Properties

Label 48.12.a.f.1.1
Level $48$
Weight $12$
Character 48.1
Self dual yes
Analytic conductor $36.880$
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] = [48,12,Mod(1,48)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(48, 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("48.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 48.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: \(36.8804726669\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 48.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+243.000 q^{3} -5370.00 q^{5} +27760.0 q^{7} +59049.0 q^{9} +O(q^{10})\) \(q+243.000 q^{3} -5370.00 q^{5} +27760.0 q^{7} +59049.0 q^{9} -637836. q^{11} +766214. q^{13} -1.30491e6 q^{15} +3.08435e6 q^{17} +1.95114e7 q^{19} +6.74568e6 q^{21} -1.53124e7 q^{23} -1.99912e7 q^{25} +1.43489e7 q^{27} +1.07513e7 q^{29} +5.09374e7 q^{31} -1.54994e8 q^{33} -1.49071e8 q^{35} +6.64741e8 q^{37} +1.86190e8 q^{39} +8.98833e8 q^{41} +9.57947e8 q^{43} -3.17093e8 q^{45} +1.55574e9 q^{47} -1.20671e9 q^{49} +7.49498e8 q^{51} +3.79242e9 q^{53} +3.42518e9 q^{55} +4.74127e9 q^{57} -5.55307e8 q^{59} +4.95042e9 q^{61} +1.63920e9 q^{63} -4.11457e9 q^{65} -5.29240e9 q^{67} -3.72090e9 q^{69} +1.48311e10 q^{71} +1.39710e10 q^{73} -4.85787e9 q^{75} -1.77063e10 q^{77} -3.72054e9 q^{79} +3.48678e9 q^{81} -8.76845e9 q^{83} -1.65630e10 q^{85} +2.61256e9 q^{87} -2.54728e10 q^{89} +2.12701e10 q^{91} +1.23778e10 q^{93} -1.04776e11 q^{95} -3.90925e10 q^{97} -3.76636e10 q^{99} +O(q^{100})\)

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
\(5\) −5370.00 −0.768492 −0.384246 0.923231i \(-0.625539\pi\)
−0.384246 + 0.923231i \(0.625539\pi\)
\(6\) 0 0
\(7\) 27760.0 0.624281 0.312141 0.950036i \(-0.398954\pi\)
0.312141 + 0.950036i \(0.398954\pi\)
\(8\) 0 0
\(9\) 59049.0 0.333333
\(10\) 0 0
\(11\) −637836. −1.19412 −0.597062 0.802195i \(-0.703665\pi\)
−0.597062 + 0.802195i \(0.703665\pi\)
\(12\) 0 0
\(13\) 766214. 0.572350 0.286175 0.958177i \(-0.407616\pi\)
0.286175 + 0.958177i \(0.407616\pi\)
\(14\) 0 0
\(15\) −1.30491e6 −0.443689
\(16\) 0 0
\(17\) 3.08435e6 0.526860 0.263430 0.964679i \(-0.415146\pi\)
0.263430 + 0.964679i \(0.415146\pi\)
\(18\) 0 0
\(19\) 1.95114e7 1.80777 0.903886 0.427773i \(-0.140702\pi\)
0.903886 + 0.427773i \(0.140702\pi\)
\(20\) 0 0
\(21\) 6.74568e6 0.360429
\(22\) 0 0
\(23\) −1.53124e7 −0.496066 −0.248033 0.968752i \(-0.579784\pi\)
−0.248033 + 0.968752i \(0.579784\pi\)
\(24\) 0 0
\(25\) −1.99912e7 −0.409420
\(26\) 0 0
\(27\) 1.43489e7 0.192450
\(28\) 0 0
\(29\) 1.07513e7 0.0973353 0.0486677 0.998815i \(-0.484502\pi\)
0.0486677 + 0.998815i \(0.484502\pi\)
\(30\) 0 0
\(31\) 5.09374e7 0.319556 0.159778 0.987153i \(-0.448922\pi\)
0.159778 + 0.987153i \(0.448922\pi\)
\(32\) 0 0
\(33\) −1.54994e8 −0.689428
\(34\) 0 0
\(35\) −1.49071e8 −0.479755
\(36\) 0 0
\(37\) 6.64741e8 1.57595 0.787976 0.615706i \(-0.211129\pi\)
0.787976 + 0.615706i \(0.211129\pi\)
\(38\) 0 0
\(39\) 1.86190e8 0.330446
\(40\) 0 0
\(41\) 8.98833e8 1.21162 0.605812 0.795608i \(-0.292848\pi\)
0.605812 + 0.795608i \(0.292848\pi\)
\(42\) 0 0
\(43\) 9.57947e8 0.993722 0.496861 0.867830i \(-0.334486\pi\)
0.496861 + 0.867830i \(0.334486\pi\)
\(44\) 0 0
\(45\) −3.17093e8 −0.256164
\(46\) 0 0
\(47\) 1.55574e9 0.989462 0.494731 0.869046i \(-0.335267\pi\)
0.494731 + 0.869046i \(0.335267\pi\)
\(48\) 0 0
\(49\) −1.20671e9 −0.610273
\(50\) 0 0
\(51\) 7.49498e8 0.304183
\(52\) 0 0
\(53\) 3.79242e9 1.24566 0.622829 0.782358i \(-0.285983\pi\)
0.622829 + 0.782358i \(0.285983\pi\)
\(54\) 0 0
\(55\) 3.42518e9 0.917674
\(56\) 0 0
\(57\) 4.74127e9 1.04372
\(58\) 0 0
\(59\) −5.55307e8 −0.101122 −0.0505612 0.998721i \(-0.516101\pi\)
−0.0505612 + 0.998721i \(0.516101\pi\)
\(60\) 0 0
\(61\) 4.95042e9 0.750461 0.375230 0.926932i \(-0.377564\pi\)
0.375230 + 0.926932i \(0.377564\pi\)
\(62\) 0 0
\(63\) 1.63920e9 0.208094
\(64\) 0 0
\(65\) −4.11457e9 −0.439846
\(66\) 0 0
\(67\) −5.29240e9 −0.478896 −0.239448 0.970909i \(-0.576966\pi\)
−0.239448 + 0.970909i \(0.576966\pi\)
\(68\) 0 0
\(69\) −3.72090e9 −0.286404
\(70\) 0 0
\(71\) 1.48311e10 0.975556 0.487778 0.872968i \(-0.337808\pi\)
0.487778 + 0.872968i \(0.337808\pi\)
\(72\) 0 0
\(73\) 1.39710e10 0.788773 0.394386 0.918945i \(-0.370957\pi\)
0.394386 + 0.918945i \(0.370957\pi\)
\(74\) 0 0
\(75\) −4.85787e9 −0.236379
\(76\) 0 0
\(77\) −1.77063e10 −0.745469
\(78\) 0 0
\(79\) −3.72054e9 −0.136037 −0.0680185 0.997684i \(-0.521668\pi\)
