386\pi\)
\(158\) 0 0
\(159\) −2.18667e49 −0.00404530
\(160\) 0 0
\(161\) −1.12421e51 −0.155039
\(162\) 0 0
\(163\) −4.02579e51 −0.415377 −0.207688 0.978195i \(-0.566594\pi\)
−0.207688 + 0.978195i \(0.566594\pi\)
\(164\) 0 0
\(165\) −1.12195e50 −0.00869164
\(166\) 0 0
\(167\) −1.17072e52 −0.683308 −0.341654 0.939826i \(-0.610987\pi\)
−0.341654 + 0.939826i \(0.610987\pi\)
\(168\) 0 0
\(169\) −1.00394e52 −0.442968
\(170\) 0 0
\(171\) −1.88470e52 −0.630717
\(172\) 0 0
\(173\) −2.17798e52 −0.554587 −0.277293 0.960785i \(-0.589437\pi\)
−0.277293 + 0.960785i \(0.589437\pi\)
\(174\) 0 0
\(175\) 3.30295e52 0.641959
\(176\) 0 0
\(177\) 2.93523e50 0.00436788
\(178\) 0 0
\(179\) 9.95625e52 1.13776 0.568879 0.822421i \(-0.307377\pi\)
0.568879 + 0.822421i \(0.307377\pi\)
\(180\) 0 0
\(181\) −6.46484e52 −0.568998 −0.284499 0.958676i \(-0.591827\pi\)
−0.284499 + 0.958676i \(0.591827\pi\)
\(182\) 0 0
\(183\) 9.66552e51 0.0657090
\(184\) 0 0
\(185\) 3.07832e53 1.62098
\(186\) 0 0
\(187\) 7.69638e51 0.0314781
\(188\) 0 0
\(189\) −1.34704e53 −0.429072
\(190\) 0 0
\(191\) −7.79937e53 −1.93988 −0.969942 0.243337i \(-0.921758\pi\)
−0.969942 + 0.243337i \(0.921758\pi\)
\(192\) 0 0
\(193\) 6.70136e52 0.130487 0.0652433 0.997869i \(-0.479218\pi\)
0.0652433 + 0.997869i \(0.479218\pi\)
\(194\) 0 0
\(195\) 7.06157e52 0.107916
\(196\) 0 0
\(197\) −1.22925e54 −1.47804 −0.739019 0.673685i \(-0.764711\pi\)
−0.739019 + 0.673685i \(0.764711\pi\)
\(198\) 0 0
\(199\) −1.40028e54 −1.32790 −0.663952 0.747775i \(-0.731122\pi\)
−0.663952 + 0.747775i \(0.731122\pi\)
\(200\) 0 0
\(201\) −6.76139e52 −0.0506908
\(202\) 0 0
\(203\) 3.60707e54 2.14288
\(204\) 0 0
\(205\) 1.10971e54 0.523591
\(206\) 0 0
\(207\) 2.32514e53 0.0873265
\(208\) 0 0
\(209\) 1.28505e53 0.0385018
\(210\) 0 0
\(211\) −7.49292e53 −0.179479 −0.0897394 0.995965i \(-0.528603\pi\)
−0.0897394 + 0.995965i \(0.528603\pi\)
\(212\) 0 0
\(213\) −1.85710e53 −0.0356381
\(214\) 0 0
\(215\) −1.05054e55 −1.61849
\(216\) 0 0
\(217\) −1.41600e55 −1.75493
\(218\) 0 0
\(219\) −1.05441e54 −0.105335
\(220\) 0 0
\(221\) −4.84410e54 −0.390836
\(222\) 0 0
\(223\) −7.65012e54 −0.499464 −0.249732 0.968315i \(-0.580342\pi\)
−0.249732 + 0.968315i \(0.580342\pi\)
\(224\) 0 0
\(225\) −6.83128e54 −0.361587
\(226\) 0 0
\(227\) −1.58232e55 −0.680277 −0.340138 0.940375i \(-0.610474\pi\)
−0.340138 + 0.940375i \(0.610474\pi\)
\(228\) 0 0
\(229\) 2.86045e55 1.00069 0.500347 0.865825i \(-0.333206\pi\)
0.500347 + 0.865825i \(0.333206\pi\)
\(230\) 0 0
\(231\) 4.55689e53 0.0129953
\(232\) 0 0
\(233\) −3.61119e55 −0.840980 −0.420490 0.907297i \(-0.638142\pi\)
−0.420490 + 0.907297i \(0.638142\pi\)
\(234\) 0 0
\(235\) −3.83219e55 −0.730050
\(236\) 0 0
\(237\) 1.54904e54 0.0241812
\(238\) 0 0
\(239\) −5.47124e55 −0.701027 −0.350514 0.936558i \(-0.613993\pi\)
−0.350514 + 0.936558i \(0.613993\pi\)
\(240\) 0 0
\(241\) −8.92529e55 −0.940204 −0.470102 0.882612i \(-0.655783\pi\)
−0.470102 + 0.882612i \(0.655783\pi\)
\(242\) 0 0
\(243\) 4.18975e55 0.363447
\(244\) 0 0
\(245\) −3.36060e56 −2.40444
\(246\) 0 0
\(247\) −8.08808e55 −0.478043
\(248\) 0 0
\(249\) 3.44871e55 0.168644
\(250\) 0 0
\(251\) −1.98048e56 −0.802487 −0.401244 0.915971i \(-0.631422\pi\)
−0.401244 + 0.915971i \(0.631422\pi\)
\(252\) 0 0
\(253\) −1.58535e54 −0.00533080
\(254\) 0 0
\(255\) −2.70955e55 −0.0757186
\(256\) 0 0
\(257\) −1.71029e55 −0.0397778 −0.0198889 0.999802i \(-0.506331\pi\)
−0.0198889 + 0.999802i \(0.506331\pi\)
