Properties

Label 192.12.a.i.1.1
Level $192$
Weight $12$
Character 192.1
Self dual yes
Analytic conductor $147.522$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [192,12,Mod(1,192)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(192, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("192.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 192 = 2^{6} \cdot 3 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 192.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: \(147.521890667\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 192.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-243.000 q^{3} +7130.00 q^{5} +19536.0 q^{7} +59049.0 q^{9} +O(q^{10})\) \(q-243.000 q^{3} +7130.00 q^{5} +19536.0 q^{7} +59049.0 q^{9} -196148. q^{11} -361414. q^{13} -1.73259e6 q^{15} -130942. q^{17} +1.85167e7 q^{19} -4.74725e6 q^{21} -2.15609e7 q^{23} +2.00878e6 q^{25} -1.43489e7 q^{27} -1.91664e8 q^{29} -2.07934e8 q^{31} +4.76640e7 q^{33} +1.39292e8 q^{35} +2.00785e8 q^{37} +8.78236e7 q^{39} -1.43526e9 q^{41} +7.12703e8 q^{43} +4.21019e8 q^{45} +4.96082e8 q^{47} -1.59567e9 q^{49} +3.18189e7 q^{51} +3.35011e9 q^{53} -1.39854e9 q^{55} -4.49956e9 q^{57} +4.58322e9 q^{59} -3.42750e9 q^{61} +1.15358e9 q^{63} -2.57688e9 q^{65} +1.70794e10 q^{67} +5.23929e9 q^{69} +7.91508e9 q^{71} +3.15597e10 q^{73} -4.88132e8 q^{75} -3.83195e9 q^{77} -4.10236e10 q^{79} +3.48678e9 q^{81} -1.99747e10 q^{83} -9.33616e8 q^{85} +4.65743e10 q^{87} -1.06402e10 q^{89} -7.06058e9 q^{91} +5.05279e10 q^{93} +1.32024e11 q^{95} +6.44111e9 q^{97} -1.15823e10 q^{99} +O(q^{100})\)

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
\(3\) −243.000 −0.577350
\(4\) 0 0
\(5\) 7130.00 1.02036 0.510181 0.860067i \(-0.329578\pi\)
0.510181 + 0.860067i \(0.329578\pi\)
\(6\) 0 0
\(7\) 19536.0 0.439336 0.219668 0.975575i \(-0.429503\pi\)
0.219668 + 0.975575i \(0.429503\pi\)
\(8\) 0 0
\(9\) 59049.0 0.333333
\(10\) 0 0
\(11\) −196148. −0.367218 −0.183609 0.982999i \(-0.558778\pi\)
−0.183609 + 0.982999i \(0.558778\pi\)
\(12\) 0 0
\(13\) −361414. −0.269971 −0.134985 0.990848i \(-0.543099\pi\)
−0.134985 + 0.990848i \(0.543099\pi\)
\(14\) 0 0
\(15\) −1.73259e6 −0.589107
\(16\) 0 0
\(17\) −130942. −0.0223671 −0.0111836 0.999937i \(-0.503560\pi\)
−0.0111836 + 0.999937i \(0.503560\pi\)
\(18\) 0 0
\(19\) 1.85167e7 1.71561 0.857805 0.513975i \(-0.171828\pi\)
0.857805 + 0.513975i \(0.171828\pi\)
\(20\) 0 0
\(21\) −4.74725e6 −0.253651
\(22\) 0 0
\(23\) −2.15609e7 −0.698495 −0.349247 0.937031i \(-0.613563\pi\)
−0.349247 + 0.937031i \(0.613563\pi\)
\(24\) 0 0
\(25\) 2.00878e6 0.0411397
\(26\) 0 0
\(27\) −1.43489e7 −0.192450
\(28\) 0 0
\(29\) −1.91664e8 −1.73521 −0.867603 0.497258i \(-0.834340\pi\)
−0.867603 + 0.497258i \(0.834340\pi\)
\(30\) 0 0
\(31\) −2.07934e8 −1.30448 −0.652238 0.758015i \(-0.726170\pi\)
−0.652238 + 0.758015i \(0.726170\pi\)
\(32\) 0 0
\(33\) 4.76640e7 0.212013
\(34\) 0 0
\(35\) 1.39292e8 0.448282
\(36\) 0 0
\(37\) 2.00785e8 0.476016 0.238008 0.971263i \(-0.423506\pi\)
0.238008 + 0.971263i \(0.423506\pi\)
\(38\) 0 0
\(39\) 8.78236e7 0.155868
\(40\) 0 0
\(41\) −1.43526e9 −1.93472 −0.967360 0.253404i \(-0.918450\pi\)
−0.967360 + 0.253404i \(0.918450\pi\)
\(42\) 0 0
\(43\) 7.12703e8 0.739319 0.369660 0.929167i \(-0.379474\pi\)
0.369660 + 0.929167i \(0.379474\pi\)
\(44\) 0 0
\(45\) 4.21019e8 0.340121
\(46\) 0 0
\(47\) 4.96082e8 0.315512 0.157756 0.987478i \(-0.449574\pi\)
0.157756 + 0.987478i \(0.449574\pi\)
\(48\) 0 0
\(49\) −1.59567e9 −0.806984
\(50\) 0 0
\(51\) 3.18189e7 0.0129137
\(52\) 0 0
\(53\) 3.35011e9 1.10038 0.550190 0.835040i \(-0.314555\pi\)
0.550190 + 0.835040i \(0.314555\pi\)
\(54\) 0 0
\(55\) −1.39854e9 −0.374696
\(56\) 0 0
\(57\) −4.49956e9 −0.990508
\(58\) 0 0
\(59\) 4.58322e9 0.834613 0.417306 0.908766i \(-0.362974\pi\)
0.417306 + 0.908766i \(0.362974\pi\)
\(60\) 0 0
\(61\) −3.42750e9 −0.519593 −0.259797 0.965663i \(-0.583656\pi\)
−0.259797 + 0.965663i \(0.583656\pi\)
\(62\) 0 0
\(63\) 1.15358e9 0.146445
\(64\) 0 0
\(65\) −2.57688e9 −0.275468
\(66\) 0 0
\(67\) 1.70794e10 1.54547 0.772735 0.634729i \(-0.218888\pi\)
0.772735 + 0.634729i \(0.218888\pi\)
\(68\) 0 0
\(69\) 5.23929e9 0.403276
\(70\) 0 0
\(71\) 7.91508e9 0.520636 0.260318 0.965523i \(-0.416173\pi\)
0.260318 + 0.965523i \(0.416173\pi\)
\(72\) 0 0
\(73\) 3.15597e10 1.78179 0.890895 0.454209i \(-0.150078\pi\)