−0.0680185 + 0.997684i \(0.521668\pi\)
\(80\) 0 0
\(81\) 3.48678e9 0.111111
\(82\) 0 0
\(83\) −8.76845e9 −0.244339 −0.122170 0.992509i \(-0.538985\pi\)
−0.122170 + 0.992509i \(0.538985\pi\)
\(84\) 0 0
\(85\) −1.65630e10 −0.404888
\(86\) 0 0
\(87\) 2.61256e9 0.0561966
\(88\) 0 0
\(89\) −2.54728e10 −0.483539 −0.241769 0.970334i \(-0.577728\pi\)
−0.241769 + 0.970334i \(0.577728\pi\)
\(90\) 0 0
\(91\) 2.12701e10 0.357307
\(92\) 0 0
\(93\) 1.23778e10 0.184496
\(94\) 0 0
\(95\) −1.04776e11 −1.38926
\(96\) 0 0
\(97\) −3.90925e10 −0.462220 −0.231110 0.972928i \(-0.574236\pi\)
−0.231110 + 0.972928i \(0.574236\pi\)
\(98\) 0 0
\(99\) −3.76636e10 −0.398041
\(100\) 0 0
\(101\) 9.31078e9 0.0881492 0.0440746 0.999028i \(-0.485966\pi\)
0.0440746 + 0.999028i \(0.485966\pi\)
\(102\) 0 0
\(103\) −4.85751e10 −0.412865 −0.206433 0.978461i \(-0.566185\pi\)
−0.206433 + 0.978461i \(0.566185\pi\)
\(104\) 0 0
\(105\) −3.62243e10 −0.276987
\(106\) 0 0
\(107\) −2.25596e11 −1.55496 −0.777482 0.628906i \(-0.783503\pi\)
−0.777482 + 0.628906i \(0.783503\pi\)
\(108\) 0 0
\(109\) −6.94512e10 −0.432348 −0.216174 0.976355i \(-0.569358\pi\)
−0.216174 + 0.976355i \(0.569358\pi\)
\(110\) 0 0
\(111\) 1.61532e11 0.909876
\(112\) 0 0
\(113\) −3.59665e11 −1.83640 −0.918198 0.396122i \(-0.870356\pi\)
−0.918198 + 0.396122i \(0.870356\pi\)
\(114\) 0 0
\(115\) 8.22274e10 0.381222
\(116\) 0 0
\(117\) 4.52442e10 0.190783
\(118\) 0 0
\(119\) 8.56217e10 0.328909
\(120\) 0 0
\(121\) 1.21523e11 0.425931
\(122\) 0 0
\(123\) 2.18417e11 0.699532
\(124\) 0 0
\(125\) 3.69560e11 1.08313
\(126\) 0 0
\(127\) −2.50273e11 −0.672192 −0.336096 0.941828i \(-0.609107\pi\)
−0.336096 + 0.941828i \(0.609107\pi\)
\(128\) 0 0
\(129\) 2.32781e11 0.573726
\(130\) 0 0
\(131\) 9.78918e10 0.221694 0.110847 0.993837i \(-0.464644\pi\)
0.110847 + 0.993837i \(0.464644\pi\)
\(132\) 0 0
\(133\) 5.41637e11 1.12856
\(134\) 0 0
\(135\) −7.70536e10 −0.147896
\(136\) 0 0
\(137\) −1.55015e11 −0.274416 −0.137208 0.990542i \(-0.543813\pi\)
−0.137208 + 0.990542i \(0.543813\pi\)
\(138\) 0 0
\(139\) 1.12627e12 1.84102 0.920512 0.390715i \(-0.127772\pi\)
0.920512 + 0.390715i \(0.127772\pi\)
\(140\) 0 0
\(141\) 3.78045e11 0.571266
\(142\) 0 0
\(143\) −4.88719e11 −0.683457
\(144\) 0 0
\(145\) −5.77343e10 −0.0748014
\(146\) 0 0
\(147\) −2.93230e11 −0.352341
\(148\) 0 0
\(149\) −1.38458e12 −1.54452 −0.772261 0.635306i \(-0.780874\pi\)
−0.772261 + 0.635306i \(0.780874\pi\)
\(150\) 0 0
\(151\) 6.98601e11 0.724196 0.362098 0.932140i \(-0.382061\pi\)
0.362098 + 0.932140i \(0.382061\pi\)
\(152\) 0 0
\(153\) 1.82128e11 0.175620
\(154\) 0 0
\(155\) −2.73534e11 −0.245576
\(156\) 0 0
\(157\) 2.13127e12 1.78316 0.891580 0.452863i \(-0.149597\pi\)
0.891580 + 0.452863i \(0.149597\pi\)
\(158\) 0 0
\(159\) 9.21557e11 0.719181
\(160\) 0 0
\(161\) −4.25071e11 −0.309684
\(162\) 0 0
\(163\) 1.63564e11 0.111342 0.0556708 0.998449i \(-0.482270\pi\)
0.0556708 + 0.998449i \(0.482270\pi\)
\(164\) 0 0
\(165\) 8.32319e11 0.529819
\(166\) 0 0
\(167\) 8.80943e9 0.00524816 0.00262408 0.999997i \(-0.499165\pi\)
0.00262408 + 0.999997i \(0.499165\pi\)
\(168\) 0 0
\(169\) −1.20508e12 −0.672416
\(170\) 0 0
\(171\) 1.15213e12 0.602591
\(172\) 0 0
\(173\) −7.30852e11 −0.358571 −0.179286 0.983797i \(-0.557379\pi\)
−0.179286 + 0.983797i \(0.557379\pi\)
\(174\) 0 0
\(175\) −5.54956e11 −0.255593
\(176\) 0 0
\(177\) −1.34940e11 −0.0583830
\(178\) 0 0
\(179\) −3.92371e12 −1.59590 −0.797950 0.602724i \(-0.794082\pi\)
−0.797950 + 0.602724i \(0.794082\pi\)
\(180\) 0 0
\(181\) 2.27931e12 0.872110 0.436055 0.899920i \(-0.356375\pi\)
0.436055 + 0.899920i \(0.356375\pi\)
\(182\) 0 0
\(183\) 1.20295e12 0.433279
\(184\) 0 0
\(185\) −3.56966e12 −1.21111
\(186\) 0 0
\(187\) −1.96731e12 −0.629136
\(188\) 0 0
\(189\) 3.98326e11 0.120143
\(190\) 0 0
\(191\) −3.38709e12 −0.964147 −0.482073 0.876131i \(-0.660116\pi\)
−0.482073 + 0.876131i \(0.660116\pi\)
\(192\) 0 0
\(193\) −4.92921e12 −1.32499 −0.662494 0.749067i \(-0.730502\pi\)
−0.662494 + 0.749067i \(0.730502\pi\)
\(194\) 0 0
\(195\) −9.99840e11 −0.253945
\(196\) 0 0
\(197\) 6.35785e12 1.52667 0.763337 0.646001i \(-0.223560\pi\)
0.763337 + 0.646001i \(0.223560\pi\)
\(198\) 0 0
\(199\) 3.78554e12 0.859875 0.429938 0.902859i \(-0.358536\pi\)
0.429938 + 0.902859i \(0.358536\pi\)
\(200\) 0 0
\(201\) −1.28605e12 −0.276491
\(202\) 0 0
\(203\) 2.98455e11 0.0607646
\(204\) 0 0
\(205\) −4.82674e12 −0.931123