\(258\) 0 0
\(259\) −1.25028e57 −2.42361
\(260\) 0 0
\(261\) −7.46028e56 −1.20699
\(262\) 0 0
\(263\) −3.32579e56 −0.449712 −0.224856 0.974392i \(-0.572191\pi\)
−0.224856 + 0.974392i \(0.572191\pi\)
\(264\) 0 0
\(265\) 3.37998e55 0.0382504
\(266\) 0 0
\(267\) −1.95618e56 −0.185521
\(268\) 0 0
\(269\) −1.74613e57 −1.38963 −0.694815 0.719189i \(-0.744514\pi\)
−0.694815 + 0.719189i \(0.744514\pi\)
\(270\) 0 0
\(271\) −1.96258e56 −0.131235 −0.0656174 0.997845i \(-0.520902\pi\)
−0.0656174 + 0.997845i \(0.520902\pi\)
\(272\) 0 0
\(273\) −2.86811e56 −0.161352
\(274\) 0 0
\(275\) 4.65777e55 0.0220729
\(276\) 0 0
\(277\) 1.40234e57 0.560503 0.280252 0.959927i \(-0.409582\pi\)
0.280252 + 0.959927i \(0.409582\pi\)
\(278\) 0 0
\(279\) 2.92862e57 0.988472
\(280\) 0 0
\(281\) −3.15409e57 −0.900068 −0.450034 0.893011i \(-0.648588\pi\)
−0.450034 + 0.893011i \(0.648588\pi\)
\(282\) 0 0
\(283\) −6.43417e57 −1.55422 −0.777108 0.629368i \(-0.783314\pi\)
−0.777108 + 0.629368i \(0.783314\pi\)
\(284\) 0 0
\(285\) −4.52406e56 −0.0926135
\(286\) 0 0
\(287\) −4.50715e57 −0.782848
\(288\) 0 0
\(289\) −4.91926e57 −0.725773
\(290\) 0 0
\(291\) 1.08011e57 0.135513
\(292\) 0 0
\(293\) −7.47274e57 −0.798162 −0.399081 0.916916i \(-0.630671\pi\)
−0.399081 + 0.916916i \(0.630671\pi\)
\(294\) 0 0
\(295\) −4.53705e56 −0.0413006
\(296\) 0 0
\(297\) −1.89958e56 −0.0147531
\(298\) 0 0
\(299\) 9.97819e56 0.0661879
\(300\) 0 0
\(301\) 4.26685e58 2.41989
\(302\) 0 0
\(303\) 2.94510e57 0.142954
\(304\) 0 0
\(305\) −1.49402e58 −0.621312
\(306\) 0 0
\(307\) 2.04451e58 0.729183 0.364592 0.931168i \(-0.381209\pi\)
0.364592 + 0.931168i \(0.381209\pi\)
\(308\) 0 0
\(309\) 7.95820e57 0.243664
\(310\) 0 0
\(311\) −3.46247e58 −0.910997 −0.455499 0.890236i \(-0.650539\pi\)
−0.455499 + 0.890236i \(0.650539\pi\)
\(312\) 0 0
\(313\) 7.50527e58 1.69853 0.849267 0.527963i \(-0.177044\pi\)
0.849267 + 0.527963i \(0.177044\pi\)
\(314\) 0 0
\(315\) 1.03306e59 2.01292
\(316\) 0 0
\(317\) −9.59709e58 −1.61156 −0.805781 0.592213i \(-0.798254\pi\)
−0.805781 + 0.592213i \(0.798254\pi\)
\(318\) 0 0
\(319\) 5.08663e57 0.0736800
\(320\) 0 0
\(321\) −1.17880e58 −0.147425
\(322\) 0 0
\(323\) 3.10342e58 0.335414
\(324\) 0 0
\(325\) −2.93160e58 −0.274060
\(326\) 0 0
\(327\) −1.42326e58 −0.115189
\(328\) 0 0
\(329\) 1.55647e59 1.09154
\(330\) 0 0
\(331\) 9.45783e58 0.575219 0.287610 0.957748i \(-0.407139\pi\)
0.287610 + 0.957748i \(0.407139\pi\)
\(332\) 0 0
\(333\) 2.58588e59 1.36511
\(334\) 0 0
\(335\) 1.04512e59 0.479307
\(336\) 0 0
\(337\) 4.62637e59 1.84475 0.922375 0.386295i \(-0.126245\pi\)
0.922375 + 0.386295i \(0.126245\pi\)
\(338\) 0 0
\(339\) −1.33814e58 −0.0464310
\(340\) 0 0
\(341\) −1.99682e58 −0.0603408
\(342\) 0 0
\(343\) 7.01170e59 1.84677
\(344\) 0 0
\(345\) 5.58129e57 0.0128229
\(346\) 0 0
\(347\) −4.68114e59 −0.938878 −0.469439 0.882965i \(-0.655544\pi\)
−0.469439 + 0.882965i \(0.655544\pi\)
\(348\) 0 0
\(349\) −1.32904e59 −0.232884 −0.116442 0.993197i \(-0.537149\pi\)
−0.116442 + 0.993197i \(0.537149\pi\)
\(350\) 0 0
\(351\) 1.19560e59 0.183176
\(352\) 0 0
\(353\) −7.06798e59 −0.947523 −0.473762 0.880653i \(-0.657104\pi\)
−0.473762 + 0.880653i \(0.657104\pi\)
\(354\) 0 0
\(355\) 2.87056e59 0.336977
\(356\) 0 0
\(357\) 1.10050e59 0.113211
\(358\) 0 0
\(359\) −8.86439e59 −0.799704 −0.399852 0.916580i \(-0.630939\pi\)
−0.399852 + 0.916580i \(0.630939\pi\)
\(360\) 0 0
\(361\) −7.44875e59 −0.589745