0.890895 + 0.454209i \(0.150078\pi\)
\(74\) 0 0
\(75\) −4.88132e8 −0.0237520
\(76\) 0 0
\(77\) −3.83195e9 −0.161332
\(78\) 0 0
\(79\) −4.10236e10 −1.49998 −0.749988 0.661451i \(-0.769941\pi\)
−0.749988 + 0.661451i \(0.769941\pi\)
\(80\) 0 0
\(81\) 3.48678e9 0.111111
\(82\) 0 0
\(83\) −1.99747e10 −0.556609 −0.278304 0.960493i \(-0.589772\pi\)
−0.278304 + 0.960493i \(0.589772\pi\)
\(84\) 0 0
\(85\) −9.33616e8 −0.0228226
\(86\) 0 0
\(87\) 4.65743e10 1.00182
\(88\) 0 0
\(89\) −1.06402e10 −0.201978 −0.100989 0.994888i \(-0.532201\pi\)
−0.100989 + 0.994888i \(0.532201\pi\)
\(90\) 0 0
\(91\) −7.06058e9 −0.118608
\(92\) 0 0
\(93\) 5.05279e10 0.753139
\(94\) 0 0
\(95\) 1.32024e11 1.75054
\(96\) 0 0
\(97\) 6.44111e9 0.0761581 0.0380790 0.999275i \(-0.487876\pi\)
0.0380790 + 0.999275i \(0.487876\pi\)
\(98\) 0 0
\(99\) −1.15823e10 −0.122406
\(100\) 0 0
\(101\) −1.06774e11 −1.01088 −0.505439 0.862863i \(-0.668669\pi\)
−0.505439 + 0.862863i \(0.668669\pi\)
\(102\) 0 0
\(103\) 2.79038e10 0.237169 0.118585 0.992944i \(-0.462164\pi\)
0.118585 + 0.992944i \(0.462164\pi\)
\(104\) 0 0
\(105\) −3.38479e10 −0.258816
\(106\) 0 0
\(107\) −1.49670e11 −1.03163 −0.515817 0.856699i \(-0.672511\pi\)
−0.515817 + 0.856699i \(0.672511\pi\)
\(108\) 0 0
\(109\) −2.05730e11 −1.28072 −0.640358 0.768077i \(-0.721214\pi\)
−0.640358 + 0.768077i \(0.721214\pi\)
\(110\) 0 0
\(111\) −4.87907e10 −0.274828
\(112\) 0 0
\(113\) 6.03547e8 0.00308163 0.00154081 0.999999i \(-0.499510\pi\)
0.00154081 + 0.999999i \(0.499510\pi\)
\(114\) 0 0
\(115\) −1.53729e11 −0.712718
\(116\) 0 0
\(117\) −2.13411e10 −0.0899902
\(118\) 0 0
\(119\) −2.55808e9 −0.00982667
\(120\) 0 0
\(121\) −2.46838e11 −0.865151
\(122\) 0 0
\(123\) 3.48767e11 1.11701
\(124\) 0 0
\(125\) −3.33822e11 −0.978385
\(126\) 0 0
\(127\) −6.94981e11 −1.86661 −0.933303 0.359090i \(-0.883087\pi\)
−0.933303 + 0.359090i \(0.883087\pi\)
\(128\) 0 0
\(129\) −1.73187e11 −0.426846
\(130\) 0 0
\(131\) 5.01509e11 1.13576 0.567880 0.823111i \(-0.307764\pi\)
0.567880 + 0.823111i \(0.307764\pi\)
\(132\) 0 0
\(133\) 3.61742e11 0.753729
\(134\) 0 0
\(135\) −1.02308e11 −0.196369
\(136\) 0 0
\(137\) 3.96827e11 0.702486 0.351243 0.936284i \(-0.385759\pi\)
0.351243 + 0.936284i \(0.385759\pi\)
\(138\) 0 0
\(139\) −5.87790e11 −0.960817 −0.480409 0.877045i \(-0.659512\pi\)
−0.480409 + 0.877045i \(0.659512\pi\)
\(140\) 0 0
\(141\) −1.20548e11 −0.182161
\(142\) 0 0
\(143\) 7.08906e10 0.0991381
\(144\) 0 0
\(145\) −1.36656e12 −1.77054
\(146\) 0 0
\(147\) 3.87748e11 0.465913
\(148\) 0 0
\(149\) −5.49939e11 −0.613466 −0.306733 0.951796i \(-0.599236\pi\)
−0.306733 + 0.951796i \(0.599236\pi\)
\(150\) 0 0
\(151\) −1.09467e10 −0.0113478 −0.00567390 0.999984i \(-0.501806\pi\)
−0.00567390 + 0.999984i \(0.501806\pi\)
\(152\) 0 0
\(153\) −7.73199e9 −0.00745570
\(154\) 0 0
\(155\) −1.48257e12 −1.33104
\(156\) 0 0
\(157\) 1.98667e12 1.66218 0.831088 0.556141i \(-0.187718\pi\)
0.831088 + 0.556141i \(0.187718\pi\)
\(158\) 0 0
\(159\) −8.14078e11 −0.635304
\(160\) 0 0
\(161\) −4.21213e11 −0.306874
\(162\) 0 0
\(163\) 5.08774e11 0.346332 0.173166 0.984893i \(-0.444600\pi\)
0.173166 + 0.984893i \(0.444600\pi\)
\(164\) 0 0
\(165\) 3.39844e11 0.216331
\(166\) 0 0
\(167\) 1.66583e12 0.992405 0.496203 0.868207i \(-0.334727\pi\)
0.496203 + 0.868207i \(0.334727\pi\)
\(168\) 0 0
\(169\) −1.66154e12 −0.927116
\(170\) 0 0
\(171\) 1.09339e12 0.571870
\(172\) 0 0
\(173\) −2.93849e12 −1.44169 −0.720843 0.693098i \(-0.756245\pi\)
−0.720843 + 0.693098i \(0.756245\pi\)
\(174\) 0 0
\(175\) 3.92434e10 0.0180741
\(176\) 0 0
\(177\) −1.11372e12 −0.481864
\(178\) 0 0
\(179\) 8.48882e11 0.345267 0.172634 0.984986i \(-0.444772\pi\)
0.172634 + 0.984986i \(0.444772\pi\)
\(180\) 0 0
\(181\) −5.03014e12 −1.92463 −0.962317 0.271932i \(-0.912338\pi\)
−0.962317 + 0.271932i \(0.912338\pi\)
\(182\) 0 0
\(183\) 8.32883e11 0.299987
\(184\) 0 0
\(185\) 1.43160e12 0.485709
\(186\) 0 0
\(187\) 2.56840e10 0.00821361
\(188\) 0 0
\(189\) −2.80320e11 −0.0845502
\(190\) 0 0
\(191\) 1.40796e12 0.400781 0.200390 0.979716i \(-0.435779\pi\)
0.200390 + 0.979716i \(0.435779\pi\)
\(192\) 0 0
\(193\) −1.94985e11 −0.0524127 −0.0262063 0.999657i \(-0.508343\pi\)
−0.0262063 + 0.999657i \(0.508343\pi\)
\(194\) 0 0
\(195\) 6.26182e11 0.159041
\(196\) 0 0
\(197\) −5.16140e12 −1.23938 −0.619688 0.784848i \(-0.712741\pi\)
−0.619688 + 0.784848i \(0.712741\pi\)