\(206\) 0 0
\(207\) −9.04180e11 −0.165355
\(208\) 0 0
\(209\) −1.24451e13 −2.15870
\(210\) 0 0
\(211\) −1.79494e11 −0.0295458 −0.0147729 0.999891i \(-0.504703\pi\)
−0.0147729 + 0.999891i \(0.504703\pi\)
\(212\) 0 0
\(213\) 3.60395e12 0.563237
\(214\) 0 0
\(215\) −5.14418e12 −0.763668
\(216\) 0 0
\(217\) 1.41402e12 0.199493
\(218\) 0 0
\(219\) 3.39495e12 0.455398
\(220\) 0 0
\(221\) 2.36328e12 0.301548
\(222\) 0 0
\(223\) −2.22568e12 −0.270263 −0.135132 0.990828i \(-0.543146\pi\)
−0.135132 + 0.990828i \(0.543146\pi\)
\(224\) 0 0
\(225\) −1.18046e12 −0.136473
\(226\) 0 0
\(227\) 4.15848e11 0.0457923 0.0228961 0.999738i \(-0.492711\pi\)
0.0228961 + 0.999738i \(0.492711\pi\)
\(228\) 0 0
\(229\) 1.81248e13 1.90186 0.950928 0.309414i \(-0.100133\pi\)
0.950928 + 0.309414i \(0.100133\pi\)
\(230\) 0 0
\(231\) −4.30264e12 −0.430397
\(232\) 0 0
\(233\) −1.87641e10 −0.00179007 −0.000895033 1.00000i \(-0.500285\pi\)
−0.000895033 1.00000i \(0.500285\pi\)
\(234\) 0 0
\(235\) −8.35433e12 −0.760394
\(236\) 0 0
\(237\) −9.04092e11 −0.0785410
\(238\) 0 0
\(239\) 1.76252e13 1.46200 0.730999 0.682379i \(-0.239054\pi\)
0.730999 + 0.682379i \(0.239054\pi\)
\(240\) 0 0
\(241\) −8.90117e11 −0.0705267 −0.0352633 0.999378i \(-0.511227\pi\)
−0.0352633 + 0.999378i \(0.511227\pi\)
\(242\) 0 0
\(243\) 8.47289e11 0.0641500
\(244\) 0 0
\(245\) 6.48003e12 0.468990
\(246\) 0 0
\(247\) 1.49499e13 1.03468
\(248\) 0 0
\(249\) −2.13073e12 −0.141069
\(250\) 0 0
\(251\) 2.42280e13 1.53502 0.767508 0.641040i \(-0.221497\pi\)
0.767508 + 0.641040i \(0.221497\pi\)
\(252\) 0 0
\(253\) 9.76677e12 0.592364
\(254\) 0 0
\(255\) −4.02480e12 −0.233762
\(256\) 0 0
\(257\) 7.80492e12 0.434246 0.217123 0.976144i \(-0.430333\pi\)
0.217123 + 0.976144i \(0.430333\pi\)
\(258\) 0 0
\(259\) 1.84532e13 0.983837
\(260\) 0 0
\(261\) 6.34851e11 0.0324451
\(262\) 0 0
\(263\) −1.65956e13 −0.813272 −0.406636 0.913590i \(-0.633298\pi\)
−0.406636 + 0.913590i \(0.633298\pi\)
\(264\) 0 0
\(265\) −2.03653e13 −0.957278
\(266\) 0 0
\(267\) −6.18988e12 −0.279171
\(268\) 0 0
\(269\) 2.85236e13 1.23471 0.617357 0.786683i \(-0.288203\pi\)
0.617357 + 0.786683i \(0.288203\pi\)
\(270\) 0 0
\(271\) −2.33800e13 −0.971658 −0.485829 0.874054i \(-0.661482\pi\)
−0.485829 + 0.874054i \(0.661482\pi\)
\(272\) 0 0
\(273\) 5.16863e12 0.206291
\(274\) 0 0
\(275\) 1.27511e13 0.488898
\(276\) 0 0
\(277\) −3.03641e13 −1.11872 −0.559361 0.828924i \(-0.688953\pi\)
−0.559361 + 0.828924i \(0.688953\pi\)
\(278\) 0 0
\(279\) 3.00780e12 0.106519
\(280\) 0 0
\(281\) 1.59749e13 0.543942 0.271971 0.962306i \(-0.412325\pi\)
0.271971 + 0.962306i \(0.412325\pi\)
\(282\) 0 0
\(283\) −2.87045e13 −0.939993 −0.469997 0.882668i \(-0.655745\pi\)
−0.469997 + 0.882668i \(0.655745\pi\)
\(284\) 0 0
\(285\) −2.54606e13 −0.802089
\(286\) 0 0
\(287\) 2.49516e13 0.756394
\(288\) 0 0
\(289\) −2.47587e13 −0.722419
\(290\) 0 0
\(291\) −9.49948e12 −0.266863
\(292\) 0 0
\(293\) 4.81754e13 1.30333 0.651663 0.758508i \(-0.274071\pi\)
0.651663 + 0.758508i \(0.274071\pi\)
\(294\) 0 0
\(295\) 2.98200e12 0.0777117
\(296\) 0 0
\(297\) −9.15225e12 −0.229809
\(298\) 0 0
\(299\) −1.17325e13 −0.283923
\(300\) 0 0
\(301\) 2.65926e13 0.620362
\(302\) 0 0
\(303\) 2.26252e12 0.0508930
\(304\) 0 0
\(305\) −2.65838e13 −0.576723
\(306\) 0 0
\(307\) −2.57350e13 −0.538597 −0.269298 0.963057i \(-0.586792\pi\)
−0.269298 + 0.963057i \(0.586792\pi\)
\(308\) 0 0
\(309\) −1.18037e13 −0.238368
\(310\) 0 0
\(311\) −3.46043e13 −0.674446 −0.337223 0.941425i \(-0.609488\pi\)
−0.337223 + 0.941425i \(0.609488\pi\)
\(312\) 0 0
\(313\) 1.26066e13 0.237194 0.118597 0.992942i \(-0.462160\pi\)
0.118597 + 0.992942i \(0.462160\pi\)
\(314\) 0 0
\(315\) −8.80251e12 −0.159918
\(316\) 0 0
\(317\) −8.18243e13 −1.43568 −0.717838 0.696210i \(-0.754868\pi\)
−0.717838 + 0.696210i \(0.754868\pi\)
\(318\) 0 0
\(319\) −6.85754e12 −0.116230
\(320\) 0 0
\(321\) −5.48198e13 −0.897758
\(322\) 0 0
\(323\) 6.01801e13 0.952443
\(324\) 0 0
\(325\) −1.53176e13 −0.234332
\(326\) 0 0
\(327\) −1.68766e13 −0.249616
\(328\) 0 0
\(329\) 4.31874e13 0.617703
\(330\) 0 0
\(331\) 3.40115e13 0.470513 0.235256 0.971933i \(-0.424407\pi\)
0.235256 + 0.971933i \(0.424407\pi\)
\(332\) 0 0
\(333\) 3.92523e13 0.525317
\(334\) 0 0
\(335\) 2.84202e13 0.368028
\(336\) 0 0
\(337\) −5.99439e13 −0.751244 −0.375622 0.926773i \(-0.622571\pi\)
−0.375622 + 0.926773i \(0.622571\pi\)
\(338\) 0 0
\(339\) −8.73985e13 −1.06024
\(340\) 0 0