\(362\) 0 0
\(363\) −1.77200e59 −0.123214
\(364\) 0 0
\(365\) 1.62982e60 0.995994
\(366\) 0 0
\(367\) 1.87716e60 1.00890 0.504451 0.863440i \(-0.331695\pi\)
0.504451 + 0.863440i \(0.331695\pi\)
\(368\) 0 0
\(369\) 9.32186e59 0.440943
\(370\) 0 0
\(371\) −1.37280e59 −0.0571901
\(372\) 0 0
\(373\) −2.92907e59 −0.107541 −0.0537703 0.998553i \(-0.517124\pi\)
−0.0537703 + 0.998553i \(0.517124\pi\)
\(374\) 0 0
\(375\) 2.82584e59 0.0914982
\(376\) 0 0
\(377\) −3.20153e60 −0.914820
\(378\) 0 0
\(379\) −1.26439e60 −0.319051 −0.159525 0.987194i \(-0.550996\pi\)
−0.159525 + 0.987194i \(0.550996\pi\)
\(380\) 0 0
\(381\) −3.73003e59 −0.0831716
\(382\) 0 0
\(383\) −7.65652e60 −1.50959 −0.754797 0.655958i \(-0.772265\pi\)
−0.754797 + 0.655958i \(0.772265\pi\)
\(384\) 0 0
\(385\) −7.04368e59 −0.122877
\(386\) 0 0
\(387\) −8.82487e60 −1.36301
\(388\) 0 0
\(389\) −7.85479e60 −1.07478 −0.537389 0.843335i \(-0.680589\pi\)
−0.537389 + 0.843335i \(0.680589\pi\)
\(390\) 0 0
\(391\) −3.82867e59 −0.0464401
\(392\) 0 0
\(393\) −7.29959e59 −0.0785368
\(394\) 0 0
\(395\) −2.39439e60 −0.228645
\(396\) 0 0
\(397\) −1.21556e61 −1.03086 −0.515430 0.856932i \(-0.672368\pi\)
−0.515430 + 0.856932i \(0.672368\pi\)
\(398\) 0 0
\(399\) 1.83748e60 0.138471
\(400\) 0 0
\(401\) 1.13229e61 0.758692 0.379346 0.925255i \(-0.376149\pi\)
0.379346 + 0.925255i \(0.376149\pi\)
\(402\) 0 0
\(403\) 1.25680e61 0.749198
\(404\) 0 0
\(405\) −2.10291e61 −1.11591
\(406\) 0 0
\(407\) −1.76313e60 −0.0833327
\(408\) 0 0
\(409\) −1.93004e61 −0.812958 −0.406479 0.913660i \(-0.633244\pi\)
−0.406479 + 0.913660i \(0.633244\pi\)
\(410\) 0 0
\(411\) −4.22742e60 −0.158779
\(412\) 0 0
\(413\) 1.84276e60 0.0617507
\(414\) 0 0
\(415\) −5.33073e61 −1.59462
\(416\) 0 0
\(417\) 2.39589e60 0.0640132
\(418\) 0 0
\(419\) 3.12082e60 0.0745141 0.0372571 0.999306i \(-0.488138\pi\)
0.0372571 + 0.999306i \(0.488138\pi\)
\(420\) 0 0
\(421\) 6.12499e61 1.30760 0.653802 0.756666i \(-0.273173\pi\)
0.653802 + 0.756666i \(0.273173\pi\)
\(422\) 0 0
\(423\) −3.21916e61 −0.614814
\(424\) 0 0
\(425\) 1.12487e61 0.192292
\(426\) 0 0
\(427\) 6.06807e61 0.928957
\(428\) 0 0
\(429\) −4.04456e59 −0.00554786
\(430\) 0 0
\(431\) 7.83169e61 0.963030 0.481515 0.876438i \(-0.340087\pi\)
0.481515 + 0.876438i \(0.340087\pi\)
\(432\) 0 0
\(433\) 2.94510e61 0.324814 0.162407 0.986724i \(-0.448074\pi\)
0.162407 + 0.986724i \(0.448074\pi\)
\(434\) 0 0
\(435\) −1.79077e61 −0.177232
\(436\) 0 0
\(437\) −6.39262e60 −0.0568022
\(438\) 0 0
\(439\) 8.39024e60 0.0669666 0.0334833 0.999439i \(-0.489340\pi\)
0.0334833 + 0.999439i \(0.489340\pi\)
\(440\) 0 0
\(441\) −2.82301e62 −2.02491
\(442\) 0 0
\(443\) 7.70512e61 0.496926 0.248463 0.968641i \(-0.420075\pi\)
0.248463 + 0.968641i \(0.420075\pi\)
\(444\) 0 0
\(445\) 3.02370e62 1.75420
\(446\) 0 0
\(447\) 3.65939e61 0.191065
\(448\) 0 0
\(449\) 1.71627e62 0.806857 0.403429 0.915011i \(-0.367818\pi\)
0.403429 + 0.915011i \(0.367818\pi\)
\(450\) 0 0
\(451\) −6.35592e60 −0.0269172
\(452\) 0 0
\(453\) −3.16662e61 −0.120862
\(454\) 0 0
\(455\) 4.43329e62 1.52566
\(456\) 0 0
\(457\) 4.06573e62 1.26214 0.631070 0.775726i \(-0.282616\pi\)
0.631070 + 0.775726i \(0.282616\pi\)
\(458\) 0 0
\(459\) −4.58755e61 −0.128524
\(460\) 0 0
\(461\) 2.67924e62 0.677704 0.338852 0.940840i \(-0.389961\pi\)
0.338852 + 0.940840i \(0.389961\pi\)
\(462\) 0 0
\(463\) 5.44459e62 1.24397 0.621987 0.783027i \(-0.286325\pi\)
0.621987 + 0.783027i \(0.286325\pi\)