\(198\) 0 0
\(199\) 1.02078e12 0.231867 0.115934 0.993257i \(-0.463014\pi\)
0.115934 + 0.993257i \(0.463014\pi\)
\(200\) 0 0
\(201\) −4.15029e12 −0.892278
\(202\) 0 0
\(203\) −3.74434e12 −0.762338
\(204\) 0 0
\(205\) −1.02334e13 −1.97412
\(206\) 0 0
\(207\) −1.27315e12 −0.232832
\(208\) 0 0
\(209\) −3.63201e12 −0.630003
\(210\) 0 0
\(211\) 9.12166e12 1.50148 0.750742 0.660596i \(-0.229696\pi\)
0.750742 + 0.660596i \(0.229696\pi\)
\(212\) 0 0
\(213\) −1.92336e12 −0.300589
\(214\) 0 0
\(215\) 5.08157e12 0.754374
\(216\) 0 0
\(217\) −4.06219e12 −0.573102
\(218\) 0 0
\(219\) −7.66900e12 −1.02872
\(220\) 0 0
\(221\) 4.73243e10 0.00603846
\(222\) 0 0
\(223\) 4.68108e11 0.0568419 0.0284210 0.999596i \(-0.490952\pi\)
0.0284210 + 0.999596i \(0.490952\pi\)
\(224\) 0 0
\(225\) 1.18616e11 0.0137132
\(226\) 0 0
\(227\) −1.32196e13 −1.45571 −0.727857 0.685729i \(-0.759483\pi\)
−0.727857 + 0.685729i \(0.759483\pi\)
\(228\) 0 0
\(229\) 8.47551e11 0.0889346 0.0444673 0.999011i \(-0.485841\pi\)
0.0444673 + 0.999011i \(0.485841\pi\)
\(230\) 0 0
\(231\) 9.31163e11 0.0931451
\(232\) 0 0
\(233\) −1.77637e13 −1.69464 −0.847318 0.531086i \(-0.821784\pi\)
−0.847318 + 0.531086i \(0.821784\pi\)
\(234\) 0 0
\(235\) 3.53707e12 0.321936
\(236\) 0 0
\(237\) 9.96873e12 0.866012
\(238\) 0 0
\(239\) −1.10633e13 −0.917690 −0.458845 0.888516i \(-0.651737\pi\)
−0.458845 + 0.888516i \(0.651737\pi\)
\(240\) 0 0
\(241\) −1.31452e13 −1.04153 −0.520765 0.853700i \(-0.674353\pi\)
−0.520765 + 0.853700i \(0.674353\pi\)
\(242\) 0 0
\(243\) −8.47289e11 −0.0641500
\(244\) 0 0
\(245\) −1.13771e13 −0.823416
\(246\) 0 0
\(247\) −6.69219e12 −0.463164
\(248\) 0 0
\(249\) 4.85385e12 0.321358
\(250\) 0 0
\(251\) 2.49029e13 1.57777 0.788887 0.614539i \(-0.210658\pi\)
0.788887 + 0.614539i \(0.210658\pi\)
\(252\) 0 0
\(253\) 4.22912e12 0.256500
\(254\) 0 0
\(255\) 2.26869e11 0.0131766
\(256\) 0 0
\(257\) −1.90572e13 −1.06030 −0.530148 0.847905i \(-0.677864\pi\)
−0.530148 + 0.847905i \(0.677864\pi\)
\(258\) 0 0
\(259\) 3.92253e12 0.209131
\(260\) 0 0
\(261\) −1.13176e13 −0.578402
\(262\) 0 0
\(263\) −2.58113e13 −1.26489 −0.632447 0.774604i \(-0.717949\pi\)
−0.632447 + 0.774604i \(0.717949\pi\)
\(264\) 0 0
\(265\) 2.38863e13 1.12279
\(266\) 0 0
\(267\) 2.58556e12 0.116612
\(268\) 0 0
\(269\) 1.60114e13 0.693093 0.346547 0.938033i \(-0.387354\pi\)
0.346547 + 0.938033i \(0.387354\pi\)
\(270\) 0 0
\(271\) 2.14306e13 0.890644 0.445322 0.895371i \(-0.353089\pi\)
0.445322 + 0.895371i \(0.353089\pi\)
\(272\) 0 0
\(273\) 1.71572e12 0.0684782
\(274\) 0 0
\(275\) −3.94017e11 −0.0151072
\(276\) 0 0
\(277\) −2.25644e13 −0.831352 −0.415676 0.909513i \(-0.636455\pi\)
−0.415676 + 0.909513i \(0.636455\pi\)
\(278\) 0 0
\(279\) −1.22783e13 −0.434825
\(280\) 0 0
\(281\) 1.45643e13 0.495912 0.247956 0.968771i \(-0.420241\pi\)
0.247956 + 0.968771i \(0.420241\pi\)
\(282\) 0 0
\(283\) −2.43495e13 −0.797380 −0.398690 0.917086i \(-0.630535\pi\)
−0.398690 + 0.917086i \(0.630535\pi\)
\(284\) 0 0
\(285\) −3.20818e13 −1.01068
\(286\) 0 0
\(287\) −2.80392e13 −0.849992
\(288\) 0 0
\(289\) −3.42548e13 −0.999500
\(290\) 0 0
\(291\) −1.56519e12 −0.0439699
\(292\) 0 0
\(293\) 4.32740e11 0.0117073 0.00585363 0.999983i \(-0.498137\pi\)
0.00585363 + 0.999983i \(0.498137\pi\)
\(294\) 0 0
\(295\) 3.26784e13 0.851607
\(296\) 0 0
\(297\) 2.81451e12 0.0706712
\(298\) 0 0
\(299\) 7.79240e12 0.188573
\(300\) 0 0
\(301\) 1.39234e13 0.324809
\(302\) 0 0
\(303\) 2.59461e13 0.583630
\(304\) 0 0
\(305\) −2.44381e13 −0.530174
\(306\) 0 0
\(307\) −5.89356e13 −1.23344 −0.616718 0.787185i \(-0.711538\pi\)
−0.616718 + 0.787185i \(0.711538\pi\)
\(308\) 0 0
\(309\) −6.78062e12 −0.136930
\(310\) 0 0
\(311\) −1.29359e13 −0.252125 −0.126062 0.992022i \(-0.540234\pi\)
−0.126062 + 0.992022i \(0.540234\pi\)
\(312\) 0 0
\(313\) −2.56949e13 −0.483452 −0.241726 0.970345i \(-0.577714\pi\)
−0.241726 + 0.970345i \(0.577714\pi\)
\(314\) 0 0
\(315\) 8.22503e12 0.149427
\(316\) 0 0
\(317\) −1.33665e13 −0.234527 −0.117263 0.993101i \(-0.537412\pi\)
−0.117263 + 0.993101i \(0.537412\pi\)
\(318\) 0 0
\(319\) 3.75945e13 0.637199
\(320\) 0 0
\(321\) 3.63699e13 0.595614
\(322\) 0 0
\(323\) −2.42461e12 −0.0383732
\(324\) 0 0
\(325\) −7.25999e11 −0.0111065
\(326\) 0 0
\(327\) 4.99925e13 0.739422
\(328\) 0 0
\(329\) 9.69147e12 0.138616
\(330\) 0 0
\(331\) −6.25245e12 −0.0864960 −0.0432480 0.999064i \(-0.513771\pi\)