\(341\) −3.24897e13 −0.381590
\(342\) 0 0
\(343\) −8.83888e13 −1.00526
\(344\) 0 0
\(345\) 1.99813e13 0.220099
\(346\) 0 0
\(347\) −9.78685e13 −1.04431 −0.522157 0.852850i \(-0.674872\pi\)
−0.522157 + 0.852850i \(0.674872\pi\)
\(348\) 0 0
\(349\) −1.42790e14 −1.47624 −0.738120 0.674670i \(-0.764286\pi\)
−0.738120 + 0.674670i \(0.764286\pi\)
\(350\) 0 0
\(351\) 1.09943e13 0.110149
\(352\) 0 0
\(353\) 1.44246e14 1.40069 0.700346 0.713804i \(-0.253029\pi\)
0.700346 + 0.713804i \(0.253029\pi\)
\(354\) 0 0
\(355\) −7.96429e13 −0.749707
\(356\) 0 0
\(357\) 2.08061e13 0.189896
\(358\) 0 0
\(359\) 1.24349e14 1.10059 0.550293 0.834972i \(-0.314516\pi\)
0.550293 + 0.834972i \(0.314516\pi\)
\(360\) 0 0
\(361\) 2.64205e14 2.26804
\(362\) 0 0
\(363\) 2.95301e13 0.245911
\(364\) 0 0
\(365\) −7.50243e13 −0.606165
\(366\) 0 0
\(367\) 1.60110e14 1.25532 0.627659 0.778488i \(-0.284013\pi\)
0.627659 + 0.778488i \(0.284013\pi\)
\(368\) 0 0
\(369\) 5.30752e13 0.403875
\(370\) 0 0
\(371\) 1.05277e14 0.777641
\(372\) 0 0
\(373\) −3.47258e13 −0.249031 −0.124516 0.992218i \(-0.539738\pi\)
−0.124516 + 0.992218i \(0.539738\pi\)
\(374\) 0 0
\(375\) 8.98031e13 0.625344
\(376\) 0 0
\(377\) 8.23777e12 0.0557099
\(378\) 0 0
\(379\) 1.46500e14 0.962327 0.481163 0.876631i \(-0.340214\pi\)
0.481163 + 0.876631i \(0.340214\pi\)
\(380\) 0 0
\(381\) −6.08164e13 −0.388090
\(382\) 0 0
\(383\) −6.43419e13 −0.398933 −0.199467 0.979905i \(-0.563921\pi\)
−0.199467 + 0.979905i \(0.563921\pi\)
\(384\) 0 0
\(385\) 9.50830e13 0.572887
\(386\) 0 0
\(387\) 5.65658e13 0.331241
\(388\) 0 0
\(389\) −3.98900e13 −0.227061 −0.113530 0.993535i \(-0.536216\pi\)
−0.113530 + 0.993535i \(0.536216\pi\)
\(390\) 0 0
\(391\) −4.72287e13 −0.261357
\(392\) 0 0
\(393\) 2.37877e13 0.127995
\(394\) 0 0
\(395\) 1.99793e13 0.104543
\(396\) 0 0
\(397\) 1.06552e14 0.542268 0.271134 0.962542i \(-0.412601\pi\)
0.271134 + 0.962542i \(0.412601\pi\)
\(398\) 0 0
\(399\) 1.31618e14 0.651573
\(400\) 0 0
\(401\) 3.41445e13 0.164447 0.0822236 0.996614i \(-0.473798\pi\)
0.0822236 + 0.996614i \(0.473798\pi\)
\(402\) 0 0
\(403\) 3.90289e13 0.182898
\(404\) 0 0
\(405\) −1.87240e13 −0.0853880
\(406\) 0 0
\(407\) −4.23996e14 −1.88188
\(408\) 0 0
\(409\) 5.33349e13 0.230427 0.115213 0.993341i \(-0.463245\pi\)
0.115213 + 0.993341i \(0.463245\pi\)
\(410\) 0 0
\(411\) −3.76686e13 −0.158434
\(412\) 0 0
\(413\) −1.54153e13 −0.0631288
\(414\) 0 0
\(415\) 4.70866e13 0.187773
\(416\) 0 0
\(417\) 2.73682e14 1.06292
\(418\) 0 0
\(419\) 1.01288e14 0.383159 0.191580 0.981477i \(-0.438639\pi\)
0.191580 + 0.981477i \(0.438639\pi\)
\(420\) 0 0
\(421\) −1.57928e14 −0.581981 −0.290991 0.956726i \(-0.593985\pi\)
−0.290991 + 0.956726i \(0.593985\pi\)
\(422\) 0 0
\(423\) 9.18650e13 0.329821
\(424\) 0 0
\(425\) −6.16600e13 −0.215707
\(426\) 0 0
\(427\) 1.37424e14 0.468499
\(428\) 0 0
\(429\) −1.18759e14 −0.394594
\(430\) 0 0
\(431\) −5.13171e14 −1.66202 −0.831012 0.556254i \(-0.812238\pi\)
−0.831012 + 0.556254i \(0.812238\pi\)
\(432\) 0 0
\(433\) −7.49248e13 −0.236560 −0.118280 0.992980i \(-0.537738\pi\)
−0.118280 + 0.992980i \(0.537738\pi\)
\(434\) 0 0
\(435\) −1.40294e13 −0.0431866
\(436\) 0 0
\(437\) −2.98766e14 −0.896773
\(438\) 0 0
\(439\) 3.68335e14 1.07817 0.539086 0.842250i \(-0.318770\pi\)
0.539086 + 0.842250i \(0.318770\pi\)
\(440\) 0 0
\(441\) −7.12550e13 −0.203424
\(442\) 0 0
\(443\) 1.11248e14 0.309793 0.154896 0.987931i \(-0.450496\pi\)
0.154896 + 0.987931i \(0.450496\pi\)
\(444\) 0 0
\(445\) 1.36789e14 0.371595
\(446\) 0 0
\(447\) −3.36453e14 −0.891730
\(448\) 0 0
\(449\) −8.83314e13 −0.228434 −0.114217 0.993456i \(-0.536436\pi\)
−0.114217 + 0.993456i \(0.536436\pi\)
\(450\) 0 0
\(451\) −5.73308e14 −1.44683
\(452\) 0 0
\(453\) 1.69760e14 0.418115
\(454\) 0 0
\(455\) −1.14220e14 −0.274588
\(456\) 0 0
\(457\) −9.42094e12 −0.0221083 −0.0110541 0.999939i \(-0.503519\pi\)
−0.0110541 + 0.999939i \(0.503519\pi\)
\(458\) 0 0
\(459\) 4.42571e13 0.101394
\(460\) 0 0
\(461\) −6.97134e14 −1.55941 −0.779706 0.626146i \(-0.784631\pi\)
−0.779706 + 0.626146i \(0.784631\pi\)
\(462\) 0 0
\(463\) 1.87941e14 0.410513 0.205256 0.978708i \(-0.434197\pi\)
0.205256 + 0.978708i \(0.434197\pi\)
\(464\) 0 0
\(465\) −6.64687e13 −0.141784
\(466\) 0 0
\(467\) −3.20007e14 −0.666678 −0.333339 0.942807i \(-0.608175\pi\)
−0.333339 + 0.942807i \(0.608175\pi\)
\(468\) 0 0
\(469\) −1.46917e14 −0.298966
\(470\) 0 0
\(471\) 5.17899e14 1.02951
\(472\) 0 0