\(464\) 0 0
\(465\) 7.02989e61 0.145146
\(466\) 0 0
\(467\) −4.82071e62 −0.899839 −0.449919 0.893069i \(-0.648547\pi\)
−0.449919 + 0.893069i \(0.648547\pi\)
\(468\) 0 0
\(469\) −4.24484e62 −0.716637
\(470\) 0 0
\(471\) 7.46841e61 0.114087
\(472\) 0 0
\(473\) 6.01705e61 0.0832045
\(474\) 0 0
\(475\) 1.87816e62 0.235197
\(476\) 0 0
\(477\) 2.83928e61 0.0322127
\(478\) 0 0
\(479\) −1.74319e63 −1.79251 −0.896253 0.443543i \(-0.853721\pi\)
−0.896253 + 0.443543i \(0.853721\pi\)
\(480\) 0 0
\(481\) 1.10971e63 1.03467
\(482\) 0 0
\(483\) −2.26688e61 −0.0191722
\(484\) 0 0
\(485\) −1.66955e63 −1.28135
\(486\) 0 0
\(487\) 2.19497e63 1.52932 0.764661 0.644433i \(-0.222907\pi\)
0.764661 + 0.644433i \(0.222907\pi\)
\(488\) 0 0
\(489\) −8.11767e61 −0.0513657
\(490\) 0 0
\(491\) 4.81659e62 0.276901 0.138451 0.990369i \(-0.455788\pi\)
0.138451 + 0.990369i \(0.455788\pi\)
\(492\) 0 0
\(493\) 1.22844e63 0.641875
\(494\) 0 0
\(495\) 1.45680e62 0.0692114
\(496\) 0 0
\(497\) −1.16590e63 −0.503832
\(498\) 0 0
\(499\) 2.59805e63 1.02161 0.510805 0.859696i \(-0.329347\pi\)
0.510805 + 0.859696i \(0.329347\pi\)
\(500\) 0 0
\(501\) −2.36066e62 −0.0844983
\(502\) 0 0
\(503\) −5.26346e63 −1.71563 −0.857816 0.513958i \(-0.828179\pi\)
−0.857816 + 0.513958i \(0.828179\pi\)
\(504\) 0 0
\(505\) −4.55229e63 −1.35171
\(506\) 0 0
\(507\) −2.02437e62 −0.0547777
\(508\) 0 0
\(509\) −1.22138e63 −0.301289 −0.150644 0.988588i \(-0.548135\pi\)
−0.150644 + 0.988588i \(0.548135\pi\)
\(510\) 0 0
\(511\) −6.61963e63 −1.48916
\(512\) 0 0
\(513\) −7.65972e62 −0.157201
\(514\) 0 0
\(515\) −1.23012e64 −2.30397
\(516\) 0 0
\(517\) 2.19492e62 0.0375310
\(518\) 0 0
\(519\) −4.39171e62 −0.0685805
\(520\) 0 0
\(521\) −9.22332e62 −0.131583 −0.0657916 0.997833i \(-0.520957\pi\)
−0.0657916 + 0.997833i \(0.520957\pi\)
\(522\) 0 0
\(523\) 1.18236e64 1.54155 0.770777 0.637105i \(-0.219868\pi\)
0.770777 + 0.637105i \(0.219868\pi\)
\(524\) 0 0
\(525\) 6.66012e62 0.0793850
\(526\) 0 0
\(527\) −4.82238e63 −0.525668
\(528\) 0 0
\(529\) −9.94900e63 −0.992135
\(530\) 0 0
\(531\) −3.81126e62 −0.0347814
\(532\) 0 0
\(533\) 4.00041e63 0.334207
\(534\) 0 0
\(535\) 1.82209e64 1.39398
\(536\) 0 0
\(537\) 2.00760e63 0.140696
\(538\) 0 0
\(539\) 1.92481e63 0.123610
\(540\) 0 0
\(541\) −1.28271e64 −0.755077 −0.377539 0.925994i \(-0.623229\pi\)
−0.377539 + 0.925994i \(0.623229\pi\)
\(542\) 0 0
\(543\) −1.30358e63 −0.0703627
\(544\) 0 0
\(545\) 2.19996e64 1.08917
\(546\) 0 0
\(547\) −3.21410e64 −1.46001 −0.730007 0.683440i \(-0.760483\pi\)
−0.730007 + 0.683440i \(0.760483\pi\)
\(548\) 0 0
\(549\) −1.25502e64 −0.523240
\(550\) 0 0
\(551\) 2.05109e64 0.785095
\(552\) 0 0
\(553\) 9.72498e63 0.341860
\(554\) 0 0
\(555\) 6.20717e63 0.200451
\(556\) 0 0
\(557\) −1.16869e64 −0.346819 −0.173409 0.984850i \(-0.555478\pi\)
−0.173409 + 0.984850i \(0.555478\pi\)
\(558\) 0 0
\(559\) −3.78713e64 −1.03308
\(560\) 0 0
\(561\) 1.55191e62 0.00389260
\(562\) 0 0
\(563\) −9.31800e63 −0.214969 −0.107484 0.994207i \(-0.534280\pi\)
−0.107484 + 0.994207i \(0.534280\pi\)
\(564\) 0 0
\(565\) 2.06839e64 0.439029
\(566\) 0 0
\(567\) 8.54113e64 1.66845
\(568\) 0 0
\(569\) 7.98965e64 1.43678 0.718388 0.695642i \(-0.244880\pi\)
0.718388 + 0.695642i \(0.244880\pi\)
\(570\) 0 0
\(571\) 7.20740e64 1.19352 0.596761 0.802419i \(-0.296454\pi\)
0.596761 + 0.802419i \(0.296454\pi\)
\(572\) 0 0
\(573\) −1.57268e64 −0.239887
\(574\) 0 0
\(575\) −2.31707e63 −0.0325645