−0.0432480 + 0.999064i \(0.513771\pi\)
\(332\) 0 0
\(333\) 1.18561e13 0.158672
\(334\) 0 0
\(335\) 1.21776e14 1.57694
\(336\) 0 0
\(337\) 6.09170e13 0.763439 0.381720 0.924278i \(-0.375332\pi\)
0.381720 + 0.924278i \(0.375332\pi\)
\(338\) 0 0
\(339\) −1.46662e11 −0.00177918
\(340\) 0 0
\(341\) 4.07858e13 0.479027
\(342\) 0 0
\(343\) −6.98021e13 −0.793873
\(344\) 0 0
\(345\) 3.73562e13 0.411488
\(346\) 0 0
\(347\) 3.85685e13 0.411548 0.205774 0.978600i \(-0.434029\pi\)
0.205774 + 0.978600i \(0.434029\pi\)
\(348\) 0 0
\(349\) −3.48853e13 −0.360664 −0.180332 0.983606i \(-0.557717\pi\)
−0.180332 + 0.983606i \(0.557717\pi\)
\(350\) 0 0
\(351\) 5.18590e12 0.0519559
\(352\) 0 0
\(353\) 9.67569e13 0.939553 0.469777 0.882785i \(-0.344335\pi\)
0.469777 + 0.882785i \(0.344335\pi\)
\(354\) 0 0
\(355\) 5.64345e13 0.531238
\(356\) 0 0
\(357\) 6.21614e11 0.00567343
\(358\) 0 0
\(359\) −1.64650e14 −1.45728 −0.728638 0.684899i \(-0.759846\pi\)
−0.728638 + 0.684899i \(0.759846\pi\)
\(360\) 0 0
\(361\) 2.26378e14 1.94332
\(362\) 0 0
\(363\) 5.99815e13 0.499495
\(364\) 0 0
\(365\) 2.25020e14 1.81807
\(366\) 0 0
\(367\) −1.19183e14 −0.934435 −0.467218 0.884142i \(-0.654744\pi\)
−0.467218 + 0.884142i \(0.654744\pi\)
\(368\) 0 0
\(369\) −8.47505e13 −0.644907
\(370\) 0 0
\(371\) 6.54478e13 0.483436
\(372\) 0 0
\(373\) 1.61183e14 1.15590 0.577949 0.816073i \(-0.303853\pi\)
0.577949 + 0.816073i \(0.303853\pi\)
\(374\) 0 0
\(375\) 8.11187e13 0.564871
\(376\) 0 0
\(377\) 6.92700e13 0.468455
\(378\) 0 0
\(379\) 1.33388e14 0.876195 0.438098 0.898927i \(-0.355652\pi\)
0.438098 + 0.898927i \(0.355652\pi\)
\(380\) 0 0
\(381\) 1.68880e14 1.07769
\(382\) 0 0
\(383\) 6.38158e13 0.395671 0.197836 0.980235i \(-0.436609\pi\)
0.197836 + 0.980235i \(0.436609\pi\)
\(384\) 0 0
\(385\) −2.73218e13 −0.164617
\(386\) 0 0
\(387\) 4.20844e13 0.246440
\(388\) 0 0
\(389\) 3.85909e13 0.219665 0.109833 0.993950i \(-0.464968\pi\)
0.109833 + 0.993950i \(0.464968\pi\)
\(390\) 0 0
\(391\) 2.82322e12 0.0156233
\(392\) 0 0
\(393\) −1.21867e14 −0.655731
\(394\) 0 0
\(395\) −2.92498e14 −1.53052
\(396\) 0 0
\(397\) 2.64317e13 0.134517 0.0672585 0.997736i \(-0.478575\pi\)
0.0672585 + 0.997736i \(0.478575\pi\)
\(398\) 0 0
\(399\) −8.79033e13 −0.435165
\(400\) 0 0
\(401\) 4.59354e13 0.221235 0.110617 0.993863i \(-0.464717\pi\)
0.110617 + 0.993863i \(0.464717\pi\)
\(402\) 0 0
\(403\) 7.51502e13 0.352170
\(404\) 0 0
\(405\) 2.48608e13 0.113374
\(406\) 0 0
\(407\) −3.93836e13 −0.174802
\(408\) 0 0
\(409\) 4.23890e13 0.183136 0.0915682 0.995799i \(-0.470812\pi\)
0.0915682 + 0.995799i \(0.470812\pi\)
\(410\) 0 0
\(411\) −9.64289e13 −0.405581
\(412\) 0 0
\(413\) 8.95378e13 0.366675
\(414\) 0 0
\(415\) −1.42419e14 −0.567943
\(416\) 0 0
\(417\) 1.42833e14 0.554728
\(418\) 0 0
\(419\) −4.41929e14 −1.67177 −0.835883 0.548907i \(-0.815044\pi\)
−0.835883 + 0.548907i \(0.815044\pi\)
\(420\) 0 0
\(421\) 3.10369e14 1.14374 0.571870 0.820344i \(-0.306218\pi\)
0.571870 + 0.820344i \(0.306218\pi\)
\(422\) 0 0
\(423\) 2.92932e13 0.105171
\(424\) 0 0
\(425\) −2.63033e11 −0.000920177 0
\(426\) 0 0
\(427\) −6.69597e13 −0.228276
\(428\) 0 0
\(429\) −1.72264e13 −0.0572374
\(430\) 0 0
\(431\) −3.06558e13 −0.0992860 −0.0496430 0.998767i \(-0.515808\pi\)
−0.0496430 + 0.998767i \(0.515808\pi\)
\(432\) 0 0
\(433\) −2.06583e14 −0.652245 −0.326122 0.945328i \(-0.605742\pi\)
−0.326122 + 0.945328i \(0.605742\pi\)
\(434\) 0 0
\(435\) 3.32075e14 1.02222
\(436\) 0 0
\(437\) −3.99236e14 −1.19834
\(438\) 0 0
\(439\) 1.97201e14 0.577238 0.288619 0.957444i \(-0.406804\pi\)
0.288619 + 0.957444i \(0.406804\pi\)
\(440\) 0 0
\(441\) −9.42228e13 −0.268995
\(442\) 0 0
\(443\) −1.98874e14 −0.553805 −0.276903 0.960898i \(-0.589308\pi\)
−0.276903 + 0.960898i \(0.589308\pi\)
\(444\) 0 0
\(445\) −7.58644e13 −0.206090
\(446\) 0 0
\(447\) 1.33635e14 0.354185
\(448\) 0 0
\(449\) 6.87389e14 1.77766 0.888828 0.458240i \(-0.151520\pi\)
0.888828 + 0.458240i \(0.151520\pi\)
\(450\) 0 0
\(451\) 2.81523e14 0.710465
\(452\) 0 0
\(453\) 2.66006e12 0.00655166
\(454\) 0 0
\(455\) −5.03420e13 −0.121023
\(456\) 0 0
\(457\) 3.89942e14 0.915085 0.457543 0.889188i \(-0.348730\pi\)
0.457543 + 0.889188i \(0.348730\pi\)
\(458\) 0 0
\(459\) 1.87887e12 0.00430455
\(460\) 0 0
\(461\) −7.08640e14 −1.58515 −0.792576 0.609774i \(-0.791260\pi\)
−0.792576 + 0.609774i \(0.791260\pi\)