\(473\) −6.11013e14 −1.18663
\(474\) 0 0
\(475\) −3.90057e14 −0.740139
\(476\) 0 0
\(477\) 2.23938e14 0.415219
\(478\) 0 0
\(479\) 1.50382e14 0.272491 0.136245 0.990675i \(-0.456496\pi\)
0.136245 + 0.990675i \(0.456496\pi\)
\(480\) 0 0
\(481\) 5.09334e14 0.901996
\(482\) 0 0
\(483\) −1.03292e14 −0.178796
\(484\) 0 0
\(485\) 2.09927e14 0.355212
\(486\) 0 0
\(487\) −1.76546e14 −0.292045 −0.146022 0.989281i \(-0.546647\pi\)
−0.146022 + 0.989281i \(0.546647\pi\)
\(488\) 0 0
\(489\) 3.97462e13 0.0642831
\(490\) 0 0
\(491\) 8.60958e14 1.36155 0.680775 0.732492i \(-0.261643\pi\)
0.680775 + 0.732492i \(0.261643\pi\)
\(492\) 0 0
\(493\) 3.31607e13 0.0512821
\(494\) 0 0
\(495\) 2.02253e14 0.305891
\(496\) 0 0
\(497\) 4.11711e14 0.609021
\(498\) 0 0
\(499\) −6.01209e14 −0.869907 −0.434953 0.900453i \(-0.643235\pi\)
−0.434953 + 0.900453i \(0.643235\pi\)
\(500\) 0 0
\(501\) 2.14069e12 0.00303003
\(502\) 0 0
\(503\) −1.09203e15 −1.51221 −0.756103 0.654453i \(-0.772899\pi\)
−0.756103 + 0.654453i \(0.772899\pi\)
\(504\) 0 0
\(505\) −4.99989e13 −0.0677420
\(506\) 0 0
\(507\) −2.92834e14 −0.388219
\(508\) 0 0
\(509\) 8.76371e14 1.13695 0.568474 0.822702i \(-0.307534\pi\)
0.568474 + 0.822702i \(0.307534\pi\)
\(510\) 0 0
\(511\) 3.87835e14 0.492416
\(512\) 0 0
\(513\) 2.79967e14 0.347906
\(514\) 0 0
\(515\) 2.60848e14 0.317284
\(516\) 0 0
\(517\) −9.92308e14 −1.18154
\(518\) 0 0
\(519\) −1.77597e14 −0.207021
\(520\) 0 0
\(521\) −3.71989e14 −0.424544 −0.212272 0.977211i \(-0.568086\pi\)
−0.212272 + 0.977211i \(0.568086\pi\)
\(522\) 0 0
\(523\) 9.73507e14 1.08788 0.543939 0.839125i \(-0.316932\pi\)
0.543939 + 0.839125i \(0.316932\pi\)
\(524\) 0 0
\(525\) −1.34854e14 −0.147567
\(526\) 0 0
\(527\) 1.57109e14 0.168361
\(528\) 0 0
\(529\) −7.18341e14 −0.753919
\(530\) 0 0
\(531\) −3.27903e13 −0.0337074
\(532\) 0 0
\(533\) 6.88699e14 0.693473
\(534\) 0 0
\(535\) 1.21145e15 1.19498
\(536\) 0 0
\(537\) −9.53463e14 −0.921393
\(538\) 0 0
\(539\) 7.69683e14 0.728741
\(540\) 0 0
\(541\) 1.74606e15 1.61985 0.809925 0.586533i \(-0.199508\pi\)
0.809925 + 0.586533i \(0.199508\pi\)
\(542\) 0 0
\(543\) 5.53872e14 0.503513
\(544\) 0 0
\(545\) 3.72953e14 0.332256
\(546\) 0 0
\(547\) −1.74624e14 −0.152467 −0.0762333 0.997090i \(-0.524289\pi\)
−0.0762333 + 0.997090i \(0.524289\pi\)
\(548\) 0 0
\(549\) 2.92317e14 0.250154
\(550\) 0 0
\(551\) 2.09772e14 0.175960
\(552\) 0 0
\(553\) −1.03282e14 −0.0849254
\(554\) 0 0
\(555\) −8.67427e14 −0.699232
\(556\) 0 0
\(557\) 1.58365e15 1.25157 0.625785 0.779995i \(-0.284779\pi\)
0.625785 + 0.779995i \(0.284779\pi\)
\(558\) 0 0
\(559\) 7.33993e14 0.568757
\(560\) 0 0
\(561\) −4.78057e14 −0.363232
\(562\) 0 0
\(563\) 9.75798e14 0.727049 0.363525 0.931585i \(-0.381573\pi\)
0.363525 + 0.931585i \(0.381573\pi\)
\(564\) 0 0
\(565\) 1.93140e15 1.41125
\(566\) 0 0
\(567\) 9.67931e13 0.0693646
\(568\) 0 0
\(569\) 1.92427e15 1.35254 0.676269 0.736655i \(-0.263596\pi\)
0.676269 + 0.736655i \(0.263596\pi\)
\(570\) 0 0
\(571\) −1.61132e15 −1.11092 −0.555461 0.831543i \(-0.687458\pi\)
−0.555461 + 0.831543i \(0.687458\pi\)
\(572\) 0 0
\(573\) −8.23063e14 −0.556650
\(574\) 0 0
\(575\) 3.06113e14 0.203099
\(576\) 0 0
\(577\) −2.53250e15 −1.64848 −0.824239 0.566242i \(-0.808397\pi\)
−0.824239 + 0.566242i \(0.808397\pi\)
\(578\) 0 0
\(579\) −1.19780e15 −0.764982
\(580\) 0 0
\(581\) −2.43412e14 −0.152536
\(582\) 0 0
\(583\) −2.41894e15 −1.48747
\(584\) 0 0
\(585\) −2.42961e14 −0.146615
\(586\) 0 0
\(587\) 3.11508e15 1.84484 0.922421 0.386186i \(-0.126208\pi\)
0.922421 + 0.386186i \(0.126208\pi\)
\(588\) 0 0
\(589\) 9.93860e14 0.577685
\(590\) 0 0
\(591\) 1.54496e15 0.881425
\(592\) 0 0
\(593\) −2.20723e15 −1.23608 −0.618040 0.786146i \(-0.712073\pi\)
−0.618040 + 0.786146i \(0.712073\pi\)
\(594\) 0 0
\(595\) −4.59788e14 −0.252764
\(596\) 0 0
\(597\) 9.19885e14 0.496449
\(598\) 0 0
\(599\) −3.41159e15 −1.80763 −0.903813 0.427927i \(-0.859244\pi\)
−0.903813 + 0.427927i \(0.859244\pi\)
\(600\) 0 0
\(601\) −1.57545e15 −0.819588 −0.409794 0.912178i \(-0.634399\pi\)
−0.409794 + 0.912178i \(0.634399\pi\)
\(602\) 0 0
\(603\) −3.12511e14 −0.159632
\(604\) 0 0
\(605\) −6.52579e14 −0.327325
\(606\) 0 0
\(607\) 5.05592e14 0.249036 0.124518 0.992217i \(-0.460262\pi\)
0.124518 + 0.992217i \(0.460262\pi\)
\(608\) 0 0
\(609\) 7.25246e13 0.0350825
\(610\) 0 0
\(611\) 1.19203e15 0.566319
\(612\) 0 0
\(613\) −1.98375e15 −0.925665 −0.462832 0.886446i \(-0.653167\pi\)