\(576\) 0 0
\(577\) 3.11163e64 0.403047 0.201523 0.979484i \(-0.435411\pi\)
0.201523 + 0.979484i \(0.435411\pi\)
\(578\) 0 0
\(579\) 1.35127e63 0.0161360
\(580\) 0 0
\(581\) 2.16512e65 2.38420
\(582\) 0 0
\(583\) −1.93591e62 −0.00196641
\(584\) 0 0
\(585\) −9.16910e64 −0.859338
\(586\) 0 0
\(587\) 1.26293e65 1.09241 0.546205 0.837652i \(-0.316072\pi\)
0.546205 + 0.837652i \(0.316072\pi\)
\(588\) 0 0
\(589\) −8.05180e64 −0.642959
\(590\) 0 0
\(591\) −2.47869e64 −0.182775
\(592\) 0 0
\(593\) −1.16829e65 −0.795726 −0.397863 0.917445i \(-0.630248\pi\)
−0.397863 + 0.917445i \(0.630248\pi\)
\(594\) 0 0
\(595\) −1.70107e65 −1.07047
\(596\) 0 0
\(597\) −2.82354e64 −0.164209
\(598\) 0 0
\(599\) 2.64716e65 1.42315 0.711573 0.702612i \(-0.247983\pi\)
0.711573 + 0.702612i \(0.247983\pi\)
\(600\) 0 0
\(601\) −1.04245e65 −0.518208 −0.259104 0.965849i \(-0.583427\pi\)
−0.259104 + 0.965849i \(0.583427\pi\)
\(602\) 0 0
\(603\) 8.77934e64 0.403650
\(604\) 0 0
\(605\) 2.73902e65 1.16505
\(606\) 0 0
\(607\) −2.13825e65 −0.841635 −0.420818 0.907145i \(-0.638257\pi\)
−0.420818 + 0.907145i \(0.638257\pi\)
\(608\) 0 0
\(609\) 7.27336e64 0.264989
\(610\) 0 0
\(611\) −1.38148e65 −0.465990
\(612\) 0 0
\(613\) −9.41515e64 −0.294108 −0.147054 0.989129i \(-0.546979\pi\)
−0.147054 + 0.989129i \(0.546979\pi\)
\(614\) 0 0
\(615\) 2.23763e64 0.0647475
\(616\) 0 0
\(617\) 3.62608e65 0.972156 0.486078 0.873916i \(-0.338427\pi\)
0.486078 + 0.873916i \(0.338427\pi\)
\(618\) 0 0
\(619\) 1.97673e65 0.491154 0.245577 0.969377i \(-0.421023\pi\)
0.245577 + 0.969377i \(0.421023\pi\)
\(620\) 0 0
\(621\) 9.44972e63 0.0217654
\(622\) 0 0
\(623\) −1.22810e66 −2.62279
\(624\) 0 0
\(625\) −6.22185e65 −1.23236
\(626\) 0 0
\(627\) 2.59119e63 0.00476115
\(628\) 0 0
\(629\) −4.25801e65 −0.725966
\(630\) 0 0
\(631\) 1.83594e65 0.290515 0.145257 0.989394i \(-0.453599\pi\)
0.145257 + 0.989394i \(0.453599\pi\)
\(632\) 0 0
\(633\) −1.51089e64 −0.0221945
\(634\) 0 0
\(635\) 5.76558e65 0.786431
\(636\) 0 0
\(637\) −1.21148e66 −1.53475
\(638\) 0 0
\(639\) 2.41136e65 0.283786
\(640\) 0 0
\(641\) 9.34817e64 0.102226 0.0511132 0.998693i \(-0.483723\pi\)
0.0511132 + 0.998693i \(0.483723\pi\)
\(642\) 0 0
\(643\) −6.98717e65 −0.710139 −0.355070 0.934840i \(-0.615543\pi\)
−0.355070 + 0.934840i \(0.615543\pi\)
\(644\) 0 0
\(645\) −2.11833e65 −0.200143
\(646\) 0 0
\(647\) −8.30387e65 −0.729508 −0.364754 0.931104i \(-0.618847\pi\)
−0.364754 + 0.931104i \(0.618847\pi\)
\(648\) 0 0
\(649\) 2.59863e63 0.00212321
\(650\) 0 0
\(651\) −2.85524e65 −0.217015
\(652\) 0 0
\(653\) −7.01610e65 −0.496176 −0.248088 0.968737i \(-0.579802\pi\)
−0.248088 + 0.968737i \(0.579802\pi\)
\(654\) 0 0
\(655\) 1.12831e66 0.742606
\(656\) 0 0
\(657\) 1.36910e66 0.838780
\(658\) 0 0
\(659\) −2.56897e66 −1.46539 −0.732693 0.680559i \(-0.761737\pi\)
−0.732693 + 0.680559i \(0.761737\pi\)
\(660\) 0 0
\(661\) 1.01472e66 0.539028 0.269514 0.962996i \(-0.413137\pi\)
0.269514 + 0.962996i \(0.413137\pi\)
\(662\) 0 0
\(663\) −9.76774e64 −0.0483311
\(664\) 0 0
\(665\) −2.84023e66 −1.30932
\(666\) 0 0
\(667\) −2.53041e65 −0.108701
\(668\) 0 0
\(669\) −1.54258e65 −0.0617640
\(670\) 0 0
\(671\) 8.55710e64 0.0319409
\(672\) 0 0
\(673\) 1.27088e66 0.442334 0.221167 0.975236i \(-0.429013\pi\)
0.221167 + 0.975236i \(0.429013\pi\)
\(674\) 0 0
\(675\) −2.77634e65 −0.0901225
\(676\) 0 0
\(677\) 2.17082e66 0.657341 0.328671 0.944445i \(-0.393399\pi\)