\(462\) 0 0
\(463\) 6.00543e14 1.31174 0.655871 0.754873i \(-0.272301\pi\)
0.655871 + 0.754873i \(0.272301\pi\)
\(464\) 0 0
\(465\) 3.60264e14 0.768475
\(466\) 0 0
\(467\) −2.95277e13 −0.0615159 −0.0307579 0.999527i \(-0.509792\pi\)
−0.0307579 + 0.999527i \(0.509792\pi\)
\(468\) 0 0
\(469\) 3.33663e14 0.678980
\(470\) 0 0
\(471\) −4.82760e14 −0.959658
\(472\) 0 0
\(473\) −1.39795e14 −0.271492
\(474\) 0 0
\(475\) 3.71959e13 0.0705797
\(476\) 0 0
\(477\) 1.97821e14 0.366793
\(478\) 0 0
\(479\) −1.00961e15 −1.82940 −0.914702 0.404128i \(-0.867575\pi\)
−0.914702 + 0.404128i \(0.867575\pi\)
\(480\) 0 0
\(481\) −7.25665e13 −0.128510
\(482\) 0 0
\(483\) 1.02355e14 0.177174
\(484\) 0 0
\(485\) 4.59251e13 0.0777088
\(486\) 0 0
\(487\) 8.21837e14 1.35949 0.679745 0.733448i \(-0.262090\pi\)
0.679745 + 0.733448i \(0.262090\pi\)
\(488\) 0 0
\(489\) −1.23632e14 −0.199955
\(490\) 0 0
\(491\) −8.64714e14 −1.36749 −0.683745 0.729721i \(-0.739650\pi\)
−0.683745 + 0.729721i \(0.739650\pi\)
\(492\) 0 0
\(493\) 2.50968e13 0.0388115
\(494\) 0 0
\(495\) −8.25821e13 −0.124899
\(496\) 0 0
\(497\) 1.54629e14 0.228734
\(498\) 0 0
\(499\) 6.07789e14 0.879427 0.439714 0.898138i \(-0.355080\pi\)
0.439714 + 0.898138i \(0.355080\pi\)
\(500\) 0 0
\(501\) −4.04796e14 −0.572965
\(502\) 0 0
\(503\) 4.90522e14 0.679257 0.339629 0.940560i \(-0.389699\pi\)
0.339629 + 0.940560i \(0.389699\pi\)
\(504\) 0 0
\(505\) −7.61299e14 −1.03146
\(506\) 0 0
\(507\) 4.03754e14 0.535271
\(508\) 0 0
\(509\) 1.07479e15 1.39436 0.697179 0.716897i \(-0.254438\pi\)
0.697179 + 0.716897i \(0.254438\pi\)
\(510\) 0 0
\(511\) 6.16549e14 0.782804
\(512\) 0 0
\(513\) −2.65694e14 −0.330169
\(514\) 0 0
\(515\) 1.98954e14 0.241999
\(516\) 0 0
\(517\) −9.73056e13 −0.115862
\(518\) 0 0
\(519\) 7.14053e14 0.832358
\(520\) 0 0
\(521\) −8.25510e13 −0.0942139 −0.0471069 0.998890i \(-0.515000\pi\)
−0.0471069 + 0.998890i \(0.515000\pi\)
\(522\) 0 0
\(523\) −6.87128e14 −0.767854 −0.383927 0.923363i \(-0.625429\pi\)
−0.383927 + 0.923363i \(0.625429\pi\)
\(524\) 0 0
\(525\) −9.53615e12 −0.0104351
\(526\) 0 0
\(527\) 2.72273e13 0.0291773
\(528\) 0 0
\(529\) −4.87939e14 −0.512105
\(530\) 0 0
\(531\) 2.70635e14 0.278204
\(532\) 0 0
\(533\) 5.18722e14 0.522318
\(534\) 0 0
\(535\) −1.06715e15 −1.05264
\(536\) 0 0
\(537\) −2.06278e14 −0.199340
\(538\) 0 0
\(539\) 3.12988e14 0.296339
\(540\) 0 0
\(541\) −1.39934e15 −1.29819 −0.649093 0.760709i \(-0.724851\pi\)
−0.649093 + 0.760709i \(0.724851\pi\)
\(542\) 0 0
\(543\) 1.22232e15 1.11119
\(544\) 0 0
\(545\) −1.46686e15 −1.30679
\(546\) 0 0
\(547\) 1.41257e15 1.23333 0.616667 0.787224i \(-0.288483\pi\)
0.616667 + 0.787224i \(0.288483\pi\)
\(548\) 0 0
\(549\) −2.02391e14 −0.173198
\(550\) 0 0
\(551\) −3.54898e15 −2.97694
\(552\) 0 0
\(553\) −8.01437e14 −0.658993
\(554\) 0 0
\(555\) −3.47878e14 −0.280424
\(556\) 0 0
\(557\) −1.02686e15 −0.811539 −0.405769 0.913975i \(-0.632996\pi\)
−0.405769 + 0.913975i \(0.632996\pi\)
\(558\) 0 0
\(559\) −2.57581e14 −0.199595
\(560\) 0 0
\(561\) −6.24121e12 −0.00474213
\(562\) 0 0
\(563\) −1.08846e15 −0.810992 −0.405496 0.914097i \(-0.632901\pi\)
−0.405496 + 0.914097i \(0.632901\pi\)
\(564\) 0 0
\(565\) 4.30329e12 0.00314438
\(566\) 0 0
\(567\) 6.81178e13 0.0488151
\(568\) 0 0
\(569\) −1.95900e15 −1.37695 −0.688474 0.725261i \(-0.741719\pi\)
−0.688474 + 0.725261i \(0.741719\pi\)
\(570\) 0 0
\(571\) 1.58327e15 1.09158 0.545791 0.837921i \(-0.316229\pi\)
0.545791 + 0.837921i \(0.316229\pi\)
\(572\) 0 0
\(573\) −3.42134e14 −0.231391
\(574\) 0 0
\(575\) −4.33109e13 −0.0287359
\(576\) 0 0
\(577\) 2.69378e14 0.175346 0.0876728 0.996149i \(-0.472057\pi\)
0.0876728 + 0.996149i \(0.472057\pi\)
\(578\) 0 0
\(579\) 4.73814e13 0.0302605
\(580\) 0 0
\(581\) −3.90225e14 −0.244538
\(582\) 0 0
\(583\) −6.57118e14 −0.404079
\(584\) 0 0
\(585\) −1.52162e14 −0.0918226
\(586\) 0 0
\(587\) −7.20364e14 −0.426621 −0.213311 0.976984i \(-0.568425\pi\)
−0.213311 + 0.976984i \(0.568425\pi\)
\(588\) 0 0
\(589\) −3.85025e15 −2.23797
\(590\) 0 0
\(591\) 1.25422e15 0.715554
\(592\) 0 0
\(593\) −1.27192e15 −0.712294 −0.356147 0.934430i \(-0.615910\pi\)
−0.356147 + 0.934430i \(0.615910\pi\)
\(594\) 0 0
\(595\) −1.82391e13 −0.0100268
\(596\) 0 0
\(597\) −2.48049e14 −0.133869
\(598\) 0 0
\(599\) −4.83469e14 −0.256166 −0.128083 0.991763i \(-0.540882\pi\)
−0.128083 + 0.991763i \(0.540882\pi\)
\(600\) 0 0