−0.462832 + 0.886446i \(0.653167\pi\)
\(614\) 0 0
\(615\) −1.17290e15 −0.537584
\(616\) 0 0
\(617\) −1.93099e15 −0.869382 −0.434691 0.900580i \(-0.643142\pi\)
−0.434691 + 0.900580i \(0.643142\pi\)
\(618\) 0 0
\(619\) −9.47689e14 −0.419148 −0.209574 0.977793i \(-0.567208\pi\)
−0.209574 + 0.977793i \(0.567208\pi\)
\(620\) 0 0
\(621\) −2.19716e14 −0.0954679
\(622\) 0 0
\(623\) −7.07124e14 −0.301864
\(624\) 0 0
\(625\) −1.00840e15 −0.422955
\(626\) 0 0
\(627\) −3.02415e15 −1.24633
\(628\) 0 0
\(629\) 2.05030e15 0.830306
\(630\) 0 0
\(631\) −1.00593e15 −0.400318 −0.200159 0.979763i \(-0.564146\pi\)
−0.200159 + 0.979763i \(0.564146\pi\)
\(632\) 0 0
\(633\) −4.36170e13 −0.0170583
\(634\) 0 0
\(635\) 1.34397e15 0.516574
\(636\) 0 0
\(637\) −9.24597e14 −0.349290
\(638\) 0 0
\(639\) 8.75761e14 0.325185
\(640\) 0 0
\(641\) 1.13144e15 0.412966 0.206483 0.978450i \(-0.433798\pi\)
0.206483 + 0.978450i \(0.433798\pi\)
\(642\) 0 0
\(643\) 8.68306e14 0.311539 0.155769 0.987793i \(-0.450214\pi\)
0.155769 + 0.987793i \(0.450214\pi\)
\(644\) 0 0
\(645\) −1.25003e15 −0.440904
\(646\) 0 0
\(647\) 1.37636e15 0.477265 0.238633 0.971110i \(-0.423301\pi\)
0.238633 + 0.971110i \(0.423301\pi\)
\(648\) 0 0
\(649\) 3.54195e14 0.120753
\(650\) 0 0
\(651\) 3.43607e14 0.115177
\(652\) 0 0
\(653\) 2.55276e15 0.841370 0.420685 0.907207i \(-0.361790\pi\)
0.420685 + 0.907207i \(0.361790\pi\)
\(654\) 0 0
\(655\) −5.25679e14 −0.170370
\(656\) 0 0
\(657\) 8.24974e14 0.262924
\(658\) 0 0
\(659\) −9.26792e14 −0.290478 −0.145239 0.989397i \(-0.546395\pi\)
−0.145239 + 0.989397i \(0.546395\pi\)
\(660\) 0 0
\(661\) −1.90332e15 −0.586683 −0.293342 0.956008i \(-0.594767\pi\)
−0.293342 + 0.956008i \(0.594767\pi\)
\(662\) 0 0
\(663\) 5.74276e14 0.174099
\(664\) 0 0
\(665\) −2.90859e15 −0.867288
\(666\) 0 0
\(667\) −1.64627e14 −0.0482847
\(668\) 0 0
\(669\) −5.40841e14 −0.156037
\(670\) 0 0
\(671\) −3.15756e15 −0.896143
\(672\) 0 0
\(673\) 4.92990e15 1.37643 0.688217 0.725505i \(-0.258394\pi\)
0.688217 + 0.725505i \(0.258394\pi\)
\(674\) 0 0
\(675\) −2.86852e14 −0.0787930
\(676\) 0 0
\(677\) 4.30293e14 0.116286 0.0581429 0.998308i \(-0.481482\pi\)
0.0581429 + 0.998308i \(0.481482\pi\)
\(678\) 0 0
\(679\) −1.08521e15 −0.288555
\(680\) 0 0
\(681\) 1.01051e14 0.0264382
\(682\) 0 0
\(683\) 3.81244e15 0.981498 0.490749 0.871301i \(-0.336723\pi\)
0.490749 + 0.871301i \(0.336723\pi\)
\(684\) 0 0
\(685\) 8.32429e14 0.210887
\(686\) 0 0
\(687\) 4.40432e15 1.09804
\(688\) 0 0
\(689\) 2.90580e15 0.712952
\(690\) 0 0
\(691\) −4.03729e15 −0.974901 −0.487450 0.873151i \(-0.662073\pi\)
−0.487450 + 0.873151i \(0.662073\pi\)
\(692\) 0 0
\(693\) −1.04554e15 −0.248490
\(694\) 0 0
\(695\) −6.04805e15 −1.41481
\(696\) 0 0
\(697\) 2.77232e15 0.638356
\(698\) 0 0
\(699\) −4.55967e12 −0.00103350
\(700\) 0 0
\(701\) −4.71267e15 −1.05152 −0.525761 0.850632i \(-0.676219\pi\)
−0.525761 + 0.850632i \(0.676219\pi\)
\(702\) 0 0
\(703\) 1.29700e16 2.84896
\(704\) 0 0
\(705\) −2.03010e15 −0.439013
\(706\) 0 0
\(707\) 2.58467e14 0.0550299
\(708\) 0 0
\(709\) 8.66706e14 0.181684 0.0908422 0.995865i \(-0.471044\pi\)
0.0908422 + 0.995865i \(0.471044\pi\)
\(710\) 0 0
\(711\) −2.19694e14 −0.0453457
\(712\) 0 0
\(713\) −7.79972e14 −0.158521
\(714\) 0 0
\(715\) 2.62442e15 0.525231
\(716\) 0 0
\(717\) 4.28293e15 0.844085
\(718\) 0 0
\(719\) −2.62666e13 −0.00509795 −0.00254897 0.999997i \(-0.500811\pi\)
−0.00254897 + 0.999997i \(0.500811\pi\)
\(720\) 0 0
\(721\) −1.34844e15 −0.257744
\(722\) 0 0
\(723\) −2.16298e14 −0.0407186
\(724\) 0 0
\(725\) −2.14931e14 −0.0398511
\(726\) 0 0
\(727\) 5.68018e15 1.03734 0.518672 0.854973i \(-0.326427\pi\)
0.518672 + 0.854973i \(0.326427\pi\)
\(728\) 0 0
\(729\) 2.05891e14 0.0370370
\(730\) 0 0
\(731\) 2.95465e15 0.523552
\(732\) 0 0
\(733\) −1.62312e15 −0.283320 −0.141660 0.989915i \(-0.545244\pi\)
−0.141660 + 0.989915i \(0.545244\pi\)
\(734\) 0 0
\(735\) 1.57465e15 0.270771
\(736\) 0 0
\(737\) 3.37568e15 0.571861
\(738\) 0 0
\(739\) −3.64794e15 −0.608840 −0.304420 0.952538i \(-0.598463\pi\)
−0.304420 + 0.952538i \(0.598463\pi\)
\(740\) 0 0
\(741\) 3.63283e15 0.597372
\(742\) 0 0
\(743\) −6.76045e15 −1.09531 −0.547655 0.836704i \(-0.684479\pi\)
−0.547655 + 0.836704i \(0.684479\pi\)
\(744\) 0 0
\(745\) 7.43520e15 1.18695
\(746\) 0 0
\(747\) −5.17768e14 −0.0814465
\(748\) 0 0
\(749\) −6.26254e15 −0.970734
\(750\) 0 0
\(751\) −2.64833e13 −0.00404532 −0.00202266 0.999998i \(-0.500644\pi\)