0.328671 + 0.944445i \(0.393399\pi\)
\(678\) 0 0
\(679\) 6.78098e66 1.91581
\(680\) 0 0
\(681\) −3.19061e65 −0.0841234
\(682\) 0 0
\(683\) −7.17341e64 −0.0176538 −0.00882692 0.999961i \(-0.502810\pi\)
−0.00882692 + 0.999961i \(0.502810\pi\)
\(684\) 0 0
\(685\) 6.53440e66 1.50133
\(686\) 0 0
\(687\) 5.76787e65 0.123746
\(688\) 0 0
\(689\) 1.21846e65 0.0244151
\(690\) 0 0
\(691\) 4.47891e66 0.838374 0.419187 0.907900i \(-0.362315\pi\)
0.419187 + 0.907900i \(0.362315\pi\)
\(692\) 0 0
\(693\) −5.91690e65 −0.103482
\(694\) 0 0
\(695\) −3.70338e66 −0.605278
\(696\) 0 0
\(697\) −1.53497e66 −0.234493
\(698\) 0 0
\(699\) −7.28166e65 −0.103996
\(700\) 0 0
\(701\) −3.84070e66 −0.512905 −0.256452 0.966557i \(-0.582554\pi\)
−0.256452 + 0.966557i \(0.582554\pi\)
\(702\) 0 0
\(703\) −7.10949e66 −0.887949
\(704\) 0 0
\(705\) −7.72730e65 −0.0902784
\(706\) 0 0
\(707\) 1.84895e67 2.02101
\(708\) 0 0
\(709\) 6.56172e64 0.00671168 0.00335584 0.999994i \(-0.498932\pi\)
0.00335584 + 0.999994i \(0.498932\pi\)
\(710\) 0 0
\(711\) −2.01136e66 −0.192555
\(712\) 0 0
\(713\) 9.93343e65 0.0890216
\(714\) 0 0
\(715\) 6.25176e65 0.0524579
\(716\) 0 0
\(717\) −1.10323e66 −0.0866894
\(718\) 0 0
\(719\) 2.12438e67 1.56352 0.781760 0.623580i \(-0.214322\pi\)
0.781760 + 0.623580i \(0.214322\pi\)
\(720\) 0 0
\(721\) 4.99620e67 3.44478
\(722\) 0 0
\(723\) −1.79971e66 −0.116266
\(724\) 0 0
\(725\) 7.43437e66 0.450092
\(726\) 0 0
\(727\) −2.27873e67 −1.29311 −0.646554 0.762868i \(-0.723791\pi\)
−0.646554 + 0.762868i \(0.723791\pi\)
\(728\) 0 0
\(729\) −1.70946e67 −0.909414
\(730\) 0 0
\(731\) 1.45314e67 0.724850
\(732\) 0 0
\(733\) −2.32774e66 −0.108891 −0.0544454 0.998517i \(-0.517339\pi\)
−0.0544454 + 0.998517i \(0.517339\pi\)
\(734\) 0 0
\(735\) −6.77639e66 −0.297335
\(736\) 0 0
\(737\) −5.98601e65 −0.0246406
\(738\) 0 0
\(739\) 1.90706e65 0.00736579 0.00368290 0.999993i \(-0.498828\pi\)
0.00368290 + 0.999993i \(0.498828\pi\)
\(740\) 0 0
\(741\) −1.63090e66 −0.0591151
\(742\) 0 0
\(743\) −3.96691e67 −1.34963 −0.674816 0.737986i \(-0.735777\pi\)
−0.674816 + 0.737986i \(0.735777\pi\)
\(744\) 0 0
\(745\) −5.65639e67 −1.80662
\(746\) 0 0
\(747\) −4.47798e67 −1.34291
\(748\) 0 0
\(749\) −7.40056e67 −2.08421
\(750\) 0 0
\(751\) 4.47498e67 1.18373 0.591865 0.806037i \(-0.298392\pi\)
0.591865 + 0.806037i \(0.298392\pi\)
\(752\) 0 0
\(753\) −3.99348e66 −0.0992360
\(754\) 0 0
\(755\) 4.89471e67 1.14281
\(756\) 0 0
\(757\) 1.35751e67 0.297845 0.148922 0.988849i \(-0.452420\pi\)
0.148922 + 0.988849i \(0.452420\pi\)
\(758\) 0 0
\(759\) −3.19673e64 −0.000659210 0
\(760\) 0 0
\(761\) 3.22776e67 0.625693 0.312846 0.949804i \(-0.398717\pi\)
0.312846 + 0.949804i \(0.398717\pi\)
\(762\) 0 0
\(763\) −8.93529e67 −1.62848
\(764\) 0 0
\(765\) 3.51822e67 0.602947
\(766\) 0 0
\(767\) −1.63558e66 −0.0263621
\(768\) 0 0
\(769\) −6.92939e67 −1.05057 −0.525287 0.850925i \(-0.676042\pi\)
−0.525287 + 0.850925i \(0.676042\pi\)
\(770\) 0 0
\(771\) −3.44865e65 −0.00491894
\(772\) 0 0
\(773\) 1.00550e68 1.34947 0.674733 0.738062i \(-0.264259\pi\)
0.674733 + 0.738062i \(0.264259\pi\)
\(774\) 0 0
\(775\) −2.91845e67 −0.368606
\(776\) 0 0
\(777\) −2.52109e67 −0.299705
\(778\) 0 0
\(779\) −2.56291e67 −0.286815
\(780\) 0 0
\(781\) −1.64413e66 −0.0173236
\(782\) 0 0
\(783\) −3.03197e67 −0.300831
\(784\) 0 0
\(785\) −1.15441e68 −1.07875
\(786\) 0 0
\(787\) 2.41454e66 0.0212534 0.0106267 0.999944i \(-0.496617\pi\)
0.0106267 + 0.999944i \(0.496617\pi\)