\(601\) 3.68175e15 1.91533 0.957667 0.287880i \(-0.0929503\pi\)
0.957667 + 0.287880i \(0.0929503\pi\)
\(602\) 0 0
\(603\) 1.00852e15 0.515157
\(604\) 0 0
\(605\) −1.75995e15 −0.882768
\(606\) 0 0
\(607\) 2.30279e15 1.13427 0.567135 0.823625i \(-0.308052\pi\)
0.567135 + 0.823625i \(0.308052\pi\)
\(608\) 0 0
\(609\) 9.09875e14 0.440136
\(610\) 0 0
\(611\) −1.79291e14 −0.0851789
\(612\) 0 0
\(613\) −1.23428e14 −0.0575947 −0.0287973 0.999585i \(-0.509168\pi\)
−0.0287973 + 0.999585i \(0.509168\pi\)
\(614\) 0 0
\(615\) 2.48671e15 1.13976
\(616\) 0 0
\(617\) −2.94167e15 −1.32442 −0.662210 0.749318i \(-0.730381\pi\)
−0.662210 + 0.749318i \(0.730381\pi\)
\(618\) 0 0
\(619\) −3.62405e15 −1.60286 −0.801430 0.598088i \(-0.795927\pi\)
−0.801430 + 0.598088i \(0.795927\pi\)
\(620\) 0 0
\(621\) 3.09375e14 0.134425
\(622\) 0 0
\(623\) −2.07866e14 −0.0887360
\(624\) 0 0
\(625\) −2.47824e15 −1.03945
\(626\) 0 0
\(627\) 8.82579e14 0.363732
\(628\) 0 0
\(629\) −2.62912e13 −0.0106471
\(630\) 0 0
\(631\) −1.12294e15 −0.446884 −0.223442 0.974717i \(-0.571729\pi\)
−0.223442 + 0.974717i \(0.571729\pi\)
\(632\) 0 0
\(633\) −2.21656e15 −0.866882
\(634\) 0 0
\(635\) −4.95522e15 −1.90461
\(636\) 0 0
\(637\) 5.76698e14 0.217862
\(638\) 0 0
\(639\) 4.67377e14 0.173545
\(640\) 0 0
\(641\) 3.60734e15 1.31664 0.658321 0.752738i \(-0.271267\pi\)
0.658321 + 0.752738i \(0.271267\pi\)
\(642\) 0 0
\(643\) −4.13436e15 −1.48337 −0.741683 0.670751i \(-0.765972\pi\)
−0.741683 + 0.670751i \(0.765972\pi\)
\(644\) 0 0
\(645\) −1.23482e15 −0.435538
\(646\) 0 0
\(647\) −5.00765e15 −1.73644 −0.868221 0.496177i \(-0.834737\pi\)
−0.868221 + 0.496177i \(0.834737\pi\)
\(648\) 0 0
\(649\) −8.98990e14 −0.306485
\(650\) 0 0
\(651\) 9.87113e14 0.330881
\(652\) 0 0
\(653\) 5.80675e15 1.91386 0.956931 0.290314i \(-0.0937598\pi\)
0.956931 + 0.290314i \(0.0937598\pi\)
\(654\) 0 0
\(655\) 3.57576e15 1.15889
\(656\) 0 0
\(657\) 1.86357e15 0.593930
\(658\) 0 0
\(659\) 3.09336e15 0.969529 0.484765 0.874645i \(-0.338905\pi\)
0.484765 + 0.874645i \(0.338905\pi\)
\(660\) 0 0
\(661\) 3.99504e14 0.123144 0.0615719 0.998103i \(-0.480389\pi\)
0.0615719 + 0.998103i \(0.480389\pi\)
\(662\) 0 0
\(663\) −1.14998e13 −0.00348631
\(664\) 0 0
\(665\) 2.57922e15 0.769076
\(666\) 0 0
\(667\) 4.13244e15 1.21203
\(668\) 0 0
\(669\) −1.13750e14 −0.0328177
\(670\) 0 0
\(671\) 6.72298e14 0.190804
\(672\) 0 0
\(673\) −2.16111e15 −0.603384 −0.301692 0.953405i \(-0.597551\pi\)
−0.301692 + 0.953405i \(0.597551\pi\)
\(674\) 0 0
\(675\) −2.88237e13 −0.00791734
\(676\) 0 0
\(677\) −4.72622e15 −1.27725 −0.638626 0.769518i \(-0.720497\pi\)
−0.638626 + 0.769518i \(0.720497\pi\)
\(678\) 0 0
\(679\) 1.25833e14 0.0334590
\(680\) 0 0
\(681\) 3.21236e15 0.840456
\(682\) 0 0
\(683\) −4.37911e15 −1.12738 −0.563692 0.825985i \(-0.690620\pi\)
−0.563692 + 0.825985i \(0.690620\pi\)
\(684\) 0 0
\(685\) 2.82938e15 0.716791
\(686\) 0 0
\(687\) −2.05955e14 −0.0513464
\(688\) 0 0
\(689\) −1.21078e15 −0.297070
\(690\) 0 0
\(691\) 7.37871e15 1.78177 0.890884 0.454230i \(-0.150086\pi\)
0.890884 + 0.454230i \(0.150086\pi\)
\(692\) 0 0
\(693\) −2.26273e14 −0.0537773
\(694\) 0 0
\(695\) −4.19094e15 −0.980382
\(696\) 0 0
\(697\) 1.87935e14 0.0432741
\(698\) 0 0
\(699\) 4.31659e15 0.978399
\(700\) 0 0
\(701\) −2.97202e15 −0.663136 −0.331568 0.943431i \(-0.607578\pi\)
−0.331568 + 0.943431i \(0.607578\pi\)
\(702\) 0 0
\(703\) 3.71787e15 0.816658
\(704\) 0 0
\(705\) −8.59507e14 −0.185870
\(706\) 0 0
\(707\) −2.08594e15 −0.444114
\(708\) 0 0
\(709\) 3.84420e15 0.805845 0.402923 0.915234i \(-0.367994\pi\)
0.402923 + 0.915234i \(0.367994\pi\)
\(710\) 0 0
\(711\) −2.42240e15 −0.499992
\(712\) 0 0
\(713\) 4.48323e15 0.911169
\(714\) 0 0
\(715\) 5.05450e14 0.101157
\(716\) 0 0
\(717\) 2.68838e15 0.529829
\(718\) 0 0
\(719\) −3.49014e15 −0.677383 −0.338691 0.940898i \(-0.609984\pi\)
−0.338691 + 0.940898i \(0.609984\pi\)
\(720\) 0 0
\(721\) 5.45128e14 0.104197
\(722\) 0 0
\(723\) 3.19427e15 0.601328
\(724\) 0 0
\(725\) −3.85009e14 −0.0713859
\(726\) 0 0
\(727\) −1.26477e15 −0.230978 −0.115489 0.993309i \(-0.536844\pi\)
−0.115489 + 0.993309i \(0.536844\pi\)
\(728\) 0 0
\(729\) 2.05891e14 0.0370370
\(730\) 0 0
\(731\) −9.33228e13 −0.0165364
\(732\) 0 0
\(733\) 7.50679e15 1.31033 0.655167 0.755484i \(-0.272598\pi\)
0.655167 + 0.755484i \(0.272598\pi\)
\(734\) 0 0
\(735\) 2.76464e15 0.475400