−0.00202266 + 0.999998i \(0.500644\pi\)
\(752\) 0 0
\(753\) 5.88741e15 0.886242
\(754\) 0 0
\(755\) −3.75149e15 −0.556539
\(756\) 0 0
\(757\) 7.37364e15 1.07809 0.539045 0.842277i \(-0.318785\pi\)
0.539045 + 0.842277i \(0.318785\pi\)
\(758\) 0 0
\(759\) 2.37333e15 0.342001
\(760\) 0 0
\(761\) 6.20983e15 0.881991 0.440996 0.897509i \(-0.354625\pi\)
0.440996 + 0.897509i \(0.354625\pi\)
\(762\) 0 0
\(763\) −1.92796e15 −0.269907
\(764\) 0 0
\(765\) −9.78027e14 −0.134963
\(766\) 0 0
\(767\) −4.25484e14 −0.0578774
\(768\) 0 0
\(769\) −8.86195e15 −1.18832 −0.594162 0.804346i \(-0.702516\pi\)
−0.594162 + 0.804346i \(0.702516\pi\)
\(770\) 0 0
\(771\) 1.89659e15 0.250712
\(772\) 0 0
\(773\) −2.95047e15 −0.384507 −0.192254 0.981345i \(-0.561580\pi\)
−0.192254 + 0.981345i \(0.561580\pi\)
\(774\) 0 0
\(775\) −1.01830e15 −0.130833
\(776\) 0 0
\(777\) 4.48413e15 0.568019
\(778\) 0 0
\(779\) 1.75375e16 2.19034
\(780\) 0 0
\(781\) −9.45980e15 −1.16493
\(782\) 0 0
\(783\) 1.54269e14 0.0187322
\(784\) 0 0
\(785\) −1.14449e16 −1.37034
\(786\) 0 0
\(787\) −1.35545e16 −1.60038 −0.800190 0.599746i \(-0.795268\pi\)
−0.800190 + 0.599746i \(0.795268\pi\)
\(788\) 0 0
\(789\) −4.03272e15 −0.469543
\(790\) 0 0
\(791\) −9.98429e15 −1.14643
\(792\) 0 0
\(793\) 3.79308e15 0.429526
\(794\) 0 0
\(795\) −4.94876e15 −0.552685
\(796\) 0 0
\(797\) −1.59922e16 −1.76152 −0.880762 0.473558i \(-0.842969\pi\)
−0.880762 + 0.473558i \(0.842969\pi\)
\(798\) 0 0
\(799\) 4.79846e15 0.521308
\(800\) 0 0
\(801\) −1.50414e15 −0.161180
\(802\) 0 0
\(803\) −8.91121e15 −0.941892
\(804\) 0 0
\(805\) 2.28263e15 0.237990
\(806\) 0 0
\(807\) 6.93123e15 0.712863
\(808\) 0 0
\(809\) 9.45859e15 0.959643 0.479821 0.877366i \(-0.340701\pi\)
0.479821 + 0.877366i \(0.340701\pi\)
\(810\) 0 0
\(811\) 1.07996e16 1.08092 0.540461 0.841369i \(-0.318250\pi\)
0.540461 + 0.841369i \(0.318250\pi\)
\(812\) 0 0
\(813\) −5.68134e15 −0.560987
\(814\) 0 0
\(815\) −8.78341e14 −0.0855651
\(816\) 0 0
\(817\) 1.86909e16 1.79642
\(818\) 0 0
\(819\) 1.25598e15 0.119102
\(820\) 0 0
\(821\) −1.10411e16 −1.03306 −0.516529 0.856270i \(-0.672776\pi\)
−0.516529 + 0.856270i \(0.672776\pi\)
\(822\) 0 0
\(823\) −1.26104e16 −1.16420 −0.582101 0.813117i \(-0.697769\pi\)
−0.582101 + 0.813117i \(0.697769\pi\)
\(824\) 0 0
\(825\) 3.09852e15 0.282266
\(826\) 0 0
\(827\) 7.86172e14 0.0706704 0.0353352 0.999376i \(-0.488750\pi\)
0.0353352 + 0.999376i \(0.488750\pi\)
\(828\) 0 0
\(829\) −5.92589e15 −0.525659 −0.262829 0.964842i \(-0.584656\pi\)
−0.262829 + 0.964842i \(0.584656\pi\)
\(830\) 0 0
\(831\) −7.37848e15 −0.645894
\(832\) 0 0
\(833\) −3.72192e15 −0.321528
\(834\) 0 0
\(835\) −4.73066e13 −0.00403317
\(836\) 0 0
\(837\) 7.30896e14 0.0614986
\(838\) 0 0
\(839\) −2.10209e16 −1.74566 −0.872831 0.488022i \(-0.837718\pi\)
−0.872831 + 0.488022i \(0.837718\pi\)
\(840\) 0 0
\(841\) −1.20849e16 −0.990526
\(842\) 0 0
\(843\) 3.88189e15 0.314045
\(844\) 0 0
\(845\) 6.47126e15 0.516746
\(846\) 0 0
\(847\) 3.37348e15 0.265901
\(848\) 0 0
\(849\) −6.97520e15 −0.542705
\(850\) 0 0
\(851\) −1.01788e16 −0.781775
\(852\) 0 0
\(853\) −1.98378e16 −1.50409 −0.752046 0.659110i \(-0.770933\pi\)
−0.752046 + 0.659110i \(0.770933\pi\)
\(854\) 0 0
\(855\) −6.18693e15 −0.463086
\(856\) 0 0
\(857\) 7.96639e15 0.588663 0.294332 0.955703i \(-0.404903\pi\)
0.294332 + 0.955703i \(0.404903\pi\)
\(858\) 0 0
\(859\) 1.59749e16 1.16540 0.582701 0.812687i \(-0.301996\pi\)
0.582701 + 0.812687i \(0.301996\pi\)
\(860\) 0 0
\(861\) 6.06324e15 0.436704
\(862\) 0 0
\(863\) 4.02926e15 0.286527 0.143264 0.989685i \(-0.454240\pi\)
0.143264 + 0.989685i \(0.454240\pi\)
\(864\) 0 0
\(865\) 3.92467e15 0.275559
\(866\) 0 0
\(867\) −6.01635e15 −0.417089
\(868\) 0 0
\(869\) 2.37310e15 0.162445
\(870\) 0 0
\(871\) −4.05511e15 −0.274096
\(872\) 0 0
\(873\) −2.30837e15 −0.154073
\(874\) 0 0
\(875\) 1.02590e16 0.676176
\(876\) 0 0
\(877\) 9.76958e15 0.635884 0.317942 0.948110i \(-0.397008\pi\)
0.317942 + 0.948110i \(0.397008\pi\)
\(878\) 0 0
\(879\) 1.17066e16 0.752476
\(880\) 0 0
\(881\) 1.62583e16 1.03207 0.516033 0.856569i \(-0.327408\pi\)
0.516033 + 0.856569i \(0.327408\pi\)
\(882\) 0 0
\(883\) −1.66061e16 −1.04108 −0.520540 0.853837i \(-0.674269\pi\)
−0.520540 + 0.853837i \(0.674269\pi\)
\(884\) 0 0
\(885\) 7.24626e14 0.0448669
\(886\) 0 0
\(887\) 1.91314e16 1.16995 0.584975 0.811051i \(-0.301104\pi\)
0.584975 + 0.811051i \(0.301104\pi\)