\(788\) 0 0
\(789\) −6.70618e66 −0.0556116
\(790\) 0 0
\(791\) −8.40090e67 −0.656415
\(792\) 0 0
\(793\) −5.38584e67 −0.396583
\(794\) 0 0
\(795\) 6.81544e65 0.00473006
\(796\) 0 0
\(797\) −2.29036e68 −1.49842 −0.749210 0.662332i \(-0.769567\pi\)
−0.749210 + 0.662332i \(0.769567\pi\)
\(798\) 0 0
\(799\) 5.30079e67 0.326957
\(800\) 0 0
\(801\) 2.54000e68 1.47730
\(802\) 0 0
\(803\) −9.33490e66 −0.0512029
\(804\) 0 0
\(805\) 3.50397e67 0.181283
\(806\) 0 0
\(807\) −3.52094e67 −0.171842
\(808\) 0 0
\(809\) 7.74628e66 0.0356700 0.0178350 0.999841i \(-0.494323\pi\)
0.0178350 + 0.999841i \(0.494323\pi\)
\(810\) 0 0
\(811\) 2.32748e68 1.01134 0.505669 0.862727i \(-0.331246\pi\)
0.505669 + 0.862727i \(0.331246\pi\)
\(812\) 0 0
\(813\) −3.95739e66 −0.0162286
\(814\) 0 0
\(815\) 1.25476e68 0.485689
\(816\) 0 0
\(817\) 2.42626e68 0.886584
\(818\) 0 0
\(819\) 3.72410e68 1.28484
\(820\) 0 0
\(821\) 4.26945e68 1.39094 0.695470 0.718556i \(-0.255196\pi\)
0.695470 + 0.718556i \(0.255196\pi\)
\(822\) 0 0
\(823\) −1.89793e68 −0.583961 −0.291981 0.956424i \(-0.594314\pi\)
−0.291981 + 0.956424i \(0.594314\pi\)
\(824\) 0 0
\(825\) 9.39201e65 0.00272955
\(826\) 0 0
\(827\) 8.12864e67 0.223171 0.111586 0.993755i \(-0.464407\pi\)
0.111586 + 0.993755i \(0.464407\pi\)
\(828\) 0 0
\(829\) −6.84615e68 −1.77589 −0.887943 0.459953i \(-0.847866\pi\)
−0.887943 + 0.459953i \(0.847866\pi\)
\(830\) 0 0
\(831\) 2.82770e67 0.0693122
\(832\) 0 0
\(833\) 4.64848e68 1.07684
\(834\) 0 0
\(835\) 3.64893e68 0.798975
\(836\) 0 0
\(837\) 1.19024e68 0.246368
\(838\) 0 0
\(839\) 4.43097e67 0.0867147 0.0433574 0.999060i \(-0.486195\pi\)
0.0433574 + 0.999060i \(0.486195\pi\)
\(840\) 0 0
\(841\) 2.71501e68 0.502418
\(842\) 0 0
\(843\) −6.35996e67 −0.111303
\(844\) 0 0
\(845\) 3.12911e68 0.517951
\(846\) 0 0
\(847\) −1.11247e69 −1.74193
\(848\) 0 0
\(849\) −1.29740e68 −0.192195
\(850\) 0 0
\(851\) 8.77091e67 0.122942
\(852\) 0 0
\(853\) 1.15923e69 1.53768 0.768842 0.639439i \(-0.220833\pi\)
0.768842 + 0.639439i \(0.220833\pi\)
\(854\) 0 0
\(855\) 5.87428e68 0.737481
\(856\) 0 0
\(857\) −1.19885e69 −1.42467 −0.712334 0.701840i \(-0.752362\pi\)
−0.712334 + 0.701840i \(0.752362\pi\)
\(858\) 0 0
\(859\) 1.22015e69 1.37269 0.686345 0.727276i \(-0.259214\pi\)
0.686345 + 0.727276i \(0.259214\pi\)
\(860\) 0 0
\(861\) −9.08830e67 −0.0968074
\(862\) 0 0
\(863\) −5.27188e68 −0.531755 −0.265878 0.964007i \(-0.585662\pi\)
−0.265878 + 0.964007i \(0.585662\pi\)
\(864\) 0 0
\(865\) 6.78835e68 0.648464
\(866\) 0 0
\(867\) −9.91930e67 −0.0897495
\(868\) 0 0
\(869\) 1.37140e67 0.0117544
\(870\) 0 0
\(871\) 3.76760e68 0.305941
\(872\) 0 0
\(873\) −1.40247e69 −1.07909
\(874\) 0 0
\(875\) 1.77408e69 1.29355
\(876\) 0 0
\(877\) −1.25938e69 −0.870296 −0.435148 0.900359i \(-0.643304\pi\)
−0.435148 + 0.900359i \(0.643304\pi\)
\(878\) 0 0
\(879\) −1.50682e68 −0.0987011
\(880\) 0 0
\(881\) 1.05379e69 0.654365 0.327182 0.944961i \(-0.393901\pi\)
0.327182 + 0.944961i \(0.393901\pi\)
\(882\) 0 0
\(883\) −1.51174e69 −0.890021 −0.445010 0.895525i \(-0.646800\pi\)
−0.445010 + 0.895525i \(0.646800\pi\)
\(884\) 0 0
\(885\) −9.14859e66 −0.00510725
\(886\) 0 0
\(887\) −2.55796e69 −1.35422 −0.677108 0.735883i \(-0.736767\pi\)
−0.677108 + 0.735883i \(0.736767\pi\)
\(888\) 0 0
\(889\) −2.34173e69 −1.17583
\(890\) 0 0
\(891\) 1.20446e68 0.0573674
\(892\) 0 0
\(893\) 8.85059e68 0.399911
\(894\) 0 0
\(895\) −3.10318e69 −1.33035
\(896\) 0 0