\(736\) 0 0
\(737\) −3.35009e15 −0.567525
\(738\) 0 0
\(739\) 7.20650e15 1.20276 0.601381 0.798962i \(-0.294617\pi\)
0.601381 + 0.798962i \(0.294617\pi\)
\(740\) 0 0
\(741\) 1.62620e15 0.267408
\(742\) 0 0
\(743\) 1.39426e15 0.225894 0.112947 0.993601i \(-0.463971\pi\)
0.112947 + 0.993601i \(0.463971\pi\)
\(744\) 0 0
\(745\) −3.92107e15 −0.625958
\(746\) 0 0
\(747\) −1.17948e15 −0.185536
\(748\) 0 0
\(749\) −2.92396e15 −0.453233
\(750\) 0 0
\(751\) 7.66385e15 1.17065 0.585326 0.810798i \(-0.300967\pi\)
0.585326 + 0.810798i \(0.300967\pi\)
\(752\) 0 0
\(753\) −6.05141e15 −0.910928
\(754\) 0 0
\(755\) −7.80503e13 −0.0115789
\(756\) 0 0
\(757\) −2.12920e15 −0.311307 −0.155654 0.987812i \(-0.549748\pi\)
−0.155654 + 0.987812i \(0.549748\pi\)
\(758\) 0 0
\(759\) −1.02768e15 −0.148090
\(760\) 0 0
\(761\) 2.44003e14 0.0346561 0.0173280 0.999850i \(-0.494484\pi\)
0.0173280 + 0.999850i \(0.494484\pi\)
\(762\) 0 0
\(763\) −4.01915e15 −0.562664
\(764\) 0 0
\(765\) −5.51291e13 −0.00760752
\(766\) 0 0
\(767\) −1.65644e15 −0.225321
\(768\) 0 0
\(769\) 7.87277e15 1.05568 0.527840 0.849344i \(-0.323002\pi\)
0.527840 + 0.849344i \(0.323002\pi\)
\(770\) 0 0
\(771\) 4.63090e15 0.612162
\(772\) 0 0
\(773\) 1.12137e16 1.46138 0.730691 0.682709i \(-0.239198\pi\)
0.730691 + 0.682709i \(0.239198\pi\)
\(774\) 0 0
\(775\) −4.17692e14 −0.0536657
\(776\) 0 0
\(777\) −9.53176e14 −0.120742
\(778\) 0 0
\(779\) −2.65762e16 −3.31923
\(780\) 0 0
\(781\) −1.55253e15 −0.191187
\(782\) 0 0
\(783\) 2.75017e15 0.333940
\(784\) 0 0
\(785\) 1.41649e16 1.69602
\(786\) 0 0
\(787\) 1.34408e16 1.58695 0.793475 0.608603i \(-0.208270\pi\)
0.793475 + 0.608603i \(0.208270\pi\)
\(788\) 0 0
\(789\) 6.27215e15 0.730286
\(790\) 0 0
\(791\) 1.17909e13 0.00135387
\(792\) 0 0
\(793\) 1.23875e15 0.140275
\(794\) 0 0
\(795\) −5.80437e15 −0.648241
\(796\) 0 0
\(797\) 1.64398e16 1.81082 0.905408 0.424542i \(-0.139565\pi\)
0.905408 + 0.424542i \(0.139565\pi\)
\(798\) 0 0
\(799\) −6.49580e13 −0.00705709
\(800\) 0 0
\(801\) −6.28291e14 −0.0673259
\(802\) 0 0
\(803\) −6.19036e15 −0.654306
\(804\) 0 0
\(805\) −3.00325e15 −0.313122
\(806\) 0 0
\(807\) −3.89077e15 −0.400158
\(808\) 0 0
\(809\) 1.37074e16 1.39072 0.695358 0.718664i \(-0.255246\pi\)
0.695358 + 0.718664i \(0.255246\pi\)
\(810\) 0 0
\(811\) −4.58825e15 −0.459232 −0.229616 0.973281i \(-0.573747\pi\)
−0.229616 + 0.973281i \(0.573747\pi\)
\(812\) 0 0
\(813\) −5.20764e15 −0.514213
\(814\) 0 0
\(815\) 3.62756e15 0.353385
\(816\) 0 0
\(817\) 1.31969e16 1.26838
\(818\) 0 0
\(819\) −4.16920e14 −0.0395359
\(820\) 0 0
\(821\) −1.03921e16 −0.972331 −0.486165 0.873867i \(-0.661605\pi\)
−0.486165 + 0.873867i \(0.661605\pi\)
\(822\) 0 0
\(823\) −5.90788e15 −0.545422 −0.272711 0.962096i \(-0.587920\pi\)
−0.272711 + 0.962096i \(0.587920\pi\)
\(824\) 0 0
\(825\) 9.57462e13 0.00872217
\(826\) 0 0
\(827\) 6.40538e15 0.575791 0.287895 0.957662i \(-0.407044\pi\)
0.287895 + 0.957662i \(0.407044\pi\)
\(828\) 0 0
\(829\) 6.31696e15 0.560348 0.280174 0.959949i \(-0.409608\pi\)
0.280174 + 0.959949i \(0.409608\pi\)
\(830\) 0 0
\(831\) 5.48315e15 0.479981
\(832\) 0 0
\(833\) 2.08940e14 0.0180499
\(834\) 0 0
\(835\) 1.18773e16 1.01261
\(836\) 0 0
\(837\) 2.98362e15 0.251046
\(838\) 0 0
\(839\) −1.09652e16 −0.910598 −0.455299 0.890339i \(-0.650468\pi\)
−0.455299 + 0.890339i \(0.650468\pi\)
\(840\) 0 0
\(841\) 2.45345e16 2.01094
\(842\) 0 0
\(843\) −3.53912e15 −0.286315
\(844\) 0 0
\(845\) −1.18468e16 −0.945994
\(846\) 0 0
\(847\) −4.82222e15 −0.380092
\(848\) 0 0
\(849\) 5.91694e15 0.460368
\(850\) 0 0
\(851\) −4.32910e15 −0.332495
\(852\) 0 0
\(853\) −3.39896e15 −0.257707 −0.128854 0.991664i \(-0.541130\pi\)
−0.128854 + 0.991664i \(0.541130\pi\)
\(854\) 0 0
\(855\) 7.79589e15 0.583515
\(856\) 0 0
\(857\) 1.05370e16 0.778614 0.389307 0.921108i \(-0.372715\pi\)
0.389307 + 0.921108i \(0.372715\pi\)
\(858\) 0 0
\(859\) −1.29250e16 −0.942903 −0.471451 0.881892i \(-0.656270\pi\)
−0.471451 + 0.881892i \(0.656270\pi\)
\(860\) 0 0
\(861\) 6.81352e15 0.490743
\(862\) 0 0
\(863\) 1.45598e16 1.03537 0.517685 0.855571i \(-0.326794\pi\)
0.517685 + 0.855571i \(0.326794\pi\)
\(864\) 0 0
\(865\) −2.09514e16 −1.47104
\(866\) 0 0
\(867\) 8.32390e15 0.577061
\(868\) 0 0
\(869\) 8.04669e15 0.550819
\(870\) 0 0
\(871\) −6.17273e15 −0.417232
\(872\) 0 0
\(873\) 3.80341e14 0.0253860
\(874\) 0 0
\(875\) −6.52155e15 −0.429839