\(888\) 0 0
\(889\) −6.94758e15 −0.419637
\(890\) 0 0
\(891\) −2.22400e15 −0.132680
\(892\) 0 0
\(893\) 3.03547e16 1.78872
\(894\) 0 0
\(895\) 2.10703e16 1.22644
\(896\) 0 0
\(897\) −2.85101e15 −0.163923
\(898\) 0 0
\(899\) 5.47641e14 0.0311041
\(900\) 0 0
\(901\) 1.16972e16 0.656287
\(902\) 0 0
\(903\) 6.46201e15 0.358166
\(904\) 0 0
\(905\) −1.22399e16 −0.670209
\(906\) 0 0
\(907\) 1.07252e16 0.580182 0.290091 0.956999i \(-0.406314\pi\)
0.290091 + 0.956999i \(0.406314\pi\)
\(908\) 0 0
\(909\) 5.49792e14 0.0293831
\(910\) 0 0
\(911\) 2.82249e16 1.49032 0.745162 0.666883i \(-0.232372\pi\)
0.745162 + 0.666883i \(0.232372\pi\)
\(912\) 0 0
\(913\) 5.59284e15 0.291771
\(914\) 0 0
\(915\) −6.45985e15 −0.332971
\(916\) 0 0
\(917\) 2.71748e15 0.138399
\(918\) 0 0
\(919\) −1.83551e16 −0.923679 −0.461839 0.886964i \(-0.652810\pi\)
−0.461839 + 0.886964i \(0.652810\pi\)
\(920\) 0 0
\(921\) −6.25361e15 −0.310959
\(922\) 0 0
\(923\) 1.13638e16 0.558359
\(924\) 0 0
\(925\) −1.32890e16 −0.645227
\(926\) 0 0
\(927\) −2.86831e15 −0.137622
\(928\) 0 0
\(929\) −1.10125e16 −0.522155 −0.261078 0.965318i \(-0.584078\pi\)
−0.261078 + 0.965318i \(0.584078\pi\)
\(930\) 0 0
\(931\) −2.35446e16 −1.10323
\(932\) 0 0
\(933\) −8.40884e15 −0.389392
\(934\) 0 0
\(935\) 1.05645e16 0.483486
\(936\) 0 0
\(937\) 8.36357e15 0.378289 0.189145 0.981949i \(-0.439429\pi\)
0.189145 + 0.981949i \(0.439429\pi\)
\(938\) 0 0
\(939\) 3.06340e15 0.136944
\(940\) 0 0
\(941\) 2.22942e16 0.985028 0.492514 0.870305i \(-0.336078\pi\)
0.492514 + 0.870305i \(0.336078\pi\)
\(942\) 0 0
\(943\) −1.37633e16 −0.601045
\(944\) 0 0
\(945\) −2.13901e15 −0.0923289
\(946\) 0 0
\(947\) 3.78038e16 1.61291 0.806457 0.591293i \(-0.201382\pi\)
0.806457 + 0.591293i \(0.201382\pi\)
\(948\) 0 0
\(949\) 1.07048e16 0.451454
\(950\) 0 0
\(951\) −1.98833e16 −0.828888
\(952\) 0 0
\(953\) 4.79568e15 0.197624 0.0988119 0.995106i \(-0.468496\pi\)
0.0988119 + 0.995106i \(0.468496\pi\)
\(954\) 0 0
\(955\) 1.81887e16 0.740939
\(956\) 0 0
\(957\) −1.66638e15 −0.0671057
\(958\) 0 0
\(959\) −4.30321e15 −0.171313
\(960\) 0 0
\(961\) −2.28139e16 −0.897884
\(962\) 0 0
\(963\) −1.33212e16 −0.518321
\(964\) 0 0
\(965\) 2.64698e16 1.01824
\(966\) 0 0
\(967\) −3.37420e16 −1.28329 −0.641645 0.767002i \(-0.721748\pi\)
−0.641645 + 0.767002i \(0.721748\pi\)
\(968\) 0 0
\(969\) 1.46238e16 0.549893
\(970\) 0 0
\(971\) −3.58587e16 −1.33318 −0.666590 0.745425i \(-0.732247\pi\)
−0.666590 + 0.745425i \(0.732247\pi\)
\(972\) 0 0
\(973\) 3.12651e16 1.14932
\(974\) 0 0
\(975\) −3.72217e15 −0.135291
\(976\) 0 0
\(977\) −1.20023e16 −0.431366 −0.215683 0.976463i \(-0.569198\pi\)
−0.215683 + 0.976463i \(0.569198\pi\)
\(978\) 0 0
\(979\) 1.62474e16 0.577405
\(980\) 0 0
\(981\) −4.10102e15 −0.144116
\(982\) 0 0
\(983\) 4.42687e15 0.153834 0.0769170 0.997038i \(-0.475492\pi\)
0.0769170 + 0.997038i \(0.475492\pi\)
\(984\) 0 0
\(985\) −3.41417e16 −1.17324
\(986\) 0 0
\(987\) 1.04945e16 0.356631
\(988\) 0 0
\(989\) −1.46684e16 −0.492951
\(990\) 0 0
\(991\) 3.79167e16 1.26016 0.630080 0.776530i \(-0.283022\pi\)
0.630080 + 0.776530i \(0.283022\pi\)
\(992\) 0 0
\(993\) 8.26479e15 0.271651
\(994\) 0 0
\(995\) −2.03283e16 −0.660807
\(996\) 0 0
\(997\) 5.76003e13 0.00185183 0.000925915 1.00000i \(-0.499705\pi\)
0.000925915 1.00000i \(0.499705\pi\)
\(998\) 0 0
\(999\) 9.53830e15 0.303292
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 48.12.a.f.1.1 1
3.2 odd 2 144.12.a.l.1.1 1
4.3 odd 2 3.12.a.a.1.1 1
8.3 odd 2 192.12.a.q.1.1 1
8.5 even 2 192.12.a.g.1.1 1
12.11 even 2 9.12.a.a.1.1 1
20.3 even 4 75.12.b.a.49.1 2
20.7 even 4 75.12.b.a.49.2 2
20.19 odd 2 75.12.a.a.1.1 1
28.27 even 2 147.12.a.c.1.1 1
36.7 odd 6 81.12.c.a.28.1 2
36.11 even 6 81.12.c.e.28.1 2
36.23 even 6 81.12.c.e.55.1 2
36.31 odd 6 81.12.c.a.55.1 2
60.23 odd 4 225.12.b.a.199.2 2
60.47 odd 4 225.12.b.a.199.1 2
60.59 even 2 225.12.a.f.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.12.a.a.1.1 1 4.3 odd 2
9.12.a.a.1.1 1 12.11 even 2
48.12.a.f.1.1 1 1.1 even 1 trivial
75.12.a.a.1.1 1 20.19 odd 2
75.12.b.a.49.1 2 20.3 even 4
75.12.b.a.49.2 2 20.7 even 4
81.12.c.a.28.1 2 36.7 odd 6
81.12.c.a.55.1 2 36.31 odd 6
81.12.c.e.28.1 2 36.11 even 6
81.12.c.e.55.1 2 36.23 even 6
144.12.a.l.1.1 1 3.2 odd 2
147.12.a.c.1.1 1 28.27 even 2
192.12.a.g.1.1 1 8.5 even 2
192.12.a.q.1.1 1 8.3 odd 2
225.12.a.f.1.1 1 60.59 even 2
225.12.b.a.199.1 2 60.47 odd 4
225.12.b.a.199.2 2 60.23 odd 4