\(897\) 2.01202e67 0.00818483
\(898\) 0 0
\(899\) −3.18717e69 −1.23042
\(900\) 0 0
\(901\) −4.67527e67 −0.0171307
\(902\) 0 0
\(903\) 8.60376e68 0.299244
\(904\) 0 0
\(905\) 2.01497e69 0.665315
\(906\) 0 0
\(907\) 1.67267e69 0.524373 0.262186 0.965017i \(-0.415557\pi\)
0.262186 + 0.965017i \(0.415557\pi\)
\(908\) 0 0
\(909\) −3.82406e69 −1.13835
\(910\) 0 0
\(911\) 2.16759e68 0.0612767 0.0306383 0.999531i \(-0.490246\pi\)
0.0306383 + 0.999531i \(0.490246\pi\)
\(912\) 0 0
\(913\) 3.05322e68 0.0819775
\(914\) 0 0
\(915\) −3.01257e68 −0.0768318
\(916\) 0 0
\(917\) −4.58272e69 −1.11031
\(918\) 0 0
\(919\) 1.42756e69 0.328610 0.164305 0.986410i \(-0.447462\pi\)
0.164305 + 0.986410i \(0.447462\pi\)
\(920\) 0 0
\(921\) 4.12258e68 0.0901712
\(922\) 0 0
\(923\) 1.03482e69 0.215092
\(924\) 0 0
\(925\) −2.57690e69 −0.509057
\(926\) 0 0
\(927\) −1.03333e70 −1.94029
\(928\) 0 0
\(929\) 8.84307e68 0.157846 0.0789231 0.996881i \(-0.474852\pi\)
0.0789231 + 0.996881i \(0.474852\pi\)
\(930\) 0 0
\(931\) 7.76144e69 1.31712
\(932\) 0 0
\(933\) −6.98179e68 −0.112654
\(934\) 0 0
\(935\) −2.39882e68 −0.0368066
\(936\) 0 0
\(937\) −4.16313e69 −0.607491 −0.303746 0.952753i \(-0.598237\pi\)
−0.303746 + 0.952753i \(0.598237\pi\)
\(938\) 0 0
\(939\) 1.51338e69 0.210042
\(940\) 0 0
\(941\) 1.24869e70 1.64853 0.824266 0.566203i \(-0.191588\pi\)
0.824266 + 0.566203i \(0.191588\pi\)
\(942\) 0 0
\(943\) 3.16183e68 0.0397113
\(944\) 0 0
\(945\) 4.19849e69 0.501702
\(946\) 0 0
\(947\) 6.06340e69 0.689433 0.344716 0.938707i \(-0.387975\pi\)
0.344716 + 0.938707i \(0.387975\pi\)
\(948\) 0 0
\(949\) 5.87539e69 0.635741
\(950\) 0 0
\(951\) −1.93518e69 −0.199287
\(952\) 0 0
\(953\) −3.62312e69 −0.355140 −0.177570 0.984108i \(-0.556824\pi\)
−0.177570 + 0.984108i \(0.556824\pi\)
\(954\) 0 0
\(955\) 2.43092e70 2.26826
\(956\) 0 0
\(957\) 1.02568e68 0.00911131
\(958\) 0 0
\(959\) −2.65400e70 −2.24472
\(960\) 0 0
\(961\) 9.50929e67 0.00765858
\(962\) 0 0
\(963\) 1.53061e70 1.17394
\(964\) 0 0
\(965\) −2.08869e69 −0.152575
\(966\) 0 0
\(967\) −2.47757e70 −1.72387 −0.861933 0.507023i \(-0.830746\pi\)
−0.861933 + 0.507023i \(0.830746\pi\)
\(968\) 0 0
\(969\) 6.25780e68 0.0414775
\(970\) 0 0
\(971\) 1.69313e70 1.06915 0.534574 0.845122i \(-0.320472\pi\)
0.534574 + 0.845122i \(0.320472\pi\)
\(972\) 0 0
\(973\) 1.50415e70 0.904983
\(974\) 0 0
\(975\) −5.91133e68 −0.0338904
\(976\) 0 0
\(977\) −1.88519e70 −1.02999 −0.514994 0.857193i \(-0.672206\pi\)
−0.514994 + 0.857193i \(0.672206\pi\)
\(978\) 0 0
\(979\) −1.73185e69 −0.0901812
\(980\) 0 0
\(981\) 1.84803e70 0.917250
\(982\) 0 0
\(983\) −1.50691e69 −0.0712982 −0.0356491 0.999364i \(-0.511350\pi\)
−0.0356491 + 0.999364i \(0.511350\pi\)
\(984\) 0 0
\(985\) 3.83136e70 1.72823
\(986\) 0 0
\(987\) 3.13850e69 0.134980
\(988\) 0 0
\(989\) −2.99326e69 −0.122753
\(990\) 0 0
\(991\) 3.55474e70 1.39020 0.695099 0.718914i \(-0.255361\pi\)
0.695099 + 0.718914i \(0.255361\pi\)
\(992\) 0 0
\(993\) 1.90709e69 0.0711319
\(994\) 0 0
\(995\) 4.36441e70 1.55268
\(996\) 0 0
\(997\) −1.08915e70 −0.369618 −0.184809 0.982774i \(-0.559167\pi\)
−0.184809 + 0.982774i \(0.559167\pi\)
\(998\) 0 0
\(999\) 1.05094e70 0.340243
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 16.48.a.d.1.3 4
4.3 odd 2 1.48.a.a.1.1 4
12.11 even 2 9.48.a.c.1.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1.48.a.a.1.1 4 4.3 odd 2
9.48.a.c.1.4 4 12.11 even 2
16.48.a.d.1.3 4 1.1 even 1 trivial