\(876\) 0 0
\(877\) 6.57548e15 0.427986 0.213993 0.976835i \(-0.431353\pi\)
0.213993 + 0.976835i \(0.431353\pi\)
\(878\) 0 0
\(879\) −1.05156e14 −0.00675919
\(880\) 0 0
\(881\) −2.17457e16 −1.38040 −0.690202 0.723617i \(-0.742478\pi\)
−0.690202 + 0.723617i \(0.742478\pi\)
\(882\) 0 0
\(883\) 1.89298e15 0.118676 0.0593378 0.998238i \(-0.481101\pi\)
0.0593378 + 0.998238i \(0.481101\pi\)
\(884\) 0 0
\(885\) −7.94085e15 −0.491676
\(886\) 0 0
\(887\) −2.03579e16 −1.24495 −0.622475 0.782639i \(-0.713873\pi\)
−0.622475 + 0.782639i \(0.713873\pi\)
\(888\) 0 0
\(889\) −1.35772e16 −0.820066
\(890\) 0 0
\(891\) −6.83926e14 −0.0408020
\(892\) 0 0
\(893\) 9.18581e15 0.541295
\(894\) 0 0
\(895\) 6.05253e15 0.352298
\(896\) 0 0
\(897\) −1.89355e15 −0.108873
\(898\) 0 0
\(899\) 3.98534e16 2.26353
\(900\) 0 0
\(901\) −4.38671e14 −0.0246123
\(902\) 0 0
\(903\) −3.38338e15 −0.187529
\(904\) 0 0
\(905\) −3.58649e16 −1.96382
\(906\) 0 0
\(907\) −2.08169e16 −1.12610 −0.563049 0.826424i \(-0.690372\pi\)
−0.563049 + 0.826424i \(0.690372\pi\)
\(908\) 0 0
\(909\) −6.30490e15 −0.336959
\(910\) 0 0
\(911\) −1.56740e15 −0.0827616 −0.0413808 0.999143i \(-0.513176\pi\)
−0.0413808 + 0.999143i \(0.513176\pi\)
\(912\) 0 0
\(913\) 3.91799e15 0.204397
\(914\) 0 0
\(915\) 5.93846e15 0.306096
\(916\) 0 0
\(917\) 9.79748e15 0.498980
\(918\) 0 0
\(919\) 1.45715e15 0.0733278 0.0366639 0.999328i \(-0.488327\pi\)
0.0366639 + 0.999328i \(0.488327\pi\)
\(920\) 0 0
\(921\) 1.43213e16 0.712124
\(922\) 0 0
\(923\) −2.86062e15 −0.140556
\(924\) 0 0
\(925\) 4.03332e14 0.0195832
\(926\) 0 0
\(927\) 1.64769e15 0.0790564
\(928\) 0 0
\(929\) −1.70074e16 −0.806400 −0.403200 0.915112i \(-0.632102\pi\)
−0.403200 + 0.915112i \(0.632102\pi\)
\(930\) 0 0
\(931\) −2.95466e16 −1.38447
\(932\) 0 0
\(933\) 3.14343e15 0.145564
\(934\) 0 0
\(935\) 1.83127e14 0.00838086
\(936\) 0 0
\(937\) −3.71528e16 −1.68044 −0.840222 0.542242i \(-0.817576\pi\)
−0.840222 + 0.542242i \(0.817576\pi\)
\(938\) 0 0
\(939\) 6.24387e15 0.279121
\(940\) 0 0
\(941\) −2.09136e16 −0.924031 −0.462016 0.886872i \(-0.652874\pi\)
−0.462016 + 0.886872i \(0.652874\pi\)
\(942\) 0 0
\(943\) 3.09454e16 1.35139
\(944\) 0 0
\(945\) −1.99868e15 −0.0862718
\(946\) 0 0
\(947\) 3.53699e16 1.50907 0.754533 0.656262i \(-0.227863\pi\)
0.754533 + 0.656262i \(0.227863\pi\)
\(948\) 0 0
\(949\) −1.14061e16 −0.481031
\(950\) 0 0
\(951\) 3.24807e15 0.135404
\(952\) 0 0
\(953\) 1.45761e16 0.600661 0.300331 0.953835i \(-0.402903\pi\)
0.300331 + 0.953835i \(0.402903\pi\)
\(954\) 0 0
\(955\) 1.00388e16 0.408941
\(956\) 0 0
\(957\) −9.13545e15 −0.367887
\(958\) 0 0
\(959\) 7.75241e15 0.308627
\(960\) 0 0
\(961\) 1.78280e16 0.701655
\(962\) 0 0
\(963\) −8.83789e15 −0.343878
\(964\) 0 0
\(965\) −1.39024e15 −0.0534800
\(966\) 0 0
\(967\) −1.57202e16 −0.597878 −0.298939 0.954272i \(-0.596633\pi\)
−0.298939 + 0.954272i \(0.596633\pi\)
\(968\) 0 0
\(969\) 5.89181e14 0.0221548
\(970\) 0 0
\(971\) 1.34980e16 0.501837 0.250918 0.968008i \(-0.419267\pi\)
0.250918 + 0.968008i \(0.419267\pi\)
\(972\) 0 0
\(973\) −1.14831e16 −0.422121
\(974\) 0 0
\(975\) 1.76418e14 0.00641235
\(976\) 0 0
\(977\) −3.80905e15 −0.136898 −0.0684489 0.997655i \(-0.521805\pi\)
−0.0684489 + 0.997655i \(0.521805\pi\)
\(978\) 0 0
\(979\) 2.08705e15 0.0741698
\(980\) 0 0
\(981\) −1.21482e16 −0.426905
\(982\) 0 0
\(983\) −5.42344e16 −1.88465 −0.942325 0.334698i \(-0.891366\pi\)
−0.942325 + 0.334698i \(0.891366\pi\)
\(984\) 0 0
\(985\) −3.68008e16 −1.26461
\(986\) 0 0
\(987\) −2.35503e15 −0.0800297
\(988\) 0 0
\(989\) −1.53665e16 −0.516411
\(990\) 0 0
\(991\) −1.20030e16 −0.398919 −0.199459 0.979906i \(-0.563919\pi\)
−0.199459 + 0.979906i \(0.563919\pi\)
\(992\) 0 0
\(993\) 1.51934e15 0.0499385
\(994\) 0 0
\(995\) 7.27815e15 0.236589
\(996\) 0 0
\(997\) −2.90012e15 −0.0932379 −0.0466189 0.998913i \(-0.514845\pi\)
−0.0466189 + 0.998913i \(0.514845\pi\)
\(998\) 0 0
\(999\) −2.88104e15 −0.0916093
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 192.12.a.i.1.1 1
4.3 odd 2 192.12.a.s.1.1 1
8.3 odd 2 24.12.a.a.1.1 1
8.5 even 2 48.12.a.e.1.1 1
24.5 odd 2 144.12.a.m.1.1 1
24.11 even 2 72.12.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.12.a.a.1.1 1 8.3 odd 2
48.12.a.e.1.1 1 8.5 even 2
72.12.a.d.1.1 1 24.11 even 2
144.12.a.m.1.1 1 24.5 odd 2
192.12.a.i.1.1 1 1.1 even 1 trivial
192.12.a.s.1.1 1 4.3 odd 2